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Renormalization: A number theoretical model. (English) Zbl 1165.81035
Coproducts of the Dirichlet convolution ring of arithmetic functions are studied. They are biassociative, biunital and antipodal convolutions, but not satisfy the homomorphism axiom and named Hopf gebra. Then a method to reestablish a Hopf algebra structure from a Hopf gebra is presented. The author says this method can be regarded as a number theoretical model of renormalization.
The Dirichlet convolution ring of arithmetic functions is reviewed in §1 as follows: Let \(f: \mathbb{N}\to\mathbb{C}\) be an arithmetic function. Then its generating function is introduced by \(f(s)=\sum_{n\geq 1}{f(n)\over n^s}\), \(s\in\mathbb{C}\), \(f(s)g(s)\) is the convolution \[ (f* g)(s)= \sum_{n\geq 1}\,\sum_{d|n} {f(d)g(n/d)\over n^s}. \] An arithmetic function \(f\) is called complete multiplicative if \(f(n\cdot m)= f(n)\cdot f(m)\) for all \(n\), \(m\). \(f\) is called multiplicative if \((n, m)= 1\), then \(f(n\cdot m)= f(n)\cdot f(m)\). As examples of multiplicative, but not completely multiplicative functions, the Möbius function, Euler totient function and von Mangoldt function are explained. Number theoretical model of renormalization in this paper uses difference of multiplicative and complete multiplicative functions.
In §2, using the Kronecker duality \((|): \mathbb{N}\times\mathbb{N}\to \mathbb{Z}_2\), \((n|m))= \delta_{n,m}\), the coproducts of additions and multiplication \[ \begin{aligned} \Delta^+(n) &= \sum_{n_1+ n_2=m} n_1\oplus n_2= n_{(1)}\oplus n_{(2)},\\ \Delta^.(n) &= \sum_{d|n} d\times{n\over d}= n_{[1]}\times n_{[2]},\end{aligned} \] are introduced (cf. B. Fauser and P. D. Javis [J. Knot Theory Ramifications 16, No. 4, 379-438 (2007; Zbl 1124.16032)], hereafter referred to as [1]). \(\Delta^.\) is a multiplicative function but not a complete multiplicative function (Prop. 2.9). As a consequence, the coproduct map does not satisfy the homomorphism axiom. Algebras with coproducts \(\Delta^+\) and \(\Delta^.\) become Hopf gebras. These are shown as Prop.2.5 and 2.6, in this section.
In §3, Hopf gebra is defined and discusses how to deform a Hopf gebra to a Hopf algebra. One of such a plan is the modified crossing [Z. Oziewicz, Czech. J. Phys. 47, No. 12, 1267–1274 (1997; Zbl 0948.16029) and B. Fauser and Z. Oziewicz, Clifford Hopf gebra for two-dimensional space. Misc. Alg. 2, 31–42 (2001)]. But the author says this plan is tedious for the practical use and the plan unrenormalization is proposed. In this plan, the coproduct \(\Delta^.\) is renormalized to \(\underline\Delta^.\); \(\underline\Delta^.(p)= \underline\Delta(p)\), if \(p\) is a prime number and \[ \underline\Delta^.(n\cdot m)= \underline\Delta^.(n)\cdot \underline\Delta^.(m), \] for any \(n.m\). It is stated the pairing \[ (n|m)= \prod_i \delta_{r_i,s_i} r_i!,\quad n= \prod_i p^{r_i}_i,\quad m= \prod_j p^{s_j}_j, \] dualizes the multiplication \(\cdot\) into \(\underline\Delta^.\) (Corol. 3.14. cf. [1]). To illustrate renormaliztion of coproduct by this procedure can be seen as a number theoretical model of traditional renormalization, the Bell series \(f_p(x)=\sum_{n\geq 0} f(p^n)x^n\), \(p\) is a prime number, is used (§4). If \(f\) is a complete multiplicative function, then \[ f_p(x)= {1\over 1-f(p)x}. \] Let \(g\) be a complete multiplicative function and \(g(1)= 1\), \(f\) a multiplicative function such that \(f(p^{n+1})= f(p) f(p^n)- g(p) f(p^{n-1})\). Then we have \[ f_p(x)= {1\over 1-f(p)x+ g(p)x^2} \] (cf. T. M. Apostol, Introduction to analytic number theory. New York etc.: Springer (1976; Zbl 0335.10001)]). Consequently, we obtain \[ f(m\cdot n)= f(m)f(n)- \sum_{d|\text{gcd}(n, m)} g(d) f\Biggl({m\cdot n\over d^2}\Biggr). \] The author says the second term of this right-hand side can be regarded as the counter term.
This paper has two Appendices which deal with characterization of complete multiplicativity, groups and subgroups of Dirichlet convolution and relation of renormalization group analysis in quantum field theory and number theory (cf. J. Lambek [Am. Math. Mon. 73, 969–973 (1966; Zbl 0152.03105)]; P.-Q. Dehaye [Bull. Belg. Math. Soc.-Simon Stevin 9, No. 1, 15–21 (2002; Zbl 1168.11301); A. Peterman [The so-called renormalization group method applied to the specific prime numbers logarithmic decrease, Eur. Phys. J. C17, 367–369 (2000)]).
MSC:
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
11M41 Other Dirichlet series and zeta functions
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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