Archive for May, 2006

Categorical spacetime

May 17, 2006

John Baez is preparing a colloquium at the Perimeter Institute where the application of Category Theory (CT) to quantum spacetime will be discussed. This page contains links to (a few) introductory and (many) advanced papers on how the ideas of CT (and its generalization to n-categories) can be used to analyze the category of Hilbert spaces.

I’ve developed a lively interest in CT during my computer science wanderings , and even wrote an elementary introduction (geared to programmers) that you may find useful (if you’re new to the field and don’t mind digressions into computer programming).

Categories are deceptively simple: they abstract the notion of sets and their morphisms, or, if you prefer, of objects and their transformations. Thus, a category is defined as a set of objects related by arrows that are composable (in the functional sense). Composition is associative and there’s an identity arrow for each object in the category. As you can easily see, virtually any mathematical (and, by extension, physical) structure can be modeled as a category. For instance, your objects can be groups, with arrows representing group morphisms. Or you could model any discrete physical system’s evolution, by taking as objects its states and as arrows its transitions. With the latter example, we begin to see how CT relates to physics: think of arrows as processes or object transformations. CT somehow captures their essence: one moves from the basic category definition to an exploration of morphisms (called functors) between categories, and from here to constructing categories whose objects are categories, with functors as arrows. Iterate and you’re soon talking about processes consisting of processes consisting of processes consisting of… But of course that’s not all: one also studies morphisms between functors (called natural transformations), and obtains a very precise statement of how any object is equivalent to the set its transformations: objects just disappear on behalf of their relationships! This result, called Yoneda’s Lemma, is beautifully presented in Barry Mazur’s When is a thing equal to some other thing? and further explored in Brown and Porter’s Category Theory: an abstract setting for analogy and comparison (both of them being also an excellent introduction to CT). Although i’m not privy with the applications of CT to LQG and other ‘relationist’ approaches, i think that this blurring of objects in favor of their transformations are at the heart of CT’s appeal to some physicists.

Even discounting its (highly tentative) applications to physics, CT raises deep issues in pure mathematics and philosophy: for instance, this very interesting entry of the Stanford Encyclopedia of Philosophy ends its introduction with these words:

Category theory is both an interesting object of philosophical study, and a potentially powerful formal tool for philosophical investigations of concepts such as space, system, and even truth. It can be applied to the study of logical systems in which case category theory is called “categorical doctrines” at the syntactic, proof-theoretic, and semantic levels. Category theory is an alternative to set theory as a foundation for mathematics. As such, it raises many issues about mathematical ontology and epistemology. Category theory thus affords philosophers and logicians much to use and reflect upon.

All that said, i’m a bit skeptical on the usefulness of CT as a means of synthesizing new laws. For what i’ve seen, it is very powerful for analyzing and discovering properties of already known structures, but i have yet to find a convincing case where it is used to reveal brand new laws. For instance, CT is used in Axiomatic QFT to formalize the theory axioms (see, e.g., Buchholz’s Algebraic quantum field theory: a status report or any other of the review papers on Local Quantum Physics), but it’s not clear to me whether CT is, in this context, more than a convenient mathematical language. Let me, however, rush to say that i’m by no means an expert, and that reading Baez’s writings above may well change my mind!

At any rate, i do like Category Theory, if only for its mathematical beauty, and i think it’s very worthwhile spending a little time learning a bit about it if you are interested in mathematical physics (see here for more of my recommended readings).


Update: Over at A Neighborhood of Infinity, there’s a discussion about “two kinds of maths” (namely, structural and content-providing) which somehow coincides with my half-baked feelings about CT (and, as a bonus, provides a link to yet another proof of Yoneda’s lemma):

Most branches of mathematics have a mixture of the two types of theorem. Typically the structure theorems are used as a tool to discover content. In some sense the content is the end and the structure is the means. But category theory seems different in this regard – it seems to be mainly about structure. Every time I read category theory I see ever more ingenious and abstract tools but I don’t really see what I think of as content. What’s hard in category theory is getting your head around the definitions and often the theorems are trivial consequences of these. For example consider the Yoneda lemma as described here. This is a pretty challenging result for beginning category theorists. And yet these notes say: “once you have thoroughly understood the statement, you should find the proof straightforward”. This exactly fits with what I’m getting at: the theorem is an immediate consequence of the definitions and the proof essentially shows that the machinery does what you’d expect it to do.

