## Tuesday, December 6, 2011

### Theory Update 130

Let's get back to the basic reason that String Theory is obviously wrong. As neatly summarised in Furey's paper, we can list all particle states using the normed division algebras from triality: the reals, complex numbers, quaternions and octonions. Let us call this large collection of numbers $RCHO$. Note that dimensionally, $RCHO$ resembles $O \cdot O$, octonions with octonion coefficients. M theorists like kneemo have been studying Jordan algebras over such large collections.

Consider $J_{3}(O \cdot O)$, the algebra of $3 \times 3$ matrices over $O \cdot O$. Since $O \cdot O$ has (real) dimension $64$, $J_{3}(O \cdot O)$ has dimension $216$, from three copies of $64$ off the diagonal and $3 \times 8$ along the diagonal. Note that $216 = 2^3 3^3$, the information dimension for three qubits and three qutrits. The simple prime factorization allows us to parse matrix components in many ways, for both $2 \times 2$ and $3 \times 3$ matrices. For instance, we could use a $3 \times 3$ pseudo-algebra built with off diagonal copies of $J_{3}(C \cdot O)$, the $54$ dimensional bioctonion algebra of the doubled tetractys. The diagonal would then require objects of dimension $18$, twice the dimension for the two qutrit hexagon path diagram.

Compare this with particles. There are $54$ left and right handed leptons and quarks, including color and generation. Then $216 = 4 \times 54$, allowing for fermions, antifermions and mirror versions. This full list is easily expressed in terms of ribbon diagrams with three strands. Fourier supersymmetry creates all bosonic states. No fairies or zombies appear anywhere.

1. Somewhat amusingly, even string theory now provides grounds for being deleted at the PF forum - if you use it to talk about FTL neutrinos...

I realized this morning that the tetractys is a transect of the three-dimensional Koide space of sqrt-mass values, perpendicular to Foot's (1,1,1) vector, with the vertices on each edge marking where the cone crosses over into negative values for one of the coordinates. The triangular surface in your drawing of path space, the other day, must have given it away.

2. Yes, the qudit paths form bases for our NC and NA algebras.

But many thanks for explicitly noting the Foot vector connection and the sign crossing thing. I have not been thinking about it the same way as Alejandro et al, and of course the meaning of it all is still unclear.

3. So that identifies the Foot point $(1,1,1)$ with the central point on the tetractys in $X$, $Y$, $Z$ space (note, this is not path space).

4. Hmmm. Doesn't that identify the scale factor $\sqrt{\mu}$ (or $\sqrt{M}$ in Alejandro's notation) with distance along a cone vector? Then the factor of $3$ in $\mu_{q} = 3 \mu_{l}$ would come from a $\sqrt{3}$, which is the diagonal on a cube.

So what does it mean to fit a cube with sides $1/ \sqrt{3}$ (which has diagonal equal to $1$) inside this larger quark cube?

5. Oh, probably we are looking at a cube rotated by $\pi /4$ angles, which is like the $\pi /4$ in quark lepton mixing complementarity. That is, we line one diagonal up along an edge on another cube, so that the ratio of diagonals is $\sqrt{3}$.

6. This associates the factor of $3$ directly with directions in $XYZ$ space, which we should have, because color and generation are closely related.

7. So a new vertex might look like $(X \pm Y)/2$. This defines a tiling of space by shrunken, rotated cubes.

8. Well, I'm glad you're making something of this observation. :-) I thought this must have been how you constructed the tetractys in the first place!

9. Well, it is, more or less, but I find it helps to say things over and over again using different words. It's just the way my autistic brain works, not knowing what the other parts of my brain are doing.