An Exceptionally Technical Discussion of AESToE

[strangerep]

[...] Kaon physics, where you mix two "strong eigenstates" and mix them to the two "weak eigenstates" to get the REAL physical eigenstates. [...]

I'm wondering how do you define "real physical eigenstates"? If you mean mass eigenstates,
then... what makes mass (or energy-momentum) more "real" than other quantum numbers?

This reminds me,... what are the Casimirs of E8? And is there a list somewhere of
maximal set(s) of commuting generators? Does a complete classification and analysis
of unitary irreducible representations of E8 exist? If so, could some please give me
a reference?

TIA.

 

[Tony Smith]

Berlin asks "... what are the Casimirs of E8? ..."

In their paper at http://arxiv.org/abs/hep-th/0702024 Cederwall and Palmkvist say:
"... The orders of Casimir invariants are known for all finite-dimensional semi-simple Lie
algebras. ... In the case of e8, the center ... of the universal enveloping subalgebra is generated by elements of orders 2, 8, 12, 14, 18, 20, 24 and 30.
The quadratic and octic invariants correspond to primitive invariant tensors in terms of which the higher ones should be expressible.
While the quadratic invariant is described by the Killing metric,
the explicit form of the octic invariant is previously not known ...
We thus consider the decomposition of the adjoint representation of E8 into representations of the maximal subgroup Spin(16)/Z2. The adjoint decomposes into the adjoint 120 and a chiral spinor 128. ...
The final result for the octic invariant is, up to an overall multiplicative constant:
...[ their equation 2.3 ]... ".

As to "... a complete classification and analysis of unitary irreducible representations of E8 ...", see
http://aimath.org/E8/
and
http://www.liegroups.org/AIM_E8/technicaldetails.html which says
"... In principle, the set of irreducible representations of the split real form of E8 are known. How many are there? Before the software was written, we didn't know. We expected about 696,729,600 the order of the Weyl group. (This is what small examples suggest.) In fact the number is 453,060. ..."
and
http://aimath.org/E8/computerdetails.html
and
http://www.liegroups.org/kle8.html
and
a post by John Baez at
http://golem.ph.utexas.edu/category/2007/03/news_about_e8.html

Tony Smith

 

[Tony Smith]

Sorry - it was strangerep not Berlin who asked the questions I quoted in my previous reply.
Tony Smith

 

[rntsai]

Hi Berlin/Jan,

You asked if anyone confirmed the quantum numbers. I did, and for the first two generations (8-S+ and 8-S- in Garrretts paper)) everything is ok.

In some ways this adds to my confusion since Garette only claims
that only one generation not two are ok (page 22 of his paper).

The interesting thing happens with the third generation. ALL of the assignments for third generations of quarks and leptons have two quantum numbers wrong.

However, it is possible to arrange this third generation into a nice pattern. I think it was you to call it numerology.

I found the following: for the quantum numbers B2, B1 and W-3 all third generation particles have two of those numbers wrong for an amount of 1/2 or -1/2. The third is right. The table:

1/2, 1/2, 0 wrong offset for: 6 leptons, 12 anti-quarks
-1/2, -1/2, 0 wrong offset for: 6 anti-leptons, 12 quarks

1/2, -1/2, 0 wrong offset for: 2 leptons, 6 anti-quarks
-1/2, 1/2, 0 wrong offset for: 2 anti-leptons, 6 quarks

-1/2, 0, 1/2 wrong offset for 3 quarks, 3 anti-quarks
1/2, 0, -1/2 wrong offset for: 3 quarks, 3 anti-quarks

In these sets the left particles are opposite to the anti-right particles. The pattern seems too nice to be ignored. How can quantum numbers be wrong by a fixed amount? The nicest example I know of is Kaon physics, where you mix two "strong eigenstates" and mix them to the two "weak eigenstates" to get the REAL physical eigenstates. If you do that with the above you should mix a lepton and an anti-lepton "root" to get the physical state. So, both the third gen of leptons and quarks are kind of "kaon" physical states.

other than B_2, all other quantum numbers are spins which only have
-1/2,0,+1/2 values; I think what you note above is probably just a simple
consequence of that...that's my opinion (prejudice?) until I can actually
verified how unique this error pattern really is.

What I don't understand (among others) how to translate this into the groups you guys are strugling with. Could anybody try to put the sets of particles I identified above into a right group structure?

