The standard model from loop quantum gravity via spinors octonions

[kodama]

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/

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. Gresnigt

We 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

 

[mitchell porter]

The standard model contains fermions built from spinor variables. But most of these papers are either using spinors to construct bosonic variables (e.g. all the LQG papers at the start), or else they are "identifying" various algebras with the standard model fermions, without actually making them fermionic. From what I can see, only the papers by Krasnov construct true fermions.

As for LQG itself, I have previously expressed my opinion that LQG's "canonical quantization" (which is radically different from what that usually means in QFT) is the wrong way to quantize Ashtekar variables. Instead, the more orthodox approach taken by some Russians in 1998 is much more promising. It doesn't do the things that people want LQG to do (provide a nonperturbative completion, discretize space), so in a sense it's just another low-energy effective field theory of quantum gravity, but nonetheless I would give it much much better odds of being the right path. So I would like to first see this Russian method (which is just an application of the well-known BRST method of quantization) applied to "standard model coupled to Ashtekar variables", and then we would be much better placed to evaluate these more speculative algebraic proposals.

 

[kodama]

The standard model contains fermions built from spinor variables. But most of these papers are either using spinors to construct bosonic variables (e.g. all the LQG papers at the start), or else they are "identifying" various algebras with the standard model fermions, without actually making them fermionic. From what I can see, only the papers by Krasnov construct true fermions.

As for LQG itself, I have previously expressed my opinion that LQG's "canonical quantization" (which is radically different from what that usually means in QFT) is the wrong way to quantize Ashtekar variables. Instead, the more orthodox approach taken by some Russians in 1998 is much more promising. It doesn't do the things that people want LQG to do (provide a nonperturbative completion, discretize space), so in a sense it's just another low-energy effective field theory of quantum gravity, but nonetheless I would give it much much better odds of being the right path. So I would like to first see this Russian method (which is just an application of the well-known BRST method of quantization) applied to "standard model coupled to Ashtekar variables", and then we would be much better placed to evaluate these more speculative algebraic proposals.

mitchell porter
I encourage you personally to write papers that expand on "this Russian method (which is just an application of the well-known BRST method of quantization) applied to "standard model coupled to Ashtekar variables",

Other method of quantization applied to "standard model coupled to Ashtekar variables" besides canonical quantization applied to loop quantum gravity
I think that's a great idea and suggestion.

I wonder if you can have unbroken supersymmetry, only applying to spinors, but not to standard model particles, which are composites of spinors.

my other issue though is any theory of quantum gravity and quantum spacetime, could you combine both Noncommutative Geometry Spectral Triple and octonions and spinors,

you mention LQG and Russian methods of BRST to Ashketar variables, but what about spinfoam?

to what extent does spinfoam exist as a viable theory of quantum gravity and quantum spacetime independent of canonical LQG? even if canonical LQG is "wrong" does that imply spinfoam is also wrong or could spinfoam exist independent of LQG?

if spinfoam can exist on its own and independent of LQG, these papers on spinfoam could also show a merger of these major theories of octonions and spinors, noncommutative geometry, quantum gravity and spacetime, and topology change and exotic smoothness and braiding

there is this paper combining spin foam with

arXiv:1005.1057 (math-ph)

[Submitted on 6 May 2010]

Spin Foams and Noncommutative Geometry​

Domenic Denicola (Caltech), Matilde Marcolli (Caltech), Ahmad Zainy al-Yasry (ICTP)

We extend the formalism of embedded spin networks and spin foams to include topological data that encode the underlying three-manifold or four-manifold as a branched cover. These data are expressed as monodromies, in a way similar to the encoding of the gravitational field via holonomies. We then describe convolution algebras of spin networks and spin foams, based on the different ways in which the same topology can be realized as a branched covering via covering moves, and on possible composition operations on spin foams. We illustrate the case of the groupoid algebra of the equivalence relation determined by covering moves and a 2-semigroupoid algebra arising from a 2-category of spin foams with composition operations corresponding to a fibered product of the branched coverings and the gluing of cobordisms. The spin foam amplitudes then give rise to dynamical flows on these algebras, and the existence of low temperature equilibrium states of Gibbs form is related to questions on the existence of topological invariants of embedded graphs and embedded two-complexes with given properties. We end by sketching a possible approach to combining the spin network and spin foam formalism with matter within the framework of spectral triples in noncommutative geometry.

Comments:	48 pages LaTeX, 30 PDF figures
Subjects:	Mathematical Physics (math-ph); General Relativity and Quantum Cosmology (gr-qc)
MSC classes:	81T75
Cite as:	arXiv:1005.1057 [math-ph]
	(or arXiv:1005.1057v1 [math-ph] for this version)
	https://doi.org/10.48550/arXiv.1005.1057

and spin foam with spinors

arXiv:gr-qc/0609040 (gr-qc)

[Submitted on 12 Sep 2006 (v1), last revised 22 Dec 2006 (this version, v2)]

Fermions in three-dimensional spinfoam quantum gravity​

Winston Fairbairn (CPT)

We study the coupling of massive fermions to the quantum mechanical dynamics of spacetime emerging from the spinfoam approach in three dimensions. We first recall the classical theory before constructing a spinfoam model of quantum gravity coupled to spinors. The technique used is based on a finite expansion in inverse fermion masses leading to the computation of the vacuum to vacuum transition amplitude of the theory. The path integral is derived as a sum over closed fermionic loops wrapping around the spinfoam. The effects of quantum torsion are realised as a modification of the intertwining operators assigned to the edges of the two-complex, in accordance with loop quantum gravity. The creation of non-trivial curvature is modelled by a modification of the pure gravity vertex amplitudes. The appendix contains a review of the geometrical and algebraic structures underlying the classical coupling of fermions to three dimensional gravity.

Comments:	40 pages, 3 figures, version accepted for publication in GERG
Subjects:	General Relativity and Quantum Cosmology (gr-qc)
Cite as:	arXiv:gr-qc/0609040

arXiv:1201.2120 (gr-qc)
[Submitted on 10 Jan 2012 (v1), last revised 15 Apr 2012 (this version, v2)]
Spinors and Twistors in Loop Gravity and Spin Foams
Maite Dupuis, Simone Speziale, Johannes Tambornino
Spinorial tools have recently come back to fashion in loop gravity and spin foams. They provide an elegant tool relating the standard holonomy-flux algebra to the twisted geometry picture of the classical phase space on a fixed graph, and to twistors. In these lectures we provide a brief and technical introduction to the formalism and some of its applications.



