From Eternity to Book Club: Chapter Eleven

By Sean Carroll | March 23, 2010 10:27 am

Welcome to this week’s installment of the From Eternity to Here book club. Part Three of the book concludes with Chapter Eleven, “Quantum Time.”


This distinction between “incomplete knowledge” and “intrinsic quantum indeterminacy” is worth dwelling on. If the wave function tells us there is a 75 percent chance of observing the cat under the table and a 25 percent chance of observing her on the sofa, that does not mean there is a 75 percent chance that the cat is under the table and a 25 percent chance that she is on the sofa. There is no such thing as “where the cat is.” Her quantum state is described by a superposition of the two distinct possibilities we would have in classical mechanics. It’s not even that “they are both true at once”; it’s that there is no “true” place where the cat is. The wave function is the best description we have of the reality of the cat.

It’s clear why this is hard to accept at first blush. To put it bluntly, the world doesn’t look anything like that. We see cats and planets and even electrons in particular positions when we look at them, not in superpositions of different possibilities described by wave functions. But that’s the true magic of quantum mechanics: What we see is not what there is. The wave function really exists, but we don’t see it when we look; we see things as if they were in particular ordinary classical configurations.

Title notwithstanding, the point of the chapter is not that there’s some “quantum” version of time that we have to understand. Some people labor under the impression that the transition from classical mechanics to quantum mechanics ends up “quantizing” everything, and turning continuous parameters into discrete ones, perhaps even including time. It doesn’t work that way; the conventional formalism of quantum mechanics (such as the Schrödinger equation) implies that time should be a continuous parameter. Things could conceivably change when we eventually understand quantum gravity, but they just as conceivably might not. In fact, I’d argue that the smart money is on time remaining continuous once all is said and done. (As a small piece of evidence, the context in which we understand quantum gravity the best is probably the AdS/CFT correspondence, where the Schrödinger equation is completely conventional and time is perfectly continuous.)

However, we still need to talk about quantum mechanics for the purposes of this book, for one very good reason: we’ve been making a big deal about how the fundamental laws of physics are reversible, but wave function collapse (under the textbook Copenhagen interpretation) is an apparent counterexample. Whether it’s a real counterexample, or simply an artifact of an inadequate interpretation of quantum mechanics, is a matter of much debate. I personally come down on the side that believes that there’s no fundamental irreversibility, only apparent irreversibility, in quantum mechanics. That’s basically the many-worlds interpretation, so I felt the book needed a chapter on what that was all about.

Along the way, I get to give my own perspective on what quantum mechanics really means. Unlike certain parts of the book, I’m pretty happy with how this one came out — feel free to correct me if you don’t completely agree. Quantum mechanics can certainly be tricky to understand, for the basic reason that what we see isn’t the same as what there is. I’m firmly convinced that most expositions of the subject make it seem even more difficult than it should be, by speaking as if “what we see” really does reflect “what there is,” even if we should know better.

Two-slit kitty

So I present a number of colorful examples of two-state systems involving cats and dogs. Experts will recognize very standard treatments of the two-slit experiment and the EPR experiment, but in very different words. Things that seem very forbidding when phrased in terms of interference fringes and electron spins hopefully become a bit more accessible when we’re asking whether the cat is on the sofa or under the table. I did have to treat complicated macroscopic objects with many moving parts as if they could be described as very simple systems, but I judged that to be a worthwhile compromise in the interests of pedagogy. And no animals were harmed in the writing of this chapter! Let me know how you think the strategy worked.

  • Tom Allen

    Although hardly an expert, I did recognize your cat analogy to the two-slit experiment. That’s too much of a classic to be unfamiliar even to duffers.

    I am especially grateful to this chapter for putting me at ease with the “Many Worlds” interpretation. I was up until now completely put off by its name and the extravagance it implies. The decoherence concept feels much better to me as an explanation of what happens upon “observation”. “Collapse” now seems like the more vague concept.

  • Sean

    If it makes even one person more comfortable with the Many-Worlds interpretation, the whole book-writing thing was worth it.

