Spohn H. Dynamics of Charged Particles and their Radiation Field
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Notes and references 143
Section 11.2
The Darwin Lagrangian is discussed in Jackson (1999). In Kunze and Spohn
(2000c) the errors in (11.29) are estimated. Kunze and Spohn (2001) extend their
analysis to include radiation reaction. The major novel difficulty is to properly
match the initial conditions of the comparison dynamics (11.31). The next post-
Coulombic correction, of order |v/c|
4
,iscomputed formally by Landau and Lif-
shitz (1959), Barker and O’Connell (1980a, 1980b), and Damour and Sch
¨
afer
(1991). It contains quadrupole corrections to the Coulomb interaction and terms
proportional to
...
v.Itwould be of interest to compare these results with the system-
atic expansion presented here.
A qualitatively rather similar problem arises in general relativity. The object
of interest is a binary pulsar, like the famous Hulse–Taylor pulsar PSR 1913 +
16. It consists of two neutron stars, each with a mass of roughly 1.4 solar mass
and a diameter of 10 km. They rotate around their common center of mass with
a period of 7 h 45 min. The neutron stars move slowly with |v/c|
∼
=
10
−3
. Since
one of the neutron stars is rotating, it emits radio waves through which the orbit
can be tracked with very high precision, in fact so precise that damping through
the emission of gravitational waves can be verified quantitatively. I refer to Hulse
(1994) and Taylor (1994). As in the case of charges, the theoretical challenge is to
obtain the orbits of the two neutron stars in an expansion in |v/c|.For gravitation
there is no dipole radiation and damping appears only at order |v/c|
5
, with |v/c|
0
being the Newtonian orbit. Since experimental accuracy is expected to increase
further (Will 1999) various groups have taken up the challenge with the present
order at |v/c|
7
(Jaranowski and Sch
¨
afer 1998).
Section 11.3
The relativistic Vlasov–Maxwell equations already appear in the original 1938
paper of Vlasov, see Vlasov (1961). The existence of solutions is studied at in-
creasing level of generality in Glassey and Schaeffer (1991, 1997, 2000). In the
nonretarded Vlasov–Poisson approximation the existence of solutions is now well
understood (Pfaffelmoser 1992; Schaeffer 1991) and the link to the N -particle sys-
tem has been established for a mollified potential (Neunzert 1975; Braun and Hepp
1977), a review being Spohn (1991). Physically the natural requirement is to have
the charge diameter much smaller than the interparticle distance. Since this case is
somewhat singular, a satisfactory derivation of the Vlasov–Poisson approximation
is open, with a partial step towards its solution in Batt (2001).
As in the case of N charges, the solution to the Vlasov–Maxwell system can
be expanded in powers of 1/c. The leading order is then Vlasov–Poisson, as
144 Many charges
established by Schaeffer (1986), which is corrected
`
alaDarwin at order c
−2
,
as proved by Bauer and Kunze (2003). A one-component system can dissipate
energy only through quadrupole radiation, which first appears at order c
−5
.A
two-component system emits dipole radiation at order c
−3
. Properties of the for-
mally derived Vlasov equation including radiative friction are studied by Kunze
and Rendall (2001).
Section 11.4
The statistical mechanics of charges plus Maxwell field is usually treated only
on the level of thermodynamics (Alastuey and Appel 2000). Lebowitz and Lieb
(1969) and Lieb and Lebowitz (1972) prove the existence of the, in fact shape-
dependent, thermodynamic limit for Coulomb systems. A very readable review
is Lieb and Lebowitz (1973). The existence of the infinite-volume limit of the
correlation functions in the case of charge-symmetric systems is proved by
Fr
¨
ohlich and Park (1978). For the Debye–H
¨
uckel theory I recommend the excellent
survey by Brydges and Martin (1999).
12
Summary and preamble to the quantum theory
Within the framework of specific models for the coupling between charges and the
electromagnetic field we have presented a fair amount of rather detailed arguments
and computations. Thus before embarking on the quantized theory, it might be
useful to summarize our main findings.
• Extended charge.Tohaveawell-defined dynamics, a smeared charge distri-
bution has to be used. This can be done either on the semirelativistic level
of the Abraham model or in the form of a relativistically covariant theory,
i.e. the Lorentz model. In the latter case internal rotation must be included by
necessity.
• Adiabatic regime. Situations for which the classical electron theory can be exper-
imentally tested fall in the adiabatic regime with a remarkable level of accuracy.
Quantum mechanics must be used way before one leaves the domain of validity
of the adiabatic approximation. A good example is the hydrogen atom in a bound
state. Sufficiently far from the nucleus, which is certainly satisfied when at least
a Bohr radius away from it, the assumptions for the adiabatic approximation are
fulfilled and the dynamics of the electron is well governed by Eq. (9.14). On
the other hand, it is known that the fluorescent spectrum of the hydrogen atom
is accounted for only by quantum mechanics. To test the classical electron the-
ory on the basis of this system is simply not feasible. Thus, in the range where
the classical electron theory is applicable by necessity one is inside its adiabatic
regime. In this regime the particle becomes point-like and is characterized by a
charge, an effective mass, and, in the case of internal rotation, by an effective
magnetic moment; compare with sections 4.2 and 10.1. From the full charge and
mass distribution, which in principle constitute an infinite number of free param-
eters, only a few of their low-order moments are retained. They then enter in the
Landau–Lifshitz equation (9.10), which governs the motion of the charge with
great precision and properly accounts for friction through radiation. In addition,
145
146 Summary and preamble to the quantum theory
the electromagnetic fields are determined from the Li
´
enard–Wiechert potentials
as generated by the motion of a point charge.
