Why do we only sum probability amplitudes of Feynman diagrams, never subtract?

Let us consider an electron-electron collision, Møller scattering.

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

   ^              ^
     \           /
       \       /
        |~~|
       /       \
     /           \
   e-            e-

If the energy of the collision is less than 1.022 MeV, then no new electron-positron pair can be produced.

If the energy is larger, then new pairs will be created.

Intuitively, the production of a pair should reduce the probability amplitude of an elastic collision where the electrons exchange a large amount p of 4-momentum.

But the possibility of pair production is not explicitly apparent in the Feynman formula which sums the t- and u-channels of elastic scattering.

Could it be that the Feynman formula for elastic scattering in some implicit way "knows" about the possibility of pair production? That is probably not true. We may imagine new physics where a lighter variant of electron exists and can produce a large number of new pairs. How could the Feynman formula for ordinary electrons be aware of such new physics?

Furthermore, the electron-electron collision may produce a photon. That is, the collision is not elastic. How could the elastic collision formula be aware of the (complex) process of photon radiation in the collision?

The simplest pair production diagram in Møller scattering is the following:


 e-      e+             e-     e+
  ^     ^               ^     ^
    \     \              /     /
      \     \          /     /
       |~~\____/~~|
      /                      \
    /                          \
   e-                          e-

The diagram contains four photon-electron vertices:

        |~~

while the elastic scattering only has two such vertices. If the integral formula for the pair production has a much smaller probability amplitude (or cross section) than the formula for elastic scattering, then in the first approximation we can ignore the effect which pair production has on the elastic probability amplitude.

We have not yet found the pair production cross section from literature.

A semiclassical treatment of pair production

Let us consider electrons and positrons as classical objects which obey special relativity.

For each classical trajectory of particles we associate an integral over a lagrangian density. The integral gives the phase, or the probability amplitude, of that history.

If an electron and a positron come to the distance 3 * 10^-15 m from each other, then their combined energy is zero, assuming that they have no kinetic energy.

We assume that we can add such a zero-energy pair to any history where the existing other particles bump into the particles in the pair, giving the pair a 4-momentum which makes them real, a 511 keV electron and positron.

That is, classical collisions can create new pairs by tearing apart an electron and a positron which exist as a zero-energy pair.

Our assumption is somewhat similar to the hole theory of Dirac. In Dirac's hole theory, an electron with an energy -511 keV gets excited to a state of an energy +511 keV, leaving behind a hole, which is the positron.

The zero-energy pair can be considered a zero-energy state of a positronium "atom". The atom gets excited by other particles, and goes into a state where the pair will annihilate again (= virtual pair), or goes into a state where the electron and the positron escape as real particles.

Is our model deterministic? Suppose that the electrons exchange more than 1.022 MeV of energy in a collision. What determines if a pair is produced, or if the collision is elastic?

Or should we make the model probabilistic?

The two electrons which enter the experiment can be considered as uncorrelated. Pair production can be seen as a positron moving backward in time, colliding with both electrons, and scattering forward in time.

But is the positron moving backward in time uncorrelated with the two electrons?

Suppose that the initial state of electrons is such that they would collide and exchange more than 1.022 keV of kinetic energy. Is there always some positron trajectory which will rob some of the energy and produce a real pair?

We believe that empirical tests show that elastic collisions are possible at large energies. Pairs are not always produced.

Why destructive interference does not cancel high 4-momentum in the vacuum polarization loop?

In our blog we have previously claimed that a physical phenomenon with an associated 4-momentum p cannot produce any phenomena of a higher absolute absolute value of the 4-momentum.

In the wave interpretation, p is associated with a wavelength λ = 1 / |p| in a spacetime diagram.

time
^      wavelength λ
|    \     \
|      \     \   -----> 4-momentum p
|        \     \
 ------------------------------> space

We can develop the wave forward in time and space in the diagram through the Huygens principle: let each point act as a point source of new waves, and calculate the interference of the new waves at a new point.

Let

       D(ψ) = 0

be a wave equation. If the right side is strictly zero at all spacetime points, we say that it is a homogeneous equation.

If the right side is not zero everywhere, then we call the non-zero part a source.

The Green's function method calculates the response of a wave equation to a Dirac delta source in the equation. That is, we assume that the wave equation

         D(ψ) = 0

has on the right side, instead of 0, a Dirac delta term at a certain spacetime point x. The Green's function is the solution of the new equation. We say that it is the response to an impulse source of the wave equation.

If we have a tense string, then pressing the string briefly at a certain point with a finger with a force F, is an impulse F × Δt, and the resulting wave is the response to an impulse source.

If we have a source which is not concentrated to one spacetime point, but is continuous, we can build an approximate solution by summing the response to an impulse source at each each spacetime point x.

Suppose that the source is cyclic and has a certain wavelength λ in the spacetime diagram.

The response to a Dirac delta impulse contains waves for all kinds of 4-momenta p.

It is obvious that there tends to be a destructive interference for all waves where p does not match the cycle (wavelength λ) of the source.

In particular, all high 4-momenta p will have a total destructive interference.

This is the reason why tree-like Feynman diagrams have strictly restricted 4-momenta p at every part of the diagram.

However, if we allow an imagined wave, as in the previous blog post, to have any 4-momentum p, then the imagined wave introduces an arbitrarily high p, or an arbitrarily short wavelength λ, to the diagram. Destructive interference does not cancel it.

Diverging of the vacuum polarization loop integral

          q + p -->
     ~~~O~~~~~~~~~~~~
q -->   <-- p            q -->

The vacuum polarization loop carries the photon 4-momentum q, as well as an arbitrary 4-momentum p which circles around the loop.

The impulse response to a Dirac delta impulse at a spacetime point x contains a certain spectrum (= propagator) of various 4-momenta. The intensity depends on the sum q + p. That is, the probability amplitude of the diagram above depends on both q and p.

If we allow any p, then there exists no sensible probability distribution for p. The integral of probability amplitudes over all p diverges, or alternatively, we may say that the integral is not defined.

The causality of a Feynman diagram

The imagined wave with an arbitrarily high 4-momentum p does not follow "causally" from the input waves to the Feynman diagram.

The diverging of the integral seems to be the result of this acausality.

The vacuum polarization loop

The ideas of Gordon in Compton scattering have helped us forward with the analysis of the vacuum polarization loop in Feynman diagrams.

                               wave of an

                                emitted photon

    electron wave         \   \

                      ---------        \   \

^ time           ---------

|

|             ____________

           \    ____________    positron wave

        \   \   ____________

     \   \   \

       \   \   \

      photon wave

1. Let us imagine that there is a positron around. The positron is a solution of the Dirac equation with no electromagnetic field.

2. A (virtual) photon causes a disturbance in the positron field. The disturbance is a source term in the Dirac equation.

3. We try to remedy the solution of the Dirac equation by using Green's functions of the Dirac equation to cancel the source term.

4. Green's functions produce an electron wave. We may interpret that the positron traveling backward in time absorbed the photon and turned into an electron. 

5. Next we imagine that there is an electromagnetic wave which corresponds to the electromagnetic wave which would be produced by the electron emitting the photon which it absorbed earlier.

6. The imagined wave disturbs the wave of the electron. The disturbance produces a positron wave which matches the original positron solution. An emitted photon wave is also produced.

The loop is complete! The positron, which we first just imagined, was "produced" by the scattering of the electron backwards in time, and the scattering also produced the emitted photon wave, which we originally only imagined to exist.

It is like trying to find solutions for the perturbed Dirac equation by assembling Lego blocks. We can use a block where an incoming photon produces an electron-positron pair.

If we turn that block around, we have a block where an incoming electron and a positron produce a photon.

As long as we can assemble a diagram which obeys certain rules, we are free to "imagine" the existence of whatever particle.

Note that in the diagram, all the waves really span the entire diagram area, and are overlapped. There is a large spatial uncertainty about the location of each particle.

What if the waves were classical waves?

Classically, we cannot just imagine the existence of any non-zero wave. In the diagram, there would be no positron wave present. The photon wave would proceed undisturbed.

What about the magnitudes of each wave? Let us use classical mechanics. Let us assume that the imagined waves do exist.

The electron flux is typically very small compared to the positron flux. It cannot "produce" the entire positron flux which exists in the diagram.

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

Baron Münchhausen told the story where he pulls himself out of a swamp by his own pigtail.

The Baron Münchhausen type trick of creating an electron-positron loop from (almost) nothing cannot work in classical mechanics if the disturbance is small. The "feedback" of the loop should be strictly equal to one, to allow a Münchhausen type of a process.

We know that pairs are produced in high-energy collisions of electrons. In quantum mechanics, a disturbance seems to have the ability to "concentrate" its effect on a very small spatial area, such that the feedback of a loop becomes strictly 1.