Comments: 16 pages; to appear in the Proceedings of the 3rd Quantum Gravity and Quantum Geometry School, February 28 - March 13, 2011 Zakopane, Poland. v2: minor amendments


Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th)

Cite as: arXiv:1201.2120 [gr-qc]
(or arXiv:1201.2120v2 [gr-qc] for this version)

https://doi.org/10.48550/arXiv.1201.2120

spinfoam contains spinors, spinfoam supports "spin foam formalism with matter within the framework of spectral triples in noncommutative geometry" and spinfoam and spinors plus octonions gives standard model coupled to gravity


also worth mentioning

spin networks/spin foam supports topology change

arXiv:1111.1252 (gr-qc)
[Submitted on 4 Nov 2011 (v1), last revised 24 Apr 2012 (this version, v2)]
Topspin Networks in Loop Quantum Gravity
Christopher L. Duston

We discuss the extension of loop quantum gravity to topspin networks, a proposal which allows topological information to be encoded in spin networks. We will show that this requires minimal changes to the phase space, C*-algebra and Hilbert space of cylindrical functions. We will also discuss the area and Hamiltonian operators, and show how they depend on the topology. This extends the idea of "background independence" in loop quantum gravity to include topology as well as geometry. It is hoped this work will confirm the usefulness of the topspin network formalism and open up several new avenues for research into quantum gravity.

Comments: 33 pages, significantly updated and improved
Subjects: General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)
Cite as: arXiv:1111.1252 [gr-qc]
(or arXiv:1111.1252v2 [gr-qc] for this version)

https://doi.org/10.48550/arXiv.1111.1252

arXiv:2106.14188 (gr-qc)
[Submitted on 27 Jun 2021]
Including topology change in Loop Quantum Gravity with topspin network formalism with application to homogeneous and isotropic cosmology
Mattia Villani


We apply topspin network formalism to Loop Quantum Gravity in order to include in the theory the possibility of changes in the topology of spacetime. We apply this formalism to three toy models: with the first, we find that the topology can actually change due to the action of the Hamiltonian constraint and with the second we find that the final state might be a superposition of states with different topologies. In the third and last application, we consider an homogeneous and isotropic Universe, calculating the difference equation that describes the evolution of the system and which are the final topological states after the action of the Hamiltonian constraint. For this last case, we also calculate the transition amplitudes and probabilities from the initial to the final states.

Comments: Accepted for the pubblication on Class. Quantum Grav
Subjects: General Relativity and Quantum Cosmology (gr-qc)
Cite as: arXiv:2106.14188 [gr-qc]
(or arXiv:2106.14188v1 [gr-qc] for this version)

https://doi.org/10.48550/arXiv.2106.14188
Related DOI:
https://doi.org/10.1088/1361-6382/ac0e1a

and topology change can lead to particles via Bilson Thompson braiding and octonions

[Submitted on 16 Oct 2019]
Braids, 3-manifolds, elementary particles: number theory and symmetry in particle physics
Torsten Asselmeyer-Maluga

n this paper, we will describe a topological model for elementary particles based on 3-manifolds. Here, we will use Thurston's geometrization theorem to get a simple picture: fermions as hyperbolic knot complements (a complement C(K)=S3∖(K×D2) of a knot K carrying a hyperbolic geometry) and bosons as torus bundles. In particular, hyperbolic 3-manifolds have a close connection to number theory (Bloch group, algebraic K-theory, quaternionic trace fields), which~will be used in the description of fermions. Here, we choose the description of 3-manifolds by branched covers. Every 3-manifold can be described by a 3-fold branched cover of S3 branched along a knot. In case of knot complements, one will obtain a 3-fold branched cover of the 3-disk D3 branched along a 3-braid or 3-braids describing fermions. The whole approach will uncover new symmetries as induced by quantum and discrete groups. Using the Drinfeld--Turaev quantization, we will also construct a quantization so that quantum states correspond to knots. Particle properties like the electric charge must be expressed by topology, and we will obtain the right spectrum of possible values. Finally, we will get a connection to recent models of Furey, Stoica and Gresnigt using octonionic and quaternionic algebras with relations to 3-braids (Bilson--Thompson model).

Comments: 37 pages, 9 Figures The paper was planned to be self-contained. Therefore, I used only minor results and material from the papers 1812.08158, 1801.10419, 1709.03314, 1502.02087, 1006.2230 and 1601.06436
Subjects: General Physics (physics.gen-ph); High Energy Physics - Theory (hep-th)
Cite as: arXiv:1910.09966 [physics.gen-ph]
(or arXiv:1910.09966v1 [physics.gen-ph] for this version)

https://doi.org/10.48550/arXiv.1910.09966

topology change can also give rise to particles in a spinfoam type gravity theory

 

[kodama]

I agree that it's an interesting idea that should be explored. However, my personal circumstances are utterly destructive and I am lucky if I get to think about physics, let alone write papers about it.

there are theories connecting Ashtekar variable to the weak force SU(2)

does "this Russian method (which is just an application of the well-known BRST method of quantization) also included ?

are there other accepted ideas for connecting Ashtekar variable to other method of quantization, joined with the weak force ?

 

[mitchell porter]

there are theories connecting Ashtekar variable to the weak force SU(2)

These theories are based on taking a space-time symmetry (e.g. the Lorentz group in Minkowski space-time, the rotation group SO(4) or its double cover Spin(4) in Euclidean four-space), complexifying it, then factorizing this complexified group into a right-handed and a left-handed factor. The right-handed group is gives rise to a gauge theory of gravity, the left-handed group is a complexified version of the gauge field for the weak force.

All of them except Woit work with the Lorentz group, e.g.

"Gravi-Weak Unification" (Nesti, Percacci 2007)

"Gravitational origin of the weak interaction's chirality" (Alexander, Marciano, Smolin 2012)

Woit proposes the Euclidean group in

"Euclidean Twistor Unification" (Woit 2021)

in the context of working in twistor space, which he hopes can give him the other gauge groups of the standard model too.

The Lorentzian graviweak unifications all seem to posit an unbroken phase of the theory in which space is purely topological (no metric), and then a broken phase in which the right-handed group gives rise to a gauge theory of gravity (e.g. Lorentz Poincare gauge gravity, or general relativity in Ashtekar variables) on a space-time manifold with metric, and the left-handed group gives rise to the weak-force gauge field, propagating within that emergent metrical space-time.

It's not clear to me that any of these are well-defined as quantum theories, at least in the metric phase. Gauge theories of gravity are based on noncompact gauge groups, and that's already a problem quantum-mechanically. Complexification also makes a gauge group noncompact, so in graviweak unification the weak gauge field also becomes noncompact (LQG employs "reality conditions" which are meant to restrict to a compact real-valued subgroup, and the first two papers above want to un-complexify the complexified weak force in the same way).

