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Introduction

In his second-most upvoted post, called "Why quantum mechanics?" (second only to his post on fibre bundles & gauge theory) in the physics SE community, Urs Schreiber, in the setting of classical mechanics on a connected symplectic manifold $(X,\omega)$, introduces the quantomorphism group $\text{QuantMorph}(X,\omega)$ as the nontrivial Lie integration of the Kostant-Souriau extension:

\begin{align*} \mathfrak{u}(1)\simeq i\mathbb{R}\xrightarrow{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \mathfrak{poiss}(X,\omega)\xrightarrow[f\to X_f]{\,\,\,\,\,\,\,\,\,\,\,\,} \mathfrak{ham}(X,\omega),\\ \end{align*} It is then said that this group "...seamlessly leads to the quantum mechanics of the system". The wave of interest behind this post seems to stem from an ensuing demonstration where Urs considers the simplest possible example of $X\simeq \mathbb{C}^n$, and produces the famous Heisenberg group $\text{Heis}(X,\omega)$ as a special case of this group extension. This leads to my question:

$$$$ Why isn't the WKB approximation mentioned in that post or the nLab page?

For the rest of this post, let $X$ be a vector space. Proper (strict) quantization of the ring $\mathbb{A}_0\simeq C^\infty(X,\mathbb{C})$ of classical observables produces a family of non-commutative (Weyl) algebras: $\mathbb{A}_\hbar:=(C^\infty(X,\mathbb{C}),\,\star_\hbar)$, where $\hbar\geq 0$ and the non-commutative product $\star_\hbar$ is the standard Moyal star-product. Naively, from this perspective, to form the group which seamlessly leads to the quantum dynamics of the system, one should integrate the algebra with the Moyal bracket $\{\{f,g\}\}:=f\star_\hbar g-g\star_\hbar f$, not the Poisson bracket: $$$$ \begin{align*} \mathfrak{quantmorph}(X,\omega)&:=(\mathbb{A}_\hbar,\{\{,\}\}_\hbar)\\ &\neq \mathfrak{poiss}(X,\omega):=(\mathbb{A}_0,\{,\}) \end{align*} $$$$ Of course, none of this affects the validity of Urs' post, since his examples, including the ones on the nlab page for the quantomorphism group (e.g. the metaplectic extension $\text{Mp}(X,\omega)$), involve polynomial observables of sufficiently low-degree, in which case the extensions produce the same results.

However, at higher order, especially in interacting theories, the groups will start to diverge. In this sense, the nLab definition of the quantomorphism group seems to "approximate" the group which integrates the Moyal bracket. Of course, this is the well-known WKB/semiclassical approximation where we discard observables of order $O(\hbar^2)$. This makes me wonder: shouldn't we then write $$\mathfrak{poiss}(X,\omega) \xrightarrow{\text{Lie integration}}\text{WKBQuantMorph}(X,\omega),$$ In which case we then say that the group "...seamlessly leads to the semiclassical dynamics of the system"?

Or is there a deeper point that Urs meant, which I missed?

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  • $\begingroup$ What is “WKBQuantMorph(X,ω)”? (And does it, like QuantMorph, actually depend on not just (X,ω) but also the choice of a bundle with connection (L,ϖ) over X?) $\endgroup$ – Francois Ziegler May 1 '18 at 6:15
  • $\begingroup$ Sorry about the wording of the question: "WKBQuantMorph" is just a silly renaming of (i.e. is the exact same thing as) the nLab definition of QuantMorph, and so yes it does depend on the choice of prequantum line bundle. What I called "QuantMorph" would be the Lie integration of the Lie algebra of smooth functions equipped with the Moyal bracket. $\endgroup$ – David Roberts May 1 '18 at 6:25
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    $\begingroup$ Oh, I see. But you must be after more than just a proposed change in terminology...? (Does your putative other QuantMorph exist in the same generality as Kostant and Souriau’s? “The” integration of a Lie algebra is a problematic notion in infinite dimensions.) $\endgroup$ – Francois Ziegler May 1 '18 at 6:39
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    $\begingroup$ Oh, hello, mirror me ^_^ $\endgroup$ – David Roberts May 1 '18 at 8:19
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    $\begingroup$ More seriously, if something isn't mentioned on an nLab page that deserves to be there even if only a mention and a supporting reference, please add it! Omission merely means the people who have edited the page either didn't know about the link, or were too busy elsewhere to do a literature search and write stuff up fully. If you do edit, you may need to use a disambiguating user name, else our edit histories will get conflated. $\endgroup$ – David Roberts May 1 '18 at 8:22

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