An Exceptionally Technical Discussion of AESToE

[Kea]

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.

When I say 'topos theory' I am thinking of the elementary axiomatisation, which allows toposes that are not necessarily Grothendieck ones, but it would be nice to see how coding examples of sheaf categories fit in with, say, the twistor string picture, which is one of the 'nice' parts of string theory.

The easiest metric to work with in this setting is ++--, because it allows one to look at real points and sort out some operad combinatorics using well understood examples.

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 is interesting. The 24 cell comes up naturally in my thinking, too, although not in a way directly related to coding. I would like to spend more time trying to understand how code lattices and operad polytopes fit together. See also Tony Smith's work.

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.

Right, we want to replace the {0,1} with {0,1,2} etc., but not as a set, because this is just a 0-category, and Set is an instance of a classical (in all senses of the word) 1-topos. We need to generalise the axioms of a topos, to higher categorical dimensions, so that the basic model for an n-topos uses a classifier based on (n-1)ary logic. Classical toposes can easily have three truth values, but there are other reasons why the classical structure isn't good enough. Eg., the lattices are always distributive.

Self-reference is to be avoided at all costs!

We probably mean different things by this term. I am trying never to think of classical spaces, even fractal ones, except as emergent structures in omega-categorical geometry. In a sense, nothing gets renormalised out, but the way one phrases physical questions based on experimental constraints hopefully means that only the right things get counted. By the way, the modern understanding of renormalisation, a la Connes, Kreimer et al, is very category theoretic in nature.

In our approach there is no 'fixed Planck scale', so we agree with you there. There is a whole heirarchy of $\hbar$ associated to the heirarchy mentioned above. Cosmologically, a varying $\hbar$ and $c$ replaces the cosmological constant, which classically must be zero.

Cheers
Kea

 

[Berlin]

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.

Hi Garrett,

Don't think so. I am not satisfied with it. It's just no physics. All my 'interpretations' are shaky or plain wrong. I am sorry for earlier statements. No dynamics, no predictions, no logic, no beauty. Ah! Report back when I can do better. This work has to succeed! It is too attractive not to support it. Nice to see that Lawrence is thinking of a second supersymmetric E8 too. I said that earlier, but did not work it out. Will be away for a week. After that I will try again!

Jan

 

[rntsai]

I'll add some random thoughts on my current state in reading the paper :

(1)Why e8? I still haven't seen what "exceptional" property forces the
choice of e8 over others? For example e6,d5,... also has very nice
structures that can be manipulated to give similar patterns.

(2)Basis choice of e8. Ideally the results shouldn't matter,
in that case the more "natural" the basis the better. As
far as this step is concerned, I think the basis does seem
to matter "more than it should"; you can pick other sets of
vectors (roots) such that their inner product is compatible
with e8. More on this later.

(3)Assignment of roots to particles. You can use subspaces
invariant under subalgebras as a rule or guide. Garrett
uses 8 dim subspaces invriant under a d4 for example.
This choice is fairly elegant and ends up collecting the
three generations into three sets of 64. One complication
is that these three sets are associated with the 8V,8S+,and
8S- reps of d4. These don't sit inside e8 in a fully symmetric
way as far as I can tell. Triality really goes with d4, it's a
a little better with e6 than e8. With a certain choice of (2)
I could make any two of the three "look the same" as far
quantum numbers. The one that fails, fails in a nice way.
More on this too later if I manage to characterise it better.

(4)Besides the quantum numbers, e8 should also tell how all these
paricles/fields interact. This I think is a bigger claim than
being able to get the correct sets of quantum numbers. Garrett,
you can ignore my ramblings above, but for this part did you
actually check whether the commutation relations for any of
the three generations give the right results?

 

[rntsai]

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

I haven't read much about quantum gravity, but I'm fairly experienced with
coding theory; if you think there's a link between the two, then maybe that's
a good motivator to get more familiar with the subject. Let me ask some basic
question, I hope they're not too elementary.

Is this a "classical" or "quantum" correction code? It is possible to treat
both in the same setting; especially if you work with linear classical codes
(and their counter part stabilizer quantum codes). To me you don't give up
much by doing this. With these, the code can be seen as a subspace invariant
under a certain group (its stabilzer). The "channel" or "errors" live in a group
that includes the stabilizer. These don't leave the code invariant, but change
it in a detectable or correctable way. In both cases, the channel (or error group)
has to be explicitely described.

So what is the underlying space that these qubits live in? what are the allowed
errors?,...

 

[Lawrence B. Crowell]

Hi Jan,

The first rather minor change that might be of interest to me is to have a conformal gravity in here. This might involve generalizing the how the Higgs particles are framed. Another possibility is with the addition of 8 more "0-roots" for the extended 248 rep for E_8. The Hamming code $H_8$ can be used to construct E_8, with the Weyl group matrix $W(E_8)$. This construction involves the Kissing number (minimal sphere packing condition in 8-dim) with the icosian pairing of quaterions $diag[Q_4,~Q_4]$. This gives [itex]2^4\times 14~=~224[/tex] minimal vectors $((\pm1/2)^4,~0^4)$, and just as in the Grossett polytope these pair up in the 240 case they pair up in this case in 248. So this "buys" us eight additional roots. This is one too short, but ... we can still frame the other framed sixteen in some ways to define a $\phi_{dil}$. It is interesting if we defined an additional framing similar to how I suggested yesterday that the summation of $e^{\vee,~\wedge}\phi$ leads to a net zero hypercharge and zero $1/2i\omega_{\Gamma}^3$ and $1/2\omega_S^3$, as well as I think the other q-numbers. This will be the central dilaton component and the other 8 will define the remaining components to fill out the 9 + 6 = 15 dimensions of $SU(4)~\sim~spin(6)$ conformal gravity.
This will then give a theory with AdS/CFT. It will also in my opinion begin a linking of string theory, which is more of a particle theory, with LQG, which in turn is more of a spinor gravity than a particle theory.

For supersymmetry, it would be nice to define superfields

$$\Phi^i~=~\phi^i~+~\theta{\bar\psi}^i~+~{\bar\theta}\psi^i~+~{\bar\theta}\theta F^i$$

for $\phi^i$ and $\psi^i$ related by a graded algebra, gauge fields defined by anti-commutators of supergenerators and the whole thing. These two superfield components might then exist in their own E_8 and transformation between the two are given by super generators. If we get conformal gravity in the picture this might be possible. For in getting a conformal gravity and if it can be shown that $AdS~\simeq~S^5$, for the AdS an $SO(3,2)~\subset~SU(4)$ this theory then exist in an N = 4 SUSY theory.

Cheers,

Lawrence B. Crowell

 

[Lawrence B. Crowell]

Kea,

The propsect for Topos theory is interest, though it is a bit far afield here with roots of E_8. Yet this may play a role with my Leech lattice approach that contain three E_8's. One E_8 defines particles, the other their superpartners and the last one has a Cartan matrix for the whole set of possible $M^4$ manifolds. This last bit leads into quantization via the modular functions for the $\Lambda_{24}$. We might think of the set of four manifolds so defined as the configuration variables for Hawking-Hartle type of wave functionals.

