Distance between physical theories based on information theory

Jacques Calmet1∗ and Xavier Calmet2† ∗

Institute for Cryptography and Security (IKS) Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany † Physics and Astronomy, University of Sussex, Falmer, Brighton, BN1 9QH, UK

Abstract We introduce a concept of distance between physical theories described by an action. The definition of the distance is based on the relative entropy. We briefly discuss potential physical applications. 1 2

email:[email protected] email:[email protected]

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Introduction

Our understanding of nature is based on models which provide an approximation of the physical reality. In many areas of science, such as, for example, physics or some areas of chemistry, models are often defined in terms of an action which allows to derive the equations of motion of the model using the least action principle. A model is uniquely defined through its action which is a functional of the fields introduced in the model. It also depends on potential coupling constants and mass parameters. Equivalently, the model can be given in terms of a Hamiltonian which can be derived from to the action using a Legendre transformation. Building a Hamiltonian or equivalently an action requires to specify the symmetries (e.g. space-time or gauge symmetries) of the theory as well as the values of the parameters of the theory such as the number of fields, the spin-statistics of these fields and the value of the mass and coupling parameters at some energy scale. One also needs to impose boundary conditions on the fields in order to obtain the field equations of the theory and to fix the vacuum around which one develops the theory. However, once this has been done the theory is usually uniquely determined. Observables can then be computed within a certain approximation and compared to experiments. This allows to compare a theory to experiments and thus to determine if the theory under consideration is giving an adequate description of nature within the precision of the calculation done using the theory and of the performed experiments. One could however consider a different question and ask how different are two theories. In other words, we are interested in introducing the notion of a distance between two models described by an action. In the present work, we extend our considerations to quantum field theories. One option would be to compute all possible observables and to do a chi-square fit, but this would be an extremely cumbersome and for most practical cases impossible task. Furthermore, one might try to compare theories that do not even have the same number of observables in which case the method proposed previously would fail. We propose to introduce the notion of a distance between Hamiltonians, and thus physical theories, based on the Kullback-Leibler relative entropy which is frequently used in information theory as a concept of a distance. However, this is not a distance in the usual sense. A distance d(P1 , P2 ) between two points P1 and P2 has to satisfy the following three axioms: 1. Positive definiteness: ∀P1 , P2 : d(P1 , P2 ) ≥ 0 2. Symmetry d(P1 , P2 ) = d(P2 , P1 ) 3. Triangle inequality: ∀P1 , P2 , P3 : d(P1 , P2 ) ≤ d(P1 , P3 ) + d(P2 , P3 ). 1

However, it is often useful to introduce a concept of distance between elements of more abstract sets in any field of knowledge as very well illustrated by the recently published encyclopedia [1]. An even more recent attempt is to define a distance between dialects in linguistics. In a domain closer to our interests, one could ask for example what is the distance between two distributions between e.g. the Gaussian and binomial distributions. It is useful to introduce the concept of entropy as a mean to define such distances. In information theory, Shannon entropy [2] represents the information content of a message or, from the receiver point of view, the uncertainty about the message the sender produced prior to its reception. It is defined as X − p(i) log p(i), (1) i

where p(i) is the probability of receiving the message i. The unit used is the bit. The relative entropy can be used to define a “distance” between two distributions p(i) and g(i). The Kullback-Leibler [3] distance or relative entropy is defined as D(g||p) =

X

g(i) log

i

g(i) p(i)

(2)

where p(i) is the real distribution and g(i) is an assumed distribution. Clearly the KullbackLeibler relative entropy is not a distance in the usual sense: it satisfies the positive definiteness axiom, but not the symmetry or the triangle inequality axioms. It is nevertheless useful to think of the relative entropy as a distance between distributions. The Kullback-Leibler distance is relevant to discrete sets. It can be generalized to the case of continuous sets. For our purposes, a probability distribution over some field (or set) X is a distribution p : X ∈ R, such that 1.

R X

d4x p(x) = 1

2. For any finite subset S ⊂ X,

R S

d4x p(x) > 0.

Let us now apply this definition to models described by an action or a Hamiltonian. For a given Hamiltonian there exists a density matrix ρ defined by X ρ= |φn iwn hφn | (3) n

where |φn i, n ∈ {1...N } are the states of the system and wn is the probability of finding the system in the state |φn i. The von Neumann entropy is then given by S = −Tr(ρ log ρ). 2

(4)

Given two Hamiltonians H1 and H2 and the corresponding density matrices ρ1 and ρ2 , we can introduce the relative entropy of ρ1 with respect to ρ2 : D(ρ1 ||ρ2 ) = Tr(ρ1 (log ρ1 − log ρ2 ))

(5)

