1998/07/31 by S. Carlip, J. E. Nelson · 1 citation
Mathematics · Physics and Astronomy · #Algebra over a field #Black Holes and Theoretical Physics #Combinatorics #Computer science #Cosmology and Gravitation Theories #Geometry #Group (periodic table) #Hilbert space #Holonomy #Loop quantum gravity #Mathematics #Modular design #Modular group #Noncommutative and Quantum Gravity Theories #Operator (biology) #Physics #Pure mathematics #Quantum #Quantum gravity #Quantum mechanics #Representation (politics) #Topology (electrical circuits) #Torus #Transformation (genetics) #gr-qc #hep-th
paper · pdf · doi:10.1103/physrevd.59.024012
published as Phys.Rev.D59:024012,1999 · 23 pages, LaTeX, no figures; minor changes and clarifications in response to referee (basic argument and conclusions unaffected)
arxiv created 1998/10/05 · openalex publication_date 1998/12/14 · arxiv updated 2010/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The role of the modular group in the holonomy representation of (2+1)-dimensional quantum gravity is studied. This representation can be viewed as a ``Heisenberg picture,'' and for simple topologies, the transformation to the ADM ``Schr"odinger picture'' may be found. For spacetimes with the spatial topology of a torus, this transformation and an explicit operator representation of the mapping class group are constructed. It is shown that the quantum modular group splits the holonomy representation Hilbert space into physically equivalent orthogonal ``fundamental regions'' that are interchanged by modular transformations.