## Unsolved Problems in Physics

New ideas for solving old problems

A professor keeps improving his PhD thesis for the rest of his career.

This is a collection of theory projects addressing long-standing questions in physics. The following essays and papers are meant to inject fresh thoughts into problems that have resisted conventional thinking for a long time. Some of these ideas may seem far-fetched. Critique and suggestions are welcomed (e-mail: fhimpsel@wisc.edu).

## The Stability of the Electron

My longtime favorite is the puzzle about the internal stability of the electron, the prototype of all charged particles. If one considers a tiny charge cloud, it would explode due to electrostatic repulsion ("Coulomb explosion"). This seemingly simple problem has fascinated physicists for more than a century, including Poincare, Lorentz, Fermi, Dirac, Feynman, as well as my Diplom thesis advisor Fritz Bopp (the successor of Arnold Sommerfeld at the University of Munich).

Analyzing the forces that act on the electron might also shed light onto the strength of the electromagnetic interaction. It is quantified by the electromagnetic coupling constant (the square of the elementary charge). This is the famous fine structure constant alpha, discovered by Sommerfeld in 1916. Feynman once wrote that all good theoretical physicists put the value of alpha up on their wall and worry about it. Balancing electromagnetic against non-electromagnetic forces might determine alpha and thereby solve both of these century-old problems in one stroke.

### Force Balance inside the Hydrogen Atom

In order to get started with the stability problem of the electron, it helps to analyze the stability of the hydrogen atom and dig a little deeper. In that case the electron is attracted to the positive charge of the proton, threatening to collapse rather than explode. Solving this puzzle played an important role in establishing quantum mechanics. Today we would say that the hydrogen atom is stable because it represents a minimum of the total energy. The negative electrostatic energy is balanced by the positive kinetic energy generated by squeezing the electron. However, such global reasoning does not reveal anything about the local stability of the electron's charge density at every point in space.

The total energy balance is often explained qualitatively in terms of forces by saying that the electrostatic attraction is balaced by the Pauli repulsion, i.e., the quantum-mechanical repulsion due to Pauli's exclusion principle. To express this statement mathematically, one can use the Lagrangian formalism which serves as the basis of particle theory. Technically, the force density is obtained from the divergence of Maxwell's stress tensor. Applying this formalism to the hydrogen atom, I found that the electrostatic attraction exactly balances the kinetic repulsion at every point inside the H atom (see the two links below).

At a first glance, such a local force density balance might seem incompatible with the uncertainty relation, which implies that one cannot localize an electron exactly without applying an infinitely large momentum. But that holds only for a single experiment. If the experiment is repeated many times with many momenta, one obtains a probability density for the location of the electron inside the hydrogen atom. This can be accomplished by repeated scattering experiments. Such experiments have been used to determine the charge distribution inside the proton (its "form factor").

Force Density Balance inside the Hydrogen Atom, arXiv:1112.6216 [physics.atom-ph]

Hyperfine Wave Functions and Force Densities for the Hydrogen Atom, arXiv:1702.05844 [physics.atom-ph]

### Stability of an Electron in the Dirac Sea

The H atom contains only a single electron, but there are situations where an electron is surrounded by many other electrons, all the way to an infinite number. The mobile electrons in a metallic solid are an example. They are frequently modeled by a gas of free electrons in a homogeneous positive background charge. This is the Fermi sea. Its analog in quantum electrodynamics is the Dirac sea, where the vacuum becomes populated by an infinitate number of "virtual" electrons and positrons.

A prototypical example is the vacuum polarization, a faint halo of opposite charges surrounding every charged particle. That creates a situation reminiscent of the H atom, where the electrostatic attraction needs to be balanced by the repulsion from squeezing. But the vacuum polarization charges are squeezed much harder, down to the reduced Compton wavelength. That is a factor of alpha smaller than the Bohr radius of the H atom (alpha ~ 1/137).

