The speed of sound inside a neutron star CAN exceed the vacuum speed of light

https://www.nature.com/articles/s41567-020-0914-9

Evidence for quark-matter cores in massive neutron stars by Eemeli Annala, Aleksi Vuorinen et al. in Nature Physics June 1, 2020 studies the mass distribution of a neutron star, assuming an arbitrary function f:

       pressure = f (energy density of matter).

The authors mention that most hadronic models predict that the speed of sound squared, c_s^2, is equal to 0.5 or larger for high densities (we use natural units where c = 1).

Is it possible for the speed of sound to exceed the vacuum speed of light?

The answer is definitely yes, if we define the speed of sound as the phase speed of a well-formed periodic pressure wave (a "sine" wave). That is, as the speed of the crest of wave. A crest of an infinite sine wave does not transmit any information from a point A to a point B. The speed of the crest is not constrained by the universal signal speed limit, that is, the vacuum speed of light c.

https://motls.blogspot.com/2020/10/a-fun-calculation-of-maximum-speed-of.html

https://arxiv.org/abs/gr-qc/0703121

Lubos Motl in his blog post (2020), as well as George Ellis et al. in their arxiv paper (2007) claim that the "causal limit" for the speed of sound is the speed of light, c. But they fail to define what they exactly mean by the speed of sound.

When considering the stiffness properties of matter, the natural definition for the speed of sound is the phase speed, not the signal speed. The phase speed depends on the stiffness, and can exceed the speed of light.

                          string

      wall |--------------------------| wall

As a practical example, consider a tense string which is attached to walls at its endpoints. If we pluck the string, we can create a standing wave into it. There is no speed limit for the phase speeds of the two sine wave components of the standing wave. The standing wave does not transport any information and is not constrained by the speed of light.

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.