Complexified Path Integrals and the Phases of Quantum Field Theory

TL;DR

The paper shows that quantum-field-theoretic phase structure can be understood by considering the complete set of solutions to the Schwinger-Dyson and action-principle equations, including complexified path integrals over inequivalent contours. It analyzes zero-dimensional models to connect contour choices with symmetry-breaking and theta vacua, establishing a link between non-analyticities from Lee-Yang zeros and Stokes phenomena, and demonstrates that Borel-resummed perturbation theory around different saddles yields inequivalent, physically meaningful sectors. It further argues that in the thermodynamic (large-N) limit the action principle can emerge from SD equations via collapse of the solution set, and extends the discussion to non-polynomial actions and lattice-like settings with potential physical relevance of exotic complex sectors. Overall, the work provides a unified framework connecting complexified path integrals, non-perturbative phase structure, and resummation techniques, with implications for gauge theories and beyond.

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.