radiative corrections for extended charged particles in classical electrodynamics

ANNALS OF PHYSICS 142, 284-298 (1982)

Radiative Corrections for Extended Charged Particles in Classical Electrodynamics

W.T.GRANDY,JR., AND ALI AGHAZADEH

Department of Physics and Astronomy, Universitv of Wvoming, Laramie, W.voming 82071

Received December 9, 198 1

Radiative effects on classical nonrelativistic charged particles are discussed in complete analogy with those arising in quantum electrodynamics. The effects originate in the interaction of the particle with its own electromagnetic fields. In particular, we derive an expression for the anomalous magnetic moment which provides both an intuitively appealing explanation of its origin and a reasonable relation to experimental values.

I. INTRODUCTION

During the early development of quantum electrodynamics (QED) a number of classical and semi-classical studies of the point electron were carried out in order to obtain some intuitive pictures of the new phenomena being observed, such as the Lamb shift and anomalous moment [l-3]. Subsequent development has led to what is perhaps the most precisely verified theory in the history of science, as well as to a great deal of physical insight into electrodynamic processes. Nevertheless, there exist a number of conceptual difficulties with the present formulation of QED, such as infinite renormalization parameters and vacuum interactions, which give rise to uncertainties regarding its ultimate soundness [4].

Upon further reflection it is realized that the classical theory of a point charge is beset with many of the same problems, despite the fact that the classical Lorentz- Dirac equation has been studied extensively for over 45 years [S, 61. Aside from infinite mass renormalization, the classical theory encounters a number of further difficulties, many of which persist in the quantum field-theoretic formulation. The existence of runaway solutions is well known, of course, as are the teleological attempts to eliminate them by specification of final rather than initial conditions [5]. This latter procedure results in pre-acceleration of the particle over very small, but finite, time intervals. Although the Lorentz-Dirac equation can be derived in a relativistically-invariant way [ 71, it is not clear that the associated energy-momentum tensors possess a similar satisfactory interpretation [6]. Because the combined Maxwell-Lorentz theory contains exact conservation laws, so that runaway solutions and pre-accelerations are formally inadmissible, one is compelled to believe that a

0003.4916/82/100284-15$05.00/O Copyright 0 1982 by Academic Press, Inc. All rights of reproduction in any form reserved.

284

CLASSICAL RADIATIVE CORRECTIONS 285

charged particle simply cannot occupy a mathematical point as presumed in both QED and the Lorentz-Dirac theory.

Historically the electron was envisioned as an extended particle by Lorentz [8], Herglotz [9], Abraham [lo], and others, but the implied rigid structure was not able to survive the requirements of relativity. An inability to solve this problem, which essentially continues until today, led the founders of the theory to invoke the point- charge limit as the residue of the Lorentz series, and subsequently Dirac reformulated these ideas ab initio as the theory of a point charge [ 111. It seems that only Schott maintained a serious interest in the extended rigid particle [ 12, 131. More recently there has been renewed interest in relativistic models with deformable structure which show some promise [ 14, 151. Nevertheless, after 80 years of work with the point charge the problems mentioned earlier still persist, giving rise to the feeling that their origins may have little to do with relativity and quantum mechanics, but more with the presumption of a structureless particle. Just what structure there may be is not yet evident, of course, but it seems that there is still something to be learned from the model of a classical nonrelativistic extended charged particle. Much of the earlier history of the subject has been discussed in the review by Erber [ 161, and there has been recent interest in the model in connection with QED [ 17, 181.

In a remarkable article that has received much less attention than it deserves, Bohm and Weinstein reformulated the model of a classical extended charge by replacing the Lorentz expansion with a Fourier analysis [ 191. We review this formulation (in a form suggested by Jaynes [20]) in Section 2, along with the associated equation of motion and its general solutions. A finite mass renormalization presents itself immediately and a classical analogue of the Lamb shift emerges. In Section 3 we are able to derive an anomalous magnetic moment for the spinning charge in a surprisingly straightforward way, an effect arising from the interaction of the normal moment with its own fields. Some perspective with regard to this model is attempted in Section 4.

