# Arcadian Functor

occasional meanderings in physics' brave new world

## Sunday, May 31, 2009

Although I will have four wonderful months over the summer to focus on work, I am naturally wondering where I should go when my UK visa runs out in October. Should I hide in a Perimeter cupboard? Having seen homeless people in Toronto in winter, I suspect that would not be a great option. Maybe I could head to Switzerland once again, and hopefully this time avoid that nasty crevasse on the upper Grosseraletsch glacier. Camping out at altitude is free there, but internet connections might be problematic. Hmm. New Zealand is looking quite appealing compared to the current alternatives.

## Friday, May 29, 2009

## Thursday, May 28, 2009

### Quantum Cosmology

The cosmology conference at Perimeter finished with a very interesting series of talks, the new consensus definitely tending towards a quantum explanation for the Dark Force. Although nobody mentioned Louise Riofrio's work, Starkman's results were displayed, the WMAP guy said primordial black holes were consistent with some difficult tests of dark matter theories, and even the string theorists seemed keen to move away from a cosmological constant.

Particularly enjoyable was the talk of Raphael Sorkin, who predicted an apparent cosmic acceleration from quantum causal set theory 20 years ago! He started the talk with a few entertaining jokes, before settling on the title Everpresent Lambda, referring to this paper. This model assumes that fluctuations are nonlocal and due to quantum discreteness.

A causal set is a network of atoms of spacetime with a partial order. Assuming an effective gravitational path integral one considers the uncertainty relation

$\Delta \Lambda \Delta V \simeq \hbar$

and under the assumption that the expectation value for $\Lambda$ is zero one finds that $\Lambda$ should be related to $\sqrt{V}^{-1}$. The model lets the volume $V$ be the volume to the past of some event. Although Sorkin did not discuss it, one could also rearrange the uncertainty relation to obtain

$c \simeq \frac{\sqrt{V}}{\Delta V}$

using $\hbar c = 1$. In this form it more closely resembles Riofrio's observationally successful quantum cosmology rule $R=ct$, since a time parameter is related to the number of nodes in the network which measures the volume.

Particularly enjoyable was the talk of Raphael Sorkin, who predicted an apparent cosmic acceleration from quantum causal set theory 20 years ago! He started the talk with a few entertaining jokes, before settling on the title Everpresent Lambda, referring to this paper. This model assumes that fluctuations are nonlocal and due to quantum discreteness.

A causal set is a network of atoms of spacetime with a partial order. Assuming an effective gravitational path integral one considers the uncertainty relation

$\Delta \Lambda \Delta V \simeq \hbar$

and under the assumption that the expectation value for $\Lambda$ is zero one finds that $\Lambda$ should be related to $\sqrt{V}^{-1}$. The model lets the volume $V$ be the volume to the past of some event. Although Sorkin did not discuss it, one could also rearrange the uncertainty relation to obtain

$c \simeq \frac{\sqrt{V}}{\Delta V}$

using $\hbar c = 1$. In this form it more closely resembles Riofrio's observationally successful quantum cosmology rule $R=ct$, since a time parameter is related to the number of nodes in the network which measures the volume.

## Wednesday, May 27, 2009

### Perimeter

I am feeling quite guilty about the number of air miles I am ticking off this year. Arrived Waterloo this afternoon, just in time for two lectures at Perimeter on the subject of Dark Energy, where the large mug of readily available coffee kept my head from hitting the desk.

The new building is far more impressive than I imagined it would be, since I clearly remember the site from my stay here in 2003. It must be a wonderful place to work. Sadly, the schedule for this conference is still not up.

The new building is far more impressive than I imagined it would be, since I clearly remember the site from my stay here in 2003. It must be a wonderful place to work. Sadly, the schedule for this conference is still not up.

## Monday, May 25, 2009

### Fairy Update

Tommaso Dorigo reports on a new study of different event generators which strengthens the Tevatron case for fairy field exclusion in the 158-180 GeV range.

## Saturday, May 23, 2009

### M Theory Lesson 279

Unfortunately, I haven't managed to catch many twistor theory talks here yet, but this week L. Mason drew a nice picture on the whiteboard, which looked something like this: Observe the closed polygons in the dual conformal space, representing momentum conservation. Mason was discussing Yangian symmetries that arise from an integrable system associated to a harmonic map.

## Friday, May 22, 2009

### Everett Today

This afternoon I went to a fascinating and informative seminar by Peter Byrne on the life of Hugh Everett III, the originator of the Many Worlds interpretation. The story is based on a seemingly exhaustive search of papers, notes and letters, some discovered only recently in LA. A long sequence of these documents, along with photographs, were flung briefly onto the projector during a lightening fast hour and a half summary of Everett's life. Amongst the gems was a short personal reply that Everett received as a child, to a lost letter, in which Einstein states that there is no irresistable force and no immovable object.

