Three generations of Fermions from octonions Clifford alegbras

[kodama]

interesting.

is the flavor of a particle in any way connected to they yukawa couplling of the particle to the Higgs field and three generations? is flavor similar to electric or color charge or mass? is the flavor of a first generation electron the same flavor as a first generation up or down quark, and if so why are their yukawa coupling differ (with the quarks being more massive than an electron)

 

[ohwilleke]

interesting.

is the flavor of a particle in any way connected to they yukawa couplling of the particle to the Higgs field and three generations?

Trivially, yes. Particle generation and type determine the Yukawa coupling of that particle. But there isn't any deeper formula except that higher generation particles of the same type are more massive than lower generation particles of the same type, in a magnitude which Koide's rule predicts exactly to the limits of experimental measurement for charged leptons.

is flavor similar to electric or color charge or mass?

Flavor, as noted above, is related to mass, but discrete (also due to the mass-energy relation E=mc², flavor is not the only source of mass which can also come, for example, from the energy of gluon fields in hadrons).

It is not similar to electric charge or color charge, which are perfectly conserved in the SM. Mass-energy is perfectly conserved (as is CPT), but flavor and mass are not perfectly conserved except incident to conservation of baryon number, lepton number, electromagnetic charge, and mass-energy.

 

[mitchell porter]

All of the fundamental particles in the Standard Model that interact via the weak force have rest mass, all of the fundamental particles in the Standard Model that don't interact via the weak force (i.e. photons and gluons) are massless. This would be true in a theory of everything that added a massless graviton as well. This suggests that fundamental particle mass is a weak force mediated process which the Higgs field mechanism of mass generation in the Standard Model basically is

There is a completely orthodox sense in which you could say that mass generation in the standard model is a "weak isospin mediated process" - which is very close to being weak-force-mediated (though not exactly the same), since weak isospin is the property that is gauged to produce the weak force.

What I'm referring to: the Higgs field has weak isospin and hypercharge - as do all standard model fermions - and the Higgs-yukawa interaction terms in the standard model, are precisely those combinations of Higgs field and fermion fields that are invariant under weak and hypercharge gauge transformations (it is a general principle of gauge theories that the allowed terms are all and only those field combinations which are invariant under the gauge group of the theory).

However, note that these Higgs-yukawa interaction terms, while containing fields charged under the gauge charges, do not contain the gauge fields themselves. The Higgs field "H" and the various fermions "psi", have weak quantum numbers, but the weak gauge field W does not make an appearance in the yukawa terms.

From a conventional field-theoretic perspective, if asked to explain why mass is distributed the way it is in the standard model, one might say: The default state in a gauge theory is for all the elementary fields to be massless, for reasons to do with renormalization. The Higgs mechanism is the one way to add mass at an elementary level in a gauge theory, and in the standard model, this involves particles with electroweak charges because the Higgs field of the standard model happens to carry electroweak charge, but this is a contingent thing; if the Higgs field had carried some other kind of charge, the yukawa couplings would have involved particles with that other charge.

From an unorthodox perspective: several times it has been suggested that the weak gauge field SU(2)_left can be combined with an SU(2)_right from the Ashtekar variables for gravity, into a single gauge field. It was suggested here, and it's also part of Woit's unification proposal. So one might be tempted to suppose that standard model mass always involves an SU(2) charge, because that SU(2) charge is somehow partnered with an SU(2) hidden in the gravitational field. The problem with this concept is that from a conventional perspective, you can couple gravity to any field theory and there's no need to have an SU(2) gauge field included. So probably this particular connection is just a coincidence.

 

[ohwilleke]

Graviweak Unification

Graviweak unification is also discussed in this previous PF thread that cites a number of additional sources. https://www.physicsforums.com/threa...tion-at-ilqgs-online-talk-by-marciano.670299/

Graviweak unification also competes with QCD squared approaches relating gravity in functional form and mathematical properties as a non-Abelian theory to QCD.

This said, unifying gravity and a SM force be it the weak force or the strong force, still doesn't really get you far on the path of explaining why there are exactly three generations of SM fermions.

Other Reasons To Think That There Are More Than Three SM Fundamental Fermion Generations

In addition to the W boson width v. top quark width argument for no more than three generations, there are a couple of strong experimental hints that there indeed aren't more than three generations of SM fermions for whatever the reason.

There is something to the extended Koide's formula which seems to apply quite accurately to the heavier quark masses and the charged leptons. If one extends the formula based upon recent data on the mass of the bottom and top quarks and presumes that there is a b', t, b triple, and uses masses of 173.4 GeV for the top quark and 4.190 GeV for the bottom quark, then the predicted b' mass would be 3.563 TeV (i.e. 3,563 GeV) and the predicted t' mass would be about 83.75 TeV (i.e. 83,750 GeV). These possibilities haven't been experimentally ruled out. But, iof the relationship between decay time for fundamental fermions and mass were extrapolated in any reasonable way to these masses, they would have decay times far shorter than that of the W boson that facilitates this process. Thus, the bar to fourth generation quarks is similar to the physics that prevents top quarks from hadronizing.

Of course, even if Koide's formula is not correct in this domain, it is suggestive of the kinds of masses for fourth generation quarks that one would expect and the estimated masses need not be very precise to give rise to the same conclusion.

Similarly, the extension of a Koide triple for charged leptons (to a muon, tau, tau prime triple) would imply a 43.7 GeV tau prime, which has been excluded at the 95% confidence level for masses of less than 100.8 GeV and with far greater confidence at 45.09 GeV and below, which would otherwise be produced at a significant and easy to measure frequency in Z boson decays and Higgs boson decays and have many other global implications for the SM.

And, since there are no active neutrinos of less than 45,090,000,00 eV, as determined by Z boson decays, and all three of the neutrino masses are very likely all under 0.1 eV (given cosmology limits), the likelihood of a fourth generation of SM neutrino more massive than that seems very low indeed.

