2012/10/25 by Casey Tomlin, Madhavan Varadarajan · 3 citations
Mathematics · Medicine · Physics and Astronomy · #Black Holes and Theoretical Physics #Hamiltonian (control theory) #Hamiltonian constraint #Lie algebra #Loop quantum gravity #Mathematical physics #Mathematics #Neonatal Health and Biochemistry #Noncommutative and Quantum Gravity Theories #Physics #Poisson bracket #Quantum #Quantum field theory #Quantum gravity #Quantum mechanics #gr-qc #hep-th
paper · pdf · doi:10.1103/physrevd.87.044039
published as Phys.Rev. D87 (2013) 4, 044039 · 57 pages, 9 figures
arxiv created 2012/10/25 · openalex publication_date 2013/02/19 · arxiv updated 2015/01/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The GNewton\ensuremath→0 limit of Euclidean gravity introduced by Smolin is described by a generally covariant U(1)3 gauge theory. The Poisson-bracket algebra of its Hamiltonian and diffeomorphism constraints is isomorphic to that of gravity. Motivated by recent results in parametrized field theory and by the search for an anomaly-free quantum dynamics for loop quantum gravity, the quantum Hamiltonian constraint of density weight 4/3 for this U(1)3 theory is constructed so as to produce a nontrivial loop quantum gravity type representation of its Poisson brackets through the following steps. First, the constraint at finite triangulation and the commutator between a pair of such constraints are constructed as operators on the ``charge'' network basis. Next, the continuum limit of the commutator is evaluated with respect to an operator topology defined by a certain space of ``vertex smooth'' distributions. Finally, the operator corresponding to the Poisson bracket between a pair of Hamiltonian constraints is constructed at finite triangulation in such a way as to generate a ``generalized'' diffeomorphism and its continuum limit is shown to agree with that of the commutator between a pair of finite-triangulation Hamiltonian constraints. Our results, in conjunction with the recent work of Henderson, Laddha and Tomlin in a (2+1)-dimensional context, constitute the necessary first steps toward a satisfactory treatment of the quantum dynamics of this model.