T. D. Palev

Many-body Wigner quantum systems

We present examples of many-body Wigner quantum systems. The position and the momentum operators RA and PA, A=1,…,n+1, of the particles are noncanonical and are chosen so that the Heisenberg and the Hamiltonian equations are identical. The spectrum of the energy with respect to the center of mass is equidistant and has finite number of energy levels. The composite system is spread in a small volume around the center of mass and within it the geometry is noncommutative. The underlying statistics is an exclusion statistics.We present examples of many-body Wigner quantum systems. The position and the momentum operators RA and PA, A=1,…,n+1, of the particles are noncanonical and are chosen so that the Heisenberg and the Hamiltonian equations are identical. The spectrum of the energy with respect to the center of mass is equidistant and has finite number of energy levels. The composite system is spread in a small volume around the center of mass and within it the geometry is noncommutative. The underlying statistics is an exclusion statistics.

Finite-dimensional representations of the quantum superalgebra

Explicit expressions for the generators of the quantum superalgebraUq[gl(n/m)] acting on a class of irreducible representations are given. The class under consideration consists of all essentially typical representations: for these a Gelfand-Zetlin basis is known. The verification of the quantum superalgebra relations to be satisfied is shown to reduce to a set ofq-number identities.

U_q[{\rm gl}(n/m)]

The properties of A-statistics, related to the class of simple Lie algebras sl(n + 1), n ∈ Z+ (Palev, T.D.: Preprint JINR E17-10550 (1977); hep-th/9705032), are further investigated. The description of each sl(n + 1) is carried out via generators and their relations (see eq. (2.5)), first introduced by Jacobson. The related Fock spaces Wp, p ∈ N, are finite-dimensional irreducible sl(n+ 1)-modules. The Pauli principle of the underlying statistics is formulated. In addition the paper contains the following new results: (a) The A-statistics are interpreted as exclusion statistics; (b) Within each Wp operators B(p) ± ,...,B(p) ± , proportional to the Jacobson generators, are introduced. It is proved that in an appropriate topology (Definition 2) lim p→∞ B(p) ± i = B ± i , where B ± i are Bose creation and annihilation operators; (c) It is shown that the local statistics of the degenerated hard-core Bose models and of the related Heisenberg spin models is p = 1 A-statistics.

and related

A noncanonical quantum system, consisting of two nonrelativistic particles, interacting via a harmonic potential, is considered. The center‐of‐mass position and momentum operators obey the canonical commutation relations, whereas the internal variables are assumed to be the odd generators of the Lie superalgebra sl(1,2). This assumption implies a set of constraints in the phase space, which are explicitly written in the paper. All finite‐dimensional irreducible representations of sl(1,2) are considered. Particular attention is paid to the physical representations, i.e., the representations corresponding to Hermitian position and momentum operators. The properties of the physical observables are investigated. In particular, the operators of the internal Hamiltonian, the relative distance, the internal momentum, and the orbital momentum commute with each other. The spectrum of these operators is finite. The distance between the constituents is preserved in time. It can take no more than three different valu...

q

We present three groups of non-canonical quantum oscillators. The position and momentum operators of each group generate basic Lie superalgebras, namely sl(1/3), osp(1/6) and osp(3/2). The sl(1/3) oscillators have finite energy spectrum and finite dimensions. The osp(1/6) oscillators are related to the para-Bose statistics. The internal angular momentum s of the osp(3/2) oscillators takes no more than three (half)integer values. In a particular representation s=1/2.

-identities

An n-particle three-dimensional Wigner quantum oscillator model is constructed explicitly. It is non-canonical in that the usual coordinate and linear momentum commutation relations are abandoned in favour of Wigners suggestion that Hamiltons equations and the Heisenberg equations are identical as operator equations. The construction is based on the use of Fock states corresponding to a family of irreducible representations of the Lie superalgebra sl(1|3n) indexed by an A-superstatistics parameter p. These representations are typical for p ≥ 3n but atypical for p < 3n. The branching rules for the restriction from sl(1|3n) to gl(1) ⊕ so(3) ⊕ sl(n) are used to enumerate energy and angular momentum eigenstates. These are constructed explicitly and tabulated for n ≤ 2. It is shown that measurements of the coordinates of the individual particles give rise to a set of discrete values defining nests in the three-dimensional configuration space. The fact that the underlying geometry is non-commutative is shown to have a significant impact on measurements of particle separation. In the atypical case, exclusion phenomena are identified that are entirely due to the effect of A-superstatistics. The energy spectrum and associated degeneracies are calculated for an infinite-dimensional realization of the Wigner quantum oscillator model obtained by summing over all p. The results are compared with those applying to the analogous canonical quantum oscillator.

Jacobson generators, Fock representations and statistics of sl(n+1)

The properties of a non-canonical 3D Wigner quantum oscillator, whose position and momentum operators generate the Lie superalgebra sl(13), are further investigated. Within each state space W(p), p = 1, 2, ..., the energy Eq, q = 0, 1, 2, 3, takes no more than four different values. If the oscillator is in a stationary state ψq W(p) then measurements of the non-commuting Cartesian coordinates of the particle are such that their allowed values are consistent with it being found at a finite number of sites, called nests. These lie on a sphere centred on the origin of fixed, finite radius q. The nests themselves are at the vertices of a rectangular parallelepiped. In the typical cases (p > 2) the number of nests is 8 for q = 0 and 3, and varies from 8 to 24, depending on the state, for q = 1 and 2. The number of nests is less in the atypical cases (p = 1, 2), but it is never less than 2. In certain states in W(2) (respectively in W(1)) the oscillator is polarized so that all the nests lie on a plane (respectively on a line). The particle cannot be localized in any one of the available nests alone since the coordinates do not commute. The probabilities of measuring particular values of the coordinates are discussed. The mean trajectories and the standard deviations of the coordinates and momenta are computed, and conclusions are drawn about uncertainty relations.

Wigner quantum systems. Two particles interacting via a harmonic potential. I. Two‐dimensional space

Abstract:The paper contains essentially two new results. Physically, a deformation of the parastatistics in a sense of quantum groups is carried out. Mathematically, an alternative to the Chevalley description of the quantum orthosymplectic superalgebra in terms of

Wigner quantum oscillators

m

A non-commutative n-particle 3D Wigner quantum oscillator

pairs of deformed parabosons and n pairs of deformed parafermions is outlined.