2. THE EXTENDED CHARGED PARTICLE

We adopt as a model a spherically-symmetric and rigid charge distribution,

P(Xl r) = et-Cl x - z(t)l), (1)

of mechanical mass m, and total charge e (of unspecified sign), where z(t) is the position vector of both the center of mass and center of charge. The form factor has normalization

[f(x) d3x = 1, (24

286 GRANDYANDAGHAZADEH

with Fourier transform

F(k) = 1 eik* y(x) d3x. Pb)

The particle current density is

.i(x, 4 = p(x, 0 +I. (3)

In the Coulomb gauge the scalar potential is not retarded and produces no net force on the particle itself, so that only the transverse current and vector potential need be considered. Although the particle will generally be thought to spin uniformly, in the nonrelativistic approximation necessitated by rigidity there is no coupling between rotational and translational motion. Thus we ignore spin for the moment and write for the retarded transverse vector potential [ 191

A(xy I)= (q3 --co ‘-II dl’/qF(k)exp[ik. (X--Z(P)}]

x sin[ck(t - t’)]k x [z(P) x k]/k2. (4)

The force on the particle due to its own fields is just the Lorentz force constructed from those fields. In the nonrelativistic approximation, however, the contribution from the magnetic field can be omitted, there will be no self torque in the absence of external field coupling to the spin, and we obtain a linear approximation. When the exponential in Eq. (4) is approximated carefully [21] we obtain the total self force

F,(f)=j’ &(f-t’)i(P)dt’, -02

in terms of the basic memory function of the theory:

Q(t) = 2 f= IF( cos(ckt) dk, t > 0. 0

(6)

The equation of motion under an external force is then m, z(t) = F, + F,, . A more revealing equation of motion is obtained by integrating twice by parts in

Eq. (5). Although the integrated terms would appear to vanish at the lower limit only through a presumption that the particle was at rest sometime in the remote past, we shall presently uncover a remarkable property of Q(f) that causes these terms to vanish without such initial conditions. From Eq. (6) we verify that Q(0) = 0 and

6m = Q(0) = 2 Iv (I;(k)/ 2 dk. 0

(7)


We thus obtain the renormalized equation of motion

(m, + 6m) Z(t) = F,.,(t) + F,, (8)

in terms of the force of radiation reaction

F,,(t) = f Q(t - t’) :(t’) dt’, (9) --cc

describing the renormalized effect of the particle’s fields on itself. In the Appendix we indicate how Eqs. (8) and (9) follow directly from the Dirac equation in the appropriate limits.

Note that the renormalization constant 6m is finite and entirely associated with the electromagnetic structure of the particle. It is simply the necessary mass shift owing to this structure that is generated by accelerating the particle. Curiously, expression (7) is precisely that obtained in quantum electrodynamics [22, p. 5271, so that it apparently has little to do with quantization or relativity. If Eq. (2b) is substituted into Eq. (6), and W denotes the total energy stored in electrostatic fields, then one readily finds that 6m = (4/3) W/c’. Thus, it appears that the infamous factor of 4/3 actually provides the correct mass shift arising from a redistribution of field energy when the particle is accelerated. This factor is also observed as part of the coefficient of the infinite renormalization term in the Lorentz-Dirac theory of the point charge [23], and therefore only becomes evident dynamically for an extended particle with m, = 0. Note that F, contains not only 6m, but also a remainder which for F,, = 0 induces other dynamical effects (see below).

‘Earlier we, alluded to a rather remarkable property of Q(t), which we now illustrate by choosing a spherical-shell distribution of radius a:

f(x) = (4na*)-’ 6(x-a), F(k) = sin(ka)/ka. (10)

Equation (6) then yields

Q(t) = n(a - ct/2)/2,

= 0,

t < 2a/c,

t > 2a/c, (11)

so that Q(t - t’) in Eq. (9) is nonzero only over a very small time interval of order r0 = (2/3)(e2/mc3). This is, in fact, a rather general property of spherically-symmetric charge distributions, and the entire effect of the particle’s own fields on its motion at time t takes place over this short interval prior to t.

