What is the quantum of arbitrary energy flow?

The quantum of periodic motion contains hf of energy, where h is the Planck constant and f is the frequency of the oscillation. The motion can be the oscillation of a harmonic oscillator or the oscillation of the electric field in a plane wave.

When we studied a collision of particles in the previous blog posts, we identified that the flow of energy within an electric, or rather, electromagnetic field, is a key concept.

Energy can flow between the field and the kinetic energy of charged particles, or energy can flow spatially. To be significant, a spatial flow has to happen regardless of the inertial coordinate system. An electron flying freely is static in the coordinates that move along with it - there is no energy flow.

An electromagnetic wave is clearly a classical phenomenon at least if the field is strong enough to be measured. What is not classical is that a quantum oscillator can absorb a whole quantum h f of  energy in one shot from the field. A weak classical field would only give a minuscule energy to each oscillator. But a quantum field serves the energy in big portions, quanta.

It is as if the whole energy of the electromagnetic wave would be concentrated at a few tiny spots, the photons of the field.

One may imagine that the whole undulating wave is actually an image which these small dots draw like in an old cathode ray TV tube.

But when we have a non-periodic process like the energy flow in a collision, what is the quantum and how big is its energy?

If there is real pair production, then the quantums are the electron and the positron. We may consider the pair production as the excitation of a positronium atom from a virtual zero-energy state, to a state of > 1.022 MeV of energy. One would think that the energy quantum of the energy flow in the electromagnetic field has to be > 1.022 MeV to produce a real pair.

In a Feynman diagram, a virtual photon can carry any amount of energy and momentum which it can receive from a particle. In the simplest diagram, we typically assume that there is just one virtual photon which carries all the momentum and the energy which is exchanged in the process between two particles. But in a more complex diagram we may have two or more virtual photons doing the exchange.

The Feynman way of thinking is that we may assume that the whole energy exchange is one big virtual photon. It can transform to a pair if its energy is > 1.022 MeV.

The classical limit of the Feynman thinking would be one huge quantum which can be absorbed if two charged objects collide. Obviously, the energy of the quantum has to be restricted by the spacetime dimensions of the collision process.

When two electrons of combined kinetic energy 511 keV collide, they may approach to within 3 * 10^-15 m of each other. That is much less than the wavelength 2 * 10^-12 m of a 511 keV photon. Apparently, we may think that the whole collision process happens in an essentially pointlike zone compared to the wavelengths of the the particles emitted in the collision. This may be the reason why we can treat the whole collision energy as just one quantum.

If the collision process would occupy several wavelengths of the photon, then we might be forced to assume several quantums.

We have thought of the collision process as classical. The flow of energy happens in a tiny spot of the order 3 * 10^-15 m. Maybe it is a general principle that the quantum mechanical behavior of such tiny spot will allow all energy to be taken as a single quantum?

Classically, the collision process leads to an increase of the electric field in various zones in spacetime. A created pair would draw energy from such a zone. If there are many zones where the electric field increases, then the pair may be able to draw energy only from a tiny portion of the field. As an example, when two electrons approach each other, the electric field strength grows in most of space.

    e- ---->  <---- e-

A new created electron-positron pair can counteract the field strengthening only in a sector of space? Or, actually, can it counteract in the whole space?
               e-
               ^
               |
               |
               v
 e- ---->  e+ <---- e-

The positron will fly close to the colliding electrons, screening the charge of one electron. The created electron will fly away. The net result is that the two electron charges never came as close to each other as they would have come without the positron. The electric field never gained the energy it would have gained without the positron.

Thus, also classically, a pair can draw on much of the energy of the collision.

We do not need to speculate about the quantum of energy flow if the pair can also classically absorb most of the energy in the flow.

Another example is an electron flying close to a nucleus.
        e- ----->
        |
        |
        Z

The vertical line in the Feynman diagram symbolizes the energy flow in the combined electric field of the nucleus Z and the electron.

As the electron approaches the nucleus, it draws on the energy of the electric field. It comes to screen some of the positive charge in the nucleus, that is the fundamental reason for the attraction.

However the field in the zone between the nucleus and the electron grows. What if a pair is created in that zone? Then the electron of the pair can fly to screen the charge of the nucleus and the positron can fly to screen the charge of the original electron. The end result is that the original electron will draw less energy from the field to its kinetic energy.

        e- ------>
         |  e+
         |  e-
        Z

If we return to the rod model of electric attraction, pait production is like inelastic stretching of the rod. The rod transfers kinetic energy to the approaching electron. But some of the energy may go to inelastic (permanent) stretching of the rod. Then the electron will feel less pull and will gain less kinetic energy.

When the electron moves further from the nucleus, what if pair production happens in the more remote electric field which gains strength as the electron stops screening the nucleus? The created electron would fly to screen the nucleus and the positron would screen the original electron. The receding electron will pump less energy to the field and retain more kinetic energy. In the rod model, this corresponds to the rod stretching inelastically in the receding phase of the collision.

In the case of the nucleus and the electron it is not clear how much of the available energy can the pair production consume. Is it 50% or 1%?

A quantum of the energy flow is the produced pair? Mini-electrons?

It seems that there is no need to introduce a quantum to the energy flow in the electric field. The Dirac field, on the other hand, will absorb energy in produced pairs plus the kinetic energy of the pair.

If space has an infinite density of virtual pairs floating around, which virtual pair will absorb the energy and become real?

Maybe it is better to imagine that the energy flow produces a flux of electrons and positrons in a causal way. The pairs are quantums of this flux. Then we can discard the idea of virtual pairs floating around.

Suppose that there would exist mini-electrons who would have just 10^-9 of the charge and the energy of the electron. Then pair production would be less quantized and more "fluid". Can we write an "inverse" Maxwell equation for such charged fluid? A changing electric field produces a flux of charged fluid which tries to minimize the energy flow in the field?

The inverse Maxwell equation actually exists in a metal, where there is a lot of loose charges floating around. But in a metal, positive charges cannot move. Another analogy is a semiconductor or plasma, where we have post positive and negative carriers of charge.

Since the mass of an electron-positron pair is large, 1.022 MeV, real pairs cannot always form to reduce the energy flow.

Suppose that we had a light mini-electron whose mass would be smaller in proportion to the charge than in the electron. The classical radius of a mini-electron would be largish, otherwise its electric field would outweigh its rest mass.

A light mini-electron would pop up easily. We would observe charges appearing from empty space, reducing modest changes in an electric field. Such light pair production would be an everyday phenomenon. There would be less room for virtual pairs.

Is there some limit how small the charge of a mini-electron can be? Electromagnetic waves in the vacuum cannot be converted to any particle with a non-zero rest mass. Mini-electrons would not block electromagnetic waves. But they would make collisions of charged particles very inelastic and they would also draw energy from any process which produces electromagnetic waves.

Is Michael Atiyah right about the fine structure constant, after all?

We have in this blog tossed the idea that electromagnetic waves are in reality polarization waves of virtual pairs in space. That is, the fundamental field is the Dirac field, and photons are just phonons of the Dirac field.

The value of the fine structure constant can be expressed as

     α = k e^2 / h-bar c

The numerator in the ratio is the force between two electrons at a unit distance from each other. The denominator is the energy of a 1 Hz photon multiplied by the photon speed and divided by 2π.

If a photon is a phonon of the Dirac field, can we derive an exact mathematical formula for the above expression?

It seems to boil down to the question if the photon energy is somehow determined by the electron. The Planck constant would then depend on the properties of the electron.

Michael Atiyah this week claimed that he has derived the exact mathematical value of the fine structure constant. The author of this blog has not checked his proof, but is skeptical of its correctness.

Pair production is the process where the energy of a photon and the energy of an electron meet. If we are able to analyze pair production, we may be able to solve the question if the fine structure constant has a precise mathematical value.

In the previous blog post we raised the question what is the quantum of arbitrary energy flow in an electric field in the presence of charges. The photon is the quantum of free plane waves, it is not the quantum of energy flow in a more complex configuration.

We have coined the term quasi-photon for a photon propagating in a polarizable medium. The quasi-photon moves at a speed less than light, and, consequently, appears to have a non-zero rest mass.

What speaks against Atiyah's conjecture is that to allow the existence of atoms, there has to be flexibility in the system the electron + Planck's constant + the Coulomb force. Atoms obviously must have many possible orbits for electrons. If the physics were so tightly constrained that the Coulomb force is strictly determined, the whole system might not allow many different orbits for electrons. This argument is by no means exact. We should analyze the physical machinery more thoroughly.