I can help translate these into group (or better yet algebra) constructs but I would need
a more concrete description of which roots of e8 you're working with. For example, to
me 8S+ corresponds to 8 roots only; in Garrett's paper (Table 9) 8S+ is associated
with 64 roots. The details of this correspondance are not described in enough detail
in the paper for me to crack.

 

[Tony Smith]

rntsai said "... to me 8S+ corresponds to 8 roots only; in Garrett's paper (Table 9) 8S+ is associated with 64 roots.
The details of this correspondance are not described in enough detail in the paper for me to crack. ...".

It seems to me that in Garrett's Table 9 the first-generation lepton and quark particles and antiparticles correspond to 8S+ and indeed have 8+8+24+24 = 64 elements:

spin +1/2:
8 left-handed particles
8 right-handed particles
8 left-handed antiparticles
8 right-handed antiparticles

spin -1/2:
8 left-handed particles
8 right-handed particles
8 left-handed antiparticles
8 right-handed antiparticles

That would be 4x8 = 32 spin +1/2 plus 4x8 = 32 spin -1/2 for the total of 64.

I think that Garrett's Table 9 uses 8S- for second generation (another 64)
and 8V for the third generation (another 64).

It seems to me that in Garrett's paper an 8-element 8S+ lives only in the 52-dimensional F4 subalgebra of 248-dimensional E8 and that when you put the F4 inside E8 the 8 elements of the F4 8S+ correspond to 8x8 = 64 elements of the E8 8S+.

That corresponds roughly to E8 being an octonification of F4, sort of multiplying some of the F4 elements (such as 8S+ and 8S- and 8V) by 8 so that each of them become 8x8 = 64-dimensional, for a total of 3x64 = 192 of the E8 dimensions.

As to the other 248-192 = 56 dimensions of E8, they seem to correspond to two copies of the 28-dimensional D4 subalgebra of F4,
so that (as has been discussed in earlier posts) E8 contains two copies of D4,
one for gravity and the other for the Standard Model.

In rough equations:

52-dim F4 = 28-dim D4 + 8-dim 8S+ + 8-dim 8S- + 8-dim 8V

248-dim E8 = 2 x ( 28-dim D4 ) + 8 x ( 8-dim 8S+ + 8-dim 8S- + 8-dim 8V )

Note that the ( 8-dim 8S+ + 8-dim 8S- + 8-dim 8V ) is closely related to the exceptional 27-dimensional Jordan algebra J(3,O) of 3x3 Hermitian octonion matrices.

I hope I got that right, but maybe I did not, because in my picture the physical meanings of 8S+ and 8S- and *V are different, and I could be confused when talking about Garrett's physical meanings.

Tony Smith

PS - Sorry for using capital letters for Lie algebras, but I am not good at being consistent with upper case for groups and lower case for algebras, and I hope that what I say is clear from context.

 

[Berlin]

First two generations Quantum numbers

One of you asked me to show the first two generations quantum numbers. Attached is a doc file (excel was not allowed) for my choices. I checked with Garrett's paper and (as far as I can see) made only one change in the lepton sector compared to his choices:

- I did not use w=-1/2 for all gen 1 and 2 leptons, but used w=-1/2 for gen 1 and w=1/2 for gen 2. Visa versa for the anti-leptons.

Did I made a mistake somewhere? Does anyone of you understand what impact the change in w-number will have on the group structure?

And Strangerep: I do not know how to work out the "kaon" idea. Maybe the two 'oscillating' states for gen three particles can only show up in the form we know because they are not allowed to show up otherwise in the interactions we study.. For example: as far as I know gen three quarks only show up in color neutral particles with short lifetime.

and about mass:

"OH WHATS MASS GOT TO DO WITH IT, GOT TO DO WITH IT
WHAT`S MASS BUT A SECOND HAND EMOTION
WHAT`S MASS GOT TO DO WITH IT
WHO NEEDS F4
WHEN F4 CAN BE BROKEN"

Jan

 

[rntsai]

One of you asked me to show the first two generations quantum numbers. Attached is a doc file (excel was not allowed) for my choices. I checked with Garrett's paper and (as far as I can see) made only one change in the lepton sector compared to his choices:

It looks like a fair amount of work went into producing this
table; it will take some time to go through it in detail.

For G1 leptons, first two columns, these don't look right.
their average should be +/-1/2 for left handed particles and
0 for right handed ones (see page 9, eq 2.9). Either the
table entry is wrong or the identification with the particle
isn't right.

