[Paper Review] Geometry and physics of today

Anastasios Mallios|ArXiv.org|May 20, 2004

Relativity and Gravitational Theory29 references3 citations

TL;DR

This paper proposes a reformulation of fundamental physics using Abstract Differential Geometry (ADG), which eliminates the need for a background spacetime manifold by deriving differential structures directly from algebraic and categorical frameworks. It argues that physical laws—especially gauge theories and field equations—generate geometry relationally, not through geometric manifolds, but via connections and differential triads in an algebraic setting, offering a background-independent approach to quantum gravity.

ABSTRACT

The ``geometry'', in the sense of the classical differential geometry of smooth manifolds (CDG), is put under scrutiny from the point of view of Abstract Differential Geometry (ADG), along with resulting, thereby, potential physical consequences, in what, in particular, concerns physical ``gauge theories'', when the latter are viewed as being, anyway, of a ``geometrical character''. Yet, ``physical geometry'', in connection with physical laws and the associated with them, within the context of ADG, ``differential'' equations (whence, no background spacetime manifold is needed thereat), are also under discussion.

Motivation & Objective

• To reframe the foundations of physics by replacing classical differential geometry (CDG) with Abstract Differential Geometry (ADG), thereby removing reliance on a background spacetime manifold.
• To demonstrate that physical laws, particularly gauge theories, are inherently geometric in nature but derive their geometric structure from connections and differential equations, not from manifolds.
• To provide a framework for quantum gravity by formulating differential equations and geometric structures in a space-independent, relational manner, avoiding singularities and background structures.
• To show that the traditional notion of 'geometry' in physics is not fundamental but a model derived from physical laws, with curvature arising from connections rather than manifolds.
• To establish a new foundation for unified field theory by combining ADG with generalized functions (Rosinger’s theory), enabling a singularity-free treatment of physical fields.

Proposed method

• Formalizes physical laws as ${\cal A}$-connections on sheaf-theoretic modules, replacing the need for smooth manifolds in differential geometry.
• Uses the concept of a 'differential triad'—an algebraic structure encoding derivation, module, and connection—within the ADG framework to define differential operators without coordinates.
• Applies the relational principle: geometry emerges from physical laws via connections, not from spacetime geometry, thus achieving a background-independent formulation.
• Employs category-theoretic and sheaf-theoretic tools to define differential structures abstractly, allowing for non-smooth or singular configurations typical in quantum gravity.
• Integrates Rosinger’s theory of generalized functions to handle singularities and distributions in field equations without requiring classical differentiability.
• Reinterprets curvature as a consequence of the ${\cal A}$-connection, not of a Riemannian metric, thereby decoupling geometry from manifold structure.

Experimental results

Research questions

• RQ1 Can physical geometry be derived from connections and differential equations without assuming a background spacetime manifold?
• RQ2 How can gauge theories be reformulated as purely algebraic-geometric structures within ADG, independent of smooth manifolds?
• RQ3 What is the role of the ${\cal A}$-connection in generating curvature and physical laws in a relational, background-independent framework?
• RQ4 How does ADG enable a consistent formulation of quantum gravity by avoiding spacetime singularities?
• RQ5 In what way does the absence of a manifold structure in ADG resolve the foundational issues of classical differential geometry in fundamental physics?

Key findings

• Physical geometry is not a pre-existing structure but emerges from physical laws via ${\cal A}$-connections, which generate curvature without requiring a manifold.
• The classical notion of differential geometry, based on smooth manifolds, is shown to be a limiting case; ADG generalizes it by deriving differential structures from algebraic data.
• The framework allows for the formulation of differential equations and field theories without any reference to coordinates or background spacetime, fulfilling a Leibnizian vision of geometry.
• Singularities in classical field theories—such as those in general relativity or quantum field theory—are avoided because the differential machinery in ADG does not depend on manifold structure.
• The unification of gauge theories and gravity becomes more natural in ADG, as both are treated as connections on algebraic modules, suggesting a deeper structural unity.
• The use of generalized functions (Rosinger) within ADG enables the treatment of non-smooth or distributional fields, providing a path toward a quantum theory of gravity without divergences.