Polarization procedure in geometric quantization The geometric quantization can be considered as an approach the formalize the way of
associating a quantum theory corresponding to a given classical theory.
Suppose we start with a sympetic manifold $(M,w)$ with symplectic form
$w$. The  geometric quantization procedure falls into the following
three steps: prequantization, polarization, and metaplectic correction.
In the Polarization we have choose at each point of $M$ a
Lagrangian subspace of the complexified tangent bundle
$TM \otimes \mathbb{C} \to M$. Therefore it's a kind of choice of a
distiguished subbundle $P \subset TM \otimes \mathbb{C}$. The quantum
is then defined as space of sections $s: M \to P$.
Now from physical viewpoint the term Polarization suggest that the
polarization procedure makes something geometrical, but up to now
I haven't any idea what idea hides behind the polarization. I only
accept is as a formal step in order to obtain a new associated object
with interesting properties, but purely abstractractly.
Does anybody know any didactically valuable example which probably
vizualize what is roughly going on there or a way how it should be recomended to
to think about the polarization
intuitively? Why calling this procedure polarization is a meaningful name?
 A: In pre-quantization, wave-functions are functions of all of the phase space variables. There is a general formula to write pre-quantum operators, using the symplectic potential of the phase space (This is why the procedure is called "geometric quantization"; it is done using the phase space geometry). These operators are reducible representations of the commutator algebra. Polarizations give you irreducible representations of the commutator algebra. So the quantum wave-function cannot be a simultaneous eigenstate of non-commuting variables. The pre-quantum Hilbert space is simply too large. Choosing a polarization means, choosing a direction in the phase space, in which the pre-quantum wave-function does not change. This leads to wave-functions to be functions of half of the phase space variables. Which half? It depends on the polarization choice.
There are three possible polarizations: 1) choosing the position Hilbert space as the subspace (wave function is a function of position), 2) choosing the momentum Hilbert space (wave-function is a function of momentum) 3) choosing the Segal-Bargmann space (complex polarization, wave-function is a function of either x+ip or x-ip).
About visualizing polarizations:
One can think of choosing these three polarizations like combing the Hilbert space in three different ways: horizontally, vertically, or diagonally. Or like looking at the Hilbert space through a polarizer lens, where rotating the lens 45 or 90 degrees gives you different polarizations. I guess this is why it was called polarizations. It filters out the excessive parts of the pre-quantum wave-function.
Useful References:
V.P. Nair. Quantum Field Theory: A modern perspective. Springer, 2005. (CH 3 and 20)
B. C. Hall. Quantum Theory for Mathematicians. Springer, 2013. (CH 21, 22 and 23)
