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\begin{document}

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\section*{quantum observable}

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

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{algebraic_quantum_field_theory}{}\paragraph*{{Algebraic Quantum Field Theory}}\label{algebraic_quantum_field_theory}

[[!include AQFT and operator algebra contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition_in_geometric_quantization}{Definition in geometric quantization}\dotfill \pageref*{definition_in_geometric_quantization} \linebreak
\noindent\hyperlink{OnASymplecticManifold}{On a symplectic manifold}\dotfill \pageref*{OnASymplecticManifold} \linebreak
\noindent\hyperlink{on_an_plectic_smooth_groupoid}{On an $n$-plectic smooth $\infty$-groupoid}\dotfill \pageref*{on_an_plectic_smooth_groupoid} \linebreak
\noindent\hyperlink{irreducible_representations_and_superselection_sectors}{Irreducible representations and superselection sectors}\dotfill \pageref*{irreducible_representations_and_superselection_sectors} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{in_algebraic_quantum_theory}{In algebraic quantum theory}\dotfill \pageref*{in_algebraic_quantum_theory} \linebreak
\noindent\hyperlink{in_geometric_quantization}{In geometric quantization}\dotfill \pageref*{in_geometric_quantization} \linebreak


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

An [[observable]] in [[quantum physics]].

\hypertarget{definition_in_geometric_quantization}{}\subsection*{{Definition in geometric quantization}}\label{definition_in_geometric_quantization}

We consider the notion of quantum observables in the the context of [[geometric quantization]]. See also [[quantum operator (in geometric quantization)]].

\hypertarget{OnASymplecticManifold}{}\subsubsection*{{On a symplectic manifold}}\label{OnASymplecticManifold}

Let $(X, \omega)$ be a ([[presymplectic manifold|pre]]-)[[symplectic manifold]], thought of as the [[phase space]] of a [[classical mechanics|physical system]].

Assume that $\omega$ is prequantizable (integral) and let $\nabla : X \to \mathbf{B} U(1)_{conn}$ be a [[prequantum bundle]] $E \to X$ [[connection on a bundle|with connection]] for $\omega$, hence with [[curvature]] $F_\nabla = \omega$. Write $\Gamma_X(E)$ for the space of smooth [[sections]] of the [[associated bundle|associated]] [[complex line bundle]]. This is the \emph{prequantum space of states}.

\begin{defn}
\label{}\hypertarget{}{}
For $f \in C^\infty(X, \mathbb{C})$ a function on phase space, the corresponding \textbf{pre-quantum operator} is the linear map on prequantum states

\begin{displaymath}
\hat f : \Gamma_X(E) \to \Gamma_X(E)
\end{displaymath}
given by

\begin{displaymath}
\psi \mapsto -i \nabla_{v_f} \psi + f \cdot \psi
  \,,
\end{displaymath}
where

\begin{itemize}%
\item $v_f$ is the [[Hamiltonian vector field]] corresponding to $f$;


\item $\nabla_{v_f} : \Gamma_X(E) \to \Gamma_X(E)$ is the [[covariant derivative]] of sections along $v_f$ for the given choice of prequantum connection;


\item $f \cdot (-) : \Gamma_X(E) \to \Gamma_X(E)$ is the operation of degreewise multiplication pf sections.



\end{itemize}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
In terms of [[schreiber:Higher geometric prequantum theory]] we may, as discussed there, identify the [[Poisson bracket]] [[Lie algebra]] $\mathfrak{Poisson}(X,\omega)$ with the Lie algebra of the group of automorphism $\exp(O) \colon \nabla \stackrel{\simeq}{\to} \nabla$ regarded in the [[slice (∞,1)-topos|slice]] over $\mathbf{B}U(1)_{conn}$. Moreover, the space of sections is equivalently the space of maps $\Psi \colon \nabla \to \mathbb{C}//U(1)_{conn}$ in the slice from $\nabla$ into the differential refinement of the smooth universal line bundle $\mathbb{C}//U(1) \to \mathbf{B}U(1)$. In this formulation the action of prequantum operators is just the precomposition action

\begin{displaymath}
\widehat{\exp(O)} \colon (\nabla \stackrel{\Psi}{\to} \mathbb{C}//U(1)_{conn})
  \mapsto 
  (\nabla \stackrel{exp(O)}{\to} \nabla \stackrel{\Psi}{\to} \mathbb{C}//U(1)_{conn})
  \,.
\end{displaymath}
\end{remark}
Now after a choice of [[polarization]] a [[quantum state]] is a prequantum [[wave function]] which is covariantly constant along the [[Lagrangian submanifolds]] of the foliation. Not all prequantum operators will respect the space of such quantum states inside all quantum states. Those that do become genuine quantum operators.

\begin{defn}
\label{GeometricQuantumOperator}\hypertarget{GeometricQuantumOperator}{}
Let $\mathcal{P}$ be a [[polarization]] of the [[symplectic manifold]] $(X,\omega)$. then a [[quantum state]] or [[wavefunction]] is a prequantum state $\psi$ such that $\nabla \Psi$ vanishes along the [[leaves]] of the polarization.

