Vectors and spinors

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.

How much is 1/3?

Amazon has three reviews of my book “The finite simple groups”, one each with 3, 4 and 5 stars. They helpfully translate this into percentages for me, and tell me I have 34% at 5 star, 35% at 4 star, and 31% at 3 star. Now I wonder how they worked that out?

Maximal subgroups of classical groups

The long-awaited book `The maximal subgroups of the low-dimensional finite classical groups’ by John Bray, Derek Holt, and Colva Roney-Dougal has just appeared in the LMS Lecture Notes Series. It contains, in marvellously complete form, with proofs, the material which appeared in Peter Kleidman’s PhD thesis (1987), without proof, on maximal subgroups of simple classical groups in dimensions up to 12, and extends this to the almost simple groups. The latter is what really makes the achievement so impressive. The book satisfies a long-felt need in finite group theory, and will be an important work of reference for a long time to come.

solving yesterday’s problems tomorrow

The University of Birmingham has a very clever slogan, which it advertises widely, that is “Solving tomorrow’s problems today”.  As a mathematician, I immediately thought of the possibilities of replacing these temporal references by arbitrary combinations of “yesterday”, “today” and “tomorrow”. As a slogan for our Government, how about “Solving yesterday’s problems tomorrow”? Or, as a slogan for mathematics, how about “Solving tomorrow’s problems yesterday”?

How to write LaTeX in this blog

Write a expression in between signs in the usual way, but write the word latex immediately after the opening sign. Just for the fun of it, here is an example: This works equally well in posts or in comments.