A correction in the electron’s self-energy calculation?

Sergio A. Prats López, October 2016

SergioPL81@gmail.com

Abstract The aim of this paper is to show a possible mistake in quantum electrodynamics (QED) when modelling the potential created by a unit of volume of a particle (i.e. the density of the particle at that point).

The error occurs because in QED’s formulation the particle’s density | | is omitted, and therefore the density of charge is always normalized in order to work with “condensed plane waves”, normalized to a particle per cubic centimeter over the region where there is the perturbation. Such normalization has no relevant impact on the interactions with other particles but it does have in the calculus of the self-energy.

It is neither on the scope of this paper to evaluate the impact that this change would have on all the predictions and experiments achieved with QED nor to explain how QED has been able to give so good predictions with it.

At the end of the article, a theoretical test is proposed in order to compare the self-energy of a wave function with the energy that it would have as a free charge distribution in the electromagnetic classical model, taking | | as a free charge density. This test could prove that the correction raised here is right, because for extensive wave functions the classical and quantum model should give similar results.

This paper is based on Richard Feynman’s book “QUANTUM ELECTRODYNAMICS” book, specifically on the “Advanced Books Classics” edition, ISBN 0-201-36075-6. The aim of this paper is to generate a reflection and, if appropriate, to refute the correction explained at the beginning of this abstract.

In order to understand and to be able to evaluate this article, knowledge of quantum electrodynamics is needed.

KEYWORDS

QED, quantum-electrodynamics, self-energy, divergent self-energy.

Introduction

This paper, like the work that it is based, will show its results on electrons treated mathematically as spinors following the Dirac equation:

∇ − =

Where the terms are:

is a spinor-like wave function.

‘m’ is the electron mass and ‘e’ is its charge ∇ is the four-gradient operator, ∇≡ ∇ = [ , , , ] A is the four-potential. ≡ = [ , ]. Notice I use the bold letter “A” to refer a spatial

vector. The accent multiplies each four-vector by the Dirac matrices:

∇= ∇ =

QED uses the propagator to evaluate the effect of a perturbation over an incoming wave function. Since the propagator has a much simpler expression in momentum-energy domain, QED results are obtained in this domain using plane waves. In space-time domain, a plane wave takes the following shape: = · . In momentum-energy domain it is just a delta: = ( − ).

= [ , _0] is the momentum-energy four-vector. Energy takes into account rest mass energy as well as kinetic energy.

= [ , ] is the space-time four vector. ‘ ’ is the spinor that solves Dirac’s equation for a plane wave. Every plane wave has two

different solutions depending on its spin.

Working with plane waves and Feynman diagrams it is easy to get the amplitude to move from an incoming plane wave with momentum to an outgoing wave with momentum , as consequence of an interaction with a potential, radiated electromagnetic field or another electron.

Any wave packet not only contains a single plain wave but a ( ) distribution, as Heissenberg uncertainty principle shows. In order to get the total amplitude to get a plain wave, the amplitudes from every waves must be integrated.

Interaction with external potentials In order to show some issues when working with plain waves, I will evaluate the elastic scattering (i.e. ignoring bremsstrahlung) of an electron against an atomic nucleus, considered a point charge Z*e. This is a process that does not require neither the use of Feynman diagrams nor propagators but it shows how waves are normalized in QED.

We have an incident plain wave · and we want to know the probability of having an outgoing plain wave · , the amplitude can be obtained through this integral:

= − · ·

The ‘~’ char multiplies any four vector by the matrix. = ∗ = ∗

Integration over space is equivalent to a Fourier Transform over the variable = − , obtaining this amplitude:

= − ( )4

Finally, after normalization of the spinors and multiplying by the density of states, the following probability of occurrence per second is obtained:

./ =2

(2 )(2 )| |

Ω(2 )

Where:

Terms (2 ) y (2 ) are the result of normalizing the spinors, since for a plane wave ( ) = 2 .

( )

is the outgoing wave’s density of states, i.e. how many plain wave-states we

have for each outgoing energy, that is, ( )

for each solid angle.

These results are generally correct but a comment must be done. The incoming and outgoing waves are normalized to a particle per cubic centimeter1 which would force to limit the integration

region to a sphere with radius = 3/4 “truncating” the Fourier transform.

