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Electronic Journal of Differential Equations

The spin-statistics connection; some pedagogical remarks in response to Neuenschwander's question.

Arthur S. Wightman

Published Jul 1, 2000• 6 citations
OpenAlex

Abstract

Q: Is there a simple proof of the Spin-Statistics Theorem? A: Not if all that is assumed is invariance under the Euclidean Group. When this hypothesis is supplemented by invariance under restricted Lorentz transformations, the key move is to disregard Henry David Thoreau's advice: "Simplify, simplify" and instead: complexify, complexify.

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Q: Is there a simple proof of the Spin-Statistics Theorem? A: Not if all that is assumed is invariance under the Euclidean Group. When this hypothesis is supplemented by invariance under restricted Lorentz transformations, the key move is to disregard Henry David Thoreau's advice: "Simplify, simplify" and instead: complexify, complexify.
The proof given by Pauli (Pauli, Relativistic Theory of Elementary Particles'' IIL, 1947, Supplement) for a theorem concerning the connection between the spin and the statistics is based on a consideration of irreducible representations of the tensor and spinor quantities in transformations of a Lorentz group with a determinant equal to unity. A proof is given of the above theorem tn which it is shown that it is enough to restrict oneself to a consideration of the transformation of the quantities with general inversion (inverston of all four co-ordinate axes).
We respond to a recent request from Neuenschwander for an elementary proof of the Spin-Statistics Theorem. First, we present a pedagogical discussion of the results for the spin-0 Klein–Gordon field quantized according to Bose–Einstein statistics; and for the spin-12 Dirac field quantized according to Fermi–Dirac statistics and the Pauli Exclusion Principle. This discussion is intended to make our paper accessible to students familiar with the matrix solution of the quantum harmonic oscillator. Next, we discuss a number of candidate intuitive proofs and conclude that none of them pass muster. The reasons for their shortcomings are fully discussed. Then we discuss an argument, originally suggested by Sudarshan, which proves the theorem with a minimal set of requirements. Although we use Lorentz invariance in a specific and limited part of the argument, we do not need the full complexity of relativistic quantum field theory. Motivated by our particular use of Lorentz invariance, if we are permitted to elevate the conclusion of flavor symmetry (which we explain in the text) to the status of a postulate, one could recast our proof without any explicit relativistic assumptions, and thus make it applicable even in the nonrelativistic context. Such an argument, presented in the text, sheds some light on why it is that the ordinary Schrödinger field, considered strictly in the nonrelativistic context, seems to be quantizable with either statistics. Finally, an argument starting with ordinary-number valued (commuting), and with Grassmann-valued (anticommuting), oscillators shows in a natural way that these must relativistically embed into Klein–Gordon spin-0 and Dirac spin-12 fields, respectively. In this way, the Spin-Statistics Theorem is understood at the expense of admitting the existence of the simplest Grassmann-valued field.