⚛️ nuclear theory

Formulation of fully covariant Quantum-Molecular Dynamics for an N-body system with scalar and vector potentials

This paper presents a fully covariant Quantum-Molecular Dynamics framework for relativistic N-body systems interacting via scalar and vector potentials, deriving exact equations of motion to resolve fundamental issues regarding time constraints, non-relativistic limits, and frame independence while offering new insights through scattering analyses.

Original authors: Jiaxing Zhao, Joerg Aichelin, Elena Bratkovskaya

Published 2026-09-11

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CC0 1.0

Original authors: Jiaxing Zhao, Joerg Aichelin, Elena Bratkovskaya

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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

In the subatomic world, particles do not merely bounce off one another like billiard balls; they exist in a complex, shifting landscape where space and time are woven together. When scientists try to understand how groups of these particles move and interact at speeds approaching the speed of light, they face a profound difficulty: the rules that govern a single particle moving alone do not easily translate to a crowded room of many particles all influencing each other. For decades, physicists have relied on two main ways to describe this chaos. One approach treats the particles as a smooth, flowing fluid, which is excellent for calculating average behaviors but misses the individual quirks of specific collisions. The other approach tracks every single particle as a distinct object, capturing the messy details of how they clump together or fly apart, but this method has historically struggled to remain accurate when the particles move at relativistic speeds. The challenge has been to build a model that tracks individual particles with perfect precision while respecting the strict laws of Einstein's relativity, ensuring that the description of their motion looks the same to every observer, regardless of how fast they are moving.

A team of researchers has now constructed a new mathematical framework that achieves this balance, creating a fully consistent way to simulate the evolution of many interacting particles at high speeds. They focused on a system where particles interact through two distinct types of forces: one that acts like a change in the particle's effective mass, and another that acts like a push or pull on its motion, similar to how electric and magnetic fields influence a charged object. In previous models, these two forces were often mixed together or treated with approximations that broke down at high energies. The authors derived a set of rules that keep these forces separate and distinct, allowing them to see exactly how each one shapes the path of a particle. They tested their new equations by simulating collisions between two and four particles, carefully checking that the results remained consistent no matter which angle or speed the observer was viewing the simulation from.

The researchers discovered that the choice of how to synchronize time between particles is not just a technical detail, but a fundamental part of the physics. In a relativistic system, there is no single "now" that applies to everyone; different observers see events happening at different times. The team showed that while different ways of defining this shared time can make the mathematical description of the particles' paths look different on paper, the actual physical outcome—the real trajectory the particles take through space—remains identical. This confirmed that their framework is robust and does not depend on an arbitrary choice of perspective. They also found that the two types of forces they modeled play very different roles. The force that changes the particle's mass affects its motion in a way that is distinct from the force that pushes it, and these differences become more pronounced as the particles move faster.

To verify their findings, the team ran detailed computer simulations of particles colliding. They started with simple scenarios, such as two particles moving toward each other, and then moved to more complex interactions involving four particles, which can be thought of as a small-scale model of two atomic nuclei colliding. In these simulations, they observed that when the particles interacted through the "push" force, they scattered at wide angles, almost bouncing off each other like magnets repelling. In contrast, when they interacted through the "mass-changing" force, the particles tended to pass through each other with less dramatic deflection. The difference was so significant that at high speeds, the non-relativistic models, which had been used for years, failed to predict the correct scattering angles, often missing the mark entirely. The new model, however, correctly predicted that the "push" force would cause a much sharper turn in the particle's path than the mass-changing force, a result that held true whether the simulation was run from a stationary viewpoint or from a moving one.

The study also addressed a long-standing concern in physics known as cluster separability. This principle states that if a large group of particles splits into two separate, non-interacting groups, the motion of one group should not be influenced by the existence of the other. The researchers found that their method, which uses a global clock to track the system, works well for most practical purposes, though they noted that strictly speaking, a perfectly isolated group would ideally need its own internal clock to be mathematically exact. Despite this minor theoretical nuance, the simulations showed that the method is highly effective for describing real-world scenarios, such as the collisions that occur in heavy-ion experiments. By providing a clear, covariant description of how scalar and vector forces shape the behavior of matter, this work offers a powerful new tool for understanding the extreme conditions found in the cores of neutron stars and the aftermath of high-energy particle collisions. The framework does not just solve a mathematical puzzle; it provides a reliable map for navigating the complex, relativistic dance of the subatomic world, ensuring that the story of how matter evolves remains consistent for every observer.

Technical Summary: Formulation of Fully Covariant Quantum-Molecular Dynamics for an N-body System with Scalar and Vector Potentials

Problem Statement
The description of the time evolution of interacting many-body systems with relativistic velocities remains a central challenge in nuclear, hadronic, and particle physics. While the Boltzmann-Uehling-Uhlenbeck (BUU) approach offers computational efficiency for single-particle observables, it loses higher-order correlations due to truncation. Conversely, Quantum Molecular Dynamics (QMD) retains correlations and fluctuations essential for cluster formation and multifragmentation but has historically relied on non-relativistic formulations. Existing relativistic extensions (e.g., RQMD, JAM, UrQMD) often employ approximations, lack manifest covariance, or treat scalar and vector potentials in a manner that obscures their distinct dynamical roles. Specifically, at beam energies where the Lorentz factor $\gamma \gg 1$ , the non-relativistic formulation of QMD is inadequate. A consistent relativistic framework is required to address the synchronization of particle time coordinates, the distinction between Lorentz-scalar and baryon four-current densities, and the separate dynamical effects of scalar ( $U_S$ ) and vector ( $V^\mu$ ) potentials, which are crucial for determining the Equation of State (EoS) of strongly interacting matter.

