[Paper Review] Deformation principle and further geometrization of physics

Yuri A. Rylov|ArXiv.org|Apr 23, 2007

Relativity and Gravitational Theory9 references3 citations

TL;DR

This paper proposes a geometric foundation for physics based on the deformation principle, where physical geometries are defined entirely by a world function σ, generalizing Riemannian geometry to include multivariant structures. By replacing the Euclidean world function σ_E with a physical world function σ, the framework naturally describes quantum effects and discrete/continuous geometries uniformly, offering a geometric explanation for quantum behavior and elementary particles as geometric objects in a multivariant space-time.

ABSTRACT

The space-time geometry is considered to be a physical geometry, i.e. a geometry described completely by the world function. All geometrical concepts and geometric objects are taken from the proper Euclidean geometry. They are expressed via the Euclidean world function σ_E and declared to be concepts and objects of any physical geometry, provided the Euclidean world function σ_E is replaced by the world function σof the physical geometry in question. The set of physical geometries is more powerful, than the set of Riemannian geometries, and one needs to choose a true space-time geometry. In general, the physical geometry is multivariant (there are many vectors which are equivalent to a given vector, but are not equivalent between themselves). The multivariance admits one to describe quantum effects as geometric effects and to consider existence of elementary particles as a geometrical problem, when the possibility of the physical existence of an elementary geometric object in the form of a physical body is determined by the space-time geometry. Multivariance admits one to describe discrete and continuous geometries, using the same technique. A use of physical geometry admits one to realize the geometrical approach to the quantum theory and to the theory of elementary particles.

Motivation & Objective

• To develop a comprehensive geometric framework for physics that transcends Riemannian geometry by incorporating multivariance.
• To resolve the limitations of conventional geometry in describing microcosmic phenomena such as quantum behavior and discrete structures.
• To establish a geometric basis for quantum mechanics and elementary particle theory by treating quantum effects as intrinsic geometric properties.
• To provide a method for constructing both discrete and continuous geometries using a single formalism based on the world function.
• To identify the true space-time geometry among a family of T-geometries by matching physical observables, particularly those related to particle motion and mass.

Proposed method

• Define all geometric objects and concepts via the world function σ, generalizing Euclidean geometry to any physical geometry.
• Apply the deformation principle to generate new geometries by deforming the Euclidean world function σ_E into a physical world function σ.
• Use tubular geometries (T-geometries) as a class of physical geometries that naturally admit multivariance, especially for timelike vectors.
• Construct space-time geometries where multiple vectors are equivalent to a given vector but not equivalent to each other, introducing multivariance.
• Derive the metric tensor g^{ik}(x) from the world function σ, even when σ does not satisfy the standard Riemannian equation (8.1), allowing non-Riemannian geometries.
• Investigate how multivariant geometries can reproduce quantum-like behavior without postulating wave functions or force fields, by attributing such effects to geometric structure.

Experimental results

Research questions

• RQ1 Can quantum effects be explained as geometric consequences of multivariant space-time geometry rather than as fundamental postulates?
• RQ2 How can a unified geometric framework describe both discrete and continuous geometries using the same formalism?
• RQ3 What conditions must a world function σ satisfy to generate a space-time geometry that reproduces observed quantum behavior?
• RQ4 Can the existence and properties of elementary particles be derived purely from geometric constraints in a multivariant space-time?
• RQ5 How does the failure of the standard Riemannian equation (8.1) for the world function σ affect the description of metric fields and their physical interpretation?

Key findings

• The deformation principle enables the construction of multivariant geometries that naturally describe quantum effects as geometric phenomena, without invoking wave functions or probabilistic postulates.
• A family of plane uniform isotropic space-time geometries exists, parameterized by a function of one variable, among which the Minkowski geometry is only one member, and most are multivariant for timelike vectors.
• The world function σ(7.7) generates a space-time geometry that is not Riemannian, as it does not satisfy the standard equation (8.1), yet it yields a metric tensor identical to Minkowski's up to a constant factor.
• In the limit |σ| ≫ |σ₀|, the world function σ satisfies a modified equation (8.2) involving a non-trivial correction term, indicating a complex, non-local metric field structure.
• The resulting geometry cannot be imitated by a single-point metric tensor field; instead, it requires a complex field system that inherently encodes multivariance, resembling quantum behavior.
• The geometric approach eliminates the need for fitting in theoretical physics: once the correct world function σ is chosen, predictions follow from logic and geometry alone, not empirical adjustment.