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\section*{polarization}

\begin{quote}%
This entry is about polarization of [[phase spaces]] (or of any [[symplectic manifold]]) into [[canonical coordinates]] and [[canonical momenta]]. Different concepts of a similar name include the \emph{[[polarization identity]]} (such as in an [[inner product space]] or a [[Jordan algebra]]) or \emph{[[wave polarization]]} (such as polarized [[light]]). On the other hand, the concept of \emph{[[polarized algebraic variety]]} is closely related.


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

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

[[!include geometric quantization - contents]]

\hypertarget{symplectic_geometry}{}\paragraph*{{Symplectic geometry}}\label{symplectic_geometry}

[[!include symplectic geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{OfASymplecticManifold}{Of a symplectic manifold}\dotfill \pageref*{OfASymplecticManifold} \linebreak
\noindent\hyperlink{OfAPoissonLieAlgebroid}{Of a Poisson Lie algebroid}\dotfill \pageref*{OfAPoissonLieAlgebroid} \linebreak
\noindent\hyperlink{OfACourantLie2Algebroid}{Of a Courant Lie 2-algebroid}\dotfill \pageref*{OfACourantLie2Algebroid} \linebreak
\noindent\hyperlink{OfASymplecticLienAlgebroid}{Of a symplectic Lie $n$-algebroid}\dotfill \pageref*{OfASymplecticLienAlgebroid} \linebreak
\noindent\hyperlink{of_an_plectic_smooth_groupoid}{Of an $n$-plectic smooth $\infty$-groupoid}\dotfill \pageref*{of_an_plectic_smooth_groupoid} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{real_polarizations}{Real polarizations}\dotfill \pageref*{real_polarizations} \linebreak
\noindent\hyperlink{K&#228;hlerPolarization}{K\"a{}hler polarizations}\dotfill \pageref*{K&#228;hlerPolarization} \linebreak
\noindent\hyperlink{convex_polarizations}{Convex polarizations}\dotfill \pageref*{convex_polarizations} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{quantum_states_and_wave_functions}{Quantum states and wave functions}\dotfill \pageref*{quantum_states_and_wave_functions} \linebreak
\noindent\hyperlink{bohrsommerfeld_leaves}{Bohr-Sommerfeld leaves}\dotfill \pageref*{bohrsommerfeld_leaves} \linebreak
\noindent\hyperlink{liouville_integrability}{Liouville integrability}\dotfill \pageref*{liouville_integrability} \linebreak
\noindent\hyperlink{refinement_to_higher_geometric_quantization}{Refinement to higher geometric quantization}\dotfill \pageref*{refinement_to_higher_geometric_quantization} \linebreak
\noindent\hyperlink{refinement_to_higher_quantization_by_pushforward}{Refinement to higher quantization by push-forward}\dotfill \pageref*{refinement_to_higher_quantization_by_pushforward} \linebreak
\noindent\hyperlink{other}{Other}\dotfill \pageref*{other} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

For a [[symplectic manifold]] $(X, \omega)$ regarded as the [[phase space]] of a [[physical system]], a choice of \emph{polarization} is, locally, a choice of decomposition of the [[coordinates]] on $X$ into ``[[canonical coordinates]]'' and ``[[canonical momenta]]''.

The archtypical example is that where $X = T^* \Sigma$ is the [[cotangent bundle]] of a [[manifold]] $\Sigma$. In this case the canonical [[canonical coordinates]] are those parameterizing $\Sigma$ itself, while the canonical [[canonical momenta]] are coordinates on each [[fiber]] of the cotangent bundle.

But for general symplectic manifolds there is no such canonical choice of coordinates and momenta. Moreover, in general there is not even a global notion of canonical momenta. Instead, a choice of (real) polarization is a [[foliation]] of phase space by [[Lagrangian submanifolds]] and then

\begin{itemize}%
\item the ``[[canonical coordinates]]'' are coordinates on the corresponding [[leaf space]] (parameterizing the leaves);


\item the ``[[canonical momenta]]'' are coordinates along each [[leaf]]. If there is no \emph{typical leaf} then this is not a globally defined notion, only the polarization itself is.



