nLab Bargmann-Segal transform

Idea

The classical Bergmann-Segal transform is a unitary transformation from the Hilbert space of Lebesgue square integrable functions $L^{2}(\mathbb{R}^n)$ to the Bargmann-Fock space of analytic functions with certain growth condition.

Vectors in the Hilbert space may be represented in the coherent state representation: $|f\rangle = \int |z\rangle\langle z|f\rangle d\mu$; if $f$ is in $L^2(\mathbb{R}^n)$ then $\langle z|f\rangle$ is a holomorphic function and this passage is called the Bargmann-Segal transform. This way a Hilbert space of holomorphic function appears, the Bargmann-Fock space.

The Bergmann-Segal transform is therefore an integral transform whose kernel is the overlap between the projective measure corresponding to the coherent states and a measure corresponding to an orthonormal basis coming from some polarization for $L^2$-sections. The kernel is a special case of a Bergman kernel in complex analysis.

Related concepts

• coherent state

Literature

Named after Valentine Bargmann and Irving Segal.

The classical case of the Heisenberg group:

• Valentine Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Communications on Pure and Applied Mathematics 14 (1961) 187-214 [doi:10.1002/cpa.3160140303, MR0157250]

Further generalization is to Perelomov coherent states.

More recent generalized Segal-Bargmann transform of Hall:

• Brian Charles Hall, The Segal-Bargmann coherent state transform for Lie groups. J. Funct. Anal. 122 (1994) 103–151 [doi:10.1006/jfan.1994.1064;

• Brian Charles Hall, Geometric quantization and the generalized Segal-Bargmann transform for Lie groups of compact type, Comm. Math. Phys. 226 (2002) 233-268 [doi:10.1007/s002200200607]