Dismal science

By Daniel Holz | October 7, 2009 11:06 am

As Sean highlighted, Paul Krugman recently wrote a piece on the state of the economic profession. Krugman was brutal; economic “science” failed to anticipate, much less predict, the current economic crisis. Krugman describes how economists had become infatuated with beautiful theories, and became increasingly removed from the real world. A world in which there are indeed real estate bubbles, financial speculators, and complete and catastrophic financial collapse. The current recession hit, demonstrating the failure of economics as a discipline.

Sean was intrigued by the search for an Economic “Theory of Everything”. He points out that this theory may indeed exist, and when found, will be proclaimed beautiful. Of course, the analogies to String Theory are irresistible. But I will resist. As Sean suggests, General Relativity is a beautiful theory. In fact, it’s stupendously gorgeous. (Unlike quantum mechanics, which although it certainly possesses its charms, is not something I would call ravishing.) General Relativity also happens to be right, so far as we can tell. It is a significant improvement over Newtonian gravity, which itself is no ugly duckling. We’ve come a long way from Ptolemy. Indeed, aesthetics plays a major role in science. The Nobel Laureate himself singles out Sean’s post, and mentions it on his blog. Beauty and truth are not in opposition. But beauty is not a replacement for truth.

As I was reading through Krugman’s essay, however, I confess to a certain amount of envy. His description of the field of economics made it sound awfully familiar. People have different theories. They argue. They write papers. There are influential thinkers. There’s lots of math. It’s just like physics, with one crucial difference:
You’re probably expecting me to say that economics is essentially gobbledygook, while physics is a “pure” science. That may be true, but it’s an argument for another day. What struck me is the fact that what economists do and say really matters, in an immediate and tangible way. They engage in abstruse arguments about the money supply and the subprime market, but at the end of the day, someone somewhere listens to them, and makes a decision about the interest rate, or whether to bailout a troubled bank. Suddenly, millions of people may be out of work. Trillions of dollars may evaporate. A large fraction of the population of the planet may be affected.

Of course, it’s not that physics is irrelevant. It is almost impossible not to look around and see the role of physicists in one’s daily life (starting with the computer in front of you). But if I miss a factor of two, a family in Detroit does not go hungry. Although economics often seems like a “lightweight” discipline, when compared to harder sciences, it has the advantage of immediate relevancy. And at first blush, this seems highly desirable. On second thought, however, maybe I’d rather not have to worry about destroying Iceland while looking for a bug in my code.

CATEGORIZED UNDER: Science and Society
  • Metre

    As one witt put it “Economics is useful chiefly as a means of employment for economists.” Other than that, its utility is questionable.

  • Ryano

    Maybe the problem with the extending GR to a theory of everything is the same as with economics: it is too beautiful, and every extension everyone takes tries to preserve that beauty. In the details, Nature may not be a kind, symmetric thing.

  • http://togroklife.com greg

    What struck me is the fact that what economists do and say really matters, in an immediate and tangible way. They engage in abstruse arguments about the money supply and the subprime market, but at the end of the day, someone somewhere listens to them, and makes a decision about the interest rate, or whether to bailout a troubled bank. Suddenly, millions of people may be out of work. Trillions of dollars may evaporate. A large fraction of the population of the planet may be affected.

    You envy the ability to ruin a large portion of the human population? I think it requires either complete blindness to this possibility or overweening arrogance to consider yourself to be right about something that, if you’re not, has the potential to ruin so many lives. People still point to the consciences of the physicists in the 30s and 40s and 50s who feared the potential (and actual) destructiveness of the nuclear weapons that their work helped to create. Where are the economists who show such awareness of the possible negative impact of their ideas and work?

  • Ja Muller

    “Maybe the problem with the extending GR to a theory of everything is the same as with economics: it is too beautiful, and every extension everyone takes tries to preserve that beauty. In the details, Nature may not be a kind, symmetric thing”

    Good point. It would sort of be like if physicists insisted that the universe is supersymmetric and chose to ignore all of the seemingly random and ugly features of the standard model.

  • http://www.edge.org/3rd_culture/bios/weinstein.html Eric Weinstein

    Hi Daniel,

    A great post. But, oddly, I believe that economics in 2009 needs physicists desperately.

    It is not so much what physicists do that currently matters (like it once did at Los Alamos), but what they refuse to do. In general, physicists refuse to get involved. It is not much different for mathematicians and evolutionary biologists. But, right now, the House Science and Technology committee (rather than finance committee) is looking at the *scientific* foundations of the field. Physicists are absolutely necessary to solve some of these problems in the core theory.

    Economic theory, as you write above, is not ‘essentially gobbledygook'; it is some kind of fascinating amalgam of total anti-scientific nonsense, experiment, technically intimidating one upsmanship and pure gorgeous theory. Lamentably, the silly components are so off putting that most of us cannot bear that the underlying structures since the marginal revolution and the Pareto’s are mathematically those of Gauge theory and have been so since the 1920s at least (e.g. work of the Algerian economist F. Divisia).

    Let me put it another way. The field is still worried about eliminating the Ahranov-Bohm effect in measurement. They are often trying to propogate point prices for illiquid securities so there is always a definite value at all times even though we see a dynamic distribution of possible prices that suddenly collapses under ‘price discovery’. The basic theory further denies the essential attributes of humans as decision makers because the gauge theory needed to handle what are called ‘ordinal preferences’ requires something akin to the Virasoro algebra (seen as reparameterizations of the space of positive real ‘utils’).

    Physics isn’t irrelevant at all to policy. What is going on is really a monastic retreat by the experts in the dynamics of extended objects (seasonal preferences), uncertain objects (illquid assets, instantaneous preferences), calculus on function spaces (tastes) and non-linear gauged objects (welfare, trade). Now, perhaps, there is little to do here. But at the level of ‘expected value’ physicists, mathematicians and biologists are necesary. We are needed, but we find every excuse to avoid having impact as if we’ve been trained to thow the game.

    Check out a few who are doing something different from our recent conference on this topic:

    D. Farmer: http://pirsa.org/09050020/ Physicists attempt to scale the towers of finance
    P. Malaney: http://pirsa.org/09050022/ A New Marginalism: Gauge Theory in Economics
    Perimeter Conference: http://pirsa.org/C09006

    You’ve clearly got Krugman’s ear. Think about what’s keeping you from using it.

    Best regards and keep up the great blogging,

    Eric Weinstein

  • http://www.edge.org/3rd_culture/bios/weinstein.html Eric Weinstein

    P.S. As for the economics version of the theory of everything, brace yourself. Pretty, it ain’t: http://users.drew.edu/dlawson/research/degustibus.pdf

  • Ijon Tichy

    On “destroying Iceland”. It wasn’t economists who destroyed Iceland. It was the Icelanders themselves. The banks, the government *and* the common people. They were warned before the financial crisis hit, but their response was, “we’re special, there is nothing to worry about”. Well, Iceland, like all Nordic countries, *is* special: it has one of the most socially advanced societies on the planet. But that did not stop them from succumbing to greed and stupidity.

