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Gazeau, Jean Pierre; Murenzi, Romain

Covariant affine integral quantization(s). (English) Zbl 1407.81111

J. Math. Phys. 57, No. 5, 052102, 21 p. (2016).
The manuscript is a continuation of work on covariant integral quantization on a Lie group which was introduced earlier by H. Bergeron and the first author [Ann. Phys. 344, 43–68 (2014; Zbl 1343.81157)]. The general setting is as follows: Let \(G\) be a Lie group with left Haar measure \(d\mu\). Suppose \(U\) is an irreducible representation of \(G\) on a Hilbert space \(\mathcal{H}\). Then we define an operator associating to a function or distribution on \(G\) by \(f\mapsto A_f=\int_G f(g)U(g)MU(g)^*d\mu(g)\) for a given bounded operator \(M\) on \(\mathcal{H}\) that satisfies \(c_M\int_G U(g)MU(g)^*d\mu(g)=I\) for a normalization constant \(c_M\).
If \(G\) is the Heisenberg group, \(U\) the Schrödinger representation and \(M\) a rank-one operator, then the operator \(A_f\) is a localization operator and if \(M\) is a density operator, then \(A_f\) is a so-called mixed-state localization operator, see the results in [the reviewer and E. Skrettingland, J. Math. Pures Appl. (9) 118, 288–316 (2018; Zbl 1486.47086)].
The main goal of this paper is to define covariant integral quantization for affine groups.
Reviewer: Franz Luef (Trondheim)

Cited in 8 Documents

MSC:

81S10 Geometry and quantization, symplectic methods
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
81R30 Coherent states

Keywords:

quantization; affine group; affine coherent states

Citations:

Zbl 1486.47086; Zbl 1343.81157
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Full Text: DOI arXiv

References:

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