## Saturday, September 18, 2021

### Quantum Field Theory And Quantum Gravity In Twistor Space

Peter Woit (author of the "Not Even Wrong" blog and a math professor at Columbia University) is working on developing a Twistor space based unification of the Standard Model and gravity. He feels he has made some breakthroughs and has provided to his blog readers a 48 page pre-pre-print working draft of his latest paper on the topic. I'll lay out what he's doing as simply as possible below, and the below the fold provide some additional quoted details about it.

In the real world of physics we directly measure only real number valued quantities (the real numbers are all of the integers, i.e. ... -3, -2, -1, 0, 1, 2, 3 . . . , and all of the numbers in between the integers).

But, mathematically, complex number valued quantities, in which each number has a real number part and an imaginary number part, have a lot of attractive properties (among other things, it is more "complete" in that it is harder to do mathematically operations that produce undefined quantities like taking the square root of negative numbers in the real numbers, and can store the same amount of information in half as many numbers).

Twistor physics is basically a way to describe physics by transforming real number valued point-like physical properties and objects (scalars) and real number valued directional physical properties (vectors), into a closely related parallel description in terms of complex valued scalars and vectors.

Euclidian Twistor space transforms the space-time of classical physics into a complex valued twistor space. Minkowski Twistor space transforms the space-time of quantum field theory, in which special relativity is observed, into a complex valued twistor space.

There is a way to generalize concepts in Euclidian Twistor space into Minkowski Twistor space, although for a variety of technical reasons, this is much trickier than it seems because the underlying mathematical structure of Euclidian Twistor space is not parallel in a crude way to Minkowski Twistor space that is formulated in a quite different way.

To make this transformation from Euclidian Twistor space to Minkowski Twistor space, you have to identify an imaginary time direction in Twistor space, and the key to doing that in physics applications involves the Higgs field.

It turns out that Standard Model physics can be described in a very natural and comparatively easy to mathematically manipulate form by describing it in a non-physical Minkowski Twistor space. Spinors (which are complex valued objects sometimes crudely described as the square root of vectors) in Twistor space are a very compact and useful way to describe fundamental particles in Standard Model physics. It may even be that the non-physical Twistor space-spinor description of the world is more fundamental than the world we actually observe and that the world we actually observe is emergent from this more fundamental description of it.

In particular, Peter Woit has observed that, in a Twistor space formulation:

• Exactly the internal symmetries of the Standard Model occur.
• The intricate transformation properties of a generation of Standard Model fermions correspond to a simple construction.
• Conformal symmetry is built into the picture in a fundamental way.

A Twistor space formulation, among other things, tames some of the mathematical wildness and intractability of the general case of quantum gravity by imposing a conformal symmetry (which does not a priori hold in four dimensional general relativity, although a conformal symmetry version of general relativity has been actively explored in the classical case, and some authors have argued that this may even address some dark matter and dark energy issues).

For a given spacetime and metric, a conformal transformation is a transformation that preserves angles, and a conformal symmetry is closely related to properties of a physical law that apply without respect to scale (i.e. scale invariance), but in a more mathematically sublime manner that is relevant to preserving renormalizability in quantum field theories.

So, by imposing a very subtle additional symmetry upon general relativity, it can be made to be much better behaved mathematically, which is a big deal because one of the main problems with quantum gravity has been its non-renormalizability and its general mathematical intractability.

Peter Woit has also concluded that in a Twistor space approach that "One gets a new chiral formulation of gravity, unified with the Standard Model."

The group theory formulation of the Standard Model is SU(3) x SU(2) x U(1), where SU(3) is the color (strong force) part, the SU(2) is the weak (left handed) part, and the U(1) is the electromagnetic part.

The approach Peter Woit is taking to unify the Standard Model and Quantum Gravity is basically to observe that you can describe gravity with the right handed SU(2) group that is an exact counterpart to the left handed SU(2) group that describes the Standard Model weak force, into a combined real valued SU(4) group, which corresponds to, in the necessary ways, the complex valued SL(2,C) group that can be found in the appropriate Twistor space. The right handed SU(2) group has the same number of degrees of freedom and the proper interrelationships among its parts necessary to describe gravity.

