Publication: Positivity in Lorentzian Barrett-Crane Models of Quantum Gravity

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Title	Positivity in Lorentzian Barrett-Crane Models of Quantum Gravity
Authors/Editors*	J. Wade Cherrington, J. Daniel Christensen
Where published*	Classical and Quantum Gravity
How published*	Journal
Year*	2006
Volume	23
Number	-1
Pages	721-736
Publisher
Keywords
Link	http://www.iop.org/EJ/abstract/0264-9381/23/3/011/
Abstract	The Barrett-Crane models of Lorentzian quantum gravity are a family of spin foam models based on the Lorentz group. We show that for various choices of edge and face amplitudes, including the Perez-Rovelli normalization, the amplitude for every triangulated closed 4-manifold is a non-negative real number. Roughly speaking, this means that if one sums over triangulations, there is no interference between the different triangulations. We prove non-negativity by transforming the model into a ``dual variables'' formulation in which the amplitude for a given triangulation is expressed as an integral over three copies of hyperbolic space for each tetrahedron. Then we prove that, expressed in this way, the integrand is non-negative. In addition to implying that the amplitude is non-negative, the non-negativity of the integrand is highly significant from the point of view of numerical computations, as it allows statistical methods such as the Metropolis algorithm to be used for efficient computation of expectation values of observables.