So, since fundamental fermion generations, by virtue of symmetries in the equations of the SM, are all or nothing affairs and that one cannot have just three generations of quarks, while having four generations of leptons, for example, the non-detection of fourth generation leptons strongly disfavors the existence of fourth generation SM particles in general.

What are the second and third generations good for? Not much.

A world without second and third generation Standard Model fermions wouldn't be all that different from our own world. A world without third generation Standard Model fermions would be even less different and difficult to distinguish without advanced scientific equipment designed to look for them.

The longest lived of them is the muon at 1O^-6 seconds of mean lifetime. The tau lepton is much shorter lived. Hadrons containing second and third generation fermions are mostly shorter lived, or similar in mean lifetime. Particles, fundamental and composite alike, that last for a microsecond or less are never going to make up a large share of the universe by mass or by number of particles, vis-a-vis particles that are either stable (protons, bound neutrons, electrons, electron neutrinos), or metastable (a free neutron has a mean lifetime of about 15 minutes).

Charm quarks, bottom quarks, top quarks, and tau leptons all have masses of more than 1 GeV which takes lots of collider energy to produce on shell, and there are very few natural processes that give rise to them at any time in the last 13.6 billion years or so.

Strange quarks and muons take less energy to create and are created in nature, but still play a very minor role. Atomic nuclei would decay slightly different because kaons, which contain second generation quarks (strange quarks), are one of the many meson mediators of the residual strong force that binds nucleons into atomic nuclei that is http://www.scholarpedia.org/article/Nuclear_Forces. but to which kaons make a minor contribution (see also here). There are practical applications of muons that were developed in the last fifty years, but not really any natural processes involving muons that would rock our world if they didn't exist.

The differences between a world with three active neutrinos and just one kind of active neutrino would be almost impossible to discern without very advanced scientific equipment and a reason to look for more than one. It would tweak a few cosmology parameters every so slightly, but that isn't anything you could know without state of the art telescopes like Planck and an immense amount of computational power to apply to the data it collects.

Usually physics is reductionist. But, to explain the things that the Standard Model explains it was necessary to create a greatly more complex model of the Universe, which in the case of second and third generation fermions, the hadrons they create, and the CKM and PMNS matrixes that they necessitate, do very little to explain what is going on in Nature when we don't poke and prod it with particle accelerators, and have very little engineering applications (except for the muon).

We could have a perfectly sensible universe with an up quark, down quark, electron, electron neutrino, W boson, Z boson, gluons, photons, and a Higgs boson. Indeed, it is dirty little secret that a lot of BSM theories find a way to do just that while giving no serious thought other than hand waving to how to bring their theories from pre-falsified one generation of fermions theories to real life replicating three generations of fermion theories.

 

[ohwilleke]

A new preprint discusses a flavon theory of the three generations and discusses what would be necessary to detect it. I don't endorse it, but it is one explanation that is out there.

 

[kodama]

Is flavor the sole reason the top quark binds to the higgs field more strongly than a up quark?

does a tau and a top quark have the same flavor or different flavor, as both are same 3rd generation?

 

[ohwilleke]

A very different approach to three generations.

arXiv:2206.02557 (cross-list from physics.gen-ph) [pdf, ps, other]
The Three Faces of U(3)
J. LaChapelle

SU(3)×U(1) as a gauge symmetry is known to be phenomenologically problematic unless the U(1) factor lives in a "dark" or "hidden" sector. By contrast, U(3) is a semi-direct product group that is characterized by non-trivial homomorphisms mapping U(1) into the automorphism group of SU(3). Since the semi-direct product includes mutual coupling among all gauge field mass-energy, U(3) at least has a chance of being phenomenologically sensible as a gauge symmetry outside of any dark or hidden sector. For U(3), there are three different non-trivial homomorphisms that induce three separate defining representations. In a toy model of U(3) Yang-Mills (endowed with a suitable inner product) coupled to massive fermions, this renders three distinct covariant derivatives acting on a single matter field. By employing a mod3 permutation of the vector space carrying the defining representation induced by a "large" gauge transformation, the three covariant derivatives and one matter field can alternatively be expressed as a single covariant derivative acting on three distinct species of matter fields possessing the same U(3) quantum numbers but different renormalized masses. One can interpret this as three generations of matter fields.

Also, from the conclusion, is this little tidbit:

"We did not consider U(2) as a replacement for SU(2) × U(1), but off hand the same mechanism would appear to apply and it should be studied in the context of spontaneous symmetry breaking."

 

[mitchell porter]

the three covariant derivatives and one matter field can alternatively be expressed as a single covariant derivative acting on three distinct species of matter fields

Doesn't make sense to me. Just to be a little clearer about what's going on: instead of considering the usual product of gauge groups SU(3) x U(1) - which is arguably what you have in the standard model after the Higgs breaks the electroweak group - the author wants the gauge group to be the semi-direct product SU(3) ⋉ U(1). Part of making a gauge theory is to construct a covariant derivative based on the gauge group, and the author says there are three ways to do it for SU(3) ⋉ U(1), each of which involves a different representation of the group. (Anyone seriously checking whether this paper makes sense, should also see if the construction of a positive definite inner product in 2.1 also works, but I'll skip that for now.)

The author's idea seems to be that you can start with "one matter field" (as in the quote above) but have it interact with the SU(3) ⋉ U(1) gauge field in three different ways, corresponding to the three different representations. But how is that magically the same as having three matter fields? I don't get it. The way I see it, either it's equal to one matter field interacting with three copies of the gauge field (so the matter field has a different representation under each copy of the gauge group), or it's equal to one matter field interacting with one gauge field in a peculiarly redundant way.

edit: Maybe the idea could be, that the three generation copies of the matter field, would be three different ways that the matter field could be "dressed" with a cloud of virtual particles, according to the three different couplings?? ... More work is required to show that this makes QFT sense.