The nature of the transition from topological gravity to metrical gravity is also unclear to me (Vafa and other string theorists have also worked on this, without any widely accepted results).

You might also worry that the Coleman-Mandula theorem is somehow being violated, but that involves having spin-2 gravitons and I'm not even clear on how they arise within these Yang-Mills-like theories of gravity (since Yang-Mills gives rise to spin-1 gauge bosons).

In Woit's case, starting with Euclidean space introduces additional technical problems to do with the very definition of chiral quantum fields in Euclidean space (see the appendices to his paper).

So I have a lot of questions and misgivings about what is being proposed in these papers. I don't feel I can endorse any of them as definitely being mathematically well-defined theories, whereas I do have confidence that the basic constructions of the usual perturbative gravity (i.e. gravitons) and perturbative string theory (i.e. the "trouser diagrams" and so forth that you see even in popularizations) are mathematically well-defined (which is a separate question from whether they describe physical reality).

Having doubts is not the same as being sure that nothing here works. And the idea is interesting enough that I came up with yet another angle on it - instead of working with spin foams, which is the usual way to quantize the Ashtekar variables in LQG, start with the BRST quantization in

"Path integral for the Hilbert-Palatini and Ashtekar gravity" (Alexandrov, Vassilievich 1998)

BRST quantization is a completely standard method for nonabelian quantum field theory. So the bottom-up approach here would be as follows. Consider general relativity coupled to an SU(2) gauge field. Normally one would write general relativity using the metric, and end up with a perturbative theory of gravitons and massless weak bosons. The plan instead would be to write the gravitational part in Ashtekar variables, and quantize it in the fashion of Alexandrov and Vassilievich. This would still be incomplete at high energies, and I further suggest that one could try to complete it via ambitwistor strings, a version of string theory which can reproduce the chiral scattering formulas for Yang-Mills and gravity, but whose embedding into standard string theory is a bit uncertain.

That's my graviweak research program, and if it worked, one could try e.g. to obtain the standard model a la Woit, maybe from octonionic supertwistors as we have discussed in other threads. But I have plenty of doubts here too. I'm not even sure that Alexandrov and Vassilievich's method works. No one has built on that paper of theirs, so it hasn't had the reality check that comes when new people try to apply someone's idea.

 

[kodama]

These theories are based on taking a space-time symmetry (e.g. the Lorentz group in Minkowski space-time, the rotation group SO(4) or its double cover Spin(4) in Euclidean four-space), complexifying it, then factorizing this complexified group into a right-handed and a left-handed factor. The right-handed group is gives rise to a gauge theory of gravity, the left-handed group is a complexified version of the gauge field for the weak force.

All of them except Woit work with the Lorentz group, e.g.

"Gravi-Weak Unification" (Nesti, Percacci 2007)

"Gravitational origin of the weak interaction's chirality" (Alexander, Marciano, Smolin 2012)

Woit proposes the Euclidean group in

"Euclidean Twistor Unification" (Woit 2021)

in the context of working in twistor space, which he hopes can give him the other gauge groups of the standard model too.

The Lorentzian graviweak unifications all seem to posit an unbroken phase of the theory in which space is purely topological (no metric), and then a broken phase in which the right-handed group gives rise to a gauge theory of gravity (e.g. Lorentz Poincare gauge gravity, or general relativity in Ashtekar variables) on a space-time manifold with metric, and the left-handed group gives rise to the weak-force gauge field, propagating within that emergent metrical space-time.

It's not clear to me that any of these are well-defined as quantum theories, at least in the metric phase. Gauge theories of gravity are based on noncompact gauge groups, and that's already a problem quantum-mechanically. Complexification also makes a gauge group noncompact, so in graviweak unification the weak gauge field also becomes noncompact (LQG employs "reality conditions" which are meant to restrict to a compact real-valued subgroup, and the first two papers above want to un-complexify the complexified weak force in the same way).

The nature of the transition from topological gravity to metrical gravity is also unclear to me (Vafa and other string theorists have also worked on this, without any widely accepted results).

You might also worry that the Coleman-Mandula theorem is somehow being violated, but that involves having spin-2 gravitons and I'm not even clear on how they arise within these Yang-Mills-like theories of gravity (since Yang-Mills gives rise to spin-1 gauge bosons).

In Woit's case, starting with Euclidean space introduces additional technical problems to do with the very definition of chiral quantum fields in Euclidean space (see the appendices to his paper).

So I have a lot of questions and misgivings about what is being proposed in these papers. I don't feel I can endorse any of them as definitely being mathematically well-defined theories, whereas I do have confidence that the basic constructions of the usual perturbative gravity (i.e. gravitons) and perturbative string theory (i.e. the "trouser diagrams" and so forth that you see even in popularizations) are mathematically well-defined (which is a separate question from whether they describe physical reality).

Having doubts is not the same as being sure that nothing here works. And the idea is interesting enough that I came up with yet another angle on it - instead of working with spin foams, which is the usual way to quantize the Ashtekar variables in LQG, start with the BRST quantization in

"Path integral for the Hilbert-Palatini and Ashtekar gravity" (Alexandrov, Vassilievich 1998)

BRST quantization is a completely standard method for nonabelian quantum field theory. So the bottom-up approach here would be as follows. Consider general relativity coupled to an SU(2) gauge field. Normally one would write general relativity using the metric, and end up with a perturbative theory of gravitons and massless weak bosons. The plan instead would be to write the gravitational part in Ashtekar variables, and quantize it in the fashion of Alexandrov and Vassilievich. This would still be incomplete at high energies, and I further suggest that one could try to complete it via ambitwistor strings, a version of string theory which can reproduce the chiral scattering formulas for Yang-Mills and gravity, but whose embedding into standard string theory is a bit uncertain.