There was a transcript from a conference involving Wheeler (Everett's advisor), deWitt, Podolsky, Feynman and others, which concluded with Feynman's criticism of the concept of universal wave function. As history shows, this criticism was largely ignored by the relativists, and others. There was a brief account of the interactions between Wheeler and Bohr, and Everett's friendship with Misner. Everett himself, the proud son of a military man and a brilliant (but forgotten) mother, was destined for an illustrious career at the Pentagon during the cold war.

There was a transcript from a conference involving Wheeler (Everett's advisor), deWitt, Podolsky, Feynman and others, which concluded with Feynman's criticism of the concept of universal wave function. As history shows, this criticism was largely ignored by the relativists, and others. There was a brief account of the interactions between Wheeler and Bohr, and Everett's friendship with Misner. Everett himself, the proud son of a military man and a brilliant (but forgotten) mother, was destined for an illustrious career at the Pentagon during the cold war.

## Wednesday, May 20, 2009

### More Seminars

Starting tomorrow we have the new Interdisciplinary Seminar in Fundamental Physics here in the Comlab. Term time really is busy. There is also a regular OASIS seminar, a twistor seminar, a string seminar, a theory seminar, a QFT seminar, philosophy seminars and many more!

Today we heard a very interesting talk from Boris Zilber, a logician, involving something like a Combescure $R$ matrix for time evolution, and yesterday Fay Dowker spoke about the causal sets multiple history interpretation of quantum mechanics.

Today we heard a very interesting talk from Boris Zilber, a logician, involving something like a Combescure $R$ matrix for time evolution, and yesterday Fay Dowker spoke about the causal sets multiple history interpretation of quantum mechanics.

## Sunday, May 17, 2009

### Angels and Demons

The film Angels and Demons, although based on the novel by Dan Brown, alters the original plot in fascinating ways. Spoiler warning.

The explosive antimatter from the LHC belongs to Vittoria, elevated from her original role as an adopted daughter (who needs rescuing later in the story) to an esteemed collaborator (and active experimentalist) of the murdered physicist running the containment experiment. Tom Hanks also arrives at the Vatican, where the ambiguously evil boss of the Vatican, supposedly dealing with the recent death of his beloved pope, is horrified by the arrogance of the scientists who think they can say something about the moment of creation. The God Particle features several times in the script. And we get a fair dose of Tom Hanks expertly studying an unknown manuscript by Galileo, who is supposed to be one of the founders of the secret society of The Illuminati, whose modern members are bent on destroying the Catholic Church in revenge. In the end, the hero and Vittoria manage to save the Church from certain doom, returning power to some old white man (who magically avoided being murdered and whose colleagues somehow don't manage to look all that innocent).

Highly recommended entertainment. In true Oxford fashion, my local cinema advertises a special screening tonight, at which local physicists will explain the science of antimatter and fairy fields.

The explosive antimatter from the LHC belongs to Vittoria, elevated from her original role as an adopted daughter (who needs rescuing later in the story) to an esteemed collaborator (and active experimentalist) of the murdered physicist running the containment experiment. Tom Hanks also arrives at the Vatican, where the ambiguously evil boss of the Vatican, supposedly dealing with the recent death of his beloved pope, is horrified by the arrogance of the scientists who think they can say something about the moment of creation. The God Particle features several times in the script. And we get a fair dose of Tom Hanks expertly studying an unknown manuscript by Galileo, who is supposed to be one of the founders of the secret society of The Illuminati, whose modern members are bent on destroying the Catholic Church in revenge. In the end, the hero and Vittoria manage to save the Church from certain doom, returning power to some old white man (who magically avoided being murdered and whose colleagues somehow don't manage to look all that innocent).

Highly recommended entertainment. In true Oxford fashion, my local cinema advertises a special screening tonight, at which local physicists will explain the science of antimatter and fairy fields.

### M Theory Lesson 278

MUBs tell us to focus on the finite fields $F_{q}$, where $q = p^{n}$ is a prime power. For a fixed $p$, the inverse limit of these fields is the ring of $p$-adic integers. This gadget has the annoying property of being uncountable, and is responsible for the beautiful fractals that naturally describe embeddings of $p$-adic numbers in the complex plane.

From a logos perspective, the axioms for fields are rather messy, and they should not be considered in the context of ordinary sets. Set theory doesn't even know the difference between the continuum cardinality and other choices, so why do we use it to inspire definitions of categories? Actually, category theorists have thought about this for a long time, and there are many kinds of category capable of all the important things that sets are capable of, but which aren't at all like the usual category of sets.

