Φ⁴ on Rough Spaces: Constructing a Quantum Field Measure Beyond Euclidean Geometry

The Φ⁴ model is a fundamental random field from mathematical physics, but in high dimensions it is ill-defined without renormalization. In this talk I’ll explain how to build the renormalized Φ⁴ stochastic dynamics on rough metric spaces (including fractal-type geometries) and how this yields the associated Φ⁴ invariant measure. The main message is that stochastic quantization—originally developed in Euclidean settings—can be extended to genuinely non-smooth geometries, producing a rigorous interacting field measure on spaces where (heat) diffusion behaves very differently from the classical case.