CKM and New Physics III

At kneemo's request, let us recall how triality forces the non standard CKM phase of $\textrm{sin} 2 \beta = 0.649$. In the CKM matrix $V_{ij}$ we are interested in phases of the form

$\beta = \textrm{arg} \frac{- V_{cd} V_{cb}^{*}}{V_{td} V_{tb}^{*}}$

Observe that multiplying the complex matrix $V_{ij}$ by any phase $\phi$ does not alter these characteristic phases. Now triality tells us that the $3 \times 3$ matrix should be built from three $2 \times 2$ factors. Since $V_{ij}$ is magic (along rows and columns) the $2 \times 2$ blocks must all be magic. That is, we start with general $2 \times 2$ complex circulants. Multiplication by a phase $\phi$ allows us to make the diagonal elements real. After division by a normalization constant, there is still potentially an arbitrary phase in the off diagonal slots. However, by scaling the diagonal one can always adjust this phase to some fixed value. We choose the fixed phase $i$, so that the $2 \times 2$ factors look like:

This phase is motivated by the $R_2$ MUB circulant, given by the eigenvectors of the Pauli matrix $\sigma_{Y}$. This is the case $a = 1$, and this factor appears in the very elementary tribimaximal neutrino mixing matrix. For three parameters $a$, $b$ and $c$ in a triple product of distinct $3 \times 3$ factors $R_{ij}$, we now fit the nine real CKM entries. The parameters required are $-0.231$, $24.00$ and $0.0035$.

All CKM phases are now fixed by the triple product. The value of $\sin 2 \beta$ is $0.649$, which any educated child may easily verify.

CKM and New Physics II

From Soni's table below we see that the clean penguin estimates suggest a value less than the $0.66$ of current measurement, which is even further from the incorrect QCD prediction. Taking this constraint seriously, we are led to a value for $\textrm{sin} 2 \beta$ that is very close to the $0.649$ predicted by triality in quantum information.

CKM and New Physics

The reviewer who said there was nothing new in the mixing matrix paper would be advised to watch this seminar by Amarjit Soni. It discusses the likelihood that $\textrm{sin} 2 \beta$ takes a non Standard Model value ... namely very likely. Soni appears to be unaware that a numerical prediction for $\textrm{sin} 2 \beta$ exists and is consistent with experimental bounds. The talk is enjoyable also for the audience participation.

Theory Update 70

Observe that the three qutrit tripled trefoil quandles are specified by two out of the three words in the quandle, since the third word is given by the quandle rules. A triple may be listed in any order, and this defines a larger set of $162$ triples. By construction, this set is invariant under permutations in $S_3$. A set with these properties is known as a symmetric quasigroup.

Manin wants to reconstruct projective surfaces from combinatorial information, and he shows that an example of an Abelian symmetric quasigroup is given by collinearity for plane cubic curves. A quasigroup is like a group, in that it has left and right inverses, but it may not have an identity element and it may not be associative. The multiplication table for a quasigroup is a Latin square. For example, the elements $X$, $Y$ and $Z$ of one trefoil quandle define the familiar circulant $3 \times 3$ matrix Latin square.

A quasigroup with identity is called a loop. A loop is a Moufang loop if it satisfies the rule $(xy)(zx) = x((yz)x)$. The non zero octonions form a nonassociative Moufang loop under ordinary multiplication. For the associative product of trefoil quandle triples, the Moufang rule is also obviously obeyed. There are not many nonassociative Moufang loops of low finite order. There is one of order $12$ and there are $5$ of order $81$. Associativity depends on the prime factorization of the loop order.

Theory Update 69

On Phil's post, M theorist kneemo provides this link. I have been waiting for kneemo to mention it again, so that I can post a couple of its diagrams, namely one $M2$ brane and one $M5$ brane, in the stringer's jargon. Here they are:
The triangle is used to describe the complex projective plane. To each interior point, imagine gluing a little torus. Exterior points get circles, which are degenerate tori. An interior trivalent vertex is then like a (circle over a) pair of pants, which is a projective line. We recognise the dual tetractys on the right hand diagram, where all $4$-valent vertices have been resolved into trivalent ones.

