Resonance Absorption by a Charged Damped Oscillator:
Part 2
By Patrick Bruskiewich
©2012
Abstract
In part 1 , the resonance absorption of a charged damped oscillator was derived, In part 2
a relativistic invariant form factor is introduced and a model for the mass for a dressed
electron and a quark is presented. The only parameters in this model is a characteristic
time b, the time light takes to travel a distance on the order of the size of the particle, as
well as the measured charge e and the speed of light c. An interpretation of the Form
Factor sech(bco) and its relativistic invariance is that the electron spontaneously dresses
itself with an electromagnetic field that imparts the bare charge it with its mass.
1.0 The Classical Abraham and Lorentz Electron
In 1881 J.J. Thomson presented a theory describing the electroodynamic forces on a
charged sphere moving through a material with inductive capacity. Due to self-induction
the electromagnetic energy behaves as an apparent electromagnet mass on the sphere, an
effect known as electromagnetic self-energy.
Subsequent work by O. Heaviside, M. Abraham and H. Lorentz in the period 1890 to
1910 expanded the Thomson Theory of self-energy to describe the electromagnetic mass
of an electron. Abraham was a mathematical physicist at Gottingen and provided a first
model of the electron by assuming it was a charged and rigid sphere of some radius a.
Abraham's model, though reasonable for the era, was deficient. In Abraham's model the
electron would remain a rigid sphere irrespective of the relative motion of the electron.
A key experiment of this era was the measurement by J.J. Thomson of the charge to mass
ratio e/m for the electron. Exact measurement showed that this ratio depended on the
velocity of the electron.
According to the Special Theory of Relativity the charge of the electron is invariant,
while the measured mass of the electron is dependent upon the velocity the particle has
relative to the observer. There is an agreed on the proper attributes, the "rest mass" (or
proper mass) and "rest charge" (or proper charge), for the electron. [1]
Efforts to understand how the mass of an electron may depend on its relative motion led
to an appreciation of the role that electromagnetic effects played on the mass of the
electron. The earliest model viewed the electron as a miniature sphere with charge on
its surface.
In terms of pre-quantum electrodynamics, the self-energy E of the electron is made up by
a number of contributions, namely
1.1
E = T + -j{^ 2 +H 2 )dV
where T is the kinetic energy, £, is the electric field strength, H is the magnetic field
strength due to the electron, and V is volume..
Together Abraham and Lorentz derived a simple expression for the electromagnetic self-
energy of an electron at rest, namely [2]
1.2
where e is the charge of the electron uniformly distributed over the surface of a sphere of
radius a. [3] The classical electron is at rest and without spin or intrinsic magnetic field.
1
e 2
em
~2
a
Using the familiar Einstein mass-energy expression
E = m&
1.3
we find an expression for the electromagnetic self-energy
m„ m = —
2
f 2\
\ a J
e
\ c J
c
J_
ac
1.4
The radius a is known as the Lorentz electron radius, or equivalently the classical
electron radius ( r c i ass ) or the Thomson scattering length, and has a value of [4]
a*r class =2.82x10 15 m
1.5
This value is found by using the proper or rest mass of the electron in the Abraham and
Lorentz expression for electromagnetic self-energy, and solving for the radius.
We can define a characteristic time for the electron x
_a _ 2.82xl0" 15 m
1 ' c " 2.99xl0*m/s
1.6
= 9.43xl(T 24 s
which is the time it takes light to travel the distance of the Lorentz electron radius.
Modern day experiments in high-energy particle physics have probed the size of the
electron and it appears that the electron's charge distribution is consistent with a particle
with a radius considerably smaller than the Lorentz electron radius.
