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Exceptional algebraic structures in physics

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Tony Smith

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Jan 2, 2001, 10:47:09 PM1/2/01
to
[Moderator's note: again, this is an edited version of an email
from Tony Smith. - jb]

Freudenthal algebras are not written up in many places.
The book of Gursey and Tze is indeed not easy reading,
but there is a nice definition of Fr3(O) in
section 1.5.4 (pages 91-92) of Boris Rosenfeld's book,
Geometry of Lie Groups (Kluwer 1997), which I think that
I learned about from you, so you probably have a copy
(it is a Kluwer book and is indeed too expensive,
but I bought one anyway).

------------------------------------------------------------------

As to having a conversation about my stuff on s.p.r.,
I would be happy to do that, if you would put my messages up
for me. As I have said, one of my shortcomings is difficulty
in posting on usenet groups (I know it ought to be easy,
but I never even figured out how to configure a newsreader,
which is why I usually only lurk using deja).

Therefore, I will say for the record that you always have
my permission to put up any message that you get from me on s.p.r.
(I will probably see it when I lurk.)
Also, you have my permission to edit any such messages (most
of them would probably be much improved by any editing).

I don't expect that you would agree with all my stuff,
especially since (as always) it is a work in progress and is
subject to revision (hopefully a process of improvement),
so any substantive criticism (from you or anybody else) would
be appreciated and considered by me to be helpful.

Even though I do very much appreciate your willingness to
take a further look at my model, please do not feel obligated
to do so. One thing that I do not want is for anybody study
such stuff as a chore rather than because it is fun.

--------------------------------------------------------------------

Also,
thanks for stating your program.

If I understand it correctly, it is sort of the inverse of mine.
You start with h_3(O) and go down to Spin(8) and then to the Standard Model,
whereas (at least historically)
I started with Spin(8) and built it up to h_3(O) (and on to Fr3(O) etc).

Here is how I understand how the two ideas may be related:

You start with h3_(O) as an algebra of 3x3 matrices

d S+ V

S+* e S-

V* S-* f

where d, e, and f are real numbers
and S+, V, and S- are Octonions,
and * denotes conjugation.

You (same as I do) identify 8-dim V as a vector space containing
4-dim physical spacetime,
and
8-dim S+ as half-spinor fermion particles
and
8-dim S- as half-spinor fermion antiparticles.

Then you say that there is
a conventional 1-1 supersymmetry between bosons and fermions
that is broken when triality symmetry is broken,
which
coincides with dimensional reduction by which 4-dim physical spacetime
is projected out (or selected) from 8-dim V,
thus
leaving only h_2(O) symmetry between S+ and S- particles and antiparticles.

The main difference that you and I have over that is
that

You have a conventional 1-1 fermion-boson supersymmetry

while

I have an 8-8-28 relation (not a 1-1 fermion-boson supersymmetry)
between fermion particles, fermion antiparticles and gauge bosons.
That is
there are 8 types of fermion particles
and 8 types of antiparticles,
and
there are 28 (adjoint rep of Spin(8)) gauge bosons,
which correspond to antisymmetric fermion particle/antiparticle pairs.

Note that there is a level at which
you can say that my symmetry is a triality-based 1-1-1,
and
that is the level of the terms of the Lagrangian over 8-dim V base
manifold (unbroken 8-dim spacetime). It works this way:
In the 8-dimensional spacetime,
the dimension of each of the 28 gauge bosons in the Lagrangian is 1,
and
the dimension of each of the 8 fermion particles is 7/2,
so that
the total dimension 28x1 = 28 of the gauge bosons
is equal to
the total dimension 8x(7/2) = 28 of the fermion particles
and also is equal to
the total dimension 8x(7/2) = 28 of the fermion antiparticles.

I call the above 8-8-28 relation a subtle triality supersymmetry,
and
it is useful because the related 1-1-1 Lagrangian triality supersymmetry
can be useful in cancellations for ultraviolet finiteness at the
level of unbroken 8-dim vector spacetime,
so that
the lower energy theory (with broken V and 4-dim spacetime)
is really an effective theory that should inherit nice cancellations
from its more symmetric high-energy unbroken form.

---------------------------------------------------------------------

One more thing: the book to which I referred just for fun:
Bruce Hunt,
The Geometry of some special Arithmetic Quotients (Springer 1996)
is
actually an example of how a publisher and the web CAN and DO coexist
(so far, at least). See the beautiful images, etc. at URL

http://www.mathematik.uni-kl.de/~wwwagag/Galerie.html

Perhaps it may be significant that both the publisher and the web page,
in the case of harmonious coexistence, are German.


Tony 2 Jan 2001


Tony Smith

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Jan 2, 2001, 1:31:11 AM1/2/01
to
[Moderator's note: Tony Smith has allowed me to post his email on
s.p.r., so I will post this, and then my reply, and then his reply
to that. Tony has allowed me to edit his email, so I'll freely
delete stuff that's not very relevant to the matter at hand. - jb]

I very much enjoyed your discussion of OP2 etc geometry,
particularly since I have gotten involved in a discussion
with Jack Sarfatti and Gary Bekkum about geometry and physics.
I just sent to them an e-mail, but I will not copy you with it
because you need not be drawn into that discussion. On the
other hand, so that you can see what it says, I will put
the text of it at the end of this message.
Please note that my notations differ slightly from yours,
such as your h_3(O) is my J3(O).

I will make one remark here that is not in the message quoted below:

There is a chain of symmetric spaces / fibrations:

E7 / E6 x U(1)

E6 / D5 x U(1)

D5 / D4 x U(1)

that (since Spin(9,1) is a form of D5)
is related to your comments about
"... a nice embedding of Spin(9,1) in this form of E6 ...",

so that the E6 inside E7 structure that I am describing below
is related to the D5 inside E6 structure that you mentioned.

--------------------------------------------------------------------

Here is the [edited by jb] text of my message to Jack Sarfatti and
Gary Bekkum:

--------------------------------------------------------------------

OK - I will describe for you what I mean
by M (material spacetime and states)
and
by Q (the multiverse of possible states/worlds of the Many Worlds).

Although my model is fundamentally a Planck-scale lattice model,
I will use its continuum approximation in this description of M and Q.

Most of what I am going to say can be found in these 4 books:
Sigurdur Helgason,
Differential Geometry, Lie Groups, and Symmetric Spaces (Academic 1978);
Arthur L. Besse (a pseudonym for a group of French mathematicians),
Einstein Manifolds (Springer-Verlag 1987);
Boris Rosenfeld,
Geometry of Lie Groups (Kluwer 1997); and
Feza Gursey and Chia-Hsiung Tze,
On the Role of Division, Jordan and Related Algebras in
Particle Physics (World 1996).

-----------------------------------

M is the 56-dimensional Freudenthal algebra Fr3(O) of 2x2 vector-matrices

a X

Y b


where a and b are real numbers
and X and Y are elements of the 27-dimensional Jordan algebra H3(O),
which itself is an algebra of 3x3 matrices


d S+ V

S+* e S-

V* S-* f


where d, e, and f are real numbers
and S+, V, and S- are Octonions,
and * denotes conjugation.

----------------------------------

Physically,

each X and Y of the 27-dim Jordan algebra H3(O) are interpreted as:

the 8-dimensional octonion V splits into two 4-dimensional subspaces,
that correspond to
4-dimensional physical spacetime
and
4-dimensional internal symmetry space;

the 8-dimensional octonion S+ corresponds to the 8 first-generation
fermion particles (+half-spinors);

the 8-dimensional octonion S- corresponds to the 8 first-generation
fermion antiparticles (-half-spinors);

the 3 real numbers d, e, and f are like "algebraic glue"
holding V, S+, and S- together.

and

each element of the Freudenthal algebra Fr3(O) is interpreted as:

X and Y together (27+27 = 54 dimensions) are a complexification of J3(O);

the 2 real numbers a and b are like "algebraic glue"
holding X and Y together.

---------------------------------------------------------------------

Remarks -

A reason for using the Freudenthal algebra Fr3(O)
instead of tensor products of octonions is that
the Freudenthal algebra has nicer properties. -
For example,
in Freudenthal algebras X(XX) = (XX)X which is not
true for the tensor product OxO of octonions.

A reason for going up to the Freudenthal algebra and not just
using the Jordan algebra is so that related symmetric spaces
have complex structure, and the math of Hermitian symmetric spaces
and bounded complex domains can be used in calculations of
force strength constants and particle masses. (Armand Wyler was the
first to do this, as far as I know, but he did not do it entirely
correctly and his physical interpretations (being pre-standard model)
were not very clear or convincing.)

--------------------------------------------------------------------------

Now that we have defined M, the space of material states,

the next thing to do is to define Q, the space of all possible worlds
of the Many-Worlds.

First,
note that the symmetry group of M = Fr3(O) is the group
of automorphisms of M = Fr3(O), which is the Lie group E6.

E6 is a 78-dimensional Lie algebra,
whose lowest dimensional non-trivial representations
are 27-dimensional, corresponding to the two 27-dim Jordan
subspaces of the 56-dimensional Freudenthal algebra Fr3(O).
Note that the two 27-dim representations are complex conjugates
of each other, and can be denoted by 27 and 27*.

Second,
represent the (first-order, or first approximation to) geometry of Q
by a symmetric space G / K (for Lie groups G and K)
where K is the local symmetry (isotropy) group that should
represent the symmetry of material states M,
which we have seen is the automorphism group E6 of Fr3(O).
That is,
the symmetric space for geometry of Q should be of the form G / E6

A natural candidate for G is E7, the next highest rank group in
the E series of Lie groups, so look at E7 / E6.

E7 / E6 is not a good symmetric space, because it also needs a
a complex symmetry U(1) as a local symmetry (as it is naturally
a complex symmetric space), so use

E7 / E6 x U(1) for the geometry of Q (to first order approximation).

Since E7 is 133-dimensional, and E6 is 78-dimensional, and U(1) is 1-dim,
the symmetric space

Q = E7 / E6 x U(1) has 133-78-1 = 54 real dimensions, or 27 complex
dimensions.

Finally (for this e-mail) note that,

just as E6 is the automorphism group of the (binary) Freudenthal
algebra Fr3(O) with 56 real dimensions,

E7 is the automorphism group of a (ternary) algebra
that is 56-complex-dimensional.

-----------------------------------

Here is a thought about how to proceed beyond simply describing
the geometry of Q to actually calculating stuff.

Note that Q is 27-complex-dimensional,
and that the 27-dim of Q are related to the 27-dim of the
Jordan algebra J3(O)
and that J3(O) has a 26-dim traceless subalgebra J3(O)o
and
that world-lines in Q (lines of states that could form a world-line
succession of states)
look sort of like bosonic strings in 26-dim J3(O)o.

Then use bosonic string theory
to construct a concrete first approximation to a Bohmian landscape,
thus
unifying Bohm theory and Deutsch's ManyWorlds.

As you can see, there is a lot more work to do,
but I think the structure is very promising,
especially since I can (and have) used the math structures
to calculate force strengths and particle masses for the
4 forces and the observed quarks and leptons,
with the (first-order) results being pretty accurate
(better than 10%), except that I have a disagreement
with current interpretations of the T-quark mass,
in that I interpret existing data to be 10% consistent
with my calculated figure of 130 GeV,
as opposed to the currently fashionable figure of about 170 GeV.
For details of that, see
http://arXiv.org/abs/physics/0006041
For even more details,
see my web pages such as
http://www.innerx.net/personal/tsmith/TCZ.html

---------------------------------------------------

Finally (as to the math),
I will give a math reference that might not be so directly
relevant to the above, but it has so many beautiful things
related to E6 and geometry that I cannot fail to mention it:

Bruce Hunt,
The Geometry of some special Arithmetic Quotients (Springer 1996).


Tony 2 Jan 2001


ba...@galaxy.ucr.edu

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Jan 2, 2001, 2:18:12 PM1/2/01
to
Tony Smith wrote:

> There is a chain of symmetric spaces / fibrations:
>
> E7 / E6 x U(1)
>
> E6 / D5 x U(1)
>
> D5 / D4 x U(1)
>
> that (since Spin(9,1) is a form of D5)
> is related to your comments about
> "... a nice embedding of Spin(9,1) in this form of E6 ...",

I'm a bit confused about this. I want to describe Spin(9,1) in
a geometrically motivated way as a subgroup of the collineations
of OP^2, and clearly it should be something like what I said:
the subgroup that preserves an OP^1 in OP^2, and perhaps also
a point in OP^1 - remember, an inclusion h_2(O) -> h_3(O)
gives a dimension-2 projection p in h_3(O) (i.e. a line in OP^2)
but also a dimension-1 projection 1 - p in h_3(O) (i.e. a point in OP^2).
However, I'm not quite sure exactly how much this Spin(9,1) preserves,
nor where the U(1) comes into the game, from this particular viewpoint.

And of course this matters to me because in my secret theory of
everything I want to start with h_3(O) and all its symmetries, and
then by some symmetry-breaking process get Spin(9,1) to pop out.
Basically this symmetry-breaking should be the process of picking out
a distinguished h_2(O) in h_3(O).

This symmetry-breaking process should also be the stage at which fermions
become different from bosons, since picking an h_2(O) inside h_3(O)
amounts to taking the 3 copies of O inside h_3(O) and declaring
two of them to be the left and right spinor reps of Spin(8), and
the third to be the vector rep. I.e., triality gets broken!

I think it's cool that breaking triality could simultaneously
make 10d spacetime appear out of h_3(O) and distinguish between
fermions and bosons. In my creation myth this is a very primordial
stage in the fall from symmetry. The next big disaster would be
the picking out of a distinguished imaginary octonion, which cuts
10d spacetime down to 4d spacetime (sl(2,O) to sl(2,C)) and makes
the Standard Model gauge group appear a la Dixon.

I must admit I have never gotten too far in understanding your
model, but I guess it's time to make another stab at it.
If you would be willing to have a conversation about it on
sci.physics.research, that would be great. (I like s.p.r. better
than email because more people get involved and more unexpected
connections get made.)

