This work comments and praises Dirac's work on the reaction force theory, it is based on his 1938 'Classical theory of radiating electrons' paper. Some comments from the author are added.

1. An Apology to Dirac's Radiation Reaction Force
Theory
Sergio A. Prats López, June 2021
Abstract
The aim of this note is to defend the virtues of Dirac’s electromagnetic reaction force
theory, published in 1938 in his “Classical theory of radiating electrons” paper [1] to
convince the reader that this theory offers a cause for, the already at that time, well-know
reaction force formula and also adapts that formula to non-single point particles giving
and provides a covariant expression for that force.
I also will provide arguments to show the fact that in this theory there is an influence from
the future caused by the advanced potential should not be considered a reason to
discard the theory but an opportunity to join the classical electromagnetic world with the
quantum world and some effects as the spontaneous emission which are not possible to
explain in the light of the classical theory.
What is the Reaction Force
The electromagnetic radiation reaction force, also named self-force or damping force,
has been studied by multiple authors for more than one Century and is still considered
an unresolved topic for many physicists.
Kirk T. McDonald’s "On the History of the Radiation Reaction" paper [2] offers a good
summary of the multiple efforts that have been done to solve this topic which was first
detected by Lorentz and Planck in the 1890s and it is still subject of research in the 21st
Century.
The expression of the reaction force for a non-relativistic point particle is:
𝑭𝒔𝒆𝒍𝒇 = 𝒗̈ (1)
This expression has a particular property which is that, when integrated over a period of
time in which the initial and final accelerations are zero, the amount of work done by the
reaction force on the particle is equal to the amount of energy radiated by the particle:
𝑊 = ∫ 𝑭𝒔𝒆𝒍𝒇 · 𝒗 𝑑𝑡 = 𝒗 · 𝒗̇ | − ∫ 𝑃 𝑑𝑡 (2)
Where the term 𝑃 (2) is the power radiated by a point charge because of its
acceleration, this term is given by the Larmor formula:
𝑃 = 𝑣̇ (3)
The Larmor formula can be obtained by integrating the flow of electromagnetic
momentum that crosses a spherical surface surrounding the particle 𝒑 =
𝑺
𝒄𝟐 = 𝑬 × 𝑯.

2. Where S can be calculated considering only the acceleration component of the
electromagnetic fields. The electromagnetic field that a particle creates can be derived
from the Lienard-Wiechert potentials and is:
𝑬(𝒓𝟎, 𝑡) =
̂
𝒗
̂·
𝒗
+
̂× ̂
𝒗
×
̇
̂·
𝒗
(4)
𝑯(𝒓𝟎, 𝑡) =
∗
𝒗
× ̂
̂·
𝒗
+
̂× ̂× ̂
𝒗
×
̇
̂·
𝒗
=
̂
× 𝑬(𝒓𝟎, 𝑡) (5)
The fields (4) and (5) have one component that do not depend on acceleration and
another that depends on it, these components are often referred as induced and
accelerated components of the EM field. The acceleration component decays with the
distance, therefore the Poynting vector decays with the squared distance and that means
that the power of the accelerated field does not decay as it moves away from the charge.
This suggests that the radiation must be subtracting energy from the particle to make it
fly away to the infinite. It is not like the electrostatic field energy which decays with 𝑟
and it is always following the particle.
Evidence supporting the Reaction Force
From an experimental point of view, it seems that the most remarkable effect that was
associated to the reaction force was the radiation resistance of an antenna, which in my
point of view is not yet clear that may be caused by the reaction force or by the combined
effect of many charges radiating harmonically. However, this is not an argument against
the radiation reaction, what happens is that the radiation reaction is only relevant in the
scale of mid energy phenomena like Compton Scattering or Bremsstrahlung and
therefore it is usually ignored in favor of the Quantum Electromagnetic Theory, however
examining those effects under the light of the classical theory could bring an interesting
point of view.
