The equality of 3-manifold invariants

The invariants of 3-manifolds defined by Kuperberg for involutory Hopf algebras and those defined by the authors for spherical Hopf algebras are the same for Hopf algebras on which they are both defined.Comment: 8 pages, definition of state sum invariant improved for clarity, plus minor typos corrected. With 3 postscript figures. further change: BoxedEPSF macro now include

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.Comment: 31 pages, latex. v3: minor clarification

Quantum Gravity as Topological Quantum Field Theory

The physics of quantum gravity is discussed within the framework of topological quantum field theory. Some of the principles are illustrated with examples taken from theories in which space-time is three dimensional.Comment: 23 pages, amstex, JMP special issue (deadline permitting). (Text not changed

State sum models for quantum gravity

This paper reviews the construction of quantum field theory on a 4-dimensional spacetime by combinatorial methods, and discusses the recent developments in the direction of a combinatorial construction of quantum gravity.Comment: amslatex, 7 pages, ICMP conference tal

A Lorentzian version of the non-commutative geometry of the standard model of particle physics

A formulation of the non-commutative 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 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 see-saw mechanism of neutrino mass generation.Comment: Approx. 14 pages. v2: minor corrections; conclusions unaffecte