Elsevier

Physics Letters A

Volume 267, Issue 1, 6 March 2000, Pages 40-44
Physics Letters A

The classical counterpart of the Goldstone theorem

https://doi.org/10.1016/S0375-9601(00)00066-9Get rights and content

Abstract

The classical counterpart of the Goldstone theorem associated to the spontaneous breaking of a continuous symmetry for infinite dimensional classical systems is discussed. Solutions of the free wave equation as analogs of the Goldstone bosons can be proved to exist in one space dimension. In space dimension two and three the Goldstone modes appear as asymptotic states or as free wave behaviour in bounded regions.

Introduction

The mechanism of spontaneous symmetry breaking (SSB) seems to be at the basis of most of the recent developments in theoretical physics (from statistical mechanics to many-body theory and to elementary particle theory) and its relevance also stems on its innovative philosophical content.

By the general wisdom of Classical Mechanics, codified in the classical Noether theorem, one learns that the symmetries of the Hamiltonian or of the Lagrangean are automatically symmetries of the behaviour of the physical system described by it. This belief therefore precludes the possibility of describing systems with different physical properties in terms of the same Hamiltonian. The realization that this obstruction does not a priori exists, that one may unify the description of apparently different systems in terms of a single Hamiltonian and to account for the different behaviours by the mechanism of SSB, is a real revolution in the way of thinking in terms of symmetries and corresponding properties of physical systems. It is in fact non-trivial to understand how the conclusions of the Noether theorem can be evaded and how a symmetry of the dynamics cannot be realized as a mapping of the physical configurations of the system which commutes with the time evolution.

The standard cheap explanation does not emphasize the need of infinite degrees of freedom as a crucial ingredient and identifies the phenomenon with the existence of a degenerate ground (or equilibrium) state, unstable under the symmetry operation, a feature often present even in simple finite dimensional mechanical models (like e.g. a particle on a plane, each point of which defines a ground state unstable under translations or even the popular mechanical model of the double well potential), but which is usually not accompanied by a non-symmetric behaviour.

The phenomenon is rather related to the fact that, for non-linear (dynamical) systems with infinite degrees of freedom, the solutions of the dynamical problem generically fall into classes or `islands', stable under time evolution and with the property that they cannot be related by physically realizable operations. This means that starting from the configurations of a given island one cannot reach the configurations of a different island by physically realizable modifications. The different islands can then be interpreted as the realizations of different physical systems or different phases of a system or as disjoint physical worlds.

The spontaneous breaking of a symmetry (of the dynamics) in a given island (or phase or physical world) can then be explained as the result of the instability of the given island under the symmetry operation. In fact, in this case one cannot realize the symmetry within the given island, namely one cannot associate to each configuration the one obtained by the symmetry operation.

The existence of such structures is not obvious and in general it involves a mathematical control of the non-linear time evolution of systems with infinite degrees of freedom and the mathematical formalization of the concept of physical disjointness of different islands. For quantum systems, where the mathematical basis of SSB has mostly been discussed, the physical disjointness has been ascribed to the existence of inequivalent representations of the algebra of local observables.

For classical infinite dimensional systems the mathematical formalization of physical disjointness relies on the constraint of essential localization in space of any physically realizable operation (so that configurations with different limits at infinity belong to disjoint islands). For the specific case of nonlinear scalar field equations, to which for concreteness we shall restrict our discussion, one can in fact show [1], [2] that an island can be characterized by some reference bounded (locally `regular') configuration, having the meaning of the `ground state', and its H1 perturbations. Each island is therefore isomorphic to a Hilbert space (Hilbert space sector).

The stability under time evolution is guaranteed by the condition that the reference configuration satisfies a generalized stationarity condition, i.e. it solves some elliptic problem. Such a condition is in particular satisfied by the time independent solutions and a fortiori by the minima ϕ of the potential and the corresponding Hilbert space sectors Hϕ are of the form ϕ+χ,χ∈H1, i.e. χ,χ∈L2 (for a more precise mathematical definition and statement, see Refs. [1], [2], and [3] for a simple account.). The existence of minima of the potential unstable under the symmetry gives therefore rise to islands (or phases or disjoint physical worlds) in which the symmetry cannot be realized or, as one says, it is spontaneously broken.

