Archive for May, 2006

The cyclist team

May 29, 2006

 Fieldtheory Figs FeynmancovGeorgia Tehch’s Pedrag Cvitanovic’ and friends write physics and maths books under the nome de guerre of The cyclist team. These books are interesting in many ways. First of all, they are comprehensive and of excellent quality, although, fortunately, these are not extremely rare in the field. What is not so common is the dose of humour and engaging wittiness distilled in their pages. And, besides, they’re being written over a span of many years in a totally public manner: you can view and download them in PDF of Postscript and are frequently updated. As they explain in their webbook rationale, they don’t even plan to ever publish them:

The relevant parts of a good text will be printed and perused, no less than a good electronic preprint. A bad text should be junked anyway. If a student in Buenos Aires or Salamanca reads a chapter and is wiser for it, that is all it takes to make us happy. The webbook has done something to further little piece of wisdom that we know and love.

The oldest book is Pedrag’s (Quantum) Field Theory, and its companion (and more modern) lecture notes: Quantum Field Theory, a cyclist tour. In Pedrag’s own words:

Relax by reading Classics Illustrated, diagrammatic, Predragian vision of field theory. The exposition assumes no prior knowledge of anything (other than Taylor expansion of an exponential, taking derivatives, and inate knack for doodling). The techniques covered apply to QFT, Stat Mech and stochastic processes.

As is a norm, the book’s site contains many other bits of additional information, including the delicious fable of Quefithe.

Next comes the Group Theory Book, which, under the subtitle of Birdtracks, Lie’s and Exceptional Groups and spanning almost 300 pages, will tell you all you’ll ever need about Lie Groups and Algebras. This nice PDF presentation makes for a good summary of its contents, or, as Pedrag says, of “most of the Webbook at a cyclist pace, in 50 overheads” (see also here for more short intros). In case you’re wondering, birdtracks are to Lie Groups what Feynman’s diagrams to QFT, and then more. As you can see, Pedrag loves diagrams and pictures, in a way that reminds me of Penrose’s fondness for geometrical descriptions (actually, birdtracks have many a point in common with Penrose’s diagrammatic tensor notation, who even wrote a letter claiming his precedence on it). And, again, don’t miss the book’s site for lots of additional goodies.

Finally, there’s the Chaos Book, probably my favourite. Again, the authors introduce it far better than i would:

Quite a few excellent mathematics monographs on nonlinear dynamics and ergodic theories have been published in last three decades. On the whole, they are unreadable for non-mathematicians, and they give no hint that the theory is applicable to problems of physics, chemistry and other sciences.
By now, there are also many physics textbooks on “chaos”. Most lack depth, and many of them are plain bad, emphasizing pictorial and computer-graphics aspects of dynamics and short changing the student on the theory. That’s a pity, as the subject in its beauty and intellectual depth ranks alongside statistical mechanics and quantum field theory, with which it shares many fundamental techniques. The book represents authors’ attempt to formulate the subject as one of the basic cornerstones of the advanced graduate physics curriculum of future.

The amount of additional information for this book is almost overwhelming, including computer programs, additional exercises (the book itself contains many) and a long list of projects written by students. I won’t try to summarize the wide range of themes covered by the book (here you have the table of contents of its three volumes–classical chaos, quantum chaos and appendices), but a very good way of getting a glimpse of its scope and fun style is reading its Overture (PS). An amazing way to become acquainted with an amazing subject!

The cyclist team

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Starlight

May 28, 2006

Starlight

(from an inspiring little piece over at the Science Musings Blog)

The third policeman

May 28, 2006

I’ve just finished The Third Policeman, my discovery of Flann O’Brien‘s hilarious and extremely witty work. I’ve enjoyed so much this novel that i had to find an alibi for posting it here. But that was easy. Let me introduce to you the prolific and sadly forgotten Irish physicist and philosopher de Selby, whose highly original theories constitute a reference frame of sorts in The Third Policeman’s plot. We learn in there, for instance, how de Selby foretold modern ideas about the problem of time:

Human existence de Selby has defined as ‘a succession of static experiences each infinitely brief’ […] From this premise he discounts the reality or truth of any progression or serialism in life, denies that time can pass as such in the accepted sense and attributes to hallucinations the commonly experienced sensation of progression as, for instance, in journeying from one place to another or even ‘living’.