Vacuum polarization is caused by the "dynamics" of the system, not by a static field

Our blog posts in the past days have pointed into just one direction: real pair production only happens when a system of charges is under a dynamic, changing, electromagnetic field. The same holds for virtual pairs or vacuum polarization - they only are relevant in such a dynamic system.

Problems of virtual pairs in empty space

Postulating virtual pairs in "empty" space leads to many problems:

1. Lorentz invariance: if space is filled with virtual pair matter, what is our speed relative to it?

2. What is the density of virtual pairs per cubic meter? If it is infinite, then tunneling  should happen at an infinite rate and vacuum polarization should render all charges effectively zero.

3. What is the energy density of the virtual pair matter? If it is large, then space should immediately collapse into a black hole.

4. If a virtual pair can absorb a virtual photon like in the simple Feynman diagram, physics becomes "non-causal" in the sense that a potentially infinite number of virtual pairs can take part in the process.

5. Virtual pair loops cause divergences in Feynman integrals.

Our blog aims to show that processes of real particles create their own environment - there is no virtual pair matter floating around in empty space. What may look like virtual pairs popping out from empty space is really phenomena which are produced by the real particles that entered the physical experiment.

Energy flow between particles and the electromagnetic field

e- ~~~~~~~ e+

In a Feynman diagram, no arrow is drawn into a line which describes the interaction of two particles via a virtual photon. That makes sense because the interaction is mutual, it equally happens to both directions.

An exception to this is when system A emits a real photon which later interacts with system B. In that case we could specify the direction of the interaction.

When an electron approaches a positron, the pair draws energy from the electric field. The pair acquires kinetic energy from the field. We cannot uniquely say how the extra kinetic energy is split between the particles because the numerical value depends on our choice of the inertial coordinate system. It is the pair which acquires kinetic energy relative to each other and not an individual particle. Similarly, it is the pair whose electric field gives up that energy.

Since we are dealing with a pair, maybe we should work in a 6-dimensional space plus the time coordinate? Then it is a single particle moving in a potential which has an infinitely deep well at the diagonal x = x', y= y', z = z'.

Using the 6-dimensional space, the energy flow is now between the potential energy and the kinetic energy of the single particle.

How does energy flow create real pairs?

Real pair production in quantum field theory has a classical analogue in an electric breakdown of a medium under an intense electric field. In a dynamic system, an intensifying electric field may produce an electric breakdown.

When an electron and a positron approach each other they receive kinetic energy from the electric field. That means that the electric field grows weaker in most of the space. Looking from far away, the charges of the electron and the positron cancel out each other and the electric field becomes almost zero.

But the electric field does grow in magnitude in the zone between the electron and the positron. At the middle point it doubles relative to just having one of the particles present. Nature has an option to reduce the growing electric field by creating a pair which cancels out some of the electric field. The electric field behaves then nonlinearly because the resulting field is not the sum of the two original component fields.

When the electron and the positron recede from each other, the extra kinetic energy that they drew from the electric field returns back to the electric field. The electric field grows stronger almost everywhere except in the zone in between the electron and the positron.

Suppose that we have a growing electric field:

      ------------->   E electric field

Pair production reduces the electric field:

      ------------->  E
     e-  <----   e+ field of the pair

We assume that the electron and the positron are born close to each other. They need to tunnel far enough of each other so that the energy reduction of the electric field offsets the energy of the pair. The energy of the original electric field E flows into the energy of the pair. The creation of the pair opens a new degree of freedom into the system and energy flows to that new degree of freedom.

It is the spatial energy flow which is important in breaking in a new degree of freedom. It is not the static electric field.

If we have a plane wave, then conservation of momentum prevents the conversion of electromagnetic energy to real pairs. There is no energy flow in a plane wave. There is a uniform energy density throughout the whole space.

Measuring the magnitude of energy flow

How can we quantify the magnitude of the "dynamics" of a system? If we are working in the center of mass coordinates of a colliding electron and a positron, then we can divide space into cells and determine the electromagnetic field energy in a cell as a function of time.

Another way is to do a Fourier decomposition of the field against time. What kind of a decomposition does a "dynamic" field have? What is the decomposition of a static field? Since local flow of energy is important in pair production, a global decomposition into Fourier modes may not help us in any way?

If we have several electrons flying at random directions far away from each other, there is energy flow in the center of mass coordinates, but no pair production. What is the difference from a collision? Why the energy flow in a collision produces pairs?

In the far away case, we can ignore interactions. For each flying electron, there is an inertial coordinate system where its field is static close to the electron. And a static field does not produce pairs.

In the collision case, there is no inertial coordinate system where the field is static. There is flow of energy regardless of the inertial coordinate system.

Self-energy as energy flow

If an electron is in an accelerating motion, then there is no inertial coordinate system where its electric field is static.

Conjecture 1. The self-energy Feynman diagram, where an electron seems to send a virtual photon to itself, describes the energy flow in the electric field of an accelerating electron.

We will address later the divergence of the self-energy Feynman integral. The self-energy loop can circulate an arbitrary amount of momentum p, which causes a logarithmic divergence in the integral value.

A freely flying electron does not have the self-energy phenomenon, because it is static in an inertial coordinate system. People sometimes draw an electron flying inside a cloud of self-energy photons. In our opinion, that is wrong.

Quantizing the energy flow process

We now have some classical intuition about the energy flow in a collision. A great conundrum is how on earth can Feynman simplify the complex process into the flight of a few virtual particles, and get accurate numerical results!

Quantum mechanics does exhibit similar simplicity in the hydrogen atom. When a hydrogen atom becomes excited, a very complex orbital change can be summarized as absorption of one photon.

The simplest Feynman diagram that describes an electron flyby of a nucleus, contains just a virtual photon line which transfers momentum p to the electron but no energy.

But the process does involve flow of energy. The kinetic energy of the electron draws on the energy of the field. Also, there is flow of energy into the zone between the electron and the nucleus. When the electron recedes, the extra kinetic energy flows back to the field.

A potentially forming real electron-positron pair tries to take its toll on the increasing energy in various places in the field.

The energy flow in the flyby happens slower than the speed of light. If we conjecture that the flow is transferred by quasi-particles, those particles have a non-zero rest mass. This is the reason why the conversion to a real pair (which has a non-zero rest mass) can happen, while it cannot happen when energy is carried by light-speed photons.

We have found a new interpretation for a virtual photon: it is a quasi-photon with a non-zero rest mass. Because of the rest mass, it can move at a slow speed and carry more momentum than a photon.

Real pair production can be interpreted as a tunneling process where an energy flow into a zone of space tunnels into the creation of a real pair. The energy must arrive to the zone at a speed less than light because otherwise we cannot conserve momentum in pair creation.

A virtual pair is a failed creation of a real pair.

What is the energy quantum of a quasi-photon? To create a real pair, it should be larger than 1.022 MeV.

Classical limit

Our analysis has finally brought us some understanding of the classical limit of pair production.

The virtual photon in the simplest Feynman diagram signifies a complex energy flow between the kinetic energy of the particles and the electric field, as well as energy flow within the field.

Those zones where the electric field grows will tend to produce pairs. Classically, we may imagine that the increase of the electric field somehow destabilizes a local electric dipole and may cause it to break apart, releasing a free electron and a positron. A static electric field, on the other hand, cannot break the dipole. As if an electron would be in a hole. A static electric field cannot pull it out, but an increasing electric field can.

---------------------> E electric field

______         ______

          |_e-_|

              ^

              |   disturbance

A growing electric field is a disturbance which can pop the electron out of the hole. The hole is actually the hole of the Dirac hole theory. It is immune to a static electric field. It does not polarize under the static electric field, in contrast to a more familiar electron "hole", the hydrogen atom.

If electron holes have an infinite density per cubic meter, then any small polarization of a hole under an electric field would cancel out all electric fields altogether.

Zitterbewegung through tunneling in the field of the electron?

Pair production reduces the increase of an electric field. But why it cannot reduce the field which was there before the increase started, the static electric field?

We have been claiming in this blog that a static electric field cannot produce pairs. Suppose that we are wrong.

Let us calculate an example.

The electric field strength of an electron reaches the Schwinger limit at a radius of 3 * 10^-14 m. The classical electron radius is 3 * 10^-15 m and the Compton wavelength is 2 * 10^-12 m.

Suppose that we have an electron at a distance 1 meter from a positron. How close should we move the electron to the positron to recover its rest mass of 511 keV in mechanical energy?

The potential of the positron-electron system is

      k e^2 / r.