This is an approach that I've previously described as a gravito-weak unification.

There are details to be worked out, but it is one of the more promising approaches to quantum gravity, along with the QCD squared approach, emergent gravity approaches (with a thermodynamic/entropy/holographic orientation), the string theory concept of a graviton (and Twister theory and string theory are not necessarily inherently incompatible), supersymmetric supergravity theories, and  loop quantum gravity oriented approaches.

Woit's approach, if it worked, would eliminate the inherent theoretical inconsistencies between classically formulated general relativity, and the quantum mechanically formulated Standard Model of Particle Physics, in a mathematically tractable way, which would be a big deal.

But, it would not, on its own, do anything that existing classical general relativity (possibly refined to include a conformal symmetry) does not do, to address issues like dark matter and dark energy. It would solve those problems only if general relativity, or general relativity with conformal symmetry, can already do that in the classical case (as some investigators, who are minority in the scientific community, have claimed).

Woit's approach also does not include a "grand unified theory" that describes all of the forces of nature in terms of a single unified group of which the four known forces of nature are merely parts, or with a unification of all of the fundamental forces as aspects of the same underlying force at high enough energies.

More Technical Background

Roger Penrose is commonly credited with the developing the concept of a "twistor" in physics applications (in 1967).

Mathematically, projective twistor space ${\displaystyle \mathbb {PT} }$ is a 3-dimensional complex manifold, complex projective 3-space ${\displaystyle \mathbb {CP} ^{3}}$. It has the physical interpretation of the space of massless particles with spin. It is the projectivisation of a 4-dimensional complex vector space, non-projective twistor space ${\displaystyle \mathbb {T} }$ with a Hermitian form of signature (2,2) and a holomorphic volume form. This can be most naturally understood as the space of chiral (Weylspinors for the conformal group ${\displaystyle SO(4,2)/\mathbb {Z} _{2}}$ of Minkowski space; it is the fundamental representation of the spin group ${\displaystyle SU(2,2)}$ of the conformal group. This definition can be extended to arbitrary dimensions except that beyond dimension four, one defines projective twistor space to be the space of projective pure spinors for the conformal group.

In its original form, twistor theory encodes physical fields on Minkowski space into complex analytic objects on twistor space via the Penrose transform.

These are in turn related to spinors.

In geometry and physics, spinors /spɪnər/ are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation. However, when a sequence of such small rotations is composed (integrated) to form an overall final rotation, the resulting spinor transformation depends on which sequence of small rotations was used. Unlike vectors and tensors, a spinor transforms to its negative when the space is continuously rotated through a complete turn from 0° to 360° (see picture). This property characterizes spinors: spinors can be viewed as the "square roots" of vectors (although this is inaccurate and may be misleading; they are better viewed as "square roots" of sections of vector bundles – in the case of the exterior algebra bundle of the cotangent bundle, they thus become "square roots" of differential forms).

It is also possible to associate a substantially similar notion of spinor to Minkowski space, in which case the Lorentz transformations of special relativity play the role of rotations. Spinors were introduced in geometry by Élie Cartan in 1913. In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum, or "spin", of the electron and other subatomic particles.

Spinors are characterized by the specific way in which they behave under rotations.

. . .

To obtain the spinors of physics, such as the Dirac spinor, one extends the construction to obtain a spin structure on 4-dimensional space-time (Minkowski space). Effectively, one starts with the tangent manifold of space-time, each point of which is a 4-dimensional vector space with SO(3,1) symmetry, and then builds the spin group at each point. The neighborhoods of points are endowed with concepts of smoothness and differentiability: the standard construction is one of a fibre bundle, the fibers of which are affine spaces transforming under the spin group. After constructing the fiber bundle, one may then consider differential equations, such as the Dirac equation, or the Weyl equation on the fiber bundle. These equations (Dirac or Weyl) have solutions that are plane waves, having symmetries characteristic of the fibers, i.e. having the symmetries of spinors, as obtained from the (zero-dimensional) Clifford algebra/spin representation theory described above. Such plane-wave solutions (or other solutions) of the differential equations can then properly be called fermions; fermions have the algebraic qualities of spinors. By general convention, the terms "fermion" and "spinor" are often used interchangeably in physics, as synonyms of one-another.

