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This Week's Finds in Mathematical Physics (Week 206)
Also available at http://math.ucr.edu/home/baez/week206.html May 10, 2004 This Week's Finds in Mathematical Physics  Week 206 John Baez I just got back from Marseille, where Carlo Rovelli, Laurent Freidel and Phillipe Roche held the first really big conference on loop quantum gravity and spin foams since the 2nd Warsaw workshop run by Jerzy Lewandowski back in 1997: 1) Non Perturbative Quantum Gravity: Loops and Spin Foams, 37 May 2004, CIRM, Luminy, Marseille, France, http://w3.lpm.univmontp2.fr/~philip...ravitywebsite/ It was good to see old friends and talk about quantum gravity near the "Calanques"  the rugged limestone cliffs lining the Mediterranean coastline. It was good to meet lots of young people who have recently entered this difficult field: about 100 people attended, considerably more than at any previous meeting. But most of all, it was good to see some progress on the tough problem of understanding dynamics in nonperturbative quantum gravity. Can we get the 4dimensional spacetime we know and love, whose geometry is described by general relativity, to emerge from some theory that takes quantum physics into account? And can we do it *nonperturbatively*? In other words, can we do quantum physics without choosing some fixed spacetime geometry from the start, a "background" on which small perturbations move like tiny quantum ripples on a calm preestablished lake? A background geometry is convenient: it lets us keep track of times and distances. It's like having a fixed stage on which the actors  gravitons, strings, branes, or whatever  cavort and dance. But, the main lesson of general relativity is that spacetime is *not* a fixed stage: it's a lively, dynamical entity! There's no good way to separate the ripples from the lake. This distinction is no more than a convenient approximation  and a dangerous one at that. So, we should learn to make do without a background when studying quantum gravity. But it's tough! There are knotty conceptual issues like the "problem of time": how do we describe time evolution without using a fixed background to measure the passage of time? There are also practical problems: in most attempts to describe spacetime from the ground up in a quantum way, all hell breaks loose! We can easily get spacetimes that crumple up into a tiny blob... or spacetimes that form endlessly branching fractal "polymers" of Hausdorff dimension 2... but it seems hard to get reasonably smooth spacetimes of dimension 4. It's even hard to get spacetimes of dimension 10 or 11... or *anything* remotely interesting! It almost seems as if we need a solid background as a bed frame to keep the mattress of spacetime from rolling up or otherwise misbehaving. Unfortunately, even *with* a background there are serious problems: we can use perturbation theory to write the answers to physics questions as power series, but these series diverge and nobody knows how to resum them. String theorists are pragmatic in a certain sense: they don't mind using a background, and they don't mind doing what physicists always do: approximating a divergent series by the sum of the first couple of terms. But this attitude doesn't solve everything, because right now in string theory there is an enormous "landscape" of different backgrounds, with no firm principle for choosing one. Some estimates guess there are over 10^{100}. Leonard Susskind guesses there are 10^{500}, and argues that we'll need the anthropic principle to choose the one describing our world: 2) Leonard Susskind, The Landscape, article and interview on John Brockman's "EDGE" website, http://www.edge.org/3rd_culture/suss...ind_index.html This position is highly controversial, but my point here shouldn't be: developing a backgroundfree theory of quantum gravity is tough, but working *with* a background has its own difficulties. And let's face it: we haven't spent nearly as much time thinking about backgroundfree or nonperturbative physics as we've spent on backgrounddependent or perturbative physics. So, it's quite possible that our failures with the former are just a matter of inexperience. Given all this, I'm delighted to see some real progress on getting 4d spacetime to emerge from nonperturbative quantum gravity: 3) Jan Ambjorn, Jerzy Jurkiewicz and Renate Loll, Emergence of a 4d world from causal quantum gravity, available as hepth/0404156. This trio of researchers have revitalized an approach called "dynamical triangulations" where we calculate path integrals in quantum gravity by summing over different ways of building spacetime out of little 4simplices. They showed that if we restrict this sum to spacetimes with a wellbehaved concept of causality, we get good results. This is a bit startling, because after decades of work, most researchers had despaired of getting general relativity to emerge at large distances starting from the dynamical triangulations approach. But, these people hadn't noticed a certain flaw in the approach... a flaw which Loll and collaborators noticed and fixed! If you don't know what a path integral is, don't worry: it's pretty simple. Basically, in quantum physics we can calculate the expected value of any physical quantity by doing an average over all possible histories of the system in question, with each history weighted by a complex number called its "amplitude". For a particle, a history is just a path in space; to average over all histories is to integrate over all paths  hence the term "path integral". But in quantum gravity, a history is nothing other than a SPACETIME. Mathematically, a "spacetime" is something like a 4dimensional manifold equipped with a Lorentzian metric. But it's hard to integrate over all of these  there are just too darn many. So, sometimes people instead treat spacetime as made of little discrete building blocks, turning the path integral into a sum. You can either take this seriously or treat it as a kind of approximation. Luckily, the calculations work the same either way! If you're looking to build spacetime out of some sort of discrete building