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Journal of Mathematical Physics

Irreducible Representations of Generalized Oscillator Operators

Thomas F. Jordan, Narasimhaiengar Mukunda, Stephen V. Pepper

Published Aug 1, 1963• 43 citations
Publisher

Abstract

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.

DOI
10.1063/1.1704038

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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.
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.
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).
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.
We consider the relationship between the unitary quantization scheme and the para-Fermi statistics of order 2. We propose an appropriate generalization of Green’s ansatz, which has made it possible to transform bilinear and trilinear commutation relations for the creation and annihilation operators for two different para-Fermi fields φ a and φ b into identities. We also propose a method for incorporating para-Grassmann numbers ξ k into the general unitary quantization scheme. For the parastatistics of order 2, a new fact has been revealed: the trilinear relations containing both para-Grassmann variables ξ k and field operators ak and bm are transformed under a certain reversible mapping into unitary equivalent relations in which commutators are replaced by anticommutators, and vice versa. It is shown that this leads to the existence of two alternative definitions of the coherent state for para-Fermi oscillators. The Klein transformation for Green’s components of operators a k and b m is constructed in explicit form, which enabled us to reduce the initial commutation rules for the components to the normal commutation relations for ordinary Fermi fields. We have analyzed a nontrivial relationship between the trilinear commutation relations of the unitary quantization scheme and the so-called Lie supertriple system. The possibility of incorporating the Duffin–Kemmer–Petiau theory into the unitary quantization scheme is discussed briefly.
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.