An Exceptionally Technical Discussion of AESToE

[patfla]

OK I’m trying to understand the actual procedure here.

Garrett may have given me much (in combination with what I had already) of what I need here

We can rotate the coordinate axes of the root system however we wish, describing the same algebra. This just corresponds to a different choice of basis elements for the same Lie algebra -- still E8. I think rntsai has done a good job of explaining this in his previous post. He (or she?) is also correct that the roots alone aren't enough to tell you which Cartan subalgebra particle/field we get when two roots add to give one at the origin. To describe this, we would have to work in a specific representation, or at least write down these structure constants.

First you pick some simpler subalgebra of E8 – say G2 or F4. This will make the problem vastly more tractable. You can ‘see’ things more clearly and the operations will be simpler and/or faster. Next you need to find a basis within that subalgebra where two of the vectors will use the one-and-only Lie algebra operator – the bracket or commutator – and give a result back at the origin. The origin, in 3D, being x=0,y=0,z=0. What does this mean? Linear algebra I and the dot product (so here we’re talking second yr undergraduate mathematics [imo]). The dot product (in 2D – I guess it’s the wedge product in higher dimensions) will act upon two vectors and give a result of 0 when the two vectors are perpendicular. Perpendicular is the important part. So what does it mean ‘in the real world’ that the vectors are perpendicular? Well that depends. We’ll leave it as an exercise of the reader (but at this point, you are quite close to ‘the real world’).

Back up. In a (probably special, unitary [except that unitary implies group and not algebra]) Lie subalgebra when the commutator is applied to two (root) vectors) and you end up at 0 you have a particle/field. Yowza! A first big success (you found the right basis). So how do you find the right basis? Random doesn’t seem like a good idea (there are a lot of them). Hunches and intuition, if you’re so provided (meaning a professional, practitioner or very talented amateur), can go a long ways. But better yet, some more systematic way of a) try a new (likely) basis b) compute root vectors and see if any computation lands at the origin. Doing this suggests programming.

Doubtless, there’s much more to be said about finding the right basis. What that is that could be said: I don’t know.

But, as Garrett said you still have a (probably) very thorny issue before you. You’ve got a particle, but you don’t yet have enough information to figure out which one. Most particles are ‘known’ (outside) the theory, but, according to this theory, there’s a small number (18 I believe) that are not. As regards identifying the hot particle now in your hand, Garrett said:

He (or she?) [rntsai] is also correct that the roots alone aren't enough to tell you which Cartan subalgebra particle/field we get when two roots add to give one at the origin. To describe this, we would have to work in a specific representation, or at least write down these structure constants.

So what procedure is involved here (identifying the particle/field)? With more time, patience (and probably help) I’ll figure it out, but at the moment I don’t know. Although I feel I have, more-or-less, figured out what a representation is. Representation is a keyword and there’s a whole field (of study) corresponding to representation theory. In CS object oriented terms, I’d like to think: you write an OO class in your text editor. It’s a specification - so far it ‘does’ nothing. Then you run a program (containing that class) and the class, as we say, ‘instantiates’. It’s an object in memory and now it’s actually doing something as a part of the program. Which for many people is still somewhat abstract, but is something I’ve been doing for yrs, so for me it’s intuitive. Anyway the instantiation (or object [running in memory]) is the representation. The analogy breaks down quickly though. For a class there’s basically one instantiation (one might use polymorphism to claim that’s there more than one – it’s certainly the case that you can get different ‘behaviors’ out of the object depending on polymorphism). But there are an infinity of representations for E8. Fortunately though, every member of this infinity can be generated from the basic, unitary representations. There’s an enormous number of these, but that number is finite.

The explanation of the last procedure (identifying the particle/field that you’ve just computed) may lie somewhere right before our eyes if we look upthread and know how to recognize what we’re looking at. Certainly somewhere out on the web (again you have to know what you’re looking at).

I’m done (for now) but of course ‘the problem’ isn’t. Is it a boson or fermion (well, you’ve sort of determined that already)? More importantly: computing its actions. Again I’m somewhat guessing here, but an example of an action would be how the W and Z bosons combine to produce the weak interaction.

So what was all that? I’m trying to check my understanding of things (so it’s sort of a big, long question). Berlin told us that he’s using an Excel spreadsheet. Well I know only too well how cumbersome (for me) these become at a point (look up OLAP – extremely cool), so while what I should use popped up as a question in my mind, I think it’s still better that I use the GAP software. Garrett apparently uses Mathematica. And, it goes with saying, these are tools. Then there’s understanding. And (according to Einstein) beyond understanding (knowledge) lies imagination.

Sorry too long. A single, specific question. This depends on some small part of the above being correct. Is it always two roots that combine to produce one particle/field? But wait I’ve probably gotten something (major) wrong. E8 has 240 symmetries, therefore particle/fields. And so 240 roots also. Well if you could only use each root once, then that would give only 120 particles. Maybe you can use a given root more than once?

slipped just saw Garrett’s latest post as I was writing this up.

Pat,
You belong in New Jersey.

Everyone, obviously, is impressed with your IQ. But what’s also struck me from the time I read your first posts (Backreaction) was your EQ as well. And so I’m puzzled: you might have played the academic politics game quite well. There are secondhand descriptions of academic politics all over the place, but I’ve had the opportunity to watch academic politics (and so the application of EQ in this regard) up close in the form of my physicist brother-in-law who made it onto the tenure track a couple of yrs ago. He’s an astrophysicist (currently one project is the polarization of the CMB) and there are web photos of him somewhere both at ESO, high up in the Chilean mountains, as well as at the South Pole. Oh yes, in your neck of the woods, he's also been to the top of Mauna Kea and the Keck installation (or whatever the whole facility is called).

But as you say yourself: “I didn’t want to do string theory”.

I’m from Boston originally and much of my family has lived in or around New York (but not me). I came to California in 1984 to finish up my undergrad at Berkeley. And have stayed in CA ever since, minus 6 yrs in Tokyo that is. We’re in the Bay Area. I’ve always wondered about this. If there’s a SoCal, do we live in NoCal? I finished at Berkeley Phi Beta Kappa with degrees in Japanese and Computer Science but grad school was sort of foreclosed upon by my having lost several yrs to surviving cancer in my early twenties. So all things considered I’m quite happy to be where I am (as opposed to, say, dead [or in New Jersey]).

pat

 

[Toastus]

Could someone describe AESToE in simple terms for me? I don't understand most of this, but I do get that it describes everything with one geometric shape.

 

[rntsai]

OK I’m trying to understand the actual procedure here.

