Fractal theory space: Spacetime of noninteger dimensionality

We construct matter field theories in a “theory space” that is fractal, and invariant under geometrical renormalization group (RG) transformations. We treat in detail complex scalars, and discuss issues related to fermions, chirality, and Yang-Mills gauge fields. In the continuum limit these models describe physics in a noninteger spatial dimension which appears above a RG invariant “compactification scale” M. The energy distribution of KK modes above M is controlled by an exponent in a scaling relation of the vacuum energy (Coleman-Weinberg potential), and corresponds to the dimensionality. For truncated-$s$-simplex lattices with coordination number s the spacetime dimensionality is $1+[3+2\ln \mathrm{}(s)/\ln \mathrm{}(s+2\mathrm{}\left)\mathrm{}\right].$ The computations in theory space involve subtleties, owing to the $1+3$ kinetic terms, yet the resulting dimensionalites are equivalent to thermal spin systems. Physical implications are discussed.