If the inertial mass of the electron does not change, that breaks newtonian mechanics?

Let us think more about the van de Graaff box and the electron inertial mass inside it.

V. F. Mikhailov. Influence of an electrostatic potential on the inertial
electron mass. Annales de la Fondation Louis de Broglie, 26:33–38, 2001.

V. F. Mikhailov observed a change in the oscillation frequency of electrons in a Barkhausen-Kurz oscillator which is placed inside a charged spherical shell. But subsequent experiments have not confirmed his result.

https://www.researchgate.net/publication/316716539_Experimental_Investigation_of_the_Influence_of_Spatially_Distributed_Charges_on_the_Inertial_Mass_of_Moving_Electrons_as_Predicted_by_Weber's_Electrodynamics

Our thought experiments, on the other hand, suggest that the inertial mass has to change. If not, the center of mass of the system would not be conserved.

Mikhailov and others have thought they are testing Wilhelm Weber's electrodynamic hypothesis from 1848. We did not know of Weber when we designed our thought experiments. The thought experiments use just the basic electrodynamics, Newton's law, and Einstein's mass-energy equivalence.

1. Very basic electrodynamics claims that since the electron in the charged sphere sees a zero electric and zero magnetic field, there are no forces on it. This does not yet determine what is the inertia of the electron, but people seem to think the inertia is the same as for a free electron.

2. Poynting's vector, on the other hand, claims that the movement of the electron causes energy flow in the electric field surrounding the sphere. That energy flow should exert inertia on the movement of the electron.

3. The Aharonov-Bohm effect says that also a constant potential affects electron behavior. There is no need for the field strength vector to be non-zero at the electron.

Sources of error in the experiments

Why were several experimenters unable to measure the change in the inertial mass? The obvious suspect is the influence of the electron on the charge distribution of the metal sphere surrounding it.

The metal sphere tries to keep its electric field uniform and normal to the surface. If we move an electron slowly inside the shell, then the electric field outside the shell does not change at all. There is no flow of energy in the electric field outside the shell, and thus no extra inertia from the electric potential of the shell.

The charge distribution in the shell polarizes to cancel any change in the outside electric field. We may model the polarization with a "mirror electron" which moves to the opposite direction from the test electron. The effective inertial mass of the test electron should thus be constant, twice the inertial mass of a free electron.

There is an electric current in the shell. That will cause resistive energy loss. In the oscillator, there is energy loss from electromagnetic radiation. Can we discern these losses? Did the experimenters calculate these?

At high frequencies of 1 GHz or more, the field of the oscillator will "mostly" be electromagnetic radiation. How does that affect the model?

The electron in the hydrogen atom has a frequency of some 10^18 Hz. Hydrogen does not show a distorted spectrum inside a metal shell. Maybe the electric neutrality of the atom cancels the effects on the inertial mass of the electron in the atom. This is probably a quantum effect. The mirror electron model would make the inertial mass double.

The shell should be made of an insulator. Even in that case, can we be sure that there is no current or significant polarization in the insulator?

There are several metal parts around the oscillating electrons inside the shell. The electron will polarize charges in these, and the influence tends to reduce the temporal change of the electron's electric field. Did the experimenters calculate these?

Polarization of air and non-metallic parts will shield some of the changes of the electric field.

We need to check the articles of the experiments.

A negative potential adds to the rest mass, does not reduce it?

Both the old and the new energy-momentum relation claim that a static electric potential affects the kinetic energy of an electron. The effect is big even at moderate potentials of a few kV.

The Stark effect is measured under an electric field, not in a constant potential.

What is the effect on the hydrogen spectrum?

According to our new energy-momentum relation, the electron has an imaginary mass if the potential is > +511 kV, and should behave in a weird way.

A van de Graaff generator can create voltages up to 25 MV.

What about observers inside the static electric field? How do they measure the inertial mass of the electron?

An electron inside a positively charged van de Graaff generator

If positive voltages over +511 kV can be generated, how can we explain that no weird physics appears? Could it be that the geometry of the situation assigns the negative inertial mass to something else than the electron? Or is the inertial mass measured by people inside the shell of the van de Graaff generator different than what people outside measure?

Let us put a man standing inside a van de Graaff generator which has a positive voltage V_b. The voltage can be low, too.


           +   +    +    +
          ____________
   +    |                       |   +
         |                       |
   e- --->  \O        ----------->
         |       |              |
   +    |___/\________|   +
           +    +     +    +
         x                    y
          <--------------- E

The man lets the electron come in and fills an energy store at x with energy E. Then he moves the electron to y and uses an energy E to push the electron out.

We assume that the man is weightless.