That's my graviweak research program, and if it worked, one could try e.g. to obtain the standard model a la Woit, maybe from octonionic supertwistors as we have discussed in other threads. But I have plenty of doubts here too. I'm not even sure that Alexandrov and Vassilievich's method works. No one has built on that paper of theirs, so it hasn't had the reality check that comes when new people try to apply someone's idea.

there is this

arXiv:2407.13807 (hep-th)
[Submitted on 18 Jul 2024]
Quantizing the Bosonic String on a Loop Quantum Gravity Background
Deepak Vaid, Luigi Teixeira de Sousa

With the goal of understanding whether or not it is possible to construct a string theory which is consistent with loop quantum gravity (LQG), we study alternate versions of the Nambu-Goto action for a bosonic string. We consider two types of modifications. The first is a phenomenological action based on the observation that LQG tells us that areas of two-surfaces are operators in quantum geometry and are bounded from below. This leads us to a string action which is similar to that of bimetric gravity. We provide formulations of the bimetric string for both the Nambu-Goto (second order) and Polyakov (first order) formulations. We explore the classical solutions of this action and its quantization and relate it to the conventional string solutions.
The second is an action in which the background geometry is described in terms of the pullback of the connection which describes the bulk geometry to the worldsheet. The resulting action is in the form of a gauged sigma model, where the spacetime co-ordinates are now vectors which transform under ISO(D,1). We find that for the particular case of a constant background connection the action reduces to the bimetric action discussed above. We discuss classical solutions and quantization strategies for this action and its implications for the broader program of unifying string theory and loop quantum gravity.

Comments: Comments/criticisms very welcome; 23 pages, no figures
Subjects: High Energy Physics - Theory (hep-th)
Cite as: arXiv:2407.13807 [hep-th]
(or arXiv:2407.13807v1 [hep-th] for this version)

https://doi.org/10.48550/arXiv.2407.13807

This would still be incomplete at high energies, and I further suggest that one could try to complete it via ambitwistor strings, a version of string theory which can reproduce the chiral scattering formulas for Yang-Mills and gravity, but whose embedding into standard string theory is a bit uncertain.

unifying string theory and loop quantum gravity

 

[kodama]

These theories are based on taking a space-time symmetry (e.g. the Lorentz group in Minkowski space-time, the rotation group SO(4) or its double cover Spin(4) in Euclidean four-space), complexifying it, then factorizing this complexified group into a right-handed and a left-handed factor. The right-handed group is gives rise to a gauge theory of gravity, the left-handed group is a complexified version of the gauge field for the weak force.

All of them except Woit work with the Lorentz group, e.g.

"Gravi-Weak Unification" (Nesti, Percacci 2007)

"Gravitational origin of the weak interaction's chirality" (Alexander, Marciano, Smolin 2012)

Woit proposes the Euclidean group in

"Euclidean Twistor Unification" (Woit 2021)

in the context of working in twistor space, which he hopes can give him the other gauge groups of the standard model too.

The Lorentzian graviweak unifications all seem to posit an unbroken phase of the theory in which space is purely topological (no metric), and then a broken phase in which the right-handed group gives rise to a gauge theory of gravity (e.g. Lorentz Poincare gauge gravity, or general relativity in Ashtekar variables) on a space-time manifold with metric, and the left-handed group gives rise to the weak-force gauge field, propagating within that emergent metrical space-time.

It's not clear to me that any of these are well-defined as quantum theories, at least in the metric phase. Gauge theories of gravity are based on noncompact gauge groups, and that's already a problem quantum-mechanically. Complexification also makes a gauge group noncompact, so in graviweak unification the weak gauge field also becomes noncompact (LQG employs "reality conditions" which are meant to restrict to a compact real-valued subgroup, and the first two papers above want to un-complexify the complexified weak force in the same way).

The nature of the transition from topological gravity to metrical gravity is also unclear to me (Vafa and other string theorists have also worked on this, without any widely accepted results).

You might also worry that the Coleman-Mandula theorem is somehow being violated, but that involves having spin-2 gravitons and I'm not even clear on how they arise within these Yang-Mills-like theories of gravity (since Yang-Mills gives rise to spin-1 gauge bosons).

In Woit's case, starting with Euclidean space introduces additional technical problems to do with the very definition of chiral quantum fields in Euclidean space (see the appendices to his paper).

So I have a lot of questions and misgivings about what is being proposed in these papers. I don't feel I can endorse any of them as definitely being mathematically well-defined theories, whereas I do have confidence that the basic constructions of the usual perturbative gravity (i.e. gravitons) and perturbative string theory (i.e. the "trouser diagrams" and so forth that you see even in popularizations) are mathematically well-defined (which is a separate question from whether they describe physical reality).

Having doubts is not the same as being sure that nothing here works. And the idea is interesting enough that I came up with yet another angle on it - instead of working with spin foams, which is the usual way to quantize the Ashtekar variables in LQG, start with the BRST quantization in

"Path integral for the Hilbert-Palatini and Ashtekar gravity" (Alexandrov, Vassilievich 1998)

BRST quantization is a completely standard method for nonabelian quantum field theory. So the bottom-up approach here would be as follows. Consider general relativity coupled to an SU(2) gauge field. Normally one would write general relativity using the metric, and end up with a perturbative theory of gravitons and massless weak bosons. The plan instead would be to write the gravitational part in Ashtekar variables, and quantize it in the fashion of Alexandrov and Vassilievich. This would still be incomplete at high energies, and I further suggest that one could try to complete it via ambitwistor strings, a version of string theory which can reproduce the chiral scattering formulas for Yang-Mills and gravity, but whose embedding into standard string theory is a bit uncertain.

That's my graviweak research program, and if it worked, one could try e.g. to obtain the standard model a la Woit, maybe from octonionic supertwistors as we have discussed in other threads. But I have plenty of doubts here too. I'm not even sure that Alexandrov and Vassilievich's method works. No one has built on that paper of theirs, so it hasn't had the reality check that comes when new people try to apply someone's idea.

What are the physics implications of a BRST quantization of Ashketar variables for quantum gravity, and why hasn't it attracted more attention from the Loop Gravity community?

 

[mitchell porter]

What are the physics implications of a BRST quantization of Ashketar variables for quantum gravity, and why hasn't it attracted more attention from the Loop Gravity community?

Because it is more in the spirit of the usual approach to quantum gravity. The usual approach starts with the metric variables of general relativity and treats the metric as just another field on a continuum background. LQG starts with Ashtekar's connection variables, and then tries to recover the space-time continuum from the combinatorics of finite spin foams. Alexandrov and Vassilievich also start with Ashtekar's connection, but they are treating it as a field on space-time.

It was actually Ashtekar and some of his colleagues who first worked out the BRST algebra for the Ashtekar variables, in papers around 1990. But as far as I know, that has never played a role in LQG, and perhaps it simply can't, given the paths to quantization that the LQG community favored.

Incidentally, a BRST-quantized Ashtekar gravity does not imply that space-time remains a smooth 4d continuum all the way down; such a theory is probably "UV incomplete", the same as perturbative quantum gravity based on the metric, meaning that it runs out of validity at the highest energies. The question is what the UV completion should be. Strings are a UV completion of the usual perturbative gravity, which is why I suggested that ambitwistor strings might be a UV completion for BRST-quantized Ashtekar gravity (ambitwistors are good for the chiral- or helicity-based approach to physics, of which the Ashtekar variables are an example).

