A paper just appeared in Physical Review Letters with a provocative title: “A Quantum Solution to the Arrow-of-Time Dilemma,” by Lorenzo Maccone. Actually just “Quantum…”, not “A Quantum…”, because among the various idiosyncrasies of PRL is that paper titles do not begin with articles. Don’t ask me why.
But a solution to the arrow-of-time dilemma would certainly be nice, quantum or otherwise, so the paper has received a bit of attention (Focus, Ars Technica). Unfortunately, I don’t think this paper qualifies.
The arrow-of-time dilemma, you will recall, arises from the tension between the apparent reversibility of the fundamental laws of physics (putting aside collapse of the wave function for the moment) and the obvious irreversibility of the macroscopic world. The latter is manifested by the growth of entropy with time, as codified in the Second Law of Thermodynamics. So a solution to this dilemma would be an explanation of how reversible laws on small scales can give rise to irreversible behavior on large scales.
The answer isn’t actually that mysterious, it’s just unsatisfying. Namely, the early universe was in a state of extremely low entropy. If you accept that, everything else follows from the nineteenth-century work of Boltzmann and others. The problem then is, why should the universe be like that? Why should the state of the universe be so different at one end of time than at the other? Why isn’t the universe just in a high-entropy state almost all the time, as we would expect if its state were chosen randomly? Some of us have ideas, but the problem is certainly unsolved.
So you might like to do better, and that’s what Maccone tries to do in this paper. He forgets about cosmology, and tries to explain the arrow of time using nothing more than ordinary quantum mechanics, plus some ideas from information theory.
I don’t think that there’s anything wrong with the actual technical results in the paper — at a cursory glance, it looks fine to me. What I don’t agree with is the claim that it explains the arrow of time. Let’s just quote the abstract in full:
The arrow of time dilemma: the laws of physics are invariant for time inversion, whereas the familiar phenomena we see everyday are not (i.e. entropy increases). I show that, within a quantum mechanical framework, all phenomena which leave a trail of information behind (and hence can be studied by physics) are those where entropy necessarily increases or remains constant. All phenomena where the entropy decreases must not leave any information of their having happened. This situation is completely indistinguishable from their not having happened at all. In the light of this observation, the second law of thermodynamics is reduced to a mere tautology: physics cannot study those processes where entropy has decreased, even if they were commonplace.
So the claim is that entropy necessarily increases in “all phenomena which leave a trail of information behind” — i.e., any time something happens for which we can possibly have a memory of it happening. So if entropy decreases, we can have no recollection that it happened; therefore we always find that entropy seems to be increasing. Q.E.D.
But that doesn’t really address the problem. The fact that we “remember” the direction of time in which entropy is lower, if any such direction exists, is pretty well-established among people who think about these things, going all the way back to Boltzmann. (Chapter Nine.) But in the real world, we don’t simply see entropy increasing; we see it increase by a lot. The early universe has an entropy of 1088 or less; the current universe has an entropy of 10101 or more, for an increase of more than a factor of 1013 — a giant number. And it increases in a consistent way throughout our observable universe. It’s not just that we have an arrow of time — it’s that we have an arrow of time that stretches coherently over an enormous region of space and time.
This paper has nothing to say about that. If you don’t have some explanation for why the early universe had a low entropy, you would expect it to have a high entropy. Then you would expect to see small fluctuations around that high-entropy state. And, indeed, if any complex observers were to arise in the course of one of those fluctuations, they would “remember” the direction of time with lower entropy. The problem is that small fluctuations are much more likely than large ones, so you predict with overwhelming confidence that those observers should find themselves in the smallest fluctuations possible, freak observers surrounded by an otherwise high-entropy state. They would be, to coin a pithy phrase, Boltzmann brains. Back to square one.
Again, everything about Maccone’s paper seems right to me, except for the grand claims about the arrow of time. It looks like a perfectly reasonable and interesting result in quantum information theory. But if you assume a low-entropy initial condition for the universe, you don’t really need any such fancy results — everything follows the path set out by Boltzmann years ago. And if you don’t assume that, you don’t really explain our universe. So the dilemma lives on.
Thanks for joining in, Lorenzo. I don’t want to give a point-by-point detailed response, as it will get quickly out of hand. But I think the main sticking point is this question of whether entropy is objective or subjective.
