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Novel Regge-like trajectories for spinning, dilating, hadronic particles

The paper investigates the motion of spinning, dilating particles in complex geometric backgrounds to demonstrate that their rest mass is not necessarily constant, leading to the discovery of new Regge-like trajectories that relate mass to hypermomentum currents.

Original authors: Damianos Iosifidis

Published 2026-04-28

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CC BY 4.0

Original authors: Damianos Iosifidis

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are playing a video game where the characters are usually just simple, solid dots moving across a screen. In standard physics (General Relativity), we treat particles like those dots: they have a position, a speed, and a mass that stays the same no matter what.

But this paper suggests that real-world particles—specifically hadrons (the tiny building blocks inside atoms, like protons)—are not just dots. They are more like complex, squishy, spinning fidget spinners. They don't just move; they can expand, contract, twist, and deform.

Here is a breakdown of the paper’s big ideas using everyday analogies.

1. The "Squishy" Particle (Hypermomentum)

In standard physics, we mostly care about Spin (how a particle rotates). This paper says that for hadrons, we need to care about two more "squishy" behaviors:

Dilation: Imagine a balloon inflating and deflating. This is the particle breathing in and out.
Shear: Imagine taking a ball of dough and squeezing it so it turns into an oval. This is the particle changing its shape.

The author calls these combined behaviors "Hypermomentum." Instead of just being a rigid marble, the particle is a dynamic, living shape.

2. The "Rough" Road (Non-Riemannian Geometry)

Most physics assumes the "road" (spacetime) is smooth and predictable. This paper explores a much "rougher" road called Metric-Affine Gravity.

On this road, the ground isn't just curved (like a hill); it’s also twisted (torsion) and stretchy (nonmetricity). Imagine driving a car not just on a winding mountain road, but on a road made of trampoline fabric that stretches and twists under your tires as you drive.

3. The Big Discovery: Changing Mass

This is the "Eureka!" moment of the paper. In standard physics, a particle's mass is like its "identity"—it’s a constant number.

However, the author shows that when a "squishy" particle travels on a "stretchy" road, its mass actually changes.

The Analogy:
Imagine you are running a race while holding a heavy backpack.

In standard physics, the backpack is a solid weight. It stays the same.
In this paper’s physics, the backpack is a magic bag of sand. As you run over bumpy, stretching terrain, the bag automatically expands or contracts. Because the bag is changing shape and size based on the ground beneath you, the "weight" you feel (your dynamical mass) fluctuates.

4. Regge Trajectories: The "Musical Scale" of Particles

The paper mentions "Regge-like trajectories." In music, if you change the tension of a guitar string, the pitch (frequency) changes in a predictable way.

The author found that the particle's mass changes in a predictable pattern related to its "breathing" (dilation) and "squeezing" (shear). This creates a "scale" of mass states. This is a huge deal because it might explain things like the Roper Resonance—a mysterious, heavier version of a proton that scientists have struggled to fully explain. The author suggests this "heavy version" is just a proton that has been "stretched" by the geometry of space.

Summary

The Old View: A solid marble rolling on a smooth, curved hill. Its weight never changes.

The New View: A squishy, breathing balloon rolling on a twisting, stretching trampoline. As the balloon breathes and the trampoline stretches, the balloon's weight shifts up and down in a beautiful, mathematical pattern.

This paper, titled "Novel Regge-like trajectories for spinning, dilating, hadronic particles" by Damianos Iosifidis, explores the motion of microstructured particles within the framework of Metric-Affine Gravity (MAG). It investigates how the non-Riemannian features of spacetime—specifically torsion and nonmetricity—affect the dynamical mass of particles possessing hadronic properties (spin, dilation, and shear).

1. The Problem

In standard General Relativity (Riemannian geometry), the mass of a test particle is typically treated as a constant of motion. However, hadrons (like protons or neutrons) are not point particles; they possess internal structures characterized by hypermomentum.

