View 1: SCG as a Mean-Field Description in Low-Energy Quantum Gravity

The first reading holds that classical GR succeeds under ordinary circumstances because they reside in a regime in which it provides a good approximation to an as-yet unknown fundamental theory of QG (which need not be a final theory of physics). More particularly, models of GR are taken to be “mean-field” solutions of the unknown theory, and quantum perturbations around those solutions are taken to provide an effective field theory (EFT) for the underlying unknown theory: A ubiquitous implementation of this EFT is quantized linearized general relativity.Footnote ⁴ This picture is discussed and its many concrete applications described in, among other places, Reference BurgessBurgess (2004) and Reference WallaceWallace (2022), both of who call it “low-energy quantum gravity” (LEQG).Footnote ⁵ Their point (also made by Reference Crowther and LinnemannCrowther and Linnemann [2019]) is that such a theory is both empirically successful and constitutes a quantum theory of gravity in any reasonable sense: indeed, in just the sense that we have a quantum theory of the electromagnetic, weak, and strong interactions (viz., a UV-incomplete EFT of some higher energy physics).

In the LEQG framework, SCG amounts to the low-energy limit of the gravitational EFT, when quantum fluctuations in the gravitational field may be ignored. For instance, in the case of quantized linear gravity, Reference Hartle and HorowitzHartle and Horowitz (1981) derive (2.1) as the lowest order quantum matter corrections, in the large $N$ limit of $N$ gravitating quantum systems (a fact to which we will return). But one also expects SCG as a limit on more general grounds than that derivation: Sakharov’s “induced gravity” program (Reference VisserVisser 2002) begins with the observation that (2.1) holds in the limit for any theory that dynamically couples a Lorentzian metric to a quantum field.

Within the approach described by Burgess and Wallace, SCG then solves the problem of incorporating quantum matter into classical dynamical spacetime theory, provided that we restrict attention to an appropriate regime in LEQG.Footnote ⁶ (Indeed, one could take SCG as showing that there was no real “problem” in the first place.) On this picture, given the strength of the gravitational interaction relative to others, one would expect in this regime leading order corrections to the expectation value of stress-energy, for a given state of matter, to come from quantum fluctuations of the matter field itself, and not from gravitational fluctuations (at least away from strong curvature). Then, even though one expects deviations from classical gravity in the long term because of the nonlinear character of (2.1), for short durations of time, it is sufficient to model such a system in terms of accumulating effects of back-reaction by matter corrections on the spacetime curvature, which would otherwise determine the left-hand side of the equation. Such a modeling program is known as “stochastic gravity,” and has been successfully developed (see, e.g. Reference Hu and VerdaguerHu and Verdaguer [2008]). As noted by Reference WallaceWallace (2022, p. 39), stochastic gravity may be derived from LEQG, indicating no tension between the present view of SCG and the expectation that stochastic noise due to the quantum nature of matter influences the effective classical description of gravity.

What is crucial to this view is that gravity is understood as fundamentally quantum in nature, and only effectively treated as classical, that is, for the purposes of approximation. This approximation is summarized by the dictates of SCG, interpreted in terms of LEQG, perhaps improved with corrections from stochastic gravity.

View 2: SCG as a Candidate Fundamental Theory of Gravity

According to a second possible view, (2.1–2.2) constitutes an alternative to string theory, loop quantum gravity, and so on for a fundamental theory in its own right, not approximation. This view is not taken as a serious possibility by the QG community, yet there have long been efforts to rule it out definitively: Reference Huggett and CallenderHuggett and Callender (2001) critique theoretical arguments and also the experimental approach that we discuss later. Likely, that this view is not taken seriously in part reflects the fact that it is beset by mathematical difficulties. Namely, it is not clear that any spacetime and quantum field could simultaneously satisfy both equations. One can define a QFT satisfying (2.2) in a given curved background spacetime (and perhaps solve the equations), but once one has, likely its stress-energy will not satisfy (2.1) in the background. One might then introduce a new background for which (2.1) is satisfied, but now (2.2) will likely not hold, and a new QFT must be defined. And so on. As we say, perhaps the equations can be solved simultaneously, perhaps the process described even converges on a solution (and perhaps merely approximately so), but it is far from sure.

Still, physicists are no strangers to mathematical difficulties in the course of theoretical research. Arguably, what is more conceptually troubling is the disunity involved in accepting that some parts of the world are classical while others are quantum even at the most fundamental level. Moreover, there are difficulties contemplating what even some basic physical models of such a theory would look like, beyond maybe the vacuum sector. For these reasons, in conjunction with the mathematical difficulties, it is perhaps not surprising that this view is given little credence by physicists. At the same time, it is considered worth eliminating as a live possibility.

Moreover, it is not even clear that Eqs. (2.1–2.2) adequately capture what is claimed: a meshing together of classical gravity and quantum matter, as we currently understand them. Recall in view 1 that corrections due to matter fluctuations, as studied in stochastic gravity, plausibly are relevant to the gravitational properties of a classically gravitating quantum system, so that it is a virtue of that view that LEQG recovers familiar stochastic gravity techniques. As just stated, view 2 categorically denies the role for any such corrections. The result is that there is rampant loss of physical information in coupling the material quantum system to gravity: After all, expectation values are insensitive to all higher-order correlators in the QFT.

View 3: SCG as A Mean-Field Description of $X$

Now, taking SCG as an approximation does not in itself commit one to the view that SCG is an approximation to LEQG; perhaps the low-energy approximation to a fundamental theory of QG is not LEQG, but some other theory, so that SCG is a “mean-field description” of that. View 3 is thus an epistemically careful deviation from view 1: SCG is a mean-field description but one stays uncommitted to the specific microstructure $X$ . Now, there are few serious advocates for such alternatives, but some have been mooted. First, Reference Carney, Stamp and TaylorCarney et al. (2019) discuss a toy model in which a mesoscopic “ancilla” continuously weakly monitors the fluctuations in the microscopic quantum fields, feeding the inevitably noisy record of those fluctuations directly into a classical gravitational interaction channel.Footnote ⁷ More prominently, thermodynamic derivations of GR proposed by Reference JacobsonJacobson (1995) and Reference PadmanabhanPadmanabhan (2014) render the degrees of freedom as collective ones; such derivations are often taken by these authors and the wider ‘emergent gravity’ community as suggestive that GR’s degrees of freedom are therefore not to be quantized. (Though, we note, such a strong conclusion seems too quick, given that the quantization of collective degrees of freedom can sometimes result in a better approximation, as is arguably the case for sound waves whose quantization leads to phonons.) And a final non-LEQG viewpoint on SCG, which is often taken as expressing a similar view of spacetime geometry as collective degrees of freedom, is Sakharov’s induced gravity (Reference VisserVisser (2002) for a review): SCG is seen as a consequence at lower energies for any sufficiently rich quantum field theory scenarios, just provided that one assumes a dynamical metric at lower energies. Of course, such results supporting the induced gravity program do not rule out SCG being an approximation to LEQG, but they do indicate a (conceivable) universality of SCG: that it could arise in a suitable limit in many conceivable theories.

As the range of examples illustrate, the point is not to advocate for a particular theory (contrast this with LEQG in view 1). Rather, the coexistence of the different examples demonstrates that one could, for the time being, remain neutral on the exact relationship between QG and SCG. This would be to accept that SCG approximates some theory $X$ , which in turn approximates fundamental QG (or even something wilder), but meanwhile to take no stance on whether $X$ is LEQG, or some theory in which gravity is classical, or some alternative, considered or unconsidered.

In the discussion so far, introducing view 3 may seem unmotivated; what is the point of singling out this particular epistemic stance? The vast majority of theorists seem to adopt view 1, for instance. One theme through the remainder of this Element is that the differences across these three different views of SCG, given our current best physics, really matter to how we understand experiments in tabletop quantum gravity. Take for instance the Page and Geilker tabletop experiment to be presented in the next section. We will show how the extremely strong interpretation of the experimental results given by the original authors, in contrast to the typically deflationary attitude found in response to those results, reflects the difference between SCG view 1 and SCG view 3, where each is understood as opposed to SCG view 2. Getting clear on the Page and Geilker experiment in this manner will later pay dividends in clarifying the terms of the disagreement regarding the significance of the proposed GIE experiments.

3.1 COW

Consider first the COW experiment (named after Colella, Overhauser (1974) and Werner (Reference Colella, Overhauser and WernerColella et al. 1975)): Neutrons pass through a beam splitter and the different components of their wavefunction follow paths at different heights in the Earth’s gravitational field, producing a relative phase shift between the components, which can be observed as interference on recombination. Our discussion of this scenario will largely be within NRQM, but it can also be modeled covariantly to little difference, as we shall briefly see. As with all the cases we discuss, the question of whether the effect is quantum is delicate.

The basic setup is shown in Figure 3.1, with the Earth understood to be at the bottom of the diagram: The neutron beam enters from the source S at left, is split at A, with the two components following lower and upper paths and recombining at D, with interference due to any relative phase of the components observed at T.Footnote ⁸ How do we now calculate the resulting shift?

Figure 3.1 Schematic diagram of the COW neutron interferometry experiment

Consider a nonrelativistic description first: Let us suppose that the neutron beam is well described by a plane wave, $\psi (r,t)=C{e}^{-i(pr-Et)/\hslash }$ as dictated by the Schrödinger equation, so that the phase is $pr-ET$ in $\hslash =1$ units (here $r$ is the distance along a path). Then one might suppose that the effect is the result of CD and AB being at different heights in the Earth’s gravitational field, and hence corresponding to different gravitational potential energy (GPE) for each component along those path segments. But this would be a mistake. First, neutron energy is of course conserved, with changes in GPE canceling with changes in kinetic energy (KE). (Note that in the experiments with gravcats discussed later, the KE will be zero, and differences in GPE can affect the phase through the $Et$ component.) Of course, a neutron will decelerate along AC and hence take longer to traverse CD than AB, and so the lower path ABD represents less time than the upper path ACD, even though the energy is the same along both. Does this lead to a phase shift of $E\cdot (t_{ACD}-t_{ABD})$ ? No. Because the part of the wave reaching D along the upper path ACD takes an extra $t_{ACD}-t_{ABD}$ to get there, it must have left S a time $t_{ACD}-t_{ABD}$ earlier than the wave along lower path, in order for them to arrive simultaneously. Hence they would have been out of phase on emission by exactly $-E\cdot (t_{ACD}-t_{ABD})$ , exactly canceling the difference picked up around the path.Footnote ⁹ In other words, the $Et$ part of the phase contributes no relative phase, and the whole observed effect is due to the $pr$ part of the phase. Before we go on to calculate this quantity, note that when considering the deceleration of neutrons due to gravity, we have treated them as classical particles. Indeed, the entire calculation of $pr$ will proceed in this way, with implications for the quantum nature of the phenomenon.

