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2021 Circle and Ellipse? Click here for an interesting story. Physics of the Lorentz Group, Second Edition: Beyond High-energy Physics and Optics, with Sibel Baskal and Marilyn Noz, to be published by the IOP (British Institute of Physics). Maquis Who's Who Professional Narraitive Features Hemisphere Magazine Milestones His home page University of Maryland Photo-biography Top Educators Click here to expand this image. Wall Street Journal Millenium Magazine 2021. Research Features Webpage. What do they say about me? Click here for the magazine article. IAOTP (International Association of Top Professionals) Video. Top Professor of the Year. Fearless Leaders. Press Release Distribution. Lifetime Achievement PR Underground. List of Distributions. 2020 Integration of Dirac’s Efforts to Construct a Lorentz-covariant Quantum Mechanics, with Marilyn E. Noz. Symmetry [12(8)], 1270 (2020), doi:10.3390/sym12081270, arXiv:2008.00551, pdf. Kant and Hegel in Physics, arXiv:https:2009.06198, pdf with sharper images. webpage, recommended. 2019 Einstein's E = mc2 derivable from Heisenberg's Uncertainty Relations, with Sibel Baskal and Marilyn Noz, Quantum Reports [1(2)], 236 - 251 (2019), doi:10.3390/quantum1020021, arXiv:abs/1911.03818, For pdf with sharper images, click here. Mathematical Devices for Optical Sciences. with S. Baskal and M.E. Noz IOP Book available from the amazon.com Role of Quantum Optics in Synthesizing Quantum Mechanics and Relativity, Invited paper presented at the 26th International Conference on Quantum Optics and Quantum Information (Minsk, Belarus, May 2019), arXiv.org/abs/1911.01642, pdf with sharper images. Poincaré Symmetry from Heisenberg's Uncertainty Relations, with S. Baskal and M. E. Noz, Symmetry [11(3)], 236 - 267 (2019), doi:10.3390/sym11030409, arXiv: abs/1903.05348"> ArXiv. Click here for my Einstein page. Click here for the Lorentz group applicable to optical sciences. 2018 New Perspectives on Einstein's E = mc2, with Marilyn E. Noz, World Scientific, availabble from various sources. For the city of Bern, the birth place of E = mc2, click here. 2017 In physics, harmonic oscillators and two-by-two matrices never become out-dated. Harmonic Oscillators and Two-by-two Matrices in Symmetry Problems in Physics, Collection of the papers published in the Symmetry journal. Click here available from the amazon.com. Photons in the Quantum World, with S. Baskal and M. E. Noz, Invited paper presented at the 15th International Conference on Squeezed States and Uncertainty Relations (Jeju, South Korea, August 12-17, 2017). ArXiv. Loop Representation of Wigner's Little Groups, with S. Baskal and M. E. Noz, Symmetry [9](7), 97:1-22 (2017), doi:10.3390/sym9070097. ArXiv. 2016 Princeton's Einstein Genealogy Eugene Paul Wigner's Nobel Prize, pdf. ArXiv. Symmetries of Massive and Massless Neutrinos, Invited paper presented at the International Conference on Hadron Structure and QCD: from Low to High Energies (Gatchina, Russia, 2016). ArXiv. Do Small-Mass Neutrinos Participate in Gauge Transformations? with G. Q. Maguire Jr. and M. E. Noz, Advances in High Energy Physics (2016), Article ID 1847620; doi:10.1155/2016/1847620;1-7. ArXiv. Entangled Harmonic Oscillators and Space-Time Entanglement, Symmetry [8](7), 55:1 - 26 (2016) doi: 10.3390/sym8070055. ArXiv. 2015 Physics of the Lorentz Group, with S. Baskal and M. E. Noz, IOP Concise Physics pdf, available from the amazon.com. The title page of the book contains this figure. The O(3,2) Symmetry derivable from the Poincaré Sphere Chapter in the Nova Editorial Book: Relativity, Gravitation, Cosmology: Foundations, edited by Valeriy Dvoeglazov (2015). ArXiv. Lorentz Coherence and the Proton Form Factor, Phys. Scr. [90], 074037:1-7 (2015), Special Issue dedicated to 150 Years of Margarita and Vladimir Man'ko. ArXiv. 