It appears that all fundamental particles in nature that are spin-1/2 are described by the Dirac equation, with the possible exception of the neutrino. There does not seem to be any a priori reason why this would be the case. A perfectly valid choice for spinors would be the non-complexified version of Cℓ2,2(${\displaystyle \mathbb {R} }$), the Majorana spinor. There also does not seem to be any particular prohibition to having Weyl spinors appear in nature as fundamental particles.

The Dirac, Weyl, and Majorana spinors are interrelated, and their relation can be elucidated on the basis of real geometric algebra. Dirac and Weyl spinors are complex representations while Majorana spinors are real representations.

Weyl spinors are insufficient to describe massive particles, such as electrons, since the Weyl plane-wave solutions necessarily travel at the speed of light; for massive particles, the Dirac equation is needed. The initial construction of the Standard Model of particle physics starts with both the electron and the neutrino as massless Weyl spinors; the Higgs mechanism gives electrons a mass; the classical neutrino remained massless, and was thus an example of a Weyl spinor. However, because of observed neutrino oscillation, it is now believed that they are not Weyl spinors, but perhaps instead Majorana spinors. It is not known whether Weyl spinor fundamental particles exist in nature.

Peter Woit has been active in further developing it. In a recent post, he sums up what he finds promising about this path to unifying the quantum field theory of the Standard Model and quantum gravity in Twistor space:

There are several (to me at least…) attractive aspects:

• Spinors are tautological objects (a point in space-time is a space of Weyl spinors), rather than complicated objects that must be separately introduced in the usual geometrical formalism.
• Analytic continuation between Minkowski and Euclidean space-time can be naturally performed, since twistor geometry provides their joint complexification.
• Exactly the internal symmetries of the Standard Model occur.
• The intricate transformation properties of a generation of Standard Model fermions correspond to a simple construction.
• One gets a new chiral formulation of gravity, unified with the Standard Model.
• Conformal symmetry is built into the picture in a fundamental way.

There’s more in this version about how quantum gravity fits into this, when formulated in terms of chiral variables (i.e. Ashtekar variables). This gives a new context for old questions about quantizing in these variables (this is in Eucldean signature, the other chirality is not space-time geometry but internal Yang-Mills geometry, and the imaginary time component of the vierbein is distinguished and given the dynamics of a Higgs field). I haven’t spent much time on this yet, but suspect this new context may help overcome problems that people trying to pursue quantum gravity in this chiral connection framework have run into in the past.

One common reaction I’ve gotten to these ideas is the one I myself had in the past: analytic continuation relates expectation values of field operators in Euclidean and Minkowski signature, so my left-handed SU(2) after analytic continuation gives part of Lorentz symmetry, not an internal symmetry. What took me a long time to realize is just how different Euclidean and Minkowski signature QFT is. Yes, Schwinger functions and Wightman functions can be related by analytic continuation (in a rather subtle way, the Wightman functions aren’t functions, but boundary values of holomorphic functions). But at the level of states and operators things are very different. It’s just not true that there is some holomorphic formulation of QFT states and operators, with Euclidean and Minkowski space restrictions related by analytic continuation. There’s a lot of explanation about this in the paper.

