Carlo Rovelli, one of the fathers of Loop Quantum Gravity, has just published a short, non-highly-technical overview of the conceptual issues to be faced by any theory merging Quantum Field Theory and General Relativity. I found specially enlightening his observations about time in General Relativity:
In general relativity, when we describe the dynamics of the gravitational field (not to be confused with the dynamics of matter in a given gravitational field), there is no external time variable that can play the role of observable independent evolution variable. The field equations are written in terms of an evolution parameter, which is the time coordinate x0, but this coordinate, does not correspond to anything directly observable. The proper time τ along spacetime trajectories cannot be used as an independent variable either, as τ is a complicated non-local function of the gravitational field itself. Therefore, properly speaking, GR does not admit a description as a system evolving in terms of an observable time variable. This does not mean that GR lacks predictivity. Simply put, what GR predicts are relations between (partial) observables, which in general cannot be represented as the evolution of dependent variables on a preferred independent time variable.
A deep point, easily overlooked even by experts in General Relativity.
Carlo’s article also contains an interesting bibliography for further study. I found there his A dialog on quantum gravity, a fun comparison of our two main contenders for a merger of quantum mechanics and GR. Although it is probably too jargon laden for casual readers (and, in my opinion, gives a totally biased view of the subject), the final remarks in this dialog prettily summarise a position which i tend to favour:
I think that string theory is a wonderful theory. I have a tremendous admiration for the people that have been able to build it. Still, a theory can be awesome, and physically wrong. The history of science is full of beautiful ideas that turned out to be wrong. The awe for the math should not blind us. In spite of the tremendous mental power of the people working in it, in spite of the string revolutions and the excitement and the hype, years go by and the theory isn’t delivering physics. All the key problems remain wide open. The connection with reality becomes more and more remote. All physical predictions derived from the theory have been contradicted by the experiments. I don’t think that the old claim that string theory is such a successful quantum theory of gravity holds anymore. Today, if too many theoreticians do strings, there is the very concrete risk that all this tremendous mental power, the intelligence of a generation, is wasted following a beautiful but empty fantasy. There are alternatives, and these must be taken seriously. Loop gravity is pursued by a far smaller crowd; has problems as well, as you pointed out, but is succeeding in places where strings couldn’t get, and is closer to reality. And if you think at the quantum excitations bulding up physical space, you truly see quantum mechanics and general relativity talking to one another. And is beautiful. I have an immense respect for string theorists, but I think it is time to explore something else. Don’t you think, to say the least, that both theories are worthwhile exploring?
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