SzegoKernel

Szegő Kernel and Coherent States

The Szegő kernel is an important mathematical object in the context of coherent states, complex analysis, and quantum mechanics. It is often defined for various spaces of analytic functions and plays a crucial role in defining a reproducing kernel Hilbert space (RKHS) associated with coherent states.

In quantum mechanics, coherent states are specific types of quantum states that maintain their shape under time evolution and are eigenstates of the annihilation operator. They are particularly useful in the context of quantum optics and are often used to describe states of light fields.

In the context of the Fock space of analytic functions, the Szegő kernel $S(z,w)$ often takes the form:

$$ S(z,w) = e^{z\bar{w} - \frac{1}{2}( |z|^2 + |w|^2 )} $$

The kernel serves as a reproducing kernel, meaning that it allows one to reconstruct a function in the Hilbert space from its values at specific points. Mathematically, the Szegő kernel $S(z, w)$ on a space of holomorphic functions $\mathcal{H}$ is defined such that for any function $f$ in $\mathcal{H}$, the function value $f(z)$ can be recovered as:

$$ f(z) = \int S(z, w) f(w) d\mu(w) $$

where $d\mu(w)$ is some measure on the domain of $f$.

In summary, the Szegő kernel is a crucial tool for working with spaces of analytic functions and has important applications in quantum mechanics, particularly in the theory of coherent states.