On the Necessity of Interaction Cross-Terms in a Fundamental Geometric Framework

J. Rogers, SE Ohio 

Abstract Standard physical models rely on the principle of linear superposition, wherein the total influence of distinct forces is treated as the arithmetic sum of independent components (

$F_{total}=\sum F_i$

). This paper argues that if the fundamental structure of the universe is geometric—as established by General Relativity—such linear addition is an approximation valid only in low-energy regimes. A strictly geometric framework implies that interaction terms must be multiplicative, necessitating the existence of significant cross-terms (

$2AB$

) representing the coupling between distinct force sectors. We explore the possibility that phenomena such as the "running" of the fine-structure constant (

$\alpha$

) are evidence of these geometric cross-terms, specifically the coupling between relativistic mass-energy and charge, rather than effects solely attributable to vacuum polarization.

1. Introduction Since the advent of General Relativity, gravity has been understood not as a force, but as a geometric property of spacetime curvature. However, the other fundamental interactions (electromagnetism, the strong, and the weak nuclear forces) are largely treated as quantum fields operating upon a fixed geometric background.

This dichotomy allows for a mathematical convenience where forces are summed linearly. If we accept the premise of a unified geometric universe—where all forces are manifestations of the geometry of a single manifold—this linear treatment becomes theoretically inconsistent. Geometry is inherently non-linear; the combination of two curvatures cannot be described merely by the sum of their parts.

2. The Failure of Linear Superposition in Geometry In a geometric universe, the "intensity" or "charge" of a force represents a specific curvature or vector within the metric. If we consider two distinct geometric influences,

$I_A$

(e.g., mass-gravity) and

$I_B$

(e.g., electric charge), the total interaction geometry is not the distance between points

$A+B$

. Rather, the interaction magnitude is determined by the product of these geometric states.

The total interaction intensity

$\mathcal{I}$

should logically follow a binomial expansion:

$\mathcal{I}=(I_A+I_B{)}^2=I_A^2+I_B^2+2I_A{I}_B$

In this equation:

• $I_A^2$
  represents the pure gravitational self-interaction.
• $I_B^2$
  represents the pure electromagnetic self-interaction.
• $2I_A{I}_B$
  represents the Geometric Cross-Term.

Standard physics effectively ignores the

$2I_A{I}_B$

term in low-energy environments. However, in a true geometric theory, this term must exist. It represents the mechanical mixing of the "charge dimension" and the "mass dimension."

3. Relativistic Velocity as the Trigger for Cross-Term Dominance The term

$I_A$

(mass) is traditionally treated as a constant invariant rest mass. However, in high-energy particle collisions, the relevant geometric factor is the total energy-momentum, which scales with the Lorentz factor (

$\gamma$

).

As a particle accelerates:

$I_{A_{dynamic}}\propto \gamma m$

Consequently, the geometric cross-term evolves:

$2I_B\cdot (\gamma m)$

At low velocities (

$\gamma \approx 1$

), the cross-term is negligible compared to the electromagnetic term

$I_B^2$

. However, as

$\gamma$

increases significantly, the cross-term becomes a measurable contributor to the total interaction strength.

4. Phenomenological Implications: The Running of

$\alpha$

Experimental data confirms that the effective fine-structure constant (

$\alpha$

) increases with collision energy (the "running of

$\alpha$

"). The Standard Model attributes this to "vacuum polarization"—a screening effect by virtual particle-positron pairs.

We propose an alternative geometric interpretation: The observed increase in

$\alpha$

is the result of the relativistic mass-energy of the particle coupling with its charge via the cross-term derived above.

• Standard Model:
  $\alpha$
  changes because the "screening" of the vacuum is penetrated.
• Geometric Model:
  $\alpha$
  changes because the "Mass" term (
  $\gamma m$
  ) has increased, amplifying the
  $2\cdot Charge\cdot Mass$
  interaction.

This model eliminates the requirement for unobservable "virtual particles" to explain the energy-dependence of coupling constants, replacing them with a kinematic, geometric feedback loop.

5. Extension to the Strong and Weak Forces If this geometric logic is applied to the Standard Model in its entirety, the total universal force is an expansion of all fundamental "intensities":

${\mathcal{I}}_{total}=(I_{grav}+I_{em}+I_{weak}+I_{strong}{)}^2$

The resulting expansion contains 16 terms: 4 "pure" terms (the known forces) and 12 cross-terms. We suggest that these cross-terms are not zero, but represent the complex interactions currently observed in particle physics (such as the electroweak unification or the fractional charges of quarks) without the need for additional ad-hoc particles.

6. Conclusion While linear superposition is a useful approximation for low-energy mechanics, it is insufficient for a fundamentally geometric universe. The existence of interaction cross-terms is a mathematical necessity of a unified geometric framework. We submit that the "running" of coupling constants provides empirical evidence for these cross-terms, driven specifically by the coupling of relativistic kinetic energy with static charge. Further investigation into these geometric cross-terms may yield a more "mechanical" foundation for high-energy physics, independent of perturbative virtual particle theories.