On the Derivation of Equations of Motion from Symmetries in Quantum-Mechanical Systems via Heisenberg's Uncertainty

Abstract

We propose the construction of equations of motion based on symmetries in quantum-mechanical systems, using Heisenberg's uncertainty principle as a minimal foundation. From canonical operators, two spaces of conjugate operators are constructed, along with a third space derived from the former, which includes the ``Symmetry-Dilation'' operator. When this operator commutes with the main equation of motion, it defines the set of observables compatible with a complete basis of operators (symmetry generators), organized into a Lie algebra dependent on Heisenberg's uncertainty principle within Minkowski spacetime. Furthermore, by requiring the dilation operator to commute with the central operator, the wavefunction is constrained, thereby constructing known structures. Specific cases are derived -- relativistic, non-relativistic, and a lesser-studied case: ``ultra-relativistic (Carroll-Schr\"odinger)''. Our work may open new avenues for understanding and classifying symmetries in quantum mechanics, as well as offer an alternative method for deriving equations of motion and applying them to complex scenarios involving exotic particles.