I may have been a bit hasty to dismiss GR in my most recent paper, posted here a couple of weeks ago at https://robwilson1.files.wordpress.com/2024/04/camps3.pdf. What happens in that paper is that I work with 3×3 matrices over the split quaternions H’, in order to implement phase space, and I find that they split as 1+14+21, which is compatible with QM, and not as 1+15+20, as would be required for GR. However, when I tried to embed the model into E8, in order to try and get the E8 crowd interested, I found that the embedding I wanted was A1+A2+A5 (which I’ve written about before, but not managed to get onto the arxiv), and that I had got the wrong real form of A5.
I’m not sure exactly how many real forms A5 has, but it is at least seven, of which two occur in the semisplit magic square that underlies most of the E8 models that people consider, including our “octions” model https://arxiv.org/abs/2204.05310. The real form I was considering occurs in the split magic square only. The 3×3 matrices over ordinary (non-split) quaternions were studied in some detail in the octions paper, as they formed one of the main stepping stones to get to E8. But I am now convinced that the octions paper got it wrong, and should have transposed the magic square, and used the other real form, which is SU(3,3). And I also got it wrong, and should have used SU(3,3).
The reason is that SU(3,3) not only contains Sp_6(R), which is the symmetry group of phase space in Hamiltonian mechanics, and is, as I have recently shown, the formalism that underlies the Dirac algebra in (relativistic) quantum mechanics, but also contains SO(3,3), which is how the group of general covariance in GR acts on phase space. In other words, we can combine the two provided we complexify phase space. This is not even a crazy idea, it was already done by Maxwell in the 19th century. Complex phase space allows one to treat momentum and current at the same time. If you can’t treat momentum and current at the same time, then you can’t do electrodynamics at all! And it was done by Dirac in 1928 in the context of QM. So it is known to be necessary, and you may well ask, why I thought I could get away without it? Good question, to which I do not have an answer.
Anyway, now that I know it is necessary, I know that I have to study the action of SU(3,3) on complex phase space. This group has got everything in it that is needed for physics. It’s got the Dirac algebra for QM, it’s got general covariance and the Riemann curvature tensor for GR, it’s got twistors if you want to follow Penrose, it’s got Hamiltonian symmetries of phase space if you want to follow Bohm/Hiley/de Gosson, and its centralizer in E8 is SU(3) x SL(2,R), which is the correct real form for the gauge group of the Standard Model of Particle Physics. What more could you want? The Moon?
Maybe you want to see the Riemann curvature tensor, because that’s the bit that no-one believes can be there in quantum mechanics. As a representation of SO(3,3), the RCT is usually constructed by first converting to compact SO(6), then taking the symmetric 6×6 matrices and subtracting off the identity matrix. If you want to do this with SU(3,3), then what you get is a 21-dimensional irreducible complex representation, in place of the real 1+20. In other words, we get a significant generalisation of the RCT, and a corresponding extension or correction to GR.
On the other hand, it may be worth noting that SU(3,3) has got a real 20-dimensional irreducible representation, that is constructed in quite a different way, as the anti-symmetric cube of the natural 6-dimensional representation (instead of the symmetric square). Is this what the RCT is really trying to be? I have no idea. BUT this representation plays a prominent role in the E8 model, and represents (among other things) the left-handed leptons. There are two copies of this representation, one for neutrinos, one for electrons. So does this mean that the RCT is trying to describe neutrinos? Can this possibly make sense? Instead of working in the fourth power of spacetime, we work in the cube of phase space? The cube of phase space breaks up as 1+9+9+1 if we separate position from momentum, whereas the RCT breaks up as 1+9+10 if we restrict to SO(3,1). So we get the Einstein tensor coming out, but not the (spin 2) Weyl tensor. So we get something like the field equations, but not the spin 2 graviton. Instead of the spin 2 graviton we get three directions of momentum times three directions of distance-squared. Well, if that isn’t quantum gravity travelling at the speed of light then I’ll eat my hat!
But, did you notice that there are also 3 dimensions of momentum squared times 3 dimensions of distance? That means that, in addition to the 1/r^2 ordinary Newtonian gravity, there is a 1/r type of gravity, which is what is needed for MOND. Moreover, there is a transition between the two that occurs when the momentum/distance relation swaps over. Why does this occur empirically at a particular acceleration scale? I have no idea. But one thing is for sure, this isn’t caused by Dark Matter. It is caused by neutrinos interacting with each other, because it has a momentum-squared term – one momentum for each neutrino.
