Representations of the Commutation Relations Describing a Non-Relativistic Infinite Free Bose Gas.

Alain Joye, Marco Merkli

H. Araki, E. J. Woods

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 E(f, g)=(Ψ, eiφ(f) eiπ(g)Ψ) 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.

Evgenii Igorevich Zelenov

The standard C*-formalism of representations of commutations relations for systems of infinitely many degrees of freedom over a real numbers field is extended to the case of a p-adic numbers field. It is proved that any positive function on the Weyl algebra over a p-adic symplectic space is regular and restriction of the generating function of a state of the Weyl algebra on any nongenerate finite-dimensional subspace of p-adic symplectic space is continuous. Some nontrivial class of nonequivalent representations of commutations relations is described and analogies of Fock representations and representations with number operators are constructed. The criteria for the existence of free-field unitary quantization of symplectic transformations is given.

Gerhard C. Hegerfeldt, O. Melsheimer

S. Chaturvedi, Ashok Kumar Kapoor, R. Sandhya +2 more

A generalized commutation relation of a single-mode oscillator is proposed. Bose-Einstein, Fermi-Dirac, and Greenberg's infinite statistics are shown to be special cases of this commutation relation. The Fock states, coherent states, and analogs of squeezed coherent states for this commutation relation are constructed. Finally, the quantization of a free infinite-statistics field is discussed using the Umezawa-Takahashi procedure [H. Umezawa and Y. Takahashi, Prog. Theor. Phys. 9, 14 (1953); Nucl. Phys. 51, 193 (1964)].