Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Tuesday, July 31, 2007

Solar Superorbits

At last, a new paper by A. L. Kholodenko on the Quantum Signatures of Solar System Dynamics. From the abstract:

"In this work new uses of Heisenberg's ideas are found. When superimposed with the equivalence principle of general relativity, they lead to quantum mechanical treatment of observed resonances in the Solar system. To test correctness of our theoretical predictions the number of allowed stable orbits for planets and for equatorial stable orbits of satellites of heavy planets is calculated resulting in surprisingly good agreement with observational data."

The first part of the paper is an excellent outline of the modern form of Heisenberg's honeycomb quantum mechanics along with an historical account of its motivation. Happy reading.

Update: Matti Pitkanen has some comments about the paper.

Monday, July 30, 2007

Tour De Force

Warning: make sure you are sitting down before you attempt to look at this 642 page draft of the new book, Noncommutative Geometry, Quantum Fields and Motives, by Matilde Marcolli and Alain Connes.

Not content with introducing QFT, NCG, Connes' Standard Model, the Riemann hypothesis, motives and the kitchen sink, they finish up (from page 611) with a section entitled The analogy between QG and RH. The preface makes it abundantly clear that the book is primarily about this, as yet mysterious, correspondence. Unfortunately, I suspect I will find most of the book extremely mysterious as long as I live.

M Theory Lesson 79

Event Symmetry has also long pondered the j-invariant, and now has a post relating this invariant to a hyperdeterminant of type 4, which is to say a function of a 2x2x2x2 hypermatrix. He points out that the degree of the polynomial is the magical number 24, the dimension of the Leech lattice. Note that Michael Rios has also been keen on hyperdeterminants for some time because of their appearance in the quantum computational approach to Black Hole entropy.

Sunday, July 29, 2007

M Theory Lesson 78

The last Riemann post raised again the issue of defining numbers in terms of diagrams. In logos theory one cannot simply pull infinite sets, or real numbers, out of a hat. The rabbit prefers to live in a burrow, where any number represents an enormous variety of objects.

The Euler characteristic of a category was one natural way of assigning rational numbers to finite type diagrams. What about irrational numbers? Legend has it that Pythagoras treated Hippasus rather badly (perhaps by killing him) after Hippasus demonstrated that the square root of 2 was irrational. A proof begins by assuming that $\sqrt{2}$ is rational and hence expressible as an indecomposable ratio of two integers $a$ and $b$, and then derives a contradiction. This relies on the concept of primeness for the ordinals. From the classical assumption that, if P is false, (not P) must be true it is deduced that $\sqrt{2}$ is irrational, the only alternative to being rational. Ah, hang on a minute. In logos theory we don't want to assume that complement is an involution, so the proof doesn't quite work, but there clearly needs to be more than one kind of number. The geometric definition of $\sqrt{2}$ uses two squares of side length 1, which are both cut in half and glued together as shown. This assumes that the correct units for area are $L^{2} = L.L$ where $L$ is a unit of length, but of course $L$ could be any unit. It seems there are an awful lot of assumptions being made to define irrational numbers. In three dimensions in M Theory the complement is a triality. Thus we should probably assign a different numerical status to $\sqrt{2}$ and $\sqrt{3}$. And only when we get to weak $\omega$ categories do we get all the reals.

If we tried to measure a real edge of a square of unit area we could never show it was exactly $\sqrt{2}$ because that would require subdividing the interval further and further until the atomic, or subatomic, structure of the environment altered our sense of the edge, or altered the edge itself so that there was no longer anything to measure.

On the other hand, zeta values show up as soon as one starts thinking about knots and QFT, although the zeta values are treated as abstract basis elements for an algebra over the rational numbers. It is convenient to assign real number values to them in order to evaluate physical quantities, but this comes from the rules of ordinary complex analysis, which we would like to move beyond.

Saturday, July 28, 2007

M Theory Lesson 77

Posts here often discuss operads, for which compositions are represented by a directed rooted tree and the root is the single output. Dually one can consider cooperads with multiple outputs. These give rise naturally to coalgebras rather than algebras. And yes, people have considered vertex operator coalgebras and so on. Recall that coalgebras are essential to the concept of Hopf algebra. An example of a Hopf algebra is a universal enveloping algebra of a semisimple Lie algebra. Deformations of these, also Hopf algebras, are the notorious quantum groups.

Now Ross Street has published a book, Quantum Groups: a Path to Current Algebra, which I would love to get my hands on. The book blurb says, "A key to understanding these new developments is categorical duality." That simply means the duality given by turning a tree upside down, as drawn, so we need operad structures that combine upward branches and downward branches. Leaping ahead to triality, one expects branchings in three directions and no chosen root. Such diagrams are associated to cyclic operads, and we actually consider these with polygonal tilings, since the dual to a polygon is a branching tree with no real root.

Friday, July 27, 2007

Higgs Blogging II

The Higgs mechanism is without a doubt one of the most beautiful features of the Standard Model. This may, however, be an argument in favour of considering it a natural effective mass generator, rather than an argument for taking its particle nature seriously. Mathematicians such as N. Hitchin have run with Higgs fields and constructed fantastic spaces. Alain Connes has even reformulated the Standard Model using NCG. But again, this seems to be an argument in favour of the Higgs being a stopgap for something deeper. For those of us who work on a QFT not based on symmetry or Lagrangians, the Higgs can only potentially exist as a derived concept, in which case there is no need to posit it a priori.

But the biggest problem with the Higgs is its attachment to the vacuum, and its pervasion of space. That's simply not something that particles do. So just because nobody has yet fully explained the particle masses any other way, doesn't mean the Higgs boson exists. How can we possibly hope to understand the Higgs mechanism without understanding mass?