The original CO (Reference Overhauser and ColellaOverhauser and Colella 1974) prediction is based then on lower momentum along CD than AB, because of gravitational deceleration along AC. Basic mechanics tells us that a particle of mass $m$ and initial vertical speed $u$ will have a final speed $\sqrt{u^2-2gh}$ after traveling a distance $h$ ; or to first order in $g$ , a speed of $u-gh/u$ . Assuming that the reflection is perfect, the phase change along CD is thus $m(u-gh/u)h$ , while that along AB is simply $muh$ since no deceleration is involved. The momenta along the vertical legs should be the same, and so CO predict a relative phase shift of $mg{h}^2/u$ , which is indeed observed.Footnote ¹⁰

However, since CO use classical particle considerations to calculate the motion of the neutron beam, they should do so consistently, for instance, taking into account that paths will not be horizontal but parabolic as the neutrons fall during their flight: They reach the plates not at B and D, but slightly lower. This perturbation affects the time, magnitude, and direction of the paths, and at first order so must be accounted for. CO reference this effect, but the analysis of Reference MannheimMannheim (1998) shows that they do not correctly compute it: Indeed, he shows that there is no relative phase along the closed part of the paths! Instead, the particle considerations that CO invoke, when applied consistently, reveal a different source for the observed interference – which happens to agree with their original computation! (With hindsight, the coincidence is perhaps not a great surprise given that $mg{h}^2/u$ is the natural dimensionless quantity in the problem.)Footnote ¹¹

Though the details of Mannheim’s calculation are somewhat off the main line of this Element, the bottom line is, first, that there is no phase difference at all between a point on the splitter and the recombining screen along the two paths. However, second, a point on the wavefront at the splitter will not reach the same point on the recombining screen; rather, wavefront points that are slightly offset on the splitter reach the same point of recombination. Finally, since the splitter is not perpendicular to the neutron beam, such wavefront points will travel different distances to the splitter, and so already be out of phase at the splitter, completely accounting for the effect!

One might thus conclude that the effect is not gravitational at all! But this would be too hasty: Although the phase changes along the paths cancel, their explanations involve two quite different combinations of speed and trajectory shifts, and the Earth’s gravity is the cause in both. Indeed, in the actual experiment, when the apparatus is rotated about a horizontal axis through its plane, so that the “vertical” paths are no longer vertical, and the horizontal paths change their relative heights, the phase changes no longer cancel, and the observed relative phase changes.

As an aside to our main topic, we note that observing a phase shift $mg{h}^2/u$ amounts to a measurement of the neutron mass, since the size of the experiment and the speed of the neutron are known. This is remarkable, because in the vicinity of the equivalence principle is the idea that it is impossible to measure the mass of a body using purely gravitational effects – but that is exactly what is accomplished by the COW experiment.Footnote ¹² Indeed, one might therefore worry that the experiment reveals an incompatibility between the equivalence principle and NRQM. However, reflection on the equation of motion, the Schrödinger equation, of course reveals the identity of gravitational and inertial mass that constitutes the Newtonian equivalence principle: The same mass $m$ appears in the kinetic term that governs inertial motion, and in the potential term that determines the gravitational force. (The same points hold, mutatis mutandis, in a covariant analysis (Reference MannheimMannheim 1998, p. 57).)Footnote ¹³

What though does the experiment show about the relation between Newtonian gravity and NRQM? Clearly the effect is quantum in the sense that in the $\hslash \to 0$ limit the neutron is purely classical, and has no phase at all. Additionally, in Mannheim’s analysis, the effect depends on the finite spatial (and temporal) spread of the wavefunction. However, our question regards the nature of the interaction of gravity with quantum matter, the neutrons in this case. So a more relevant consideration is that the GPE has no direct effect on the phase, as one might have naively thought: As we discussed, the energy is conserved (and the time from source to target is irrelevant) so we find no $Et$ component of the phase. Moreover, the calculations of $p$ and $r$ , which do contribute to the phase, depend on purely classical properties of the neutrons: their trajectories determined by Newtonian mechanics in a constant gravitational field. So in that regard, gravity and matter are related purely classically in the experiment. (Note that all these considerations – except the finite spatial extent of the wavefunction – apply equally to CO’s analysis.)

As we mentioned earlier, we have focused on an NRQM model of the experiment, in order to simplify a fairly complex situation. However, Mannheim also outlines a covariant model, which it is worth briefly discussing to see that it produces the same result, and also as a warm-up for later calculations. First, for the metric describing the Earth’s gravitational field we use the weak-field approximation to the Schwarzschild solution to the Einstein field equation:

$d{s}^2=(1+2V(\vec{r}\left)/c^2\right)d{t}^2-d{\vec{r}}^2,$(3.1)

where $V\left(\vec{r}\right)$ is the Newtonian potential, in this case $-gx$ when the Earth is the source of the field. Timelike geodesics then satisfy $\ddot{y}=\ddot{z}=0$ , $\ddot{x}=g$ , so that in this approximation neutrons will follow the same paths that we computed in the Newtonian analysis. Moreover, treating the neutrons as excitations of a Klein–Gordon quantum field reveals that the phase change is again dependent on the momentum along each path. Hence overall, Mannheim’s Newtonian calculation of neutron interference applies equally to the covariant treatment.

That said, the covariant situation is different, insofar as the effect also has an entirely classical realization, unlike a Newtonian analysis. This is because covariantly light will follow null geodesics, and not the straight and vertical paths expected in a Newtonian treatment. Remarkably, the very same considerations that apply to massive particles apply to massless classical waves: Interference is predicted for a classical light beam COW experiment, for exactly the same reasons as for the quantum neutron experiment. Thus, one can envision a gravitational Michelson–Morley experiment to observe the bending of light in the Earth’s gravitational field!Footnote ¹⁴ Thus, there is a purely classical realization of the effect covariantly, though not in the Newtonian treatment.

Since one expects the analog of the COW experiment even in this classical system, it seems that the COW experiment’s credentials as an observation of the quantum nature of gravity are at best weak. But looking at things in this way misses the main point of the experiment. Instead, one is interested in treating the gravitational interaction channel as opaque: to be studied with increasingly sophisticated quantum probes, in terms of the effects registered in those probes. In doing so, one hopes that physicists might develop good experimental control over the gravitational behaviors of material quantum systems that are increasingly poorly described as classical, providing new knowledge (a “causal manipulation” kind of knowledge) that is relevant in the course of theoretical research on the problem of QG. Here, for instance, one may study the dynamics of individual quantum probes isolated from their (known) quantum environments, but which are nonetheless subject to gravity for the duration. Later, we will evaluate GIE experiments similarly, as expanding the range of experimental quantum probes of gravity. So, simply to emphasize this sense of continuity between COW and GIE, we speak of a “control tradition.”

3.2 Page and Geilker

The second experiment is that conducted by Reference Page and GeilkerPage and Geilker (1981), and reported as “indirect evidence for quantum gravity.” So begins, in contrast with what we have just said about COW, the tradition of experiments in tabletop quantum gravity that claim to search for increasingly sophisticated witnesses of the underlying quantum nature of gravity.

The Page and Geilker experiment is a modification of the famous Cavendish experiment. In the original, two smaller test masses are placed on the ends of a hanging torsion balance, and a larger source mass is placed close to each, but on opposite sides of each end: The gravitational attraction between the test and source masses is thereby measured by the deflection of the balance. If the weights are placed in, say, position A in Figure 3.2, then the balance will deflect counterclockwise. In the authors’ modification, quantum mechanical radioactive decay is used to randomize between two classical configurations: the weights at A exerting a counterclockwise torque, and the weights at B a clockwise torque. Since radioactive decay is a quantum process, Reference Page and GeilkerPage and Geilker reason that in each run the weights will be in a superposition of classical positions A and B. The expected position of the weights in such a state is between the two positions, co-located with the test masses, and so SCG – which depends on the expected distribution of matter – predicts that no net gravitational force is exerted on those test masses, and the torsion should vanish in every run.

Figure 3.2 Schematic diagram of the quantum Cavendish experiment viewed from above. A and B represent the two possible locations of the weights; on each run, the actual position is decided by the reading of a Geiger counter

Of course, it was no shock that when Page and Geilker performed the experiment, they observed instead that the balance always deflected, with equal frequency in each direction: Although SCG predicts the correct expected displacement, it is wildly incorrect with respect to the individual runs. As Reference BallentineBallentine (1982) dryly puts it, “a less surprising experimental result has scarcely, if ever, been published.” (The observed result is of course compatible with an entirely different setup, one in which the source weights are randomly prepared in each trial in just one of the two classical configurations. So if the experiment is to rule out SCG, it is crucial that the choice of position is determined quantum mechanically.)

In the first place, Page and Geilker intend that the experiment serves as a refutation of SCG, construed as a candidate fundamental theory of gravity, as in view 2. As such, the result is of course unsurprising – we do not expect macroscopic source mass positions to coherently superpose! (Thinking of our earlier discussion of the Møller–Rosenfeld equation specifically, we expect the torsion in any one run to depend on the position of the sources in that run, not their average position over a series of runs.) However, caution is merited. As the experiment depends on the outcomes of individual runs rather than a statistical average for an ensemble, one simply cannot interpret the experiment as a test of SCG without some answer to the “measurement problem” in mind.

Indeed, Page and Geilker assume an Everettian interpretation of quantum mechanics (QM): The experiment is a massively open system, since the sources are placed in position by a (human) experimenter who observes the outcome of the decay. Thus, instead of being in superposition, the A-placement and B-placement branches decohere. Such decoherence makes no difference to the expected positions of the weights, so the SCG prediction is still no deflection. The observed deflections are understood instead as a random (with respect to the Born Rule) sampling of branches over an ensemble of experiments, in each of which the deflection is toward the position of the weights in that branch. Granted then, the experiment refutes SCG as a candidate fundamental theory, given an Everett interpretation.

Of course, standard interpretations of quantum mechanics predict the same outcome if SCG fails. As Reference WallaceWallace (2022, Section 5) points out, if an interpretation allows only unitary dynamics, then it will appeal to decoherence in the way just described. While if an interpretation invokes some kind of collapse, real or effective, it will occur when the experimenter measures whether a decay occurred as part of the procedure, “collapsing” the system into one of the branches. Thus, no one would plausibly expect a macroscopic superposition of the source masses in the experiment, so at the macroscopic scales relevant here, classical descriptions of (fundamentally) quantum physics suffice to capture, in each run, the gravitational dynamics of the experiment.

