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arXiv:math-ph/0410049 · Infinite Dimensional Analysis Quantum Probability and Related Topics

ULTRACOHERENCE AND CANONICAL TRANSFORMATIONS

Joachim Kupsch, Subhashish Banerjee

Published Sep 1, 2006• 7 citations
arXivHTMLPDF

Abstract

The (in)finite dimensional symplectic group of homogeneous canonical transformations is represented on the bosonic Fock space by the action of the group on the ultracoherent vectors, which are generalizations of the coherent states.
DOI
10.1142/s0219025706002469

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When solving problems involving quantum mechanical systems, it is frequently desirable to find the matrix elements of a unitary representation TTT of a real algebraic Lie group GGG. This requires defining an inner product on the Hilbert space mathbbH\\mathbb{H}mathbbH that carries the representation TTT. In the case where the representation is determined by a representation of a subgroup containing the lowest weight vector of TTT, this can be achieved through the coherent state construction. In both the scalar and vector coherent state methods, the process of finding the overlaps can be simplified by introducing the coherent state triplet (mathfrakFmathbbHD\\mathfrak{F}_{\\mathbb{H}_D}mathfrakFmathbbHD​​, mathbbHD\\mathbb{H}_DmathbbHD​, mathfrakFmathfrakHD\\mathfrak{F}^{\\mathfrak{H}_D}mathfrakFmathfrakHD​) of Bargmann spaces. Coherent state wave functions -- the elements of mathfrakFmathbbHD\\mathfrak{F}_{\\mathbb{H}_D}mathfrakFmathbbHD​​ and of mathfrakFmathbbHD\\mathfrak{F}^{\\mathbb{H}_D}mathfrakFmathbbHD​ -- are used to define the inner product on mathbbHD\\mathbb{H}_DmathbbHD​ in a way that simplifies the calculation of the overlaps. This inner product and the group action Gamma\\GammaGamma of GGG on mathfrakFmathbbHD\\mathfrak{F}^{\\mathbb{H}_D}mathfrakFmathbbHD​ are used to formulate expressions for the matrix elements of TTT with coefficients from the given subrepresentation.\n\nThe process of finding an explicit expression for Gamma\\GammaGamma relies on matrix factorizations in the complexification of GGG even though the representation TTT does not extend to the complexification. It will be shown that these factorizations are, in fact, justified, that the overlaps and Gamma\\GammaGamma action can be expressed in terms of the given subrepresentation, and that it is possible to find numerical values for the inner product in mathbbH\\mathbb{H}mathbbH. The scalar and vector coherent state methods will both be applied to Sp(nnn) and Sp(n,mathbbRn,\\mathbb{R}n,mathbbR).
A one-parameter symplectic group {etÂ}t∈ℝ derives proper canonical transformations indexed by t on a Boson–Fock space. It has been known that the unitary operator Ut implementing such a proper canonical transformation gives a projective unitary representation of {etÂ}t∈ℝ on the Boson–Fock space and that Ut can be expressed as a normal-ordered form. We rigorously derive the self-adjoint operator Δ(Â) and a local exponent [Formula: see text] with a real-valued function τÂ(·) such that [Formula: see text].
Canonical transformations are defined and discussed along with the coherent and the ultracoherent vectors. It is shown that the single-mode and the n -mode squeezing operators are elements of the group of canonical transformations. An application of canonical transformations is made, in the context of open quantum systems, by studying the effect of squeezing of the bath on the decoherence properties of the system. Two cases are analysed. In the first case, the bath consists of a massless bosonic field with the bath reference states being the squeezed vacuum states and squeezed thermal states while in the second case a system consisting of a harmonic oscillator interacting with a bath of harmonic oscillators is analysed with the bath being initially in a squeezed thermal state.