Beyond complex Langevin equations II: a positive representation of Feynman path integrals directly in the Minkowski time

TL;DR

This paper introduces a positive representation for Minkowski path integrals using doubled real variables, enabling a statistical formulation directly in Minkowski time, with applications to quantum mechanics examples.

Contribution

It develops a novel positive representation for complex Gaussian weights in Minkowski time by doubling variables, allowing direct statistical treatment of path integrals.

Findings

Successfully applied to quantum mechanical examples including a particle in a magnetic field

Constructed a continuum limit with specific coupling tuning

Discussed potential generalizations and interpretations

Abstract

Recently found positive representation for an arbitrary complex, gaussian weight is used to construct a statistical formulation of gaussian path integrals directly in the Minkowski time. The positivity of Minkowski weights is achieved by doubling the number of real variables. The continuum limit of the new representation exists only if some of the additional couplings tend to infinity and are tuned in a specific way. The construction is then successfully applied to three quantum mechanical examples including a particle in a constant magnetic field -- a simplest prototype of a Wilson line. Further generalizations are shortly discussed and an intriguing interpretation of new variables is alluded to.