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Using a family of coherent state representations we obtain in a natural and coordinate independent wayan explicit realization of a projective unitary representation of the symplectic group. Dequantization of these operators gives us the corresponding classical functions.

Ingrid C. Daubechies, Theoretische Natuurkunde

Published Jan 1, 2012• 0 citations

Abstract

cause the substitution has taken us out of it) is, up to some constant factor, a unitary operator. The family of these oper ators gives us our projective representation. We work with intrinsic and coordinate-free notations differing from the no tations used in Refs. 2, 3, or 4. At the end of the paper we rewrite some of the results in the more familiar x-p notations. Following the prescription given in Ref. 11 for the de quantization of these operators, we proceed then to compute the classical functions corresponding to the symplectic transformations. This calculation of classical functions for symplectic transformations has been done for one-param eter subgroups of the symplectic group.8.9 One then only

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cause the substitution has taken us out of it) is, up to some constant factor, a unitary operator. The family of these oper ators gives us our projective representation. We work with intrinsic and coordinate-free notations differing from the no tations used in Refs. 2, 3, or 4. At the end of the paper we rewrite some of the results in the more familiar x-p notations. Following the prescription given in Ref. 11 for the de quantization of these operators, we proceed then to compute the classical functions corresponding to the symplectic transformations. This calculation of classical functions for symplectic transformations has been done for one-param eter subgroups of the symplectic group.8.9 One then only

We present an explicit construction of coherent states for an arbitrary irreducible representation of the unitary symplectic group USp (4). Three different families of coherent states are obtained, corresponding to the subgroups U (1) × U (1), U (2) and SU (2) × SU (2). The symplectic structure on the manifold of coherent states is obtained, and canonical coordinates are used to express the classical limit of quantum observables. One of the families is seen to provide a trivial classical limit.

The conventional quantization of the harmonic oscillator in terms of operators Q and P can be implemented with the help of irreducible unitary representations of the Heisenberg-Weyl group which acts transitively and effectively on the simply connected classical phase space S q,p ≊ 2 . In the description of the harmonic oscillator in terms of angle and action variables ϕ and I the associated phase space S ϕ,I corresponds to the multiply connected punctured plane 2 – {0}, on which the 3-dimensional symplectic group Sp(2, ) acts transitively, leaving the origin invariant. As this group contains the compact subgroup U (1) it has infinitely many covering groups. In the here relevant irreducible unitary representations (positive discrete series) the self-adjoint generator K o of U (l) represents the classical action variable I . It has the possible spectra n + k, n = 0,1,...; k > 0, where k depends on the covering group. This implies different possible spectra for the action variable Hamiltonian ℏωK 0 of the harmonic oscillator. On the other hand, expressing the operators Q and P (non-linearly) in terms of the three generators K 0 etc. of Sp (2, ) leads to the usual framework. Possible physical (experimental) implications and generalizations to higher dimensions are discussed briefly.