### Light Speed Returns

Our poor friend turbo-1 at PF tried to support the consideration of a varying speed of light by quoting Einstein as follows:

"In the second place our result shows that, according to the general theory of relativity, the law of the constancy of the velocity of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity and to which we have already frequently referred, cannot claim any unlimited validity. A curvature of rays of light can only take place when the velocity of propagation of light varies with position."

Alas, nobody agreed with turbo-1 (or Einstein for that matter) and it was said that there is no evidence in favour of a varying $c$ in a cosmological setting. As Tony Smith says on Louise's blog, "Nature spreads the Big Lie" (that's the magazine, not the bitch, Tommaso).

On a lighter note, Urs Schreiber has discovered the virtues of dimension raising Gray tensor products in particle physics (skip the stuff about membranes) and there are now too many mathematicians blogging about Category Theory to keep up with them. Gone are the days when a physicist could greet the term functor with a blank expression. I'm going to have to find a new branch of mathematics so I can be a rebel again.

"In the second place our result shows that, according to the general theory of relativity, the law of the constancy of the velocity of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity and to which we have already frequently referred, cannot claim any unlimited validity. A curvature of rays of light can only take place when the velocity of propagation of light varies with position."

Alas, nobody agreed with turbo-1 (or Einstein for that matter) and it was said that there is no evidence in favour of a varying $c$ in a cosmological setting. As Tony Smith says on Louise's blog, "Nature spreads the Big Lie" (that's the magazine, not the bitch, Tommaso).

On a lighter note, Urs Schreiber has discovered the virtues of dimension raising Gray tensor products in particle physics (skip the stuff about membranes) and there are now too many mathematicians blogging about Category Theory to keep up with them. Gone are the days when a physicist could greet the term functor with a blank expression. I'm going to have to find a new branch of mathematics so I can be a rebel again.

## 18 Comments:

Your supportive posts are always appreciated. The fact that people are discussing varying c is a victory for our side. I still wish to have been in Sydney for GRG18.

Someone who promotes c change gets to party in Chicago with astronauts and NASA administrators, and that blogger who erases Kea's comments is stuck at the Daily Kos convention. This may show who is really dedicated to science.

Lol! I would love to go to some of those parties!

08 04 07

Great post Kea. It is so interesting to see how orthodoxies infiltrate scientific thinking and how it becomes a sin to question them. Michael Duff was championing a changing c and other dynamical 'constants' for a while now. And until someone can PROVE why the speed of light in vacuum should always be constant, this is something that should be questioned.

Its sorta like Blacks living in segregation. When I lived in Atlanta Georgia for a while there were places that a Black did not go. Why? I asked. 'Oh because they aren't too friendly to Blacks there'. So I couldn't visit some location because no other Black person could? Silliness pure silliness!

And while my analogy may be a bit abstract, the point I am getting at is that the scientific community has become complacent and almost like a cultist religion. If one dares question the COMMANDMENTS of the cult, one is ostrasized regardless of the veracity of the question.

It is GOOD to see more physicists attack category theory, but I believe that you have such a unique way of thinking that you will always be a rebel anyway;)

Mahndisa, your Georgia racist analogy does not seem "abstract" to me.

In addition to being blacklisted because of my physics views,

when my church (multiracial) here in Georgia submitted expansion plans to the local zoning board, most of the relatively affluent white neighbors objected, and a lot of personal animosity was directed at me.

Fortunately, with the backing of the NAACP (it is NOT an obsolete organization, no matter what NPR radio might say), our church won the zoning board vote,

but

my direct personal experience is that attacks by physics "establishment" figures

and attacks by affluent white church-objectors

are very similar, including use of lies and resistance to honest discussion of substantive issues,

so

Mahndisa's analogy seems to me more concrete than abstract.

As Mahndisa said, such behaviour is indeed "Silliness pure silliness!",

however,

it is sad that such "silliness" is not only obstructive to progress,

both scientific and social,

but also that it does inflict real pain of hurt feelings on many people, with long-term consequences (to all parties involved) that may not be obvious.

Tony Smith

PS - Some details of the church situation can be seen at

www.valdostamuseum.org/hamsmith/Aug2005Update.html#gnfmbc

which has links to some relevant pdf documents.

Hi Kea!

