Ashtekar-Kodama Gravity as a Classical and Quantum Extension of Loop Quantum Gravity

Abstract

This paper presents a new theory of gravity, called here Ashtekar-Kodama (AK) gravity, which is based on the Ashtekar-Kodama formulation of loop quantum gravity (LQG), yields in the limit the Einstein equations, and in the quantum regime a full renormalizable quantum gauge field theory. The three fundamental constraints (hamiltonian, gaussian and diffeomorphism) were formulated in 3-dimensional spatial form within LQG in Ashtekar formulation using the notion of the Kodama state with positive cosmological constant &#923;<i>.</i><i> </i>We introduce a 4-dimensional covariant version of the 3-dimensional (spatial) hamiltonian, gaussian and diffeomorphism constraints of LQG. We obtain 32 partial differential equations for the 16 variables <i>E</i><i>_mn</i> (E-tensor, inverse densitized tetrad of the metric) and 16 variables <i>A</i><i>_mn</i> (A-tensor, gravitational wave tensor). We impose the boundary condition: for large distance the E-generated metric <i>g</i>(<i>E</i>) becomes the GR-metric <i>g</i> (normally Schwarzschild-spacetime). The theory based on these Ashtekar-Kodama (AK) equations, and called in the following Ashtekar-Kodama (AK-) gravity has the following properties. &#8226; For &#923;<i> </i>=<i> </i>0 the AK equations become Einstein equations, A-tensor is trivial (constant), and the E-generated metric <i>g</i>(<i>E</i>)<i> </i>is identical with the GR-metric <i>g</i>. &#8226; When the AK-equations are developed into a &#923;<i>-</i>power series, the &#923;<i>-</i>term yields a gravitational wave equation, which has only at least quadrupole wave solutions and becomes in the limit of large distance <i>r</i> the (normal electromagnetic) wave equation. &#8226; AK-gravity, as opposed to GR, has no singularity at the horizon: the singularity in the metric becomes a (very high) peak. &#8226; AK-gravity has a limit scale of the gravitational quantum region 39 &#956;m, which emerges as the limit scale in the objective wave collapse theory of Gherardi-Rimini-Weber. In the quantum region, the AK-gravity becomes a quantum gauge theory (AK quantum gravity) with the Lie group extended SU(2) = <i>&#949;</i>-tensor-group(four generators) as gauge group and a corresponding covariant derivative. &#8226; AK quantum gravity is fully renormalizable, we derive its Lagrangian, which is dimensionally renormalizable, the normalized one-graviton wave function, the graviton propagator, and demonstrate the calculation of cross-section from Feynman diagrams.