Jonathan Oppenheim and Andrea Russo, "Anomalous contributions to galactic rotation curves due to stochastic spacetime."
I'm kind of enjoying this paper despite the fact I don't really understand it. There's lots of things in this paper that I do not understand. But what can I get out of this, despite that? That is the question. And that's what this little essay is about.
Equation (1) is an equation that is at the heart of Modified Newtonian Dynamics or MOND theory.
It's really kind of an empirical observation. We could use equation (1) to explain the distribution of observable matter in almost all of the galaxies that we can see.
But what is equation (1) saying? It's really saying that there are two laws of gravity. One is the law that Newton gave us and the other is a different law that only applies when the gravitional attraction between two objects is below a certain level.
Or another way to express this is to think of it as accelerations. And this posited second law of gravity only applies to gravitationally induced accelerations that are below something like 10^(-10) m/s^2.
Equation (2) points out there is a connection, possibly coincidental, between 10^(-10) m/s^2 and Brownian motion.
Brownian motion was explained by Einstein. See Wikipedia.
The theory that this paper describes was not motivated by an attempt to explain MOND. Nor was it motivated by an attempt to explain dark matter. The theory builds on the work of a lot of people, but the authors, and these other people, were, and still are, trying to reconcile quantum mechanics with Einstein's General Theory of Relativity.
So it was a surprise when something very like MOND, ie. equation (1), popped out of the equations.
This theory is not a Theory Of Everything. This is a Theory Of Everything.
But it is a new theory of quantized gravity. This is a theory that gravity is not quantized.
[Edit: I didn't realize it the first time I read this paper but this theory, Post Quantum Gravity, aka Post Quantum Classical Gravity Theory, assumes that Einstein's Theory of General Relativity is an accurate description of the universe. All the equations of general relativity are a part of Post Quantum Gravity Theory.
Likewise Post Quantum Gravity assumes that Quantum Field Theory is an accurate description of the universe (apart from some modifications). Thus all the equations of quantum field theory (with some modification) are also part of Post Quantum Gravity Theory.
It isn't possible to reconcile Quantum Field Theory with the Theory of General Relativity if we assume the universe is determinate. Thus Post Quantum Gravity implies that the universe is indeterminate.
So this paper is not a description of Post Quantum Gravity. That was done in a previous paper that I have not yet read. What this paper explores is some of the consequences of believing that both the Theory of General Relativity and Quantum Field Theory are simultaneously true.]
In this theory, spacetime is not quantized. But all matter fields that exist within that spacetime are quantized.
I do not understand equation (3). I get that it's fundamental to this gravity theory, but it is not the theory itself.
This equation is generated by pushing things in a certain direction until you hit a limit that is simple enough to analyze. Or in the paper's own words,
"Here, we only study the classical limit of the theory. In this limit, the quantum matter degrees of freedom will have decohered, but in the dynamics of the classical-quantum framework, the classical degrees of freedom still undergo stochastic evolution. We will also not concern ourselves with the evolution of the matter degrees of freedom, and thus only represent them by their mass density m(x), neglecting the Hamiltonian term which governs their evolution."
There are a whole lot of equations that appear in the paper that are like that. They are the product of pushing things in different directions to the limit.
Equation (4) is another equation I don't understand, but again it is a product of pushing things to some limit and ignoring certain complications. And they note that equation (4) is consistent with "the Onsager-Machlup function."
Now the Onsager-Machlup function comes from a paper called "Fluctuations and irreversible processes." And this is going to be a constant theme of this paper. They are going to note all these equations that they can dervive that are consistent with the work that other people have done.
Equation (5) gives a "global maximum" that is pleasingly simple and that is consistent with Poisson's equation, and something else that I don't understand.
Equation (6), which would be applicable in a vacuum, is even more simple.
Equation (7) could come straight from Newtonian mechanics except that it has three terms added. And those three terms are a quadratic equation of the absolute value of x.
