Poisson brackets of the exterior forms

Question

Let $(M,\omega)$ be a symplectic manifold. Poisson brackets of the exterior forms are defined by the following formulas:

$$\{ \alpha, \{ \beta , \gamma \} \}= \{ \{ \alpha, \beta \},\gamma \}+ \{ \beta ,\{ \alpha , \gamma \} \}$$

$$ \{ \alpha ,\beta \}= (-1)^{deg(\alpha) deg(\beta )+1}  \{ \beta , \alpha \}$$

$$\{ f \alpha,\beta \}= \alpha \wedge \nabla_{X_f} \beta + f \{ \alpha, \beta \}$$

with $\alpha,\beta,\gamma \in \Lambda^* (TM)$ and $f \in {\cal C}^{\infty}(M)$, $X_f=(df)^*$. $\nabla$ is a symplectic connection.

Can we quantize the structure?

Answer 1

Yes. Quantization gives the algebra generated by the free Dirac field operators over the manifold. A construction is in

• J. Dimock, Dirac quantum fields on a manifold. Transactions of the American mathematical Society, 269 (1982), 133-147.