The diverging of the Feynman integral over a loop

The diverging of the Feynman integral indicates that something is wrong with the assumption that quantum mechanics can conjure up Baron Münchhausen type loops without any restriction. Feynman's rules allow the loop to carry any 4-momentum around, without any restriction.

In previous blog posts we developed the particle model of a photon as a rotating electric dipole.

If we assume that all the particles, including photons, obey certain restrictions of classical mechanics, then it is impossible for a loop to carry an arbitrarily large 4-momentum. No diverging of integrals is possible.

But does that restrict Feynman diagrams too much, so that they would no longer agree with empirical data?

Why does Feynman use a Green's function to describe the electric field of an electron?

In the electron-electron collision diagram, one electron sends a virtual photon, carrying some 4-momentum. The other electron absorbs this photon and receives a push.

Feynman assumes that the distribution of various 4-momenta in the photon is the Green's function for the massless Klein-Gordon equation. Why?

Let us consider the drum skin analogy of the static electric field of the electron. If I press the drum skin with my finger, it creates a depression into the skin. That depression is analogous to the static electric field of a particle.

We may imagine that instead of pressing with a constant force F, I keep tapping the skin with my finger at a very rapid pace.

The tapping creates a depression. A single tap is equivalent to applying an "impulse source" to the wave equation of the drum skin. The Green's function for the skin wave equation, by definition, is the response of the skin to that impulse.

That is, we may imagine that the static electric field of a particle consists of a very rapid pace of Green's functions emanating from the particle. The electric field does not carry energy away. There has to be a total destructive interference for the "on-shell" waves in the decomposition of the Green's function.

On the other hand, waves carrying just linear momentum p, can progress. Those waves apparently are responsible for the static depression in the drum skin or the static electric field of a particle.

The decomposition for the various p obeys the decomposition of the Green's function.

If there is a planar wave describing another electron nearby, the photon waves for various p disturb the free Dirac equation of that other electron. That is, the equation no longer is equal to zero, but a (small) source term appears.

Each wave p creates a source term. If we perturb the planar wave solution to find a more accurate solution for the source term associated with p, then another wave appears. That wave is interpreted as the wave of an electron which absorbed the photon with a momentum p.

Relationship to the classical scattering from a static Coulomb potential

If we calculate the scattering distribution, assuming that the electrons are charged particles of classical mechanics, the result is the same, or almost the same as when we use the Feynman diagram formula.

Classically, the momentum p which the electrons exchange is roughly proportional to 1 / r, where r is the minimum distance between the electrons. The number of electrons receiving a push > |p| is proportional to

       1 / |p^2|,

which is derived from the fact that the area for passing at a distance < r is proportional to r^2.

There is probably some general mathematical theorem which shows that an 1 / r potential for an incoming flux of particles can be implemented through the absorption of quanta of the Green's function for the massless Klein-Gordon wave equation.

If a photon is an orbiting virtual electron-positron pair, does that explain Compton scattering?

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

Thomson scattering means that a low-energy photon is scattered by an electron at rest.

Compton scattering is the same phenomenon with a high-energy (> 511 keV) photon.

http://www.mso.anu.edu.au/~geoff/HEA/5_Compton_Scattering_I.pdf

The cross section of Thomson scattering is of the order of the electron classical size. The classical electron radius is 3 * 10^-15 m. That is also the distance where the potential energy of two close electrons is equal to 511 keV, that is, the mass of the electron.

The cross section of Compton scattering is of the order of the electron classical size divided by the energy of the photon (given in units of 511 keV).

Let us assume that a "photon" moves in a medium of coupled electron-positron dipoles. Oscillation of such a dipole spreads to the neighbor dipole through the electric force. The photon is really a phonon of this medium. We do not assume the existence of any electromagnetic waves. The oscillation is strictly in the dipoles.

Suppose then that we have a free electron in the medium. What is the cross section of its collision with a phonon?

We may model a phonon as a moving oscillation of a single dipole. The oscillation of a single dipole jumps to the neighboring dipole at (almost) the speed of light. The phonon moves fast through the medium.

If the free electron happens to be within 3 * 10^-15 meters from the positron or the electron in the oscillating dipole of the phonon, then there is very strong interaction between the free electron and the phonon. This might explain why the cross section of a photon-electron collision is of the order of that length.

The free electron robs energy and momentum from the oscillation of the dipole.

We may assume that the dipole has before the collision assumed an equilibrium position in the electric field of the electron.

When the dipole starts to oscillate, what is the effect on the free electron? If the electron is not close to the ends of the dipole, the momentum transfer is inversely proportional to the distance to the ends of the dipole, and the periodically changing field probably cancels away most of the momentum transfer to the free electron.

Why is the cross section inversely proportional to the energy of the photon in Compton scattering?

The history of the Klein-Nishina formula

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5709540/

In 1928, Klein and Nishina were able to derive the correct differential cross section formula for Compton scattering, based on the brand-new Dirac equation. Yuji Yazaki in the link (2017) tells about the history of the discovery.

In 1926, Dirac treated scattering as a state transition of the system electron & an oscillating electromagnetic field. The apparent "collision of a photon" is a state transition which happens at a certain probability per second. Dirac derived the correct formula for a "spinless" electron. Klein and Nishina included the magnetic field of the electron in the formula.

We need to find out what is the relationship between the Feynman approach to scattering and the Klein-Nishina approach.

https://arxiv.org/abs/1501.06838

Waller and Tamm (1930), and in unpublished notes, Ettore Majorana, modified the Klein-Nishina semiclassical approach to a quantum field theoretical framework. It turned out that the electron goes through intermediate states. Thomson scattering is produced by negative-energy, that is, positron, intermediate states.

We need to compare the ideas of Waller, Tamm, Majorana, and Feynman.

If Navier-Stokes allows simulation of a Turing machine, then Gödel may make existence of a solution undecidable

Let us consider the Navier-Stokes equation of perfect fluid, with no atoms or other type of a cutoff at very short distances.

Solutions of the Navier-Stokes equation easily develop turbulence. Turbulence is a fractal-like phenomenon.

Can we harness turbulence to do complex digital calculations, like on a Turing machine?

If yes, then we might get analogues of the Gödel incompleteness theorem for solutions of the Navier-Stokes equation.

Suppose that a proof of a contradiction from the Peano axioms is equivalent to proving that a certain solution of the Navier-Stokes equation develops a singularity. Then it would be an undecidable problem if a singularity appears.

https://terrytao.wordpress.com/2007/03/18/why-global-regularity-for-navier-stokes-is-hard/

Terence Tao in his 2007 blog post refers to a possible connection of problems of complexity theory (e.g., P = NP) to solutions of the Navier-Stokes equation. Turbulence develops complex, pseudorandom structures. (See his note "6. Understanding pseudorandomness".)

If we can build a digital computer from turbulence, then complexity theory will pop up.

Real fluid has a cutoff at the atomic scale. We do not expect a turbulence-based Turing machine to have any relevance in the real physical world.

Quantum fields probably have a cutoff at the scale of the Planck length, because mini black holes may turn up. It is unlikely that we can harness microscopic quantum fields to make a Turing machine, but this deserves further thought.

A swarm of virtual quanta: a possible solution to the renormalization problem of QED

UPDATE October 27, 2019: If there is just a single vacuum polarization loop, the loop length is 2, and the propagator for large k is roughly 1 / k^2. Since k can have any value in R^4, the integral diverges badly, like k^2.

But if we have a swarm of virtual photons, and some of the virtual photons create virtual pairs, and the pairs recombine in such a way that the loop length for a momentum loop is 6 or more, then the propagator for large k is 1 / k^6 or less. Then the integral converges.

If we could modify the Feynman method in such a way that the momentum loop length is always at least 6, we would get rid of the ultraviolet divergence.

---

Let us assume that two electrons are colliding in a particle accelerator.

         ^                   ^
             \     p      /
              |~~~~|
             /             \
         e-                 e-

The simplest Feynman scattering diagram contains just one virtual photon which is exchanged between the electrons. The photon carries a four-momentum p.

The propagator of a photon is

       i g_μν / p^2.

Let us think about a classical scattering of two electrons. If the distance is larger than the Compton wavelength 2 * 10^-12 m, then classical Coulomb scattering is a good approximation.

In the classical scattering, the electrons move in curved paths. If we want to use the quantum mechanical particle interpretation, the electrons exchange a large number of virtual photons on their path.