However, for this very reason, interpretations that go beyond unitary dynamics may also predict the same outcomes even if view 2 of SCG is correct. For example, since collapse interpretations such as that of Reference Ghirardi, Rimini and WeberGhirardi, Rimini and Weber (1986) entail a collapse once the decay (or its absence) is observed by the experimenter, they predict that the source masses are never in superposition: So, the expected mass distribution is always either in position A or in position B. Similarly, a hidden variable theory, such as Bohm’s, could trace a definite configuration to some uncontrollable – hence effectively random – epistemic uncertainty in the initial state, leading to “effective collapse” (Reference Dürr, Goldstein and ZanghiDürr et al. 1992, Section 3.2). In other words, the implications of Page and Geilker’s experiment are sensitive to the interpretation of QM: According to the Everett interpretation, it does serve to rule out SCG, while for collapse or hidden variables it may not do so. One would like to close such loopholes; the experiments with gravcats to be discussed later would achieve this goal (and maybe rather more).Footnote ¹⁵

Page and Geilker not only take – according to their Everettian lights – the experiment to refute view 2 of SCG, they also take it to provide indirect evidence for the quantum nature of gravity. This point can be readily understood in terms of Bayesian updating of probabilistic degrees of belief (“credences”), a framework on which we will draw at several points. It is a trivial theorem of probability that the probability of a hypothesis $H$ , conditional on a possible piece of evidence $E$ is given by $p\left(H|E\right)=p\left(E|H\right)p\left(H\right)÷p\left(E\right)$ . Bayesian updating proposes that if $E$ becomes known to be true – if, that is, an experiment or observation produces outcome $E$  – then, on pain of irrationality, one’s credences should change so that $p_{new}\left(H\right)=p\left(H|E\right)$ , the “posterior (to the updating) credence.”Footnote ¹⁶ Now suppose that there are a number of hypotheses $H_i$ , all but one of which, $\bar{H}$ , entail $E$ ; while $\bar{H}$ entails $\neg E$ . Then suppose that outcome $E$ is observed, so that $p_{new}\left(\bar{H}\right)=0$ . Because probability is conserved, $p\left(\bar{H}\right)$ must be redistributed over the remaining hypotheses; from the theorem (since $p\left(E|H_i\right)=1$ ), this will be in proportion to $p\left(H_i\right)$ , the “prior” for $H_i$ . The standard picture is a quantity of sand evenly spread over a collection of boxes of different sizes; if one box has to be emptied, its contents are spread over the other boxes, in proportion to their sizes.

This situation, roughly speaking, describes the Page and Geilker experiment. If it rules out view 2, then the prior probability gets spread across the remaining alternatives including view 1: The quantum nature of gravity is confirmed in the sense that the credence for SCG as a mean-field description in LEQG increases.

As we explained, whether this conclusion holds depends at least in part on attitudes toward quantum measurement, and so whether view 2 is refuted.Footnote ¹⁷ But even supposing that it is, how seriously one takes the refutation of view 2 to count as meaningful evidence for the quantum nature of gravity – say, substantive shifts between prior and posterior credences – also depends on downplaying the possibility of some alternative to LEQG producing SCG in a low-energy limit, namely view 3. On this understanding of SCG, the observed classicality of the gravitational interaction within the individual runs of their experiment would not necessarily mean that gravity is fundamentally quantum. Since the prior for view 2 is spread across views 1 and 3 in proportion to their priors, then the degree to which view 1 is confirmed depends both on one’s prior for view 2 (how likely it is that SCG is the final word on QG), and the relative priors for views 1 and 3 (how much of the available probability accrues to each). One can then understand the muted reactions of physicists to Page and Geilker’s paper. Whether one already has a low prior for view 2, or has comparable priors for views 1 and view 3, or adopts a collapse interpretation of QM, not much is to be learned from performing the experiment.

4.1 What Makes a Witness

Describing such a “gravitationally induced entanglement” (GIE)Footnote ¹⁸ experiment, Marletto and Vedral write: “the entanglement between the positional degrees of freedom of the masses is an indirect witness of the quantization of the gravitational field” (our emphasis): Such experiments are distinct from (and far easier than) “direct” observations of the quantum nature of gravity, for example, from the observation of individual gravitons or their effects, perhaps some gravitational analog of the photoelectric effect.Footnote ¹⁹

Let us remark briefly on the “direct/indirect” distinction, as the GIE literature seems to be at cross purposes on this point. Namely, Marletto and Vedral’s classification of the experiment as indirect (also Reference Carney, Stamp and TaylorCarney et al. [2019]) is at odds with the discussion found in the supplementary materials of Reference Bose, Mazumdar and MorleyBose et al. (2017), where one reads: “it is fair to say that there are no feasible ideas yet to test [QG]’s quantum coherent behavior directly in a laboratory experiment. Here, we introduce an idea for such a test $\dots$ .” Likely, this clash is terminological. For instance, Reference Christodoulou and RovelliChristodoulou and Rovelli (2019, p. 65) cite both as supporting the same conclusion: “As emphasised by Bose et al. and by Marletto and Vedral, the main reason for the interest of the experiment is precisely to provide direct evidence that gravity is quantised” (our emphasis). So perhaps there is consistency across these claims, but some competing intuitions about how to draw a direct/indirect distinction, with the notion of “witness” absorbing tension between direct evidence and a lingering sense that the results of the GIE experiments would still somehow be “indirect.” One philosophical inclination is, of course, to attempt precisification: definitions of the terms that are consonant with familiar uses, or intuitions of “direct/indirect” in other physics contexts (e.g. Reference FranklinFranklin [2017]; Reference ElderElder [2022]).

However, after our unsuccessful attempts to obtain a useful clarification of the terms, it is our conclusion that there is simply insufficient “data” on how they are used in the context of GIE. Meanwhile, as we aim to show, the most interesting philosophical question concerns the nature and possibility of disagreement over the significance of a positive GIE result, and that disagreement does not at all turn on the question of a direct/indirect distinction. For instance, none of those quoted question that it would “witness” the quantum nature of gravity; and none of those disputing the latter would be placated by being told that the witness was not “direct,” but merely “indirect.” For this reason, to clean up actors’ terms, we will here set aside the “direct/indirect” language, and assert for the sake of moving forward: All players mentioned intend the experiment as a “witness” of the quantum nature of gravity. We maintain that mere stipulation of this kind does not diminish the task at hand, namely, of evaluating the stakes of the new experiment from within the witness tradition; as we shall soon see, there is a real question of whether it would be a witness at all.

As for “witness,” the term itself comes from an analogy with the term “entanglement witness” from quantum information theory: An observable acting on a given tensor-product system, whose expectation value for an entangled state of the system is incompatible with the range of expectation values for a separable state (Reference VedralVedral 2006). Measuring such a value in the laboratory then constitutes a “witness” of entanglement, in contrast to a measurement of Bell-inequality violating correlations (the latter closer to an observation of entanglement itself). Analogously, in a GIE experiment instead of observing quantum behavior of gravity itself, observed entanglement (itself perhaps observed via an entanglement witness) is incompatible with purely classical gravity. In both cases, the broad idea is to measure a proxy for the feature of interest. Still, we prefer an even broader understanding of the term, divorced from technical notions: empirically accessing a specific (theory-laden) feature of the world. It is this totally nontechnical use of the term, which licensed us previously to identify the Page and Geilker experiment as the beginning of the “witness” tradition in tabletop quantum gravity experiments.

We will return to the question of what to make of GIE experiments beyond witnessing in Section 7. For now, with the previous discussion in mind, let us turn to a first, naive pass at the experiment itself, to get the basic ideas in hand.

4.2 A Naive Account of the Experiments

In the experiment, two identical gravcats are prepared in position superpositions along the x-axis. Let us say that the first gravcat is the sum of Gaussians located at $-D\pm \Delta$ , while the second is a sum of Gaussians located at $+D\pm \Delta$ ; let us set $D\underset{˜}{>}\Delta$ . Given their macroscopic size, we can take the gravcats to have no initial motion along the x-axis. Moreover, provided that the duration of the experiment is short, the gravcats will not have time to accelerate due to mutual gravitational attraction; they are effectively stationary. The arrangement is shown in Figure 4.1, and is described by the following state (ignoring overall normalization):

$\begin{array}{cc}\Psi \left(0\right)& =(|L⟩+|R⟩)\otimes (|L⟩+|R⟩)\\ & =|L⟩\otimes |L⟩+|L⟩\otimes |R⟩+|R⟩\otimes |L⟩+|R⟩\otimes |R⟩,\end{array}$(4.1)

with $|L⟩$ representing the left wavepacket of gravcat 1 in the first tensor product slot, and the left wavepacket of gravcat 2 in the second slot, and so on.

Figure 4.1 The two gravcat arrangement (the central packets are closer than shown)

Because the distances between the packets of the two gravcats are different, different terms in (4.1) will have different gravitational potential energy (GPE), and so will develop relative phases.Footnote ²⁰ For example, for $|L⟩\otimes |L⟩$ the GPE will be $G{m}^2/2D$ , while for $|L⟩\otimes |R⟩$ , it will be $G{m}^2/(2D+2\Delta )$ . Since the GPE is the only contribution to the energy (kinetic energy is zero), these different potentials will produce a different phase in each term according to the Schrödinger equation: $\Phi \left(t\right)=e^{-iEt}\Phi \left(0\right)$ for energy eigenstates such as these (recalling that we set $\hslash =1$ ). By linearity then:

$\begin{array}{cc}\Psi \left(t\right)=e^{\frac{-iG{m}^{2}t}{2D}}|L⟩\otimes |L⟩+e^{\frac{-iG{m}^{2}t}{2D+2\Delta }}|L⟩\otimes |R⟩& +\\ e^{\frac{-iG{m}^{2}t}{2D-2\Delta }}|R⟩\otimes |L⟩& +e^{\frac{-iG{m}^{2}t}{2D}}|R⟩\otimes |R⟩.\end{array}$(4.2)

Or, since $D\approx \Delta$ , for short times ( $t\ll 2D/G{m}^2$ )

$\Psi \left(t\right)\approx |L⟩\otimes |L⟩+|L⟩\otimes |R⟩+e^{\frac{-iG{m}^{2}t}{\delta }}|R⟩\otimes |L⟩+|R⟩\otimes |R⟩,$(4.3)

with $\delta =2(D-\Delta )$ the separation of the closest pair in (4.5). This wavefunction does not factorize except for special values of $t$ (and neither does the un-approximated wavefunction (4.2)), and so the mutual gravitational attraction “induces” entanglement between the gravcats. It is this entanglement that is claimed to provide an indirect witness of the quantum nature of gravity.Footnote ²¹ (In contrast to such an experiment, note that the COW experiment involves “gravitationally induced interference.”) All of this is still beyond current technology, but not so far beyond that experimentalists aren’t attempting to develop existing techniques to make such a measurement, or one along similar lines.Footnote ²²

Now, given our description of the experiment, the claim that observing such entanglement witnesses the quantum nature of gravity should seem puzzling: After all, it appears that we simply appealed to classical gravity, just as in the COW experiment (whether by their lights or Mannheim’s). Yet, Reference Colella, Overhauser and WernerColella et al. (1975) did not claim that the neutron interference which was observed provided a witness of the “quantum nature” of gravityFootnote ²³ – indeed, we even suggested that the experiment as self-described ultimately represents work in a different tradition of experiment, one that instead emphasizes control. How is the present appeal to classical gravity relevantly different? Of course, in the COW experiment, the source of the field was considered as classical, namely the Earth; while in the GIE experiment it is quantum, the gravcats themselves. But that is a point about the gravcats, not gravity. (The same, of course, can be said of the sources in the Page and Geilker experiment, noting that gravcats remain in coherent superposition.)