A guest poster on The Everything Seminar today points out that most of our baryonic mass comes from the binding of quarks and other partons in atomic nuclei, and not from the rest mass of quarks themselves. In other words, an unseen seething relativistic swamp of matter, for which we don't even have a real theory, makes up most of the mass we care about, unless we want to consider Dark Matter as well, in which case we clearly don't know very much about anything. This is the realm of quantum gravity, not the Standard Model. Assuming Dark Matter requires a whole new category of observables, obliterating our trust in the old ones, and leads to the expectation of new physics at the TeV scale.

Thursday, July 26, 2007

Higgs Blogging

While checking out the latest New York Times story on the fairy field, I found an anti-establishment physics blog that's way more interesting than Not Even Wrong: check out this blog by R. Mirman, complete with links to error messages for the cosmological constant. This old guy is angry.

Clifford seemed a little bothered by my continued vocal disbelief in the existence of fairy fields, and he said, "Hi Kea! I see you’re still ahead of us fools who wait for experiments before concluding what we believe about Nature."

Well, actually, as it happens, I was, and still am, of the opinion that theoretical science is at least partly about trying to understand the outcomes of experiments before they are performed. It is therefore my duty, as a professional (albeit unpaid) quantum gravity theorist, to make sensible predictions about events at the LHC, and elsewhere. And then there is the not so minor issue of all the previous experiments that were supposed to see certain fairy particles, but didn't. What are all those experimentalists doing, anyway?

Meanwhile Amanda, at Tommaso Dorigo's blog, said (in disparagement as usual), "Almost as bad: Famous Physicist says: Here is a really interesting idea: let’s all use category theory/twistors/moonshine/-fill in badly motivated or frankly unphysical idea which is clearly going nowhere HERE-."

Wow! Isn't that cool! Category Theory and Twistors and Moonshine are getting mocked all in one breath! Most of the physicists I know still haven't thought about these things. Maybe physics is progressing as it should, after all.

So why does the Higgs not exist? For starters, there never was any reason to think it should. As everyone knows, QFT likes particles to be massless. QFT does not solve mass generation: it includes masses as parameters. Secondly, our best bets for observables in a theory which does actually describe mass, doesn't include in any obvious way a Higgs particle. In other words, it doesn't exist. Other stuff probably exists, but it's different stuff. Now many people keep insisting on telling me that physical observables are about symmetry, and this friendly advice is usually accompanied by much hand-waving about Lagrangians and representation theory and other mathematics from QFT, which never had anything to do with gravity, as it is observed.

By the way, those infernal twistors also say interesting things about massless particles, like about how they might be viewed in a unified setting using cohomology. And like how higher non-Abelian cohomology is needed to understand mass. Now I realise that QFT doesn't usually deal with twistors, but Nature (the bitch) isn't going to let QFT reign supreme forever, folks, especially when it comes to the issue of mass generation.

It seems obvious that the fairy field is unlikely to exist, even without formally extending the notion of observable into the language of motivic cohomology and moonshine (but that sounds like fun). Maybe I'm just stupid and I never understood QFT and there really is a Higgs, which Fermilab will find any day now. I guess we'll wait and see, just like we'll keep waiting for protons to decay, or gravitational waves to magically appear in the M31 GRB event.

Update: Tony Smith just posted an article on moonshine monsters, Jordan octonions, and the Standard Model here. Enjoy.

Tuesday, July 24, 2007

M Theory Lesson 76

Recall that Postnikov computed the volumes and number of lattice points for general permutohedra in a variety of ways. In particular, volumes for standard permutohedra can be computed using Weyl group lattices. For $S_{3}$ in dimension 2 one compares the area of 6 (the order of the group) triangles, such as the blue triangle shown, to the sum of the areas of the three hexagons (permutohedra). The hexagons are drawn via a specific algorithm. First pick a point in the blue triangle, the central vertex. This point has an image in the other triangles. The convex hulls of these points give the collection of hexagons. Note that these hexagons look like the central piece of a honeycomb for 3x3 matrices. Sorry about the wonky picture.

Monday, July 23, 2007

Riemann Rocks

If we don't worry about complex numbers, various forms of the Riemann Hypothesis have known to be true for some time now. Bombieri came up with a simple proof for algebraic curves over finite fields [1] in the early 1970s.

Let $k$ be the finite field with $q$ elements and $C$ a curve over this field. The zeta function is about divisors on $C$, which we can think of as formal collections of points on the curve. Prime divisors will be labelled $P$. Now let $d(P) = [k_{P} : k]$, which is the degree of a field extension $k_{P}$, the so-called residue field. The variables $t$ and $s$ will be used, where $t = q^{-s}$. The Euler product formula for the zeta function takes the form

$\sum_{D} N(D)^{-s} = \prod_{P} (1 - N(P)^{-s})^{-1}$

where the sum is over divisors and the product over prime divisors, and $N(P) = q^{d(P)}$. So it looks just like the usual Riemann zeta formula, except that numbers have been replaced by (numbers associated to) geometric objects. If we let

$L(t) = \zeta (t) (1 - t) (1 - qt)$

and let $a_{i}$ be the inverses of the roots of $L(t)$, then the Riemann hypothesis has the following simple form: the $a_{i}$ all satisfy $| a_{i} | = \sqrt{q}$. There are $2g$ such $a_{i}$ and the set can be ordered into two parts so that

$a_{i} a_{g + i} = q$

The functional equation is

$\zeta (1 - s) = (q^{2g - 2})^{s - 0.5} \zeta (s)$

and in all this $g$ is basically the genus of the curve. Do these kind of primes $P$ look more like respectable categorical objects?

