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Journal of Mathematical Physics

Infinite-dimensional representations of the graded Lie algebra (Sp(4):4). Representation of the para-Bose operators with real order of quantization

I. Inaba, Takayoshi Maekawa, TAKUO YAMAMOTO

Published Jun 1, 1982• 6 citations

Abstract

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).

DOI: 10.1063/1.525462

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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).

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.

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.

In this paper we reexamine the definition of parafermions and parabosons by means of Green's triple relations, and extend these relations by including a parity operator

P

which is also determined by means of triple relations. As a consequence, we are dealing with new algebraic structures. It is shown that the algebra underlying a set of

n

parafermions together with

P

is the orthogonal Lie algebra

so(2n+2)

. The Fock spaces correspond to particular irreducible representations of

so(2n+2)

, and the action of

P

in these spaces leads to interesting observations. Next, we show that the algebra underlying a set of

n

parabosons together with

P

is the orthosymplectic Lie superalgebra

osp(2|2n)

. In this case, the Fock spaces correspond to certain irreducible infinite-dimensional representations of

osp(2|2n)

. Both for parafermions and parabosons the spectrum of

P

is closely related to the so-called order of statistics

p

, introduced by Green.

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.