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International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 890523, 26 pages
http://dx.doi.org/10.1155/2010/890523
Research Article

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

1Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland
2School of Mathematics, West Pomeranian University of Technology, Szczecin, al. Piastów 17, 70-310 Szczecin, Poland

Received 17 May 2010; Accepted 5 December 2010

Academic Editor: Asao Arai

Copyright © 2010 Wiktor Radzki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.