The electron does not have an intuitive physical model at all?

For the past four months we have been trying to find an "intuitive" physical model for the Pauli and Dirac equations, but we have failed. The biggest obstacle is the gyromagnetic ratio g = 2. Any classical point charge which moves in a circular orbit will have the gyromagnetic ratio g = 1.

What does an "intuitive" physical model mean? We have concentrated on studying models which contain one or more point particles. The wave of a classical point particle would be determined using a path integral approach: the lagrangian density over a path would determine the phase of the wave.

The electron as a classical point particle

Maybe the hypothesis that the electron is, in some way, a classical point particle is wrong? We can certainly measure the position of an electron with a very great precision. There are claims on the Internet that experiments set the limit of the electron radius at most to 10^-22 m.

In an earlier blog post we calculated that a point-like electron must move at the speed of light in a circle of radius 2 * 10^-13 m, to explain its spin angular momentum.

The electron in some aspects looks point-like, but its spin and magnetic moment suggest that it should quite a large sphere, or a point particle moving around quite a large circle, to explain its spin and magnetic moment.

A classical point charge involves the paradox of the infinite energy of its electric field. The electric field of the electron contains its mass 511 keV of energy outside the classical radius of the electron 3 * 10^-15 m.

Maybe it is best to reject the idea that the electron is a classical point particle, in any sense.

What is the correct kinetic energy term in the Pauli equation?

The kinetic energy operator in the Pauli equation

     (1/ (2m) (σ ∙ (p - qA))^2, + qϕ) ψ
     = (i h-bar ∂ / ∂t) ψ

is

       (σ ∙ (p - qA))^2,

where σ is the vector of the three Pauli matrices, and the exponent 2 means that we take the product of 2 × 2 matrices, that is, the composite mapping.

One may ask why the operator should not be

       (p - qA)^2,

where the exponent 2 means taking the inner product of two vectors? If the vector potential A is zero, then the kinetic energy operator does, indeed, take a simpler form, the Laplace operator,

       p^2,

where the exponent 2 means taking the inner product of the vector p with itself.

An inner product p^2 in a 3-dimensional space is simple in the sense that "cross terms" of p_x, p_y of the vector p components in the x, y, z directions do not affect the value of the product. It is just the sum

       p_x^2 + p_y^2 + p_z^2

which matters.

However, when A is not zero, the magnetic field determines a preferred coordinate system in space, and it is not at all obvious that the simple inner product form of the kinetic term is the right one then.

In classical electrodynamics with the electron a point particle, in the Hamiltonian formulation, the kinetic term is an inner product also when A is not zero. But in wave mechanics, it might not be the right kinetic term.

The Cauchy stress tensor in classical mechanics is a full 3 × 3 matrix. Elastic energy can be stored in the normal stresses of a solid object, but shear stresses also contribute to elastic energy. That is, "cross terms" in that case are important in the determination of energy of the system.

We conclude that maybe the right kinetic operator in the Pauli equation should also involve cross terms, in one way or another.

If Nature has decided that the kinetic term still should have the form of being a "square" of something, then the "factorization" of p^2 with the Pauli matrices is a good candidate as the kinetic term.

Adding cross terms takes us away from a classic point particle model.

http://hitoshi.berkeley.edu/221a/landau.pdf

When A is not zero, then operators p_x - qA_x and p_y - qA_y, in general, do not commute. This might be an intuitive reason why the inner product of vectors no longer is the correct kinetic energy operator. In free space, with A zero, operators p_x, p_y, p_z are orthogonal in the sense that they have a common eigenbase, and there is an eigenfunction for which p_x is zero and p_y non-zero, and so on. Maybe it only makes sense to calculate the kinetic energy as a simple inner product if the operators involved are orthogonal? The metric in the underlying space is flat in that case?

If the electron has no point particle model, there are problems for hidden variable interpretations

The de Broglie-Bohm interpretation assumes that particles at all times have a precise position and that they move according to what the "pilot wave" tells them to do. The position of the particle is a hidden variable of the system.

