Can we prove the quantization of gravity with the Casimir effect? Probably not.

Quantum gravity phenomenology has hit the news again. This time the headline is that we can supposedly use the gravitational Casimir effect to demonstrate the existence of gravitons, and thereby the quantization of the gravitational field. You can read this on New Scientist or Spektrum (in German), and tomorrow you’ll read it in a dozen other news outlets, all of which will ignore what I am about to tell you now, namely (surpise) the experiment is most likely not going to detect any quantum gravitational effect.
The relevant paper is on the arxiv

Gravitational Casimir effect
James Q. Quach
Phys. Rev. Lett. 114, 081104 (2015)
arXiv:1502.07429 [gr-qc]

I’m here for you. I went and read the paper. Then it turned out that the argument is based on another paper by Minter et al, which has a whooping 60 pages. Don’t despair, I’m here for you. I went and read that too. It’s only fun if it hurts, right? Luckily my attempted martyrdom wasn’t put to too much test because I recalled after the first 3 pages that I had read the Minter et al paper before. So what is this all about?

The Casmir effect is a force that is normally computed for quantum electrodynamics, where it acts between conducting, uncharged plates. The resulting force is a consequence of the boundary conditions on the plates. The relevant property of the setup in quantum electrodynamics is that the plates are conducting, which is what causes the boundary condition. Then, the quantum vacuum outside the plates is different from the vacuum between the plates, resulting in a net force. You can also do this calculation for other geometries with boundary conditions; it isn’t specific to the plates, this is just the simplest case.

The Casimir effect exists for all quantized fields, in principle, if you have suitable boundary conditions. It does also exist for gravity, if you perturbatively quantize it, and this has been calculated in the context of many cosmological models. Since compactified dimensions are also boundary conditions, the Casmir effect can be relevant for all extra-dimensional scenarios, where it tends to destabilize configurations.

In the new paper now, the author, James Quach, calculates the gravitational Casimir effect with a boundary condition where the fields do not abruptly jump, but are smooth, and he also takes into account a frequency-dependence in the reaction of the boundary to the vacuum fluctuations. The paper is very clearly written, and while I haven’t checked the calculation in detail it looks good to me. I also think it is a genuinely new result.

To estimate the force of the resulting Casmir effect one then needs to know how the boundary reacts to the quantum fluctuations in the vacuum. The author for this looks at two different case for which he uses other people’s previous findings. First, he uses an estimate for how normal materials scatter gravitational waves. Then he uses an estimate that goes back to the mentioned 60 pages paper how superconducting films supposedly scatter gravitational waves, due to what they dubbed the “Heisenberg-Coulomb Effect” (more about that in a moment). The relevant point to notice here is that in both cases the reaction of the material is that to a classical gravitational wave, whereas in the new paper the author looks at a quantum fluctuation.

Quach estimates that for normal materials the gravitational Casimir effect is ridiculously tiny and unobservable. Then he uses the claim in the Minter et al paper that superconducting materials have a hugely enhanced reaction to gravitational waves. He estimates the Casimir effect in this case and finds that it can be measureable.

The paper by Quach is very careful and doesn’t overstate this result. He very clearly spells out that this doesn’t so much test quantum gravity, but that it tests the Minter et al claim, the accuracy of which has previously been questioned. Quach writes explicitly:

“The origins of the arguments employed by Minter et al. are heuristical in nature, some of which we believe require a much more formal approach to be convincing. This is echoed in a review article […] Nevertheless, the work by Minter et al. do yield results which can be used to falsify their theory. The [Heisenberg-Coulomb] effect should enhance the Casimir pressure between superconducting plates. Here we quantify the size of this effect.”

Take away #1: The proposed experiment does not a priori test quantum gravity, it tests the controversial Heisenberg-Coulomb effect.

So what’s the Heisenberg-Coulomb effect? In their paper, Minter et al explain that a in a superconducting material, Cooper pairs aren’t localizable and thus don’t move like point particles. This means in particular they don’t move on geodesics. That by itself wouldn’t be so interesting, but their argument is that this is the case only for the negatively charged Cooper pairs, while the positively charged ions of the atomic lattice move pretty much on geodesics. So if a gravitational wave comes in, their argument, the positive and negative charges react differently. This causes a polarization, which leads to a restoring force.

You probably don’t feel like reading the 60 pages Minter thing, but have at least a look at the abstract. It explicitly uses the semi-classical approximation. This means the gravitational field is unquantized. This is essential, because they talk about stuff moving in a background spacetime. Quach in his paper uses the frequency-dependence from the Minter paper not for the semi-classical approximation, but for the response of each mode in the quantum vacuum. The semi-classical approximation in Quach’s case is flat space by assumption.

Take away #2: The new paper uses a frequency response derived for a classical gravitational wave and uses it for the quantized modes of the vacuum.

These two things could be related in some way, but I don’t see how it’s obvious that they are identical. The problem is that to use the Minter result you’d have to argue somehow that the whole material responds to the same mode at once. This is so if you have a gravitational wave that deforms the background, but I don’t see how it’s justified to still do this for quantum fluctuations. Note, I’m not saying this is wrong. I’m just saying I don’t see why it’s right. (Asked the author about it, no reply yet. I’ll keep you posted.)

We haven’t yet come to the most controversial part of the Minter argument though. That the superconducting material reacts with polarization and a restoring force seems plausible to me. But to get the desired boundary condition, Minter et al argue that the superconducting material reflects the incident gravitational wave. The argument seems to be basically that since the gravitational wave can’t pull apart the negative from the positive charges, it can’t trespass the medium at all. And since the reaction of the medium is electromagnetic in origin, it is hugely enhanced compared to the reaction of normal media.

I can’t follow this argument because I don’t see where the backreaction from the material on the gravitational wave is supposed to come from. The only way the superconducting material can affect the background is through the gravitational coupling, ie through its mass movement. And this coupling is tiny. What I think would happen is simply that the superconducting film becomes polarized and then when the force becomes too strong to allow further separation through the gravitational wave, it essentially moves as one, so no further polarization. Minter et al do in their paper not calculate the backreaction of the material to the background. This isn’t so surprising because backreaction in gravity is one of the thorniest math problems you can encounter in physics. As an aside, notice that the paper is 6 years old but unpublished. And so

Take away #3: It’s questionable that the effect which the newly proposed experiments looks for exists at all.

My summary then is the following: The new paper is interesting and it’s a novel calculation. I think it totally deserves publication in PRL and I have little doubt that the result (Eqs 15-18) is correct. I am not sure that using the frequency response to classical waves is good also for quantum fluctuations. And even if you buy this, the experiment doesn’t test for quantization of the gravitational field directly, but rather it tests for a very controversial behavior of superconducting materials. This controversial behavior has been argued to exist for classical gravitational waves though, not for quantized ones. Besides this, it’s a heuristic argument in which the most essential feature – the supposed reflection of gravitational waves – has not been calculated.

For these reasons, I very strongly doubt that the proposed experiment that looks for a gravitational contribution to the Casimir effect would find anything.