[1] C. J. Moreno, Algebraic Curves over Finite Fields, Cambridge (1991)

Sunday, July 22, 2007

M Theory Lesson 75

So it seems the Morava paper belongs to a productive train of thought, leading to yet more incomprehensible papers by Manin et al. This is the honey that led Manin into a place hard to follow.

But the notion of extended modular operad is clearly a good one. Manin et al solve the problem of the missing 2-punctured moduli $M_{0,2}$ (which makes the operad messy) by replacing all the $M_{g,m}$ with the extended (compactified) spaces $L_{g,m,n}$. The index $n$ labels a second collection of $n$ marked points on the genus $g$ surface. There is a surjective morphism from $M_{0,m+2}$ to the $L_{0,2,m}$, which are toric varieties associated to permutahedra: phew, something we can understand!

So instead of looking for tilings of the ordinary complex moduli $M_{0,m}$ with 2-operad polytopes, we can tile the extended moduli spaces. Loday's geometric realizations of associahedra and permutohedra (from cubes) may come in handy at last. Recall that a 5-leaved labelling of the usual Stasheff associahedron in three dimensions becomes a 4-leaved labelling for permutations in a permutohedron, so Loday's shift in the number of marked points from $M$ to $L$ may clarify the abovementioned surjective mapping.

Saturday, July 21, 2007

M Theory Lesson 74

While in Sydney I managed to pick up my notes from the Streetfest in 2005. Getzler spoke about open and closed modular operads.

Moduli spaces of genus $g$ curves with the Deligne-Mumford compactification form an example of a modular operad. In modular operad theory one replaces planar rooted trees by more general graphs, but we want to use surfaces with boundaries instead of graphs. For $n$ punctures (boundary circles) on a surface, $m$ boundary arcs and $h$ holes it turns out that the dimension of $M(g,n,h,m)$ is $6g - 6 + 3h + 2n + m$, and this compactified moduli is good enough to completely classify the TCFTs (meaning TFTs based on moduli spaces). Getzler ended with a reference to this paper...oh my, ribbon graphs again. Ribbon graphs form a modular operad under gluing of their edges. Note that this is orthogonal to the usual gluing of trees and more like the pasting of surfaces along boundary elements.

Note that in this paper, Costello carefully distinguishes the rooted trees from cyclic forests, which more correctly describe ribbon graphs. Operads are viewed as monoidal functors from the basic structure type: rooted trees, cyclic forests, or graphs.

Such musings led, as usual, to a google search, this time on the terms modular operad and 2-operad, which resulted in precisely one hit, namely a paper by Morava which I intend to take home and read.

Friday, July 20, 2007

Star Formations

Courtesy of Bad Astronomy, here is an image of the very young star HD 15115. The disc of planet forming dust is seen to be very lopsided. The conjecture is that a nearby star may have disturbed the system and upset its symmetry, but the readers of this blog can probably think of an alternative explanation.

Thursday, July 19, 2007

M Theory Lesson 73

Twistor space is 6 dimensional (over the reals). If we ask ourselves which 2-level polytopes might be used to model such a space we find that there are quite a few, namely those labelled by the trees shown, along with obvious permutations and variants on the (2,5) leaved case. The 8 leaved rooted tree will be familiar as the Stasheff associahedron sitting in $\mathbb{R}^6$. The first 6 leaved example is a typical element of the notorious counterexamples discovered by Tamarkin, and presumably plays a role in describing the 6 dimensional moduli $M_{2,0}$ and $M_{0,6}$ which feature so prominently in M Theory.

Wednesday, July 18, 2007

Around the B'Sphere

Jacques Distler has finally figured out why Condensed Matter physicists are keen on pentagons and hexagons these days. Louise Riofrio continues with a series of informative posts on new spacesuit designs, for the serious traveller. And the wonderful David Corfield links to a great paper on operads by Manin et al. One has to wonder what Manin is thinking about these days: working on operads one minute, and with Marcolli, an expert on the Riemann hypothesis, the next.

Tuesday, July 17, 2007

GRG18 5c

There were a number of interesting talks on Friday last week. In the morning we heard from Rob Myers on Quark Soup at RHIC. The header slide included this cool image. Myers than made a quip about a string theorist talking about nuclear physics at a gravity meeting, dispelling the fears ignited by the talk title that grand claims would be made for string theory. On the other hand, the introduction included a mention of both the 2004 Nobel prize and the Clay Institute Yang-Mills problem, neither of which seem particularly relevant except in the context of a potential leap in our understanding of QCD. Ashtekar asked the inevitable question at the end of the talk, but Myers simply admitted that he didn't really know what it all meant.

This story is about the use of 5d AdS gravity to compute in N=4 SUSY Yang-Mills, which is taken to be close enough to real QCD in certain regimes of the temperature density phase diagram. RHIC data indicates that high temperature (just above $T_{c}$) quark-gluon plasmas display fluid like behaviour. This hydrodynamic model correctly predicts the elliptic flow. A necessary assumption is that the shear viscosity $\eta$ is small and it was found theoretically that

$\frac{\eta}{S} \simeq 1$

where $S$ is entropy density. This suggests the need for a non-perturbative approach. For $L$ a length scale in the AdS metric, it turns out that the entropy density is given by

$S = \frac{1}{4 G_{5}} \frac{r_{0}^{3}}{L^{3}} = 0.75 S_{free}$

where $r_{0}$ is at the AdS horizon. The 0.75 factor transfers to the observed 0.75 value for normalised energy density, characteristic of fluids. This is thought to be a universal characteristic of gauge theories, in which case the string techniques are computationally impressive but devoid of new physical content. Under AdS/CFT, fluctuations in gravitational horizons become deviations from equilibrium in plasmas. The $\frac{\eta}{S}$ ratio is stable under higher order corrections.

