Spohn H. Dynamics of Charged Particles and their Radiation Field

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10.2 The Abraham model with spin 123

to determine the charge solitons. We set

q(t) = vt, ω(t) = ω, E(x, t) = E(x − vt), B(x, t) = B(x − vt) (10.24)

and have to determine the solutions of

−v ·∇B =−∇×E, − v ·∇E =∇×B − (v + ω × x)eϕ,

∇·E = eϕ, ∇·B = 0 , (10.25)

0 =



xeϕ(x)



E(x) + (v + ω × x) × B(x)



, (10.26)

0 =



xeϕ(x)x ×



E(x) + (v + ω × x) × B(x)



, (10.27)

for which we turn to Fourier space. The inhomogeneous Maxwell equations

(10.25) are then solved by



E =



B =



(10.28)

with



(k) =−i



− (k · v)



−1

(k − (k · v)v)eϕ(k), (10.29)



(k) =−



− (k · v)



−1

(ω × k)(v ·∇

)eϕ(k), (10.30)

and



(k) = i



− (k · v)



−1

(k × v)eϕ(k), (10.31)



(k) =−



− (k · v)



−1

(k × (ω ×∇

))eϕ(k). (10.32)

Note that



are odd, and



are evenink.

Since the integral over an odd term vanishes, a zero Lorentz force results in the

condition

−



kϕ

∗



− (v · k)



−1

(ω × k)(v ·∇

)ϕ

−



kϕ

∗



− (v · k)



−1

v × (k × (ω ×∇

))ϕ





− (v · k)



−1

((ω ×∇

)ϕ

∗

) × (k × v)ϕ

=−



kϕ

∗



− (v · k)



−1

|k|

−1

ϕ





(ω × k)(v · k) + v × (k × (ω × k)) − ((ω × k) · v)k



= 0

(10.33)

for every v and ω, using the fact that ϕ is radial.

124 Spinning charges

The Lorentz torque requires more work. Using again the fact that the integral

overanodd term vanishes, we have



keϕ

∗



∇



+∇

× (v ×



) +∇

× ((ω × i∇

) ×



)



(10.34)

=−e



k|k|

−1

ϕ

∗



− (k · v)



−1

ϕ



k × (k − (k · v)v)

− k × (v × (k × v))



+ e



k|k|

−1

ϕ

∗

k ×



(ω ×∇

) ×





= e



k|k|

−1

ϕ

∗

k × (∇

(ω ·



) − ω∇



)

=−e



k|k|

−1

ϕ

∗

× (k × ω)∇



For the divergence of



we obtain

∇



= 2



− (k · v)



−2

(ω ·∇

− (v · ω)(v ·∇

))eϕ (10.35)

and therefore zero Lorentz torque results in the condition



k|∇

ϕ|



− (k · v)



−2

(k × ω)(ω · k − (v · ω)(v · k)) = 0 . (10.36)

Taking into account that ϕ is radial, the torque vanishes only if either ω  v or

ω⊥v. If v = 0, the torque always vanishes. For ω oblique to v Eqs. (10.17)–

(10.20) have no soliton-like solution.

Physically the charge distribution is rigid, but the electromagnetic ﬁelds are

Lorentz contracted along v. This mismatch yields a nonvanishing torque unless

ω  v, respectively ω⊥v. Clearly, the mismatch is an artifact of the semirelativis-

tic Abraham model. As discussed in the previous section, for a relativistic ex-

tended charge distribution there is a charged soliton for every v and ω.Because in

the Abraham model some charge solitons are “missing”, an analysis of the adia-

batic limit is hampered at an early stage and we do not really know what happens.

Through radiation damping the spin could be forced to remain parallel to v(t).

There could be an effective dynamics separately for the parallel and perpendicular

components of ω(t). Only one particular case lends itself to a more detailed analy-

sis. We simply make sure that q(t) = 0 for all t, e.g. by taking E

= 0, B

= εB

with B a spatially constant, possibly time-dependent vector, and suitable initial

conditions for the Maxwell ﬁeld. Then the Abraham model without external forces

has a stationary solution for every ω and the adiabatic limit is meaningful and of

interest. We take up this problem in the following section.

In the quantized version of the Abraham model, the Pauli–Fierz Hamiltonian to

be discussed in chapter 13, the spin couples differently and the Lorentz torque is

10.3 Adiabatic limit and the gyromagnetic ratio 125

not the quantization of the right-hand side of (10.20). The Pauli–Fierz model has

atwo-fold degenerate ground state for every ﬁxed total momentum (smaller than

some critical value p

). Associated to this subspace there is an adiabatic evolution

which admits an arbitrary spin orientation. Thus through quantization one regains

some features of the relativistic model.