block, a handy candidate is the "4simplex": the 4dimensional analogue of a tetrahedron. This shape is rigid once you fix the lengths of its 10 edges, which correspond to the 10 components of the metric tensor in general relativity. There are lots of approaches to the path integrals in quantum gravity that start by chopping spacetime into 4simplices. The weird special thing about dynamical triangulations is that here we usually assume every 4simplex in spacetime has the same shape. The different spacetimes arise solely from different ways of sticking the 4simplices together. Why such a drastic simplifying assumption? To make calculations quick and easy! The goal is get models where you can simulate quantum geometry on your laptop  or at least a supercomputer. The hope is that simplifying assumptions about physics at the Planck scale will wash out and not make much difference on large length scales. Computations using the socalled "renormalization group flow" suggest that this hope is true *IF* the path integral is dominated by spacetimes that look, when viewed from afar, almost like 4d manifolds with smooth metrics. Given this, it seems we're bound to get general relativity at large distance scales  perhaps with a nonzero cosmological constant, and perhaps including various forms of matter. Unfortunately, in all previous dynamical triangulation models, the path integral was *NOT* dominated by spacetimes that look like nice 4d manifolds from afar! Depending on the details, one either got a "crumpled phase" dominated by spacetimes where almost all the 4simplices touch each other, or a "branched polymer phase" dominated by spacetimes where the 4simplices form treelike structures. There's a transition between these two phases, but unfortunately it seems to be a 1storder phase transition  not the sort we can get anything useful out of. For a nice review of these calculations, see: 4) Renate Loll, Discrete approaches to quantum gravity in four dimensions, available as grqc/9805049 or as a website at Living Reviews in Relativity, http://www.livingreviews.org/Article...1/199813loll/ Luckily, all these calculations shared a common flaw! Computer calculations of path integrals become a lot easier if instead of assigning a complex "amplitude" to each history, we assign it a positive real number: a "relative probability". The basic reason is that unlike positive real numbers, complex numbers can cancel out when you sum them! When we have relative probabilities, it's the *highly probable* histories that contribute most to the expected value of any physical quantity. We can use something called the "Metropolis algorithm" to spot these highly probable histories and spend most of our time worrying about them. This doesn't work when we have complex amplitudes, since even a history with a big amplitude can be canceled out by a nearby history with the opposite big amplitude! Indeed, this happens all the time. So, instead of histories with big amplitudes, it's the *bunches of histories that happen not to completely cancel out* that really matter. Nobody knows an efficient generalpurpose algorithm to deal with this! For this reason, physicists often use a trick called "Wick rotation" that converts amplitudes to relative probabilities. To do this trick, we just replace time by imaginary time! In other words, wherever we see the variable "t" for time in any formula, we replace it by "it". Magically, this often does the job: our amplitudes turn into relative probabilities! We then go ahead and calculate stuff. Then we take this stuff and go back and replace "it" everywhere by "t" to get our final answers. While the deep inner meaning of this trick is mysterious, it can be justified in a wide variety of contexts using the "OsterwalderSchrader theorem". Here's a pretty general version of this theorem, suitable for quantum gravity: 5) Abhay Ashtekar, Donald Marolf, Jose Mourao and Thomas Thiemann, OsterwalderSchrader reconstruction and diffeomorphism invariance, preprint available as quantph/9904094. People use Wick rotation in all work on dynamical triangulations. Unfortunately, this is *not* a context where you can justify this trick by appealing to the OsterwalderSchrader theorem. The problem is that there's no good notion of a time coordinate "t" on your typical spacetime built by sticking together a bunch of 4simplices! The new work by Ambjorn, Jurkiewiecz and Loll deals with this by restricting to spacetimes that *do* have a time coordinate. More precisely, they fix a 3dimensional manifold and consider all possible triangulations of this manifold by regular tetrahedra. These are the allowed "slices" of spacetime  they represent different possible geometries of space at a given time. They then consider spacetimes having slices of this form joined together by 4simplices in a few simple ways. The slicing gives a preferred time parameter "t". On the one hand this goes against our desire in general relativity to avoid a preferred time coordinate  but on the other hand, it allows Wick rotation. So, they can use the Metropolis algorithm to compute things to their hearts' content and then replace "it" by "t" at the end. When they do this, they get convincing good evidence that the spacetimes which dominate the path integral look approximately like nice smooth 4dimensional manifolds at large distances! Take a look at their graphs and pictures  a picture is worth a thousand words. Naturally, what *I'd* like to do is use their work to develop some spin foam models with better physical behavior than the ones we have so far. Now that Loll and her collaborators have gotten something that works, we can try to fiddle around and make it more elegant while making sure it still works. In particular, I'm hoping we can get wellbehaved models that don't introduce a preferred time coordinate as long as they rule out "topology change"  that is, slicings where the topology of space changes. After all, the OsterwalderSchrader theorem doesn't require a *preferred* time coordinate, just *any* time coordinate together with good behavior under change of time coordinate. For this we mainly need to rule