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arXiv:1103.3227

How is spontaneous symmetry breaking possible?

David John Baker, Hans Halvorson

Published Mar 16, 2011• 5 citations
arXivHTMLPDF

Abstract

We pose and resolve a seeming paradox about spontaneous symmetry breaking in the quantum theory of infinite systems. For a symmetry to be spontaneously broken, it must not be implementable by a unitary operator. But Wigner's theorem guarantees that every symmetry is implemented by a unitary operator that preserves transition probabilities between pure states. We show how it is possible for a unitary operator of this sort to connect the folia of unitarily inequivalent representations. This result undermines interpretations of quantum theory that hold unitary equivalence to be necessary for physical equivalence.

DOI
10.48550/arxiv.1103.3227

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We pose and resolve a seeming paradox about spontaneous symmetry breaking in the quantum theory of infinite systems. For a symmetry to be spontaneously broken, it must not be implementable by a unitary operator. But Wigner's theorem guarantees that every symmetry is implemented by a unitary operator that preserves transition probabilities between pure states. We show how it is possible for a unitary operator of this sort to connect the folia of unitarily inequivalent representations. This result undermines interpretations of quantum theory that hold unitary equivalence to be necessary for physical equivalence.
This essay expounds the algebraic framework describing general physical the-ories, within which the phenomenon of spontaneous symmetry breaking (SSB) makes its appearance in infinite quantum systems. This is in contrast with the fact that a large class of theories- both classical and quantum, finite and infinite- are termed, in the conventional account of classical and quantum mechanics, as exhibiting SSB. This discrepancy will be understood in the light of an interpretation that finds the symmetry breaking to be in some respects stronger in the algebraic account than is generally the case in the conventional picture. The case of SSB in the standard account of quantum field theory (QFT) will then be discussed, and it will be argued that, although one would expect a connection with the algebraic account to be possible, this turns out to be problematic. Finally the role of the idealisation of infinite systems, crucial to algebraic SSB, will be dis-cussed.
Spontaneous symmetry breaking (SSB) in quantum systems, such as ferromagnets, is normally described as (or as arising from) degeneracy of the ground state; however, it is well established that this degeneracy only occurs in spatially infinite systems, and even better established that ferromagnets are not spatially infinite. I review this well-known paradox, and consider a popular solution where the symmetry is explicitly broken by some external field which goes to zero in the infinite-volume limit; although this is formally satisfactory, I argue that it must be rejected as a physical explanation of SSB since it fails to reproduce some important features of the phenomenology. Motivated by considerations from the analogous classical system, I argue that SSB in finite systems should be understood in terms of the approximate decoupling of the system's state space into dynamically-isolated sectors, related by a symmetry transformation; I use the formalism of decoherent histories to make this more precise and to quantify the effect, showing that it is more than sufficient to explain SSB in realistic systems and that it goes over in a smooth and natural way to the infinite limit.
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A cornerstone of quantum mechanics is the characterization of symmetries provided by Wigner's theorem. Wigner's theorem establishes that every symmetry of the quantum state space must be either a unitary transformation or an antiunitary transformation. Here we extend Wigner's theorem from quantum states to quantum evolutions, including both the deterministic evolution associated with the dynamics of closed systems and the stochastic evolutions associated with the outcomes of quantum measurements. We prove that every symmetry of the space of quantum evolutions can be decomposed into two state space symmetries that are either both unitary or both antiunitary. Building on this result, we show that it is impossible to extend the time-reversal symmetry of unitary quantum dynamics to a symmetry of the full set of quantum evolutions. Our no-go theorem implies that any time-symmetric formulation of quantum theory must either restrict the set of the allowed evolutions or modify the operational interpretation of quantum states and processes. Here we propose a time-symmetric formulation of quantum theory where the allowed quantum evolutions are restricted to a suitable set, which includes both unitary evolution and projective measurements but excludes the deterministic preparation of pure states. The standard operational formulation of quantum theory can be retrieved from this time-symmetric version by introducing an operation of conditioning on the outcomes of past experiments.
This chapter evaluates whether unitary equivalence is an appropriate criterion of physical equivalence for quantum theories that fall outside the scope of the comforting uniqueness results. It describes the Stone-von Neumann and Jordan-Wigner uniqueness results, provides examples of unitarily inequivalent representations, and proposes an alternative to unitary equivalence as a criterion for physical equivalence. The chapter also offers an impressionistic introduction to the rudiments of operator algebra theory.
The roots of the modern understanding of symmetries in physics can be traced to the work of Sophus Lie on transformations of differential equations in the nineteenth century. In this chapter, the history of the subject is used to emphasize the distinction between the symmetry of the dynamics of a physical system and the symmetry of its specific state. The central concept of spontaneous symmetry breaking is then introduced as the discrepancy between the two notions of symmetry. It is shown how the presence of spontaneous symmetry breaking can be detected through the sensitivity of measurements on the physical state to external perturbations. This naturally leads to the concepts of the order parameter and the vacuum manifold. The rest of the chapter is devoted to highlighting some subtle features of spontaneous symmetry breaking. These include the properties of the ground state and the tower of low-lying excited states in a finite volume, and the absence of well-defined unitary operators representing broken symmetry in the thermodynamic limit.