When we first met the Higgs field, we didn’t say much about it, but it’s very important and there’s a lot we need to know about it. So let’s talk Higgs.
As we learned here, the fermionic points in our spacetime have some probability of remaining stationary from one time tick to the next, and some probability of moving to an adjacent point cell. A field, in physics, is simply something that takes a value at every point of spacetime. The Higgs field takes a value of the Planck energy, 1019 GeV, at a stationary point and a value of zero at a moving point. The average value, or vacuum expectation value, of the Higgs field is known to be 246 GeV, and from this we learn that there are about 1017 moving points for each stationary point.
The Higgs field has been in the news a lot lately. For a long time it was the only part of the standard model of particle physics that hadn’t been confirmed experimentally. That changed in 2012 when the Higgs boson was observed at the Large Hadron Collider at CERN in Geneva. In the above description of the Higgs field there’s nothing about a Higgs boson, but in quantum field theory any field can fluctuate, and a fluctuation of a field behaves like a particle. When the fluctuation dies out the “particle” decays. At the LHC the decay products were found to be exactly what was expected from a Higgs boson.
The Higgs field is credited with being the source of the masses of the elementary subatomic particles like the electron and quarks. Ask a physicist what mass is and how the Higgs field makes electrons massive and you’ll get a lot of hand-waving. Physicists don’t know what the Higgs field is, and when it comes to the electron and similar particles, they can’t even tell you what mass is. In composite particles like the proton, most of the mass comes from the energy required to bind the quarks together (m = E/c2). But for the most elementary particles, they see the mass terms involving the Higgs field in their equations, but they can’t tell you what the Higgs or mass might be. We know what the Higgs field is, but what’s its connection with mass?
What is mass? While the mass of a composite particle or a complex object is influenced by many factors, such as binding energy, defining mass for a basic particle like the electron, which is a resonance of a single fermionic point, is actually quite simple. Mass is resistance to acceleration—the larger the mass, the greater the force required for a given acceleration. Remember f = ma? But factors other than mass can impede acceleration—for example, the object may be constrained in some way. The equivalence principle of general relativity says that we can’t tell the difference between gravity and acceleration. A similar equivalence principle is at work here. If an object resists acceleration, we can’t tell if that’s because it’s massive or because its position is constrained. In fact, a particle has mass if and only if its position can be known. A massless particle like the photon moves at the speed of light and has an infinite position uncertainty.
In our spacetime model, the mass of a particle that is a resonance of a single stationary fermionic point, that is, an electron, is the same as the degree to which its position can be known—it has mass precisely because its position uncertainty is finite. Its mass is inversely proportional to the precision with which its position is constrained, that is, the more precisely we can locate it, the greater its mass.
Our fermionic points are pulled together by gravity and held apart by degeneracy pressure that comes from the Pauli exclusion principle—no two identical fermions can have the same position. Thus, each point is confined to a tiny point cell whose walls are other points and whose radius is most likely the Planck length, 10-33 centimeter. If it were a particle in a spherical box it would have the Planck energy, 1019 GeV. The Higgs field takes this value for a stationary point.
This is the ground state energy of the point. To make an electron, the point has to be excited. The first excited state of a particle in a spherical box has an energy that is twice the ground state energy. If we consider the particle to be the energy above the ground state, then the electron has an energy (or mass, since E =mc2) that is equal to the ground state energy, which is the Planck energy.
Now, wait a minute! The electron isn’t anywhere near that massive. Well, no. This is called the bare mass of the electron, or the mass you would see if you knew exactly where it was. If you look up the mass of the electron, what you find is called the dressed mass. This is the mass determined by the Higgs field. The Higgs says that there is one stationary point for every 1017 moving points, so the best you can say about the electron’s position is that it is somewhere in a volume of space equal to 1017 point cell volumes. The electron mass calculated on the basis of a point cell of this size agrees with the experimental value, 0.511 MeV. You can find the calculation in my physics paper. So this is how the Higgs field “gives mass” to the elementary particles.
If the Higgs vacuum expectation value, or vev, were equal to the Planck energy, only electrons would exist, their mass would be the Planck mass, and nothing could move. This is a state of very high potential energy, so the field evolves towards a state of lower potential energy, vev = 0. If vev = 0, only massless neutrinos would exist and nothing could have a defined position. It would be possible for fermionic points to be in the same quantum state, which is impossible. Therefore, the Higgs potential has an infinite barrier at vev = 0 and the vev settles at 246 GeV. Thus, fermions have mass precisely because the Higgs vev is greater than zero.
The hierarchy problem asks why there is such a large “desert” between the Planck scale (1019 GeV) or the grand unification scale (1016 GeV) and the electroweak scale (246 GeV). The reason is that the Higgs field is headed for zero potential, and it’s only because it can’t get there that the electroweak scale exists at all. On the other side of the same coin, the higher the Higgs vev, the more rigid the universe, so life might not be possible without such a desert.
Here’s another little tidbit the physicists don’t know. The Higgs field has a linear potential energy between zero and the Planck energy, not the wavy one usually assumed. Since they didn’t know the exact shpe of the potential, they’ve never been able to calculate the Higgs boson’s mass.The wave function associated with a linear, or triangular, potential is known, so it’s straightforward to calculate the mass of the Higgs boson when you know the vev as well. When I did this I got a mass of 120 GeV for the Higgs boson. That’s lower than the LHC value of 125 GeV because of a simplifying assumption I made, but it’s close enough to prove the concept.