Arcadian Functor

occasional meanderings in physics' brave new world

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Location: New Zealand

Marni D. Sheppeard

Saturday, March 31, 2007


Some people are having difficulty understanding Louise, even when she draws very simple Space Time diagrams. Let's use a slightly different diagram. We see that an observer at Now looks back to the past, and the infinite future looks back to Now. How much simpler can it be?

Upcoming Seminar

Next Thursday, on the 5th of April, I will give a seminar for the in-house gravity meeting at UC. Everybody is welcome. If I am mysteriously detained by authorities before the talk, which is at 4pm in room 701, I will endeavour to put my slides on this blog.

M Theory Lesson 35

One day, many years ago, I wanted to catch a bus from Sikkim down to the plains. It was my first lesson in the theory of order-under-chaos. A major landslide had blocked the only road out of the mountains, and yet in no time at all a relay of buses was setup. Travel between buses simply required a short walk along the goat track over the landslide debris.

It looks like M Theory needs to discuss the concepts of operadification and cooperadification. Don't worry, I am not suggesting that we adopt this cumbersome terminology. Instead, let's name the 2 functors that describe these processes entropy and information respectively. Entropy describes the process of leaping up to the next quantum level, which is far more complex and intricate. Information is the dual process of dropping down whilst investigating a question. It might not be humans asking questions. The galaxy might want to ask Computer Earth a question now and again.

Baez et al have been describing a program of groupoidification. This lives in the realm of n-category theory, and we expect that such categorical structures will arise as algebras of master operads.

Thursday, March 29, 2007

M Theory Lesson 34

What is a Machian principle for inertia in a quantum world? Inertia is a property of moving bodies in a classical reality, but this reality must emerge from quantumness at all scales. Once we understand such a reality, inertia is more fundamental than mass or distance or time, because it is a statement that the dynamics of any given body should be related to those of all others.

In M Theory, the classical reality is built with levels of quantumness. That the levels themselves are quantized is clear by looking at the world: the atoms and molecules, the planets and solar systems, the galaxies and clusters. The lowest interesting level describes the observed particle spectrum of light particles. This level is a universal manifestation of the prime number 3. Matti Pitkanen has described such a reality in detail. There is no vacuum concept but nothingness itself, as Schwinger described it, and this primitive generator of the Number Theory Universe is also everything.

Louise Riofrio, who is speaking at Imperial College today, has amounted a large body of evidence in support of a Machian cosmology.

As Schrodinger said in the 1944 book What is Life?
"From the early great Upanishads the recognition ahtman=brahman … was in Indian thought considered, far from being blasphemous, to represent the quintessence of deepest insight into the happenings of the world."


The bloodhound commenter nosy snoopy has found this paper by Schlesinger. It is a remarkably clear exposition of the notion of a tower of quantisations, in terms of deformed Turing machines. All of String theory fits onto the lowest rungs of this tower. On a related note, Never Ending Books informs us of a new paper on the Manin-Marcolli cave, inspired by Plato.

The physical principles of M theory also include the n-alities that arise from the Machian nature of inertia. Carl Brannen has made great progress undertanding inertia for the first rung. See this PF post about the particle masses. They are organised according to the primes 2 and 3. This number 3 is the 3 that we have been discussing in recent posts. In the world of 3, one would work with trialgebras, rather than bialgebras; trioperads rather than bioperads. Naturally that is a master idempotent-swapping trioperad.

Physically, we observe that the classical reality that emerges from considering all scales is nothing short of a Mathematical God. The unreasonable effectiveness of mathematics is no longer unreasonable.

Behold, I come quickly: hold that fast which thou hast, that no man take thy crown.

Wednesday, March 28, 2007

Countdown III

This new kneemo (Michael Rios) paper should make interesting reading. And don't forget Louise's talk!

M Theory Lesson 33

Let's have some more fun with pictures. A ribbon can be drawn on a boundary tube, even if the tube has to twist to keep up with the edges of the ribbon. Using the edges of the ribbon as boundaries for square pieces of tube, we could draw swiss cheese pictures on the cylinder. But instead of disc operad circles we could add in points and make a swiss cheese cubes operad picture. I apologise if I'm making up inappropriate names for these diagrams. According to Aaronson, M Theory will probably not be able to show that P = NP. This hypothesis is about packing cubes inside bigger cubes in polynomial time.

