M Theory Lesson 274

Tony Smith brings our attention to a recent set of slides by C. Vafa of F theory fame. Vafa is discussing some simple $3 \times 3$ matrices for neutrino and charged lepton masses and mixings. The final slide indicates a mass hierarchy of the form

$m_1 : m_2 : m_3 = c^4 : c^2 : 1$

where $c$ is a characteristic GUT coupling. This expression should be familiar to AF readers, at least in the context of MUB operations.

Recall that a $3 \times 3$ symmetric matrix is the sum of a symmetric 1-circulant and a 2-circulant, which is automatically symmetric. Thus a symmetric matrix $M$ with first row $(c^2 , c, 1)$ may be expressed as where the second factor is itself a sum of circulants. This makes explicit the contribution from dimension $2$ that we see in the Fourier decompositions for the mixing matrices. Vafa's (neutrino) Yukawa matrix is just of this form for $a=0$ and $c$ a cubed root of unity.

Visa Update

Is it almost May now? Yes, I thought so. Well, today I tried to contact the correct visa section at the Border Agency, but the first phone call, after some minutes, ended up with a voice message giving me another number, and on phoning that number I was told that they were too busy to take calls. Anyway, as far as I can tell somebody there still has my passport etc, along with a perfectly valid application to extend my stay beyond some weeks ago.

M Theory Lesson 273

Abtruse Goose tells us that for a matrix $A$, the exponential satisfies

$e^{F^{-1} A F} = F^{-1} e^{A} F$

We can easily apply this to Koide matrices, which are diagonalised by the Fourier transform matrix $F_3$. It follows that a Koide matrix is an exponential of the matrix The first entry of the circulant $A$ is $\textrm{log} (\sqrt{m_1 m_2 m_3})$. Expressed in terms of the natural scale

$\mu = 25.054309435 \sqrt{\textrm{MeV}}$,

the charged lepton case takes the value

$\textrm{log} (\sqrt{2} + \textrm{cos}(\frac{2}{9}))(\sqrt{2} + \textrm{cos}(\frac{2}{9} + \frac{2 \pi}{3}))(\sqrt{2} + \textrm{cos}(\frac{2}{9} - \frac{2 \pi}{3}))$

which gives us more crazy numbers to play with! The ease of swapping addition for multiplication in the circulant Fourier transform is a sign that the Fourier transform might have something to do with basic arithmetic.

Problem with the Matrix

As many AF readers know, Carl Brannen has been diligently trying to solve the magic matrix decomposition problem for $3 \times 3$ unitary matrices. Now Lubos Motl very helpfully decided to solve this problem for Carl, also providing witty commentary along the lines of

Let me give you some examples because they’re easier for you than mathematics. Math class is hard, Barbie.

Unfortunately, he made a very elementary mistake and failed to solve the problem after all, but was somewhat annoyed, it would seem, when Carl pointed this out to him. I am informed that Carl has now been banned from Motl's blog. It is a shame that we cannot continue this fruitful collaboration because, as Carl often points out, Motl has made lasting contributions to physics with his concept of tripled Pauli statistics. We will just have to continue playing with our simple matrices alone.

Quote of the Week

In Week 201, which came up recently at the cafe, John Baez said, in relation to Galois theory,

The moral is this: you can't solve a problem if the answer has some symmetry, and your method of solution doesn't let you write down a correct answer that has this symmetry! An old example of this principle is the medieval puzzle called Buridan's Ass. Placed equidistant between two equally good piles of hay, this donkey starves to death because it can't make up its mind which alternative is best.

Good advice for string theorists and particle physicists, perhaps? Also check out some of Baez's cool new links.

M Theory Lesson 272

Today at morning tea we talked, amongst other things, about neutrino symmetries.

Recall that the symmetry group $A_4$ can be used to describe the tribimaximal mixing matrix, $T$, for neutrinos. We have probably neglected to point out before that $A_4$ is secretly a $2$-group, where the $2$ refers to categorical $2$-arrows. That is, $A_4$ is the group $C_{3}$ acting on the group $C_{2} \times C_{2}$. There is another order 12 group, which we will call $G$, where $C_{2} \times C_{2}$ is replaced by the only other four element group, $C_{4}$.

