Exact calculations with Moyal product by "Bopp Shift"

Question

I'm now working on my Phd thesis on the area of deformation quantization and field theory. After doing all the "ground work" (definitions, motivations, basics of the theory etc) I have now to do some "dirty work" with many and many Moyal Products. So I have searched for some tools to make life easier. In particular I found this MO question with a nice comment by user @Comaszachos saying that we can "Bopp shift away". I look at the pdf linked in the comment and on page 27 we have a lemma stating:

$$f(x,p)*g(x,p)=f(\hat x,\hat p)g(x,p)=f(x+\frac{ih}{2} \partial_p , p-\frac{ih}{2} \partial_x)g(x,p)$$

So I tried to apply this to some simple examples and that is what I found: take $f(x,p)=g(x,p)=e^x e^p$. Than

$$f(x,p)*g(x,p)=e^{x+\frac{ih}{2} \partial_p}e^{p-\frac{ih}{2} \partial_x}e^xe^p$$

as $[x,\frac{ih}{2} \partial_p]=[p,\frac{ih}{2} \partial_x]=0$, we have

$$f(x,p)*g(x,p)= (e^x e^{ \frac{ih}{2} \partial_p} e^p e^{-\frac{ih}{2} \partial_x}) (e^x e^p)$$

using Lagrange's formula for the shift operator

$$f(x,p)*g(x,p)=e^xe^{\frac{ih}{2} \partial_p}(e^pe^{x-\frac{ih}{2}}e^p) =e^xe^{p+\frac{ih}{2}}e^{x-\frac{ih}{2}}e^{p+\frac{ih}{2}} =e^{2x+2p+\frac{ih}{2}}$$

and this is wrong! We can see it by the fact that $f*g=fg+\frac{ih}{2}\{f,g\}+O(h^2)$ and $\{e^xe^p,e^xe^p\}=0$, but $$e^{2x+2p+\frac{ih}{2}}=e^{2x+2p}+e^{2x+2p}\frac{ih}{2}+O(h^2)$$

As a comment, the exact correct answer is $$f(x,p)*g(x,p)=e^{2x+2p}$$ where I used $$f(x,p)*g(x,p)=\lim_{(x',p')\to (x,p)}e^{\frac{ih}{2}(\partial_x \partial_{p'}-\partial_{x'} \partial_p)}f(x,p)g(x',p')$$

with the fact $e^{\frac{ih}{2}(\partial_x \partial_{p'}-\partial_{x'} \partial_p)}=e^{\frac{ih}{2}\partial_x \partial_{p'}}e^{-\frac{ih}{2}\partial_{x'} \partial_p}$ and expanding and collecting terms.

My question is: Where is the error? How do I properly apply the Bop shift? Is there any restriction to use it?

This post imported from StackExchange MathOverflow at 2023-11-30 17:41 (UTC), posted by SE-user Diego Santos