Unpacking this issue is a major task of the remainder of the Element. We will see how different theoretical starting points can lead one to different conclusions. For instance, the naive account tacitly assumes a bipartite state of two gravcats, acting directly on each other, through an interaction Hamiltonian

$\hat{H}=\frac{G{m}^2}{|{\hat{x}}_1-{\hat{x}}_2|},$(4.7)

in which gravity appears as a classical potential (except, of course, it is, formally, an operator).Footnote ²⁴ But the claim that the experiment would witness the quantum nature of gravity requires that it is something about the quantum nature of whatever the classical potential in (4.7) actually describes that ultimately explains the predicted outcome. What we will now work out explicitly is that whether or not one takes this view – namely, whether the ultimate explanation for the predicted outcome of the experiment concerns facts about an underlying gravitational field, or (just) the classical potential, fashioned into an operator – amounts to a metatheoretical choice of modeling framework, or “paradigm” in a “lite” sense (compared to various stronger senses also found in the literature): No rational argument alone can make the proponent of one paradigm swap to the other (Section 5.3).

To foreshadow: We will present two paradigms relative to which one can understand the GIE experiments. The first paradigm starts from principles of our best current physics to vindicate the naive account: That is, it turns out not to be so naive after all. We shall see that there is at least one sense in which a nonlocal interaction may be seen to fall directly out of the fundamental field dynamics of GR, within the context of these GIE experiments. In this paradigm then, gravcats do not witness the quantum nature of gravity. The claim to witness comes within a second paradigm, spurred by nearly 200 years of physics, including relativity, telling us that bodies do not act at a distance on one another, but rather their interactions are mediated by fields (cf. Reference HesseHesse [2005]). Exactly how to understand the gravitational field is of course equivocal, and indeed here it will mean different things: In some contexts it means the metric field of GR, in others the spin-2 massless field that appears in the linearization of GR on a Minkowski background. But at the simplest level, to treat gravity as itself a dynamical system responsible for mediating between bodies is to model the experiments with gravcats as tripartite, rather than bipartite (as in the naive model), with the field joining the two gravcats in the labeling of the full quantum state.

However, before discussing further how claims that gravcats witness the quantum nature of gravity are dependent on the paradigm adhered to, in the next subsection we will briefly rehearse an uncontroversial yet significant consequence of the GIE experiment: namely, that it would rule out semiclassical gravity, as understood on view 2 in Section 2, in such a way as goes further than the QM interpretation-dependent refutation claimed by Reference Page and GeilkerPage and Geilker.

4.3 Ruling Out Semiclassical Gravity

Quite simply, the observation of post-experiment gravcat entanglement would be in direct experimental conflict with semiclassical gravity, defined by Eqs. (2.1–2.2), taken as a fundamental theory of nature, because such an interaction cannot result in entanglement. (As far as we are aware, the only other phenomena purporting to show the same thing are the two that have already been flagged: the controversial Page and Geilker experiment and the cosmological theory-laden explanation mentioned earlier of CMB structure by fluctuations in an inflaton field.) Let us explain.

To simplify matters, we work in the Newtonian limit of Eq. (2.1), in which the stress-energy of each gravcat is merely its mass density, $m|\psi \left(x\right){|}^2$ ; for each gravcat, a mass density zero almost everywhere, but with equal small Gaussian peaks at the locations of the two wavepackets. The effect of such a distribution is (Reference Anastopoulos and HuAnastopoulos and Hu 2014, Section 1.2)Footnote ²⁵ to introduce a potential into the Hamiltonian of the form:

${\hat{H}}_{gravity}=-G{m}^2\int d{x}^{\prime }\frac{|\psi \left(x\right){|}^2}{|x-x^{\prime }|}.$(4.8)

Inserting this term into Eq. (2.2) yields what is known as the “Newton–Schrödinger equation.” This approach has often been criticized, and while not plausible, arguably has not been refuted empirically; hence it remains to date as a possible theory of quantum matter and classical gravity. However, we can readily see that witnessing entanglement in the proposed way would amount to direct empirical evidence against the theory.

Since gravcats only interact gravitationally, ${\hat{H}}_{matter}$ is merely the kinetic energy, which vanishes for stationary particles. Hence, assuming that the experiment does not last long enough for an appreciable change in velocity, Eq. (2.2) will only introduce phases according to this gravitational potential, which we can see will not produce entanglement. Suppose that the potential, implicit in (4.8), in which the left packet of the first gravcat sits is $V_{L1}$ , then for an initial state $|L⟩$ the time-dependent Schrödinger equation dictates that the state is given by:

${\psi }_{L1}=e^{-i{V}_{L1}t}|L⟩,$(4.9)

and similarly for the other packets. Then, by linearity, if the initial state is again (4.1), then the Schrödinger equation yields as the time-dependent state:

$\begin{array}{cc}\Psi \left(t\right)& =e^{-i{V}_{L1}t}|L⟩\otimes e^{-i{V}_{L2}t}|L⟩+e^{-i{V}_{L1}t}|L⟩\otimes e^{-i{V}_{R2}t}|R⟩\\ & +e^{-i{V}_{R1}t}|R⟩\otimes e^{-i{V}_{L2}t}|L⟩+e^{-i{V}_{R1}t}|R⟩\otimes e^{-i{V}_{R2}t}|R⟩\\ & =(e^{-i{V}_{L1}t}|L⟩+e^{-i{V}_{R1}t}|R⟩)\otimes (e^{-i{V}_{L2}t}|L⟩+e^{-i{V}_{R2}t}|R⟩),\end{array}$(4.10)

so that the wavefunction factorizes, and there is no entanglement according to SCG.

It’s easy to understand why: Because it is determined by the expected matter distribution, the gravitational field is the same in each of the four terms; thus, in any term, the potential acting on any gravcat depends only on its position, not that of the other gravcat; so the $|L⟩$ (and $|R⟩$ ) state of each gravcat picks up the same phase in any term in which it appears, as can be seen in (4.10). But then it is a simple mathematical fact that the state remains factorizable.

Hence the tabletop experiments offer an unequivocal empirical test of the quantum nature of gravity insofar as they rule out a classical treatment in the form of SCG, if gravcat entanglement is observed as hypothesized. According to LEQG and view 1, this failure is quite understandable; as Reference Anastopoulos and HuAnastopoulos and Hu (2014, p. 5) note (and we mentioned earlier), SCG is a good effective theory in the limit of many bodies, and the gravcats are simply outside that regime. (It’s worth noting that SCG fails to explain CMB structure for an entirely different reason, namely that contributions from quantum fluctuations must be included as corrections to the mean-field approximation in order to match the empirical data.)

That said, a real puzzle remains about the enthusiasm of the community for carrying out the experiment, given that it will require considerable effort and expense. After all, it is almost universally believed in the community that view 2 of SCG is in fact false (not withstanding the co-authorship of Reference Bose, Mazumdar and MorleyBose et al. (2017) by members of the quantum collapse community), so appears hardly worth refutation! (Similarly for any view that holds that the gravitational field remains classical.) Equivalently, experimentally ruling out SCG cannot meaningfully increase one’s credence that gravity is indeed quantum. We will return to this question in Section 7.

5.1 The Newtonian Model

We have already seen that, on a naive account, the Hamiltonian (4.7) will produce entanglement, without assigning quantum states to the gravitational field. As we noted, such a view treats gravity as a direct, instantaneous interaction at a distance; if one takes seriously finite relativistic propagation, and indeed the dynamical nature of the gravitational field in GR, then such an interaction cannot be allowed strictly speaking. But one might come at things from a rather different starting point, and conclude that the naive account is vindicated. Here, the thought is that the treatment of Newtonian gravity as a direct, instantaneous action-at-a-distance interaction, as in the naive model, is merely an apt stand-in for the “true” (more on this momentarily) physical degrees of freedom, given that the experiment is assumed to take place entirely in the Newtonian regime. Specifically, Reference Anastopoulos and HuAnastopoulos and Hu (2014, Section 3.1) start with the GR action for a scalar field

$S[g,\varphi ]=\frac{1}{8\pi G}\int d{x}^4\sqrt{-g}(R-\frac{1}{2}(\nabla \varphi {)}^2-\frac{1}{2}{m}^2{\varphi }^2),$(5.1)

where $R$ is the Ricci scalar, and $\nabla$ the covariant derivative. They take a 3+1 Minkowski background, assume linear perturbations, and work in the Hamiltonian framework. They make a standard gauge choice – the longitudinal component of the metric perturbation and the transverse components of its conjugate momentum both vanish – to obtain the Hamiltonian

$H=\int dr{H}_{KG}-G\int dr\int d{r}^{\prime }\frac{ϵ\left(r\right)ϵ\left(r^{\prime }\right)}{|r-r^{\prime }|}+\dots$(5.2)

where $H_{KG}$ is the field action in Minkowski spacetime, and $ϵ\left(r\right)$ is the energy density of the field. In the static situation of the GIE experiment, the matter field has no kinetic energy, and the mass-energy contributes only an irrelevant overall phase, so the first term is ignored. The second term yields the Newtonian potential once the nonrelativistic limit is taken, but at this stage of the analysis can be seen to represent, in the chosen gauge, the first-class “scalar” constraint of GR. Finally, the higher-order terms indicated by ellipses include the contribution to the Hamiltonian from the transverse-traceless components of the gravitational field, the so-called “true” degrees of freedom in the chosen gauge; quantizing these yields gravitational quanta, gravitons in this approach. However, to the relevant order, static gravcats produce no perturbations and such terms can also be ignored in the derivation of the interference effect (though not absent from a full description of the experiment, as we discuss shortly). In short, on quantization, the whole Hamiltonian reduces to (4.7).

Following Reference Anastopoulos and HuAnastopoulos and Hu (2014), we extract two points from this discussion. The first is simply that this derivation justifies treating gravity as an immediate interaction between gravcats in the proposed experiments: The Newtonian model is simply what is obtained from standard quantization procedures applied to GR with scalar fields, which is itself a strategy well motivated by the success of GR and QFT. The more interesting second point, however, is that the derivation provides us with an analysis of the nature of the degrees of freedom at play: In particular, in the derivation of (5.2), all “true degrees of freedom” of the gravitational field contribute to terms dropped as negligible. By contrast, the only significant term in the Newtonian model is the Newtonian potential, which is not associated to any dynamically propagating parts of the field: The Hamiltonian is nothing but the scalar gauge constraint, and so is “pure gauge.” Reference Anastopoulos, Lagouvardos and SavvidouAnastopoulos et al. (2021, Section 6.1) put the idea quite starkly: “Our analysis shows that, according to GR, the two parts of a quantum bipartite system that interact gravitationally in the Newtonian regime do so without an intermediate degree of freedom” (emphasis in original).

