Ashtekar Formulation with Temporal Foliations

We heuristically discuss the Ashtekar type canonical analysis of gravity using temporal foliations instead of spacial foliations and analyze its implications on classical and quantum gravity in general. The constraints and the constraint algebra are essentially the same as before. The gauge group of the real gravity theory is now SO(2,1) instead of SU(2) on the phase of canonical gravity restricted to real triads. We briefly discuss the classical and quantum aspects of the theory. We propose a generalisation of the Ashtekar formalism in which the type of the foliation is included in the initial conditions. The gauge group and the phase space of Ashtekar's canonical (real) gravity is clarified. The area spectrum is unchanged if we use only the finite dimensional representations of SL(2,C) to construct the quantum Hilbert space of the theory. If we use the infinite dimensional representations of SL(2,C) to construct the Hilbert space of Loop quantum gravity, we suggest the possibility that the Ashtekar formulation using spacial foliations is related to the Lorentzian Barrett-Crane model based on SL(2,C)/SU(2) and the Ashtekar formulation with temporal foliations is related to the Rovelli-Perez model based on SL(2,C)/SU(1,1)Z_2. This work has implications for loop quantum gravity