Abstract
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.
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References
Apostol, T.M.: Introduction to Analytic Number Theory. New York: Springer-Verlag, 1979 [fouth printing 1995]
Bourbaki N. (1989). Elements of Mathematics: Algebra I Chapters 1–3. Springer-Verlag, Berlin
Brouder C., Schmitt W. (2007). Renormalization as a functor on bialgebra. J. Pure Appl. Alg. 209: 477–495
Brüdern J. (1995). Einführung in die analytische Zahlentheorie. Springer-Verlag, Berlin
Carlitz L. (1971). Problem E 2268. Amer. Math. Monthly 78: 1140
Dehaye P.-O. (2002). On the structure of the group of multiplicative arithmetical functions. Bull. Belg. Math. Soc. Simon Stevin 9(1): 15–21
Ebrahimi-Fard K., Kreimer D. (2005). The Hopf algebra approach to Feynman diagram calculations. J. Phys. A: Math. Gen. 38: R385–R407
Epstein H., Glaser V. (1973). The role of locality in perturbation theory. Ann. Inst. Henri Poincaré 19: 211–295
Fauser B. (2001). On the Hopf-algebraic origin of Wick normal-ordering. J. Phys. A: Math. Gen. 34: 105–115
Fauser, B.: A Treatise on Quantum Clifford Algebras. Konstanz, 2002, Habilitationsschrift, available at http://arxiv.org/list/math.QA/0202059, 2002
Fauser B., Jarvis P.D. (2004). A Hopf laboratory for symmetric functiuons. J. Phys. A: Math. Gen. 37(5): 1633–1663
Fauser B., Jarvis P.D. (2007). The Dirichlet Hopf algebra of arithmetics. J. Knot Theor., its Ramif. 16(4): 379–438
Fauser, B., Jarvis, P.D.: The Hopf algebra of plethysms. Work in progress, 2007
Fauser B., Jarvis P.D., King R.C., Wybourne B.G. (2006). New branching rules induced by plethysm. J. Phys A: Math. Gen. 39: 2611–2655
Fauser B., Oziewicz Z. (2001). Clifford Hopf gebra for two dimensional space. Misc. Alg. 2(1): 31–42
Lambek J. (1966). Arithmetical functions and distributivity. Amer. Math. Monthly 73: 969–973
Lawvere F.W., Rosebrugh R. (2003). Sets for Mathematics. Cambridge Univ. Press, Cambridge
Leroux P. (1975). Les Catégories Möbius. Cahiers Top. Géom. Différ. Catég. 16: 280–282
Leroux P. (1990). Reduced matrices and q-log-concavity properties of q-Stirling numbers. J. Combin. Theory Ser. A 54: 64–84
Oziewicz Z. (1997). Clifford Hopf gebra and biuniversal Hopf gebra. Czech. J. Phys. 47(12): 1267–1274
Petermann A. (2000). The so-called Renormalization Group method applied to the specific prime numbers logarithmic decrease Eur. Phys. J. C 17: 367–369
Schwab E.D. (2004). Characterizations of Lambek-Carlitz type. Arch. Math. (Brno) 40: 295–300
Selberg A. (1949). An elementary proof of the prime number theorem. Ann. Math. 50: 305–313
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Communicated by A. Connes
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Fauser, B. Renormalization: A Number Theoretical Model. Commun. Math. Phys. 277, 627–641 (2008). https://doi.org/10.1007/s00220-007-0392-2
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DOI: https://doi.org/10.1007/s00220-007-0392-2