One objection I’ve run into is that by distinguishing a direction in Euclidean space I’m breaking Lorentz symmetry. What’s true is quite the opposite: having such a distinguished direction is needed to get Lorentz symmetry after analytic continuation. If you want to start in Euclidean space and get Lorentz symmetry, you have to do something like distinguish a direction and get an Osterwalder-Schrader reflection in that direction, which you need to get from SO(4) to SL(2,C). From the other direction, if you start in Minkowski space-time and analytically continue, you have a choice of lots of possible Euclidean slices to analytically continue to. You need to pick one, and that will distinguish an imaginary time direction. This is most easily seen in the twistor formalism, where the Minkowski space-time geometry is determined by a quadratic form that picks out a 5-dimensional hypersurface in PT. This will project down to an imaginary time = 0 subspace of Euclidean space-time, which picks out the imaginary time direction.

neo said...

interesting write up, thanks.

what role does Ashtekar variables play in Woit's theory?

Mitchell said...

"Euclidian Twistor space transforms the space-time of classical physics into a complex valued twistor space. Minkowski Twistor space transforms the space-time of quantum field theory, in which special relativity is observed, into a complex valued twistor space."

Euclidean space here does not refer to classical (non-relativistic) space-time, it refers to a four-dimensional manifold in which there is no time at all. All directions are space-like.

This is not a quantum-vs-classical thing; it is about replacing elements of a quantum calculation done in Minkowski space (3 space dimensions, 1 time dimension), with a quantum calculation done in a purely Euclidean 4-space (4 space dimensions, 0 time dimensions). Also known as Wick rotation; mathematically, a form of analytic continuation, i.e. extending a function beyond its originally envisaged domain.

The original twistor space or twistor transform, popularized within physics by Roger Penrose, complexifies Minkowski space and then performs a change of variables. That's the "Minkowski Twistor space". Euclidean Twistor space results from performing the twistor transform on the Euclidean 4-space mentioned previously.

"To make this transformation from Euclidian Twistor space to Minkowski Twistor space, you have to identify an imaginary time direction in Twistor space, and the key to doing that in physics applications involves the Higgs field."

Not quite. The Euclidean space has four-dimensional rotational symmetry, which locally can be expressed in terms of a "vierbein" or "tetrad" of four orthogonal tangent vectors which are all interchangeable. Picking a time direction means singling out one of those vectors, and reducing the rotational symmetry to three dimensions.

The four-dimensional rotational symmetry factorizes as SU(2) x SU(2), and twistor space also has a local U(1) x SU(3) symmetry. Apparently the twistor counterpart (image under the twistor transform?) of an individual tangent vector, is invariant under one of those SU(2)s, as well as the local U(1). Peter's idea is that it has the right degrees of freedom (two complex numbers?) and right transformation properties (charged under SU(2) x U(1)) to be the standard model Higgs field.

He then supposes that the other SU(2) factor, from the four-dimensional rotational symmetry, can be used to define general relativity, based on an SU(2) connection rather than a metric. This is Ashtekar's choice of variables (connection rather than metric).

Mitchell said...

Some further comments as I try to get the concepts in view. I may make some mistakes about the details of twistor space, but I'll charge on ahead for now.

The significance of Wick rotations and similar techniques, may be compared to the way that virtual particles are allowed to have unphysical momenta ("off-shell"). For a virtual particle, you're integrating over a range of possibilities in a Feynman sum, to get a contribution to the final probability amplitude. Allowing the virtual particle to go off-shell doesn't matter physically, so long as the energy-momentum in initial and final states respects the mass of the particles involved. Going off-shell is then just a calculus technique, in which you increase the range of an integral in order to be able to compute it, but in a way designed to not change the final result.

In the same way, physical processes are occurring in Lorentzian space, but it's often easier to calculate probability amplitudes in Euclidean space, and then analytically continue the resulting expressions, back to metrics with Lorentzian signature.

A twistor transform is a complexification of space-time coordinates, followed by a kind of half Fourier transform - you Fourier transform (I think) only the left-handed components of a field, not the right-handed ones. So Minkowski twistor space and Euclidean twistor space result from performing the twistor transform on Minkowski space and Euclidean space respectively.

