I have to make a break on writing these posts, maybe a month or two. It is because I started reading the book of Julius Wess and Jonathan Bagger, Supersymmetry and Supergravity (2. edition 1992), which I bought and once read some 25 years ago, still at the time thinking that I will do some research on theoretical physics on free time. Well, there was never any free time, but now I am on pension and actually can do what I want.
The book is great, really. First when I started it, there was no explanations of any symbols used. The authors just put a Lie algebra for a quantum field theory. There were many claims on every page and no proofs. So I read again the back cover where the book is praised for clarity and its pedagogical approach. I thought that this must be some insider joke, my students would have had much to say of this pedagogical approach. But on the second reading of the first pages it become clear: the first page does advice to read Annex A and there you find the definitions, and while there are no proofs, there are exercises. If you do those exercises, then you do verify all claims (or some at least) in the book.
Later, I do not think all claims are explained by exercises, like in the book a tensor is given Lorentz and Einstein indices and you can change them by summing over the vielbein (the indices are just component representations of the same tensor, so of course you can change from one representation to another), but why should there be a representation where all components transform in a covariant way? But I am sure it is all correct. Julius Wess seems to have been a real genius. He invented supersymmetry algebras and supergravity with late Bruno Zumino.
Just in case you wonder what is supersymmetry, it is that there is a theorem which shows that under some rather weak conditions (e.g. there is a mass gap, the condition is not necessarily fulfilled in Young-Mills fields, I showed that for classical Young-Mills fields is is certainly not fulfilled and these can be quantized in some sense) the Lie algebra for every quantum field theory fulfills some conditions, basically equations of commutators. Wess and Zumino noticed that you can also have theories that satisfy more general conditions involving anticommutators. Thus, supersymmetric fields are quantum fields, but a bit more general type. Supergravity is what you get when you take a supersymmetric field theory and add there curvature, connection and torsion. (They indeed have torsion. In General Relativity the spaces do not have torsion, one uses the Levi-Civita connection).
I was surprised that I still could follow the book. (Just the main line, I have not worked exercises.) All these gauge fields, Lie algebras, connections, curvatures, propagators and so on were still familiar. But what was the best is that in the beginning, page 23, there were field equations for a supersymmetric quantum field of a scalar. The equations immediately reminded me of Nordstrom’s scalar gravitation theory. You just have to add the spinor field to Nordstrom’s theory to get Nordstrom’s quantum gravity. Or so it may be. I am thinking of working out some of the formular of Wess and Zumino on two concrete cases: Nordstrom’s and Einstein’s gravitation theory, and see if there pops up graviton, gravitino, a scalar (chiral) field, a vector field and some auxiliary fields, like in the book. This will take me a long while. Just to calculate curvature for Nordstrom’s theory (to be convinced that in Einstein’s theory terms cancel in a very odd way, as in Schwarzschild’s solution) took two months.
But as this blog only has some ten revisiting readers, I do not think it is a big loss for humanity if I do not post for a while. Especially as the later chapters of Tegtmeier’s book (E. R. Carmin: Das Schwarze Reich) get to be more and more dubious and I still cannot figure out what the great conspiracy it today. But I will return to that one also, somewhat later.