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Comment by millstone | Hacker News Reader

Armchair physicist here, and yet:

> The probabilistic predictions of quantum theory are conventionally obtained from a special probabilistic axiom. But that is unnecessary because all the practical consequences of such predictions follow from the remaining, non-probabilistic, axioms of quantum theory, together with the non-probabilistic part of classical decision theory.

Well this seems crazy! We can drop the Born rule axiom, because you (a game theorist) will make the same decisions whether the universe is deterministic or probabilistic?

The difference is crucial, right? A PRNG requires hidden state: exfiltrate the state and predict all future results. But there is no room in the wavefunction + Schrödinger equation for such state: you either augment it (Bohmian mechanics) or accept the essential probabilistic nature.

Probabilistic predictions cannot be obtained from deterministic axioms, just like PRNGs cannot produce true randomness.

How does Deutsch resolve this?

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Deutsch is a many-worlder so he would say there is nothing to resolve because there is no randomness. Everything in the time-evolution of the multiverse is deterministic.

Could I ask a question? As you are clearly someone with a deep knowledge QM.

Why is Bohmian mechanics so frequently dismissed? It's such a straight forward premise - there is an actually physical guiding wave and everything else falls out "normally". Instead pages are written about MWI and hypothetical agents optimizing their overall decision in universes they'll never see. And undefinable concepts of trillions of universes splitting simultaneously. And I'm not trying to be snarky at all, as I feel what you wrote above is one of the best presentations od the topic I've seen.

To be perfectly frank QM is starting to scare me about physics. To a layman which I am, Bohmian mechanics is so simple and straightforward it's almost "obviously right". The macroscopic analogy of Farady waves is almost a nail in the coffin (to a layman). Allow for non-locality of guiding waves, just like ripples on a lake and everything else is deterministic and matches all scientific observations. Events have probability for the same reason the weather has probability: because we can't measure with the sensitivity required to account for the complexity of the system and the difference in initial conditions.

And yet, I've never seen any write up against Bohmian. I've never seen anyone with deep knowledge on the subject discuss why Bohmian isn't the best interpretation. It's just dismissed as an "also-ran". What scares me is 20 years from now something will emerge making it clear that Bohmian is the "correct" interpretation. And then the question in my mind will be, "what took it so long to even be seriously considered?". It would be proof to me that physicis is being shifted from investigating and following up on the best explanation for data to instead becoming "lawyers" trying to find data to support their favorite pre-decided argument (ie their favorite QM interpretation).

Now all of that said, there is a reason why laymen don't make meaningful contributions to a field. Their are deeper complexities that make their intuitions wrong. But why are these flaws in Bohmian never discussed?

Would you mind taking even a few sentences to write up "what's wrong with Bohmian" that would make the infinitely more complex MWI a more likely candidate? I'm lost.

[1] - http://www.tcm.phy.cam.ac.uk/~mdt26/tti_talks/deBB_10/bush_t...

The best place to find the answer to this question that I know of it David Z. Albert's excellent book, "Quantum Mechanics and Experience." But the short answer to your question is that Bohmian mechanics has two problems:

1. In order to account for the outcome of Bell-type experiments on entangled particles it has to assign a temporal ordering to space-like separated events. The technical term for this is that you have to choose a "preferred foliation of space-time". There has to be a preferred reference frame. But you can never actually know what the preferred reference frame actually is.

2. Yes, particles "have positions", but you can never actually know what those position are (which is why I put "have positions" in scare quotes). This is where all the quantum randomness hides in Bohmian mechanics. It's all "pre-computed" in the infinite precision of a particle's position, but that position is necessarily hidden from observation. I call this an IPU, an Invisible Pink Unicorn. It's exactly the same thing as universe-weights in MWI -- a set of numbers that are part of the theory but rendered immune from observation not by practical limitations on technology, but by the theory itself.

This is the fundamental problem with all attempts to make quantum mechanics look deterministic. The simple fact of the matter is that it's not deterministic, so any attempt to make it look deterministic that makes the same predictions as QM has to hide the randomness somewhere. Bohm hides it in particle positions, and MWI hides it in universe weights. But it doesn't matter what you call the place in the theory where you've hidden the randomness. What matters is that there is a place in the theory where you've hidden the randomness, where it must forever remain hidden from the prying eyes of experiment. So the claims that both Bohm and MWI make of being deterministic are misleading at best.