Home Sweet Home III

Here goes: an ad for WordPress on Blogger! As Carl points out, WordPress now allows latex in comments, and the formatting is lovely. I'm a little busy at present, but I may set up a new blog on WordPress soon, unless Blogger shows signs of better accommodating scientific blogging.

MoonshineMath continues with a remarkable series of posts on codes and hexagons and other goodies of interest to M theorists. Whilst on that topic, it has come to my attention that some people are still under the impression that this is a fringe string theory blog. I don't know how to explain in a short sentence that this has never been, and never will be, any such thing (all right, so I may have joked elsewhere that I was working on string theory). Yes, I did mention the Veneziano amplitude on a number of occasions, but that was merely to point out that string theory physics could be almost entirely avoided. Of course, the majority who persist in ignoring our existence whilst at the same time being avid followers of the blog, are under no such illusions. They are quite certain that I am nowhere near as respectable as a professional string theorist, or any other professional for that matter, professionalism (as defined arbitrarily by their cronies) being a requirement of respectability.

I continue to be lectured about such matters by my 'superiors' even having passed my 40th birthday recently. I continue to receive 'career' advice and knowing nods from women who have always known I was misguided to follow a vocation meant for male minds, and I continue to get advice on 'anger management' and no amount of pointing out the sort of things I have to put up with has any effect whatsoever on peoples' opinions. They have made up their minds. They know. Not one of them really knows anything about my life (their opinions are based on gossip, not on the testimony of people who were actually present) but they must know better than me how I should live it, otherwise why would they constantly give me advice?

The difficulty, of course, is that I often am rather stupid. The ill defined subject matter in my thesis is based loosely on ideas I was mulling over at the start of 2004, but I was actively prevented from following this line of reasoning for two and a half years, under the justification that I couldn't possibly know what was good for me to work on and I should follow an alternative line of development based on ideas of my betters. Of course, that line of development would go nowhere, as I tried to explain for over 6 months. Well, actually, I went as far as to scream some blatant facts every now and again, having exhausted all other attempts to be heard, but this rarely decreases the level of blind condescension and is better avoided. The main problem was that many points required an hour's explanation, but I was only ever given 5 minutes (although people would happily listen to hour after hour of my lectures on established results of interest) after which, having failed to make my point, the subject was roundly dismissed. One cannot expect the status quo to change soon.

Monday, July 16, 2007

Home Sweet Home II

I just found out that the department decided not to take my application for a tutoring job seriously this term, since I might not be here all year (um, where exactly would I be going?) so if posts are a little infrequent this week it is only because I will have to be very busy applying for waitressing jobs.

Home Sweet Home

All right, so I came straight to the office, but that's close enough.

Following up on results from GRG18, Carl Brannen has a post with a tentative prediction, "Suffice it to say that I expect that the gravity wave people will eventually detect gravity waves, but that these will not correlate to any simultaneous astronomical events." That is to say, the $\sqrt{3}$ factor which often arises in discussions is used in a comparison between the (faster) GWs and standard model particles. Now if this was true maybe one could search for events in the detector data, corresponding to nearby objects such as GRB 070201, for which one has a precise prediction for the arrival time, but no, oh dear, maybe the detector wasn't operating then. 800 kpc is nearby in universal terms, but still some distance compared to the age of human technology. Hmmm. Yes, this could be tricky to sort out!

In general, something like the proposed Southern Hemisphere AIGO telescope will be necessary to pinpoint locations in the sky, via triangulation with respect to the Northern Hemisphere telescopes (which unfortunately lie almost in a plane).

Saturday, July 14, 2007

Penrose's Landscape

The friday night public lecture by Sir Roger Penrose was held in the large Convention Centre auditorium. Penrose has a remarkable ability to convey complex ideas with real clarity. He used his traditional colourful handdrawn overheads, and although the talk only contained a couple of simple equations, one cannot say it was aimed at the general public. In fact, what he did was outline his latest view of cosmology, which he considers an improvement on current fashionable ideas.

Brace yourself for this: the basic elements of this view, motivated by the second law of thermodynamics, include slowly decaying masses over cosmic time (so we cannot compute them), a Dark Energy and genuine information loss into black holes. Penrose gave a wonderful explanation of what entropy is, using the interaction of the sun and earth as an example of how important entropy is for understanding the cosmos. He explained how the CMB was black body radiation and that this meant a high entropy state, whereas the Big Bang should be a low entropy state. The resolution is to realise that gravity behaves very differently to other matter, and that gravity is also needed to describe the initial singularity. That is, entropy increases with gravitational clumping and so a smooth initial state can correspond to a low entropy state. Finally, the new ingredient is the idea that time loses its meaning in the early universe. Treated as a conformal boundary, matter should be considered massless and since such boundaries may be attached to earlier eons (he assumes a cosmic cyclic time) the singularity effectively disappears.

The mathematical argument hinges on the concept of conformal boundary, so Penrose went to the trouble of explaining Weyl curvature by showing images of distorted galaxies in gravitational lensing rings. The extra scalar factor needed to describe clocks in the massive regime arises as structure internal to all the light cones, namely a sequence of internal hyperboloids for ticks of a clock. In other words, clocks only make sense when there is mass, as indicated by frequency = m.constant (assuming hbar and c constant). It is necessary here for the late universe to also lose its concept of time, as everything forms black holes which only evaporate away incredibly slowly. Penrose's solution was to say, well, there will be no humans, just a bunch of photons, which don't see time anyway.

