Since you asked in the comments, let me provide some references to the geometric quantization of Chern-Simons theory. This does of course not answer your second qustion about how to deal with spinors in geometric quantization and I would be interested in an answer to this myself. However, it goes in the direction of applying geometric quantization in field theory, although the field theory in question might be rather exotic.
So, the field theory in question is 3d-Chern Simons theory, which is a topological gauge theory, defined as follows: Consider the following data:
- A compact, orientable smooth manifold $\mathcal{M}$.
- A principal $G$-bundle $\pi:P\to\mathcal{M}$ with Lie group $G$, whose Lie algebra $\mathfrak{g}$ is equipped with a non-degenerate $\mathrm{Ad}$-invariant symmetric bilinear form $\langle\cdot,\cdot\rangle$.
Now, lets take a connection 1-form $A\in\Omega^{1}(P,\mathfrak{g})$, i.e. a "gauge field" in physics terminology. Then, we define the "Chern-Simons form" by
$$\mathrm{CS}[A]:=\operatorname{tr}(A\wedge\mathrm{d}A)+\frac{2}{3}\mathrm{tr}(A\wedge [A\wedge A])\in\Omega^{3}(P)$$
where "$\operatorname{tr}(\cdot\wedge\cdot)$" denotes the induced wedge-product defined using the inner product $\langle\cdot,\cdot\rangle$ on $\mathfrak{g}$, i.e.
$$\mathrm{tr}(\omega\wedge\eta):=\sum_{a,b=1}^{\mathrm{dim}(G)}(\omega^{a}\wedge\eta^{b})\langle T_{a},T_{b}\rangle\in\Omega^{k+l}(P)$$
for all $\omega\in\Omega^{k}(P,\mathfrak{g})$ and for all $\eta\in\Omega^{l}(P,\mathfrak{g})$, where $\{T_{a}\}_{a=1}^{\mathrm{dim}(G)}$ denotes a basis of $\mathfrak{g}$ and where $\omega^{a}\in\Omega^{k}(P)$ and $\eta^{b}\in\Omega^{l}(P)$ denote the coordinate forms with respect to this basis. One can easily verify that this definition is independent of the choice of basis. Note that if $G$ is compact and simple and $\langle\cdot,\cdot\rangle$ positive definite, then $\langle\cdot,\cdot\rangle$ is necessarily a negative multiple of the Killing form of $\mathfrak{g}$, which is also the reason why we use the notation "$\mathrm{tr}$" above.
As a next step, note that the bundle $P$ is trivial if and only if it admits a smooth global section. In particular, this is the case if $G$ is compact and simply-connected and if $\mathcal{M}$ has dimension $\leq 3$. The "Chern-Simons action'' is then defined to be the functional
$$S_{\mathrm{CS}}[s,A]:=\int_{\mathcal{M}}\,s^{\ast}\mathrm{CS}[A],$$
where $s:\mathcal{M}\to P$ is a global gauge. At this point, the definition clearly depends on the choice of chosen gauge. Let $f\in\mathcal{G}(P)$ be a gauge transformation (i.e. a bundle automorphism of $P$). Then, after a straight-forward calculation, one finds that
$$S_{\mathrm{CS}}[f\circ s,A]-S_{\mathrm{CS}}[s,A]=S_{\mathrm{CS}}[s,f^{\ast}A]-S_{\mathrm{CS}}[s,A]=-\frac{1}{6}\int_{\mathcal{M}}\,s^{\ast}\mathrm{tr}(\theta\wedge [\theta\wedge\theta]),$$
with $\theta:=\sigma_{f}^{\ast}\mu_{G}$, where $\mu_{G}\in\Omega^{1}(G,\mathfrak{g})$ denotes the Maurer-Cartan form on $G$ and where $\sigma_{f}\in C^{\infty}(P,G)$ is the map defined by $f(p)=p\cdot\sigma_{f}(p)$ for all $p\in P$. One can show that the integral on the right-hand side is always an element in $6\cdot\mathbb{Z}$. To sum up, the Chern-Simons action $S_{\mathrm{CS}}[s,A]$ is independent of the choice of global gauge modulo $\mathbb{Z}$. Hence, we have a well-defined action of the type
$$S_{\mathrm{CS}}[A]:=S_{\mathrm{CS}}[s,A]\in\mathbb{C}/\mathbb{Z}.$$
As a quantum theory, we hence get a well-defined theory, since for example in the formal path integral, we consider the complex exponential of the action and, after choosing a convenient normalization, different choices of gauges leave the exponential invariant.
More details about Chern-Simons theory can be found in the papers by D. S. Freed:
- D. S. Freed: Classical Chern-Simons Theory, Part 1. Advances in Mathematics,
113(2):237–303, 1995. Preprint.
- D. S. Freed: Classical Chern-Simons theory, Part 2. Houston Journal of Mathematics, 28(2):293–310, 2002. See here.
The first part covers simply connected gauge groups and the second part covers arbitrary compact Lie groups.
Now, lets turn to the question of quantization: The geometric quantization of 3d-Chern Simons theory goes back to the seminal paper by Edward Witten from 1989:
- E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121(3):351–399, 1989. See here.
This is basically one of the Fields-Medaille winning works of E. Witten establishing a link between QFT and knot theory. A summary of the geometric quantization of Chern-Simons theory with a lot of references can be found in this nlab article. I also remember some very good (hand-written) lecture notes on the Geometric Quantization of Chern-Simons theory, namely at the university of Munich (Germany) by M. Schottenloher. See here (last chapter).
Note that there is also an alternative way of quantizing 3d-Chern Simons theory, namely the 3-dimensional topological field theory defined by the Reshetikhin–Turaev model. However, it was shown for example in
- J. Roberts: Skein theory and Turaev-Viro invariants, Topology 34(4):771-787, 1995. See here.
that the two approaches are equivalent.
As a last remark, althout Chern-Simons theory might seem to be a little bit exotic, let me just say the general relativity in 3-dimensions is in fact a Chern-Simons theory with gauge group $\mathrm{SU}(2)$ (Euclidean case) and $\mathrm{SO}(1,2)$ (Lorentzian case) (without a cosmological constants; otherwise you have to add a volume term). Hence, 3d-Quantum Chern-Simons Theory can be taken as a theory of 3-dimensionl quantum gravity. In fact, it has been shown that this theory is related to other quantum gravity theories in 3d, like the Ponzano-Regge spin foam model, the Boulatov group field theory model, LQG and quantization of 3d-BF theory.
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.
Best Answer
It is said (I forget by whom) that the process of of quantization -- of turning a classical phase space into a Hilbert space -- is not a functor. There is no set recipe, but there is instead a variety of recipes.
Canonical quantization, for example, consists of making the classical system look like a bunch of quantized harmonic oscilators. This works for for any system undergoing small (hence linear) oscillations and so is good for many systems, and in particular weakly interacting fields.
A cartoon description of geometric quantization is that it tries to make eveything look like the quantization of spin. In particular it nicely generates the representation spaces of compact groups from the Kirillov-Constant symplectic structure --- and of course this representation theory is everywhere in particle physics.
I think it fair to say that nearly all physicists working on actual physical systems are familiar with the first of these processes but very few are familiar with the second. Therefore the motivation for geometric quantization comes mostly from the mathematics side.
Some of the ideas of Moyal algebras and star products are used in the quantum Hall effect as a low energy/high-magnetic-field approximation, but again this is a niche area.