>but that position is necessarily hidden from observation. I call this an IPU, an Invisible Pink Unicorn.

But why expect that all state of the universe be open to observation? This seems counter-intuitive to me. It seems far more reasonable that there necessarily are facts about an implementation that no supervening system can determine from within that system. For example, there are facts about a physical computer that no software running on that computer could deduce. So the fact that a QM theory posits state that is in principle off-limits to observation doesn't seem like a reductio, but the expected case.

That's a good point, but remember, this is about rhetoric, not physics. The question is not whether hidden state exists (it clearly does) but what kind of story you want to tell about it. If you find it enlightening to think about hidden state as position, and you don't mind accepting all of the difficulties that entails (like having to choose a preferred foliation), then by all means go for it. But that is very different from saying that this story is actually true. The only reason to prefer Bohm over a similar story that ascribes the randomness to a literal invisible pink unicorn making decisions about experimental outcomes is aesthetics.

What EPR and Bell's arguments showed is that if you have definite results of experiments at the space-time locations where/when we think the results happened, then there has to be something non-local going on and a foliation is the simplest way to orchestrate that. So either give up a certain kind of definiteness (MW) or introduce a foliation (BM). A foliation is somewhat unpleasant, but there are ways to tease them out of the existing structures of relativity + wave function [0].

As for determinism, that is not the main point for many proponents of BM. Rather, it was about having a clear theory. Let's start at the beginning. We decide to have a theory about particles. What does that mean? Well, there is some stuff with positions and those positions change in time. And that's what BM gives. It explains, immediately, why a wave function is on a configuration space of particles. Having this leads to a variety of important notions. For example, having a position leads to clarity on identical particles, namely, use a space without labels on the particles and the wave functions just work out[1] (disclaimer: I am an author on that one and did part of my PhD thesis on that [4]; my thesis also derives spin as well as the Dirac equation from a Bohmian perspective).

Another example is QFT and divergences. From a Bohmian perspective, QFT is best thought of as about wave functions over a configuration space consisting of different disjoint sectors involving different number of particles. To do this, there is a random jump process created (not deterministic!). Thinking about wave functions that work with that, one is led to wave functions where the probability moves from one sector to another appropriately based on this jumping. And that solves, at least in some simple cases, the UV divergence of QFT.[2]

The randomness of BM is actually quite interesting. It is all about the Quantum Equilibrium Hypothesis, something which is justified in similar ways to thermodynamics kinds of arguments. In fact, it makes it clear that what is interesting is not why we can't know some things, but why we can know stuff at all. [3]

Also, in case you have not seen it, you might want to take a look at a version of MW by some of the Bohmians behind the papers.[4] As with most things, there is a lot of clarity in their perspective.

Finally, as for why BM is not more popular, well, let's just say it is rather hard to stay in academia as a Bohmian. I barely tried, largely because academia is unpleasant for a variety of other reasons, but it simply is very hard to get hired when working on unfashionable material. Grants and all that. This is on top of the problem of getting people to change some fundamentally long held beliefs. This goes hand in hand with the fact that standard QM is the thermodynamics version of BM and so QM agrees with BM empirically to the extent that QM makes predictions. That is to say, BM provides the rigorous foundation for the collapse rules.

[0]: Can Bohmian Mechanics be Made Relativistic? http://arxiv.org/pdf/1307.1714 [1]: Fermionic Wavefunctions on Unordered Configuration Space. http://arxiv.org/pdf/1403.3705 [2]: Bohmian Trajectories for Hamiltonians with Interior–Boundary Conditions. https://arxiv.org/pdf/1809.10235.pdf [3]: Quantum Equilibrium and the Origin of Absolute Uncertainty. http://arxiv.org/pdf/quant-ph/0308039 [4]: Many Worlds and Schrodinger's First Quantum Theory. http://arxiv.org/pdf/0903.22111 [5]: Connections with Bohmian Mechanics. http://jostylr.com/thesis.pdf

> We decide to have a theory about particles.

I would say that's exactly where Bohm runs off the rails. The fact of the matter is that quantum systems are not particles. They are waves [1]. They can sometimes bunch up into very small spaces and behave to a very good approximation as if they were particles, but they aren't.

