Mass and infinite dimensional geometry

Abstract

I unravel an elegant geometric meaning of the mass of the lowest energy excited state of a renormalizable quantized field theory by studying the weighted geometry of the classical configuration space of the theory. A suitably defined regularized Bakry-Emery Ricci curvature of these infinite dimensional spaces controls the spectra of the corresponding quantum Hamiltonians. The Ricci curvature part of the full Bakry-Emery Ricci curvature appears to be purely quantum in nature. This geometric contribution to the spectra in the context of quantum field theory has not been studied previously to my knowledge. Assuming the existence of rigorous quantization, I present a few problems starting from massive free particles to the non-abelian Yang-Mills theory. A remarkable property is observed in the large $N$ Yang-Mills theory, where a non-trivial mass gap is preserved. This occurs due to the fact that the regularized Bakry-Emery Ricci curvature that is responsible for the gap of the configuration space scales as $g^{2}_{YM}N=\lambda$ ('t Hooft coupling) that remains invariant.