Natural cutoffs via compact symplectic manifolds. (English) Zbl 1332.83044
Summary: In the context of phenomenological models of quantum gravity, it is claimed that ultraviolet (UV) and infrared (IR) natural cutoffs can be realized from local deformations of the Hamiltonian systems. In this paper, we scrutinize this hypothesis and formulate a cutoff-regularized Hamiltonian system. The results show that while local deformations are necessary to have cutoffs, they are not sufficient. In fact, the cutoffs can be realized from globally-deformed Hamiltonian systems that are defined on compact symplectic manifolds. By taking the universality of quantum gravity effects into account, we then conclude that quantum gravity cutoffs are global (topological) properties of the symplectic manifolds. We justify our results by considering three well-known examples: the Moyal, Snyder and polymer-deformed Hamiltonian systems.
MSC:
| 83C45 | Quantization of the gravitational field |
| 53Z05 | Applications of differential geometry to physics |
| 54F65 | Topological characterizations of particular spaces |
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
| 81T15 | Perturbative methods of renormalization applied to problems in quantum field theory |
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