PBS Space Time

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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 isnt 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.

Theres general relativitythe theory of space and time and gravity and the largest scales of the universe.

And there's quantum mechanicsthe 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.

Weve 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 dont know how to make gravity quantum.

Well, what if weve been going about it wrong all along?

What if gravity simply isnt quantum?

Today we will ask whether its 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, lets discuss what happens when one naively tries to couple classical gravity as we know it to quantum theory.

Our starting point is Einsteins 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 objectsits really a set of 10 partial differential equations.

But despite the name its ok to think of it as a single equationit 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 dynamicshow 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 vectorsfor 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 classicalit does not describe precise properties of objects but rather tracks the evolution of the wavefunctionthe fuzzy distribution of probabilities, representing what might be observed when a measurement is made.

Its subjectssystems of quantum particles and quantum fieldsare 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 interactionswed 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 alsoeffectively 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 askwhat 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.

Its like flipping a coin a kajillion timesyou tend towards roughly end up with equal numbers ofheads versus tails.

In the same way, each time you measure the center of mass of the Earth youll get the same positionaccounting 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 classicalfor example, its smoothly varying, without discrete, quantized states.

Alternatively, we could say that theres 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.

Lets start with the second one I describeda 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.

Its called semiclassical gravity, and its 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 doesnt depend on the quantum uncertainty of its component atoms.

Thats 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 objectsay, 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 geometriesone for each of the two mass-energy distributions.

In this case, our falling apple wouldnt 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 theres 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 particles wavefunction, which is equivalent to knowing its momentum.

But according to Heisenberg's uncertainty principle this means one shouldnt be able to determine the particle position.

So if we measure the interference pattern we cant 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 classicali.e.

the field really does pass through one slit or the other.

This has been taken as a solid argument for why gravity cant be truly classical, even if its 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 doesnt 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 Oppenheims theory, which he calls post-quantum gravitywe still have a singular spacetimeone gravitational fieldbut 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 others location based on the direction of the gravitational field.

Its that perfect learning of position that can violate the uncertainty principle.

But in post-quantum gravity, gravitationally interacting objects dont learn precise positions, but rather a probabilistic distribution of the possible positions that depend on the position wavefunction of the sources of gravity.

Lets 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 cant at first tell whether its 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 Oppenheims theory, theres 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 superpositionthey cause decoherencebasically, they collapse the wavefunction of the quantum Earth.

This causes theEarth to end up at a single location.

In a sense, with every interaction between the apple and the Earths gravitational field, the apple conducts a very gentle measurement of the Earths 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 Earths position becomes defined.

If the apple fell more to the left, thats where the Earths 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 Oppenheims 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 mechanicsthat 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 Oppenheims 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 isnt likely to be our final theory, and it may or may not be on the right track.

But its very interesting to see that there are still new directions to solving this century-old problem.

It give us hope that well eventually unify our two great theories.

Will it be quantum or post-quantum gravity?

As long as we figure it out, Im cool living in either a quantized or a random spacetime.

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