  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 $d_s^{\rm UV}\leq D$ in the ultraviolet
(UV). We apply this general result to Stelle theory of renormalizable gravity,
which attains the universal value $d_s^{\rm UV}=2$ for any dimension $D$.
