[Paper Review] Discrimination of particle masses in multivariant space-time geometry

Yuri A. Rylov|ArXiv.org|Dec 9, 2007

Relativity and Gravitational Theory7 references3 citations

TL;DR

This paper proposes that multivariant space-time geometry—where vectors at a point can be equivalent to a given vector elsewhere but not to each other—can lead to particle mass discrimination due to zero-variance conditions, explaining the discrete nature of elementary particle masses. Using T-geometry with a world function formalism, it shows that finite discreteness scale $ k = \lambda_0^2 / \sigma_0 $ induces constraints on particle masses, even in partly discrete geometries.

ABSTRACT

Multivariance of geometry means that at the point $P_{0}$ there exist many vectors $P_{0}P_{1}$, $¶_{0}P_{2}$,... which are equivalent (equal) to the vector $\Q_{0}Q_{1}$ at the point $Q_{0}$, but they are not equivalent between themselves. The discrimination capacity (zero-variance) of geometry appears, when at the point $P_{0}$ there are no vectors, which are equivalent to the vector $Q_{0}Q_{1}$ at the point $Q_{0}$. It is shown, that in some multivariant space-time geometries some particles of small mass may be discriminated (i.e. either they do not exist, or their evolution is impossible). The possibility of some particle discrimination may appear to be important for explanation of the discrete character of mass spectrum of elementary particles.

Motivation & Objective

• To explain the discrete character of elementary particle mass spectra using geometric principles rather than quantum postulates.
• To develop a geometric framework—T-geometry—where dynamics arises from space-time structure, not differential equations.
• To show that multivariance and zero-variance (discrimination capacity) in space-time geometry generate quantum-like effects and mass quantization.
• To introduce a partly discrete geometry model with a finite discreteness degree $ k = \lambda_0^2 / \sigma_0 $, bridging continuous and fully discrete geometries.
• To demonstrate that particle mass is a geometric property, determined by the length of a vector in a skeleton-based world chain.

Proposed method

• Uses T-geometry (world function formalism) to describe space-time, replacing vector spaces with a single primitive element: points and the world function $ \sigma(P_0, P_1) = \frac{1}{2}\rho^2(P_0, P_1) $.
• Modifies the world function to break Euclideaness conditions, enabling multivariance and zero-variance in geometry.
• Introduces a discreteness scale $ \lambda_0 $ and a finite degree of discreteness $ k = \lambda_0^2 / \sigma_0 $, interpolating between continuous (Minkowski) and fully discrete geometries.
• Constructs world chains from skeletons $ \{P_0, P_1, ..., P_n\} $ and a leading vector $ \vec{P_0P_1} $, with evolution determined geometrically, not by differential equations.
• Applies the formalism to analyze vector equivalence: when equations for vector equality (e.g., $ \sigma(P_0,P_1) = \sigma(Q_0,Q_1) $) have no solution, particles are discriminated.
• Derives particle mass as the length of a geometric vector, linking mass to the world function structure.

Experimental results

Research questions

• RQ1 Can the discrete mass spectrum of elementary particles emerge from geometric constraints in multivariant space-time?
• RQ2 How does zero-variance geometry (lack of vector equivalence) lead to particle discrimination and mass quantization?
• RQ3 What role does the finite discreteness degree $ k = \lambda_0^2 / \sigma_0 $ play in constraining particle masses in partly discrete geometries?
• RQ4 Can particle dynamics be fully derived from geometric structure without differential equations or quantum postulates?
• RQ5 How does the world function formalism in T-geometry enable the unification of geometric dynamics and quantum-like phenomena?

Key findings

• In multivariant space-time geometry, the absence of solutions to vector equivalence equations (zero-variance) leads to particle discrimination, meaning certain particles cannot exist or evolve.
• Even in partly discrete geometries with finite $ k = \lambda_0^2 / \sigma_0 $, constraints on particle mass arise due to geometric discreteness and zero-variance conditions.
• The particle mass is geometrically defined as the length of a vector in the world chain, making mass a direct consequence of space-time structure.
• Quantum effects and mass discreteness arise from different aspects of multivariance: large $ \sigma $ values cause multivariance (quantum effects), small $ \sigma $ values cause zero-variance (discrimination and discreteness).
• Geometrical dynamics in T-geometry is inherently discrete, with evolution steps determined by the leading vector length, and does not require continuous manifolds or differential equations.
• The framework explains the discrete mass spectrum as a result of the skeleton and vector structure of elementary geometrical objects in a partly discrete, multivariant space-time.