One question: where can I learn about "Fr3(O)"? I find Gursey
and Tze's book very frustrating. Do I need to brush up on my
German and read Freudenthal's stuff?


gons...@us.ibm.com

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Jan 4, 2001, 2:03:50 PM1/4/01
to
Tony Smith and John Baez in concert... in a more perfect world of the many
worlds you could charge super bowl prices for this... this model seems to
start with a vector spacetime and half spinors matter/antimatter and end
with a D5 aka Spin(9,1) bivector spacetime and E6 5-vector (I think)
matter/antimatter, is this right? Let me make sure I understand what E6
adds to D5. From Tony Smith's website it describes it as adding (instead of
just more bivectors to get to D6) half of the 5 dimensional hypercube
vertices to the positive end of the 6th axis and the other half of the 5
dimensional hypercube vertices to the negative end of the 6th axis. In
coordinates this seems like <.5,.5,.5,.5,0,1> with all 2^5=32 possibile
sign changes. This nicely produces the same distance from the origin as the
D5 bivectors and involves only multiples of .5 (which is nice since when
one gets to E8 it is described as D8s interlaced by .5 along each axis).
The 5th dimension being zero seems oddly asymmetrical to me so somebody
correct me if it's needed. Also if anyone has all the E8 coordinates above
D5 sitting handy in a file, I'd enjoy seeing them. The triality thing both
fascinates and confuses me. I suspect I'm confused but my pet idea is that
there's something interesting about projecting the D5 bivector spacetime,
E6 5-vector matter and E6 5-vector antimatter on to the three planes (XZ,
YZ, XY) of the D3 cuboctahedron enclosed in E6. John


John Baez

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Jan 6, 2001, 12:52:38 AM1/6/01
to
Tony Smith wrote:

>Freudenthal algebras are not written up in many places.
>The book of Gursey and Tze is indeed not easy reading,
>but there is a nice definition of Fr3(O) in
>section 1.5.4 (pages 91-92) of Boris Rosenfeld's book,
>Geometry of Lie Groups (Kluwer 1997), which I think that
>I learned about from you, so you probably have a copy
>(it is a Kluwer book and is indeed too expensive,
> but I bought one anyway).

I'll have to check it out from the library again. I get
the vague feeling that one needs to read Freudenthal's
papers in German to really understand Freudenthal algebras.

By now I feel I understand how simple Jordan algebras give
rise to projective spaces, or in other words, lattices of
propositions suitable for doing quantum logic. (I sketched
this in "week162", and I'll explain it in more detail in
my octonion review article.) This is good enough for
understanding how the exceptional Jordan algebra relates
to the octonionic projective plane. But it's not good enough
to understand Rosenfeld's "bioctonionic", "quateroctonionic"
and "octooctonionic" projective planes - which are not even
strictly projective planes in the usual axiomatic sense!
For these, one probably needs a generalization of Jordan algebras,
and some generalized concept of projective plane.

When people talk about generalizing Jordan algebras, they
tend to talk about Jordan pairs, Jordan triples and Freudenthal
triples. But I don't really understand these concepts and
how they fit together. I guess I should reread this article:

Kevin McCrimmon, Jordan algebras and their applications,
AMS Bulletin 84 (1978), 612-627.

It explains Jordan triples and Jordan pairs quite nicely,
but it hasn't really sunk in yet.

I should also reread this:

G. Sierra, An application of the theories of Jordan algebras and
Freudenthal triple systems to particles and strings, Class. Quant.
Grav. (4) 1987, 227.



>Also, thanks for stating your program.

Perhaps I should emphasize that it's pretty vague at this point.

>If I understand it correctly, it is sort of the inverse of mine.
>You start with h_3(O) and go down to Spin(8) and then to the Standard Model,
>whereas (at least historically)
>I started with Spin(8) and built it up to h_3(O) (and on to Fr3(O) etc).
>
>Here is how I understand how the two ideas may be related:
>

>You start with h_3(O) as an algebra of 3x3 matrices


>
>d S+ V
>
>S+* e S-
>
>V* S-* f
>
>where d, e, and f are real numbers
>and S+, V, and S- are Octonions,
>and * denotes conjugation.
>
>You (same as I do) identify 8-dim V as a vector space containing
> 4-dim physical spacetime,
>and
> 8-dim S+ as half-spinor fermion particles
>and
> 8-dim S- as half-spinor fermion antiparticles.
>
>Then you say that there is
>a conventional 1-1 supersymmetry between bosons and fermions
>that is broken when triality symmetry is broken,
>which
>coincides with dimensional reduction by which 4-dim physical spacetime
>is projected out (or selected) from 8-dim V,
>thus
>leaving only h_2(O) symmetry between S+ and S- particles and antiparticles.

Actually, I don't think that's quite what I want to do. Sometime maybe
I should say more about what I really want to do. But I'm getting
quite tired all of a sudden, so I'll just ask you a question:

>I have an 8-8-28 relation (not a 1-1 fermion-boson supersymmetry)
>between fermion particles, fermion antiparticles and gauge bosons.
>That is
>there are 8 types of fermion particles
>and 8 types of antiparticles,
>and
>there are 28 (adjoint rep of Spin(8)) gauge bosons,
>which correspond to antisymmetric fermion particle/antiparticle pairs.

How do all these particles match up with those in the Standard Model?

Also, are you talking about one generation of fermions here, or what?

I guess I'd really like a nice simple description of your model, or
some simple version of your model, since I sometimes get the impression
you have several. I'd really like to see a version where on the one hand
we clearly see the Standard Model particles, and on the other hand, we
see them fitting inside some bigger, more symmetrical scheme.

I guess I'll start by rereading some of your papers. I'll print a
couple out, and then go home and sleep with them under my pillow.

John Baez

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Jan 6, 2001, 1:11:08 AM1/6/01
to
In article <936br6$1gso$1...@mortar.ucr.edu>,
John Baez <ba...@galaxy.ucr.edu> wrote:

>By now I feel I understand how simple Jordan algebras give
>rise to projective spaces, or in other words, lattices of
>propositions suitable for doing quantum logic.

>This is good enough for

>understanding how the exceptional Jordan algebra relates
>to the octonionic projective plane. But it's not good enough
>to understand Rosenfeld's "bioctonionic", "quateroctonionic"
>and "octooctonionic" projective planes - which are not even
>strictly projective planes in the usual axiomatic sense!
>For these, one probably needs a generalization of Jordan algebras,
>and some generalized concept of projective plane.
>
>When people talk about generalizing Jordan algebras, they
>tend to talk about Jordan pairs, Jordan triples and Freudenthal
>triples. But I don't really understand these concepts and
>how they fit together. I guess I should reread this article:
>
>Kevin McCrimmon, Jordan algebras and their applications,
>AMS Bulletin 84 (1978), 612-627.

And I guess I should have read this, too!!!

Frank (Tony) Smith, Jr., Hermitian Jordan Triple Systems, the
Standard Model plus Gravity, and alpha_E = 1/137.03608,
Phys.Rev. D49 (1994) 3779-3782, available as hep-th/9302030.

For some silly reason this is the paper of yours I never
got round to reading, when it's clearly the one I needed to
read most of all.

It's silly for me to talk about my ideas until I understand
this stuff!


Tony Smith

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Jan 5, 2001, 10:28:25 PM1/5/01
to gons...@us.ibm.com, ba...@math.ucr.edu
John Gonsowski asks for "... all the E8 coordinates above D5
sitting handy in a file ...".

My web page at
http://www.innerx.net/personal/tsmith/E8.html
has 7 sets of coordinates for the 240 first-layer vertices of E8 lattices.

My web page at
http://www.innerx.net/personal/tsmith/Weyl.html#E8cartan
has an attempt (maybe not too clear, but 8-dim is hard to draw in 2-dim)
at illustrating it with pictures.
It may be useful to look at other similar pictures on the same web page
that attempt to illustrate E7, E6, D5, and D4. Their specific URLs are
http://www.innerx.net/personal/tsmith/Weyl.html#E7cartan
http://www.innerx.net/personal/tsmith/Weyl.html#E6cartan
http://www.innerx.net/personal/tsmith/Weyl.html#D5cartan
http://www.innerx.net/personal/tsmith/Weyl.html#D4cartan

Some of the images are stereo pairs, so if your eyes don't fuse them,
you will see twice too many points. In that case, just ignore half of them.

The D3 cuboctahedron is shown at
http://www.innerx.net/personal/tsmith/Weyl.html#D3cartan

Since each link of the chain D3-D4-D5-E6-E7-E8 can be built up
from its predecessor, it might be useful to look at them in sequence.

John Gonsowski also says
"The triality thing both fascinates and confuses me.".

Me, too, which is why I keep working with it.

There is a nice 1998 book with triality-related material,
and even a section on Freudenthal algebras:

The Book of Involutions,
by M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol
(with a preface by J. Tits),
American Mathematical Society Colloquium Publications,
vol. 44, American Mathematical Society,
Providence, RI, 1998

You can see a table of contents and more information at URL

http://www.math.ohio-state.edu/~rost/BoI.html


Tony 5 Jan 2001


zir...@my-deja.com

unread,
Jan 5, 2001, 5:12:44 PM1/5/01
to
In article <l03102802b67716fa1e93@[38.30.118.178]>,
Tony Smith <tsm...@innerx.net> wrote:

> A reason for going up to the Freudenthal algebra and not just

> using the Jordan algebra [....]

Earlier in this forum, John Baez mentioned the work of M. Gunaydin and I
looked at a recent paper by M. Gunaydin et al. called "Conformal and
Quasiconformal Realizations of Exceptional Lie Groups" (hep-th/0008063).
You might be interested in this paper because, like your approach, their
model also "goes beyond the framework of Jordan algebras". Their
construction "contains all previous examples of generalized space-times
based on exceptional Lie groups" and is supposedly relevant for
supergravity and M-theory although it is not clear how. You might also
be interested in some of the references in this paper. What is a
Freudenthal algebra and how does it compare to Jordan or Lie algebras?


james dolan

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Jan 6, 2001, 8:25:44 PM1/6/01
to
[Moderator's note: James Dolan has allowed me to post these emails. - jb]

John Baez wrote:

|When people talk about generalizing Jordan algebras, they tend to
|talk about Jordan pairs, Jordan triples and Freudenthal triples.

are "jordan triples" the same as "jordan triple systems"??? (maybe
not??)

jordan triple systems are the things that correspond pretty nicely to
"bounded symmetric domains". in my opinion it's ridiculously easy to
understand how this works. the correspondence is just a slight extra
flourish atop the correspondence between "lie triple systems" and
"symmetric spaces", which in turn is just a slight extra flourish atop
the correspondence between "lie algebras" and "lie groups".

trying to dredge up old memories here, i think it's roughly like this:

a "symmetric space" situation is essentially just a lie group equipped
with a suitable involution; the "symmetric space" itself being the
sub-space of the group skew-fixed under the involution.

thus a "lie triple system" situation is essentially just a lie algebra
equipped with a suitable involution; the "lie triple system" itself
being the sub-space of the lie algebra skew-fixed under the
involution. (a tri-linear product naturally lives here, hence "lie
triple system".)

a "bounded symmetric domain" (actually i'm not even sure that's the
exact right terminology, come to think of it) is essentially a
symmetric space equipped with a complex structure for which the
reflection operation at each point is holomorphic. (i think that also
means that instead of just z/2 = o(1) acting around each point, u(1)
acts around each point. or something like that.)

thus a "jordan triple system" is essentially just a lie triple system
with a slight bit of extra structure amounting to a nice complex
structure, or something like that. that makes sense if you think
about it, in that having i around to turn skew-adjoint-flavored things
into self-adjoint-flavored things is just about the sort of thing
you'd want to have in order to be able to convert a lie-flavored thing
into a jordan-flavored thing.


(furthermore, if i remember correctly, a jordan triple system is only
a hair's breadth away from being the same thing as a jordan algebra; i
think it's something like, if you supply a jordan triple system with a
special element to be the "identity element", then you can reconstruct
the _bi_-linear jordan product from the _tri_-linear jordan product.
or something like that.)


don't think i ever heard of any "freudenthal triple" anything.

----------------------------------------------------------------------

by the way, another thing i forgot to mention: the theory of lie
algebras and the theory of jordan algebras are both "sub-theories" of
the theory of associative algebras, in obvious ways; let's call these
sub-theories t1 and t2, respectively. furthermore the theory of lie
triple systems is a sub-theory of the theory of lie algebras, again in
a very obvious way; thus it's also a sub-theory t3 of the theory of
associative algebras. all that is obvious. what's perhaps less
obvious is that, if i remember correctly, in the lattice of
sub-theories of the theory of associative algebras, not only is t3
contained in t1, but also t3 is contained in t2.

exercise: prove this. that is, express the "lie triple product" in
terms of the (bilinear) jordan product.

Ralph E. Frost

unread,
Jan 6, 2001, 8:49:56 PM1/6/01
to
<ba...@galaxy.ucr.edu> wrote in message
news:200191021918...@math-cl-n04.ucr.edu...

> Tony Smith wrote:
>
> > There is a chain of symmetric spaces / fibrations:
> >
> > E7 / E6 x U(1)
> >
> > E6 / D5 x U(1)
> >
> > D5 / D4 x U(1)
> >
> > that (since Spin(9,1) is a form of D5)
> > is related to your comments about
> > "... a nice embedding of Spin(9,1) in this form of E6 ...",
>
> I'm a bit confused about this. ...


Me too, but I'll bet for different reasons. Plus, I'll bet YOU are just
pretending...

Can either of you refer to some specific visual representations on the web
that approximate what the jargon points at, please?
..

> And of course this matters to me because in my secret theory of
> everything I want to start with h_3(O) and all its symmetries, and
> then by some symmetry-breaking process get Spin(9,1) to pop out.
> Basically this symmetry-breaking should be the process of picking out
> a distinguished h_2(O) in h_3(O).
>
> This symmetry-breaking process should also be the stage at which fermions
> become different from bosons, since picking an h_2(O) inside h_3(O)
> amounts to taking the 3 copies of O inside h_3(O) and declaring
> two of them to be the left and right spinor reps of Spin(8), and
> the third to be the vector rep. I.e., triality gets broken!

Is this, "picking an h_2(O) inside h_3(O)", a process where one picks a
particular "algebraic/geometric structure" that is inside some other
structure?

Also, where you refer to breaking triality, aren't you just switching to a
different association? Also, to get to "left-right", somehow didn't you
have to impose some structure to be able to make that (left-right)
differientiation?


>
> I think it's cool that breaking triality could simultaneously
> make 10d spacetime appear out of h_3(O) and distinguish between
> fermions and bosons. In my creation myth this is a very primordial
> stage in the fall from symmetry. The next big disaster would be
> the picking out of a distinguished imaginary octonion, which cuts
> 10d spacetime down to 4d spacetime (sl(2,O) to sl(2,C)) and makes
> the Standard Model gauge group appear a la Dixon.

Triality.... Now there is a cool-sounding term.... a very thoughtful and
thought-provoking, very cool term.....

Can you elaborate on it, please?