Tropical geometers know that lines are trivalent vertices. Their lines come from considering the maximum (or minimum) function as a product on the positive real number set. This product is distributive with respect to addition, giving us two products to define a semiring. Polynomials are then defined using these two products. Hexagons appear for higher degree curves.

In ternary geometry, there is no need to stop at two products. Since distributivity can be weakened, we retain the multiplicative product. Then the positive real numbers come with three natural operations. $M$-branes can be taught to children.

Is it String Theory?

Today Phil writes about why he likes String Theory. He is clearly attempting to discuss physics, as well as mathematics, but one can only assume that the term String Theory alludes to the physics. No one doubts that stringy mathematics is of great interest, and I cannot wait to see it's M theory derivation applied further.

But is it OK to call any physical idea that uses this mathematics String Theory? Even if the physics is totally different? The stringers appear to think so. One gets the impression that even without sparticles, fairy fields or unobservable universes, that it's all still String Theory! So some of them tell me. If it's right, it must be String Theory! The stringers knew about the MINOS results all along! They must have predicted the downfall of the standard cosmology!

MINOS in March

If May is too far away, take a look at this must see seminar in March!

Theory Update 68

Here we see how the $24$ area pieces of the Fano star generate a permutohedron. The grey edges of the diagonal triangle $I$ are ignored, and other edges have two green lines crossing them.

One hexagon on the outside creates the spherical polytope, which tiles three dimensional space. This hexagon contains three Fano star vertices $J_i$, which may be glued to form the $7$ vertices of another Fano object on the tetrahedron.

Quote of the Day

At this rate Tommaso will soon be able to lodge his bet as triple A rated security. It certainly looks a lot more solid than what most banks have in their vaults.

DB at Woit's blog

Theory Update 67

We can fit other objects into circulant loops and lines. Kneemo tells me to think about the Fano star diagram. The star has $9$ vertices as shown, with lines of three vertices. With the unmarked central triangle $(j_1, j_2, j_3)$, there are many triangles in the diagram. Choosing $2$ separated sets of three vertices, we can form the off diagonal $1$-circulant positions of a $3 \times 3$ matrix. The $2$-circulant entries also form lines in the diagram.
Alternatively, we could have filled the matrix $1$-circulants with the three obvious triangles: small $j_i$, large $J_i$ and $I$. This can be done so that the $2$-circulants fit the remaining three long lines.

Theory Update 66

Now let us write out $54$ trefoil quandles from the tetractys path set, graded according to the number of central neutrino paths in the quandle. The total number of each quandle type is noted in brackets.

zero(25)
$XXX,YYY,ZZZ$
$YYX,XXZ,ZZY$;$YXY,XZX,ZYZ$;$XYY,ZXX,YZZ$
$XXY,YYZ,ZZX$;$XYX,YZY,ZXZ$;$YXX,ZYY,XZZ$
$XXX,YYZ,ZZY$;$XXX,YZY,ZYZ$;$XXX,ZYY,YZZ$
$YYY,ZZX,XXZ$;$YYY,ZXZ,XZX$;$YYY,XZZ,ZXX$
$ZZZ,YYX,XXY$;$ZZZ,YXY,XYX$;$ZZZ,XYY,YXX$
$XXX,YYX,ZZX$;$XXX,YXY,ZXZ$;$XXX,XYY,XZZ$
$YYY,ZZY,XXY$;$YYY,ZYZ,XYX$;$YYY,YZZ,YXX$
$ZZZ,YYZ,XXZ$;$ZZZ,YZY,XZX$;$ZZZ,ZYY,ZXX$