But if the electron does not have an intuitive model where it is a point particle with a precise position, how can we make the de Broglie-Bohm interpretation to work? What are the hidden variables then?

https://en.wikipedia.org/wiki/De_Broglie–Bohm_theory

The Bohm model in the non-relativistic case is simply that we imagine the point particle to be a water molecule which travels along the probability current of the wave function.

According to Wikipedia, the relativistic case is problematic because then in spacetime there are no fixed planes of equal time. What does it then mean that the probability is conserved?

Hrvoje Nicolic and others have developed Bohmian models that handle the Dirac equation.

We may study probability conservation in a future blog post. Probability conservation has further problems in curved spacetime.

What is the correct kinetic energy term in the Dirac equation?

The Pauli equation comes from a non-relativistic energy momentum relation:

       E = p^2 / (2m) + m.

The energy E can be divided in a simple way into the energy of the rest mass m, and the kinetic energy term p^2 / (2m).

The relativistic energy-momentum relation

       E^2 = p^2 + m^2

does not admit such a simple formula for the kinetic energy. The formula for E would involve a square root, which is awkward.

The quantum mechanical operator for E involves a partial derivative on time, and p involves partial derivatives on spatial coordinates. A general principle of special relativity is that time and space should be treated in a unified way. That suggest that the temporal and spatial operators should be combined somehow:

       E^2 - p^2 = m^2.

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

The operator on the left side of the equation is a d'Alembert operator or a wave operator. In the case of the Pauli equation, we can split the wave operator into the kinetic energy part (the Laplace operator) and the partial derivative on time. In the relativistic case, such a split may not be possible.

The electron-positron pair is the origin of the 720 degree spin 1/2 rotation symmetry?

This blog post is a continuation for the pipe model that we have developed for the spinning of a created electron-positron pair.

The idea is that the wave function (relevant for the rotation) of the system electron & positron has a 360 degree rotation symmetry. The wave function of the system completes one wavelength when we rotate coordinates by 360 degrees.

The wave function of the system is the product of the wave functions of the electron and the positron.

The individual wave function of a particle does not behave in an intuitive way for our 3-dimensional space: the phase of the wave function is inverted when we rotate coordinates by 360 degrees.

That inversion does not make much sense for a complete quantum system, but does happen if we break up the system into parts.

We may require that the wave function of a complete quantum system must return to the original value after we rotate coordinates by 360 degrees.

If we try to treat the electron as an independent quantum system, we end up with the strange 720 degree rotation symmetry of the spin.

The SU(2) geometry of spinors gets an explanation from this.

What about the gyromagnetic ratio 2? In quantum mechanics, we do not have rules about how to determine the momentum of an individual part of a quantum system. The rotation of the electron lives in a strange 720 degree geometry. How should we map it to the ordinary 3D space to determine the velocity v of the electron and calculate the magnetic force it exerts?

It might be that the observer must treat the electron rotation as having a double velocity compared to the "real" velocity, so that he sees the electron wave function having a 360 degree rotation symmetry. Then he will get a double value for v, and will see the electron exert a magnetic force which is 2X of what we would expect based on the spin angular momentum 1/2 h-bar.

What about the spin angular momentum, the 1/2 h-bar? Why he does not see the electron spin as double? One may conjecture that since quantum mechanics conserves angular momentum, one must see the spin to have its correct value. On the other hand, there is no conservation of magnetic moment. Nothing breaks if an observer sees and feels the magnetic moment of an electron as double.

The spin angular momentum has been measured directly by flipping the spin of many electrons in a block of solid material. We know that the angular momentum is 1/2 h-bar. The flipping obviously has to use the magnetic moment of the electron as a handle which we use to flip the orientation of the spin. Why does the magnetic handle have a double strength, compared to what we would expect?

The Dirac equation offers sort of an explanation. We still need to find the link from the model we tried to sketch, to the Dirac equation.

The Dirac equation does not have a reasonable speed operator v. There is also the mystery of zitterbewegung. These are probably connected to the fact that the electron only forms half of the complete quantum system, the electron-positron pair.

The concept of the reduced mass may offer a clue. A two-particle system is represented by the center of mass and the vector from particle 1 to particle 2. The treatment of the complete electron-positron quantum system might involve similar ideas.

