Vacuum polarization in QED: the virtual pair is a dipole particle

We found a new way to interpret the QED vacuum polarization diagram.

   e-  ------------------------------------

                            |  q  dipole interaction

                            |

                  ~~~~~~~~~    virtual pair (dipole)

                /          |            \       

              / k         | q           \ k

        ------------------------------------

   proton

The virtual pair (wavy line) in the diagram is an electric dipole particle, and the interactions k and q in the diagram are dipole interactions with an electric charge. The dipole particle is a boson.

The first 4-momentum transfer k creates the dipole particle. The second transfer k destroys it.

The Fourier decomposition of a dipole potential is of the form

       p • P / p²,

where p is a spatial momentum vector and P is the dipole momentum of the dipole.

Let us assume that the polarization vector always aligns parallel to the momentum vector. Then the propagator of the dipole potential is of the form

       1 / |p|.

In the diagram, the dipole absorbs and emits the spatial momentum q at the same time. Why at the same time?

Otherwise, the dipole would gain angular momentum which the proton cannot absorb. The proton interacts with the dipole through its symmetric 1 / r Coulomb potential. The proton cannot absorb angular momentum through that potential.

In classical polarization, the dipole in the diagram would exert forces on the electron and the proton simultaneously.

In the diagram we only count the propagator of the dipole particle once because the absorption and the emission of the momentum q is simultaneous. The motion of the dipole itself does not change at all.

We assume that the propagator of the proton is 1. It is almost on-shell in the diagram. In Feynman diagrams, outgoing on-shell particles are conceptually assigned a "propagator" value 1, that is, no propagator is written for them.

The propagator of the dipole particle is the massive Klein-Gordon propagator for a tunneling particle:

       1 / ( |k|² + m² ),

where |k|² is the euclidean square of the 4-momentum k. See our previous blog post for an explanation why a tunneling particle must live in the euclidean metric.

The high-momentum q case

For high momenta q, the value of the Feynman integral over all 4-momenta|k| < q for our diagram is something like

       ~ 1 / q * 1 / q * log(q² / m²).

But what is the power of the electron charge e in the integral?

There are 8 vertices. The electron and the proton couple to the dipole through their charge e. The dipole moment depends on e and the length of the axis of the dipole. We have not specified any length in the diagram.

Let us assume that polarization is only slightly superlinear. Then doubling e approximately doubles the force. The vertices which connect to the dipole do not carry the factor e on them. The power of e in the integral is only 4.

The low-momentum q case

The integral is over

        1 / q * 1 / q * 1 / |k| * 1 / |k| * 1 / m².

We integrate over all 4-momenta |k| < q.

The result is

       ~ 1 / m²,

like it should be.

Conclusions

We assumed that the virtual pair behaves like a massive boson electric dipole particle which is tunneling, and that its dipole moment is slightly superlinear on the electric field strength.

We assumed that the dipole absorbs and emits the momentum q simultaneously. We claimed that the propagator of the dipole particle must therefore be counted only once.

Our model produces formulae which have the right form in terms of q and m, save some constant coefficients. We do not even try to calculate the coefficients now. We have to be able to derive the properties of the dipole from basic Feynman diagrams, or from classical arguments, first.

The virtual pair is a boson which is tunneling: this explains the tricks in vacuum polarization regularization?

UPDATE September 10, 2021: We removed the erroneous claim that assigning q to just one side of the virtual pair loop is a restriction.

----

We have been studying the calculations of Hagen Kleinert before Formula (12.454) at

http://users.physik.fu-berlin.de/~kleinert/kleiner_reb6/psfiles/index.html

Kleinert estimates the low-momentum behavior of the vacuum polarization diagram. He does a Wick rotation to get to use the euclidean metric.

The formula seems to describe the following process:

          e- -----------------------------------------------

                                  |  virtual photon q

                                  |                     

                           ________ 

                          |              |  tunneling

                     k   |              |  boson

                          |              |  k + q

  proton ------------------------------------------------

The propagator of the boson is

       1 / ( |p|² + m² ),

where | ... | is the square of the euclidean norm of the 4-momentum p, and m is the mass of the electron. That is,

        |p|² = E² + s²,

where E is the energy and s the spatial momentum in the 4-momentum p.

The boson is the virtual pair. It is a boson because its spin is the sum of the electron and the positron half-integer spins.

