Advances in Mathematical Physics
Volume 2011 (2011), Article ID 191083, 14 pages
doi:10.1155/2011/191083
Research Article

The Homological Kähler-De Rham Differential Mechanism part I: Application in General Theory of Relativity

Department of Mathematics, National and Kapodistrian University of Athens, Panepistimioupolis, 15784 Athens, Greece

Received 7 March 2011; Accepted 12 April 2011

Academic Editor: Shao-Ming Fei

Copyright © 2011 Anastasios Mallios and Elias Zafiris. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The mechanism of differential geometric calculus is based on the fundamental notion of a connection on a module over a commutative and unital algebra of scalars defined together with the associated de Rham complex. In this communication, we demonstrate that the dynamical mechanism of physical fields can be formulated by purely algebraic means, in terms of the homological Kähler-De Rham differential schema, constructed by connection inducing functors and their associated curvatures, independently of any background substratum. In this context, we show explicitly that the application of this mechanism in General Relativity, instantiating the case of gravitational dynamics, is related with the absolute representability of the theory in the field of real numbers, a byproduct of which is the fixed background manifold construct of this theory. Furthermore, the background independence of the homological differential mechanism is of particular importance for the formulation of dynamics in quantum theory, where the adherence to a fixed manifold substratum is problematic due to singularities or other topological defects.