Toward an understanding of the spinstatistics theorem
Abstract
We respond to a recent request from Neuenschwander for an elementary proof of the SpinStatistics Theorem. First, we present a pedagogical discussion of the results for the spin0 KleinGordon field quantized according to BoseEinstein statistics; and for the spin1/2 Dirac field quantized according to FermiDirac 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 ordinarynumber valued (commuting), and with Grassmannvalued (anticommuting), oscillators shows in a natural way that these must relativistically embed into KleinGordon spin0 and Dirac spin1/2 fields, respectively. In this way, the SpinStatistics Theorem is understood at the expense of admitting the existence of the simplest Grassmannvalued field.
 Publication:

American Journal of Physics
 Pub Date:
 April 1998
 DOI:
 10.1119/1.18860
 Bibcode:
 1998AmJPh..66..284D
 Keywords:

 01.50.i;
 05.30.d;
 11.10.Cd;
 11.10.Lm;
 Educational aids;
 Quantum statistical mechanics;
 Axiomatic approach;
 Nonlinear or nonlocal theories and models