First you pick some simpler subalgebra of E8 – say G2 or F4. This will make the problem vastly more tractable. You can ‘see’ things more clearly and the operations will be simpler and/or faster. Next you need to find a basis within that subalgebra where two of the vectors will use the one-and-only Lie algebra operator – the bracket or commutator – and give a result back at the origin. The origin, in 3D, being x=0,y=0,z=0. What does this mean? Linear algebra I and the dot product (so here we’re talking second yr undergraduate mathematics [imo]). The dot product (in 2D – I guess it’s the wedge product in higher dimensions) will act upon two vectors and give a result of 0 when the two vectors are perpendicular. Perpendicular is the important part. So what does it mean ‘in the real world’ that the vectors are perpendicular? Well that depends. We’ll leave it as an exercise of the reader (but at this point, you are quite close to ‘the real world’).

Hi Pat,
The algebra involved here is actually simpler than this.

There's a lot of jargon in both Lie algebras/groups and in
their application here to elementary particles that make things
even more confusing. There are a lot of constructs that go
by different names; slight variants are sometimes also referred
to by the same name many times sometimes without distinction.
I can try to simplify things to the best of my knowledge (and time).

1 question mark = I know the theory, but don't know what Garrett's saying.
2 question marks = I'm not sure myself because lack of knowledge in the area

- there are no dot products, cross products, ... here at all.
you just have one algebra over the reals(?). You can think of
commutation as its operation or just that it has a product
that satissfies the lie algebra axioms.

- E8 is a group; e8 is an algebra. You can associate several
"E8's" with e8. All the particle assignments and their quantum
can be done working with e8 only. I don't know if e8 over reals
or e8 over complexes is enough; I think reals(?) are enough
in spite of the appearance of complex numbers in some parts.
This importance of this distinction will come up later.

- real and complex e8 is 248 dimensional (over its repective field) and
you can find a basis for it such that 240 of the 248 basis vectors satisfy
certain conditions. You can then call these 240 basis vectors "roots" of the
algebra. These "roots" can be further divided into 120 positive and 120 negative
ones. Again here you're just picking names for basis elements.

I'm beginning to think I'm adding to the jargon rather than clarifying it.
Maybe I'll stop and rethink this.

 

[patfla]

Hello rntsai

The dot product thing was just an analogy. Working my way upwards, as it were, from Linear Algebra I.

The operator in the Lie algebra is the commutator (generic term I think) or Lie bracket. There seem to be several (mathematical) formulations of the Lie bracket on Wikipedia. I think I like this one best

http://en.wikipedia.org/wiki/Lie_bracket_of_vector_fields

pat

 

[Berlin]

Hello Jan, I've been lurking in this topic for a while now and am sorry to pollute it with an off-topic post, but every time you mention your "Joppe conjecture" it makes me smile, as the only Joppe I know is my taijiquan teacher :) So I was wondering, how did you arrive at the name?

PS: good luck with your work on both theories!

Hi Emanual,
Joppe is my family name (Netherlands). I will of course not say to whom your name reminds me of :-)

Jan

 

[rntsai]

The dot product thing was just an analogy. Working my way upwards, as it were, from Linear Algebra I.

OK. There really us just one definition of "Lie bracket" (more jargon);
I prefer $$x \star y$$ over [x,y] : this is R-bilinear and
$$x \star y = -y \star x$$
$$z \star (x \star y) = (z \star x) \star y + x \star (z \star y)$$

the last one is "product rule" or "Jacobi Identity" (more jargon);
other "definitions" using differentials, commutation, ... are just
representations of it; the above is more fundemental.

 

[rntsai]

Sorry too long. A single, specific question. This depends on some small part of the above being correct. Is it always two roots that combine to produce one particle/field? But wait I’ve probably gotten something (major) wrong. E8 has 240 symmetries, therefore particle/fields. And so 240 roots also. Well if you could only use each root once, then that would give only 120 particles. Maybe you can use a given root more than once?

It's one root per particle. There are 240 roots, so exactly 240 elementary particles (in this
theory at least); these include some 20 odd new particles not in the standard model
e8 has a lot more symmetries than 240; its reflection
group or "Weyl group" has order 696,729,600. This is a discrete group different
than the Lie group E8...

 

[Tony Smith]

Garrett said that the "... torsion question is especially interesting ...".

Here are some possibly useful things about torsion:
In hep-th/9601066, "Geometric Interpretation of Electromagnetism in a Gravitational Theory with Torsion and Spinorial Matter", Keniche Horie wrote:
"... General relativity is enlarged by allowing for an arbitrary complex linear connection and by constructing an extended spinor derivative based on the complex connection. Thereby the spacetime torsion not only is coupled to the spin of fermions and causes a four-fermion contact interaction, but the non-metric vector-part of torsion is also related to the electromagnetic potential. However, this long-standing relation is shown to be valid only in a special U(1) gauge ...".

Since Horie got U(1) gauge bosons from a complex connection with torsion,
you can speculate about whether:
a quaternionic connection with torsion could have produced SU(2) gauge bosons
and
an octonionic connection with torsion could have produced SU(3) (a subgroup of G2) gauge bosons.

Then you might think of looking at a CxQxO connection (2x4x8 = 64-dim)
and
see that is what Geoffrey Dixon has been using for model-building - see his book "Division algebras ..." to which there is a link at his web page at
http://www.7stones.com/Homepage/AlgebraSite/algebra0.html
which also has links to some of his other works.

As to how that might be related to E8 models,
consider that the 64-dim CxQxO thing might correspond to spinors ( Geoffrey Dixon writes about that in his book ),
and that two copies of it might correspond to the 64+64 = 128-dim Spin(16) half-spinors
inside E8.

Tony Smith

 

[rntsai]

Since Horie got U(1) gauge bosons from a complex connection with torsion,
you can speculate about whether:
a quaternionic connection with torsion could have produced SU(2) gauge bosons
and
an octonionic connection with torsion could have produced SU(3) (a subgroup of G2) gauge bosons.

Then you might think of looking at a CxQxO connection (2x4x8 = 64-dim) ...

As to how that might be related to E8 models,
consider that the 64-dim CxQxO thing might correspond to spinors ( Geoffrey Dixon writes about that in his book ),

Hi Tony,
I don't wish to offend in any way; I think you're pretty sharp guy and have one
of the most interesting websites around. That being said, why one would want to
bother with dealing with octanions for this and other problems completeley
escapes me. There's enough complexity/symmetry/structure with lie algebras
over reals to handle things...I can give up commutativity without problems,
but you lose so much by giving up associativity...is it really worth it?
(I know you lose associativity when you go to Lie algebras, but in that
case it really is worth it : the Jacobi identity is still primitive enough plus
you have the very natural commutation as standard model...)

