Interactive Games and Representation Theory. Ii. a Second Quantization

math.RT/9808098 This short note is devoted to a second quantization of a classical picture described in [1]. 1.1. Interactive games and intention fields. Definition 1 [1]. An interactive system (with n interactive controls) is a control system with n independent controls coupled with unknown or incompletely known feedbacks (the feedbacks, which are called the behavioral reactions, as well as their couplings with controls are of a so complicated nature that their can not be described completely). An interactive game is a game with interactive controls of each player. Below we shall consider only deterministic and differential interactive systems. For symplicity we suppose that n = 2. In this case the general interactive system may be written in the form: (1) ˙ ϕ = Φ(ϕ, u 1 , u 2), where ϕ characterizes the state of the system and u i are the interactive controls: u i (t) = u i (u • i (t), [ϕ(τ)]| τ ≤t), i.e. the independent controls u • i (t) coupled with the feedbacks on [ϕ(τ)]| τ ≤t. One may suppose that the feedbacks are integrodifferential on t. Proposition [1]. Each interactive system (1) may be transformed to the form (2) below (which is not, however, unique): (2) ˙ ϕ = ˜ Φ(ϕ, ξ), where the magnitude ξ (with infinite degrees of freedom as a rule) obeys the equation (3) ˙ ξ = Ξ(ξ, ϕ, ˜ u 1 , ˜ u 2), where˜u i are the interactive controls of the form˜u i (t) = ˜ u i (u • i (t); ϕ(t), ξ(t)) (i.e. the feedbacks are on ξ(t) as well as on ϕ(t) and are differential on t). Remark 1. One may exclude ϕ(t) from the feedbacks in the interactive controls˜u i (t).