I should have said that at the time of the phase transition from the inflationary era to the present era the transtion was

from E6 to gravity plus Higgs plus Standard Model.

It could well be that the inflationary era began with an even higher (still octonionic nonassociative) symmetry of E8,

which during inflation decayed to E7,

which later during inflation decayed to the E6.

The higher transitions are probably connected to our earlier blog discussions about 2-time, 3-time, 4-time etc.

There could be even more “topological defect” things (beyond the E6 stuff I mentioned in my previous comment) due to the E8 to E7 and E7 to E6 transitions.

As to whether artifacts from such defects are still around, or whether the inflation process effectively erased them, is an interesting question.

Tony Smith

]]>“… If we assume that a high temperature phase of the Universe has an exceptional symmetry group G … the use of an 8D nonassociative division algebra is physically justified. …”.

That is what I am suggesting:

That the inflationary phase has E6 exceptional symmetry and complexified J(3,O) structure (that is, Freudenthal) and 8-dim Octonion spacetime.

As you also said:

“… in a low-temperature phase, such as the one we live in now, the use of an 8D nonassociative algebra to describe physics does not seem justified at all. …”.

That is also what I am suggesting:

That after the inflationary phase we have

a phase transition “freezing out” of a preferred quaternionic structure,

which forms our 4-dim associative spacetime with gravity (MacDowell-Mansouri) and Higgs (geometric as done by Meinhard Mayer, now at UC Irvine)

plus a 4-dim coassociative internal symmetry space (with CP2 structure), effectively a Batakis-type Kaluza-Klein Standard Model structure.

As you also said:

“… if our universe did have an E8 phase,

there were likely some interesting topological defects produced during the early phase transitions. …”.

In my E6 view, the dominant topological defect at the end of inflation was the “frozen-out” quaternionic structure.

If the structure “froze-out” consistently all over our univers with the same quaternionic spacetime, then our 4-dim spacetime itself would be the “topological defect”.

If there were several different quaternionic subspaces freezing-out in different regions of the 8-dim Octonionic inflationary spacetime, then there would be interesting “topological defect” things at the boundaries of those regions, perhaps with interesting 3-sphere flow-knot-braid characteristics.

How long such interesting “topological defects” would live and how they would decay and what artifacts might still be around is worth investigating.

Tony Smith

]]>… the end of the inflationary phase corresponds to the “freezing out” of a preferred quaternionic subspacetime in Octonionic 8-dim spacetime,

producing

a 4-dim Minkowski-like spacetime plus a 4-dim CP2 internal symmetry space in which Unitarity is preserved in experimental observations.

If we assume that a high temperature phase of the Universe has an exceptional symmetry group G, such as say G=E8, the use of an 8D nonassociative division algebra is physically justified. However, in a low-temperature phase, such as the one we live in now, the use of an 8D nonassociative algebra to describe physics does not seem justified at all.

In studying cosmological phase transitions, many people look to the nematic liquid crystal analogy, which helps shed light on the formation and behavior of topological defects, such as cosmic strings, domain walls, etc. Such topological defects would have occurred early in the universe, after a transition to a more ordered lower-temperature phase with symmetry group H, such that M=G/H has non-trivial homotopy groups. So if our universe did have an E8 phase, there were likely some interesting topological defects produced during the early phase transitions.

]]>On your blog a while back (a post about Jordan algebras), you linked to a PF discussion in which Carl Brannen asked about

“…octonions … a physical justification for non associativity …”.

My best guess about that is that the inflation phase of universe expansion seems to me to involve producing a lot of new stuff (unless it comes from an ad-hoc inflaton field that I don’t like)

so

producing new stuff from nothing seems to me to violate Unitarity

and

Octonion nonassociativity violates Unitarity

(see the book Quaternionic Quantum Mechanics and Quantum Fields ((Oxford 1995),

where Stephen L. Adler says at pages 50-52, 561:

“… If the multiplication is associative,

as in the complex and quaternionic cases,

we can remove parentheses in … Schroedinger equation dynamics …

to conclude that … the inner product … is invariant …

this proof fails in the octonionic case,

and hence one cannot follow the standard procedure to get a unitary dynamics. …

[so there is a]… failure of unitarity in octonionic quantum mechanics…”.

