1996/01/09 by Henri Waelbroeck, Jose A. Zapata, José A Zapata
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Covariant Hamiltonian field theory #Covariant transformation #Euclidean geometry #Hamiltonian (control theory) #Hamiltonian lattice gauge theory #Homotopy and Cohomology in Algebraic Topology #Lattice (music) #Lattice field theory #Noncommutative and Quantum Gravity Theories #Observable #Superintegrable Hamiltonian system #gr-qc
paper · pdf · doi:10.1088/0264-9381/13/7/009
published as Class.Quant.Grav. 13 (1996) 1761-1768 · 10 pages of text. One figure available from J.A. Zapata upon request
arxiv created 1996/01/09 · openalex publication_date 1996/07/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We show that 't Hooft's representation of (2 + 1)-dimensional gravity in terms of flat polygonal tiles is closely related to a gauge-fixed version of the covariant Hamiltonian lattice theory. 't Hooft's gauge is remarkable in that it leads to a Hamiltonian which is a linear sum of vertex Hamiltonians, each of which is defined modulo . A cyclic Hamiltonian implies that `time' is quantized. However, it turns out that this Hamiltonian is constrained . If one chooses an internal time and solves this constraint for the `physical Hamiltonian', the result is not a cyclic function. Even if one quantizes following Dirac, the `internal time' observable does not acquire a discrete spectrum. We also show that in Euclidean three-dimensional lattice gravity, `space' can be either discrete or continuous depending on the choice of quantization. Finally, we propose a generalization of 't Hooft's gauge for Hamiltonian lattice formulations of topological gravity dimension four.