The above should be equal to 511,000 e. We get

r = 9 * 10^9 * 1.6 * 10^-19 / 511,000 m
   = 3 * 10^-15 m.

It is the classical electron radius.

The zitterbewegung of an electron has an amplitude equal to the Compton wavelength 2 * 10^-12 m.

If there exists tunneling phenomena in the electron field, it should happen at the scale of the classical radius 3 * 10^-15 m. Could tunneling at such short distances explain the light-speed zitterbewegung at a distance scale 700 times larger?

Suppose that a virtual pair is created close to an electron. The virtual positron comes within less than 3 * 10^-15 m of the electron and annihilates. Then the virtual electron becomes real. The location of a 511 keV electron cannot be determined with a better accuracy than the Compton wavelength. The electron appears to have jumped a distance of approximately one Compton wavelength.

That could be the explanation of the zitterbewegung. But zitterbewegung is not present in Dirac solutions which contain purely positive frequency waves. How would we explain the absence of zitterbewegung in that case?

In this blog we have been claiming that a static electric field cannot produce pairs. But if we have an electron present, then it is a system of an electron plus its static electric field. Maybe we could allow pair production in such a case?

Can we avoid non-causality if we allow pair production close to a free electron?

Static electric field = laser beam of zero frequency, high-energy collisions

Let us study more carefully the interpretational dichotomy we found in our previous blog post.


              ^  ^  ^
  |   |   |   |   |   |   |   |   |
  v  v  v              v  v  v

-----------> observer

The arrows denote a static electric field in a tube. An observer flies in the tube at almost the speed of light.

Alternatively, electromagnetic waves propagate in the tube and cause polarization that progresses at almost the speed of light, and the observer is static in the tube.

The observer thinks that he is flying in a medium against a coherent laser beam.

If the observer has an electron flying along with him, he will see the electron move up and down, propelled by the electric field of the laser beam.

He thinks that the laser beam consists of real photons. The photons scatter from the electron and cause it to move up or down. The (small) energy to oscillation comes from the laser beam.

The electron will slow down slightly as it collides with photons coming from the right in the diagram.

We have another observer who is static relative to the tube.

If the static observer believes in a Feynman-style model, he thinks that the static electric field consists of virtual photons that transfer momentum. Virtual photons are absorbed by the electron flying past him. Those virtual photons transfer momentum to the electron and cause it to oscillate up and down. The energy to the oscillation comes from the original kinetic energy of the electron. The electron slows down slightly.

Here we have two observers. One thinks that he is in a dynamic field of a laser beam and another thinks that it is a static electric field.

What about the vacuum polarization loop in the Feynman diagram where we have a virtual photon transferring momentum? Does it have a counterpart in the dynamic interpretation as a laser beam?

The Feynman diagram which describes an electron colliding with a photon is:

----------- e- -------------->
   /                        \
 /                            \

The electron absorbs a photon and then emits it again. We cannot put a virtual pair loop into this diagram because there is no internal photon line.

If we add a self-energy photon for the electron, then we can put a virtual pair loop into it.

The Feynman diagram which describes static electric field interaction is something like:

----------- e- --------->
               |
               |
----------- e- ---------->

The vertical line is a virtual photon which carries a momentum p but little energy. We can add a virtual pair loop to the virtual photon line. The Feynman integral of the loop diverges badly. There are renormalization tricks to remove the divergence.

Conjecture 1. The laser beam diagram describes the physical system as well as the static field diagram. We can do away with the diverging Feynman integral by replacing the static electric field with a laser beam of a very low frequency.


Conjecture 1 probably implies that we can discard the vacuum polarization loop altogether from a Feynman diagram where the internal virtual photon transfers just momentum but no energy. That is the case if we have two electron beams moving at opposite directions at a considerable distance:

e- e- e- e- e- ----->

<--- e- e- e- e- e-

Conjecture 1 probably implies that the Schwinger effect does not exist.

High-energy collisions

What about the case where there is a virtual photon line between the electrons, and this virtual photon carries substantial energy? If a high-speed electron collides with a slow electron, there will often be a substantial kinetic energy transfer.

If the collision is high-energy, even a real pair can be produced.

e- ---------------------->
                |
                |
<------------------------ e-
Diagram A.

Above is a diagram in the case where no real particle is produced, just some momentum and energy is transferred. We draw the lower electron moving to the left to make the diagram more realistic.

                  /
                /
e- ------------------------>
                    |
                    |
<-------------------------- e-
Diagram B.

Above the upper electron converts some of its kinetic energy to a real photon and then gives its extra momentum to the lower electron.

e- ----------------------------->
                |  ----------- e- --->
                |/
                 \
                    ------------ e+ --->
                          |
<--------------------------------- e-
Diagram C.

Above the upper electron gives up some of its kinetic energy to a virtual photon. The photon must carry away a lot of momentum. The photon transforms to a real pair, which transfers its extra momentum and energy to the lower electron.

Classical limit of high-energy collisions

We need to understand the high-energy collision mechanisms intuitively. Studying the classical limit is one way of understanding what is going on.

If the electrons would have a very large rest mass and a very big charge, then the collision should resemble the collision of two classical charged bodies.

Diagram A corresponds to a very mundane exchange of momentum and energy, mediated by the static electric field of the charges.

Diagram B describes the radio waves emitted in the collision. The electric field changes in a very complex way as the charges accelerate or decelerate. Since there is no positive charge present, there is no dipole and the intensity of emitted real photons should be very low or zero. The converse process is a photon absorption. If we try to absorb a photon with a system which contains just two negative charges, it is hard to conserve momentum, energy, and angular momentum.

Diagram C is the problematic one in the classical limit. Classically, pairs cannot jump out of empty space. What about a classical collision in a medium which contains elementary electron-positron dipoles which the changing electric fields can separate?

https://en.m.wikipedia.org/wiki/Electrical_breakdown

An electrical breakdown is the natural classical analogue of pair production. When colliding charges come very close together, charges of dipoles in the medium separate and try to reduce the increased electric field.

If we have a zone in space where the electric field E has a large absolute value, then, classically, the energy density is E^2. An electric breakdown will reduce the energy density and convert some of the energy of the electric field into the energy of the separated charges.

From this point of view, pair production is a tunneling phenomenon where energy of the electric field is converted into energy of the created pairs.

In a medium, the density of atoms restricts the rate of tunneling. But if we assume that empty space contains an infinite density of virtual pairs, then the tunneling rate should be infinite. This paradox is at the heart of non-causality, and probably at the heart of Feynman integral divergences.

Our goal is to show that production of pairs happens only under dynamic conditions where there is flow of energy between the kinetic energy of the system and the energy of the electric field (= potential energy). Then we probably can tame the divergences. This also implies that the Schwinger effect does not exist.

Does the Schwinger effect exist for a constant electric field?

UPDATE Sept 28, 2018: Julian Schwinger's 1951 paper treats two cases: a constant electric field and a plane wave field. Schwinger concludes that in a plane wave field there is no nonlinear behavior and that the Maxwell equations hold. But for a constant field, pair production occurs. Schwinger does not say anything about the paradox that a constant electric field can be interpreted as a plane wave of an infinite wavelength. Why does one field produce pairs and the other not?

Schwinger uses renormalization to remove a logarithmic divergence in the constant field case. Maybe the divergence is a result of an underlying assumption that empty space contains an infinite density of virtual pairs which may tunnel into real pairs? That infinite density is the origin of "non-causality" in physical processes and we aim to make physics causal in this blog.

----

The Schwinger limit is the electric field strength above which pair production of real electrons and positrons becomes significant. Consequently, the Maxwell equations of electromagnetic waves become significantly nonlinear.

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

Julian Schwinger calculated the limit to be around 1.32 * 10^18 V / m.

https://journals.aps.org/pr/abstract/10.1103/PhysRev.82.664

Julian Schwinger in his 1951 paper estimated the pair production rate for a constant electric field.

Static field of a capacitor

Let us do some thought experiments. Suppose that we would be able to collect enough static electric charges in a capacitor, so that the electric field between the plates would approach the Schwinger limit.
             
________|________
 ^        ^        ^       ^
 |         |         |        |
_________________
               |

The arrow lines in the diagram represent the electric field. We should observe real electrons and positrons materializing between the plates from time to time, if the Schwinger effect exists for a static electric field.

The materialization process can be seen as a tunneling process where a virtual electron-positron pair is created and the particles move in the electric field far enough to gain the combined energy 1.022 MeV and become real.

