Geometric modular action and spontaneous symmetry breaking

TL;DR

This paper investigates spontaneous symmetry breaking in quantum field theories on Minkowski space under the geometric modular action condition, revealing constraints on internal symmetries and invariance properties.

Contribution

It establishes that internal symmetries commute with Poincare transformations and that translation-invariant states are Poincare invariant under the CGMA.

Findings

Internal symmetry groups commute with Poincare group representations.

Translation-invariant vectors are Poincare invariant.

The sector decomposition preserves the CGMA.

Abstract

We study spontaneous symmetry breaking for field algebras on Minkowski space in the presence of a condition of geometric modular action (CGMA) proposed earlier as a selection criterion for vacuum states on general space-times. We show that any internal symmetry group must commute with the representation of the Poincare group (whose existence is assured by the CGMA) and each translation-invariant vector is also Poincare invariant. The subspace of these vectors can be centrally decomposed into pure invariant states and the CGMA holds in the resulting sectors. As positivity of the energy is not assumed, similar results may be expected to hold for other space--times.