But, did you also notice that there is a momentum-cubed term? Independent of distance? Is that what Dark Energy really is? Interactions of three neutrinos at once?! No wonder the cosmological constant is so small! No wonder it is 120 orders of magnitude smaller than particle physicists think it is!
So, are we any nearer to a resolution of this conundrum? Is Einstein’s GR like Eric Morecambe’s piano concerto? All the right notes, but not necessarily in the right order?
Hidden assumptions 
April 19, 2024 at 9:46 am |
Hubble Tension, anyone? Why is Dark Energy, or the Cosmological Constant, not constant? In my model it’s obvious – it is a momentum-cubed term, so it depends on large scale motion with respect to the whole of the rest of the universe.
April 19, 2024 at 10:40 am |
Adjoint SU(3,3) consists of adjoint Sp_6(R) together with a 14-dimensional representation with signature (6,8), compared to the signature (8,6) that occurs in SL(3,H’). The change of signature converts it from fermions into bosons. It still splits as 6+8 for U(3), but now the 8 is compact, so presumably massless, so can be interpreted as gluons, while the 6 are boosts, so massive, and form a complex 3-dimensional representation, so can be perhaps be interpreted as the three weak force mediators: the Z, W+ and W- bosons. If so, then we don’t need any of the rest of E8, as all the nuclear forces are included already in SU(3,3). So all I need to do to the “CAMPS” model is adjust the real form from SL(3,H’) to SU(3,3), and leave everything else alone.
April 19, 2024 at 10:53 am |
An individual neutrino mediates “ordinary” Newtonian gravity, at a strength roughly 40 orders of magnitude weaker than electromagnetism (+/-3 orders of magnitude for the difference between electron and proton masses). A pair of neutrinos interacts therefore at ~80 orders of magnitude, and a triplet at ~120. The fact that the (observed) “Dark Energy” is 120 orders of magnitude smaller than particle physicists expect is down to the fact that particle physicists count *all* particles in the vacuum energy, and I count *only* the neutrinos. This therefore accounts for the 120 orders of magnitude discrepancy.
Moreover, it says that this “vacuum energy” is essentially gravitational in nature, and hence it implies that the large “vacuum energy” that particle physicists calculate has nothing to do with a true vacuum, but everything to do with the gravity of the Solar System.
April 19, 2024 at 9:12 pm |
And, I think, in order to get 120 orders of magnitude rather than only 80, we have to assume that the pair production of a neutrino and antineutrino is not spontaneous, but is caused by another neutrino.
April 19, 2024 at 1:25 pm |
Ha, ha, I got the two 20’s mixed up. The anti-symmetric cube of the 6 splits as 10a+10b, so this is where the Einstein field equations come from. The Riemann tensor comes from the adjoint SU(3,3): it is exactly the part of SU(3,3) that lies outside SO(3,3). So now I have exactly what is needed to generalise GR. In fact it is exactly what I said years ago, when using Cl(3,3) to classify the representations of SL(4,R), where again it was clear that the two 10’s in the Einstein field equations are not self-dual, but are dual to each other.
But now the embedding in E8 gives us another bombshell: the Einstein field equations (done properly) have massless solutions consisting of neutrinos, i.e. gravitational waves travelling at the speed of light.
April 19, 2024 at 5:39 pm |
I mean, the bombshell isn’t the gravitational waves per se, they were in GR from the beginning, the bombshell is that they consist of neutrinos.
April 19, 2024 at 3:19 pm |
Maybe Einstein had the notes in the right order to begin with, in 1911, when he proposed a variable speed of light before jumbling the notes with curved spacetime
April 19, 2024 at 3:30 pm |
Curved Woit-handed Spacetime to boot
April 19, 2024 at 3:37 pm
if spacetime is not actually curved, can it still be Woit-handed?
April 19, 2024 at 3:43 pm
I guess it can.
April 19, 2024 at 5:29 pm |
I don’t think Einstein thought in terms of curved spacetime. That seems to come from later “textbook” interpretations.
April 19, 2024 at 6:20 pm
But isn’t that what Riemannian geometry is about?
