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I already asked this in the physics forum but without much attention, so I thought it might attract more attention here.

Is there an integral expression for the Poisson bracket that can be derived from the integral expression of the Moyal bracket? this is, given that the Moyal bracket can be written as $$\lbrace f,g \rbrace_M(p,q)=\frac{2}{\pi^ 2\hbar^ 3}\int dp_1dp_2dq_1dq_2 f(q+q_1,p+p_1)g(q+q_2,p+p_2) \sin\left(\frac{2}{\hbar}\left(p_1q_2-q_1p_2\right)\right),$$ and the fact that $$\lim_{\hbar\rightarrow 0}\frac{1}{i\hbar}\lbrace f,g\rbrace_M=\lbrace f,g\rbrace_P,$$ what expression do I get?

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    $\begingroup$ if you take the limit $\hbar\rightarrow 0$ of the phase space integral, by a stationary phase approximation, you get the expected answer for the Poisson bracket, $\{f,g\}_P=(\partial_x f)(\partial_p g)-(\partial_x g)(\partial_p f)$; there is no integral left in that limit. $\endgroup$
    – Carlo Beenakker
    Commented Apr 24, 2023 at 11:31
  • $\begingroup$ Can you explain in a more detailed way how you perform that limit? $\endgroup$
    – Nicolas Medina Sanchez
    Commented Apr 24, 2023 at 12:22
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    $\begingroup$ Crossposted from physics.stackexchange.com/q/758317/2451 $\endgroup$
    – Qmechanic
    Commented Aug 16, 2023 at 18:13

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