The tension between quantum mechanics and relativity, and what it signifies for the standard model of particle physics?

The principle of quantum linear superposition is well-tested for elementary particles such as electrons. The theory of special relativity is also well tested. However, the two are not consistent with each other, except in an approximate sense.

We realise this when we ask the question: what kind of spacetime geometry and gravitation is produced by an electron? It is tempting to think of the produced gravitation as a very tiny (quantum) perturbation of the flat spacetime of special relativity. However this cannot be correct, precisely because of the validity of the principle of quantum linear superposition.

Imagine for simplicity that the electron state is a superposition of two localised position states A and B. Had the electron been at A, it will produce a perturbation of flat space time, with the perturbation peaked at A. Similarly if it had been at B, the produced perturbation would have been peaked at B. If we now consider a point X in space, the field there will be a superposition of the fields due to the location A and the location B. Such a field is not a perturbation of flat spacetime ! If we want to write down the Schrodinger equation for a test particle, then what time parameter to use at location X? That determined by the clock rate fixed by location A or by location B? There is ambiguity. The concept of a classical spacetime is lost [even in a perturbative sense] the moment we consider the implications of quantum superposition for spacetime geometry. In order to describe quantum spacetime - the one produced by an electron - we must entirely give up on the flat spacetime of relativity, even at low energies.

But quantum field theory on a flat spacetime works extremely well. We are able to construct the highly successful standard model of particle physics, assuming the flat spacetime of relativity, and assuming Lorentz invariance. Does that not imply that flat spacetime is an excellent approximation in the limit of low gravity? No, it does not imply that. There are 26 free parameters in the standard model, which have to be put in by hand, after measuring them experimentally. Could it be that these parameters get determined uniquely, and are not arbitrary, if we describe the standard model, not on flat spacetime, but on that spacetime which elementary quantum particles produce. Gravity is tiny, but it is not a perturbation of flat spacetime. What then is it a perturbation of?

The gravity is a perturbation of a [non-commutative] spinor spacetime, from which the flat spacetime of relativity is derived as an approximation. When we describe the standard model on this spinor spacetime, we find evidence that the parameters of the standard model are not free, but take values as measured in experiments. Such a noncommutative spinor spacetime is compatible with the quantum superposition principle. We could say that when we take the square root of the Klein-Gordon equation to arrive at the Dirac equation, we should also take the square root of flat spacetime so as to arrive at the noncommutative spinor spacetime. Then we write down the Dirac equation on this spinor spacetime. Right away we find that electric charge is quantised [0, 1/3, 2/3, 1 : neutrino, down quark, up quark, electron]. And that the low energy fine structure constant is 1/137. It cannot be any other value.