 

[Tony Smith]

rntsai asked "... why one would want to bother with dealing with octanions ... you lose so much by giving up associativity...is it really worth it? ...".

Sorry that I gave that impression by assuming and not quoting details of how Geoffrey Dixon uses CxQxO which he denotes by T as a spinor space in his book cited in my comment.
As he says ( here I modify some of his notation, such as by denoting the 8x8 real matrix algebra by M(R,8), etc ) (pages 68, 40, 66-67, 84-85, 128)
"... T is not only nonassociative ... but ... also nonalternative ...
The left adjoint algebras are
CL = C
QL = Q
OL = M(R,8)
which imply the Clifford algebra isomorphisms
CL = Cl(0,1)
QL = Cl(0,2)
OL = Cl(0,6)
...
Let T = CxQxO and TL = CLxQLxOL
...
we view TL as the "Pauli" algebra ... from which the "Dirac" algebra ... will be built ...
TL = M(C,16) so we may identify it with the Clifford algebra Cl(0,9) ...
The object space of Cl(0,9) = M(C,16) is the space of 16x1 complex spinors ...
[ 2x2 matrices with entries in TL ]... M(TL,2) = C(32), which is the complexification of Cl(1,9)
So TL is, so-to-speak, the "Pauli" algebra to the ... "Dirac" algebra M(TL,2) ...
The object space of M(TL,2) is T2, 2x1 matrices over T. ...
The spinors of M(TL,2) are elements of the 128-dimensional T2,
the space of 2x1 columns over 64-dimensional T ...".

So, Geoffrey Dixon uses T2, two copies of nonassociative nonalternative T,
as a 128-dimensional spinor space analogous to the 128-dim part of E8
that is the fermionic/spinor half-spinor of Spin(16),
and
he operates on it with an associative Clifford algebra Cl(1,9),
the bivector Lie algebra of which is Spin(1,9) = SL(2,O) as described, for example, in John Baez's paper on Octonions at
http://math.ucr.edu/home/baez/octonions/

By looking at that Clifford algebra, Geoffrey Dixon gets representations of standard model things that may be usefully related to the standard model things that appear in E8 models.

At this point I refer to Geoffrey Dixon's book and papers for more details,
but
the basic point that I want to make is that the nonassociativity of octonions does not render them useless in physics model building because you can work with related associative Clifford algebras.

If you are going to build models based on E8 (or any other exceptional Lie group/algebra), you are effectively using octonions whether you explicitly acknowledge it or not.
For example, the basic structure E8 / Spin(16) = 128-dim half-spinor of Spin(16)
is the symmetric space (OxO)P2
which is the projective plane of the tensor product of two copies of the octonions,
which symmetric space is described by Boris Rosenfeld in his book "Geometry of Lie Groups" (Kluwer 1997).

As to whether or not it is "really worth it",
the worth of any physics model is what you can calculate with it,
and
it permits me to calculate particle masses, force strengths, etc that are substantialy consistent with experimental results, using similar model-building techniques.

Tony Smith

PS - It is not offensive at all to ask such questions. I learned most of what little I know by asking such questions over many years.

 

[rntsai]

If you are going to build models based on E8 (or any other exceptional Lie group/algebra), you are effectively using octonions whether you explicitly acknowledge it or not.
For example, the basic structure E8 / Spin(16) = 128-dim half-spinor of Spin(16)
is the symmetric space (OxO)P2
which is the projective plane of the tensor product of two copies of the octonions,
which symmetric space is described by Boris Rosenfeld in his book "Geometry of Lie Groups" (Kluwer 1997).

Hi Tony,
I'm glad to see you have this attitude about things, but what you write above actually
is what makes my point. By working with these exceptional groups we're picking
all the symmetry/structure of octanions without having to deal with them!
Lie algebras are much easier to work with and their knowledge database is
much bigger. I can't imagine doing calculations in (OxO)P2 for example being
easy at all, whereas I can get everything I need to get about any rep of e8.

 

[Tony Smith]

rntsai said "... we're picking all the symmetry/structure of octanions without having to deal with them! Lie algebras are much easier to work with ...".

You should always use whatever approach you are more comfortable with,
as long as you realize that the 240 root vectors of E8 are 240 units that close under an octonion multiplication, so that octonions are really there anyway,
and
that when you go from local Lie algebra to global Lie group structure you inevitably encounter things like (OxO)P2.

However, the goal of model building is to do physics,
so if you can make a physics model that let's you calculate particle masses and force strengths etc by using a more limited set of math-structure-tools then that is a good thing and should be worked out and written up.

Tony Smith

 

[starkind]

Hi
I am somewhat embarrassed to mention this here, since it is probably just a trivial result. However, I believe I have discovered and can demonstrate that there are 240 possible two dimensional projections of a cube undergoing rotations. These projections are all unique and do not include any trivial rotations in the plane of observation.

Probably this result is well-known, and it is only coincidence that E8 has 240 roots. But if my result is correct, then would it not be a simplification to talk about rotations of a three dimensional cube rather than a seemingly more complicated E8 object?

 

[rntsai]

Hi
I am somewhat embarrassed to mention this here, since it is probably just a trivial result. However, I believe I have discovered and can demonstrate that there are 240 possible two dimensional projections of a cube undergoing rotations. These projections are all unique and do not include any trivial rotations in the plane of observation.

Probably this result is well-known, and it is only coincidence that E8 has 240 roots. But if my result is correct, then would it not be a simplification to talk about rotations of a three dimensional cube rather than a seemingly more complicated E8 object?

No need to be embarrassed, you never know where a good idea can come from so it's
always to keep an open mind about things. There must be some equivalence imposed
on these projections; it seems to me there are infinitely many : take a cube rotate
by any degree and project; if there is a "natural" way to define these equivalence
classes, how do you "add" and "multiply" them? it might end up that these are
actually harder to work with than lie algebras in spite of their definition

 

[starkind]

“There must be some equivalence imposed
on these projections; it seems to me there are infinitely many : take a cube rotate
by any degree and project; if there is a "natural" way to define these equivalence
classes, how do you "add" and "multiply" them? it might end up that these are
actually harder to work with than lie algebras in spite of their definition.”

Exactly so. I have been trying to follow the ongoing discussion in this thread and have to admit I am not up to speed on the terms. I have been studying and have some ideas about how it all fits together, but am far from fluent. So I have to use language that I do understand, in a more general way.

For example, there are some obvious choices for placement of axis of rotation. First, all rotation axies to be considered are through the center point, or origin of the cube. Then, there are pairs of points on the surface of the cube which are defined by the shape of the cube.