It turns out that the force balance remains intact for the vacuum polarisation surrounding an electron. One can show that each spherically-symmetric shell of vacuum electrons and positrons is balanced. The subsequent sum over the infinite number of such shells in the Dirac sea is tricky, but can be performed by adapting a method from calculations of the vacuum polarization itself. Since every charged particle is surrounded by the same electron/positron vacuum polarization at large distances, this force balance is common to all charged particles. The figure illustrates the summation over the charge densities of all electrons and positrons in the Dirac sea surrounding a positive point charge. These are spherical wave solutions of the Dirac equation in the Coulomb potential of the reference electron. They are characterized by two quantum numbers, the radial momentum (plotted along the x-axis) and the angular momentum (which increases in steps of 1 from black to magenta). The distance from the point charge is fixed at 1/10 of the reduced Compton wavelength in this plot. Notice the many near-cancellations which complicate the calculation.

The Stability of the Vacuum Polarization Surrounding a Charged Particle, arXiv:1512.08257 [quant-ph]

### The Exchange Force

The problem with Coulomb explosion becomes obsolete after applying the exclusion principle to the self-interaction. In that case the Coulomb interaction involves the same electron twice, once as the source of its electric field and a second time as the particle that interacts with the field. That would violate the exclusion principle.

This argument does not apply to the H atom, where the Coulomb interaction occurs between a single electron and the proton. But it applies to any situation involving more than one electron, such as the interaction of an electron with the Dirac sea. In addition to the Coulomb interaction discussed in the previous section one has to include the exchange interaction.

These arguments can be quantified by the Hartree-Fock method which includes separate terms for the Coulomb and the exchange interaction. The self-Coulomb term is canceled automatically by the self-exchange term.

To illustrate this concept it is useful to return to the Fermi sea, the analog of the Dirac sea in metals. The exclusion principle generates a depletion of electrons in the Fermi sea near a specified reference electron. This so-called exchange hole exactly compensates the negative charge of the reference electron and thereby strongly reduces its Coulomb interaction with the remaining Fermi sea.

This well-studied phenomenon suggests the existence of an exchange hole not only in the Fermi sea, but also in the Dirac sea. Such an object can be defined, as shown in the figure and the reference below. The exchange hole in the Dirac sea extends over the reduced Compton wavelength, which is 100-1000 times smaller than typical exchange holes in the Fermi sea.

The size difference can be traced to different cutoffs in a sum over all pairs. The Fermi sea contains electrons with wavelengths ranging from infinity down to the Fermi wavelength (which depends on the electron density). The Dirac sea, on the other hand, contains electrons and positrons with all possible wavelengths (which leads to infinite densities of electrons and positrons). The resulting infinities cancel out between electrons and positrons and leave a finite result for the size of the exchange hole. It is comparable to the reduced Compton wavelength, the only available length scale.

This scheme can be extended to multiple exchange interactions using the Hartree-Fock method. Including double-exchange leads to an exchange hole surrounded by an exchange electron (the "exchange hole of the exchange hole"). That solves a problem with charge conservation, because the reference electron is now surrounded by a neutral electron-hole pair, the exchange exciton. Above: Radial electron densities of the exchange hole, the exchange electron, and their sum (also shown in 3D as inset). The switch from hole to electron character occurs near the reduced Compton wavelength.

After figuring out the exchange interaction in the Dirac sea, one can return to the original goal of balancing the forces acting on the reference electron, the exchange hole, and the exchange electron. This object resembles the negative ion of positronium, which has been investigated both theoretically and experimentally.

The Exchange Hole in the Dirac Sea, arXiv:1701.08080 [quant-ph]

## A Higgs Boson Composed of Gauge Bosons

The Higgs boson, the Higgs potential, and the Brout-Englert-Higgs mechanism of symmetry breaking are central to the standard model of particle physics (therefore a well-deserved Physics Nobel Prize in 2013). This mechanism explains how a broken symmetry of the universe is able to generate mass for fundamental particles. (Composite particles, such as the proton, can acquire mass by a different mechanism where the kinetic energy of their constituents is equivalent to mass according to Einstein.)

