Archive for July, 2006

Students and quantum mechanics

July 28, 2006

The latest issue of Physics Today has a freely available article that, under the title Improving students’ understanding of quantum mechanics, gives a very interesting analysis of student’s difficulties when faced with university courses in Quantum Mechanics:

Extensive testing and interviews demonstrate that a significant fraction of advanced undergraduate and beginning graduate students, even after one or two full years of instruction in quantum mechanics, still are not proficient at those functional skills. They often possess deep-rooted misconceptions about such features as the meaning and significance of stationary states, the meaning of an expectation value, properties of wavefunctions, and quantum dynamics. Even students who excel at solving technically difficult questions are often unable to answer qualitative versions of the same questions.

The article goes on describing little problems posed to students and how they revealed fundamental misconceptions (if you know a bit about QM you may find interesting to try your hand at them too), and proposing ways of improving their understanding by means of interactive software and tutorials (more fun ahead). Recommended.


Three Feynmans walk into a bar…

July 28, 2006

The first two vanish into the back room to play a game of darts, leaving the third to chat up the barmaid. After emptying a stein, he pulls a bit of ribbon from his pocket and entertains her with a trick hereby he ties the ribbon to the handle, twists the stein around two full turns, and then magically untwists the ribbon without moving the stein. He tells her how particles he studies have a property called “spin”, and that particles whose spin is 1/2 actually behave like the stein with the ribbon tied to it: coming back where they started only after two, but not one, full turns. Feynman then leans closer, and conspiratorially whispers to her why (in language a barmaid can understand), just spin 1/2 particles act this way, and not other particles.

What story does the Feynman tell the barmaid?

(Posted by Edward Green to sci.phys.research. Hints here and here.)

Feynman spinning

Electronic referees

July 22, 2006

Judging a paper’s quality may be hard for human referees, and people are looking for alternatives. For instance, this recent PhysicsWeb news gives an overview of P. Chen et al. article Finding Scientific Gems with Google, where the authors take advantage of Google’s page rating algorithm to assess the relative importance of all publications in the Physical Review family of journals from 1893 to 2003. Since the rating algorithm weights pages by number of referrers [0], there’s in principle no value added to traditional citation indexes: both popularity measures are linearly correlated. The catch is that there are exceptions: papers that are not widely cited but that, judging for the number of web pages linking to them, seem to be much more influential than one would think (the article mentions quite a few, Feynman, Murray and Gell-Mann’s one on Fermi interactions being an example). Amusing; although i must confess that this kind of democratic assessments of our scientific endeavours remind me somewhat of a well-known Planck dixit:

Max PlanckA new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.

Max Planck, 1858–1947

(I remember i jotted down this same quotation some twenty years ago, together with a note showing my skepticism… nowadays i think i’m much more of a planckian than i used to be.)

Returning to our electronic referees, over at PhysOrg there’s a story about how computer science may help us in detecting bogus papers, where by bogus i mean automatically generated ones (looks like our human referees do sometimes find their task really hard!). Probably the most popular case of such a prank was the article accepted at WMSCI 2005 whose author came out to be SCIgen, an automatic paper generator created by the guy in portrait on the right. And our field is not immune to similar problems, as exemplified by the amusing Bogdanoff Affair (besides, as you’ll see, most probably no computer program would be of much help in this case).

[0] Of course, i’m oversimplifying: see here for the complete history behind Google’s PageRank.

Update: Andrew Jaffe, in his excellent blog (recommended, but you probably already knew it), has some interesting thoughts about peer review, and a recent initiative by Nature to open a debate on the issue and looking for ways of improvement.

