A Three generations of Fermions from octonions Clifford alegbras

[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?