Consider then the vacuum polarization loop for a single virtual photon:

         p           p + k
   r_0 ~~~~~O~~~~~ r_3
                       -k     

The photon starts from a spacetime point r_0 and ends up at r_3. The electron-positron pair are born at r_1 and annihilate at r_2. The contribution in the phase of the Feynman diagram is something like:

       exp(i *[(r_1 - r_0) • p
                    + (r_2 - r_1) • (p + k - k)
                     + (r_3 - r_2) • p
               ])
       =
       exp(i (r_3 - r_0) • p).

We see that there is a perfect constructive interference if we let r_1, r_2, and k vary.

The constructive interference causes the integral over all k to diverge badly.

We have in our blog stressed that momenta k >> p should have a negligible contribution to natural phenomena where a momentum p is the input. That is, if the process is fuzzy at a length scale L, phenomena with a length scale << L should not affect much.

But in the vacuum polarization loop, high k contribute greatly.

A possible solution is to require that in the scattering, the electrons exchange a very high number of low-momentum virtual photons.

Then the corresponding virtual electrons and positrons form large "swarms", where they can annihilate with any of a large number of opposite charges. There is no longer perfect constructive interference because an electron of a momentum p + k can annihilate with a positron of an arbitrary momentum k', and the positron was not born at the same spacetime point as the electron.

We conjecture that there is an almost perfect destructive interference for high momenta k. Only if k is of the same order of magnitude as p, is the contribution considerable.

Let the electrons pass by at a distance L.

We conjecture that the classical limit of the process is that the EM field as well as the reaction to it, the vacuum polarization electron field, are fuzzy at the length scale L. The fields have little contribution from high momentum k planar waves.

Our suggestion has an obvious problem: how do we model the creation of an energetic real photon or a real electron-positron pair in a collision? We should allow a high momentum p photon to produce a 1.022 MeV pair.

So far we have not found in literature practical examples of how large a cutoff Λ one should use in a collision of a momentum p, so that the Feynman formulas predict the outcome well. Is 2 × p a suitable cutoff? Or should it be much larger?

The Feynman diagram with just one virtual photon is a perturbation diagram where we approximate the perturbation as a single Dirac delta impulse on the other field. That sounds like a very crude way to calculate a solution for the QED lagrangian (whatever the correct lagrangian is). The divergence in the Feynman integral might be an artifact of the very crude approximation.

If we calculate with a swarm of virtual quanta, we might be closer to solving the fields non-perturbatively. That is, closer to the correct solution. If large momenta k have a very low weight in the correct solution, then there is no divergence problem. Then the solution to divergences is to calculate correct, non-perturbative solutions.

What is the bare charge of an electron?

The scattering experiment in our previous blog post can be interpreted as a way to measure the electric repulsion between two electrons, and therefore, their charge.

Some people think that the electron is surrounded by a cloud of virtual electron-positron pairs which screen part of the negative bare charge of the electron. For an unknown reason, there is less screening if real electrons come close to each other. That would explain the stronger coupling constant. But this does not help much because we have to refer to an "unknown reason".

We in this blog have the philosophy that only measurable quanta exist as "particles" and the rest is classical fields which obey classical field equations.

We do not know the correct QED lagrangian. Let us assume that Feynman diagrams indirectly describe the lagrangian correctly, whatever it is.

The trigger to create a virtual pair in the one-loop Feynman diagram is the exchange of the large momentum virtual photon between the colliding electrons.

The reaction in the electron field is to the momentum exchange. That suggests that the reaction is a dynamic phenomenon.

Suppose that we somehow attach two electrons very close to each other. They exchange a lot of momentum in a second. Is there a similar virtual pair loop present in this case?

We need to analyze the Feynman formulas for the loop. What kind of a reaction do the formulas describe in the electron field?

Scattering of two electrons is a classical field phenomenon - is renormalization really needed?

In last fall we promised in this blog that we will show that the divergences in the Feynman loop diagrams are an artifact, which is a result of a wrong integration order.

Thus, no renormalization is needed if the scattering amplitudes are calculated in the correct way.

Our analysis of of the QED lagrangian brought this question up again.

Scattering of two electrons with low energy

Let us consider the scattering of two electrons which possess much less than 1.022 MeV of kinetic energy. Let us assume the bounce is symmetric.

    ^                ^
      \            /
        \        /
         |~~~| virtual photon
        /         \
      /             \
   e-                e-

There is electric repulsion between the two electrons which makes them to bounce off each other. The virtual photon marks the repulsion. The virtual photon is not a particle in any way.

An analogue for the repulsion is a spring:

  e-  ---|\/\/\/\/\/\/\/\/|---  e-

The electrons push each other with a rod containing a spring.

What is analogous to the virtual electron-positron loop in the Feynman diagram for the virtual photon?

       ~~~~O~~~~
             loop

An analogue is that the spring can give way not just in the spring /\/\/\ part but also in the straight rod parts ---.

If we want to calculate the bounce very precisely, we need to take into account all other degrees of freedom where the kinetic energy of electrons can be stored temporarily, not just the electric field.

In the case of the spring, the straight rods could store a little energy. It is similar for a virtual electron-positron loop: it can make the repulsive potential between the two electrons a little less steep. The loop will store some energy for the time when the electrons pass by. The loop will return the energy back to the kinetic energy of the electrons when they start to recede.

The virtual loop, which is also called vacuum polarization, might create a temporary charge distribution like this:

       -   e-  ++  e-    -

The positive charge density between the electrons makes the repulsion a little weaker in the bounce.

But the running coupling constant makes the electron repulsion stronger at short distances. What can cause that?

We remarked in our earlier post about a radio transmitter that if an EM wave is created by a disturbance of the EM field, and the wave is guaranteed to get absorbed soon again by another disturbance, then the wave may store a lot of momentum relative to the energy. A free plane wave in an EM field stores less momentum per energy.

The same is probably true for a short-lived wave in the electron field. We would need a correct QED lagrangian to analyze this in detail. The temporary field is born by interaction from the rapidly changing electric field between the bouncing electrons. We should show from the correct QED lagrangian that the electron field, indeed, is disturbed by the rapidly changing electric field, and stores some energy and momentum for a short time.

Let us assume that a temporary wave in the electron field is able store a little energy and considerable momentum. The temporary field does not need the 1.022 MeV of energy which would be needed for a real pair.

Is our analysis of the waves fully classical? Where does quantum mechanics enter the picture?

Note that our qualitative analysis did not refer to quanta anywhere except that the colliding electrons in the pictures were assumed to be particles.

A more precise analysis would assume that the electrons are waves obeying the Dirac equation when moving free of interactions.

The waves are smooth. The fuzziness of the waves in space it at least of the order of the Compton wavelength of the electrons.

If we solve the waves fully classically, where does quantum mechanics enter the picture? Maybe only at the measuring device. It will measure particles whose probability distribution can be derived from the classical wave solution.

What is the divergence in the Feynman loop diagram?

The well-known problem in the Feynman diagram formulas is the divergence of the calculation of the loop contribution. If we integrate over all possible momenta carried by the loop, the integral diverges.

How does that divergence show up in our classical analysis?

It does not because when two classical waves of a wavelength λ meet, we do not need to consider detail much smaller than λ in the reaction of the electron field. We can use a cutoff at λ, and intuitively know that finer detail has very little effect.

Using large momenta in the electron field would involve fine spatial detail in the electron field.

A Feynman diagram calculates all possible paths of the bouncing electrons. Electrons are point particles in the Feynman diagram and can come to a very close distance from each other. Very fine detail in the reaction of the electron field to that close encounter does have an effect on the Feynman calculation formula. If we try to calculate an intermediate result after the loop, the result may well diverge.

However, that intermediate result is not what we measure from the experiment. It makes no sense to calculate such.

If we only calculate the end results of the experiment, and if our intuition that fine detail has a vanishing effect is right, then the end results will not diverge.

If we are right, the divergence in Feynman diagrams is just an artifact from a wrong integration order. We must not calculate the diverging intermediate result.

The general problem of divergences in partial differential equations

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

A Millennium problem is to prove the existence and the smoothness of solutions for the Navier-Stokes equation. The problem is in turbulence. Does its infinitely fine detail have a large effect on the solution?

The QED classical wave equations might have a similar problem. We need to prove that no turbulence-like phenomenon can appear. If we cannot prove the existence and smoothness, then the divergence of Feynman intermediate results is a symptom of a real mathematical problem.

If we can prove the smoothness and existence of QED wave solutions, then the divergence is just an artifact.

We have in this blog post shown the connection between the existence of smooth solutions for a classical partial differential equation and the need to renormalize a Feynman diagram calculation.

To recapitulate:

1. If the classical equation has smooth solutions, then renormalization is really not needed. The apparent need for renormalization is a result of trying to calculate nonsensical intermediate results.

