The Relativistic Dynamics of the Combined
Particle–Field System in Renormalized Classical
Electrodynamics
∗
Hans-Peter Gittel
Department of Mathematics, University of Leipzig,
Augustusplatz 10/11, D - 04109 Leipzig, Germany
Jerzy Kijowski
†
Centrum Fizyki Teoretycznej PAN,
Aleja Lotników 32/46, PL - 02-668 Warsaw, Poland
Department of Mathematical Methods in Physics, University of Warsaw,
ul. Hoża 74, PL - 00-682 Warsaw, Poland
Eberhard Zeidler
‡
Department of Mathematics, University of Leipzig,
Augustusplatz 10/11, D - 04109 Leipzig, Germany
Max–Planck–Institute for Mathematics in the Sciences,
Inselstr. 22 – 26, D - 04103 Leipzig, Germany
May 13, 1998
∗
E-mail: [email protected]
†
E-mail: [email protected]
‡
E-mail: [email protected]
1
Abstract
This paper develops a general theory for the nonlinear, renormalized interaction
between charged particles and electromagnetic fields. For the combined “particle +
field” system, a fundamental relativistically invariant dynamical equation is derived
from first principles. This theory was first obtained in an alternative way by one of
us (J.K.) in an earlier paper. Here, we prove that the initial–value problem for the
“particle + field” system is well–posed. The existence and uniqueness result is based
on a careful analysis of the singularites of the electromagnetic field along the trajectory of the moving charged particle. Furthermore, the Banach fixed–point principle is
used. The theory improves the classical Dirac theory for the motion of electrons. In
particular, it is shown how to deal with the Dirac paradox of runaway solutions.
1
Introduction
From first principles, we rigorously derive the relativistic equation of motion for a charged
particle in an electromagnetic field, taking fully into account the self–interactions of the particle with its radiation field. Our nonlinear equation of motion is formulated in a relativistic
covariant way.
In addition, we formulate and solve a well–posed initial–value problem for the combined
“particle + field” system. We show that, for given external electromagnetic field together
with initial data of the particle and of the electromagnetic field at time t = 0, there exist
locally both a unique trajectory of the particle and a unique total electromagnetic field
produced by the moving particle. We want to stress the important fact that the initial data
of the particle and of the electromagnetic field cannot be given independently. In particular,
we show that in our approach the value of the acceleration of the particle at time t = 0
is uniquely encoded in the initial data of the electromagnetic field. This is a crucial point
which was first discussed in the paper [9].
The derivation of the equation of motion for the “particle + field” system is based on
the following four facts.
(i) In the particle’s comoving frame Σ+ , the particle rests. Naturally enough, we assume
that the leading singularity of the electromagnetic field on Σ+ is equal to the Coulomb
2
singularity.
(ii) By a careful mathematical analysis, we show that the leading singularity uniquely
determines the total singularity of the electromagnetic field near the trajectory of the moving
particle. We explicitly compute this singularity.
(iii) Using (ii), we calculate the renormalized, total energy and momentum of the combined “particle + field” system (cf. [9]) in a simple way.
(iv) We postulate conservation of renormalized energy and momentum.
Throughout the paper we use the decomposition F = Fsing + Freg of the electromagnetic
field F . This decomposition is defined locally, in a neighbourhood of the particle’s trajectory.
The singular part Fsing is uniquely determined by our method within a reasonable large class
of functions. The regular part Freg is continuous across the trajectory of the particle. Our
decomposition is justified by the observation that the field F − Fsing is continuous for each
solution F of the initial–value problem for the Maxwell system in the sense of generalized
functions.
Our approach improves the classical nonlinear Dirac theory for the motion of electrons.
However, in contrast to the ill–posed Dirac theory, we get an initial–value problem which is
well–posed from the physical point of view. In particular, runaway solutions do not appear
for initial data which are consistent with our theory.
In contrast to the classical paper [5] by Dirac, our derivation of the equation of motion
is completely independent of the use of the retarded or advanced solutions of Maxwell equations. This way, the symmetry of both the electrodynamics and mechanics with respect to
time reversal is not violated by our theory. However, we shall see that the retarded (resp.
advanced) Green function for Maxwell equations provide a technical tool for solving the
initial–value problem. Using this tool, the equation of motion can be reduced to a nonlinear
operator equation for the particle’s trajectory. The existence and uniqueness proof follows
then by means of the Banach fixed–point theorem. Here, a detailed analysis of the field
singularities in the vicinity of the particle’s trajectory is necessary.
The paper is based on ideas published in [9, 10]. It is organized as follows:
3
2
1
Introduction
2
Notation
3
The basic equations of motion for the particle–field system
4
The main result
5
The retarded electromagnetic field
6
An equivalent formulation of the equation of motion
7
Fermi–Walker transport
8
Computation of the singular part of the electromagnetic field
9
Derivation of the equation of motion for the particle
10
Solution of the initial–value problem for the basic equations.
Notation
Tensors. We will work on a 4–dimensional flat (oriented) Minkowski space–time manifold
M4 with the metric tensor G := gαβ dxα ⊗ dxβ . Greek (resp. latin) indices run from 0 to 3
(resp. from 1 to 3). Moreover, greek (resp. latin) indices are lowered by gαβ (resp. gjk ).
We assume that the metric (gαβ ) has signature (−, +, +, +). We shall use also curvilinear
coordinates, but the coordinate x0 is always time–like, whereas the coordinates xj are space–
like. The natural basic vectors ∂0 , ∂1 , ∂2 , ∂3 are also denoted by b0 , b1 , b2 , b3 . There exists a
natural inner product vw on M4 . If v = v α bα and w = wα bα , then
vw = gαβ v α wβ ,
bα bβ = gαβ .
The vector v is called space–like (resp. time–like) iff v 2 > 0 (resp. v 2 < 0).
Let F = F αβ bα ⊗ bβ be an antisymmetric tensor, i.e., F αβ = −F βα . Since bα ∧ bβ :=
bα ⊗ bβ − bβ ⊗ bα ,
1
F = F αβ bα ∧ bβ .
2
The dual tensor is given by
1
∗F = (∗F )αβ bα ⊗ bβ = (∗F )αβ bα ∧ bβ .
2
Let T = T αβ bα ⊗ bβ . Then the tensor T becomes a linear operator if we define
¡
¢
T n := nγ gγα T αβ bβ
4
for all vectors n = nα bα . This definition is independent of the chosen basis. Hence the trace
of T equals tr T = gαβ T αβ . Let a, b, c be 4–vectors. Then
(a ⊗ b)c = (ac)b,
(a ∧ b)c = (ac)b − (bc)a.
(2.1)
Inertial systems. By an inertial (or laboratory) system Σ of coordinates we understand
a system for which the metric tensor is given by (gαβ ) = diag(−1, 1, 1, 1). Here, the natural
basis vectors will be denoted by e0 , e1 , e2 , e3 . Then eα eβ = gαβ .
In particular, the three position vectors e1 , e2 , e3 form an orthonormal system. In the
system Σ, the 3–dimensional position vector and time are denoted by x = xj ej and t = x0 ,
respectively. Furthermore, we use the 4–position vector
x = xα eα = x + te0 .
Maxwell equations. The classical vacuum Maxwell equations read as follows in the
inertial system Σ :
div E = ρ,
div B = 0,
curl B = Et + j,
curl E = −Bt .
Here,
E
= electric field vector,
B
ρ
= electric charge density,
= magnetic field vector,
j
= electric current density vector.
We use a system of units where ε0 = c = 1, and hence µ0 = 1.
The tensor of the electromagnetic field is given by F = 21 F αβ bα ∧ bβ . In an inertial system
Σ, we get bα = eα , E = E j ej , B = B j ej along with

0 −E 1 −E 2
 1
0
−B 3
¡ αβ ¢ 
E
F
=
 E2 B3
0

E3
−B 2
B1
−E 3


B2 

 .
1
−B 
0
Moreover, Fjk = F jk as well as F0j = −F 0j . This means that
F = E ∧ e0 − Alt(B),
5
(2.2)
where we introduce the convenient notation
Alt(B) := B 1 (e2 ∧ e3 ) + B 2 (e3 ∧ e1 ) + B 3 (e1 ∧ e2 ).
(2.3)
The dual tensor ∗F is obtained from F by replacing E with −B and B with E. Hence
∗F = −B ∧ e0 − Alt(E). If the magnetic field can be expressed by a vector product, i.e., if
B = c × d, then Alt(B) = c ∧ d and
F = E ∧ e0 − c ∧ d.
