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?