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Introduction to Path Integrals in Field Theory U. Mosel1 Institut fuer Theoretische Physik, Universitaet Giessen D-35392 Giessen, Germany SS 02 July 6, 2002

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http://theorie.physik.uni-giessen.de

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This manuscript on path integrals is based on lectures I have given at the University of Giessen. If you ﬁnd any conceptual or typographical errors I would like to learn about them. In this case please send an e-mail to [email protected]

Contents I

Non-Relativistic Quantum Theory

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1 PATH INTEGRAL IN QUANTUM THEORY 1.1 Propagator of the Schr¨odinger Equation . . . . 1.2 Propagator as Path Integral . . . . . . . . . . . 1.3 Quadratic Hamiltonians . . . . . . . . . . . . . 1.3.1 Cartesian metric . . . . . . . . . . . . . 1.3.2 Non-Cartesian metric . . . . . . . . . . . 1.4 Classical Interpretation . . . . . . . . . . . . . .

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7 7 10 13 14 15 17

2 PERTURBATION THEORY 20 2.1 Free propagator . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Perturbative Expansion . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Application to Scattering . . . . . . . . . . . . . . . . . . . . . 27 3 GENERATING FUNCTIONALS 3.1 Groundstate-to-Groundstate Transitions . . . . . . . . . . . . 3.1.1 Generating functional. . . . . . . . . . . . . . . . . . . 3.2 Functional Derivatives of Transition Amplitudes . . . . . . . . . . . . . . . . . . . . .

32 33 37

II

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Relativistic Quantum Field Theory

4 CLASSICAL RELATIVISTIC FIELDS 4.1 Equations of Motion . . . . . . . . . . . . . 4.1.1 Examples . . . . . . . . . . . . . . . 4.2 Symmetries and Conservation Laws . . . . . 4.2.1 Geometrical Space–Time Symmetries 4.2.2 Internal Symmetries . . . . . . . . .

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CONTENTS

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5 PATH INTEGRALS FOR SCALAR FIELDS 56 5.1 Generating Functional for Fields . . . . . . . . . . . . . . . . . 57 5.1.1 Euclidean Representation . . . . . . . . . . . . . . . . 59 6 EVALUATION OF PATH INTEGRALS 6.1 Free Scalar Fields . . . . . . . . . . . . . 6.1.1 Generating functional . . . . . . . 6.1.2 Feynman propagator . . . . . . . 6.1.3 Gaussian Integration . . . . . . . 6.2 Stationary Phase Approximation . . . . 6.3 Numerical Evaluation of Path Integrals . 6.3.1 Imaginary time method . . . . . 6.3.2 Real time formalism . . . . . . .

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62 62 62 64 68 71 74 74 75

7 S-MATRIX AND GREEN’S FUNCTIONS 7.1 Scattering Matrix . . . . . . . . . . . . . . . 7.2 Reduction Theorem . . . . . . . . . . . . . . 7.2.1 Canonical ﬁeld quantization . . . . . 7.2.2 Derivation of the reduction theorem .

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78 78 80 80 82

8 GREEN’S FUNCTIONS 8.1 n-point Green’s Functions . . . . 8.1.1 Momentum representation 8.1.2 Operator Representations 8.2 Free Scalar Fields . . . . . . . . . 8.2.1 Wick’s theorem . . . . . . 8.2.2 Feynman rules . . . . . . . 8.3 Interacting Scalar Fields . . . . . 8.3.1 Perturbative expansion . .

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87 87 88 89 91 91 92 94 96

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99 99 101 102 103 103 104 107 109 109 110

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9 PERTURBATIVE φ4 THEORY 9.1 Perturbative Expansion of the Generating Function 9.1.1 Feynman rules . . . . . . . . . . . . . . . . . 9.1.2 Vacuum contributions . . . . . . . . . . . . 9.2 Two-Point Function . . . . . . . . . . . . . . . . . . 9.2.1 Terms up to O(g 0 ) . . . . . . . . . . . . . . 9.2.2 Terms up to O(g) . . . . . . . . . . . . . . . 9.2.3 Terms up to O(g 2 ) . . . . . . . . . . . . . . 9.3 Four-Point Function . . . . . . . . . . . . . . . . . 9.3.1 Terms up to O(g) . . . . . . . . . . . . . . . 9.3.2 Terms up to O(g 2 ) . . . . . . . . . . . . . .

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CONTENTS

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10 DIVERGENCES IN n-POINT FUNCTIONS 10.1 Power Counting . . . . . . . . . . . . . . . . . 10.2 Dimensional Regularization of φ4 Theory . . . . . . . . . . . . . . . . . . . 10.2.1 Two-point function . . . . . . . . . . . 10.2.2 Four-point function . . . . . . . . . . . 10.3 Renormalization . . . . . . . . . . . . . . . . . 10.3.1 Renormalization of φ4 Theory . . . . .

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116 117 118 122 125

11 GREEN’S FUNCTIONS FOR FERMIONS 11.1 Grassmann Algebra . . . . . . . . . . . . . . 11.1.1 Derivatives . . . . . . . . . . . . . . 11.1.2 Integration . . . . . . . . . . . . . . 11.2 Green’s Functions for Fermions . . . . . . . 11.2.1 Generating Functional for fermions . 11.2.2 Green’s Functions . . . . . . . . . . .

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128 128 130 132 138 138 141

12 INTERACTING FIELDS 12.1 Feynman Rules . . . . . . . . . 12.1.1 Fermion Loops . . . . . 12.2 Wick’s Theorem . . . . . . . . . 12.3 Removal of Degrees of Freedom: Yukawa Theory . . . . . . . . . 12.3.1 Perturbative Expansion

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113 . . . . . . . . . 114

144 . . . . . . . . . . . . . . . . . 144 . . . . . . . . . . . . . . . . . 145 . . . . . . . . . . . . . . . . . 148 . . . . . . . . . . . . . . . . . 150 . . . . . . . . . . . . . . . . . 152

13 PATH INTEGRALS FOR GAUGE FIELDS 13.1 Gauge invariance in Abelian theories . . . . . 13.2 Non-abelian gauge ﬁelds . . . . . . . . . . . . 13.3 Path integrals . . . . . . . . . . . . . . . . . . 13.3.1 Gauge Fixing . . . . . . . . . . . . . . 13.4 Feynman Rules . . . . . . . . . . . . . . . . . 13.4.1 Faddeev-Popov Determinant . . . . . . 13.4.2 Ghost ﬁelds . . . . . . . . . . . . . . . 13.4.3 Feynman Rules . . . . . . . . . . . . .

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157 . 158 . 162 . 164 . 169 . 171 . 171 . 175 . 176

14 EXAMPLES FOR GAUGE FIELD THEORIES 14.1 Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . 14.2 Quantum Chromodynamics . . . . . . . . . . . . . . . . . . 14.3 Electroweak Interactions . . . . . . . . . . . . . . . . . . . .

184 . 184 . 184 . 186

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CONTENTS

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A Units and Metric 189 A.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 A.2 Metric and Notation . . . . . . . . . . . . . . . . . . . . . . . 190 B Functionals B.1 Deﬁnition . . . . . . . . . B.2 Functional Integration . . B.2.1 Gaussian integrals B.3 Functional Derivatives . .

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192 . 192 . 193 . 193 . 196

C RENORMALIZATION INTEGRALS

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D GRASSMANN INTEGRATION FORMULA

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E BIBLIOGRAPHY

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Part I Non-Relativistic Quantum Theory

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Chapter 1 PATH INTEGRAL IN QUANTUM THEORY In this starting chapter we introduce the concepts of propagators and path integrals in the framework of nonrelativistic quantum theory. In all these considerations, and the following chapters on nonrelativistic quantum theory, we work with one coordinate only, but all the results can be easily generalized to the case of d dimensions.

1.1

Propagator of the Schr¨ odinger Equation

We start by considering a nonrelativistic particle in a one-dimensional potential V (x). The Schr¨odinger equation reads Hψ(x, t) = −

∂ψ(x, t) h ¯ 2 ∂ 2 ψ(x, t) + V (x)ψ(x, t) = i¯ h . 2m ∂x2 ∂t

(1.1)

This equation allows us to calculate the wavefunction ψ(x, t) at a later time, if we know ψ(x, t0 ) at the earlier time t0 < t. For further calculations we rewrite this equation into the following form

∂ i¯ h − H ψ(x, t) = 0 . ∂t

(1.2)

Next, we consider the function K (x, t; xi , ti ) which is deﬁned as solution of the equation

∂ i¯ h − H K (x, t; xi , ti ) = i¯ hδ(x − xi )δ(t − ti ) . ∂t 7

(1.3)

CHAPTER 1. PATH INTEGRAL IN QUANTUM THEORY

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K is the “Green’s function” of the Schr¨odinger equation (K is also often called “propagator”) with the initial condition K(x, ti + 0; xi , ti ) = δ(x − xi ) .

(1.4)

The solution of the Schr¨odinger equation (1.2) can be written as

ψ(x, t) =

K (x, t; xi , ti ) ψ(xi , ti ) dxi

(1.5)

for t > ti (Huygen’s principle). Relation (1.5) can be proven by inserting the lhs into the Schr¨odinger equation

∂ i¯ h −H K (x, t; xi , ti ) ψ(xi , t) dxi ∂t

= i¯ h

δ (t − ti ) δ (x − xi ) ψ (xi , t) dxi

= i¯ hδ (t − ti ) ψ(x, t) = 0

for t > ti .

(1.6)

Thus the ψ deﬁned by (1.5) is indeed a solution of the Schr¨odinger equation for all times t > ti . K (x, t; xi , ti ) is the probability amplitude for a transition from xi , at time ti , to the position x, at the later time t. The restriction to later times preserves causality. We can ﬁnd an explicit form for the propagator, if the solutions of the stationary Schr¨odinger equation, ϕn (x), and the corresponding eigenvalues, En , are known. Since the ϕn form a complete system, K can certainly be expanded in this basis (for t ≥ ti ) K (x, t; xi , ti ) =

an ϕn (x)e− h¯ En t Θ (t − ti ) . i

(1.7)

n

Here the stepfunction Θ(t) = 0 for t < 0 and Θ(t) = 1 for t ≥ 0 takes explicitly into account that we propagate the wavefunction only forward in time. The expansion coeﬃcients obviously depend on xi , ti an = an (xi , ti ) .

(1.8)

Because of the initial condition K(x, ti + 0; xi , ti ) = δ (x − xi ) we have δ (x − xi ) =

i

an (xi , ti )ϕn (x)e− h¯ En ti .

(1.9)

n

The lhs is time-independent; thus we must have i

an (xi , ti ) = an (xi )e+ h¯ En ti ,

(1.10)

CHAPTER 1. PATH INTEGRAL IN QUANTUM THEORY and consequently δ (x − xi ) =

an (xi ) ϕn (x) .

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(1.11)

n

This can be fulﬁlled by

an (xi ) = ϕ∗n (xi )

(1.12)

(closure relation). Thus we have a representation of K (x, t; xi , ti ) in terms of the eigenfunctions and eigenvalues of the underlying Hamiltonian K (x, t; xi , ti ) = Θ(t − ti )

ϕ∗n (xi ) ϕn (x)e− h¯ En (t−ti ) . i

(1.13)

n

It is easy to show that this propagator fulﬁlls (1.3). In Dirac’s bra and ket notation this result can also be written as K (x, t; xi , ti ) =

ϕ∗n (xi ) ϕn (x)e− h¯ En (t−ti ) Θ (t − ti ) i

n

=

n|xi e− h¯ En (t−ti ) x|n Θ (t − ti ) i

n

=

ˆ

ˆ

n|e+ h¯ Hti |xi x|e− h¯ Ht |n Θ (t − ti ) i

i

n ˆ

= x|e− h¯ H(t−ti ) |xi ≡ x|Uˆ (t, ti ) |xi Θ (t − ti ) .(1.14) i

Thus the propagator is nothing else than the time development operator i ˆ Uˆ (t, ti ) = e− h¯ H(t−ti )

(1.15)

for t > ti in the x representation. It is also often written as ˆ

K (x, t; xi , ti ) = x|e− h¯ H(t−ti ) |xi Θ (t − ti ) ≡ xt|xi ti . i

(1.16)

The notation here is that of the Heisenberg representation of quantum mechanics. In this representation the physical state vectors are time-independent and the operators themselves carry all the time-dependence whereas this is just the opposite for the Schr¨odinger representation. For example, for the position operator xˆ in the Schr¨odinger representation we obtain the timedependent operator i ˆ i ˆ (1.17) xˆH (t) = e h¯ Ht xˆe− h¯ Ht and xˆH (t)|xt = x|xt with

i

ˆ

|xt = e h¯ Ht |x .

(1.18) (1.19)

The state |xt is thus the eigenstate of the operator xˆH (t) with eigenvalue x and not the state that evolves with time out of |x; this explains the sign of the frequency in the exponent.

CHAPTER 1. PATH INTEGRAL IN QUANTUM THEORY

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t

t

t1

ti x xi

x

Figure 1.1: Possible paths from xi to x, corresponding to (1.22).

1.2

Propagator as Path Integral

We start by dividing the time-interval between ti and t by inserting the time t1 . The wavefunction is ﬁrst propagated until t1 and then, in a second step, until t

ψ (x1 , t1 ) = ψ(x, t) =

K (x1 , t1 ; xi , ti ) ψ (xi , ti ) dxi

(1.20)

K(x, t; x1 , t1 )ψ (x1 , t1 ) dx1 .

Taking these two equations together we get

ψ(x, t) =

K (x, t; x1 , t1 ) K (x1 , t1 ; xi , ti ) ψ(xi , ti ) dxi dx1 .

(1.21)

Comparing this result with (1.5) yields

K (x, t; xi , ti ) =

K (x, t; x1 , t1 ) K (x1 , t1 ; xi , ti ) dx1 .

(1.22)

We can thus view the transition from (xi , ti ) to (x, t) as the result of a transition ﬁrst from (x, t) to all possible intermediate points (x1 , t1 ), which is then followed by a transition from these intermediate points to the endpoint. We could also say that the integration in (1.22) is performed over all possible paths between the points (xi , ti ) and (x, t), which consist of two straight line segments with a bend at t1 . This is illustrated in Fig. 1.1.

CHAPTER 1. PATH INTEGRAL IN QUANTUM THEORY

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We now subdivide the time interval further into (n + 1) equal parts of length Δt = η. We then have in direct generalization of the previous result

K (x, t; xi , ti ) =

...

dx1 dx2 . . . dxn

(1.23)

× [K (x, t; xn , tn ) K (xn , tn ; xn−1 , tn−1 ) . . . K (x1 , t1 ; xi , ti )] . The integrals run here over all possible paths between (xi , ti ) and (x, t) which consist of (n + 1) segments with boundaries that are determined by the time steps ti , t1 , . . . , tn , t. We now calculate the propagator for a small time interval Δt = η from tj to tj+1 . For this propagation we have according to (1.16) ˆ

K (xj+1 , tj+1 ; xj , tj ) = xj+1 |e− h¯ Hη |xj (1.24) i ˆ ∼ = xj+1 |1 − Hη|x j h ¯ i ˆ j = δ (xj+1 − xj ) − ηxj+1 |H|x h ¯ 1 i p(xj+1 −xj ) iη ˆ j = e h¯ dp − xj+1 |H|x 2π¯ h h ¯ i

with the representation for the δ-function δ(x − x ) =

1 ik(x−x ) e dk . 2π

(1.25)

ˆ is given by We now assume that H ˆ = Tˆ(ˆ H p) + Vˆ (ˆ x) .

(1.26)

Here Tˆ, pˆ, Vˆ , xˆ are all operators; we assume that T (ˆ p) and V (ˆ x) are Taylorexpandable. In this case, where the p- and x-dependences separate, we can also bring the last term in (1.24) into an integral form. We have ˆ j = xj+1 |Tˆ + Vˆ |xj . xj+1 |H|x

(1.27)

First, we consider the ﬁrst summand xj+1 |Tˆ|xj =

=

=

dp dpxj+1 |p p |Tˆ(ˆ p)|pp|xj dp dpxj+1 |p δ(p − p)T (p)p|xj dpxj+1 |pT (p)p|xj .

(1.28)

CHAPTER 1. PATH INTEGRAL IN QUANTUM THEORY

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With the normalized momentum eigenfunctions x|p = √

i 1 e h¯ px 2π¯ h

(1.29)

we thus obtain i 1 xj+1 |Tˆ(ˆ p)|xj = T (p) e h¯ p(xj+1 −xj ) dp . 2π¯ h

(1.30)

While there is an operator pˆ on the lhs of this equation there are only numbers p on its rhs. For the potential part an analogous transformation can be performed xj+1 |Vˆ |xj = V (xj ) δ (xj+1 − xj ) 1 i p(xj+1 −xj ) e h¯ dp V (xj ) = 2π¯ h

(1.31)

Again, on the lhs Vˆ is an operator, while the rhs of this equation contains no operators. In summary, we have for the propagator over a time-segment η i 1 dp e h¯ p(xj+1 −xj ) 2π¯ h i i iη 1 1 − dp T (p)e h¯ p(xj+1 −xj ) + dp e h¯ p(xj+1 −xj ) V (xj ) h ¯ 2π¯ h 2π¯ h i 1 iη = dp e h¯ p(xj+1 −xj ) 1 − H (p, xj ) 2π¯ h h ¯ 1 i −→ dpj exp (1.32) [pj (xj+1 − xj ) − ηH (pj , xj )] . η→0 2π¯ h h ¯

K (xj+1 , tj+1 ; xj , tj ) =

Here H = T + V is a function of the numbers x and p and no longer an operator! In the last step we have renamed the integration variable into pj to indicate that it may be viewed as the momentum of a classical particle moving from xj to xj+1 between times tj and tj+1 . We now insert (1.32) into (1.23) and obtain K (x, t; xi , ti ) =

lim

n→∞

n k=1

dxk

n dpl l=0

2π¯ h

⎧ n ⎨i

exp ⎩

h ¯ j=0

(1.33) ⎫ ⎬

[pj (xj+1 − xj ) − ηH (pj , xj )]⎭ .

(with x0 = xi and xn+1 = x). The asymmetry in the range of the products over x- and p-integrations comes about because with n intermediate steps between xi and x there are n + 1 intervals and corresponding momenta.

CHAPTER 1. PATH INTEGRAL IN QUANTUM THEORY

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The integrand here is, for ﬁnite n, a complex function of all the coordinates x1 , x2 , . . . , xn and the momenta p1 , p2 , . . . , pn . In the limit n → ∞ it depends on the whole trajectory x(t), p(t). Here we note that p is not the momentum canonically conjugate to the coordinate x, but instead just an integration variable. In the limit n → ∞ we obtain for the exponent n

[pj (xj+1 − xj ) − ηH (pj , xj )]

j=0

= −→

xj+1 − xj − H (pj , xj ) η pj η j=0 n

t

n→∞

dt [p(t )x(t ˙ ) − H(p(t ), x(t ))] .

(1.34)

ti

With this result we rewrite (1.33) in an abbreviated, symbolic form

K (x, t; xi , ti ) =

Dx

Dp e

i h ¯

t

dt [p(t )x(t ˙ )−H(p(t ),x(t ))]

ti

,

(1.35)

h). The integrals here are where Dx stands for dxk and Dp for dpl /(2π¯ limits of n-dimensional integrals over x and p for n → ∞, they are integrals over all functions x(t) and p(t) and are deﬁned by (1.33). Equation (1.35) represents an important result. It allows to calculate the propagator and thus the solution of the Schr¨odinger equation in terms of a path integral over classical functions.

1.3

Quadratic Hamiltonians

Even though the propagator (1.35) looks like a path integral over an exponential function of the action, this is in general not the case, because ˙ p) px˙ − H(p, x) = L(x, x,

(1.36)

is not equal to the classical Lagrange function since p is not the canonical momentum as already stressed above. Therefore, in general one cannot express the path integral (1.35) in terms of the action. Such a simpliﬁcation, however, is possible for a special p-dependence of the Hamiltonian. If H depends at most quadratically on p, then the path integration over the momentum p can be performed and the action appears in the exponent. This will be discussed in the next 2 sections.

CHAPTER 1. PATH INTEGRAL IN QUANTUM THEORY

1.3.1

14

Cartesian metric

In the last section we have made the special ansatz H = T (p) + V (x) in which the momenta and coordinates are separated. For the special case, in which H depends only quadratically on p with constant coeﬃcient, e.g. p2 + V (x) , 2m

H=

(1.37)

we can further simplify the path-integral (1.33) K (x, t; xi , ti ) =

lim n→∞

n

dxk

k=1

⎧ ⎨i

n

n dpl

× exp ⎩ η h ¯ j=0

l=0

(1.38)

2π¯ h

⎫

⎬ p2j xj+1 − xj pj − − V (xj ) ⎭ . η 2m

Using the integral relation +∞ −∞

e

−ap2 +bp+c

dp =

π b2 +c e 4a a

(1.39)

for Gaussian integrals, discussed in more detail in App. B.2.1, we obtain by performing the p-integration

K =

lim n→∞ ×

m i2π¯ hη

n+1 2

(1.40) ⎧ ⎨i

n

n

⎡

m ⎣ dxk exp ⎩ η h ¯ j=0 2 k=1

xj+1 − xj η

2

⎤⎫ ⎬ − V (xj )⎦⎭ .

Thus in this special case (H = p2 /2m + V ) the propagator K is given (again in abbreviated notation) by

K (x, t; xi , ti ) = N

Dx e

i h ¯

t

=N

with the Lagrangian L(x, x) ˙ = and the action

t

S[x(t)] =

L(x,x)dt ˙

ti

i

Dx e h¯ S[x(t)] ,

m 2 x˙ − V (x) 2

L(x(t ), x(t ˙ )) dt .

(1.41)

(1.42) (1.43)

ti

N represents the factor in front of the integral in (1.40). There is a problem with N : the factor N is complex and becomes inﬁnite for n → ∞, η →

CHAPTER 1. PATH INTEGRAL IN QUANTUM THEORY

15

0. We will see, however, later, for example in Sect. 2.1, that the whole pathintegral leads to a well-deﬁned expression. In addition, this problem will be bypassed in later developments where we show that physically relevant is only a normalized propagator in which N has been removed. The propagator K has thus been reduced to a one-dimensional pathintegral, which is only possible for Hamiltonians which are quadratic in p. This is a quite important result that we will use throughout all the following sections. Eq. (1.41) shows that the propagator is given by the phase exp ¯hi S[x(t)] summed over all possible trajectories x(t) with ﬁxed starting and end points. This is reminiscent of the partition function in statistical mechanics which is obtained by summing the Boltzmann factor exp (−En /T ) over all possible states of the system. At this point we should realize that in going from (1.38) to (1.40) we have integrated an oscillatory integrand (eif (p) ) over an inﬁnite interval. This was only possible by a mathematical trick: in applying the Gaussian integration formula (B.18) we have in eﬀect used the quantity iη in (1.38) as if it were real. In other words: we have analytically continued expression (1.38) into the complex plane by setting the time interval η → −iη with η real and positive. Then (1.38) becomes a well-behaved Gaussian integral. After performing the integration we have then gone back to the original η. This analytical continuation is in general possible only if no singularities are encountered while going to the real variable η . Also, one has to worry about phase ambiguities connected with the appearance of the squareroot of a complex number.

1.3.2

Non-Cartesian metric

The momentum integration is even possible for Hamiltonians of a much more general form. As an example we consider H=

1 f1 (x)p2 + f2 (x)p + f3 (x) . 2m

(1.44)

Here a problem arises because the canonical quantization of such a Hamiltonian is ambiguous. This is so because the classical coordinates and momenta commute, so that H can be brought into various forms that are classically all equal, but diﬀer after quantization because the operators pˆ and xˆ do not commute. Even though the time development operator (1.24) could still be evaluated for any of these various forms, these would in general not lead to a path integral of the form (1.35). We, therefore, turn the question around and ask:

CHAPTER 1. PATH INTEGRAL IN QUANTUM THEORY

16

Given a path integral

Dx

Dp e

i h ¯

t

dt (px−H(p,x)) ˙

ti

(1.45)

with the classical Hamiltonian (1.52), can this still be identified as a propagator and, if so, for which Hamiltonian? The answer to this question is given here without proof1 :

Dx

Dp e

i h ¯

t

dt [px−H(p,x)] ˙

ti

i

ˆ

= x |e− h¯ (t−ti )HW |x .

(1.46)

ˆ W is the “Weyl-ordered” Hamiltonian Here H ˆW = H

1 2 pˆ f1 (x) + 2ˆ pf1 (x)ˆ p + f1 (x)ˆ p2 8m 1 + [ˆ p] + f3 (x) . pf2 (x) + f2 (x)ˆ 2

(1.47)

In this form the momentum- and coordinate-dependent terms are symmetrized. Note that the path integral in (1.46) is well determined because all quantities on the rhs of this equation are classical, commuting quantities. The lhs of (1.46) can be simpliﬁed by performing the integration over the momenta in (1.33) K =

lim n→∞

n

dxk

k=1

⎧ n ⎨i

× exp ⎩

h ¯ j=0

n

dpl l=0

(1.48)

2π¯ h

⎫ ⎬

p2j pj (xj+1 − xj ) − η f1 (xj ) + f2 (xj )pj + f3 (xj ) 2m

⎭

.

Using again the Gaussian integral formula (1.39) this gives

K =

lim n→∞

m i2π¯ hη

n+1 n 2

⎧ ⎪ n ⎨i

⎛

m × exp ⎪ η⎜ ⎝ ¯ j=0 2 ⎩h 1

dxk

k=1

!

n

l=0

(xj+1 −xj ) η

The proof can be found in [LEE], p. 475

1

(1.49)

f1 (xl )

"2

− f2 (xj )

f1 (xj )

⎞⎫ ⎪ ⎬ − ηf3 (xj )⎟ ⎠ . ⎪ ⎭

CHAPTER 1. PATH INTEGRAL IN QUANTUM THEORY

17

The exponent (in round brackets) is !

"2

xj+1 −xj − f2 (xj ) m η (. . .) = − f3 (xj ) 2 f1 (xj ) 2 ˙ 2 (x) + f22 (x) −→ m x˙ − 2xf − f3 (x) . η→0 2 f1 (x)

(1.50)

This last expression is just the Lagrangian L=

m g1 (x)x˙ 2 + g2 (x)x˙ + g3 (x) 2

(1.51)

(a kinetic term of this form appears, for example, when rewriting the kinetic energy of a particle from cartesian into polar coordinates). For this L the corresponding classical Hamiltonian is given by H = px˙ − L = (mg1 x˙ + g2 ) x˙ −

m g1 x˙ 2 − g2 x˙ − g3 2 2

m m p − g2 = − g3 g1 x˙ 2 − g3 = g1 2 2 mg1 1 1 2 g2 1 g22 = p − p+ − g3 . 2m g1 mg1 2m g1

(1.52)

With

1 g2 g22 , f2 = − , f3 = − g3 g1 mg1 2mg1 this is just the Hamiltonian (1.44) that we started out with. We thus have for the complete propagator in this case f1 =

K (x, t; xi , ti )

=

lim

n→∞

m i2π¯ hη

n+1 n 2

k=1

dxk

n

l=0

⎧ ⎨i

(1.53)

(1.54)

n

⎫ ⎬

g1 (xl ) exp ⎩ η L(xj , x˙ j )⎭ . h ¯ j=1

Thus, in this case the path integral is changed. The square root of a function that determines the metric of the system appears in the integrand. For Hamiltonians even more general than (1.44) an additional potential term appears in the Lagrangian [Grosche/Steiner].

1.4

Classical Interpretation

The simple form (1.41) for the path integral allows a very physical interpretation of the classical limit to quantum mechanics. For the classical path the

CHAPTER 1. PATH INTEGRAL IN QUANTUM THEORY

18

variation of the action is, according to Hamilton’s principle, equal to zero, i.e. the action is stationary2 t

δS =

L (xcl + δx, x˙ + δ x˙ cl ) dt −

ti

t

L (xcl , x˙ cl ) dt = 0 .

(1.55)

ti

This implies that all the paths close to the classical path give about equal contributions to the path integral. For each path (C1 ) somewhat more removed from the classical one there will also be another one, (C2 ), whose h. Then we have action diﬀers from that on C1 just by π¯ i

i

i

e h¯ S(C1 ) = e h¯ S(C2 )+iπ = −e h¯ S(C2 ) ,

(1.56)

so that the contributions from these two paths cancel each other. Sizeable contributions to the path integral thus come from paths close to the classical one. Quantum mechanics then describes the ﬂuctuations of the action in a narrow range around the classical path. This observation forms the basis for a semiclassical approximation. This can be formulated by expanding the action functional S[x(t), x(t)] ˙ in terms of ﬂuctuations δx around the classical path xcl (t). This gives

1 δ2L δ2L δ2L 2 S[x, x] ˙ = Scl + (δx) + 2 ˙ 2 + ..... δx δ x˙ + 2 (δ x) 2 2 δx δxδ x˙ δ x˙ 2 (1.57) ≡ Scl + δ S + . . . . Here all the derivatives have to be taken at the classical path. Because S is stationary at the classical path, there is no ﬁrst derivative in this equation. The propagator (1.41) now becomes

K (x, t; xi , ti ) = N

Dx e

i S h ¯

= Ne

i S h ¯ cl

&

'

1i 2 Dx exp δ S + .... . 2h ¯

(1.58)

Note that this result (without higher order terms) is exact for Lagrangians that depend at most quadratically on x. The second factor gives the eﬀects of quantum mechanical ﬂuctuations around the classical path. An interesting observation on the character of these ﬂuctuations can be made. The main contribution to the integrand in (1.40) for η → 0 comes from exponents 2 η m xj+1 − xj

1, (1.59) h ¯ 2 η 2

For an explanation of functionals and their derivatives see App. B

CHAPTER 1. PATH INTEGRAL IN QUANTUM THEORY

19

i.e. from velocities vj ≈ 2¯ h/(ηm) which diverge with η → 0. This implies √ that the average displacement d within a time-step η is proportional to η, so that d2 ∼ η (not d2 = v 2 dt2 = v 2 η 2 !), just as for a random walk. The main contribution to the path integral, and therefore to the quantum mechanical ﬂuctuation around the classical trajectory, thus comes from paths that are continous, but have no ﬁnite derivative.

Chapter 2 PERTURBATION THEORY In this chapter we discuss ﬁrst how to calculate the propagator of a free particle and derive its analytic form. In most cases, however, with a potential included the exact propagator cannot be calculated. Thus one has to resort to perturbation theory which will also be developed in this chapter.

2.1

Free propagator

We start with the free propagator K0 , given by

i

Dx e h¯ S0

K0 = N

=

lim

n→∞

m 2πi¯ hη

⎡

n+1 +∞

n 2 −∞

n i m dxk exp ⎣ η h ¯ j=0 2 k=1

xj+1 − xj η

2 ⎤ ⎦ . (2.1)

This path-integral can be performed exactly. With (B.19) we obtain for the free propagator

K0 =

lim

n→∞

m 2πi¯ hη

n+1 ⎛ 2

⎝

(n + 1)

⎞1 2

in π n

i m × exp (x − xi )2 n + 1 2¯ hη

m 2¯ hη

n ⎠

(2.2)

,

since xn+1 = x, x0 = xi . With (n + 1)η = t − ti this becomes (

K0 (Δx, Δt) =

2 i m(x−xi ) m h ¯ 2(t−ti ) = e 2π¯ hi (t − ti )

20

i mΔx2 m e h¯ 2Δt 2π¯ hiΔt

(2.3)

CHAPTER 2. PERTURBATION THEORY

21

with Δx = x−xi , Δt = t−ti . This is the propagator of a free particle for Δt ≥ 0; for Δt < 0 it has to be supplemented by the condition K = 0. Because of Galilei invariance and time homogeneity the free particle propagator depends only on the space- and time-distances. The last step, from (2.2) to (2.3), shows nicely how in the limit n → ∞ the inﬁnite normalization factor combines with another equally ill deﬁned factor from the path integral into a well deﬁned product in (2.2). Since a free particle has conserved momentum, it is advantageous to transform K0 into the momentum representation 1 − i pΔx e h¯ K0 (Δx, Δt) dΔx 2π¯ h m − i pΔx i m Δx2 1 e h¯ e h¯ 2Δt dΔx . = √ hiΔt 2π¯ h 2π¯

K0 (p, Δt) = √

(2.4)

We can now use again the integral relation (B.18) in the form +∞

e

−aΔx2 +bΔx

dΔx =

−∞

π b2 e 4a a

(2.5)

(with a = −im/(2¯ hΔt) and b = −ip/¯ h) to write

(

1 m π2¯ hΔt − i p2 Δt e h¯ 2m K0 (p, Δt) = 2π¯ h iΔt −im 2 i p 1 e− h¯ 2m Δt , = √ 2π¯ h

(2.6)

so that we obtain 1 i pΔx K0 (x, t; xi , ti ) = √ e h¯ K0 (p, Δt) dp 2π¯ h p2 Δt 1 h¯i pΔx− 2m e dp Θ(Δt) , = 2π¯ h

(2.7)

where in the last line the causality condition Θ(Δt) has been written explicitly; it takes care of the boundary condition K = 0 for Δt < 0. Eq. (2.7) is just the Fourier representation of the propagator (2.3). The step function can be rewritten using the relation +∞ 1 eiωΔt Θ(Δt) = dω 2πi ω − iε −∞

(ε > 0) ,

(2.8)

CHAPTER 2. PERTURBATION THEORY

22

which follows directly from the residue theorem: for Δt > 0 the integral can be closed in the upper half of the complex ω plane; Cauchy’s integral theorem then gives 2πi in the limit ε → 0 so that Θ(t) = 1. If Δt < 0, on the other hand, then the loop integration can only be closed in the lower half-space thus missing the pole at ω = +iε). Multiplying (2.6) with (2.8) gives

K0 (x, t; xi , ti ) =

1 (2π)2 h ¯i

+∞

dp dω

e

i h ¯

!

−∞

We now substitute E=

pΔx−

p2 −¯ hω 2m

" Δt

ω − iε

.

p2 −h ¯ω 2m

(2.9)

(2.10)

and obtain K0 (x, t; xi , ti ) =

i 1 i¯ h (pΔx−EΔt) h ¯ dp dE e 2 p (2π¯ h)2 E − 2m + iε

(2.11)

(the “−” sign coming from the substitution is cancelled by another sign obtained by inverting the integration boundaries). Since we will need these expressions later on in three space dimensions we give them here in a straightforward generalization of (2.7) and (2.11) i p2 −t) p ·( x − x )− (t 1 2m K0 (x , t ; x, t) = e h¯ d3p Θ(t − t) , (2π¯ h)3

(2.12)

and K0 (x, t; xi , ti ) =

i 1 3 i¯ h ( p·Δ x−EΔt) h ¯ . d p dE e 2 p (2π¯ h)4 E − 2m + iε

(2.13)

The integrand on the right-hand side of (2.11) is the free propagator in the energy-momentum representation. Since p and E are independent variables in the integral, we see that propagation also takes place at energies E = p2 /2m. The classical dispersion relation does show up as a pole in the propagator.

2.2

Perturbative Expansion

We now assume that the unperturbed particle moves freely and that the perturbing interaction is given by V (x, t). We furthermore assume that H

CHAPTER 2. PERTURBATION THEORY

23

has the special form H = p2 /2m + V (x, t), so that the propagator is given by i (2.14) K (x, t; xi , ti ) = N Dx e h¯ S with

t

t

S=

L(x, x) ˙ dt = ti

ti

m 2 x˙ − V (x, t) dt . 2

(2.15)

Since the integrand here is a classical function, we have

e

i h ¯

i S h ¯

=e

t ti

m 2 x˙ 2

i dt − h ¯

e

t

V (x,t) dt

ti

.

(2.16)

The second factor can now be expanded in powers of the potential, thus yielding a perturbative expansion of the action i h ¯

e

t

V (x,t) dt

ti

⎛ t

⎞2

1 1 ⎝ i ∼ V (x, t) dt − V (x, t) dt⎠ + . . . . =1− 2 h ¯ 2! h ¯ t t t

i

(2.17)

i

When we substitute this expansion into the expression (2.14) we obtain

K=N

) i

Dx e h¯ S0

⎡ ⎢

× ⎣1 −

i h ¯

t

⎤ ⎛ t ⎞2 , 1 1 ⎝ ⎥ V (x, t) dt − V (x, t) dt⎠ + . . .⎦ 2

2! h ¯

ti

ti

= K0 + K1 + K2 + . . . ,

(2.18)

i.e. a sum ordered in powers of the interaction. First order propagator. We next determine the ﬁrst-order propagator K1 . According to (2.18) it is given by i i = − N Dx e h¯ S0 V (x, t) dt h ¯ tf

K1

ti

= − ×

m i lim h ¯ n→∞ 2π¯ hiη +∞

n

−∞ i=1

dxi

(2.19)

n k=1

n+1 2

⎛

V (xk , tk ) η exp ⎝i

n m

2¯ hη

j=0

⎞

(xj+1 − xj )2 ⎠ .

CHAPTER 2. PERTURBATION THEORY

24

Here the time-integral has been written as a sum. In a next step we now split the sum in the exponent into two pieces, one running from j = 0 to j = k − 1 and the other from j = k to j = n and separate the corresponding integrals. This gives

K1

n i = − n→∞ η dxk V (xk , tk ) lim h ¯ k=1 ⎛

⎜

k

2 ×⎜ ⎝N

dx1 dx2 · · · dxk−1 e

⎛ n−k+1 ×⎜ ⎝N 2

dxk+1 · · · dxn e

with N≡

(2.20)

m i 2¯ hη

m i 2¯ hη

-

k−1 j=0

n j=k

(xj+1 −xj )

(xj+1 −xj )2

⎞ 2

⎟ ⎟ ⎠

⎞ ⎟ ⎠

m . 2π¯ hiη

(2.21)

The term in the ﬁrst bracket is nothing else than the propagator from ti to tk (K0 (xk , tk ; xi , ti )) , and that in the second bracket is that from tk to t (K0 (x, t; xk , tk )). Thus we have K1 (xf , tf ; xi , ti ) = +∞ tf i dx dt K0 (xf tf ; x, t) V (x, t)K0 (x, t; xi , ti ) . − h ¯ −∞

(2.22)

ti

The time integral over the interval from ti to tf can be extended to ∞ by noting that K0 (x, t; xi , ti ) = 0 K0 (xf , tf ; x, t) = 0

for t < ti for tf < t .

(2.23)

This gives K1 (xf , tf ; xi , ti ) = +∞ +∞ i − dt dx K0 (xf , tf ; x, t) V (x, t)K0 (x, t; xi , ti ) . h ¯ −∞

−∞

(2.24)

CHAPTER 2. PERTURBATION THEORY

25

xf tf

x2 t2 =

+

+ x1 t1

+ ... x1 t1

xi ti Figure 2.1: Born series for the propagator Higher order propagators. Similarly, the higher order terms in the perturbation expansion (2.18) can be obtained. This yields ﬁnally K (xf , tf ; xi , ti ) =

i K0 (xf , tf ) − K0 (xf , tf ; x1 , t1 ) V (x1 , t1 ) K0 (x1 , t1 ; xi , ti ) dx1 dt1 h ¯ ! 1 − 2 K0 (xf , tf ; x1 , t1 ) V (x1 , t1 ) K0 (x1 , t1 ; x2 , t2 ) h ¯ " × V (x2 , t2 ) K0 (x2 , t2 ; xi , ti ) dx1 dt1 dx2 dt2 + ··· .

(2.25)

Note that the time-integrations in (2.25) are eﬀectively time-ordered because of the implicit Θ (tf − ti ) function in each of the propagators, leading to tf > t1 > t2 > · · · > ti . This fact explains why, for example, the factor 1/(2!) in the second order term of (2.18) no longer appears in (2.25) 1 2!

2

dtV (x, t)

1 = V (x, t)V (x, t ) dt dt 2!

1 = [V (x, t)V (x, t )Θ (t − t) + V (x, t)V (x, t )Θ(t − t )] dt dt 2! =

V (x, t )Θ (t − t) V (x, t) dt dt .

(2.26)

The last line follows from a simple change of variables; the Θ function in it is absorbed into the propagator in (2.25). A similar argument holds for all

CHAPTER 2. PERTURBATION THEORY

26

the higher-order terms in the interaction. In each case the prefactor (1/n!) stemming from the expansion of the exponential in (2.17) is cancelled by the time ordering inherent in the propagators. Equation (2.25) is the Born series for the propagator; it can be represented graphically as shown in Fig. 2.1. A time axis runs here from bottom to top. The straight lines denote the free propagation of the particles (K0 ) and the dots stand for the interaction vertices (−iV /¯ h) (the circles mark the interaction range) and a space and time integration appears at each vertex. Bethe-Salpeter equation. The expansion (2.25) can be formally summed. This can be seen as follows (in obvious shortterm notation) K = K0 + K 0 U K 0 + K 0 U K 0 U K 0 + · · · = K0 + K0 U (K0 + K0 U K0 + · · ·)

(2.27) with U =

− ¯hi V

.

The expression in parentheses is just again K, so that we obtain the BetheSalpeter equation (2.28) K = K0 + K0 U K . The Bethe-Salpeter equation is an integral equation for the full interacting propagator K as can be seen most easily from its space-time representation K (xf , tf ; xi , ti ) = K0 (xf , tf ; xi , ti ) (2.29) i − K0 (xf , tf ; x, t) V (x, t)K (x, t; xi , ti ) dx dt . h ¯ We can also represent the Bethe-Salpeter equation in a diagrammatic way. If we denote the so-called “dressed propagator” K, that includes all the eﬀects of the interactions, by a double line = K (x2 , t2 ; x1 , t1 ) ,

(2.30)

then this equation can be graphically represented as shown in Fig. 2.2. Each graph in Fig. 2.2 ﬁnds a one-to-one correspondence in (2.28): the single lines represent the free propagator, the double line the dressed propagator and the dot stands for the interaction U . (2.28) shows that the factors in the second term of this equation have to be written from left to right against the time-arrow in Fig. 2.2. The Bethe-Salpeter equation can also be written in an equivalent form for the interacting wavefunction

Ψ (xf , tf ) =

K (xf , tf ; xi , ti ) Ψ (xi , ti ) dxi

CHAPTER 2. PERTURBATION THEORY

=

27

+

Figure 2.2: Bethe-Salpeter equation (2.28). The arrows indicate the time direction.

=

K0 (xf , tf ; xi , ti ) Ψ (xi , ti ) dxi

i K0 (xf , tf ; x, t) V (x, t)K (x, t; xi , ti ) Ψ (xi , ti ) dxi dx dt ¯ h = K0 (xf , tf ; xi , ti ) Ψ (xi , ti ) dxi (2.31) −

i − K0 (xf , tf ; x, t) V (x, t)Ψ(x, t) dx dt . h ¯ This constitutes an integral equation for the unknown wavefunction Ψ(x, t).

2.3

Application to Scattering

Let us now apply the results of the last section to a scattering process. In this case the particle is free at t = −∞, then undergoes the scattering interaction and then, at t = +∞, is free again. We treat this problem as usual by adiabatically switching on and oﬀ the interaction V (x, t). The initial condition (for t → −∞) for the wavefunction then is (2.32) Ψin (x, t) = N ei(ki ·x−ωi t) . We choose here √ a box normalization with periodic boundary conditions so that N = 1/ V . The scattering state that evolves from this incoming state is denoted by Ψ(+) (x, t). The superscript (+) indicates that the state evolves forward in time, starting from Ψin at t = −∞; it thus fulﬁlls the boundary condition (2.33) Ψ(+) (x, t → −∞) = Ψin (x, t) . In a scattering experiment one looks at t → +∞ for a free scattered particle

CHAPTER 2. PERTURBATION THEORY

28

with deﬁnite momentum; the corresponding ﬁnal state is denoted by

Ψout (x, t) = N ei(kf ·x−ωf t) .

(2.34)

The probability amplitude for the presence of Ψout in the scattered state Ψ(+) is given by

Sf i =

Ψ∗out (xf , tf ) Ψ(+) (xf , tf ) d3xf

for

tf → ∞ .

