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arXiv:hep-th/9808112 · Acta Phys. Polon.

Representations of the Heisenberg algebra by difference operators

A. Z. Górski, Jacek Szmigielski

Published Aug 19, 1998• 3 citations
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Abstract

We construct a class of representations of the Heisenberg algebra in terms of the complex shift operators subject to the proper continuous limit imposed by the correspondence principle. We find a suitable Hilbert space formulation of our construction for two types of shifts: (1) real shifts, (2) purely imaginary shifts. The representations involving imaginary shifts are free of spectrum doubling. We determine the corresponding coordinate and momentum operators satisfying the canonical commutation relations. The eigenvalues of the coordinate operator are in both cases discrete.

DOI
10.48550/arxiv.hep-th/9808112

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We construct a class of representations of the Heisenberg algebra in terms of the complex shift operators subject to the proper continuous limit imposed by the correspondence principle. We find a suitable Hilbert space formulation of our construction for two types of shifts: (1) real shifts, (2) purely imaginary shifts. The representations involving imaginary shifts are free of spectrum doubling. We determine the corresponding coordinate and momentum operators satisfying the canonical commutation relations. The eigenvalues of the coordinate operator are in both cases discrete.
A large class of continuous representations of separable Hilbert spaces is constructed with the aid of representations of the canonical commutation relations (CCR) for a scalar boson field φ(f) and its canonical conjugate π(g). A representation of the CCR for a scalar boson field consists of two operator-valued functions V[g] and W[f], defined for all f, g in Schwartz's space 𝒮 of real test functions, where V[g] and W[f] are unitary operators defined on some separable Hilbert space ℌ, and which satisfy the commutation relations V[g]W[f] = e−i(f,g) W[f]V[g]. These unitary operators are related to the field and its momenta by V[g] = e−iπ(g), W[f] = eiφ(f). We explicitly construct a family of such representations with the help of von Neumann's theory of infinite direct products of Hilbert spaces, the pertinent parts of which are reviewed. A continuous representation ℌ of the Hilbert space ℭ is composed of a linear vector space of complex, bounded, continuous functionals defined on 𝒮 × 𝒮. These functionals are defined for all Ψ∈H by ψ(f, g)=(V[g]W[f]Φ0, Ψ). In this definition, Φ0 is a fixed unit vector in ℌ. The properties of the functions in ℭ depend on the choice of the representation of the CCR and on the choice of Φ0. When ℭ is constructed with the aid of an irreducible representation of the CCR, an inner product can be defined for all pairs of functionals in ℭ by an intuitively meaningful, rigorously defined integral in the sense of Friedrichs and Shapiro. With this inner product, ℭ is a complete Hilbert space congruent with ℌ. As in all continuous representations, a reproducing kernel exists and determines the functions in the continuous representation. One such space is closely related to a space of analytic functionals introduced by Segal and Bargmann. The representation of various operators as kernels and as functional derivatives is discussed. Finally, the construction of a vast number of unitary invariants for a representation of the CCR is used to establish the unitary inequivalence of uncountably many of the representations that we construct.
A deformation of Heisenberg algebra induces among other consequences a loss of Hermiticity of some operators that generate this algebra. Therefore, these operators are not Hermitian, nor is the Hamiltonian operator built from them. In the present paper, we propose a position deformation of Heisenberg algebra with both maximal length and minimal momentum uncertainties. By using a pseudo-similarity transformation to the non-Hermitian operators, we prove their Hermiticity with a suitable positive-definite pseudo-metric operator. We then construct Hilbert space representations associated with these pseudo-Hermitian operators. Finally, we study the eigenvalue problem of a free particle in this deformed space and we show that this deformation curved the quantum levels allowing particles to jump from one state to another with low energy transitions.
We consider Hilbert space representations of a generalization of canonical commutation relations , where 's are the elements of an algebra with identity , is the imaginary unit, and is a real number with antisymmetry . Some basic aspects on Hilbert space representations of the generalized CCR (GCCR) are discussed. We define a Schrödinger-type representation of the GCCR by an analogy with the usual Schrödinger representation of the CCR with degrees of freedom. Also, we introduce a Weyl-type representation of the GCCR. The main result of the present paper is a uniqueness theorem on Weyl representations of the GCCR.
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We present, for the isospectral family of oscillator Hamiltonians, a systematic procedure for constructing raising and lowering operators satisfying any prescribed “distorted” Heisenberg algebra (including the q-generalization). This is done by means of an operator transformation implemented by a shift operator. The latter is obtained by solving an appropriate partial isometry condition in the Hilbert space. Formal representations of the nonlocal operators concerned are given in terms of pseudo-differential operators. Using the new annihilation operators, new classes of coherent states are constructed for isospectral oscillator Hamiltonians. The corresponding Fock–Bargmann representations are also considered, with specific reference to the order of the entire function family in each case.