# nLab Bargmann-Segal transform

See coherent state.

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

Classical case of Heisenberg group:

• V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Communications on Pure and Applied Mathematics 14 (1961) 187-214 MR0157250 doi

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

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. 122:103–151, 1994, doi; Geometric quantization and the generalized Segal–Bargmann transform for Lie groups of compact type, Comm. Math. Phys., 226:233–268, 2002. doi

Created on November 12, 2012 23:55:59 by Zoran Škoda (193.51.104.23)