If you insist that your theory talk about particles, then Bohm is a not-entirely-unreasonable place to end up. But that's kind of like saying that if you insist that your theory tell you how many angels can dance on the head of a pin that 42 is a not-entirely-unreasonable answer.

Thanks for the references, those look interesting.

P.S. The reference 4 link is broken. I'm guessing you meant https://arxiv.org/abs/0903.2211

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[1] https://arxiv.org/abs/1204.4616

Yes, that article is what I meant.

I read the article you cited. While enjoyable, it does not make its case for me. It very much feels like presuming the wrong ideas of what a particle theory is.

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First, double slit. Points on the screen, wave pattern built up. The natural conclusion is a wave guiding a particle. That's what BM gives. BM does not deny the existence of the wave function. Rather, it gives it a reason for being. The article talks about viewing the wave function collapse as a ballon hitting a needle and popping. But it doesn't really explain why a world full of waves should have anything point-like. It certainly doesn't come from the dynamics of the Schrodinger equation. It just is a statement that the wave should localize when it encounters something already localized. Now, this being the wave function, it is not multiple waves on real space, but rather waves on R^3N space where N is the number of particles which includes those of the detecting screen. But what particles, right? An electron does not have a wave function. There is only one universal wave function. This is not N waves rolling around in 3-space. It is one wave in 3N space and its relation directly to our experience in 3-space is rather obscure.

This does not mean you can't have waves. But I think your angel analogy applies to saying "everything is a wave" and MW is not an unreasonable place to end up in.

The article also says that particles are not logically consistent with the 2-slit experiment. That is simply false as BM demonstrates. That theory has been mathematically proven to exist and agree with the standard QM predictions. It works. The double slit is explained as particle AND wave, two separate entities.

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Another piece of the article was about a confined wave that instantly expands. It seems to get at the heart of the misconception. There is no notion in BM of having to confine waves to make them particle like. Rather, BM allows waves to expand as they do. They can be their own thing, doing whatever they like. The particle aspect is handled by the particles which can do what the wave tells them to do. In the Dirac version of BM, the velocity can never be greater than the speed of light for the particle, basically by construction. To the extent that a narrowly confined wave function leads to problems, this would say, those would not be the relevant wave functions. One can always replace such a sharp thing with a close enough in L2 approximation to not have that kind of sharp behavior. This gets into the domain of the Hamiltonian which can have some pretty important aspects to the evolution.

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The vacuum and movement. This is just me spitballing, but in the ground state, according to BM, particles will not move. They just sit there. So it is quite possible for the vacuum to look empty (nothing moving), but there is plenty of stuff out there in terms of particles. Then when someone moves, they see the particles moving. Not sure if this is reasonable or not, but it is my thoughts on how one can have a vacuum with "no particle" and then have particles present with a relative motion.

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The final part I will comment on is that of QFT. This would presumably be the strongest of the arguments. But here, the field of operators is fundamentally different than the wave function. As presented in the article, we have a field of operators based on EM. They operate on wave functions over Fock space, which is the union of R^3N spaces (removing collisions and replacing that with boundary conditions). Is the claim that the operator field's expectation values are the reality of our experience? I am guessing the claim is that if I want to verify my chair's placement then I am supposed to compute the expectation values of those operators at the space-time point and out pops a chair?

The Bohmian version of this is that we use the wave function, evolving according to a Hamiltonian which may contain an operator valued-field, to guide the particles, including controlling the creation and annihilation of the particles. We can then do an analysis of the theory and come up with predictions, generally in the form of operators as observables. These are deduced, not postulated. A chair is where it is because the chair's particles are where they are. It is a conceptually simple process to map my experience to what the theory is talking about. Having a simple map from my experience to some state of something in the theory is a really nice feature. It is not crucial, but verifiability is greatly helped by this.

--- A very important part of BM is that operators as observables is not postulated, but deduced. This has major advantages in that it is practically trivial to write down a Bohmian theory on a manifold. The standard stuff has problems, such as what the momentum operator becomes. In BM, you write down the theory and then one can analyze to see what would emerge, if anything, to take the place of that observable.

An application of this is creating a quantum theory on shape space, namely, the space of relative configurations that Julian Barbour likes to work with. It is a very natural space to consider and Bohmian mecahnics can accommodate it easily: https://arxiv.org/pdf/1808.06844.pdf

> The natural conclusion is a wave guiding a particle.