Tony Smith!? Yeah I have seen his web site, though not recently. He seems to have a sort of mystic-physics based on math. As I recall his website was an interesting resource on some topics.

Penrose employs Cech cohomology in twistors, which is a sort of presheaf theory. I was unaware of twistor string theory theory.

We do need to be careful not to fall into the trap of chasing after mathematics to define the physics. That is always a sort of sanity check we have to keep making. Isham I believe has embraced Topos theory as a possible way of doing quantum gravity, because Topos theory weakens the Boolean excluded middle rule. It is best to have the mathematics "fall in your lap," instead of chasing after mathematics.

Cheers,

Lawrence B. Crowell

 

[Lawrence B. Crowell]

Hi rhtsai,

I have to make this rather brief. To be honest I am a bit new to the theory of codes. The main codes I am concerned about are Golay codes which correspond to sphere packing in lattices. The minimal sphere packing define roots which from a coding perspective are "letters." The spheres are Planck units of volume, and so the volume defined by the polytope of spheres is the minimal volume for the theory.

The noise comes from fluctuations or quantum noise on a smaller scale. Physically the idea is to define the S-matrix in a noisy channel. This channel could be a quantum black hole. Again physically the idea is that the quanta emitted by a black hole contain all the quantum information of the quanta that went in. They have just been "encrypted." These codes have a Hamming distance, and this is what defines the errors which are detectable or correctable.

I will write more on this next week if there is an interest. I think that this type of system defines a quantization of E_8 physics. I will also write more in the future about my sense of what is meant by quantization as well. This is more of a unification of two relationship systems for particles. One is a geometric model system we call classical mechanics, which in its most complete form is general relativity plus classical gauge fields. The other is a non-geometric relationship theory called quantum mechanics. Nonlocality and Bell inequalities for entangled states indicate that while quantum states have a representation in space and spacetime, quantum mechanics is fundamentally "blind" to space. I think the two relationships are specific cases of a more general relationship system.

Cheers,

Lawrence B. Crowell

 

[John G]

Why e8? I still haven't seen what "exceptional" property forces the choice of e8 over others? For example e6,d5,... also has very nice structures that can be manipulated to give similar patterns...This choice is fairly elegant and ends up collecting the
three generations into three sets of 64. One complication is that these three sets are associated with the 8V,8S+,and 8S- reps of d4. These don't sit inside e8 in a fully symmetric way as far as I can tell. Triality really goes with d4... Besides the quantum numbers, e8 should also tell how all these particles/fields interact. This I think is a bigger claim than being able to get the correct sets of quantum numbers...

The choice of stopping at F4 or E6 or E8 kind of depends on how many roots you need to fit your model into. Garrett needs E8 to fit 3 generations (at E6 he only has 3 generations of leptons I think). Tony can fit one full generation into E6 but needs to go up to E8 for quantization related ideas (Dirac Gammas, M/F-theory). Full symmetry seems to be an E8/D8 thing which gives you the general boson/adjoint vs. fermion/spinor relationship. D8 is Spin(16) thus the bivectors of Cl(8) x Cl(8). That shows that the 120-dim D8 comes from the product of two Cl(8) vectors (8x8) and a 28-dim bivector from one Cl(8) and a 28-dim bivector from another Cl(8)... 64+28+28=120. So that breakdown is up at D8 but there is a symmetry break to get down to D4+D4 instead of D8. It actually does seem like an 8-dim down to 4-dim spacetime symmetry break.

 

[garrett]

rntsai,
"Why E8?"... I actually didn't just pick E8 and break it up into the standard model. I started with the algebraic structure of the standard model and gravity and managed to shape that into two d4's acting on an 8x8 block of the first generation fermion components:
http://arxiv.org/abs/gr-qc/0511120
If you look at that paper, you'll see a big matrix -- a representation of the two d4's and a block of 64. I didn't know then that this exact structure is part of E8. You can imagine how happy I was to find out that adding two more blocks of 64, possibly for the second and third generations of fermions, gives the algebraic structure of E8. I didn't write this last paper as a bottom up construction from the standard model, but it may be good to keep in mind that that's how it was originally found.

Lawrence,
Regarding conformal gravity: in this E8 theory the frame and Higgs are literally multiplied in the frame-Higgs part of the connection. Because of this, the scale of the Higgs and the scale of the frame are a shared degree of freedom -- the conformal degree of freedom is described redundantly by the frame and Higgs scale.

John G,
Yes, I haven't totally given up on Kaluza-Klein type theories -- they're certainly worth thinking about.

 

[rntsai]

The choice of stopping at F4 or E6 or E8 kind of depends on how many roots you need to fit your model into. Garrett needs E8 to fit 3 generations (at E6 he only has 3 generations of leptons I think). Tony can fit one full generation into E6 but needs to go up to E8 for quantization related ideas ...

This seems like a natural way of including more and more particles with larger algebras;
one complication that I ran into is that these 8 dimensional subpaces do not behave
the same way when they're part of different algebras. Here are three examples :

f4 under d4 : 28 + 3x8
e6 under d4 : 28 + 6x8 + 2x1
e8 under d4 : 28 + 24x8 + 28x1

so f4 has 3 8's, these do not commute with each others or within themselves.
e6 has 6 8's, each one commutes with 3 other 8's (including itself)
e8 has 24 8's, each one commutes with 9 other 8's (including itself)

if the commutation of these subspaces describes how the particles interact
with each other, I don't see how the above different behavior can be accommodated.

 

[Tony Smith]

rntsai mentioned three examples :
f4 under d4 : 28 + 3x8
e6 under d4 : 28 + 6x8 + 2x1
e8 under d4 : 28 + 24x8 + 28x1

I would look at them as follows:

f4 under d4 : 28 + 3x8 = 3x(8x1)

e6 under d4 : 28 + 8x2 + 2x(8x2) + 2x1 = 28 + 2x1 + 3x(8x2)

e8 under d4+d4 : 28 + 28 + 64 + 128 = 28 + 28 + 3x(8x8)

In each case the 3 times 8xn part is made up of 3 copies of 8xn related by triality.

For n=1, the f4 case, you have a triality relating real 8-dim vector (Kaluza-Klein spacetime) and 8 fermion particles and 8 fermion antiparticles.

For n=2, the e6 case, you have a triality relating a comples 8-dim vector (Kaluza-Klein spacetime) and 8 complex fermion particles and 8 complex fermion antiparticles,
with the real versions appearing as Shilov boundaries of the corresponding bounded complex domains, which allows you do to Armand Wyler-type calculations force strengths and particle masses.

