A symmetry trip from Caldirola to Bateman damped systems

TL;DR

This paper explores the symmetry structures of damped quantum harmonic oscillators, revealing how extending their algebraic symmetries naturally leads to Bateman's dual system with a reservoir particle, and discusses its quantization.

Contribution

It demonstrates that including time evolution symmetry in damped systems extends their algebra to encompass Bateman's dual system, providing a new perspective on their symmetry and quantization.

Findings

Symmetry algebra of Caldirola-Kanai system extended to include time evolution.

Extension leads to Bateman's dual system with a reservoir particle.

Quantization results in a first-order Schrödinger equation.

Abstract

For the Caldirola-Kanai system, describing a quantum damped harmonic oscillator, a couple of constant-of-motion operators generating the Heisenberg algebra can be found. The inclusion of the standard time evolution symmetry in this algebra for damped systems, in a unitary manner, requires a non-trivial extension of this basic algebra and hence the physical system itself. Surprisingly, this extension leads directly to the so-called Bateman's dual system, which now includes a new particle acting as an energy reservoir. The group of symmetries of the dual system is presented, as well as a quantization that implies, in particular, a first-order Schr\"odinger equation. The usual second-order equation and the inclusion of the original Caldirola-Kanai model in Bateman's system are also discussed.