Commun. Math. Phys. 291, 347–356 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0791-7

Communications in

Mathematical Physics

Proof of the Julia–Zee Theorem Joel Spruck1 , Yisong Yang2 1 Department of Mathematics, Johns Hopkins University, Baltimore,

MD 21218, USA. E-mail: [email protected]

2 Department of Mathematics, Yeshiva University, New York, NY 10033, USA.

E-mail: [email protected] Received: 7 October 2008 / Accepted: 24 December 2008 Published online: 1 April 2009 – © Springer-Verlag 2009

Abstract: It is a well accepted principle that finite-energy static solutions in the classical relativistic gauge field theory over the (2 + 1)-dimensional Minkowski spacetime must be electrically neutral. We call such a statement the Julia–Zee theorem. In this paper, we present a mathematical proof of this fundamental structural property. 1. Introduction Consider the Maxwell equations ∂ν F µν = −J µ

(1.1)

defined over a Minkowski spacetime of signature (+ − · · · −), where Fµν = ∂µ Aν − ∂ν Aµ

(1.2)

is the electromagnetic tensor induced from the gauge vector field Aµ , µ = 0 designates the temporal index, µ = i, j, k denote the spatial indices, and J µ = (J 0 , J i ) = (ρ, j) is the current density in which ρ = J 0 expresses the electric charge density. As spatial vector fields, the electric field E = (E i ) and magnetic field B = (B i ) are given by F 0i = −E i ,

F i j = −εi jk B k .

(1.3)

In view of (1.3), the µ = 0 component of (1.1) relating electric field and charge density reads div E = ∂i E i = ρ,

(1.4)

which is commonly referred to as the Gauss law (constraint). In the static situation, we have E i = −F 0i = F0i = −∂i A0 .

(1.5)

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Thus, a trivial temporal component of the gauge field, A0 = 0,

(1.6)

implies that electric field is absent, E = 0. The condition (1.6) is also known as the temporal gauge condition, which makes the static solution electrically neutral. In their now classic 1975 paper [10], Julia and Zee studied the Abelian Higgs gauge field theory model. Using a radially symmetric field configuration ansatz and assuming a sufficiently fast decay rate at spatial infinity, they were able to conclude that a finite-energy static solution of the equations of motion over the (2 + 1)-dimensional Minkowski spacetime must satisfy the temporal gauge condition (1.6), and thus, is necessarily electrically neutral. This result, referred to here as the Julia–Zee theorem, leads to many interesting consequences. For example, it makes it transparent that the static Abelian Higgs model is exactly the Ginzburg–Landau theory [6] which is purely magnetic [9,12]. Since the work of Julia and Zee [10], it has been accepted [4,7,8,11,13,20] that, in order to obtain both electrically and magnetically charged static vortices, one needs to introduce into the Lagrangian action density the Chern–Simons topological terms [2,3], which is an essential construct in anyon physics [22,23]. See also [5]. On the other hand, it is well known that electrically and magnetically charged static solitons, called dyons by Schwinger [17] (see also the related work of Zwanziger [24, 25]), exist as solutions to the Yang–Millis–Higgs equations over (3 + 1)-dimensional spacetime [10,14,16]. Therefore, the Julia–Zee theorem is valid only in (2 + 1) dimensions. The importance of the Julia–Zee theorem motivated us to carry out this study. In Sect. 2, we make a precise statement of the Julia–Zee theorem in the context of the original Abelian Higgs model and present a rigorous proof. In Sect. 3, we extend the Julia–Zee theorem to the situation of a non-Abelian Yang–Mills–Higgs model. In Sect. 4, we prove a non-Abelian version of the theorem. Fortunately our method works almost exactly as in the simpler Abelian Higgs model. In Sect. 5, we consider further extensions and applications of our results.

2. The Julia–Zee Theorem Recall that, in normalized units, the classical Abelian Higgs theory over the (2 + 1)dimensional spacetime is governed by the Lagrangian action density 1 1 L = − Fµν F µν + Dµ φ D µ φ − V (|φ|2 ), 4 2

(2.1)

where D µ φ = ∂ µ φ + iAµ φ defines the gauge-covariant derivative, φ is a complex scalar (Higgs) field, the spacetime indices µ, ν run through 0, 1, 2, the spacetime metric takes the form η = (ηµν ) = diag(1, −1, −1), which is used to lower and raise indices, and V ≥ 0 is the potential density of the Higgs field. The associated equations of motion are Dµ D µ φ = −2V  (|φ|2 )φ, ∂ν F µν = −J µ ,  i
µ φ D φ − φ Dµφ . Jµ = 2

(2.2) (2.3) (2.4)

Proof of the Julia–Zee Theorem

349

In the static situation, the operator ∂0 = 0 nullifies everything. Hence the electric charge density ρ becomes ρ = J0 =

i (φ D 0 φ − φ D 0 φ) = −A0 |φ|2 2

(2.5)

and a nontrivial temporal component of the gauge field, A0 , is necessary for the presence of electric charge. On the other hand, the µ = 0 component of the left-hand side of the Maxwell equation (2.3) is ∂ν F 0ν = ∂i (Fi0 ) = ∂i2 A0 = A0 .

