J.-B. Bru and W. de Siqueira Pedra
Universal Bounds for Large Determinants from Non Commutative H older Inequalities in Fermionic Constructive Quantum Field Theory
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ABSTRACT. Efficiently bounding large determinants is an essential step in non relativistic fermionic constructive quantum field theory, because, together with the summability of the interaction and the covariance, it implies the absolute convergence of the perturbation expansion of all correlation functions in terms of powers of the strength u 2 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 by
the decay properties of the covariance and the interparticle interaction alone. Our proofs use H older inequalities for general non commutative Lp spaces derived by Araki and Masuda.