Equation (7) is connected to the language of "diffusive dynamics," ie. "Most Probable Paths."
And I think they are saying that these most probable paths are favored stochastic deviations from Poisson's equation.
They connect this to Brownian motion. And then they start to connect this to general relativity.
They push equation (7) to some limit in the context of a vacuum region and get equation (8).
And equation (8) resembles some things in quantum field theory.
Pushing things further they get a partition function, equation (9), from equation (8).
And from that equation (9) they get a "two-dimensional normal distribution in κ1 and κ2, in which the two random variables are anti-correlated, with a conditional mean which scales with inverse distance, and standard deviation which scales like GN / √D0V where V is the spatial volume of the region we are considering."
I take it that's a good thing!
They move back to relativity with equation (10), which again is a special case, "a spherically symmetric metric." And this is consistent with Einstein's theory and the "Schwarzschild solution."
I'm not sure where equation (11) comes from, but it has something to do with Brownian motion.
Appendixes A and B of the paper explain it in more detail, but taking a result from those appendixes you can show that γ3 to γ(infinity) can be ignored as they are too minor, and only γ0, γ1, and γ2 need to be tracked.
We get to equation (15), which leads to the observation,
"Since we are considering large-scale fluctuations which exist over all space, they are naturally suppressed by a volume element, so it’s interesting to see that γ2 is only suppressed by this amount. We also see here, or by explicit calculation, that higher powers in the expansion of Eq. (14) would be more suppressed, which motivated us to integrate out such higher powers."
And that gives us equation (16).
They integrate equation (16) over "4-geometries" and that gives equation (17).
Equation (17) matches "galactic rotation curves."
In this context, "γ2 corresponds to the cosmological constant term of Schwarzschild deSitter, while γ1 contributes to the geodesic equation of stars far from the galactic centre."
Equation (18) is "the covariance matrix of the normal distribution determined from" equation (15).
Equation (19) is the conditional mean of γ1, given an observation of γ2.
And finally they calculate γ1 ≈ 10^(−10) m/s^2 by plugging in the cosmological constant and the Hubble radius into equation (19).
And that's MOND. This is where they get MOND's theory empirical observation as a result.
The next paragraph in this essay begins with an interesting sentence,
"Secondly, since we expect that we live in a typical universe, this tells us that the variance in γ2 should be of the order of Λ2, so that the value of γ2 we witness is typical."
This implies that there could be different universes.
But if we are in a "typical universe," then through a chain of logic that I don't follow, they get that this "explains why general relativity is modified at low acceleration. Crucially, the fact that γ1 and γ2 are anti-correlated comes out of the path integral. We have also not needed to fine-tune β."
And then they have a section where they explore possible problems with the theory. I'll skip that except to say that a lot of it boils down to a need to do a lot more mathematical work to explore what this means in other directions than what they've looked at so far.
I made a partial transcript of Curt Jaimangul's interview of Jonathan Oppenhiem. This transcript is less than a third of that interview, but this is it seems to me a particularly interesting part. (The full one and a half hour interview is at )
Jonathan Oppenheim:
I wouldn't say I don't like the current approaches. I think I'm more of the view that there's been this assumption, and I think based on some incorrect arguments, there's been this assumption that we have to quantize gravity.
We started off by starting with the Einstein equation where you have matter on one side, which is quantum, and it should be equal to something which is classical, and that doesn't make sense, so people have thought we have to quantize gravity.
And there's been a bit of discussion about that, I think in part because we've spent more than 100 years failing to quantize gravity. And so because there's been about this 100 years of failure, well, it's useful to think, well, is this really the right approach, or is it possible that we've gone down, and are taking, a wrong direction?
So I don't know what the correct answer is. I don't know if gravity is quantum or classical or something else.
I suppose my perspective is just that we should -- because of the arguments that people marshaled claiming that we had to quantize gravity, because those have turned out to be incorrect, that I think it's important to revisit the issue and to explore the possibility that maybe gravity could somehow be fundamentally classical.
