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New strings for old Veneziano amplitudes. II: group-theoretic treatment. (English) Zbl 1116.81066

Part I, see the author, Int. J. Geom. Methods Mod. Phys. 2, No. 4, 563–584 (2005; Zbl 1081.81093), for Part III see the author, ibid. J. Geom. Phys. 56, No. 9, 1433–1472 (2006; Zbl 1117.81115).
The original amplitude proposed by Veneziano was to explain the \(2\to2\) scattering of scalar (spin-0) pions. This amplitude is essentially the Euler Beta function and has a remarkable duality property (quite unlike amplitudes in quantum field theory), in the so-called \(s\)- and \(t\)-channels. The pions satisfy equations of motion (they are “on mass-shell”) and hence a set of functions of their momenta, called the Mandelstam variables, satisfy the relation that their sum adds to \(-1\). Starting with this property, a generalized relation is written as a homogeneous equation. The present paper is one in a series of papers in which the author explores mathematical properties of Veneziano-like amplitudes that satisfy this type of relation. It may also be of interest to note that the Veneziano amplitude has been generalized by considering its quantum extension (e.g. D. Coon, Phys. Lett., B 29, 669 ff. (1969); M. Baker and D. Coon, Phys. Rev. D 2, 2349 (1970)]) (\(q\)-deformation) as well as to the field of \(p\)-adic numbers [B. Grossmann, Phys. Lett., B 197, 101 ff. (1987; Zbl 0694.22006); I. Volovich, Classical Quantum Gravity 4, No. 4, L83–L87 (1987)].

MSC:

81T99 Quantum field theory; related classical field theories
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
13A50 Actions of groups on commutative rings; invariant theory
20F55 Reflection and Coxeter groups (group-theoretic aspects)
20G45 Applications of linear algebraic groups to the sciences
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