The homological Kähler-de Rham differential mechanism models the dynamical behavior of physical fields by purely algebraic means and independently of any background manifold substratum. This is of particular importance for the formulation of dynamics in the quantum regime, where the adherence to such a fixed substratum is problematic. In this context, we show that the functorial formulation of the Kähler-de Rham differential mechanism in categories of sheaves of commutative algebras, instantiating generalized localization environments of physical observables, induces a consistent functorial framework of dynamics in the quantum regime.

The basic conceptual and technical issue pertaining to the current research attempts towards the construction of a viable quantum theory of the gravitational field, refers to the problem of independence of this theory from a fixed spacetime manifold substratum. In relation to this problem, we have argued about the existence and functionality of a homological schema of functorial general relativistic dynamics, constructed by means of connection inducing functors and their associated curvatures, which is, remarkably, independent of any background substratum [

The absolute representability principle of the classical general theory of relativity is based on the set-theoretic conception of the real line, as a set of infinitely distinguished points coordinatized by means of the field of real numbers. Expressed categorically, this is equivalent to the interpretation of the algebraic structure of the reals inside the absolute universe of

According to the functorial Kähler-de Rham general relativistic dynamical mechanism [

From a physical viewpoint, the construction of a sheaf of algebras of observables constitutes the natural outcome of a complete localization process [

In the sequel, we consider the localization environment of the category of sheaves of sets

A topological covering system on

For every open reference domain

If

If

As a consequence of the conditions above, we can check that any two

If we consider the partially ordered set of open subsets of a topological measurement space

Furthermore, we can show that if

It is instructive to emphasize that the algebraic functorial Kähler-de Rham framework of dynamics is based for its conceptualization and operative efficacy, neither on the methodology of real analysis nor on the restrictive assumption of smoothness of observables, but only, on the functorial expression of the inverse processes of infinitesimal scalars extension/restriction. Nevertheless, it is instructive to apply this algebraic framework for the case of algebra sheaves of smooth observables, in order to reproduce the differential geometric mechanism of smooth manifolds geometric spectra, interpreted in the localization environment of the category

A complete settlement of this issue, addressing the principle of covariance as above, comes from the mathematical theory of the abstract differential geometry (ADG) [

For our physical purposes, we conclude that any cohomologically appropriate sheaf of algebras

Conclusively, it is instructive to recapitulate and add some further remarks on the physical semantics associated with the preceding algebraic cohomological dynamical framework by invoking the sheaf-theoretic terminology explicitly. The basic mathematical objects involved in the development of that framework consists of a sheaf of commutative unital algebras

At a further stage of development, the implementation of the notion of functorial dynamical connectivity requires the functorial modeling of the notion of a physical field in terms of a connection expressing the algebraic process of infinitesimal scalar extensions of the algebra sheaf of local observables. Thus, we conclude that a pair (

A significant observation has to do with the fact that if

If we consider a coordinatizing open cover

The behavior of the curvature

According to the above dynamical framework, applications of ADG include the reformulation of Gauge theories in sheaf-theoretic terms [

The basic defining feature of the quantum theory according to the Bohrian interpretation [

The resolution of valuation vagueness in the quantum theory can be algebraically comprehended through the notion of local relativization of representability of the valuation algebra with respect to commutative algebraic contexts that correspond to prepared measurement environments [

The decisive fact that implies the relativization of quantum representability of the valuation algebra with respect to commutative algebraic contexts is due to the observation that a globally noncommutative or even partially commutative algebra of quantum observables determines an underlying categorical diagram (presheaf) of commutative observable algebras. Then, each one of the latter can be locally identified with a commutative algebra of classical observables. Consequently, the physical information contained in a quantum observable algebra can be recovered by a gluing sheaf-theoretic construction referring to its local commutative subalgebras [

More precisely, we make the basic assumption that there exists a coordinatization functor,

As a consequence, we depict the category of presheaves

Now, it is important to notice that the counit of the adjunction

Conclusively, it is worthwhile to emphasize that discussions, of background manifold independence pertaining to the current research focus in quantum gravity, should take at face value the fact that the fixed manifold construct in general relativity is just the byproduct of fixing physical representability in terms of the field of real numbers. Moreover, it is completely independent of the possibility of formulating dynamics, since the latter can be developed precisely along purely algebraic lines, that is, by means of functorial connections. Hence, the usual analytic differential geometric framework of smooth manifolds, needed for the formulation of classical General Relativity, is just a special coordinatization of the universal functorial mechanism of infinitesimal scalars extension and thus should be substituted appropriately, in case a merging with quantum theory is sought. The substitution is guided by the principle of relativized representability, which forces the topos