Reality conditions and Ashtekar variables: A different perspective

Abstract

We give in this paper a modified self-dual action that leads to the SO(3)-ADM formalism without having to face the difficult second class constraints present in other approaches (for example, if one starts from the Hilbert-Palatini action). We use the new action principle to gain some new insights into the problem of the reality conditions that must be imposed in order to get real formulations from complex general relativity. We derive also a real formulation for Lorentzian general relativity in the Ashtekar phase space by using the modified action presented in the paper.

Emergent Photons and Gravitons: The Problem of Vacuum Structure

“…The coefficient of the Holst term in the action is inversely proportional to what is known as [13] the Barbero-Immirzi parameter γ. In the first order formalism, especially as elucidated by the loop-gravity community, this Holst term creates mixing between torsion degrees of freedom and ordinary metric degrees of freedom, but in such a way that the Einstein equations for ordinary macroscopic applications remain unaffected.…”

Loop quantum gravity and light propagation

“…Some proposals to come to terms with difficulty (ii) were: to consider real connection variables [28], to implement a Wick transform [29] and to define tractable reality constraints [30]. All of these left open (i).…”

“…Now we are ready to calculate the different contributions to the magnetic sector of the Hamiltonian (22), which we parameterize in terms of the tensorW a 1 ...am rr 1 ...rn introduced in (28).…”

Effective gauge group of pure loop quantum gravity is SO(3): New estimate of the Immirzi parameter

“…Within this context, the gauge group is made up of SO(3) spatial rotations. In a later development, Barbero [18] suggested that the real connection A ia ≡ γk ia + Γ ia can be used instead to arrive at a real phase space formulation of General Relativity if γ, the Immirzi parameter, is real.…”