M theorists need to learn more about these alternatives. For example, today David Corfield brings our attention to the concept of pretopos. If one delves a little into this idea, the rationals and finite fields start to look even more remote from the (not uniquely defined) reals than they do in a topos!

From a logos perspective, the axioms for fields are rather messy, and they should not be considered in the context of ordinary sets. Set theory doesn't even know the difference between the continuum cardinality and other choices, so why do we use it to inspire definitions of categories? Actually, category theorists have thought about this for a long time, and there are many kinds of category capable of all the important things that sets are capable of, but which aren't at all like the usual category of sets.

M theorists need to learn more about these alternatives. For example, today David Corfield brings our attention to the concept of pretopos. If one delves a little into this idea, the rationals and finite fields start to look even more remote from the (not uniquely defined) reals than they do in a topos!

## Friday, May 15, 2009

### Force of Gravity

Congratulations to Carl Brannen for winning an honourable mention in the prestigious Gravity Research Foundation essay contest, with a paper entitled The Force of Gravity in Schwarzschild and Gullstrand-Painleve Coordinates. And good luck with the publishing ... hmmm.

Of course even I shudder automatically at the idea of gravity as a Force, but that is why this work is a perfect example of the triumph of pragmatism over obtuse abstraction, a philosophical subject that will be discussed in an interesting seminar here this afternoon by Harvey Brown. In the end, it's the job of a physicist to make things straightforward, and the picture of graviton exchange is something that any schoolkid could understand.

Of course even I shudder automatically at the idea of gravity as a Force, but that is why this work is a perfect example of the triumph of pragmatism over obtuse abstraction, a philosophical subject that will be discussed in an interesting seminar here this afternoon by Harvey Brown. In the end, it's the job of a physicist to make things straightforward, and the picture of graviton exchange is something that any schoolkid could understand.

## Thursday, May 14, 2009

### M Theory Lesson 277

The close connection between MUBs and finite fields makes one wonder how to properly state categorical axioms for modular arithmetic. As far as I can tell, this issue is far from resolved in the literature. For example, in the topos Set the natural number object contains finite sets as subsets, but the axioms of arithmetic rely on the infinite object.

Recall that one dream for logoses is to understand an ordinal $n$ as an elementary category, independently of larger numbers. Just taking oriented simplices, for instance, doesn't say anything at all about modular arithmetic, basically because one never imagines pieces of space disappearing under addition! How can we hope to understand the complex numbers if we don't even understand finite fields?

Recall that one dream for logoses is to understand an ordinal $n$ as an elementary category, independently of larger numbers. Just taking oriented simplices, for instance, doesn't say anything at all about modular arithmetic, basically because one never imagines pieces of space disappearing under addition! How can we hope to understand the complex numbers if we don't even understand finite fields?

## Tuesday, May 12, 2009

### Twistor Seminar

Today David Skinner entertained us with a talk about recent work on recursion relations in twistor theory, being careful to point out that the new geometrical aspects of this viewpoint on scattering amplitudes are quite different to those of the original string theory setting. For those of us who don't like integrals, there were some nice diagrams of triangulated polygons. Apparently Andrew Hodges put a new paper on the arxiv today explaining how amplitudes could be related to simple polytopes in a twistor space, but it hasn't materialised yet. Perhaps tomorrow.

Meanwhile, I have a nice pair of new shoes!

Meanwhile, I have a nice pair of new shoes!

## Saturday, May 09, 2009

### Taxicabs

A while back, Lieven Le Bruyn linked to an article on the story of Ramanujan and Hardy and the number 1729. In 1919, Ramanujan knew instantly that 1729 was an interesting number, because it may be expressed in two ways as

$1^{3} + 12^{3} = 9^{3} + 10^{3}$.

On being pressed further, Ramanujan did not know the smallest number that may be expressed as a sum of two fourth powers, although Euler had solved this problem. But there is a good reason why 1729 is more natural. Recall that Pythagorean triples solve equations of the form

$x^{2} + y^{2} = z^{2}$

because they form three sides of an irregular triangle, whose lengths are expressed as areas of squares. So the simplest cubic analogue should be about volumes on the faces of a tetrahedron. And as every category theorist knows, tetrahedra are naturally marked by face pairs, with a three dimensional arrow going between face compositions.

Let's try to match the Ramanujan quadruple $(1,12,9,10)$ to volumes associated to the faces of a tetrahedron. Each edge of the tetrahedron must correspond to a distinct pair of numbers, like $(9,12)$. The opposite edge corresponds to the conjugate pair, which for $(9,12)$ is $(1,10)$. We could choose the tetrahedron with edge lengths equal to, say, the average of the numbers in the edge pair, but there ought to be a

$(1,3, \sqrt{10})$

$(3,4,5)$

$(5, \sqrt{10}, \sqrt{17})$

$(1,4, \sqrt{17})$

which is kind of cute, since $10 - 1 = 9$, $17 - 16 = 1$, $10 + 17 - 25 = 2$ and so on.

