An Algebraic-Recursive Approach to Generate Higher-Order Symmetry Operators for Schrödinger and Klein-Gordon equations

TL;DR

The paper develops an algebraic-recursive framework to generate higher-order symmetry operators that commute with a central Hamiltonian $\hat{H}$, anchored in a Jordan–Lie algebra decomposition of first-order generators for the Schrödinger equation. It then systematically constructs the centralizers $Z_n(\hat{H})$ by projecting products of lower-order generators, yielding explicit bases (e.g., $\mathbf{A}_4$, $\mathbf{A}_{10}$, $\mathbf{A}_{20}$) and a general counting formula $a_4(n)=\tfrac{(n+1)(n+2)(n+3)}{6}$; these constructions extend to the Klein–Gordon equation in Minkowski space, where relativistic symmetry operators arise and relate to Poincaré/Lorentz structures. The work also explores fractional symmetry operators via Dunford contour integrals and perturbative deformations of the symmetry algebra (e.g., 1D harmonic oscillator perturbations and a fourth-order KG extension), revealing Weyl-algebra–type structures in perturbed regimes. Altogether, the method provides a compact, algebraic route to classify and generate higher-order centralizers, with potential applications to relativistic wave equations and nonlocal symmetry analyses. The results offer a practical framework to study symmetry hierarchies and their perturbations in quantum mechanics and quantum field–like settings.

Abstract

This article explores an algebraic-recursive approach to construct differential operators that commute with a central operator $\hat{H}$ in quantum mechanics. Starting from the Schrödinger equation for a free particle, the work derives first-order symmetry generators, such as translations, rotations, and boosts, and examines their algebraic basis encompassing Lie and Jordan algebras. The analysis is then extended to higher-order operators, demonstrating how they can be constructed from the first-order ones through algebraic operations and Lie algebra simplification. This methodology is applied to the Klein-Gordon equation in Minkowski space-time, yielding relativistic symmetry operators. Furthermore, we defined an approximation to fractional symmetry operators of the Schrodinger equation, and a perturbative approach is employed for a case where the commutation is more general, illustrated with a one-dimensional harmonic oscillator and the fourth-order Klein-Gordon equation. The results include a general formula for the number of operators as a function of the order and the dimension of the algebraic basis, providing a reduced-form development of the differential higher-order centralizers' basis.