10.3 Adiabatic limit and the gyromagnetic ratio

We consider a spinning charge sitting forever at the origin and hence choose

= 0, B

= εB

with a constant B

, the initial E ﬁeld odd, and the initial

B ﬁeld even in x. Then the equations of motion simplify. We recall them for com-

pleteness,

∂

B(x, t ) =−∇×E(x, t ), ∂

E(x, t) =∇×B(x, t) − (ω(t) × x)eϕ(x),

(10.37)

∇·E(x, t ) = eϕ(x), ∇·B(x, t ) = 0 , (10.38)

together with Newton’s rotational equations of motion

ω = e



xϕ(x)x ×



E(x, t) + (ω(t) × x) × (ε B

+ B(x, t))



. (10.39)

To obtain the effective dynamics let us ﬁrst argue statically. The angular mo-

mentum, s,ofthe charge soliton is the sum s = s

+ s

with s

= I

ω and



x x × (E × B) (10.40)

for E, B the charge soliton ﬁeld at v = 0 and ω. Inserting from (10.28)–(10.32)

we obtain

= I

ω with I



k|∇

ϕ|

|k|

−2

. (10.41)

Therefore

s = (I

+ I

)ω . (10.42)

The external torque is µ × B

with the magnetic moment

µ = µω ,µ=



xϕ(x)x

, (10.43)

and thus the spin precession reads

s = µ × B

,(I

+ I

)

ω = µω × B

. (10.44)

126 Spinning charges

The conventional deﬁnition of the gyromagnetic ratio g is through

ω = g

ω × B

, (10.45)

where m is the mass of the particle; compare with the BMT equation (10.7) for

small velocities. Equating (10.44) and (10.45) we deduce the effective g-factor of

the Abraham model as

g =

(µ/e)2m

+ I

1 +

)



k|ϕ|

|k|

−2

1 + (e

)



k|∇

ϕ|

|k|

−2



k|∇

ϕ|

. (10.46)

For e → 0weobtain g = 1, as it has to be. In the opposite limit, m

→ 0, only

the second summands survive. We did not discover any simple bounds, but for a

uniformly charged sphere and ball the integrals have already been computed at the

end of section 10.1. One obtains with R = R

the radius of the sphere, respectively

ball,

sphere

1 + (e

/4π Rm

)(2/3)

1 + (e

/4π Rm

)(1/3)

, g

ball

1 + (e

/4π Rm

)(4/5)

1 + (e

/4π Rm

)(2/7)

. (10.47)

Thus g

sphere

→ 2, respectively g

ball

→ 14/5, for Rm

→ 0. For g = 2 the spin

and orbital precession are exactly in phase, whereas for g = 1 the spin turns once

during two cyclotron revolutions.

To provide dynamical support we follow the scheme of chapter 7. One integrates

(10.37), (10.38) and inserts in the Lorentz torque taking into account that the initial

ﬁelds decay quickly. Then

ω(t) = εµω(t) × B

+ N

self

(t), (10.48)

where, after some rearrangement, the retarded torque simpliﬁes to

self

(t) =



k|∇

ϕ|



− (cos |k|(t − s))ω(s) +

|k|

(sin |k|(t − s))ω(t) × ω(s)



(10.49)

10.3 Adiabatic limit and the gyromagnetic ratio 127

Let us denote the solution to (10.48) by ω

(t) = ω(εt).Weinsert this ansatz in

(10.49) and Taylor-expand. Then

self

(ε

−1

t) =



−1



k|∇

ϕ|



− (cos |k|(ε

−1

t − s))ω(εs)

|k|

(sin |k|(ε

−1

t − s))ω(εt) × ω(εs)



∼



−1



k|∇

ϕ|



− (cos |k|s)



ω(t) − εs ˙ω(t)

¨ω(t)



|k|

(sin |k|s)ω(t)



ω(t)

−εs ˙ω(t) +

¨ω(t)





. (10.50)

Let



∞

dtt



k|∇

ϕ|

|k|

sin |k|t , J



∞

dtt



k|∇

ϕ|

cos |k|t .

(10.51)

Then, using the fact that ϕ is radial,



k|∇

ϕ|

|k|

−2

, I

= 0 ,

4π





ϕ(x)ϕ(x



)x · x



|x − x



|=−

2π



k|∇

ϕ|

|k|

−4

(10.52)

and

= 0, J

=−pI

p−1

, p = 1, 2,.... (10.53)

Therefore to order ε

self

(t) =−ε

˙ω(t) + ε

ω(t) ×¨ω(t), (10.54)

and inserted in (10.48)

ε ˙ω(t) = εµω(t) × B

− εI

˙ω(t) + ε

ω(t) ×¨ω(t), (10.55)

where I

= 2e

/3inagreement with the static result (10.41).

