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

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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?

This post imported from StackExchange Physics at 2014-03-17 04:21 (UCT), posted by SE-user user1247
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).

This post imported from StackExchange Physics at 2014-03-17 04:21 (UCT), posted by SE-user Ron Maimon
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).

This post imported from StackExchange Physics at 2014-03-17 04:21 (UCT), posted by SE-user Dilaton
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.

This post imported from StackExchange Physics at 2014-03-17 04:21 (UCT), posted by SE-user Dilaton
@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.

This post imported from StackExchange Physics at 2014-03-17 04:21 (UCT), posted by SE-user user1247
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.

This post imported from StackExchange Physics at 2014-03-17 04:21 (UCT), posted by SE-user Dilaton
If you want to see a real 'crackpot', check S. J. Crothers out. Some of the most basic errors that are possible to make are made in his ramblings. He has contacted myself, my Ph.D. supervisor and many really senior guys [which I am not one of] in the field of GR, Astrophysics and Cosmology and is nothing but disgracefully rude to them (e.g. Kerr et al.). You can read his story and the letters on his site (which he uses (somehow) in support for his crap.

This post imported from StackExchange Physics at 2014-03-17 04:21 (UCT), posted by SE-user Killercam

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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.

This post imported from StackExchange Physics at 2014-03-17 04:21 (UCT), posted by SE-user Philip Gibbs
answered May 30, 2013 by (650 points)
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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.

This post imported from StackExchange Physics at 2014-03-17 04:21 (UCT), posted by SE-user R S Chakravarti
answered Apr 28, 2013 by (20 points)
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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.

answered May 28, 2013 by (4,305 points)
well I'm an active professional physicist, and this is hardly obvious to me (though admittedly I'm a lowly experimentalist). Your statement, taken literally, that quantum is "fundamental" is so wrong I actually don't believe you mean it. I have read a couple of texts on quantum gravity and one of the biggest takeaways was that there are a very long list of canonical reasons why one might be lead to suspect that both QM and GR are emergent from a deeper theory rather than one being fundamental. Given that even the likes of 't Hooft is working on such an example should give you extreme pause.

This post imported from StackExchange Physics at 2014-03-17 04:21 (UCT), posted by SE-user user1247
you keep appealing to popularity. I don't think it's true that you speak for the entire theoretical physics community here. Or at least at this point I see no reason to trust you to that high a degree. I saw 't Hooft give a lecture at CERN a month or so ago and the theorists I talked to weren't quite so dismissive. They may have disagreed with him, but I don't at all think they were quite so sure of themselves that QM is absolutely fundamental. (continued)

This post imported from StackExchange Physics at 2014-03-17 04:21 (UCT), posted by SE-user user1247
(continued) Besides, there is a ton of mainstream work here in quantum foundations. While I'm not a theorist it is clear to me you are a bit off base here. BTW Gibbs gave an answer above that seems more reasonable to me. Take a look and see what you think.

This post imported from StackExchange Physics at 2014-03-17 04:21 (UCT), posted by SE-user user1247
@user1247 yes, I like Phil Gibbs's answer, I have already upvoted it ;-). Maybe it would be interesting for you to check related questions on quantum foundations on this site and see, what for example Lubos Motl (a theoretical physicist) himself says concerning these issues and look at the votes he get to estimate what people think. Not everybody likes his sometimes hm ... stingy style of writing when he proofs somebody wrong, but about the fact that he knows what he is talking about concering (quantum and other) physics, most serious physicists agree. So you dont have to trust me :-). Cheers

This post imported from StackExchange Physics at 2014-03-17 04:21 (UCT), posted by SE-user Dilaton

One can get non-linear behavior from purely linear phenomena, if they happen in large enough spaces. An ordinary differential equation $\dot x(t)=f(t,x(t))$ can be rewritten as a linear partial differential equation $\partial_t\phi(t,x)=f(x)\partial_x \phi(t,x)$, and the theories are essentially equivalent.

