Methods used to prove the existence of superselection rules are studied in detail. It is shown that the known methods of proving superselection rules are applicable only if the superselecting operator is an observable. In particular, place permutations are not observables, so the fact that place permutations commute with all observables does not lead to a superselection rule between vectors of different symmetry types. It is shown that if states not having a definite symmetry type can exist, then it is possible to have several different states which are eigenstates of the same observables with the same eigenvalues. In this case a maximal set of observables does not exist. Therefore a proof of the Symmetrization Postulate (SP) which assumes a maximal set of observables actually presupposes part of what is to be proved. A rigorous proof of the SP must depend only on the weaker assumption of a non-degenerate set of observables. The usual formulation of the Transition Probability Postulate (TPP) is used to construct a trivial proof of the assertion that physical states are represented by unique rays in Hilbert space; however the feature of the TPP which is essential to this proof is not essential to the TPP itself. The features of the TPP which are essential for computing transition probabilities are identified. These essential features of the TPP are then used to prove that states are represented by unique rays. The difficulties involved in rigorously defining the interchange operator Pij are discussed and a careful definition of Pij is given. The uniqueness of the ray is then used to show that states of systems containing several indistinguishable particles must be eigenstates of Pij and must satisfy the Symmetrization Postulate. Finally, the SP is used to prove the slightly stronger result that a given state must have the same symmetry type for each pair of the same species of particles.
Methods used to prove the existence of superselection rules are studied in detail. It is shown that the known methods of proving superselection rules are applicable only if the superselecting operator is an observable. In particular, place permutations are not observables, so the fact that place permutations commute with all observables does not lead to a superselection rule between vectors of different symmetry types. It is shown that if states not having a definite symmetry type can exist, then it is possible to have several different states which are eigenstates of the same observables with the same eigenvalues. In this case a maximal set of observables does not exist. Therefore a proof of the Symmetrization Postulate (SP) which assumes a maximal set of observables actually presupposes part of what is to be proved. A rigorous proof of the SP must depend only on the weaker assumption of a non-degenerate set of observables. The usual formulation of the Transition Probability Postulate (TPP) is used to construct a trivial proof of the assertion that physical states are represented by unique rays in Hilbert space; however the feature of the TPP which is essential to this proof is not essential to the TPP itself. The features of the TPP which are essential for computing transition probabilities are identified. These essential features of the TPP are then used to prove that states are represented by unique rays. The difficulties involved in rigorously defining the interchange operator Pij are discussed and a careful definition of Pij is given. The uniqueness of the ray is then used to show that states of systems containing several indistinguishable particles must be eigenstates of Pij and must satisfy the Symmetrization Postulate. Finally, the SP is used to prove the slightly stronger result that a given state must have the same symmetry type for each pair of the same species of particles.
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.
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.
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
We present in this paper a set of postulates for a physical system and deduce from these the main general features of the quantum theory of stationary states. Our theory is strictly operational in the sense that only the observables of the physical system are involved in the postulates. The collection of all bounded self-adj oint operators on a Hilbert space, which has previously been used as a mathematical model for the observables in quantum mechanics, satisfy the postulates, as do a variety of considerably more general mathematical structures. We show that for any two observables there exists a pure state of the system (this term being defined essentially as done by von Neumann and Weyl) in which they have different (expectation) values. Each observable has a spectral resolution, and a state of the system induces, in a natural way, a probability distribution on the range of spectral values of each observable. A number of observables are simultaneously observable if and only if they commute (although the product of two observables is not defined in general, a reasonable notion of commutativity can be introduced). Inasmuch as Hilbert space plays no role in our theory, our proofs are necessarily of a different character from the proofs of these results for the case of the system of all bounded selfadjoint operators. Actually, Hilbert space appears to be somewhat inadequate as a state space even for the latter system, in that there exist pure states of the system which cannot be represented in the usual way by rays of the Hilbert space. The postulates are partly algebraic and partly metric. The algebraic postulates require essentially that an observable can be multiplied by real numbers and raised to integral powers, and that any two observables can be added. It is assumed that the usual algebraic laws are satisfied so that (1) the observables can be treated like the elements of a linear space, and (2) the usual rules for dealing with polynomials in one variable with real coefficients remain valid when the variable is replaced by an observable. It is not assumed that two observables have a product. The metric postulates require that for each observable there be defined a kind of maximum numerical value, which plays the part of a norm, and has various properties in accord with its physical significance. While this norm is quite essential to the development of the theory, an interesting consequence of the theory is that the norm can be (uniquely) defined in a purely algebraic fashion. This shows that the objective features of a physical system,-the spectral values and probability distributions of the observables, and the pure states,-are completely determined by the algebra of observables, i.e., by the rules for addition, scalar multiplication, and powers, of observables.