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Generalized functions for quantum fields obeying quadratic exchange relations. (English) Zbl 0986.81071

Proc. Steklov Inst. Math. 228, 81-92 (2000) and Tr. Mat. Inst. Steklova 228, 90-100 (2000).
Summary: The axiomatic formulation of quantum field theory (QFT) of the 1950’s in terms of fields defined as operator valued Schwartz distributions is re-examined in the light of subsequent developments. These include, on the physical side, the construction of a wealth of (2-dimensional) soluble QFT models with quadratic exchange relations, and, on the mathematical side, the introduction of the Colombeau algebras of generalized functions. Exploiting the fact that energy positivity gives rise to a natural regularization of Wightman distributions as analytic functions in a tube domain, we argue that the flexible notions of Colombeau theory which can exploit particular regularizations is better suited (than Schwartz distributions) for a mathematical formulation of QFT.
For the entire collection see [Zbl 0952.00070].

MSC:

81T05 Axiomatic quantum field theory; operator algebras
46F30 Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.)
46N50 Applications of functional analysis in quantum physics
81T10 Model quantum field theories
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