Hamiltonian dynamics of gauge theories of gravity

Abstract

We investigate the Hamiltonian structure of gauge theories of gravity based on arbitrary gravitational and matter field Lagrangians. The gauge group of the theory in question is the semisimple product of the local Lorentz group and the group of diffeomorphisms of spacetime (the local Poincar\'e group). We derive formulas for the symplectic two-form $\ensuremath{\Omega}$, the translational $E$, and the rotational $J$ generators. The Hamilton equations expressed in terms of $\ensuremath{\Omega}$, $E$, and $J$ are equivalent to the variational Euler-Lagrange equations. The ten constraint equations of the theory are closely related both to properties of the symplectic two-form $\ensuremath{\Omega}$ and to an action of the gauge group in the space of solutions. The dynamical generators $E$ and $J$ can be expressed by the left-hand sides of the constraint equations, that is, the constraints generate the dynamics by means of the Hamilton equations for the functions $E$ and $J$. On the other hand, the action of the gauge group in the set of initial data determines their "time" evolution. We show that this evolution is in a one-to-one correspondence with that generated by the Hamilton equations.