Complexified Path Integrals and the Phases of Quantum Field Theory

TL;DR

This paper explores complexified path integrals in quantum field theory, revealing additional solutions that can describe physical phenomena like phase transitions and offering new insights beyond traditional methods.

Contribution

It introduces the concept of complexified path integral solutions and demonstrates their relevance to phase transitions and non-perturbative effects in quantum field theories.

Findings

Additional solutions relate to phase transitions and symmetry breaking.

Relation between Lee-Yang zeros and Stokes phenomena is established.

Borel resummation yields inequivalent solutions depending on contours.

Abstract

The path integral by which quantum field theories are defined is a particular solution of a set of functional differential equations arising from the Schwinger action principle. In fact these equations have a multitude of additional solutions which are described by integrals over a complexified path. We discuss properties of the additional solutions which, although generally disregarded, may be physical with known examples including spontaneous symmetry breaking and theta vacua. We show that a consideration of the full set of solutions yields a description of phase transitions in quantum field theories which complements the usual description in terms of the accumulation of Lee-Yang zeroes. In particular we argue that non-analyticity due to the accumulation of Lee-Yang zeros is related to Stokes phenomena and the collapse of the solution set in various limits including but not restricted to, the thermodynamic limit. A precise demonstration of this relation is given in terms of a zero dimensional model. Finally, for zero dimensional polynomial actions, we prove that Borel resummation of perturbative expansions, with several choices of singularity avoiding contours in the complex Borel plane, yield inequivalent solutions of the action principle equations.