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\section*{off-shell Poisson bracket}

\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{on_paths_in_a_symplectic_manifold}{On paths in a symplectic manifold}\dotfill \pageref*{on_paths_in_a_symplectic_manifold} \linebreak
\noindent\hyperlink{the_trajectory_space_of_a_symplectic_manifold}{The trajectory space of a symplectic manifold}\dotfill \pageref*{the_trajectory_space_of_a_symplectic_manifold} \linebreak
\noindent\hyperlink{hamiltonian_evolution}{Hamiltonian evolution}\dotfill \pageref*{hamiltonian_evolution} \linebreak
\noindent\hyperlink{TheOffShellPoissonBracketOnSpaceOfPathsInSymplecticManifold}{The off-shell Poisson bracket}\dotfill \pageref*{TheOffShellPoissonBracketOnSpaceOfPathsInSymplecticManifold} \linebreak
\noindent\hyperlink{the_symplectic_leaves}{The symplectic leaves}\dotfill \pageref*{the_symplectic_leaves} \linebreak
\noindent\hyperlink{BoundaryFieldTheoryInterpretation}{Boundary field theory interpretation}\dotfill \pageref*{BoundaryFieldTheoryInterpretation} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Given a [[symplectic manifold]] $(X,\omega)$ and given a [[Hamiltonian]] [[function]] $H \colon X \longrightarrow \mathbb{R}$, there is a [[Poisson bracket]] on an [[algebra of functions]] on the [[smooth space|smooth]] [[path space]] $[I,X]$ -- the ``space of histories'' or ``space of [[trajectories]]'' -- for $I = [0,1]$ the closed [[interval]], which is such that its [[symplectic leaves]] are each a copy of $X$, but regarded as the space of initial conditions for evolution with respect to $H$ with a [[source]] term added.

The first statement was first observed for the [[Peierls bracket]] of [[local prequantum field theory]] in (\hyperlink{Marolf93}{Marolf 93, section II}), but the construction there is not specific to the [[Peierls bracket]]. That the construction provides a [[foliation]] of trajectory space by [[symplectic leaves]] labeled by [[level sets]] of the [[Euler-Lagrange equation|Euler-Lagrange function]] was explicitly pointed out at (\hyperlink{BrunettiFredenhagenRibeiro12}{Brunetti-Fredenhagen-Ribeiro 12, top of p. 4}). (Again for the [[Peierls bracket]], but the statement holds more generally.) These references and that this means a symplectic foliation by [[source]] terms was highlighted out by (\hyperlink{Khavkine13}{Khavkine 13}).

\hypertarget{on_paths_in_a_symplectic_manifold}{}\subsection*{{On paths in a symplectic manifold}}\label{on_paths_in_a_symplectic_manifold}

We describe here the [[off-shell]] Poisson bracket in the context of [[mechanics]], hence for [[mechanical systems]] with [[finite set|finite]]-[[dimension|dimensional]] [[phase space]]. The basic idea is that sketched in (\hyperlink{Marolf93}{Marolf 93, section II}), but we try to make it precise. Then we similarly look into the [[foliation]] by [[symplectic leaves]] as suggested by (\hyperlink{Khavkine13}{Khavkine 13}).

\hypertarget{the_trajectory_space_of_a_symplectic_manifold}{}\subsubsection*{{The trajectory space of a symplectic manifold}}\label{the_trajectory_space_of_a_symplectic_manifold}

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

\begin{displaymath}
\{-,-\} \;\colon\; C^\infty(X)\otimes C^\infty(X) \longrightarrow C^\infty(X)
\end{displaymath}
for the [[Poisson bracket]] induced by the [[symplectic form]] $\omega$, hence by the [[Poisson bivector]] $\pi \coloneqq \omega^{-1}$.

For notational simplicity we will restrict attention to the special case that

\begin{displaymath}
X = \mathbb{R}^2 \simeq T^\ast \mathbb{R}
\end{displaymath}
with canonical [[coordinates]]

\begin{displaymath}
q,p \;\colon\; \mathbb{R}^2 \longrightarrow \mathbb{R}
\end{displaymath}
and symplectic form

\begin{displaymath}
\omega = \mathbf{d}q \wedge \mathbf{d}p
  \,.
\end{displaymath}
The general case of the following discussion is a straightforward generalization of this, which is just notationally more inconvenient.

Write $I \coloneqq [0,1]$ for the standard [[interval]] regarded as a [[smooth manifold]] [[manifold with boundary|with boundary]]. The [[mapping space]]

\begin{displaymath}
P X \coloneqq [I, X]
\end{displaymath}
canonically exists as a [[smooth space]], but since $I$ is [[compact topological space|compact]] this structure canonically refines to that of a [[Fréchet manifold]] (see at \emph{[[manifold structure of mapping spaces]]}). This implies that there is a good notion of [[tangent space]] $T P X$. The task now is to construct a certain [[Poisson bivector]] as a [[section]] $\pi \in \Gamma^{\wedge 2}(T P X)$.

