Generalizing the Kodama State I: Construction

Abstract

The Kodama State is unique in being an exact solution to all the ordinary constraints of canonical quantum gravity that also has a well defined semi-classical interpretation as a quantum version of a classical spacetime, namely (anti)de Sitter space. However, the state is riddled with difficulties which can be tracked down to the complexification of the phase space necessary in its construction. This suggests a generalization of the state to real values of the Immirzi parameter. In this first part of a two paper series we show that one can generalize the state to real variables and the result is surprising in that it appears to open up an infinite class of physical states. We show that these states closely parallel the ordinary momentum eigenstates of non-relativistic quantum mechanics with the Levi-Civita curvature playing the role of the momentum. With this identification, the states inherit many of the familiar properties of the momentum eigenstates including delta-function normalizability. In the companion paper we will discuss the physical interpretation, CPT properties, and an interesting connection between the inner product and the Macdowell-Mansouri formulation of general relativity.

First part in two part series, 20 pages

References in corpus (6)

• Real Ashtekar Variables for Lorentzian Signature Space-times
• Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action
• Quantum gravity with a positive cosmological constant
• A Note On The Chern-Simons And Kodama Wavefunctions
• A Note on Parity Violation and the Immirzi Parameter
• A Generalization of the Kodama State for Arbitrary Values of the Immirzi Parameter

Cited by in corpus (11)

• SL(2,C) Chern-Simons Theory, a non-Planar Graph Operator, and 4D Loop Quantum Gravity with a Cosmological Constant: Semiclassical Geometry
• Equivalence of the Chern-Simons state and the Hartle-Hawking and Vilenkin wave-functions
• Finite states in four dimensional quantized gravity
• Barbero-Immirzi parameter, manifold invariants and Euclidean path integrals
• Quantum cosmology of a dynamical Lambda
• A Mesoscopic Quantum Gravity Effect
• In Search of Quantum de Sitter Space: Generalizing the Kodama State
• Complex Ashtekar variables, the Kodama state and spinfoam gravity
• Some notes on the Kodama state, maximal symmetry, and the isolated horizon boundary condition
• Canonical analysis of covariant unimodular gravity and an extension of the Kodama state
• Superspace worldline formalism approach to Quantum Gravity: dimensional reduction and Holography