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The other day in my string theory class, I asked the professor why we wanted to quantize gravity, in the sense that we want to treat the metric on space-time as a quantum field, as opposed to, for example, just leaving the metric alone, and doing quantum field theory in curved space-time. Having never studied it, it's not obvious why, for example, the Standard Model modified to work on a space-time with non-trivial metric wouldn't work.

The professor replied in a way that suggested that, once upon a time, this was actually a controversial point in the physics community, and there was a debate as to whether one should head in the direction of quantizing the metric or not. Now, he said, the general consensus is that quantizing the metric is the right way to go, but admitted he didn't have time to go into any of the reasons that suggest this is the route to take.

And so I turn here. What are the reasons for believing that in order to obtain a complete and correct quantum theory of gravity, we must quantize the metric?

EDIT: I have since thought about this more, and I have come up with an extension to the original question. The answers already given have convinced me that we can't just leave the metric as it is in GR untouched, but at the same time, I'm not convinced we have to quantize the metric in the way that the other forces have been quantized. In some sense, gravity isn't a force like the other three are, and so to treat them all on the same footing seems a bit strange to me. For example, how do we know something like non-commutative geometry cannot be used to construct a quantum theory of gravity. Quantum field theory on curved non-commutative space-time? Is this also a dead end?

EDIT: At the suggestion of user markovchain, I have asked the previous edit as a separate question.

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Possibles duplicates: physics.stackexchange.com/q/6980/2451 , physics.stackexchange.com/q/10088/2451 and links therein. –  Qmechanic Jan 25 '13 at 22:12
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You should start a new question rather than editing the original one :) –  markovchain Feb 26 '13 at 21:31
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6 Answers 6

Same as @SamRoelants answer but not restricted to gravitational waves. Given $$G_{\mu\nu}=8\pi T_{\mu\nu}$$ $T_{\mu\nu}$ is constructed from the matter fields (Klein Gordon, Dirac or whatever). These are operators (or operator-valued distributions if you like), hence so is the gravitational source $T_{\mu\nu}$. So the right hand side obeys the rules of quantum theory, with all its machinery of superposition etc.

There seem to be two options: (1) take the expectation value of the right hand side and use that to define the LHS. This is a "QFT in curved space" approach. (2) accept that the LHS is quantized too i.e we need a quantum gravity theory.

Quantum matter and classical gravity just doesn't fit.

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I've since added to my original question, and I would be interested if you have anything additional to say. –  Jonathan Gleason Jan 26 '13 at 13:37
    
Re the edit: In string theory (which I have only a poor understanding of), I think gravity is a force just like the other three. There has been some thought ( physics.stackexchange.com/questions/33428/…) given to how spacetime itself could emerge from string theory, but I have to defer to the experts to explain it. –  twistor59 Jan 26 '13 at 13:56
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Not in any way a hard proof, but here's the intuition my thesis supervisor once told me.

Imagine for a moment we had a firm grasp on gravitational waves (the fact that we can't produce them is only a technical hindrance, but they're part of Einstein's theory none the less.) This would allow us to use a new kind of way to probe quantum phenomena: we probe using gravitational waves (rather than electromagnetic radiations or electrons etc...).

Now you reach a conflict: either you say the gravity wave doesn't couple to any other fields (good luck convincing yourself of that, the metric field couples to every field in your Lagrangian), or you somehow need the gravitational field to be quantized, in order to be consistent with the rest of the already quantized fields in the standard model.

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I've since added to my original question, and I would be interested if you have anything additional to say. –  Jonathan Gleason Jan 26 '13 at 13:36
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Jonathan, the extension to your question makes your original question far more clear. Your ideas are in line with what many people have thought and are thinking about quantum gravity. My take on this is this: whether there is need to touch the metric or not, depends on whether one takes GR (General Relativity), or some extensions of it, to be the fundamental theory of gravity or not at the quantum level. Lagrangians with higher powers in the curvature tensor R, or high order derivatives, for example, where of the former type, and the hope was that they would ‘soften’ the divergences of their quantum versions. These models treated the metric as fields of the theory. However, they were abandoned mainly because their spectrum contained non-physical excitations – ghosts in particular. String theories and the loop quantum gravity are theories of the quantum-geometrical type you are suggesting. They don’t take GR to be the fundamental theory at the high energy scale, but the hope is that GR will be recovered at the low energy limit of these theories. Neither of them do this convincingly at the moment. Also, it remains to be seen whether the spectra of sparticles/particles predicted by these theories, actually make any physical sense! I hope this adds some more clarity to this absolutely fascinating discussion.

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I've since added to my original question, and I would be interested if you have anything additional to say. –  Jonathan Gleason Jan 26 '13 at 13:37
    
I disagree with some points: Einstein s vacuum equation are intrinsically built into ST, so there is actually no problem in recovering GR from ST. In addition, quantum gravities are intended to explain physics at the quantum gravity (Planck) scale, so any effects observable at our accessible energy scales would be a highly welcome bonus, but the absence of such effects in at present doable experiments would by no means be sufficient to disqualify any of the available quantum gravity theories. –  Dilaton Jan 29 '13 at 0:05
    
Well, you have said it yourself differently: "Einstein's vacuum equations are built into ST, ..." In other words ST do not 'prove' GR at the low energy scale. I did not disqualify any theory in what I said. Theories usually disqualify themselves by failing to make contact with experiment. As the great Richard Feynman said in the character of the physical law : "...if the theory is not verified by experiment, it doesn't matter how beautiful it is or what your name is ..." –  JKL Jan 29 '13 at 22:37
    
@John: Einstein's vacuum equations are a consequence of string theory's assumptions. They are not 'built in' in the sense of being assumed. –  user1504 Feb 27 '13 at 18:35
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I view the problem as follows. We know that both general relativity and quantum field theory are tremendously successful at describing our world in certain limits. Given this observation, it seems natural to conjecture that there should exist an underlying theory, which in the relevant limits should be able to reproduce both GR and QFT. Given that the metric contains the natural degrees of freedom of GR and that QFT are quantum theories, one can expect that in such a unifying theory, the metric should in some way be quantized.