three(2)
$XYZ,YZX,ZXY$;$XZY,ZYX,YXZ$

two(9)
$XYZ,YYY,ZYX$;$YZX,YYY,YXZ$;$ZXY,YYY,XZY$
$XZY,ZZZ,YZX$;$ZYX,ZZZ,ZXY$;$YXZ,ZZZ,XYZ$
$ZXY,XXX,YXZ$;$XYZ,XXX,XZY$;$YZX,XXX,ZYX$

one(18)
$XYZ,XZX,XXY$;$YZX,ZXX,XYX$;$ZXY,XXZ,YXX$
$XZY,XXZ,XYX$;$ZYX,XZX,YXX$;$YXZ,ZXX,XXY$
$XYZ,YYX,ZYY$;$YZX,YXY,YYZ$;$ZXY,XYY,YZY$
$XZY,ZYY,YXY$;$ZYX,YYZ,XYY$;$YXZ,YZY,YYX$
$XYZ,YZZ,ZXZ$;$YZX,ZZY,XZZ$;$ZXY,ZYZ,ZZX$
$XZY,ZZX,YZZ$;$ZYX,ZXZ,ZZY$;$YXZ,XZZ,ZYZ$

Theory Update 65

We can break up the $21$ outer paths of the tetractys for three qutrits using three trefoil quandles. Recall that the trefoil rules may be expressed in the cyclic form $\{ a \circ b = c, b \circ c = a, c \circ a = b \}$, and this is applied to the multiplication $(123)(456) = ((14)(25)(36))$ on three qudit words. The three quandles are coloured black, red and green.
These $21$ paths can be mapped to a $3 \times 3$ planar matrix by interpreting the trefoil cycles as $1$-circulant components.
This is just a two qutrit matrix with the first coordinate doubled. There are also three trefoil quandles of the form $\{ XYZ, YYY, ZYX \}$, using the central $6$ path neutrino vertex along with the corner charged lepton vertices. A diagonal element $YYY$ belongs to two trefoil quandles, whereas an off diagonal element $XXY$ belongs to only one.

Note also that the other three trefoil quandles form the $2$-circulant pieces of the above matrix. Each $2$-circulant uses one path of each colour (black, red and green). Thus there are at least $9$ trefoil quandles in the tetractys path set.

MINOS in May

It's still some time off, but anyone near California should put this in their diary ... a MINOS talk scheduled for May 1, 2011. The abstract states:

MINOS has previously reported the results of $\bar{\nu}_{\mu}$ disappearance from a direct observation of muon antineutrinos ... In the present analysis we have a (forward horn current) $\bar{\nu}_{\mu}$ data sample with $7.1e20$ protons on target, which will be used to improve the previous measurements. This talk summarizes the agreement between data and simulation in the Near Detector at Fermilab.

Meanwhile

Susy may be a rotting corpse, but professionals continue to favour it. Someone just told me that nobody here was interested in my work, but he does not speak for everyone. In fact, there is a small chance that I might be off overseas again soon, meaning that someone who is interested in quantum gravity might buy me a ticket and give me somewhere to stay. Any other offers? I would quite like to have a home one day. I am now working on my next paper, but I will put off telling you what it is about!

LHC Update

Thanks to Resonaances for this plot from Alessandro Strumia's talk. Even for an unconstrained fairy field mass, the LHC will leave only a few percent of the parameter space. Goodbye Susy.

Zeta Infinities

Fortunately, with categorical quantum arithmetic one can follow Euler in completely ignoring all inconvenient axioms of infinity. Consider the Riemann zeta function for the ordinary counting numbers. Recall that the Pauli exclusion principle selects out ordinals $n = p_1 p_2 p_3 \cdots p_r$ which are square free. This introduces the Mobius function $\mu (n)$, so that

$Z(s) = \sum_{n} \frac{\mu (n)}{n^{s}}$

This is the inverse of $\zeta (s)$, which we can see as follows. Consider $Z \zeta (s)$, a sum over all square free $n$. Each term is labeled by a finite string of primes $p_i$. Now using a prime $p$, split the entire sum into a part with terms that include $p$ and a part with terms that do not. Then

