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L. Crane

Publications and source records attributed to L. Crane.

5 recordsLinked to original sources
arXiv2003

2-categorical Poincare Representations and State Sum Applications

This is intended as a self-contained introduction to the representation theory developed in order to create a Poincare 2-category state sum model for Quantum Gravity in 4 dimensions. We review the structure of a new representation 2-category appropriate to Lie 2-group symmetries and discuss its application to the problem of finding a state sum model for Quantum Gravity. There is a remarkable richness in its details, reflecting some desirable characteristics of physical 4-dimensionality. We begin with a review of the method of orbits in Geometric Quantization, as an aid to the intuition that the geometric picture unfolded here may be seen as a categorification of this process.

math.QA↗
arXiv2003

Measurable Categories and 2-Groups

Using the theory of measurable categories developped by Yetter in work in preparation, we provide a notion of representations of 2-groups more well-suited to physically and geometrically interesting examples than that proposed in unpublished work of Barrett and Mackaay using 2-VECT of Kapranov and Voevodsky. Using this theory we sketch a 2-categorical approach to the state-sum model for Lorentzian quantum gravity proposed in by the authors in previous work and suggest state-integral constructions for 4-manifold invariants.

math.QA↗
arXiv1997

On the Classical Limit of the Balanced State Sum

The purpose of this note is to make several advances in the interpretation of the balanced state sum model by Barrett and Crane in gr-qc/9709028 as a quantum theory of gravity. First, we outline a shortcoming of the definition of the model in pointed out to us by Barrett and Baez in private communication, and explain how to correct it. Second, we show that the classical limit of our state sum reproduces the Einstein-Hilbert lagrangian whenever the term in the state sum to which it is applied has a geometrical interpretation. Next we outline a program to demonstrate that the classical limit of the state sum is in fact dominated by terms with geometrical meaning. This uses in an essential way the alteration we have made to the model in order to fix the shortcoming discussed in the first section. Finally, we make a brief discussion of the Minkowski signature version of the model.

gr-qc↗