Reality Conditions and Ashtekar Variables: a Different Perspective

TL;DR

The paper proposes a modified self-dual action that yields the $SO(3)$-ADM formalism without the problematic second-class constraints and uses this to study reality conditions and real Lorentzian formulations in the Ashtekar framework. By adding a total-derivative term, it derives an $SO(3)$-ADM Hamiltonian with a canonical pair $( ilde{eta}^a_i,e_{ai})$, clarifying how to connect to Ashtekar variables. Reality conditions are shown to be transformable into the choice of Hamiltonian constraint form, enabling real Lorentzian or Euclidean realizations within a complex theory. A real Lorentzian Ashtekar formulation is explicitly constructed, with a more intricate scalar constraint that preserves the Gauss and vector constraints and remains amenable to loop-variable methods. These results highlight how signature and reality conditions interact at the Lagrangian and Hamiltonian levels and open avenues for further quantum-gravity investigations using real Ashtekar variables.

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.