 

[kodama]

what about using Batalin–Vilkovisky formalism for Ashtekar gravity?

 

[mitchell porter]

what about using Batalin–Vilkovisky formalism for Ashtekar gravity?

It shouldn't be necessary. BRST adds "ghost fields" to Yang-Mills. BV also adds "ghosts for ghosts" to a quantized theory, so BRST can be extended to theories with non-polynomial interactions, like general relativity. But the Ashtekar variables allow the interactions of general relativity to be expressed in polynomial form, so I would think BRST is enough.

 

[kodama]

It shouldn't be necessary. BRST adds "ghost fields" to Yang-Mills. BV also adds "ghosts for ghosts" to a quantized theory, so BRST can be extended to theories with non-polynomial interactions, like general relativity. But the Ashtekar variables allow the interactions of general relativity to be expressed in polynomial form, so I would think BRST is enough.

standard LQG used Dirac quantization of Ashketar variables
so what is "wrong" with Dirac quantization?

does BRST solve problems with LQG, for example recovery of the semi classical limit ?

btw are there any other methods for quantization of Ashketar variables besides Dirac quantization, BRST or Batalin–Vilkovisky formalism that succeeds ?

 

[mitchell porter]

There's nothing wrong with Dirac quantization (usually called canonical quantization), but what was done in canonical LQG was different in a variety of ways from the Dirac quantization of other quantum fields. In non-gravitational QFT, you would consider the background space-time to be fixed (usually Minkowski space), and then quantize e.g. the Fourier modes of the field, corresponding to plane waves of different wavelengths.

In LQG, you don't even start with a metric, instead you just have the Ashtekar complex gauge field. LQG's canonical quantization procedure then focuses on "Wilson loop" observables (how much the field vectors "rotate" as you follow a particular closed loop), builds a Hilbert space from these that is infinitely larger than usual (with an uncountably infinite number of basis vectors); then defines a subset of that Hilbert space as the physical states, in a way that seems to lose crucial information, and should also make impossible certain effects (anomalous symmetry breaking) which we know actually occur in nature.

If we want to compare canonical quantization in LQG to something else, I think it would be how quantization is done in "topological QFTs", because they also are about quantizing fields on a manifold without a metric. But TQFTs end up with a finite-dimensional Hilbert space rather than an uncountably large one. Understanding the reasons for this difference would probably shed some light on why canonical LQG developed as it did.

 

[kodama]

could you

There's nothing wrong with Dirac quantization (usually called canonical quantization), but what was done in canonical LQG was different in a variety of ways from the Dirac quantization of other quantum fields. In non-gravitational QFT, you would consider the background space-time to be fixed (usually Minkowski space), and then quantize e.g. the Fourier modes of the field, corresponding to plane waves of different wavelengths.

In LQG, you don't even start with a metric, instead you just have the Ashtekar complex gauge field. LQG's canonical quantization procedure then focuses on "Wilson loop" observables (how much the field vectors "rotate" as you follow a particular closed loop), builds a Hilbert space from these that is infinitely larger than usual (with an uncountably infinite number of basis vectors); then defines a subset of that Hilbert space as the physical states, in a way that seems to lose crucial information, and should also make impossible certain effects (anomalous symmetry breaking) which we know actually occur in nature.

If we want to compare canonical quantization in LQG to something else, I think it would be how quantization is done in "topological QFTs", because they also are about quantizing fields on a manifold without a metric. But TQFTs end up with a finite-dimensional Hilbert space rather than an uncountably large one. Understanding the reasons for this difference would probably shed some light on why canonical LQG developed as it did.

does BRST-quantized Ashtekar gravity approach have these issues ?

could Ads/Cft bring up another quantization Ashtekar gravity ?

 

[mitchell porter]

There are numerous techniques of quantization, like canonical quantization, BRST or BV, symplectic quantization, deformation quantization, and so on. These are ways to construct a quantum theory, and very often there are several techniques capable of giving you the same theory. And on the other hand, often there are multiple quantum theories that can be constructed from the same beginning via the same technique, e.g. there can be families of deformation quantizations which have different values of a deformation parameter.

The LQG canonical quantization has been dubbed "polymer quantization", and this is a method which apparently always produces something distinct from other methods of quantization, even for a system as simply as the harmonic oscillator. This is considered a point heavily against it, since there is a standard quantum theory of the harmonic oscillator, it's ubiquitous in applications of quantum mechanics and quantum field theory, whereas I'm not aware of any use for the "polymer-quantized harmonic oscillator".

So canonical LQG is its own exercise, the product of a distinct quantization technique which has no connections to other parts of physics that I know. Then there are the spin foams, which should arise as a perturbative calculation in some quantum theory, but we don't know what theory.

Then as I've mentioned, there are the topological QFTs, and the mainstream perturbative quantum gravity, based on canonical quantization of the GR metric. I'll also add that within all these categories, you also get different quantum theories based on different choices of the fundamental fields and their interactions. Thus general relativity, unimodular gravity, conformal gravity, and so on. These are all distinct field theories of gravity which can be quantized canonically, and which have various interrelations.

Canonical LQG came about by starting with general relativity, rewriting the fields in Ashtekar variables, and then quantizing in a way which was inspired by topological QFTs, but which is actually what we might now call polymer quantization. The Russian paper that I keep touting, also begins with the Ashtekar change of variables, but instead constructs a BRST path integral, which is one of those "mainstream" methods of quantization that gives you quantum field theories which are e.g. compatible with the usual quantum harmonic oscillator. So it should give you a theory much closer to the usual perturbative quantum gravity, which is calculable to some extent, but which is also "UV-incomplete", i.e. not a final or ultimate theory of gravity. The most common idea is that string theory provides the UV completion of quantum GR, and I have suggested that the ambitwistor strings might provide a UV completion for quantum GR in Ashtekar variables.

Regarding AdS/CFT, it's only the AdS side which is gravitational. So you could consider your favorite quantum gravity theory in AdS, and ask what its CFT dual is. You could presumably do that for Ashtekar variables in the AdS, and an answer would tell you something, but it might not help you for real-world applications, where we are in flat or De Sitter space.

 

[kodama]

so what happens if you try symplectic quantization, deformation quantization of Ashtekar variables?
Is there any resources for BRSTof Ashtekar variables? what precludes it from a UV completion for quantum GR in Ashtekar variables.

why'd you think the ambitwistor strings might provide a UV completion

 

[kodama]

would Ryu–Takayanagi conjecture apply for Ashtekar variables?

