Area Metric Reality Constraint in Classical and Quantum General Relativity Area Metric Reality Constraint in Classical and Quantum General Relativity Suresh K Maran Classical Quantum

Plebanski Action for GR Starting point for Back ground Independent Models: Loop Quantum Gravity and Spin Foams *

Plebanski Constraint Solution I Solution II SO(4,C) General Relativity *

Reality Conditions in Loop Quantum Gravity: Loop Quantum Gravity: Self +anti-self split of Plebanski action Canonical Quantization of selfdual SO(4,C) Plebanski Action and Imposing Certain Reality Conditions A. Ashtekar, Lectures on non Perturbative Canonical Gravity, Word Scientific, 1991 and Signature Reality **

Area Metric Space-time Metric g ab =η ij θ a i θ b j =θ a ⋅ θ b M. P. Reisenberger, arXiv:gr-qc/ ***

Area Metric XaXa YbYb ***

Non-Zero only if both metric and density b are simultaneously real or imaginary 1)Metric imaginary and Lorentzian. 2)Metric is real and it is Riemannian or Kleinien. **

Discretization of Plebanski Action Discretization of the BF Part: **

Discretization of Plebanski Constraints: Barrett-Crane Constraints J. W. Barrett and L. Crane J.Math.Phys., 39: , Simplicity Constraints: The bivectors B i associated with triangles of a tetrahedron must satisfy B i ∧ B j =0 ∀ i,j ->Set of Constraints Bivectors of the triangles of a flat four simplex satisfy ->Contains: Discretization of the Plebanski Constraint * Work by: Baez, Barrett, Crane, Freidel, Krasnov, Reissenberger, Barbieri

Discretization of an Area Metric Flat Four Simplex: Flat Four Simplex: Let B ij be the complex bivector associated with the triangle 0ij where i and j denote one of the vertices other than the origin and i< j. Let B ij be the complex bivector associated with the triangle 0ij where i and j denote one of the vertices other than the origin and i< j. Let B i denote the bivector associated to the triangle made by connecting the vertices other than the origin and the vertex i Let B i denote the bivector associated to the triangle made by connecting the vertices other than the origin and the vertex i The Barrett-Crane constraints for SO(4,C) general relativity imply that The Barrett-Crane constraints for SO(4,C) general relativity imply that B ij =a i ∧ a j B i =-∑ ik B jk ***

Discretization of an Area Metric choose the vectors a i to be the complex vector basis inside the four simplex choose the vectors a i to be the complex vector basis inside the four simplex The area metric is given by ***

Real Four Simplex: The necessary and sufficient conditions for a four simplex with real non-degenerate flat geometry 1) The SO(4,C) Barrett-Crane constraints and 2) The reality of all possible bivector scalar products. ***

Spin Foam Models: Barrett-Crane Models Associate Group Representation Space to each triangle bivectors -> Lie Operators Impose Barrett-Crane Constraints at quantum level The Model is a Path Integral Quantization of the discrete action. *

SO(4,C) Quantum Tetrahedron An Unitary Irreducible Representation (irrep) of SL(2,C) is labelled by  = n/2+iρ Gelfand et al: Genereralized functions Vol.5 Unitary Irreps of SO(4,C) (  L,  R ) n L +n R =even An unitary Irreducible Representation (irrep) of SO(4,C) is assigned to each triangle ***

SO(4,C) Quantum Tetrahedron Barrett-Crane Intertwiner **

Alternative Formulae CS³ defined by CS³ defined by **

Quantum Four Simplex *

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In my formulation ρn=0 is a reality constraint and in the Original Barrett-Crane formulation it is a Lorentzian simplicity constraint. i≠j => Internal irrep ρn=0 ***

The Formal Structure of BC Intertwiners A homogenous space X of G, A homogenous space X of G, A G invariant measure on X and, A G invariant measure on X and, T-functions which are maps from X to the Hilbert spaces of a subset of unitary irreps of G T-functions which are maps from X to the Hilbert spaces of a subset of unitary irreps of G where R labels an irrep of G. The T-functions are complete in the sense that on the L² functions on X they define invertible Fourier transforms. The T-functions are complete in the sense that on the L² functions on X they define invertible Fourier transforms. Formal quantum states Ψ : Formal quantum states Ψ : *** Reisenberger :gr-qc/ Freidel&Krasnov hep-th/ Maran gr-qc/

Quantum Tetrahedron for Real General Relativity The Real models can be considered as reduced versions of SO(4,C) model using area reality constraints.  n=0 implies  or n is zero *** GX

Conclusion General Relativity = SO(4,C) BF theory + Plebanski (simplicity) Constraint + Reality Constraint General Relativity = SO(4,C) BF theory + Plebanski (simplicity) Constraint + Reality Constraint The formulation is signature independent. The formulation is signature independent.  An opportunity to show that Lorentzian is special from other signatures.  Stephan Hawking splices various signatures.  Implications for Ashtekar formalism. ***