Spontaneous Symmetry Breaking in the Phase Space

TL;DR

This paper investigates spontaneous symmetry breaking in the phase space of the $$ theory, revealing novel behaviors such as stability changes and instabilities related to tachyonic fields, in both Minkowski and Euclidean frameworks.

Contribution

It introduces the study of SSB in phase space derived from Poincare maps, highlighting the effects of discretization and discovering new stability phenomena and instabilities.

Findings

Stable fixed points become unstable in phase space.

Unique Euclidean-time instability at the critical point.

Potential hosting of tachyonic fields in Euclidean space.

Abstract

In this brief, the spontaneous symmetry breaking (SSB) of the $\varphi^4$ theory in phase space, is studied. This phase space results from the appropriate system of Poincare maps, produced in both the Minkowski and the Euclidean time. The importance of discretization in the creation of phase space, is highlighted. A series of interesting, novel, unknown behaviors are reported for the first time; among them the most characteristic is the change in stability. In specific, the stable fixed points of the $\varphi^4$ potential appear as unstable ones, in phase space. Additionally, in the Euclidean-time phase space a unique instability in the position of the critical point, can be created. This instability is further proposed to host tachyonic field in Euclidean space.