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Journal of Physics Conference Series

Sudarshan's non-relativistic approach to the spin-statistics connection

Anil Shaji

Published Nov 1, 2009• 0 citations
Publisher

Abstract

George Sudarshan was quite a bit ahead of the times when he sketched out a non-relativistic proof of the spin-statistics connection forty years back. A recent revival of interest in the role of relativistic arguments in proving the spin-statistics connection prompted us to revisit Sudarshan's original arguments and present them in simpler fashion. The proof is designed to work in three space dimensions. Sudarshan's observation was that if we impose SU (2) invariance, along with a few other restrictions, on the kinematic part of the Lagrangian of a field theory which is derivable from a Weiss-Schwinger type of the Principle of Least Action, then the spin-statistics connection automatically follows. The assumptions and simplifications that go into constructing the proof will be carefully laid out so as to identify the limitations of this approach.

DOI
10.1088/1742-6596/196/1/012013

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George Sudarshan was quite a bit ahead of the times when he sketched out a non-relativistic proof of the spin-statistics connection forty years back. A recent revival of interest in the role of relativistic arguments in proving the spin-statistics connection prompted us to revisit Sudarshan's original arguments and present them in simpler fashion. The proof is designed to work in three space dimensions. Sudarshan's observation was that if we impose SU (2) invariance, along with a few other restrictions, on the kinematic part of the Lagrangian of a field theory which is derivable from a Weiss-Schwinger type of the Principle of Least Action, then the spin-statistics connection automatically follows. The assumptions and simplifications that go into constructing the proof will be carefully laid out so as to identify the limitations of this approach.
A recent paper by Peshkin [1] has drawn attention again to the problem of understanding the spin statistics connection in non-relativistic quantum mechanics. Allen and Mondragon [2] has pointed out correctly some of the flaws in Peshkin's arguments which are based on the single valuedness under rotation of the wave functions of systems of identical particles. We examine carefully the claim that is made in the title of [2] that there can be ``no spin-statistics connection in non-relativistic quantum mechanics''. We show that we can derive the spin statistics connection for non-relativistic quantum field theories which have equations of motion of the Hamiltonian type based on SU(2) invariance of the Lagrangian. The formalism and machinary of non-relativistic quantum field theory as opposed to usual quantum mechanics is necessary for constructing our proof.
After some brief historical remarks we recall the first demonstration of the Spin-Statistics theorem by Pauli, and some further developments, like Streater-Wightman's, in the axiomatic considerations. Sudarshan's short proof then follows by considering only the time part of the kinetic term in the lagrangian: the identity SU(2)=Sp(1,C) is crucial for the argument. Possible generalization to arbitrary dimensions are explored, and seen to depend crucially on the type of spinors, obeying Bott's periodicity 8. In particular, it turns out that for space dimentions (7, 8, or 9) modulo 8, the conventional result does not automatically hold: there can be no Fermions in these dimensions.
Here we explore the possibility to obtain a non-relativistic proof of the spin-statistics theorem. First, we examine the structure of axioms and theorems involved in a relativistic Schwinger-like proof of the spin-statistics relation. Second, starting from this structure we identify the relativistic assumptions. Last, we reformulate these assumptions in order to obtain a Galilean proof. We conclude that the spin-statistics theorem cannot be deduced into the framework of Galilean quantum field theories for three space dimensions because two of the assumptions needed to prove the theorem are incompatible. We analyze, however, the conditions under which a non-relativist proof could still be deduced.
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.