Spinor mathematics and understanding

Now for something completely different, the Standard Model all over again. Readers of my internet page and this blog will know that electroweak unification is mathematically OK, but quantum chromodynamics and quantum gravity/M-theory (11 dimensions) is horses***. I have sympathy with quantum mechanics and general relativity, at least the physical aspects of them (not the lengthy mathematical horses*** like some interpretative Copenhagen horses*** about ‘parallel universes’; I have sympathy with just the predictions which can be tested, i.e., I’m interested in science).

Spin in quantum field theory is described by ‘spinors’, which are more sophisticated than vectors. The story of spin is that Wolfgang Pauli, inventor of the phrase ‘not even wrong’, in 1924 suggested that an electron has a ‘two-valued quantum degree of freedom’, which in addition to three other quantum numbers enabled him to formulate the ‘Pauli exclusion principle’. (I use this on my home page to calculate how many electrons are in each electron shell, which produces the basic periodic table.)

Because the idea is experimentally found to sort out chemistry, Pauli was happy. In 1925, Ralph Kronig suggested that the reason for the two degrees of freedom: the electron spins and can be orientated with either North Pole up or South Pole up. Pauli initially objected because the amount of spin would give the old spherical model of the electron (which is entirely false) an equatorial speed of 137 times the speed of light! However, a few months later two Dutch physicists, George Uhlenbeck and Samuel Goudsmith, independently published the idea of electron spin, although they got the answer wrong by a factor (the g-factor) of 2.00232 (this is just double the 1.00116 factor for the magnetic moment of the electron). The first attempt to explain away this factor of 2 was by Llewellyn Thomas and was of the abstract variety (put equations together and choose what you need from the resulting brew). It is called the ‘Thomas precession’. Spin-caused magnetism had already been observed as the anomalous Zeeman effect (spectral line splitting when the atoms emitting the light are subjected to an intense magnetic field). Later the Stern-Gerlach experiment provided further evidence. It is now known that the ordinary magnetism of iron bar magnets and magnetite is derived from electron spin magnetism. Normally this cancels out, but in iron and other magnetic metals it does not completely out in each atom, and this fact allows magnets. Anyway, in 1927 Pauli accepted spin, and introduced the ‘spinor’ wave function. In 1928, Dirac introduced special relativity to Pauli’s spinor, resulting in ‘quantum electrodynamics’ that correctly predicted antimatter, first observed in 1932.

The Special Orthogonal group in 3 dimensions, or SO(3), allows spinors. It is traced back to Sophus Lie who in 1870 introduced special manifolds to study the symmetries of differential equations. The Standard Model, symmetry unitary groups SU(3)xSU(2)xU(1) is a development and application of spinor mathematics to physics. SU(2) is not actually the weak nuclear force despite having 3 gauge bosons. Contrary to what I said a couple of posts back, the weak force arises from the mixture SU(2)xU(1), which is of course the electroweak theory. Although U(1) described aspects of electromagnetism and SU(2) aspects of the weak force, the two are unified and should be treated as a single mix, SU(2)xU(1). Hence there are 4 electroweak gauge bosons, not 1 or 3. One whole point of the Higgs field mechanism is that it is vital to shield (attenuate) some of those gauge bosons, so that they have a short range (the weak force), unlike electromagnetism.

On the other hand, for interactions of very high energy, say 100-GeV, the weak force influence SU(2) vanishes and SU(3)xU(1) takes over, so the strong nuclear force and electromagnetism then dominate. Now the question is how far I go into the mathematics of the Standard Model? Should I study spinors and symmetry unitary groups? How far has that got other people? More likely, you should compromise and try working a bit from both ends of the same problem. Look into the abstract maths by all means, but also look out for any clues that can be deduced from reality, hard experimental fact.