• Point-charge limit.Asjudged from the context of chapter 28 of the Feynman
Lectures, “consistency” in the 1963 opinion of R. Feynman, compare with the
citation at the end of section 3.3, refers to the point-charge limit R
ϕ
→ 0. I agree,
butasargued at length there is no need ever to take this limit. Letting the size
R
ϕ
of the extended charge distribution shrink to zero yields objects of infinite
mass. While the mere mathematical operation is admissible, it would result in
a theory with very little physical content. The attempt to compensate through a
proper adjustment of the bare mass fails, since the electromagnetic mass merely
adds to the bare mass. Thus, the bare mass necessarily becomes negative which
results in an unstable Hamiltonian.
The transition to the quantum theory of photons, electrons, and nuclei could
hardly be less spectacular. I find it truly amazing that the simple rules of canonical
quantization work so well for the Abraham model and thus open the gateway to
a theory describing a vast territory of physical experience. Of course, just as the
Abraham model, the theory is semirelativistic, and is thus also known as nonrela-
tivistic quantum electrodynamics. No quantization of the Lorentz model seems to
be available. In the relativistic domain one has to rely on conventional quantum
electrodynamics.
We continue to adhere to the principle of restricting our attention to dynamical
problems, for which the interaction between the charged particles and the photon
field must be included. In particular, the emission and absorption of light by atoms
and the scattering of photons from charges will play an important role. Adiabatic-
type limits will be studied again. They show up in the limit of slow motion, where
the photon field is approximated by the static Coulomb interaction, and in the
derivation of the effective mass and the effective magnetic moment. To keep the
topics within manageable size, many things had to be left out. In terms of applica-
tions the most serious omission is macroscopic electrodynamics, where the photon
field is treated in the classical limit and matter is taken into account in a continuum
description in terms of suitable electric and magnetic susceptibilities. Needless to
say, they must be based on an atomistic quantum model of matter.
Part II
Quantum theory
13
Quantizing the Abraham model
Classical theories must emerge from quantum mechanics and there is no reason to
expect a simple recipe which would yield the physically correct quantum theory
from the classical input. On the other hand, at least in the nonrelativistic domain,
the rules of canonical quantization have served well and it is natural to apply them
to the Abraham model. There is one immediate difficulty. Canonical quantization
starts from identifying the canonical variables of the classical theory. Thus we first
have to rewrite the equations of motion for the Abraham model in Hamiltonian
form. For this purpose we adopt the Coulomb gauge, as usual, so as to eliminate
the constraints. In the quantized version we thereby obtain the Pauli–Fierz Hamil-
tonian which has an obvious extension to include spin.
We have to ensure that the Pauli–Fierz Hamiltonian generates a unitary time
evolution on the appropriate Hilbert space of physical states. Mathematically this
means that we have to specify conditions under which the Pauli–Fierz Hamiltonian
is a self-adjoint operator, an issue which can be satisfactorily resolved. Still, the
true physical situation is more subtle and in fact not so well understood. It is related
to the abundance of very low-energy photons, i.e the infrared problem, and to the
arbitrariness of the cutoff at high energies, i.e. the ultraviolet problem. There are
many items of interest before these, and it will take us a while to start discussing
these subtleties.
Some words on our notation: In the beginning we keep c,
, and later set them
equal to one, mostly without notice. The vector notation, like x, tends to be a little
heavy, in particular since some of the objects become either operators or random
variables. Therefore we stick with x, whose vector character has to be inferred
from the context.
149
150 Quantizing the Abraham model
13.1 Lagrangian and Hamiltonian rewriting of the Abraham model
We consider N charges coupled to the Maxwell field. Their motion is governed by
(11.1), (11.2), which we repeat with the only difference that the relativistic kinetic
energy is replaced by its Galilean cousin.
(Inhomogeneous Maxwell–Lorentz equations)
c
−1
∂
t
B =−∇×E , c
−1
∂
t
E =∇×B −c
−1
j , (13.1)
∇·E = ρ, ∇·B = 0 , (13.2)
where the charge and current density are given by
ρ(x, t) =
N
j=1
e
j
ϕ(x − q
j
(t)) , j(x, t) =
N
j=1
e
j
ϕ(x − q
j
(t))v
j
(t) (13.3)
satisfying charge conservation by fiat.
(Newton’s equations of motion)
m
j
d
dt
v
j
(t) = e
j
E
ϕ
(q
j
(t), t) + c
−1
v
j
(t) × B
ϕ
(q
j
(t), t)
, (13.4)
j = 1,...,N . ϕ is the charge distribution. It satisfies Condition (C), Eq. (2.38).