I looked at three equivalence classes of polar pairs defined on the surface. First, there are opposite vertices. Eight of them, the corners of the cube, and so four axis choices. Then there are the faces of the cube, six of them, each with a center point. Using opposite center points as poles gives three more axis lines, the usual three dimensional orthogonal basis. Then there are the edges, each with a center point to use as a pole with the opposite pole being on the parallel edge. There are twelve edges and so six axis lines in this equivalence class.

So, three face centered axies, four vertex centered axies, and six edge centered axies.

Now observe the cube from some fixed point at a sufficient distance in the equatorial plane. The cube presents the observer with a visible surface, which is one of its projections onto the two dimensional plane. Rotation of the cube about an axis produces a sequence of symmetrical views. For example, a face centered axis has four symmetrical projections in which it presents a single face, and four in which it presents two faces. If the faces are labeled A,B,C,D, then the rotation sequence presents the following series: A, AB, B, BC, C, CD, D, DA, and repeat. Three axies, eight projections each, twenty four possible projections.

The vertex centered axies when rotated produce four symmetry groups with three members each. The projection sequence alternates between two visible faces and three visible faces. So there are twelve possible projections per axis, and four axies, so forty-eight possible projections.

The edge centered axies have a sequence of eight elements each, showing one, two, or three faces. The sequence is one face, three faces, two faces, three faces, one face, three faces, two faces, three faces, and so on. So six axies, eight projections each, is forty-eight possible projections.

I have listed one hundred and twenty unique projections. Now if we consider the two possible directions of rotation, we have two hundred and forty unique projections.

I suspect that all of this floral verbiage can be stated simply and clearly using a few lines of Lie algebra, but I don’t know how to do that.

So that’s the bones. Still have to think about how they might fit together.

S.

 

[gdixon]

Octonions

Objections to the use of the octonions in physics because they are
not associative closes off the objector to a very useful perspective.
Although I don't deeply follow contemporary approaches to unification
and TOEs, every approach I have some awareness of can be linked
to the octonions and other division algebras. That is not to say that
the octonions are necessarily required in the development of any given
theory, but that they are an example of resonant mathematics, and
the mathematical underpinnings of viable physical theories tend to
accrete around the mathematically resonant. Consider the space-time
dimensions 4, 10 and 26, which have played such a great part in
string theories. The three corresponding transverse dimensions
are 2, 8 and 24. These are also the only known dimensions n for which
the n-dimensional laminated lattices (A2, E8, Λ24) simultaneously provide
the tightest sphere packings, give the best kissing numbers, and the
kissing spheres lock into place. These dimensions are mathematically
resonant. (They are also linked to the complex numbers, quaternions
and octonions.)

It is important to understand that the suggestion here is that any theory,
even if shown ultimately to be very wrong, will find its best results
cropping up when its mathematics align with such resonances, and
that a fair amount of time can be saved by accepting this notion from
the outset.

As to the octonions specifically, they are much more than just a
nonassociative algebra. As Conway and Sloane in their marvelous
"Sphere Packings" book point out, and as I and others have pointed out,
the octonions are a spinor space. They are intimately linked to Spin(8),
triality, Bott periodicity, all the exceptional Lie groups, space-times
of dimensions 10, 18, 26, ..., SU(3), and on and on. It is not possible
to produce a useful theory without the hint of octonionic influence, and
doubtful that one could be constructed that would not benefit in added
insight by recouching its amenable parts using the octonions.

 

[sooperdooper]

“There must be some equivalence imposed
on these projections; it seems to me there are infinitely many : take a cube rotate
by any degree and project; if there is a "natural" way to define these equivalence
classes, how do you "add" and "multiply" them? it might end up that these are
actually harder to work with than lie algebras in spite of their definition.”

Exactly so. I have been trying to follow the ongoing discussion in this thread and have to admit I am not up to speed on the terms. I have been studying and have some ideas about how it all fits together, but am far from fluent. So I have to use language that I do understand, in a more general way.

For example, there are some obvious choices for placement of axis of rotation. First, all rotation axies to be considered are through the center point, or origin of the cube. Then, there are pairs of points on the surface of the cube which are defined by the shape of the cube.

I looked at three equivalence classes of polar pairs defined on the surface. First, there are opposite vertices. Eight of them, the corners of the cube, and so four axis choices. Then there are the faces of the cube, six of them, each with a center point. Using opposite center points as poles gives three more axis lines, the usual three dimensional orthogonal basis. Then there are the edges, each with a center point to use as a pole with the opposite pole being on the parallel edge. There are twelve edges and so six axis lines in this equivalence class.

So, three face centered axies, four vertex centered axies, and six edge centered axies.

Now observe the cube from some fixed point at a sufficient distance in the equatorial plane. The cube presents the observer with a visible surface, which is one of its projections onto the two dimensional plane. Rotation of the cube about an axis produces a sequence of symmetrical views. For example, a face centered axis has four symmetrical projections in which it presents a single face, and four in which it presents two faces. If the faces are labeled A,B,C,D, then the rotation sequence presents the following series: A, AB, B, BC, C, CD, D, DA, and repeat. Three axies, eight projections each, twenty four possible projections.

The vertex centered axies when rotated produce four symmetry groups with three members each. The projection sequence alternates between two visible faces and three visible faces. So there are twelve possible projections per axis, and four axies, so forty-eight possible projections.

The edge centered axies have a sequence of eight elements each, showing one, two, or three faces. The sequence is one face, three faces, two faces, three faces, one face, three faces, two faces, three faces, and so on. So six axies, eight projections each, is forty-eight possible projections.

I have listed one hundred and twenty unique projections. Now if we consider the two possible directions of rotation, we have two hundred and forty unique projections.

I suspect that all of this floral verbiage can be stated simply and clearly using a few lines of Lie algebra, but I don’t know how to do that.

So that’s the bones. Still have to think about how they might fit together.

S.

I have been working on something similar; although not with cubes. Right now it explains proton to eletron mass ratio, and it gives very stong indications of "why" we will never have more than 3 generations of matter in any form and predicts the fine structure constant. Please message me I would like to talk further about how you are developing your model. Right now I'm trying to map Lisi's work on to my own, and see if they generally agree.

 

[starkind]

Objections to the use of the octonions in physics because they are
not associative closes off the objector to a very useful perspective.
Although I don't deeply follow contemporary approaches to unification
and TOEs, every approach I have some awareness of can be linked
to the octonions and other division algebras. That is not to say that
the octonions are necessarily required in the development of any given
theory, but that they are an example of resonant mathematics, and
the mathematical underpinnings of viable physical theories tend to
accrete around the mathematically resonant. Consider the space-time
dimensions 4, 10 and 26, which have played such a great part in
string theories. The three corresponding transverse dimensions
are 2, 8 and 24. These are also the only known dimensions n for which
the n-dimensional laminated lattices (A2, E8, Λ24) simultaneously provide
the tightest sphere packings, give the best kissing numbers, and the
kissing spheres lock into place. These dimensions are mathematically
resonant. (They are also linked to the complex numbers, quaternions
and octonions.)

