A new post with this title has appeared on Peter Woit’s blog, with his typically inane content that has very little to do with either symmetry or physics. He doesn’t allow comments from anyone who actually knows anything about symmetry, because they will show up the fact that he doesn’t know much about symmetry. So he has deleted three of my comments so far, and will no doubt continue deleting as many as I submit.

My main objection to what he has written is that he thinks all (interesting) representations of all (interesting) groups are unitary. The classification of representations into orthogonal, unitary and symplectic goes back to the last decade of the 19th century, and the underlying linear algebra is older still. Of these, the unitary ones are the least interesting. If you want to understand classical physics, you need orthogonal representations and groups, and if you want to understand quantum physics you need symplectic representations and groups.

It is the stupid belief of Woit and others that unitary representations and groups describe quantum physics that is the single most important reason why they have not made any progress in 50 years. It is no good Woit pontificating about the ills of string theory, when he is just as much a part of the problem as everyone else.

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August 10, 2022 at 3:18 pm |

Woit is really excited about his own work that uses SO(2,4) instead of SO(1,3) to describe spacetime, no doubt because the double cover is SU(2,2) so he can have unitary spinor representations. (That is, unitary in the mathematical sense, not the physical sense, which is more restrictive.) I’ve looked at it, and I know why it doesn’t do the things he’d hoped for – three generations, for example. It is because he is obsessed with unitary representations.

Einstein’s mass equation tells us what group to use – it is SO(1,4). No ifs, no buts, that is the actual group that actually describes the actual symmetry of the actual spacetime that we actually live in in this actual universe. And its double cover is symplectic, not unitary. Unitary representations give us a complex spacetime. Only symplectic representations can give us a real spacetime.

God, why are they so stupid?

August 10, 2022 at 3:42 pm |

I’ve said it before, and I’ll say it again. If you want three generations of elementary fermions, as experiment demands, then you have to have symplectic spinors. There is no choice. That is what the mathematics requires. If you insist on having unitary spinors, you will never ever find the three generations. It simply can’t be done. Wake up and smell the coffee. Listen to the mathematicians who understand this mathematics. Don’t censor them and ignore them. The universe can and will change. Experimental results can and will change. Mathematics cannot and will not change.

August 10, 2022 at 4:30 pm |

Woit continues to censor and ignore me. He will never understand the three generations of fermions, that I can predict with confidence.

August 10, 2022 at 7:00 pm |

A commenter on Woit’s blog asks about the distinction between “symmetries” and “invariances”. Woit rambles on semi-incoherently for several paragraphs without getting to grips with the issue.

The difference between symmetries and invariances is something that anyone who has ever taught linear algebra or group theory has to get straight. I wrote text-books on both, and I know how it is done. Woit doesn’t appear to know how it is done, and doesn’t appear to be interested in learning how it is done.

But you don’t need to be a mathematician or a physicist to understand the issue. A symmetry is an alibi, and an invariance is an alias. That’s all.

August 10, 2022 at 7:20 pm |

Gauge groups in physics provide aliases, not alibis. That is why those physicists who actually understand these things will insist that a gauge group is not a symmetry group.

Some physicists do not understand this, and will insist on trying to use the Lorentz group to provide alibis. The Lorentz group does not provide alibis, and if you try to use it in this way, it will twist and bend your spacetime. If you use it in the way Einstein used it, to provide aliases, your spacetime remains perfectly flat.

August 10, 2022 at 7:57 pm |

Of course, if spacetime is flat then the Strong Equivalence Principle is false. But this is confirmed by experiment beyond reasonable doubt, so that’s OK.

August 10, 2022 at 10:55 pm |

The idea that you can get three generations from the symplectic spinors of a spacetime with SO(4,1) symmetry sounds somewhat closer to mainstream theoretical physics than many of your other propositions. After all, SO(4,1) describes 4d de Sitter space, and “Families from Spinors” is a well-cited old paper by Wilczek and Zee.

But these spinor unification proposals are usually a kind of Kaluza-Klein theory, in which one has a giant spinor in higher-dimensional space-time, with enough components to contain all three standard model generations. Eric Weinstein (author of “Geometric Unity”) is the only person I can think of, who proceeded differently – he tried to define a spinor on the 14-dimensional bundle of metrics of 4d space-time (10 metrical degrees of freedom + 4 space-time degrees of freedom), from which he would obtain two generations, with the heavier third generation to arise as part of a Rarita-Schwinger field on that metric bundle.

August 11, 2022 at 6:22 am |

Yes, it sounds mainstream, until you look closely. It entangles the generation with the momentum, so you get neutrino oscillations in a changing gravitational field. It has two different masses, that transform differently under non-inertial transformations, so that the gravitational mass of the electron drifts away from its inertial mass (or vice versa). It explains the chirality of beta decay as a chirality of the motion of the experiment, and nothing to do with an “intrinsic” chirality (which is in any case a contradiction in terms).

