M Theory Lesson 289

The rest mass of the electron appears in the formula for the Rydberg constant $R$:

$m = \frac{2 h R}{c \alpha^{2}}$

Describing $m$ as the eigenvalue of a Koide matrix with angle parameter $\theta = \delta + 2 \pi /3$, we find that

$\textrm{cos} \theta = \frac{1}{\sqrt{2}} (( \frac{2 \times 13.6056923}{7.29735254^{2} \times 313.85949})^{0.5} - 1)$

in terms of $R$ and $\alpha$, and in agreement with the value $\delta = 2/9$ for Brannen's natural scale $313.86$ MeV. Since both $R$ and $\alpha$ have been measured extremely accurately, the first relation shows that errors in the known electron mass are related to errors in Planck's constant. Conversely, an exact value for $\delta$, along with an accurate value of the natural scale, could be used to predict more accurate values of $\hbar$.

Multi Muon Fairies

Back in the exciting days of the multi muon discovery, we discussed the incredibly simple mass triplet $(1,2,4)$, where $1 \simeq 15$ GeV.

Now in Carl Brannen's Koide relation analysis of mass triplets, there is a natural scale given by

$( \sqrt{m_1} + \sqrt{m_2} + \sqrt{m_3} ) \sqrt{\textrm{GeV}}$

and for the multi muon triplet this evaluates to $292$ GeV, which happens to be in exact agreement with Tony Smith's old estimate for the mass of a pair of Higgs scalars.

Biological Theory

The comments at Physics World this month contain a link to the work of Nottale et al, on fractal spacetime with scale relativity. Quoting a recent paper:

[Nottale] has suggested that the observation scale, i.e., the space and time resolution at which a system is observed or experimented, should also be considered as characterizing the state of reference systems. It is an experimental fact known since Greek philosophers that the scale of a system can be defined only in a relative
way: only scale ratios do have a physical meaning, never absolute scales.

To be honest, the mathematics seems rather dull, but the thing to note is that this paper is published:
Ref: Progress in Biophysics and Molecular Biology 97 (2008) 79–114

Oxford Continued

Having been here long enough now to know the good cheap eating places, the quietest college gardens, and to have met the world's worst hair cutter, I have been making the most of my annual pass to the spectacular Blenheim Palace, which is a short bus ride away. I tend to avoid the palace itself, which is usually jam packed with people, and instead explore the extensive parks. And I have discovered a difficult topological dilemma, not unlike the bridge problem.

Certain areas of the grounds are barred to the public, but the boundaries of these areas are unclear, since they are only marked out with intermittent Private signs. I diligently obey all the Private signs, out of a genuine respect for people's privacy. However, to my horror, yesterday I found myself coming upon a large Private sign from behind! Moreover, I was then forced to retrace my steps, since the alternative was to take another path right around the palace, many kilometers long. Now at no point did I ignore a Private sign going forward. They had simply neglected to place any signs at all at the start of the meagre path that I had followed into the forest (and this was not in the region covered by the large do not stray from paths sign).

In other words, boundaries defined by points rather then lines turn islands in a river into buoys in the sea.

Gina Says II

Gil Kalai has now made available Part Two of Gina Says: Adventures in the Blogsphere String War. I must apologise to Gil for mistakenly thinking that he was Gina. Gina might really be a Gina, after all. Enjoy!

Apollo 11

For the anniversary of Apollo 11, I am remembering the contribution of Parkes observatory in Australia, where I once briefly worked in this same control room (sufficiently long ago that it didn't look that different to this).

Fairy Update

After Tommaso Dorigo's July 14 report on the W/top ellipse, our friend Lubos Mottle calculated that the MSSM was now 13 times more likely than the Standard Model. In a more recent post, Tommaso points out that the MSSM ellipse used in this computation unfortunately appears to lie in the excluded zone of $M_{H} < 114$ GeV. In other words, is there any reason to have much confidence in either scenario?

M Theory Lesson 288

I suspect that there are still a few people out there who wonder why some bloggers occasionally ramble on about twistors, although it is mind boggling to think that the advances in twistor theory made by stringy people over the last few years have entirely escaped their attention.