I didn’t say that entropy was objective rather than subjective; I said that there are different definitions of entropy. One way of defining entropy is to focus on our subjective ignorance of the precise microstate of the system; that’s what Gibbs did, and the von Neumann entropy in quantum mechanics basically does the same thing. That’s fine for certain purposes, but not for others. In particular, for a closed system that entropy never changes! So you wouldn’t believe in the Second Law if that’s all you had to go on.
Alternatively, you can define entropy without any mention of our subjective information, following Boltzmann. You do have to do something, namely you have to choose a coarse-graining on the space of states, typically in a way that conforms to your ability to do macroscopic observations. But once you choose a coarse-graining, the entropy is completely objective, and it’s given by the formula on Boltzmann’s tombstone, S = k log W. There’s nothing in that formula that makes reference to our knowledge of the system. And that entropy can happily increase, in accordance with the Second Law.
My point is that it’s this second notion of entropy that is relevant for the arrow of time in the real world. If you like, you can completely forget about entropy, information, our knowledge of the pure state of the universe, etc., and just say this: the early universe was in a state that is extremely finely tuned, one of only a very small number of states with those macroscopic characteristics. The shortcut way of saying that is “the early universe had a low entropy,” but you don’t need to use those words.
And that fact about the fine-tuning of the early universe is (1) perfectly objective, given a sensible macroscopic coarse-graining; (2) all you need to account for the arrow of time; (3) unexplained by the physics we know and love, including what’s in your paper.
So, again, I think your results are fine, but fall short of explaining the arrow of time.
Ahcuah– The particles look random, but the gravitational field was smooth to an extraordinarily finely-tuned degree. A higher-entropy state would look something like a time-reversed Big Crunch — full of singularities and wild inhomogeneities.
Okay, I have been thinking of past and future as going from “negative infinity” to “positive infinity” on the parametrization of some world line.
However, is it the case to say the limit of our past as perceived by an observer is really just the lowest entropy state of the universe?
In other words, if I marched back along my worldline (assuming it extended to negative infinity) would it appear as if I was moving into my past, *until I reached the lowest entropy state of the universe*, at which time after I pass through this state, continuing along my worldline toward negative infinity would it appear as if I was marching into some new unknown future?
I don’t know if I worded this clearly. Maybe another way to say it is: say my worldline is parametrized between negative infinity < T < positive infinity where T is the point of lowest entropy in the universe. When I am at a point x such that: negative infinity < x < T, I perceive time going from x to the negative infinity direction but when I pass through T so that now T < x < positive infinity my perception of time swaps going from x to the positive infinity direction?
Thanks.
Sean–
The fundamental flaw in the paper seems to be that it merely buries Loschmidt’s paradox beneath another layer of formalism. It is quite obviously true that people only learn about systems in a forward-time-direction when the entropy of those systems is non-decreasing in a forward-time-direction, as the paper argues. But, just as obviously, the previous sentence is also true if you replace both instances of the word “forward” with “backward”!
Indeed, it is easy to imagine a universe—or, perhaps, a distant, causally disconnected corner of our own universe—in which people learn backwards in time about systems whose entropies grow large going backwards in time. Of course, to these people, “backwards in time” would mean “fowards in time.”
But, in any event, we’ve solved nothing! A universe with time-reversal-invariant fundamental laws that allows people to learn forwards in time about forward-time-entropy systems obviously also allows “opposite people” somewhere else to learn backwards in time about backward-time-entropy systems. So how have we evade Loschmidt’s paradox, even in the seemingly modest way that you grant the paper? Isn’t the paper actually useless?
Joseph– If I understand you correctly, I think that’s basically right. The only nitpick is that it’s hard to imagine “you” traveling backwards in time and actually experiencing these things. You can never experience “time running backwards,” because our experience of time is tied to the growth of entropy.
Wanu– I don’t think that’s quite fair. Loschmidt’s paradox rests on the fact that there are just as many decreasing-entropy trajectories through phase space as there are increasing-entropy trajectories, which is true (assuming reversible dynamics). Therefore you can’t mathematically prove that entropy increases along most trajectories, which is what Boltzmann was trying to do at the time, and L’s criticism was fair.
This paper isn’t trying to do that. The claim here is simply that memories accumulate in the direction of increasing entropy, which is fine. I think it’s a claim that most people in the field already accept, but it’s always nice to see an explicit demonstration.
