Contractions of Product Density Operators of Systems of Identical Fermions and Bosons

Wiktor Radzki

doi:10.1155/2010/890523

Abstract

Recurrence and explicit formulae for contractions (partial traces) of antisymmetric and symmetric products of identical trace class operators are derived. Contractions of product density operators of systems of identical fermions and bosons are proved to be asymptotically equivalent to, respectively, antisymmetric and symmetric products of density operators of a single particle, multiplied by a normalization integer. The asymptotic equivalence relation is defined in terms of the thermodynamic limit of expectation values of observables in the states represented by given density operators. For some weaker relation of asymptotic equivalence, concerning the thermodynamic limit of expectation values of product observables, normalized antisymmetric and symmetric products of density operators of a single particle are shown to be equivalent to tensor products of density operators of a single particle.

Highlights

This paper see preprint 1, presenting the results of a part of the author’s thesis 2, deals with contractions partial traces of antisymmetric and symmetric product density operators representing mixed states of systems of identical noninteracting fermions and bosons, respectively.If H is a separable Hilbert space of a single fermion boson, the space of the nfermion resp. n-boson system is the antisymmetric resp. symmetric subspace H∧n resp
Density operators of n-fermion resp. n-boson systems are identified with those defined on H⊗n and concentrated on H∧n resp
The first objective of this paper is to find the recurrence and explicit formulae for LknK and LknG for K and G being, respectively, antisymmetric and symmetric products of identical trace class operators, including operators 1.3

Summary

Introduction

This paper see preprint 1 , presenting the results of a part of the author’s thesis 2 , deals with contractions partial traces of antisymmetric and symmetric product density operators representing mixed states of systems of identical noninteracting fermions and bosons, respectively. K-particle observables of n-fermion and n-boson systems k < n are represented, respectively, by operators of the form. The first objective of this paper is to find the recurrence and explicit formulae for LknK and LknG for K and G being, respectively, antisymmetric and symmetric products of identical trace class operators, including operators 1.3. The problems described above have been solved for k 1, 2 by Kossakowski and Mackowiak , and Mackowiak The formulae they derived were exploited in calculations of the free energy density of large interacting n-fermion and n-boson systems. The main results of this paper are Theorems 3.1, 3.4, 4.9, and 4.14

Basic Notation

Contractions of Operators

Then n k

Findings

Asymptotic Form for Contractions of Product States