Universal bounds for large determinants from non-commutative Hölder inequalities in fermionic constructive quantum field theory

Abstract

Efficiently bounding large determinants is an essential step in non-relativistic constructive quantum field theory to prove the absolute convergence of the perturbation expansion of correlation functions in terms of powers of the strength [Formula: see text] of the interparticle interaction. We provide, for large determinants of fermionic covariances, 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 [Formula: see text]. In particular, the convergence of perturbation series at [Formula: see text] 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 Hölder inequalities for general non-commutative [Formula: see text]-spaces derived by Araki and Masuda [Positive cones and [Formula: see text]-spaces for von Neumann algebras, Publ. RIMS[Formula: see text] Kyoto Univ. 18 (1982) 339–411].