The net result is that a 511 keV electron moved from x to y and an energy E moved the other way.

The inertial mass of the electron from the point of view of the person moving it can be positive, though.

                  rod
                 --------------------------W   weight
                   --> lever
                  |
                 /|\      supporting
               /  |  \    structure
             /    E   \
          -/----------\------ box floor

If there is a lever which moves a mass E to the left when the man pushes it to the right, the lever certainly has an inertial reaction but still moves mass to the opposite direction. The lever is attached to the box. What if the man uses a rod which is attached to a weight outside the box, to push on the lever? He does not touch the box at all but uses the rod to win the inertia of the lever. The supporting structure of the lever pushes the box to the right.

The end result is that the weight W has moved to the left, as well as the mass E, and the box has moved to the right. The head of the lever "borrowed" some inertia from the box.

If the electron can borrow inertia from the the electric field of the box, then the electron can behave quite normally also in a potential which is higher than +511 kV.

Let the box have a voltage V_b > 0. We conjecture that the man inside the box will feel that the electron has an inertia of

       511 keV + |e V_b|,

that is, we need to add the absolute value of the (negative) potential to the inertial mass of the electron. Also an outside observer will think that the electron has that same inertia.

The conjecture assumes that the forces between the electron, the box, and the fields form a kind of a lever system which moves E.

If there are unknown forces between the electron and the box, then the inertial mass of the electron can be arbitrarily high. The electric field of the box might be very "viscous", such that moving the energy hole caused by the electron would require great force.

We definitely need empirical experiments to determine how the electric field behaves.

Classical solution

Classical electrodynamics claims that there are no forces between the box and the electron because the electric field is zero inside the box. Let us calculate what would happen if the man inside the box would feel that the inertia of the electron is M.

The electron performs a displacement of s m, where s = y - x. The man pushes the box left with his feet to win the inertia M. If there are no forces at all between the electron and the box, then the displacement of the box is -s M.

The displacement of E is -s E. We get an equation

       s m - s M - s E = 0
      <=>
       s M = s m - s E.

The formula is not sensible if E > m.

Another way to calculate things classically is to use the Poynting vector. Let us assume that the positively charged box is surrounded by a negatively charged box, such that its charge exactly cancels the field far from the box. We can then restrict ourselves to calculating the electric field energy inside the box and in its immediate vicinity.

The electric field energy of the electron inside the box is essentially 511 keV, because it is the only field there. When the electron moves from x to y, the Poynting vector describes the field energy flow from the right side of the outside of the box to the left side. Thus, the electron also acts as a kind of lever which moves the energy E from right to left.

We see that the Poynting approach of calculating the field energy gives roughly the same result as our conjecture in the previous section: the inertia is

       511 keV + |V|.

The inertia depends very much on the geometry of the setup. In the hydrogen atom, the Poynting approach says that the inertia is almost zero when the electron is in a potential -1.022 MeV close to the proton, because the energy of the combined electric field is close to zero. Here we assume that the whole rest mass of the electron is in its field and the field does not gain more energy when the electron comes close to the proton.

Experimental tests of the inertial mass of the electron under a potential

Let us check what literature says about the inertial mass of an electron under a potential.

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

Wikipedia says that Maxwell electrodynamics does not conserve particle momentum if there is radiation out. That is reasonable. Wilhelm Weber's 1848 theory is claimed to conserve momentum.

http://www.nrcresearchpress.com/doi/10.1139/p04-046#.W-28t_ZuLb0

Mikhailov (1999) measured the dependency of the electron inertial mass on the potential and confirmed that it changes. But further experimenters have disputed his result.

People are trying to break newtonian mechanics with Weber's theory. Since the classical theory, the Poynting approach, and our thought experiments give conflicting results, more experiments are needed.

Electron spin is a consequence of an uncertainty principle?

Dirac derived the existence of the electron spin from his relativistic equation. But, actually, does nonrelativistic quantum mechanics already imply the existence of the spin?

The electron has an electric field. If we would know that the field does not rotate, that would probably break an uncertainty principle of the rotation of an object.

That is, there has to exist a spin for all objects which have some kind of spatial extension, small and large. The spin is not constrained to elementary particles.

The electron is a point particle. One might conjecture that it cannot rotate, because it is a point. However, the electric field of the electron is not a point, and has an orientation in space. When we measure the electron spin, we measure the rotation state of its electric field.

https://aapt.scitation.org/doi/10.1119/1.11806

David Hestenes has a similar idea in his 1979 paper.

The rotation axis is confined to the surface of a unit sphere. Does the spin 1/2 correspond to half a wave circling the sphere? That might explain the strange 720 degree rotation symmetry of the spin.

