Instanton Representation of Plebanski Gravity: Hilbert Space Structure and Proposed Kodama State Resolution

1. Instanton representation of Plebanski gravity Hilbert space structure and proposed resolution of the Kodama state Eyo Eyo Ita III GR19 Conference, Mexico City 2010

2. Motivation and Outline • Motivation: • Quantization of the physical D.O.F. for general relativity • Construction of a Hilbert space of states for gravity with a well-defined semiclassical limit • Addressal of the issues surrounding the Kodama state • Plebanski starting action • Instanton representation • Canonical structure and quantization • Hilbert space of states • Proposed resolution of the Kodama state Email: ita@usna.edu, eei20@cam.ac.uk

3. Starting Plebanski action • Variables: CDJ matrix, SO(3,C) connection and 2-forms • Define spatial components of SO(3,C) curvature and 2-forms • Plebanski equations of motion imply Einstein equations • Equation #1 is simplicity constraint: time gauge implies that • Put back into action and perform 3+1 decomposition

4. Plebanski action in time gauge

5. Covariant form of the action • Diffeomorphism constraint implied symmetric CDJ matrix • Physical interpretation of inverse CDJ matrix • Self-dual part of Weyl curvature plus trace part • Fixes Petrov classification of spacetime • Principal null directions • Radiation properties • CDJ matrix admits polar decomposition (θ are complex SO(3) angles) • Would like to use eigenvalues as basic momentum space variables • But canonically conjugate `coordinates’ do not generically exist!

6. Reduction to kinematic phase space • Canonically conjugate coordinates can exist only for • Reduced phase space under Gauss’ and diffeomorphism constraints • This defines the kinematic phase space • Six distinct configurations with 3 D.O.F. (Ex. Diagonal connection) • Perform polar decomposition of configuration space • This implies the 3+1 decomposition • Note spatial gradients and SO(3,C) angles have disappeared • Yet this is still the full theory with 6 phase space D.O.F. per point • Apply temporal gauge (as in Yang-Mills theory)

7. Define densitized variables • Dimensionless configuration space variables • Momentum space: Uses densitized eigenvalues of CDJ matrix • This yields an action with globally defined coordinates

8. Quantization of the Kinematic Phase space • Symplectic two Form on each 3-D spatial hypersurface • Canonical Commutation Relations • Holomorphic functional Schrodinger Representation

9. Auxilliary Hilbert space • Discretization of into lattice cells of characteristic volume • States are normalizable in Gaussian measure • Overlap between two states inversely related to Euclidean distance in C2 • Continuum limit (Direct product of states at each lattice site) • States are eigenstates of momentum operator

10. Hamiltonian constraint on • Classical constraint on densitized eigenvalues • Quantum constraint in polynomial form • Act on auxilliary Hilbert space • Rescaled to dimensionless eigenvalues

11. Hilbert space for Λ=0 • Two-to-one correspondence with C2 manifold • for all Λ=0 states • Plane wave evolution w.r.t. time T

12. Hilbert space for Λ≠0 • Three-to-one correspondence with C2 manifold • only for Kodama state • Hypergeometric evolution w.r.t

13. Main Results • Regularization independent Cauchy complete Hilbert space in continuum limit for Λ=0 (e.g. same as discretized version) • States are labelled by the densitized eigenvalues of the CDJ matrix (encode algebraic classification of spacetime: This characterizes the semicalssical limit of the quantum theory) • For Λ≠0, the Kodama state is the only regularization independent state in the continuum limit (same as discretized version). Semiclasical limit of Petrov Type O spacetimes • The remaining Λ≠0 states (e.g. Petrov Type D and I) are regularization dependent, which implies that space is discrete on the scale of the inverse regulating function. Discretized version of the state annihilated by the constraint but the continuum limit is not part of the solution space • These state are labelled by two independent eigenvalues of CDJ matrix • Instanton representation appears well-suited to quantization procedure

14. Conclusion • Performed quantization of the kinematic phase space of instanton representation • Quantized physical degrees of freedom of GR • These encode the algebraic classification of spacetime • Constructed Hilbert space of states for Type I,D,O • Proposed resolution of Kodama state normalizability: • Quantum state with semiclassical limit of Petrov Type O • Purely a time variable on configuration space • One does not normalize a wavefunction in time • New series of papers on instanton representaion

15. Special Acknowledgements US Naval Academy. Annapolis, MD Division of Mathematics and Sciences USNA Physics Department Department of Applied Mathematics and Theoretical Physics, Center for Mathematical Sciences. Cambridge, United Kingdom

16. QUESTIONS? • Email addresses • ita@usna.edu • eei20@cam.ac.uk