Tchavdar D. Palev

Para‐Bose and para‐Fermi operators as generators of orthosymplectic Lie superalgebras

It is shown that the relative commutation relations between n pairs of para‐Fermi operators and m pairs of para‐Bose operators can be defined in such a way that they generate the simple orthosymplectic Lie superalgebra B(n,m). In a case of ordinary statistics this leads to mutually anticommuting Bose and Fermi fields.

Wigner approach to quantization. Noncanonical quantization of two particles interacting via a harmonic potential

Following the ideas of Wigner, we quantize noncanonically a system of two nonrelativistic point particles, interacting via a harmonic potential. The center of mass phase‐space variables are quantized in a canonical way, whereas the internal momentum and coordinates are assumed to satisfy relations, which are essentially different from the canonical commutation relations. As a result, 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. In particular, the distance between the constituents is preserved in time and can take at most four different values. The orbital momentum is either zero or one (in units ℏ/2). The operators of the coordinates do not commute with each other and, therefore, the position of any one of the constituents cannot be localized; the particles are smeared with a certain probability in a finite space volume, which moves together with the center of mass. In t...

Fock space representations of the Lie superalgebra A(0,n)

An infinite class of finite‐dimensional irreducible representations and one particular infinite‐dimensional representation of the special linear superalgebra of an arbitrary rank is constructed. For every representation an orthonormal basis in the corresponding representation space is found, and the matrix elements of the generators are calculated. The method we use is similar to the one applied in quantum theory to compute the Fock space representations of Bose and Fermi operators. For this purpose we first introduce a concept of creation and annihilation operators of a simple Lie superalgebra and give a definition of Fock‐space representations.

Finite‐dimensional representations of the Lie superalgebra gl(2/2) in a gl(2)⊕gl(2) basis. II. Nontypical representations

All finite‐dimensional irreducible representations of the general linear Lie superalgebra gl(2/2) are written down in matrix form. The basis within each representation space is chosen in such a way that it makes evident the decomposition of gl(2/2) into irreducible representations of its even subalgebra gl(2)⊕gl(2). Special attention is devoted to the analysis of all nontypical representations and some indecomposible representations.

Irreducible finite‐dimensional representations of the Lie superalgebra gl(n/1) in a Gel’fand–Zetlin basis

All finite‐dimensional irreducible representations of the general linear Lie superalgebra gl(n/1) are studied. For each representation, a concept of a Gel’fand–Zetlin basis is defined. Expressions for the transformation of the basis under the action of the generators are written down.

Finite‐dimensional representations of the special linear Lie superalgebra sl(1,n). II. Nontypical representations

In a series of two papers all finite‐dimensional irreducible representations of the special linear Lie superalgebra sl(1,n) are written down in a matrix form. This paper develops a background for constructing the representations. Expressions for the transformation of the basis under the action of the generators are given for all induced and, hence, for all typical sl(1,n) modules.

Highest weight irreducible unitarizable representations of Lie algebras of infinite matrices. The algebra A

Two classes of irreducible highest weight modules of the general linear Lie algebra gl(∞), corresponding to two different Borel subalgebras, are constructed. Both classes contain all unitary representations. Within each module a basis is introduced. Expressions for the transformation of the basis under the action of the algebra are written down.

Canonical realizations of Lie superalgebras: Ladder representations of the Lie superalgebra A(m,n)

A simple formula for realizations of Lie superalgebras in terms of Bose and Fermi creation and annihilation operators is given. The essential new feature is that Bose and Fermi operators mutually anticommute. The Fock representation of these operators is used in order to construct a class of irreducible finite‐dimensional representations of the simple Lie superalgebra A(m,n). The matrix elements of the generators are written down. For m≳0 all representations turn out to be nontypical.

THE HOLSTEIN-PRIMAKOFF AND DYSON REALIZATIONS FOR THE LIE SUPERALGEBRA GL(M/N+1)

The known Holstein - Primakoff and Dyson realizations for gl(n+1), in terms of Bose operators (Okubo S 1975 J. Math. Phys. 16 528) are generalized to the class of the Lie superalgebras gl(m/n+1) for any n and m. Formally the expressions are the same as for gl(m+n+1), however, both Bose and Fermi operators are involved.

Transformations of some induced osp(3/2) modules in an so(3)○+sp(2) basis

The modules of the orthosymplectic Lie superalgebra osp(3/2), induced from finite‐dimensional irreducible submodules of the stability subalgebra so(3)⊕gl(1) are investigated. The corresponding infinite‐dimensional irreducible or indecomposable modules, the Kac modules, and the related typical and atypical modules are studied in detail. Every such module is decomposed into a direct sum of either indecomposable or irreducible modules of the even subalgebra so(3)⊕sp(2). For each of these (infinite‐dimensional or finite‐dimensional, irreducible or indecomposable) modules relations are written down, giving the transformations of the basis under the action of the algebra generators.