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arXiv:hep-th/9611167

Generalized statistics and the algebra of observables

Jon Magne Leinaas

Published Nov 21, 1996• 6 citations
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Abstract

A short review is given of how to apply the algebraic Heisenberg quantization scheme to a system of identical particles. For two particles in one dimension the approach leads to a generalization of the Bose and Fermi description which can be expressed in the form of a 1/x^2 statistics interaction between the particles. For an N-particle system it is shown how a particular infinite-dimensional algebra arises as a generalization of the su(1,1) algebra which is present for the two-particle system.

DOI
10.48550/arxiv.hep-th/9611167

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A short review is given of how to apply the algebraic Heisenberg quantization scheme to a system of identical particles. For two particles in one dimension the approach leads to a generalization of the Bose and Fermi description which can be expressed in the form of a 1/x^2 statistics interaction between the particles. For an N-particle system it is shown how a particular infinite-dimensional algebra arises as a generalization of the su(1,1) algebra which is present for the two-particle system.
We show that the algebraic quantization method of Heisenberg and the analytical method of Schrödinger are not necessarily equivalent when applied to systems of identical particles. Heisenberg quantization is a natural approach, but inherently more ambiguous and difficult than Schrödinger quantization. We apply the Heisenberg method to the examples of two identical particles in one and two dimensions, and relate the results to the so-called fractional statistics known from Schrödinger quantization. For two particles in d dimensions we look for linear, Hermitian representations of the symplectic Lie algebra sp(d, R). The boson and fermion representations are special cases, but there exist other representations. In one dimension there is a continuous interpolation between boson and fermion systems, different from the interpolation found in Schrödinger quantization. In two dimensions we find representations that can be realized in terms of multicomponent wave functions on a three-dimensional space, but we have no clear physical interpretation of these representations, which include extra degrees of freedom compared to the classical system.
A few years ago, Agarwal (1991 Phys. Rev. A 44 8398) showed that the Susskind–Glogower phase operators, expressible in terms of Bose operators, provide a realization of the algebra for particles obeying infinite statistics. In this paper we show that the SU (1, 1) phase operators, constructed in terms of the elements of the su (1, 1) Lie algebra, also provide a realization of the algebra for infinite statistics. There are many realizations of the su (1, 1) algebra in terms of single or multimode bose operators, three of which are discussed along with their corresponding phase states. The Susskind–Glogower phase operator is a special case of the SU (1, 1) phase operator associated with the Holstein–Primakoff realization of su (1, 1).