Contrary to the usual view that quantum mechanics in configuration space has, in general, only one algebaic representation( the Heisenberg algebraic representation), we have proved that quantum mechanics in configuration space has, in general, alternative algebraic representations: (i) In a finite domain of configuration space, it can be expressed in terms of su(2) algebra; (ii) in an infinite domain of configuration space, it can be expressed in terms of su(l,l) algebra.The above results open a new possibility to reformulate quantum mechanics and provide more mathematical tools to solve diverse physical problems.Nonlinear relations of different Lie algebas may imply a unification of Lie algebras and their physical implications ( the quantum motion modes ) in a deeper nonlinear domain.This observation raises challenging mathematical and physical questions.
We show the use of the theory of Lie algebras, especially their oscillator realizations, in the context of quantum mechanics. One can construct oscillator realizations from matrix realizations. In the case of symplectic and special orthogonal algebra, we demonstrate an alternative method of obtaining oscillator realizations from symmetric or exterior power of a vector space of annihilation and creation bosonic or fermionic operators. We find Lie algebra of polynomials of degree at most two in phase space of a mechanical system, which form the semi-direct product of the Heisenberg algebra and symplectic algebra. It is shown that a classical system with Hamiltonian function in this algebra can be quantized by two equivalent representations - Schrödinger or Bargmann-Fock representation. The second mentioned representation generates the same operators of symplectic algebra as we got from their previous formal construction from symmetric power of a vector space of bosonic operators. Quantization is demonstrated on the bosonic harmonic oscillator. We use the similarities between bosonic and fermionic oscillator realizations to define the fermionic harmonic oscillator. Some properties of spinor representations of special orthogonal algebra are illustrated on its state space. Powered by TCPDF (www.tcpdf.org)
The aim of this paper is to construct coherent states à la Perelomov and à la Barut–Girardello for a polynomial Weyl–Heisenberg algebra. This generalized Weyl–Heisenberg algebra, denoted by , depends on r real parameters and is an extension of the one-parameter algebra (Daoud and Kibler 2010 J. Phys. A: Math. Theor. 43 115303) which covers the cases of the su (1, 1) algebra (for κ > 0), the su (2) algebra (for κ < 0) and the h 4 ordinary Weyl–Heisenberg algebra (for κ = 0). For finite-dimensional representations of and , where is a truncation of order s of in the sense of Pegg–Barnett, a connection is established with k -fermionic algebras (or quon algebras). This connection makes it possible to use generalized Grassmann variables for constructing certain coherent states. Coherent states of the Perelomov type are derived for infinite-dimensional representations of and for finite-dimensional representations of and through a Fock–Bargmann analytical approach based on the use of complex (or bosonic) variables. The same approach is applied for deriving coherent states of the Barut–Girardello type in the case of infinite-dimensional representations of . In contrast, the construction of coherent states à la Barut–Girardello for finite-dimensional representations of and can be achieved solely at the price of replacing complex variables by generalized Grassmann (or k -fermionic) variables. Some of the results are applied to su (2), su (1, 1) and the harmonic oscillator (in a truncated or not truncated form). This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Coherent states: mathematical and physical aspects'.
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 expression of Hermite polynomials via Laguerre ones allows to connect the standard representations of the Heisenberg algebra with representations of generalized oscillator algebra, related to the Laguerre polynomials. We give the connection between the measures participating in the orthogonality relations for these polynomials and the connection between energy levels of related oscillators. Such considerations can be extended to a general case in which Hermite polynomials are replaced by polynomials orthogonal with respect to a symmetric measure. This work is partially supported by RFBR grant No 09-01-00504-a.