Konzeptionelle Fragen der Quantengravitation und mathematischer Strukturen in der dreidimensionalen Gravitation
Zusammenfassung der Projektergebnisse
Part 1 - Hamiltonian quantisation formalisms and state sum models: • General mathematical framework for defects and observables in 3d state sum models and TFQT: theory of Gray categories with duals and development of diagrammatical calculus); generalisation to the tricategory of fusion categories, bimodule categories bimodule functors and bimodule natural transformations. • Generalisation of the notion of a group-valued lattice gauge theory to Hopf algebras: axiomatic description and construction of local Hopf algebra gauge theories on ribbon graphs; construction of the algebra of observables and proof of topological invariance; explicit relation to existing models and quantisation approaches. • Quantum deformation of two four-dimensional spin foam models: Construction of a prominent 4d state sum model based on quantum groups. • The extended and the gauge invariant Hilbert space of 3d gravity: detailed investigation of the Hilbert space of 3d gravity; explicit relation between two quantisation approaches; implementation of constraint operators and quantum group symmetries. Part 2 - Geometrical and physics aspects of the classical theory: • Observables of 3d gravity and measurements by observers: characterisation of the nonlocal diffeomorphism invariant observables of 3d gravity (Wilson loops) in terms of measurements with returning lightrays (return time, direction, frequency shift); reconstruction of the spacetime geometry from such measurements. • Initial singularities of flat 3d and 4d spacetimes in terms of light signals: description of the initial singularity of globally hyperbolic 3d and 4d spacetimes in terms of light signals emitted near the initial singularity (background radiation); geometrical properties, classification. • Stationary flat 3d spacetimes with particles: construction of stationary flat spacetimes with massive particles with spin; investigation of causality structure and classification results. Part 3 - 3d quantum gravity as quantised hyperbolic geometry: • generalisation of shear coordinates on Teichmüller space to moduli spaces of 3d spacetimes by u analytic continuation techniques; explicit description of symplectic structure and relation to the cotangent bundle of Teichmüller space; explicit expressions for mapping class group actions and description of generating Hamiltonians. Part 4 - Quantisation of Lorentzian 3d gravity via Dirac’s constraint quantisation formalism: • Phase space reduction for Lorentzian 3d gravity with vanishing cosmological constant: 3d gravity as a constrained system; gauge fixing and Dirac bracket; gauge fixing as selection of a reference frame; residual symmetries and physical interpretation. • Mathematical structures associated with gauge fixing: dynamical r-matrices via a gauge fixing procedure; transition between different gauge fixing conditions and dynamical gauge transformations. Part 5 - Quantum group symmetries in 3d gravity: • quantum group symmetries of 3d gravity as a multi-parametric quantum deformations of sl(2, R) with the cosmological constant and the speed of light as a deformation parameters; classification of multi-parametric deformations; construction of the associated quantum algebras and non-commutative spaces; classical and cosmological limit.
Projektbezogene Publikationen (Auswahl)
- Cosmological measurements, time and observables in (2+1)-dimensional gravity, Class. Quantum Grav. 26 (2009) 055006
Catherine Meusburger
- Combinatorial quantisation of the Euclidean torus universe, Nucl. Phys. B 841, Issue 3 (2010) 463-505
Catherine Meusburger, Karim Noui
- Three-dimensional gravity and Drinfel’d doubles: spacetimes and symmetries from quantum deformations, Physics Letters B 687 (2010) 375-381
Angel Ballesteros, Francisco J. Herranz, Catherine Meusburger
- Gauge fixing in (2+1)-gravity: Dirac bracket and spacetime geometry, Class. Quantum Grav. 28 (2011) 125008
Catherine Meusburger, Torsten Schönfeld
- Particles with spin in stationary flat spacetimes, Geom. Dedicata Volume 161, Issue 1 (2012) 23-50
Thierry Barbot, Catherine Meusburger
(Siehe online unter https://doi.org/10.1007/s10711-011-9692-y) - Quantum deformation of two four-dimensional spin foam models, J. Math. Phys. 53 (2012) 022501
Winston J. Fairbairn, Catherine Meusburger
(Siehe online unter https://doi.org/10.1063/1.3675898) - Drinfel’d doubles for (2+1)- gravity, Class. Quantum Grav. 30 (2013) 155012
Angel Ballesteros, Francisco J. Herranz, Catherine Meusburger
- Recovering the geometry of a flat spacetime from background radiation, Annales Henri Poincaré (2013) 1-67
Francesco Bonsante, Catherine Meusburger, Jean-Marc Schlenker
- Gauge fixing and classical dynamical r-matrices in ISO(2,1)-Chern-Simons theory, Commun. Math. Phys. 327.2 (2014) 443-479
Catherine Meusburger, Torsten Schönfeld
(Siehe online unter https://doi.org/10.1007/s00220-014-1938-8) - Generalised shear coordinates on the moduli spaces of three-dimensional spacetimes, 40 pages, J. Differ. Geom.
Catherine Meusburger, Carlos Scarinci
- Gray categories with duals and their diagrams, 156 pages, Adv. Math.
John W. Barrett, Catherine Meusburger, Gregor Schaumann
- Hopf algebra gauge theory on a ribbon graph. 68 pages
Catherine Meusburger, Derek Wise
