Symplectic form on covariant phase space

Question

Usually, the phase space of a physical system is defined as the cotangent bundle of the configuration space at some fixed time slice $t = t_0$, conveniently co-ordinatized by $\{q^a, p_a\}$ where $$ p_a = \frac{\partial \mathcal{L}}{\partial \dot{q}^a}. $$

This $p$ co-ordinates (called canonical momenta) are convenient, because the symplectic structure on the phase space is of very simple form: $$ \left\{ q^a, p_b \right\} = \delta^a_b, \\ \{q,q\}=\{p,p\}=0. $$

A moment of reflection will convince you that the phase space is nothing more than the space of solutions of the equations of motion, together with a suitable topology making it into a differential manifold.

This second definition seems far more natural and far-reaching than the first one: it makes sense even in exotic cases, e.g. with degenerate Hessian, with discrete time, with non-determinism of the equations of motion, etc. Also, this definition does not single out a specific value of the time parameter $t = t_0$, thus making independence of $t_0$ in the canonical formalism manifest. In fact I would go further and say that this second definition does not make any assumptions about existence of time at all!

I would like to understand how to define the symplectic structure (the Poisson bracket) for this second definition of the phase space, and to which extend it is possible.

I expect this structure to be generated by the action functional $$ S(t_i, t_f) = \intop_{t_i}^{t_f} dt \mathcal{L} $$ taken as a function of phase space (i.e. a function on the space of solutions to the equations of motion, parametrized by $t_i$ and $t_f$).

However, I don't know how to write the general definition of the Poisson bracket between two functions of the phase space, defined by the action functional.

Answer 1

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.

Answer 2

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 $$\{H_\xi,H_\zeta\}=\Omega (\delta_\zeta\phi,\delta\xi\phi)$$

Answer 3

Ok, I did some digging around and here's what I found (based on Qmechanic's answer, but a little more general).

Define the off-shell phase space to be simply the space of all field configuration, not necessarily satisfying the equations of motion. Off-shell classical observables are, by analogy, functions over the off-shell phase space. We define the Peierls bracket between two such functionals to be

$$ \left\{ F[x], G[x] \right\} = \int dt' \int dt \, \frac{\delta F}{\delta x(t')} G_F(t', t) \frac{\delta G}{\delta x(t)}, $$

where $G_F(t', t)$ is the Feynman propagator (retarded minus advanced). The crucial point that I didn't understand before is that $G_F$ is actually a functional, which depends on $x(t)$. And it doesn't describe propagation of the entire field $x$, just a linear propagation of its infinitesimal fluctuation. Thus, the highly nontrivial dependence on $x$ is encoded in the Peierls bracket.

I still don't understand one thing though. The resulting structure of off-shell phase space with the Peierls bracket is not equivalent to the usual phase space (and is in fact infinitely larger). How do I pull-back this algebra on the phase space?