Tuesday, March 27, 2007

Countdown II

Oh, all right, I can't resist. For the conference countdown, let us ponder the fact that the correspondence between the prime three and the simple particles may give us an M theory derivation of the baryonic mass fraction, which would agree with Louise Riofrio's result of $\frac{\pi - 3}{\pi}$.

M Theory Lesson 32

Well, I can't compete with Louise's, Tommaso's and Mottle's blogs for conference blogging this week, so let's continue with a few remarks on p-adic logic. Mahndisa made an interesting comment on kneemo's blog. She said:

Any three vertex shape could represent three adicity in context and the lines that connect each vertex could be seen as a swapping morphism of sorts. I played around with this geometry a while ago and you can extend it up to n rational dimensions. A four adic system is represented by a trapezoid, rectangle, rhombus or square, each of the vertices are connected via lines and each of these lines represent swapping.

Mahndisa has been studying Matti Pitkanen's p-adic physics for quite some time now, and has a unique perspective on where all this is heading. From a categorical point of view, it is interesting to note that we are discussing new structures. I recall Batanin discussing a mysterious kind of operad which swaps sources and targets. Recall that sources and targets are idempotents for us.

Thus the usual 2-adic situation is the tip of the iceberg. And since categories can arise as algebras for operads, the p-adic logic would give rise to new kinds of category.

Sunday, March 25, 2007


As the countdown commences, let us ponder for a moment the meaning of a new cosmology for the human vision of its place in the cosmos.

Nowadays, the Newtonian universe feels cold and mean. One often hears the sentiment that we are no more than fluff on the edge of a random galaxy. With Dark Matter thrown in, people say we are not even that. After pondering a question from M. Porter I thought of coming up with some new slogans. Let's try this one out for size first: we are Hawking radiation.

In the 20th century view of the cosmos, this is really no better than thinking of ourselves as garbage. But the slogan is meant as a quantum gravitational one, and the concept of Hawking radiation is immensely rich. By considering only the lightest particles of the standard model we have trusted a false picture of nothingness. But the complexity of life has increased with time. The black hole at the centre of the earth is a part of what we are.

M Theory Lesson 31

Ternary logic, as introduced on kneemo's blog, is based on functions between three objects: 0, 1 and 2. Stepping back to the two object case, we see that there are four possible ways of mapping 0 and 1 into the two element set, which plays the role of subobject classifier in the topos Set. The values 00, 01, 10 and 11 form the four vertices of the parity square. It was the combinatorics of this simple square that led Gray to develop 2-categories and the Gray tensor product in the 1960s.

A unary arrow $0 \rightarrow 1$, thought of as a category, is used to lift morphisms in any 1-category to functors. Similarly, we would expect the square to form pseudofunctors for morphisms. This is what happens when the Mac Lane pentagon is lifted up to the sides of the parity cube.

Given that Gray based his whole theory of 2-categories on the parity square, it is natural to ask what would happen with the parity cube. After all, QCD prefers ternary logic. As kneemo noted however, things are immediately different for the cube. There is only one possible square on the two letters 0 and 1, but there are three possible cubes on the letters 0, 1 and 2. This brings a notion of triality into this higher categorical structure, which we expect will clarify the division algebras.

So it seems that all of 20th century physics comes from understanding nothing more than the low prime number 3. Then there are more primes...

Saturday, March 24, 2007

It's Time

As a relatively new blog, kneemo's U Duality is not as popular as other excellent physics sites. It is time this changed. He has some truly amazing concise, easy to read posts on M theory. Was that the parity cube he mentioned today? Goodness me, it does appear so. I suspect that it won't take long now for our more astute readers to figure out that his ternary logic picture has something to do with (a) string theory symmetries and (b) the stuff Carl has been talking about.

So Carl's derivation of the particle masses (sorry, I can't say lepton anymore, because there are more of them now) fits into a language capable of producing M theory? Well, yes folks, indeed. Stay tuned to Arcadian Functor, your up-to-date news on unpublishable ideas.

Friday, March 23, 2007

Bohmian Bloom

Carl Brannen appears to have posted nibbles all over the place regarding a new way of looking at the delta parameter in the 3x3 mass matrices, which should bring a smile to the face of the Bohmian mechanics. The nice thing about a good theory is that everybody is happy.