On the other hand, some “mental experiments” suggest that the reaction force is
necessary:
1. Let a couple of point particles at relativistic speed approach one another, the
particles have the same charge, with the same sign so that they repel, and the
same mass.
Because of the repulsion force the particles will get closer until they stop and then
will start accelerating away. Simulation shows that when the reaction force is not
applied, the final speed of the particles is slightly superior to the original speed.
The kinetical energy increase is approximately the Larmor energy that the
particles have radiated.
The explanation is this: since the particles in this experiment are always
accelerating with opposite direction and equal intensity, their contributions to the
far field are destructive, therefore the overall field energy radiated at far distances
will be close to zero.
This suggests that the energy that each of the particles radiated to their proximity
but does not get to far due to the destructive interference between the two particle
fields ends up returning to the particles and therefore, since the reaction force

3. has not been taken into account in the simulation, their total energy has
increased.
2. Another similar mental experiment can be done using a couple of point particles
with same charge and sign but one of them being super massive so that it does
not accelerate at all. The lighter particle comes straight towards the massive one
with relativistic speed. It will slow down until it reaches a point where it stops and
then it is repelled away.
Without the reaction force the kinetical energy is being converted in potential
energy and then it is converted back in kinetical energy. Simulation confirms that
the final speed is equal to the initial speed although the lighter particle has been
radiating power. Since the heavier particle has not radiated at all, the far field will
contain the Larmor energy that was radiated by the lighter particle, increasing the
total energy of the system.
Therefore, we can argue that the electromagnetic reaction force is an effect that is in
general negligible in a the low-energy scale and in the medium-energy scale it might be
better understood under the lens of QED, however it is still a necessary component to
make the Classical Electrodynamics (CED) theory coherent when one introduces
“particles” in this theory instead of just working with independent densities of charge and
current, 𝐽 = [𝜌, 𝑱], in this sense pure classical electrodynamics do not require the
reaction force, the reaction force only becomes necessary when particles come into play
at CED.
Properties of a CED system with particles
Before jumping on the Dirac reaction force theory, I will comment the differences
between a CED system without particles and with them.
When we are not considering particles, we have a single ℝ space that represents the
space time. Inside that ℝ space-time coordinates we have four-vector 𝐽 representing
all the charge and current. This 𝐽 generates the electromagnetic field, which is
represented by the electromagnetic field tensor 𝐹 applied to the ℝ space. The 𝐹
interacts with the charge 𝐽 at each point and accelerates it according to the Lorentz
force. 𝐹 may also have a component that is not generated by any of the particles which
we may call 𝐹 and it is a homogenous solution of the Maxwell equations in our region
under observation.
On a CED system with particles, each particle is a 𝐽 four-vector deployed on its own ℝ
space. For simplicity in many problems the particles are treated as single points particles
so that they can be described as 𝐽 = 𝐽 (𝑡)𝛿(𝒙 − 𝒙𝟎(𝒕)), this way all the particles can be
represented in the same ℝ space and what it is done is to consider that a particle is
affected by the field the rest of the particles create, plus an external field if it exists.
Let it be 𝐹 the actual electromagnetic field affecting the i-th particle:
𝐹 = 𝐹 − 𝐹 (6)
Where 𝐹 is the overall field, created by all the charges plus the external field, and 𝐹
is the field created by the i-th particle.

4. I want to clarify that since (6) cancels completely the effect of the self-field, it does not
lead to the reaction force, a “small” tweak done by Dirac on this expression leads to the
reaction force, but I consider necessary to explain the whereabouts of this simpler
expression prior to jumping into the Dirac theory.
Expression (6) is the sum of two electromagnetic fields:
 𝐹 , which is the field created by all the particles, 𝐽 = ∑ 𝐽 , plus an external
field.
 −𝐹 field that it is created by the i-th particle with changed sign.
