Instanton representation of Plebanski gravity Eyo Eyo Ita III Physics Department, US Naval Academy Spanish Relativity Meeting September 10, 2010

Motivation Quantization of the physical degrees of freedom for general relativity Classical dynamics of these degrees of freedom Construction of a Hilbert space of states for gravity with a well-defined semiclassical limit Proposed resolution of the Kodama state –Issues of normalizability raised by Ed Witten –Interpretation and role within new Hilbert space

Outline Plebanski starting action Instanton representation Canonical structure and quantization Hilbert space of states Proposed resolution of the Kodama state Future and in-progress Directions

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

Plebanski action in time gauge

Covariant form of the action After Diffeomorphism constraint, CDJ matrix is symmetric Physical interpretation of inverse CDJ matrix –Self-dual part of Weyl curvature plus trace part –Fixes algebraic classification of spacetime for Petrov Types I, D, O –Determines principal null directions and radiation properties of spacetime 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 on the full phase space! This presents an obstacle to quantization

Quantizable configurations Can exist only on reduced phase space after implementation of Gauss’ law and diffeomorphism constraints There are six distinct configurations of configuration space which can be canonically conjugate to eigenvalues of CDJ matrix. Three D.O.F. per point for each configuration

Reduction to kinematic phase space After application of the temporal gauge (as in Yang-Mills theory) Perform polar decomposition of configuration space Rotates the variables into SO(3,C) frame solving Gauss’ law constraint This yields a 3+1 decomposition of –Note that the SO(3,C) angles θ have disappeared from the action –Also the canonical one form is free of spatial gradients –Yet this is still the full theory with 6 phase space degrees of freedom per point (there are spatial gradients in the Hamiltonian) –Action almost in canonical form: just need globally defined coordinates

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

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

Auxilliary Hilbert space Discretization of into lattice cells of characteristic size States are normalizable in Gaussian measure Overlap between two states inversely related to Euclidean distance in C 2 Continuum limit (Direct product of states at each lattice site) States are eigenstates of momentum operator

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

Hilbert space for Λ=0 Two-to-one correspondence with C 2 manifold for all Λ=0 states Plane wave evolution w.r.t. time T

Hilbert space for Λ≠0 Three-to-one correspondence with C 2 manifold only for Kodama state Hypergeometric evolution w.r.t

Main Results Regularization independent Cauchy complete Hilbert space in continuum limit for Λ=0 (e.g. same as discretized version) annihilated by the Hamiltonian constraint of the full theory States are labelled by the densitized eigenvalues of the CDJ matrix (encodes the algebraic classification of spacetime: This characterizes the semiclassical limit of the quantum theory) For Λ≠0, the Kodama state is the only regularization independent state in the continuum limit (if and are unrelated). 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 But if then Λ≠0 states in continuum limit are also annihilated by the the Hamiltonian constraint of the full theory. These state are labelled by two independent eigenvalues of CDJ matrix

Future/in-progress Directions Computation of Hamiltonian dynamics on reduced phase space (e.g. solution of `reduced Einstein equations’ Coherent state interpretation of states Hilbert space for theory coupled to matter fields (standard model)

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

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

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