## Archive for the ‘Quantum Gravity’ Category

### Inflated claims

December 2, 2009

The Institute of Physics’ Isaac Newton medal was awarded this year to Alan Guth, of inflation fame, and his talk is available at the IoP’s website. The talk is devoted to explaining how great and awesome the inflationary theory is, including the usual reasons it’s believed by many to be correct: it explains the large-scale homogeneity of the universe (by causally connecting in the past regions that are unconnected after the inflationary period) and supposedly predicts a smoothed out geometry with an average mass density close to its critical point (that is, a universe with flat spatial sections). There’s also some talk about eternal inflation and pocket universes, which i didn’t understand well enough to comment on: this article by Guth himself is a good way to learn more.

Dr Guth is extremely happy about the fact that estimates for the ratio of the observed over the critical density ($\Omega$) have jumped during the last decade from the 0.2-0.3 range to a value almost exactly equal to 1, which corresponds to the flat model. The correction comes from taking bringing dark energy into the picture, and the second half of the talk is devoted to some speculations as to its mysterious origin. To Guth, the most plausible explanation is that this dark energy corresponds to the vacuum energy density ($\Lambda$), but there’s this little problem that quantum field theory predicts an infinite value for it. Even if you try to introduce an (arbitrary) ultraviolet cut in the calculation at the usual Plank energy, the value obtained is some 120 orders of magnitude greater than the observed one. String theory and the landscape to the rescue! Which of course explains nothing, in my opinion.

Guth’s last resort is the anthropic principle, according to which $\Lambda$ is so low because it’s the only way intelligent (?) beings would be here to observe it, and then uses an analogy so broken that i’m sure i’m missing something obvious. It goes as follows: Kepler initially thought that the radius of the orbits of planets in the solar system should have values deducible from geometrical considerations alone, but, as we all know, that’s not the case: their concrete values could be different just by changing the initial conditions leading to the formation of the solar system. So it is with $\Lambda$: since we have no way of computing its unexpectedly low value, it must be that the reason for it is that we exist. This argument is so plainly wrong that, as i said, i’m sure i’m misunderstanding (perhaps one of my two or three readers will set the record straight in the comments).

Other than that, and the fact that it feels at times like a commercial (including some arguments pretty close to straw men when he shows the curves matching the observations of the microwave background (without any mention, by the way, to the possible discrepancies for low terms of the multipolar expansion)), the talk is entertaining and gives a good, if quick, overview of commonly accepted wisdom on the field these days. So, if you have a grain of salt handy and don’t mind a bit of hand-waving, just ignore my rants and go for it.

(While you’re at it, i’ll be giving a try to last year’s winner, Anton Zeilinger, and his talk on quantum information and the foundation of quantum mechanics).

### Taking issue with String Theory

October 7, 2006

With the recent publication of Lee Smolin’s and Peter Woit’s books on the troubles of our theories of everything, every blogger in town seems to be talking about the crisis of modern fundamental physics (a.k.a. string theory and, so to speak, friends). Christine has just posted a list of recent posts on the subject. Like her, i’m reading Smolin’s book, courtesy of the publisher, and a review will eventually follow (once i find something to say about it that has not already been said!). In the meantime, i just wanted to add a few links to articles that i like on this pesky matter:

• Jim’s Stab at String Theory is a very interesting discussion by Jim Weatherall on why it doesn’t really matters whether string theory is right. There you’ll find also a video interview with Peter Woit, by John Horgan (who is not specially happy with ST, either).
• Among the free contents of the latest Physics Today issue, Burton Richter takes no prisoners when it comes to describe what’s wrong with all this super stuff.

I find Richter’s stabs, er, criticisms particularly compelling: his writing is clear and to the point, and his arguments are all but crisp and pungent. It’s curious that, by contrast, Smolin’s delicacy has actually augmented my curiosity on string theory (but i’m just halfway reading his book, so let’s better wait until i’m done).

On the other hand, i’m starting to be more and more in agreement with Weatherall’s arguments on the irrelevance of this whole business. At the very least, i’m trying to keep in mind that there’s arguably much more to fundamental physics than this debate. Maybe it’s time for some fresh air.

### Nobody expects the Strings Inquisition

June 13, 2006

The recent comments by you-know-who against the positive press that Peter Woit’s book Not Even Wrong is receiving just reminded me (by some weird association of ideas) of this epoch-making gag by Monty Python. I was about to write an entry on my recently ordering this book and planning to read it on a trip next week, but now i’m scared of confessing having bought it! (and, anyway, Christine Dantas has recently put quite nicely almost my exact feelings on this matter). Imagine, i might even like it, and got immediately classified as a crackpot and a nincompoop by the theoretical physics community!

I’m trying hard to respect string theory and theorists (and even plan on reading Zee, Weinberg and Zwiebach’s books, already waiting on my shelf). Why, i even admire several string theorists. But it would help if someone told me that Motl is not the string community’s spokesman. Or to read every now and then a bit of self-criticism from said community (something in the vein of Smolin’s comments in, say, Three Roads to Quantum Gravity). For if the (so to speak) dialectic battles between those two are to be taken as the kind of discussion we theoretical physicists favour these days, poor Monty Python are just out of business. Paraphrasing Erwin Schrodinger, i wouldn’t like it, and i would be sorry i ever had anything to do with it.

Update: Christine’s again right on the spot. And this is much, much closer to the kind of discussion i was asking for!

### 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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### Connes and quantum statistics

May 22, 2006

I 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.

### 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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### 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?

### 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…

April 15, 2006