However, the standard model uses an ad-hoc potential for its Higgs boson that blemishes the elegance of the theory. It contains a quadratic term that mimics a mass term, but has the wrong sign (corresponding to an imaginary mass). A fourth-order term is added, even though it is absent for all other fundamental particles. The standard Higgs potential contains two adjustable parameters, one of them being the mass of the Higgs boson. This construct looks rather artificial. Its lack of elegance has been a concern to many theorists. As a result, models have been proposed where the Higgs particle is not fundamental, but composed of pairs of fundamental fermions (similar to the electron pairs in superconductors). But such models are outperformed by the standard Higgs potential, which matches the data with the help of adjustable parameters.

My contribution to the extensive search for a suitable Higgs model is the idea of constructing a Higgs from fundamental bosons instead of fermions. These are the gauge bosons that are characteristic of each symmetry. The theory of symmetry groups assigns gauge bosons to a unique representation of each group, i.e., the adjoint. The construction of this Higgs boson is also uniquely determined by symmetry. As a result, one obtains a result for the Higgs mass in terms of the gauge boson masses. The standard model, as well as extensions with additional Higgs bosons, are not able to explain the Higgs mass. That's why it took so long to find the Higgs boson.

The gauge bosons mediate interactions and thereby become an essential part of particle theory. In the standard model, the photon mediates the electromagentic interaction, the W and Z mediate the weak interaction, and the gluons the strong interaction. Each interaction corresponds to a characteristic symmetry. If the symmetry is preserved, the corresponding gauge bosons are massless (such the photon). If the symmetry is broken, the gauge bosons acquire mass (such as the W and Z). Since the Higgs boson breaks the symmetry of the weak interaction and thereby creates masses for the W and Z, one might expect that the Higgs mass and the W,Z masses are related.

According to the Brout-Englert-Higgs mechanism, the Higgs boson needs to acquire a finite "vacuum expectation value" (VEV), in order to break the weak symmetry. Viewing the Higgs particle as a wave, its VEV corresponds to a constant amplitude around which it oscillates -- rather than symmetrically around zero. If the Higgs boson is composed of gauge bosons, those must have finite VEVs too. At a first glance this concept raises concerns, because the gauge bosons are vector particles. If they had a fixed vector as VEV, that would define a preferred direction in space (like a bar magnet pointing in a specific direction). Consequently the rotational symmetry of empty space would be broken. That may be the reason why no one seems to have considered this concept before.

But there is a loophole: one can choose an individual direction for the expectation value for each of the virtual gauge bosons that populate empty space in particle theory. This direction can be either parallel or perpendicular to the momentum of a particular gauge boson. (Perpendicular is preferred because of gauge symmetry.) Isotropy of the vacuum requires that the momenta of all the virtual gauge bosons in vacuum must average out to zero. Likewise, the directions of these gauge bosons must average out to zero. On the other hand, the scalar product of a gauge boson with itself does not average out to zero. This corresponds to a gauge boson pair. Homogeneity of the vacuum then requires all pairs to exhibit the same scalar product.

Such considerations suggest to define a pair of composite Higgs bosons by a linear combination of W and Z pairs. The relation becomes unique when demanding both Lorentz and gauge invariance. The former is satisfied by scalar products of gauge boson pairs.The latter is preserved by having both the gauge boson pairs and the Higgs pair in the SU(2) singlet configuration. The remaining proportionality constant is determined by setting the mass Lagrangians of the gauge and Higgs bosons equal to each other. Since mass is an observable, the mass Lagrangians must be gauge invariant.

The resulting relation between gauge and Higgs bosons immediately produces a result for the mass of the Higgs boson. It becomes simply half of its VEV. The latter can be obtained experimentally from the measured value of Fermi's coupling constant GF. The result matches the observed Higgs mass within 2% , the accuracy expected within the "tree-level" approximation used in this work. This result goes beyond the standard model.

