Behaviour of Electromagnetic Waves in Different Media and Structures Part 13 doc

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17
The Influence of Vacuum Electromagnetic
Fluctuations on the motion of Charged Particles
Guozhong Wang
Wenzhou University
China
1. Introduction
Lorentz-Dirac equation (LDE) is the widely accepted classical equation to describe the
motion of a scalar point charge acted by external electromagnetic fields and its own

radiating fields. Because LDE is highly nonlinear, the ubiquitous electromagnetic fluctuating
fields of the vacuum would produce a nonzero contribution to the motion of charged
particles, which provides a promising way to understand the century-old problems and
puzzles associated with uniformly accelerating motion of a point charge. The vacuum
fluctuations also have an intimate relationship with the Unruh effect. In this chapter we will
restrict our main stamina to investigate the influence of the vacuum electromagnetic
fluctuating fields on the motion of a point charge. Because the discussion upon problems,
such as the Unruh effect etc., would greatly digress from the motif of this book, we just
disperse brief remarks on these problems at suitable places.
2. New reduction of order form of LDE
In 1938, Dirac for the first time systematically deduced the relativistic equation of motion for
a radiating point charge in his classical paper [Dirac, 1938]. Being a singular third order
differential equation, the controversy about the validity of LDE has never ceased due to its
intrinsic pathological characteristics, such as violation of causality, nonphysical runaway
solutions and anti-damping effect etc. [Wang et al., 2010]. All these difficulties of LDE can be
traced to the fact that its order reduces from three to two as the Schott term is neglected.
However, LDE derived by using the conservation laws of momentum and energy is quite
elegant in mathematics and is of Lorentz invariance. Furthermore, many different methods
used to derive the equation of motion for a radiating point charge lead to the same equation,
and all pathological characteristics of LDE would disappear in its reduction of order form.
Plass invented the backward integration method for scattering problems, and H. Kawaguchi
et al. constructed a precise numerical integrator of LDE using Lorentz group Lie algebra
property [Plass, 1961; Kawaguchi et al., 1997]. These methods are enough to numerically
study the practical problems. On the other hand, Landau and Lifshitz obtained the
reduction of order form of LDE [Landau & Lifshitz, 1962], which fully meets the
requirements for dynamical equation of motion and is even recommended to substitute for
the LDE. But one should keep in mind that Landau and Lifshitz equation (LLE) gaining the

Behaviour of Electromagnetic Waves in Different Media and Structures

354
advantages over LDE is at the price of losing the orthogonality of four-velocity and four-
acceleration of point charges. This complexion means that LDE is still the most qualified
equation of motion for a radiating point charge. To be clear, we make the assumption that
LDE is the exact equation of motion for a radiating point charge.
2.1 Description of reduction of order form of LDE
For a point charge of mass
m
and charge
e
, LDE reads

2
0
()=++
    
e
xFxxxx
m
μμ
ν
μμ
ν
τ
, (1)
where
()x
μ
τ
is the spacetime coordinates of the charge at proper time

τ
,
2
0
2/3= em
τ
the
characteristic time of radiation reaction which approximately equals to the time for a light to
transverse across the classical radius of a massive charge. The upper dots denote the
derivative with respect to the proper time, Greek indices
μ
,
ν
etc. run over from 0 to 3 .
Repeated indices are summed tacitly, unless otherwise indicated. The diagonal metric of
Minkowski spacetime is
(1,1,1,1)−−−. For simplicity, we work in relativistic units, so that
the speed of light is equal to unity. The second term of the right side of Eq. (1) is referred to
as the radiation reaction force, and

x
μ
is the so called Schott term.
We have assumed in Eq. (1) that charged particles interact only with electromagnetic
fields
F
μ
ν
which has the matrix expression:
123

132
23 1
321
0
0
0
0
−−−




−


−




−


EEE
EBB
EB B
EBB
.
The Lorentz force is

0
(, ) ( , )=⋅ +×= ⋅ +×



   




dx dx
eF x e E x x E x B e E E B
dt dt
μν
ν
γ
, (2)
where
21/2
(1 )
−
=−

x
γ
is the relativistic factor.
By replacing the acceleration in the radiation reaction force with that produced only by
external force, Landau and Lifshitz obtained the reduction of order form of LDE

2

0
(,,) [ ]=++
  
df
x
f
xx
f
x
d
μ
μ
μμ
ττ
τ
, (3)
where ( , , ) /
=

f
xx eF x m
μμν
ν
τ
. Eq. (3) is nonsingular and gets rid of most pathological
characteristics of LDE, but the applicable scope is also slightly reduced. LLE is quite
convenient to numerically study macroscopic motions of a point charge.
If one does not care about the complexity, there exists another more accurate reduction of
order form of LDE than LLE, which also implies a corresponding reduction of order series
form of LDE. In this section, we will present this reduction of order form of LDE. To do so,

the most important step is using the acceleration produced only by external forces to
approximate the Schott term, namely

The Influence of Vacuum Electromagnetic Fluctuations on the motion of Charged Particles

355

22
00
(,,) [ ] [ ]
∂
=++≈++
∂


      

f
x
f
xx x xx F x xx
x
μ
μ
μμμμνμ
ν
ττ τ
, (4)
where
0

()
∂∂
=+ +
∂∂

ff
Ff x
x
μ
μ
μμ ν
ν
τ
τ

is the analytical function of proper time, spacetime coordinates and four-velocity. Letting
2
=

kx，Eq. (4) becomes
00
/−∂∂=+
   
xx
f
xF kx
μ
ν
μ
ν

μμ
ττ
,
which can be put into the matrix form
0
=+
 
Ax F kx
τ
, with
0123
(, , , )=
T
FFFFF
and
0123
(, , , )=

T
xxxxx
. The symbol “
T “ denotes transpose operation of a matrix. The explicit
expression of matrix A is
00 01 02 03
0000
10 11 12 13
0000
20 21 22 23
0000
30 31 32 33

0000
1/ / / /
/1/ / /
//1//
///1/


−∂ ∂ −∂ ∂ −∂ ∂ −∂ ∂


−∂∂ −∂∂−∂∂ −∂∂




−∂∂ −∂∂ −∂∂−∂∂


−∂∂ −∂∂−∂∂ −∂∂








f
xfxfxfx
f

xfxfxfx
f
xfxfxfx
f
xfxfxfx
ττττ
ττττ
ττττ
ττττ
,
each element of matrix
A
is the analytical function of
τ
,
x
and

x . We define generalized
four-velocity and four-acceleration vectors as
0123 0123
(,,,), (,,,)
ΔΔΔΔ ΔΔΔΔ
==
ΔΔΔΔ ΔΔΔΔ


xxxx FFFF
XD
μμ

,
where
Δ is the determinant of matrix A , and Δ
F
μ
and Δ

x
μ
are determinants of matrices
obtained by replacing the -th
μ
column of A with column matrices F and

x respectively. So
four-acceleration can be expressed as

0
=+


xD kX
μμ μ
τ
. (5)
Because the square of the four-acceleration k is involved, Eq. (5) is still not the explicit
expression of acceleration. However, k can be expressed as

