Abstract Quantum theory stipulates that if two particles are identical in all physical aspects, the allowed states of the system are either symmetric or antisymmetric with respect to permutations of the particle labels. Experimentally, the symmetry of the states can be inferred indirectly from the fact that neglecting the correct exchange symmetry in the theoretical analysis leads to dramatic discrepancies with the observations. The only way to directly unveil the symmetry of the states for, say, two identical particles is through the interference of the two-particle state and the physically permuted one, and measuring the phase associated with the permutation process, the so-called particle exchange phase. Following this idea, we have measured the exchange phase of indistinguishable photons, providing direct evidence of the bosonic character of photons.
A remarkable feature of quantum theory is that particles with identical intrinsic properties must be treated as indistinguishable if the theory is to give valid predictions in all cases. In the quantum formalism, indistinguishability is expressed via the symmetrization postulate (Dirac P 1926 Proc. R. Soc. A 112 661 , Heisenberg W 1926 Z. Phys. 38 411 ), which restricts a system of identical particles to the set of symmetric states ('bosons') or the set of antisymmetric states ('fermions'). However, the physical basis and range of validity of the symmetrization postulate has not been established. A well-known topological derivation of the postulate implies that its validity depends on the dimensionality of the space in which the particles move (Laidlaw M and DeWitt C 1971 Phys. Rev. D 3 1375–8 , Leinaas J M and Myrheim J 1977 Il Nuovo Cimento B 37 1–23 ). Here we show that the symmetrization postulate can be derived by strictly adhering to the informational requirement that particles which cannot be experimentally distinguished from one another are not labelled. Our key novel postulate is the operational indistinguishability postulate, which posits that the amplitude of a process involving several indistinguishable particles is determined by the amplitudes of all possible transitions of these particles when treated as distinguishable. The symmetrization postulate follows by requiring consistency with the rest of the quantum formalism. The derivation implies that the symmetrization postulate admits no natural variants. In particular, the possibility that identical particles generically exhibit anyonic behavior in two dimensions is excluded.
We consider the implications of the Revised Symmetrization Postulate (see quant-ph/9908078) for states of more than two particles. We show how to create permutation symmetric state vectors and how to derive alternative state vectors that may be asymmetric for any pair by creating asymmetric interdependencies in their state descriptions. Because we can choose any pair to create such an asymmetry, the usual generalized exclusion rules which result, apply simultaneously to any pair. However, we distinguish between simultaneous pairwise exclusion rules and the simultaneous pairwise anti-symmetry of the conventional symmetrization postulate. We show how to construct a variety of state vectors with multiple interdependencies in their state descriptions and various exchange asymmetries - including one which is anti-symmetric under exchange of two bosons - all without violating the spin-statistics theorem. We conjecture that it is possible to construct state vectors for arbitrary mixes of bosons and fermions that emulate the conventional symmetrization postulate in a limited way and give examples. We also prove that it is not possible to define a single state vector that simultaneously obeys the conventional symmetrization postulate in its standard form (in which the exchange phase does not depend on the spins of additional particles that are present) for every pair that can be interchanged.
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
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 $\ensuremath{\mu}$.
Identical particles have dynamically invariant labels in all setups where their degrees of freedom can be partitioned into two parts, which we call internal and visible, with the visible part subject to unitary evolution, while the dynamically invariant internal part accounts for the partial distinguishability of particles. We give the explicit form of the visible state in terms of the indistinguishability function, generalizing the standard symmetrization/antisymmetrization postulate for bosons/fermions with internal modes. The indistinguishability function on the symmetric group accounts for the permutation symmetry of the dynamically invariant label state and gives a self-contained description of the symmetry spectrum of the visible state. It is shown that the indistinguishability of single particles emitted independently from a stable source is completely characterized by the projective measures on the symmetry spectrum. We reveal a hierarchy of successive upper bounds on the projective measure of indistinguishability in terms of marginal projective measures. We point out an experimentally feasible way to get the upper bounds on the indistinguishability of an arbitrarily large number of single photons by direct experimental readout of their marginal projective measures. We also analyze the indistinguishability of coherent superpositions and convex mixtures and resolve a recently posed problem.