Arcadian Functor

occasional meanderings in physics' brave new world

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Location: New Zealand

Marni D. Sheppeard

Wednesday, January 31, 2007

Roast Yam

One of the more common introduced mammals in the Southern Alps is the YAM (young Australian male). These mammals usually congregate in pairs, or in packs up to ten. I met an uncharacteristically charming and intelligent specimen yesterday in Christchurch. We got to talking about his recent climb of Mt Sefton and then he spotted me reading some papers and said he was very interested in physics, and he was even thinking of going back to university to study it (his background was in the Arts).

It transpired that he had been listening to radio shows on the String Wars, and in particular he remembered an episode on a certain book, the title of which he could not quite remember. Anyway, to cut a long story short, he was very keen to hear all about Category Theory! He said that there had to be something very wrong with physics when it was perfectly clear to a layman such as himself that the entrenched concepts being discussed were so plainly inadequate. We agreed that neuroscientists, for instance, had a better conceptual picture of the measurement of space than many physicists. I must confess, I had no idea this topic had become so mainstream.

Mount Cook has had a number of fatal accidents over the years, but very few lucky escapes. Last week, however, a group of three climbers headed down towards the Linda glacier from the summit. They attached themselves via a sling to a large rock, which then gave way (the rock, that is) as two of the climbers began to abseil. The guy at the top survived the fall, because a small falling rock cut cleanly through the anchor sling, separating him from his colleagues.

Tuesday, January 30, 2007

M Theory Lesson 12

One of Devadoss's many papers on the real points of the moduli of punctured spheres is Cellular Structures determined by Polygons and Trees. In this paper he considers how the real case may be extended to the compactification of the complex moduli of $n$ points.

We saw that labels on trees were not necessary for the real case, which is described by associahedra. Devadoss points out that by labelling the leaves of trees with the numbers $1,2, \cdots n$ one can describe the complex moduli. Points on $\mathbb{CP}^1$ determine non-planar trees, because diffeomorphisms of the sphere can permute any two points without collision (recall that collisions were the basis for understanding the real case). This cyclicity is allowed on branch labellings, because the leaves should really be considered identical.

Now consider that we really want 2-dimensional operads to describe complex moduli. As it happens, 2-ordinal trees are exactly like 1-level trees with an integral number of branches on each leaf of the base 1-tree. However, 2-ordinals may have any collection of numbers $a_1, a_2, a_3, \cdots a_m$ of branches. So the simple labelled trees needed for the complex moduli are only a subset of the 2-ordinals. It remains to understand, therefore, in what sense the complex moduli is really a substructure of something, in this higher dimensional setting. Recall that in topos theory thinking, number fields are never to be considered fundamental in themselves.

Max Kelly

It is with great sadness that we hear of the sudden death of Max Kelly in Sydney, a few days ago.

Max Kelly was truly one of the founders of category theory, having worked actively in the field since the 1960s. He was knowledgable about many diverse subjects. Until the end of his life he was a keen researcher and active member of the Aus Cat group, often attending seminars despite difficulties with his sight. We will all miss him.

Thursday, January 25, 2007

The Time Machine II

In Book XI of The Confessions, St Augustine (354-430) investigates the perception of Time in his mortal attempt to understand the eternity of God. He was the first philosopher to appreciate the special significance of the present in a world of measurement, so I will quote him at some length:

"Who shall hold it and fix it, that it may rest a little, and by degrees catch the glory of that everstanding eternity, and compare it with the times which never stand, and see that it is incomparable; and that a long time cannot become long, save from the many motions that pass by, which cannot at the same instant be prolonged; but that in the Eternal nothing passeth away, but that the whole is present...

Thy years are one day, and Thy day is not daily, but today; because Thy today yields not with tomorrow, for neither doth it follow yesterday. Thy today is eternity; therefore didst Thou beget the Co-eternal, to whom Thou saidst, "This day have I begotten Thee." Thou hast made all time; and before all times Thou art, nor in any time was there not time...

What, then, is time? If no one ask of me, I know; if I wish to explain to him who asks, I know not. Yet I say with confidence, that I know that if nothing passed away, there would not be past time; and if nothing were coming, there would not be future time; and if nothing were, there would not be present time. Those two times, therefore, past and future, how are they, when even the past now is not; and the future is not as yet? But should the present be always present, and should it not pass into time past, time truly it could not be, but eternity. If, then, time present - if it be time - only comes into existence because it passes into time past, how do we say that even this is, whose cause of being is that it shall not be, namely, so that we cannot truly say that time is, unless because it tends not to be?