And I think there's some motivation for that. I don't think it is just a random idea that, okay, maybe gravity is special and classical. I think there's some good arguments you can make in favor of not quantizing gravity.
So that's my perspective, it's just that it's possible that it could be classical gravity. And there's some reason to believe that gravity is special and different to the other forces and therefore should remain classical.
Curt Jaimungal: What would some of those reasons be?
Jonathan Oppenheim:
So I think that gravity is different from other forces in that what Einstein's general relativity tells us is that matter causes spacetime to bend, and it's that curvature of spacetime that is the manifestation of gravity. So we're just following a straight line, we're just free-falling in a geodesic, and that's because spacetime is curved. That gives us the appearance of a force, which we call gravity.
But it's the only force which can be described universally as a geometry.
So in that way gravity is special. It universally describes this background causal structure -- which all the other fields live in this arena of this curved background geometry.
And so there are reasons to imagine that that background structure has to be fundamentally classical.
And I think one of the main reasons I would give for that is that the causal structure, which gravity gives us -- so it gives us this causal structure. I don't think that we really know how to do quantum theory without that causal structure.
At least I don't know any way of doing it.
And we can try and quantize it. But then in some sense I feel like we lose our legs. We lose all this background structure which we needed in order to perform quantum theory.
Curt Jaimungal: And is this related to -- you make a non-canonical choice at some point?
Jonathan Oppenheim: Right.
Curt Jaimungal: And so you lose something that's special about general relativity.
Jonathan Oppenheim:
That's right. Let's just even start with how physicists -- Like what is physics?
Usually what we do is we specify some system at some initial time. And then we ask how does it evolve? And predict what's going to happen to it at some future time.
And in a quantum theory of gravity, merely those statements are difficult to really make precise.
So for example, if we want to specify the initial state of the system, well, in a curved spacetime that's not so easy because you have to find some hypersurface across all of space, and you label that and say this represents an initial time slice.
That choice is not -- you know there's a number of ways that you can slice up spacetime into some family of hypersurfaces. These spatial Cauchy surfaces that are kind of evolving in time like this. And you can do that in quantum field theory because you have a definite causal structure.
So if I want to say that this curve slice here represents the state of the system at time t = 0 then that's a well-defined statement. But I don't know how to make that statement if the geometry itself is the thing I'm quantizing.
Because then I don't have that causal structure.
Now I can imagine just choosing one. And I just choose some particular slice and I then just quantize it as I would any other field theory. But as far as we can, as best we understand, the quantum theory of gravity that we would come up with will be dependent on this choice of how we chose to slice up our spacetime.
Curt Jaimungal:
What if the loop quantum theorist would say, hey, the constraints on such a system in our theory are such that it's independent of this even though we chose a Cauchy surface.
The observables are independent of such foilations.
So, does that not get over the objection?
Jonathan Oppenheim:
Well, so I guess the problem with -- so there [are] two approaches you could do.
One is you can just hope that your theory will be independent of the choices.
You can try and make -- have a completely background independent approach -- which is what loop quantum gravity tries to do. And in particular with the spin foam. You use something called spin foam networks, which is the object of interest, and take a background independent approach. And the problem then they face is that they have no idea how to recover the classical geometry at the end.
So there's no way for them to know, or they haven't at least been able to show, that in some low energy limit they will recover gravity, classical gravity.
On the other hand in string theory they kind of take the other approach where they have a background dependent theory and hope that they'll be able to either show that it's independent of that choice, or come up with a background independent approach.
And some would say that a lot of string theory now is something called ADS-CFT or holography, and there's some claim that that is to some extent, or to a larger extent, background independent.
Curt Jaimungal: So this Post-Quantum Theory of yours. Can you please describe it?
Jonathan Oppenheim:
So the main idea is that you somehow take seriously the idea that maybe we need this classical background structure of spacetime. And that it has to be classical.
You can start simply by saying: Can we consistently couple classical systems and quantum systems?