(2.4)
Note that Alt(E)b = E × B for b = B 0 e0 + B.
In order to translate expressions for the electromagnetic field tensor from the special
comoving system to general systems of reference, we will use (2.4) later on. Furthermore,
let jext denote the 4–vector of the external sources. In an inertial system Σ, we put jext =
jext + ρext e0 , where ρext and jext denote the external electric charge density and the external
electric current density vector, respectively.
Trajectories of particles. On M4 , the 4–trajectory denotes a time–like curve x = q(τ )
which is parametrized by the eigentime τ . Furthermore, the 4–velocity u(τ ) and the 4–
acceleration a(τ ) are defined through u(τ ) := q̇(τ ) and a(τ ) := q̈(τ ). In the sequel, the dot
will always indicate the derivative with respect to τ .
In particular, let x = q(t) describe the motion of a particle in an inertial system Σ.
Suppose that the particle moves with underlight speed, i.e., |q0 (t)| < 1. Then the eigentime
τ of the particle is given by
Zt q
1 − q0 (σ)2 dσ.
τ (t) :=
0
The inverse function is denoted by t = t(τ ). Set q(τ ) := q(t(τ )). Then
q̇(τ ) = q0 (t(τ ))ṫ(τ ) ,
2
q̈(τ ) = q00 (t(τ ))ṫ(τ ) + q0 (t(σ))ẗ(τ ).
The 4–trajectory of the particle is given by x = q(τ ) = q(τ ) + t(τ )e0 . Furthermore,
u(τ ) = q̇(τ ) + ṫ(τ )e0 ,
a(τ ) = q̈(τ ) + ẗ(τ )e0 .
(2.5)
In particular, if q(0) = q0 (0) = 0, then u(0) = e0 , and a(0) = q̈(0). The relations above
immediately imply
u2 = −1 (i.e., u is time–like), au = 0 (i.e., a is space–like), ȧu = −a2 .
6
(2.6)
The comoving system Σ+ . To simplify substantially our analysis, we will use an
(accelerated) comoving system of reference, uniquely definedmby the particle’s trajectory.
For this purpose, we consider the rest–frame hyperplane Σ+ (τ ), i. e. the 3–dimensional
plane through the point q(τ ) of the trajectory which is orthogonal to the tangent vector
u(τ ) (Fig. 1). Analytically, the hyperplane Σ+ (τ ) consists of precisely all the points x ∈ M4
which satisfy the equation
(x − q(τ ))u(τ ) = 0.
(2.7)
Definition 2.1. The comoving system Σ+ consists of all the points (τ, y) with y ∈ Σ+ (τ ).
That is, the time coordinate in Σ+ equals the eigentime of the particle. By the Fermi
transformation, we understand the passage from (τ, y) to the corresponding event x, i.e.
x = q(τ ) + y.
(2.8)
Figure 1: The Fermi system
The map (τ, y) 7→ x is not bijective. However, the following argument shows that this
map is locally injective. Let us fix a point x0 := q(τ0 ) of the trajectory at a fixed eigentime
τ0 .
Proposition 2.2. Let k ≥ 2. Suppose that the 4–trajectory x = q(τ ) is C k on a
neighbourhood of τ0 . Then, the following are true.
(i) There exist both a neighbourhood U of the point x0 in M4 and a neighbourhood V of
τ0 in R such that, for each given point x ∈ U, equation (2.7) possesses a unique solution
τ (x) ∈ V.
The function τ = τ (x) is C k−1 on V.
7
(2.9)
(ii) For all points and all vectors h,
τ 0 (x)h = −u(τx )h + (q(τx ) − x)a(τx )τ 0 (x)h.
(2.10)
Here, we set τx := τ (x). In particular,
τ 0 (x0 ) = −u(τ0 ) .
(2.11)
Proof. Ad (i). Define g(x, τ ) := (x − q(τ ))u(τ ). The real function g is C k−1 on U × V.
∂
g(x0 , τ0 )
∂τ
Obviously, g(x0 , τ0 ) = 0. Moreover,
= −u(τ0 )2 = 1. Thus, statement (i) follows
from the implicit function theorem (cf. [20], Theorem 4.B).
Ad (ii). Replace x with x + σh in equation (2.7), and differentiate with respect to the
real parameter σ at the point σ = 0.
2
Definition 2.3. Let τx := τ (x). The inverse Fermi transformation denotes the local
transformation
τ = τx ,
y = x − q(τx ),
(2.12)
given by by Proposition 2.2(i). We call (τx , y) the local Fermi coordinates of the event x.
By Proposition 2.2(i), the local map x 7→ (τx , y) is C k−1 . From the physical point of
view, the event x ∈ M4 is observed in the comoving system at the point y at time τx .
3
The Basic Equations of Motion for the Particle–Field
System
In this section we formulate results which will be derived later. In particular, points (i) —
(iii) are consequences of the Maxwell equations (see Section 8), whereas point (iv) is implied
by conservation of the renormalized energy-momentum (see Section 9).
3.1
The Basic Equations in an Arbitrary System of Reference
Let F denote the electromagnetic tensor field, and let jext = (ρext , jext ) be a regular 4–vector
field of the external sources. We assume that the continuity equation
∂t ρext + div jext = 0
8
(3.1)
is satisfied. The total electromagnetic current is a sum of jext and a singular δ–like current
carried by the particle. It will be proved that the singular current uniquely implies the
singular part of the field in the vicinity of the particle.
Then, both the field and the particle’s trajectory have to fulfil the following basic equations.
(i) The Maxwell equations for the electromagnetic field
Div F = jext ,
Div(∗F ) = 0
(3.2)
must be valid outside the trajectory.
(ii) The electromagnetic field in a neighbourhood of the trajectory may be decomposed as
F = Fsing + Freg ,
(3.3)
where the regular part Freg is continuous across the trajectory.
(iii) The singular part Fsing of the electromagnetic field has the following structure
Fsing = F−2 + F−1 + F0 ,
(3.4)
QN
∧ u(τx ),
4πR2
Q
F−1 (x) := −
((aN)N + a) ∧ u(τx ),
8πR
Q
Q
(3(aN)2 N + 6(aN)a − 3a2 N) ∧ u(τx ) −
N ∧ ȧ(τx ).
F0 (x) :=
32π
8π
(3.5)
where
F−2 (x) :=
For the sake of brevity, we write
R :=
p
y2 ,
N := y/R,
and a := a(τx ),
(3.6)
(y and τx are as in the previous section).
(iv) The following equation of motion for the particle
m0 a(τ ) = QFreg (q(τ ))u(τ )
(3.7)
must be satisfied. Here, m0 and Q denote the rest mass and the charge of the particle,
respectively. In addition, x = q(τ ) represents the particle’s trajectory, and τ denotes its
eigentime.
9
We stress that all the expressions above have an invariant (geometric) meaning. Therefore, the basic equations (i) through (iv) do not depend on the choice of a reference system.
In the present paper, we consider the following
Problem: Given an external 4–current jext and an appropriate set of initial conditions,
find both
(i) the electromagnetic field tensor F , and
(ii) the 4-trajectory x = q(τ ) of the particle,
fulfilling the above system of equations and satisfying the initial conditions.
Remark 3.1. The equation of motion (3.7) is equivalent to
m0 a(τ ) = Q (F − Fsing ) (q(τ ))u(τ ).
(3.8)
This can be regarded as a renormalized Lorentz equation.
Remark 3.2 (regularity). Both the singular part Fsing and the regular part Freg are
smooth outside the trajectory. The continuity of Freg across the trajectory will be proved in
Section 10.
Remark 3.3 In coordinate language, we have:
(i) The Maxwell equations for the electromagnetic field
³ 1
´
1
β
,
|g|− 2 ∂α |g| 2 F αβ = jext
³ 1
´
1
|g|− 2 ∂α |g| 2 (∗F )αβ = 0 for xα 6= q α (τ ).
(ii) The local structure of the electromagnetic field in a neighbourhood of the trajectory
αβ
αβ
F αβ = Fsing
+ Freg
.
(iii) The singular part of the electromagnetic field
αβ
αβ
αβ
+ F0αβ .
+ F−1
Fsing
= F−2
The expressions of Fkαβ for k = −2, −1, 0 follow from (3.5).
(iv) The equation of motion for the particle
αβ
uα
m0 aβ = QFreg
along the trajectory xα = q α (τ ).