  • mark a. thomas

    How could the economists have missed the impending disaster? The market curves were there for all to see and one could see that the curve was to steep and too fast and that a ‘bubble’ was going to readjust. No, there were predictions and the governments knew too. It is typical, put your hands in your pockets and hope that the thing goes away without you having to bring unwelcome attention to yourself. Helpless and spineless. Wait for disaster then chime in with everyone else so that you are not the ruination of a party where everyone goobles down wealth-leaving no one to be responsible. Typical american-european response of call to arms afterwards. Same thing happened during the Dot-com bubble of 2000. Money was sucked from a straw then as now.

  • Low Math, Meekly Interacting

    People listen to economists, and not to physicists. That pretty much sums up just how hosed we are, doesn’t it.

  • joe

    Too bad Krugman kept referring to this blog as “Discover” instead of “Cosmic Variance”. That’s what you guys get for selling out ;-)

  • Ellipsis

    GR is — possibly — such a beautiful theory because of the relative lack of good data on cosmology until recently (and thus some bias toward a simple explanation of the little data available). When we have (usually smaller) things in our hand/lab that we can probe more easily, eventually I think we see that they are not so simple. Like you said, beauty is absolutely no substitute for truth.

    While we can complain about economics and economists, one should remember that some of it might be because of the inherent unpredictability of human behavior (without, clearly, complete knowledge of everyone’s neurons) and, thus, that they may have a fundamentally harder job than physicists. Of course, that is probably only part of the story.

  • SFJP

    My feeling is that contemporary economists are like the physicians of Molière’s time, just infatuated Diafoirus like doctors (I write this for people acquainted with classical French culture), who cannot practically do or predict anything, but who have a lot of gibberish discourses to make you think they can speak about anything!!! As a matter of fact, this presently has just led to a kind of a worldwide disaster, in the same way as if theoretical physicists were asked to definitively tell the power that be about what will happen and what to do in the next ten years on our poor little earth! The economical world is doing a lot of misuses of scientific and mathematical jargon just to justify unbelievable behavior or personal and highly dubious theories.

    With respect to theoretical physics, the difference is that no one would use the latest BH or stringy theories to dictate the behavior of governments, while it is the case of economy theories!

    True scientists have to denounce this state of fact and to make the world know that economists are no more true scientists than astrologists!

  • Gary

    Krugman, himself, failed to anticipate and predict the current economic crisis and adequately act to avert it.

    Krugman is Economics’ Ptolemy with a happy meal Nobel.

  • Matt

    Daniel–

    I’m afraid I must challenge you here. It has become fashionable to describe general relativity as being “the beautiful theory” of fundamental physics and quantum mechanics as being “the ugly theory,” or, at least the way you put it, not a “ravishing” theory.

    I think this rather provocative assertion of yours requires a defense. I suspect that this all-to-common sentiment stems more from the dichotomy between how general relativity and quantum mechanics are typically taught, the former being introduced in a unified way from the principle of equivalence and the latter as an ad hoc mixture of facts and techniques.

    But quantum mechanics really stems from far simpler (and more general and logically rigid) basic principles than general relativity, namely, the superposition principle and conservation of information (unitarity). You get ladder operators, path integrals, and decoherence, just to name a few profound and beautiful ideas that emerge from the theory. And, for heaven’s sake, the theory is, quite famously, linear! (What could possibly be more elegant than that?)

    General relativity, meanwhile, is only an approximate description of Nature, and is as nonlinear and messy as any theory one can find in classical physics. (It should also be noted that string theory modifies general relativity, not quantum mechanics.) Just deriving its simplest nontrivial solution, the Schwarzschild solution, takes pages of calculations! And try writing down the solution to the two-body problem!

    I must respectfully express my disagreement.

  • Matt

    Oh, and you’re totally wrong, Gary. Krugman was griping about how the housing bubble was going to doom us all, for nearly two years before the crash. Everyone called him a histrionic Cassandra, not realizing the irony.

  • Low Math, Meekly Interacting

    The trouble I have is not knowing whether or not Krugman, Roubini, and the like would be right about something else. Even when they are right, I can’t say quite why, because if I ask someone else from a different economic school, they’ll draw completely different conclusions from the same indicators and trends. If I’m indefinite enough about the time frame, I can easily imagine anyone being right about any up or down market one could imagine, à la the stopped clock. I predicted the collapse of the housing bubble, as did half the people I work with. You watch house prices inflate at the rate they did in the greater Boston area, and it doesn’t take a genius to surmise such a trend is unsustainable. After all, none of our wages increased at the same rate. Only financiers enjoyed such non-linearity.

    What took genius, actually, was keeping that trend swinging upward for as long as it did, in clear defiance of what we non-experts refer to as common sense. The financial prestidigitation required to hide all that risk from view for as long as they managed to is a testament to human ingenuity and the mother of invention that is avarice. The cleverest confidence man could only stand in awe of the swindle that fueled the modern securitization of the mortgage industry.

    Is beautiful economics something to be yearned for or feared then, I wonder.

  • Chris W.

    Low Math, Meekly Interacting: People listen to economists, and not to physicists. That pretty much sums up just how hosed we are, doesn’t it.

    Perhaps, if people listened as much to physicists as economists, physicists would become as intellectually dishonest and mercenary as economists.

    To punish me for my contempt for authority, fate made me an authority myself. — Albert Einstein

  • Metre

    Recall the difficulty that Boltzmann had in developing statistical physics. Now replace each of those gas particles with a human being making making self-interested decisions and a government constantly changing the laws of interaction and you have economics. Not an easy system to model mathematically.

  • chris

    Hi Matt,

    what you describe just about sums up the beauty of GR! its fundamental equation is so extremely simple to write down (forget about the fact that nasty, huge tensors lurk behind the shiny R and T) but the consequences are staggering. i think this is what many scientists deem beautiful.

    oh, and economy should for once prove its basic assumption: that you can treat the elements (humans) statistically in large numbers. as long as that’s not proven all the phenomenology is baseless speculation.

  • Low Math, Meekly Interacting

    I dunno, Chris. Say for instance I stubbornly refused to acknowledge that God does not play dice, yet all experimental evidence indicated otherwise. I’m not sure how much intellectual dishonesty I could get away with. We have no rigorous mechanism, so far as I can tell, to test the predictions of macroeconomists. We can’t, for instance, go back to the 1930s and see how well things would have worked without the New Deal.