This phenomenon is deeply rooted in the non-linearity of the problem and the fact that infinite degrees of freedom are involved. A simple prototype is given by the non-linear wave equation for a Klein–Gordon field ϕ:RsRn, with `potential' U(ϕ)=λ(ϕ2a2)2. The model displays some analogy with the mechanical model of a particle in Rn subject to the potential U(q)=λ(q2a2)2, which can be regarded as the higher-dimensional version of the double well potential in one dimension. But the differences are substantial: in the infinite dimensional case of the Klein–Gordon field, each point of potential has actually become infinite dimensional and in fact each absolute minimum ϕ, with |ϕ|=a describes the infinite set of configurations which have this point as asymptotic limit, equivalently the Hilbert space of configurations which are H1 modifications of ϕ. Whereas in the finite dimensional case there is no physical obstruction or `barrier', which prevents to move from one minimum to the other, in the infinite dimensional case there no physically realizable operation, which leads from the Hilbert space sector defined by one minimum to that defined by another minimum, because this would require to change the asymptotic limit of the configurations and this is not possible by means of essentially localized operations, the only ones which are physically realizable.

The realization of the above structures allows to evade part of the conclusions of the classical Noether theorem and to obtain an improved version [3] which takes into account the integrability condition of the charge density, overlooked in the standard treatments (see, e.g., Refs. [4], [5]), and in this way accounts for SSB [3]. In fact, one may prove that the local conservation law, μjμ(x)=0, associated to a given symmetry, which for simplicity in the following will be taken to be internal, i.e. commuting with space and time translations, gives rise to a global conservation law or to a conserved charge, in a given island, only if the symmetry leaves the island stable. Thus, the improved version of Noether theorem still yields the local conservation laws corresponding to the generators of the symmetry group G of the dynamics, but in a given phase or physical world, described by the Hilbert space sector Hϕ, one has the global conservation law only for the generators of the stability subgroup of the given island or Hilbert space sector, namely for the generators of the subgroup Gϕ such that Gϕϕ=ϕ. In this case one says that in Hϕ the symmetry group G of the dynamics is broken down to Gϕ.

The mechanism of SSB does not only provide a general strategy for unifying the description of apparently different systems, but it also provide information on the energy spectrum of an infinite dimensional system, by means of the so-called Goldstone theorem [6], according to which to each broken generator T of a continuous symmetry there corresponds a massless mode, i.e. a free wave. The quantum version of such a statement has been turned into a theorem [7], [8], whereas, as far as we know, no analogous theorem has been proved for classical (infinite dimensional) systems and the standard accounts rely on heuristic arguments.

The aim of the present note is to critically revisit the folklore about the classical version of the Goldstone theorem, to propose a mathematically acceptable substitute of the heuristic arguments and to correct the conclusions based on the quadratic approximation of the potential.

The standard heuristic argument, which actually goes back to Goldstone [6], considers as a prototype the nonlinear equation□ϕ+U′(ϕ)=0,where the multi-component real field ϕ transforms as a linear representation of a Lie group G and the potential U is invariant under the transformations of G. This implies that for the generator Tα one has0=δαU(ϕ)=U′j(ϕ)Tαjkϕk,∀ϕand therefore the derivative of this equation at ϕ=ϕ givesUjk′′(Tαϕ)k=0.Thus, in an expansion of the potential around ϕ, the quadratic term, which has the meaning of a mass term, has a zero eigenvalue in the direction Tαϕ. This is taken as evidence that there is a massless mode. In our opinion, the argument is not conclusive, and as we shall see not really correct, since it involves an expansion and one should in some way control the effect of higher order terms; moreover, it is not clear that there are (physically meaningful) solutions in the direction of Tαϕ for all times, so that for them the quadratic term disappears. In any case, the argument does not show that there are massless solutions as in the quantum case.

Another heuristic argument appeals to the finite dimensional analogy, where the motion of a particle along the bottom of the potential, i.e. along the orbit {gα(λ)ϕ}, where gα(λ),λ∈R, is the one parameter subgroup generated by Tα, does not feel the potential, since U′(gαϕ)=0, and therefore the motion is like a free motion. This is considered as evidence that, correspondingly, in the infinite dimensional case there are massless modes. Again the argument does not appear complete, since it is not at all clear that there are physically meaningful solutions, i.e. belonging to the physical sector of ϕ and therefore of the form ϕ=ϕ+χ,χ∈H1(Rs), s=space dimension, of zero mass.

Section snippets

A classical counterpart of the Goldstone theorem

We propose the following version of the Goldstone theorem for classical fields. We consider the case of space dimension s=3, unless otherwise stated and for simplicity the case of compact semi-simple Lie group G of internal symmetries. The potential is assumed to be sufficiently regular, i.e. it is at least of class C3, bounded from below and satisfies supz(1+|z|2)−1|U′′(z)|<∞, so that one can rely on the established existence theorems (see e.g. Refs. [1], [2]).

Theorem 1

Under the above assumptions, let G

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