Neutrinos and the CMB II

It is good to see that Graham D is rather busy with his new astrophysics program. This post is just a quick reminder that Louise Riofrio's varying $c$ cosmology derives the baryonic matter fraction as

$\Omega_{b} = 1 - \frac{3}{\pi} = 4.507034%$

and we can now compare this value to fractions obtained by alternative means.

Neutrinos and the CMB

If one asks an astronomer in which direction the solar system moves with respect to the Milky Way, they will point towards the star Vega in the constellation Lyra. This is the direction that the DAMA experiment considered under the hypothesis of a dark matter aether in the galactic halo, with respect to which the Earth's motion either adds or subtracts to the solar system motion depending on the time of year.

But if one looks at the location of Vega in the sky, it is roughly correlated with the dipole anisotropy in the cosmic microwave background radiation. This was pointed out in 2008 by Richard Saam (and no doubt others) and it means that we could consider a CMB origin for the DAMA results. This is fortunate, because we were already wondering about the coincidence between the CMB temperature and the (lightest) antineutrino rest mass.

The error bars on the DAMA annular modulation phase are surely large enough to accomodate this alternative origin for their observation. As Saam notes, the maximum of the DAMA cosine corresponds to a minimum in the Earth's motion with respect to the CMB. This would be in agreement with larger cross sections with the wondrous neutrino sector around early June. Thus there is no need to posit a galactic aether.

Antineutrons

Charged particles and their antiparticles, such as the electron and positron, are known to have remarkably similar masses. But what about other neutral particles? According to the Particle Data Group, the neutron and antineutron have different masses. That is, they report the result

$\frac{m_n - \overline{m}_n}{m_n} = 9 \pm 6 \times 10^{-5}$

where the neutron mass is $m_n = 939.565346 \pm 0.000023$ MeV. This amounts to a difference that is many orders of magnitude greater than the neutrino mass scale! Further evidence of CPT violation.

Neutrinos Again II

As can be seen in the period $\omega$ graph below, which searches for solutions to the new mass relation, there is a second solution for the new $\theta$ at the fixed Koide parameter value of $r = \sqrt{2}$. That is, the mass relation for neutrinos and antineutrinos also holds at $\theta = 0.8857$. Using the phase $\pm \pi/12$ there is a corresponding set of six new masses associated to this $\theta$, given by

$m_{i}: 0.00166, 0.0250, 0.0333$ eV
$\overline{m}_{i}: 0.00084, 0.0130, 0.0461$ eV

These all lie within the mass range of the neutrinos and antineutrinos but have smaller $\Delta m^2$ values. The mass triplet sums are still $0.06$ eV, as for the usual Koide rule. Now however, the Koide rule holds when (i) the lightest antiparticle mass (at $0.00084$ eV) occurs with a negative eigenvalue and (ii) the lightest particle mass ($0.00166$ eV) also occurs with a negative eigenvalue. This is like exchanging a basic $SO(2,4)$ metric light cone (for a six dimensional $\nu$ $\overline{\nu}$ square root eigenvalue space) for an $SO(3,3)$ metric. So we do not expect these particles (assuming they exist) to participate in the weak interaction with ordinary matter, like the neutrino sector, but if localisable then their detection would in principle be similar to that of the neutrino sector. This brings the total count of neutral CPT violating chiral Koide particles to $24$, and no more are possible.

Neutrinos Again

Returning to the six masses of the neutrinos and antineutrinos, we can search for further approximate relations between the values and their ratios. In particular, observe that the scale independent relation holds exactly for a universal phase not of $2/9$, but quite close at $\theta = 0.222535$. This phase does not substantially alter the predicted neutrino mass values or the $\Delta m^2$ of the MINOS experiment, for a similar scale choice. However, this $\theta$ ruins the correct $2/9$ rule for the charged lepton masses. For the charged leptons the rule above could never hold, because particles have the same mass as antiparticles.

Neutrinos and Supernovae V

Those of you who have been following Graham D's noble effort to educate the general public about the new cosmology, before a single stringer or loopy cottons onto it, may be beginning to ponder the implications for dark matter. One wonders how anyone can take whackalinos seriously anymore.

But we must be careful not to view this neutrino (and black hole) mass mechanism in terms of an aether. One heavily criticised experiment to have reported a positive low energy result for dark matter is the DAMA experiment, which has searched for an annular modulation in dark matter flux. The results are often discussed in what sounds like an aether of dark matter particles, but this betrays a classical mode of thinking that does not capture the other worldliness of the neutrino interactions. The Earth is simply in a different gravitational configuration within the galaxy at different times of the year. DAMA is so far unique in carrying out this kind of test. See their newest results.

