Gamma convergence and renormalization group: Two sides of a coin?

Abstract

We discuss, both from the point of view of Gamma convergence and from the point of view of the renormalization Group, the zero range strong contact interaction of three non-relativistic massive particles. Formally, the potential term is g (delta (x_3-x_1) + delta (x_3 -x_2)), ;, g < 0 and is the limit epsilon rightarrow 0 of approximating potentials V_epsilon (|x_i -x_3|) = epsilon ^{-3} V ( frac{|x_i - x_3|}{epsilon }) , V( x) in L^1(R^3) cap L^2 (R^3) . The presence of a delta function in the limit does not allow the use of standard tools of functional analysis. In the first approach (European Phys. J. Plus 136-363, 2021), (European Phys. J. Plus 1136-1161, 2021), we introduced a map mathcal{K}, called Krein Map , from L^2 (R^9) to a space (Minlos space) mathcal{M}) of more singular functions. In { mathcal M}, the system is represented by a one parameter family of self-adjoint operators. In the topology of L^2 (R^9), the system is an ordered family of weakly closed quadratic forms. By Gamma convergence, the infimum is a self-adjoint operator, the Hamiltonian H of the system. Gamma convergence implies resolvent convergence (An Introduction to Gamma Convergence Springer 1993) but not operator convergence!. This approach is variational and non-perturbative. In the second approach, perturbation theory is used. At each order of perturbation theory, divergences occur when epsilon rightarrow 0. A finite renormalized Hamiltonian H_R is obtained by redefining mass and coupling constant at each order of perturbation theory. In this approach, no distinction is made between self-adjoint operators and quadratic forms. One expects that H = H_R , i.e., that “renormalization” amounts to the difference between the Hamiltonian obtained by quadratic form convergence and the one obtained by Gamma convergence. We give some hints, but a formal proof is missing. For completeness, we discuss briefly other types of zero-range interactions.