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\begin{document}

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\section*{quantomorphism group}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{geometric_quantization}{}\paragraph*{{Geometric quantization}}\label{geometric_quantization}

[[!include geometric quantization - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{traditional_construction}{Traditional construction}\dotfill \pageref*{traditional_construction} \linebreak
\noindent\hyperlink{InHigherGeometry}{In higher geometry}\dotfill \pageref*{InHigherGeometry} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{smooth_structure}{Smooth structure}\dotfill \pageref*{smooth_structure} \linebreak
\noindent\hyperlink{group_extension}{Group extension}\dotfill \pageref*{group_extension} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{CoveringAnAffineSymplecticGroup}{Covering an affine symplectic group}\dotfill \pageref*{CoveringAnAffineSymplecticGroup} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{SmoothManifoldStructure}{Smooth manifold structure}\dotfill \pageref*{SmoothManifoldStructure} \linebreak
\noindent\hyperlink{examples_2}{Examples}\dotfill \pageref*{examples_2} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Given a ([[presymplectic form|pre]])[[symplectic manifold]] $(X,\omega)$, its \emph{quantomorphism group} is the [[Lie group]] that [[Lie integration|integrates]] the [[Lie algebra|Lie bracket]] inside the [[Poisson algebra]] of $(X, \omega)$. This is a [[circle group]]-[[central extension]] of the group of [[Hamiltonian symplectomorphisms]]. It extends and generalizes the [[Heisenberg group]] of a [[symplectic vector space]].

(Warning on terminology: A more evident name for the quantomorphism group might seem to be ``Poisson group''. But this already means something different, see \emph{[[Poisson Lie group]]}.)

\hypertarget{traditional_construction}{}\subsubsection*{{Traditional construction}}\label{traditional_construction}

Over a [[symplectic manifold]] $(X, \omega)$ an explicit construction of the corresponding quantomorphism group is obtained by choosing $(P \to X, \nabla)$ a [[prequantum circle bundle]], regarded with an [[Ehresmann connection]] 1-form $A$ on $P$, and then defining

\begin{displaymath}
QuantomorphismGroup \hookrightarrow Diff(P)
\end{displaymath}
to be the [[subgroup]] of the [[diffeomorphism group]] $P \stackrel{\simeq}{\to} P$ on those [[diffeomorphisms]] that preserve $A$. In other words, the quantomorphism group is the group of equivalences of [[connection on a bundle|bundles with connection]] that need not cover the identity [[diffeomorphism]] on the base manifold $X$.

Notice that the tuple $(P,A)$ is a [[regular contact manifold]] (see the discussion there), and so the quantomorphism group is equivalently that of [[contactomorphisms]] $(P,A) \to (P,A)$ of weight 0.

This is an infinite-dimensional [[Lie group]]. References discussing its [[infinite-dimensional manifold]]-structure are collected \hyperlink{SmoothManifoldStructure}{below}. But the group has immediately the structure of a group in [[diffeological spaces]] (making it a [[smooth group]]) (\hyperlink{Souriau79}{Souriau 79}).

\hypertarget{InHigherGeometry}{}\subsubsection*{{In higher geometry}}\label{InHigherGeometry}

This perspective lends itself to a more abstract description in [[higher differential geometry]]: we may regard the [[prequantum circle bundle]] as being modulated by a morphism

\begin{displaymath}
\nabla : X \to \mathbf{B} U(1)_{conn}
\end{displaymath}
in the [[cohesive (∞,1)-topos]] $\mathbf{H} =$ [[Smooth∞Grpd]], with [[domain]] the given symplectic manifold and [[codomain]] the smooth [[moduli stack]] for [[circle n-bundle with connection|circle bundles with connection]]. This in turn may be regarded as an object $\nabla \in \mathbf{H}_{/\mathbf{B}U(1)_{conn}}$ in the [[slice (∞,1)-topos]]. Then the quantomorphism group is the [[automorphism group]]

\begin{displaymath}
\mathbf{QuantMorph}(X,\nabla) \coloneqq 
  \underset{\mathbf{B}U(1)_{conn}}{\prod} \mathbf{Aut}(\nabla)
\end{displaymath}
in $\mathbf{H}$, or rather its [[differential concretification]] (\hyperlink{FRS13}{FRS 13}).