2. If the classical equation does not have smooth solutions, then it is not a proper physical model. We need to modify it, for example, by introducing a cutoff, which is equivalent to renormalization with a cutoff.

We need to look at the literature about the existence of smooth solutions for various partial differential equations. Coupled field equations are probably nonlinear in most cases. Little is known about the existence of solutions for nonlinear equations.

Linearized general relativity is not renormalizable. How does that show up when we try to solve the classical Einstein equations?

Conclusions

Renormalization may be unnecessary in QED.

The need for renormalization, or new physics at the Landau pole scale or the Planck scale, is connected to the existence of solutions for the corresponding classical field equations.

If smooth solutions exist, no renormalization nor any new physics is required. In such a case, the concept of an effective field theory is unnecessary.

A drum skin inspired QED lagrangian density

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

The lagrangian density is

       L = ψ-bar (iγ_μ (∂_μ + ieA_μ + ieB_μ) - m) ψ
           - F_μν F^μν.

Wikipedia says that A is the EM field of "the electron itself" and B is the external field.

The first question is how many fields there are in the lagrangian? It looks like each electron has its own field ψ, and also its own EM field A.

Apparently, there is also a free EM field present as well as an external field whose origin may be charges far away.

The basic idea of a lagrangian density is to find a local extreme value in a spacetime integral of that density.

Suppose that we have just a slow-moving single electron in a static external electric field B.

The lagrangian above treats the energy of the combined electric fields A + B as potential energy. The mass of the electron is potential energy, and is constant.

But the potential of the electron in the field A + B is strange. If B is the static potential of other electrons, then the potential in B should have a minus sign in the lagrangian. That is correct in the above formula.

And what is the potential energy of the electron in its own electric field? If the potential energy is negative, then the sign in the above formula is wrong.

The above formula seems to mention the total energy of an electron three times as potential energy:

1. potential energy in its own field,
2. mass,
3. -2 × energy of the electric field of the electron, in the last term.

A drum skin QED lagrangian

How could we write a lagrangian which makes sense?

We had the analogy where a drum skin is pressed with a finger. The static electric field of the electron is the depression in the skin.

The depression is like a negative electric field and the finger is a positive charge sitting at the bottom of the depression.

The variables are the skin depression and the finger position.

       L = eA - F^2,

where A is the skin depression as a positive value, F^2 is the skin deformation energy and e is the downward force on the finger as a positive value.

What if we add another pressing finger? We have to introduce another field A_2 for it. The other finger stays in its own depression.

Note that the pressing force for each finger i only releases energy for its own depression A_i and ignores other depressions. We need separate terms F_i^2 for each A_i.

We can implement a repulsive force between the fingers by adding the energy of the total field F^2. If the fingers move closer, then F^2 grows.

There is a problem, though. Each A_i should be independent of the other A_j. But the term F^2 couples them. What to do? We may multiply the energy associated with each i with a large number M. The private field of each i is then "rigid".

To accommodate drum skin vibrations as well as an external field, we need to introduce yet another field B which is not visible for any of the electrons. Drum skin vibrations live in the global field B.

The total QED lagrangian is

       L = Σ_i M (e A_i(x_i) - F_i^2) - F^2,

where x_i is the position of the i'th electron and F^2 is the positive energy of the total field, which is the sum of all A_i and B. M is the large positive number which makes the static field of each electron rigid.

The mass of an electron is

       M (F_i^2 - e A_i(x_i) + F_i^2.

We should still add kinetic energies to the lagrangian.

The classical lagrangian

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

The lagrangian is

       L = L_field + L_interaction

           = -1/(4μ) F^αβ F_αβ - A J,

where Wikipedia says A J means "many terms expressing the electric currents of other charged fields in terms of their variables".

If there are no pointlike charges but a finite charge density, then the lagrangian above might be similar to the lagrangian of an attractive force, and it would calculate correctly the field inside and around a uniform charge density ball.

But if we have two such charge balls, the lagrangian then attracts them together, which is wrong.

We need to check the literature. Has anyone got the EM lagrangian right?

Is the blowup problem of classical fields related to the renormalization problem of quantum field theory?

https://www.physicsforums.com/threads/validity-of-theoretical-arguments-for-unruh-and-hawking-radiation.978501/page-3#post-6243209

https://terrytao.wordpress.com/2007/03/18/why-global-regularity-for-navier-stokes-is-hard/

There might exist a classical field problem which resembles the renormalization problem. In the Navier-Stokes equation, a Millennium problem is to prove that solutions do not "blow up" because of turbulence.

In a realistic fluid there is a natural scale, the scale of molecules, at which the Navier-Stokes equation stops working. The blowup cannot happen. This sounds like an energy cutoff which is used to eliminate the divergence in renormalization.

The concept of an "effective theory" contains the idea that at very short distances there is new physics which provides the necessary cutoff.

If we try to model an electromagnetic field in a gravitational field, and consider the backreaction of the two fields when they interact, the renormalization problem may appear in the classical fields as a blowup problem. For example, the solution might not be stable under small perturbations.

After all, Feynman diagrams are perturbation calculations. If the perturbations diverge, then a classical solution might be unstable.

Which brings us to the old topic if general relativity has any solutions at all under realistic matter fields.

Christodoulou and Klainerman (1990) proved the "nonlinear stability" of the Minkowski metric under general relativity.

This is a very interesting question: if Feynman diagrams with gravitons diverge, how could Christodoulou and Klainerman show the stability in the (very restricted) case of the Minkowski metric?

In physics, if we are calculating with two fields, we usually ignore the backreaction. If we calculate the behavior of a laser beam which climbs out of a gravitational field, we assume that the backreaction on the gravitational field is negligible. But it might be that a precise calculation shows the the solution is not stable.

Is the magnetic field just a Lorentz-transformed Coulomb electric field?

Suppose that we have a charge which is static in the laboratory frame. We know that it will interact with a moving test charge only through its static Coulomb electric field.

The movement of the test charge is explained solely through the Coulomb interaction. There is no magnetic field.

Suppose that we switch to a moving frame. Clocks which were placed at various spatial coordinates of the laboratory frame appear to go slower, and they are no longer in synchrony when viewed from the moving frame.

If an observer in the laboratory frame measures the test charge to move from a position x_1 to x_2, and measures the elapsed time as t, then the observer in the moving frame will measure a different time interval.

Furthermore, the measured time interval depends on the spatial distance x_2 - x_1. The speed of the charge appears to affect the force on the test charge, when the force is observed in the moving frame. The moving observer interprets that a magnetic field is affecting the path of the test charge. The force depends on the field strength and the speed of the test charge.

We see that in this simple case, we can say that the magnetic force is just an illusion which appears when the test charge path is Lorentz transformed over to a moving frame. A human has problems understanding the transformation and explains the surprising movement with an imagined magnetic force.

The simple model can explain the magnetic field which we see around a wire carrying an electric current.

http://www.feynmanlectures.caltech.edu/II_13.html

Richard P. Feynman calculated the relativistic effect of the electrons moving inside a wire, relative to the protons. He considers a negative test charge moving at the same (very slow) velocity v as the electrons inside the wire.

Feynman writes that there is a magnetic field which will cause the test charge to curve toward the wire. If we move to a frame moving along the test charge, then, he writes, the protons in the wire will appear length-contracted, and the negative test charge, in this moving frame, is pulled by their electric attraction.

Let the density of protons in the laboratory frame be P.

The density of electrons in the laboratory frame is γE, where γ > 1 (because of the length contraction) and E is the density of the electrons in the rest frame of the electrons. If there is no electric Coulomb force in the laboratory frame, then it must be that

        P = γE.

Let us then switch to the moving frame. The density of electrons is E, and the density of protons in this frame is γP. The density of protons is γ^2 times the density of electrons. There, indeed, is a pulling electric force on the negative test charge.

---

Feynman writes that, "of course", we cannot reduce magnetic fields solely to the Coulomb electric field and a Lorentz transformation. He does not specify the reason, though.

When we consider configurations like the radio transmitter of our previous blog post, it is not clear at all how we could get rid of the magnetic field.

If we move a charge back and forth, some of the spherical waves which are produced will reflect back. If we just believe in a Coulomb field which spreads at the speed of light out from the moving charge, what then causes the reflection? There has to be a wave equation acting, if there is a reflection. And it is hard to write a wave equation without a magnetic field.

A clearer picture of how a radio transmitter works

For concreteness, let us assume that we want to send 1 GHz radio waves. Let us assume that we have a device which moves an electric charge back and forth a distance significantly less than 15 cm. The cycle time is one nanosecond.

   <--------- charge --------->

                < 15 cm

We feed mechanical energy to the device, and the device converts the energy into photons which can be observed far away.