\end{itemize}
Locally this is a choice of [[coordinate patch]] $\phi : \mathbb{R}^{2n} \to X$ such that the [[symplectic form]] takes the form

\begin{displaymath}
\phi^* \omega = \sum_{i = 1}^n \mathbf{d} q^i \wedge \mathbf{d}p_i
\end{displaymath}
where the $\{q^i : \mathbb{R}^{2n} \simeq \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}\}$ are the canonical coordinates on the first $\mathbb{R}^n$-factor of the [[Cartesian space]] $\mathbb{R}^{2n}$, and where $\{p_o : \mathbb{R}^{2n} \simeq \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}\}$ are canonical coordinates on the second $\mathbb{R}^n$-factor.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

The traditional notion of polarization applies to a [[symplectic manifold]].

\begin{itemize}%
\item \href{OfASymplecticManifold}{Of a symplectic manifold}

\end{itemize}
Symplectic manifold are the lowest step in a tower of notions in [[higher symplectic geometry]] which proceeds with [[n-plectic geometry]] for all $n$ and manifolds refined to [[smooth infinity-groupoids]]. The next simplest cases in this tower are [[symplectic Lie n-algebroids]], which for $n=1$ are [[Poisson Lie algebroids]] and for $n = 2$ are [[Courant Lie 2-algebroids]]:

\begin{itemize}%
\item \hyperlink{OfAPoissonLieAlgebroid}{Of a Poisson Lie algebroid}


\item \hyperlink{OfACourantLie2Algebroid}{Of a Courant Lie 2-algebroid}


\item \hyperlink{OfASymplecticLienAlgebroid}{Of a symplectic Lie n-algebroid}



\end{itemize}
\hypertarget{OfASymplecticManifold}{}\subsubsection*{{Of a symplectic manifold}}\label{OfASymplecticManifold}

Let $(X, \omega)$ be a [[symplectic manifold]].

\begin{defn}
\label{ByLagrangianFoliation}\hypertarget{ByLagrangianFoliation}{}
A \textbf{real polarization} of $(X, \omega)$ is a [[foliation]] by [[Lagrangian submanifolds]].

\end{defn}
For instance (\hyperlink{Weinstein}{Weinstein, p. 9}).

More generally

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{polarization} of $(X,\omega)$ is a choice of involutive [[Lagrangian subspace|Lagrangian subbundle]] $\mathcal{P} \hookrightarrow T_{\mathbb{C}} X$ of of the [[complexification|complexified]] [[tangent bundle]] of $X$.

\end{defn}
For instance (\hyperlink{BatesWeinstein}{Bates-Weinstein, def. 7.4})

\hypertarget{OfAPoissonLieAlgebroid}{}\subsubsection*{{Of a Poisson Lie algebroid}}\label{OfAPoissonLieAlgebroid}

A [[Poisson Lie algebroid]] $\mathfrak{P}$ is a [[symplectic Lie n-algebroid]] for $n = 1$. Regarding its [[Chevalley-Eilenberg algebra]] as the algebra of functions on a [[dg-manifold]], that dg-manifold carries a graded [[symplectic form]] $\omega$. One can then say

\begin{defn}
\label{ForPoissonLieAlgebroidyByLagrangianFoliation}\hypertarget{ForPoissonLieAlgebroidyByLagrangianFoliation}{}
A dg-[[Lagrangian submanifold]] of $(\mathfrak{P}, \omega)$ is also called a \textbf{$\Lambda$-structure}. (\hyperlink{Severa}{\v{S}evera, section 4}).

Hence we might say \textbf{real polarization} of $(\mathfrak{P}, \omega)$ is a foliation by dg-Lagrangian submanifolds.

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
For $(X, \pi)$ the [[Poisson manifold]] underlying a [[Poisson Lie algebroid]] $(\mathfrak{P}, \omega)$, a dg-Lagrangian submanifold of $(\mathfrak{P}, \omega)$ corresponds to a [[coisotropic submanifold]] of $(X, \pi)$.

