Sunday, February 27, 2005

Veneziano Amplitude for Winding Strings

It seems I am caught in strange world where topological functions are happening and if such tubes could contract and then expand then what energy amplitudes on tree models would say the string should have this much energy, and then as the energy grows that the amplittude of the string is also changing? Here the loops would have varying energy determinations that would allow the loop to twist and turn?

Ramzi R. Khuri March 11 1993

String configurations with nonzero winding number describe soliton string states. We compute the Veneziano amplitude for the scattering of arbitrary winding states and show that in the large radius limit the strings always scatter trivially and with no change in the individual winding numbers of the strings. In this limit, then, these states scatter as true solitons.

In demonstration Greg Egan's site for the use of Animations this particular link was strange to me if something was considered in this link.

When you play with the coordinates you realize the energy changes that can take place in the loop. Equally important was when you observe the faces of the directions when these coordiante are selected. To me something was triggered when it was understood the the euclidean directions actually could been view from these faces, six in all if held in context of the higher dimenisons.

Unified treatment: analyticity, Regge trajectories, Veneziano amplitude, fundamental regions and Moebius transformations
Abdur Rahim Choudhary

In this paper we present a unified treatment that combines the analyticity properties of the scattering amplitudes, the threshold and asymptotic behaviors, the invariance group of Moebius transformations, the automorphic functions defined over this invariance group, the fundamental region in (Poincaré) geometry, and the generators of the invariance group as they relate to the fundamental region. Using these concepts and techniques, we provide a theoretical basis for Veneziano type amplitudes with the ghost elimination condition built in, related the Regge trajectory functions to the generators of the invariance group, constrained the values of the Regge trajectories to take only inverse integer values at the threshold, used the threshold behavior in the forward direction to deduce the Pomeranchuk trajectory as well as other relations. The enabling tool for this unified treatment came from the multi-sheet conformal mapping techniques that map the physical sheet to a fundamental region which in turn defines a Riemann surface on which a global uniformization variable for the scattering amplitude is calculated via an automorphic function, which in turn can be constructed as a quotient of two automorphic forms of the same dimension.


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