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Bootstrapping

May 16, 2006

As we all know, one cannot pull oneself up by the bootstraps, except for Baron Münchhausen, who is told to have lifted himself out of a swamp by the simple device of pulling himself up by his own hair. As for myself, i’ve pulled my hair lots of times to no avail and, as a result, i have some empirical reasons to question the Baron’s deeds!. You probably know the that the reason for our boostrapping inability is, basically, Newton’s Third Law, from which it is easily derived that the center of mass of a set of bodies can only accelerate as the result of external forces. Thus, no matter how much wriggling, you won’t be able to swim in empty space. Or will you?

Center of mass in a sphereWell, actually there’s a twist: you won’t be able to swim in empty, flat space. As it happens, curved space gives you rope enough to boostrap yourself! This well known phenomenon is explored in detail in the delicious Swimming in curved space or the Baron and the cat, recently published by the New Journal of Physics (which, by the way, is an open-access publication, although it maybe falls a bit short of what every scientific journal should be). The authors provide a detailed account of how Riemannian curvature can be used as a sort of handle to pull you around. Even if you don’t feel like following all the mathematical details, you can see movies of the swimming in action over a sphere or a hyperboloid, showing how the periodic motion of three particles on constant curvature spaces gives rise to a net displacement of their center of mass (which carries the particles around in the process). As the article explains, the gist of this effect hinges on the fact that the center of mass is only well-defined in Euclidean space: if you take for instance three equidistant (yellow) points on a sphere, their center of mass would be located at the red point, but if you move the two bodies on the right to the red dot, the new ‘center of mass’ is displaced to the blue dot.

Swimming in a circleThere are other simple examples of boostrapping. For instance, in spaces with non-trivial topologies like a circle. In there, a particle (red) can split in two parts (blue) that recombine in the opposite side (hollow red) without violating Newton’s law. A similar effect is possible in spaces with intersecting geodesics, as i’m sure you can convince yourself with a little thought (or just cheating and looking at the article’s appendices). Note also that this fun magic works on the basis of geometry alone: there’s no need of relativity or anything, just plain Newtonian mechanics.

Over at MathTrek, Ivars Peterson is talking about Edwin Abbot’s classic Flatland. Taking into account how fun this little book is as it is, just imagine how much more interesting could the history have been had flatlanders lived in a curved two-dimensional space. Any takers? :)

Boldness

May 15, 2006

This morning i’ve been a bit surprised by the number of new articles in the arXiv.org feed claiming what would be very significant advancements in our understanding of quantum gravity and related issues. Not that i’ve had the time to read them, or that i’ve got the expertise to quickly sift wheat from chaff: i’m just listing them here for those of you with better criteria (with the hope of reading some insightful comment):

Too good to be true, right?

The LISA Newsletter

May 14, 2006

The LISA International Science Community has just announced its LISC web portal. It will hopefully become the place for coordinating all LISA related resources, and, although it’s still much under construction, you’ll already find there the pretty neat first issue of the LISA Newsletter, which covers, in a very accessible way, a lot of interesting topics:

  • Introduction to Extreme-Mass-Ratio Inspirals. EMRIs (the capture of a stellar-mass body by a super-massive black hole) are one of the most important gravitational wave sources detectable in the frequency range (peaked at 3 mHz) covered by LISA. Here you’ll find a short and well-written introduction to their properties. If they capture your fancy, Eric Poisson‘s living review and this review article by Kostas Glampedakis are a good way of learning more. For non-experts, see also the less specialized presentations by Poisson.
  • Status Report on LISA Pathfinder. Pathfinder (the project i work on) was started in 2004 and aims at testing some of the technologies to be used in the real LISA experiment. The article is written by the mission’s principal investigator Stephano Vitale and (besides a good overview of our current efforts) contains nice pictures of some of the widgets that we plan to put up there by late 2009.
  • A Year of Breakthroughs in Numerical Relativity. This article gives an overview of the impressive progress last year in calculating waveforms of gravitational waves (see, e.g., Goddard’s black hole mergers). Knowing what kind of signals we are looking for will be extremely helpful in data-analysis activities once LISA is operational. Warm-up work in the latter area is also reviewed.

And there’s more! As i said, the newsletter is beautifully edited and aimed at non-specialists. Thus, it’s an excellent way of getting acquainted with the current status and prospects of our quest for these elusive waves.

