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The second-order differential equation for a damped harmonic oscillator canbe converted to two coupled first-order equations, with two two-by-two matricesleading to the group $Sp(2)$. It is shown that this oscillator system containsthe essential features of Wigner's little groups dictating the internalspace-time symmetries of particles in the Lorentz-covariant world. The littlegroups are the subgroups of the Lorentz group whose transformations leave thefour-momentum of a given particle invariant. It is shown that the damping modesof the oscillator correspond to the little groups for massive andimaginary-mass particles respectively. When the system makes the transitionfrom the oscillation to damping mode, it corresponds to the little group formassless particles. Rotations around the momentum leave the four-momentuminvariant. This degree of freedom extends the $Sp(2)$ symmetry to that of$SL(2,c)$ corresponding to the Lorentz group applicable to the four-dimensionalMinkowski space. The Poincar\'e sphere contains the $SL(2,c)$ symmetry. Inaddition, it has a non-Lorentzian parameter allowing us to reduce the masscontinuously to zero. It is thus possible to construct the little group formassless particles from that of the massive particle by reducing its mass tozero. Spin-1/2 particles and spin-1 particles are discussed in detail.

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