We can say that both −𝐹 and −𝐹 are electromagnetic fields since each obey the
Maxwell equations everywhere, this is consistent as long as −𝐹 only causes force on
the i-th particle while 𝐹 causes force on any particle. This means that 𝐹 is invisible
to all the other particles than i-th (and the other particles are invisible to this field). This
fact, together with the minus sign, describe the corrective behavior of this field.
Obviously, each particle has its own −𝐹 field.
Any electromagnetic field exchanges energy and momentum with the charges according
to the well-known conservation equations:
𝜕 𝑢 + ∇ · 𝑺 + 𝑱 · 𝑬 = 0 (7)
𝜕 𝑺 − ∇ · 𝜎 + 𝜌𝑬 + 𝑱 × 𝑩 = 0 (8)
Since our two fields 𝐹 and −𝐹 obey the Maxwell equations and are independent one
another, equations (7) and (8) must apply to each of them individually, this means that
each of them will have its own electromagnetic energy-stress tensor, whose expression
is:
𝑇 = [𝐹 𝐹 − 𝜂 𝐹 𝐹 ] (9)
Where 𝜂 is the Minkowski metric tensor and 𝐹 is the electromagnetic tensor.
Since the contribution of 𝐹 is multiplied by -1 this factor is present in the energy and
momentum that the field transfers to the charge, (7) and (8), and therefore it must be too
in the “particle energy-stress tensor” 𝑇 . Therefore, when we have a CED system with
particles, each of them will contain a corrective term −𝑇 transforming the actual
energy-stress tensor into this:
𝑇 = 𝑇 − ∑ 𝑇 (10)
Expression (10) contains the overall tensor for the electromagnetic field and it is
independent of the observed particle, it contains the real electromagnetic energy and
momentum, which it is inferior to 𝑇 because there is usual electromagnetic interaction
on a charge with itself, note that the reaction force shows that there is some interaction
between a charge and itself but the model based on (6) is not considering it and also the
reaction force is much weaker than the electrostatic force.

5. A result of this is that for a lonely particle the total energy of the EM field is zero, this
result refutes the that the energy-mass of the electron may have electromagnetic origin.
Another way to see this is that, in absence of an external field the force done on the
particle is zero, therefore the energy being transferred to the field is zero and because
of this, according to (6) the amount of work needed to bring the charge together will be
zero.
I have included as an annex the calculations that shows for the Hydrogen atom two
simplest energy levels, how the difference in their electromagnetic potential energy is
equal to the difference of their electromagnetic field energy when we apply the correction
on the field energy shown in (10). Of course, if we do not apply (10) the difference in
potential energy does not coincide with the difference in the field energy.
Rejecting counter arguments based on the use of stresses
One may argue that for a point particle at rest there is no need to exclude the force of
the own particle since, when the particle is at rest, that force cancels when integrated
over all the directions. One may even say that this contribution is the one that at second
order causes the reaction force but all this has an insurmountable problem: let the
particle has some internal structure, it can be so small that can be considered a point for
the external fields and yet its charge is not really concentrated in a single point but on a
tiny region. If we use 𝐹 instead of 𝐹 , the particle charge would repel itself intensely,
unless we figure out some very strong tensions are keeping the particle together. If we
use 𝐹 this will not happen since 𝐹 has been removed. It is still true that the shape
of the particle might change, but as long as changes on 𝐹 are small over the size of
the particle, it is a good approximation to say that the Lorentz force is applied uniformly
on the particle and therefore this can keep its shape. (6) can be applied to extensive
particles, in that case, unless the external field is uniform through the particle, the shape
of the particle will change over time.
Any “fictitious stress” applied to keep the particle immutable will likely alter the outcome
of the calculation and there are no experimental or theorical arguments supporting that
kind of stress, we can establish a relation between quantum electrodynamics (QED) and
quantum mechanics (QM) by assuming that the density of probability and current of
probably are, after being multiplied by the electron charge e, the densities of charge and
current:
𝜌 = 𝑒|𝜓| (11)
𝑱 =
∗
(𝜓∗
∇𝜓 − 𝜓∇𝜓∗) (12)
We may notice that only the external potentials are involved in the Hamiltonian, which is
in accordance with (6) and also there is one stress, the one caused by the Quantum
potential:
𝑄 = −
∇| |
(13)
The Quantum potential is derived from the Schrödinger wave equation but the Pointcaré
stress is something “necessary” to hold a particle together but it has never been
detected.

