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

Abstract: The second-order differential equation for a damped harmonic oscillator can beconverted to two coupled first-order equations, with two two-by-two matrices leading to thegroup Sp (2). ItisshownthatthisoscillatorsystemcontainstheessentialfeaturesofWigner’slittlegroupsdictatingtheinternalspace-timesymmetriesofparticlesintheLorentz-covariantworld. The little groups are the subgroups of the Lorentz group whose transformationsleave the four-momentum of a given particle invariant. It is shown that the damping modesof the oscillator correspond to the little groups for massive and imaginary-mass particlesrespectively. When the system makes the transition from the oscillation to damping mode,it corresponds to the little group for massless particles. Rotations around the momentumleave the four-momentum invariant. This degree of freedom extends the Sp (2) symmetryto that of SL (2 ,c ) corresponding to the Lorentz group applicable to the four-dimensionalMinkowski space. The Poincare sphere contains the

[1]  A. Leggett,et al.  Dynamics of the dissipative two-state system , 1987 .

[2]  O. J. Marstrand,et al.  Linear Representations of the Lorentz Group , 2014 .

[3]  V. V. Sinev.  Joint analysis of spectral reactor neutrino experiments , 2011 .

[4]  John Stillwell,et al.  Symmetry , 2000, Am. Math. Mon..

[5]  H. A. Alhendi,et al.  Textures with two traceless submatrices of the neutrino mass matrix , 2007, 0708.2007.

[6]  Young S. Kim,et al.  E(2)-like little group for massless particles and neutrino polarization as a consequence of gauge invariance , 1982 .

[7]  C. Brosseau.  Fundamentals of Polarized Light: A Statistical Optics Approach , 1998 .

[8]  Eugene P. Wigner,et al.  Cylindrical Group and Massless Particles , 1987 .

[9]  Young S. Kim,et al.  de Sitter group as a symmetry for optical decoherence , 2006 .

[10]  Y. S. Kim,et al.  One analytic form for four branches of the ABCD matrix , 2010, 1002.1234.

[11]  Johannes Bergström,et al.  Combining and comparing neutrinoless double beta decay experiments using different nuclei , 2013 .

[12]  Tao Han,et al.  Lepton number violation and W ′ chiral couplings at the LHC , 2012, 1211.6447.

[13]  Nadine Gottschalk,et al.  Fundamentals Of Photonics , 2016 .

[14]  Steven Weinberg,et al.  Photons and Gravitons in S-Matrix Theory: Derivation of Charge Conservation and Equality of Gravitational and Inertial Mass , 1964 .

[15]  Marilyn E. Noz,et al.  Symmetries Shared by the Poincar´ e Group and the Poincar´ e Sphere , 2013, 1306.2017.

[16]  V. Bargmann,et al.  Irreducible unitary representations of the Lorentz group , 1947 .

[17]  Y. S. Kim,et al.  Lorentz Group in Ray and Polarization Optics , 2012, Mathematical Optics.

[18]  Young S. Kim,et al.  Eulerian parametrization of Wigner’s little groups and gauge transformations in terms of rotations in two‐component spinors , 1986 .

[19]  Y. S. Kim,et al.  Lens optics and the continuity problems of the ABCD matrix , 2014, 1401.1205.

[20]  E. Wigner,et al.  On the Contraction of Groups and Their Representations. , 1953, Proceedings of the National Academy of Sciences of the United States of America.

[21]  Y. S. Kim.  Cylindrical Group and Massless Particles , 2014 .

[22]  Adrian Palcu.  Neutrino Mass as a Consequence of the Exact Solution of 3-3-1 Gauge Models Without Exotic Electric Charges , 2006 .

[23]  I. Satake,et al.  On some types of topological groups: Ann. of Math., (2) 50 (1949), 507–558 , 2001 .

[24]  Ernest Ma,et al.  Gauge model of generation nonuniversality revisited. , 1988 .

[25]  P. W. Higgs.  Broken Symmetries and the Masses of Gauge Bosons , 1964 .

[26]  L. Infeld.  Quantum Theory of Fields , 1949, Nature.

[27]  Eugene P. Wigner,et al.  On unitary representations of the inhomogeneous lorentz group , 1989 .

[28]  I. P. Ivanov,et al.  Higgs boson masses of the general two-Higgs-doublet model in the Minkowski-space formalism , 2009, 0910.4492.

[29]  Y. S. Kim.  Space-time geometry of relativistic particles , 2014 .

[30]  M. Drewes.  The Phenomenology of Right Handed Neutrinos , 2013, 1303.6912.

[31]  Jun Cao,et al.  Unambiguous determination of the neutrino mass hierarchy using reactor neutrinos , 2013, 1303.6733.

[32]  D. Han,et al.  Stokes Parameters as a Minkowskian Four-vector , 1997, physics/9707016.

[33]  S. Sternberg,et al.  Symplectic Techniques in Physics , 1984 .

[34]  Kenkichi Iwasawa,et al.  On Some Types of Topological Groups , 1949 .

[35]  J. L. Lopes,et al.  A model for leptons , 1977 .

[36]  N. Sasao,et al.  Observables in Neutrino Mass Spectroscopy Using Atoms , 2012, 1209.4808.

[37]  A. Barut,et al.  The four states of the Massless neutrino with pauli coupling by Spin-Gauge invariance , 1986 .

[38]  Sibel Başkal,et al.  Little groups and Maxwell-type tensors for massive and massless particles , 1997 .

[39]  J. Bergstrom,et al.  Combining and comparing neutrinoless double beta decay experiments using different nuclei , 2012, 1212.4484.

[40]  G. Guralnik,et al.  Global Conservation Laws and Massless Particles , 1964 .

[41]  Young S. Kim,et al.  Symmetries Shared by the Poincaré Group and the Poincaré Sphere , 2013, Symmetry.

[42]  R. Shaw,et al.  Unitary representations of the inhomogeneous Lorentz group , 1964 .

[43]  Y S Kim,et al.  Iwasawa effects in multilayer optics. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[44]  T. S. Kosmas,et al.  Exotic Lepton Flavour Violating Processes in the Presence of Nuclei , 2013 .

[45]  S. Bas,et al.  One analytic form for four branches of the ABCD matrix , 2010 .

[46]  Marilyn E. Noz,et al.  Theory and Applications of the Poincaré Group , 1986 .

[47]  Lino Miramonti,et al.  Advancements in solar neutrino physics , 2013, 1304.6304.

[48]  Young S. Kim,et al.  Photons, Neutrinos and Gauge Transformations , 1986 .

[49]  D. Malacara-Hernández,et al.  PRINCIPLES OF OPTICS , 2011 .