This property of the memory function also provides insight into the point-particle theory. By changing variables in Eq. (9) we observe that

‘_

F,,(t) = j= Q(t) i(t - t’) dt’, (12) 0

5 m Q(r) dt = 2e2/3c3 = more.

0 (13)


Definition of the weighted average

(i), EC /.a Q(f) 2(t - t') dt' / jm Q(f) dr' (14)

0 0

then allows us to rewrite Eq. (12) as

F,.,(t) = (2e2/3c3)(i),.

That is, F,, does not depend on the value of Z at time t, but is just an average of Z over a very short time interval prior to t. When Eq. (15) is employed in Eq. (8) the result is seen to be the so-called modified Abraham-Lorentz equation of motion [S], but with g(t) replaced by (Z),. It is just the neglect of this small time delay that causes runaway solutions and introduces unphysical pre-accelerations into the point- particle theory. Moreover, if one expands the average in Eq. (14) in powers of the retardation time it is possible to regain the original series expansion of Lorentz [20]. The Fourier analysis carried out here has led to an exact summation of that series in the nonrelativistic limit. Although it appears that the series can always be summed for cutoff distributions [ 181, it usually diverges otherwise. Further evidence that a series expansion is not the proper tool for studying this problem emerges from deeper analysis of the equation of motion.

2.1. The Equation of Motion

Laplace transformation of the equations of motion (8) and (9), under initial conditions maintaining the particle quiescent at the origin prior to t = 0, produces the algebraic equation

z(s)[ms’ - s3q(s)] = F,,(s), m=m,+6m, (16)

where the dynamical mass is given by

sq(s) = s lm e-“‘Q(t) dt 0

4e2 J

m s* (F(k)/’ dk =37cc20 s2 + k2c2 ’ (17)

Although Eq. (16) can be solved directly for z(s), and z(t) obtained by inverse Laplace transformation,

(18)

this is not the general solution. If the quantity in brackets in Eq. (16) possesses zeros in the complex s-plane there will then exist solutions to the homogeneous equation in terms of J-functions and their derivatives.


It is evident from Eq. (18) that a necessary and sufftcient condition for no runaway solutions is that

sq(s) = m, + 6m (19)

have no roots in Re s > 0. From Eq. (17) we see that if it has such roots these must lie on the real axis, and from Eq. (7) it is clear that sq(s) increases monotonically from zero at s = 0, to 6m as s -+ co. As a consequence, there are no roots of Eq. (19) in Re s > 0 if the bare mass m, > 0, and we can take a in Eq. (18) just to the right of the imaginary axis. If -6m < m, < 0 there is precisely one positive real root leading to a runaway solution, and for m, < -6m this root moves into the left half-plane where it describes a stable damped solution. There are, of course, no pre- accelerations.

One concludes that a necessary and sufficient ‘condition for runaway solutions to arise is that -6m < m, < 0. But m, < 0 would yield runaway solutions from Newtonian mechanics alone, independent of electromagnetic considerations, so that the origin of these difficulties can not be ascribed to Maxwell-Lorentz electrodynamics. Indeed, were m, + 6m < 0 electrodynamic interactions would remove the instabilities. Thus, it is not possible to ignore higher-order terms in the Lorentz series, nor is it possible to pass to the point-charge limit and still maintain the total mass m fixed and positive. In order to maintain stability in such a case the infinite 6m must be balanced by an infinite negative mechanical mass, which almost cancels 6m and provides energy for the runaway solutions. Although these observations are not at all new [8, 19, 241, they serve to re-emphasize the perils in presuming that a physical charge can occupy a mathematical point in space.

As mentioned above, there may exist solutions to the homogeneous equation (16) when F,, = 0. In order to study these we first record the following two Fourier transforms,

(.a Q(t) cos(ckt) dt = (2e2/3c3) IF(k)(*. ‘0

I .m Q(l)sin(wr)dt=$-$-PIm 0 0

,F’k!‘:;, dk, Wb)

where the second integral is a Cauchy principal value. We then rewrite Eq. (19) as

(.COi)(l)e-SLdt=mo, Res>O, JO

and investigate possible solutions on the imaginary axis, s = io. Separation into real and imaginary parts and use of Eqs. (20) yields the set of conditions

m, + 6m + 6m’(o) = 0, (224

F(o/c) = 0, Pb)

290 GRANDY AND AGHAZADEH

where

(23)

Equations (22) are the Bohm-Weinstein conditions under which the charged particle can execute force-free nonradiating self-oscillations, or Schott modes [ 121. The absence of radiation is guaranteed to O(v/c) by Eq. (22b), for then the transverse fields vanish. Hence, the entire effect of the radiation-reaction term in these cases is to adjust the motion so that radiation is absent.