222 2
0000

()()2=+ + = + +
 
kD kXD kX Xk kDXD
μμ μ
μμ μ
ττττ
, (6)
which is a quadratic algebra equation of k . The physical solution of k is

2
2222
000
2
(1 2 ) (1 2 ) 4
=
−+− −

D
k
DX DX XD
μμ
μμ
τττ
. (7)
Thus we obtained the expression of acceleration, and the result is

2
0
2222
000

2
(1 2 ) (1 2 ) 4
=+
−+− −



DX
xD
DX DX XD
μ
μμ
μμ
μμ
τ
τττ
, (8)

Behaviour of Electromagnetic Waves in Different Media and Structures

356
which is now the explicit function of proper time
τ
, spacetime coordinates x and velocity

x .
As a corollary, we can discuss the applicable scope of LDE from the existing condition of the
solution of
k , namely, the quantity under the square root appeared in Eq. (7) must be
nonnegative. It is often taken for granted that LDE would be invalid at the scale of the

Compton wavelength of the charge.
Following the above procedure, we can construct an iterative reduction of order form of
LDE, which is a more accurate approximation to the original LDE. As the first step, we
approximately expressed LDE as
2
00
(,,) [ ]=++≈+
     
x
f
xx x xx G kx
μμ μ μ μ μ
ττ τ
,
where
0
()
∂∂ ∂
=+ + +
∂∂ ∂
  


known
xx x
Gf x x
xx
μμ μ
μμ ν ν
νν

τ
τ

is the function of
τ
, x and

x owing to the four-acceleration

known
x
ν
being taken as that given by
Eq. (8). From its definition, the square of four-acceleration
k can be worked out
2
222
000
2
(12)(12)4
=
−⋅+−⋅−

G
k
Gx Gx G
τττ
,
and the four-acceleration is
2

0
222
000
2
(1 2 ) (1 2 ) 4
=+
−⋅+−⋅−



Gx
xG
Gx Gx G
μ
μμ
τ
τττ
.
By repeating this procedure, we can obtain the n-th iterative expression of four-acceleration:

2
0
222
000
2
(1 2 ) (1 2 ) 4
=+
−⋅+−⋅−




n
nn
nnn
Gx
xG
Gx Gx G
μ
μμ
τ
τττ
, (9)
where
1111
00 1
()
−−−−
−
∂∂ ∂
=+ =+ + +
∂∂ ∂
   


nnnn
n n
dx x x x
Gffxx
dxx
μμμμ

μμ μ ν ν
νν
ττ
ττ
.
We emphasize again that
0

x
μ
is taken as that given by Eq. (8). Hereto we have obtained the
iterative self-contained reduction of order form of LDE.
As an example, we apply the new reduction of order iterative form of LDE to a special case,
a point charge undergoing one-dimensional uniformly accelerating motion along
1
x
direction acted by a constant electric field
E . Assuming that the ratio of charge e to mass m is
one, the equations of motion are
01 020 1 10
00
10 121 0 01
00
[] []
[] []
=+ + ≈+ +
=+ + ≈+ +
       
       
xEx xxx Ex Exkx

x Ex x x x Ex Ex kx
ττ
ττ
.

The Influence of Vacuum Electromagnetic Fluctuations on the motion of Charged Particles

357
It is easy to obtain the expressions of
0

x and
1

x , they are
120 0 1
000
0
2
0
021 1 0
000
1
2
0
()
,
1( )
()
1( )

+++
=
−
+++
=
−




Ex E x k x Ex
x
E
Ex E x k x Ex
x
E
τττ
τ
τττ
τ
.
From its definition
20212
() ()== −
  
kx x x ，we obtain
2
=−kE
. The expression of four-
acceleration is extremely simple, namely

0110
,==
   
xExxEx. It is obvious that further
iterative procedures will not bring any changes, which completely coincides with LDE for
this motion. It is astonishing that a point charge undergoing one-dimensional uniformly
accelerating motion emitting energy and momentum does not suffer radiation reaction force
at all, which induces a series of puzzling problems lasted for over one hundred years since
the radiation fields of this motion had been calculated [Born, 1909, as cited in Fulton &
Rohrlich, 1960; Lyle, 2008].
We would like to make a brief remark on Eq. (8). Our approximate procedure contains all
effects of the second term of radiation reaction force, so our result is more accurate than
LLD. This conclusion is also embodied by its Taylor series form on
0
τ
which includes infinite
terms, while LLE only includes the linear term of
0
τ
.
It is easy for one to utilize the method of Landau and Lifshitz to construct a reduction of
order iterative form of LDE [Aguirregabiria, 1997]. To compare two different reduction of
order forms of LDE is the main content of the next subsection.
2.2 Reduction of order form of LDE up to
2
0
τ
term
We know that the quantity
0

τ
characterizing the radiation reaction effect is an extremely
small time scale
24
(10 )
−
s , so every piece involved in Eq. (8) could be expanded as power
series of the parameter
0
τ
. We are just interested in the first three terms of this series form of
LDE, which is accurate enough to study practical problems and making the comparison
between two series forms of LDE obtained respectively by Landau and Lifshitz’s method
and ours meaningful. To get this series form up to
2
0
τ
term, we first expand matrix
A to
2
0
τ
term, and the result is

2
00
1( )
∂∂∂∂∂
Δ≈ − + − +
∂∂∂∂∂



fffff
xxxxx
μμνμν
μμννμ
ττ
. (10)
For the calculation of the four-vector D
μ
，it is adequate to calculate its zero-th component
and retain the result to
2
0
τ
term
00
00 2
00
∂∂∂
≈+ + +
∂∂∂


fff
DF F F
xxx
ν
μμ
μμν

ττ
.
The space component expressions of four-vector D
μ
can be obtained from the Lorentz
covariance, and the result is

μ
μβμ
μαα
ααβ
ττ
∂∂∂
≈+ + +
∂∂∂


2
00
fff
DF F F
xxx
. (11)

Behaviour of Electromagnetic Waves in Different Media and Structures

358
Due to the same footing as
μ
D , the generalized four-velocity vector

μ

X in Eq. (8) can be
immediately written out by changing
μ
F in Eq. (11) to
μ

x , namely

μβμ
μμ α α
ααβ
ττ
∂∂∂
=+ + +
∂∂∂

 