From this it is clear what the quantomorphism [[∞-group]] of an [[n-plectic ∞-groupoid]] should be: for

\begin{displaymath}
\nabla : X \to \mathbf{B}^n U(1)_{conn}
\end{displaymath}
the morphism modulating a [[prequantum circle n-bundle]], the corresponding quantomorphism $n$-group is again $Aut(\nabla)$, now formed in $\mathbf{H}_{/\mathbf{B}^n U(1)_{conn}}$

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{smooth_structure}{}\subsubsection*{{Smooth structure}}\label{smooth_structure}

The quantomorphism group for a [[symplectic manifold]] may naturally be equipped with the structure of a [[group object]] in [[ILH manifolds]] (\href{Omori}{Omori}, \href{RatiuSchmid}{Ratiu-Schmid}), as well as in [[convenient manifolds]] (\hyperlink{Vizman}{Vizman, prop.}).

\hypertarget{group_extension}{}\subsubsection*{{Group extension}}\label{group_extension}

\begin{prop}
\label{}\hypertarget{}{}
For $(X,\omega)$ a [[connected topological space|connected]] [[symplectic manifold]] there is a [[central extension of groups]]

\begin{displaymath}
1 \to U(1) \to QuantomorphismGroup(X,\omega) \to HamiltonianSymplectomorphisms(X,\omega) \to 1
  \,.
\end{displaymath}
\end{prop}
This is due to (\hyperlink{Kostant}{Kostant}). It appears also (\hyperlink{Brylinski93}{Brylinski, prop. 2.4.5}).

[[!include geometric quantization extensions - table]]

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{CoveringAnAffineSymplecticGroup}{}\subsubsection*{{Covering an affine symplectic group}}\label{CoveringAnAffineSymplecticGroup}

Given a [[symplectic vector space]] $(V,\omega)$ one may consider the restriction of its [[quantomorphism group]] to the [[affine symplectic group]] $ASp(V,\omega)$ (\hyperlink{RobbinSalamon93}{Robbin-Salamon 93, corollary 9.3})

\begin{displaymath}
\itexarray{
     ESp(V,\omega) &\hookrightarrow& QuantMorph(V,\omega)
     \\
     \downarrow && \downarrow
     \\
     ASp(V,\omega)
     &\hookrightarrow&
     HamSympl(V,\omega)
  }
\end{displaymath}
Sometimes (e.g. \hyperlink{RobbinSalamon93}{Robbin-Salamon 93, p. 30}) this $ESp(V,\omega)$ is called the \emph{extended symplectic group}, but maybe to be more specific one should at the very least say ``[[extended affine symplectic group]]'' or ``extended inhomogeneous symplectic group'' (\hyperlink{ARZ06}{ARZ 06, prop. V.1}).

Notice that the further restriction to $V$ regarded as the [[translation group]] over itself is the [[Heisenberg group]] $Heis(V,\omega)$

\begin{displaymath}
\itexarray{
     Heis(V,\omega)
     &\hookrightarrow&
     ESp(V,\omega) &\hookrightarrow& QuantMorph(V,\omega)
     \\
     \downarrow && \downarrow && \downarrow
     \\
     V
     &\hookrightarrow&
     ASp(V,\omega)
     &\hookrightarrow&
     HamSympl(V,\omega)
  }
\end{displaymath}
The group $ESp(V,\omega)$ is that of those [[quantomorphisms]] which come from [[quadratic Hamiltonians]]. Those elements covering elements in the [[symplectic group]] instead of the [[affine symplectic group]] come from [[homogeneously quadratic Hamiltonians]] (e.g. \hyperlink{RobbinSalamon93}{Robbin-Salamon 93, prop. 10.1}). In fact $ESp$ is the [[semidirect product]] of the [[metaplectic group]] $Mp(V,\omega)$ with the [[Heisenberg group]] (\hyperlink{ARZ06}{ARZ 06, prop. V.1}, see also \hyperlink{Low12}{Low 12})

\begin{displaymath}
ESp(V,\omega) \simeq Heis(V,\omega) \rtimes Mp(V,\omega)
  \,.
\end{displaymath}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[conserved current]]


\item [[Hamiltonian action]], [[classical anomaly]]



\end{itemize}
[[!include slice automorphism groups in higher prequantum geometry - table]]

[[!include higher Atiyah groupoid - table]]

\hypertarget{References}{}\subsection*{{References}}\label{References}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

Original accounts are

\begin{itemize}%
\item [[Jean-Marie Souriau]], \emph{Structure des systemes dynamiques} Dunod, Paris (1970)

Translated and reprinted as (see section V.18 for the quantomorphism group):

[[Jean-Marie Souriau]], \emph{Structure of dynamical systems - A symplectic view of physics}, Brikh\"a{}user (1997) doi:\href{https://doi.org/10.1007/978-1-4612-0281-3}{10.1007/978-1-4612-0281-3}


\item [[Bertram Kostant]], \emph{Quantization and unitary representations}, in \emph{Lectures in modern analysis and applications III}. Lecture Notes in Math. 170 (1970), Springer Verlag, 87---208 doi:\href{https://doi.org/10.1007/BFb0079068}{10.1007/BFb0079068}