6. Dirac’s Reaction force explanation
At this point I have talked broadly about the reaction force and spent most of the time
arguing that a particle does not interact with itself in the same way that it interacts with
the other particles, in fact if (6) were right, the particle would not be with itself at all, but
if that were true, the works from Planck about damping radiation would be wrong, the
braking radiation should be a banned phenomena in CED and also would mean that
CED leads to situations in which the total energy is not conserved since particles can
end up increasing its energy after an interaction. Notice that all these effects require
high accelerations typically only possible with relativistic energies but that is not a valid
argument to make us think that the reaction force is unnecessary. As a final
consequence, the refutation of (6) makes (10) to be refuted too. The refutation of (6)
does not mean it is completely useless, since for a particle under “weak” accelerations
both (6) and (10) are valid.
Dirac solved the Reaction Force problem by proposing an immensely creative solution,
he noticed that for a particle at rest the advanced potentials could also balance the
effect of a particle on itself and then proposed a correction which was a mix between
self-advanced and self-retarded field which reads like this:
𝐹 = 𝐹 − 𝐹 + 𝐹 _ (14)
𝐹 = 𝐹 + 𝐹 − 𝐹 _ (14B)
Where 𝐹 _ is the electromagnetic tensor based on the particle advanced potentials
(The other two components, 𝐹 and 𝐹 , are based on the retarded potentials).
We can see that the reaction force is the Lorentz force caused by the 𝐹 − 𝐹 _
field, which obviously only affects the i-th particle. This makes the reaction force nature
to be manifestly electromagnetic. The presence of advanced fields in (14) makes clear
that the reaction force includes an effect from the future.
The advanced potentials created by a charge 𝐽 depends on the charge position and
velocity in the advanced time, in the future. When dealing with advanced potentials, a
point (r,t) will be affected by a charge which is at position |𝒓 − 𝒓′| at the time 𝑡 +
𝒓 𝒓
,
that is in a time later than t.
For an inertial particle the advanced and retarded fields are equal, therefore it turns out
that (14) and (6) are equal when a particle is at rest, which means that a particle at rest
will not interact with itself according to Dirac solution.
For a non-point particle, the advanced and retarded fields will differ (and therefore
there will be reaction force) if the particle has accelerated during its light cone time
period, therefore if the particle has a radius R, the reaction force will not be nil if the
particle has accelerating between ± .
Some attempts have been done to explain this force, the most remarkable although
unsuccessful, might be the one that Feynman and Wheeler did [3]. The author opinion

7. is that the self-force term does not need any external mechanism to justify it but it
should be taken as a principle on how particles work.
Dirac, also provided in [1] the covariant expression for the reaction force (1) on a point
particle:
𝐹𝑠𝑒𝑙𝑓 = (𝑣̈ − 𝑣̇ 𝑣 ) (15)
In the previous expression all the objects are four vectors, 𝑣 is the 4-velocity, 𝑣̇ the
squared module of the four-acceleration and 𝑣̈ the four-vector, for the acceleration
proper time derivative.
Dirac derived (15) from (14), therefore we can consider that both (1) and (15) are just
the non-covariant and covariant approximations to the real reaction force for a point
particle, while (14) is the exact covariant form for the reaction force and it is valid also
for non-point particles, however, the differences between (14) and (15) are only
appreciable when the acceleration in the ± allows the particle to reach relativistic
velocities or when the force is very different over the region where the particle exists so
expression (15) is expected to work almost in any case.
