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.

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.

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.

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”!

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.

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.

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.

“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.

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.

I suspect he had string theory in mind.

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.

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.