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Progress of Theoretical Physics

Spin and Statistics

Lowell Brown, Julian Schwinger

Published Dec 1, 1961• 5 citations
Publisher

Abstract

A general discussion of the vacuum expectation value for the product of two fields shows how the structure depends upon the integral or half-integral nature of the spin. When this is compared with the statement of statistics as the requirement that fields commute or anticommute for points in space-like relation, at distances greater than any specified amount, the connection between spin and statistics follows.

DOI
10.1143/ptp.26.917

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A general discussion of the vacuum expectation value for the product of two fields shows how the structure depends upon the integral or half-integral nature of the spin. When this is compared with the statement of statistics as the requirement that fields commute or anticommute for points in space-like relation, at distances greater than any specified amount, the connection between spin and statistics follows.
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 ${\ensuremath{\epsilon}}^{2}=1$.
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 this article, we begin with a review of Pauli's version of the spin-statistics theorem and then show, by re-defining the parameter associated with the Lie-Algebra structure of angular momentum, that another interpretation of the theorem may be given. It will be found that the vanishing commutator and anticommutator relationships can be associated with independent and dependent probability events respectively, and not spin value. Consequently, it gives a more intuitive understanding of quantum field theory and it also suggests that the distinction between timelike and spacelike events might be better described in terms of local and non-local events. Pacs: 3.65, 5.30, 3.70.+k
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.