20071117, 02:26  #1 
Dec 2005
2^{2}·23 Posts 
An Exceptionally Simple Theory of Everything
A physicist in Nevada created a theory that explains everything:
http://science.slashdot.org/article..../11/15/2322225 http://uk.youtube.com/watch?v=xHw9zcCvRQ http://en.wikipedia.org/wiki/An_Exce..._of_Everything http://arxiv.org/abs/0711.0770 Now, would someone explain Group theory, Symmetry groups, Differentiable manifolds and Lie groups to me in that order, so I can understand E8 and hopefully have some chance of understanding "An Exceptionally Simple Theory of Everything?" Someone on slashdot said that people need to learn them in that order to understand it, although I think Differentiable manifolds do not need to be learned as the third item and could be learned as the first or second instead, but I do not know anything about differentiable manifolds, so I am probably wrong. Last fiddled with by ShiningArcanine on 20071117 at 02:28 
20071117, 15:34  #2  
"Ben"
Feb 2007
2·1,789 Posts 
Quote:
I don't know enough to begin to address that request, but I do know that those are pretty deep topics that would require quite a bit of study to properly understand. Start in the usual places (wikipedia, mathworld, etc), if you don't have access to the right books. Then get the right books. Last fiddled with by bsquared on 20071117 at 15:51 

20071117, 15:57  #3  
Cranksta Rap Ayatollah
Jul 2003
641_{10} Posts 
Quote:
A set G with an associative operation * such that: i) There exists an identity element in G, i.e. x * e = e * x = x ii) If x is an element of G, then there exists a y such that x * y = y * x = e (i.e. the inverse of x) iii) If x, y are in G, then x * y is in G (closure) Examples of groups are the integers with addition, real numbers with addition, the symmetries of a polygon (for example, given a square with vertices labeled, there are 8 ways to pick up the square, rotate it, flip it, whatever, and put it back down in the same place it was) A manifold is a space that is locally euclidean at each point. For example, surfaces are 2dimensional manifolds. The surface of a sphere is not euclidean (the fifth postulate is not satisfied) but since we are so small compared to the earth, we consider our neighborhood to be on a flat, 2d surface. So a sphere is a manifold. With a bit more of a headache, you might be able to see that all invertible 2x2 matrices with real entries are a 4dimensional manifold A differentiable manifold is a space that is not only locally euclidean, but "smooth" enough to do calculus on. A Lie group is a differentiable manifold that is also a group. The example of all invertible 2x2 matrices with real entries (It's called GL2(R) ) can be considered a group under matrix multiplication. Therefore, GL2(R) is a Lie group. 

20071118, 11:52  #4 
Dec 2005
2^{2}·23 Posts 
Is it a coincidence that the properties you listed for groups correspond to the reflexive, symmetric and transitive closures of relations on sets (i.e. equivalence relations)?
If all invertible 2x2 matrices with real entries are 4dimensional manifolds, why do they claim that the circle be represented by a 2x2 matrice on the following website: http://aimath.org/E8/representation.html The only way that I can see that the circle can be represented by that would be to take its determinant, which gives sin^2(x) + cos^2(x), which resembles the equation sin^2(x) + cos^2(x) = 1, which I believe is the sum of the absolute values of its x and y coordinates of a circle with a radius of 1 as a function of its angle measure, although the specialness of that does not seem to really jump out at me and it does not seem to be four dimensional. Do you mean that each entry in the matrix represents a different coordinate in a four dimensional space according to some kind of coordinate system? Are differentiable manifolds related to Gauss's theorem? Last fiddled with by ShiningArcanine on 20071118 at 11:54 
20071118, 20:54  #5  
Cranksta Rap Ayatollah
Jul 2003
281_{16} Posts 
Quote:
Quote:
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No more questions. You don't seem to be thinking for yourself anymore, and I'm too tired to do it for both of us. 

20071119, 05:40  #6 
Aug 2002
Ann Arbor, MI
433 Posts 
I don't like their definition of a representation.
A representation is just a map from a group into the space of nxn matrices so that composition of group elements corresponds to multiplication of their respective matrix representations. The circle is a group under addition of angles (where everything is working modulo 2pi). You map an angle x to the matrix you have above. Then if you take the matrix associated to an angle x, and multiply it by the matrix associated to an angle y, the result is that same as the matrix you'd get by putting in x+y. 
20071119, 21:20  #7 
Feb 2007
2^{4}·3^{3} Posts 
just 3 comments :
1/ obviously (?!), the words "exceptional" and "simple" are a "clin d'oeil" to the precise meaning of these terms in group theory, rather than referring to an extremly simple theory. 2/ astonishing that the author does not cite any published paper of himself in the references but at least, without attempting any other judgement whatsoever, I think it is a nice pedagogical review of the quite well known (group) structures of the usual GUT's, of course assuming the reader has some background on the subject.  I don't know another as complete and as nicely & colorful illustrated compilation of all this information, which although certainly otherwise available, is so only in very widely scattered sources. 
20071120, 13:37  #8  
"Lucan"
Dec 2006
England
2×3×13×83 Posts 
Quote:
"The truth is rarely pure and never simple" or words to that effect. (Oscar Wilde). 

20071126, 05:17  #9 
Nov 2005
182_{10} Posts 
At least that's better then "The truth is purely simple, and rarely pure." as the more cynical think.

20071129, 16:40  #10  
Feb 2007
2^{4}·3^{3} Posts 
Quote:
PS: didn't you rather mean "... never pure" ?! Last fiddled with by m_f_h on 20071129 at 16:41 

20100402, 17:40  #11  
A Sunny Moo
Aug 2007
USA (GMT5)
3×2,083 Posts 
Quote:


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