This reminds us of the Fourier decomposition $T = F_{3} F_{2}$. The group $C_{2}$ is generated by the Pauli swap, $\sigma_{X}$, and $C_{3}$ may be generated by the basic three dimensional permutation $(231)$. Any discrete Fourier expansion is expressed as an element of the group algebra for one of these groups. Note that these groups are also generated by the circulant operators $R_{d}$, and in dimension two $F_{2}$ also gives $C_{2}$. Then $T$ may be defined directly in terms of $A_4$ generators as $T = R_{3}F_{2}$.

Now since $R_{2}$ does not generate $C_2$, but $R_{2}^{2}$ generates $C_{4}$, we can consider the alternative mixing matrix $S = R_{3} R_{2}$ as a $G$ (non local) version of the mixing matrix. This was the matrix that vaguely resembled a root of the CKM matrix.

Aside: A new paper by Harrison et al discusses the nearness of the CKM matrix to unitarity.

Breakfast Ideas

Breakfast at my new house is very civilised. We sit down together at the table, chat a little and eat slowly. This morning a visiting materials physicist lamented about previous unsuccessful attempts to study category theory, the usefulness of which he was anyways greatly in doubt. I promised to come up with a recommended reading list, tailored to a materials scientist, but having thought about it further today I must confess to being totally stumped. Clearly there are still some gaps to fill in the introductory category theory genre. Now we can all find the good arxiv papers and standard textbooks. Any other recommendations?

These days one often comes across people in other (ie. usually not physics) departments who have some interest in category theory. I already mentioned Lawvere and Rosebrugh's book, Sets for Mathematics, to the lovely young philosopher who is studying Frege, amongst other things. And I suspect that Ross Street's book on Quantum Groups is also useful. Poor Carl is presently struggling with the ubiquitous text by the late Mac Lane. This book is very good, but perhaps not for the beginner. My favourite is Sheaves in Geometry and Logic, but that betrays a bias towards topos theory.

Aside: Now you can support Abtruse Goose with this groovy cap. I thought of merchandising for funds, but unfortunately it is entirely against my green anti-materialist ethos. Maybe I'll buy a cap though. AG deserves it.

M Theory Lesson 271

Having committed ourselves to finite field arithmetic for now, we remember that MUB operators use only a small set of truth values. The qubit case requires the field with five elements, whereas qutrits only require the four element field.

But the factor of $i$ required in the qubit case means that one does not automatically encounter (at least in quantum mechanics) the mod $7$ multi muon arithmetic, because six dimensions uses $12$th roots of unity. However, the main obstacle to using square roots in the qubit case, namely the smallness of the number $2$, no longer applies in dimension $6$. Thus there should be a measurement operator set based on the seven element field for the important dimension six case.

The interesting case from the point of view of modular mathematics is the number $24$, which initially appears for (stringy $F$ theory) dimension $12$, combining mass and spin quantum numbers. Recall that these dimensions are also counted by the triple of Riemann moduli spaces, $M_{0,6}$, $M_{1,3}$ and $M_{2,0}$, each of twistor space dimension.

Oxford Life XII

Today I visited the museum across the road. There is an exhibition on the life of Darwin, focusing on the voyage of the Beagle. Unlike most such exhibitions, the building itself is part of the exhibition, with a plaque marking the place of the great debate between Bishop Wilberforce and T. H. Huxley in 1860.

In Darwin's time, ideas about natural evolution had already occurred to a number of people. The truly new feature of Darwin's theory was the idea of a common ancestor, entailing both a diversification of life over time and an increase in complexity, as a larger variety of organisms adapt to their environment.

Various physicists have discussed the application of evolution to the cosmos. Unfortunately, these discussions usually involve multiverses of an alarmingly Boolean character, and a 19th century Darwinism that would have modern fans of Hegel rolling their eyes. Moreover, one could argue forever on an appropriate measure of complexity (monotone in epoch) for a classical cosmos. At the end of the day, this total denial of quantum cosmology ruins all attempts to correlate the existence of life with the special parameters of fundamental physics.

Evolution would look quite different from the centre of the universe. My past is continually constructed from a complex collection of local propositions, most of which I share with other humans, although they too are but figments of the imagination. A past ancestor is a being about which I have, in some sense, more knowledge. Moreover, since your local universe is entirely your own, and not at all mine, any omnipotent creature that happens to inhabit all universes would have to be a shallow beast, its existence relying only on the thinnest common threads of our ideas.

Stringy Appeal

The most frustrasting element of the String Wars is the brick wall of the quark gluon plasma. As Mottle quite rightly points out:

The minimum ratio of viscosity and the entropy density can be translated in another way: it is actually the maximum ratio of the entropy density to viscosity. For a fixed viscosity (and volume), which physical system has the highest entropy density (and therefore the net entropy)? Well, in the gravitational context we know the answer. Black holes maximize the entropy. They're the ultimate bound state of matter into which the matter collapses into, and by the second law of thermodynamics, they must maximize the entropy among all such bound systems.

