On Peter Woit’s blog a couple of days ago, he wrote on the subject of Nima Arkadi-Hamed’s lectures “His third talk emphasises thinking of space-time vectors as two by two matrices (see section 40.4 of my QM book). This is a truly fundamental idea about space time geometry that gets too little attention in most physics courses.” Now there is no point in me trying to comment on this on his blog, because he won’t allow it. So I’ll write my comments here.
First, I agree it is a truly fundamental idea. Second, it is completely bungled, so that most of the benefits of this mathematical breakthrough are lost.
To explain: physicists only know of one kind of matrix, namely Hermitian matrices. Mathematicians know three: Hermitian, anti-Hermitian and unitary. Hermitian matrices generate Jordan algebras (think fermions), anti-Hermitian matrices generate Lie algebras (think bosons), and unitary matrices generate Lie groups (think gauge groups). To work with Hermitian matrices, physicists first multiply them by i (to make them anti-Hermitian) and then exponentiate them (to make them unitary). Hence it becomes impossible in practice to distinguish the three concepts.
Spacetime, then, is expressed in terms of Hermitian 2×2 matrices, acted on by SL(2,C) by multiplying on the left by any 2×2 complex matrix with determinant 1, and on the right by its complex conjugate transpose. This gives you the Lorentz group SO(3,1). Or you can do exactly the same with anti-Hermitian matrices. But not with unitary matrices.
So what happens if you try to convert spacetime to unitary matrices? Well, first you have to multiply the time coordinate by i, which converts Minkowski spacetime to Euclidean spacetime. Then you have quantised spacetime, because you’ve now got a compact Lie group U(2). You’ve lost the action of SO(3,1), but you’ve still got the action of SO(3), which is the conjugation action of SU(2) on itself. But you have gained a left-multiplication action of U(2) on itself, which is of huge importance in particle physics, because it is the gauge group of electro-weak interactions.
That is how to do it properly.
Hidden assumptions 
June 23, 2021 at 8:50 am |
In other words, electro-weak interactions *are* quantum spacetime. But the gauge group acts by multiplication, not by conjugation, so that spacetime is fermionic, not bosonic (as already discussed here on a number of occasions). And the shape of spacetime is determined by the neutrinos (as already discussed here on a number of occasions).
June 23, 2021 at 8:09 pm |
I haven’t read Peter Woit’s blog for quite some time. Is there something particular going on with all this that makes a discussion there not allowed? Also, thank you for discussing it here, I find it very interesting and enlightening.
There was a comment on the last post with a question about the vacuum energy on a large scale volume of spacetime. I held back on replying because I don’t have a real answer to it. But the trivia I was going to blather on about kind of fits here as well. And so now I am throwing this reply together for both said comment and this post.
The Higgs boson, essential to vacuum energy, has equations using an imaginary mass term. Now the Higgs does not have imaginary mass whatsoever. Just that it as a symmetry breaking mechanism is kind of ridiculuos and sloppy.
So the best answer I have to the question relies on General Relativity. With the generalized field equation Einstein gave us, Gij = Gji, he says spacetime can not ever be empty. An “empty spacetime” is meaningless and impossible. And this minimum energy value is what we call the vacuum.
The question then becomes, does GR describe the entire universe perfectly? No, it is not a ToE. It is merely the best rendering of gravity we currently have. But incredibly useful, with a good number of solutions to be found in the Ehlers group to describe stationary vacuums. So the question, wondering about differing energy values, all I can offer to it is the Ehlers group and solutions like it, where we only care about the minimum.
Back to the Higgs- the imaginary mass terms is not too unusual… One can also represent Fermions with Quaternions, or matter fields with Octonions. The Octonions here refer to the So(10) model, of which I mentioned earlier that no other gauge group has these matter fields. The Octonions are the reason for this ten-dimensional weirdness.
June 25, 2021 at 6:50 pm
No, there’s nothing particular going on, it’s just that he controls comments very closely, and does not allow anything that has the slightest hint of a “pet theory” about it. I don’t understand the rest of your comment, so I’ll stick to the things I think I do understand.
June 23, 2021 at 9:21 am |
Two years or so ago I wrote a short paper explaining all this, including the mathematical proof that the standard version is self-contradictory and therefore wrong. Needless to say, the paper was rejected both by the arXiv and by the journal(s), without refereeing.
