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: $\mid f\u27e9=\int \mid z\u27e9\u27e8z\mid f\u27e9d\mu $; if $f$ is in ${L}^{2}$ then $\u27e8z\mid f\u27e9$ 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
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