I am familiar with the giant spinor paradigm, and to some extent I am working within it to try and make my ideas more acceptable. But the important question is why does the symmetry of the giant spinor break up so much? No-one ever considers that question, which I know how to answer: there is a finite group that controls all the symmetry. So I start with a giant spinor for so(12,4) coming from one of the various E_8 models, and break the symmetry from the Lie group to a finite group of order 120, after which I examine the debris. Which turns out to be the Standard Model extended to three generations.

August 11, 2022 at 7:10 pm

I gather that this will be a kind of “oction” model then.. I might defer further comment until I can see in more detail, what you’re doing with the available ingredients.

August 11, 2022 at 7:19 pm |

Yes, it is not entirely unrelated to “octions”. I throw the ingredients up in the air and re-assemble them in different ways until I think I see something interesting…

August 11, 2022 at 6:30 pm |

Just for fun, I tried to post the following comment to Woit’s blog, under his post on “Symmetry and physics”:

“Surely the most interesting question about symmetry and physics is how to understand the generation symmetry of the three generations of fermions? What does your approach have to say about this problem?”

I have no doubt that he will delete it, just as he deleted my other five attempts to post something relevant. He really doesn’t like questions he can’t answer.

August 12, 2022 at 2:47 pm |

Surprise, surprise, he’s deleted it. People don’t like questions that poke at their hidden assumptions.

August 12, 2022 at 8:28 am |

Since Woit is obviously not going to explain how symmetry really works in physics, I’d better do it instead. I’ll leave out all the discrete groups, and just tell you how all the classical and quantum fields work. OK?

For classical physics you need orthogonal groups, and for quantum physics you need their double covers, the spin groups. The orthogonal group SO(6,4) and the spin group Spin(6,4) are big enough to contain everything.

The compact part of Spin(6,4) is the gauge group of the Pari-Salam model, that contains the gauge group of the Standard Model, plus 9 extra dimensions for the 9 elementary particles of matter (3 electrons, 3 up quarks, 3 down quarks – the neutrinos are *not* matter, whatever particle physicists think). The usual interpretation of Pati-Salam is that they extend 3 colours to 4, but what the group actually does is add quantum numbers for the 3 generations – but without mass, so it’s hard to distinguish generations from colour.

To add mass, and all the 24 parameters of the Standard Model, you just have to add the 24 boosts of Spin(6,4). Now you need a Lorentz group Spin(3,1) or Spin(1,3) – the former fixes Spin(3,3), which I used to think was correct, while the latter fixes Spin(5,1), which I now believe is correct. Spin(5,1) has five boosts, which give masses to the five fundamental particles – proton, neutron, and three generations of electron. But you need a gauge group for mass, which must be Spin(1,1), leaving SO(4), so there are only four linearly independent masses here. There is also a quadratic form here, so there should be another mass formula – like the Koide formula, but with a correction for the proton/neutron mass difference.

The Lorentz group has complex (Weyl) spinors, but the Standard Model uses quaternionic (Dirac) spinors, to include mass, which means we have to extend the Lorentz group to Spin(1,4), generated by the Dirac gamma matrices. Now we have reduced the effective gauge group down to Spin(5), splitting into Spin(2) for proton/neutron and Spin(3) for 3 electrons, with the 6 cross-terms for the 6 quarks. That’s all you need for putting the 3 generations into the Standard Model.

So now I can return to classical physics, where the group is now SO(5) x SO(1,4). The classical fields belong to the adjoint representation of the gauge group (well, this is the rule in quantum field theory, so it must be true in classical field theory too). Adjoint SO(5) is 10-dimensional, of which 6 dimensions are electromagnetic, leaving 4 for gravity. NB 4, not 3. The extra dimension is required for corrections to Newtonian gravity. Adjoint SO(1,4) is also 10-dimensional. Special Relativity is the theory of electromagnetism you obtain by identifying adjoint SO(4) with adjoint SO(1,3). General Relativity ought to be the theory of gravity you obtain from the other four dimensions, by identifying vector SO(4) with vector SO(1,3).

But it isn’t. The reason is that GR uses the wrong representation of the wrong group: the 10-dimensional tensors (stress-energy, Ricci) are symmetric tensors on 4-dimensional spacetime, when they should be anti-symmetric tensors on 5-dimensional spacetime. The Bianchi identities reduce the 10 Einstein equations to 4, by removing the 6 electromagnetic terms (Maxwell’s equations). By doing so, all the boosts have been taken out, so that there is nothing left for gravity to do: it cannot make things move. There is also the far worse problem that symmetric tensors create singularities (black holes etc) that do not occur in anti-symmetric tensors.

