Archive for December, 2006

M Theory Lesson 8

By now hopefully we suspect that the categorical concept of monad is important for probing possible definitions of observable. A monad $T$ naturally defines $T$-algebras. Let’s look at an example from Mac Lane’s classic Categories for the Working Mathematician (p 138, 1st edition).

Define a functor $P$ on Set as follows. On sets, $P$ sends $X$ to the set of all subsets of $X$. A function $f$ gets sent to $Pf$, which sends $S$ to the direct image of $S$ under $f$, as a subset of $X$. There is a natural transformation whose components are arrows from $X$ to $PX$ which take elements of $X$ to one point sets, and yet another natural transformation with arrows from $PPX$ to $PX$ which takes sets of sets to a union of sets. This data makes $P$ a monad, called the power set monad.

Recall that a complete semi-lattice $C$ satisfies that every subset $S$ has a least upper bound in $C$. A $P$-algebra is a complete semi-lattice with $x \leq y$ given by $h \{ x,y \} = y$ where $h$ is part of the data for a $P$-algebra, and it also gives the least upper bound for $S$. So the category of $P$-algebras is the category of all complete semi-lattices along with the appropriate arrows.

This has been mentioned a number of times before, so I hope I’m not boring you to death. Alas, I must run again.

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A quick and jolly hello to my dear blogosphere friends and other friends who are far away. Happy New Year! Special wishes to those still struggling with the effects of the tsunami, which occurred exactly two years ago today, the 26th. I’m sitting in an air-conditioned internet cafe near the beach – can’t complain. And there was snow in the mountains down south yesterday.

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Awaiting Fairies

We all await the new constraints on the SM Higgs mass, which will follow from the new measurement of W mass. Tommaso has already given away that the first four digits are 8040 which can only mean $m_W$ = 80.40 GeV. Meanwhile, the no Higgs vote has stabilised at just above 50%. Will all CDF worker family members please refrain from feeding the CDF worker too much food over the next couple of weeks. They have a lot of work to do.

Update: It’s now official …
$m_W = 80.413 \pm 0.048$ GeV

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Posts on this blog will probably be less frequent over the next few weeks. This is not due to the contagious lethargy of the season, but rather to the lack of reliable computing facilities in the neighbourhood.

For a good laugh, take a look at Scott Aaronson’s post on becoming a mercenary in the String Wars after a cushy visit to Stanford. Personally, I’d settle for a bug infested hovel and some chips to eat, if anyone feels like flying me somewhere. I’d love to talk about quantum topos theory, M Theory and operads and the calculation of LHC amplitudes, but no one seems interested in such things.

Oh, my! I’ve been tagged by Mahndisa.
Six Weird Things About Me:
1. I own more ice tools than handbags.
2. I am extremely anxious about being even one minute late.
3. It’s unusual amongst my peers that I don’t think it’s unusual that I don’t have any substantial assets, such as a car.
4. I tend to be succinct.
5. When I was barely two years old I went missing and was found in a pantry, having just polished off a large jar of black olives.
6. I’ve been missing a few times. Last time it involved quite a number of search teams and an airforce iroquois, but I’ll tell that story some other time.

Hmmm. I guess I’ll tag Louise, Carl B, Matti, Nigel, Tommaso and Jonathon (but I don’t mind if you ignore it).

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Machian Madness

In mentally separating a body from the changeable environment in which it moves, what we really do is to extricate a group of sensations on which our thoughts are fastened and which is of relatively greater stability than the others, from the stream of all our sensations.

These are the words of Ernst Mach (1838-1916). Amongst physicists, Mach is known as the last anti-atomist, persisting in his view well after the 1905 papers of Einstein. But Mach’s views were not philosophically trivial, based on a line of reasoning going back to the Monadology of Leibniz. In his point of view Relationalism was being neglected in favour of the classical reductionism long in vogue.

From The Analysis of Sensations, published in 1897: The popular notion of an antithesis between appearance and reality has exercised a very powerful influence on scientific and philosophical thought. We see this, for example, in Plato’s pregnant and poetical fiction of the Cave, in which, with our backs turned towards the fire, we observe merely the shadows of what passes (Republic, vii 1). But this conception was not thought out to its final consequences, with the result that it has had an unfortunate influence on our ideas about the universe. The universe, of which nevertheless we are a part, became completely separated from us, and was removed an infinite distance away.

Mach greatly influenced Einstein’s thinking about Relativity. Ironically, the unfinished task of understanding Mach’s principle for inertia must bring together both Atomism (in a more monadic guise) and Relativity.