It involves exchanging a classical function $\psi$ for an exponentiated (quantum) version. For us, as always, this is about the profound interaction of addition and multiplication thought of as monads. (This came up recently in our discussion of the Riemann hypothesis).

And for some fun, here is my stylised version of Kuperberg's generalised 6j symbol, from the spider paper. The mushy rectangles are Jones-Wenzl idempotents.

Thursday, March 22, 2007

Time and Again

Connes has a good post on emergent Time at NCG. Recall that Jones' subfactors for a Galois theory for the $II_{1}$ case gave rise to the planar operads that we are interested in. Apparently Connes' thesis was on the reduction of type $III$ automorphisms to the $II_{1}$ ones. This naturally brings to mind our discussion with Matti Pitkanen on the appearance of braids in the boundary of his light-like 3-spaces. The convergence of ideas between these different NCG approaches is beginning to look promising.

For those in the UK, don't forget to hear Louise Riofrio talk about emergent time at the cosmology conference at Imperial.

Wednesday, March 21, 2007


Louise Riofrio informs us that efforts are being made to remove her talk from the program of the cosmology conference at Imperial College. She is scheduled to speak on March 29 at 12.30 pm.

If you would like to express support for Louise, the organising committee may be contacted as follows (thanks to Louise for the information). If, on the other hand, you wish to complain bitterly about the falling standards of such an august institution, you are also most welcome to contact the organising committee, in whom we have complete faith.

Richard Lieu:
Carlo Contaldi:
T. Kibble:

M Theory Lesson 30

A paper by D. Thurston introduces the idea of a knotted trivalent (ribbon) graph (KTG). It turns out that these gadgets may be generated from three simple graphs, namely the unknotted tetrahedron and two unknotted Mobius strips, one with a left twist and one with a right twist. The moves allowed on single graphs are the bubble move and the unzip move There is also a connected sum operation, which splices two graphs together along an edge. Any knot may be represented by a string of KTG operations. Bar-Natan's important paper on non-associative tangles includes a pentagon relation, which Thurston encodes via KTG moves. The sequence of three moves begins with three unknotted tetrahedra, which are connected via sum and then unzipped to obtain the final triangular prism This is of course reminiscent of the gluing of three sides of the parity cube on which the categorified Mac Lane pentagon appears.

Tuesday, March 20, 2007

Background Independence

Let's get back to basics. When one starts thinking about the mathematical manifestation of background independence, whatever that means, it is very easy to get in a muddle. Maybe one picks a collection of nice principal bundle spaces and then tries deleting as much as possible. Spacetime points? OK, gone. Oops! Everything else seems to have gone out the window as well. Internal symmetries? Disappeared. Poof. Is there anything left? What about the spacetime of quantum mechanics? Well, we'd better figure out a way to do quantum mechanics without the usual wavefunction paraphenalia.

Are Feynman diagrams all right? Well, they might be, in the context of diagrammatic reasoning, but then that pesky Minkowski background is still lurking there somewhere... Maybe if we turned that into twistor geometry things would finally look up. Points look like spheres now, which in itself doesn't sound like a huge advance, but the beauty of it is that causality now has a language capable of stepping outside of set theory. Category theory isn't just there to put a whole lot of complicated geometry into a neat package. It allows foundational mathematics and real physical calculations from new axiomatics.

Sigh. If only it was all a little bit easier...

Monday, March 19, 2007

Monday Morning

Around the blogosphere today: for some reason Carl Brannen has been looking at the baryon PDG data. Tommaso Dorigo continues to write entertaining posts about a hypothetical Higgs particle, and Louise Riofrio must be up to some mischief, because she isn't usually this quiet...

Beautiful morning here in the South. Yesterday's lahar on Mt Ruapehu is thought to have stabilised the crater lake for the time being. The emergency response was swift and no casualties were reported.

Sunday, March 18, 2007

M Theory Lesson 29

Morrison and Nieh's paper on su(3) knot homology uses some relations of Kuperberg's, namely which involves trivalent vertices that we wish to add to the complexes of the homology. That is, instead of a smooth cobordism between the pieces on the right hand side of the third relation, we allow a 4 vertex diagram >-< at the top or bottom of the bordism. In the skein relation, whereas the smooth pair of lines has a coefficient of $q^2$, the 4 vertex diagram has a coefficient of $q^3$. The objects of their bordism category are webs generated by trivalent vertices with directed edges, either all ingoing or all outgoing.