In short, in my view,

the inflationary phase involves an 8-dim octonionic spacetime

in which a lot of stuff in created by Octonionic violation of Unitarity,

and

the end of the inflationary phase corresponds to the “freezing out” of a preferred quaternionic subspacetime in Octonionic 8-dim spacetime,

producing

a 4-dim Minkowski-like spacetime plus a 4-dim CP2 internal symmetry space in which Unitarity is preserved in experimental observations.

Tony Smith

]]>I forgot to mention a few interesting notes about the spin foam topic and the relation to J(3,O) eigenmatrices. The Jordan eigenvalue problem gives us the well-known spectral decomposition of an element of the Jordan algebra, when the eigenvalues are distinct. This is because the eigenmatrices for an element A of J(3,O) are of the form Q_i=(A-Ic_i)x(A-Ic_i), so equivalent eigenvalues c_i=c_j yield the same eigenmatrix.

In the nondegenerate case, we get the usual spectral decomposition A=c_1P_1+c_2P_2+c_3P_3, where P_i are orthonormal versions of the eigenmatrices Q_i. By hitting A with 3×3 SO(9) transformations, we can transform two qutrits, while leaving a third invariant. The three types of SO(9) transformations that leave each qutrit invariant, respectively, are merely the three embeddings of SO(9) in F4. By combining transformations from each SO(9) embedding, we recover a general F4 transformation. So in a very real sense, SO(9) describes the octonionic analog of SU(2) spin foam.

]]>although I would prefer to see foam nodes (instead of SU(2) things) more like the traceless part J(3,O)_0 of exceptional Jordan algebra J(3,O).

Yes, exactly. The ternary logic Kea refers to goes back to our discussions of J(3,O) and OP^2. Typically, a pure qutrit is represented as an element of C^3, which can be written as a normalized 3-component column matrix. Asserting that two qutrits that differ only by a scalar multiple are equivalent enables us to map our qutrits to points of CP^2.

In trying to generalize this construction for octonionic qutrits, one encounters ambiguity in the C^3 equivalence class construction and must define pure qutrits in terms of 3×3 projectors (or in general, equivalence classes of elements Q in J(3,O) satisfying Q x Q=0, where ‘x’ here means the Freudenthal cross product). Such Q are actually the eigenmatrices of the exceptional Jordan eigenvalue problem, and conveniently E6(-26) preserves their characteristic equation.

To see the connection to extremal black holes read: E_6 and the bipartite entanglement of three qutrits by Duff and Ferrara. Note that Duff and Ferrara base their construction on J(3,O_s), where O_s are the split octonions. For J(3,O), one can use the positive definiteness of the trace norm to show the eigenvalues for the eigenmatrices are real. However, this trick doesn’t work for J(3,O_s). In this sense, qutrits for J(3,O) can be described quite elegantly, and this should facilitate an OP^2 qutrit description of extremal black holes in N=2 SUGRA.

]]>“… trying to be even more abstract than loopoids, and get codes out of logic …”

maybe you are looking at quantum information theory and quantum game theory.

My possibly relevant web page is at

http://www.tony5m17h.net/info.html

Some things mentioned there might be useful

(I apologize if they are already so well-known that

my references to them here are not useful):

1 – Cerf and Adami in quant-ph/9512022 show that quantum information theory is similar (maybe completely isomorphic) to fermion particle-antiparticle pairs of particle physics.

2 – Vlasov in quant-ph/0010071 shows that Clifford algebras can be used to construct computationally universal sets of quantum gates for n-qubit systems.

3 – Perhaps my favorite quantum code is discussed by Steane in quant-ph/9608026

the Quantum Reed-Muller code [[ 256, 0, 24 ]]

where 256 is the dimension of my favorite Clifford Algebra Cl(8).

4 – Zizzi In gr-qc/0304032 sees our universe in its inflationary era as a conscious quantum computer, in something like a spin-foam model,

although I would prefer to see foam nodes (instead of SU(2) things) more like the traceless part J(3,O)_0 of exceptional Jordan algebra J(3,O).

Tony Smith

]]>Being blacklisted by the Cornell arXiv, I have no recent publications, but I do have a web page at

http://www.tony5m17h.net/loopoids.html

about Loopoids.

It mentions a lot of things, but does not describe all of them in detail, including something with one of my favorite mathematical names:

“the Hilton-Roitberg Criminal”,

and I hope some things there might be of interest.

Tony Smith

]]>Tony, I completely agree about Loopoids! That’s great! In a sense this *is* exactly what I’ve been doing, trying to understand the logic of tri and tetra categories *before* dropping down to special associative structures. Often I slip back into familiar territory, but only as part of the learning process.