2
00
fff
Xx x x
xxx
. (12)
Then we need to calculate the square of four-acceleration
k . Eqs. (8) and (9) show that it is
enough for the expansion of

k to retain the
τ
0
term,

2
2222
000
2
2
0
0
22
0
2
(1 2 ) (1 2 ) 4
(1 2 )
(1 2 )
2[ ( )]
=
−+− −
≈≈+
−
∂∂∂
=+ + + +
∂∂ ∂






D
k
DX DX XD
D
DDX
DX
fff
fffxfxf
xx
μμ
μμ
μ
μ
μ
μ
μμ μ
μνν
μμ
νν
τττ
τ
τ
τ
τ
. (13)
So we obtian the four-acceleration accurate to
τ
0
term is

μμ μ
μμ μμ
νν
μ
νν
μμ μ
μννμ
νν
ττ τ
τ
τ
τ
∂∂∂
=+ =+ + + +
∂∂ ∂
∂∂∂
=+ + + + +
∂∂ ∂

  




2
00 0
2
0
()

()
fff
xD kX
f
x
ff
x
xx
fff
fxffx
xx
, (14)
which is the same as LLE.
We need to calculate the result of once iteration to obtain the
2
0
τ
term of the series form of the
acceleration. All involved calculation is straightforward but cumbersome. Retaining to
the
τ
2
0
term, the four-vector
μ
G is

μμμ
μμ ν ν
νν

μμ μμ μμ
μννν
νν
μμ μ μ
νμμ
μνν ννν
νν νν
τ
τ
ττ τ
ττ
τ
ττ
ττ
∂∂∂
=+ + +
∂∂ ∂
∂+ ∂+ ∂+
=+ + + +
∂∂ ∂
∂∂∂ ∂
∂∂ ∂
=+ + + + + + +
∂∂ ∂ ∂∂ ∂ ∂
  


  



  


0
01 01 01
001
2
11 1
001
()
()() ()
[()]
[][
xxx
Gf x x
xx
fx fx fx
fxfx
xx
f
ff f
xx x
fxf xfx
xx xxx
ν
]
. (15)
We then calculate the square of four-acceleration k and the result is

μμμ

μνν
νν
μμ μ
νν
μμμ
νν
ττ τ
τ
τ
τ
∂∂∂
=+⋅=+ + + +⋅
∂∂∂
∂∂ ∂
=+ + + + ⋅
∂∂ ∂
 



2 2
00 0
22
0
(1 2 ) ( [ ]) (1 2 )
2[ ]
fff
kG Gx f x f Gx
xx
ff f

ffxfffffx
xx
. (16)
At last, we arrive at the targeted expression of

x
μ
, which can be expressed as

2
000
01 2
=+ =+ +
    
xG kxx x x
μ
μμμμμ
τττ
, (17)

The Influence of Vacuum Electromagnetic Fluctuations on the motion of Charged Particles

359
and the quantities
0

x
μ
,
1


x
μ
and
2

x
μ
are
0
=

xf
μμ


2
1
1
11
1
2
2
2( )
∂∂∂
=+ + +
∂∂ ∂
∂
∂∂ ∂
=+ + +

∂∂ ∂ ∂
∂∂ ∂
++ + +⋅
∂∂ ∂
  

  
  



fff
xxffx
xx
f
xx x
xxfx
xxx
ff f
f
xff fffxx
xx
μμ μ
μνν μ
νν
μ
νμμ
μννν
ννν
αα α

ν
νμ
ααα
νν
τ
τ
τ
, (18)
which is quite forbidding on the first face compared with the original LDE!
As stated before, by repeating the method of Landau and Lifshitz to reduce the order of
LDE, one can also construct the series form of order reduced approximation of LDE, which
is quite simple and the result is

0
2
1
01
()
−
−
=
=+ +


  
i
i
i
xf
dx

xf xx
d
μμ
μ
μ
μμ
τ
τ
. (19)
One needs to iterate two times to get the series expression up to
2
0
τ
term using Eq. (19), and
the result is

2
22 2
00
[][()]=+ + + + +
   
L
df df df
d
x f fx fx x
dddd
μμ
μ
μμ μμ
ττ

ττττ
. (20)
We just point out that Eqs. (17) and (20) obtained through two different ways are completely
consistent with each other, which shows that the reduction of order form of LDE has unique
expression so long as the external force is orthogonal to four-velocity and depends only on
proper time, spacetime coordinate and four-velocity. This fact also supports that LDE is the
correct equation describing the motion of charged particles.
The general reduction of order series form of LDE can be expressed as

0
0
=
=+=+

  
N
n
n
N
n
xxRxR
μ
μμμμ
τ
, (21)
where
R
μ
is of the order
1

0
+N
τ
. Apart from the first term
0

x
μ
, all other single term is not
orthogonal to

x
μ
as the original LDE does, which can be seen by the calculation

22 22
2222
()
()
() 2
−
+ = += += +−
=− + − =− + − − = − ⋅ −
   
         

       
NN
NN NN
N

dx dx d x R dx dR
xxxx xx xx xx
ddddd
dx dR dR dR
xxx xxRx RxRx
dd d d
μμμμμμ
μ
μμ μ μ μ
μμμ
μ
μ
μμμ
τττττ
ττ τ τ
, (22)
which is of order
1
0
+N
τ
. To guarantee the orthogonal property between four-velocity and
four-acceleration, the reduction of order series form of LDE must contain infinite terms. The

Behaviour of Electromagnetic Waves in Different Media and Structures

360
present method reducing the order of LDE contains infinite terms of
0
τ

at each iterative
process indicating that it is more accurate than that of Landau and Lifshitz.
In subsection 1, we have known that the uniformly accelerating motion of a point charge is
quite special. According to the Maxwell’s electromagnetic theory, the accelerated charge
would for certain radiate electromagnetic radiation which would dissipate the charge’s
energy and momentum. W. Pauli observed that at 0=t when the charge is instantaneously
at rest [Pauli, 1920, as cited in Fulton & Rohrlich, 1960], the magnetic field of its radiating
field is zero everywhere in the corresponding inertial frame shown by Born’s original
solution which means that the Poynting vector is zero everywhere in the rest frame of the
charge at that instant of time and came to the conclusion that the uniformly accelerated
charge does not radiate at all. Whether or not a charge undergoing uniformly accelerating
motion emits electromagnetic radiation is still an open question. Nowadays, most authors of
this area think that it does emit radiation. But one is faced another difficult question, namely
what physical processes taking place near the neighborhood of the charge are able to give
precise zero radiation reaction force. These problems have been extensively studied for a
long time, but the situation still remains in a controversial status [Ginzburg, 1970; Boulware,
1980].
If one measures the macroscopic external field at the vicinity of the charge performing one-
dimensional motion, what conclusions would he/she obtain? Just as one can not distinguish
the gravitation field from a uniform acceleration by local experiments done in a tiny box,
which is called Einstein equivalence principle, any macroscopic external fields felt by the
charge are almost constant and the radiation reaction force would vanish according to LDE
for charge’s one-dimensional macroscopic motions! What is the mechanism of radiation
reaction? There are various points of view for charges’ radiation reaction. Lorentz regarded
charges as rigid spheres of finite size, and the radiation reaction force comes from the
interactions of the retarded radiation fields of all parts of the charge; Dirac regarded
electrons as point charges and regarded the radiation reaction field as half the difference of
its retarded radiation field minus its advanced field; Teitelboim obtained the radiation
reaction force just using retarded radiating field of accelerated charges [Teitelboim , 1970;
Teitelboim & Villarroel, 1980]; while Feynman and Wheeler thought that the radiation