Oh, that reminds me. I fixed up the Terence Tao hexagons: These trivalent vertices satisfy a triangle condition $a + b + c = 0$ on the numbers assigned to each edge of the honeycomb. The semi-infinite edges correspond to eigenvalues of the triangular set of 3x3 Hermitean matrices.

Saturday, March 17, 2007

M Theory Lesson 28

I distinctly remember learning about the complex number $i$ when I was about 13. The teacher alluded briefly to some new class of number for solving equations, which she then promptly told us we shouldn't worry our little heads over. Of course I approached her after class, having put away the book I was reading during her lesson, and I begged her to tell me about these new numbers. She told me that it really wasn't a good idea, even though I was very smart, to jump too far ahead of what I was supposed to know. But I whined and whined until she relented and told me about the square root of -1.

At first I was a little disappointed. Only when I found out about Euler's relation did I decide that it was pretty cool. But even now, the geometry enters as a tool, rather than as a means of defining the number itself. Numbers should come from Euler characteristics of categories. But as Khovanov homology tells us, even Euler characteristics are derived from higher dimensional (categorified) invariants.

The Jones polynomial is an Euler characteristic for Khovanov homology. The parameter $q$ of the Jones polynomial is usually taken to be complex valued in quantum group machinations. We saw that the factor $q + q^{-1}$ arose naturally as a label for a single loop. One way to view the number $i$ is as a solution to the equation $i + i^{-1} = 0$. What if we needed a zero to arise in this manner? First we would need a concept of imaginary number.

M Theory Lesson 27

The planar algebras of Vaughan Jones arise from a coloured operad of empty discs in a larger disc, with string pieces (open or closed) in the surface, such that there are an even number of boundary points on each disc. The theory of Jones' subfactors is a kind of Galois theory for $II_{1}$ factors. Matti Pitkanen has considered the relevance of this to physics.

In M theory we would also like to consider higher dimensional operads. For quaternionic number theory it is appropriate to start with 3-sphere discs, plus further structure. Rather than marked boundary points, for instance, a 3-sphere can contain knots and links and also surface boundary components. If the surface pieces are punctured spheres they can be made to look like 2-disc diagrams. We could pack enormous amounts of algebraic information into such an operadic structure. Batanin's 2-level tree composition is a guide to horizontal and vertical compositions in the 2-operad case.

Friday, March 16, 2007

M Theory Lesson 26

In Bar-Natan's introduction to Khovanov homology he draws the parity cube based on the (left handed) trefoil smoothings, discussed in the last lesson. The indices of the cube correspond to a crossing of the knot. Recall that these knot crossings also correspond to the three squares in the 2-dimensional K4 Stasheff polytope of 9 sides. The interior of this polytope tiled the real moduli of the six punctured Riemann sphere.

We can therefore think of these knot crossings as being embedded in a canonical manner in the real moduli space. That is, one crossing for each generation of the particle zoo. The trefoil knot traces out a loop through these idempotents. The non-planarity of knots is thus associated with the non-planarity of complexification in the tiling of moduli.

But we need to get to quaternions and octonions before all this makes sense. Just remember that the numbers don't get put in by hand. They always come from diagrams. Bar-Natan points out that integers can come from Khovanov morphisms for empty links.

Thursday, March 15, 2007

On the Third Road

Sometimes when children squabble, the few who are uninterested in the argument will move quietly away to enjoy a happier game. They will be undisturbed by the others, fighting as they are over a great prize. And so the Third Road, after a long lonely stretch through the hills, comes upon a crossroad.

The lone travellers find others on the road, all with stories to tell. I want to thank Matti Pitkanen, Louise Riofrio, Carl Brannen, Michael Rios, Mahndisa Rigmaiden and our other friends (like Tommaso), for all the fun we've had, and will keep on having, even if the scenery changes.

And it will change.

Wednesday, March 14, 2007

M Theory Lesson 25

In Bar-Natan's talk about Khovanov homology he works out the Jones polynomial for a trefoil knot.