One of the important results about entanglement is this twistor geometry paper by Peter Levay. Recall the exchange relation from lesson 285:

$P_{13}P_{24} = P_{12}P_{34} + P_{23}P_{14}$

for Plucker coordinates. In terms of twistor geometry, we can write these variables as

$P_{\mu \nu} = Z_{\mu} W_{\nu} - Z_{\nu} W_{\mu}$

where I am not going to worry about whether the indices should be up or down. The twistor geometry is extremely helpful, because now we can write the (norm of the) hyperdeterminant for three qubit entanglement as

$\frac{1}{2} | P^{\mu \nu} P_{\mu \nu} | = | (Z \cdot Z) (W \cdot W) - (Z \cdot W)^{2} |$

which could hardly be simpler. Moreover, the other three qubit measure is

$\frac{1}{4} \tau_{A(BC)} = \| Z \|^{2} \| W \|^{2} - | \langle Z | W \rangle |^{2}$

The $W$ state condition corresponds to null twistors. So associahedra type polytopes really are beginning to look nice in twistor spaces.

See also Levay's more recent papers on black hole entropy and finite geometries.

M Theory Lesson 287

The categorical diagram calculus for observables and basis structures involves algebra objects, meaning that there are trivalent vertices labelled by a multiplication $m$, since the strings represent the object in the category. There is a dual notion called comultiplication, which we will label by $c$. For the example of finite dimensional Hilbert spaces, we imagine that the strings represent qudits, for some dimension $d$. The parallel inputs stand for the tensor product of qudit spaces. Typical diagrams will therefore contain hexagons, as in Now Andrei Akhvlediani has been giving an excellent series of talks on PROs, PROPS and generalised spider theorems, so yesterday I found myself wondering about an alternative bialgebra morphism, which draws out the paths on the hexagon and looks like: The hexagon has become a little loop. The thing to notice is that all the $c$ labels have moved to the top and all the $m$ labels to the bottom. This process is a lot like what physicists call a normal product in quantum field theory, where all creation operators are put on the left of the annihilation operators. But annihilation acts first, so we should read real time upwards in the diagram, although that is somehow backwards from what is happening in the category.

Soon we will look at the simple ordinal matrices that count the paths on a diagram, and thereby represent a bialgebra morphism. In this case we are considering a $3 \times 3$ matrix, indexed by inputs and outputs, just like the $3 \times 3$ matrices for entanglement. The product of two such matrices forms another path counting matrix of the same kind. These matrices are symmetric, since paths run two ways. Antisymmetry may be introduced by orienting the paths.

News from home

Fortunately, it appears that nobody was injured in today's large earthquake, which measured 7.8 on the Richter scale. The epicentre was some distance from population centres, but I suspect that almost everyone in the South Island felt this one.

Cats in London

The next Categories, Logic and Physics one day meeting is in London on August 6. As you can see, there is a really interesting line up of talks. See you there!

Volume One is Out

Issue number 1 of Volume 1 of Rejecta Mathematica is now available online! This excellent journal includes a short note by yours truly from quite some time ago, so apologies for the fact that it has now been superceded by our more recent, but of course unpublished, work. Anyway, this is the first official publication on mass matrices and the Fourier transform.

Fairy Interlude

Higgs is the traveller and narrator in the adventures of Erewhon. He arrives in this mythical land, as often in such stories, by crossing a mountainous region, which in its description is, not coincidentally, very much like a beautiful region of New Zealand.

One of Butler's main reasons for creating the character of Higgs was his interest in Darwin's Origin of the Species, first published in the same year that Butler moved to New Zealand. Higgs discovers that the Erewhonians have banished machines, on the basis that the eventual evolution of humans into superior machines seemed inevitable, following the profoundly convincing arguments of earlier Erewhonian philosophers. Far from being critical of Darwin's work, as some believed he was with this satire, Butler admired it greatly. His satire is directed more at a society that would shun the consequences of evolution, rather than embrace them, no matter how frightening they may appear. And the frightening idea was not that you are descended from a monkey, but rather that your demise is inevitable.

Quote of the Month

Courtesy of Abtruse Goose:

Just a run of the Yang-Mills black hole.

M Theory Lesson 286

Recall that the tribimaximal mixing matrix may be expressed in the form $F_3 F_2$, that is, as a product of quantum Fourier operators. This expression is asymmetric in the choice of two directions out of three. What if we combine different $F_2$ operators? For example, the old bimaximal mixing matrix is obtained from the product of two $F_2$ matrices: The product of $F_3$ with the bimaximal matrix is not tribimaximal, but its form is similar, with the value of $2/3$ being replaced by a value $0.85477$.