    Modeling seems to suffer from the problems endemic to the field, namely the role of ideology in its construction. With such a diversity of models, one is bound to be right at any given time, but that doesn’t make it very easy to pick signals out of the noise.

  • Matt

    chris,

    Yeah, but look how much more comes out of the postulates of quantum mechanics. You can start with the superposition principle and information conservation, and derive the Schrodinger equation, quantization, thermodynamics, path integrals, decoherence, and a string of deep and profound theorems, from the no-cloning theorem to the no-communication theorem.

    Throw in nothing more than a little Lorentz symmetry, and out come the necessity of particles, the existence of antiparticles, particle creation/annihilation, spin, CPT and crossing symmetry, and even the inevitability of fields. Put in a massless spin 1 particle, and Maxwell’s gauge theory is inevitable. Plug in a massless spin 2, and general relativity is the almost unique consequence. Tell me, can one go the other way and derive quantum mechanics from general relativity? I think not!

    My point here isn’t that GR is not beautiful. It is, in its own way. But quantum mechanics is just as elegant, at least when you see it presented correctly, which is sadly a rare thing in our contemporary undergraduate approaches to the subject. (And this has an impact on the way that many grown-up physicists remember the subject.) This is partly an accident of history; GR was devised by far fewer minds than QM, after all, and this messier history for QM is often mirrored in the way the subject gets taught. I hope things change some day soon.

    So don’t argue merely by standing up for GR. Explain why QM deserves to have its reputation so besmirched.

  • http://danielholz.com daniel

    @ Matt 14. One way to think about the beauty inherent in the theories is to look at their development. General relativity, in many ways, is the product of thought experiments. You start with some simple assumptions (e.g., the laws of physics should be frame independent, free fall is universal), and eventually you end up with this beautiful, geometric, all-encompassing theory. When asked what he would have done if his theory was observationally disproved, Einstein could state “Then I would have felt sorry for the dear Lord. The theory is correct.” That is an aesthetic statement. Now look at quantum mechanics. It seems to me we were dragged, kicking and screaming, to the development of quantum theory. Given a blank slate, and some basic principles (linearity?), I don’t see any way to come up with anything resembling quantum mechanics. At the very least, you need to postulate something akin to wavefunctions. Why do this? To wit, here’s a quote from Werner Heisenberg: “I remember discussions with Bohr which went through many hours till very late at night and ended almost in despair; and when at the end of the discussion I went alone for a walk in the neighbouring park I repeated to myself again and again the question: Can nature possibly be so absurd as it seemed to us in these atomic experiments?” I am not trying to malign quantum mechanics. It’s an incredible theory, and a wonderful playground. But, at least to my (jaundiced) eye, General Relativity has a certain irresistible fundamental sparkle.

  • Matt

    daniel–

    Ah, but you missed my most recent comment, where I specifically addressed the consequences of QM’s historical development. So your quoting of historical figures like Heisenberg precisely proves my point.

    Indeed, were it not for Einstein, GR probably would have been discovered in a totally messy way by at least as many hands as QM, as even Feynman later attested. It’s hard to get to GR from just the equivalence principle, but it was possible, as Einstein demonstrated. And quantum mechanics likewise follows from simple principles, although in that case those principles weren’t realized until many of the consequences of QM had been identified. The reason is not because those underlying principles were hard to understand or because the theory doesn’t flow out from them, but just because they weren’t as observationally obvious at the beginning. I can see elevators. I can’t see abstract vector spaces. But vector spaces are elegant and simple mathematical constructs.

    The standard QM curriculum developed at the same time as the subject itself, and so its messiness reflects that same messiness of QM’s historical provenance. But the subject doesn’t have to be taught that way. It is possible to write textbook for quantum mechanics where you start with the principles of superposition (the kinematical postulate) and information conservation (the dynamical postulate) on the first page and derive (with sufficient effort) the entire theory as a nontrivial consequence. That’s almost the very definition of beauty in physics.

    And the very fact that you seem to think that the principle of linearity and the existence of wave functions are separate postulates leads me to question whether you actually understand how to get quantum mechanics starting from first principles. (Not that I doubt in any way your expertise in how to use the techniques of quantum mechanics in its many applications.) That may be why you, and so many others, still think it’s so fundamentally messy and ad hoc. And it’s an artifact of how we still teach the subject, which is a real shame.

    chris–

    I’ll see your G=8piG T and raise you an H=0, which encodes the entire dynamical content of quantum mechanics when you work in a reparametrization invariant formalism. (If you don’t work in a reparametrization invariant formalism, then GR is actually far messier than QM!)

  • Matt

    I think it’s worth adding that general relativity is not a self-consistent physical theory; indeed, the theory predicts its own breakdown in numerous contexts. This is, to my mind, a pretty compelling knock against its elegance. The theory is broken.

    Quantum mechanics, by contrast, is not broken, and can stand alone as a self-consistent framework. There are, after all, countless quantum-mechanical systems (including all systems containing only a finite number of degrees of freedom) that exhibit no singularities or breakdowns at all. Of course, we may have to consider increasingly complicated quantum-mechanical systems as we probe Nature at its deepest levels (much as we have to consider increasingly complicated metrics in GR), but even string theory presumes that quantum mechanics itself needs no direct modification.

    By contrast, everyone expects that general relativity will have to be modified in some (likely) ugly way. Indeed, the Einstein equation is itself ad hoc in the sense that higher order terms in R can be added, and the only classical argument against doing so is that it makes the problem harder to solve. (And that, by dimensional analysis, the higher order terms are small in many contexts.) But we do add such higher order terms when we derive GR as an effective quantum theory from more fundamental theories.

    One of the beautiful things about quantum mechanics, by contrast, is that it’s so hard to modify it even slightly in a logically consistent way; that logical rigidity, absent from GR, is a very elegant feature of QM.

  • http://whenindoubtdo.blogspot.com Eugene

    Hey Daniel,

    Nice post.

    Although I think there is a danger in looking for “beauty” when we should be focused on the “truth”. I do understand that you are not saying that things which are right must be beautiful, but sometimes I do wish we as physicist focus less on the prettiness of our theories, and more on how cool it is that we actually understand how things work even when they are fugly.

    Eugene

  • Low Math, Meekly Interacting

    My post should have said “God does play dice”. For some reason, my attempts to edit generated error messages.

  • Eric Habegger

    I think there may eventually be a mathematical foundation to economics that is sound but the state of the discipline is far from that. It seems like the biggest the thing that economics has to take into account is the fact that humans are often irrational and that they tend to come to a concensus prematurely in some cases and way too late in other cases. It seems to be a case where individuals and groups easily become correlated in their opinions, sort of like entanglement in quantum physics.