The virtual pair cannot live long. It is presumably described by a wave of a type

        ψ = exp(-i (i E t + s x)).

That is a typical wave for a tunneling particle. The energy i E in the wave is imaginary, which makes the wave to decay exponentially.

The massive Klein-Gordon equation is

       d²ψ / dx² - d²ψ / dt² + m² ψ  =  0.

Its propagator is

       1 / (p² + m²).

Note that we use the east coast sign convention (- + + +) when we calculate the square of the 4-momentum p.

For low speeds,

        E = m c² + s² / (2 m).

We see that if E is imaginary, we must make m, too, imaginary.

That is, a tunneling boson lives in a euclidean world. It is not in the Minkowski metric. It lives in imaginary time.

Question. Should we use 2 m as the pair mass?

If yes, then we can change the variable, make p = 2 p', and the propagator looks formally the same.

Question. Is the force in the diagram an attractive force?

The force in the diagram is attractive if it does not change the phase of the electron wave. Then the diagram will have constructive interference with the plain Coulomb scattering diagram. We do not see why the phase of the electron would change in the diagram.

For the original vacuum polarization diagram we argued that the photon cannot on its own change its phase. It has to be an attractive force.

How to fix the low momentum behavior?

The low-momentum q behavior of our diagram is not right. The product of the two massive boson propagators then becomes

       1 / m⁴,

while it should be ~ 1 / m². Also, the diagram is missing a coefficient 1 / q².

The propagator for low momentum maybe has to be something like

       1 / ( p² + m q ),

where q is the momentum exchange.

When a photon moves in a medium, its apparent "rest mass" depends on its energy and the refractive index. The mass is tiny for blue light, and even tinier for red light. It makes sense to let the effective mass in the propagator to depend on the momentum.

In the low-momentum regime no real pairs can be produced. Could that imply that we must use a model which is different from the high-momentum regime?

Why is the virtual pair a boson which is tunneling?

How do we motivate the claim that the virtual pair is a boson, and it is tunneling?

We wrote in an earlier blog post that the pair looks like a "bound state". Feynman diagrams do not handle bound states. The propagators for the pair in the Feynman diagram are for independent particles. A bound state from outside looks like a single particle. It makes sense to model it as a single particle.

The virtual pair clearly is doing tunneling, because its lifetime is short. In an earlier blog post we remarked that the Uehling potential looks like a force which is mediated by a massive particle. The force only reaches to a distance which is the reduced Compton wavelength of the electron.

We need to check if our new interpretation of the calculation of Hagen Kleinert allows us to keep the cutoff at its natural place |q|, and still reproduce his numerical results.

The proton can send massive virtual photons?

In the diagram above, we may interpret the boson as a photon with (rest) mass m. This opens up the possibility that a charged particle can send such particles, not just massless photons.

A massive photon is a virtual pair. It can couple to charges like a charged particle.

Above we referred to the index of refraction of a transparent material. A photon in such a material travels at less than the speed of light. This is explained by the fact that the photon causes polarization and acquires a positive mass. Since the photon makes electrons in the material to move, it obviously can couple to charged particles in the same way as a charged particle can.

How does the classical analogue behave?

If we embed the proton and the electron in a medium where polarization is superlinear on the electric field strength, we will see a classical process. The classical process is complicated: energy flows into polarization, and flows back when the electron recedes. This classical process must have a description with Feynman type diagrams. What are the diagrams like?

Feynman diagram phase rules: lots of errors?

We suspect that the sign of the probability amplitude in the Feynman vacuum polarization diagram is flipped. We have lots of open questions about the phase of the probability amplitude which various Feynman diagrams calculate.

http://www.fe.infn.it/~bettoni/particelle/lecture21.pdf

In the link are listed the Feynman rules from Bjorken and Drell.

http://users.physik.fu-berlin.de/~kleinert/b6/psfiles/Chapter-11-qed.pdf

Let us look at the derivation of the vacuum polarization correction in the book by Hagen Kleinert (2016). It is in Section 12.16.

                           e+    q - p

                              _____

                            /           \

              ~~~~~                ~~~~~

                            \______/

   virtual           e-     q + p     virtual photon q

   photon q

We assume that q is a pure spatial momentum exchange, for example, in the direction of the x axis.