This distance is clearly not a distance as understood in metric spaces but it is nevertheless useful to think of the relative entropy as a distance between density matrices and thus Hamiltonians. Let us illustrate how to use this new concept through a concrete example which comes from statistical mechanics. The density matrix can also be defined in terms of the partition function Z. In statistical mechanics the partition function is defined as   H Z = Tr exp − , (6) kT is the Boltzmann constant and T the temperature. The density where k = 8.6 × 10−5 eV K matrix is then given by   1 H ρ(T ) = exp − . (7) Z kT The Kullback-Leibler relative entropy can be expressed in terms of the partition functions. In that case eq.(5) reads D(ρ1 ||ρ2 ) = −β1

d ln Z1 − ln Z1 + ln Z2 − β2 Tr(ρ1 H2 ), dβ1

(8)

where βi = (kTi )−1 . We interpret this distance as a distance between the Hamiltonian H1 corresponding to ρ1 and the Hamiltonian H2 corresponding to ρ2 . For models with no overlap, i.e., Tr(ρ1 H2 ) = Tr(H2 ), the distance is just the entropy of the model (up to a factor −k) determined by the Hamiltonian H1 . If H1 = H2 and for the same temperature then the distance is vanishing. We shall calculate the relative distance between one-dimensional harmonic oscillator and a one-dimensional free particle and compared it to the distance between an one-dimensional anharmonic oscillator and a one-dimensional free particle. All three theories are assumed to be in contact with a thermal bath having a temperature T and we shall assume that the overlap between the models is small and negligeable. The partition function of a onedimensional free particle in a system of length L is given by [4] r mkT Zfree = L . (9) 2π~ 3

The magnitude of L is of no particular importance as we shall use Zfree as a comparator for both the harmonic and anharmonic oscillators. The partition function for a one-dimensional harmonic oscillator is given by [4] 1 ZHO = ~ω 2 sinh 2kT

(10)

where ω is the frequency of the oscillator. Finally for an anharmonic oscillator (H = H0 +λx4 , the partition function is given by [5]   3λ~ ~ω 2 ~ω coth . (11) ZAHO ≈ ZHO 1 − 4m2 ω 3 kT kT The distance between the Hamiltonian of the one dimensional harmonic oscillator and that of the one-dimension free particle can now easily be computed. One finds: d(HHO ||Hfree ) = D(ρHO ||ρfree )      1 ~ωβ ~ωβ 1 = ~ωβ coth csch − log 2 2 2 2

(12)

In order to get numbers, we set all parameters of this distance to unity in their respective units. We find d(HHO ||Hfree ) = 1.123. The distance between the Hamiltonian of the one dimensional anharmonic oscillator and that of the one-dimension free particle is then given by      1 ~ωβ ~ωβ 1 d(HAHO ||Hfree ) = ~ωβ coth csch − log (13) 2 2 2 2 3λ(~2 ωβ 2 coth(~ωβ) csch2 (~ωβ) − ~β coth2 (~ωβ) + O(λ) − 2m2 ω 2 Setting again all parameters with the exception of λ to unity, we find d(HAHO ||Hfree ) = 1.124 for λ = 0.001. It is easy to see that one obtains d(HHO ||Hfree ) = d(HAHO ||Hfree ) in the limit λ → 0. Our result matches the intuition that the anharmonic oscillator is further distant from the one dimensional free particle model than the harmonic oscillator is from the one dimensional free particle model. It is easy to convince oneself that this observation is a generic feature which is not dependent on our choice of parameters. Note that λ is positive, this is to insure a stable potential. Thus, the difference between the two distances is always positive. To illustrate that point, another numerical example is as follows, taking, again in appropriate units, ω = 1.5, β = 4, m = 2 and λ = 0.01, we find d(HHO ||Hfree ) = 6.01 and d(HAHO ||Hfree ) = 6.02. The same definition of a distance can be applied to any physical theory which can be defined in terms of an action. In particular it is particularly interesting to apply this notion 4

to quantum field theories. Once an action I[fields] which is a functional of all the fields introduced in the model as well as a function of the coupling constants is known, one can introduce the partition function Z Z Z = D[fields] exp (−βI[fields, renormalized parameters]) (14) where β is the imaginary time. Generally speaking, it is not always possible to calculate the partition function exactly for a quantum field theory, but this can often be done in perturbation theory. To the best of our knowledge, the concept of distance between physical theories described by an action has never been proposed. It fits well however into the present trend to quantify any sort of ontological knowledge [6]. Another potential application is that of the landscape scenario in string theory where the notion of a distance between theories could be important [7] to probe the separation from a given vacuum to that of the standard model of particle physics. Finally, note that we have considered here the case of the Kullback-Leibler entropy. We can easily extend our considerations to continuous sets of models. In other words, we can introduce Fisher’s metric over a continuous space of physical models. This allows us to define a local metric on theory spaces with a physical distance ds2 = gij dxi dxj where xi are local coordinates. Furthermore, it has been shown that Fisher’s metric can be derived from an action [8]. It would be interesting to see if imposing symmetries at the level of the action can lead to interesting relations between physical models.

References [1] M.M. Deza and E. Deza, “Encyclopedia of Distances,” Springer, Heidelberg, 2009. [2] C. E. Shannon, “A Mathematical Theory of Communication,” Bell System Technical Journal 27:379-423, 623-656, July and October 1948. [3] S. Kullback, “Information Theory and Statistics,” John Wiley, New York, 1959. [4] R. P. Feynman, “Statistical Mechanics,” Advanced Book Classics, ISBN 0-201-36076-4. [5] G. Parisi, “Statistical Field Theory,” Addison Wesley, 1988. [6] J. Calmet, “An Introduction to the Quantification of Culture,” Proc. of Advances in Multiagent Systems, Robotics and Cybernetics: Theory and Practice, Volume III, G. Lasker and J. Pfalzgraf eds., InterSym-2009, Baden-Baden, forthcoming, 2010. [7] M. R. Douglas, “Spaces of Quantum Field Theories,” arXiv:1005.2779, 2010. 5

[8] X. Calmet and J. Calmet, Phys. Rev. E 71, 056109, 2005.

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