 

[mitchell porter]

My starting assumption for all of these is that they will produce the same theory of quantum gravity that was first worked out in the 1960s. That theory was found by starting with GR's metric and treating it as just another quantum field, and that resulted in Feynman diagrams of gravitons, and a belief that the theory is UV-incomplete because non-renormalizable. The Ashtekar approach starts with a connection rather than a metric, but still, it's just a change of variables. If you plot a 2d curve in Cartesian coordinates and then switch to polar coordinates, the curve itself doesn't change, and it is reasonable to expect that quantizing general relativity in Ashtekar variables should just produce the same theory, expressed differently. As I've discussed, LQG took a different path, treating it as a kind of polymer-quantized topological field theory. If we don't do that, but instead use a standard quantization technique, we shouldn't expect a theory that's fundamentally different from standard quantum gravity. Instead, we might hope for new insights about that standard theory, perhaps including new perspectives on how to complete it.

So, to recapitulate, there's a standard theory of quantum gravity, obtained by perturbatively quantizing the metric of general relativity. The most common belief is that it is UV-incomplete and that it is completed by string theory. An interesting alternative idea is that it is non-perturbatively UV-complete; that's asymptotic safety. And then we have a zillion other ideas, with varying degrees of concreteness and credibility. Incidentally, my position on string theory is that it is basically on the right track, but one, maybe two things need to change: the phenomenologists need to explain certain things that they presently ignore, like MOND and the Koide formula (and that will require them to focus on different kinds of stringy constructions); and, with less certainty, the theorists may need to develop parts of string theory that are currently neglected or even unknown, in order for the phenomenologists to build the right kinds of model. So I am a kind of string partisan, but hopefully I can still rationally evaluate proposals that are completely non-stringy.

Hopefully this makes it clear why I would say that gravity in Ashtekar variables, when quantized in a normal way, would lead to a UV-incomplete theory that is completed by strings. It's just a simple extrapolation from the paradigm I just outlined. As for why those would be "ambitwistor strings" specifically - as Peter Woit could tell you, twistors are deeply related to the "chiral" perspective on physics, of which the Ashtekar variables are an important example. Innumerable formulas for the scattering of particles that are mostly spinning in the same direction, in both Yang-Mills gauge theory and general relativity, have been found by using twistor variables. Strings in ambitwistor space seem to be the ones that are closest to this side of field theory.

Nonetheless, all this is just a hypothesis. People have to do the work, of quantizing Ashtekar gravity in these various ways, and see what comes of it. Maybe there will be obstructions, mathematical barriers, that prevent certain forms of quantization from being carried through at all. Maybe there will be quantization ambiguities which imply the existence of several distinct quantum theories. Maybe other issues will arise.

I've already mentioned the 1998 Russian paper in which a BRST path integral for Ashtekar variables is studied. There are works from 1997 and 2008 which study deformation quantization using Ashtekar variables. Ashtekar himself has coauthored works on symplectic quantization and on the symplectic structure of general relativity. (Everything symplectic is about "phase space", the doubling of coordinates in which you specify position and momentum, or analogous pairs like angle and angular momentum.) I can't testify as to their quality, but they offer a starting point for each of these possibilities.

 

[kodama]

My starting assumption for all of these is that they will produce the same theory of quantum gravity that was first worked out in the 1960s. That theory was found by starting with GR's metric and treating it as just another quantum field, and that resulted in Feynman diagrams of gravitons, and a belief that the theory is UV-incomplete because non-renormalizable. The Ashtekar approach starts with a connection rather than a metric, but still, it's just a change of variables. If you plot a 2d curve in Cartesian coordinates and then switch to polar coordinates, the curve itself doesn't change, and it is reasonable to expect that quantizing general relativity in Ashtekar variables should just produce the same theory, expressed differently. As I've discussed, LQG took a different path, treating it as a kind of polymer-quantized topological field theory. If we don't do that, but instead use a standard quantization technique, we shouldn't expect a theory that's fundamentally different from standard quantum gravity. Instead, we might hope for new insights about that standard theory, perhaps including new perspectives on how to complete it.

So, to recapitulate, there's a standard theory of quantum gravity, obtained by perturbatively quantizing the metric of general relativity. The most common belief is that it is UV-incomplete and that it is completed by string theory. An interesting alternative idea is that it is non-perturbatively UV-complete; that's asymptotic safety. And then we have a zillion other ideas, with varying degrees of concreteness and credibility. Incidentally, my position on string theory is that it is basically on the right track, but one, maybe two things need to change: the phenomenologists need to explain certain things that they presently ignore, like MOND and the Koide formula (and that will require them to focus on different kinds of stringy constructions); and, with less certainty, the theorists may need to develop parts of string theory that are currently neglected or even unknown, in order for the phenomenologists to build the right kinds of model. So I am a kind of string partisan, but hopefully I can still rationally evaluate proposals that are completely non-stringy.

Hopefully this makes it clear why I would say that gravity in Ashtekar variables, when quantized in a normal way, would lead to a UV-incomplete theory that is completed by strings. It's just a simple extrapolation from the paradigm I just outlined. As for why those would be "ambitwistor strings" specifically - as Peter Woit could tell you, twistors are deeply related to the "chiral" perspective on physics, of which the Ashtekar variables are an important example. Innumerable formulas for the scattering of particles that are mostly spinning in the same direction, in both Yang-Mills gauge theory and general relativity, have been found by using twistor variables. Strings in ambitwistor space seem to be the ones that are closest to this side of field theory.

Nonetheless, all this is just a hypothesis. People have to do the work, of quantizing Ashtekar gravity in these various ways, and see what comes of it. Maybe there will be obstructions, mathematical barriers, that prevent certain forms of quantization from being carried through at all. Maybe there will be quantization ambiguities which imply the existence of several distinct quantum theories. Maybe other issues will arise.

I've already mentioned the 1998 Russian paper in which a BRST path integral for Ashtekar variables is studied. There are works from 1997 and 2008 which study deformation quantization using Ashtekar variables. Ashtekar himself has coauthored works on symplectic quantization and on the symplectic structure of general relativity. (Everything symplectic is about "phase space", the doubling of coordinates in which you specify position and momentum, or analogous pairs like angle and angular momentum.) I can't testify as to their quality, but they offer a starting point for each of these possibilities.

what theory would result from combining the asymptotic safety with BRST path integral for Ashtekar variables? would the non-perturbatively UV-complete; that's asymptotic safety be compatible with with BRST path integral for Ashtekar variables creating a viable theory of quantum gravity ?