Figure 1. Density of probability for a Gaussian wave function (wave packet).

Apart from the truncation, this integral requires the incoming wave to be centered over the nucleus. If the wave is centered at a distance + from the nucleus, a phase factor · should be added on each amplitude.

For a plain wave, this phase does not affect the probability, but since any packed wave is decomposed in many plain waves, they will interfere between them because of this phase term.

At big distances even for closed plain waves the phase term ·( ) may cause their amplitudes to interfere destructively.

Therefore, if wave packet is far away from the nucleus or if it is diluted over a big area, the effect of an electrostatic potential drops drastically. I would like to modify the expression of amplitude like this:

′ = − ·( ) ·( )

Ignoring the objections I have just done, in QED the amplitude to pass from a wave with momentum to due to a electroestatic field can be expressed as:

1 In the book, normalizations are done to a particle by cubic centimeter, however, using the basic unit in the international system (meter) we would have a particle per cubic meter.

= − ( ( ) )

Where ‘a(Q)’ is the Fourier transform to the electrostatic potential ‘1/r’ and ‘e’ is the electron charge.

Thus, the electron normalization per cubic centimeter is masked and the fact that the electron should pass through the nucleus (what makes sense, on the other hand) is obviated.

Interactions with virtual states Most of QED processes use virtual states either on electrons or in photons. In a virtual state the Shell condition is violated. That is, identity ( − ) − ( − ) = fails, (m is the particle’s rest mass, m=0 for photons). Another simpler expression of the Shell condition with no potentials is − = .

A virtual state is always an intermediate state within the process. The amplitude in that process depends on the value the propagator takes for the virtual state, propagators tend to infinite when the Shell condition is fulfilled.

The propagator in the momentum dominion is:

( ) =1

−

Where m is the rest mass and = .

QED processes may be represented by Feynman’s diagrams, in them, one or several particles suffer intermediate (virtual) states before reaching their final states.

For example, if we study the Compton radiation, we have these two diagrams that represent the possible order in which may occur the actions in the Compton radiation process:

Figure 2: Feynman for Compton effect.

If we take the first diagram, the interpretation to it is the following: we have a plain wave electron with momentum that absorbs a photon with momentum = [ , ]. After absorbing this photon, the electron happens to have a momentum ′ = + that will violate the Shell condition for any value of , therefore, the electron will come into a virtual state. The electron will propagate in such state to send afterwards a photon. After that, the electron will reach its final state with a momentum = + − that must respect the Shell condition. Notice that for a given and . When you chose , value is determined.

The amplitude of this process for an incoming photon with momentum and polarization that results in an outgoing photon with momentum and polarization is:

= − (4 ) 1

+ −

Where the terms of this expression are:

is the interaction between the incident photon and the electron in its initial state.

is the electron’s virtual state propagator.

is the interaction between the scattered photon and the electron in its final state.

As a last comment, the electron’s density of probability is one electron per cubic centimeter and the photon’s electromagnetic potential is also normalized to a photon per cubic centimeter.

Interaction among electrons

In the interaction among electrons a new element comes to play: the field sources are the electrons themselves and therefore they evolve with time. In the previous scenarios, the potential was a punctual source or it was a normalized plain wave that represented a photon.

Let it be “A” and “B” two interacting electrons, in the space the first order propagator is expressed by:

( )(3,4; 1; 2) = − (3,5) (4,6) ( − ) (5,1) (6,2)

I will describe this integral:

Numbers “1”, “2”, “3”, “4”, “5” and “6” inside the propagator represent specific space-time points, the meaning of each of these numbers is:

o “1”: position of the electron “A” at the beginning of the experiment. o “3”: position of the electron “A” at the end of the experiment. o “5”: position of the electron “A” at an intermediate point over which, the

integration is done. This is the point where “A” causes a perturbation over “B”. o “2”: position of the electron “B” at the beginning of the experiment. o “4”: position of the electron “B” at the end of the experiment. o “6”: position of the electron “B” at an intermediate point where “B” is perturbed

by “A”.

