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kodama

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- TL;DR Summary
- loop quantum gravity as a theory of everything

in an interview Michio Kaku was asked about Loop Quantum gravity and he replied that it is only a theory of pure gravity, but that the universe also contains particles, the particles of the standard model, and only superstring theory unifies both quantum gravity and the standard model.

the literature on loop quantum gravity is that it is a theory of quantum gravity only, unlike string theory

but lqg contains spinors

Spinor representation for loop quantum gravity

E Livine, J Tambornino - Journal of mathematical physics, 2012 - pubs.aip.org

3d Lorentzian loop quantum gravity and the spinor approach

F Girelli, G Sellaroli - Physical Review D, 2015 - APS

U (N) tools for loop quantum gravity: the return of the spinor

EF Borja, L Freidel, I Garay

Quantum gravity in three dimensions, Witten spinors and the quantisation of length

W Wieland

independent line of research connects octonions with the standard model via spinors

John Baez discusses theory here

Octonions and the Standard Model

https://www.physicsforums.com/threads/octonions-and-the-standard-model.995505/

other papers connecting spinors and octonions with the standard model

Are octonions necessary to the Standard Model?

P Rowlands, S Rowlands - Journal of Physics: Conference …, 2019 - iopscience.iop.org

… , particularly because of the significance of the octonions in creating the E8 symmetry. The …

of Standard Model physics lies somewhere within them. We aim to show that, while octonions

Octonion internal space algebra for the standard model

I Todorov - Universe, 2023 - mdpi.com

Standard Model particles from split octonions

M Gogberashvili - Prog. Phys, 2016 - books.google.com

Octonions in Particle Physics through Structures of Generalised Proper Time

DJ Jackson - arXiv preprint arXiv:1909.05014, 2019 - arxiv.org

Standard model physics from an algebra?

C Furey - arXiv preprint arXiv:1611.09182, 2016 - arxiv.org

… the Lorentz representations necessary to describe the standard model. Then, in Chapter 4,

… characteristics of the standard model. In Chapter 6 we introduce the complex octonions

and octonions and spinors

Spin (11, 3), particles, and octonions

K Krasnov - Journal of Mathematical Physics, 2022 - pubs.aip.org

… -spinor representation S+ of the group Spin(11, 3). We describe an octonionic model for

Spin(11, 3) in which the semi-spinor …

Notes on spinors and polyforms II: quaternions and octonions

N Bhoja, K Krasnov - arXiv preprint arXiv:2205.05447, 2022 - arxiv.org

… The link to split octonions arises if we consider Majorana-Weyl spinors.

Using octonions to describe fundamental particles

T Dray, CA Manogue - Clifford Algebras: Applications to Mathematics …, 2004 - Springer

… octonionic description of the lO-dimensional massless Dirac equation. We extend this formalism

to 3-component octonionic "spinors", … , consisting of 3 x 3 octonionic Hermitian matrices

a rough sketch of this research program is that Weyl spinors combined with octonions, clifford algebras and exception Jordan algebras, give rise to the gauge groups of the standard model, including 3 generations

Three fermion generations with two unbroken gauge symmetries from the complex sedenions

Adam B. Gillard, Niels G. Gresnigt

Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra

N. Furey

The Standard Model Algebra - Leptons, Quarks, and Gauge from the Complex Clifford Algebra Cl6

Ovidiu Cristinel Stoica

to be clear this is a research program that still has many issues that need to be worked on, perhaps with other ideas like non commutative geometry

there are some elements missing in octonions and standard model research program and perhaps Alain Connes Noncommutative Geometry and the Standard Model could fill in some of the gaps, and there are also approaches that combine noncommutative geometry with Loop quantum gravity such as

Intersecting Quantum Gravity with Noncommutative Geometry

by J Aastrup ·

Noncommutative Geometry and Loop Quantum Gravity

Mathematisches Forschungsinstitut Oberwolfach

https://publications.mfo.de › OWR_2010_09

by C Fleischhack — The workshop “Noncommutative Geometry and Loop Quantum Gravity: Loops,. Algebras and Spectral Triples” has been organized by Christian Fleischhack

perhaps Noncommutative Geometry over an almost commutative C∗-algebra Spectral Triples over loop quantum gravity space of connections is needed to fill in the gaps of octonions and spinors

since loop quantum gravity contains spinors, is it possible to derive the standard model from LQG via spinors and octonions for a unification of loop quantum gravity with the standard model via octonions and spinors, creating a theory of everything?

are there any theoretical problems with introducing octonions and the standard model research program into loop quantum gravity spinors, resulting in the standard model of particle physics unified with quantum gravity and thus a unified theory of everything?

since there are missing pieces in the octonions and standard model research program, could combining octonions with Alain Connes Noncommutative Geometry and the Standard Model spectral triples fill in some of the missing pieces, octonions in Noncommutative Geometry and spectral triples. there are papers combining Noncommutative Geometry with loop quantum gravity

loop quantum gravity is the best develop theory I know of that

contains spinors and is a quantum gravity, however this is

not unique to loop quantum gravity, any quantum spacetime theory that obeys general relativity and contains spinors, if compatible with octonions, could also be a unified theory of everything

