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 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:

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

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

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 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

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:

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

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