For n=3, the e8 case, you have a triality relating real 8-dim vector Kaluza-Klein spacetime times 8 Dirac gammas,
and 8 fermion particles times 8 Dirac gammas,
and 8 fermion antiparticles times 8 Dirac gammas,

Also, for the e8 case, the 28 + 28 gives you two d4

one d4 includes a conformal d3 = a3 = SU(2,2) = Spin(2,4) Conformal MacDowell-Mansouri gravity
that acts on the 4-dim physical spacetime part of the 8-dim Kaluza-Klein

the other d4 includes a SU(4) with a color SU(3) in its U(3) subgroup,
plus a SU(4) / U(3) = CP3 twistor space of 15-9 = 6 dimensions
which twistor space gives electroweak U(2) because
CP3 contains CP2 = SU(3) / U(2) which is the 4-dim internal symmetry part of the 8-dim Kaluza-Klein on which the U(2) acts as a local symmetry gauge group.

Tony Smith

 

[rntsai]

I would look at them as follows:

f4 under d4 : 28 + 3x8 = 3x(8x1)

e6 under d4 : 28 + 8x2 + 2x(8x2) + 2x1 = 28 + 2x1 + 3x(8x2)

e8 under d4+d4 : 28 + 28 + 64 + 128 = 28 + 28 + 3x(8x8)

In each case the 3 times 8xn part is made up of 3 copies of 8xn related by triality.

For n=1, the f4 case, you have a triality relating real 8-dim vector (Kaluza-Klein spacetime) and 8 fermion particles and 8 fermion antiparticles.

For n=2, the e6 case, you have a triality relating a comples 8-dim vector (Kaluza-Klein spacetime) and 8 complex fermion particles and 8 complex fermion antiparticles,
with the real versions appearing as Shilov boundaries of the corresponding bounded complex domains, which allows you do to Armand Wyler-type calculations force strengths and particle masses.

For n=3, the e8 case, you have a triality relating real 8-dim vector Kaluza-Klein spacetime times 8 Dirac gammas,
and 8 fermion particles times 8 Dirac gammas,
and 8 fermion antiparticles times 8 Dirac gammas,

Yes of course there are many ways to collect these spaces but there's a key
point that I haven't seen addressed directly : what's the expected relation
between these spaces? Garette's treatment is different than most other
settings in that the spaces live in a Lie algebra so you can actually also
"multiply" them (Lie algebra product); and this should
correspond to the way the particles interact physically. If I misundertood
this key point, please correct me.

For example, take f4 (n=1), the above would associate one of the 8's with
Kaluza-Klein spacetime. I don't know how to impose "commutativity" on this,
but according to my calculations this space is not commutative.
Now for e6 (n=2) two 8's are collected into a complex 8-dim vector Kaluza-Klein spacetime;this time the space should be commutative.
The physics should rule out one of the two.
This might have been a bad example, substitute fermions and complex fermions, the situation is the same.

 

[Lawrence B. Crowell]

Lawrence,
Regarding conformal gravity: in this E8 theory the frame and Higgs are literally multiplied in the frame-Higgs part of the connection. Because of this, the scale of the Higgs and the scale of the frame are a shared degree of freedom -- the conformal degree of freedom is described redundantly by the frame and Higgs scale.

And Tony Smith wrote:

the other d4 includes a SU(4) with a color SU(3) in its U(3) subgroup,
plus a SU(4) / U(3) = CP3 twistor space of 15-9 = 6 dimensions
which twistor space gives electroweak U(2) because
CP3 contains CP2 = SU(3) / U(2) which is the 4-dim internal symmetry part of the 8-dim Kaluza-Klein on which the U(2) acts as a local symmetry gauge group.

Tony Smith

So my original idea is maybe more appropo --- just describe the dilaton by an appropriate framing, where the scale of the two are mutually dependent. Tony's comment with the universal bundle theorem, which BTW I think is more

$$CP^3~=~\frac{U(4)}{U(3)\times U(1)},$$

though I won't quibble, seems to describe null congruences or massless twistors and pull out "gravity" from the $U(4)$ or $SU(4)$ conformal group. This does seem to put a color group into a D_4 which did not previously have such.

The dilaton field also probably needs to be connected to the YM field (ala Kaluza-Klein) by some DE of the form

$$\nabla_a\nabla^a\phi~=~m^2\phi~+~g F^{ab}F_{ab}\phi|\phi|^2$$

so their might be an occurrence of color, weak & hypercharge in each D_4.

I'll have to think a bit more on this.

Cheers,

Lawrence B. Crowell

 

[Tony Smith]

rntsai said "... the spaces live in a Lie algebra ... and this should correspond to the way the particles interact physically ...".

To see how that works for f4 and e6 you need to see that the 8-dim Kaluza-Klein spacetime and the 8 fermion particles and 8 fermion antiparticles live in a part of f4 that corresponds to 24 dim of the 27-dim Jordan algebra J(3,O),
or for e6 in a complex version of that Jordan algebra.
I won't go into those details here because there is another way to see how it works for the e8 case relevant for Garrett-type e8 physics.

Consider the 120-dim so(16) subalgebra of 248-dim e8.

Since so(16) corresponds to the 120 bivectors (grade 2) of the Cl(16) Clifford algebra,
and since by periodicity factors by tensor product into Cl(16) = Cl(8) (x) Cl(8)
you can construct the so(16) bivectors in terms of the vectors and bivectors of two Cl(8) algebras (one for the d4 used to make gravity, denoted by _grav subscript
and
the other for the other d4 used to make the standard model, denoted by _sm subscript)

Since Cl(8) graded structure is 1 + 8 + 28 + 56 + 70 + 28 + 8 + 1
we make the bivectors of Cl(16) from tensor product of two Cl(8) as
( 1_ grav + 8_grav + 28d4_grav + ...) (x) ( 1_ sm + 8_sm + 28d4_sm+ ...)
which gives you for grade 2 bivector part:

0-grade scalar times 2-grade bivectors (0+2 = 2)
1_grav (x) 28d4_sm = 28d4_sm
28d4_grav (x) 1_sm = s8d4_grav

and 1-grade vectors times 1-grade vectors (1+1 = 2)
8_grav (x) 8_sm = 64

So,
the 120 dim so(16) = 28d4_sm + 28-dim + 64

Since the 8_grav is acted upon as a vector space by the 28d4_grav,
and since d4_grav acts as gravity on the 8-dim Kaluza-Klein spacetime,
the 8_grav represents the 8-dim Kaluza-Klein spacetime so that its e8 algebra structure is consistent with its physical interactions.

Since the 8_sm is acted upon as a vector space by the 28d4_sm,
and since d4_sm acts as the standard model gauge groups on the Dirac gammas of the Dirac operator,
the 8_sm represents the 8 Dirac gammas of the Kaluza-Klein spacetime so that its e8 algebra structure is consistent with its physical interactions.

Therefore, the 64 inside so(16) = 28 + 28 + 64 is represented as
8_KKspacetime (x) 8_DiracGammas = 8_grav (x) 8_sm

Tony Smith

PS - With respect to fermions related to the 64+64 =128-dim half-spinors of so(16) inside 248-dim e8 = 120-dim adjoint so(16) + 128-dim half-spinors of so(16),
you can look at the symmetric space
E8 / Spin(16) = (OxO)P2
which is Rosenfeld's projective plane of octo-octonions OxO.
One OxO dimension of the plane corresponds to fermion particles
and
the other OxO dimension of the plane corresponds to fermion antiparticles.

