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exceptional group symmetry breaking as a binary entropy process
*To*: mathgroup at smc.vnet.net
*Subject*: [mg75483] exceptional group symmetry breaking as a binary entropy process
*From*: Roger Bagula <rlbagula at sbcglobal.net>
*Date*: Wed, 2 May 2007 03:56:32 -0400 (EDT)
Continuation of my work on entropy in the early universe and information
theory;
Thought experiment:
Suppose we had a worm hole from the remote past when the universe was E8
symmetry to the present and you could represent the state as a graph:
GE8->ME8 ( matrix for the graph)
Gpresent-> Mpresent
Then you would necessarily have a transform T that would take place in
the worm hole as:
GE8->Gpresent
as
Mpresent=T*ME8
For the information to be conserved T would have to be a unitary
Jacobian like transform:
Information Entropy as
H(present)=H(E8)+Log[Measure[T]]/Log[2]
I get the the Limit :
Limit[Measure[T], t-> Large]=0
This single approach gave me two copies of a group with 98 elements.
A symmetry breaking linear approach with two "target" groups also gave
an unexpected result:
E8-> -37*"7" +39*"13"
507->507
E8*E8 and SO(32) are 496 =507-11
Entropy excess is ( inflation's heating origin?) :
0.003310557481995602
which is less than alpha/2.
"7"-> U(1)*SO(4)->(in hyperbolic terms) U(1)*SO(3,1)
I'm having trouble getting the {x,y} out of the "a" array.
Mathematica:
Clear[En,a,b,x,y,n,m]
(*Binary Information Entropy for a group with n elements:*)
En[n_]=Sum[-(m/n)*Log[m/n]/Log[2],{m,1,n}]
a = Flatten[Table[Table[ Flatten[{n, m, x /. NSolve[{x*N[En[n]] +
y*N[En[m]] - En[248] == 0, x*n + y*m - 248 == 0}], y /.
NSolve[{x*N[En[n]] + y*N[En[m]] - En[248] == 0, x*n + y*m - 248 ==
0}]}], {n, 1, m}], {m, 1, 50}], 1]
x = 100; y = 100;
b = Table[Abs[Round[a[[n,
3]]]*N[En[a[[n, 1]]]] + Round[a[[n, 4]]]*N[
En[a[[n, 2]]]] - N[En[248]]], {n, 1, Length[a]}]
Min[b]
0.003310557481995602`
Flatten[Table[If[b[[n]] - Min[b] == 0, a[[n]], {}], {n, 1, Length[a]}]]
{7, 13, -36.996957416987826`, 38.99836168607037`}
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