Abstract
We describe here the basic idea of fractional statistics in low dimensions. In quantum mechanics, multiplication of the wave function by a complex phase factor does not change the physical state described by the wave function. Another such physical identity transformation is a permutation of arguments in the wave function if they refer to identical particles. In three dimensions, a swapping of arguments gives rise to a phase factor of + 1 for two bosons and − 1 for two fermions. In two dimensions, fractional statistics appears as a possibility, where an interchange of two particles via a continuous path in the configuration space may give rise to a more general phase factor of e±iθ, with the sign of the phase depending on the direction of the path. In one dimension, two particles can be interchanged only by passing through each other. This introduces boundary conditions on the wave functions that may lead to other kinds of fractional statistics. Physical applications of these ideas are only briefly touched upon.
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