Midsummer Fairies

At the Summertown laundromat this morning we ran into a newcomer in town, namely the friendly string phenomenologist Stuart Raby, recognisable from a recent conference T-shirt. After sorting out, with some difficulty, which coins one should use in which washing machines, Raby expressed suitable horror at the idea that the Higgs might not exist. Anyway, after some discussion about the recovery of a weakly constrained MSSM from a heterotic string compactification, he admitted to having also considered mass matrices and mixing data using more interesting GUT models, in particular in this paper from 2005.

In contrast with Connes' failed prediction of the Higgs mass, this paper still meets experimental constraints, with a prediction of around 120 GeV. The parameters and best fit fermion masses are given on page 12, and neutrino masses on page 13. The neutrino mass sum is much less than 1 eV, in agreement with Carl Brannen's estimate, but the neutrino mixing deviates from tribimaximal.

Strings 09

The Strings 09 slides are now online. Get your fix of Arkani-Hamed twistors here! The Grassmanians and Schubert maths starts on slide 42, which contains an example that has the answer 42!

A New Archive

Phil Gibbs is developing a new archive, vixra.org, now ready to take your submissions! Please note the handy link on the left sidebar.

Is Everyone Dead Yet?

It appears that a new stage of The Wars has begun. Today I learned from our friend Mottle that string theory has officially taken over condensed matter physics and its views on quantum gravity, with the AdS/CMT correspondence. Here is one of the crucial papers under discussion. Unsurprisingly, this has generated comments elsewhere. Mottle says:

If you don't know, string theory has won the string wars

Given it's ability to completely swallow competing ideas, like a Taniwha, one can only conclude that Mottle is basically correct, but the accuracy of his statement would probably be improved by inventing a new term for string theory, to capture its latest metamorphosis into a background independent, holographic, information theoretic fermi liquid theory.

An Idea

Tommaso Dorigo had a bizarre idea ... an idea that I am not sure makes any sense, since I cannot imagine that there would be any takers ... but here it is. Tommaso suggests that I (alone, or along with Carl, if he is interested) find a highly respectable person to put their name on a paper, entirely written by me (us), as an experiment to see if the paper might be publishable. Not a bad idea. The implementation of this proposal would have to be confidential, so any potential takers must email me privately at the email address in the comments section. For this experiment, I will write a short paper on any subject that is discussed in this blog, and which I have spent some time thinking about.

At Your Leisure II

AF posted this a while back, but since hardly anybody seems to have considered it, let me post it once again: excerpts from the Declaration of Academic Freedom.

Article 2: Who is a scientist

A scientist is any person who does science. Any person who collaborates with a scientist in developing and propounding ideas and data in research or application is also a scientist. The holding of a formal qualification is not a prerequisite for a person to be a scientist.

Article 4: Freedom of choice of research theme

Many scientists working for higher research degrees or in other research programmes at academic institutions such as universities and colleges of advanced study, are prevented from working upon a research theme of their own choice by senior academic and/or administrative officials, not for lack of support facilities but instead because the academic hierarchy and/or other officials simply do not approve of the line of inquiry owing to its potential to upset mainstream dogma, favoured theories, or the funding of other projects that might be discredited by the proposed research. The authority of the orthodox majority is quite often evoked to scuttle a research project so that authority and budgets are not upset. This commonplace practice is a deliberate obstruction to free scientific thought, is unscientific in the extreme, and is criminal. It cannot be tolerated.

A scientist working for any academic institution, authority or agency, is to be completely free as to choice of a research theme, limited only by the material support and intellectual skills able to be offered by the educational institution, agency or authority. If a scientist carries out research as a member of a collaborative group, the research directors and team leaders shall be limited to advisory and consulting roles in relation to choice of a relevant research theme by a scientist in the group.

Article 8: Freedom to publish scientific results

A deplorable censorship of scientific papers has now become the standard practice of the editorial boards of major journals and electronic archives, and their bands of alleged expert referees. The referees are for the most part protected by anonymity so that an author cannot verify their alleged expertise. Papers are now routinely rejected if the author disagrees with or contradicts preferred theory and the mainstream orthodoxy. Many papers are now rejected automatically by virtue of the appearance in the author list of a particular scientist who has not found favour with the editors, the referees, or other expert censors, without any regard whatsoever for the contents of the paper. There is a blacklisting of dissenting scientists and this list is ommunicated between participating editorial boards. This all amounts to gross bias and a culpable suppression of free thinking, and are to be condemned by the international scientific community.