1. 1 Classical Radiative Energy and Acceleration
From the Larmour formula, if an external force field causes a charged particle to have an
acceleration a for a time T the energy E rad radiated is on the order of [9]
Rad
f 2\
\ C J
2 rri
a 1
c
1.7
2( fi
3U37y
a 1
c
From dimensional analysis we see that
a 1
D
1
second
1.8
For a particle initially at rest, after undergoing a uniform acceleration a for a short period
x will have a kinetic energy Eke of order
E KE U-m(ary
1.9
The radiative effects would become important when the radiated energy E ra d is
comparable to the Eke
1.10
f r f
that is
1 ( Vn 2
—m[ar) U —
r „i\
\ c j
a 2 T
c
l.ll
In terms of the acceleration and radiation time
T
D
r 2\
\ c j
1.12
mc
Using an adiabatic approximation
1.13
We arrive at some measure of a characteristic time x for the acceleration of the system,
namely
tU
f „2\
\ C J
1
mc
1.14
In terms of the particle self-energy 5E becomes
SE = mc □ —
f _2\
\ C J
1.15
If we represent the size of the electron by setting a =c x,
«-2
3
^ 2 ^
v^y
1.16
which is similar in form to the Abraham-Lorentz expression.
We can consider this energy to be the energy needed to assemble an electron given its
interaction with its own electric field. This means that we must take into consideration
the field surrounding the electron has being intrinsically tied to the electron itself.
2.0 The Quantum Electrodynamic Mass of a Electron
In classical terms it was hypothesized that an electron could exist in a space free of light
quanta, however, quantum effects have to be taken into consideration. Fluctuations in
space, and the electromagnetic characteristics of the electron itself, breaks the
homogenity and symmetry of space into a region adjacent to and a region far from the
bare electric charge.
This symmetry breaking sees the polarization of the vacuum. This symmetry breaking
also sheaths the bare electron mass in a dressing of virtual photons. The total self-energy
of an electron in space includes its electromagnetic coupling with surrounding quanta.
In non-relativistic Quantum Electrodynamics the electromagnetic mass of an electron is
given by a simplistic integral
8m =
2.1
where e is the measured charge of the electron, c is the speed of light and co is the angular
frequency of the surrounding photons coupling with the electron.
This integral is inherently flawed in that there is, in fact, as outlined below, in this
integral there is no actual coupling between the electron and a photon field. If one
considers the electron to be an oscillator interacting with photons it becomes self evident
how this integral is deficient.
Missing is also a back reaction of the electron to the surrounding photons. In this regards
the integral neither conserves momentum nor energy and is the basis for numerous
misconceptions.
A simple model of the electrodynamic interaction between an electron and photons is
presented in the next section. This simple model outlines some of the basic features of
the proper coupling between an electron and a surrounding field of blackbody photons.
This simplistic integral is also clearly divergent. A basic problem with Quantum
Electrodynamics (QED) is the predicted infinite self-energy for the electron. Infinities
where they appear in QED, are dealt with either through subtraction techniques which
together are considered renormalizing techniques or through the use of cut-offs or
regularization techniques. These approaches are somewhat artificial.
There are a number of ways to approach an upper cut-off, including arguing on heuristic
grounds that this integration should be cut off at an upper limit on the order of a
maximum angular frequency
f 2\
mc
max
ft
2.2
A lower cut-off is also found to be necessary.
2. 1 Limitations to the Simplistic Integral
With cut-offs the simplistic integral becomes
dm =
Ue 2
3
\ ~) J Li I w t *'min
Inc
r »max ,
dco
J(»„„
2.3
This simplistic integral is clearly not relativistic invariant. The calculated value for 5m is
both frame dependent, and dependent on the choice of cut-off values.
It has been shown that a self-energy expression made finite by any formal cut-off
procedure is not sufficient to make the energy and momentum transform as a four-vector.
Having no cut-off or suppression of the higher angular frequencies makes no sense in that
such high frequencies in turn point to distances far smaller than the particle or system
interacting with the frequency. The interaction or back reaction of the system needs to
be considered.
In terms of defining an electromagnetic vacuum state, if we define it as the state of lowest
energy in which there are no photons in the modes, an infinite sum over empty modes
should be equal to zero.
In some sense defining a sum over decoupled states, which is what the simplistic integral
implies, leads to a divergence that makes no sense either. Therefore, coupling between
the field and electron needs to be explicitly defined to make the integration meaningful.
2.2 A Simple Model of the Electrodynamic Interaction between
an Electron and Photons
A charged particle near a position of stable equilibrium may be treated as a simple
harmonic oscillator. In electrodynamics the motion of a charged damped oscillator can
be described by the expression
f2
Ci Jv _ -. LtJv
m
2km h mco\x - F (t)
2.4
dt dt
where x is the displacement from equilibrium, m is the mass of the charged oscillator, k is
some damping factor, and coo is the natural angular frequency of the undamped oscillator.