Is there a way to make the electron electric field rotate more rapidly? The spin 1/2 may correspond just to the lowest rotation energy state.

In an electric potential, the electric field of the electron may have much more energy than 511 keV, and it would be natural if new rotation states appear.

What about the spin of the muon? The muon is 200 times heavier than the electron. Could we make the muon to spin more rapidly?

What is the spectrum of hydrogen like under an electric potential?

https://en.m.wikipedia.org/wiki/Aharonov–Bohm_effect

The Aharonov-Bohm effect for an electric potential has not been measured yet.

But, according to our new energy-momentum relation as well as the old one, a constant electric potential changes the inertial mass of the electron. It should be easy to measure under a potential of, say, a few kilovolts.

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

In the Stark effect, the spectrum of an atom is affected by a static non-zero electric field. What about a constant potential? Our Nov 15, 2018 post studies this question.

Space and time are interchangeable at short distances; antiparticles come from this

If we have a relativistic electron, then the uncertainty in its energy E is of the same order of magnitude as the uncertainty in the momentum p.

We can combine the Heisenberg uncertainty principles into a spacetime uncertainty principle:

       Δ sqrt( t^2 + x^2 ) ΔE ≥ h / (4π).

Time and space appear completely symmetric in the relation and we can use the euclidean metric.

This is probably the reason why a "natural" wave packet of a Dirac electron always contains also negative frequencies, that is, the positron. The positron is an electron traveling back in time. If time and space are interchangeable, we cannot ban such paths, just like we cannot ban paths that go to the negative x direction.

We cannot ban superluminal paths either. The path integral for the electron should contain all kinds of paths, zigzagging in any direction in spacetime.

Richard P. Feynman in his 1949 papers stressed the symmetry of his diagrams when t and x are switched.

Large objects are, for some reason, confined inside the light cone (why?), but particles are not when the distance is less than their Compton wavelength. This allows the wave functions of particles be smooth in spacetime, which in turn may remove the need for regularizarion in the vacuum polarization loop of QED.

If a particle has a zero rest mass, or m + V is zero, then p = E, and the Heisenberg spacetime uncertainty principle is perfectly symmetric in time and space. Photons, too, can zigzag in time. Is there zitterbewegung of photons?

The symmetry of space and time at short distances explains why particles have to have antiparticles: there is no mechanism which prevents a particle from colliding in a way which sends it back in time, that is, it becomes an antiparticle.

Does the new energy-momentum relation break relativistic gauge theories?

Our new energy-momentum relation is equivalent to an old version when |p| and |V| are small.

       (E - V)^2 = (p + A)^2 + m^2
      <=> (|p|, |V| small)

       E - V = m
             sqrt((p + A)^2 / m^2 + 1)
                = (p + A)^2 / (2m) + m
      <=> (|p|, |V| small)

        E = (p + A)^2 / (2 (m + V)) + m + V
           = (m v + V v)^2 / (2 (m + V))
              + m + V.

The choice A = v V gives the same result as treating V as rest mass.

However, in relativistic settings, the two energy-momentum relations are inequivalent. A question is if relativistic gauge field theories are flawed as they use the old relation?

A brief study of the material on the Internet reveals that gauge field theories are usually nonrelativistic, or are expressed through a lagrangian density and Feynman diagrams. Feynman diagrams have interactions only at vertexes, and the particles are assumed to be free in the lines between them. The question whether potential energy and rest mass should be identified never comes up.

Rest mass in the Standard model is potential energy in the Higgs field. Thus, the Standard model does identify at least the Higgs potential energy with rest mass. Before the Higgs field assumes its vacuum expectation value, fermions have a zero rest mass. The physics would be weird in such a setting because the value m - V would often be negative and the fermion would be a tachyon. Maybe the Higgs field always has had the same expectation value? Is there evidence that fermions can be rest-massless at high energies? If the rest mass of the electron would be zero, would its acceleration under a potential be infinite?

The new energy-momentum relation and Lorentz transformation of the potential

We were perplexed in our Nov 4, 2018 post about how to Lorentz transform the potential and the rest mass of a particle.

The electric scalar potential ϕ and the magnetic vector potential A are prime examples.

We work in 1+1 dimensions.

The Lorentz transformation (ignoring the gamma parameter, we assume v is small) is

       ϕ = ϕ - v A

       A = A - v ϕ.

Our new energy-momentum relation is

       E^2 = p^2 + (m + V)^2,

where V is the scalar potential of the particle. When we Lorentz transform the relation, should we Lorentz transform m and V, too?

