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Philosophical Issues in Quantum Gravity

16 July 2023

It is well known that general relativity (GR) and quantum mechanics (QM) do not play nicely together. Their mathematical descriptions, and our physical conceptions of them, are so disparate that it's hard to imagine what their unification should look like (or even what "unification" should mean). That's not to say we can't unify them even approximately—we can, by studying quantum field theories (and their back-reactions) on curved spacetimes. But such "semi-classical" approximations only get us so far: quantum theories in gravity. What we want is a quantum theory of gravity. But is this what we need?

I think yes, and I'd wager that that's an uncontroversial opinion. By their nature, neither GR nor QM is universal in scope, since GR only describes the gravitational dynamics and effects of classical matter, and QM only describes the (non-gravitational) evolution and interactions of quantum matter. Based on this (and also that no known observations require a quantum theory of gravity), one might posit that the matter content of our Universe exists in a perfect and exhaustive disunion of quantum and non-quantum (i.e., classical) parts. This may seem reasonable prima facie (especially to devout subscribers of the Copenhagen "interpretation"), since quantum theory coincides with observations to exquisite precision without any gravitational considerations whatsoever (e.g., theorists at CERN disregard gravity entirely when calculating scattering cross-sections) and vice versa (scientists as NASA chart trajectories to Mars without ever using the Schrödinger equation). However, this perspective is manifestly flawed, since the field equations of GR intricately couple the energy content of matter fields to spacetime curvature. Since quantum matter has nonzero energy, GR and QM must overlap, and this begs for a quantum description of gravity.

String theory, loop quantum gravity, supergravity, causal set theory, and the rest of the panoply are all different approaches to this same problem: how do we describe gravity quantum mechanically? One way forward, largely inspired by the unprecendented successes of the Standard Model, is to quantize gravity. In this picture, the gravitational field is composed of many massless spin-2 quanta called gravitons, like how the electromagnetic field is composed of many massless spin-1 quanta called photons. However, to date there is no known observation that corroborates the view that gravity is quantized like the electromagnetic field. All we know (\(\gtrsim 5.1 \sigma\)) is that if gravitons exist, then they have an incredibly small mass of order \(10^{-22}\) eV (this is from the inaugural LIGO paper). Moreover, unlike photons, it is very unlikely that we will ever espy individual gravitons using today's technology. Indeed, Dyson argued in 2013 that a LIGO-like, single-graviton-detector is doomed to collapse to a black hole before a measurement is ever made. What, then, is the origin of the widespread belief that gravitons exist?

Perhaps the answer lies in the successes of the quantum gravity theories we have so far. String theory, for example, tells us that there exists a mathematical framework in which the quantum fields we know and love, and a new and massless spin-2 field—the graviton—happily coexist. This is an important fact, and probably one of the greatest accomplishments of string theory so far, so I do not mean to downplay its significance. However, this mathematical statement does not prove that gravitons exist anymore than, e.g., the Banach-Tarski paradox proves that energy conservation is flawed. Like all science, only through the empirical verification of the predictions of string theory (or any other theory conditionalized on the existence of gravitons) will abductive and Bayesian reasoning tell us to increase our bets that gravity is quantized. And it is this critique that applies across the broad spectrum of quantum gravity theories: no matter a theory's mathematical successes, the burden of proof lies in experimental science.

Let's talk experiments, then, which does not just mean experiments going on at state-of-the-art laboratories like LIGO and CERN. By "experimental science" I mean the blanket term that covers all experiments that are in principle possible. This includes, for instance, thought experiments with the (sometimes) logical laboratory inside our carbon heads, as well as programmatic experiments going on inside the silicon heads of even more logical computers. It also includes experiments that we humans might never be able to carry out because of a technological (or some other non-physical) limitation.

You might think, "If we can never carry out an experiment for real (not just on a computer or in our heads), then surely there is no sense in which the alleged 'experiment' is an accurate telling of what would happen if, say, some super advanced, posthuman species were to actually carry it out." It's ostensibly a reasonable position, but I think it's moonshine insofar as mathematics accurately portrays our Universe. In my Platonic view, if one conditionalizes an experiment on a set of true physical premises (by which I mean premises that are experimentally verified beyond all doubt, say \(5\sigma\)), then, so long as the implication arrows flow unidirectionally from the premises to the conclusion, the conclusion is at least as strong as the \(5\sigma\) premises on which it is based. In other words, if one uses deductive reasoning (which is logically certain), and neither abductive nor inductive reasoning (which are statistically uncertain), then the deduced conclusions will be at least as certain as the conditionals. In this way, the outcome is still falsifiable, as Popper would like it to be, since one can test the premises on which the outcome is based. This is a profound realization, and it is why thought experiments are such an empowering tool in physics (and why a theoretical physicist invariably uses the Latin phrase "a priori" when describing their research).

One such empowering experiment is due to Eppley and Hannah. According to Mattingly, it is their experiment that is responsible for the widespread suspicion that gravity is, in fact, quantized. It goes as follows.

Suppose gravity is classical. This can mean a whole host of different things. In Eppley and Hannah's paper, it means that a gravitational wave can have arbitrarily low momentum and, simultaneously, arbitrarily short wavelength. This is in stark contradiction to de Broglie's relation, \(p = h / \lambda\), but the point is that this is to be expected, because the classical gravity wave need not abide by the familiar rules of QM. Now, with this low momentum/small wavelength gravity wave, Eppley and Hannah show that one can in principle isolate a quantum particle to within one wavelength of the gravity wave, while introducing vanishingly small momentum uncertainty into the particle. This, in turn, implies the quantum particle's position and momentum can simultaneously be known to arbitrary precision, thus violating Heisenberg's uncertainty principle. It's as if the classical wave infects the quantum particle with its "classicality," and the particle's characteristic "quantumness" dies away.

