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Journal of Mathematical Physics

A rigorous approach to the Feynman-Vernon influence functional and its applications. I

Sergio A Albeverio, Laura Cattaneo, Sonia Mazzucchi, Luca Di Persio

Published Oct 1, 2007• 9 citations
Publisher

Abstract

A rigorous representation of the Feynman-Vernon influence functional used to describe open quantum systems is given, based on the theory of infinite dimensional oscillatory integrals. An application to the case of the density matrices describing the Caldeira-Leggett model of two quantum systems with a quadratic interaction is treated.
DOI
10.1063/1.2799881

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Abstract We establish a formal connection between the Feynman–Vernon path integral formalism for open quantum systems and the Wiener path integral used in classical stochastic dynamics, further exploring its applications. By considering a generalized influence functional in the strong decoherence limit, we demonstrate that integrating over the quantum coherence length leads to stochastic Langevin dynamics. Specifically, under our model, we show that the quantum Feynman differential transforms into an underdamped multiplicative non-Markovian Langevin dynamics. Applying this framework to the Wigner function representation, we show that the system follows a stochastic path interpretable via classical probability theory, allowing for the investigation of decoherence through classical dynamics. Finally, we address the inverse problem: constructing an equivalent quantum influence functional from a given classical Langevin equation.
The effective dynamics of a system interacting with a bath or environment is presented in two ways, (1) the (LGKS) replacement of the von Neuman equation for the density matrix and (2) the Feynman-Vernon path-integral derivation, by integrating out the bath degree of freedom, to arrive at a system's density matrix. In this paper, I connect the two methods by deriving a Lindbladian in a mechanical example, a point particle interacting with a bath of harmonic oscillators, previously considered by Feynman and Vernon (FV) and expounded on later by Caldeira and Leggett (CL). But the (FV)/(CL) results only in non-Markov effect, memory terms from the bath interaction. To derive a Lindbladian, I changed the interaction term they considered to take into account the point particle interacting with the bath harmonic oscillators to something more realistic. From the resulting path-integral expression of the system's propagator for the density matrix, the Lindbladian and non-Markov terms are read for this simple problem. I also point out the causes of these terms, the Markov Lindbladian from the very local interaction of the point particle with the classical solutions of the harmonic oscillator and the non-Markov term from the global interaction of the point particle with the fluctuation of the classical solutions.