In each case, one O of the OxO represents 8 first-generation fermion particles (or antiparticles)
and the other O of the OxO represents the 8 Dirac Gammas of 8-dim KKspacetime
somewhat like a generalization of the fermion / Dirac Gamma ideas of David Hestenes in his spacetime algebra.

PPS - Of course, all the structural stuff in this message can be seen as being derived from the product rules of the 240 root vectors of e8,
so
it shows that
"... the spaces [representing KKspacetime, fermions, and Dirac gammas, as well as the gravity d4 and standard model d4, do all] live in ...[the e8]... Lie algebra ...
and this ...[does indeed]... correspond to the way the particles interact physically ...".

 

[Tony Smith]

In my previous message I only dealt with first-generation fermions, which can be seen as being represented by octonions,
because (in my view) the second and third generation fermions can be regarded as composites of those octonions
and
that is (in my view) related to the post by Lawrence Crowell who said
"... we ... have one E_8 ...
another for the supersymmetric pairs of these fields
and a third ... three copies of E_8 into the Leech lattice ...".

My view of that would be
"... we ... have one E_8 ... for the first generation of fermions represented by Octonions in terms of the E8 lattice (the lattice being made from Octonionic space by orbifolding as in my E6 string model)

another for ... pairs of ... Octonions OxO or in lattice terms the Barnes-Wall 16-dim lattice to represent second-generation fermions

and
a third ... triples OxOxO or three copies of E_8 into the Leech lattice ... to represent third generation fermions ...".

The combinatorics of E8 and pairs and triples of E8 work as in my physics model to give realistic particle masses for the second and third generation fermions - details are on my website at
www.valdostamuseum.org/hamsmith/[/URL]

It is also interesting that Lawrence Crowell said (I have changed his word 240-cell to 240-vertex, because the 8-dim Witting polytope with 240 vertices has a lot more than 240 7-dim cells - It has 17,280 7-simplexes 4_20 and 2,160 7-cross-polytopes 4_11):

"... The 120 icosian (half the 240-vertex) supports the M_{12} Mathieu group, which under a double cover defines the 240-vertex and the Leech lattice error correction code M_{24} ...".

In their paper "Finite Simple Groups Which Projectively Embed in an Exceptional Lie Group are Classified" (Bulletin (New Series) AMS v. 36 no.1 January 1999 pages 75-93) Griess and Ryba showed that M_12 is projectively embedded in E8(C),
and that "... Any sporadic group ... which contains ... The Mathieu group M_23 ... is ... eliminated from consideration as a subgroup of E8(C) ...".
Since M_24 is related to the automorphism group of the Leech lattice, going to 3 copies of E8 and then to the Leech lattice seems to be interesting from many points of view.

Tony Smith

 

[rntsai]

Hi Tony,
You gave a number of interesting examples, I'll comment on this for now :

Consider the 120-dim so(16) subalgebra of 248-dim e8.

Since so(16) corresponds to the 120 bivectors (grade 2) of the Cl(16) Clifford algebra,
and since by periodicity factors by tensor product into Cl(16) = Cl(8) (x) Cl(8)
you can construct the so(16) bivectors in terms of the vectors and bivectors of two Cl(8) algebras (one for the d4 used to make gravity, denoted by _grav subscript
and
the other for the other d4 used to make the standard model, denoted by _sm subscript)

Since Cl(8) graded structure is 1 + 8 + 28 + 56 + 70 + 28 + 8 + 1
we make the bivectors of Cl(16) from tensor product of two Cl(8) as
( 1_ grav + 8_grav + 28d4_grav + ...) (x) ( 1_ sm + 8_sm + 28d4_sm+ ...)
which gives you for grade 2 bivector part:

0-grade scalar times 2-grade bivectors (0+2 = 2)
1_grav (x) 28d4_sm = 28d4_sm
28d4_grav (x) 1_sm = s8d4_grav

and 1-grade vectors times 1-grade vectors (1+1 = 2)
8_grav (x) 8_sm = 64

So,
the 120 dim so(16) = 28d4_sm + 28-dim + 64

This basically corresponds to d8/d4 decomposition. I haven't looked
at this combination before, but I tried it today :

d8/d4 = 1x28 + 8x8 + 28x1

the 1x28 is d4 itself (adjoint rep), the 28x1 works out to be a second d4 inside
d8 (the first d4 acts trivially on it). The 8x8 are 8 8-dim spaces. Looking at their
weights, they're all 8V reps (8S+ and 8S- don't occur here). Each space commutes
within itself but with none of the others. In general these sit inside d8 in a much
more symmetric way than the 8-dim spaces of e8...their quantum numbers "look
the same", probably becase they're all 8V's.

Since the 8_grav is acted upon as a vector space by the 28d4_grav,
and since d4_grav acts as gravity on the 8-dim Kaluza-Klein spacetime,
the 8_grav represents the 8-dim Kaluza-Klein spacetime so that its e8 algebra structure is consistent with its physical interactions.

I'll feel better after I verify this directly myself. To be honset, I'm not too familar
with 8-dim KK; any good references? but if the other d4 corresponds to sm, then
it should be a little bit easier to check that. This should be just looking on the
product d4_sm * 8's; one thing I haven't been able to come to grips with is whether
the product of the 8's among themselves : 8's * 8's matters.

 

[Lawrence B. Crowell]

In my previous message I only dealt with first-generation fermions, which can be seen as being represented by octonions,
because (in my view) the second and third generation fermions can be regarded as composites of those octonions
and
that is (in my view) related to the post by Lawrence Crowell who said
"... we ... have one E_8 ...
another for the supersymmetric pairs of these fields
and a third ... three copies of E_8 into the Leech lattice ...".

My view of that would be
"... we ... have one E_8 ... for the first generation of fermions represented by Octonions in terms of the E8 lattice (the lattice being made from Octonionic space by orbifolding as in my E6 string model)
Tony Smith

The three copies of E_8 might be thought of as two related by SUSY $E_8~\leftarrow\rightarrow~E_8$ on a multiplet by muliplet basis. So for example with the Lisi construction, each element in the Grossett polytope on page 17 would be an element of the SUSY multiplet defined by

$$\Psi~=~\psi~+~\theta{\bar\phi}~+~{\bar\theta}\phi~+~F$$

for $\theta,~{\bar\theta}$ the SUSY Grassmann terms and $\phi$ the SUSY pair which is defined on the second E_8. So for particle, anti-particle on one E_8 we have the corresponding SUSY pairs of particles and anti-particles on a second E_8.

