Is Gravity RANDOM Not Quantum?

Hey everyone, we have brand new product at the  merch store. More info at the end of the episode. The holy grail of theoretical physics  is to find the long-sought theory of quantum gravity.

But what if this  theory is as mythical as the grail of legend? What if gravity isn’t weirdly  quantum at all, but rather … just a bit messy? Or random?

So says the postquantum  gravity hypothesis of Jonathan Oppenheim. We have two great theories in physics that  together appear to explain almost everything. There’s general relativity–the theory of space  and time and gravity and the largest scales of the universe.

And there's quantum mechanics–the  theory of atoms, matter, and the smallest scales. These theories are each verified to astonishing  precision, and yet seem to contradict each other at the most fundamental level. We’ve talked about  these conflicts in the past, and the efforts to resolve them by unifying quantum mechanics  and general relativity into a single theory.

We usually think of this master theory as “quantum  gravity”, but that name hides an assumption: the assumption that the solution is to “quantize  gravity” in order to work correctly alongside quantum mechanics. But after nearly 100 years  of very smart people thinking about this, we still don’t know how to make gravity quantum.  Well, what if we’ve been going about it wrong all along?

What if gravity simply isn’t quantum?  Today we will ask whether it’s possible to come up with classical theory of gravity which is  compatible with all the weirdness of quantum mechanics? According to physicist Jonathan  Oppenheim, the answer is basically yes, but some additional weirdness must be added to  gravity and its interactions: it must be random.

Before we get into the type of “randomness”  gravity needs to make such a theory work, let’s discuss what happens when one naively  tries to couple classical gravity as we know it to quantum theory. Our starting point is  Einstein’s famous equations of general relativity. Here we have the Einstein field equations.

We have the Einstein tensor, which describes the geometry of spacetime, equalling a bunch of  constants times the stress-energy tensor, which describes the matter and energy  inside that spacetime. By the way, the Einstein field equations are pluralized  because these tensors are multi-part objects–it’s really a set of 10 partial differential  equations. But despite the name it’s ok to think of it as a single equation–it equates  the geometry of spacetime with its contents.

As John Archibald Wheeler put it, space tells  matter how to move, matter tells space how to curve. In other words, the right side  defines the dynamics–how matter will move, while the left is the mass-energy that gets moved.  So let's just call it the Einstein equation.

The Einstein equation is a classical  equation. Despite being complicated, its parts are regular numbers and vectors–for  example, the stress-energy tensor has things like momenta, energy, and mass of objects  in the universe, while the Einstein tensor describes a smoothly varying field with a  well-define and singular value everywhere. The Einstein equation of general relativity  describes spacetime and gravity, while the rest of physics is described by quantum mechanics and  the Schrodinger equation.

The Schrodinger equation is distinctly not classical–it does not describe  precise properties of objects but rather tracks the evolution of the wavefunction–the  fuzzy distribution of probabilities, representing what might be observed when a  measurement is made. Its subjects–systems of quantum particles and quantum fields–are  quantized, in that they tend to jump between discrete states rather than move or vary  smoothly like a classical object or field would. We tend to think that our classical, macroscopic  world arises as the combination of countless quantum interactions–we’d say that classical  behavior is a statistical limit of large numbers of quantum interactions.

In that spirit, many  physicists believe the general relativity and the Einstein equation are not fundamental,  but rather emerge from quantum foundations. And that's understandable, because we do know that  matter and energy are fundamentally quantum. There must be quantum fields and quantum particles that  somehow work together to produce the classical stress-energy tensor.

Those quantum entities  do all the weird quantum stuff, like existing in superpositions of being in multiple states  at once and having discrete energy levels. So the classical stress-energy tensor on the  right of the Einstein equation should emerge from some quantum analog. Standard approaches to  unifying quantum mechanics and general relativity have been to try to make the left side of  the equation quantum also–effectively to find a quantum version of the Einstein tensor  representing a quantum spacetime geometry.

But our long difficulties in making that work is  what led Jonathan Oppenheim to ask–what if only the right side of the Einstein equation  is quantum but not the left? What if matter and energy can be quantum, but the spacetime  they live in really is fundamentally classical? The challenge here is that both sides of  the Einstein equation have to be the same type of mathematical object.

So if the spacetime  represented by the Einstein tensor is classical, then we should probably understand how a  classical distribution of mass and energy defined by a classical stress-energy  tensor can arise from quantum parts. Take the Earth for example. Each of its  countless atoms has a bit of quantum uncertainty in its position.

