Renormalization : A number theoretical model
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Submission date: 25. Jan. 2006
published in: Communications in mathematical physics, 277 (2008) 3, p. 627-641
DOI number (of the published article): 10.1007/s00220-007-0392-2
MSC-Numbers: 16W30, 30B50, 11A15, 81T15, 81T16
Keywords and phrases: hopf algbera, renormalization, dirichlet series, dirichlet convolution, multiplicative arithmetic functions
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We analyse the Dirichlet convolution ring of arithmetic number theoretic functions. It turns out to fail to be a Hopf algebra on the diagonal, due to the lack of complete multiplicativity of the product and coproduct. A related Hopf algebra can be established, which however overcounts the diagonal. We argue that the mechanism of renormalization in quantum field theory is modelled after the same principle. Singularities hence arise as a (now continuously indexed) overcounting on the diagonals. Renormalization is given by the map from the auxiliary Hopf algebra to the weaker multiplicative structure, called Hopf gebra, rescaling the diagonals.