It is true that string theory correctly retrodicted this behaviour for plasmas. And it is also true that, under this observation, it must be well nigh impossible for a brainwashed clever string theorist to buy the idea that the physics of string theory is mostly wrong. In M Theory, we agree wholeheartedly that black hole physics dictates the behaviour of such plasmas. We disagree that classical geometry, classical symmetry principles and unobserved SUSY partners have much to do with it.

The black hole entropy is described by an entanglement measure which, using category theory, may be given meaning entirely outside the world of complex geometry. Even Hilbert spaces and spectral triples disappear. In this brave new world, naive stringy extra dimensions simply count operator sets. Lagrangians are emergent. So there is a place, on the other side of that brick wall, where people are standing and shouting back, finding it impossible to believe that they could (relatively speaking) be wrong. For string theorists, there is one question: do you seriously believe that your so called theory is crazy enough?

M Theory Lesson 270

In M Theory, we often talk about classical structures in the topos Set and quantum (and classical) structures in a category of vector spaces, Vect.

Thus it shouldn't be surprising that quantum mechanics might have something to do with arithmetic. Consider finite dimensional vector spaces over the field with two elements. If we were to sum a plane with another plane, the four dimensional resultant space would have $2 + 2 = 4$ basis vectors. On the other hand, there would be $4 \times 4 = 16$ elements in the full space. The basis cardinality was added while the vector space cardinality was multiplied. This is not so obvious when working with spaces over infinite fields.

Now recall that the adjunction from Set to Vect was well behaved for finite fields. And we can talk about Set as the category of vector spaces over the field with one element. The forgetful functor from Vect takes the abovementioned product into a set with that many elements. So if we started with two two element sets, and chose two two dimensional spaces over the field with two elements, then we would end up with a $16$ element set of vectors in Set, along with a natural map to the four element set $2 + 2$ with which we started.

A Stringy Yarn

Who would have thought this day would come? Not content to ignore Louise's latest post, The String Enterprise Unravels, our friend Mottle joins the comment section at his most charming. Although Carl and others are merely mammals, Louise and I actually earn the status of inferior humanhood! Maybe he's miffed that our blog ratings are improving.

Oxford Life XI

The mornings are still quite cold here, but the daffodils and exotic squirrels are everywhere. Conference time is over. Two prominent members of the group, Bob and Mehrnoosh, are heading off to Canada for a while. (Hmmm, I wonder what mischief we can get up to while they're gone.) Meanwhile, I am planning to move into a nice house next week, not far from work. And needless to say, the Home Office are still working on my visa application.

QPL 09 II

Jamie Vicary bet me a bottle of wine that the fairy field would be found in the next few years. Actually, he was willing to bet on the next two years, but I let him off the hook on that one. (Unfortunately, my total stakes on this question include a little wine and around ten dollars, so even I can afford to lose.) At QPL, Vicary spoke about his characterisation of the complex numbers using natural structures in dagger monoidal categories with superposition (see this paper).

Superposition says that given two morphisms $f,g: A \rightarrow B$, there exists a morphism called $f+g$ and addition is commutative and associative. The crucial notion is that of a $\dagger$ limit for a diagram $D$, defined to be a limit $L$ such that the arrows $f_{S}: L \rightarrow D(S)$ satisfy

$\sum f_{s} \circ f_{s}^{\dagger} = 1_{L}$,

where the sum is over a set of source objects in $D$. This is a normalisation condition for superpositions. When all objects in a discrete diagram $D$ are sources, this reduces to the categorical biproduct $\oplus$. Given a category with a zero object and all finite biproducts (such as the category of Hilbert spaces) it turns out that there is a unique superposition rule.

One of the things that Vicary shows is that, for a category with tensor unit $I$ and all finite dagger limits, the semiring of scalars $I \rightarrow I$ has a natural embedding into a characteristic zero field. This relies on the decomposition of any non-zero ordinal $p: I \rightarrow I$ into a diagonal arrow

$I \oplus I \oplus \cdots \oplus I \rightarrow I$

and its adjoint codiagonal. So, if we want to work with finite fields of characteristic $p$, we can now identify exactly which pieces of complex number structure break down. For instance, there might be a zero map constructed from a finite diagonal and codiagonal on the unit object.