$Z \zeta (s) = - p^{- \zeta (s)} S + S$

for $S$ the terms that do not include $p$. We take a minus sign out of the first term to account for the change of sign in the Mobius function when the number of prime factors is altered by $1$. So we have

$Z \zeta (s) = \prod_{p} (1 - p^{- \zeta (s)}) = s$

at least when $s > 1$. Since $\zeta (s)$ is a bosonic partition function, this inversion is a form of supersymmetry. In the fermionic case, we are summing over all subsets of the prime numbers. As in M theory, but not string theory, spaces and numbers are built from primes. Knowing the fermions is enough to generate the bosons.

Theory Update 64

Carl Brannen now likes Hopf algebras. What are Hopf algebras, and why should we care? As categorical physicists, we will not worry so much about elements of an algebra, but will define the concept in terms of diagrams. An algebra is a set with a multiplication operation. A category theorist says that the multiplication is an arrow $m: A \otimes A \rightarrow A$ from two copies of the algebra back to the algebra, which we call $A$. A Hopf algebra requires maps as follows:
Now $A$ is both an algebra (with multiplication $m$) and a coalgebra (with comultiplication $\Delta$). An algebra must have an identity element, and this is the unit map $\eta: I \rightarrow A$ from the scalars $I$ into $A$. Similarly, a coalgebra has a counit. Observe that the duality between algebras and coalgebras is represented by a flip of diagrams across a horizontal axis on the page.

The one strange element that we need to add is the antipode map. Consider an example. The algebra of functions on a finite group is a Hopf algebra. The comultiplication is defined by $\Delta (f) (x,y) = f(xy)$, using group multiplication. Multiplication of functions $m(f,g)$ is given pointwise. The antipode satisfies $S(f)(x) = f(x^{-1})$, relying on the existence of inverse elements in the group. Thus the group structure naturally puts a Hopf algebra structure on its function space. The antipode obeys two laws of the form
Hopf algebras became important in physics in the late 1970s when a group of Russian physicists, who were studying algebras for quantized non linear partial differential equations, stumbled upon a set of commutation relations using sinh functions. The mathematician Drinfeld soon understood the significance of these rules, and the theory of Quantum Groups was born. A Quantum Group is not really a group. It is a Hopf algebra, namely a deformation of the universal enveloping algebra of a Lie algebra. The deformation parameter $q$ is like a quantum parameter $e^{\hbar}$.

All the Weil

The proven Weil conjectures for varieties over finite fields outline the properties of zeta functions for such spaces. For example, consider the projective line over the finite field with $p^n$ elements. It's zeta function is

$\zeta (s) = \frac{1}{(1 - p^{-s})(1 - p^{1-s})}$

and higher dimensional projective spaces just have more factors on the bottom. Grothendieck and others eventually understood the Weil conjectures using an $l$-adic cohomology theory, where $l$ is a prime different from $p$. The zeta function was expressed in terms of determinants for $l$-adic cohomology groups, and a Poincare duality explains the famous duality symmetry of zeta functions.

In quantum gravity we find fields with $p^n$ elements in the mutually unbiased bases for dimension $p^n$. As for a projective line, there are always $p^n + 1$ elements in a set of MUBs. Considering $n$ qupits for all $n$, we are naturally led to a generalisation of $p \times p$ matrix entries to the $p$-adic numbers. But constructing the complex numbers correctly in this axiomatic setting does not mean taking the complex numbers as they are usually given to us, complete with an unmeasurable continuum and a set theoretic continuum hypothesis. As any topos theorist knows, the real and complex numbers are not even uniquely defined in a topos. Quantum gravity will use the numbers it requires, and no more. A useful complicated space is, after all, described by a finite amount of data.

