In quantum mechanics a state is described by a ray in a Hilbert space, and an observable by an Hermitian operator acting on that Hilbert space. After measurement, the state collapses to an eigenstate of the operator representing the measurements. The probability of any outcome is given by the Born rule.

Erm, that might not have been all that clear. Let me rephrase that: in quantum mechanics, the state of the system represents, well, the state of the system (a rare occasion in physics when the term used makes any sense, and even has some relation to the idea it signifies). This means basically all the information specifying the contingent facts about the particular history of the system in question. This leaves only the question of accessing that information.

Famously, quantum mechanical systems are very delicate, the act of measurement tends to disturb the system, changing the state in drastic ways. The words used differ, depending on your religious affiliation (a.k.a. interpretation), for example people talk about the collapse of the wavefunction, or something even more obscure and violent-sounding involving multiple worlds, etc. . Either way, traditionally one contemplates accessing information about the system by coupling it to an external classical system. The act of measurement then translates some aspect of the state into a classical variable, for example the position of a dial, or anything tangible enough for us to access and process. Such classical measuring device is called an *observable*; quantum mechanics tells us the probabilities of getting different outcomes when using strictly classical measuring devices, and how those disturb the system they probe. Perhaps there are other types of possible measurements, we might at some point need to understand measuring devices that are not strictly classical.

So, for a physical system to represent a traditional observable, a measuring device, one has to suppress its quantum fluctuations. One necessary condition for that is that the system is macroscopic, it is made out of many ingredients. This also means the system is massive, it is heavy, at least compared to the microscopic physics it is supposed to be measuring.

Now, we can go a bit further, a *local *observable is a classical measuring device localized in space, so it probes some local aspect of the system. The more localized you want your measurement to be, the smaller the space your measuring device probes.

We have reached a fundamental trade off: local observable is a classical (macroscopic) system sharply localized in space. If that system is allowed to gravitate, it will affect the geometry of spacetime. For example, if we try to measure fine details of some spatial structure, we will need to squeeze a macroscopic measuring device into a small space. Instead of providing us with a precise measurement, the device will undergo gravitational collapse and create a horizon, preventing us from observing anything at all.

(anyone with experience with quantum gravity knows that almost all of the qualitative arguments have the same punchline, “…and then a black hole forms!”)

Observables in quantum gravity then, as long as we insist they are strictly classical measuring devices, cannot be localized in space. This is not all bad, delocalized observations can be quite useful. For example, you can try probing the system you are interested in from the outside. Suppose you are interested in the process of collapsing some distribution of matter and watching the resulting black hole evaporate, then you can situate your lab far away from that system, and just look at all the signals you can gather from it (this is the so-called Schwartzschild observer). The advantage is that you can use traditional quantum mechanics, with its sharp observables, to describe the process from this viewpoint. The disadvantage is that you cannot ask detailed questions about the internal structure and construct a local narrative of the process, or even be certain that such a narrative uniquely exists.

This description from the outside is also related to the idea of *holography*, and to duality, and to a lot of fascinating stuff which I may get to at some point. The fly in the ointment is that sometimes there is no “outside”. If we are trying to discuss the dynamics of the whole universe, which (say) is finite in its spatial extent, we need some new ideas. The problem with quantum gravity in such circumstances is very basic: we don’t know which questions make sense, never mind finding the answers.

Note to the experts, if anyone is reading: there is a slightly more technical (and more foolproof) argument along those lines involving diffeomorphism invariance. It must be equivalent to the above, but right now I don’t immediately see the connection. In any event, conversations are part of the idea of blogs, I’m told. So, help anyone?

(Thanks Sean Carroll)

**Update**: Lubos follows with interesting discussion.

on October 9, 2008 at 3:50 pmSean CarrollThanks, Moshe. This argument, which relies on gravitational backreaction in the presence of some big, heavy classical observing apparatus, seems conceptually pretty different than the argument based on diffeomorphism invariance. I understand the argument that you just gave — although it is consistent with LIGO, for example, “observing” a passing gravitational wave, and therefore the nonlocality doesn’t seem to have to be

thatnonlocal.But the diffeomorphism-based argument seems more robust, and (in versions I’ve heard) would appear to banish all possible observables to infinity. And it doesn’t seem obviously connected to heaviness, or to the value of Newton’s constant; indeed, we can always choose to formulate non-gravitational theories in diffeomophism-invariant ways if we so choose, so it doesn’t even seem to have that much to do with gravity. That’s why I’m confused.

on October 9, 2008 at 4:39 pmUncle AlDoes any quantized theory of gravitation reduce to General Relativity or evince its validated predictions? Theory is sourced by symmetries. Observables are sourced by symmetry breakings. Diffeomorphism arguments and expecially perturbative treatments exclude symmetry breakings. What can you hope to observe?

