Quantum spectral dimension in quantum field theory

Abstract

We reinterpret the spectral dimension of spacetimes as the scaling of an effective self-energy transition amplitude in quantum field theory (QFT), when the system is probed at a given resolution. This picture has four main advantages: (a) it dispenses with the usual interpretation (unsatisfactory in covariant approaches) where, instead of a transition amplitude, one has a probability density solving a nonrelativistic diffusion equation in an abstract diffusion time; (b) it solves the problem of negative probabilities known for higher-order and nonlocal dispersion relations in classical and quantum gravity; (c) it clarifies the concept of quantum spectral dimension as opposed to the classical one. We then consider a class of logarithmic dispersion relations associated with quantum particles and show that the spectral dimension $d_s$ of spacetime as felt by these quantum probes can deviate from its classical value, equal to the topological dimension $D$. In particular, in the presence of higher momentum powers it changes with the scale, dropping from $D$ in the infrared (IR) to a value $d_s^{\mathrm{UV}}\le D$ in the ultraviolet (UV). We apply this general result to Stelle theory of renormalizable gravity, which attains the universal value $d_s^{\mathrm{UV}}=2$ for any dimension $D$.

26 pages, 3 figures; v2: discussion clarified and improved at several points, typos corrected, results unchanged; v3: some material confined to an appendix, discussion streamlined, results unchanged

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• ABC of multi-fractal spacetimes and fractional sea turtles
• Thermal dimension of quantum spacetime
• Spontaneous Dimensional Reduction in Quantum Gravity
• On the UV dimensions of Loop Quantum Gravity
• Deformed symmetries in noncommutative and multifractional spacetimes
• Standard Model in multiscale theories and observational constraints
• Dimensional reduction in manifold-like causal sets
• Dimensional flow in the kappa-deformed space-time
• Ultraviolet dimensional reduction of spacetime with zero-point length