June 23, 2021 at 8:24 pm |
The U(2) group looks to me like the most fundamental thing of all. I commented earlier about the isomorphism 2T = SL(2,3) = 24-cell. Then mentioned how the 24-cell taken along with its dual give the 48 root vectors of F4. The model of GL(2,3) is a natural extension from SL(2,3) = 2T. And so I am certain that GL(2,3)- which has order 48- and F4’s 48 root vectors composed of the 24-cell with its dual- are the same model. Only GL(2,3) stops short of all of the symmetry one can use in F4. So kind of the same model, anyway. But all this symmetry with GL(2,3) is a subgroup of U(2). Thus the U(2) is the truly fundamental thing amidst it all.
June 25, 2021 at 7:01 pm
There are certainly some very fundamental connections between GL(2,3) and F4, that cannot be adequately conveyed in a paragraph or two. Which part of this vast web of mathematics you regard as most fundamental is really a matter of taste. For my taste, it is GL(2,3) that is most fundamental.
June 24, 2021 at 4:27 pm |
It is evident that Schroedinger and Clifford believed matter to a) have wave properties, b) exchange energy, and c) have spherical symmetry. It is Einstein who wrote matter must be spherical and spaically extended. Here is a quote from Einstein from before the days he turned to the discrete, instead holding a worldview like Schroedinger and Clifford believed in —
Physical objects are not in space, but these objects are spacially extended. In this way, the concept of empty space loses its meaning. Since the theory of GR implies the representation of physical reality by a continous field, the concept of particles or material points cannot play a fundamental part, nor can the concept of motion. The particle can only appear as a limited region in space in which the field strength or energy density is particulary high.
June 25, 2021 at 12:18 am |
This quote is from 1950. Four years later, by 1954, Einstein changed his mind. Not only about this, but about GR. The 1954 quote, as you are familiar with, no doubt, is the one where he calls GR his “castle in the air”.
June 25, 2021 at 7:06 pm
What one gets from this, I think, is that Einstein was always in doubt, as a good scientist should be. Lesser scientists are sometimes too apt to confuse theories with facts.
June 25, 2021 at 1:19 am |
John Stewart Bell wrote this in 1977, apparently oblivious to Einstien’s call for a discrete model—
The discomfort I feel is associated with the fact that the observed perfect quantum correlations seem to demand something like the genetic hypothesis… This is so rational I think Einstein saw that, and the others refused to see it… I feel that Einstein’s intellectual superiority over Bohr, in this instance, was enormous, a vast gulf between the man who saw clearly what was needed, and the obsurantist. So for me, it is a pity that Einstein’s idea doesn’t work. The reasonable thing just doesn’t work.
…….
…The guiding wave, in the general case, propagates not in ordinary three-space but in a multi-dimensional configuration space is the origin of the notorious nonlocalty of QM. It is a merit of the deBrogle Bohm version to bring this out so explicitly that it cannot be ignored…
June 25, 2021 at 7:15 pm |
The “notorious nonlocality of QM” arises, in my opinion, only because it is an incomplete theory, that does not include a theory of gravity. It is incompatible with GR, but does not provide any alternative gravitational theory. Gravity is capable of providing a large amount of essentially non-local (for terrestrial QM experiments) information, and is therefore (in principle) capable of resolving the paradoxes of non-locality. They are only paradoxes if one insists on trying to understand QM in isolation from its gravitational environment.
June 25, 2021 at 2:34 pm |
The ironic thing about DeBrogle Bohm Theory is that it violates Speacil Relativity. The realist intrepretation finds itself in trouble because of its own hidden variables! It is not Lorentz covariant. Bell, in 1989, still leaning towards the call of DeBrogle Bohm, noted the lack of Lorentz covariance and said: One cannot talk about Quantum Theory without talking about Lorentz covariance.
Any basic discrete theory gets itself into the same trouble, but instead because of reference frames. Some take this as *any* discrete theory will not work because of the need to be Lorentz covariant. It does certainly present a challenge.
The question about the vacuum on the previous post comes back, but worded differently: How is a discrete vacuum supposed to be Lorentz covariant?
Pure speculation here, but if GR is a castle in the air, then maybe the tether, anchoring it on the Earth, is SR, Einstien’s earlier theory and what it says about the speed of light…
June 25, 2021 at 7:31 pm |
These questions are deep, important, philosophical and mathematical. No discrete theory can be Lorentz-covariant. This isn’t a problem with the theory, it is a basic mathematical fact. You have to choose whether you want a discrete theory, or a Lorentz-covariant theory. I choose the former, and then I have to accept Lorentz-covariance as an approximation only. That, in essence, is what this post was about in the first place.
