Frame transforms, star products and quantum mechanics on phase space

Abstract

Using the notions of frame transform and of square integrable projective representation of a locally compact group $G$, we introduce a class of isometries (tight frame transforms) from the space of Hilbert-Schmidt operators in the carrier Hilbert space of the representation into the space of square integrable functions on the direct product group $G\times G$. These transforms have remarkable properties. In particular, their ranges are reproducing kernel Hilbert spaces endowed with a suitable 'star product' which mimics, at the level of functions, the original product of operators. A 'phase space formulation' of quantum mechanics relying on the frame transforms introduced in the present paper, and the link of these maps with both the Wigner transform and the wavelet transform are discussed.

48 pages

References in corpus (8)

• Alternative commutation relations, star products and tomography
• Spin tomography
• Tomograms and other transforms. A unified view
• Star products, duality and double Lie algebras
• A tomographic setting for quasi-distribution functions
• Generalized Phase Space Representation of Operators
• Applications of the group SU(1,1) for quantum computation and tomography
• Characterization of tomographically faithful states in terms of their Wigner function

Cited by in corpus (7)

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• Discovering the manifold facets of a square integrable representation: from coherent states to open systems
• Quantum Tomography and the Quantum Radon Transform
• Groupoids and Coherent states