In the diagram, p is arbitrary 4-momentum. We must integrate over all possible values of p up to some cutoff

       |p| <  Λ,

where the norm |p| is the euclidean norm, not the Minkowski space norm. That is,

       |(t, x, y, z)| = sqrt( t² + x² + y² + z² ).

Let us then look at how the 99 page document in the link determines the phase of the correction. Is it 1, i, -1, or -i?

The photon propagator (12.92) is

       -g^μν * i / q²,

where g is the Minkowski metric. In our blog we use the "east coast" sign convention, where g is

                -1      0      0     0

                 0      1      0     0

                 0      0      1     0

                 0      0      0     1.

In the matrix, the first coordinate is the time coordinate. Note that the square of the 4-momentum q² is the square in the Minkowski metric, not in the euclidean metric. In Feynman diagrams, one always uses the Minkowski metric, unless explicitly mentioned.

The square in the Minkowski metric, with our east coast sign convention, is

       (t, x, y, z)² = -t² + x² + y² + z².

The propagator of the second photon line in the diagram brings the coefficient

       -i

to the Feynman formula for the vacuum polarization diagram. In literature, the calculated correction is real. Thus, the loop has to get an imaginary value. Let us find out from where does the coefficient i come to the loop.

The imaginary value of the vacuum loop integral

Formula (12.446) contains no gamma matrices and is easy to grasp. We believe that M there is the real electron mass, and

       M²

is a positive real number.

In Formulae (12.452) and (12.453) we see that the imaginary factor i comes from a contour integral around the poles of

       1 / ( p² - M² )².

Does this make sense? A real-valued integral got an imaginary value.

We have suggested to put the cutoff at

        |p| < |q|.

If |q| is small, the pole is not even in the integration 4-volume. The real-valued integral should stay real.

Could we define that the integral in the Feynman calculus is always made imaginary, either through a contour integral or through explicitly multiplying it by i?

The pole corresponds to a real electron. Its propagator in the Feynman calculus is infinite. If q is small, there is not enough energy to create a real electron. The role of almost real electrons needs to be studied. Are they important in vacuum polarization?

The sign flip: negative or positive contribution?

This is a harder problem than the extra i in the value of the integral.

For example, the sign of the contour integral depends on the way we travel around the poles: clockwise or counter-clockwise.

https://en.wikipedia.org/wiki/Propagator

Wikipedia lists different contours: retarded, advanced, and Feynman propagators.

It is suspicious that the sign of a physical entity, the wave function, depends on an ad hoc choice of the integration contour. We need to find a more robust way to calculate the limits of the integrals close to the poles.

Finding correct phase factors for Feynman diagrams

In the Schrödinger equation, with various potentials V, the phase of a wave may change any amount. Reflection from a hard wall makes a 180 degree shift to the phase. Reflection from a soft wall may cause any phase shift.

Feynman rules determine the phase in a way which is probably too crude to be true. Multiplication by -1 or i does not bring all the possibilities.

We need to develop new rules for calculating the phase in Feynman diagrams. We can already suggest one rule:

The phase does not change over a virtual pair loop or any diagram which does not have any external input. The phase cannot change if no external factor affects the system. There is no Baron Munchausen trick.

We may think of the system as isolated if no external lines come in. In quantum mechanics we assume that the phase of a whole "system" does not change if there is no external interaction. The system may be a molecule, for example. We can perform the double slit experiment with molecules. Their phase behaves in a predictable way.

We have conservation of momentum and conservation of the speed of the center of mass. Conservation of the phase might be a similar law of nature.

Question. Can we derive conservation of the phase from other conservation laws of nature?

Why did Richard Feynman and many others think that the phase could change over a virtual pair loop? Maybe they had in mind the Dirac sea of negative energy electrons. Those electrons, if they exist, would be an external factor which would make the phase to change.

The bare charge of the electron is the same as the apparent charge?

Let us assume that there really is a sign error in the traditional way that the vacuum polarization diagram is added to the plain photon propagator. Instead of weakening the electric attraction, the diagram strengthens it.

   virtual photon p

             ~~~~~~~~~~~ O ~~~~~~~~~~~

                                       virtual pair loop

Let us study the consequences.