The close analogy (e.g. Reference Chen, Giacomini and RovelliChen et al. [2022]) with the more familiar case of quantum electrodynamics may be helpful to reinforce the picture. One way to quantize the electromagnetic field is to perturb around the Coulomb field of the charges, representing the Gauss constraint (the perturbations describing the transverse parts of the potential). On quantization then, the perturbations become photon excitations above a vacuum state represented by the Coulomb potential (e.g. Reference KleinertKleinert [2016, Section 12.3]). In the electromagnetic analog of the GIE experiments, the whole effect comes from the Coulomb – pure gauge – part, the zero excitation state of the true degree of freedom, the photon. Again, it seems that the interestingly quantum part is irrelevant. (Indeed, because of the close formal correspondence, it makes sense to speak of the scalar constraint as a gravitational Gauss constraint. And so we will.)

However, one must be cautious not to lean too heavily on the (widely accepted) technical distinctions between “true degrees of freedom” and (mere) “pure gauge degrees of freedom.” Physical significance, even in gauge theories, need not be reserved just for the former (for which one may derive equations of motion). That is to say, “true” degree of freedom, in contrast to “pure gauge,” is a standard, well-defined formal notion in gauge theory. Nonetheless, the locution “true” is dangerous: In the quoted remark previously discussed, Anastopoulos et al. appear to move from “true” degrees of freedom in the technical sense to an ontologically thick use of “degrees of freedom” in describing the relevant physics. This move is too hasty. Granted, in linear gravity, the Newtonian potential is pure gauge; but to infer from that formal fact that it is somehow unreal or unphysical is to say that Newtonian gravity – which after all was our best theory of the gravitational field before relativity – itself is therefore “unreal” or “unphysical.” Such a view seems implausibleFootnote ²⁶ (though one could envision it having defenders). Thus, we reject the inference from not “true” in the technical sense to “unreal” in the ontological sense; although we will see later that the split has some relevance to the debate. To avoid any possible conflation of a technical term with an ontological concept, we will thus simply avoid talk of “(true) degrees of freedom” in the following, except on a couple of occasions in which we specifically refer back to this discussion. Indeed, in the paradigm to which we now turn, one simply does not proceed in analyzing the experiments by means of the interpretational tools used in gauge theories.

Now, as Reference Chen, Giacomini and RovelliChen et al. (2022) demonstrate, it is not correct that the static Newtonian potential suffices for a full description of a GIE experiment. Our treatment focuses on the stage of the experiment in which superposed gravcats interact gravitationally while remaining (approximately) stationary (Reference Marletto and VedralMarletto and Vedral [2019, Question 5] make the same point). However (footnote 21 for more), before that stage they must evolve into superposition, and after that stage they must evolve out of superposition. During those phases, because the packets are moving, the static field no longer suffices to describe the gravitational interaction, and graviton modes are involved. This is true even though the motion is slow (as it must be, to avoid emitting gravitons, which would destroy gravcat entanglement, as discussed in Section 6).

While we are sympathetic to this point, it does not establish that the Newtonian model is ill-suited (or even that the GIE experiment indisputably witnesses QG, as the tripartite analysis to be discussed shortly has it). For no one doubts that any analysis involves some idealization of the situation. Certainly this is true of the interaction stage: The proponent of the Newtonian model paradigm could take the attitude that we should idealize the previous and following stages as involving no interaction at all. The relevant issue is rather which idealizations to make – a matter of theoretical perspective, which in turn is precisely what is under contention.Footnote ²⁷ Of course, if the experiments become sufficiently sensitive that one cannot idealize in this way, then the issue will become empirical. But in that case, one would be in the regime in which one is witnessing gravitons, precisely the benchmark for positive claims of witness that we understand proponents of the Newtonian model paradigm advocate. (Moreover, by measuring that effect one would have moved beyond the GIE experiment.)

5.2 Tripartite Models

Notwithstanding the derivation of the nonlocal interaction term in the Newtonian model in a gauge theoretic approach to GR, there is another view that insists on gravity, in the light of GR, as mediating interactions between massive systems such as the gravcats: Specifically, the Hamiltonian should contain no interaction term directly between gravcats, but only interactions between the gravcats and a third “gravity” subsystem. Sometimes this assumption is called “locality”; it can be secured by insisting that gravitational interactions propagate at a finite velocity. However, because considerations of causal connectibility will be relevant in the next subsection, and because the assumption could hold even if the gravcats were spatially coincident (or if effects propagated instantaneously), we will avoid that terminology. What matters here is that the only interaction terms are between gravcat and gravity subsystems, and no gravcat-gravcat interaction terms.

Under these assumptions, the state of the system after splitting is in fact not (4.1), but

$\begin{array}{cc}|L⟩\otimes |{\gamma }_{LL}⟩\otimes |L⟩& +|L⟩\otimes |{\gamma }_{LR}⟩\otimes |R⟩+|R⟩\otimes |{\gamma }_{RL}⟩\otimes |L⟩\\ & +|R⟩\otimes |{\gamma }_{RR}⟩\otimes |R⟩.\end{array}$(5.3)

$|{\gamma }_{XY}⟩$ represents the gravity subsystem for a pair of gravcats located at $X$ and $Y$ , respectively, so each gravitational state is that appropriate to the gravcat positions in the corresponding term. After $t$ , the state is not approximated by (4.3) but by

$\begin{array}{cc}|L⟩\otimes |{\gamma }_{LL}⟩\otimes |L⟩& +|L⟩\otimes |{\gamma }_{LR}⟩\otimes |R⟩+e^{\frac{-iG{m}^{2}t}{\delta }}|R⟩\otimes |{\gamma }_{RL}⟩\otimes |L⟩\\ & +|R⟩\otimes |{\gamma }_{RR}⟩\otimes |R⟩.\end{array}$(5.4)

The key thingFootnote ²⁸ to recognize is that in states (5.3) and (5.4), the gravity subsystem is in a superposition of classical gravitational states. This is of course because the gravcats are in a spatial superposition and the gravity subsystem couples to the wavepackets through the Hamiltonian.Footnote ²⁹ Thus, in this model, the gravity subsystem exhibits a quantum nature, in the sense of superposing classical states.

Exactly what the mediating quantum gravitational system is taken to be is not the same across theoretical treatments: For Reference Marletto and VedralMarletto and Vedral (2017a), it is an information channel; for Reference Bose, Mazumdar and MorleyBose et al. (2017) and Reference Chen, Giacomini and RovelliChen et al. (2022), a quantum field; and for Reference Christodoulou and RovelliChristodoulou and Rovelli (2019), a metric field capable of quantum superpositions, and so on. Further, Reference AdlamAdlam (2022), referring to our discussion of tripartite models, distinguishes “thin” and “thick” variants: the former capturing minimal aspects of the nature of gravity, and the latter including central features of GR. Of course, ultimately, they all take these to be different representations of the same physical object.

We consider some of these ideas in more detail in Section 5.2.2, but first we will see that the production of gravcat entanglement witnesses the quantum nature of gravity (in tripartite models) on quite general grounds. All that needs to be assumed in fact is that it is a mediating subsystem, in line with the general conception of gravity in GR. We already saw in Section 4.3 that if one takes view 2 of SCG – SCG as a candidate fundamental theory – then we should expect no entanglement in the experiments with gravcats. However, we will now see that this result can be generalized to the conclusion that no classical mediator can produce entanglement between quantum systems with no direct interaction.

5.2.1 Quantum and Information Theoretic Considerations

In this section, we present two results, showing from general quantum mechanical and general information theoretic considerations, respectively, that interactions with a classical mediator cannot lead to entanglement between two systems. Two preliminary notes. First, of course, the very setup of two local subsystems and a mediating subsystem means that the tripartite paradigm is assumed from the beginning, so these results cannot settle the tripartite-Newtonian question. What they can do is show why the observation of entanglement in a GIE experiment means that the tripartite system must have a quantum mediator – or rather, as we shall see, that the mediator not be classical. Second, results like those we discuss presently are given for simple, finite-dimensional systems. One might question the generality of such analyses in the context of infinite-dimensional gauge field theories – a question we set aside.

First then, a proof in quantum theoretic terms.Footnote ³⁰ We will assume the simplest kind of system, in which the subsystems are two-dimensional, but as Reference Marletto and VedralMarletto and Vedral (2017a) discuss, the extension to $N$ -dimensional systems seems straightforward, and thence to quantum fields and the GIE experiment in the Fock representation (although this claim has been challenged: Reference Anastopoulos and HuAnastopoulos and Hu [2022, Section 2]).

Claim: Suppose that $H_2$ is a two-dimensional Hilbert space, and consider the tripartite system $H_2\otimes H_2\otimes H_2$ . (a) Let the first and last slots describe qubits that interact “locally”: the Hamiltonian $H=H_{12}\otimes I+I\otimes H_{23}$ (where $H_{12}$ acts on the first and second terms of the tensor product only, etc.). And (b) let the second slot represent a classical bit: The only observable for this subsystem is ${\sigma }_z$ (and the identity $I$ ). Given the locality of the qubits as expressed by (a) and the classicality of the bit as expressed by (b), no entanglement can arise between the qubits.

Proof.

1. (1) From (a) and (b) we have

   $\begin{array}{ccc}{H}_{12}& =& A\otimes I+B\otimes {\sigma }_z\end{array}$(5.5)
   $\begin{array}{ccc}{H}_{23}& =& I\otimes C+{\sigma }_z\otimes D\end{array}$(5.6)
   assuming that the Hamiltonian is an observable. It follows that
   $\begin{array}{ccc}[H_{12}\otimes I,I\otimes H_{23}]& =& 0\end{array}$(5.7)
   As the two terms in the Hamiltonian $H$ commute, the unitary operator describing the time translations factorizes, that is,
   $\begin{array}{ccc}{e}^{-iHt}=e^{i(H_{12}\otimes I+I\otimes H_{23})t}& =& e^{-i{H}_{12}\otimes It}\cdot e^{-I\otimes H_{23}t}.\end{array}$(5.8)

   That is, since the two factors commute, we can treat the evolution as an interaction first between one of the qubits and the bit, and then between the other qubit and the bit, in either order.

