# Arcadian Functor

occasional meanderings in physics' brave new world

## Monday, August 31, 2009

This is a lot of fun, so I downloaded the extrasolar data and found several systems with four or more known planets in them. Here are the plots of planet period vs $n$, as in $n = 1,2,3,4$.

## Sunday, August 30, 2009

## Friday, August 28, 2009

### Quote of the Week

Scientists have discovered a planet that shouldn't exist. The finding, they say, could alter our understanding of orbital dynamics, a field considered pretty well settled since the time of astronomer Johannes Kepler 400 years ago.So says an article about the discovery of the hot Jupiter Wasp-18b, by C. Hellier et al. Thanks, Lobo7922.

### Twistor Buzz

Lectures 4A and 4B are compulsory viewing. In the most deliciously outrageous display of presumption I have ever witnessed, Arkani-Hamed introduces twistor QFT to a group of beginning graduate students who have never even taken a course in old fashioned QFT. Moreover, he carefully explains to them why Feynman diagrams are redundant and quantum gravity has a better way of doing things. Wow. If I tried that, I'd be lynched.

## Thursday, August 27, 2009

### The Fermi Debate

The Fermi debate appears to have convinced some people that any kind of variation in $c$ has been ruled out, or will soon be ruled out as more GRBs are observed.

But consider the Riofrio cosmology. If one must insist (sigh) on a classical picture of photons whizzing through an objective vacuum from GRB 090510 to us, then the Riofrio cosmology would say that all photons shift their speed in unison, as the cosmic epoch changes. They would all be measured locally at a speed $c$. This quantum cosmology is consistent with all known data.

The Loopie picture, on the other hand, defends its new position on the fence by pointing out that DSR breaks Lorentz symmetry through deformation, a supposedly subtle kind of symmetry (actually, it's pretty simple and there is no physical motivation for it). The linked paper begins with the line:

But consider the Riofrio cosmology. If one must insist (sigh) on a classical picture of photons whizzing through an objective vacuum from GRB 090510 to us, then the Riofrio cosmology would say that all photons shift their speed in unison, as the cosmic epoch changes. They would all be measured locally at a speed $c$. This quantum cosmology is consistent with all known data.

The Loopie picture, on the other hand, defends its new position on the fence by pointing out that DSR breaks Lorentz symmetry through deformation, a supposedly subtle kind of symmetry (actually, it's pretty simple and there is no physical motivation for it). The linked paper begins with the line:

What is the fate of Lorentz symmetry at Planck scale?There is a problem with this statement. Traditional string theorists make similar statements. The Riofrio cosmology does not impose a fixed scale on the non local theory of quantum gravity, which contains all possible values of $c$, $\hbar$ and $M_{\textrm{pl}}$. It is capable of recovering local Lorentz invariance precisely because it considers a fixed Planck scale an approximation.

## Wednesday, August 26, 2009

### Computing Masses

In the Scholars International series yesterday there were three sessions (A, B, C) by Arkani-Hamed, nominally on

For example, lecture B discusses the impossibility of precision local observables in quantum gravity, such as rest mass, using the following argument. Quantum mechanics gives us the limit of the uncertainty principle, which is to say that an infinite precision measurement of position requires an infinite amount of energy. On top of this, gravity tells us that the infinite apparatus required to measure a mass would confront the limits of Planck scale physics. With finite resources, infinite precision is clearly impossible.

What is wrong with this argument? Firstly, no one seriously denies that, in practice, finite resources are all we really have. This does not mean that there exists no theory capable of computing the rest masses to high precision, but this theory must circumvent the argument above. Observe that it was first demonstrated that quantum gravity could not be a local spacetime theory, and then we discussed experiments taking place in a classical spacetime. So logically, the argument cannot hold as it stands, once we have abandonned the local point of view, no matter how compelling it sounds.

The computation of rest masses is an important aspect of quantum gravity. It is true that the description of such observables should not impose a unique and universal spacetime. So in quantum gravity, when we measure the rest mass of a particle, we carry with us several strict experimental conditions that limit our capacity to draw resources from the apparently objective spacetime. An example:

Observer spacetime construction: our status as an observer living roughly $13.5$ billion years after the big bang, a cosmic epoch by which the varying $c$ cosmology sets a mass scale that limits our ability to probe vastly different scales (note that this does not imply that humans have a special status, only that they must be considered as observers with limitations).