Peter Woit uses as his ingredients, the complex coordinates of Euclidean twistor space, and a spinor field/bundle/connection on that space too. As mentioned, Euclidean twistor space has a four-dimensional rotational symmetry SO(4), which is extended to Spin(4) to describe spinors (recall that a 360-degree rotation multiplies a spinor by -1, so you need two full rotations to restore the original state), as well as a local U(1) x SU(3) symmetry (this might be a redundancy of the complex coordinates?).

Then as described, he picks one direction as a time direction. He defines this via a timelike vector field which is invariant under the local U(1) and one of the SU(2) factors of the Spin(4). His proposal is that this field is the Higgs field, that this SU(2) x U(1) is the electroweak gauge field, and the local SU(3) is the gluon field, that the other SU(2) is the gravitational field expressed as an Ashtekar connection, and that the Euclidean 4-spinor is a single standard model generation, with its components having the same transformation properties under the SU(3) x SU(2)L x U(1) as the standard model fermions.

So just to sum up again, the standard model gauge group and the gauge group of the Ashtekar formulation of general relativity all come from symmetries of the complex coordinates of Euclidean twistor space; one fermion generation comes from the 4-spinor bundle of Euclidean twistor space; and the Higgs field comes from the vector field he uses to specify a time direction.

The ingredients are, Euclidean twistor space, spinor bundle, choice of time direction.

I have undoubtedly misidentified some of the objects along the way, but I think I am right that these are his ingredients.

Mitchell said...

The claim then must be, that if you quantize (Euclidean twistor space + 4-spinor bundle + 1-vector field) in the right way, you get standard model with higgs field and one fermion generation, coupled to gravity.

neo said...

@ Mitchell

overall what do you think of Woit's proposal ?

sounds like a TOE but simpler than string theory

andrew said...

@Mitchell

I had considered discussing the Fourier transform issue both as used in a Twister transform and as an analogy to the Twister paradigm, but decided that doing so would increase the demands the post made on a reader who hadn't had post-calculus level math too far.

andrew said...

@neo

Abhay V. Ashtekar is best known for his quantum gravity work (mostly in loop quantum gravity) and especially the Ashtekar Variables:

"In the ADM formulation of general relativity, spacetime is split into spatial slices and a time axis. The basic variables are taken to be the induced metric q_{ab}(x) on the spatial slice and the metric's conjugate momentum K^{ab}(x), which is related to the extrinsic curvature and is a measure of how the induced metric evolves in time. These are the metric canonical coordinates. In 1986 Abhay Ashtekar introduced a new set of canonical variables, Ashtekar (new) variables to represent an unusual way of rewriting the metric canonical variables on the three-dimensional spatial slices in terms of an SU(2) gauge field and its complementary variable. Ashtekar variables provide what is called the connection representation of canonical general relativity, which led to the loop representation of quantum general relativity and in turn loop quantum gravity and quantum holonomy theory."

Often Ashtekar variables, much like imaginary numbers, appear in necessary intermediate calculations, but vanish in the end result (although this isn't always the case). Many loop quantum gravity papers at some point in their discussion basically ask "what are the Ashtekar variables doing here?", and reach an inconclusive result.

"The ADM formalism (named for its authors Richard Arnowitt, Stanley Deser and Charles W. Misner) is a Hamiltonian formulation of general relativity that plays an important role in canonical quantum gravity and numerical relativity. It was first published in 1959." (Including two of the three authors of one of the leading but dreaded GR textbooks called Gravitation that I have on my bookshelf as a reference despite the fact that it is a poor self-study reference and that I don't like the way it handles some key issue.

It basically helps overcome difficulties arising from the fact that Einstein's equations in GR are formulated in a conceptually different way than the Hamiltonians and Langrangians of quantum mechanics, which creates a stylistic issue to overcome in formulating quantum gravity in addition to the substantive difficulties involved.

andrew said...

