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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{extended affine symplectic group} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory} [[!include group theory - contents]] \hypertarget{symplectic_geometry}{}\paragraph*{{Symplectic geometry}}\label{symplectic_geometry} [[!include symplectic geometry - contents]] \hypertarget{geometric_quantization}{}\paragraph*{{Geometric quantization}}\label{geometric_quantization} [[!include geometric quantization - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{example}{Example}\dotfill \pageref*{example} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} 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 [[Hamiltonians]] that are [[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{example}{}\subsection*{{Example}}\label{example} Let $(V,\omega) = (\mathbb{R}^2, \mathbf{d}p\wedge \mathbf{d}q)$ be the 2-dimensional [[symplectic vector space]]. Write \begin{displaymath} q,p \colon \mathbb{R}^2 \longrightarrow \mathbb{R} \end{displaymath} for its two canonical coordinates functions (the ``[[canonical coordinates]] and [[canonical momenta]]''). Write \begin{displaymath} i \colon \mathbb{R}^2 \longrightarrow \mathbb{R} \end{displaymath} for the [[constant function]] with value 1. The [[Poisson bracket]] is \begin{displaymath} \{p.q\} = i \,. \end{displaymath} Any [[smooth function]] $H \colon \mathbb{R}^2 \to \mathbb{R}$ we may call a [[Hamiltonian]]. Given a Hamiltonian $H$, its [[Hamiltonian flow]] is the [[flow]] given by the [[vector field]] (the [[Hamiltonian vector field]]) corresponding to the [[derivation]] $\{H,-\}$ on $C^\infty(\mathbb{R}^2)$. Those Hamiltonians whose Hamiltonian flows are [[linear functions]] on $\mathbb{R}^2$ are precisely the [[homogeneously quadratic Hamiltonians]]: \begin{displaymath} \exp(t\{\tfrac{1}{2}p^2,-\}) \colon \left[ \itexarray{ q \\ p } \right] \mapsto \left[ \itexarray{ q +t p \\ p } \right] \end{displaymath} \begin{displaymath} \exp(\{\tfrac{1}{2}q^2,-\}) \colon \left[ \itexarray{ q \\ p } \right] \mapsto \left[ \itexarray{ q \\ p + t q } \right] \end{displaymath} \begin{displaymath} \exp(t \{q p,-\}) \colon \left[ \itexarray{ q \\ p } \right] \mapsto \left[ \itexarray{ e^t q \\ e^{-t} p } \right] \end{displaymath} The general element of the [[metaplectic group]] $Mp(\mathbb{R}^2,\mathbf{d}q \wedge \mathbf{d}p)$ is hence \begin{displaymath} \exp(t_1 \tfrac{1}{2}p^2 + t_2 \tfrac{1}{2}q^2 + t_3 q p) \end{displaymath} By [[differentiation|differentiating]] this by $t$ at $t = 0$ we obtain a [[basis]] for the [[Lie algebra]] $\mathfrak{sp}(V,\omega)$ of, both, the [[symplectic group]] $Sp(V,\omega)$ as well as its [[metaplectic group]] $Mp(V,\omega)$ \begin{displaymath} \left[ \itexarray{ 0 & 0 \\ 1 & 0 } \right] \,, \;\;\; \left[ \itexarray{ 0 & 1 \\ 0 & 0 } \right] \,, \;\;\; \left[ \itexarray{ 1 & 0 \\ 0 & -1 } \right] \end{displaymath} The Hamiltonians that generate translations are precisely the homogeneously linear Hamiltonians: \begin{displaymath} \exp(t\{p,-\}) \colon \left[ \itexarray{ q \\ p } \right] \mapsto \left[ \itexarray{ q + t \\ p } \right] \end{displaymath} \begin{displaymath} \exp(t\{q,-\}) \colon \left[ \itexarray{ q \\ p } \right] \mapsto \left[ \itexarray{ q \\ p - t } \right] \end{displaymath} This is the key point where the extension appears: While these two linear translation operations themselves (i.e. the underlying [[symplectomorphisms]]) of course commute with each other, their generating Hamiltonians do not Poisson commute but instead form the [[Heisenberg algebra]] extension of the [[translation group]]. The general element of the extended affine symplectic group $ESp(\mathbb{R}^2, \mathbf{d}q \wedge \mathbf{d}p)$ is \begin{displaymath} \exp(t_1 \tfrac{1}{2}p^2 + t_2 \tfrac{1}{2}q^2 + t_3 q p + t_4 p + t_5 q + t_6 i) \,. \end{displaymath} \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item [[Joel Robbin]], [[Dietmar Salamon]], \emph{Feynman path integrals on phase space and the metaplectic representation}, Math. Z. \textbf{221} (1996), no. 2, 307---335, (\href{http://www.ams.org/mathscinet-getitem?mr=98f:58051}{MR98f:58051}, \href{http://dx.doi.org/10.1007/BF02622118}{doi}, [[RobbinSalamonMetaplectic.pdf:file]]), also in [[Dietmar Salamon]] (ed.), \emph{Symplectic Geometry}, LMS Lecture Note series 192 (1993) \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 extended inhomogeneous symplectic group]] \end{document}