nLab Bargmann-Fock space

Idea

The Bargmann–Fock space (also Bargmann-Segal space or Segal–Bargmann space) is a Hilbert space of holomorphic square integrable functions with respect to the Gaussian weighted Lebesgue measure.

By taking the Taylor series of these square integrable holomorphic functions $f(z)$ in $z$, we see how they can be regarded as elements of the usual bosonic Fock space.

Sometimes the Bargmann–Fock space is taken to be the space of square integrable antiholomorphic functions (ie. holomorphic in $\bar z$). This different convention corresponds to the choice of complexification of the configuration space of states $\mathbb{R}^n$, and this is connected to the usual bosonic Fock space by taking Taylor series of $f(z)$ in $\bar z$.

The Bargmann–Fock space can be generalized further to state spaces of compact, connected Lie groups by forming the space of holomorphic functions on its complexification as a Lie group which are square integrable with respect to a measure given by an appropriate heat kernel. See Hall coherent state.

Related concepts

• coherent state

• Bargmann-Segal transform

Literature

• Wikipedia, Bargmann space

The original reference is

• 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]

A generalization

• M. Arik, D. D. Coon, Hilbert spaces of analytic functions and generalized coherent states, J. Math. Phys. 17, 524–527 (1976) doi