Does the Dirac equation work by chance?

Despite three months of hard work we have not been able to find an intuitive physical model which would explain the Dirac equation. Richard P. Feynman wrote that no one has understood the Dirac equation "directly".

Is it possible that the Dirac factorization of the Klein-Gordon operator just by chance adds the necessary degree of freedom that can be used to describe the spin 1/2 of the electron?

Does the Dirac derivation of his equation predict the spin 1/2?

The Klein-Gordon equation is Lorentz covariant, and Dirac derives his own equation in a way which makes it Lorentz covariant, too.

But what properties of the electron does the Dirac equation actually predict?

The eigenfunctions of the equation can be defined as a standard plane wave times a 4-component Dirac spinor.

We then note that if we define an operator S in a way similar to the Pauli equation, S obeys rules which emulate the simple commutation rules we expect from a particle of spin 1/2. That is, if we know the spin projection in the z direction, we know nothing of the projection in the x direction, and so on.

Did the Dirac equation predict the spin 1/2? One might say: no - we just picked an arbitrary operator S which emulates the known rules of spin 1/2. But the Dirac equation made it possible to use a very simple operator S. At least in that sense, the Dirac equation predicted the spin 1/2.

The Dirac hamiltonian H commutes with L + S, where L is the usual orbital angular momentum operator. That fact suggests that S really describes some kind of angular momentum.

But H does not commute with S if the electron is relativistic. Does that make sense? If we have a free electron, why would its spin change during its flight?

Does the Dirac equation predict the gyromagnetic ratio 2?

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

The Pauli equation is the non-relativistic limit of the Dirac equation.

In the general form of the Pauli equation, the interaction with a magnetic field B is hidden in the minimal coupling

       (σ • (p - qA))^2

kinetic term.

But in the standard form, the interaction is shown explicitly as

       σ • B.

The Dirac equation does predict the correct interaction strength and the gyromagnetic ratio 2.

Is it possible that the Dirac equation by chance gets the ratio 2 right, even if the equation does not "really" describe the physical system? That looks unlikely.

The Dirac factorization of the Klein-Gordon operator is a general way to add a spin degree of freedom?

The Klein-Gordon equation with the minimal coupling describes the behavior of a spinless electrically charged massive point particle.

The Dirac factorization trick of the Klein-Gordon operator adds a spin degree of freedom, and furthermore, the minimal coupling gives the right interaction strength with a magnetic field for the spin, too. The minimal coupling term was designed to describe the behavior of a spinless point particle, but it magically produces the right interaction also for the spin angular momentum.

Is it a general rule that factorization of an operator adds a spin-like degree of freedom? If yes, why?

Does the factorization always give a sensible strength for the interaction of the spin with an external field?

The spin 1/2 h-bar of an electron is a remnant of the orbital angular momentum of positronium?

In the hydrogen atom, the electron orbital angular momentum in the z direction is an integer multiple of h-bar. Classically (that is, in Newtonian mechanics), almost the entire orbital angular momentum is in the movement of the electron, and only a tiny fraction in the movement of the proton.

In the positronium "atom", the orbital angular momentum is divided evenly between the electron and the positron. This may be the origin of the strange electron spin 1/2 h-bar. An electron and a positron are always created together. If they form some kind of a primitive positronium atom where the particles do not yet possess a spin, then after flying away, the electron and the positron will evenly share the 1 h-bar of orbital angular momentum of the positronium atom. In this model, the electron and the positron would have parallel spins. In the real world, they seem to have opposite spins.

Classically, the distance between the electron and the positron is 2r in the positronium atom, where r is the Bohr radius of the hydrogen atom. Both particles move around the center of mass in a circular orbit whose radius is r. Since the electric pull on the electron in positronium is only 1/4 of the pull in a hydrogen atom, the orbital velocity of the electron has to be 1/2 of that in a hydrogen atom, so that the centripetal acceleration agrees with the electric pull.

The de Broglie wavelength is defined as

       λ = h / p.