Quote of the Week

In this paper a fractal SUSY QM model is proposed to prove the Riemann Hypothesis. It is based on a quantum inverse scattering method related to a fractal potential given by a Weierstrass function (continuous but nowhere differentiable) ... It requires using suitable fractal derivatives and integrals ...

Castro and Mahecha

Canonical Coordinates

Back in M Theory Lesson 2 we looked at Mulase and Penkava's canonical coordinates for Riemann surfaces. As is well known, there are two important lattices in the plane: the square lattice, where a square can be glued to give a torus, and a hexagonal lattice.

For $\tau$ a sixth root of unity, there is a natural triangle with point $p$ given by the average of $\omega_1$, $\omega_2$ and $\omega_3$. A Strebel differential $Q$ is expressed in terms of the Weierstrass function as

$Q = \frac{1}{4 \pi^2} \frac{(\textrm{d}P^3)^2}{P^3 (1 - P^3)} $

exhibiting an obvious triality symmetry. Now given $n$ points $p_i$ on a Riemann surface, along with a set of $n$ positive real numbers, we get a unique Strebel differential $Q$. From this one generates a canonical triangulation of the surface. The real numbers $r_i$ are associated to contour integrals $\int \sqrt{Q}$ and one may choose a branch of the square root to make $r_i$ positive.

Theory Update 63

Recall that the fractal honeycomb with $27$ outputs arises from blowing up the ten points of the tetractys to larger hexagons.

There are now $81$ internal nodes and $9$ smaller hexagons. Fractal honeycombs nicely keep track of increasing powers of $3$, a basis for $3$-adic numbers. We can either zoom in or zoom out, introducing a process of division by $3$. The central tetractys vertex was marked with the six paths given by the permutations of $XYZ$. These paths are now the separated nodes of a hexagon. In creating this hexagon, we needed a fourth qutrit to form the $81$ paths of a four qutrit word.

Diamonds in the Sky

A ring of black holes from Chandra:

Mirror Neutrinos

After several months of attempting (and failing) to publish a note on neutrino cosmology, Graham Dungworth and I have now posted the paper online.

On the Streets of String City

Black Holes as Qudits

Oh, look. Check out kneemo's new paper on the arxiv.

Quote of the Day

And today Baez said:

If we got the bioctonions, quateroctonions and octooctonions into the game, we might account for the other exceptional compact simple Lie groups: $E_6$, $E_7$ and $E_8$. That would be quite magical.

Indeed. Michael Rios calls it M Theory computronium. Stringers and loopies don't yet know what M Theory computronium is, because they don't yet appreciate that Spaces and Algebras are made of quantum information. Mathematicians don't yet know what M Theory computronium is, because they don't yet appreciate that Spaces and Algebras are made of quantum information.

Theory Update 62

As with the associahedron, we may pull some faces of the permutohedron up onto three faces of a tetrahedron.

This puts the $7$ points of a Fano plane inside $7$ out of the $8$ hexagonal faces. Note how the upper triangles form a shadow Fano plane for the Fano plane on the base triangle.

Rejection

There were two paper rejections in my mail box this morning. The first one was fairly typical: one reasonable, positive review and one highly negative review. The negative reviews are usually very short. The second rejection came with a two line review. The first sentence was:

This paper sound like a mixing of subjects, even updated, without any physical ground.

Hmm. I am having some trouble understanding what this means. Presumably the reviewer is confusing the word updating with the phrase modifying like a crackpot.

Theory Update 61

In this supercool paper, the authors define a tripled Fano plane (yes, that's right, three copies of Furey's particle zoo). It describes a set of $21 = 3 \times 7$ (left cyclic) modules over a noncommutative ring on eight elements. The ring is given by the upper triangular $2 \times 2$ matrices over the field with two elements. Similarly for right cyclic modules.

The authors are familiar with the connection between octonion physics and so called stringy black holes. They find it odd that this structure is not studied in physics.

Theory Update 60

We had fun putting half an associahedron onto a triangle, so let us also put half a permutohedron onto a triangle. This time one requires the $4$ qutrit paths.