Seems to me that Einstein had two choices: either consider that the velocity of light was not constant (eg, varies significantly in the vicinity of a large mass like the sun) as he did in his first papers on gravitation OR that the velocity of light is constant but that spacetime is curved so that light follows a curved path in the vicinity of said mass.
Einstein chose the latter interpretation, but the two are really equivalent in many regards, except that (as Dicke suggested) the variable light speed interpretation also implies that G is a reflection of all the matter in the universe (ie, Machs principle)
April 20, 2024 at 2:45 am
I think you are on pretty safe grounds there, after all he tried his hands at teleparallelism
https://arxiv.org/abs/physics/0503046
In general, I think it’s safe to say he was far less orthodox about GR than later generation of theorists. He considered singularities as a clear breakdown and a strong indication that GR had to be regarded as provisional.
April 19, 2024 at 6:44 pm |
I just listened to the first two of Unzicker’s talks in the series that you posted above. I will listen to some more later, but first I want to point out what he actually said. “Space and time are different.” That’s what he said. That is what I am saying. Spacetime as invented by Minkowski in 1908 is a nonsense – that’s what Unzicker said, and that is what I say. Reducing the number of physical constants is what defines progress – that’s what he said, that’s what I say. I have written down loads and load of equations that reduce the number of arbitrary constants in particle physics. Does anyone listen? No. Maybe I need to try to talk to Unzicker. He is the only person I have ever heard talk sense about fundamental physics.
April 19, 2024 at 7:34 pm
I don’t disagree that space and time are different. It’s pretty clear that they are fundamentally different, despite the efforts to treat them on equal footing with Spacetime (either Minkowskian OR Euclidean)
The talk of imaginary time, which I consider to be nonsense, comes from the concept of Spacetime and an effort to make it Euclidean to get rid of the different sign on the time.
but although Einstein initially considered the work of Minkowski to be “mere mathematics”, he later adopted the concept of Spacetime in his development of GR.
some (even The likes of Paul Dirac) considered that to be a huge mistake.
April 19, 2024 at 7:49 pm
Yes, and yet it seems that Dirac used SL(2,C) as a double cover of SO(3,1). Or is that a misunderstanding of what Dirac actually did? The Dirac equation actually does not use SL(2,C), despite what the textbooks say. The Dirac gamma matrices actually generate Spin(3,2) = Sp(4,R). And they have nothing to do with spacetime.
I am starting to have a new respect for Dirac, whose ideas appear to have been completely mangled by the textbooks that the so-called giants of modern physics have been taught from.
April 19, 2024 at 9:27 pm
Another physicist who thinks lumping time and space together as “spacetime” was a big mistake is Julian Barbour.
in fact he is the one who highlighted Dirac’s comment to the same effect.
Barbour notes that Diracs comment made him completely rethink all of gravitational theory.
Barbour has also commented that Einstein “almost” implemented Machs ideas in GR.
im not clear specifically what he meant by that but I have little doubt that Barbour has thought more deeply about the fundamentals of GR and Machs principle than most physicists.
April 19, 2024 at 9:40 pm
Interesting. The wikipedia article on Julian Barbour is very revealing – it quotes Lee Smolin and Sean Carroll as “criticising” his ideas, but both criticisms are entirely content-free, and childish in the extreme.
April 19, 2024 at 9:48 pm
There seem to be different opinions about the extent to which Einstein adopted Mach’s Principle in GR, largely because there seems to be a wide range of ideas as to what Mach’s principle actually says. I seem to remember reading somewhere that Einstein himself was dissatisfied with GR because he did not feel that he had fully incorporated Mach’s Principle. I share this opinion, whether or not it was Einstein’s opinion. Yet I have received many emails in recent days from the MOND crowd asserting that GR does incorporate Mach’s Principle – but only in the high-acceleration regime, not the low-acceleration regime. Well, that to me means that Einstein did NOT incorporate Mach’s Principle into GR.
April 19, 2024 at 10:39 pm
I have a great deal of respect for Julian Barbour.
He is a very creative thinker.
He has made some controversial claims (eg, about time) but he backs them up with very clear, logical arguments.
and he doesn’t hide his basic ideas behind a blizzard of math, unlike lots of other physicists today.
April 20, 2024 at 2:51 pm
It’s unclear how GR could supposedly implement Machs principle in the high acceleration but not low acceleration regime (whatever high and low refer to)
it’s also not clear how it could “almost” implement Machs principle.