Since the electron momentum p in a positronium atom is just half of the hydrogen atom, the electron only completes half of a de Broglie wavelength in its circular orbit. In previous blog posts we have said that a "natural" periodic movement of a single particle must contain an integer number of wavelengths, to avoid destructive interference. But we noted that if two particles are moving "in unison", then we may consider them as a single system, and it is enough that the system completes a full number of wavelengths in one period. Now we realize that the positronium atom is just such a system.

In a previous blog post we developed the "pipe model" of an electron-positron pair. The particles rotate in unison at the opposite ends of the pipe. A problem is to find a way how the particles can preserve their tandem movement even when one of the particles is accelerated or its spin is rotated.

If we think of a positronium atom in a laboratory, both the electron and the positron have a spin 1/2 h-bar, and they may also have a multiple of h-bar of orbital angular momentum. How do we model the annihilation then?

Sabine Hossenfelder on good problems in the foundations of physics

http://backreaction.blogspot.com/2019/01/good-problems-in-foundations-of-physics.html

Sabine Hossenfelder has written an interesting post about which problems might be fruitful for research in the foundations of physics. Since our blog is about foundations of physics, let us comment on her program.

https://motls.blogspot.com/2019/01/why-competent-physicists-cant-remain.html

Lubos Motl wrote a harsh criticism of Hossenfelder.

The thesis of Hossenfelder is that when experiments conflict theoretical predictions, that is a fruitful experiment-led problem.

If theory itself is inconsistent, that makes a fruitful theory-led problem.

Hossenfelder analyzes 12 problems, if they are fruitful and if they are experiment-led or theory-led. Our blog has touched several of those problems. Let us go through the list of problems.

Dark matter

There is no principle in particle physics which prohibits weakly interacting particles. A priori, the existence of dark matter is more probable than its non-existence. The competing hypotheses, like MOND, suffer from the fact that it is hard to modify newtonian mechanics or general relativity without breaking conservation of energy, momentum, and angular momentum, or equivalence principles. We have not seen anyone developing a MOND model where conservation laws would hold.

Dark energy

Dark energy can be accommodated to general relativity through a cosmological constant. It does not break anything. But what is the origin of the cosmological constant? In our blog, we hold the view that empty space is truly empty - it does not contain energy or vacuum fluctuations. We would not explain dark energy by "vacuum energy" which is present in empty space. Dark energy might be an unknown force.

Hierarchy problem

Why is gravity much weaker than other forces? An anthropic argument is that a strong gravity would make everything collapse into black holes, and humans would not exist.

The weakness itself is not mysterious about gravity. But gravity does have mysterious properties: why does it affect all mass-energy, why does it appear to modify spacetime geometry, why the force is always attractive, and why the gravitating mass is equivalent to the inertial mass?

Grand unification

There is nothing in particle physics that requires the electroweak and strong interactions to be unified at high energies. However, the unification of electromagnetism with the weak interaction hints at that possibility.

At high energies, the relative strength of different interactions seems to converge. We will study this phenomenon in spring 2019 when we will analyze vacuum polarization loops and running coupling constants. Is there such a thing as a "bare charge", or is it an artifact caused by higher level Feynman diagrams?

Quantum gravity

One of the goals of our optical gravity model, and also our renormalization/regularization study, is to find a way to integrate gravity into ordinary quantum mechanics.

There are problems with the geometry of black holes in classical general relativity. We do not know if the Kerr solution is stable.

We do not know at what speed does information about mass-energy distribution spread in general relativity.

To build a model of quantum gravity, we need to clarify classical general relativity.

Black hole information loss

Our view is that Hawking radiation probably does not exist. Therefore, there is no information loss problem.

In quantum mechanics, systems develop in a unitary way and there is no information loss. The fact that the hypothetical Hawking radiation would break this principle is one of the symptoms which show that Hawking used flawed quantum field theory. Other symptoms include problems with energy conservation, momentum conservation, and the classical limit of his hypothesis.

We do not understand why some physicists hold a religious view that Hawking radiation "must" exist. The derivations of Hawking radiation rest on a very shaky, and probably flawed, use of quantum field theory.