    When that happens patterns of irrationality in finance continue way past their natural “decay” deadline. It then becomes like a string of dominoes where the unhealthy consensus all becomes destroyed in an equally unhealthy way. So it would seem that economics would have to somehow incorporate quantum ideas on a macroscopic scale to become mathematically valid.

  • TomS

    @Matt – Can you cite specific references that show how QM (the Schrodinger equation in particular) is derivable from the superpostion principle and information conservation with no other assumptions? I took QM in the late 60’s and and conservation of “information” was never mentioned. I don’t even know how “information” is defined :( So, I’m well behind the times. Thanks.

  • Belizean

    Eric Habegger wrote:
    “It seems like the biggest the thing that economics has to take into account is the fact that humans are often irrational…”

    What has always amazed me is that it seems that it would be completely straight-forward to create a mathematical model of markets that incorporates the peculiar irrationality to which markets are susceptible — the herd mentality.

    I just don’t get why this is a hard problem.

    Perhaps I should take a couple of weeks off, devise a model of bubbles that incorporates the herd mentality, publish it, and collect my Nobel prize in a couple of decades.

  • Matt

    TomS–

    I’ll see if I can find a nice derivation online. In the meantime, here’s a sketch.

    The concept of information as a precisely defined physical entity goes back at least to the work of Jaynes, who used information theory as his starting point for statistical mechanics. In modern language, one defines information to be a probability scheme—that is, a set of possible states or outcomes with corresponding probabilities that all add up to one. Information is what you know about the state of a system, and it increases to a maximum when one of these probabilities is one and the rest are zero. (By contrast, your information is zero when all the probabilities are equal.) This is the definition of information used, for example, back sixty years ago by the father of modern information theory, Claude Shannon, who went on to become a seminal player in the development of communications theory and computer science.

    In quantum mechanics, the superposition principle directly implies that information, if it is to have any invariant meaning, must be encoded in a manner that is insensitive to arbitrary changes of basis—namely, as the eigenvalue spectrum of a certain matrix called the density matrix of the system. Density matrices go back at least to Landau, and much work with them is due to von Neumann.

    Information conservation (which only applies to closed systems) then implies that the only smooth time evolution allowed for the system must leave the eigenvalue spectrum of the system’s density matrix unchanged. The only such transformation is unitary, and when you consider infinitesimal time intervals—in order to obtain the associated differential equation—you find that the density matrix must evolve according to an equation called the Liouville (or von Neumann) equation.

    In the special, idealized case in which the initial probability scheme encoded in the system’s density matrix is trivial, with a single state vector singled out with unit probability, the Liouville equation trivially reduces to the Schrodinger equation for that state vector.

    What’s especially miraculous is that if you couple a (formerly) closed system to a larger apparatus, you can use the Liouville equation to show that measurements by the apparatus cause the density matrices of both systems to transition (“decohere”) rapidly to diagonality in the eigenbasis of the measured observable, with the eigenvalue spectrum of each system’s density matrix conforming precisely to the Born probability formula. (One can show that for an apparatus with N degrees of freedom, this decoherence occurs exponentially fast with N, and for a typical macroscopic system like a table or chair happens in a time interval of order 10^-40 seconds.) So you actually get the phenomena of probabilistic measurements and state-vector collapse for free.

    But, fortunately, you can also derive what’s called the no-communication theorem, which shows that if the dynamics of the theory—as encoded in the particular system’s unitary time evolution operator—is local in space, then no actual information ever gets instantaneously transmitted between spatially separated systems, despite the rapid collapse.

    It’s truly remarkable that all these things come out correctly, even getting the typical time scale of collapse correct, without having to put in drastic modifications to the theory by hand. One could certainly have imagined that we might not have been so lucky!

    These are just a few examples of what I was talking about when I said that you get so many rich consequences in quantum mechanics from so few starting principles. But the subject is not taught this way at all, at least at the undergraduate level (and, in all too many places, even at the graduate level). This means that many working physicists never see this stuff done right, and so, despite being very good at using the techniques of quantum mechanics, go on in their careers with a rather messy picture of what quantum mechanics is at its core. Like I said, that’s a shame.

  • GeorgeRic

    Check out T.E.A.F.S. (available at Amazon.com) TEAFS shows that problem to be our efficiency in production, and as a result we do not need everyone working. In America now that means starvation, even though we have a surfeit of goods. So: Distribute the goods by figuring their value and then issueing checks to all. The shares should be fair, rewarding most those who contribute most to our surfeit of goods.
    Keep the capitalist procedure of requiring companies to show a profit. Profits prove they are doing what they should: providing customers with the things they want at a price that people want to pay.
    No, it is not inflationary. The stores send their receipts to their bank, and the next check is reduced by the amount still in people’s pockets.
    GeorgeRic

  • TomS

    @Matt – Thanks very much for the very nice overview. Although I’m tempted to ask one or two follow-up questions, this is probably not the place. I will search out further references on my own. If you or anyone else happens to think of a nice reference, it will be greatly appreciated.

  • Matt

    TomS—

    I’m happy to share. I’ll be sure to post a link here if I find a good reference. (Most of what I know I’ve just absorbed from the other high energy people I work with.) And I don’t think anyone would mind if you posted your follow-up questions. I’d be glad to try and answer them if I can. Maybe we’ll all learn something interesting along the way.

  • TomS

    Well, ok. (The blog leaders can snuff this if it’s inappropriate.) Take the case of a pure state (“trivial density matrix”). Then you say the Liouville equation reduces to the Schrodinger equation. I take it that the conservation of “information” requires the time evolution operator of the state to be unitary and thus we get a corresponding Hermitian “Hamiltonian” that pushes the state infinitesimally forward in time, and thus a Schrodinger type equation. My question is: does this general argument provide any details as to the form of the Hamiltonian? For example, can we see why it often takes the form of a kinetic energy operator plus a potential energy operator as is familiar from the usual presentation of QM?

  • Matt

    TomS–

    Your explanation preceding your question is perfect. And your question is likewise an excellent one.

    The answer is no. Quantum mechanics alone doesn’t tell you what specific form the Hamiltonian should take. Different systems, after all, are precisely distinguished not just by their different degrees of freedom, but also by their different Hamiltonians, so there isn’t going to be some universal principle that governs all Hamiltonians. The Hamiltonian for a spin degree of freedom will not necessarily have an obvious decomposition in terms of kinetic versus potential energy, and, in any event, it will generally look very different from the Hamiltonian for a free particle, for example.