One enjoyable feature of these excellent (albeit stringy) lectures was the stress on the interconnectedness of the outstanding problems, over all physical scales. Arkani-Hamed says, for instance, that a leap in our understanding of quantum mechanics, or any theory that supercedes it, must involve cosmology and other domains of physics. I wholeheartedly agree.

*Research Skills*, but actually a grand overview (for beginning graduate students) of his current picture of physical reality.For example, lecture B discusses the impossibility of precision local observables in quantum gravity, such as rest mass, using the following argument. Quantum mechanics gives us the limit of the uncertainty principle, which is to say that an infinite precision measurement of position requires an infinite amount of energy. On top of this, gravity tells us that the infinite apparatus required to measure a mass would confront the limits of Planck scale physics. With finite resources, infinite precision is clearly impossible.

What is wrong with this argument? Firstly, no one seriously denies that, in practice, finite resources are all we really have. This does not mean that there exists no theory capable of computing the rest masses to high precision, but this theory must circumvent the argument above. Observe that it was first demonstrated that quantum gravity could not be a local spacetime theory, and then we discussed experiments taking place in a classical spacetime. So logically, the argument cannot hold as it stands, once we have abandonned the local point of view, no matter how compelling it sounds.

The computation of rest masses is an important aspect of quantum gravity. It is true that the description of such observables should not impose a unique and universal spacetime. So in quantum gravity, when we measure the rest mass of a particle, we carry with us several strict experimental conditions that limit our capacity to draw resources from the apparently objective spacetime. An example:

Observer spacetime construction: our status as an observer living roughly $13.5$ billion years after the big bang, a cosmic epoch by which the varying $c$ cosmology sets a mass scale that limits our ability to probe vastly different scales (note that this does not imply that humans have a special status, only that they must be considered as observers with limitations).

One enjoyable feature of these excellent (albeit stringy) lectures was the stress on the interconnectedness of the outstanding problems, over all physical scales. Arkani-Hamed says, for instance, that a leap in our understanding of quantum mechanics, or any theory that supercedes it, must involve cosmology and other domains of physics. I wholeheartedly agree.

## Tuesday, August 25, 2009

## Monday, August 24, 2009

### Big Jupiters

More really cool websites: the extrasolar planet encyclopaedia has an interactive tool for both extrasolar planet and star data. For example, I created this plot of star metallicity against planet period:

## Sunday, August 23, 2009

### Ultra Deep

Thanks to Gizmodo for the link to the zoomable Hubble Ultra Deep Field image. Galaxies in this remarkable image existed possibly as long ago as 400 million years after the Big Bang. Check out the 3D version.

## Friday, August 21, 2009

### Gravity Goose

Recall that, at GRG18, the LIGO collaboration reported null results for GRB 070201, which was coincident with the spiral arms of the Andromeda galaxy. That is, if this burst was located in Andromeda, it could not have been generated by a compact binary under the usual theoretical assumptions, most notably the assumption that gravitons (if they exist, which I suspect not) travel at speed $c$.

Nature reports on LIGO's latest null results with the heading: Gravity waves 'around the corner'. You gotta love those quotation marks. The new paper, for those fortunate enough (including me, at present) to have access, is here. Part of the abstract reads:

Nature reports on LIGO's latest null results with the heading: Gravity waves 'around the corner'. You gotta love those quotation marks. The new paper, for those fortunate enough (including me, at present) to have access, is here. Part of the abstract reads:

Our result constrains the energy density of the stochastic gravitational wave background normalized by the critical energy density of the Universe, in the frequency band around $100$ Hz, to be < $6.9 \times 10^{-6}$ at $95$% confidence. The data rule out models of early Universe evolution with relatively large equation of state parameter, as well as cosmic (super)string models with relatively small string tension that are favoured in some string theory models.

## Thursday, August 20, 2009

### Blog Highlights

Check out these wonderful posts from this week:

The Supernova Condensate talks about relativistic chemistry and the discovery of amino acids in comets. Carl Brannen discusses the proton spin puzzle. Dynamics of Cats links to the latest null results from LIGO. Finally, thanks to Motivic Stuff for mentioning a cool conference in October.

The Supernova Condensate talks about relativistic chemistry and the discovery of amino acids in comets. Carl Brannen discusses the proton spin puzzle. Dynamics of Cats links to the latest null results from LIGO. Finally, thanks to Motivic Stuff for mentioning a cool conference in October.

