Why is Mendel Sachs's work not taken seriously? Or is it?

Question

Back in college I remember coming across a few books in the physics library by Mendel Sachs. Examples are:

General Relativity and Matter

Quantum Mechanics and Gravity

Quantum Mechanics from General Relativity

Here is something on the arXiv involving some of his work.

In these books (which I note are also strangely available in most physics department libraries) he describes a program involving re-casting GR using quaternions. He does things that seem remarkable like deriving QM as a low-energy limit of GR. I don't have the GR background to unequivocally verify or reject his work, but this guy has been around for decades, and I have never found any paper or article that seriously "debunks" any of his work. It just seems like he is ignored. Are there glaring holes in his work? Is he just a complete crackpot? What is the deal?

Answer 1

There are many formalisms that relate general relativity to quaternions in the literature and it would be a huge task to entangle their interelations and see who cited each other. Quaternions or split quaternions or biquaternons can be related to the Pauli matrices so it is easy to see how someone might then relate GR to QM. (This does not mean that QM needs to be based on quaternions rather than complex numbers) All theory that uses twistors or spinor formalisms for quantisation of gravity have a similar flavour and could probably be related to the work of MS in some way.

It is unlikely that MS had derived Quantum Field Theory from GR because GR is a local theory and QFT is non-local. It is possible that he related some formulation of GR to "first quantised" local equations such as the Dirac equation. Notice that in the modern view the Dirac Equation is regarded as classical even though it includes spin half variables and the Planck constant. The distinction between classical and quantum is not as clean as some people like to believe.

I have not studied his work but I will hazard a guess that his work was not really ignored or debunked. It was just incorporated into other approaches with different interpretations that may have made it non-obvious that some of his ideas were included. One day when we know the final theory of physics there will be lots of science historians who dig through old papers and work out who really had the important ideas first, then perhaps MS will get more credit (if his ideas are part of the final answer and he thought of them first). Until then there is just a big melting pot of ideas that often get reinvented and the shear quantity of papers means that if you spend your time reading everything that anyone else has done you will never make any progress yourself.

Answer 2

Mendel Sachs may have been blacklisted, which would certainly be wrong. But his theory has a fatal error. His derivation depends on the assumption that certain 2x2 complex matrices, standing for quaternions, approach the Pauli spin matrices in the limit of zero curvature. This is impossible; the Pauli matrices are not quaternions and the argument collapses.

Answer 3

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.

Answer 4

First of all, if Mendel Sachs does things like deriving QM as a low-energy limit of GR, he has things completely upside down. The fundamental laws of physics are quantum, so quantum mechanics can not be derived from something else. It is rather the case that general relativity is derievable as the classical low energy limit from a high energy quantum mechanics theory of gravity (or quantum gravity for short). This works for example for string theory.

In addition, the only reasonable number system to describe quantum mechanics in are complex numbers. Some arguments why quantum mechanics has to use complex variables (instead of real variables) are given here. Complex numbers are needed for the Schrödinger equation to work, to conserve total probabilities, to describe commutators between non commuting operators (observables), to have plane wave momentum eigenstates, etc ... Generally, important physical operations in quantum mechanics demand that probability amplitudes obey the rules for addition and multiplication for complex numbers, they themself have to be complex numbers.

In this article describing why quantum mechanics can not be different from the way it is, some explanations are given why using larger number systems than complex numbers to describe quantum mechanics are no good either. Using quaternions, the quaternionic wave function can be reduced to complex building blocks for example, so going from a complex number description of quantum mechanics to octanions introduces nothing new from a physics point of view. Using octanions would be really bad, since octanions have the lethal bug that they are not associative.

So in summary, my reasons for being suspicious or more honestly even dismissive of Mendel Sachs's work as described here is that he seems to fundamentally misunderstand the relationship between quantum theories and their classical limits. In addition, the only reasonable number system to describe quantum mechanics are complex numbers, so I agree with Ron Maimon that introducing quaternions would at best be empty formalism.

Answer 5

Good question! (I have wondered the same.)

I hold Mendel Sachs (deceased 05/05/12) to have been the most astute theoretical physicist since Einstein. His quaternion formalism was, no doubt, exactly what Einstein sought over his last thirty years, to complete GR. And its spinor basis induces me to suspect that Sachs' interpretation of QM, via Einstein's Mach principle, as a covariant field theory of inertia, is also right on the mark.

Considering Sachs' volume of output, after much mulling, I finally had to conclude that he was "blacklisted," the establishment not permitting any discussion if they can have anything to do with it! I can see no other way that that quantity -- much less, quality -- of work could have been ignored.