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.

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 gμν=diag(+++) and set c=1=μ0=ϵ01. The particle coordinate four-vector is zμ(τ), where τ is the proper time. An overdot indicates differentiation with respect to the proper time: żμdzμ/dτvμ,z¨μaμ, etc. We will often abbreviate inner products, e.g., aλbλa·b,aλaλa2. When a space-time point xμ 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 “τ02” factor in the “O” sign is caused by the O(ϵ) piece of the self-field, which, when multiplied by ephys and divided by mphys, provides a second factor of τ0 that combines with the explicit factor of ϵτ0.
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
).
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