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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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