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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?

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closed as not constructive by dmckee Jun 1 '13 at 16:49

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There are many questions also about the related Geometric Algebra. This type of thing is not physics, but formalism, and I have seen the claims about "QM from GR", they derive a quantization rule similar to Bohr Sommerfeld from a GR looking thing, and this is total rubbish from the point of view of physics. This part is crackpot, but the part about quaternions is probably empty formalism rather than wrong (although I didn't review it). –  Ron Maimon Jan 17 '12 at 5:00
He does things that seem remarkable like deriving QM as a low-energy limit of GR ... This sentence looks suspicious, I thought it is the other way round, GR is derievable as the classical low energy limit from a high energy quantum mechanics theory of gravity (or quantum gravity for short). –  Dilaton May 28 '13 at 14:25
In addition, see here for arguments why quantum mechanics has to use complex variables instead of anything else, so a quantum gravity can not be based on quaternions either. Physicists know this, and that is probably among other things why they ignore such approaches to quantum gravity. –  Dilaton May 28 '13 at 14:34
@Dilaton, I didn't type the sentence wrong. That is what he does in his books: QM as the low-energy limit of GR. I'm an experimentalist, so I probably don't have the background to dig into it enough, but I've just never been able to find anything wrong in his books and found it strange I never found any refutation or critical reviews of his works. His logic appears OK by my eye and he seems to have been an actual physicist at a real college, and his books seem to be in all the physics libraries... it's just odd. –  user1247 May 28 '13 at 17:56
Just for the notes, I did neither say nor mean that it is user1247 who typed the sentence wrong, but the sentence IS wrong from a physics point of view. –  Dilaton May 30 '13 at 17:02

5 Answers 5

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.

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Hi akhmeteli, can you elaborate a bit more exactly than just saying it is not "watertight" generally, about what arguments I explained you exactly disagree with and why from a physics (or mathematical) point of view? To me, the reasoning in the articles I linked too looks perfectly clear and right, I see no error therein. –  Dilaton May 30 '13 at 9:40
@Dilaton: Please see the edit to my answer. –  akhmeteli May 31 '13 at 6:42

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.

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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.

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I disagree that it is obvious that QM is more fundamental than GR. I think you are tautologically assuming as an axiom that QM cannot be some emergent property of GR. While it is de rigueur to quantize classical theories, it is a mistake to assume that classical theories cannot have QM as an emergent property (on the other hand modern extensions of Bell's inequalities are increasingly constraining these lines of thought). The parenthetical constraints excepted, I see no a priori reason why QM cannot emerge from GR. In fact the converse has maybe more obvious fundamental problems. –  user1247 May 29 '13 at 20:57
About complex #s vs quaternions, I agree with akhmeteli. I remember when I learned QM from Bohm's book he re-wrote QM without complex #s. Maybe not as compact a formalism, but certainly allowed. You seem to arrive at this understanding in the latter part of your answer, when you agree the quaternion stuff may just be empty formalism. On the other hand it is not obvious to me that the formalism must be completely empty. Perhaps the quaternion formalism allows a bit more freedom, being homomorphic rather than isomorphic to the complex one, leading to some additional structure. –  user1247 May 29 '13 at 21:07
@user1247 you and Mendel Sachs have it completely wrong. The real world community of active professional physicists knows that the fundamental laws of nature are quantum and that the classical theoreries are derieved from the them as a limit. The question how QM can be represented mathematically, and what does not work can be objectively and rigorously evaluated too. To bat that the voting pattern on this thread converges to represent the opinions and prejudices of people who know stuff no well enough instead of representing the knowledge of the real world active physicist community ... –  Dilaton May 30 '13 at 16:41
funnily enough, I can't stand many of Lubos' answers. Surely he is competent, I won't dispute that. But he definitely represents one very hard-line perspective about certain things that is not shared by everyone of his caliber. I don't dislike his answers just because he is so arrogant, but more because he doesn't attempt to understand where the questioner is coming from, often seemingly almost purposefully obtuse. It would please him more to insult rather than inform. –  user1247 May 30 '13 at 22:06
In any case despite your appeal to one person's opinion (Lubos), I don't think there is any a priori reason QM is fundamental. It is de rigueur to use that language, and I would use it myself when teaching a class. But that doesn't mean everyone literally thinks it must be fundamental. In fact, that is kind of stupid. Is there some logical proof it must be fundamental? Of course not. And as I pointed out there are people with Nobel's (does Lubos have one?) who work on this stuff (there are also many others in the mainstream in QM foundations). –  user1247 May 30 '13 at 22:09

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.

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Could you elaborate: the Pauli matrices are indeed isomorphic to the unit quaternions. I'm sure you're right about the details, but given the isomorphism this answer is bound to look a bit odd to a nonspecialist like me. –  WetSavannaAnimal aka Rod Vance 23 hours ago

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.

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Quantity of work is a poor measure of the work's value. –  Guy Gur-Ari Aug 10 '12 at 1:46
For the users who downvote SJRubenstein's opinion, have the elegance to motivate your vote. He has honestly answered user1247's question and I see no reason to downvote him but to confirm his view that there are some fanatics out there willing to censor anyone who is not mainstream. –  Shaktyai Sep 9 '12 at 6:20
To evade downvotes you should probably base your answer on physics arguments instead of sociological fuss and personal prejudices. Terms like "establishment" etc are often used in the internet by crackpots and trolls advertising their own physically not consistent personal pet theories, to attack professional physicists who know exactly what they are doing. –  Dilaton May 28 '13 at 14:45
To compare this guy who comes across as having a relatively weak understanding of actual physics, to Einstein, is comical to me... –  Killercam May 31 '13 at 7:36

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