--
Ralph E. Frost
Frost Low Energy Physics
http://www.dcwi.com/~refrost/index.htm


Tony Smith

unread,
Jan 6, 2001, 11:25:09 PM1/6/01
to
Zirkus wrote:
"... I looked at a recent paper by M. Gunaydin et al. ..".

Gunaydin writes papers with math that I like,
In fact, I was motivated to write one of my earlier papers
http://xxx.lanl.gov/abs/hep-th/9302030
by reading a paper by Gunaydin at
http://xxx.lanl.gov/abs/hep-th/9301050
which paper was about Jordan Algebras.
The paper by Gunaydin also dealt with Super-Jordan Algebras,
but since supersymmetry of the 1-1 fermion-boson type is not
to my taste, I don't read so much about the supersymmetric things.


-----------------------------------------------------------------------

Zirkus also wrote:
"... What is a Freudenthal algebra and
how does it compare to Jordan or Lie algebras? ...".

Very roughly (NOT rigorously), Jordan and Lie algebras are sort
of complementary.
If you consider a matrix algebra with matrices A and B,
and their product AB,
then
you can decompose AB = (1/2)(AB + BA) + (1/2)(AB - BA)
Then
the symmetric part (1/2)(AB + BA) forms a Jordan algebra
and
the antisymmetric part (1/2)(AB - BA) forms a Lie algebra.

The nature of the entries of the matrix algebra
generally determine what kind of algebra you get:

Real matrices - B and D type Lie algebras (orthogonal)
Complex matrices - A type Lie algebras (unitary)
Quaternion matrices - C type Lie algebras (symplectic)
Octonion matrices - E, F, G type Lie algebras (exceptional)
(the non-associativity of octonions
means that there are only a few execptionals:
G2, F4, E6, E7, and E8 are the only ones).

You get a similar classification of Jordan algebras.

As to Freudenthal algebras,
they can be roughly (not completely rigorous here)
seen as being complexifications of Jordan algebras.

For instance (this is relevant to the thread),
there is a 27-dim Jordan algebra h_3(O) of 3x3 Hermitian Octonion matrices.
The automorphism group of h_3(O) is the 52-dim F4 exceptional Lie algebra.

If you complexify the 27-dim Jordan algebra, and add 2 more dimensions,
you get a 2x27 + 2 = 54+2 = 56-dim algebra
that is the Freudenthal algebra Fr3(O),
whose automorphism group is the 78-dim E6 exceptional Lie algebra.


-----------------------------------------------------------------------


James Dolan wrote about bounded symmetric domains and
symmetric space equipped with a complex structure.

That is exactly why I use the Freudenthal algebras instead
of the Jordan algebras, because there are two series of symmetric spaces:

Jordan h_3(O) has automorphism Freudenthal Fr3(O) has
group F4 automorphism group E6

The symmetric space F4 / B4 = OP2 The symmetric space E6 / D5xU(1)
the Cayley projective plane. has complex structure (note the
local U(1) group).

The symmetric space B4 / D4 = OP1 = S8 The symmetric space D5 / D4xU(1)
the 8-sphere. has complex structure.

These symmetric spaces are real, These symmetric spaces are
and not Hermitian, so that Hermitian, and do correspond
they do not correspond to 1-1 to bounded symmetric
bounded symmetric complex complex homogeneous domains.
homogeneous domains.
Physically, I like this
because I can look at
Shilov Boundary of the domain,
and, in the case of D5 / D4xU(1)
I can get (instead of S8)
the space S7 x RP1,
which is topologically like
S7 x S1,
and I can take that as
an 8-dim Minkowski space.

If I project out 4-dim of the S7,
I then have a physical Minkowski
spacetime like S3 x S1 for
my physics model.


Further, I can use the math of bounded homogeneous domains,
and related kernels etc, to calculate the Electromagnetic
Fine Structure Constant to be 1 / 137.03608... by a method
motivated by the work by Armand Wyler in the 1970s,
although I like to think that I have a sounder physical
intuition for what I am doing than Wyler expressed in his papers.


As to the term Freudenthal triple,
it is used in standard math works, such as
the paper by R. Skip Garibaldi
http://www.math.ucla.edu/~skip/
in his paper
Groups of Type E7 Over Arbitrary Fields,
http://xxx.lanl.gov/abs/math.AG/9811056


-----------------------------------------------------------------------

Ralph Frost wrote:
"... Triality.... Can you elaborate on it, please? ...".

There are a lot of ways to look at triality,
but one way is to look at the D4 Lie algebra,
and notice that it has 3 representations that are 8-dimensional:

8-dim vector space (for SO(8) rotations)
8-dim + half-spinors (from its 8-dim Clifford Algebra)
8-dim - half-spinors (also from its 8-dim Clifford Algebra).

You can show that each of these 3 representations are isomorphic to
each other, so that if you designate them as spokes (Dynkin diagram):

*
|
|
*
/ \
/ \
* *

arranged around a central (28-dim adjoint) representation like
a Mercedes-Benz sign, then you can rotate and transform each into
the other. These transformations are the Triality Automorphisms
of the D4 Lie algebra, and they have a lot of interesting consequences,
particularly since by them you can show that a property of the
vector space is also a property of the half-spinor spaces,
and vice versa.


Also, Ralph Frost wrote:
"... Can either of you refer to some specific
visual representations on the web ...".

I mentioned some in a reply to John Gonsowski's posting.


Here is what I said:

My web page at
http://www.innerx.net/personal/tsmith/Weyl.html#E8cartan
has an attempt (maybe not too clear, but 8-dim is hard to draw in 2-dim)
at illustrating it with pictures.
It may be useful to look at other similar pictures on the same web page
that attempt to illustrate E7, E6, D5, and D4. Their specific URLs are
http://www.innerx.net/personal/tsmith/Weyl.html#E7cartan
http://www.innerx.net/personal/tsmith/Weyl.html#E6cartan
http://www.innerx.net/personal/tsmith/Weyl.html#D5cartan
http://www.innerx.net/personal/tsmith/Weyl.html#D4cartan

Some of the images are stereo pairs, so if your eyes don't fuse them,
you will see twice too many points. In that case, just ignore half of them.

Since each link of the chain D3-D4-D5-E6-E7-E8 can be built up
from its predecessor, it might be useful to look at them in sequence.


-----------------------------------------------------------------------

Tony 6 Jan 2001


Tony Smith

unread,
Jan 7, 2001, 12:26:07 AM1/7/01
to
John Baez wrote:

>"... I'll just ask you a question:


>
>How do all these particles match up with those in the Standard Model?
>

>Also, are you talking about one generation of fermions here, or what? ...".


I have at very high energies (i.e. before 8-dim space is broken
into 4-dim physical spacetime and 4-dim internal symmetry space)
only the first generation fermion particles,
and I have the 8 of the standard model
neutrino
red, green, blue down quarks
red, green, blue up quarks
electron,

as well as the corresponding antiparticles.

Breaking the 8-dim space produces 2 more generations,
for 3 generations (no more and no less).


Of the 28 gauge bosons,
16 form a U(2,2) that contains a conformal SU(2,2) - Spin(2,4) that,
when gauged, produces gravity by the MacDowell-Mansouri mechanism.


The 12 remaining become the 12 generators of the Standard Model
SU(3) x SU(2) x U(1) (8 gluons + 3 weak bosons + 1 photon).


Details of how this works can be lengthy, especially if you
go into enough detail about combinatorial stuff and
geometry of bounded complex homogeneous domains
in order to calculate particle masses and force strength constants.

The calculations are pretty much consistent with accepted values,


except that I have a disagreement
with current interpretations of the T-quark mass,
in that I interpret existing data to be 10% consistent
with my calculated figure of 130 GeV,
as opposed to the currently fashionable figure of about 170 GeV.
For details of that, see
http://arXiv.org/abs/physics/0006041
For even more details,
see my web pages such as
http://www.innerx.net/personal/tsmith/TCZ.html

Since the T-quark mass calculation leads to interesting (to me)
analysis of experimental results, it is probably my favorite.

For what it may be worth, all the details are on my web site.


For the calculations to be widely (i.e., beyond me) accepted,
somebody independent of me is going to have to work through
all the details and either verify them or find errors, whatever.

Historically,
the only way that Feynman diagrams really got accepted
(beyond Feynman) was when Freeman Dyson worked through them
independently and showed that Feynman diagrams were indeed
equivalent to the more conventional calculations of Schwinger,
and
the only way that 't Hooft's calculations of renormalizability
of the ElectroWeak model was (since his advisor Veltman did not
understand, or really believe, in what 't Hooft had done)
for Ben Lee to work independently through the calculations
and then announce to the world that 't Hooft was indeed correct.


Realistically, I don't expect anybody to work through my stuff,
but I like it and I find it to be beautiful, so I keep working on
it for fun.


Tony 6 Jan 2001


zir...@my-deja.com

unread,
Jan 8, 2001, 10:33:30 PM1/8/01
to
In article <l03102802b67d5c6d7004@[38.30.210.62]>,
Tony Smith <tsm...@innerx.net> wrote:

> Very roughly (NOT rigorously), Jordan and Lie algebras are sort

> of complementary [...]

Thanks for your reply, but I already knew or had deduced from your
first post what you wrote above. I was hoping for a more rigorous
description of what a Freudenthal algebra is (e.g., Is there a structure
or representation theory of these algebras?). However, when I have the
time, I will look at the references you have kindly suggested. I've
started reading about your model on your website but there may be a
problem with having a many-worlds interpretation (MWI) of quantum
computation (QC). This paper makes, IMHO, a compelling argument that
MWIs contradict QC:

http://arxiv.org/abs/quant-ph/0005069


> These transformations are the Triality Automorphisms

> of the D4 Lie algebra, [...]

Another way of realizing the triality automorpisms of Spin(8) is on
"classes of gamma-matrices representations which furnish a Majorana-Weyl
basis for Majorana-Weyl spinors. Next, triality transformations can be
lifted to connect spacetimes supporting Majorana-Weyl spinors sharing
the same dimensionality, but different signatures. Recursive formulas
for gamma-matrix representations allow to extend the 8-dimensional
properties to higher dimensional cases as well. Dualities induced by
triality are found connecting even-dimensional Majorana-Weyl spacetimes
(and odd-dimensional Majorana ones).... Indeed, higher dimensional
supersymmetric theories admit formulations in different signatures which
are all interrelated by triality induced transformations." From:

http://arxiv.org/abs/hep-th/0005034

For info on Majorana-Weyl spinors see TWF #93


------------------------------------------------------


Sent via Deja.com
http://www.deja.com/

Tony Smith

unread,
Jan 10, 2001, 12:16:27 AM1/10/01
to zir...@my-deja.com, ba...@math.ucr.edu

Zirkus wrote:
"... I was hoping for a more rigorous description

of what a Freudenthal algebra is (e.g., Is there a structure
or representation theory of these algebras?).
However, when I have the time,
I will look at the references ...".

My apologies for describing Freudenthal algebras too
roughly and without sufficient details. As you suggest,
the details may be described in such materials as
the paper of Skip Garibaldi
http://www.math.ucla.edu/~skip/


Groups of Type E7 Over Arbitrary Fields

http://xxx.lanl.gov/abs/math.AG/9811056
and
another paper (where the relevant algebras are
called Brown algebras) by Skip Garibaldi,
Structurable Algebras and Groups of Type E6 and E7
http://xxx.lanl.gov/abs/math.AG/9811035
as well as
the book


The Book of Involutions,
by M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol
(with a preface by J. Tits),
American Mathematical Society Colloquium Publications,
vol. 44, American Mathematical Society,
Providence, RI, 1998

http://www.math.ohio-state.edu/~rost/BoI.html

I think that those materials will give the details
much more completely and accurately than I could,
so I won't confuse matters further by trying.

-----------------------------------------------------------

Zirkus wrote about the triality paper of F. Toppan at
http://arxiv.org/abs/hep-th/0005034

Thanks for referring me to an interesting paper.

-----------------------------------------------------------

Zirkus also wrote:
"... there may be a problem with having a many-worlds


interpretation (MWI) of quantum computation (QC).
This paper makes, IMHO,
a compelling argument that MWIs contradict QC:
http://arxiv.org/abs/quant-ph/0005069

..."

As to paper by Giuseppe Castagnoli, Dalida Montiy, and
Alexander Sergienkoz about Quantum Computing and Many-World at
http://arxiv.org/abs/quant-ph/0005069
the authors are entitled to their opinion,
but their paper is not convincing to me.
They say
"... This work conflicts (Castagnoli et al., 2000)
with the many worlds interpretation.
As well known, this interpretation aims to restore
the principle of causality by denying the
objectivity of the (non-causal) principle
that there is always a single measurement outcome
(denoted by S in the following).
The fact we experience S , would be subjective in character
{ ascribable to a limitation of our perception.
Thus,
from the one hand, S would be subjective in character;
from the other, it yields the speed-up,
an objective consequence in a most obvious way.
To avoid this contradiction,
we are obliged to accept the objectivity of S . ...".

I do not view the ManyWorlds model as subjective
rather than objective. I think that all outcomes are
objective, and that my observation in this World of
outcome A does not preclude the existence of another World
in which I observe outcome B.

When I looked at their reference (Castagnoli et al., 2000)
for more details on why they think many worlds is wrong,
I saw that the reference was
"Castagnoli, G., Ekert, A. (2000). Private discussion."
which was to me quite unhelpful.

Of course, they are entitled to whatever opinion they want to hold,
as am I. In this World at least, they and I have differing
opinions about many worlds.

My opinions seem to me to be closer to those of David Deutsch,
such as (but not limited to) the following:
http://www.qubit.org/people/david/Articles/Frontiers.html


------------------------------------------------------------

Tony 9 Jan 2001


John Baez

unread,
Jan 10, 2001, 7:47:23 PM1/10/01
to
Hi. Sorry to be slow in continuing this thread, but I've
been reading a bit, trying to understand all these strange
algebraic structures, and especially their relation to quantum
logic.

In an attempt to lure the nonexperts into thinking about this
subject, I'll start out gently. Experts should forgive the
pedagogical tone.

In article <l03102802b67d5c6d7004@[38.30.210.62]>,
Tony Smith <tsm...@innerx.net> wrote:

>Very roughly (NOT rigorously), Jordan and Lie algebras are sort
>of complementary.
>If you consider a matrix algebra with matrices A and B,
>and their product AB,
>then
>you can decompose AB = (1/2)(AB + BA) + (1/2)(AB - BA)
>Then
>the symmetric part (1/2)(AB + BA) forms a Jordan algebra
>and
>the antisymmetric part (1/2)(AB - BA) forms a Lie algebra.

Of course this is only true in general when we work with matrices
having entries lying in some associative algebra. The weird stuff
starts when we work with the octonions.

