Basic question in similarities and difference on quantizations

Question

In physics, usually quantization means canonical quantization. i.e., which we treat classical objects to quantum operators. i.e., For the association $Q:f \mapsto \hat{f}$ from functions on the classical space of states to operators on quantum states: $$[Q(f), Q(g)] = -i\hbar Q(\{f,g\})$$ where $[-,-]$ is the commutator of linear maps and $\{-,-\}$ is the Poisson bracket.

In terms of mathematics (or mathematical physics society) the quantization can be split into three subjects:

1. Deformation quantization,

2. Batalin-Vilkovisky (BV) formalism, and

3. Geometric quantization.

Can anyone give me a nice explanation of the difference between three three quantizations? What are the similarities and differences between them?

Naively, I noticed quantization is related to Hamiltonian and so symplectic geometry but not have much deeper thoughts on this topic.

Answer 1

Deformation quantisation focuses on quantising the algebra of observables. Hence it is in the Heisenberg picture. In contrast, geometric quantisation focuses on quantising the space of states and so in the Schrodinger picture. The former can quantise fields and so its relevant to QFT whilst the latter - so far - is only applicable to mechanical systems - so its only relevant to QM.

Now, Fedesov in '94 established formal deformation quantisation on a symplectic manifold, originally of finite dimension though Karabegov-Schlichenmaier found a variant applicable to infinite dimension. They are both referred to as Fedesov quantisation.

Kontesevich establishes formal deformation quantisation of all Poisson manifolds of finite dimension. This is referred to as Kontsevich quantisation.

It turns out that Kontsevich's quantisation is typically not applicable to field theory. However, Fedesov quantisation in infinite dimension yields the quantisation in perturbative QFT (pQFT). This is the quantisation that is used typically in traditional QFT books.

BV quantisation is a generalisation of BRST quantisation of Yang-Mill theories to Lagrangian gauge theories where the constraint algebra can be expressed via algebras other than Lie algebras. The Costello-Gwilliam formalism for this and applicable to pQFT shows that it is another version of deformation quantisation.

Geometric quantisation was established by Kirrilov, Kostant and Souriau. It focuses on a finite-dimensional symplectic manifold representing a mechanical system. Its geometric quantisation gives the quantisation of this mechanical system.