It is important to understand that the suggestion here is that any theory,
even if shown ultimately to be very wrong, will find its best results
cropping up when its mathematics align with such resonances, and
that a fair amount of time can be saved by accepting this notion from
the outset.

As to the octonions specifically, they are much more than just a
nonassociative algebra. As Conway and Sloane in their marvelous
"Sphere Packings" book point out, and as I and others have pointed out,
the octonions are a spinor space. They are intimately linked to Spin(8),
triality, Bott periodicity, all the exceptional Lie groups, space-times
of dimensions 10, 18, 26, ..., SU(3), and on and on. It is not possible
to produce a useful theory without the hint of octonionic influence, and
doubtful that one could be constructed that would not benefit in added
insight by recouching its amenable parts using the octonions.

This will require me some study.

 

[starkind]

...it explains proton to eletron mass ratio, and it gives very stong indications of "why" we will never have more than 3 generations of matter in any form and predicts the fine structure constant.

This will be very powerful stuff, then.

I was going to append the thought of a cubic lattice collapsing into a dense packed sphere lattice as a model for directional time in an infall universe. But I have a lot of other stuff to think about right now.

index 14962

 

[starkind]

gdixon

You wrote:
“the mathematical underpinnings of viable physical theories tend to
accrete around the mathematically resonant.”

This is beautiful use of language, but I am not sure it can be evaluated for validity. ‘Resonant’ is a fine word, and I know what it means in general use, but have been unable to find a special meaning for it in mathematics. If it has no special meaning, then the quoted sentence, lovely as it is, probably reduces logically to something like “Good theories use good math.” Of course I cannot disagree with such a statement without descending into semantics.

Nevertheless, I have given some hours to this reply, partly because the quoted statement rolls around the tongue like colorful flavored marbles, as opposed to clunking on my teeth like dusty rocks. Thanks!

I spent quite a bit of Wiki time with the rest of it too, and got the feeling you know what you are saying. I should like to ask, if you are feeling generous, about the “n-dimensional laminated lattices (A2, E8, Λ24).” I didn’t find any results for the use of ‘laminated’ with the idea of lattices, but I think it means that the lattice structure can be sliced up into layers of similar structure. Also, I danced around with the last term in parenthesis, not sure what the letter before the number 24 should be. Is it lambda? What does it mean, in the sense that E in E8 means exceptional?

The idea of transverse dimensions gave me more stuff to gnaw at. I think I get it that the four dimensional space-time in which we are comfortable can be sliced up into two dimensional planes on which we can show state space, phase space, Hilbert space, and such. Wiki has a page on transversality which I find tantalizing. Pity poor Tantalus.

I was delighted to find the comment “These are also the only known dimensions n for which the n-dimensional laminated lattices (A2, E8, Λ24) simultaneously provide
the tightest sphere packings, give the best kissing numbers, and the
kissing spheres lock into place.” I like packing spheres and understand that the kissing numbers refer to the places where packed spheres touch each other. That would be twelve places on the surface of each sphere internal to a dense packing of like-sized three dimensional spheres.

I like the dense packing of like-sized three dimensional spheres for its simplicity and because there is only one likely axis on which to perform rotations. Or rather, there are six such axies, but they are indistinguishable one from another unless an observer chooses to count and label them. And it is cool that there are only two facial symmetries, one being the triangle and the other, the square.

About that idea of the kissing spheres locking into place: I wonder if it would be useful to think of time as excluded from such regions?

Thanks for a good read.

S.
15197

 

[gdixon]

A resonance in experimental physics is where the signal jumps
above the noise indicating the presence of something interesting.
A resonance in mathematics is a dimension, for example,
at which many complex notions become suddenly simple,
and this simplicity leads to enormous richness (1,2,4,8,24,
in particular). Little of what is accounted "good mathematics"
is resonant, and in the worst case it is intellectual onanism.
A2 = Λ2, E8 = Λ8, Λ24 = the Leech lattice, and Λ = Lambda,
in lattice theory indicating laminated, a technical term. Buy
Conway and Sloane's Sphere Packing book, just for the fun of it.

Two dimensions are required for motion: one space; one time.
Everything else is transverse, at least in the context of Lorentz
spaces.

One can not argue that some mathematics/dimensions are
richer (more resonant in my terminology) than others.
The division algebras, parallelizable spheres, all the classical
Lie groups, arise out of the special nature of dimensions 1,2,4,8.
Assuming that richness of that sort (and beyond) is required
of the mathematical underpinning of our (or any) physical reality
is my opinion. I don't do much research anymore, but all of
what I have done has been informed by this perspective - and
all of what I am doing now, and shall ever do.

Cheers, signing off now.

 

[Thomas Larsson]

More info about octonions can be found on Geoffrey's home page, http://www.7stones.com/Homepage/AlgebraSite/algebra0.html .

 

[starkind]

I am thankful for the reference and links. Conway and Sloane's Sphere Packing book will be on my list, even though it is expensive for us fixed income types. On the same topic, I read George Szpiro's book some years ago, Kepler's Conjecture, on the topic of sphere packings, and found it inspirational. It has the advantage of being accessible to ordinary readers, both in price and language, for those who are not interested in the exceptionally technical.

Gdixon wrote:

"The division algebras, parallelizable spheres, all the classical
Lie groups, arise out of the special nature of dimensions 1,2,4,8.
Assuming that richness of that sort (and beyond) is required
of the mathematical underpinning of our (or any) physical reality
is my opinion. I don't do much research anymore, but all of
what I have done has been informed by this perspective - and
all of what I am doing now, and shall ever do."

I was astonished at the certainty, as well as the finality, expressed in this statement. After many readings, I began to insert a catch of the breath before "is my opinion." Do I detect an edit? Was there a tentative thought covered by the catch phrase? What does one conclude, assuming that richness?

I welcome Geoffrey Dixon to PF, altho I myself have no official standing here. The link to Dr. Dixon's home page gives me lots to think about. Thanks!

15458

 

[patfla]

Unless I’m mistaken, we may have gone somewhat off-course from what Garrett originally intended for AESToE.

Which isn’t to say that I’ll necessarily steer things back in some suitable direction, but I’ve been off doing a bunch of stuff and want to return and ask some questions.