2014 QCD. What else is needed for the proton structure function, presented at the Hadron Structure and QCD: from Low to High Eenegies (Gatchina, Russia, 2014). ArXiv. Wigner’s Space-Time Symmetries based on the Two-by-Two Matrices of the Damped Harmonic Oscillators and the Poincaré Sphere, with S. Baskal and M. E. Noz, Symmetry [6], 473 - 515 (2014), doi: 10.3390/sym6030473. ArXiv. Poincaré Sphere and a Unified Picture of Wigner's Little Groups, Proceedings of Wigner 111, Colorful & Deep Symposium, edited by S. Varro, P. Adam, T.S. Biro, G. G. Barnafoldi, and P. Levai, EPJ Web of Conferences, [78], 01005:1 - 8 (2014), doi10.151/ejpcon/20147801005. ArXiv. Entropy and Temperature from Entangled Space and Time, with M. E. Noz, Phys. Sci. Int. J. [4], 1015 - 1039 (2014). ArXiv. Lens optics and the continuity problems of the ABCD matrix, with S. Baskal , J. Mod. Opt. [61], 161 - 166 (2014). ArXiv. 2013 Historical Approach to physics according to Kant, Einstein, and Hegel, presented at the 32nd Congress of the Italian Society of Historians of Physics and Astronomy (Rome, Italy, September 2012), publshed in the "Physics, Astronomy, and Engineering. Critical Problems in the History of Science and Society", edited by Raphael Pisano, Danilo Cappechi, and Anna Lukesova (Scientia Socialis Press, Scientia Socialis, Siauliai, Lithuania, 2013), pages 113 - 122. ArXiv: https://arxiv.org/abs/1301.6091 pdf with sharper images. Symmetries shared by the Poincaré Group and Poincaré Sphere, with Marilyn E. Noz, Symmetry [5], 223 - 252 (2013). doi:10.3390/sym5030233, ArXiv. Lorentz Group in Ray and Polarization Optics, with Sibel Baskal, Chapter 9 in "Mathematical Optics: Classical, Quantum and Computational Methods" edited by Vasudevan Lakshminarayanan, Maria L. Calvo, and Tatiana Alieva (CRC Taylor and Francis, New York 2013), pages 303 - 349. ArXiv. 2012 Dirac Matrices and Feynman's Rest of the Universe, with M. E. Noz, Symmetry [4], 626 - 643 (2012). ArXiv. Poincaré Sphere and Decoherence Problems, presented at the Fedorov Memorial Symposium: International Conference "Spins and Photonic Beams at Interface," dedicated to the 100th anniversary of F.I.Fedorov (Minsk, Belarus, 2011). ArXiv. 2011 Lorentz Harmonics, Squeeze Harmonics and Their Physical Applications, with M. E. Noz, Symmerty [3], 16 - 36 (2011). ArXiv. Click on each figure to enlarge it. Einstein's Hydrogen Atom 2nd International Conference on Multimedia Technology (July 2011) in Hangzhou, China; Workshop on Physics and Mathematics. IEEE Catalong No. CFP1153K-PRT, [7], Part II, 6324 - 6329 (2011), pdf. ArXiv. Time separation as a hidden variable in the Copenhagen school of quantum mechanics, in Advances in Quantum Theory, with M. E. Noz, AIP Conference Series, No.1327, 138 - 147 (2011). ArXiv. Internal space-time symmetries of particles derivable from periodic systems in optics, Optics and Spectroscopy [111], 731 - 737 (2011). ArXiv. The Language of Two-by-two Matrices spoken by Optical Devices, presented at the 19th International Conference on Composites or Nano Engineering, Shanghai (China 2011). ArXiv. 2010 Internal Space-time Symmetries according to Einstein, Wigner, Dirac, and Feynman, with M. E. Noz, presented at the Petrov Centennial Conference (Kazan 2010). ArXiv. QCD's Partner needed for Mass Spectra and Parton Structure Functions, Acta Physica Polonica B [2], 325 - 330 (2010). ArXiv. Optical activities as computing resources for space-time symmetries, Journal of Modern Optics [57], 17 - 22 (2010). AriXiv Possible Minkowskian Language in Two-level Systems, Journal of Optics and Spectroscopy [108], 297 - 300 (2010). ArXiv. One analytic form for four branches of the ABCD matrix, with S. Baskal, J. Mod. Opt. [57], 1251 - 1259 (2010). ArXiv. Tangential Continuity, a new word in mathematics? 2009 ABCD matrices as similarity transformations of Wigner matrices and periodic systems in optics, with S. Baskal, J. Opt. Soc. Am. A [26], 3049 - 2054 (2009). ArXiv. Can the Quark Model Be Relativistic Enough to Include the Parton Model? with M. E. Noz, The Physics of Quarks: New Research (Horizons in World Physics, Volume 265), edited by Nicolas L. Watson and Theo M. Grant, Nova Science Publishers, Inc. (Hauppauge, NY, 2009). pages 139 - 162. ArXiv. 2007 Coupled oscillators and Feynman's three papers, IOP Conference Series [70],012010:1-19 (2007). ArXiv. Can you do quantum Mechanics without Einstein? with M. E. Noz, Foundations of probability and physics - 4, AIP Conf. Proc. [889], 152 - 161 (2007). ArXiv. 