### EPSRC

The English Research Council has three year postdoctoral fellowships for early career scientists and engineers. Their precise criteria for assessment are:

• Quality, originality and potential impact of the research proposed

• Qualities of the individual as an independent researcher

• Intellectual ability of the candidate

• Awareness of the broader context surrounding the proposed research

• Timeliness of the Fellowship and how it will aid the Fellow’s career development

• The candidate’s ability to plan and manage resources

• Ability to communicate to a generalist audience

Some time ago I sent off an initial abstract to the research office and I have been told, on more than one occasion since, that they really liked the abstract. So I went to the meetings where they explained the application proceedure. I drafted the required documents and discussed them in detail with a departmental research officer. A second departmental research officer also looked over them and suggested helpful modifications. I completed the drafts and uploaded the documents to the online system, all according to the detailed specifications.

Only one thing was missing: the host department letter of support. Previously Bob had told me not to apply, because the EPSRC fellowships were prestigious. That is, not for people like me. Bob is the only person in the department with any hope of understanding my proposal (modulo the particle physics). The internal deadline is now past and I doubt I will find another university (anyone?) to host me before the final deadline next week.

After all this, I get the distinct impression that the government is, on good advice, desperately trying to diversify activities in scientific research. But the system is doomed to fail. For instance, I would like to know how the independent researcher criterion can possibly be met. It turns out that the crucial support letter cannot be obtained without a PI endorsement.

Anyway, I wish all the applicants here the best of luck. I'll be thinking of you while I lie by the pool drinking pina coladas this southern summer. Go Goose.

• Quality, originality and potential impact of the research proposed

• Qualities of the individual as an independent researcher

• Intellectual ability of the candidate

• Awareness of the broader context surrounding the proposed research

• Timeliness of the Fellowship and how it will aid the Fellow’s career development

• The candidate’s ability to plan and manage resources

• Ability to communicate to a generalist audience

Some time ago I sent off an initial abstract to the research office and I have been told, on more than one occasion since, that they really liked the abstract. So I went to the meetings where they explained the application proceedure. I drafted the required documents and discussed them in detail with a departmental research officer. A second departmental research officer also looked over them and suggested helpful modifications. I completed the drafts and uploaded the documents to the online system, all according to the detailed specifications.

Only one thing was missing: the host department letter of support. Previously Bob had told me not to apply, because the EPSRC fellowships were prestigious. That is, not for people like me. Bob is the only person in the department with any hope of understanding my proposal (modulo the particle physics). The internal deadline is now past and I doubt I will find another university (anyone?) to host me before the final deadline next week.

After all this, I get the distinct impression that the government is, on good advice, desperately trying to diversify activities in scientific research. But the system is doomed to fail. For instance, I would like to know how the independent researcher criterion can possibly be met. It turns out that the crucial support letter cannot be obtained without a PI endorsement.

Anyway, I wish all the applicants here the best of luck. I'll be thinking of you while I lie by the pool drinking pina coladas this southern summer. Go Goose.

## Wednesday, August 19, 2009

### Quote of the Week

Thanks to Dynamics of Cats for an article about physics publishing, by Rick Trebino. My favourite line:

Assure the senior editor that, if anyone even considered asking about this, you would immediately and emphatically confirm under oath, on a stack of Newton’s Principia Mathematicas, and under penalty of torture and death that, in this matter, the journal was most definitely not biased in your favor in any way, shape, or form in the current geological epoch or any other and in this universe or any other, whether real or imagined.

### Deligne at Cambridge

Although a recent Cambridge lecture by Deligne introduces the cohomology of algebraic varieties as gently as possible, by the end of the hour he is amusingly talking about, as he puts it,

The lecture carefully describes the differences between three types of cohomology:

1. Betti: topological, eg. the windings of a path around a circle

2. de Rham: about differential forms (associated to the variables in the polynomial that define the space)

3. $p$-adic cohomology: this is really nice when considering base number fields that are not necessarily $\mathbb{C}$.

For the de Rham cohomology functor, one ends up with vector spaces over some number field $k$. For $p$-adic cohomology, one instead has modules over the ring $\mathbb{Z}_{p}$ of $p$-adic integers.

This is just what happens in quantum mechanics when we stop worrying about Hilbert spaces. The mutually unbiased bases in dimension $d$, for $d$ a prime power, are given by the structure of the finite field on $d$ elements. The qubit component of quantum mechanics, for example, uses all dimensions $d = 2^{n}$, where $n$ is the number of qubits. A qubit set of observables therefore only needs these finite fields. If we throw in an appropriate categorical limit, we end up with the $2$-adic integers. Let us say that qubits are not about vector spaces then, because they are more naturally about modules over the $2$-adic integers.

