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Physical Review

Selection Rules for Parafields and the Absence of Para Particles in Nature

Oscar Wallace Greenberg, Albert Messiah

Published Jun 7, 1965• 367 citations
Publisher

Abstract

Green's parafield quantization is reviewed. It is shown, both for a single field and for sets of fields, that all Fock-like representations of Green's trilinear commutation rules are realized by Green's ansatz with anticommuting (commuting) Bose (Fermi) component fields for para-Bose (para-Fermi) fields. Restrictions on the form of the interaction Hamiltonian density ${H}_{I}(x)$ are derived from the requirement that ${H}_{I}(x)$ be a paralocal operator. From these restrictions on ${H}_{I}$ selection rules on the $S$ matrix are proved to all orders of perturbation theory. The most important such rule prohibits all reactions in which the total number of para particles of order $p>1$ in the initial and final states is one. This last selection rule, together with experimental information, leads to the conclusion that no presently known particle can be para.

DOI
10.1103/physrev.138.b1155

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All of the irreducible representations are found for a single pair of creation and annihilation operators which together with the symmetric or antisymmetric number operator satisfy the generalized commutation relation characteristic of para-Bose or para-Fermi field quantization. The procedure is simply to identify certain combinations of these three operators with the three generators of the three-dimensional rotation group in the para-Fermi case, and with the three generators of the three-dimensional Lorentz group in the para-Bose case. The irreducible representations are then easily obtained by the usual raising and lowering operator techniques. The applicability of these techniques is demonstrated by a simple argument which shows that the commutation relations require that the generator to be diagonalized have a discrete spectrum.

