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arXiv:math-ph/0008035 · Fortschritte der Physik

Representations of the Schrödinger algebra and Appell systems

Philip Feinsilver, Jerzy Kocik, René Schott

Published Mar 15, 2004• 31 citations
arXivHTMLPDF

Abstract

Abstract We investigate the structure of the Schrödinger algebra. Two constructions are given that yield the physical realization via general methods starting from the abstract Lie algebra. Representations are found on a Fock space with basis given by a canonical Appell system. Generalized coherent states are used in the construction of the Hilbert space of functions on which certain commuting elements act as self‐adjoint operators. This yields a probabilistic interpretation of these operators as random variables. An interesting feature is how the semidirect product structure of the Lie algebra is reflected in the probability density function. A Leibniz function and orthogonal basis for the Hilbert space are found. Then certain evolution equations connected with canonical Appell systems on this algebra are shown.

DOI
10.1002/prop.200310124

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Abstract We investigate the structure of the Schrödinger algebra. Two constructions are given that yield the physical realization via general methods starting from the abstract Lie algebra. Representations are found on a Fock space with basis given by a canonical Appell system. Generalized coherent states are used in the construction of the Hilbert space of functions on which certain commuting elements act as self‐adjoint operators. This yields a probabilistic interpretation of these operators as random variables. An interesting feature is how the semidirect product structure of the Lie algebra is reflected in the probability density function. A Leibniz function and orthogonal basis for the Hilbert space are found. Then certain evolution equations connected with canonical Appell systems on this algebra are shown.
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Abstract This chapter develops the fundamental notions and concepts of the probabilistic and statistical interpretation of quantum mechanics. It begins with a brief review of the mathematical structure given by a Hilbert space of state vectors and an algebra of observables represented by selfadjoint operators. Special emphasis is laid on the connection between the spectral representation of a selfadjoint operator and a corresponding random variable that describes the possible measurement outcomes. The chapter further introduces the concepts of a composite quantum system, of the reduced density matrix, and of entangled quantum states. A section on quantum entropies is included, as well as the mathematical formulation of the generalized theory of quantum measurements in terms of completely positive quantum operations and effects.
We examine the possibility of a direct Fock representation of the recently obtained non-trivial central extensions CEHeis of the Heisenberg algebra, generated by elements a, a†, h and E satisfying the commutation relations [a, a†]CEHeis = h, [h, a†]CEHeis = z E and [a, h]CEHeis = ̄z E, where a and a† are dual, h is self-adjoint, E is the non-zero selfadjoint central element and z 2 C \ {0}. We define the exponential vectors associated with the CEHeis Fock space, we compute their Leibniz function (inner product), we describe the action of a, a† and h on the exponential vectors and we compute the moment generating and characteristic functions of the classical random variable corresponding to the self-adjoint operator X = a + a† + h.
The self adjoint linear operators which represent the observables of a physical system are in general not an irreducible system. Because a complete set of commuting observables must determine the state of a physical system unambiguously the observables generate an algebra of operators which must contain a maximal abelian subalgebra. The structure of such algebras is investigated and it is shown by applying the theory of the direct integral of Hilbert spaces that there exists always a unique canonical representation of the Hilbert space as a direct integral in such a way that the transformations which are induced by the observables in the component subspaces are irreducible.
This chapter provides a simple general introduction to some basic mathematics necessary for a subsequent discussion of Lie algebras, operator methods, and their applications to theoretical physics and chemistry. It also provides some useful mathematical tools to derive important theoretical results and expressions in a simple and elegant way. In quantum mechanics the state of a system is represented by an element of a particular vector space, in which one defines an appropriate scalar or inner product, known in mathematics as Hilbert space. Hermitian operators with a complete set of eigenvectors belonging to that state space represent physical observables. Moreover, in certain cases it proves useful to work with vector spaces formed by linear operators. Hermitian operators are particular cases of linear operators and physical observables are particular cases of Hermitian operators. In many physical applications of operator methods one needs to reorder products of exponential operators or to rewrite one such operator as a product of simpler ones.