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Physical Review

The Connection Between Spin and Statistics

Wolfgang Ernst Friederich Pauli

Published Oct 15, 1940• 813 citations
Publisher

Abstract

In the following paper we conclude for the relativistically invariant wave equation for free particles: From postulate (I), according to which the energy must be positive, the necessity of Fermi-Dirac statistics for particles with arbitrary half-integral spin; from postulate (II), according to which observables on different space-time points with a space-like distance are commutable, the necessity of Einstein-Bose statistics for particles with arbitrary integral spin. It has been found useful to divide the quantities which are irreducible against Lorentz transformations into four symmetry classes which have a commutable multiplication like $+1,\ensuremath{-}1,+\ensuremath{\epsilon},\ensuremath{-}\ensuremath{\epsilon}$ with ${\ensuremath{\epsilon}}^{2}=1$.

DOI
10.1103/physrev.58.716

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In the following paper we conclude for the relativistically invariant wave equation for free particles: From postulate (I), according to which the energy must be positive, the necessity of Fermi-Dirac statistics for particles with arbitrary half-integral spin; from postulate (II), according to which observables on different space-time points with a space-like distance are commutable, the necessity of Einstein-Bose statistics for particles with arbitrary integral spin. It has been found useful to divide the quantities which are irreducible against Lorentz transformations into four symmetry classes which have a commutable multiplication like +1,\ensuremath−1,+\ensuremathϵ,\ensuremath−\ensuremathϵ+1,\ensuremath{-}1,+\ensuremath{\epsilon},\ensuremath{-}\ensuremath{\epsilon}+1,\ensuremath−1,+\ensuremathϵ,\ensuremath−\ensuremathϵ with \ensuremathϵ2=1{\ensuremath{\epsilon}}^{2}=1\ensuremathϵ2=1.
We define statistics of quantum mechanical particles, obtaining the Bose–Einstein and Fermi–Dirac varieties of indistinguishable particles. After finding the rotation and Lorentz matrices in different Lorentz representations, we describe the spin-statistics theorem, relating fermions with half-integer spin and bosons with integer spin. We explain two simple proofs and say some words on two others. We end by discussing symmetries in more generality, and we discuss the fact that internal symmetries must commute with spacetime ones, due to the Coleman–Mandula theorem.
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.
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).
It is not possible to use the Schrödinger equation to describe the motion of a particle with relativistic effects since it is not invariant relative to the Lorentz transformation. However, the attempt to generalize quantum mechanics in the relativistic domain on the basis of the one particle theory is somehow difficult. So we find the appropriate equations on the basis of the relativistic invariance. To obtain the non-relativistic equations, we take the limit as the speed of light tends to infinity. This obviously recovers the Schrödinger equation. Under this heading, we extend the non-relativistic Schrödinger equation in the description of particles with speeds comparable with that of light. This extension should be consistent with the Lorentz transformation [ 12 ]. The characteristic feature of the relativistic wave equations is the spin which provides a useful gauge in the applicability of particular equations that describe particular kinds of particles. We have two types of relativistic equations, i.e., the equation with spin-zero due to Schrödinger and spin-half due to Dirac that describes the electron.