2. (2) So let the tripartite system start in a fully factorizable pure state, $\Psi$ , and apply the unitary evolution:

   $\begin{array}{ccc}{e}^{-iHt}|\Psi ⟩& =& e^{-i{H}_{12}\otimes It}\cdot e^{-I\otimes H_{23}t}|{\psi }_1⟩\otimes |{\psi }_2⟩\otimes |{\psi }_3⟩\\ & =& e^{-i{H}_{12}\otimes It}|{\psi }_1⟩\otimes \sum _{j=0}^1{\alpha }_j|{\chi }_j⟩\otimes |{\varphi }_j⟩,\end{array}$(5.9)
   where in the last step, we have used the fact that the bit is classical (b), so there is only one orthonormal basis for the mediating subsystem, namely $\{|0⟩,|1⟩\}$ , respectively, the $-$ 1 and $+$ 1 eigenstates of ${\sigma }_z$ . Hence we can continue
   $=e^{-i{H}_{12}\otimes It}|{\psi }_1⟩\otimes \sum _{j=0}^1{\alpha }_j|j⟩\otimes |{\varphi }_j⟩.$(5.10)

3. (3) Since (b) the only observables for the bit are (multiples of) the identity and ${\sigma }_z$ , there is a superselection rule: All observables on the system commute with $I\otimes {\sigma }_z\otimes I$ . In particular, restricting attention to the bipartite 2–3 subsystem, the most general form of an observable is $I\otimes A+{\sigma }_z\otimes B$ , which commutes with ${\sigma }_z\otimes I$ . As a result (which can also be readily checked), the pure state $\sum _{j=0}^1{\alpha }_j|j⟩\otimes |{\varphi }_j⟩$ is indistinguishable from the mixture $\sum _{j=0}^1|{\alpha }_j{|}^2|j⟩⟨j|\otimes |{\varphi }_j⟩⟨{\varphi }_j|$ : The expectation values of the states agree for all observables of the general form.

   Therefore, we can replace the pure state in (5.10) with a physically equivalent density matrix,

   $|{\psi }_1⟩\otimes \sum _{j=0}^1{\alpha }_j|j⟩\otimes |{\varphi }_j⟩\to |{\psi }_1⟩⟨{\psi }_1|\otimes \sum _{j=0}^1|{\alpha }_j{|}^2|j⟩⟨j|\otimes |{\varphi }_j⟩⟨{\varphi }_j|.$(5.11)

   This state is separable between 2 and 3 (and indeed between 1): In general, any density matrix of the form $\sum _j|c_j{|}^2{\rho }_j^a\otimes {\rho }_j^b$ can be expressed as a mixture of factorizable pure states. In short, by rewriting the state in this way, we have made manifest the absence of entanglement between 2 and 3, ultimately a consequence of the classical nature of the bit.

4. (4) Now, it should be clear that since 3 and 2 are not entangled, then – even if 2 were quantum – an interaction between 1 and 2 alone cannot entangle 1 and 3. (Of course, if 2 and 3 were entangled, that entanglement could be “exchanged” between 2 and 1.) But for completeness, we will show this to be the case. So consider the interaction between 1 and 2, producing the state

   $\begin{array}{ccc}& & e^{-i{H}_{12}\otimes It}|{\psi }_1⟩⟨{\psi }_1|\otimes \sum _{j=0}^1|{\alpha }_j{|}^2|j⟩⟨j|\otimes |{\varphi }_j⟩⟨{\varphi }_j|e^{+i{H}_{12}\otimes It}\\ & =& \sum _{j=0}^1|{\alpha }_j{|}^2{e}^{-i{H}_{12}t}|{\psi }_1⟩⟨{\psi }_1|\otimes |j⟩⟨j|e^{+i{H}_{12}t}\otimes |{\varphi }_j⟩⟨{\varphi }_j|.\end{array}$(5.12)
   This state is a mixture of pure states in which the 1–2 subsystem factorizes with the 3 subsystem; hence it is separable between the two qubits. (Of course, by appeal to the classicality of the bit, we can as before show that it is also separable between the 1 and 2 subsystems.)

   Thus, we have shown using the standard assumptions of QM and the locality of the qubits (a), that no entanglement can arise between the qubits, given the classicality of the bit as expressed by (b).

Now, as we noted in Step 3, there is a superselection rule, so that the bit state is always a mixture of ${\sigma }_z=\pm 1$ states. But (since we assume the Hamiltonian to be an observable), the superselection rule implies a selection rule. That is,

$[I\otimes {\sigma }_z\otimes I,H]=0$(5.13)

so that the value of the bit is a constant. Thus not only is the bit a mixture of $+$ 1 and $-$ 1 ${\sigma }_z$ states, it is a constant mixture. In other words, in this framework, the qubits have no effect on the bit, although the bit state does affect the final states of the qubits. The one-way nature of the interaction seems to refute the claim that the bit mediates any interaction between the qubits, after all – so it is no surprise that no entanglement arises between them! (Of course, if entanglement does occur between local systems in the sense of (a), as expected in the GIE experiment, then the proof shows that (b) fails, so the mediator is nonclassical. In particular, there are then no selection or superselection rules, and so the mediator is not frozen in some mixed state.)

This situation makes one wonder whether the quantum framework is actually apt for representing a classical bit in the first place. Perhaps the conclusion that the mediator cannot be classical depends on adopting a hobbled description for it. One would like to see the same result in a broader setting that doesn’t already suppose the quantum formalism. Moreover, the proof assumes in Step 1 that the Hamiltonian is an observable, and one might somehow question that assumption. For these and other reasons, it is worth sketching an alternative, more general information theoretic result.

Reference Marletto and VedralMarletto and Vedral (2020) use the general information theoretic framework of “constructor theory” (Reference Deutsch and MarlettoDeutsch and Marletto 2015) to prove that if interactions with a mediating subsystem produce entanglement between mutually isolated systems, then the mediator cannot be classical. Both the exact statement of the theorem and its proof rely on some new and unfamiliar machinery, so we will just sketch some central ideas to give a feeling for the result; for details, the reader is referred to the original papers.Footnote ³¹

Constructor theory works at a very high level of generality, in which various subsystems are posited, each with some set of mutually exclusive states $s_i$ , thought of as providing a full description of the subsystem. In addition, state transitions for individual and composite systems – “tasks” – $[s_i\to s_j]$ are specified as possible or impossible. For instance, suppose we have a subsystem $S$ with states $\{0,1,y,z\}$ (the numerals and letters should be understood here simply as state labels). Sets of states correspond to determinables (or “variables” in constructor theory) and individual states to their determinate values (“attributes”).Footnote ³² Let us suppose then that $B=\{0,1\}$ is one variable, with attributes 0 and 1 (so that $B$ is for ‘bit’); and that $X=\{y,z\}$ is another.

Simplifying considerably, let us say that a variable $\{x_1,x_2,\dots ,x_n\}$ of a system is an “observable” when there is a second system – the measuring apparatus – with states $\{s_r,s_1,\dots ,s_n\}$ , and a possible task that for any $1\le i\le n$ has the effect $\left[\right(x_i,s_r)\to (x_i,s_i\left)\right]$ . That is, $s_r$ is the apparatus-ready state, the $s_i$ are its various possible meter readings, correlated with states of the subsystem, so that the task amounts to a nondestructive measurement of the variable (the state of the measured subsystem is unchanged).

To continue our example, let us thus further suppose that $B$ and $X$ are both observables. In that case, we have a further choice: Their union may or may not also be an observable – in describing a system, we choose whether or not the necessary process is possible. If $\{0,1,y,z\}$ is an observable, then $S$ is simply a classical system, with a four-dimensional state space. But if it is not an observable, then the situation is like that in a qubit, in which $B$ and $X$ are incompatible observables, ${\sigma }_z$ and ${\sigma }_x$ say, which cannot be observed simultaneously. Broadly speaking then, this situation abstracts the notion of “complementarity” without assuming the quantum formalism. So we see that constructor theory can accommodate classical and quantum on an equal footing; the general framework of states and tasks is neutral between them, and the difference arises only from which tasks are possible. (Of course, quantum is also distinguished from classical by the capacity for entanglement, beyond any classical statistical correlation, and this distinction too can be represented.) Moreover, the generality of the framework allows for possibilities between the fully classical and the fully quantum (and indeed possibilities orthogonal to that distinction); in particular, the generalizations of complementarity and entanglement, while clearly nonclassical, do not by themselves entail the full quantum formalism.

Suppose, then, two mutually isolated qubits (or “superinformation media”, as they are represented in constructor theory), which can each interact with a mediating subsystem: Of course, this state of affairs is cashed out in terms of which two subsystem tasks are and are not possible. Then, Marletto and Vedral show, if initially separable qubits end up entangled, then the mediator must be nonclassical, possessing complementary variables, and entangling with the qubits, in their generalized senses.Footnote ³³ Thus, given the tripartite model, we again conclude that a positive result in the GIE experiment would witness nonclassical behavior of gravity. In this framework, we don’t shoehorn a classical system into QM, and there is no requirement that the mediator be static if classical.

Note that strictly the theorem shows that mediating entanglement requires a nonclassical mediator, not that it requires a quantum one: indeed nonclassical, nonquantum models accounting for entanglement mediation exist (Reference Marletto and VedralMarletto and Vedral (2019) refers to Reference Hall and ReginattoHall and Reginatto [2018]). In this regard, Marletto and Vedral reasonably compare their result with Bell’s theorem: In that case too, experiment can only show that a system is nonclassical, not that it is quantum.

Now that we have seen how entanglement mediation rules out classicality, we should note that GIE experiments also bear on gravitational collapse theories (often associated with Reference DiósiDiósi (1989) and Penrose [1994]). Such proposals could be thought of as falling into the tripartite camp, but claiming that the gravitational field abhors a superposition, and induces a collapse whenever it deviates from a classical state (by a prescribed amount), and thus is effectively always classical. Here, the implication of observing entanglement is more equivocal than for SCG: As Reference Bose, Mazumdar and MorleyBose et al. (2017, p. 1) put it, such theories imply “the breakdown of quantum mechanics itself at scales macroscopic enough to produce prominent gravitational effects.” The question of course is what counts as “prominent.” On the one hand, by Penrose’s estimates, the proposed experiment, with gravcats of ${10}^{-14}$ kg separated by $100\mu$ m, the gravitational collapse time should be of the order of a second, which would be fast enough for the classicality of the field to affect any observed entanglement. And so it seems it is a “prominent” effect: The quantum state will collapse, and no entanglement will be observed. However, on the other hand, should entanglement be observed, the theories do have a tunable parameter, which could be set to prevent collapse in the currently envisioned GIE experiments, although they would place a new bound on it. But so doing is to accept that the experiment witnesses a quantum superposition of the gravitational field, which is at least against the spirit of Penrose’s position, and quite possibly falls afoul of the very arguments by which he motivates it.

In this context, we also observe that the GIE experiments are of a different character from that envisioned in Reference Marshall, Simon, Penrose and BouwmeesterMarshall et al. (2003) to explore (inter alia) collapse theories, including the Penrose–Diósi type. At the heart of the setup is a single gravcat in the form of a mirror: It is put into spatial superposition by reflecting the two packets of a beam split photon off it; and one observes whether it remains in that state, or collapses because of its sizable gravitating mass. If collapse were observed, that would be evidence that GIE would not be observed, because the gravcats in the GIE experiments would not remain in superposition either. But in the alternative case, it would not require any quantum behavior of gravity to explain an observed absence of collapse; that outcome would be compatible with SCG even. In this regard, then, the experiment asks a very different question to the GIE experiments.