Granted, other ideas of his were much more debatable, as his theory about night being caused by black air accumulation, but to err is the mark of genious. Other characters are also prone to philosophical digressions. For instance, one would say that policeman MacCruiskeen is well acquainted with some of our modern theories of quantum gravity:

That is the real point, said MacCruiskeen, but it is so thin that it could go into your hand and out in the other extremity externally and you would not feel a bit of it and you would not see nothing and hear nothing. It is so thin that maybe it does not exist at all and you could spend half an hour trying to think about it and you could put no thought around it in the end. The beginning part of the inch is thicker than the last part and is nearly there for a fact but i don’t think it is if it is my private opinion that you are anxious to enlist.

And there’s more, including a theory of everything based on a single, possibly relational, entity: the omnium. But i won’t spoil the fun by giving up the plot, which, to tell the truth, has nothing to do with physics, but rather with the Carollian travels of an unnamed murderer through a surrealist, almost quantum world. (In case you’re not yet convinced, here you have yet another excerpt from the novel; or see here for more about O’Brien.)

Update: MacCruiskeen’s needle seems to have been finally found!

Leibniz space-times

May 27, 2006

More often than not, Lee Smolin’s essays are engaging and thought provoking. I specially appreciate his willingness to tackle conceptual issues, often dismissed as philosophical or uninteresting by a great deal of the physics community (which, in my opinion, should know better). Also of note are his efforts to convey to non-specialists the key ideas and problems faced by modern physics, without unduly over-simplifications or dishonest hype.

A case in point is his recent essay The Case for Background Independence, where the meaning, virtues and drawbacks of relationist theories of quantum gravity are explored in detail. More concretely, Smolin describes the close relationship between three key issues in fundamental physics, to wit:

  • Must a quantum theory of gravity be background independent, or can there can be a sensible and successful background dependent approach?
  • How are the parameters of the standard models of physics and cosmology to be determined?
  • Can a cosmological theory be formulated in the same language we use for descriptions of subsystems of the universe, or does the extension of physics from local to cosmological require new principles or a new formulation of quantum theory?

The article begins with a brief historical review of relationism, as understood by Leibniz and summarized in his principles of sufficient reason (there’s always a rational cause for Nature’s choices) and the identity of the indiscernible (entities with exactly the same properties are to be considered the same) [1]. These principles rule out absolute space-times (like Newton’s) or a fixed Minkowskian background (like perturbative string theory), since they single out a preferred structure ‘without reason’, as do theories posing any number of free parameters (think of the much debated landscape) [2]. As is well known, Newton won the day back in the seventeenth century, until Mach’s sharp criticism marked the resurgence of relationist ideas. Mach rejected Newtonian absolute space-time, favouring a purely relational definition of inertia [3], which ultimately would inspire Einstein in his quest for the general theory of relativity [4].

Smolin’s article continues with a careful definition, in modern terms, of relational space and time, and follows with a discussion of some current theories featuring background independence: general relativity, causal sets, loop quantum gravity, causal dynamical triangulation models and background independent approaches (by Smolin himself) to M-theory. In a nutshell, it is argued that any self-respecting relational theory should comply to three principles:

  • There is no background.
  • The fundamental properties of the elementary entities consist entirely in relationships between those elementary entities.
  • The relationships are not fixed, but evolve according to law. Time is nothing but changes in the relationships, and consists of nothing but their ordering.

None of the theories above passes without problems this litmus test of pure relationsm. Take for instance general relativity. To begin with the dimension, topology and differential structure of space-time are givens, and thus play the role of a background. And, on the other hand, only when we apply GR to a compact universe without boundary can we aspire to a relational view, since otherwise we would have arbitrary boundary conditions (partially) determining the structure of space-time. Once you abide to these preconditions, a proper interpretation of general covariance (in which you identify space-times related by arbitrary coordinate transformations) provides a relational description of space-time (for an in-depth discussion of the subtle interplay between gauge invariance and relationsm, see also this excellent article by Lusanna and Pari, and references therein). As a second example, loop quantum gravity is also background dependent: in this case, the topological space containing the spin-networks of the theory. Other than that, loops are an almost paradigmatic case of a relational description in terms of graphs, with nodes being the entities and edges representing their relationships.