2006 Einstein's coordinate in philosophy. My own philosophy. Einstein, Kant, and Taoism, ArXiv. Lorentz Group for Finite Crystals, with S. Baskal and E. Georgieva, ArXiv. The Question of Simultaneity in Relativity and Quantum Mechanics, with M. E. Noz, in Quantum Theory: Reconsideration of Foundations III, American Institute of Physics 0-7354-0301/06, pages 168-178 (2006). ArXiv. de Sitter group as a symmetry for optical decoherence, with S. Baskal, Journal of Physics A [39], 7775 - 7788 (2006). pdf. ArXiv with a different title. 2005 Standing Waves in the Lorentz-covariant World, with M. E. Noz, Foundations of Physics [35], 1289 - 1305 (2005). ArXiv. Rotations associated with Lorentz boosts, with S. Baskal, J. Phys. A [38], 6545 - 6556 (2005). ArXiv. Coupled oscillators, entangled oscillators, and Lorentz-covariant Oscillators, with M. E. Noz, Jounal of Optics B: Quantum and Semiclassical [7], s 459 - 467 (2005). ArXiv. Harmonic Oscillators as Bridges between Theories: Einstein, Dirac, and Feynman, with M. E. Noz, presented at the International Symposium on Interdisciplinary Science (Natchitoches, Louisiana, October 2004), Proceedings edited by A. Ludu, AIP Conference Proceedings, CP755, 61 - 69 (2005). ArXiv. The language of Einstein spoken by optical instruments, Optics and Spectroscopy [99] (3), 443 - 446 (2005). ArXiv. 2004 Einstein, Wigner and Feynman: From E = mc 2 to Feynman’s decoherence via Wigner’s little groups, presented at the Wigner Centennial Conference held in Pecs, Hungary (July 2002), Acta Physica Hungarica A [19], 317-318 (2004). ArXiv. Dashen-Frautschi Fiasco and Historical Roadmap for Strings, ArXiv. Is it possible to construct the Proton Structure Function by Lorentz-boosting the Static Quark-model Wave Function? with M. E. Noz, Int. J. Modern Phys. A [19], 5435 - 5442 (2004). ArXiv. Lorentz group in ray optics, with S. Baskal, E. Georgieva, and M. E. Noz, Journal of Optics B: Quantum and Secmiclassical Optics [6] S455 - S472 (2004). AriXiv. 2003 Feynman liked the Brazilian city of Rio de Janeiro, and he went there very often. I went there in 2000 to see why he liked this city so much. Click here for my Feynman page. Feynman's Branes and Feynman's Oscillators, presented at the Workshop on Branes and Generalized Dynamics (Argonne, Illinois, USA, October 2003). ArXiv. Is the concept of quantum probability consistent with Lorentz covariance? with M. E. Noz, presented at the Second International Conference on Foundations of Probability in Physics (Vaxjo, Sweden, June 2002). Proceedings edited by Andrei Khrennikov, Vaxjo University Press (Vaxjo 2002). pages 345-358. ArXiv. Feynman's decoherence, Optics and Spectroscopy [94], 733 - 740 (2003). ArXiv. Lens optics as an optical computer for group contractions, with S. Baskal, Phys. Rev. E [67], 056601:1 - 6 (2003). ArXiv. Slide-rule-like property of Wigner’s little groups and cyclic S matrices for multilayer optics, with E. Georgieva, Phys. Rev. E [68], 026606: 1 -7 (2003). ArXiv. Wigner's new physics frontier: Physics of two-by-two matrices, including the Lorentz group and optical instruments, presented at various conferences during the year 2003. ArXiv. 2002 Lens optics and group contractions, with S. Baskal, Phys. Rev. E [67], 056601:1 - 7 (2002). ArXiv. Wigner's spins, Feynman's partons, and their common ground, based on one of the lectures delivered at the Advanced Study Institute on Symmetries and Spin (Prague, Czech Republic, July 2001), Czech.J.Phys. 52: C433 - C442 (2002). ArXiv. Internal spacetime symmetries of massless particles and neutrino polarization as a consequence of gauge invariance, based on one of the lectures delivered at the Advanced Study Institute on Symmetries and Spin (Prague, Czech Republic, July 2001), Czech.J.Phys. 52: C353 - C364 (2002). ArXiv. Quark-parton phase transitions and the entropy problem in quantum mechanics, presented at International Conference on Quantum Electrodynamics and Statistical Physics, dedicated to the 90th anniversary of Alexander I. Akhiezer (Kharkov, Ukraine, Oct. 30 - Nov. 3, 2001), The Problems of Atomic Science and Technology, Special Issue dedicated to the 90th Birthday Anniversary of A. I. Akhiezer, No. 6 (1), 149 - 153 (2001). ArXiv. Wigner rotations in laser cavities. with S. Baskal, Phys. Rev. E [66], 26604:1 - 6 (2002). ArXiv. 