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p-adic Feynman and string amplitudes. (English) Zbl 0671.22011
It is shown that the scalar models of the field theory over p-adic numbers are the analogs of Dyson’s hierarchical models (after discretization). The standard methods of quantum field theory, in particular Feynman diagrams, renormalization and Wilson’s renormalization group are proven to have generalizations to the p-adic case. The explicit representation for p-adic Feynman amplitudes and for the string scattering Koba-Nielsen amplitudes is given. It appears that in the p- adic case the exact representation as a sum of elementary functions can be obtained.
Reviewer: P.Maslanka

MSC:
22E70 Applications of Lie groups to the sciences; explicit representations
11F33 Congruences for modular and \(p\)-adic modular forms
81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
83E15 Kaluza-Klein and other higher-dimensional theories
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
22E50 Representations of Lie and linear algebraic groups over local fields
53C80 Applications of global differential geometry to the sciences
74K05 Strings
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
81T17 Renormalization group methods applied to problems in quantum field theory
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