Abstract
Quantum fields, from a mathematical point of view, are highly singular. These fields are believed to describe the interactions of elementary particles. For the interaction of electrons with light (photons), the quantum field description is exact within the limits of experimental accuracy (5 significant figures). For these reasons, i.e. the mathematical difficulties and the importance to physics, the problem of formulating the mathematical foundations of quantum field theory has attracted the attention of both mathematicians and physicists over a period of several decades. On the side of the physicists, the most striking achievements were the calculation in the late 1940’s and early 1950’s of the Lamb shift and the anomalous magnetic moment of the electron together with the development of the renormalization method on which these calculations were based. Of the mathematicians, J. von Neumann was the first to realize that new mathematical theories would be required to formulate quantum field theory correctly and this realization was one of the motives for developing the theory of operator algebras.
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