out topology change. Moreover, Loll and her collaborators have argued in 2d toy models that topology change is one thing that makes models go bad: the path integral can get dominated by spacetimes where "baby universes" keep branching off the main one: 6) Jan Ambjorn, Jerzy Jurkiewicz and Renate Loll, Nonperturbative Lorentzian quantum gravity, causality and topology change, Nucl. Phys. B536 (1998) 407434. Also available as hepth/9805108. Renate Loll and W. Westra, Spacetime foam in 2d and the sum over topologies, Acta Phys. Polon. B34 (2003) 49975008. Also available as hepth/0309012. By the way, it's also reading about their 3d model: 7) Jan Ambjorn, Jerzy Jurkiewicz and Renate Loll, Nonperturbative 3d Lorentzian quantum gravity, Phys.Rev. D64 (2001) 044011. Also available as hepth/0011276. and for a general review, try this: 8) Renate Loll, A discrete history of the Lorentzian path integral, Lecture Notes in Physics 631, Springer, Berlin, 2003, pp. 137171. Also available as hepth/0212340. All this is great, but don't get me wrong  there were a lot of *other* cool talks at the conference besides Loll's. I'll just mention a few. Laurent Freidel spoke on his work on spin foam models. Especially exciting is how David Louapre and he have managed to "sum over topologies" in 3d Riemannian quantum gravity with vanishing cosmological constant  otherwise known as the PonzanoRegge model He has to subtract out a counterterm that would otherwise lead to a bubble divergence, but then he gets a beautiful theory where the sum over spin foams is Borel summable: 9) Laurent Freidel and David Louapre, Nonperturbative summation over 3D discrete topologies, Phys.Rev. D68 (2003) 104004. Also available as hepth/0211026. Their work on gaugefixing and the inclusion of spinning point particles in the PonzanoRegge model is also very impressive, especially given how long this model has been studied. It shows we have lots left to learn! 10) Laurent Freidel and David Louapre, PonzanoRegge model revisited I: Gauge fixing, observables and interacting spinning particles, available as hepth/0401076. The title suggests we're in for more treats to come. Kirill Krasnov gave a talk entitled simple "ln(3)"  it was all about the appearance of this constant in the work of Hod, Dreyer, Motl and Neitzke on black hole entropy and the ringing of black holes. I've discussed all this at length in "week198", but Krasnov has given an elegant new proof of Hod's conjecture using Riemann surface theory. One can even think of this as a "stringy" explanation of the quasinormal modes of black holes  but much remains mysterious he 11) Kirill Krasnov, Black hole thermodynamics and Riemann surfaces, Class. Quant. Grav. 20 (2003) 22352250. Also available as grqc/0302073. Kirill Krasnov and Sergey N. Solodukhin, Effective stringy description of Schwarzschild black holes, available as hepth/0403046. While I'm at it, I can't resist mentioning Krasnov's work on including point particles in 3d Lorentzian quantum gravity with negative cosmological constant, since it has close connections with that of Freidel and Louapre, though the context is a bit different: 12) Kirill Krasnov, Lambda0 quantum gravity in 2+1 dimensions I: quantum states and stringy Smatrix, Class. Quant. Grav. 19 (2002) 39773998. Also available as hepth/0112164. Kirill Krasnov, Lambda0 quantum gravity in 2+1 dimensions II: black hole creation by point particles, Class. Quant. Grav. 19 (2002) 39994028. Also available as hepth/0202117. If I could duplicate myself, I'd have one copy write a book on 3d quantum gravity that would synthesize all these wonderful results in a nice big picture. It's not realistic physics; it's just a toy model. But the math is *so* nice, and so enlightening for realworld physics in some ways, that it's hard to resist pondering it! TQFTs, Riemann surfaces, hyperbolic geometry, spinning point particles colliding and creating black holes  a wonderful stew! Alas, I don't have time to savor it. There were a lot of other interesting talks  but I don't have time to go through and describe all of them, either. So, I'll wrap up with something very different! Lee Smolin told me some neat stuff about MOND  that's "Modified Newtonian Dynamics", which is Mordehai Milgrom's way of trying to explain the strange behavior of galaxies without invoking dark matter. The basic problem with galaxies is that the outer parts rotate faster than they should given how much mass we actually see. If you have a planet in a circular orbit about the Sun, Newton's laws say its acceleration is proportional to 1/r^2, where r is its distance to the Sun. Similarly, if almost all the mass in a galaxy were concentrated right at the center, a star orbiting in a circle at distance r from the center would have acceleration proportional to 1/r^2. Of course, not all the mass is right at the center! So, the acceleration should drop off more slowly than 1/r^2 as you go further out. And it does. But, the observed acceleration drops off a lot more slowly than the acceleration people calculate from the mass they see. It's not a small effect: it's a HUGE effect! One solution is to say there's a lot of mass we don't see: "dark matter" of some sort. If you take this route, which most astronomers do, you're forced to say that *most* of the mass of galaxies is in the form of dark matter. Milgrom's solution is to say that Newton's laws are messed up. Of course this is a drastic, dangerous step: the last guy who tried this was named Einstein, and we all know what happened to him. Milgrom's theory isn't even based on deep reasoning and beautiful math like Einstein's! Instead, it's just a blatant attempt to fit the experimental data. And it's not even elegant. In fact, it's downright ugly. Here's what it says: the usual Newtonian formula for the acceleration due to gravity is correct as long as the acceleration is bigger than a = 2 x 10^{10} m/sec^2 But, for accelerations less than this, you take the geometric mean of the acceleration Newton would predict and this constant a. In other words, there's a certain value of acceleration such that above this value, the Newtonian law of gravity