F(t) is a forcing term.
Consider a simple periodic forcing term from an external source of photons,
F(t) = eE cos (cot)
2.5
with e the charge of the oscillator and E is an average electric field strength.
The power absorption P(co) by a charged damped oscillator is described by the expression
Uke 2 ^
dP(co) = u(u)
y 3m j
CO
(co 2 -col) +(2koof
dco
2.6
This expression has the form
dP(a>) = gS(co)R(co)dco
2.7
where g(a) is the coupling constant
g = (2ef
2.8
S(a>) is the Source term
S(co)- — u(co)
2.9
and R(co) is the Response function for the damped oscillator, namely
R(co) =
[mj
CO
(col -co 2 ) +{2kej) :
2.10
By the correspondence principle there should be similarity between the classical and
quantum analogue of a charged particle oscillator.
It is evident therefore that the integral
dm
r 4e 2
3ttc 3
\[dco
2.11
is incomplete in its formulation in that it does not take into account a response and back
reaction in the interaction between an electron and photons.
2.3 Power Absorbed by a Charged Damped Oscillator
For a charged particle oscillator immersed in blackbody radiation the total power
absorbed is
p H
f 2e^
\ 3c J
tr (u ^
2.12
where the blackbody energy density is given by
V a
u
c
2.13
with s representing the intrinsic or internal energy of the charged oscillator.
The total power absorbed is by the damped charged oscillator is
P(a>) =
f 2e 2 ^
V J J
f 2e^
n
j c
2 2
An v,
V J J
f 2e^
V 3c J
mc
f
C0 t
£
\mc )
2.14
Given that power is defined as energy per unit time, and time can be measured in terms of
the angular frequency of the oscillator
we find that
P(co)
AE
At
AEco
2n
AEco n 1
2n 2n
f
v 3c /
f
CO n
\mc
K 3c j
f
b
\mc
2.15
2.16
where b is some measure of time characteristic to the system.
The intrinsic energy of the oscillator includes both kinetic energy T and potential energy
U, as well as it rest mass-energy
T i TT , 2 2.17
8 = T + U +mc
For a free electron at rest and away from a potential field, and without discernable
internal structure, we can assume that
( o A
\mc J
2.18
This means that the energy absorbed by the system is given by
AE x —
3
f e^
V c )
2.19
We can let a = cb and recover an expression similar to the Abraham and Lorenz self-
energy expression,
SE= 2 -
3
V^
\ a J
2.20
2.4 Renormalizing and Regularizing the Integral
Returning to the simplistic divergent integral we, in fact, find that a more telling form for
this integral is
(•GO /•(
dco =
Jo Jo
CO
'doA
[co 2 J
2.21
To subtract out the divergent part of the integral, this expression can be reformulated to
( l
2 2 2
\co co +a J
1^ 2 a 2
co =
2 2
co +a
2.22
It is worth noting that
I
co 2 dco
2 2
co +a
= co -a tan
= \dco-i
a tan
co
a
-i
co
a
2.23
This means that
2 2 2
\—rda>-\— -dco = f— -dco
J /si J /SI -L- rt J Srt -L- rt
2 2
co +a
co +a
= a tan"
co
a
2.24
which is finite. Through this technique we have done away with the divergence by both
renormalizing and regularizing the integral.
We can use this technique because we do not measure absolute energies in physical
measurements but relative measurements of energy.
3.0 Reformulation of the Divergent Integral
One form of regularization is to use a Form Factor in the integral which acts as a
convergence factor which "cuts-off ' or suppresses the divergent contributions of the
higher energies of the intermediate vector bosons mediating the electromagnetic
interaction, with the additional proviso that the function is also well behaved at co = 0.
In terms of the self-energy of the electron, instead of setting artificial limits to this
integration, it seems reasonable to incorporate a convergent Form Factor F(co) into the
integration, so that the electromagnetic mass can be expressed in a form
f
dm' =
Ae z
3ttc 3
\[F(co)dco
3.1
In essence there is no reason why F(co) = 1, which, while simple, is divergent. It is
possible that F(co) has a less pathological form.
