# Is quantum gravity, ignoring geometry, the theory of a fictitious force?

This question is motivated by this question and this one, but I will try to write it in such a way that it is not duplicate. In short, I don't understand the motivation for a "quantum theory of gravity" that predicts a particle such as the graviton, because it stands in opposition to the entire framework of general relativity. Let me explain.

First, a personal comment: I am a mathematician with a general understanding of what GR is (I took an undergraduate course from Sean Carroll, which used Hartle's book because his own was not yet published). I know slightly less than the corresponding amount of QM, which I do understand at the level of "an observable is a Hermitian operator acting on a Hilbert space of which the wave function is an element"; I know nothing technical about QFT, though I believe that "a particle is a representation of the gauge group" is true. However, I do have a popular-science understanding of elementary particles (at the precise level of Asimov's Atom, which I read a lot in high school).

Now for the framing of the question. GR in a nutshell is that that spacetime is fundamentally a 4-manifold with a metric locally isomorphic to Minkowski space and that particles are timelike geodesics in this manifold. The force of gravity only appears as such when these geodesics (i.e. curves) are parametrized in a coordinate patch, where "geodesic" and "coordinatewise unaccelerated" are different concepts; the apparent acceleration seems to reflect a force (via Newton's second law) that we experience as gravity. This phenomenon is identical to the appearance of a centrifugal force in a rotating coordinate system where none exists in a stationary one (it is identical in that this is a particular instance of it).

I understand that the standard model is formulated in a flat spacetime, and its use for the hypothetical graviton is to mediate the force of gravity. I also understand that quantum gravity is a (shall we say) paradigm in which the standard model is made consistent with GR, and that the hallmark of a theory of quantum gravity is that it predicts a graviton. But this seems to be barking up the wrong tree: you cannot remove the geometry from GR without changing it entirely, but you cannot consider gravity a force (and thus, have a corresponding quantum particle) without removing the geometry.

It does seem from this question that I have oversimplified somewhat, however, in that some theories, which I had thought were not very mainstream, do treat gravity in an "emergent" manner rather than as an interaction.

My question: What I want to know is: rather than pursuing a "quantum theory of gravity", why not pursue a "gravitational quantum theory"? That is: since the formulation of standard quantum physics is not background independent in that it requires a flat spacetime, it cannot be compatible with GR in any complete way. At best it can be expanded to a local theory of quantum gravity. Why is it that (apparently) mainstream opinion treats GR as the outlier that must be correctly localized to fit the standard model, rather than as the necessary framework that supports a globalization of the standard model? One in which the graviton is not a particle in the global picture, but a fictitious object that appears from formulating this global theory in local coordinates?

PS. I have seen this question and this one, which are obviously in the same direction I describe. They discuss the related question of whether GR can be derived from other principles, which is definitely a more consistent take on the unification problem. I suppose this means that part of my question is: why must Einstein's spacetime be replaced? Why can't the pseudo-Riemannian geometry picture be the fundamental one?

Edit: I have now also seen this question, which is extremely similar, but the main answer doesn't satisfy me in that it confirms the existence of the problem I'm talking about but doesn't explain why it exists.

PPS. I'd appreciate answers at the same level as that of the question. I don't know jargon and I'm not familiar with any theoretical physics past third-year undergrad. If you want to use these things in your answer you'll have to unravel them a bit so that, at least, I know what explicit theoretical dependencies I need to follow you.

• The first question you link is the key one. For Einstein's equation to make sense we have to either quantise the left side or classicalise the right side. It's hard to quantise gravity but at least we have some promising avenues (string theory). The prospect of converting QM to an underlying classical theory seems exceedingly unlikely. Commented Jun 5, 2014 at 7:37
• @JohnRennie: "Dear President Nixon: Your tenure has witnessed the rise of the VHS tape and string theory as a promising avenue to quantum gravity. Which one do you think will be persued longer?" :) Commented Jun 5, 2014 at 9:11
• @NikolajK: I mean no comment on string theory other than at least it's a known avenue (even if it eventually turns out to be a dead end). Any ideas for reformulating QM are gratefully received - I know of none that have entered the mainstream. Commented Jun 5, 2014 at 9:14
• @JohnRennie According to the first answer there, the issue with the literal Einstein equations with $T_{\mu\nu}$ quantized is that it then "obeys quantum mechanics" while the Einstein tensor does not. It then presents two options, and I'm advocating the first one, and I'm also claiming that the formulation implicit in the second one (quantize the Einstein tensor) is actually a local one that ignores geometry, and therefore, doesn't cut to the heart of the matter and can't be ultimately a generalization of GR. Why not do "QFT in curved space"? Commented Jun 5, 2014 at 17:33
• @RyanReich: this isn't my area, but as I recall if you use the expectation value of <T> you get the wrong answers if the matter is in a superposition of states. I think this is what this Wikipedia article is getting at. Commented Jun 5, 2014 at 17:44

First of all, there is a quantum field theory on a curved background, even though it is not perfect. There are problems with global definitions of spinors, vacua, particle numbers etc. and this all seems to be a consequence of the core properties of GR such as no privileged definition of time or "global God observer".