works as usual, while below this value the law suddenly changes. Any physicist worth his salt who hears this modification of Newton's law should be overcome with a feeling of revulsion! There just *aren't* laws of physics that split a situation in two cases and say "if this is bigger than that, then do X, but if it's smaller, then do Y." Not in fundamental physics, anyway! Sure, water is solid below 0 centigrade and fluid above this, but that's not a fundamental law  it presumably follows from other stuff. Not that anyone has derived the melting point of ice from first principles, mind you. But we think we could if we were better at big messy calculations. Furthermore, you can't easily invent a Lagrangian for gravity that makes it fall off more *slowly* than 1/r^2. It's easy to get it to fall off *faster*  just give the graviton a mass, for example! But not more slowly. It turns out you can do it  Bekenstein and Milgrom have a way  but it's incredibly ugly. So, MOND should instantly make any decent physicist cringe. Esthetics alone would be enough to rule it out, except for one slight problem: it seems to fit the data! In some cases it matches the observed rotation of galaxies in an appallingly accurate way, fitting every wiggle in the graph of stellar rotation velocity as a function of distance from the center. So, even if MOND is wrong, there may need to be some reason why it *acts* like it's right! Apparently even some proponents of dark matter agree with this. But: take everything I'm saying here with a grain of salt. I'm no expert on this stuff, so if you know any astrophysics you should read the literature and make up your own mind. Here are two reviews that Smolin especially recommended: 13) Robert H. Sanders and Stacy S. McGaugh, Modified Newtonian Dynamics as an Alternative to Dark Matter, available as astroph/0204521. 14) Anthony Aguirre, Alternatives to dark matter (?), available as astroph/0310572. Here's McGaugh's website with links to many papers on MOND, including Milgrom's original papers: 15) The MOND pages, http://www.astro.umd.edu/~ssm/mond/litsub.html McGaugh is a strong proponent of MOND  though he didn't start out that way  so the selection may be biased. Does anyone know an intelligent detailed critique of MOND? If so, I want to see it! We can't throw out Newton's law of gravity (or more precisely, general relativity, which has Newtonian gravity as a limiting case for low densities and low velocities) unless we have *very* good reasons! So we have to think about things carefully, and weigh the evidence on both sides. If I could duplicate myself, I'd have one copy try to get to the bottom of this dark matter / MOND puzzle. But I can't... .... so if you're an expert who knows a lot about this, let me know what you think  or better yet, post an article about this to sci.physics.research!  Previous issues of "This Week's Finds" and other expository articles on mathematics and physics, as well as some of my research papers, can be obtained at http://math.ucr.edu/home/baez/ For a table of contents of all the issues of This Week's Finds, try http://math.ucr.edu/home/baez/twf.html A simple jumpingoff point to the old issues is available at http://math.ucr.edu/home/baez/twfshort.html If you just want the latest issue, go to http://math.ucr.edu/home/baez/this.week.html 
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This Week's Finds in Mathematical Physics (Week 206)

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This Week's Finds in Mathematical Physics (Week 206)
Modified Newtonian Dynamics
MOND basically says that if you double the distance of a star from the galactic centre, then you half the force of gravity, instead of quartering it as Newton's inverse square law would say. This is what a physical theory needs to explain. 
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This Week's Finds in Mathematical Physics (Week 206)

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This Week's Finds in Mathematical Physics (Week 206)
In sci.astro.research John Baez wrote:
So, even if MOND is wrong, there may need to be some reason why it *acts* like it's right! Apparently even some proponents of dark matter agree with this. Try this: M. Kaplinghat and M. S. Turner, "How Cold Dark Matter Theory Explains Milgrom's Law," astroph/0107284, Astrophys.J. 569 (2002) L19. Note that this analysis also explains why the ``critical acceleration'' in MOND does *not* apply at cluster scales. There is some debate over these results, but the paper is certainly worth reading. Steve Carlip 
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This Week's Finds in Mathematical Physics (Week 206)
In article , alistair wrote: Modified Newtonian Dynamics MOND basically says that if you double the distance of a star from the galactic centre, then you half the force of gravity, instead of quartering it as Newton's inverse square law would say. This is what a physical theory needs to explain. This is not accurate. John Baez's description of MOND is much closer to what the theory actually says. John's description was essentially this: if you define a_N to be the Newtonian acceleration a_N = F / m then the actual acceleration of an object is a = a_N if a_N a_0 a = sqrt(a_N a_0) if a a_0 Here a_0 is some fundamental constant. He pointed out, quite correctly, that it's ugly for a fundamental law to be split into cases like that. Last time I checked, the MOND people weren't dogmatic about this exact form. They considered smooth functional relationships between a and a_N. I think that as long as the relationship approaches the above behavior in the limits, a  a_N when a_N a_0 a  sqrt(a_N a_0) when a_N a_0 the MOND people are satisfied. So I guess something like a = sqrt(a_N (a_N + a_0)) might do the trick. Personally, I can't get past my theorist's objections to MOND. It doesn't play well at all with general relativity, and I just don't believe that general relativity is completely on the wrong track. But of course the issue should be settled observationally, not based on theoretical prejudice (however welljustified!). Ted  [Email me at , as opposed to .] 