It's good to be a rebel in one's younger years, but you need to take stock now. If category theory is going to be all the rage of the next ten years, you have the moral obligation to be all over the place giving interviews and cheering to camera flashes. It would be a shame to seclude oneself to something else, just because it has good odds to be unsuccessful.

On a different note, you did good in specifying... I would have believed Tony referred to the Bitch (but note the capitals).

Cheers,

T.

If category theory is going to be all the rage of the next ten years, you have the moral obligation to be all over the place giving interviews and cheering to camera flashes.Lol! But the cameras won't be after me. There are no shortage of people out there with their hands up, saying how they've known for a long time that this was a promising direction (even though many of them are so young they were practically in diapers when I started out on this track).

Tommaso and Kea, my view is (to paraphrase Einstein)

NATURE may be subtle, but she is not malicious.

In short, NATURE NEVER LIES.

(The same cannot be said (truthfully) about the magazine.)

In a for-some-reason-deleted post Kea said that she "... had to struggle for 40 years to earn the right to stand beside them as a 'professional' thinker. ...".

Kea, I think that you are far above the "'professional' thinker" category.

The "'professional' thinkers" belong to "the Amagamated Union of Philosophers, Sages, Luminaries and Other Thinking Persons" and are represented by

Majikthise and Vroomfondel, whose "... brains ..[are]... too highly trained ..." to see how things really work.

Tommaso, you said "... If category theory is going to be all the rage ...".

The usefulness of category theory is easy to see.

It not only is a very high level of abstract generalization, it is also connected with some very specific interesting structures.

To grossly oversimplify:

Category Theory gives Homotopy Theory gives 8-fold Periodicity

which

gives Real Clifford Algebras, Spin Lie Algebras, Octonions, and Exceptional Lie Algebras,

and

even "'professional" thinker" Ed Witten has been quoted (by John Baez) as

"... suggest[ing] that the correct theory of our universe could be an exceptional structure of some sort. ...".

Speaking of Octonions, they are not Associative, but are Alternative, which leads me to differ a bit from John Baez, who has applied Groupoids (which are associative and therefore not in line with Octonion structures) to Category Theory, while I would prefer to use nonassociative Loopoids

(Loops having been called "nanassociative groups"

by Lohmus, Paal, and Sorgsepp in their 1994 book Nonassociative Algebras in Physics).

When (back in 2003) I asked John Baez (by e-mail)

"... How much of the groupoid structures that you like would survive

if associativity (which octonions do not have)

were to be replaced by left-adjoint and right-adjoint associativity and alternativity and power-associativity (which octonions have) ... ?"

he replied

"Dunno! You are proposing a theory of "loopoids", but it would be a major project to develop this theory and see if it's worthwhile. ...".

So, Kea, if you want

"... a new branch of mathematics so ...[you]... can be a rebel again ..."

and

at the same time really be doing Category Theory,

you might consider Loopoids

(among other things - there are lots of ways to see the same elephant).

Tony Smith

08 07 07

Thanks for the response to my rambling Tony. The statements about groupoids versus loopoids is a curious one. In the end it seems as though one would work with loopoids as the general case, then groupoids will fall out as special cases.

Funny that. Have you written any papers on this approach? I realize that non associative groups are a rage right now, but the application to category theory doesn't seem to be...

Kea, I like your approach but you think at such an abstract level that it is difficult to keep up sometimes;) Keep on trucking Kea and Tony:)

"You are proposing a theory of loopoids, but it would be a major project to develop ...."Tony, I completely agree about Loopoids! That's great! In a sense this

isexactly what I've been doing, trying to understand the logic of tri and tetra categoriesbeforedropping down to special associative structures. Often I slip back into familiar territory, but only as part of the learning process.Mahndisa asked whether I have written any papers about Loopoids.

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

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

Cool, thanks. I really like the Gray code! But I'm trying to be even more abstract than loopoids, and get codes out of logic and higher categorical thinking. In that sense, loopoids maybe aren't a big enough kind of non-associativity. Anyhow, it's really nice to see connections with what you are thinking.

Kea, since you are

"... 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

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

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).

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 3x3 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.

...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)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 3x3 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.

Kneemo, thanks for very interesting comments about Octonionic structures.

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

... 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.

Kneemo, as you said:

"... 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

Kneemo, I apologize for oversimplifying and omitting mention of E7 and E8 in my immediately previous comment.

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

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