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nonrelativistic ultrarelativistic
(1-215)
In this quick survey of radiation problems we have neglected the interaction of the particle with its own emitted field. The same was true in the discussion of the motion in Sec. 1-3. As a matter of principle this is, of course, wrong, but except in extreme situations it is nonetheless a perfectly justified approximation. Indeed, from Larmor's formula (1-194) the radiated energy grad ~ 1(e 2 /4n) X (dv/dt (!::.t/c 3 ). As long as grad is small as compared to typical energies in the problem, for instance, the particle energy go ~ m[(dv/dt) dtY for a moving charge,
QUANTUM FIELD THEORY
we can neglect classical radiative corrections. This leads to the condition
2 e2 2 re Lit TO = - - - = - 34nmc 3 3 c
(1-216)
where a characteristic radiation time TO has been introduced. On the contrary, this should become relevant if the forces vary appreciably in a time TO or on distances of the order CTo. The classical point charge theory due to Lorentz, including radiative corrections, was the subject of much controversy before the advent of quantum mechanics, which has changed the perspective considerably. Even though in most situations it is not the relevant approximation, it is instructive to understand the limitations of classical mechanics in its most elaborate treatment of point charges. Let us present an heuristic derivation of the correction fll to the Lorentz force law to take into account the self-interaction. We write
dUll mull == mTr= eF~;tuv
+ jIl
(1-217)
The extra term fll should be a four-vector such that for small velocities one obtains from (1-217) the Larmor energy loss 2 dlffrad - - = mduo - 2 e (dV)2 -= -- dt dt 3 4n dt in the nonrelativistic limit. Furthermore, from translation invariance it should only depend on u and its derivatives with respect to proper time. No new quantity independent of e and m with the dimension of a length should arise if the particle is structureless. Finally, we want to keep the definition of proper time, so that u 2 remains equal to one, from which we deduce that u -f has to vanish. The four-vector - (ii - u )u ll = (U 2 )UIl has a fourth component reducing to - (dv /dt)2 in the nonrelativistic limit. The requirementf- u = 0 forces us to construct the combination iill - (ii - u)u ll orthogonal to u. Hence the above requirements lead to the classical Lorentz-Dirac equation in the form:
2 e2 mull = eFllv uv + - 4n [iill - (u -ii)ull ] ext 3 -
(1-218)
where F ext represents the contribution of all external charges. The time component of this equation gives a relativistic generalization of the energy balance 2 dlff F Ov 2 e (')2 ..0 (1-219) dT = e extUv + :3 4n u Uo + mToU The first term on the right-hand side obviously corresponds to the work of external forces; the second is the dissipative (u 2 < 0) Larmor term. The third one (the socalled Schott's energy term) is a total derivative. It can be neglected when taking averages over (almost) periodic motions or, more generally, when the variation of the acceleration is small during time intervals of order To. The original derivation
CLASSICAL THEORY 43
of (1-218) by Lorentz used a spherical model of the charge and was not free of objections from the relativistic standpoint. Dirac obtained the same equation in a fully relativistic way using local conservation of energy and momentum. Both, however, had to incorporate in the inertial mass an infinite (positive) contribution equal to the electrostatic Coulomb energy created by the charge, a typical renormalization effect. If the observed mass were only electromagnetic in origin we would have to introduce in this Coulomb energy a short-distance cutoff a such that e 2 /4na ~ mc 2 , that is, a ~ cro ~ r e. This is therefore the limit on distances where classical physics comes into difficulty. It is much smaller than the Compton wavelength A = h/mc where quantum effects become important (re/A = (J( ~ 10- 2 ). Therefore the latter always hides the small-distance classical effects. We shall see that rather than diverging linearly (with inverse length) as here, the "bare" mass of a spin i electron diverges only logarithmically. Even ignoring these infinities, we can expect difficulties at short distances or short times. Thus let us give a closer look at the equation of motion, omitting even dissipative and relativistic effects. We rewrite it in three-dimensional notation as
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