2001 Internal space-time symmetries of massive and massless particles and their unification, prsented at the International Conference on Supersymmetry and Quantum Field Theory, in commemoration of Dmitri Volkov's 75th Birthday, Kharkov, Ukraine, 2000), Nuclear Physics (B - Proc. Suppl.) [102/103], 369 - 176. ArXiv. Lorentz group in Feynman's world: Wignr's Little Groups and their Applicationns, with M. E. Noz, in "Lorentz and Poincaré Invariance - 100 Years in the Development of Relativity," Pages 351 - 376, edited by J. P. Hsu and P. Zhang (World Scientific, 2001). ArXiv. Symmetries shared by particle physics and quantum optics, presented at the 23rd International Colloquium on Group Theoretical Methods in Physics (Dubna, 2000). ArXiv. Iwasawa effect in multilayer optics, with E. Georgieva, Phys. Rev. E [64], 026602: 1-6 (2001). ArXiv. Shear representations of beam transfer matrices, with S. Bakal, Phys. Rev. E [63], 056606:1 - 6 (2001). ArXiv. (with a different title). 2000 Feynman's entropy and decoherence problems, presented at the 20th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, (Gif-sur-Yvette, France 3-13 July 2000). Proceedings edited by Ali Mohammad-Djafari, AIP Conference Proceedings, Volume 568 (2001): pages 409-417. ArXiv. Further contents of Einstein's E = mc2 , presented at the 4th International Conference on Geometrization of Physics (Kazan, Russia, October, 1999). ArXiv, webpage. Interferometers and decoherence matrices, with D. Han and M. E. Noz, Phys. Rev. E [61], 5907 - 5913 (2000). ArXiv. 1999 Illustratative Example of Feynman's Rest of the Universe, with D. Han and M. E. Noz, Am. J. Phys. [67], 61 - 66 (1999), pdf. ArXiv. Gauge Transformations on Massless Spin-1/2 Particles and Neutrino Polarization as a Consequence of Gauge Invariance, contribution to the Proceedings of International Workshop on "Lorentz Group, CPT and Neutrinos" (Zacatecas, Mexico, June 23-26, 1999). ArXiv. Space-time Symmetry Transformations of Elementary Particles realized in Optics Laboratories, presented at various conferences during the year 1999. ArXiv. Wigner Rotations and Iwasawa Decompositions in Polarization Optics, with D. Han and M. E. Noz, Phys. Rev. E. [60], 1036 - 1041 (1999). ArXiv. Two-pearl Strings: Feynman's Oscillators (Protvino) presented at the 22nd International Workshop on the Fundamental Problems of High Energy Physics and Field Theory (Protvino, Russia, June 1999). ArXiv. 1998 Does Lorentz boost destroy coherence? Fortschr. der Physik [46], 713 - 724 (1998). ArXiv. Lorentz Covariance and Internal Space-time Symmetry of Relativistic Extended Particles, Contribution to the Nova Collection of papers on the Lorentz group (edited by Valeri Dvoeglazov). ArXiv. Covariant Model of Relativistic Extended Particles based on the Oscillator Representation of the Poincaré Group, contribution to the Volume dedicated to Prof. T.Y.Wu on his 90th Birthday edited by J. P. Hsu and L. Hsu (World Scientific, Singapore, 1978). ArXiv. Separable Four-dimensional Harmonic Oscillators and Representations of the Poincaré Group, 3rd International Workshop on Classical and Quantum Integrable Systems (Yerevan, Armenia, 1998). ArXiv. Wigner's photons, presented at the 5th International Conference on Squeezed States and Uncertainty Relations (Lake Balatonfured, Hungary, May 27-31, 1997). ArXiv. Squeezed States as Representations of Symplectic Groups, presented at the 5th International Conference on Squeezed States and Uncertainty Relations (Lake Balatonfured, Hungary, May 27-31, 1997). ArXiv. 1997 Little groups and Maxwell-type tensors for massive and massless particles, Euro. J. Phys. [47], 357 - 380 (1997). ArXiv. Jones matrix formalism as a representation of the Lorentz group J. Opt. Soc. Am. A [14], 2290 - 2298 (1997). ArXiv. Stokes parameters as a Minkowskian four-vector, with D. Han and M. E. Noz, Phys. Rev. E [56], 6055 - 6076 (1997). ArXiv. I met Erdal Inonu at a conference held in Instanbul (1997). The younger person in this photo is Nikolaj Gromov. He was very enthusiastic about group contractions at that time. Click here for my Wigner page, and Einstein page. Group Contractions: Inonu, Wigner, and Einstein, Int. J. Mod. Phys. A [12], 71 - 78 (1997). ArXiv. High-speed Contraction of Transverse Rotations to Gauge Transformations presented at the International Workshop on Quantum Groups, Deformations, and Contractions, Istanbul, Turkey (September 1997). ArXiv. Wavlets and information preserving transformations, Quantum