Is there any way to do that?
At the moment, we have only two frameworks for physics. We have quantum theory and we have classical mechanics.
We don't really have, at the moment, any credible other frameworks which are beyond quantum theory, for example.
And so the first question you can just ask is: Is it possible to consistently couple a quantum system with a classical system?
And there's been a huge number of approaches to that. Where people have tried to consistently couple a quantum system with a classical system. And almost all of those have been unsuccessful.
But actually in the early or the mid-90s, there were actually a few examples of such consistent coupling due to Dioshi and due to two people named Blanchard and Yajic. And they found some examples where you can consistently couple a quantum system with a classical system.
And so the first thing we had to do was just to say: Okay, what is the most general form of dynamics that we can come up with, which couples a classical system with a quantum system? So that's the first thing you do. You derive the most general dynamics that can do that.
Curt Jaimungal: Before we go to the second thing that you do, can you describe what it means to couple?
Jonathan Oppenheim:
Right. So we're used to the following coupling between a classical system and a quantum system.
Imagine that we are doing a double-slit experiment.
So we have, say we fire a bunch of electrons or some photons through two slits, and they form a diffraction pattern at the far end. And they interfere. So we see a nice interference pattern at the screen behind these two slits.
That's a coupling between a quantum system and a classical system. Because we treat the two slits as classical. We treat the screen at the back as classical, and there's this little photon gun, which is firing photons or firing electrons, and that's what we treat classically.
Or a partical in a potential, we treat this potential that's sitting there -- that's like a classical potential, which is produced by a magnetic field or something like that.
We treat that as classical. And the particle moves in --
Curt Jaimungal: I see. I see.
Jonathan Oppenheim:
So that's an example where the classical system exerts a force onto the quantum system. And we do that all the time in physics.
So we know very well how to couple a classical system which acts onto a quantum system.
What we didn't know how to do, except for these examples that, you know, there were these examples in the 90s, which, you know, curiously were not, I don't think, really known, except in a very small community.
So the question that you want to address now is: Can a quantum system backreact onto a classical system? Can it exert a force onto a classical system without causing a contradiction?
And there were arguments that were given as to why that will always result in a contradiction. And I think the most famous argument is due to Feynman at one of the first Chapel Hill conferences -- these famous conferences that were organized to discuss general relativity. And the argument was as follows. He imagines a double-slit experiment. And he imagines this particle, which, you know, sometimes you could imagine a particle which goes through slit number one, and sometimes it goes through slit number two.
And then he says, well, imagine that this particle has a gravitational field. And imagine that we measure this gravitational field. And imagine that we can measure the gravitational field to arbitrary accuracy. Then we could, by measuring the gravitational field to arbitrary accuracy, we can discover where this particle is, because we measure its gravitational field.
And then we would know if it went through the left slit or the right slit.
And if we know whether it went through the left slit or the right slit then we shouldn't have an interference pattern.
Yet we see interference patterns and therefore Feynman argued that we would have to quantize the gravitational field.
Curt Jaimungal:
Why can't someone say, look, we've never done that experiment where we've detected gravitationally whether its gone through A or B.
And if we were to do that, we would see that there would be no interference pattern.
Jonathan Oppenheim:
Right. That's a good question. And it turns out that if you merely try to write down the state of a quantum system and a classical system such that the classical system knows which slit the particle went through, then that's enough for you to not have an interference pattern.
So whether or not you measure the gravitational field, the mere fact of the gravitational field knowing which slit the particle went through would be enough to cause the interference pattern to not be there.
And that's the same with like, you know, people often ask about quantum theory: Do I have to look at the cat, whether it's dead or alive, in order to collapse it? Does a person have to look at it?
Well, no, just the environment measuring, you know, the environment measures the cat. Here the gravitational field is measuring the cat and [it] is measuring which slit the particle went through.
So if your environment is classical, then just the fact that in theory it could be measured to arbitrary accuracy to determine which slit the particle went through, that would be enough to destroy your interference pattern.
