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International Journal of Modern Physics B

QUANTUM THEORIES FOR IDENTICAL PARTICLES

Jon Magne Leinaas, Jan Myrheim

Published Oct 1, 1991• 27 citations
Publisher

Abstract

We recall some of the arguments which led us to the concept of fractional statistics in one and two dimensions. This concept has been much studied and is well understood in terms of Schrödinger quantization. Heisenberg quantization is an equally natural approach to the same problem, it is not equivalent, and it leads to many interesting open questions.

DOI
10.1142/s0217979291001024

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We recall some of the arguments which led us to the concept of fractional statistics in one and two dimensions. This concept has been much studied and is well understood in terms of Schrödinger quantization. Heisenberg quantization is an equally natural approach to the same problem, it is not equivalent, and it leads to many interesting open questions.
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.