One might ask how the W and Z gauge bosons develop finite VEVs without introducing an artificial Higgs-like potential. It turns out that they interact with each other and even with themselves, a property following from the "non-abelian" character of the SU(2) symmetry that characterizes the weak interaction. These "self-interactions" lead to quadratic and quartic potentials which can be calculated from a collection of Feynman diagrams (given in the references below). They are fairly complex, but remain at the single-loop level in leading order. Today's computer codes can handle them with ease.

If such calculations of the W,Z self-interactions yield a potential that breaks the SU(2) symmetry of the weak interaction, the concept of a Higgs boson composed of the W,Z would lead to a new standard model with fewer adjustable parameters. A critical experimental test will require energies near the theshold for producing Higgs pairs.

To get started without calculations, one can construct an empirical Higgs potential in terms of gauge boson pairs. Similar to the standard model, this involves adjustable parameters which are matched to observables (for example the masses of the gauge bosons and Fermi's coupling constant). The first publication below shows that this is indeed possible. The resulting potential can then be plotted as a function of the scalar products WW and ZZ, as shown in the figure below. There are two topologically distinct cases, with the minimum occuring either along a line or at a point.

The second publication below reveals an interesting connection to Yang-Mills theories, a general class of gauge theories. They play a dominant role in the standard model and its extensions. The simplest example is the SU(2) model of the weak interaction. Among the fundamental questions are the mechanism of symmetry breaking, the resulting generation of mass, and the very existence of Yang-Mills theories (by rigorous mathematical standards). One of the seven Millenium Prizes of the Clay Mathematics Institute is dedicated to these questions. Above: The two possible topologies for the potential of the composite Higgs boson, plotted in terms of the W and Z gauge boson potentials (in units of GeV). Such potentials replace the traditional Higgs potential. They can be calculated from the gauge boson self-interactions.

A Higgs Boson Composed of Gauge Bosons, arXiv:1502.06438 [hep-ph]

Dynamical Symmetry Breaking by SU(2) Gauge Bosons, arXiv:1801.04604 [hep-ph]

## Anharmonic Waves in (Quantum) Field Theory

This project started with the notion that any realistic theory of the four fundamental interactions has inherently nonlinear character. When fields/particles interact with each other, they violate the superposition principle by generating scattered waves/particles. Most notably, Einstein's equations for gravity in curved space-time are highly nonlinear. Nevertheless, classical and quantum fields are typically expanded into non-interacting sinusoidal waves when describing scattering. Would it be possible to develop an expansion into a more general class of interacting waves. That might be helpful for unifying gravity with the other three interactions. What are the criteria for such waves to satisfy basic priciples of physics?

In laser physics and acoustics, sinusoidal waves are characteristic of linear media. Nonlinear media generate harmonics. These are described by a Fourier series of sinusoidal waves which contain integer multiples of a fundamental frequency. It turns out that three simple constraints make anharmonic waves compatible with relativistic field theory and quantum physics. Some familiar concepts have to be abandoned, such as the superposition principle (of course), orthogonality, and the existence of single-particle eigenstates. The figure shows three generic symmetries of anharmonic waves that satisfy the ground rules of relativity and quantum physics. The parameter q represents the strength of the anharmonicity, increasing from zero (red) to the maximum (black). The two sets of curves in each panel are the anharmonic generalizations of sine and cosine. In all cases the familiar relation ( sin2x + cos2x ) = 1 remains valid. This is achieved by a periodic phase modulation of sine and cosine. For the waves with lowest symmetry (top row) either the sine or cosine develops a non-zero average. That corresponds to a non-zero vacuum expectation value in quantum field theory (see the section about the Higgs boson above).

Anharmonic Waves in Field Theory, arXiv:1108.1736 [hep-th]

Quantum Electrodynamics with Anharmonic Waves, arXiv:1112.6216 [hep-th]