The problem is that the overwhelming majority of trajectories have the feature that entropy doesn’t change at all — it starts high, and stays high — and deviations from that tend to be very small. Our observable universe is a large deviation, a fact that remains to be explained.
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Dear Sean, thank you for your reply. I think you are not agreeing with my premises. Then it’s useless to go on discussing about the consequences: it’s clear that if
we use different premises we will generally disagree on the result.
My premise (clearly stated in my paper) is basically only ONE: I assume that quantum mechanics is valid at all scales.
If you accept that, then it’s nonsensical to write about a “sensible macroscopic coarse-graining” as you do in your answer above. There is no such thing as a sensible coarse graining anywhere in quantum mechanics: a sufficiently powerful observer can keep track of all the unitary evolution of all microscopic degrees of freedom.
If you consider my premise invalid, then you should say so explicitly (and our debate will presumably end there), but if you want to prove that my argument is fallacious, you should USE my premise and logically derive a different conclusion.
Thank you for the interesting debate.
Sincerely,
Lorenzo
I shouldn’t jump in without reading the paper a bit more carefully, but it apparently proceeds from T-reversal invariant dynamics, so by symmetry it seems it should equally allow for past entropy increasing processes (which it clearly does) AND future entropy decreasing processes, BOTH leaving information in the present. Otherwise it some arrow, some asymmetry is already presupposed. So we are left with the question of why we have no record of the future. (It seems the paper does not contemplate literal collapses to an eigenstate, because it discusses measurements being undone. So it seems the only dynamics is of T-symmetric Hamiltonian/unitary kind. It also does not seem to consider the T-reverse of its own analysis. But I was a bit cursory, so I apologize if I am wrong.)
Lorenzo– I have no trouble agreeing with your premise that quantum mechanics is valid at all scales. Nor do I disagree with any of the results in your paper. I just disagree that they solve the arrow-of-time problem.
I don’t understand what you are saying about coarse-graining in quantum mechanics. We do it all the time; we consider the set of states with certain expectation values for certain observables, etc. Of course if you know the exact pure state, and the exact Hamiltonian, you can evolve the state unitarily in either direction of time. But if you apply that to a closed system, the entropy is strictly constant. (In an open system it can of course change, due to entanglement.) So that’s clearly not what we have in mind when we talk about the arrow of time. For that, you need to coarse-grain. (The universe is, to a good approximation, a closed system.)
Nick– Thanks for joining in. I think the paper does assume reversible dynamics, and that both entropy-increasing and entropy-decreasing processes are allowed. But the point is that we would only remember the lower-entropy direction of time, so that any entropy-changing process is experienced as an entropy-increasing process. My objection is that this doesn’t explain why the entropy has been increasing consistently for billions of years over billions of light-years.
My sleep tank is down to the fumes right now, so this might make no sense at all, but I might as well ask anyway:
Why is the direction of increasing entropy also the axis picked out by the different sign in the Minkowski metric?
Basically because the time direction is the one in which you have a well-posed initial value problem. From “the entropy was low in the past” we can derive “the entropy will increase with time,” whereas from “the entropy is low on your left” you can’t derive “the entropy will increase as you move to the right.”
That gives me a wonderful idea for a science-fiction story. . . .
(Greg Egan has probably gotten there first. Or, rather, leftwards.)
Given Minkowski geometry – why isnt time measured with a ruler instead of a
clock ? If it were ++++ we would expect rulers.
Dear Sean, thank you for continuing the debate. Sure WE do coarse-graining all the time, but that’s hardly a FUNDAMENTAL limitation in our theory. It’s just due to the limitation of the information WE (subjectively) can access. If you assume quantum mechanics is valid at all scales, this limitation CAN be dropped (by a sufficiently powerful observer).
What I’m saying is that a powerful observer will see a constant entropy evolution whereas a “normal” observer INSIDE the universe will see the SAME evolution as a coarse-grained/entropy-increasing one. The subjectivity of the entropy is the key in understanding why our observer time orientation coincides with the entropy-increasing direction, even though the physics (on a universal superobserver scale) is time symmetrical.
In other words, our coarse graining is due to the fact that we are losing part of the information due to the correlations that build up between us and our environment (these are the ONLY INTRINSIC limitations that we have: I’m neglecting the plain surpassable ignorance here). A superobserver (that remains factorized from us and our environment) wouldn’t have these limitations even when he’s assisting to our same time evolution. What he sees as a time-symmetrical entropy preserving evolution, we see as an entropy increasing one.