Curt Jaimungal:
Thus the conclusion is that the particle is in a superposition -- like even a gravitational field is in a superposition?
Jonathan Oppenheim:
Yeah, that's what Feynman concluded -- that the gravitational field had to also be in a superposition with the particle. That was the only way to consistently think of the double slit experiment in which the particle produced a gravitational field, because it was a massive particle.
But if you (and maybe this is in part comes from thinking about things from a quantum information perspective) -- one thing we've learned about quantum theory is that the state of the wave function, the quantum state, is more analogous to a classical probability distribution than it is to, say, a single C number or a single position of momentum of a particle.
So one way to think of the cat [being in] a quantum state is a bit analogous to a probability distribution. And because we think about probability distributions all the time in quantum information theory, it's quite natural to think about Feynman's no-go argument and to just say: Well, wait a second, what if the gravitational field is a probability distribution of different configurations?
Then measuring the gravitational field will not determine which slit the particle went through.
So, for example, if the particle goes through the left slit, it might produce some random, slightly random, gravitational field. And if it goes through the right slit, it will produce a slightly different random distributio of gravitational fields.
But because we have a random distribution of two different gravitational fields, measuring the gravitational field does not determine which slit the particle went through.
Curt Jaimungal:
Is a C number supposed to be thought of as a scalar? Or a complex number? Like, C stands for what?
Jonathan Oppenheim:
Oh sorry. Yeah. So I guess in this case, when I talk about a C number, I just mean that this is in the context of looking at Einstein's equation, where you have an operator on one side, and then it's just a number, the Einstein tensor, on the other. So it's a single number versus an operator or say, a vector, which is, you know in quantum theory observables are operators which act on quantum states, which are vectors.
Whereas in classical mechanics, we just have, you know, the result of a measurement, or sorry, a particle is just described by a number. This is the particle's position and momentum.
Curt Jaimungal:
I've heard the term C number. I've heard it but I've never read it. So I don't know what the definition. What is the definition of C number?
So what is the difference between classical randomness and quantum randomness?
You mentioned it before, but can you briefly outline it once more? Because you're about to make the connection between, well, you're about to explain how randomness solves a harmony issue.
Jonathan Oppenheim:
Right, right. So a classical system can have a probability. So we can imagine, for example, a particle as a position and a momentum, and we can also imagine that we have a probability distribution of different positions and momentums. So, you know, there's some probability that the particle has position x = 0 and the momentum, you know, 10 units. And we can imagine such a distribution, and the probabilities, both of its particular values that the position can take and the values that the momentum can take. Those are all positive. And they all sum to one, if we to sum them up.
On the other hand, [for] a quantum state, we cannot ascribe probabilities to a particular outcome of a position and a momentum measurement. That just doesn't exist.
Because we're either going to perform the position measurement or we're going to perform the momentum measurement, but we don't perform both.
And so the quantum state does not need to be described by a probability distribution over position and momentum, whose values are all positive. It's described by something else, which is, you know it can be described, for example, by something called a Wigner distribution, which looks a lot like a probability distribution, except its values are not always positive.
And that's okay because I will never measure and get a negative probability because I can't measure both the position and the momentum of the particle. So it's okay if a quantum state, if its distribution has a negative values.
That's not okay for a classical distribution. It has to always have positive values because, you know, the probability that a particle has a particular position and momentum needs to be positive.
Curt Jaimungal:
So now what's meant by the coupling between gravity, or some gravitational system, some classical system, and
some quantum system. [Does it have] to be stochastic?
Also can you outline what the difference between randomness and stochasticity is?
Jonathan Oppenheim:
Okay. I use those words interchangeably. Maybe there's a more technical terminology and maybe they do mean different things, but I tend to use them interchangeably.
