I don't know much about general relativity, so I have little or nothing to say about M. Sachs' work. However, I'd like to make some remarks on some answers here where Sachs is criticized, and this is how the following is relevant to the question. For example, I don't quite understand @R S Chakravarti's critique:"the Pauli matrices are not quaternions". It is well-known that the Pauli matrices are closely related to quaternions (http://en.wikipedia.org/wiki/Pauli_matrices#Quaternions ), so maybe this critique needs some expansion/explanation. I also respectfully disagree with some of @Dilaton's statements/arguments, e.g., "the only reasonable number system to describe quantum mechanics in are complex numbers" Dilaton refers to L. Motl's arguments, however the latter can be less than watertight - please see my answer at QM without complex numbers . Maybe eventually we cannot do without complex numbers in quantum theory, but it looks like one needs more sophisticated arguments to prove that.
EDIT(05/31/2013) Dilaton requested that I elaborate why I question the arguments that seem to prove that one cannot do without complex numbers in quantum theory.
Let me describe the constructive results that show that quantum theory can indeed be described using real numbers only, at least in some very general and important cases. I’d like to strongly emphasize that I don’t have in mind using pairs of real numbers instead of complex numbers – such use would be trivial.
Schrödinger (Nature (London) 169, 538 (1952)) noted that you can start with a solution of the Klein-Gordon equation for a charged scalar field in electromagnetic field (the charged scalar field is described by a complex function) and get a physically equivalent solution with a real scalar field using a gauge transform (of course, the four-potential of electromagnetic field will also be modified compared to the initial four-potential). This is pretty obvious, if you think about it. Schrödinger made the following comment: “"That the wave function ... can be made real by a change of gauge is but a truism, though it contradicts the widespread belief about 'charged' fields requiring complex representation." So it looks like either at least some arguments Dilaton mentioned (referred to) in his answer and comment are not quite watertight, or Schrödinger screwed up somewhere in his one- or two-page-long paper:-) I would appreciate if someone could enlighten me where exactly he failed:-)
L. Motl offers some arguments related to spin. Furthermore, Schrödinger’s approach has no obvious generalization for equations describing a particle with spin, such as the Pauli equation or the Dirac equation, as, in general, one cannot simultaneously make two or more components of a spinor wavefunction real using a gauge transform. Apparently, Schrödinger looked for such generalization, as he wrote in the same short article: “One is interested in what happens when [the Klein-Gordon equation] is replaced by Dirac’s wave equation of 1927, or other first-order equations. This … will be discussed more fully elsewhere.” As far as I know, Schrödinger did not publish any sequel to his note in Nature, but, surprisingly, his conclusions can indeed be generalized to the case of the Dirac equation in electromagnetic field - please see my article http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf or http://arxiv.org/abs/1008.4828 (published in the Journal of Mathematical Physics). I show there that, in a general case, 3 out of 4 components of the Dirac spinor can be algebraically eliminated from the Dirac equation, and the remaining component (satisfies a 4th-order PDE and) can be made real by a gauge transform. Therefore, a 4th-order PDE for one real wavefunction is generally equivalent to the Dirac equation and describes the same physics. Therefore, we don’t necessarily need complex numbers in quantum theory, at least not in some very important and general cases. I believe the above constructive examples show that the arguments to the contrary just cannot be watertight. I don’t have time right now to consider each of these arguments separately.
This post imported from StackExchange Physics at 2014-03-17 04:21 (UCT), posted by SE-user akhmeteli