- I did not use w=-1/2 for all gen 1 and 2 leptons, but used w=-1/2 for gen 1 and w=1/2 for gen 2. Visa versa for the anti-leptons.

Did I made a mistake somewhere? Does anyone of you understand what impact the change in w-number will have on the group structure?

As far as I could tell the new quantum number w
doesn't enter the picture when calculating any physical quantum numbers.
So I don't know why you had to change its value at all.

It's possible you started from the 240 roots of e8 and tried to
associate these with particles along a path sperate from the one
Garrett took. In that case you're free to move things around as
long you don't create a new structure. Rearranging the 240 roots
in any particular order doesn't matter, you can also replace the
columns (eigenvalues of cartan elements or quantum numbers) with
linear combinations, as long as you process entire columns... How
a new arrangement affects dynamics,... is another matter.

I wish Garrett had included a similar table as an appendix; or better
yet attach it here (hint to Garrett...BTW too bad about AAPL today!)...
this would clarify things a lot.

 

[garrett]

I almost included such an appendix, but I decided to just abbreviate it (as Table 9) and include it in the main body of the paper. I thought people might enjoy the game of piecing together the details from previous tables. ;) Also, I wasn't happy about publishing the explicit list when the second and third generations don't work quite right. But, since you're asking, I'll attach a table of the explicit roots (the other 120 are the antiparticles of these).

AAPL certainly did take a dive. Fortunately, I didn't have much to lose.

 

[rntsai]

Thanks Garrett,

You can tell fairly quickly that Berlin's table is different.

The pagebreak in the pdf file took out one line. Would it be
too much trouble to generate a text file with the same data?

I could offer stock recommendations in return, but based on
recent performance, you're better off settling for my gratitude.

 

[garrett]

Sorry, it's dumped from Mathematica and doesn't convert well to normal text because of all the sub and superscripts. The line wasn't removed, just split, so it should be clear what it is. I don't currently have it as a text file, or I'd send it.

 

[rntsai]

I'll attach a table of the explicit roots (the other 120 are the antiparticles of these).

Quick question : Is the breakup of particle/antiparticle along the lines of positive/negative roots?

 

[blogder]

thank you..

 

[Berlin]

remarks

Rntsai: you are correct that my gen1 leptons needed some correction to get the w-L and w-R right. Was not difficult to do. My gen-2 however has "wrong" w-l and w-R numbers consistently. They do have the right charge and hypercharge. Garrett's gen-2 neutrino's seem to have charge 1... What's wissdom? I more and more suspect that there is something wrong with the w-L, w-R, B1 and B2 sector. To get all quantum numbers correct for all three generations I get things like:
- rotate 90 degrees in the w-L w-R plane (gen 2, B1 and B2 correct)
- parallel shift in w-L, w-R, B1 and B2 of (0.5, 0.5, 0.5, 0.5) (gen 3 leptons)
- shift (-0.5, -0.5, -0.5, -0.5) for gen 3 anti-leptons.

Have not all details correct, but my suggestion would be to 'demand' charge and hypercharge to be correct and try to reverse engineer the theory from there.

You also asked if anti-particles are minus in all roots. In my scheme that is not the case.

Jan

 

[garrett]

rntsai,
No, there is not a direct correlation, at least not in this assignment.

Jan,
Yes, I agree with this strategy, and am working on it a bit now. There is this nice paper by Chang and Soo,
http://arxiv.org/abs/hep-th/9406188
which might help in building a manifestly chiral description, with only a w-L.

 

[staf9]

Just a couple things after being away from this discussion for a while,

Could anyone tell me the roots for the tau neutrinos? This is what I got for the roots, which I don't think is the same as in garrett's pdf:

[0 0 0 1 1 0 0 0]
[0 0 1 0 1 0 0 0]
[0 1 0 0 1 0 0 0]
[1 0 0 0 1 0 0 0]

Jan - I don't think I completely understand what you mean by 'demand'.

 

[Emanuel]

I believe he means 'go with the assumption' that they are correct and work backwards from there to find a configuration in agreement with scientific observations/the standard model. (but being a layman, I could be wrong or might not have phrased that correctly)

 

[patfla]

'set' or 'fix' (the values) might be better word choices (than 'demand'). And, of course, then work backwards.

 

[staf9]

Ahh, thanks for clearing that up. I feel kind of dumb now

Chang and Soo's paper could be adapted for w-R in addition to w-L, right? It details two-component spinors in the paper.