Postmodern Prometheus

July 22, 2006

Hiroshi, isn't he?I know i’m wandering a bit off-topic here, but this post over at Mind Hacks was too fun to let it pass unnoticed: according to recent news, Hiroshi Ishiguro, director of the Intelligent Robotics Lab (IRL) at Osaka University in Japan, has created an android double of himself. You can see Hiroshi in the figure on the left (stolen from this recent Scientific American article). If you didn’t recognise the android at first, don’t worry: Ishiguro’s team have been conducting a variation of Turing’s test where androids are shown to humans for around two seconds; 70% of times people take the androids for humans. The experiment is part of a study trying to explain the so-called uncanny valley effect: the emotional response of humans to robots increases as the robot is made more human-like, up to a point where a too close resemblance causes the empathy to abruptly fall. A sort of too good to be true effect. According to the IRL experiments, this may be caused by the stillness of androids: it’s impossible for humans to stand perfectly still, and that’d be what makes robots uncanny. To prove their hypothesis, Ishiguro’s team has added micro movements to their toys. As the result of the experiment with 20 subjects, 70% of the subjects did not become aware they were dealing with an android when the android had micro movements, but 70% became aware with a static android.

SensorsOf course, there’s much more to Android Science (as they call their interdisciplinary endeavours) than micro movements. This beautiful paper gives an introduction to some of the challenges faced by these intrepid researchers, ranging from an understanding of cognition and intelligent behaviour in humans to the technical challenges of creating an adequate sensor system for the robot: besides the piezo-electric films under their silicon skin, the androids use a sensor network distributed over their environment that includes video, sound and infrared motion sensors. (Yes, that’s a bit like cheating, but the results are impressive: you can get a glimpse of the science and technology behind sensors and data processing in these posters by the IRL researchers) The paper also discusses in length the Turing tests mentioned above.

I just wonder, why do the androids look so angry?

Pebble physics

July 20, 2006

From the Curiosities Department comes this news over at PhysicsWeb (see also a previous piece at about recent advancements in our understanding of pebbles:

Aristotle's peeblesA question that has been around since the time of Aristotle — what shape is a pebble? — has now been solved by physicists in France and the US. Douglas Durian of the University of Pennsylvania and colleagues in Strasbourg say that a pebble is “a nearly round object with a near-Gaussian distribution of curvatures”. All pebbles, regardless of their original shape, end up with a similar shape that depends solely on how the pebble was eroded over time. The results could help geologists determine the history of a pebble simply by looking at its geometry (Phys. Rev. Lett. 97 028001).

You can also take a look at this nice presentation for more details, and even see some movies by the people of Strasbourg. For all the nitty-gritty details, the articles can be found in the arXiv, here and here.

steiner.pngThis news reminded me of an older one, The Mistery of the Skipping Stone, where the physics of bouncing stones in water is (more or less) explained (unsurprisingly, the determining factor seems to be the initial velocity of the stone: see this very readable paper, to appear in AJP, by the same author, Lyderic Bocquet). A piece of amazing trivia included in the article: In 2002 an American called Kurt Steiner set a new world record when he threw a stone across a river in Pennsylvania and made it bounce… 40 times. Unbelievable? I thought so, but here’s the proof. By comparison, the team of physicists writing the article were using a specially designed catapult for their experiments, but they got just 20 bounces. What’s your mark?

Update: And, when it comes to talk about physics and throwing, nobody more apt (one would say) than a physicist let loose at the Baseball Major League: in A Magnus Force on the Mound,  Major league pitcher Jeff Francis brings an educated insight to the physics of baseball (besides giving me an excellent excuse to publicize the excellent Symmetry Magazine, a joint SLAC/Fermilab publication about particle physics for the rest of us).

The dimensionality of the world

July 17, 2006

Although Bee has recently written an amazing and thorough article over at Backreaction with virtually everything one needs to know about extra dimensions in physics, let me add a sort of footnote in the form of some naive musings, a couple links and a Hertzian digression in this somewhat iffy post.

