Area Metric Reality Area Metric Reality ConstraintConstraint

in Classical and Quantum in Classical and Quantum General RelativityGeneral Relativity

Suresh K Maran

Classical Quantum

Plebanski Action for GRPlebanski Action for GR

Starting point for Back ground Independent Models: Loop Quantum Gravity and Spin Foams

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Plebanski ConstraintPlebanski Constraint

Solution I

Solution II

SO(4,C) General RelativitySO(4,C) General Relativity

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Reality Conditions in Loop Quantum Reality Conditions in Loop Quantum Gravity:Gravity:

Loop Quantum Gravity:

•Self +anti-self split of Plebanski action

Canonical Quantization of selfdual SO(4,C) Plebanski Action and Imposing Certain Reality ConditionsA. Ashtekar, Lectures on non Perturbative Canonical Gravity, Word Scientific, 1991

and and SignatureSignatureRealityReality

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Area MetricArea MetricSpace-time Metric

gab =ηijθaiθb

j

=θa⋅θb

M. P. Reisenberger, arXiv:gr-qc/980406

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Area MetricArea Metric

Xa

Yb

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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.

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Discretization of Plebanski Discretization of Plebanski ActionAction

Discretization of the BF Part:

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Discretization of Plebanski Discretization of Plebanski Constraints: Barrett-Crane Constraints: Barrett-Crane ConstraintsConstraints J. W. Barrett and L. Crane J.Math.Phys., 39:3296--3302, 1998.J. W. Barrett and L. Crane J.Math.Phys., 39:3296--3302, 1998.

Simplicity Constraints:Simplicity Constraints:

The bivectors BThe bivectors Bii associated with associated with triangles of a tetrahedron must satisfytriangles of a tetrahedron must satisfy

BBii ∧B ∧Bjj =0 ∀i,j =0 ∀i,j

->Set of Constraints Bivectors of the triangles of a flat four simplex satisfy

->Contains: Discretization of the Plebanski Constraint

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Work by: Baez, Barrett, Crane, Freidel, Krasnov, Reissenberger, Barbieri

Discretization of an Area MetricDiscretization of an Area Metric

Flat Four Simplex:Flat Four Simplex:• Let BLet Bijij be the complex bivector associated with be the complex bivector associated with

the triangle 0ij where i and j denote one of the the triangle 0ij where i and j denote one of the vertices other than the origin and i< j. vertices other than the origin and i< j.

• Let BLet Bii denote the bivector associated to the denote the bivector associated to the triangle made by connecting the vertices other triangle made by connecting the vertices other than the origin and the vertex ithan the origin and the vertex i

• The Barrett-Crane constraints for SO(4,C) general The Barrett-Crane constraints for SO(4,C) general relativity imply thatrelativity imply that

BBijij =a =aii ∧a ∧ajj BBii =-∑ =-∑ikik B Bjkjk

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3

4

0

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Discretization of an Area Discretization of an Area MetricMetric choose the vectors achoose the vectors aii to be the complex to be the complex

vector basis inside the four simplexvector basis inside the four simplex

The area metric is given byThe area metric is given by

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Real Four Simplex:Real Four Simplex:

The necessary and sufficient The necessary and sufficient conditions for a four simplex with conditions for a four simplex with real non-degenerate flat geometryreal non-degenerate flat geometry

1) The SO(4,C) Barrett-Crane 1) The SO(4,C) Barrett-Crane constraints andconstraints and

2) The reality of all possible bivector 2) The reality of all possible bivector scalar products.scalar products.

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Spin Foam Models: Spin Foam Models: Barrett-Crane ModelsBarrett-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.

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SO(4,C) Quantum Tetrahedron

•An Unitary Irreducible Representation (irrep) of SL(2,C) is labelled by= n/2+iρGelfand et al: Genereralized functions Vol.5

•Unitary Irreps of SO(4,C) (L,R)nL+nR=even

•An unitary Irreducible Representation (irrep) of SO(4,C) is assigned to each triangle

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SO(4,C) Quantum Tetrahedron

Barrett-Crane Intertwiner

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Alternative FormulaeAlternative Formulae

• CS³ defined byCS³ defined by

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Quantum Four SimplexQuantum Four Simplex

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In my formulation ρn=0 is a reality constraint and in the Original Barrett-Crane formulation it is aLorentzian simplicity constraint.

i≠j => Internal irrep ρn=0***

The Formal Structure of BC The Formal Structure of BC IntertwinersIntertwiners• 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 T-functions which are maps from X to the

Hilbert spaces of a subset of unitary irreps of Hilbert spaces of a subset of unitary irreps of GG

where R labels an irrep of G. where R labels an irrep of G. • The T-functions are complete in the sense The T-functions are complete in the sense

that on the L² functions on X they define that on the L² functions on X they define invertible Fourier transforms. invertible Fourier transforms.

• Formal quantum states Ψ :Formal quantum states Ψ :

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Reisenberger :gr-qc/9809067

Freidel&Krasnov hep-th/9903192

Maran gr-qc/0504092

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

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G X

Conclusion Conclusion

• General Relativity = SO(4,C) BF theory + General Relativity = SO(4,C) BF theory + Plebanski (simplicity) Constraint + Reality Plebanski (simplicity) Constraint + Reality Constraint Constraint

• The formulation is signature The formulation is signature independent.independent.

An opportunity to show that Lorentzian is special An opportunity to show that Lorentzian is special from other signatures. from other signatures.

Stephan Hawking splices various signatures. Stephan Hawking splices various signatures. Implications for Ashtekar formalism.Implications for Ashtekar formalism.

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