...Behold, the present time, which alone we found could be called long, is abridged to the space scarcely of one day. But let us discuss even that, for there is not one day present as a whole. For it is made up of four-and-twenty hours of night and day, whereof the first hath the rest future, the last hath them past, but any one of the intervening hath those before it past, those after it future. And that one hour passeth away in fleeting particles. Whatever of it hath flown away is past, whatever remaineth is future. If any portion of time be conceived which cannot now be divided into even the minutest particles of moments, this only is that which may be called present; which, however, flies so rapidly from future to past, that it cannot be extended by any delay. For if it be extended, it is divided into the past and future; but the present hath no space...

And yet, O Lord, we perceive intervals of times, and we compare them with themselves, and we say some are longer, others shorter. We even measure by how much shorter or longer this time may be than that; and we answer, "That this is double or treble, while that is but once, or only as much as that." But we measure times passing when we measure them by perceiving them; but past times, which now are not, or future times, which as yet are not, who can measure them? Unless, perchance, any one will dare to say, that that can be measured which is not. When, therefore, time is passing, it can be perceived and measured; but when it has passed, it cannot, since it is not.

...In whatever manner, therefore, this secret preconception of future things may be, nothing can be seen, save what is. But what now is is not future, but present."

Wednesday, January 24, 2007

The Time Machine

I immediately thought of Louise Riofrio when I saw this post on light cones at Asymptotia. In 1895, H. G. Wells wrote The Time Machine.

"Scientific people," proceeded the Time Traveller, after the pause required for the proper assimilation of this, "know very well that Time is only a kind of Space".

Rod Taylor then promptly dives into the leather seat and pushes the lever to accelerate into Time. Popular conceptions of time travel have evolved little since H. G. Wells' story. The fact is that 20th century physics, after Relativity, did little to alter our notions of time. Moreover, even Relativity could not cure us of this charming delusion, in which The Time Traveller is almost always depicted hopping into a futuristic vehicle, as if about to drive down the road. Is this not itself a hint that GR might fail when it comes to cosmological scales?

In M-Theory, the picture must change. When quantum information content defines Epoch, there is the idea that humans (or things of a similar complexity) can only have arisen in this environment, in this now. Now must become a time, a time we understand better than a distant time, like a Planck time we cannot see.

Tuesday, January 23, 2007

Nelson to Battle

Ernest Rutherford was born in Nelson in 1871 and graduated with a degree in the physical sciences from Canterbury College in 1893. He took a second degree there in geology and chemistry. Rutherford left New Zealand in 1895 to work for J. J. Thompson at the Cavendish Laboratory. In 1911, after working at McGill and Manchester, Rutherford deduced from the work of Marsden and Geiger that most of an atom's mass is contained in the nucleus, thousands of times smaller. Such a discrepancy between the masses and characteristic lengths of the basic building blocks of nature had never been observed, or even remotely conceived, before.

These days physicists blithely discuss interactions from the Planck scale, to nuclear scales, to planetary scales, to the size of the Universe itself! One might be forgiven for thinking that they have been a little cursory in their consideration of some of the scales in between. After all, chemists know quite a lot about the molecular scale, but we know absolutely nothing whatsoever about what happens at the Planck scale, or anywhere near it.

Monday, January 22, 2007

Parrot Party

Sorry, Carl, but if you keep sending me cool photos I'm just going to have to post some of them, such as this shot of you with Escher the Grey Parrot.
...and here is a photo from home:

Saturday, January 20, 2007

007 Marches On

Just a quick hello, today. Those experimental guys have actually been busy over the holidays. Tommaso reports on a new Higgs prediction from recent improved EW data. Take a look!

The search terms on this blog are getting more interesting. For example, who would be googling Broken Pentagon or Operad + Jordan Algebra? But my all time favourite would have to be M-Theory hogwash. There appear to be a few of brands of M-Theory out there these days. Who knows what will show up on google?

Thursday, January 18, 2007

M Theory Lesson 11

Let's try another picture formatting option on squashed cube diagrams, which are what we get when we start thinking about higher dimensional categories. A natural transformation is an arrow between functors between 1-categories. What if we had a kind of 2-functor between 2-categories? These can have pseudonatural transformations between them, and then of course there has to be yet another level of arrow, and these are called modifications.

Upcoming Events

On Thursday Jan 25, John Conway will be speaking at the University of Auckland about his Game of Life. He's a good speaker, so make it if you can.

A new regular Kiwi event is the Victoria-Canterbury Gravity workshop, which will be held on Feb 7-8 this year at the University of Canterbury. There will be a talk entitled Gravitational Charge in Ribbon Graph M-Theory and other goodies. If you're thinking of coming, I promise that the summer weather is generally much better than the weather we had for the Kerrfest, namely three days of wind and freezing rain.