Communication, Computing, and Measurement, edited by O. Hirota, A. S. Holevo, and C. M. Caves (Plenum, New York, 1997). pages 119 -126, presented at the 3rd International Conference on Quantum Communications and Measurements (Fuji-Hakone Land, Japan, September, 1996). ArXiv. Eugene Wigner and translation invariance (Poznan 96), presented at the 4th International School on Theoretical Physics: Symmetry and Structural Properties of Condensed Matter (Zajaczkowo, Poland, August-September, 1996). Proceedings edited by T. Lulek, W. Florek, and B. Lulek, World Scientific (Singapore, 1997): pages 1-12. ArXiv. Lorentz group in condensed matter physics (Poznan 96) presented at the 4th International School on Theoretical Physics: Symmetry and Structural Properties of Condensed Matter (Zajaczkowo, Poland, August-September, 1996). Proceedings edited by T. Lulek, W. Florek, and B. Lulek, World Scientific (Singapore, 1997): pages 116-127. ArXiv. 1996 Click here for my Wigner page. Wigner's Last Papers on Spacetime Symmetries, presnted at the 4th Wigner Symposium (Gudalajara, Mexico, 1995). Proceedings edited by N. M. Atakisiev, T. H. Seligman, K. B. Wolf (World Scientific, Singapore 1996): pages 1-10. ArXiv. Wigner's Influence on Particle Physics: Unification of Spacetime Symmetries of Massive and Massless Particles, Acta Physica Polonica B [27], 2741 - 2746 (1996). ArXiv. Lorentz Group derivable from Polarization Optics, with M. E. Noz, presented at the 21st International Colloquium on Group Theoretical Methods in Physics (Goslar, Germany, July, 1996). ArXiv. Polarization Optics and Bilinear Representations of the Lorentz Group, with D. Han and M. E. Noz, Phys. Lett. A [219], 26 - 32 (1996). Quantization and Spontaneous Emission in Circular Cylindrical Cavities, with K. Kakazu, Prog. of Theor. Phys. [96], 883 - 899 (1996). ArXiv. Lorentz Boosts as Squeeze Transformations and the Parton picture J. Moscow Phys. Soc. [6], 45 - 58 (1996). ArXiv. Paul A. M. Dirac and his wife. She was Wigner's younger sister. Wigner's Sisters, based on the preface of the Proceedings of the 4th Wigner Symposium, edited by N. M. Atakisiev, T. H. Seligman, K. B. Wolf (World Scientific, Singapore 1996). Symposium (World Scientific, Singapore, 1996). Webpage. ArXiv. Neutrino Polarization as a Consequence of Gauge Invariance pesented at the 2nd International Workshop on Quantum Systems (Minsk, Belarus, May, 1996). ArXiv. Lorentz Boosts as Squeeze Transformations and Coherence Problems, with D. Han presented at the 4th Wigner Symposium (Guadalajara, Mexico, 1995). Proceedings edited by N. M. Atakisiev, T. H. Seligman, K. B. Wolf (World Scientific, Singapore 1996): pages 458-461. ArXiv. Lorentz Boosts as Squeeze Transformations and Coherence Problems, with D. Han, presented at the 4th Wigner Symposium (Gudalajara, Mexico, 1995). ArXiv. 1995 Dirac's mathematical poem for two oscillators. Click here for larger image. O(3,3)-like Symmetries of Coupled Harmonic Oscillators, with D. Han and M. E. Noz, J. Math. Phsys. [36], 3940 - 3954 (1995). ArXiv. Wavelets, Windows, and Photons, with D. Han and M. E. Noz, Phys. Lett. A [206], 299 - 304 (1995). Two Different Squeeze Transformations, with D. Han, presented at the 4th International Conference on Squeezed States and Uncertainty Relations (Taiyuan, Shanxi, China, June 5-8, 1995). ArXiv. 1994 Quantization of Electromagnetic Fields in Cavities and Spontaneous Emission, with K. Kakazu, Phys. Rev. A [50], 1830 - 1839 (1994). Correspondence between the Classical and Quantum Canonical Transformation Groups from an Operator Formulation of the Wigner Function, with L. Yeh, Foundations of Physics [24], 873 - 882 (1994). 1993 Symmetries of Two-mode Squeezed States, with D. Han, M. E. Noz, and L. Yeh, J. Math. Phys. [34], 5493 - 5508 (1993). 1992 E(2)-symmetric two-mode sheared states, with L. Yeh, J. Math. Phys. [33], 1237 - 1246 (1992). O(2,1)-like Little Group for Space-like Momenta and Localized Light Waves in Continous Media, with L. Yeh, J. Math. Phys. [33], 3190 - 3196 (1992). 1991 Space-time Geometry of Relativistic Particles in Four-Dimensional Phase Space, with C. H. Kim, J. Math. Phys. [32], 1998 - 2006 (1991). Time-dependent Mode Coupling and Generation of Two-mode Squeezed States, with V. I. Man'ko, Phys. Lett. A [157], 226 - 228 (1991). Phase Space Picture of Quantum Mechanics wirh M. E. Noz, World Scientific Publishing Company (Singapore}. 