TITLE: Prequantization and Hilbert space
QUESTION [6 upvotes]: In the definition of pre-quantization on a symplectic manifold $(M,\omega)$, we represent a function $f\in C^{\infty}(M)$(with Lie algebra structure) to $\hat{f}$ in the Hilbert space $L^2(M,L,\mu)$ associated to $M$. But we always assume that the Hilbert space should not be too big. But why do we assume this?. Also we consider only irreducible representations for definition of prequantization ? 
PS: see page 1 , line 9 of this reference

REPLY [4 votes]: I'll add a "general abstract" reason to the "phenomenological reasons".
Phenomenologically, as others pointed out here, the processes of cutting down the space of sections of the prequantum line bundle on phase space to only those which respect a polarization corresponds precisely to wave functions depending only on canonical coordinates and not on canonical momenta.
This is known since the 1930s to yield the correct description of observable physics, and that "proves" it in as far as we regard geometric quantization as a theory of physics.
But of course one may ask why this step is natural just from the point of view of mathematics? Is there maybe a deeper mathematical reason for this?
And, yes, that turns out to be the case. In the modern understanding, geometric quantization in the case that there exists a Kähler polarization is really the push-forward of generalized cohomology of the prequantum line bundle in complex topological K-theory, hence the (de-categorified) index of the spin^c-Dirac operator twisted by the prequantum bundle.
For pointers to details of how this works see on the nLab the page geometric quantization by push-forward.
And from this perspective, the fact that geometric quantization "cuts down" the space of sections in the hallmark of index theory: from a space of sections of some bundle we pass to just the graded kernel of some graded differential operator on these. 
By another way: some references relating geometric quantization to deformation quantization are listed here.