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Segal–Bargmann transform: the q-deformation

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Abstract

We give identifications of the q-deformed Segal–Bargmann transform and define the Segal–Bargmann transform on mixed q-Gaussian variables. We prove that, when defined on the random matrix model of Śniady for the q-Gaussian variable, the classical Segal–Bargmann transform converges to the q-deformed Segal–Bargmann transform in the large N limit. We also show that the q-deformed Segal–Bargmann transform can be recovered as a limit of a mixture of classical and free Segal–Bargmann transform.

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Acknowledgements

The first author was partially funded by the ERC Advanced Grant “NCDFP” held by Roland Speicher. The second author was funded by the same ERC Advanced Grant “Non-commutative distributions in free probability” (Grant No. 339760). The second author would like to thank Roland Speicher for allowing his stay in Saarbrücken, Germany, so the authors had a chance to collaborate.

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 Route de Narbonne, 31062, Toulouse, France

    Guillaume Cébron

  2. Department of Mathematics, University of California, San Diego, La Jolla, CA, 92093-0112, USA

    Ching-Wei Ho

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  1. Guillaume Cébron
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  2. Ching-Wei Ho
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Correspondence to Ching-Wei Ho.

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Cébron, G., Ho, CW. Segal–Bargmann transform: the q-deformation. Lett Math Phys 108, 1677–1715 (2018). https://doi.org/10.1007/s11005-017-1039-7

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  • DOI: https://doi.org/10.1007/s11005-017-1039-7

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