Mathematics: solving tomorrow's problems yesterday
It is well known that Dirac predicted antiparticles. It is also well known that the properties he predicted for them turned out not to be correct. It is well known that his theory of the electron does a pretty good job of describing what happens when an electron and a proton get together to form […]
The phrase “intrinsic spin” is, of course, an oxymoron. Spin is always relative, always measured with respect to something else, and can never be intrinsic. Nevertheless, “intrinsic spin” is an important concept in quantum mechanics, and apparently it has some real physical meaning. So I want to examine how such an obvious oxymoron can be […]
Back in the Good Old Days of E8 theories, when I first met Corinne Manogue and Tevian Dray, they began their construction of a model by putting the Lorentz group (in the form Spin(3,1)) in the middle of Spin(12,4), leaving Spin(9,3) which they split into Spin(6)=SU(4) and Spin(3,3)=SL(4,R). Now if you take the compact part […]
Or, to put it another way, I am a pure mathematician. You see, that’s what pure mathematicians do. They hoard interesting things, in the hope that they might come in useful one day. Pure mathematicians are essential parts of the ecosystem, because applied mathematicians throw away everything (mathematical) that they don’t think is useful. But […]
If it really is true that the group SL(4,R) describes the symmetries of spacetime on all scales, then it is necessary to quantise SL(4,R). Certainly not everyone agrees with me that it is possible to quantise a non-compact Lie group, as quantisation usually implies compactness in some way or other. That, I think, is why […]
I have a lot of trouble with conventions. In mathematics and in physics “convention” means a decision that was taken at some point about a notation or a definition, that people generally adhere to in order to make communication easier. But sometimes, different groups of people make different decisions about the same thing, and that […]
At a young age, perhaps around 10, my brother got interested in bird-watching. Two specific memories that stay with me are these two exchanges (they may not be verbatim): At the time, of course, I was just a by-stander, without the experience to know who was right. But with more than half a century of […]
Weyl groups are the standard method in physics of relating continuous and discrete groups. They are named after Hermann Weyl, whose work in the first half of the 20th century was of enormous importance for both mathematics and physics. He was the first to propose the gauge principle for physical theories, and is particularly known […]
The special theory of relativity established the principle that a unified force field could in at least one case (electromagnetism) be represented by a Lie algebra, that is the adjoint representation of a Lie group, in this case the Lorentz group. The unification here established that the magnetic field is a subalgebra, but the electric […]
The trouble with quantum mechanics, as I have had occasion to remark before, lies in the identification of SU(2) with Spin(3). It is undeniable that these two groups are mathematically equivalent, but it is not undeniable that they are physically equivalent. The assumption that they are physically equivalent was made in 1925, and has never […]