The particle structure of the quantum mechanical Bose and Fermi gas

Abstract

In the framework of von Neumann’s description of measurements of discrete quantum observables we establish a one-to-one correspondence between symmetric statistical operators W of quantum mechanical systems and classical point processes κ_W, thereby giving a particle picture of indistinguishable quantum particles. This holds true under irreducibility assumptions if we fix the underlying complete orthonormal system. The method of the Campbell measure is developed for such statistical operators; it is shown that the Campbell measure of a statistical operator W coincides with the Campbell measure of the corresponding point process κ_W. Moreover, again under irreducibility assumptions, a symmetric statistical operator is completely determined by its Campbell measure. Themethod of the Campbell measure then is used to characterize Bose-Einstein and Fermi-Dirac statistical operators. This is an elementary introduction into the work of Fichtner and Freudenberg [10, 11] combined with the quantum mechanical investigations of [2] and the corresponding point process approach of [30]. It is based on the classical work of von Neumann [22], Segal, Cook and Chaiken [7, 8, 28] as well as Moyal [18].