 

[Berlin]

New three generations of leptons

I followed the strategy to search for root assignments, consistent with the color and EW quantum numbers, but with a flexible choice for w-T, w-S and w. Look for 'logical' and elegant assemblees of roots.

A managed to get the three generation of leptons into a new and elegant choice of roots vectors. My starting point was that the up/down lepton particles have Q# w=+/- 0.5, contrary to Garrett. The gen 3 leptons get w=+/- 1. I interpreted that that the root vectors for the third gen leptons are spin 1 bosons coupled to a gen 1 or gen 2 lepton to get the third gen physical leptons with the right quantum numbers. It looks as if there are 16 fields that couple to gen 1 or gen 2 leptons to create that third generation.

Physical gen 3 leptons = root gen3 (boson) + lepton from gen 1 or 2 with:
spin 0.5 up = spin 1 up +spin 0.5 down etc.

Whether this relates to the Chang/Soo paper I have not figured out. Can you Garrett? 16 is a beautiful number. And the quarks involve 3 x 16 = 48 fields...

Look at the table and you see how. I did not have the time to check if other combinations are possible. Quarks are next to check. Nice puzzle.

Jan

 

[Berlin]

third generation of quarks

A managed to get the third generation of quarks as well in the same way as the leptons. My starting points were:
use w=+/- 0.5 for the up/down quarks of gen 1 and 2 (not difficult to find). For the third gen quarks I used the assignments Garrett used. They work out well. The third gen quarks all have w=0 roots, which I interpreted that that the root vectors for the third gen quarks are spin 0 bosons and couple to a gen 1 or gen 2 lepton to get the right quantum numbers.

48 fields can be identified. Like the leptons:

Physical gen 3 quark = root gen3 (spin 0 boson) + lepton from gen 1 or 2 with:
spin 0.5 up = spin 0 + spin 0.5 up etc. See attachment.

So, my conclusion for this moment would be that the third generation of quarks and leptons can be assigned to a E8 rootvector (with boson properties) "coupled" with a gen 1 or 2 lepton. Why the physical states are 'coupled' particles I don't know. Why we don't see the 'bare' bosons I don't know either.

Jan

 

[Lawrence B. Crowell]

E_8 and quantum error correction

I read your paper "An exceptionally simple ...," several times last month. I have a fair number of questions, but I will keep the more technical ones until later. One question I have is whether the Higgs vev in equation 3.8 that determines the cosmological constant is related to the dilaton field, such as one in the SU(4) conformal theory.

I read you paper with considerable interest since Vogan & deCloux group numercially computed the Kazhdan-Lusztig polynomials for the split real group E_8 with the ATLAS program. The exceptional E_8 plays a role in string theory and there are some indications it may operate with LQG as well. (BTW, I am not a partisan of either theory and suspect these are two different perspectives on the same problem).

I have been pondering whether quantum gravity is most fundamentally an error correction code for a sphere packing. Physically the idea is that quantum bits are preserved through all possible channels, such as noisy quantum gravity channels like black holes. So my idea is that quantum states are preserved, or their The kepler problem and the 24-cell are the minimal sphere packings in three and four dimensions and the 240-cell (E_8 polytope representation) is "probably" the minimal sphere packing in eight dimensions, at least estimated by Elkies. Sphere packing defines Golay codes, where each vertex is a "letter" in a code, such as the octahedral C_6 is the GF(4) hexacode. The 120 icosian (half the 240-cell) supports the M_{12} Mathieu group, which under a double cover defines the 240-cell and the Leech lattice error correction code M_{24}.

The theta function realization of the Leech lattice involves three E_8's, or polynomials over them. These lead to a modular system of theta-functions, which interestingly obey Schrodinger equations. The heterotic string of course has two E_8's. I have pondered whether the role of the third E_8 is with the Cartan center description of "fake" M^4s in Donaldson's theorem on four dimensional moduli.