Each atom is  in a superposition of multiple places at once until measured. If you were to  measure one atom, its location would resolve randomly within a small range  defined by the uncertainty principle. But each different possible location for that one  atom gives a different stress-energy tensor for that atom, and so a different spacetime geometry  due to the gravitational field of that atom.

So what if you try to measure the position of  the entire Earth by finding its center of mass? Think of that as measuring the position  of every single atom in unison. If those positions all get defined randomly, then any  significant variations cancel out.

It’s like flipping a coin a kajillion times–you tend  towards roughly end up with equal numbers of heads versus tails. In the same way, each  time you measure the center of mass of the Earth you’ll get the same position–accounting  for its own movement of course. You find the same stress-energy tensor and so the  same gravitational field each time you look.

This is why the stress-energy tensor for  macroscopic objects can be treated as classical. What about describing a classical gravitational  field for a truly quantum object? Let's look at two options: we can either write down all the  possible stress-energy tensors for all possible locations of that object.

The Einstein equation  could then be used to find a whole family of possible gravitational fields for each of  those. A superposition of many mass-energy distributions leading to a superposition of  possible spacetimes. While the superposition part of this is a quantum phenomenon,  each spacetime in the superposition is otherwise classical–for example, its smoothly  varying, without discrete, quantized states.

Alternatively, we could say that  there’s just one spacetime which is somehow uniquely defined by the superposition  of all of those mass-energy distributions. As though the effect of matter on the  fabric of spacetime is determined by every possible configuration  that quantum matter might be in. Both of these approaches are problematic.

Let’s  start with the second one I described–a singular spacetime shaped by the quantum superposition  of mass and energy that it contains. In fact, this is historically the most successful approach  to introducing quantum mechanics into general relativity. It’s called semiclassical gravity,  and it’s how Stephen Hawking figured out that radiation leaks out of black holes.

The idea is  that spacetime curvature is defined by something called the expectation value of the stress-energy  tensor. The expectation value is just the average measurement value you would get if you were  to measure a quantum system multiple times. It better reflects the global wavefunction than  does a single, random measured realization of it.

In our example of the Earth, we saw that  the location of the center of mass doesn’t depend on the quantum uncertainty of  its component atoms. That’s basically the same as saying its classical  stress-energy tensor is equal to the expectation value of the total quantum  stress-energy tensor of all of its atoms. But consider a truly quantum object–say, a quantum  version of the Earth that really could exist in a superposition in which there are two locations  for its center of mass.

If you were to measure the location of the quantum Earth it would resolve  to, say, left or right. The expectation value of its location prior to measurement would be  halfway in between. In semiclassical gravity, that in-between location defines the singular  spacetime geometry and the gravitational field.

In regular classical gravity, objects like apples  fall towards the center of mass. In semiclassical gravity apples fall towards the expectation  value of the center of mass. In the case of the quantum Earth in superposition it falls halfway  between the two possible locations.

In fact, each version of Earth should also fall  towards that inbetween point. For those living in one of those superpositions  unable to see the other superposition, the Earth would appear to be attracted towards  nothing. Needless to say, this seems odd.

Semiclassical gravity was always  meant to be an approximation, valid only in specific circumstances.  But this thought experiment rules it out as a consistent unification of quantum  mechanics and classical general relativity. OK, moving right along.

The other option I  mentioned for gravity being classical-ish is for there to be a different classical  spacetime corresponding to each possible mass-energy distribution in our  quantum stress-energy tensor. If our quantum Earth is in a superposition of  two locations, then instead of having a single, classical spacetime geometry representing the  average of the two, we might have a superposition of two spacetime geometries–one for each of the  two mass-energy distributions. In this case, our falling apple wouldn’t fall towards  the average center of mass.

Instead, it would fall to one or the other  possible locations randomly. This sounds more promising because now  we have apparently random motion under gravity that tracks the quantum  randomness in the position of the source of that gravity. But there’s  actually a big problem here also.

This time we violate a sacred  tenet of quantum theory: Heisenberg's uncertainty principle. The  argument is due to Feynman, Aharonov, and Rohrlich, and in somewhat  simplified form goes like this. We run our good-ol’ fashioned double slit experiment.

A  beam of quantum particles is shot at a pair of slits. The quantum properties of the matter allow  each particle to be in a superposition of paths, passing through “both” slits at the same time.  The wavefunctions of the particles interfere with themselves and produce an interference  pattern on the screen placed after the slits.

