Bargmann transform

The Bargmann transform of a function, $f$, is a linear map $B:X(ℝ)\to Y(ℂ)$ defined by

$$Bf(z)=\sqrt{2}{\int }_{ℝ}f(t)e^{2\pi tz-\pi t^2-\frac{\pi }{2}{z}^2}𝑑t$$

Theorem.

The Bargmann transform on $L^{\mathrm{2}}\mathit{}\mathrm{(}\mathrm{R}\mathrm{)}$, $B\mathrm{:}{L}^{\mathrm{2}}\mathit{}\mathrm{(}\mathrm{R}\mathrm{)}\mathrm{\to }{\mathrm{F}}^{\mathrm{2}}\mathit{}\mathrm{(}\mathrm{C}\mathrm{)}$, is a unitary transformation. Here ${\mathrm{F}}^{\mathrm{2}}\mathit{}\mathrm{(}\mathrm{C}\mathrm{)}$ is the Fock space.