### 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*the additional condition that it leaves the value of action invariant: call this group

_{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

_{D,S}*S*or whether several orbits are required. For the more complex option zero energy states would be superposition of states corresponding to several orbits of

_{D}*S*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

_{D}*S*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.

_{D}