Constrained analysis of topologically massive gravity

Abstract

We quantize the Einstein gravity in the formalism of weak gravitational fields by using the constrained Hamiltonian method. Special emphasis is given to the 2+1 spacetime dimensional case where a (topological) Chern-Simons term is added to the Lagrangian.

New canonical analysis for higher order topologically massive gravity

“…as expected [14,15,22]. From the second class constraints we can construct the Dirac brackets between the canonical variables.…”

“…We have observed in previous sections, that the GLT analysis for the higher-order derivative Chern-Simons theory allowed us to know the complete structure of the constraints. Hence, in this section, we shall perform the GLT analysis for TMG theory and we shall report the complete structure of the constraints; our study will complete those results reported in [22]. Now, the action under study is given by…”

“…we can observe that the constraints are identified by means of the rank and the null vectors; it is not necessary to fixing them by hand such as in Ostrogradski formalism is done [22]. For instance, one null vector is given by w = (0, 0, ∂ i ω, 0, η i j ω, 0, 0, 0), where ω is an arbitrary function.…”

“…where R is the Ricci tensor, g is the determinant of the metric tensor and μ is a coupling constant [15]. If we consider the week gravitational field formalism, the action ( 53) is reduced to the well known Lagrangian for TMG which it is given by [22]…”

“…In this manner, with all commented above, the purpose of the present work is to report a detailed GLT study of TMG theory. Our analysis will follow a different procedure to that presented in [22]. We will develop a detailed GLT framework 0123456789().…”