(2.6)

Consequently, the static version of the equations of motion (2.2)–(2.4) may be written as Di2 φ = 2V  (|φ|2 )φ − A20 φ,  i
φ Di φ − φ Di φ , ∂ j Fi j = 2 A0 = |φ|2 A0 ,

(2.7) (2.8) (2.9)

in which (2.9) is the Gauss law. Moreover, since the energy-momentum (stress) tensor has the form  

Tµν = −ηµ ν Fµµ Fνν  +

 1
Dµ φ Dν φ + Dµ φ Dν φ − ηµν L, 2

(2.10)

the Hamiltonian density is given by H = T00 =

1 1 1 1 |∂i A0 |2 + A20 |φ|2 + Fi2j + |Di φ|2 + V (|φ|2 ), 2 2 4 2

so that the finite-energy condition reads  H dx < ∞. R2

(2.11)

(2.12)

With the above formulation, the Julia–Zee theorem may be stated as follows. Theorem 2.1 (The Julia–Zee Theorem). Suppose that (A0 , Ai , φ) is a finite-energy solution of the static Abelian Higgs equations (2.7)–(2.9) over R2 . Then either A0 = 0 everywhere if φ is not identically zero or A0 ≡ constant and the solution is necessarily electrically neutral. Our proof of the theorem is contained in the following slightly more general statement. Proposition 2.1. Let A0 be a solution of A0 = |φ|2 A0 over R2 . Suppose that  |∇ A0 |2 dx < ∞. (2.13) R2

Then A0 =constant. Furthermore, if φ is not identically zero, then A0 ≡ 0.

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Proof. Let 0 ≤ η ≤ 1 be of compact support and define for M > 0 fixed the truncated function ⎧ if A0 > M, ⎨M A0M = A0 if |A0 | ≤ M, (2.14) ⎩ −M if A < −M. 0 Then, multiplying (2.9) by η A0M and integrating, we have  R2

[η∇ A0 · ∇ A0M + A0M ∇ A0 · ∇η + η|φ|2 A0M A0 ] dx = 0.

(2.15)

Using (2.14) in (2.15), we find 



{|A0 |


+

η|φ|

supp(η)

2

A20 dx

+M

2 {|A0 |>M}∩

supp(η)

η|φ|2 dx

η|∇ A0 |2 dx supp(η)  1  1 2 2 |∇ A0 |2 dx |∇η|2 dx .

{|A0 |


≤M

R2

(2.16)

R2

For R > 0, we now choose η to be a logarithmic cutoff function given as ⎧ ⎨1 η = 2− ⎩ 0

log |x| log R

if |x| < R, if R ≤ |x| ≤ R 2 , if |x| > R 2 .

(2.17)

2π . log R

(2.18)

Then  R2

|∇η|2 dx =

Using (2.18) in (2.16) gives 



{|A0 |
|φ|

2

A20 dx



≤

{|A0 |
|∇ A0 |2 dx 



{|A0 |
≤M

+

|φ|

2

A20 dx


1 2π R2 |∇ A0 |2 dx 2 1

(log R) 2

+M .

2

|φ| dx + 2

{|A0 |>M}∩B R

{|A0 |
|∇ A0 |2 dx (2.19)

The right-hand side of (2.19) tends to zero as R tends to infinity. Letting M tend to infinity proves the proposition.

Proof of the Julia–Zee Theorem

351

3. A Non-Abelian Julia–Zee Theorem In this section, we consider the simplest non-Abelian Yang–Mills–Higgs theory for which the gauge group is SU (2) or S O(3). For convenience, we work with the gauge group in the adjoint representation so that the gauge field and Higgs field are all real 3-vectors expressed as Aµ = (A1µ , A2µ , A3µ )T , = (φ 1 , φ 2 , φ 3 )T . Then the Yang–Mills field curvature tensor and gauge-covariant derivative are given by Fµν = ∂µ Aν − ∂ν Aµ + Aµ × Aν ,

Dµ = ∂µ + Aν × ,

so that the Lagrangian density is written as 1 1 L = − Fµν · Fµν + Dµ · D µ − V ( ) 4 2 1 1 1 1 = − Fi j · Fi j + F0i · F0i − Di · Di + D0 · D0 − V ( ). (3.1) 4 2 2 2 As a consequence, the equations of motion, or the Yang–Mills–Higgs equations, are D µ Fµi + × Di = 0, δV ( ) = 0, D µ Dµ + δ coupled with the Gauss-law constraint

(3.2)

D µ Fµ0 + × D0 = 0.