Multiple dimensions and the problem of time
As a student, i was in love with Kaluza-Klein theory and its extremely elegant explanation of electromagnetism as the purely geometrical effect of a fourth spatial dimension. The really magic thing is that the electromagnetic energy momentum tensor (in four dimensions) arises as a consequence of an empty five-dimensional space where particles follow geodesics; in other words, photons are purely geometry, just as gravitational forces. The problem, of course, was to explain why we don’t measure that fifth dimension. Kaluza just prescribed that no physical quantity depended on it, while Klein tried a somewhat more satisfactory solution by compactifying it to an unobservable size, and making it periodic, just as the second dimension of a long hose, which becomes one-dimensional when seen from a distance. Unfortunately, this beautiful picture seemed to lead to insurmountable difficulties with chirality or the mass of the electron, unless one goes the string way and adds more compact dimensions to our universe. I thought Kaluza-Klein theories were all but abandoned in their original 5-dimensional form these days, but following some links in the recent review article by Orfeu Bertolami, The Adventures of Spacetime, proved me utterly wrong. There’s been quite a lot of activity in the area during the last decade, leading even to a Space-Time-Matter consortium, a sort of physicists’ club promoting 5-dimensional gravity theories without compactification. The consortium is coordinated by P.S Wessan, and has quite a few members and interesting publications: see for instance this comprehensive review of KK theories of gravity for an introduction to Wessan and friend’s ideas. What i find compelling about their approach (and what, at the same time, of course reveals my prejudices) is that they tackle multidimensional physics from the point of view of general relativity, rather than particle physics. However, i guess that a word of caution is in order: i’ve read very little about these (to me) novel approaches to KK theories, and i’m not yet ready to endorse them; if they were right (and i definitely wish they were), they’d be quite revolutionary: for instance, they explain quantum indeterminacy as a result of particles travelling in higher dimensions… that’d be extremely cool (and actually make real one of my silly ideas of old), but perhaps too cool to be true? Well, i’ll leave it for you to decide (as for me, i think i’m going to read Wessan’s book, Five Dimensional Physics, lest student dreams can really come true!).

Returning to Bertolami’s paper, let me mention that it is part of a forthcoming book entitled Relativity and the Dimensionality of the World, the good news being that the above link points to freely available versions of many of its chapters, written by various authors, including Wessan and G.F.R. Ellis. The latter writes about his rather original ideas on time in General Relativity, and the Block Universe idea, familiar to all relativists, of a world represented as a frozen 4-dimensional whole. Ellis observes that such a representation clearly suggests that time is an illusion: the entire universe just is. The problem is that such a view seems incompatible with irreversible, macroscopic phenomena, as well as with the fundamental indeterminism inherent to quantum mechanics. To take into account these facts of life, Ellis proposes an Evolving Block Universe: time passes; the past is fixed and immutable, and hence has a completely different status than the future, which is still undetermined and open to influence; the kinds of `existence’ they represent are quite different: the future only exists as a potentiality rather than an actuality. The point being that our regular, predictable universe models are based on too simplistic assumptions and oversimplified systems, and that taking into account realistic, emergent ones renders the future under-determined. Although very interesting from a philosophical point of view, Ellis ideas need much fleshing out before becoming a solid theory of anything. But still, he makes many a fine point, and quite a lot of good questions worth thinking about.

A digression: Hertz’s mechanics
Finally, Bertolami’s paper draw my attention to a facet of Heinrich Hertz‘s work i was totally unaware of, namely, his contributions to the interpretation of classical mechanics. After gaining a place in the history of physics with his experimental confirmation of the existence of electromagnetic waves, and before his tragic death when he was only 37, Hertz wrote a book, The Principles of Mechanics Presented in a New Form, where he proposed a formulation of Newtonian physics freed of forces, using instead a variational principle. According to Hertz’s principle, particles move along paths of least curvature, where the (three dimensional) metric is defined by constraints instead of forces. Similar principles were proposed by Gauss and d’Alembert before Hertz, but the latter was notorious (if only ephemerally) for pushing to the forefront a view of space-time defined by matter in a purely relational, Leibnizian fashion: Hertz tries to derive his system of the world from material particles alone. Unfortunately, i’ve found little information on-line on Hertz’s ideas, which seem to be better known to philosophers due to their influence on Wittgenstein (who directly mentions Hertz in his Tractatus). For those of you with a philosophical soft spot, this paper presents a re-interpretation of some of Wittgenstein’s ideas under a Hertzian perspective. As a physicist, i find Hertz’s ideas interesting almost only as a historical curiosity, and don’t know how relevant they really are to modern epistemology: comments welcome! ;)

