Scattering Amplitudes & Orbits of Cognitive Representations under Subgroup of Symplectic Group Respecting the Extension of Rationals

Abstract

In this article the ideas inspired by the work of number theorist Minhyong Kim are  applied to the construction of scattering amplitudes   with finite cognitive precision  in terms of cognitive representations and their orbits under subgroup S_D of symplectic group  respecting  the extension of rationals defining the adele. One could pose to  S_D the additional condition that it  leaves the value of action invariant: call this group S_D,S: this would  define what I have called micro-canonical ensemble (MCE). The obvious question is whether the simplest  zero energy states could correspond to single  orbit of S_D or whether several orbits are required. For the more complex option zero energy states would be superposition of states corresponding to several orbits of S_D with coefficients constructed of  symplectic invariants. The following arguments lead to the conclusion that MCE and single orbit orbit option are non-realistic, and raise the question whether the orbits of S_D could combine to an orbit of its Yangian analog.  A generalization of the formula for scattering amplitudes in terms of n-point functions emerges  and  somewhat surprisingly one finds that the unitarity  is an  automatic  consequence of state orthonormalization in zero energy ontology.