Particle masses

There is no principle in particle physics that requires particle rest masses to have a deeper explanation. But there may exist a model, a string model, for example, which might cast more light on the problem.

Quantum field theory

Our hypothesis is that both the infrared and ultraviolet divergences of Feynman integrals are a result of a wrong integration order. We will study that hypothesis in spring 2019.

The Landau pole means that higher order Feynman diagrams will contribute more to the process than lower order diagrams. It is a complexity explosion. The energy is so high that a black hole would form before Landau pole energies are reached. The black hole may save us from a Landau pole.

The measurement problem

Our view is that the many worlds interpretation, where the "branch" for an observing "subject" is chosen with the Bohmian hidden variable method, is the most sensible interpretation of quantum mechanics.

It is not clear if we can ever devise experiments which would differentiate between interpretations. The problem may remain a philosophical one.

The flatness problem

If empty space is truly empty of energy, then flatness is expected.

In optical gravity, we have a hypothesis that the true geometry of spacetime is the flat Minkowskian geometry. But we would need a model to explain the Big Bang. If spacetime is flat, why does the universe appear to expand?

Magnetic monopoles

Some GUTs imply the existence of magnetic monopoles. However, in ordinary particle physics there is no principle that would dictate that they should exist.

A deeper understanding of quantum electrodynamics may resolve this problem. An electron is a source of the electric field. Why there is no source particle for the magnetic field?

Baryon asymmetry

We pointed out that if there are superheavy particles and antiparticles, then a single particle might decay into a whole visible universe which contains just matter. The asymmetry is probably just a local phenomenon.

Why is the cosmic microwave background so isotropic?

Why is the temperature so uniform in areas which are not causally connected in a standard Big Bang model? There may be unknown laws of physics which create a nearly uniform energy distribution in a phase change of the universe. There is no need for the patches to be causally connected if the same mechanism creates the mass-energy in each patch.

The inflation hypothesis explains the uniformity, but Paul Steinhardt has criticized it because it requires fine-tuning which may be even harder than the problem it tries to explain.

Another explanation would be a Big Bounce model. But we do not know laws which would cause the universe to contract after a Big Bang.

The gyromagnetic ratio is 2 because the spin lives in a 1+2-dimensional space?

In the Pauli equation, the effect of the vector potential can be modeled with the flow vector of water.

Close to the wire carrying the current, water flows to the same direction as the current. The velocity vector of water is the vector potential A.

          --->      --->        A

        ------>   ----->

      --------------------- wire

          I -->

If we have an electron doing a circle in the plane of the page anti-clockwise, then the flow of the water "helps" it in its movement close to the wire. The water flow will inflate the wavelength of the electron close to the wire.

Suppose that we have a stationary wave solution in a 2-dimensional (the time is the 3rd dimension) round cavity. Suppose that we increase the area of half of the cavity by 1% and decrease the other half by 1%. This corresponds to modifying the wave equation slightly inside the cavity. The wavelength in one half will decrease 0.5%. That corresponds to a kinetic energy increase of 1%.

But if the cavity would be 1-dimensional, and we would decrease the length of one half by 1%, that would correspond to a 2% increase in kinetic energy.

The angular momentum of the translational movement of an electron,

       r × v,

lives in a 1+3-dimensional space. There are three orthogonal axes of rotation. But the spin angular momentum seems to live in a 1+2-dimensional space. The eigenvectors of the Pauli matrices are in a 2-dimensional space.

The difference of dimensionality may be the origin of the strange gyromagnetic ratio g = 2 for the spin. If the effect of a magnetic field is to change the volume (or the area in 2D or the length in 1D) of a half of the the stationary wave system by a ratio which depends on B, then its effect on the kinetic energy depends on the dimensionality of the system.