Lords of the Ring

July 12, 2006

As part of the series of articles it is running on CERN’s LHC, Seed Magazine has just released Lords of the Ring, a very nice short film featuring some of the physicists (Brian Cox, Jon Butterworth and Albert de Roeck) working on the collider and, of course, the collider itself. It’s only five minutes, but worth viewing (with congratulations to SM for using a GNU/Linux/PPC compatible format! ;)).

Update: And a new premier, this time Seed’s first audio slideshow, where  Luis Alvarez-Gaume and Ulrich Fuchs discuss the magnitude of the LHC, how it works and why future CERN experiments could revolutionize our understanding of how the universe works.

Nature’s nifty tricks

July 10, 2006

Herman BondiA few weeks ago, and thanks to this post over at A Neighborghood of Infinity, i discovered a little jewel: Herman Bondi‘s book Relativity and Common Sense, an insightful and original introduction to special relativity for the layman. The above mentioned post gives an overview of Bondi’s approach, based on what is known as k-calculus, arguably simple enough to be taught at highschools. Bondi makes the case for an understanding of SR as an evolution of Newtonian ideas, rather than as the revolution we all thought it was. Whether successful or not, this attempt leads Bondi to start his book with a delicious review of classical physics, illustrating the concepts and principles with very well chosen common life phenomena and their corresponding explanations. You know, trying to come to grips with quantum mechanics interpretations or finding a unified theory of everything is all very well, but it’s the marvel of being able to explain those other nature’s nifty tricks what draw me to physics in the first place. These little and easy to understand principles convey in a very real and fun sense the magic of ours world’s understandability. Bondi touches a lot of these magic things in the first fifty pages of his book, and i thought some of you may amuse yourselves finding (or just remembering) the explanations to these down-to-earth phenomena:

  • Conservation of momentum. Your baby is sleeping in her pram. Would you put the brakes on or off to ensure she will still be there when you come back and find that she awakened in the meantime?
  • Spinning cats. Can you think of a way of exploiting angular momentum conservation to explain how a cat manages to land always on all fours? (For extra score, how’s this problem related to Yang-Mills theory?)
  • Coriolis force. Can you explain it in simple terms (involving, say, a disk and a little ball)? Does it have anything to do with kitchen sinks? And what about weather? How does it explain cyclones and anticyclones?
  • Wave phenomena: the sonic boom. Can you explain how the explosion associated with a plane travelling at ultrasonic velocities is produced?
  • More on waves: Doppler effect. You surely know about the proverbial band on a train wagon, but what about putting the band on the station and yourself in the train? Will be the frequency shift identical? Why or why not?

These look like simple, even basic, questions, but their solution does not lack subtle points. I’m with Bondi in that one should better understand everyday physics before jumping to the unification of forces or even, more modestly, special relativity. At least in my case, and with the benefit of hindsight, i regret having jumped too early into mathematical physics and abstract stuff, only later learning about funny things like those above: i’m sure i’d be a better physicist had i spent more time in the wonderful world of understandable experiments… but, oh well, you already know the line: you’re never too old… ;-)

Turning back to Bondi’s book, never mind some of the unfavourable comments at Amazon. Even if you don’t buy his claims about SR being common sense (i don’t), the book is just excellent.