June 26, 2021 at 1:09 am
Perhaps there is a loophole?
So(3,3) has three copies of So(3,1) and three of So(1,3).
And O(3,3) has three copies of O(3,1) and three of O(1,3).
The spin-1/2 particles in So(3,1) are joined by another set in So(1,3). These new particles have one space dimension, and are the only ones that are explicitly only right or left handed.
The 6 subgroups of O(3,3) means we have 6 spinors. Two sets, each with three.
Each lepton has a unique Su(2) × Su(2) algebra corresponding directly to it. It is the Su(2) × Su(2) that forces them to obey Speacil Relativity.
June 27, 2021 at 7:17 am |
To put it another way, when you write spacetime as a 2×2 Hermitian or anti-Hermitian matrix, time is a *scalar*, and therefore commutes with the space coordinates, which anti-commute with each other. So why on Earth would Dirac want to make time *anti-commute* with space? Yet that is what he did, and what everyone still does today. It makes no sense.
June 27, 2021 at 5:30 pm |
I made a serious error in a previous comment that I’ll try to fix here. The equation I mentioned there is supposed to be Gik=Gki. It is introduced in a 1945 paper by Einstein. The one Mendel Sachs built his Quaternion theory from. I know nothing of Sachs, other than his ideas are not widely accepted, and graduate students sometimes get homework problems telling them to explain in detail why Sachs theory is a dead end. Such homework I have not done myself. It’s completely safe to take their word for it. At least I think so… perhaps I should do the work… But the equation Einstein put forward, and I wrote wrong on this thread, is in fact Gik=Gki. Here is the introduction on the paper, the paper itself being a footnote in the appendix of Einstien’s Relativity book.
Every attempt to establish a unified field theory must start, from my opinion, from a group of transformations which is no less general than that of the continuos transformations of the four coordinates. For we should hardly be successful in looking for the subsequent enlargement of the group for a theory based on a narrower group. It is further reasonable to attempt the establishment of a unified theory by a generalization of the relativistic theory of gravitation. Such a generalization, which does not seem to be discovered so far, is described in [this paper… and is] to be considered unified only in a limited sense.
June 28, 2021 at 5:05 pm
Beyond Einstein, Sachs, and even Heim, there is a generalization of GR for a 3×3 spacetime. In one paper, you mentioned CL(4,4). The Split Octonions have signature (4,4). The Zorn matrices are composed of the Split Octonions and can be used to describe QCD via SU(3).
The 2×2 Zorn matrices are able to extend to a 3×3 representation…
June 29, 2021 at 3:08 am
The octonionic connections are a far too large symmetry, probably a good example of what Einstein called the subsequent enlargement of the symmetries in GR. With regular octonions, the 3×3 hermitian octonionic matrices are represented by M(8)3 = F4. The complexified version of it is E6.
The other division algebras give-
M(1)3 = O(3)
M(2)3 = Su(3)
M(4)3 = Sp(3,H)
M(8)3 is directly related to the thread here about the 3×3 spacetime GR I mentioned. These kinds of symmetries strike me as way too large. I agree most definitely that there is no use in going to CL(4,4)…
June 29, 2021 at 3:38 pm |
For that last comment, I need mention that the equivalences are by automorphisms, and that M(1,2,4,8)3 are the exceptional Jordan algebras.
My main concern with it however, is the appearance of F4. Being connected to the 3×3 octonionic Jordan algebra, it is secretly present in models of which invoke that… and, in a sense, complexified F4 is E6, quaternions crossed with F4 is E7, and octonions crossed with F4 is E8. Split Octonions, or CxO, have a connection to E6 in this same limited sense… No doubt the 3×3 Octonionic GR is too large.