Cohomology Revisited

Modern physicists and mathematicians like to draw diagrams with strings and ribbons. An object in a category is often depicted by a strand with an arrow (as in (a)). Recognising that an arrow represents two colours, we might like to use three colours instead, as in (b). A tensor product of objects is given by placing their strands side by side. In (c) we thicken the strand to a ribbon, which allows twists in our diagram networks. But why stop there? We might as well use the three dimensions at our disposal and thicken the ribbon backwards, as in (d). Finally, taking multiple coloured beams, we can form braided networks from them.

One of the simplest diagrams in a $1$-dimensional category is a triangle. With beams we could draw this:
This is Penrose's triangle. He uses it to discuss the first cohomology group, where the numbers are interpreted as positive distances. Elements of a cohomology group are classes of cocycles, and in one dimension a category theorist would draw the cocycle condition as a triangle
where the $d_{ij}$ are distances of observation. We have a coboundary when the cycle composition results in the number $1$, where an inverted arrow corresponds to taking the reciprocal positive number. That is, each $d_{ij}$ is a ratio $x_{i}/ x_{j}$. Reciprocation is the multiplicative analog of a minus sign for addition. In a higher dimensional category, more complicated polytopes are used to specify cocycle conditions. When these polytopes are canonical axioms for a suitable class of categories, the cohomology should be universal.

Theory Update 59

Given a line of three words, one cannot necessarily impose the trefoil quandle rules. However, under the product $(123)(456) = ((14)(25)(36))$ the Pauli quandle rules will apply to any cyclic triple of words. That is, we can use Pauli type quandles to fill in lines in a Fano plane. Now on the tetractys there are three sets of natural trefoil triples. One is given by the triple of lines:

Thus a true octonion contains three trefoil knots, whereas a split octonion may be specified by mixing a pair of quaternion trefoil lines.

Theory Update 58

Inside the path cube sits the lepton hexagon, punctured orthogonally by the lepton line. In the commutative tetractys, the whole hexagon becomes a point. If we turn the hexagon into a cube, as in three dimensional categories, then the distances from the cube source to the hexagon vertices are not all equal. The distances would belong to the set $\{1 , \sqrt{2} \}$. These are the two parameters of the neutrino tribimaximal mixing matrix.

Theory Update 57

The tetractys for three trits came from the commutative projection of the $27$ noncommutative paths. We can place all $27$ paths on a three dimensional torus $T^3$.
The front face contains paths starting with $X$ and the rear face paths starting with $Z$. Everything is cyclic, so opposite faces are glued together by extra edges. Now the vertical diagonal faces (colour coded) correspond to the trefoil quandle $B_3$ generators $a$, $b$ and $c$. Ignoring the words $XXX$, $YYY$ and $ZZZ$, each such face has eight vertices on an octagon. There are always two neutrino words of type $XYZ$, which we take to be the source and target of a cube. Each colour could encode a Fano plane, with a trit word for each octonion unit in the plane.

However, we will have to play more with the quandle rules to find the optimal choice of quaternion lines for the Fano plane. We would like to create Fano planes from the hexagon on the commutative tetractys. So there is a correspondence between the $27$ paths and the dimensions of the $3 \times 3$ exceptional octonion Jordan algebra. Let us check how quandle rules might express the quaternion rules, along one line in the Fano plane. Define a two word product $(123)(456) = ((14)(25)(36))$. Use the quandle rules $XX = X$ and $XY = Z$ etc. Then we see that $(YXY)(XXZ) = (ZXX)$, as required. Each unit is given by a word of length three. A child could understand quantum gravity.

It from bit and trit. Space from knots. Algebra from knots.

Theory Update 56

A natural way to turn rooted trees into braid elements is with a pseudoline arrangement. For a chorded pentagon we obtain pseudolines on three strands, as shown.

