  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 H over the space 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 H -> 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.
