Not to normal order - Notes on the kinetic limit for weakly interacting quantum fluids

TL;DR

This paper explores the derivation of the quantum Boltzmann equation for weakly interacting quantum fluids, proposing a novel approach inspired by nonlinear Schrödinger equation techniques that avoids normal ordering in expansions.

Contribution

It introduces a new method for deriving the quantum Boltzmann equation by leveraging techniques from nonlinear Schrödinger theory, bypassing traditional normal ordering constraints.

Findings

Term-by-term convergence of the Duhamel expansion

Analysis of equilibrium time correlation functions

Extension of classical kinetic methods to quantum fluids

Abstract

The derivation of the Nordheim-Boltzmann transport equation for weakly interacting quantum fluids is a longstanding problem in mathematical physics. Inspired by the method developed to handle classical dilute gases, a conventional approach is the use of the BBGKY hierarchy for the time-dependent reduced density matrices. In contrast, our contribution is motivated by the kinetic theory of the weakly nonlinear Schrodinger equation. The main observation is that the results obtained in the latter context carry over directly to weakly interacting quantum fluids provided one does not insist on normal order in the Duhamel expansion. We discuss the term by term convergence of the expansion and the equilibrium time correlation <a(t)* a(0)>.