Wigner's Space-time Symmetries based on the Two-by-two Matrices of the Damped Harmonic Oscillators and the Poincar\'e Sphere

TL;DR

This paper links damped harmonic oscillators to Wigner's space-time symmetries, showing how oscillator damping modes correspond to particle little groups, and explores the continuous transition from massive to massless particles within the Lorentz group framework.

Contribution

It demonstrates the connection between oscillator damping modes and Wigner's little groups, extending the $Sp(2)$ symmetry to $SL(2,c)$ and relating it to particle mass transitions.

Findings

Damping modes correspond to little groups for massive and imaginary-mass particles.

Transition from oscillation to damping mode models massless particle symmetries.

The Poincaré sphere encodes $SL(2,c)$ symmetry and allows continuous mass reduction.

Abstract

The second-order differential equation for a damped harmonic oscillator can be converted to two coupled first-order equations, with two two-by-two matrices leading to the group $Sp(2)$. It is shown that this oscillator system contains the essential features of Wigner's little groups dictating the internal space-time symmetries of particles in the Lorentz-covariant world. The little groups are the subgroups of the Lorentz group whose transformations leave the four-momentum of a given particle invariant. It is shown that the damping modes of the oscillator correspond to the little groups for massive and imaginary-mass particles respectively. When the system makes the transition from the oscillation to damping mode, it corresponds to the little group for massless particles. Rotations around the momentum leave the four-momentum invariant. This degree of freedom extends the $Sp(2)$ symmetry to that of $SL(2,c)$ corresponding to the Lorentz group applicable to the four-dimensional Minkowski space. The Poincar\'e sphere contains the $SL(2,c)$ symmetry. In addition, it has a non-Lorentzian parameter allowing us to reduce the mass continuously to zero. It is thus possible to construct the little group for massless particles from that of the massive particle by reducing its mass to zero. Spin-1/2 particles and spin-1 particles are discussed in detail.