$1^{3} + 12^{3} = 9^{3} + 10^{3}$.

On being pressed further, Ramanujan did not know the smallest number that may be expressed as a sum of two fourth powers, although Euler had solved this problem. But there is a good reason why 1729 is more natural. Recall that Pythagorean triples solve equations of the form

$x^{2} + y^{2} = z^{2}$

because they form three sides of an irregular triangle, whose lengths are expressed as areas of squares. So the simplest cubic analogue should be about volumes on the faces of a tetrahedron. And as every category theorist knows, tetrahedra are naturally marked by face pairs, with a three dimensional arrow going between face compositions.

Let's try to match the Ramanujan quadruple $(1,12,9,10)$ to volumes associated to the faces of a tetrahedron. Each edge of the tetrahedron must correspond to a distinct pair of numbers, like $(9,12)$. The opposite edge corresponds to the conjugate pair, which for $(9,12)$ is $(1,10)$. We could choose the tetrahedron with edge lengths equal to, say, the average of the numbers in the edge pair, but there ought to be a

*right angled*tetrahedron for the Ramanujan numbers. This tetrahedron would have three Pythagorean faces and one skew face. Now the smallest Pythagorean triple $(3,4,5)$ provides a right angled tetrahedron with edge lengths $(1,3,4,5, \sqrt{10} , \sqrt{17})$. The Ramanujan triangles would have to be$(1,3, \sqrt{10})$

$(3,4,5)$

$(5, \sqrt{10}, \sqrt{17})$

$(1,4, \sqrt{17})$

which is kind of cute, since $10 - 1 = 9$, $17 - 16 = 1$, $10 + 17 - 25 = 2$ and so on.

### GRB 090423

The first article I saw on GRB 090423 was dated April 1, but as outrageous as it seems from the point of view of Dark Age cosmology, I knew it was real. After all, Louise has been explaining for years that the early universe contained large black holes. This new GRB boggles the mind, smashing previous records with a whopping redshift of 8.2.

## Tuesday, May 05, 2009

### M Theory Lesson 276

Thanks to Phil for commenting about magic matrices and MUB operators. The interaction of the matrix $R_3$ with the Fourier operator $F_3$ is expressed in relations such as where all entries have the same norm, and under normalisation are just phases. Note that $2 + \overline{\omega} = 1 - \omega$. The odd phase differs from the other phase by a special angle. The moduli of these matrices are all permutation matrices, which are also trivially magic with row sum $1$. The special angle, in radians, is given by $\theta = 0.2928428$, which corresponds to a sin squared of $1/12$. Actually, the phase difference is $\pi - 2 \theta$. By cubing $1 - \omega$, we see that the basic phase here is just $\pi/6$, a $12$th root. Unsurprisingly, the value of $\textrm{sin} \theta$ also turns up in the Fourier transform of the neutrino tribimaximal mixing matrix in circulant form.

### M Theory Lesson 275

Last time we saw how a symmetric magic matrix could be decomposed into one three dimensional and one two dimensional piece. Sadly, AF has neglected to mention some lovely properties of Combescure matrices, acting via conjugation. For example, on the two dimensional circulant piece we have so $R_3$ simply permutes the indices. And recall that $R_3$, being a $1$-circulant, fixes any $1$-circulant. On three dimensional $2$-circulants it acts like the basic permutation operator $(231)$. That is, $R_3$ naturally encodes several permutation actions.

## Monday, May 04, 2009

## Saturday, May 02, 2009

### Quick Update

Hat tip to Carl for the link to this nice new talk by Bob Coecke at PI. Not long til this conference now. And they're letting me attend this conference too!

Back in Oxford, there are still plenty of seminars to attend. This afternoon, Steve Simon gave a beautiful talk (on TQFTs, the Kauffman invariant and why string theory is wrong ... OK, so he didn't use those words exactly), starting with the story of Lord Kelvin and Peter Tait, who first developed knot theory on the motivation that knots in the aether should have fundamental physical significance.

And today being May Day, I was up at 5am to join the crowds under the Magdalen tower.

Back in Oxford, there are still plenty of seminars to attend. This afternoon, Steve Simon gave a beautiful talk (on TQFTs, the Kauffman invariant and why string theory is wrong ... OK, so he didn't use those words exactly), starting with the story of Lord Kelvin and Peter Tait, who first developed knot theory on the motivation that knots in the aether should have fundamental physical significance.

And today being May Day, I was up at 5am to join the crowds under the Magdalen tower.