Although I guess when I think of a stochastic process, I think of a dynamical process. So, you know, the particle is going through the left slit, and in a deterministic theory, it would bend spacetime in a particular way. And in a stochastic theory, it would -- it almost like flips a coin and depending on the value of the coin, it bends spacetime in a slightly different way.
So you can imagine that there's these coins being tossed all the time, which determines if the particle goes to the left slit and it produces some, you know, different gravitational fields with different probabilities. And if it goes through the other slit, the right slit, it will produce some -- it'll bend spacetime in some other way.
But the way in which it bends spacetime is determined not just by which slit it went through, but also by it flipping a coin. Now I want to be careful when I say flipping a coin, because that one is almost imagining that there is some physical process by which it determines which of these gravitational fields to produce. But actually I don't believe that there is actually any physical process which is determining which gravitational field is produced.
Curt Jaimungal:
Let's make it simple and imagine that someone's walking through two doors.
Does that mean they are constantly carry with them a coin? And even before they encounter those two doors, they're flipping it and making some other decision, and then when they get to the door, then they flip it and then it's a left-right decision?
Is this coin just being flipped for them? How does this work?
There's no physical process?
Jonathan Oppenheim:
Yeah. I mean, maybe even another way of saying it is imagine that someone's going through the right door, you know, someone goes to the right door or the left door. If I'm far away, I can sit there with a pendulum and I could actually figure out, I could try and figure out which door they went through by trying to measure the gravitational field.
But now imagine that at every point in space, the gravitational field is just fluctuating randomly. So it's as if, you know, it could have one value, but someone just goes and randomly makes that value be slightly higher or slightly lower. And so at every point in space, you can kind of imagine that there's this noise, this randomness, which is effecting the gravitational field and making it very difficult to measure and determine exactly what the gravitational field [is]. Whether it's being produced by the person going through the right door or the left door.
Curt Jaimungal:
Okay. Then the way that I'm imagining it is that you have a pendulum and it's (I don't know how the pendulum apparatus is supposed to be when you actually measure) -- but I'm just going to say that it's completely still.
And then you see, does it move slightly to the left or move slightly to the right? Because it's attracted to the person who goes to the left or to the right.
Okay. You're saying that actually if you were to look at that pendulum it would be constantly jittery because just even without anyone going through the doors --
Jonathan Oppenheim: Right.
Curt Jaimungal:
Oh, because what I was about to say is because this cannot be solved with more precision -- but then you would have to do a series of measurements.
Jonathan Oppenheim:
Yeah. So, I mean the most famous gravity experiment is probably the Cavendish experiment, where the sit there with this beam with two weights on the end and it's held by a string and it kind of rotates like this. And you use that to measure the gravitational field of the Earth or of two balls of a kilogram mass, for example.
And if you ever have seen that experiment or you've tried to do it in, say, an undergraduate physics lab, you'll see that the torsion pendulum kind of moves about quite a lot and is jiggling much in the way that we just described. And the reason it's jiggling is mostly because air molecules are hitting it and the system is very noisy and there's heat and we don't have very good control of gusts of air, which push and pull the pendulum.
But imagine that we got rid of all of those gusts of air and had everything in a perfect vacuum and didn't have any stray electromagnetic fields or gravitational fields around. But we cleaned up everything. Would there still be some fundamental noise?
And this theory predicts that there will be.
And it's this noise which somehow allows interference patterns because the gravitational field, because it's random and noisy, it doesn't allow us to really determine which slit the particle went through.
Curt Jaimungal:
Now, the quantum field theorist would say there is noise anyhow, because there's some fluctuations. So is there a way you [are] distinguishing the noise from the fluctuations versus, I don't know what type of noise this is called, this post-quantum noise?
Jonathan Oppenheim:
Yeah. That's a very good question. And there is disagreement. So, you know, I've had disagreement with some of my colleagues about this. But we've calculated how much noise there has to be. And there has to be a lot more noise. If gravity is fundamentally classical, there needs to be a lot more noise in the gravitational field in comparison to the quantum case.