Neutrinos and Supernovae IV

Well, I really am a bit slow. The last time I mentioned Graham D's analysis of the neutrino cosmology, I failed to notice one of his main points: since a given neutrino has a different mass to the antineutrino, and they cannot annihilate, one supposes that the Koide masses are exactly halved into left and right handed annihilating particles. This allows us to view the CMB photons at $2.7$ K straightforwardly as annihilating pairs. The six Koide masses are thus assumed to label $12$ chiral neutral particles.

Neutrino and antineutrino beams at MINOS, and elsewhere, are thus viewed as mixed particles. The CMB coincidence then strengthens the premise that mass differences will not be found for the charged lepton antiparticles, which only occur with chirality. It also makes the neutrinos look more like the quarks, each coming in $12$ types, and is thus less surprising that neutrino and quark mixing shows similarities.

It seems that Graham D has been busy, trying to recover a number of basic astronomical parameters, so we should wish him and the other astronomers the best of luck with this endeavour! Other people I know are also busy studying the new cosmology, in an earthly laboratory setting.

Spring Day

At last it seems that spring is here. Tonight our summer time begins, a cause of great confusion amongst many supposedly intelligent northern hemisphere acquaintances who like to phone antipodeans in the middle of the night. Yes, we are officially $12$ hours ahead of GMT ... BUT ... since we are now going into summer, there will be periods of time in spring when we are either $11$, $12$ or $13$ hours ahead of time in the UK, depending on whether or not we, or they, have summer or winter time. Please check the clock!

What Century?

Without much thought, I recently made a couple of brief feminist remarks on a post about a new postdoc position (the kind I don't get offered) on Dave Bacon's blog. I would have completely forgotten it, except that apparently certain anonymous cowards have been telling the blog owner to delete my comments, so in true blogger fashion I figure that now is the time to advertise the post more widely!

I told Dave that I was most happy for him to leave any comments up on his blog, no matter how many anonymous cowards told him which comments he should delete, and so on. That rather unique species, anonymous cowardus, appears to be under the misconception that this is still the last millenium. If they are in a position of authority in their local ass licking environment, they unconsciously assume they can apply the same authority more widely in the world at large, even across cultural and international borders. Bloggers are fascinated that such creatures exist, but feel little need to cater to their self serving whims.

Perhaps we should feel a little sorry for these particular well fed anonymous cowards. They probably don't know many women, and they may need some lessons in how to discriminate more intelligently. The internet can help! How about the excellent short guide, Derailing for Dummies.

Theory Update 4

Young twistor particle physicists are turning up everywhere these days. Mathew Bullimore has a recent paper on twistor diagram techniques, which is well worth reading. New papers by Mason, Skinner, Arkani-Hamed, Cachazo and others are continually appearing. We still await the official appearance of associahedra in a description of these structures, but at least the concept of information dimension is now thoroughly entrenched at the heart of modern scattering techniques.

The News Today

From the CMS experiment, and reported at vixra log and TRF, a stunning new image from the LHC. The report of particle correlations resembling quark gluon plasma results is here. On the CMS seminar slides by Roland, we have a comparison of the short range correlations with Monte Carlo simulations, showing an effective cluster size of $K$ particles for the minimum bias results. The open circles and squares are simulation and the red dots are CMS results. The interesting high multiplicity events have long range (near side) correlations that peak in the transverse momentum range of $1$ to $3$ GeV$/c^2$. The important plot, that shows the effect after a high multiplicity trigger is applied:

Theory Update 3

Using cospan objects (edge wedges) in cubical path diagrams is a fun way to study arrow compositions. Recall the difference between horizontal and vertical composition for two arrows, where the final picture looks quite different in each case. After a few steps, further compositions lead to a wide range of diagrams.

Like in entropy triangulations, we can normalise the square areas to integral values $d$, and then the single edge lengths around a square go as $1$, $\sqrt{2}^{-1}$, $\sqrt{3}^{-1}$ and so on, just like the normalisation factors for the MUBs in dimension $d$. That's nice, because we want to think of the square areas as designating a dimension, namely that of the composed arrow. Observe that even for two $2$-arrow compositions one can have a dimension $3$ object. This is what happens for the Crans-Gray tensor product.

Theory Update 2

Manin begins his essay on Cantor with a quote by Tasic:

God is no geometer, rather an unpredictable poet.

In mathematics today, the spirit of set theory has been subsumed by the hierarchy of weak $n$-categories and their generalisations. Instead of an infinity of set elements, assuming an axiom of infinity, there is an infinite tower of increasingly complex equivalences and relations between relations. A mere set is but a $0$-category, sitting at the bottom of the ladder, and yet many constructions to date take their intuition from the land of sets.