 

[weirdoguy]

what theory would result from combining the asymptotic safety with BRST path integral for Ashtekar variables?

How can anyone tell if no one did it yet?

 

[mitchell porter]

what theory would result from combining the asymptotic safety with BRST path integral for Ashtekar variables?

In principle, asymptotic safety is not a postulate that you can freely add to a theory; it is a property that a theory inherently either has or doesn't have.

For example, the usual attitude toward the theory of quantum gravity that results from quantizing the GR metric, is that it has infinitely many undetermined couplings (arbitrarily large numbers of gravitons can interact at a single vertex, and each different such interaction can have an independent coupling constant), and that this makes it unpredictive.

But if it is asymptotically safe, that means that in the UV, it reduces to a predictive theory with a finite number of couplings. Asymptotic safety theorists are therefore saying that quantized GR, when considered in its totality, does actually behave differently than perturbation theory would lead you to believe. They are unable to demonstrate this for the full theory. Instead, they focus on just a few of the infinitely many interaction terms, and show that if you work with this radically truncated version of the full theory, sometimes it is unexpectedly renormalizable.

I have been proposing that quantum gravity in Ashtekar variables, done correctly, is just the usual metric-based theory of quantum gravity. If this is so, then the answer to whether it is asymptotically safe should be exactly the same too. Maybe one could look at one of those truncated versions of quantum gravity in the asymptotic safety literature, using Ashtekar variables. On the other hand, the study of asymptotic safety requires specialized methods of renormalization like the functional renormalization group, and one would have to see whether the special features of Ashtekar variables (the use of complex variables, the need for reality conditions) cause any problems for such techniques.

I hope it's clear that a lot of these questions about what happens when you combine theory X with method Z, could define an entire research program. To really know what happens may require nothing less than PhD-level calculations, or even technical insights that no one yet possesses. I'm just trying to give an idea of what would be "sensible" or "reasonable" to expect, given what we do already know.

 

[kodama]

In principle, asymptotic safety is not a postulate that you can freely add to a theory; it is a property that a theory inherently either has or doesn't have.

For example, the usual attitude toward the theory of quantum gravity that results from quantizing the GR metric, is that it has infinitely many undetermined couplings (arbitrarily large numbers of gravitons can interact at a single vertex, and each different such interaction can have an independent coupling constant), and that this makes it unpredictive.

But if it is asymptotically safe, that means that in the UV, it reduces to a predictive theory with a finite number of couplings. Asymptotic safety theorists are therefore saying that quantized GR, when considered in its totality, does actually behave differently than perturbation theory would lead you to believe. They are unable to demonstrate this for the full theory. Instead, they focus on just a few of the infinitely many interaction terms, and show that if you work with this radically truncated version of the full theory, sometimes it is unexpectedly renormalizable.

I have been proposing that quantum gravity in Ashtekar variables, done correctly, is just the usual metric-based theory of quantum gravity. If this is so, then the answer to whether it is asymptotically safe should be exactly the same too. Maybe one could look at one of those truncated versions of quantum gravity in the asymptotic safety literature, using Ashtekar variables. On the other hand, the study of asymptotic safety requires specialized methods of renormalization like the functional renormalization group, and one would have to see whether the special features of Ashtekar variables (the use of complex variables, the need for reality conditions) cause any problems for such techniques.

I hope it's clear that a lot of these questions about what happens when you combine theory X with method Z, could define an entire research program. To really know what happens may require nothing less than PhD-level calculations, or even technical insights that no one yet possesses. I'm just trying to give an idea of what would be "sensible" or "reasonable" to expect, given what we do already know.

why don't Asymptotic safety theorists create a theory of gravity that reproduced GR in the ir and is manifest Asymptotic safety in uv reduces to a predictive theory with a finite number of couplings. could for example spin foam or CDT or a connection based gravity reduces to a predictive theory with a finite number of couplings

what about gravity that uses complex variables or even space-time

 

[mitchell porter]

why don't Asymptotic safety theorists create a theory of gravity that reproduced GR in the ir and is manifest Asymptotic safety in uv

Let me clarify a few things about what asymptotic safety means... As you know, in quantum field theory, there is this procedure called renormalization that is part of making calculations. One aspect of renormalization is that it is anchored by the use of observed quantities like mass and charge. A theory may have all kinds of calculations that need renormalization; if all of them can be made meaningful, by the use of a finite number of such quantities, the theory is called "renormalizable".

Quantum general relativity is considered "non-renormalizable" because you can have any number of gravitons interacting at a point, and all these infinitely many different interactions appear to need independent renormalization constants. This form of quantum gravity still has some value as an "effective field theory", i.e. at "low energies", because only a finite number of the possible interactions are relevant there; but in the full theory, in the UV, all possible graviton interactions are relevant.

This would make it very hard to calculate anything in the UV, because no matter how many kinds of graviton interaction you had included in your calculation, there are infinitely many others that contribute to an unknown degree... The inability to calculate doesn't mean that it's impossible for reality to be like that, but it makes it hard to approach the topic scientifically.

Asymptotic safety (AS) is the idea that quantum general relativity might be fully renormalizable after all, that the apparently infinite number of distinct coupling constants is just an illusion arising from the "perturbative" approach of working in terms of gravitons, and that if you work instead in terms of the wavefunctional of the gravitational field, there's only finitely many coupling constants after all, and thus only finitely many renormalizations needed.

The dominant hypothesis in AS research is that GR itself is asymptotically safe; they just can't prove it. They can show UV fixed points in various highly truncated forms of quantum GR, but the critics don't believe those arguments apply to the full theory (as well as making other critiques of method 1 2).

I take you to be suggesting, what if you make the provability of AS, your priority in choosing a UV theory? That is, look for a UV theory which reduces to GR in the IR, and which by construction is easier to prove to be AS... It seems a fair idea, but then we have to ask, for what kind of theories can AS be proven? Approaches like spin foams and CDT seem like they would have the same problem as perturbative quantum GR, with its infinitely many kinds of graviton vertices - there are infinitely many distinct ways in which the spin foam or the dynamical triangulation could change its local topology, each with an independent coupling. Working in Ashtekar variables, on the other hand, seems like it has a chance of leading somewhere new.

This review lists a few "non-gravitational examples" of asymptotically safe theories (section 3).

Meanwhile, there is an argument (section VI here) that any kind of asymptotic safety is incompatible with black hole thermodynamics, since it suggests that n-dimensional quantum gravity in the UV is an n-dimensional conformal field theory (that would be the behavior at the UV fixed point), whereas black hole thermodynamics says that n-dimensional quantum gravity in some sense behaves like an (n-1)-dimensional field theory (since entropy depends on the area of the event horizon, not the volume of the black hole).