    You need some input. In general, a system’s experimentally accessible states all consist of superpositions of low-energy eigenstates above some ground state. (This is why all practical quantum theories are really only “effective” theories.) If the ground state is Poincare-invariant, and the dynamics of the low-energy eigenstates is likewise Poincare-invariant and local (and not too strong), then the low-energy states will inevitably look like particles, as Wigner first showed. And the Hamiltonian of these particles, being the P^0 operator of the Poincare group, will be constrained to have the usual form sqrt(c^2 P^2 + m^2 c^4) + V, where V is the potential energy. In the non-relativistic limit, of course, you get the familiar Hamiltonian P^2/2m + V.

    So you need a little input. But, as I mentioned before, you also get a great many other consequences from this little bit of Lorentz-invariance.

  • Chris W.

    Back to economics: See this new post by Brad DeLong, and the spirited and articulate exchanges in the comments.

  • Chris W.

    PS: I should credit Peter Woit (on N.E.W.) for pointing to Brad DeLong’s post.

    I suspect he had string theory in mind. :)

  • http://www.shaky.com Timon of Athens

    Matt said: “Plug in a massless spin 2, and general relativity is the almost unique consequence. ”

    No, it isn’t a consequence at all, let alone an almost unique one.
    [a] You have a spin-2 field on a space with R^4 topology. Prove that this particular object is the metric tensor, ie actually measures lengths and angles. Bear in mind that there are infinitely many distinct rank-2 tensors on any manifold. Only one of them is the metric tensor.
    [b] Consider a cosmological model with topology [3-sphere] x [a line]. Derive this from spin-two particles propagating on Minkowski space. Repeat with models having the topology [compact 3-manifold accepting a metric of constant negative curvature] x [a line].

    When you can do these things, come back and claim that GR is a “consequence” of that ugly QM, including all that “measurement” nonsense. And by the way, the fact that string theory modifies GR but not QM is not the most convincing argument in the world. To put it mildly.

  • Scott

    I don’t know. Once a field is recognized as relevant, it becomes politicized. Politicians don’t listen to economists, they look over their options and pick the most self-serving one.

  • Matt

    Timon of Athens–

    “No,” you say, as though you know it for a fact. Look, I didn’t invent the spin-2 argument—it goes back all the way to the 50s and 60s to independent work by Kraichan, Deser, Feynman, and Weinberg. And it’s at the heart of why string theory has attracted so much interest, precisely because the theory must contain a massless spin-2 particle. Why do you think Witten got interested? You really think Witten is an idiot?

    Don’t you suspect even for a moment that your dispute with the argument might just possibly be because you don’t understand it? Look up their papers. For Weinberg, for example, go read his papers from the 60s on gravitons and photons.

    There’s no manifold a priori. The starting assumption is merely that the quantum-mechanical ground state of the system is, to a good approximation, Poincare-invariant, and that the dynamics among the low-lying energy eigenstates is likewise Poincare-invariant, as well as being local and weak. Then if you presume that one of these excitations is massless and has spin-2 (the masslessness is important), the claim is that its coherent states are precisely gravitational fields.

    This is in the weak-coupling regime. The system will appear to look like flat R^4, by the assumption that the ground state (vacuum state) was Poincare-invariant. Because we’re at weak coupling, gravitational fields are weak and go to zero at infinity. But there are no other spin-2 massless fields that couple to anything else. Why? Because unitarity and Lorentz-invariance mandate that a spin-2 must couple to a conserved rank-2 current. Simple S-matrix arguments show that this conserved rank-2 current must be the energy-momentum tensor. (It comes down to showing that the gravitational “charge” must satisfy a certain conservation equation that rules out nontrivial interactions unless that charge is rest-mass.) Then you can either use Deser’s boot-strap expansion or Weinberg’s gauge-theoretic construction to show that the full nonlinear extension of the spin-2 theory must governed by the Einstein-Hilbert Lagrangian, up to possible Pauli interaction term corrections and higher order terms in the curvature tensor. The geometric interpretation comes afterward, when you see that the gravitational field tensor couples to everything precisely as a metric tensor should. (So that it gives you lengths and angles.) This is old stuff, and, like I said, it’s been around for a while.

    A simpler set of arguments shows that if a massless spin-1 is among your low-energy excitations, then the theory must be a Maxwellian gauge theory, and if there are multiple interacting spin-1 massless particles, they must comprise a non-Abelian gauge theory based on a Lie algebra that closes.

    Now, you can always imagine that there are additional rank-2 fields on your effective spacetime manifold, but they cannot physically couple to anything else, because that would violate unitarity. So they have no influence on the dynamics and therefore have no physical meaning. Similar arguments explain why we don’t expect to find massless spin-3 or higher either, at least around a Lorentz-invariant vacuum; there’s nothing they could couple to that wouldn’t violate unitarity.

    That’s the story at weak coupling. At weak coupling, the field and particle descriptions are both useful. As we tune the theory to strong coupling, just like with all QFTs, the particle picture breaks down, and coherent field states become the proper degrees of freedom to employ, at least unless you can find another weak coupling regime to use. Actually, that’s what we often do. If you turn on a strong gravitational field, you might be able to find an effective ground state with different symmetry properties, and the low-energy excitations will break up according to some other symmetry group. This leads to fascinating phenomena, such as the singleton particles of AdS vacua. Witten’s review of the AdS/CFT correspondence from the 90s paints a lovely picture.

    But while quantum mechanics is widely expected to remain a part of Nature’s description at all scales, many believe based on various thought experiments that the manifold/geometry picture breaks down eventually when we start getting near the Planck length scale. One of the lovely things about string-theoretic models is we can actually see how this can happen. In string theory, spacetime is the target space of various nonlinear sigma models, and that target space can change in nongeometric ways at strong coupling. You can even model nongeometric topology transitions. It’s really cool stuff.

    GR is beautiful. QM is beautiful. I’d be happy if people would just acknowledge the beauty in both these powerful and profound theories of Nature, and not disrespect QM just because they don’t understand where it comes from. Enough with the vitriol, please.