reaction of accelerated charges comes from the advanced radiating fields of all other charges
in the Universe coherently superimposed at the location of the radiating charge [Wheeler &
Feynman, 1945; 1949]. The common point of these different viewpoints is that the radiation
reaction is represented by the variation of external force field with time, which gives zero
radiation reaction for one-dimensional uniformly accelerating motion of charges showing
that the general mechanism of radiation reaction is not complete. It has been proved that the
electron of a hydrogen atom would never collapse using the nonrelativistic version of LDE
which is quite contrary to the conventional idea of classical electrodynamics that the atoms
would fall into the origin within rather a short time interval, which is called the theorem
[Eliezer, 1947; Carati,2001]. Cole and Zou studied the stability of hydrogen atoms using LLE
and obtained similar results with that of quantum mechanics [Cole & Zou, 2003].
It seems that the problems associated with the radiation reaction have little possibility to be
resolved without introducing new factors. According to quantum field theory, the vacuum
is not empty but full of all kinds of fluctuating fields. To investigate how the
electromagnetic fluctuating fields of the vacuum influence the motion of a charge is the
main content of this chapter.

The Influence of Vacuum Electromagnetic Fluctuations on the motion of Charged Particles

361
3. Electromagnetic fluctuating fields of the vacuum
The development of quantum field theory (QFT) shows that the vacuum is not an empty
space but an extremely complicated system. All kinds of field quanta are created and then
annihilated or vice versa. The effects of electromagnetic fluctuating fields of the vacuum
have been verified by experiments
，such as Lamb shift of hydrogen atoms and Casimir
force etc Unruh effect, which has been a very active area of physics and has not been
verified by experiments as yet, shows that the vacuum state is dependent on the motion of
observers. The quantized free fields of QFT are operator expressions, which can not be used
directly in classical calculation. T. W. Marshall proposed a Lorentz invariant random

classical radiation to model the corresponding fluctuating fields of the vacuum which was
carried forward by T. H. Boyer [Boyer, 1980]. A number of phenomena associated with the
vacuum of quantum electrodynamics can be understood in purely classical electrodynamics,
provided we change the homogeneous boundary conditions on Maxwell’s equations to
include random classical radiation with a Lorentz invariant spectrum. This section briefly
introduces this classical model of the vacuum fluctuating fields, which is borrowed heavily
from T. H. Boyer’s paper.
3.1 Random classical radiation fields for massless scalar cases
The introduced random radiation is not connected with temperature radiation but exist in
the vacuum at the absolute zero of temperature; hence it is termed classical zero-point
radiation, which is treated just as fluctuations of classical thermal radiation. The only special
aspect of zero-point radiation is Lorentz invariant indicating there is no preferred frame.
Thermal radiation involves radiation above the zero-point spectrum and involves a finite
amount of energy and singles out a preferred frame of reference.
A spectrum of random classical radiation can be written as a sum over plane waves of
various frequencies and wave vectors with random phases. For the massless scalar field, the
spatially homogeneous and isotropic distribution in empty space can be written as an
expansion in plane waves with random phases

(,) ( )cos[ ()]
1
()[()exp( ) ()exp( )]
2
∗
=⋅−−
=−⋅+⋅







 
rt dkf kr t k
dk f a k ik x a k ik x
ϕωωθ
ω
, (23)
where the
()

k
θ
is the random phase distributed uniformly on the interval (0, 2 )
π
and
independently for each wave vector

k . The Lorentz invariance of Eq. (23) requires
that
()f
ω
must be proportional to 1/
ω
. The average over the random phases are

'''
'
1
cos[ ( )]cos[ ( )] sin[ ( )]sin[ ( )] ( )

2
cos[ ( )]sin[ ( )] 0
<>=<>=−
<>=



kk kk kk
kk
θθ θθ δ
θθ
. (24)
Eqs. (23) and (24) have immediate connection with the counterparts of QFT, which are free
field
†
ˆˆ
() [()exp( ) ()exp( )]
2
=−⋅+−⋅




dk
xakikxakikx
φ
πω
,

Behaviour of Electromagnetic Waves in Different Media and Structures

362
and the commutators for creation and annihilation operators
†
ˆ
()

akand
ˆ
()

ak satisfying
†
††
ˆˆ
[ ( ), ( ')] ( ')
ˆˆ ˆ ˆ
[ ( ), ( ')] [ ( ), ( ')] 0
=−
==


  
ak a k k k
ak ak a k a k
δ
.
Compared with the free field of QFT one can immediately get the Lorentz invariant spectral
function, that is
1

()
2
=

f
ω
πω
.
These similarities make the quantitative calculations using random classical radiation fields
comparable to the results given by full quantum theory.
As an example, we will deduce the Unruh effect using random classical radiation field. For a
charge undergoing one-dimensional uniformly accelerating motion, the world line is

11
cosh( ), sinh( )==xata
aa
ττ
, (25)
where
a is the constant acceleration seen at charge’s rest frame. The uniform accelerating
motion of a charge will be investigated in section 4.
We would like to evaluate the average value
(, /2)(, /2)<− +>


rs t rs t
ϕϕ
,
which characterizes the random classical field. By a Lorentz transformation, the point


r can
be changed to the origin of another inertial frame, which will be taken as the laboratory
frame. Thus we just need to calculate

(0, /2) (0, /2)<− +>
ϕστ ϕστ
, (26)
where
/2±
στ
have two different interpretations; the results of Lorentz transformation or
just two different proper times. The charge undergoes the uniformly accelerating motion
along the x direction of the laboratory frame, and we calculate Eq. (26) at two positions of
the charge at proper times
/2±
στ
, namely

cosh( ( /2)) sinh( ( / 2))
(0, /2) ( , )
±±
±=
aa
aa
στ στ
ϕστ ϕ
. (27)
By the expression of field
ϕ
, the correlation function Eq. (26) becomes

3
2
(0, /2) (0, /2) cos{ [cosh( ( /2))
4
cosh( ( /2))] [sinh( ( /2)) sinh( ( / 2))]}
<− +>= −
−+− −−+


x
k
dk a
a
aaa
a
ϕστ ϕστ στ
πω
ω
στ στ στ
. (28)
The stationary character of the correlation function is not exhibited since the free parameter
σ
is included. However, the physical argument that there is no preferred time for uniformly