  • Aaron Sheldon

    What was the difference you intended between saying ‘observing the cat on the sofa’ and ‘the cat is on the sofa’. In particular what difficulty in the interpretation of quantum mechanics were you trying to elucidate?

    I suspect there is a nuance in your quote that I am failing to grasp.

  • Blake Stacey

    I personally come down on the side that believes that there’s no fundamental irreversibility, only apparent irreversibility, in quantum mechanics. That’s basically the many-worlds interpretation,

    Or something closely related to it, like the Rovelli relational interpretation.

  • Sean

    Aaron– I’m not sure what you are asking that isn’t addressed in the quote in the original post (or in the book chapter). Quantum mechanics says that there is generically no such thing as “where the cat is.”

  • Ray

    You alluded to some problems in recovering the probability formulas from the MWI. What exactly are those? I assume mathematically, the only place squares can come from is unitarity. Is the milder statement proven, that if a probability formula makes sense (i.e. obeys Bayes law etc.) it has to be the one from the copenhagen interpretation.

  • Sean Rutledge

    An extremely lucid description of quantum mechanics. An enormous national savings in energy consumption could be claimed if we contented ourselves with the obvious description of nature that the formalism provides. Whatever “interpretation” problems remain are purely human. As experimenters entangle ever larger systems I don’t see any realistic alternative remains.
    Thanks, Sean

  • Sean

    Sean– Thanks! And good name.

    Ray– I’m not an expert on the state of play, this is an area of current research. But the basic idea is simple — so the MWI explains how the wave function evolves into two non-interacting “worlds.” It’s clear what the coefficient of each branch is in the full wave function. But why are we supposed to interpret the squares of those coefficients as the “probability” that we’re going to observe that outcome? I personally think this is just a matter of getting some details right, but other people take it very seriously.

  • Aaron Sheldon

    Are you trying to point out that the observable states depend on the measurement instruments? Or are you trying to caution readers away from thinking there are naturally preferred states.

    • Sean

      Neither, really. I’m trying to nudge people away from identifying “what we observe” with “what is real.” Wave functions are real, but we don’t observe them directly.

  • Sebastien

    I think this may well be my favorite chapter of the whole book. It was the most lucid explanation of quantum mechanics I’ve read so far, and for the first time, I feel like I can explain at least the basics concepts to my friends.

    So thanks!

  • Ahmed

    >I personally come down on the side that believes that there’s no fundamental irreversibility, >only apparent irreversibility… basically the many-worlds interpretation..

    That’s probably where the bulk of the debate will lie.

    George Ellis wrote ‘On the Flow of Time’ for the Foundational Questions essay contest. With your entry and his together, I think the case was almost made clear for non-reversibility. But there are subtle differences like these. He went a little further with the many worlds issue (in that essay), and though it was a brief treatment, I think he summarized the opposing view succinctly and with clarity.

    Anyway, most people will not realize that this ‘quantum indeterminacy’ business is actually far easier to understand than the ‘regular’ probability theory, as applied today in assigning values of uncertainty to macroscopic events. No matter what formulation of probability is used (bayesian, frequentist..etc), the entire field, from what I experience every day, can be summarized as a ‘science of ignorance’. This basically describes both the formulation and the affliction, and it is hard to decide which is worse. That even mildly useful inferences can result from it, in any realistic setting, is incredible – and I say that as someone who has written accurate programs driven by the calculation of millions of probabilities per minute, with various underlying theories.

    Every statistician should be made to read Feynman’s classical talk, “Simulating physics with Computers”. Esp. the questions he was asked at the end. It really hammers home how *real*, how non-problematic and beautiful quantum probability is, with the unfolding of time, in contrast to the ‘normal’ uncertainty people are supposedly comfortable with.

  • Clifford

    Hey, I really liked the new creative ways of expressing QM herein; but you lost me with decoherence. I hope you can take that as constructive criticism to hopefully improve that part of your explanations. Kudos!

  • Aaron Sheldon

    I tend to stay away from etiological arguments in physics about what is real or not real. I’m of more of a what is observed is observed kind of guy myself.

