Universal Bounds for Large Determinants from Non–Commutative Ho ̈lder Inequalities in Fermionic Constructive Quantum Field Theory
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Efficiently bounding large determinants is an essential step in non–relati- vistic fermionic constructive quantum field theory, because, together with the summability of the interaction and the covariance, it implies the abso- lute convergence of the perturbation expansion of all correlation functions in terms of powers of the strength u ∈ 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 pertur- bation series at u = 0 of any fermionic quantum field theory is ensured by the decay properties of the covariance and the interparticle interaction alone. Our proofs use Ho ̈lder inequalities for general non–commutative Lp–spaces derived by Araki and Masuda [AM].