 

[ohwilleke]

A couple of new Octonion papers:

arXiv:2206.06911 [pdf, other]
An E8⊗E8 unification of the standard model with pre-gravitation, on an octonion-valued twistor space
Priyank Kaushik, Vatsalya Vaibhav, Tejinder P. Singh
Comments: 8 pages, 1 figure, to be presented at the conference 'When ℏ meets G', 27 June-1 July 2022, Institut d'Astrophysique Paris, this http URL
Subjects: High Energy Physics - Phenomenology (hep-ph)
We propose an E8⊗E8 unification of the standard model with pre-gravitation, on an octonionic space (i.e. an octonion-valued twistor space equivalent to a 10D space-time). Each of the E8 has in its branching an SU(3) for space-time and an SU(3) for three fermion generations. The first E8 further branches to the standard model SU(3)c⊗SU(2)L⊗U(1)Y and describes the gauge bosons, Higgs and the left chiral fermions of the standard model. The second E8 further branches into a right-handed counterpart (pre-gravitation) SU(3)grav⊗SU(2)R⊗U(1)g of the standard model, and describes right chiral fermions, a Higgs, and twelve gauge bosons associated with pre-gravitation, from which general relativity is emergent. The extra dimensions are complex and they are not compactified, and have a thickness comparable to the ranges of the strong force and the weak force. Only classical systems live in 4D; quantum systems live in 10D at all energies, including in the presently observed low-energy universe. We account for 208 out of the 496 degrees of freedom of E8⊗E8 and propose an interpretation for the remaining 288, motivated by the trace dynamics Lagrangian of our theory.

AND

[Submitted on 14 Jun 2022]

Octonion Internal Space Algebra for the Standard Model​

Ivan Todorov

Our search for an appropriate notion of internal space for the fundamental particles starts with the Clifford algebra Cℓ10 with gamma matrices expressed as left multiplication by octonion units times a pair of Pauli matrices. Fixing an imaginary octonion unit allows to write 𝕆=ℂ⊕ℂ3 reflecting the lepton-quark symmetry. We identify the preserved unit with the Cℓ6 pseudoscalar, ω6=γ1⋯γ6. It is fixed by the Pati-Salam subgroup of Spin(10), GPS=Spin(4)×Spin(6)/ℤ2, which respects the splitting Cℓ10=Cℓ4⊗̂ Cℓ6, while =12(1−iω6) is the projector on the 16-dimensional particle subspace (annihilating the antiparticles). We express the generators of the subalgebras Cℓ4 and Cℓ6 in terms of fermionic oscillators describing flavour and colour, respectively. The standard model gauge group appears as the subgroup of GPS that preserves the sterile neutrino (identified with the Fock vacuum). The ℤ2-graded internal space algebra  is then included in the projected tensor product: ⊂Cℓ10=Cℓ4⊗Cℓ06. The Higgs field appears as the scalar term of a superconnection, an element of the odd part, Cℓ14, of the first factor. As an application we express the ratio mHmW of the Higgs to the W-boson masses in terms of the cosine of the theoretical Weinberg angle.

Comments:	32 pages, Extended version of a lecture presented at the Workshop Octonions and the Standard Model, Perimeter Institute, Waterloo, Canada, February-May 2021, and at the 14th International Workshop Lie Theory and Its Applications to Physics (LT 14), Sofia, June 2021
Subjects:	High Energy Physics - Theory (hep-th); High Energy Physics - Phenomenology (hep-ph); Mathematical Physics (math-ph)
Report number:	IHES/P/22/01
Cite as:	arXiv:2206.06912 [hep-th]

 

[ohwilleke]

Is flavor the sole reason the top quark binds to the higgs field more strongly than a up quark?

does a tau and a top quark have the same flavor or different flavor, as both are same 3rd generation?

To state the obvious, in the SM, a particle's coupling strength of the Higgs field is proportional to its rest mass, and there are twelve SM rest masses that are derived from Higgs field interactions (six quark masses, three charged lepton masses, the W mass, the Z mass, and the Higgs mass).

For SM fundamental fermions, rest mass is a function of generation, but also of whether a particle is an up-type quark, down-type quark, charged lepton, or neutrino.

By definition, each of the twelve different kinds of SM fermions has a different flavor: "The Standard Model counts six flavours of quarks and six flavours of leptons."

So, a tau and a top quark have different flavors even though they in the same 3rd generation.

Since fermion rest mass in the SM is entirely and uniquely determined by flavor, it is fair to say that flavor is the sole reason that the top quark binds to the Higgs field more strongly than an up quark.

Generation v. Flavor In Quarks v. Charged Leptons

Also, while the tau in the SM is truly just a heavy electron or muon, which is identical in all respects except mass to the electron and muon (a SM rule which is being questioned by some experimental data from the LHC ), this simple description of the differences between generations isn't quite true in the case of the quarks.

Up quarks and charm quarks have the same electromagnetic and strong force properties and couplings as the top quark controlling for mass (and for that matter the same gravitational force properties and couplings as the top quark controlling for mass).

But the CKM matrix elements implicated the up quark, the charm quark, and the top quark, respectively, differ from each other in ways that can only be determined experimentally in the SM. Top quarks almost always transform into bottom quarks. Up quarks very infrequently transform to bottom quarks (even in the absence of conservation of mass-energy limitations), often transform into down quarks, and occasionally transform into strange quarks. Charm quarks most often transform into strange quark, but sometimes transform (mass-energy conservation permitting) into bottom or down quarks.