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Freezing Over

It might not happen as in the movie, but a possible consequence of changing ocean currents is a fast onset ice age. It is well known that rapid glaciation has occurred in the past. Who will be ready for that?

One respected colleague appears to think that girls will be on the beach in their bikinis, while the hardier men will be stuck in a crevasse on a large glacier. Well, the former would be preferable, I can assure you. It does not take long to make oneself some clothing. Sea levels will fall, but glaciers will quickly thicken and bury anyone who happens to find themselves near a current surface. Having myself fallen 20 metres to the bottom of a large crevasse on the Grosser Aletsch glacier, I can testify to the fact that infalling water and snow is unhelpful in fighting off the effects of hypothermia.

For those who are wondering, the new background photo is the view of Mt Cook from the terminal lake of the Hooker glacier. This glacier has been retreating rapidly for some time.

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M Theory Lesson 7

Craig Pastro says it should be called $M^2$ Theory (after the two conspirators) but the conventional name will suffice. Recall that in Lesson 6 we discussed points and how one really should worry about one’s concept of point in thinking about quantum gravity. The star student charged ahead to think about generalised idempotents in relation to parity.

So let’s go back to the relation $T^2 = T$. The reason for the capital $T$ (besides our swanky new latex capabilities) is that rather than sources or targets for arrows in a category, we would now like to weaken the relation and talk about monads.

A monad is a functor $T: C \rightarrow C$ with natural transformations $\mu: TT \rightarrow T$ and $\eta: 1 \rightarrow T$. Think of these as multiplication and unit. They satisfy an associativity and unit law. The square that represents associativity may have its vertices labelled by signs –, -+, +- and ++ where the source — is the composition TTT before bracketing. Such parity cubes appear naturally in higher categorical contexts.

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Testing Testing

Thanks to the helpful gebar and Asymptotia I can now attempt …

$\int_0^1 f(u_{ij}) \omega = \zeta(1,2) + \zeta(3)$

$H^{2}(\mathbf{T}^{+} \times \mathbf{T}^{+} , S_{m,n}(- \mu – 2 , – \eta – 2))$

Yipee!!! That was fun. Unfortunately, I haven’t got categorical diagrams going yet, and only moduli integrals are interesting, but it’s a start. Let’s toast a great day: the day Blogger nerds start free publishing in mathematics! In other news, the no Higgs vote is growing, now at 57%.

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Closer to the Top

Fermilab have announced evidence for the single top quark. In the blogosphere, Tommaso Dorigo and Tony Smith have recently discussed single top production at D0. The parameter determined was the Vtb, which lies between 0.68 and 1.0. This is consistent with the Standard Model, which predicts the blue line in the diagram.

In other news today, a story on sea level rise, which may be more imminent than many appreciate. There were also some nice pictures in New Scientist last week from Antarctica.

Update 16.12: Tommaso has some further cautionary comments to make about the D0 diagram above.

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Weak Bimonoids

Without going into the diagrams, which unfortunately is most of the story, I will say a little about the recent AusCat talk of Ross Street on weak bimonoids in braided monoidal categories, which he also spoke about at ANU recently. The study of these new diagrams was partly motivated by the work of the mathematical physicist Robert Coquereaux.

A monoid is given by an object A and arrows m and eta for multiplication and unit. The braiding enters in considering AoB to be a monoid. Dually, a comonoid has a comultiplication and counit. A bimonoid is a monoid A which also has a comonoid structure. The trick to defining a weak bimonoid is to carefully choose a self dual set of diagrams such that bimonoids are always weak bimonoids.

The aim is to relate this definition to a concept of quantum category. In this setting the term quantum involves a linearisation process (so this is not really a quantum gravity kind of quantum). A category is usually specified by source and target maps from C1 to C0, the arrow and object sets. On linearisation one moves into a category of vector spaces rather than sets, and the objects C1 and C0 are comonoids in this category.

How about working with more general monoidal categories? A quantum category is by definition such a category V, with maps s and t into opp(C0) and C0 respectively. Note that the condition of oppositeness is null in the case of vector spaces. This is a natural definition because it looks like the dual of a diagram defining a bialgebroid (I looked on Wiki but they haven’t quite got this far). In this dual diagram we consider objects A and R, where A is a weak bimonoid and R is the analogue of C0.

The nice diagrams allow one to show that R can be recovered from the structure of A. The proof crucially uses the idempotents t and s, and the splitting of idempotents in the sense of a Karoubi envelope.

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