Exploring Quantum Gravity: Why Quantize Gravity?

[Sumo]

I was wondering the other day why we need to find a quantum explanation for gravity? Why not just try to take the QFT and put it on a curved spacetime, and see what happens. That was how I came upon quantum field theory on a curved spacetime and semiclassical gravity.

My question is (because I'm only a first year physics student, I don't really understand a lot of this), does this way of doing things actually work? Why can we not just consider gravity as described by Einstein to be fundamental, as we consider the axioms of QFT to be fundamental?

 

[Haelfix]

You take a theory with the EH action, and try to quantize it in the usual way(s). Easy to write down, hard to solve.

Depending on the quantization method used, you run into different obstructions. One of which is the problem that the field theory in question is nonrenormalizable in the power counting sense. You can truncate the series, in which case you get semiclassical gravity (the whole qft in curved spacetime program you might have heard about), but that's expected to fail or becomes incomplete at very high curvatures or high energy densities.

Why do you need to quantize gravity in the first place? Well that's a good question. The problem is that Einsteins field equations aren't just about gravity, but also is a theory of all matter and energy (parametrized by the stress energy tensor). We know that a lot of that matter and energy in the real world is in fact quantum, so at the very least we probably want to promote the stress energy tensor into a bonafide quantum operator (eg something with a hat or an expectation value around, that lives in a hilbert space and satisfies the usual axioms), perhaps keeping the curvature tensor and ricci tensor classical.

But then you immediately run into a problem. Einsteins equations are nonlinear, and you will not be able to solve for the metric to then construct a linear evolution operator in the usual way. Quantum mechanics does not work at all with such nonlinearities, or at least, no one knows how to make it work. So you are back to having to solve the whole thing at once, which as previously stated, is a hard problem.

 

[Demystifier]

In semiclassical gravity, the metric is determined by the AVERAGE energy-momentum of quantum matter. However, the average energy-momentum mat be very different from the ACTUAL energy-momentum. The actual energy-momentum obeys the uncertainty relations, while non-quantized gravity should not obey any uncertainty relations. That makes the semiclassical gravity inconsistent. In fact, there are experiments showing that gravity is NOT determined by the average energy-momentum, but by the actual one.

 

[marcus]

... In fact, there are experiments showing that gravity is NOT determined by the average energy-momentum, but by the actual one.

Can you give us some links/details?

 

[javierR]

I was wondering the other day why we need to find a quantum explanation for gravity? Why not just try to take the QFT and put it on a curved spacetime, and see what happens.

There are the issues arising from the disjoint treatments of the stress-energy tensor and the curvature of spacetime as touched upon above. But there are other problems with (1) general relativity and (2) quantum field theory that are a common basis for searches of a quantum theory for gravity.
In general relativity there are black hole solutions, which have curvature singularities. That's technically a break-down for a physical theory: the curvature becomes larger and larger until it is not longer well-defined. So it's natural to expect that our notion of spacetime must be modified at such scales.
In quantum field theory, we work on a background spacetime, whether flat or curved. You can find special theories that are "finite", which means that physical quantities don't have infinities, and therefore are technically well-defined. But the majority of quantum field theories, including those of the electromagnetic and other interactions in our universe, are not finite. At best, they may only be renormalizable, which means the following: Since we have limited capabilities getting exact solutions in quantum field theory, we often have to work with series expansions to calculate physical quantities. "Renormalizable" means that we can make the calculation give us a finite result if we consider a finite number of terms in the series. But even if the individual terms in the series are all expected to be finite, the series may not converge to a finite result. Since the renormalization and series business involves spacetime at increasingly smaller spacetime scales, we again naturally ask whether these problems are indicating that we need to modify our notion of spacetime at those scales.

 

[Haelfix]

<Tuv> is a very formal object in quantum gravity. Taken at face value, its badly divergent (including log and quadratic divergences). Both the full thing, as well as the object that appears in a semiclassical treatment suffer from this problem.

In the semiclassical treatment, we need to renormalize the quantity to obtain something finite. This is done in several ways (see chapter 6 in Birrel and Davies). Of course its still cutoff dependant (even if we hide it in other terms) and we are ignoring higher order gravitational and matter corrections, but in principle there are several regularization schemes available to use that all give equivalent results (amongst others, Pauli -Villars, adiabatic regularization, pointsplitting etc etc)

The biggest problem is that no one knows exactly what the 'real' <Tuv> actually is, since it has never been calculated before. You can try to guess by bootstrapping it, in analogy with some treatments with the classical case, but that afaik has met with limited success.

 

[Demystifier]

Can you give us some links/details?

http://prola.aps.org/abstract/PRL/v47/i14/p979_1