In contrast, the probability of transformations one one charged lepton into another charged lepton, or of a W or Z boson into a particular generation of charged lepton, are identical (i.e. there is charged lepton universality), even though this doesn't hold for transformations of one kind of neutrino into another generation of neutrino. There is also (as I understand it) a too small to observe SM correction to charged lepton universality predicted transition probabilities, at loop level, to account for indirect transformations of charge lepton flavor via virtual neutrinos, which could oscillate according to the PMNS matrix, mediated by virtual W bosons.

An Aside Re Neutrinos

The SM is agnostic in the source and nature of the neutrino masses. Indeed, there is some pedantic debate over whether neutrino mass is really part of the SM although from a practical perspective, most people would say that it is when they are talking about SM physics predictions.

If neutrinos did couple to the Higg field, they would have such a weak coupling that we would probably never observe Higgs boson decays to neutrinos even if they did, like other SM fundamental particles, have a coupling to the Higg field proportional to their rest mass. I'll illustrate this conclusion with the following back of napkin class calculations.

The branching faction of the charged leptons is for tau-lepton pairs, 6.27% (observed), for muon pairs, 0.021 8% (observed), and for electron-positron pairs, 0.000 000 5% (not yet observed).

Roughly speaking branching fractions ratios are on the same order of magnitude of the square of mass ratios.

Electrons are 511,000 eV v. something on the order of 0.050 eV for the largest neutrino mass eigenstate, a ratio of 10⁷.

This implies that if neutrinos got their mass via the same SM Higgs mechanism that applies to other SM fermions (whether or not that makes sense for other reasons), that a branching fraction from Higgs boson decays for neutrinos on the order of 10¹⁴ smaller than that of the roughly 5*10^-7 for electrons.

This would imply a Higgs boson branching fraction on the order of not more than 10^-21 for any kind of neutrino, when muon pairs with a branching fraction of about 2*10^-2 are at the current experimental detection threshold.

In addition, a smaller proportion of neutrinos passing through a detector are actually seen than a proportion of charged leptons passing through a detector, since neutrinos interact more weakly with other matter, so a Higgs boson decay into neutrinos would not be possible at the same branching fraction that a charge lepton Higgs boson decay can be detected.

Realistically, detector precision would have to improve by a factor of about 10²² or more over current technology to directly observe neutrino decays from Higgs bosons in a statistically significant way.

I don't expect that to happen anytime during the lives of anyone who ever encounters me alive.

 

[kodama]

arXiv:2206.06911 [pdf, other]
An E8⊗E8 unification of the standard model with pre-gravitation, on an octonion-valued twistor space
Priyank Kaushik, Vatsalya Vaibhav, Tejinder P. Singh

I wonder if some variation of this idea could be used with Woit's 4D Euclidean Twistor space.

 

[kodama]

so what determine the flavor of a particle. and how is flavor tied in with generations, since the second and third generation are heavier than first.

 

[mitchell porter]

arXiv:2206.06911 [pdf, other]
An E8⊗E8 unification of the standard model with pre-gravitation, on an octonion-valued twistor space
Priyank Kaushik, Vatsalya Vaibhav, Tejinder P. Singh

I wonder if some variation of this idea could be used with Woit's 4D Euclidean Twistor space.

This paper is almost an ultimate development of trends seen earlier in this thread... One of the contrasts between the mainstream of theory, and the alternatives we are discussing in this thread, is that the mainstream theories may not describe reality, but they are at least mathematically well-defined as full quantum theories; whereas these alternative theories involve wishful thinking, even just at the mathematical level.

Here one should also note a curious superficial similarity with one of the best-known approaches to a theory of everything, the E8xE8 heterotic string in ten dimensions. In this theory, one also has E8xE8 and ten dimensions, but the hoped-for mechanisms are almost completely different.

For the sake of comparison, let me say a bit about how the standard model and gravity were to be obtained from the heterotic string... First of all, in that model, gravity has nothing to do with E8xE8, it comes from a separate gravitational sector of string states. E8xE8 is the gauge sector of the string, all the observed gauge fields must come from within this. But the heterotic string also has N=1 supersymmetry, meaning that there are fermionic superpartners to those E8xE8 gauge bosons, and the observed fermions must come from among those. Finally, an effect associated with compactification reduces one of those E8s to E6, one of the known candidate groups for grand unification. So in the field theory limit, you end up with a GUT with E6 symmetry group and representations (which then must be higgsed down to the standard model), and a second, unbroken E8 sector which might contribute to dark matter. This was the high hope of unification in the late 1980s (e.g. it's the topic of Brian Greene's PhD thesis), before string theorists came up with several other approaches (braneworlds, F-theory), and before the LHC threw the usual conceptions of supersymmetry into doubt.

OK, so what's going on in this paper? It would take quite a few hours of work to chase down everything they are saying, what follows is just what I can make out already.

First, this paper is the work of Tejinder Singh's group in India. I already cited another paper of theirs in comment #25 in this thread. Their work is distinctive and ambitious enough to warrant being singled out. They're trying to do everything from explain quantum mechanics per se (via a version of Adler's "trace dynamics"), to explaining the actual particle masses (see #25).

So, what are they doing with E8xE8? I think at heart it's similar to the chiral graviweak unification mentioned in #38-#39 in this thread. This is the idea that the weak force and gravity can be described together by an SU(2)_L x SU(2)_R gauge theory. The first SU(2) is the weak force, the second SU(2) is gravity in Ashtekar variables... Here, Singh et al want to get SU(3)xSU(2)xU(1) from one E8, and gravity from the other E8. So this is the first significant divergence from string theory's use of E8xE8: in string theory, gravity has a separate origin.

Part II of this paper is called "Branching of E8 x E8", so this is where their plans for the group are described in detail. One E8 is broken to an SU(3) symmetry among generations, times the standard model gauge group. The other one is broken the same way, except that SU(3)xSU(2)xU(1) is here interpreted as a weird "pre-gravitational" gauge group. There's supposed to be an SU(3) of "gravi-gluons" which only matter for gravity on a subatomic scale, and a "U(1)_g" which apparently gives rise to ordinary gravity but I don't see how.

