Searcharxiv⌕ Search

arXiv subjects

N. I. Stoilova

Publications and source records attributed to N. I. Stoilova.

At least 55 records · Page 3Linked to original sources
arXiv2005

Solutions of the compatibility conditions for a Wigner quantum oscillator

We consider the compatibility conditions for a N-particle D-dimensional Wigner quantum oscillator. These conditions can be rewritten as certain triple relations involving anticommutators, so it is natural to look for solutions in terms of Lie superalgebras. In the recent classification of ``generalized quantum statistics'' for the basic classical Lie superalgebras [math-ph/0504013], each such statistics is characterized by a set of creation and annihilation operators plus a set of triple relations. In the present letter, we investigate which cases of this classification also lead to solutions of the compatibility conditions. Our analysis yields some known solutions and several classes of new solutions.

math-ph↗
arXiv2005

A classification of generalized quantum statistics associated with basic classical Lie superalgebras

Generalized quantum statistics such as para-statistics is usually characterized by certain triple relations. In the case of para-Fermi statistics these relations can be associated with the orthogonal Lie algebra B_n=so(2n+1); in the case of para-Bose statistics they are associated with the Lie superalgebra B(0|n)=osp(1|2n). In a previous paper, a mathematical definition of ``a generalized quantum statistics associated with a classical Lie algebra G'' was given, and a complete classification was obtained. Here, we consider the definition of ``a generalized quantum statistics associated with a basic classical Lie superalgebra G''. Just as in the Lie algebra case, this definition is closely related to a certain Z-grading of G. We give in this paper a complete classification of all generalized quantum statistics associated with the basic classical Lie superalgebras A(m|n), B(m|n), C(n) and D(m|n).

math-ph↗
arXiv2004

All fundamental fermions fit inside one su(1|5) irreducible representation

We present the mathematical framework for a unified theory based upon su(1|5). The Lie superalgebra su(1|5) has irreducible representations of dimension 32, in which the 32 fundamental fermions of one generation (leptons and quarks, of left and right chirality, and their antiparticles) can be accommodated. The branching of these su(1|5) representations with respect to its subalgebra su(3) x su(2) x u(1) reproduces precisely the classification of these fundamental fermions according to the gauge group of the Standard Model. Furthermore, a simple construction of the relevant representations is given, and some consequences are discussed.

hep-th↗
arXiv2004

A classification of generalized quantum statistics associated with classical Lie algebras

Generalized quantum statistics such as para-Fermi statistics is characterized by certain triple relations which, in the case of para-Fermi statistics, are related to the orthogonal Lie algebra B_n=so(2n+1). In this paper, we give a quite general definition of ``a generalized quantum statistics associated to a classical Lie algebra G''. This definition is closely related to a certain Z-grading of G. The generalized quantum statistics is then determined by a set of root vectors (the creation and annihilation operators of the statistics) and the set of algebraic relations for these operators. Then we give a complete classification of all generalized quantum statistics associated to the classical Lie algebras A_n, B_n, C_n and D_n. In the classification, several new classes of generalized quantum statistics are described.

math-ph↗
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↗
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↗
arXiv1998

Irreducible highest weight representations of the quantum algebra $U_h(A_\infty)$

A class of highest weight irreducible representations of the algebra $U_h(A_\infty)$, the quantum analogue of the completion and central extension $A_\infty$ of the Lie algebra $gl_\infty$, is constructed. It is considerably larger than the class of representations known so far. Within each module a basis is introduced and the transformation relations of the basis under the action of the Chevalley generators are explicitly given. The verification of the quantum algebra relations is shown to reduce to a set of nontrivial $q$-number identities. All representations are restricted in the terminology of S. Levendorskii and Y. Soibelman (Commun. Math. Phys. {\bf 140}, 399-414 (1991)).

math.QA↗
arXiv1997

Eigenvalues of Casimir operators for $gl(m/\infty)$

A full set of Casimir operators for the Lie superalgebra $gl(m/\infty)$ is constructed and shown to be well defined in the category $O_{FS}$ generated by the highest weight irreducible representations with only a finite number of non-zero weight components. The eigenvalues of these Casimir operators are determined explicitly in terms of the highest weight. Characteristic identities satisfied by certain (infinite) matrices with entries from $gl(m/\infty)$ are also determined.

math-ph↗
arXiv1997

Highest weight irreducible representations of the quantum algebra $U_h(A_\infty)$

A class of highest weight irreducible representations of the algebra $U_h(A_\infty)$, the quantum analogue of the completion and central extension $A_\infty$ of the Lie algebra $gl_\infty$, is constructed. It is considerably larger than the known so far representations. Within each module a basis is introduced and the transformation relations of the basis under the action of the Chevalley generators are explicitly written. The verification of the quantum algebra relations to be satisfied is shown to reduce to a set of nontrivial $q$-number identities. All our representations are restricted in the terminology of S. Levendorskii and Y. Soibelman (Commun. Math. Phys. 140, 399-414 (1991)).

q-alg↗