1990 Space-time geometry of relativistic particles with E. P. Wigner, J. Math. Phys. [31], 55 - 60 (1990). pdf. Canonical transformation in quantum mechanics, with E. P. Wigner, Am. J. Phys. [58], 439 - 447 (1990). Linear Canonical Transformations of Coherent and Squeezed States in the Wigner Phase Space III. Two-mode States with D. Han and M. E. Noz, Phys. Rev. A [41], 6233 - 6244 (1990). Entropy and Lorentz Transformations, with E. P. Wigner, Phys. Lett. A [147], 343 - 346 (1990). pdf. Lorentz-squeezed Hadrons and Hadronic Temperature, eith D. Han and M. E. Noz, Phys. Lett. A [144], 111 - 115 (1990). pdf. 1989 Observable Gauge Transformations in the Parton Picture Phys. Rev. Lett. [63], 348 - 351 (1989). pdf. Einstein's World Massive/Slow between Massless/Fast Energy Momentum E = p2/2m Einstein's E=(m2 + p2)1/2 E = cp Helicity Spin, Gauge S3 S1S2 Wigner's Little Group S3 Gauge Trans. Hadrons, Bound States Gell-Mann's Quark Model One Lorentz-Covariant Entity Feynman's Parton Picture This table is from my PRL paper published in 1989. Since I made a contribution to the 3rd row of the above table, I am known as Wigner's student in the physics community. New observable effects of the Lorentz group, with D. Han and E. Hardekopf, Nucle. Phys. B (Proc. Suppl.) [6], 321 - 326 (1989). Squeezed states and thermally excited states in the wigner phase-space picture of quantum mechanics, with M. Li, Phys. Lett. A [139], 445 - 448 (1989). Covariant phase-space representation and overlapping distribution functions, with E. P. Wigner, Phys. Rev. A [39], 2829 - 2834 (1989). Linear canonical transformations of coherent and squeezed states in the Wigner phase space. II. Quantitative analysis . with D. Han, and M. E. Noz, Phys. Rev. A [40], 902 - 912 (1989), 1988 Linear canonical transformations of coherent and squeezed states in the Wigner phase space, with D. Han and M. E. Noz, Phys. Rev. A [37], 807 – 814, (1988). Covariant phase-space representation for harmonic oscillators. with E. P. Wigner, Phys. Rev. A [38], 1159 - 1167 (1988). Special relativity and interferometers, with D. Han, Phys. Rev. A. [37], 4494 - 4495 (1988). Special Relativity and Quantum Theory, with M. E. Noz, Kluwer Academic Publshers (1988), available from the amazon.com 1987 Cylindrical group and massless particles, with E. P. Wigner, J. Math. Phys. [28], 1175 - 1179 (1987). pdf. Covariant phase-space representation for localized light waves, with E. P. Wigner, E. P. Wigner, Phys. Rev. A [36], 1293 – 1297 (1987). Decomposition of Lorentz transformations, with D. Han and D. Son, J. Math. Phys. [28], 2373 - 2378 (1987). Thomas precession, Wigner rotations and gauge transformations, with D. Han and D. Son, Class. Quantum Grav. [4], 1777 - 1783 (1987). Uncertainty relations for light waves and the concept of photons, with D. Han and M. E. Noz, Phys. Rev. A [35], 1682 - 1691 (1987). 1986 In 1986, I was able to tell Eugene Wigner the stories he wanted to hear, and he used to visit the Univ. of Maryland. John S. Toll was the chancellor of the University at that time. He became very happy to meet Wigner. Photons, neutrinos, and gauge transformations, with D. Han and D. Son, Am. J. Phys. [54], 818 - 821 (1986). Eulerian Parametrization of Wigner's Little Groups and Gauge Transformations in term of Rotations of Two-component Spinors, with D. Han and D. Son, J. Math. Phys. [27], 2228 - 2236 (1986). pdf. Theory and Applications of the Poincaré Group, with M. E. Noz, Reidel, Dordrecht (1986), Math. Rev. available from the amazon.com. 1985 Unitary transformations of photon polarization vectors, with D. Han and D. Son, Phys. Rev. D [31], 328 - 330 (1985). Time-energy uncertainty relation and Lorentz covariance, with P. Hussar and E. E. Noz, Am. J. Phys. [53], 142 - 147 (1985). 1984 Internal space-time symmetries of massive and massless particles, with D. Han, M. E. Noz, and D. Son, Am. J. Phys. [52], 1037 - 1043 (1984). 