The ADM classical constraint equation H = 0 becomes H*Y[g] = 0, and where time enters into the picture it is something the analyst inputs. The lapse functions N are determined by a coordinate condition, analogous to a gauge. One way in which we can do this is to impose a field on the metric g. For that field F defined on each g there exists a wave equation and it is not hard to introduce a phase on the wave functional Y[g, F] so that the W-D equation is extended to

<br /> i\frac{\partial Y}{\partial t}~=~HY~\rightarrow~iK_tY~=~HY, <br />

for K_t a Killing vector. Now remember, this field is defined within some scaling or conformal setting. We can just as well chose another field conformally scaled otherwise. This wave equation is perfectly time reverse invariant, even if this "time" is in a sense fake. If we have another metric g' it has a similar wave functional X[g', F'] and wave equation

<br /> i\frac{\partial X}{\partial t}~=~HX~\rightarrow~iK&#039;_tX~=~HX. <br />

Yet covariance requires that K_t =/= K'_t and so we can't describe a superposition of states, and a path integration over possible states

<br /> Z~=~\int \delta[g]e^{iS}, <br />

where S includes NH, is not defined in the usual sense as some parameterization of states in a time ordered sense. There is no single definition of time.

The course graining of these metric configurations leads to an energy uncertainty functional

<br /> \delta E_g~\sim~ |\nabla\delta g|^2, <br />

which describes a coarse graining over many metric configurations by the violation of general covariance imposed by the implicit coordinate map between the two. Most of these wave functionals are over metric configuration variables which have no classical description, or in fact have no possible dynamical (diffeomorphic) description. These 4-manifolds are "fake" and this course graining of possible metric configurations, with these as well, introduces this error functional. The Cartan center of E_8 describes the set of possible M^4's and these "fake" manifolds. This is in part why I think quantum gravity requires the S^3xSL_2(7) \subset M_{24} or more fundamentally M_{24} as a quantum error correction code, which embeds three E_8's --- an E_8xE_8 for the graded heterotic supergravity field theory and the third for this configuration of all possible spacetimes. In the restricted S^3xSL_2(7) this is a thee dimensional Bloch sphere where each point on it is a "vector" in a three space spanned by the Fano planes associated with these three E_8's. S^3xSL_2(7) has 1440 roots and is itself a formidable challenge, but this represents a best first approach. M_{24} has 196560 roots and clearly an explicit calcuation of those is not possible at this time.

What is interesting is that if this is the case this has a triality to it with 3 copies of E_8. There are also three Jacobi theta functions which are modular forms (functions) with a range of interesting properties. At any rate this is my main question at this time, whether you or anyone else has pondered this sort of hypothesis for quantum supergravity.

Lawrence B. Crowell

 

[Kea]

codes and QG

What is interesting is that if this is the case this has a triality to it with 3 copies of E_8. There are also three Jacobi theta functions which are modular forms (functions) with a range of interesting properties. At any rate this is my main question at this time, whether you or anyone else has pondered this sort of hypothesis for quantum supergravity.

Lawrence, thanks for some interesting remarks. Of course, many M theorists are now interested in such codes in exactly this context. I have blogged a little about it. But personally, I view the codes as secondary to the underlying logic, interpreted as observables in a category theoretic language. Triality utilises qutrits as well as qubits, by the way, and this kind of triality was associated to mass by Carl B and others in a completely different context to Garrett's paper.

 

[garrett]

Hi Lawrence,
The Higgs comes in as part of the frame-Higgs, which is part of the E8 connection -- so it isn't naturally associated with a dilaton. However, since the Lagrangian is currently assembled by hand, I think it would be possible to cook up an alternative Lagrangian in which this field acts as a dilaton.

I agree there are many interesting different directions this E8 theory could go, and ways it connects up with other similar ideas. But I have tried to stick as closely as possible to the minimal mathematics necessary to match the standard model and gravity, so haven't worked with some of the wilder possibilities. And I also agree that quantum mechanics should be brought into this theory in some natural and interesting way, but I certainly haven't figured out how yet.

 

[garrett]

Jan,
If you've got an interesting new assignment of particles to E8 roots, that's great. You might wish to write it up as a paper.

 

[John G]

One question I have is whether the Higgs vev in equation 3.8 that determines the cosmological constant is related to the dilaton field, such as one in the SU(4) conformal theory... Golay codes, where each vertex is a "letter" in a code... the Leech lattice involves three E_8's, or polynomials over them. These lead to a modular system of theta-functions, which interestingly obey Schrodinger equations. The heterotic string of course has two E_8's. I have pondered whether the role of the third E_8 is...

