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. The polarization is a choice 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 geometrically or a way how it should be recomended to to think about the polarization intuitively? Why calling this procedure polarization is a meaningful name? In physicsStackexchange I asked some days ago an identical question. Following Adolfo Holguin's hint this terminology arise also seemingly independently from mathematical physics as an important concept in symplectic geometry without be immediately connected to ma phy and I would like to understand if there is an motivating reason why this procedure is called polarization.