Examining the objections on the Reaction Force
The main objection against the reaction force, in both (1) and (14) forms, is that it
allows pre-accelerated or runaway solutions in which the particle starts accelerating
before the external source of the acceleration reaches the particle. That is something
that one can expect after having seen that the advanced potentials are involved in the
reaction force.
In order to study runaway solutions, Dirac showed in [1] the effect of an
electromagnetic pulse in a point particle. In sake of simplicity the pulse was a delta that
propagates in the Y axis, 𝐸 = 𝑘𝛿(𝑦 − 𝑡) and reaches the particle at t=0. While one
may argue that this pulse has infinite energy, it is a simplification for a sharp pulse, the
result Dirac arrived by working on (15) is:
𝑥̇ = 𝑘 ∗ 𝑒 , 𝑡 < 0 (16)
𝑥̇ = 𝑘, 𝑡 ≥ 0 (17)
One can see that because of the reaction force, all the acceleration happened before
the pulse reached the electron, this can be interpreted this way: the advanced field
makes the particle to accelerate before the pulse reaches it, this pre-acceleration will
cause in turn the particle to accelerate at previous time and so on, the intensity of this
acceleration will decay exponentially as we move backwards in time. Because is
very small, the period in which this acceleration happens will be very small and 𝑒
will become insignificant very quickly.
We can conclude from the previous example that, as far as the reaction force become
relevant, the motion of a particle at a time will not be fully known until it reaches a time
in which it will be undisturbed. From a time-evolution point of view the system is still
deterministic but needs to move backward and forward in time to reach the real motion,

8. fortunately the pre-acceleration effect decays so fast that the correction on the motion
usually could be neglected, however for sharp accelerations this pre-acceleration will
“soften” the acceleration reducing the amount of radiated energy so it must be taken
into account. Notice that for a sharp acceleration 𝑎 = 𝛿(𝑦 − 𝑡) the amount of
radiated energy would be infinite.
At this point, I state that the reaction force theory, with backward effects, it is the
solution for the classical electromagnetism with charges and the origin of it is the
combination of retarded and advanced field self-field that it is used to cancel the own
contribution to 𝐹 , as Dirac proposed.
I state that the difficulties on this theory arise because of grouping packets of charge
into particles than do not interact with themselves the same way that the do with the
rest of the particle.
While one may feel these difficulties are “unsatisfactory”, one must not forget that
Quantum Mechanics only is able to give a statistical description of the events that may
happen and the wave functions cannot represent where the particle is more than in a
statistical sense. On the other hand, the theories that deal with creation and destruction
operators such as QED are only able to explain the probability for an event to happen,
without explaining how the system evolves in time. What is more, the Feynman
diagrams often consider that the parts of a process such as the emission and
absorption of a virtual photon may happen in any sequence, while this order is not a
time-bounded sequence, it suggests the existence of advanced potentials.
As conclusion, the author considers that after Dirac’s contribution, the reaction
force should be regarded as a completed theory. The existence of a backward
action, related to the advanced potentials is a hypothesis that needs to be done to
describe some of the aspects of nature and should not be refuted because it
contradicts our common sense. It is well known that many things in agreement with our
common sense have been proved to be wrong, such as the independence between
time and space. In this case, however, there has not been identified any experimental
evidence that enforces us to accept this backward action, however it seems to be the
only way to give coherence to the necessary reaction force. I assert using this
hypothesis could bring light to the many still not well explained quantum world
phenomenon.
The energy correction in the Dirac theory
As the author has been so focused in measuring the energy in QED systems with
particles, it is necessary to review the energy-stress tensor under the Dirac theory, it
should equal than (10):
𝑇 = 𝑇 − ∑ 𝑇 (18)
However, here the fields used in 𝑇 should be half-retarded, half-advanced:
𝑇 = [ (𝐹𝑟𝑒𝑡 + 𝐹𝑎𝑑𝑣 )(𝐹𝑟𝑒𝑡 + 𝐹𝑎𝑑𝑣 ) −
𝜂 (𝐹𝑟𝑒𝑡 + 𝐹𝑎𝑑𝑣 )(𝐹𝑟𝑒𝑡 + 𝐹𝑎𝑑𝑣 )] (19)

9. For a particle that has been always at rest, this is exactly the same than the result
obtained when we simply subtract the retarded field as in (10), but when the particle
accelerates the advanced and retarded fields emit a far field that becomes an issue for
the theory.