The way I see it is this: quantum mechanics works well when
we use complex matrices. Then the self-adjoint n x n matrices
are OBSERVABLES, and they form a Jordan algebra with

A o B = (1/2)(AB + BA)

Since this Jordan algebra is "simple" and "formally real", the
projection operators act like PROPOSITIONS. In particular,
we can define "and", "or", and "not" of these propositions, and
we can ask whether one "implies" another. In technojargon, we
have an orthocomplemented lattice.

We can then talk about symmetries of our Jordan algebra.

There are two symmetry groups. One preserves all the Jordan
algebra structure. This acts on our propositions in a way
that preserves the "and", "or" and "not" operations, as well
as whether one proposition "implies" another. This group
turns out to be PU(n) = U(n)/U(1). We call it the AUTOMORPHISM
GROUP of our Jordan algebra.

Another, bigger group preserves only the "and" and "or" operations,
but not the "not". It also preserves whether one proposition
"implies" another, since P implies Q precisely when (P and Q) = P.
This group is called the STRUCTURE GROUP of our Jordan algebra.
I believe this group is PGL(n,C) = GL(n,C)/GL(1,C).

Most of these concepts can be generalized to all the Jordan algebras
considered by Jordan, Wigner and von Neumann in their famous paper,
namely the "simple, formally real" ones:

selfadjoint n x n real matrices
selfadjoint n x n complex matrices
selfadjoint n x n quaternionic matrices
selfadjoint 3 x 3 octonionic matrices

and last but not least:

spin factors

(which are explained in http://math.ucr.edu/home/baez/week163.html)

[Digression: It's particularly interesting that the structure
groups of the spin factors are just the Lorentz groups of Minkowski
*spacetimes*, while their automorphism groups are the rotation groups
of the corresponding Euclidean *spaces*. Taking the quotient

(structure group)/(automorphism group)

we get homogeneous spaces of the sort used to construct spin
foam models of quantum gravity. It is natural to wonder what
happens if we use other Jordan algebras and mimic this construction.
This is one of the secret reasons I'm interested in this stuff.]

Now, Jordan algebras are good for describing observables, but
Lie algebras are good for describing the generators of symmetries,
which are equally important in physics. Again, things work well
with complex matrices. The skew-adjoint n x n matrices are SYMMETRY
GENERATORS, and they form a Lie algebra with

[A,B] = AB - BA

This Lie algebra is called u(n), and the corresponding group
is U(n). U(n) acts as automorphisms of the Jordan algebra of
self-adjoint n x n complex matrices, so the Lie side of things
fits together nicely with the Jordan side of things.

So far, this also works if we replace the complex numbers by
reals or quaternions in our n x n matrices. The octonionic
case is not quite so simple.

But the complex numbers have a distinct advantage over the
reals and quaternions. Only in this case can we turn any
self-adjoint complex matrix into a skew-adjoint one, and
vice versa, by multiplying by i. I.e., only in this case
can we naturally identify the Jordan algebra of OBSERVABLES
with the Lie algebra of SYMMETRY GENERATORS.

Of course, it was Emmy Noether who told us this was a good idea:
observables should generate symmetries.

So the question is: in what other situations can we mimic
this "best of all possible worlds"? We don't just want a
Jordan algebra... we don't just want a Lie algebra... we want
something that's both - or at least a reasonable facsimile of both!

I'm guessing is that we can do a trick like this whenever
we have a "Hermitian Jordan triple system". I got this notion
from your remarks together with these comments by Jim Dolan:

>trying to dredge up old memories here, i think it's roughly like this:

>a "symmetric space" situation is essentially just a lie group equipped
>with a suitable involution; the "symmetric space" itself being the
>sub-space of the group skew-fixed under the involution.

>thus a "lie triple system" situation is essentially just a lie algebra
>equipped with a suitable involution; the "lie triple system" itself
>being the sub-space of the lie algebra skew-fixed under the
>involution. (a tri-linear product naturally lives here, hence "lie
>triple system".)

>a "bounded symmetric domain" (actually i'm not even sure that's the
>exact right terminology, come to think of it) is essentially a
>symmetric space equipped with a complex structure for which the
>reflection operation at each point is holomorphic. (i think that also

>means that instead of just Z/2 = O(1) acting around each point, U(1)


>acts around each point. or something like that.)

[Yes, I put in some capital letters here. - jb]

>thus a "jordan triple system" is essentially just a lie triple system
>with a slight bit of extra structure amounting to a nice complex
>structure, or something like that. that makes sense if you think
>about it, in that having i around to turn skew-adjoint-flavored things
>into self-adjoint-flavored things is just about the sort of thing
>you'd want to have in order to be able to convert a lie-flavored thing
>into a jordan-flavored thing.

So how does this work, exactly? I have my ideas, which I would like
to explain, just for my own benefit. But I think I'll stop here, 'cause
this post is long enough already.


gons...@us.ibm.com

unread,
Jan 10, 2001, 9:08:06 PM1/10/01
to
zir...@my-deja.com wrote:
I've started reading about your model on your website but there may be a

problem with having a many-worlds interpretation (MWI) of quantum
computation (QC). This paper makes, IMHO, a compelling argument that
MWIs contradict QC:
http://arxiv.org/abs/quant-ph/0005069

Tony Smith can of course speak for himself and in much more detail if needed
but I just had to remark that this is a kind of possible criticism for this model that I
would have never expected to see. The paper you cite mentions Deutsch
being too conservative in sticking to rigid causality. Tony Smith's model
definitely includes a future effects present non-causality so apparently Tony
Smith's merger of a "Bohmian landscape" with "Deutsch's Many-Worlds"
does not include Deutsch's conservatism with respect to causality. Actually
until today I was not aware that Deutsch was rigidly stuck on causality and
I've now seen this mentioned in two places today. Since I'm here let me add
some comments related to Tony Smith's response to Ralph Frost and I
about questions we had related to visualizing the geometry of these
exceptional algebra root vectors. I had already seen the web page with the
E8 coordinates Tony Smith referred us to but I hadn't recognize it as being E8
cause I couldn't see D5 in it anywhere (plus having 7 different sets through me
off). Looks to me like the best rotation of the E8 axes for visualizing its
coordinates isn't one that lets you see the D5 vertices as bivectors. The
human brain is not well suited for 8 dimensional visualizing... John

Tony Smith

unread,
Jan 11, 2001, 4:58:40 PM1/11/01
to tsm...@innerx.net
In trying to understand octonionic groups related to Jordan algebras,
I ran across the article Jordan Algebras and their Applications
by Kevin McCrimmon (Bull. A.M.S. 84 (1978) 612.

In it is the following table (table 5.1, with a reference
citation to the book Exceptional Lie Algebras,
by N. Jacobson, Dekker, New York, 1971) (due to typgraphical
limitations I have used + for direct sum and * for overbar,
and I will use x sometimes to mean tensor product):

----------------------------------------------------------------------

Type Lie Algebra Lie (or Algebraic) Group Dimension

G2 Derivations of O Automorphisms of O 14

F4 Derivations of H3(O) Automorphisms of H3(O) 52

E6 Reduced structure Reduced structure group 52+(27-1)=78
algebra Strlo(J)= Strl(J)/R Id of H3(O)
= Der J + VJo

E7 Superstructure Superstructure group 27+79+27=133
algebra Strlo(J)= of H3(O)
J + Strl(J) + J*
of H3(O)

E8 ? ? 248


----------------------------------------------------------------------


The above table may be more detailed and clearer than the table
I used in my previous message, but it still leaves unanswered
my questions about E8. However, McCrimmon goes on to write:

"... when A = O [octonions], J = J3(O), the Lie algebra

Der A + ( Ao x Jo ) + Der J

will have dimension 14 + (7x26) + 52 = 248. ...".

It seems to me that what McCrimmon calls the Superstructure algebra

J + Strl(J) + J*

may be what Tony Sudbery calls conformal Con J = Str J + J^2
in his paper Division algebras, (pseudo)conformal groups and spinors,
J. Phys. A: Math. Gen. 17 (1984) 939-955.


If you consider the bottom 4 rows of the above table
to be about the last (4th) column in a Freudenthal-Tits magic square,
then, from pp. 351-352 of the book Geometry of Lie Groups
by Boris Rosenfeld (Kluwer 1997) (here I have omitted some
overbar notations and used Q instead of H for quaternions):

Type Compact Groups
of motions in
the Elliptic Planes

F4 OS2

E6 (CxO)S2

E7 (QxO)S2

E8 (OxO)S2

Rosenfeld says: "... If we replace the elliptic planes ... by lines
in these planes and use the interpretation of these lines in
real elliptic spaces ...
[such as the theorem ...
The Hermitian elliptic lines (QxO)S1 and (OxO)S1 admit
interpretations as the manifold of 3-planes in the space S11,
respectively as the manifold of 7-planes in the space S15. ..]
we obtain ... Theorem 7.24.
The groups of motions in lines in the planes whose groups of
motions are the compact groups in the Freudenthal magic square
are locally isomorphic to the groups of motions in the
following real elliptic spaces:

[Type] [Real Elliptic Spaces]

[F4] S8

[E6] S9

[E7] S11

[E8] S15 ...".

If you look at the full magic square,
you see that the diagonal entries are related to Hopf fibrations:

[Real Elliptic Spaces]

S1 S2 S4 S8

S2 S3 S5 S9

S4 S5 S7 S11

S8 S9 S11 S15


It is interesting to me that E8, which is related to
"... the manifold of 7-planes in the space S15 ...",
is related to the geometry of the last Hopf fibration.


I realize that I have not tied up all the above
into any coherent whole,
but I am just presenting them as clues as to how
to understand the question presented by John Baez in his
"... [Digression:


It's particularly interesting that the structure groups
of the spin factors are just the Lorentz groups
of Minkowski *spacetimes*,
while their automorphism groups are the rotation groups
of the corresponding Euclidean *spaces*.
Taking the quotient (structure group)/(automorphism group)
we get homogeneous spaces of the sort used
to construct
spin foam models of quantum gravity.
It is natural to wonder what happens
if we use other Jordan algebras and mimic this construction.

This is one of the secret reasons I'm interested in this stuff.] ...".


Tony 11 Jan 2001


zir...@my-deja.com

unread,
Jan 11, 2001, 1:50:43 PM1/11/01
to
In article <l03102800b681920a01a4@[38.30.210.56]>,
Tony Smith <tsm...@innerx.net> wrote:

> I do not view the ManyWorlds model as subjective
> rather than objective. I think that all outcomes are
> objective, and that my observation in this World of
> outcome A does not preclude the existence of another World
> in which I observe outcome B.

I'm not an expert on Many Worlds models (MW) but, IIUC, MW are local and
deterministic such that local measurements split local systems
(including observers) in a subjectively random manner. Also, distant
systems only split once the causally transmitted effects of the local
interactions reach them. Are these assumptions part of your MW?
Regarding quantum computation, there may be another reason to *not*
prefer MW. The Fourier transforms in Shor's algorithms can be
significantly simplified using a semi-classical approach (rather than,
say, a MW):

http://arxiv.org/abs/quant-ph/9511007

> When I looked at their reference (Castagnoli et al., 2000)
> for more details on why they think many worlds is wrong,
> I saw that the reference was
> "Castagnoli, G., Ekert, A. (2000). Private discussion."
> which was to me quite unhelpful.

From looking at their websites I notice that A. Ekert and D. Deutsch
work at the same centre at Oxford U. and discuss these kind of issues:

http://www.qubit.org/people/artur

I didn't find out what they would say about the two papers I referred to
above (and I wouldn't want to speak for them anyways). BTW, J.A. Wheeler
was one of the coauthors (with H. Everett) of the first MW paper. But he
eventually rejected MW because "It required too much metaphysical
baggage to carry around". Some wit once put up a sign in front of a
house near Harvard U. that read "Institute for High Energy Meta-Physics"
which could also be IHEMP (notice too the "hemp" reference). The people
who lived at that house were probably many-worlders !-))) Just kidding!!


Tony Smith

unread,
Jan 12, 2001, 1:00:45 AM1/12/01
to
This is a further comment on John Baez's "... Digression:
... Taking the quotient

(structure group)/(automorphism group)

we get homogeneous spaces of the sort used
to construct

spin foam models of quantum gravity. ...".


In light of part of McCrimmon's (and Jacobson's) table:

Type Lie Algebra Lie (or Algebraic) Group Dimension

F4 Derivations of H3(O) Automorphisms of H3(O) 52

E6 Reduced structure Reduced structure group 52+(27-1)=78

algebra Strl_o(J)= Strl(J)/R Id of H3(O)
= Der J + VJ_o


it seems that, in the case of H3(O), the quotient

(structure group)/(automorphism group) = E6 / F4

Also, since

78-dim E6 is made up of 52-dim F4 plus 26-dim H3(O)_o

where 26-dim H3(O)_o is the traceless subalgebra
of the 27-dim octonionic Jordan algebra H3(O)

you should have

(structure group)/(automorphism group) = E6 / F4 = 26-dim H3(O)_o

which can then be used to construct a spin foam model.

I would like very much to be told how such a construction goes,
because
in my opinion such an H3(O)_o spin foam model
should lead to,
not just quantum gravity, but a Theory Of Everything.

The way I see it, there are three possibilities:


1 - Superstring Theory is right.
In that case, the 26-dim H3(O)_o spin foam things
will be the bits of 26-dim space through which
superstrings/membranes/whatever vibrate,
and H3(O)_o Spin Foam Theory will give
Superstring Theory a good foundation.

2 - Superstring Theory is wrong, but my D4-D5-E6 etc model is right.
In that case (this is obviously the case I like best),
the H3(O)_o spin foam things will be neighborhoods,
each containing a representation in 4-dim spacetime,
a representation in 4-dim internal symmetry space,
an 8-dim representation for fermion particles, and
an 8-dim representation for fermion antiparticles,
for 4+4+8+8 = 24 of the 26 dimensions of H3(O)_o,
the other 2 being like algebraic glue holding
them together consistently, thus giving a nice
concrete spin foam structure to my D4-D5-E6 etc model.

3 - Superstring Theory is wrong, and so is my D4-D5-E6 etc model.
This is the case that neither Superstring people nor I like,
so I will ignore it and hope that it doesn't happen (the ostrich
approach).


Therefore: In 2 out of 3 cases,

E6 / F4 = H3(O)_o Spin Foam Theory will be a big success.


It won't be the end of theoretical physics, though,
because we will still have to figure out how E7 and E8 work,
so
it will be the best of all possible worlds for theoretical physicists:

a model that not only unifies and works;
but also
a model that shows ways to do a lot more interesting work,
so that grants can keep on being granted for years and years.