(2.35)

This is just the transition amplitude from the initial state i to the ﬁnal state f (S-matrix). Expansion (2.29) yields the Bethe-Salpeter equation for the scattering wavefunction

Ψ(+) (xf , tf ) = −

K0 (xf , tf ; xi , ti ) Ψin (xi , ti ) d3xi

(2.36)

i K0 (xf , tf ; x, t) V (x, t)K(x, t; xi , ti )Ψin (xi , ti ) d3xi d3x dt . h ¯

Inserting this into (2.35) gives for the S-matrix

Sf i =

Ψ∗out (xf , tf ) K0 (xf , tf ; xi , ti ) Ψin (xi , ti ) d3xi d3xf

−

i h ¯

(2.37)

Ψ∗out (xf , tf ) K0 (xf , tf ; x, t) V (x, t)

× K(x, t; xi , ti )Ψin (xi , ti ) d3xi d3x dt d3xf . Since Ψin is a plane wave, we know that

φ (xf , tf ) =

K0 (xf , tf ; xi , ti ) Ψin (xi , ti ) d3xi

(2.38)

is also a plane wave state, since K0 is the free propagator so that no interaction takes place. φ is actually the same wavefunction as Ψin , only taken at a later time and space point

φ (xf , tf ) = N ei(ki ·xf −ωi tf ) .

(2.39)

This means that the ﬁrst integral in (2.37) can be easily evaluated

Ψ∗out (xf , tf ) φ (xf , tf ) d3xf = δ 3 kf − ki

.

(2.40)

We thus have i 3 Sf i = δ kf − ki − d xf d3x d3xi dt (2.41) h ¯ ×[Ψ∗out (xf , tf ) K0 (xf , tf ; x, t) V (x, t)K(x, t; xi , ti )Ψin (xi , ti ) ] . 3

CHAPTER 2. PERTURBATION THEORY

29 xf tf

xt

xi ti Figure 2.3: First order scattering diagram, corresponding to the second term in (2.41). Time runs from left to right. The amplitude for a scattering process is given by the second term. If K on the rhs of (2.41) is now represented by the Born series expansion of the Bethe-Salpeter equation (2.27) this amplitude can be graphically represented as shown in Fig. 2.3. This diagram can be translated into the amplitude just given by writing for each straight-line piece xt t1

x2 t2

= K0 (x2 , t2 ; x1 , t1 )

(2.42)

and for each interaction vertex xt

i = − V (x, t) + integration over x, t . h ¯

(2.43)

The rules are completed by multiplying Ψin and Ψ∗out at the corresponding sides of the diagram and then integrating over the spatial variables of these wavefunctions and over all intermediate times. The ordering of factors is such that in writing the various factors from left to right one goes against the ﬂow of time in the ﬁgure. These rules are illustrated for a second-order scattering process in Fig. 2.4. In this ﬁgure the time runs from ti to tf from left to right. The corresponding amplitude is given by (2)

A

=

&

d xi d xf d x1 d x2 dt1 dt2 Ψ∗out (xf , tf ) K0 (xf , tf ; x2 , t2 ) 3

3

3

3

'

i i − V (x2 , t2 ) K0 (x2 , t2 ; x1 , t1 ) − V (x1 , t1 ) K0 (x1 , t1 ; xi , ti ) Ψin (xi , ti ) . h ¯ h ¯ (2.44)

CHAPTER 2. PERTURBATION THEORY

30 xf tf

x1 t1

x2 t2 xi ti Figure 2.4: Second order scattering diagram, corresponding to (2.44). We now evaluate the ﬁrst order scattering amplitude (cf. (2.41)) A

i ! ∗ = − Ψout (xf , tf ) K0 (xf , tf ; x, t) V (x, t)K0 (x, t; xi , ti ) h ¯ " × Ψin (xi , ti ) d3x dt d3xi d3xf (2.45)

(1)

somewhat further by using expression (2.6) for the free propagator (extended to three dimensions) p2 (t −t) 1 h¯i p·(x −x)− 2m K0 (x , t ; x, t) = e d3p Θ(t − t) . 3 (2π¯ h)

(2.46)

Inserting this into the expression for A(1) gives A(1) = −

e

i −h ¯

i 1 1 3 d x dt d3xi d3xf d3p d3q 6 h ¯ V (2π¯ h)

(pf ·xf −Ef tf ) e

i h ¯

2

p p ·( xf − x)− 2m (tf −t)

V (x, t)e

(2.47) i h ¯

2

q q·( x− xi )− 2m (t−ti )

e

i ( pi · xi −Ei ti ) h ¯

,

where the time-ordering tf > t > ti is understood and we have abbreviated Ei = p2i /(2m) and Ef = p2f /(2m). We integrate ﬁrst over xi and xf . This yields

d3xi −→

(2π¯ h)3 δ 3 (pi − q)

d3xf −→ (2π¯ h)3 δ 3 (pf − p) .

Performing next the integrals over p and q gives for the amplitude (1)

A

i 3 =− d x dt h ¯V

(2.48)

CHAPTER 2. PERTURBATION THEORY

× e

i +h E t ¯ f f

e

i h ¯

p2

31

f − pf · x− 2m (tf −t)

V (x, t)e

i h ¯

p2

i (t−t ) p i · x− 2m i

i i h¯i (−pf ·x+Ef t) = − e V (x, t)e h¯ (+pi ·x−Ei t) d3x dt . h ¯V

e

i −h Et ¯ i i

(2.49)

The time-integration is performed next. For that we assume that V (x, t) acts only during a ﬁnite, but long time-interval ([−T, T ]), in which it is timeindependent; at the boundaries of the interval it is adiabatically, i.e. without any signiﬁcant energy-transfer, turned on and oﬀ. We then obtain

dt V (x, t)e (Ef −Ei )t dt = i h ¯

T

T →∞

V (x)ei(ωf −ωi )t dt −→ V (x) 2πδ (ωf − ωi ) ,

−T

(2.50) with ωf − ωi = (Ef − Ei )/¯ h. Thus A(1) becomes A(1) = −

i i 2πδ (ωf − ωi ) e h¯ (pi −pf )·x V (x) d3x . h ¯V

This is the well-known lowest-order Born-approximation result.

(2.51)

Chapter 3 GENERATING FUNCTIONALS In this section we consider the transition amplitude in the presence of an external “source” J(t), so that the Hamiltonian reads ¯ J(t)x . HJ (x, p) = H(x, p) − h

(3.1)

A classical example is that of a harmonic oscillator with externally driven equilibrium position x0 (t). Its Hamiltonian is p2 1 + k(x − x0 (t))2 2m 2 = H − kx0 (t)x + O(x20 ) .

H x0 =

(3.2)

With h ¯ J(t) = kx0 (t) and for small amplitudes x0 we just have the form of (3.1). It is evident that the states of this system will change as time develops, because of its changing equilibrium position. Suppose now that the system was in its groundstate at t → −∞ and that x0 (t) acts only for a limited time period. We could then calculate the transition probability for the system to be still in its groundstate at t → +∞ by using the techniques developed in sections 2.2 and 2.3. There we saw (cf. (2.25) and (2.44)) that this probability is determined by matrixelements of time-ordered products of the interaction, i.e. of the operator xˆ in the present case. In this chapter we will show that these matrixelements can all be generated once only the functional dependence of the probability for the system to remain in its groundstate on an external source is known.

32

CHAPTER 3. GENERATING FUNCTIONALS

3.1

33

Groundstate-to-Groundstate Transitions

Generalizing the special example of the introduction we assume that an arbitrary physical system is at ﬁrst (at ti ) stationary and then changes under the inﬂuence of an external source J(t)x of ﬁnite duration. After the source has been turned oﬀ, the groundstate of the system is still the same (up to a phase), but the system may be excited. The propagator for this system, is quite generally, given by

xf , tf |xi , ti J =

Dx

Dp e

i h ¯

tf

[px−H(x,p)+¯ ˙ hJ(t)x] dt

ti

.

(3.3)

The value of this propagator depends obviously on the source function J(t); it is a functional of the source J(t). In the following paragraphs we will now discuss this functional dependence for tf → +∞ and ti → −∞. We will also show that it determines the vacuum expectation values of time-ordered xˆ operators. We start by calculating the propagator (3.3) of a system under the inﬂuence of a source, i.e. for a system described by (3.1). We ﬁrst assume that the source is nonzero only for a limited time between −T and +T J(t) = 0

for |t| > T .

(3.4)

We can then write for the propagator (ti < −T , tf > T ) xf tf |xi ti J =

dx dx xf tf |x T x T |x −T J x −T |xi ti .

(3.5)

Note that the two outer propagators are taken with the sourceless Hamiltonian (J = 0) because of condition (3.4). They are given by, e.g., ˆ

x −T |xi ti = x|e− h¯ H(−T −ti ) |xi i

=

ϕn (x)ϕ∗n (xi ) e− h¯ En (−T −ti ) . i

(3.6)

n

Similarly we get ˆ

xf tf |x T = xf |e− h¯ H(tf −T ) |x i

=

ϕn (xf )ϕ∗n (x )e− h¯ En (tf −T ) . i

(3.7)

n

ˆ without a source; we Here the ϕn are eigenstates of the Hamiltonian H ˆ is bounded from below with eigenvalues assume that the spectrum of H En ≥ E0 . The dependence on ti and tf is now isolated.

CHAPTER 3. GENERATING FUNCTIONALS

34

Taking now the limits ti → −∞ and tf → +∞ is not straightforward because both times appear as arguments of oscillatory functions. These functions oscillate the more rapidly the higher the eigenvalues En are. We can thus expect that the dominant contribution will come from the lowest eigenvalue E0 . Wick Rotation. This can indeed be shown by by a mathematical trick, the so-called Wick rotation. In this method one looks at the propagators (3.6) and (3.7) as functions of ti and tf , respectively, and continues these variables analytically from the real axis into the complex plane. One then performs the limits and after that rotates back to real times. Mathematically this is achieved by replacing the physical Minkowski time t by a complex time τ (3.8) t −→ τ = te−iε . The direction of the rotation is mandated by the requirement that there are no singularities of the integrand encountered in the rotation of the timeaxis. This is the case for 0≤ε<π . (3.9) For larger angles the rhs of (3.6) and (3.7) develops an essential singularity for ti/f → −/ + ∞. This rotation corresponds to an analytic continuation in the variable ε; the real, four-dimensional Minkowski space can be viewed as a subspace of a four-dimensional complex space (or of a ﬁve-dimensional space with 3 real space-coordinates and 1 complex time-coordinate). The substitution (3.8) corresponds to a clockwise rotation of the time-axis out of the real subspace into the larger complex space. For ε = +π/2 this corresponds to a rotation t = +∞ → τ = −i∞, t = −∞ → τ = +i∞; only this special case is often called a Wick rotation. Physical results are obtained by evaluating the relevant expressions in the enlarged, complex space and then, at the end, rotating the result back to the real time-axis (which works only if no singularities are encountered on this way). We now apply this rotation to the expression e− h¯ E0 ti x −T |xi ti = i

ϕn (x)ϕ∗n (xi ) e h¯ En T e h¯ (En −E0 )ti i

i

(3.10)

n

and thus perform the limit ti → −∞ on the rotated time axis τi →−∞(cos ε−i sin ε)

=

e− h¯ E0 τi x −T |xi τi i

lim

lim

τi →−∞(cos ε−i sin ε) n

(3.11)

ϕn (x)ϕ∗n (xi ) e h¯ En T e+ h¯ (En −E0 )τi . i

i

CHAPTER 3. GENERATING FUNCTIONALS

35

Because we have separated out the lowest frequency the last factor vanishes for ε > 0 , except for n = 0. Thus we get e− h¯ E0 τi x −T |xi τi = ϕ0 (x)ϕ∗0 (xi ) e h¯ E0 T . i

lim

τi →−∞(cos ε−i sin ε)

i

(3.12)

Analogously we also obtain e+ h¯ E0 τf xf τf |x T = ϕ0 (xf )ϕ∗0 (x ) e h¯ E0 T . i

lim

τf →+∞(cos ε−i sin ε)

i

(3.13)

Both expressions, (3.12) and (3.13), are the analytical continuations of the corresponding limits on the real time axis from ε = 0 to ε > 0. Since the right hand sides of these equations do not depend on ε they can be continued back to the real time axis (ε = 0) without any change. By means of the Wick rotation we have thus been able to make the expression (3.12) and (3.13) convergent; in this process we have found that only the groundstate contributes to the transition probability. We now insert these expressions into (3.5) and get lim

=

e h¯ E0 (τf −τi ) xf τf |xi τi J i

τi →−∞(cos ε−i sin ε) τf →∞(cos ε−i sin ε)

i ϕ0 (xf )ϕ∗0 (xi ) e h¯ E0 2T

(3.14)

dx dx ϕ∗0 (x ) x T |x −T J ϕ0 (x)

The integral in the last line can be rewritten, using ϕ0 (x) = x|0. This gives

dx dx

ϕ∗0 (x ) x T |x

−T J ϕ0 (x) =

dx dx 0|x x T |x −T J x|0

= 0T |0 −T J

(3.15)

so that we obtain for (3.14) e h¯ E0 (τf −τi ) xf τf |xi τi J i

lim

τi →−∞(cos ε−i sin ε) τf →∞(cos ε−i sin ε)

= ϕ0 (xf )ϕ∗0 (xi ) e h¯ E0 2T 0T |0 −T J . i

(3.16)

We obtain by Wick-rotating back e h¯ E0 (tf −ti ) e− h¯ E0 2T 0T |0 −T J = t lim xf tf |xi ti J . i →−∞ ϕ∗0 (xi )ϕ0 (xf ) t →+∞ i

i

(3.17)

f

With (in the limit ti → −∞, tf → +∞) xf tf |xi ti J=0 = ϕ∗0 (xi )ϕ0 (xf )e− h¯ E0 (tf −ti ) i

(3.18)

CHAPTER 3. GENERATING FUNCTIONALS

36

(3.17) becomes 0T |0 −T J = t lim →−∞

i tf →+∞

xf tf |xi ti J − i E0 2T e h¯ . xf tf |xi ti 0

(3.19)

This immediately gives for the free (J = 0) transition amplitude 0T |0 −T 0 = e− h¯ E0 2T , i

(3.20)

as it should. Equation (3.17) implies that the groundstate-to-groundstate transition amplitude is – up to a factor – given by a path integral from arbitrary xi to arbitrary xf and thus does not depend on these quantities, if only the corresponding times are taken to inﬁnity. Gs-to-gs transition rate. The vacuum transition rates 0T |0 −T J deserve some explanation. Formally, they are given by 0T |0 −T J = 0|e− h¯ (H−¯hJx)2T |0 . i

(3.21)

The vacuum state |0 is assumed to be unique, if there is no source J present, and normalized to 1. It is the vacuum of the theory before and after the i action of the source. On the other hand, e− h¯ (H−¯hJx)2T |0 is the state that the vacuum at t = −T has evolved into at t = +T under the inﬂuence of the external source J(t). If this external perturbation has acted adiabatically (very gently turned on and oﬀ again) the vacuum at t > T diﬀers from that of the source-free theory only by a phase. The matrixelement (3.21) is then the probability amplitude to ﬁnd the original vacuum in the time-evolved vacuum state. The absolute value of this probability amplitude is surely 1, so that the two states can diﬀer only by a J-dependent real phase i

0T |0 −T J = e h¯ (S[J]−E0 2T )

(3.22)

where we have taken out the free propagation contribution. This phase is the quantity determined by (3.17). To conclude these considerations we note that instead of using the Wickrotation to make the transition rates well behaved for very large times we could also have added a small negative imaginary term −iεEn to all eigenvalues En . This would have given a damping factor to the oscillating exponentials that becomes larger with n and thus would have led to a suppression of all higher excitations. This becomes apparent by looking at expressions (3.6) and (3.7). At the end of the calculation the limit ε → 0 would have to be performed.

CHAPTER 3. GENERATING FUNCTIONALS

3.1.1

37

Generating functional.

This groundstate-to-groundstate transition rate is a functional of the source J(t) which we denote by xf tf |xi ti J − i E0 2T e h¯ . xf tf |xi ti J=0

W [J] = 0 +∞|0 −∞J = t lim →−∞

i tf →+∞ T →∞

(3.23)

W [J] is called a generating functional for reasons that will become clear in the next section. In order to get rid of the phase that is produced already by a source-free propagation (exp − ¯hi E0 2T ) we deﬁne now a normalized generating functional Z[J] =

xf tf |xi ti J 0 +∞|0 −∞J W [J] = t lim = i →−∞ xf tf |xi ti J=0 W [0] 0 +∞|0 −∞J=0 t →+∞

(3.24)

f

with Z[0] = 1. The functional Z[J] describes the processes relative to the unperturbed (J = 0) time-development. The numerator in (3.24) is a transition amplitude and can therefore be written as a path integral xf + ∞|xi − ∞J =

Dx

Dp e

i h ¯

+∞

[px−H(p,x)+¯ ˙ hJ(t)x] dt

−∞

.

(3.25)

If H is quadratic in p and of the form H = p2 /(2m)+V (x), the propagator can be rewritten as (see Sect. 1.3) xf + ∞|xi − ∞J = N

Dx e

i h ¯

+∞

[L(x,x)+¯ ˙ hJ(t)x] dt

−∞

.

(3.26)

In the normalized functional Z[J] the (inﬁnite) factor N cancels out because it is independent of J Z[J] =

W [J] 0 +∞|0 −∞J = W [0] 0 +∞|0 −∞J=0

=

Dx e

i h ¯

+∞

[L(x,x)+¯ ˙ hJ(t)x] dt

−∞

Dx e

i h ¯

+∞ −∞

[L(x,x)] ˙ dt

(3.27)

CHAPTER 3. GENERATING FUNCTIONALS

3.2

38

Functional Derivatives of Transition Amplitudes

In this section we will show – by using the methods outlined in App. B – that the groundstate expectation value of a time-ordered product of interaction operators – in this present case of the operators xˆ(t) – can be obtained as functional derivatives of the functional W [J] with respect to J. We start with the deﬁnition of a path integral as a limit of a ﬁnite dimensional integral (see (1.33)) xf tf |xi ti J =

n

lim

n→∞

dxk

n dpl

k=1

l=0

⎛

(3.28)

2π¯ h

⎞

n i [pj (xj+1 − xj ) − ηH (pj , xj ) + h ¯ Jj xj ]⎠ . × exp ⎝ h ¯ j=0

and calculate its functional derivative with respect to J. In order to become familiar with functional derivatives we do this in quite some detail. Using the deﬁnition (B.26) for the functional derivative we get (with the abbreviation F (xj , pj ) = pj (xj+1 − xj ) − ηH(pj , x¯j )) δxf tf |xi ti J δJ(t1 ) )

n n 1 dpl = lim lim dxk ε→0 ε n→∞ h k=1 l=0 2π¯ ⎡

(3.29)

⎛

⎞

n i × ⎣exp ⎝ {F (xj , pj ) + h ¯ xj [Jj + εδ(tj − t1 )]}⎠ h ¯ j=0

⎞⎤⎫

⎛

n ⎬ i − exp ⎝ (F (xj , pj ) + h ¯ xj Jj )⎠⎦ ⎭ h ¯ j=0

=

lim

n→∞

n

dxk

k=1

⎛

l dpl

⎞

n i (F (xj , pj ) + h ¯ xj Jj )⎠ , ix1 exp ⎝ 2π¯ h h ¯ j=0 l=0

which can be written as δxf tf |xi ti J δJ(t1 )

= i

Dx

Dp x(t1 )e

i h ¯

tf ti

[px−H(x,p)+¯ ˙ hJ(t)x] dt

.

(3.30)

We now want to relate this derivative of a classical functional to quantum mechanical expressions and thus understand its physical signiﬁcance and

CHAPTER 3. GENERATING FUNCTIONALS

39

meaning. In order to do so, we go back to the deﬁnition of the propagator. There we had (in (1.23)) xf , tf |xi ti

(3.31)

dx1 . . . dxn xf tf |xn , tn xn tn |xn−1 tn−1 . . . x1 t1 |xi ti .

=

Now, in (3.30) for J = 0, we have one factor more on the righthand side of this equation

dx1 . . . dxn xf tf |xn tn . . . x1 t1 |xi ti x1

=

dx1 . . . dxn xf tf |xn tn . . . x1 t1 |ˆ x(t1 )|xi ti ;

(3.32)

the last step is possible because |x1 is an eigenstate of the xˆ operator with eigenvalue x1 (cf. (1.18)). The last integral is obviously equal to xf tf |ˆ x(t1 )|xi ti ,

(3.33)

i.e. to a matrixelement of the position operator. We thus have . δxf tf |xi ti J .. . δJ(t1 ) .

= i

Dx

Dp x(t1 )e

i h ¯

tf

[px−H(x,p)] ˙ dt

ti

J=0

= ixf tf |ˆ x(t1 )|xi ti .

(3.34)

For the case that H is separable in x and p and quadratic in p, this relation reads . δxf tf |xi ti J .. . δJ(t1 ) .

= iN

Dx x(t1 )e

i h ¯

tf

L(x,x) ˙ dt

ti

J=0

= ixf tf |ˆ x(t1 )|xi ti .

(3.35)

The functional derivative of the propagator with respect to the source thus gives the transition matrix element of the coordinate x. The higher order functional derivatives yield δ n xf tf |xi ti J δJ(t1 )δJ(t2 ) . . . δJ(tn ) n

= (i)

Dx

Dp x(t1 )x(t2 ) . . . x(tn ) e

(3.36) i h ¯

tf ti

[px−H(x,p)+¯ ˙ hJ(t)x] dt

.

CHAPTER 3. GENERATING FUNCTIONALS

40

One might guess that .

. δ n xf tf |xi ti J . . = in xf tf |ˆ x(t1 )ˆ x(t2 ) . . . xˆ(tn )|xi ti , δJ(t1 )δJ(t2 ) . . . δJ(tn ) .J=0

(3.37)

but this equation is not quite correct. We see this by considering explicitly the second derivative. We can proceed there exactly in the same way as for the ﬁrst. We have for the rhs of (3.36) in the case of the second derivative . δ xf tf |xi ti J .. . δJ(tα )δJ(tβ ) .

2

= i2

J=0

= i2

Dx

Dp x(tα )x(tβ )e

i h ¯

tf

[px−H(x,p)] ˙ dt

ti

dxi . . . dxn xf tf |xn tn · · · xl tl |ˆ x(tα )|xl−1 tl−1

x(tβ )|xk−1 tk−1 · · · x1 t1 |xi ti . · · · xk tk |ˆ

(3.38)

Here we have assumed that tα > tβ , since each of the inﬁnitesimal Green’s functions propagates only forward in time. In this case (3.38) is indeed equal to x(tα )ˆ x(tβ )|xi ti . (3.39) i2 xf tf |ˆ However, if tα < tβ , then these two times appear in a diﬀerent ordering in the rhs of (3.38) and thus of the matrix element. The two cases can be combined by introducing the time-ordering operator T )

x(t2 )] = T [ˆ x(t1 )ˆ

xˆ(t1 )ˆ x(t2 ) xˆ(t2 )ˆ x(t1 )

t1 > t2 t2 > t1 .

(3.40)

With the time-ordering operator we have . δ 2 xf tf |xi ti J .. . δJ(t1 )δJ(t2 ) .

2

= i

Dx

Dp x(t1 )x(t2 )e

i h ¯

tf

[px−H(x,p)] ˙ dt

ti

J=0

= i2 xf tf |T [ˆ x(t1 )ˆ x(t2 )] |xi ti .

(3.41)

The same reasoning leads to the following result for higher-order derivatives n 1

i

=

.

. δ n xf tf |xi ti J . . δJ(t1 )J(t2 ) . . . δJ(tn ) .J=0

Dx

Dp x(t1 )x(t2 ) . . . x(tn ) e

i h ¯

tf

[px−H(x,p)] ˙ dt

ti

= xf tf |T [ˆ x(t1 )ˆ x(t2 ) . . . xˆ(tn )] |xi ti .

(3.42)

CHAPTER 3. GENERATING FUNCTIONALS

41

This is the generalization of (3.41). We thus have n

1 i

n

1 i

=

=

.

. δn Z[J]... δJ(t1 )δJ(t2 ) . . . δJ(tn ) J=0

Dx Dp x(t1 )x(t2 ) . . . x(tn ) e

=

.

δn xf tf |xi ti J .. . δJ(t1 )δJ(t2 ) . . . δJ(tn ) xf tf |xi ti J=0 .J=0

Dx Dp e

i h ¯

tf

i h ¯

tf

[px−H(x,p)] ˙ dt

ti

[px−H(x,p)] ˙ dt

ti

x(t1 )ˆ x(t2 ) . . . xˆ(tn )] |xi ti xf tf |T [ˆ , xf tf |xi ti

(3.43)

where the limit ti → −∞, tf → +∞ is understood and all the times t1 , . . . , tn lie in between these limits. If the Hamiltonian is quadratic in p and separates in p and x, then we have x(t1 )ˆ x(t2 ) . . . xˆ(tn )] |xi ti xf tf |T [ˆ = xf tf |xi ti

i

Dx x(t1 )x(t2 ) . . . x(tn ) e h¯ S[x(t)]

i

Dx e h¯ S[x(t)]

,

(3.44) where S[x(t)] is the action that depends functionally on the trajectory x(t). We now rewrite this equation. The numerator of the lhs becomes (in the limit ti → −∞, tf → +∞, indicated by the arrow) xf tf |T [. . .]|xi ti J=0 −→ xf tf |00|T [. . .]|00|xi ti J=0 = 0|T [. . .]|0 ϕ0 (xf )e− h¯ E0 tf ϕ∗0 (xi )e+ h¯ E0 ti , i

i

(3.45)

while the denominator can be written as (c.f. (3.18)) ˆ

ˆ

xf tf |xi ti = xf |e− h¯ Htf e+ h¯ Hti |xi −→ ϕ0 (xf )e− h¯ E0 tf ϕ∗0 (xi )e+ h¯ E0 ti i

i

i

i

(3.46)

where we have used in both cases (1.19), inserted a complete set of states and – through the limit of inﬁnite times – projected out the groundstate. The gs wavefunctions and the time-dependent exponentials cancel out so that we obtain (for quadratic Hamiltonians)

x(t2 ) . . . xˆ(tn )] |0 = 0|T [ˆ x(t1 )ˆ

i

Dx x(t1 )x(t2 ) . . . x(tn ) e h¯ S[x(t)]

i

Dx e h¯ S[x(t)] . n . 1 δn = Z[J]... i δJ(t1 )δJ(t2 ) . . . δJ(tn ) J=0 (3.47)

CHAPTER 3. GENERATING FUNCTIONALS

42

+∞ with S[x(t)] = −∞ L(x(t), x(t)) ˙ dt. This is a very important result. It shows that the groundstate expectation value of a time-ordered product of position operators, the so-called correlation function, can be obtained as a functional derivative of the functional Z[J] ˆ as can be seen deﬁned in (3.24). Note that |0 is the groundstate of H, from (3.45). Thus the groundstate appearing on the lhs of (3.47) and the propagator Z[J] are linked together: if Z[J] contains, for example, only a free Hamiltonian, then |0 is the groundstate of a free theory. If, on the other hand, Z[J] contains interactions, then |0 is the groundstate of the full interacting theory. Since all the correlation functions can be generated from Z[J] this functional is called a generating functional. For the remainder of this book we will be concerned with these generating functionals. Coming back to our example of the driven harmonic oscillator, discussed at the start of this section, we see that the time-ordered vacuum expectation values of xˆ are just the matrix elements that would appear in a timedependent perturbation theory treatment of the groundstate of this system. Thus, if all the correlation functions are known, then the perturbation series expansion is also known.

Part II Relativistic Quantum Field Theory

43

Chapter 4 CLASSICAL RELATIVISTIC FIELDS In this chapter a few essential facts of classical relativistic ﬁeld theory are summarized. It will ﬁrst be shown how to derive the equations of motion of a ﬁeld theory, for example the Maxwell equations of electrodynamics, from a Lagrangian. Second, the connection between symmetries of the Lagrangian and conservation laws will be discussed1 .

4.1

Equations of Motion

The equations of motion of classical mechanics can be obtained from a Lagrange function by using Hamilton’s principle that the action for a given mechanical system is stationary for the physical space–time development of the system. The equations of motion for ﬁelds that determine their space–time dependence can be obtained in an analogous way by identifying the ﬁeld amplitudes at a coordinate x with the dynamical variables (coordinates) of the theory. Let the functions that describe the ﬁelds be denoted by Φα (x) with xμ = (t, x) ,

(4.1)

where α labels the various ﬁelds appearing in a theory. The Lagrangian L of the system is expressed in terms of a Lagrange density L, as follows:

L=

L(Φα , ∂μ Φα ) d3 x

1

(4.2)

The units, the metric and the notation used in this and the following chapters is explained in Appendix A.

44

CHAPTER 4. CLASSICAL RELATIVISTIC FIELDS

45

where the spatial integration is performed over the volume of the system. The action S is then deﬁned as usual by t1

L dt =

S= t0

L(Φα , ∂μ Φα ) d4 x

(4.3)

Ω

with the Lorentz-invariant four-dimensional volume element d4 x = d3 x dt. The, in general ﬁnite, space–time volume of the system is denoted by Ω. As pointed out before the ﬁelds Φα (x) play the same role as the generalized coordinates qi in classical mechanics; the analogy here is such that the ﬁelds Φα correspond to the coordinates q and the points x and α to the classical indices i. The corresponding velocities are given in a direct analogy by the time derivatives of Φα : ∂t Φα . Lorentz covariance then requires that also the derivatives with respect to the ﬁrst three coordinates appear; this explains the presence of the four-gradients ∂μ Φα in (4.2). In order to derive the ﬁeld equations from the action S by Hamilton’s principle, we now vary the ﬁelds and their derivatives Φα → Φα = Φα + δΦα ∂μ Φα → (∂μ Φα ) = ∂μ Φα + δ(∂μ Φα ) .

(4.4)

This yields δL = L(Φα , (∂μ Φα ) ) − L(Φα , ∂μ Φα ) ∂L ∂L δΦα + δ(∂μ Φα ) = ∂Φα ∂(∂μ Φα ) ∂L ∂L = δΦα + ∂μ (δΦα ) . ∂Φα ∂(∂μ Φα )

(4.5)

According to the Einstein convention a summation over μ is implicitly contained in this expression. In going from the second to the third line diﬀerentiation and variation could be commuted because both are linear operations. The equations of motion are now obtained from the variational principle

δS = Ω

∂L ∂L ∂L − ∂μ δΦα + ∂μ δΦα ∂Φα ∂(∂μ Φα ) ∂(∂μ Φα )

= 0

d4 x (4.6)

for arbitrary variations δΦα under the constraint that δΦα (t0 ) = δΦα (t1 ) = 0 ,

CHAPTER 4. CLASSICAL RELATIVISTIC FIELDS

46

where t0 and t1 are the time-like boundaries of the four-volume Ω. The last term in (4.6) can be converted into a surface integral by using Gauss’s law; for ﬁelds which are localized in space this surface integral vanishes if the surface is moved out to inﬁnity. Since the variations δΦα are arbitrary the condition δS = 0 leads to the equations of motion ∂ ∂xμ

∂L ∂L − =0. ∂(∂μ Φα ) ∂Φα

(4.7)

The relativistic equivalence principle demands that these equations have the same form in every inertial frame of reference, i.e. that they are Lorentz covariant. This is only possible if L is a Lorentz scalar, i.e. if it has the same functional dependence on the ﬁelds and their derivatives in each reference frame. In a further analogy to classical mechanics, the canonical ﬁeld momentum is deﬁned as ∂L ∂L = . (4.8) Πα = ˙ ∂(∂0 Φα ) ∂ Φα From L and Πα the Hamiltonian H is obtained as

H=

Hd x= 3

(Πα Φ˙ α − L) d3 x .

(4.9)

The Hamiltonian H represents the energy of the ﬁeld conﬁguration.

4.1.1

Examples

The following sections contain examples of classical ﬁeld theories and their formulation within the Lagrangian formalism just introduced. We start out with the probably best-known case of classical electrodynamics, then generalize it to a treatment of massive vector ﬁelds and then move on to a discussion of classical Klein–Gordon and Dirac ﬁelds that will play a major role in the later chapters of this book.

Electrodynamics The best-known classical ﬁeld theory is probably that of electrodynamics, in which the Maxwell equations are the equations of motion. The two homogeneous Maxwell equations allow us to rewrite the ﬁelds in terms of a four-potential , (4.10) Aμ = (A0 , A)

CHAPTER 4. CLASSICAL RELATIVISTIC FIELDS

47

deﬁned via = ∇ ×A , B = −∇A 0 − ∂A . E (4.11) ∂t Note that the 2 homogeneous Maxwell equations are now automatically fulﬁlled. The two inhomogeneous Maxwell equations2 ·E = ∇ ×B − ∂E = ∇ ∂t with external density ρ and current j can be ∂μ F μν =

ρ, j

(4.12)

rewritten as

∂F μν = jν ∂xμ

,

(4.13)

with the four-current j ν = (ρ, j)

(4.14)

and the antisymmetric ﬁeld tensor F μν =

∂Aν ∂Aμ − = ∂ μ Aν − ∂ ν Aμ . ∂xμ ∂xν

(4.15)

The tensor Fμν is the dyadic product of two fourvectors and thus a Lorentztensor. Equation (4.13) is the equation of motion for the ﬁeld tensor or the four-vector ﬁeld Aμ . It is easy to show that (4.13) can be obtained from the Lagrangian 1 L = − Fμν F μν − j ν Aν 4

(4.16)

by using (4.7); the ﬁelds Aν here play the role of the ﬁelds Φα in (4.7). We have ∂L = −j ν , ∂Aν 1 ∂L (4.17) = − 2(+F μν − F νμ ) = −F μν . ∂(∂μ Aν ) 4 The last step is possible because F is an antisymmetric tensor. The equation of motion is therefore ∂ ∂xμ 2

∂L ∂F μν ∂L − =− + jν = 0 , μ ∂(∂μ Aν ) ∂Aν ∂x

Here the Heaviside units are used with c = 1 and 0 = μ0 = 1.

(4.18)

CHAPTER 4. CLASSICAL RELATIVISTIC FIELDS

48

in agreement with (4.13). It is now easy to interpret the two terms in L (4.16): the ﬁrst one gives the Lagrangian for the free electromagnetic ﬁeld, whereas the second describes the interaction of the ﬁeld with charges and currents. The two homogeneous Maxwell equations can also be expressed in terms of the ﬁeld tensor by ﬁrst introducing the dual ﬁeld tensor 1 F˜ μν = μνρσ Fρσ ; 2

(4.19)

here μνρσ is the Levi–Civita antisymmetric tensor which assumes the values +1 or −1 according to whether (μνρσ) is an even or odd permutation of (0,1,2,3), and 0 otherwise. In terms of F˜ the homogeneous Maxwell equations read (4.20) ∂μ F˜ μν = 0 . Eqs. (4.13) and (4.20) represent the Maxwell equations in a manifestly covariant form. The Lagrangian (4.16) is obviously Lorentz-invariant since it consists of invariant contractions of two Lorentz-tensors (the ﬁrst term) and two Lorentz-vectors (the second term). It is also invariant under a gauge transformation (4.21) Aμ −→ Aμ = Aμ + ∂ μ φ because F itself is gauge-invariant by construction and the contribution of the interaction term to the action is gauge-invariant for an external conserved current. The same then holds for the equation of motion (4.18). Symmetry (Lorentz-invariance), gauge-invariance and simplicity (there are no higher order terms in (4.16)) thus determine the Lagrangian of electrodynamics. The gauge freedom can be used to impose constraints on the four components of the vector ﬁeld Aμ . In addition, for free ﬁelds this gauge freedom can be used, for example, to set the 0th component of the four-potential equal to zero. Thus, a free electromagnetic ﬁeld has only two degrees of freedom left.

Massive Vector Fields Vector ﬁelds in which – in contrast to the electromagnetic ﬁeld – the ﬁeld quanta are massive are described by the so-called Proca equation: ∂μ F μν + m2 Aν = j ν .

(4.22)

Operating on this equation with the four-divergence ∂ν gives, because F is antisymmetric, (4.23) m2 ∂ν Aν = ∂ν j ν .

CHAPTER 4. CLASSICAL RELATIVISTIC FIELDS

49

For m = 0 and a conserved current, this reduces the equation of motion (4.22) to 2 + m2 Aν = j ν ; ∂ν Aν = 0 . (4.24) Thus for massive vector ﬁelds the freedom to make gauge transformations on the vector ﬁeld is lost. The condition of vanishing four-divergence of the ﬁeld reduces the degrees of freedom of the ﬁeld from 4 to 3. The space-like components represent the physical degrees of freedom. The Lagrangian that leads to (4.22) is given by 1 1 L = − F 2 + m2 A2 − j · A . 4 2

(4.25)

Klein–Gordon Fields A particularly simple example is provided by the so-called Klein–Gordon ﬁeld φ that obeys the equation of motion

∂μ ∂ μ + m 2 φ = 2 + m 2 φ = 0 ;

(4.26)

such a ﬁeld describes scalar particles, i.e. particles without intrinsic spin. The Lagrangian leading to (4.26) is given by L= since we have

and

1 (∂μ φ) (∂ μ φ) − m2 φ2 2

(4.27)

∂L = ∂ μφ ∂ (∂μ φ)

(4.28)

∂L = −m2 φ . ∂φ

(4.29)

It is essential to note here that the Lagrangian density (4.27) that leads to (4.26) is not unique; unique is only the action S = L d4x. For localized ﬁelds for which the surface contributions vanish we can perform a partial integration of the kinetic term in (4.26)

so that we obtain

d4x (∂μ φ) (∂ μ φ) = −

d4x φ2φ

(4.30)

1 (4.31) L = − φ 2 + m2 φ . 2 The Lagrangians (4.31) and (4.27) are equivalent. Since both give the same action they also lead to the same equation of motion (4.26).

CHAPTER 4. CLASSICAL RELATIVISTIC FIELDS

50

An interesting case occurs if we consider two independent real scalar ﬁelds, φ1 and φ2 , with the same mass m. The total Lagrangian is then simply given by a sum over the Lagrangians describing the individual ﬁelds, i.e. L=

1 1 (∂μ φ1 ) (∂ μ φ1 ) − m2 φ21 + (∂μ φ2 ) (∂ μ φ2 ) − m2 φ22 . 2 2

(4.32)

On the other hand, we can also construct two complex ﬁelds from the two real ﬁelds φ1 and φ2 , namely 1 φ = √ (φ1 + iφ2 ) 2

(4.33)

and its complex conjugate. In terms of these the Lagrangian (4.32) can be rewritten to L = (∂μ φ)∗ (∂ μ φ) − m2 φ∗ φ =

− φ∗ 2 + m2 φ .

(4.34)

Dirac Fields A particularly simple example is provided by the Dirac ﬁeld Ψ for which the equation of motion is just the Dirac equation (iγ μ ∂μ − m) Ψ = 0 ,

(4.35)

where the γμ are the usual (4 × 4) matrices of Dirac theory. Ψ itself is a (4 × 1) matrix of four independent ﬁelds, a so-called spinor. The corresponding Lagrangian is given by ¯ (iγ μ ∂μ − m) Ψ . L=Ψ

(4.36)

This can be seen by identifying the ﬁelds Φα in (4.7) with the four components ¯ the equation ¯ = Ψ† γ0 . Since L does not depend on ∂μ Ψ of the Dirac spinor Ψ of motion is simply given by ∂L μ ¯ = (iγ ∂μ − m) Ψ = 0 . ∂Ψ

4.2

(4.37)

Symmetries and Conservation Laws

As in classical mechanics there is also in ﬁeld theory a conservation law associated with each continuous symmetry of L. The theorem which describes

CHAPTER 4. CLASSICAL RELATIVISTIC FIELDS

51

the connection between the invariance of the Lagrangian under a continuous symmetry transformation and the related conserved current is known as Noether’s theorem. In the following, this will be illustrated for diﬀerent types of symmetries which then lead to the well-known conservation laws. The common expression in the arguments to follow is the change of the Lagrangian density under a change of the ﬁelds and their derivatives (see (4.4)). According to (4.5) and the Lagrange equations of motion (4.7) this change is given by ∂L (4.38) δΦα . δL = ∂μ ∂(∂μ Φα )

4.2.1

Geometrical Space–Time Symmetries

In this section we investigate the consequences of translations in four-dimensional space–time, i.e. inﬁnitesimal transformations of the form xν → xν = xν + ν

,

(4.39)

where ν is a constant inﬁnitesimal shift of the coordinate xν . Under such transformations the change of L is given by δL = ν

∂L = ν ∂ ν L , ∂xν

(4.40)

since L is a scalar. If now L is required to be form-invariant under translations, it does not explicitly depend on xν . In this case, δL is also given by (4.38). The changes of the ﬁelds Φα appearing there are for the space–time translation considered here given by ∂Φα = ν ∂ ν Φα . (4.41) δΦα = ν ∂xν Inserting (4.41) into (4.38) yields

δL = ν ∂μ

∂L ∂ ν Φα ∂(∂μ Φα )

.

(4.42)

=0 ,

(4.43)

Equating (4.42) and (4.40) ﬁnally gives

∂μ

∂L ∂ ν Φα − L g μν ∂(∂μ Φα )

since the ν are arbitrary. By deﬁning the tensor T μν as T μν ≡

∂L ∂ ν Φα − L g μν ∂(∂μ Φα )

(4.44)

CHAPTER 4. CLASSICAL RELATIVISTIC FIELDS

52

this equation reads ∂μ T μν = 0 .

(4.45)

Relation (4.45) has the form of a continuity equation. Spatial integration over a ﬁnite volume yields ⎛

⎞

/ d ⎝ 0μ ∂T iμ 3 (μ) · n dS . S T (x)d3 x⎠ = − d x = − dt ∂xi V

V

(4.46)

S

(μ) Here n is a unit vector vertical on the surface S pointing outwards and S is a three-vector: (μ) = (T 1μ , T 2μ , T 3μ ) . S (4.47) The surface integral on the rhs of (4.46) is taken over the surface S of volume V. For localized ﬁelds it can be made to vanish by extending the volume towards inﬁnity. It is then evident that the quantities

Pμ =

T 0μ d3 x

(4.48)

are conserved. These are the components of the four-momentum of the ﬁeld, as can be veriﬁed for the zeroth component,

P

0

=

T

=

00

3

d x=

∂L ∂ 0 Φα − L d3 x ∂(∂0 Φα )

(Πα Φ˙ α − L) d3 x = H

,

(4.49)

according to (4.8) and (4.9). The spatial components of the ﬁeld momentum are ∂L k 0k 3 P = T d x= (4.50) ∂ k Φα d3 x . ∂(∂0 Φα ) T μν , as deﬁned in (4.44), has no speciﬁc symmetry properties. It can, however, always be made symmetric in its Lorentz-indices because (4.45) does not deﬁne the tensor T uniquely. We can always add a term of the form ∂λ Dλμν , where Dλμν is a tensor antisymmetric in the indices λ and μ, such that T becomes symmetric.3 3

In classical mechanics the form invariance of the Lagrangian under rotations leads to the conservation of angular momentum. Analogously, in a relativistic field theory the form invariance of L under four-dimensional space–time rotations (Lorentz covariance) leads to the conservation of a quantity that is identified with the angular momentum of the field. To obtain the same form for the angular momentum as in classical mechanics it is essential that Tμν is symmetric.

CHAPTER 4. CLASSICAL RELATIVISTIC FIELDS

53

Comparing (4.46) with (4.50) and assuming T to be symmetric we see that (μ) in (4.47) describe the momentum the normal components of the vectors S ﬂow through the surface S of the volume V and thus determine the “radiation pressure” of the ﬁeld.4 These properties allow us to identify Tμν as the energymomentum tensor of the ﬁeld. For the Lagrangian (4.16) of electrodynamics Tμν is just the well-known Maxwell’s stress tensor. As already mentioned at the beginning of this chapter these conservation laws are special cases of Noether’s theorem, which can be summarized for the general case as follows: Each continuous symmetry transformation that leaves the Lagrangian invariant is associated with a conserved current. The spatial integral over this current’s zeroth component yields a conserved charge.

