Louis Crane

Relativistic spin networks and quantum gravity

Relativistic spin networks are defined by considering the spin covering of the group SO(4), SU(2)×SU(2). Relativistic quantum spins are related to the geometry of the two-dimensional faces of a 4-simplex. This extends the idea of Ponzano and Regge that SU(2) spins are related to the geometry of the edges of a 3-simplex. This leads us to suggest that there may be a four-dimensional state sum model for quantum gravity based on relativistic spin networks that parallels the construction of three-dimensional quantum gravity from ordinary spin networks.

Four‐dimensional topological quantum field theory, Hopf categories, and the canonical bases

A new combinatorial method of constructing four‐dimensional topological quantum field theories is proposed. The method uses a new type of algebraic structure called a Hopf category. The construction of a family of Hopf categories related to the quantum groups and their canonical bases is also outlined.

Clock and category: Is quantum gravity algebraic?

We investigate the possibility that the quantum theory of gravity could be constructed discretely using algebraic methods. The algebraic tools are similar to ones used in constructing topological quantum field theories. The algebraic structures are related to ideas about the reinterpretation of quantum mechanics in a general relativistic context.

ON ALGEBRAIC STRUCTURES IMPLICIT IN TOPOLOGICAL QUANTUM FIELD THEORIES

We show that any 3D topological quantum field theory satisfying physically reasonable factorizability conditions has associated to it in a natural way a Hopf algebra object in a suitable tensor category. We also show that all objects in the tensor category have the structure of left-left crossed bimodules over the Hopf algebra object. For 4D factorizable topological quantum filed theories, we provide by analogous methods a construction of a Hopf algebra category.

Measurable Categories and 2-Groups

Using the theory of measurable categories developed in [10], we provide a notion of representations of 2-groups better suited to physically and geometrically interesting examples than that using 2-VECT (cf. [8]). Using this theory we sketch a 2-categorical approach to the state-sum model for Lorentzian quantum gravity proposed in [6], and suggest state-integral constructions for 4-manifold invariants.

Causal sites as quantum geometry

We propose a structure called a causal site to use as a setting for quantum geometry, replacing the underlying point set. The structure has an interesting categorical form, and a natural “tangent 2-bundle,” analogous to the tangent bundle of a smooth manifold. Examples with reasonable finiteness conditions have an intrinsic geometry, which can approximate classical solutions to general relativity. We propose an approach to quantization of causal sites as well.

An algebraic interpretation of the Wheeler - DeWitt equation

We make a direct connection between the construction of three-dimensional topological state sums from tensor categories and three-dimensional quantum gravity by noting that the discrete version of the Wheeler - DeWitt equation is exactly the pentagon for the associator of the tensor category - the Biedenharn - Elliott identity. A crucial role is played by an asymptotic formula relating 6j-symbols to rotation matrices given by Edmonds.