Fundamental Theory

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

A longtime favorite has been the question about the internal stability of the electron. If one models an electron as tiny charge cloud, it would explode due to electrostatic repulsion ("Coulomb explosion"). This problem has puzzled 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 stabilize the electron might also shed light onto the strength of the electromagnetic force, which 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 with non-electromagnetic forces would determine alpha and thereby solve both of these century-old problems in one strike.

Overview Talk



Force Balance inside the Hydrogen Atom

Chasing the stability problem led me to many dead ends. Eventually a simple warm-up exercise was successful, where I analyzed the stability of hydrogen atom. The electron is attracted to the positive charge of the proton, threatening to collapse in this case rather than explode. Explaining this puzzle played an important role in establishing quantum mechanics. Today we would say that it is stable, because it represents a minimum of the total energy. The attractive electrostatic energy is balanced by the repulsive kinetic energy generated by squeezing the electron down to the Bohr radius. But such global reasoning does not reveal anything about the local stability of the electron's charge density at a specific point in space.

One can define the local force density acting on the charge cloud of the electron using the Lagrangian formalism (which also serves as the basis of particle theory). It turns out that the electrostatic attraction exactly balances the repulsion from squeezing the electron at every point inside the H atom. At a first glance, such a local force density balance might seem incompatible with the uncertainty relation. But that holds only for a single experiment. If the experiment is repeated many times, one obtains a probability density for the location of the electron -- similar to the force density calculated here. For example, one can determine the probability of finding the electron at a certain location inside the H atom by repeated scattering experiments. Such experiments have been used to determine the charge distribution inside the proton (the so-called 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 a Charged Particle

The H atom contains only a single electron, but there are many situations where one has to deal with the infinite number of "virtual" electrons and positrons that populate the vacuum of quantum electrodynamics (often labeled the Dirac sea). 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 is squeezed much harder, down to the reduced Compton wavelength which is a factor of alpha smaller than the Bohr radius (alpha ~ 1/137). Demonstrating the force balance for the infinite number of electrons and positrons in the Dirac sea requires special techniques from quantum electrodynamics to get rid of infinitely large expressions. Since every charged particle is surrounded by the same electron/positron vacuum polarization at large distances, this force is universal.


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The figure illustrates the summation of the charge density over all electrons and positrons in the Dirac sea surrounding a positive point charge. These are the spherical wave solutions of the Dirac equation in the Coulomb potential. 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.

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



The Exchange Force

The problem with Coulomb explosion is solved when going from classical physics to quantum physics. The exclusion principle forbids an electron from interacting with itself. That would imply that the same electron appears twice at the same location, once as the source of its electric field and a second time as the particle that interacts with that field. In Hartree-Fock theory, the resulting self-Coulomb interaction (which causes Coulomb explosion) is canceled by the self-exchange interaction.

Such reasoning introduces the exchange interaction as a major player for determining the stability of the electron. This interaction entangles an electron with the vacuum electrons that form the Dirac sea. Those are allowed to interact with the electron.

There is an analog to the Dirac sea in solid state physics: the Fermi sea of electrons in a metal. The exchange interaction with the Fermi sea can be explained (and mathematically quantified) by the concept of an exchange hole. It is again the exclusion principle that generates a depletion of electrons in the Fermi sea around a given electron. The positive charge of the exchange hole compensates the negative charge of the electron and thereby removes the problem with Coulomb explosion.

The analogy with the Fermi sea suggests the existence of an exchange hole in the Dirac sea. Indeed, one can define such an object, as shown in the reference below. This exchange hole extends over the reduced Compton wavelength, which is 100-1000 times smaller than a typical exchange hole in metals (determined by the Fermi wavelength).

In the Dirac sea the exchange hole is surrounded by an exchange electron (see the figure). It consists of the vacuum electrons displaced by the exchange hole. In the metallic Fermi sea these displaced electrons can be extracted to infinity without any energy barrier (for example by a long wire). But the Dirac sea is an insulator with a band gap of 1 MeV. The resulting barrier prevents the extraction of the displaced electrons.

Having figured out the exchange interaction in the Dirac sea, one can return to the original goal of establishing an internal force balance for the electron. Judging from the results for the hydrogen atom and the vacuum polarization, the major players will be the confinement repulsion versus the electrostatic forces between the the reference electron, its exchange hole, and the displaced electron. Magnetic forces become significant only at distances smaller than the size of the exchange hole.


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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.

The Exchange Hole in the Dirac Sea, arXiv:1701.08080 [quant-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. Those would not exist without nonlinearity. Nevertheless, quantum mechanics is carried out in Hilbert space, a linear vector space describing the various states of a particle. And the current description of the electromagnetic, weak, and strong interactions by quantum field theory operates in Fock space, a collection of linear vector spaces. Gravity, on the other hand, is described by general relativity -- an intrinsically nonlinear theory. That may be the fundamental reason why a theory of quantum gravity in four-dimensional space-time has remained elusive. Would it be possible to formulate our well-tested quantum field theories via a nonlinear version of Fock space that includes gravity?