I also note Noncommutative Geometry Spectral Triplesin connection with both octonions and loop quantum gravity

there is also included

which cited Sundance Bilson-Thompson "Quantum gravity and the standard model".

the literature on loop quantum gravity is that it is a theory of quantum gravity only, unlike string theory

but lqg contains spinors

Spinor representation for loop quantum gravity

E Livine, J Tambornino - Journal of mathematical physics, 2012 - pubs.aip.org

3d Lorentzian loop quantum gravity and the spinor approach

F Girelli, G Sellaroli - Physical Review D, 2015 - APS

U (N) tools for loop quantum gravity: the return of the spinor

EF Borja, L Freidel, I Garay

Quantum gravity in three dimensions, Witten spinors and the quantisation of length

W Wieland

independent line of research connects octonions with the standard model via spinors

John Baez discusses theory here

Octonions and the Standard Model

https://www.physicsforums.com/threads/octonions-and-the-standard-model.995505/

I've slowly been writing a thread on octonions and particle physics, just to explain some facts in a self-contained way, with all the proofs. I don't know where this will lead. I'm certainly not presenting a theory of physics, much less advocating one. Mainly it's just fun.

Octonions and the Standard Model 1. How to define octonion multiplication using complex scalars and vectors, much as quaternion multiplication can be defined using real scalars and vectors. This description requires singling out a specific unit imaginary octonion, and it shows that octonion multiplication is invariant under SU(3).

Octonions and the Standard Model 2. A more polished way to think about octonion multiplication in terms of complex scalars and vectors, and a similar-looking way to describe it using the cross product in 7 dimensions.

Octonions and the Standard Model 3. How a lepton and a quark fit together into an octonion - at least if we only consider them as representations of SU(3), the gauge group of the strong force. Proof that the symmetries of the octonions fixing an imaginary octonion form precisely the group SU(3).

Octonions and the Standard Model 4. Introducing the exceptional Jordan algebra: the 3×3 self-adjoint octonionic matrices. A result of Dubois-Violette and Todorov: the symmetries of the exceptional Jordan algebra preserving their splitting into complex scalar and vector parts and preserving a copy of the 2×2 adjoint octonionic matrices form precisely the Standard Model gauge group.

other papers connecting spinors and octonions with the standard model

Are octonions necessary to the Standard Model?

P Rowlands, S Rowlands - Journal of Physics: Conference …, 2019 - iopscience.iop.org

… , particularly because of the significance of the octonions in creating the E8 symmetry. The …

of Standard Model physics lies somewhere within them. We aim to show that, while octonions

Octonion internal space algebra for the standard model

I Todorov - Universe, 2023 - mdpi.com

Standard Model particles from split octonions

M Gogberashvili - Prog. Phys, 2016 - books.google.com

Octonions in Particle Physics through Structures of Generalised Proper Time

DJ Jackson - arXiv preprint arXiv:1909.05014, 2019 - arxiv.org

Standard model physics from an algebra?

C Furey - arXiv preprint arXiv:1611.09182, 2016 - arxiv.org

… the Lorentz representations necessary to describe the standard model. Then, in Chapter 4,

… characteristics of the standard model. In Chapter 6 we introduce the complex octonions

and octonions and spinors

Spin (11, 3), particles, and octonions

K Krasnov - Journal of Mathematical Physics, 2022 - pubs.aip.org

… -spinor representation S+ of the group Spin(11, 3). We describe an octonionic model for

Spin(11, 3) in which the semi-spinor …

Notes on spinors and polyforms II: quaternions and octonions

N Bhoja, K Krasnov - arXiv preprint arXiv:2205.05447, 2022 - arxiv.org

… The link to split octonions arises if we consider Majorana-Weyl spinors.

Using octonions to describe fundamental particles

T Dray, CA Manogue - Clifford Algebras: Applications to Mathematics …, 2004 - Springer

… octonionic description of the lO-dimensional massless Dirac equation. We extend this formalism

to 3-component octonionic "spinors", … , consisting of 3 x 3 octonionic Hermitian matrices

a rough sketch of this research program is that Weyl spinors combined with octonions, clifford algebras and exception Jordan algebras, give rise to the gauge groups of the standard model, including 3 generations

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.

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 realized 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.

The Standard Model Algebra - Leptons, Quarks, and Gauge from the Complex Clifford Algebra Cl6

Ovidiu Cristinel Stoica

A simple geometric algebra is shown to contain automatically the leptons and quarks of a family of the Standard Model, and the electroweak and color gauge symmetries, without predicting extra particles and symmetries. The algebra is already naturally present in the Standard Model, in two instances of the Clifford algebra Cℓ6, one being algebraically generated by the Dirac algebra and the weak symmetry generators, and the other by a complex three-dimensional representation of the color symmetry, which generates a Witt decomposition which leads to the decomposition of the algebra into ideals representing leptons and quarks. The two instances being isomorphic, the minimal approach is to identify them, resulting in the model proposed here. The Dirac and Lorentz algebras appear naturally as subalgebras acting on the ideals representing leptons and quarks. The resulting representations on the ideals are invariant to the electromagnetic and color symmetries, which are generated by the bivectors of the algebra. The electroweak symmetry is also present, and it is already broken by the geometry of the algebra. The model predicts a bare Weinberg angle θW given by sin2θW=0.25. The model shares common ideas with previously known models, particularly with Chisholm and Farwell, 1996, Trayling and Baylis, 2004, and Furey, 2016.