After his review of quantum gravity theories, Smolin takes issue with string theory. His subsequent train of thought heavily relies on the fact that relationism, or, more concretely, Leibniz’s principle of the indiscernible, rules out space-times with global symmetries. For if we cannot distinguish this universe from one moved 10 feet to the left, we must identify the two situations, i.e., deny any meaning or reality to the underlying, symmetric structure. But, as is happens, the M-theory programme consists, broadly speaking, in maximizing the symmetry groups of the theories embodied in the desired unified description. More concretely, in background-dependent theories, the properties of elemental entities are described in terms of representations of symmetries of the background’s vacuum state. Each of the five string theories embodied by M-string (should it exist!) has its own vacuum, related with each other via duality transformations (basically, compactifying spatial dimensions one way or the other one is able to jump from one string theory to the next). Thus, M-theory should be background independent (i.e., encompass different backgrounds), but, on the other hand, one expects that the unique unified theory will have the largest possible symmetry group consistent with the basic principles of physics, such as quantum theory and relativity. Smolin discusses some possible solutions this contradiction (which a lack, er, background to comment intelligently), including some sort of (as yet unknown) dynamical mechanism for spontaneous symmetry breaking (which would result in a Leibniz-compliant explanation for the actual properties–such as masses and coupling constants–that we find in our universe).

After all the fuss, there is disappointingly little to be said about relationist unified theories [5]. Invoking again the principle of the indiscernible, Smolin rules out symmetries that would make (unified) identities undistinguishable (if two entities have the same relationships with the rest, they are the same entity). By the same token, a universe in thermal equilibrium is out of the question. Reassuringly, our universe is not, and the negative specific heat of gravitationally bound systems precludes its evolution to such an state. The case is then made (after casting evolution theory as a relationist one, which is OK by me) for Smolin’s peculiar idea of cosmological natural selection. To my view, it is an overly speculative idea, if only for the fact that it depends on black holes giving rise to new universes when they collapse [6]. If that were the case, and provided that each new universe is created with random values for the free parameters of our theories, one would expect that a process similar to natural selection would lead to universes with its parameters tuned to favour a higher and higher number of black-holes (which seems to be the case in our universe). Nice as the idea is, i think we’re a little far from real physics here.

The article closes with a short section on the cosmological constant problem (with the interesting observation than only casual set theory has predicted so far a realistic value) and relational approaches to (cosmological) quantum theory. Again, the author adheres to non-orthodox ideas. This time, to recent proposals (see here and here) of hidden-variable theories, albeit they have far better grounds than the reproducing universes idea. The possibility of a relational hidden-variable theory is argued for with a simple and somewhat compelling line of thought. In classical physics, the phase space of a system of N particles is described by a 6N variables, while a quantum mechanical state vector would depend on 3N variables. On the other hand, in a purely relational theory one would need to use N^2 variables, as these are the number of possible relations between N particles. These would be the hidden-variables completely (and non-locally) describing our particles, which would need statistical laws when using just 3N parameters.

An amazing journey, by all accounts.


[1] See here for excellent (and free) editions of all relevant Leibniz works, including his Monadology, and here for commented excerpts of the Leibniz-Clarke correspondence.

[2] See also here for an interesting take on Leibniz’s principle under the light of Gödel’s and Turing’s incompleteness theorems as further developed by Gregory Chaitin.

[3] Julian Barbour’s “The Discovery of Dynamics: A Study from a Machian Point of View of the Discovery and the Structure of Dynamical Theories” is the definitive reference to know more about the history of the absolute/relative divide. (Another amazing book by Barbour on these issues is “The End of Time : The Next Revolution in Physics”, thoroughly reviewed by Soshichi Uchii here. Smolin himself has many an interesting thing to say about Barbour’s timeless Platonia.)

