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Nov. 16th, 2007 12:06 am**elvum**

Any fans out there of theories of everything?

I've never been good enough at HEP theory to do more than skim this stuff, but I particularly liked the end of section 5: "The theory has no free parameters. The coupling constants are unified at high energy, and the cosmological constant and masses arise from the vacuum expectation values of the various Higgs fields..." - the guy is simply reeling off desirable features from the GUT ticklist.

I eagerly await the results of some peer review, and about a decade of LHC running. :-)

I've never been good enough at HEP theory to do more than skim this stuff, but I particularly liked the end of section 5: "The theory has no free parameters. The coupling constants are unified at high energy, and the cosmological constant and masses arise from the vacuum expectation values of the various Higgs fields..." - the guy is simply reeling off desirable features from the GUT ticklist.

I eagerly await the results of some peer review, and about a decade of LHC running. :-)

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Date: 2007-11-16 08:32 am (UTC)gaspodog.livejournal.comAnd if it manages to predict something correctly, it'll have one up on string theory...

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Date: 2007-11-16 10:43 am (UTC)pozorvlak.livejournal.comobviouslya crank. Though he spends a somewhat surprising amount of space rehearsing basic Lie algebra theory (which, from a quick skim, he gets right - beyond that point, my grasp of the mathematics falters). He includes some serious people in his acknowledgements, but then anyone can do that.*googles*

OK, John Baez (who is to be taken seriously) thinks there might be something in it. See here from some background. Here's another post about it, suggesting that it may well be right, but probably only by pushing existing problems (unification) somewhere else (where the symmetry-breaking comes from), and not making them much easier in the process. Oh, and this looks like a great explanation of the physics behind it.

Would anyone like me to explain about Lie algebras, root systems, E8 etc?

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Date: 2007-11-16 10:59 am (UTC)elvum.livejournal.comRe your offer, go for it. :-)

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Date: 2007-11-16 12:11 pm (UTC)pozorvlak.livejournal.comRecall that if you take any kind of object, its symmetries form a mathematical object called a "group" - a set whose elements can be "multiplied" together (perform one rotation/reflection/permutation/

Take some Lie group G. Because it's a manifold, we can consider the space of tangents to G at some point x: this forms an object called a "vector space" (which we can think of as, yes, a space of vectors). It's particularly interesting to consider the tangent space of G at the multiplicative identity element (which is a point in G). Call this tangent space g. The multiplication of G induces a multiplication-like structure on g called the Lie bracket, but it's not much like ordinary multiplication. It satisfies the rules

[a,b] = -[b,a]

[[a,b],c] + [[b,c],a] + [[c,a],b] = 0

Remember, g is a space of vectors, and if a and b are vectors in g, then so is [a,b].

We call such a structure a Lie algebra. In fact, we call every vector space with such a Lie bracket a Lie algebra. Examples include R^3 with vector product, and (as described above), every Lie group gives rise to a Lie algebra, which is usually referred to as the name of the group in lowercase. So the Lie group SU(3) (unital 3x3 matrices with determinant 1) gives rise to a Lie algebra su(3), O(2) (orthogonal 2x2 matrices) gives rise to a Lie algebra o(2), and GL(3) (invertible 3x3 matrices) gives rise to a Lie algebra gl(3).

Because we're mathematicians, we like to classify any new type of object we're handed. It turns out that any (complex?) Lie algebra can be decomposed into a "solvable" bit and a "semisimple" bit. Classification of solvable Lie algebras is both open and extremely hard. Classification of semisimple Lie algebras is easier, as every semisimple complex Lie algebra is a direct sum of simple Lie algebras, where a "simple" Lie algebra is one which is not abelian (abelian means that [a,b] = 0 for all a and b), and has no nontrivial ideals (something like normal subgroups).

By a frankly tedious and unenlightening process, the classification of complex simple Lie algebras can be reduced to the classification of sets of vectors satisfying various restrictive geometric constraints. Such sets are called "root systems". The classification of root systems can be further reduced to the classification of graphs (called Dynkin diagrams) satisfying some purely combinatorial restrictions, and at this point we can solve the problem. It turns out that there are four infinite families of Dynkin diagrams/root systems/semisimple complex Lie algebras, called A_n, B_n, C_n and D_n, and five "exceptional" diagrams that don't fit into any family, called E_6, E_7, E_8, F_4 and G_2. Of these, E_8 gives rise to the highest-dimensional and most complicated Lie algebra. What it has to do with the Standard Model, I couldn't tell you.

Does that help? :-)

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Date: 2007-11-16 12:45 pm (UTC)elvum.livejournal.comIt was certainly informative and explanatory, and I thank you.

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Date: 2007-11-16 12:51 pm (UTC)pozorvlak.livejournal.comI don't entirely understand how to interpret the diagrams, but roughly, every vertex corresponds to a copy of sl2, and the number of edges between two vertices tells you how to multiply elements from those copies of sl2 together.

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Date: 2007-11-16 12:54 pm (UTC)elvum.livejournal.com## no subject

Date: 2007-11-16 02:05 pm (UTC)pozorvlak.livejournal.comsimpleLie algebra, so in some sense it's a reasonable choice. I believe it has some other desirable properties, too. But you're right about the predictions: to his credit, Lisi agrees with you :-)## no subject

Date: 2007-11-16 02:19 pm (UTC)pozorvlak.livejournal.com## no subject

Date: 2007-11-16 04:12 pm (UTC)gaspodog.livejournal.com## no subject

Date: 2007-11-17 06:27 pm (UTC)gaspodog.livejournal.comAgain, I am no expert, so I bow t those who know more about it, but Ars Technica reckon it's bollocks, and you can see the blog post they are working from as well as a forum thread discussing the paper.

The main guy discrediting it (the blog poster)

isa string theorist, but his reasoning seems sound, so I don't think it's just jealousy. His track record as a theorist looks ok though - if his former jobs and body of published work are anything to go by.Apparently he does like to argue with the Quantum Loop Gravity people, so maybe he just likes to cause a fuss.

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Date: 2007-11-19 12:47 pm (UTC)pozorvlak.livejournal.com