Thank you for your debate, and I’m looking forward to hearing back from you soon.
Lorenzo
Hi Sean, it’s been awhile, I hope you are well.
I’m going to have to agree with Lorenzo here. I think the connection between the two premises you are using is that the manner of chosen course-graining is equivalent to stipulating your knowledge of the system. Classically, identifying the degrees of freedom at all scales is equivalent to stating complete knowledge of the dynamics. This is the classical equivalent to the super-observer in quantum mechanics.
On the other hand, specifying a macroscopic course graining is equivalent to stating ignorance about what happens beyond the specified cut-off. The second law in an information context states that one’s ignorance about small scale degrees of freedom will “infect”, via interactions, the larger scale degrees of freedom, ergo entropy increases. Lorenzo’s result seems to be the quantum mechanical analog of this fact, that quantum entanglement is responsible for the apparent increase in entropy of a sub-system of a larger pure state.
I can give you an explicit construction for a completely classical, time reversible system from a paper I’ve written with Carson Chow (http://arxiv.org/abs/0704.1650). We studied the Kuramoto model of coupled oscillators at finite size by constructing a path integral for the population statistics. The inverse size of the population (1/N) serves as a loop expansion parameter. The relevant fact for the present discussion is that, although the dynamics is completely time reversible the population statistics are not. It depends upon whether you ask questions about the past or the future. In fact, the neat result is that the loop corrections are responsible for the appearance in the population dynamics of a diffusion operator. The interpretation of this is that in the mean field limit, an individual oscillator decouples from the system, i.e. information about it at one time provides perfect information about it in the future and the past, whereas at finite size one’s ignorance about other oscillators produces a “random walk” behavior. The system relaxes to a higher entropy state, conditioned on our knowledge at a previous time. This is exactly the result of Boltzmann’s H-Theorem, without an assumption of “molecular chaos”, it falls out exactly, but it depends upon having a degree of ignorance about the system’s detailed dynamics. Knowing the entire configuration at one time restores the time reversal symmetry.
Sure, there is a very close connection between the “subjective” and “objective” notions of entropy. If I tell you not only that a system is in a certain macrostate at the present moment, but also that it was in a lower-entropy macrostate at some previous moment, you have a lot more information about the current state than if I had only specified the current macrostate.
All of which is beside the point I am trying to make, which is in danger of getting lost among the interesting discussions of different notions of entropy. Namely: the early universe was in an extremely finely-tuned state, one that (given the coarse-graining implied by our macroscopic observables) we label “low entropy.” If you explain that, you explain the observed phenomenological arrow of time; if you don’t, you don’t. And nothing about quantum mechanics implies that the early universe had to have such a low entropy — it could have been much larger!
Michael Buice: maybe I’m misinterpreting, you’re saying you can predict the future population statistics but not the past ones? Or visa verse?
Sean – it is nice to converse again! Anyway, I didn’t intend my remarks towards yours, but towards the original paper, to make a distinct argument that it doesn’t solve all the problems of the arrow of time (a low entropy initial condition does solve the problem, I believe). I’d be very interested to hear what Professor Maccone has to say, since it’s such a neat paper, so maybe if I spell it out a bit more he will be tempted to respond.
Basically, the paper shows that processes ABC (with time increasing left to right) in which entropy increases are ones which leave information behind after C while processes XYZ with decreasing entropy leave none behind after Z. But if the dynamics is T-reversal invariant – as the unitary dynamics is, unless there’s an assumption about the Hamiltonian I’m not seeing – then it also follows that processes CBA (again with time increasing left to right) in which entropy decreases are ones which leave information ‘behind’ BEFORE C, while processes ZYX with increasing entropy leave none ‘behind’ BEFORE Z. (Technical aside: of course I should use the time reverse of state A, B, etc, but for brevity did not – in QM, the time reverse of a state is its complex conjugate, so more carefully I should have written, e.g., C*B*A*.)
Just to be clear, my point so far just applies T-reversal to the result of the paper. Disputing what I just said requires demonstrating that the time reversal of the process is not possible, and hence explaining why the dynamics is not T-symmetric, and hence showing that an arrow is built into the dynamics – which it isn’t in standard unitary QM.
