Last time I had time to write a detailed physics post, I got to the point of declaring fields as the fundamental entities of nature. Since we can get particles from quantizing (certain kinds of) fields, and since the pointlike particles are confusing and paradoxical, we don’t really need them. Our basic theory of nature is then a (quantum) field theory.
Now, if you are contrarian, which I think everybody should be, you probably immediately think this is a rather dogmatic statement. We don’t really have a complete theory of nature yet, we haven’t unified gravity with quantum mechanics or explained consciousness, not to mention even finding the Higgs. As far as we know the universe may well be a computer, or it is human shaped, perhaps it is elephants (or turtles) all the way down, or angles dancing on a head of a pin. Wasn’t there also that theory of everything, something with a letter and a number? Once we get foundational and stuff, the possibilities are endless really…
The point is that it doesn’t really matter. Consider any example of continuous medium that we are all familiar with, say a rubber band or a surface of a pond. We all know these are all not really continuous, they are made of many small ingredients and the continuous description is only approximately true when we don’t look too closely. Be that as it may, if you want to describe the motion of a fluid, you’d better use the Navier-Stokes equations, rather than the quantum mechanics of the gazillion molecules making up that fluid.
Extending this logic leads to the idea of Effective Field Theory (EFT): when we don’t look too closely, anything at all will look like a field theory. Trivial as it sounds, it turns out that we can actually say quite a bit about the type of field theory we get, without knowing the precise details of microscopic physics. We can even get signs when that approximation stops making sense, without knowing much about the underlying structure. All of which makes EFT an extremely useful tool, one that dominates huge parts of modern theoretical physics, in fact.
If the description is only approximate, when do we have to replace it with something better? this is simply a matter of length scales. If we are interested in questions involving only long length scales, we are in effect averaging out all the small scale details, resulting in continuous description of the physics. If we look closer we may (or may not) see something else, which may itself be continuous or discrete (or involving turtles, or not).
For example, we describe the electrons and their electromagnetic interactions by a field theory called Quantum Electrodynamics (QED). This has to be a quantum field theory when we are interested in length scales for which quantum mechanics is important, for example when we want to discuss atomic transitions and emission and absorption spectra . It turns out that when we look at electrons at shorter and shorter distances (by smashing them together at higher and higher energies), the description by QED (or more accurately, by the full standard model of particle physics) is holding up to very high accuracy. Maybe in the future we will see some substructure, maybe not, in the meantime we call the electrons “fundamental” fields (just to tweak our condensed matter colleagues).
On the other hand, when we look at protons and neutrons- the basic ingredients of the atomic nucleus, we have a different situation. If we don’t have too much resolving power, we describe them by an EFT involving proton and neutron fields, interacting in specific way as to reproduce details of nuclear physics. However, we now do have more resolving power (also known as particle accelerators), and we know that when we probe the nucleus at even shorter distances, we find further substructure. The protons and neutrons are really made out of quarks, which interact (predominantly) via the strong force. The new structure includes quark and gluon (the strong interaction version of the photon) fields. Yes, it is still a quantum field theory, called Quantum Chromodynamics (QCD), but the fields visible at short distances are different from the ones visible when only probing long distance physics.
Now, finally coming to gravity. The classical physics of the gravitational force is explained by Einstein’s general theory of relativity (GR), one of the most beautiful and exquisitely tested theories in physics. In the center of the theory is the metric tensor, the basic field which explains the gravitational force, and describes the geometry of spacetime. There are currently no experimental results deviating from classical GR, it is certainly the correct description of gravitational physics at long distance scales.
Alas, there are many indications (which I may get to at some stage) that quantizing the gravitational field is problematic: when going down to length scales for which the quantum mechanics of gravity is important, we simply don’t know what to do. Unfortunately, this distance scale, the Planck length, is tiny. If we are not extremely lucky, we probably cannot probe this length scale directly. Nevertheless, we have now an incomplete story, full of interesting puzzles and paradoxes, and making that story coherent is the problem of quantum gravity.
So, what can happen when we go to such short distances? in other words, what is the quantum structure underlying classical general relativity? There are exactly two possibilities:
1. The description of gravitational physics by a field theory involving the metric tensor holds all the way down to the Planck length. This is similar to what happens in QED: this field theory describes the physics accurately all the way down to distance scales for which quantum mechanics is relevant. In this case, in order to describe quantum gravitational phenomena, one has to quantize the metric tensor, something we don’t really know how to do.
2. The metric field shows some substructure already at distance scales larger than the Planck scale. That substructure is what underlies the physics of classical gravity. In order to describe physics at extremely short distance scales, such as the Planck length, one has to quantize that substructure. This is similar to nuclear physics, when we are interested in very short scales we look at the quantum mechanics of the quark and gluon fields, QCD.
That is the basic dichotomy of quantum gravity (and not background independence, as is often claimed). There are many clues, about which I will write at some point, that it is the second option that is more likely. Among those clues are the myriad of conceptual and practical problems we encounter when we try to follow the first route, and the fact that nearly 80 years of attempts have not resulted in much progress. On the other hand, after countless failed attempts, we do now have in string theory a few working examples of general relativity emerging as long distance approximation to something more fundamental, lending us confidence this is indeed the right scenario.
As a result, most researchers interested in quantum gravity (almost all of whom are labeled string theorists) have abandoned the attempt to quantize the metric tensor directly. There are still a few holdouts, who take at least some features of classical general relativity seriously all the way down to extremely short distances, I hope everyone can join me in wishing them good luck in their quest.
Update: Lubos lists some of the reasons why quantizing the metric tensor directly is unlikely to work.