I am reading a paper by Karasev and quite confused with the abstract definition of quantization.
To begin with, we define a map from the phase space $\mathbb{R}^n_x\times\mathbb{R}^n_p$ to $\mathbb{C}^n$ by $$ \mathcal{Z}:(x,p)\to x+ip. $$ Let $\Lambda$ be a Lagrangian submianifold of the phase space and $T\Lambda^{\mathbb{C}}$ the complexification of its tangent bundle. Then, the restriction of the differential $$ d\mathcal{Z}:T\Lambda^{\mathbb{C}}\to\mathbb{C}^n $$ is nondegenerate.
My first question is how to understand $T\Lambda^{\mathbb{C}}$. Since the tangent bundle $T\Lambda$ contains all tangent space of $\Lambda$, then we complexify these linear spaces. But is there any manifold structure for $T\Lambda^{\mathbb{C}}$? Is it sufficient to prove that $d\mathcal{Z}$ is nondegenrate for each $(x,p)\in\Lambda$? Can we represent $d\mathcal{Z}$ in the matrix form by using the Catesian basis of the phase space and the projection map to $\Lambda$?
Next, we define two functions $$ \mathcal{X}( \alpha, \eta)=X( \alpha)- id \mathcal{Z}( \alpha)^{-1*} \eta,\quad \mathcal{P}( \alpha, \eta)=P( \alpha)+d \mathcal{Z}( \alpha)^{-1*} \eta, $$ where $$ \alpha\in \Lambda,\quad \eta\in T^*_ \alpha \Lambda^{\mathbb{C}},\quad X(\alpha)+iP(\alpha)=\mathcal{Z}(\alpha). $$ Let $d\sigma$ be a smooth measure on $\Lambda$. For a differential operator $A$, we define its transpose with respect to the measure by $A^\sigma$.
Then we can define the quantization of $X(\alpha)$ and $P(\alpha)$ by an ordered calculus. $$ \hat{\mathcal{X}}=\mathcal{X}( \alpha^{(2)},-\frac{ih}{2}(d^{(1)}-{d^{ \sigma}}^{(3)})),\quad \hat{\mathcal{P}}=\mathcal{P}(\alpha^{(2)},-\frac{ih}{2}(d^{(1)}-{d^{ \sigma}}^{(3)})). $$ The index in the () refers to the order of operator. I think that this kind of formulation is a generalizaiton of Weyl quantization. But I hope to express it in a more concrete way. I still have several questions.
First, since the differential operator $d$ maps $n$-form to $n+1$-form, is $d^{\sigma}$ maps $n+1$-form to $n$-form? If so, how can we sum them up? Second, is there any other way to define the quantized operator $\hat{\mathcal{X}}$ and $\hat{\mathcal{P}}$ by using coordinate chart of $\Lambda$? Is there any reference about this topic? I am new to quantization theory and the topics of differential geometry, especially with complex manifold. Any suggestion will be appreciated!
Thanks in advance.