@neo

Re the pertinent issue in canonical quantum gravity:

"The Wheeler-DeWitt equation (sometimes called the Hamiltonian constraint, sometimes the Einstein-Schrödinger equation) is rather central as it encodes the dynamics at the quantum level. It is analogous to Schrödinger's equation, except as the time coordinate, t, is unphysical, a physical wave function can't depend on t and hence Schrödinger's equation reduces to a constraint:

H*Psi = 0

Using metric variables lead to seemingly insurmountable mathematical difficulties when trying to promote the classical expression to a well-defined quantum operator, and as such decades went by without making progress via this approach. This problem was circumvented and the formulation of a well-defined Wheeler-De-Witt equation was first accomplished with the introduction of Ashtekar-Barbero variables and the loop representation, this well defined operator formulated by Thomas Thiemann.

Before this development, the Wheeler-De-Witt equation had only been formulated in symmetry-reduced models, such as quantum cosmology.

Many of the technical problems in canonical quantum gravity revolve around the constraints. Canonical general relativity was originally formulated in terms of metric variables, but there seemed to be insurmountable mathematical difficulties in promoting the constraints to quantum operators because of their highly non-linear dependence on the canonical variables. The equations were much simplified with the introduction of Ashtekar's new variables. Ashtekar variables describe canonical general relativity in terms of a new pair of canonical variables closer to that of gauge theories. In doing so, it introduced an additional constraint, on top of the spatial diffeomorphism and Hamiltonian constraint, the Gauss gauge constraint.

The loop representation is a quantum hamiltonian representation of gauge theories in terms of loops. The aim of the loop representation, in the context of Yang-Mills theories, is to avoid the redundancy introduced by Gauss gauge symmetries allowing work directly in the space of Gauss gauge invariant states. The use of this representation arose naturally from the Ashtekar-Barbero representation, as it provides an exact non-perturbative description, and also because the spatial diffeomorphism constraint is easily dealt with within this representation.

Within the loop representation, Thiemann has provided a well defined canonical theory in the presence of all forms of matter and explicitly demonstrated it to be manifestly finite! So there is no need for renormalization. However, as LQG approach is well suited to describe physics at the Planck scale, there are difficulties in making contact with familiar low energy physics and establishing it has the correct semi-classical limit."

neo said...

Standard LQG does not include the standard model,

but if Woit's Euclidian Twistor space is correct,

it sounds like a TOE, with both gravity - Ashektar's variables and SM gauge groups and fermions, i.e LQG can be embedded in a larger theory that includes the SM.

Mitchell said...

@andrew

I'll thank you in turn for having a go at dissecting Woit's theory, since until this post, I didn't really get how it's supposed to work.

@neo

At this point the main thing I'm curious about, is whether this is somehow a twistorial version of the idea that you can get a single generation from an SO(10) spinor. Not just in the sense of getting a single generation from a spinor on a particular space; but whether this might literally be a transposition of an SO(10) spinor-generation to twistor variables.

We should also not jump to the conclusion that using Ashtekar variables implies using LQG methods to quantize. Thiemann's method in particular is just wrong, since by construction it preserves classical symmetries (axial symmetry of electromagnetism) that are quantum-mechanically broken in the real world (ABJ chiral anomaly, relevant to pion decay). It would be better to start with existing ideas from conventional field theory or from twistor theory.

neo said...

@Mitchell

what do you think is the correct way to quantize Ashtekar variables, perhaps in this framework, with conformal symmetry and twistor theory.

does the Coleman-Mandula theorem apply here?

overall is this a promising line of research into TOE?

neo said...

his response

Peter Woit says:
September 24, 2021 at 6:44 pm

Anonymous,
No, I’m not embedding both space-time and internal symmetries in a larger symmetry group and having that act on the states. That’s what Coleman-Mandula says you can’t do (because you’ll end up with a trivial theory).

For me a lot of the appeal of this picture is exactly that it gives a sort of unification that does not involve embedding everything in a bigger grand unification group, that you then have to break somehow to get back to the SM and usual space-time symmetries.

Mitchell said...