In earlier blog posts we have asked the question where does the momentum go? The device moves at a speed considerably less than light. Its kinetic energy is converted into photons which can only carry away a fraction of the momentum lost by the device. Where is the momentum stored?

We earlier introduced the fishing float analogy for the charge. When we move the float, a complex pattern of water waves forms around the float. Waves reflect also back toward the float. The force which water exerts on the float at each moment is a very complex phenomenon. We should make a computer simulation to calculate the force.

The radio transmitter, similarly, has a "self-force" which the field of the charge exerts on the charge itself.

Now we see that some of the momentum which the device lost while accelerating forth, can be reabsorbed by the device when it moves back. The momentum was temporarily stored in the electromagnetic field of the moving charge.

In half a nanosecond, momentum and energy in the field can move at most 7.5 cm away, if they want to be reabsorbed by the device. Thus, it is the immediate vicinity of the antenna which stores the extra momentum.

The Larmor formula estimates the average energy flux from a charge doing a periodic motion. The Larmor formula has been tested empirically on radio transmitters.

If the motion of the charge is non-periodic, then the energy flux is hard to calculate  and there is no reason to assume that the Larmor formula would be correct in such a case.

Quantization of the electromagnetic wave

Far away from our radio transmitter, the EM wave is very close to a plane wave. More precisely, it is a sum of two circularly polarized plane waves that rotate in opposite directions.

A plane wave is easy to quantize. We may imagine that it consists of an integer number of coherent photons. The Fourier decomposition stays almost constant. The photons are long-lived.

The waveform close to the transmitter is much more complex. It can be calculated using classical Maxwell's equations. Does the complex wave consist of photons and "virtual photons"?

We can, of course, calculate a Fourier decomposition for the complex waveform. Since the wave is interacting with the moving charge, the Fourier decomposition changes fast.

Can we meaningfully Fourier decompose the static electric field of the charge? If it is really static, the decomposition is strange. Maybe the decomposition makes sense if the charge moves?

We could interpret that the modes in the Fourier decomposition consist of photons or virtual photons which are being created and absorbed at a fast pace.

A spherical wave will always have some of the wave energy reflected back toward the center.

If a Fourier mode is a plane wave filling the entire space, then every mode is "approaching", as well as "receding" from the center.

We cannot describe the functioning of a radio transmitter just by photons moving away from the transmitter. There are always virtual photons which will be reabsorbed by the transmitter.

Where are the photons and virtual photons born in a radio transmitter?

Where are the photons or virtual photons "born" in space? The Coulomb electric field of the transmitter charge oscillates in a wide area of space. Are the quanta born at the charge? Or are they born farther away from the movements of its static electric field?

Let us consider a simple analogy. We have a drum skin which we disturb by pressing a finger agains it and moving the finger back and forth.

Where are the waves in the drum skin born? The energy comes through the finger.

Away from the finger, the drum skin obeys some kind of a partial differential equation.

As the finger moves back and forth, parts of the skin oscillate up and down. That is probably the main origin of the vertical waves in the skin.

Only part of the energy and the momentum which the finger inputs is carried away as vertical, ring-form waves. The rest is absorbed back into the finger in a backreaction.

Is there conversion from longitudinal waves to vertical waves? Suppose that the drum skin has zero friction.

All energy is at the lowest level transferred as horizontal stretching of the skin. We conjecture that we can ignore longitudinal waves.

Let us return to the radio transmitter.

1. The moving charge acts as a source in Maxwell's equations. It produces disturbance to the EM field.

2. The waves are born at the charge, but much of the energy and the momentum in the waves is reabsorbed back into the charge in a backreaction. Specifically, some of the energy in spherical waves is always reflected back.

3. Some of the waves escape as well-formed spherical waves.

4. Photons as well as virtual photons and longitudinal photons are born at the moving charge.

5. Virtual photons and longitudinal photons are quickly reabsorbed by the moving charge.

6. Virtual photons and longitudinal photons can store a considerable amount of momentum and return it back to the moving charge during the next half-cycle. The magnetic field of the moving charge resists changes in the speed of the charge. Most of the momentum is probably stored in that magnetic field and returned back to the charge when the charge changes the direction.

Is there a unique decomposition of an EM field into virtual and real photons?

We defined above a virtual photon as a quantum of a wave which is quickly reabsorbed to the moving charge in the backreaction.

For a plane wave, we have a unique decomposition into photons: we just specify the wave and the number of photons.

Is there a similar decomposition into virtual photons?

In Feynman diagrams, virtual photons can carry any momentum and energy. They have a "continuous spectrum". That suggests that there is no unique decomposition.

However, the success of Feynman diagrams suggests that virtual photons are, indeed, emitted and absorbed as discrete quanta. If we take that literally, then the decomposition of an EM field is those virtual and real photons which have been emitted but not yet absorbed. We probably cannot measure what exact quanta are present, though.

How can a moving charge know that the virtual photons that it emitted will be reabsorbed soon?

Let us consider an electromagnetic coil. It is well known that a current will create a magnetic field which, in turn, will keep the current running even if the voltage is switched off. There is considerable momentum stored in the field.

How does the coil know that it can store momentum into the magnetic field, and that the momentum will be reabsorbed? If we remove the coil, the momentum cannot linger in thin air.

The solution probably is that we cannot remove the charges which are present in the magnetic field, without making the charges to reabsorb the momentum in the magnetic field. The magnetic field of a moving charge resists changes in the motion of the charge. The momentum stored in the field will always be returned back to the charge.

In an earlier blog post we had the model where the field of a charge is an elastic object attached to the charge. Then it is clear that any momentum stored in the movements of the elastic object will always be returned back to the charge.

Thus, the short-lived electromagnetic waves close to a moving charge are such that they inevitably will get reabsorbed if we try to remove the charge.

Summary

The nature of virtual photons was revealed, and we explained the functioning of a radio transmitter with the new concepts.

Virtual photons correspond to classical waves which the moving charge will reabsorb soon after they were emitted. The reabsorption is the backreaction of the charge on its own field. It is also called the "self-force" of the charge on itself.

The charge can only emit short-lived waves which certainly will get reabsorbed. Virtual photons cannot live for long.

The momentum stored in short-lived waves can be considerable. The stored momentum explains how a radio transmitter can convert kinetic energy of its charges to real photons, even though the real photons cannot carry away all the momentum.

Some of the waves which the moving charge emits will survive to a large distance. Those waves consist of real photons which the charge emitted. Far away, spherical waves look like plane waves, and there is negligible reflection back toward the source of the waves.

What is the relevance of gauge symmetries in a gauge theory?

https://arxiv.org/abs/1107.4664

Simon Friederich (2012) writes that the ontology of gauge theories has received considerable attention from philosophers in recent years. There is a debate about spontaneous symmetry breaking: is it a genuine phenomenon of nature or is it just imagination of humans.

The author of this blog has had a feeling that Baron Munchausen has written the "derivation" of a gauge field from a global symmetry of a lagrangian.

For example, in the Dirac equation we can multiply a solution by any complex number whose absolute value is 1, and we get a new solution. We say that there in the lagrangian is a gauge symmetry, whose group is U(1), which are the rotations of the complex plane around the origin.

Turning a global gauge symmetry into a local symmetry

Let us then rotate our coordinate axes of the complex plane by a different amount at different points of spacetime, but in a continuous way. If we had a solution of the Dirac equation, that solution is spoiled by the rotations. To restore the lagrangian to its original value, we introduce a 4-vector potential A. If we choose A in a suitable way and replace the partial derivatives in the lagrangian with gauge covariant derivatives

       D_μ = ∂ / ∂x_μ - A_μ,

then we can restore the original value of the lagrangian.

We say that we have turned the global symmetry into a local symmetry by introducing A. We are free to rotate the local complex plane, as long as we compensate with a suitably chosen A.

We then claim that the new lagrangian describes an electron under a potential A.

A philosophical question is, if in nature there existed a "global symmetry" which was turned into a "local symmetry", or is this just a story which we construct in our human minds?

As we have noted in various blog entries in the past year, the new lagrangian with a 4-vector potential A does not describe the interaction of an electron correctly because it fails to take into account the change of the inertial mass of the electron under a static electric field potential. It looks like the story with symmetries is just human imagination, and furthermore, the story is incorrect.

The role of degrees of freedom in a field

Let us think about waves in a tense string. A degree of freedom in the string is the vertical position of a point in the string, at various spacetime locations. The position is given as a real number z(t, x).

We start the construction of a wave equation from this degree of freedom.

Let us imagine that the string has an infinite length. The vertical elevation of whole string is not important for waves. What counts is the differences of the vertical position at various spacetime points. There is a global symmetry

       z(t, x) -> z(t, x) + C

for the solutions, where C is a constant real number.