And it's true that in the quantum case, you also need some noise there because you can imagine the same experiment, the double slit experiment that I just gave. You can imagine the same argument being made about the electromagnetic field. How is it that if the particle goes through the left or the right slit, I can measure the electromagnetic field. And, you know, what is it about the electomagnetic field which doesn't allow me to determine which slit the particle went through?
Curt Jaimungal:
So what is it about the electromagnetic field?
Jonathan Oppenheim:
Right, right. Well, it's, as you say, it's the fact that there is, we can't measure the electromagnetic field to arbitrary accuracy.
I can't measure because [since] the electromagnetic field has a quantum nature then I can't measure with exact precision, you know, the electromagnetic field and say it's conjugate degrees of freedom.
So because we can't measure the electromagnetic field to arbitrary accuracy, we're not able to determine which slit the particle went through.
Or another way of saying it is this. In quantum mechanics, you can have two different states which cannot be distinguished. In other words, the electromagnetic field will be in a different state depending on whether the particle went through the left slit or the right slit. But even though the state of the electromagnetic field is different, I still can't tell it apart. It has some overlap. Both states, there's overlap between the two states, and those two states are not orthoganal, we would say.
In other words, they can't be distinguished perfectly.
And it's because those two different states of the electromagnetic field are not distinguishable, it's that which allows you to still have an interference pattern.
Now sometimes you don't. Sometimes the particle will go through the left slit and it will emit a photon because it happened to hit the wall in a certain way. And if I were to measure that photon I would be able to determine which slit the particle went through.
So there is some decoherence. The interference pattern does get disturbed a little bit by the electromagnetic field, but not by very much.
Curt Jaimungal:
This theory of yours, does it have predictions?
So one prediction I see is that it's a null prediction. Namely that there is no graviton.
Jonathan Oppenheim: Right.
Curt Jaimungal:
Does it have other predictions?
Am I even correct by saying that there is no graviton?
Jonathan Oppenheim:
Right. There's gravitational waves, but there is no quantized particle which is responsible for carrying the gravitational force.
So one of the big predictions is this noise in the gravitational field.
So we predict that you should go and do the Cavendish experiment and you will have to see a large amount of noise in the gravitational field.
Now the problem is that there is already a large amount of noise in the gravitational field. If you ask the people at NIST who are responsible for keeping the one kilogram mass and doing these precise measurements of, say, a one kilogram mass, they will tell you that it's actually a very difficult experiment to perform and that their measurements do have quite a large variance and inaccuracy.
So the experiment we're proposing is, you know, higher precision tests of those measurements of, say, a one kilogram mass in order to put a bound on how much noise there is in the system.
I mean what's exciting [is] I feel like this question of whether or not gravity is fundamentally quantum or classical is a real one. And I think what's exciting is that this actually allows us, through these precision Cavendish measurements, to actually, you know, determine whether gravity has a quantum or classical nature.
Curt Jaimungal:
Would the noise, if you were to, with precision, measure the gravitational field, would that noise still be there even in other approaches to harmonizing gravity with quantum mechanics like string theory, where you sum over metrics?
So there is some uncertainty as to what the gravitational field is in string theory.
So would they say that that also would produce noise? Or is this noise distinguishable from the noise that you're talking about?
Jonathan Oppenheim:
So there will be some noise, and it has in some ways a similar form, but the amount of it is just much less in a quantum theory.
And the reason is that in a classical theory, you can perform in some sense two different experiments.
You can do a precision test of gravity and see if there's noise in the gravitational field.
And the other thing you can do is you can do an inteference experiment. So I can take a gold atom, for example, a very heavy atom, and see if I get an interference pattern.
And if you get an interference pattern, you can keep pushing how coherent you can make a gold atom. So I might imagine a gold atom that can follow two different paths and be in a superposition of these two different paths. And if I can keep the gold atom in superposition for a very long time, then it would mean that I would need a large amount of noise in the gravitational field in order to keep that coherence.
