As with the boson field, the creation and annihilation operators for the paraboson field are unbounded, and the usual paraboson relations therefore, cannot be regiorously satisfied. There are no Weyl relations for parabosons, and so the paraboson relations must be treated in unbounded form. A new approach for treating such unbounded relations is developed, and it is found that the paraboson field is determined by the requirement that the appropriate operator is the one-dimensional number operator. A similar result is shown th hold for the boson field.
As with the boson field, the creation and annihilation operators for the paraboson field are unbounded, and the usual paraboson relations therefore, cannot be regiorously satisfied. There are no Weyl relations for parabosons, and so the paraboson relations must be treated in unbounded form. A new approach for treating such unbounded relations is developed, and it is found that the paraboson field is determined by the requirement that the appropriate operator is the one-dimensional number operator. A similar result is shown th hold for the boson field.
The inverse of paraboson annihilation and creation operators is introduced. The properties of these operators, their number state representation and commutation relations are discussed. Also introduced are the family of eigenstates of bilinears in paraboson operators and their inverses; and study the nonclassical properties, namely, squeezing and fluctuations in number of parabosons in these eigenstates. It is observed that a family of the eigenstates of a+−1a is the parasqueezed vacuum, S(σ)‖0,α〉. Similarly, a family of the eigenstates of aa+−1 is the parasqueezed first state, S(σ)‖n=1,α〉.
In the previous chapter we discussed the usual realization of many-body quantum mechanics in terms of differential operators (Schrödinger picture). As in the case of the two-body problem, it is possible to formulate many-body quantum mechanics in terms of algebraic operators. This is done by introducing, for each coordinate r1,r2,... and momentum p1, p2, . . . , boson creation and annihilation operators, b†iα, biα. The index i runs over the number of relevant degrees of freedom, while the index α runs from 1 to n + 1, where n is the number of space dimensions (see note 3 of Chapter 2). The boson operators satisfy the usual commutation relations, which are for i ≠ j, . . . [biα, b†jα´] = 0, [biα, bjα´] = 0,. . . . . .[bjα, b†iα´] = 0, [b†jα, b†iα´] = 0,. . . . . . [biα, b†iα´] = ẟαα´, [biα, b†iα´] = 0, [b†iα, b†iα´] = 0. . . .
Recently, we have observed that the bosonized supersymmetric quantum mechanics leads naturally to the R-deformed Heisenberg algebra (RDHA) related to parabosons and parafermions. Here we show that RDHA, in turn, gives rise not only to the bosonized supersymmetry associated with Witten's supersymmetric quantum mechanics but also supplies us with a supersymmetry characterized by a nonlinear superalgebra. Such nonlinear supersymmetry may be specified alternatively by a central extension of the usual N=1 superalgebra with grading operator included nontrivially as an even generator. The systems under consideration are described by the Hamiltonians of the simplest form Hn=a+a− and Ha=a−a+ and reveal supersymmetries at the special values of the deformation parameter corresponding to parabosons of odd order. The peculiar nature of parabosonic supersymmetries is encoded in essentially nonlinear structure of their supercharges realized in terms of parabosonic creation-annihilation operators $a^\\pm$. At the values of deformation parameter corresponding to the even order parabosons the spectra of the Hamiltonians Hn and Ha have no supersymmetry but reveal `holes' in comparison with bosonic spectra, whose number is correlated with the parastatistics' order. It is shown that RDHA and the associated linear and nonlinear supersymmetries are realizable in the system of two identical fermions. The nonlinear supersymmetry of pure parabosonic systems possesses the structure of a reduced parasupersymmetry and may also be realized by appropriate modification of the classical analog of Witten's supersymmetric quantum mechanics.
In this paper we reexamine the definition of parafermions and parabosons by means of Green's triple relations, and extend these relations by including a parity operator P which is also determined by means of triple relations. As a consequence, we are dealing with new algebraic structures. It is shown that the algebra underlying a set of n parafermions together with P is the orthogonal Lie algebra so(2n+2). The Fock spaces correspond to particular irreducible representations of so(2n+2), and the action of P in these spaces leads to interesting observations. Next, we show that the algebra underlying a set of n parabosons together with P is the orthosymplectic Lie superalgebra osp(2∣2n). In this case, the Fock spaces correspond to certain irreducible infinite-dimensional representations of osp(2∣2n). Both for parafermions and parabosons the spectrum of P is closely related to the so-called order of statistics p, introduced by Green.
The number operator a*a, where a is an annihilation operator, plays a fundamental role in the statistics of bosons and fermions. However, it is possible for other statistics to have the same number operator. We have previously shown that for one degree of freedom there is a type of statistics having this number operator corresponding to each p which is a positive integer or ∞. Fermions are obtained when p = 1 and bosons are obtained when p = ∞. No state can have more than p particles. In order to treat many degrees of freedom it is necessary to first consider the case of one degree of freedom differently. We show here that for more degrees of freedom a similar situation occurs but for each case other than bosons and fermions there is a positive integer q, such that no state can have more than q particles, even when the number of degrees of freedom is infinite. Thus these statistics are probably not physically realizable except in an approximate way.