This sounds like a strong argument, but as always the details matter. One may find in the AS literature, claims that the effective dimensionality of space-time itself reduces to 2 dimensions in the UV; and on the other hand, there was some recent work in string theory on fixed points above the decompactification energy scale. The most profound way to approach this issue might be in a spirit similar to the "swampland" program, which is an attempt to identify what is and is not logically possible in quantum gravity. E.g. to what extent is the existence of a UV fixed point in quantum gravity, incompatible with holographic black-hole thermodynamics? And further progress would involve greater precision about what is meant by those two concepts.

 

[kodama]

Let me clarify a few things about what asymptotic safety means... As you know, in quantum field theory, there is this procedure called renormalization that is part of making calculations. One aspect of renormalization is that it is anchored by the use of observed quantities like mass and charge. A theory may have all kinds of calculations that need renormalization; if all of them can be made meaningful, by the use of a finite number of such quantities, the theory is called "renormalizable".

Quantum general relativity is considered "non-renormalizable" because you can have any number of gravitons interacting at a point, and all these infinitely many different interactions appear to need independent renormalization constants. This form of quantum gravity still has some value as an "effective field theory", i.e. at "low energies", because only a finite number of the possible interactions are relevant there; but in the full theory, in the UV, all possible graviton interactions are relevant.

This would make it very hard to calculate anything in the UV, because no matter how many kinds of graviton interaction you had included in your calculation, there are infinitely many others that contribute to an unknown degree... The inability to calculate doesn't mean that it's impossible for reality to be like that, but it makes it hard to approach the topic scientifically.

Asymptotic safety (AS) is the idea that quantum general relativity might be fully renormalizable after all, that the apparently infinite number of distinct coupling constants is just an illusion arising from the "perturbative" approach of working in terms of gravitons, and that if you work instead in terms of the wavefunctional of the gravitational field, there's only finitely many coupling constants after all, and thus only finitely many renormalizations needed.

The dominant hypothesis in AS research is that GR itself is asymptotically safe; they just can't prove it. They can show UV fixed points in various highly truncated forms of quantum GR, but the critics don't believe those arguments apply to the full theory (as well as making other critiques of method 1 2).

I take you to be suggesting, what if you make the provability of AS, your priority in choosing a UV theory? That is, look for a UV theory which reduces to GR in the IR, and which by construction is easier to prove to be AS... It seems a fair idea, but then we have to ask, for what kind of theories can AS be proven? Approaches like spin foams and CDT seem like they would have the same problem as perturbative quantum GR, with its infinitely many kinds of graviton vertices - there are infinitely many distinct ways in which the spin foam or the dynamical triangulation could change its local topology, each with an independent coupling. Working in Ashtekar variables, on the other hand, seems like it has a chance of leading somewhere new.

This review lists a few "non-gravitational examples" of asymptotically safe theories (section 3).

Meanwhile, there is an argument (section VI here) that any kind of asymptotic safety is incompatible with black hole thermodynamics, since it suggests that n-dimensional quantum gravity in the UV is an n-dimensional conformal field theory (that would be the behavior at the UV fixed point), whereas black hole thermodynamics says that n-dimensional quantum gravity in some sense behaves like an (n-1)-dimensional field theory (since entropy depends on the area of the event horizon, not the volume of the black hole).

This sounds like a strong argument, but as always the details matter. One may find in the AS literature, claims that the effective dimensionality of space-time itself reduces to 2 dimensions in the UV; and on the other hand, there was some recent work in string theory on fixed points above the decompactification energy scale. The most profound way to approach this issue might be in a spirit similar to the "swampland" program, which is an attempt to identify what is and is not logically possible in quantum gravity. E.g. to what extent is the existence of a UV fixed point in quantum gravity, incompatible with holographic black-hole thermodynamics? And further progress would involve greater precision about what is meant by those two concepts.

Asymptotic safety (AS) is the idea that quantum general relativity might be fully renormalizable after all, that the apparently infinite number of distinct coupling constants is just an illusion arising from the "perturbative" approach of working in terms of gravitons, and that if you work instead in terms of the wavefunctional of the gravitational field,

are there any approaches that emphasizes terms of the wavefunctional of the gravitational field instead of gravitons?

The dominant hypothesis in AS research is that GR itself is asymptotically safe; they just can't prove it. They can show UV fixed points in various highly truncated forms of quantum GR, but the critics don't believe those arguments apply to the full theory (as well as making other critiques of method 1 2).

I take you to be suggesting, what if you make the provability of AS, your priority in choosing a UV theory?

yes. couldn't you create a full theory of gravity out of those various highly truncated forms of quantum GR where they can show UV fixed points

Working in Ashtekar variables, on the other hand, seems like it has a chance of leading somewhere new.

sounds exciting, what's special about Working in Ashtekar variables that seems like it has a chance of leading somewhere new?

can you do AS with just curvature rather than gravitons?

Meanwhile, there is an argument (section VI here) that any kind of asymptotic safety is incompatible with black hole thermodynamics, since it suggests that n-dimensional quantum gravity in the UV is an n-dimensional conformal field theory (that would be the behavior at the UV fixed point), whereas black hole thermodynamics says that n-dimensional quantum gravity in some sense behaves like an (n-1)-dimensional field theory (since entropy depends on the area of the event horizon, not the volume of the black hole).

black hole thermodynamics hasn't been experimentally tested. holographic black-hole thermodynamics is just a theory.

I have earlier pointed out the apparent contradiction that there are string theorists who claim that string theory reproduces Hawking entropy for extremel black holes correctly without any fudge factors, and other physicists who claim that extremel black holes entropy is zero, a contradiction

 

[weirdoguy]

black-hole thermodynamics is just a theory.

Um, quite a weird statement. You should know by now that "theory" in science means something else than in everyday language, and eg. principles by which your computer has been built are also "just a theory". I think "hypotheses" or "hypothetical models" would be a better wording.

The reason I'm saying all of this is that "just a theory" is a slogan used a lot by anti-science people. Let's not give them fuel. And I know that the name "string theory" is not helping

 

[berkeman]

Thread is closed temporarily for Moderation...

 

[PeterDonis]

black hole thermodynamics hasn't been experimentally tested.

Neither has anything else being discussed in this thread.

holographic black-hole thermodynamics is just a theory.

The way you are using the term "theory", everything else being discussed in this thread is also "just a theory".

 

[PeterDonis]

The topic has been discussed sufficiently. Thread will remain closed.