I also don't see where they get their fermions from. In the first paragraph of the paper they say "An important aspect of E8 is that the adjoint representation is the fundamental representation, therefore we are able to write the fermions and the bosons using the same rep". Well, it's true enough that E8's adjoint rep is also its fundamental rep; but a bosonic field is still a different thing from a fermionic field. That's why the heterotic string uses E8 superfields, with fermionic and bosonic components. (This was also a problem of Garrett Lisi's E8 theory: he tried to get bosons and fermions from the same E8 gauge field, but if the field components have different particle statistics, it's no longer an E8 object but instead something simpler.) Singh et al have a section on triality and maybe they think they get fermions that way somehow, but I don't see it, and in general, it seems to be another example (see comment #9) in which alleged "fermions" aren't, and even can't be, true fermions.

There are other peculiar details, e.g. everything is ultimately made of 2-branes in octonionic twistor space, and there's more numerology relating masses to the degrees of gauge groups. I return to my introductory comment, that a theory like this involves a lot of mathematical wishful thinking. Rather than exhibiting a mathematical framework clear enough to be predictive, instead we are given pieces of a jigsaw puzzle, with the hope that the pieces can be assembled into a single well-defined object, while retaining all their desired properties. There's no law against this kind of theorizing, but in this case, the large number of peculiar propositions and leaps of faith make it apriori very unlikely that the whole thing really does hang together.

 

[kodama]

So, what are they doing with E8xE8? I think at heart it's similar to the chiral graviweak unification mentioned in #38-#39 in this thread. This is the idea that the weak force and gravity can be described together by an SU(2)_L x SU(2)_R gauge theory. The first SU(2) is the weak force, the second SU(2) is gravity in Ashtekar variables... Here, Singh et al want to get SU(3)xSU(2)xU(1) from one E8, and gravity from the other E8. So this is the first significant divergence from string theory's use of E8xE8: in string theory, gravity has a separate origin.I also don't see where they get their fermions from. In the first paragraph of the paper they say "An important aspect of E8 is that the adjoint representation is the fundamental representation, therefore we are able to write the fermions and the bosons using the same rep".

There are other peculiar details, e.g. everything is ultimately made of 2-branes in octonionic twistor space, and there's more numerology relating masses to the degrees of gauge groups. I return to my introductory comment, that a theory like this involves a lot of mathematical wishful thinking. Rather than exhibiting a mathematical framework clear enough to be predictive, instead we are given pieces of a jigsaw puzzle, with the hope that the pieces can be assembled into a single well-defined object, while retaining all their desired properties. There's no law against this kind of theorizing, but in this case, the large number of peculiar propositions and leaps of faith make it apriori very unlikely that the whole thing really does hang together.

Do you find Peter Woit's derivation of fermionic fields in his proposal to be satisfactory?

The reason I ask is that this proposal makes use of twistor space and so does Woits, both are Twistor.

Could the twistor theory of Woit theory be some way combined with the use of octonionic twistor space in this paper?

Maybe some combination of Woit's proposal with some form of octonions to get 3 generations, in twistor space.

 

[mitchell porter]

Do you find Peter Woit's derivation of fermionic fields in his proposal to be satisfactory?

The reason I ask is that this proposal makes use of twistor space and so does Woits, both are Twistor.

Could the twistor theory of Woit theory be some way combined with the use of octonionic twistor space in this paper?

Maybe some combination of Woit's proposal with some form of octonions to get 3 generations, in twistor space.

Well, first let's review what's involved in making a fermion field. I can only think of two ways to do it. One is that you take an appropriate classical quantity, treat it as an operator, and then postulate that it is anticommuting. This is standard textbook quantization. The other approach is to have a classically anticommuting quantity and do a path integral over it. This is the approach that requires Grassmann numbers.

Woit wants to start in Euclidean space, and then obtain formulae appropriate for our empirical Minkowski space-time, through analytic continuation of the Euclidean formulae. OK, that's a standard thing, except that in the case of fermion fields, there are technical problems and Woit doesn't like the standard approach to them - see appendices B.2 and B.3 of his 2021 paper. Then on top of this, Woit wants to work in Euclidean twistor space, and you want him to try the octonionic version of this. These seem to be domains where even the basics of quantum field theory haven't been studied much. But at least they are well-defined. The most efficient course of action may be to ask him directly.

I will mention that in mainstream theory, there are objects called supertwistors, which are twistors with some Grassmann components. This must be the standard way to get a fermion field from a twistor path integral. But I don't know if it's compatible with Woit's construction or philosophy.

 

[kodama]

Well, first let's review what's involved in making a fermion field. I can only think of two ways to do it. One is that you take an appropriate classical quantity, treat it as an operator, and then postulate that it is anticommuting. This is standard textbook quantization. The other approach is to have a classically anticommuting quantity and do a path integral over it. This is the approach that requires Grassmann numbers.

Woit wants to start in Euclidean space, and then obtain formulae appropriate for our empirical Minkowski space-time, through analytic continuation of the Euclidean formulae. OK, that's a standard thing, except that in the case of fermion fields, there are technical problems and Woit doesn't like the standard approach to them - see appendices B.2 and B.3 of his 2021 paper. Then on top of this, Woit wants to work in Euclidean twistor space, and you want him to try the octonionic version of this. These seem to be domains where even the basics of quantum field theory haven't been studied much. But at least they are well-defined. The most efficient course of action may be to ask him directly.

I will mention that in mainstream theory, there are objects called supertwistors, which are twistors with some Grassmann components. This must be the standard way to get a fermion field from a twistor path integral. But I don't know if it's compatible with Woit's construction or philosophy.

