Scattering amplitudes and orbits of cognitive representations under subgroup of symplectic group respecting the extension of rationals

Number theorist Minhyong Kim has speculated about very interesting general connection between number theory and physics (see this). The reading of a popular article about Kim's work revealed that number theoretic vision about physics provided by TGD has led to a very similar ideas and suggests a concrete realization of Kim's ideas (see this). The identification of points of algebraic surface with coordinates, which are rational or in extension of rationals, gives rise to what one can call identification problem. In TGD framework the imbedding space coordinates for points of space-time surface belonging to the extension of rationals defining the adelic physics in question are common to reals and all extensions of p-adics induced by the extension. These points define what I call cognitive representation, whose construction means solving of the identification problem.

Cognitive representation defines discretized coordinates for a point of "world of classical worlds" (WCW) taking the role of the space of spaces in Kim's approach. The symmetries of this space are proposed by Kim to help to solve the identification problem. The maximal isometries of WCW necessary for the existence of its K\"ahler geometry provide symmetries identifiable as symplectic symmetries. The discrete subgroup respecting extension of rationals acts as symmetries of cognitive representations of space-time surfaces in WCW, and one can identify symplectic invariants characterizing the space-time surfaces at the orbits of the symplectic group.

This picture could be applied to the construction of scattering amplitudes with finite cognitive precision in terms of cognitive representations and their orbits under subgroup SD of symplectic group respecting the extension of rationals defining the adele. One could pose to SD the additional condition that it leaves the value of action invariant: call this group SD,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 SD or whether several orbits are required. For the more complex option zero energy states would be superposition of states corresponding to several orbits of SD 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 SD could combine to an orbit of its Yangian analog.

See the chapter The Recent View about Twistorialization in TGD Framework or the article Scattering amplitudes and orbits of cognitive representations under subgroup of symplectic group respecting the extension of rationals.