Spontaneous symmetry breaking in models with second-class constraints

TL;DR

This paper investigates how spontaneous symmetry breaking affects models with second-class constraints, revealing that constraints can behave as first-class and cause pathologies like acausal modes.

Contribution

It demonstrates that second-class constraints can act as first-class on certain regions, altering degrees of freedom and leading to potential physical pathologies.

Findings

Constraints become degenerate at critical Hamiltonian values

Second-class constraints can behave as first-class during symmetry breaking

Models may exhibit acausal propagation modes

Abstract

In this work the spontaneous symmetry breaking in certain nonlinear theories with second-class constraints is explored. Using the Dirac's method we perform an analysis of the constraints and the counting of the degrees of freedom. The corresponding effective Hamiltonian is constructed explicitly. It is shown that on the surfaces where the effective Hamiltonian takes critical values the symplectic structure becomes degenerate. In particular, we demonstrate that under the condition of spontaneous symmetry breaking, which implies non trivial vacuum surfaces, second-class constraints behave as first-class ones on certain regions of the phase space, leading to undefined Dirac's brackets and to the modification of the number of degrees of freedom. As a physical consequence, these models can suffer from certain pathologies such as the existence of modes with an acausal propagation. Concrete examples where this phenomena occurs are described in detail.