This means considering all smoothings of the 3-crossing knot, which are labelled by the ordered triplets 000, 100, 010, 001, 110, 101, 011 and 111. The 111 stands for the final result of three separate unlinked loops. The term 110 is a two loop diagram, contributing $q^{2} (q + q^{-1})^{2}$ to the invariant. The power weighting of the $q^{2}$ factor out the front goes with the number of disjoint loops. The 000 term is a bit different, because it corresponds to one loop inside another. It contributes $(q + q^{-1})^{2}$ to the invariant. There is a factor of $(q + q^{-1})$ for each loop.

Recall that Pfeiffer and Lauda studied this homology theory in the context of a 2-categorical TFT for open and closed strings.

Tuesday, March 13, 2007

Riemann Rolls On

Since Never Ending Books and Noncommutative Geometry seem to have initiated a carnival of posts on the Riemann hypothesis, let us continue also in this vein.

Shifting from drawing trees with lines to drawing them with ribbons, we find that some alterations are permitted. A Y vertex with lines looks the same upward (product) as it does downward (coproduct). With ribbons, we might have diagrams like this which were considered by Kauffman et al in studying invariants for templates. An example of a template is the one for the Lorenz attractor, namely where the ribbons are permitted to curl around via cap elements. The study of templates and knots took off with the work of Joan Birman et al on the periodic orbits of the Lorenz system. Knots correspond to words in the 2 letters defined by the holes in the template, around which one draws knots. Robert Ghrist came up with a 4 holed template, which is universal for knots and links.

Cosmological Conundrum

Whilst some cosmologists are talking about raisin bread and balloons, yet others will be attending a conference at Imperial on Outstanding questions for the standard cosmological model. That sounds like a lot to pack into a short conference, from March 26 to 29. If you're about, make sure you catch the talk by Louise Riofrio around lunchtime on March 29. Some people from UC will be attending, so we will be hearing more later about this interesting event.

Around the blogs, Never Ending Books has a post on the Riemann hypothesis, and kneemo reminds us to look at the PDG data. I wonder why?

Monday, March 12, 2007

M Theory Lesson 24

As with ordinary primes, there is a unique way to construct knots from prime knots, but only if one specifies the order of the decomposition. In other words, knots are a kind of non-commutative prime.

The product operation for a torus knot, for example, involves a connecting cylinder surface containing two strands, which replace a small segment from each of the two knots. At the boundaries of this cylinder, the strands form two marked points on the loop of the knot, which are the ends of the cut out segment.

As Bar-Natan showed, one can also consider non-associative tangles. This paper contained the first attempt at defining the polytopes of Batanin's higher operads.

Consider this lovely image of the Seifert surface of a trefoil, created using SeifertView by Jarke J. van Wijk.

Ribbon Review

For readers who have yet to look at the 2005 Bilson-Thompson paper on a ribbon diagram classification for one generation of the Standard Model, I shall reproduce the pictures here: The signs indicate twists in the ribbon. This preon model was inspired by earlier ones. Carl Brannen has made some remarkable discoveries based on another preon approach. It looks as though any unified theory that incorporates these elements will probably predict no Higgs boson (or no Dark Energy either). That's OK, because we haven't actually seen one of those.

Sunday, March 11, 2007

M Theory Lesson 23

Observe that the knots of Lesson 22 are not the same as the knots that appear inside 3-manifolds in Jones-Witten CSFT. Rather they are like the objects in the 1-operad case, which are the boundary circles for Riemann surfaces. The full 2-operad diagram, bounded by 3-spheres, is a 4-dimensional space.

Some people spend a lot of time worrying about 4-manifold invariants. For example, will we ever find a combinatorial formulation of the Seiberg-Witten invariants? The usual idea here is to consider a spin foam formulation along the lines of the Crane-Yetter classical invariant for 4-manifolds. A more careful use of higher categorical structures should improve the performance of the spin foam geometry. What if we used our 2-operad combinatorics?

The 3-spheres alone form the usual 4-discs 1-operad, which is well understood. Think of the internal 3-spheres as the leaves of a 1-level Batanin tree. The second level leaves must effectively attach labels to each internal 3-sphere. This must be associated with the embedded link.

Saturday, March 10, 2007

Higher and Higher

Thanks to n-Category Cafe for pointing out the availability of Gurski's thesis on tricategories, probably destined to become a useful reference. In reply to John Baez, Tod Trimble (well known for his handwritten diagrams for tetracategories) immediately commented: if your work takes you beyond weak 3-categories and into the land of weak 4-categories, then you’ll especially need to read this!

