geometric quantization

#basement

Release the von Neumann condition in the Dirac canonical quantization. Alternatively, modifying the relation between Poisson bracket and Lie bracket gives rise to the deformation quantization.

We may have different (or even no; $e x a m p l e s ? M e a n i n g o f n o q u a n t i z a t i o n ?$ ) prequantizations, which assembles the phase space (for gauge fields, $t h e m o d u l i s p a c e o f c o n n e c t i o n s ?$ ) a complex line bundle and map Poisson algebra (spanned by real smooth functions on the phase space) to Lie brackets.

Then, polarization is somewhat more harsh.
When the phase space is coincidently the cotangent bundle $T^{*} M$ , we can just employ the canonical quantization (real); but when it's not, such as when its volume is finite, we have to apply the Kahler (complex) polarization.
“量子化是个谜”[2/6] 几何量子化简介 - 知乎
 “量子化是个谜”[3/6] Kähler流形的量子化 - 知乎