Few-body hierarchy in non-relativistic functional renormalization group equations and a decoupling theorem

Stefan Floerchinger

doi:10.1016/j.nuclphysa.2014.04.013

Abstract

For non-relativistic quantum field theory in the few-body limit with instantaneous interactions it is shown within the functional renormalization group formalism that propagators are not renormalized and that the renormalization group equations of one-particle irreducible vertex functions are governed by a hierarchical structure. This hierarchy allows to solve the equations in the n-body sector without knowledge or assumptions about the m-body sectors where m>n.

Highlights

In classical mechanics as well as non-relativistic quantum mechanics it is an evident fact that one can solve the equations governing n interacting massive particles without having to have any knowledge or making any assumptions about the solution to the corresponding equations for n + 1 or more particles
In this paper we review the general arguments for the decoupling between n-particle problems in non-relativistic quantum field theory and discuss explicitly how it is realized in the formalism of the functional renormalization group
We introduce briefly the functional renormalization group formalism and the particular adaptations that must be made to apply it in our situation

Summary

Introduction

In classical mechanics as well as non-relativistic quantum mechanics it is an evident fact that one can solve the equations governing n interacting massive particles without having to have any knowledge or making any assumptions about the solution to the corresponding equations for n + 1 or more particles This important and fundamental feature is not given for relativistic quantum field theory, . In this paper we review the general arguments for the decoupling between n-particle problems in non-relativistic quantum field theory and discuss explicitly how it is realized in the formalism of the functional renormalization group. The prime example for this is non-relativistic quantum field theory with instantaneous interactions in the few-body limit but there exist other examples such as statistical reaction–diffusion systems [2].

Perturbative loop expansion

Classification and decomposition of correlation functions

Non-perturbative renormalization group formalism

The decoupling theorem and its implications

Proof of the decoupling theorem

Conclusions