Quantum field theory on multifractal space–time: Varying dimension and ultraviolet completeness

Inspired by various quantum gravity approaches, we explore quantum field theory where space–time exhibits scaling properties and dimensional reduction with changing energy scales, effectively behaving as a multifractal manifold. Working within canonical quantization, we demonstrate how to properly quantize fields in such a multifractal space–time. Our analysis reveals that a nondifferentiable nature of space–time is not merely compatible with quantum field theory but significantly enhances its mathematical foundation. Most notably, this approach ensures perturbative UV finiteness and improved behavior of the series expansion and enables rigorous construction of the S-matrix in the interaction picture by breaking vacuum translational invariance. The multifractal structure tames dominant, large-order divergence sources in the perturbative series and resolves the Landau pole problem through asymptotic safety, substantially improving the theory’s behavior in the deep ultraviolet regime. Our formulation preserves all established predictions of standard quantum field theory at low energies while offering novel physical behaviors at high energy scales.