Looking at the path table of monomials, in the spirit of Cantor many would see a partition of all possible words on an alphabet $\{ X, Y, Z, \cdots, \}$ grading the free monoid functor on the category of sets. The table stretches off to infinity without question.

But as physicists, we stare in amazement at the first few entries of the table and wonder how many infinities can be built with just these few words, the qubits and qutrits. What are these information sets? They live not in any ordinary category of sets, but in some other world. With qubits alone, dualities reside. As words grow longer, so do the possibilities of categorical composition. What was once a Hilbert space of dimension $n^d$, is now a small collection of words in the symbols of measurement.

Theory Update 1

For years now this blog has posted M Theory lessons, but my original reason for using this title (which I leave to the reader's imagination) has somewhat diminished in importance and so instead of Lesson $360$, today we begin with our theory update number $1$. It is just a lazy choice of post title, and I am not about to change direction.

In the two dimensional path table, or monomial table, we can consider maps (arrows) that embed one group of paths into a larger group, either to the right or above it. Observe that such a map always exists for objects that are adjacent, either horizontally or vertically, and these maps can be composed along a line. So instead of a single map between path sets, we can define a category with $1$-morphisms between two objects, given by the pair of arrows that end in a rectangle corner. Moreover, given a pair of such $1$-morphisms, we can similarly define a $2$-morphism using a larger rectangle, as shown by the medium arrows in the diagram.


In a typical $2$-category, these $2$-morphisms have two types of composition, vertical and horizontal. This is usually drawn as a globule picture, but the diagram illustrates the point. Now given two $2$-morphisms there is a $3$-morphism, given by the paths around the big arrows in the diagram. Observe how a vertical composition of $2$-arrows results in a different kind of $3$-morphism to the horizontal composition.

For example, consider the maps discussed in the last lesson. There are two sets of maps of interest into the qutrit tetractys: one from the three qubit path set and one from the two qutrit path set. The former selects $8$ noncommuting paths and imbeds them into the $27$ paths, whereas the latter embeds $9$ paths. The former is associated to the Jordan algebra over the bioctonions and the latter to the $57$ dimensional nonlinear version of $E_8$. In general, we have maps $n^{d} \rightarrow m^{t}$ between the $d$ $n$-dit space and the $t$ $m$-dit space.

M Theory Lesson 359

As kneemo has discussed, in 2000 Gunaydin et al described $E_8$ in a nonlinear fashion, on a space of dimension $57$. Where do we get the number $57$ from path spaces? The $56$ dimensional $2 \times 2$ FTS matrix was a circulant of type $(1,27)$. A natural ternary analogue would be a $3 \times 3$ operator of circulant form $(1,9,9)$, which has total dimension $57$.

We can map the $9$ length $2$ qutrit paths into the $27$ path qutrit tetractys by concatenating on the left or right with an $X$, $Y$ or $Z$, giving a total of six maps. Note how the extra dimension ($57$ compared to $56$) essentially arises from the extra diagonal term. From a categorical perspective, this reflects the replacement of an abstract duality by triality. The length $2$ paths were associated to an associahedron secondary polytope, and the concatenations embed this into a higher dimensional polytope associated to the three qutrits. However, the noncommutative and nonassociative geometry will take us beyond mere $1$-ordinal polytopes.

M Theory Lesson 358

A nice paper by Michel Planat shows how the Weyl group $W(E_8)$ of $E_{8}$ may be generated using gates for three qubits, namely the Toffoli gate and the two gates $I \otimes S$ and $S \otimes I$, where $S$ is a $4 \times 4$ matrix with entries $\pm 1$. The matrix $S$ may be related to a simple braiding matrix $R$ by the relation $RS = F_2 \otimes I$, for $F_2$ the usual Hadamard gate, or Fourier transform. Similarly, $W(E_7)$ is given by three simple three qubit gates, including the Toffoli gate. As Planat remarks:

Indeed, the unitary realization of $W(E_8)$ with quantum gates (of the GHZ type) is much different from the Weyl group one gets from the Lie algebra of $E_8$.

He also defines two gates to realise $W(E_6)$.

M Theory Lesson 357

Mixing up the Freudenthal triple system for three qubits and the tetractys qutrits we obtain a $56$ (complex) dimensional system, namely the one discussed in Rios' paper on Jordan $C^{*}$ algebras. The source and target paths are the complex diagonal elements of the FTS matrix. These bioctonion Jordan algebras allow us to consider complex mixing matrices, and other interesting operators, as Jordan algebra objects. A $C^{*}$ algebra may also be viewed as a certain kind of category with only one object, so that its elements are arrows $C \rightarrow C$. Automorphisms for this category are then endofunctors, just like morphisms ought to be, making the Jordan algebra quite a lot like a group.

