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Journal of Mathematical Physics

Representations and properties of para-Bose oscillator operators. I. Energy position and momentum eigenstates

Narasimhaiengar Mukunda, E. C. G. Sudarshan, J. K. Sharma, C. L. Mehta

Published Sep 1, 1980• 101 citations
Publisher

Abstract

Para-Bose commutation relations are related to the SL(2,R) Lie algebra. The irreducible representation 𝒟α of the para-Bose system is obtained as the direct sum Dβ⊕Dβ+1/2 of the representations of the SL(2,R) Lie algebra. The position and momentum eigenstates are then obtained in this representation 𝒟α, using the matrix mechanical method. The orthogonality, completeness, and the overlap of these eigenstates are derived. The momentum eigenstates are also derived using the wave mechanical method by specifying the domain of the definition of the momentum operator in addition to giving it a formal differential expression. By a careful consideration in this manner we find that the two apparently different solutions obtained by Ohnuki and Kamefuchi in this context are actually unitarily equivalent.

DOI
10.1063/1.524695

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Para-Bose commutation relations are related to the SL(2,R) Lie algebra. The irreducible representation 𝒟α of the para-Bose system is obtained as the direct sum Dβ⊕Dβ+1/2 of the representations of the SL(2,R) Lie algebra. The position and momentum eigenstates are then obtained in this representation 𝒟α, using the matrix mechanical method. The orthogonality, completeness, and the overlap of these eigenstates are derived. The momentum eigenstates are also derived using the wave mechanical method by specifying the domain of the definition of the momentum operator in addition to giving it a formal differential expression. By a careful consideration in this manner we find that the two apparently different solutions obtained by Ohnuki and Kamefuchi in this context are actually unitarily equivalent.
The energy, position, and momentum eigenstates of a para-Bose oscillator system were considered in paper I. Here we consider the Bargmann or the analytic function description of the para-Bose system. This brings in, in a natural way, the coherent states ‖z;α〉 defined as the eigenstates of the annihilation operator ?. The transformation functions relating this description to the energy, position, and momentum eigenstates are explicitly obtained. Possible resolution of the identity operator using coherent states is examined. A particular resolution contains two integrals, one containing the diagonal basis ‖z;α〉〈z;α‖ and the other containing the pseudodiagonal basis ‖z;α〉〈−z;α‖. We briefly consider the normal and antinormal ordering of the operators and their diagonal and discrete diagonal coherent state approximations. The problem of constructing states with a minimum value of the product of the position and momentum uncertainties and the possible α dependence of this minimum value is considered.
Infinite-dimensional representations for the system of the two para-Bose operators, which generate the graded Lie algebra (GLA) (Sp(4):4), are constructed by using the irreducible representations (discrete series) of the Lie algebra Sp(4)≊SO(3,2). It is shown that there are four kinds of the irreducible representations of the GLA’s (Sp(4):4), i.e., cases (I), (II), (III), and (IV). Case (I) is described by the three irreducible representations of Sp(4) and corresponds to the para-Bose quantization with real order of the quantization greater than 1 and case (IV) corresponding to the ordinary Bose quantization by the two irreducible representations of Sp(4). Cases (II) and (III), which are described by the four and two irreducible representations of Sp(4), respectively, and cannot be obtained by the method of Fock space, express the representations of the graded Lie algebra (Sp(4):4).
It is known that para-particles, together with fermions and bosons, of a single mode can be described as an irreducible representation of the Lie (super) algebra 𝔰𝔩2(ℂ) (2-dimensional special linear algebra over the complex number ℂ), that is, they satisfy the equation of motion of a harmonic oscillator. Under the equation of motion of a harmonic oscillator, we obtain the set of the commutation relations which is isomorphic to the irreducible representation, to find that the equation of motion, conversely, can be derived from the commutation relation only for the case of either fermion or boson. If Nature admits of the existence of such a sufficient condition for the equation of motion of a harmonic oscillator, no para-particle can be allowed.
A detailed study is made of the wave-mechanical representation of a one-dimensional Bose-like oscillator whose canonical variables satisfy the general commutation relations first proposed by Wigner. The eigenvalue problems of the momentum and Hamiltonian operators are completely solved, and this is made possible only when wavefunctions in general are allowed to be hyperfunctions. The equivalence between the wave- and matrix-mechanical representations is thereby established for any value of c (a characteristic parameter of the theory), contrary to the conclusion reached previously by Yang. It is also found that for the case −1/2<c<0 or 0<c<1/2 there exist two classes of eigenfunctions that are mutually separated by a superselection rule.
In place of the usual commutation relation [â,â†]=1 we consider the generalized commutation relation characteristic of the para-Bose oscillators, viz, [â, Ĥ]=Ĥ where Ĥ is the Hamiltonian (1/2)(ââ†+â†â). The number states and the representation of various operators in the basis formed by these states are obtained. We then introduce the para-Bose coherent states defined as the eigenstates of â for this generalized case. We consider some of the properties of these coherent states and also show that the uncertainty product 〈 (Δq̂)2〉 〈 (Δp̂)2〉 in this case could be made arbitrarily small.