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Quantum field-theory models on fractal spacetime. I: Introduction and overview. II: Hierarchical propagators. (English) Zbl 0713.58051

Commun. Math. Phys. 125, No. 4, 613-636 (1989); ibid. 126, No. 1, 85-101 (1989).
Summary: Part I explores the possibility of giving a nonperturbative definition of the quantum field-theory models in noninteger dimensions, which have been previously studied by Wilson and others using analytic continuation of dimension in perturbation integrals. The method employed here is to base the models on fractal point-sets of noninteger Hausdorff-Besicovitch dimension. Two types of scalar-field models are considered: the one has its propagator \((=covariance\) operator kernel) given by a proper-time or heat-kernel representation and the other has a hierarchical propagator. The fractal lattice version of the proper-time propagator is shown to be reflection-positive. The hierarchical models are introduced and their properties discussed on an informal basis.
Part II applies rigorous renormalization group methods to the hierarchical models to establish the existence of field theories with non-Gaussian ultraviolet renormalization group fixed points in \(4- \epsilon\) dimensions.

MSC:

37N99 Applications of dynamical systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
81T16 Nonperturbative methods of renormalization applied to problems in quantum field theory
81T17 Renormalization group methods applied to problems in quantum field theory
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