Tony 12 Jan 2001


John Baez

unread,
Jan 12, 2001, 1:01:01 PM1/12/01
to
In article <l03102803b683cf3b6fa1@[38.30.210.28]>,
Tony Smith <tsm...@innerx.net> wrote:

>In trying to understand octonionic groups related to Jordan algebras,
>I ran across the article Jordan Algebras and their Applications
>by Kevin McCrimmon (Bull. A.M.S. 84 (1978) 612.

Me too. It's a great article. Each time I read it, I like it better.
I keep it by my bed these days and ponder it each night.
It's one of the main keys I'm using in my attempts to unlock the
secrets of Jordan algebras, Jordan pairs, and their relation to
quantum logic.

All the stuff you quoted from this paper has been on my mind
constantly, especially the table you quoted, which I present in
somewhat simplified form here:

Type Lie Algebra Lie Group Dimension

F4 Derivations of J = H_3(O) Automorphisms of J = H_3(O) 52
Der(J)

E6 Reduced structure Reduced structure group 52+(27-1)=78

algebra str_0(J)= Str(J)/R* of H_3(O)
= Der(J) + J_0

E7 Superstructure Superstructure group 27+79+27=133

algebra sstr_0(J)= of H_3(O)
J + str_0(J) + J*
of H_3(O)

E8 ? ? 248

[For the non-expert: some of these terms are defined below, while
others are defined in http://math.ucr.edu/home/baez/week163.html]

I want, if possible, to understand this table in terms of quantum logic.
It would be cool if these various Lie groups appeared as symmetries of
various sorts of "quantum logic".

Of course, the basic definitions should work for a fairly general class
of Jordan algebras, not just the exceptional Jordan algebra J = H_3(O).
To understand this exceptional case, we should understand the generalities.

So we have, for any formally real Jordan algebra J:

Lie Algebra Lie (or Algebraic) Group

Derivations of J Automorphism group of J
Der(J) Aut(J)

Reduced structure Reduced structure group of J
algebra str_0(J)= Str(J)/R*
= Der(J) + J_0

Superstructure Superstructure group
algebra sstr_0(J) = of J
J + Str_0(J) + J*

Hyperstructure Hyperstructure group of J???
algebra???

As I hinted in my last post, any formally real Jordan algebra J
determines a quantum logic: a bunch of propositions together with
"and", "or" and "not" operations and an "implication" relation.
The propositions are just the projections in J. "Implication" is just
the partial order that any formally real Jordan algebra has. "And"
and "or" can defined using implication. "Not" is defined like this:
if p is a projection, not(p) is the projection 1 - p.

Technically, the "and", "or" and "implication" make the projections
into a LATTICE. With the help of the "not", this becomes an
ORTHOCOMPLEMENTED LATTTICE.

I explained how the automorphism group Aut(J) acts as symmetries of
this orthocomplmented lattice.

I also explained how the structure group Str(J) acts as symmetries of the
lattice, but does not preserve the negation.

Actually, I'm lying. I didn't really explain Str(J): I didn't say
what this group was! I just defined it as the symmetries of the
lattice of propositions, and this is not quite right. The symmetries
of the lattice form a group which contains Str(J)/R*, which McCrimmon calls
the "reduced structure group". Here R* is the group of nonzero real
numbers. But what's Str(J) in the first place?!

(I've been using the notation Str(J) all over the place, but just now
finally I get around to talking about what it means! So I get a really
bad grade when it comes to clarity of exposition this time.)

McCrimmon gives a couple of definitions of Str(J) which he claims are
equivalent in favorable cases.

First he says any finite-dimensional semisimple Jordan algebra has
something analogous to a determinant, which he calls the "generic norm",
but I'll call "det". I wish I understood why this is true. I know
it's true in lots of examples, but that's all.

Anyway, he then says that in this case, Str(J) consists of all
linear maps T: J -> J which preserve the determinant up to a fixed
constant factor:

det(Tx) = c det(x) for all x in J

Str(J) acts as symmetries of the lattice of propositions, but multiplication
by constants acts trivially, so Str(J)/R* is what really matters for
the purposes of quantum logic. This is what McCrimmon calls the
"reduced structure group". There are also other symmetries of the lattice
of propositions not coming from Str(J)/R*... I need to understand this
better.

Now, I don't have a clue how the "superstructure group" relates to
the quantum logic defined by the Jordan algebra J. This groups seems
to act as nonlinear maps on J. In the case when J is a spin factor,
so we can identify it with Minkowski spacetime, I think these nonlinear
maps are precisely the conformal transformations. Because of this
nonlinearity, I don't see what the "superstructure group" has to do
with quantum logic. I can only get anywhere if I switch from Jordan
algebras to hermitian Jordan triples at this point.

And apparently nobody has a clue about this "hyperstructure group"
business. (I made up this silly name to fill in the question marks
in McCrimmon's chart, but it doesn't actually mean anything yet.)

I intend to post an article continuing this, in which I talk about
how hermitian Jordan triples get into the game. And someday I hope
to explain all this stuff much more clearly and make a This Week's
Finds out of it.

I'm gonna stop for now, though.

Tony Smith

unread,
Jan 11, 2001, 12:34:48 PM1/11/01
to

Thanks very much to John Gonsowski for explicitly stating
a difference between my point of view and that of David Deutsch.
As John Gonsowski said:
"... Tony Smith's model definitely includes

a future effects present non-causality so apparently
Tony Smith's merger of a "Bohmian landscape" with "Deutsch's Many-Worlds"
does not include Deutsch's conservatism with respect to causality.
Actually until today I was not aware
that Deutsch was rigidly stuck on causality ...".


====================================================================


Thanks also to John Baez for a clear description of
Automorphism groups and Structure groups.

John Baez says:
"... U(n) acts as automorphisms of the Jordan algebra of


self-adjoint n x n complex matrices, so the Lie side of things
fits together nicely with the Jordan side of things.

So far, this also works if we replace the complex numbers by
reals or quaternions in our n x n matrices. The octonionic

case is not quite so simple. ...".


The octonionic case intrigues and puzzles me.


If the complex situation is a uniform series

------------------------------------------------------------------
Automorphism Algebra

U(n) Jordan h_n(C) of nxn C matrices
------------------------------------------------------------------


then how do you interpret these octonionic things
where G2, F4, E6, E7, E8 are all of the exceptional Lie groups:

------------------------------------------------------------------
Automorphisms Algebra

14-dim G2 8-dim octonions O

52-dim F4 27-dim Jordan h_3(O) of 3x3 O matrices

78-dim E6 56-dim Freudenthal CxO algebra

133-dim E7 QxO algebra

248-dim E8 OxO algebra

------------------------------------------------------------------


Is the series some sort of
complexifications based on CxO,
quaternionifications based on QxO,
octonionifications based on OxO
of
the octonionic h_3(O),
with
some sort of modification of the h_3(O) product ?

Does the quaternionified QxO algebra have twice the dimension of
the complexified CxO Freudenthal algebra, i.e., is it 112-dim ?

What about the octonionified OxO algebra ?
Since, as far as I know, E8 is the only large Lie group whose
adjoint representation is its smallest representation,
then
is there some sense in which
E8 is where an algebra and its automorphism group sort of coincide ?

Is that yet another "explanation" of "why" the E-series terminates ?

(Compare such other possible "explanations" as

8-dim is the lowest dim before
D-series spinor reps get bigger than vector reps;

8 is the (real) Clifford periodicity. )

Tony 11 Jan 2001

Tony Smith

unread,
Jan 13, 2001, 12:20:44 AM1/13/01
to
John Baez wrote:

>[...] "and", "or" and "implication" make the projections into a LATTICE.

>
>With the help of the "not", this becomes an ORTHOCOMPLEMENTED LATTTICE.

>[...] the automorphism group Aut(J) acts as symmetries
>of this orthocomplemented lattice.

>[...] the structure group Str(J) acts as symmetries of the lattice,


>but does not preserve the negation.

>[...] we have, for any formally real Jordan algebra J:


>
> Lie Algebra Lie (or Algebraic) Group
>
> Derivations of J Automorphism group of J
> Der(J) Aut(J)
>
> Reduced structure Reduced structure group of J
> algebra str_0(J)= Str(J)/R*
> = Der(J) + J_0
>
> Superstructure Superstructure group
> algebra sstr_0(J) = of J

> J + str(J) + J*


>
> Hyperstructure Hyperstructure group of J???

> algebra??? ...".


Therefore,
ignoring subtleties such as Structure vs. Reduced Structure,
the corresponding lattice table would be,
using the octonion case for concreteness:

Lie Algebra Lattice Operations

F4 Derivations of J and, or, implication, not
Der(J)

E6 Reduced structure and, or, implication
algebra str_0(J)=
= Der(J) + J_0

E7 Superstructure ????
algebra sstr_0(J) =
J + str(J) + J*

E8 Hyperstructure ????
algebra???


This raises in my mind the queston of what Lattice Operations
if any are lost in going to the Superstructure algebra E7
or to the Hyperstructure algebra E8.

One possibly useful fact is that the
Superstructure algebra corresponding to E7
does not have a full Jordan product structure.

In fact (see page 223 of Gursey and Tze, On the Role of Division,
Jordan and Related Algebras in Particle Physics (World 1996),
the Superstructure algebra corresponding to E7 does
not have a binary product, but only a ternary product.

Does that mean that the binary relation of implication
is no longer present in the Superstructure algebra ?

What sort of lattice structure would
corespond to a ternary product structure ?


Tony 12 Jan 2001


John Baez

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Jan 13, 2001, 9:22:03 PM1/13/01
to
Here Tony and I continue our quest to understand the exceptional
Lie groups as symmetries of some sort "quantum logic" built using
the octonions, with an eye towards better understanding the role
these groups play in various "theories of everything", like superstring
theory, Tony's own theory, and the more ambitious sorts of spin foam
models....

In article <l03102800b6858fd11eb5@[38.30.118.98]>,
Tony Smith <tsm...@innerx.net> wrote:

>Therefore,
>ignoring subtleties such as Structure vs. Reduced Structure,
>the corresponding lattice table would be,
>using the octonion case for concreteness:
>
> Lie Algebra Lattice Operations
>
> F4 Derivations of J and, or, implication, not
> Der(J)
>
> E6 Reduced structure and, or, implication
> algebra str_0(J)=
> = Der(J) + J_0
>
> E7 Superstructure ????
> algebra sstr_0(J) =
> J + str(J) + J*
>
> E8 Hyperstructure ????
> algebra???

>This raises in my mind the question of what Lattice Operations


>if any are lost in going to the Superstructure algebra E7
>or to the Hyperstructure algebra E8.

Yes, this is exactly the kind of thing I'm wondering about!
By now, I suspect that the pattern in the above table does
not continue in any simple way down to where the question
marks start showing up. The reasons are:

1. I don't see too many options for how it could continue.

"And", "or" and "implication" are not really separate:
if you have a lattice, a transformation that preserves
one of these 3 things preserves them all. It's not hard
to see this. As I noted a while back, in a lattice we
can define "implication" in terms of "and" by saying that
P implies Q iff

(P and Q) = P

Thus any map which preserves the "and" operation on a
lattice automatically preserves the "implies" relation.

Similarly, we can define "implication" in terms of "or"
by saying that P implies Q iff

(P or Q) = Q

Thus any map which preserves the "or" operation on a
lattice automatically preserves the "implies" relation.

Conversely, in a lattice we can define "and" in terms
of "implication" by saying that (P and Q) = R iff

a. R implies P
b. R implies Q
c. R is the weakest proposition with properties 1 and 2, i.e.:
if R' implies P and R' implies Q, then R' implies R.

Similarly, in a lattice we can define "or" in terms
of implication - I leave this as an exercise for those
wishing to flex their logic muscles. So any transformation
of a lattice that preserves implication must preserve the
"and" and "or" operations.

2. I said before that the superstructure group of a Jordan
algebra acts as conformal transformations of this algebra, and
that these transformations are nonlinear, which is so terrible
that I couldn't see how any of the lattice structure of projections
in this Jordan algebra would be preserved. But it's even worse
than that: these conformal transformations are only *partially
defined* - just as with the "action" of the conformal group on
Minkowksi space, which is not really an action, since it's only
partially defined. With a little more thought one sees how terrible
the situation is: the superstructure group doesn't even even act as
transformations of the projections in our Jordan algebra, much less
preserve any sort of structure on them!!!

Now, the second point makes the first point pretty much irrelevant, but
I couldn't resist making the first point anyway, since it certainly
is interesting in *general* to consider the transformations of a
lattice of propositions that preserve more or less structure, even
if it turns out to be irrelevant for understanding this "superstructure
group" stuff.

Indeed, I find it fascinating and mysterious how the automorphism
group Aut(J) of a Jordan algebra of observables acts to preserve *all*
logical operations of the corresponding lattice of propositions,
while the structure group Str(J) preserves implication, but not
negation. What does this mean? In the case where our Jordan
algebra J is the n x n hermitian complex matrices, we have

Aut(J) = U(n)/U(1)

while

Str(J) = GL(n,C)/C*

so we see that the extra feature of *preserving negation* is
closely related to *unitarity*. In short, there's a weakened
version of the logic of quantum mechanics, where we drop negation,
which admits nonunitary operators as symmetry!

Moreover, in the case when J is a spin factor, we see that the big
group Str(J) is the group of *Lorentz transformations*, while Aut(J)
is the subgroup preserving the split into space and time, i.e.
the group of *rotations*.

Why should *preserving the inner product* (i.e. unitarity) and
*preserving negation* be related to *preserving the split into
space and time* this way? I don't know; it's curious. All I
know is that it makes sense in the context of spinors, where only
the rotations act as unitary operators, not the Lorentz transformations.
And of course spinors arise naturally as representations of the spin
factors.

Anyway, you can see that there are some funny things going on here,
possibly worth pondering. But now I don't think we'll understand the
conceptual meaning of this "superstructure group" and (still more
mysterious) "hyperstructure group" business by merely pondering
the various amounts of structure that a transformation of a lattice
can preserve. We need something else...