I would love to hear you and Woit discuss this, specifically earlier papers here on octonions and the standard model, octonions and 3 generations, octonionic version of twistor theory, and whether Woit Euclidean Twistor Unification can incorporate these hypothesis mathematically. also discussion on supertwistors and Euclidean Twistor Unification but Woit isn't much of a fan of supersymmetry.

btw in addition to Woit, if you know Priyank Kaushik, Vatsalya Vaibhav, Tejinder P. Singh, perhaps you can refer them to Woit and see if they can come up with a combination theory

for those who are new

Euclidean Twistor Unification​

Peter Woit

Taking Euclidean signature space-time with its local Spin(4)=SU(2)xSU(2) group of space-time symmetries as fundamental, one can consistently gauge one SU(2) factor to get a chiral spin connection formulation of general relativity, the other to get part of the Standard Model gauge fields. Reconstructing a Lorentz signature theory requires introducing a degree of freedom specifying the imaginary time direction, which will play the role of the Higgs field.
To make sense of this one needs to work with twistor geometry, which provides tautological spinor degrees of freedom and a framework for relating by analytic continuation spinors in Minkowski and Euclidean space-time. It also provides internal U(1) and SU(3) symmetries as well as a simple construction of the degrees of freedom of a Standard Model generation of matter fields. In this proposal the theory is naturally defined on projective twistor space rather than the usual space-time, so will require further development of a gauge theory and spinor field quantization formalism in that context.

Comments:	48 pages, 1 figure
Subjects:	High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc)
Cite as:	arXiv:2104.05099 [hep-th]

arXiv:2206.06911 [pdf, other]
An E8⊗E8 unification of the standard model with pre-gravitation, on an octonion-valued twistor space
Priyank Kaushik, Vatsalya Vaibhav, Tejinder P. Singh

I wonder if some variation of this idea could be used with Woit's 4D Euclidean Twistor space.

 

[ohwilleke]

I would love to hear you and Woit discuss this, specifically earlier papers here on octonions and the standard model, octonions and 3 generations, octonionic version of twistor theory, and whether Woit Euclidean Twistor Unification can incorporate these hypothesis mathematically. also discussion on supertwistors and Euclidean Twistor Unification but Woit isn't much of a fan of supersymmetry.

btw in addition to Woit, if you know Priyank Kaushik, Vatsalya Vaibhav, Tejinder P. Singh, perhaps you can refer them to Woit and see if they can come up with a combination theory

for those who are new

Euclidean Twistor Unification​

Peter Woit

Comments:	48 pages, 1 figure
Subjects:	High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc)
Cite as:	arXiv:2104.05099 [hep-th]

arXiv:2206.06911 [pdf, other]
An E8⊗E8 unification of the standard model with pre-gravitation, on an octonion-valued twistor space
Priyank Kaushik, Vatsalya Vaibhav, Tejinder P. Singh

I wonder if some variation of this idea could be used with Woit's 4D Euclidean Twistor space.

Woit has a blog (Not Even Wrong) that take comments where would could ask in a post about his Twistor work, and he also has a public academic email. Doesn't hurt to ask.

 

[kodama]

Woit has a blog (Not Even Wrong) that take comments where would could ask in a post about his Twistor work, and he also has a public academic email. Doesn't hurt to ask.

Actually I received personal messages from a theoretical physicist on just this idea, and I told him exactly what you said, contact Woit, Priyank Kaushik, Vatsalya Vaibhav, Tejinder P. Singh et al, specifically he expressed considerable research interest in combining octionions papers with Woit's Euclidean Twistor Unification, octononic Twistor theory which has the standard model, with Woit's specific proposal on fermions in his proposal.

there's a lot of overlap in these 2 proposals including use of twistor space and SU(2) Ashketar variables for gravity. there are differences to be sure.

the paper

An E8⊗E8 unification of the standard model with pre-gravitation, on an octonion-valued twistor space​

"The extra dimensions are complex and they are not compactified, and have a thickness comparable to the ranges of the strong force and the weak force."

"Only classical systems live in 4D; quantum systems live in 10D at all energies, including in the presently observed low-energy universe"

https://arxiv.org/abs/2206.06911

so extra 6 dimensions are described in terms of imaginary numbers.

maybe string theory could try this as well.

I told him it'd be great to see a paper from him about this.

 

[mitchell porter]

there's a lot of overlap in these 2 proposals including use of twistor space and SU(2) Ashtekar variables for gravity. there are differences to be sure.

The Singh et al paper is a bizarre sprawling patchwork compared to Woit's (see my #48 in this thread). Woit may want to interpret it in a way that is ultimately untenable, but at least his paper is based on a mathematical object that ought to make sense, spinors in Euclidean twistor space; and so it's not a big leap to ask about Euclidean spinors for the "octonionic" 10d twistor space, and what the counterpart of his "one-generation" object (see pages 11-12 of "Euclidean Twistor Unification") is for that larger twistor space, and whether it is now "three generations".

 

[kodama]

The Singh et al paper is a bizarre sprawling patchwork compared to Woit's (see my #48 in this thread). Woit may want to interpret it in a way that is ultimately untenable, but at least his paper is based on a mathematical object that ought to make sense, spinors in Euclidean twistor space; and so it's not a big leap to ask about Euclidean spinors for the "octonionic" 10d twistor space, and what the counterpart of his "one-generation" object (see pages 11-12 of "Euclidean Twistor Unification") is for that larger twistor space, and whether it is now "three generations".

yes that was what I was asking, whether an octonic version of Woit's proposal would contain 3 generations based on earlier papers in this thread. Not sure about using E8 since the paper acknowledges many particles unobserved.
The Singh paper states the 6 dimensions are complex, 4 real, so apparently does not need to be compactified. Do you really think 6 dimensions described by imaginary numbers is physical?

Could you have octonic twistor space in only 4 dimensions, with the earlier papers expanding 1 generation to 3?

