On constraints of pure-connection formulation of general relativity for nonzero cosmological constant

Abstract

The authors show that the recently proposed pure-connection action for general relativity with a cosmological constant by Capovilla, Dell and Jacobson (1989) correctly yields the usual constraints of general relativity as given by Ashtekar (1986). They also point out alternative possibilities for the Capovilla-Dell-Jacobson action.

Gauge connection formulations for general relativity

“…For the treatment of the Λ = 0 case, it is natural to try to apply the same strategy. However, this time the field ρ cannot be solved from (5), which is easy to see by setting Λ = 0 in (17). Hence (2), like the CDJ action [3], fails to reduce to pure connection action principles in this case.…”

Alternative derivation of Krasnov’s action for general relativity

“…In this section we perform the canonical analysis of the action principle (11), which sets up the first step towards a canonical quantization of this formulation. For such a purpose, we perform the 3+1 decomposition of the action (11). We foliate the spacetime by 3-manifolds Ω t at a constant global time function t, so that the spacetime has the topology R × Ω, where Ω is a spacial compact 3-manifold without a boundary.…”

“…Although Ashtekar variables were originally obtained through a canonical transformation performed on the ADM variables, they naturally arise from the Hamiltonian analysis of the Plebanski action [10]. The Hamiltonian formulation of the CDJ action with Λ = 0 also leads to a phase space described by the variables and constraints of the Ashtekar formalism (the same holds for the intermediate step towards it in the case of Λ = 0 [11]), which shows again the equivalence of the CDJ formulation with general relativity, and even the generalizations proposed in [8] have the same number of physical degrees (DOF) of freedom of gravity when they are written in terms of such variables.…”

Towards a path integral for pure spin connection formulation of gravity

“…This action is obtained by solving the classical equation of motion for the 'metric-variable' Σ, from the self-dual two form action for a SL(2, C) connection A, a non-dynamical matrix field ψ and two-forms Σ [3]. The equivalence of the pure-connection theory to that of Ashtekar can be shown by a 3 + 1 decomposition [4], as well as by comparing the constraints arising due to diffeomorphism and gauge invariances of the theory in the two formulations [5]. In an interesting alternative approach, Peldán [6] performed inverse Legendre transform on the Hamiltonian comprising purely of constraints (characteristic of diffeomorphism invariant theories) and obtained a pure-spin connection action.…”

“…(trφ) 2 − trφ 2 = 0 (8) which follows from the characteristic equation satisfied by a non-degenerate 3 × 3 matrix. We use this equivalent form also because it is this form which leads to the agreement between the actions of Capovilla et al and Peldán [5,7]. The delta function imposing this constraint can be promoted to the action by using its functional representation via the introduction of a (complex) auxiliary field µ [9], as…”