10
3.2
The Basic Equations in an Inertial System
Let Σ be an inertial system with position vector x and time t = x0 . Then, the basic equations
read as follows.
(i) The Maxwell equations for the electromagnetic field
div E = ρext ,
div B = 0 ,
−∂t E + curl B = jext ,
∂t B + curl E = 0
(3.9)
for x 6= q(t).
(ii) The local structure of the electromagnetic field in a neighbourhood of the trajectory
E = Esing + Ereg ,
B = Bsing + Breg .
(iii) The singular part of the electromagnetic field
Esing = E−2 + E−1 + E0 ,
Bsing = B−2 + B−1 + B0 ,
where
E−2 (t, x) :=
¢
Q ¡
ṫ(τ
)(x
−
q(τ
))
−
(t
−
t(τ
))
q̇(τ
)
,
x
x
x
x
4πR3
B−2 (t, x) := −
E−1 (t, x) := −
B−1 (t, x) :=
E0 (t, x) :=
Q
(x − q(τx )) × q̇(τx ) ,
4πR3
Q
ṫ(τx ) ((aN)(x − q(τx )) + Rq̈(τx ))
8πR2
¢
Q ¡
+
(aN)(t
−
t(τ
))
+
R
ẗ(τ
)
q̇(τx ) ,
x
x
8πR2
Q
((aN)(x − q(τx )) + Rq̈(τx )) × q̇(τx ) ,
8πR2
... ¢
Q ¡
ṫ(τx )(3(aN)2 − 3a2 ) − 4t(τx ) (x − q(τx ))
32πR
¢
Q ¡
(3(aN)2 + a2 )(t − t(τx )) + 6R(aN)ẗ(τx ) q̇(τx )
−
32πR
¢
Q ¡
...
+
6R(aN)ṫ(τx )q̈(τx ) + 4(t − t(τx ))q(τx ) ,
32πR
11
(3.10)
B0 (t, x) := −
¢
Q ¡
(3(aN)2 + a2 )(x − q(τx )) + 6R(aN)q̈(τx ) × q̇(τx )
32πR
Q
...
+
(x − q(τx )) × q(τx ) .
8πR
Here, we use the notation introduced in (3.6). Moreover, x := x + te0 .
(iv) The equation of motion for the particle
d m0 v(t)
q
= QEreg (t, q(t)) + Qv(t) × Breg (t, q(t)) ,
dt
2
1 − v(t)
(3.11)
where x = q(t) (resp. v := q0 (t)) denotes the trajectory (resp. the 3–dimensional laboratory–frame velocity) of the particle. The above explicit expressions of the fields Ek , Bk for
k = −2, −1, 0 follow from (3.5) along with (2.2) through (2.5).
Remark 3.4. Let ρext , jext be smooth everywhere. Moreover, suppose that Eext (0, x) =
Bext (0, x) = 0 for all x ∈ R3 . In this special case, the unique (smooth) solution to (3.9) is
given by the following well–known formulas
Zt
Eext (t, x) = −
£
¤
t−σ
Mt−σ
x {jext }(σ) + (t − σ)Mx {∂r jext + grad ρext }(σ) dσ ,
0
(3.12)
Zt
(t − σ)Mt−σ
x {curl jext }(σ) dσ
Bext (t, x) = −
0
(cf., e.g. [18], §§15,46). Here, for a function f = f (t, x), we use the mean value over a sphere
of radius R centered at x:
MR
x {f }(σ)
1
:=
4π
Z
f (σ, x + Rz) dSz ,
(3.13)
S
where S := {z : |z| = 1}. The radial derivative is given by ∂r j = (z grad)j. The expressions
(3.12) follow from the fact that each component of the fields Eext and Bext satisfies the
inhomogeneous wave equation. Solutions of this equation are given by a mean value formula.
For our purposes, solution (3.12) is not satisfactory because
1. it does not take into account the singular current carried by the particle, and
2. it does not satisfy the (usually inhomogeneous) initial conditions for the field.
Nevertheless, the integral representation (3.12) of the field will be an important tool for
the construction of the solution of our problem.
12
4
The Main Result
Let us consider a fixed inertial system Σ. Suppose that, at the initial time t = 0, we are
given the field initial data E(0, x) = E0sing (x) + E0reg (x) and B(0, x) = B0sing (x) + B0reg (x)
compatible with our basic equations formulated above. Under specific regularity conditions
imposed on E0reg and B0reg we will prove that the initial–value problem has a unique local
solution.
Observe that the singular part of the field data contains the information about the initial
position, velocity, and acceleration of the particle. Hence, no additional initial conditions
for the particle’s trajectory are necessary.
To simplify the formulation of the regularity conditions, we will pass to the particle’s
co–moving frame in which the initial velocity of the particle vanishes. By a suitable choice
of the origin, the initial position of the particle also may be trivialized: q(0) = q0 (0) = 0.
Putting r := |x| and n := x/r, we thus assume the following:
(i) There exist two vectors a0 and ȧ0 such that the singular part of the initial data has
the structure
E0sing (x) =
¢ 3Q ¡ 0 2
¢
Q ¡ 0
Qn
−
(a n)n + a0 +
(a n) n + 2(a0 n)a0 − (a0 )2 n ,
2
4πr
8πr
32π
(4.1)
B0sing (x) =
Q
n × ȧ0 .
8π
Note that a0 = q00 (0) = a(0) and ȧ0 = q000 (0) = a0 (0).
(ii) The regular part of the initial data E0reg and B0reg is C 2 in a neighbourhood U of the
origin and satisfies the compatibility conditions
div E0reg (x) = ρext (0, x),
div B0reg (x) = 0 for all x ∈ U ,
(4.2)
for given external sources ρext , jext . They are assumed to be C ∞ on [0, t0 ] × R3 for a certain
t0 > 0.
(iii) The following constraint between the singular and the regular part of the initial data
m0 a0 = QE0reg (0) ,
is fulfilled. This follows from (3.7).
13
(4.3)
Theorem 4.1. The above initial–value problem for the basic equations of motion has a
unique local solution. This solution has the following properties:
(a) The trajectory x = q(t) of the particle is C 4 on a sufficiently small time interval
[0, t1 ].
(b) The regular part E−Esing , B−Bsing of the electromagnetic field is continuous across
the trajectory for all times t ∈ (0, t1 ].
Corollary 4.2. In addition, suppose that the initial data E0reg , B0reg are C m , m ≥ 3.
Then the trajectory is C m+2 .
The complete proof of this theorem will be given in the sequel.
5
The Retarded Electromagnetic Field
Formula (3.12) enables us to deal with external sources. To handle singular (δ-like) sources
of the field, carried by the particle’s trajectory, we may use e.g. the classical retarded field
Fret or the advanced field Fadv . For physical background see, for instance, the monograph
[15] by Rohrlich (Chapters 4 and 6) or and the monograph [16] by Ruei (Chapter XVII).
Those fields are solutions of the inhomogeneous Maxwell system
´
³
Div ∗F ret = 0
Div F ret = Qδζ ,
adv
adv
(5.1)
where Q is the electric charge of the particle, ζ = {x = q(τ ), τ ∈ R} its trajectory, and δζ
denotes the Dirac distribution concentrated on ζ.
For our purposes, the specific form of both the solutions (cf. [15],(4-98) and [16],(XVII26)) will not be necessary. What we need is the following approximation (cf. [15],(6-62)):
¢
Q
Q ¡
Q
N∧u−
((aN)N + a) ∧ u +
3(aN)2 N + 6(aN)a − 3a2 N ∧ u
2
4πR
8πR
32π
Q
Q
− N ∧ ȧ ∓
u ∧ ȧ + O(R)
8π
6π
Q
= Fsing (x) ∓
u ∧ ȧ + O(R) , as R → 0 .
(5.2)
6π
F ret (x) =
adv
This formula contains the standard singular part, common to any solution of (5.1) (cf. (3.4),
(3.5), and the proof in Section 8). But formula (5.2) contains also the value of the regular
14
part of the field on the particle’s trajectory Freg (q(τ )) = ∓ (Q/6π) u∧ ȧ, being characteristic
for the special solution chosen here. These properties may be summarized as follows.
Lemma 5.1. (a) Let the trajectory x = q(τ ) with q(0) = 0 , u(0) = e0 be C m , m ≥ 2, on
a neighbourhood of τ = 0. Then, the fields F ret are C m−2 outside the trajectory.
adv
(b) The field Fret − Fsing can be extended to a field which is continuous across the trajectory.