But before leaping ahead to dimension 4, recall that a while back we came across sesquicategories, which are 2-dimensional but without an interchange law. Leinster's book talks about a 2-operad for sesquicategories, which has sesquicategories for algebras. Sorry, perhaps I should have mentioned that categories can arise from operads just like algebras. On page 295, Leinster mentions an induced 3-operad, arising from the sesquicategory 2-operad. This is a 3-operad called Gray, because its algebras are Gray categories. We need to understand these, and there is a lot about them in Gurski's thesis!

Friday, March 09, 2007

M Theory Lesson 22

The little disc diagrams of the last post are elements of the 2-discs 1-operad. To each $n \in \mathbb{N}$ we associate the diagram with $n$ distinct little discs inside the bigger disc. There is a $d$-discs 1-operad in any dimension $d$.

It is natural to wonder whether or not the boundary nature of operad mathematics can tell us anything about holography (this is Cambridge?). This has been discussed, such as in this article by Zois, which is motivated by Kontsevich's original idea that the physical principle should be related to the higher dimensional Deligne conjecture. This conjecture now has many proofs (apparently), most notably via the operad methods of Batanin.

In other words, what we really care about are $n$-operad analogues of the 2-discs operad. Recall that 2-categorical TFTs were studied by Pfeiffer and Lauda. With objects, 1-arrows and 2-arrows, there is room for both open and closed strings. But rather than their stringy diagrams, we would like to invent higher operads of discs with extra data.

Moving up in dimension to 3-spheres (for building quaternions) we would naturally consider 3-sphere diagrams with embedded knots and links. The whole diagram is a 2-arrow, whereas the 3-sphere pieces are 1-arrows between the links as their local boundary. The nice thing about knots is that they have primes.

It sounds like we're doing homotopy theory of spheres, doesn't it? Well, that is in fact one way of looking at higher category theory, only we need to attach arrows to our diagrams.

Thursday, March 08, 2007

M Theory Lesson 21

Apparently kneemo has been reading Tom Leinster's free book on Higher Operads. This text takes a fresh approach to weak n-categories from the point of view of operads and multicategories. It also has some wonderful quotes, such as, from James R. Brown: some 'pictures' are not really pictures, but rather are windows to Plato's heaven. On this blog, we like to consider simple pictures.

After a meandering introduction, mostly written for topologists, Leinster begins with a picture of an arrow for a multicategory:Of course, this is just the kind of picture that computer scientists like to draw, and indeed many computer scientists work with multicategories. One way of thinking of an operad is as a multicategory with one object.

In operad land, only boundaries really matter. A cylinder, for instance, changes into and a pair of pants into the face diagram

Wednesday, March 07, 2007

Neutrinos Abound

On the Resonaances blog there is a nice post about cosmological bounds for neutrino masses. Apparently the data can set an upper bound

$\sum m_{\nu} < 0.68$eV

which fits with Carl Brannen's prediction of the masses, as expected.

In other news, Valencia and colleagues may have found a new particle with mass 214.3 MeV, which they are calling the HyperCP particle. Unbelievably, there are already reports that it might be a Higgs. Cough.

M Theory Lesson 20

Kholodenko's paper 'Heisenberg Honeycombs Solve Veneziano Puzzle' shows how important the honeycomb concept is for physics. Take a look at Tao's cool honeycomb applet. Your left and right mouse buttons will shrink/expand the hexagons in the diagram.

For a $3 \times 3$ Hermitean matrix a typical honeycomb looks like: Er, no. That's a bit messy. The Ys should look alike. Anyway, note that the infinite lines go off to the north east, north west and south.

Now why would such hexagon diagrams be so important for physical ampitudes? Remember we tiled the real moduli $M(0,4)(\mathbb{R})$ with a line segment corresponding to the 1-dimensional Stasheff polytope for the associator, labelled by a 1-level 3-leaved tree. The 2-dimensional analogue is one of two kinds of hexagon, each labelled by a 2-level tree. Could this be what this is really about?

