Quantum spectral dimension in quantum field theory

We reinterpret the spectral dimension of spacetimes as the scaling of an effective self-energy transition amplitude in quantum field theory, 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 non-relativistic diffusion equation in an abstract diffusion time; (b) it solves the problem of negative probabilities known for higher-order and non-local 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 $\ds$ 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 $\ds^{\rm UV}\leq D$ in the ultraviolet (UV). We apply this general result to Stelle theory of renormalizable gravity, which attains the universal value $\ds^{\rm UV}=2$ for any dimension $D$.