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T. D. Palev

Publications and source records attributed to T. D. Palev.

At least 19 recordsLinked to original sources
arXiv2015

Do New Quantum Statistics Exist in Nature?

We recall the definitions of a Wigner quantum system (WQS) and of A-, (resp B-, C- and D-) quantum (super)statistics. We outline shortly the relation of these new statistics to the classes A, (resp B, C and D) of Lie (super)algebras. We describe in some more details some of the properties of A-oscillator and A-superoscillator. Both of them fall into the category of finite quantum systems and both of them have quite unusual properties. For example let R and P be the position and the momentum operators of an 3D A-superoscillator along the x (or y or z) axes. Then contrary to the canonical uncertainty relations the product of the standard deviations of R and P of the oscillating particle reads: $$ ΔR .ΔP \leq {\hbar/2}. $$ i.e., the inequality above $\le$ is just the opposite to the one that appears in the canonical quantum mechanics.

math-ph↗
arXiv2007

A Note on the Statistics of Hardcore Fermions

It is shown that the statistics of the hardcore fermions is A-superstatistics of order one [see T.D.P. J. Math. Phys. 21, 1293 (1980)]. The Pauli principle for these particles is formulated. The Hubbard operators, which constitute a basis in the Lie superalgebra gl(1|n), are expressed via the creation and the annihilation operators of the hardcore fermions.

cond-mat.str-el↗
arXiv2006

SL(3|N) Wigner quantum oscillators: examples of ferromagnetic-like oscillators with noncommutative, square-commutative geometry

A system of $N$ non-canonical dynamically free 3D harmonic oscillators is studied. The position and the momentum operators (PM-operators) of the system do not satisfy the canonical commutation relations (CCRs). Instead they obey the weaker postulates for the oscillator to be a Wigner quantum system. In particular the PM-operators fulfil the main postulate, which is due to Wigner: they satisfy the equations of motion (the Hamiltonian's equations) and the Heisenberg equations. One of the relevant features is that the coordinate (the momentum) operators do not commute, but instead their squares do commute. As a result the space structure of the basis states corresponds to pictures when each oscillating particle is measured to occupy with equal probability only finite number of points, typically the eight vertices of a parallelepiped. The state spaces are finite-dimensional, the spectrum of the energy is finite with equally spaced energy levels. An essentially new feature is that the angular momenta of all particles are aligned. Therefore there exists a strong interaction or correlation between the particles, which is not of dynamical, but of statistical origin. Another relevant feature is that the standard deviations of, say, the $k$-th coordinate and the momenta of $α$-th is $ Δ\hat R_{αk} Δ\hat P_{αk} \le {p \hbar/{|N-3|}}~(~N\ne 3,~ p -$fixed positive integer), namely instead of uncertainty relations one has "certainty" relations. The underlying Lie superalgebraic structure of the oscillator is also relevant and will be explained in the context.

hep-th↗
arXiv2003

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

An n-particle 3-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 Wigner's suggestion that Hamilton's 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\geq 3n but atypical for p<3n. The branching rules for the restriction from sl(1|3n) to gl(1) \oplus so(3) \oplus sl(n) are used to enumerate energy and angular momentum eigenstates. These are constructed explicitly and tabulated for n\leq 2. It is shown that measurements of the coordinates of the individual particles gives rise to a set of discrete values defining nests in the 3-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 realisation 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.

hep-th↗
arXiv2003

Microscopic and macroscopic properties of A-superstatistics

The microscopic and the macroscopic properties of A-superstatistics, related to the class A(0,n-1)\equiv sl(1|n) of simple Lie superalgebras are investigated. The algebra sl(1|n) is described in terms of generators f_1^\pm, >..., f_n^\pm, which satisfy certain triple relations and are called Jacobson generators. The Fock spaces of A-superstatistics are investigated and the Pauli principle of the corresponding statistics is formulated. Some thermal properties of A-superstatistics are constructed under the assumption that the particles interact only via statistical interaction imposed by the Pauli principle. The grand partition function and the average number of particles are written down explicitly in the general case and in two particular examples: 1) the particles have one and the same energy and chemical potential; 2) the energy spectrum of the orbitals is equidistant.

math-ph↗
arXiv2003

The non-commutative and discrete spatial structure of a 3D Wigner quantum oscillator

The properties of a non-canonical 3D Wigner quantum oscillator, whose position and momentum operators generate the Lie superalgebra sl(1|3), are further investigated. Within each state space W(p), p=1,2,..., the energy E_q, q=0,1,2,3, takes no more than 4 different values. If the oscillator is in a stationary state ψ_q\in 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 centered on the origin of fixed, finite radius \varrho_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 two. In certain states in W(2) (resp. in W(1)) the oscillator is ``polarized'' so that all the nests lie on a plane (resp. 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.

hep-th↗
arXiv2002

Deformed Clifford Cl_q(n|m) and orthosymplectic U_q[osp(2n+1|2m)] superalgebras and their root of unity representations

It is shown that the Clifford superalgebra Cl(n|m) generated by m pairs of Bose operators (odd elements) anticommuting with n pairs of Fermi operators (even elements) can be deformed to Cl_q(n|m) such that the latter is a homomorphic image of the quantum superalgebra U_q[osp(2n+1|2m)]. The Fock space F(n|m) of Cl_q(n|m) is constructed. At q being a root of unity (q=\exp ({iπl/k})) q-bosons (and q-fermions) are operators acting in a finite-dimensional subspace F_{l/k}(n|m) of F(n|m). Each F_{l/k}(n|m) is turned through the above mentioned homomorphism into an irreducible (root of unity) U_q[osp(2n+1|2m)] module. For q being a primitive root of unity (l=1) the corresponding representation is unitary. The module F_{1/k}(n|m) is decomposed into a direct sum of irreducible U_q[sl(m|n)] submodules. The matrix elements of all Cartan-Weyl elements of U_q[sl(m|n)] are given within each such submodule.