1983 Gauge transformations as Lorentz-Boosted rotations, with D. Han and D. Son, Phys. Lett. B [131], 327 - 329 (1983). Einstein's World Massive/Slow between Massless/Fast Energy Momentum E = p2/2m Einstein's E=(m2 + p2)1/2 E = cp Helicity Spin, Gauge S3 S1S2 Wigner's Little Group S3 Gauge Trans. Massless composite particles and space-time description of gauge transformations, with D. Han and D. Son, Phys. Rev. D [27], 2348 - 2353 (1983). c-number time-energy uncertainty relation in the quark model, with D. Han, M. E. Noz, and D. Son, Phys. Rev. D [27], 3032 - 3035 (1983). Illustrative examples of the symplectic group, with M. E. Noz, Am. J. Phys. [51], 368 - 375 (1983). 1982 Polarizations of massless neutrinos. Did you know they are consequences of the gauge invariance on massless particles? E(2)-like Little Group for Massless Particles and Polarization of Neutrinos, with D. Han and D. Son, Phys. Rev. D [26], 3717 - 3725 (1982). Photon Spin as a Rotation in Gauge Space, with D. Han and D. Son, Phys. Rev. D [25], 461 - 463 (1982). Space-time Symmetries of Confined Quarks, with D. Han, M. E. Noz, and D. Son, Phys. Rev. D [25], 1740 - 1743 (1982). Little groups, the quark model and gauge transformations, with D. Han, M. E. Noz, and D. Son, Physica A [114], 197 - 199 (1982). Dirac’s light-cone coordinate system, with M. E. Noz, Am. J. Phys. [50], 721 - 724 (1982). 1981 How would Bohr's atom look to a moving obserber? Dirac's Form of Relativistic Quantum Mechanics with D. Han, Am. J. Phys. [49], 1157 - 1161 (1981). The Little Group for Photons and Gauge Transformations, with D. Han, Am. J. Phys. [49], 348 - 351 (1981). Symplectic Formulation of Relativistic Quantum Mechanics, with M. E. Noz, J. Math. Phys. [22], 2289 - 2293 (1981). Physical Principles in Quantum field Theory and Covariant Harmonic Oscillators, with M. E. Noz, Found. of Phys. [11], 895 - 903 (1981). 1980 Quark model in the quantum mechanics curriculum, with P. E. Hussar and M. E. Noz, Am. J. Phys. [48], 1043 - 1049 (1980). Lorentz deformation in the O(4) and light-cone coordinate systems, with M. E. Noz, J. Math. Phys. [21], 1224 - 1228 (1080). Yukawa's Approach and Dirac's Approach to Relativistic Quantum Mechanics, with M. E. Noz, Progr. Theor. Phys. [64], 1852 - 1860 (1980). pdf. Lorentz Deformation and the Jet Phenomenon II: Explanation of the Nearly Constant Transverse Jet Phenomenon with M. S. Noz and S. H. Oh, Found. of Phys. [10], 635 - 639 (1980). Three-particle symmetry classifications according to the method of Dirac, with P. E. Hussar and M. E. Noz, Am. J. Phys. [48], 1038 - 1042 (1980). 1979 Representations of the Poincaré group for relativistic extended particles, with M. E. Noz and S. H. Oh, J. Math. Phys. [20], 1341 - 1344 (1979). pdf, pdf plus Physics Auxiliary Publication Service. The Gaussian entanglement is contained in this publicaction, and was re-introduced in the following paper. A simple method for illustrating the difference between the homogeneous and inhomogeneous Lorentz groups, with M. E. Noz and S. H. Oh, Am. J. Phys. [47], 892 - 897 (1979). Physical basis for minimal time-energy uncertainty relation, with M. E. Noz, Found. of Phys. [9], 375 - 387 (1979). Lorentz Deformation and the Jet Phenomenon, with M. E. Noz and S. H. Oh, Found. of Phys. [9], 947 - 954 (1979). Lorentz Deformation of the Bethe-Salpeter Wave Function, with M. E. Noz, Hadronic J. [2], 460 - 480 (1979). 1978 Relativistic harmonic oscillators and hadronic structures in the quantum-mechanics curriculum, with M. E. Noz, Am. J. Phys. [46], 484 - 488 (1978). Quarks, Partons and Lorentz-Deformed Hadrons, with M. E. Noz, Progr. Theor. Phys. [60], 801 - 816 (1978). Group theory of covariant harmonic oscillators, with M. E. Noz, Am. J. Phys. [46], 480 - 483 (1978). Covariant Form of the Uncertainty Relations, J. Kor. Phys. Soc. [11], 1 - 5 (1978). 1977 Covariant harmonic oscillators and the parton picture, with M. E. Noz, Phys. Rev. D [15], 335 - 338 (1977). Covariant Harmonic Oscillators and Excited Baryon Decays, with M. E. Noz, Progr. Theor. Phys. [57], 1373 - 1386 (1977). Covariant Harmonic Oscillators and the Chiral Configuration Mixing, with M. E. Noz, Phys. Rev. D [15], 2032 - 2035 (1977). 1976 Model for relativistic bound-state perturbation theory Phys. Rev. D [14], 273 - 279 (1976). Lorentz-invariant Minimum Uncertainty Products J. Kor. Phys. Soc. [9], 54 - 58 (1976). 1975 S-matrix, Wave Functions, and Foundations of Quantum Mechanics, New Physics [15], 46 - 54 (1975). Covariant Harmonic Oscillators and Diffractive Excitations, with M. E. Noz, Phys. Rev. D [12], 122 - 128 (1975). Covariant Harmonic Oscillators and Excited Meson Decays, with M. E. Noz, Phys. Rev. D [12], 129 - 138 (1975). 