The author addresses a fundamental gap: how does the interaction between a particle's internal hypermomentum (spin, dilation, and shear) and a non-Riemannian spacetime (containing curvature, torsion, and nonmetricity) influence the particle's physical properties? Specifically, the paper asks whether the dynamical rest mass ( $M$ ) remains constant when a particle moves through a background with nonmetricity.

2. Methodology

The study employs a classical field-theoretic approach using the Metric-Affine Geometry framework. The methodology can be broken down into several steps:

Geometric Framework: The author utilizes a generic $n$ -dimensional manifold defined by a metric $g_{\mu\nu}$ and a general linear connection $\Gamma^\lambda_{\mu\nu}$ . This allows for the independent existence of curvature ( $R$ ), torsion ( $S$ ), and nonmetricity ( $Q$ ).
Equations of Motion: The author starts from the generalized equations of motion for microstructured bodies (derived from MAG conservation laws), which describe the evolution of canonical momentum ( $P_\mu$ ) and hypermomentum ( $H_{\mu\nu}$ ).
Hypermomentum Decomposition: The hypermomentum is decomposed into three irreducible components:
1. Spin ( $S_{\mu\nu}$ ): Antisymmetric part.
2. Dilation ( $H$ ): The trace part.
3. Shear ( $\Sigma_{\mu\nu}$ ): The symmetric traceless part.
Supplementary Conditions: To close the system of equations and fix the particle's centroid, the author evaluates the necessity of "Supplementary Conditions." He extends the well-known Spin Supplementary Conditions (SSC) (like Pirani or Tulczyjew) to propose a new Shear Supplementary Condition (SSC).

3. Key Contributions

Discovery of Mass Non-Conservation: The paper proves that in the presence of nonmetricity, the dynamical mass $M$ is generally not a constant of motion, even if standard spin supplementary conditions are applied.
Introduction of the Shear Supplementary Condition: The author introduces a formal mathematical constraint ( $\xi_{\mu\nu}P^\mu = 0$ ) to handle the shear component of hypermomentum, analogous to how the Tulczyjew constraint handles spin.
Derivation of Novel Regge-like Trajectories: The paper establishes new mathematical relationships (trajectories) that link the particle's mass to its internal dilation and shear charges.
Identification of Duality: The author demonstrates a mathematical duality between the coupling of dilation to the trace of nonmetricity and the coupling of shear to the traceless part of nonmetricity.

4. Results

The paper presents three primary mathematical results regarding mass evolution:

Pure Dilation Case: For a particle with only dilation charge, the mass evolves according to:
$\dot{M} = -\frac{Q_\mu u^\mu}{4l} \dot{H}$
This leads to a Regge-like trajectory where mass is a function of the dilation charge $M(H)$ . The author suggests this could physically explain the Roper resonance (a radial excitation of the proton) as a result of spacetime "micro-stretching."
Pure Shear Case: For a particle with only shear charge, the mass evolution is:
$\dot{M} = -\frac{1}{4l} \hat{\mu}_{\mu\nu} \dot{\Sigma}^{\mu\nu}$
This shows that anisotropic deformations (shear) also cause mass fluctuations.
General Case: When both dilation and shear are present, the mass evolution is governed by a combined equation. However, if the Shear Supplementary Condition is met, the momentum remains parallel to the velocity, simplifying the dynamics.

5. Significance

The significance of this work is both theoretical and phenomenological:

Theoretical Physics: It extends the Mathisson-Papapetrou equations (standard for spinning particles in GR) into the much more complex realm of Metric-Affine Gravity, providing a rigorous way to study how "internal" particle properties interact with "external" non-Riemannian geometry.
Hadronic Physics: It provides a geometric interpretation for hadronic excitations. Instead of viewing mass shifts solely through the lens of Quantum Chromodynamics (QCD) color-force interactions, this paper suggests that the geometry of spacetime itself (via nonmetricity) may play a role in the mass spectra of particles.
Unified View: By linking Regge trajectories (a staple of particle physics) to nonmetricity (a staple of advanced gravity theories), the paper bridges the gap between the microscopic properties of matter and the macroscopic structure of spacetime.