There are some very good other threads here on physicsforums (that relate). For one ‘DeSitter group SO(4,1) intro’ much of which is Garrett discussing deSitter space and other matters with no less than John Baez. John’s writing style is very different than Garrett’s. In post #10 in the above topic John has a very intuitive, geometric way of working our way ‘upwards’ (in dimensionality) to deSitter space. One neat trick is to get is a cartan connection via one tangent sphere rolling along a bumpy second tangent sphere from one point on the second sphere to another (John explains it much better). Very interesting. For one, as sphere A rolls, because B is bumpy, A twists a bit here-and-there which is a manifestation of basic SO(3,1) (rotation group of course rntsai – I will henceforth pay more attention to the lower-case algebra vs. upper-case group notation).

And actually some part of the early discussion between JB and Garrett is groups vs. algebras – respectively. So this goes back-and-forth until John kind of throws up his hands saying you need both simultaneously. And then ups the ante by saying that’s really not the right way of looking at things at all.

Also within ‘DeSitter group SO(4,1) intro’ there’s a pointer off to another topic that Garrett started called ‘Lie group geometry, Clifford algebra, symmetric spaces, Kaluza-Klein, and all that’. With some really gnarly looking math in it if you’re so inclined.

In Garrett’s first post he says:

The best way to understand the DeSitter group, SO(4,1), is to first understand the DeSitter algebra, so(4,1). This is the algebra of rotations for a five dimensional space with signature (4,1) -- that's four directions with positive norm and one direction with negative norm. (The same way Minkowski space has signature (1,3) or (3,1).) The DeSitter algebra has 10 generators which have commutation relations (structure coefficients) between them under the antisymmetric product (bracket).

Just prior to that marcus expresses deSitter space as w2 + x2 + y2 + z2 - t2.

So my first really dumb question. I still don’t properly understand the group notation (or signature as I think Garrett calls it). In SO(4,1) the 4 refers to dimensions with a positive norm – the 1 to a single dimension with a negative norm.

One can only assume that the 4 are spatial dimensions and the 5th, t, is time. So how come t = time has a negative norm? (something to do perhaps with the fact that t travels in only one direction [at anything larger than the Planck scale]).

I'm eventually making my way back here to so(3,1) gravity. So this is 3 spatial dimensions (w positive norm) and a single time dimension, as above with a negative norm? Obviously I'm scratching my head over the negative norm stuff (and that's presuming that t = time).

rntsai: ran your GAP program. Pretty straightforward really. And of course one doesn’t need a debugger or anything (thinking too much like that programmer I am). After executing a given statement (when running step-by-step), you simply display the resulting object to the screen and have some sort of algebraic structure to then decipher. And I’m making progress with these.

The more I play with it the more I like the GAP software (for doing this kind of math) and there are, in general, excellent resources at http://www.gap-software.com.

<STRIKE>I’m trying to keep my posts shorter (I may not succeed)</STRIKE>. (is physicsforums back at something like HTML 2.0?) I’ve heard of sphere packing (and its wonders) of course, but if there’s a single parallel line of study I’ll take, it’s that pointed to by Tony Smith and Geoffrey Dixon gdixon. The resonance of the octonions. I take it on faith that a great deal of what’s expressed here in Lie algebras and groups can be done with octonions. rntsai questions the utility of that – but that’s another matter.

I was intrigued with this expression (from Tony) upthread:

T = CxQxO

I would have normally thought that C was $$\mathbb{C}$$ – the complex numbers. Q is $$\mathbb{Q}$$ or the rationals and O - well – maybe the octonions or $$\mathbb{O}$$. 1 and 3 are right, but the middle term is actually the quaternions.

Which is explained here:

http://www.7stones.com/Homepage/su3.html

as

which is just the complexification of the quaternionization of the octonions.

! So what is that? You seem to be shoe-horning 8 dimensions first into 4 and then into 2. (here I make not a dumb question, but rather a stupid statement).

One last item. If the first part of a Wikipedia entry baffles the reader, see if it has an Examples section. The initial description of a Cartan subalgebra is not immediately penetrable.

However example #2 says:

A Cartan subalgebra of the Lie algebra of n×n matrices over a field is the algebra of all diagonal matrices.

Ah – that’s much better. Well our simple Lie algebras do consist of sq matrices (populated by quantum numbers). And a diagonal matrix simply serves ‘disentangle’ the various components of the linear transformation. Or more technically (I believe) orthogonalize them.

pat

 

[Tony Smith]

patfla mentions the ambiguity in mathematics notation of the lettter "Q".
It is used for the rational numbers, and also for the Quaternions.
However,
some people use the letter "H" (the "H" coming from "Hamilton") for the Quaternions to avoid that ambiguity,
and
others (including me) often use "Q" for the Quaternions hoping that context will make clear what is meant.

Also, some people don't like to use "O" for Octonions ( because "O" is also used for "orthogonal" as in the orthogonal group O(p,q) ) and so instead they use "Ca" (the "Ca" coming from Cayley).

patfla also says "... In SO(4,1) the 4 refers to dimensions with a positive norm – the 1 to a single dimension with a negative norm.
One can only assume that the 4 are spatial dimensions and the 5th, t, is time. So how come t = time has a negative norm? ...".

SO(p,q) and Clifford algebra Cl(p,q) notations are also not unanimously followed in the math community. Some people use (p,q) to denote signature with p + dimensions and q - dimensions, and some use it the other way around. Since both notations are found not only in papers but also in textbooks, you need to figure out which convention is used in whatever you are reading.

As to whether SO(3,1) or SO(1,3) should represent transformations of physical spacetime, you can look at the Clifford algebras Cl(3,1) or Cl(1,3) from whence they come (their Lie algebras are the bivector parts of those Clifford algebras, using the Lie bracket product).
Here I will use +++- and ---+ to be clear instead of (p,q) type notation.
(see for example F. Reese Harvey's book "Spinors and Calibrations" (Academic 1990))
Cl(+++-) = M(2,Q) = 2x2 matrices of Quaternions
Cl(---+) = M(4,R) = 4x4 matrices of Real numbers

So, the question of +++_ versus ---+ signature of physical spacetime becomes: do you want Quaternionic or Real structure?
John Baez has a web page about that question at
http://math.ucr.edu/home/baez/symplectic.html
Here are a few quotes from that web page:
"... spin-1/2 particles in nonrelativistic quantum mechanics are naturally quaternionic if we take time reversal into account! ...
[quoting Toby Bartels] "... we want an operator T: H -> H
...[where]... H ... is a 2-dimensional complex vector space ... the Hilbert space of a non-relativistic spin-1/2 particle ...
to describe the effect of time reversal on our spin-1/2 particle ...
an antiunitary operator with these properties does exist, and is unique up to phase.
It satisfies T2 = -1
Now what have we got? A quaternionic structure on H.
[in terms of Quaternion basis elements {1,i,j,k}] T = j.
... so H becomes a 1-dimensional quaternionic Hilbert space!
... So: spin-1/2 particles in nonrelativistic quantum mechanics are naturally quaternionic if we take time reversal into account!
... f the spin is an integer, T2 = 1 is a real structure, making the Hilbert space the complexification of a real Hilbert space. ..." [end of quote of Toby Bartels]

Even better, it turns out that the same stuff applies to representations of the Poincare group: the reps corresponding to fermions are quaternionic, while the reps corresponding to bosons are real - and the operator j turns out to be nothing other than the CPT operator! ...".

