https://doi.org/10.1140/epjp/s13360-022-02939-6
Regular Article
Gamma convergence and renormalization group: Two sides of a coin?
1
Math. Dept Sapienza, Rome, Italy
2
Mathematics Area Sissa, Trieste, Italy
Received:
26
February
2022
Accepted:
9
June
2022
Published online:
28
June
2022
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 
and is the limit 
of approximating potentials 
, 
. 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 
, called Krein Map , from 
to a space (Minlos space) 
) of more singular functions. In 
, the system is represented by a one parameter family of self-adjoint operators. In the topology of 
, 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 . A finite renormalized Hamiltonian 
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 
, 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.
This paper is dedicated to Elliott Lieb, an outstanding scientist and a friend, in celebration of his 90th birthday.
Missing Open Access funding information has been added in the Funding Note.
Copyright comment corrected publication 2022
© The Author(s) 2022. corrected publication 2022
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
