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

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Area Metric Reality Constraint in Classical and
Quantum General RelativitySuresh K Maran
Quantum
Classical
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Plebanski Action for GR
Starting point for Back ground Independent
Models Loop Quantum Gravity and Spin Foams

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

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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

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Area Metric
Space-time Metric gab ?ij?ai?bj ?a?b
M. P. Reisenberger, arXivgr-qc/980406

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Area Metric

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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 Action
Discretization of the BF Part

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Discretization of Plebanski Constraints
Barrett-Crane Constraints J. W. Barrett and L.
Crane J.Math.Phys., 393296--3302, 1998.
Work by Baez, Barrett, Crane, Freidel, Krasnov,
Reissenberger, Barbieri
-gtSet of Constraints Bivectors of the triangles
of a flat four simplex satisfy

• Simplicity Constraints
• The bivectors Bi associated with triangles of a
  tetrahedron must satisfy
• Bi ?Bj 0 ?i,j

-gtContains Discretization of the Plebanski
Constraint

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Discretization of an Area Metric

• Flat Four Simplex
• Let Bij be the complex bivector associated with
  the triangle 0ij where i and j denote one of the
  vertices other than the origin and ilt j.
• Let Bi 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
• Bij ai ?aj Bi -?ik Bjk

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Discretization of an Area Metric

• choose the vectors ai to be the complex vector
  basis inside the four simplex
• The area metric is given by

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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.

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Spin Foam Models Barrett-Crane Models
Associate Group Representation Space to each
triangle bivectors -gt 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/2i?
• Gelfand et al Genereralized functions Vol.5
• Unitary Irreps of SO(4,C) (?L,?R)
• nLnReven
• 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 Formulae

• CS³ defined by

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Quantum Four Simplex

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In my formulation ?n0 is a reality constraint
and in the Original Barrett-Crane formulation it
is a Lorentzian simplicity constraint.
i?j gt Internal irrep ?n0

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The Formal Structure of BC Intertwiners
Reisenberger gr-qc/9809067 FreidelKrasnov
hep-th/9903192 Maran gr-qc/0504092

• A homogenous space X of G,
• 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
• 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.
• Formal quantum states ?

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Quantum Tetrahedron for Real General Relativity
?n0 implies ? or n is zero
G
X
The Real models can be considered as reduced
versions of SO(4,C) model using area reality
constraints.

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Conclusion

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