BargmannFockSpace 2007-02-19 06:02:37 2007-02-21 14:45:43 Definition Fock space % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here The Bargmann-Fock space (or simply Fock space) is the Hilbert space of entire functions, $\mathcal{F}^2(\mathbb{C})$ s.t. $$ \int_\mathbb{C}|F(z)|^2 e^{- \pi |z|^2}dx dy<\infty$$ with associated inner product $$ \int_\mathbb{C}F(z)\overline{G(z)}e^{- \pi |z|^2}dx dy$$ where $z=x+i y$ \begin{thebibliography}{2} \bibitem{vb} V. Bargmann, ``Remarks on a Hilbert Space of Analytic Function'' {\it Proceedings of the National Academy of Sciences of the United States of America} {\bf 48} (1962): 199 - 204 \bibitem{vbt} V. Bargmann \& I. T. Todorov, ``Spaces of analytic functions on a complex cone as carriers for the symmetric tensor representations of SO(n)'' {\it Journal of Mathematical Physics} {\bf 18} 6 (1977): 1141 - 1148 \end{thebibliography}