Spontaneous Symmetry Breaking in Quantum Systems. A review for Scholarpedia

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.

💡 Research Summary

The paper provides a concise yet rigorous review of spontaneous symmetry breaking (SSB) in quantum systems, aiming to correct several oversimplifications that have become part of the standard textbook narrative. It begins by recalling the classical picture of SSB—where a symmetric Hamiltonian admits a set of degenerate minima and the system selects one non‑symmetric ground state—and then points out that the quantum version is far more subtle because it typically involves infinitely many degrees of freedom, continuous spectra, and the need to work with operator algebras rather than individual wavefunctions.

A central theme of the work is the distinction between symmetry as a unitary representation (U(g)) on a Hilbert space and symmetry as an automorphism of a *‑algebra of observables (\mathcal{A}). The author stresses that the statement “the symmetry is broken” should be interpreted in terms of the failure of the symmetry generators to belong to the local observable algebra, not merely as a non‑invariant expectation value in some particular state. In the infinite‑volume limit, the domain and continuity properties of the generators (Q) become delicate, and naïve manipulations can lead to contradictions.

To place the discussion on firm mathematical ground, the paper adopts the algebraic quantum field theory (AQFT) framework. Local algebras (\mathcal{A}(\mathcal{O})) are assigned to bounded spacetime regions (\mathcal{O}), and the dynamics is encoded in a one‑parameter group of automorphisms (\alpha_t). The crucial observation is that the Goldstone theorem requires the dynamics to preserve locality: for any local observable (A\in\mathcal{A}(\mathcal{O})) the evolved operator (\alpha_t(A)) must remain in a (possibly enlarged) local algebra (\mathcal{A}(\mathcal{O}_t)). When this condition holds and the symmetry generator (Q) is itself a local observable (i.e., (Q\in\mathcal{A}(\mathcal{O})) for some region), the commutator (