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•Kodama State is one of the only known solutions to all constraints with well
defined semi-classical interpretation. •Exact form in connection and spin network.
Generalizing the Kodama State Andrew Randono Center for Relativity University of Texas at Austin
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
•Exact form in connection and spin network basis:
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 good thing) •Loop transform in a complex variables not as rigorous as with real variables •Reality constraint must be implemented
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 •Answer is surprising: •Opens up a large Hilbert space of states •de-Sitter/Chern-Simons is one element of this class
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
Interpretation •Generalized Kodama state is like NR momentum state
•Infinite set of states parameterized by curvature •Expect different states to be orthogonal
Levi-Civita Curvature Operator •Momentum operator in momentum basis:
•Similarly can construct gauge covariant curvature operator
•Using this definition of curvature operator:
Properties
Delta-function normalizable Invariance under large gauge t-forms Solve Hamiltonian constraint Semi-classical interpretation (WKB) CPT invariance
?? Negative Energies
References •gr-qc/0504010, “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