Then the argument of the paper goes, therefore of processes in the past, we can only ever see records of those which are entropy increasing, any in which it decreased leave no trace. (Aside: to be honest, it’s hard to believe that. It means that if I ever were to watch a cup of water spontaneously boiling, or spontaneously freezing, then I couldn’t remember it. Really? I suppose the suggestion of the paper is that such things are happening all the time, but never leaving a trace.)
But by symmetry we can say the T-symmetric thing about the T-reversed processes. Of processes in the future, we can only ever see ‘records’ of those which are entropy decreasing, any in which it increases leave no trace! Well that seems right about the entropy increasing processes, which we expect in the future, but of course there is in experience a complete and utter absence of ‘records’ of the future and hence of the entropy decreasing processes – but nothing has been said to explain why, as far as I can see. (Using natural language, we would generally call ‘records’ of the future ‘portents’, but I prefer to stick with ‘records’ to make the use of T-symmetry manifest.) By the T-symmetry of QM, for every entropy increasing process there is an entropy decreasing process, its T-reverse; by the proofs of the paper, if the former leaves traces after, the latter leaves traces before. Until it is explained why there is only one but not the other, an arrow remains. Perhaps in the future, but not in the past, we appeal to Boltzmannian considerations – entropy decrease is unlikely? But (a) the asymmetry of explanations is unpleasant, and (b) that seems to give ground to Sean’s views.
A couple more points. (i) Basically I’m pointing out a hidden asymmetric assumption – ignoring the T-reverse of the entropy increasing, information leaving processes. The philosopher Huw Price is has done much to unearth these in other contexts. (ii) Despite what I said earlier, it’s of course not true that we know nothing about future processes, we know a great deal about what will happen, we just don’t ‘remember’ it even in a T-reversed sense. For example, I know that the coffee in my cup will be cooling and heating up the room, leading to entropy increase. So if such knowledge is possible, why then shouldn’t the T-reverse be possible? That would be knowledge of a process in which the coffee spontaneously warmed up, decreasing the entropy. Of course we don’t think such processes occur, but if they did, why shouldn’t we know about them? But that is contrary to the claim of the paper, that because they leave no traces, such processes are unknowable — but by symmetry they are just as knowable or unknowable as future entropy increasing processes. (iii) It seems that information about the past shows that processes satisfy various conservation laws. It seems that could only be the case if either (a) we have information about all the processes that did occur, or (b) they were energetically (etc) isolated from any other processes, otherwise if there were transfer between the processes we see, and those we don’t, then there wouldn’t be conservation in those we see.
Ok, that turned into much more than I intended. The bottom line is: what about the T-reverse of the entropy increasing record leaving processes? And, I also agree with Sean’s points, which do take care of the issues I raise.
Thanks, Nick
[I will try to keep this short!]
Dear Dr. Caroll,
I’m a regular reader of CV and specially of your posts that I really enjoy. Maybe this would be an interesting moment to comment on your idea of large/small entropy in the early universe and the arrow of time problem, I would like to have your feedback if possible. I’ve seen one of your lectures on this topic available on Google Videos, so I draw information about your view from there mostly. I have a criticism regarding your stantement that once one accepts that the entropy is small in the early universe, then the problem is solved by Boltzmann’s physics. I believe you may be referring to the H theorem:
dS/dt >= 0
However, Boltzmann H theorem is not a statement that S always increases, as was pointed out immediatly after the publication of the theorem by Loschmidt. Suppose that one observer sets his coordinate system and experiences an increase in entropy for what he calls positive time intervals. Then another observer related to the first by a time reversal can still satisfy Boltzmann H theorem by detecting decrease in entropy. So if we allow time reversal in the underlying mechanics, in a single universe there can be both observers seeing increase in entropy and observers that see decrease in entropy. It is a physical requeriment not contained in Boltzmann H theorem that spacetime must be time orientable. Therefore, when one concludes that the entropy always increases or remains constant in a physical process from Boltzmann H theorem one is already assuming that no time reversals are allowed in the fundamental theory.
