Discrete quantum gravity: The Lorentz invariant weight for the Barrett-Crane model
نویسنده
چکیده
Abstract. In a recent paper [1] we have constructed the spin and tensor representations of SO(4) from which the invariant weight can be derived for the Barrett-Crane model in quantum gravity. By analogy with the SO(4) group, we present the complexified Clebsch-Gordan coefficients in order to construct the Biedenharn-Dolginov function for the SO(3,1) group and the spherical function as the Lorentz invariant weight of the model.
منابع مشابه
Discrete Quantum Gravity : II . Simplicial complexes , irreps of SL ( 2 , C ) , and a Lorentz invariant weight in a state sum model
Abstract. In part I of [1] we have developed the tensor and spin representation of SO(4) in order to apply it to the simplicial decomposition of the Barrett-Crane model. We attach to each face of a triangle the spherical function constructed from the Dolginov-Biedenharn function. In part II we apply the same technique to the Lorentz invariant state sum model. We need three new ingredients: the ...
متن کاملProjected Spin Networks for Lorentz connection: Linking Spin Foams and Loop Gravity
In the search for a covariant formulation for Loop Quantum Gravity, spin foams have arised as the corresponding discrete space-time structure and, among the different models, the Barrett-Crane model seems the most promising. Here, we study its boundary states and introduce cylindrical functions on both the Lorentz connection and the time normal to the studied hypersurface. We call them projecte...
متن کاملDiscrete Quantum Gravity: I. Zonal spherical functions of the representations of the SO(4,R) group with respect to the SU(2) subgroup and their application to the Euclidean invariant weight for the Barrett-Crane model
Abstract. Starting from the defining transformations of complex matrices for the SO(4, R) group, we construct the fundamental representation and the tensor and spinor representations of the group SO(4, R). Given the commutation relations for the corresponding algebra, the unitary representations of the group in terms of the generalized Euler angles are constructed. The crucial step for the Barr...
متن کاملDiscrete Quantum Gravity : II . Simplicial complexes , irreps of SL ( 2 , C ) , and
Abstract. In part I of [1] we have developed the tensor and spin representation of SO(4) in order to apply it to the simplicial decomposition of the Barrett-Crane model. We attach to each face of a triangle the spherical function constructed from the Dolginov-Biedenharn function. In part II we apply the same technique to the Lorentz invariant state sum model. We need three new ingredients: the ...
متن کاملIntegrability for Relativistic Spin Networks
The evaluation of relativistic spin networks plays a fundamental role in the Barrett-Crane state sum model of Lorentzian quantum gravity in 4 dimensions. A relativistic spin network is a graph labelled by unitary irreducible representations of the Lorentz group appearing in the direct integral decomposition of the space of L functions on three-dimensional hyperbolic space. To ‘evaluate’ such a ...
متن کامل