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MathGroup Archive 2006

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Update on Weinberg-Sallam model in supr symmetry as E8xE8

  • To: mathgroup at smc.vnet.net
  • Subject: [mg70316] Update on Weinberg-Sallam model in supr symmetry as E8xE8
  • From: Roger Bagula <rlbagula at sbcglobal.net>
  • Date: Thu, 12 Oct 2006 05:38:14 -0400 (EDT)

I spent yesterday learning about super symmetric Higgs symmetry breaking 
theory.
Instead of {W(-),W(+) ,Z0,H0} in the U(1)*SU(2) theory there are:
{W(-),W(+) ,Z0,H0,H(-),H(+) ,h0,A0}
That's right 8 particles... all gauge bosons.
I came up with a "between" theory:
{W(-),W(+) ,Z0,H0,H(-),H(+)}
A D3<-A3*A3 symmetry breaking of an E8*E8 symmetry.
E8*E8-> (A3*D5)2
 >From the Dynkin Diagram/ graphs for the groups.

 The bonding matrix for Dynkin Diagram of E8xE8 is:
M = {{0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{1, 0, 1, 0, 0, 0, 0,  0, 0, 0, 0, 0, 0, 0, 0, 0},
 {0,  1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0},
 {0, 0, 1, 0, 1, 0, 0, 0, 0,  0, 0, 0, 0, 0, 0, 0},
 {0,  0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0},
 {0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0},
 {0,  0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
 {0, 0, 0, 0, 0, 0, 1, 0, 0,  0, 0, 0, 0, 0, 0, 0},
{0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0},
{0, 0, 1, 0, 0, 0, 0, 0, 0,  0, 0, 1, 0, 0, 0, 0},
{0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0},
 {0, 0, 0, 0, 0, 0, 0, 0, 1,  0, 1, 0, 1, 0, 0, 0},
{0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0},
 {0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0},
{0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1},
{0, 0, 0, 0, 0, 0, 0, 0, 0,  0, 0, 0, 0, 0, 1, 0}}
Characteristic Polynomial: ( equivalent to what is called a secular 
determinant
in quantum chemical bonding theory):
x(1 + x)(3 - 6 x^2 - 12 x^3 + 3 x^4 + 19 x^5 +
        15\x^6 + 16 x^7 - 44x^8 - 25 x^9 + 34
          x^10 + 9 x^11 - 10x^12 - x^13 + x^14)
Roots are :
{-2.01654, -1.76713, -1.48269, -1., -0.767533, -0.402218 - 0.477129 \
\[ImaginaryI], -0.402218 + 0.477129 \[ImaginaryI], -0.0739029 - 0.752057 \
\[ImaginaryI], -0.0739029 + 0.752057 \[ImaginaryI], 0., 0.674423\
\[InvisibleSpace] - 0.123061 \[ImaginaryI], 0.674423\[InvisibleSpace] + \
0.123061 \[ImaginaryI], 1.35865\[InvisibleSpace] - 0.0546097 
\[ImaginaryI], \
1.35865\[InvisibleSpace] + 0.0546097 \[ImaginaryI], 1.84408, 2.07591}

The three real roots which probably correspond to the symmetry breaking 
A3 are:
{-1.48269, -1., -0.767533}
The rest are relatively symmetrical about zero positive and negative.
These three super symmetrical solutions appear to be the tail that wags 
the dog.
You might look at them as the SU(2) three of Weinberg and Salam model
except in a Fermion type model there are two states for each energy 
level ( 32 in all).
They are the "between" theory:
{W(-),W(+) ,Z0,H0,H(-),H(+)}
To be clear  , this is my own invention based on looking at Dynkin 
diagrams as graphs
and taking the graphs as being like chemical bonding models.
In older terms they call it an "analog" model.
It is more like category theory than group theory.
It gives a visualization of the Higgs scalar symmetry breaking models in 
terms of the Dynkin diagrams.
This breaking leaves the D5 unit that corresponds to the fivefold 
symmetry that is seen in the fine structure constant intact.
Mathematica code:
Clear[M, v, F]
M = {{0, 1, 0, 0, 0, 0,
   0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0,
      0, 0}, {0, 1, 0, 1, 0, 0,
    0, 0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 
0, 0,
      0, 0}, {0, 0, 0, 1, 0, 1,
    0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 
0, 0,
      0, 0}, {0, 0, 0, 0, 0, 1,
    0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 
0, 0,
      0, 0}, {0, 0, 0, 0, 1, 0,
    0, 0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 
0, 0,
      0, 0}, {0, 0, 0, 0, 0, 0,
    0, 0, 0, 1, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 
1, 0,
      0, 0}, {0, 0, 0, 0, 0, 0,
    0, 0, 0, 0, 0, 1, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 
1, 0,
      1, 0}, {0, 0, 0, 0, 0, 0,
    0, 0, 0, 0, 0, 0, 0, 1, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0,
      1, 0}}
v[1] = Table[If[n == 1, 1, 0]1, {n, 1, 16}]
v[n_] := v[n] = M.v[n - 1]
a = Table[Floor[v[n][[1]]], {n, 1, 50}]
Det[M - x*IdentityMatrix[16]]
Factor[%]
aaa = Table[x /. NSolve[Det[M - x*IdentityMatrix[
      16]] == 0, x][[n]], {n, 1, 16}]
a4 = Table[{Re[aaa[[n]]], Im[aaa[[n]]]}, {n, 1, Length[aaa]}]
ListPlot[a4, PlotRange -> All, Axes -> False]
Abs[aaa]
a1 = Table[N[a[[n]]/a[[n - 1]]], {n, 13, 50}]

Roger Bagula



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