*motivic reasons*for things.The lecture carefully describes the differences between three types of cohomology:

1. Betti: topological, eg. the windings of a path around a circle

2. de Rham: about differential forms (associated to the variables in the polynomial that define the space)

3. $p$-adic cohomology: this is really nice when considering base number fields that are not necessarily $\mathbb{C}$.

For the de Rham cohomology functor, one ends up with vector spaces over some number field $k$. For $p$-adic cohomology, one instead has modules over the ring $\mathbb{Z}_{p}$ of $p$-adic integers.

This is just what happens in quantum mechanics when we stop worrying about Hilbert spaces. The mutually unbiased bases in dimension $d$, for $d$ a prime power, are given by the structure of the finite field on $d$ elements. The qubit component of quantum mechanics, for example, uses all dimensions $d = 2^{n}$, where $n$ is the number of qubits. A qubit set of observables therefore only needs these finite fields. If we throw in an appropriate categorical limit, we end up with the $2$-adic integers. Let us say that qubits are not about vector spaces then, because they are more naturally about modules over the $2$-adic integers.

## Sunday, August 16, 2009

### Fermi on GRB 090510

We are (amusingly) told, once again, that string theory has been proven correct. Translation: observable naive Lorentz violation via frequency dependent photon speeds has been beautifully ruled out by GRB 090510.

But the new report, from the GMB and LAT collaborations, details other interesting features of the short gamma ray burst, such as:

But the new report, from the GMB and LAT collaborations, details other interesting features of the short gamma ray burst, such as:

We find no lags below 1 MeV (in agreement with the thus far known short GRB lags in that energy range), and above 30 MeV; however, we find that the bulk of the photons above 30 MeV arrive 258±34 ms later than those below 1 MeV.which is in agreement with previous GRB results indicating some energy dependent time delay. The conservative explanation would be that the unknown details of the emission processes accounts for the apparent delay, but such event dependent delays could be ruled out once sufficiently many GRB spectra are obtained to test the correlation between spectra, redshift and delay times.

## Friday, August 14, 2009

### Saturn Equinox

Thanks to Bad Astronomy for pointing to Cassini's wonderful new revelations of Saturn. In particular, check out these images of the F ring.

## Thursday, August 13, 2009

## Wednesday, August 12, 2009

### Disclaimer

Dear obnoxious individual (this applies to more than one person)

Guess what? This is my blog. When you told me how wonderful you were, and how I had better behave myself, I heard you. The first time. If you are as wonderful as you say you are, why do you need my ear or sympathy? After all, there is nothing I can do for you. Surely your time would be better spent writing papers for, say, viXra, which is open to everyone. Then everyone can see how wonderful you are.

All the best

Dumb buxom blonde

Guess what? This is my blog. When you told me how wonderful you were, and how I had better behave myself, I heard you. The first time. If you are as wonderful as you say you are, why do you need my ear or sympathy? After all, there is nothing I can do for you. Surely your time would be better spent writing papers for, say, viXra, which is open to everyone. Then everyone can see how wonderful you are.

All the best

Dumb buxom blonde

### viXra Rules

Poor Carl. All through the arxiv saga, he wanted to give the arxiv moderators the benefit of the doubt. He never complained when the gravity paper was moved to General Physics. But now, Carl has tried to post another paper on General Physics. It is most assuredly a completely innocent paper about certain aspects of quantum information theory, and contains no offensive material of any kind. Nonetheless, the paper was removed from General Physics, and Carl received an email, which starts:

From: Don Beyer [www-admin]

Subject: arXiv:0908.1209 Removed

To: carl@brannenworks.com

Date: Monday, August 10, 2009, 10:19 AM

Your submission has been removed upon a notice from our moderators, who determined it inappropriate for arXiv.

From: Don Beyer [www-admin]

Subject: arXiv:0908.1209 Removed

To: carl@brannenworks.com

Date: Monday, August 10, 2009, 10:19 AM

Your submission has been removed upon a notice from our moderators, who determined it inappropriate for arXiv.

## Sunday, August 09, 2009

### Changing Light Speed

Thanks to Carl Brannen for pointing out a new paper by Sanejouand, Empirical evidences in favor of a varying speed of light. The paper summarises results from lunar laser ranging, the Pioneer anomaly, supernovae redshifts and the known fixed constants, namely fine structure and Rydberg. He finds that the varying speed of light hypothesis is (a) consistent with all these results and (b) explains the results that the Dark Force cannot.

