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Journal of Physics A Mathematical and General

Homotopy and statistics: of spin and diagonals

Lawrence S. Schulman

Published Oct 21, 1993• 1 citation
Publisher

Abstract

The path integral treatment of statistics for identical particles is extended to the case where they possess spin by virtue of a continuous internal coordinate. In this case the restriction to bosons or fermions begins at space dimension two. The role of the 'diagonal' (points of coordinate coincidence) in these arguments is clarified.

DOI
10.1088/0305-4470/26/20/006

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The path integral treatment of statistics for identical particles is extended to the case where they possess spin by virtue of a continuous internal coordinate. In this case the restriction to bosons or fermions begins at space dimension two. The role of the 'diagonal' (points of coordinate coincidence) in these arguments is clarified.
Abstract We introduce the idea of the path integral and use it to study the properties of identical particles under exhange. We explain why the topology of paths through space-time, the fundamental group of the configuration space (the braid group in 2+1 dimensions and the permutation group in 3+1 dimensions) determines the possible particle types. We argue that in 3+1 dimensions one only has bosons and fermions but in 2+1 dimensions other particles, known as anyons, can exist. We describe both the abelian and nonabelian type of anyon and argue why nonabelian anyons could be useful for quantum computation. We introduce the notion of the “quantum dimension” of an anyon which describes the size of the many anyon Hilbert space.
We present a description of the quantum mechanical states of a system of n indistinguishable particles moving on a manifold M by C/S2- valued functions ψ′ defined on the configuration space Mn/Sn. These functions satisfy one of two homotopy conditions, which characterize the particles as bosons or fermions. Any closed curve of Mn/Sn which does not intersect the diagonals can be classified as even or odd according to whether its lifts to Mn have end points in Mn which are even or odd permutations of each other. For bosons, ψ′ must map any such closed curve which also avoids kerψ′ onto an even number of basic loops of (C−{0})/S2. For fermions, ψ′ must map such even loops of Mn/Sn onto an even number, and odd loops onto an odd number, of basic loops of (C−{0})/S2.
We show that there are novel generalizations of statistics for identical particles in one dimension. These arise due to possible boundary conditions on wave functions, or equivalently due to δ-function interactions, at points of coincidence of particle coordinates. Special choices of these boundary conditions describe bosons, fermions or paraparticles. The general solution for the boundary conditions involves vector-valued wave functions and statistics with non-Abelian features, even though the classical configuration space has an Abelian fundamental group. Physical models leading to such non-Abelian statistics, involving for example δ-function interactions of spins, are constructed. Properties under parity and time reversal of the new boundary conditions are studied. It is shown that the Bethe ansatz does not always give eigenstates of energy. When it does, the wave numbers in the ansatz for three or more particles must satisfy a weaker form of the Yang-Baxter equations and certain additional equations. The energy spectrum and wave functions for identical particles on R1 or S1 are discussed in some simple cases. Our work generalizes the previous work of Lieb and Liniger, Lieb, Yang and Leinaas and Myrheim.
A generalized framework for particle statistics is developed within which it becomes clear how bosons can combine to form fermions without violating the normal connection between spin and statistics. In the situation where internal and external particle degrees of freedom are separable, the possible statistical types are found and their combining rules derived. Here the statistical "interaction" between two particles is characterized either by the familiar sign or by an integer ("twist number"). However, further possibilities arise if the separability of external from internal degrees of freedom fails, as it can in gauge theories. In spatial dimensions other than three the "fermion from boson" phenomenon does not occur.