Self-Energy Renormalization for Inhomogeneous Nonequilibrium Systems and Field Expansion via Complete Set of Time-Dependent Wave Functions

TL;DR

This paper develops a renormalization framework for inhomogeneous quantum fields in equilibrium and nonequilibrium states, deriving quantum transport equations and demonstrating relaxation to equilibrium in a triple-well model.

Contribution

It extends self-energy renormalization to inhomogeneous, nonequilibrium systems using Thermo Field Dynamics and introduces a field expansion with time-dependent wave functions.

Findings

Derived quantum transport equations from renormalization conditions.

Numerical simulations show relaxation to equilibrium in a triple-well model.

Determined all matrix elements of the energy counter term.

Abstract

The renormalization conditions of inhomogeneous systems of a quantum field under an external potential are studied, for both equilibrium and nonequilibrium scenarios and based on Thermo Field Dynamics. Extending the concept of the on-shell self-energies to these systems, we impose the renormalization conditions upon them. All the matrix elements of the energy counter term are determined. In the nonequilibrium case, in which the field operator is expanded to time-dependent wave functions so as to satisfy the appropriately chosen differential equation, the quantum transport equation is derived from the renormalization condition. Through numerical calculations of a triple-well model with a reservoir, we show that the number distribution and the time-dependent wave functions are relaxed to the correct equilibrium forms at the long-term limit.