Has This Mysterious Mathematical Group Been Identified in Physics Research?

[AJPsi]

While investigating various aspects of generalised least action principles over the last several years I have come across an algebraic mathematical group that I am finding hard to classify but whose root vectors should relate to the standard model (no it is not E8 ! nor any exceptional group I am aware of.)

I would be very surprised if the mathematicians have not already investigated and named it, so I am looking for two things :

a) A resource/contact by which I can find the name of the group, given its algebraic root vectors, and from that of course anything that is already known about it by other researchers.
b) Help from a physicist who knows how to make a formal comparison between the numbers in the standard model and the properties of a proposed group for a GUT.

I have the equation that generates the set of root vectors for the group, and of course the generated set of root vectors themselves, but that is all the knowledge I have about the group. It seems to have some similarities to the SO and Spin groups. Curiously the equation seems to be a rather oddball way of generalising the Lorentz transform of special relativity.

There are some pretty neat symmetries in this group, and it has already led a few years back to a proposed new dimensional grouping of physical constants from which a more accurate value of G (Newtons's constant) can be calculated (G = 6.673870 92(30) x10-11 m3kg-1s-2 using CODATA 2010 values). The value is holding up very well to new more accurate experiments of G published last year.

There are also internal symmetries that should relate the standard model to dark matter, indicating an exact 1/6 visible 5/6 dark ratio that remains in the empirical bounds of the latest CMB results. This six fold symmetry should I suspect show up as a resonance at exactly 6 times the observed Higgs energy at CERN. Finally this group would indicate dark matter should interact not only gravitationally, but with some residual weak force effects with visible matter, but no electromagnetic or strong force interactions. But then my understanding of this group remains far from complete, and all this could be a crock of the proverbial.

 

[Demystifier]

Is the group simple?
https://en.wikipedia.org/wiki/Simple_Lie_group

 

[haushofer]

A former colleague of mine, Teake Nutma, developed the program "Simplie". Maybe you can use that:

https://github.com/teake/simplie

 

[AJPsi]

A former colleague of mine, Teake Nutma, developed the program "Simplie". Maybe you can use that:

https://github.com/teake/simplie

Thanks, this is almost what I need. But I have the root system and want to go the Dynkin diagram, and from first read, this program goes the other way. Nonetheless I will give it a go and report back!

 

[AJPsi]

Is the group simple?
https://en.wikipedia.org/wiki/Simple_Lie_group

I am slowly learning about group theory, but this is a new topic of study for me (the root system for this group came out of other considerations in physics which I do know more about!) Since this problem is a means to an end for me, and not an end in itself, I thought asking for help might make it go a bit faster if I can find the right person to look at it.

What I know about group theory would fit on the head of a pin. When I hear terms like simple (and then read its definition!) , "almost simple" and "semi-simple" I know I am out of my depth. How do I test if a group is simple, knowing its roots? I would be surprised if it is not a simple group, given its symmetric root structure that can be calculated from a single equation.

 

[fresh_42]

But I have the root system and want to go the Dynkin diagram, ...

I don't get this. You have a root system but don't know much about groups? Where did it come from?

 

[AJPsi]

I don't get this. You have a root system but don't know much about groups? Where did it come from?

In short, in a roundabout way. Bear with me, or better yet beer with me.

As part of my research into the unification problem I took a long hard look at a rather old fashioned topic in physics; that of principles of least action. In particular I asked the question: "Is there a mathematical framework that can guarantee that all solutions in that framework are of least action." The answer for non-trivial field theories is no.

But there is a really clever way to bypass the normal restrictions imposed by calculus of variations when dealing with discrete geometries. And in the process this bypass technique also proved to have a deep connection with the principle of scale-invariance and local/non-local conformal distance maps (explored with Euler graph theory) This led to a rather weird (OK really weird) constraint equation on these geometries that seemed to have no obvious solutions.

It was not until I realized the Lorentz transform of special relativity led to a special case solution of what I was seeking that I came to a generalised version that did have solutions. By comparing it with a rather obscure paper by Lucien Hardy on the derivation of quantum mechanics, I also worked out it also obeyed the axioms of quantum mechanics. At which point I started to get serious about exploring it to the end.

This constraint equation has 672 vector solutions that lie on the unit 8D sphere, and these list out exactly like the permutations/combinations of a https://en.wikipedia.org/wiki/Root_system of a mathematical group. Just that I cannot find a reference anywhere to the actual root system that I have!