Let’s consider the particle had a small acceleration over t=0 which radiated 𝜉 joules of
energy in the overall field. The corrected field will emit a component whose energy is
𝜉/4 both towards the past and the future. Since the corrective field only sees the
charge created by the i-th particle, the far field will travel eternally both backward and
forward, never interacting and therefore never losing its energy.
If we examine the corrective field energy flow from 𝑡 = −∞ to 𝑡 = +∞, it may seem that
the field is travelling from the far-away regions to the particle, it converges on it at t=0
and then leaves it towards the infinite. This means that 𝑇 has a radiated field whose
distance to the particle is just |𝑡|/𝑐 and whose energy is 𝜉/4. This field has existed
always in both past and future, therefore we may consider that the overall energy of the
system needs to be reduced on 𝜉/4 because this correction, that is because the
particle accelerated at some moment.
Well, the previous hypothesis is not sustainable, a particle in the real world will be
always interacting with the particles surrounding it and hence accelerating. All this
acceleration would cause negative contributions to the energy which finally would
overcome the energy of the particle making the whole system to have negative energy.
What is more, this far field correction is like debris that will not be seen by any other
body. While the induced field energy is bounded to the particle and needs to be
subtracted, there is no need to balance the far field since the reaction force already
subtracted that energy from the particle to pay for the Larmor energy.
The solution to this issue is still unclear to the author, but the simplest approach would
be to establish a threshold distance from the particle from where the corrective field is
ignored, the rules to stablish this threshold have not been defined.
Conclusion
The electromagnetic reaction force is an intermediate step between classical
electrodynamics and the quantum theories, the origin of this force is the self-half
retarded field and self-half advanced field as shown in (14) which makes it
electromagnetic nature evident, expressions (1) and (15) are the non-relativistic and
relativistic approximations to (14) which are more than enough accurate for most of the
classical problems, in fact, there are not many classical problems that require the
electromagnetic reaction force but it is necessary to consider it for the integrity of the
classical theory when using charged particles instead of density of charge.
The fact that a particle induced field does not affect to itself is often a bypassed topic,
the implication that it has in the stress-energy tensor is something that as far as I know,
it has never been discussed.
As a personal opinion, the idea of the advanced action on oneself introduced in the
Dirac’s theory is absolutely revolutionary and has not been taken in sufficient
consideration, while many argue that this is un-physical it may be the key to
understand from a theorical point of view many of the results that happens in the
quantum world, such as the wave-function collapse or the quantum entanglement in

10. which the detection of one particle could not travel fast enough to reach the other
particle in time, but this problem would be solved if some effect travelled backwards…
Annex – Difference of EM energy in the Hydrogen atom
Let it be 𝜓 the wavefunction for quantum number (n=1, l=0, m=0) and 𝜓 for (n=2,
l=0, m=0), their expressions are:
𝜓 (𝑟) =
√ / 𝑒 /
(20)
𝜓 (𝑟) =
√ / (2 − )𝑒 /
(21)
The energy of these levels can be calculated with the non-relativistic Hamiltonian as:
𝐸 =< 𝜓∗|𝐻|𝜓 >= − < 𝜓∗|∇ |𝜓 > +𝑒 < 𝜓∗|𝑉|𝜓 > (22)
The two terms at the right-hand side of (22) can be identified as the “quantum potential”
energy and the electromagnetic potential energy, it turns out that for the quantum level
(1,0,0) the quantum potential energy is +13.6 eV and the EM potential energy is -27.2
eV summing a total of -13.6 eV, for level (2,0,0) those values are exactly 1 fourth of the
values for the (1,0,0) level, giving the well-know difference of energy of 10.2 eV between
these levels.
