1. Generalizing the Kodama State
Andrew Randono
Center for Relativity
University of Texas at Austin

2. Introduction
Kodama State is one of the only known solutions to all constraints with well defined semi-classical interpretation
Represents quantum (anti)de-Sitter space
Cosmological data suggest we are in increasingly lambda dominated universe
World appears to be asymptotically de-Sitter
Introduction:
--K State is only (??) known solution to
All constraints with well defined semi-classical
Interpretation
--State represents quantum de-Sitter space
--Has well defined expansion in spin network basis
(q-deformed K-bracket)
--K State is riddled with difficulties
Including CPT problems, non-normalizability
Not invariant under large gauge t-forms,
Negative energies (??)
Exact form in connection and spin network basis:

3. But… Not normalizable under kinematical inner product
Not known to be under physical inner product
Linearized solutions not normalizable under linearized inner product
Not invariant under CPT
Violates Lorentz invariance?
CPT inverted states have negative energy?
Not invariant under large gauge transformation
(This could be a good thing)
Problems:
-Not invariant under large gauge t-forms (could be a problem, but maybe not)
-Not normalizable under kinematical inner product (not known to be normalizable
under true inner product
-Not ostensibly invariant under CPT (show what CPT does)
-Inverted state represents negative energy solutions (??)
-Loop transform, although believed to be Kauffman bracket has not been shown
At the level of rigour of real variables
-Reality constraints (??ask Lee about this)
Loop transform in complex variables not as rigorous as with real variables
Reality constraint must be implemented

4. Resolution Problems can be tracked down to complexification
Immirzi is pure imaginary:
Need to extend state to real values of Immirzi parameter
This can be done
Resolution to problem:
Complexification corresponds to choosing Immirzi Parameter to be imaginary. Need
To extend state to real values of Immirzi parameter
We show that this can be done
Answer is surprising: infinite # of states, one of which is de-Sitter (Chern Simons)
Answer is surprising:
Opens up a large Hilbert space of states
de-Sitter/Chern-Simons is one element of this class

5. The Generalized States
States are WKB states corresponding to first order perturbations about de-Sitter spacetime
Write action as functional of Ashtekar-Barbero connection,
States are pure phase WKB states when Immirzi is real
States appear to depend on position and momentum
This requires careful interpretation

6. Interpretation Generalized Kodama state is like NR momentum state
Infinite set of states parameterized by curvature
Expect different states to be orthogonal
Similarities with momentum eigenstate:
-Point out variables: time is Cherns-Simons, position is connection, momentum is
Levi-Civita curvature
-Can label states by curvature
-Kinematical inner product shows states are orthogonal:
(Show this, but discuss issue of gauge invariance)
-Need to formailze this

7. Levi-Civita Curvature Operator
Momentum operator in momentum basis:
Similarly can construct gauge covariant curvature operator
Curvature operator:
-We can now define curvature operator from these states
-Our operator (will not come from canonical construction)
-Analagous to momentum operator: p=\int p|p><p| dp
-Slight modifications to account for gauge covariance
(Show here)
-States are eigenstates of curvature operator
Using this definition of curvature operator:

8. Properties      ?? Delta-function normalizable
Invariance under large gauge t-forms
Solve Hamiltonian constraint
Semi-classical interpretation (WKB)
CPT invariance
Negative Energies     
Discussion ??

9. References
gr-qc/ , “A Generalization of the Kodama State for Arbitrary Values of the Immirzi Parameter”
Upcoming papers:
Generalizing the Kodama State I: Construction
Generalizing the Kodama State II: Properties and physical interpretation