1. Keep the cutoff very large. Then the diagram greatly enhances the electric attraction. The "bare charge" of the electron has to be minuscule in the absolute value. We see that little charge greatly magnified when we measure the electron from a macroscopic distance, for example, in the Millikan-Fletcher experiment.

2. Put the cutoff at the natural value |p|, where p is the exchanged momentum. In this case, the natural interpretation is that the bare charge is what we measure from a macroscopic distance. At short distances, creation of virtual pairs enhances the attraction.

In Case 1, we do "renormalization": the bare charge is hidden from us. We use the charge measured from a macroscopic distance. This is an ugly complication.

In Case 2, we have to find good grounds for using |p| as the cutoff. In this blog we have studied throughly destructive interference of the created virtual pairs. The interference is not trivial, because we have two particles, not just one. Case 2 is the beautiful solution. There is no need for an arbitrary cutoff or renormalization.

We can, of course, take as an axiom that there is destructive interference in this case. But it is better to prove the claim from general principles of quantum mechanics.

http://meta-phys-thoughts.blogspot.com/2021/08/feynman-vacuum-polarization-and.html

See the note at the start of the link. We got the correct Uehling potential with a 7% accuracy by putting the cutoff at |p|.

Is there destructive interference in virtual pairs?

We could model the two particles in the nonrelativistic case as a single particle in 6 + 1 dimensions. Then we probably get destructive interference for large 7-momenta.

What is the 7-dimensional wave function of the single particle? 

The Dirac wave function of the positron rotates to the opposite direction to the wave function of the electron. If we take the product of the two, we get a strange non-rotating wave function. Note that the classic Dirac problem of negative energy states of the electron resurfaces here. There probably is destructive interference for this strange wave, too, for large 7-momenta.

Maybe we should interpret that the Dirac wave is a classical wave, and calculate the product from abstract complex waves? Then we could make both waves to rotate to the same direction. Then there is destructive interference.

Yet another model is that the electron wave in the loop "turns back in time". Then we have just one particle which travels back in time as a positron. In this case, there is destructive interference.

Feynman vacuum polarization and the "complement" problem

UPDATE August 29, 2021: Our method, when putting a sharp cutoff for |k| at |p|, gives for small |p| << M the correction term

      α p² / (16 π M²),

to the photon propagator, while in the correct formula 16 is replaced with 15.

http://users.physik.fu-berlin.de/~kleinert/b6/psfiles/Chapter-11-qed.pdf

See Formulae (12.446) and (12.465) in the book by Hagen Kleinert (2016) (there q takes the role of our p).

If we put a smooth cutoff, we might be able to get the exact right result. The calculation shows that we get a fine estimate for the Uehling potential with our method.

We need to study large |p| next.

----

Our previous blog post was left at an intriguing problem.

        e- ----------------------------------------

                                |

                               O  vacuum polarization

                                |   virtual pair, 4-momenta

                                |   k, p - k

                                |

                                |  virtual photon, spatial

                                |  momentum p

        Z+ ----------------------------------------

There probably is a sign error in the Feynman vacuum polarization integral.

If the photon would scatter from a real pair, there might be a 180 degree phase shift, but we do not agree that a photon can change its phase by scattering from "empty space", or from a virtual pair which is created by the photon itself. That would be a Baron Munchausen trick. Literature skips over the phase determination discussion and immediately assumes that the vacuum polarization diagram should be subtracted from the plain photon propagator, not added to it.

We believe that large-4-momentum virtual pairs are wiped out by destructive interference. The natural cutoff is roughly  |p|, which is the momentum exchange of the electron and the proton. For larger 4-momenta, there is almost total destructive interference.

Or, we can set a smooth cutoff function

       |p| / |k|,

where k is the 4-momentum circling the virtual pair loop.

But why does the Feynman integral seem to calculate the "complement" of the effect of virtual pairs with small 4-momentum?

We want to calculate the value of the Feynman integral

       F(p²)

of the process if the 4-momentum in the integration is restricted to be smaller than a certain cutoff Λ.

Case A. Let us first set the cutoff Λ to a very large value.

The difference

       F(0) - F(p²) = D

is the correction that we have to add to the plain Coulomb scattering probability amplitude if we let the exchanged momentum be p. This is according to the usual Feynman calculus.

Case B. Let us set the cutoff Λ to |p|. We are interested in the value

       F(p²) = D'.

