Schwinger-Keldysh path integral for the quantum harmonic oscillator

Abstract

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 $i\epsilon$ 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.

112 pages, 1 figure

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• Chaos in the black hole S-matrix
• Breakdown of the coherent state path integral: two simple examples
• Renormalisation in Open Quantum Field theory II: Yukawa theory and PV reduction