[Paper Review] Spontaneous Symmetry Breaking in Quantum Systems. A review for Scholarpedia

Franco Strocchi|arXiv (Cornell University)|Jan 26, 2012

Quantum Mechanics and Applications4 references3 citations

TL;DR

This paper reviews spontaneous symmetry breaking (SSB) in quantum systems, emphasizing that the Goldstone theorem's conclusion relies critically on the localization properties of time evolution in infinite systems. It clarifies that SSB is possible in quantum field theories due to inequivalent representations of local algebras, not through explicit symmetry-breaking terms, and corrects misconceptions in standard treatments by grounding the mechanism in the algebraic structure of infinite systems and the failure of the Stone–von Neumann theorem in this context.

ABSTRACT

The mechanism of spontaneous symmetry breaking in quantum systems is briefly reviewed, rectifying part of the standard wisdom on logical and mathematical grounds. The crucial role of the localization properties of the time evolution for the conclusion of the Goldstone theorem is emphasized.

Motivation & Objective

• To clarify the mechanism of spontaneous symmetry breaking in quantum systems, correcting misconceptions in standard treatments.
• To establish that SSB is possible in quantum field theories due to the existence of inequivalent representations of local algebras in infinite systems.
• To demonstrate that the Goldstone theorem's conclusion depends crucially on the localization properties of time evolution, not just on symmetry breaking per se.
• To show that the standard argument that tunneling prevents symmetry breaking in finite systems does not apply to spin systems, where a finite-dimensional analog of the Stone–von Neumann theorem holds.
• To provide a rigorous foundation for SSB using the algebraic approach to quantum field theory, based on $C^*$-algebras of observables and their representations.

Proposed method

• Uses the algebraic approach to quantum field theory, modeling infinitely extended systems via $C^*$-algebras of local observables.
• Applies the Haag–Kastler framework, where observables are bounded operators localized in spacetime regions, ensuring mathematical rigor and operational clarity.
• Defines a symmetry as an automorphism of the observable algebra and considers its breaking when not implementable by a unitary operator in a given representation.
• Employs the cluster property and asymptotic abelianess to characterize physically acceptable states and ensure decorrelation at spacelike separation.
• Relies on the failure of the Stone–von Neumann theorem in infinite systems, which allows for multiple inequivalent representations and thus enables SSB.
• Analyzes the role of time evolution localization in the derivation of the Goldstone theorem, showing it is essential for the existence of massless modes.

Experimental results

Research questions

• RQ1 Why does spontaneous symmetry breaking occur in quantum field theories despite the equations of motion being symmetric?
• RQ2 How does the algebraic approach to quantum field theory resolve ambiguities in the standard treatment of spontaneous symmetry breaking?
• RQ3 What is the role of the localization of time evolution in the validity of the Goldstone theorem?
• RQ4 Why does the standard argument based on tunneling fail to prevent symmetry breaking in finite-dimensional spin systems?
• RQ5 How do inequivalent representations of local algebras in infinite systems allow for physically distinct phases and broken symmetry?

Key findings

• Spontaneous symmetry breaking in quantum systems arises not from explicit symmetry-breaking terms, but from the existence of multiple inequivalent representations of the local algebra in infinite systems.
• The Goldstone theorem's conclusion — the existence of massless modes — depends crucially on the localization properties of the time evolution, not merely on the presence of a continuous symmetry.
• Infinite systems described by $C^*$-algebras of local observables admit multiple physically distinct phases (i.e. inequivalent representations), which is the mathematical origin of SSB.
• The Stone–von Neumann theorem does not apply to infinite systems, allowing for multiple irreducible representations and thus enabling spontaneous symmetry breaking even when the dynamics are symmetric.
• The cluster property, which ensures decorrelation at spacelike separation, is equivalent to the uniqueness of the translationally invariant ground state, characterizing a pure phase.
• The failure of the standard tunneling argument to prevent symmetry breaking in finite spin systems is explained by the existence of a finite-dimensional analog of the Stone–von Neumann uniqueness theorem, which rules out SSB in such cases.