  • chris

    Hi Matt,

    thanks for your detailed reply. it is certainly true that QM has a messy history – but (being an advocate of the historical approach in learning it) i see it as almost inevitable. and, by the way, GR was not so linear too. true, Einstein wrote a fantastic first paper that can almost be taken as a textbook, but consider this:
    a) how long did it take to establish the modern concept of black hole? or is it even established? i know of at least one extremely respectable physicist who still denies them (Veltman). Or think of the cosmological constant that famously took 80 years to be discovered and when it was it send shock waves through the physics community. so i challenge your opinion that it took longer to establish QM than GR. it just took longer to get to the core equations – which is a sign of the beauty of GR to me and possibly many others.
    b) it seems to me that you think of QM more like a principle and not like a theory. yes, you can throw in this and that and get here and there – but that even diminishes its status as a theory. it seems to me, that whay you understand as QM is just the superposition principle more or less. then you should compare that not to GR but maybe to the equivalence principle.
    c) about consistency: sure, QM with a 1/r potential is free of nasty divergences – but come on, the difficulties are hidden in the potential! i can equally point to the nice metric you can derive given a homogenous matter distribution in GR. and so what? the only candidate selfconsistent theory i know of is QCD by the marvels of assymptotic freedom. that one might qualify as beaytiful i guess – but it’s far, far away from QM (being nonlinear and all).
    d) on a related note: you quoted the natural emergence of the particle concept from QM as a triumph. i see it quite differently. the essential ingredient that gives you particles is poincare invariance, which of course emerges form the flat space solution of GR :-). the really beautiful quantum field theories are only obtained once you add at least SR to QM. yet another sign that there is still a lot of content hidden in GR that begs to be discovered (and is that not a sign of beauty?)
    e) throw in spin-2 into QM and obtain gravity? i don’t think so. 30 years of attempting this have failed so far. on the other hand, there are very promising recent developments that make it plausible that you can quantize GR along the lines of QCD (assymptotic safety scenario). it is a renormalization group argument that, if true, will give you uniquely the GR lagrangean. how beautyful this GR based unification attempt is compared to the clumsy GR=flat_space_QFT+spin_2 that has repeatedly failed over the last decades.

    it’s a surprising thing that i notice over and over in the physics community. GR – for all its merits – is treated by most physicists as a freak theory somewhat removed from the main stream. and regardless of how many unexpected triumphs it gave to bordering theories that incorporated this or that aspect into the current mainstream it still to this day is not taken seriously.

  • Matt

    chris–

    I just want to point out that the comments of Timon of Athens demonstrate exactly what I was driving at with my first posting. Things have gotten so political these days that even the statement “GR is beautiful, but so is QM” has become a source of controversy. Again, I’m not denying GR’s beauty. I’m just saying that QM is beautiful, too, and that both theories have flaws that mire their beauty in certain respects.

    I have some real work I need to get to, so this will be my last posting here. But let me address each of your points in turn.

    a) As you well know, the Schwarzschild solution was discovered in the trenches during WWI. Although realistic stellar collapse wasn’t modeled until a bit later, the key problem wasn’t with the solution per se, but its physical interpretation. The same was true for the cosmological constant, which should have been in Einstein’s equation from the beginning, and was inserted by Einstein himself early on, but was not supported by empirical evidence until much later.

    GR, as a theoretical framework, was finished remarkably quickly, and by a remarkably small number of people, even though experimental support and physical interpretation took longer to establish. And we got very, very lucky. We could have lived in a solar system without our moon and without the planet Mercury, in which case it might have taken far longer to establish GR as an experimentally defensible proposition about Nature.

    QM had a harder time. The experiments came in first, and they were all so varied and surprising that it seemed hopeless at first even to look for a single underlying set of principles. Also, the key conceptual obstacle to GR was the notion that space and time could bend and stretch, but SR had already convinced the right people that it was okay. The physics community was ready for GR when it arrived.

    QM, by contrast, violated numerous classical assumptions that many people were loath to discard, even today! Decoherence was discovered by accident by a 50s paper by Bohm (and in the context of his own interpretation of the theory), but wasn’t revived and turned mainstream until decades later. And modern information theory didn’t exist until decades after the Schrodinger equation. The experiments that brought us QM came before most people were ready, and that had a lot to do with the utter mess that resulted. The “old quantum theory” that began with Planck and Einstein’s first arguments for the quantized nature of light and the ad hoc Bohr-Sommerfeld quantization rules lingered for decades until the theory was put on the strong theoretical footing (which included the Schrodinger equation) that we recognize today. Don’t you remember how long it took just for people to agree on whether it was a wave theory or a matrix theory? It was a crazy time!

    And the reason it took so long to get the core equations, compared to GR, is because for GR, Einstein identified the key principle at the beginning. In QM, the key principle was so different from our classical experience of the world that it only came much later. For QM, the experiments came first, and only later did people realize that the framework rested on simple principles like superposition and information conservation. Who knows how things would have turned out if QM had been discovered fifty years later, when more of this conceptual groundwork was already in place? That’s why I argue that historical contingency is not an ideal way to present a subject—history is a big experiment that only has one trial, after all, and we don’t usually place so much credence on single-trial experimental runs!

    b) My point is not that QM is “just a principle,” anymore than GR is “just a principle.” My point was that each theory can be derived from simple starting principles. What comes out in each case is very interesting and complicated. If you add an additional principle here and there in either theory, you get vastly more structure. Getting huge amounts of output from small numbers of starting assumptions is part of what makes a theory elegant. But the way QM is usually presented, not starting from its underlying principles, this arrangement is obscured, and that’s a shame.

    c) I’m not saying that all of the systems that can be described by QM are simple. And, just like GR, many systems with large numbers of degrees of freedom, like QFTs, predict that they have only a finite regime of validity. But there are many, many idealized systems in QM that are perfectly well-defined. One of the first systems anybody studies is the simple harmonic oscillator, which is the height of simplicity and elegance. It’s an idealized system, just as the pure Schwarzschild solution is highly idealized. But there is no counterpart in GR of such a simple but nontrivial system that is self-consistent. And the simple harmonic oscillator appears everywhere in QM, from QFT even to string theory. That’s beautiful!

    d) We can argue whether flat space arises from Poincare symmetry or whether Poincare symmetry arises from flat space. But Poincare symmetry is vastly simpler than general manifolds. That you can start with just Poincare symmetry, even without knowing the equivalence principle beforehand, and get the equivalence principle and GR just from looking at the allowed self-consistent interactions of massless spin-2 excitations is remarkable.

    What I will also say is that if you presuppose geometry as a necessary assumption, then you are limited to the kinds of symmetry groups (e.g., isometries, certain kinds of gauge symmetries) that make sense in geometric terms. But QM is far more general. Our ground state and the low-energy dynamics can have myriads of other kinds of symmetries as well. Extended supersymmetry is one important hypothetical example that is extremely difficult to handle geometrically. (Although N=1 SUSY can be modeled nicely as an extension of manifolds to fermionic coordinates.)

    And at very high energies, the geometric picture may well break down. There’s no ultimate reason for geometry, except that it’s elegant and it works at low energies. Geometry can be tweaked, and even destroyed, without necessarily introducing logical contradictions. That’s one of the useful ideas that we get from some of the toy models we find in string theory, for example. Various nonlinear sigma models, and also matrix models, let us see just that. It’s possible to destroy geometry at very high energies and still have a model that doesn’t violate logical principles. So there’s no reason to demand that geometry must be an ultimate feature of Nature.