(3.4)

(3.3)

This is the equation that governs the A0 field and is our main concern. The actual form of (3.4) is: − Di Fi0 + × D0 = 0, i = 1, 2.

(3.5)

In the static case, ∂0 = 0. So D0 = A0 × , Fi0 = ∂i A0 + Ai × A0 .

(3.6)

Inserting (3.6) into (3.5), we get A0 + ∂i (Ai × A0 ) + Ai × ∂i A0 + Ai × (Ai × A0 ) − × (A0 × ) = 0. (3.7) On the other hand, the Hamiltonian density of the theory is H = F0i · F0i + D0 · D0 − L 1 1 1 1 = Fi j · Fi j + F0i · F0i + Di · Di + D0 · D0 + V ( ), 4 2 2 2 which is positive definite. The terms containing A0 give us the A0 energy,  

1 1 F0i · F0i + D0 · D0 dx E(A0 ) = 2 R2 2  

1 1 2 2 |∂ dx. = i A0 + (Ai × A0 )| + |A0 × | 2 R2 2

(3.8)

(3.9)

It can be checked that the governing equation for A0 , Eq. (3.7), is the Euler–Lagrange equation of (3.9).

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Theorem 3.1 (A non-Abelian extension of the Julia–Zee Theorem). Let A0 be a solution of (3.4) with finite energy E(A0 ) < ∞. Then E(A0 ) = 0. In particular, F0i ≡ 0, D0 ≡ 0, and |A0 | ≡constant. Moreover, if the nonnegative potential density V is such that V ( ) = U (| |2 ) and ( , Ai , A0 ) is a finite-energy solution of the Yang–Mills –Higgs equations (3.2)–(3.4), then A0 ≡ 0. Otherwise the solution triplet ( , Ai , A0 ) must be trivial, i.e., | | ≡ θ, Fi j ≡ 0, E(A0 ) = 0,

(3.10)

where s = θ 2 ≥ 0 is a certain zero of the function U (s). To see that the absence of the electric sector in the non-Abelian Yang–Mills–Higgs model is implied by the above theorem, recall that the ’t Hooft electromagnetic tensor [19] (see also [15,18] for related discussion and extension) may be written as Fµν =

1 1 · Fµν − · (Dµ × Dν ). | | | |3

(3.11)

Hence E i = F0i ≡ 0 if E(A0 ) = 0. 4. Proof of the Non-Abelian Julia–Zee Theorem Let 0 ≤ η ≤ 1 be of compact support and define for M > 0 fixed the truncated vector field,

A0M

=

A0 M |A0 | A0

if |A0 | ≤ M, if |A0 | > M.

(4.1)

Then, using ηA0M as a test function, we obtain from (3.7) the expression 

η ∂i A0M · ∂i A0 − A0M · ∂i (Ai × A0 ) − A0M · (Ai × ∂i A0 ) R2   −A0M · (Ai ×(Ai × A0 ))+A0M · ( × (A0 × )) +∂i η A0M · ∂i A0 dx = 0. (4.2)

Using the definition of A0M in (4.1), we see that A0M · (∂i Ai × A0 ) = 0, 2 |A0 |   (Ai × A0M ) · (Ai × A0 ) = Ai × A0M  in {|A0 | > M}, M 2 |A0 |  M  (A0M × ) · (A0 × ) = A0 ×  in {|A0 | > M}, M |A0 | (∂i A0M )2 in {|A0 | > M}, ∂i A0M · ∂i A0 = M |A0 | ∂i A0M · (Ai × A0M ) in {|A0 | > M}. −2A0M · (Ai × ∂i A0 ) = 2 M

Proof of the Julia–Zee Theorem

353

We then obtain from (4.2) that  η{|∂i A0 + (Ai × A0 )|2 + |A0 × |2 } dx {|A0 |≤M}  |A0 | + η{|∂i A0M + Ai × A0M |2 + |A0M × |2 } dx {|A0 |>M} M  {∂i η A0M · ∂i A0 } dx =− R2  =− {∂i η A0M · (∂i A0 + Ai × A0 )} dx. R2

(4.3)

We again choose η according to (2.17). Using (2.18), we have   |∂i A0 + (Ai × A0 )|2 + |A0 × |2 dx {|A0 ≤M}∩B R



≤M  ≤M

2π log R

1

 1  2

|∂i A0 + Ai × A0 | dx 2

R2

4π E(A0 ) log R

2

1 2

.