The Influence of Vacuum Electromagnetic Fluctuations on the motion of Charged Particles

363
accelerating motions indicates that Eq. (28) must be independent of
σ

. In fact, by
performing the Lorentz transformation
'
'
cosh( ) sinh( ),
cosh( ) sinh( )
=−
=−
x
xx
ak a
kk a a
ωω σ σ
σω σ
,
from unprimed laboratory frame to the primed frame in which the charge is instantaneously
at rest at proper time
σ
, the correlation function becomes

3
22
2
(0, /2) (0, /2)
'2'
cos[ [sinh( )] ( ) csc ( )
4' 2 2 2
<− +>
==−


dk a a a
h
a
ϕ
στ
ϕ
στ
ωτ τ
πω π
, (29)
which is the function of the proper time
τ
.
The result Eq. (29) will be compared with the correlation function

(0, /2) (0, /2)<− +>
TT
st st
ϕϕ
(30)
for a charge at the origin of an inertial frame where the random thermal radiation is
2
22
11
() ( ) coth( )
2exp( /)1 2 2
=+ =
−



T
c
f
kT kT
ω
πω
ωω ω
,
where
k is the Boltzmann constant, T the temperature. The calculation of correlation
function Eq. (30) is straightforward and the result is

22
0
(0, /2) (0, /2)
coth( )cos ( ) csc ( )
2
∞
<− +>
==−



TT
BB
st st
kT kTt
dth
ckT
φφ

ωππ
ωω ω
ππ
. (31)
The time parameter
s
is automatically counteracted without employing the Lorentz
transformation.
If we compare the correlation function of the charge accelerated through the vacuum Eq.
(29) with the correlation function Eq. (31) for a stationary charge in random classical scalar
zero point radiation plus a Plank thermal spectrum, we find that they are identical in
functional form provided that the acceleration and the temperature are related by

2
=
a
T
k
π
. (32)
It is in this sense that one speaks of an observer accelerated through the inertial vacuum as
finding himself in a thermal bath which is called the Unruh effect [Unruh, 1976].
3.2 Random classical radiation fields for electromagnetic fields case
We just list the expressions used in the following sections of the classical model of the
vacuum electromagnetic fluctuating fields, which can be written as

Behaviour of Electromagnetic Waves in Different Media and Structures

364

3
2
1,2
3
2
1,2
ˆ
(,) (, )cos[ (, )]
2
ˆ
ˆ
(,) (, )cos[ (, )]
2
=
=
=⋅−−
=×⋅−−






 




ert dk k k r t k
brt dk k k k r t k

λ
λ
ω
ελ ωθλ
π
ω
ελ ωθλ
π
, (33)
where
λ
denotes the polarization degree of freedom and
ˆ
(,)

k
ελ
the polarization vector
which satisfies the sum rule

2
2
1
(,) (,) /
=
=−



ij ijij

kk kkk
λ
ελελδ
. (34)
The average over random phases satisfies the rules:

'
1
sin[ ( , )]sin[ ( ', ')] cos[ ( , )]cos[ ( ', ')] ( ')
2
cos[ ( , )]sin[ ( ', ')] 0
< >=< >= −
<>=



kk kk kk
kk
λλ
θλ θ λ θλ θ λ δδ
θλ θ λ
. (35)
The average values of two components of random classical electromagnetic radiation fields
can be obtained by using Eqs. (33)-(35). For example, the average value of two electric
components is

33
22
1,2 ' 1,2
33

'
22
, ' 1,2 ' 1,2
3
2
1,2
'
ˆ
(,) ' (', ')
22
cos[ ( , )]cos[ ' ( ', ')]
'1
ˆ
(,) ' (', ') ( ')
222
ˆˆ
(,) (,)
4
==
==
=
=
×< ⋅− ⋅− >
=−
=
















ij i j
ij
ij
ee d k k d k k
kx k kx k
dk k dk k k k
dk k k
λλ
λλ
λλ λ
λ
ωω
ελ ε λ
ππ
θλ θ λ
ωω
ελ ε λδδ
ππ
ω
ελελ

π
33
22
34
2
44
66
=−
=⋅=




ij
ij
ij ij cutoff
kk
dk dk
k
dk k k
δω
ππ
δδ
ππ
, (36)
where
cuto
ff
k is the cutoff of wave vector and is usually taken as that corresponding to the
Compton wavelength of massive charges. The other two average values are

4
,0
6
< >=< >= < >=

i j i j ij cutoff i j
bb ee k eb
δ
π
. (37)
The preparation of the vacuum electromagnetic fluctuating fields is finished. However, we
strongly recommend serious-minded readers to read the original papers of T. H. Boyer. In
the next section, we will investigate the possible effects of electromagnetic fluctuating fields
of the vacuum on the radiation reaction of a radiating charge using the reduction of order
series form of LDE obtained in section 2.
4. Nonzero contribution of vacuum fluctuations to radiation reaction
In section 2, we have obtained the reduction of order series form of LDE up to
2
0
τ
term, and
here we present it again

The Influence of Vacuum Electromagnetic Fluctuations on the motion of Charged Particles

365

22
11

00
2
1
1
[][
2( ) ]
∂∂∂
∂∂
=+ + + + + +
∂∂ ∂ ∂∂
∂∂∂ ∂
∂
+++ + + +⋅
∂∂ ∂∂ ∂
 
   


   
 
fff
xx
xf x f fx x
xx x
fff f
x
f
xfxfffffxx
xx x x
μμ μ

μμ
μμ ν ν μ ν
νν ν
μαα α
μ
ν
νννμ
ααα
νν ν ν
ττ
ττ
τ
, (38)
where
2
0
1
()
∂∂∂
=+ + + +
∂∂ ∂
  

fff
xf x f fx
xx
μμ μ
μ
μννμ
νν

τ
τ
.
In getting this equation, we have assumed that the external fields are electromagnetic fields
which can be expressed by the functions of the proper time and spacetime coordinates.
Therefore the external force has the Lorentz-force form:

0
[( ) , ( ) ( )]==+⋅++×+







e
f
Fx EexxEe x Bb
m
μμν
ν
, (39)
where
(,)

Ex
τ
and
(,)


Bx
τ
are external electromagnetic fields, ()

ex and ()

bxthe vacuum
electromagnetic fluctuating fields expressed in Eq. (33). For the sake of simplicity, we have
these electromagnetic fields absorb the factor
/em
. The component expressions of the force
field are

000
111
01 1 23 3 32 2
222
02 2 31 1 13 3
333
03 3 12 2 21 1
()
()[()()]
()[()()]
()[()()]
=⋅ + = +
=++ +−+=+
=++ +−+=+
=++ +−+=+