There are also several other important criticism to this view that the arrow of time problem is reduced to a question of the amount of entropy in the early universe. First, entropy to some extent is a subjective concept, because it is the amount of missing information in a chosen description of a system. If I solve Newton’s equations for 1 mol of atoms, assuming they obey such an equation, then the total entropy is zero. However, if I decide to average over all velocities and positions keeping the total kinetic energy fixed E and omit all information on the individual positions and velocities, then on maximizing the amount of missing information I get the entropy of the system and a Lagrange multiplier, the temperature. If I also want to keep the number of particles fixed I get an extra parameter, the chemical potential. There is nothing original in this, I’m just describing the work of Jaynes on Statistical Mechanics. Hence, entropy is a probabilistic artifact of a choice of description. It is hard to see why time would be strictly related to the fact that I’ve chosen to omit certain variables. It’s much more likely that there is no such thing as time reversal in the underlying laws, which also is what you need in order to conclude that S can only increase (or decrease, but not both in the same universe) from Boltzmann H theorem.
Secondly, we see that gravity does have a time arrow. Consider a Schwarszchild black hole for instance. You have geodesics that can start in the interior of the black hole and end in the exterior, just write the radial infall geodesic with the time parametrization reversed. However, by a *physical assumption* (or experimental fact!) if one observer can fall inside the black hole, we *postulate* that the time-reversed geodesic is not allowed in the same space-time. Hence, one gets a black hole. But even with such restriction, Einstein’s equations allow you to write the time-reversed sector, the white hole in the continuation of Schwarszchild spacetime, but we see no astrophysical white holes, only black holes. From GR you would be forced to conclude that if fundamental physics does really have a T symmetry, for every astrophysical black hole in the universe there should be a white hole, but this is not what is seen. So time arrow must be inserted, built, even in fundamental physics. This is such a simple conclusion to draw from Schwarszchild spacetime that I don’t see how it could be wrong, but maybe I am.
Thirdly, Boltzmann H theorem, and my understanding of Statistical Mechanics and equilibrium systems, says that entropy can stay constant. It is just not true that “time stops” if a system is in equilibrium: individual particles still collide, and one still have to apply to these constituents the scattering theory in which “past” and “future” are defined (in fact, such a scattering theory with sharp definition of past and future is the starting point for proving Boltzmann H theorem). It is hard to see theoretically how one would justify the arrow of time or even understand what time is if one insists that the passage of time is related exclusively to the observation of entropy increase since we can perform gedankenexperiments in which S stays constant.
Cheers
Leonardo
Dear Sean, I have done exactly what you’re asking for. I have given you a good reason why the universe should indeed be in a zero entropy state (which is the fine-tuned state you want).
In addition, I have shown that (thanks to QM) this zero entropy state of the universe appears SUBJECTIVELY to US to be in a much higher entropy state (because we are entangled subsystems of the universe). It is in a zero entropy state only from the SUBJECTIVE point of view of a superobserver.
Then, I have given you a reason why, even though the superobserver sees a constant entropy evolution, we see an entropy-increasing evolution (which is the arrow-of-time).
This is exactly what you’re asking for. I don’t see what is the point on which we are disagreeing.
Thank you for keeping up with the debate! I’m hopeful that we are converging to a common understanding, and I appreciate that.
All the best,
Lorenzo
Lorenzo,
But the fine-tuning isn’t that the universe is in a pure state, is it? Does “universe in a pure state” always imply “any observer looking at the early history of the universe will see a low entropy state”? Couldn’t there be pure states of the universe where the entopy seen by observers in the early universe was high?
if space and time are on the same footing then how come we do not
have to apply entropy to space ?
Dear Lorenzo,
You claim that only two people are capable of having the information needed to conclude that the entropy in the universe is constant, a superobserver, and the mathematician who proved it is so. But as we well know from the plethora of phenomenon predicted by string theories it is all too easy to jot down an equation without being able to prove, in a more rigorous and scientific sense, that the theory is correct. So I would like to ask, is your theory falsifiable? How might I go about proving it to be false? Also, does your theory make any new predictions about the universe? Lastly if I may, in what sense do you consider this result to be science?
Thank you,
Hi Sean
At risk of bringing even more considerations into play, here, with Lorenzo’s permission, is the text of three emails we’ve exchanged over the past couple of days. It is my turn to reply, and I’ll do that here, when I’ve figured out which issue it might be most useful to concentrate on.
Cheers
Huw.
1. MACCONE TO PRICE
Dear Prof. Price,
the scientific journalist Michael Slezak has recently contacted you about my theory on the thermodynamic arrow of time arising from quantum mechanics [PRL, 103, 080401 (2009)].