    As far as reversibility, consider the conjugate observables of momentum and position, the eigen states of each are superpositions of the other, so successive alternating observations of position and momentum will alternate between “collapsing” of the wave-function in each observable. In effect each alternate observation undoes the effect of the last. The only problem comes in when one restricts the definition of wave function to being a eigen state of a particular observable, rather than an arbitrary element of the space of all states. But that is a hard idea to express without resorting to Hilbert Spaces and Fourier Analysis.

  • paul valletta

    Does not the Sofa and Table have their own wavefunctions also?..and these should be equated therein?

    Neat, if you ask where the cat is by giving co-ordinates (sofa and table are just co-ordinates), then the question is as meaningless as asking if single wavefunctions are “real” ?
    best p.v.

  • Claire C Smith

    It seems lots of theoretical physics is about inverse square laws and things that go at right angles to other things, much like real physical terms in classical physics, like electromagnetism that does. But look at this. I have always thought that the difference between maths and physics is that applied maths in physics, is mainly deduction sideways, whereas physics is deduction longways and side ways. I’ll explain.

    The first deduction is sideways to get the coefficient – then the wave function that is (at right angles to) the coefficient squared to get probability, is longways. I suppose it’s like seeing around the quantum corner. There can be as many sideways ones as long ways ones but the longways ones are rare and meddle with the observer influence and the problem with classical time. In the collapse of the function, the deduction to form the coefficients are equally just as part of the system as the others, but we see them as part of the same whole, but hidden and assume it’s just any deductive method. The thing is to see the difference between the two methods and why we assume only one is about observer infuence. The maths deduction system as an intervention could also be observing our universe and we not know. This makes time it self quite appleaing to think about, let alone many worlds, whcih is just as interetsing of course.

    Could time turn around corners?

  • Metre

    I admit to some confusion. Decoherence loses information to the environment, so doesn’t that make it irreversible; i.e. make you unable to reconstruct previous coherent state?

  • TimG

    @Metre: It basically means you’d have to act on the whole thing — the system and its environment — to reconstruct the previous state. But in practice a macroscopic environment has millions of billions of billions of particles, so that’s for all intents and purposes totally impossible. In technical terms, the evolution of System+Environment is “unitary” (reversible, basically) but actually reversing it is an impossibility. Whereas in the old-school “Copenhagen Interpretation” of Quantum Mechanics, the measurement process actually makes a non-unitary change (i.e., irreversible *even in principle*) . . . which raises all sorts of questions about why measurement doesn’t follow the same rules as everything else (since ordinary evolution of states *is* unitary).

  • Will

    Forgive me if this has been addressed, but in what way is this question linked to CP (or T) violation?

  • Corey

    I have been looking forward to this chapter. Although I’ve read about the Copenhagen and “many worlds” interpretations before, the concepts were explained more clearly in this book.

    To extend the idea of entangled systems, let’s take two entangled systems that are independent of one another, each with an observer. For the observer within each system, their own system appears decoherent, with one outcome assigned 100% probability. My question is this. Does one system appear coherent (uncollapsed) to an observer in the other system? I would think so. So if the cat/dog example applied to a particular house in the U.S. and also to a particular house in Australia, each with their own cat/dog system, an observation in the U.S. would not collapse the superposition state that exists in Australia. Is this correct? In effect, the cat/dog system only appears collapsed to its own observer, plus anything else entangled with that observer. The rest of the universe could continue to treat the cat/dog system as a coherence wave function.

    Also, I was thinking through how you identified which of the “many worlds” the observer sees, by essentially naming which state is observed (sofa, yard). For the space of states described in previous chapters, such as a box with x number of gas particles, we assumed each value could be quantified. Each particle had a specific value for position, momentum, and time. With the uncertainty principle, it seems that we can never specify each value precisely so we are never able to assign a particular value 100% probability to any state and we are always in some way in a coherent state, no matter how macroscopic or entangled we are. Is that correct?