\end{itemize}
A textbook account is in

\begin{itemize}%
\item [[Jean-Luc Brylinski]], section II.4 \emph{Loop spaces, characteristic classes and geometric quantization}, Birkh\"a{}user (1993)

\end{itemize}
and in

\begin{itemize}%
\item Rudolf Schmid, \emph{Infinite-dimensional Lie groups with applications to mathematical physics}, J. Geom. Symmetry Phys., Volume 1 (2004), 54-120. \href{https://projecteuclid.org/euclid.jgsp/1495505067}{Project Euclid}

\end{itemize}
The description in terms of automorphism in the slice $\infty$-topos over the moduli stack of (higher) connections is in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], \emph{[[schreiber:Higher geometric prequantum theory]]} (\href{http://arxiv.org/abs/1304.0236}{arXiv:1304.0236})

\end{itemize}
and in section 4.4.17 of

\begin{itemize}%
\item \emph{[[schreiber:differential cohomology in a cohesive topos]]}

\end{itemize}
\hypertarget{SmoothManifoldStructure}{}\subsubsection*{{Smooth manifold structure}}\label{SmoothManifoldStructure}

The [[diffeological space]]-structure ([[diffeological group]], [[smooth group]] structure) on the quantomorphism group is at least implicit in

\begin{itemize}%
\item [[Jean-Marie Souriau]], \emph{Groupes diff\'e{}rentiels}, in \emph{Differential Geometrical Methods in Mathematical Physics} (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lecture Notes in Math. 836, Springer, Berlin, (1980), pp. 91--128. (\href{http://www.ams.org/mathscinet-getitem?mr=607688}{MathSciNet})

\end{itemize}
The [[ILH manifold|ILH group]] structure on the quantomorphism group is discussed in

\begin{itemize}%
\item H. Omori, \emph{Infinite dimensional Lie transformation groups}, Springer lecture notes in mathematics 427 (1974)


\item T. Ratiu, R. Schmid, \emph{The differentiable structure of three remarkable diffeomorphism groups}, Math. Z. 177 (1981)



\end{itemize}
The [[convenient manifold|regular convenient Lie group]] structure is discussed in

\begin{itemize}%
\item Cornelia Vizman, \emph{Some remarks on the quantomorphism group} ([[VizmanQuantomorphism.pdf:file]])

\end{itemize}
A [[metric]]-structure on quantomorphism groups is discussed in

\begin{itemize}%
\item Y. Eliashberg,; L. Polterovich, \emph{Partially ordered groups and geometry of contact transformations}. Geom.Funct.Anal.10(2000),no.6, 1448-1476. doi:\href{https://doi.org/10.1007/PL00001656}{10.1007/PL00001656}, arXiv:\href{https://arxiv.org/abs/math/9910065}{math/9910065}

\end{itemize}
\hypertarget{examples_2}{}\subsubsection*{{Examples}}\label{examples_2}

The quantomorphisms over elements of the [[symplectic group]] of a [[symplectic vector space]] are discussed in

\begin{itemize}%
\item [[Irving Segal]], \emph{Transforms for operators and symplectic automorphisms over a locally compact abelian group}, Math. Scand. 13 (1963) 31-43


\item [[Joel Robbin]], [[Dietmar Salamon]], \emph{Feynman path integrals on phase space and the metaplectic representation} in [[Dietmar Salamon]] (ed.), \emph{Symplectic Geometry}, LMS Lecture Note series 192 (1993) ([[RobbinSalamonMetaplectic.pdf:file]])


\item [[Sergio Albeverio]], J. Rezende and J.-C. Zambrini, \emph{Probability and Quantum Symmetries. II. The Theorem of Noether in quantum mechanics}, Journal of Mathematical Physics 47, 062107 (2006) (\href{http://gfm.cii.fc.ul.pt/people/jczambrini/JMathPhys-47-062107.pdf}{pdf})


\item Stephen G. Low, \emph{Maximal quantum mechanical symmetry: Projective representations of the inhomogenous symplectic group}, J. Math. Phys. 55, 022105 (2014) (\href{http://arxiv.org/abs/1207.6787}{arXiv:1207.6787})



\end{itemize}
[[!redirects quantomorphism]] [[!redirects quantomorphisms]]

[[!redirects quantomorphism groups]]

[[!redirects quantomorphism ∞-group]] [[!redirects quantomorphism ∞-groups]] [[!redirects quantomorphism infinity-group]] [[!redirects quantomorphism infinity-groups]]

[[!redirects quantomorphism n-group]] [[!redirects quantomorphism n-groups]]

[[!redirects higher quantomorphism group]] [[!redirects higher quantomorphism groups]]



\end{document}