Theory Update 55

For the ribbon braid group $B_3$, we can twist ribbons and also perform braiding within the three strands of $B_3$. One cyclic set of generators is shown.
The generators $(12,23,31)$ are those given by the trefoil quandle. The ribbon strands are labeled $1$, $2$, $3$. We see that two qutrits may be used to label the generators, under the correspondence $1 = XX$ and $12 = \{ XY , YX \}$. In other words, the three ribbons are specified by the letters $X$, $Y$ and $Z$.

Observe that the trefoil quandle rule is now naturally associated to braids of the form $\sigma_{1} \sigma_{2}^{-1} = (12)(23)^{-1}$ $= 12 3^{-1} 2^{-1}$, which are used to specify the Bilson-Thompson particle spectrum. Since qutrits and triality are used to specify braid information, a $3 \times 3$ Koide mass matrix now lives in an exceptional bioctonion Jordan algebra, as do the neutrino and CKM mixing matrices.

Yet Again

Theory Update 54

On the other hand, the hexagon piece of a tetractys neatly cuts the associahedron into two pieces. It does so by separating the inner loop of Loday's trefoil from the outer three sections, which form another loop.

Associahedra Again

The $9$ faces of the associahedron provide endless entertainment. We would like to glue three octagons together to form the polytope, but recall that two vertices of the associahedron are special: those at the centre of three pentagons. We can glue two octagons together, but then we find a polygon boundary with a different number of edges.

Cosmological Interlude

Since we have been playing a lot with the dimension $27$, let us recall the one universal Koide phase, in the form $\phi = 1/9 \pi$. This is the usual $2/9$ phase divided by $2 \pi$. As it happens, the fraction of baryonic matter in our universe is exactly

$\Omega_b = 1 - 27 \phi = 1 - 3/ \pi$

as predicted by Louise Riofrio many years ago, and recently confirmed by WMAP. The $3/ \pi$ is a dark matter fraction. It may be thought of as an inverse sixth root of unity, where $6$ labels the dimension of the (dual) tetractys neutrino path space.

Theory Update 53

As kneemo will no doubt elaborate upon, we can now use the Pauli quandle and tripled Fano hexagon to discuss twisting ribbons in all three dimensions.
A directed Fano line is now either an internal triangle for the hexagon (not the triangles shown) or an internal chord parallel to a tetractys edge (for color). For example, in Furey's scheme for the octonion units, the line $(4,5,7)$ involves one set $(u,d,e,\nu)$. Only the full tetractys, however, can resolve the charged leptons and neutrinos, and cover all three generations. The original Bilson-Thompson braids use only one plane in space, namely the plane of the page, but space is three dimensional because there are three generations.

Theory Update 52

Recall that three copies of the Fano vertices can fit into the dual tetractys. The Furey triangle is now deformed into a hexagon.

As usual, the unlabeled hexagon gives us the associahedron. More structure leads to more complicated, but still perfectly logical, polytopes.

Theory Update 51

Thanks to a commenter at the Cafe for Cohl Furey's wonderful 2010 paper on the numbers $R \cdot C \cdot H \cdot O$. Here is a picture from the paper: Hmm, now where were those CKM parameters ...

Motive Madness

In his lectures on motives, Manin begins with the central question:

If numbers are similar to polynomials in one variable over a finite field, what is the analogue of polynomials in several variables? Or, in more geometric terms, does there exist a category in which one can define absolute Descartes powers $\textrm{Spec} Z \times \cdots \times \textrm{Spec} Z$?

Here $\textrm{Spec} Z$ refers to the so called spectrum of the integers $Z$. It is thought of as a space built out of prime elements. But everyone knows that prime spaces should be described in terms of knots and tangles. Bits and trits are associated to spaces of prime dimension. If we built spaces from bits and trits and other $p$-dits, where there are always $d = p + 1$ MUBs, we would be building spaces from knots. A Cartesian product takes strings of prime elements. But let's not forget that dits have ever increasing dimensions, and that $\omega$ categories are the most natural spaces to create out of paths.