quaternions can be used for QM because complex numbers are special quaternions, like the way a square is a special kind of rectangle. further, high energy QM involves special relativity, which is a linear approximation of a non-linear general relativity, i.e. we approache the lorentz transformation when 6 of the elements of Reimann space metric approach 0 in the non-relativistic limit. The reason QM is considered non-relativistic is because it does not adhere to the equivalence principle so that the laws of physics are completely objective w.r.t. all frames (instead, QM takes a subjective view where we only consider the effects of the macro on the micro, but not vice versa-- hence an unreciprocated, incomplete approximate view of matter). QM is a philosophy of measurement, not a philosophy of matter. it is posed on the tenets of Logical Positivism rather than Abstract Realism.

i don't think you can get non-linear behavior from purely linear phenomena ...so it makes no sense that QM can be a limiting factor of GR since QM is linear by definition (allowing it's use of the linear operator and eigenfunction framework tools, which won't behave in a non-linear system). it's just logical. if one attempts to build a universal model with QM alone, one will eventually run into logical paradoxes because there isn't enough functional substance in complex numbers alone. the standard model would then be interpreted not as a collection of fundamental particles, but instead as a collection of modes that represent fundamental interactions within the framework of a quaternion formalized GR. Indeed, M.Sachs approach is able to account for everything that QM already does simply due to the fact that complex numbers ARE quaternions [i.e. a complex number s0 + is1 is a quaternion s0 + is1 + js2 + ks3 with s2 = s3 = 0]. if theoretical physicists today are fundamentally using QM to predict the existence of fundamental particles, what they are really doing is building the wholeness of GR from a bunch of fuzzy puzzle pieces using a powerful statistical probablity calculus like the way boltzmann did to make the study of millions of individual particles a practical case. in time, this may become more clear methinks. i don't even think most of the mainstream really agrees with the Copenhagen Interpretation anymore either.

when one really ponders about it, QM probably is the more "scientific" theory, but that is only because it is based on "we can only completely positively know that whatever we measure is truly real" while GR has the power to predict things we have yet to observe (we are quantum in scale compared to these predictions, and since QM does not consider the effects of the micro on the macro then how can it logically have any substance to explain these cosmic scaled forms in the universe? it would have to be extended into a new framework, but that frame work already exists -- it's GR). in reality, only time will tell which theory will end up engulfing the other...but i wager on GR only because our instrumentation technology will only improve over time (it has improved immensely since the Copenhagen Interpretation) and will reveal the clear simplicity of nature that GR and the quaternion frameworks provide.

please provide examples i'm not considering if i am wrong, because i may not have studied them.

Arnold, thanks for that information. That makes sense. Is there by any chance a proper name for that equivalence between these two formulations?

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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
answered May 29, 2013 by (40 points)
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.

This post imported from StackExchange Physics at 2014-03-17 04:21 (UCT), posted by SE-user Dilaton

This post imported from StackExchange Physics at 2014-03-17 04:21 (UCT), posted by SE-user akhmeteli

One can rewrite every complex number $a+ib$ as a real 2 x 2 matrix $\pmatrix{ a & b \cr -b & a}$, hence complex numbers can be avoided everywhere if one absolutely wants.

The real reason why QM uses complex numbers is that it makes everything much simpler as when working with real numbers only (or with quaternions). Unless anothe approach produces significantly superior results, QM with omplex numbers is the way to think of it.

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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.

This post imported from StackExchange Physics at 2014-03-17 04:21 (UCT), posted by SE-user SJRubenstein
answered Aug 9, 2012 by (-10 points)
Quantity of work is a poor measure of the work's value.

This post imported from StackExchange Physics at 2014-03-17 04:21 (UCT), posted by SE-user Guy Gur-Ari
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.

This post imported from StackExchange Physics at 2014-03-17 04:21 (UCT), posted by SE-user Shaktyai
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.

This post imported from StackExchange Physics at 2014-03-17 04:21 (UCT), posted by SE-user Dilaton
To compare this guy who comes across as having a relatively weak understanding of actual physics, to Einstein, is comical to me...

This post imported from StackExchange Physics at 2014-03-17 04:21 (UCT), posted by SE-user Killercam

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