Hence I have the roots, but not the group name! Go figure! I do not really want to publish the equation yet, but would not mind finding the right person to do some collaborative research on it with to better explore it's properties, hence this posted question.

 

[fresh_42]

Well, one of the first things you could do is to compute its Lie algebra and calculate whether the Killing form is non-degenerate or not. This at least will show you whether you have a semisimple group or not and whether there is a chance of a scalar product. And if not, the radical will show how to split the group. It's easier to do calculations on the Lie algebra level than on the group level.

 

[samalkhaiat]

While investigating various aspects of generalised least action principles over the last several years I have come across an algebraic mathematical group that I am finding hard to classify but whose root vectors should relate to the standard model (no it is not E8 ! nor any exceptional group I am aware of.)

I would be very surprised if the mathematicians have not already investigated and named it, so I am looking for two things :

a) A resource/contact by which I can find the name of the group, given its algebraic root vectors, and from that of course anything that is already known about it by other researchers.
b) Help from a physicist who knows how to make a formal comparison between the numbers in the standard model and the properties of a proposed group for a GUT.

I have the equation that generates the set of root vectors for the group, and of course the generated set of root vectors themselves, but that is all the knowledge I have about the group. It seems to have some similarities to the SO and Spin groups. Curiously the equation seems to be a rather oddball way of generalising the Lorentz transform of special relativity.

There are some pretty neat symmetries in this group, and it has already led a few years back to a proposed new dimensional grouping of physical constants from which a more accurate value of G (Newtons's constant) can be calculated (G = 6.673870 92(30) x10-11 m3kg-1s-2 using CODATA 2010 values). The value is holding up very well to new more accurate experiments of G published last year.

There are also internal symmetries that should relate the standard model to dark matter, indicating an exact 1/6 visible 5/6 dark ratio that remains in the empirical bounds of the latest CMB results. This six fold symmetry should I suspect show up as a resonance at exactly 6 times the observed Higgs energy at CERN. Finally this group would indicate dark matter should interact not only gravitationally, but with some residual weak force effects with visible matter, but no electromagnetic or strong force interactions. But then my understanding of this group remains far from complete, and all this could be a crock of the proverbial.

This is a root system extended by real(ly) (de)graded non-compact infinite-dimensional super-gibberish Lie algebra.
Dear Sir, Don’t waste people’s time in here just because you have time to waste.
The staff should warn you and close this thread.

 

[AJPsi]

This is a root system extended by real(ly) (de)graded non-compact infinite-dimensional super-gibberish Lie algebra.
Dear Sir, Don’t waste people’s time in here just because you have time to waste.
The staff should warn you and close this thread.

Well nobody has warned me. All I have done is ask for some help with a problem and Physicsforums seems such a smiley face "Fusion of science and community" This is my first post and I had not been able to judge the impolite culture of this community. Consider this thread closed.

 

[fresh_42]

Let me add a word of honesty. You awoke the impression of being someone who has some layman view and maybe a good idea on a very complex subject. I might be wrong, but that has been at least my impression. Professors at my university called people who sent in "proofs" of Fermat's theorem or similar "tri-sectionists" referring to the antique problem on how to divide an angle into three equal parts only with the help of compass and ruler. It is proven not to be possible. However, this didn't reduce the number of mails which were sent in.

I don't want to assume your idea is of the described form. Nevertheless, the subject is really, really complex and many scientist worked on it for decades. The probability to find a good GUT through a brilliant idea is simply almost zero. And almost certainly zero if someone has difficulties in basic concepts like simplicity of groups. For that reason PF's politics is not to debate on unpublished and non-peer-reviewed concepts. These politics are public and advised to read at the registration.

As I said, I might be wrong, but it is the impression I got.

 

[AJPsi]

Let me add a word of honesty. You awoke the impression of being someone who has some layman view and maybe a good idea on a very complex subject. I might be wrong, but that has been at least my impression. Professors at my university called people who sent in "proofs" of Fermat's theorem or similar "tri-sectionists" referring to the antique problem on how to divide an angle into three equal parts only with the help of compass and ruler. It is proven not to be possible. However, this didn't reduce the number of mails which were sent in.