Proof. Ad (a). The assertion follows from the expressions for retarded and advanced
fields containing second derivatives of the trajectory (cf. [15],(4-98)).
Ad (b). This follows from (5.2) above.
2
In the existence and uniqueness proof for the initial-value problem below, we will use the
following construction.
C 3 –Prolongation of the trajectory. Let us suppose that the trajectory is given only
for nonnegative times t. We prolong the trajectory to the past in such a way that the
prolonged trajectory is C 3 at the initial time t = 0. For this purpose, let us make the ansatz
¡
¢
q̃(t) = f + gt + ht2 + kt3 exp(λt) if t < 0 .
(5.3)
Here, the four vectors f, g, h, k, and the scalar λ are chosen in such a way that they match
the corresponding derivatives of the trajectory at t = 0 and that the trajectory remains
time–like. Explicitly, in our fixed inertial system, the prolonged trajectory is given by the
formula
(
x=
q(t)
for
¢
¡1 0 1 0
a + 6 (ȧ − 3λa0 ) t t2 exp(λt) for
2
t≥0
t<0 .
(5.4)
Depending on the given vectors a0 and ȧ0 , the value of λ > 0 can be calculated from the
relation q̃0 (t)2 < 1 for all t < 0 . (In fact, we need the prolongation of the trajectory only for
small negative times. Thus, the trajectory can be appropriately altered for negative t where
|t| is sufficiently large.)
To solve the initial value problem, we shall use the retarded field associated with the
0
represents the
prolonged trajectory (5.4). We denote this field by F̃ret . In addition, F̃ret
initial value of F̃ret at time t = 0. This initial value depends only on the part x = q̃(t) of
0
is uniquely
the trajectory corresponding to times t < 0 . Therefore, (5.4) implies that F̃ret
15
determined by the initial acceleration a0 and its derivative ȧ0 . Furthermore, by (5.2), we get
0
0
F̃ret
(x) = Fsing
(x) −
Q
e0 ∧ ȧ0 + O(r) ,
6π
as r → 0 ,
(5.5)
0
where r := |x| and Fsing
(x) = Fsing (0, x) for x 6= 0.
6
An Equivalent Formulation of the Equation of Motion
6.1
Derivation of a Dirac–Lorentz Type Equation
For solving the initial–value problem, let us express the regular part of the field in equation
(3.7) in terms of the field F̃ which arises in the following decomposition
F = F̃ret + F̃ .
(6.1)
Using (3.3) and(5.2) we get
F̃ = Freg +
Q
u ∧ ȧ + O(R) ,
6π
as R → 0 ,
R=
p
y2
for time τ > 0. Hence, equation (3.7) can be rewritten in the form
Q2
(u ∧ ȧ) u(τ ) + QF̃ ((q(τ ))u(τ )
6π
¢
Q2 ¡
=
ȧ − a2 u (τ ) + QF̃ ((τ ))u(τ ) ,
6π
m0 a(τ ) = −
(6.2)
by (2.1) and (2.6). Equation (6.2) represents an equation of the so–called Dirac–Lorentz type
1
with the nonlinear term QF̃ ((τ ))u(τ ) which is sometimes called “external force”, whereas
the first term is often called a “self–interaction force”. We stress that only the sum of the
two has a physical interpretation as a necessary constraint which must be fulfilled by the
“particle + field” system. The decomposition (6.2), based on (6.1), has been chosen only
because it is useful in solving the initial value problem. To solve the final–value problem,
(6.1) should be replaced by an analogous decomposition using the advanced field. Moreover,
given a trajectory, any combination αFadv + βFret , α + β = 1, may be used. This proves
1
Such equations are considered in [15],(6-57).
16
that the notion of a “self–interaction force” and an “external force” is purely conventional
and has no physical meaning (these problems were thoroughly discussed in [9] and [10]).
From the point of view of the initia– value problem, the “external force”, given by the
last term of (6.2), depends in a nonlocal way on the particle initial data q, u, a, and the field
initial data at τ = 0. The mathematical structure of equation (6.2) will be carefully studied
in Section 10.
6.2
The Solution of the Dirac Paradox
Let us consider the very special case where the electromagnetic field is given by
F = F̃ret .
That is, F̃ ≡ 0. From (6.2) we obtain the classical (nonlinear) Dirac equation of motion
m0 a(τ ) =
¢
Q2 ¡
ȧ − a2 u (τ ),
6π
q(0) = 0,
q̇(0) = e0 .
(6.3)
The initial condition tells us that the particle rests at time τ = 0. Equation (6.3) is identical
to the following equation
m0 q̈(τ ) =
¢
Q2 ¡...
q(τ ) − q̈(τ )2 q̇(τ ) ,
6π
q(0) = 0,
q̇(0) = e0 .
(6.4)
In the purely mechanical framework (i.e. as the dynamical equation for the particle alone),
this equation has an ill–posed initial value problem because the initial data for q̈ are missing.
Moreover, this equation admits self–accelerated (the so–called runaway) solutions which
approach the speed of light as time goes to infinity. In order to avoid such unphysical
solutions, Dirac added ad hoc an additional, artificial condition.
2
In the context of our theory, the missing initial data q̈(0) for the third order differential
equation (6.4) are provided by the field initial data (in fact, all the initial data are carried
by the field). Consider, for example, the following quite natural initial condition
F = F−2
2
at time t = 0 outside the point x = 0 .
(6.5)
In this connection, we refer to [5, 7, 15] and [16], Vol. 2. A detailed, very critical discussion of the Dirac
theory can be found in the monograph [14] by Parrott.
17
This condition means that the total electromagnetic field equals the Coulomb field at the
initial time t = 0. Let us discuss the consequences of condition (6.5). From (3.3),(3.4) it
follows that F−1 = F0 = Freg = 0 at t = 0 outside the point x = 0. Hence
E(0, x) =
Qn
,
4πr2
B(0, x) = 0 .
By (4.1) and (4.3), this implies a0 = 0, i.e., q00 (0) = 0. Thus, we have to add the condition
q̈(0) = 0 to the Dirac equation (6.3). The modified equation (6.4) has the unique solution
q(τ ) ≡ τ e0 (particle at rest for all times). This way, runaway solutions have been excluded.
7
Fermi–Walker Transport
In order to simplify the Maxwell equations in the comoving system, let us introduce a special
orthonormal frame on the Fermi hyperplane.
Let Σ be a fixed inertial system along with the orthonormal vector basis e0 , e1 , e2 , e3 (cf.
Section 2). For m ≥ 2, let
x = q(t),
0 ≤ t ≤ t0 ,
(7.1)
q0 (0) = 0.
(7.2)
be a C m –trajectory of a particle such that
q(0) = 0,
Suppose that |q0 (t)| < 1 on [0, t0 ], i. e., the velocity of the particle is less then the speed of
light.
Definition 7.1. By Fermi–Walker transport along the trajectory x = q(τ ), we un+
+
+
derstand four vectors b+
0 , b1 , b2 , b3 depending of τ which satisfy the ordinary differential
equation
¡ +¢
¡ +¢
ḃ+
α = abα u − ubα a
(7.3)
+
+
along with the initial condition b+
α (0) = eα . We call b0 (τ ), . . . , b3 (τ ) the Fermi frame at
eigentime τ .
+
Proposition 7.2. The Fermi frame is orthonormal, i.e., b+
α (τ )bβ (τ ) = eα eβ for all τ .
Proof. By (7.3),
d ¡ + +¢
+
+ +
= ḃ+
b b
α bβ + bα ḃβ
dτ α β
¢ ¡ +¢ ¡ +¢ ¡ +¢ ¡ +¢ ¡ +¢ ¡ +¢ ¡ +¢
¡
ubβ − abβ ubα + abβ ubα − abα ubβ = 0.
= ab+
α
18
+
In addition, b+
α (0)bβ (0) = eα eβ .
2
Proposition 7.3. The 4–vector b+
0 (τ ) of the Fermi frame is identical to the 4–velocity
u(τ ). In addition, we obtain the differential equation
¡ +¢
ḃ+
j = abj u
(7.4)
along with the initial condition b+
j (0) = ej .
Proof. By (2.6), au = 0 and u2 = −1. Hence, u̇ = (au)u − u2 a. Furthermore, u(0) = e0 .
Since the solution of (7.3) is unique for α = 0, b+
0 (τ ) = u(τ ). Finally, it follows from
+ +
Proposition 7.2 that ub+
j = b0 bj = 0. This yields (7.4).