Beyond the renormalization of I

we have also obtained the radiation reaction

ω(t) ×¨ω(t).Asfor the translational degrees of freedom only the solution on the

center manifold is of physical relevance. To compute the effective dynamics we

128 Spinning charges

regard (10.44) as the unperturbed dynamics and reinsert in (10.55). To be some-

what more general let us take B

to be time dependent and varying on the slow

time scale. One obtains

+ I

) ˙ω = µω × B

+ εe



µI

/3(I

+ I

)



˙ω(ω · B

) + (ω × (

× ω))



(10.56)

Since ω

is conserved under (10.56), the radiation reaction only modiﬁes the fre-

quency of gyration to order ε.Asecond-order term like ¨ω would lead to friction in

the effective equation. As can be seen from (10.53), its prefactor J

vanishes and

radiation damping appears only at order ε

through I

....

ω .

Notes and references

Section 10.1

BMT is an acronym for Bargmann, Michel and Telegdi (1959). The BMT equa-

tion is explained in Jackson (1999). Bailey and Picasso (1970) is an informative

article on how the BMT equation is used in the analysis of the high-precision mea-

surements of the electron and muon g-factor. The BMT equation with g = 2is

the semiclassical limit of the Dirac equation (Rubinow and Keller 1963; Bolte

and Keppeler 1999; Spohn 2000b; Panati et al. 2002a). Appel and Kiessling

(2001) compute the effective parameters for a charge distribution concentrated on

a sphere.

Just as for translational degrees of freedom, one way to guess the effective spin

dynamics is to impose Lorentz invariance. In addition, one could require that the

equations of motion come from a Lagrangian action. In full generality, including

an electric dipole moment, this program is carried out by Bhabha (1939), Bhabha

and Corben (1941) with earlier work by Frenkel (1926). Alternative approaches

are compared in Corben (1961) and Nyborg (1962). Concise summaries are Barut

(1964), who discusses also how the BMT equation ﬁts into the general scheme,

Teitelbom et al. (1980), and Rohrlich (1990). A more microscopic approach would

be to carry out the adiabatic limit for the Lorentz model of section 2.5. In Nodvik’s

version of the model such an expansion is pushed to the order where translational

and rotational degrees of freedom couple (Nodvik 1964).

The Lorentz model simpliﬁes if initial data are assumed such that the particle

moves at constant velocity. Then translational and rotational degrees of freedom

decouple. Appel and Kiessling (2002) study the existence of solutions and their

long-time limit. In the adiabatic limit, compare with section 10.3, the angular mo-

mentum responds to an external torque through the effective gyromagnetic ratio

of (10.9).

Notes and references 129

Section 10.2

The nonrelativistic model of a rotating charge is introduced by Abraham (1903)

and studied by Herglotz (1903), Schwarzschild (1903), and Thomas (1927).

Schwarzschild (1903) notes that a stationary solution exists only if ω is either

parallel or orthogonal to v. Kiessling (1999) remarks that the standard form of

the total angular momentum is conserved only if the inner rotation of the charged

particle is included.

Section 10.3

Grandy and Aghazadeh (1982) compute the gyromagnetic ratio to order e

. The

validity of the equations of motion (10.56) is proved in Imaikin et al. (2004).

Many charges

There is little effort in extending the Abraham model to several particles. We label

their positions and velocities as q

(t), v

(t), j = 1,... ,N .The j-th particle has

bare mass m

b j

and charge e

, where for simplicity the form factor ϕ is assumed

to be the same for all particles. The motion of each particle is governed by the

Lorentz force as before, and the current in the Maxwell equations now becomes

the sum over the single-particle currents. Therefore the equations of motion read

−1

∂

B(x, t ) =−∇×E(x, t ),

−1

∂

E(x, t) =∇×B(x, t) −



j=1

ϕ(x − q

(t))c

−1

(t),

∇·E(x, t ) =



j=1

ϕ(x − q

(t)) , ∇·B(x, t) = 0 , (11.1)



(t)



= e



(t), t) + c

−1

(t) × B

(t), t)



, (11.2)

where i = 1,... ,N with γ

= (1 − (v

/c)

)

