Generalised spectral dimensions in non-perturbative quantum gravity. (English) Zbl 1522.83077
Summary: The seemingly universal phenomenon of scale-dependent effective dimensions in non-perturbative theories of quantum gravity has been shown to be a potential source of quantum gravity phenomenology. The scale-dependent effective dimension from quantum gravity has only been considered for scalar fields. It is, however, possible that the non-manifold like structures, that are expected to appear near the Planck scale, have an effective dimension that depends on the type of field under consideration. To investigate this question, we have studied the spectral dimension associated to the Laplace-Beltrami operator generalised to \(k\)-form fields on spatial slices of the non-perturbative model of quantum gravity known as causal dynamical triangulations. We have found that the two-form, tensor and dual scalar spectral dimensions exhibit a flow between two scales at which an effective dimension appears. However, the one-form and vector spectral dimensions show only a single effective dimension. The fact that the one-form and vector spectral dimension do not show a flow of the effective dimension can potentially be related to the absence of a dispersion relation for the electromagnetic field, but dynamically generated instead of as an assumption.
MSC:
| 83C45 | Quantization of the gravitational field |
| 83C15 | Exact solutions to problems in general relativity and gravitational theory |
| 62D20 | Causal inference from observational studies |
| 37C45 | Dimension theory of smooth dynamical systems |
| 47A10 | Spectrum, resolvent |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 52C15 | Packing and covering in \(2\) dimensions (aspects of discrete geometry) |
| 83C50 | Electromagnetic fields in general relativity and gravitational theory |
Keywords:
non-perturbative quantum gravity; causal dynamical triangulations; scale-dependent dimension; running spectral dimension; Hodge Laplacian; discrete geometryReferences:
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