... and I think *part* of that something else is a better understanding
of the conformal group and its relation to the Lorentz group. Instead
of tackling the tricky example of the exceptional Jordan algebra, it's
probably easier to ponder the case when J is the 2 x 2 hermitian complex
matrices. After all, these are observables for a spin-1/2 particle, and
everyone loves the spin-1/2 particle, especially people (like me) who work
on spin foam models. In this case we have:

Aut(J) = automorphism group of J = rotation group SO(3)
Str(J) = structure group of J = Lorentz group SO(3,1)
Sstr(J) = superstructure group of J = conformal group SO(4,2)

The papers by Gunaydin that you referred me to make it clear that
this is part of a general pattern that works for lots of fancier
Jordan algebras and even Jordan triple systems, so it's worth
pondering. At the level of *geometry* it seems well understood
(by others, not yet me). But I'd feel happier if I also understood
it at the level of *quantum logic*. And here's my idea for how to
do it:

SO(3,1) is double covered by SL(2,C), so it has a projective
representation on C^2, so it becomes the symmetries of the
lattice of propositions about a spin-1/2 particle.

SO(4,2) is double covered by SU(2,2), so it has a projective
representation on C^4, so it becomes (some of) the symmetries
of the lattice of propositions about a "doubled" spin-1/2 particle.

Actually, I believe this idea goes back to von Weiszacker's "ur theory",
and it's also strongly reminiscent of Penrose's "twistor theory", but
here I'm hoping it generalizes to other Jordan algebras. Something
like this: could the superstructure group of a Jordan algebra act
naturally on some "doubled" version of the Jordan algebra? I'm
not sure what sense of "doubling" I mean here... but I've given a
tiny bit of hope by your remarks on that Freudenthal algebra which
seems somehow like a "doubled" version of the exceptional Jordan algebra.

Anyway, that's one direction I want to pursue, mainly by rereading the
literature and seeing if any of it makes more sense if I keep this idea
in mind.


Tony Smith

unread,
Jan 14, 2001, 10:21:14 PM1/14/01
to
As John Baez has pointed out,
since "... "And", "or" and "implication" are not really separate ...",
so in my efforts to try to understand

Lie Algebra Lattice Operations

F4 Derivations of J and, or, implication, not
Der(J)

E6 Reduced structure and, or, implication
algebra str_0(J)=
= Der(J) + J_0

E7 Superstructure ????
algebra sstr_0(J) =
J + str(J) + J*

E8 Hyperstructure ????
algebra???

I will give up on the lattice/logic approach and try something else.

=======================================================================

John Baez says
"... the complex numbers have a distinct advantage
... Only in this case can we turn any


self-adjoint complex matrix into a skew-adjoint one,
and vice versa, by multiplying by i.
I.e., only in this case
can we naturally identify the Jordan algebra of OBSERVABLES
with the Lie algebra of SYMMETRY GENERATORS.

... We don't just want a Jordan algebra


... we don't just want a Lie algebra

... we want something that's both ...".


In other words, for complex 3x3 matrices
(the number entries denote dimension,
and * denotes an entry that by symmetry is not independent
of other entries with numbers):

Jordan Lie
Hermitian Skew-Hermitian (Anti-Hermitian)
Self-adjoint Skew-adjoint


1 2 2 1 2 2
J3(C) = * 1 2 L3(C) = * 1 2 = 9-dim U(3)
* * 1 * * 1

Here J3(C) is a nice 9-dim Jordan algebra,
and
L3(C) is 9-dim U(3) = SU(3) x U(1).

Since U(3) reduces, by cutting out 1-dim U(1), to 8-dim SU(3)
and
since J3(C) has a nice traceless 8-dim subspace J3_0(C),
you get a hint that a useful way to rewrite the relation is


Traceless Jordan Irreducible Lie
Hermitian Skew-Hermitian (Anti-Hermitian)
Self-adjoint Skew-adjoint


1 2 2 1 2 2
J3_0(C) = * - 2 L3(C) = * - 2 = 8-dim SU(3)
* * 1 * * 1

The - marks the dimension lost due to the trace zero condition.


==================================================================


In the octonion case, the correspondence is not as simple.
for example (using the traceless version):


Jordan Lie
Hermitian Skew-Hermitian (Anti-Hermitian)
Self-adjoint Skew-adjoint


1 8 8 7 8 8
J3(O) = * - 8 L3(O) = * - 8
* * 1 * * 7

Here J3_0(O) is the 26-dim subalgebra of the 27-dim Jordan algebra J3(O),
but
L3(O) is a 38-dim thing that is NOT an exceptional Lie algebra.
To get L3(O) to be a Lie algebra,
you have to add the 14-dim automorphism group G2 of the Octonions O,
thus getting the 38+14 = 52-dim exceptional Lie algebra F4.

[Moderator's note: actually J3_0(O), the 3x3 traceless hermitian
octonionic matrices, is not a Jordan subalgebra of J3(O). It is
just a subspace. I have taken the liberty of correcting a similar
slip below. - jb]

In comparing the Complex and Octonionic cases,
you see that
the Jordan algebra can be made only of the 3x3 matrices,
but
the Lie algebra also needs the derivations of the Division Algebra
that is used in the 3x3 matrices.
Also,
you see that (as John Baez had noted) the Hermitian condition
and the skew(anti)Hermitian condition lead to different
dimensionalities on the diagonal of the 3x3 matrices,
because only in the Complex case
does the real dimension equal the imaginary dimension.


For the octonion case on which I am fixated, the book
Geometry of Lie Groups by Boris Rosenfeld (Kluwer 1997),
gives on pages 79-80 a theorem of Vinberg (I use Q instead of H,
and I use * and ' for conjugations in writing the theorem):

"... The ... Lie algebras ... F4, E6, E7, and E8
are direct sums of
the linear spaces of skew-Hermitian 3x3 matrices
whose entries are elements in the algebras O, CxO, QxO, and OxO,
respectively, with zer traces
and of
the ... Lie algebras of ... automorphisms in these algebras. ...
The condition of skew-Hermiticity is
a_ij = a_ij* for octonionic matrices
and
a_ij = a_ij*' for matrices with entries from tensor products ..."


Vinberg Lie Algebra Constructions in Rosenfeld:


7 8 8
52-dim F4 = * - 8 + 14
* * 7

8 2x8 2x8
78-dim E6 = * - 2x8 + 14
* * 8


10 4x8 4x8
133-dim E7 = * - 4x8 + 14 + 3
* * 10


14 8x8 8x8
248-dim E8 = * - 8x8 + 14 + 14
* * 14


Compare the Vinberg constructions with the
Freudenthal-Tits Lie Algebra Constructions in McCrimmon
in which the exceptional Lie algebras F4, E6, E7, and E8
also correspond to the pairs
(A,J) = (R,O), (C,O), (Q,O), and (O,O)
but according to the formula

Lie Algebra = Der A + ( A_0 x J3_0(O) ) + Der J3(O)


Freudenthal-Tits Lie Algebra Constructions in McCrimmon:

52-dim F4 = 0 + (0x26) + 52

78-dim E6 = 0 + (1x26) + 52

133-dim E7 = 3 + (3x26) + 52

248-dim E8 = 14 + (7x26) + 52


which can be written as:


7 8 8
52-dim F4 = 0 + (0x26) + * - 8 + 14
* * 7


1 8 8 7 8 8
78-dim E6 = 0 + * - 8 + * - 8 + 14
* * 1 * * 7


3 24 24 7 8 8
133-dim E7 = 3 + * - 24 + * - 8 + 14
* * 3 * * 7


7 56 56 7 8 8
248-dim E8 = 14 + * - 56 + * - 8 + 14
* * 7 * * 7


which in turn can be written so that the Vinberg relation is clear:


7 8 8
52-dim F4 = 0 + * - 8 + 14
* * 7


8 16 16
78-dim E6 = 0 + * - 16 + 14
* * 8


10 32 32
133-dim E7 = 3 + * - 32 + 14
* * 10


14 64 64
248-dim E8 = 14 + * - 64 + 14
* * 14


How do these Octonionic Lie algebras correspond to
the corresponding Jordan algebras, which I will designate
by J3(O), J3(CxO), J3(QxO), and J3(OxO) ???

[Moderator's note: only the first two of these become Jordan
algebras with the product a o b = (1/2)(ab + ba); the others
are at best "Jordan-like", which is one reason they are harder
to understand. - jb]

That is what I am working on now, and will try to write about
in a later post. For now, I will say that it seems that you
do NOT just take the non-skew3x3 Hermitian KxO matrices.

For example,
it is known that the Freudenthal algebra whose automorphism
group is E6 is 56-dimensional.
Since the 3x3 matrices of CxO are 9x2x8 = 144-dimensional,
and since the Vinberg skew part of E6 is 48-dimensional,
it seems to me that the non-skew part is 144-48 = 96-dimensional,
which is bigger than the 56-dimensional Freudenthal algebra.


From Rosenfeld's book, pages 91 and 56, it seems to me
that 56-dim Fr3(O) with automorphisms E6 (structure group in
the physics interpretation we are studying) should
be written as a 2x2 Zorn-type array:

1 8 8
1 * 1 8
* * 1


1 8 8
* 1 8 1
* * 1

---------------------------------------------------------------

John Baez asks
"... could the superstructure group [such as E7 for octonions]


of a Jordan algebra act naturally
on some "doubled" version of the Jordan algebra?
I'm not sure what sense of "doubling" I mean here..."

Here is my best guess as of now about the doubling:


If you try to "think like a Vegan" and go to 3-dim 2x2x2 array
for the 112-dim Brown "algebra-like thing" corresponding to E7,
you get a picture like this:


1 8 8
1 ------------- * 1 8
/ | * * 1
/ | / |
/ | / |
/ | / |
1 8 8 | / |
* 1 8 ------------ 1 |
* * 1 | |
| | | |
| | | |
| | | |
| 1 8 8 |
| * 1 8 ----|------- 1
| * * 1 | /
| / | /
| / | /
| / | /
| / 1 8 8
1 ------------ * 1 8
* * 1


Note that the J3(O) corners and the 1 corners correspond
to two tetrahedra within the cube.

You can go to a 4-dim 2x2x2x2 array to get a 224-dim "thing" that
would look like a tesseract and might be what is needed for E8,
but my ASCII drawing ability is too limited to try that now.


Tony 14 Jan 2001


John Baez

unread,
Jan 16, 2001, 12:05:30 PM1/16/01
to
In article <l03102800b681920a01a4@[38.30.210.56]>,
Tony Smith <tsm...@innerx.net> wrote:

>Zirkus wrote:
>"... I was hoping for a more rigorous description
>of what a Freudenthal algebra is (e.g., Is there a structure
>or representation theory of these algebras?).
>However, when I have the time,
>I will look at the references ...".

>My apologies for describing Freudenthal algebras too
>roughly and without sufficient details. As you suggest,
>the details may be described in such materials as
>the paper of Skip Garibaldi
>http://www.math.ucla.edu/~skip/
>Groups of Type E7 Over Arbitrary Fields
>http://xxx.lanl.gov/abs/math.AG/9811056

This paper has the definition of a Freudenthal triple system.

It's pretty far-out: a Freudenthal triple system is

1) a 56-dimensional vector space V
2) a nondegenerate skew-symmetric bilinear form b: V x V -> R
3) a trilinear product t: V x V x V -> V

such that if we define the quadrilinear form

q(x,y,z,w) = b(x,t(y,z,w))

then

1) q is symmetric
2) q is nonzero
3) t(t(x,x,x),x,y) = b(y,x)t(x,x,x) + q(y,x,x,x)x for all x,y in V

Clearly this is aimed at nothing other than getting ahold
of the natural structure on the smallest nontrivial irreducible
representation of E7, which happens to be 56-dimensional.

It happens that we can think of this as 2x2 matrices

a x
y b

where a,b are real and x,y lie in the exceptional Jordan algebra, h_3(O),
which (in case one has lost ones scorecard!) itself consists of 3x3
hermitian octonionic matrices.

I'm not sure how much light this sort of axiomatic description
sheds on the deep inner meaning of E7. It's nice to know that E7
is exactly the group of symmetries of the main example of the above
structure, but then one must ask: what's so great about this structure?

Personally I find the descriptions of E7 as the isometry group of
the projective plane over the quateroctonions (H tensor O), or as
the "conformal group" of the exceptional Jordan algebra, to be more
evocative - but still a bit puzzling. In any event, I'll keep gnawing
away at these issues until they make sense to me.

I have a lot more to say about Tony's other posts, but I must resist
making a full-time occupation out of this thread - much as I'd enjoy it!

John Baez

unread,
Jan 16, 2001, 12:06:49 PM1/16/01
to
Tony Smith writes:

>John Baez wrote:

>>... Taking the quotient
>>
>>(structure group)/(automorphism group)
>>
>>we get homogeneous spaces of the sort used
>>to construct
>>spin foam models of quantum gravity. ...".

>It seems that, in the case of H3(O), the quotient


>
>(structure group)/(automorphism group) = E6 / F4

Yes, that's certainly true, with a certain real form of E6 and
the compact real form of F4. (One must be careful about real
forms here!)

>Also, since
>
>78-dim E6 is made up of 52-dim F4 plus 26-dim H3_0(O)
>
>where 26-dim H3_0(O) is the traceless subalgebra


>of the 27-dim octonionic Jordan algebra H3(O)
>
>you should have
>

>(structure group)/(automorphism group) = E6 / F4 = 26-dim H3_0(O)


>
>which can then be used to construct a spin foam model.

Well, you're mixing up the Lie algebras and the Lie groups here.
The quotient of Lie algebras e6/f4 is a vector space that can be
naturally identified with H3_0(O). That's really cool!

But the quotient of Lie groups E6/F4 is what matters for the spin foam
models, and this is a bit "curvier" - it has a natural metric that's
not flat.

They are closely related, however: e6/f4 can be viewed as a tangent
space of E6/F4.

A baby example of the same phenomenon is this:

sl(2,C)/su(2) = R^3, 3d flat space
SL(2,C)/SU(2) = H^3, 3d hyperbolic space.

This is what we get if we replace the Jordan algebra H3(O) by
the smaller Jordan algebra H2(C).

>I would like very much to be told how such a construction goes,
>because

>in my opinion such an H3_0(O) spin foam model


>should lead to,
>not just quantum gravity, but a Theory Of Everything.

Shhh! That's supposed to be secret. :-)

Yes, of course something like this is my goal, but I'm not eager
to count my chickens before they are hatched, nor take my ideas to
market while they're still half-baked.

Anyway, to see a theory of quantum gravity in 3+1 dimensions
coming from a spin foam model based on the homogeneous space
SL(2,C)/SU(2), read these:

John Barrett and Louis Crane, A Lorentzian signature model for quantum
general relativity, preprint available as gr-qc/9904025.

Alejandro Perez and Carlo Rovelli, Spin foam model for Lorentzian general
relativity, available as gr-qc/0009021.

and for background, this stuff in 4 dimensions (Riemannian rather
than Lorentzian quantum gravity):

Roberto De Pietri, Laurent Freidel, Kirill Krasnov, and Carlo Rovelli,
Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeneous
space, preprint available as hep-th/9907154.

