Spectral dimension and diffusion in multi-scale spacetimes

Abstract

Starting from a classical-mechanics stochastic model encoded in a Langevin equation, we derive the natural diffusion equation associated with three classes of multiscale spacetimes (with weighted, ordinary, and "q-Poincaré" symmetries). As a consistency check, the same result is obtained by inspecting the propagation of a quantum-mechanical particle in a disordered environment. The solution of the diffusion equation displays a time-dependent diffusion coefficient and represents a probabilistic process, classified according to the statistics of the noise in the Langevin equation. We thus illustrate, also with pictorial aids, how spacetime geometries can be more completely catalogued not only through their Hausdorff and spectral dimension, but also by a stochastic process. The spectral dimension of multifractional spacetimes is then computed and compared with what was found in previous studies, where a diffusion equation with some open issues was assumed rather than derived. These issues are here discussed and solved, and they point towards the model with q-Poincaré symmetries.

23 pages, 3 multiple figures, 1 table. v2: introduction considerably expanded to highlight the main results, typos corrected, references added

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• Diffusion in multiscale spacetimes
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• Quantum scalar field theories with fractional operators
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• Fractal Spacetimes in Stochastic Gravity? -- Views from Anomalous Diffusion and the Correlation Hierarchy
• Spectral dimensions from the spectral action
• Varying electric charge in multiscale spacetimes
• Dimensional flow and fuzziness in quantum gravity: emergence of stochastic spacetime
• Quantum spectral dimension in quantum field theory
• Quantum field theory with varying couplings
• Deformed symmetries in noncommutative and multifractional spacetimes
• Standard Model in multiscale theories and observational constraints
• Renormalization of lattice-regularized quantum gravity models II. The case of causal dynamical triangulations
• Particle-physics constraints on multifractal spacetimes
• Finite entanglement entropy and spectral dimension in quantum gravity
• On observational signatures of multi-fractional theory