We could now let C vary according to the spacetime point (t, x), and introduce a gauge field A to compensate for the difference in the lagrangian of our string.

The existence of a global symmetry was the result of two things:

1. We chose a real number z as the degree of freedom.

2. We wrote a typical wave equation where only the differences in the value of the degree of freedom z matter.

Rather than saying that we constructed A from a global gauge symmetry, we could say that we constructed the gauge field A from the degree of freedom which we chose.

We were lucky to find a suitable degree of freedom which describes the string well. The existence of a global symmetry is an automatic consequence from this degree of freedom and the normal way how waves behave. The global symmetry is not interesting in itself.

Can a scalar field grow its energy as the universe expands?

Here is an update to a question we have been looking at this fall. This text is has been copied from a discussion at:

https://www.physicsforums.com/threads/does-a-fields-vacuum-density-violate-conservation-of-energy.977279/

The fields which we know from our everyday experience, for example, the photon field and the electron-positron field, have a constant number of quanta as the universe expands, if we ignore reactions with other particles.

 
            photon
A ~~~~~~~~~~~~~~> B
   <--------------------------------->
      expanding universe

A photon will gain a redshift when it moves in an expanding universe, because if it is emitted by an event A and absorbed in another event B far away, then B "sees" A receding at a great speed - B sees the photon redshifted. If we take the energy of the photon to be what B sees, then the photon has lost energy.

The redshift mechanism reduces the kinetic energy of a electron, too, when it moves in an expanding universe.

What about a scalar field which some people claim, is causing the acceleration of the expansion right now?

The first thing to note is that a field whose energy increases in an expanding universe is "exotic matter", that is, it has a negative pressure and negative gravity. It breaks various energy conditions of General Relativity. We do not know such matter from our everyday experience. It is highly speculative to assume that such matter could exist.

What about the energy content in the hypothetical scalar field? Is that energy contained in quanta of some kind? If yes, does the number of such quanta increase as the universe expands?

Our everyday experience is that energy in a weakly interacting system is, indeed, divided into well-defined quanta which can carry mass and kinetic energy.

The question is harder in a strongly interacting system, say, a crystal. Is the vibrational energy in a crystal divided into quanta of some kind? Certainly not in any unique way.

Suppose that we work in the Minkowski spacetime. The universe is not expanding. We want to excite the Higgs scalar field which has the famous Mexican hat potential. We use a huge particle collider to produce a vast density of Higgs particles. In that way we store a lot of energy in the Higgs field. Can we say that the excited Higgs field has its energy stored in quanta of some kind? It is a strongly interacting system.

We expect the excited Higgs field to release just the same amount of energy as we pumped into it. What if the universe is expanding at the same time? Can the released energy be bigger than the pumped energy?

That would be surprising. Rather, we would expect the Higgs field lose some of the kinetic energy of the Higgs particles as the universe expands.

The hypothesis that a scalar field can grow its energy in an expanding universe is highly speculative. It is at odds with what we know about other fields.

The Higgs 1964 paper is correct

In the previous two blogs we had doubts if the Goldstone boson really is eliminated when we couple the gauge field A to the Higgs field. How can a massive field A "simulate" a Goldstone boson which is supposed to move at the speed of light?

It turns out that the lagrangian is fundamentally changed by the addition of the coupling. The field φ_1 becomes "redundant" in the sense that only the value of

        B = A - gradient(φ_1)

is relevant in the equation, and we can freely choose φ_1 without affecting the physics of other fields. We say that φ_1 has become redundant.

How is it possible that φ_1 was very relevant before the coupling, but we can discard it after the coupling is done?

/\/\/\/\/\  a
/\/\/\/\    b

A somewhat similar thing happens if we have two springs A and B which exert a constant push force F. Let the length of A be a and the length of B be b. If we squeeze the springs separately, then both a and b are relevant in the lagrangian of the system.

/\/\/\/\/\/\/\/\/\/\/\ a + b

But if we attach the springs together, then only the value of a + b is relevant, not a or b any more. We can freely choose a and b, as long as a + b stays the same.

The Goldstone boson of the uncoupled Higgs model simply does not exist in the coupled model. It would still exist if the coupling were very weak. But the coupling is strong enough, so that the Goldstone field is not relevant at all.

That is what "eating" the Goldstone boson means. The value B ate both the field A and the Goldstone field.

What if the gauge field A is coupled to a fermion?

Let the field A in the Higgs 1964 paper be coupled to a fermion, say, to the electron.

Let us replace A by B in the lagrangian of the fermion, let that be L(B). Let us find a solution ψ for the wave function of the fermion in L(B).

We can make a solution for the old lagrangian with A, by multiplying ψ with a phase factor:

        exp(i e φ_1) ψ

for an arbitrary φ_1. What did we show? We showed that we still can choose φ_1 arbitrarily and have a solution, without affecting the physics seen by outside observers. An observer cannot measure the absolute phase factor of ψ. The Goldstone field stays redundant even when A is coupled to a fermion.

Some analysis on the Higgs 1964 paper Broken Symmetries and the Masses of Gauge Bosons

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.13.508

A freely readable copy of the Peter Higgs 1964 paper (just 2 pages) is available at the link.

The paper introduces two scalar fields and a gauge field A.

A somewhat analogous system would have the electromagnetic 4-potential A as the gauge field, and an electrically charged boson particle field as the scalar fields. An electron is a spinor field in the Dirac equation.

We know that if we have a free electromagnetic 4-potential A (of electromagnetic waves), then we can add the gradient of an arbitraty differentiable function f to A, and nothing changes in the physics. It is called a gauge symmetry.

Our August 24, 2019 post showed that there is no gauge symmetry if an electric charge is present: the inertial mass of the charge varies if its electric potential changes relative to far-away space. The electric potential of far-away space defines a preferred frame.

The perturbation in the Higgs paper

The fields are classical at this point. Higgs introduces a Mexican hat potential which causes the scalar fields to assume a constant non-zero value at a minimum energy configuration. We have defined the coordinates of the scalar fields such that the constant value of the two scalar fields is

        (0, v).

That is, the minimum energy happens when the fields have that constant value everywhere in the Minkowski space.

Higgs studies a small perturbation of the field values

        (Δφ_1, Δφ_2)

around the minimal energy value (0, v).

We have not seen a proof anywhere that the perturbation series converges. Thus, we do not know if the quadratic approximation by Higgs approximates the original system of equations.

The equations are rather complicated. It might be hard to prove the convergence.

One may ask if it matters at all if the perturbation series converges. The perturbation equations by Higgs look nice and we do not really need to care about the original equations.

Introducing the "Unitary gauge": why does the Goldstone boson disappear?

Higgs defines a new 4-vector potential B by subtracting the gradient of Δφ_1 from A and writes equivalent equations using B. We say that he moved to the "unitary gauge" for the 4-vector potential.

He writes that the new equations describe a massive scalar boson and a massive gauge boson.

The Goldstone boson, which travels at the speed of light, disappeared. Where did it go?

The Goldstone boson is not directly visible in the new equations. But it still exists in the system. A mathematical manipulation of the equations does not change the physics. There still exist waves that carry energy at the speed of light.

What are the "real" physical fields described by the equations? We cannot answer that question by defining a new variable and removing an old variable.

Does the Higgs change of a variable prove that there is no speed-of-light energy transfer in the system? Or does it just hide the obvious speed-of-light waves behind complex equations?

A way to study this is to keep using the old field A and not move to B. The physics have to be the same. Is there some mechanism in the equations which couples the light-speed waves tightly to slower waves and prevents energy from being transferred at the speed of light?

Light within a medium moves slower than the light speed in the vacuum. If an electron is shot at a high speed into the medium, it will move faster than the local speed of light, and will emit Cherenkov radiation. Could something similar happen in the Higgs model? A Goldstone wave would lose energy to the gauge field and soon disappear? Since a Goldstone boson can have a low energy, it cannot produce massive gauge bosons. In the Higgs paper there are no massless gauge bosons, but in the Standard Model there is the photon.

The Goldstone bosons of the Higgs field may quickly decay to photons, and would not be noticed in the LHC instruments.

The analogue of light within a medium

Light moves slower in a medium, for example, inside glass than in vacuum.

There are charges in glass which can oscillate in synchrony with the light waves.  The massless photon field is coupled to a massive "field" of the charges. The photon "acquires a mass" through the interaction, and moves thus slower than the light speed in vacuum.

This effect is clearly analogous to the model in the Higgs 1964 paper.

When a wave of light enters glass, the wave quicly loses some energy to the oscillation of the charges. The end result is an oscillation wave moving slower than light in vacuum.

We may imagine that a photon has been converted to a quantum of an oscillation wave. A photon became a phonon.