\end{prop}
(\hyperlink{Severa}{\v{S}evera, section 4})

\begin{remark}
\label{}\hypertarget{}{}
The dg-Lagrangian submanifolds also correspond to [[branes]] in the [[Poisson sigma-model]] (see there) on $(\mathfrak{P}, \omega)$.

\end{remark}
\hypertarget{OfACourantLie2Algebroid}{}\subsubsection*{{Of a Courant Lie 2-algebroid}}\label{OfACourantLie2Algebroid}

A [[Courant Lie algebroid]] $\mathfrak{C}$ is a [[symplectic Lie n-algebroid]] for $n = 2$. Regarding its [[Chevalley-Eilenberg algebra]] as the algebra of functions on a [[dg-manifold]], that dg-manifold carries a graded [[symplectic form]] $\omega$. One can then say

\begin{defn}
\label{ForPoissonLieAlgebroidyByLagrangianFoliation}\hypertarget{ForPoissonLieAlgebroidyByLagrangianFoliation}{}
A dg-[[Lagrangian submanifold]] of $(\mathfrak{C}, \omega)$ is also called a \textbf{$\Lambda$-structure}. (\hyperlink{Severa}{\v{S}evera, section 4}).

Hence we might say \textbf{real polarization} of $(\mathfrak{C}, \omega)$ is a foliation by dg-Lagrangian submanifolds.

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
The dg-Lagrangian submanifolds of a Courant Lie 2-algebroid $(\mathfrak{C}, \omega)$ correspond to [[Dirac structures]] on $(\mathfrak{C}, \omega)$.

\end{prop}
(\hyperlink{Severa}{\v{S}evera, section 4})

\hypertarget{OfASymplecticLienAlgebroid}{}\subsubsection*{{Of a symplectic Lie $n$-algebroid}}\label{OfASymplecticLienAlgebroid}

??

\hypertarget{of_an_plectic_smooth_groupoid}{}\subsubsection*{{Of an $n$-plectic smooth $\infty$-groupoid}}\label{of_an_plectic_smooth_groupoid}

A simple notion of a real polarization for [[2-plectic manifolds]] is considered within the context of higher geometric quantization in \hyperlink{Rogers}{Rogers, Chap. 7}.

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

\hypertarget{real_polarizations}{}\subsubsection*{{Real polarizations}}\label{real_polarizations}

(\ldots{})

\hypertarget{K&#228;hlerPolarization}{}\subsubsection*{{K\"a{}hler polarizations}}\label{K&#228;hlerPolarization}

If the [[symplectic manifold]] $(X,\omega)$ lifts to the structure of a [[Kähler manifold]] $(X, J, g)$, hence with [[Riemannian metric]] $g(-,-) = \omega(-,I(-))$, then the holomorphic/antiholomorphic decomposition induced by the [[complex manifold]] structure is a polarization of $(X,\omega)$. Polarizations of this form are therefore called \textbf{[[Kähler polarizations]]}.

\hypertarget{convex_polarizations}{}\subsubsection*{{Convex polarizations}}\label{convex_polarizations}

\begin{itemize}%
\item [[p-convex polarization]]

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

\hypertarget{quantum_states_and_wave_functions}{}\subsubsection*{{Quantum states and wave functions}}\label{quantum_states_and_wave_functions}

Upon ([[geometric quantization|geometric]]) [[quantization]] of the physical system described by the [[symplectic manifold]] $(X, \omega)$ a [[quantum state]] is supposed to be a function on $X$ -- or rather a [[section]] of a [[prequantum line bundle]] which is a ``wave-function that only depends on the canonical coordinates'', not on the canonical momenta. In terms of polarizations this is formalized by saying that a [[space of states (in geometric quantization)|quantum state]] is a section which is [[covariant derivative|covariant constant]] along the [[leaves]] of the polarization (along the ``momentum direction'').

\hypertarget{bohrsommerfeld_leaves}{}\subsubsection*{{Bohr-Sommerfeld leaves}}\label{bohrsommerfeld_leaves}

After a choice of [[prequantum line bundle]] $\nabla$ lifting $\omega$, a \textbf{[[Bohr-Sommerfeld leaf]]} of a (real) polarization is a [[leaf]] on which the prequantum line bundle is not just flat, but also trivializable as a [[circle bundle]] with connection.

