Three generations of Fermions from octonions Clifford alegbras

[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?