4.2.2

Internal Symmetries

Relativistic ﬁeld theories may contain conservation laws that are not consequences of space–time symmetries of the Lagrangian, but instead are connected with symmetries in the internal degrees of freedom such as, e.g., isospin or charge. We therefore now allow for a mixture of the diﬀerent ﬁelds under the transformation (4.51) Φα (x) → Φα (x) = e−iεqαβ Φβ , where is an inﬁnitesimal parameter and the qαβ are ﬁxed c-numbers. We then have (4.52) δΦα (x) = Φα (x) − Φα (x) = −iεqαβ Φβ (x) . The change of the Lagrangian is given by (4.38)

δL = ∂μ

∂L δΦα ∂(∂μ Φα )

.

(4.53)

If L is invariant under this variation δΦα , then we have

δL = ∂μ

∂L δΦα = 0 . ∂(∂μ Φα )

(4.54)

Equation (4.54) is in the form of a continuity equation for the “current” j μ (x) = 4

1 ∂L δΦα . ∂(∂μ Φα ) ε

(4.55)

More precisely, S k(μ) denotes the flux of the μth component of the field momentum in the direction xk .

CHAPTER 4. CLASSICAL RELATIVISTIC FIELDS

54

Inserting the ﬁeld variations δΦα yields for the current j μ (x) = −i

∂L qαβ Φβ . ∂(∂μ Φα )

(4.56)

Equations (4.54) and (4.56) imply that the “charge”

Q=

j (x) d x = −i 0

3

∂L qαβ Φβ d3 x ∂(∂0 Φα )

(4.57)

of the system is conserved. The physical nature of these “charges” and “currents” has to remain open. It depends on the speciﬁc form of the symmetry transformation (4.63) and can be determined only by coupling the system to external ﬁelds.

Example: Quantum Electrodynamics To illustrate this conservation law, the theory of electromagnetic interactions is used as an example. However, in contrast to the considerations in Sect. 4.1.1 we now consider a coupled system of a fermion ﬁeld Ψ(x) and the electromagnetic ﬁeld Aμ (x) to determine the physical meaning of the conserved current. Together with a quantization procedure this theory is called Quantum Electrodynamics (QED). The Lagrangian is given by 1 ¯ [iγ μ (∂μ + ieAμ ) − m] Ψ . L = − Fμν F μν + Ψ 4

(4.58)

L contains a part that describes the free electromagnetic ﬁeld (ﬁrst term). The second term describes the fermion Lagrangian; it is obtained from the free particle Lagrangian of (4.36) by replacing the derivative ∂μ by the covariant derivative (4.59) Dμ = ∂μ + ieAμ (minimal coupling). Here e is the electron’s charge (e = −|e|). The Lagrangian (4.58) is obviously invariant under a variation of the fermion ﬁelds of the form Ψ → Ψ = e−ie Ψ .

(4.60)

Comparison with (4.51) gives qαβ = e δαβ so that the conserved “current” given by (4.56) is: ¯ μΨ . (4.61) jμ (x) = e Ψγ Note that this conserved current is exactly the quantity that couples to the electromagnetic ﬁeld in (4.58). This property allows one to identify the current (4.61) as the electromagnetic current of the electron ﬁelds.

CHAPTER 4. CLASSICAL RELATIVISTIC FIELDS

55

Example: Scalar Electrodynamics The Lagrangian for the case of a complex scalar ﬁeld interacting with an electromagnetic ﬁeld is given by 1 L = − Fμν F μν + (Dμ φ)∗ (Dμ φ) − m2 φ∗ φ . 4

(4.62)

This Lagrangian is simply the sum of the free electromagnetic Lagrangian (4.16) and the Lagrangian for a complex scalar ﬁeld (4.34), where in the latter again the derivative ∂μ has been replaced – through minimal substitution – by the covariant derivative Dμ (4.59). The Lagrangian (4.62) is obviously invariant under the phase transformations φ(x) −→ e−iεe φ(x) φ∗ (x) −→ e+iεe φ∗ (x)

(4.63)

The conserved current connected with this invariance can be obtained from the deﬁnition (4.56)

j

μ

∂L ∂L = −i eφ + (−e)φ∗ ∗ ∂ (∂μ φ) ∂ (∂μ φ ) ∗ μ μ = ie (φ D φ − φ (D φ)∗ ) .

(4.64)

The conserved charge is then given by

Q=

3

0

d x j (x) = ie

∗

d3 x φ∗ D0 φ − φ D0 φ

.

(4.65)

It is remarkable that now the electromagnetic ﬁeld Aμ appears in the conserved current (through the covariant derivative Dμ = ∂ μ + ieAμ ). Again the conserved current provides the coupling to the electromagnetic ﬁeld. If the scalar ﬁeld is real then it is invariant under the transformation (4.63) only for e = 0. Eq. (4.65) then shows that in this case the conserved charge Q = 0.

Chapter 5 PATH INTEGRALS FOR SCALAR FIELDS In this chapter we apply the methods developed in chapter 3 to the case of scalar ﬁelds. There we showed that the vacuum expectation values of timeordered products of xˆ operators could be obtained as functional derivatives of a generating functional. All these results can be taken over into ﬁeld theory remembering that ﬁelds play the role of the coordinates of the theory and the spatial locations x correspond to the indices of the classical coordinates. This implies that we can obtain the vacuum expectation values of time-ordered ﬁeld operators by performing the derivatives on an appropriate functional. To discuss this functional is the main purpose of this chapter. It is easy to see that these vacuum expectation values of time-ordered products of ﬁeld operators play an important role in quantum ﬁeld theory. Each ﬁeld operator creates or annihilates particles and a time-ordered product of ﬁeld operators can thus describe the probability amplitude for a physical process in which particles are created and annihilated. The quantitative information about such a process is contained in the S matrix which can be obtained from the vacuum expectation values of time-ordered ﬁeld operators by means of the so-called reduction theorem which we will derive in a later chapter. The rest of this manuscript will therefore be concerned with calculating these expectation values and with developing perturbative methods for their determination when an exact calculation is not possible.

56

CHAPTER 5. PATH INTEGRALS FOR SCALAR FIELDS

5.1

57

Generating Functional for Fields

We assume that the system is described by a Lagrangian of the form L (φ, ∂μ φ) =

1 μ ∂ φ∂μ φ − m2 φ2 − V (φ) . 2

(5.1)

In order to obtain the functional W [J] for ﬁelds we note that the ﬁelds play the role of the coordinates of the theory and that sums over the diﬀerent coordinates have to be replaced by integrals over the space-time coordinates. In order to deﬁne more stringently what is actually meant by a path integral for ﬁelds we write it down here in detail for a free ﬁeld Lagrangian (V = 0). In order to do so we bring it into a form as close as possible to the classical deﬁnition (1.38). We ﬁrst Fourier-expand the ﬁeld 1 qk (t)eik·x . φ(x, t) = √ V k For a real ﬁeld φ we have

∗ q− k = qk .

(5.2)

(5.3)

Inserting this expansion into the free Lagrangian gives1

1 μ L = Ld x = ∂ φ∂μ φ − m2 φ2 d3x 2 " 1 ! = qk qk k · k − m2 + q˙k q˙k ei(k+k )·x d3x . 2V kk 3

(5.4)

Integration over d3x gives V δk,−k so that we have 1 q˙k q˙−k − (k 2 + m2 )qk q−k 2 k 1 = (q˙k q˙k∗ − ωk qk qk∗ ) 2 k

L =

(5.5)

with ωk = k 2 + m2 . Using (5.3), writing qk = Xk + iYk , and grouping then the coordinates Xk and −iYk into a new vector xk gives L= 1

1 2 x˙ k − ωk2 x2k . 2 k

For ease of notation we drop the vector arrows in the indices

(5.6)

CHAPTER 5. PATH INTEGRALS FOR SCALAR FIELDS

58

In the general spirit of ﬁeld theory we introduce a source density J which we also expand 1 jk (t)eik·x . (5.7) J(x, t) = √ V k With this expansion we obtain for the source term

d3x J(x)φ(x) =

jk (t)q−k (t) =

k

Jk (t)xk (t)

(5.8)

k

where we have grouped jk and −ijk into a new vector Jk . The Lagrangian (5.6) has the structure of a Lagrangian with quadratic momentum dependence and constant coeﬃcient, discussed in Sect. 1.3.1; also the souce term reads formally just the same as for the nonrelativistic systems treated in chapter 3. We can thus again integrate the momentum dependence out and obtain for the vacuum-to-vacuum transition amplitude (cf. (3.23)) W [J] =

lim 0tf |0ti J

tf →+∞

ti →−∞

1 = lim η→0 2π¯ hiη

n+1 n 2

with Ll = L (xl , x˙ l ) =

iη

dxkj e

n -

(Ll +Jk xkl )

l=0

,

(5.9)

k j=1

1 2 x˙ kl − ωk2 x2kl , 2 k

(5.10)

where the ﬁrst index (k) denotes the coordinate and the second (j) the time interval. If we now identify the integration measure as

1 Dφ = lim η→0 2π¯ hiη

n+1 2

n

dxkj ,

(5.11)

k j=1

we can write the generating functional also as

" 1! W [J] = Dφ exp i ∂μ φ∂ μ φ − m2 − iε φ2 + Jφ d4x 2 & ' ε 2 4 = Dφ exp i L (φ, ∂μ φ) + Jφ + i φ d x . (5.12) 2

Here the volume element d4x is given by the Lorentz-invariant expression d4x = d3x dt .

(5.13)

The term iεφ2 /2 with positive ε has been introduced in an ad hoc manner to ensure the convergence of W when taking the ﬁelds to inﬁnity with the

CHAPTER 5. PATH INTEGRALS FOR SCALAR FIELDS

59

understanding that ultimately ε has to be taken to 0 (cf. the discussion at the end of Sect. (3.1)). The second line of (5.12) also gives the generating functional for an interacting theory with the interaction V included in L. This can be easily proven by including an additional interaction in the Lagrangian (5.4) and Fourier-expanding it. The generating functional for a scalar ﬁeld theory is thus given by

W [J] =

Dφ e−i

d4x[ 12 φ(2+m2 −iε)φ+V (φ)−Jφ]

(5.14)

where the Lagrangian in the form (4.31) has been used. In analogy to (3.24) we deﬁne a normalized functional Z[J] =

W [J] . W [0]

(5.15)

Again the inﬁnite normalization factor inherent in W [J] cancels out in this deﬁnition. Since W [J] involves an exponential it will often be convenient in the following discussions to introduce its logarithm iS[J] which is itself a functional of J W [J] ≡ eiS[J] =⇒ S[J] = −i ln W [J] = −i ln Z[J] − i ln W [0] .

5.1.1

(5.16)

Euclidean Representation

In the preceding section we have ensured convergence of the generating functional by introducing the ε-dependent term in the energies. In this subsection we go back to the alternative method that relies on the Wick rotation that we introduced in Sect. 3.21 and discuss the Euclidean representation of the generating functional. This discussion also illustrates in some more detail the remarks on integrating oscillatory functions made at the end of Sect. 1.3.1. The real Euclidean space is obtained from Minkowski space by rotating the real axis in the x0 plane by δ = −π/2 into the negative imaginary axis (Wick rotation). We denote a space-time point in Euclidean space by xE ; it is related to the usual space-time point x in Minkowski space by xE = (x, x4 )

with x4 = ix0 = it .

(5.17)

Under the Wick rotation t → −it and x4 thus becomes real. With this deﬁnition we can extend the usual Minkowski-space deﬁnitions of volume element and space-time distance to Euclidean space d4 xE ≡ d3 x dx4 = d3 x idt = id4 x dx2E =

3 j=1

dx2j + x24 = −dx2 .

(5.18)

CHAPTER 5. PATH INTEGRALS FOR SCALAR FIELDS

60

The d’Alembert operator is then given by 2=

4 2 ∂2 ∂2 2 =− ∂ −∇ 2=− − ∇ ≡ −2E . 2 ∂t2 ∂x24 a=1 ∂xa

(5.19)

With these transformations the generating functional for a free scalar ﬁeld in Minkowski space (5.14) becomes in its Euclidean representation

WE0 [J] =

Dφ e−

d4xE { 12 [(∂E φ)2 +m2 φ2 ]−Jφ}

,

(5.20)

with (∂E φ)2 = − (∂φ)2 . Because x4 is now real, (∂E φ)2 is always positive and the exponent is negative deﬁnite; the integral thus converges and is welldeﬁned even without adding in the ε-dependent term. Since the exponent is furthermore quadratic in the ﬁelds, WE0 [J] can be evaluated by using the techniques for Gaussian integrals that are explained in App. B. Physical results are then obtained by rotating backwards after all integrations have been performed. Remembering that in ﬁeld theory the ﬁelds play the role of the coordinates of a Lagrangian theory we can now directly generalize some of the results of Chapt. 3 to ﬁeld theory. In particular, we have that WE [J] of eq. (5.20) is the transition amplitude from the vacuum state of the free theory at t → −∞ to that at t → +∞ under the inﬂuence of the external source J (cf. (3.27) so that the normalized transition amplitude ZE is given by ZE0 [J] =

WE0 [J] 0 + ∞|0 − ∞J , = 0 WE [0] 0 + ∞|0 − ∞0

(5.21)

where |0 is the vacuum state of the free theory. Eq. (5.20) shows that the normalized transition amplitude can be understood as an integration of the source action exp (+ d4xE Jφ) with the weights wE0 (φ) = e− 2 1

over all ﬁelds φ

ZE0 [J] =

d4xE [(∂E φ)2 +m2 φ2 ]

(5.22)

4 Dφ wE0 (φ)e d xE Jφ Dφ wE0 (φ)

.

(5.23)

Eq. (3.47) then shows that for a function of the ﬁelds O(φ)

Dφ wE0 (φ) j Oj (φ(xj )) ˆ j ))]|0 . = 0|T [ Oj (φ(x Dφ wE0 (φ) j

The ﬁeld operators on the rhs here are those of free ﬁelds.

(5.24)

CHAPTER 5. PATH INTEGRALS FOR SCALAR FIELDS

61

In the interacting case all these relations still hold if wE0 is replaced by a weight function for the interacting theory and the vacuum state is now that of the interacting theory which we will denote by |˜0. To obtain the weight function of the interacting theory we use a perturbative treatment of the interaction V . By writing in analogy to (5.22) wE (φ) = e− = e−

d4xE { 12 [(∂E φ)2 +m2 φ2 ]+V (φ)}

1

V (φ) d4xE − 2

e

[(∂E φ)2 +m2 φ2 ] d4xE ≡ e−

(5.25) V (φ) d4xE

0 (φ) wE

we get with (5.24)

˜0|T [

ˆ ˜0 = Oj (φ)]|

j

=

Dφ wE (φ)

j

Dφ wE0 (φ)e−

0|T [

Oj (φ) V (φ) d4xE

Dφ wE0 (φ)e−

=

ˆ − j Oj (φ0 )e

0|T [e−

V

j

(φ) d4x

Oj (φ) E

V (φˆ0 ) d4xE

V (φˆ0 ) d4xE

]|0

]|0

.

(5.26)

Here |0 on the rhs is the vacuum state of the non-interacting free theory (V = 0) and all the ﬁeld operators on the rhs are free ﬁeld operators φˆ0 at all times if they were free at t → −∞. This can be seen from the path integral which contains the free weight wE0 connected with free propagation. In the following chapters it is often useful to think in terms of this probabilistic interpretation even when we work in Minkowski metric. This is possible by relating Euclidean and Minkowski space by analytic continuation.

Chapter 6 EVALUATION OF PATH INTEGRALS Only a limited class of path integrals can be evaluated analytically [GroscheSteiner] so that one is forced to use either numerical or perturbative methods. In this section we ﬁrst calculate the generating functional for free scalar ﬁelds, both by a direct reduction method and, to gain familiarity with this technique, with the the methods of Gaussian integration developed in App. B.2.1. We then show how more general types of path integrals can approximately be reduced to the Gaussian form. In this reduction we ﬁnd a systematic method for a semiclassical expansion in terms of the Planck constant h ¯ . Finally, we brieﬂy discuss methods for the numerical evaluation of path integrals.

6.1

Free Scalar Fields

The generating functional for a free scalar ﬁeld theory plays a special role in the theory of path integrals. This is so because of two reasons: ﬁrst, a free-ﬁeld theory is the simplest possible ﬁeld theory and, second, as we have just seen in the last section the eﬀects of the interaction term V (φ) can be described with the help of perturbation theory in which the functional of the full, interacting theory is expanded around that of the free theory.

6.1.1

Generating functional

The representation of the generating functional of a free ﬁeld theory (V = 0 in (5.14)) as a path integral is still quite cumbersome for practical applications. For these it would be very desirable if we could factorize out the functional dependence on J in form of a normal integral; the remaining path integral 62

CHAPTER 6. EVALUATION OF PATH INTEGRALS

63

would then disappear in the normalization. In the following we will therefore separate the path integral (5.9) into 2 factors, one depending on J and the other one being an integral over φ, i.e. just a number. For this purpose we start with the generating functional

W [J] =

Dφ e−i

d4x[ 12 φ(2+m2 −iε)φ−Jφ]

(6.1)

where the Lagrangian in the form (4.31) has been used. The ﬁeld φ here is an integration variable; it does not fulﬁll a Klein-Gordon equation! We now introduce a ﬁeld φ0 that does just that !

"

2 + (m2 − iε) φ0 (x) = J(x) .

(6.2)

J thus plays the role of a source to φ0 . We now take this ﬁeld φ0 as a reference ﬁeld and expand φ around it, setting φ = φ0 + φ . We thus obtain for the integrand in the exponent in (6.1) 1 (φ0 + φ ) 2 (φ0 + φ ) + (φ0 + φ ) m2 − iε (φ0 + φ ) − J (φ0 + φ ) 2 " 1 ! = φ 2 + m2 − iε φ 2 " " 1 ! 1 ! + φ0 2 + m2 − iε φ0 + φ0 2 + m2 − iε φ 2 2 " 1 ! 2 + φ 2 + m − iε φ0 − Jφ0 − Jφ . (6.3) 2

The third and the fourth terms on the rhs give the same contribution when integrated over. The second term gives, according to (6.2), 12 Jφ0 . Collecting all terms we therefore have for the action in (6.1)

" 1 ! S[φ, J] = − d x φ 2 + m2 − iε φ − Jφ 2 " 1 ! 1 = − d4x φ 2 + m2 − iε φ + Jφ0 2! 2 " 0 2 + φ 2 + m − iε φ0 − Jφ0 − Jφ . 4

(6.4)

In the last line of this equation we can again apply (6.2) to obtain " 0 1 4 1 ! d x φ 2 + m2 − iε φ − Jφ0 S[φ, J] = − 2

(6.5)

We are now very close to our aim to factorize out the J dependence. To do so we solve (6.2) by writing φ0 (x) = −

DF (x − y)J(y) d4y ,

(6.6)

CHAPTER 6. EVALUATION OF PATH INTEGRALS

64

where DF , the so-called Feynman propagator, fulﬁlls the equation !

2 + m2 − iε

"

DF (x) = −δ 4 (x)

(6.7)

in complete analogy to the nonrelativistic propagator K in chapter 1. Using

1 δ (x) = 2π

4

4

we obtain DF (x − y) =

d4k e−ikx

1 e−ik(x−y) d4k . (2π)4 k 2 − m2 + iε

(6.8)

(6.9)

Substituting (6.6) into the action (6.5) gives

S[φ, J] = =

! " 1 4 − d x φ 2 + m2 − iε φ + J(x) DF (x − y)J(y) d4y 2 " 0 1 4 1 ! − d x φ 2 + m2 − iε φ 2 1 − J(x)DF (x − y)J(y) d4x d4y . (6.10) 2

The exponential of the last term no longer depends on φ and can, therefore, be pulled out of the pathintegral (6.1). The pathintegral involving the exponential of the ﬁrst term appears also in the denominator of the normalized generating function (5.15) and thus drops out. We thus obtain now for the normalized generating functional Z0 [J] =

i W [J] 4 4 = e− 2 J(x)DF (x−y)J(y) d x d y . W [0]

(6.11)

This is the vacuum-to-vacuum transition amplitude for a free scalar ﬁeld theory. Note that it no longer involves a path integral.

6.1.2

Feynman propagator

The imaginary part of the mass, originally introduced to achieve convergence for the path integrals, appears here now in the denominator of the propagator DF 1 e−ikx DF (x) = (6.12) d4k . (2π)4 k 2 − m2 + iε It determines the position of the poles in DF , which are, in the k0 -integration, at (6.13) k02 = k 2 + m2 − iε

CHAPTER 6. EVALUATION OF PATH INTEGRALS

65

Im k0

–ω k+iδ Re k0 +ω k–iδ

Figure 6.1: Location of the poles in the Feynman propagator. or

k0 = ± k 2 + m2 ∓ iδ .

(6.14)

The poles are therefore located as indicated in Fig. 6.1. The location of the poles, originally introduced only in an ad-hoc way to achieve convergence of the path integrals, determines now the properties of the propagator of the free Klein-Gordon equation, the Feynman propagator. This can be seen by rewriting the Feynman propagator DF in the following form 1 4 e−ikx d k (6.15) (2π)4 k 2 − m2 + iε e−ikx 1 3 = d k dk 0 (2π)4 k02 − k 2 − m2 + iε 1 1 1 3 −ikx 1 = d k dk0 e − (2π)4 2ωk k0 − ωk + iδ k0 + ωk − iδ

DF (x) =

with ωk2 = k 2 + m2 . We now ﬁrst perform the integration over k0 . Since the exponential contains a factor e−ik0 t , the path can be completed in the upper half plane for negative times and in the lower half for t > 0. Cauchy’s theorem then gives 2πi × the sum of the residues of the enclosed poles i 3 eik·x +iωk t −iωk t d k −Θ(−t)e − Θ(t)e DF (x) = (2π)3 2ωk

=

i 3 eik·x +iωk t −iωk t − d k Θ(−t)e + Θ(t)e (2π)3 2ωk

(6.16)

CHAPTER 6. EVALUATION OF PATH INTEGRALS

66

(in the second term here an extra “−” sign appears because of the negative direction of the contour integral). It can now be shown that DF propagates free ﬁelds with negative frequencies backwards in time and those with positive frequencies forwards. To demonstrate this we write i 3 eik·x +iωk x0 −iωk x0 d k Θ(−x )e + Θ(x )e DF (x) = − 0 0 (2π)3 2ωk i 1 ikx = − Θ(−x0 ) d3k e 3 (2π) 2ωk i 3 1 −ikx − Θ(x0 ) dk e . (6.17) (2π)3 2ωk

Here we have changed k → −k in the ﬁrst integral, which does not change its value under this substitution. The integrands are products of orthogonal, normalized solutions of the Klein-Gordon equation (±) ϕk (x) =

1 (2π)3 2ωk

e∓ikx

(6.18)

which fulﬁll the normalization and orthogonality conditions appropriate for a scalar ﬁeld (cf. (4.65) without an A ﬁeld)

i

d3x ϕk (x)ϕ˙ k (x) − ϕ˙ k (x)ϕk (x) = ±δ 3 (k − k) . (±)∗

and i

(±)

(±)∗

(±)∗

(∓)

(±)

(±)∗

(∓)

(6.19)

d3x ϕk (x)ϕ˙ k (x) − ϕ˙ k (x)ϕk (x) = 0 .

(6.20)

We can thus write 1 3 1 1 √ dk √ eikx iDF (x) = Θ(−x0 ) 3 (2π) 2ωk 2ωk 1 3 1 1 √ + Θ(x0 ) dk √ e−ikx 3 (2π) 2ωk 2ωk

= Θ(−x0 )

d

(−)∗ (−) k ϕk (0)ϕk (x)

3

+ Θ(x0 )

(6.21)

(+)∗

d3k ϕk

(+)

(0)ϕk (x) .

Comparison with (1.13) shows that negative-frequency solutions are propagated backwards in time, and positive-frequency solutions forward. This particular behavior is a consequence of the location of the poles relative to the integration path. This location is ﬁxed by the sign of the ε term which in turn was needed to achieve convergence for the generating functional.

CHAPTER 6. EVALUATION OF PATH INTEGRALS

67

Euclidean representation. The Feynman propagator can also be given in a Euclidean representation. We can deﬁne a corresponding Euclidean momentum space by requiring that k 0 x0 = k4E xE4 ; this condition ensures that a plane wave propagating forward in time does so both in Euclidean and in Minkowski space. We thus get with k4 = −ik0 . (6.22) kE = (k, k4 ) The eigenvalues of 2E become k 2 = k02 − k 2 = −(k42 + k 2 ) = −kE2 < 0. This gives for the momentum space volume element and the energy-momentum distance in Euclidean space d4 kE ≡ d3 k dk4 = −d3 k idk0 = −id4 k dkE2

=

3

dkj2 + dk42 = −dk 2 .

(6.23)

j=1

Combining both of these deﬁnitions yields for the typical exponent of a plane wave kx = kμ xμ = k0 x0 − k · x = ik4 (−i)x4 − k · x = k4 x4 − k · x = kE xE .

(6.24)

Note that this is not equal to the Euclidean scalar product. However, in Fourier transforms, where this expression often appears, we always have an integration over d3 k and can thus change k → −k; thus in these Fourier integrals – and only there – we can replace kx by kE xE . The Feynman propagator can now be rewritten. For that purpose we choose a diﬀerent path for the integration over k0 after we have Wick-rotated the time axis. Instead of integrating along the real energy (k0 ) axis we integrate along the imaginary energy axis. We then close the integration path in the right half of the k0 plane for t > 0 and in the left half for t < 0. Since in this way the same poles are included as on the original path the value of the integral does not change. This fact makes this rotation in the k0 -space diﬀerent from the one in the t-space where the imaginary time-axis has to be rotated back to the real one after the calculations in order to obtain physical results. The Euclidean propagator then reads i e−ikE xE 4 d kE (2π)4 kE2 + m2 i e−ikE xE = − d4kE . (6.25) 4 2 2 2 (2π) k + k4 + m Because k4 is real, the integral no longer contains any poles on its integration path and is therefore well deﬁned. DF (x) =

−

CHAPTER 6. EVALUATION OF PATH INTEGRALS

6.1.3

68

Gaussian Integration

In Sect. 1.3 we have already used a Gaussian integral relation to integrate out the p-dependence of the path integral. In many cases the generating functions appearing in ﬁeld theory are of a form that contains the ﬁelds and their derivatives only in quadratic form so that again a Gaussian method can be used. In order to gain familiarity with this technique we derive in this section again the generating functional for the free scalar ﬁeld theory. We thus apply the Gaussian integration formulas of Sect. B.2.1 to the generating functional (6.1)

Dφ e

W [J] =

− 2i

φ(2+m2 −iε)φ d4x i

e

Jφ d4 x

.

(6.26)

In order to make the matrix structure of the exponent more visible we use two-fold partial integration and write

W [J] =

= →

Dφ e− 2 i

Dφ e

− 2i

φ(x)δ 4 (x−y)(2y +m2 −iε)φ(y) d4x d4y i

e

φ(x)[(2y +m2 −iε)δ 4 (x−y)]φ(y) d4x d4y i

e

Jφ d4 x

Jφ d4 x

1 E 2 4 4 Dφ e− { 2 φ(xE )[(−2y +m )δ(xE −yE )]φ(yE ) d yE −Jφ} d xE . (6.27)

In the last step we have gone over to the Euclidean representation of the generating functional (cf. (5.20)), using 2E = −2 and δ 4 (xE −yE ) = −iδ 4 (x− y) (cf. Sect. 5.1.1). The integrals appearing here are now of Gaussian type and can thus be integrated by using the expressions developed in the last section. We ﬁrst identify the matrices A and B in the Gaussian integration formula (B.18) as 2 4 AE (x, y) = −(2E y − m )δ (xE − yE )

B(x) = −J T (x) .

(6.28)

AE is real with positive eigenvalues (kE2 +m2 > 0) as required by the derivation in section B.2.1. It is also symmetric as can be seen by writing the d’Alembert operator in a discretized form, e.g. .

. 1 d2 . . φ(x) = lim [φ(xi + h) − 2φ(xi ) + φ(xi − h)] 2 . h→0 h2 dx xi

= Aij φ(xj ) with φ(xi + h) = φ(xi+1 ) etc. and Aij = δi,j−1 − 2δi,j + δi,j+1 .

(6.29)

CHAPTER 6. EVALUATION OF PATH INTEGRALS

69

Thus the necessary conditions for the application of (B.13) are fulﬁlled. Applying now (B.18) gives !

"− 1

W E [J] = det(−2E + m2 )δ(xE − yE )

2

1

e2

−1

J(xE )(AE (xE ,yE ))

J(yE ) d4 xE d4 yE

. (6.30) The Wick rotation back to real times is easily performed by the transformation (5.19) 2E → −2 − iε; it reintroduces the term +iε to guarantee the proper treatment of the poles. In this case A becomes AE → A = (2 + m2 − iε)iδ(x − y) .

(6.31)

and the volume element d4xE d4yE → −d4 xd4 y

(6.32)

so that we ﬁnally obtain1 W [J] = √

−1 1 2 4 4 4 1 e− 2 J(x)[i(2y +m +iε)δ (x−y)] J(y) d x d y . det A

(6.33)

The determinant of a matrix A is in general given by the product of its eigenvalues αi . Therefore we have

ln(det A) = ln

αi =

i

ln αi = tr ln A .

(6.34)

i

In the present case, though, A contains an operator and the notation det A deserves some explanation. The matrix is given by A(x, y) =

2 + m2 iδ(x − y) = 2 + m2 i

1 4 ik(x−y) dke (2π)4

i 4 2 2 d k −k + m eik(x−y) (2π)4 1 1 4 = i d k d4k eik x −k 2 + m2 δ 4 (k − k ) e−iky . 4 4 (2π) (2π)

=

Thus the momentum representation of the operator A is given by

A(k, k ) = −k 2 + m2 δ 4 (k − k ) .

(6.35)

We now evaluate the trace of the logarithm of this matrix where – in accordance with (6.34) – the logarithm of a matrix is explained by taking the 1

Note that formally we could have obtained this also by using (B.18) with A = i(2y + m − iε)δ 4 (x − y), B = −iJ, C = 0. 2

CHAPTER 6. EVALUATION OF PATH INTEGRALS

70

logarithm of each of the diagonal elements, i.e. the eigenvalues, after the matrix has been diagonalized. This yields in the present case

d4k 4 4 tr ln A(x, y) = d xd y d k δ (x − y)ei(k x−ky) ln(−k 2 + m2 )δ 4 (k − k ) 4 (2π) 4 dk d4x ln(−k 2 + m2 ) (6.36) = 4 (2π) 4

4

Now we use that the inverse operator appearing in (6.33) is just the Feynman propagator. This can be seen by deriving the equation of motion for the inverse operator δ (x − y) =

A(x, z)A−1 (z, y) d4z

4

=

!

= i

"

i 2z + m2 − iε δ 4 (x − z) A−1 (z, y) d4z

δ 4 (x − z) 2z + m2 − iε A−1 (z, y) d4z

= i 2x + m2 − iε A−1 (x, y) .

(6.37)

This is just the deﬁning equation for −DF (6.7), so that we have A−1 = iDF .

(6.38)

The propagator can thus be obtained as the inverse of the operator between the two ﬁelds in the Lagrangian for the free ﬁeld (4.31). For the normalized generating functional we thus obtain Z0 [J] =

i W [J] 4 4 = e− 2 J(x)DF (x−y)J(y) d x d y . W [0]

(6.39)

Equation (6.39) is the result derived earlier in section 5.1 (cf. (6.1.1)). The propagator that appears here is just given by the inverse of the Klein-Gordon Operator −1 DF (x − y) = − 2 + m2 δ(x − y) , (6.40) i.e. of that operator that appears between the two ﬁelds in the Lagrangian for a free Klein-Gordon ﬁeld 1 L = − φ(2 + m2 )φ . 2

(6.41)

CHAPTER 6. EVALUATION OF PATH INTEGRALS

6.2

71

Stationary Phase Approximation

If the path integral in question is not that over a Gaussian, it can be approximately brought into a Gaussian form by using the so-called stationary phase or saddle point method. In this method one ﬁrst looks for the stationary point of the exponent in the path integral. As explained earlier this will give a major contribution to the path integral. The remaining contributions are approximated by expanding the exponent around the stationary point. We illustrate this method here for the case of a scalar ﬁeld with selﬁnteractions. The Lagrangian is given by " 1 ! L = − φ 2 + m2 φ − V (φ) 2

(6.42)

and the action S[φ, J] =

d4x (L + Jφ)

(6.43)

is a functional of the ﬁeld φ and the source J. We next determine the stationary point by looking for the zero of the functional derivative .

δS[φ, J] .. ! . = − 2 + m2 φ0 (x) − V (φ0 (x)) + J(x) = 0 ; . δφ(x) φ0

(6.44)

this is the classical equation of motion corresponding to the action S[φ, J]. The stationary ﬁeld is just the classical ﬁeld; the corresponding classical action is & ' 1 4 2 (6.45) S[φ0 , J] = − d x φ0 2 + m φ0 + V (φ0 ) − Jφ0 . 2 We now expand S[φ, J] around this stationary ﬁeld (cf. (B.35),(B.37)) S[φ, J] = S[φ0 , J] (6.46) . . 1 δ2S . . [φ(x1 ) − φ0 (x1 )] [φ(x2 ) − φ0 (x2 )] + · · · . d4x1 d4x2 + 2 δφ(x1 )δφ(x2 ) .φ0 The second functional derivative appearing here can be obtained by varying the ﬁrst derivative (6.44). We thus get from the deﬁnition (B.26) .

.

. " . δ ! δ2S . . 2 . = − . (2 + m )φ + V (φ) − J 1. δφ(x2 )δφ(x1 ) .φ0 δφ(x2 ) φ0

(6.47)

Using now (B.27) we get .

. ! " δ2S . . = − 2 + m2 + V (φ0 ) δ 4 (x2 − x1 ) 1 δφ(x2 )δφ(x1 ) .φ0

(6.48)

CHAPTER 6. EVALUATION OF PATH INTEGRALS

72

which is an operator. The index 1 here means that the corresponding expressions are to be taken at the point x1 . The action (6.45) is calculated at the ﬁxed classical ﬁeld φ0 . It can, therefore, be taken out of the path integral so that we ﬁnally obtain

W [J] =

i

Dφ e h¯

i

= e h¯ S[φ0 ,¯hJ]

d4x (L+Jφ)

Dφ exp −

1

× [φ(x1 ) − φ0 (x1 )]

1!

(6.49) i 2¯ h

d4x1 d4x2 "

0

2 + m + V (φ0 ) δ (x2 − x1 ) [φ(x2 ) − φ0 (x2 )]} 2

4

1

+ ... .

(6.50)

In order to facilitate the following discussion we have put the unit of action, h ¯ , explicitly into this expression by setting i → i/¯ h and J → h ¯ J. The path integral remaining here is now in a Gaussian form. It can be evaluated after a Wick rotation, just as in the developments leading to (6.33). After a “coordinate transformation” φ → φ = φ − φ0 and after scaling the √ ¯ φ we get for the generating functional ﬁelds by φ → h 1

!

"0− 1

W [J] = e h¯ S[φ0 ,¯hJ] det i 2 + m2 + V (φ0 ) δ 4 (x2 − x1 ) i

2

.

(6.51)

We now perform a normalization with respect to the free case (6.33), i.e. to 1

!

"0− 1

det i 2 + m2 δ 4 (x2 − x1 )

W0 [0] =

2

= {A(x1 , x2 )}− 2 1

(6.52)

with A from (6.31); the index 0 on W denotes V = 0. This gives for the normalized generating functional ˜ [J] = W [J] = e h¯i S[φ0 ,¯hJ] W W0 [0]

× det

&

(6.53)

−1

' − 1

d z A (x2 , z) (A(z, x1 ) + iV (φ0 (x1 ))) δ (z − x1 ) 4

4

2

With A−1 = iDF (6.38) we get 1

!

"0− 1

˜ [J] = e h¯i S[φ0 ,¯hJ] det δ 4 (x1 − x2 ) − DF (x2 − x1 )V (φ0 (x1 )) W

2

(6.54)

Our aim is now to write the inverse root of the determinant as a correction term to the classical action. For this purpose we use (6.34) − 12

{det[. . .]}

1 − tr ln[. . .] =e 2 .

(6.55)

.

CHAPTER 6. EVALUATION OF PATH INTEGRALS

73

The matrix is given by !

"

x1 |1 − DF V (φ0 )|x2 = δ 4 (x1 − x2 ) − DF (x2 − x1 )V (φ0 (x1 )) .

(6.56)

The trace of its logarithm is then given by

tr ln[. . .] = We can now write

d4x ln [1 − DF (0)V (φ0 (x))] .

(6.57)

˜ [J] = e h¯i S[φ0 ,J] W

(6.58)

with S[φ0 , J] =

−

&

d4x

'

1 φ0 2 + m2 φ0 + V (φ0 ) + h ¯ d4x Jφ0 2

i 4 + h h2 ) . ¯ d x ln [1 − DF (0)V (φ0 (x))] + O(¯ 2

(6.59)

The ﬁrst line is just the classical action S[φ0 , J]. The two terms can be summed with a resulting action S[φ0 , J] = −

&

1 d x φ0 2 + m2 φ0 + Veﬀ (φ0 ) + h ¯ Jφ0 2

'

4

(6.60)

with the eﬀective potential i Veﬀ (φ(x)) = V (φ(x)) − h ¯ ln [1 − DF (0)V (φ0 (x))] . 2

(6.61)

Expression (6.59) shows that the saddle point approximation amounts to an expansion of the action in powers of h ¯ . This is in accordance with the discussion in Sect. 1.4 that quantum mechanics describes the ﬂuctuations of the action around the classical path. The potential Veﬀ incorporates the eﬀects of these ﬂuctuations into a classical potential. Eq. (6.59) suggests a perturbative treatment through an expansion of the logarithm (ln(1 + x) = x − x2 /2 + x3 /3 − . . .) in terms of DF V , i.e. the strength of the potential. The trace corresponds to an integration over x such that the initial and ﬁnal space-time points in the individual terms in the expansion are identical, i.e. to a closed loop integration. With higher orders in the expansion of the logarithm more and more vertices appear, but they are always located on this one closed loop. This loop expansion approximates the quantum mechanical behavior, whereas the perturbation treatment takes interactions into account. If we take φ0 as a constant ﬁeld, i.e. if we neglect its space-time dependence through the d’Alembert operator in (6.44), then the operator 1 +

CHAPTER 6. EVALUATION OF PATH INTEGRALS

74

DF V (φ0 ) becomes local in momentum space. In this case we can evaluate the trace of its logarithm by integrating over the eigenvalues x1 | ln [1 − DF V (φ0 )] | 1 d4k −ik(x2 −x1 ) e ln 1 − 2 V (φ0 ) . = (2π)4 k − m2 + iε

(6.62)

This equation writes the original matrix in x-space as a result of a unitary transformation of a diagonal matrix in k-space. The logarithm is taken of this diagonalized matrix which is then transformed back to x-space.

6.3

Numerical Evaluation of Path Integrals

An alternative method for the evaluation of path integrals is that of direct numerical computation; with rapidly increasing computer power this method becomes more and more important nowadays.

6.3.1

Imaginary time method

The generating functional is in general given by ˆ

W [J] = 0|e−i(H+J)(tf −ti ) |0

(6.63)

for ti → −∞ and tf → +∞, as we have seen in chapter 3. After a Wick rotation this becomes ˆ

W [J] = lim 0|e−β(H+J) |0 ,

(6.64)

β→∞

where β denotes the real Euclidean time. It is immediately obvious that (6.64) also equals the groundstate expectation value of the statistical operator of quantum statistics if we identify β with the inverse temperature, i.e. β = 1/T . Inserting the explicit deﬁnition (5.9) and performing the Wick rotation gives

n+1 W [J] = lim n→0 2π¯ hβ

n+1 n 2

with Hl = Hl (xl , x˙ l ) =

dxkj e

β − n+1

n l=0

(Hl +Jk xkl )

,

(6.65)

k j=1

" 1 ! 2 x˙ kl + ωk2 x¯2kl + V (xkl ) , 2 k

(6.66)

CHAPTER 6. EVALUATION OF PATH INTEGRALS

75

so that W [J] can be written as

β n + 1

W [J] = lim dxkj ρ(xkj )e− n+1 l V (xkl ) n→0 2π¯ hβ k j

with

β

ρ(x) = e− n+1

-

1 (x˙ 2kl +ωk2 x2kl )+Jk xkl l 2

[

].

(6.67)

(6.68)

The function ρ(x) is of Gaussian shape and can, therefore, analytically be normalized into ρ(x) P (x) = . (6.69) dx ρ(x) The multiple integrals appearing here can be evaluated by a Monte-Carlo technique which samples the integrand at a large number of points, where each ‘point’ really corresponds to a full path x(t). Given a certain point x = x1 , . . . , xn one randomly chooses a new point x = x1 , . . . , xn , often by just changing one single coordinate. One then evaluates r=

P (x ) . P (x)

(6.70)

If r is larger than 1, the new point is accepted. If r < 1, on the other hand, then a random number ρ between 0 and 1 is picked. If ρ < r, then the new point is also accepted, otherwise it is rejected. This method is repeated until a large enough number of points is sampled. In this way the most important regions in x space are sampled, thus generating ﬁnally M accepted points xm . The integral is then approximated by W [J] =

M -n β 1 e− n+1 l=0 V ((xkl )m ) . M m=1

(6.71)

The sampling algorithm just described is known after its inventor as the Metropolis algorithm; it plays an important role in numerical evaluations of statistical physics expressions. In the present case of evaluating the generating functional for gs to gs transitions one has to choose a ﬁxed value of β, i.e. the Euclidean time. The calculations then have to be performed for several values of β with a subsequent extrapolation to β → ∞ (or T → 0).

6.3.2

Real time formalism

The major diﬃculty in evaluating a path integral numerically stems from the oscillatory character of the integrand. The discretized form of W [J] (5.9) can

CHAPTER 6. EVALUATION OF PATH INTEGRALS

76

– in an obvious abbreviation – be written as the limit of a multidimensional integral (6.72) W = dx eiS(x) , where x is a n-dimensional vector. Importance sampling such as the one just discussed in the last section cannot directly be used because there is no positive probability weight function in the integrand. A weight function can, however, be inserted by a mathematical trick. Using (B.18) in the form

we can write

W =

dx0 det(A/2π)e− 2 (x−x0 ) 1

T A(x−x ) 0

dxdx0 det(A/2π)e− 2 (x−x0 ) 1

=1

T A(x−x ) 0

(6.73)

eiS(x) .

(6.74)

The Gaussian factor under the integral ensures that values of x close to x0 will contribute the most to the integral. The function S(x) can, therefore be expanded around x0 1 S(x) = S(x0 ) + S1 (x)(x − x0 ) + (x − x0 )T S2 (x0 )(x − x0 ) + . . . 2 .

with

dS .. . S1 (x0 ) = dx .x0

(6.75)

.

d2 S .. . S2 (x0 ) = . dx2 .x0

and

(6.76)

After inserting this expansion we can perform the x-integration and obtain from (B.18)

W =

dx0 e

iS(x0 )

det(A) det[A − iS2 (x0 )]

1 2

T

e− 2 S1 (x0 )[A−iS2 (x0 )] 1

−1

S1 (x0 )

. (6.77)

If we now use that (6.73) is still approximately valid even if A is a function of x0 , we can choose A = A(x0 ) = iS2 (x0 ) + c−1 1

(6.78)

with c > 0 and obtain

W =

!

"− 1

dx0 eiS(x0 ) det(1 − iS2 (x0 )A−1 )

The function

c

T

2

c

T

e− 2 S1 (x0 )S1 (x0 ) .

P (x0 ) = e− 2 S1 (x0 )S1 (x0 )

(6.79)

(6.80)

CHAPTER 6. EVALUATION OF PATH INTEGRALS

77

thus provides a probability distribution for sampling the remaining integrand and the expression can be evaluated with the Monte-Carlo method discussed in the last section. We obtain, therefore, M 1 1 W = eiS(xi ) [det(1 + icS2 (xi )]− 2 , M i=1

(6.81)

where the M points xi are taken randomly from the distribution P (x). For large values of c the points with S1 ≈ 0 will contribute the most to W . This is just the stationarity condition discussed in section 6.2 which corresponds to the classical solution. Since only a few points close to this conﬁguration will contribute signiﬁcantly, a good sampling with good statistics can be obtained. On the other hand, for small c the probability distribution becomes broad and the statistics correspondingly worse; however, quantum mechanical eﬀects beyond the one-loop approximation now start to contribute. Thus, by choosing the proper value of c quantum mechanics can be switched on.

