Phase spaces and deformation theory (Q5894211)

There is a physical model where the space-time of classical physics is a section of a universal fibre space \(\tilde E\), defined on the moduli space \(\underline H=\text{Simp}(H)\) of the physical system composed of an observer and an observed, both sitting in Euclidean 3-space. This moduli space is called the time-space. In this mathematical model, time is defined to be a metric \(\rho\) on the time-space, measuring all possible infinitesimal changes of the state of the objects in the family that is studied. Into this picture, dynamics is introduced via the phase space \(\text{Ph}(A)\), constructed in general for any associative algebra \(A\). The phase space \(\text{Ph}(A)\) is a universal pair consisting of a homomorphism of \(k\)-algebras \(A\rightarrow\text{Ph}(A)\) together with a derivation \(d:A\rightarrow\text{Ph}(A)\), such that for any homomorphism of \(A\) into a \(k\)-algebra \(R\), the derivations of \(A\) in \(R\) are induced by unique homomorphisms \(\text{Ph}(A)\rightarrow R\) composed with \(d\). This process is iterated, and one obtains the limit morphism \(\iota(n):\text{Ph}^n(A)\rightarrow\text{Ph}^\infty(A)\). The image of this morphism is denoted \(\text{Ph}^{(n)}(A)\), and there is a universal derivation \(\delta\in\text{Der}_k(\text{Ph}^\infty(A),\text{Ph}^\infty(A))\) called the Dirac derivation. A general dynamical structure of order \(n\) is a two-sided ideal \(\delta\) in \(\text{Ph}^\infty(A)\) inducing a surjective homomorphism \(\text{Ph}^{n-1}(A)\rightarrow\text{Ph}^\infty(A)/\sigma=:A(\sigma)\). Now, let \(A\) be the \(k\)-algebra \(H\) of the moduli \(\underline H\). It is proved that to such a time-space with a fixed dynamical structure, there exists a kind of quantum field theory. Because \(H\) is the ring of the moduli space of the essential objects under study, the ring \(\text{Ph}^\infty(H)\) 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 purpose of the article is to study the phase-space construction in greater detail. There is a natural descending filtration of two-sided ideals \(\{\mathcal F_n\}_{0\leq n}\) of \(\text{Ph}^\infty(A)\), and the corresponding quotients \(\text{Ph}^n(A)/\mathcal F_n\) are finite dimensional vector spaces. Considered as affine varieties, these are the noncommutative Jet-spaces. The article gives a framework for the study of general systems of (noncommutative) PDE's. An introduction to the noncommutative deformations of families of modules, and the generalized Massey products is given. These subjects are connected by using the Massey products based on the phase-space of a resolution of the finitely generated \(k\)-algebra \(A\). The \(\text{Ph}^\infty(A)\) is a noncommutative analogue of the notion of higher differentials treated elsewhere. A nice result of the article is the use of the phase-space to construct the Massey products of the (finitely generated) \(k\)-algebra \(A\).