  • TimG

    Corey, you lost me at “Let’s take two entangled systems that are independent of one another.” If they’re entangled, then surely they’re not independent, no?

  • Dave

    Sean, you are the effing man. That is all.

  • uncle sam

    There are several problems with MWI and the idea that we can handle quantum time in easy reversal. First, the “splitting” problem and separation of the “worlds”. Just aside from the basic issue of why the various alternatives would be effectively “separated” in any sense instead of being fully “superposed” in all effect – there are internal contradictions IMHO. Just take the question of where the paths must “part ways” in some sense. They can’t always be separated, since we need their mutual action to show interference effects (or else you’re really going out on a limb.) So the “split” has to take place at some further level, like where “detectors” are going to click on way or the other. This is fishy. The whole point is supposed to be (if you read up on MWI and its handmaiden, the decoherence interpretation of waving away (pardon the pun) the measurement problem), that “detectors” are not special.

    1. OK. So let’s say I have a MZ interferometer with beamsplitter BS1 to “split” an incoming photon. Then we recombine the waves in another, BS2. We know we can arrange to get all hits in channel 1 at detector D1, etc, showing (to a realist) that some waves went through both paths in order to interfere. But there is still choice after BS2. We can do more splitting of the wave towards other detectors, or adjust path differences so it’s e.g. 70/30 chance for D1 v. D2 instead of 100%, or even just exactly where on D1’s counter does the photon “hit.” Only then does MWI imagine “splitting” of some sort. (Oh, people say evasive things like “the options don’t interfere anymore” but interference is just a way to get nice patterns instead of sloppy ones, it doesn’t change the basic point of *there being present* a superposition or not.)

    But consider that if detectors aren’t special, then BS1 should have sent the waves into separate “worlds.” After all, by itself BS1 serves to instigate options, of either going one way or the other. Let’s have BS1 by itself and alternate detectors D1′ and D2′ outside BS1 instead of recombination later. Then we could say the “choice” and split consists of one world where the photon was sent through to D1′, and another world (or whatever the equivalent is) in which it reflected to D2′. So the big stinker here for MWI is: BS1 is already a quantum choice juncture. It should *already* split the wave into worlds (or whatever the hell they are) before they ever get to BS2. Hence, there should be no interference. We should be in either the world where the photon took the choice to go one path after BS1, or the other path, but not any world in which it took “both.”

    2. Time-reversal: Yeah, Schroedinger equation per se is time reversible but that doesn’t lead to reasonable time reversal of events, even in QM. Consider a light-emitter in front of a screen. OK, the wave functions travel from the emitter to the screen, and somehow end up hitting at select spots. Maybe all the possible hits are represented “somewhere” and in proper proportions etc. But try to run it backwards: WFs proceed “from” the screen, and – why the hell should they converge on the emitter if there’s no real causal requirement? You could say the universe was just lucky to keep having backwards light converge on the same spot. But the irony is, if the prior “receiving” atoms are now “emitters” of equal standing, then the WF proceeding from them should be an expanding shell and not reconverging back to a lucky spot. Actually it’s easier in classical mechanics since you just imagine the particles all moving backwards – however absurd, it’s just as “doable” and will continue to work “right.” But the WFs are not inherently equivalent from emitter and absorber (?) In QM, you have to not only start the reversed world out “just right”, but it has to continue to be “lucky” after that …

    To find more such arguments, Google for “decoherence” + “circular argument” as well as “quantum measurement paradox.”

  • Peter Coles

    I like the cat example. Is the litter tray in another part of the multiverse?

  • Rafael Greenblatt


    I think you have a point in one case (1), that you get into trouble if you think about the wave function for the whole universe splitting up, which is how many worlds tends to be talked about (though also it seems to me that it isn’t fair to insist that the splitting happen at a clearly specified instant; we seem to be talking about an emergent property, it should be a little fuzzy in the same way as critical temperatures are for finite systems for example). I suspect that this is part of the motivation for Rovelli’s interpretation (which someone mentioned a ways back), and I also feel I ought to mention that it seems not to be a problem (at least a problem of the same significance) in Bohmian mechanics, which is also time reversible. Looking back, it seems like Corey’s question brings up the same issue.