To flesh out these somewhat vague ideas, I started looking into nonlinear plane waves, first for classical fields and then for quantum fields (see the two links below). Sinusoidal plane waves have served as basis for both classical and quantum fields using Fourier expansions. They are characteristic of linear media, but nonlinear media generate harmonics, a well-known phenomenon in laser physics and acoustics. Therefore my first attempts focused on nonlinear plane waves. Three simple constraints make anharmonic waves compatible with relativistic field theory and quantum physics. But some familiar concepts have to be abandoned, such as the superposition principle, orthogonality, and the existence of single-particle eigenstates.


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The figure shows three possible symmetries of anharmonic waves, with the strength q of the anharmonicity increasing from zero (red) to the maximum (black). The two sets of curves in each plot represent anharmonic generalizations of sine and cosine. In all cases the familiar relation sin2x+cos2x=1 remains valid. This is achieved by a periodic modulation of the phase of sine and cosine. For the lowest symmetry (top row) either the sine or cosine develops a non-zero average which corresponds to a non-zero vacuum expectation value in quantum field theory. This phenomenon enables the Higgs field to give masses to fundamental particles, including itself.

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

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



A Higgs Boson Composed of Gauge Bosons

The Higgs boson and the Brout-Englert-Higgs mechanism of symmetry breaking are central to the standard model of physics (therefore a well-deserved Physics Nobel Prize in 2013). However, the Higgs potential of the standard model is an ad-hoc construct that blemishes the elegance of the theory. It contains a quadratic term that mimics a mass term, but with an imaginary mass. A fourth-order term is added, even though no other fundamental particle has such a term. Combined with two adjustable parameters this construct looks rather clumsy. This lack of elegance has been a concern to many theorists. It led to models where the Higgs particle is not fundamental, but composed of fermion pairs (analogous to the electron pairs in superconductors). But such models are outperformed by the standard Higgs potential, which matches the data with the help of its adjustable parameters.

Any Higgs model needs to produce a finite "vacuum expectation value" (VEV) of the Higgs field. It can be viewed as a wave oscillating around a finite amplitude, rather than symmetrically around zero. The VEV then generates masses for fundamental particles. These are either fermions (quarks and leptons) or gauge bosons. The latter are an essential ingredient of particle theory, since they mediate interactions. The photon mediates the electromagentic interaction, the W and Z mediate the weak interaction, and the gluons the strong interaction. Each interaction is characterized by its "gauge" symmetry using group theory. If the symmetry is preserved, the gauge bosons are massless (such the photon). If it is broken, they acquire mass (such as the W and Z). Thus the Higgs boson is connected to a broken gauge symmetry. In the standard model this is the SU(2) symmetry of the weak interaction.

My contribution to the extensive search for a suitable Higgs model is the idea of constructing a composite Higgs boson from boson pairs, instead of fermion pairs. It employs the gauge bosons of a broken symmetry. They are generated automatically by group theory using the "adjoint representation" of the symmetry group. That is a rather general concept which applies to any theory involving gauge symmetry.

At a first glance, the concept of gauge bosons with a finite VEV raises concerns, since they 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 violated. That may be the reason why no one seems to have considered this idea before. But there is an escape: One can choose an individual direction for the expectation value of each gauge boson, referenced to the direction of its momentum (either parallel or perpendicular to it). The vacuum of quantum field theory contains an infinite number of virtual gauge bosons, each with a different momentum. To obtain the VEV, one has to sum over their individual expectation values. Since their momenta average out to zero, their expectation values do so, too. That eliminates any preferred direction in space. The VEVs of the gauge boson pairs forming the composite Higgs boson do not average out to zero. They form a scalar (like the Higgs boson) and thus are able to play the role of the Higgs boson.

Consequently, the connection between the standard Higgs boson and the gauge bosons is established by comparing pairs (which are both scalars). This concept produces a formula 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 for the "tree-level" approximation used in this work. The standard model, as well as other Higgs models are not able to explain the Higgs mass. That's why it took so long to find the Higgs boson. The new Higgs model also allows the Higgs potential to be calculated from the fundamental self-interactions of the gauge bosons. That provides additional tests, but such a program still needs to be implemented. It would require an expert in writing sophisticated computer programs for calculating quadratic and quartic self interactions of the W and Z bosons.

As it stands, this model can serve as testbed for replacing the artificial Higgs potential by fundamental gauge interactions. Before using it as alternative to the standard model, one will have to incorporate fermions into the model. To get started, one could use the Higgs-fermion interaction of the standard model, but there might be more elegant options that avoid introducing all the fermion masses as adjustable parameters. As long as the Higgs-fermion interaction remains an open question, one can try to construct an empirical Higgs potential from observables, such as the masses of the gauge bosons and Fermi's coupling constant. As shown in the first publication below, this is indeed possible. Since the Higgs boson consists of the W and Z gauge bosons, the Higgs potential can be plotted as a function of the the scalar products WW and ZZ, as shown in the figure. There are two topologically distinct cases, with the minimum occuring either along a line or at a point.

The second publication below points out an interesting connection to Yang-Mills theories, a general class of gauge theories which comprise the SU(2) model as simplest example. Yang-Mills theories play a dominant role in the standard model and its extensions. A fundamental question has been the mechanism of symmetry breaking and the resulting generation of mass, as well as 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.

This paper also derives the coupling g of the weak interaction by normalizing the relation between the composite Higgs boson and the SU(2) gauge bosons. The resulting value g=2/3 matches the experimental result with "tree-level" accuracy (like the value of the Higgs mass derived from the same relation).

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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]



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