Real-time diagram technique for instantonic systems

The Schwinger-Keldysh diagram technique is commonly used in calculation of in-in correlation functions in real time. In the case of a thermal state, such correlation functions can be obtained by analytical continuation of the Matsubara correlation functions from imaginary time to real one. However, usually not all in-in correlation functions can be obtained using such a procedure. In addition, analytic continuation using numerical methods is an ill-posed problem. Thus, even in the case of a thermal state, the Schwinger-Keldysh technique may be inevitable. If the potential of the system allows degenerate minima, instanton effects come into play, therefore, when calculating the correlation function, it is necessary to integrate over instanton moduli space, which includes the one corresponding to translational invariance in imaginary time. However, the closed Schwinger-Keldysh time contour explicitly violates this invariance. In this talk I will demonstrate that this invariance must be restored and show how this can be done. After this, I will build a generalization of the Schwinger-Keldysh technique to instanton systems and demonstrate it using the example of the first few-point correlation functions. Finally, I will explain why calculating in-in correlation functions in instanton systems is technically difficult and discuss the physical applications of such calculations.