# Is the quantization of gravity necessary for a quantum theory of gravity?

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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You should start a new question rather than editing the original one :) – markovchain Feb 26 at 21:31

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 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 at 13:56

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 at 13:36

I am not sure, if this contribution can help you and adds new info here. It is an excerpt from notes that I took for myself. It is a prosaic collection of arguments, that I found. I wanted to motivate myself properly for the stuff, that I was studying at the end of my undergrad time, at the level of understanding that I had back then. I read somewhere else, that you are sometimes irritated by the imprecision of the physicists. Mathematical rigour will never compensate for the lack of empirical basis. I would like to encourage you to absorb as much as you can from the meta-theoretical vantage point that philosophers of science especially physics takes. Certainly, they are not physicists themselves, but they add an important ingredient tot he soup. They are unbiased and challange the physicists assumptions, convictions, ideology and so on.

A careful analysis of current fundamental theoretical physics leads to the observation that we have two mutually incompatible theories, Quantum Theory and General Relativity and together with a third framework, Thermodynamics and Statistical Mechanics, one is rather successfully able to describe a vast domain of physical phenomena.

Despite this empirical effectiveness, we have to state that fundamental physics is in a conceptual confusion regarding the above mentioned theories’ elementary and universal assumptions.

On the one side stands a manifold of Quantum Theories and Quantum Field Theories, which are used to describe the phenomena of physics on a small scale. For example, QFT is applied in the Standard Model of elementary particle physics, where it describes the fundamental interactions of the electromagnetic, weak and strong forces. On the other side is General Relativity, which is used to depict the structure of the universe on a large scale. It describes the fundamental interaction between matter and spacetime geometry by means of Einstein's field equations

Both these theories are empirically well supported within their own domains of application and description of physical phenomena but the question arises if an overlapping domain exists and if so, how we could describe and observe it. At the so-called Planck scale it is expected that an interface of both theories is needed to provide a satisfying description of the microstructure of spacetime (together with matter). A theory, which would be able to do so, is called Quantum Gravity.

As of today, there is still no complete and consistent quantum theory of gravity available and all candidates suffer from formal and conceptual problems. Their biggest common problem is the difficulty to test their predictions by experiments, since the relevant scale is (probably) too far away for current experimentation. This is certainly not a strong counterargument against research in this field, because future data from cosmological and astrophysical observations become sooner or later available and will provide the needed guidance to modify or dismiss the approaches, which we have or construct overall new ones.

Let us conduct an anamnesis of the problem of the mutual incompatibility of GR and QT.

Observe at first, that in QT one uses an external time parameter to describe the evolution of states through e.g. the Schr\"odinger equation. In QFT one assumes a fixed non-dynamical spacetime background as a part of its ontological basis, being independent of the fields, which are dwelling on it. In GR one dispenses with the notions of external time and fixed spacetime background. Instead a dynamical spacetime geometry is introduced, which is identical with the gravitational field. The coordinate time becomes a non-observable gauge variable and the physical variable called 'time', is theorized by a function of the gravitational field.

There is an interplay between this dynamical background structure and the matter fields. This leads to their interwoven co-evolution, captured by the field equations of Einstein

$G_{\mu\nu}(g)=8\pi T_{\mu\nu}(g)$.

There one observes that the left hand side deals with the spacetime geometry, which has a smooth classical structure and on the other side the energy-momentum tensor of matter fields is given. These fields are fundamentally quantum mechanical.

This stipulates that at small scales all dynamical fields and physical quantities should have quantum properties. This means that the fields are composed by discrete quanta, they follow the superposition principle and that they obey probabilistic laws. Physical quantities don't have sharply defined values in general, which is captured by Heisenberg’s uncertainty principle.

GR with its smooth Riemannian metric field and deterministic form violates this. Physical quantities are depicted by tensor fields, which usually have well-defined values. The energy-momentum tensor should be an operator, revealing an inconsistency of the formalism. One would have to replace the operator by an expectation value, depending on a fixed spacetime background, though the idea of the field equations is to have the metric dynamical.

In addition, gravity couples universally to all forms of energy. Since this energy is quantized the coupling should be so, too.

A quantum theory of gravity should therefore be capable of synthesizing both incommensurable frameworks and resolve these conceptual problems by giving a correct description of quantum geometry and matter.

Penrose and Hawking have proven that there are inevitable spacetime singularities under reasonable conditions on causality and energy, which cannot be prevented. This is articulated in the singularity theorems, referring to the assumed initial singularity of the cosmos and black holes. The consequence is, that General Relativity cannot be a valid theory without restrictions and one expects that in domains of strong gravitational fields as singularities, a theory of quantum gravity should replace General Relativity and explain e.g. the evolution of black holes. Hence, there seem to be a class of phenomena, which should be explained by means of generally relativistic and quantum effects.

