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Journal of Mathematical Physics

Proof of the Symmetrization Postulate

Marvin D. Girardeau

Published Jul 1, 1969• 16 citations
Publisher

Abstract

It is proved within the framework of nonrelativistic quantum mechanics that identical particles are either boson or fermions. The starting assumptions are: (a) if ψ(x1 … xn) is in the space H of allowed states, then so is Pψ for every permutation P; (b) |Pψ|2 = |ψ|2 for all ψ∈H, all allowed configurations (x1 … xn), and all ψ∈H; (c) H is a vector space (principle of superposition); (d) every ψ∈H is continuous along every path in the n-particle configuration space C; and (e) there is at least one physical observable connecting each pair of irreducible components of C.

DOI
10.1063/1.1664967

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•Jan 1, 1971•0 citations
It is proved within the framework of nonrelativistic quantum mechanics that identical particles are either boson or fermions. The starting assumptions are: (a) if ψ(x1 … xn) is in the space H of allowed states, then so is Pψ for every permutation P; (b) |Pψ|2 = |ψ|2 for all ψ∈H, all allowed configurations (x1 … xn), and all ψ∈H; (c) H is a vector space (principle of superposition); (d) every ψ∈H is continuous along every path in the n-particle configuration space C; and (e) there is at least one physical observable connecting each pair of irreducible components of C.
On the basis of reasonable physical assumptions, it is proved that for systems of identical spinless particles, the permissible nonrelativistic Schr\"odinger wave functions are either all symmetric or all antisymmetric. The proof is a generalization of that given by Girardeau. Although connectivity properties of the configuration space play an important role in the proof, it is not true, as was previously believed, that a connected space is always necessary. In particular, for one-dimensional systems, connectivity of the configuration space is not necessary for the proof to hold.
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 3n3n3n-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.
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
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.
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.