Segal–Bargmann space
In mathematics, the Segal–Bargmann space (for Irving Segal and Valentine Bargmann), also known as the Bargmann space or Bargmann–Fock space, is the space of holomorphic functions F in n complex variables satisfying the square-integrability condition:
where here dz denotes the 2n-dimensional Lebesgue measure on Cn. It is a Hilbert space with respect to the associated inner product:
The space was introduced in the mathematical physics literature separately by Bargmann and Segal in the early 1960s; see Bargmann (1961) and Segal (1963). Basic information about the material in this section may be found in Folland (1989) and Hall (2000) . Segal worked from the beginning in the infinite-dimensional setting; see Baez, Segal & Zhou (1992) and Section 10 of Hall (2000) for more information on this aspect of the subject.
A basic property of this space is that pointwise evaluation is continuous, meaning that for each a in Cn, there is a constant C such that
It then follows from the Riesz representation theorem that there exists a unique Fa in the Segal–Bargmann space such that
The function Fa may be computed explicitly as
where, explicitly,
The function Fa is called the coherent state with parameter a, and the function
is known as the reproducing kernel for the Segal–Bargmann space. Note that
meaning that integration against the reproducing kernel simply gives back (i.e., reproduces) the function F, provided, of course that F is an element of the space (and in particular is holomorphic).
Note that
It follows from the Cauchy–Schwarz inequality that elements of the Segal–Bargmann space satisfy the pointwise bounds
One may interpret a unit vector in the Segal–Bargmann space as the wave function for a quantum particle moving in Rn. In this view, Cn plays the role of the classical phase space, whereas Rn is the configuration space. The restriction that F be holomorphic is essential to this interpretation; if F were an arbitrary square-integrable function, it could be localized into an arbitrarily small region of the phase space, which would go against the uncertainty principle. Since, however, F is required to be holomorphic, it satisfies the pointwise bounds described above, which provides a limit on how concentrated F can be in any region of phase space.
...
Wikipedia