Spin-Statistics and Topology

Abstract

The spin-statistics theorem is quite a remarkable one since it correlates two very distinct kinds of symmetries, namely spin symmetry, which is a property of single particles, and exchange symmetry, which is a property of sets of particles, and is trivial for single particles. The theorem is, of course, well-established within the framework of local relativistic field theory¹, but for a number of reasons the time would seem to be ripe for placing it in a broader context. Among these reasons are the following:

• (a) The theorem is now known to be valid for a variety of objects, notably extended objects such as solitons² and strings³, which are not described by local relativistic field theory.

• (b) With the advent of topological considerations in field theory it has been realized that the spin-statistics theorem has a topological basis⁴. Furthermore, it has been realized that it is only the first in a serties of topological properties, the next in the series being the existence of a Wess-Zumino (closed but not exact) 2-form⁵.

• (c) There is a general feeling that, even within the context of local relativistic field theory, it should not be necessary to use the full machinery to establish the spin-statics correlation, but that it should follow directly from some key topological assumptions.