Integration is done over the variables and , each of them represent the space time, = , the integration affects the intermediate points “5” and “6”.

is the zeroth order propagator for the Dirac equation.

is the delta function with only positive energy. It is defined so that it will only take the points where the relativistic four-distance is zero. To understand its relationship with the electromagnetic potential we have to take into account this identity:

12

( ( − ) + (− − )) = ( − )

On the other hand, the relationship between the positive-energy delta function and the “normal” delta is this:

( ) = ( ) +1

∗1

y contains both the electric and magnetic interactions, the second one is the scalar product of the electron’s velocities.

Figure 3: Feynman diagram for the interaction among electrons.

At this point, the error I’m warning it is clearly shown. We are evaluating point to point the effect of a charged particle over the Hamiltonian of another charged particle through its potential, but the potential a point causes over another particle must depend on the amount of charge at that point, that is, = | | , where must be normalized so that | | = 1 in our reference frame.

In the standard representation of Dirac matrices, the charge and current densities are obtained this way:

= | |

= ∗

Another way perhaps more clear to view the previous error is looking at the Born Series and the way the propagator is obtained:

= + + + ⋯

If we calculate V integrating every point contributions, the potential contributed by each point is the potential caused by the charge and current density in that point. In the classical model we would have the Lienard-Wiechert potential generated by that point.

Plain waves have the same charge density and current density everywhere, therefore, if we decompose the electrons “A” and “B” into plain waves, we only need to add a “de-normalization” factor that turns “e” into “e/V” to correct the absence of the | | factor in the integral, where V is the volume where the wave is normalized.

The electron’s self-energy First of all, I will define the physical sense of the self-energy. I define the self-energy of a charged particle as the energy contained in the electromagnetic induced field that this particle would create in absence of other external electromagnetic fields.

In the classical model, let a spatial density of charge ‘dq’ have a density of mass ‘dm’. This charge density generates an electric field. If isolated, the overall energy contained in this electric field,

‘dE’ will be an order of magnitude smaller than , so = 0, which means that for an

infinitesimal particle, the self-energy is zero.

The electron does not have an infinitesimal charge, therefore, if it is concentrated in a spatial region, its self-energy will not be zero, however it is clear that it should not be infinite (in a classical approach), except if the electron is concentrated in a single point, but in that case, its kinetical energy would be infinite also (in a quantum approach).

With a classical approach, if we have a density of charge spread by the space, and we assume that the ratio between the density of charge and density of mass (rest mass) is the same everywhere, then every point will follow these equations in respect of charge, current, momentum and energy:

[ ( , ), ( , )] = ( , )[ ( , ), 1]

[ , )] = ( ), [ , ]

Where ( ) = and ( ) is the ratio between the mass and charge.

According to Poynting theorem, the momentum earned or lost by a charge is compensated with the momentum earned or lost by the electromagnetic field surrounding this charge. Therefore, if we have a charge distribution with all the charge the same sign, we can obtain the classical self-energy (SE) by comparing the energy our charge has at present with the energy that it will have in the far future, when the charge has been dispersed infinitely and therefore the field energy is nearly zero, having returned the energy from the field to the charge density. The expression to get the classical self-energy is this:

= ( , ∞) ∗ ( , ∞) − ( , ) ( , ) [A]

Where ( , ∞) is the charge density in the far future and ( , ) is the density now, ( , ) is the

velocity now and ( , ) ( , ) contains the density of energy at any point.

Under this definition, the self-energy of a real plain wave is zero, it can be easily confirmed by integrating its electric density of energy.

Feynman calculates the electron self-energy from its self-interaction. In a self-interaction the final state must be the same as the initial state, any other transaction is prohibited since no photons may be emitted.

Figure 4: Feynman diagram for self-interaction.

The interpretation of this diagram is this: at “3” the electron emits a virtual photon that it is absorbed by the same electron at “4”, since this is a first order diagram, it is assumed that the electron propagates freely between “3” and “4”.

Self-interaction may be modeled as a time-independent perturbation where the self-states remain unaltered, so this self-interaction only causes a change in the energy of each self-state and therefore a phase shift with factor ∆ where ∆ is the electron self-energy and T the analyzed period of time.