The third E_8 is a bit strange to argue for. Whether this involves actual particle states or some sort of quasi particle states or ... , I am unsure of. The matrix exists for quantization. A quantum wave functional over all possible metric configurations will not describe a classical manifold for each of these configuration variables. Most of these spaces, or their spacetimes are "strange" or they are homeomorphic but not diffeomorphic. Remember, the Wheeler-DeWitt equation provides no meaning to a "time," but this is something which is assigned by an analyst. Most of these metrics are for spaces where no workable diffeomorphic system can be assigned. This result on the set of all possible moduli for four manifolds is given by a the Cartan center of E_8. I am not sure what connection exists with $F_4~=~C_{E_8}(G_2)$, but if there it might suggest some third generation of particles. I will remain agnostic on this front for the time being.

another for ... pairs of ... Octonions OxO or in lattice terms the Barnes-Wall 16-dim lattice to represent second-generation fermions

and
a third ... triples OxOxO or three copies of E_8 into the Leech lattice ... to represent third generation fermions ...".

Tony Smith

The Barnes-Wall Lattice $\Lambda_{16}$ consists of 4320 vectors as given by the minimal kissing condition. 480 of these vectors are the minimal vectors of the form $1/\sqrt{2}(\pm2^2,~0^{14})$, which are two copies of the 240-cell or $E_8$ lattice, with an additional 3840 vectors $1/\sqrt{2}(\pm1^8,~0^8)$. The two $E_8$ lattices are long and short roots. The outer or longer roots are

$$E_8^L~=~\{\pm e_i\}\times\{\pm e_i, \pm e_k, \pm e_k, \pm e_e_l\}$$

which is for the indices running from 1 to 8 (or 0 to 7) is 16x14 plus 16 = 240. The other shorter roots are

$$E_8^S~=~\{(\pm 1/2f_i,\pm 1/2 f_i)\}\times\{1/4\sum_i \pm f_i\}$$

and the 2840 elements are determined by products $e_if_j~=~g_a$, of which there are 8 on the 480 elements of the E_8's.

The question is then whether this can support supersymmetry. So then we have for the two $E_8s$ superfields with pairs corresponding to each element. So the two lattices share elements in common with supermultiplets. The products between elements in different SUSY mulitplets define the rest of this lattice. To be honest, I am not sure what these would correspond to.

An error correction code for the B-W system appears to naturally admit supersymmetry in some ways. For the Galois field F_q acting in $C^{q^n}$ we may define the error operator $E_{ij}~=~A_iB_j$, for the A and B vectors in $F_q$, and these act on a quantum state

[tex]
E_{ij}|x_1,x_2,\dots,x_n\rangle~=~(-1)^{f(x)}|x'_1,x'_2,\dots,x'_n\rangle
[/itex]

with $$f(x)~=~xAx^{\dagger}$. For A a diagonal matrix and x fermionic valued this defines a Witten index. For the off-diagonal terms in A this may then refer to these strange "cross-products" between elements in the two E_8s. Without going into details, which are somewhat considerable, this defines an error correction code $[n,~k,~d]$.

The combinatorics of E8 and pairs and triples of E8 work as in my physics model to give realistic particle masses for the second and third generation fermions - details are on my website at
www.valdostamuseum.org/hamsmith/[/URL]

It is also interesting that Lawrence Crowell said (I have changed his word 240-cell to 240-vertex, because the 8-dim Witting polytope with 240 vertices has a lot more than 240 7-dim cells - It has 17,280 7-simplexes 4_20 and 2,160 7-cross-polytopes 4_11):

"... The 120 icosian (half the 240-vertex) supports the M_{12} Mathieu group, which under a double cover defines the 240-vertex and the Leech lattice error correction code M_{24} ...".

In their paper "Finite Simple Groups Which Projectively Embed in an Exceptional Lie Group are Classified" (Bulletin (New Series) AMS v. 36 no.1 January 1999 pages 75-93) Griess and Ryba showed that M_12 is projectively embedded in E8(C),
and that "... Any sporadic group ... which contains ... The Mathieu group M_23 ... is ... eliminated from consideration as a subgroup of E8(C) ...".
Since M_24 is related to the automorphism group of the Leech lattice, going to 3 copies of E8 and then to the Leech lattice seems to be interesting from many points of view.

Tony Smith[/QUOTE]
Yes this is an interesting prospect. We do need to make sure we have the physical idea right. My sense is that the third E_8 is required because as a theory of quantum gravity, particularly in the language of ADM and LQG, we have a wave functional over four manifolds (or a path integral which gives constraints for a wave functional over foliations of spaces). There is an interesting connection between the centralizer of E_8 and the moduli space for four manifolds. This leads to a rich theory on the classification of all four manifolds (which quantum gravity might be as a quantum wave functional over M^4s). So this suggests at least a third E_8, which might define another set of particle states and which leads to a geometric quantization.

Lawrence B. Crowell

 

[rntsai]

An error correction code for the B-W system appears to naturally admit supersymmetry in some ways. For the Galois field F_q acting in $C^{q^n}$ we may define the error operator $E_{ij}~=~A_iB_j$, for the A and B vectors in $F_q$, and these act on a quantum state

[tex]
E_{ij}|x_1,x_2,\dots,x_n\rangle~=~(-1)^{f(x)}|x'_1,x'_2,\dots,x'_n\rangle
[/itex]

with $$f(x)~=~xAx^{\dagger}$. For A a diagonal matrix and x fermionic valued this defines a Witten index. For the off-diagonal terms in A this may then refer to these strange "cross-products" between elements in the two E_8s. Without going into details, which are somewhat considerable, this defines an error correction code $[n,~k,~d]$.

The notation here is a little ambiguous. How are the [tex],x,x',x^{\dagger}[/itex] related?
Is the A in [tex]f(x)=x A x^{\dagger}[/itex] the same A in [tex]E_{ij} = A_i B_j[/itex]
In general, this looks very close to the definition of errors in a quantum
code setting. There are basically two types of primitive errors (phase+translation;
some places use [tex] \sigma_z,\sigma_x [/itex] since they're associated with pauli matrices) :

[tex]
|x\rangle \to (-1)^{q*x}|x+p\rangle
[/itex]

there's an obvious link to harmonic analysis, Fourier transforms,...
The -1 goes with finite field GF(2); it is replaced by an r-th root of
unity if another field is used. I think you're building a quantum code
here with one copy of e8 for the q-part and a second for the p-part.
I can say more once I understand the terminology better.

 

[Lawrence B. Crowell]

Hi Tony,
You gave a number of interesting examples, I'll comment on this for now :


This basically corresponds to d8/d4 decomposition. I haven't looked
at this combination before, but I tried it today :

d8/d4 = 1x28 + 8x8 + 28x1

the 1x28 is d4 itself (adjoint rep), the 28x1 works out to be a second d4 inside
d8 (the first d4 acts trivially on it). The 8x8 are 8 8-dim spaces. Looking at their
weights, they're all 8V reps (8S+ and 8S- don't occur here). Each space commutes
within itself but with none of the others. In general these sit inside d8 in a much
more symmetric way than the 8-dim spaces of e8...their quantum numbers "look
the same", probably becase they're all 8V's.