Since your objection to my theory is very non-trivial but can be easily overcome, I wanted to answer to your objection. I’m quite familiar with your book and with various other publications of yours, and I know you ferociously criticize the “no collapse” interpretation of quantum mechanics. However, you can certainly admit (as a pure hypothesis) that unitary quantum mechanics can be valid at all scales. This is the explicit (and unique) starting point of my theory. You might not agree that this is the case, but please indulge with me (consider it as a purely artificial hypothesis, if you want).
You wrote to Michael Slezak:
The answer to this lies in the fact that, if quantum mechanics is valid at all scales, a measurement is nothing else than an entanglement between the observer and the system to be observed. Namely, consider the unitary evolution that describes the observer that is performing a measurement: such evolution will start from factorized states of the observer and the system, and will evolve them into an entangled state of the two:
(|spin up>+|spin down>)/sqrt(2) |Alice>
evolves into (|spin up>|Alice sees up>+|spin down>|Alice sees down>)/sqrt(2)
If a second observer looks at the measurement result (i.e. when multiple observers get entangled among themselves) their global entangled wave function (of the system + two observers) is such that they both agree on the results of any measurement:
(|spin up>|Alice sees up>|Charlie sees up>+ |spin down>|Alice sees down>|Charlie sees down>)/sqrt(2)
[This was pointed out by Hugh Everett in his PhD thesis. Although I don’t agree with many of his other conclusions, I think that this is quite uncontroversial.]
This argument is based ONLY on the UNITARY (i.e. time-symmetrical) part of quantum mechanics, so that there’s no circularity in my explanation, namely, I’m not using a hidden underlying time arrow to explain the thermodynamic time arrow.
That’s why, even though the quantum theory is entirely time symmetric, ALL the observers agree in which direction time is flowing, simply because of the way they get entangled by the unitary time evolution when they interact among themselves!!
I hope this answers to your objection to my paper and I hope to be hearing back from you soon.
All the best,
Lorenzo Maccone
2. PRICE TO MACCONE
Dear Lorenzo
Thanks for your email, and for taking the trouble to clarify your argument for me. Let me try to develop my concerns a bit.
First, let’s go back to the classical case. In your paper, you mention Borel’s argument, about the rate at which correlations accumulate as a result of interactions.
Question: If the dynamics are time-symmetric, why should the accumulation of correlations be time-asymmetric? Why should interactions increase correlations ‘to the future’ but not ‘to the past’?
As you may know, I think that these issues are relevant to the so-called interactionist proposal concerning the origin of the thermodynamic asymmetry, namely that the irreversible increase in entropy is caused by uncontrollable influences from the environment. I say to the interactionist: Why is it that influences coming in from the past result in entropy increase towards the future, whereas influences coming in from the future do not result in entropy increase towards the past? They say: It is because influences coming in from the future are precisely correlated, in just the way they need to be for entropy to decrease towards the past. I say: There’s the double standard — your argument that entropy increases towards the future needs to assume that influences coming in from the past are not correlated in the precise way they need to be for entropy to decrease, but that’s just assuming the conclusion you want to derive. And to cut a long story short, I want to say that the right answer in the classical case is that there is no relevant asymmetry in, or due to, the dynamics of interacting systems: the asymmetry of the 2nd law rests entirely on special initial conditions.
Now let’s consider the QM case, and I set aside any objection to “no collapse” views.
Your argument relies on the assumption that unitary evolution of interacting systems produces entanglement, but this is puzzling in precisely the same way as in the classical case: given that the dynamics itself is time-symmetric, why should it produce increasing entanglement in one direction but not the other? Where does decoherence get its time-asymmetry from? (In as sense, this is a QM version of Loschmidt’s puzzle: where does the asymmetry come from, given that the dynamics is time-symmetric?) I suspect that as for the 2nd law in the classical case, the answer must be that there is a very special initial state — and that it is simply being ASSUMED that there are not special final states of the same kind.
If I understand you correctly, your proposal is that observers break the symmetry. I have several concerns about this idea, but let me concentrate on the issue about where the asymmetry of agents comes from, on your view. It seems to me that the argument from Everett you mention below assumes that the two observers have the same time sense — i.e., they do not disagree about which direction is the past and which is the future. I would like to see an analysis which doesn’t take this for granted, and which doesn’t assume that QM interactions cannot produce recoherence, as well as decoherence.