For the electromagnetic potential energy, we might expect this energy to be stored in the
electrostatic field, let’s notice that since these levels have the same phase everywhere
in the space, their currents are zero and therefore their electromagnetic field is purely
electric. Of course, the field will have infinite energy because the proton is considered a
point charge but if we correct the energy of the individual fields the energy would be not
infinite. If we do not apply (10) to calculate the electrostatic energy but use simple the
overall field we can still get the difference of energy between (2,0,0) and (1,0,0) but as it
will be shown, that difference is not equal to the EM potential energy:
𝑒(< 𝜓∗|𝑉|𝜓 > −< 𝜓∗|𝑉|𝜓 >) (23)
Let it be 𝑬𝟏 that the electron in the (1,0,0) level is creating at some point and 𝑬𝟐 the field
that the electron at (2,0,0) is causing, if we apply (10) the difference of electrostatic
density of energy between these two levels would be:
∆𝑢 = 𝑬𝒑 + 𝑬𝟐 − 𝑬𝒑 + 𝑬𝟏 − (𝐸 − 𝐸 ) = 2𝑬𝒑(𝑬𝟐 − 𝑬𝟏) (24)
On the other hand, if we do not apply (10) the difference off energy between these two
levels is:
∆𝑢 = 𝑬𝒑 + 𝑬𝟐 − 𝑬𝒑 + 𝑬𝟏 = 2𝑬𝒑(𝑬𝟐 − 𝑬𝟏) + (𝐸 − 𝐸 ) (25)
By integrating ∆𝑢 and ∆𝑢 over the volume we can get the electrostatic field energy and
compare it with (23).

11. We can use (11) to obtain the density of charge and current from the wave function,
because of the spherical symmetry in these levels the field created at a point will be
radial and created by the inner charge:
𝑬𝒆 =
𝒓
(26)
It turns out that the inner charge for the (1, 0, 0) is:
𝑒 (𝑟) = 𝑒 ∗ (1 − 1 + + 𝑒 /
) (27)
Where 𝑎 is the Bohr radius, for (2,0,0) the inner charge is:
𝑒 (𝑟) = 𝑒 ∗ 1 − 1 + + + 𝑒 (28)
In order to get the field energy, we need to calculate the integral of 𝐸 𝐸 and 𝐸 𝐸 over
the radius, since the proton field is just −
𝒓
(considering e is the electron charge),
and the volume factor is 4𝜋𝑟 , the whole thing is like multiplying 𝐸 or 𝐸 by − and then
integrating. The result of this is:
∫ 2𝐸 𝐸 ∗ 4𝜋𝑟 𝑑𝑟 = −27.2 𝑒𝑉 (29)
∫ 2𝐸 𝐸 ∗ 4𝜋𝑟 𝑑𝑟 = −6.8 𝑒𝑉 (30)
Therefore, if we apply (24) we obtain the difference in the potential energy between the
Hydrogen atom (2,0,0) and (1,0,0) levels, what is more, the terms 2𝐸 𝐸 and 2𝐸 𝐸 are
equal to the potential energy term 𝑒 < 𝜓∗|𝑉|𝜓 > for both levels.
On the other hand, if we choose to ignore (18) we need to include the terms (𝐸 − 𝐸 )
whose value is 8.50 eV for the (1,0,0) level and 2.04 eV for the (2,0,0) level meaning that
the potential energy will not longer be the same than the electromagnetic field energy.
This stands as a proof that: 1. The electrons electron rest mass should not be regarded
as electromagnetic energy-mass and 2. The field energy must subtract the individual
energy of each charged particle, as shown in (18)
References
[1] Classical theory of radiating electrons – Dirac – 1938
[2] On the History of the Radiation Reaction - Kirk T. McDonald’s - 2020
[3] Interaction with the Absorber as the Mechanism of Radiation – John
Archibald Wheeler and Richard Philips Feynman - 1945