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Communications in Theoretical Physics

θ Statistics and Quantum Group

Yi-Xin Chen, Guang-jiong Ni

Published Jun 1, 1992• 2 citations
Publisher

Abstract

General theory of quantum statistics with internal degree of freedom in two dimensions is studied. Because of the topological nontrivial properties in the configuration space of indistinguishable particles, the intrinsic part of many-body wave function is associated with the representation of quantum group, which describes θ statistics. The statistical potential and the effective Hamiltonian of general two-dimensional system can be derived from the Kohno connection.

DOI
10.1088/0253-6102/17/4/499

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arXiv:math-ph/0302037•Journal of Mathematical Physics•Mar 17, 2004•4 citations
General theory of quantum statistics with internal degree of freedom in two dimensions is studied. Because of the topological nontrivial properties in the configuration space of indistinguishable particles, the intrinsic part of many-body wave function is associated with the representation of quantum group, which describes θ statistics. The statistical potential and the effective Hamiltonian of general two-dimensional system can be derived from the Kohno connection.
The standard topological approach to indistinguishable particles formulates exchange statistics by using the fundamental group to analyze the connectedness of the configuration space. Although successful in two and more dimensions, this approach gives only trivial or near-trivial exchange statistics in one dimension because two-body coincidences are excluded from configuration space. Instead, we include these path-ambiguous singular points and consider configuration space as an orbifold. This orbifold topological approach allows unified analysis of exchange statistics in any dimension and predicts possibilities for anyons in one-dimensional systems, including non-Abelian anyons obeying alternate strand groups. These results clarify the nontopological origin of fractional statistics in one-dimensional anyon models.
This chapter describes how, in two-dimensional systems, the scalar fractional statistics may occur besides Bose–Einstein and Fermi–Dirac statistics. One-dimensional irreducible unitary representations of the full braid group for selected manifolds (Euclidean plane, sphere, torus, and Euclidean three-dimensional space) and its relation with symmetry of the wave function of the system are presented. The idea of anyons — particles obeying fractional statistics — is discussed in detail. The relation between spin and statistics in two- and three-dimensional systems and its topological nature is discussed. The chapter also shows how magnetic flux attached to particles in planar systems can change their statistical properties.
A treatment of the spin-statistics relation in nonrelativistic quantum mechanics due to Berry and Robbins [Proc. R. Soc. London Ser. A 453, 1771–1790 (1997)] is generalized within a group-theoretical framework. The construction of Berry and Robbins is reformulated in terms of certain locally flat vector bundles over n-particle configuration space. It is shown how families of such bundles can be constructed from irreducible representations of the group SU(2n). The construction of Berry and Robbins, which leads to a definite connection between spin and statistics (the physically correct connection), is shown to correspond to the completely symmetric representations. The spin-statistics connection is typically broken for general SU(2n) representations, which may admit, for a given value of spin, both Bose and Fermi statistics, as well as parastatistics. The determination of the allowed values of the spin and statistics reduces to the decomposition of certain zero-weight representations of a (generalized) Weyl group of SU(2n). A formula for this decomposition is obtained using the Littlewood–Richardson theorem for the decomposition of representations of U(m+n) into representations of U(m)×U(n).