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This Week's Finds in Mathematical Physics (Week 206)
Lee Smolin told me some neat stuff about MOND  that's "Modified
Newtonian Dynamics", which is Mordehai Milgrom's way of trying to explain the strange behavior of galaxies without invoking dark matter. The basic problem with galaxies is that the outer parts rotate faster than they should given how much mass we actually see. From http://www.astro.ucla.edu/~wright/density.html#MOND it seems MOND fails on the scale of galaxy clusters. also I don't know if MOND can explain an accelerating universe, predict correct elemental abundances, provide a good fit to CMB anisotropy spectrum. anyhow I think the fantastic fit of WMAP and SDSS,2DF to the standard Lambda+CDM is more or less a nail in the coffin for all alternative models. Nevertheless the standard model requires BOTH nonbaryonic dark matter and dark energy for which there is no laboratory evidence. So I guess as good scientists it makes sense to think of alternatives and try to test them experimentally :) anyhow I have a question to experts on gravitation & cosmology on this forum. what do you think about Conformal gravity which is cited in the paper by Aguirre (mentioned by John above?) As far as I know unlike MOND this cannot be ruled out at cluster scales. It also provides a natural explanation for an accelerating universe. I however don't know if Conformal gravity is consistent with predictions of GR at solarsystem and binary pulsar distance scales. Thanks [Moderator's note: Excessively quoted text truncated by moderator. Please quote with care. See http://wwwstud.uniessen.de/~sb0264/HowToPost.html usc] 
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This Week's Finds in Mathematical Physics (Week 206)
I checked out this article: http://www.astro.umd.edu/~ssm/mond/astronow.html
and there's an example of a graph of rotation velocity vs. radius showing one of the wiggles JB mentions, with the note "even the kink observed in the gas distribution is reflected in the rotation". Forgive me for being a little sceptical but the Newtonian prediction has exactly the same kink. In fact, the MOND curve is just the Newtonian curve scaled up. Just about any reasonably well behaved modification of the Newtonian formula that has a scaling effect that brings the Newtonian curve roughly in alignment with measured results is going to have that kink. So while it may be impressive that MOND predicts this overall scaling correctly, I'm not in the least bit impressed with it managing to match this wiggle. Are there examples where MOND predicts a wiggle that simply isn't present in the Newtonian case. That'd be impressive. Or am I misinterpreting the graph?  Torque "John Baez" wrote in message ... Also available at http://math.ucr.edu/home/baez/week206.html May 10, 2004 This Week's Finds in Mathematical Physics  Week 206 John Baez I just got back from Marseille, where Carlo Rovelli, Laurent Freidel and Phillipe Roche held the first really big conference on loop quantum gravity and spin foams since the 2nd Warsaw workshop run by Jerzy Lewandowski back in 1997: 1) Non Perturbative Quantum Gravity: Loops and Spin Foams, 37 May 2004, CIRM, Luminy, Marseille, France, http://w3.lpm.univmontp2.fr/~philip...ravitywebsite/ It was good to see old friends and talk about quantum gravity near the "Calanques"  the rugged limestone cliffs lining the Mediterranean coastline. It was good to meet lots of young people who have recently entered this difficult field: about 100 people attended, considerably more than at any previous meeting. But most of all, it was good to see some progress on the tough problem of understanding dynamics in nonperturbative quantum gravity. Can we get the 4dimensional spacetime we know and love, whose geometry is described by general relativity, to emerge from some theory that takes quantum physics into account? And can we do it *nonperturbatively*? In other words, can we do quantum physics without choosing some fixed spacetime geometry from the start, a "background" on which small perturbations move like tiny quantum ripples on a calm preestablished lake? A background geometry is convenient: it lets us keep track of times and distances. It's like having a fixed stage on which the actors  gravitons, strings, branes, or whatever  cavort and dance. But, the main lesson of general relativity is that spacetime is *not* a fixed stage: it's a lively, dynamical entity! There's no good way to separate the ripples from the lake. This distinction is no more than a convenient approximation  and a dangerous one at that. So, we should learn to make do without a background when studying quantum gravity. But it's tough! There are knotty conceptual issues like the "problem of time": how do we describe time evolution without using a fixed background to measure the passage of time? There are also practical problems: in most attempts to describe spacetime from the ground up in a quantum way, all hell breaks loose! We can easily get spacetimes that crumple up into a tiny blob... or spacetimes that form endlessly branching fractal "polymers" of Hausdorff dimension 2... but it seems hard to get reasonably smooth spacetimes of dimension 4. It's even hard to get spacetimes of dimension 10 or 11... or *anything* remotely interesting! It almost seems as if we need a solid background as a bed frame to keep the mattress of spacetime from rolling up or otherwise misbehaving. Unfortunately, even *with* a background there are serious problems: we can use perturbation theory to write the answers to physics questions as power series, but these series diverge and nobody knows how to resum them. String theorists are pragmatic in a certain sense: they don't mind using a background, and they don't mind doing what physicists always do: approximating a divergent series by the sum of the first couple of terms. But this attitude doesn't solve everything, because right now in string theory there is an enormous "landscape" of different backgrounds, with no firm principle for choosing one. Some estimates guess there are over 10^{100}. Leonard Susskind