The only theory currently able to encompass GR and QFT is string theory, which is a theory of quantum gravity. However it is certainly not constructed by "quantizing the metric", and I do not really understand why your string theory teacher said that there is a consensus that quantizing the metric is the right thing to do.

In its perturbative formulation, it is constructed by quantizing a 2-dimensional quantum field theory living on the string worldsheet, and the Einstein equations arise as consistency conditions for this QFT. In the two non-perturbative formulations which are AdS/CFT and Matrix theory, non-gravitational systems (a QFT and a matrix model) are quantized, and in certain limits one can show that they are approximated by classical gravity.

The fact that string theory does not proceed by "quantizing the metric" is also obvious from its history. It was developped for a completely different purpose, namely describing strong interactions. People had much trouble getting rid of an annoying spin 2 particle until they realized that it could be interpreted as the graviton in a theory of quantum gravity.

As far as I am aware, direct approaches to quantizing the metric allow to define certain quantum theories, but so far it cannot be shown that these theories have semi-classical saddle points correponding to smooth space-times obeying the Einstein equations. So nobody really know if they are really quantizing gravity or doing something else.

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Perhaps I was unclear about what my professor had said. I don't think he meant that the starting point for string theory is 'promoting' the metric to a quantum field, but rather, at the end of the day, this is how you wind up thinking of the metric. To be quite honest, I'm still in the course, and so quite inexperienced, so I expect my interpretation of his meaning is not so accurate. I would suspect that anything that does not make sense is a result of my ignorance, not his. –  Jonathan Gleason Feb 28 '13 at 10:40
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This preprint by Carlip might be of interest.

http://arxiv.org/abs/0803.3456

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An idea I have long had and might eventually start working on (actually, I just did) is based on the bad taste that «quantisation» à la Kostant, Vergne, and Souriau left in my mouth long ago. Now it used to be said that first quantisation is not even defined (So let's toss out Kostant and Souriau) but second quantisation is a functor. But there are foundational reasons for thinking that QFT is fundamentally wrong: just another useful asymptotic approximation like the Law of Large Numbers.

The first reason is that quantum measurement suggests that the axioms about observables are mere approximations. (So we can toss out Irving Segal too.) Furthermore, no satisfactory relativistic theory of measurement has ever been accepted. QFT avoids the whole issue, but then if observables are not fundamentally physical, why should algebras of observables be any better? Why should operator-valued fields be any better? So I no longer worry about renormalisation or QFT: a useful approximation can have divergences when one tries to apply it to some situation outside the range of validity of the approximation, without that amounting to a foundational crisis (This is what people in Stat Mech learned long ago, in fact it is practically a quote from Sir James Jeans's poo-pooing the whole H-theorem controversy). Which rather undermines the main motivation for string theory, too...

The second reason is that a quantum field, like a classical field, assumes there can be an infinite number of harmonic oscillators. But we've weighed the universe so there is a top energy level. And the effective universe is finite in size, so Planck's Law suggests there is also a minimum energy level. So there are only a finite number of harmonic oscillators in the Universe, that number is bounded (for a given time-slice), and so every Hilbert space is finite dimensional and every spectrum is discrete, just like my physics teacher told us all long ago. («Now remember, every particle is a particle in a box.») So we can toss out Reed and Simon too. (There might be something wrong about my using Planck's law and a finite effective size of the Universe...)

You say, but there are no finite dimensional irreducible unitary representations of the Lorentz group with dimension bigger than 1. But Gen Rel makes that less important, does it not?

Therefore the arguments that go back to Bohr and Rosenfeld about using non-quantised gravity to probe quantum systems is not so decisive: it is a proof by contradiction, but if their use of observables and measurement axioms can only be considered approximate, there is no longer anything decisive about their contradiction.

Non-commutative geometry rests on that whole Dixmier--Souriau thing, so toss Alain Connes, too.

Fifty years from now, all this Quantum Gravity thing will look like the luminiferous ether looks to us today.

The real obstacles to reconciling Quantum Theory with Gen Rel are bad enough without imagining phony obstacles. Bell sensed it and worried about it: Quantum Mechanics lives on phase space, but Relativity of any kind (special or general) lives on configuration space, i.e. space--time. (four dimensions, not 2^256...) (That was one advantage of QFT: it returned to actual space--time...) I feel that even so, the most promising approach is still to take Quantum Theory and make it generally covariant (and this might not involve anything much worse than Yang-Mills theory), rather than start with Gen Rel and «quantise» it. But even if one could overcome these difficulties, there does not seem to be any practical way to experimentally confirm such a theory without going into cosmology, where the observed facts are hardly as precisely established as the advance in the perihelion of Mercury was....

I would be curious to know what Alan Guth would have to say about this.

Assume that the Universe was one spherically symmetric particle in its ground state...

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