Contractions of product density operators of systems of identical fermions and bosons

TL;DR

This paper derives formulas for contractions of symmetric and antisymmetric products of density operators in quantum systems, showing their asymptotic equivalence to single-particle density operators in the thermodynamic limit.

Contribution

It provides explicit recurrence and formulae for contractions of product density operators and establishes their asymptotic equivalence to single-particle operators for fermions and bosons.

Findings

Contractions are asymptotically equivalent to antisymmetric and symmetric products of single-particle density operators.

Explicit recurrence relations for contractions are derived.

Asymptotic equivalence holds in the thermodynamic limit for expectation values of observables.

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. This paper presents the results of a part of the author's thesis [W. Radzki, "Kummer contractions of product density matrices of systems of $n$ fermions and $n$ bosons" (Polish), MS thesis, Institute of Physics, Nicolaus Copernicus University, Toru\'{n}, 1999].