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World Scientific series in 20th century physics

Influence functional and closed-time-path Green's function

Zhao-bing SU, Liaoyuan Chen, Yu Xiaotong, KUANG-CHAO CHOU

Published May 1, 2009• 0 citations
Publisher

Abstract

In this Brief Report, we explicitly show the equivalence of the Feynman-Vernon influence-functional approach and the path-integral formulation of the Schwinger-Keldysh closed-time-path Green's function. The latter simplifies the practical calculations considerably with a systematic diagrammatic technique and is expected to have a broader application.

DOI
10.1142/9789814280389_0094

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In this Brief Report, we explicitly show the equivalence of the Feynman-Vernon influence-functional approach and the path-integral formulation of the Schwinger-Keldysh closed-time-path Green's function. The latter simplifies the practical calculations considerably with a systematic diagrammatic technique and is expected to have a broader application.
I review the generating function for quantum-statistical mechanics, known as the Feynman-Vernon influence functional, the decoherence functional, or the Schwinger-Keldysh path integral. I describe a probability-conserving iεi\varepsiloniε prescription from a path-integral implementation of Lindblad evolution. I also explain how to generalize the formalism to accommodate out-of-time-ordered correlators (OTOCs), leading to a Larkin-Ovchinnikov path integral. My goal is to provide step-by-step calculations of path integrals associated to the harmonic oscillator.
We present a comprehensive review of the most fundamental and practical aspects of thermo-field dynamics (TFD), including some of the most recent developments in the field. To make TFD fully consistent, some suitable changes in the structure of the thermal doublets and the Bogoliubov transformation matrices have been made. A close comparison between TFD and the Schwinger-Keldysh closed time path formalism (SKF) is presented. We find that TFD and SKF are in many ways the same in form; in particular, the two approaches are identical in stationary situations. However, TFD and SKF are quite different in time-dependent nonequilibrium situations. The main source of this difference is that the time evolution of the density matrix itself is ignored in SKF while in TFD it is replaced by a time-dependent Bogoliubov transformation. In this sense TFD is a better candidate for time-dependent quantum field theory. Even in equilibrium situations, TFD has some remarkable advantages over the Matsubara approach and SKF, the most notable being the Feynman diagram recipes, which we will present. We will show that the calculations of two-point functions are simplified, instead of being complicated, by the matrix nature of the formalism. We will present some explicit calculations using TFD, including space-time inhomogeneous situations and the vacuum polarization in equilibrium relativistic QED.