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\begin{document}

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\section*{reduced phase space}

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

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

[[!include symplectic geometry - contents]]

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

[[!include physicscontents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{for_local_lagrangians}{For local Lagrangians}\dotfill \pageref*{for_local_lagrangians} \linebreak
\noindent\hyperlink{for_extended_local_lagrangians}{For extended local Lagrangians}\dotfill \pageref*{for_extended_local_lagrangians} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In [[physics]], a [[local Lagrangian]] induces a [[covariant phase space]] equipped with a canonical [[presymplectic form]]. The [[quotient]] of this by [[symmetries]] that, in good cases, make the pre-symplectic form a genuine [[symplectic form]], is called the \emph{reduced phase space}.

Generally, given a [[symplectic manifold]] or [[presymplectic manifold]] or [[Poisson manifold]] regarded as a [[phase space]] equipped with a suitable ([[Hamiltonian action|Hamiltonian-]]) [[action]] by a [[Lie group]], the corresponding [[symplectic reduction]] or [[presymplectic reduction]] or [[Poisson reduction]] is, if it exists, the corresponding reduced phase space.

[[!include symplectic reduction - table]]

\hypertarget{details}{}\subsection*{{Details}}\label{details}

\hypertarget{for_local_lagrangians}{}\subsubsection*{{For local Lagrangians}}\label{for_local_lagrangians}

Given a [[local Lagrangian]] (we display it in [[codimension]] 1 [[mechanics]] for simplicity of notation)

\begin{displaymath}
L \;\colon\; \Omega^{0,1}\left(\mathbf{Fields}(I) \times I\right)
\end{displaymath}
the corresponding [[covariant phase space]] is the space of solutions of the [[Euler-Lagrange equations]]

\begin{displaymath}
\{EL = 0\}
\end{displaymath}
equipped with the [[presymplectic form]]

\begin{displaymath}
\omega = \delta \frac{\delta L}{\delta \dot \phi} \wedge \delta \phi
\end{displaymath}
which is exact, with [[potential]]

\begin{displaymath}
\theta = \frac{\delta L}{\delta \dot \phi} \wedge \delta \phi
\end{displaymath}
as discussed in detail at \emph{[[covariant phase space]]}.

Together, this is a [[prequantum bundle]], hence a [[circle bundle with connection]] whose [[curvature]] is $\omega$. It so happens that the underlying U(1)-[[principal bundle]] of this is trivial, and hence the [[connection on a bundle|connection]] is given by the globally defined [[differential 1-form]] $\theta$.

But this trivialility is only superficial: the [[symmetry]] [[group]] $G$ of the Lagrangian is supposed to act by [[Hamiltonian flows]] and the prequantum connection $\theta$ is to be equipped with $G$-[[equivariant connection]] structure for it to count as a connection on the [[reduced phase space]].

Another way to say this, using the [[higher differential geometry]] of [[smooth groupoids]]: the above [[prequantum bundle]] is modulated by a map $\theta \;\colon\; \{EL = 0\} \longrightarrow \mathbf{B}U(1)_{conn}$ to the [[smooth groupoid|smooth]] [[moduli stack]] of [[circle n-bundles with connection|circle bundles with connection]], and the $G$-[[Hamiltonian action]] induced the [[action groupoid]] $\{EL = 0\} \longrightarrow \{EL = 0\}//G$; and the construction of the reduced [[prequantization]] is the construction of the diagonal [[morphism]] $\nabla_{red}$ in the following [[diagram]] (of [[smooth groupoids]])

\begin{displaymath}
\itexarray{
    \{EL = 0\} &\stackrel{\theta}{\longrightarrow}& \mathbf{B}U(1)_{conn}
    \\
    \downarrow & \nearrow_{\nabla_{red}}
    \\
    \{EL = 0\}//G
  }
  \,.
\end{displaymath}
\hypertarget{for_extended_local_lagrangians}{}\subsubsection*{{For extended local Lagrangians}}\label{for_extended_local_lagrangians}

For $n$-dimensional [[field theory]], the [[local Lagrangian]] of the above dsicussion arises as the [[transgression]] of an $n$-form Lagrangian down in [[codimension]] $n$, a refinement discussed in more detail at \emph{[[local prequantum field theory]]}. Such a Lagrangian may analogously be understood as being the [[principal infinity-connection|higher connection]] on a [[prequantum n-bundle]] which, in turn, maps to the ordinary [[prequantum bundle]] under [[transgression]]. Hence one may ask for a \emph{universal} or \emph{extended} [[equivariant structure]] already on this [[prequantum n-bundle]] which is such that under [[transgression]] (to any [[Cauchy surface]]) it induces an equivariant structure and hence a reduced phase space as above.

Examples of such extended reduced phase space structures include:

\begin{itemize}%
\item the [[universal Chern-Simons circle 3-connection]]


\item the [[universal Chern-Simons circle 7-connection]]



\end{itemize}
and other examples discussed at \emph{[[local prequantum field theory]]}.

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

\begin{itemize}%
\item [[symplectic reduction]]


\item [[quantization commutes with reduction]]


\item [[equivariant cohomology]]


\item [[BV-BRST formalism]]



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

See the references at \emph{[[quantization commutes with reduction]]}.

[[!redirects reduced phase spaces]]



\end{document}