BTW John Baez who I've seen here also works on octonions and the standard model, and has worked on Ashketar variables. He might also be worth contacting.

 

[kodama]

Submitted on 9 Jun 2022]

Gauge flavour unification: from the flavour puzzle to stable protons​

Joe Davighi

The idea of unification attempts to explain the structure of the Standard Model (SM) in terms of fewer fundamental forces and/or matter fields. However, traditional grand unified theories based on SU(5) and Spin(10) shed no light on the existence of three generations of fermions, nor the distinctive pattern of their Yukawa couplings to the Higgs. We discuss two routes for unifying the SM gauge symmetry with its flavour symmetries: firstly, unifying flavour with electroweak symmetries via the group SU(4)×Sp(6)L×Sp(6)R; secondly, unifying flavour and colour via SU(12)×SU(2)L×SU(2)R. In either case, all three generations of SM fermions are unified into just two fundamental fields. In the larger part of this proceeding, we describe how the former model of `electroweak flavour unification' offers a new explanation of hierarchical fermion masses and CKM angles. As a postscript, we show that gauge flavour unification can have unexpected spin-offs not obviously related to flavour. In particular, the SU(12)×SU(2)L×SU(2)R symmetry, when broken, can leave behind remnant discrete gauge symmetries that exactly stabilize protons to all orders.

Comments:	7 pages. Contribution to the proceedings of "La Thuile 2022, Les Rencontres de Physique de la Vallée d'Aoste"
Subjects:	High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Theory (hep-th)
Cite as:	arXiv:2206.04482 [hep-ph]

 

[mitchell porter]

I am still in the process of learning enough about twistor basics to understand Woit's construction. However, I see at least two pathways to generalizing it.

One step towards twistor space is to represent points in Minkowski space as certain 2x2 matrices. For the usual 4d twistor, these are complex matrices, but for the 10d "twistor", the matrices need to be octonionic. So that's the starting point for one way to mimic the construction with octonions.

As for the other... I think I mentioned that the variables which transform like a generation, are bosonic in Woit's present construction. That's a problem, but perhaps if those variables had superpartners, then those partner variables would still have the same transformation properties, while now being fermions. Of course you'd still have the original bosonic variables too, so, squarks and sleptons along with your quarks and leptons. But, one step at a time.

I mentioned supertwistors. A supertwistor is just an ordinary bosonic twistor, with some fermionic variables added. Maybe a d=4 supertwistor with n fermionic variables will give you n fermionic "Woit generations". Maybe a d=10 N=1 octonionic supertwistor can give you several Woit generations in 4 dimensions; maybe there's some relationship to d=4 N=4 super-Yang-Mills, which can be obtained from d=10 N=1 super-Yang-Mills (I am sure I have seen deformations of the d=4 N=4 theory in which there are three fermion generations).

 

[kodama]

I am still in the process of learning enough about twistor basics to understand Woit's construction. However, I see at least two pathways to generalizing it.

One step towards twistor space is to represent points in Minkowski space as certain 2x2 matrices. For the usual 4d twistor, these are complex matrices, but for the 10d "twistor", the matrices need to be octonionic. So that's the starting point for one way to mimic the construction with octonions.

As for the other... I think I mentioned that the variables which transform like a generation, are bosonic in Woit's present construction. That's a problem, but perhaps if those variables had superpartners, then those partner variables would still have the same transformation properties, while now being fermions. Of course you'd still have the original bosonic variables too, so, squarks and sleptons along with your quarks and leptons. But, one step at a time.

I mentioned supertwistors. A supertwistor is just an ordinary bosonic twistor, with some fermionic variables added. Maybe a d=4 supertwistor with n fermionic variables will give you n fermionic "Woit generations". Maybe a d=10 N=1 octonionic supertwistor can give you several Woit generations in 4 dimensions; maybe there's some relationship to d=4 N=4 super-Yang-Mills, which can be obtained from d=10 N=1 super-Yang-Mills (I am sure I have seen deformations of the d=4 N=4 theory in which there are three fermion generations).

For the usual 4d twistor, these are complex matrices, but for the 10d "twistor", the matrices need to be octonionic. So that's the starting point for one way to mimic the construction with octonions.Could you use the other papers on octionions and the standard model to get 3 generations?
How would you deal with the extra 6 dimensions to get the 4 we observe?

That's a problem, but perhaps if those variables had superpartners, then those partner variables would still have the same transformation properties, while now being fermions. Of course you'd still have the original bosonic variables too, so, squarks and sleptons along with your quarks and leptons. But, one step at a time.

since SUSY hasn't been observed and requires SUSY breaking, and apparenlty Woit's claim is you get both SU(3)xSU(2)xSU(1) and his fermions without SUSY, isn't his proposal simpler?

btw is this a paper you plan to write? perhaps you could also email John Baez, esp octonions and the standard model

https://math.ucr.edu/home/baez/standard/

 

[mitchell porter]

since SUSY hasn't been observed and requires SUSY breaking, and apparenlty Woit's claim is you get both SU(3)xSU(2)xSU(1) and his fermions without SUSY, isn't his proposal simpler?

He does not have fermions. He doesn't have gauge bosons either, for that matter. He has a geometric object ("the space of linear maps from C^4 to itself") which has the group transformation properties of a generation, if you decompose it in a certain way; and he has a fiber bundle ("at each point on PT, a principal bundle...") valued in the standard model group. To get the standard model, he has to somehow turn these objects into quantum fields with the right statistics (fermion, boson), in a way which doesn't spoil their original identity in terms of twistor space. (This is one of the problems of Lisi's E8 theory, that if you treat some elements of E8 as fermions and others as bosons, as required if it's going to give you the standard model - then you no longer have E8.)

btw is this a paper you plan to write?