>The way I see it, there are three possibilities:
>
>1 - Superstring Theory is right.
>

>2 - Superstring Theory is wrong, but my D4-D5-E6 etc model is right.
>

>3 - Superstring Theory is wrong, and so is my D4-D5-E6 etc model.
>

>In 2 out of 3 cases,
>

>E6 / F4 = H3_0(O) Spin Foam Theory will be a big success.

At those odds, I guess I'd better get cracking. :-)

Tony Smith

unread,
Jan 15, 2001, 8:56:06 PM1/15/01
to tsm...@innerx.net
First, I did make an effort at drawing a tesseract representation
of a Jordan-like thing that might corresponding to E8:

1 8 8
* 1 8 ----------------------------------------- 1
* * 1 //
/ | \ // |
/ | \ // |
/ | \ // |
/ | \ // |
/ | \ // |
/ | \ // |
/ | \ // |
/ | \ // |
/ | \ 1 8 8 |
1 ------------------------------------------ * 1 8 |
\ | \ * * 1 |
| \ | \ / /| |
| \ | \ / / | |
| \ | \ /1 8 8 | |
| \ | 1 ------------/ * 1 8 | |
| \ | / / * * 1 | |


| \ | / | / / | | |
| \ | / | / / | | |

| \| / | / / | | |
| \ 1 8 8 | / / | | |


| \* 1 8 ------------ 1 | | |
| | * * 1 | | | |
| | | | | | | |
| | | | | | | |
| | | | | | | |
| | | 1 8 8 | | |
| | | * 1 8 ----|------- 1 | |
| | | * * 1 | / \ | |
| | | / / | / \| |
| | | / / | / | |
| | | / / | / |\ |

| | / / 1 8 8 / | \ |
| | 1 ------------ * 1 8 | \ |
| | / / * * 1 | \ |


| |/ / \ | \ |
| / / \ | \ |
| / / \ | \1 8 8

| / 1 ----------------------\---------|-------- * 1 8
| / / \ | * * 1


| / / \ | /
| / / \ | /

| / / \ | /
| / / \ | /

| / / \ | /


| / / \ | /
| / / \ | /

| / / \| /
1 8 8 /

* 1 8 ----------------------------------------- 1
* * 1


It has 8x1 + 8x(1+1+1+8+8+8) = 224 dimensions,
which is twice the 112 dimensions of the Br3(O) structure of E7
and is 4 times the 56 dimensions of Fr3(O) of E6.

==================================================================

Now for more text-oriented matters:

As John Baez says:
"... the complex numbers have a distinct advantage ..."
and here I will take that remark out of context to mention a point,
made by Stephen Adler in his (Oxford 1995) book
Quaternionic Quantum Mechanics and Quantum Fields (pp.10-11):
"...
standard quantum mechanics [is formulated] in a complex Hilbert space
... The special Jordan algebras are equivalent ... to the
Dirac formulation in ... real, complex, or quaternionic Hilbert space
...
the ... exceptional Jordan algebra ... of the 27-dimensional
non-associative algebra of 3x3 octonionic Hermitian matrices
... corresponding to a quantum mechanical system over
a two- (and no higher) dimensional projective geometry that
cannot be given a Hilbert Space formulation ...
...
Zel'manov (1983) ... proved that in the infinite-dimensional case
one finds no new simple exceptional Jordan algebras ...".

Very roughly and non-rigorously,
Adler is stating the conventional wisdom that:

You can't do serious physics with Octonions,
because
Octonion non-associativity prevents you from building
a nice big Hilbert space, with high-order tensor products.

However,
although my model has a lot of octonionic structure,
it does not fail due to those conventional objections.
Here is a rough outline (ignoring things such as signature)
of how my model gets high-order tensor products:

My model is based on the D4 Lie algebra,
which is the bivector Lie algebra of the Cl(8) Clifford algebra,
which has graded structure:

1 8 28 56 70 56 28 8 1

and total dimension 2^2 = 256 = 16x16 = (8+8)(8+8)
with 8-dimensional +half-spinors, 8-dimensional -half-spinors,
8-dimensional vectors, and 28-dim bivector adjoint representation.

Therefore Cl(8) contains all 4 fundamental representations
of the D4 Lie algebra, with Dynkin diagram

8
|
28
/ \
8 8

and I can (and do) embed all my D4 structures into Cl(8),
which is nice and real and associative.
Even further, for any value of N, no matter how large,
Clifford periodicity
lets me decompose Cl(8N) as the tensor product of N copies of Cl(8):

Cl(8N ) = Cl(8) x ...(N times tensor product)... x Cl(8)

Therefore, with my model,
I can build nice big spaces for real physics using Cl(8N).

=================================================================


That having been said,
I will put the quote of John Baez back into context:
"... the complex numbers have a distinct advantage ...
... Only in this case can we turn any


self-adjoint complex matrix into a skew-adjoint one,
and vice versa, by multiplying by i.

... We don't just want a Jordan algebra


... we don't just want a Lie algebra

... we want something that's both ...".


With Octonion/D4/Cl(8) structures, how close can we come to that goal ?


Consider the Lie algebra E8,
and just look at its dimensional structural skeleton,
ignoring Lie-to-Jordan changes of product rules (just as you ignore
the effect of multiplication by i when you do a Lie-to-Jordan conversion
of complex matrices).

-------------------------------------------------------------------

If the Jordan-like algebra corresponding to the Lie algebra E8,
which E8 related Jordan-like algebra is denoted here by JE8,
is represented by a 16-vertex 4-dimensional hypercube tesseract,
with 8 vertices being 1-dimensional
and 8 vertices being 27-dimensional J3(O),
for a total of 8 + 8x27 = 8 + 216 = 224 dimensions,
then
the 248-dim Lie algebra E8 might be seen to be made up of
the 224-dim JE8 plus
the 24-dim Chevalley algebra Chev3(O) of 3x3 Hermitian Octonion matrices
with zero diagonal.

That is,

248-dim E8 = 224-dim JE8 + 24-dim Chev3(O) =


8 8x8 8x8 * 8 8
= * 8 8x8 + 8 + * * 8
* * 8 * * *

This can be rearranged to see the relation to
the Vinberg construction of E8 given in Rosenfeld's book:


8 8x8 8x8
248-dim E8 = * 8 8x8 + 8 + 24
* * 8


14 8x8 8x8


= * - 8x8 + 14 + 14
* * 14

----------------------------------------------------------------

In other words,
248-dim E8 does not have the same structural skeleton as 224-dim JE8,
but must have an additional 24-dim Chev3(O).

For the other exceptional cases:

133-dim E7, the automorphism group of a 112-dim thing Br3(O),
does not have the same structural skeleton as Br3(O),
but needs to have an additional 21-dim structure.

78-dim E6, the automorphism group of the 56-dim Fr3(O),
does not have the same structural skeleton as Fr3(O),
but needs to have an additional 22-dim structure.


52-dim F4, the automorphism group of 27-dim J3(O),
does not have the same structural skeleton as J3(O),
but
has the structural skeleton of 2 copies of 26-dim traceless J3(O)o.

The F4 doubling of J3(O)o may be related to the fact
that the 48-vertex F4 root vector diagram is made up
of two copies of the 4-dim 24-cell, which are dual to each other.


If F4 looks like a doubling of the traceless J3(O)o
and also has a root vector diagram that doubles the 24-cell,
it might be useful, in order to try to get
"... something that's both ...
a Jordan algebra ...[and]... a Lie algebra ...",
to look at a Lie algebra whose root vector diagram
is only one copy of a 24-cell, which Lie algebra is 28-dim D4.

In fact, (including the finite symmetry group S3), 28-dim D4,
which is represented by antisymmetric 8x8 real matrices,
is
the automorphism group of the 24-dim Chevalley algebra Chev3(O):


8
| 0 8 8
28 = automorphisms of * 0 8
/ \ * * 0
8 8


If you take the three 8s of Chev3(O) as corresponding to
the three 8-dim fundamental representations of D4,
that is the vector and two half-spinor representations,
you see that,
not only is D4 (up to finite S3 outer automorphism)
the automorphism group of Chev3(O),
but also
the three 8s of Chev3(O) describe D4.

For example, you can build the 28-dim adjoint of D4 by
taking the wedge product of two of the 8s, as 8/\8 = 28,
which is the same procedure that J. F. Adams used
to construct representations of E8 in his paper
The Fundamental Representations of E8,
Contemporary Mathematics 37 (1985) 1-10,
reprinted in vol. 2 of The Selected Works of J. Frank Adams,
ed. by. J. P. May and C. B. Thomas (Cambridge 1992).

Therefore, in my opinion:


D4 and Chev3(O) are Octonionic Lie and Jordan-like structures
that are substantially equivalent to each other.


--------------------------------------------------------------------
My view of D4 as exceptional is reinforced by the table
from page 540 of The Book of Involutions,


by M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol
(with a preface by J. Tits),
American Mathematical Society Colloquium Publications,
vol. 44, American Mathematical Society,

Providence, RI, 1998:

dim A F FxFxF H3(F,a) H3(K,a) H3(Q,a) H3(C,a)
1 0 0 A1 A2 C3 F4
2 0 U A2 A2xA2 A5 E6
4 A1 A1xA1xA1 C3 A5 D6 E7
8 G2 D4 F4 E6 E7 E8

The authors say:
"... Here ... Q for Quaternion algebra and C for a Cayley algebra;
U is a 2-dimensional abelian Lie algebra.

The fact that D4 appears in the last row is one more argument for
considering D4 as exceptional. ...".
-------------------------------------------------------------------

If you start with D4,
you can build a sequence of larger groups with fibrations
that have either real or complex structure:

Real Complex

D4 D4

B4 B4/D4 = OP1 = S8 (real) D5 D5/D4xU(1) = (CxO)P1

F4 F4/B4 = OP2 (real) E6 E6/D5xU(1) = (CxO)P2

E6 E6/F4 = J3(O)o =
= complex parts for OP1 and OP2

E7 E7/E6xU(1) E7 E7/E6xU(1) (54-dim)

E8 E8/E7xSU(2) E8 E8/E7xSU(2) (112-dim) = Br3(O)


Note that after E6, the two series merge.

In the complex series, the D5 and E6 fibrations give you
Hermitian symmetric spaces that correspond to bounded complex
homogeneous domains, with nice Shilov boundaries
and harmonic structure from Bergman kernels, Poisson kernels,

In the real series, at the F4 level everything is real,
with no nice Shilov boundaries and stuff,
but the J3(O)o structure added at the E6 level
combines with the D4-B4-D4 chain to effectively
convert it to the complex structure of the D4-D5-E6 chain.


E7 contains, for both chains, the Superstructure,
at which level the lattice interpretation becomes difficult,
(I interpret E7 as describing the overall structure
of the Worlds of the ManyWorlds of quantum theory),

and

E8 contains, for both chains, the Hyperstructure.

Note that:

E8/E7xSU(2) is 112-dim and I conjecture that
it has the structure of Br3(O),
the Jordan-like algebra of which E7 is the automorphism group,
so that:
248-dim E8 contains 112-dim Br3(O)

Since E7 is a local symmetry group of E8/E7xSU(2),
it is also true that
248-dim E8 contains 133-dim E7.


Therefore, E8 Hyperstructure should be seen as
an amalgam of both the Lie and Jordan-like algebras E7 and Br3(O)
also containing as glue the quaternionic SU(2):

E8 = E7 + Br3(O) + SU(2)
248 133 112 3


That leaves only E7 Superstructure
(and its Jordan-like algebra Br3(O)
to be given physical interpretation
(since the E7 and Br3(O) interpretation
will automatically give the E8 interpretation).

E6 is interpreted in my model as representing
spacetime plus internal symmetry space,
gauge bosons,
fermion particles, and
fermion antiparticles,
so
what remains of E7 to be interpreted is
E7/E6xU(1) = 54-real-dim = 27-complex-dim = Hermitian symmetric
space corresponding to the bounded symmetric domain of type EVII,
which is "... represented ...
by the 3x3 Hermitian matrices over the Cayley numbers ..."
according to
Differential Geometry, Lie Groups, and Symmetric Spaces,
by Sigurdur Helgason (Academic 1978), page 527.


---------------------------------------------------------------


Since I see the Hyperstructure of E8 as containing
both E7 Lie and Br3(O) Jordan-like structure,
and
since I use Clifford algebras like Cl(8), which contains D4,
to get large tensor products,
I should say how to fit E8 into Clifford structure.

The most straighforward way is to use

248-dim E8 = 120-dim bivector adjoint of D8 + 128-dim D8 half-spinor

and so embed E8 in the Clifford algebra Cl(16), with graded structure

1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1

and total dimension 2^16 = 65,536 = (128+128)(128+128)

Since Cl(16) = Cl(2x8) = Cl(8)xCl(8)
( For example, 120 = 1x28 + 8x8 + 28x1 and 128 = 8x8 + 8x8. )

E8 can be represented in a tensor product of Cl(8) algebras,
which is consistent with the structures of my physics model.

--------------------------------------------------------------

Tony 15 Jan 2001

gons...@us.ibm.com

unread,
Jan 18, 2001, 3:52:29 PM1/18/01
to

Tony Smith wrote:
If you start with D4,
you can build a sequence of larger groups with fibrations
that have either real or complex structure:
Real Complex
D4 D4
B4 B4/D4 = OP1 = S8 (real) D5 D5/D4xU(1) = (CxO)P1
F4 F4/B4 = OP2 (real) E6 E6/D5xU(1) = (CxO)P2

E6 E6/F4 = J3(O)o = complex parts for OP1 and OP2


E7 E7/E6xU(1) E7 E7/E6xU(1) (54-dim)
E8 E8/E7xSU(2) E8 E8/E7xSU(2) (112-dim) = Br3(O)

...E7 contains, for both chains, the Superstructure
...E8 contains, for both chains, the Hyperstructure


E8/E7xSU(2) is 112-dim and I conjecture that

i t has the structure of Br3(O),


the Jordan-like algebra of which E7 is the automorphism group,
so that: 248-dim E8 contains 112-dim Br3(O)

...248-dim E8 does not have the same structural skeleton as 224-dim JE8,


but must have an additional 24-dim Chev3(O).

...D4 and Chev3(O) are Octonionic Lie and Jordan-like structures


that are substantially equivalent to each other.


