  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].

Quantum Physics