Holes on a Roll

Tommaso Dorigo's interest in Black Holes in globular clusters has turned into a lovely series of posts on his blog. The ongoing study into globular clusters finds

... that a black hole's mass is proportional to the mass of the stellar environment it inhabits. Supermassive black holes found by Hubble in the centers of galaxies represent about 0.5 percent of the galaxies' mass. Amazingly, the black holes now found in star clusters, which are 10000 times less massive than a galaxy, also obey this trend.

This finding makes far more sense if the black holes were there from the beginning, as Louise Riofrio often likes to remind us. Note that the term the beginning in this context is categorically (in the mathematical sense) not a beginning in the classical sense, and similarly with the word there, which we use freely with the assurance that our readers will not be tempted to picture quantum gravitational degrees of freedom as big black billiard balls floating in the aether.

Wednesday, January 17, 2007

Blazing Sunset

This is Colin Thomas's view of the McNaught comet from Western Australia last night. Sunset is almost upon us, and I am hoping to run outside right now with my nephew to take a look tonight. The cloud might obscure the view, but let's give it a go. We have a small toy 180x telescope with us.

Monday, January 15, 2007

Swinging Schwinger

Sometimes I mention the volume The Physicist's Conception of Nature (ed. Mehra), a collection of lectures given at the 70th birthday celebrations for Dirac in 1972. Schwinger's contribution is interesting, because he seems decidedly irritated at the seriousness with which modern particle physicists take the concept of the vacuum.

...the vacuum is the state in which no particles exist. It carries no physical properties; it is structureless and uniform. I emphasise this by saying that the vacuum is not only the state of minimum energy, it is the state of ZERO energy, ZERO momentum, ZERO angular momentum, ZERO charge, ZERO whatever.

Quite adamant, isn't he? Carl Brannen places much importance on Schwinger's ideas in his derivation of the lepton masses, in particular the measurement algebra. This is conceptually important, because it stresses experiment and the link between input and output states, rather than the usual arbitrary independence of these. Observe that this is more than a mere recognition of the interdependence of states, because it sees more reality in the measurement itself than in intermediate states.

Saturday, January 13, 2007

Random Thoughts

Where is Mahndisa? I hope she is industriously writing papers, or at least enjoying a winter holiday. A while back, Mahndisa mentioned the helpful Riemann Hypothesis page. It is full of interesting information on various attempts to prove either the original hypothesis or modern variants.

For example, it would suffice to prove some form of the GUE hypothesis. Recall that GUE stands for Gaussian Unitary Ensemble, just as in Random Matrix theory. Yes folks, that's the same random matrix theory of the matrix models that kneemo was discussing with us a while back. The idea is that the zeroes of the zeta function (rescaled) are distributed like the eigenvalues of large ($N \rightarrow \infty$) random matrices in the ensemble.

In fact, many attempts to prove the Riemann Hypothesis involve matching the zeroes to a physical eigenvalue problem. Now what physical system could possibly have something to say about such a number theoretic problem? One can't help but wonder.

Thursday, January 11, 2007

Dark Mysteries Update

There have been many wonderful posts lately from Louise Riofrio, regarding new discoveries of so-called Dark Matter. What will we find next? Will the Concorde Cosmology pick up the pieces yet again, and continue enthusiastically to censor opposition? We eagerly await the next installment!

M Theory Lesson 10

Let's take a brief look at the history of solitons. In August 1834 Scott Russell observed a strange wave on a narrow channel of water near Edinburgh. After this he built wave tanks in his garden in order to study these solitary waves. Although he himself thought they were of fundamental importance, enthusiasm for solitons took quite a while to catch on.

In 1895 the KdV equation appeared. It describes non-linear shallow water waves. The modern Inverse Scattering method for solving non-linear PDEs was not developed until the 1960s, in the work of Kruskal et al. But earlier it was observed that the KdV equation is solved using the Weierstrass P function. That is, $u(x,t) = -P (x + ct) + \frac{c}{3}$.

Another application of the ISM is to the sine-Gordon equation. By the early 1980s physicists were studying quantum lattice analogues to such equations, and Kulish and Reshitikhin published an article (J. Sov. Math. 23 (1981) 2435) with some funny looking commutation relations. This was the birth of Quantum Groups. In no time at all the mathematicians were attacking these new algebras. In 1986 Drinfeld presented his seminal work at the ICM.

By the early 1990s there was already Kassell's fat yellow book, complete with an introduction to tensor categories! I remember having it on my desk at ANU in early '94. Apparently one day, when I wasn't around, Kulish walked past the desk and paused as he spotted Kassell's book. I'm told he was absolutely flabbergasted to discover that the QISM had become a huge industry.