Quote of the Week

Why do people care? Because this is how gender norms are enforced, maintained, and handed down to the next generation. If they didn’t show how much they cared, you might think it didn’t matter whether or not you took hubby’s name, or popped out babies ... you might get to thinking that heteronormativity as a way of life was optional, and not an option you were particularly interested in. People care, because telling each other the story of How The Tribe Is Supposed To Be is how we make people behave, how we socially shame and norm each other into doing what we think ought to be done even if none of us like it very much. When people behave differently it is terrifying ...

Thus Spake Zuska

Neutrinos and Supernovae III

We see that Graham's idea is basically correct, but not strictly about antineutrino annihilation, as follows. Taking one single (electron) antineutrino of mass $0.001169$ eV and turning it into a photon, Wien's law for a black body spectrum (which keeps $hc$ constant) gives us the temperature of $2.732$ K.

On the other hand, the observed CMB temperature of $2.725$ K does not require a great scale change in the Koide triplet. The accuracy of this scale is currently limited by the neutrino experiments. So there is potentially a play off between neutrino mass observations and the precisely observed CMB temperature, assuming that $\overline{m}_{1}$ is a good generator for it. However, it seems more likely that the discrepancy is due to competing effects from other species.

Neutrinos and Supernovae II

Graham D at Galaxy Zoo has a classical, but extremely good, understanding of the new cosmology. Here is a quote from Aug 9:

The implication of mass charge difference for chiral neutrinos is staggering. We have to go back to Aristotle, Newton and then on to Einstein to understand what we mean by mass, mass energy. Galileo didn't have to actually do the Tower of Pisa experiment and Aristotle knew why. Different balls in weight fall at the same rate. You know it and the lunar experiment with a feather and a lunar rock confirm it. Even today the force of gravity is examined at tinier distances in the sub millimetre range; no effect is apparent ... for all matter, even elements made of electrons and nuclei with quark compositions, and let me say all of the same (universal?) lopsided or cack handed L chiral form. Only in former supersymmetric models are their yet undiscovered counterparts theoretically predicted. There's a lot to unlearn. For every reaction there is an equal and opposite reaction. Well if we made those balls from either a L and R neutrino counterpart, and not as a mix, or placed them as billiard balls on the table, we would be dumbstruck.

Everywhere, every nook and cranny of the universe is stuffed full of these vacuum particles; real particles and not virtual particles, although they are in dynamic equilibrium with them. They form and annihilate incessantly, in your blood and in the depths of space, constantly emitting a black body spectrum of radiation. With just enough energy to create and annihilate each other they are created with very low velocities, ie. they are cold neutrinos, and are not to be confused with hot neutrinos or fast movers with great kinetic energy in earlier and hotter times, that are exchanging this kinetic energy with increasing potential energy during universal expansion. At all times the total energy of the system is conserved, whether at universal scale or a thermodynamic system on a much more local scale. How local? Down to $10^{-18}$m, and that's a small distance scale compared to a millimetre, or the confines of a single atom. All matter types, whether normal matter or dark matter, flow through this false vacuum residue, that is nominally whatever it was conceived to be: the quark antiquark supercondensate, metric, ether.

Vacuum matter is created and annihilates more or less at rest. All other matter flows through it and is subjected to an inertial drag. We see the glow of this creation and annihilation; the electron neutrinos annihilate to produce a photon with an energy equivalent to a thermal bath at $0.89$ Kelvin. These are $8$ fold more abundant in number than our L antineutrino world. However, each of our less numerous L antineutrinos pack in a total $90$ fold more energy, with a glow of a thermal bath everywhere. Photons don't penetrate far, acquiring mass in a superconductor analogy, but this asymmetric glow from R/L antineutrinos is everywhere, at the centre of the Earth as well as the bloodstream and it glows at a thermal bath temperature which hopefully ... the penny has dropped? Of $2.737$ Kelvin.

That's not quite $2.725$ K, but I have no doubt that Graham is thinking hard about this!

From Down South

Numerous images from the September 4 earthquake are now available online. My favourite ones show the new fault offset, like this:

M Theory Lesson 356

We see that noncommutative paths add one dimension to number spaces, just as the quaternions are one complex dimension larger than the complex numbers. Similarly, the diagram shows how nonassociativity adds two (complex) dimensions to the quaternions. It maps the path $XXX$ to the path $XXY$, which is just an edge in the lattice of noncommutative paths.

The square of bracketings is a categorical associator square, and the four object version will break the monoidal structure of a higher category. This square is a natural transformation between functors, so we should not discuss nonassociativity outside of a higher categorical context.