guesses there are 10^{500}, and argues that we'll need the anthropic principle to choose the one describing our world: 2) Leonard Susskind, The Landscape, article and interview on John Brockman's "EDGE" website, http://www.edge.org/3rd_culture/suss...ind_index.html This position is highly controversial, but my point here shouldn't be: developing a backgroundfree theory of quantum gravity is tough, but working *with* a background has its own difficulties. And let's face it: we haven't spent nearly as much time thinking about backgroundfree or nonperturbative physics as we've spent on backgrounddependent or perturbative physics. So, it's quite possible that our failures with the former are just a matter of inexperience. Given all this, I'm delighted to see some real progress on getting 4d spacetime to emerge from nonperturbative quantum gravity: 3) Jan Ambjorn, Jerzy Jurkiewicz and Renate Loll, Emergence of a 4d world from causal quantum gravity, available as hepth/0404156. This trio of researchers have revitalized an approach called "dynamical triangulations" where we calculate path integrals in quantum gravity by summing over different ways of building spacetime out of little 4simplices. They showed that if we restrict this sum to spacetimes with a wellbehaved concept of causality, we get good results. This is a bit startling, because after decades of work, most researchers had despaired of getting general relativity to emerge at large distances starting from the dynamical triangulations approach. But, these people hadn't noticed a certain flaw in the approach... a flaw which Loll and collaborators noticed and fixed! If you don't know what a path integral is, don't worry: it's pretty simple. Basically, in quantum physics we can calculate the expected value of any physical quantity by doing an average over all possible histories of the system in question, with each history weighted by a complex number called its "amplitude". For a particle, a history is just a path in space; to average over all histories is to integrate over all paths  hence the term "path integral". But in quantum gravity, a history is nothing other than a SPACETIME. Mathematically, a "spacetime" is something like a 4dimensional manifold equipped with a Lorentzian metric. But it's hard to integrate over all of these  there are just too darn many. So, sometimes people instead treat spacetime as made of little discrete building blocks, turning the path integral into a sum. You can either take this seriously or treat it as a kind of approximation. Luckily, the calculations work the same either way! If you're looking to build spacetime out of some sort of discrete building block, a handy candidate is the "4simplex": the 4dimensional analogue of a tetrahedron. This shape is rigid once you fix the lengths of its 10 edges, which correspond to the 10 components of the metric tensor in general relativity. There are lots of approaches to the path integrals in quantum gravity that start by chopping spacetime into 4simplices. The weird special thing about dynamical triangulations is that here we usually assume every 4simplex in spacetime has the same shape. The different spacetimes arise solely from different ways of sticking the 4simplices together. Why such a drastic simplifying assumption? To make calculations quick and easy! The goal is get models where you can simulate quantum geometry on your laptop  or at least a supercomputer. The hope is that simplifying assumptions about physics at the Planck scale will wash out and not make much difference on large length scales. Computations using the socalled "renormalization group flow" suggest that this hope is true *IF* the path integral is dominated by spacetimes that look, when viewed from afar, almost like 4d manifolds with smooth metrics. Given this, it seems we're bound to get general relativity at large distance scales  perhaps with a nonzero cosmological constant, and perhaps including various forms of matter. Unfortunately, in all previous dynamical triangulation models, the path integral was *NOT* dominated by spacetimes that look like nice 4d manifolds from afar! Depending on the details, one either got a "crumpled phase" dominated by spacetimes where almost all the 4simplices touch each other, or a "branched polymer phase" dominated by spacetimes where the 4simplices form treelike structures. There's a transition between these two phases, but unfortunately it seems to be a 1storder phase transition  not the sort we can get anything useful out of. For a nice review of these calculations, see: 4) Renate Loll, Discrete approaches to quantum gravity in four dimensions, available as grqc/9805049 or as a website at Living Reviews in Relativity, http://www.livingreviews.org/Article...1/199813loll/ Luckily, all these calculations shared a common flaw! Computer calculations of path integrals become a lot easier if instead of assigning a complex "amplitude" to each history, we assign it a positive real number: a "relative probability". The basic reason is that unlike positive real numbers, complex numbers can cancel out when you sum them! When we have relative probabilities, it's the *highly probable* histories that contribute most to the expected value of any physical quantity. We can use something called the "Metropolis algorithm" to spot these highly probable histories and spend most of our time worrying about them. This doesn't work when we have complex amplitudes, since even a history with a big amplitude can be canceled out by a nearby history with the opposite big amplitude! Indeed, this happens all the time. So, instead of histories with big amplitudes, it's the *bunches of histories that happen not to completely cancel out* that really matter. Nobody knows an efficient generalpurpose algorithm to deal with this! For this reason, physicists often use a trick called "Wick rotation" that converts amplitudes to relative probabilities. To do this trick, we just replace time by imaginary time! In other words, wherever we see the variable "t" for time in any formula, we replace it by "it". Magically, this often does the job: our amplitudes turn into relative