Getting Schwarzschild right

July 9, 2006

A recent post over at sci.physics.research mentioned some papers (available at the arxiv) that claim that we’ve been misinterpreting Schwarzschild’s solution all these years; more concretely, that the event horizon (and, therefore, static black holes, among other things) is just a mirage produced by ill-chosen coordinates. As it happens, it’s these article’s authors who are misinterpreting coordinates changes, as shown in the answers to the original post. In particular, i strongly recommend T. Essel’s post, Flogging the Xprint to all students of General Relativity, as a beautiful tutorial on the meaning of coordinates in GR, a really tricky issue (as shown by the elementary errors disclosed even in published papers). The thread includes also a bit about the interplay between coordinates and topology and, come to think of it, is a good reading also for those of you privy with the field, if only for the fun of it.

Essel finishes his long post with a reflection worth mulling over:

The fact that such obviously wrong papers continue to be produced, published, and cited is dismaying, because one major goal of providing electronic archives is to make it easier to find/obtain/study relevant previous work, yet this kind of rampant repetition of old errors suggests that some “researchers” have forgotten that -reading- is the most important part of library research!

This raises a disturbing question: by drastically lowering the threshold of pain involved in simply -finding- and -obtaining- relevant prior work, while leaving unaltered the threshold of pain involved in -reading- what one has obtained, has the advent of the arXiv had the unexpected and paradoxical effect of -decreasing- knowledge of the research literature
among researchers? If we make “the easy part” of library research -too easy-, will the next generation fail to take the trouble to read the contents of our libraries (that’s “the hard part” of library research), on the grounds that actually -reading- the literature would constitute an unacceptable burden on the time and energy of busy scholars?

I have no easy answers to these questions, other that my feeling that keeping up with current research in anything but an extremely narrow and specialised area is an overwhelming (if not plain impossible) task. As a side effect, i’ve noted that i lowered my crackpot-detection threshold by a considerable degree, which bothers me a bit because i think that really new ideas will probably be, on a superficial reading, on the verge of crackpotism. Sifting wheat from chaff has always been a problem, but who ordered so much chaff?

Update: Malcolm MacCallum has published yet another refutation of Antoci et al claims about event horizons. Very instructive.

Geometrically speaking

July 6, 2006

While a was a full-time physics and maths student, i seldom, if ever, thought of proving anything using a diagram, or any kind of non-algebraic method, for that matter. One could make a couple of drawings every now and then to help understanding, but that was all. Not even after learning differential geometry did my view change. As a matter of fact, with the emphasis on (and the beauty of) abstract representations (as in abstract tensor notations), using drawings of surfaces embedded in Euclidean space felt like cheating. To make things even worse, my first serious physics book had been Landau and Lifshitz’s Classical Field Theory, where even words are scarce, let alone drawings or diagrammatic reasoning [1]. In a nutshell, i would have felt at home reading Lagrange’s introduction to his Méchanique Analytic [2]:

No figures will be found in this work. The methods like i set forth require neither constructions nor geometrical or mechanical arguments, but only algebraic operations, subject to a regular and uniform procedure.

Counting squares

Brown picturesI’m stealing the quote above from a talk entitled Proofs and Pictures [3], which started me re-thinking about diagrams in physics (and maths) in the first place. It was given at the Perimeter Institute by James Brown, a professor of Philosophy of Science at the University of Toronto. In this fun talk, professor Brown explores the use of geometrical reasoning in maths and physics as a means of actually proving results. Some simple but instructive (and, to me, somewhat surprising and definitely amusing) examples of such “proving by diagrams” are given in the figure on the left (click to enlarge), which shows how getting general formulas for arithmetic and geometric sums may be as easy as counting squares. I’m giving away just two of them, so that you can try your hand with the other two and have a little fun (you can also try to invent your own, maybe going to 3- or even n-dimensional cubes, in which case, please, don’t forget to post your discoveries below! :)). Although elementary, these proofs are intriguing: would you accept them as such? Brown argues that they do, since they can be used to show the validity of the induction step in the usual algebraic proofs. I’m not sure i buy the argument, but it’s a very interesting one.