We have in this case fixed the Feynman integral sign error. We add D' to the plain Coulomb scattering probability amplitude to get the corrected amplitude. This is according to our new calculus.

If D = D', then the "complement" problem is solved. Both methods give the same numerical correction.

The value of D' corresponds to what effect is still left, D corresponds to what effect fell out. It would not be a big surprise if D = D'.

Calculations with Feynman integrals are cumbersome. We need to study detailed calculations to get numerical values out of our new cutoff procedure. Our new cutoff could be called "regularization", but the big difference is that our cutoff has a physical motivation in destructive interference, while regularization methods are ad hoc.

Vacuum polarization makes a strong electric field stronger through a dipole?

Vacuum polarization in the Dirac sea comes from the fact that a charge deforms the paths of negative energy electrons in the sea.

For example, the density of negative energy electrons is larger close to a proton. The density of negative energy electrons is imagined to be lower somewhere far away, perhaps at the infinity.

We do not like the idea of the vacuum being full of particles like negative energy electrons. We would like the vacuum to be totally empty. How could we simulate Dirac's idea without assuming a Dirac sea?

Dirac has to put a cutoff on the momenta |p| of negative energy electrons in, say, one cubic meter of empty space. Then the number of negative electrons in a stationary state in that cubic meter is finite, and we can calculate a finite amount of polarization caused, for example, by a proton.

The cutoff is an ugly feature. The second ugly problem is that negative energy electrons have to be able to store momentum to be able to interact with real particles. What guarantees that empty space does not soak up a macroscopic amount of momentum?

Feynman diagrams solve the second ugly problem by requiring that energy and momentum are always conserved and exit the process in real particles.

Though, as we remarked in an earlier blog post, Feynman diagrams do not conserve the speed of the center of mass. As if empty space could contain mass-energy which our process can displace. Clearly, we must add conservation of the speed of the center of mass to the rules which a Feynman diagram must obey.

The cutoff problem exists in Feynman diagrams, just as in Dirac sea vacuum polarization. 

Uehling and Serber in 1935 calculated vacuum polarization from the Dirac model, and got the correct result. The Lamb shift has been measured very precisely, and it must include the effect from the Uehling potential.

The Feynman diagram

In modern literature, the Uehling potential is calculated from the vacuum polarization Feynman diagram. The result is the same as Uehling got from Dirac's model. Clearly, the Feynman diagram has to calculate the same thing as Dirac's model, though it is not yet clear to us how it does that trick.

        e- -------------------------------------

                              |

                              |

                             O  vacuum polarization loop

                              |

                              | virtual photon p

        Z+ -------------------------------------

The Feynman diagram at the first sight does not assume that the vacuum is full of particles. No such particles are drawn into the diagram. But a vacuum full of particles creeps in when we set a cutoff on the virtual pair 4-momenta in the vacuum polarization loop.

Maybe the solution is to set the cutoff at the nucleus energy?

https://www.cambridge.org/fi/academic/subjects/physics/quantum-physics-quantum-information-and-quantum-computation/collected-works-p-m-dirac-19241948-volume-1?format=HB&isbn=9780521362313

Dirac in the papers which can be found in the book in the link, suggested that the cutoff should be something like 1 GeV. That is the mass of the proton.

We have in this blog suggested that destructive interference wipes off virtual pairs whose 4-momentum is large. In wave phenomena, long waves typically cannot create shorter waves. The nucleus in the diagram has lots of energy. Its wavelength in the time direction is very short. It could create wave phenomena whose 4-momentum is large.

                                   virtual pair loop

large virtual photon

                           ~~~ O ~~~

                         /                     \

        Z+ ------------------------------------------

In the Feynman diagram above, the nucleus creates a large virtual photon, which in turn can create a virtual pair with large 4-momentum. The electron might interact with the pair. A problem is that the diagram will have more vertices than the simpler vacuum polarization diagram, and consequently, its amplitude is smaller in the Feynman calculus.

This idea does not work, though. If the cutoff would depend on the mass of the nucleus, then the charge visible to a far-away observer would depend on that mass.

A possible solution: vacuum polarization is temporary pairs which "conduct" electric field lines better?

The Uehling potential shows that the proton attracts the electron stronger than we would expect if the electron comes within the reduced Compton wavelength distance 4 * 10⁻¹³ m from the proton.

