Reasonable requirements of (a) physical invariance under particle permutation and (b) physical completeness of state descriptions [1], enable us to deduce a Symmetric Permutation Rule(SPR): that by taking care with our state descriptions, it is always possible to construct state vectors (or wave functions) that are purely symmetric under pure permutation for all particles, regardless of type distinguishability or spin. The conventional exchange antisymmetry for two identical half-integer spin particles is shown to be due to a subtle interdependence in the individual state descriptions arising from an inherent geometrical asymmetry. For three or more such particles, however, antisymmetrization of the state vector for all pairs simultaneously is shown to be impossible and the SPR makes observably different predictions, although the usual pairwise exclusion rules are maintained. The usual caveat of fermion antisymmetrization—that composite integer spin particles (with fermionic constituents) behave only approximately like bosons—is no longer necessary.
Abstract. Reasonable requirements of (a) physical invariance under particle permutation and (b) physical completeness of state descriptions [1], enable us to deduce a Symmetric Permutation Rule(SPR): that by taking care with our state descriptions, it is always possible to construct state vectors (or wave functions) that are purely symmetric under pure permutation for all particles, regardless of type distinguishability or spin. The conventional exchange antisymmetry for two identical half-integer spin particles is shown to be due to a subtle interdependence in the individual state descriptions arising from an inherent geometrical asymmetry. For three or more such particles, however, antisymmetrization of the state vector for all pairs simultaneously is shown to be impossible and the SPR makes observably different predictions, although the usual pairwise exclusion rules are maintained. The usual caveat of fermion antisymmetrization – that composite integer spin particles (with fermionic consitituents) behave only approximately like bosons – is no longer necessary. I TERMINOLOGY
We examine historic formulations of the spin-statistics theorem from a point of view that distinguishes between the observable consequences and the ``symmetrization postulate''. In particular, we make a critical analysis of concepts of particle identity, state distinguishability and permutation, and particle ``labels''. We discuss how to construct unique state vectors and the nature of the full state descriptions required for this -- in particular the elimination of arbitrary
2π rotations on fermion spin quantization frames and argue that the failure to do this renders the conventional symmetrization postulate (and previous ``proofs'' of it) at best {\em incomplete}. We discuss particle permutation in a general way for any pairs of particles, whether identical or not, and make an essential distinction between exchange and pure permutation. We prove a revised symmetrization postulate that allows us to construct state vectors that are naturally symmetric under pure permutation, {\em for any spin}. The significance of particle labels (which, in the exchange operation, are not permuted along with other variables) is then that they stand in for any asymmetry (order dependence) that is present in the full state descriptions necessary for unique state vectors but not explicit in the regular state variables. {\em The exchange operation is then the physical transformation that reverses any asymmetry implicit in the labels}. We point out a previously unremarked geometrical asymmetry between all pairs of particles that is present whenever we choose a common frame of reference. We compute the exchange phase for various state vectors using different spin quantization frames, and prove the Pauli Exclusion Principle and its generalization to arbitrary spin.
We discuss the conditions under which identical particles may yet be distinguishable and the relationship between particle permutation and exchange. We show that we can always define permutation-symmetric state vectors. When the particles are completely indistinguishable, then exchange is equivalent to permutation and therefore the exchange eigenvalue for such permutation-symmetric state vectors is always +1. Exchange asymmetry arises when the particles are physically distinguishable, even though otherwise identical, and can be computed from the transformations that arise when the distinguishing features are reversed. There is a fundamental spatial asymmetry between the relative orientations of any two vectors in a common frame of reference that persists even in the limit that the vectors coincide. For a pair of particles this asymmetry between their spin quantization frames renders them distinguishable even when otherwise identical. In the conventional construction, this distinction is not properly accounted for. Particle exchange is then equivalent to reversing this relative orientation --- which requires a relative rotation by 2pi on the spin quantization frame of one particle with respect to the other, thus resulting in the conventional exchange phase.
The symmetrization postulate (SP) that states of more than one identical particle are either symmetric or antisymmetric under permulations is studied from the theoretical and experimental points of view. The theoretical analysis is carried out within the framework of particle quantum mechanics; the field-theory approach to identical particles using Bose, Fermi, para-Bose and para-Fermi quantization is not considered in this article. Particles not obeying SP can be accommodated in quantum mechanics, provided some modifications are made in the usual quantum-mechanical formalism. The main modification is to replace the usual ray by a many-dimensional "generalized ray" as the representative of a physical state. The properties of one-body measurements of systems having several identical particles are discussed, and the unobservability in such measurements of interferences between states having different irreducible permutation symmetries is pointed out. The condition of indistinguishability of identical particles is formulated precisely, and is analyzed both for interactions which conserve particle number and for general interactions which do not. For such general interactions, with the additional assumptions of time-reversal invariance and of coherence of the states having given values of charge, baryon number, and lepton number, it is shown that there is an absolute selection rule forbidding transitions between states ${\mathcal{F}}^{\ifmmode\times\else\texttimes\fi{}}$ which contain any number of particles of species which obey SP but at most one particle of a species not obeying SP, and states which violate SP. Since only states in ${\mathcal{F}}^{\ifmmode\times\else\texttimes\fi{}}$ are now avaiable as initial states of experiments, this selection rule forbids production of SP-violating states in any experiment which is feasible at present. Because of this, presently proposed experimental tests of SP are in fact tests of the quantum-mechanical description of identical particles together with time-reversal invariance and coherence of states in a given superselecting sector. The inclusion of internal variables, such as isospin, for particles violating SP is discussed. A comprehensive discussion is given of direct experimental tests of the SP selection rules. Such tests are more difficult to perform than appears at first sight, because in many cases the indistinguishability of identical particles or the conservation laws already imply the consequences of SP. Criteria for valid tests of SP are given. Several different types of tests are described, with illustrative examples of each. A survey is given of the direct experimental evidence for SP for the various particles. The Fermi character of electrons and positrons and of nucleons is accepted, as is the Bose character of photons. There is good evidence for the Bose nature of pions, especially from the absence of $2\ensuremath{\pi}$ decay of
K20. There is no direct evidence for the statistics of
K, $\ensuremath{\Lambda}$, $\ensuremath{\Sigma}$, $\ensuremath{\Xi}$, or $\ensuremath{\mu}$. Feasible tests are proposed for the statistics of
K and of those hyperons which have an asymmetric decay; but no such tests were found for the other hyperons or for $\ensuremath{\mu}$.
The symmetrization postulate (SP) states that wave functions are either completely symmetric or completely antisymmetric under permutations of identical particles. It is shown by one-dimensional counter-examples that SP is not demanded by the usual physical interpretation of the mathematical formalism of wave mechanics unless one makes use of further physical properties of real systems; the error in a standard proof of SP which ignores these properties is pointed out. It is then proved that SP is true for those systems of spinless particles which have the following properties: (a) probability densities are permutation-invariant, (b) allowable wave functions are continuous with continuous gradient, (c) the
3n-dimensional configuration space is connected, (d) the Hamiltonian is symmetric, and (e) the nodes of allowed wave functions have certain properties. The counterexamples show that SP is not a necessary property of those systems which do not have property (c). The proof is extended to particles with internal degrees of freedom (including spin) by noting that any observable commutes with every permutation and making use of a particular observable acting only on internal variables. Extraneous mathematical assumptions, such as that of the existence of a "complete" set of commuting observables, criticized by Messiah and Greenberg, are not employed. Some comments are made on the conventional nature of the concept of identity for similar particles; the equivalence between the usual formulation in which different species of similar particles are treated as distinct, and that in which they are regarded as identical particles in different internal states, is emphasized.