    But QM, by contrast, is extremely hard to modify, even at just a logical level. If you tweak it just slightly, you can end up with violations of unitarity, negative probabilities or probabilities that don’t add up to one, acausal information propagation, and so forth. That logical rigidity is one reason why so many people expect QM to be a part of whatever ultimate theory of Nature we eventually find. And the fact that you can get GR from QM and Poincare invariance (which, again, is a rather miraculous and undeservedly lucky fact) adds to the suspicion that QM is ultimately the more framework.

    e) Throw in spin-2 into QM and obtain gravity? Yes, I know so. You’re confusing classical GR with a full theory of quantum gravity. You’re correct that we don’t get the full theory of quantum gravity, valid at all energy scales, just from putting a massless spin-2 into QM, and you’re right that we’re still working on that problem after decades of effort.

    What I was actually explaining is that we get a consistent *effective* field theory for gravity, which reduces to classical gravity and some first-order hbar corrections when you work at low energies. (Donoghue has famously done some arduous one-loop calculations and worked out some of these first-order QM corrections, which turn out to be extremely tiny effects that are sometimes as small as 10^-71!)

    But getting classical gravity from the tree level QM of massless spin-2 particles is all I was ever arguing for, because classical GR is what people mean when they say GR. That’s the GR people are talking about when they talk about the beauty of GR. And that’s the theory whose beauty I was comparing with the beauty of QM.

    Of course, to get a theory that incorporates GR but is also valid in all regimes, you need to add more stuff. There are many proposals, from supergravity to string theory and beyond, and even some recent hopes that adding just a few more terms to to the Einstein-Hilbert action will result in a UV fixed point under RG flow when you include QM corrections. But all of these proposals leave QM intact, and typically modify GR, sometimes even destroying its smooth geometric foundations at high energies.

    Now to your final comments. GR is not a freak theory. It is taken seriously. It is valuable and makes lots of useful predictions about our universe. We incorporate its formalism into much of modern theoretical physics. And it’s a beautiful theory. But so is QM, and that’s my whole point.

    And it simply pains me to see that things have gotten so political that simply stating these facts is considered controversial and offensive. People are so defensive about their own fields of study that they feel this strange need to disrespect other subjects, and it really needs to stop.

    As someone who uses both GR and QM, I like both subjects and think everyone should just be a little more kind to each other and less full of bile. I see it in some of the interactions between condensed matter people and high energy people, between relativity people and particle physicists, and so forth.

    It’s like it’s not enough for a subject to be interesting and elegant. People demand that they subject they study must be the most fundamental subject in all of physics. Some condensed matter people insist that emergent phenomena are the most fundamental concept in physics. Some relativists insist that manifolds are fundamental features of physics, and anyone who challenges that metaphysical claim (a claim that has nothing to do with the triumphs of GR or its elegance) is a terrible person. And on it goes.

    There is a good kind of competition and debate in a scientific community, but much of the ill will I’m referring to here is wasteful and unproductive. It’s a source of friction, not a source of energy. It makes people intolerant of each other, and it turns off impressionable students. And I hope it changes someday.

  • http://www.dentinmud.org Dennis Towne

    I think that one very important idea has been left out of this discussion. That idea is that economics is an optimization problem, not a discovery problem.

    It’s an optimization problem because there are things that everyone desires, and there are well known ways to get each of them, but you can’t have them all at the same time. The market, and economists, do their best to be ‘good enough’ at the things deemed important.

    Unfortunately, the things deemed important are in constant flux, and there are active, intelligent operators constantly working to compromise the system. It’s not like physics where you can expect your framework to be consistent from day to day.

    If you’ve ever done any signal processing work, you’ll recognize this same issue when working with voice processing: there is no universe-mandated official vocal tract speech model to which all voices adhere. You do your best and you’re constantly working to improve it, but there will always be voices, sounds, and modes of operation that will give you bogus results.

    Following this reasoning, I find it incredibly unfair for people to think that something as complex as a global economy is going to have an obvious ‘clearly correct’ answer, and blame economists for not finding it. You might as well blame doctors for not finding a magical cure for cancer, though that would be a lot more reasonable: cancers don’t have intelligent actors working to exploit the system. Financial systems do, by their very nature.

  • Chris W.

    Matt, et al: All in all, your comments on QM and GR are really wonderful overviews, even if y’all get a bit testy at times. I’m sorry that they are off-topic relative to this post, because they’ll probably get overlooked by many interested readers as a result.

  • Chris W.

    PS: Matt, would you care to identify yourself? Just a hint, maybe?

  • http://www.shaky.com Timon of Athens

    “This is in the weak-coupling regime. The system will appear to look like flat R^4, by the assumption that the ground state (vacuum state) was Poincare-invariant. ”

    I was going to say that this, and most of your other baseless assertions, are mere hand-waving. But that would be an insult. To hand-waving.

    What is this “system” if it is not defined on a manifold? How can a “system” look like flat R^4? You’re right about one thing: assertions like this are indeed “old”. Old and tired.

    “The geometric interpretation comes afterward, when you see that the gravitational field tensor couples to everything precisely as a metric tensor should. (So that it gives you lengths and angles.)”

    Prove it.

    “This is old stuff, and, like I said, it’s been around for a while.”

    Yes, and people like Roger Penrose have been telling you for decades that there are many many problems with it. That’s why the only people who buy this junk are those with a vested interest.

    “If you turn on a strong gravitational field, you might be able to find an effective ground state with different symmetry properties, and the low-energy excitations will break up according to some other symmetry group. ”

    So this is your response to my request for a derivation of topologically non-trivial cosmological models from spin-2 excitations propagating on Minkowski space — which is all you really have. Again, we can’t really call this “hand-waving”, can we? It’s not even that. We really need a new terminology for the way you people argue. Any suggestions? “Higher-order hand-waving?” I’m not asking for a rigorous proof of any of your assertions — what used to be called a “physicist’s derivation” would be perfectly acceptable. Instead of that you just blandly produce a long string of unsupported assertions, each one of which would require years of work to substantiate. Work which nobody is doing, because proof by assertion has ousted all other kinds.

    “In string theory, spacetime is the target space of various nonlinear sigma models, and that target space can change in nongeometric ways at strong coupling. You can even model nongeometric topology transitions

    He said gleefully. And that’s the whole point of this extended legend you have re-told here for the umpteenth time: “Hey, we string theorists are *real* men! We don’t need no stinkin geometry!” The whole objective of the game is to drive out precisely that which makes GR so beautiful — the geometric aspect, the *explanation* of gravitation in terms of spacetime geometry. Make all the grandiose claims you want, just don’t tell us that anything you are doing is “beautiful”!

  • Matt

    I said I wasn’t going to post anything else, but I left my browser open to this page by mistake, and I saw this most recent posting by Timon of Athens. So it seems appropriate for me to follow up once more. But the temptation to get sucked in to these kinds of endless internet discussions has killed far too many of my waking hours over the years, so I’ll leave it at this, and hope people will read this final posting from me and my previous postings carefully.

