Symmetry 2014, 6, 473-515; doi:10.3390/sym6030473 OPEN ACCESS symmetry ISSN 2073-8994 www.mdpi.com/journal/symmetry Article Wigner’s Space-Time Symmetries Based on the Two-by-Two Matrices of the Damped Harmonic Oscillators and the Poincaré Sphere Sibel Ba¸skal 1, Young S. Kim 2;* and Marilyn E. Noz 3 1 Department of Physics, Middle East Technical University, Ankara 06800, Turkey; E-Mail:
[email protected] 2 Center for Fundamental Physics, University of Maryland, College Park, MD 20742, USA 3 Department of Radiology, New York University, New York, NY 10016, USA; E-Mail:
[email protected] * Author to whom correspondence should be addressed; E-Mail:
[email protected]; Tel.: +1-301-937-1306. Received: 28 February 2014; in revised form: 28 May 2014 / Accepted: 9 June 2014 / Published: 25 June 2014 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.