Journal of Mathematical Physics•Aug 1, 1963•43 citations
Green's parafield quantization is reviewed. It is shown, both for a single field and for sets of fields, that all Fock-like representations of Green's trilinear commutation rules are realized by Green's ansatz with anticommuting (commuting) Bose (Fermi) component fields for para-Bose (para-Fermi) fields. Restrictions on the form of the interaction Hamiltonian density HI(x){H}_{I}(x)HI​(x) are derived from the requirement that HI(x){H}_{I}(x)HI​(x) be a paralocal operator. From these restrictions on HI{H}_{I}HI​ selection rules on the SSS matrix are proved to all orders of perturbation theory. The most important such rule prohibits all reactions in which the total number of para particles of order p>1p>1p>1 in the initial and final states is one. This last selection rule, together with experimental information, leads to the conclusion that no presently known particle can be para.
Although usually only two kinds of statistics, namely Bose-Einstein and Fermi-Dirac statistics, are considered in Quantum Mechanics and in Quantum Field Theory, other kinds of statistics, called collectively parastatistics, are conceivable. We critically review theoretical studies of parastatistics to date, point out and clarify several confusions. We first study the "proofs" so far proposed for the symmetrization postulate which excludes parastatistics, emphasizing their ad hoc nature. Then, after exploring in detail the structure of the quantum mechanical theory of paraparticles, we clarify some confusions concerning the compatibility of parastatistics with the so-called cluster property, which has been an issue of controversy for several years. We show, following a suggestion of Greenberg, that the quantum mechanical theory of paraparticles can be formulated in terms of density matrix compatibly with the cluster property. We also discuss such topics as selection rules for systems with variable numbers of paraparticles, the connection between statistics and permutation characters, and the classification of paraparticles. For the quantum field theory of paraparticles, we study discrete representations of the para-commutation relations and illustrate in detail Greenberg and Messiah's theorem concerning Green's ansatzes. Also, fundamental topics such as the spin-statistic theorem, the TCP theorem and the observability of parafields are discussed on the basis of Green's ansatzes. Finally, we point out that the so-called particle permutation operators do not always define multi-dimensional representations of the permutation group both in first and second quantization theories. This questions the validity of the correspondence between the two theories which has recently been proposed.
The nonrelativistic theory of a single para-Fermi field of order ppp is investigated. General properties of state vectors are studied in detail, and it is shown that the state-vector space can be spanned by what we shall call standard state vectors. A restriction on the form of interaction Hamiltonians is derived from the requirement that our formalism be described by local Lagrangian field theory. This restriction on interaction Hamiltonians gives rise to a conservation law of a physical quantity to be called AAA, which resembles the magnitude of angular momentum with respect to its rule of addition. The conservation law of AAA leads then to absolute selection rules for reactions, which are a generalization of those obtained elsewhere. The problem of bound states made up of our para-Fermi field is also studied, and all bound states are classified into (p+1p+1p+1) categories according to their statistical behaviors. It is found that for p< 3p<~3p< 3 all bound states can be described by ordinary parafield theory, whereas for p> 4p>~4p> 4 such is no longer the case. Furthermore, it can be shown that in the theory of p=2p=2p=2 no fermion bound states are possible. In this sense it may be said that para-Fermi fields of p=1p=1p=1 and 3 occupy a very privileged position in para-Fermi theory in general. The main results in this paper are stated as 12 theorems. It is expected that the whole argument will be valid in a relativistic theory as well.
A connection between a unitary quantization scheme and para-Fermi statistics of order 2 is considered. An appropriate extension of Green's ansatz is suggested. This extension allows one to transform bilinear and trilinear commutation relations for the annihilation and creation operators of two different para-Fermi fields ϕa\phi_{a}ϕa​ and ϕb\phi_{b}ϕb​ into identity. The way of incorporating para-Grassmann numbers ξk\xi_{k}ξk​ into a general scheme of uniquantization is also offered. For parastatistics of order 2 a new fact is revealed, namely, the trilinear relations containing both the para-Grassmann variables ξk\xi_{k}ξk​ and the field operators aka_{k}ak​, bmb_{m}bm​ under a certain invertible mapping go over into the unitary equivalent relations, where commutators are replaced by anticommutators and vice versa. It is shown that the consequence of this circumstance is the existence of two alternative definitions of the coherent state for para-Fermi oscillators. The Klein transformation for Green's components of the operators aka_{k}ak​, bmb_{m}bm​ is constructed in an explicit form that enables us to reduce the initial commutation rules for the components to the normal commutation relations of ordinary Fermi fields. A nontrivial connection between trilinear commutation relations of the unitary quantization scheme and so-called Lie-supertriple system is analysed. A brief discussion of the possibility of embedding the Duffin-Kemmer-Petiau theory into the unitary quantization scheme is provided.
We consider the relationship between the unitary quantization scheme and the para-Fermi statistics of order 2. We propose an appropriate generalization of Green’s ansatz, which has made it possible to transform bilinear and trilinear commutation relations for the creation and annihilation operators for two different para-Fermi fields φ a and φ b into identities. We also propose a method for incorporating para-Grassmann numbers ξ k into the general unitary quantization scheme. For the parastatistics of order 2, a new fact has been revealed: the trilinear relations containing both para-Grassmann variables ξ k and field operators ak and bm are transformed under a certain reversible mapping into unitary equivalent relations in which commutators are replaced by anticommutators, and vice versa. It is shown that this leads to the existence of two alternative definitions of the coherent state for para-Fermi oscillators. The Klein transformation for Green’s components of operators a k and b m is constructed in explicit form, which enabled us to reduce the initial commutation rules for the components to the normal commutation relations for ordinary Fermi fields. We have analyzed a nontrivial relationship between the trilinear commutation relations of the unitary quantization scheme and the so-called Lie supertriple system. The possibility of incorporating the Duffin–Kemmer–Petiau theory into the unitary quantization scheme is discussed briefly.
A description of the Fock space of a parabose oscillator of order p based on commutation relations bilinear in creation and annihilation operators rather than trilinear ones is developed. A new statement of the well-known Green’s ansatz is introduced to help understand the significance of the bilinear commutation relations. It is also applied to show how all parabose oscillators of even order (respectively, odd order) can be obtained as irreducible constituents of the Fock space associated with a Green’s ansatz of two (three) terms. Analogs of the form in which the parabose Green’s ansatz is presented are provided for one parafermi oscillator and for systems of many parabose and parafermi oscillators of the same order. Methods used in the paper allow various new results for q-deformed parabose oscillators to be derived.
All of the irreducible representations are found for a single pair of creation and annihilation operators which together with the symmetric or antisymmetric number operator satisfy the generalized commutation relation characteristic of para-Bose or para-Fermi field quantization. The procedure is simply to identify certain combinations of these three operators with the three generators of the three-dimensional rotation group in the para-Fermi case, and with the three generators of the three-dimensional Lorentz group in the para-Bose case. The irreducible representations are then easily obtained by the usual raising and lowering operator techniques. The applicability of these techniques is demonstrated by a simple argument which shows that the commutation relations require that the generator to be diagonalized have a discrete spectrum.