Indeed. But the natural conclusion can be wrong. Case in point: hold an object in your hand and let it go. It falls. The natural conclusion is that there was a force pulling it down. But this is not actually true. (I gather I don't have to explain this to you. AFAICT, you know physics better than I do.)

In fact, if you think about it, "points on the screen" is the only reason we have to believe in particles, which is to say, in spatially-localized quanta. But this is not probative. It only shows that the spatial localization is of the same order as the size of an atom. But we already know that atoms aren't particles, so this is manifestly not slam-dunk evidence that whatever is tickling those atoms is a particle, nor than an atom's constituent parts are particles.

> So it is quite possible for the vacuum to look empty (nothing moving), but there is plenty of stuff out there in terms of particles.

That doesn't seem reasonable to me. Particles exert forces on each other via gravity and electromagentism. I don't see how you're going to get a stable static vacuum configuration out of that without special pleading.

> Is the claim that the operator field's expectation values are the reality of our experience? I am guessing the claim is that if I want to verify my chair's placement then I am supposed to compute the expectation values of those operators at the space-time point and out pops a chair?

That's a bit of a caricature, but yes, if you want a completely accurate answer, that is what you have to do. Just as if you want a completely accurate answer about what happens when you drop an apple you have to solve Einstein's field equations. F=Gm1m2/r^2 is a damned good approximation, but it's deeply wrong about the physics.

You might find this interesting:

http://blog.rongarret.info/2018/05/a-quantum-mechanics-puzzl...

> "points on the screen" is the only reason we have to believe in particles

There are also the cloud chamber paths. But more generally, I would say most people's experiences correspond closer to stuff having well-defined localization which is more in the ballpark of particles than waves. Waves spread.

Also, this is not a matter of what is true, but rather what is plausible. A natural conclusion which works would be a good candidate to continue to pursue. My opposition to the paper cited was simply that it wants to argue that everything is a wave, something which seems to be an assumption not supported by evidence. That reality can be described that way is one thing, and a fine thing if it leads to interesting notions, but to say it must be a certain way is a completely different kind of claim.

> That doesn't seem reasonable to me. Particles exert forces on each other via gravity and electromagentism. I don't see how you're going to get a stable static vacuum configuration out of that without special pleading.

Particles in BM do not exert forces on each other. The wave function moves the particles about directly by specifying the velocity, not the acceleration. The forces are all in the wave function evolution (gravity is a bit of a mystery, but what else is new). Keep in mind there is a single wave function that represents the universe. It has a complicated dynamics which is where the forces are at work.

Maybe the short paper Are All Particles Identical [0] might help. It describes a version of BM in which all particles are identical and electron, quark, etc., are different states of a single particle type with the mass being incorporated into the wave function itself. Particles in that theory really are just points with nothing else intrinsic about them.

So the typical vacuum state might be just a small part of the story with a non-interacting sector. I don't know, but it certainly does not seem to rule out particles to me as being impossible.

> Just as if you want a completely accurate answer about what happens when you drop an apple you have to solve Einstein's field equations.

Yes, but if you present me with the solved system, I understand immediately what it is describing: a path through space-time of the apple. In a solved version is easy to see the correspondence.

This is not true of quantum wave stuff. It is true of BM. Relativity might mess with our intuition and be difficult to compute, but it is easy to understand how the elements correspond to our experience. That's the crucial difference.

Now, there is no reason to believe that the fundamental theory has to have that property. It might not. But it is extremely important than to be very clear about how the elements of the theory, the stuff it cares about saying what it state is, does correspond to our experience.

[0]: https://arxiv.org/pdf/quant-ph/0405039.pdf

> There are also the cloud chamber paths.

Those amount to the same thing.

> Particles in BM do not exert forces on each other.

Right. That's part of the problem IMHO. Particles in BM don't really do anything except get pushed around by the wave function. So they don't really correspond to what most people intuitively think of when they think of particles: electrons and protons (and neutrons), which combine not only a spatial location but also mass and electric charge into a single unified package. BM particles only have the spatial location part.

> it is easy to understand how the elements correspond to our experience

I think it's not so hard to see how QM corresponds to our experience if you look at it the right way. Not quite as easy as relativity, but not nearly as hard as it's commonly made out to be.

