An Algebraic-Recursive Approach to Generate Higher-Order Symmetry Operators for Schr\"odinger and Klein-Gordon equations

TL;DR

This paper introduces an algebraic-recursive method to systematically generate higher-order symmetry operators for Schrödinger and Klein-Gordon equations, expanding the understanding of their algebraic structures and symmetries.

Contribution

It presents a novel recursive algebraic approach to construct higher-order symmetry operators for quantum equations, including relativistic cases, and introduces approximations for fractional symmetries.

Findings

Derived first-order symmetry generators for Schrödinger equation

Extended the method to Klein-Gordon equation in Minkowski space

Provided a formula for the number of symmetry operators based on order and algebraic basis

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 Schr\"odinger 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.