Abstract
Representations of the celebrated Heisenberg commutation relations in quantum mechanics (and their exponentiated versions) form the starting point for a number of basic constructions, both in mathematics and mathematical physics (geometric quantization, quantum tori, classical and quantum theta functions) and signal analysis (Gabor analysis). In this paper we will try to bridge the two communities, represented by the two co-authors: that of noncommutative geometry and that of signal analysis. After providing a brief comparative dictionary of the two languages, we will show, e.g. that the Janssen representation of Gabor frames with generalized Gaussians as Gabor atoms yields in a natural way quantum theta functions, and that the Rieffel scalar product and associativity relations underlie both the functional equations for quantum thetas and the Fundamental Identity of Gabor analysis.
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Acknowledgments
Large parts of this manuscript were written during a stay of F. Luef at the Max Planck Institute for Mathematics at Bonn. F. Luef would like to express his gratitude for hospitality and excellent working conditions. In addition F. Luef acknowledges the support from the EU-project MEXT-CT-2004-517154 and the Marie Curie Outgoing Fellowship PIOF 220464.
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luef, F., Manin, Y.I. Quantum Theta Functions and Gabor Frames for Modulation Spaces. Lett Math Phys 88, 131–161 (2009). https://doi.org/10.1007/s11005-009-0306-7
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DOI: https://doi.org/10.1007/s11005-009-0306-7