M Theory Lesson 355

Recall the entry $n = 2$, $d = 3$ of the path table, with its noncommutative path weights:
As shown by Duff et al, this diagram encodes the hyperdeterminant invariant for three qubits via the Freudenthal triple system for the Jordan algebra $C \oplus C \oplus C$. The FTS is built from two Jordan alegbra elements, corresponding to the triplets $XXY$ and $XYY$, and two complex numbers, corresponding to the unique paths $XXX$ and $YYY$, otherwise known as $a_{000}$ and $a_{111}$. The FTS neatly manifests the triplet symmetry $SL(2)^{3}$ of entanglement classification. We see this in the grading of (permutation reduced) hyperdeterminant terms. There are three three dimensional representations in the nine terms, given by

$a_{100}^{2} a_{011}^{2}$, $-2 a_{000} a_{111} a_{100} a_{011}$, $-2 a_{100} a_{010} a_{011} a_{101}$

plus permutations on the bits. The three remaining terms of Cayley's hyperdeterminant $\Delta$, fixed under permutations, are

$a_{000}^{2} a_{111}^{2}$, $4 a_{000} a_{011} a_{101} a_{110}$, $4 a_{111} a_{100} a_{010} a_{001}$

Note that all $12$ terms have a total bit weight of $6$, reflecting the invariance of the expression under bit flipping and qubit swapping. The invariant $\Delta$ may be expressed as the determinant of a symmetric $2 \times 2$ matrix with terms of degree $2$ in the $a_{ijk}$. The off diagonal entries of this matrix have coefficient $1$ and the diagonal has coefficient $2$, just like the qutrit matrix from the general theory, which was a reduction of the associahedron (secondary) polytope.

The Tetractys

Here is a poem by Sir William Rowan Hamilton:

THE TETRACTYS

Or high Mathesis, with her charm severe,
Of line and number, was our theme; and we
Sought to behold her unborn progeny,
And thrones reserved in Truth's celestial sphere:
While views, before attained, became more clear;
And how the One of Time, of Space the Three,
Might, in the Chain of Symbol, girdled be:
And when my eager and reverted ear
Caught some faint echoes of an ancient strain,
Some shadowy outlines of old thoughts sublime,
Gently he smiled to see, revived again,
In later age, and occidental clime,
A dimly traced Pythagorean lore,
A westward floating, mystic dream of FOUR.

Neutrinos and Supernovae

Over at the astronomers' forum Galaxy Zoo, member Graham D discusses implications of the unequal $\nu$ and $\overline{\nu}$ Koide masses for cosmology and astrophysics.

In particular, he notes that the very small masses indicate appearance threshold temperatures of relatively recent cosmic epochs. The smallest mass (for the electron neutrino) is $0.0038$ eV and this corresponds to a temperature of $4.41$ K at a cosmological redshift of $z = 0.60$. Graham argues that such recent universal phase changes ought to have observable effects on stellar processes at these special epochs. For instance, supernovas emitting significant energy in massive neutrinos would appear to be dimmed in the optical. It is noted that the supernovae data (see plot below) are beginning to reach a point where kinks at certain redshifts may be detected. Similarly for other stellar processes.

This example merely scratches the surface of ramifications of this Koide sector for astrophysics. Graham rightly chastises me for not elaborating on the possible significance of the $24$th root of unity, namely the phase $\pi /12$. Like Brannen, he appears to be fond of a $12$ neutrino description of mass generation, but personally I see no reason to describe quantum gravitational degrees of freedom with the old language when the terminology of M theory black hole physics is available.

That there is an arithmetic significance to the information dimension $24$ is obvious to mathematicians. For example, the modular discriminant is the $24$th power of the Dedekind eta function on the upper half plane. The Leech lattice in dimension $24$ exists basically because $1^2 + 2^2 + \cdots + 24^2 = 70^2$, and $24$ is the only integer $> 1$ with this square property. The Leech lattice may be given as three copies of the E8 lattice in dimension $8$. Ultimately, this is why I am happy to think about the eighth roots (namely the $\pi /4$ of the basic $R_2$ matrix), because they are also fundamental.

M Theory Lesson 354

Taking noncommutative paths of length $d$ in $n$ variables to their endpoint monomials results in a planar complex of measured simplices. The standard simplices form one column, as shown in the table. If we replace $X$ by a zero and $Y$ by a $1$, the row of measured intervals condenses the $d$ qubit objects. The secondary polytope in this row (see the green book) is the cube. For example, the interval of length $3$ maps to the square, because the triangulations of this simplex are given by the partitions of $3$, namely the partitions $(1,1,1)$, $(2,1)$, $(1,2)$ and $(3)$. Moreover, this square secondary polytope also comes from the noncommutative paths for the interval of length two, since we can draw Young diagrams around the nodes of the planar lattice of noncommutative paths. In higher rows, these standard partition maps may still be applied to the edges of a higher dimensional simplex.