    On (2), I think you’re missing the point. Part of the premise of many worlds is that the wave function evolves deterministically according to Schrodinger’s equation alone (or some equivalent); there’s no need to be lucky. Of course you need to specify a lot more about the “initial” wave function than just “photons are emitted from these spots”, but that’s just because there is a lot more going on (more degrees of freedom, to be technical) than one might think at first. The thing is actually to take the complex conjugate of the complete final wave function describing the whole system (universe? multiverse?), which is (includes?) a very particular superposition of states in which photons are being emitted in different patterns.

  • Juan R. González-Álvarez

    1) About cats. The theory of decoherence and other advances precisely show that the cat has not wave-function associated due to entanglement (the cat is a dissipative structure, not an isolated hydrogen atom :-). Of course, the cat is still in some quantum state (one indistinguishable from a classical state FAPP), but this state is not pure and cannot be described by a wave-function. For the description of the mixed quantum states we cannot use wave-function theory.

    The collapse of wave-function is a real process caused by the interactions by the measurement apparatus. As correctly pointed by Landau & Lifschitz in their celebrated manual on quantum mechanics, the collapse is irreversible. Using decoherence and similar models, the collapse of wave-functions has been already tested in computers.

    2) About quantum time, you are completely right that quantum gravity does not work by quantizing everything. In particular, it is a mistake to try to quantize the evolution parameter found in the Schrödinger or similar quantum equation.

  • uncle sam

    Rafael: Well, maybe the many worlds is not extremely precise but advocates can’t just blow off the problem of when it happens – because we clearly have “superpositions” and both WFs together in some sense for awhile, and then later we don’t. There is still a problem as I long-windedly explained, why the first BS in a MZ interferometer doesn’t do that, but the second one (or detectors outside one or the other BS) can make that happen.

    I still don’t see why the requirement of reconvergence of WFs toward what should be an emitter isn’t a problem. The WFs coming from “an emitter” are going to expand outward in shells and this is not reasonable if they were supposed to be emitted (in forward time) by a source. Just imagine playing a movie of photon WFs backward, and ask what should happen to the WFs from an “emitter” – it just doesn’t run backward. (Even better, REM the issue of what are we even saying, “run backwards” – if no absolute time, then relative to what is the time running “backwards”? It just shows how muddy the concepts are.

    @Juan: Decoherence: no, it doesn’t really solve the measurement problem. (Again, search out decoherence + “circular argument” and check Stanford Ency. of Phil (of Science issues) on that. The WFs (as of “alive” + “dead”) are still there, they are just not well ordered in phase. But the model has them both staying around until something extraordinary intervenes. Sure, the detection statistics of such decohered waves won’t show interference patterns, but REM that we wouldn’t even have “detection statistics” in the first place if the wave didn’t “collapse.” The collapse intervention turns well-ordered waves into well-ordered statistics; and sloppy waves into mixture-type statistics, but that isn’t the question. Why are there any statistics at all, rather than endless waves.

    It is ironic you mention computer simulations, since they actually can’t accomplish the dodge accomplished in the verbalized decoherence argument based on comparing statistics. Suppose we had a simulation with a superposition of two WFs extended in space, and I modeled them as as red and green distributions. Let them undergo various distortions and entanglements in the environment. So I have a complicated, shifting pattern of red and green (maybe, representing where a photon might be, or two polarization states it started as) and yellows (if colors mix like light.) But in order to localize the photon or “collapse” it into one of the states, I would have to “cheat” (as said in philosophy of a fallacious argument or subterfuge) by just switching off on of the colors.

