Abstract
We work on a spacetime manifold foliated by timelike leaves. In this setting, we explore the solution of the second-class constraints arising during the canonical analysis of the Holst action with a cosmological constant. The solution is given in a manifestly Lorentz-covariant fashion, and the resulting canonical formulation is expressed using several sets of real variables that are related to one another by canonical transformations. By applying a gauge fixing to this formulation, we obtain a description of gravity as an $SU(1,1)$ gauge theory that resembles the Ashtekar-Barbero formulation.
Highlights
The quest for a consistent quantum theory of gravity is perhaps the greatest endeavor of modern theoretical physics
We explore the solution of the second-class constraints arising during the canonical analysis of the Holst action with a cosmological constant
The loop approach is based upon the Ashtekar-Barbero variables [6], which arise from the canonical analysis–in the time gauge—of the Holst action for general relativity [7]
Summary
The quest for a consistent quantum theory of gravity is perhaps the greatest endeavor of modern theoretical physics. The loop approach is based upon the Ashtekar-Barbero variables [6], which arise from the canonical analysis–in the time gauge—of the Holst action for general relativity [7]. For the sake of completeness, in the Appendix we report the canonical formulation that emerges when the second-class constraints are solved in a nonexplicitly Lorentz-covariant fashion (but preserving full Lorentz invariance), and in Sec. V we adapt the space gauge to this framework and show that the canonical formulation of Sec. IV follows.
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