If we reverse the direction of time, then we see a real electron and a real positron tunnel against the electric field and meet to annihilate each other, in such a way that the energy is zero, and no photon is created.

How can we calculate the electron-positron production rate of the Schwinger process? That is a problem because the pair seems to appear "non-causally" from the space between the plates. If we assume that the space between the plates contains virtual electron-positron pairs that try to tunnel into real pairs, how do we know the number of virtual pairs per cubic meter? Equivalently, if we assume that between the plates there is a Dirac sea filled with electrons in a negative energy state, how do we know the density of these negative energy electrons? The more there are such electrons, the larger the flux of tunneling pairs should be.

We need to study the Schwinger 1951 paper and check how he restricts the number of negative energy electrons or virtual pairs.

If there are electrons in one plate, then a single electron can tunnel through the potential wall which is holding it in the plate. The electron tunnels and appears in the space between the plates. One might interpret this process as a virtual pair forming and the positron flying to annihilate the electron in the plate. The tunneling process is causal because it is a process of a real electron in the plate.

A very slowly changing electric field of a laser beam

We may create an almost static electric field if we have a very strong laser which sends radio waves of an immense wavelength. If the cycle of waves is longer than, say, 1 second, then we might treat the electric field created by the laser beam at some point of space as essentially constant, or static.

The laser beam consists of real photons. Conservation of momentum and energy does not allow the conversion of photons to real electron-positron pairs.

Lorentz invariance, too, implies that no pairs can be produced, since if we switch to a suitable reference frame which moves to the direction of the photons, we can make the power flux per square meter of the laser beam appear to be arbitrarily low, and consequently, it cannot create real pairs. This Lorentz invariance argument is mentioned in Wikipedia.

How would Julian Schwinger explain why the slowly varying electric field of a laser does not produce pairs, but a constant electric field does?

Zitterbewegung

There is a hypothesis that the strange zitterbewegung - trembling motion - of most solutions of the Dirac equation is really due to a process where the static electric field of the electron creates a virtual pair, and the electron annihilates with the virtual positron, making the virtual electron real.

If the above explanation is correct, then we might consider zitterbewegung a Schwinger process in the static field of a single electron.

Conservation of momentum and conservation of the center of mass is a problem in zitterbewegung. If we would be able to measure zitterbewegung, how could we ensure that conservation laws are obeyed? Classically, a point particle cannot jump around unless other particles are involved.

In our capacitor thought experiment, if a virtual pair is created between the plates, and the virtual positron annihilates with an electron on a plate, then we might see this as zitterbewegung where the electron jumped a macroscopic distance to the position where the virtual electron was.

Schwinger process only occurs with dynamic fields?

Our thought experiment of a laser beam suggests that there really is no Schwinger process with static electric fields. The tunneling of electrons from one plate is a real phenomenon, but it is not the Schwinger process where a pair appears non-causally between the plates.

We do witness creation of real pairs all the time, but that happens in dynamic situations, in particle colliders and in nature.

According to Wikipedia, people are trying to confirm the Schwinger process by shooting two laser beams at each other. Then the creation of pairs may be seen as photons colliding. The situation is very dynamic.

Conjecture 1. There is no Schwinger process with static or very slowly changing electric fields. Thus, there is no non-causal production of pairs. In dynamic situations, real pairs are created causally.

A general goal in this blog is to prove that there are no vacuum fluctuations. All that we observe is a causal consequence of real particles. If there were a Schwinger process in a static electric field, that would break the causality principle.

An observer flying in a tube of an alternating electric field

We need to study more carefully what exactly is different between a static electric field and a laser beam with a very low frequency.

Let us consider another thought experiment. Suppose that we have an observer flying at almost the speed of light in a tube where we have a static electric field pointing alternately up and down.

 ^  ^  ^                       ^  ^  ^

 |   |   |   |   |   |   |   |   |   |   |

             v  v  v  v  v

------->  observer

Let the observer carry a vertical dipole antenna. He will observe the charges in the  antenna to start oscillating. He may interpret this in the way that he is static in a beam of a low-frequency laser and the "quasi-photons" in the laser beam excite electrons and nuclei in the antenna and make the antenna to oscillate. He sees the photons as quasi-photons in the sense that he can measure the speed of the electromagnetic waves and conclude that they propagate at a speed slightly less than light in the vacuum.

A static observer, on the other hand, interprets that the kinetic energy of the observer is converted to the oscillation of the antenna.

Here we have an interesting dichotomy: the moving observer sees the antenna absorbing quasi-photons, while the static observer sees the charges in the antenna start moving because of their own kinetic energy.

What about pair production? The moving observer sees the quasi-photons move at a speed slightly less than light. He may interpret that they have a rest mass > 0. Since they have a non-zero rest mass, one can conserve energy and momentum and convert them into a real pair with a non-zero rest mass. But can the pair have a rest mass = 1.022 MeV?

Light propagates in an optically dense medium at a speed less than light in the vacuum. We may think of photons in a dense medium as quasi-photons with a non-zero rest mass. We know that such photons will create real pairs if the energy of the photon is > 1.022 MeV.

What if we make our alternating tube such that the quasi-photons have a very low energy? It just requires that the alternating sections if the tube are very long.

If there is a Schwinger effect in a static electric field, then our observer will encounter those pairs moving at almost the speed of light. How does he interpret what he sees? He thinks that he is in a flux of very low-energy quasi-photons. How can that flux create 1.022 MeV real pairs?

This must have something to do with the fact that the virtual photons in Feynman diagrams are not quantized: we can always split a virtual photon carrying a momentum p to two virtual photons each carrying a momentum p / 2. If we have an observer moving relative to these virtual photons, then his hypothetical collisions with virtual photons are not "quantized".

The Schwinger effect would create pairs which are very much quantized: they are point particles with a 511 keV rest mass. If there is really no quantum of a static electric field, then how can it give rise to quantums of the Dirac field?

Light propagating in a polarizable tube

Suppose that we send an electromagnetic wave down an electrically polarizable tube. Then the tube at any instant of time will look like the tube in the diagram of the previous section. There will be alternating sections of tube where the electric field points up and down.

If we have a static observer inside the tube, he will see roughly the same things as the moving observer in the previous section of our blog post.

The electromagnetic wave propagating through the tube is like light propagating in an optically dense medium.

Suppose that in an optically dense medium, we have an observer who moves as fast as light moves in that medium. For him, a beam of light moving to the same direction looks like he would be static in a static electric field. If the Schwinger effect for a static field would exist, he should see electron-positron pairs materializing.

The Schwinger effect requires the static electric field to be immensely strong. A static observer would see a very intense laser beam in a medium. The density of individual photons is large. Can these photons somehow create electron-positron pairs if the energy of a single photon is low?

A Feynman diagram which builds a 1.022 MeV pair from a collision of atoms and 1 eV photons would contain over a million vertexes. It must be extremely unlikely that pairs are created if the photons are low energy.

In this last thought experiment, we found two ways of looking at the experiment:

1. a fast moving observer who sees the electromagnetic wave standing still; if Schwinger is right he should also see pairs being created;

2. a static observer who sees low-frequency light propagating in a medium; no pairs should be produced because the photon is low-energy.

Who is right, 1 or 2?

A photon in a medium has a "rest mass"?

Our observation about light propagating in an optically dense medium opens some interesting points of view.

Since the speed of the photon is less than the speed of light in the vacuum, we can interpret the photon as a non-zero rest mass particle whose energy is split between the rest mass and the kinetic energy.

If the medium is optically denser, then the rest mass of the photon is larger. The photon makes the charges in the medium to vibrate and in that way "adopts" some of the rest mass of the charges.

In our example of a polarizable tube, the heavier are the polarizable charges are in the tube, the slower the electromagnetic wave propagates.

In our blog we have raised the hypothesis that electromagnetic waves are mechanical waves of "virtual electron-positron pairs" of a zero rest mass, where the coupling between the charges is the Coulomb force. In the tube example, if we make the charges to have a zero rest mass, then the speed of electromagnetic waves is probably the speed of light in the vacuum. What is the density of these "virtual pairs" per cubic meter? Does the density matter?

When a photon comes close to a nucleus, it can "adopt" some of the rest mass of the nucleus. Because the photon now has a rest mass, it can convert itself to a real pair and conserve energy and momentum. This is one way of viewing pair production.

When a photon comes close to an electron, it can adopt some of the rest mass of the electron, but it cannot get a rest mass high enough to convert itself into a pair.

Did Michael Atiyah derive the value of the fine structure constant?

