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[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. :-)

Comments

[gaspodog.livejournal.com]

I read about this one in New Scientist - it sounded vaguely interesting (not that I comprehend the symmetries / mathematics involved). Sort of like "I dropped all the numbers into this pretty pattern and most of them kind of fitted, so hey, let's see if it predicts something".

And if it manages to predict something correctly, it'll have one up on string theory...

[pozorvlak.livejournal.com]

He's not obviously a 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?

[elvum.livejournal.com]

Lee Smolin (ditto) also thought it sounded promising, IIRC.

Re your offer, go for it. :-)

[pozorvlak.livejournal.com]

OK, here goes... I'll pitch it a bit below your level, so more people can understand it)

Recall 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/whatever, then the other) or inverted, and which contains an "identity" element, which when multiplied by any other element gives you back the other element. Now, some groups (the symmetries of a featureless n-dimensional space, for instance) have geometric structure: in particular, some groups have the structure of a manifold (space that looks locally like Euclidean n-space for some n). A group that is also a manifold, and whose multiplication and inversion functions are infinitely differentiable, is called a Lie group (named after the mathematician Sophus Lie).

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? :-)

[elvum.livejournal.com]

For certain values of "help"... :-)

It was certainly informative and explanatory, and I thank you.

[pozorvlak.livejournal.com]

Cool.

I 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.

[elvum.livejournal.com]

Intuituvely, I feel that there should probably be a large number of mathematical entities that have symmetries that correspond to those of the Standard Model, plus some spares. Continuing in that vein, I think that the elegance of this new theory is due in part to the relatively small number of spares. The fact that Lisi had to use the most complicated Lie algebra is only mildly concerning, given that. But theories are only as good as their successful predictions, so let's see what new particles Lisi's theory requires, and whether we can find them with the LHC. :-)

[pozorvlak.livejournal.com]

Well, it's the most complicated simple Lie 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 :-)

[pozorvlak.livejournal.com]

A further word of explanation: a "representation" of a Lie algebra is a model for it in terms of matrix multiplication. In more detail: suppose you have a vector space A, with an associative linear multiplication operation on it (such things are called associative algebras). Then you can define a Lie bracket on it, by setting [a,b] = ab - ba. The space of nxn matrices forms an associative algebra under matrix multiplication, so we can form a Lie algebra of nxn matrices (equivalently, we can form a Lie algebra of linear transformations on an n-dimensional vector space V). A representation of a Lie algebra g is then a Lie algebra homomorphism from g to such a Lie algebra of matrices: equivalently, it's a choice of matrix A for each vector a in g so that the Lie bracket is realized by AB - BA. You can learn a lot about Lie algebras by studying their representations, but I can't tell you exactly what, because my Lie algebra lecturer used to schedule all his lectures for 9am on the grounds that undergraduates would be bright and fresh at that time. Consequently, I was unusually vulnerable to attacks of somnolence, and representation theory sends me to sleep at the best of times :-)

[gaspodog.livejournal.com]

I have the measure of this Lisi chap. My proposed symmetry is clearly far superior to his... :)

[gaspodog.livejournal.com]

It seems that my theory may be just as good as his after all...

Again, 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) is a 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.

[pozorvlak.livejournal.com]

Lubos is an unpleasant troll. He may also be a brilliant physicist: I'm not qualified to judge. More moderate physicists seem to think that what Garrett has done is very interesting but suffers from some unresolved (but not necessarily unresolvable) problems. There's a long thread here, which goes into the issues in some detail.