The first instance of a simplex with central vertices is the tetractys. This simplex came from the $27$ length $3$ paths on a cubic lattice. The central vertex stands for the six paths in three distinct variables, namely the six permutations in $S_3$. This is the first vertex weight that is the product of two prime numbers. Volumes of pieces in a triangulation (of a simplex) are also naturally expressed in terms of prime factors, once a minimal volume is normalised to $1$. Such volumes determine coefficients for generalised entropy measures.

Quote of the Week

They have become heroes to the stupid and laughing stocks to those who know better.

from Chris Kennedy

Up North II

If feminists complain in public, there are usually several thousand well fed white guys around ready to tell them they should stop complaining and be grateful they didn't fight in some war. I've been told to behave myself for over 40 years, and it really is getting quite tiresome. The idea that maybe I have actually earned some respect is, well, well off their radar. So why do we complain? It makes us hated everywhere, as we well know, and it doesn't help us in any way whatsoever. We do it for the younger women. There was no internet when we were young, and nobody told us what we would be up against, so we learned slowly. The younger women will be better prepared. Now the fat boys can keep on moaning, but they can no longer silence the voice of Big Sister.

Up North

Things are slowly looking up, here in the north. I have a nice place to live in the Hutt valley, a short train ride from the city. Most days I walk up the hill to the busy campus from the station, and on rare occasions the sun actually comes out. In general, people have been helpful and friendly. I have a library to use, and better software with which to write papers.

M Theory Lesson 353

On the surface across the corner of the cube we can draw the points of the path space up to permutations. There are nine points around the triangle and one central point representing $XYZ$. Let us include an edge whenever a trit is flipped. If an outer edge is weighted by $1/2$ and an inner edge by $1$ we recover a Pythagorean tetractys labeled by both path types and weighted valencies at a node, the latter corresponding to the path count on the cube. This is a more cyclic picture than the use of projective degree that we saw for the associahedron ternary triangle, reflecting the need to replace duality by triality.

M Theory Lesson 352

Some years ago we were considering noncommutative path integrals and higher categories. Now one version of Kapranov's classic work appears in this 2009 book. Here Hermitian matrices appear as a fundamental domain for the exponential map. (Kapranov promises further work on this subject, but all we find is one old paper).

Recall that the paths on a cube could be placed on a hexagon not unlike the three qutrit hexagons of recent posts. In what way do the $27$ possible paths for three qutrits get split up on the cube? We see that these $27$ paths are somewhat unnatural on the big cube, because they vary in length from $\sqrt{3}$ to $3$. However, they neatly fit into one corner of the cube. The point labeled $XYZ$ is reached by the six possible paths (of the degree $3$ hexagon) and stands for all six paths of length $\sqrt{3}$. Similarly, $XXY$ stands for three possible paths. The partition of the $10$ vertices is now $(1,6,3)$ for length squares of $(3,5,9)$ and path multiplicities $(6,3,1)$. That is, $27 = 6 + 18 + 3$. (This puts the real diagonal of a Jordan Hermitian object onto the vertices $XXX$, $YYY$ and $ZZZ$). A face of the cube is selected by eliminating one variable, picking out four three qutrit vertices and a total of eight paths (a basis for off diagonal octonions).

God and M Theory

Lubos Motl informs us that the new book by Hawking and Mlodinow, titled The Grand Design, is headed for the top of the bestseller list ... for ALL books. This is not surprising. Even here in Wellington it took up the front page of the world news last Friday, since everyone wants to see how the book hands over God's job to M Theory. When the hubbub dies down, one suspects that Hawking's popularity will henceforth diminish.

Now given that nobody actually understands M Theory, and one modern God spits on the rotten Patriarchy that abuses her name, this book is bound to generate much outdated dribble. To appreciate that M Theory describes the universe in terms of observers is to realise that a classical universal observer must be abolished, because Jill's universe cannot be John's. At the same time it forces John to allow Jill a God axiom, and Jill to accept John's disbelief. But then John's understanding that Jill's reality is posited in his own image is close to an acceptance that he himself is God, in his own universe, although his conscious self is hardly the observer that surrounds him. And if he cannot know himself then he will never know the Mind of God.

The old picture is as absurd as a flat Earth. One does not need equations to say that the Earth is a sphere, or that John and Jill inhabit the Earth, the centre of their universe. The book, however, seems to focus on the premise of the Big Bang and the existence of an essentially (oh no, not again) classical multiverse. As such, it fails to be as deliciously heretical as one would hope. And for God's sake, a leap forward in physical theory does not a Theory of Everything make.