UPDATE Jan 14, 2019. Michael Atiyah passed away on January 11, 2019. He was born on April 22, 1929. Rest in peace.
----
Short answer: no. The derivation is almost certainly wrong in many ways.
----
According to Lubos Motl, the Riemann hypothesis proof that Michael Atiyah gave out today:

https://drive.google.com/file/d/17NBICP6OcUSucrXKNWvzLmrQpfUrEKuY/view

is flawed in many ways:

https://motls.blogspot.com/2018/09/nice-try-but-i-am-now-99-confident-that.html

Also, several anonymous commentators on the Internet have claimed that the definition of a weakly analytic function in the paper is wrong.

In another paper, Michael Atiyah claims that he can derive the exact value of the fine structure constant α from renormalization principles and geometric mathematical principles:

https://drive.google.com/file/d/1WPsVhtBQmdgQl25_evlGQ1mmTQE0Ww4a/view

The α paper is of interest to our blog. The fine structure constant can be defined, for example, as

                 α = k_e e^2 / h-bar c,

in SI units, where k_e is the Coulomb constant. In natural units,

                 α = e^2 / 4π.

In natural (Planck) units, α is the force between two electrons when they are placed at a 1 natural unit distance from each other.

Our goal in this blog is to show that the correct way to calculate scattering amplitudes is to use a cutoff for the energy or the momentum of virtual pairs. No renormalization should be used. In our approach, the force between two electrons at one unit distance of each other can be set anything we like - it is truly a constant of nature and not a geometric mathematical constant like π is. That is, the physics is sensible in all models regardless of the value of α.

What about π? Can we construct sensible physical models with different values of π? The mathematical value of π is based on an idealized plane geometry which happens to be approximately the geometry which we observe in physical space. Can we construct a physical model where there are no idealized planes at all, at any level of abstraction?

If we have a physical model where the space is smooth at small distances, then we can define π as the limit of the circumference / diameter when we study smaller and smaller circles.

If our physical model would be a discrete model, then the mathematical π might have no bearing on our physical model.

If our physical model would be a fractal, then the mathematical value of π might not be relevant.

Thus, in smooth and continuous models of physics, the mathematical value of π will be very relevant. In other kinds of models, it might not be.

If we return back to the case of α, Michael Atiyah would have hard time proving that the physics will not work with any other value of α than what we have in this universe. A priori, the result of Michael Atiyah is most probably wrong, but we have not yet studied his paper in detail. Michael Atiyah does not seem to use much physics in his paper. It would be strange if one could derive the value of the fine structure constant without a physical analysis.

Summary of our virtual photon models so far

We are stuck with the problem that we cannot find an intuitive quantum mechanical model for the electron flyby of a nucleus.

The trivial classical model is the static electric field of the electron and the nucleus. If we assume a heavy electron a heavy nucleus, they both move at a constant velocity and the finite communication speed of special relativity does not need to come to play.

Rod model

We made a quantum mechanical model of the static field by assuming a massless elastic rod on which the electron tugs on. But the problem with the rod model is that it does not give us an intuitive model of virtual pair production in association with the tugging.

Z ----------------------- e-

In the rod model, the virtual pair should appear like this:

Z ------ e- --- e+ ----- e-

The momentum p which the nucleus Z is sending the original electron is for a short time absorbed by the virtual pair. Since the virtual pair must disappear quickly, it has to give the momentum back to Z or to the original electron. However, the rod model does not give us a natural mathematical formula which should govern the appearance of the virtual pair.

Exchange of two real photons

In another model we made, the nucleus Z and the electron both send real photons whose magnetic field creates the pull. The nucleus and the electron have to move relative to each other so that the magnetic force is present.

This model is not intuitive from the classical point of view. Why should a simple pull be implemented in a complex way through magnetic fields? Why cannot we observe these real photons if they exist? Why should the charges move?

Production of real pairs

One way to study virtual pair production is to study real pair production. A virtual pair is like a real pair which failed to materialize because of lack of energy.

For a real pair to appear, it should tap on the kinetic energy of the electron or the nucleus. If we are working in coordinates where the nucleus Z is static, then the nucleus cannot send out the real photon which would turn into a real pair.

The pair must use the kinetic energy of the electron. The Feynman diagram is something like this:

----- e- -------------->

           |    ----- e+ -->

           | /

             \
                ------ e- -->

                   |

                   |

------- Z ------------->

The electron sends a photon which contains at least 1.022 MeV energy and a lot of momentum p. The momentum is large because the energy had to come from the kinetic energy of the electron. The photon is transformed to a real pair. The pair gives its excessive momentum to Z in a virtual photon.

If the electron would send out a real photon, the diagram would be like this:

              ~~~~~~~
           /
         /
----- e- -------------->
              |
              |
----- Z --------------->

The electron transforms some of its kinetic energy to a real photon. Then the electron gives its excessive momentum to the nucleus Z. This is called bremsstrahlung.

Production of real photons is a dynamic process. The electron cannot be static. It must have kinetic energy which it radiates away. On the other hand, the static electric force is present even if the electron and the nucleus are static. One wonders if virtual pairs can be produced at all if they are static.

Question 1. Is there empirical evidence that the loop correction is needed in the exchange of a virtual photon which has a negligible energy? The Feynman formula does not contain the energy of the virtual photon, just the momentum it carries. Or do the loop corrections cancel each other out if there is only a negligible energy transfer in the flyby?

A virtual particle is a loop of two particles? Feynman cannot handle tunneling

In the previous blog post we conjectured that the virtual photon which mediates the electric pull between the nucleus and the electron is actually two real photons which move to opposite directions:

e- --------------->---------->-------
                    ^             |
                     |             | photons
                     |            v
Z --------------- >---------->-------

We make the photons real by adding enough energy to them so that the energy E and the momentum p carried by the photon match. From where do these photons get the energy? If the electron moves slowly, then it may not have enough kinetic energy to donate for the photons. Also, a slow electron spends more time near the nucleus and receives more momentum p. It looks like we have to borrow the energy from the vacuum.

The loop of photons bears some resemblance to a loop of a virtual electron and a virtual positron:
         ______
       /             \
      ^               |
e-   |               v    e+
       \_______/

If we want to make the pair real, we need to borrow energy from the vacuum.

In an earlier blog post, we defined a virtual electron as a real electron which has entered a high potential wall and has a negative kinetic energy.

One aspect is that when we have a system of particles, we cannot really define the energy of an individual particle. It is the system, whose energy we can calculate. Should we consider all particles real then and call the system virtual when the potential energy of the particles exceeds the total energy we gave to the system?

If we have a positron which tries to tunnel through the potential wall of a nucleus Z, it is not the positron which becomes virtual but the system Z e+ which becomes virtual. How do we handle this in a path integral? If we use the Schrödinger equation, the wave function of the positron will decay exponentially if it comes too close to the nucleus. Feynman propagators do not contain such exponentially decaying terms because they are free particle propagators.

We have found one of the reasons why the Feynman integrals for virtual particles diverge:

Feynman formulas do not take into account the fact that the probability amplitude of a path should decay exponentially if its kinetic energy is negative at some point.

An example is a photon whose energy is less than 1.022 MeV. Let the photon scatter from a nucleus Z.

Any path which contains an electron-positron pair has a negative kinetic energy (we interpret the rest mass of the pair potential energy) and the integral over the lagrangian should decay exponentially. The system is trying to tunnel through a potential wall.

But the Feynman formulas contain the product of the propagators of various photons, the nucleus, an elecron, and a positron. The propagators do not take into account the amount of energy that the system has initially available.

Feynman diagrams are about scattering processes where particles do not need to tunnel through ordinary potential walls like an electric potential. They may do that, though.

But in scattering there is tunneling through walls imposed by the energy of particle creation. No wonder Feynman integrals diverge in those cases as they cannot handle tunneling at all. The real question is why the formulas after renormalization give exquisitely accurate answers?

A virtual particle is the combined effect of two real ones?

The conjecture of the previous blog post suggests that we can replace a virtual particle with two real ones, and get an equivalent system.

Suppose that we want to push a boat away from a pier. We can use a rod and a virtual phonon, that is, the push of the rod, to accomplish the task. The boat recedes only slightly during the several seconds we push. We transfer a large amount of momentum to the boat but just a little bit of kinetic energy.

boat

  ____________

/                       |

\____________|

      |

      |  rod

      o

     /|\ man

     /\

The man has also another way of transferring a large amount of momentum but just a little bit of energy. He can throw a heavy football at the boat. The ball bounces and the man catches it.

     boat
_________

       /\

     /    \

   /        \

O           ball      

The Feynman formula for the phase change in the wave function caused by a virtual photon is exactly like for a real photon. It depends on the transferred momentum p, not on the energy. The phase change is thus equivalent to the situation where two real photons are used to transfer the momentum. One can carry p / 2 and another -p / 2 to the other direction.

