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


2022/07/12 10:00 – 11:00


Online (Zoom) 


Dr. Nobuyuki Matsumoto ( RIKEN BNL Research Center )





Katsuta SAKAI /


Recently, Hoshina, Fujii, and Kikukawa [1] 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 [2] 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 [3,4]. 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. This talk is based on [5].
[1] H. Hoshina, H. Fujii and Y. Kikukawa, “Schwinger-Keldysh formalism for Lattice Gauge Theories,” PoS LATTICE2019, 190 (2020)
[2] G. Kanwar and M. L. Wagman, “Real-time lattice gauge theory actions: Unitarity, convergence, and path integral contour deformations,” Phys. Rev. D 104, no.1, 014513 (2021) [arXiv:2103.02602 [hep-lat]]
[3] W. Langguth and A. Inomata, “Remarks on the Hamiltonian path integral in polar coordinates,” J. Math. Phys. 20, 499-504 (1979)
[4] M. Bohm and G. Junker, “Path integration over compact and noncompact rotation groups,” J. Math. Phys. 28, 1978-1994 (1987)
[5] N. M. “Comment on the subtlety of defining real-time path integral in lattice gauge theories,” [arXiv:2206.00865 [hep-lat]]

Release date 2022/06/30 Updated 2024/03/02