The role of holomorphic functions (and their generalizations in the form of holomorphic sections of vector bundles) in physics is invaluable. Please see for example the following
review by B.C. Hall, discussing holomorphic methods in mathematical physics, especially in quantum mechanics.

It should be emphasized that these theories cover important parts of
quantum theory but they are not exclusive. Still more general function
spaces are needed to describe other systems in quantum mechanics.

The Hilbert spaces describing the states of many very important systems
appearing in almost all areas of physics have a holomorphic realization
as (reproducing kernel) Hilbert spaces of holomorphic functions (or
sections). This is true for the harmonic oscillator, the electron spin, the
hydrogen atom and even in very advanced applications like the Hilbert
spaces corresponding to Chern-Simons theories.

The intuition behind the major role played by holomorphic functions in
quantum theory is the following:

Imagine $\mathbb{R}^2$ to be the phase space of a particle moving on a line $(x, p) \in \mathbb{R}^2$, where $x$ is the position and $p$ is the momentum. since $\mathbb{R}^2 \equiv \mathbb{C}$ is a complex manifold, we can use the "shorthand”:

$$z = x+ ip$$

Consider the Hilbert space of functions on $ \mathbb{C}$ corresponding to the Gaussian inner product:

$(\psi, \phi) = \int \overline{\psi(z)} \phi(z) e^{-\bar{z}z} d\mathrm{Re}z d\mathrm{Im}z$

Taking the whole Hilbert space would allow construction of wave functions
arbitrarily concentrated around any point $(x_0, p_0)$ in phase space,
this is in contradiction with the Heisenberg uncertainty principle. On
the other hand restricting the Hilbert space to holomorphic functions

$$ \frac{\partial \psi}{\partial \bar{z}} = 0$$

will restrict the "width" of the functions due to the square
integrability condition.

This space (of holomorphic functions) is the Hilbert space of the
harmonic oscillator. Another way to look at the holomorphicity restriction is to notice that without this restriction the full Hilbret space corresponds to an infinite number of copies of the harmonic oscillator, each closed under the action position and momentum operators.

The meaning of the restriction in this respect is that restricted Hilbert space carries an irreducible representation of the observable algebra.

This is a basic property required by Dirac in his axiomatic formulation
of quantum theory. An irreducible representation which describes a single system is the right choice because we started classically from a single system.

This post imported from StackExchange Physics at 2014-03-30 15:19 (UCT), posted by SE-user David Bar Moshe