Penguins and lollypops

Turning our attention to physics, probably the most famous diagrams in the field are Feynman’s. As i’m sure you know, they offer a convenient notation for manipulating terms in QED’s perturbative expansions. Taken at face value, or, one might say, analytically, the represent just algebraic combinations of functions (propagators) entering a power series expansion in a small parameter (the interaction coupling constant, alpha). But they’re usually interpreted as providing the actual physical mechanism for the interaction of real particles by means of exchanges of virtual, unobservable photons. Albeit intuitive and appealing, this interpretation has always bothered me. After reading about it in popular science books, i expected QED being somehow based on photon exchanges from scratch. Instead, what one has is a principle of least action which leads to differential equations unsolvable in exact analytical form. Then, when calculating an approximate solution to a scattering problem using a power series, one obtains (the analytical equivalent) of Feynman diagrams and interprets them, so to speak, after the fact. I would somehow feel more comfortable if the process were the other way around: start with the (supposedly) physical underlying process (the photon exchange) and derive the scattering amplitude. Each Feynman diagram would then represent an actually possible scenario, in the same sense that an electron choosing one slit in the two-slit experiment is possible: one can break the superposition and observe the electron in its way through the slit. But this is of course impossible: virtual photons are unobservable, if only because they travel faster than light and violate energy conservation. To add to my uneasiness, a plain Feynman series leads to divergences to be cured, non-diagrammatically, by renormalisation. Yet, everyone since Feynman discusses this spooky photon ping-pong as the right interpretation [4], so probably i’m just showing off my lack of understanding! And, besides, one could arguably point to measurable vacuum polarisation effects like Casimir’s as an experimental proof of the existence of virtual particles (see for instance this recent, accesible account at PR Focus). Or one could even see the situation as a derivation of the interaction underlying mechanism from first principles, an stunning testament to their power [5]. At any rate, and specially if one accepts the mainstream interpretation, Feynman diagrams appear as a good example of how diagrammatic tools can be more than just a picture, and not only in mathematics. For more on Feynman diagrams and pointers to further reading, see their WikiPedia entry, or get Diagrammar a CERN report by ‘t Hooft and Veltman with all the gory details with a deliciously retro (as in written in 1973 using a typewritter) flavour.

PenguinBefore leaving the subject of Feynman diagrams, let me mention two bits of diagrammatic folklore stolen from Peter Woit’s latest book. Naturally enough, recurring diagrams have got pet names over the years. The first one seems to have been the tadpole (for a diagram shaped, well, like a tadpole), coined by Sidney Coleman and resignedly accepted by the Physical Review editors after he proposed lollypop and spermion as alternatives. The second anecdote involves a diagram (depicted above) known as penguin since Melissa Franklin won a dart match over John Ellis: Tommaso Dorigo has recently recounted the story in his blog.

Tensors and birds

Roger Penrose’s thought is all but geometrical, and it comes as no surprise that he has made many a contribution to the physics by drawing camp. Every decent course on General Relativity touches conformal diagrams [6], a nifty method envisioned by Penrose and Brandon Carter (back in the sixties) to bring infinity back into your drawing board. The trick consists on scaling your metric by a global function that vanishes quickly enough when your original coordinates go to infinite. Such scaling is known as a conformal transformation, and has the virtue of preserving angles; in particular, null geodesics are mapped into null geodesics and, therefore, the causal structure (represented by null cones) is untouched. While beautiful and handy, i think that conformal diagrams do not add anything really new from a computational standpoint (as Feynman diagrams do), let alone serving as the basis for actual proofs.

