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Physical Review

Connection between Spin and Statistics

Gerhart Lüders, Bruno Zumino

Published Jun 15, 1958• 149 citations
Publisher

Abstract

The proof of the connection between spin and statistics for interacting fields is divided into two parts: commutation relations involving components of a single field, and commutation relations between different fields. The first problem is treated in this paper: the connection between spin and statistics is shown to follow from a few simple postulates. The explicit discussion is limited to the cases of spin zero and of spin one-half.

DOI
10.1103/physrev.110.1450

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Toward an understanding of the spin-statistics theorem

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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.

American Journal of Physics•Apr 1, 1998•107 citations
The proof of the connection between spin and statistics for interacting fields is divided into two parts: commutation relations involving components of a single field, and commutation relations between different fields. The first problem is treated in this paper: the connection between spin and statistics is shown to follow from a few simple postulates. The explicit discussion is limited to the cases of spin zero and of spin one-half.
A nonrelativistic proof of the spin-statistics theorem is given in terms of the field operators satisfying commutation and anticommutation relations, which are introduced here in the coordinate space as a means to build the permutation symmetry into the brackets of identical particles. An eigenvalue problem of a πππ-rotation for a product of two annihilation operators is combined with an analysis on its rotational property to prove the connection that the field operators for integral-spin and half-integral-spin particles obey the commutation and anticommutation relations, respectively.
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 ${\ensuremath{\epsilon}}^{2}=1$.
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.
Abstract. The spin-statistics connection is derived in a simple manner under the postulates that the original and the exchange wave functions are simply added, and that the azimuthal phase angle, which defines the orientation of the spin part of each single-particle spin-component eigenfunction in the plane normal to the spinquantization axis, is exchanged along with the other parameters. The spin factor (−1) 2s belongs to the exchange wave function when this function is constructed so as to get the spinor ambiguity under control. This is achieved by effecting the exchange of the azimuthal angle by means of rotations and admitting only rotations in one sense. The procedure works in Galilean as well as in Lorentz-invariant quantum mechanics. Relativistic quantum field theory is not required.
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.