I don't want to assume your idea is of the described form. Nevertheless, the subject is really, really complex and many scientist worked on it for decades. The probability to find a good GUT through a brilliant idea is simply almost zero. And almost certainly zero if someone has difficulties in basic concepts like simplicity of groups. For that reason PF's politics is not to debate on unpublished and non-peer-reviewed concepts. These politics are public and advised to read at the registration.

As I said, I might be wrong, but it is the impression I got.

I am just really bad at social media. What you infer is a long way from my intent, and my background (I can PM you some personal and professional references so you need not take my word for it).

Since I cannot close a thread myself, I expect the moderator will send a warning in due course and close it. Anyway this is my last post, and so apologies to not replying to others as to outcome of some of their suggestions.

 

[haushofer]

I'm not sure why you should leave because of just one comment of a user; this doesn't need to be representative of the whole forum. But that's of course your own decision. :) Whatever you decide, good luck, and if you have any questions concerning group theory, PF is here to help you.

 

[arivero]

Before closing, let me google

https://www.google.com/search?num=30&espv=2&q=SO(8)+representations+672&oq=SO(8)+representations+672&gs_l=serp.3...1562.3144.0.3335.4.4.0.0.0.0.63.245.4.4.0...0...1c.1.64.serp..0.0.0.TRCyGlGnsx4

From your description, SO(8) or SO(9) or perhaps Spin(9) is the thing you are searching; these groups were well hammered in the supergravity research around 1984, before the string revolution.

 

[AJPsi]

Before closing, let me google

https://www.google.com/search?num=30&espv=2&q=SO(8)+representations+672&oq=SO(8)+representations+672&gs_l=serp.3...1562.3144.0.3335.4.4.0.0.0.0.63.245.4.4.0...0...1c.1.64.serp..0.0.0.TRCyGlGnsx4

From your description, SO(8) or SO(9) or perhaps Spin(9) is the thing you are searching; these groups were well hammered in the supergravity research around 1984, before the string revolution.

Ok, given Arivero's post subsequent to my last post, that effort needs to be acknowledged so I will break my word on that basis.

The roots of SO(8) and the roots of SU(4) are both members of a sub-group of 48 vectors of the total 672 vectors I have. Perhaps I need to find a list of known root systems (the internet seems very hit and miss on how groups/properties are enumerated) and the correct ways to combine/split them to form other groups. So maybe a text reference (I am not asking you to solve this for me, just pointers to the right material)

I think I am slowly seeing a way to the Cartan matrix if that would help any.Once again I apologise for not having any post-grad training in group theory. My years have been full enough as it is.

I will change my mind about posting till the promised moderator warning comes (as a group you sure seem quite conflicted in this regard about my OP) It seems to me the entire forum topic "beyond the standard model" pretty much breaks the site rules all on it's own, let alone any contributions to it.

 

[haushofer]

Well, we are a community, but luckily we differ as persons :P Maybe also Teake's PhD-thesis can help you:

http://www.rug.nl/research/portal/files/14628607/15_thesis.pdf!null

If not, you can still enjoy the pretty pictures in it ;)

For some literature about Lie groups, I can highly recommend the following:

http://www.staff.science.uu.nl/~hooft101/lectures/lieg07.pdf
http://phyweb.lbl.gov/~rncahn/www/liealgebras/texall.pdf
http://www.cmls.polytechnique.fr/perso/renard/Hall_Group.pdf

Hope this helps.

 

[AJPsi]

Well, we are a community, but luckily we differ as persons :P Maybe also Teake's PhD-thesis can help you:

http://www.rug.nl/research/portal/files/14628607/15_thesis.pdf!null

If not, you can still enjoy the pretty pictures in it ;)

For some literature about Lie groups, I can highly recommend the following:

http://www.staff.science.uu.nl/~hooft101/lectures/lieg07.pdf
http://phyweb.lbl.gov/~rncahn/www/liealgebras/texall.pdf
http://www.cmls.polytechnique.fr/perso/renard/Hall_Group.pdf

Hope this helps.

Many thanks. That program by Teake is proving to be easy to use and really useful even if it cannot quite do what I need "out of the box". And understanding algorithms and programming is trivial for me even if the more arcane theorems in group theory are not. More precisely it is the difficulty understanding a large obtuse body of terminology. I love a topic that takes a whole degree to work through three layers deep of terminology just to discover what "almost simple" actually means!

I think I have all I need for the moment to answer part a) of my OP, and will report back in due course on the results assuming this thread has not been closed. A methodology for part b) of my question has finally been answered elsewhere.