I'll feel better after I verify this directly myself. To be honset, I'm not too familar
with 8-dim KK; any good references? but if the other d4 corresponds to sm, then
it should be a little bit easier to check that. This should be just looking on the
product d4_sm * 8's; one thing I haven't been able to come to grips with is whether
the product of the 8's among themselves : 8's * 8's matters.

A few potentially related comments. My Dynkin diagrams didn't work out, but ...

F_4 is the a D_4 plus three 8's, with dim 48, and under D_4 this is dim = 52. E_6 has rank 6 and is dimension 78 under D_4, so we have more space to play in that with the 52 dimensions of F_4. The covering group is Z_2 and Z_3, where Z_3 leads to something called the Haguchi-Hanson metric. The Z_2 is the automorphism group and Z_3 is the fundamental group. There is then a fundamental representation of 27-dimensions, where three copies of these modulo three dimension for the Z_3 gives the 78 dimensions of E_6. These 27 dimensional representations give Jordan matrices J^3(V).

E_6 also preserves the projective lines in OP^2, and is then a group of collineations. The 27 dimensional Jordan algebra J^3(V) is defined by the 3x3 octonionic matrix

(e_1 e_4 e_5)
(e_7 e_2 e_6)
(e_8 e_9 e_3)

which is a nine element matrix over 8 dimensional basis elements, and is thus 9 x 8 dimensional. The matrix is overdetermined and the restriction on hermiticity gives the 3 x 8 + 3 x1 matrix

(Re(e_1) e_4 e_5)
(e^*_4 Re(e_2) e_6)
e^*_5 e^*_6 Re(e_3) ),

in 27 dimensions, which is equivalent to the 27-dim representation of E_6, and is the matrix of the E_6 roots. The roots of E_6 may be found by as 27 combinations of (3; 3; 3) and are

(1,-1,0;0,0,0;0,0,0), (-1,1,0;0,0,0;0,0,0),
(-1,0,1;0,0,0;0,0,0), (1,0,-1;0,0,0;0,0,0),
(0,1,-1;0,0,0;0,0,0), (0,-1,1;0,0,0;0,0,0),
(0,0,0;1,-1,0;0,0,0), (0,0,0;-1,1,0;0,0,0),
(0,0,0;-1,0,1;0,0,0), (0,0,0;1,0,-1;0,0,0),
(0,0,0;0,1,-1;0,0,0), (0,0,0;0,-1,1;0,0,0),
(0,0,0;0,0,0;1,-1,0), (0,0,0;0,0,0;-1,1,0),
(0,0,0;0,0,0;-1,0,1), (0,0,0;0,0,0;1,0,-1),
(0,0,0;0,0,0;0,1,-1), (0,0,0;0,0,0;0,-1,1),

The Dynkin diagram for E_6 is

O
|
O---O---O---O---O,

and by decompositions on this diagram it can be seen that the vertical O ~ SU(2) may be removed to give

O
| -----------> O---O---O---O---O plus O = SU(6)xSU(2)
O---O---O---O---O

or that

O
O \
| -----------> O---O---O---O = SO(10) xU(1)
O---O---O---O---O /
O

or that the six O's can be broken up into three O---O ~ SU(3) and the group contain SU(3)xSU(3)xSU(3). This structure is involved with the Haguchi-Hanson metric for the 7 sphere embedding in the S^8 in the Hopf fibration.

Clearly the SO(10) is important to physics as this covers su(5) and this in turn breaks into the standard model twisted bundle group SU(3)xSU(2)xU(1). For the SO(10) the adjoint 78 dim representation decomposes into a 45-adjoint, a 16-spinor and its dual bar-16-spinor plus the SO(10) (as a singlet from the 78). The 78 then decomposes as

78 ----> 45_0 + 16_{-3} + bar-16_3 + 1_0,

where the subscript is the u(1) charge on the respective sector. The 45_0 is the anti-Hermitian portion of the octonions (3 x 8 + 3 x 7) with the Jordan matrix

(Im(e_1) e_4 e_5)
-e^*_4 Im(e_2) e_6)
-e^*_5 -e^*_6 Im(e_3)),

This portion may be further reduced to a 14 -dimensional group that contain the smallest exceptional group G_2.

E_6 also admits the group spin(8), which can be decomposed into the dS and AdS group so(4,1) and so(3,2) under their respective non-euclideanization. However, we can't get the group for elementary particles, say the gut so(10) and the (Anti-)DeSitter spacetime symmetry groups in at the same time. To extend this to D_8 ~ spin(16) we need to "pack" two D_4's. E_8 containes spin(16), which is a representation for the closed string. For the spin group decomposition of these heterotic groups consistently contain the DeSitter group. The spin(16) group has 128 generators. The additional 112 roots (from the total 240 in E_8) define a D_8 group (in terms of root system not lattice), which is an acceptable gauge theoretic model SO(8), which also contains the SO(3,2) under suitable change of signature. Similarly E_6 and E_7 sit inside E_8. E_6xsu(3)/(Z/3Z) and E_7xsu(2)/(Z/2Z) are maximal subgroups of E_8, where both E_7 and E_6 under signature changes contain the Desitter group.

The E_8 transforms differently under these two subroups differently. Under E_7xSU(2) E_8 has the representation

(3,1) + (1,133) + (2,56),

as one copy of the 133 E_7 roots (eg the 133), 2 copies of 56 and one copy of 3. We can similarly define E_8 under E_6 and d_4 + d_4.

 

[Lawrence B. Crowell]

The notation here is a little ambiguous. How are the [tex],x,x',x^{\dagger}[/itex] related?
Is the A in [tex]f(x)=x A x^{\dagger}[/itex] the same A in [tex]E_{ij} = A_i B_j[/itex]
In general, this looks very close to the definition of errors in a quantum
code setting. There are basically two types of primitive errors (phase+translation;
some places use [tex] \sigma_z,\sigma_x [/itex] since they're associated with pauli matrices) :

[tex]
|x\rangle \to (-1)^{q*x}|x+p\rangle
[/itex]

there's an obvious link to harmonic analysis, Fourier transforms,...
The -1 goes with finite field GF(2); it is replaced by an r-th root of
unity if another field is used. I think you're building a quantum code
here with one copy of e8 for the q-part and a second for the p-part.
I can say more once I understand the terminology better.

Sorry about the ambiguity with the "A" symbol. No these are different. The error correction operator is a binary product of spinors in a clifford basis. So A_i and B_j is the exterior product of sigma_i 's which form a Clifford basis. That ambiguity comes from typing this stuff up while watching football. And your observation is right, this theory is essentially one of Fourier transforms and harmonic analysis.

Lawrence B. Crowell

 

[John G]

I'll feel better after I verify this directly myself. To be honset, I'm not too familar
with 8-dim KK; any good references? but if the other d4 corresponds to sm, then
it should be a little bit easier to check that. This should be just looking on the
product d4_sm * 8's; one thing I haven't been able to come to grips with is whether
the product of the 8's among themselves : 8's * 8's matters.