As for Mendel Sachs, he ignores a huge number of developments in the history of science and says there is only a very limited unification of the fields. He read Einsteins 1945 paper, but not Einsteins later attempts around 1954… the discrete QM suggestion. Sachs throws out the idea of quantum gravity. He has, however, found a Lie group, dubbed the “Einstein group”, that generalizes the theory and replaces the Poincare group. In the Pauli matrices, there is a set of Complex numbers he replaces with Quaternions. Hence the rejection of his ideas and the legend of undergrads doing homework to debunk it all in one night. Sachs did make serious mathematical mistakes and dropped some fundamental rotation units whilst changing around some algebras. That should not, in my opinion, ruin the merit of his Einstein group. However, this group is the consequence of Einsteins paper where he said the unification was limited severly. Where Einstein says it is limited, despite that he never found the Einstein group, there is an ominous hint at Sach’s fatal flaw: he misinterprets all of Maxwell’s equations. The U(1) field is not respected.
Maxwell’s equations started out as 20, written in Quaternions. Heaviside reduced it to 4 equations without any Quaternions. Sachs brings back quaternions. But he does not have the correct interpretation of the U(1) gauge invariance. Since then, instead of backtracking to the discovery of the Einstein group, the whole of Sachs work is ignored. But perhaps he makes a very important point in replacing the Poincare group.
Eric Weinstein matched So(10) to the 10 Einstein equations. The Poincare group is 10 dimensional, and Sachs replaced it with 16 dimensions with the Einstein group, the extra 6 in there correspond to Einsteins 16 equations before they were simplified to 10. Except these 6 are inherent in the Einstein group and nontrivial this time. This is where he places the Maxwell equations. The 16 is inherent in his Einstein group because of his use of the Quaternions.
June 29, 2021 at 9:23 pm
There is no doubt at all that these Lie groups F4, E6, E7 and E8 are very interesting for mathematics. But there is a great deal of doubt as to whether they are interesting for physics. I have been listening for nearly 15 years to physicists who have tried to persuade me that they are, but I remain to be convinced.
June 30, 2021 at 2:51 am
Spin groups, on the other hand, seem more practical.
If I may keep being general instead of using technical detail:
Spin(1) = O(1)
Spin(2) = U(1)
Spin(3) = SU(2)
Spin(4) = Lorentz Group (6 dim)
Spin(5) = Poincare Group (10 dim)
Spin(6) = 15 dim Conformal Group
The Lorentz group gives 3 spacial rotations and 3 boosts. Tony Smith, through some math I am not well versed with enough to try and explain, builds a theory using the spin(n) groups of n=4,5,6… And gets an S4 group, whose 24 elements have a 1 to 1 correspondence with D4.
Then the ADE series is taken in full and he goes up to E8. I think where he gets to A3=D3, in his theory taken as Su(4) = Spin(0,6), he has no need for the extension. But he couldn’t resist the call of the exceptional series.
With Mendel Sachs, the Poincare group is replaced. But not the Lorentz group. The Poincare group is a symptom of the Lorentz group. Sachs generalized GR but if the Lorentz group is errorneous, then his generalization isn’t a fix for GR at all. The merits of the Einstein Group lose their standing if the Lorentz group is the ultimate problem.
July 1, 2021 at 2:50 am
There are many papers by Y.S. Kim about O(3,3). He starts with an observation by Dirac–
Sp(4) = O(3,2) ((the DeSitter spacetime)),
And extends the group to the Sl(4,R) = O(3,3), for the structure of 15 Dirac matrices.
Aside from the Dirac matrices, he says Sl(4,R) is unphysical. I think he is concerned with some trivial subgroups of Sl(4,R). Despite his work and papers, he called it unphysical, and if he offers any reasoning at all for it, it’s lost to me because my phone won’t open the pdf on it. Somehow, I think it’s merely a matter of his opinion(s)…
July 1, 2021 at 6:25 pm
The main ideas of YS Kim’s ideas are actually this:
– Einstein deserved a Nobel Prize for E=MC2
– Kim met with Eugene Wigner and upon discovering Wigner’s “little groups”, which are internal spacetime symmetries hidden in a Lorentz-covariant theory, decided Wigner also deserved a Nobel Prize, the work just as important, if not more than, Einstein’s equivalence principle.
The mainstream never took notice. YS Kim stayed relatively unknown. He wrote a number of papers about SL(4,R). I am still unable to comment on those, however.
