Phase spaces and deformation theory (Q5900273)

In an earlier paper the author has sketched a physical toy model where the space-time of classical physics became a section of a universal fiber space \(\tilde{E}\) defined on the moduli space \(\underline{H}=\text{Simp}(H)\) of the physical system under consideration. In this case systems composed of an observer and an observed, both sitting in Euclidean 3-space. This moduli space was then called the \textit{time-space}. Time was defined to be a metric \(\rho\) on the time-space, measuring all possible infinitesimal changes of \textit{the state} of the objects in the family under study. This gave a model of relativity where all velocities turned out to be a projective space. Dynamics where introduced via the general construction of a \textit{phase space Ph(A)} for any associative algebra \(A\). This is a universal pair of a homomorphism of algebras, \(\iota:A\rightarrow Ph(A)\) and a derivation \(d:A\rightarrow Ph(A)\) such that for any homomorphism of \(A\) into a \(k\)-algebra \(R\), the derivations of \(A\) in \(R\) are induced by unique homomorphisms \(Ph(A)\rightarrow R\) composed with \(d\). The \(Ph(-)\)-construction can be iterated, obtaining a direct limit with morphism \(\iota(n):Ph^n(A)\rightarrow Ph^{\infty}(A)\) and a universal derivation \(\delta\in Der_k(Ph^{\infty}(A),Ph^{\infty}(A))\) called the \textit{Dirac}-derivation. A general \textit{dynamical structure of order} \(n\) is defined as a two-sided \(\delta\)-ideal \(\sigma\) in \(Ph^\infty(A)\) inducing a surjective homomorphism \(Ph^{(n-1)}(A)\rightarrow Ph^\infty(A)/\sigma=:A(\sigma).\) Associated to any \textit{time space} \(H\) with a fixed dynamical structure \(H(\sigma)\) there is a kind of quantum field theory. In particular, if \(H\) is the affine ring of a moduli space of the objects we want to study, \(Ph^\infty(H)\) is the complete ring of observables containing the parameters not only of the iso-classes of the objects, but also of all dynamical parameters. The choice of \(\sigma\) correspond to the introduction of a parsimony principle, e.g. to the choice of some Lagrangian. In this paper, the author study this phase-space construction in greater detail, and considering \(Ph^n(A)\), \(n\geq 1\) as noncommutative Jet-spaces, this can be used to study noncommutative partial differential equations (Noncommutative PDE`s) and their solutions. This is an extension of the prolongation-projection procedure of Elie Cartan for formally solving noncommutative PDE`s. In an example, the author shows that in the commutative case this generalizes the classical theory.