Michael,
The link does not seem to work.
The J3(O) matrix is diagonalized by F4 into real eigenvalues. The F4 is the automorphism group of the Jordan 3x3. The sequence
F4: 0 --- > A_4 --- > F_{52/16} --- > OP^2 -- > 0
Means the eigenvalues found with F4 can be found in part with the subgroup A_4 = SO(9) or SO(8,1).
The E6 contains SO(10) or SO(9,1) . As with spacetime and the Lorentz group SO(3,1) the rotations SO(3) on a spatial surface that in turn foliate spacetime. These foliations define a thickened small region of spacetime with some lapse function, and the two spatial boundaries define two SO(3)s. In this way the SO(9,1) embeds SO(9) in a pairing. This is a physical reason why the E6 ÔèéSO(9,1) is over the CxO. Similarly the E7 is over HxO, the quarter-octonions.
For E6 the eigenvalues are not necessarily real. These complex numbers may appear perplexing, and further violate unitarity. However, in this C* algebraic form of the Weyl-Heisenberg algebra (a, a^ÔÇá, N, c):
[a, a^ÔÇá] = 2c
[N,a] = Ôê'a
[N,a^ÔÇá] =aÔÇá
[c,Ôï...] = 0,
where the standard theory has c = ›. We can see this does not result in a larger phase space volume or demolish the Hessian transformation principles of conservative physics. The Hopf algebra coproduct is introduced:
╬"a = aÔè--I + IÔè--a,
which is a coproduct "duplication" of the operator into left and right forms of the operator. Now the coproduct can be deformed forming a quantum group:
╬"aq = aqÔè--qc + qÔê'cÔè--aq
where q is the deformation parameter related to a geometric-like series:
[x]q = q^x Ôê' q^{-x}/(qÔê'q^{Ôê'1})
The elements of the q-deformation can be functions of any form. In standard QM these would be complex variables or functions, but they can also be real valued.
The deformation ╬"a(q) = a(q)Ôè--q^c + q^{Ôê'c}Ôè--a(q) is related to Bogoliubov operators. We may write
q^c = Ie^{2¤Çi¤å} = I(cos(2¤Ç¤å) + i sin(2¤Ç¤å)),
for ¤å a parameter related to acceleration. Of course in addition
(q^c)^{-1} = Ie^{-2¤Çi¤å} = I(cos(2¤Ç¤å) - i sin(2¤Ç¤å)).
This gives
╬"a(q) = a(q)Ôè--I(cos(2¤Ç¤å) + i sin(2¤Ç¤å)) + (cos(2¤Ç¤å) - i sin(2¤Ç¤å))IÔè--a(q)
= (a(q)Ôè--I + IÔè--a(q))cos(2¤Ç¤å) + i(a(q)Ôè--I - IÔè--a(q))sin(2¤Ç¤å)
We may of course form the Hermitian conjugate
╬"a^ÔÇá(q) = a^ÔÇá(q)Ôè--q^{-c} + qÔè--a^ÔÇá(q)
= (a^ÔÇá(q)Ôè--I + IÔè--a^ÔÇá(q))cos(2¤Ç¤å) + i(a^ÔÇá(q)Ôè--I - IÔè--a^ÔÇá(q))sin(2¤Ç¤å).
The commutator [╬"a(q), ╬"a^ÔÇá(q)] is
[╬"a(q), ╬"a^ÔÇá(q)] = ([a(q), a^ÔÇá(q)] Ôè--I + IÔè--[a(q), a^ÔÇá(q)])(cos^2(2¤Ç¤å) + sin^2(2¤Ç¤å)),
which gives a unit product.
The hyperbolic trigonometric functions obtained from ¤å --- > i¤å are squeeze parameters and we have a C* form of the Bogoliubov operators
b = au + a^ÔÇáv, b^ÔÇá = a^ÔÇáu + av
for u = cosh(2¤Ç¤å) and v = sinh(2¤Ç¤å). The commutator of b and b^ÔÇá is
[b, b^ÔÇá] = u^2 - v^2 = 1.
The q-deformed algebra has the same u^2 - v^2 = 1. This then means we have the operators employed in computing Hawking and Unruh radiation.
Cheers LC
