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)

  • $\begingroup$ Not sure if this will address your question, but you may find some general insights watching P. Dirac's lecture as he explains how he introduced the Poisson brackets into QM and the thinking around it. The relevant YouTube portion is here: youtu.be/ma7TSAq87lg?t=1085 (it starts around 18:00). I hope this helps. $\endgroup$ – ad2004 Nov 25 '20 at 17:52

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.

  • $\begingroup$ I took a look at Harlow & Wu paper, and seems nice but overkill for the moment at first glance, altough it could be useful. On the other hand the QFT book you suggest is really detailed and I appreciate it as it will be useful for other topics also. In the paper I only saw a quick proof of the independence on the choice of slice of integration, given one particular foliation. In the book, pag. +- 19, they introduce Poisson brackets. At the level of the book, my doubt is at the level of prooving the independece on the foliation of the symplectic structure eq. (3.9), as for instance $\pi$ is ... $\endgroup$ – Coltrane8 Nov 25 '20 at 23:13
  • $\begingroup$ ... a component of a four vector. So I'm stuck in the explicit proof in charts. (I think it would suffice to show the independence of the canonical one form, but I don't know how to connect to the independ. of the sympl. struct.). Anyway I would be settled just if I could show explicitly the independence of the Poisson Bracket within a choise of foliation in the Minkowski spacetime. $\endgroup$ – Coltrane8 Nov 25 '20 at 23:17
  • $\begingroup$ @Coltrane8 Observe that $\delta L=EL\delta\phi +d\theta$ where $\theta$ is the symplectic current. The symplectic current is defined by $\omega=\delta\theta$, the symplectic form begin defined as $\Omega=\int_\Sigma\omega$ for some Cauchy slice $\Sigma$. Note that by $\delta^2=0$, it follows that $\delta^2 L=0$ and hence $\delta EL\wedge\delta\phi+d\omega=0$. Evaluating on-shell includes the demand $\delta EL=0$, and hence $d\omega=0$ on-shell. This implies that $\Omega$ is independent of the bulk portion of $\Sigma$, but this does not show that $\Omega$ is independent of the spatial bounary.. $\endgroup$ – Richard Myers Nov 26 '20 at 2:27
  • $\begingroup$ @Coltrane8 of $\Sigma$. For this we require the arguments of Harlow and Wu, which imply that the pullback of $\omega$ to the spatial boundary vanishes when properly defined. Together these two facts imply that $\Omega$ is independent of the Cauchy slice chosen. Of course, what I have just written is the special case with boundary Lagrangian $\ell$ and boundary term $C$ set to zero. It is straightforward to generalize these arguments to include them. $\endgroup$ – Richard Myers Nov 26 '20 at 2:31

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