 

[lpetrich]

AJPsi, would it be possible for you to describe that group for us? Also how you get that group's root vectors. Seems like you need that group's Lie algebra for that, and if you could describe that, it would be most helpful. It seems to me that you need the Lie algebra to get the root vectors, because those vectors are the eigenvalues of the Cartan-subalgebra operators for all the algebra operators. A Cartan subalgebra is a maximal commuting subalgebra of an algebra.

If you've ever worked with the ladder-operator formalism for quantum-mechanical angular momentum, you've worked with a Lie algebra. It's the smallest simple one, SU(2) / SO(3) / Sp(2). One of the angular-momentum components, by convention the z component, is that algebra's Cartan subalgebra.

I tried finding some SU(n) or some SO(n) algebra that gives 672 root vectors, and I failed. It would take a lot of trial and error to do that for product groups, so I decided to pass on that.

 

[AJPsi]

Thank you for your thoughts and time on this.

The equation simply generates 672 permutations of vectors expressed as coordinates in 8 dimensions (actually 4C is a more convenient way of expressing it) ; Each co-ordinate lies on the unit circle (unit vector), and is a root of the the equation we are working with i.e. of the form f(x0,x1,x2,x3,x4,x5,x6,x7)=0. The equation itself was derived from other arguments which are the subject of original research not at all related to group theory, but given the equation does generate these permutations, that appear just like roots of a group, the associated group properties would likely have implications for interpreting this other research.

The difficulty we have is we do not want to disclose this equation being used in the context of this research prior to publication. But neither do we want to publish an "unknown" group that is actually known to mathematicians. And we do not have contacts with mathematicians expert in this area so it is hard to find someone to collaborate with. So a dilemma. I think perhaps I should be trying some math forums instead and retry the maths departments at universities in our country. But of course you would understand how well cold calls go down attempting that...

 

[arivero]

f(x0,x1,x2,x3,x4,x5,x6,x7)=0.

Uh. Besides math, try to chech the 1981 - 1986 work in 11D Kaluza Klein, where the KK manifold is seven dimensional. After Witten's hint of the quotient of S3xS5 by some U(1) action, an effor to classify seven dimensional manifolds and their groups of isometries was done in the physicists literature. Some of this effort was mirrored by mathematicians in the start of the century.

And of course, with x0...x7 you are alway near of any quaternionic/octonionic object. It is amusing how paranoid the octonionists/quaternionists are. It could be related to the turbulent history of the topic during the XIXth century.

Have luck! Please update the thread when you publish your preprint.

 

[lpetrich]

Permutations of a set of vectors or permutations of the elements of each vector?

The first factorial that 672 evenly divides is 8!, so a 672-element permutation group might be possible for a set of 8 symbols. However, I tried experimenting with various combinations of length-8 permutations, and I could not find any combination that generated a 672-element group. I recently found Lists of subgroups of various groups, which links to Subgroups of Sym(8). That group is the group of all length-8 permutations, and that table mentions no subgroups with order 672.

Factoring 672 gives 2^5 * 3 * 7 and that implies that there exists a 672-element group: Z2^5 * Z3 * Z7. It can be realized as 20-element permutations, making it a subgroup of Sym(20).

 

[fresh_42]

How about $G= <(1,2,3),(1,2)(3,4),(1,2,3,4,5,6,7),(1,2,3,4,5,6,7,8)>$?
(I haven't done any calculations, just a thought.)

 

[lpetrich]

I expanded that group in brute-force fashion, and I found that it's Sym(8). I had to write some C++ code for that, since Mathematica and Python were far too slow for that. In the C++ code, I had to put in a chained hash table to speed up the membership checking for each product that I calculate. But with an Intel Core 2 Duo iMac, I succeeded in showing that it's Sym(8) in 4 minutes of CPU time.

 

[AJPsi]

I expanded that group in brute-force fashion, and I found that it's Sym(8). I had to write some C++ code for that, since Mathematica and Python were far too slow for that. In the C++ code, I had to put in a chained hash table to speed up the membership checking for each product that I calculate. But with an Intel Core 2 Duo iMac, I succeeded in showing that it's Sym(8) in 4 minutes of CPU time.

Thanks all, I have found a prof in a maths department not too far away who is up on this stuff, so will take this info and the last couple of replies and report back in a couple of weeks. Again, thanks all.