Some things seem easier to see with the split real form of E8 (E8/D8) and others with the compact real form (E8/E7xSU(2)). Tony's 8-dim KK shows up easiest for me as D5/D4xU(1) where D5 is the somewhat popular SO(10) spacetime. So you can see it as a 10-dim spacetime or as the D5/D4xU(1) = 16-dim = 8 complex-dim spacetime. After the symmetry break you have D3/D2xU(1) = 8-dim = 4 complex-dim spacetime. From the 10-dim perspective Tony breaks it up into CP3 and CP2. Tony's D4_grav becomes a D3 conformal gravity after the symmetry break and his D4_sm also then fits into D3. For Tony, the product of 8s describes the connection between spacetime and the gammas so I would think it also describes the connection between gravity and the standard model (before the symmetry break).

 

[CarlB]

The Koide mass formula people are working overtime. They've just found out that the mathematics they've been using has already been explored by people interested in "Mutually unbiased bases". Two bases for quantum states are "unbiased" if all transition probabilities between states of one basis and the other are equal. Paper on MUBs:
http://arxiv.org/abs/quant-ph/0610216

For the case of a Hilbert space with dimension 3, one can choose 4 bases that are mutually unbiased. Each of these four bases has 3 elements, the equivalent of "spin up" and "spin down" for qutrits. The possible relationship with E8 is that the Koide mass formula was found this way, with the 3 basis elements corresponding to the 3 generations.

The three basis elements of a state can always be written (in pure density matrix form) as (1 + wJ + wwJJ)/3, where J is a matrix that is a cube root of unity, and w is a complex cubed root of unity. The three generations come from the three complex cubed roots of unity.

To get the other three sets of basis states, one chooses a matrix M that also cubes to unity, and satisfies JM = w MJ. Then one finds that four mutually unbiased bases for 3-Hilbert are generated by the four sets of 3 pure density matrices:

{ (1 + wJ + wwJJ)/3 },
{ (1 + wM + wwMM)/3 },
{ (1 + wJM + wwJJMM)/3 },
{ (1 + wJMM + wwJJM)/3 },

where each row correspond to 3 quantum states by putting w as the three complex cubed roots of unity. A canonical choice for J and M are:
$$J = \left(\begin{array}{ccc}0&1&0\\0&0&1\\1&0&0\end{array}\right)$$
$$M = \left(\begin{array}{ccc}1&&\\&w&\\&&w^2\end{array}\right)$$



To fit this into the standard model, perhaps one could take the (1 + wJ + wwJJ)/3 set as the three generations of the charged leptons (neutrinos), and the other three sets as the three colors of the down quark (up quark). This follows the Koide mass formula. Don't know if this will work with E8, but then triality corresponds to a choice of the cubed root of unity w.

 

[Tony Smith]

carlB said "... The three generations come from the three complex cubed roots of unity. ...".

That sounds to me like the Eisenstein integers of the complex plane, which form a triangular pattern based on "complex cubed roots of unity":
w = ( - 1 + i sqrt(3) ) / 2
w^2 = ( 1 - i sqrt(3) - i sqrt(3) - 3 ) / 4 = ( - 1 - i sqrt(3) ) / 2
w^3 = ( 1 + i sqrt(3) - i sqrt(3) + 3 ) / 4 = 1

I don't understand the details of the Koide mass formula,
but
using w, w^2, and w^3 for first, second, and third generations of leptons
seems
to have some similarity to my use of octonions O to describe
first, second, and third generations of fermions as O, OxO, and OxOxO
as I mentioned in post number 225 (on page 15 here).
As I said there, "... The combinatorics ... work as in my physics model to give realistic particle masses for the second and third generation fermions - details are on my website at www.valdostamuseum.org/hamsmith/[/URL] ...".

Tony Smith

 

[Lawrence B. Crowell]

carlB said "... The three generations come from the three complex cubed roots of unity. ...".

That sounds to me like the Eisenstein integers of the complex plane, which form a triangular pattern based on "complex cubed roots of unity":
w = ( - 1 + i sqrt(3) ) / 2
w^2 = ( 1 - i sqrt(3) - i sqrt(3) - 3 ) / 4 = ( - 1 - i sqrt(3) ) / 2
w^3 = ( 1 + i sqrt(3) - i sqrt(3) + 3 ) / 4 = 1

Tony Smith

I agree, and as I have looked at this paper it appears to be of some importance. The Galois field is $GF(4)~=~(0,~1,~z,~z^2)$ with $z~=~{1\over 2}(i\sqrt{3}~-~1)$ with $z^2~=~z^*$. GF(4) is the Dynkin diagram for the Lie Algebra D_4 = spin(8). The properties of the basis elements that produce a commutator are

$$z^2~=~z~+~I,~z^3~=~I,~{\bar z}~=~z^2,$$

and defines the hexcode system $C_6$

As kea wrote"... we want to replace the {0,1} with {0,1,2} etc., but not as a set, because this is just a 0-category, and Set is an instance of a classical (in all senses of the word) 1-topos. " This hexacod system as a system of Eisenstein integers may be extended to a general cyclotomic field for the $C_{24}$ system. The simplest extension is one for the 24 minimal vectors for the polytope $\{3,~4,~3\}$ or the 24-cell. The cyclotomic field $Z(z)$ are

$$z^a,~z^a(1~-~z),~z^a(1~-~\sqrt{2}),~a~=~1,~\dots,~7.$$

This then leads to the [24, 12, 8] extended binary Golay code $C_{24}$ consists of 4096 binary words of length 24. The Hamming weight of a binary word is the number of letters that consist of "ones." The [24, 12, 8] Golay code contain 759 words of weight 8 and an equal number of weight 16. This is in addition to the zero word and the word consisting of 24. The remaining 2576 words are of weight 12. The weight 8 words and weight 12 words are called octads and dodecads respectively.

Lawrence B. Crowell

 

[patfla]

Hi Garrett,

Any response to the Jan 23 "Symmetry Issues??" posting as found here?

http://exceptionallysimpletheoryofeverything.blogspot.com/

Which I guess is actually a link to an article on telegraph.co.uk (haven't clicked through and read yet - but the blog explanation says that, aside from the main claim, the telegraph article is pretty much content free - we have to wait for a soon-to-be-released book or something).

pat

 

[Tony Smith]

pat mentions a Telegraph article by Marcus du Sautoy entitled
"Garrett Lisi: This surfer is no Einstein"
in which du Sautoy says
"... the consensus, after investigation, is that it is impossible to use E8 in the way Lisi was hoping and produce a consistent model that reflects reality. Lisi has been riding a wave - but it is time to knock him off his board ...",
and
the web page of the article refers to du Sautoy's "... new book 'Finding Moonshine: A Mathematician's Journey Through Symmetry' which is published by Harper Collins on Feb 4 ...".
According to Amazon.com the book will be released in the USA on 11 March 2008 under the title "Symmetry: A Journey into the Patterns of Nature".