So let’s do it properly: first let’s express electromagnetism as an identity between SO(4) on the left and SO(4) on the right. This incorporates the weak force and radioactivity into classical electromagnetism, and ensures that these forces have only `local’ effects, and leaves only gravity for the large-scale effects. Then gravity is an identification of the vector SO(4) on the left with the vector SO(4) on the right. This is a great idea, because it removes the catastrophic error of putting in an “isomorphism” between SO(4) and SO(1,3) – which are not isomorphic. This error pervades all of modern physics, and is an absolute disaster at all scales from the atomic nucleus to the entire universe.

So, anyway, gravity does 4 things – 3 of them are accelerations in 3 independent direction. What is the fourth? The symmetry group tells you: SO(4) is the symmetry group of mass-momentum, and acceleration (or, strictly speaking, the force) is a change in momentum. So the fourth dimension is a change in mass. This is the point where I am declared insane, but I really don’t understand why. All I have done is gone through the whole theory of fundamental physics, taking out the mistakes as I go. Moreover, the quantum mechanism for changing mass is well-known – it is called the weak force. There *must* be a classical mechanism for changing mass as well – it is the fourth dimension of gravity. And Einstein nearly got it, in the form of the Ricci scalar, or the cosmological constant. He would have got it, if he’d used Euclidean spacetime instead of Minkowski spacetime.

Woit is obsessed with Euclidean spacetime, and nearly gets it too. But he’s not there yet.

Anyway, electromagnetism in adjoint SO(4) is mediated by vectors, so that by the same token, gravity in vector SO(4) must be mediated by spinors, i.e. by neutrinos and anti-neutrinos. These cause changes in mass via the weak interaction, as well as changes in momentum which we call gravity. There’s just a couple more things I need to explain – one is the Riemann Curvature Tensor, and what is wrong with it, and the other is why dark matter had to be invented (to deal with the fact that the curvature tensor is wrong).

It is all to do with the difference between real matter described by SO(5), and “ordinary” matter described by SO(3) – that is, with only “ordinary” electrons, not the three generations. If we want to describe gravity in terms of ordinary matter alone, then we split the group as SO(3) x SO(3,4), and adjoint SO(3,4) has to be used to describe the rest of gravity. This is the Riemann Curvature Tensor – well, it isn’t but it should be. The RCT is a symmetric tensor on adjoint SO(1,3), identified with 6 of the 7 coordinates of the SO(3,4) vector, and is wrong for the same reason the Ricci tensor is wrong. It is also wrong because it assumes mass is a constant, which it isn’t.

Anyway, what this tensor does is to try to compensate for the fact that the three mass eigenstates of the electron have been reduced to one. In other words, it tries to create muons and tau particles out of electrons, because our model of stars and galaxies cannot see the muons and tau particles. So what it includes is the *difference* between a muon and an electron, and the *difference* between a tau particle and an electron. What it does not include is any charge, or any particle state, or any interactions except gravity. In other words, it creates “dark matter”. But the dark matter has no physical reality, it is a figment of physicists’ imagination designed to make up for the fact that they haven’t got enough imagination to imagine that (gravitational) masses of elementary particles are not necessarily constant.

OK, have you got that? That is how physics and symmetry *actually* work together.

August 12, 2022 at 10:29 pm |

I have seen many theories of everything, both mainstream and alternative. For me, this post marks the emergence of “your” theory of everything. A degree of mathematical and philosophical specificity has been reached which makes it concrete, comprehensive, and unique. In some sense I suppose it is an “E8 graviGUT”, but with so many unusual features that, when they are all taken into account, it might really be something else.

I do have one immediate question – when you talk about “break[ing] the symmetry from the Lie group to a finite group”, does that involve something like, considering the algebra over a finite field?

August 12, 2022 at 10:36 pm

(I mean the Lie algebra.)

August 13, 2022 at 6:50 am

I meant Lie group, not Lie algebra, and no, it is nothing like reducing to a finite field. What I do is take a suitable copy of the binary icosahedral group 2I inside the Weyl group of E8, and use that to split up the Lie algebra. What I get (in the enveloping algebra) is more or less the same as the complex group algebra of 2I. But in fact the real group algebra seems to have all the necessary structures – which are all orthogonal or symplectic, never unitary.

August 13, 2022 at 8:03 am

Thanks for the vote of confidence! But at the end of the day, I do not believe that this Spin(6,4) model is the real “theory of everything”. The splitting so(5) + so(1,4) is really a splitting of representations of the finite group 5+1+4, and the Einstein equations (or what they become) are equating the anti-symmetric squares of 5 and 1+4, not the adjoint representations of so(5) and so(1,4). Both these anti-symmetric squares split as 3a+3b+4, which gives the splitting into EM 3a+3b (photons) and gravity 4 = 2a x 2b (neutrinos x antineutrinos) at the fundamental level.

But it is impossible to explain all this at once, so I’m trying to write up the QFT version on its own, with one or two quantitative predictions if I can manage it.

