The fact that the spin group cannot be equal to the double cover of the rotation group, but can only be canonically isomorphic to it, implies that the spinors that are used in quantum mechanics have a different relationship to spacetime from the one that has been assumed for the past 90+ years. We have a chain of subgroups, SU(2) contained in SL(2,C) contained in SL(4,R). The spinors are 2-dimensional complex representations of SU(2) and SL(2,C), which become 4-dimensional real when we forget the complex structure. Now SL(4,R) has two 4-dimensional real representations, one on spacetime itself, and one on the dual of spacetime. Position in spacetime is denoted by a vector. The dual describes momentum-energy, and is denoted by a co-vector. This duality goes back to Hamilton’s 19th century formulation of Newton’s 17th century mechanics.

The spinors for SU(2) – the spin 1/2 representation – are self-dual. Even the spinors for SL(2,C) – the spin (1/2,0) and (0,1/2) representations – are self-dual. This is important, and must be stressed: the dual is obtained by taking the transpose-inverses of the matrices, and this is NOT the same as complex conjugation, that swaps the spin (1/2,0) and (0,1/2) representations. Therefore the spinors are not vectors, and they are not co-vectors, but instead are some strange mixture of the two. Hence the dire warnings in textbooks on quantum mechanics, that spinors are not vectors, however much they may look like vectors.

To make the left- and right-handed Weyl spinors out of vectors and co-vectors, therefore, we must first put a complex structure onto spacetime, so that it becomes a 2-dimensional complex space. Then the dual becomes another copy of the same representation (NOT the complex conjugate), so we can mix the two together in arbitrary complex combinations. At this point we can split the representation apart again, and then we have to replace one half by its complex conjugate in order to get the left/right formalism to agree with the standard formalism.

All this can be done easily, of course, so that the mathematical formalism of quantum mechanics still works. But the process depends on the observer’s choice of complex structures, and in particular how a particular direction in space matches up with a particular direction of momentum. In other words, it depends on the motion of the laboratory in which the experiment is done.

It follows, therefore, that quantum mechanical experiments can detect all the important features of the motion of the laboratory. By setting up suitable experiments, therefore, one can measure the tilt of the Earth’s axis, count the number of days in a year, or in a month, measure the inclination of the Moon’s orbit, and its eccentricity, and so on. This is not a conjecture, or wild speculation, or arrant nonsense, as most physicists seem to believe. It is not even a prediction. It is a mathematical theorem, and therefore it is *true.*