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Ghostly syllabus for new degree

May 11, 2006

Are you thinking about starting a Ph.D.? Podering string vs. loops? Condensed matter you say? Nah, doubt no more: finally, Coventry University offers you the degree you always wanted. Over at OmniBrain, all the, uh, gory details.

Uncle Al was all too right:

Only two things are infinite, the universe and human stupidity, and I’m not sure about the former.

Albert Einstein

A problem of hierarchy

May 10, 2006

One of the many puzzles (a.k.a. Mysteries of Life) faced by modern theoretical physics is the so-called hierarchy problem: when one compares [1] the relative strength of the four fundamental forces, two widely separated scales are evident:

Interaction Coupling constant
Strong 1
Electromagnetic 1/137
Weak 1/10^6
Gravitational 1/10^39

Or, as Lisa Randall puts it in this interview:

The gist of it is that the universe seems to have two entirely different mass scales, and we don’t understand why they are so different. There’s what’s called the Planck scale, which is associated with gravitational interactions. It’s a huge mass scale, but because gravitational forces are proportional to one over the mass squared, that means gravity is a very weak interaction. In units of GeV, which is how we measure masses, the Planck scale is 10 to the 19th GeV. Then there’s the electroweak scale, which sets the masses for the W and Z bosons. These are particles that are similar to the photons of electromagnetism and which we have observed and studied well. They have a mass of about 100 GeV. So the hierarchy problem, in its simplest manifestation, is how can you have these particles be so light when the other scale is so big.

As you probably know, Randall’s response to this conundrum implies a long detour through multiple dimensions, as recently reviewed over at Backreaction, which was predated by a proposal by Arkani-Hamed, Dimopoulos and Dvali, nicely explained for the rest of us in this Physics Today article. (As a warmup for higher-dimensional physics, you may find entertaining this recent pedagogical review of Kaluza-Klein theories.)

An alternative solution has been put forward by the supersymmetry proponents. As explained (hyped?) in this beatiful review of particle physics:

According to supersymmetry, every “ordinary” particle has a companion particle — differing in spin by half a unit, but with otherwise identical properties. Furthermore, the strengths of the interactions of the superpartners are identical to those of the corresponding ordinary particle. Supersymmetry so simplifies the mathematics of quantum field theory and String Theory that it allows theoriests to obtain solutions that would otherwise be far beyond their calculating ability.

For reasons too complex to explain here (even if i really understood them: see here and here for some of the nitty-gritty details), supersymmetry is claimed to lead to a unification of fundamental forces at very high energies (some 10^28K, or 10^{-39} seconds after the Big Bang), somehow making natural the wild differences in scale of the (energy-dependent) coupling constants in our current universe. As mentioned, the theory also makes easier to define renormalizable QFTs (due to some magical cancellations), and has become one of the main ingredients of String theory, although there is at least another extension to the standard model of particle physics that seem to share these magic cancellation virtues, solving in the process the hierarchy problem: this Physics World article gives an introduction to this so-called ‘little Higgs’ theory.

Personally, i find all these untestable super-theories and multiple dimensions rather unconvincing, and would prefer some old good 4-dimensional solution. Alas, no one seems to be avaialable… maybe it’s time to embrace Compactified Dementia and be done with that.


[1] The excellent little comparison of coupling constants for the fundamental forces pointed to by the above link is part of a nifty site called HyperPhysics, an amusing experiment combining HyperCards, Javascript experiments and similar online tricks with well written contents. Visit it for fun, hierarchy problem or not.

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Arxiv Structure

May 7, 2006

I bet this is old hat for many of you, but just in case: i’ve discovered a new, nifty way of searching and browsing arxiv.org papers: Xstructure. The exciting part is browsing: articles are classified by theme and can be viewed in a variety of trees and listings. For instance, here‘s the entry page for the recently added gr-qc archive: there you’ll find submission statistics and some useful links, including Review Articles and Authority Articles, which lists the most cited ones. Interestingly, comparing the authority articles in gr-qc with those of hep-th clearly shows what we could call a quantum gravity divide: the former consists almost exclusively of papers on Loop Quantum Gravity (Smolin, Rovelli, Ashtekar and friends), while the latter is monopolized by the String and M-Theory guys (Witten, Polchinski, Randall…). Hardly surprising, i know, but still…

Needle in a haystack

May 7, 2006

I’m well on my way reading Penrose’s Road to Reality, and (as you may expect) having a great time. The first part, devoted to maths, was just delicious and, to me, an excellent refresher that has prompted me to dust from the shelves some books on rediscovered topics like complex analysis (“Visual Complex Analysis” (Tristan Needham)) or fiber bundles &co. (“Gauge Theory and Variational Principles” (David Bleecker)).