July 2, 2021 at 8:40 pm
YS Kims work leans on the assumptions of Paul Dirac. If time and space anticommute, as Dirac declared they *should*, then this hidden assumption, if wrong, will render these papers trivial. Kim follows Dirac in using the harmonic oscillator and models the hydrogen atom with it. He considers it in motion, like Einstein’s thought experiments, and applies the Lorentz transformation. So his spacetime is continous. He fixes some errors by Dirac, to get Lorentz covariance. Dirac had 3 papers where he lost the Lorentz covariance, and Kim manages to combine them as well as fix that problem. So in the end, we have a representation of the harmonic oscillator with a working Lorentz transformation inducing refrence frame. It is really a generalization of the Dirac equation, I would say.
But Dirac’s equation is far from the unification of everything in the Quantum realm as well as some large-scale phenomenon like dark matter.
With discrete spacetime, models like Lattice Gauge Theory have tried to use it, but they turn back to continuous variables more often then not. There is only one model that really needs mention here that succesfully champions discrete spacetime.
HP Noyes and his bit-string model. The “string” term refers to computer science and not String Theory with its continuous universe and higher dimensional branes and all that madness now commonly associated with the word. His work is incomplete, like reading what could be formulated into a potentail ToE, instead of being currently any kind of ToE. That needs to be said first, but what I really what to point out is that he has discovered some mass ratios. These mass ratios are close to expirement, but not entirely, but certainly to a degree where they shouldn’t be so ignored. There would be critisism, as with any proposal, on the matter, had the work not been presented as philosophy.
July 2, 2021 at 9:24 pm
Kim brings Lorentz Covariance to a Hydrogen Atom described by a standing wave, thus closing a disparity between Einstein and Bohr.
The work involves SR. If one wants to get gravity, the curvature of spacetime in the theory of GR, to work with the Dirac equation, then one should look instead at the Spin(n) groups. They give even more interesting results and you don’t need these papers by Kim.
Noyes doesn’t invoke the Spin groups as far as I know, just a 256-bit “string” which, in a sense, is his player of choice for Conway’s Game of Life.
I still hold the belief, as stated earlier, that the Spin groups will prove fundamental in some way, invaluable for a ToE. But I can only speculate, just like these other theories I mention…
July 4, 2021 at 8:19 am |
Some of the comments above do not distinguish properly between a Lie group and its double cover. This is a common failing in physics, where it is usual to work with the Lie algebra so that you don’t have to worry about making this distinction. Unfortunately, it is an important distinction, and ignoring it is the fundamental reason why the incompatibility of GR and QM is not properly understood.
The group SO(3,3) and its double cover Spin(3,3) = SL(4,R) are used in two places in physics: one is in GR, and the other is in various GUTs in particle physics. There are three representations that need to be understood first: the 4-dimensional representation and its dual, and the (self-dual) 6-dimensional representation. In GR, the two 4s hold spacetime and 4-momentum, and the 6 holds the field-strength tensor. In all the GUTs I have seen, the 4s are various types of spinors, and spacetime lives in the 6. This is an incompatibility that physicists try to get around by throwing more mathematics at the problem. This strategy will never work, because there is no group in which both can be true at the same time.
Hence, *if* the group SL(4,R) can be used to unify the theories of GR and QM, *then* it is necessary to make a decision as to whether the theory has 4-dimensional spacetime or 6-dimensional spacetime. Now if you are devising a new theory, you can make any decisions you like, but for my money a 4-dimensional spacetime is far preferable to a 6-dimensinal spacetime. That means that all the GUTs, string theories, SUSYs etc. are wrong.
No-one who knows anything about experimental high-energy physics can seriously disagree with this conclusion, and yet theoreticians are still trying to tinker with these theories, ending up with yet more theories that are wrong for the same reasons as the old ones. The reason they are doing this is because they cannot face the awful truth, that the error is not in the development of the GUTs etc, but in the standard interpretation of the fundamental work of Dirac. The group SL(2,C) that is called the Lorentz group in QM is not physically or mathematically related to the group SO(3,1) that is called the Lorentz group in relativity. The fact that they have isomorphic Lie algebras is a red herring that has led physicists astray for 90 years.
July 4, 2021 at 2:37 pm |
I have a serious criticism to throw at YS Kims work. It’s a small little case of what you described here.
He extends O(3,2) to O(3,3). I say that doesn’t make sense from what he is really trying to do.
I say what he should’ve done is extend O(3,2) into O(4,2). There is a huge difference.
If only he had used O(4,2) in a spefic paper I am thinking about… He would have the full of what he’s been looking for/at. The Su(2,2) of which it is isomorphic to… it contains the Poincare group. Now why do I say that is so very important for YS Kim?
