Bargmann-Fock space

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}}dxdy<\infty

with associated inner product

\int_{\mathbb{C}}F(z)\overline{G(z)}e^{-\pi|z|^{2}}dxdy

where $z=x+iy$

References

1 V. Bargmann, “Remarks on a Hilbert Space of Analytic Function” Proceedings of the National Academy of Sciences of the United States of America 48 (1962): 199 - 204
2 V. Bargmann & I. T. Todorov, “Spaces of analytic functions on a complex cone as carriers for the symmetric tensor representations of SO(n)” Journal of Mathematical Physics 18 6 (1977): 1141 - 1148