In our example, an electron circles in a plane close to the wire. There are 1+2 dimensions. If the spin would live in a 1+1-dimensional system, then the effect of B on the kinetic energy of a stationary solution might be 2X in the spin case.

https://en.wikipedia.org/wiki/Pauli_matrices#Eigenvectors_and_eigenvalues

If we stretch the plane by 1 % in the direction of the x axis, then a vector which is at a 45 degree angle to the x axis will get stretched by 0.5 %. Many angles between the eigenvectors of various σ_i are 45 degrees or 135 degrees. If the effect of a magnetic field would be to stretch one eigenvector by 1%, then the volume of a 3D cube might grow by 2 %. This might be the origin of the gyromagnetic ratio g = 2.

Why does the spin live in 1+2 dimensions?

Why does the spin live in one less spatial dimensions than the ordinary angular momentum? Maybe some uncertainty relation drops one spatial dimension from the description of a 1/2 h-bar angular momentum?

Is there a "quantum state" of an individual electron in a multiple electron system

The Pauli exclusion principle claims that in a non-hydrogen atom, each electron occupies a separate "quantum state". Similarly, in a piece of metal, free electrons fill a "Fermi sea" of states, each falling to its own pigeonhole which is determined by the "quantum state" of the electron.

We criticized the Pauli exclusion principle because there is no definition of what that "quantum state" is, or means.

Let us look at the helium atom. The usual way of modeling it is to assume that we have a single particle moving in a 6-dimensional space. There is a central potential due to the nucleus and a "planar potential" which has a very high energy when the single particle position is equivalent to the two electrons being very close.

If we find a solution of the Schrödinger equation, is there any way to "factorize" it into two parts where each part describes the state of a single electron?

If we have two electrons in different hydrogen atoms, there exists such a factorization. It is trivial.

Let us look at a simpler problem. Suppose that we have two particles in an external potential well in a space with a time dimension and one space dimension. There is a repulsion between the particles. The Schrödinger equation then is about a single particle in a 1+2-dimensional space.

The external potential makes a square well for the single particle. In addition to that, there is the interaction potential of the original two particles. That potential is concentrated on the line y = x, where x and y are the spatial coordinates of the single particle.
                 ________
                |         / |
                |      /    |
                |   /       |
                |/______|
  ^ y
  |
  |
   ---------> x

The diagram is not in scale. The rectangle depicts the square potential well. The diagonal depicts the interaction potential wall of the original two particles.

A solution of the Schrödinger equation is a stationary wave inside the rectangle. It is like a rectangular drum skin vibrating in a resonance pattern within that square.

Now, is there any reason why the solution could be factorized into solutions of two individual particles?

If the particles do not interact, then the factorization is trivial. If there is a weak interaction, we may get some results with perturbation methods. But what if the interaction is strong?

Assume that each solution is determined uniquely by a set of quantum numbers of each electron

Let us assume that we have a strongly interacting electron system. Let us assume that each solution of the Schrödinger equation is uniquely (except by a phase factor) determined by some "quantum numbers" that we attach to each individual electron.

Can we derive the Pauli exclusion principle, for example, from the antisymmetricity of the fermion wave function? The antisymmetry means that the sign of the wave function is flipped if we replace coordinate values of, say, x_1, y_1, z_1 with x_2, y_2, z_2, and conversely.

Now, if electrons 1 and 2 have the exact same quantum numbers, we have:

       Ψ_switched = Ψ_original,

because the sequence of quantum numbers specifying Ψ did not change.

But, the antisymmetry of the fermion wave function implies

       Ψ_switched = -Ψ_original.

We have that Ψ must be zero. We can derive the Pauli exclusion principle from the assumptions:

1. The wave function solution is uniquely determined by a set of "quantum numbers" which can be "assigned" to the coordinate triplet of each electron.

2. The wave function is antisymmetric under the switch of two coordinate triplets.

Is there a mathematical proof that helium atom solutions have property 1 above?

A brief Internet search does not lead us to any such proof. A related question is in which cases a wave equation has a discrete spectrum of stationary states or "resonant" states.

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

The spectral theorem states that all the solutions of the Schrödinger equation can be written as sums of eigenfunctions of the hamiltonian. Each eigenfunction is associated with an energy eigenvalue.

In which cases is the spectrum of energy eigenvalues discrete?

Does the spectral theorem imply anything about a multiple electron system? Could the theorem give some factorization of the solution for individual electrons?