How to make electric attraction stronger? Create a temporary electric dipole between the electron and the proton. The dipole will make the attraction stronger. It "conducts" electric field lines. The energy for the creation of the dipole would come from the approaching electron.

In this blog we have thus far used electric polarization of a solid as the model of polarization. If polarization grows superlinearly with the field strength, then the attraction between the electron and the proton will appear stronger.

Since a very strong (changing?) electric field can create a real pair, we suspect that polarization really is superlinear.

But how to reconcile this very different model with the idea of the Dirac sea?

The Uehling "force" is mediated by an electron-positron pair?

The Uehling potential becomes important at the distance of the electron reduced Compton wavelength, and it drops off exponentially at larger distances.

This immediately brings into mind that the Uehling potential is an attractive force caused by a dipole virtual electron-positron pair. The mediator "particle" of the Uehling force is a pair.

     proton

           ●        -    +       •  e-

                     dipole

A dipole is born between the proton and the electron which is passing it. The pair makes the attraction between the proton and the electron stronger. A dipole "conducts" the electric line of force and makes attraction stronger.

The energy-time uncertainty principle tells us that the pair, whose energy as a real system would be 1.022 MeV, can at most reach to the distance 1/2 times the reduced Compton wavelength of the electron.

How do we calculate the probability of a dipole forming?

When the electron passes the proton, it produces a sharp pulse in the electric field of the electron. The pulse contains frequencies which are high enough to create a 1.022 MeV pair, but the total energy of the pulse is too small. A real pair cannot be created, but maybe we can create a very short-lived virtual pair?

There is a sign error in the Feynman diagram and it calculates the "complement" of the correct effect?

The following could explain things.

The Feynman vacuum polarization diagram erroneously reduces the calculated cross section. It should increase it.

http://meta-phys-thoughts.blogspot.com/2020/12/does-vacuum-polarization-exist-for.html

We wrote on December 25, 2020 that a particle cannot change its phase on its own. The photon (momentum p) in the diagram should keep its phase, not change it by 180 degrees.

The Feynman formula, when we increase the exchanged spatial momentum |p| between the proton and the electron (make the electron go closer), leaves out the effect of virtual pairs with low 4-momentum. Think of Dirac's model. Negative energy electrons with low momentum have the polarization spread over a large spatial area. Decreasing the distance between the electron and the proton leaves out their effect.

But in the "Uehling force" model above, low 4-momentum is necessary for the pair, because otherwise destructive interference wipes it out. In that model, it is the low-4-momentum pairs which create the attractive force.

Thus, when we increase |p|, the Feynman integral (with its negative sign, or 180 degree phase change) leaves out the probability amplitude contribution, say, P, of low-4-momentum pairs.

      e- ---------------------------------------------

                                 |  virtual

                                 |  photon

                                 |  p

     Z+ ---------------------------------------------

Above we have the diagram of plain Coulomb scattering.

Since P in the Feynman calculus has a 180 degree phase shift relative to the plain Coulomb scattering probability amplitude, adding P to it would reduce the cross section. Leaving P out increases the cross section: the Coulomb force looks a little bit stronger, which is the right end result.

We suggest that in the correct calculation, P should not have a 180 degree phase shift, and it should be added to the plain Coulomb scattering amplitude. We need to study this in detail.

P is kind of a "complement" of the Feynman integral.

If this is true, empty space is truly empty. It is the proton and the electron themselves who create virtual pairs, which make the electric line of force stronger.

Open problems: why does Dirac's model calculate the Uehling potential right, even though in his model it is the bending paths of negative energy electrons which cause the polarization? It is not virtual pair dipoles like in our model.

Why does the Feynman integral formula calculate the same thing as Dirac's model? In Dirac's model the negative energy electron enters the process as an independent particle. In the Feynman diagram, it is one half of a virtual pair.

Coulomb focusing for the 2s orbital of hydrogen: it cannot explain vacuum polarization

We claimed in an earlier blog post that electron on the 2s orbital moves radially toward the proton or away from it. Then Coulomb focusing would have no effect.

http://hyperphysics.phy-astr.gsu.edu/hbase/hydwf.html

But when we look at the radial probability of the electron on the 2s orbital, we see that the probability very close to the proton is much less than it would be if the electron would move exactly radially.

