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Journal of Mathematical Physics

Coherent states and projective representation of the linear canonical transformations

Ingrid C. Daubechies

Published Jun 1, 1980• 23 citations
Publisher

Abstract

Using a family of coherent state representations we obtain in a natural and coordinate-independent way an explicit realization of a projective unitary representation of the symplectic group. Dequantization of these operators gives us the corresponding classical functions.
DOI
10.1063/1.524562

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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.

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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.
Classical linear transformations in coordinate-momentum phase space are mapped to unitary quantum-mechanical operators in Hilbert space to produce a generalization of squeeze operators. The classical to quantum transition is manifestly apparent in the derivation. The unitary operators are evaluated in the coherent-state representation using the ``integration within ordered products'' technique in the one- and two-mode cases. Application of these operators results in a new generalization of coherent states.