Among the [[smooth functions]] on $P X$ are the [[evaluation maps]]

\begin{displaymath}
ev \;\colon\; P X \times I = [I,X] \times I \stackrel{}{\longrightarrow} X
\end{displaymath}
whose components we denote, as usual, for $t \in I$ by

\begin{displaymath}
q(t) \coloneqq q \circ ev_t \;\colon\; P X \longrightarrow \mathbb{R}
\end{displaymath}
and

\begin{displaymath}
p(t) \coloneqq p \circ ev_t \;\colon\; P X \longrightarrow \mathbb{R}
  \,.
\end{displaymath}
Generally for $f \colon X \to \mathbb{R}$ any [[smooth function]], we write $f(t) \coloneqq f \circ ev_t \in C^\infty(P X)$. This defines an embedding

\begin{displaymath}
C^\infty(X) \times I \hookrightarrow C^\infty(P X)
  \,.
\end{displaymath}
Similarly we have

\begin{displaymath}
\dot q(t) \;\colon\; P X \longrightarrow \mathbb{R}
\end{displaymath}
and

\begin{displaymath}
\dot q(t) \;\colon\; P X \longrightarrow \mathbb{R}
\end{displaymath}
obtained by [[differentiation]] of $t \mapsto q(t)$ and $t \mapsto p(t)$.

\hypertarget{hamiltonian_evolution}{}\subsubsection*{{Hamiltonian evolution}}\label{hamiltonian_evolution}

Let now

\begin{displaymath}
H \;\colon\; X \times I \longrightarrow \mathbb{R}
\end{displaymath}
be a [[smooth function]], to be regarded as a time-dependent [[Hamiltonian]]. This induces a time-dependent function on trajectory space, which we denote by the same symbol

\begin{displaymath}
H
  \;\colon\;
  P X \times I
   \stackrel{(ev,id)}{\longrightarrow}
  X \times X
   \stackrel{H}{\longrightarrow}
  \mathbb{R}
  \,.
\end{displaymath}
Hence for $t \in I$ we write

\begin{displaymath}
H(t) 
  \;\colon\;
  P X \times \{t\}
  \stackrel{(ev, id)}{\longrightarrow}
  X \times \{t\}
  \stackrel{H}{\longrightarrow}
  \mathbb{R}
\end{displaymath}
for the function that assigns to a trajectors $(q(-),p(-)) \colon I \longrightarrow X$ its [[energy]] at (time) parameter value $t$.

Define then the [[Euler-Lagrange equation|Euler-Lagrange density]] induced by $H$ to be the functions

\begin{displaymath}
EL(t)
  \;\colon\;
  P X
  \longrightarrow 
  \mathbb{R}^2
\end{displaymath}
with components

\begin{displaymath}
EL(t)
  = 
  \left(
    \itexarray{
      \dot q(t) - \frac{\partial H}{\partial p}(t)
      \\
      \dot p(t) + \frac{\partial H}{\partial p}(t)
    }
  \right)
  \,.
\end{displaymath}
The [[trajectories]] $\gamma \colon I \to X$ on which $EL(t)$ vanishes for all $t \in I$ are equivalently those

\begin{itemize}%
\item for which the [[tangent vector]] $\dot \gamma \in T_{\gamma}X$ is a [[Hamiltonian vector field|Hamiltonian vector]] for $H$;


\item which satisfy [[Hamilton's equations]] [[equations of motion|of motion]] for $H$.



\end{itemize}
Since the [[differential equations]] $EL = 0$ have a unique solution for given initial data $(q(0), p(0))$, the evaluation map

\begin{displaymath}
\left\{
    \gamma \in P X | \forall_{t \in I}\; EL_\gamma(t) = 0 
  \right\}
  \stackrel{\gamma \mapsto \gamma(0)}{\longrightarrow}
  X
\end{displaymath}
is an [[equivalence]] (an [[isomorphism]] of [[smooth spaces]]).

\hypertarget{TheOffShellPoissonBracketOnSpaceOfPathsInSymplecticManifold}{}\subsubsection*{{The off-shell Poisson bracket}}\label{TheOffShellPoissonBracketOnSpaceOfPathsInSymplecticManifold}

Write

\begin{displaymath}
Poly(P X) \hookrightarrow C^\infty(P X)
\end{displaymath}
for the [[subalgebra]] of [[smooth functions]] on [[path space]] which are

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


\item of [[integrals]] over $I$


\item of the smooth functions in the image of $C^\infty(X) \times I \hookrightarrow C^\infty(P X)$


\item and all their [[derivatives]] along $I$.