Methodology
The authors develop a fully covariant transport framework based on constrained Hamiltonian dynamics (Dirac formalism). The methodology proceeds as follows:

Constraint Formulation: The system is described in an $8N$ $8 N$ -dimensional phase space reduced to $6N+1$ $6 N + 1$ dimensions via constraints.
- First-class constraints ( $H_i$ ): These represent the mass-shell conditions for $N$ particles interacting via scalar ( $\Phi_i$ ) and vector ( $A_{i,\mu}$ ) potentials. The total Dirac Hamiltonian is $H = \sum \lambda_i H_i$ .
- Second-class constraints ( $\chi_i$ ): These are time constraints introduced to fix the relation between particle time coordinates and a Lorentz-invariant evolution parameter $\tau$ . The authors utilize a global synchronization scheme where the time constraints define a hypersurface orthogonal to the system's total four-velocity $U^\mu$ .
Derivation of Equations of Motion (EoM): Using Dirac brackets, the authors derive manifestly covariant EoMs for the particle 4-vectors $q^\mu_i(\tau)$ $q_{i}^{μ} (τ)$ and $p^\mu_i(\tau)$ $p_{i}^{μ} (τ)$ . The derivation explicitly includes momentum-dependent derivative terms arising from the vector potential, which were often neglected in previous works.
- The compatibility of the mass-shell constraints ( $\{H_i, H_j\} = 0$ ) imposes specific forms on the potentials. For scalar interactions, the potential depends on the transverse relative distance $q_T$ . For vector interactions, a specific form is required to satisfy the compatibility condition, leading to momentum-dependent vector potentials.
Numerical Implementation: The framework is tested on two-body and four-body systems. The authors solve the EoMs numerically for various initial conditions, comparing:
- Scalar-only vs. Vector-only interactions.
- Relativistic vs. Non-relativistic limits.
- Different choices of time constraints (gauge choices).
- Frame independence (calculating in the Center-of-Mass frame vs. the Lab frame).

Key Contributions

Manifestly Covariant EoMs: The paper presents the first derivation of relativistic transport equations for an $N$ -body system interacting with both scalar and vector two-body potentials without approximations. The equations explicitly account for the momentum dependence of the vector potential.
Analysis of Time Constraints: The authors systematically investigate the implications of different time constraint choices. They demonstrate that while different constraints lead to different parametrizations of the evolution (different $\tau$ vs. $t_i$ ), the physical world-lines remain identical provided the "world-line invariance" condition is met. They clarify that differences in trajectories arise from synchronization prescriptions rather than distinct physical dynamics.
Distinct Roles of Potentials: The study disentangles the dynamical roles of scalar and vector potentials. It shows that while scalar potentials modify the effective mass, vector potentials shift the 4-momentum and generate Lorentz-like forces. This distinction leads to qualitatively different scattering behaviors, particularly at higher energies.
Validation of Covariance: The framework is numerically verified to be frame-independent. Trajectories calculated in the Center-of-Mass frame and boosted to the Lab frame match those calculated directly in the Lab frame.
Extension to N-body Systems: The formalism is generalized from two-body to $N$ -body systems (specifically demonstrated for a four-body "deuteron-deuteron" collision), providing a pathway for simulating heavy-ion collisions.

Results

Non-Relativistic Limit: The derived relativistic equations correctly reduce to non-relativistic equations in the limit $m \to \infty$ or low momentum.
Scalar vs. Vector Dynamics: Numerical simulations of two-nucleon scattering reveal significant differences between scalar and vector interactions.
- For a vector potential, the interaction acts similarly to a Lorentz force, proportional to velocity, leading to large deflection angles even at moderate energies.
- For a scalar potential, the force is driven by density gradients.
- The difference in scattering angles and final momenta between scalar and vector cases increases with initial beam momentum, challenging the validity of the Schrödinger Equivalent Potential (SEP) approximation at GeV energies.
Frame Independence: The numerical results confirm that the physical trajectories $x_i(t_i)$ are independent of the reference frame used for the calculation, validating the Lorentz invariance of the implementation.
Cluster Separability: The authors acknowledge a limitation in their current global synchronization strategy. While the framework works well for interacting systems, exact cluster separability (where isolated clusters evolve with their own synchronization vectors) is not automatically satisfied in the current implementation without manual reconstruction of constraints, a known issue in global synchronization schemes.

Significance and Claims
The paper claims to provide the "first step" of a fully covariant framework capable of serving as an analysis tool for upcoming relativistic heavy-ion experiments (e.g., CBM at FAIR, HIAF, NICA, RHIC-BESII). The authors assert that a fully covariant treatment including both scalar and vector potentials is necessary to consistently extract the high-density Equation of State from experimental data, as the distinct dynamical roles of these fields become crucial at intermediate beam energies.

The work establishes a theoretical backbone for extending QMD models to relativistic energies, enabling the study of collective phenomena, cluster production, and the microscopic origin of the EoS within a unified transport framework. The authors explicitly state that the extension to nuclear cluster formation and the detailed discussion of corresponding observables will be presented in a forthcoming paper, maintaining a modest scope regarding the immediate application to complex heavy-ion collisions.

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