Finally, given this relevance of GIE experiments to collapse interpretations, and indeed to SCG in a more interpretation-neutral way than Page and Geilker, it is interesting to note the claims of Reference AdlamAdlam (2022) that GIE experiments can function as evidence for interpretations of QM, given the tripartite paradigm. Adlam starts with a standard distinction between a $\psi$ -complete interpretation of matter theories (which only includes a quantum sector), and a $\psi$ -incomplete interpretation (which also includes “non-quantum be-ables”). $\psi$ -incomplete interpretations come in further variants: Either the matter theory’s ontology includes the quantum sector but also some other nonquantum sector ( $\psi$ -supplemented); or its ontology includes only a nonquantum sector, so the quantum sector is more than epistemic but not ontological (say lawlike structure); or purely epistemic ( $\psi -$ epistemic). Her main argument is then that, given that on $\psi$ -incomplete views gravity could also be sourced by a nonquantum sector (and, in the case of $\psi -$ epistemic interpretations, may even have to be thus sourced), there seems a correlation between a negative result on spacetime superpositions in the context of tabletop experiments, and increased confidence in a $\psi -$ incomplete interpretation. Conversely, a positive result on spacetime superpositions from the tabletop experiments would seem to boost confidence in a $\psi -$ complete interpretation.

5.2.2 The Quantum Description of Gravity

In this subsection, we elaborate further on the quantum description of the “gravity” subsystem in these experiments, within the tripartite models paradigm, which explicitly views gravity as a physical subsystem on a par with the gravcats. The questions are, what, formally and materially, should the states $|\gamma ⟩$ be, what are the gravcat states, and what is their interaction? It is of course natural to start with GR: Our current best physical theory of gravity after all does theorize gravity as a dynamical system with a nontrivial vacuum sector. So, within the tripartite models paradigm, GR itself would seem to justify the assumption in (5.3) that we have a tripartite system, something that seems ad hoc in Newtonian gravity proper. And indeed, this is the starting point that theorists have generally taken, in a few different ways.

For instance, the supplementary materials to Reference Bose, Mazumdar and MorleyBose et al. (2017) derive the phases of the various terms in the gravcat superposition by quantizing linearized GR: Briefly, one again starts by representing the classical GR gravitational field as linear perturbations in a 3+1 Minkowski background, using the stationary approximation for the gravcats, but because the gravcats are nonrelativistic, only the energy-momentum, $T_{00}$ , component of the stress energy contributes. So only the $h_{00}$ components of the field are relevant, and hence quantized. The experiment can then be modeled in terms of an interaction between localized mass excitations, mediated by coherent states of the modes of the $h_{00}$ field.Footnote ³⁴ The appropriate Hamiltonian has been solved, and introduces the same phases as predicted by the Newtonian potential.

Another approach is taken by Reference Christodoulou and RovelliChristodoulou and Rovelli (2019), who model the experiment using quantum superpositions of the metric field, locating the experimental effect as a consequence of superpositions of gravitational redshifts. Of course, from a quantum field theory perspective, these are presumably superpositions of coherent states of the perturbatively quantized (linearized) gravitational field, but in their presentation they are simply treated as quantum states labeled by classical geometries.Footnote ³⁵ Starting classically, again assuming that the gravcats are heavy enough, and the duration short enough that they remain (approximately) stationary during the experiment, the metric field due to a body of mass $m$ and radius $R$ can be approximated by

$d{s}^2=(1-2\varphi (\overrightarrow{r}\left)\right)d{t}^2-d{\overrightarrow{r}}^2.$(5.14)

Outside the body, $\varphi \left(\overrightarrow{r}\right)=Gm/r$ is just the Newtonian potential; while inside the body, one takes $\varphi \left(\overrightarrow{r}\right)=Gm/R$ , a constant. (Compared with (3.1), we now set $c=1$ and consider the interior as well as exterior field.) As always, we are in the regime of linearized gravity, so that for a pair of gravcats, the classical geometry will be the sum of two such solutions. It is then straightforward to calculate the proper time elapsed along the worldline of (a point in) one gravcat at a distance $\delta \gg R$ from another:

$\tau ={\int }_0^{T}ds={\int }_0^{T}dt\sqrt{1-2Gm/R-Gm/\delta }\approx T\cdot (1-Gm/R-Gm/\delta ),$(5.15)

showing the usual GR time dilation effect of a nearby mass. If $d\gg R$ , most of the effect will come from the mass of the gravcat itself, but we also see that there is a contribution dependent on the distance to the other gravcat.

Thus, the components of the superposition (4.1), corresponding to different gravcat separations, correspond to different metrics, and hence different proper times. Now assuming that the (classical) gravitational field labels what is actually some coherent state in a quantum treatment of gravity, which can superpose, the full state is therefore (5.3), with $|{\gamma }_{LR}⟩$ and so on describing different states of the metric field of GR. And thus the different terms of the superposition are associated with different proper times and will develop relative phases (relative to a common lab time, $T$ ). In particular, a gravcat pair has a phase $e^{-im\tau }$ from its mass energy, or (ignoring a common phase) $e^{-iG{m}^{2}T/\delta }$ from (5.15) – the very phase one expects (4.3) from the Newtonian potential, but now seen to be a relativistic redshift effect. Thus, the very same relative phase between the components of the gravcat superposition, and ultimately the very same entanglement, is predicted by treating metric states as superposable quantum states (to leading order).

Both of these treatments, which in different ways “quantize” the metric field of GR, relate gravity to a physical quantity (a metric field!), which naturally should be treated as a third subsystem in addition to the gravcats, and which then must be quantum (or at least nonclassical) by the general quantum and information theoretic arguments, if the gravcats become entangled as expected. Thus, from this point of view, the derivation of the effect by appeal to the Newtonian potential with which we presented it on the naive first pass is an approximate heuristic. (Recall for contrast that the analysis by Anastopoulos and Hu in chapter 5 – or rather the interpretation thereof in favor of the Newtonian model view – explicitly renders the Newtonian potential as uninformative of the physical parts of the gravitational field.)

A word of caution: Because modeling the system in this way involved an appeal to GR, it is tempting to think that the finite speed of propagation of the gravitational interaction in GR is necessary and sufficient for the system to be tripartite. But while a finite propagation speed seems to require that the system be tripartite, we would nonetheless now like to stress that it is by no means necessary. This point can be seen clearly by adopting a geometrized formulation of Newtonian gravity: Newton–Cartan theory.Footnote ³⁶ In fact, as we demonstrate in the appendix, whereas the gravitational redshift effect noted in Reference Christodoulou and RovelliChristodoulou and Rovelli (2019) is specific to their relativistic setting, the surrounding argument – namely, that a superposition of spacetime geometries demonstrates the quantum nature of gravity – is not. In particular, one can perform an independently motivated analysis of the experiments with gravcats in the framework of Newton–Cartan theory, which is analogous to their relativistic analysis: In both, gravcat pairs develop different phase factors with respect to different, superposed, spacetime geometries, leading to entanglement. Again, all that is important is the decision to treat gravity as a mediating subsystem, a decision whose subsequent execution proceeds analogously across both theoretical contexts.Footnote ³⁷

Moreover, this example emphasizes that the mediating subsystem need not be a dynamically propagating field for the entanglement theorems to apply, and hence for the GIE experiment to witness QG.Footnote ³⁸ In other words, even though the theorems entail that a successful GIE experiment witnesses the nonclassical nature of gravity, it does not, without further assumptions, witness perturbatively quantized linearized gravity. Of course, it is possible that these further assumptions are supplied by the concurrent commitment to GR in the background of an analysis of the Newtonian-regime experiment. This is a point we will discuss in Section 6, in the context of a thought experiment that has been analyzed by Reference Belenchia, Wald and GiacominiBelenchia et al. (2018).

Finally, it is worth pointing out that the locality assumption behind the tripartite view is only to hold approximately at the relevant energy level of the experiment – underlying physics (say, at a string scale) may just as well be nonlocal. More precisely, so it has been explicitly shown by Reference Marshman, Mazumdar and BoseMarshman et al. (2020, Section VII) for modifications of gravity in the UV based on the most general quadratic actions in four dimensions that are invariant under parity and torsion-free – such actions generally simply include nonlocal interactions. In a sense, this point is not very surprising: Even beyond the effective field theory paradigm, we tend to hold physics at some domain to be independent (i.e., decoupled) from that at lower scales (higher energies). Interestingly, for future experimentation, however, a disagreement in underlying physics (including an actual nonlocal nature of gravity) becomes eventually distinguishable through the relevant entanglement entropy in the experiments with gravcats when performed with ever smaller probe separations.

5.3 Why “Paradigms”?

At this point, we should clarify and justify our description of the Newtonian model and tripartite models as, respectively, not just views of the GIE experiments, but paradigms. First, by this term, we do not mean to commit to a bundle of all the senses and aspects of “paradigm” inaugurated in Reference KuhnKuhn (1962); and certainly we do not claim that there is incommensurability between the paradigms, understood as an incommensurability in either meaning or standards of experiment, let alone in the very nature of rationality. Rather, we have in mind a sense of paradigm developed by the Kuhn of Objectivity, Value Judgement, and Theory Choice (in Reference KuhnKuhn [1977a]), who emphasizes that scientists can agree about the criteria of rational choice, yet come to different decisions because they disagree about how to implement them. For instance:

Here, Kuhn argues that no calculus of theory choice will universally determine the unique correct theory to adopt in the light of evidence and extra-empirical criteria. The canons of scientific rationality, that is, leave room for reasonable people not only to disagree, but to recognize one another’s rationality in reaching different conclusions. (Of course, in actuality, they might not, either because they do not appreciate Kuhn’s point, or because they abandon the scientific canons.)

However, Kuhn’s specific analysis is not a perfect fit for the present case, because in the GIE experiments we are not faced with disagreement over basic theory, but over how that theory should be used to model a particular proposed experimental phenomenon. That is, as we have emphasized, both models take as their starting points the methodological principles that a model of GIE should start from our best theory of gravity, GR, and the application of generally accepted methods of quantization, namely quantized linear gravity, understood as an EFT. Such assumptions are very widely (though not universally) held in the QG theoretical community, so the different models do not represent a different understanding of what is “really” going on, at a more fundamental level. For instance, the situation is quite different from the choice between Ptolemaic and Copernican systems in the sixteenth century (to take Kuhn’s example). GIE is even disanalogous to the choice between Lorentzian and Einstein–Minkowski accounts of the null outcome of the Michelson–Morley experiment. Even though these cases are alike insofar as their competing paradigms make exactly the same predictions so that the choice is strictly underdetermined by the data, they differ crucially because in the case of GIE, they do not disagree about the basic underlying physics.Footnote ³⁹

Instead, the GIE paradigms disagree with respect to how to extract and approximate a model for the experiment, from a shared physics. It is striking when distinct theories predict exactly the same data, but not so remarkable that two different models, extracted by established methods from a successful theory, do. Rational indeterminacy arises because each model embodies a different aspect of GR as central: In one case, it is the lesson that gravity is causally propagated, while in the other, it is the way in which one takes the Newtonian limit. Which aspect one attaches the greater weight to, given the particulars of the GIE experiment, determines which model one concludes is the most faithful to the experiment; but one can reasonably take either to be more important, and so shared rational criteria leave the model indeterminate. And while proponents of both models can be brought to understand the analysis of the other, they will fail to accept the complete relevance it has for the other – the degrees of freedom analysis of when to call putative gravitational systems physical is simply mutually exclusive from viewing the gravitational field as propagating and vice versa. In this sense, the models are “incommensurable.”