Ee
Ee
Ee
Ee
fxEeff
f
xE e xB b xB b f f
f
xE e xB b xB b f f
f
xE e xB b xB b f f
. (40)
We would like to emphasize again that Eq. (38) is accurate enough for any macroscopic
motions of a charge due to
0
τ
24
(10 )
−
 s
being an extremely small time scale. Because the
atomic nucleus is of finite size, Eq. (38) could even be used to study the electron’s motion of
a hydrogen atom.
It is impossible or meaningless to trace the effects of vacuum electromagnetic fluctuations by

directly solve the equation of motion of charged particles due to its stochastic property.
Therefore, we should perform the average calculation for each term of the acceleration over
the random phases of vacuum fluctuating fields, which is a coarse grained process. The
nonlinear terms of Eq. (38) will produce nonzero contribution of vacuum electromagnetic
fluctuating fields to the radiation reaction. According to the rules of average manipulation
over random phases Eq. (35), The result of the average for the first term of charge’s
acceleration is just the external force
E
f
μ
. We will focus on the calculation of the average
results for the next two terms of the acceleration, which is quite cumbersome.
4.1 Effects of the vacuum fluctuations on the radiation reaction in the order of
0
τ

The part of acceleration linear with
0
τ
is

2
1
∂∂∂
=+ + +
∂∂ ∂
  

fff
xxffx

xx
μμ μ
μ
νν μ
νν
τ
. (41)

Behaviour of Electromagnetic Waves in Different Media and Structures

366
Because the average value of the product of odd number vacuum electromagnetic
fluctuating components is zero, the average results of the first two terms of Eq. (41) are just
∂∂
+
∂∂

EE
f
f
x
x
μ
μ
ν
ν
τ
.
The third term
∂

∂

f
f
x
μ
ν
ν

is possible to produce nonzero contribution to radiation reaction. Firstly, we calculate the
average value of its zero component
00000
00000
0123
0123
0
01 23 32 1 02 31 13 2 03 12 2
()
()
()()(
∂∂+∂∂
< >=< + >= + < >
∂∂∂∂
∂ ∂∂∂∂
=+<+++>
∂∂∂∂∂
∂
=+<+−++−++−
∂



  

Ee E e
Ee E e
E eeee
Eeeee
E
E
fffff
fff ff
xxxx
fffff
fffff
xxxxx
f
fxexbxbexexbxbexexbxb
x
ννν νν
νννν
ν
ν
ν
ν
13
0
011
)
3
>

∂
=+<>
∂


E
E
e
f
fxee
x
ν
ν
.
Then we calculate the average value of the first space component and the result is
11111
11111
0123
0123
1
102311330312212
1
()
()
()()
∂∂+∂∂
< >=< + >= + < >
∂∂∂∂
∂∂∂∂∂
=+<+++>

∂∂∂∂∂
∂
=+<++−−+−>
∂
∂
=+
∂


 


Ee E e
Ee E e
E
Eeeee
E
Eii
E
E
fffff
fff ff
xxxx
fffff
fffff
xxxxx
f
f
xee x e xb xb b xe xb xb b
x

f
f
x
ννν νν
νννν
ν
ν
ν
ν
ν
ν
1
222
11 13 12 1 11
∂
<−−>= −<>
∂
 

E
E
f
xe xb xb f x ee
x
ν
ν
.
Because all three space components are equivalent, the final result is

011

[3 , ]
∂∂
<>=+−<>
∂∂




E
E
ff
f
fxxee
xx
μμ
νν
νν
. (42)
The average value of the fourth term of Eq. (41) can be directly calculated and the result is
222
,,
222 2 2
01 23 32 02 31 13
2
03 12 21
2 22 222222 222222
01 23 32 02 31 13
()( )
()()
()

()()
<>=< + + >= +<>
=+<−+− −+−
−+− >
=+<− ++ − ++

  
 
    
eE e E e
E
Eii
Eii
f
xffffxfxfx
f
xxexexbxb xexbxb
xe xb xb x
f
x xe xexbxb xexbxb
μ
μμ μ μ μ
μμ
μ
μ
μ
22 22 22
03 12 21
22
11

()
(3 4 )
−++ >
=−+ <>
 



E
xe xb xb x
fx x x ee
μ
μμ
.

The Influence of Vacuum Electromagnetic Fluctuations on the motion of Charged Particles

367
Therefore the complete expression linear with
0
τ
of the contribution of vacuum fluctuations
to the radiation reaction is

22
000 11
4( , )−<>





xx xx ee
τ
. (43)
4.2 Effects of the vacuum fluctuations on the radiation reaction in the order of
2
0
τ

The part of acceleration that is proportional to
2
0
τ
is

μ
μμ μ
μννν
ννν
αα α
ν
ν
μ
ααα
νν
τ
τ
∂
∂∂∂
=+ + +

∂∂ ∂ ∂
∂∂ ∂
++ + +⋅
∂∂ ∂
  
  



11 1
21
2
2( )
f
xxx
xxfx
xxx
ff f
f x ff fffxx
xx
. (44)
We had better expand it into explicit expression of external electromagnetic fields and
vacuum fluctuating fields and then collect the same terms together. The expansion of the
first term of Eq. (44) is
μμ μ
μ
νν μ
νν
μμμνμ μ
ν

ννν
μμ
ν
νν ν ν
τττ
ττ τ τ τ
∂∂∂
∂∂
=+++
∂∂∂ ∂ ∂
∂∂∂∂∂∂ ∂
=+ + + + + +
∂∂∂∂∂∂∂∂ ∂





2
1
22 2
2
2
()
2
fff
x
xf fx
xx
fffffff

xf f fxff
xx x x
.
The second term of Eq. (44) is
μμ μ
μ
νν αα μ
νν α α
μμαμμα
ν
αα
μ
α
ννανα να ν
τ
τ
∂∂ ∂
∂∂
=+++
∂∂∂∂ ∂
∂∂∂∂∂ ∂
=+ ++ +
∂∂ ∂∂ ∂ ∂ ∂∂ ∂

  

 

2
1

22 2
()
(2)
ff f
x
xx x f fx
xx x x
ffffff
xx f fx
xxxxx xx x
.
The third term of Eq. (44) is
2
1
22 2
2
22
()
(2)
∂∂ ∂
∂∂
=+++
∂∂∂∂∂
∂∂∂∂∂∂ ∂
=++ ++ + +
∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
∂∂∂ ∂∂
=++ + +
∂∂ ∂ ∂ ∂ ∂ ∂



 




ff f
x
ff xffx
xx xx
ff fff f f
fx ffxf
x x xx x x xx x
ff f ff
ffxff
xxxx xx
μμ μ
μ
νν αα μ
νν αα
μμ μαμ μ α
ναα α μμ
ν αν
ν α αν ν α αν ν
μμ μ αμ
ναανν
νααν να
τ
δδ
τ

