A Generalization of the Kodama State for Arbitrary Values of the Immirzi Parameter

Abstract

The Kodama State for Lorentzian gravity presupposes a particular value for the Immirzi-parameter, namely $\beta=-i$. However, the derivation of black hole entropy in Loop Quantum Gravity suggests that the Immirzi parameter is a fixed value whose magnitude is on the order of unity but larger than one. Since the Kodama state has de-Sitter spacetime as its classical limit, to get the proper radiation temperature, the Kodama state should be extended to incorporate a more physical value for $\beta$. Thus, we present an extension of the Kodama state for arbitrary values of the Immirzi parameter, $\beta$, that reduces to the ordinary Chern-Simons state for the particular value $\beta=-i$. The state for real values of $\beta$ is free of several of the outstanding problems that cast doubts on the original Kodama state as a ground state for quantum general relativity. We show that for real values of $\beta$, the state is invariant under large gauge transformations, it is CPT invariant (but not CP invariant), and it is expected to be delta-function normalizable with respect to the kinematical inner product. To aid in the construction, we first present a general method for solving the Hamiltonian constraint for imaginary values of $\beta$ that allows one to use the simpler self-dual and anti-self-dual forms of the constraint as an intermediate step.