Chapter 7 S-MATRIX AND GREEN’S FUNCTIONS The ultimate aim of all our ﬁeldtheoretical developments in this book is to calculate reaction and transition rates for processes involving elementary particles. In this chapter, we therefore, now derive a connection between these transition rates and expectation values of ﬁeld operators, the so-called reduction theorem.

7.1

Scattering Matrix

The typical initial state of a scattering experiment is that of widely separated on-shell particles at t → −∞. On-shell here means that these particles fulﬁll the free energy-momentum dispersion relation. The groundstate of the system is the state of lowest-energy, i.e. the state with no particles present. At large times t → +∞ the ﬁnal state is again that of free, non-interacting on-shell particles. The vacuum state of the theory is unique and is therefore the same as the initial vacuum state. The transition rate of any quantum process, be it a scattering process m + n → m + n is determined by the so-called S-matrix. We deﬁne the Smatrix as the probability amplitude for a process that leads from an ingoing state |α, in to an outgoing state |β, out . The particles are assumed to move freely in these asymptotic states outside the range of the interaction; both of these states can therefore be characterized by giving the momenta of all participating particles and possibly other quantum numbers as well all of which are denoted by α and β. The S-matrix is thus given by Sβα = β, out|α, in . 78

(7.1)

CHAPTER 7. S-MATRIX AND GREEN’S FUNCTIONS

79

We can then also introduce an operator Sˆ that transforms in-states (bras) into out-states (bras) β, out| = β, in|Sˆ (7.2) so that we have ˆ in . Sβα = β, in|S|α,

(7.3)

Sˆ is a unitary operator. This can be seen by taking the hermitean conjugate of (7.2) and writing β, in|SˆSˆ† |α, in = β, out|α, out = δα,β .

(7.4)

SˆSˆ† = 1.

(7.5)

Thus we have

The ﬁeld operators transform in the standard way under the unitary transformation S (7.6) φout = Sˆ† φˆin Sˆ . In Sect. 2.3 we have expressed the matrix element Sβα in terms of wavefunctions. There we showed that S (see (2.35)) could be written as

Sβα =

Ψ∗β (x, t → +∞)Ψ(+) x, t → +∞) d3x . α (

(7.7)

Here Ψ(+) was a wavefunction that fulﬁlled an “in” boundary condition, i.e. it evolved forward in time, starting from an incoming free plane wave at t → −∞. Ψβ , on the other hand, was a free plane wave for t → +∞. We now generalize these considerations to ﬁelds and introduce the free asymptotic ﬁelds φin and φout φin = φout =

lim φ(x, t)

t→−∞

lim φ(x, t) .

(7.8)

t→+∞

Using now the time-development operator U on φin gives for the S- matrix Sβα = φout |U (+∞, −∞)|φin

(7.9)

with the time-development operator (1.1). With U (+∞, −∞) = U (+∞, t0 )U (t0 , −∞) we can write the S-matrix also as (−)

(+)

Sβα = φout |U (+∞, t0 )U (t0 , −∞)|φin = φout (t0 )|φin (t0 ) .

(7.10)

CHAPTER 7. S-MATRIX AND GREEN’S FUNCTIONS

80

This form corresponds to (7.7). As is obvious from its deﬁnition the S-matrix determines all the transition rates possible within the ﬁeld theory. For example, a cross-section for a 1 + 1 → 1 + 1 collision is simply given by the absolute square of S, multiplied with the available phase-space of the outgoing particles and normalized to the incoming current. The Reduction Theorem to be derived in the following section provides a link between the S-matrix and expectation values of timeordered products of ﬁeld operators that can be calculated as derivatives of generating functionals.

7.2

Reduction Theorem

Since the asymptotic states appearing in the S matrix are those of free, onshell particles we can describe them as non-interacting quantum excitations of the vacuum of the theory with free dispersion relations. We, therefore, ﬁrst review the basic properties of free ﬁeld creation and annihilation operators in the next subsection.

7.2.1

Canonical field quantization

The ﬁeld operators are obtained by quantizing the free asymptotic ﬁelds φin and φout . This is done in the usual way by imposing commutator relations for the ﬁelds and their momenta. We impose the canonical commutator relations of quantum mechanics for coordinates and corresponding canonical momenta. Remembering that in ﬁeld theory the ﬁelds play the role of the classical coordinates we thus impose1 [Π(x, t), φ(x , t)] = − iδ 3 (x − x ) " ˙ x , t) = 0 Π(x, t), φ(

!

[φ(x, t), φ(x , t)] = 0 .

(7.11)

We now employ the normal mode expansion of the ﬁelds (see the discussion at the start of section 5.1) and momenta 1 1 √ φ(x, t) = √ ak (t) eik·x + a†k (t) e−ik·x 2ωk V k

i ωk √ Π(x, t) = − √ ak (t) eik·x − a†k (t) e−ik·x 2ωk V k 1

(7.12)

From here on all the fields φ in this section are operators; only for ease of notation we do not write the ’operator-hats’ explicitly

CHAPTER 7. S-MATRIX AND GREEN’S FUNCTIONS

81

with ωk = k 2 + m2 . Since the ﬁelds φ and the momenta Π are now operators, the ’expansion coeﬃcients’ a and a† are operators as well. The inverse Fourier transformation is given by 1 = √ eik·x (ωk φ(x, t) + iΠ(x, t)) d3x 2ωk V 1 † √ eik·x (ωk φ(x, t) − iΠ(x, t)) d3x ak (t) = 2ωk V

a†k (t)

(7.13)

Using the commutator relations (7.11) we obtain also the commutator relations for the operators ak and a†k ! !

"

ak (t), a†k (t)

= δk,k

ak (t), ak (t)

=

"

!

"

a†k (t), a†k (t) = 0 .

(7.14)

The asymptotic in and out ﬁelds are free ﬁelds so that the time-dependence of their operators a and a† is harmonic ak (t) = ak (0)e−iωk t a†k (t) = a†k (0)e+iωk t

(7.15)

as can be obtained from the Heisenberg equation of motion. The asymptotic in ﬁelds φin are then given by 1 1 √ φin (x, t) = √ ak (0) e−ikx + a†k (0) e+ikx ; 2ωk V k

(7.16)

with ikx = i(ωk t − k · x). The ﬁelds φout can be represented in the same way. ˙ the free ﬁeld annihilation and Remembering that for free ﬁelds Π = Φ, creation operators for the in and out states are given in terms of the ﬁelds and momenta as −i 3 −ikx d xe (∂t φ(x, t) + iωk φ(x, t)) 2ωk V V i ak (0) = √ d3x e+ikx (∂t φ(x, t) − iωk φ(x, t)) 2ωk V V

a†k (0) = √

(7.17)

with kx = ωk t − k · x. The operators (7.17) are the same as the ones used in the well known algebraic treatment of the harmonic oscillators for the normal ﬁeld modes. The a† are the creation and the a the annihilation operators for free ﬁeld quanta and the vacuum (groundstate) of the free ﬁeld theory is given by a|0 = 0.

CHAPTER 7. S-MATRIX AND GREEN’S FUNCTIONS

7.2.2

82

Derivation of the reduction theorem

In a realistic physics situation the scattering or decay processes that we aim to describe involve interactions between the particles. Simulating the situation in a scattering experiment we, therefore, now assume that the interactions between the particles are adiabatically being switched on and oﬀ; adiabatically here means without energy transfer. The asymptotic “in” and “out” states are thus free states which can be described by the free ﬁeld operators (7.17). The operators a† (t) and a(t) in (7.13) have a harmonic time-dependence (7.15) only for times t → ±∞. The free ﬁeld operators (7.17) acting on the vacuum of the full, interacting theory create and annihilate ﬁeld quanta only at times t → ±∞ while they do so at all times when acting on the vacuum state of the noninteracting theory. They can thus be used to describe the asymptotic states. We can thus write for the S matrix element (7.1) Sβα = β, out|α, in = β, out|a†in (k)|α − k, in ,

(7.18)

where we have assumed that the in-state |α, in contained a free particle with three-momentum k ; |α−k, in is then that in state in which just this particle is missing. We can further write this as β, out|α, in = β, out|a†out (k)|α − k, in + β, out|a†in (k) − a†out (k)|α − k, in .

(7.19)

In the ﬁrst term β, out|a†out (k) = β − k, out| (= 0, if β, out| does not contain a particle with momentum k). We now rewrite the in and out creation operator into a more compact form ↔ † (7.20) ain,out (k) = −i d3x fk (x) ∂ t φin,out (x, t) with

↔

f (t) ∂ t φ(x, t) = f and

∂φ ∂f − φ ∂t ∂t

(7.21)

1 fk (x) = √ e−ikx . 2ωk V

With this expression we can get the S-matrix element into the form β, out |α, in = β − k, out|α − k, in − iβ, out|

d3x fk (x)

(7.22) ↔ ∂t

[φin (x) − φout (x)] |α − k, in .

CHAPTER 7. S-MATRIX AND GREEN’S FUNCTIONS

83

For only 2 particles in the in and out states the ﬁrst term on the rhs represents a single particle transition matrixelement. In this case it can obviously only contribute if both particles do not change their energy and momentum, i.e. if |out and |in are identical. It is then just the forward scattering amplitude. The rhs of (7.22) is time-independent. We can see this by calculating the time derivative of its integrand

↔

∂t f (x) ∂ t φin (x) = f ∂t2 φin − ∂t2 f φin .

(7.23)

Here f (x) = e−ikx and φin both solve the free Klein-Gordon equation with the same mass. We therefore have

∂t f (x)

↔ ∂t

2 φin − (∇ 2 f )φin . φin (x) = f ∇

(7.24)

The integral over this expression vanishes after twofold partial integration of one of the terms. The same holds, of course, for the term involving φout in (7.22). The rhs of (7.22) is thus indeed time-independent. We can, therefore, take it at any time and in particular also at t → ±∞. Then we can replace the in and out ﬁelds at these times by the limits of the ﬁeld φ(x). This gives for the S-matrix element β, out|α, in = β − k, out|α − k, in

↔

d3x fk (x) ∂ t φ(x)|α − k, in

lim iβ, out|

+

t→+∞

−

↔

d3x fk (x) ∂ t φ(x)|α − k, in .

lim iβ, out|

t→−∞

(7.25)

We now write this expression in a covariant form by using

lim − lim

t→+∞

=

=

↔

! " ↔ ∂ dx f (x) ∂ t φ = d4x f ∂t2 φ − ∂t2 f φ ∂t 3

dt −∞

d3x f (x) ∂ t φ

t→−∞

+∞

!

2 − m2 )f d4x f ∂t2 − (∇

"

φ.

(7.26)

Note that here φ is an interacting ﬁeld, since we integrate now over all times. Thus, in contrast to (7.23) φ does not solve the free Klein-Gordon equation and, consequently, this integral does not vanish. Twofold partial integration in the second term on the rhs allows us now to roll the Laplace operator from f over to φ. This gives

4

dx

f ∂t2

φ−

(∂t2 f )φ

=

d4x f (x)(2 + m2 ) φ(x) .

(7.27)

CHAPTER 7. S-MATRIX AND GREEN’S FUNCTIONS

84

We thus have β, out|α, in = β − k, out|α − k, in

(7.28)

d4x fk (x)(2 + m2 )β, out|φ(x)|α − k, in .

+i

In this expression we have removed one particle from the in state. We now continue by removing one particle with the momentum k from the out state by going through exactly the same steps as before. Disregarding the ﬁrst term in (7.28), that contributes only to forward scattering, we get β, out|φ(x)|α − k, in = β − k , out|aout (k )φ(x)|α − k, in = β − k , out|φ(x)ain (k )|α − k, in (7.29) + β − k , out|aout (k )φ(x) − φ(x)ain (k )|α − k, in . We next replace the annihilation operators by the corresponding ﬁeld operators as in (7.20) (here we have to take the hermitian conjugate operator) and obtain β, out|φ(x)|α − k, in = β − k , out|φ(x)|α − k − k , in

+i

↔

(7.30)

d3x β − k , out| fk∗ (x ) ∂ t [φout (x )φ(x) − φ(x)φin (x )] |α − k, in .

Taking now the limits t → ±∞ gives, as above, for this expression β − k , out|φ(x)|α − k − k , in ↔ 4 ∂ ∗ + iβ − k , out| d x fk (x ) ∂ t T [φ(x )φ(x)] |α − k, in ∂t = . . . + iβ − k , out|

d4x fk∗ (x )∂t2 T [φ(x )φ(x)]

− ∂t2 fk∗ (x ) T [φ(x )φ(x)] |α − k, in .

(7.31)

We now use again (7.27) and obtain

= · · · + i β − k , out|

d4x fk∗ (x )(2 + m2 )T [φ(x )φ(x)] |α − k, in .

Combining this result with (7.28) ﬁnally gives (neglecting the forward scattering amplitude) β, out|α, in = i2

d4x d4x fk∗ (x )fk (x)

(7.32)

×(2 + m2 )(2 + m2 )β − k , out|T [φ(x )φ(x)] |α − k, in .

CHAPTER 7. S-MATRIX AND GREEN’S FUNCTIONS

85

This reduction can obviously be continued on both sides until we have (with n particles with momenta k in the out state and m particles with momenta k in the in state) Sβα = βn k , out|αm k, in = im+n

m

d4xi

n

i=1

(2j

j=1

(7.33)

d4xj fk∗j (xj )fki (xi ) ×

+ m )(2i + m )0|T [φ(x1 )φ(x2 ) . . . φ(xn )φ(x1 )φ(x2 ) . . . φ(xm )] |0 . 2

2

This is the so-called Reduction Theorem that enables us to express the Smatrix in terms of the (n + m)-point Green’s function, somestimes also called correlation function G(x1 , x2 , . . . , xn , x1 , x2 , . . . , xm ) = 0|T [φ(x1 )φ(x2 ) . . . φ(xn )φ(x1 )φ(x2 ) . . . φ(xm )] |0 .

(7.34)

Note that in the reduction theorem (7.33) the information about the interaction of the particles is contained in the (interacting) ﬁeld operators φ. Also the vacuum appearing here is that of the full, interacting theory. The n+m-point function appearing there is, therefore, also that of the interacting theory! The physical process described by the reduction theorem is that of m onshell particles in the initial state at the asymptotic space-time coordinates x1 , . . . , xm and n on-shell particles at the space-time coordinates x1 , . . . , xn with an interaction region in between these sets of coordinates. As we will see later, the Klein-Gordon operators 2 + m2 when acting on G just remove the propagators from the interaction region out to the asymptotic points, creating so-called vertex functions Γ(x1 , . . . , xn , x1 , . . . , xm ). The reduction theorem (7.33) then gives the transition rate Sβα as the Fourier transform of this vertex function. The Fourier transform in (7.33) contains factors of the form exp(ik x ) for the outgoing particles and exp(−ikx) for the incoming ones. This could be symmetrized by changing all outgoing momenta k → −k . This gives Sβα = βn − k , out|αm k, in m+n

= i

m

i=1

4

d xi

n

j=1

d4xj

1 1 e−i(kj xj +ki xi ) 2ωkj V 2ωki V

×Γ(x1 , x2 , . . . , xn , x1 , x2 , . . . , xm )

.

(7.35)

The connection between the S-matrix and the Green’s function as expressed by the reduction theorem has been derived here only for scalar ﬁelds,

CHAPTER 7. S-MATRIX AND GREEN’S FUNCTIONS

86

but it is valid in general. The only formal diﬀerence is that the Klein-Gordon operator 2 + m2 has to be replaced by the corresponding free-ﬁeld operator. It is also important to note that the method of canonical quantization used here to derive the reduction theorem has been used only for the asymptotic states. Thus, even for ﬁelds where this method runs into diﬃculties when interactions are present, like, e.g., the gauge ﬁelds to be treated in chapter 13, the reduction theorem holds in the form given above. In the remainder of this book we will be concerned with calculating the correlation functions by using path integral methods. Once these correlation functions are known the reduction theorem allows us to calculate any reaction rate or decay probability.

Chapter 8 GREEN’S FUNCTIONS In chapter 7 we have found that all the S-matrix elements can be calculated once the correlation, or Green’s, functions are known. In this chapter we discuss how these functions can be obtained as functional derivatives of the generating functionals of the theory.

8.1

n-point Green’s Functions

In chapter 7 we have seen that the correlation function, i.e. the vacuum expectation value of time-ordered ﬁeld operators, !

"

ˆ 1 )φ(x ˆ 2 ) . . . φ(x ˆ n ) |0 . G(x1 , x2 , . . . , xn ) = 0|T φ(x

(8.1)

determines the transition rate for all physical processes. Remembering that in ﬁeld theory the ﬁeld operators play the role of the coordinates in classical quantum theory we can now directly use the results obtained in Chapt. 3 and Sect. 5.1.1 and write using (3.47) !

"

ˆ 2 ) . . . φ(x ˆ n ) |0 ˆ 1 )φ(x G(x1 , x2 , . . . , xn ) = 0|T φ(x

=

with S[φ] =

+∞ −∞

Dφ φ(x1 )φ(x2 ) . . . φ(xn )eiS[φ]

Dφ eiS[φ]

(8.2)

L(φ, ∂μ φ)d4x. The vacuum here is that of the full, interacting

Hamiltonian. The latter expression can also be obtained as a functional derivative of the generating functional of the theory (cf. (3.47)) so that we can also equiv87

CHAPTER 8. GREEN’S FUNCTIONS

88

alently deﬁne the n-point Green’s function by G(x1 , x2 , . . . , xn ) =

n 1

i

.

. δ n Z[J] . . . δJ(x1 )δJ(x2 ) . . . δJ(xn ) .J=0

(8.3)

We note that G(x1 , . . . , xn ) is a symmetric function of its arguments. Therefore, according to (B.37) the following relation holds Z[J] =

1 n

n!

dx1 . . . dxn in G(x1 , x2 , . . . , xn )J(x1 )J(x2 ) . . . J(xn ) . (8.4)

Connected Green’s Functions. Guided by (3.22) we deﬁne a functional S[J] by the relation (8.5) Z[J] = eiS[J] and introduce the so-called connected Green’s functions Gc in terms of S[J] deﬁned by the relation n−1

Gc (x1 , . . . , xn ) =

1 i

.

. δnS . . . δJ(x1 ) . . . δJ(xn ) .J=0

(8.6)

The name of this correlation function and its physics content will become clear later in this section.

8.1.1

Momentum representation

Very often it is advantageous to work in momentum space because the external lines of Feynman graphs represent free particles with good momentum. In general the transformation of the Green’s function into the momentumrepresentation is given by

e−i(p1 x1 +p2 x2 +...+pn xn ) G(x1 , x2 , . . . , xn ) d4x1 d4x2 . . . d4xn = (2π)4 δ 4 (p1 + p2 + . . . + pn ) G(p1 , p2 , . . . , pn )

(8.7)

The δ-function here reﬂects the momentum conservation due to translational invariance. This can be seen by performing pairwise transformations of two space-time points to their cm. point and their relative coordinate. If we then assume that G depends only on the latter, the integral over the c.m. coordinate can be performed and yields the δ-function. As in our discussion around (7.35) we take all the momenta as pointing into the vertex (see Fig. 8.1).

CHAPTER 8. GREEN’S FUNCTIONS p1

89 p4 p5

p2

p6 p7

p3

Figure 8.1: Momentum representation of the n-point function. Note that all momenta are pointing into the shaded interaction region.

8.1.2

Operator Representations

Operator representation of the generating functional. For completeness, we now derive an alternative expression for the generating functional Z[J]. We start by deﬁning the operator functional ˆ = T ei Z[J]

ˆ J(x)φ(x) d4x

(8.8)

ˆ where φˆ is an operator! If we form the functional derivatives of Z[J] we get, in analogy to (3.43), n

1 i

! " δ n Zˆ ˆ n )Z[J] ˆ 1 ) . . . φ(x ˆ , = T φ(x δJ(x1 ) . . . δJ(xn )

(8.9)

ˆ = 1, so that, because of Z[0] n

1 i

.

. ! " ˆ δ n 0|Z[J]|0 . ˆ 1 ) . . . φ(x ˆ n ) |0 . φ(x = 0|T δJ(x1 ) . . . δJ(xn ) .J=0 n

=

1 i

.

. δnZ . . . (8.10) δJ(x1 ) . . . δJ(xn ) .J=0

ˆ Thus all the functional derivatives of 0|Z[J]|0 agree with those of Z[J] at J = 0. According to (8.3) and (8.4) the two expressions therefore have to be equal ˆ Z[J] = 0|Z[J]|0 . (8.11) Functional form of the scattering operator. After having seen that the Green’s functions can be obtained as functional derivatives of a generating functional in this section we show that the scattering operator Sˆ can also be

CHAPTER 8. GREEN’S FUNCTIONS

90

written in a functional form as δ i φˆin (x)(2+m2 ) δJ(x) d4 x ˆ S = :e : Z[J]|J=0 (8.12) ∞ k k i δ = d4x1 . . . d4xk : φˆin (x1 ) . . . φˆin (xk ) : Z[J] |J=0 . δJ(x1 ) . . . δJ(xk ) k=0 k! Here the : : symbol denotes the so-called normal ordered product of ﬁeld operators. This normal-ordered product is deﬁned in such a way that all operators in it are reordered so that all the annihilation operators are moved to the right. This reordering takes place without a sign change for boson ﬁelds and with a sign-change for each pairwise exchange for fermion ﬁelds. δ in (8.12) acts only on Z[J]. The operator (2 + m2 ) δJ(x) The matrix elements of the operator (8.12) indeed agree with (7.33). This can be seen by considering again a matrixelement with n particles in the out states and m particles in the in state. In the expansion of the exponential in (8.12) only that term can contribute that contains exactly m + n powers of the ﬁelds. We thus have ˆ n, out|S|m, in = im+n

m

i=1

4

d xi

m+n

d4xj

(8.13)

j=m+1

1 n| : φin (x1 ) . . . φin (xm+n ) : |m (m + n)! × (21 + m2 )(22 + m2 ) . . . (2m+n + m2 ) im+n G(x1 , x2 , . . . , xm+n ) .

×

Here the in ﬁeld operator φin (x) can simply be replaced by operators of the free ﬁeld (7.8). The normal product reorders the expansion (8.13) such that all the annihilation operators are on the right. Since each ﬁeld contains two independent sums over positive and negative energy eigenstates, respectively, we have in total 2m+n operator products; of these only the term with n creation operators and m annihilation operators can contribute. This gives with the expansion (7.12) n| : φin (x1 ) . . . φin (xm+n ) : |m n n+m

(m + n)! n| = fk∗k (xk )a†k fkl (xl )al |m m! n! k=1 l=n+1 n n+m

(m + n)!

∗ fkk (xk ) fkl (xl ) . = m! n! k=1 l=n+1

(8.14)

The degeneracy factor in front of the matrix element follows from the binomial expansion of the individual terms in the normal mode expansion (fk ak + fk∗ a†k )m+n . The integration in (8.13) over the xi and xj just gives an extra degeneracy factor m!n!. Taking this result together with (8.13) is the same as (7.33).

CHAPTER 8. GREEN’S FUNCTIONS

8.2

91

Free Scalar Fields

We consider ﬁrst the case of free ﬁelds. In this case the generating functional can be given analytically (6.11) Z0 [J] = e− 2 i

J(x)DF (x−y)J(y) d4x d4y

.

(8.15)

The ﬁrst functional derivative vanishes at J = 0 because the integral (8.15) is Gaussian. For the second functional derivative we obtain i δ 2 Z0 [J] 4 4 = −iDF (x1 − x2 ) e− 2 J(x)DF (x−y)J(y) d x d y δJ(x1 )J(x2 )

2

+ (−i)

d4x d4y DF (x1 − x)DF (x2 − y)J(x)J(y) e− 2 i

so that we have

(8.16) J(x)DF (x−y)J(y) d4x d4y

,

.

δ 2 Z0 [J] .. . = iDF (x1 − x2 ) . G(x1 , x2 ) = − δJ(x1 )δJ(x2 ) .J=0

(8.17)

The two-point function is thus just the Feynman propagator. It is therefore also a solution of (cf. (6.7)) !

2 + m2 − iε

8.2.1

"

G(x1 , x2 ) = −iδ 4 (x)

(8.18)

Wick’s theorem

The higher order derivatives can be most easily obtained by expanding (8.15) ∞ 1

n i J(x)DF (x − y)J(y) dx dy (8.19) 2 n=0 n! ∞ i n 1 = 1+ − dx1 . . . dx2n D12 D34 . . . D2n−1 2n J1 J2 . . . J2n 2 n=1 n!

Z0 [J] =

−

with the shorthand notation Dij = DF (xi − xj ), Jk = J(xk ). Noting that Z always contains even powers of J, it is immediately evident that all n-point functions with odd n vanish because an odd functional derivative of an even function always vanishes at J = 0. Taking now the 2k-th functional derivative of Z0 and using (8.3) and (B.36) we obtain 2k

1 δ 2k Z0 | i δJ1 . . . δJ2k J=0 (i)k = k Dp1 p2 . . . Dp2k−1 p2k 2 k! P

G(x1 , x2 , . . . , x2k ) =

(8.20)

CHAPTER 8. GREEN’S FUNCTIONS

92

where the sum runs over all permutations (p1 , p2 , . . . , p2k ) of the numbers (1, 2, . . . , 2k). The factor in front of the sum removes the doublecounting because of the symmetry Dp2 p1 = Dp1 p2 (2k ) and because of the random order of factors under the sum (k!). Equation (8.20) states that the n-point function of a system of free bosons can be written as a properly normalized and symmetrized product of twopoint functions. This is the so-called Wick’s theorem. As an example we consider the case n = 4. We then have G(x1 , x2 , x3 , x4 ) = −

1 Dp p Dp p . 8 P ∈S4 1 2 3 4

(8.21)

Among the 24 terms in the sum, 12 are pairwise equal because the product of the two propagators commutes. Furthermore, the propagator Dp1 p2 and Dp2 p1 are pairwise equal. Thus, there are only 24 : 2 : 2 : 2 = 3 essentially distinct terms in the sum; the factor 1/8 just takes care of all the others. We thus have G(x1 , x2 , x3 , x4 ) =

− DF (x1 − x2 )DF (x3 − x4 ) − DF (x1 − x3 )DF (x2 − x4 ) − DF (x1 − x4 )DF (x2 − x3 ) .

(8.22)

The ﬁrst few n-point Green’s functions are therefore – according to (8.17), (8.20) and (8.22) – given by G(x1 ) G(x1 , x2 ) G(x1 , x2 , x3 ) G(x1 , x2 , x3 , x4 )

= = = = =

etc.

0 0|T [φ(x1 )φ(x2 )] |0 = iDF (x1 − x2 ) 0 0|T [φ(x1 )φ(x2 )φ(x3 )φ(x4 )] |0 − DF (x1 − x2 )DF (x3 − x4 ) − DF (x1 − x3 )DF (x2 − x4 ) − DF (x1 − x4 )DF (x2 − x3 )

(8.23)

(8.24)

Here |0 is the vacuum state of the free Hamiltonian because it was obtained as a functional derivative of the non-interacting functional Z0 (8.19). All n-point functions with odd n vanish.

8.2.2

Feynman rules

As in the classical case (cf. Sect. 2.3) we can again represent these results in a graphical way. The Feynman rules, that establish the connection between

CHAPTER 8. GREEN’S FUNCTIONS

93

the algebraic and the graphical representation, are for the case of free ﬁelds still rather trivial. They are given by 1) each Feynman propagator is represented by a line: = iDF (x − y) . x y 2) each source is represented by a cross:

= iJ(x) .

x

3) There is an integration over all space-time coordinates of the currents 4) Each diagram has a factor that takes its symmetry into account. For example, if there is an integration over the endpoints x and y, these could be exchanged without changing the result; the symmetry factor is, correspondingly, 1/2. With rule 1) we get, for example, for the fourpoint function (8.22), i.e. the two-particle Green’s function 1 2 G(x1 , x2 , x3 , x4 ) =

1 3

2 4

+

1

2

+

(8.25)

3 4 3 4 Each line connecting the two points x and y denotes the free propagator and gives a factor iDF (x − y). We now set (8.26) Z0 [J] = eiS0 [J] with

i 4 4 d x d y J(x)DF (x − y)J(y) iS0 [J] = − 2 By using rules 2), 3) and 4) we ﬁnd immediately iS0 = Now we expand Z0 [J] = eiS0 [J] = e i = 1+ − J(x)DF (x − y)J(y) d4x d4y 2 2 i 1 − J(x)DF (x − y)J(y) d4x d4y + · · · + 2! 2

(8.27)

(8.28)

CHAPTER 8. GREEN’S FUNCTIONS = 1 + iS0 [J] + = 1+

94

1 1 (iS0 [J])2 + (iS0 [J])3 + . . . 2! 3! 1 1 + + + ... . 2! 3!

(8.29)

We now call all graphs that hang together connected graphs and all the others unconnected graphs. In our simple case here S0 [J] is represented by only one connected graph (8.28), whereas Z0 [J] – through the power expansion (8.29) of the exponential function – generates unconnected graphs as well. In the simple case of a free ﬁeld discussed here there is only one connected graph (see (8.25)). The connected Green’s function can be obtained from its deﬁnition (8.6) as Gc (x1 , x2 ) =

1 δ 2 S0 = iDF (x1 − x2 ) . | i δJ(x1 )δJ(x2 ) J=0

(8.30)

All higher functional derivatives of S0 [J] vanish. Gc is in this free case thus just given by the Feynman propagator.

8.3

Interacting Scalar Fields

In this section we consider Lagrangians of the form L = L0 − V (φ) ,

(8.31)

where L0 is the free scalar Lagrangian (5.1) and V represents a selﬁnteraction of the ﬁeld. For such a Lagrangian the generating functional for the n-point functions can no longer be given in closed form. In order to obtain the n-point function one has to resort to perturbative methods. The n-point function then follows from its deﬁnition (8.2)

G(x1 , x2 , . . . , xn ) =

Dφ φ(x1 )φ(x2 ) . . . φ(xn ) eiS[φ]

Dφ e

iS[φ]

(8.32)

with the exponential in the generating functional of the form eiS[φ] = ei

d4x (L0 −V +i 2ε φ2 )

.

(8.33)

The action exponential can be Taylor-expanded in the interaction strength eiS[φ] =

∞ 1 N =0 N !

−i

d4x V

N

eiS0 [φ]

(8.34)

CHAPTER 8. GREEN’S FUNCTIONS

95

with the free action

S0 [φ] =

ε d x L0 + i φ2 2

4

.

(8.35)

Inserting this into (8.32) gives the n-point function of the interacting theory in terms of powers of the interaction and the free-ﬁeld action

G(x1 , x2 , . . . , xn ) =

Dφ φ(x1 )φ(x2 ) . . . φ(xn ) eiS[φ] Dφ e

Dφ φ(x1 )φ(x2 ) . . . φ(xn )

=

Dφ

∞ 1

∞ 1 N =0 N !

−i

N =0 N !

(8.36)

iS[φ]

−i

d4x V

N

N

4

d xV

eiS0 [φ]

eiS0 [φ]

By using (8.2) and the developments in Sects. 3.2 and 5.1.1 we can rewrite this equation also in terms of normalized vacuum expectation values. The last line of (8.36) involves the free action S0 and thus free propagation. Therefore, if the ﬁeld operators are those of free in ﬁelds at asymptotic times they remain so even during propagation. Also the vacuum appearing in the quantum mechanical vacuum expectation value is then that of the non-interacting theory so that we get !

"

ˆ 2 ) . . . φ(x ˆ n ) |˜0 ˆ 1 )φ(x G(x1 , x2 , . . . , xn ) = ˜0|T φ(x

0|T φˆin (x1 ) . . . φˆin (xn ) = 0|T

∞ 1 N =0 N !

∞ 1 N =0

N!

−i

−i

Vˆ d4x

Vˆ d x

N

4

N

|0 .

(8.37)

|0

˜ denotes here the vacuum state of the full, interacting As in Sect. 5.1.1 |0 Hamiltonian, whereas |0 is that of the free Hamiltonian. With (8.37) we have achieved a remarkable result: (8.37) expresses the expectation value of the time-ordered product of ﬁeld operators in the vacuum state of the full, interacting theory by a perturbative expansion over free ﬁeld expectation values. The latter can be calculated as path integrals over products of classical ﬁelds and powers of the interaction (8.36). This enables us to calculate G, and ultimately also the scattering matrix S (Chapt. 7), perturbatively up to any desired order in the interaction.

CHAPTER 8. GREEN’S FUNCTIONS

8.3.1

96

Perturbative expansion

Eq. (8.32) allows us to calculate the perturbative expansion of the full Green’s function up to any desired order in V . Alternatively, these higher order terms can also be obtained as functional derivatives of the free generating functional Z0 [J] which is known. According to our general considerations the functional for the Lagrangian (8.31) is given by Z[J] = Z0 = Z0

Dφ ei

Dφ e−i

d4x (L0 −V (φ)+Jφ+i 2ε φ2 )

d4x V (φ) i

(8.38)

d4x (L0 +Jφ+i 2ε φ2 )

e

.

Here Z0 is just the inverse of the path integral for J = 0 Z0−1 =

Dφ ei

d4x (L0 −V (φ)+i 2ε φ2 )

.

(8.39)

We now use the relation 4 4 ε 2 ε 2 1 δ ei d x (L0 +Jφ+i 2 φ ) = φ(y) ei d x (L0 +Jφ+i 2 φ ) . i δJ(y)

(8.40)

This relation, read from right to left, will also be true for any function V (φ), as can be seen by expanding V into a series in powers of φ. We thus have also i

V [φ(y)] e

d4x (L0 +Jφ+i 2ε φ2 )

4 ε 2 1 δ ei d x (L0 +Jφ+i 2 φ ) =V i δJ(y)

(8.41)

and consequently, after exponentiation, also e−i

d4y V [φ(y)] i −i

= e

e

d4y V ( 1i

d4x (L0 +Jφ+i 2ε φ2 ) δ δJ(y)

) ei

d4x

L0 +Jφ+i 2ε φ2

(

(8.42) ).

This relation allows us to take the V -dependent factor in (8.38) out of the path-integral Z[J] = Z0 e

−i

δ d4y V ( 1i δJ(y) )

Dφ e

i

d4x (L0 +Jφ+i 2ε φ2 )

.

(8.43)

The last factor in (8.43) has been expressed in terms of the free two-particle propagator introduced in the last section. We thus have Z[J] = Z0 e−i

δ d4z V ( 1i δJ(z) ) − 2i

4

4

J(x)DF (x−y)J(y) d x d y e 4 δ 1 = Z0 e−i d x V ( i δJ(x) ) eiS0 [J] 4 1 δ = Z0 e−i d x V ( i δJ(x) ) Z0 [J] .

(8.44)

CHAPTER 8. GREEN’S FUNCTIONS

97

Expanding the exponential that contains the interaction V then gives the perturbative expansion for Z[J] Z[J] = Z0

∞ 1 N =0

−i

N!

4

d xV

1 δ i δJ(x)

N

Z0 [J] .

(8.45)

Since we will later on be mostly interested in connected graphs we do not need Z[J] directly but instead its logarithm. We, therefore, now expand the functional iS[J] = ln Z[J] in powers of the interaction V . We start by inserting a factor 1 = exp (+iS0 ) exp (−iS0 ) between Z0 and the exponential in (8.44) and taking the logarithm

ln Z[J] = ln Z0 + ln 1 · e−i

= ln Z0 + iS0 + ln e

d4x V ( 1i

δ δJ

−iS0 −i

e

&

) eiS0 [J]

d4x V iS0

e

−iS0 [J]

= ln Z0 + iS0 [J] + ln 1 + e

e

−i

δ d4x V ( 1i δJ )

−1 e

= iS[J] .

' iS0 [J]

(8.46)

A perturbation theoretical treatment is now based on a Taylor expansion of the logarithm. For that purpose we abbreviate

ε[J] = e−iS0 [J] e−i

d4x V

− 1 eiS0 [J]

(8.47)

and obtain iS[J] = ln Z[J] = ln Z0 + iS0 [J] + ln(1 + ε[J]) 1 2 3 = ln Z0 + iS0 [J] + ε[J] − ε [J] + O(ε ) . 2

(8.48)

Equation (8.48) represents an expansion of S in powers of the (for V → 0) small quantity ε. In order to obtain a perturbative expansion in terms of the potential V we now rearrange the expansion (8.48). First we expand ε[J] of (8.47) in terms of the strength of the interaction. This gives ) −iS0 [J]

−i

ε[J] = e

+

1 −i 2!

4

d xV

d4x V

1 δ i δJ

1 δ i δJ

(8.49) ⎫ ⎬

2

+ · · ·⎭ e+iS0 [J] .

We now insert this expression into (8.48) and obtain

1 iS[J] = ln Z0 + iS0 [J] + ε[J] − ε2 [J] + . . . 2

CHAPTER 8. GREEN’S FUNCTIONS

98

−iS0 [J]

= ln Z0 + iS0 [J] + e

1 + e−iS0 [J] −i d4x V 2! )

−i

1 −iS0 [J] − e −i d4x V 2

4

d xV

1 δ i δJ

2

1 δ i δJ

eiS0 [J]

eiS0 [J]

1 δ i δJ

= ln Z0 + iS0 [J] + iS1 [J] + iS2 [J] −

,2

e

iS0 [J]

+ O(V 3 )

1 (iS1 [J])2 + O(V 3 ) , 2 (8.50)

with i J(x)DF (x − y)J(y) d4x d4y 2 1 δ −iS0 [J] 4 iS1 [J] = e (−i) d x V e+iS0 [J] i δJ

iS0 [J] = −

1 −iS0 [J] iS2 [J] = −i d4x V e 2!

1 δ i δJ

2

e+iS0 [J] .

(8.51)

Equation (8.50) represents a perturbative expansion of S in powers of V . Note that the term of second order in the interaction V receives contributions both from the linear and the quadratic term in the original expansion (8.48) in ε. The term ∼ S1 obviously just contains an iteration of the linear term.

Chapter 9 PERTURBATIVE φ4 THEORY In this chapter we apply the formalism developed in the preceding chapter to the so-called φ4 theory whose Lagrangian is given by L = L0 − V (φ) = L0 −

g 4 φ . 4!

(9.1)

Here g is a coupling constant. This φ4 theory is a prototype of a ﬁeld theory with selﬁnteractions. It serves as a didactical example which exhibits all phenomena of more complex ﬁeld theories.

9.1

Perturbative Expansion of the Generating Function

We start with the generating functional for connected Green’s functions (8.50) iS[J] = ln Z0 + iS0 [J] + iS1 [J] + iS2 [J] −

1 (iS1 [J])2 + O(V 3 ) . 2

(9.2)

Inserting the interaction of φ4 theory (9.1) into (8.51) we obtain i 4 4 iS0 [J] = − d z d y J(z)DF (z − y)J(y) , 2 ig −iS0 [J] 4 δ4 iS1 [J] = − e d x 4 eiS0 [J] 4! δJ (x) and 1 −ig 2 −iS0 [J] 4 4 δ4 δ4 iS2 [J] = e d xd y 4 eiS0 [J] . 2! 4! δJ (x) δJ 4 (y)

99

(9.3)

CHAPTER 9. PERTURBATIVE φ4 THEORY

100

For notational convenience in the following we now introduce the Si deﬁned by ig iS1 [J] = − S˜1 [J] 4! −ig 2 ˜ S2 [J] . iS2 [J] = 4!

(9.4)

The generating function for the connected Green‘s functions in φ4 theory reads then iS[J] = ln Z0 + iS0 [J] +

−ig ˜ −ig S1 [J] + 4! 4!

2

2

1 −ig ˜ S1 [J] S˜2 [J] − 2 4!

+ ... .

(9.5) Since each of the Si [J] contains functional derivatives of the known S0 [J] we can now evaluate the functional derivatives of S[J] and obtain all the Green’s functions. First, we calculate the terms linear in g

δ4 e+iS0 [J] 4 δJ (x) i δ4 4 4 = e−iS0 [J] d4x 4 e− 2 J(z)DF (z−y)J(y) d z d y . δJ (x)

S˜1 [J] = e−iS0 [J]

d4x

(9.6)

The fourth functional derivative is most easily obtained by going to a discrete representation. Noting that ∂ 4 − i Ji Dij Jj e 2 = [−3Dkk Dkk + 6iDkk (DJ)k (DJ)k ∂Jk4

(9.7)

+ (DJ)k (DJ)k (DJ)k (DJ)k ] e− 2 Ji Dij Jj i

we obtain (k =x) ˆ S˜1 [J] = −3

+ 6i

+

DF (x − x)DF (x − x) d4x DF (y − x)DF (x − x)DF (x − z)J(y)J(z) d4x d4y d4z

[DF (x − y)DF (x − z)DF (x − v)DF (x − w) "

× J(y)J(z)J(v)J(w) d4x d4y d4v d4w d4z .

(9.8)

The ﬁrst term has no sources and will, therefore, not contribute to any Green’s function, the second term is quadratic in J and thus contributes to the two-point function and the last term here has the structure of a point interaction of four ﬁelds generated by independent sources at y, z, v, and w and thus contributes only to the four-point function.

CHAPTER 9. PERTURBATIVE φ4 THEORY

9.1.1

101

Feynman rules

We can again represent these results in a graphical form by using the rules given in section 8.2.2. These were iDF (x − y) =

1) propagator: 2) source:

iJ(x) =

x

y

x

3) Integration over the space-time coordinates of the sources 4) Symmetry factor for each diagram We supplement these now for the interacting theory by the additional rules −ig = 4!

5) Each interaction is represented by a dot:

6) Integration

d4x for each loop.

If we represent the interacting connected functional iS[J] by a double line (9.9) iS[J] = we can draw the graphs for iS[J] = ln Z0 + iS0 [J] +

−ig ˜ S1 [J] + O(g 2 ) 4!

(9.10)

as = ln Z0 + ⎛

+⎜ ⎝

w x

+

+ y

x

z

y

x

z

⎞

(9.11)

⎟ 2 ⎠ + O(g )

v

The ﬁrst graph on the right-hand side is again the zeroth order term (9.3), whereas the graphs in the parentheses represent all the terms of ﬁrst order in the interaction (9.8). The ﬁrst one of these describes a process without any external lines; this is a vacuum process that takes place regardless if physical particles are present or not. It constitutes a background to all physical processes. The second graph with the single loop describes a mass change due to the selﬁnteraction that we will discuss in the next section. The third graph, ﬁnally, describes a true interaction process.

CHAPTER 9. PERTURBATIVE φ4 THEORY

102

The symmetry factors in these graphs are the factors in front of the integrals in (9.8); they can be obtained as follows. The ﬁrst graph carries the factor 1/2 as explained in section 8.2.2. To construct the vacuum graph we pick one of the 4 legs of the interaction vertex and then connect with any of the other three free legs; there are always 2 pairwise equal loops. Thus in total we get a weight of 4 × 3/(2 × 2) = 3. For the second term in parentheses in (9.11) we have four legs of the vertex to connect with the external line to y; this gives 4 possibilities. The external line to z can then still be connected with 3 remaining vertex legs. Since there is an exchange symmetry between y and z we get an additional factor 1/2 (as in S0 ), so that the weight of this vertex becomes 6. The last graph, ﬁnally, is obtained by joining one of the four legs of the vertex to one of the external points, say z. This generates 4 possibilities. Next we join any one of the 3 remaining free legs of the vertex to the external point y; there are obviously 3 ways to do this. The remaining 2 legs can be joined in 2 diﬀerent ways with the two external points v and w. Thus, there are in total 4! = 24 possibilities. However, since all the external points v,w,y and z can be exchanged without changing any of the physics (v,w,y and z are integration variables), there are also 4! identical terms so that the last graph in the parentheses in Fig. 9.11 carries the weight 1. These weights (symmetry factors) have to be multiplied for each graph to the analytical expression obtained by following the rules given above for the translation of the pictorial representation into an analytical one. Indeed, using the symmetry factors just given and following the Feynman rules for the graphs (9.11) gives the expression (8.50) (together with (9.4) and (9.8)).