Of course, this idea is full of holes. The first thing one is tempted to do is to pile all the eons on top of each other and try to find some multiverse picture that still respects the second law in this elegant way. Perhaps this cosmology is useful in that sense. Penrose also predicted that conformal structure would be detected in the WMAP data, and perhaps this carries over to a more quantum viewpoint also. He did mention that a serious problem might arise from issues in the foundation of quantum mechanics. I think the biggest conceptual issue is the lack of a concept of 'NOW'. In a cosmology with time asymmetry, any observer must have a means of distinguishing the past from the future, and this suggests a time mark with the meaning of now, but in Penrose's picture there is nothing special about any given era within the cosmic epoch, and there is no quantitative means of measuring the complexity of interesting physical systems, such as life.

Friday, July 13, 2007

GRG18 5b

Earlier this afternoon there were 2 notable short talks by Steve Giddings in the Quantum Cosmology session. First, he reconsidered the original Hawking radiation calculation by asking to what extent higher quantum corrections actually matter, in the light of a resolution to the information paradox via the new concensus that macroscopic locality is abandonned. In this talk he discussed a breakdown of the original argument at time scales around M^3, and in the second talk (on a de Sitter background) he mentioned several independent analyses that point to a break down of perturbative QFT at time scales of the order of R.S where R is a deSitter radius and S its entropy. This is based on joint work with Marolf, based on the idea that non-locality must be considered, very fundamentally, in a relational setting, and in fact the talk began with a slide listing historical work on such approaches to gravity. He also made the point that strings could not address the kind of non-locality that seemed to be required.

GRG18 5a

Jumping ahead to the 3:30 session, as promised Marka spoke about LIGO results triggered by em observations. Note again that the S5 run has been operating at design sensitivity for some time now. Triggered events include gamma ray bursters, short hard GRBs and neutrinos. 39 GRBs were checked during the S2, S3 and S4 runs with a bandwidth of 40 to 2000 Hz. With the longer S5 run, 157 GRBs were checked from Nov 2005 to Mar 2007. For 70% of these events 2 detectors were in operation and for 40% there were three. For soft gamma ray flares such as SGR 1806-20 from Dec 2004, at a distance of around 10 kpc and energy around 10^46 erg, there were signatures for quasi-periodic signals in satellite observations, but no gravitational waves detected. Upper limits are beginning to look interesting.

The new, preliminary S5 result of most interest is for GRB 070201 in the galaxy M31. Images show the source region overlapping the spiral arms of M31. This burst duration was around 0.15 sec with a secondary pulse of 0.08 sec, and it was observed using two Hanford LIGO detectors. No GWs detected. The conclusion is that it is quite unlikely the source can be associated to M31 at all, and an upcoming paper will include minimum distance estimates for the source (based on the assumption that gravitational waves exist).

GRG18 Day 5

The weather has improved considerably, but most people are still sitting inside all day today, planning to stay late for R. Penrose's public lecture tonight.

The first plenary this morning was by Maria A. Papa on data analysis for ground based gravitational wave detection, with a few brief comments on LISA plans. She focussed on the most sensitive current results for different observation types, namely inspiralling compact objects, bursts, contnuous quasi-periodic phenomena and the stochastic background. The most recent results for the S3 and S4 science runs with LIGO for inspiralling objects are given in gr-qc/0704.3368. No plausible events were found. What is the expected detection rate? This is estimated using a unit of cumulative luminosity $L$ which is 10^10 times the value for the sun. For binary systems of (two times) 0.5, 1.4 or 10 solar masses respectively, the maximum distance goes like 5.7, 16.1 or 77 Mpc using the S4 analysis. This corresponds to a rate of about 1 per 1000 years, or a bit better, for black hole binaries at 100 Mpc. This should be greatly improved with the advanced detector setups previously discussed. Moreover, for S5 data (which has been taken for 20 months now) one expects roughly 1 per 10 to 800 years for a 2x10 solar mass binary.

A detection efficiency analysis was also discussed and included in the analysis. Then Papa went onto summarise the intriguing cases of searches which were triggered by electromagnetic observations. In particular, in February this year, there was a gamma ray burst GRB070201 at only 1.1 degree from the centre of M31, which is only 800 kpc from the Milky Way. Unfortunately, it appears that Hanford LIGO was the only GW facility operating at the time, but 2 detectors were in science mode. Theory suggests that GRBs are associated to either neutron star mergers or neutron star / black hole mergers. There will be another session this afternoon discussing this event with the latest data set. Another triggered search was in 2004, for the hyperflare object SGR 1806-20, which is thought to be due to magnestar quakes. The reference here is astro-ph/0703419. It is expected that a two times improvement in detector sensitivity will begin to reach interesting energy regimes for these objects.

Other sources discussed were long-lived signals from stochastic background, and continuous signals from pulsars. For one particular pulsar (Crab) the upper limit set by the first year of S5 data is lower than the limit from the spin down rate of the source. However, not all energy is emitted in gravitational waves, as discussed for example in Palomba et al (AA 354 (2000)). The S5 data can already constrain moment of inertia and ellipticity for such objects. In summary, GW observations are beginning, but just beginning, to contribute interesting astrophysical information. Papa concluded with a nice quote attributed in spirit to Kip Thorne, which says that if gravitational waves are finally observed "no cherished belief would be challenged".

Thursday, July 12, 2007

GRG18 4a

Two fantastic talks in the Quantum Cosmology session this afternoon. First, Mairi Sakellariadou spoke about inflation in Loop Quantum Cosmology. Gibbons et al showed some time ago that in FRW (ie. classically) the probability of inflation onset goes as exp(-3N) where N is the number of e-foldings. In other words, it is exponentially unlikely. The LQC analysis offers the possibility of corrections to the exponent which might greatly increase the likelihood of inflation, but this was shown to be the case only for <20 e-foldings, so even in LQC inflation is highly unlikely, and so it appears that inflation (if it exists at all - er, yeah, right) can only be addressed fully within a proper theory of QG.