Does a photon "exist" within glass? If we have no way of creating a photon inside the glass without at the same time creating the corresponding oscillation wave, then we might say that a photon cannot exist. It has been "eaten" by the oscillation wave.

We may imagine that charges in the glass are attached to the solid matter with springs. There are rubber bands between adjacent charges.

The electromagnetic field has the role of the gauge field.

The lagrangian in the Higgs paper

It is easiest to analyze the model in the Higgs paper by looking at the lagrangian.

We first analyze the fields as classical.

It is best to keep the original variables. Let

       φ_1 = 0

       φ_2 = v

approximately, where v is constant and non-zero. We will look at small deviations of the φ_i fields from those values.

Let us first forget A. Then a disturbance in φ_1 moves at the speed of light.

A disturbance in φ_2 moves much slower because a deviation from v stores a lot of energy in the lagrangian formula. It corresponds to a massive particle.

Let us then couple A to the system. We keep φ_2 a constant v. If we let φ_1 = 0, we see that a deviation of A from 0, A^2 stores energy in the lagrangian. That means that A is massive and a disturbance in A moves slowly.

We have now coupled A. What happens if we first set A = 0 in all space, and then disturb φ_1 at a point in space?

The lagrangian shows an interaction between φ_1 and A. Obviously, energy will start leaking from φ_1 to A immediately. The disturbance in φ_1 keeps moving at the speed of light, and it keeps leaking energy to A for its whole trip. The energy stored in A moves much slower.

Let us then use quantum mechanics. A disturbance in φ_1 corresponds to a massless boson. The energy E of a quantum may be very small.

The mass of a quantum of A may be larger than E. Then the quantum of φ_1 cannot decay into a quantum of A.

The reasoning above suggests that a massless Goldstone boson cannot be eaten by a massive gauge boson.

Can we excite φ_1 without exciting the field A at the same time? In the electroweak interaction, the gauge field A interacts with many different particles. We can probably excite A separately from the Goldstone boson field. Then the Goldstone boson field cannot be merged to the gauge field A.

The coupling "constant" (not really a constant) between φ_1 and A in the lagrangian is

       2e φ_2 = 2ev.

If we set the constant e very small, then we clearly have two separate fields, φ_1 and A which interact very little. How fast does energy leak to A?

Decoupling the Goldstone bosons?

If the only equations of our physical system are the ones given in the Higgs paper, then moving to the new variable B can bee seen as decoupling the free degree of freedom in A and the Goldstone field.

Once one has solved B, one is free to choose any A and the Goldstone field φ_1, as long as 

       B = A - 1 / (ve) * gradient(φ_1).

This requires that all the communication to these fields go through the "interface" given by B. Then we may say that we decoupled a degree of freedom.

However, in more complicated lagrangians, one can generally communicate with A in many ways. Then we cannot do a decoupling in the way given.

What fields "exist" if some fields are decoupled?

One may argue that the field φ_1 does not exist at all if we use the new equations with B. Then the claim that there was a symmetric potential for φ_i and the symmetry was broken is vacuous. The system where the symmetry broke does not exist.

If we claim that φ_1 really existed, and exists right now, then we probably can excite φ_1 separately from A, and the Goldstone boson exists.

Conclusion

If the scalar fields existed, and do exist now, and the fields fell into an energy minimum of the Mexican hat, then the Goldstone boson does exist, contrary to the claim in the Higgs 1964 paper. The gauge field cannot eat the Goldstone boson.

Maybe it is hard to excite the Goldstone field, and that is the reason why the boson has not been found.

It might also be that φ_1 does not exist at all. There was no symmetry breaking nor a Mexican hat potential. There is just the field φ_2 and its potential curve which has the minimum at a non-zero value v.

Can one really "eat" the Goldstone bosons of the Higgs field after a symmetry breaking?

https://www.theorie.physik.uni-muenchen.de/lsfrey/teaching/archiv/sose_09/rng/higgs_mechanism.pdf

When the Higgs field finds its minimum at a certain vacuum expectation value, the field can still oscillate around that minimum.

One direction x of the oscillation has a potential term of a form

       V(x) = k x^2,

where x is the displacement from the minimum.

Other directions y of oscillation have the respective

       V(y) = 0.

Each such direction y corresponds to the massless Klein-Gordon equation, and a massless scalar particle, a Goldstone boson.

A Brout-Englert-Higgs trick is to do away with the motion in the directions y by "rotating" the local frame so that all the displacements y stay zero. This requires adjusting the gauge fields in such a way that they "simulate" the effect that a Goldstone field would have had on the value of the lagrangian. The trick is described as "eating" the Goldstone bosons.

If a gauge field is only coupled through its derivatives to other fields, then one has a large freedom to transform the gauge field, and the physics stays the same. An example is the electromagnetic field, where one has a very large freedom to modify the 4-vector potential (= the gauge field). However, when the gauge field is coupled to, e.g., the Higgs field, then transformations are very much restricted.

Some people seem to think that one can carelessly transform the gauge fields and do away with the Goldstone bosons, and still keep the physics the same. That is a mistake.

http://philsci-archive.pitt.edu/10962/1/Sebastien_Rivat_-_Spontaneous_symmetry_breaking_-_2.pdf

Sebastien Rivat has observed that there is a problem in eating the Goldstone bosons.

As we wrote in our blog on August 24, 2019, the gauge symmetry of the electric potential does not work if an electron is present: the inertial mass of the electron depends on its potential difference relative to far-away space. The interaction with the electron spoils the gauge symmetry. A similar thing happens in the above analysis with the Higgs field: the interaction with the Higgs field spoils the gauge symmetry in the gauge fields.

It is not clear to us yet, what implications the various problems in gauge symmetry have for the standard model.

https://physicstoday.scitation.org/doi/full/10.1063/PT.3.2196

Eating up the Goldstone bosons was the revolution of 1964 which earned a Nobel prize for Englert and Higgs.

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.13.508

Above is a link to a freely readable copy of the original Higgs paper published in October 1964. The paper is just two pages.

Peter Higgs first defines two real scalar fields which interact with a vector field A.

In equation (3) he defines a vector field B which is based on A and the Goldstone field

       Δφ_1.

He claims that B in equation (4) describes a vector field whose quanta have a non-zero mass.

But a disturbance in the Goldstone field moves at the speed of light. How can we model it with a new field B where the quantum moves slower than light?

Also, the vector field A is usually understood as an external field. For example, in the Dirac equation with the minimal coupling, A is an external field. Does it make sense to define a field B which is a mix of the particle fields (the scalar fields) and the vector field?

The Aharonov-Bohm effect

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

The Aharonov-Bohm effect shows that the absolute value of the electromagnetic vector potential A has observable effects. It is not just the electric field E and the magnetic field B which affect the observable behavior of electrons.

One cannot carelessly transform the electromagnetic vector potential A and still keep the observable physical behavior of the system same. The coupling between the electron field and the electromagnetic field spoiled the gauge symmetry.

Similarly, the coupling of the Higgs field to the gauge field probably spoils the gauge symmetry. The trick by Higgs and others of eating the Goldstone bosons by moving the contribution of the Goldstone field to the gauge field probably does not work.

Why have we not observed the massless Goldstone bosons? Would they show up in the LHC accelerator?

Or is the Z and W boson mass creation mechanism very different from what the standard model claims?

The gauge symmetry of electromagnetism plus charges does not hold in a relativistic setting

Much of modern physics rests on the assumption that electromagnetism has a gauge symmetry, and that the electromagnetic field is a gauge field.

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

The very first gauge symmetry introduced in Wikipedia is the option to set the zero electric potential at any value we like.

Let us then consider the electromagnetic field with charges, say, electrons and positrons.

As long as the inertial mass of an electron does not change substantially with the potential, we can indeed fix V = 0 at any voltage we like.

However, this is not true if the potential, relative to far-away space, is significant compared to the rest mass 511 keV of the electron.

We can say that the electric potential of far-away space is a preferred electric potential. All potentials were not created equal.

The Klein-Gordon equation is derived from the energy-momentum relation

        E^2 = p^2 + m^2,

(where we have set c = 1).

The relation holds in vacuum. But it does not hold if the particle has significant potential energy.

If we in a potential write the relation as

       (E - V)^2 = p^2 + m^2,

that relation holds approximately if the potential |V| is small. It does not hold in a relativistic setting.

The "minimal coupling", which is used to introduce an electromagnetic vector potential A to the Klein-Gordon equation or the Dirac equation, assumes that |V| is small.

