Electric charge is not conserved in reflection from an accelerating potential

UPDATE: Our post on Oct 23, 2018 about proton decay contains a numerical estimate of how quickly quarks will turn into antiparticles because of reflections inside the proton: in 10^2,500 years.

http://meta-phys-thoughts.blogspot.com/2018/10/the-proton-decays-because-positive.html

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The spider model of an accelerating mirror raised an interesting question. If reflection converts right-hand waves to a mixture of right-hand and left-hand waves, is it possible that electric charge is not conserved?

Or is the flux of positrons balanced by a flux of new electrons from the acceleration motor?

A symmetric collision of two electrons does not produce a chirp wave. The reduced mass method which is used in two-particle collisions transforms the problem to a single charge moving in a static electric field. Reflection from a static field does not produce a chirp.

If there is an inertial coordinate system where the electron appears to move in a static field, then there is no chirp.

The accelerating mirror has to be propelled by something else than the collision of just 2 particles to uncover phenomena that is associated with chirps.

The mathematical question is: if a flux of positive frequency waves is reflected from an accelerating mirror, does that create negative frequency waves which are not balanced by an equal amount of new positive frequency waves? The wave equation in question is the Dirac equation.

                               p
           e- ------------------------>                | <---- accelerating potential

Let us assume that we have a flux of electrons of momentum p hitting an accelerating potential. It is like a mirror. An observer sitting on the mirror will see some of the electrons as positrons of various momenta. This is because the electron wave will appear as a chirp, and a chirp contains negative frequencies.

Let us draw a spacetime diagram.

t
^              ^
|                 \
|                    \    \
|                      ^     \
|                    /         |
|                  /            |
|                /              |  mirror
|              e-   p
|
----------------------------------> x

It would be a little miracle if a wave can reflect from an accelerating mirror without acquiring negative frequencies. But is it possible that an equal amount of positrons and electrons are produced in the complicated reflection?

If we again think of the spider, it will mostly produce right-hand waves and a little flux of left-hand waves. Since the spider is accelerating, also the right-hand waves contain negative frequencies in the laboratory frame. If the spider thinks that it produces a flux F of left-hand waves, the observer in the laboratory frame will think it produces a flux 2F. Is there any way to show that the left-hand waves are not balanced by an equal amount of new right-hand waves?

Let us think of a simpler example. First the mirror is static and then suddenly starts to move at some speed v.

The question is what is the Fourier decomposition if the wave

        exp(i (E t - p x))

is changed to exp(i (E' t - p' x)) at a certain point.

If we have an arbitrary point z in spacetime, it will receive the first flux in a time interval t_1 to t_2. It will receive the second flux from some time t_3 to t_4. What is the Fourier decomposition of the flux that z sees?

The integral of
   
       exp(i (E t - p x)) exp(i (-E'' t - q x))

over an interval will differ a little bit from zero for most combinations of E'', q. Thus, there is almost always some negative frequency flux mixed with positive frequencies. Almost always is an electron mixed with a little bit of positron.

It is hard to analyze a single reflection, but by iterating the process we found a proof of charge nonconservation. The process of converting electrons to positrons is extremely slow: the flux is comparable to the (nonexistent) Unruh radiation. But in principle, we can change the charge, if the wave equation of the electron is not very exotic.

Theorem 1. Electric charge is not conserved if we allow arbitrary accelerating motion for electric potentials. This holds for the Dirac equation and most wave equations.

Proof. Suppose that we have a vessel where a mirror performs oscillation.

                     motor
   -------------- /--------
  |               /            |  vessel
  |       <--  |  -->       |
  |                            |
   ------------------------

Let us put an electron into the vessel. It will bounce from the accelerating or decelerating mirror many times. Most wave equations will create some negative frequencies from a positive frequency flux which bounces from an accelerating mirror. That is, a fixed frequency wave will reflect as a chirp. A chirp contains negative frequencies.

The wave function of the electron will slowly grow into a mixture of 50% electron and 50% positron. Does that prove that charge is not conserved? The electron may by accident gain so much energy that a pair is created. The pair may get annihilated later.  Probably there is a steady state where the number of electrons or positrons does not grow any more. Let this number be N.

What if we initially put N electrons into the vessel? After some time we will have N / 2 electrons and N / 2 positrons. Charge is not conserved.

There is still the possibility that the driving motor of the oscillating mirror might produce electrons or positrons that balance the created charges. But how would the motor know if we are turning electrons into positrons or the other way around inside the vessel? Thus, the motor cannot save conservation of charges. QED.


Theorem 1 is in conflict with the hole theory of Dirac. If the positron were just a hole in the Dirac sea of electrons, charge would be conserved. Paul Dirac apparently did not notice that his wave equation is not compatible with the hole theory.

Corollary 2. It is possible to change the baryon number, the electric charge, or any quantum number of a system where particles are subject to arbitrary irregular accelerations. QED.

Fermions turn into a boson soup?