A \textbf{quantum operator} is a prequantum operator which preserves quantum states among all prequantum states.

\end{defn}
\begin{prop}
\label{CompatibiltyOfFlowWithPolarization}\hypertarget{CompatibiltyOfFlowWithPolarization}{}
A prequantum operator given by a [[Hamiltonian]] function $f$ with [[Hamiltonian vector field]] $v_f$ is a quantum operator, def. \ref{GeometricQuantumOperator}, with respect to a given [[polarization]] $\mathcal{P}$ precisely if its flow preserves $\mathcal{P}$, hence precisely if

\begin{displaymath}
[v_f, \mathcal{P}] \subset \mathcal{P}
  \,.
\end{displaymath}
\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
If $\mathcal{P}$ is a [[Kähler polarization]] then its underlying [[almost complex structure]] it induces a [[spin{\tt \symbol{94}}c structure]], as discussed there. If $\rho \colon G \to QuantMorph(X,\nabla)$ is a [[Hamiltonian action]] (a homomorphism to the [[quantomorphism group]]) such that each prequantum operator $\rho(g)$ is a quantum operator in that it preserves the polarization, by prop. \ref{CompatibiltyOfFlowWithPolarization}, then the corresponding [[spin{\tt \symbol{94}}c structure]] is $G$-invariant. Accordingly the [[index]] of the [[spin{\tt \symbol{94}}c Dirac operator]] which gives the \href{http://ncatlab.org/nlab/show/geometric%20quantization#AsIndexOfSpinCDiracOperator}{geometric quantization by cohomological quantization} exists not just in [[K-theory]], where it yields the [[space of quantum states]], but even in $G$-[[equivariant K-theory]], exhibiting a [[representation]] of $G$ on the Hilbert space. This is the action of the quantum observables given by $\rho$ from the point of view of cohomological quantization.

\end{remark}
\begin{example}
\label{}\hypertarget{}{}
Over a phase space which is a [[cotangent bundle]] and with respect to the corresponding canonical vertical polarization, a Hamiltonian function is a quantum operator precisely if it is at most linear in the [[canonical momenta]].

\end{example}
See for instance (\hyperlink{Blau}{Blau, around p. 35})

\hypertarget{on_an_plectic_smooth_groupoid}{}\subsubsection*{{On an $n$-plectic smooth $\infty$-groupoid}}\label{on_an_plectic_smooth_groupoid}

(\ldots{})

\hypertarget{irreducible_representations_and_superselection_sectors}{}\subsubsection*{{Irreducible representations and superselection sectors}}\label{irreducible_representations_and_superselection_sectors}

The [[space of quantum states]] forms a linear [[representation]] of a given [[algebra of observables]]. The decomposition of that into [[irreducible representations]] is physically the decomposition into [[superselection sectors]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[observable]]


\item [[Bogoliubov's formula]]



\end{itemize}
[[!include products in pQFT -- table]]

\begin{itemize}%
\item [[causally local net of observables]]


\item [[order-theoretic structure in quantum mechanics]]

\begin{itemize}%
\item [[Jordan algebra]], [[Bohr topos]]


\item [[Harding-Döring-Hamhalter theorem]]



\end{itemize}

\item [[effect algebra]]


\item [[Hamiltonian operator]]


\item [[self-adjoint operator]], [[daseinisation]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{in_algebraic_quantum_theory}{}\subsubsection*{{In algebraic quantum theory}}\label{in_algebraic_quantum_theory}

Comprehensive discussion is in

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[geometry of physics -- perturbative quantum field theory]]} -- \emph{\href{geometry+of+physics+--+perturbative+quantum+field+theory#Observables}{Observables}}

\end{itemize}
\hypertarget{in_geometric_quantization}{}\subsubsection*{{In geometric quantization}}\label{in_geometric_quantization}

See also the references at [[geometric quantization]].

Standard facts are recalled for instance around p. 35 of

\begin{itemize}%
\item [[Matthias Blau]], \emph{Symplectic Geometry and Geometric Quantization} ([[BlauGeometricQuantization.pdf:file]])

\end{itemize}
Computation of quantum observables by [[index]] maps in [[equivariant K-theory]] is in (see specifically around p. 8 and 9)

\begin{itemize}%
\item [[Michèle Vergne]], \emph{Geometric quantization and equivariant cohomology} (\href{http://www.math.jussieu.fr/~vergne/pageperso/articles/CONGEURO.pdf}{pdf})

\end{itemize}
[[!redirects quantum observables]]

[[!redirects quantum operator]] [[!redirects quantum operators]]



\end{document}