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Journal of Mathematical Physics

Translations of the Discrete Bose-Einstein Operators

David Shelupsky

Published Jan 1, 1966• 4 citations
Publisher

Abstract

The effects on a certain family of representations of the commutation relations of the operator-translations ak → ak + ck, ak* → ak* + ck*, are investigated. Here ak and ak* are the annihilation and creation operators of the representations, the ck are complex numbers (scalar operators), and the representations considered are the discrete representations described by Gårding and Wightman, and Schweber and Wightman, generalizations of the usual Fock representation. Necessary and sufficient conditions on the translations are obtained for the resultant representations to remain in the family of discrete representations, and the resultant translation groups are investigated for their relevant structure.

DOI
10.1063/1.1704804

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The effects on a certain family of representations of the commutation relations of the operator-translations ak → ak + ck, ak* → ak* + ck*, are investigated. Here ak and ak* are the annihilation and creation operators of the representations, the ck are complex numbers (scalar operators), and the representations considered are the discrete representations described by Gårding and Wightman, and Schweber and Wightman, generalizations of the usual Fock representation. Necessary and sufficient conditions on the translations are obtained for the resultant representations to remain in the family of discrete representations, and the resultant translation groups are investigated for their relevant structure.
All of the irreducible representations are found for a single pair of creation and annihilation operators which together with the symmetric or antisymmetric number operator satisfy the generalized commutation relation characteristic of para-Bose or para-Fermi field quantization. The procedure is simply to identify certain combinations of these three operators with the three generators of the three-dimensional rotation group in the para-Fermi case, and with the three generators of the three-dimensional Lorentz group in the para-Bose case. The irreducible representations are then easily obtained by the usual raising and lowering operator techniques. The applicability of these techniques is demonstrated by a simple argument which shows that the commutation relations require that the generator to be diagonalized have a discrete spectrum.
Abstract: \n In this paper we consider general commutative relations and construct its irreducible representations. Also we prove classified theorems. These commutative relations appear, in particle, in connection with nonstandard Poisson brackets. As applications we consider quantum Hamiltonians, which are written for creation and annihilation operators with arbitrary commutative relations. For this Hamiltonians we prove a criterion of presence of conservation laws depending on particle number operators.
Representations of discrete symmetry operators (DSO's) connected with space (𝒫), time (T), and generalized charge (𝒞) are considered. It is shown that if one writes a DSO as exp (iπΩ) × a phase transformation, then (under certain conditions on Ωs) to each DSO there corresponds a set of Ωs which is closed with respect a Lie algebra, which is isomorphic to the Lie algebra of generators of rotation in an n-dimensional Euclidean space; where n is the number of commuting observables that changes sign under this DSO in a given representation (e.g. linear momentum representation). In the particular case of the (T𝒞𝒫) operation, there are six Ωs, of which two are diagonal, viz. the generalized charge Q, and spin projection along the z axis Sz; corresponding Euclidean group is four-dimensional. For the sake of completeness, the representations are also given for the following cases: (i) nonrelativistic quantum mechanics, (ii) quantum theory of free fields, in terms of field operators.
We consider Fock representations of the Q-deformed commutation relations ∂s∂t†=Q(s,t)∂t†∂s+δ(s,t) for s,t∈T. Here T:=Rd (or more generally T is a locally compact Polish space), the function Q:T2→C satisfies |Q(s,t)|≤1 and Q(s,t)=Q(t,s)¯, and ∫T2h(s)g(t)δ(s,t) σ(ds)σ(dt):=∫Th(t)g(t) σ(dt), σ being a fixed reference measure on T. In the case, where |Q(s,t)|≡1, the Q-deformed commutation relations describe a generalized statistics studied by Liguori and Mintchev. These generalized statistics contain anyon statistics as a special case (with T=R2 and a special choice of the function Q). The related Q-deformed Fock space F(H) over H:=L2(T→C,σ) is constructed. An explicit form of the orthogonal projection of H⊗n onto the n-particle space Fn(H) is derived. A scalar product in Fn(H) is given by an operator Pn≥0 in H⊗n which is strictly positive on Fn(H). We realize the smeared operators ∂t† and ∂t as creation and annihilation operators in F(H), respectively. Additional Q-commutation relations are obtained between the creation operators and between the annihilation operators. They are of the form ∂s†∂t†=Q(t,s)∂t†∂s†, ∂s∂t=Q(t,s)∂t∂s, valid for those s,t∈T for which |Q(s, t)| = 1.
We construct a new class of representations of the canonical commutation relations, which generalizes previously known classes. We perturb the infinitesimal generator of the initial Fock representation (in other words, the free quantum field) by a function of the field which is square-integrable with respect to the associated Gaussian measure. We characterize perturbations which lead to representations of the canonical commutation relations. We provide conditions entailing the irreducibility of such representations, show explicitly that our class of representations subsumes previously studied classes, and give necessary and sufficient conditions for our representations to be unitarily equivalent, or just quasi-equivalent, with Fock, coherent or quasifree representations. 1991 Mathematics Subject Classification: 81S05, 46L60, 81T05.