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so(4) Plebanski Action and Relativistic Spin Foam Model

R. De Pietri, L. Freidel

Abstract

In this note we study the correspondence between the ``relativistic spin foam'' model introduced by Barrett, Crane and Baez and the so(4) Plebanski action. We argue that the $so(4)$ Plebanski model is the continuum analog of the relativistic spin foam model. We prove that the Plebanski action possess four phases, one of which is gravity and outline the discrepancy between this model and the model of Euclidean gravity. We also show that the Plebanski model possess another natural dicretisation and can be associate with another, new, spin foam model that appear to be the $so(4)$ counterpart of the spin foam model describing the self dual formulation of gravity.

so(4) Plebanski Action and Relativistic Spin Foam Model

Abstract

In this note we study the correspondence between the ``relativistic spin foam'' model introduced by Barrett, Crane and Baez and the so(4) Plebanski action. We argue that the Plebanski model is the continuum analog of the relativistic spin foam model. We prove that the Plebanski action possess four phases, one of which is gravity and outline the discrepancy between this model and the model of Euclidean gravity. We also show that the Plebanski model possess another natural dicretisation and can be associate with another, new, spin foam model that appear to be the counterpart of the spin foam model describing the self dual formulation of gravity.

Paper Structure

This paper contains 6 sections, 1 theorem, 44 equations.

Table of Contents

  1. Introduction
  2. The Baez-Barrett-Crane model
  3. The $so(4)$ Plebanski action
  4. The discretization of the constraints
  5. Conclusion
  6. Notation for the SO(4) and SO(3,1) groups.

Key Result

Theorem 1

If then equation (ELsix) is equivalent to equation Moreover, in this case, (ELsix) and (ELsixbis) are fullfilled iff there exists a real tetrad field $e^I= e^I_\mu dx^\mu$ such that one among the following equalities is satisfied :

Theorems & Definitions (1)

  • Theorem 1