One such choice is exponential in some power of co
F(co) = e aa 3.2
The fact that there are a number of ways to approach regularization means that it is a
formalist approach and not a complete nor a consistent theory.
If we stipulate ancillary requirements like relativistic invariance we reduce the possible
regularization functions to a few candidate expressions.
3. 1 Regularization through the use of a Form Factor
In the work of Feynmann, Schwinger et al., concepts of gauge invariance and Lorentz
covariance were incorporated in Relativistic Quantum Electrodynamic so that many of
the problems with divergences can be circumvented, however a complete does not
presently exist.
A simple approach to deal with the divergent would be to incorporate a well-behaved
Form Factor into the integral, such as
F(m)^ Ql
co 2 +a 2 3.3
The choice of this type of form factor is related to the shape of the Lorentz absorption
expression for a damped oscillator at resonance, namely
\™dP(co)ccco 2 J (
d(co 2 -co 2 )
(co 2 -co 2 ) +(2kco o y
3.4
where
> P(co) is the power absorbed by the damped oscillator at angular frequency co,
> coo is the resonant frequency of the undamped oscillator, and
> k is the damping coefficient.
This choice of form function would change the form of the integral to
[ do) Ml
a 2 dco
\
co 2 +a 2
J
3.5
This particular choice of form factor avoids the singularity at co =0, that is
lim
a
co->0 2 . 2
cd +a
-^1
3.6
while introducing a parameter a. In the limit as co goes to infinity this function
approaches a zero value. The divergence is dealt with by the form function.
Such a function is similar to retaining the first two terms in a Taylor series expansion of
an exponential convergence function (n=2)
where
F(oo)
1
2
a
2 . 2
co +a
3.7
2
a =
f 1 A
\a)
3.8
The integral for this form factor yields a finite value in closed form, namely
I,
00 a 2 dco i
/
co 2 +a 2
a
1
— tan
a
fmW
W
\a ))
100
lo
f
= a
71
-0
V
71
a
3.9
However, this particular choice of form factor is not explicitly relativistic invariant.
Mindful of the general requirements of electrodynamics, it is thought that the form factor
might be exponential in form. It would be nice if we could retain the nature of this form
factor and make it relativistic invariant.
3. 1 Dressing the Electron by Means of a Contact Transformation
Consider a contact transformation of the form
z = ( co 2 - col ) = ae&
3.10
We find that
J,
l dz \
z 2 +a 2
)
HI
'ds }
2j , i2
^+1
y
3.11
This transformed integral becomes
• 00 Q
a\ I — — — ds
1
o e 2s +\
J
= — j sech( l s , )<i l s'
3.12
where sech(s) is the hyperbolic sech(s) function,
sech(s) =
cosh(5)
3.13
s . —s
e +e
We are free to choose the form of the parameter s.
The simplest choice is
s = b(co-co )
3.14
4.0 The Form Factor F(uj) = sech(buj)
Consider then the Form Factor F(co) is
F (&>) = sech (b\co - co Q |j
4.1
the shape of which is shown in Fig. 1: the Sech(x) Hyperbolic Function.
sechx
-4.0 -2.0
2.0 4.0
Fig. 1: the Sech(x) Hyperbolic Function
The nature of the function Sech(x) is such that it is unnecessary to set artificial lower and
upper limits to the angular frequency co in the model.
The integral from zero to infinity for the Sech(x) Hyperbolic Function yields
^ sech(bm)dcD = (-)tan l (e bco )
fn\
2
Kb)
{tan" 1 (oo)-tan" 1 (l)} 4.2
(2\\n n
(n\\
\^J
which is clearly not divergent.
4. 1 Regularization and the Sech(bcj) Form Factor
Given that
sech(bco) = -
2e
-bco
+ e
-2bco
4.3
we find that
j SQch(bco)dco =
2jb
!)£
e *V<»
4.4
This equality seems to indicate that the denumerator in the sech(bco) form function
*M = T
(i + e" 2to )
4.5
plays a role in regularization, that is
^ SQch(bo))d(0 = j X R(o))e~ bco dco
4.6
5.0 Relativistic Covariance of the Sech(bcj) Form Factor
Since each inertial frame is equivalent under relativity, the form of the self-energy
integral should be such that it yields the same value for the rest mass in all equivalent
inertial frames.