An axiomatic system is no stronger than its weakest axiom (Euclid’s parallel postulate). A simplifying postulate (massed sector isotropic vacuum) that does not obtain real world births self-consistent mathematical models that are unphysical. GR drips validations. Seek allowed exceptions (GR vs. teleparallelism).

Nobody knows if left and right shoes violate the Equivalence Principle. Pharmacology knows optical isomers fit a given receptor site differently. Screws and sockets, chemistry, crystallography, pure math… glovers and cobblers. Chirality and diastereotopic interaction based upon mass configuration are not mysteries to anybody except physicists. Somebody should look. At worst, the problem hides elsewhere. At best, Einstein’s elevator has an empirical exception and “there’s your problem.”

on October 9, 2008 at 5:22 pmMosheSean, I think the two arguments are related, though as you point out the diffeomorphism based one does not seem to have an associated length scale. But, if you are outside the action, you have a non-fluctuating background spacetime, say flat space, and that breaks diffeomorphisms, allowing you to define observables. If you are pedantic, the metric always fluctuates just a little bit (since gravity is long ranged) and everything is strictly well-defined only at infinity. However, there should be a notion of “far enough”- for example the charge of the electron, or any scattering matrix element, are also strictly well-defined only at infinity, but we do measure them at finite distance without being too worried.

Let me try to make this more quantitative later (which is probably not today…). BTW, I am curious about your statement “we can always choose to formulate non-gravitational theories in diffeomophism-invariant ways”, not sure what you mean.

on October 9, 2008 at 7:40 pmMosheAlso, look at the nice discussion in section 2.1 of this paper by Nima and friends, it has specifically an answer to the question in your second paragraph.

on October 10, 2008 at 5:51 amRobertI still think the diffeomorphism argument is not really relevant, at least in my experience I am constantly observing local gravity in a quantum world and I do not have access to spacial infinity. Of course, you should use the proper notion of locality, namely relative to the matter fields.

on October 10, 2008 at 8:27 amSean CarrollGreat, I will have a look at that paper.

Diffeomorphisms can be thought of as coordinate transformations; there’s no reason why you couldn’t write down a theory without gravity (i.e., in flat spacetime) in a perfectly coordinate-invariant way, with covariant derivatives etc. The only difference is that the geometry is specified by “Riemann = 0” rather than by some differential equation. Of course we generally “choose a gauge” by picking some particular coordinate system adapted to the flat spacetime, but that’s our choice, not a statement about the theory.

on October 10, 2008 at 8:59 amMosheRobert, we live in a world described very well by classical linearized gravity, where diffeomorphism invariance is spontaneously broken. So, the concerns regarding quantum mechanical observables (whether they are local or not) is irrelevant for all practical purposes. These are not the circumstances discussed here. Not sure what you mean by your second sentence.

Sean, I am now convinced the issue has to do with exactly what it means for diffeomorphism invariance to be spontaneously broken, which is a subtle issue. The region “outside” can be described by a classical background metric that break diff. spontaneously, allowing local observables, only if quantum fluctuations of the metric are not too large. Strictly speaking you’d really have to go to “infinity” if you want that to happen, only there the gauge invariance is truly broken (in finite volume you don’t break either global or local symmetries). However, normally you’d declare the world to be “classical enough” at some finite distance, whose value does depend on Newton’s constant. That will determine the scale of non-locality, which under normal circumstances should be tiny. Let me know if this makes sense.

on October 10, 2008 at 9:01 amMosheCrossed comments, I now understand the issue with introducing redundant variables to write any theory in a formally diff. invariant way. This is dealt with in the reference I gave, basically in that case you can get observables which are local in the physical metric.

on October 10, 2008 at 12:19 pmArunIt is a wee bit confusing this morning, Moshe, previously you wrote (I think) that gauge redundancy arises from writing some physics in a manifestly local manner. But paradoxically in quantum gravity the gauge redundancy – diffeomorphism invariance – makes at least local observables, if not locality – not exist any more.

