Star Product and Poisson Brackets

Question

I have the following definition of star product, \begin{equation} \star=\exp\left[\frac{i\hbar}{2}\left(\frac{\overleftarrow{\partial}}{\partial Q^{I}}\frac{\overrightarrow{\partial}}{\partial P_{I}}-\frac{\overleftarrow{\partial}}{\partial P_{I}}\frac{\overrightarrow{\partial}}{\partial Q^{I}}\right)\right];\,\,\,I=1,\ldots,M \end{equation} So if $A(P,Q)$ and $B(P,Q)$ are matrix observables, whose poisson brackets are written as, \begin{equation} \left\{A,B\right\}=\frac{\partial A}{\partial Q^{I}}\frac{\partial B}{\partial P_{I}}-\frac{\partial A}{\partial P_{I}}\frac{\partial B}{\partial Q^{I}}, \end{equation} How can I write an expression for $\{A,B\}_{\star}$?

Answer 1

The quantum extension (deformation) of the PB is the scaled commutator expressed in phase space, conventionally dubbed the Moyal bracket, $$ \frac{1}{i \hbar} \left(A \star B - B \star A \right) \equiv \{\{A,B\}\} = \frac{2}{\hbar} A ~~ \sin \left ( {{\frac{\hbar }{2}}(\overset{\leftarrow}{\partial_x} \overset{\rightarrow}{\partial_p}-\overset{\leftarrow}{\partial_p}\overset{\rightarrow}{\partial_x})} \right )~~ B = \{A,B\} + O(\hbar^2),$$ as expected from the correspondence principle, the limit ħ → 0.

Many of its properties related to associativity are more easily proved in Baker's integral representation, $$ \{ \{ A,B \} \}(x,p) = {2 \over \hbar^3 \pi^2 } \int\! dp' \, dp'' \, dx' \, dx'' A(x+x',p+p') B(x+x'',p+p'')\sin \left( \tfrac{2}{\hbar} (x'p''-x''p')\right)~. $$

The $O(\hbar^2)$ higher derivatives over and above the PB often probe nonlinearity in the potential of the relevant problem and deform classical Liouville flows into dramatically different characteristic quantum configurations in phase space. In sharp contrast to classical mechanics, they render the quantum probability fluid compressible.