Assuming fine structure, Rydberg and electron charge to be truly constant, one must have constants

$\epsilon \hbar c$ and $\frac{m_e c^{2}}{\hbar}$.

There are then two natural alternatives to consider under the varying $c$ hypothesis. First, if the electron mass is constant in cosmic time, we find that

$\frac{c^{2}}{\hbar}$

must be constant, forcing $\hbar$ to vary, but not as in the usual description of Louise Riofrio's cosmology. Because $\hbar c$ cannot then be constant, fine structure depends on a variation in $\epsilon$. Secondly however, if we assume that $\hbar c$ is constant, it follows that the electron mass $m_e$ must go as $\hbar^{3}$. That is, electron mass starts out at zero and grows larger with Riofrio's cosmological law $M = t$. This is also reminiscent of Penrose's thermodynamic cosmology.

Note that the latter alternative would not prevent, in principle, the computation of local mass relations; only the computation of absolute scales. Such a cosmology, at least initially, therefore relies on precisely one parameter, which is a measure of our epoch.

Assuming fine structure, Rydberg and electron charge to be truly constant, one must have constants

$\epsilon \hbar c$ and $\frac{m_e c^{2}}{\hbar}$.

There are then two natural alternatives to consider under the varying $c$ hypothesis. First, if the electron mass is constant in cosmic time, we find that

$\frac{c^{2}}{\hbar}$

must be constant, forcing $\hbar$ to vary, but not as in the usual description of Louise Riofrio's cosmology. Because $\hbar c$ cannot then be constant, fine structure depends on a variation in $\epsilon$. Secondly however, if we assume that $\hbar c$ is constant, it follows that the electron mass $m_e$ must go as $\hbar^{3}$. That is, electron mass starts out at zero and grows larger with Riofrio's cosmological law $M = t$. This is also reminiscent of Penrose's thermodynamic cosmology.

Note that the latter alternative would not prevent, in principle, the computation of local mass relations; only the computation of absolute scales. Such a cosmology, at least initially, therefore relies on precisely one parameter, which is a measure of our epoch.

## Saturday, August 08, 2009

### A Sunny Day

As one of my housemates put it, there is an unusual yellow fiery ball in the sky today. I shall make the most of it, and enjoy the ~~tourist~~ cultural experience of jousting knights at Blenheim palace. Yesterday I learned a little more local history, which goes back to ancient times. Richard I was born in Oxford, and Rosamund, the mistress of his father Henry II, was known in the area of Blenheim palace.

## Friday, August 07, 2009

### Fun at Fermilab

Tommaso Dorigo, Matti Pitkanen and others discuss possible Fermilab evidence for non standard physics. The mass triplets involved, including the multi muon results $h_i$, are:

$(W, Z'_{1}, Z'_{2}) = (1, 3, 9)$

$(h_1, h_2, h_3) = (1, 2, 4)$

where the scales selected differ roughly by a factor of $5$. This looks far more elegant than the unobserved proposed fairy field of unknown mass. In Matti's approach, these are known as $p$-adic base triplets for $p=3$ and $p=2$, which are also the primes involved in the quantum information approach to particle mixing matrices.

$(W, Z'_{1}, Z'_{2}) = (1, 3, 9)$

$(h_1, h_2, h_3) = (1, 2, 4)$

where the scales selected differ roughly by a factor of $5$. This looks far more elegant than the unobserved proposed fairy field of unknown mass. In Matti's approach, these are known as $p$-adic base triplets for $p=3$ and $p=2$, which are also the primes involved in the quantum information approach to particle mixing matrices.

## Thursday, August 06, 2009

### Happy CLAP

Tomorrow I will be off to CLAP at Imperial. The latest* news is that Andreas Doering of Imperial will soon be taking up a lectureship in the group here at Oxford. Too bad for Perimeter, where he was thinking of taking up a position.

*Blog release permission was obtained.

*Blog release permission was obtained.

## Wednesday, August 05, 2009

### M Theory Lesson 292

In the March 09 paper, The S-matrix in Twistor Space, we find Hodges diagrams related to the BCFW recursion rules. Ignoring edge and vertex information, the butterfly identity looks like and the Poincare dual of the right hand graph gives which brings the rule down to the familiar exchange relation: On the other hand, a Poincare dual of the left hand graph does not immediately return the other graph, up to a square diagonal, but we see that the resulting graph is closely related as in the usual cyclic operad correspondence between polygons and trees.