9.1.2

Vacuum contributions

We now consider the normalization term ln Z0 . Since we are working with normalized generating functionals, Z0 is given by W [0]−1 Z0−1

=

Dφ e

= e−i

i

d4x(L0 −V +i 2ε φ2 )

δ d4xV ( 1i δJ(x) ) − 2i

e

(9.12)

J(x)DF (x−y)J(y)d4x d4y |

J=0

.

We can now treat this expression in exactly the same way as we just did for S[J]; the only change being that we have to take the ﬁnal result at J = 0. This gives (see (9.10)) ln Z0 = −iS[0] = −iS0 [0] −

(−ig) ˜ S1 [0] + O(g 2 ) . 4!

(9.13)

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103

In the graphical representation this reads, using S0 [0] = 0 and S1 [0] = −3DF2 (0) d4x (cf. (9.8)), ln N = −

(9.14)

The normalization constant, or – in other words – the denominator of the generating functional, thus contains just the vacuum graph. Inserting (9.14) into (9.11) then removes the vacuum contribution giving, ﬁnally, for Z[J] the graphical representation up to terms of ﬁrst order in the interaction =

+

+

(9.15)

Although we have shown here only for ﬁrst-order coupling that the denominator in Z[J] (see (5.15),(5.16)) just removes the vacuum contributions, this is a general result that holds to all orders of perturbation theory.

9.2

Two-Point Function

The connected n-point function can now be obtained from its deﬁnition in (8.6) . n−1 . 1 δnS . . . (9.16) Gc (x1 , . . . , xn ) = i δJ(x1 ) . . . δJ(xn ) .J=0 In this section we work out the connected two-point function in the lowest orders of the coupling constant.

Terms up to O(g 0 )

9.2.1

There is only one connected Green’s function in the free case which is just given by the Feynman propagator (8.30). 9.2.1.1

Momentum representation

We now evaluate the lowest order (in g) two-point function in momentum space (cf. (8.7)). Taking the Fourier-transform of the two-point function (8.30) gives

e−i(p1 x1 +p2 x2 ) Gc (x1 , x2 ) d4x1 d4x2

=

e−i(p1 x1 +p2 x2 )

i (2π)4

(9.17)

d4q

e−iq(x1 −x2 ) d4x1 d4x2 . 2 2 q − m + iε

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104

−p

p

Figure 9.1: Two-point function of a scalar theory. We ﬁrst perform the integrations over x1 and x2 and obtain, according to the deﬁnition (8.1.1), for the rhs (2π)4 δ 4 (p1 + p2 )Gc (p1 , p2 ) = (2π)4 δ 4 (p1 + p2 )

p21

i . − m2 + iε

(9.18)

From this equation we can read oﬀ the momentum representation of the propagator. The momenta p1 and p2 are incoming momenta that point towards a vertex. Thus, the momentum representation of the free propagator is given by i G0 (p, p = −p) = 2 , (9.19) p − m2 + iε pictured in Fig. 9.1. Note that here the second momentum appears with a negative sign. This is due to our notation to take all momenta as incoming (see Fig. 8.1).

9.2.2

Terms up to O(g)

Up to terms linear in the coupling strength we obtain from (9.5) .

. δ2 S . . Gc (x1 , x2 ) = −i δJ(x1 )δJ(x2 ) .J=0

(9.20) .

. δ 2 S0 (−ig) δ 2 S1 . . = −i + O(g 2 ) . − δJ(x1 )δJ(x2 ) 4! δJ(x1 )δJ(x2 ) .J=0

With S0 [J] from (9.3) and S1 [J] from (9.8) this gives for the two-point function Gc (x1 , x2 ) = iDF (x1 − x2 ) ig − − 12i d4xDF (x − x)DF (x − x1 )DF (x − x2 ) + O(g 2 ) 4! g 4 d xDF (x2 − x)DF (x − x)DF (x − x1 ) = iDF (x1 − x2 ) − 2 + O(g 2 ) . (9.21)

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105

This is the connected propagator, up to terms of O(g), of the interacting theory. We can represent this equation in the following graphical form, where denotes the “dressed” propagator x1

x2

=

x1

x2

+

(9.22)

x1 x2 with the rules developed above. Since the n-point functions involve derivatives with respect to the source current, taken at zero source, the external lines of all Feynman graphs do not contain crosses, that depict sources, anymore. They are instead given by free propagators. The weight factors for these diagrams have been explained at the end of section 8.2.2. Since we deal here with Green’s functions with deﬁnite, ﬁxed external points, the exchange symmetry factors must not be divided out here. Thus, the ﬁrst diagram on the rhs in (9.22) carries the weight 1 and the second, the so-called tadpole diagram, the weight 12. 9.2.2.1

Momentum representation

In Sect. 8.1.1 we have introduced the momentum representation of the Green’s function and in Sect. 9.2.1.1 we have already evaluated it for the free case. Here we now determine it for the φ4 theory up to terms of order O(g). Taking the Fourier-transform of the two-point function (9.21) gives

e−i(p1 x1 +p2 x2 ) Gc (x1 , x2 ) d4x1 d4x2

=

e−i(p1 x1 +p2 x2 ) ⎡

i (2π)4

(9.23)

d4q

e−iq(x1 −x2 ) d4x1 d4x2 2 2 q − m + iε

g ⎣ −i(p1 x1 +p2 x2 ) 4 1 − e dx 2 (2π)4 ×

3

e−iq2 (x2 −x) e−iq1 (x−x1 ) d q1 d q2 d q3 2 d4x1 d4x2 . (q1 − m2 + iε)(q22 + m2 − iε)(q32 − m2 + iε) 4

4

4

As in Sect. 9.2.1.1 we ﬁrst perform the integrations over x1 , x2 and x and obtain for the rhs i (2π)4 δ 4 (p1 + p2 )Gc (p1 , p2 ) = (2π)4 δ 4 (p1 + p2 ) 2 p1 − m2 + iε g 1 1 − (2π)4 δ 4 (p1 + p2 ) 2 2 2 2 p1 − m + iε p2 − m2 + iε d4q 1 × . (9.24) 4 2 (2π) q − m2 + iε

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106

This equation is used to read oﬀ the momentum representation of the propagator (see (8.7)). The momenta p1 and p2 are incoming momenta that point towards a vertex. Thus, the momentum representation of (9.21) is given by i (9.25) − m2 + iε −ig d4q i i i +S 2 2 4 2 2 2 p − m + iε 4! (2π) q − m + iε p − m2 + iε

Gc (p, p = −p) =

p2

where the symmetry factor is S = 12. Equation (9.25) gives the momentum representation of the two-point function up to terms of order g. The ﬁrst term on the rhs gives the free propagator (9.19) already obtained in Sect. 9.2.1.1, whereas the second term gives the contribution of the interaction to this two-point function. Momentum space Feynman rules. The Feynman rules for (9.25) are now i . 1) each line gives a factor 2 q − m2 + iε 2) each vertex gives a factor

−ig . 4!

3) there is four-momentum conservation for the sum of all momenta ﬂowing into a vertex.

4) each internal line gives an integration

d4q . (2π)4

5) to each diagram a weight factor has to be multiplied as explained above. Selfenergy. With the abbreviation Σ=

g d4 q i 4 2 2 (2π) q − m2 + iε

(9.26)

and the free two-point function G0 from (9.19 we can write the two-point function (9.25) as

Σ Σ ΣG0 Gc (p, −p) = G0 + G0 G0 = G0 1 + G0 ≈ G0 1 − i i i i 1 = 2 2 p − m + iε 1 − Σ p2 −m1 2 +iε i . = 2 2 p − m − Σ + iε

−1

(9.27)

CHAPTER 9. PERTURBATIVE φ4 THEORY

107

h

a)

b)

c)

Figure 9.2: Feynman graphs for the two-point function up to O(g 2 ). Here we have consistently kept terms up to O(g). The quantity Σ appears like an additional mass term in the ﬁnal result. It is, therefore, called a selfenergy and the second graph on the rhs in (9.22) is called a selfenergy insertion. The appearance of this selfenergy is a ﬁrst indication that the mass m appearing in the Lagrangian is the mass of the particle only in a classical theory. In quantum theory it gets changed by the interactions.

9.2.3

Terms up to O(g 2 )

As noted at the end of Sect. 8.3.1 there are two distinct contributions to the second order term, one being a genuine term of second order in V and the other just being an iteration of the ﬁrst order term. With the help of the Feynman rules we can now construct the corresponding Feynman graphs up to terms of O(g 2 ). For the two-point function these are given in Fig. 9.2. Graph (a) in Fig. 9.2 obviously just represents an iteration of the ﬁrst order tadpole graph in (9.22). Its contribution to the two-point function is given by

i −ig d4q i (9.28) Ga (p, −p) = Sa 2 2 4 2 p − m + iε 4! (2π) q − m2 + iε i −ig d4q i i × 2 , 2 4 2 2 2 p − m + iε 4! (2π) q − m + iε p − m2 + iε Sa is the symmetry factor; it is simply given by the product of the corresponding factors for the one-loop graphs, Sa = 12 · 12 = 144. It is evident from Fig. 9.2a, as well as from its algebraic representation in (9.28), that the graph can be cut into two parts, each representing a ﬁrst

CHAPTER 9. PERTURBATIVE φ4 THEORY

108

order process. This reﬂects the appearance of the last term in (8.50) that is simply the square of the ﬁrst order term. Such a graph that falls apart into 2 unconnected parts, if one internal line is cut, is called one-particlereducible; otherwise it is one-particle-irreducible (1PI). The reducible graph here is generated by the square of the ﬁrst order term ∼ S˜12 in (9.5). In order to facilitate the following discussions we introduce now the vertex function Γ(p1 , p2 , . . . , pn ), sometimes also called connected proper vertex function, which describes only 1PI graphs and in which the propagators for the external lines are missing. The n-point vertex function is, therefore, given by (9.29) Γ(p1 , p2 , . . . , pn ) −1 −1 −1 = G (p1 , −p1 )G (p2 , −p2 ) . . . G (pn , −pn )Gc (p1 , p2 , . . . , pn ) . The free 1PI 2-point function is deﬁned by Γ(p, −p) = p2 − m2 .

(9.30)

Note that the product of inverse two-point Green’s functions and the n-body function is just the combination that appears in the reduction theorem (cf. (7.35)). The 1PI part of the Green’s function for the graph 9.2a reads Γa (p, −p) = G−1 (p, −p)G−1 (−p, p)Ga (p, −p) −ig d4q i = Sa 4 2 4! (2π) q − m2 + iε

(9.31)

with Sa = 144. We then get for the graph in Fig. 9.2b, which is one-particle irreducible,

−ig 2 d4q d4u i i(2π)4 δ 4 (q − u) 4! (2π)4 (2π)4 q 2 − m2 + iε u2 − m2 + iε 4 dr i (9.32) × 4 2 (2π) r − m2 + iε

Γb (p, −p) = Sb

with Sb = 12 · 12 = 144. For the graph in Fig. 9.2c (also 1PI) we ﬁnally obtain

−ig 4!

2

d4q d4r d4s δ(p − (q + r + s)) (2π)4 (2π)4 (2π)4 i i i , (9.33) × 2 2 2 2 2 q − m + iε r − m + iε s − m2 + iε

Γc (p, −p) = Sc

with Sc = 4 · 4! = 96.

CHAPTER 9. PERTURBATIVE φ4 THEORY 3

109

4

x

1

2

Figure 9.3: Feynman graph for the four-point function.

9.3

Four-Point Function

It is easy to see that the three-point function vanishes for the model considered here since the third functional derivative (see (9.16)) of the action (8.50) at J = 0 vanishes.

9.3.1

Terms up to O(g)

The four-point function up to terms of O(g) is given by 3

1 i

Gc (x1 , x2 , x3 , x4 ) =

.

. δ4 S . . δJ(x1 ) . . . δJ(x4 ) .J=0 .

(9.34) .

. . δ 4 S0 (−ig) δ 4 S1 . . . . = i + δJ(x1 ) . . . δJ(x4 ) .J=0 4! δJ(x1 ) . . . δJ(x4 ) .J=0

+ O(g 2 ) where S is given by (9.5) and Fig. 9.2. Because Gc involves the fourth derivative with respect to the source and S0 depends on J only quadratically (see (9.3)), only the last term of S1 in (9.8) can contribute to the Green’s function. Thus we get Gc (x1 , . . . , x4 ) = −ig

d4xDF (x − x1 )DF (x − x2 )DF (x − x3 )DF (x − x4 ) .

(9.35) In momentum space this is simply the product of the four propagators (9.19) times the factor −ig. The corresponding Feynman graph is given in Fig. 9.3. It carries the symmetry factor 4!, corresponding to a symmetry under permutation of all external legs. Unconnected graphs. From the connected graphs calculated so far, we could reconstruct also the unconnected graphs. Up to terms linear in the

CHAPTER 9. PERTURBATIVE φ4 THEORY

110

coupling constant g we get in symbolic notation Z[J] = e

= 1+

+

= 1+

1 ( 2!

+

1 + 2!

)2 + . . .

+

2

+

+

+O(g 2 ) .

(9.36)

The four-point function is generated by diagrams with four external legs (each external leg corresponds to a factor J in the generating functional), because G is given by a fourth functional derivative. Therefore, only the last diagram in the second line of (9.36) and the square of the ﬁrst term in the last line can contribute to the four-point function in this order. We thus have for the four-point function up to terms linear in the coupling G(x1 , x2 , x3 , x4 ) =

+

+

(9.37)

In both of the ﬁrst two diagrams the two particles just move by each other, without interaction. These unconnected graphs thus do not contribute to any interaction processes.

9.3.2

Terms up to O(g 2 )

. In order to become more familiar with Feynman graphs, we now construct the connected four-point function up to terms of order g 2 in a graphical way. This four-point function is shown in Fig. 9.4. The ﬁrst line in Fig. 9.4 gives the four-point function just constructed, with a symmetry factor S1 = 4!. The four diagrams on the second line are just the basic vertices with self-energy insertions on each of the external legs; these insertions carry the extra symmetry factor S2 = 12, as we have seen in section 9.1.1. The last three diagrams in the third line are of a new topological structure. They represent modiﬁcations of the basic interaction vertex through the insertion of internal lines. Each one of these graphs has the same external lines. We thus have an overall factor for all graphs S1

4

2 k=1 pk

i , − m2

(9.38)

CHAPTER 9. PERTURBATIVE φ4 THEORY

3

4

3

4

1

2

111

=

1

2 3

4

+

3

4

+ 1

2

3

4

4

+ 1

2

3 +

3

3

4

1

2

+ 1

2 3

4

q1

4 + O(g 3 )

+

+ q2 1 1

2

2

1

2

Figure 9.4: Four-point function of φ4 theory up to terms ∼ g 2 . with the external symmetry factor S1 = 4! so that the full four-point function is given by Gc (p1 , p2 , p3 , p4 ) = S1

4

2 k=1 pk

i (G1 + G2 + G3 ) , − m2

(9.39)

where Gi denotes the contribution from the i-th line in Fig. 9.4 without the external line symmetry factor. The ﬁrst basic vertex then just gives the contribution G1 =

−ig . 4!

(9.40)

The graphs of the second line have one loop in addition. We thus have G2 = (

4 i i −ig 2 d4q ) S2 2 4! (2π)4 q 2 − m2 l=1 pl − m2

(9.41)

CHAPTER 9. PERTURBATIVE φ4 THEORY

112

for their contribution, with S2 = 12. The internal momenta pl here are those between the loop and the four-point vertex. Since the loop carries no momentum away they are the same as the corresponding incoming momenta. The three graphs of the third line, ﬁnally, have two internal lines, which are, however, related by energy - and momentum conservation at the incoming vertex. They give i i −ig 2 d4q1 d4q2 ) S3 G3 = ( 2 2 4 4 2 4! (2π) (2π) q1 − m q2 − m2 × (2π)4

δ 4 (q1 + q2 − (pk + pl ))

(9.42)

kl

where the last sum runs only over the pairs of numbers (1,2), (1,3) and (1,4), i.e. the external legs at one of the vertices in each of the three graphs. The δfunction appears because the net momentum running into the dressed vertex has to be zero (see (8.7)). The symmetry factor S3 can, for example for the middle graph of Fig. 9.4, be obtained as follows. The external leg 1 can be connected with the left vertex in 4 diﬀerent ways; the same holds for the leg 2 with the right vertex. Once these connections (4 × 4 possibilities) have been done, each of the 3 free legs of the left vertex can be connected with the external leg 3; the same holds for the connections of the right vertex to the external point 4 (3 × 3 possibilities). Finally, each of the remaining 2 legs of the left vertex can be connected to the right vertex; for the remaining leg there is then no freedom left (2 possibilities). Thus, the symmetry factor for the last 3 graphs in Fig. 9.4 is (4 · 4)(3 · 3)2 (4!)2 4! = = . (9.43) S3 = S1 2S1 2

Chapter 10 DIVERGENCES IN n-POINT FUNCTIONS Many of the expressions obtained in the preceding sections for two- and fourpoint functions are actually ill-deﬁned because they diverge, as we will show in this section. We start with a rather general discussion of divergences in φ4 theory and then evaluate explicitly the two- and four-point functions. To illustrate the divergence of the Green’s functions obtained we consider, as an example, the two-point function Gc (p, −p) =

i − m2 + iε i g i + −i iD . (0) F 2 p2 − m2 + iε p2 − m2 + iε

p2

(10.1)

The loop contribution between the two Feynman propagators in the second term on the rhs is given by g g d4q i Σ = iDF (0) = . 4 2 2 2 (2π) q − m2 + iε

(10.2)

The integral here diverges: after integrating over q0 we obtain integrals of the form (cf. section 6.1.2)

d3q √ 2 . q + m2

(10.3)

By introducing an upper bound Λ for the integral over |q| and then taking Λ → ∞ we see that the integral diverges as Λ2 ; this is called a quadratic divergence. Because this divergence happens here for large q one also speaks of a quadratic ultraviolet divergence. 113

CHAPTER 10. DIVERGENCES IN N -POINT FUNCTIONS

114

Another way to see the degree of divergence is the so-called “powercounting”: There are 4 powers of q in the integration measure, but only two powers of q in the denominator of (10.2); the degree of divergence is then given by the net power (2) of q.

10.1

Power Counting

The power-counting just illustrated for the case of the tadpole diagram can be generalized to any Feynman graphs with an arbitrary number of loops. In order to see this we consider a theory with an interaction ∼ φp in n dimensions. Since each loop contributes according to the Feynman rules an integral dnq to the total expression and since each internal propagator gives a power q −2 , we have for the degree of divergence D in a diagram with L loops and I internal lines D = nL − 2I . (10.4) Note that here each loop has also at least 1 internal line. For example, the tadpole diagram has L = 1 and I = 1 , giving D = 2 in four dimensions. D > 0 clearly diverges, D = 0 corresponds to a logarithmic divergence, and D < 0 seems to be convergent. If a graph has V interaction vertices, then the total number of lines in φp theory is pV , since each vertex has p legs. These legs can be either external or internal lines. If they are internal, they count twice because each internal line originates and disappears at a vertex. Thus we have pV = E + 2I ,

(10.5)

where E is the number of external lines. In addition, the number of loops, L, is related to the number of vertices, V , by L=I −V +1 .

(10.6)

Combining equations (10.5) and (10.6) with (10.4) allows us to eliminate L and I to obtain

n(p − 2) p −p V − −1 E . D =n+ 2 2

(10.7)

The degree of divergence of a connected graph in this φp theory in n dimensions thus depends on the number of external legs and, in general, also on the number of vertices. In the perturbative treatment of ﬁeld theory derived in Chap. 8.3 we have seen that the order of perturbation theory directly equals the number of

CHAPTER 10. DIVERGENCES IN N -POINT FUNCTIONS

115

D = 4 · 1 − 2 · 1 = +2

D =4·1−2·2=0

D = 4 · 0 − 2 · 1 = −2

D = 4 · 1 − 2 · 3 = −2

Figure 10.1: Examples for graphs with diﬀerent degree of divergence. On the right equation (10.4) is illustrated for n = 4. vertices, V , in a Feynman diagram. Thus, the degree of divergence becomes larger and larger with increasing order of a perturbative treatment, if the factor of V in (10.7) is positive. On the other hand, D is independent of this order if that factor is zero and D becomes even smaller when going to higher orders of perturbation theory if the factor is negative. Thus the perturbative treatment leads to a ﬁnite number of divergent terms if and only if n(p − 2) −p≤0 . 2

(10.8)

If this factor is zero, then the total number of divergent diagrams in a perturbative expansion can still be inﬁnite, but in each order of perturbation theory only the same ﬁnite number of divergent diagrams appears. In this case the theory is called renormalizable. If the factor is negative, then even the total number of divergent terms is ﬁnite; in this case the theory is called superrenormalizable. In both cases one can add a ﬁnite number of so-called counter terms to the Lagrangian that just remove these divergences.

CHAPTER 10. DIVERGENCES IN N -POINT FUNCTIONS

116

In order to become familiar with this power counting for φ4 theory in four dimensions we give three examples in Fig. 10.1. According to our earlier considerations we expect that D ≥ 0 diverges. However, this does not guarantee that graphs with D < 0 actually converge. This is illustrated by the lowest example, in Fig. 10.1, that has L = 1, I = 3, and thus D = 4 · 1 − 2 · 3 = −2, but diverges, because of the loop on the externals legs. Only when each possible subgraph has also D < 0, then the whole expression converges (Weinberg’s Theorem). Note that in the physical case n = 4 and p = 4 D does not depend on V , i.e. it is independent of the order of perturbation theory. φ4 theory is thus renormalizable.

10.2

Dimensional Regularization of φ4 Theory

Regularization serves to make all divergent expressions convergent, at the expense of introducing a parameter that has no physical meaning and ultimately has to be removed. If the integrals can be performed and evaluated as a function of this parameter then the inﬁnite contributions can be separated from the ﬁnite ones. We show this procedure by evaluating now the divergent integrals by a modern technique called dimensional regularization. This technique starts by considering the theory in n dimensions. We ﬁrst consider the dimensions in the Lagrangian L=

" g 1! μ ∂ φ∂μ φ − m2 φ2 − φ4 . 2 4!

Since the action S=

L dnx

(10.9)

(10.10)

is a dimensionless quantity (in units in which h ¯ = 1), L must have the −n dimension ( is a length). The kinetic energy term in L thus has also dimension −n and – since [∂μ ] = −1 – we get [φ] = 1−n/2 . Any potential term of the form gφp must have the same dimension as the Lagrangian, i.e. n −n ; we thus get [g] = −n /p(1−n/2) = −(n+p(1− 2 )) . The mass term (p = 2) thus has [m2 ] = −2 in n dimensions, as it should. Note that the dimension of g is just the factor of the number of vertices, V , in (10.7). Thus, the dimension of the coupling constant and the renormalizability of a theory are closely connected: if the dimension of g is that of a positive or zeroth power of mass, then the theory is superrenormalizable or renormalizable, respectively.

CHAPTER 10. DIVERGENCES IN N -POINT FUNCTIONS

117

These considerations show that for p = 4 [g] = n−4 = mass4−n ;

(10.11)

i.e. in four dimensions the coupling constant of a φ4 theory is dimensionless and the theory is, therefore, renormalizable; in fact, only theories with p ≤ 4 are renormalizable. Therefore, according to the discussion in the preceding section the φ4 theory contains only a ﬁnite number of divergent vertices. In n = 4 dimensions, however, this is no longer true. If we want to keep the coupling constant dimensionless in order to ensure renormalizability also in n dimensions we have to modify the φ4 term in the Lagrangian (8.51) such that an additional factor with the dimension of (mass)4−n absorbs the dimension and g becomes dimensionless L = L0 −

g 4−n 4 μ φ . 4!

(10.12)

Note that μ here is an arbitrary mass.

10.2.1

Two-point function

The two-point function is completely determined once we know the selfenergy. We start by calculating this quantity in lowest order in the coupling by evaluating explicitly the contribution of the tadpole diagram. To do so we ﬁrst go to n dimensions so that (10.2) becomes dnq i g . Σ = μ4−n 2 (2π)n q 2 − m2 + iε

(10.13)

This integral can be obtained analytically by going into a space of n-dimensional polar coordinates (see Appendix. C). The result is

n g μ4−n n−2 n m π2Γ 1− Σ= n 2 (2π) 2

.

(10.14)

The divergence of this expression is now manifest, since the Γ-function has poles at 0 and the negative integers, and thus also for n = 4. We now expand Γ around this pole. For that purpose we write

n ε = Γ −1 + Γ 1− 2 2

(10.15)

with ε = 4 − n and expand in powers of ε (cf. (C.4))

Γ −1 +

ε 2 = − − 1 + γ + O(ε) , 2 ε

(10.16)

CHAPTER 10. DIVERGENCES IN N -POINT FUNCTIONS

118

where γ is the Euler-Mascheroni constant (γ ≈ 0.577..). We thus get for n close to 4

2 g με 2−ε 4−ε 2 m π − − 1 + γ + O(ε) Σ = 2 (2π)4−ε ε m2 = g 32π 2

4πμ2 m2

ε 2

We now use

2 − − 1 + γ + O(ε) ε

(10.17)

.

ε→0

xε = eε ln x −→ 1 + ε ln x

(10.18)

and obtain

gm2 ε 2 4πμ2 1 + − − 1 + γ + O(ε) ln Σ −→ 2 2 32π 2 m ε 4πμ2 gm2 2 = − − 1 + γ − ln + O(ε) 32π 2 ε m2 gm2 1 gm2 4πμ2 = − 1 − γ + ln + O(ε) . − 16π 2 ε 32π 2 m2 ε→0

(10.19)

We have thus split Σ into two parts. The ﬁrst one is clearly diverging as n approaches 4 (ε → 0). The second term is ﬁnite and depends on the arbitrary mass μ that was originally introduced only to keep the coupling constant free of dimension. The appearance of the arbitrary mass μ in the ﬁnite part is related to the arbitrariness in separating an overall inﬁnite expression into a sum of an inﬁnite and a ﬁnite contribution. Note that Σ is independent of p. In next higher order the same is true for the selfenergy contribution of Fig. 9.2b on page 107 whereas that of Fig. 9.2c depends quadratically on p [Ramond].

10.2.2

Four-point function

In this section we will now evaluate the four-point function. By looking at Fig. 9.4 on page 111 we expect that the four graphs in the second line just contribute again to the selfenergy. On the other hand, we expect that the three diagrams in the lowest line can all graphically be contracted such that they contain only one interaction point with possibly modiﬁed interaction strength. As an example, we now evaluate the middle graph in the last line of Fig. 9.4. According to the Feynman rules its contribution to the 1PI vertex is

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(see (9.42))

−ig 4−n ΔΓ(p1 , p2 , p3 , p4 ) = μ 4!

2

d4q i i S3 S1 4 2 2 (2π) q − m (p − q)2 − m2 (10.20) with p = p1 + p3 . The symmetry factor is S1 S3 = 12 · 4! (9.42). The two denominators in the integrand can be combined into one by a mathematical trick due to Feynman that is very often used for the evaluation of such expressions. This trick starts from the elementary integral relation b a

1 1 1 dx b−a = − |ba = − + = . 2 x x b a ab

(10.21)

By substituting now x = az + b(1 − z) we obtain

b a

dx dz = (a − b) 2 . 2 x [az + b(1 − z)] 1

(10.22)

0

(10.23)

Combining (10.21) and (10.23) gives 1 dz = 2 . ab [az + b(1 − z)] 0 1

(10.24)

We now apply this to the integrand in (10.20) and obtain 1 1 1 dz = 2 2 2 2 2 2 q − m (p − q) − m {(q − m )z + [(p − q)2 − m2 ] (1 − z)}2 0 1

= 0

dz . [q 2 − 2pq(1 − z) + p2 (1 − z) − m2 ]2 (10.25)

The ﬁrst three terms in the denominator can be combined by substituting q = q − p(1 − z) .

(10.26)

This gives for expression in the denominator q 2 − 2pq(1 − z) + p2 (1 − z) − m2 = [q − p(1 − z)]2 − m2 − p2 (1 − z)2 + p2 (1 − z) = q 2 − m2 − p2 z(z − 1) .

(10.27)

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The 1PI vertex now reads in n dimensions 1 1 2 2 4−n dnq 1 ΔΓ(p1 , p2 , p3 , p4 ) = g (μ ) dz 2 n 2 (2π) [q − m2 − sz(z − 1)]2 0 (10.28) with s = p2 = (p1 + p3 )2 . We now interchange the order of integration and evaluate the integral over q ﬁrst with the help of (C.18) in Appendix C

dnq 1 (2π)n [q 2 − m2 − sz(1 − z)]2

" n−4 n Γ 2 − n i ! 2 2 = m + sz(1 − z) 2 π 2 n (2π) Γ(2)

.

(10.29)

The four-point function thus reads

n

1 2 2 4−n π 2 n Γ 2− ΔΓ(p1 , p2 , p3 , p4 ) = g (μ ) i n 2 (2π) 2 ×

1

!

" n−4

dz m2 + sz(1 − z)

2

.

(10.30)

0

We now introduce again ε = 4 − n. This gives − ε 1 2 1 2 ε 1 ε m + sz(1 − z) 2 ΔΓ(p1 , p2 , p3 , p4 ) = g μ i Γ dz . (10.31) 2 16π 2 2 4πμ2 0

Now the four-point function is in a form that allows to take the limit ε → 0. Using again xε = eε ln x → 1 + ε ln x (10.32) and (C.2)

Γ

ε 2 = − γ + O(ε) 2 ε

(10.33)

gives &

ΔΓ(p1 , p2 , p3 , p4 ) =

ig 2 με 2 − γ + +O(ε) 32π 2 ε ⎡

'

ε m2 + sz(1 − z) × ⎣1 − ln 2 4πμ2 1

0

⎡

⎤

dz ⎦

ig 2 με 1 ig 2 με ⎣ m2 + sz(1 − z) = γ + ln − 16π 2 ε 32π 2 4πμ2 1

⎤

dz ⎦

0

+ O(ε)

with

s = (p1 + p3 )2 .

(10.34)

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Here the four-point function has been separated into a divergent part and a convergent term (for ε → 0). The integral appearing is a function of p, m, and μ; the latter dependence remains even when ε → 0. So far, we have only calculated the middle graph in the last line of Fig. 9.4 on page 111. It is evident, however, that the result can be directly taken over also to the other 2 graphs in the last line by taking for p the appropriate total momentum at one of the vertices. We, therefore, introduce now the three Lorentz-invariant Mandelstam variables s = (p1 + p3 )2 = p2 t = (p1 + p2 )2 u = (p1 + p4 )2 ,

(10.35)

with the property s + t + u = m21 + m22 + m23 + m24 . These variables represent the total squared four-momentum at the vertex that involves p1 in each of the graphs in the last line of Fig. 9.4. We thus get for the sum of all three diagrams in the last line of Fig. 9.4, the vertex correction diagrams, Γv (p1 , p2 , p3 , p4 ) =

3ig 2 με 1 (10.36) 16π 2 ε ig 2 με [3γ + F (s, m, μ) + F (t, m, μ) + F (u, m, μ)] . − 32π 2

Here F (s, m, μ) denotes the integral in (10.34). Of the diagrams in Fig. 9.4 only the ﬁrst and the three last ones are 1PI graphs. The four diagrams in the second line are all 1P reducible; they just diﬀer by the propagators on the external legs that are left out when we consider the 1PI four-point function. The complete 1PI four-point function is, therefore, given by Γ(p1 , p2 , p3 , p4 ) = −igμε + Γv (p1 , p2 , p3 , p4 ) 3ig 2 με 1 = −igμε + 16π 2 ε 2 ε ig μ − [3γ + F (s, m, μ) + F (t, m, μ) + F (u, m, μ)] 32π 2 3g 1 = − igμε 1 − (10.37) 16π 2 ε g [3γ + F (s, m, μ) + F (t, m, μ) + F (u, m, μ)] . + 32π 2 Equation (10.37) gives the eﬀective, “dressed” interaction vertex. By comparison with the free 1PI four-point function, its forms suggests to absorb all

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122

eﬀects of the loop graphs contained in the curly brackets into a new eﬀective coupling constant g −→ g {1 + δg(s, t, u, m, ε, μ)} .

(10.38)

This eﬀective coupling constant depends on s, t, u, m and μ and, as the selfenergy, separates into a divergent and a ﬁnite term. It contains all the eﬀects of the loops.

10.3

Renormalization

In the preceding two subsections we have seen that both the 1PI two-point and the 1PI four-point functions are (for ε = 0) regular functions of ε. The in four dimensions diverging quantities have thus been regularized. For n → 4 both have divergent and ﬁnite contributions from higher order diagrams. In this section we now show how to handle these divergences by the renormalization technique. Any renormalization procedure requires ﬁrst a regularization, either by a cut-oﬀ for the upper bounds of diverging integrals or by going to n = 4 dimensions, to be followed by a procedure in which the dependence on these artifacts is removed. Since the separation of a divergent quantity into a ﬁnite and an inﬁnite contribution is arbitrary there are various so-called renormalization schemes. They all have in common that they either add terms to the Lagrangian or scale the ﬁelds and coupling constants such that the original form of the Lagrangian is maintained. The most obvious scheme is the so-called minimal subtraction scheme that removes just the pole contributions in the dimensional regularization, i.e. the terms that go like powers of 1/ε. This scheme is straighforward and well-suited for the dimensional regularization, but it leads to expressions in which the parameters m (mass) and g (coupling) have no direct relation to measurable quantities. Here we discuss another scheme, in which we require that the parameters of the Lagrangian assume their physical, measured values, i.e. m is the physical mass and g the physical coupling constant. We start by looking at the structure of the most general two-point function. When we go to terms that depend on g 2 and higher orders of the interactions we always encounter 1P reducible graphs, such as the one in Fig. 9.2a. The general structure of the (reducible) two-point function is of the form Σ Gc (p, −p) = G0 (p, −p) + G0 (p, −p) G0 (p, −p) i Σ Σ + G0 (p, −p) G0 (p, −p) G0 (p, −p) + . . . , (10.39) i i

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where Σ is the so-called proper self-energy. Note that (10.39) is an expansion in terms of irreducible diagrams. Equation (10.39) can be summed and gives

Σ Σ Σ G0 (p, −p) + G0 (p, −p) G0 (p, −p) + . . . i i i −1 Σ = G0 (p, −p) 1 − G0 (p, −p) i Σ −1 −1 = G0 (p, −p) − i i = 2 . (10.40) p − m2 − Σ + iε

Gc (p, −p) = G0 (p, −p) 1 +

The self-energy Σ is in general a function of the momentum p of the particle. We can therefore expand Σ(p2 ) around the on-shell point p2 = m2 , where m is the physical, observable mass Σ(p2 ) = Σ(m2 ) + (p2 − m2 )Σ1 + Σ2 (p2 ) . Here Σ1 =

∂Σ |2 2 ∂p2 p =m

Σ2 (m2 ) = 0 .

and

(10.41)

(10.42)

In (10.41) only the ﬁrst two terms of the expansion of Σ have been written out explicitly; Σ2 (p2 ) denotes the whole remainder of the expansion. Σ, being the selfenergy insertion of a two point function, is quadratically divergent. Consequently, Σ1 as the ﬁrst derivative of Σ with respect to p2 is logarithmically divergent and Σ2 as a second derivative is convergent since taking the derivate always adds one more power of q 2 in the denominator of the Feynman propagators. Inserting this expansion into (10.40) gives for the propagator Gc (p, −p) =

i

− − − − m2 )Σ1 − Σ2 (p2 ) + iε i 1 . (10.43) = 2 )+Σ (p2 ) Σ(m 2 2 2 1 − Σ1 p − m − + iε 1−Σ1 p2

m2

Σ(m2 )

(p2

The pole of this propagator should be at the physical mass and its residuum should be i. However, this is in general not the case, if we start with the observable, physical mass in the Lagrangian, because the selﬁnteractions contribute to the self-energy.

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124

Counter terms. In order to get the pole to the correct, physical location we therefore have to change the mass in the Lagrangian by adding a so-called counterterm to it 1 1 with Z= . (10.44) Lcm = − δm2 Zφ2 2 1 − Σ1 Z is usually called “ﬁeld renormalization constant” for reasons that will become obvious a little later (see (10.56)). Note that this counterterm has exactly the same form as the mass term in the original Lagrangian. It will thus also add a term −Zδm2 in the denominator of the dressed propagator (10.43). We determine the unknown δm2 by the requirement δm2 + Σ(m2 ) = 0

(10.45)

so that the new term just cancels the selfenergy contribution at the on-shell point. With the counterterm added, the propagator then becomes 1 i G(p, −p) = . (10.46) 2 2 1 − Σ1 p − m − ZΣ2 (p2 ) + iε ˜ 2 (m2 ) = 0 by deﬁnition, this propagator has the correct pole at the Since Σ physical mass m. Its residue, however, is – instead of being simply i – i = iZ . (10.47) 1 − Σ1 This deﬁciency can be cured by adding another, additional counterterm 1 Lcφ = (Z − 1) ∂μ φ∂ μ φ − m2 φ2 (10.48) 2 to the Lagrangian. Then the propagator becomes i G(p, −p) = Z 2 2 2 p − m − ZΣ2 (p ) + (Z − 1)(p2 − m2 ) + iε i = 2 . (10.49) 2 p − m − Σ2 (p2 ) + iε This propagator has the pole at the correct, physical mass m (because of Σ2 (m2 ) = 0) and the correct residue i. By adding the given counterterms we have thus removed the divergent quantities Σ(m2 ) and Σ1 from the propagator. The one remaining quantity Σ2 (p2 ) involves a second derivative of the selfenergy with respect to p2 and is convergent; it vanishes at the on-shell point. Note that the counterterms all have the structure of terms already present in the original Lagrangian. If this is the case, in general a theory is called renormalizable.

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125

Renormalization of φ4 Theory

10.3.1

We now specify all of these general considerations to the example of φ4 theory. As we have discussed in Sect. 10.2 the φ4 theory is renormalizable, i.e. only a ﬁnite number of elementary vertices diverges. These are just the terms we have calculated in the last two subsections, namely the selfenergy contribution (10.19) and the vertex function (10.37). For the selfenergy we have, up to one-loop diagrams, (cf. (10.19)),

gm2 1 gm2 4πμ2 1 − γ + ln − Σ(p ) = − 16π 2 ε 32π 2 m2 2

+ O(ε) ,

(10.50)

i.e. Σ1 = Σ2 = 0 and Z = 1. Equation (10.45) reduces to δm2 = −Σ(0) = −Σ =⇒ Z = 1 .

(10.51)

Thus in this one-loop approximation, there is no ﬁeld renormalization, but already in the two-loop approximation we would get Σ1 = 0 because the graph Fig. 9.2c is p-dependent. To be general we, therefore, keep the factor Z in the following expressions. So far we have not taken the change of the coupling constant due to higher order loop diagrams into account. In (10.37) we have indeed already seen that also the interaction vertex gets modiﬁed due to higher order loop corrections. The structure there was such that the coupling constant g was replaced by an eﬀective coupling constant g(1 + δg(s, t, u, m, ε, μ)). If we again want to have the physical, observable coupling constant in our Lagrangian we must get rid of the modiﬁcation by a proper counter term. We, therefore, deﬁne a vertex renormalization constant Zg by Zg = (1 + δg(s, t, u, m, ε, μ))−1 |r ,

(10.52)

where r denotes an in principle arbitrary renormalization point. Since we want to relate the coupling to a measurable quantity we use the so-called symmetric point 1 2 4 pα · pβ = m (10.53) δij − 3 3 for i, j = 1, . . . , 4. At this point we have s = t = u = −4m2 /3. We can then introduce the additional counter term Lcv = −

gμε (Zg − 1) φ4 . 4!

(10.54)

In this way we can ensure that the coupling constant g in the Lagrangian has its physical, observable value at r.

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126

The Lagrangian with all the counterterms now reads L =

!

"

(∂μ φ)2 − m2 φ2 −

gμε 4!

! " 1 1 gμε (Zg − 1) φ4 + (Z − 1) ∂ μ φ∂μ φ − m2 φ2 − δm2 Zφ2 − 2 2 4! " Z! 1 gμε = (∂μ φ)2 − m2 φ2 − δm2 Zφ2 − (10.55) Zg φ4 . 2 2 4!

This Lagrangian leads to the correct physical behaviour for the 2-point Green’s function with the physical mass m and the proper residue. The ﬁelds φ, in terms of which the propagator is deﬁned, are thus the physical ﬁelds, which include already the eﬀects of selﬁnteractions. By introducing the “bare ﬁeld” φ0 and the “bare mass” m0 by √ φ0 = Zφ 2 (10.56) m0 = m2 + δm2 and a “bare coupling constant” by g0 = gμε

Zg , Z2

(10.57)

then the whole Lagrangian can be expressed in terms of bare quantities only L=

" 1! g0 (∂μ φ0 )2 − m20 φ20 − φ40 . 2 4!

(10.58)

This bare Lagrangian has the same form as the original one, because all the counter terms had the same form as terms already appearing in the original Lagrangian. As a consequence it leads to ﬁnite physical quantities in all orders of perturbation theory. If this is the case, then the theory is well deﬁned, i.e. it is said to be renormalizable. The bare Lagrangian is really considered to be the ‘true’ Lagrangian of the theory because it leads only to ﬁnite physical quantities. In the preceding considerations we have chosen 2 arbitrary renormalization points; we have required the propagator to have a pole at p2 = m2 , where m is the physical mass (cf. (10.49)), and for the vertex we have chosen the symmetric point r (s = t = u = −4m2 /3) (cf. (10.52)). Choosing other renormalization points will lead to other values for the mass and the coupling. This arbitrariness in the renormalization point reﬂects the fact that the divergent expressions for the selfenergy and the vertex all separate into a divergent and a ﬁnite term. Any scheme, that removes the inﬁnite terms, will lead to ﬁnite expressions for the physical observables, independent of

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127

what it does with the ﬁnite contributions. It will also lead to the same npoint function, and – with the help of the reduction theorem – to the same observable transition rates. In the dimensional renormalization discussed here all the counter terms and renormalization factors also depend on the arbitrary mass μ, even after ε → 0 and after the inﬁnite terms have been removed. This μ was originally introduced to keep the coupling constant g dimensionless also for n = 4. It is obvious from our considerations above that – for speciﬁed renormalization points – choosing diﬀerent values for μ will lead to diﬀerent values for the wavefunction renormalization Z and the coupling constant g. This arbitrariness, however, is nothing else than the arbitrariness we have already encountered in the choice of the renormalization point. The physics must be independent of μ. This observation is the starting point for the development of the renormalization group method. It is essential to realize that even in a massless theory, which contains no dimensional scale parameters, μ introduces a scale that determines the momentum dependence of the coupling constant. Thus, at the quantum level, a bare Lagrangian is not enough to specify a theory, but a renormalization scheme must be added that introduces necessarily a scale into the theory.

Chapter 11 GREEN’S FUNCTIONS FOR FERMIONS For simplicity of notation we have so far in this book discussed only the path integrals and generating functions for scalar ﬁelds. All this formalism can be easily generalized also to vector ﬁelds which obey the Proca equation (4.24) and thus fulﬁll component by component the Klein-Gordon equation. For fermion ﬁelds, however, there is a problem. The main idea in using path integrals is to express quantum mechanical transition amplitudes by integrals over classical ﬁelds; the values of these ﬁelds at the discrete coordinatesites were taken to be commuting numbers. Such a formalism can, however, not “know” about the Pauli principle. For example, with the formalism developed so far, a fermion could be propagated to a point in conﬁguration space which is already occupied. In nature, however, this propagation is Pauli-forbidden. For the description of fermions it is, therefore, necessary to extend the theory developed so far such that the Pauli principle is taken into account. This can be achieved by using an anticommuting algebra for the classical ﬁelds, the so-called Grassmann algebra.

