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Springer eBooks

Realizations of Lie Algebras Through Rational Functions of Canonical Variables

Heinz Dietrich Doebner, T. D. Palev

Published Jan 1, 1970• 5 citations
Publisher
DOI
10.1007/978-3-7091-5835-7_17

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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.
A fully consistent realization of the quantum operators corresponding to the canonically conjugate phase and number variables is proposed, resorting to the positive discrete series of the irreducible unitary representation of the Lie algebra su (1, 1) of the double covering group of SO ↑ (1, 2). The realization holds in subspace , the system Fock space minus the vacuum state. A possible way to extend it to the full space of states based on recourse to a dilated extension of Hilbert space is discussed.