Hamiltonian analysis of general relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach

Abstract

We show how to systematically apply the Faddeev-Jackiw symplectic method to General Relativity (GR) and to GR extensions. This provides a new coherent frame for Hamiltonian analyses of gravitational theories. The emphasis is on the classical dynamics, uncovering the constraints, the gauge transformations and the number of degrees of freedom; but the method results are also relevant for canonical quantization approaches. We illustrate the method with three applications: GR and to two Brans-Dicke cases (the standard case ω = −3/2 and the case with one less degree of freedom, ω = −3/2). We clarify subtleties of the symplectic approach and comment on previous symplectic-based Hamiltonian analyses of extended theories of gravity, pointing out that the present approach is systematic, complete and robust.

Cosmological framework for renormalization group extended gravity at the action level

“…Numerical analysis on the CMB power spectrum constitute a relevant piece of information for addressing this issue, which is a work in progress. Further developments on the theoretical side, as a Hamiltonian formulation (e.g., [70]), are also being considered.…”

Symplectic analysis for the Holst action with Dirac fields

“…Essentially, there are two degrees of freedom relative to the gravitational theory classically equivalent to General Relativity [21] and four more degrees of freedom associated with the fermions [22].…”

“…Once the Lagrangians of all iterations must be equivalent. If the number of coordinates in the last, say k-th, iteration is N (k) , and if M unsolved constraints were found, then the number of degrees of freedom must be [21] NDF = 1 2…”

On constrained analysis and diffeomorphism invariance of generalised Proca theories

“…In fact, it can already be seen from the Faddeev-Jackiw iterative process, without computing Poisson's bracket, that there exists first-class constraints for Maxwell theory but there is no first-class constraint in Proca theory. The criteria is provided by [30], which is summarised as follows. If all of the zero modes in the Faddeev-Jackiw process give rise to independent constraints, then there is no first-class constraint.…”

“…However, if there are zero modes which do not give rise to new constraints, then there are first-class constraints. In fact, [30] also provides the way of counting number of degrees of freedom from the number of zero modes and of constraints. However, we will not discuss this way of counting in this paper.…”