All scientists shall have the right to present their scientific research results, in whole or in part, at relevant scientific conferences, and to publish the same in printed scientific journals, electronic archives, and any other media. No scientist shall have their papers or reports rejected when submitted for publication in scientific journals, electronic archives, or other media, simply because their work questions current majority opinion, conflicts with the views of an editorial board, undermines the bases of other current or planned research projects by other scientists, is in conflict with any political dogma or religious creed, or the personal opinion of another, and no scientist shall be blacklisted or otherwise censured and prevented from publication by any other person whomsoever. No scientist shall block, modify, or otherwise interfere with the publication of a scientist's work in the promise of any presents or other bribes whatsoever.

At Your Leisure

In the next installment of the arxiv adventures, let us look at what happened to the paper by Carl that won an honourable mention in the Gravity Research Foundation competition this year. Here it is:

C. A. Brannen
The force of gravity in Schwarzschild and Gullstrand-Painlevé coordinates
Physics > General Physics
0907.0660

That is, on the crackpot physics arxiv, rather than the gravity arxiv where it was submitted, and where poor Carl, after some difficulty in trying to upload the paper, was told that he might now upload a version at his leisure. That's right, folks. Now papers that are sanctioned by professionals are being put onto this blacklist arxiv, which everybody knows that nobody reads. Are you going to complain about it? Do something about it? No, you're not, are you.

M Theory Lesson 285

After several weeks of wondering where it was, I finally managed to track down a paper by Sergey Fomin and Nathan Reading, Root systems and generalized associahedra. Read chapter 3.

On page 38 they consider Grassmannian spaces, in a way not unlike that currently popular among twistor (ex-string) theorists. Consider the example Gr$(2,4)$. For any complex $2 \times 4$ matrix, we can define $2 \times 2$ submatrices of the form $M_{kl} = (z_{1k}, z_{1l}; z_{2k}, z_{2l})$. Letting $P_{kl} = \textrm{det} M_{kl}$ for all allowed $k$ and $l$, we have the relation

$P_{ik} P_{jl} = P_{ij} P_{kl} + P_{il} P_{jk}$

Fomin and Reading call this an exchange relation, because in the form

$xy = ac + bd$

it describes a relation between different chorded square pieces of a polygon, just like in the associahedra diagrams. Each exchange relation describes an edge in an associahedron. There are as many variables as one needs to label the sides of a square, and the diagonals, namely $2n + 3$, where $n = 1$ in the case of the basic square.

Aside: Of course, I tried googling exchange relation and BCFW, but there were, unfortunately, zero hits.

M Theory Lesson 284

Recall that there are three two-vertex ternary trees. Since binary trees (vertices of associahedra) are described by fully chorded (ie. triangulated) polygons, we now expect to cut up polygons into squares, since the four sides correspond to the $4$-valent vertex of a ternary tree. Only polygons with an even number of sides appear. For the hexagon, the three ternary trees are described by three single chord diagrams. But these particular diagrams also have an interpretation in terms of binary trees! That is, they represent the three square faces of our favourite associahedron, which Loday matched to the crossings of a trefoil knot in three dimensional space. These binary trees each have five leaves, just like the ternary trees in the diagram above. The difference is that each ternary vertex has been resolved into two binary vertices using the rule:

Summertime

My yahoo page constantly bombards me with calls to escape the English weather and head to Spain. Is it cooler there? Because we are all dying of heat stroke here. Nonetheless, Tommaso, Filippo and I enjoyed a wander about Oxford yesterday afternoon, and dinner on Banbury Road. This morning, finally, the welcome rain arrived, but it is still quite warm.

M Theory Lesson 283

Whilst perusing the literature on Cayley's hyperdeterminant, I kept coming across a reference to the textbook by Gelfand, Kapranov and Zelevinsky. The library here does not have this book, but recently it occured to me that I might order it online. Imagine that! After only a week, a beautiful hard bound dark green copy arrived, well wrapped, from Ohio.

The front cover has a diagram that looks something like this: This is an example of a secondary polytope. In the section Examples of secondary polytopes, on page 237, we find the associahedra. I can't wait to read more.