M Theory Lesson 351

In the 1992 paper, Kapranov et al finish by detailing the example of three ternary quadrics. The full degree $12$ resultant is listed as $68$ bracket terms grouped according to $28$ different weights. The $14$ vertices of the associahedron appear as the singleton terms. In terms of brackets, these are

$-[145][246][356][456], [146][156][246][356], [145][245][256][356],$
$[145][246][346][345], [126]^2[156][356], [125]^2[256][356], [134]^2[246][346],$
$[136]^2[146][246], [145][245][235]^2, [145][345][234]^2,$
$[136]^2[126]^2, [125]^2[235]^2, [134]^2[234]^2, [123]^4$

where we observe how a zero area, like $[135]$, occurs as an unseen $1$ due to the zero volume exponent. These vertices thus depend on the metric structure of the triangle with vertices at $1$, $2$ and $3$. The so called toric specialisation takes bracket terms to monomials in $c_i$, the six coefficients in a quadratic form $f(X,Y,Z)$, via

$[i_1 i_2 \cdots i_k] \mapsto \textrm{det}(i_1, i_2, \cdots, i_k) c_1 c_2 \cdots c_k$

Here we see the volume factor go to an entropic coefficient, as in the term $[123]^4$ giving $256 c_{1}^{4}c_{2}^{4}c_{3}^{4}$. The sum of all $28$ such determinant terms turns out to be the product of principal minors for a $3 \times 3$ symmetric matrix with diagonal $2c_1, 2c_2, 2c_3$ and off diagonal terms $c_4$, $c_5$ and $c_6$. Thus a $3 \times 3$ matrix is associated to the six degree $2$ three qutrit functions.

Observe how the bracket monomials above could actually resemble associations of five objects, which usually label the associahedron by trees dual to the triangulated polygon. There are four nodes on a five leaved tree, and a total exponent of four on the bracket monomials. A tree node is labeled by a bracket, but a given bracket is allowed to repeat itself. For instance, if we imagine that $[123]$ labels the bottom node of the special tree for $[123]^4$, then $[123]$ stands for the fact that all subtrees above and to the left of the node are full binary trees. Then the mirror tree would be given by the term $-[145][246][356][456]$, and the spreading out of the $1,2,3$ reflects the fact that the left branch is now a minimal tree. Here we see metric structure being associated to tree shapes.

Thinking of Christchurch

Dear Christchurch, I hope you are all OK. Initial reports say magnitude 7.2.

M Theory Lesson 350

A general hyperdeterminant begins with an $n$ dimensional matrix of shape

$(k_1 + 1) \times (k_2 + 1) \times \cdots \times (k_n + 1)$

and the special set $A$ is now the full set of entries $a_{i_{1} i_{2} \cdots i_{n}}$. The first main theorem (page 446) says that the hyperdeterminant of this matrix is non trivial if and only if

$k_{j} \leq \sum_{i \neq j} k_{i}$

for all possible $j$. For a hypercubic matrix, where all the $k_{i}$ are the same, this is certainly true. That is, the hypercube for $n$ qudits always gives us something. The next proposition says that this hyperdeterminant is invariant (in a suitable sense) under what could be called a SLOCC group: $SL(V_1) \times \cdots \times SL(V_{n})$. Another cool result (page 454) tells us that the degree of our hyperdeterminant is given by a sum over partitions $\lambda$

$N(k_1, \cdots, k_{n}) = \sum_{\lambda} (d_{\lambda k}) (1 + m_2 + \cdots + m_p)! \prod_{i = 2}^{p} \frac{(i - 1)^{m_i}}{(m_i)!}$

where the $m_i$ are the indices of the partition $\lambda$ (so that $m_1 = 0$) and $d_{\lambda k}$ counts binary matrices with row sums in the ordered partition $k$ (of the $k_i$) and column sums in $\lambda$. The cubic case $(k + 1)^{3}$ is given by

$N(k,k,k) = \sum 2^{k - 2j} \frac{(j + k + 1)!}{(j!)^{3} (k - 2j)!}$

for a sum over $0 \leq j \leq k/2$. We see that for three qubits, where $k = 1$, there is only one term and it gives the degree $4$ of the classical polynomial.

M Theory Lesson 349

Woit has been reasonably bashing the Imperial press office for hyping work of Duff et al about entanglement for multiple qudits. Now Philip Gibbs blogs about his own 2001 paper linking multidimensional determinants to properties of elliptic curves. These tori are secretly what the green book is about!

Hyperdeterminants are special cases of the general discriminants. Recall that typical integer coordinates for a set $A$ lie on a lattice in the continuum. The dimension depends on the number of variables, and by limiting the digits to qudits we limit the allowed degree of terms in our initial polynomials. So for the triangular associahedron, we had qutrit objects $0$ and $1$ and $2$ and monomials of degree two, like $Y^2$ for the point $(0,2,0)$ in the plane. We can do this for any type of qudit and any number of particles $n$. The dimension of the secondary polytope will go up with $n - d$ and its shape encodes the resultant (entanglement measure). It quickly becomes too lengthy to write down, but the algorithm is completely understandable.