Note that our experience is at odds with "reality" long before you get to QM. Even in a purely classical model of an atom, it's mostly empty space, and what we perceive as "solid" objects are really just electrons in outer shells trying to push each other out of the way.

I'm not qualified to answer your question, but I'm guessing it'll boil down to "Bohmian mechanical interpretation is more mathematically complicated than the others. Therefore, even though it is the least counter-intuitive of the possibilities, Occam's Razor still requires us to discount it because of the complexity."

Thanks for the writeup, I enjoyed it very much. How does Deutsch connect this view to experimental results? Is the idea that all results are realized, or is my choice of experiment itself predetermined?

> How does Deutsch connect this view to experimental results?

MWI makes exactly the same predictions as regular QM, so it's connected to experimental results in the exact same way that regular QM is. The only difference is rhetoric. The canonical interpretation (Copenhagen) says, "The outcome is random, we don't know why, and we can't know why." MWI says, "The outcome is not random, all possible outcomes actually happen in point of physical fact. It appears random because when you make a quantum observation you split into multiple copies of yourself, and every one of those copies has the subjective sensation of seeing a random outcome even thought the actual outcomes were deterministic." So...

> Is the idea that all results are realized,

Yes.

> or is my choice of experiment itself predetermined?

No. This is yet another interpretation of QM called superdeterminism.

https://en.wikipedia.org/wiki/Superdeterminism

My layman’s understanding is that all results are realized many times. The values that we usually interpret as probabilities are really the proportion of universes in which the event occurs; a 95% probably of an experimental outcome means that if we sampled 100 worlds that are successors of the one in which the experiment occurred, 95 of them would have produced that outcome and 5 wouldn’t.

We perceive this as randomness due to the weak anthropoic principle: from our perspective, there is only one universe and things actually occurred in only one way, because there’s no interaction between worlds after the split; at every opportunity, we effectively draw one random result from the distribution because all of the other worlds don’t contain us, they contain copies of us.

Yes, that is almost exactly right. The only issue is that what distinguishes one universe from another is not actually well-defined, so you can't actually "count universes". You have to base the calculation of what proportion of universes experience one outcome versus another on something else. That something else turns out to be "branch weight", which, it turns out, is where the quantum randomness hides in MWI because branch weights can't be measured, not even in principle.

Both you and the other reply refer to counting universes/branches, which is obviously a term of art that I’m not understanding. Where I talk about sampling successor universes, are you just emphasizing that we’re drawing from a continuous probability distribution instead of a discrete one, or is there something else?

> a term of art that I’m not understanding

Actually, it's a hand-wavy term that is not well-defined, because "universe" and "branch" and "world" are not well-defined.

> sampling successor universes

The problem with this is that in order to sample something you have to specify the procedure by which you are going to do the sampling because your results are going to depend on that procedure. So even leaving aside the fact that the thing that you're trying to sample is not even well-defined, there are other issues.

Sampling finite sets is straightforward, but for infinite sets it gets tricky [1], and for infinite sets without a total ordering it gets very tricky. Consider, for example, generating a random complex number. There are at least two plausible ways to do this:

1. Generate two random reals x and y, and combine them to produce the random complex number x + iy.

2. Generate two random reals, r and theta and combine them to produce the random complex number r * (sin(theta) + i cos(theta)).

Depending on which of these methods you choose, you'll get different-looking distributions. Neither one is "correct".

If you're going to talk about sampling worlds you have all these difficulties in spades because worlds are rays in Hilbert spaces, i.e. vector spaces with an infinite number of dimensions. There are an infinite number of ways to generate random distributions over a Hilbert space. The trick is to make an argument for why one particular way is better than all the others.

The obvious argument is to present a way of sampling that reproduces the Born rule and argue that it's better because it reproduces the experimental results. But in order to not be begging the question, this sampling procedure can't have the Born rule hidden within it. That is what Wallace and Deutsch claim to have done, but which I (and many others) dispute.

[1] https://math.stackexchange.com/questions/997173/can-you-pick...

Thanks; I had assumed that drawing a sample was a fundamental operation on a well-defined probability distribution, and that existing QM tools were sufficient to define that distribution, if not distinguish between the various interpretations.

That means branch counting matters, which means there has to be some way for the universes to communicate the results of branch counting with each other. Otherwise there's no mechanism for probability to arise.