It is this commonality with Kuhn’s argument that then leads us to conclude that talk of “paradigms” is appropriate here. (At least paradigms “lite,” because stronger forms of incommensurability are not in evidence.) The analysis of the models that we have offered in Section 5 has been aimed at making clear the different modeling assumptions behind the models, to reveal the different theoretical stances embodied in the models, and to demonstrate that indeed they are all reasonable, just weighed differently. We hope that doing so, plus our discussion of paradigms, will go some way to helping clarify – though not resolve, obviously! – disagreements over the correct model.Footnote ⁴⁰

Shortly after this Element appeared on the arXiv preprint repository, so did a parallel analysis (Reference Fragkos, Kopp and PikovskiFragkos et al. [2022]), which identifies the choice of gauge as the “assumption” leading to divergence on the question of Newtonian versus tripartite models. That is, the different choices of gauge made in Reference Anastopoulos and HuAnastopoulos and Hu (2014) and Reference Bose, Mazumdar, Schut and TorošBose et al. (2022) lead to different models: one in which a Newtonian potential comes from a gauge constraint (as discussed earlier), and a tripartite model in which modes of the field are quantized degrees of freedom (somewhat different from the $h_{00}$ quantization discussed earlier).Footnote ⁴¹ Technically, this point is correct, and indeed highlights a key decision point; but it is also too narrow, missing the many (other) ways one might give a tripartite model of gravity. Indeed, as Reference Fragkos, Kopp and PikovskiFragkos et al. indicate, a more basic disagreement than any choice of preferred gauge involves a whole package of views one might hold about the nature of Newtonian gravity – views about the importance of causality, constraints, and so on. Our emphasis thus far has been to articulate how these latter views come together as distinct packages, and that these packages are the ultimate difference-makers in interpretations of the GIE experiments.

Moving on, suppose that one accepts our application of Kuhn’s conception to GIE experiments, one can still worry that we have missed some argument, relying on deeper principles agreed to by the partisans of both paradigms, which would after all rationally require one group to abandon their commitments, leading to the collapse of our talk of paradigms. Our presentations of this workFootnote ⁴² have indeed encountered such resistance to calling the positions “paradigms” in our sense, some rehearsing considerations already discussed, and some raising new arguments to be considered here. It is worth noting, however, that while those arguing against paradigm talk have done so because they believe that one of the models is clearly the right one, they have not agreed about which one in fact is the right one. So this datum is (at least) compatible with the thesis that our interlocutors are adopting different paradigms in our sense! Nonetheless, we take their arguments seriously, and indeed addressing them will help clarify our claim.

Objection: By denying gravity the status of a causal field, the Newtonian model is committed to such an unreasonable refusal of physical background knowledge that it is disqualified as sensible physical theorizing by any paradigm-independent standard of modern physics.Reply: The Newtonian model does take into account physical background knowledge – arguably, just the same knowledge, but it weighs the relevance of the analysis given in Section 5.1 completely differently. Indeed, that the Newtonian model takes into account the same physical background knowledge is emphasized, ironically, in a recent article by Reference Christodoulou, Di Biagio and AspelmeyerChristodoulou et al. (2022). In the article, which is intended to reinforce that GIE experiments indicate quantum superposition of spacetimes, the authors distinguish a “slow-motion” approximation (sources moving at nonrelativistic speeds), where the gravitational interaction is still local (i.e. causal), and a “near-field approximation” (sources are much closer together than time elapsed through the experiment, in natural units). The “Newtonian limit” that reproduces the naive model, claim the authors, denotes the overlapping regime where both approximations are satisfied. Now, as they show, in the slow-motion, not near-field approximation regime, locality considerations show up as corrections between the employ of laboratory time function (as in the Newtonian limit) and a retarded time function. But, setting up retarded GIE experiments in the slow-motion, not near-field regime is “not realistic for the foreseeable future” and seeing the effects of slow-motion correction terms to the Newtonian limit analysis of the GIE experiments is “unlikely reachable” (p. 4). Thus, although there are retarded GIE protocols that they point to on a further horizon of experimentation (p. 5) that, given physical background knowledge, would preclude a Newtonian model description, the GIE experiments under scrutiny here do not: precisely because they are in the Newtonian limit regime of our fundamental gravitational physics.Footnote ⁴³Objection: Both paradigms take GR as their starting point, thus accepting that gravity is well described by a dynamical classical field in the appropriate limit. What then is the state of this field supposed to be if we have superposed gravcats, and their consequent observable entanglement? The general information theoretic theorems show that it cannot be a classical state if the interaction is mediated by gravity in the appropriate sense, and no other mechanism in which gravity remains in a classical field state has been proposed (if any be possible). Doesn’t then entanglement require a quantum superposition according to the tenets of both paradigms?Footnote ⁴⁴

Reply: First, one might accept GR in its regime, but not accept that gravity is a “field” (broadly speaking), classical or quantum, more fundamentally. One might have a specific alternative theory in mind (gravity as ancilla, perhaps), but more likely, one might simply be neutral on the topic. That would be an unusual stance for a theorist, but would be less implausible for an experimentalist, whose goal of probing the quantum + gravity intersection requires only very broad theoretical commitments (and a far more detailed understanding of the causal powers of the various elements within any specific experiment, e.g. Reference Hacking and TauberHacking [1984]). Either way, one could resist the objection, because one is not committed to any field state at all.

However, as we noted, most theorists accept the EFT approach to QG (including of course anyone accepting the Newtonian analysis of Reference Anastopoulos and HuAnastopoulos and Hu [2014]), and thus that “really” gravity is a field, and so a third subsystem as conceived in the tripartite approach. Isn’t any such a person thus bound to accept that paradigm, and conclude that gravcat entanglement would witness a nonclassical state of the field? This thought leads to the second reply: “yes and no.” The “really” provides important wiggle room, because it means “(more) fundamentally”; in which case yes, in a more complete physical description, gravity is in a superposition, but also perhaps no, in the model most apt for the experiment, gravity is not represented as quantum, and hence such behavior is not witnessed. That is, the dichotomy between Newtonian and tripartite paradigms is not (or need not be) one concerning the fundamental nature of gravity, but one concerned with the best model of a specific situation, and hence where one sets the bar for experimental observation of quantum behavior.

Finally, the proponent of the Newtonian model that we are imagining can articulate clearly the kind of experiment in which they would agree that the quantum nature of gravity was observed, maybe one in which no mere potential could do the job: for instance, the kind of graviton decoherence experiment that we have mentioned, as well as perhaps the retarded GIE protocols discussed by Reference Christodoulou, Di Biagio and AspelmeyerChristodoulou et al. (2022) that we have already referenced, or perhaps some observation of the different proper times read by clocks in different, superposed geometries.Footnote ⁴⁵ Such other experiments might, that is, claim epistemic virtue in having set higher the bar for witnessing the quantum behavior of gravity.Objection: If gravity is necessarily quantum in a full description, which, as we just emphasized, (nearly) all hands agree, isn’t that reason to accept the tripartite model, in order to represent the fact?Reply: Surely it is a reason, but there are other reasons not to accept the model. First, one could adhere to the maxim that empirically equivalent models which include fewer details are to be preferred: Why retain features that are superfluous to an empirically successful idealization? Such a maxim arguably prefers the Newtonian model precisely because it eliminates relativistic causality.

Second, does adopting the tripartite model in fact succeed in representing gravity as quantum? For the model is in the Newtonian regime, in which the only contribution to gravity is the Gaussian constraint, not any dynamical part of the field. (This point is true even in the specific models given by Reference Bose, Mazumdar and MorleyBose et al. (2017); Reference Christodoulou and RovelliChristodoulou and Rovelli (2019); and Reference Chen, Giacomini and RovelliChen et al. (2022); if the $|\gamma ⟩$ states were not the specific ones given, then the total state would violate the gauge constraint.Footnote ⁴⁶) In that case, in the tripartite models, the gravitational state is not describing a dynamical quantum mediator in the way intended in the analysis of Section 5.2.1. But then, the desire to represent the underlying quantum nature of gravity cannot be a reason to adopt the tripartite model, since that model cannot represent it properly after all! And yet, including a quantum state for gravity, even in such a hobbled way, represents something of its dynamical nature….

Stepping back from this to-and-fro, the arguments originally based in the physics have surely left the realm of unequivocal reasons for model choice, and degenerated to value judgments: the question of which model to select has come down to a question of whether some part of the formalism alone well represents a physical property such as the quantum nature of gravity. Our point, then, is that there are compelling physical arguments that accrue to both sides, even granting agreement about LEQG, yet the considerations are not of the kind that can be settled empirically, or by more fundamental principles. They rather “influence decisions without specifying what those decisions must be” (Reference KuhnKuhn [1977b, 362]). Which is why we described the positions as “paradigms.”Objection: The analysis given in Section 5.1 leading to the Newtonian model paradigm depends on a specific choice of gauge. But conclusions about the quantum nature of gravity ought to be gauge-independent; put another way, the conclusion that gravity is classical depends on an arbitrary choice.

Reply: There is nothing per se inappropriate about fixing a gauge to construct a model; one does it all the time. And that is exactly what is done here, as a means to take the Newtonian limit, as one must to describe the GIE experiment.Footnote ⁴⁷ (In line with the previous reply, one can of course agree that “really” the field is quantum, yet it is most appropriate to construct a gauge fixed model with gauge-relative observables.)

Taking stock: Having shown the failure of various attempts to find common ground to settle which of the Newtonian model and class of tripartite models, explicated in the first part of the section, is the better description of the GIE experiment, we provisionally conclude that they are paradigms in the sense explained. The merits of this conclusion will be clear in the following section, in which we critically assess the implications of a crucial and much discussed paper in the GIE literature, for interpretations of the experiments. Instead of a practical experiment, Reference Belenchia, Wald and GiacominiBelenchia et al. (2018) propose a thought experiment in which gravity putatively has to be treated as quantized mediator in order to avoid a paradox. With this claim, we essentially agree. However, the introduction to the paper can be read to claim that this conclusion further settles the question of the best account of the GIE experiment (something we have also heard suggested informally):

If that were correct, then there would be no rational indeterminacy after all; so, we are obligated to consider this objection as well. It will take a dedicated section to do so.