# Symplectic form on covariant phase space

The sought-for covariant Poisson bracket for Lagrangian theories is known as the Peierls bracket $$\{ F,G \}~:=~\iint_{[t_i,t_f]^2}\!dt~dt^{\prime}~\sum_{I,K=1}^{2n} \frac{\delta F }{\delta z^I(t)}~G^{IK}_{\rm ret}(t,t^{\prime})~\frac{\delta G }{\delta z^K(t^{\prime})} - (F\leftrightarrow G),$$ where $G^{IK}_{\rm ret}(t,t^{\prime})$ is the retarded Green's function, see e.g. various textbooks by Bryce S. DeWitt, and this & this Phys.SE answers by user Urs Schreiber.

This is exactly the essence of the "Covariant phase space" approach, developed by Ashtekar, Wald, Witten, etc. See e.g. Lee & Wald (1990). It can be applied to particle or field theories and in particular gauge theories. The construction is briefly as follows:

1- Consider the *solution space* $\cal S$ defined as the collection of solutions to your given theory with Lagrangian $\cal L[\phi]$ and constrained possibly by some boundary conditions.

2- The variation of $\cal L$ gives the equations of motion and a total derivative term $$\delta \cal L=E[\phi]\delta \phi+\partial_\mu \theta^\mu(\delta \phi)$$

3- A *tangent vector* at a point $\phi\in \cal S$ is represented by a field perturbation $\delta \phi$ which solves the linearized field equations. One can also define a differential form $d_V\phi$ as the exterior derivative on $\cal S$.

4- To build the *canonical structure* on $\cal S$, take an arbitrary Cauchy surface $\Sigma$ in the spacetime and define the *(pre)symplectic* form as follows
$$\Omega(\delta_1 \phi,\delta_2 \phi)=\int_\Sigma d\Sigma \,n_\mu \delta_1 \theta^\mu(\delta_2 \phi)- (1\leftrightarrow 2)$$
where $n$ is the normal to the hypersurface and $d\Sigma$ denotes the volume form. Equivalently one can write
$$\Omega=\int_\Sigma d_V \theta$$
which is a 2 form with respect to the exterior derivative $d_V$ on $\cal S$. The above 2 form have degenecaries in case of gauge theories. In that case one should quotient $\cal S$ by the group $\cal G_0$ of pure gauge transformations, i.e. all gauge transformations that act locally in the bulk. Large gauge transformations (those acting nontrivially at the boundary survive and comprise the symmetries of the phase space).

5- The formalism is covariant since no explicit decomposition in fields are required and the choice of $\Sigma$ is arbitrary. The independence of $\Omega$ from $\Sigma$ is a result of the fact that $\partial_\mu \omega^\mu=0$ using the equations of motion. The pair $(\cal S, \Omega)$ is called the *covariant phase space*.

6- The analysis of (asymptotic) symmetries and conservation laws are very straightforward in this formalism. To build the generator of a symmetry transformation $\delta_\xi \phi$, simply take $$\delta H_\xi\equiv \Omega (\delta \phi,\delta_\xi\phi).$$ The charge $H_\xi$ exists if $\delta H_\xi$ is integrable, which is equivalent to the condition $\cal{L}_{\delta_\xi}\Omega=0$, i.e. that $\delta_\xi\phi$ is a symplectic symmetry (canonical transformation). The poisson bracket between two charges is $$\{H_\xi,H_\zeta\}=\Omega (\delta_\zeta\phi,\delta\xi\phi)$$