CKM Again

The standard Euler parameters for the CKM matrix are (in degrees) very close to $13.01$, $2.39$ and $0.201$. Observe that these values are essentially the same as the three $R_2$ parameters $(a,b,c)$ $=$ $(0.231, 1/24, 0.0035)$, since these take the form $\tan \phi$. The difference is that the fourth phase parameter is now given as the product $abc$ of $R_2$ factors.

M Theory Lesson 348

This green book should be popular amongst physicists working on black hole entropy measures.

Recall that the set $A$ of points in the continuum of dimension $k - 1$ represented monomials in $k$ variables. A major theorem in the book shows that the secondary polytope of the (generalised) discriminant of $A$ may be expressed in terms of $A$ and its convex hull via triangulations, just as for the associahedra. In general, a vertex of this magic polytope, along with the simplex volumes of the triangulation, gives us a monomial of the discriminant, with its coefficient. The logarithm of the coefficient takes the form

$\sum V_{i} \textrm{log} V_{i}$

which is clearly an entropy in terms of the simplex volumes. For example, the triangles inside a (triangular) hexagon give us the coefficient for a $3d$ associahedron vertex. The authors point out that this entropy explains classical numbers such as the $27$ in the cubic discriminant of a polynomial. Later on in the book we find other instances of this non statistical entropy. In connection to discriminant hypersurfaces in tori, we see an entropy with the property that the sum $\sum V_{i}$ of volume terms equals zero. This suggests the possibility of negative volumes, or areas.

The final chapter covers general hyperdeterminants, including of course Cayley's classic example, now well known in studies of multiple qubit entanglement. It is encouraging to know that mathematicians have thought hard about computational tools associated to such general matrix objects.

M Theory Lesson 347

This triangle version of the associahedron is given coordinates of homogeneous degree $2$. That is, it lies in the plane

$x + y + z = 2$.

The other ternary functions have varying degree, according to the partition of the total number $27$, where the six point triangle uses $6/27$ points. Note that the degree $3$ case might resemble the seven points of a Fano plane. However, the six outside points are the mixed functions, like $012$, and the centre point is the function $111$, so now six lines are required to cover the outside points. The point $111$ also lies on a line $y + z = 2$, when $x = 1$ ,and also the line $x + y = 2$. Thus the hexagon actually has a self intersection at $111$.

M Theory Lesson 346

On page 436 of the remarkable book about multi dimensional determinants, by Gelfand, Kapranov and Zelevinsky, they describe how the equivalence between a deformed triangle with marked midpoints and a hexagon

is related to the description of the $3d$ associahedron (labelled as usual by chorded hexagons) as a secondary polytope. To the triangle one associates a set $A$ of six (three dimensional) coordinates, namely

$(2,0,0),(0,2,0),(0,0,2),(1,1,0),(1,0,1),(0,1,1)$

and these stand for three quadratic forms, namely

$f_1 = a_{11} X^2 + a_{12} Y^2 + a_{13} Z^2 + a_{14} XY + a_{15} XZ + a_{16} YZ$
$f_2 = a_{21} X^2 + a_{22} Y^2 + a_{23} Z^2 + a_{24} XY + a_{25} XZ + a_{26} YZ$
$f_3 = a_{31} X^2 + a_{32} Y^2 + a_{33} Z^2 + a_{34} XY + a_{35} XZ + a_{36} YZ$

in terms of a $3 \times 6$ matrix. The associahedron secondary polytope is associated to the set $A$, and there is a resultant $R(f_1,f_2,f_3)$ which is a degree $12$ polynomial in the $a_{ij}$. The $14$ allowed triangulations of the hexagon, and actual volumes associated to the polytope, correspond to the structure of $R(f_1,f_2,f_3)$. That is, if we let $[j_1 j_2 j_3]$ be the ordinary determinant of the $3 \times 3$ submatrix $a_{ij_{k}}$ then these $14$ expressions include, for example

$[145][245][256][356], [134]^{2}[246][346].$

Twistor theorists will recognise the (Grassmann) process of taking square submatrices for nice coordinate systems. Each digit $j_{k}$ stands for a point on the triangle. If we choose to label the three vertices $1,2,3$ and the midpoints $4,5,6$ then the two internal triangle configurations correspond to the polynomials

$-[145][246][356][456]$ and $[123]^{4}$

since these contain the terms $[456]$ and $[123]$. These terms look different to the other $12$ terms: the first because it is negative and the second because it only involves one triangle.