math.QA↗
arXiv2002

Discrete Space Structure of the 3D Wigner Quantum Oscillator

The properties of a noncanonical 3D Wigner quantum oscillator, whose position and momentum operators generate the Lie superalgebra sl(1|3), are further investigated. Within each state space W(p), p=1,2,..., the energy E_q, q=0,1,2,3, takes no more than 4 different values. If the oscillator is in a stationary state ψ_q \in 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 centered on the origin of fixed, finite radius \varrho_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 two. In certain states in W(2) (resp. in W(1)) the oscillator is ``polarized'' so that all the nests lie on a plane (resp. 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. The rotational invariance of the system is also discussed.

hep-th↗
arXiv2001

Jacobson generators of the quantum superalgebra $U_q[sl(n+1|m)]$ and Fock representations

As an alternative to Chevalley generators, we introduce Jacobson generators for the quantum superalgebra $U_q[sl(n+1|m)]$. The expressions of all Cartan-Weyl elements of $U_q[sl(n+1|m)]$ in terms of these Jacobson generators become very simple. We determine and prove certain triple relations between the Jacobson generators, necessary for a complete set of supercommutation relations between the Cartan-Weyl elements. Fock representations are defined, and a substantial part of this paper is devoted to the computation of the action of Jacobson generators on basis vectors of these Fock spaces. It is also determined when these Fock representations are unitary. Finally, Dyson and Holstein-Primakoff realizations are given, not only for the Jacobson generators, but for all Cartan-Weyl elements of $U_q[sl(n+1|m)]$.

math.QA↗
arXiv2001

Wigner quantum systems (Lie superalgebraic approach)

We present three groups of examples of Wigner Quantum Systems related to the Lie superalgebras $osp(1/6n)$, $sl(1/3n)$ and $sl(n/3)$ and discuss shortly their physical features. In the case of $sl(1/3n)$ we indicate that the underlying geometry is noncommutative.

hep-th↗
arXiv2001

Macroscopic properties of A-statistics

A-statistics is defined in the context of the Lie algebra sl(n+1). Some thermal properties of A-statistics are investigated under the assumption that the particles interact only via statistical interaction imposed by the Pauli principle of A-statistics. Apart from the general case, three particular examples are studied in more detail: (a) the particles have one and the same energy and chemical potential; (b) equidistant energy spectrum; (c) two species of particles with one and the same energy and chemical potential within each class. The grand partition functions and the average number of particles are among the thermodynamical quantities written down explicitly.

hep-th↗
arXiv2000

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

The properties of A-statistics, related to the class of simple Lie algebras sl(n+1) (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, first introduced by Jacobson. The related Fock spaces W_p (p=1,2,...) 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 W_p operators B(p)_1^\pm, ..., B(p)_n^\pm, proportional to the Jacobson generators, are introduced. It is proved that in an appropriate topology the limit of B(p)_i^\pm for p going to infinity is equal to B_i^\pm, where B_i^\pm 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.

hep-th↗
arXiv2000

Quasiboson representations of sl(n+1) and generalized quantum statistics

Generalized quantum statistics will be presented in the context of representation theory of Lie (super)algebras. This approach provides a natural mathematical framework, as is illustrated by the relation between para-Bose and para-Fermi operators and Lie (super)algebras of type B. Inspired by this relation, A-statistics is introduced, arising from representation theory of the Lie algebra A_n. The Fock representations for A_n=sl(n+1) provide microscopic descriptions of particular kinds of exclusion statistics, which may be called quasi-Bose statistics. It is indicated that A-statistics appears to be the natural statistics for certain lattice models in condensed matter physics.

hep-th↗
arXiv2000

Fock representations of the Lie superalgebra q(n+1)

For the Lie superalgebra $q(n+1)$ a description is given in terms of creation and annihilation operators, in such a way that the defining relations of $q(n+1)$ are determined by quadratic and triple supercommutation relations of these operators. Fock space representations $V_p$ of $q(n+1)$ are defined by means of these creation and annihilation operators. These new representations are introduced as quotient modules of some induced module of $q(n+1)$. The representations $V_p$ are not graded, but they possess a number of properties that are of importance for physical applications. For $p$ a positive integer, these representations $V_p$ are finite-dimensional, with a unique highest weight (of multiplicity 1). The Hermitian form that is consistent with the natural adjoint operation on $q(n+1)$ is shown to be positive definite on $V_p$. For $q(2)$ these representations are ``dispin''. For the general case of $q(n+1)$, many structural properties of $V_p$ are derived.

math.QA↗
arXiv1999

A description of the quantum superalgebra $U_q[sl(n+1|m)]$ via creation and annihilation generators

A description of the quantum superalgebra $U_q[sl(n+1|m)]$ and in particular of the special linear superalgebra $sl(n+1|m)$ via creation and annihilation generators (CAGs) is given. It provides an alternative to the canonical description of $U_q[sl(n+1|m)]$ in terms of Chevalley generators. A conjecture that the Fock representations of the CAGs provide microscopic realizations of exclusion statistics is formulated.

math.QA↗