τ
2
2
2
∂∂
+
∂∂ ∂
+

 
ff
f
ff fx
xx x
ff
μα
αν νμ
α
αν ν
μ
,
and the fourth term is
2
1
∂∂∂ ∂∂ ∂∂∂
=+ + +
∂∂∂ ∂∂ ∂∂∂
  
 
fff ff fff

xxffx
x x xx xx x
μνμ νμ νμμ
ν
αα ν
ν ν αν αν ν
τ
.
Now we collected all these terms together and found that there are three terms contained
in
2

x
μ
which are linear with force field:

Behaviour of Electromagnetic Waves in Different Media and Structures

368
22 2
2
2
∂∂ ∂
++
∂∂∂ ∂∂

f
ff
xxx
xxx

μμ μ
ννα
ν
να
ττ
.
The average of these three terms over the random phases of vacuum fluctuating fields just
leaves the contribution of the external force field, and the result is
22 222 2
22
22
∂∂ ∂∂∂ ∂
<+ + >=+ +
∂∂∂∂∂∂∂∂ ∂∂
 
EE E
f
fffff
xxx xxx
xxx xxx
μμ μμμ μ
ννα ννα
ν
να ν να
ττ ττ
.
There are seven terms contained in
2

x

μ
which are quadratic in force fields. Because the
vacuum fluctuating field just the function of spacetime coordinates, the derivatives with
respect to proper time
τ
or four-velocity

x
μ
equal to zero. Furthermore, the derivative with
respect to spacetime coordinates would turn the cosine function into sine function, which
combining with another cosine function of a vacuum fluctuating field would make the
average value zero. Therefore, the average value of all quadratic terms of force fields
contained in
2

x
μ
over the random phases of vacuum fluctuating fields is
2
2
2
2
22 2 4
22 4
22 2 4
22
∂∂∂ ∂ ∂
<+ + +
∂∂∂ ∂∂ ∂

∂∂ ∂ ∂
++ + >
∂∂ ∂∂ ∂
∂∂∂ ∂ ∂
=+ + +
∂∂∂ ∂∂ ∂
∂∂ ∂
++
∂∂ ∂∂


 





EE
EE E
EEE
E
EE
E
fff f f
f
ffx
xx x
ff f f
xxfxfx
xx xx x

fff f f
f
ffx
xx x
ff f
xxf
xx xx
μνμ μ
ν
ν
ννμ
νν ν
αμ μ α
ννανμ
α
να να ν
μνμ μ ν
ν
ννμ
νν ν
αμ μ
ννα
να ν
τττ
τττ
,
4
∂
+
∂


E
E
f
xf x
x
α
νμ
α
αν
.
There are six terms contained in
2

x
μ
which are trinomial of force fields
2
2 22
22 4 2
∂∂ ∂ ∂ ∂
++ + ++⋅
∂∂ ∂∂ ∂ ∂

   
ff f f f
f
ff f ff fx fxxffx
xx xx x x
αμ μ α μ

μν α ν νμ νμ
α
να αν ν ν
.
These terms are sequently denoted as
()a
,
()b
,
()c
,
()d
,
()e
and
()f
. We just present the
calculation process for the first term, only final results of other five terms are given.
We first expand term
()a into
2
,,
,, ,,
22
,
22
,
22()()()
2 ( )( ) ( )( )
2( ) 2

22 4
<>=<+ + +>
=< + + + + + >
=< + + >
=+<>+<>
EeE e e
E
EeE e EeE ee
E
Ee Eee
E
Ee Eee
EE
ff f f f f f f
f
ff ff f ff ff
fff fff
ff f f f f f
μνν μμ
νν
νν μ νν μ
νν νν
μνμ
ν
μμνμ
ν
,
and last two terms are denoted as
(.2)a an(.3)a . We will calculate the average values of these
two terms separately.

For the average value of term
(.2)a
, the result is
22222
,0 ,1 ,2 ,3
22<>= < − − − >
eeeee
EE
ff ff f f f
μμ


The Influence of Vacuum Electromagnetic Fluctuations on the motion of Charged Particles

369
222
01 23 32 02 31 13 03 12 21
222 222 222
02311 01311 02111
222 22
023 01
2 ( )( )( )
2 ( )( )( )
2[ ( )(
=< − +− − +− − +− >
=< −++ −++ −++ >
=−++−+
         
   
     

ii j j
E
ii ii
E
ii
E
fxexexexbxb xexbxb xexbxb
f xexe xxxee xxxee xxxee
fxx xxx xx
μ
μ
μ
2222
302111
2
011
)( )]
2(4 1)
+−++ < >
=− − < >


E
x xxx ee
xfee
μ
.
As for the term
(.3)a
, we first calculate its zero-th component and the result is

00 01 02 03 0
,,0,1,2,3
012 3
11 01 02 03
01 2 3
11 0 1 11 0 2 11 0 3
44444
4444
44 4 4
<>=<>−<>−<>−<>
=< >−< >−< >−< >
=<>−<>−<>−<
      
   
Eee Eee Eee Eee Eee
E i i E ii E ii E ii
Eii E E E
f
ff f ff f ff f ff f ff
f xxee f x exe f xexe f xexe
f xx ee f xx ee f xx ee f xx
ν
ν
11
02
011
4( )
>
=−⋅<>





EE
ee
fx xf x ee
,
then we calculate its first space component, and the result is
10 11 12 13 1
,,0,1,2,3
012222 3
01 0 2 311 1211 1311
012123
01 0 11 12
44444
44()4 4
[4 4 (2 1) 4 4 4
<>=<>−<>−<>−<>
= < >− < + + >+ < >+ < >
=−−+++
     
   
Eee Eee Eee Eee Eee
Eii E E E
EE EEE
fff fff fff fff fff
f
xxee f x x x ee f xxee f xxee
fxx f x fxx fxx f
ν

ν
13 11
012
01 011
]
[4( ) 4 (2 1)]
<>
=+⋅− −<>





EE E
xx ee
fx f xx f x ee
.
The other two space components of term
(.3)a
can be obtained by the isotropy of the
vacuum, and the final average result of term
(.3)a
is
22 2
,
22 02 0 2
011 0 0 011
220 0 2
00 0 0
222 4

2 2(4 1) [4( ), 4( ) 4 (2 1)]
2[(42)4),4( )(166)
<>= +<>+< >
=−−<>+ −⋅ +⋅− −<>
=−++⋅−+⋅+−

 
 




  
Ee Eee
EE
EEEEEE
EE
EEEEE
E
ff ff f f f f f
f
fxfee fxxfxfxfxxfx ee
ff x f xf x fx f xx x
μμ μνμ
ν
μμ
μ
11
2 200 2
00011

