Real-time diagram technique for instantonic systems

TL;DR

This paper develops a real-time diagram technique for instantonic systems within the Schwinger-Keldysh formalism, addressing the challenge of incorporating instantonic effects and restoring translational invariance in the presence of degenerate minima.

Contribution

It introduces an extension of the Schwinger-Keldysh diagram method to include instantonic effects and restores broken invariance, enabling better analysis of instantonic systems.

Findings

Constructed a modified Schwinger-Keldysh formalism for instantonic systems.

Demonstrated the method on low-order correlation functions.

Showed how to recover translational invariance in the presence of instantons.

Abstract

The Schwinger-Keldysh diagram technique is usually involved in the calculation of real-time in-in correlation functions. In the case of a thermal state, one can analytically continue imaginary-time Matsubara correlation functions to real times. Nevertheless, not all real-time correlation functions usually can be obtained by such procedure. Moreover, numerical analytic continuation is an ill-posed problem. Thus, even in the case of a thermal state one may need for the Schwinger-Keldysh formalism. If the potential of a system admits degenerate minima, instantonic effects enter the game, so one should also integrate over the instantonic moduli space, including the one, corresponding to the imaginary time translational invariance. However, the Schwinger-Keldysh closed time contour explicitly breaks such invariance. We argue, that this invariance must be recovered, and show, how it can be done. After that, we construct an extension of the Schwinger-Keldysh diagram technique to instantonic systems and demonstrate it on the example of the first few-point correlation functions.