  We construct a theory of fields living on continuous geometries with
fractional Hausdorff and spectral dimensions, focussing on a flat background
analogous to Minkowski spacetime. After reviewing the properties of fractional
spaces with fixed dimension, presented in a companion paper, we generalize to a
multi-fractional scenario inspired by multi-fractal geometry, where the
dimension changes with the scale. This is related to the renormalization group
properties of fractional field theories, illustrated by the example of a scalar
field. Depending on the symmetries of the Lagrangian, one can define two
models. In one of them, the effective dimension flows from 2 in the ultraviolet
(UV) and geometry constrains the infrared limit to be four-dimensional. At the
UV critical value, the model is rendered power-counting renormalizable.
However, this is not the most fundamental regime. Compelling arguments of
fractal geometry require an extension of the fractional action measure to
complex order. In doing so, we obtain a hierarchy of scales characterizing
different geometric regimes. At very small scales, discrete symmetries emerge
and the notion of a continuous spacetime begins to blur, until one reaches a
fundamental scale and an ultra-microscopic fractal structure. This fine
hierarchy of geometries has implications for non-commutative theories and
discrete quantum gravity. In the latter case, the present model can be viewed
as a top-down realization of a quantum-discrete to classical-continuum
transition.