It is straightforward to show that the sech(bco) function is relativistic covariant. The
transformation of the angular frequency co from one frame into another frame travelling
at some constant relative speed is given by [5]
co' = yco (l - j3 cos (a))
5.1
where a is the angle between the direction of the ray of light and the direction of relative
motion of the two frames of reference, and (3 = v/c.
We find for the differential that
dco' = y(\-j3 cos (or)) d co 5.2
It is simple to show that
where
bco' _ byco(l-j3cos(a))
b'to
= e
b' = y(l-ficos(a))b
5.3
5.4
We can now easily demonstrate that the integral is relativistic covariant, namely
j sech(bco')da)' = j sech (b' co)y (l - /3 cos (af) dco
= /(l-/?cos(ei:)) j SQch.{b'co)dco
= r (l-/?eo.(a))(f)i
= \ sech(bco)da)
It is worth noting that the integral of the Form Factor sech(bco) runs from zero to infinity
in all relativistic inertial frames.
5.0 Positive and Negative Energy Symmetry
Mindful that positive energy states represent photon-electron coupling and negative
energy states represent photon-positron coupling we find that
j sech (bco)d co = j sech (bco)dco
5.6
which implies, all things being equal, that the electron and positron have similar self-
energies. This means the Form Function is also C-invariant.
6.0 The "Dressed" Electron
In Quantum Field Theory the measured values of mass and charge are not the bare values
of the electron mass and charge.
In the model presented in this paper, the form factor F(co) in the mass integral is said to
"dress" the electron with a surrounding electromagnetic field. This self-generated
electromagnetic field surrounds and "dresses" the bare electron. It is this "dressed"
electron that is measured.
Using the Form Factor sech(bco), the electromagnetic mass of the dressed electron is
8 m
r Ae 2
3ttc
J sech(bco)da)
4e 2
n
3ttc
\ c j
6.1
The pre-factor is a reasonable value of 2/3 for a self-energy, and the expression for the
dressed electromagnetic mass is also of familiar form. [6]
The only parameters in this model of a dressed electron is a characteristic time b, which
appears to be the time light takes to travel a distance on the order of the size of the
electron, as well as the measured charge e and the speed of light c.
6. 1 The Electromagnetic Self-Energy of a Dressed Electron
In terms of the electromagnetic energy 5E for our candidate Form Factor we find
SE = Sm'c
\ c J
6.2
Evidently the denominator cb is a measure of distance a, comparable to the Lorentz
electron radius, namely
r □ cb
6.3
If we assume that the fine structure constant is not a running scale at MeV energies
1
6.4
tic 137
we find an expression for electromagnetic self-energy 5E that depends on only one
parameter, the characteristic time b
se=*(JL)L
3{l37)b
6.5
Using phenomenological arguments it is possible to estimate a reaction threshold and its
relationship to the characteristic time b. If we use a measure of the radius of an electron
on the order of
r = 1.75xl0" 15 m 6.6
we arrive at a value for the characteristic time b ern of
24
b «5.83xl0" /4 5
6.7
This yields a value for the electromagnetic self-energy 5E of the dressed electron of
= 0.550MeV
which is comparable to the measured 0.51 1 MeV mass-energy of the electron.
6.8
6.2 General Form for Self-Energy Expression
The form of the integral is such that
E&hi SQch(^bo))do)
In general terms we can express the self-energy expression as
6.9
AGO
= g(a)
~i]b
where g(a) is some coupling term for the interaction under consideration.
For the electromagnetic coupling term
s( a ).
r 4e 2 '
y37TC j
6.10
6.11
the self-energy of the electron is on the order of 0.550 MeV.