The possibility of such radiationless self-oscillations in the absence of an external force has long been known [25,26], and was studied extensively by Schott [ 1 l-151. Not all spherically-symmetric charge distributions will satisfy Eqs. (22), of course, but there are a number that do. The spherical shell of Eq. (10) has no Schott modes unless m, = 0, in which case it possesses infinitely many at definite eigenfrequencies [25]. A uniform sphere has a single Schott mode, and only if m, # 0. at frequency w2 = 2e2/m,a3. It is somewhat interesting that in this latter model one can take e and a as fixed and obtain a mass spectrum through the special frequency o. Equation (22b) yields

tan x = x, x = (2e2/m,c2a)1’2, (24)

where x2 is essentially the ratio of electrostatic energy at the surface to the rest energy. The roots of this equation are the zeros of the spherical Bessel function ji(x), so that introduction of Planck’s constant yields the excitation energies AE, = hc x,/a. If we set a = 3r0, where r0 = e’/mc’ = aX, is the renormalized classical electron radius and AC the electron Compton wavelength, the total energy of the particle in the first exicted state is, to a very good approximation,

E,zmc*[l+$-‘I. (25)

With mc* = 0.5 11 MeV, this gives a rest energy of 105.52 MeV, which is very close to that of the muon. The empirical formula (25) is, of course, very well known [27].

In addition to Schott modes in the absence of external forces, there also exist damped, and therefore radiating, self-oscillations. These modes correspond to roots of Eq. (19) located in the half-plane Re s < 0, and cannot be obtained from the Bohm- Weinstein conditions (22). Rather, Eq. (19) provides the analytic continuation of these conditions into the left half-plane. The existence of this type of solution had been noted earlier [ 161. Although both the Schott modes and those which are damped suggest a classical Zitterbewegung, it is clear that the associated frequencies describe highly relativistic motions. They are therefore inconsistent with our basic approximation, and can be taken seriously only if they persist in a fully relativistic model. Nevertheless, this qualitive feature is strongly supportive of an inner structure to the charge.


2.2. Simple Harmonic Oscillator

We have not considered here the importance of energy balance and the important relation of particle motion to radiated energy. A complete understanding of this problem is still lacking for the point-charge theory [28], and the notion of an extended particle as a transducer [29] remains to be completely exploited (although, see the next section). We shall therefore defer further discussion of this problem and focus here on a simple model having some relation to QED. The basic idea has been suggested elsewhere [20].

Consider a pure harmonic oscillator with force constant K = m,w2 and the equations of motion (8) and (9). The solutions can be re-analyzed as in Eq. (16) and we study the zeros of [ms’ + K - s3q(s)]. One again concludes that there are no runaway solutions, so that it may be useful to approximate the equation of motion subject to this constraint. Because the radiation-reaction term is very small it can be neglected in first approximation, and a first solution to Eq. (8) is

z(t) = A cos(wt) + B sin(wt), (26a)

which yields

i(t - z) = i(t) cos(wz) + it(t) sin(ws). Wb)

Substitution of this last result into Eq. (8) yields the identity

6 Q(r) i(t - r) dt = Z(t) .i,” Q(t) cos(or) ds

+ w%(t) 1

m Q(r) sin(or) dr.

(27)

The integrals have already been evaluated in Eqs. (20), so that the next approximation to the equation of motion is

(1 + 6m/m, + 6m’/m,) s(t) = --w’z(t) + r0 [F(o/c)l’ Z(t), (28)

where 6m’ is defined in Eq. (23). This is just the Abraham-Lorentz equation for a point charge, but with two modifications arising from higher-order terms in the Lorentz series: a new electromagnetic mass shift emerges, and the strength of the dissipative term is reduced by the form factor.