@neo

It is too early to say if the existing construction can be modified to give three generations (not one), actual gauge symmetries (at this point I don't think the "symmetries" have been gauged), yukawa couplings, etc, or whether it is stuck in a form that cannot get any closer to the real world.

neo said...

@Mitchell

how difficult is gauging the symmetries ?

is there any theory - that explains three generations and calculation for yukawa couplings

for example, does string theory also explain 3 generations and provide yukawa couplings

he suggests in his slides S7 for 3 generations

https://www.math.columbia.edu/~woit/twistorunification/brown9-23-21.pdf

andrew said...

@neo Nobody has pulled off that feat yet.

neo said...

a twistorial version of the idea that you can get a single generation from an SO(10) spinor.

what is special about SO(10) spinor?

neo said...

@Mitchell,

would any of these papers

[Submitted on 20 May 2019]
The Cℓ(8) algebra of three fermion generations with spin and full internal symmetries
Adam B. Gillard, Niels G. Gresnigt

In this paper, the basis states of the minimal left ideals of the complex Clifford algebra Cℓ(8) are shown to contain three generations of Standard Model fermion states, with full Lorentzian, right and left chiral, weak isospin, spin, and electrocolor degrees of freedom. The left adjoint action algebra of Cℓ(8)≅C(16) on its minimal left ideals contains the Dirac algebra, weak isopin and spin transformations. The right adjoint action algebra on the other hand encodes the electrocolor symmetries. These results extend earlier work in the literature that shows that the eight minimal left ideals of C(8)≅Cℓ(6) contain the quark and lepton states of one generation of fixed spin. Including spin degrees of freedom extends Cℓ(6) to Cℓ(8), which unlike Cℓ(6) admits a triality automorphism. It is this triality that underlies the extension from a single generation of fermions to exactly three generations.

Subjects: General Physics (physics.gen-ph); High Energy Physics - Theory (hep-th)
Cite as: arXiv:1906.05102 [physics.gen-ph]

Submitted on 5 Apr 2019]
Three fermion generations with two unbroken gauge symmetries from the complex sedenions
Adam B. Gillard, Niels G. Gresnigt

We show that three generations of leptons and quarks with unbroken Standard Model gauge symmetry SU(3)c×U(1)em can be described using the algebra of complexified sedenions C⊗S. A primitive idempotent is constructed by selecting a special direction, and the action of this projector on the basis of C⊗S can be used to uniquely split the algebra into three complex octonion subalgebras C⊗O. These subalgebras all share a common quaternionic subalgebra. The left adjoint actions of the 8 C-dimensional C⊗O subalgebras on themselves generates three copies of the Clifford algebra Cℓ(6). It was previously shown that the minimal left ideals of Cℓ(6) describe a single generation of fermions with unbroken SU(3)c×U(1)em gauge symmetry. Extending this construction from C⊗O to C⊗S naturally leads to a description of exactly three generations.

Subjects: High Energy Physics - Theory (hep-th)
Journal reference: Eur. Phys. J. C (2019) 79: 446
DOI: 10.1140/epjc/s10052-019-6967-1
Cite as: arXiv:1904.03186 [hep-th]

[Submitted on 17 Oct 2019]
Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra
N. Furey

A considerable amount of the standard model's three-generation structure can be realised from just the 8C-dimensional algebra of the complex octonions. Indeed, it is a little-known fact that the complex octonions can generate on their own a 64C-dimensional space. Here we identify an su(3)⊕u(1) action which splits this 64C-dimensional space into complexified generators of SU(3), together with 48 states. These 48 states exhibit the behaviour of exactly three generations of quarks and leptons under the standard model's two unbroken gauge symmetries. This article builds on a previous one, [1], by incorporating electric charge.
Finally, we close this discussion by outlining a proposal for how the standard model's full set of states might be identified within the left action maps of R⊗C⊗H⊗O (the Clifford algebra Cl(8)). Our aim is to include not only the standard model's three generations of quarks and leptons, but also its gauge bosons.