Traditionally, m is considered a scalar which stays the same in a Lorentz transformation. If we identify V with m, then V should not be transformed either. The energy-momentum relation transforms then like:

       (E - v p)^2 = (p - v E)^2 + (m + V)^2.

If we try to Lorentz transform also m and V, we end up with a relation like:

       (E - v p)^2 = (p - v E - v m - v V)^2

                              + (m + V)^2.

If we have p = 0, that relation would indicate that the inertial mass of the particle is double of what we are used to. It may be more logical to think that the transformation of E in the momentum term already includes the transformation of m and V.

We are left with the question in which frame should we measure m or V. The simplest case is when we measure in the rest frame of the object which produces the electric potential V. We assume there is no magnetic field.

Conjecture 1. The correct energy-momentum relation is

       E^2 = p^2 + (m + V)^2,

and we must not Lorentz transform m or V when we move to a new frame. V is the work we need to do to bring the particle from infinity to its current position, in the rest frame of the object that produces the field. We assume V does not change with time and that we move the particle under a constant potential V.


Suppose that we have an electron at rest. Then

       E = m + V.

If we Lorentz transform, we get

       E^2 = (-mv - v V)^2 + (m + V)^2.

We can interpret -v V either as the vector potential of V or as the momentum of the rest mass V.

How does Conjecture 1 differ from a traditional way to handle the potential in the energy-momentum relation?

Some people write

       (E - V)^2 = (p + A)^2 + m^2,

where A is the vector potential associated with the scalar potential V. The value E' = E - V is considered the energy of the particle.

If A = 0 and p = 0, and we Lorentz transform the above, we get

       (E - V)^2 = (-v (E - V) - v V)^2 + m^2
                       = (-v E)^2 + m^2,

where we Lorentz transformed V to get a vector potential A = -v V.

The Lorentz transformation of our new relation is

       E^2 = (-v E)^2 + (m + V)^2.

The kinetic term -v E is the same as in the traditional case. Our new energy-momentum relation gives roughly the same results as a traditional one when |V| and |p| are small.

All rest mass is potential energy?

Can we consider also the rest mass as potential energy? We can bring an electron from nonexistence by creating a pair. What is the role of the antiparticle in this?

                e+
             
                e-
                   \O
                     |
              E    /\  ----------> E
     =========================
     -    -    -    -    -    -    -     -    -     -    -
               x                        y

Let us introduce once again the man standing on a finite charged plane, who we had in our Nov 3, 2018 blog post. This time, the man uses an energy store E to create the electron far away from the plane by "detaching" it from the corresponding positron. That is, the man uses energy to create a pair.

We claim that both particles have an inertial mass 511 keV before the man brings the electron close to the plane, where the inertial mass of the electron is 511 keV plus the potential V.

The man moves the positron and the electron horizontally from x to y. After that, he fills an energy store at y by letting the electron move away and annihilate.

The inertial mass of the positron was 511 keV, that we know. The inertial mass of the electron had to be 511 keV plus V, otherwise the center of mass would have moved.

In this example, it makes sense to say that the rest mass of the electron is potential energy.

If the universe contains electrons that do not have an antiparticle, is it still reasonable to say that their rest mass is potential energy?

The new Dirac equation and a potential step; Klein paradox solved again

We learned about the conserved current in our previous blog post, as well as about Lorentz covariance. But it was not clear if the potential of the electron should be Lorentz transformed.

Let us analyze once again what happens at a potential step.

Potential step of the Schrödinger equation

The standard plane wave solution for the Schrödinger equation is

       exp(-i (E t - p x)).

Let us match the solutions for a potential step of height V. The incoming wave is 1, the reflected wave B, and the transmitted wave is 1 + C.

         p                q
     1 -->      1 + C -->
    -p <-- B __________
   ________| V
              x = 0

From the continuity we get:

       1 + B = 1 + C => B = C.

From the continuous derivative d/dx:

       p - B p = q + B q
      =>
       B = (p - q) / (p + q).

The "current" is proportional to p times the square of the amplitude. We should have

       (1 - B^2) p = (1 + B)^2 q
      <=>
       (1 - B) p = (1 + B) q

which is true.

If V is high, then q is imaginary and

       B  = exp(-i * 2 φ),

where φ is the phase of the complex number (p, q). The plane wave solutions of the Schrödinger equation work very well over a potential step.

Potential step of the new Dirac equation

The standard plane wave solution for our new 1+1-dimensional Dirac equations is

      (1, p / (E + m + V)) exp(-i (E t - p x)),

and its conserved current (divided by 2) is

      J = p / (E + m + V).

Let us solve the potential step in the diagram of the previous section. From the continuity of component 1 we get

       1 + B = 1 + C.

