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2018 | 96 | 59-82
Article title

Maxwell-Lorentz Matrix

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EN
Abstracts
EN
Lorentz invariance of Maxwell electromagnetic equations is demonstrated in two complementary ways: first, we give a pedestrian review with three-vector equations, and we then express Maxwell equations in a four-vector matrix form (the Maxwell-Lorentz matrix) which demonstrates the intimate connection of Maxwell equations with the Lorentz group. Each Maxwell-Lorentz matrix component is the product of three matrices: a derivative matrix, a 4x4 Lorentz group generator matrix, and an electromagnetic field matrix. We obtain rotary Lorentz transformations of the electromagnetic field matrix from Lorentz equation matrices. We then transform the derivative and electromagnetic matrices and obtain an explicit matrix demonstration of Lorentz invariance of Maxwell equations. To obtain this result, we express all transformation matrices in exponential form to facilitate the application of simple Lorentz group algebra. The pedestrian approach illustrates what the Lorentz group matrix approach actually accomplishes and helps one to gain some appreciation of group theory methods.
Discipline
Year
Volume
96
Pages
59-82
Physical description
Contributors
  • Centro de Investigación en Computación, Instituto Politécnico Nacional, CDMX, México
author
  • 1241 N. Wildwood Drive, Stephenville, Texas 76401 – 2327, USA
  • ESIME-Zacatenco, Instituto Politécnico Nacional, Edif. 4, Col. Lindavista 07738, CDMX, México
  • ESIME-Zacatenco, Instituto Politécnico Nacional, Edif. 4, Col. Lindavista 07738, CDMX, México
References
  • [1] O. D. Jefimenko, Z Naturforsch A 54 (1999) 637-644.
  • [2] Various complex combinations of electrical and magnetic fields (and related four vectors) have been discussed by many autors: Misner [3] and Baylis [4] mention the Faraday, while Ivezic [5-7] refers to the Majorana four-vector, and Anastasovski [8] refers to the Barut [9] four-vector. Most recently, Armour [10] gives many references related to Lorentz group generators and the Riemann-Silberstein vector.
  • [3] C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation, Freeman, San Francisco (1973) p. 73.
  • [4] W. E. Baylis, Electrodynamics: A Modern Geometric Approach, Birkhauser, Boston (1988) p. 70.
  • [5] T. Ivezic, Found. Phys. Lett. 12 (1999) 105-125.
  • [6] T. Ivezic, Found. Phys. 31 (2001) 1139-1183.
  • [7] T. Ivezic, Found. Phys. 33 (2003) 1339-1347.
  • [8] P. K. Anastosovski, T. E. Bearden, C. Ciubotariu, et al, Optik 111 (2000) 246-248.
  • [9] A. O. Barut, Electrodynamics and Classical Theory of Fields and Particles, Dover, New York (1980) p. 98.
  • [10] R. S. Armour, Jr. Found. Phys. 34 (2004) 815-842.
  • [11] M. Acevedo, J. López-Bonilla, M. Sánchez-Meraz, Apeiron 12(4) (2005) 371-384.
  • [12] P. Lorrain, D. Corson, Electromagnetic Fields and Waves, Freeman, San Francisco (1970).
  • [13] P. Lorrain, D. Corson, F. Lorrain, Electromagnetic Fields and Waves, 3rd ed. Freeman, San Francisco (1970).
  • [14] W. Tung, Group Theory in Physics, World Scientific, Philadelphia (1985) p. 176.
  • [15] M. Carmeli, Group Theory and General Relativity, World Scientific, River Edge, NJ (2000) p. 22.
  • [16] L. H. Ryder, Quantum Field Theory, Cambridge, New York (1996) p. 36-38.
  • [17] H. E. Moses, Phys. Rev. 113 (1959) 1670-1679.
  • [18] P. Strange, Relativistic Quantum Mechanics, Cambridge, New York (1998) p. 18.
  • [19] J. D. Jackson, Rev. Mod. Phys. 73 (2001) 663-680.
  • [20] J. D. Jackson, Am. J. Phys. 70 (2002) 917-928.
  • [21] W. I. Fushchich, A. G. Nikitin (Eds.), Symmetries of Maxwell´s Equations, Kluwer, Norwell, MA (1987).
Document Type
article
Publication order reference
Identifiers
YADDA identifier
bwmeta1.element.psjd-a489cf12-82cf-493a-bbfe-84f68fcbb856
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