From the diagram at the link we see that the electron on the 2s orbital "typically" swings to a distance of 6 Bohr radii, and then dives back toward the proton again.

Let us calculate what would the total probability be for a sphere whose radius is the reduced Compton wavelength

        r = 4 * 10⁻¹³ m = 1 / 137 Bohr radii,

if a classical electron would go almost directly toward the proton.

The electron moves slowly at 6 Bohr radii, and spends most of its time there. The potential at 6 radii is -4.5 eV, and the kinetic energy there is

       4.5 eV - 13.6 eV / 4 = 1.1 eV.

The kinetic energy at r is

       511 keV / 137 = 4 keV.

We see that the classical electron moves 60 times faster at r than at its typical distance.

The probability of the classical electron being in the r-sphere is

       P' = 1 / (137 * 6 * 60) = 1 / 50,000.

However we calculated in the previous blog post that the quantum mechanical probability is

       P = 1 / 1,500,000.

The quantum mechanical probability is 1 / 30 of the classical. We may interpret this that the quantum electron misses the r-sphere in most cases, and instead goes through a sphere whose radius is

        R = 4 * 10⁻¹² m = 10 r.

The speed of an electron in that sphere is just 20 times faster than at the typical distance.

In quantum mechanics, everything is diffuse about the location of particles. It is not surprising that the electron does not swing exactly through the nucleus.

The far electric field of the electron lags behind and does not have time to take part in the sudden movement of the electron in the R-sphere. The effective mass of the electron is slightly reduced, and the electron will pass the proton slightly closer. It is as if the potential which is felt by the electron would be slightly lower. This is kind of extra Coulomb focusing, which is not taken into account in the Schrödinger equation and its solution.

Let us calculate the effect of the mass reduction, on an electron which flies past the proton at the distance R.

                 v

        e-  ---------->   

                                  R = impact parameter

                 

                             ● Z+ 

The kinetic energy of the electron is

       511 keV / (10 * 137) = 400 eV.

The speed of the electron is

       v = 0.04 c = 1.2 * 10⁷ m / s.

The fly-by lasts

       t = 2 * R / v = 7 * 10⁻¹⁹ s.

The force is

       F = k e² / r² = 2 * 10⁻⁵ N,

and the acceleration

 

       a = F / m_e = 2 * 10²⁵ m / s².

The acceleration makes the electron to come closer to the proton the distance:

       s = 1/2 a (t / 2)² = 10⁻¹² m.

We see that the electron turns substantially when close to the proton, tens of degrees.

The field which is farther than

       D = c t / 2 = 10⁻¹⁰ m

does not have time to react. The effective mass of the electron is

      m_e' = m_e (1 - 1 / 70,000).

Since the mass is reduced, the electron will go

       s / 70,000

closer than it would otherwise. Thus, the effective potential is reduced

       400 eV / 280,000 = 0.0014 eV.

The quantum mechanical probability of finding the electron in the R-sphere is

       1,000

times the probability of finding it in the r-sphere, or

       1 / 1,500.

We get an effective potential reduction of

       0.0014 eV / 1,500 = 0.9 μeV.

The reduction is not of the same order of magnitude as the Uehling potential reduction 0.1 μeV.

The Coulomb focusing effect certainly exists in a classical model. Does it show up in a quantum model?

Dirac's sea of negative energy electrons

Mass reduction and Coulomb focusing cannot explain vacuum polarization effects.

Paul Dirac in his 1934 paper sketched a gas of negative energy electrons whose average location is slightly displaced by the electric field of the proton. This displacement is the vacuum polarization. E. A. Uehling and Robert Serber in 1935 calculated the correct, empirically tested, potential which is caused by vacuum polarization.

Let us have a cubic meter of empty space. If we put a ceiling on the momentum |p| that a negative energy electron can possess, we can fit just a finite number of those electrons in the cube. This is because the Pauli exclusion principle prohibits the electrons having the exact same momentum p and spin. It is like the Fermi sea of electrons in metal.

Vacuum polarization in Dirac's model is a collective phenomenon. In a Feynman diagram, vacuum polarization is a dynamic process with just a few particles present. One can derive the Uehling potential formula from a Feynman diagram (but one must use regularization).

Question. Why is the Feynman model equivalent to the Dirac sea model?