Dirac and Lagrangian reductions in the canonical approach to the first-order form of the Einstein–Hilbert action

Abstract

It is shown that the Lagrangian reduction, in which solutions of equations of motion that do not involve time derivatives are used to eliminate variables, leads to results quite different from the standard Dirac treatment of the first order form of the Einstein-Hilbert action when the equations of motion correspond to the first class constraints. A form of the first order formulation of the Einstein-Hilbert action which is more suitable for the Dirac approach to constrained systems is presented. The Dirac and reduced approaches are compared and contrasted. This general discussion is illustrated by a simple model in which all constraints and the gauge transformations which correspond to first class constraints are completely worked out using both methods in order to demonstrate explicitly their differences. These results show an inconsistency in the previous treatment of the first order Einstein-Hilbert action which is likely responsible for problems with its canonical quantization.

Darboux Coordinates for the Hamiltonian of First Order Einstein-Cartan Gravity

“…This suggests (because the Hamiltonian and Lagrangian formalisms must lead to the same result, of course, if the reductions are performed correctly [27]) that such a reduced …”

“…When D = 3 (and only when D = 3) equation (29) can be solved for ω m(pq) and in equation (26) terms proportional to the connections ω m(pq) (with all "space" indices) cancel out, leading to separation of these two equations, (26) and (27), into equations containing only ω m(pq) and ω k(p0) , respectively. In addition, when D = 3, some terms in the Lagrangian disappear (see [1]).…”

Algebraic analysis of a model of two-dimensional gravity

“…In D = 2 the field equations cannot be solved for Γ λ µν in terms of g µν [5][6][7], which is why equation (2) does not provide an equivalent first-order formulation of EH action in 2D. For the Hamiltonian treatment of real two-dimensional gravity in second-order form see [8,9].…”

“…First of all, the action (2) is indeed equivalent to the original second-order EH action (1) in dimensions D > 2 [5]. Second, the structure of constraints in the 2DG model is much closer to the higher dimensional first-order gravity (2) (see [5,[10][11][12]) than the structure of constraints of the real 2D gravity (see [8,9]).…”

“…Second, the structure of constraints in the 2DG model is much closer to the higher dimensional first-order gravity (2) (see [5,[10][11][12]) than the structure of constraints of the real 2D gravity (see [8,9]). As it was shown in [8], the Hamiltonian formulation of the second-order EH action (or real 2D gravity) in two dimensions leads to three primary first class constraints which generate the gauge transformations consistent with zero degrees of freedom and triviality of the Einstein equations in 2D.…”