Fluctuation-dissipation theorem and quantum tunneling with dissipation at finite temperature

TL;DR

This paper reformulates the fluctuation-dissipation theorem to include dissipation and quantum tunneling at finite temperature, using an oscillator model without explicit oscillators, and extends it to fermionic systems for broader physical applications.

Contribution

It presents a new reformulation of the fluctuation-dissipation theorem that incorporates dissipation and quantum tunneling without explicit oscillator models, and generalizes it to fermionic systems.

Findings

Reproduces tunneling formulas with dissipation at finite temperature

Generalizes fluctuation-dissipation theorem to fermionic systems

Provides a physical interpretation in terms of second quantized fermionic oscillators

Abstract

A reformulation of the fluctuation-dissipation theorem of Callen and Welton is presented in such a manner that the basic idea of Feynman-Vernon and Caldeira -Leggett of using an infinite number of oscillators to simulate the dissipative medium is realized manifestly without actually introducing oscillators. If one assumes the existence of a well defined dissipative coefficient $R(\omega)$ which little depends on the temperature in the energy region we are interested in, the spontanous and induced emissions as well as induced absorption of these effective oscillators with correct Bose distribution automatically appears. Combined with a dispersion relation, we reproduce the tunneling formula in the presence of dissipation at finite temperature without referring to an explicit model Lagrangian. The fluctuation-dissipation theorem of Callen-Welton is also generalized to the fermionic dissipation (or fluctuation) which allows a transparent physical interpretation in terms of second quantized fermionic oscillators. This fermionic version of fluctuation-dissipation theorem may become relevant in the analyses of, for example, fermion radiation from a black hole and also supersymmetry at the early universe.