So there's a tradeoff in some sense between how long I can keep a gold atom in superposition and how much noise there needs to be in the gravitational field in order for the gold atom to keep being in a coherent superposition.
And there's a tradeoff between those two things. And so I can perform both those experiments. And the longer I'm able to extend the coherence time of [the] gold atom, the more noise I know there must be in the gravitational field if the gravitational field is classical.
So between those two experiments you could potentially, say, rule out the classical theory of gravity.
Whereas in the quantum case, there's no such tradeoff. There's a related tradoff, but it's not quite the same. And so it turns out that for a quantum system, if the gravitational field is quantized, then there doesn't need to be nearly as much noise in it.
So I've been thinking about J.S. Bell's
"Introduction to the hidden-variable question" (1971).
See
https://cds.cern.ch/record/400330/files/CM-P00058691.pdfIt's only a few pages long. But in those few pages Bell manages to say quite a lot.
I think this is also the second paper where he suggests doing an experiment that would prove either that there is something wrong with quantum mechanics or that there are no local hidden variables in quantum mechanics.
The paper is divided into four sections.
The first part explains what he's trying to do, ie. why he thinks hidden-variables are so important.
He gives three motivations for being interested in hidden-variable interpretations of quantum mechanics.
(a) He points out the great difficulty we are having defining the boundary between the classical world and the quantum mechanical world. And he guesses that most likely the real world is either all classical or all quantum mechanical. And that would mean that one of the two is an approximation of reality. And he points out, by the way, how much trouble we are in, in terms of understanding the world around us, if the world is all quantum mechanics.
So for instance if there are hidden-variables then that would remove the observer from the central role in physics that the observer plays in the Copenhagen interpretation of quantum mechanics.
(b) He points out that if the hidden-variables exist then we can basically turn quantum mechanics into something like a classical theory.
(c) He points out that there are parts of quantum mechanics, ie. specifically entangled particles, that look they might be, in a sense, hidden-variables.
In the second part of the paper he criticizes J von Neumann's reasoning in one part of von Neumann's paper,
"Mathematische Grundlagen der Quantenmechanik" (1932). Bell starts by objecting to how von Neumann reasoned about some of the math. To oversimplify, Bell thinks that Neumann oversimplified the issue and that von Neumann's logic doesn't prove von Neuman's conclusion that a hidden-variables don't work, or to say it another way that one can't build a consistent interpretation of quantum mechanics from hidden variables.
Bell points out that it's trivially easy to construct an ad hoc hidden-variable scheme for any given experiment. But it's very hard to come up with a comprehensive and consistent interpretation along these lines.
In the third part of his essay, Bell attempts to come up with a comprehensive and consistent local hidden-variable interpretation. Up to a certain point, it goes well, although as usual there's the problematic lack of definition of what a 'measurement' is. Or to quote J.S. Bell:
But it is just at this point that the notoriously vague "reduction of the wave packet" intervenes, at some ill-defined time, and we come up against the ambiguities of the usual theory, which for the moment we aim only to reinterpret rather than to replace. It would be very interesting to go beyond this point. But we will not make the attempt here, for we will find a very striking difficulty at the level to which the scheme has been developed already.
And so we arrive at part four of Bell's essay where he explains the difficulty with a consistent and comprehensive local hidden-variable interpretation for quantum mechanics.
Skipping the math, if you assume the existence of local hidden-variables and also that the math of quantum mechanics is correct, then you get a contradiction.
And it should be possible to actually see this contradiction or not, in a sense, by experiment.
The problem with actually conducting the experiment at that time was that it required a very low error rate. Or to put it another way, there seems to be a lot of noise in these actual experiments.
In any case there is a certain parallel between the experiment that J.S. Bell suggested in 1966 and the experiment that Jonathan Oppenheim suggested more recently. It's not the same experiment, but both are kind of about the boundaries between the classical and quantum. And note it's an unusual situation to have, in this context, where the theorists are able to suggest an actual experiment.