]
2[(82),8 (166)]
<>
=−+−+−<>





E
EEE E
E
fee
ff x f fxx x f ee
μ
.
In the following, we just list the final results for terms
() ( )−bf
, which would facilitate the
check processes for serious-minded readers.
The average result for term
()b is
0
011
226[,2]
∂∂ ∂∂
<>= +−<>
∂∂ ∂∂




 
E
E
EEE
ff ff
f
ffxEfee
xx xx
νμ νμ
αα
αν αν
,
and the result for term
()c
is
2
0
∂
<>=
∂∂

f
ff
xx
μ
αν
αν
，
which is the result of the facts that external electromagnetic fields are linear function of four-

velocity and vacuum fluctuating fields are the functions of spacetime coordinates.
The average result of term
()d is
40
∂
<>=
∂


f
ffx
x
α
νμ
α
ν
,

Behaviour of Electromagnetic Waves in Different Media and Structures

370
which is closely dependent on the structure of the Lorentz force , and the average result for
term
()e is
2202 22
00,0 0 11
[(41),4 (53)]
∂∂
<>=+−+ −+<>
∂∂




 

E
EE EE
ff
xfxffx xfxfxxee
xx
μμ
νν
νν
.
The last term
()f
vanishes due to the fact that the Lorentz force is orthogonal to four-velocity
vector. Collecting these pieces together, we arrive at the final expression proportional
to
2
0
τ
of the contribution of vacuum fluctuating fields to the radiation reaction of a radiating
charge:

220 2
00 ,00 0 0 11
[( 12 3) , 12 12 (24 3) )]−+ + − − <>






EE E
xf fxxxExfee
τ
. (45)
It is easy to check that the contribution of vacuum fluctuating fields to the radiation reaction
is orthogonal to the four-velocity of the radiating charge. The only undetermined quantity
contained in Eqs. (43) and (45) is the average value of two electric components of vacuum
fluctuating fields, which can not be determined precisely. However, the cutoff of wave
vector is often taken as that corresponding to the Compton wavelength of the massive
charge. For electrons, this quantity is
452
11
/6 9.3378 10
−
<>= =Ω= ×
C
ee k T
π
, here “T “
denotes the unit of magnetic field-Tesla.
It is necessary to illustrate the justification of our procedure, first performing the average
calculation before solving the equation of motion, used to treat Eq. (38). It seems that the
opposite procedure is more in line with general ideas. However, if one obtained the solution
by directly solving Eq. (38) without performing the average calculation over random phases
beforehand, we are justified to do the Taylor expansion about the vacuum fluctuations and
then do average calculation for each term of this series. We can expect that two Taylor series
obtained from two opposite procedures would coincide with each other at least for the first

several terms.
In the following two sections, we will study the one-dimensional uniformly accelerating
motion and the planar motion using the equation obtained in this section, which includes
the radiation reaction up to
2
0
τ
term with the contribution of the external field and the
vacuum fluctuating fields.
5. One-dimensional motion of a charge acted by a constant electric field
In section 1, we have obtained the equation of motion of the one-dimensional uniformly
accelerating motion of a point charge in a constant electric field. Although the radiation is
emitted out during the charge undergoing this motion, the radiation reaction is
bewilderingly zero! Much effort has been devoted to this problem, but there is no
reasonable interpretation of this problem as yet. We propose a viewpoint that the radiation
reaction of one-dimensional macroscopic motions of charged particles mainly comes from
vacuum electromagnetic fluctuations expressed by Eqs. (43) and (45). Otherwise, it is very
hard to understand the zero radiation reaction for the uniformly accelerating motion of a
charge.
Assuming the constant electric field
E
is along x-axis, the radiation reaction of external field
is zero, and the equation of motion of this case can be easily written out

The Influence of Vacuum Electromagnetic Fluctuations on the motion of Charged Particles

371

22
000

220 0 2
00 0 0 0
[4 ,4 ]
[(12 3) , 12 12 (24 3) ]
=−Ω
−Ω − − − + −



  





E
EE E
x f xx xx
xf xEfxxxf
μμ
τ
τ
, (46)
where all the derivatives are with respect to proper time
τ
, and Ω denotes
11
<>ee . In this
case, the Lorentz force is
0

( , ,0,0)=

E
fxExE
μ
. Eq. (46) shows that the magnitude of
transverse motion exponentially decreases with proper time, namely
22
2,3 0 0
exp( 8 )−Ω

xx
ττ
,
which is reasonable because the mass of the charge increased with its velocity making it
harder and harder for vacuum fluctuations to jolt it. However, the decay time
21
00
(8 )
−
Ω

 x
τ
is
very large showing that transverse motion induced by vacuum fluctuations could remain
considerable magnitude for a long time and triggers extra radiation which is interpreted as
Unruh radiation by some authors [Chen & Tajima, 1999; Thirolf et al., 2009; Iso et al., 2010].
The transverse motion of the charge is neglected in the following calculation.
The four-velocity and acceleration are

00
(,), (,)==

xxxxxx
μμ
respectively，and the
component form of equation of motion can be expressed as

22 2
00000
22 2
0000 00
4(123)
4(123)
=−Ω −Ω −
=−Ω−Ω −
     
     
xxE xx Ex x
xxE xx E x x
ττ
ττ
. (47)
Letting
0
cosh ( ), sinh ( )==

xx
β
τβτ

, Eq. (47) is transformed into

22
00 0
(1 3 ) 2 sinh(2 ) 6 cosh(2 )=−Ω−Ω −Ω

EE
β
ττ
β
τ
β
, (48)
which can be further expressed as

0
0
0
0
22
00 0
22
00 0
24 22 2
00 0 00
42
(1 3 ) 2 sinh(2 ) 6 cosh(2 )
(2 )
(1 3 ) (2 ) [ (4 ) 1] 3 [ (4 ) 1]
(3)(13) 3

−=
−Ω−Ω −Ω
=
−Ω −Ω −−Ω +
=
−Ω+ Ω + − Ω +Ω− Ω
=
−++




o
d
EE
exp d
Eexp exp Eexp
ydy
Ey E y E
ydy
My Ny L
β
β
β
β
β
β
β
β
β

ττ
ττ βτ β
ββ
τβτβτ β
ττ τ ττ
, (49)
where
() 1=>yexp
β
, and
222
00 0 00
3, (13), 3=Ω+ Ω = − Ω =Ω− Ω
M
EN E L E
ττ τ ττ
. The initial
value of proper time
o
τ
should not be confused with the characteristic time of radiation
reaction
0
τ
.
The time coordinate
0
()x
τ
can be obtained from

0
cosh=

x
β
, namely
00
00
22
00 0
0
cosh
() (0) cosh
cosh
(1 3 ) 2 sinh(2)6 cosh(2)
−= =
=
−Ω−Ω −Ω



xx d d
d
EE
ττ
τ
β
τβτβ
β
ββ

ττ
β
τ
β