0
$\begingroup$

I don't know if this is a valid question to ask, but I am wondering about the following: We are given a set of $\mathcal{c}$-number Lie-Brackets

$$ [q_i,q_j] = 0= [p_i,p_j] \\ [q_i,p_j] = c_{ij}, $$

with $c_{ij} \in \mathbb{C}$. Is it possible to obtain a quantum theory from this algebra? If yes, is there some kind of recipe how this is done? If no, why not, and what's the best way to approach such a question?

This post is related to Quantization of $c$-number Dirac-Bracket, but kept more general.

$\endgroup$
3
$\begingroup$

For what it's worth, given a $z$-independent Poisson structure $$\begin{align}\{z^I,z^J\}_{PB}~=~&\omega^{IJ}~\in~\mathbb{R}, \cr I,J~\in~&\{1,\ldots,2n\}, \end{align}\tag{1} $$ where $\omega^{IJ}$ is an arbitrary (not necessarily invertible) real skewsymmetric matrix, one may define an associative non-commutative Groenewold-Moyal star product $$\begin{align}f\star g~:=~& f \exp\left(\stackrel{\leftarrow}{\partial_I}\frac{i\hbar}{2} \omega^{IJ} \stackrel{\rightarrow}{\partial_J}\right) g\cr ~=~&fg+\frac{i\hbar}{2}\{f,g\}_{PB} +{\cal O}(\hbar^2). \end{align}\tag{2} $$ The functions/symbols $f,g$ with the star product $\star$ is a representation of corresponding operators $\hat{f},\hat{g}$ with the composition $\circ$, cf. e.g. my Phys.SE answer here. It's a quantization in that sense.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.