As for the physics chapters afterwards, i find their quality more uneven. In general, my impression is that one needs a pretty good basis on the topics discussed by Penrose to really appreciate and fully understand many a discussion in the book. I’d even say that pretty good means a university background in a technical area. Albeit, admittedly, it’s hard to judge from my standpoint, i don’t buy the claim (made at the preface) that even someone don’t getting rational numbers can understand (parts of) the book. For instance, i’ve found that the chapters on Relativity provide an insightful overview for people in the know, but, in my opinion, will fail to convey the deep principles of the theory to newbies. This impression was reinforced after reading the chapters on Quantum Field Theory: i’ve already forgotten most of what i learned about QFT at the university, and, besides, i never really understood it in depth. As a result (i think), those chapters of the book devoted to QFT have been frequently hard to understand to me, and i’ve finished them with a strong feeling of being missing many important points. Oh well, maybe it’s just me.

GodDespite the above knit-picks, the book is absolutely worth reading. In particular, i’ve enjoyed immensely the chapters dedicated to Cosmology and the problem of the low entropy of the Big Bang. If we are to believe in the second law of thermodynamics, the entropy in the universe has been increasing since its origin some 13 thousand million years ago. In other words, its path through the phase space has been traversing regions of greater and greater volume, where a region stands for a set of microscopic configurations giving rise to the same macroscopic behaviour (for instance, there are virtually an infinite number of possible positions and velocities for the atoms in the air of my room which gives raise to macroscopic values for pressure, temperature or smell that are indistinguishable by means of macroscopic measurements). Taking into account that the most entropic objects in the current universe are black holes (using the famous Bekenstein-Hawking relationship between a black hole’s entropy and the area of its horizon; see here and here for details) and an estimation of the number of black holes in the current universe, Penrose concludes that the volume of the phase space compatible with our Big Bang is about one part in 10^{10^{123}} of the total available. As shown in the figure, god must have had a hard time finding the exact point to start all this! If you don’t have Penrose’s book (or want to read a nice summary), you can read about these intriguing issues in his excellent survey (Space-time and Cosmology (PDF), part of the freely available Tanner Lectures on Human Values from the University of Utah [1], or hear and see his three lectures at Princeton University. There you’ll find why inflation and the anthropic principle, fashionable cures to everything as they may be, are not a solution to this conundrum (at least according to Penrose; you may find amusing to think about it a bit before reading the answer ;-)). Enlightening.


[1] As an aside, you’ll find lots of other interesting Tanner Lectures in their site, including this one by Richard Dawkins on Science and Religion.

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Albert the robot

May 5, 2006

And now for a bit of fun, courtesy of GeekLand :

This amazing robot has been developed at the Hubo Labs. More links and fun with robots at Roborama. Enjoy!

Cosmic Variance’d

May 3, 2006

To my great surprise and pleasure, Sean has been so kind as to mention physics musings in his last post at Cosmic Variance. First of all, I just wanted to say thanks: i’m sure that in many cases reading Cosmic Variance is one of the reasons one starts thinking of writing a physics blog in the first place. At least it was for me. I’m also happy because i’m fond of Sean’s writings in more than a way. A few months ago, i spent some weeks getting up to speed, again, in General Relativity, and carried home several textbooks. Some from my university days, and others published in the interim. And of all those books, the only one that i finally read (and enjoyed) from cover to cover was Sean’s “Spacetime and Geometry: An Introduction to General Relativity”. I’m sure it needs no presentation or recommendation, but, just in case, you can get a feeling of how good it is by reading the notes it’s based on or visiting the book’s website.

Sean’s post has also enlarged my list of monitored blogs with several interesting new entries. Among them, there’s one i’ve been enjoying specially during the last couple of hours. Lest you miss it in the post’s comments, here’s my recommendation: Alejandro’s Reality Conditions is just excellent. He has a lot to say about quantum gravity and the interpretation of quantum mechanics (two of my favourite areas), but also frequently touches more philosophical themes i find absorbing, like cognitive science and the problem of consciousness. With a dose of humor and many interesting links for a good measure. Highly recommended!


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