\end{itemize}
Define a [[bilinear function]]

\begin{displaymath}
\{-,-\}
  \;\colon\;
  Poly(P X) \otimes Poly(P X)
  \longrightarrow Poly(P X)
\end{displaymath}
as the unique function which is a [[derivation]] in both arguments and moreover is a solution to the [[differential equations]]

\begin{displaymath}
\frac{\partial}{\partial t_2}
  \left\{f(t_1), q(t_2)\right\}
  =
  \left\{
    f(t_1), \frac{\partial H}{\partial p}(t_2)
  \right\}
\end{displaymath}
\begin{displaymath}
\frac{\partial}{\partial t_2}
  \left\{f(t_1), p(t_2)\right\}
  =
  -
  \left\{
    f(t_1), \frac{\partial H}{\partial q}(t_2)
  \right\}
\end{displaymath}
subject to the initial conditions

\begin{displaymath}
\{f(t), q(t)\} = \{f,q\}
\end{displaymath}
\begin{displaymath}
\{f(t), p(t)\} = \{f,p\}
\end{displaymath}
for all $t \in I$, where on the right we have the original [[Poisson bracket]] on $X$.

This bracket directly inherits skew-symmetry and the [[Jacobi identity]] from the [[Poisson bracket]] of $(X, \omega)$, hence equips the [[vector space]] $Poly(P X)$ with the structure of a [[Lie bracket]]. Since it is by construction also a [[derivation]] of $Poly(P X)$ as an [[associative algebra]], we have that

\begin{displaymath}
\left(
    Poly\left(P X\right),
    \;
    \left\{
      -,-
    \right\}
  \right)
  \;\;\;
  \in P_1 Alg
\end{displaymath}
is a [[Poisson algebra]]. This is the ``off-shell Poisson algebra'' on the space of [[trajectories]] in $(X,\omega)$.

\hypertarget{the_symplectic_leaves}{}\subsubsection*{{The symplectic leaves}}\label{the_symplectic_leaves}

Observe that by construction of the off-shell Poisson bracket, specifically by the [[differential equations]] defining it, the [[Euler-Lagrange equation|Euler-Lagrange function]] $EL$ generate a [[Poisson reduction|Poisson ideal]].

For instance

\begin{displaymath}
\left(
  \itexarray{
  \frac{\partial}{\partial t_2}
  \left\{f(t_1), q(t_2)\right\}
  &=&
  \left\{
    f(t_1), \frac{\partial H}{\partial p}(t_2)
  \right\}
  \\
  \frac{\partial}{\partial t_2}
  \left\{f(t_1), p(t_2)\right\}
  &=&
  -
  \left\{
    f(t_1), \frac{\partial H}{\partial q}(t_2)
  \right\}
  }
  \right)
  \;\;\;
  \Leftrightarrow
  \;\;\;
  \left(
    \left\{
      f(t_1), \;
      EL(t)
    \right\}
    =  
    0
  \right)
  \,.
\end{displaymath}
Moreover, since $\{EL(t) = 0\}$ are [[equations of motion]] the [[Poisson reduction]] defined by this Poisson idea is the subspace of those [[trajectories]] which are solutions of [[Hamilton's equations]], hence the ``on-shell trajectories''.

As remarked above, the initial value map canonically identifies this on-shell trajectory space with the original [[phase space]] manifold $X$. Moreover, by the very construction of the off-shell Poisson bracket as being the original Poisson bracket at equal times, hence in particular at time $t = 0$, it follows that restricted to the [[zero locus]] $EL = 0$ the off-shell Poisson bracket becomes [[symplectic manifold|symplectic]].

All this clearly remains true with the function $EL$ replaced by the function $EL - J$, for $J \in C^\infty(I)$ any function of the (time) parameter (since $\{J,-\} = 0$). Any such choice of $J$ hence defines a symplectic subspace

\begin{displaymath}
\left\{
    \gamma \in P X
    \;|\;
    \forall_{t \in I}\; EL_\gamma(t) = J
  \right\}
\end{displaymath}
of the off-shell Poisson structure on trajectory space. Hence $\left(O X, \left\{-,-\right\}\right)$ has a [[foliation]] by [[symplectic leaves]] with the [[leaf space]] being the [[smooth space]] $C^\infty(I)$ of [[smooth functions]] on the interval.