See also John Baez's week156 at
http://math.ucr.edu/home/baez/week156.html
where he says in a footnote "... Squark found in Volume 1 of Weinberg's "Quantum Field Theory" that the CPT operator on the Hilbert space of a spin-j representation of the Poincare group is an antiunitary operator with (CPT)^2 = -1^2 j. So indeed we do have (CPT)^2 = 1 in the bosonic case, making these representations real, and (CPT)^2 = -1 in the fermionic case, making these representations quaternionic. ...".

Note that all the above is consistent with the general approaches of
Geoffrey Dixon (T = CxQxO acting as a spinor space, with fermionic Quaternions acting like spinors, and generalizing the above to include Octonions)
and
Garrett Lisi using
248-dim E8 = 120-dim bosonic adjoint SO(16) + 128-dim fermionic half-spinor SO(16)
where, on the Lie Group level,
E8 / SO(16) = 128-dim fermionic half-spinor SO(16) = (OxO)P2 = the octo-octonionic projective plane known as Rosenfeld's elliptic projective plane. Rosenfeld is at Penn State and has a web page at
http://www.math.psu.edu/katok_s/BR/init.html

patfla also quotes Geoffrey Dixon as describing T = CxQxO as "... the complexification of the quaternionization of the octonions ..."
and then asks:
"... So what is that?
You seem to be shoe-horning 8 dimensions first into 4 and then into 2 ...".

It is not so much shoe-horning O into Q into C as it is
starting with 8-dim O
then expanding to QxO by letting each element of O be 4-dim Q to go to 4x8 = 32-dim
and
finally expanding again to CxQxO by letting each element of QxO be 2-dim complex to go to 2x32 = 64-dim.

Geoffrey Dixon then (as described in his book) uses two copies of T = CxQxO as his basic 64+645 = 128-dim spinor space
which seems to correspond to Garrett Lisi's E8 / SO(16) = (OxO)P2 = 128-dim spinor-type fermion space.

What makes me think that E8 physics is realistic and probably true is that so many different points of view (octonion, Clifford algebra, Lie algebra, symmetric space geometry, ...) all seem to fit with it consistently and to describe what we observe in the physics of gravity and the standard model.

Tony Smith

PS - I should also note the ambiguity of notation using T for time and T for CxQxO.
All in all, I think that you have to pay close attention to context when reading math/physics literature.

 

[patfla]

Thanx Tony

t is not so much shoe-horning O into Q into C as it is
starting with 8-dim O
then expanding to QxO by letting each element of O be 4-dim Q to go to 4x8 = 32-dim
and
finally expanding again to CxQxO by letting each element of QxO be 2-dim complex to go to 2x32 = 64-dim.

Aaahhh. Actually if I’d thought a minute or two about what the tensor product does, I probably should have been able to arrive at the correct interpretation.

I think I was distracted by this:

the complexification of the quaternionization of the octonions

I’m good at languages; speak several; and there was a way in which I thought this was really quite funny (which is not to diminish its truth/usefulness/whatever).

And as for context, your first problem in Japanese is that the word order is completely inverted (from English). The main verb arrives only at the very end. However I feel the real problem is that Japanese contains a huge (by English standards) of homonyms. How to disambiguate them?

 

[rntsai]

The more I play with it the more I like the GAP software (for doing this kind of math) and there are, in general, excellent resources at http://www.gap-software.com.

I think you meant to include another link (the above has nothing to do with GAP :
Groups Alogorithms and Programming). Here's a link for that :

http://www-gap.mcs.st-and.ac.uk/.

Some (final?) words on where I got in the calculations :
I tried duplicating what's in the paper, but I think I hit a wall.
I'll leave here the furthest I got in case someone will find it
useful and maybe unclock the next step :

This is a table of decomposing f4 under its subalgebras d4>a3>a2.
Each subalgebra breaks up subspaces into finer and finer components,...

The columns are :

(1) dimension of d4 rep
(2) dimension of a3 rep
(3) dimension of a2 rep
(4) eigenvalue of cartan generator #1 of d4
(5) eigenvalue of cartan generator #2 of d4
(6) eigenvalue of cartan generator #3 of d4
(7) eigenvalue of cartan generator #4 of d4
(8) eigenvalue of centralizer of a3 in d4
(9) eigenvalue of centralizer of a2 in a3

the last two look very close to physically meaningful values. The
problem is that there are a lot of 3 dimensional subspaces here.
Only 4 of the last 6 are independant; it's possible to get linear
combinations that give all sorts of patterns, but that seems like
an excercise in numerology. I couldn't find any "natural" way to associate
these with the quark/leptons... of the three generations... and I couldn't
find in the paper how that association is done.

Anyway, I'll leave this here (attachement); hopefully the above is descriptive
enough : f4 = d4 + 8V + 8S+ + 8S-

 

[patfla]

Hi rntsai - yes you're right. Not the .com site.

I did get the software originally from the .uk site you mention.

Later though I found this (and this is what I should have put up instead of the .com url):

http://www.gap-system.org/

Lots of interesting looking, and possibly useful stuff.

So much time, so little to do.

pat

 

[John G]

...it's possible to get linear combinations that give all sorts of patterns, but that seems like an excercise in numerology. I couldn't find any "natural" way to associate these with the quark/leptons... of the three generations... and I couldn't
find in the paper how that association is done.

Anyway, I'll leave this here (attachement); hopefully the above is descriptive
enough : f4 = d4 + 8V + 8S+ + 8S-

Conventional string theory (and Tony's unconventional string theory) use 8S+ and 8S- for one generation of fermions/antifermions. Does this seem more natural to you? Garrett I think goes up to E6 to get antifermions while conventionally E6 gets you complex fermions. I personally think this is the one area where Garrett might have to become more conventional but even with Garrett's model, any particular group of 8 multiplet should break up into quarks and leptons in a conventional way.

 

[rntsai]

Conventional string theory (and Tony's unconventional string theory) use 8S+ and 8S- for one generation of fermions/antifermions. Does this seem more natural to you?

