John W. Barrett

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.

Asymptotic analysis of the Engle―Pereira-Rovelli―Livine four-simplex amplitude

The semiclassical limit of a four-simplex amplitude for a spin foam quantum gravity model with an Immirzi parameter is studied. If the boundary state represents a nondegenerate four-simplex geometry, the asymptotic formula contains the Regge action for general relativity. A canonical choice of phase for the boundary state is introduced and is shown to be necessary to obtain the results.

Lorentzian spin foam amplitudes: graphical calculus and asymptotics

The amplitude for the 4-simplex in a spin foam model for quantum gravity is defined using a graphical calculus for the unitary representations of the Lorentz group. The asymptotics of this amplitude are studied in the limit when the representation parameters are large, for various cases of boundary data. It is shown that for boundary data corresponding to a Lorentzian simplex, the asymptotic formula has two terms, with phase plus or minus the Lorentzian signature Regge action for the 4-simplex geometry, multiplied by an Immirzi parameter. Other cases of boundary data are also considered, including a surprising contribution from Euclidean signature metrics.

INVARIANTS OF PIECEWISE-LINEAR 3-MANIFOLDS

In this paper we develop a theory for constructing an invariant of closed oriented 3-manifolds, given a certain type of Hopf algebra. Examples are given by a quantised enveloping algebra of a semisimple Lie algebra, or by a semisimple involutory Hopf algebra. The invariant is defined by a state sum model on a triangulation. In some cases, the invariant is the partition function of a topological quantum field theory.

Lorentzian version of the noncommutative geometry of the standard model of particle physics

A formulation of the noncommutative geometry for the standard model of particle physics with a Lorentzian signature metric is presented. The elimination of the fermion doubling in the Lorentzian case is achieved by a modification of Connes’ internal space geometry [“Gravity coupled with matter and the foundation of non-commutative geometry,” Commun. Math. Phys. 182, 155–176 (1996)] so that it has signature 6 (mod 8) rather than 0. The fermionic part of the Connes-Chamseddine spectral action can be formulated, and it is shown that it allows an extension with right-handed neutrinos and the correct mass terms for the seesaw mechanism of neutrino mass generation.

The Ponzano–Regge model

The definition of the Ponzano–Regge state-sum model of three-dimensional quantum gravity with a class of local observables is developed. The main definition of the Ponzano–Regge model in this paper is determined by its reformulation in terms of group variables. The regularization is defined and a proof is given that the partition function is well defined only when a certain cohomological criterion is satisfied. In that case, the partition function may be expressed in terms of a topological invariant, the Reidemeister torsion. This proves the independence of the definition on the triangulation of the 3-manifold and on those arbitrary choices made in the regularization. A further corollary is that when the observable is a knot, the partition function (when it exists) can be written in terms of the Alexander polynomial of the knot. Various examples of observables in S3 are computed explicitly. Alternative regularizations of the Ponzano–Regge model by the simple cut-off procedure and by the limit of the Turaev–Viro model are discussed, giving successes and limitations of these approaches.

Asymptotics of relativistic spin networks

The stationary phase technique is used to calculate asymptotic formulae for SO(4) relativistic spin networks. For the tetrahedral spin network this gives the square of the Ponzano–Regge asymptotic formula for the SU(2) 6j-symbol. For the 4-simplex (10j-symbol) the asymptotic formula is compared with numerical calculations of the spin network evaluation. Finally, we discuss the asymptotics of the SO(3, 1) 10j-symbol.

Holonomy and path structures in general relativity and Yang-Mills theory

This article is about a different representation of the geometry of the gravitational field, one in which the paths of test bodies play a crucial role. The primary concept is the geometry of the motion of a test body, and the relation between different such possible motions. Space-time as a Lorentzian manifold is regarded as a secondary construct, and it is shown how to construct it from the primary data. Some technical problems remain. Yang-Mills fields are defined by their holonomy in an analogous construction. I detail the development of this idea in the literature, and give a new version of the construction of a bundle and connection from holonomy data. The field equations of general relativity are discussed briefly in this context.

A note on area variables in Regge calculus

We consider the possibility of setting up a new version of Regge calculus in four dimensions with areas of triangles as the basic variables rather than the edge lengths. The difficulties and restrictions of this approach are discussed.

Geometrical measurements in three-dimensional quantum gravity

A set of observables is described for the topological quantum field theory which describes quantum gravity in three space-time dimensions with positive signature and positive cosmological constant. The simplest examples measure the distances between points, giving spectra and probabilities which have a geometrical interpretation. The observables are related to the evaluation of relativistic spin networks by a Fourier transform.