11.1

Grassmann Algebra

In this section we outline the basic mathematical properties of the Grassman algebra as far as we will need them in the later developments. We deﬁne the n generators i , . . . , n of an n-dimensional Grassmann algebra by the anticommutation relations {i , j } ≡ i j + j i = 0 . 128

(11.1)

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129

Let us now consider series expansions in these variables. Since 21 = 22 = . . . = 2n = 0 ,

(11.2)

because of (11.1), any series in i must have the form φ() = φ0 +

φ1 (i)i +

i

+

φ2 (i, j)i j

(11.3)

i
φ3 (i, j, k)i j k + . . . + φn (1, 2, . . . n)1 2 . . . n .

i
Here the φi are ordinary, commuting c-numbers. Note that the expansion actually terminates because of (11.2). Every element of the algebra can be written in the form (11.3); thus, the various powers of in this expansion constitute a basis of the algebra. If we choose the function φi to be totally antisymmetric in all their variables, then we can also write φ() = φ0 +

φ1 (i)i +

i

1 φ2 (i, j)i j 2! ij

1 + φ3 (i, j, k)i j k + . . . . 3! ijk

(11.4)

From now on we can assume that the φi are indeed antisymmetric since any symmetric part of φ would not contribute anyway. As a consequence of this relation any analytical function in 1 dimension has the form (11.5) φ(1) () = φ0 + φ1 , and in 2 dimensions φ(2) () = φ0 + φ1 (1)1 + φ1 (2)2 + φ2 (1, 2)1 2 .

(11.6)

These relations imply for the Taylor expansion of a Gaussian e− = 1

(11.7)

e−i j = 1 − i j .

(11.8)

2

and

CHAPTER 11. GREEN’S FUNCTIONS FOR FERMIONS

11.1.1

130

Derivatives

Since functions of Grassmann variables can, because of (11.2), be at most linear in ε, the derivative operator is deﬁned as an algebraic operation that, when applied to a Grassmann variable, simply replaces that variable by a 1; when applied to a product the variable ﬁrst has to be commuted to a position next to the derivative before it can be removed. We then get, for example, )

,

∂ ∂1 ∂ , i 1 = (i 1 ) + i ∂j ∂j ∂j ∂i ∂1 ∂1 = 1 − i + i = δij 1 . ∂j ∂j ∂j

(11.9)

The “−” sign in the second line appears, because the factor 1 must be ﬁrst brought to the left before the derivative can act. This result is valid in general. We thus can deﬁne the derivative also by the anticommutation relation ) , ∂ , i = δij . (11.10) ∂j This relation can be used to obtain the derivative of a function φ(). Applying the lhs of (11.10) to φ, which can always be written as polynomial (see (11.4)), yields )

,

∂ ∂ ∂ , i φ() = [i φ()] + i φ() (11.11) ∂j ∂j ∂j ∂ 1 = δij φ() + φ0 − φ1 (k)k + φ2 (k, l)k l ∂j 2! kl k 1 − φ3 (k, l, m)k l m + . . . 3! klm

i + i

∂ φ() . ∂j

Here all the terms in φ() with an odd number of s have changed sign when they were brought to the left so that the derivative can act on them. With

and

∂ k l = δkj l − δlj k ∂j

(11.12)

∂ k l m = δkj l m − δlj k m + δmj k l ∂j

(11.13)

we get )

,

∂ 1 1 , i φ() = δij φ() + −φ1 (j) + φ2 (j, l)l − φ2 (k, j)k ∂j 2! l 2! k

CHAPTER 11. GREEN’S FUNCTIONS FOR FERMIONS −

131

1 1 φ3 (j, l, m)l m + φ3 (k, j, m)k m 3! lm 3! km

∂ 1 − φ3 (k, l, j)k l + . . . i + i φ( ) 3! kl ∂j = δij φ() 1 φ2 (j, k)k + φ3 (j, k, l)k l − . . . i − φ1 (j) − 2! kl k ∂ + i φ( ) . (11.14) ∂j Because of (11.10) this has to be equal to δij φ(). Commuting i through to the left side of the square parentheses changes again the signs of all odd terms. The last two terms in (11.14) then cancel each other if the expression in the square parentheses equals the derivative of φ 1 ∂ φ() = φ1 (j) + φ2 (j, k)k + φ3 (j, k, l)k l + . . . . ∂j 2! kl k

(11.15)

This gives the derivative of a general function φ(). Eq. (11.15) could also have been obtained by diﬀerentiating the expansion (11.4) directly. The second derivatives can also be deﬁned ∂ ∂ φ() = φ2 (j, i) ∂i ∂j 1 1 + φ3 (j, i, l)l − φ3 (j, k, i)k + . . . 2! l 2! k = φ2 (j, i) +

φ3 (j, i, l)l + . . . .

(11.16)

l

Because the φi are antisymmetric this gives immediately ∂2 ∂2 φ() + φ() = 0 , ∂i ∂j ∂j ∂i or

)

∂ ∂ , ∂i ∂j

(11.17)

,

=0.

(11.18)

Thus we have, in particular, ∂2 =0. ∂2i

(11.19)

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132

This relation implies that there is no inverse to diﬀerentiation. This can be seen by multiplying the deﬁning equation for the inverse ∂ ∂ε from the left by

∂ . ∂ε

∂ ∂ε

−1

φ() = φ()

(11.20)

This gives

∂ ∂ ∂ε ∂ε

∂ ∂ε

−1

∂ φ() = 0 · ∂ε

−1

φ() =

∂ φ() ∂ε

(11.21)

and thus the inverse does not exist.

11.1.2

Integration

Integration in the space of Grassmann variables can, therefore, be deﬁned only in an operational sense. This is achieved by the following relations

di = 0

di i = 1 .

(11.22)

The symbols di obey the commutation relations {di , dk } = {di , k } = 0 ,

(11.23)

but note that d is not an inﬁnitesimal interval and, in particular, not a member of the Grassmann algebra of the i . Integration over Grassmann variables has thus the same eﬀect as differentiation. The integral in (11.22) is the only nonvanishing integral over functions of i . It has the property of translational invariance

(1)

d φ () =

d φ(1) ( + α) = φ1

(11.24)

with the deﬁnition of φ(1) in (11.5); α is also a Grassmann number. For Grassmann variables we can also deﬁne a δ-function. In 1 dimension we have −

d ( − )φ( ) = −

d ( − )(φ0 + φ1 )

= φ0 + φ1 = φ() .

(11.25)

We can thus identify the δ function as δ( − ) = −( − ) ;

(11.26)

CHAPTER 11. GREEN’S FUNCTIONS FOR FERMIONS

133

note that δ( − ) is an odd function. There is also a Fourier-representation for the δ-function

δ( − ) = −( − ) =

dζ [1 − ζ( − )] =

dζ e−ζ(− ) ;

(11.27)

here (11.8) has been used. 11.1.2.1

Multiple integrals

In multiple integrals the integration variable ﬁrst has to be commuted to a position next to the integration measure. For example, we get

di dj e

−i j

=

di dj (1 − i j )

= 0−

di dj i j = +

di

dj j i

=

di i = 1 .

(11.28)

We now determine the Jacobian J that appears when we perform a linear transformation (11.29) = O of the integration variables in n dimensions; O is a general matrix. J is deﬁned by

d1

. . . dn 1

. . . n

=

=

d1 . . . dn J (O)1 (O)2 . . . (O)n

(11.30)

d1 . . . dn J O1α O2β . . . Onν α β . . . ν .

Note that this equality holds for general matrices O. The product of the Grassmann variables α , . . . , ν vanishes, if any of the indices appears twice. Thus only the permutations of 1, . . . , n survive. We can therefore bring each of them into the ordered form by writing α β . . . ν = (−)P 1 . . . n ,

(11.31)

where (−)P is the sign of the permutation. We thus have

d1 . . . dn 1 . . . n =

d1 . . . dn J

=

(−)P O1α . . . Onν 1 . . . n

P

d1 . . . dn J 1 . . . n det(O) .

(11.32)

CHAPTER 11. GREEN’S FUNCTIONS FOR FERMIONS

134

On the other hand, we can also evaluate the integrals over the ’s directly. This gives d1 . . . dn 1 . . . n = d1 . . . dn 1 . . . n , (11.33) because of (11.22). Comparison with (11.33) yields J = [det(O)]−1 .

(11.34)

Note that this is the inverse of the ordinary result! 11.1.2.2

Gaussian integrals

We next consider the Gaussian integral

T M

d1 . . . dn e−

I(n) =

(11.35)

where M is an antisymmetric (n×n) matrix with ordinary c-number elements and is a vector with n elements. Expanding the exponential function we obtain 2 1 T T M − · · · . (11.36) e− M = 1 − T M + 2! For simplicity let us consider the case n = 2 ﬁrst. We have for I(2)

I(2) =

d1 d2 e

−T M

&

=

d1 d2

2 1 T 1 − M + M − . . . 2

'

T

2 1 = 0 − d1 d2 M + d1 d2 T M − . . . 2 ≡ −I1 (2) + I2 (2) − . . . T

(11.37)

For the ﬁrst integral we obtain

I1 (2) =

=

=

d1 d2 T M =

d1 d2 i Mij j

d1 d2 (1 M11 1 + 1 M12 2 + 2 M21 1 + 2 M22 2 ) d1 d2 (1 M12 2 + 2 M21 1 ) ,

(11.38)

because the integrands of the other two terms contain squares of Grassmann variables which vanish. We therefore have (because M is antisymmetric)

I1 (2) = −M12 + M21 = −2M12 = −2 det(M ) .

(11.39)

CHAPTER 11. GREEN’S FUNCTIONS FOR FERMIONS

135

The second integral in (11.37) is

I2 (2) = =

d1 d2 T M

2

(11.40)

d1 d2 i Mij j k Mkl l .

The indices i, j, k and l all have to be diﬀerent, because otherwise the product of the s vanishes. This, however, is impossible because we have only 2 diﬀerent variables; the integral I2 (2) thus vanishes. The same holds for all higher order terms in (11.37). In summary we obtain for the integral (11.37)

2

I(2) = −I1 (2) = +2 det(M ) = 2 2 det(M ) .

(11.41)

For 3 variables we obtain immediately

T M

d1 d2 d3 e−

=0,

(11.42)

because the exponent is quadratic in and, therefore, the series expansion involves only even powers. The term of second order in gives 0, because of d = 0; the term of fourth order also vanishes, because, with only three variables, it has to contain a square of one of them, which vanishes. A similar reasoning leads one to the conclusion that with four variables only the term of fourth order in can contribute. Thus we have

I(4) =

T M

d1 d2 d3 d4 e−

2 1 d1 . . . d4 T M 2 1 = d1 . . . d4 i Mij j k Mkl l 2 4 1 = Mij Mkl d1 . . . d4 i j k l . 2 ijkl=1

=

(11.43)

Here only the terms with all four indices diﬀerent can contribute. We thus get 4! = 24 nonvanishing terms. Of these 24 terms many are equal to each other: the antisymmetry Mij = −Mji gives Mij Mkl = Mji Mlk and Mij Mlk = Mji Mkl = −Mij Mkl and thus a factor 22 , the symmetry Mij Mkl = Mkl Mij another factor 2, so that only 3 essentially diﬀerent terms remain. We then have

I(4) = 4M12 M34

d1 . . . d4 1 2 3 4

CHAPTER 11. GREEN’S FUNCTIONS FOR FERMIONS

136

+ 4M13 M24 + 4M14 M23

d1 . . . d4 1 3 2 4 d1 . . . d4 1 4 2 3

= 4 (M12 M34 − M13 M24 + M14 M23 ) 4

= 2 2 det(M )

(11.44)

The last step can be seen by explicitly expanding the (4 × 4) determinant of M in terms of its elements in the ﬁrst row. The result just obtained holds in general )

d1 . . . dn e

−T M

=

n

22

det(M ) 0

for

n even n odd

.

(11.45)

The proof is given in Appendix D. Equation (11.45) is the equivalent of (B.10) for commuting numbers. Note that here the squareroot of the determinant appears in the numerator whereas for commuting numbers it is in the denominator. The other integral relations for commuting variables, derived in section B.2.1, can also be obtained for Grassmann variables. One gets

1 T M +ηi i

d1 . . . dn e− 2

=

1 T M −1 η

det(M ) e 2 η

,

(11.46)

where the ηi are also Grassmann variables with {ηi , ηj } = 0 = {ηi , j }

(11.47)

(there is no factor 2(2n)/2 here because the exponent contains an extra factor 1/2). Equation (11.46) corresponds directly to (B.10). Complex integration. We now enlarge the algebra of the n by adding the conjugate elements. Here conjugation is deﬁned by the properties (i )∗ (∗i )∗ (λi )∗ (1 2 · · · n )∗

= = = =

∗i i λ∗ ∗i ∗n ∗n−1 · · · ∗1 .

(11.48)

The latter condition ensures that ∗ is real. The n Grassmann generators ∗1 , . . . , ∗n are then combined with the n generators 1 , . . . , n into a 2ndimensional algebra with generators κ1 , . . . , κ2n such that {κ1 , . . . , κ2n } = {∗1 , 1 , ∗2 , 2 , . . . , ∗n , n } .

(11.49)

CHAPTER 11. GREEN’S FUNCTIONS FOR FERMIONS

137

All the κi anticommute with each other. With this deﬁnition we evaluate a Gaussian integral containing also conjugate Grassmann numbers Z∼

†M

d∗1 d1 d∗2 d2 . . . d∗n dn e−

.

(11.50)

The matrix M in (11.50) can now be a general, complex matrix without any speciﬁc symmetries. In contrast to the c-number integrals discussed in section B.2.1 it does not have to have special properties since convergence of the integrals is guaranteed here by the Grassmann algebra. In order to be able to apply the results obtained above (see (11.45)) to this integral we now enlarge the general (n × n)-dimensional matrix M into an antisymmetric (2n × 2n)-dimensional matrix N , so that the exponent in (11.50) can be rewritten n 2n 1 1 ∗i Mij j = κk Nkl κl = κT N κ . (11.51) † M = 2 k,l=1 2 i,j=1 For example, for the case n = 2 the matrix N reads ⎛ ⎜ ⎜ ⎝

N =⎜

0 M11 0 M12 −M11 0 −M21 0 0 M22 0 M21 −M12 0 −M22 0

⎞ ⎟ ⎟ ⎟ . ⎠

(11.52)

One then gets in general det(N ) = [det(M )]2 . Using now (11.45) gives Z∼

dκ1 . . . dκ2n e− 2 κ 1

T Nκ

(11.53)

=

det(N ) ,

(11.54)

= det(M ) .

(11.55)

and with (11.53) we ﬁnally get Z∼

†M

d∗1 d1 . . . d∗n dn e−

This result can also easily be obtained for a diagonal matrix M . Even though the integral is a 2n-dimensional one, the matrix M on the rhs is only ndimensional. Note again that the matrix M here is a general complex matrix. A slightly more general form follows when we add linear terms in the exponent and write an expressively imaginary M . Then we obtain

† M +iη † +i† η

d∗n dn e−i

= det(iM ) eiη

† M −1 η

.

(11.56)

n

In all these formulas for Grassmann integrals the determinant appears in the numerator in contrast to the Gaussian integrals over ordinary numbers (B.22).

CHAPTER 11. GREEN’S FUNCTIONS FOR FERMIONS

11.2

138

Green’s Functions for Fermions

In this section we now apply the Grassmann algebra to a path integral description of fermions. We do so by introducing classical Grassmann ﬁelds ψ for the fermions. These appear in the continuum limit when the number of generators n of the Grassmann algebra goes to inﬁnity. The relations corresponding to (11.1),(11.10) and (11.22) are, respectively, )

{ψ(x), ψ(y)} = 0 , δ = δ(x − y) , ψ(y) δψ(x)

dψ(x) = 0

dψ(x) ψ(x) = 1 .

(11.57)

With these ﬁelds we can construct path integrals of the type Z∼

Dψ¯ Dψ ei

¯ ψ d4x ψM

(11.58)

where the ψ¯ and ψ are Grassmann ﬁelds.

11.2.1

Generating Functional for fermions

The Lagrangian for Dirac ﬁelds is ¯ ψ) ¯ (iγμ ∂ μ − m) Ψ − V (ψ, L=Ψ

(11.59)

¯ are to be considered as independent ﬁelds. They are now where Ψ and Ψ taken to be elements of an inﬁnite-dimensional Grassmann algebra. The path integral is then – in direct generalization of our results for boson ﬁelds – given by Z[η, η¯] = Z0

¯ DΨ ei DΨ

¯ d4x [L+¯ η (x)Ψ(x)+Ψ(x)η(x) ]

.

(11.60)

Here Z0 is the normalization factor, equal to the inverse of the path integral without sources. The functions η(x) and η¯(x) are four-component source ¯ and Ψ, respectively. They functions corresponding to the classical ﬁelds Ψ are also taken to be Grassmann ﬁelds that anticommute among themselves ¯ . as well as with the ﬁelds Ψ and Ψ ¯ )} = {ψ(x), ¯ ¯ )} = 0 ψ(x {ψ(x), ψ(x )} = {ψ(x), ψ(x

CHAPTER 11. GREEN’S FUNCTIONS FOR FERMIONS {η(x), η(x )} = {η(x), η¯(x )} = {¯ η (x), η¯(x )} = 0 ¯ )} = {¯ η (x), ψ(x )} = 0 . {η(x), ψ(x )} = {η(x), ψ(x

139

(11.61)

While (11.60) is an obvious generalization of our earlier results obtained in Chap. 5 for scalar ﬁelds, its actual (numerical) evaluation may not seem to be straightforward, because of the Grassmann nature of the ﬁelds. In order to deﬁne more stringently what is actually meant by the path integral in (11.60) we can proceed as in chapter 5.1 for scalar ﬁelds and Fourier-expand the ﬁelds. This is carried through in detail in [LEE]. For our present purpose ¯ can be expanded in terms of it is enough to mention that the ﬁelds Ψ and Ψ Grassmann numbers α and ¯α such that Ψ(x) =

φα (x)α (t)

α

¯ Ψ(x) =

φ¯α (x)¯α (t) ,

(11.62)

α

where the φα are x-dependent Dirac spinors (u(pα , sα )eipα ·x or v(pα , sα )e−ipα ·x , for positive and negative energies, respectively, see [LEE], p. 517-521). Equation (11.62) just amounts to an x-dependent change of the basis from ¯ The fermionic action, e.g. for α , ¯α to the Grassmann numbers Ψ and Ψ. free ﬁelds, then reads in terms of the Grassmann numbers (with k ≡ γμ k μ )

S = =

¯ (i ∂ − m) Ψ d4x Ψ

φ¯α ¯α (i ∂ − m) φβ β d4x ,

(11.63)

αβ

and the path-integral integration measure is really ¯ DΨ = DΨ ∼

k

j

α

j

¯ k , tj ) dΨ(xk , tj ) dΨ(x d¯α (tj ) dα (tj ) .

(11.64)

Here the ﬁrst index (α) denotes the Grassmann generator, the second (j) the time interval. The last step here was possible because of the orthonormality of the functions φα and φ¯α . 11.2.1.1

Feynman propagator for fermions

In the case of free boson ﬁelds we succeeded in writing the generating functional as a term involving only space-time integrals over the sources (see

CHAPTER 11. GREEN’S FUNCTIONS FOR FERMIONS

140

(6.11) and (6.39)). This had the advantage that the two-point function could be directly read oﬀ from that expression. We achieve here the same for fermions by using (11.56). This gives Z0 [η, η¯] = Z0

¯ DΨDΨ ei

= Z0 det(iM ) ei¯ηM

¯ d4x(L+¯ η Ψ+Ψη)

−1 η

,

(11.65)

where the matrix M is obtained by identifying −i† M ←→ i =

!

¯ ∂ − m)Ψ(x) d4x Ψ(x)(i

−i

"

(11.66)

!

"

¯ d4 x d4 y Ψ(x) (i ∂y + m)δ 4 (x − y) Ψ(y)

after partial integration. Since Z0 [0, 0] = 1, we have Z0 = [det(iM )]−1 . We thus get −1 4 4 Z0 [η, η¯] = ei η¯(x)M (x,y)η(y)d x d y . (11.67) The inverse operator appearing here can be obtained as at the end of section (6.1.3)

M (x, z)M −1 (z, y) d4z =

!

"

(i ∂ z + m) δ 4 (x − z) M −1 (z, y) d4z

δ 4 (x − z) (i ∂ z − m) M −1 (z, y) d4z

=

−

=

− (i ∂ x − m) M −1 (x, y) = δ 4 (x − y) . (11.68) !

The last equation is fulﬁlled by M −1 (x, y) = − (i ∂ + m) DF (x − y) ,

(11.69)

−(i ∂ − m)(i ∂ + m)DF (x − y) = (2 + m2 )DF (x − y) = − δ 4 (x − y) ,

(11.70)

since

because of (6.7). Here we have used {γμ , γν } = 2gμν . We therefore have now the desired result. For free ﬁelds the normalized generating functional is Z0 [η, η¯] = e−i

η¯(x)SF (x−y)η(y) d4xd4y

,

(11.71)

CHAPTER 11. GREEN’S FUNCTIONS FOR FERMIONS

141

with the Feynman propagator for fermions SF (x − y) = (i ∂ + m)DF (x − y)

(11.72)

being the propagator of the Dirac equation (i ∂ − m) SF (x − y) = δ 4 (x − y) .

(11.73)

Note that SF is a (4 × 4) matrix because of the Dirac matrix structure of γμ . For SF we can also ﬁnd an integral representation 1 4 e−ik(x−y) d k (2π)4 k 2 − (m − i)2 1 k + m 4 −ik(x−y) = d k e (2π)4 k 2 − (m − i)2 1 1 = d4k e−ik(x−y) 4 (2π) k − (m − i)

SF (x − y) = (i ∂ + m)

(11.74)

because of ( k + m)( k − m) = k 2 − m2 . Note that SF (x−y) is not symmetric under exchange x ↔ y, while DF (x− y) is. However, it is antisymmetric under an operation that corresponds in Dirac theory to simultaneous x ↔ y exchange and hermitean conjugation since (11.75) γ0 SF (y − x)† γ0 = SF (x − y) , which can easily be derived with the help of the relation γ0 γμ† γ0 = γμ .

(11.76)

As we have discussed in Sect. 6.1.2 for scalar ﬁelds the Feynman propagator moves positive-energy solutions forward in time and those with negative energy backwards. The same is true for the propagator for fermions as can be shown following steps very similar to those leading to (6.21).

11.2.2

Green’s Functions

The n-point functions for fermions can now be obtained, as in the case of bosons, as functional derivatives of the functional Z[J]. For this purpose we modify the deﬁnition (8.3) such that the presence of two fermion ﬁelds, Ψ ¯ is taken into account. and Ψ, We deﬁne the 2n-point function for fermions by (11.77) Gα,β,...,2ν (x1 , x2 , . . . , x2n ) . 2n 2n . 1 δ Z . . = , i δη2ν (x2n ) · · · δην+1 (xn+1 )δ η¯ν (xn ) · · · δ η¯α (x1 ) .η=¯η=0

CHAPTER 11. GREEN’S FUNCTIONS FOR FERMIONS

142

where Z[η, η¯] is given by (11.60) Z[η, η¯] = Z0

¯ DΨ ei DΨ

¯ d4x [L+¯ η (x)Ψ(x)+Ψ(x)η(x) ]

.

(11.78)

Here the order of the functional derivatives with respect to η and η¯, respectively, is ﬁxed by convention (see discussion below). Note that the Green’s function now depends also on the Dirac spinor-indices, in addition to the space-time coordinates. As in (8.2) for bosons, the Green’s function can be written as a vacuum expectation value of a time-ordered product of ﬁeld operators G(x1 , x2 , . . . , x2n )

=

(11.79)

¯ DΨ Ψ(xn ) · · · Ψ(x1 )Ψ(x ¯ 2n ) · · · Ψ(x ¯ n+1 ) eiS DΨ ¯ DΨ eiS DΨ !

"

ˆ¯ ˆ¯ ˆ n−1 ) · · · Ψ(x ˆ 1 ) Ψ(x ˆ n )Ψ(x = 0|T Ψ(x 2n ) · · · Ψ(xn+1 ) |0 , where S denotes the action. The special ordering of ﬁelds in (11.79) is a consequence of our ordering of the derivatives in (11.77); it ensures that there is no additional phase present. This can be seen as follows: In calculating the derivatives with respect to η¯ no phase appears, because ﬁrst the derivative is anticommuted through all ﬁeld operators already pulled down from the exponential, then acts without a further sign change on the exponential and then is anticommuted back, thus creating the same phase backwards as on the way forward. The extra phase (−) that appears from the derivative with respect to η due to the necessary reordering in the exponent is cancelled by the same phase caused by anticommuting the derivative operator δ/δη with ¯ that were already pulled down, and by all the Grassmann ﬁelds Ψ and Ψ, reordering them. Thus, the n-point Green’s function, originally introduced as a functional derivative of a generating functional, can in general also be deﬁned as the ˆ vacuum expectation value of a time-ordered product of ﬁeld operators Φ(x) that can describe either boson or fermion ﬁelds !

"

ˆ n )Φ(x ˆ n−1 ) · · · Φ(x ˆ 1 ) Φ(x ˆ 2n ) · · · Φ(x ˆ n+1 ) |0 . G(x1 , x2 , . . . , x2n ) = 0|T Φ(x (11.80) The only diﬀerence between boson and fermion ﬁelds appears in the reordering caused by the T operator; here a ”(−)” sign appears whenever a pair of fermions changes its order whereas there is no such sign when the boson operators are reordered.

CHAPTER 11. GREEN’S FUNCTIONS FOR FERMIONS

143

2-point function. The two-point function is given by .

δ 2 Z[η, η¯] .. . Gαβ (x1 , x2 ) = − . δηβ (x2 )δ η¯α (x1 ) .η=¯η=0

(11.81)

Using (11.65) for the generating functional of the free theory we obtain

Gαβ (x1 , x2 ) =

¯ ¯ β (x2 )ei DΨDΨ Ψα (x1 )Ψ 4 ¯ DΨDΨ ei L d x

L d4x

(11.82)

Here α and β are Dirac-Spinor indices. This expression can, as for bosons, be written as an expectation value over a time-ordered product of ﬁeld operators !

"

ˆ¯ ) |0 ˆ 1 )Ψ(x G(x1 , x2 ) = 0|T Ψ(x 2

(11.83)

where, now for fermions, we have !

"

⎧ ⎨

ˆ¯ ) ˆ 1 )Ψ(x Ψ(x t > t2 2 ˆ¯ ) = ˆ 1 )Ψ(x T Ψ(x for 1 2 ˆ¯ )Ψ(x ⎩ −Ψ(x ˆ t 2 > t1 2 1)

.

(11.84)

Note the extra minus sign compared to the boson case. This sign appears because of the Grassmann nature of the fermion ﬁelds when we reorder the ﬁelds as in the developments leading to (3.47). We could have obtained the two-point function also from (11.71). This gives (11.85) Gαβ (x1 , x2 ) = i[SF (x1 − x2 )]αβ . Again, as in scalar ﬁeld theory, the 2-point function is – up to a phase – just equal to the propagator of the relevant equation of motion. The propagator of our fermionic theory is thus just the inverse of the ¯ and Ψ in the Lagrangian operator appearing between the ﬁelds Ψ ¯ ∂ − m)Ψ L = Ψ(i G(x, y) = i(i ∂ − m)−1 = iSF (x − y) .

(11.86)

For boson ﬁelds this was also the case (see the discussion at the end of section 6.1.3) 1 L = − φ(2 + m2 )φ 2 G(x, y) = − i(2 + m2 )−1 = iDF (x − y) .

(11.87)

We can thus read oﬀ the propagator directly from the Lagrangian1 . 1

The factor 1/2 in the Lagrangian for the bosons is a special feature of the uncharged field and would not be there if we were working with charged fields.

Chapter 12 INTERACTING FIELDS So far we have treated only the case of free fermion ﬁelds. In the case of an interacting theory, expression (8.44) can directly be taken over for the case of fermions so that we have Z[η, η¯] = Z0 e

−i

d4x V ( 1i

δ 1 δ , ¯ δη i δ η

) e−i

η¯(x)SF (x−y)η(y)d4x d4y

.

(12.1)

Here the interaction V is deﬁned by the Lagrangian of the interacting theory ¯ Ψ) . L = L0 − V (Ψ,

(12.2)

If the theory involves both boson and fermion ﬁelds and their couplings, i.e. if the Lagrangian is given by φ ¯ ¯ L = LΨ 0 + L0 − V1 (Ψ, Ψ) − V2 (φ) − Vint (Ψ, Ψ, φ) ,

(12.3)

then Z[J, η, η¯] is given simply by (cf. (8.44)) Z[J, η, η¯] = Z0 e−i

δ 1 δ 1 δ d4x Vint ( 1i δη , i δη¯ , i δJ )

1 δ 1 δ 1 δ 4 4 ×e−i d x V1 ( i δη , i δη¯ ) e−i d x V2 ( i δJ ) 4 4 ×e−i η¯(x)SF (x−y)η(y)d x d y i 4 4 ×e− 2 J(x)DF (x−y)J(y)d x d y .

(12.4)

This functional can generate all the n-point functions of the interacting theory and these can again be graphically represented in the form of Feynman diagrams.

12.1

Feynman Rules

The Feynman rules of Sect. 9.1 can be extended by the following rules (in momentum space) 144

CHAPTER 12. INTERACTING FIELDS

145

Figure 12.1: Fermion-boson vertex (see (12.6)). The solid line denotes the fermion, the dashed one the boson. The dot denotes the interaction vertex. 1) each internal fermion line gives a factor i . k − (m − i)

(12.5)

The fermion lines carry now an arrow that points into the direction into which the fermion’s charge ﬂows. This is a consequence of the fact that the fermion propagator is no longer symmetric in its arguments (cf. (11.75)) so that we have to give the direction of motion explicitly. ¯ that carry diﬀerent For fermions we deal with two ﬁelds, ψ and ψ, charges. In a Feynman graph, in which time runs from left to right, particle lines are always rightwards directed whereas antiparticle lines move leftwards. In a loose way of speaking the fermion lines move from ψ¯ to ψ, because ψ¯ creates particles and ψ annihilates them. Since the opposite holds for antiparticles their lines run leftwards, reﬂecting the fact that the Feynman propagator propagates them backwards in time. In a coupled theory a new class of diagrams can appear because of the coupling of fermion and boson ﬁelds. For example, a coupling term of the form ¯ ¯ α Ψα φ , = gΨ (12.6) Vint = g ΨΨφ the so-called Yukawa coupling, will generate vertices of the form shown in Fig. 12.1. We, therefore, have a new Feynman rule dealing with fermionboson vertices: 2) each boson-fermion vertex gives a factor −ig. If the coupling term ¯ μ Ψ, then an involves other bilinear forms of the Dirac spinors, e.g. Ψγ additional factor γμ would appear at each vertex.

12.1.1

Fermion Loops

A change in the rules appears when we consider fermion loop insertions in the boson propagator, e.g. the graph shown in Fig. 12.2, where the solid

CHAPTER 12. INTERACTING FIELDS

(x α)

146

(y β)

Figure 12.2: Fermion loop contribution to the boson propagator. line denotes the fermions and the dashed line the bosons. Such a term is obviously of second order in the fermion-boson coupling constant (indicated by the two vertices) and contributes to the bosonic 2-point function. It can thus be generated by the second order term in the expansion of (12.4), properly generalized to fermion ﬁelds. Because the graph in Fig. 12.2 is oneparticle irreducible only the true second order term (∼ S2 in (8.50),(8.51)) can contribute. This term has the structure iS[η, η¯, J] = . . .

1 −iS0 [η,¯η,J] 1 δ 1 δ 1 δ + e (−i) d4x Vint , , 2 i δη(x) i δ η¯(x) i δJ(x) + ... .

2

e+iS0 [η,¯η,J] (12.7)

Here S0 [η, η¯, J] is given by iS0 [η, η¯, J] =

−

i 4 4 d v d w J(v)DF (v − w)J(w) 2

−i

d4x d4y η¯(x )SF (x − y )η(y ) .

(12.8)

We now remember that only those terms in the generating functional can contribute to the boson propagator in Fig. 12.2 that are of second order in the bosonic current J and of zeroth order in the fermionic currents η and η¯. Therefore, the only contributing term in (12.7) with the interaction (12.6) is iS2loop = − × =

(−ig)2 −iS0 e 2 d4x d4y

δ6 e+iS0 δηα (x)δ η¯α (x)δJ(x)δηβ (y)δ η¯β (y)δJ(y)

(−ig)2 −iS0 4 4 4 4 d xd y d vd w e 2 &

(12.9)

'

δ4 × DF (x − v)J(v)DF (y − w)J(w) e+iS0 , δηα (x)δ η¯α (x)δηβ (y)δ η¯β (y)

CHAPTER 12. INTERACTING FIELDS

147

after performing the derivative with respect to J. We now concentrate on the fermionic derivative (for notational convenience we drop here the index F for the propagators) δ3 δ 4 4 e−i η¯(x )S(x −y )η(y ) d x d y δηα (x)δ η¯α (x)δηβ (y) δ η¯β (y) δ δ2 (−i) Sβγ (y − y )ηγ (y ) d4y e−i ··· = δηα (x)δ η¯α (x) δηβ (y) δ δ = − iSββ (y − y) δηα (x) δ η¯α (x)

−

⊕

4

Sβγ (y − y )ηγ (y ) d y

δ = δηα (x)

− Sββ (y − y)

4

η¯δ (x )Sδβ (x − y) d x

e

−i

···

Sαγ (x − y )ηγ (y ) d4y

4

Sβγ (y − y )ηγ (y ) d y Sαβ (x − y) + O(η η¯) e

−i

···

. (12.10)

Here the symbol ⊕ denotes a ‘plus’ sign that is due to the Grassmann nature of the source functions and would have been opposite, if we had worked with bosons instead of fermions. The terms denoted by O(η η¯) stand for expressions that involve products of the two source functions and would thus be linear in η or η¯ after the last functional derivative has been taken. Since the n-point function is given by a functional derivative at vanishing source, these terms do not contribute to the 2-point function we are after. Performing now the last derivative gives for the expression (12.10) =

[−Sββ (y − y)Sαα (x − x) ⊕ Sβα (y − x)Sαβ (x − y) + O(η η¯)] × e−i

η¯(x )SF (x −y )η(y ) d4x d4y

.

(12.11)

The sign of the second term would have been opposite, if we had worked with bosons. We now combine this result with (12.9). Since we are interested in particular in the fermion loop insertion to the boson propagator shown above, we need to construct a bosonic 2-point function with no external fermion lines. Therefore, only those terms of the fermionic part (12.11) can contribute to this graph that contain no sources η or η¯, except in the exponential. All other terms ∼ η η¯ would vanish after setting η = η¯ = 0. Disregarding the vacuum contributions (ﬁrst term in (12.11)) we thus have as the only term of (12.11) that contributes to the loop diagram ⊕ tr [SF (y − x)SF (x − y)] .

(12.12)

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148

Inserting this result into (12.9) gives for the relevant part of S2 iS2loop = ⊕

(−ig)2 4 4 4 4 d x d y d v d w tr [SF (y − x)SF (x − y)] 2 × DF (x − v)DF (y − w)J(v)J(w) (12.13)

and, correspondingly, for the 2-point function Gloop boson (x1 , x2 )

δ 2 (iS2loop ) =− δJ(x1 )δJ(x2 )

(−ig)2

=

(12.14)

d4x d4y DF (x1 − x)tr [SF (x − y)SF (y − x)] DF (y − x2 ) .

When we retrace this detailed calculation we ﬁnd that the positive sign in (12.12) and thus the negative sign in (12.14) is due to the fact that η and η¯ are Grassmann ﬁelds. Thus, we get the additional Feynman rule, which holds in general, 3) Any fermionic loop graph is associated with an additional (−) sign and a trace over the Dirac indices.

12.2

Wick’s Theorem

In Sect. 8.2.1 we had expressed the n-point function for bosons G(x1 , . . . , xn ) in terms of a symmetrized product of n/2 2-point functions (Wick’s theorem). The same can now be done also for the general theory that involves bosons and fermions. Mixed n-point functions. First, it is clear that boson and fermion ﬁelds commute; the corresponding ﬁeld operators act in diﬀerent Hilbert spaces so that the groundstate expectation value separates into a product of vacuum expectation values over fermion and boson operators separately, e.g. ˆ¯ ˆ¯ ˆ φ(3) ˆ Ψ(4)|0 ˆ φ(3)|0 ˆ ˆ φ(2) ˆ Ψ(4)|00| 0|Ψ(1) = 0|Ψ(1) φ(2) .

(12.15)

We, therefore, need to formulate here Wick’s theorem only for the fermions. Wick’s theorem for fermions. We now derive a simple method, known as Wick’s Theorem that we encountered for bosons already in Sect. 8.2.1, for the evaluation of a vacuum expectation value of a time-ordered product of free ﬁeld operators. In that case we can use the form (11.67) of the generating

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functional for the evaluation of the functional derivative. Performing ﬁrst the functional derivatives with respect to η¯ on that expression gives1 δ 2n 4 4 e−i η¯(x)SF (x−y)η(y) d x d y |η=¯η=0 δη(x2n ) · · · δη(xn+1 )δ η¯(xn ) · · · δ η¯(x1 )

δn n = (−i) SF (xn − y)η(y) d4y δη(x2n ) · · · δη(xn+1 )

×

×e

SF (xn−1 − y)η(y) d4y · · ·

−i

η¯(x)SF (x−y)η(y) d4x d4y

. . . . .

SF (x1 − y)η(y) d4y .

(12.16)

n=¯ η =0

Now the diﬀerentiation with respect to η is performed. Since the whole expression has to be taken at η = η¯ = 0, only the prefactors of the exponential can contribute. This gives = (−i)n

(−)P SF (x1 − xp2n )SF (x2 − xp2n−1 ) . . . SF (xn − xpn+1 ) ,

(12.17)

P

where the sum over P runs over all permutations of the indices n + 1, . . . , 2n. The sign of the permutations is such that (−)P = +, if always the outermost operators in (11.77) are combined into the 2-point function, i.e. if (1, 2n), (2, 2n − 1) · · · (n, n + 1) are combined. The Dirac indices, that have not been written down here, have to be combined in the same way. Inserting this result into the deﬁnition of the Green’s function in (11.77) gives G(x1 , x2 , . . . , x2n ) =

2n 1

= in

i

δ 2n Z δη(x2n ) . . . δη(xn+1 )δ η¯(xn ) . . . δ η¯(x1 )

(−)P SF (x1 − xp2n ) · · · SF (xn − xpn+1 ) .

P

(12.18) Equation (12.18) is Wick’s theorem for fermions. Written as a vacuum expectation value of time-ordered ﬁeld operators it reads !

"

ˆ¯ ˆ¯ ˆ n ) · · · Ψ(x ˆ 1 ) Ψ(x 0|T Ψ(x 2n ) · · · Ψ(xn+1 |0 = in

!

"

!

(12.19)

ˆ¯ ˆ¯ ˆ 1 )Ψ(x ˆ (−)P 0|T Ψ(x p2n ) |0 · · · 0|T Ψ(xn )Ψ(xpn+1 ) |0 .

P 1

"

To facilitate the notation the Dirac indices are not written down here.

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Wick’s theorem for fermions (12.18) has a form that is analogous to that for bosons (8.20). The essential diﬀerence is the appearance of the sign of the permutation that reﬂects the antisymmetric of fermionic states. In addition the degeneracy factor 1/2n is missing here because for fermions we have no longer a symmetry under exchange: SF (x − y) = SF (y − x) (see (11.75)). This form also exhibits clearly the important consequence of using anticommuting Grassmann ﬁelds for the fermions. The exchange of any two of ¯ gives the coordinates appearing in the ﬁelds Ψ or of any 2 in the ﬁelds Ψ a “−” sign to the Green’s function. In particular, the 2-particle, 4-point Green’s function has the property Gαβγδ (x1 , x2 , y1 , y2 ) = −Gβαγδ (x2 , x1 , y1 , y2 ) = −Gαβδγ (x1 , x2 , y2 , y1 ) = Gβαδγ (x2 , x1 , y2 , y1 ) . (12.20) This implies that for G not to vanish we must have x1 = x2 and y1 = y2 , if all the Dirac-indices are the same, thus reﬂecting the Pauli principle.

12.3

Removal of Degrees of Freedom: Yukawa Theory

In the beginning of Chap. 12 we have ﬁrst met a coupled fermion-boson theory with a simple Yukawa coupling. We now consider a theory that contains free fermions coupled to a self-interacting boson ﬁeld by means of a Yukawa interaction. Its Lagrangian reads 1 ¯ L = − φ2φ − V (φ) + ψ¯ (i ∂ − mF ) ψ − g ψψφ 2 1 = − φ2φ − V (φ) + ψ¯ (i ∂ − mF − gφ) ψ . (12.21) 2 The term V (φ) may contain both the mass term and further selﬁnteraction terms. Since the coupling constant g here is dimensionless we expect that this theory is renormalizable (cf. the discussions in Sect. 10.2). The second line of (12.21) shows that the Yukawa coupling contributes the fermion mass and, for mF = 0, can even generate the whole mass of a fermion. This mechanism plays an important role in the theory of electroweak interactions. In the preceding sections we have discussed the perturbative treatment of such a Lagrangian. Here we consider now an alternative treatment that is based on the fact that the generating functional is given by

Z[η, η¯, J] = N

&

Dψ¯ Dψ Dφ exp i

'

¯ ψ, φ] + Jφ + η¯ψ + ψη ¯ d4x . L[ψ, (12.22)

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151

It is thus quadratic in the fermion ﬁelds and, therefore, of Gaussian form in the fermion ﬁelds. The path integral over the fermionic degrees of freedom can thus be performed. According to (11.56) and (11.65) this gives

Z[η, η¯, J] = N

i¯ η M −1 η −i

Dφ det(M ) e

e

[ 12 φ2φ−V (φ)−Jφ] d4x

(12.23)

with (see (11.66)) M = i ∂ − m∗

with

m∗ = mF + gφ ;

(12.24)

the factor ‘i’ before M i/n the determinant has been absorbed into the normalization constant N . Note that the fermion determinant M is a function of φ. Equation (12.23) is completely equivalent to (12.22). We now concentrate on processes that contain only bosons on the external legs and thus require only bosonic n-point functions for the calculation of transition rates etc. We can then set η and η¯ to zero. In doing so we give up the possibility to calculate any n-point function with fermionic external legs because these are obtained as functional derivatives with respect to η. Our procedure thus amounts to keeping only fermionic loops in our theory for the scalar ﬁeld φ. More speciﬁcally, as we will see below, our procedure keeps fermionic eﬀects only on the one-loop level. Using now (6.55) (12.25) det(M (φ)) = etr ln M (φ) gives

Z[J] = N

1 4 Dφ e−i [ 2 φ2φ+V (φ)−Jφ] d x+tr ln M (φ) .

(12.26)

Equation (12.26) shows that now formally all the fermion degrees of freedom have been removed. Alternatively, we could also say that we are now working with a new, eﬀective action for the bosonic sector SB = −

1 φ 2φ + V (φ) d4x − i tr ln (i ∂ − m∗ (φ)) . 2

(12.27)

The trace here has to be performed over Dirac indices and over space-time coordinates (or, equivalently, in momentum space). The corresponding eﬀective Lagrangian density reads 1 LB = − φ2φ − V (φ) − i Tr ln (i ∂ − m∗ (φ)) , 2 where we have used tr = Tr d4 x ;

(12.28) (12.29)

Tr is a trace over the Dirac (and possibly other internal) indices. In this form, which is still exact, the original Lagrangian (12.21) is said to be bosonized.