[4] Barbour argues in his book that Einstein seems to have misunderstood Mach’s discussions on the concept of inertia, taking it for the dynamical quantity entering Newton’s second law instead of the inertial motion caused by space-time according to Newton’s first law.

[5] I’m also a bit surprised by Smolin’s uncritical acceptance of reductionism, which he simply considers, “to a certain degree”, as common-sense.

[6] Tellingly, the only reference where this theory is developed is Smolin’s popular science book “The Life of the Cosmos”.

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A magic cloak

May 26, 2006

This PhysicsWeb article describes a very fun proposal for invisibility cloaks. Come to think of it, the principle is simple: cover your object with a material than bends light coming from its rear around the body and forwards it afterwards reversing the deviation.
Snell-1Not surprisingly, the idea was “inspired by the geometry of curved space — a discipline that is normally in the firm hands of researchers in general relativity.” The tricky part is, of course, finding a material with the right behaviour. The feat seems to be possible thanks to the so-called metamaterials, which may have negative-index refraction. As shown in the figure on the right (taken from the link above), according to Snell’s law, a material with a negative index of refraction is able to reflect light through negative angles with respect to the surface’s normal. As you can see, with an apt enough arrangement of such prisms, you can obtain the needed light twist around the cloaked object. Curious? More on negative index refraction in this list of publications.

Heaven and Earth

May 23, 2006

The NASA/ESA Hubble Space Telescope has captured the first-ever picture of a distant quasar lensed into five images. In addition the picture holds a treasure of lensed galaxies and even a supernova.

Heic0606A410

More details on this beauty over at ESA’s site. Almost as beautiful, in its own way, is the soon-to-be-operative Large Hadron Collider at CERN. Seed Magazine has a nice page giving some numbers to put LHC’s dimensions in perspective.

LHC

Connes and quantum statistics

May 22, 2006

Alain ConnesI just noticed (hat tip Not even wrong) that Alain Connes has made available his book on non-commutative geometry, one of the third roads to quantum gravity. Not an easy reading, by any account, but surely an interesting one. Not only that. He’s also making available (in his downloads page) most of his recent articles and lecture notes, which make for an impressive list. A very interesting and enjoyable reading i’ve found there is Connes’ highly original View of Mathematics (PDF), which makes for a good introduction to NCG. And there is also a brief essay, Advice to the beginner, where, besides guidelines to young mathematicians, Connes gives his particular view of the physics community:

I was asked to write some advice for young mathematicians. The first observation is that each mathematician is a special case, and in general mathematicians tend to behave like “fermions” i.e. avoid working in areas which are too trendy whereas physicists behave a lot more like “bosons” which coalesce in large packs and are often “overselling” their doings, an attitude which mathematicians despise.

The bit about overselling rings a bell, doesn’t it?

Update: Also of note is this interview with Alain Connes (PDF), mentioned, again, over at Not Even Wrong:

The interview also contains quite a few amusing stories. In one of them Connes tells about a well-known string theorist who walked out of his talk at Chicago because he wasn’t very interested, but two years later was paying rapt attention to the same talk when Connes gave it at Oxford. When Connes asked him about this, the physicist told him that the difference was that in the meantime he had heard that Witten had been seen reading Connes’s book in the library at Princeton.

Quantum probability

May 22, 2006

I just stumbled upon John Baez’s page on Bayesian probability and Quantum Mechanics, which nicely summarizes one of the first difficulties i had with the latter: Born’s interpretation of the wave function as a probability. The problem hinged on my naive (frequentist) interpretation of probabilities, and the conclusion that QM describes only ensembles, not individual systems. For to compute the probability of an experiment’s outcome, i reasoned, you need to repeat the experiment a large number of times. Then, counting the number of times your outcome happens and dividing by the total number of repetitions one obtains the sought for probability. Problem is, what is large? Well, nothing short of infinite, it seems. Because, with this frequentist definition of probability, nothing prevents your tossing a coin a hundred times and getting a hundred tails. And such a situation may still be compatible with a half and half probability for heads and tails! My unsettling conclusion was that QM predicts nothing at all about individual systems! Come to think of it, it doesn’t even predict anything about finite ensembles.

