Time-reversal violation as loop-antiloop symmetry breaking: the Bessel equation, group contraction and dissipation

TL;DR

This paper explores how the Bessel equation can be transformed into damped or amplified oscillators, revealing that group contraction leads to time-reversal symmetry breaking and connecting loop-algebras with Euclidean algebras.

Contribution

It introduces a novel link between Bessel equations, group contraction, and symmetry breaking, providing insights into time-reversal violation and algebraic structures.

Findings

Bessel equation can be transformed into damped/amplified oscillators.

Group contraction causes loop-antiloop symmetry breaking.

Time-reversal symmetry is violated due to group contraction.

Abstract

We show that the Bessel equation can be cast, by means of suitable transformations, into a system of two damped/amplified parametric oscillator equations. The relation with the group contraction mechanism is analyzed and the breakdown of loop-antiloop symmetry due to group contraction manifests itself as violation of time-reversal symmetry. A preliminary discussion of the relation between some infinite dimensional loop-algebras, such as the Virasoro-like algebra, and the Euclidean algebras e(2) and e(3) is also presented.