    I think it’s useful that everyone gets to see the tone and language employed by Timon. It’s precisely what I’ve been trying to describe in my previous postings. When one’s comments include phrases like “…the way you people…” etc., it’s providing useful insight for observers into the person’s frame of mind.

    Where Timon learned that it was acceptable to address other human beings this way, I don’t know. I certainly wish him the best.

    This is the comments section to a blog page. That’s why I tried to present the arguments in an accessible way, in a manner that would hopefully be as clear as possible to an interested lay audience. That’s why I said things like “The system will appear to look like flat R^4, by the assumption that the ground state (vacuum state) was Poincare-invariant.” To get technical would, in this instance, require talking about the irreducible unitary representations of the Poincare symmetry group on vector spaces, and how they have precisely the degrees of freedom that correspond to points in a flat spacetime. (Other symmetry groups imply other kinds of structures.) You don’t have to start with a manifold in the first place.

    In general, a quantum-mechanical system does not require, a priori, a classical spacetime manifold at all. Obvious, of course, are examples of QM systems having finite-dimensional state spaces, like simple two-level systems. However, in some important cases, most especially the kinds of QM systems important when relativistic effects are important, an effective manifold structure can arise. But it’s not necessary to the logical consistency of a QM theory.

    String theory may or may not ultimately describe Nature, but as a system unto itself, it provides a test case that shows that there exist relativistic systems that can be generalized beyond the requirement of having a classical spacetime manifold. It’s a proof of principle.

    But must we? Does Nature truly allow classical geometry to break down? It’s an excellent question. But the answer is not yet known for certain. And there’s no reason to impose upon Nature the requirement that it must be geometrical, just because it’s very beautiful. (Although it is!) Nature doesn’t care for human ideology. Apart from insisting on some minimum level of logical self-consistency and agreement with known observations and experiments in accessible regimes, it’s best to keep an open mind.

    In particular, string theorists are not trying to “drive out precisely what makes GR so beautiful,” as Timon says. They’re trying to make physics consistent in a larger regime of validity in manner that is logically consistent and also consistent with the low-energy observations and experiments that we can perform today, and they’re only modifying what they have to modify. The motivation isn’t some kind of malice toward Einstein. There’s no grand conspiracy among string theorists driven by a hatred of geometry; considering how much differential geometry one uses everyday in string theory, that would seem a rather surprising attitude. And most string theorists talk about Einstein in rather glowing terms.

    At weak coupling, particle excitations can be superimposed to form what are called coherent states. For a single, 1D, simple harmonic oscillator, coherent states are the Gaussian states that survive in the noisy, classical limit, and behave precisely like a sharp, pointlike particle oscillating as would be expected classically. For systems of relativistic particles, coherent states are what we see as classical fields. Bound states involving classical force fields, for example, contribute to scattering amplitudes as infinite summations that precisely correspond to the constituent force-carrying particles giving rise to classical coherent states. The math behind this is simple and very beautiful, but not easily amenable to a text-only-interface.

    In any event, that’s a sketch of the kind of technical language one must employ to discuss some of the very simplest statements I made. I’m not going to write down technical expressions and derive formulas for the rest of it here. Timon’s request that I do so is an unreasonable argumentation tactic.

    For those who wish to see how the technical stuff goes, there are plenty of excellent resources going back many years, some of which I’ve already cited in my previous postings. The argument that a massless spin-2 excitation must give rise to a diff-invariant rank-2 traceless tensor field that couples to the conserved energy momentum tensor (and thus can only appear in the action precisely as a metric tensor) goes back a very long way, but independent discussions from the 1960s can be found in papers by Weinberg (his series “Feynman rules for any spin,” and his paper “Photons and gravitons” in particular are nice) and Deser (his completion of the infinite-series approach to deriving the full Einstein-Hilbert Lagrangian that was first started by Gupta in the 1950s), as well as in Feynman’s Lectures on Gravitation. (There are some more good authors, but their names escape me at the moment.) This stuff has been streamlined over the years and there are cleaner, more modern references. We also use these arguments frequently in papers and talks. It’s the language of effective field theory. (For a good modern example, see Donoghue’s recent papers.) but digging up more recent references is something I’ll leave to the interested reader.

    But that’s enough of this. Best wishes to everyone, and good night. We all have real work to do.

  • Matt

    P.S. One more thing to add. Timon asks earlier that I “derive” various spacetime topologies from spin-2 massless particles on Minkowski spacetime. That’s a bit like asking one to derive Stokes’ theorem from the Cauchy definition of the real line. And, as he probably knows, changing from one spacetime topology to another is a nontrivial affair in classical GR! But, all that said, we can actually do it with QM. Like I mentioned earlier, there are fascinating constructions in string theory, but we don’t even have to go that far.

    I point the interested reader, just as one example, to some recent papers on the arXiv on the Standard Model landscape that I just happened to see the other day. The model considered in that paper, in particular, is one where you compactify one spatial dimension in a universe governed at low energies by the Standard Model of particle physics, together with an effective field theory for gravity. One finds a stable vacuum with the compactified dimension having a finite radius, assuming the minimal Standard Model with certain assumptions about neutrinos.

    Whether this particular situation is ever realized in Nature is not the issue. What’s important is that in this toy model, the authors show is that there is a nonzero transition amplitude from our own uncompactified vacuum to this hypothetical compactified vacuum. (These ground states differ by having different symmetry groups. The symmetry group of a ground state that gives rise to flat Minkowski space is the Poincare group; by contrast, the ground state giving rise to what looks like a compactified spacetime, which breaks Lorentz invariance by having a definite size, has a different symmetry group, which ends up being the particular isometry group of that spacetime.)

    This sort of construction happens a lot. Witten’s “bubble of nothing” is perhaps the most adorable example, and explains why the simplest compactification set-ups of Kaluza and Klein are actually highly unstable. It’s fun stuff, and quite lovely. I suggest that curious readers check it out!

    Okay, now I’m really done. So easy to get sucked in…

    Anyway, best wishes to all.

  • Spence

    Robert Shiller, an economist at Yale actually predicted the current recession.
    http://www.yalealumnimagazine.com/issues/2009_09/shiller032.html
    The main problem is that nobody listened to him.

    In fact, another noted economist formerly at yale, now at princeton, Avidit Acharya, also predicted the current recession, but Acharya didn’t publish. Too bad.

    I think the bottom line here is this:

    You can’t get around the fact that borrowing a lot of money is risky, and the fact is that high risk debt was fraudulently categorized as a safe investment.

  • Count Iblis
  • mariana

    umm seriously? economics is not a lightweight discipline

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