2003/05/21 by Thomas Thiemann · 10 citations
Mathematics · Physics and Astronomy · #Algebra over a field #Black Holes and Theoretical Physics #Canonical quantum gravity #Commutator #Constraint (computer-aided design) #Constraint algebra #Cosmology and Gravitation Theories #General relativity #Hamiltonian (control theory) #Hamiltonian constraint #Lie algebra #Loop quantum gravity #Mathematical optimization #Mathematics #Noncommutative and Quantum Gravity Theories #Observable #Physics #Pure mathematics #Quantum #Quantum gravity #Quantum mechanics #Spin foam #Theoretical physics #gr-qc #hep-th #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1088/0264-9381/23/7/002
published as Class.Quant.Grav.23:2211-2248,2006 · LATEX, uses AMSTEX
arxiv created 2003/05/21 · openalex publication_date 2006/03/14 · arxiv updated 2011/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The Hamiltonian constraint remains the major unsolved problem in loop quantum gravity (LQG). Some time ago, a mathematically consistent candidate Hamiltonian constraint was proposed but there are still several unsettled questions which concern the algebra of commutators among smeared Hamiltonian constraints which must be faced in order to make progress. In this paper, we propose a solution to this set of problems based on the so-called master constraint which combines the smeared Hamiltonian constraints for all smearing functions into a single constraint. Due to a harmonic interplay of several mathematical facts, the problems with the commutator algebra disappear and chances are good that one can control the solution space and the (quantum) Dirac observables of LQG. Even a decision on whether the theory has the correct classical limit and a connection with the path integral (or spin foam) formulation could be in reach.