Positivity and Convergence in Fermionic Quantum Field Theory

TL;DR

This paper establishes norm bounds ensuring the convergence of perturbation theory in fermionic quantum field theory, linking positivity properties of matrices to the applicability of Gram bounds, with implications for renormalization group analysis.

Contribution

It provides a simple, elementary proof connecting positivity of matrices to the convergence of fermionic perturbation theory, enhancing understanding of Gram bounds in this context.

Findings

Norm bounds imply perturbation convergence under certain conditions

Positivity properties are key to applying Gram bounds with uniform constants

Results support stability and decoupling in fermionic interactions

Abstract

We derive norm bounds that imply the convergence of perturbation theory in fermionic quantum field theory if the propagator is summable and has a finite Gram constant. These bounds are sufficient for an application in renormalization group studies. Our proof is conceptually simple and technically elementary; it clarifies how the applicability of Gram bounds with uniform constants is related to positivity properties of matrices associated to the procedure of taking connected parts of Gaussian convolutions. This positivity is preserved in the decouplings that also preserve stability in the case of two-body interactions.