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Taming nonrenormalizability. (English) Zbl 1176.81077
Summary: Nonrenormalizable scalar fields, such as $\varphi^{4}_n, n \geqslant 5$, require infinitely many distinct counterterms when perturbed about the free theory, and lead to free theories when defined as the continuum limit of a lattice regularized theory restricted only to arbitrary mass and coupling constant renormalization. Based on the proposal that functional integrals for interacting nonrenormalizable models do not reduce to the expression for the free field functional integral as the coupling constant vanishes-a proposal supported by the fact that even the set of classical solutions for such models does not reduce to the set of free field solutions as the coupling constant vanishes-it has been conjectured that for nonrenormalizable models the interaction term acts partially as a hard core eliminating certain fields otherwise allowed by the free theory. As a consequence, interacting models are continuously connected to a pseudofree theory that takes into account the hard core as the coupling constant vanishes, and this general view is supported by simple quantum mechanical examples as well as soluble but nonrelativistic nonrenormalizable models. This paper proposes a pseudofree model for relativistic nonrenormalizable models about which it is argued that a perturbation expansion of the interaction is term-by-term divergence free.

81T10Model quantum field theories
81T15Perturbative methods of renormalization (quantum theory)
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