Perhaps I need to go back and read that again.

on October 10, 2008 at 3:50 pmMosheArun, I think the two statements are consistent. In gauge theory there are gauge invariant non-local observables, in gravity all gauge invariant observables are in fact non-local. In both cases, one introduces unphysical, redundant variables to make things seem local (i.e. represent the observables as an accumulation of many local, albeit individually unphysical, effects). The way I presented it, the reason for non-locality is the back-reaction of the measurement process, nothing to do with gauge invariance, which is only a language. If you insist using the gauge variables, you are temporarily using a local language, but you are bound to find the same kind of non-locality of observables when discussing physical, language independent quantities.

on October 10, 2008 at 4:16 pmPeter MorganQFT in Minkowski space is not properly local, in view of the Reeh-Schlieder property. By measuring enough local observables in a finite region of space-time, we can tie down the state of the whole universe as closely as we like. This is consistent with local causality expressed by the commutation relation because it’s just to say, for example, that a civilization 10 light-years away, say, could be observing all emissions from earth (now, in our rest frame, they’re observing 1998) and deduce that, with probability 30%, 80%, 100%-epsilon, depending on how good their observations and modeling are, there would be a stock market crash in 2008. No messaging, just good observation of the past and modeling of the future. Better than ours.

If you “suppress quantum fluctuations”, you break quantum field theory, for which Planck’s constant is the Universal scale of quantum fluctuations.

“Famously, quantum mechanical systems are very delicate, the act of measurement tends to disturb the system, changing the state in drastic ways” is problematic. Placing and operating different measurement apparatuses does not change the density matrix that is used to describe the ensemble of observed systems (or, more instrumentally, the preparation apparatus that generates the ensemble). If you couldn’t use the same density matrix with different measurement apparatuses, you couldn’t work out what the density matrix of the observed systems is (a question of straightforward linear algebra). To paraphrase Bohr, placing and operating a measurement apparatus affects the very possibility of placing and operating other measurement apparatuses.

Yourclaim is essentially an appeal to a classical concept of action and reaction, which is denied by quantum theory (but only in the context of measurement). If you say that the effects of the (uncontrollable) quantum fluctuations of different measurement apparatuses on the same preparation apparatus/ensemble-of-quantum-objects are different, you essentially revert to classical modeling. [Note that the different changes of the density operator after the event is a separate matter; the measurement apparatus is always present in the past of the preparation apparatus, FAPP, so classically the before measurement density matrix should be different.]Erm, that might not have been all that clear. “quantum mechanics tells us the probabilities of getting different outcomes when using strictly classical measuring devices, and how those disturb the system they probe.” You’re apparently thinking in Copenhagen interpretation terms in the first part; then I take it the “disturbance” is the projection of the density matrix to the post-measurement density matrix, von Neumann style. The pre-measurement density matrix, however, is the same for all the different measurements we make. It’s OK for QM to be constructed this way, but it’s conceptually different from classical ways of thinking. Action and reaction is fundamental to classical modeling, but here QM makes a different choice.

If you introduce quantum fluctuations, and enforce action and reaction in a classical way, quantum fluctuations are much like thermal fluctuations, but necessarily different (see papers of mine, if you will, for the different Lorentz invariance properties). Changes of quantum fluctuations from place to place, and hence quantum fluctuation gradients, presumably must have a universal effect on excitations of the field away from the vacuum state, just as gravity does.

If there is no outside, I see no distinction between quantum modeling and classical modeling of just the inside. Nor will you if you can get over the Bell inequalities, Kochen-Specker, etc.

For all that, and my critique of your duality post, I like the tone of your approach.

on October 10, 2008 at 5:22 pmAaron BergmanI have to say that I never understood the argument from diffeomorphism invariance. It’s an old joke, but when someone says that there are no local observables in quantum gravity, the proper response is to throw something at them. Diffeomorphism invariance seems to be a red herring; everything is diffeomorphism invariant simply by acting on the coordinates with the diffeomorphism. What may be physical is the existence of a distinguished set of coordinates.

on October 10, 2008 at 6:27 pmMosheThere is something in common to most of the comments I am getting: it is obviously unphysical that we have to go all the way to infinity to define observables, regardless of the situation, the strength of quantum effects etc. . I think the reason is the unphysical assumption that measurements give the outcome with infinite precision (which requires a strictly classical measuring device). If we instead demand only finite precision, those more fuzzy type of observables will have scale of non-locality which is related to all the physical variables of the situation: Newton’s constant, typical action of the process observed and the precision of the measurement. I’m not sure how those fuzzy observables, corresponding to fluctuating measuring devices, could be defined. Those are probably what we need in case of compact spatial slices.

on October 10, 2008 at 11:10 pmSean CarrollI’m with Aaron. And I think, Moshe, that you are probably right, that the apparent discrepancy likely arises from the attempt to define “infinitely sharp” observables, and perhaps it could be overcome if we allowed “fuzzy” observables. But if that’s true, it seems potentially like a crucial loophole. One hears “observables in quantum gravity are only S-matrix elements,” or “are only defined at infinity.” But fuzzy observables are … observables. It might be an interesting and difficult and worthwhile technical project to define exactly what they mean, perhaps by some smoothing procedure. (The problems are obvious: it’s hard to define a “smoothing” in a diffeo-invariant way.)

on October 10, 2008 at 11:56 pmMosheSean, I am also with you and Aaron, the problem is with the formalism, not with the physics (OK, I’m just trying to avoid things thrown at me…). The formal definition of observables in QM does not correspond to actual observation, just to an idealized limit thereof. I think that the project of defining “fuzzy” observables, if it makes sense, amounts to a modification of QM, a notoriously difficult thing to do. Maybe people who worry about measurement and the classical-quantum correspondence already have some ideas.

