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

Recently, Hoshina, Fujii, and Kikukawa in PoS LATTICE2019, 190 (2020) 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 in PRD 104, no.1, 014513 (2021) 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 by W. Langguth and A. Inomata [J. Math. Phys. 20, 499-504 (1979)]. The essential ingredient for defining the appropriate continuum theory is the iε prescription, which we show is applicable also for the Wilson action. It is here important that the iε should be implemented for both timelike and spacelike plaquettes. We then argue that such iε can be given a physical meaning that they remove singular paths having nontrivial winding for an infinitesimal time evolution that do not have corresponding paths in the continuum. Such point of view is only apparent in systems with compact variables as lattice gauge theories.