There is more to the argument that I won't go into, but the key point is that this implies that gravity cannot be classical in the sense defined (at least, it cannot be classical insofar as Heisenberg's uncertainty principle is true). However, on closer look, Mattingly, in a gloriously titled paper "Why Eppley and Hannah's Experiment Isn't" (although PRD didn't let this title slide), realized that their experiment is not possible, not even in principle. By carefully tracking all that would be involved to make such an experimental apparatus, Mattingly shows that, as a result of the fundamental physical constants in our Universe—namely, \(\hbar\), \(c\), and \(G\)—the detector would have to be so large as to be contained inside its own Schwarzschild radius. Thus, Eppley and Hannah's detector is fundamentally doomed to collapse to a black hole before any experiment can be done! This puts Eppley and Hannah's experiment into a head-scratching epistemological status. Should we accept the conclusions of a thought experiment if, even in principle, the experiment cannot be done? I, of course, will loudly voice "no, not at all" because we are deducing on physically defective premises. It's like asking if you will accept the conclusions of an experiment that presupposes a particle that can move faster than light. The mathematical deductions in such an experiment might be true, but the physical premises, and thus the physical deductions, are certainly not.

All this shows is that the main experiment that underlies the suspicion that gravity is quantized is fundamentally flawed. To me, this is worrisome, because there have been decades of work toward developing a quantum theory of gravity that assumes gravity is quantized, and now the foundation for this work seems to have collapsed. Fortunately, there are other thought experiments that seem to salvage this idea, but the verdict is still out. In my thinking about this, I formulated the following thought experiment, which has, at least for now, convinced me that it is reasonable to suspect gravity to be quantum in nature. In this way, in doing, for example, calculations in string theory, I feel less concerned about whether I think gravity is quantized (as string theory assumes it is) and more about whether I got the indices in my solution in the right place! My variation goes as follows.

Suppose gravity is classical. By this I do not mean what Eppley and Hannah meant. In my mind, what differentiates the "classical" from the "quantum" are the information-theoretic resources that set the quantum and classical worlds apart. The main one is entanglement. Therefore, by saying gravity is classical, I mean that gravitational degrees of freedom cannot be entangled. With this in mind, consider now a massive object in the spatially separated superposition \[\psi(x) = \alpha\psi_L(x) + \beta\psi_R(x),\] where the \(\psi_L\) and \(\psi_R\) wavefunctions have support in disjoint neighborhoods. We assume the object is so massive that it has a non-negligible gravitational effect. Given this setup, my question is simple: what does the gravitational field look like? I see two possibilities, based on whether gravity significantly contributes to decoherence or not.

If gravity does not contribute to decoherence (or at least does so insignificantly), then, because gravity is classical, there can only be a gravitational influence emanating from one branch of the superposition. Otherwise, if, say, Alice makes a measurement in the \(\psi_L\) neighborhood and does not find the object, then whatever gravitational influence was emanating from the \(\psi_L\) neighborhood must have to either disappear at least as fast as Alice makes her measurement (which is nonclassical and could allow for superluminal communication if the branches are far enough apart) or a gravitational influence would remain under both branches, despite there not being a source of energy to cause the gravitational influence in Alice's vicinity (besides the long-range influence from the other branch). The only conclusion I see is that exactly one branch has a gravitational influence—namely, that branch that "contains" the object. But this means that gravity does induce significant decoherence, which is a contradiction.

So maybe gravity does contribute to decoherence. But then, I claim, that the gravitational decoherence must be so prominent that it destroys the superposition very quickly after it is created. Suppose this wasn't the case, so that there is time to spatially separate the superposition. Then we are back to the contradictory case above. Hence, the superposition must collapse very quickly after it is created. That is the only way to prevent gravity from entangling with the two branches of the massive objects. But this violates the expectation in quantum measurement theory that it's possible, at least in principle, to create a spatially separated superposition of a massive object.

This simple argument suggests to me that gravity is either not classical or massive objects cannot exist in spatially separated superpositions. I suspect we will be able to do the latter eventually. Beyond technological impediments associated with isolating massive physical systems, it's not clear to me why Nature would forbid us from spatially separating a superposition of tennis balls. So, I bet on the former—that gravity is quantized. I am very interested to further investigate this experiment.

Incidentally, there have been a few papers related to this thought experiment (notably Penrose's); however, I get the impression that many in the physics community simply do not care about these papers and the issues they call out. They exist, albeit not as ubiquitously, in a similar state-of-affairs to the early papers about interpretations of QM. Like late Copenhagen subscribers, who, in earnest, would cite Bohr and von Neumann to disabuse others that there is no other way to think about QM, so too in the quantum gravity community, where many cite Eppley and Hannah as evidence that gravity is quantized. But that Eppley and Hannah's paper (and von Neumann's "proof" that Copenhagen is the only way to think about QM) is flawed is a testament to how misguided these "appeals to authority" can be.

We must not be so enamored with the status and pursuits of those before us that we mindlessly adopt their pursuits while remaining blind to the foundational issues that underlie them. In my opinion, this is a vacuous and inefficient approach to science that is to be avoided, especially when it comes to pursuits as important as quantum gravity. In Feynman's words, "In constructing theories of gravitation, we should be wary about accepting too glibly many of the prejudices of the present scientific thinking."

Posted in: Philosophy, Quantumness
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