2
+
+
+
Since b+
1 (τ ), b2 (τ ), b3 (τ ) are orthogonal to b0 (τ ), these vectors form an orthonormal
basis of the Fermi hyperplane Σ+ (τ ).
Proposition 7.4. The natural (holonomic) basis of the comoving system at the point
(τ, y) is given by
b0 = (1 + ay)u,
bj = b+
j (τ ).
Proof. Note that x = q(τ ) + y j b+
j . Thus, from bα =
∂x
∂y α
we get bj = b+
j , and
+
j
b0 = q̇(τ ) + y j ḃ+
j = u + y (abj )u = u + (ay)u, where (7.4) is used.
2
+
Corollary 7.5. In the comoving system Σ+ , the metric tensor gαβ
is given by
¡
¢
+
gαβ
= diag(−(1 + ay)2 , 1, 1, 1) .
Proof. Relation gαβ = bα bβ and Proposition 7.4 imply (7.5).
(7.5)
2
For a given vector d = dj (τ, y)bj , define ∂τ+ d = (∂τ dj ) bj . Physically, this is the time
derivative of d observed in the comoving system. We have
∂τ+ d = ḋ − (ad)u.
(7.6)
This follows from ḋ = ∂τ+ d + dj ḃj along with ḃj = (abj )u. In particular, we define
a0F := ∂τ+ a,
8
¡ ¢n
(n)
aF := ∂τ+ a.
(7.7)
Computation of the Singular Part of the Electromagnetic Field
In this section, we work in the comoving system Σ+ . Set R := |y|, and N := y/R.
19
Proposition 8.1. In the comoving system Σ+ , the Maxwell equations read as follows:
div E = Lρext ,
div B = 0 ,
−∂τ+ E + curl(LB) = Ljext ,
∂τ+ B + curl(LE) = 0
q
for R > 0, where L :=
(8.1)
+
− det(gαβ
) = 1 + ya(τ ).
Proof. This follows from F = 12 F αβ bα ∧ bβ , (3.2) and Corollary 7.5. The system (8.1)
corresponds to the tensor equation (3.2), where jext = jext + ρext L−1 b0 ,
Ã
!
1
2
3
F ij = −εijk Bk , F 0j = −E j L−1 , εijk = sgn
.
i j k
2
(8.2)
Remark 8.2. In the comoving system Σ+ , the tensor of the electromagnetic field F αβ
is explicitly given by

¡
F
¢
αβ
0
 1 −1
E L

=
 E 2 L−1

E 3 L−1
−E 1 L−1
−E 2 L−1
−E 3 L−1
0
−B 3
B2
B3
0
−B 1
−B 2
B1
0




 .


This follows immediately from (8.2). Furthermore,
F = L−1 E ∧ b0 − Alt(B) = E ∧ u − Alt(B).
(8.3)
Suppose that the electromagnetic field in Σ+ has the following expansion as a sum of
homogeneous components:
E = E−2 + E−1 + . . . + En + . . . ,
B = B−2 + B−1 + . . . + Bn + . . . ,
(8.4)
where Ek = fk (N, a, a0F , a00F , . . .)Rk and Bk = gk (N, a, a0F , a00F , . . .)Rk for all k. We assume
that the functions fk , gk , and a = a(τ ) are sufficiently smooth with respect to all their
arguments. By (3.5) and (8.3),
E−2 =
QN
,
4πR2
B−2 = 0.
Observe that div E−2 = 0 and curl E−2 = 0 for R 6= 0.
Definition 8.3. By F we denote the smallest space of functions of comoving variables
(τ, y), |y| 6= 0, which satisfies the following conditions:
20
(n)
(i) The functions N, a, aF , n = 1, 2, 3, . . . , belong to F.
(ii) If A ∈ F , then Rk A ∈ F for k = ±1, ±2, . . . .
(iii) If A, B ∈ F, then A × B ∈ F and αA + βB ∈ F for all α, β ∈ R.
(iv) If A, B, C ∈ F , then (AB)C ∈ F.
Furthermore, let Fk denote the linear subspace of F consisting of precisely all the functions from F which have the form
fk (N, a, a0F , . . .)Rk ,
k = 0, ±1, ±2, . . . .
Lemma 8.4. (a) If A ∈ Fk , then ∂τ+ A ∈ Fk .
(b) If A ∈ Fk , B ∈ F0 , then the functions curl A, (B grad)A, and B div A belong to
Fk−1 .
(c) If A ∈ Fk , then R(aN)A ∈ Fk+1 .
(d) If A ∈ Fk , then A is C k−1 with respect to the space variables provided k ≥ 1.
Proof. The proof proceeds by induction. Note that N2 = 1 and use vector calculus. For
example, ∂j Rk = kRk−1 Nj and ∂j Nk = R−1 (δjk − Nj Nk ). Hence
grad Rk = kRk−1 N,
div N = 2R−1 ,
curl N = 0.
Furthermore, use the product rule. For example, div(Rk N) = N grad Rk + Rk div N.
2
Let jext = 0 and ρext = 0. Observing the structure of the Maxwell system (8.1) along
with Lemma 8.4, it is quite natural to assume that
Ek , Bk ∈ Fk for all k .
Substituting the ansatz (8.4) into the Maxwell equations (8.1), a comparison of coefficients
with respect to Rk yields the following recursive system for k = −1, 0, . . . , n:
div Ek = 0 ,
div Bk = 0 ,
curl Bk = −curl ((RaN)Bk−1 ) − ∂τ+ Ek−1 ,
(8.5)
curl Ek = −curl ((RaN)Ek−1 ) + ∂τ+ Bk−1 .
This system can be solved for all k (which are not too large) via computer algebra 3 .
3
We thank Bernd Fiedler (Department of Mathematics, University of Leipzig) who did the calcula-
21
Motivated by the decomposition (3.3) of the electromagnetic field, we make the ansatz
E = E−2 + E−1 + E0 + Ereg ,
(8.6)
B = B−2 + B−1 + B0 + Breg ,
where the regular field is assumed to be at least continuous for all y in a neighbourhood
of y = 0. Again, Ek and Bk are determined by the recursive system (8.5) for k = −1 and
k = 0. Explicitly,
E−2 =
QN
,
4πR2
E−1 = −
E0
=
B−2 = 0,
Q
((aN)N + a),
8πR
B−1 = 0,
3Q
((aN)2 N + 2(aN)a − a2 N), B0
32π
=
(8.7)
Q
N × a0F .
8π
One checks easily by hand that these fields satisfy (8.5) for R 6= 0. Note the rules given in
Lemma 8.4.
Remark 8.5. To get the expressions (3.5) from (8.7), set
Fk := Ek ∧ u − Alt(Bk ) , k = −2, −1, 0 ,
by (2.3) and (8.3). Observe that a0F = ȧ − a2 u (cf. (7.6), (7.7)).
9
Derivation of the Equations of Motion
The energy–momentum vector of the pure field system (no particles!) is given by
Z
P(τ ) := −
T u dV ,
Σ+ (τ )
tions using the computer algebra system ’MATHEMATICA’. He developed a package ’VECTAN’ and
two associated extension packages to determine Ek ,Bk from (8.5).
In particular, solutions for k =
−1, 0, . . . , 6 were calculated. Some details are contained in the description of ’VECTAN’ which is presented at the World Wide Web pages http://www.mathematik.uni-leipzig.de/MI/fiedler/vectan.html and
http://www.wolfram.com/cgi-bin/msitem?0208-594.
22
where u is the normal unit vector of the Fermi hyperplane Σ+ (τ ). In addition, dV denotes
the volume element on Σ+ (τ ) induced by the metric on M4 . By (7.5), this is the Euclidean
metric.
The energy–momentum tensor of the electromagnetic field equals
T := −F 2 +
¢
1¡
tr F 2 I ,
4
(9.1)
where F is regarded as a linear operator. Explicitly, tr F 2 = Fγµ F µγ , and
1
Tαβ = −Fαγ F γβ + Fγµ F µγ δαβ .
4
(9.2)
Proposition 9.1. If E, B denotes the electromagnetic field observed in the comoving
system, then the energy–momentum vector is given by
P(τ ) = H(τ )u(τ ) + P(τ ),
(9.3)
where
Z
H(τ ) :=
Z
¢
1¡
E(τ, y)2 + B(τ, y)2 dy ,
2
P(τ ) :=
R3
(E(τ, y) × B(τ, y)) dy .
