A concise review of the notions of elliptic functions, modular forms, and θ–functions is provided, devoting most of the paper to applications to Conformal Field Theory (CFT), introduced within the axiomatic framework of quantum field theory. Many features, believed to be peculiar to chiral 2D (= two dimensional) CFT, are shown to have a counterpart in any (even dimensional) globally conformal i...

The perturbative treatment of quantum field theory is formulated within the framework of algebraic quantum field theory. We show that the algebra of interacting fields is additive, i.e. fully determined by its subalgebras associated to arbitrary small subregions of Minkowski space. We also give an algebraic formulation of the loop expansion by introducing a projective system A of observables “u...

The perturbative treatment of quantum field theory is formulated within the framework of algebraic quantum field theory. We show that the algebra of interacting fields is additive, i.e. fully determined by its subalgebras associated to arbitrary small subregions of Minkowski space. We also give an algebraic formulation of the loop expansion by introducing a projective system A(n) of observables...

The aim of the following lecture is to give some rough overview over most essential structures underlying all working quantum theoretical models as well as axiomatic and algebraic quantum field theory. Before specializing to ordinary Hilbert space quantum theory it will be explained why common sense reasoning cannot be applied naively and the pragmatic procedure (quantum reasoning) of getting a...