Corollary 2 opens the possibility that all falling matter is reflected back from a black hole, according to optical gravity. We had the problem that baryons cannot be reflected back. But if there is intense heat in the falling matter, particles will feel intense random acceleration. Corollary 2 implies that some baryons turn into antibaryons and annihilate. The output from such a hot soup would be bosons, like photons. Zero rest mass bosons can always reflect back from the forming horizon.

But are falling baryons close to the forming horizon subject to intense heat for a long time? They will bathe in the heat of the reflecting bosons.

Noether's theorem and conservation of charge

On the Internet there are vague claims that the U(1) gauge symmetry of the QED lagrangian implies the conservation of electric charge through Noether's theorem. The U(1) symmetry means that the phase of the wave function can be rotated by an arbitrary amount, locally, and the physics does not change.

https://en.m.wikipedia.org/wiki/Noether%27s_theorem

Noether's theorem looks exact, at least in classical physics where we can assume an infinite signal speed. In classical physics, there is no paradox if positive frequency waves turn into negative frequency waves. Noether's theorem cannot forbid it.

It seems that on the Internet no one claims having an exact proof for charge conservation in QED, through Noether's theorem or otherwise.

The free Dirac equation apparently conserves the number of particles, and consequently, also charge. But we are interested in interacting systems.

In Feynman's diagrams, charge conservation is enforced. But Feynman's diagrams are just a perturbative approximation of the physical process.

Can we write wave equations which would satisfy a charge conservation law? It looks difficult or impossible to block the leak from positive frequencies to negative frequencies.

Matter-antimatter imbalance in the universe

We found a way to turn particles to antiparticles. Can this in some way explain why there seems to be much more matter than antimatter?

The process is completely symmetric. If there is a Big Crunch, the process may equalize the amount of matter and antimatter.

There is one eminent asymmetry in the universe. Entropy is increasing with time. Could that somehow create a preferred handedness? Matter is somehow associated with low entropy?

If the universe were a giant vortex, there would be a natural handedness. But why would that handedness be associated with matter rather than antimatter?

The explanation might be spontaneous symmetry breaking. The universe forms a "crystal" and in that crystal, matter is more abundant than antimatter. In the adjacent crystal it may be the other way around.

For the symmetry to break, matter should in some way favor matter, for example, by having a stronger attracting force to it. Antimatter would favor antimatter. This might be a result of handedness. Right-handed particles prefer right-handed?

UPDATE: Look at our next blog post for the answer.

Charge non-conservation and Feynman diagrams

Feynman diagrams enforce charge conservation. An electron cannot change to a positron on-the-fly, and they are always created in pairs. How can we reconcile the charge non-conservation with Feynman diagrams?

An electron wave packet can turn into a chirp in a collision with two electrons.

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

The vertical lines mark virtual photons. The middle electron receives various momentum impulses from the other two. Since the end result is the middle electron being a chirp wave, there is a very slight chance that a positron comes out from the process. How do we model that in a Feynman diagram?

In a Feynman diagram, an electron line contains a definite momentum p. The waveform cannot be a chirp. The electron in a Feynman diagram is a classical point particle which can have a sharp momentum.

A diagram of type:

        e-
     /      \
   /          \
e- _____  e-

can symbolize the simultaneous effect of three electric fields. There would be a very small probability that an electron continues from the diagram as a positron.

If we conjecture that a positron is a negative energy electron traveling backward in time, then the conversion of an electron to positron is very natural. The electron brings to the spacetime point positive energy and the positron negative energy. Since the sum of energies is zero, they do not need to emit photons. The conversion, in a sense, is the true annihilation. The normal annihilation produces photons because there is surplus of energy at the spacetime point.

Thus, the conversion from an electron to a positron is in the spirit of Feynman diagrams, after all.

Is the electron conversion the only true annihilation? Suppose that we have a normal annihilation where the positron has the exact opposite energy to the electron. But such a process is equivalent to the electron simply moving in space.

Is the electric charge of a positron the same as of an electron?

It would be somewhat strange if the electric charge of an electron would change sign in a negative frequency component of its wave function. But is the electric charge of a positron actually the same negative elementary charge as of the electron? 

Let us think of the positron as a negative energy electron traveling back in time. Then a repelling impulse from an electron will not move the positron further away, but will attract the positron. This is because the mass of the positron is negative. A momentum pointing to the right will actually move the positron to the left.

In this interpretation, all electric charges are negative, but some particles are negative energy particles traveling back in time. If the electric charge is carried by such a particle, then it behaves like a positive electric charge.

Electric charge is not conserved in this interpretation. Pair production adds two elementary charges.

What about gravity? Why would gravity attract a negative energy particle traveling back in time? It should rather repel it. What do we know of gravity and different elementary particles?

We know that there is a gravitational pull between protons and neutrons, as well as atoms. Different atoms have a different composition of protons, neutrons, and electrons. Do we know that the inertial and gravitational mass are equal for different types of atoms? Apparently, there was a precision measurement in 1964.

Maybe the Dirac wave equation does not describe the electron and the positron correctly from the point of view of gravity? There are no negative energy particles traveling backward in time, as far as gravity is concerned.