2014/01/31 by Philipp A. Hoehn, Philipp A. Höhn · 2 citations
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Covariant transformation #Discretization #Granularity #Hilbert space #Noncommutative and Quantum Gravity Theories #Observable #Propagator #Quantum #Quantum Mechanics and Applications #Quantum state #Time evolution #gr-qc #hep-lat #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1063/1.4890558
published as J. Math. Phys. 55, 083508 (2014) · 45 pages, 1 appendix, 6 figures (additional explanations, now matches published version)
openalex publication_date 2014/08/01 · arxiv created 2014/10/02 · arxiv updated 2014/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
A temporally varying discretization often features in discrete gravitational systems and appears in lattice field theory models subject to a coarse graining or refining dynamics. To better understand such discretization changing dynamics in the quantum theory, an according formalism for constrained variational discrete systems is constructed. While this paper focuses on global evolution moves and, for simplicity, restricts to flat configuration spaces \documentclass[12pt]minimal\begindocument\mathbb RN\enddocumentRN, a Paper II [P. A. Höhn, “Quantization of systems with temporally varying discretization. II. Local evolution moves,” J. Math. Phys., e-print arXiv:1401.7731 [gr-qc].] discusses local evolution moves. In order to link the covariant and canonical picture, the dynamics of the quantum states is generated by propagators which satisfy the canonical constraints and are constructed using the action and group averaging projectors. This projector formalism offers a systematic method for tracing and regularizing divergences in the resulting state sums. Non-trivial coarse graining evolution moves lead to non-unitary, and thus irreversible, projections of physical Hilbert spaces and Dirac observables such that these concepts become evolution move dependent on temporally varying discretizations. The formalism is illustrated in a toy model mimicking a “creation from nothing.” Subtleties arising when applying such a formalism to quantum gravity models are discussed.