A Schrodinger-Type Formalism and Observable Wavefunctions in Dynamical Systems

László Árpád Gergely

X. Gutiérrez de la Cal, A. Matzkin

It is known that relativistic wavefunctions formally propagate beyond the light cone when the propagator is limited to the positive energy sector. By construction, this is the case for solutions of the Salpeter (or relativistic Schrödinger) equation or for Klein–Gordon and Dirac wavefunctions defined in the Foldy–Wouthuysen representation. In this work, we quantitatively investigate the degree of non-causality for free propagation for different types of wavepackets that all initially have a compact spatial support. In the studied examples, we find that non-causality appears as a small transient effect that can in most cases be neglected. We display several numerical results and discuss the fundamental and practical consequences of our findings concerning this peculiar dynamical feature.

R. Huegele, Z. E. Musielak, John L. Fry

A search for fundamental (Galilean invariant) dynamical equations for two- and four-component spinor wavefunctions is conducted in Galilean spacetime. A dynamical equation is considered as fundamental if it is invariant under the symmetry operators of the group of the Galilei metric and if its state functions transform like the irreducible representations of the group of the metric. It is shown that there are no Galilean invariant equations for two-component spinor wavefunctions. A method to derive the Levy-Leblond equation for a four-component spinor wavefunction is presented. It is formally proved that the Levy-Leblond and Schrodinger equations are the only Galilean invariant four-component spinor equations that can be obtained with the Schrodinger phase factor. Physical implications of the obtained results and their relationships to the Pauli–Schrodinger equation are discussed.

Leon Bess

Benno Fuchssteiner

Samir Baladi

ENTRO-QUANTUM (E-LAB-07) introduces a quantum-inspired probabilistic framework for modeling entropy dynamics in artificial intelligence systems operating near instability boundaries.Classical entropy models represent system state as a deterministic scalar. This work demonstrates that such representations fail to capture critical phenomena observed in high-dimensional AI systems, including sudden collapse events, non-local correlations, and measurement-induced instability.To address these limitations, this study defines the Entropic Wavefunction Ψ-W(s,t), a complex-valued probability amplitude over stability states. The framework extends classical entropy dynamics into a probabilistic regime governed by an Entropic Schrödinger Equation, enabling the modeling of:Superposition of stability statesMeasurement-induced collapseInformational entanglement across distributed systemsDiscontinuous entropy transitions via Quantum Jump OperatorsA formal Entropic Uncertainty Principle is derived, establishing a lower bound between measurement precision and disturbance. This result leads to the formulation of the Silent Observer Protocol, an adaptive monitoring strategy that minimizes collapse risk in near-critical systems.The framework unifies and extends previous components of the EntropyLab Research Program (E-LAB-01 through E-LAB-06), recovering classical entropy dynamics as a limiting case of wavefunction collapse.The paper provides a complete mathematical formalism, theoretical predictions (P1–P5), and a simulation-based validation strategy using Monte Carlo trajectory methods.This work is theoretical and computational in nature. All results are derived from first principles and validated through controlled simulations. No human or real-world datasets are involved.ENTRO-QUANTUM establishes a new modeling paradigm for AI stability under uncertainty, bridging deterministic control architectures and probabilistic dynamics inspired by quantum mechanics. This work is preregistered on OSF Registries: Registration DOI: 10.17605/OSF.IO/APZ7Y Registry: OSF Registries Registration Type: Preregistration Date Registered: April 10, 2026 License: CC-BY 4.0