CHAPTER 12. INTERACTING FIELDS

12.3.1

152

Perturbative Expansion

We now develop a perturbative treatment of this bosonized Lagrangian that uses an expansion around the groundstate of the bosonic sector. The true groundstate (vacuum) of the bosonic sector is, of course, determined by the potential V (φ), but it may not coincide with the minimum of this potential. Nevertheless, we may take here the constant, translationally invariant ﬁeld φ0 which minimizes V as our reference ﬁeld. In order to normalize the action, and thus the physics, to this state we subtract the action belonging to φ0 . This gives (with m∗0 = mF + gφ0 )

1 SB = − φ2φ − V (φ) d4x − i tr ln (i ∂ − m∗ ) + i tr ln (i ∂ − m∗0 ) , 2 (12.30) where a constant term has been dropped. Next we use the identity tr ln (i ∂ − m∗ ) = tr ln (i ∂ − m∗ )† = tr ln (−i ∂ − m∗ )

(12.31)

which holds because of the hermiticity of the Dirac operator i ∂ − m∗ ; here we remember from our discussion in Sect. 6.1.3 that the determinant of an operator is just the product of its eigenvalues. We thus get 1 tr ln [(i ∂ − m∗ ) (−i ∂ − m∗ )] 2 1 = tr ln 2 + m∗ 2 − ig ∂φ . 2

tr ln (i ∂ − m∗ ) =

(12.32)

Thus the eﬀective bosonic action becomes 1 φ2φ + V (φ) d4x SB = − (12.33) 2 i i − tr ln 2 + m∗ 2 − ig ( ∂φ) + tr ln 2 + m∗0 2 2 2 i m∗ 2 − m∗0 2 − ig ∂φ 1 4 = − φ2φ + V (φ) d x − tr ln 1 + 2 2 2 + m∗0 2 and the Lagrangian density now reads

LB

1 m∗ 2 − m∗0 2 − ig ∂φ i = − φ2φ − V (φ) − Tr ln 1 − 2 2 −2 − m∗0 2 1 i = − φ2φ − V (φ) − Tr ln (1 − D(0)W (x)) 2 2

(12.34)

with the operator D and the ﬁeld-dependent quantity W D=−

1 2 + m∗0 2

and

W = m∗ 2 − m∗0 2 − ig ∂φ .

(12.35)

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153

Note that D is nothing else than the Feynman propagator (cf. (6.7)). So far, no approximations have been made and the theory is still exact; note that the reference ﬁeld is in principle arbitrary. The action is now in a form that is suitable for a perturbative treatment. 2 We thus expand the logarithm (− ln(1 − x) = x + x2 + . . .) in (12.33) and obtain ∞ 1 −tr ln(. . .) = tr (12.36) (DW )n n n=1 The trace in (12.36) indicates that each term in the expansion contains one closed fermion loop (with n vertices W coupling to the ﬁeld φ). Equation (12.36) is therefore called the one-loop expansion, it represents a perturbative expansion of the eﬀective action. Integrating out the fermion degrees of freedom has thus led to an eﬀective theory that takes all fermionic one-loop diagrams into account. The expansion is expected to converge the more rapidly the closer the actual ﬁeld φ is to the chosen ﬁeld φ0 and the smaller the gradients of φ are. Our aim in the following developments is to write the trace term as an integral over a space-time density which could then be interpreted as a correction term to the Lagrangian density, thus yielding an eﬀective Lagrangian which contains the eﬀects of the fermions on the one-loop level. In order to be able to do so we have to separate the terms under the trace into a product of operators that are local in x and p. In this case the trace can be separated

d4p p|F (ˆ p)G(ˆ x)|p tr [F (ˆ p)G(ˆ x)] = Tr (2π)4 d4p d4p 4 4 = Tr dx dx (2π)4 (2π)4 ×p|F (ˆ p)|p p |x x |G(ˆ x)|xx|p 4 dp = Tr F (p) d4x G(x) . (12.37) (2π)4 The expressions in (12.36) are, however, not in a form that allows direct application of (12.37), because the ﬁeld φ, and thus also W , is dependent on x and the propagator D acts on it. We thus ﬁrst reorder each term such that all propagators are moved to the left. To do so we start from the identity !

"

W D = DW + D D−1 , W D

(12.38)

and iterate this equality by applying it again to the quantity W = [D−1 , W ] !

"

W D = DW + D D−1 , W D .

(12.39)

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154

Combining now (12.38) and (12.39) gives !

"

!

!

W D = DW + D2 D−1 , W + D2 D−1 , D−1 , W

""

D.

(12.40)

This procedure can be continued and yields the operator identity !

"

!

!

W D = DW + D2 D−1 , W + D3 D−1 , D−1 , W

""

+ ... .

(12.41)

The basic commutator appearing here is given by !

"

D−1 , W = [−2, W ] = −2W − 2 (∂ μ W ) ∂μ .

(12.42)

Since W contains a derivative of the ﬁeld this term is of order O((∂μ φ)3 ) and the next higher order term in (12.41) would be of order O((∂μ φ)4 ). Equation (12.41) thus represents an expansion in terms of gradients of the ﬁeld. We also see that it contains increasing powers of D so that we expect to encounter only a ﬁnite number of ultraviolet, i.e. for large momenta, divergent terms. We now use the identity (12.41) to rewrite the expansion (12.36). Up to terms of second order in W the expansion reads 1 (12.43) −tr ln(. . .) = tr(DW ) + tr(DW DW ) + O(W 3 ) 2 " 1 1 ! = tr(DW ) + tr(D2 W 2 ) + tr D3 D−1 , W W + O(W 3 ) . 2 2 Note that all the terms on the rhs have the momentum dependence and the x-dependence separated so that – according to (12.37) – they can be written as an integral over the momentum times a spatial integral over a Lagrangian density. Eq. (12.43) is an expansion in terms of powers of W . We now discuss the terms in the expansion (12.43) and start with the term tr(DW ). This is explicitly given by !

"

tr(DW ) = tr D m∗ 2 − m∗0 2 − ig ∂φ

.

(12.44)

Since tr γμ = 0 and

(12.45)

m∗ 2 − m∗0 2 = g 2mF (φ − φ0 ) + g φ2 − φ20

(12.46)

we get 1

!

tr(DW ) = g tr D 2mF (φ − φ0 ) + g φ2 − φ20

"0

.

(12.47)

CHAPTER 12. INTERACTING FIELDS

155

The trace operation can now be performed. This gives

! " d4k 1 4 2 2 tr(DW ) = 4g d x 2m (φ(x) − φ ) + g φ (x) − φ , F 0 0 (2π)4 k 2 − m∗0 2 (12.48) where we have used (12.37) with the trace over the Dirac indices Tr = 4. This term is now in the desired form, i.e. it is given by a space-time integral over ﬁelds. However, the momentum-integral appearing here actually diverges quadratically as power counting shows us. Also the next term in (12.43) diverges. It is given by 2

2

d4k 1 4 (2π) k 2 − m∗ 2 2 0

2

tr(D W ) = g Tr ×

&

(12.49)

d4x 2mF (φ(x) − φ0 ) + g φ2 (x) − φ20 − i ∂φ(x)

'2

and diverges logarithmically. The next term in (12.43)

!

"

!

tr D3 D−1 , W W = tr (D)3 (−2W − 2(∂ μ W )∂μ ) W

"

(12.50)

and all higher order terms are clearly ultraviolet convergent because they contain higher and higher powers of the propagator D. This means that we have only two divergent terms ((12.48) and (12.49)) in the expansion (12.36). These can be removed by introducing proper counter terms. We can thus deﬁne a ﬁnite, renormalized eﬀective Lagrangian density in the boson sector of our theory by just adding the two divergent terms (12.48) and (12.49) to the Lagrangian 1 L˜B (x) = − φ(x)2φ(x) − V (φ(x)) 2 ! " i − Tr ln 1 − D m∗ 2 (x) − m∗0 2 − ig ∂φ(x) 2 − i2g −i

d4k 2mF (φ(x) − φ0 ) + g (φ2 (x) − φ20 ) (2π)4 k 2 − m∗0 2

g 2 d4k 1 Tr 4 4 (2π) k 2 − m∗ 2 2 0 &

(12.51)

× 2mF (φ(x) − φ0 ) − i ∂φ(x) + g φ (x) − 2

φ20

'2

.

CHAPTER 12. INTERACTING FIELDS

156

Writing this expression in a somewhat more compact form gives L˜B =

=

1 − φ(x)2φ(x) − V (φ(x)) − 2 & ' i 1 2 − Tr DW + (DW ) 2 2 1 − φ(x)2φ(x) − V (φ(x)) + 2

i Tr(1 − DW ) 2 (12.52) " 1 3 ! −1 Tr D D , W W , 2

where Tr denotes a trace over the Dirac indices only and where it is understood that we have to use the expansion (??) for the Tr ln term. The two counter terms have the structure of terms already present in the original boson Lagrangian; they just remove the ﬁrst two terms in the Taylor expansion of the logarithmic term in (12.34). The Yukawa theory is thus indeed renormalizable, provided the potential V (φ) is ‘well-behaved’ and contains at most terms of fourth order in φ.

Chapter 13 PATH INTEGRALS FOR GAUGE FIELDS The fundamental interactions of nature like, e.g. the electroweak and the strong interactions are described by so-called gauge ﬁeld theories. In these theories the action is invariant under certain space-time dependent unitary “gauge” transformations. As a consequence for each ﬁeld conﬁguration there are inﬁnitely many others that can all be obtained by a gauge transformation from the original one. The physics remains unchanged under such transformations. Special problems arise when such gauge theories are quantized. The reason is quite easy to see in the framework of path integrals. If we consider a ﬁeld Aμ , the path integral would be naively written down as

DA eiS[Aμ ]

(13.1)

where the action S is a functional of Aμ . The integration measure stands for a straightforward generalization of the measure used for scalar ﬁelds DA ∼

n

4

dAμ (x, tj )

(13.2)

x j=1 μ=1

in the limit n → ∞ (cf. (5.11)); the measure so deﬁned is gauge invariant as we will see later in this chapter. For a ﬁxed ﬁeld A¯μ there are inﬁnitely many other ﬁelds, connected to A¯μ by gauge transformations, that leave the action S invariant and thus all give the same contribution to the integrand. Thus the path integral as written down here cannot converge. In this chapter we develop methods to deal with this diﬃculty. After a discussion of gauge-invariance in electrodynamics we summarize the properties of a more general class of gauge-ﬁeld theories and then develop the path integral description for them. 157

CHAPTER 13. PATH INTEGRALS FOR GAUGE FIELDS

13.1

158

Gauge invariance in Abelian theories

We start this section with a discussion of gauge invariance in electrodynamics by considering a free ﬁeld conﬁguration without currents. For the electromagnetic ﬁeld the free Lagrangian is given by 1 1 L = − Fμν F μν = − (∂μ Aν − ∂ν Aμ ) (∂ μ Aν − ∂ ν Aμ ) 4 4

(13.3)

and is clearly invariant under a gauge transformation Aμ (x) −→ Aμ (x) = Aμ (x) + ∂μ (x) .

(13.4)

Here (x) is an arbitrary diﬀerentiable function of space-time. The corresponding equation of motion for the ﬁelds Aμ is ∂ν F νμ = ∂ν (∂ ν Aμ − ∂ μ Aν ) = (δνμ 2 − ∂ν ∂ μ )Aν = 0 −→ (gμν 2 − ∂μ ∂ν ) Aν = 0 ,

(13.5)

which is also gauge invariant. If we now ﬁx a gauge, e.g. the Lorentz gauge ∂μ Aμ = 0, we obtain the free wave equation for Aμ 2Aμ = 0 ,

(13.6)

which is no longer gauge invariant. We now proceed to obtain the propagator for the electromagnetic ﬁeld. As mentioned at the end of section 11.2.2 the propagator of the theory could be read oﬀ from the term in the Lagrangian that is quadratic in the ﬁeld. The Lagrangian (13.3) can indeed also be converted into a form quadratic in the ﬁelds by writing

1 4 Ld x = − d x (∂ μ Aν − ∂ ν Aμ )(∂μ Aν − ∂ν Aμ ) (13.7) 4 1 = − d4x [ (∂ μ Aν )(∂μ Aν ) − (∂ μ Aν )(∂ν Aμ ) 4 − (∂ ν Aμ )(∂μ Aν ) + (∂ ν Aμ )(∂ν Aμ )] . 4

By partial integration we obtain

1 4 Ld x = d x (Aμ 2Aμ − 2Aμ ∂ν ∂μ Aν + Aμ 2Aμ ) 4 1 = d4x Aμ (gμν 2 − ∂μ ∂ν ) Aν , 2 4

so that L can also be written as 1 L = Aμ (gμν 2 − ∂μ ∂ν ) Aν . 2

(13.8)

(13.9)

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159

This Lagrangian is very similar to that of a scalar ﬁeld theory (4.31) so that we expect the generating functional to have the form Z[0] = ei

d4x L

.

(13.10)

Since the Lagrangian is quadratic in the ﬁelds it can be treated in analogy to the developments in Sect. 6.1.3. According to our normal procedure (cf. (11.87)) we thus expect to obtain the propagator of the electromagnetic ﬁeld as −1 ∼ gμν 2 − ∂μ ∂ν . (13.11) Dμν The problem arises when we realize that this operator D does not exist. We see this by applying its inverse (13.11) to an arbitrary four-gradient ∂ ν G −1 ν Dμν ∂ G = (2∂μ − ∂μ 2) G = 0 .

(13.12)

Thus, D−1 has a zero eigenvalue and, loosely speaking, its inverse D is inﬁnite. This inﬁnity is linked to that in the path integral over gauge ﬁelds discussed at the start of this chapter. This can be seen by a straightforward application of (B.18) to (13.10) which gives the equivalent of (6.33) with det(A) = 0 in the denominator1 . In other words: we are integrating over too many degrees of freedom when the gauge condition is not taken into account. Therefore, we expect that one way to overcome this diﬃculty is to ﬁx a gauge, for example by imposing the Lorentz gauge condition ∂ ν Aν = 0 .

(13.13)

If we then consider only potentials that fulﬁll this condition and integrate only over them, we expect the path integral to be well-behaved. This expectation is based on the fact that we can ﬁnd a Lagrangian that incorporates the gauge condition (13.13) at the price of loosing its manifest gauge invariance 1 L˜ = − Fμν F μν + LGF 4 1 1 = − Fμν F μν − (∂μ Aμ )2 . 4 2λ

(13.14)

This Lagrangian, that leads for λ = 1 to the wave equation (13.6) as we will show below, contains a so-called gauge-ﬁxing term as a quadratic constraint, 1 ). This additional term coupled in by means of a Lagrange-multiplier ( 2λ 1

Remember here that the determinant of an operator is given by a product of its eigenvalues.

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160

could be considered as a potential that becomes the more repulsive the more the actual potential Aμ diﬀers from the Lorentz-gauge (13.11); the constraint thus drives the system towards that gauge. Performing the steps leading to (13.9) once again, we ﬁnd that (13.14) is equivalent to the Lagrangian

1 1 ∂μ ∂ν Aν L˜ = Aμ gμν 2 − 1 − 2 λ

(13.15)

which leads to the same action. The equations of motion following from (13.15) are 1 gμν 2 − 1 − ∂μ ∂ν Aν = 0 . (13.16) λ We can thus either start from a manifestly covariant Lagrangian, derive the equations of motion from it and then impose a suitable gauge condition or we can, alternatively, incorporate the gauge condition into the Lagrangian, at the price of giving up its gauge invariance, and then obtain the gauge-ﬁxed equations of motion directly from it. The physics must obviously be the same in both methods. Reading again – as at the end of Chap. 11 on p. 143 – the inverse propagator oﬀ from the Lagrangian (13.15) gives

−1 = gμν 2 − 1 − Dμν

1 ∂μ ∂ν λ

(13.17)

with the momentum space representation −1 (k) Dμν

1 = −gμν k + 1 − kμ kν . λ 2

(13.18)

This operator no longer has a zero eigenvalue as we can easily see by perform−1 νλ ing again the steps in (13.12). The propagator D, deﬁned by Dμν D = δμλ , 2 thus exists. In momentum space it is given by

1 kμ kν Dμν (k) = − 2 gμν + (λ − 1) 2 k k

.

(13.19)

This is the propagator for the electromagnetic ﬁeld. The constant λ in (13.19) is arbitrary, but note that the propagator does not exist for λ → ∞; this is just the case of the unconstrained Lagrangian discussed earlier in this section (cf. (13.3) and (13.14)). Choosing λ = 1 2 The denominator k 2 from here on should really read k 2 − iε in order to ensure the proper boundary conditions for the propagator.

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gives the so-called Feynman gauge, whereas λ = 0 describes the Landaugauge. The Feynman gauge is the one that we are used to from classical electrodynamics; in it the equations of motion for the electromagnetic ﬁeld (13.16) read (13.20) 2Aν = 0 . This is the equation of motion that we obtained earlier for the Lorentz gauge. Generating functional. Now that we have constructed a propagator for the electromagnetic ﬁeld, we can write down the generating functional as

Z[J] = where DA =

4 μ=1

DA ei

(L+LGF +Jμ Aμ )d4x

(13.21)

DAμ and LGF is the gauge-ﬁxing term LGF = −1/(2λ)(∂μ Aμ )2 .

(13.22)

The gauge-ﬁxing term suppresses the contributions of ﬁelds that do not fulﬁll the Lorentz gauge condition to the path integral. We will see a little later, in section 13.3, that this functional can actually be rewritten into a form that exhibits explicitly the integration over the gauge degree of freedom. For a free ﬁeld theory the generating functional can then be obtained as usual (cf. (B.18)) − 2i

Z[J] = e

J μ (x)Dμν (x−y)J ν (y) d4 xd4 y

,

(13.23)

and the two-point function is, as usual (see (11.86), (11.87)), found to be Gμν (x, y) = iDμν (x − y)

(13.24)

The path integral (13.21) does not suﬀer from the problems mentioned at the start of this section. Now the matrix in the quadratic term of L (13.9) can be inverted and the propagator exists. Furthermore, all ﬁelds that diﬀer from a speciﬁc ﬁeld just by a gauge transformation that leads out of the covariant Lorentz gauge have a higher action and thus contribute less to the path integral. The action itself is gauge-independent, whereas the integral over the gauge-ﬁxing term and the source terms do depend on the gauge. Thus, the generating functional itself is gauge dependent and, consequently, also the Green’s functions obtained from it by functional derivatives. All of the gauges corresponding to the various values of λ, however, lead to the same physics because in calculating a transition amplitude Dμν will couple to conserved currents only which fulﬁll ∂μ j μ = 0 → kμ j μ (k) = 0. As a consequence the λ-dependent term in (13.19) always drops out and the physically observable transition amplitudes are all gauge independent.

CHAPTER 13. PATH INTEGRALS FOR GAUGE FIELDS

13.2

162

Non-abelian gauge fields

In more general gauge ﬁeld theories one demands invariance of the Lagrangian under the transformation of the particle ﬁelds (see [MOSEL], chapter 7) ψ(x) → ψ (x) = U (x)ψ(x) .

(13.25)

The particle ﬁelds are described by N -component spinors ψ where the components represent the internal degrees of freedom. Here U is a unitary, local gauge transformation k k (13.26) U (x) = e−i (x)T where the T k are the generators of a Lie group. They are Hermitian so that the Lie group is unitary and have vanishing trace so that the generated Lie algebra is special, i.e. all group elements are represented by (N × N ) matrices that have unit determinant. The latter implies that the matrix representations of the generators T k have vanishing trace. The generators form a Lie algebra with the deﬁning commutation relations [T l , T m ] = if lmn T n

(l = 1, 2, . . . N 2 − 1) .

(13.27)

The generators are normalized such that 1 tr T l T m = δ lm 2

(13.28)

in the fundamental representation; the f lmn in (13.27) are the completely antisymmetric structure constants of the group. Another important matrix representation of the generators is the so-called regular representation

Tj

kl

= −if jkl .

(13.29)

The special unitary groups in N dimensions with these properties are called SU (N ). Since the generators do not commute the group SU (N ) is a non-Abelian one. Only for the case N = 1 there is only one generator and the group becomes Abelian, called U(1). Invariance under the transformation (13.25) can be achieved only if the derivative ∂μ is replaced by the so-called covariant derivative Dμ = ∂μ + igT l Alμ

(13.30)

where g is the coupling constant and Alμ is a vector (gauge) ﬁeld; the superscript l labels the internal degrees of freedom. This “minimal substitution” ﬁxes the interaction between particle and gauge ﬁelds.

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The gauge transformation of these ﬁelds must then be given by i Aμ → AUμ = U Aμ U −1 − U ∂μ U −1 , g

(13.31)

if the Lagrangian in the particle sector is required to be invariant. Here we have introduced the ﬁelds Aμ in (13.31) as contractions of the generators with the ﬁelds Akμ Aμ = T k Akμ . (13.32) The change of A under an inﬁnitesimal gauge transformation is given by jl 1 1 − l f ljk Akμ + ∂μ j = il T k Akμ + ∂μ j g g 1 = (Dμ )j g

δAjμ =

(13.33) kl

where we have used the regular representation of SU (N ), i.e. (T j ) = −if jkl . Also for non-Abelian gauge ﬁelds the free Lagrangian is given by 1 l lμν F , L = − Fμν 4

(13.34)

but now the ﬁeld tensor also carries the quantum numbers of the internal gauge group and must have a more complicated structure l n = ∂μ Alν − ∂ν Alμ − gf lmn Am Fμν μ Aν

(13.35)

in order to be invariant under the gauge transformation (13.31). Gauge invariance also requires that the gauge ﬁelds are massless, since a mass term of the form m2 A · A would clearly violate gauge invariance. l can also be contracted with the generators to form The ﬁeld tensor Fμν l T l ). These scalars change in the a scalar in the intrinsic space (Fμν = Fμν gauge-transformation just as under any unitary transformation U = U Fμν U −1 . Fμν → Fμν

(13.36)

Equation (13.34) together with (13.35) show that the free gauge ﬁeld is selﬁnteracting, with terms ∼ A3 and ∼ A4 in L . These terms are generated by the non-Abelian piece of Fμν and are absent for an Abelian theory (f lmn = 0). Notice also that the coupling constant g, determining the interaction strength, is the same for gauge ﬁeld–gauge ﬁeld (13.35) and gauge ﬁeld– particle (13.30) interactions.

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The complete ‘generic’ Lagrangian of a non-Abelian gauge ﬁeld theory with many distinct fermion ﬁelds then reads " 1 l l μν ! ¯ F + L = − Fμν ψf (iγ μ Dμ − mf )ψf 4 f

.

(13.37)

The sum over f runs over all fermions. The fact that the coupling constant appears both in the fermion–gauge ﬁeld coupling and in the gauge ﬁeld–gauge ﬁeld coupling has an important consequence: since there is a common gauge ﬁeld for all fermions, all these diﬀerent fermions must couple with the same g to the gauge ﬁelds. This universality is a special property of non-Abelian gauge ﬁeld theories. An example for the relevance of a non-Abelian gauge ﬁeld theory is provided by Quantum Chromodynamics (QCD), the theory of the strong interactions between quarks. Here the relevant internal degree of freedom of the quarks is color and the symmetry group is SU (3). Another example is provided by the theory of the electroweak interaction where the gauge group is U (1) × SU (2); in this case the relevant degrees of freedom are the electrical charge (U (1)) and the weak isospin (SU (2)). Both of them will be discussed in some more detail in Chapt. 14. Abelian gauge field theories. For the case of QED U (x) = e−i(x) , T k = 1 and f lmn = 0; thus U is Abelian. Indeed (13.31) then reduces to the well known gauge transformation of electrodynamics 1 Aμ → Aμ = Aμ + ∂μ g

(13.38)

and also the ﬁeld tensor assumes its well known form. The case of QED is thus contained as a special case in the developments in this section. Since in this case there is no coupling constant g in the ﬁeld tensor F , universality does not hold for non-Abelian gauge ﬁeld theories.

13.3

Path integrals

We now want to develop a path integral formulation also for non-Abelian ﬁelds. We start again with the naive expression for a path integral over gauge ﬁelds (13.39) Z[0] = DA eiS[A] . Here the action is that of the original theory without any gauge-ﬁxing term

S=

d4x L

(13.40)

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165

and the integration measure DA stands for DA =

2 −1 4

N

x

j

d Alμ (x, tj )

(13.41)

l=1 μ=1

(cf. (5.11)). This integral, of course, diverges as in the case of the Abelian theory because, as discussed in the last section, the path integral runs over all ﬁeld conﬁgurations, i.e. both over essentially distinct ones and those that diﬀer only by gauge transformations from each other. It is then tempting to just add a gauge-ﬁxing term to L as we have done in (13.21). While this problem of the gauge-equivalent potentials appears for both Abelian and non-Abelian gauge ﬁeld theories, there is another diﬃculty that is speciﬁc for non-Abelian theories. The path integral in the form given above, in which an integration only over the ﬁelds appears, was shown in chapter 1 to be valid only for quadratic Hamiltonians with constant coeﬃcients in front of the kinetic term. This, however, is no longer the case for a non-Abelian gauge theory, where the ﬁeld tensor F depends not only on the derivatives of the ﬁelds, but also on the ﬁelds themselves. Eﬀectively this leads to a kinetic energy term in which the coeﬃcient of the momentumdependent terms depends on the ﬁelds themselves. When considering the nonrelativistic analogon of this case in section 1.3.2 we have started from the Hamiltonian formulation of the path integral. There we have found that a new factor involving the squareroot of the coordinate dependent coeﬃcient appears under the path integral (see (1.54)). It is therefore natural that in the treatment of non-Abelian gauge ﬁeld theories we would also have to start from the Hamiltonian representation of the path integral (1.32) (see [Lee] for a detailed treatment). At the expense of some mathematical rigor, however, the concept of a path integral over the ﬁelds only can be maintained and this is what we are going to do in this chapter. In the following we will now develop a method to split the path integral up into two factors, one containing the divergent part and the other one being convergent. The divergent part will be seen to be an inﬁnite constant that drops out when we work with the normalized functional. The divergence is here – as in the Abelian case – connected with the presence of zero eigenvalues of the matrix in the quadratic term of the gauge ﬁeld action. It is just those zero eigenvalues that we have to factor out by integrating only over those ﬁelds that correspond to nonzero eigenvalues. We start by dividing up the total conﬁguration space of all possible Aμ (x) into equivalence classes. Each of these equivalence classes contains all the possible ﬁelds AUμ that can be obtained from an initial ﬁeld Aμ (x) by gauge transformations. The integrand of (13.39) is constant along the ﬁelds AUμ , i.e. in an equivalence class. This

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166

means that the path integral (13.39) is proportional to an inﬁnite constant which is just the “volume” of the gauge group U . The physically relevant part of the path integral then just runs over the essentially distinct ﬁelds Aμ (x). This restriction can be achieved by requiring that the ﬁelds that contribute to the integration all fulﬁll a gauge condition which we write in the form (13.42) F l (Akμ (x)) = 0 . Examples of such gauge conditions are given by the Lorentz gauge Coulomb gauge Axial gauge Temporal gauge

Fk Fk Fk Fk

= ∂ μ Akμ = 0 ·A k = 0 =∇ = Ak3 = 0 = Ak0 = 0 .

(13.43)

The challenge is now to include these conditions in the path integrals such that gauge-invariance of the theory is maintained. With this aim in mind we now deﬁne a functional integration over the elements of the symmetry group SU (N ), determined by the transformation kT k

U (x) = e−i

,

(13.44)

and denote the integration measure by DU ; in Fig. 13.1 this integration over the gauge group corresponds to an integration along the solid lines. This integration measure is gauge invariant DU = D(U U ) = DU

(13.45)

because of the group-property of the transformations U ; this can also be seen by noting that we integrate over all gauge transformations U . To factor out an integral over DU is the aim in the following paragraphs. With this integration measure we now consider the integral M−1 [Alμ ] =

U

DU δ[F l (Akμ )]

(13.46)

Here F l (Akμ ) = 0 is a gauge-ﬁxing condition (13.42). The superscripts l, k stand for internal degrees of freedom of the symmetry group SU (N ); for notational convenience, we will write them out only when necessary. The δ-functional in (13.46) is really a product of Dirac δ-functions at each spacetime point (x, tj ). AU is the gauge-transformed ﬁeld. Since the δ-functional in (13.46) restricts the gauge transformations to U ≈ 1 it suﬃces to consider only inﬁnitesimal gauge transformations U (x) ≈ 1 − il (x)T l

with 1 .

(13.47)

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In this case the measure can be written as −1

N

2

DU =

dl (x, tj )

(13.48)

x,j l=1

in a discrete space-time representation. For the measure over inﬁnitesimal transformations in (13.48) we have U = U U ≈ (1 − i T l )(1 − il T l ) ≈ 1 − i( + l )T l l

l

= 1 − i T l , l

(13.49)

so that the Jacobian from to = + is

δl det δ k

=1.

(13.50)

Thus the measure (13.48) is indeed gauge-invariant.

U

AUμ

Aμ Aμ(x) Figure 13.1: Lines of constant integrands. The horizontal axis contains the physically distinct ﬁelds, the vertical one the gauge-transformation degree of freedom. The solid lines depict the ﬁelds within a given equivalence class; the dashed line represents the gauge condition F (AUμ ) = 0 for an Abelian theory, the dotted line for a non-Abelian theory exhibits the appearance of Gribov copies for large ﬁelds. The condition F (AUμ ) = 0 then deﬁnes a surface that crosses that of each equivalence class (dashed line in Fig. 13.1); we assume that for an initial ﬁeld

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168

Aμ this is the case for one group element. This is certainly the case for an Abelian gauge ﬁeld theory, like e.g. QED. There, the gauge transformation U reads (cf. (13.38)) 1 Aμ (x) −→ AUμ (x) = Aμ (x) + ∂μ (x) g

(13.51)

and the gauge condition becomes 1 F (AUμ ) = ∂μ Aμ (x) + ∂μ ∂ μ (x) = 0 g

(13.52)

where we have chosen as an example the covariant gauge ∂μ Aμ = 0. Eq. (13.52) represents an inhomogeneous wave equation for . With the boundary condition Aμ (x) → 0 as |x| → ∞ it speciﬁes a unique (x). As Gribov has pointed out this is no longer the case if we are dealing with a non-Abelian ﬁeld theory. In this case there can be several equivalent ﬁelds and all fulﬁll the same gauge condition (open points in Fig. 13.1). Thus, in this case one also has to remove these so-called Gribov copies from the path integration. However, since these copies appear only at large ﬁeld strengths we can neglect them in a perturbative treatment around either a vanishing or a classical ﬁeld conﬁguration. The integral (13.46) is gauge invariant. This can be seen by writing the Jacobian as an integral and starting from a gauge-ﬁeld AU −1

M

[AUμ ]

=

&

DU δ F

U AUμ

'

.

(13.53)

We now change the variables from U to U = U U and get, because the integration measure is gauge-invariant,

M−1 [AUμ ] =

!

DU δ F AUμ

"

= M−1 [Aμ ] ,

(13.54)

since U is just an integration variable that could be renamed into U . M−1 is thus indeed gauge-invariant. We can now transform the original path integral (13.39) by inserting the 1 from (13.46). This gives

Z[0] =

DA eiS[A] =

DA M[Aμ ] 2

!

DU δ F AUμ 34

"

eiS[A]

(13.55)

5

=1

with the action S=

d4x L .

(13.56)

CHAPTER 13. PATH INTEGRALS FOR GAUGE FIELDS

169

We now perform a gauge transformation AUμ → Aμ . Since S, M and DA are invariant under this transformation the integrand no longer depends on U and we can take the integral over U out of the remaining integral

Z[0] =

DU

DA M[Aμ ] δ [F (Aμ )] eiS .

(13.57)

This functional determines our theory in the F (Aμ ) = 0 gauge. The integral

DU

(13.58)

is just the inﬁnite factor we wanted to pull out from the expression. Since this factor is cancelled when normalizing the functional, we can from now on work with the generating functional

Z[J] =

!

DA M[Aμ ] δ F l Akμ

"

ei

d4x (L+Jμl Al μ )

.

(13.59)

This functional can be used for a perturbation theoretical treatment because it is ﬁnite. This was not possible for the original functional (13.39) which diverges because of the integration over physically equivalent gauge ﬁelds.

13.3.1

Gauge Fixing

For a derivation of the perturbation theory rules, i.e. the Feynman rules, the functional (13.59) is still not easy to handle because of the presence of the gauge-ﬁxing δ-functional. As a guideline for a simpliﬁcation we can use the easier example of an Abelian Theory, treated in Sect. 13.1. There we had succeeded in (13.21) to write down a meaningful generating functional that did not contain such a δ-functional, but instead a gauge-ﬁxing term in the Lagrangian. Taking this result as guideline we, therefore, in the following paragraphs reformulate Z[J] into a more tractable form. To do this we consider somewhat more specialized gauge conditions F m (Aμ (x)) − C m (x) = 0 ,

(13.60)

where C m (x) is an arbitrary function of space-time. Note that for given A, C m and F m this condition determines a gauge-transformation. The deﬁnition of M[A] changes correspondingly −1

M [Aμ ] =

!

"

DU δ F m (AUμ (x)) − C m (x) .

(13.61)

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170

Suppose, we looked at a ﬁeld for which the argument of the δ-functional with given C m (x) vanishes and then looked at this argument again, but with a diﬀerent function C˜ m (x), so that the argument no longer vanishes. We could then ﬁnd a gauge transformation that makes the argument of the δfunctional zero again. Since the rest of the integral is invariant under gauge transformations (and we integrate anyway over all gauge transformations) the value of the integral does not change. Thus, the FP determinant M does not change under this replacement and is even independent of the speciﬁc function C m (x)). Then, obviously, also

Z[J] =

DA M[Aμ ] δ [F m (Aμ ) − C m (x)] ei

LJ d4 x

(13.62)

(with LJ = L + J · A) is independent of C(x). We are thus free to change the integrand by a weighting factor e− 2λ i

[C m (x)]2 d4x

(13.63)

and integrate functionally over the function C; this will only change the normalization of the integral. We then get

Z[J] =

DC

1 2 4 DA M[Aμ ] δ [F m (Aμ ) − C m (x)] ei (LJ − 2λ (C(x)) ) d x .

(13.64) Now the functional integration over C can be performed since C appears in the δ-functional. This gives

Z[J] =

1 m 2 4 DA M[Aμ ] ei (LJ − 2λ [F (Aμ )] ) d x .

(13.65)

In this form L has again picked up a ‘penalty potential’ (see discussion after (13.14) on p. 159)). This generating functional is ﬁnite due to the presence of the gauge ﬁxing term and can, therefore, be used for a derivation of the Feynman rules for perturbation theory. As we have seen in the last section for the case of QED the divergence of the original path integral was linked to the zero eigenvalues of the matrix in the quadratic part of the gauge-ﬁeld action so that no one-particle propagator could be deﬁned. The vanishing eigenvalues were connected to the inﬁnite integration over the gauge transformation degree of freedom. Here now this inﬁnity has been removed by ﬁxing the gauge and the propagators are ﬁnite. Notice that the appearance of the gauge-ﬁxing term in the action is not the only change. In a non-Abelian gauge-ﬁeld theory an additional factor, the FP determinant M, appears in the integrand of the path integral. In non-Abelian theories the FP determinant in general depends on the ﬁelds and

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171

thus cannot be pulled out of the path integral. The deeper reason for this diﬀerence of Abelian and non-Abelian theories lies in the diﬀerent structure of the ﬁeld tensors in both theories. Due to the presence of the selﬁnteraction term in the non-Abelian ﬁeld tensor (13.35) the kinetic part of the ﬁeld energy becomes ﬁeld-dependent and ﬁelds and conjugate momenta no longer decouple in the Hamiltonian. This is then exactly the situation that we have encountered already in Sect. 1.3.2 in the framework of nonrelativistic quantum mechanics where a coordinate dependent term also led to an additional factor in the path integral (cf. (1.54)). While everything else in the integrand of the path integral (13.65) is gauge invariant, the source term and the gauge-ﬁxing terms break this invariance. Thus, the speciﬁc rules of the perturbation theory, e.g. the Feynman rules, will depend on the gauge. In particular also the Green’s functions obtained as functional derivatives of Z[J] are not gauge invariant. This, however, is no problem since these Green’s functions only appear at intermediate stages of the calculation. The S matrix elements, which are physical quantities, are then gauge invariant again. We have seen an example for that in the last section where we discussed the QED propagators; in calculating the S matrix the gauge-dependent propagators are always contracted with conserved currents and the gauge-dependent terms then drop out.

13.4

Feynman Rules

13.4.1

Faddeev-Popov Determinant

It now remains to evaluate M−1 [Aμ ]. In order to understand the meaning of the integral in (13.46) better and to be able to actually evaluate it we write it for n discretized space coordinates3 M−1 n =

d1 d2 . . . dn δ n [F (Aμ )]

(13.66)

with n = (xn ). In order to evaluate this integral we now change the integration variables from dn to dF . This gives d1 d2 . . . dn det Mij = dF (x1 ) dF (x2 ) . . . dF (xn ) ,

(13.67)

where det M is the Jacobian for this transformation of the integral measure. We thus get M−1 n

= 2

dF (x1 ) dF (x2 ) . . . dF (xn ) δ n [F ] (det Mij )−1 . 34

5

=1 3

To facilitate the notation we drop here the group indices.

(13.68)

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172

The integral is identically equal to 1 so that Mn is just the Jacobian

δF (xi ) Mn = det Mij = det δ(xj )

.

(13.69)

F =0

We now generalize these equations to the continuum case. Since M[Aμ ] is gauge invariant we can always choose a ﬁeld Aμ that fulﬁlls the condition F (Aμ ) = 0. This allows us to replace the condition F = 0 for the derivative in (13.69) by the simpler condition U = 1. We thus have to consider only inﬁnitesimal gauge transformations U ≈ 1 − il T l

(13.70)

for the evaluation of the derivative. The gauge condition F (A) can then be Taylor-expanded

F m (AUμ (x)) = 0 +

d4y M ml (x, y)εl (y) + O(ε2 )

with

M ml (x, y) =

(13.71)

δF m AUμ (x) δl (y)

|=0 .

(13.72)

We thus get for M−1 at a ﬁeld close to one fulﬁlling the gauge condition M−1 [Aμ ] =

!

"

DU δ F m AUμ

=

DU δ

&

'

d4y M ml (x, y)l (y) . (13.73)

For these integrations close to U = 1 we can use the integration measure (13.48) and obtain (see (13.69)) −1

!

"−1

M [Aμ ] = det M (x, y) ml

δF m (Aλ (x)) = det δl (y)

−1

.

(13.74)

=0

The determinant here, the so-called Faddeev-Popov (FP) determinant, has to be calculated with respect to both the space-time indices (x, y) and the N 2 − 1 SU (N ) group indices (m, l). Note that the FP determinant depends in general on the gauge ﬁelds Akμ . We can now replace M[Aμ ] in (13.59) by the Faddeev-Popov determinant (13.74) and obtain as the generating functional

Z[J] =

DA det M ml (x, y) δ[F l (Aμ )] ei

LJ d4x

(13.75)

with LJ = L + J · A. This generating functional is the central result of this chapter. The presence of the FP-determinant in (13.75) is a typical feature of non-Abelian gauge ﬁeld theories.

CHAPTER 13. PATH INTEGRALS FOR GAUGE FIELDS 13.4.1.1

173

Explicit forms of the FP determinant

Abelian fields. We have seen earlier that the non-Abelian gauge ﬁeld theory contains the Abelian U (1) symmetry group as a special case. We therefore start by analyzing M for an Abelian gauge ﬁeld theory such as QED. We obtain from the deﬁnition (13.72)

M (x, y) =

.

δF AUμ (x) .. . . .

δ(y)

=

.

δF Aμ (x) + g1 ∂μ (x) .. . . .

δ(y)

F =0

.

(13.76)

=0

Choosing, for example, the covariant gauge ∂μ Aμ = 0 gives M (x, y) =

1 2δ(x) 1 = 2δ 4 (x − y) . g δ(y) g

(13.77)

Thus, M is independent of Aμ , and therefore its determinant can be pulled out of the path integral in (13.75), changing only the normalization. The generating functional thus reduces to

Z[J] =

DA δ [F (Aμ )] ei

d4 x LJ

.

(13.78)

In this form the functional contains the free Lagrangian without any gaugeﬁxing terms. It is nevertheless ﬁnite because of the presence of the δfunctional. Note the diﬀerence of this functional to that written down in (13.21); we will discuss this diﬀerence in the next Section. Non-Abelian fields. For a non-Abelian ﬁeld the ﬁrst term in the gauge transformation (13.31) has a more complicated form so that it will in general contribute an Aμ -dependent factor. We start by again expanding F (AUμ ) around a ﬁeld that fulﬁlls the gauge condition, i.e. F m (Aμ ) = 0 and U ≈ 1 (cf. (13.71)). Under Aμ → AUμ = Aμ + δAμ the condition F changes into

F

m

(AUμ )

δF m (Aλ (x)) k δAμ (z) δAkμ (z) 1 4 δF m (Aλ (x)) = dz (Dμ (z))k g δAkμ (z) = 0+

d4z

(13.79)

Here we have used (13.33) to replace δA with the covariant derivative Dμ = ∂μ + igT n Anμ .

(13.80)

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174

Taking now the derivative according to (13.72) we get .

δF m (Aλ (x)) .. . M (x, y) = . δl (y) =0 m 1 δF (Aλ (x)) δ = d4z (Dμ (z))k k l g δAμ (z) δ (y) ml

.

=

. 1 4 δF m (Aλ (x)) kl . 4 . dz (z)δ (z − y) . (13.81) D μ . g δAkμ (z) F =0

This is a more general form for the FP-matrix. Its actual value depends on the gauge used. Lorentz gauge. The Lorentz gauge is given by ∂ μ Anμ = 0 .

(13.82)

The FP-matrix is then obtained from (13.81) as

.

. λ m . 1 4 δ ∂ Aλ (x) kl ml 4 Dμ (z)δ (z − y).. dz M (x, y) = k g δAμ (z) . .

=

(13.83) F =0

. 1 4 mk μ 4 . dzδ ∂x δ (x − z) Dμkl (z)δ 4 (z − y).. . g F =0

By using ∂xμ δ 4 (x − z) = −∂zμ δ 4 (x − z) and partial integration we obtain .

ml

M (x, y) =

. 1 4 mk 4 d z δ δ (x − z)∂zμ Dμkl (z)δ 4 (z − y)... g F =0 .

=

. 1 μ ml ∂x Dμ (x)δ 4 (x − y) ... . g F =0

(13.84)

Since we have to evaluate this expression for F m = ∂ μ Am μ = 0 we can interchange the order of the two derivatives and obtain M ml (x, y) =

1 ml μ 4 Dμ ∂x δ (x − y) . g

(13.85)

For the case of QED this expression indeed reduces to the form already found earlier (see (13.77)). This is so because in this case δAμ = g1 ∂μ in (13.79) and thus in this case instead of Dμ simply ∂μ appears in (13.85).

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175

Axial gauge. It is interesting to note that the FP-determinant is independent of the ﬁelds Aμ not only for Abelian gauge theories, but also in a non-Abelian theory, if the so-called axial gauge is chosen. This axial gauge is deﬁned by the constraint n·A=0 , (13.86) where n is a ﬁxed four-vector; this form comprises the last two cases in (13.43). In this case we obtain from (13.81) M ml (x, y) =

1 g

d4z

. . . kl 4 Dμ (z)δ (z − y).. .

δ nλ Am λ (x) δAkμ (z)

F =0

.

. 1 mk μ 4 4 . = d z δ (x − z) δ kl ∂μz + ig (T n )kl Anμ (z) δ 4 (z − y).. δ n g F =0 1 ml μ x 4 = (13.87) δ n ∂μ δ (x − y) , g

because F = nμ Anμ = 0. Thus in this case the FP-determinant is again independent of Aμ so that it can, as in the Abelian case, be pulled out of the path integral and put into the normalization.

13.4.2

Ghost fields

We now continue with the discussion of the general non-Abelian case (13.65). The trick now is to use (11.56) to express the Faddeev-Popov determinant by another path integral over (hypothetical) Grassmann ﬁelds η(x). We have

det(iM ) =

D¯ η Dη e−i

η¯m (x)M ml (x,y)η l (y) d4xd4y

.

(13.88)

Here the Grassmann ﬁelds η and η¯ carry the indices of the internal symmetry group SU (N ). The generating functional of the theory then becomes (cf. (13.65))

Z[J] =

=

1 m 2 4 DA det(M ) ei (LJ − 2λ [F (Aμ )] )d x

1 m 2 DA D¯ η Dη ei (LJ − 2λ [F (Aμ )] −

η¯M η d4y )d4x

.

