Proper-time formulation of classical electrodynamics. (English) Zbl 0960.78002
Summary: We show that Maxwell’s equations have a generalization associated with the proper-time of the source and a new invariance group which leaves this variable fixed for all observers. We show that the second postulate (of Einstein) depends on the anthropocentric view that the only clock to use is the proper-clock of the observer. Our work is motivated by the results of Fushchych and Shtelen (F-S), who showed that the free Maxwell equations have an additional invariance group which is Galilean. This work makes it clear that our group is distinct from but closely related to the Lorentz group, and is not Galilean. Since our present (constructive) approach requires a source, it does not apply to the free field case. In an earlier paper, we showed that the F-S transformation is an element of the proper-time (canonical) group which includes the group constructed here. This work is also related to the work of M. Wegener [Phys. Essays 8, No. 3, 427-433 (1995) (Preprint Univ. Aarhus)] who showed that use of the proper time allows the construction of Galilean transformations from Lorentz transformations.
References:
| [1] | Currie, DG, Interaction contra classical relativistic Hamiltonian particle mechanics, J. Math. Phys., 4, 12, 1470-1488, 1963 · Zbl 0125.19505 |
| [2] | Currie, DG; Jordan, TF; Sudarshan, ECG, Relativistic invariance and Hamiltonian theories of interacting particles, Rev. Mod. Phys., 35, 2, 350-375, 1963 |
| [3] | Einstein, A., Zur Elektrodynamik bewegten Körpern, Ann. Physik, 17, 891, 1905 · JFM 36.0920.02 |
| [4] | Fanchi, JR, Review of invariant time formulations of relativistic quantum theories, Foundations of Phys., 23, 3, 487-547, 1993 |
| [5] | Fushchych W.I. and Shtelen W.M., Lie invariance of Maxwell’s equations under Galilean transformations?, Dokl. AN Ukrain. SSR, 1991, N 3, 23-27 (in Russian). |
| [6] | Gill, TL, Time-ordered operators I, Trans. Am. Math. Soc, 266, 1, 161-181, 1981 · Zbl 0478.47027 |
| [7] | Gill, TL, Time-ordered operators II, Trans. Am. Math. Soc, 279, 2, 617-634, 1983 · Zbl 0544.47034 |
| [8] | Gill, TL; Zachary, WW, Time -ordered operators and Feynman-Dyson algebras, J. Math. Phys., 28, 7, 1459-1470, 1987 · Zbl 0637.58043 |
| [9] | Gill, TL; Zachary, WW, Mahmood M.F. and Lindesay J., Proper-time relativistic dynamics and a resolution of the no-interaction problem, Hadronic J., 17, 449-471, 1994 · Zbl 0831.70012 |
| [10] | Gill T.L., Lindesay J., Zachary W., Mahmood M. and Chachere G., Classical and quantum relativistic many-particle theory: A proper-time Lie-Santilli formulation, In: New Frontiers in Relativities, Editor T.L. Gill, Series on New Frontiers in Advanced Physics, Hadronic Press, Palm Harbor, FL 1996, 171-195. |
| [11] | Gill T.L., Lindesay J., Mahmood M. and Zachary W.W., Proper-time relativistic dynamics and the Fushchych-Shtelen transformation, J. Nonlin. Math. Phys., 1997, V.4, N 1-2, 12-27. · Zbl 0973.70018 |
| [12] | Lorentz, HA, Enzykl. Math. Wiss. V, 1, 188, 1903 |
| [13] | Maxwell, JC, Phil. Trans. Roy. Soc. London, 155, 459, 1865 |
| [14] | Minkowski H., Raum und Zeit, Phys. Z., 1909, N 3, 104-111. |
| [15] | Minkowski, H., Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern, Math. Ann., 68, 472-525, 1910 · JFM 41.0948.03 |
| [16] | Poincaré, H., Sur la dynamique de l’electron, C.R. Acad. Sci. Paris, 140, 1504-1508, 1905 · JFM 36.0911.02 |
| [17] | Ritz W., Phys. Z., 1912, 317. |
| [18] | Tolman, RC, The second postulate of relativity, Phys. Rev., 31, 1, 26-40, 1910 |
| [19] | Wegener M., A classical alternative to special relativity, Preprint, Univ. Aarhus. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