1974 Radial Excitations in the Symmetric Quark Model II with M. E. Noz, Nuovo Cimento [19A], 657 - 665 (1974). 1973 Covariant harmonic oscillators and the quark model, with M. E. Noz, Phys. Rev. D. [8], 3521 - 3526 (1973). Radial Effects in the Symmetric Quark Model, with T. De and M. E. Noz, Nuovo Cimento [13A], 1089 - 1101 (1973). It took 1000 years for human brains to realize the circle is a special case of the ellipse. Einstein formulated his special theory of relativity more than 100 years ago based on Lorentz transformations, yet it is not easy for physicists to see that the Lorentz boost is a squeeze transformation. 1972 Normalizable Wave Functions for Bound States and Resonances in S-matrix Theory, with K. V. Vasavada, Phys. Rev. D [5], 1002 - 1011 (1972). Radial Excitations in the Symmetric Quark Model, with M. E. Noz, Nuovo Cimento [11A], 513 - 522 (1972). Solutions of Bethe-Salpeter Equation in the Infinite-momentum Limit, with T. De, Phys. Rev. D [6], 3689 - 3695 (l972). 1971 Lorentz Contraction of the Bethe-Salpeter Green's Function, with R. Zaoui, Phys. Rev. D [4], 1764 - 1769 (1971). 1969 Spontaneously Broken SU(3) and Massive Vector Mesons. F. L. Markley, Nuovo Cimento [63A], 60 (1969) 1968 Analyticity Requirement for Regge Poles and Backward Unequal-mass Scattering, with M. Resnikoff, Phys. Rev. [169], 1288 - 1293 (1968). Analyticity Requirement for Regge Poles and Backward Unequal-mass Scattering II, with K. J. Kim, Suppl. to New Physics [7], 39 - 40 (1968). 1967 The Saturation of Equal-time Charge Commutation Relations in Perturbation Theory, with R. A. Brandt, Phys, Rev. [161], 1473 - 1482 (1967). 1966 Click here for a detailed story. Perturbed Bound-State Poles in Potential Scattering I, Phys. Rev. [142], 1150 - 1153 (1966). Perturbed Bound-state Poles in Potential Scattering II, with K. V. Vasavada, Phys. Rev. [150], 1236 - 1240 (1966). Radius of the Nucleon in a Bound-state Model, with K. V. Vasavada, Phys. Rev. [152], 1259 - 1262 (1966). 1965 Analytic Property of Three-body Unitarity Integral, with J. N. Islam, Phys. Rev. [138], B1222 - B1230 (1965). Particle-mixture Theory and Apparent CP Violation in K-meson Decay, with H. Ezawa, S. Oneda, and J. C. Pati, Phys. Rev. Lett. [14], 673 - 676 (1965). Beta-decay Coupling Constants, Nuovo Cimento [36], 523 - 532 (1965). Role of Partially Conserved Axial Vector current in $\pi$ and K Decays, with S. Oneda, Progr. Theor. Phys., Special Issue Dedicated to Professor H. Yukawa, 238 - 249 (1965). 1964 Unitary Symmetry and Non-leptonic K-decay, with S. Oneda, Phys. Lett. [8], 83 - 85 (1964). The omega-phi Mixing Angle, with S. Oneda and J. C. Pati, Phys. Rev. [135], B1076 - B1079 (1964). Final-state Interactions in the \eta to 3-\pi Decay, with S. Oneda and L. M. Kaplan, Nuovo Cimento [34], 655 - 664 (1964). Three-Particle Unitarity Integral, Phys. Rev. [135], B454 - B458 (1964). SU(3) Symmetry and the Nonleptonic K-meson Processes, with S. Oneda and D. Korff, Phys. Rev. [136], B1064 - B1073 (l964). 1963 Physics faculty photo at the Univ. of Maryland (Spring 1963). I am the youngest person in this photo. Asymptotic Behavior of the S-matrix for high Angular Momentum, with A. M. Jaffe, Phys. Rev. [129], 2818 - 2824 (1963). Anomalous Thresholds and Three-Particle Unitarity Integral, Phys. Rev. [132], 927 - 929 (1963). 1962 Pion Production in the Low-energy Limit, Phys. Rev. [125], 1771 - 1777 (l962). Complex Angular Momentum in Two-channel Problems, with A. M. Jaffe, Phys. Rev. [127], 2261 - 2266 (1962). I came to the Univ. of Maryland in July of 1962, and I was assigned to be a personal assistant to Paul A. M. Dirac who visited Maryland for one week in October of 1962. Thus, I had a God-given chance to talk and listen to Dirac. Click here for a detailed story. 1961 Beginning of a new life: prepairing papers for publication (Princeton 1961). Dispersion Relations for Production Amplitude, Phys. Rev. Lett. [6], 313 - 315 (1961). Dispersion Relations for Production Amplitude I, Phys. Rev. [124], 1241 - 1248 (1961). Dispersion Relations for Production Amplitude II, Phys. Rev. [124], 1632 - 1638 (1961). Click here for my Princeton page.