(9.4)
R3
From the physical point of view, H (resp. P) represents the total energy (resp. the total
momentum) carried by the field and observed in the comoving reference system Σ+ .
Proof. In the comoving system, we have the representations (7.5) and (8.3) which
imply Fjk = F jk , and Fj0 = −L2 F j0 . Hence, tr F 2 = 2 (E2 − B2 ). By (2.1) and (2.3),
F u = (Eu)u − u2 E = E. Consequently,
F 2 u = F (F u) = E2 u − (Eu)E − Alt(B)E = E2 u + E × B .
Hence
−T u = E2 u + E × B −
¢
¢
1¡ 2
1¡ 2
E − B2 u =
E + B2 u + E × B.
2
2
2
Proposition 9.2. For a sufficiently smooth situation, conservation of the energy–momentum vector P is equivalent to the system
∂τ+ H + Pa = 0,
∂τ+ P + Ha = 0.
23
(9.5)
Proof. Differentiation of (9.3) with respect to τ yields
¡
¢
Ṗ = Ḣ(τ )u + Ha + Ṗ = ∂τ+ H u + Ha + ∂τ+ P + (Pa)u ,
by (7.6). Thus, Ṗ = 0 is equivalent to (9.5).
(9.6)
2
Renormalization. For the “field + particle” system, the integrals (9.4) are divergent
because of the singular E−2 component of the field. Representing the total field as the sum
of the Coulomb “tail” F−2 of the particle and the remaining part F − F−2 , the integrals
(9.4) decompose as the sum of the three terms: the one quadratic in F−2 , the one quadratic
in F − F−2 and, finally, the mixed term. It is precisely the first term which is divergent.
Consequently, the total energy–momentum is a priori not well defined.
In the papers [9] and [10], a detailed analysis of this physical situation was given. Assume
that the point–like particle is merely a mathematical idealization of an extended “soliton–
like” object built from other physical matter fields strongly concentrated within a tiny region
around zero and interacting (possibly in a highly nonlinear way) with the electromagnetic
field. For the extended particle, the term quadratic in F−2 represents correctly the Coulomb
field surrounding the particle only when we are outside the particle. Inside the particle, we
have to regard the total energy–momentum tensor of all the fields–constituents of the particle.
Such an energy–momentum tensor is no longer singular. The integrals (9.4) applied to this
field give us the total energy and the total momentum of the particle at rest: H(τ ) = m0
and P(τ ) = 0. We stress the fact that the quantity m0 represents the total rest mass of
the particle (including already the mass carried by the electromagnetic field surrounding the
particle). On the other hand, the mixed terms vanish when integrated over each sphere (at
least outside the particle) because F−2 contains only the monopole part, i.e. the spherically
symmetric Coulomb singularity, whereas F − F−2 is monopole–free.
We should like to define the theory of point–like particles as an approximation of the
above situation valid in the regime when the internal structure of the extended particle may
be neglected. Hence, the above heuristics leads to the mathematically correct definition of the
renormalized energy–momentum for the “point–particle + field” system. Here, we calculate
the integrals (9.4) only for the component F − F−2 but we add the energy–momentum m0 u
carried by the particle at rest. Therefore, the renormalization consists in replacing the
24
integral containing F−2 with the “renormalized” energy–momentum m0 u. More precisely,
we put ΩR := {y : |y| ≥ R}, and SR := {y : |y| = R} and define
Z
Z
¢
1¡
2
2
HR :=
(E − E−2 ) + B dy , PR := (E × B) dy .
2
ΩR
(9.7)
ΩR
Since E − E2 behaves like O(|y|−1 ), as |y| → 0, the limits
Hren := m0 + lim HR ,
R→0
Pren := lim PR
R→0
exist and define the renormalized four–momentum vector Pren := Hren u + Pren . We may also
put PR := (HR + m0 ) u + PR . Then, Pren = lim PR . It would be quite natural to require
R→0
its conservation
Ṗren = 0.
(9.8)
For mathematical reasons, let us replace condition (9.8) with the following stronger assumption.
(A)
W e postulate that
lim ṖR = 0.
R→0
This condition would imply (9.8) if we could interchange the differentiation with the limit.
The following result is crucial.
Proposition 9.3. From (A) we obtain the equation of motion (3.7).
Remark 9.4. Since F u = E, the relativistically invariant equation (3.7) is equivalent to
m0 a(τ ) = QEreg (τ, 0)
(9.9)
in the comoving system.
Proof. Step 1. By (9.6),
¡
¢
¡
¢
ṖR = ∂τ+ HR + PR a u + ∂τ+ PR + HR a + m0 a .
We will show below that the following are met.
(i) ∂τ+ HR + PR a = Φ(R),
(ii) ∂τ+ PR + HR a + m0 a = Ψ(R),
where
Z
Φ(R) :=
L ((E − E−2 ) × B) N dS ,
SR
25
(9.10)
Q
Ψ(R) := m0 a −
4πR2
Z
Ereg dS + O(R) ,
as R → 0 .
(9.11)
SR
Taking this for granted, let us show that (i) and (ii) imply the statement. Since E − E−2 =
O(R−1 ) we get lim Φ(R) = 0. The continuity of Ereg at y = 0 gives us lim Ψ(R) =
R→0
R→0
m0 a(τ ) − QEreg (τ, 0). From these relations and assumption (A), equation (9.9) follows.
Therefore, it only remains to prove (i) and (ii). To this end, we will use the Maxwell
equation (8.1) in the comoving system along with integration by parts.
Step 2. To simplify the considerations, let us assume that ρext = 0 and jext = 0. The
general case follows by replacing F with F + Fext , where Fext is explicitly given by the
representation formulas (3.12) in the inertial system. Observe that the field Fext is smooth
everywhere.
Ad (i). Using L = 1 + ay, grad L = a , and curl E−2 = 0, we obtain the identity
41 := (E − E−2 ) curl(LB) − B curl(LE)
= −(E × B)a − div (L (E − E−2 ) × B)) .
The Maxwell equations (8.1), with ρext = 0 , jext = 0, imply 41 = (E − E−2 ) ∂τ+ E + B∂τ+ B.
Therefore, by (9.7),
Z
∂τ+ HR
=
Z
41 dy = −PR a +
ΩR
L ((E − E−2 ) × B) N dS .
SR
Ad (ii). Note the identity
42 := curl(LB) × B − E × curl(LE)
¾
½
¢
¢
1 ¡
1¡ 2
2
E + B id − a E2 + B2 .
= div L E ⊗ E + B ⊗ B −
2
2
From the homogeneous Maxwell system associated with (8.1), we deduce 42 = ∂τ+ E × B +
E × ∂τ+ B. Hence, (9.7) gives
Z
+
∂τ PR =
42 dy
ΩR
1
= − a
2
Z
ΩR
¡
2
E +B
2
¢
Z
dy +
½
¾
¢
1¡ 2
2
L (EN)E + (BN)B −
E + B N dS ,
2
SR
26
and
HR
Z
1
=
2
¡
2
2
¢
1
dy +
2
2
¢
2
1
dy +
2
E +B
ΩR
Z
1
=
2
Z
E−2 (E−2 − 2E) dy
ΩR
¡
E +B
ΩR
Z
R(E−2 N) ((E−2 − 2E) N) dS .
SR
To get the last relation, observe that E−2 = −grad(E−2 y) along with div E−2 = div E = 0.
Consequently, ∂τ+ PR + HR a = χ(R), where
Z
R
χ(R) =
(E−2 N) ((E−2 − 2E) N) a dS
2
SR Z ½
¾
¢
1¡ 2
2
−
(1 + R(aN)) (EN)E + (BN)B −
E + B N dS .
2
(9.12)
SR
Use the representations (8.6) and (8.7) of E, B, Ek , and Bk in order to collect all the terms
which are odd with respect to N. In (9.12), the integrals over these terms vanish. Finally,
we obtain
Z n
R
χ(R) = −
(E−2 N)2 a + (E−2 N) (E−1 + Ereg ) + ((E−1 + Ereg ) N) E−2
2
SR
¶
µ
o
1 2
−1
− ((E−1 + Ereg ) E−2 ) N + R(aN) (E−2 N)E−2 − E−2 N + O(R ) dS
2
Z
Q
= −
Ereg dS + O(R) , as R → 0 .
4πR2
SR
This yields the desired expression for Ψ(R) in (9.11).
10
2
Solution of the Initial–Value Problem for the Basic
Equations
In this section we prove Theorem 4.1.
