# Vacuum Decay in Real Time and Imaginary Time Formalisms

## TL;DR

This paper explores vacuum tunneling in quantum field theory using both imaginary and real time formalisms within the Wigner representation, clarifying their relationship and potential applications in cosmology.

## Contribution

It introduces a unified framework using the Wigner representation to compare instanton and classical real-time tunneling approaches in quantum field theory.

## Key findings

- Two saddle point approximations identified: instanton in imaginary time and classical dynamics in real time.
- Classical dynamics approximation justified in systems with many degrees of freedom.
- Potential applications include tunneling scenarios where instantons vanish or imaginary time deformation is impossible.

## Abstract

We analyze vacuum tunneling in quantum field theory in a general formalism by using the Wigner representation. In the standard instanton formalism, one usually approximates the initial false vacuum state by an eigenstate of the field operator, imposes Dirichlet boundary conditions on the initial field value, and evolves in imaginary time. This approach does not have an obvious physical interpretation. However, an alternative approach does have a physical interpretation: in quantum field theory, tunneling can happen via classical dynamics, seeded by initial quantum fluctuations in both the field and its momentum conjugate, which was recently implemented in Ref. [1]. We show that the Wigner representation is a useful framework to calculate and understand the relationship between these two approaches. We find there are two, related, saddle point approximations for the path integral of the tunneling process: one corresponds to the instanton solution in imaginary time and the other one corresponds to classical dynamics from initial quantum fluctuations in real time. The classical approximation for the dynamics of the latter process is justified only in a system with many degrees of freedom, as can appear in field theory due to high occupancy of nucleated bubbles, while it is not justified in single particle quantum mechanics, as we explain. We mention possible applications of the real time formalism, including tunneling when the instanton vanishes, or when the imaginary time contour deformation is not possible, which may occur in cosmological settings.