Vectors, tensors, and spinors on spacetime

Of the symmetries of spacetime, several of them are arguably not so reasonable when considering locality.

The first is time reversal. That takes time t to −t, and preserves the metric and some of the laws of mechanics. It takes the forward light cone to the backward light cone, and vice versa.

But locality is based on causality, and the past causes the future, not the other way around. There is a logical arrow of time from the past to the future. Most scientific phenomena are irreversible.

The second dubious symmetry is parity reversal. That is a spatial reflection that reverses right-handedness and left-handedness.

DNA is a right-handed helix, and there is no known life using left-handed DNA. Certain weak interactions (involved in radioactive decay) have a handedness preference. The Standard Model explains this by saying all neutrinos are massless with a left-handed helicity. That is, they spin left compared to the velocity direction. (Neutrinos are now thought to have mass.)

Charge conjugation is another dubious symmetry. Most of the laws of electricity will be the same if all positive charges are changed to negative, and all negative to positive.

Curiously, if you combine all three of the above symmetries, you get the CPT symmetry, and it is believed to be a true symmetry of nature. Under some very general assumptions, a quantum field theory with a local Lorentz symmetry must also have a CPT symmetry.

The final dubious symmetry is the rotation by 360°. It is a surprising mathematical fact that a rotation thru 720° is homotopic to the identity, but a rotation thru 360° is not. You can see this yourself by twisting your belt.

Thus the safest symmetry group to use for spacetime is not the Lorentz group, but the double cover of the connected component of the Lorentz group. That is, the spatial reflections and time reversals are excluded, and a rotation by x° is distinguished from a rotation by x+360°. This is called the spin group.

Geometric theories on spacetime are formulated in terms of vectors and tensors. Vectors and tensors are covariant, so that they automatically transform under a change of coordinates. Alternatively, you can say that spacetime is a geometric coordinate-free manifold with a Lorentz group symmetry, and vectors and tensors are the well-defined on that manifold.

If we consider spacetime with a spin group symmetry instead of the Lorentz group, then we get a new class or vector-like functions called spinors. Physicists sometimes think of a spinor as a square root of a vector, as you can multiply two spinors and get a vector.

The basic ingredients for a theory satisfying the geometry and locality axioms are thus: a spacetime manifold, a spin group structure on the tangent bundle, a separate fiber bundle, and connections on those bundles. The fields and other physical variables will be spinors and sections.

This is the essence of the Standard Model. Quarks, electrons, and neutrinos are all spinor fields. They have spin 1/2, which means they have 720° rotational symmetry, and not a 360° rotation symmetry. Such particles are also called fermions. The neutrinos are left-handed, meaning they have no spatial reflection symmetry. The Lagrangian defining the theory is covariant under the spin group, and hence well-defined on spacetime.

Under quantum mechanics, all particles with the same quantum numbers are identical, and a permutation of those identical particles is a symmetry of the system. Swapping two identical fermions introduces a factor of −1, just like rotating by 360°. That is what separates identical fermions, and keeps them from occupying the same state. This is called the Pauli exclusion principle, and it is the fundamental reason why fermions can be the building blocks of matter, and form stable objects.

Pauli exclusion for fermions is a mathematical consequence of having a Lorentz invariant quantum field theory.

All particles are either fermions or bosons. The photon is a boson, and has spin 1. A laser can have millions of photons in the same state, and they cannot be used to build a material substance.

The point here is that under general axioms of geometry and locality, one can plausibly deduce that spinor and gauge theories are the obvious candidates for fundamental physical theories. The Standard Model then seems pretty reasonable, and is actually rather simple and elegant compared to the alternatives.

In my counterfactual anti-positivist history of physics, it seems possible that someone could have derived models similar to the Standard Model from purely abstract principles, and some modern mathematics, but with no experiments.