Seems like it either does or doesn’t.
April 20, 2024 at 2:54 pm
Yeah, I tend to agree. Einstein knew he hadn’t dealt with Mach’s Principle, and who are we to argue with Einstein?
April 19, 2024 at 7:11 pm |
In fact, I was first alerted to Unzicker’s work some three or four years ago by Mike McCulloch, who also included Unzicker in the email correspondence. But I think I wasn’t ready to appreciate Unzicker’s papers at that time, and he didn’t show any interest in mine!
April 19, 2024 at 7:53 pm
I get the impression that you think I am just disagreeing with everything you post here , but let me just say that that is not at all the case.
I honestly don’t know enough about the mathematics of group theory to weigh in on any of that.
but I do know some physics which is why I point out my disagreement on comments about the coriolis effect .
What I don’t disagree with is your comments regarding inertia , Machs principle and the like.
It’s actually related to some of the stuff the Unzicker talks about, but Carver Mead (a Cal Tech scientist) has proposed an alternative gravitational theory which he calls G4V that includes a vector potential in addition to the traditional scalar potential and allows him to get the correct predictions for all the tests of GR to date. He comments that it is essentially a Machian theory, with the main novelty being that it includes the vector contribution of the momenta of all the matter in the universe in the total gravitational potential in addition to the traditional scalar contribution.
April 19, 2024 at 8:05 pm
Meads theory does disagree with GR in one regard.
it predicts a different pattern for gravitational radiation (eg, from inward spiraling black holes)
I have tried to find out if there is any evidence to decide between GR and G4V with regard to the latter difference but have not been able to find anything.
doesn’t mean it doesn’t exist, of course.
not incidentally, like Unzicker Mead starts from Einstein’s early papers on gravity which assumed a variable c.
but Mead goes on to develop the vector potential for gravity, patterning it after the vector potential from E&M.
April 20, 2024 at 3:04 pm
you should contact Unzicker again.
He has expressed interest in quaternions but says he needs a mathematician to help him out
April 19, 2024 at 8:08 pm |
Not at all. Healthy disagreements are part of the normal progress of science. If they can be resolved by correcting errors, that is one thing. But if they can’t, then agreeing to disagree is always an option.
If I say Coriolis Effect when I mean Mach’s Principle, that is probably an error that can be corrected. But the fundamental principle that they both have to do with rotations on a larger scale than the immediate physics under discussion is undeniable. And that is where and why we ultimately need to consider the entire (observable) universe at once.
Carver Mead is another person whose unconventional ideas deserve respect, because his technological innovations prove beyond a shadow of doubt that he knows exactly what he is talking about.
April 20, 2024 at 2:48 am
Very happy to see this shout-out to Carver Mead. Always thought he had a lot of interesting ideas and outstanding physics intuition.
April 19, 2024 at 8:14 pm |
Gravitational radiation in GR comes from the spin 2 Weyl tensor. That is, in my opinion, an error, arising from the fundamental mistake of uniting space with time. In my model it is a spin (1,1) representation, not spin 2, and it is a tensor product of 3 dimensions of momentum with three generations of neutrinos.
April 19, 2024 at 8:39 pm
I could be wrong, but I believe Meads theory dispenses with the need for a spin 2 graviton because it is patterned after E&M which is mediated by photons which are spin 1.
one interesting thing that Mead points out , which was actually his motivation for developing the vector potential for gravity was one of Einstein’s early papers on gravitation in which Einstein commented that an initially stationary mass placed inside a spherical shell of matter should be caused to move by the movement of the shell around it.
Mead extends that analogy to the gravity probe B experiment for which he notes that one is justified in assuming that the probe is actually stationary and the earth is moving around it, effectively sweeping out a shell of matter, similar to Einstein’s though experiment.
April 19, 2024 at 8:49 pm
One might even say that one is required to assume the probe is stationary and the earth is moving around it.
April 19, 2024 at 8:28 pm |
Gravitational Caca-phony
Einstein’s notes
Got jumbled up
By famous blokes
Who jacked his stuff
Violins
And other strings
Sound like tins
And broken springs
April 19, 2024 at 8:38 pm
Yes, if you want to play the violin well, you have to learn at a young age. The same applies, I think, to group theory. Physicists who learn about strings and groups when they are no longer young enough for their brains to adapt, never learn to play these instruments properly.