Warner Miller took a quantum computational point of view on QG: good man. He was analysing lattices built from quantum circuit information flows embedded in spacetimes with Regge calculus. The really beautiful thing was his way of factorising this calculus using Voronoi duals to the lattice, representing matter information content. So pure QG is non-sensical here, which is as it should be physically. The potentials for matter are encoded in the quantum gates. Moreover, it seems one can directly interpret the duality in terms of a principle of quantum general covariance. Miller also mentioned theorems of Cheeger et al which recover curved manifolds from the discrete Regge triangulations, so there is every reason to believe that the continuum theory behaves as desired.

GRG18 Day 4

I am told that the wine was flowing liberally on the (expensive) harbour cruise last night. Three plenary sessions this morning: H. Ringstrom on mathematical results in Cosmic Censorship, Johnathon Feng on Collider Physics for cosmology and D. Shaddock from JPL on the current status of the LISA mission, which will know around September 8 if it is scheduled for first launch under Beyond Einstein.

Let's focus on Feng's talk, which was very, very entertaining and took a relatively balanced point of view on theoretical considerations, including for instance a flowchart of possible outcomes from comparisons between LHC analyses of Dark Matter and cosmological observations. They expect only a single year's worth of LHC data to determine the existence, or not, of standard SUSY wimps based on a single species analysis. However, note that the existence of SUSY may be supergravity, or otherwise, inspired, rather than a typical string scenario.

The talk began with some recent CERN-cam pretty pictures from the LHC, which Feng noted colour-matched the design layout beautifully. The LHC is expected to yield 10^7-10^9 top quarks per year, compared to 10^2-10^4 at the TEVATRON. The current schedule for first collisions is July 2008. Feng listed some very general, but unanswered, questions in particle physics and cosmology. In particular, he asked how $\Omega_{\Lambda}$ could be around 10^{120}, and then how $\Omega_{\Lambda}$ could be zero, and then he pointed out that it was a remarkable situation when one could ask both questions side by side and sort of be making sense. The focus shifted to Dark Matter, once he pointed out that there were no (well known) compelling solutions to the DE problem. In short, DM is gravitationally interacting, not short lived, and not hot: unambiguous evidence for new physics. He gave a long list of candidates (along with a suitable jibe at the theorists) starting with primordial black holes, and then ran through the unnaturalness argument for cancellation in quantum corrections to $m_{H}^{2}$ (1 part in 10^{34}), strongly suggesting new particles, perhaps of SUSY type (again, note, that SUSY is a general term here).

The 'WIMP miracle' is the fact that by assuming a single DM species, initially in thermal equilibrium, which interacts until it freezes out to a DM relic (let's call it $\chi$) such that $\Omega_{DM}$ depends on the annihilation cross section, one can show that $\Omega_{DM}$ goes as the mass $m$ of $\chi$ squared, and miraculously this suggests (from the observed DM density) that the preferred mass for $\chi$ is in the range 100GeV - 1 TeV of the LHC. Moreover, precision constraints from LEP indicate that new particles interact in pairs, as opposed to as single components in SM particle interactions. In short, the LHC REALLY ought to see new particles. The question is, what are they?

The problem with the neutral $\chi$ particle is that it appears as missing momentum from the 2 $\chi$ at the end of the decay chain. This is not an identification of Dark Matter for cosmological purposes, although it would certainly confirm SUSY if it was found. However, a full analysis of all possible processes for $\chi$ will allow a comparison between particle physics abundance and observed DM density. On that note, Feng briefly discussed the ILC proposal for a variable E beam electron-positron collider. With constraints from WMAP, the Planck mission (2010) and the LHC, the ILC would fix the relic density as a function of $m$, to a value around 96 GeV. That is, the ILC would allow a 1% comparison between collider and cosmology DM identification.

But the most interesting part of Feng's talk was a discussion at the end of an alternative SUSY scenario, based on work of Feng et al, in which the neutral $\chi$ (usually a neutralino) is replaced as a DM candidate by the gravitino, which itself appears in a decay chain from new massive charged particles with longish lifetimes, of the order of a month. These would be seen at the LHC in water detectors built outside the other detector chambers. Such DM is only gravitationally interacting. This possibility also satisfies the WIMP miracle phenomenon, and would allow a particle physics measurement of Newton's constant $G$. Needless to say, the cosmology is radically different. The 1 month lifetime modifies Big Bang nucleosynthesis, offering a possible solution to the Li-7 anomaly which is not always discussed in the WMAP literature. It also suggests a suppression of small scale structure, such as fewer galactic halo objects.

In summary, whatever happens (short of us all killing each other) GRG19 should revolutionise cosmology.

Wednesday, July 11, 2007

GRG18 Day 3

Kip Thorne gave an excellent public lecture yesterday evening, focussing on impressive new work in numerical relativity (such as black hole collision simulations) and the search for gravitational waves. First up this morning was an informative talk by Schneider on probing cosmology with gravitational lensing techniques, covering mass determinations, direct estimates of the Hubble parameter $H_0$, substructure, the collisionless nature of Dark Matter from Bullet cluster observations and cosmic shear. Over 200 strong lensing multiple-image systems are now known. Fifteen lens systems were used to determine $H_0 = 72$ from the time delay dependent on mass distribution and background geometry, in agreement with the WMAP result and other estimates. It appears that lensing has firmly established the existence of dark halo objects for the Milky Way, expected from the visible undercount of only 20 such objects. Other examples of substructure systems included B1422+231, MG2016+112, for which one can visually see the small perturbing satellite galaxy, and the classic recent result from CL0024+17. Cosmic shear looks at the lensing effect of the 3D distribution of matter on large scales. This was detected by four groups in 2000. With more extensive surveys, it promises to further constrain cosmological parameters in the near future.