6.2 Self-Energy of a Constituent Up and Down Quark
24
A coupling value equal to Planck's constant (characteristic time b of 5.83 x 10" s),
g(a) = h
6.12
yields a characteristic energy of
ru ~fo
Char
v2;
b
1 ~
?m
h
~(n^
1 "
In
\2j
b
em
h
4b
em
xlSOMeV
6.13
which is comparable to the mass-energy of constituent up and down quarks. [7]
By phenomenological considerations, it would be logical to scale the value of the
electromagnetic characteristic time (b em ) to a constituent quark characteristic time (b quar k)
em quark
6.14
Taking into consideration that quarks comes in pairs or triplets, let
b t U-b
quark r>. e
6.15
which would yield a constituent quark's mass-energy of
' quark
f b ^
em
b
V quark J
' 'char
char
= 360MeV
6.16
6.3 Self-Energy of a "Free" Quark
The quark charge is related to the electron charge through the Weinberg electroweak
angle W , namely [8]
e^>
sin
(4r)
6.17
where
sin(<9 r )~ 0.4796
6.18
The quark coupling is related to the electromagnetic coupling term through the Weinberg
angle 0w as well, namely
f
g{oc)
quark
1
A
sm(0 w ) y
435g(a\
s( a l
6.19
In the case of a "free quark", by phenomenological considerations, it would be reasonable
to scale the value of the constituent quark characteristic time (b qua rk) to a "free" quark
characteristic time (bf ree q) such that
freeq ^ quark
6.20
By recognizing asymptotic freedom, it seems reasonable to scale bf reeq as
freeq r» \ quark J
6.21
which provides an estimate of
E
freeq
quark
em
V "freeq J
V quark J
{^g{a) em )E char
4(4.35)0.550MeK
9.57MeV
7.0 General Form of the Form Factor
In general terms, the Form Factor could be of the form
6.22
where
f
Jo
dco
bco
pe +qe
-bco
-i
CO Q
-bci) \
q
dco
( -Ibco , 2\
ye +a j
1
2
a =
bjpq
1
byfpq
(p)
/ j—
~ M
tan 1
\P_ e b<o
ViQ )\
n
1<
2
uT 1
( r~ \
—
7.1
7.2
In the case where p = q = 1 / 2 we recover the Form Factor F(co) = sech(bco) function.
8.0 A Measurement Paradox
Based on our current understanding of measurement in Quantum Field Theory,
AE*At<h
8.1
In terms of the electromagnetic self-energy 5E and the characteristic time b we find that
SE *b& —
3
V 137y
8.2
□ n
which represents a paradox.
An interpretation of this paradox is that it appears we cannot directly observe or measure
any internal structure contributing to the mass of an electron, if such a structure exists.
We are similarly constrained as far as measurements of the internal structure of quarks.
9.0 Relativistic Interpretation of Electromagnetic Self-energy 6E
In the transformation from one inertial frame to another we find for the self-energy that
8E' = ySE
9.1
and for the characteristic time b that
7
9.2
we find that
SE'b' = (ySE)\-b
\Y J
= 8Eb
9.3
is a relativistic invariant, as is outlined in the expression,
SE'b' = 8Eb
V 137y
9.4
where the right hand side is clearly is an invariant.
10.0 An Interpretation of the Dressing
An interpretation of the Form Factor sech(bco) and its relativistic invariance is that the
electron spontaneously dresses itself with an electromagnetic field that imparts the bare
charge it with its mass. Similarly, self-dressing occurs for bare quarks.
This paper deals with a phenomenological effect without elucidating any intrinsic cause.
A subsequent paper may look at the intrinsic cause of this effect.
Notes
[1] The term proper is derived from the French word "proper" which means "one's own",
hence "temps proper", or proper time, means "one's own time". This term may have
come from the writings by the French mathematician Henri Poincare.
[2] For convenience we use rationalized units so that 47i8o =1 in the expressions.
[3] If the charge is uniformly distributed over the volume of a sphere of radius r, the self-
energy is
3 e
5 r
[4] Refer to French p. 83
[5] See Bergmann p.45
[6] This pre-factor value of 2/3 indicates that if there is internal structure to the electron
that it is not be distributed on a spherical surface (1/2) nor distributed uniformly
throughout its volume (3/5) but appears to be more towards the centre of the volume, and
may be indicative of asymptotic freedom by any internal structure.
[7] Refer to Kane, p.8.
[8] Refer to Kane p. 85
[9] Refer to Jackson, p.78 1-783