John Baez wrote:
Aut(J) = automorphism group of J = rotation group SO(3)
Str(J) = structure group of J = Lorentz group SO(3,1)
Sstr(J) = superstructure group of J = conformal group SO(4,2)


One may need that E8 hyperstructure to link the threads for
the posts on this subject (grin)... OK let me make sure I
understand this well enough to talk to those who may overall
know more than me but have read less on this particular
subject... I think the above might explain why at various times
during this discussion I have regular 26-dim bosonic strings
associated with either F4,E6 or E7... From a Jordan
automorphism view the strings are an F4 thing, from an LIE
containment group view the strings are an E6 thing. If
someone complexified 26-dim bosonic string theory for the
complex chain then from the Jordan (Freudenthal)
automorphism view they are an E6 thing and from an LIE
containment group view they are an E7 thing. Does
regular non-complexified bosonic string theory only give
you rotation degrees of freedom as the above suggests
to me? Would you get the full Lorentz degrees of freedom
by going to a complexified version of 26-dim bosonic
string theory? And would you need some "quaternified"
bosonic string theory based on that 112-dim Br3(O) to
get access to the full conformal group degrees of
freedom? And does E8 not having a higher containment
LIE group explain why John Baez does not mention
degrees of freedom to go with a hyperstructure group?
At first I was thinking maybe going to hyperstructure
might add color/electroweak degrees of freedom for
bosonic string theory through SO(8) aka D4 (since D4
does this for Tony Smith's model) but Tony Smith actually
mentions some other interesting relationship between D4
and E8... Also even F4 has D4 as a subspace so why
wouldn't it get the full D4 degrees of freedom? Something
to do with LIE group subspaces vs Jordanish algebra
based bosonic string theory?... John


Tony Smith

unread,
Jan 17, 2001, 2:37:09 PM1/17/01
to
[Moderator's note: in what follows, "you" = "John Baez". - jb]

--------------------------------------------------------------------

You say "... I must resist making a full-time occupation
out of this thread - much as I'd enjoy it! ...".

Me, too, so I will try to make this comment my last for a while:

--------------------------------------------------------------------


You say, about H3(O) and generalized spin foams:
"... That's supposed to be secret. :-)
... I'm not eager to count my chickens before they are hatched ...".


My suggestion of spin foam

(structure group)/(automorphism group) = E6 / F4

was just an example, and there is in my opinion a better way.

As you say, you can look at E6 / F4
either at the Lie algebra level or at the Lie group level.

You say

"... the quotient of Lie groups E6/F4 is
what matters for the spin foam models ...".

That does exist and is a 26-dim rank-2 symmetric space of type EIV,
with a compact realization as the set of OP2s in (CxO)P2
and a
noncompact realization as the set of hyperbolic OP2s in hyperbolic (CxO)P2.

However,
if you try to buld a spin foam out of that,
I suspect that it is not as easy as the Spin(4)/Spin(3) = SU(2) = S3
case of making a foam of 3-spheres.

It seems to me that what you want for a single bubble/component of foam
would be something that has two characteristics:

1 - the 24-dim 3-octonion structure of 26-dim H3(O)o

2 - associative structure so that you can
put a lot of bubbles together nicely


My candidate for that is the Clifford algebra Cl(8),
with graded structure

1 8 28 56 70 56 28 8 1

and total dimensionality 2^8 = 256 = 16x16 = (8x8)x(8x8)

Cl(8) has

1 - 3 octonions (vector 8, and two half-spinor 8s)

2 - associative product and periodicity factorization
Cl(8N) = Cl(8) x ...(N times tensor product)... x Cl(8)

Actually,
I am not the only one using that factorization as the basis
of a fundamental physics model.

My friend and teacher David Finkelstein (who first taught me
about details of Clifford algebras back in the early 1980s)
is also working on a model in which a bunch (maybe it could
be called a foam??) of Cl(8)s is a starting point out of which
everything we see (spacetime, particles, etc) condenses.
David's web page is at URL
http://www.physics.gatech.edu/people/faculty/dfinkelstein.html
where he says, among other things:
"... The Spinorial Chessboard shows how the dynamics,
a large squad of chronons, can spontaneously break down
into a Maxwellian assembly of squads of 8 chronons each ...".
Although David's model differs from mine in some ways,
his "squad of 8 chronons" is the set of the 8 generators of Cl(8).


In both his model and my model, the "foam" is not of spacetime,
but a sort of pregeometric foam, and spacetime is derived
as a sort of "condensation".


In this picture, you are using algebra structure, not group structure,
so instead of starting with
a foam of group objects like 3-spheres
you start with
a bunch of algebra-tangents of group objects,
which you can, AFTER you put the algebra-tangents together,
THEN exponentiate up to make a big group foamy thing.


Tony 17 Jan 2001

John Baez

unread,
Jan 23, 2001, 5:31:39 PM1/23/01
to
In article <l03102800b68b97c45d71@[38.30.162.140]>,
Tony Smith <tsm...@innerx.net> wrote:

>John Baez wrote:

>> I must resist making a full-time occupation
>> out of this thread - much as I'd enjoy it!

>Me, too, so I will try to make this comment my last for a while:

I don't want to stop - just slow down a bit!

First, a clarification about something I said a while back...

I mentioned how the projections in any formally real Jordan algebra
form a lattice, and I considered two symmetry groups of this lattice:
the group which preserved the operations "not", "and", "or" and
the relation "implies", and the bigger group which only preserves
"and", "or" and "implies". I mentioned that there aren't any bigger
groups that preserve less structure, because any transformation that
preserves one of "and", "or" and "implies" automatically preserves
the other two. My proof of this consisted of showing how you could
define any of these three things in terms of any other.

Jim Dolan then emailed to emphasize that this argument only works
for *invertible* transformations. Of course those are all I was
interested in, since I was interested in getting symmetry *groups*,
not symmetry monoids. Nonetheless it's worth remembering that
the argument

"if you can define A in terms of B, any transformation that preserves
B must preserve A"

only works for invertible transformations. To take a simple example,
the lattice of all subsets of a topological space can be mapped into
the lattice of *closed* subsets by means of the "closure" operation,
and this mapping preserves "implication" (i.e. containment) but not
"and" (i.e. intersection) - since the intersection of closures can be
bigger than the closure of the intersection.

I know this is only dimly related to what we're talking about now,
but as George Washington said, "I cannot tell a lie" - so I figured
I should clear it up. In fact, the same issue came up much earlier
here on sci.physics.research, in the thread on "properties, structures
and stuff". This thread got us into some serious category theory,
but one basic thing Jim Dolan taught us was that the nice relation
between "definability" and "property preservation" works only when
we consider invertible transformations. This gets important in physics
when we move from symmetry *groups* or *groupoids* to symmetry *monoids*
or *categories*.

Second, and more relevant to what we're up to now... I think there
really is some relation between the "conformal group" of a Jordan
algebra, as defined by Gunaydin, and "doubling" the vector space on
which the Jordan algebra most naturally acts.

Consider for example a spin factor: i.e., the Jordan algebra J(V)
generated by an inner product space V with relations

v o v = <v,v>

Representations of this Jordan algebra are the same as representations
of the Clifford algebra generated by V with relations

v^2 = <v,v>

It follows that real representations of J(V) are all direct sums of
irreducible "spinor representations". There is either one or two of
these representations (depending on the dimension of V modulo 8).
Physically, V represents EUCLIDEAN SPACE, and these spinor representations
are representations of ROTATION GROUP (or really its double cover). The
rotation group is just the AUTOMORHPHISM GROUP of J(V), or equivalently,
the group preserving "not" and "implies" in the lattice of projections
in J(V).

J(V) itself is isomorphic to V + R, and physically it represents
MINKOWSKI SPACETIME. There is a bigger symmetry group acting
on the lattice of projections in J(V), which merely preserves "implies".
This turns out to be the LORENTZ GROUP of the spacetime in question.
In Jordan algebra language, it's the REDUCED STRUCTURE GROUP of J(V).
The spinor representations of this Lorentz group (or really its
double cover) are built from one or two copies of the spinor
representations for the Euclidean group (depending on the dimension
of V modulo 8).

So far I've just been setting the stage, reviewing stuff we've already
talked about. Next, let's form the CONFORMAL COMPACTIFICATION OF
MINKOWSKI SPACETIME, which we can do by adding certain points at infinity
to J(V). This has the CONFORMAL GROUP as a symmetry group. In terms
of Jordan algebras, this group is the SUPERSTRUCTURE GROUP of J(V).
The spinor representations of this conformal group (or really its
double cover) are always built from two copies of the spinor representations
for the Lorentz group.

The capitalized words do not mean I'm yelling my head off. They're
meant to emphasize a certain possible pattern.

It's tempting to continue this pattern and guess the "hyperstructure group"
of J(V). One obvious guess is this: if the Lorentz group is SO(n,1)
and the conformal group is SO(n+1,2), the next group on our list might
be SO(n+2,3). This would produce another doubling in the size of the
relevant spinor representation.

Here's another reason this "doubling" business is interesting: it's
reminiscent of the Cayley-Dickson construction. Just as one gets
R, C, H and O by repeated applications of the Cayley-Dickson construction
for algebras, maybe there's a way to get F4, E6, E7 and E8 by repeated
application of some Cayley-Dickson construction for Lie algebras!

I was encouraged to make this speculation by seeing the title of this
paper:

B. N. Allison and J. R. Faulkner, A Cayley-Dickson process for a class
of structurable algebras, Trans. Amer. Math. Soc. 283 (1984), 186-210

which is referred to in Skip Garibaldi's paper on groups of type E7.

I was also encouraged by your pictures of a square, cube, and tesseract
which might be related to E6, E7 and E8.

I don't understand this "doubling" business, but I think it's something
real.

Finally:

>"... the quotient of Lie groups E6/F4 is
> what matters for the spin foam models ...".
>
>That does exist and is a 26-dim rank-2 symmetric space of type EIV,
>with a compact realization as the set of OP2s in (CxO)P2
>and a
>noncompact realization as the set of hyperbolic OP2s in hyperbolic (CxO)P2.

Where did you learn this? Besse's book? Rosenfeld's book? Or...?

>However,
>if you try to buld a spin foam out of that,
>I suspect that it is not as easy as the Spin(4)/Spin(3) = SU(2) = S3
>case of making a foam of 3-spheres.

Of course it won't be as simple. But you know, it's not quite that the
"foam" is "made of 3-spheres". People do quantum field theory on a
3-sphere and get a spin foam model of (Riemannian) 4d quantum gravity.
The Lagrangian for this quantum field theory guarantees that Feynman
diagrams can be interpreted as 4-simplices stuck together along their
tetrahedral faces. The relevance of the 3-sphere is that L^2(S^3) can
be decomposed as a direct sum of "simple" representations of Spin(4),
which are the representations which correspond to bivectors - the right
thing for describing the quantum geometry of a triangle.

One can and should see how this generalizes to a wide class of homogeneous
spaces, and it should be especially fun for "exceptional" spaces like E6/F4.

Or maybe even bigger ones involving E7 and E8!


Tony Smith

unread,
Jan 23, 2001, 11:56:24 PM1/23/01
to tsm...@innerx.net
John Baez wrote:
"... I don't want to stop - just slow down a bit! ...".

That is fine with me, so I will just discuss some simple/easy comments
(references etc) now and try to make some deeper comments later
(which means, since tomorrow (24 Jan 2001) is the Chinese New Year,
I will no longer be writing in the Year of the Dragon - I will kind of
miss that). Anyhow:

John Baez said, about the structure of E6/F4 as


a 26-dim rank-2 symmetric space of type EIV,
with a compact realization as the set of OP2s in (CxO)P2
and a

noncompact realization as the set of hyperbolic OP2s in hyperbolic (CxO)P2:

"... Where did you learn this? Besse's book? Rosenfeld's book? ...".

I first saw the realizations in Besse's book,
where there is a very nice set of tables on pages 311-316
(For anybody who does not want to carry around the whole book,
copies of those pages are a nice convenient summary of a lot
of information.)
The realizations are also in Rosenfeld's book,
but I don't think that it has such a convenient table,
even though it does make up for it by having more detailed
discussions.

As to symmetric spaces of Type EIV, Helgason's book


Differential Geometry, Lie Groups, and Symmetric Spaces

(Academic 1978) describes them briefly (see the table on
page 518 and further materials and exercises).

It may be important to note that
some of the relevant symmetric spaces
such as BDI (p=2) for D5 / D4xU(1)
and EIII for E6 / D5xU(1)
and EVII for E7 / E6x U(1)
are Hermitian and therefore have related bounded complex
homogeneous domains
(see such references as:
Helgason;
The Encyclopedic Dictionary of Mathematics (MIT 1977 - translated
from Japanese) entries 401 G, H, I on Symmetric Riemannian Spaces;
and
L. K. Hua (in my opinion a heroic survivor of the Chinese
Cultural Revolution - see Math. Intelligencer 16, 3 (1994) 36-46)
Harmonic Analysis of Functions of Several Complex Variables
in the Classical Domains (AMS 1979) - not so much material
on exceptional cases, but very good on classical cases).

and that the harmonic structures of such complex domains,
such as Poisson kernels, Bergman kernels, etc.,
are useful (probably necessary) for building models that
do real physics.

However,
spaces like E6 / F4 (type EIV) and E8 / E7xSU(2) (type EIX)
have no Kahler structure, and no nicely directly related
bounded complex homogeneous domains.

Although the space E6 / F4 is not complex (it has no Kahler
structure), it has octonionic/complexified octonionic strucure,
and may be related to some Cayley-Dickson-doubled version of
the nice harmonic complex structures of the Hermitian spaces.

Although Rosenfeld's book has a lot of details about the geometry
of the non-Kahler cases of E6 / F4 (type EIV)
and E8 / E7xSU(2) (type EIX) - set of (QxO)P2 in (OxO)P2
it does not (as far as I saw) have a lot about harmonic structure
in those non-Hermitian cases. I have had a hard time finding
much in the way of book references about representations
of those exceptional non-Hermitian things,
but one reference is
D-Modules and Spherical Representations, by Frederic V. Bien
(Mathematical Notes, Princeton University Press 1990,
section IV.7 on pages 118-124),
which does some stuff about the cases E6 / F4 and E8 / E7xSU(2).

None of this stuff is as easy for me to understand as I wish it were,
and I don't understand it nearly as well as I should,
and I sort of hesitate to bring it up (because I don't want possibly
fruitful discussion of possibly interesting structures for physics
models to get bogged down in details of quaternionic/octonionic
harmonic functions and kernels etc),
but
I do think that such harmonic stuff is not only fun and interesting
(even if, or maybe because, it does not seem to be fully understood),
but it also may be physically important and useful.

I will try to comment on more things later,
after the New Year (of the Snake) begins.


Tony 23 Jan 2001


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