A solution to Eq. (28) in one dimension can be found in the form [30]

z(t) = A sin(@) eP Yf, y > 0, (29)

and so describes a damped self-oscillation. That is, the additional mass shift can be thought of as arising from zero-point oscillations somewhat in the manner of the intuitive picture of the Lamb shift. Of course, even though 6m’ has the apprearance of the Lamb shift, the context here is entirely different from that usually considered,

292 GRANDYAND AGHAZADEH

such as bound states in hydrogen. Nevertheless, the effect is somewhat similar. For the spherical shell of Eq. (10) we find

6~72’ = (e*/3cwa*) sin(2wa/c) - 6m, (3Oa)

6m = 2e2/3ac2, Wb)

so that 6m’ vanishes in the limit of a point charge. If we take a to be a Compton wavelength, 1, = A/m,c, and employ a frequency on the order of a Lyman-a line, o = 3a2c/8X,, then dm’/m, - a’ = (e2/Ac)5, which is the order of magnitude of the Lamb shift [31, p. 2231.

3. ANOMALOUS MAGNETIC MOMENT

“In spite of the fact that the anomalous magnetic moment of the electron has now been calculated through order (x3, it has been widely remarked that a simple intuitive explanation of the lowest-order contribution a/271 has never been found (321.” This point has also been emphasized by Weisskopf [33]. It seems more than a little interesting, therefore, that the classical model being considered here actually exhibits radiative corrections to its normal moment of the expected order.

We consider the extended particle in the rest frame of its center of mass, and now focus exclusively on the spherical-shell distribution of Eq. (10) with uniformly- distributed charge density 0 = e/47ra *. The mechanical (bare) moment of inertia is I, = (2/3) m, a’.

If the shell rotates with constant angular velocity u,,, then it has a static Coulomb electric field as well as a magnetic field

B, = P/a31 cLoy r < a,

= ]3& - r)r - ~ollrs~ (31)

r > a,

describing a static dipole magnetic moment

p. = (ea2/3c) uO. (32)

The fields themselves possess total magnetic energy and angular momentum

u, = ;z,ll;, L, =4JulJ, WI

respectively, where the electromagnetic moment of inertia is

IO E 2e2a/9c2. W)

From an experimental point of view we shall be particularly interested in observing the particle in the presence of a static and uniform external field B, directed along the positive z-axis of a fixed coordinate frame with origin at the particle center. Neglect


for the moment the self-fields of the particle. Then it is well known 1341 that the ensuing motion of the particle angular-momentum vector is given by the Larmor precession,

i=pxB, p = g(e/2mo c)L. (34)

where i = Z,;(t). If, as we pressume, the charge-to-mass ratio is constant throughout the body, then the g-factor can never be other than g = 1. This is true, in fact, regardless of the geometrical shape of the object, although as early as 1921 Compton pointed out that a g-factor greater than unity could be obtained if the particle mass were more centrally concentrated than the charge [35]. These arguments neglect the effects on the motion of the particle’s own fields.

A model of a rotating classical charge very similar to that considered here has been studied by Daboul and Jensen [36], in which they constrain the rotation axis to a fixed direction. Even with this constraint the model is quite illuminating, because one can calculate unambiguously the rate of irreversible radiation owing to angular acceleration. One interesting result is that the sphere does not emit irreversible radiation if it rotates with constant angular acceleration.

It is a simple matter to generalize this model to the case in which the sphere rotates with arbitrary unconstrained angular velocity u(f). We follow the same mathematical procedure as Daboul and Jensen 1361 to obtain for the vector potential of the self fields outside the sphere

A,@, t) = -i x j.?‘&, t’ + 7~) u(t - r- - t’) dt’, -0

(35)

where r = a/c, r+(t) = (Y f a)/c, i is a radial unit vector, and

P(r, t) = (e/4a)[ 1 + (a’/~‘)(1 - t’/r*)], 5- < 5 < t+ . (36)

