arXiv:gr-qc/0504092v1 20 Apr 2005 Spin Foams of Real General Relativity from that of Complex General Relativity using a Reality Constraint. Suresh K Maran February 8, 2020 Abstract I deduce spin foams for real general relativity (all signatures) from that of complex general relativity by imposing the reality constraint that the square of areas be real. I derive the Barrett-Crane spin foam model for complex gravity. I demonstrate how to rigorously impose the cross simplicity constraint of Barrett-Crane. By imposing the area reality con- dition at the quantum level, I deduce spin foam model for all signatures of real general relativity. I point out two interesting models: extended real gravity and a Lorentzian spin foam model. Imposing reality condition [1] is a non-trivial problem in canonical quantum general relativity. In fact many of the recent advances [2] in canonical quan- tum general relativity have been made by converting the complex formulation of the theory to a real formulation by transforming the configuration variable, a complex SL(2,C) connection, to a real SU (2) connection through a Legen- dre transformation [3]. In quantum theory Lie operators are fundamental. In general relativity bivectors (quantized) are isomorphic to the Lie algebra (oper- ators) of the relevant group. Using bivector valued 2-form fields instead of the space-time metric as a variable for general relativity [5] is one of the foundations of background independent quantum gravity ideas 1 . Since the bivectors in the background independent quantum formulations physically relate to the areas of 2- surfaces [6] as the fundamental relationship to geometry, the relationship of the value of areas to the reality of the theory is an important idea that needs to be investigated. One of the most investigated quantum model for general relativity is the Barrett-Crane spin foam model [7], [8]. Let me briefly discuss the model as given in Ref:[7]. In the Barrett-Crane model quantum amplitudes are assigned to the simplices of a triangulated manifold. The quantum state of a simplicial triangulation is described by assigning representations of the gauge group to the triangles. In case of Riemannian general relativity the proper group is 1 Visit www.qstaf.org under construnction. I refer the novice readers to the latest book by Rovelli [4] on background independent quantum gravity for startup. 1