SegalBargmanSpace

The Segal–Bargmann space is a special kind of Hilbert space that was initially studied by Irving Segal in the 1960s and further developed by Valentine Bargmann. It has applications in quantum mechanics, quantum information theory, and harmonic analysis. In these areas, it is often used to model the quantum-mechanical behavior of certain physical systems.

The Segal–Bargmann space can be defined for any locally compact group but is usually studied for the Heisenberg group and, more generally, nilpotent Lie groups. I'll start with the simplest case, which is for the real line R.

The Segal–Bargmann space on the real line is essentially the space of holomorphic functions on the complex plane that are square-integrable with respect to a Gaussian measure. More formally, it is the Hilbert space consisting of all holomorphic functions $f : \mathbb{C} \rightarrow \mathbb{C}$ for which the integral

$$\int |f(z)|² e^{-|z|²} dz < \infty$$

where the integral is over the entire complex plane, and $dz$ denotes the Lebesgue measure on the complex plane.

An important property of this space is that it carries a representation of the Heisenberg group. Specifically, let $z$ and $w$ be complex numbers. The Heisenberg group action on the Segal–Bargmann space is given by:

$$(a, b) • f(z) = e^{iaz - 1/2 ab} f(z - b)$$

for $a, b \in \mathbb{R}$.

For the general case, i.e., the case of a nilpotent Lie group $G$, the Segal–Bargmann space can be defined as follows. First, one fixes a Haar measure on $G$ and a unitary representation of $G$ on a Hilbert space $H$. This representation can be integrated to give a representation of the group algebra $L¹(G)$ on $H$. Next, one introduces a complex structure $J$ on $H$ that commutes with the action of $L¹(G)$. The Segal–Bargmann space $B(G)$ is then the space of all vectors in $H$ that are analytic vectors for the action of $L¹(G)$ and that are square-integrable with respect to a certain Gaussian measure on $H$. The Gaussian measure is chosen so that it is invariant under the action of $G$ and so that its Fourier transform is also a Gaussian.

An important aspect of the theory of Segal–Bargmann spaces is the study of the so-called Segal–Bargmann transform. This is a unitary map from the $L²$ space of a nilpotent Lie group to the Segal–Bargmann space of its complexification. For the Heisenberg group, the Segal–Bargmann transform can be computed explicitly in terms of the standard Fourier transform.

Finally, I should note that there is a version of the Segal–Bargmann space and transform for semisimple Lie groups as well, which was developed by Harish-Chandra in the 1960s. This version of the theory is considerably more complicated than the case of nilpotent Lie groups, involving representation theory of semisimple Lie groups, Harish-Chandra's Plancherel theorem, and the theory of spherical functions.

The Heisenberg group

The Heisenberg group is a foundational example in the study of Segal-Bargmann spaces and their associated transforms. For the Heisenberg group, the Segal-Bargmann transform has an explicit formulation in terms of the Fourier transform.

Let us consider the Heisenberg group $\mathbb{H}$ in $\mathbb{R}^{2n+1}$ with coordinates $(x, y, t) \in \mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R}$ and group operation

$$(x_1, y_1, t_1) \cdot (x_2, y_2, t_2) = (x_1 + x_2, y_1 + y_2, t_1 + t_2 + 2^{-1}(x_1y_2 - x_2y_1)).$$

The Segal-Bargmann transform, or heat operator, for the Heisenberg group can be defined by

$$T_tf(x) = (4\pi t)^{-n} \int_{\mathbb{R}^n} e^{-|x-y|^2/4t}f(y) dy,$$

where $f \in L^2(\mathbb{R}^n)$ and $t &gt; 0$.

Given $f \in L^2(\mathbb{R}^n)$, the Segal-Bargmann transform $B_tf : \mathbb{C}^n \rightarrow \mathbb{C}$ is then defined by

$$B_tf(z) = (2\pi t)^{-n/2} \int_{\mathbb{R}^n} e^{-|z-y|^2/4t}f(y) dy,$$

where $z \in \mathbb{C}^n$.

Finally, the Fourier transform of a function $f \in L^2(\mathbb{R}^n)$ is given by

$$\hat{f}(p) = (2\pi)^{-n/2} \int_{\mathbb{R}^n} e^{-ipy}f(y) dy,$$

where $p \in \mathbb{R}^n$.

In this context, the Segal-Bargmann transform can be computed in terms of the standard Fourier transform as follows:

$$B_tf(z) = \hat{T_tf}(z/\sqrt{2t}),$$

where $T_tf$ is the heat operator applied to $f$ and $\hat{T_tf}$ is the Fourier transform of $T_tf$.