It seems to me that du Sautoy is using a common book-marketing tactic of commenting on the web about widely-known stuff (such as Garrett's E8 model) with teaser-type criticisms (ie, "knock him off his board" with no substantive content), hoping that people will try to find some substantive content by buying the book.

I would not bet on the book having anything remotely close to the general level of content that is here on this thread, in which Garrett has already recognized that "... it is impossible to use E8 in the way Lisi was hoping ..." and has in fact gone well beyond such "consensus" criticisms by making reasonable proposals about modifying his model to satisfy such "consensus" criticisms.

Tony Smith

PS - The Telegraph web page with du Sautoy's article has a bunch of comments, which do not (to me) add much of anything to the subject, except that some followers of Mohamed El Naschie seem to be using the comments to promote the work of El Naschie.
The story of El Naschie and his work is long, convoluted, and controversial, and I will not try to get into it here.

 

[Tony Smith]

Lawrence B Crowell refers to using, for physics model-building based on E8 etc,
"... the [24, 12, 8] extended binary Golay code ...".

I agree that the classical [24, 12, 8] Golay code is very useful for E8 physics,
particularly for showing how to construct a classical Lagrangian for Gravity plus the Standard Model.

To go beyond the classical Lagrangian to quantum (say, by path-integral sum-over-histories quantization), it seems to me that it is useful to go from the Shannon-type classical-information theory [24, 12, 8] Golay code
to a corresponding quantum-type quantum information theory code.
For example, in
http://arxiv.org/abs/quant-ph/9608006
Calderbank, Rains, Shor, and Sloane
show that whereas many useful classical-error-correcting codes are binary, over the Galois field GF(2) = {0,1},
quantum-error-correcting codes are quaternary, over the
Galois field GF(4) = {0,1,w,w^2}
where w = (1/2)( - 1 + sqrt(3) i )
and w^2 = (1/2)( - 1 - sqrt(3) i ).

As to the [24, 12, 8] classical Golay code, a corrresponding quantum code seems based on Steane's paper at
http://arxiv.org/abs/quant-ph/9802061
to be
a quantum code [[ 24, 0, 8 ]] .

At the risk of belaboring the obvious:
1 - classical codes (related to Clifford algebras) give classical Lagrangians
2 - quantizing to quantum codes gives quantum Lagrangians,
which have algebraic structure of a generalized hyperfinite von Neumann algebra factor that is roughly (as John Baez said in his week 175 about a related special case) "... a kind of infinite-dimensional Clifford algebra ...".

In other words, going from classical codes to quantum codes shows a constructive link between classical Lagrangian formulations that are so useful in the Standard Model,
and quantum Algebraic Quantum Field Theory that has many useful aspects.

Tony Smith

 

[patfla]

Hi Tony,

Oh it's du Sautoy. That, for me, says a good deal right there.

In spite of the name, he's English. And looking him up now on Wikipedia, he's a mathematician at Oxford.

Heard an interview with him once and formed a fairly distinct impression. And I believe that at that time also he was trying to sell a book.

pat

 

[Kea]

The Galois field is $GF(4)~=~(0,~1,~z,~z^2)$ ... and defines the hexcode system $C_6$.

Thanks Tony, Lawrence and everybody else for interesting remarks. Hmm. We use complex 3x3 mass matrices, but in this context it seems useful to study real dimension 6 also. By the same token, thinking backwards, the 24d case of the Leech lattice might be associated to a 6d (or 12d) MUB problem, which is a famous unsolved case, because $d=6$ is not prime. And this problem of understanding how to add complexification turns up all over the place ...

 

[Kea]

Which I guess is actually a link to an article on telegraph.co.uk ...

LOL. I've also been to a talk by du Sautoy, which was largely about selling his book on the Riemann hypothesis. The article on Lisi mentions Galois! How ironic. Having said that, I think du Sautoy's writing is actually quite good, and his assessment of the Lisi paper is basically correct. Can we really expect him to hang out with the crackpots?

 

[rntsai]

GF(4) is the Dynkin diagram for the Lie Algebra D_4 = spin(8).

how?

 

[Tony Smith]

rntsai asked "how?" is "GF(4) is the Dynkin diagram for the Lie Algebra D_4 = spin(8)".

I might not do as good a job as Lawrence B Crowell would do,
here is my attempt at showing "how?":

The Galois field GF(4) = {0,1,w,w^2}
where
w = (1/2)( - 1 + sqrt(3) i )
and
w^2 = (1/2)( - 1 - sqrt(3) i )

Look at the 0 as the zero or origin as the central dot of the D_4 Dynkin diagram visualized as being in the complex plane centered on the origin

*
\
*--* ( I used two -- to make the length more nearly equal to that of \ and / )
/
*

and
look at the 1 as the outer Dynkin dot on the right-hand side
and
look at the w as the upper outer Dynkin dot on the left-had side
and
look at the w^2 as the lower outer Dynkin dot on the left-hand side.

Tony Smith

 

[rntsai]

visualized as being in the complex plane centered on the origin

*
\
*--* ( I used two -- to make the length more nearly equal to that of \ and / )
/
*

Thanks. I was looking for the edges conveying some relation between the
elements,...didn't realize that this is just plotting the points. Doesn't seem
to carry much significance.

 

[Tony Smith]

Kea said "... the 24d case of the Leech lattice might be associated to a 6d ... MUB problem, which is a famous unsolved case, because $d=6$ is not prime ...".

I think that is an important insight.
Since 6 = 2x3, MUB for 6d is sort of a hybrid of 2d and 3d.

You might look at the 3d part as related to a triality or to
the combination of 3 sets of 8d E8
to form a 24d Leech lattice thing
and
you might look at the 2d part as the 2-complex-dimensional Pauli matrix representation of MUB for 2d (see for example quant-ph/0103162 ).

Since the 2x2 complex Pauli matrices are the building blocks for the conventional complex hyperfinite II1 von Neumann factor ( see John Baez's week 175 where he calls the factor "... a kind of infinite-dimensional Clifford algebra ...", based on complex Clifford algebras which have periodicity 2, with complex Cl(2) being 2x2 Pauli-type matrices,
it seems to me a natural generalization to go to real Clifford algebras with periodicity 8 to construct a generalized real hyperfinite II1 von Neumann factor using real Cl(8),
corresponding to the 8d E8 building blocks of the 24d Leech lattice.

Maybe solving the corresponding 24d MUB problem would show how to build a nice basis for Algebraic Quantum Field Theory of such a generalized hyperfinite II1 von Neumann factor.

Maybe understanding the 6d MUB would show how to solve the 24d MUB.

Tony Smith

 

[Lawrence B. Crowell]

I will make this a quick replay to all and comment more later. I wondre if these 3x3 matrices are related to the J^3(O). This is the octonion 3x3. Then the elements of the Jordan matrix would be the J and M elements JM = wMJ which are cube roots of unity.

The paper http://arxiv.org/abs/quant-ph/9802061 appears to be doing something similar to what I posted last weekend.

More later,

Lawrence B. Crowell