Comment on the subtlety of defining real-time path integral in lattice gauge theories

TL;DR

This paper clarifies how to properly define real-time path integrals in lattice gauge theories by emphasizing the importance of the $i\,\varepsilon$ prescription, showing that the Wilson action can be used with correct implementation.

Contribution

It demonstrates that the correct implementation of the $i\varepsilon$ prescription allows the Wilson action to be used for real-time lattice gauge theory path integrals.

Findings

Proper $i\varepsilon$ implementation is crucial for continuum limit consistency.

Wilson action can be used for real-time path integrals with $i\varepsilon$.

The $i\varepsilon$ prescription suppresses singular path contributions.

Abstract

Recently, Hoshina, Fujii, and Kikukawa pointed out that the naive lattice gauge theory action in Minkowski signature does not result in a unitary theory in the continuum limit, and Kanwar and Wagman proposed alternative lattice actions to the Wilson action without divergences. We here show that the subtlety can be understood from the asymptotic expansion of the modified Bessel function, which has been discussed for path integral of compact variables in nonrelativistic quantum mechanics. The essential ingredient for defining the appropriate continuum theory is the $i\varepsilon$ prescription, and with the proper implementation of the $i\varepsilon$ we show that the Wilson action can be used for the real-time path integrals. It is here important that the $i\varepsilon$ should be implemented for both timelike and spacelike plaquettes. We also argue the reason why the $i\varepsilon$ becomes required for the Wilson action from the Hamiltonian formalism. The $i\varepsilon$ is needed to manifestly suppress the contributions from singular paths, for which the Wilson action can give different values from those of the actual continuum action.