(4.4)

The right-hand side of (4.4) tends to zero as R tends to infinity. Letting M tend to infinity proves E(A0 ) = 0. To see that |A0 | =constant, we use the result F0i = 0 to deduce that ∂i |A0 |2 = 2∂i A0 · A0 = −2(Ai × A0 ) · A0 = 0. Suppose A0 = 0. Then |A0 | = a > 0 for some constant a. Note that E(A0 ) = 0 also implies that remains parallel to A0 everywhere. So there is a scalar function u such that = uA0 . Consequently, we have Di = (∂i u)A0 + u Di A0 = (∂i u)A0 + uFi0 = (∂i u)A0 .

(4.5)

Now assume that the Higgs potential density takes the form V ( ) = U (| |2 ). Iterating (4.5), we get Di Di = (u)A0 . Hence, by (3.3) and D0 = 0, we arrive at u = U  (a 2 u 2 )u in R2 . In view of the finite-energy condition and (4.5), we have  

1 1 2 2 2 I (u) = |∇u| + 2 U (a u ) dx < ∞. 2a R2 2

(4.6)

(4.7)

It may easily be checked that, as a solution of (4.6), u is a finite-energy critical point of the functional (4.7). However, using a standard rescaling argument with x → xσ = σ x and u(x) → u σ (x) = u(xσ ) so that dI (u σ )/dσ = 0 at σ = 1, we find   U (| |2 ) dx = U (a 2 u 2 ) dx = 0, (4.8) R2

R2

which implies that U (a 2 u 2 ) ≡ 0. Since U ≥ 0, we have U  (a 2 u 2 ) ≡ 0. Inserting this into (4.6), we have u = 0 in R2 . In view of Proposition 2.1, we conclude that u ≡constant. Consequently, there is a zero, s, of U (·) of the form s = θ 2 ≥ 0 such that u ≡ ± aθ or

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θ = ± A0 , a

(4.9)

which immediately gives us Di = ± aθ Di A0 = ± aθ Fi0 = 0 over R2 . Therefore, the coupled equations (3.2) and (3.3) are reduced to the pure static Yang– Mills equations Di Fi j = 0 in R2 ,

(4.10)

which is known to have only the trivial solution, Fi j = 0, over R2 , as can easily be seen from a similar rescaling argument involving x → xσ = σ x and Ai (x) → (Aσ )i (x) = σ Ai (xσ ), i = 1, 2, in the energy functional  R2

|Fi j |2 dx.

(4.11)

The proof of the stated non-Abelian extension of the Julia–Zee Theorem is complete.

5. Extension and Application As an extension, consider a general non-Abelian gauge group, say the unitary group U (N ), with Lie algebra U(N ) consisting of N × N anti-Hermitian matrices. Then A, B = −Tr(AB),

A, B ∈ U(N ),

(5.1)

is the inner product over U(N ) which allows one to regard the Lie commutator, [ , ] on U(N ), as an exterior product so that A, [A, B] = 0, A, [B, C] = C, [A, B] = B, [C, A] ,

A, B, C ∈ U(N ). (5.2)

The U (N ) Yang–Mills–Higgs theory with the Higgs field represented adjointly has the Lagrangian action density 1 1 L = − Fµν , F µν + Dµ , D µ − V ( ). 4 2

(5.3)

In view of the method in Sect. 4 and the property (5.2), we may similarly show that a finite-energy static solution of the equations of motion of the Yang–Mills–Higgs theory in the (2 + 1)-dimensional Minkowski spacetime defined by (5.3) has a trivial temporal component, A0 . Furthermore, it is clear that our result applies to the models that contain several Higgs fields as well. As an application, consider the classical Abelian Chern–Simons–Higgs theory [13] defined by the Lagrangian action density 1 κ 1 λ L = − Fµν F µν + εµνα Aµ Fνα + Dµ φ D µ φ − (|φ|2 − 1)2 , 4 4 2 8

(5.4)

Proof of the Julia–Zee Theorem

355

over the (2 + 1)-dimensional Minkowski spacetime, where κ is the Chern–Simons coupling constant. The equations of motion governing static field configurations are λ (|φ|2 − 1)φ − A20 φ, 2 i − κ ε jk ∂k A0 = (φ D j φ − φ D j φ), 2 A0 = κ F12 + |φ|2 A0 . D 2j φ =

∂k F jk

(5.5) (5.6) (5.7)

Using the Julia–Zee Theorem stated in Sect. 2 and the existence theorem obtained in [1], we see that a finite-energy solution of the Chern–Simons–Higgs equations (5.5)– (5.7) exists which has a nontrivial temporal component A0 of the gauge field, hence a nontrivial electric sector is present in the theory, if and only if κ = 0, which switches on the Chern–Simons topological term in the model. In view of Theorem 3.1, similar applications may be made to non-Abelian Chern– Simons–Higgs vortex models [11,20,21]. Acknowledgments. The authors were supported in part by the NSF.

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