Derivation of Freund-Witten adelic formula for four-point Veneziano amplitudes
type: Item
instance of: scholarly article
title: Derivation of Freund-Witten adelic formula for four-point Veneziano amplitudes
zbMATH Open document ID: 0916.11062
DOI: 10.1007/BF01017255
published in: Theoretical and Mathematical Physics
publication date: 1994-07-13T00:00:00Z
review text: \textit{P. G. O. Freund} and \textit{E. Witten} [Phys. Lett. B 199, 191-194 (1987)] and, independently, \textit{I. V. Volovich} [Lett. Math. Phys. 16, 61-67 (1988; Zbl 0668.43006)] proposed an adelic formula for the four-point crossing-symmetric Veneziano amplitudes (for open strings): \[ B_\infty(- a(s),-a(t)) \prod^\infty_{p= 2} B_p(-a(s), -a(t))= 1,\tag{1} \] where \(B_\infty\) is the Gel'fand-Graev \(B\) function, \(B_p\) is the corresponding \(p\)-adic \(B\) function, \(\alpha(s)= 1+ s/2\), and \(s\), \(t\), \(u\) are Mandelstam variables, \(s+ t+ u= -8\). However, the infinite product in (1) diverges for all complex \(s\) and \(t\), and therefore it is not clear how (1) should be understood. Here, using the adelic group (Tate's formula), the author proposes a regularization for the divergent infinite product of \(p\)-adic \(\Gamma\) functions: \[ \Gamma_p(\alpha)= {1- p^{\alpha- 1}\over 1- p^{-\alpha}},\quad p= 2,3,5,\dots, \] and the adelic formula \[ \text{reg}\prod^\infty_{p= 2} \Gamma_p(\alpha)= {\zeta(\alpha)\over \zeta(1- \alpha)}, \] where \(\zeta(\alpha)\) is the Riemann \(\zeta\) function, is proved.
Mathematics Subject Classification ID: 11S80
Mathematics Subject Classification ID: 81T30
Mathematics Subject Classification ID: 11R56
zbMATH DE Number: 589577
zbMATH Keywords: \(p\)-adic string amplitudes
zbMATH Keywords: Tate's formula
zbMATH Keywords: Veneziano amplitudes
zbMATH Keywords: Gel'fand-Graev \(B\) function
zbMATH Keywords: \(p\)-adic \(B\) function
zbMATH Keywords: adelic group
zbMATH Keywords: regularization
zbMATH Keywords: divergent infinite product
author: V. S. Vladimirov
reviewed by: Olaf Ninnemann
MaRDI profile type: Publication
cites work: Harmonic analysis and p-adic strings
cites work: Q3346515
cites work: Q4003475
full work available at URL: https://doi.org/10.1007/bf01017255
OpenAlex ID: W2079051014
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description: scientific article; zbMATH DE number 589577