Sunday, March 8, 2015

Spin, Statistics, and Forces

While the physicists are thinking about the universe and coming up with nothing, they’re not giving me anything to blog about, so I’m going to make good on a promise I made in this post. I’m going to show how the law of spin and statistics is responsible for three of the four basic forces—gravity, electromagnetism, and the weak force.

The law of spin and statistics says that particles with half-integer spin are fermions and no two of them can ever be in the same quantum state (they obey Fermi-Dirc statistics), while particles with integer spin are bosons and seek the same state (they obey Bose-Einstein statistics). As I explained in the same post, in our model, points are mixed states, fermions on some time ticks and bosons on others. We can think of spacetime as two fields of points, one fermionic and one bosonic, but we have to account for the fact that the two fields are coupled because of the mixing. As the bosonic points seek the same state, they drag the fermionic points with them.

The quantum state of a point is described by three quantum numbers—position, spin, and time. As the bosonic points drag the fermionic points towards the same position, we see the “force” of gravity. That one is easy.

Local time ticks at a point do not have to be in phase with time at other points or with global time. Phase differences change the relativistic interval s, where ds2= dx2+ dy2+ dz2- c2dt2. Bosonic attraction tries to pull all points into time phase. The result is a force that tends to minimize ds/s, the relative effect of phase differences. The force is seen as an exchange of photons. To minimize ds/s, the force tries to maximize s. For opposite-time points the interval is timelike and the force is attractive. For same-time points the interval is spacelike and the force is repulsive. The force is the electromagnetic force, attractive for opposite charges and repulsive for like charges.

The weak interaction or weak force is a bit more subtle. It’s related to spin. Spacetime is four-dimensional and so is spin. Normal spin is in the x, y, or z direction, but points can also have spin in the t direction, and the local t axis at a point doesn’t have to line up with the global or observer’s t axis. Spin involving the t direction is called isotopic spin or isospin. The orientation of the local t axis with respect to the global t axis has two eigenstates. Either the local t axis lines up with the global t axis (t(x) = t) or the two time axes are orthogonal (t(x) = 0). If t(x) = t the point is a stationary point and if t(x) = 0 the point is a speed-of-light point. As I explained here, the Higgs field is defined by the velocity distribution of the points. The lowest energy state is for a spacetime of all speed-of-light points, so the bosonic attraction tries to drag all of the fermionic points into this state (the bosonic points are already there). We can’t have all of the fermionic points in the same state, so the Higgs field ends up with a nonzero average value, which is maintained by changing stationary points to speed-of-light points and vice versa as necessary. These changes are seen as weak interactions. For example, an electron may change to a neutrino and a W boson.  The W and Z bosons are virtual particles that mediate the weak force.

So QED. Gravity, the electromagnetic force and the weak force aren’t forces at all, but statistical tendencies imparted to fermionic points and particles by coupling to the bosonic spacetime field. In the standard model, these forces are seen as exchanges of particles—gravitons, phoyons, W and Z bosons. It works.