\hypertarget{liouville_integrability}{}\subsubsection*{{Liouville integrability}}\label{liouville_integrability}

If a polarization on an $2n$-dimensional symplectic manifold is generated from $n$ [[Hamiltonian vector fields]] whose [[Hamiltonians]] commute with each other under the [[Poisson bracket]] (and one of them is regarded as that generating time evolution of a mechanical system) then one speaks of a [[Liouville integrable system]].

\hypertarget{refinement_to_higher_geometric_quantization}{}\subsubsection*{{Refinement to higher geometric quantization}}\label{refinement_to_higher_geometric_quantization}

[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

\hypertarget{refinement_to_higher_quantization_by_pushforward}{}\subsubsection*{{Refinement to higher quantization by push-forward}}\label{refinement_to_higher_quantization_by_pushforward}

[[!include orientations in higher quantization - table]]

\hypertarget{other}{}\subsubsection*{{Other}}\label{other}

\begin{itemize}%
\item [[canonical transformation]]


\item [[polarized algebraic variety]]



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

Lecture notes include

\begin{itemize}%
\item [[Alan Weinstein]], \emph{Lectures on Symplectic manifolds} Lecture 2 \emph{Lagrangian splittings, real and complex polarizations, K\"a{}hler manifolds}, CBMS Regional Conference Series in Mathematics, AMS (1977)

\end{itemize}
\begin{itemize}%
\item Sean Bates, [[Alan Weinstein]], \emph{Lectures on the geometry of quantization}, \href{http://www.math.berkeley.edu/~alanw/GofQ.pdf}{pdf}

\end{itemize}
\begin{itemize}%
\item Kristin Shaw, \emph{An introduction to polarizations} (\href{http://www.math.toronto.edu/karshon/grad/2006-07/polarizations.pdf}{pdf})

\end{itemize}
and section 4 and 5 of

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

\end{itemize}
or section 5 of

\begin{itemize}%
\item A. Echeverria-Enriquez, M.C. Munoz-Lecanda, N. Roman-Roy, C. Victoria-Monge, \emph{Mathematical Foundations of Geometric Quantization} Extracta Math. 13 (1998) 135-238 (\href{http://arxiv.org/abs/math-ph/9904008}{arXiv:math-ph/9904008})

\end{itemize}
or

\begin{itemize}%
\item Yuichi Nohara, \emph{Independence of Polarization in Geometric Quantization} (\href{http://geoquant2007.mi.ras.ru/nohara.pdf}{pdf})

\end{itemize}
Lagrangian submanifolds of [[L-infinity algebroids]] are considered in

\begin{itemize}%
\item [[Pavol Ševera]], \emph{Some title containing the words ``homotopy'' and ``symplectic'', e.g. this one} (\href{http://arxiv.org/abs/math/0105080}{arXiv:0105080})

\end{itemize}
In the case that the polarization integrates to the [[action]] of a [[Lie group]] $G$ one may think of passing to polarized sections as equivlent to passing to $G$-[[gauge equivalence|gauge equivalence classes]]. This point of view is highlighted in

\begin{itemize}%
\item [[Gabriel Catren]], \emph{On the Relation Between Gauge and Phase Symmetries}, Foundations of Physics 44 (12):1317-1335 (2014) (\href{http://philpapers.org/rec/CATOTR}{web})

\end{itemize}
A candidate for polarizations for higher geometric quantization in $n$-plectic geometry is discussed in Chapter 7 of

\begin{itemize}%
\item [[Chris Rogers]], \emph{Higher symplectic geometry} PhD thesis (\href{http://arxiv.org/abs/1106.4068}{arXiv:1106.4068}).

\end{itemize}
[[!redirects polarizations]]

[[!redirects canonical coordinate]] [[!redirects canonical coordinates]]

[[!redirects real polarization]] [[!redirects real polarizations]]

[[!redirects complex polarization]] [[!redirects complex polarizations]]



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