If you try to fall back to the position that the probabilities are somehow inherent in the interaction between all the universes, then you may as well dispense with the universes and just say the probabilities are defined by a hidden non-local (i.e. not Bell-violating) interaction in this universe.

Because otherwise you still have to explain how a split (or differential propagation or whatever you want to call it) in one location generates an entire universe instantaneously - i.e. faster than c - while preserving state that is beyond the event horizon of the current location.

There's a kind of sleight of hand about the argument. You can just about imagine multiple copies of a single particle existing along a foliated timeline. But when MWI people say "What splits is the wavefunction of the universe" they're just handwaving.

The wavefunction of the universe doesn't exist as a testable abstraction, and even if it did, are we supposed to believe that it's the only physical entity discovered so far that can ignore relativistic limits? And even if that were true, it implies that the wavefunction of the universe is somehow perfectly deterministic - which is unproven at best.

Those are all very strong claims to be making. In fact (IMO) there's a strong whiff of handwavy nonsense about the idea.

But I have the luxury of not doing this for a living, and I'm very happy to be proved wrong by people who do.

Observer independence - and observer relevance - seems to be a separate problem.

> But I have the luxury of not doing this for a living, and I'm very happy to be proved wrong by people who do.

I'm in a similar (but likely even less knowledgable) position; I'm having trouble following your logic, presumably because it's relying on terms and results I'm unfamiliar with.

> That means branch counting matters, which means there has to be some way for the universes to communicate the results of branch counting with each other. Otherwise there's no mechanism for probability to arise.

I don't know what branch counting is, but I don't see how the various universes need to communicate with each other. Observers in most of the universes will agree with us about the probabilities, as that's what the history of those universes has shown. In a small but nonzero number of them, however, everything will have happened the way that classical physics predicts and they will have never come up with quantum mechanical theory in the first place.

> Because otherwise you still have to explain how a split (or differential propagation or whatever you want to call it) in one location generates an entire universe instantaneously - i.e. faster than c - while preserving state that is beyond the event horizon of the current location.

I don't see a problem conceptually with envisioning each split like a stress fracture that propagates at the speed of light. As information can't travel faster than that (per relativity), there's no need to envision the two new worlds separating from each other faster than that anyway.

There is no way to prove that a supposedly random source of information is not effectively a PRNG. Similarly, the Born rule must apply unless and until we learn that there is some deterministic process that QM reduces to, and even then, unless we know how the PRNG was 'seeded', we must progress _as if_ the process was in fact random.

The deep point here is, arguably, randomness is in the eye of the observer.

I'm not sure what you mean by "effectively" a PRNG, but the big objection here is the EPR paradox. A true RNG can be stateless, but a PRNG necessarily requires state. So where is that state?

It's not in the wavefunction, so we must introduce new axioms: non-local variables, or super-determinism, etc. Yuck!

The MWI program is eliminating the objectionable "part two" of QM: probabilistic measurements and wavefunction collapse. But it's no progress if the replacements are even worse!

"non-local variables, or super-determinism, etc. Yuck!"

All of these things are less philosophically objectionable to me than the idea that all outcomes (in some sense) "occur" or that the true substance of the universe is an inaccessible, linearly evolving, quantum mechanical wave function.

I mean seriously - the latter is an enormous enlargement and modification of our ontology. Non-locality and super-determinism are actually much less objectionable. Indeed, before quantum mechanics, we were essentially _certain_ that determinism would win the day.

And we only need to jettison one of them, of which non-locality seems the most plausible. In that view, spacetime emerges from some underlying non-local dynamics. This isn't particularly hard to imagine (see a Smolin paper called "Nonlocal beables" or something like that).

Thanks for calling out how the Canon interpretations of QM do not not actually save us from much of the problems that are objected to in other interpretations. Sometimes I feel like everyone just repeats what they heard as a student. Smolin's work looks interesting, thanks for the recommendation.

I knew the Phd would come in handy eventually.

> There is no way to prove that a supposedly random source of information is not effectively a PRNG.

Maybe no practical way. But a finite state PRNG will be periodic while a true RNG won’t. And the length of the non observed period gives you an estimate of the minimal size of state.

What would be the period of a 4096 bit pseudo-random number generator iterating at a time scale commensurate with the plank length?

I haven't done the math, but the answer is, a long, long, long time.

It can be periodic but unbounded.

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