And the thing about violins is, you need to know how to replace broken strings. “String theorists” have no idea how to replace broken strings. They don’t even know how to tune the strings, let alone fine tune them.
April 19, 2024 at 9:26 pm
Einstein, of course, did play the violin. Which is probably why he understood the difference between strings and elementary particles. Unlike some people I could name.
April 19, 2024 at 9:58 pm
Fundamental Things
“The universe of things
Is particles, I say
The particles make strings
It’s not the other way”
Why no, it’s really waves
Which slosh and move about
The particles are slaves
There’s really little doubt
April 19, 2024 at 10:14 pm
Newton said “it’s particles”
Witten says “it’s strings”
QM says “it’s farcical
To speak of waves as things”
April 19, 2024 at 10:32 pm
The time has come, the Walrus said,
To talk of many things:
Of groups, and waves, and QFT,
of particles and strings.
Of why the universe is hot,
And whether Higgs has wings.
April 19, 2024 at 11:57 pm
O Einstein said the Stringman
You’ve had a pleasant run!
Shall we be trotting home again?’
But answer came there none —
And this was scarcely odd, because he’d shot him with a gun”
April 20, 2024 at 12:11 am
And if I am not mistaken , Pink Floyd long ago answered the Walrus’ question with their song “Higgs on the Wing”
Saint Peter has wings
He uses to fly
To Heaven and sings
of Higgs in the sky
April 20, 2024 at 12:39 am
Oz never did give nothin to the Stringman, that he didn’t already have, and 2 never was the spin state of the graviton
Or the tropic of Sir Galahad
So, please believe in me
When I say I’m spinning ’round, ’round, ’round, ’round
Smoke glass stained bright colors
Image going down, down, down, down
Soap suds, green light bubble
April 20, 2024 at 2:50 am |
The upper limit for neutrino rest mass gets ever lower.
https://www.nature.com/articles/s41567-024-02461-9
April 20, 2024 at 2:28 pm |
Neutrinos can’t get no rest.
physicists are always bothering them
April 20, 2024 at 2:36 pm
The Neutrino’s Lament
Give me a rest
Neutrino said
Incessant test
will leave me dead
April 20, 2024 at 3:29 pm
The neutrino’s theme song
Games People Play (The Spinners)
April 20, 2024 at 3:43 pm
Can’t get no rest
Don’t know how I fly all day
When will I learn?
Physicists get in the way
I fly around
I can’t hear a sound, folks talkin’ loud, but I
Don’t see at all
I gotta get away, gotta get away, I don’t, Know where to go
It’s hopeless, I just wish they’d leave me alone
Games people play
Night or day
They’re just not stopping what they will do
Keeps me feelin’ blue
Been down too long
Right, wrong, I just can’t stop it
Spendin’ all day
Flying right through you
April 20, 2024 at 2:39 pm |
There’s no rest for the (wicked) neutrinos.
April 20, 2024 at 2:44 pm |
Neutrinos are really sinister (= left-handed).
April 20, 2024 at 2:49 pm
Woit thinks spacetime is dextrous. I think Woit is dextrous. Spacetime thinks Woit is Not Even Wrong. I think spacetime is Wrong, and so is Woit.
April 20, 2024 at 2:51 pm
Is Not Even Wrong the same as Odd Wrong?
April 20, 2024 at 5:42 pm
On Dexterity
Spacetime is dextrous
And so is P. Woit
His theory might vex us
But still, it’s adroit
April 20, 2024 at 5:48 pm
Spacetime Wars
Minkowski sang the song
That “Spacetime can’t be wrong”
Though Einstein had his doubt
The Spacetime was a rout
April 20, 2024 at 5:53 pm
Minkowski held the sway
And Spacetime rules the day
And space and time, it seems
Are figments of our dreams
April 20, 2024 at 5:54 pm
or is it nightmares?
Nightmare on Minkowski Street
April 20, 2024 at 5:56 pm
that’s what happens when you let mathematicians run amuck
April 20, 2024 at 5:57 pm
you get stuck in the muck
April 20, 2024 at 6:23 pm
Euclidean Spacetime
Minkowski’s bad enough
But Euclid struts his stuff
And time has gone awry
is multiplied by i
And real has gone complex
It really is a mess