Poisson Bracket in Relativistic Field Theory In QFT when we wrote a Lagrangian for a classical field, we switched to Hamiltonian formulation and introduced Poisson Bracket as $$\{A(x,t),B(y,t)\}^{(3)} = \int_{\Sigma_t}d^3z \left( \frac{\delta A(x)}{\delta \phi(z)}\frac{\delta B(y)}{\delta \pi(z)} -  \frac{\delta A(x)}{\delta \pi(z)}\frac{\delta B(y)}{\delta \phi(z)}\right) . $$
which I think generalizes to (but improperly)
$$ \{A(x',t),B(y',t)\}^{\Sigma_t} = \int_{\Sigma_t}dz |h|^{1/2} \left( \frac{\delta A(x')}{\delta \phi(z)}\frac{\delta B(y')}{\delta \pi(z)} -  \frac{\delta A(x')}{\delta \pi(z)}\frac{\delta B(y')}{\delta \phi(z)}\right) . $$
Where we implicitly chose a preferred spacetime slicing . Will The Poisson Brackets depend on the way I choose to slice the whole manifold? If so, is there a general covariant way to espress Poisson Bracket with explicit dependence on the choise of slicing (in the context for exemple of ADM 3+1 formalism in G.R.? (My goal is to understand how the hamiltonian formalism works and how to recover Lorentz invariance in Special Relativity and General Covariance in G.R. in this formalism)
 A: The technically correct answer to this question relies on a significant quantity of differential geometry. Essentially, the Poisson bracket can be understood as the Poisson bivector field associated to the symplectic form defined by the system. So your question about slice-independence is equivalent to the demand that the integral of the symplectic form over a Cauchy slice should be independent of which slice we choose. This is indeed the case and follows easily when phrased in the right way.
More generally, there is a covariant phase space formalism, known as the covariant phase space. To the best of my knowledge this was developed sometime around the 80s by Witten, Lee, Wald, and others whose names escape me at the moment. An old paper Crnkovic and Witten lays this approach out nicely, but some points are not described in detail there. A very nice modern paper on the topic which contains a essentially all details (so long as you're already familiar with aspects of symplectic geometry) is this paper by Harlow and Wu. In particular, both present the symplectic form (from which the bracket is in principle deducible) for GR.
A relatively approachable introduction to some aspects of symplectic geometry can be found in the final chapter of the QFT book by V. Parameswaran Nair. The third chapter of the same book contains most of the same ideas for field theories (as opposed to point particles), but you may need to read between the lines in some places to rephrase some things in a (field space) coordinate independent language.