But a problem remains: the electric field which pulls the electron towards the nucleus is radial. How could we realistically treat it as a combination of two very high energy photons whose electric field is transverse? We need to find out what the combined electric field of the two photons looks like. Could it be a radial field?

We could use two very high energy longitudinal real phonons in our rod analogue of the virtual photon. But is their combined effect really equivalent to a single push of the rod?

In the boat and rod example, how can we implement a pull with a bouncing ball? There we must fall back to the rod and a pull.

Let us study the heavy electron flyby from the point of view of dipole radio transmitters. Let us choose a coordinate system where the nucleus moves right at a velocity v and the electron left at v.

Schematically, we have first:

                -


+

and then:

-


               +

This would be like two horizontal dipole antennas transmitting half a wave, if we would have static extra charges + and - placed in the system. The the initial position is:

         +      -


+       -

and the later position is:

-       +


         -       +

We see that the upper row is a dipole which sent half a wave, and the same for the lower row.

Let us analyze the system classically.

The charges in the flyby are either static or moving at a constant speed (constant because they are very heavy).

Is it possible that the system could radiate away real radio waves? A problem is if we can use a Fourier transform to decompose the electric field? A single static charge has a static electric field. Any Fourier decomposition of a static field would suggest that it radiates energy, which does not make sense. On the other hand, a Fourier decomposition of the changing dipole field of a radio transmitter might make sense.

We return to the old Larmor radiation formula problem.

Conjecture 1. To determine the radio waves sent by a configuration of moving charges, do a dipole decomposition. Take all pairs of charges. Calculate the dipole radiation of each pair. Sum up all the dipole radiation fields.


How to calculate the dipole radiation field? The key factor seems to be the acceleration of a charge relative to the other. Static charges do not radiate.

What about the case of the heavy electron passing by a heavy nucleus? We may treat this classically. Imagine a strong coherent laser wave that tries to push the charges apart when they are already receding from each other. Can the laser field give up some of its energy to the kinetic energy of the charges?

----

Let us return to the conundrum how transverse electromagnetic waves can create a pull or a push.

The electric field of transverse waves does not create pull or push, but the magnetic field would do the trick if the charge is moving with respect to the transverse wave.

In the electron-nucleus example, we could make the nucleus to send a transverse wave at the electron, and vice versa. In the dipole transmitter diagram above, the electric field of the dipole is horizontal and the magnetic field is a circle around the dipole. The magnetic field lines point up from the computer screen or down to the screen.

Now we see that the magnetic field can be used to generate the pull between the electron and the nucleus, since the electron is moving.

If the electron and the nucleus are static, then it is harder to explain the pull with a magnetic field. We should assume that the field is moving, which sounds strange.

Our goal is to reduce the pull into two real photons that are sent between the electron and the nucleus. Then the Feynman vacuum polarization diagram starts to make sense, and we can analyze the physical configuration further.

The magnetic field of a moving charge is proportional to the velocity of the charge. We may assume that the nucleus and the electron move at opposite directions at speeds v / 2. If the speed is much less than the speed of light, then the electric pull dominates the magnetic pull.

Note that the magnetic force goes as v^2 when v increases.

If we let the nucleus do short fast spurts but stay static at other times then it creates sharp magnetic pulses. If the electron does fast spurts just at the moment it receives these pulses, then the electron will receive strong pull impulses toward the nucleus. This might be the mechanism how the electron and the nucleus, by exchanging two real photons, achieve a strong pull impulse toward each other.

Are equivalent classical systems equivalent also quantum mechanically?

We introduced a model where the electric attraction of a nucleus and an electron is replaced with a classical zero-mass elastic rod on which the nucleus and the electron grip. The rod model suggests that the mediating virtual photon "in reality" is a real phonon of the elastic rod.

This raises a general question:

Question 1. If we are studying two classical systems which, in some sense, have equivalent classical behavior, is then their quantum mechanical behavior equivalent, or even identical?


In the rod example, we replaced the classical electric attraction with a classical rod. Is the quantum mechanical behavior of the systems identical?

We can calculate the quantum mechanical behavior of a classical system by using a path integral and a lagrangian on the classical system.

S1 <--- P mapping of paths ---> S2

Conjecture 2. Suppose that we have classical systems S1 and S2. If we have

i) a one-to-one mapping P from the paths of S1 to the paths of S2, and

ii) P preserves the value of the integral of the lagrangian (= action) on each path, and

iii) P also gives some natural mapping from the observables of S1 to observables of S1,

then the quantum mechanical behavior of S1 and S2 is equivalent.

A virtual photon and a virtual phonon

Our elastic rod hypothesis makes the interaction between a nucleus and a passing electron more tangible, literally.

The Feynman integral which describes a virtual pair creation from a virtual photon diverges. But apparently, numerically right results can be acquired by putting a momentum cutoff to the virtual pair.

Our goal is to understand pair creation better. We believe that if we can model the creation in an intuitive way, then we can cure the divergence in the integral.

As we wrote in an earlier blog post, an obvious way to try pair creation is to make use of the acceleration of the electron. The electron can pull on the virtual positron and expel the virtual electron, and then accelerate away, donating them energy. If the pair stays virtual, the energy soon returns back to the electron.

In our previous blog post we modeled the flyby of an electron with classical point particles and a classical rod. What is the classical analogue of a pair creation?

e- -------- e+

A classical analogue of a virtual pair might be an electron and a positron which are tightly locked together with a very short rod. It is a positronium atom whose all mass-energy has been lost when the pair came very close together.

Alternatively, we can consider a real positronium atom and what kind of interaction might free the electron and the positron to go their own ways.

A real photon whose electric field pulls the pair apart can try pair creation. The photon tries to pull the pair apart. If the pulling does not succeed, then in the Feynman diagram we mark the real photon as temporarily decayed to a virtual pair.

A real photon can certainly pull a positronium atom apart.

What about an electron flyby of a nucleus? Can the longitudinal phonon in the rod somehow cause the virtual pair (or positronium) to separate? Is there a rapidly changing electric field that could give an energy boost to the virtual pair?

Let us make a thought experiment where the nucleus Z and the electron e- are very heavy. In that case, the flyby happens essentially at a constant speed. There is essentially no acceleration and the dynamics is low close to the electron.

Thus, the virtual pair creation by an accelerating electron cannot be responsible for the virtual pair loop in the Feynman diagram.

There is some dynamics, though. The dipole electric field of the heavy electron and nucleus does change direction and its dipole moment during the flyby. It is like a dipole antenna which sends out one half of a radio wave.

A positronium atom could harvest some energy from the changing electric field. Thus, virtual pairs will play some role. Maybe the Feynman diagram captures this dynamics and calculates the effect of virtual pairs correctly if we put a momentum cutoff?

------
An aside: Matt Strassler's take on virtual photons:
https://profmattstrassler.com/articles-and-posts/particle-physics-basics/virtual-particles-what-are-they/
According to him, they are temporary "ripples" in the electromagnetic field.
------

Above we were able to reduce the virtual photon conundrum into half of a cycle of a dipole radio transmitter. The virtual photon in some way describes what happens inside the transmitter.

The outside world will see half a cycle of radio waves. What does that look like? A dipole field turns 180 degrees.

If the electron is very heavy, then our elastic rod model has the nucleus tugging on the rod. The tugs do not alter the path of the electron significantly. It is like tugging on a rope which is attached to a heavy train passing by.

A single tug on the rod is half a cycle of a pressure wave in the rod. It is not a real phonon, because a real phonon must exist "free" and must contain one or more cycles.

A single tug is an "impulse", that is, a transfer of momentum.

Here we have a new characterization for a virtual photon or phonon: it contains only half a cycle and implements an impulse operation.

What is the lagrangian like for a tug of a rod? The lagrangian determines the speed at which the phase of the wave function rotates. The lagrangian is typically total energy minus potential energy, that is, the kinetic energy. The more kinetic energy, the faster the wave function of a particle spins.

Virtual photons are longitudinal virtual phonons of an elastic rod?

UPDATE Sept 15, 2018. The analysis below is flawed about a "real phonon". A real phonon is like a pressure wave. There have to be one or more cycles of pressure to talk about real phonons. If we use a rod to implement a single pull or push operation, then it is just half a cycle. It is not a real phonon. We can call half a cycle a "virtual phonon".