The so(3) + so(3,4) splitting is also intriguing, because that describes the “ordinary” matter that we see everyday, and in the context of E8, so(3,4) splits as adjoint+vector G2. The importance of G2 is that it acts in the same way on vectors, LH spinors and RH spinors, so it sticks the classical particles (vectors) onto their quantum wave-functions (spinors). Hence we get wave-particle duality without having to give the particles and the waves separate ontological existence. Very important for the interpretation of QM. But there are also lots of different ways to split 6 into 3+3, so which one do we want to put into G2?! My vote is to split leptons from baryons, but this takes away our Lorentz group, which is not likely to be a popular move!

Some while ago I described how it is possible to use SL(3,R) in place of the Lorentz group, but it didn’t go down well. Perhaps in this more general context it might be taken more seriously: the basic idea is that one doesn’t need a concept of energy if one has a variable concept of mass, and if we have three generations of electrons then we have a variable concept of mass that can be quantised. We still have a vector representation of SL(3,R) which can replace the adjoint representation of the Lorentz group in all calculations. One half of the vector describes three generations (of negative charge) and the other half three colours (of positive charge). So the vector describes electromagnetism, and the adjoint describes how different observers can give different mass coordinates to the four basic charged particles, and still describe the same physical theory of electromagnetism. It’s exactly the same as the way the Lorentz group is used in Special Relativity, but extended to three generations of electrons.

August 12, 2022 at 3:20 pm |

Now I’ve seen red as far as Woit is concerned. Here is my latest salvo:

I am perplexed that when you write a blog post about symmetry, you refuse to allow an expert on symmetry to comment on it. I am a mathematician with a whole career of experience in symmetry, and what you say here is not correct. It is irresponsible of you to allow these false assertions to be perpetuated.

It simply is *not* mathematically obvious to go from a Lie algebra to a unitary representation of the algebra. It *is* obvious to go to a representation of the algebra, but it is *not* obvious that the representation should be unitary.

– Of course, he’ll still delete it. But the record is here for posterity to judge.

August 12, 2022 at 5:54 pm |

He’s referring to the algebra of operators on Hilbert space, that correspond to observable properties. Since the quantum state is projected onto an eigenstate of the operator, the operator needs to be unitary in order for the new state to still satisfy the Born interpretation of quantum probabilities.

You seem to be referring to the symmetries he introduces as classical, and those do not need to be unitary groups.

August 12, 2022 at 6:04 pm

Yes, I understand that. What I don’t understand is what that has to do with physics. With those assumptions, you can’t explain entanglement, and you can’t solve the measurement problem. Therefore it is fantasy, not physics.

August 12, 2022 at 6:11 pm

I mean, you can’t even explain the polarisation of light with that model.

August 12, 2022 at 6:13 pm

And I say this really seriously, because with symplectic representations you *can* explain the polarisation of light – I know because I’ve done it.

August 12, 2022 at 6:35 pm

Or, to put it another way, the unitary representations give you the Born probabilities, but the symplectic representations tell you what is actually happening.

August 12, 2022 at 7:18 pm

Just looking at the Dirac spinors, the Dirac algebra is a complex tensor product of two Dirac spinors, and is therefore a complex 16-dimensional space, but the complex structure contains no physics, and only 16 real dimensions contain observable properties. If you simply take the correct mathematical definition of Dirac spinors as quaternionic 2-spaces, and take the quaternionic tensor product instead, then you lose the irrelevant complex structure, but at the same time you gain something much more important – you gain a measurable consequence of a relationship between the spin directions of two particles. Neither spin direction can be measured, but the relationship between them can. This is a huge gain over Born probabilities.

August 12, 2022 at 5:12 pm |

This is war. To Woit:

“Perhaps, as someone who has recently had a paper on the applications of representations of Lie algebras to physics accepted by the Journal of Mathematical Physics, and nominated as “Editor’s Pick”, I might be allowed to comment on the applications of representations of Lie algebras to physics? No? I thought not. We certainly found that the assumption that these representations were unitary was not a useful assumption, and was not in fact used in most models since Dirac. The Dirac spinor itself is a symplectic representation of the Lie algebra generated by the gamma matrices, and this fact seems to be of some importance, at least as far as our paper was concerned.”

August 13, 2022 at 1:35 pm |

The most important point of all is that the symmetry group of Einstein’s mass equation is SO(1,4), not SO(1,3). Therefore any candidate unified theory *must* be covariant under this group. That is completely independent of whether it is a quantum theory or a classical theory.

In particular, mass in a classical theory of gravity *must* be treated as a variable, not a constant. Even today, no-one takes seriously the vast amount of experimental evidence that has been accumulating for decades, that proves beyond a shadow of doubt that mass is a variable. Its variations are detected here, there and everywhere.