# Universal Bounds for Large Determinants from Non-Commutative H\"older Inequalities in Fermionic Constructive Quantum Field Theory

## TL;DR

This paper establishes universal bounds for large fermionic determinants using non-commutative H"older inequalities, ensuring convergence of perturbation series in fermionic quantum field theories under certain decay conditions.

## Contribution

It provides the sharpest known universal bounds for large fermionic determinants applicable to all one-particle Hamiltonians, including unbounded cases.

## Key findings

- Smallest universal determinant bound is exactly 1.
- Convergence of perturbation series is guaranteed under sufficient decay of matrix entries.
- Bounds are derived using non-commutative H"older inequalities by Araki and Masuda.

## Abstract

Efficiently bounding large determinants is an essential step in non-relativistic fermionic constructive quantum field theory to prove the absolute convergence of the perturbation expansion of correlation functions in terms of powers of the strength $u\in \mathbb{R}$ of the interparticle interaction. We provide, for large determinants of fermionic convariances, sharp bounds which hold for all (bounded and unbounded, the latter not being limited to semibounded) one-particle Hamiltonians. We find the smallest universal determinant bound to be exactly $1$. In particular, the convergence of perturbation series at $u=0$ of any fermionic quantum field theory is ensured if the matrix entries, with respect to some fixed orthonormal basis, of the covariance and the interparticle interaction decay sufficiently fast. Our proofs use H\"older inequalities for general non-commutative $L^{p}$-spaces derived by Araki and Masuda.