----

We could simulate the static electric pull or push between charges using an infinitely rigid rod between them. Just put the appropriate tension on the rod and there you go.

But in special relativity, there cannot be infinitely rigid rods, as they would permit faster-than-light communication. All rods have to be elastic.

In our previous blog post, one of the mysteries was why a Feynman "virtual photon" causes the correct phase shift on the electron wave function.

A possible explanation is that the pull between the nucleus and the electron is actually implemented using an elastic rod. The phase shift is due to the fact that there are degrees of freedom in the elastic rod, and, consequently, the phase of the pull phonon in the rod evolves as the phonon travels along the rod.

The rod is between the nucleus and the electron. The nucleus pulls on the rod to implement the electric attraction.

electron ------->
                          |   elastic rod
                          |
nucleus Z ----->


The rod hypothesis makes the backreaction on the electric field sensible when we switch on the interaction in some spacetime patch containing the electron. The reaction on the electron is to be pulled by the rod, and the backreaction of the rod is to get stretched by the electron.

In the QED lagrangian, it is unclear what is the backreaction on the electric field if we temporarily switch on the interaction. The effect on the electron Dirac wave function makes sense, but what is the backreaction on the electric field? If the electric field is rigidly determined by other charges in the system and does not have degrees of freedom of its own, then the backreaction is directly on those other charges. The electric field has no role in this.

Electromagnetic waves do have degrees of freedom but how could we implement a pull or a push with them? A low-energy photon which comes from the side and is absorbed by the nucleus-electron system can produce a pull or push. A problem in this model is where is the photon produced? Why would it cause the phase shift given in the Feynman formula?

If the electric field is an elastic rod, what are electromagnetic waves then? Adjacent rods interact. Electromagnetic waves are transverse waves that travel from a rod to an adjacent one. When an electron-nucleus system absorbs a photon, that means the phonon in the rod gives up its elastic energy to the kinetic energy of the electron and the nucleus.

Our blog post from spring 2018 stated that a photon is always created and absorbed by two charges of opposite sign. Contrary to the Larmor formula, an isolated accelerating charge cannot emit electromagnetic radiation. Conservation of momentum requires that a system of opposite charges must create or absorb a photon. The rod model clarifies this point.

Why we have not observed longitudinal electromagnetic waves? The wave is between two charges. Since the electric field goes as 1 / r^2, the energy of longitudinal waves probably falls very fast. Electromagnetic waves, on the other hand, are an efficient method for transmitting energy over great distances.

The rod hypothesis may also clarify momentum conservation in a QED system.

Our hypothesis has similarity to the rubber band model of QCD:

https://en.m.wikipedia.org/wiki/Color_confinement

How do real and virtual pairs come into the picture in the rod model? What is the reaction of the Dirac field to a phonon in the rod? Feynman lets a virtual photon to transform temporarily into a virtual pair. In QCD, the rubber band may break and its endpoints are new quarks.

The rod model can be used in the non-relativistic Schrödinger equation, too. There the force is mediated instantly and we do have an infinitely rigid rod which mediates the force instantly and does not contribute to the phase shift of the electron.

When we have an elastic rod, then the system electron + rod in a path integral gains a phase shift from both the electron and the rod (or, rather the phonon), since the path integral is the product of individual integrals of the rod and the electron. This shows that the non-relativistic Schrödinger equation gives a wrong prediction of the phase shift of the electron. The Feynman formula is right.

UPDATE: A fellow physicist at the University of Helsinki raised the question of what is the rod made of? If we have a rod made of iron atoms, it is the rest mass of iron atoms which makes it possible to transmit momentum and energy along the rod.

We have three ideas about the structure of the rod:

1. The rod contains some of the rest mass of the static electric field. But does the phonon move at the speed of light then?

2. The rod has no rest mass and its mass-energy comes exclusively from the mass-energy of the phonon. It is kind of a Münchhausen trick: the phonon creates the rod that it travels along. But if the rod has a zero mass-energy, then the uncertainty about the position of the rod is infinite, which does not make sense.

3. A path integral considers and calculates processes of classical systems like point particles. We can assume those particles are connected with classical objects like infinitely narrow rods. It is the probability amplitudes and their interference which gives rise to quantum phenomena. Thus, there is no problem in assuming a narrow rod. What about the mass of the rod? We certainly can use a zero mass rod to push or pull in classical mechanics. The elasticity of the rod comes from special relativity. What is characteristic of rods is that we can transmit a high amount of momentum with just a little bit of energy.  A real phonon is not bound by the mass-shell rule of a photon. Thus, a phonon is a model for a virtual photon.


Alternative 2 can be seen as a limiting case where we let the atom rest mass go to zero in the rod and let the strength of their interaction go infinite.

Alternative 3 is probably the right solution to the problem.

We can study the rod hypothesis experimentally by letting two charged objects, say marbles, interact through the electric push or pull. In theory, we could measure the speed at which the electric force propagates and measure elastic properties of the interaction.

TODO: calculate the mass-energy of a suitable "quantum mechanical" rod if we want to use the rod to simulate the electric force between the nucleus and the electron. Is the mass-energy close to the mass-energy of the electric field?

The optical theory of electron scattering and a "virtual photon"

Our previous blog post left open what exactly is the "virtual photon" which in a Feynman diagram carries momentum p from the nucleus Z to the electron which is flying by.

Let us assume that we have a heavy nucleus Z which is static and an electron flies past it quite far. The kinetic energy of the electron is almost exactly conserved (the radiation of real photons is negligible), but its momentum is slightly changed by an amount dp downward.

                 electron ------------------->


                                   Z nucleus


The wavelength of the electron stays the same after the flyby because the kinetic energy is the same. But to receive the extra momentum dp, the electron has to dive into the potential well of the nucleus, and during that dive, the wavelength of the electron is shorter.

Thus, the phase exp(i φ) of the electron wave function gets a boost, which is larger if the change in the momentum dp is larger.

Let us now look at the Feynman diagram of the process.


e- ----------->-------------->
                 | virtual photon
                 |
Z ------------>--------------->

The probability amplitude is calculated by multiplying the value for each line. In the diagram, the energy of the electron and the nucleus stays exactly the same. The flyby does not affect their phase exp(i φ) at all. But we know that the phase of the electron got a boost in the flyby. The role of the virtual photon is to add that boost exp(i φ) to the multiplied value of the diagram. When calculating the interference pattern of various electron paths, we need to have that boost taken into account.

Conjecture 1. The virtual photon in the Feynman diagram is just an obscure and misleading way to add the boost exp(i φ) to the phase of the electron.

UPDATE Sept 14, 2018: Our Conjecture 1 is wrong! See the blog post Sept 13, 2018. Also, since the non-relativistic Schrödinger equation assumes interaction which travels superluminally, also the prediction of the phase shift calculated using that equation is obviously wrong.


Since the wavelength of the electron is shorter close to the nucleus Z, we have the obvious optical analogue for the electron wave function.

Theorem 2. The vicinity of a nucleus acts like an optically dense area for the wave that describes a passing electron. QED.

UPDATE Sept 14, 2018. Optical theory gives wrong predictions for the the electron phase after the flyby! See our blog post Sept 13, 2018.


The behavior of an electron wave function that passes by heavy nuclei (but not too close) is like waves of light passing and entering grains of optically dense material. The grains have a higher optical density at the center of the grain.

Question 3. Is there such a thing as a "virtual photon"? We can define a virtual electron as a real electron which has entered a high potential wall and has a negative kinetic energy. The phase exp(i φ) of the virtual electron is frozen inside the wall. However, the phase of the "virtual photon" in the Feynman diagram changes because it is a device to add the boost to the electron phase.


A photon has a zero rest mass and can shed all its energy by redshift when it climbs up a potential wall. A photon coming from the surface of a neutron star is an example. In what way we could create photons of a negative kinetic energy?

Our simple example above does not contain anything about the possible vacuum polarization that the passing electron might cause around the nucleus. If the virtual photon is just a misleading way to add the boost to the phase of the electron, why would the virtual photon decay into a virtual electron-positron pair for a short while?

Is there experimental evidence that a virtual photon can decay into a virtual pair?

Our blog post about pair creation as tunneling suggests that a dynamic process is needed to create real or virtual pairs. There is a dynamic process in the flyby. It will produce real photons as bremsstrahlung and these real photons may decay into virtual pairs for a while. When the path of the electron changes, its electric field no longer moves at a constant speed in the rest frame of the nucleus Z. Also, the electric field comes closer to the nucleus electric field in the flyby. Thus, there is ample dynamic behavior present. We need to find out in what way that can cause vacuum polarization.

