Convergence of closed formula for Moyal product

Question

It is often stated that the Moyal product obeys the following identity $$ \left(f_{1} \star_{\hbar} f_{2}\right) = m \circ e^{\frac{i \hbar}{2} \Pi}(f_1 \otimes f_2)= \sum_{k=0}^{\infty} \frac{(-i \hbar)^{k}}{2^{k} k !} B_{k}\left(f_{1}, f_{2}\right) $$ where $\Pi$ is the bidifferential operator induced by the Poisson bivector and $m$ is the multiplication map. However, many resources do not explain how to interpret the exponential on the right and where such an expression makes sense (i.e. converges). The first question is: how should I interpret the exponential? Just as a formal expression or does it actually converge to some bilinear operator in some functional calculus?

Wikipedia and nLab state the above identity for any two smooth functions but other resources such as the book "Quantum mechanics for mathematicians" say that in general the associated power series does not converge. Instead, they say that for every $f_{1}, f_{2} \in \mathscr{S}\left(\mathbb{R}^{2 n}\right)$ and $l \in \mathbb{N}$ there is $C>0$ such that for all $\boldsymbol{p}, \boldsymbol{q} \in \mathbb{R}^{2 n}$ $$ \left|\left(f_{1} \star_{\hbar} f_{2}\right)(\boldsymbol{p}, \boldsymbol{q})-\sum_{k=0}^{l} \frac{(-i \hbar)^{k}}{2^{k} k !} B_{k}\left(f_{1}, f_{2}\right)(\boldsymbol{p}, \boldsymbol{q})\right| \leq C \hbar^{l+1} \quad \text { as } \quad \hbar \rightarrow 0 . $$ Succinctly, $$ \left(f_{1} \star_{\hbar} f_{2}\right)(\boldsymbol{p}, \boldsymbol{q})=\sum_{k=0}^{\infty} \frac{(-i \hbar)^{k}}{2^{k} k !} B_{k}\left(f_{1}, f_{2}\right)(\boldsymbol{p}, \boldsymbol{q})+O\left(\hbar^{\infty}\right) . $$ I don't really know how to interpret this either: does this just mean that the above expression converges on Schwartz functions? If not what is the most general space where it does converge? I know it does on polynomials but I was hoping to get something a bit more general than that.

Answer 1

This paper by Waldmann discusses precisely this type of question: https://arxiv.org/abs/1901.11327

Essentially there seem to be two interpretations (see section 2):

• One can view the power series as the Taylor series of an actual smooth product in the limit $h\to 0$. In this case it is often difficult to find a subspace which is closed under this smooth product.
• In this paper Waldmann takes a different approach, he wants to find a subalgebra of the space of functions where the power series actually converges.

For his approach to the Weyl Star product see section 3 of the above mentioned paper.