But the main issue of the theory is that the quantum fields are acted upon by the geometry but don't act back which is heavily against Machian principles. There is a quantum stress-energy operator $$\hat{T}^{\mu}_{\;\;\nu} = \frac{\partial \hat{\mathcal{L}}}{\partial (\partial_\mu \phi)} \partial_\nu \hat{\phi} - \delta^\mu_{\;\; \nu} \hat{\mathcal{L}}$$ But you cannot just put this equal to the left hand side of Einstein's equations $$R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R$$ Because that is a set of completely different objects. You can obviously distill a natural number from the stress-energy operator by taking it's expectation value and this gives you semi-classical gravity.

Nonetheless, the expectation value flattens out a whole spectrum of information, so the most natural thing is to try to convert the left hand side to match the quantum operator richness of the right hand side. Then you get quantum gravity.

We could argue whether gravity is treated as a force by quantization or not, but for a down-to-earth physicist that is just wordplay. When a violent cosmic event sends of a gravitational wave your detectors will vibrate upon the arrival of the wave whether you understand it as a geometric or a field effect.

When you start with a given background in classical GR and perturb it, you will get several types of deformations and certain outgoing and ingoing oscillations of the metric representing weak gravitational waves. Due to the traditional formulation of QFT concerning itself mainly with scattering events with asymptotic outgoing and ingoing states, you will also concentrate on these in quantum gravity.

You can call these perturbative ingoing and outgoing quanta "gravitons" but also "quantized gravitational wave-packets". You can call an eventual interaction of this excitation of the metric with other fields as "the graviton decaying into a positron-electron pair" or "the conversion of energy carried by the gravitational wave into the energy of a specific spinor wave by a quantum process".

The scattering picture of QFT is so stressed because it is quite well understood both intuitively and analytically, and it also has the best experimental underpinning. But this whole popular picture of flying particle-balls crumbles when you start to study non-perturbative and non-scattering effects (such as bound states). A partial understanding can then be obtained via the language of states, superposition and so on. But to be honest, I don't believe anyone really understands how does the full-fledged QFT interaction add up into something like the quark-antiquark condensate in a proton.

I believe that in a certain sense it is more useful to forget the particle picture for understanding a proton. I.e. "there are no quarks but a a certain quantum configuration of the respective fields." In the same way the quantum-gravitational field around a black-hole can be understood as "no gravitons but a certain quantum configuration of the pseudo-Riemannian geometry". So I think your intuition is basically in agreement with the current theoretical-physics picture, it's just not clear through the popular-physics portrayal.

• Let me comment that Mach's principle is false. Space-time is neither entirely absolute (it has no point) nor totally relational (it can discriminate when one is accelerating, see Newton's bucket en.wikipedia.org/wiki/Bucket_argument). Einstein actually failed to create a Machian theory. General relativity is like Newton's theory, it has an absolute inertial structure and always knows when you accelerate. It is therefore not surprising that the action of acceleration on the physics is non trivial and cause lots of troubles in order to define things like number of particles, etc etc.
– sure
Commented Dec 18, 2014 at 19:20
• @sure Yes, we could argue about the fulfillment of Mach's principle. But this answer invokes a weaker Machian requirement which is that the geometry is a dynamical variable adequately formed by everything happening in the arena.
– Void
Commented Dec 18, 2014 at 20:27

It seems to me that people have been deluded by the copenhagen interpretation of QM. I'm very excited that pilot-wave theory has resurfaced in light of experiments with dot waves and walking droplets. It seems very clear to me (tho I should mention I'm not a professional physicist) that a particle bouncing on space-time in a 4th dimensional direction is a way to easily unify quantum mechanics and gravity. The spinning geometry you mention is what would in fact enable the particle to be bouncing in the first place.

Someone mentioned that you either have to make GR quantized, or make QM classical. Pilot wave theory does the latter.

• Great article. Thanks for providing the link. It won't surprise me at all if an incorrect theory lasted for decades all because someone's personality was stronger or better timed than another person's personality. It's happened before in science. For some reason, I've always believed empty space was fluid-like in some fashion. Commented Jan 23, 2015 at 0:48