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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Axelrod S., Della Pietra S., Witten E. (1991). Geometric quantization of Chern-Simons gauge theory. J. Diff. Geom. 33:787–902
Bargmann V.(1961). On a Hilbert space of analytic functions and an associated integral transform. Comm. Pure Appl. Math. 14:187–214
Folland G.B. (1984). Real analysis Modern techniques and their applications. John Wiley & Sons, New York
Florentino C., Matias P., Mourao J., Nunes J.P. (2005). Geometric quantization, complex structures and the coherent state transform. J. Funct. Anal. 221:303–322
Ginzburg, V.L., Montgomery, R.: Geometric quantization and no-go theorems. In: Grabowski, J., Urbański, P. (eds.) Poisson geometry, Warsaw, 1998, Banach Center Publ. 51, Warsaw: Polish Acad. Sci., 2000, 69–77
Hall B.C. (2002). Geometric quantization and the generalized Segal-Bargmann transform for Lie groups of compact type. Commun. Math. Phys. 226:233–268
Lion, G., Vergne, M.: The Weil representation, Maslov index and theta series. Prog. in Math. 6, Birkhäuser Boston, MA 1980, Part I
Magneron B. (1984). Spineurs symplectiques purs et indice de Maslov de plan Lagrangiens positifs. J. Funct. Anal. 59:90–122
Robinson, P.L., Rawnsley, J.H.: The metaplectic representation, Mpc structures and geometric quantization. Mem. Amer. Math. Soc. Vol. 81, No. 410, Providence, RI: Amer. Math. Soc., 1989
Satake, I.: On unitary representations of a certain group extension (in Japanese). Sugaku 21, 241–253 (1969); Fock representations and theta-functions. In: Ahlfors, L.V. et al. (eds.) Advances in the theory of Riemann surfaces. Princeton, NJ: Princeton Univ. Press, 1971, pp. 393–405
Siegel C.L. (1943). Symplectic geometry. Amer. J. Math. 65:1–86
Stein E.M., Weiss G. (1971). Introduction to Fourier analysis on Euclidean spaces. Princeton Univ. Press, Princeton, NJ
Taylor M.E. (1996). Partial differential equations I. Basic theory. Springer-Verlag, New York
Woodhouse N.M.J. (1981). Geometric quantization and the Bogoliubov transformation. Proc. Royal Soc. London A 378:119–139
Woodhouse N.M.J. (1992). Geometric Quantization (2nd ed). Oxford Univ. Press, New York
Author information
Authors and Affiliations
Additional information
Communicated by A. Connes
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-006-0043-z