He couples 2 harmonic oscillators. Because he says QM is the study of the harmonic oscillator. And that SR is the study of lorentz covariance. And 2×2 matrices are the crux of physics. He says students don’t know how to diagonalize them. I agree with that. He has 10 forms of which model a quanta/harmonic oscillators: and the 10 transforms into the 10 of Einstein, Lorentz Covariance. QM and SR are quite literally derived from the same equation.
Then he makes the fatal extension to O(3,3). You would agree that, given his need of the Poincare group, O(4,2) = Su(2,2) makes sense? And not what he did do with O(3,3)?
July 4, 2021 at 11:16 pm
Well, I really cannot comment on YS Kim’s work, or whether or not the groups he uses are appropriate for the task at hand. Lots of people talk about the importance of the Poincare group for QM, but given that they do not (and, apparently, in most cases cannot) tell me which of the two Lorentz groups they are talking about, I am unable to assign meanings to the statements they make. In particular, I do not buy your argument (which I have heard from other people also) that O(4,2) is the “correct” signature because it contains the Poincare group. Some people prefer O(5,1), and others give up and just complexify it so they don’t have to decide. For my purposes, only O(3,3) makes sense.
July 5, 2021 at 3:06 am
There was another comment I wrote after this but WordPress didn’t put it up. Honestly, I expect it to label me as spam at this point… Though it didn’t have to erase it! I don’t know exactly what happened. I do know not to be surprised when it flags me down, the filter took quite a while to come after me so I can’t complain at all.
I wrote about Kim and Penrose both having connections to SU(2,2), and that I don’t think they were asking the right questions…. Kim did say that he thought Complex planes were not relevant for physics. So it’s odd Penrose takes the exact opposite view. Penrose said Sl(2,C) was only an analogy. Even though it’s a flawed one, people seem to have forgotten that and use the groups interchangably!
I mention this lost comment because I expressed my belief that your papers and arguements for Sl(4,R) are incredibly compelling and the most viable thing I have ever seen. And Penrose used Su(2,2) for the Poincare group, but in terms of Quantum Gravity, I do not think he is asking the right questions. Same for Kim, who’s line of thought *should* have led to the same structure as Penrose. They both have a heavy focus on the 10 **dimensions* of the Poincare group. I write that merely as an observation.
I really do think you are on the right path and that more people need to be listening. Sadly, WordPress took away the comment, which would have been like #27 or #28 on this thread alone! So it was to be expected. But anyway, it would have been nice for it to *exist* so that my full arguement/observations/musings were present. I am not really one of the supporters of the Su(2,2) and/or Poincare group line of unification. It, to me, is just a noteworthy curiosity. A strange little curiosity that is missing room for, say, 12 generators of the SM that are seen as 8+3+1, and what gravity has to do with the picture. None of these models give particle masses! What is a theory of Quantum Gravity without particle masses? Sl(4,R) has a different structure to it that the 12 generators fit into, and the Pati-Salam connection to Einstein is a truly viable way to look for Quantum Gravity.
Eric Weinstein was also to focused on the Poincare group. He used Spin(6,4) because the 10 *dimensions* of the Poincare group were connected to the theory, in a near identical manner as Dirac did to Sp(4), which was completed by Kim. I say the Poincare group is not this important, espeacily when the Pati-Salam connection is found. Not enough people talk about the Pati-Salam connection. The Arxiv moderators have either failed to see it or they just don’t care. Or maybe they think they know everything about it already. Likely, its all three of these. Same has gone on for the 2×2 matrices. Though Kim mentioned them. Well, I’m talking in circles right now, so I’ll end here. Your work is what Lee Smolin called the third road to Quantum Gravity: the new one. The right one.
July 4, 2021 at 3:02 pm |
Your case of Sl(4,R) is the most compelling thing I’ve ever seen and Sl(4,R) is certainly viable. Just that YS Kim was asking the wrong questions. And the ultimate answers to those questions shouldve been Su(2,2).
Penrose used Su(2,2) in twistor theory. The ultimate question of quantum gravity wasn’t exactly found in that group. YS Kim disagrees that complex planes are important. Penrose uses complex planes everywhere. Funny how that happens.
July 5, 2021 at 8:00 am |
I have re-instated this comment, that WordPress marked as spam for some reason that I don’t understand. Maybe phrases like “most compelling thing I’ve ever seen” ring its alarm bells.