The shape of the interference pattern tells us  the wavelength of the particle’s wavefunction, which is equivalent to knowing it’s momentum.  But according to Heisenberg's uncertainty principle this means one shouldn’t be able  to determine the particle position. So if we measure the interference pattern we can’t  know which slit the particle went through.

But what if we try to cheat by placing a test  mass betweenf the slits. When our quantum particle passes through one or the other of the  slits then it will exert a stronger gravitational pull on the test mass compared towards  that slit. So the response of the test mass should identify which slit was traversed  without disturbing the interference pattern, allowing us to simultaneously  measure both position and momentum.

This method of cheating Heisenberg only works  if the gravitational field of the particle is localized and classical–i. e. the field really  does pass through one slit or the other.

This has been taken as a solid argument  for why gravity can’t be truly classical, even if it’s allowed to be in superposition. So far so bad. We found that a singular,  classical spacetime geometry that depends on the collective quantum state of  its contents doesn’t behave sensibly for quantum superpositions of matter.

But an  otherwise-classical spacetime geometry that is in superpostion along with the  superposition of its contents will betray information about its contents in a  way that violates the uncertainty principle. But there is a way to have a singular,  classical spacetime that allows its quantum contents to still behave like quantum  objects. The trick is to add … noise.

To add a type of randomness to gravity  itself. In Jonathan Oppenheim’s theory, which he calls post-quantum gravity–we still  have a singular spacetime–one gravitational field–but now it fluctuates randomly  at every point. And the distribution of values of those fluctuations reflects the quantum  superposition of the matter generating the field.

Another way to think of it is like  this. In regular general relativity, when two objects interact gravitationally, they  sort of learn each other’s location based on the direction of the gravitational field. It’s that  perfect learning of position that can violate the uncertainty principle.

But in post-quantum  gravity, gravitationally interacting objects don’t “learn” precise positions, but rather  a probabilistic distribution of the possible positions that depend on the position  wavefunction of the sources of gravity. Let’s see how this works with the example  of quantum Earth in superposition. The apple starts falling generally towards the Earth, but  due to the fluctuations in the gravitational field you can’t at first tell whether it’s  falling more left or more right.

In fact, those fluctuations cause the falling  apple to take a sort of random walk, sometimes drawn more to the left  superposition, sometimes more to the right. Let's see what the Earth is doing as the  apple falls? Well, in Oppenheim’s theory, there’s feedback between the noisiness of the  gravitational field and the quantum matter that generates that field.

So the quantum matter of  the Earth gets its own random fluctuations. These fluctuations slowly destroy the superposition–they  cause decoherence–basically, they collapse the wavefunction of the quantum Earth. This causes  the Earth to end up at a single location.

In a sense, with every interaction between the  apple and the Earth’s gravitational field, the apple conducts a very gentle  measurement of the Earth’s position. The strength of these measurements increases as  the apple gets closer to the Earth. Over the fall, Earth and apple become correlated, and the Earth’s  position becomes defined.

If the apple fell more to the left, that’s where the Earth’s center of  mass will end up, if the the right, the right. This fundamental randomness also resolves the  issue with the uncertainty principle when we tried to use gravity to measure the path in the  double-slit experiment. If the gravitational field of the particles in the beam are noisy, and that  noisiness reflects the superposition of traveling through both paths, then we can no longer use that  field to perfectly identify which path was taken.

Perhaps the most radical aspect of  Oppenheim’s idea is that it throws away determinism. It requires that this noise  in the gravitational field is truly random. And perhaps the most radical consequence of  that is that it allows for the destruction of quantum information.

Conservation of  quantum information is thought by many to be among the most fundamental underpinnings  of quantum mechanics–that without it quantum mechanics is inconsistent. At the same time,  insisting on the preservation of information led to various challenges, like the black  hole information paradox. If you allow quantum information to be destroyed by random  fluctuations, then no such paradox exists.

So, it seems Oppenheim’s theory can describe  in a consistent way how classical spacetime evolves with superpositions of quantum matter,  and also how quantum matter itself evolves under the interaction with a classical spacetime.  Post-quantum gravity isn’t likely to be our final theory, and it may or may not be on the  right track. But it’s very interesting to see that there are still new directions to solving  this century-old problem.

It give us hope that we’ll eventually unify our two great theories.  Will it be quantum or post-quantum gravity? As long as we figure it out, I’m cool living  in either a quantized or a random spacetime.

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