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.