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.

Woit generally gives a more mathematical perspective on the whole field. He tells more details of contributions of physicists to mathematics with interesting anecdotes. He also reports varying attitudes of physicists towards mathematics.

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!

Einstein’s equations,

R_{ij} − (½)Rg_{ij} + Λg_{ij} = (8πG/c) T_{ij}

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:

- The concepts of Riemann’s curved spaces were allowed. A supposition of Euclidean geometry had been implicit in prior theories and was reluctantly abandoned for GR.
- Minkowski’s philosophical view of space time. This was a fruit of Einstein’s Special Relativity and briefly resisted by Einstein.
- Physical laws are the same now and then, here and there—thru out the 4D manifold. This is an ancient principle.
- Local adherence to special relativity and laws of forces and electro dynamics as already adapted to special relativity. This leads to covariance which says that physical laws must be locally isotropic even in the 4D sense. Electromagnetism as adapted to special relativity already showed exactly how to code the electromagnetic field and material stresses in the T tensor. I don’t know how much of this clever packaging was justified then or even now.
- The laws must be independent of the particular coordinate system in which they are expressed, despite being expressed in a coordinate system. This is a sort of methodological rather than physical stance. It has to do with what it means to express a theory using coordinates. With proper mathematical development you can expresses the theory in coordinate free terms.
- Equivalence of acceleration and gravity. This is how G gets to appear in the equations.
- “inertial motion is geodesic motion”
- The theory should be able to explain a nearly static universe. This was the blooper that Einstein later regretted. It led to the Λ term whose omission would have made the theory simpler and a better predictor of imminent experimental observations. But perhaps it is necessary after all—expressed as a 120 digit constant.

- The compact dimensions must be Ricci flat, (or nearly so). This must be a consequence of the underlying differential equations.
- The equations must be isotropic in all dimensions. This implies that the curling of dimensions must emerge from the equations.
- The geometry of the space is somehow influenced by the strings.
- The successful predictions of GR must be duplicated.
- The successful predictions of the standard model must be duplicated.