“Not Even Wrong” by Peter Woit is another sad story of today’s particle physics. On page 133 he says something mathematical that I understand and is new to me. (This is unusual in this class of books.) A sigma theory maps familiar space-time to some abstract manifold, most often some sort of Lie group. I don’t yet know whether the talk of SU(3)╳SU(2)╳U(1) theories are about sigma theories. I understand that Schrödinger’s equation for one electron is a sigma theory where the manifold is U(1) represented as complex numbers. I do not understand Schrödinger’s equation for 2 electrons as a sigma theory, however.
The rôle that U(1) plays in Schrödinger’s equation requires summing over values in U(1). When U(1) is taken to be complex numbers, this is easy. It is also easy for SU(2) but in two different ways. Some Clifford algebra models SU(2) but so to does O(3). (Bug here; I think I have answered my own objection. I must come back to this!) O(3) is not closed under addition, even with renormalization. The Clifford algebra uses only the members of the ‘Clifford group’ to identify members of SU(2). That group too is not closed under addition. I think it is not closed even after renormalization.
On page 176, Woit gives a good description of renormalization problems in terms familiar to those with calculus. The series are ‘asymptotic’ which is a mathematical euphemism for a series which gives an approximate answer. Even QED is under this cloud. This mathematical phenomenon can sometimes be used to make mathematical estimates in areas outside any physics application. It is, however, a very unsatisfactory situation for a physical theory when it is the only for the theory to produce a number.
On page 191 Woit quotes Magueijo thus:
Unfortunately, Einstein himself bears a lot of the responsibility for having brought about this state of affairs in fundamental physics . . . He became more mystical and started to believe that mathematical beauty alone, rather than experimentation could point physicists in the right direction. Regretfully, when he discovered General Relativity—employing this strategy—he succeeded!I grant there is an important point here, and that it is delivered cleverly but I must stand up for Einstein here. Einstein saw contradictions in special relativity and set out to eliminate them. He established just a few principles and challenged himself and others (Hilbert for instance) to find a theory that met these principles. It turned out that the principles led to a unique theory which was GR. This same pattern is the holy grail that particle physicists seek. I have not seen yet even an incipient list of such principles for string theory. This is the beauty of GR that shows no sign yet of being repeated in particle physics. String theory boasts only mathematical beauty so far. GR goes far beyond mathematical beauty!
Rij − (½)Rgij + Λgij = (8πG/c) Tij
were written in a language whose very syntax could only express covariant laws. Some of the principles given in the aforementioned Wikipedia article were perhaps articulated after the theory was published and collectively they can be re-aggregated several ways. I am inclined to the following division:
Woit’s Blog, xx