Renate Loll gave a clear introduction into basic problems with quantum gravity, and then outlined recent successes of the Causal Dynamical Triangulation approach, in particular the recovery of four dimensions at large scales for the Lorentzian path integral with pathological configurations removed and two dimensionality at small scales. Then Francis Everitt entertained us with his wry sense of humor, as he outlined the success of the Gravity Probe B mission. I caught a better glimpse of the unofficial frame-dragging result, and it looked possibly a little higher than the GR prediction by about 10%, but this will probably disappear in the final error analysis. He didn't really comment on this except to tentatively offer a December deadline for final results.

As expected, the poster session involved a notable lack of interest in Category Theory, but the sandwiches were yummy and the company pleasant.

Tuesday, July 10, 2007

GRG18 2a

There have been some interesting sessions this afternoon. I can only mention a couple, since the queue behind me is growing. D. Terno spoke about work with E. Livine on quantum causal histories in the context of quantum information. Causal diagrams are constrained by the requirement that one wants QM to work, and there is a natural way to associate allowable diagrams with standard quantum circuit diagrams. The relevant paper is Phys. Rev. D75 (2007) 084001. Later on M. Tajmar spoke about an observed non-classical frame dragging effect on spinning superconducting rings. Vibrational effects were dealt with using a subtraction technique between clockwise and anticlockwise modes. A question at the end prompted the remark that the observed 'magnetic' interaction between the 4 gyroscopes on Gravity Probe B may be due to this quantum effect.

GRG18 Day 2

Only Day 2 and I've started drinking coffee again, mostly to help down large quantities of cake. This morning started with a talk by B. Krishnan on Black Hole horizons, which moved quite quickly into definitions related to trapped surfaces from a short introduction on the need to move beyond event horizons. There are important applications to numerical relativity because one cannot locate event horizons. I confess to skipping the second talk for more coffee and cake and a pleasant conversation with a lady from Brisbane and the opal lady from the exhibition section.

Last up this morning we heard from Eisenstein on "Observing Dark Energy". He explained nicely his lack of theoretical prejudice as to the cause, but that wasn't until half way through the talk. The first half of the talk summarised in general terms a long list of evidence for apparent acceleration, including the supernovae, CMB acoustic scale, cluster abundances, cluster baryon mass fractions, galaxy-galaxy weak lensing and large scale structure. Having given the impression that he took all this as evidence for a cosmological constant, at question time he was bombarded with comments from D. Wiltshire, A. Vilenkin and I. Neupane.

Monday, July 09, 2007

GRG18 Day 1

After disposing of a Kauri pine's worth of paper from the conference pack, I enjoyed the champagne and food at the conference reception Sunday evening, overlooking the lights of the city. Francis Everitt confirmed that the frame dragging effect measured by Gravity Probe B does agree well with the General Relativistic predictions. Further analysis will probably be completed around December.

Monday kicked off with a welcome ceremony and then the first plenary talk, by S. Whitcomb on ground based gravitational wave detection. He covered the history of resonant bar detectors and modern interferometers, including LIGO, Geo600, Virgo, TAMA300 and the crygenic CLIO detector. I must confess that I did not realise how many of these experiments there are! In 2005 LIGO reached its target sensitivity. The measure of sensitivity is "binary neutron stars at distance x". For LIGO this is around 15 Mpc and for Virgo 3.6 Mpc. Future plans include upgrades for LIGO (called Advanced LIGO) yielding a 10 times improvement in sensitivity, with a target installation date of 2011. Virgo has 2 phases of advanced planning: the first a 2 times improvement ready for science runs in 2010 and then Advanced Virgo set for installation in 2011. The decision for the former phase is to be made later this year. The GEO experiment should remain operational whilst LIGO and Virgo undergo these plans, allowing for possible supernovae observations. Futuristic plans include a large scale cryogenic telescope in Japan (LCGT), for which the main mirror would be cooled to 10K and the baseline would be 3km; and also new acoustic wave detectors.

The second speaker was Laurent Freidel, who tried to cover as many aspects of LQG as possible in one hour. He began by stating that the key issue here is the question of background independence, and then introduced Hamiltonian quantization before defining spin networks and then discussing spin foam models and group field theory. The last part of the talk included some interesting comments on QFTs on non-commutative spacetimes, or rather the necessity of curving momentum space and using a non-commutative geometry to discuss position operators.

The week is jam packed with talks, so I'll neglect to summarise the excellent talk on Large Stellar Systems and evidence for intermediate mass black holes in globular clusters. All this was before lunch time on Monday! And now I need to grab some food before Penrose's twistor theory talk in the parallel sessions...and there may be some envious eyes looking at me hogging one of only 5 computers in the central lounge.

Friday, July 06, 2007

On The Way

Well, I'll be on the plane to Sydney in a few hours. D. Sudarsky just gave a refreshing talk here on his cosmic structure paper, which takes Penrose's gravitational collapse ideas seriously. He was not afraid to criticize common thinking in cosmology, arguing that some basic alteration to quantum mechanics really is essential. Unfortunately, he used what looked like a fairly conventional inflaton to analyse collapse, but at least this was observationally motivated and the analysis has predictive power.