The fields outside the sphere are E,(r, t) = -c-‘a,A, BS(r, t) = V x A, plus the static Coulomb field, and all quantities are to be calculated in the fixed frame. The self- force density at the surface of the sphere must be computed using the averages of the fields inside and outside the spherical shell and evaluated at the surface:

F,(a, t> = oE,,. + (a/~)[ (u x a) x IS,,,]. (37)

One then obtains the self torque on the surface as

d,(t) = j (a x F,) a* da. (38)

Calculation of the self fields from Eq. (35) is straightforward, and Eqs. (37) and (38) yield

d,(t) = -I, j,;? P(a, t’) ti(t - t’) dt’, (39)

294 GRANDY AND AGHAZADEH

P(u, t) = (3,‘2r)( 1 - t2/2rz) O(25 - t) o(t), (40)

and O(x) is a unit step function. The memory function P(a, t) plays a role analogous to that of Q(t) for the translational motion discussed earlier, so that contributions to the self torque at time t are restricted to a very short interval prior to t. It is remarkable that Eq. (39) has precisely the same form as found for the case when u(t) is directionally constrained [36]. The theory is again manifestly linear in the sense that B, does not contribute to Eq. (39). Here, of course, u(t) is arbitrary and we have a vector equation. Nevertheless, a similar analysis by means of Laplace transforms reveals force-free self oscillations, both radiationless and damped, and there are neither runaway solutions nor pre-accelerations.

We are primarily interested in the motion induced by a uniform external field B in the positive z-direction. This field will produce a torque on the body that is a direct generalization from Eq. (34):

d,,(t) = (ea2/3c) u(t) x B. (41)

From Eqs. (39) and (41) we obtain the general equation of motion in the form of angular-momentum conservation:

I, h(t) = -I, j” P(a, t’) ;(t - t’) dt’ + $ u(t) x B. 0

(42)

Now consider initial conditions such that the charge is rotating uniformly with u = u. for t < 0, in which case the self torque vanishes identically. Thus, the initial response is just the Larmor precession of Eq. (34). But this precessional motion changes the self fields so that d,(t) is no longer necessarily zero. The radiation- reaction term is very small, which allows us to iterate the equation of motion (42) about the value uo. In first approximation I,,,li(t) r (ea*/3c) u. x B, and substitution into Eq. (42) exhibits the leading-order effects of radiation reaction on the equation of motion:

I,ti(t) E (ea’/3c)( 1 - IO/Z,) u. x B

= (p. + 6~) x B. (43)

The quantities cl0 and I, are defined in Eqs. (32) and (33), respectively, I,,, = (213) m,a*, and the anomalous magnetic moment is

6p = -(ea2/3c)(Zo /I,) uo. (44)

Note that the correction is such as to reduce the total magnetic moment, as one would expect intuitively. This does not mean, however, that the g-factor is diminished.


As emphasized by Grotch and Kazes (371, mass renormalization plays a crucial role in determining the g-factor. That is, g is defined as the ratio of the total moment (with bare mass) to the physical magneton (with renormalized mass). Hence, with the moment in units of (ZmuO),

g= Ih+41z1- 144 I am ef2mc e/2m,c m, *

The renormalization constant is given by Eq. (30b), and therefore

g - 1 = f (e2/moc2a) = 2 (4/3a). (46)

Not only is the correction to the g-factor positive, in agreement with experiment, but if we take the radius of the charge distribution to be a = A,/3 we obtain the correct numerical result for the electron. This freedom of choice is equivalent to choosing an energy cutoff in the single-particle quantum treatment, and is consistent with the minimum extension required for a classical rotating particle with angular momentum h [38, p. 1731. In a completely different context Barut has obtained a similar value for the point charge [39].

4. CONCLUSIONS

The central point of this discussion has been to re-emphasize that the basic radiative effects on charged particles are already contained in nonrelativistic classical electrodynamics. In particular, the origin of the anomalous magnetic moment is rather transparent here: 6~ arises from an additional precession of the normal moment of the particle in the presence of its own fields. The crucial role of particle structure is evident.

