Geometric Quantization, Parallel Transport and the Fourier Transform

Autor:	Siye Wu, William D. Kirwin
Rok vydání:	2006
Předmět:	Geometric quantization Parallel transport Mathematical analysis Hilbert space Statistical and Nonlinear Physics Position and momentum space Symplectic vector space symbols.namesake Fourier transform symbols Connection (algebraic framework) Mathematical Physics Mathematics Vector space
Zdroj:	Communications in Mathematical Physics. 266:577-594
ISSN:	1432-0916 0010-3616
DOI:	10.1007/s00220-006-0043-z
Popis:	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.
Databáze:	OpenAIRE
Externí odkaz:	https://explore.openaire.eu/search/publication?articleId=doi_________::b32eb2eeddc223d577c9f4a491c42ada https://doi.org/10.1007/s00220-006-0043-z Zobrazit plný text záznamu Plný text ve formátu PDF