No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
The Lorentz-Dirac and Landau-Lifshitz equations from the perspective of modern renormalization theory
1. Sidney Coleman, “ Classical electron theory from a modern standpoint,” RAND Memorandum RM-2820-PR (1961), available at <http://www.rand.org/pubs/research_memoranda/RM2820.html>;
1. F. Rohrlich, Classical Charged Particles: Foundations of their Theory (Addison-Wesley, Reading, MA, 1965)
1.and A. O. Barut, Electrodynamics and Classical Theory of Fields and Particles (Macmillan, New York, 1964).
2.As developed in K. G. Wilson, “ The renormalization group and critical phenomena,” Rev. Mod. Phys. 55, 583–600 (1983).http://dx.doi.org/10.1103/RevModPhys.55.583
2.See also G. P. Lepage's “What is renormalization?” e-print arXiv:hep-ph/0506330 and “How to renormalize the Schrödinger equation,” e-print arXiv:nucl-th/9706029. Other useful perspectives are given in David B. Kaplan, “Effective field theories,” e-print arXiv:nucl-th/9506035, and in Barry R. Holstein, “ Effective interactions are effective interactions,” Prog. Part. Nucl. Phys. 50(2 ), 203–315 (2003).http://dx.doi.org/10.1016/S0146-6410(03)00013-9
3.We use the metric and set . The particle coordinate four-vector is , where is the proper time. An overdot indicates differentiation with respect to the proper time: , etc. We will often abbreviate inner products, e.g., . When a space-time point is a function argument, the index is dropped, e.g., f(x).
4. P. A. M. Dirac, “ Proceedings of the Royal Society of London. Series A, Mathematical, Physical, and Engineering Sciences,” Proc. R. Soc. Lond. A 167, 148–169 (1938).http://dx.doi.org/10.1098/rspa/1938.0124
5.The “” factor in the “O” sign is caused by the piece of the self-field, which, when multiplied by and divided by , provides a second factor of that combines with the explicit factor of .
6. H. J. Bhabha, “ On the expansibility of solutions in powers of the interaction constants,” Phys. Rev. 70, 759–760 (1946).http://dx.doi.org/10.1103/PhysRev.70.759
7. J. Z. Simon, “ Higher-derivative Lagrangians, nonlocality, problems, and solutions,” Phys. Rev. D 41, 3720–3733 (1990).http://dx.doi.org/10.1103/PhysRevD.41.3720
9.Equation (27) appears in L. Landau and E. Lifshitz, The Classical Theory of Fields (Addison-Wesley, Reading, MA, 1951), pp. 223–224. The derivation they give is strikingly modern.
10.In addition to Ref. 8, see David J. Griffiths, Thomas C. Proctor, and Darrell F. Shroeter, “ Abraham-Lorentz versus Landau-Lifshitz,” Am. J. Phys. 78, 391–402 (2010).http://dx.doi.org/10.1119/1.3269900
11.Coleman, Ref. 1, gives a full discussion of incoming and outgoing fields in the context of classical electron theory.
12. D. V. Gal'tsov, “ Radiation reaction in various dimensions,” Phys. Rev. D 66, 025016 (2002).http://dx.doi.org/10.1103/PhysRevD.66.025016
Article metrics loading...
Full text loading...