The Lagrangian for a charge subject to external potentials is discussed in every
text on classical mechanics. The Lagrangian of the coupled system, charges plus
Maxwell field, can almost be guessed on that basis. We introduce the electromag-
netic potentials through
E =−c
−1
∂
t
A −∇φ, B =∇×A , (13.5)
hence guaranteeing ∇·B = 0 and the first half of (13.1), and regard as position-
like variables {q
j
, j = 1,...,N ,φ(x), A(x), x ∈ R
3
}. Let us define the Lagrange
density
L
0
(x) =
1
2
E(x)
2
− B(x)
2
+ c
−1
j (x) · A(x) − ρ(x)φ (x), (13.6)
where, according to (13.3), ρ, j depend on the positions and velocities of the
charges. The Lagrangian of the Abraham model is then
L =
N
j=1
1
2
m
j
˙q
2
j
+
d
3
xL
0
(x). (13.7)
13.1 Lagrangian and Hamiltonian rewriting of the Abraham model 151
We only have to verify that the Euler–Lagrange equations for the action obtained
from L yield (13.1), (13.2), and (13.4). Indeed
d
dt
∂ L
∂ ˙q
j
−
∂ L
∂q
j
= 0 (13.8)
are Newton’s equations of motion. Using ‘ ˙’ for ∂
t
as concise notation, variation
with respect to φ yields
d
dt
δL
δ
˙
φ
−
δL
δφ
= 0 , (13.9)
which is equivalent to
−∇ · (c
−1
˙
A +∇φ) = ρ (13.10)
and which we recognize as the first half of (13.2). Finally
d
dt
δL
δ
˙
A
−
δL
δ A
= 0 (13.11)
amounts to
c
−1
(c
−1
¨
A +∇
˙
φ) =−∇×(∇×A) + c
−1
j , (13.12)
which is nothing but the second half of (13.1).
Since (13.9) represents only a constraint and is not an equation of motion,
clearly we are using a redundant set of dynamical variables. Let us do the counting.
We split the electromagnetic fields into longitudinal and transverse components,
E = E
+ E
⊥
, B = B
+ B
⊥
. (13.13)
Since ∇·B = 0, we have B
= 0. From ∇·E = ρ we conclude
E
=−iρk/k
2
. (13.14)
E
⊥
and B
⊥
satisfy a first-order evolution equation. Thus, in the sense of
Lagrangian mechanics, there are two independent field degrees of freedom at every
space point, while in (13.6) we employed four degrees of freedom.
We first eliminate φ through (13.10), i.e.
φ =
1
k
2
ik · c
−1
˙
A
+ ρ
. (13.15)
152 Quantizing the Abraham model
Then, using Fourier transforms and Parseval’s identity, (13.7) transforms to
L =
N
j=1
1
2
m
j
˙q
2
j
+
1
2
d
3
k
c
−2
˙
A
∗
⊥
·
˙
A
⊥
+ k
−2
ρ
∗
ρ − (k ×
A
∗
⊥
) · (k ×
A
⊥
)
+
1
2
d
3
k
c
−1
j
∗
·
A + c
−1
j ·
A
∗
− 2k
−2
ρ
∗
ρ
−ic
−1
k
−2
ρ
∗
k ·
˙
A
− ρk ·
˙
A
∗
. (13.16)
The term ρ
∗
ρ depends only on the q
j
’s and is recognized as the Coulomb potential,
1
2
d
3
kk
−2
ρ
∗
ρ =
1
2
N
i, j=1
e
i
e
j
d
3
yd
3
y
ϕ(y)(4π |q
i
− q
j
− y + y
|)
−1
ϕ(y
)
= V
ϕcoul
(q
1
,...,q
N
). (13.17)
The Coulomb potential is smeared by ϕ,which as before is indicated by the sub-
script. ϕ appears twice, since both the i-th and the j-th particle carry a charge
distribution. To simplify the last term of (13.16) we use the conservation law
˙
ρ + ik ·
j = 0. Then
L =
N
j=1
1
2
m
j
˙q
2
j
− V
ϕcoul
+
1
2
d
3
k
c
−2
˙
A
∗
⊥
·
˙
A
⊥
− k
2
A
∗
⊥
·
A
⊥
+
1
2
d
3
k
c
−1
j
∗
·
A
⊥
+ c
−1
j ·
A
∗
⊥
+c
−1
d
dt
1
2
d
3
k|k|
−1
i
ρ
A
∗
− ρ
∗
A
. (13.18)
Since A
appears only inside a total time derivative, we have identified A
as the
second redundant field. To drop the redundant degrees of freedom, the simplest
choice is to set A
= 0byexploiting the gauge freedom, which means selecting
the Coulomb gauge defined by
∇·A = 0 . (13.19)
The vector potential is purely transverse and we henceforth drop the subscript ⊥.
Transforming back to real space, the Lagrangian of the Abraham model reads
L =
N
j=1
1
2
m
j
˙q
2
j
− V
ϕcoul
+
d
3
xL(x) (13.20)
with the Lagrange density
L =
1
2
(c
−1
˙
A)
2
− (∇×A)
2
+ c
−1
j · A . (13.21)