## Tuesday, August 04, 2009

### The Farce

What counts as an explanation in physics? Consider an historical example. The energy distribution of beta decay is explained by Pauli's hypothesis, based on the principle of conservation of energy, of the existence of the neutrino.

The key words in this example are hypothesis and principle. What counts as an hypothesis? Do we allow ideas that (a) permit more paper publications, and hence career advancement, because they are based on equations that we already know, or (b) consider hypotheses that follow from the principle?

Now what counts as a principle? Do we (a) demand the data be fitted to a mathematical curve as quickly as possible, lack of explanations notwithstanding, or (b) consider carefully which established fundamental concepts are inviolable, and which are not?

The energy distribution itself is neither an hypothesis, nor a principle, but that which must be explained. Lest we forget.

The key words in this example are hypothesis and principle. What counts as an hypothesis? Do we allow ideas that (a) permit more paper publications, and hence career advancement, because they are based on equations that we already know, or (b) consider hypotheses that follow from the principle?

Now what counts as a principle? Do we (a) demand the data be fitted to a mathematical curve as quickly as possible, lack of explanations notwithstanding, or (b) consider carefully which established fundamental concepts are inviolable, and which are not?

The energy distribution itself is neither an hypothesis, nor a principle, but that which must be explained. Lest we forget.

## Monday, August 03, 2009

### M Theory Lesson 291

Many thanks to Mottle for pointing out the new S matrix paper by Arkani-Hamed et al.

Even more interesting, Mottle also points to this paper on Grassmanians. Having previously expressed some disgust at the idea of category theory in physics, he may not have noticed the intriguing references to operads.

Even more interesting, Mottle also points to this paper on Grassmanians. Having previously expressed some disgust at the idea of category theory in physics, he may not have noticed the intriguing references to operads.

## Sunday, August 02, 2009

### Imperial CLAP

Most of the abstracts are now up for this week's August 6 workshop on Categories, Logic and Physics.

### The Imperial Force

If you missed the recent London debate between the Dark Side proponent, Andrew Jaffe, and sensible theorist, Subir Sarkar, there is an Imperial podcast available. I also enjoyed the review on the Physics World blog:

Sarkar was second to take the stage and he put forward a very different view. He immediately urged us - along with all working cosmologists - to abandon this “ridiculous” notion of a mysterious repulsive fluid that allegedly fills 75% of the universe.The waning power of the Dark Force was in evidence at the debate, with onlookers asked to vote on whether or not they thought DE existed. As Physics World reports:

Despite this being just a bit of fun, it was still interesting to see Sarkar sweep to victory by such a significant margin.

### Women in Science

This post is dedicated to Zuska and Female Science Professor, whose posts I often enjoy.

I regularly hear it rumoured, amongst colleagues, that I am

Of course, in reality, one tries hard to be appreciative of any genuine attempt at assistance, rare as it is. At least, one tries for five minutes or so, before remembering that you really did earn their respect (something like, oh, 20 years ago) only they don't seem to have noticed yet. I'm feeling good today, because I have some new shoes to puke on.

*This means

I regularly hear it rumoured, amongst colleagues, that I am

*no good at taking advice**. Occasionally some helpful individual will take it upon himself to tell me this gently, as if breaking some difficult news to a small child. Perhaps they should ask my mother when I stopped taking advice, because it was before I can remember, and yes, believe it or not, I did actually notice. Gee, do you think maybe that has something to do with me being the only female physicist around here?Of course, in reality, one tries hard to be appreciative of any genuine attempt at assistance, rare as it is. At least, one tries for five minutes or so, before remembering that you really did earn their respect (something like, oh, 20 years ago) only they don't seem to have noticed yet. I'm feeling good today, because I have some new shoes to puke on.

*This means

*advice*, as opposed to helpful tips, which may of course be welcome.## Saturday, August 01, 2009

### M Theory Lesson 290

Recall that the faces of an associahedron are labelled by a polygon with one chord. In real dimension $3$ we have hexagons, for which there are two chord types. The square faces of the associahedron are marked by a hexagon of the kind shown. Since Loday's trefoil has crossings on these squares, each knot crossing corresponds to a pair of vertices on the hexagon. In M theory, we sometimes label these vertices by $(X, -X)$, or perhaps $\pm \sigma_{X}$, and so on. Note that a trefoil describes the quandle (or rack) rules for the Pauli operators.