As time passes, I am becoming more and more confused about the fact that many physicists still appear to be quite content to study quantum gravity using an old notion of observable, and their assumption that this should dictate the existence of enormous numbers of particles which have never been observed. And not just a little neutrino (predicted on the basis of conservation of energy) or not just a few quarks (predicted on the basis of lattice patterns), but ridiculous numbers of squishies which are conveniently too massive to have been observed. How can one possibly investigate new physical regimes in a framework so clearly based on guesswork? I just don't get it, and I would really like someone to explain it to me.

Thursday, July 05, 2007

Summer is Here III

I have been enjoying some of the Strings 07 talks from the comfort of my office. Everybody should take a look at the talks of David Gross and Ed Witten. I also enjoyed D. Mateos talking about RHIC physics. He thinks that the interesting results at the LHC will be in heavy ion physics, which will operate at 4 times the deconfinement temperature.

Francis Everitt from Gravity Probe B just gave an interesting seminar here in which he literally flashed up, quite unexpectedly amongst some discussion on engineering problems, the brand new preliminary results on the frame dragging effect (recall that only the geodetic effect was accounted for in the primary analysis from a few months ago). I didn't manage to jot the numbers down, since I'll be hearing a lot more details next week at GRG18, from which I will do my best to blog.

Wednesday, July 04, 2007

String Triality

String triality was considered in 1995 by M. J. Duff et al. See the hexagon on page 3 and the cube on page 9. From the abstract:

In six spacetime dimensions, the heterotic string is dual to a Type IIA string. On further toroidal compactification to four spacetime dimensions, the heterotic string acquires an SL(2,Z) strong/weak coupling duality and an SL(2,Z)×SL(2,Z) target space T,U duality ... Strong/weak duality in D = 6 interchanges the roles of S and T in D = 4 yielding a Type IIA string with fields T, S, U. This suggests the existence of a third string ... that interchanges the roles of S and U. It corresponds in fact to a Type IIB string with fields U, T, S leading to a four dimensional string/string/string triality. Since the [S dual] SL(2,Z) is perturbative for the Type IIB string, this D = 4 triality implies S-duality for the heterotic string and thus fills a gap left by D = 6 duality. For all three strings the total symmetry is SL(2,Z)×O(6,22;Z). ... In three dimensions all three strings are related by O(8,24;Z) transformations.

That last comment is quite intriguing in light of the more recent appearance of Leech lattices, moonshine math and octonionic triality. Note also the analogy with Sparling's three copies of twistor space, and the prominence of the modular group.

Shortly after this active period for string theory, Shenker suggested the existence of a third length scale, shorter than the string scale, and presumably associated to a triality. From our point of view it is quite natural to consider three scales. In terms of mass, there is the Riofrio mass of the universe, the lightest masses (of the neutrinos) and an intermediate Planck scale mass related to transitions in the spectra from particles to black holes. It is of no concern here that one is effectively invoking a scale smaller than the string scale. A zero length scale is reached at $\hbar = 0$, corresponding to the Riofrio horizon. Since particles appear to be composed of three components, and are all composed of each other, it follows that the triality probably extends to cosmic observables.

Turning this around, it makes sense to think of mass spectra from the point of view of cubic numbers. That is, non-associative cubic arrays instead of associative matrices. Unfortunately, we might have to invent these first. In the meantime, recall that numbers usually come from vector spaces (perhaps as dimensions) and even matrices usually belong to vector spaces (perhaps as algebras) so the simplest thing to do is to just continue working with operad maps into an End(V) operad, only we would like to work with a 3-dimensional analogue, which means first defining a notion of 3-End(V). But then we want to throw away linearity, so we need the higher category theory to guide us towards the right structure, because without it we are blind.

Tuesday, July 03, 2007

M Theory Lesson 72

C. S. Peirce is probably best known today for his work on semiotics, based on his notion of triple for which the symbols were but one component. Let's just recall today his simple representation of Boolean complement, indicated by a circle marking the boundary between A and not A. In Boolean logic it follows that a double circle should act as an identity operation. This looks just like the holographic cylinder, which we know is the identity in a category of cobordisms! Sticking to disc operads gives a large class of algebras (and connections to many, many things) but it may be necessary to consider also intersecting discs. Each Jones diagram, shown on the left below, includes internal lines that may not intersect, but when we overlap these big discs we make a choice as to their vertical ordering, and a line from one disc may cross over a line from another, thereby introducing the possibility of braiding for strands from different discs. Note that the three dimensionality of braids is reduced here by viewing the time direction only in a countable number of slices. Not all knots can be ordered in this way, because a strand may weave both under and over another one, which would require an ill defined ordering of discs. However, the basic $B_3$ elements of Bilson-Thompson diagrams are examples of ordered braids.

Monday, July 02, 2007

Summer is Here II

David Gross's closing talk for Strings 07 is now available at the Strings 07 blog. It's a great talk. He criticises the Landscape by simply pointing out that the solution might not be a vacuum but rather a new kind of cosmology. He did not speculate at all on what this cosmology might look like, but referred to a wonderful poem in which all the people (observers) hide from their universe in a game of hide and seek.

This is a beautiful image. Gross points out that the so-called Dark Energy might represent some balance between observers and their universe. In M Theory the Dark Energy is replaced by a tower of quantizations, and the balance principle swims throughout the categorical heirarchy in a web of n-alities that stretch across a cosmos of possible observations.

Presumably 20th century human observations are a very limited class of observation types. Only in 1897 did J. J. Thomson find that the electron was a subatomic particle, and the neutron was not discovered until 1932. The particle zoo of the Standard Model accounts for everything we know how to look at, but that doesn't seem to be very much. Wouldn't it be nice to see dark matter, other than gravitationally? What other quantum numbers can we attach to it then? If they exist, they arise from the heirarchy, from outside the realm of the forces we know. And if we do get to see something more, we are no longer what we were, and our universe can smile and offer us that cup of tea.