PenroseMore interesting for our current musings is Penrose’s graphical tensor notation. Tensor indexes (specially in its abstract flavour, also introduced by Penrose) are a quite convenient housekeeping device, ensuring almost automatically the consistency of your equations and even (once one has a bit of practice with them) suggesting their form [7]. But, convenient as they are, indexes seem to be confusing for geometrical minds like Penrose’s, who some fifty years ago devised a pictorial representation for tensor equations [8]. As you can see in the figure, the idea is simple: choose a closed polygon to represent the kernel letter of each tensor, and add an upwards leg for each contravariant index, and a downwards one for each covariant index. Index contraction is represented by joining the respective legs. A wiggly horizontal line represents symmetrisation; a straight one anti-symmetrisation. One can cross legs to indicate index shuffling. The metric gets no kernel figure (it’s just an arch), so that contractions of indexes in the same tensor are easily depicted, and raising and lowering indexes amounts to twist the requisite leg up or down. To indicate covariant differentiation, circle the tensor being differentiated and add the corresponding downwards (covariant) leg. And so on and so forth. Note also that commutative and associative laws of tensor multiplication allow your using any two dimensional arrangement of symbols that fits you, which aids in compactifying expressions. Penrose explains the many details and twists of the notation in The Road to Reality and in his (and Rindler’s) Spinors and Space-time I, where you’ll find extensions to deal graphically also with spinors and twistors. According to the latter,

The notation has been found very useful in practice as it greatly simplifies the appearance of complicated tensor or spinor equations, the various interrelations expressed being discernable at a glance. Unfortunately the notation seems to be of value mainly for private calculations because it cannot be printed in the normal way.

Besides the (not so obvious nowadays) difficulty mentioned above, i guess that the main hurdle in adopting Penrose’s notation is habit. After many years using indexes, my algebraic mind seldom finds equations confusing because of their indexes. But after a little practice it becomes easier, and i’d say that people who see equations will find it quite natural after a very little while [9]. I don’t know how popular Penrose graphics are among physicists for private use, but there’s many an example of their application and extension to related fields. A few years after its introduction, the notation was rediscovered by Pedrag Cvitanovic, who used a variation of it in an article on group theory and Feynman diagrams. More concretely, Cvitanovic uses diagrams similar to Penrose’s to represent to represent the structure constants of simple groups in the context of non-abelian gauge theories, interestingly linking them with Feynman diagrams (and closing a loop in this article!). Later on, he would use the notation very extensively in his on-line book on Group Theory, where the diagrams go by the name of bird-tracks. In a nutshell, the book is devoted to answer, in Cvitanovic words, a simple question:

“On planet Z, mesons consist of quarks and antiquarks, but baryons contain 3 quarks in a symmetric color combination. What is the color group?” If you find the particle physics jargon distracting, here is another way to posing the same question: “Classical Lie groups preserve bilinear vector norms. What Lie groups preserve trilinear, quadrilinear, and higher order invariants?”

From here, an amazing journey through the theory of Lie groups and algebras ensues, a journey conducted almost exclusively by diagrams. For, notably, Cvitanovic uses his bird-tracks (as mentioned, a very evolved kind of Feynman diagrams) to actually derive his results. We have here physics (and maths) by diagrams for real, actually replacing algebraic reasoning (and, incidentally, a proof that Penrose’s reservations about his notation not being apt for publications are unfounded nowadays–i wonder how Cvitanovic draws his diagrams).

Before leaving the subject, let me mention a couple more works inspired by Penrose’s diagrammatic notation. Yves Lafont has greatly extended it and carefully analysed its application to mathematical problems in the context of category theory and term rewriting systems. If you’re privy in the field, or simply curious, take a look at his articles Algebra and Geometry of Rewriting (PS) and Equational Reasoning With 2-Dimensional Diagrams , where Yves explores two-dimensional diagrams a la Penrose with an eye to (possibly automatic and computer-aided) derivations much in the spirit of Cvitanovic. And, turning back to physics, if there’s a theory prone to diagrammatic reasoning it must be Loop Quantum Gravity, where the basic constituents are graphs and their transformations. Arguably, LQG is the most fundamental example discussed so far of graphical reasoning applied to physics, for here graphs (and their combinations in spin foams, an evolution of another Penrose invention, spin networks) do stand for themselves, as opposed to representing some underlying algebraic mathematical entity. Wandering into the marvels of LQG would carry us too far afield, so i’ll just point out that Rovelli, Smolin and friends use not only Penrose’s spin networks, but, on occasion, also the graphical tensor notation we’ve been reviewing; see for instance their seminal paper Spin Networks and Quantum Gravity, where Rovelli and Smolin presented their famous derivation of exact solutions to the Wheeler-DeWitt equation. The notable thing is, again, the fact that graphic notation is key in many a derivation, and cannot be seen as just an aid to represent some calculations.