Notice that changing $EL \mapsto EL - J$ corresponds changing the time-dependent [[Hamiltonian]] $H$ as

\begin{displaymath}
H \mapsto H - J q
  \,.
\end{displaymath}
Such a term linear in the [[canonical coordinates]] (the [[field (physics)|fields]]) is a \emph{[[source]] term}. (The [[action functionals]] with such [[source]] terms added serve as [[integrands]] of [[generating functions]] for [[correlators]] in [[statistical mechanics]] and in [[quantum mechanics]].)

\hypertarget{BoundaryFieldTheoryInterpretation}{}\subsubsection*{{Boundary field theory interpretation}}\label{BoundaryFieldTheoryInterpretation}

Hence in conclusion we find the following statement:

The [[trajectory]] space (history space) of a [[mechanical system]] carries a natural [[Poisson manifold|Poisson structure]] whose [[symplectic leaves]] are the subspaces of those trajectories which satisfy the [[equations of motion]] with a fixed [[source]] term and hence whose symplectic [[leaf space]] is the space of possible sources.

Notice what becomes of this statement as we consider the the [[2d Chern-Simons theory]] induced by the off-shell Poisson bracket (the [[non-perturbative field theory|non-perturbative]] [[Poisson sigma-model]]) whose [[moduli stack]] of [[field (physics)|fields]] is the [[symplectic groupoid]] $SG\left(P X, \left\{-,-\right\}\right)$ induced by the Poisson structure.

By the discussion at [[motivic quantization]] in the section \emph{\href{motivic%20quantization#PoissonManifoldAtTheBoundaryOf2dChernSimonsTheory}{The Poisson manifold at the boundary of the 2d Chern-Simons theory}}, the Poisson space $\left(P X, \left\{-,-\right\}\right)$ defines a [[boundary field theory]] (in the sense of [[local prequantum field theory]]) for this [[2d Chern-Simons theory]], exhibited by a boundary [[correspondence]] of the form

\begin{displaymath}
\itexarray{
    && P X
    \\
    & \swarrow && \searrow
    \\
    \ast && \swArrow_{\xi} && SG\left(P X, \left\{-,-\right\}\right)
    \\
    & \searrow && \swarrow_{\mathrlap{\chi}}
    \\
    && \mathbf{B}^2 U(1)
    \\
    && \downarrow^{\mathrlap{\rho}}
    \\
    && KU Mod
  }
  \,.
\end{displaymath}
Notice that the [[symplectic groupoid]] is a version of the [[symplectic leaf|symplectic]] [[leaf space]] of the given [[Poisson manifold]] (its [[0-truncation]] is exactly the leaf space). Hence in the case of the off-shell Poisson bracket, the [[symplectic groupoid]] is the space of \emph{[[source field|sources]]} of a mechanical system. At the same time it is the [[moduli space]] of [[field (physics)|fields]] of the [[2d Chern-Simons theory]] of which the mechanical system is the [[boundary field theory]]. Hence the [[field (physics)|fields]] of the [[bulk field theory]] are identified with the [[sources]] of the [[boundary field theory]]. Hence conceptually the above boundary correspondence diagram is of the following form

\begin{displaymath}
\itexarray{
    && Fields
    \\
    & \swarrow && \searrow
    \\
    \ast && \swArrow_{} && Sources
    \\
    & \searrow && \swarrow_{\mathrlap{}}
    \\
    && Phases
  }
  \,.
\end{displaymath}
Such a relation

\begin{tabular}{l|l}
[[bulk field theory]]&[[boundary field theory]]\\
\hline 
[[field (physics)&field]]\\
\end{tabular}

between bulk fields and boundary sources is the characteristic feature of what is called the \emph{[[holographic principle]]} in its realization as the [[AdS-CFT correspondence]].

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

The off-shell extension of the [[Peierls bracket]] is observed in section II of

\begin{itemize}%
\item [[Don Marolf]], \emph{Poisson Brackets on the Space of Histories} Annals of Physics Volume 236, Issue 2, December 1994, Pages 374-391 (\href{http://arxiv.org/abs/hep-th/9308141}{arXiv:hep-th/9308141})

\end{itemize}
The observation that the off-shell bracket has a symplectic foliation by the level sets of the Euler-Lagrange functions appears on the top of p. 4 of

\begin{itemize}%
\item [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Pedro Ribeiro]], \emph{Algebraic Structure of Classical Field Theory I: Kinematics and Linearized Dynamics for Real Scalar Fields} (\href{http://arxiv.org/abs/1209.2148}{arXiv:1209.2148})

\end{itemize}
All this and the interpretation of the resulting symplectic foliation as a foliation by source terms has been highlighted by

\begin{itemize}%
\item [[Igor Khavkine]], personal communication

\end{itemize}
[[!redirects off-shell Poisson brackets]]



\end{document}