Quantum general relativity and the classification of smooth manifolds

Abstract

The gauge symmetry of classical general relativity under space-time diffeomorphisms implies that any path integral quantization which can be interpreted as a sum over space-time geometries, gives rise to a formal invariant of smooth manifolds. This is an opportunity to review results on the classification of smooth, piecewise-linear and topological manifolds. It turns out that differential topology distinguishes the space-time dimension d=3+1 from any other lower or higher dimension and relates the sought-after path integral quantization of general relativity in d=3+1 with an open problem in topology, namely to construct non-trivial invariants of smooth manifolds using their piecewise-linear structure. In any dimension d<=5+1, the classification results provide us with triangulations of space-time which are not merely approximations nor introduce any physical cut-off, but which rather capture the full information about smooth manifolds up to diffeomorphism. Conditions on refinements of these triangulations reveal what replaces block-spin renormalization group transformations in theories with dynamical geometry. The classification results finally suggest that it is space-time dimension rather than absence of gravitons that renders pure gravity in d=2+1 a `topological' theory.

41 pages, LaTeX2e with combined eps/PicTeX figures; v2: inaccuracies fixed

References in corpus (13)

• Spin Foam Models for Quantum Gravity
• Topological Lattice Models in Four Dimensions
• Actions for Gravity, with Generalizations: A Review
• Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity
• 3-dimensional Gravity from the Turaev-Viro Invariant
• 4-Dimensional BF Theory as a Topological Quantum Field Theory
• Quantum Gravity as Topological Quantum Field Theory
• Geometrical measurements in three-dimensional quantum gravity
• A Ponzano-Regge model of Lorentzian 3-Dimensional gravity
• Localized Exotic Smoothness
• Schroedinger's cat and the clock: Lessons for quantum gravity
• Three-dimensional Quantum Supergravity and Supersymmetric Spin Foam Models
• A State Sum Model for (2+1) Lorentzian Quantum Gravity

Cited by in corpus (8)

• General boundary quantum field theory: Foundations and probability interpretation
• Perfect discretization of reparametrization invariant path integrals
• Extended matter coupled to BF theory
• Discretisations, Constraints and Diffeomorphisms in Quantum Gravity
• Universe creation on a computer
• Diffeomorphisms from finite triangulations and absence of 'local' degrees of freedom
• Constructing a quantum field theory from spacetime
• Abelian gerbes, generalized geometries and foliations of small exotic R^4