It must be emphasized equally strongly, however, that the extended particle studied here can ltot be suggested as a viable model for the electron. What should be stressed is the real and continuing need to understand the structure of charged particles. Renewed interest in the electron Zitterbewegung is a gratifying step in this direction [401-

APPENDIX

The Dirac equation for a charged particle in the presence of electromagnetic fields is

sty= [ca.n+e#+/lmc*]v, (A.11


where 4 is a scalar potential, v/ is a positive-energy 4component bispinor, and II = p - (e/c)A. In the Heisenberg picture the equation of motion for II is

i = e(E + a X B), (A.21

and the fields are derived as usual from the vector potential A. We shall be interested primarily in the nonrelativistic limit

izeE,+f,,, (A-3)

where fex is an external Lorentz-force density. We have noted here that ca is the Dirac velocity operator, and that the linear decomposition of the fields into self and external contributions follows from writing A = A, + A,, .

Years ago Pauli [41], and subsequently Rubinow and Keller [42], demonstrated that the classical limit of the Dirac theory could be regained by writing the wave function as

ty(x, t) = a@, t) exp[iS(x, t)/h], (A.4)

such that S is just the action function of Hamilton-Jacobi theory. If one expands the bispinor a(x, t) in powers of A,

a=a,+a,(A/i) t a*., (A.5)

then substitution into Eq. (A.1) results in an infinite set of equations arising from equating the separate powers of zi. The coefficient of A0 yields

n2 t m2c2 = 7ct, (‘4.6)

which is just the relativistic Hamilton-Jacobi equation satisfied by S. The equations for a, lead to the continuity equation

V - j t B,p = 0, (A.71

where the current density and probability density are given, respectively, by

j = cuJau,, (A.8a)

p = u,tuo. (A.8b)

With the aid of the canonical equations, along with the identity noj = cpn, one can obtain the relation of the current density to the particle velocity. We find that

Cs1 i(t) = (n2 + m2c2)1/2 = c$ (A.9a)

j =pi, (A.9b)


and the solution for n is the expected result II = myi, with y = (1 - u*/c*))“*, v = i. In the nonrelativistic limit

i 2 mi(t). (A. 10)

We can now rewrite Eq. (A.3) as

m%(t) = eE, + f,, . (A.11)

If w is normalized to unity, as is the bispinor a,, and we note that z(t) does not depend on spatial coordinates, then the expectation value of Eq. (A. 11) is

mg(t) = e [ p(x, t) Es(x, t) d3x + F,,,

where

Fe, = I f&, t) P(X, t) d3x.

The vector potential of the self fields in the Dirac theory is

4(x, 0 = he I Wx -Y) KY> YW( v> d4y,

with retarded Green function

(A.12)

(A.13)

(A. 14)

(A.15)

In this limit only the transverse fields are of interest, and from Eq. (A.9b) we can immediately write

j = W(y) w(y) = ep(x’, t’> i(f), (A. 16)

which is precisely the classical form of Eq. (3) in the text. Although there is no requirement of rigidity here, if we take the bispinor a, to be spherically symmetric we regain from Eq. (A.12) precisely the form of the classical equation of motion (8) for a spherically-symmetric rigid charge distribution. Distortions of this symmetry will occur in the presence of external forces and appear in relativistic corrections, of course, so that appropriate changes in the fields and equations of motion must be treated in a self-consistent manner by means of Eqs. (A.14) and (A.16).

It should be emphasized that we are thinking of localized wavepackets for the Dirac wavefunction. On the one hand, if the bispinors a, are taken as plane waves the point-charge theory is regained, with its attendant divergences. On the other hand, localization on the order of a Compton wavelength necessarily brings in the negative- energy states. In our classical model the distribution is absolutely localized only in the sense that it possesses a sharp and discontinuous cutoff.

595/142/2-6


REFERENCES

1. T. A. WELTON, Phys. Reu. 74 (1948), 1157-1167. 2. V. F. WEISSKOPF, Rev. Mod. Phys. 21 (1949), 305-315. 3. Z. KOBA, Prog. Theor. Phys. 4 (1949), 319-330. 4. P. A. M. DIRAC, Europhys. News 8 (1977), 14. 5. F. ROHRLICH, ‘Classical Charged Particles,” Addison-Wesley, Reading, Mass., 1965. 6. C. TEITELBOIM, D. VILLARROEL, AND CH. G. VAN WEERT, Riu. Nuouo Cimento 3 (1980), l-64. 7. M. E. BRACHET AND E. TIRAPEGUI, Nuouo Cimento A 47 (1978), 210-230. 8. H. A. LORENTZ, “The Theory of Electrons,” Dover, New York, 1952. 9. G. HERGLOTZ, G&t. Nachr. (1903), 357.