(13.89)

If we remember that the FP determinant owes its existence to the gauge ﬁxing condition, then we see that the introduction of the ﬁelds η(x) amounts to a dynamical treatment of the gauge invariance. We therefore expect that this treatment generates also new Feynman graphs involving the ﬁelds η(x). In order to be able to calculate their Green’s functions we now supplement this generating functional with sources also for the so-called ”ghost ﬁeld” η

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176

and obtain for the generating functional of the full theory in the gauge-ghost sector eﬀ 4 ¯ η Dη ei LJ d x (13.90) Z[J, ξ, ξ] = DA D¯ with Leﬀ J (x) = L(x) − −

1 m [F (Aμ )]2 2λ

(13.91)

η¯(x)M η(y) d4y + Jμn (x)Anμ (x) + ξ¯n (x)η n (x) + η¯n (x)ξ n (x) .

The second term here is the gauge-ﬁxing term. As stressed earlier the gaugeﬁxing term breaks the gauge-invariance and thus causes the path integral to be well deﬁned. Note that this gauge ﬁxing term is not enough to specify the gauge for non-Abelian theories. In these the FP determinant appears in addition and manifests itself in the presence of the ﬁctitious “ghost ﬁeld”, introduced in the third term in (13.91). Since the ghosts carry the indices of the underlying symmetry group SU (N ) there are as many ghost ﬁelds as there are gauge ﬁelds. Because in general M = M (A) they couple to the gauge ﬁelds, and only to them. The last three terms are the source terms for the gauge ﬁelds and the ghost ﬁelds, respectively. The ghost ﬁeld η is clearly unphysical. It is a Lorentz-scalar, but anticommuting ﬁeld and has thus the ”wrong” spin-statistics relation. Ghosts can thus appear only on internal lines in a Feynman diagram. As we have pointed out above, there is no need to introduce a ghost ﬁeld when we are dealing with an Abelian gauge theory. In this case M is independent of the gauge ﬁeld Aμ and the ghost-ﬁeld η is totally decoupled from Aμ ; it can, therefore, be left out from the beginning in this case. It is worthwhile to remember that the same situation is encountered in a nonAbelian theory if we work in the axial gauge. In this gauge, however, the propagator is rather complicated because it involves explicitly the (arbitrary) vector nμ .

13.4.3

Feynman Rules

Equations (13.90) and (13.91) give the ﬁnal result for the generating functional of a non-Abelian gauge-ﬁeld theory. The generating functional now has the same form as the one obtained earlier for QED (cf. (13.21)), except for the presence of the ghosts which are a typical feature of non-Abelian gauge ﬁeld theories. If the system under study contains also physical fermions, then the Lagrangian would have to be supplemented by a fermion term along the lines

CHAPTER 13. PATH INTEGRALS FOR GAUGE FIELDS

177

developed in Chap. 11 together with the proper source terms. The fermiongauge ﬁeld coupling is mandated by local gauge invariance (cf. (13.30)) and appears through the covariant derivative. Since the generating functional (13.90) is ﬁnite and free of the overcounting of gauge ﬁelds discussed at the start of this chapter it can be taken as a starting point to derive the Feynman rules for non-Abelian gauge ﬁeld theories. Gauge field couplings. For the gauge ﬁeld and its couplings we obtain the Feynman-rules by isolating the selﬁnteraction terms in L. We get 1 l lμν F L = − Fμν 4 1 n = − ∂μ Alν − ∂ν Alμ − gf lmn Am ∂ μ Alν − ∂ ν Alμ − gf lop Aoμ Apν μ Aν 4 1 = − ∂μ Alν − ∂ν Alμ ∂ μ Alν − ∂ ν Alμ 4 1 lmn m n μ lν + gf Aμ Aν ∂ A − ∂ ν Alμ 2 1 n lop oμ pν − g 2 f lmn Am A A . (13.92) μ Aν f 4 The ﬁrst term alone just looks like the Lagrangian of an Abelian ﬁeld. It is the only term quadratic in the ﬁelds, so that the gauge boson propagator is just that of the photon found in (13.19), except for the additional SU(N) labels. We thus have for the gauge boson two-point function (cf. (13.24)) 1) μ l

k

ν −iδ lm kμ kν lm = 2 gμν + (λ − 1) 2 = iDμν m k + i k

.

(13.93) Note that D has to be diagonal in the intrinsic quantum numbers because L involves a trace over the group indices. The second and third line in (13.92) give the cubic self-coupling terms of the gauge ﬁeld. The cubic term 1 lmn m n μ lν n μ lν gf Aμ Aν ∂ A − ∂ ν Alμ = gf lmn Am μ Aν ∂ A 2

(13.94)

can be represented by the graph shown in Fig. 13.2. Its Feynman rules can be obtained by considering the generating functional of lowest order in the

CHAPTER 13. PATH INTEGRALS FOR GAUGE FIELDS

178

coupling constant g. Quite analogous to the situation in the φ4 model (cf. section 9.1, (9.8)) the only term of Z that contributes to the three-point Green’s function is that of order O(J 3 ); it is obtained by just replacing the ﬁelds by those generated by the external source. This gives Z[J] = −igf

lmn

×∂

mk Dμλ (x − y )J kλ (y )d4 y

nk Dνλ (x − y )J kλ (y )d4 y

Dlk νλ (x − y )Jλk (y )d4 y

μ

+ terms of lower order in J d4x

× exp −

i jκ jk J (x )Dκλ (x − y )J kλ (y )d4x d4y 2

.

(13.95)

Here the Latin superscripts refer to the intrinsic SU (N ) degrees of freedom, the Greek subscripts and superscripts are those of the Lorentz group. The three-point Green’s function can now be obtained by diﬀerentiating Z[J] Ghij πρσ (x1 , x2 , x3 ) =

3 1

i

.

. δ3Z . . . hπ iρ jσ δJ (x1 )δJ (x2 )δJ (x3 ) .J=0

(13.96)

This gives Ghij πρσ (x1 , x2 , x3 )

= igf

lmn

)

mj ni d x iDμσ (x − x3 )iDνρ (x − x2 )∂ μ iDπlh ν (x − x1 ) 4

,

+ permutations of external legs .

(13.97)

Equation (13.97) becomes in momentum space (with G = iD)

lmn ni μ lh ν Gmj Ghij πρσ (p, q, k) = igf μσ (k)Gνρ (q)(−ip )Gπ (p)

+ permutations of external legs

(13.98)

ii hh = gf hji pμ g νλ Gjj μσ (k)Gνρ (q)Gλπ (p)

+ permutations of external legs . In the Feynman gauge we obtain the gauge ﬁeld three-point function hij Ghij πρσ (p, q, k) = gf

−i −i −i k 2 p2 q 2

× (k − q)π gρσ + (q − p)σ gρπ + (p − k)ρ gπσ

(13.99)

.

CHAPTER 13. PATH INTEGRALS FOR GAUGE FIELDS

179

σ3j

k p π1h q

ρ2i Figure 13.2: Cubic gauge coupling term. The ﬁrst symbols at each gluon line give the Lorentz indices, the second number the lines and the third denote the SU (N ) labels. From this expression the vertex factor can be read oﬀ. We then have the rules 2) The vertex factor for the three-point vertex is

gf hij (k − q)π gρσ + (q − p)σ gρπ + (p − k)ρ gπσ

.

(13.100)

3) The four-point vertex in Fig. 13.3 gives similarly the factor −ig 2 [ f his f jks (gπσ gρτ − gπτ gρσ ) + f hjs f kis (gπτ gρσ − gπρ gστ ) + f hks f ijs (gπρ gστ − gπσ gρτ ) ] . τ 4k

σ3j

π1h

ρ2i

Figure 13.3: Quartic gauge coupling graph.

(13.101)

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180

l

λ

Figure 13.4: Fermion-gauge boson vertex. The fermion line is represented by a solid line. Fermion-gauge coupling. If there are also particles present, then the gauge bosons couple to them through the minimal substitution (13.30) Dμ = ∂μ + igT l Alμ

(13.102)

and the Feynman rules for these couplings can directly be obtained from our considerations in Chaps. 8.3 and 11. In particular, any fundamental fermion ﬁeld couples to the gauge ﬁelds according to the rule 4) The fermion-gauge vertex (Fig. 13.4) for a gauge boson with Dirac index λ and group index l is given by:

−ig (γλ )βα T l

fi

(13.103)

Ghost couplings. We now discuss the rules for handling those terms in the Lagrangian that involve the ghost ﬁeld. The ghost-part of the Lagrangian reads (cf. (13.91) and (13.85)) ¯ = Zghost [ξ, ξ]

=

D¯ η Dη e−i

(¯ η M η)d4x d4y+i

× exp

" i ! m − η¯ (x) Dμml (x)∂xμ δ 4 (x − y) η l (y) d4x d4y g ,

+ source terms =

¯ η ξ) d4x (ξη+¯

D¯ η Dη )

D¯ η Dη

CHAPTER 13. PATH INTEGRALS FOR GAUGE FIELDS )

, " i ! m ml μ l 4 − η¯ (x)Dμ ∂ η (x) d x + source terms g

× exp

=

181

D¯ η Dη )

× exp

−

" i ! m η¯ (x) δ ml ∂μ + ig (T n )ml Anμ (x) ∂ μ η l (x) d4x g ,

+ source terms .

(13.104)

Using the regular representation (13.29) for the generators we get for the exponent (without the source terms) i m η¯ (x)2η m (x) d4x + η¯m (x) −if nml Anμ (x) ∂ μ η l (x) d4x g i m = − η¯ (x)2η m (x) d4x + if lmn η¯m (x)Anμ (x) ∂ μ η l (x) d4x . g √ After rescaling the ghost ﬁeld (η → gη) we are thus left with the following ghost contribution to the generating functional

{. . .} =

¯ Zghost [ξ, ξ]

=

−

D¯ η Dη exp −i

!

(13.105)

d x η¯ 2η − gf 4

l

l

lmn

Anμ

η¯ ∂ η + ξ¯l η l + η¯l ξ l m

μ l

"

.

Note that the ghost ﬁeld couples only to the gauge ﬁeld and not to any of the other ﬁelds, like e.g. fermions, in the theory. Also, we remember that the ghost ﬁelds η and η¯ cannot appear on external lines because the ghosts follow the wrong spin-statistics relation. This ensures that the ghosts are integrated out so that the FP determinant is generated. The Feynman rules for the ghost ﬁelds can now be read oﬀ from (13.105). 6) The ghost propagator is given by k l

m

= δ lm

k2

i . + i

(13.106)

Note that the ghost line carries an arrow; this reﬂects the presence of ghosts and antighosts, because the FP matrix is in general not hermitean. The notation is then such that particles always run with the time axis from bottom to top and antiparticles run against it. More loosely speaking, the ghost line runs from η¯ to η.

CHAPTER 13. PATH INTEGRALS FOR GAUGE FIELDS

182

2i

q σ3j k p 1h Figure 13.5: Ghost-gauge coupling. The wavy line denotes a gauge boson whereas the dashed line denotes the ghost. The ﬁrst symbol on the gauge ﬁeld line (σ) denotes the Lorentz index, the second just numbers the particle and the third (j) gives the SU(N) label. Finally, according to (13.105) the ghost can couple to the gauge ﬁeld as depicted in Fig. 13.5. The vertex factor is obtained by considering the ghost-ghost-gauge boson three-point function. As for the gauge three-point function just discussed it can be obtained by considering the lowest order (in the gauge-ghost coupling) expression ¯ J] = −i Z[ξ, ξ, =

d4x −gf lmn

− igf

1

lmn

i

δ δJ nμ (x)

nk Dμν (x

×∂

μ

1 δ 1 δ ¯ eiS0 [ξ,ξ,J] ∂ μ ¯l m i δξ (x) i δ ξ (x)

4

− y )J (y ) d y kν

ξ¯k Dkm (x − y ) d4 y

Dlk (x − y )ξ k (y ) d4 y

¯

+ terms of lower order in J d4x eiS0 [ξ,ξ,J]

(13.107)

This generating functional determines the following rule for the gauge bosonghost coupling vertex: 7) The gauge boson-ghost vertex (Fig. 13.5) has to be associated with a factor gf hij qσ (13.108) where q is the outgoing momentum of the ghost. The four-momentum conservation (p + k = q) is understood here; it is not written explicitly.

CHAPTER 13. PATH INTEGRALS FOR GAUGE FIELDS In addition, we have, as for real, physical fermions, the rule 8) All ghost loops carry a sign (−1). We also have to remember that ghosts cannot appear on external lines.

183

Chapter 14 EXAMPLES FOR GAUGE FIELD THEORIES The Feynman rules developed in the preceding chapter are of a ‘generic’ nature: they apply to all non-Abelian gauge ﬁeld theories. The particular physics content of a gauge ﬁeld theory is determined by specifying the symmetry group SU (N ). In the preceding chapter we have referred to Quantum Electrodynamics (QED), Quantum Chromodynamics (QCD) and the theory of electroweak Interactions. In this chapter we, very brieﬂy, summarize the physical contents and the Lagrangian of these theories.

14.1

Quantum Electrodynamics

Quantum Electrodynamics is the most simple gauge ﬁeld theory. It describes the interactions between electrons and the electromagnetic ﬁeld, i.e. the Coulomb and Lorentz forces. Its ingredients are discussed at the end of appendix 4 and in Sects. 13.1 and 13.2. It is based on the U (1) gauge group and is thus an Abelian theory. Its Lagrangian is given in (4.58) in appendix 4.

14.2

Quantum Chromodynamics

Quantum Chromodynamics is the theory of the strong interactions. Its foundations, ﬁrst discussed in the 60’s, lie in the observation that the observed hadron spectrum can be understood in terms of quarks, mesons being composed of quark-antiquark pairs and baryons of three quarks; the quarks carry spin 1/2 and a so-called ﬂavor quantum number. For certain baryons, such 184

CHAPTER 14. EXAMPLES FOR GAUGE FIELD THEORIES

185

as the Δ++ , the total wavefunction, i.e. the product of space-part, spin-part and ﬂavor-part, one obtains in such a scheme (ﬂavor SU(3)) a completely symmetric wavefunction for the three quarks, in glaring contradiction to the Pauli principle. This problem was solved by the introduction of a new, additional intrinsic degree of freedom for the quarks, called color, in which the quarks can diﬀer (for a detailed discussion see [MOSEL]). The Pauli principle then requires a completely antisymmetric wavefunction in this new intrinsic space. In experiments that produce mesons, i.e. quark-antiquark pairs, through lepton-annihilation the number of colors has been found to be three. The completely antisymmetric color state has therefore been identiﬁed with the completely antisymmetric singlet state of a color SU(3) group. Gauging this group has led to the non-Abelian theory of Quantum Chromodynamics (QCD). The Lagrangian of QCD is then very simply given by 1 c cμν F + [¯ qf (iγ μ Dμ − mf )qf ] LQCD = − Fμν 4 f

.

(14.1)

Here the fundamental triplets of SU (3)C are given by the quark spinors ⎛

⎞

qrf ⎜ f ⎟ qf = ⎝ qg ⎠ qbf

,

(14.2)

which transform according to the fundamental (= lowest-dimensional) representation of SU (3)C . The indices r, g and b stand for the three colors (red, green and blue), the index f for the ﬂavor. The covariant derivative in (14.1) is given by (cf. (13.30)) Dμ = ∂μ + ig

λc c G 2 μ

.

(14.3)

Here the eight matrices λc are the so-called Gell-Mann matrices, the generators of SU (3), and the ﬁelds Gc represent the eight gauge-ﬁelds, here called the gluon ﬁelds, which form an SU (3) octett. c have the general form of a non-Abelian theory The ﬁeld tensors Fμν (13.35). The gluons thus also interact among themselves. One of the predictions of this theory is, therefore, the existence of so-called glueballs, that consist only of gauge-ﬁeld quanta. The theory, furthermore, has the property of asymptotic freedom, i.e. its running coupling constant (cf. the discussion at the end of Sect. 10.2.2) becomes larger with decreasing gluon momentum. This leads to a quarkquark potential that at small distances is Coulomb-like, but at large distances

CHAPTER 14. EXAMPLES FOR GAUGE FIELD THEORIES

186

increases linearly, and thus provides an explanation for the fact that free quarks and gluons do not seem to exist, i.e. that they are conﬁned inside the hadrons.

14.3

Electroweak Interactions

While the Lagrangian of QCD has the rather simple, generic form (14.1), that of the electroweak interactions is considerably more complicated. Its foundations go back to the early seventies when a gauge-ﬁeld theory was found that comprised both ‘classical’ Quantum Electrodynamics and the phenomena of weak interactions, responsible, for example, for the nuclear β-decay. This theory is based on a product of two gauge-groups, one an Abelian U (1) and the other one a non-Abelian SU (2). For the latter the left-handed parts of the observed fermions, quarks and leptons, ψL =

1 (1 − γ5 ) ψ 2

(14.4)

are assumed to form doublets

Ψk =

νkL ekL

or

ukL dkL

(14.5)

for the left-handed ﬁelds of the kth family of leptons and quarks. The righthanded parts (14.6) Ψk = ekR or qkR are assumed to form SU (2) singlets. In (14.5) we have used generic abbreviations νk νe νμ ντ for , or (14.7) ek e− μ− τ− and

uk dk

for

u d

,

c s

or

t b

.

(14.8)

The U (1) gauge group is very similar to QED, but it is not connected with the electric charge, but instead a so-called weak hypercharge y = 2q −2t3 where q is the electrical charge of the fermion and t3 its weak isospin ± 12 , depending on the position of the particle in the two-component spinors (14.7), (14.8), just as in the usual Pauli-spinors.

CHAPTER 14. EXAMPLES FOR GAUGE FIELD THEORIES

187

The complete Lagrangian of this theory is then given by 1 1 L = − Gμν · Gμν − F μν Fμν 4 4 ϕ 2 1 2 ϕ 2 2 † μ μ + MW W μ W 1 + + MZ Zμ Z 1 + v 2 v ¯ k iγ μ Dμ Ψk + LY Ψ + k

1 1 ϕ 1 ϕ + (∂ μ ϕ) (∂μ ϕ) − m2H ϕ2 1 + + 2 2 v 4 v

(14.9)

2

.

The boldfaced ﬁeld tensors are vectors in the internal space and their product contains a summation over the group indices. Since there are now two gauge groups present, we also have two gauge ﬁelds appearing in the Lagrangian: Gμν for the non-Abelian SU (2)L and Fμν for the Abelian U (1)Y . Thus, also the covariant derivative contains both of these ﬁelds with two diﬀerent coupling constants l

Dμ Ψk = ∂μ + igt

Wμl

1

+ ig yk Bμ Ψk , 2

(14.10)

where yk is the hypercharge quantum number of Ψk and the tl are the generators of SU (2). The three ﬁelds W l and the single ﬁeld B are the gauge ﬁelds for SU (2)L and U (1)Y , resp. By linearly combining them one obtains the 2 physical ﬁelds Wμ and Wμ† , the ﬁeld Zμ as well as the photon ﬁeld Aμ . With the covariant derivative (14.10) the ﬁrst term in the third line describes massless fermions interacting with the gauge ﬁelds in the standard way. The gauge ﬁelds W and B normally have to be massless, since an explicit mass term breaks gauge invariance. Since these ﬁelds transmit the weak interaction the latter would then be long ranged. This, however, presents an immediate problem because the weak interaction, since Fermi’s theory, is known to be very short-ranged. A way out of this diﬃculty is provided by the so-called Higgs mechanism. The main ingredient of this mass generation mechanism is the existence of a scalar Higgs ﬁeld ϕ, yet to be discovered, with non-linear self-couplings (last line in (14.9); these self-couplings lead to a non-vanishing vacuum ﬁeld. The second ingredient is a Yukawa coupling of the gauge-ﬁelds to this Higgs ﬁeld that generates their masses MW and MZ in the same way as described in Sect. 12.3. Similarly, the term LY in (14.9), generates masses for all the fermions through a Yukawa coupling to the Higgs ﬁeld. For the fermions it has the simple structure

ϕ v ¯ 1+ L = −gf √ ΨΨ v 2

,

(14.11)

CHAPTER 14. EXAMPLES FOR GAUGE FIELD THEORIES

188

from which the fermion mass can be read oﬀ as v m f = gf √ . 2

(14.12)

Appendix A Units and Metric A.1

Units

While we work in this book in the ﬁrst 3 chapters, the ’classical’ quantum mechanics part, with the usual system of units, it is customary in elementary particle physics and ﬁeld theories to choose the system of units such that the resulting expressions assume a simpler form. In particular, instead of working with the usual three mechanical units for length, mass and time one introduces instead three basic units for velocity, action and length. The choice of the velocity of light, c, as the unit for the velocity implies that c=1. (A.1) This in turn means that in such a system length and time have the same dimensions and are equivalent units. With c = 1 the relativistic energy– momentum relation assumes the simple form E 2 = p2 + m2 .

(A.2)

Choosing next the unit of action, h ¯ , such that h ¯=1

(A.3)

connects the dimensions of mass, time and length. Since time and length are equivalent, mass assumes the dimension of an inverse length; the same holds consequently for the momentum and the energy (A.2). The choice of 1 fm = 10−13 cm as a length unit, for example, leads to masses, momenta and energies all given in fm−1 . Another often-used unit for energy is that of 1 MeV, i.e. the energy that an electron acquires when it is accelerated by the voltage of 1 MV. The transformation of the expressions back to the standard MKSA system can be achieved by multiplying 189

APPENDIX A. UNITS AND METRIC

190

all quantities with the appropriate combinations of h ¯ and c. A particularly useful relation for this conversion is h ¯ c ≈ 197.3 MeV fm .

A.2

(A.4)

Metric and Notation

All vectors, both ordinary three-component vectors and those in some internal space, are distinguished by boldface italic print in this book. In relativistic expressions four-vectors are always written as

= A0 , A1 , A2 , A3 Aμ = A0 , A

(A.5)

Four-vectors with a superscript are called contravariant vectors. Covariant vectors, denoted by subscripts, are then deﬁned by Aμ = gμν Aν with the metric tensor gμν that space ⎛ 1 ⎜ 0 ⎜ ⎜ ⎝ 0 0

(A.6)

reads in Cartesian coordinates in Minkowski 0 0 0 −1 0 0 0 −1 0 0 0 −1

⎞ ⎟ ⎟ ⎟ . ⎠

(A.7)

The metric tensor with superscripts is used to raise the indices of four-vectors; it is thus the inverse of g and is given by g μν = gμν .

(A.8)

g μν gνλ = gλμ = 1

(A.9)

The product is often also abbreviated by the delta-function δλμ = gλμ .

(A.10)

A scalar product of 2 four-vectors is then deﬁned by ·B A · B = Aμ B μ = Aμ Bμ = A0 B0 − A

(A.11)

if Aμ and B μ are deﬁned by , Aμ = (A0 , A)

. B μ = (B 0 , B)

(A.12)

APPENDIX A. UNITS AND METRIC

191

The invariant square of the four-vector is thus given by 2 . A2 = Aμ Aμ = A20 − A

(A.13)

Here, as always in the text, the Einstein convention of summing over double indices, one lower and one upper, is used. The components of four-vectors are generally labeled by Greek indices, whereas roman indices are used to refer speciﬁcally to the last three (vector) components. The space-time four-vector is xμ = (x0 , x) = (t, x) ,

(A.14)

and the four-momentum is given by pμ = (E, p) .

(A.15)

Finally, the components of the contravariant four-gradient are abbreviated by ∂ ∂ μ , ∂ ≡ = (A.16) , −∇ ∂xμ ∂t so that the four-momentum operator is given by pˆμ = i∂ μ ,

(A.17)

and the Lorentz-invariant d’Alembert operator by ∂ μ ∂μ =

∂2 2=2 . −∇ ∂t2

(A.18)

Other important four-vectors are the current j μ = (ρ, j)

(A.19)

and the electromagnetic potential

. Aμ = φ, A

(A.20)

Appendix B Functionals B.1

Definition

Very generally speaking a functional is a mapping from a space of functions into the real or complex numbers. For example, the integral +∞ −∞

f (x)dx

(B.1)

is a real or complex number that depends on the special function f (x) in the integrand; other functions in general give other values for the integral. This dependence of the integral on the function f is called a functional dependence of the integral which itself is called a functional of f : +∞

F [f ] =

−∞

f (x)dx .

(B.2)

In order to stress that not a special value of f , but instead the whole function f is the argument of the functional square parentheses are used here to show this functional dependence. In physics one often deals with such functionals. For example, in classical mechanics the action ˙ (B.3) S = L(q(t), q(t))dt integrated along a trajectory q(t) plays a special role. The action depends obviously on the trajectory and is thus a functional integral of it: S = S[q(t)]. The Lagrange equations of motion are derived by investigating the changes of S with the trajectory q(t): the physical trajectory leads to a stationary value of S. To ﬁnd this stationary value, and thus the physical trajectory, requires taking the derivative of S[q] with respect to q. We thus need to deﬁne also the so-called functional derivative. 192

APPENDIX B. FUNCTIONALS

B.2

193

Functional Integration

A functional integration can be introduced as an integration over a space of functions df F [f (x)] , (B.4) which is deﬁned as a limit n → ∞ of an integration over ﬁnite n-dimensional subspaces. A typical example is given by the path integral for quadratic Hamiltonians

K (x, t; xi , ti ) = N

=

lim n→∞ ×

B.2.1

m i2π¯ hη

Dx e

i h ¯

t

L(x,x)dt ˙

ti

n+1 2

(B.5) ⎧ ⎨i

n

n

⎡

m ⎣ dxk exp ⎩ η h ¯ j=0 2 k=1

xj+1 − xj η

2

⎤⎫ ⎬ − V (¯ xj )⎦⎭ .

Gaussian integrals

In this book we are often confronted with the evaluation of path integrals over exponential functions whose exponents are quadratic in the coordinates. It is thus useful to generalize the Gaussian integral formula +∞

e

( − 12 ax2

dx =

−∞

2π a

(a > 0)

(B.6)

to this case. From the deﬁnition of the path integral it is obvious that we have to consider products of such integrals

exp −

n 1

2 k=1

ak x2k

(2π)n dx1 dx2 . . . dxn = n √ . ak k=1

(B.7)

We next assume that the n numbers ak are all positive and form the elements of a diagonal matrix A. We thus have n

ak

(B.8)

xk Akk xk = xT Ax ,

(B.9)

det(A) =

k=1

and

n k=1

ak x2k =

n k=1

APPENDIX B. FUNCTIONALS

194

where x is a column vector ⎛ ⎜ ⎜ x=⎜ ⎜ ⎝

⎞

x1 x2 .. .

⎟ ⎟ ⎟ . ⎟ ⎠

xn Thus (B.7) becomes

n

e

− 12 xT Ax

(2π) 2

n

dx=

.

(B.10)

det(A) So far, we have derived this equation only for a diagonal matrix A. It is, however, valid for a more general class of matrices. This can be seen by noting that for each real, symmetric matrix B there exists a real, orthogonal matrix O that diagonalizes B to A OT BO = A

(O real, orthogonal)

(B.11)

B = OAOT

(B real, symmetric) .

(B.12)

or, equivalently

We thus get

e

− 21 yT By

n

dy

=

e

− 12 yT OAOT y n

(2π) 2

=

e− 2 x 1

n

dy=

T Ax

dnx

n

(2π) 2

=

det(A)

,

(B.13)

det(B)

where we have substituted y = Ox and have used the fact that the Jacobian of an orthogonal transformation is 1 (because det(O) = 1). The last step is possible because the determinant of a matrix is invariant under an orthogonal transformation. Equation (B.10) is thus valid also for general symmetric matrices with positive eigenvalues; it can also be shown to hold for complex matrices with positive real parts. This result can also be extended to more general quadratic forms in the exponent. For a one-dimensional integral of such type we have +∞

e−ap

−∞

2 +bp+c

dp =

π b2 +c e 4a , a

(B.14)

APPENDIX B. FUNCTIONALS

195

We now assume an n-dimensional integral over such a form where the exponential is given by 1 T T (B.15) e−F (x) = e−( 2 x Ax+B x+C ) where A is a real symmetric matrix, B is a vector and C a constant. We bring F (x) into a quadratic form by writing 1 F (x) = (x − x0 )T A(x − x0 ) + F (x0 ) 2 where x0 is given by

(B.16)

x0 = −A−1 B

(B.17)

and F0 = F (x0 ) = C − 12 BT A−1 B. Setting now y = x − x0 gives

e

−( 12 xT Ax+BT x+C )

e− 2 y 1

n

dx =

T Ay−F 0

dny

n

(2π) 2

=

e2B 1

T A−1 B−C

.

(B.18)

det(A) Similar to the Gaussian integral (B.7) the following integral relation, which can be proven by induction from n to n + 1, holds also +∞

1

!

dx1 . . . dxn exp iλ (x1 − a)2 + (x2 − x1 )2 + . . . + (b − xn )2

−∞

=

in π n (n + 1)λn

1 2

iλ (b − a)2 exp n+1

"0

.

(B.19)

Complex integrals. We can generalize these formulas now to complex integration by noting that (B.6) can be squared and then be written as π −ax2 −ay2 2 2 dx e dy = e−a(x +y ) dx dy . = e a

(B.20)

Introducing now the complex variable z = x + iy gives π 1 −az∗ z ∗ e dz dz . = a 2i

(B.21)

This can be generalized as before to many coordinates (by replacing orthogonal matrices by unitary ones). We obtain

e

−z† Az

n ∗

n

dz dz=

∗

e−zi Aij zj dnz ∗ dnz =

(2πi)n , det(A)

(B.22)

APPENDIX B. FUNCTIONALS

196

where A is a Hermitian matrix with positive eigenvalues. Another convenient, often used relation in this context is ln det A = tr ln A ,

(B.23)

most easily proven for diagonal matrices. In this relation ln A is deﬁned by its power series expansion ln A = ln(1 + A − 1) = A − 1 −

(A − 1)2 (A − 1)3 + − ... . 2 3!

(B.24)

With the help of this relation we have

B.3

e−z

† Az

dnz ∗ dnz = (2πi)n e−tr

ln A

.

(B.25)

Functional Derivatives

Suppose that F [f ] is a functional of the function f (x). The functional derivative of F is then deﬁned by F [f (x) + εδ(x − y)] − F [f (x)] δF [f (x)] = lim . ε→0 δf (y) ε

(B.26)

For example, let us take F [f ] = 2f (x) ,

(B.27)

where 2 is the d’Alembert operator. We get from the deﬁnition (B.26) δF 1 = lim [2 (f (x) + εδ(x − y)) − 2f (x)] = 2δ(x − y) ε→0 δf (y) ε Another example is

(B.28)

+∞

F [f ] =

f (x)dx .

(B.29)

−∞

According to the deﬁnition just given we have δF 1 = lim ε→0 ε δf (y)

=

[f (x) + εδ(x − y)] dx −

f (x) dx

(B.30)

δ(x − y) dx = 1

A second example is given by F [f ] = ei

f (x)x dx

(B.31)

APPENDIX B. FUNCTIONALS

197

with δF 1 i [f (x)+εδ(x−y)]x dx e − ei f (x)x dx = lim ε→0 ε δf (y) 1 = lim ei f (x)xdx eiεy − 1 = iy ei f (x)x dx . ε→0 ε

(B.32)

A general formula that we will need quite often involves functionals F [J] of the form

F [J] =

dx1 . . .

dxn f (x1 , . . . , xn ) J(x1 )J(x2 ) . . . J(xn )

(B.33)

with f symmetric in all variables. Then the functional derivative with respect to J has the form δF [J] = n dx1 dx2 . . . dxn−1 f (x1 , x2 , . . . , xn−1 , x) δJ(x) × J(x1 )J(x2 ) . . . J(xn−1 ) .

(B.34)

In our later considerations the functional may also be deﬁned by a power series expansion φ[J] =

∞ 1 n=1

n!

dx1 . . . dxn φn (x1 , . . . , xn )J(x1 )J(x2 ) . . . J(xn ) .

(B.35)

Then the functional derivative is given by .

. δ k φ[J] 1 . . = φk (yp1 , yp2 , . . . , ypk ) , δJ(y1 )δJ(y2 ) . . . δJ(yk ) .J=0 k! p

(B.36)

where the sum runs over all permutations p1 , . . . , pk of the indices 1, . . . , k. If we assume that φk is a symmetric function under exchange of any of the coordinates x1 , . . . , xk , then the functional derivative is given by .

. δ k φ[J] . . = φk (y1 , . . . , yk ) , δJ(y1 )δJ(y2 ) . . . δJ(yk ) .J=0

just as in a normal Taylor series.

(B.37)

Appendix C RENORMALIZATION INTEGRALS In Chapt. 10 we encountered divergent integrals in the calculation of higherorder two- and fourpoint functions. To evaluate these integrals analytically in n dimensional Minkowski space is the purpose of this appendix. We start with a discussion of the Gamma function whose properties play a role in the evaluation of the integral and its expansion into n ≈ 4 dimensions. Properties of the Gamma function. The Gamma function is deﬁned by an integral representation Γ(x) =

∞

e−t tx−1 dt ;

(C.1)

0

it is single-valued and analytic everywhere except at the points z = n = 0, −1, −2, . . . where it has a simple pole with residue (−1)n /n!. It can thus be expanded around z = 0 Γ(z) =

1 − γ + O(z) , z

(C.2)

where γ is known as the Euler-Mascheroni constant (γ ≈ 0.577 . . .). Since the Gamma function also obeys the relation Γ(z + 1) = zΓ(z) = z!

(C.3)

we obtain

1 1 Γ(z) ≈ −(1 + z) −γ =− −1+γ . Γ(−1 + z) = z−1 z z 198

(C.4)

APPENDIX C. RENORMALIZATION INTEGRALS

199

It obeys the relation Γ(z) = (z − 1)Γ(z − 1) ≈ (z − 1)

−1 , z+1

(C.5)

where the second part of this equation is correct close to the pole at z = −1. Evaluation of integrals with powers of propagators in n dimensions. The typical integral to be evaluated reads

Il =

1 1 dq dq dq0 2 l = l . 2 2 (q − m + iε) (q0 − q 2 − m2 + iε) n

(C.6)

with one timelike (q 0 ) and n−1 spacelike (q k ) coordinates; the vector symbol denotes a (n − 1)-dimensional vector q = (q 1 , q 2 , . . . , q n−1 ). l is an integer parameter and m2 a real, positive number (mass squared). We ﬁrst consider the integration over q0 . The situation here is exactly as in Sect. 6.1.2. The integrand has its only poles in the second and fourth quadrant of the complex q0 plane at q0 = ± [(q 2 + m2 ) − iδ] (cf. Fig. 5.1). Since the integral behaves as 1/(q0 )2l−1 for large q0 the integration along the q0 -axis can be closed in the lower half of the complex q0 plane without changing the integral’s value. According to Cauchy’s theorem this integration contour can now be changed into one that runs along the imaginary q0 axis and closes the contour in the right half of the complex q0 plane. Because the contour still encloses the same pole (the one in the fourth quadrant of the complex plane) the value of the integral does not change. This then gives the equality +∞ −∞

dq0

+i∞

1 (q02

−

q2

−

m2

l

+ iε)

=

−i∞

dq0

1 (q02

−

q2

l

− m2 + iε)

(C.7)

since in both cases the contribution of the half-circle that closes the contour vanishes. In the integral on the rhs the integration runs along the imaginary axis in the complex q0 plane. On that axis q0 is purely imaginary; the integrand has thus been analytically continued from the originally purely real q0 to a complex one. Using the transformations (6.22), (6.23) qn = −iq0

dn qE ≡ dqn dq = −idn q ,

(C.8)

with real qn we obtain the integral

Il = (−)l i

+∞

dq

−∞

dqn

1 (qE2 +

l m2 )

= (−)l i

dn q E

1 l

(qE2 + m2 )

(C.9)

APPENDIX C. RENORMALIZATION INTEGRALS with

n i 2

qE2 =

q

200

= q 2 + qn2 .

(C.10)

i=1

(C.8) is just the Wick rotation discussed in Sect. 5.1.1. We now introduce “polar coordinates” by deﬁning an n − 1 dimensional solid angle element dΩn by the relation dnqE = qEn−1 dqE dΩn .

(C.11)

and get for the integral1 ∞

l

Il = (−) i

q n−1 dq

dΩn 0

1 . (q 2 + m2 )l

(C.12)

The integral over the solid angle can be analytically performed and yields

dΩn =

2π n/2 Γ

.

(C.13)

n 2

This can be seen by considering the n-th power of the Gaussian integral √ n

π

=

∞

=

0

dΩn

=

dx e

−x2

∞ 0

x

n

dx1 dx2 . . . dxn e−

=

n−1 −x2

e

dx =

1 n dΩn Γ 2 2

-n k=1

x2k

n−2 1 ∞ 2 dΩn d(x2 ) x2 2 e−x 2 0

.

(C.14)

Here the integral representation (C.1) of the Gamma function has been used in the last step. This gives for the integral ∞ n 2π 2

Il = (−) i n Γ 2 l

q n−1 dq

0

1 . (q 2 + m2 )l

(C.15)

The remaining integral can be evaluated with the help of Euler’s β function ∞ Γ(x)Γ(y) = 2 dt t2x−1 (1 + t2 )−x−y , B(x, y) = (C.16) Γ(x + y) 0

1

In order to simplify the notation we are no longer denoting the Euclidean vectors by the subscript E.

APPENDIX C. RENORMALIZATION INTEGRALS

201

(see Abramowitz (6.2.1)); we obtain

Γ 1 n n 1 q n−1 dq = mn−2l B( , l − ) = mn−2l l 2 2 2 2 (q 2 + m2 )

n 2

Γ l−

n 2

. (C.17)

Γ(l)

Thus the complete integral is now given by n

Γ 2π 2 1 Il = (−)l i n mn−2l 2 Γ 2

n 2

Γ l− Γ(l)

n 2

n 2

= (−)l iπ m

Γ l− n−2l Γ(l)

n 2

. (C.18)

In this expression the dependence of the integral on the dimension n is explicit; it can analytically be continued to the physical case n = 4 where it has a pole.

Appendix D GRASSMANN INTEGRATION FORMULA A more general proof than the speciﬁc examples given in section (11.1.2.2) uses the fact that any antisymmetric (n × n) dimensional matrix M can be brought into a block-diagonal form through an orthogonal transformation M −→ M = OT M O ⎛

0 M1 ⎜ 0 ⎜ −M1

⎜ ⎜ ⎜ ⎜ ⎜ M =⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

0 0

0 0

··· ···

M2 · · ·

0

0

0

0 .. .

0 .. .

−M2 .. .

0 .. .

0 0

0 0

0 0

0 0

···

(D.1) 0 0 .. . .. . .. .

··· ··· 0 · · · −Mn

0 0 .. . .. . .. .

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ Mn ⎠

(D.2)

0

for n even (for odd n the matrix M has one more row and one more column with all zeros, so that its determinant vanishes). The form of M shows clearly that (D.3) det(M ) = det(M ) = M12 M22 · · · Mn2 . We thus get

I(n) =

=

−η T M η

dη1 . . . dηn e

=

T M

d1 . . . dn J e−

dη1 . . . dηn e−η

T (OM O T )η

(D.4)

with = OT η and J being the Jacobian. In our special case here, O is orthogonal and therefore, [det(O)]−1 = J = 1. 202

APPENDIX D. GRASSMANN INTEGRATION FORMULA

203

We can now continue with the evaluation of I(n) in (D.4) and get

n n 1 d1 . . . dn (−) 2 n T M 2 , ! 2

I(n) =

(D.5)

because only that term in the expansion of the exponental can contribute in a n-dimensional integral that has exactly n factors of i . In detail, we have n (−) 2

I(n) = n 2

!

d1 . . . dn α Mαβ β

n 2

.

(D.6)

Since M has the special, block-diagonal form given above (D.2), we have = 0, except for α = 2α , β = α − 1 or α = 2α − 1, β = α + 1. Again, Mαβ there can be no two equal s under the integral, if this integral is to be nonzero. We thus have n (−) 2

I(n) =

n 2

d1 . . . dn 2α M2α ,2α −1 2α −1

!

n 2 2 (−)n

=

n 2

!

− 2α −1 M2α −1,2α 2α

d1 . . . dn 2α −1 M2α −1,2α 2α

n 2

n 2

,

(D.7)

since M is antisymmetric. Here the index α still runs from 1 to n/2; n is even so that the phase disappears in the following. We now perform the exponentiation n 22

I(n) = n 2

!

d1 . . . dn

α

2

...

β

34

n sums

ν

2α −1 2α · · · 2ν −1 2ν

5

× M2α −1,2α M2β −1,2β · · · M2ν −1,2ν . M2α −1,2α

(D.8)

· · · M2ν −1,2ν

just gives det(M ). Since The product of the factors the i appear always pairwise, they can be commuted to normal ordering (n · · · 1 ) without sign-change; the sums then give (n/2)! times the same result. Thus I(n) = 2

n 2 n

= 22

d1 . . . dn n · · · 1 det(M ) .

det(M ) (D.9)

APPENDIX D. GRASSMANN INTEGRATION FORMULA

204

The last step is possible because we have for each blockmatrix Mβα separately det(Mβα ) = − (M2α−1,2α ) (+M2α,2α−1 ) = + (M2α−1,2α )2 , and for det(M )

det(M ) =

det(Mβα ).

(D.10)

(D.11)

α

Using (D.3) we thus get the desired result

T M

d1 . . . dn e−

n

= 22

det(M ) .

(D.12)

Equation (D.12) corresponds to (B.10) for the boson case. Note that here – in contrast to the bosonic case – the determinant appears in the numerator!

Appendix E BIBLIOGRAPHY We list here a number of textbooks that provide for more information on the topics treated in this manuscript. Introductory texts 1. J.J. Sakurai, Modern Quantum Mechanics, Addison-Wesley, Redwood City, 1985; contains a very nice introduction into the non-relativistic path integral formalism. 2. U. Mosel, Fields, Symmetries, and Quarks, 2nd revised and enlarged edition: Springer, Heidelberg, 1999; contains a short introduction into fundamentals of ﬁeld theories and gauge ﬁeld theories. 3. T.D. Lee, Particle Physics and Introduction to Field Theory, Harwood, Chur, 1981; very physical treatment of ﬁeld theory and high-energy phenomenology, somewhat annoying typesetting. 4. L.H. Ryder, Quantum Field Theory, Cambridge University Press, 1985; very didactical presentation of modern ﬁeld theories. 5. C. Itzykson and J-B. Zuber, Quantum Field Theory, McGraw-Hill, New York, 1985; nearly comprehensive book on ﬁeld theory, didactically not very well done. 6. M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, Reading, 1995; also didactically very good, rather comprehensive book on quantum ﬁeld theory. The new standard book on this topic!

205

APPENDIX E. BIBLIOGRAPHY

206

Specialized Literature 1. Ta-Pei Cheng and Ling-Fong Li, Gauge Theory of elementary particle physics, Clarendon Press, Oxford, 1984; quite comprehensive book, well-structured, easy to read, but not always complete in its arguments and derivations. 2. P. Ramond, Field Theory, a modern Primer, Addison-Wesley, Redwood City, 1990; very nice introduction to ﬁeld theories, also modern aspects, builds entirely on path integral formalism. 3. P.H. Frampton, Gauge Field Theories, Benjamin-Cummings, Menlo Park, 1987; bad typesetting, not easy to read. 4. S. Pomorski, Gauge Field Theories, Cambridge University Press, 1987; not very didactical, but quite deep. 5. D. Bailin and A. Love, Introduction to Gauge Field Theory, 2nd ed., Hilger, Bristol, 1994; very comprehensive treatment of gauge ﬁeld theories based on path integral methods from the start, didactical, contains modern aspects of ﬁeld theory beyond Standard Model. 6. C. Grosche and F. Steiner, Handbook of Feynman Path Integrals, Springer, Berlin, 1998; compact presentation of theory of path integrals and their history. Unique in its description of evaluation techniques and its tables of analytically calculable path integrals. Complete list of references. 7. H. Kleinert, Path Integrals in Quantum Mechanics, Statistics and Polymer Physics, World Scientiﬁc, Singapore 1995. Application of path integrals to wide range of physics problems. 8. G. Roepstorﬀ, Path Integral Approach to Quantum Physics, Springer, Heidelberg, 1996

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