Saturday, January 06, 2007

Oh, Alaska

The following is an excerpt from a BBC news article by Patricia Cochran of the Alaska Native Science Commission.

Permafrost is melting all over Alaska as a result of rising temperatures, causing land underneath many villages to subside and softening the soil on riverbanks like the mighty Yukon River.

Mountain snow and ice melt rapidly, causing a short period when water levels in the rivers rise and move rapidly. The high, fast flowing water serves to wash away an unprecedented amount of riverbanks in villages.

The vast amount of soil taken into the river causes riverbeds to rise as eroded soil accumulates on the bottom.

River depths decrease to the point where many areas are so shallow that more and more salmon that are caught in subsistence fishing have lesions, cuts, and scrapes as they struggle to get through very shallow parts of the river.

The low levels that remain for the rest of the summer mean the water is warmer than in the past, causing further stress to the fish during the breeding season.

It may come to the stage that salmon numbers will dramatically decrease within the foreseeable future. This in turn will affect the food available for bears, land otters, eagles and people.

Less salmon carcasses taken inland and left near the rivers will decrease the fertility of land, water, and vegetation. Most "mainlanders" do not understand that we are talking about millions and millions of salmon taken by wildlife every year in Alaska, so the loss of salmon will have significant ecological impacts to land, water, wildlife and vegetation.

There are quite a lot of things that most mainlanders don't understand.

Friday, January 05, 2007

Interesting BH Found

As Louise often tells us, there is more to what we see in the sky than we realise! A black hole has been found at the centre of a globular cluster in the galaxy NGC 4472 by ESA. A quote from the BBC article:

"We were preparing for a long, systematic search of thousands of globular clusters with the hope of finding just one black hole," said Dr Maccarone. "But bingo, we found one as soon as we started the search. It was only the second globular cluster we looked at."

Thursday, January 04, 2007

Back to Basics

There is an elementary idea in Category Theory that should be more emphasised, namely the notion of a higher dimensional equation. If one looks at any equation in a random physics paper, there's a good chance it will be linear, meaning it can be written in a 1-dimensional line. Even if it's a matrix equation, such as $AB = BA$, it will be a linear equation. Now matrices are really arrows in a category Mat whose objects are ordinals $n$. An $n \times m$ matrix is then an arrow from $n$ to $m$. Thus a simple linear matrix equation of the form $AB = CD$ is actually a square diagram of sides $A, B, C$ and $D$. In general, linear equations are then 1-category diagrams.

In a higher category, with at least some non-identity $n$-arrows for $n \geq 2$, we still interpret diagrams as "equations". As an exercise, draw a square composed of four smaller squares, each containing a 2-arrow. Now label objects, 1-arrows and 2-arrows, and write out all the 1-dimensional paths as linear equations. Clearly, insisting on writing out equations this way would get tiresome very quickly.

If we came across a physical or mathematical structure indexed by three ordinal indices, as opposed to the two indices of matrices or numbers, we would naturally guess that equations in these objects would be 2-dimensional. These objects might be triangle 2-arrows in a 2-category whose objects are again the ordinals, but this could only be useful if the 2-arrows satisfied the right pasting conditions to form equations.

Wednesday, January 03, 2007

007 Moving Fast

As Louise says, this is the James Bond Year of Physics. The LHC gets underway, and whatever we see the cannons will be firing on all sides ... which is why it's important not to be identified with any side. Except the Crackpot Side, of course, which is at least a lot of fun!

Carl's book should be finished soon. It will be interesting to see how the neutrino mass predictions compare to results at KATRIN (mentioned by Sabine), which appears to have a parrot mascot.

So get those cannon balls loaded, ready for the game!

Monday, January 01, 2007

M Theory Lesson 9

A while back kneemo mentioned this paper on the geometry of $CP^{n}$ and entanglement, based on the Fubini-Study metric. It's really very nice. Think of $CP^{2}$ as a triangle, which is a manifold with corners in the jargon of 2-categories. This triangle is a projection of an octant on a 2-sphere. A generic internal point represents a 2-dimensional torus, but these tori degenerate to circles on the edges of the triangle and to points on the vertices. So an edge of the triangle is a 2-sphere, or rather a copy of $CP^{1}$. It's all very heirarchical, just like a good motivic geometry should be.

The usual reduction to $RP^{1}$ from $CP^{1}$ uses antipodal points on the 2-sphere. This reduction works just as easily in the triangle picture. For higher dimensional projective spaces, triangles become higher dimensional simplices, just as one would expect.

Now the real question is: can we take what we know about real moduli for points on $RP^{1}$ (from Brown's paper) and lift it to the complex case using this entanglement geometry, bearing in mind the operadic nature of the associahedra tilings for real moduli?