When I was a mathematician in Cambridge, we used to use the phrase “contradiction in the universe” as a joke, to describe the all-too-common situation in which we had proved two contradictory things simultaneously. Of course, it was never really a contradiction in the universe, it was always an error that was buried deeply and difficult to find.
Modern physics has lived with a “contradiction in the universe” for around a century now. Quantum mechanics is inconsistent with relativity, but both are extremely useful theories. So physicists prefer to live with a contradiction in the universe, rather than admit that there is an error in their theory. But the fact remains that the universe exists, and does not have a contradiction. Whatever physicists say, their theories are in error. The error is buried deeply and difficult to find. But it must be found, and corrected.
Well, there may be more than one error, of course, and the theories are big and complicated, so the crucial error may not be easy to find. The error I found nearly ten years ago is still the error that I regard as the fundamental error that leads directly to the contradiction. I first talked about it in a lecture to the Algebra Seminar in the University of Manchester in 2015. Yet the papers I wrote about it were completely rejected by all physicists.
The error lies in the interpretation of Dirac’s 1928 paper on the electron. Dirac used the group SL(2,C) of 2×2 complex matrices to describe the symmetries of the electron. Physicists call this group the “Lorentz group”. Unfortunately, it is not the Lorentz group. It looks like the Lorentz group, and it quacks like the Lorentz group. But it doesn’t swim like the Lorentz group. So it isn’t the Lorentz group.
Of course, finding the error is not at all the same thing as correcting the error. I’ve tried many things over the years, but none of them has so far been convincing as a “correction”. I have high hopes for my latest attempt, however, which was posted on the arxiv this morning: https://arxiv.org/abs/2404.18938. This is the same one I’ve been writing about here recently, and is based on the idea of doing away with the idea of spacetime altogether.
You see, you cannot measure space, you can only measure things. And you cannot measure time, you can only measure events. Space and time are not “things”, but abstract mathematical constructs designed to try and make sense of the things and events around us. By 1928, these mathematical constructs were no longer fit for purpose, and should have been abandoned. Yet they are still in use today.
A properly relativistic theory must not use the concept of spacetime, but must instead use the concept of phase space, that is the concepts of position and momentum. Position is not the same as space, because position is only relative, not absolute. Only differences in position are relevant. Already in the mid 19th century Hamilton explained how to do all of Newtonian physics using phase space instead of spacetime.
Now we have to interpret the group SL(2,C) in terms of phase space, instead of spacetime. It describes spin, of course – that’s what Dirac used it for. And it describes spin about a particular axis – position and momentum about that axis are vectors in the plane perpendicular to the axis. For circular orbits, the position and momentum vectors are perpendicular to each other, but for general Keplerian elliptical orbits they are not necessarily perpendicular. But it is the axis that is important here: the group SL(2,C) only makes sense relative to an axis in position-space. The Lorentz group SO(3,1) has no preferred axis. But SL(2,C) does have a preferred axis. That is why it is a serious error to consider these two groups to be the same.
It also means that the group SL(2,C) only describes 2-dimensional dynamics, not 3-dimensional. To get 3-dimensional dynamics, we need to extend the group to SL(3,C). Inside SL(3,C) is a group SO(3,C). This group SO(3,C) is the Lorentz group. And this time it actually is. Both SL(2,C) and SO(3,C) are 6-dimensional groups inside SL(3,C), and as abstract groups they look very similar – the only difference is that SL(2,C) has a + or – sign attached to its elements, that SO(3,C) does not have. But inside SL(3,C) they are completely different.
The “contradiction in the universe” arises because these two groups, SL(2,C) and SO(3,C) are assumed to be the same, and interpreted to be the same, when they are not. They look the same, mathematically, but physically they are completely different.