The integral used to obtain the self-energy is shown on page 136 of Feynman’s “Quantum Electrodynamics” book:

∆ ∗ ∗ ∗ 2 = (4) (4,3) , (3)

I would like to make some comments about this integral:

“3” represents the point of emission and “4” is the point of reception2. The integration is over the space-time in the emission, represented by the variables , as well as over the space-time in the reception, represented by the four-dimension variable .

Integrating (3) (4,3) over would lead to have ′(4) which would take the same value than (4) if the self-interaction did not alter the zeroth order propagator and if there is no other perturbation.

V is the spatial volum where the integration is done, T is the period of time between the emission and reception of the virtual photon and E is the free-electron energy.

, gives the potential when it is integrated over , however, every point

contributes with a charge density of e Coulombs / and there is no normalization in this integral, therefore, as the integration is over the infinite space, the charge integrated becomes infinite, giving an infinite energy.

2 With “point of emission” I mean the point that perturbs some other point. With “point of reception” I mean the point that it is perturbed by some other point.

So, the error in the electron self-energy calculation comes because every point generates a potential of order / without taking into account the charge density at every point.

As we have assumed that the self-energy Hamiltonian does not alter the wave eigenstates (i.e. plane waves) except by changing its energy, we can get the self-energy by quantum mechanics:

∆ =< ( )| | ( ) > [B]

Where ( ) is the wave in the reception time “4” and is the self-energy Hamiltonian:

( ) = | (3)| ,

So, in the previous Feynman’s formula a | (3)| should be included:

∆ = (4) (4,3) , (3) ∗ | (3)| [B’]

Now I will check the self-energy with this method for a plain wave = · , if I call V the volume of the space ( → ∞) and f is the wave normalized over all the space, then

=1

√2 ∗ ·

We see that the f’s bring a 1/ factor, integration over brings a V factor, so, we have to see if the potential integral is the same order than V or not.

( − )1

=1

Integrating 1/r over spherical coordinates we have:

14 =

→

2

So the potential integral is order , but V is order , so we conclude that a plain wave self-energy is zero.

In order to validate the results shown in this article, a possible test is to take a “model” wave function, for example a Gaussian, and to compare its self-energy through the electromagnetic way and through the newly proposed “QED way”, that means, to compare the results obtained for the integrals [A] and [B] shown in this paper. This test however, is excluded from this paper due to my little experience in the use of calculus methods and in the expectation of receiving more feedback about this issue.

We can define the self-potential = [ , ] as the electromagnetic potential that a particle makes over itself, this potential should not be taken into account in quantum processes in general but only for the self-interactions.

As the propagator is so complex in the space-time dominion, and QED processes are always calculated on the momentum-energy dominion, this change in the self-interaction must be brought to the momentum dominion, a way to do this may be to define a “volume” of the waves in the time they interact, based on some parameters. However, how to calculate this volume is not on the scope of this paper.

It is clear that the self-potential will depend on how much compressed the wave-function is, the more compressed, the more intense will be the self-interaction which will reduce the time between self-interactions. I think that trying to link the wave function collapse process with the self-interactions can be an extremely interesting matter, however I must exclude it from this paper because at present I don’t have an approach to start dealing with this matter.

Conclusions

When calculating the electron’s self-energy, the “emitting” wave density, | | , must be included to have the correct self-potential, by doing so the self-energy divergence disappears.

References

QUANTUM ELECTRODYNAMICS

Richard Feynman.

Edited by Advanced Books Classics, ISBN 0-201-36075-6.

Quantum Mechanics

Richard Fitzpatrick, University of Texas

http://farside.ph.utexas.edu/teaching/qm/qm.html

Classical electromagnetism

Richard Fitzpatrick, University of Texas

http://farside.ph.utexas.edu/teaching/jk1/Electromagnetism/

Classical electrodynamics Part II

Robert G Brown, Duke University

http://www.phy.duke.edu/~rgb/Class/Electrodynamics/Electrodynamics/

General Field Theory

Eduardo Fradkin, Universidad de Ilinois

http://eduardo.physics.illinois.edu/phys582/physics582.html

Acknowledgements

I want to thank specially my sister Elena all the support she has bring me which has been incredibly important and maybe without it would not have been possible to write this article.