Theory Update 50

Some years ago, mathematicians Albequerque and Majid described the octonions in terms of three bits. This is natural, since the $8$ basis elements fit onto a parity cube. Similarly, there is one bit of information in the choice of axis from a complex basis $(1,i)$. The quaternion basis looks more like a tetrahedron though, because a parity square does not put $i$, $j$ and $k$ on the same footing. But a tetrahedron fits nicely into the parity cube.

So moving onto the bioctonions, we would start with a four dimensional parity cube. Two $4$-simplices provide the vertices of the real quaternions and imaginary quaternions, and one cubic face a copy of the (real) octonions.

Using (unbracketed) words in $X$ and $Y$ of length four, we can label all $16$ vertices of a four dimensional cube. Alternatively, using left and right brackets we can also label $16$ vertices with length three words, such as $(XX)Y$. But if the words are cyclic, there should be three kinds of brackets (one around the ends), which introduces a trit index on the set of bits. In this case, there are a total of $24 = 3 \times 8$ vertices to label.

For a (cyclic) bracketing of three strands, we need to look at Bar-Natan's classic paper on nonassociative tangle diagrams. Here there are bits defined by the direction (up or down) of the arrow marked on a strand. Trit labels would instead require a $3$-coloring of each strand, omitting the arrow label. One way to do this would be to use ribbons with only three possible twist types. However, this breaks the cyclicity of three equivalent colors, much as the three cubed roots of unity contain an identity. On the other hand, the identity rule $\omega \cdot 1 = 1$ looks a bit like a Pauli quandle rule, and the Pauli quandle is neatly cyclic.

It from bit and trit.

Dying Fairies

Tommaso Dorigo posts the latest fairy field exclusion projections! Not far away!

Quote of the Day

Today at the Cafe, in a post about twisted ribbons, Baez said:

The problem is that there’s not enough overlap between people who think about time reversal and antiparticles, people who think about normed division algebras, people who think about unitary group representations and Dyson’s Threefold Way, and people who think about symmetric monoidal categories with duals.

I then did my best to point him in the direction of a large group of such individuals, but of course the comment was immediately deleted, as usual. Ex stringer Urs Schreiber has also lately been doing his best to enlighten certain people, but it seems to no avail. I therefore predict another century of sinful pseudoscience before the army of physics students finally revolts against their so called teachers.

Naughty Neutrons

Those who are paying attention will recall Graham Dungworth's thread on mirror matter at Galaxy Zoo. They may also recall that antineutrons are reported to have a mass distinct from neutrons, just as in the neutrino sector. In a lovely (albeit somewhat theoretically misguided) paper on mirror neutrons, Zurab Berezhiani demonstrates that neutron mirror neutron oscillations are not ruled out by experiment. On page $1$ it is said:

... the $n \rightarrow n'$ transition can only manifest as anomalous disappearance of the neutrons, in addition to the decay, absorption and other regular channels of their losses.

Using reported data, and with an analysis based on an hypothesised mirror magnetic field for the Earth, Berezhiani notes an anomalous delta of $- 3.5 \pm 2.5 \times 10^8$ in recorded event counts. Hmm. I wonder what effect a neutron disappearance might have on estimated cross sections for a reactor antineutrino anomaly?

Theory Update 49

Let us replace the Pauli unknot arcs with the suggestive letters $i$, $j$ and $k$. Now we can carefully draw the unknot in three dimensions, making sure that each straight line segment follows a $45$ degree line with respect to a chosen set of axes. Thus one arc moves down the $45$ degree line, one moves up the $45$ degree line, and one moves along the $135$ degree line.

If the knot arcs were thickened to ribbons, the twisting moves would look like ribbon twists, one for each of the three planes in space, as in this paper.

Theory Update 48

The Pauli spin matrices also satisfy the quandle rules under the conjugation product, but only up to a sign.