Earlier here, rntsai mentioned the dilation and conformal degrees of freedom in SU(4) and I know Tony Smith who has posted here too does relate that dilation to the Higgs VEV. Tony also is into Golay, etc. codes and the Leech lattice. I think that Triality of E8s is related to the D4 Triality so it would be a vector-half spinor-half spinor thing and thus related to the 3 generations for Garrett and one generation of particles and antiparticles plus spacetime for Tony. If Garrett went to a more Chang and Soo-like assignment, I could see his D4 + D4 housing an SU(4) conformal theory plus an SU(4) GUT plus having the vector part used to get up to SU(5) or SO(10) and the spinor part left for one generation. This is essentially what Tony wrote about after reading Garrett's paper. The relationship between the two D4s would probably still seem unusual to lots of people.

 

[Lawrence B. Crowell]

Hi Kea,

Triality from a coding perspective involves ternary bits, or what you call qutrits. There are a number of hexacode representations in F_4, where some involve ternary systems.

I tend to view physics as only being real with respect to information we obtain from it. This happens when a particle causes a detector to go "beep" and record an event. Things such as wave functions, fields, and even space and time I regard as model constructs. For example with relativity if you have the metrics g_{ab}(p) and g'_{ab}(p) for the two metrics determined in different frames then general relativity predicts that the evolute of this point is different in the two frames. General relativity is about the relationship or distance between particles, not between geometric points or the nature of manifolds. There is of course another relationship system called quantum mechanics which has some funny consequences when we think according to geometry. So I tend to think that Seth Lloyd is on the right track with the quantum computer model of the universe. The question is what is the algorithm does the quantum computer run? The quantum computer must also be the algorithm and perform computations with a code with can act as an error correction code with a Hamming distance for "bit correction."

Hi Garrett,

Whether the Higgs is a dilaton is a question that has nagged me a bit. The Higgs are "framed" with the graviton, or the gravi-weak in the F_4 or so(7,1) ~ Cl(7,1), as in equation 2.10 and the argument above. The so(8) (Euclideanized) is S^8 ~ spin(8) which contains an su(4) ~ spin(6), and where spin(6) contains the the deSitter so(4,1) and Anti-deSitter so(3,2). su(4) is a conformal model with a dilaton. So in the

so(7,1) = so(3,1) + so(4) + (4 x 4)

we have 16 framed Higgs to play with. spin(6) is 15 dimensional ---> the conformal group, which means that 9 of these Higgs (framed Higgs) would have to be dedicated to the conformal group. so(8) ~ spin(8), or spin(8) is a Z_2 cover on so(8) with center Z_2. Spin(8) embeds spin(6) as well. So it appears possible to define a conformal theory of gravity. Conformal gravity is a "good thing." I would have to think about how to apportion these framed Higgs as dilaton fields. There is a bit of a difficulty to my mind that this might bring an imbalance in how the two D_2's as gravity and weak fields act on the Higgs.

Lawrence B. Crowell

 

[Emanuel]

Jan,
If you've got an interesting new assignment of particles to E8 roots, that's great. You might wish to write it up as a paper.

Garrett,
Did you see the documents he posted containing this assignment?

Lawrence,
Wow, the technicality of this discussion just went up a few degrees. I would like to inquire about a detail: is =/= the same as !=, or put in words, the 'not equal to' operator?

 

[Kea]

Things such as wave functions, fields, and even space and time I regard as model constructs.

Yes, of course. We are working on rewriting all of QFT, for instance, in an operadic language. The quantum computation angle is only one small part of the story. It usually assumes that linear, qubit based algorithms are sufficient to discuss quantum gravitational quantities, but we strongly disagree with this idea. The ternary element enters at a very foundational level for us, namely the axiomatics of a higher topos, perhaps as a ternary extension of the Stone type duality underlying, in particular, the Fourier transform.

The question is, what algorithm does the quantum computer run?

I see the main new principle as a Machian type of holographic principle, which as you know I have mentioned before. But it is so self-referential in terms of the relation between algorithms and their implementation that I really believe it requires some heavy omega categorical machine to write down. Schreiber, for one, sort of works on this, although he is still stuck with stringy gauge theory ideas, which seem to be irrelevant.

Since Abramsky and Coecke, we know that the protocols for ordinary quantum computation can be fully implemented using a basic categorical structure, so this is all feasible.

 

[Lawrence B. Crowell]

Hello John G.

As I wrote earlier spin(6) can be contained in the graviweak or F_4. I think going to page 12 of Garrett's paper the conformal group might be obtained by "framing" nine Higgs according to something of the form

<br /> \phi_{dilaton}~=~\phi^a\Gamma_{ab}e^b<br />

I think the E_6 gives a clue as to how to partition the Higgs fields, but I will have to differ that until later when I am a more certain of things.

