This paper uses elementary techniques drawn from renormalization theory to derive the Lorentz-Dirac equation for the relativistic classical electron from the Maxwell-Lorentz equations for a classical charged particle coupled to the electromagnetic field. I show that the resulting effective theory, valid for electron motions that change over distances large compared to the classical electron radius, reduces naturally to the Landau-Lifshitz equation. No familiarity with renormalization or quantum field theory is assumed.
REFERENCES
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>;F.
Rohrlich
, Classical Charged Particles: Foundations of their Theory
(Addison-Wesley
, Reading, MA
, 1965
)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
).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
).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
).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
).7.
J. Z.
Simon
, “Higher-derivative Lagrangians, nonlocality, problems, and solutions
,” Phys. Rev. D
41
, 3720
–3733
(1990
).8.
Fritz
Rohrlich
, “Dynamics of a charged particle
,” Phys. Rev. E
77
, 046609
(2008
).See also
H.
Spohn
, “The critical manifold of the Lorentz-Dirac equation
,” Europhys. Lett.
50
, 287
–292
(2000
).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
).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
).© 2013 American Association of Physics Teachers.
2013
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