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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.

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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.

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  • $\begingroup$ But Peierls bracket assumes existence and uniqueness of the retarded Green's function... Does it still work for highly non-linear systems? Also, it seems to depend on the external time variable, which I was hoping could be avoided. $\endgroup$ – Solenodon Paradoxus Jan 28 '18 at 9:10
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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?

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