On the other hand, I think that “exactly gauge invariant” is equivalent to “infinitely sharp”, so defining smoothing in diffeo-invariant way may not be the issue.

on October 11, 2008 at 12:14 amgsIt seems to me that the “gravitational backreaction” argument and the “diffeomorphism invariance” argument are two separate things, since the latter argument can be made even in classical GR, whereas the former depends on quantum mechanics. More specifically, if we want to probe small distances in classical mechanics, we just need to use light of ever shorter wavelengths, while keeping the energy fixed. It’s only in quantum mechanics that smaller wavelengths means higher energies, and then for sufficiently high energies “… then a black hole forms” and we can’t probe any shorter distances.

on October 11, 2008 at 1:17 amDaniel GreenHi Moshe,

I think it would make a lot of sense to make the idea precise it a less complicated setting, like a U(1) gauge theory, before getting to gravity. Can you formulate your idea for these “fuzzy observables” with charged fields there?

The reason I ask is that I’m not sure I see a problem with requiring infinity to define observables. In a gauge theory, to define a gauge invariant notion of a charged field, you need a Wilson line that runs to infinity. If the theory is weakly coupled, this Wilson line doesn’t contribute much and it looks like you charged field (to leading order in perturbation theory). If you take the coupling to zero, the wilson line doesn’t do anything and you just have a local operator.

If you put this theory on a compact space, you can’t run to infinity and this doesn’t work. But, this is just another manifestation of the fact that you can’t have a net charge on a compact space. You can join two charges by a Wilson line and you have a perfectly good operator.

Of course, these are approximately just the non-gauge invariant local operators you usually talk about, as long as the theory is weakly coupled on all length scales. The interesting part is that this also tells you how to define confinement.

I would expect you could do exactly the same thing in gravity. You might have to use infinity to make observables that are precisely gauge invariant, but they are approximately just like the local things you experience every day. The black hole argument is the same as my weak coupling requirement in the gauge theory example. You should also get back exactly local things when you take Newton’s constant to 0. In fact, one can construct gauge invariant operators with exactly this property. Ideally, there would also be some nice analogy in compact case as well.

This is the philosophy of the Nima et al. paper you mentioned as well as hep-th/0512200 as far as I can tell.

on October 11, 2008 at 3:02 amiphigeniaI am also with Aaron and Sean Carroll. I think that the business about diffeomorphisms is a source of an amazing amount of confusion, eg L Motl’s blog.

As Sean points out, *Special* relativity is diffeo-invariant. We take all of the rest of physics and declare that under all circumstances

R = 0.

where R is the curvature tensor. This is a differential equation just like G = 8piT; the only difference is that it has fewer solutions. NOTE that SR is NOT a particular solution of GR, it is a full theory in its own right; and it is just as “diff-invariant” as GR. So whatever the problems with quantizing gravity may be, they have nothing to do with diffeomorphisms!

on October 11, 2008 at 7:45 amArunIn classical GR, given two (nearby) spacetime events, the simple binary yes/no of “is either of the events within the lightcone of the other?” is coordinate-change invariant. I’m inclined to call this a local observable.

In QG, naively the classical binary yes/no turns into some kind of wave function is some kind of superposition of yes and no; and which only on measurement collapses to a definite yes or definite no. I suppose the argument is that the only apparatus that can conceivably make this measurement has too strong a backreaction.

—

Would a quantum “gravity with extra large dimensions” really have quantum effects only at the scale where the measuring apparatus collapses into a black hole? If not, then the argument against sharply defined local observables because of gravitational backreaction isn’t a great one.

—

on October 11, 2008 at 7:51 amThomas LarssonThe difference between SR and GR is that in the latter, diffeos act only on physical fields. In SR you need compensating transformations on the background metric.

Being used to get things thrown at me, I cannot resist pointing out that locality and unitarity are compatible with diffeomorphism symmetry, at least on the circle, but only in the presence of an anomaly; there are unitary Virasoro reps with h > 0, but only if c > 0.