    It just doesn’t work, but send me a link to a computer simulation *that actually shows the distribution of WFs in a simulated single instance, and resolution into the likes of a measurement* and I will be impressed. I don’t think you can though, because the pretense of solving the measurement problem in these cases is done by considering an ensemble, and talking of “mixtures” and the overall statistics of an ensemble. But the model problem of Schrodinger’s Cat was always about how to model any single instance of one WF “winning” over another. How can you represent that without a slight of hand, of whisking away one of the WFs to leave the other? It can’t be “shown”, that’s much of the problem. Indeed, the simulations I see are just illustrations of how the *statistics* start to resemble that of mixtures as decoherence progresses, which is – really – beside the point of the model problem of why quantum waves create any kind of statistics at all. It’s something we find at our practical level, but can’t represent. Roger Penrose and others get this.

  • Pingback: From Eternity to Book Club: Chapter Eleven « Thoughts About Changes In Time()

  • Juan R. González-Álvarez

    To Ucle Sam:

    I think you misread me. Decoherence theory alone does not solve the measurement problem. However, I have really said:

    The theory of decoherence and other advances precisely show that the cat has not wave-function associated due to entanglement (the cat is a dissipative structure, not an isolated hydrogen atom :-)).

    I want to emphasize that the mechanical states of cats and planets are not described by wavefunctions.

    You do not give the details of your argument, but it seems that you are talking about using the Schrödinger-like propagator U for simulating the evolution of a superposition of two WFs and then cheating (your own words!) to obtain the collapse. My point was radically different. I was thinking on something close to what Penrose calls the evolution R, and on the derivation of the projection postulate of quantum mechanics.

    There is some theories and models of the collapse and we can program them in computers. We can study the transformation of a pure state into a mixed state

    ( a_u |u> + a_d |d> ) ( a_u <u| + a_d <d| ) ==> p_u |u><u| + p_d |d><d| [1]

    You talk about ensembles, but again I fail to see the details of your argument. It seems that you are talking about Gibbs ensemble theory of statistical mechanics, but here one may consider the physical ensembles of stochastic theory. We can study the outcome of a single measurement in any member of the physical ensemble as a fluctuation

    p_u |u><u| + p_d |d><d| ==> |k><k| ; where |k><k| = |u><u|, |d><d| [2]

    Probably a process as [2] is what you mean by your colorful any single instance of one WF ‘winning’ over another. But this is essentially the same kind of process associated to the typical measurements of any other stochastic variable in science: chemical concentration, electric field strength, particle momentum, temperature, etc. For instance a measurement of composition implies a process of the type

    Sum_j p_j c_j ==> c_k ; where c_k is one of {c_j} for all j

    The whole process of measurement in quantum systems in an initial pure state involve the composite process ( [1] + [2] ).

    The traditional problem of measurement is that the Schrödinger equation of quantum mechanics cannot explain neither [1] nor [2]. But we can go beyond the Schrödinger equation. You ask for a link to a computer simulations, I think that simulations of both [1] and [2] are routinely found in literature. For [1] search any standard simulation of the evolution of WF done in decoherence theory literature. For [2] search some of the usual algorithms for SSE. Also in page 10 of his work ( Adler gives basic literature references and reviews as his Adler, et al. (2001) on the stochastic reduction approach.

    Of course, there is still many room for improvement, but I see no objective reason which people would abandon detailed and consistent models by speculations as many-worlds. In his book “The large, the small, and the human mind”, Penrose consider not-really-serious-people-regarding-wavefunctions to the followers of the many-world interpretation: Everett, deWitt, Geroch, Hawking, Zureck, & Page. Penrose call really-serious-people to the authors that disagree with many-worlds and worked in realistic alternatives: De Broglie, Bohm, Griffiths, Gell-Mann, Hartle, Omnés, Hagg, Károlyhäzy, Pearle, Ghirardi, Diósi, Percival Gisin, Penrose, & others…

  • Lab Lemming

    I was disappointed that no mention of the 1992 Greg Egan novel Quarantine was made.

  • Bee

    Hi Sean,
    Sorry for being late, I’m limping behind… I’m very confused by all the cats and dogs. Could you enlighten me what is the difference between many-world decoherence due to entanglement and collapse with hidden variables. Don’t hesitate to be technical or refer me to a paper (as long as it doesn’t contain animals). Best,



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About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] .


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