Abstract
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.
Highlights
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
We show that the functorial formulation of the Kahler-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
According to this strategy, the problem of quantization of gravity is equivalent to forcing the algebraic Kahler-de Rham general relativistic dynamical mechanism of the gravitational connection functorial morphism inside an appropriate sheaf-theoretic localization environment, which is capable of incorporating the localization properties of observables in the quantum regime
Summary
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 1. According to this strategy, the problem of quantization of gravity is equivalent to forcing the algebraic Kahler-de Rham general relativistic dynamical mechanism of the gravitational connection functorial morphism inside an appropriate sheaf-theoretic localization environment, which is capable of incorporating the localization properties of observables in the quantum regime. The physical information contained in a quantum observable algebra can be recovered by a gluing construction referring to its local commutative subalgebras 7–12
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