Continuity of component 2 gives:

       (1 - B) p / (E + m) = (1 + B) q / (E + m + V).
      <=> (def.)
       (1 - B) J_p = (1 + B) J_q
      <=>
       B = (J_p - J_q) / (J_p + J_q).

The conservation of currents requires

       (1 - B^2) J_p = (1 + B)^2 J_q
      <=>
       (1 - B) J_p = (1 + B) J_q,

which is true. Thus, our new Dirac equation behaves very well.

Let us then study what happens when V is large. The energy-momentum relation is

       E^2 = q^2 + (m + V)^2.

If (m + V)^2 > E^2, then q is imaginary and also J_q is imaginary.

       B = (J_p - J_q) / (J_p + J_q)
          = exp(-i * 2 φ),

where φ is the phase of the complex number (J_p, J_q).

We have shown that our new Dirac equation is very well-behaved. There is no Klein paradox in our equation.

Could it be possible to solve the hydrogen atom with the new equation?

Is the electron very close to the proton a tachyon, a positron, or a neutral light-speed particle?

UPDATE Nov 15, 2018: In our Nov 15, 2018 post we try to analyze the impact of a negative potential energy more carefully. The inertial mass may stay positive, after all.

…

We have already discussed the case m + V < 0, in which we think an imaginary  i |m + V| is the correct interpretation for the inertial mass, and the electron is a tachyon very close to the proton. Another possibility is that the electron moves at exactly the speed of light but fails to see the force of the proton.

Yet another possibility is that the electron changes into a positron close to the proton and sees a repulsive force from the proton.

If the electron is superluminal, some observers see it right-handed, some observers left-handed (= positron). We may think that it is in an intermediate stage of turning into a positron. In pair annihilation, the electron would turn back in time and be a positron.

Does the new Dirac equation cast light on this? What happens if m + V < 0 or when m + V is imaginary?

 t
 ^
 | 511 keV    511 keV photons
 |      ^              ^
 |        \          /
 |          \ ___/
 |          /       \
 |        /           \
 |     e-              e+
 --------------------------------> x

When an electron approaches a positron to annihilate, then at some point, m + V = 0 and p = E. The plane wave in that case is

       (1, 1) exp(-i (E t - p x)).

That would be a good point to glue it to the positron plane wave, since the vector is (1, 1) for both particles?

But the electron has to emit photons. If the electron emits a 511 keV photon at that point, the plane wave after that might be something like

       (1, 2p / (i 2p)) exp(-i (0 t - 2p x)).

The mass of the electron at that stage is i 2p and it moves at an infinite speed because E = 0 and the wave fronts in a spacetime diagram "move" horizontally.

The electron emits a 511 keV photon again and the wave is

       (1, 1) exp(-i (-E t - p x)).

That is, the electron has become a positron. In the annihilation, the electron became a tachyon in the intermediate stage. If you need to turn back in time, it is not that strange to be a tachyon during one line segment.

If we flip the roles of t and x, the diagram above shows an electron in an elastic collision with a 511 keV photon.

Annihilation allows superluminal communication. That may be ok if we cannot measure the speed accurately enough to prove that superluminal communication took place. If the electron and the positron collide with a high energy, then the superluminal phase has to be shorter.

We may need a superluminal phase to make the wave function smooth in the spacetime diagram.

A basic principle of special relativity is that time and space are similar entities. Allowing superluminal travel makes them even more similar, because the path of a particle is not confined inside the light cone. Allowing traveling back in time still improves the similarity.

The diagram above suggests that at short distances, time and space become interchangeable, they have an "euclidian metric". But why at macroscopic scales we are bound inside the light cone and time and space most definitely cannot change places?

Maybe at short distances, the uncertainty of quantum mechanics allows particles to "tunnel" back in time. This might be the explanation for zitterbewegung. If we paint a fuzzy line of width one Compton wavelength to the the spacetime diagram above, it looks quite natural to allow the electron path zigzag not just in space but also in time. During the trips back in time it is a positron.

Thus, the electron is allowed to make superluminal trips as well as trips back in time as long as it stays inside the fuzzy path whose width is one Compton wavelength 2 * 10^-12 m or the corresponding time, 7 * 10^-21 s.

The annihilation case does not tell us what the electron does in the vicinity of a proton. The process might be an uneventful superluminal trip.

Note that in a head-on collision with a superluminal particle, the proton cannot exchange momentum with the particle. This is because the proton does not have time to react before the collision is already over. The potential of the proton will appear static to the colliding particle, that is, it stays in the same place as if the proton were infinitely heavy. If the superluminal particle passes from the side, then the proton will exchange momentum with it.