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International Journal of Modern Physics B

CLASSICAL TOPOLOGY AND QUANTUM STATISTICS

A. P. Balachandran

Published Oct 1, 1991• 17 citations
Publisher

Abstract

We review the theory of statistics of identical systems in terms of the properties of the classical configuration space. Special emphasis is placed on the influence of the connectivity of this space on quantum statistics. The theory shows that exotic statistical systems, such as those obeying fractional statistics, can occur in all dimensions. Particle statistics in one dimension involves novel features particular to this dimension and these are briefly described. The proof of a topological spin-statistics theorem which avoids field theory or relativity is also outlined.

DOI
10.1142/s0217979291001036

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We review the theory of statistics of identical systems in terms of the properties of the classical configuration space. Special emphasis is placed on the influence of the connectivity of this space on quantum statistics. The theory shows that exotic statistical systems, such as those obeying fractional statistics, can occur in all dimensions. Particle statistics in one dimension involves novel features particular to this dimension and these are briefly described. The proof of a topological spin-statistics theorem which avoids field theory or relativity is also outlined.
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We show that if one uses a multisheet configuration space for a system of identical particles, then fractional quantum statistics in two dimensions can be geometrically characterized as a topological quantum number for going from one sheet to the next. Furthermore, the spectrum of such a q-sheet system is shown to consist of the spectra of all species of anyons obeying different fractional statistics \ensuremath{\theta}=2\ensuremath{\pi}p/q (p=0,1, . . .,q-1) with the same denominator q. A lattice computation has been done to verify this. Various aspects of the multisheet formalism are discussed.
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