Tuesday, March 06, 2007

M Theory Lesson 19

The beta function

$B(a,b) = \frac{\Gamma (a) \Gamma (b)}{\Gamma (a + b)}$

has an integral representation, which can be found by using polar coordinates in the expression

$m! n! = \int_{0}^{2 \pi} \int_{0}^{\infty} e^{- r^2} | r \textrm{cos} \theta |^{2m + 1} | r \textrm{sin} \theta |^{2n + 1} r \textrm{d} r \textrm{d} \theta$

and pulling out the $r$ factors to obtain

$2 (m + n + 1)! \int_{0}^{\frac{\pi}{2}} \textrm{cos}^{2m+1} \theta \textrm{sin}^{2n+1} \theta \textrm{d} \theta$.

It could be fun to play around with higher dimensional analogues of such integrals using, for instance, Euler angles. In a series of papers, Kholodenko looks at multidimensional analogues, and he concludes that Veneziano-like amplitudes are capable of reproducing spectra for both open and closed strings. This involves a study of amplitudes using period integrals for Fermat hypersurfaces. On an historial note, he mentions the original paper of Chowla and Selberg on elliptic integrals. Recall that Chowla is the guy who introduced Montgomery to Dyson at tea.

The Veneziano amplitude takes the form

$V(s,t,u) = V(s,t) + V(s,u) + V(t,u)$

for $V(a,b) = B (- \alpha (a) , - \alpha (b))$. When the condition

$\alpha (s) + \alpha (t) + \alpha (u) = -2$

holds, the amplitude takes the form

$V(s,t,u) = \frac{\zeta (1 + 0.5 \alpha (s)) \zeta (1 + 0.5 \alpha (t)) \zeta (1 + 0.5 \alpha (u))}{\zeta (- 0.5 \alpha (s)) \zeta (- 0.5 \alpha (t)) \zeta (- 0.5 \alpha (u))}$.

Kholodenko discusses the generalised condition

$\alpha (s) m + \alpha (t) n + \alpha (u) l + k = 0$

where the Veneziano case is recovered when $k$ is associated to the degree $N$ of the Fermat surface, while the Shapiro-Virasoro condition arises for degree $2N$.

More recently, Kholodenko has been looking at honeycombs, which were used by Terence Tao and Allen Knutson to solve the long outstanding Horn conjecture on the spectra of $n \times n$ Hermitean matrices.

Monday, March 05, 2007

Monday Motives

Somehow we have returned to Motives! According to the illustrious wikipedia: in terms of category theory, [the theory of Motives] was intended to have a definition via splitting idempotents in a category of algebraic correspondences. Oh, yes, we were wondering about those, weren't we? Unfortunately, when I google twistor, motive and n-category, I find almost no hits. Hmm. Maybe that's the fault of the hyphen. But when I throw in Riemann hypothesis and MHV for good measure ...

Anyway, I must rush off and prepare to teach some eager young minds some good old Newtonian mechanics. It is a stunning day outside. At 7am there was a thick fog laying only a few centimetres over the ground, and the Waimakariri dust clouds have subsided.

Saturday, March 03, 2007

Moving Up

Yesterday I moved into a new place. Well, since I don't have any furniture this merely involved hopping on a different bus and picking up a set of keys, but that was fun because now I live in a hills suburb with good walks and friendly sheep for neighbours. To those who don't know me so well, I tend to move fairly regularly, and my liveliness is quite highly correlated with the proximity of hills. It didn't take me long to find the steep shortcut down into the Heathcote valley to the pub, which has excellent fish and chips.

Friday, March 02, 2007

M Theory Lesson 18

I am very grateful to kneemo for spotting this paper by Carlos Castro on the connection between the Riemann hypothesis and dual string scattering amplitudes. Castro works at a place in Atlanta with a great name: the Center for Theoretical Studies of Physical Systems.

The paper begins by looking at scattering amplitudes of the form

$\frac{\zeta (ik) \zeta (1 - ik)}{\zeta (1 + ik) \zeta (- ik)}$

which appear in studies of particles moving in the hyperbolic plane. Observe that the poles of this expression occur when $k_n = i (0.5 + i \rho_n)$, corresponding to the zeroes of the Riemann zeta function. Thus one considers complex values of $k_n$ and the question arises as to what physical interpretation the zeroes should have.

Castro starts by looking at the Veneziano four point dual string amplitude, expressed in terms of the beta function. The zeros of $\zeta$ on the critical line appear to correspond to the real poles of the tachyonic amplitude.

On another note, check out the coolest blog post ever, on Never Ending Books!