Representations of the Canonical Commutation Relations Describing a Nonrelativistic Infinite Free Bose Gas

Abstract

The existence of inequivalent representations of the canonical commutation relations which describe a nonrelativistic infinite free Bose gas of uniform density is investigated, with a view to possible applications to statistical mechanics. The functional ${\mathrm{E(f,\hspace{0.17em}g)=(\Psi ,\hspace{0.17em}e}}^{\mathrm{i\phi (f)}}{\mathrm{\hspace{0.17em}e}}^{\mathrm{i\pi (g)}}\mathrm{\Psi )}$ is used to describe the inequivalent representations. This functional is calculated for the free Bose gas in a box of volume V, and the limit is then taken as V → ∞. In this way we construct cyclic representations describing an infinite system of particles with a density distribution ρ(k) in momentum space. For a given ρ(k) the operator algebra generated by the φ(f), π(g) is reducible. For the ground‐state representation (all particles in the zero‐momentum state), the representation is a direct integral of irreducible representations (analogous to BCS theory). For finite temperatures the situation is complicated by the occurrence of representations which are not type I. The physical significance of the reducibility of the representations is discussed. It is argued that the thermal ensemble for the infinite system is a pure state at zero temperature, although there is some ambiguity as to which operators should belong to the algebra of observables. For finite temperatures, the thermal ensemble seems to be a mixture. The case of an interacting Bose gas is considered briefly.