If we grant the assumption all observers must have the same temporal orientation, then there is an interesting question about the status of this fact. Is it a kind of accident that a universe has observers with one temporal orientation rather than the other? Is it a kind of law (could physics have laws about observers)? Or can it be explained in terms of something else? The thought behind my comment to Michael was that in general, we find it plausible to explain the existence and temporal orientation of observers such as ourselves in terms of the thermodynamic properties of our environment; but this approach seems off-limits to you, since you want to explain the thermodynamic gradient in terms of observers.
Anyway, that’s enough for now. Thanks again for taking the trouble to write to me.
Best wishes
Huw.
3. MACCONE TO PRICE
Huw Price wrote:
Dear Huw, thanks to you for your interest in my paper! Being familiar with your literature, I expected this “double standard” accusation! However, in my case I think it is without merit, as my theory IS completely time symmetric. In fact, correlations are increased both ‘to the future’ and ‘to the past’. The important thing to realize is that memory is a correlation (between the observer and the observed). So we retain a memory as long as the correlation is present. I DEFINE past for the observer as “the direction in time of which we have memories”. Although subjective, it is a good definition that captures well what we “feel” is past. Then whatever is the direction of time, correlations are increased only in “our” past… [In the last paragraph of the quant-ph version of my paper arXiv:0802.0438 I elaborate on this, but I had to cut it out for the PRL version.]
Perhaps you have a different definition of “past”? Honestly, I can’t think of any other definition that doesn’t employ the 2^ law (which would then become a mere DEFINITION for the direction of time)…
Already one of the paper’s Referees had asked about the symmetry in time of my theory. This is what I had answered him/her:
Finally, regarding the Referee’s question of what would an observer traveling backwards in time see, my theory suggests that such an observer would still only see (or rather, remember) the events that lead to entropy increase. In the time-reversed universe, the information is created and erased at the same instants in which it is respectively erased and created in the direct-time universe: reversing the arrow of time, the events that lead to creation and to merging (erasure) of correlations are swapped. Observers still see only events where entropy has not decreased, namely where information has not been “erased”, although time is flowing in the opposite direction. The time symmetry of the theory requires this, and it is correctly recovered.
I do not agree with that. My theory would suggest that whatever the initial conditions, the entropy would only increase towards the future, just because even if entropy-increasing and entropy-decreasing transformations were EQUALLY likely, it’s only entropy-increasing transformations that leave trace of their having happened. This is a purely quantum effect, so that you have to assume that quantum mechanics is valid at all scales. This is my (explicitly stated) premise.
Of course if you are in a zero entropy initial state, only entropy-increasing transformations CAN happen (there’s nothing to disentangle), and viceversa if you are in a thermal equilibrium state of maximum entanglement, only entropy-decreasing transformations can happen (it is impossible to further increase the entanglement). So I’m NOT saying that initial conditions are irrelevant, I’m just saying that they do not determine the arrow of time direction.
I think I’ve already answered to this above, but let me try again: entanglement is indeed produced in both time directions. Decoherence is usually termed as a one-way process, since one always starts from your assumption of low entropy initial state, namely, that it is more probable that entanglement is created during the measurement process than it is erased. Instead, what I point out is that decoherence is reversible! (at least if one looks at the big picture of the unitary evolution of the universe). Measurements can be undone (but the measurement result must be irretrievably erased). The reason why we don’t see “undone measurements” in our past is that, since the measurement result has been erased, we don’t think that that transformation was a measurement…
Unfortunately I don’t have Hugh Everett’s thesis here with me (it’s cited as ref. [23] of my paper), but I recall that the argument that all entangled observers agree among themselves was very carefully laid out (it is essentially what I wrote you in my past mail). As I said above, I DON’T assume that QM cannot produce recoherence. This CAN certainly happen, but it won’t leave any trace in the environment (for the simple reason that any remaining trace in the environment is due to entanglement between system and environment and that will PREVENT any recoherence).
As I wrote you (following Everett’s argument) I don’t think it is an assumption, but a CONSEQUENCE of the linearity of QM.
Let’s summarize: I define the observer’s temporal orientation as “past is the direction of time where memories are”. Then I relate this to the thermodynamic gradient by pointing out that (in QM) entropy is strictly connected to entanglement (correlation), which is strictly connected with memories. I emphasize that this is not true of classical mechanics.
In addition, as you pointed out, it’s quite important that all observers agree to the temporal orientation, and I think that Everett’s argument shows just that.
Thank you again for your careful thoughts on my paper. I hope to hear from you soon!
All the best,
Lorenzo
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