−1/2

There are no external forces. The force acting on a given particle is due to the

other particles, as mediated through the Maxwell ﬁeld, and to the self-force, which

we have discussed already at length. If two particles are at a distance of only a few

times R

, then they interact strongly with forces which depend on the details of

the phenomenological and unknown charge distribution. Thus physically we trust

our model only if particles are far apart on the scale set by R

11.1 Retarded interaction

Let us take as a starting point the condition that initially particles are far

apart, thus |q

− q

|=O(ε

−1

). The velocities are less than c, not necessarily

130

11.1 Retarded interaction 131

small, and the initial ﬁelds are the linear superposition of N charge soliton

ﬁelds corresponding to the initial conditions q

, v

, i = 1,... ,N .Tounder-

stand the scales involved it is convenient to switch to macroscopic coordinates,

which simply amounts to replacing in (11.1), (11.2) e

√

εe

and ϕ by ϕ

with ϕ

(x) = ε

−3

ϕ(ε

−1

x); compare with the second half of section 6.1. Then

− q

|=O(1).

We insert the solution of the inhomogeneous Maxwell–Lorentz equations (11.1)

into the Lorentz force of (11.2). The forces are additive and the force on particle

i naturally splits into a self-force (i = j) and a mutual force (i = j).For the self-

force one uses the Taylor expansion of chapter 7. Thereby the mass is renormalized

and the next order is the radiation reaction. For the mutual force we recall that

in section 7.2 it was shown already that, to leading order, the ﬁeld generated by

charge j is the Li

enard–Wiechert ﬁeld. Thus, one obtains as retarded equations of

motion

) ˙v



j=1

j=i

εe



ret j

, t) + v

× B

ret j

, t)



+ε(e

/6π)



·¨v

+3γ

·˙v

)

+3γ

·˙v

) ˙v

+γ

¨v



(11.3)

t ≥ 0, which accounts for the effective mass m

and the radiation reaction of the

i-th particle; compare with Eq. (8.1). E

ret j

(x, t) equals (2.24) with e replaced

by e

, q replaced by q

, and t

ret

replaced by t

ret j

which is implicitly deﬁned

through

ret j

= t −|x − q

ret j

)|. (11.4)

For x = q

the retarded time is of order 1. Similarly B

ret j

(x, t) equals (2.25) with

q replaced by q

and t

ret

replaced by t

ret j

. The strength ε results from the charge,

√

εe

, and the scale factor

√

ε in (8.47). Viewed differently, on the microscopic

scale the force is of order (distance)

−2

= ε

and thus of order ε when accumulated

overatime span ε

−1

.Tosolve (11.3) one needs the trajectories for the whole past.

Our assumption of no initial slip is equivalent to

(t) + q

+ tv

, i = 1,...,N , t ≤ 0 , (11.5)

which must be added to (11.3).

Using (11.3) one can estimate the size of the various contributions. The near

ﬁelds of E

ret j

and B

ret j

are of order 1. Therefore the acceleration is of order ε,

132 Many charges

which implies that the far ﬁeld of E

ret j

and B

ret j

O(ε

). The radiation reaction

term involves ¨v

and is therefore O(ε

We see that the various contributions are well ordered in powers of ε. The

forces are weak, however, and therefore over longer times the particles will move

apart, which is of somewhat reduced interest. There are two limiting situations of

physical relevance, which will be discussed in the following sections. One pos-

sibility is to take the initial velocity |v

/c|1. Then to lowest order the

particles interact through the static Coulomb potential and post-Coulombic

corrections can be studied meaningfully. The other option is to let N →∞,

which yields a kinetic description for charge densities as commonly used in

plasma physics.

11.2 Limit of small velocities

We impose the condition that initially |v

/c|1. Then retardation effects should

be negligible and the particles interact through the static Coulomb potential. Ac-

cording to the standard textbook recipe, |v

/c|1istobeinterpreted as c →∞.

Indeed, from (11.1) one concludes B = 0 and

∇×E(x, t ) = 0 , ∇·E(x, t) =



j=1

ϕ(x − q

(t)) , (11.6)

which together with Newton’s equations of motion yields the desired result. Un-

fortunately, our argument fails on two counts. First, the interaction is obtained as

the smeared Coulomb potential. More severely, in Newton’s equations of motion

only the bare mass of charge i appears, whereas physically it should respond to

forces with its renormalized mass. Of course, the reason is that c →∞does not

ensure charges to be far apart on the scale of R

To improve we require, as in the previous section, that the initial positions sat-

isfy

− q

|=O(ε

−1

), i = j . (11.7)

Then the force is of order ε

. Under rescaling the dynamical variables should be

of order 1 as ε → 0. If in addition we demand the relation ˙q = v to be preserved,

the only choice remaining is

|=O(

√

εc) and t = ε

−3/2

/c . (11.8)