10.1
Outline of the Proof
The basic idea of the existence proof. It is sufficient to solve the equation of motion
for the particle
m0 q̈(τ ) = QFreg (q(τ ))q̇(τ ),
q(0) = 0,
27
q̇(0) = e0 ,
q̈(0) = a0 ,
(10.1)
where Freg = F − Fsing , by (3.3). The point is that we use the decomposition F = F̃ret + F̃
of the electromagnetic field from (6.1). Thus,
Freg = F̃ + F̃ret − Fsing .
(10.2)
Hence the equation of motion (10.1) is equivalent to
m0 q̈(τ ) =
¢
Q2 ¡...
q(τ ) − q̈(τ )2 q̇(τ ) + QF̃ (q(τ ))q̇(τ ),
6π
q(0) = 0 , q̇(0) = e0 , q̈(0) = a0 , (10.3)
(cf. (6.2)). Here F̃ satisfies the Maxwell system
Div F̃ = jext ,
Div(∗F̃ ) = 0 .
(10.4)
Let us consider eigentimes τ ≥ 0. The following facts are crucial for our approach.
(i) The fields Fsing and F̃ret depend upon the unknown trajectory in an explicitly known
manner (cf. (3.4), (3.5) for the first one and formula (4-98) in [15] for the latter).
(ii) By construction, F̃ret satisfies the homogeneous Maxwell system (5.1) outside the
trajectory.
(iii) The field Freg is smooth outside the trajectory and continuous across the trajectory.
The same holds true for the difference F̃ret − Fsing (cf. (5.2)). Hence the Maxwell equations
(10.4) are satisfied on a neighbourhood of the trajectory in the sense of generalized functions.
(iv) The field F̃ is uniquely determined by its initial value
³
´
0
0
0
F̃ 0 := Freg
+ Fsing
− F̃ret
on a space–like hyperplane at τ = 0. Here, F̃ 0 depends only upon the initial data F 0 ,
0
0
i. e. upon the acceleration a0 and its derivative ȧ0 contained in Fsing
, and also upon Freg
.
Explicit representation formulas for F̃ depending on F̃ 0 will be given in (10.7) below.
...
Remark 10.1. Note that the initial value ȧ0 (i.e., the value q(0)) is compatible with
equation (10.3). In fact, this equation is equivalent to (10.1), and we have assumed the latter
to be fulfilled by our initial data. We remind the reader that the first term on the right hand
side of (10.3) comes from the value of F̃ret − Fsing at zero. This value has been added here
and, at the same time, subtracted from Freg , converting it into F̃ , by (10.2).
To simplify notation, we assume that the external sources vanish, i.e., let jext = 0. By
superposition, the general case can be easily reduced to this special one. For this purpose, we
28
replace F with F + Fext , where Fext is a smooth field satisfying the inhomogeneous Maxwell
system (3.2) (cf. Remark 3.4), and observe that both the Maxwell system and the equation
of motion for the particle are linear expressions in Freg .
The uniqueness proof. Suppose that we are given two solutions of the initial–value
problem for the particle–field system having the same initial values. Furthermore, suppose
that the two solutions possess the smoothness properties as indicated in Theorem 4.1. Then,
the two trajectories satisfy the same equation of motion (10.3). The fixed–point argument
below will show that the solution of (10.3) is unique for small times t > 0. Thus, the
two trajectories coincide. Hence, by (i) and (iv) above, the two electromagnetic fields are
identical.
10.2
Mathematical Preparations
For the proof, let us use a fixed inertial system. In such a system, the equation of motion
(10.3) for the particle is transformed to the following equation:
¡
¢
¢
Q2 d ¡
d m0 v(t)
q
=
(1 − v(t)2 )−2 (1 − v(t)2 ) id + v(t) ⊗ v(t) v0 (t)
dt
6π dt
1 − v(t)2
¡
¢
Q2 (v(t)v0 (t))2 + (1 − v(t)2 )(v0 (t))2
−
v(t)
6π(1 − v(t)2 )3
(10.5)
+ QẼ(t, q(t)) + Qv(t) × B̃(t, q(t))
which we have to solve along with the initial conditions
q(0) = 0,
q0 (0) = 0,
q00 (0) = a0
(10.6)
for the unknown trajectory q = q(t), where v(t) := q0 (t). Furthermore, according to (2.2),
we set F̃ = Ẽ ∧ e0 − Alt(B̃). The fields Ẽ, B̃ are solutions of homogeneous Maxwell equations
and, therefore, may be expressed in terms of initial data as follows (cf., e.g. [18], §§15, 46):
Z ³
´
1
0
0
0
Ẽ(t, x) =
t(z grad)Ẽ + t curl B̃ + Ẽ (x + tz) dSz ,
4π
S
Z ³
´
1
B̃(t, x) =
t(z grad)B̃0 − t curl Ẽ0 + B̃0 (x + tz) dSz ,
4π
S
29
(10.7)
where S := {z : |z| = 1}. The initial values Ẽ0 , B̃0 for these fields at time t = 0 are known:
³
´
Ẽ0 = E0reg + E0sing − Ẽ0ret ,
³
´
B̃0 = B0reg + B0sing − B̃0ret ,
(10.8)
Note that the fields E0reg , B0reg are given. Moreover, E0sing , B0sing , Ẽ0ret , and B̃0ret depend on
the initial acceleration a0 of the particle and its derivative ȧ0 in an explicitly known manner
(cf. (3.4), (3.5) and formula (4-98) in [15]).
Let us first study some regularity properties of the initial values Ẽ0 , B̃0 of the field Ẽ, B̃
at time t = 0.
Proposition 10.2. Suppose that the initial values E0reg , B0reg of the electromagnetic field
are C m , m ≥ 2. Then, Ẽ0 , B̃0 are continuous in a neighbourhood of the origin x = 0.
Furthermore, they are C m outside the origin. Moreover,
∂ Ẽ0
= O(1) ,
∂xi
∂ 2 Ẽ0
= O(r−1 ) ,
∂xi ∂xj
∂ B̃0
= O(1) ,
∂xi
as r → 0 ,
∂ 2 B̃0
= O(r−1 ) ,
∂xi ∂xj
(10.9)
as r → 0 ,
(10.10)
where r := |x|.
Proof. Continuity and smoothness of Ẽ0 , B̃0 follow from Lemma 5.1 (b). To prove the
0
desired estimates for the derivatives, we use the explict formula for the retarded field F̃ret
(x)
in terms of the (prolonged backwards) trajectory (e.g. formula (4-98) in [15]).
2
Remark 10.3. The estimates (10.9) show that the integral expressions (10.7) make sense
for all (t, x) in a neighbourhood V of 0 ∈ R3 × [0, t1 ]. As it is known from the theory of linear
hyperbolic systems, jumps in the data at x = 0 are transported along the bicharacteristics
emanating from the origin, i.e., along the rays t = |x|. These rays form the forward cone
associated with the Maxwell system at x = 0. At those points of V that are away from this
cone, the fields Ẽ, B̃ are C m−1 . This can be immediately seen from formulas (10.7).
10.3
The Fixed–Point Argument
It is our aim to transform equation of motion (10.5) into an operator equation for the
acceleration
w(t) := q00 (t)
30
of the particle. To this end, we express the trajectory q = q(t) and its velocity v = q0 (t)
through the acceleration by means of
Zt
v(t) =
Zt
w(s) ds ,
q(t) =
0
v(s) ds .
(10.11)
0
Recall the initial conditions (10.6). To eliminate the third derivative q000 , we integrate the
equation of motion (10.5) on the time interval [0, t]. We obtain an equivalent operator
equation
w = G[w] .
Explictly,

(10.12)

Zt
(G3 [w] − G4 [w]) (s) ds
G[w](t) := G1 [w](t) G2 [w](t) +
(10.13)
0
for t ∈ [0, t1 ], where
¡
¢−1
G1 [w] := (1 − v2 )2 (1 − v2 ) id + v ⊗ v
,
6πm0
v
√
G2 [w] :=
+ a0 ,
2
Q2
1−v
¡
¢
2
G3 [w] := (vw) + (1 − v2 )w2 (1 − v2 )−3 v ,
´
6πm0 ³
G4 [w] :=
Ẽ + v × B̃ .
