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.
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 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 and ϕb into identity. The way of incorporating para-Grassmann numbers 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 and the field operators ak, 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 ak, 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.
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.
It is well known that Parafermi and Parabose statistics are natural extensions of the usual Fermi and Bose ones, enhancing trilinear (anti)commutation relations instead of bilinear ones. Due to this generalization, positive parameters appear: the so-called orders of paraquantization p (= 1, 2, 3, ...) and h sub 0 (= 1/2, 1, 3/2, ...), respectively, the first value leading in each case to the usual statistics. The superpostion of the parabosonic and parafermionic operators gives rise to parasupermultiplets for which mixed trilinear relations have already been studied leading to two (nonequivalent) sets: the relative Parabose and the relative Parafermi ones. For the specific values p = 1 = 2h sub 0, these sets reduce to the well known supersymmetry. Coherent states associated with this last model have been recently put in evidence through the annihilation operator point of view and the group theoretical approach or displacement operator context. We propose to realize the corresponding studies within the new context p = 2 = 2h sub 0, being then directly extended to any order of paraquantization.
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.
Coherent states are constructed in p=2-parasupersymmetric quantum mechanics in connection with the two relative para-Bose and para-Fermi sets of trilinear structure relations. They are called parasupercoherent states. Parasupersymmetric operators [such as the Hamiltonian and (two) annihilation operators] are introduced and discussed. In particular, the parasuperspectrum is determined and compared with the recent results obtained by Rubakov and Spiridonov [Mod. Phys. Lett. A 3, 1337 (1988)]. The superalgebra contents subtended by osp(2/2) and osp(3/2) are analyzed and exploited in order to get constants of motion in both contexts through parallel properties on the supersymmetric harmonic oscillator.
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) are derived from the requirement that HI(x) be a paralocal operator. From these restrictions on HI 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.