Kindergarten categories

Our final example of physics by diagrams comes from the category theory-inspired view of Quantum Mechanics invented by Samsom Abramsky, who has managed to do “quantum mechanics using only pictures of lines, squares, triangles and diamonds”. This beautiful notation (or picture language, as their authors call it) is nicely explained in Bob Coecke’s Kindergarten Quantum Mechanics, a very pedagogical set of lecture notes where it is applied to the problem of quantum teleportation. Bob’s thesis is that teleportation was not discovered until the 90’s (despite it’s being a relatively straightforward result in QM) due to the inadequacy of the commonly used, low-level mathematical language used to describe Hilbert spaces. Had lines, squares, triangles and diamonds been used from the beginning, teleportation would have followed almost immediately. Or so thinks Bob: go take a look at his article and see what’s your take. In any case, its more than sixty full-color diagrams, used instead of boring algebraic formulae, make for a fun reading (or, should i say, viewing). By the way, don’t let the mention to category theory put you off: only very basic ideas (explained in the lecture notes) are needed, if at all, in this case, and actually the author’s enthusiasm goes as far as making the bold claim that this new graphical formalism could be taught in kindergarten! Maybe that’s the gist, since i, for one, find the notation hard to follow, undoubtedly due to my old-school, algebraic upbringing. Just to give you an idea of how this preschool notation looks like and close this long post as it deserves (i.e., with a diagram), here you have how the teleportation protocol (including a correctness proof) looks like:

Kindergarten QM


[1] My copy (Spanish translation) of the fifth edition of L&L’s book has 500 pages and just 22 figures!

[2] The link above points to Volume 11 of the collection at Oeuvres de Lagrange, a site that contains what seems to be the complete Lagrange corpus, conveniently scanned and downloadable too.

[3] I would give you a direct link, did it exist. Unfortunately, PI’s website is not up to the quality of their other activities. You’ll find it by browsing to their Public Lectures Series and from there to page 2 (or search for James Brown). Another very unfortunate circumstance is that the videos are only available for those of you not using weird as in freedom operating systems :-(.

[4] That’s at least my impression. Penrose, for instance, advocates for their reality in his road. The subject is however controversial enough to grant the existence of monographs like the recent Drawing theories apart, by David Kaiser (which i cannot comment on since i’ve just added it to my wish list).

[5] But i find this argument hard to swallow. Think for instance in the interpretation of antiparticles as particles travelling backwards in time: it also follows naturally (for some definition of natural) from perturbative series and/or their diagrams, but it is not as easily accepted as the existence of virtual photons. One wonders, where’s the limit?

[6] If you haven’t your favourite textbook at hand (Hawking and Ellis being mine when it comes to anything related to causal structure), you can find a pretty good introduction on-line in this chapter of Sean Carroll’s lecture notes.

[7] There is only so many ways of combining indexes, and if you know what are the free ones on, say, your LHS and the tensors entering the RHS and its general properties (e.g. symmetries), it’s often an easy task how their indexes should be combined. It reminds me, in a way, of dimensional reasoning, where knowing the target units and the ingredients gives an often quite accurate clue of how to combine them.

[8] It was introduced in a chapter of the book Combinatorial Mathematics and its Applications (Academic Press, London, 1971), entitled Application of Negative Dimensional Tensors. But Penrose have been using it (according to this letter to Cvitanovic (PDF) since 1952.

[9] An interesting (and not too far fetched) software project would be to write a Penrose diagram editor, possibly with support for tablet input devices. Such a tool would also probably solve the publication issue. In an ideal world, one would use a stylus to draw equations which would get automatically imported as nice diagrams, regular tensor equations with indexes or both. Any takers? ;-)