In the blog posts of spring 2018 we conjectured that electromagnetic waves are phonons in a sea of virtual electron-positron pairs, or the Dirac sea of negative energy electrons. A real photon would be a real phonon. Do virtual phonons exist in solid state physics?

Let us investigate the flyby of an electron in greater detail. The electron path will dive into the potential well of the nucleus Z:


_____                                        _____
         \_____________________/
             potential energy

electron pathlength in meters  -->

Let us designate the potential of a far-away electron with 0. In the diagram above the potential is proportional to 1 / r, where r is the distance from the nucleus. The integral over the path length in meters will tell us the boost dφ that the electron got to the phase of its wave function exp(i φ) in the flyby.

If the electron is slow enough not to be relativistic, and the potential well is not deep, then there is only a marginal change in the speed of the electron as it passes by the nucleus. Then the time is approximately linear in the diagram above. The impulse force that acts on the electron is proportional to 1 / r^2 and the direction of the force changes as the electron flies past. The momentum dp which the electron receives is proportional to the integral of potential^2 ~ 1 / r^2 during the flyby.

If the electron passes closer to the nucleus, the diagram above contains a deeper potential well. In that case, dp would grow faster than dφ as we make the potential well steeper.

The Feynman propagator for the (virtual) photon has a phase factor exp(i p (x - y)) where x is the position of the electron and y is the position of the nucleus. When we go closer to the nucleus, (x - y) is shorter and the phase change is smaller. This agrees with our analysis.

In the Feynman diagram, the virtual photon may temporarily decay into a virtual pair

                          _____     e+
                        /           \
~~~~~~~~~             ~~~~~~~~~~~~~~
                         \_____/  e-


The effect of a virtual photon is to advance the phase of the electron by dφ, and the effect of vacuum polarization should generally make the effect of an electric field weaker. In Feynman diagrams, typically the next level diagram works in the opposite direction to the first level diagram.

Vacuum polarization effects are larger close to the nucleus. We thus conjecture that in a close encounter, the phase shift is less than we would expect.

Let us consider the flyby process from the viewpoint of the energy of the electric field. The combined energy of the electric fields of the nucleus Z and the electron e- can be calculated by integrating E^2 over the whole space, where E is the electric field at a point in space.

When the electron flies by, then part of the energy in the electric field is converted to the kinetic energy of the electron. We pump some energy from the electric field and convert it temporarily into energy of the Dirac field (that is, energy of the electron field).

Vacuum polarization would mean that a virtual electron-positron pair will compete with the electron for that released energy The electron would use the energy to advance its speed while the virtual pair would use the energy in a vain effort to materialize as a real pair. If the flyby is far away, then the virtual pair will fail in their effort to materialize because there is not enough energy.

How to calculate the effect of a virtual pair if we do not use the framework of a Feynman virtual photon?

For a real photon, we assumed that the energy of the electromagnetic wave will try to tunnel into a pair with a probability which is proportional to the path length of the photon and the coupling constant. What is the corresponding probability in the case of an electron flyby?

An electron flyby might create real pairs or attempt creation of virtual pairs through the following process:

1. a virtual pair is born close to the flying electron;

2. the electric field of the flying electron pulls the virtual electron and positron apart; they gain some energy by moving in the electric field of the electron;

3. the flying electron accelerates and speeds away without claiming back the energy it gave to the virtual pair.


If the electron would be flying at a constant speed in empty space, then at step 3 the virtual pair would typically lose the energy it gained by moving in the electric field of the electron. There is an exception though: if the positron of the pair reaches the (real) electron, then annihilation will free enough energy to make the virtual electron real. The annihilation may be the origin of the peculiar zitterbewegung of wave packets constructed in the Dirac equation.

Our discussion above suggests that the primary mechanism of vacuum polarization in the flyby is through the acceleration of the electron in the field of the nucleus.

People tend to think that there is some kind of static vacuum polarization around the nucleus. That would require vacuum fluctuations to exist and we would end up with the problem of the vacuum containing an infinite amount of energy. Our approach is to explain vacuum polarization through dynamic processes only. The dynamic process we are considering is the flyby.

How does vacuum polarization reduce the phase shift of the electron in the flyby? It temporarily takes some of the kinetic energy of the electron away, which makes its phase shift smaller. The Feynman diagram approach suggests that vacuum polarization actually takes all gained kinetic energy to the virtual pair, since the pair replaces the virtual photon for a while. That would be strange. Maybe the Feynman approach approximates the potential with a step function? Then all acceleration will happen at the step, and it might be a good approximation that the virtual pair steals the energy at the downward step? This reminds us of the Klein paradox at a potential wall.

The magnetic field of an electric current tries to prevent changes in the current. When the electron approaches the nucleus, its velocity increases and some of its kinetic energy is stored in the magnetic field. The field pays back the energy when the electron starts receding from the nucleus.

Energy stored in the magnetic field might be considered a "real" photon which is emitted in the speedup and absorbed in the slowdown. After some thought, the magnetic field cannot be considered a "real" photon. It is just an expression of the static electric field and follows from special relativity.

If we can show that the electron flyby can be modeled with:

1. the electron moving at a constant speed until it absorbs a real photon;

2. the real photon is similar to the virtual photon of the Feynman diagram; it can decay temporarily to a virtual pair.

Then we can replace the strange virtual photon with a more tangible real photon. We can then do as in previous blog posts and show that the production of virtual pairs is "causal". We can replace the spike functions with smooth functions in their description and can show that an energy cutoff gives the best estimate of the process. Thus, we get rid of the divergence in the Feynman integral.

There are a lot of questions about the conservation of momentum and energy in the flyby. These may bear on our model:

As the speed of light is finite, how do we make sure that momemtum and energy are  conserved? Is there a kind of a database transaction commit at the end of the flyby where nature makes sure that its bookkeeping of momentum and energy stays in balance?

The mystery of the virtual photon that mediates the electric force

In our previous blog post we raised the question what exactly is the virtual photon which in Feynman diagrams mediates electric repulsion or attraction.

      /              \
    /                  \
    ~~~~~~~~       virtual photon
   |                     |
   |                     |
  e-                  Z nucleus

Time flows upward in the diagram. The virtual photon apparently is in no way an oscillating wave of the electromagnetic field, like a real photon is.

The virtual photon symbolizes the electric attraction of the nucleus. But why should we call it a photon at all and how can it transform itself into a virtual pair in the vacuum polarization diagram?

Why does Feynman use the Klein-Gordon propagator for the virtual photon? The static electric field is present at all times. What propagates?

The answer may lie in the fact that Feynman studies what happens if the electric field of the nucleus is switched on in a small spacetime patch. We assume that the electric field is otherwise switched off. We want to find the perturbation which happens if we switch the field briefly on.

When the electric field is switched on, an electromagnetic wave will start to spread from the patch. That wave will exert an electric force on the electron. Furthermore, that wave will try to tunnel into real or virtual electron-positron pairs.

It is as if the nucleus Z would hold a capacitor from a handle, put it around the electron, and switch the electric field on for a short time interval.


                       +        -
                      |       |
                      |       |
Z ----------------|  e- |
                      |       |
                      |       |

Z uses the capacitor to simulate the electric field it would have if it were not switched off.

If e- acquires a momentum p from the capacitor, the handle will take care that Z acquires -p.

The virtual photon in the Feynman diagram symbolizes the capacitor setup and the real photon(s) which the capacitor sends when it is switched on. We kind of use real photons to reconstruct for a brief time the electric field which Z would have at the electron e-. These real photons will have vacuum polarizarion diagrams like any real photon.

What remains is that we should show our capacitor setup reproduces the Feynman formula for a virtual photon.

UPDATE: Does the capacitor approach make much sense for the static field of the nucleus Z?

We know from experiments the electric attraction that the field exerts on an electron. If our calculation of vacuum polarization would show that the field is weaker, we should just adjust the Coulomb force formula to match the measured value? Or is it so that the Coulomb field is the underlying field, but the effect of the vacuum polarization has to be added anyway?

Vacuum polarization can have an effect on a dynamically changing configuration. Specifically, the electron flies past the nucleus and its momentum will change somewhat. We could make a Fourier decomposition of the changing electric field caused by the electron. That decomposition will show real photons being produced in the flyby. And those real photons cause vacuum polarization. But how does this relate to the Feynman formula?