# Poisson brackets among derivatives in the constraints of the Hamiltonian formulation of General Relativity

The issue I have is from the Hamiltonian formulation of General Relativity (GR) and concerns calculation of certain Poisson brackets. In this formulation, we have the 3-dimensional metric $h_{ij}(x)$ as configuration variable, and $p^{ij}(x)$ as its conjugate momentum, building a phase space of GR. They satisfy the following equal time Poisson brackets:

$$\left\lbrace h_{ij}(x),p^{ab}(y)\right\rbrace=\delta_{i}^{(a}\delta_{j}^{b)}\delta^{(3)}(x-y)$$

where $\delta^{(3)}(x-y)$ is a 3D Dirac delta function, and "( )" is symmetrization of indices. I am trying to calculate Poisson brackets between Hamiltonian and momentum constraints, but I cannot obtain the result. In all texts I found on this matter, there is no intermediate steps leading to the result that they state. In particular, I am unsure how to deal with derivatives in the Poisson brackets, especially if there are covariant derivatives involved. Also what makes me feel uncomfortable is that it seems some derivatives of delta function appear, which I do not understand? Specifically, these three Poisson brackets are troubling me:

$$\left\lbrace D_{i}{p^{i}}_{j}(x),D_{a}{p^{a}}_{b}(y)\right\rbrace$$

$$\left\lbrace h_{ij}(x),D_{a}{p^{a}}_{b}(y)\right\rbrace$$

$$\left\lbrace p^{ij}(x),D_{a}{p^{a}}_{b}(y)\right\rbrace$$

where $D_{i}$ is a covariant derivative with respect to $h_{ij}$. How come the first one is not zero? How does one deal with PB among derivatives of phase space variables?

• What happens if you brute force calculate them? – gented Jul 26 '16 at 18:59

You usually get derivatives of delta functions. I like to write $$D^y_a$$ to remind myself that the derivative is with respect to $$y$$. So $$\{g_{ab}(x),D^y_e\pi^{cd}(y)\} =D^y_e\{g_{ab}(x),\pi^{cd}(y)\}=D^y\delta(x,y)\delta^{(c}_a\delta^{d)}_b$$.
To calculate the other Poisson brackets, just remember to include the metric if you've lowered an index in $$\pi^{ab}$$.