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Geometric Quantization, Parallel Transport and the Fourier Transform

Communications in Mathematical Physics volume 266, pages 577–594 (2006)Cite this article

Abstract

In quantum mechanics, the momentum space and position space wave functions are related by the Fourier transform. We investigate how the Fourier transform arises in the context of geometric quantization. We consider a Hilbert space bundle \(\mathcal{H}\) over the space \(\mathcal{J}\) of compatible complex structures on a symplectic vector space. This bundle is equipped with a projectively flat connection. We show that parallel transport along a geodesic in the bundle \(\mathcal{H} \to \mathcal{J}\) is a rescaled orthogonal projection or Bogoliubov transformation. We then construct the kernel for the integral parallel transport operator. Finally, by extending geodesics to the boundary (for which the metaplectic correction is essential), we obtain the Segal-Bargmann and Fourier transforms as parallel transport in suitable limits.

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Author notes

  1. William D. Kirwin

    Present address: Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46556, USA

Authors and Affiliations

  1. Department of Mathematics, University of Colorado, Boulder, CO, 80309-0395, USA

    William D. Kirwin & Siye Wu

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  1. William D. Kirwin
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  2. Siye Wu
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Communicated by A. Connes

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Kirwin, W.D., Wu, S. Geometric Quantization, Parallel Transport and the Fourier Transform. Commun. Math. Phys. 266, 577–594 (2006). https://doi.org/10.1007/s00220-006-0043-z

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  • DOI: https://doi.org/10.1007/s00220-006-0043-z

Keywords

  • Hilbert Space
  • Parallel Transport
  • Bergman Kernel
  • Geometric Quantization
  • Bogoliubov Transformation