Dark Energy, DeSitter Space & Catan's Forms
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This is starting something very new! Uncharted territory. Before Waldyr jumps on this:
"Anyone who has never made a mistake has never tried anything new." Einstein
I. Relation of DeSitter Group to Poincare Group
First, Klein 1872 groups define geometries.
Second, orthogonal groups of nxn matrices have n(n - 1)/2 parameters.
The invariant is x^2 = xi^2 (summation convention), xi real, i = 1 ... n
The anti-gravtating w = -1 (no boundaries) dark energy density ~ /\ > 0 corresponds to a global DeSitter group O(4,1) with invariant of Kaluza-Klein type 4 + 1 space-time with one extra space dimension.
x1^2 + x2^2 + x3^2 + x4^2 - x0^2
n(n - 1)/2 = 5x4/2 = 10
The "charges" of the Lie algebra are the 4x3/2 = 6 space-space rotations Mij = - Mji, i,j = 0,1,2,3 and the 4 space-time rotations Pi = Mi4, i = 0,1,2,3
The commutator of interest is
[Pi,Pj] = /\zpfMij
The other commutators
[Pi,Mjk] and [Mij,Mkl] are exactly the same as the Lie algebra of the 10 parameter Poincare group T4*O(3,1) in 3 + 1 space-time.
When the CONSTANT dark energy density vanishes /\zpf ---> 0 in the fermion-boson supersymmetry limit of 3 + 1 space-time then this "Wigner-Inonu contraction is from constantly curved 4 + 1 space-time to globally flat 3 + 1 Minkowski space-time, so that [Pi,Pj] = 0 where Pi is the total Energy-Momentum of 3+1 flat space-time.
Relationship to Cartan's forms.
d = dx^iPi = exterior derivative 1-form, h = c = 1 for now
d^2 = 0
d/\d = 0
That is only true when
[Pi,Pj] = 0
Therefore, when /\zpf =/= 0 we can no longer assume that d/\d = 0 in a 3 + 1 space-time with a dark energy density.
Note "/\" = Cartan exterior multiplication of forms
"/\zpf" is scalar constant curvature of 4 + 1 DeSitter space-time.
d/\d = dx^iPi/\dx^jPj = dx^i/\dx^j[Pi,Pj] = dx^i/\dx^j/\zpfMij
So for example suppose
B = dTheta
Theta is a 0-form
B is a 1-form
dB = d^2Theta = d/\d = dx^i/\dx^j/\zpfMijTheta =/= 0
This will create ANOMALIES.
For example, Maxwell's EM equations
F = dA
dF = d^2A = dx^i/\dx^jMij/\zpfA =/= 0
This begins to look like the Meissner effect in superconductors ? and also magnetic monopoles!
dF = 0 when /\zpf = 0 corresponds to
curlE + Bt = 0 i.e. induction
divB = 0 no magnetic monopoles
Then also second half of Maxwell's equations
d*F = *J
d^2*F = d*J = dx^i/\dx^jMij/\zpf*F =/= 0 current anomaly
When /\zpf = 0 this is Ampere's law with displacement current
curlH + D,t = j
divD = rho
and Gauss's law.
This is before we get /\zpf as a local dark energy/matter field including /\zpf < 0 by locally gauging all of O(4,1).