Gerrett:

The approach to embedding three copies of E_8 into the Leech lattice I propose as a quantization procedure. There is an uncertainty like principle for a myriad of four manifolds (spacetimes) under a decoherent functional. All of these four manifolds are described by the Cartan center of E_8, which indicates how many of these manifolds are homeomorphic but not diffeomorphic. I think this has a bearing on how most configuration metric variables for the Hawking-Hartle wave functional have no classical correspondence. So we then have one E_8, more or less as you have constructed, another for the supersymmetric pairs of these fields and a third for this "quantization." In string theory the E_8xE_8 is partitioned according to chiral fields on a closed string, and so this idea differs. The Jacobi theta functions obey Schrodinger equations, and have zeta function realizations.

Hello Emanuel,

Yes =/= is the same as the C code !=. This stuff is interesting and last October I was working out something similar to what Garrett completed. I had proposed a fair number of years ago on how QCD should be G_2. I also figured the vierbein approach to loop quantum gravity with the internal SO(4) was a way of including ~ SU(2)xSU(2) into gravity. Garrett's paper saved me from a lot of work.

Lawrence B. Crowell

 

[Lawrence B. Crowell]

Quantum codes and Topos theory

Yes, of course. We are working on rewriting all of QFT, for instance, in an operadic language. The quantum computation angle is only one small part of the story. It usually assumes that linear, qubit based algorithms are sufficient to discuss quantum gravitational quantities, but we strongly disagree with this idea. The ternary element enters at a very foundational level for us, namely the axiomatics of a higher topos, perhaps as a ternary extension of the Stone type duality underlying, in particular, the Fourier transform.

Topos theory!? What I propose does brush on this, or with algebraic and projective varieties. The Golay coding system has some elliptic curve structure and modularity. Coding systems defined on algebraic varieties are called Goppa codes, and projective varieties are a categorical approach to structures such as null congruences. So if we consider these as categories, then presheaves over them may define Grothendieck's category of sheaves.

I will have to admit that I am at the "101" level with Topos theory. It is a subject I am not that versed in. I would have to merinate my mind for some time in the subject before I can comment a whole lot more on this.

I will say from a physical perspective the relationship between the Golay code and Goppa quantum codes might be seen in the instanton states in Euclidean metric and the tunnelled states with a Lorentzian metric. This connection I think, again this is a big maybe, with the elliptic curve systems in the D_4 lattice with a code on the Galois field GF(9). This norm is over a cyclotomic field on the {3,4,3}, with vertices the 24 minimal vectors of D_4 ---> the 24-cell as represented by the cyclotomic field of Galois elements. This can be extended to norms over higher lattices as well.
So Topos theory might fit in this scheme, where the algebraic varieties defined on lattice norms are categories which may have presheaf constructions. Usually a classifier acts on {0,1}as functions from any set S into {0,1}, as a "code" of subsets of S. The classifier replaces the standard Boolean "on and off" or {0,1} in a "logic" over categories of sets. This system could in principle be extended to the ternary system {0, 1, omega} in the ternary system, such as often used in the GF(4) hexacode.

I see the main new principle as a Machian type of holographic principle, which as you know I have mentioned before. But it is so self-referential in terms of the relation between algorithms and their implementation that I really believe it requires some heavy omega categorical machine to write down. Schreiber, for one, sort of works on this, although he is still stuck with stringy gauge theory ideas, which seem to be irrelevant.

Since Abramsky and Coecke, we know that the protocols for ordinary quantum computation can be fully implemented using a basic categorical structure, so this is all feasible.

As for Machian holography and self-reference. Self-reference is to be avoided at all costs! However, I think this occurs at the Planck scale, and I think physics exists above that scale. The string length is sqrt{8-pi}L_p and this lattice approach (which has some 26-dim stringy stuff going on) takes things closer to L_p. In effect where things are self-referential, or where states are determined by "Godel loops" should be renormalized out of the theory.

As for string theory --- take it when needed, and ignore when it gets "funny." :-) There are some very good features to string theory, such as Veneziano amplitudes, but the whole thing fails to constrain itself effectively. String theory is like alchohol --- a little drink is good now and then, but if you drink too much you get hung over.

Lawrence B. Crowell