I have recently been studying the method of geometric quantization, and I noticed a few methods in it that seem like they could be used to create a geometric second quantization (specifically of the gauge fields). The paper I'm following (titled Geometric Quantization, written by Andrea Carosso) seems to provide a method of such a process on bosonic Fock states, however, I am wondering about the following method of doing so, which at first glance appears more straightforward, as we naturally get a field-theoretic form of the Poisson Brackets and Hamiltonian Mechanics.
Consider some scalar field $\phi:\mathbb{R^n}\to\mathbb{R}$. This mapping is a member of the (Banach) vector space $C^{\infty}(\mathbb{R}^n,\mathbb{R})$, which has a natural Hausdorff topology (resulting from the required norm and completeness in a Banach space). Thus, we are able to extend certain theorems from differential geometry to such a field, beginning first with the tangent space. We define $T_f C$, the space tangent to $f\in C^\infty(\mathbb{R}^n,\mathbb{R})$, to be the set of derivations at that point. For this vector space, the set of variational derivatives work nicely. I assume (I still need to prove this) that because the tangent space will be isomorphic to the original space itself, we have the basis:
$$\{\frac{\delta}{\delta f^i}:f^i\in\mathcal{B}(C^\infty(\mathbb{R}^n,\mathbb{R}))\}$$
Where $\frac{\delta}{\delta f}$ is a variational derivative operator and $\mathcal{B}(C^\infty(\mathbb{R}^n,\mathbb{R}))$ is a basis for $C^\infty(\mathbb{R}^n,\mathbb{R})$. Then, we naturally have our space dual to $T_f C$ (denoted by $T^*_fC$), with basis $\{\delta f^i\}$ defined such that $\delta f^i(\frac{\delta}{\delta f^j})=\delta_j^I$. It makes sense to then define the cotangent and tangent bundles (resp.) as:
$$T^*C=\bigsqcup\limits_{f\in C^\infty(\mathbb{R}^n,\mathbb{R})}T_f^*C$$
$$TC=\bigsqcup\limits_{f\in C^\infty(\mathbb{R}^n,\mathbb{R})}T_fC$$
Thinking of $C^\infty(\mathbb{R}^n,\mathbb{R})$ as potential functions, it is natural to consider our conjugate momentum fields $\pi:\mathbb{R}^n\to\mathbb{R}$ as behaving such that elements of $\Theta\in T^*C$ are of the form:
$$\Theta=\pi\hspace{.5mm}\delta\phi$$
Where $\delta$ is a variational version of the exterior derivative. Again, I have yet to prove that this works rigorously, but I assume the completeness of $C^\infty(\mathbb{R}^n,\mathbb{R})$ allows for this one form to make sense. We can then define a two-form $\Omega$ (which must be proven to be non degenerate), allowing us to construct the symplectic manifold $(T^*C,\Omega)$. Note that:
$$\Omega=\delta \pi\wedge\delta\phi$$
From here we define the Hamiltonian vector flow of some functional $F:C^\infty(\mathbb{R}^n,\mathbb{R})\to\mathbb{R}$ by:
$$X_F=\frac{\delta F}{\delta \pi}\frac{\delta}{\delta\phi}-\frac{\delta F}{\delta\phi}\frac{\delta}{\delta\pi}$$
And our Poisson Bracket arises naturally as:
$$\{F,G\}=X_F(G)=\Omega(X_F,X_G)=-\{G,F\}$$
From here I imagine one would find some sort of invariant of functionals over the potential and momentum fields and use that to construct an associated (Banach) vector bundle, using the covariant derivative on sections to create a prequantization and ultimately quantization of such functionals. This (according to my intuition) would allow for a quantization on the Fock space $\mathcal{F}$.
My question is whether this methodology is mathematically appropriate. Before I pursue this any further, I want to make sure the underlying logic is sound and whether there are any glaring problems or overcomplications of attempting to formulate second quantization (of bosonic fields) in this way.