probabilities! We then go ahead and calculate stuff. Then we take this stuff and go back and replace "it" everywhere by "t" to get our final answers. While the deep inner meaning of this trick is mysterious, it can be justified in a wide variety of contexts using the "OsterwalderSchrader theorem". Here's a pretty general version of this theorem, suitable for quantum gravity: 5) Abhay Ashtekar, Donald Marolf, Jose Mourao and Thomas Thiemann, OsterwalderSchrader reconstruction and diffeomorphism invariance, preprint available as quantph/9904094. People use Wick rotation in all work on dynamical triangulations. Unfortunately, this is *not* a context where you can justify this trick by appealing to the OsterwalderSchrader theorem. The problem is that there's no good notion of a time coordinate "t" on your typical spacetime built by sticking together a bunch of 4simplices! The new work by Ambjorn, Jurkiewiecz and Loll deals with this by restricting to spacetimes that *do* have a time coordinate. More precisely, they fix a 3dimensional manifold and consider all possible triangulations of this manifold by regular tetrahedra. These are the allowed "slices" of spacetime  they represent different possible geometries of space at a given time. They then consider spacetimes having slices of this form joined together by 4simplices in a few simple ways. The slicing gives a preferred time parameter "t". On the one hand this goes against our desire in general relativity to avoid a preferred time coordinate  but on the other hand, it allows Wick rotation. So, they can use the Metropolis algorithm to compute things to their hearts' content and then replace "it" by "t" at the end. When they do this, they get convincing good evidence that the spacetimes which dominate the path integral look approximately like nice smooth 4dimensional manifolds at large distances! Take a look at their graphs and pictures  a picture is worth a thousand words. Naturally, what *I'd* like to do is use their work to develop some spin foam models with better physical behavior than the ones we have so far. Now that Loll and her collaborators have gotten something that works, we can try to fiddle around and make it more elegant while making sure it still works. In particular, I'm hoping we can get wellbehaved models that don't introduce a preferred time coordinate as long as they rule out "topology change"  that is, slicings where the topology of space changes. After all, the OsterwalderSchrader theorem doesn't require a *preferred* time coordinate, just *any* time coordinate together with good behavior under change of time coordinate. For this we mainly need to rule out topology change. Moreover, Loll and her collaborators have argued in 2d toy models that topology change is one thing that makes models go bad: the path integral can get dominated by spacetimes where "baby universes" keep branching off the main one: 6) Jan Ambjorn, Jerzy Jurkiewicz and Renate Loll, Nonperturbative Lorentzian quantum gravity, causality and topology change, Nucl. Phys. B536 (1998) 407434. Also available as hepth/9805108. Renate Loll and W. Westra, Spacetime foam in 2d and the sum over topologies, Acta Phys. Polon. B34 (2003) 49975008. Also available as hepth/0309012. By the way, it's also reading about their 3d model: 7) Jan Ambjorn, Jerzy Jurkiewicz and Renate Loll, Nonperturbative 3d Lorentzian quantum gravity, Phys.Rev. D64 (2001) 044011. Also available as hepth/0011276. and for a general review, try this: 8) Renate Loll, A discrete history of the Lorentzian path integral, Lecture Notes in Physics 631, Springer, Berlin, 2003, pp. 137171. Also available as hepth/0212340. All this is great, but don't get me wrong  there were a lot of *other* cool talks at the conference besides Loll's. I'll just mention a few. Laurent Freidel spoke on his work on spin foam models. Especially exciting is how David Louapre and he have managed to "sum over topologies" in 3d Riemannian quantum gravity with vanishing cosmological constant  otherwise known as the PonzanoRegge model He has to subtract out a counterterm that would otherwise lead to a bubble divergence, but then he gets a beautiful theory where the sum over spin foams is Borel summable: 9) Laurent Freidel and David Louapre, Nonperturbative summation over 3D discrete topologies, Phys.Rev. D68 (2003) 104004. Also available as hepth/0211026. Their work on gaugefixing and the inclusion of spinning point particles in the PonzanoRegge model is also very impressive, especially given how long this model has been studied. It shows we have lots left to learn! 10) Laurent Freidel and David Louapre, PonzanoRegge model revisited I: Gauge fixing, observables and interacting spinning particles, available as hepth/0401076. The title suggests we're in for more treats to come. Kirill Krasnov gave a talk entitled simple "ln(3)"  it was all about the appearance of this constant in the work of Hod, Dreyer, Motl and Neitzke on black hole entropy and the ringing of black holes. I've discussed all this at length in "week198", but Krasnov has given an elegant new proof of Hod's conjecture using Riemann surface theory. One can even think of this as a "stringy" explanation of the quasinormal modes of black holes  but much remains mysterious he 11) Kirill Krasnov, Black hole thermodynamics and Riemann surfaces, Class. Quant. Grav. 20 (2003) 22352250. Also available as grqc/0302073. Kirill Krasnov and Sergey N. Solodukhin, Effective stringy description of Schwarzschild black holes, available as hepth/0403046. While I'm at it, I can't resist mentioning Krasnov's work on including point particles in 3d Lorentzian quantum gravity with negative cosmological constant, since it has close connections with that of Freidel and Louapre, though the context is a bit different: 12) Kirill Krasnov, Lambda0 quantum gravity in 2+1 dimensions I: quantum states and stringy Smatrix, Class. Quant. Grav. 19 (2002) 39773998. Also available as hepth/0112164. Kirill Krasnov, Lambda0 quantum gravity in 2+1 dimensions II: black hole creation by point particles, Class. Quant. Grav. 19 (2002) 39994028. Also available as hepth/0202117. If