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

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 u is an element of R 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 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 Holder inequalities for general non-commutative L-p-spaces derived by Araki and Masuda [Positive cones and L-p-spaces for von Neumann algebras, Publ. RIMS, Kyoto Univ. 18 (1982) 339-411].