  We propose a field theory which lives in fractal spacetime and is argued to
be Lorentz invariant, power-counting renormalizable, ultraviolet finite, and
causal. The system flows from an ultraviolet fixed point, where spacetime has
Hausdorff dimension 2, to an infrared limit coinciding with a standard
four-dimensional field theory. Classically, the fractal world where fields live
exchanges energy momentum with the bulk with integer topological dimension.
However, the total energy momentum is conserved. We consider the dynamics and
the propagator of a scalar field. Implications for quantum gravity, cosmology,
and the cosmological constant are discussed.