The error in the minimal coupling explains the Klein paradox of the Dirac equation.

https://meta-phys-thoughts.blogspot.com/2018/10/we-solved-klein-paradox.html

The minimal coupling is derived in various sources from an assumed gauge symmetry of electromagnetism. The symmetry does not hold in relativistic settings. That explains why the minimal coupling does not work reasonably with large potentials.

We have shown that relativistic electromagnetism with charges is not a gauge theory, at least in the sense which is traditionally assumed. This means that much of the standard model is based on a shaky assumption.

If the magnetic field is strong, is the minimal coupling formula wrong, in the same fashion as for the electric potential? No. The total energy of a charge is not affected by a strong magnetic field.

The Navier-Stokes millennium problem and the existence of solutions for the Einstein equations

In this blog we have brought up the question whether the Einstein field equations have a solution for any realistic physical system.

The Einstein equations are nonlinear. Therefore we cannot build complex solutions by summing simple solutions.

The Einstein equations are very strict: they imply results like the Birkhoff theorem, which dictates energy conservation for all matter fields.

https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness

The Navier-Stokes equation is nonlinear, too.

https://arxiv.org/abs/1402.0290

Terence Tao suspects in his 2015 paper that at least some smooth initial problems "blow up", that is, develop singularities.

The problem in proving the existence of solutions for Navier-Stokes is in turbulence. We currently lack tools to understand turbulent behavior mathematically. Turbulence might create a singularity very quickly for almost all initial conditions.

The (assumed) generation of singularities is another common feature of Navier-Stokes and Einstein.

We do not know if solutions of the Einstein equations might contain turbulent behavior. The problem in finding solutions for Einstein seems to lie in the strictness of the equations. Solving the equations requires that "ends meet" when we assemble a puzzle on the FLRW model finite space. How do we know there exists any configuration of pieces such that the ends meet?

Terence Tao remarks that turbulence may be analogous to problems of complexity theory, for instance, to the P = NP millennium problem. We believe that P = NP will never be solved, because the collection of all polynomial-time algorithms is too complicated a system to be tamed and analyzed.

How was the fractal-like anisotropy of the cosmic microwave background born?

http://www.astro.ucla.edu/~wright/CMB-DT.html

https://www.physicsoverflow.org/19466/why-is-the-cmb-nearly-scale-invariant

The differences in temperature are in the range 25 - 70 microkelvins if we measure a feature size between 0.2 degrees and 20 degrees in the sky. The overall temperature is 3 K.

The features are roughly "scale-invariant" in the sense that a magnified map shows similar-looking anisotropies to an unmagnified map.

The inflation hypothesis and the patterns of the microwave background

The inflation hypothesis explains the scale invariance by "random quantum fluctuations" which were inflated to a cosmic scale by the rapid expansion of the universe. There is a huge amount of energy in the anisotropies, even though the temperature difference is just around 1 / 100,000.

The energy had to come from somewhere. In the inflation hypothesis, the expansion of the universe produces energy from nothing to a scalar inflaton field. In our previous blog post we remarked that the non-conservation of energy may contradict basic principles of quantum mechanics. New energy quanta would pop up from nothing.

In classical physics, turbulence of, say hot smoke which rises from a smoke stack, creates a fractal-like structure. But it is hard to see how an expanding universe would have turbulence. In the case of the smoke, the turbulence is a result of the hot smoke colliding with cool, static air.

In a Big Bang model without inflation, there are roughly 6,000 points in the night sky which were not in a causal contact at the time when the microwave backround was born. Such points are roughly at a distance 3 degrees from each other.

How can the patterns of the anisotropies be coordinated over 20 degrees of the sky if there was no causal contact between different parts of that area? The inflation hypothesis solves this by asssuming a very fast expansion phase which caused a pattern to expand faster than light.

Could there be a mechanism which produces large patterns without a causal contact?

In a big bounce model, the causal contact would have occurred in the contraction phase.

Is there any other way to explain the large patterns? Maybe the original signal is at a constant temperature, but the space between the signal and us contains a fractal-like phenomenon which can change the apparent temperature. But how do we explain galaxy formation in that case? The fractal-like phenomenon should cause matter to collapse into galaxy clusters.

Yet another explanation is that, for an unknown reason, the matter in the Big Bang was created into a fractal-like structure. Since the Big Bang is beyond our understanding, there is no reason why the original matter content should have been smooth. If there was some process which created the original Big Bang universe, and that process was not constrained by the light speed in our universe, then the fractal-like structure could be a result of that process.

Why there are no "domain walls" or other defects in the visible universe?

It is assumed that the symmetry breaking of various fields, for example, the Higgs field and the electroweak field caused the universe to form "crystals" where a certain field has a constant value. The value in the neighbor crystal may be different.

We have not found domain walls in the visible universe. It looks like the whole visible universe is inside a single crystal.

Here we again have the problem how the whole visible universe is inside a single crystal if different parts have not been in a causal contact since the Big Bang.

The inflation hypothesis solves the problem by assuming that every crystal expanded immensely during the inflation.

Big bounce models might assume that the large crystal formed in the contraction phase.

We might also imagine that the original creation process of the Big Bang universe already had the symmetries broken, and the corresponding fields have a single value throughout the universe.

The quantum measurement problem and "quantum fluctuations" in the inflaton field

In standard quantum mechanics, if we try to measure the energy content of some object, we will get varying values because of the uncertainty principle. Some people call these varying values "quantum fluctuations".

In a measurement, the measuring apparatus interacts with the object. If we claim that the variations of temperature in the sky are "quantum fluctuations", what is the measuring apparatus in that case? Is the scalar field itself and the metric a "measuring apparatus" which measures the energy content of the scalar field?

An analogous process is crystallization of a cooled liquid. If the temperature happens to drop within a small group of molecules, they will form a small initial crystal, a seed, which will grow larger. The measuring apparatus of the temperature is the liquid itself. A small random variation of the temperature downward will start a cascading process where at the end, it is the large crystal which interacted with the original temperature variation. The crystal is the measuring apparatus.

But is the inflaton field really analogous to crystal formation? A crystal is formed from a finite number of atoms. What are the "atoms" in the case of the inflaton field?

How does a scalar field behave in an expanding spatial metric?

The inflation hypothesis assumes the existence of a scalar field whose scalar value throughout the space "rolls" to a lower energy value.

The field may initially have, say, the value 0 in all space, and it somehow moves slowly to a lower energy value of, say, 1.

What does all this mean? We do have some experience of scalar fields in classical systems, but such fields are not the lowest level primitive fields. Such scalar fields are a high-level abstraction in a system consisting of elementary particles.

For example, if we have a drum skin, its vertical displacement is a scalar value in the 2D space of the skin. The lowest level description of the skin consists of atoms. The vertical displacement is a high-level concept.

If the 2D space expands, then the vertical displacement is affected in various ways.

Does the energy of a primitive field grow as the space expands?

Why would the inflaton field in an expanding space keep its scalar value, so that the energy of the field increases as the space stretches? Such a field certainly is "exotic matter" if its energy grows as the space grows.

The energy of an individual photon decreases in an expanding universe relative to "static" matter. The decrease can be attributed to the redshift that the static matter sees relative to the matter from which the photon originated.

The number of stable elementary particles, like the proton or the electron, probably does not grow in the expansion.

The energy content of the electron field preserves the energy in the rest mass of the electrons, but loses the kinetic energy of the electrons, because an electron keeps moving to a location where its velocity is less relative to static matter.

We face the dilemma why a scalar field would behave differently. Why would its energy grow in the expansion, if the energy of the electron field does not grow?

Is a non-zero value of a field always associated with some quantum?

If we have an electric or a magnetic field which differs from zero, there is a quantum associated with it: a photon, or a quantum of some other field, like the electron field. The electromagnetic field does not have a life of its own independent of these quanta.

If we have a scalar field, why would it have a non-zero value which is not associated with a quantum of some field?

We need to study the logic behind the Higgs field. The Higgs boson is an excitation of the field. The Higgs field is assumed to be non-zero throughout the space. The non-zero value of the field is not born out from some quanta.

The vacuum is often defined as the lowest energy state of fields. There are no quanta present then.

We might then claim that quanta must be present if the fields are not in the lowest energy state. If the inflaton field initially is at a metastable higher energy state, then there must exist quanta which are responsible for that. If space expands, the number or energy of those quanta does not grow. Rather, the quanta are diluted in a larger volume of space. This would imply that the inflation mechanism cannot work. The value of the inflaton field will approach the minimum energy value as the quanta are diluted.

A charge particle of a scalar field?

The reasoning above forbids exotic matter whose energy would grow as space expands. It also predicts a new particle, the charge particle of a scalar field. The charge particle would be responsible when the field has a value which is not the minimum energy value. Can the charge particle decay to other particles, so that the charge is lost? If that is the case, then the scalar charge is not conserved.