to be clear this is a research program that still has many issues that need to be worked on, perhaps with other ideas like non commutative geometry

there are some elements missing in octonions and standard model research program and perhaps Alain Connes Noncommutative Geometry and the Standard Model could fill in some of the gaps, and there are also approaches that combine noncommutative geometry with Loop quantum gravity such as

Intersecting Quantum Gravity with Noncommutative Geometry

by J Aastrup ·

Noncommutative Geometry and Loop Quantum Gravity

Mathematisches Forschungsinstitut Oberwolfach

https://publications.mfo.de › OWR_2010_09

by C Fleischhack — The workshop “Noncommutative Geometry and Loop Quantum Gravity: Loops,. Algebras and Spectral Triples” has been organized by Christian Fleischhack

"...In fact, noncommutative geometry (NCG) provides a remarkably successful

framework for unification of all known fundamental forces. Mathematically, it

mainly grounds on the pioneering work of Connes, who related Riemannian spin

geometries to a certain class of spectral triples over commutative C∗-algebras.

Extending this formalism, Chamseddine and Connes demonstrated that the stan-

dard model coupled to gravitation naturally emerges from a spectral triple over an almost commutative C∗-algebra together with a spectral action.

And, instead of an emergent unification, matter has to be

included by hand.

Although NCG and LQG use very similar mathematical techniques – e. g., op-

erator algebras in general, or spectral encoding of geometry to be more specific –,

their conceptual problems are rather complementary. Nevertheless, only recently,

first steps to join the strengthes of both approaches have been made. In several

papers since 2005, Aastrup and Grimstrup, later with one of the organizers (RN),

have outlined how to construct a semifinite spectral triple for the full theory out

of spectral triples based on a restricted system of nested graphs.

One of the main tasks of the meeting was to bring together researchers from

different fields – first of all, noncommutative geometry and loop quantum gravity,

but also other fields like spectral triples on its own and axiomatic quantum field

theory. For this, there were several introductory talks:

• Hanno Sahlmann and Thomas Thiemann gave an overview on the..."

perhaps Noncommutative Geometry over an almost commutative C∗-algebra Spectral Triples over loop quantum gravity space of connections is needed to fill in the gaps of octonions and spinors

since loop quantum gravity contains spinors, is it possible to derive the standard model from LQG via spinors and octonions for a unification of loop quantum gravity with the standard model via octonions and spinors, creating a theory of everything?

are there any theoretical problems with introducing octonions and the standard model research program into loop quantum gravity spinors, resulting in the standard model of particle physics unified with quantum gravity and thus a unified theory of everything?

since there are missing pieces in the octonions and standard model research program, could combining octonions with Alain Connes Noncommutative Geometry and the Standard Model spectral triples fill in some of the missing pieces, octonions in Noncommutative Geometry and spectral triples. there are papers combining Noncommutative Geometry with loop quantum gravity

loop quantum gravity is the best develop theory I know of that

contains spinors and is a quantum gravity, however this is

not unique to loop quantum gravity, any quantum spacetime theory that obeys general relativity and contains spinors, if compatible with octonions, could also be a unified theory of everything

I also note Noncommutative Geometry Spectral Triplesin connection with both octonions and loop quantum gravity

there is also included

### A topological model of composite preons from the minimal ideals of two Clifford algebras

Niels G. GresnigtWe demonstrate a direct correspondence between the basis states of the minimal ideals of the complex Clifford algebras Cℓ(6) and Cℓ(4), shown earlier to transform as a single generation of leptons and quarks under the Standard Model's unbroken SU(3)c×U(1)em and SU(2)L gauge symmetries respectively, and a simple topologically-based toy model in which leptons, quarks, and gauge bosons are represented as elements of the braid group B3.

It was previously shown that mapping the basis states of the minimal left ideals of Cℓ(6) to specific braids replicates precisely the simple topological structure describing electrocolor symmetries in an existing topological preon model. This paper extends these results to incorporate the chiral weak symmetry by including a Cℓ(4) algebra, and identifying the basis states of the minimal right ideals with simple braids. The braids corresponding to the charged vector bosons are determined, and it is demonstrated that weak interactions can be described via the composition of braids.

Comments: 11 pages

Subjects: General Physics (physics.gen-ph)

Cite as: arXiv:2004.11140 [physics.gen-ph]

(or arXiv:2004.11140v2 [physics.gen-ph] for this version)

https://doi.org/10.48550/arXiv.2004.11140

which cited Sundance Bilson-Thompson "Quantum gravity and the standard model".

*Class. Quantum Grav*.**24**(16): 3975–3993. arXiv:hep-th/0603022
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