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)

• 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. – ad2004 Nov 25 '20 at 17:52

• 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 ... – Coltrane8 Nov 25 '20 at 23:13
• @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.. – Richard Myers Nov 26 '20 at 2:27
• @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. – Richard Myers Nov 26 '20 at 2:31