Q
(10.14)
Here, Ẽ and B̃ stand for Ẽ(t, q(t)) and B̃(t, q(t)), respectively. The corresponding fields are
given in (10.7). Furthermore, q and v are related to w by (10.11). We study the operator
equation (10.12) on the Banach space X := C[0, t1 ], where t1 > 0 will be chosen sufficiently
small. This space consists of all continuous vector functions w = w(t) on the time interval
[0, t1 ]. The norm on X is given by
kwk := max |w(t)| .
t∈[0,t1 ]
Consider now the ball
©
ª
B := w ∈ X : kw − a0 k ≤ R ,
(10.15)
where the radius R > 0 is fixed.
Proposition 10.4. The operator equation w = G[w] , w ∈ B has a unique solution.
Proof. This follows from the Banach fixed point–theorem (cf., e.g. [20], Vol. 1, Theorem
1.A) by using Lemma 10.6 ahead.
2
31
Corollary 10.5. The trajectory q = q(t), related to the solution w ∈ B, is C 2 on [0, t1 ].
Proof. For any given w ∈ B, set W := G[w]. By the definitions (10.13) and (10.14),
the function W = W(t)) is C 1 on [0, t1 ]. Similarly, if w is C 1 on [0, t1 ], then W is C 2 on
this interval. Finally, observe that w = G[w].
2
The higher regularity of the trajectory stated in Corollary 4.2 follows by a repeated
application of the boot–strap argument of the preceding proof.
Lemma 10.6. Let the operator G be defined by (10.13),(10.14). Then, for sufficiently
small t1 > 0, the following are met.
(a) G maps the ball B into itself.
(b) G is a contraction on B, i.e.,
kG[w] − G[ŵ]k ≤ Lkw − ŵk
for all w, ŵ ∈ B and a fixed real number L with 0 ≤ L < 1.
Proof. Let w ∈ B. From (10.11), we get the estimates
|v(t)| ≤ t(R + ka0 k) ,
|q(t)| ≤
t2
(R + ka0 k)
2
(10.16)
for t ∈ [0, t1 ]. Thus, the expression G1 [w](t) in (10.14) is well–defined for all such times t
provided t1 is sufficiently small.
Ad (a). Let w ∈ B. Set W := G[w]. By (10.13) and (10.14), the function W = W(t) is
continuous on [0, t1 ], i.e., W ∈ X . To prove that W ∈ B, use the estimate
¡
¢
|G[w](t) − a0 | ≤ C tC + tka0 k ≤ R
R
. Here and in what follows, the same symbol C denotes
C (C + ka0 k)
different positive constants. In this connection, observe that
µ
µ
¶ ¶
1
0
2 0
G1 [w]G2 [w] − a =
(1 − v )a − id +
v ⊗ v a0 O(1)
2
1−v
0
2
= a O(t ) , as t → 0 ,
for all t, 0 ≤ t ≤
by (10.16).
Ad (b). First let n = 1, 2, 3. It is easy to see that the operators Gn are Fréchet–
differentiable on B. Moreover, it follows from (10.11) and (10.14) that
|Gn [w](t) − Gn [ŵ](t)| ≤ C|v(t) − v̂(t)| ≤ tCkw − ŵk ,
32
(10.17)
for all w, ŵ ∈ B and all t ∈ [0, t1 ].
The argument for G4 is more involved. Let us consider the difference Ẽ(t, q(t))−Ẽ(t, q̂(t)).
By the representation formulas (10.7), it is sufficient to estimate the quantity
¯
¯
¯ ∂ Ẽ0
¯
0
∂ Ẽ
¯
¯
D := ¯
(q(t) + tz) −
(q̂(t) + tz)¯ ,
¯ ∂xi
¯
∂xi
where |z| = 1. Let t > 0 be fixed. For some ϑ ∈ (0, 1), the mean value theorem along with
(10.10) implies
¯−1
¯
¯
¯
D ≤ C ¯tz + ϑq(t) + (1 − ϑ)q̂(t)¯ |q(t) − q̂(t)|
¯−1 t2
¯
¯
¯
≤ C ¯|tz| − ϑ|q(t)| − (1 − ϑ)|q̂(t)|¯
kw − ŵk
2
≤ tCkw − ŵk .
Here, the relations (10.11) and (10.16) have been used. Observe that for estimating the
denominator, the time t has to be chosen small enough to ensure, say, |q(t)|, |q̂(t)| < 21 t. By
the same argument, we can derive similar inequalities for the derivatives of B̃0 and for Ẽ0
itself (use (10.9) instead of (10.10)). Hence
¯
¯
¯
¯
¯Ẽ(t, q(t)) − Ẽ(t, q̂(t))¯ ≤ tCkw − ŵk .
An analog estimate holds if Ẽ is replaced with B̃. Consequently, the definition of G4 in
(10.14) implies the validity of the inequality (10.17) for n = 4, too. Summarizing we get
|G[w](t) − G[ŵ](t)| ≤ tCkw − ŵk ≤ Lkw − ŵk
for all t ∈ [0, t1 ], where L := t1 C. Choose t1 < C −1 to obtain L < 1.
2
Remark 10.7 (mathematical generalizations). A refined analysis gives more detailed
information about the regularity of the fields under consideration. Moreover, similar results
can be derived if the function spaces C m are replaced with appropriate Sobolev spaces (cf.,
e.g. [18], §§5,40 and [20], Vol. 2A, Section 21.2).
Acknowledgement. This research was supported in part by the Deutsche Forschungsgemeinschaft (DFG) via Forschergruppe ”Nichtlineare Funktionalanalysis und Mathematische
33
Methoden der Kontinuumsmechanik”. Moreover, one of the authors (J.K.) is grateful to
the Polish National Committee for Scientific Research (KBN), Warsaw, for financial support
(Grant 2P 302 189 07).
References
[1] Born, M.: Die Thermodynamik des starren Elektrons in der Kinematik des Relativitätsprinzips. Ann. Phys. (Leipzig) 30, 1-56 (1909)
[2] Born, M., Infeld, L.: Foundation of the new field theory. Proc. Royal Soc. A 144,
425-451 (1934)
[3] Boulware, D.G.: Radiation from a uniformly accelerated charge. Ann. Phys. (N.Y.)
124, 169-188 (1980)
[4] Chruściński, D., Kijowski, J.: Variational principle for electrodynamics of moving particles. Gen. Relat. Grav. Journal 27, 267-311 (1995)
[5] Dirac, P.: Classical theory of radiating electrons. Proc. Royal Soc. A 167, 148-168
(1938)
[6] Feenberg, E.: On the Born–Infeld field theory of the electron. Phys. Rev. 47, 148-157
(1935)
[7] Hale, J., Stokes, A.: Some physical solutions of Dirac–type equations. J. Math. Physics
3, 70-74 (1961)
[8] Jackson, J.: Classical Electrodynamics. New York: Wiley 1965
[9] Kijowski, J.: Electrodynamics of moving bodies. Gen. Relat. Grav. Journal 26, 167-201
(1994)
[10] Kijowski, J.: On electrodynamical self–interaction. Acta Physica Polonica A 85, 771-787
(1994)
34
[11] Landau, L., Lifschitz, E.: The Classical Theory of Fields. New York: Pergamon Press
1962
[12] Misner, C., Thorne, K., Wheeler, J.: Gravitation. San Francisco: Freeman 1972
[13] Müller–Hoissen, F.: Higher derivative versus second order field equations. Ann. Phys.
(Leipzig) 48, 543-547 (1991)
[14] Parrott, S.: Relativistic Electrodynamics and Differential Geometry. New York: Springer
1987
[15] Rohrlich, F.: Classical Charged Particles. Reading: Addison–Wesley 1965
[16] Ruei, K.: Classical Theory of Particles and Fields. Vols. 1-3. Tapei: University Press
1972
[17] Schmutzer, E.: Grundlagen der Theoretischen Physik. Bd. 1,2. Berlin: Deutscher Verlag
der Wissenschaften 1989
[18] Triebel, H.: Höhere Analysis. Berlin: Deutscher Verlag der Wissenschaften 1972
[19] Wheeler, J., Feynman,R.: Classical electrodynamics in terms of direct inter–particle
action. Rev. Mod. Phys. 21, 425-434 (1949)
[20] Zeidler, E.: Nonlinear Functional Analysis and Its Applications. Vols. 1-4. New York:
Springer 1987
[21] Zeidler, E.: Applied Functional Analysis: Applications to Mathematical Physics. Second
edition. New York: Springer 1997
[22] Zeidler, E.: Applied Functional Analysis: Main Principles and Their Applications. New
York: Springer 1995
35