10. M. ABRAHAM, “Elektromagnetische Theorie der Strahlung,” Tuebner, Berlin, 1908. 11. P. A. M. DIRAC, Proc. Roy. Sot. London Ser. A 167 (1938), 148-169. 12. G. A. SCHO~, “Electromagnetic Radiation,” Cambridge Univ. Press, London/New York, 1912. 13. G. A. SCHOTT, Proc. Roy. Sot. London Ser. A 159 (1937), 548-570. 14. P. A. M. DIRAC, Proc. Roy. Sot. London Ser. A 268 (1962) 57-67. 15. A. 0. BARUT, Relativistic Composite Systems, in “Mathematical Physics and Physical

Mathematics” (K. Maurin and R. Raczka, Eds.), pp. 323-344, Reidel. Dordrecht, 1976. 16. T. ERBER, Fortschr. Phys. 9 (1961), 343-392. 17. E. J. MONIZ AND D. H. SHARP, Phys. Rev. D 10 (1974) 1133-1136. 18. H. LEVINE, E. J. MONIZ, AND D. H. SHARP, Am, J. Phys. 45 (1977), 75-78. 19. D. BOHM AND M. WEINSTEIN, Phys. Reu. 74 (1948). 1789-1798. 20. E. T. JAYNES, unpublished lecture notes. 21. T. ERBER AND S. M. PRASTEIN, Acta Phys. Austriaca 32 (1970), 224-243. 22. S. S. SCHWEBER, “An Introduction to Relativistic Quantum Field Theory,” Row, Peterson,

Evanston, Ill., 196 1. 23. C. TEITELBOIM, Phys. Rev. D. 4 (1971), 345-347. 24. K. WILDERMUTH, Z. Naturforsch. A 10 (1955), 45(X459. 25. A. SOMMERFELD, “Verh. d. III Internat. Mathem.-Kongr. Heidelberg,” p. 27. 1904. 26. P. EHRENFEST, Phys. Z. 11 (1910), 708-709. 27. M. TAKETANI AND Y. KATAYAMA, Prog. Theor. Phys. 24 (1960), 661-678. 28. H. STEINWEDEL, Forrschr. Phys. 1 (1953), 7-28. 29. L. MANDEL, J. Opt. Sot. Am. 62 (1972), 1011-1012. 30. G. N. PLASS, Rev. Mod. Phys. 33 (1961), 37-62. 31. W. T. GRANDY. JR., “Introduction to Electrodynamics and Radiation,” Academic Press, New

York/London, 1970. 32. H. CROTCH AND E. KAZES, Phys. Rev. D 13 (1976), 2851-2865. 33. V. F. WEISSKOPF, “Physics in the Twentieth Century: Selected Essays,” MIT Press. Cambridge.

Mass., 1972. 34. H. GOLDSTEIN, Am. J. Phys. 19 (1951), 100-109. 35. A. H. COMPTON, J. Frank&z Inst. 192 (1921), 145-155. 36. J. DABOUL AND J. H. D. JENSEN, Z. Phys. 265 (1973), 455478. 31. H. CROTCH AND E. KAZES, Am. .I. Phys. 45 (1977). 618-623. 38. C. MILLER, “The Theory of Relativity,” Oxford Univ. Press, 1962. 39. A. 0. BARUT, Phys. Lett. B73 (1978), 310-312. 40. A. 0. BARUT AND A. J. BRACKEN, Phys. Rev. D 23 (1981), 2454-2463. 41. W. PAULI, Helu. Phys. Acta 5 (1932), 179-199. 42. S. I. RUBINOW AND J. B. KELLER, Phys. Rev. 131 (1963), 2789-2796.

radiative corrections for extended charged particles in classical electrodynamics

Documents