This is just a discussion of Woit's idea, how it can be generalized, and whether it ultimately makes sense. The odds are that almost nothing in this thread corresponds to a well-formed quantum theory. (If you want to know what a well-formed quantum theory looks like... it's all those boring papers that mainstream physicists keep producing, in which they postulate particles which are then not observed.) The good thing about Woit's particular idea is that it's based on something concrete. well-defined, and already known to be physically meaningful - twistor space. So it's a little more interesting to investigate than many of the others.

 

[kodama]

He does not have fermions. He doesn't have gauge bosons either, for that matter. He has a geometric object ("the space of linear maps from C^4 to itself") which has the group transformation properties of a generation, if you decompose it in a certain way; and he has a fiber bundle ("at each point on PT, a principal bundle...") valued in the standard model group. To get the standard model, he has to somehow turn these objects into quantum fields with the right statistics (fermion, boson), in a way which doesn't spoil their original identity in terms of twistor space. (This is one of the problems of Lisi's E8 theory, that if you treat some elements of E8 as fermions and others as bosons, as required if it's going to give you the standard model - then you no longer have E8.)

This is just a discussion of Woit's idea, how it can be generalized, and whether it ultimately makes sense. The odds are that almost nothing in this thread corresponds to a well-formed quantum theory. (If you want to know what a well-formed quantum theory looks like... it's all those boring papers that mainstream physicists keep producing, in which they postulate particles which are then not observed.) The good thing about Woit's particular idea is that it's based on something concrete. well-defined, and already known to be physically meaningful - twistor space. So it's a little more interesting to investigate than many of the others.

what is needed to turn these objects into quantum fields with the right statistics in a way which doesn't spoil their original identity in terms of twistor space?


could you write a paper that does just this, namely put these objects as QFT ?

 

[mitchell porter]

could you write a paper that does just this, namely put these objects as QFT ?

The essential question is just, can it be done at all? If it can't be done, such a paper can't be written, either.

The core thing to understand here is Woit's construction. I'm happy to report minor progress in that direction. I'll refer to slide 30 from his most recent talk.

Putting aside for now the question of how he will make this a fermionic object, in his concept, a single standard-model generation corresponds to a map from C^4 to C^4. Algebraically, that's just a 4x4 complex matrix, with 16 entries. So, as in SO(10) unification, the "fermion cube", etc, the particles of a generation (including a right-handed neutrino) are organized into a single object with 16 degrees of freedom.

Realizing that is like breathing a sigh of relief. This is familiar territory. The next challenge is to understand how he motivates 4x4 complex matrices, and the desired transformation properties, in terms of twistor space.

If you look at slide 30 again, the 4x4 matrices seem to describe homomorphisms ("Hom") from one decomposition of twistor space to another. First is C x C^3, decomposing the space of complex 4-vectors into "complex lines" ... each line being made of the complex multiples of some specific 4-vector. This decomposition is used in twistor theory to construct projective twistor space, PT... Second is S_R + S_L (see his slide 6), the bispinor representation of complex vectors, which can itself be thought of as a 2x2 complex matrix mapping from one spinor to the other.

Maybe the 4x4 complex matrices can be thought of as a twistor-valued field on twistor space? e.g. a "bispinor-valued field" on a "PT of complex lines"... And then the transformations corresponding to the standard model symmetry groups, would be specific rotations, etc, in the space of field values or the geometric space.

So I don't definitely have it right yet, but I can see how this could be a kind of field theory. The remaining challenges are then (1) quantizing it, apparently one would face twistorial versions of the challenges of Euclidean field theory (2) getting the "generation" to be fermionic rather than bosonic (3) getting three generations. Regarding (3), Woit does mention a potential octonionic generalization in his talk, see slide 31.

 

[kodama]

The essential question is just, can it be done at all? If it can't be done, such a paper can't be written, either.

The core thing to understand here is Woit's construction. I'm happy to report minor progress in that direction. I'll refer to slide 30 from his most recent talk.

Putting aside for now the question of how he will make this a fermionic object, in his concept, a single standard-model generation corresponds to a map from C^4 to C^4. Algebraically, that's just a 4x4 complex matrix, with 16 entries. So, as in SO(10) unification, the "fermion cube", etc, the particles of a generation (including a right-handed neutrino) are organized into a single object with 16 degrees of freedom.

Realizing that is like breathing a sigh of relief. This is familiar territory. The next challenge is to understand how he motivates 4x4 complex matrices, and the desired transformation properties, in terms of twistor space.

If you look at slide 30 again, the 4x4 matrices seem to describe homomorphisms ("Hom") from one decomposition of twistor space to another. First is C x C^3, decomposing the space of complex 4-vectors into "complex lines" ... each line being made of the complex multiples of some specific 4-vector. This decomposition is used in twistor theory to construct projective twistor space, PT... Second is S_R + S_L (see his slide 6), the bispinor representation of complex vectors, which can itself be thought of as a 2x2 complex matrix mapping from one spinor to the other.

Maybe the 4x4 complex matrices can be thought of as a twistor-valued field on twistor space? e.g. a "bispinor-valued field" on a "PT of complex lines"... And then the transformations corresponding to the standard model symmetry groups, would be specific rotations, etc, in the space of field values or the geometric space.

So I don't definitely have it right yet, but I can see how this could be a kind of field theory. The remaining challenges are then (1) quantizing it, apparently one would face twistorial versions of the challenges of Euclidean field theory (2) getting the "generation" to be fermionic rather than bosonic (3) getting three generations. Regarding (3), Woit does mention a potential octonionic generalization in his talk, see slide 31.

how difficult are points 1- 2 to overcome?

for points 1-2 have you thought about asking a Twistor expert, or maybe recruit someone like John Baez to address these issues? I know there's Roger Penrose and Witten and M Atiyah also worked on twistors. Perimeter Institute has Simone Speziale.

for point 3, could you get 3 generations work via octonions and prior papers cited in this thread?

btw is this something you plan to work on and write a paper for?