Wigner’s Space-Time Symmetries Based on the Two-by-Two Matrices of the Damped Harmonic Oscillators and the Poincaré Sphere

Symmetry

Sibel Başkal²  Young S. Kim¹ 

[1] Center for Fundamental Physics, University of Maryland, College Park, MD 20742, USA;Department of Physics, Middle East Technical University, Ankara 06800, Turkey; E-Mail:

关键词: damped harmonic oscillators;    coupled first-order equations;    unimodular matrices;    Wigner’s little groups;    Poincaré sphere;    Sp(2) group;    SL(2;    c) group;    gauge invariance;    neutrinos;    photons;   

DOI  :  10.3390/sym6030473

来源: mdpi

【 摘 要 】

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 Poincaré 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.

【 授权许可】

CC BY   
© 2014 by the authors; licensee MDPI, Basel, Switzerland