One way out of this conundrum is Everett’s many worlds interpretation: since all possible outcomes really happen, frequentist probabilities are well-defined. I still remember being genuinely surprised when i learnt that there existed serious attempts at making sense of such an idea. I still am. John gives an excellent argument to be done with this peculiar interpretation:

Here is a sample conversation between two Everettistas, who have fallen from a plane and are hurtling towards the ground without parachutes:

Mike: What do you think our chances of survival are?

Ron: Don’t worry, they’re really good. In the vast majority of
possible worlds, we didn’t even take this plane trip.

A second way out is revising our definition of probability. We forget (initially) about frequencies, and take a Bayesian stance. In a nutshell, Bayesian probability is not measured from scratch because it is defined as a degree of belief on a given outcome. One starts with an a priori value for such a belief, and revises it (if needed) according to experiment. The gist of it is that Bayes’ theorem lets you calculate the likelihood of future outcomes based solely on your a priori probabilities. So, the tale goes, when a wave function collapses as a result of a measurement, there’s nothing real out there undergoing a physical collapse; it’s only that we have improved our knowledge of the system and must update our a priori likelihood assignments accordingly. This view mixes well, by the way, with the orthodox Copenhagen interpretation of QM, which also denies an objective reality of the wave function.

The so called relational interpretations have, i think, a clear Bayesian substrate. Probably the best known relational theory nowadays is Rovelli‘s, whose recent paper Relation EPR (nicely reviewed in Alejandro’s blog) has been widely discussed elsewhere.

While i have nothing against Bayesian probability for describing our knowledge of any system, considering it as a final interpretation of QM makes me feel uneasy. I’d rather have a theory which describes something out there, some kind of (possibly inter-subjective) reality. Atoms, stars and the whole universe seem to care little about our knowledge of them, and the quantum mechanics rules look a bit too simple to explain, out of the blue, our way of acquiring information about the world. I would rather put my money on some sort of objective, physical reduction of the state vector, maybe along the lines of some non-linear modification of Schrödinger’s equation (and probably not as fancy as Penrose’s objective reduction, but who knows!). Call me a (perhaps non-local) realist.


One last thing. One of the best ways to learn about Bayesian theory is from “Probability Theory : The Logic of Science” (E. T. Jaynes). The good news is that a draft version of it is available online. (See also Matthew Leifer’s comment below recommending Bruno de Finetti‘s work.)

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How to write papers

May 20, 2006

I’ve just found a brief but very well written Guide to Writing Papers by S. Majid (of quantum groups fame). He makes many a good point, specially for those of us whose mother tongue is not English. And you’ll find also useful advice on contents and organization.

When it comes to writing good English prose, Strunk and White’s classic “The Elements of Style” (also available and searchable online) is, of course, required reading, but i find Dupré’s “BUGS in Writing: A Guide to Debugging Your Prose” much more fun and equally insightful. Also worth a look is the online Researcher’s Bible (hat tip Jocelyn Paine), which is similar in scope to Majid’s tutorial. And for a good, thought-provoking laugh, don’t miss the indispensable How to Write a Scientific Paper, over at Improbable Research.

Let me take the opportunity to recommend two of my all-time favourite essays on scientific writing: the classic How I Write by Bertrand Russell, and the recent A parallel tradition, where Ian MacEwan beautifully makes the case for a scientific literary tradition. So much for the science/humanities divide!

Finally, lest you had not enough, this blog entry by W. Zinsser is, in my opinion, an excellent guide to improving your writing skills (scientific or otherwise), and contains links to several other interesting books on the subject.

Hooke manuscript is returned home

May 18, 2006

BBC News is reporting about a lost and found Hooke manuscript (see also this older news):

The hand-written notes are thought to contain a “treasure trove” of information about the early endeavours of the UK’s academy of science. The document, which had lain hidden in a house in Hampshire, was rescued from a public auction after a fundraising effort pulled in the £940,000 needed.

As you probably know, Hooke was famous (among other things) for his frequent disputes with other physicists (most notably Newton and Huygens) on the priority of various discoveries, and this new manuscript could help in settling (a bit late) some of them.