Fractal Theory Space: Spacetime of Noninteger Dimensionality

We construct matter field theories in “theory space” that are 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 aboveM is controlled by an exponent in a scaling relation of the vacuum energy (ColemanWeinberg potential), and corresponds to the dimensionality. For truncated-ssimplex lattices with coordination number s the spacetime dimensionality is 1+(3+2 ln(s)/ ln(s+2)). 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. ∗e-mail: [email protected]