Cosmology 101

Louise Riofrio has never given up trying to explain to adults a varying speed of light cosmology that a child could understand. Now, thanks to the theorist Marco Frasca, Carl Brannen has observed that a varying speed of light solution to Einstein's equations results from a five dimensional Kasner metric:

$\textrm{d}s^{2} = -\textrm{d}t^{2} + t(\textrm{d}x_{1}^{2} + \textrm{d}x_{2}^{2} + \textrm{d}x_{3}^{2}) + t^{-1} \textrm{d}x_{4}^{2}$

The mixture of exponents for $t$ arises from the (three dimensionally) isotropic solution conditions

$\frac{1}{2} + \frac{1}{2} + \frac{1}{2} - \frac{1}{2} = 1$
$\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = 1$.

These are like magic matrix conditions for a $4 \times 4$ matrix. The fifth dimension, which both Carl and string theorists are fond of, is necessary for these conditions to be satisfied. There is no Dark Force.

Observe that the coordinate speed of light in the three spatial dimensions goes like $c = 1/ \sqrt{t}$. When taken seriously as a solution, Riofrio's equation $R=ct$ then states that universal expansion is characterized by $R = \sqrt{t}$, or $M = R^{2}$. Since this is supposed to approximate a locally emergent cosmology with T duality properties, it indicates that string theorists are wrong to think that mass should correlate with string length, something that every 5 year old quantum mechanic knows.

A Debate

AF needs a link to this amusing conversation at Tommaso's place.

QPL 09

Day One of QPL 09 went relatively smoothly and there were a few interesting talks, but now I will just mention the beautiful talk by Joachim Kock on his recent work with A. Joyal.

They prove a (nerve) theorem characterising compact symmetric multicategories (modular operads) in terms of Feynman graphs (which is an extension to the work in this paper). Recall that whereas categories are about edges (objects) and vertices (morphisms) which compose, a multicategory allows (planar) rooted tree reductions. Unrooted trees take us to cyclic operads, and finally with loops we have (undecorated) Feynman graphs, with external legs.

The classical nerve theorem for ordinary categories looks at an adjunction between Cat and the category of directed graphs. A graph $G$ is sent to a category with objects the vertices and morphisms the paths in $G$. There is a certain factorization property for this adjunction that gives a functor $\Delta_{0} \rightarrow \Delta$, where $\Delta_{0}$ is the class of distance preserving maps, that is to say graphs with matching path lengths between distinct vertex sets. $\Delta$ includes certain extra diagrams.

In the case of the modular operads, the theorem takes a similar form. The analogue of directed graphs here is the category of presheaves on elementary graphs (the basic building blocks). A generic map is a refinement of a star graph (with one vertex) where the vertex is replaced by another graph so that the outputs match up. But there is also a class of etale maps, or covers, such that the cover of a graph is a pullback square. Anyway, there is a monad $T$ which expands a presheaf on elementary graphs to one on all Feynman graphs. The modular operads are the $T$ algebras for this monad.

Oxford Life X

MFPS has just ended and tomorrow we're into QPL, of more interest to AF readers. I agreed to help out with the videos, so I apologise in advance if the Pauli effect kicks in.

Meanwhile, the Home Office are of course taking their time deciding whether or not I should be in the country, although I don't know how I could leave without a passport, and for reasons far, far beyond my comprehension, it seems I may have to wait another week or so until I finally get paid in pounds.

Interlude

I have been busy (a) attending conferences and (b) thinking about nails in coffins.

M Theory Lesson 269

In M theory, viewing the $1$-ordinals as one level trees usually leads us to associahedra polytopes, which can be concretely embedded in a real number space.

These real spaces are useful for analysing physically interesting integrals associated to MZVs, but the question is, what are those real spaces doing there? We don't seem to need them. The association of MZVs to patterns arising from operads is quite functorial, leading one to suspect that MZVs should be defined not from the point of view of standard analysis, but as canonical numerical invariants for categorical structures. Then one wouldn't need to discuss real backgrounds. Then, if we still cared, later one could worry about whether or not these zetas were really the same as the ones that we thought we were talking about when we felt integrals were unavoidable.

Conference Heaven

Doh, I squandered the opportunity of April 1! Anyway, the conference season kicks off this week with MFPS 25 on Friday. Then next week we have QPL 09. In June, Perimeter is running Categories, Quanta, Concepts. Gee. Nowadays there are just too many categorically minded conferences to attend!