Black holes radiate with Hawking-radiation, which is a hypothetical result obtained from QFT on curved spacetime. This is a semi-classical theory, which claims that a classical and quantum sector can coexist, with which one means, that the matter fields are quantum and the gravitational field stays classical. Its descriptive power breaks down e.g. at the final stage of black hole evaporation. In addition, it is well-known that QFTs suffer from UV divergences and one expects that a theory of quantum gravity, should provide a cut-off scale and thus cure these divergences.

The natural scale, where effects of quantum gravity are expected to occur, is the Planck scale, expressed by units given by Planck in 1899. The size of the Planck scale, if it is not an extrapolation, makes it extremely difficult to design experiments, which could test the few predictions of the different proponent theories of quantum gravity.

An ontological synthesis of the principles of GR and QT could lead to a general covariant quantum field theory, assuming that the ontological principles of these theories hold up to the Planck scale.

The idea, that there might be two separate phenomenological domains for GR and QT, respectively, with an empty intersection and that there is then no need for a theory of quantum gravity is however plagued by the fact that the interaction between a classical and quantum system is inconsistent.

There are also meta-theoretical arguments for establishing such a theory. For example one could expect, that a theory that unifies the concepts of GR and QT should be stronger in explaining established facts in addition to make own new predictions.

Often the argument of unification is rolled out, which is connected to the reduction of complexity and aims at finding a single coherent framework. Aiming at unity could e.g. mean the unity of nature, i.e. that the nature has a unified structure and one expects that it allows for the systematic description of the parts, which are empirically accessible, scientific method, i.e. that there is a unique way to generate scientific knowledge and/or of theory, i.e. that scientific theories should be unified concerning the theories' terminology, ontology and nomology. These arguments are certainly sufficient for the motivation to search for a theory of quantum gravity but they are certainly not enough to imply that it is necessary to quantize gravity in order to achieve this aim. However, the necessary arguments for the quantization of gravity are provided by physical theories, which was elaborated above.

One could also take the different standpoint and argue that the gravitational field does not need to get quantized. Proponents of this perspective argue that gravity should be seen as some sort of induced/emergent force, not being fundamental. One idea could be, that gravity emerges from thermodynamical considerations, being a collective phenomenon. Considering such a stance is certainly justified, if one argues that gravity seems to be a profoundly different interaction in comparison to the other three fundamental forces, described in the Standard Model of particle physics. However, ideas like this disregard e.g. that it is possible to quantize also collective degrees of freedom (like phonons) or they are ambiguous about seeing the metric field as being dynamical.

For long time there has been ongoing research on the topic of quantum gravity resulting in various approaches, like String Theory, Loop Quantum Gravity, Causal Dynamical Triangulation, Non-commutative Geometry and Causal sets and others. This manifold of approaches can e.g. be classified by choosing the relative weight of QFT and GR as a classification criterion.

Initial attempts at quantizing gravity by means of techniques, which were developed for quantizing the other interactions, tried to covariantly quantize gravity, which needed the introduction of a splitting of the gravitational field into a fixed flat background metric and a massless perturbation, yielding a linearisation of Einstein's field equations and leading to gravitational waves as their solution. One set out to quantize these waves, calling the perturbations 'gravitons' and GR appears to look like a theory for a massless spin-$2$ field, propagating on flat Minkowski spacetime.

However, it was argued that covariant quantizations of GR are not perturbatively renormalizable, standing in contrast to Yang-Mills theories. One could argue then, that this approach did not work out, since one should have considered the full metric to be quantized and in addition that a granular structure of spacetime should circumvent the problem of renormalization.

Other ideas propose to see GR as an effective theory, not being the right one at smaller scales than the ones, where it finds its strong empirical support.

Another idea is to consider the canonical quantization of GR starting from a constrained Hamiltonian system. It was argued that bringing GR into Hamiltonian form would break the manifest general covariance/ diffeomorphism invariance due to the '$3+1$'-splitting of spacetime but others disagree about this criticism. This has led to quantum geometrodynamics and more recently to LQG. LQG attempts at constructing a solid mathematical, non-perturbative and background independent general covariant quantum theory of GR. There it is believed that the theory of quantum gravity should be a quantum theory of spacetime geometry incorporating diffeomorphism invariance. The idea is to write GR in the Hamiltonian formalism of a diff-invariant Yang-Mills field theory, with compact gauge group. One does not assume an a priori background metric. Problematic with this approach is mostly the poorly understood dynamics, which motivated the development of the so-called spinfoam models. The open question remains, whether it is possible to derive a smooth spacetime in an appropriate limit from them.

The failure of current physics to establish a unique and straightforward way to a quantum theory of gravity suggests that the formulation of such a theory requires new formalisms and physics.

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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 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 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 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 at 18:35

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 at 10:40

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....