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.”
Excerpt:
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.
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.
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.
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.
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To Ucle Sam:
I think you misread me. Decoherence theory alone does not solve the measurement problem. However, I have really said:
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 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 . 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 (http://arxiv.org/abs/quant-ph/0112095) 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…
I was disappointed that no mention of the 1992 Greg Egan novel Quarantine was made.
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,
B.