Resurgence theory, ghost-instantons, and analytic continuation of path integrals

TL;DR

This paper explores how real and complex saddle points, including ghost-instantons, influence quantum path integrals, revealing their role in non-perturbative phenomena and phase transitions in quantum theories.

Contribution

It explicitly demonstrates the impact of ghost-instantons and complex saddles on physical observables and introduces a homotopically independent cycle construction to resolve pathologies in analytic continuation.

Findings

Both instanton-anti-instanton and ghost-anti-ghost saddles affect perturbative expansions.

A homotopically independent combination of cycles yields a well-defined theory despite exponential growth.

The construction provides insights into non-perturbative effects in quantum field theories and related areas.

Abstract

A general quantum mechanical or quantum field theoretical system in the path integral formulation has both real and complex saddles (instantons and ghost-instantons). Resurgent asymptotic analysis implies that both types of saddles contribute to physical observables, even if the complex saddles are not on the integration path i.e., the associated Stokes multipliers are zero. We show explicitly that instanton-anti-instanton and ghost--anti-ghost saddles both affect the expansion around the perturbative vacuum. We study a self-dual model in which the analytic continuation of the partition function to negative values of coupling constant gives a pathological exponential growth, but a homotopically independent combination of integration cycles (Lefschetz thimbles) results in a sensible theory. These two choices of the integration cycles are tied with a quantum phase transition. The general set of ideas in our construction may provide new insights into non-perturbative QFT, string theory, quantum gravity, and the theory of quantum phase transitions.