I guess "natural" is subject to taste; mine would be if the assignment gives
the correct quantum numbers using elements that are more "distinguished";
this is why I use centralizers for example. I'll actually be happy to see how
the assignment is done (natural or otherwise). Did anyone confirm the
quantum numbers for any of the generations?

any particular group of 8 multiplet should break up into quarks and leptons in a conventional way.

Why would this be true? I assume here by 8 multiplet you mean "d4" 8 multiplet.
This is to distinguish from "a2" 8 multiplets like the one inside d4 (first 8 rows of the table
attached before) ; I think this 8 can be associated with gluons.

 

[Tony Smith]

rntsai asked, about "... us[ing] 8S+ and 8S- for one generation of fermions/antifermions ...",
"... Did anyone confirm the quantum numbers for any of the generations? ...".

Yes. For the first generation, this assignment, based on representing 8S+ by an octonion basis,

Octonion Particle

1 e-neutrino

i red up quark
j green up quark
k blue up quark

e electron

ie red down quark
je green down quark
ke blue down quark

gives color and electric charges consistent with reality, as does the corresponding assignment for 8S- and antiparticles.

It gives only left-handed particles and right-handed antiparticles,
so that the opposite handedness arises dynamically due to special relativity transformations that can switch handedness of particles that travel at less than light-speed (i.e., that have more than zero rest mass) as described by L. B. Okun in his book Leptons and Quarks (North-Holland (2nd printing 1984) page 11) where he said:

"... a particle with spin in the direction opposite to that of its momentum ...[is]... said to possesses left-handed helicity, or left-handed polarization.
A particle is said to possesses right-handed helicity, or polarization, if its spin is directed along its momentum.
The concept of helicity is not Lorentz invariant if the particle mass is non-zero. The helicity of such a particle depends oupon the motion of the observer's frame of reference. For example, it will change sign if we try to catch up with the particle at a speed above its velocity. Overtaking a particle is the more difficult, the higher its velocity, so that helicity becomes a better quantum number as velocity increases.
It is an exact quantum number for massless particles ...
The above space-time structure ... means ... that at ...[ v -> speed of light ]... particles
have only left-handed helicity, and antparticles only right-handed helicity. ...".

Tony Smith

 

[rntsai]

rntsai asked, about "... us[ing] 8S+ and 8S- for one generation of fermions/antifermions ...",
"... Did anyone confirm the quantum numbers for any of the generations? ...".

Yes. For the first generation, this assignment, based on representing 8S+ by an octonion basis,

Octonion Particle

1 e-neutrino

i red up quark
j green up quark
k blue up quark

e electron

ie red down quark
je green down quark
ke blue down quark

gives color and electric charges consistent with reality, as does the corresponding assignment for 8S- and antiparticles.

It gives only left-handed particles and right-handed antiparticles,

How does it do that? What about weak spin and hypercharge?
Does this show e_L with weak hypercharge=-1 and e_R with -2?

 

[rntsai]

The concept of helicity is not Lorentz invariant if the particle mass is non-zero. The helicity of such a particle depends oupon the motion of the observer's frame of reference.

Speaking of Lorentz, if you look in the table again, d4 breaks up as
15+6+6+1 under a3. The table also shows how the 15 breaks down
further under a2 as 8+3+3+1. Now a3 also has a d2 subalgebra, under
that the 15 breaks down as 3+3+4+4+1; the 3+3 is d2 itself (rotations/
boosts), the 4 is translation, the other 4 works out to special
conformal operations; the 1 is scaling or dilation. It's easy to verify that the
4's are commutative, and that they're the +1 and -1 eigenspaces of
dilation...just an interseting way the poincare algebra shows up in the
middle of all this!

 

[CarlB]

It's long been believed that the photon has no position operator. Margaret Hawton found a position operator some years ago but the issue is still debated. My favorite paper on this is:
http://arxiv.org/abs/quant-ph/0408017

Where this may tie in with the E8 discussion is in the method used to prove that the photon can have no position operator. It required the assumption that a position operator would look like the position operator of Schroedinger's wave equation, that is, be a simple vector. The problem arises in the assumption that one can write a photon wave function as having a specific value of spin with respect to a direction, usually taken to be z.

This assumption is a little incompatible with the fact that massless spin 1 particles take only two spin states. The result is that one misses a degree of freedom. What Hawton (and Baylis I suppose) did was to recognize that the incompatibility could be lifted if one always took spin in the direction of momentum. That is, instead of breaking the spin 1 state up using spin in the z direction, they used helicity. But that meant that the position operator needed to be a matrix; the matrix rotates the wave function to be in the +z direction.

The end result of this is that the photon wave function is most naturally written split into its left and right handed parts. This mirrors the standard model splitting of electrons into handed parts.

The usual way of doing QFT requires that one keep the left and right handed halves together and treat them as a couple. What Garrett has done is illegal mostly in that he has split right from left and treated them independently. This seems to me to have a certain resonance with what Hawton did.

Of course, being a fanatic, I think that there is a relationship between Hawton's stuff and density matrices and the consistent histories interpretation of quantum mechanics. I wrote up the details over at my blog:
http://carlbrannen.wordpress.com/2008/01/14/consistent-histories-and-density-operator-formalism/

 

[John G]

Speaking of Lorentz, if you look in the table again, d4 breaks up as 15+6+6+1 under a3. The table also shows how the 15 breaks down
further under a2 as 8+3+3+1. Now a3 also has a d2 subalgebra, under
that the 15 breaks down as 3+3+4+4+1; the 3+3 is d2 itself (rotations/
boosts), the 4 is translation, the other 4 works out to special
conformal operations; the 1 is scaling or dilation. It's easy to verify that the
4's are commutative, and that they're the +1 and -1 eigenspaces of
dilation...just an interseting way the poincare algebra shows up in the
middle of all this!

Wow, you've just pretty much described Tony's bosons. It's interesting to think what those special conformal operations (and Cartan subalgebra 1s) can do.

How does it do that? What about weak spin and hypercharge?
Does this show e_L with weak hypercharge=-1 and e_R with -2?

This is over my head but maybe this from Tony's website can help with weak spin and hypercharge until Tony returns:
http://www.valdostamuseum.org/hamsmith/su3su2u1.html

The electroweak/color charges come from the three bit structure of the octonion.

Why would this be true? I assume here by 8 multiplet you mean "d4" 8 multiplet.
This is to distinguish from "a2" 8 multiplets like the one inside d4 (first 8 rows of the table
attached before) ; I think this 8 can be associated with gluons.

It would be the vector and spinor multiplets outside the D4xD4.