I could duplicate myself, I'd have one copy write a book on 3d quantum gravity that would synthesize all these wonderful results in a nice big picture. It's not realistic physics; it's just a toy model. But the math is *so* nice, and so enlightening for realworld physics in some ways, that it's hard to resist pondering it! TQFTs, Riemann surfaces, hyperbolic geometry, spinning point particles colliding and creating black holes  a wonderful stew! Alas, I don't have time to savor it. There were a lot of other interesting talks  but I don't have time to go through and describe all of them, either. So, I'll wrap up with something very different! Lee Smolin told me some neat stuff about MOND  that's "Modified Newtonian Dynamics", which is Mordehai Milgrom's way of trying to explain the strange behavior of galaxies without invoking dark matter. The basic problem with galaxies is that the outer parts rotate faster than they should given how much mass we actually see. If you have a planet in a circular orbit about the Sun, Newton's laws say its acceleration is proportional to 1/r^2, where r is its distance to the Sun. Similarly, if almost all the mass in a galaxy were concentrated right at the center, a star orbiting in a circle at distance r from the center would have acceleration proportional to 1/r^2. Of course, not all the mass is right at the center! So, the acceleration should drop off more slowly than 1/r^2 as you go further out. And it does. But, the observed acceleration drops off a lot more slowly than the acceleration people calculate from the mass they see. It's not a small effect: it's a HUGE effect! One solution is to say there's a lot of mass we don't see: "dark matter" of some sort. If you take this route, which most astronomers do, you're forced to say that *most* of the mass of galaxies is in the form of dark matter. Milgrom's solution is to say that Newton's laws are messed up. Of course this is a drastic, dangerous step: the last guy who tried this was named Einstein, and we all know what happened to him. Milgrom's theory isn't even based on deep reasoning and beautiful math like Einstein's! Instead, it's just a blatant attempt to fit the experimental data. And it's not even elegant. In fact, it's downright ugly. Here's what it says: the usual Newtonian formula for the acceleration due to gravity is correct as long as the acceleration is bigger than a = 2 x 10^{10} m/sec^2 But, for accelerations less than this, you take the geometric mean of the acceleration Newton would predict and this constant a. In other words, there's a certain value of acceleration such that above this value, the Newtonian law of gravity works as usual, while below this value the law suddenly changes. Any physicist worth his salt who hears this modification of Newton's law should be overcome with a feeling of revulsion! There just *aren't* laws of physics that split a situation in two cases and say "if this is bigger than that, then do X, but if it's smaller, then do Y." Not in fundamental physics, anyway! Sure, water is solid below 0 centigrade and fluid above this, but that's not a fundamental law  it presumably follows from other stuff. Not that anyone has derived the melting point of ice from first principles, mind you. But we think we could if we were better at big messy calculations. Furthermore, you can't easily invent a Lagrangian for gravity that makes it fall off more *slowly* than 1/r^2. It's easy to get it to fall off *faster*  just give the graviton a mass, for example! But not more slowly. It turns out you can do it  Bekenstein and Milgrom have a way  but it's incredibly ugly. So, MOND should instantly make any decent physicist cringe. Esthetics alone would be enough to rule it out, except for one slight problem: it seems to fit the data! In some cases it matches the observed rotation of galaxies in an appallingly accurate way, fitting every wiggle in the graph of stellar rotation velocity as a function of distance from the center. So, even if MOND is wrong, there may need to be some reason why it *acts* like it's right! Apparently even some proponents of dark matter agree with this. But: take everything I'm saying here with a grain of salt. I'm no expert on this stuff, so if you know any astrophysics you should read the literature and make up your own mind. Here are two reviews that Smolin especially recommended: 13) Robert H. Sanders and Stacy S. McGaugh, Modified Newtonian Dynamics as an Alternative to Dark Matter, available as astroph/0204521. 14) Anthony Aguirre, Alternatives to dark matter (?), available as astroph/0310572. Here's McGaugh's website with links to many papers on MOND, including Milgrom's original papers: 15) The MOND pages, http://www.astro.umd.edu/~ssm/mond/litsub.html McGaugh is a strong proponent of MOND  though he didn't start out that way  so the selection may be biased. Does anyone know an intelligent detailed critique of MOND? If so, I want to see it! We can't throw out Newton's law of gravity (or more precisely, general relativity, which has Newtonian gravity as a limiting case for low densities and low velocities) unless we have *very* good reasons! So we have to think about things carefully, and weigh the evidence on both sides. If I could duplicate myself, I'd have one copy try to get to the bottom of this dark matter / MOND puzzle. But I can't... ... so if you're an expert who knows a lot about this, let me know what you think  or better yet, post an article about this to sci.physics.research!  Previous issues of "This Week's Finds" and other expository articles on mathematics and physics, as well as some of my research papers, can be obtained at http://math.ucr.edu/home/baez/ For a table of contents of all the issues of This Week's Finds, try http://math.ucr.edu/home/baez/twf.html A simple jumpingoff point to the old issues is available at http://math.ucr.edu/home/baez/twfshort.html If you just want the latest issue, go to http://math.ucr.edu/home/baez/this.week.html 
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This Week's Finds in Mathematical Physics (Week 206)

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