One of the annoying/fascinating things about quantum mechanics is the fact the world doesn’t seem to be quantum-mechanical. When you look at something, it seems to have a location, not a superposition of all possible locations; when it travels from one place to another, it seems to take a path, not a sum over all paths. This frustration was expressed by no lesser a person than Albert Einstein, quoted by Abraham Pais, quoted in turn by David Mermin in a lovely article entitled “Is the Moon There when Nobody Looks?“:
I recall that during one walk Einstein suddenly stopped, turned to me and asked whether I really believed that the moon exists only when I looked at it.
The conventional quantum-mechanical answer would be “Sure, the moon exists when you’re not looking at it. But there is no such thing as `the position of the moon’ when you are not looking at it.”
Nevertheless, astronomers over the centuries have done a pretty good job predicting eclipses as if there really was something called `the position of the moon,’ even when nobody (as far as we know) was looking at it. There is a conventional quantum-mechanical explanation for this, as well: the correspondence principle, which states that the predictions of quantum mechanics in the limit of a very large number of particles (a macroscopic body) approach those of classical Newtonian mechanics. This is one of those vague but invaluable rules of thumb that was formulated by Niels Bohr back in the salad days of quantum mechanics. If it sounds a little hand-wavy, that’s because it is.
The vagueness of the correspondence principle prods a careful physicist into formulating a more precise version, or perhaps coming up with counterexamples. And indeed, counterexamples exist: namely, when the classical predictions for the system in question are chaotic. In chaotic systems, tiny differences in initial conditions grow into substantial differences in the ultimate evolution. It shouldn’t come as any surprise, then, that it is hard to map the predictions for classically chaotic systems onto average values of predictions for quantum observables. Essentially, tiny quantum uncertainties in the state of a chaotic system grow into large quantum uncertainties before too long, and the system is no longer accurately described by a classical limit, even if there are large numbers of particles.
Some years ago, Wojciech Zurek and Juan Pablo Paz described a particularly interesting real-world example of such a system: Hyperion, a moon of Saturn that features an irregular shape and a spongy surface texture.
The orbit of Hyperion around Saturn is fairly predictable; happily, even for lumpy moons, the center of mass follows a smooth path. But the orientation of Hyperion, it turns out, is chaotic — the moon tumbles unpredictably as it orbits, as measured by Voyager 2 as well as Earth-based telescopes. Its orbit is highly elliptical, and resonates with the orbit of Titan, which exerts a torque on its axis. If you knew Hyperion’s orientation fairly precisely at some time, it would be completely unpredictable within a month or so (the Lyapunov exponent is about 40 days). More poetically, if you lived there, you wouldn’t be able to predict when the Sun would next rise.
So — is Hyperion oriented when nobody looks? Zurek and Paz calculate (not recently — this is fun, not breaking news) that if Hyperion were isolated from the rest of the universe, it would evolve into a non-localized quantum state over a period of about 20 years. It’s an impressive example of quantum uncertainty on a macroscopic scale.
Except that Hyperion is not isolated from the rest of the universe. If nothing else, it’s constantly bombarded by photons from the Sun, as well as from the rest of the universe. And those photons have their own quantum states, and when they bounce off Hyperion the states become entangled. But there’s no way to keep track of the states of all those photons after they interact and go their merry way. So when you speak about “the quantum state of Hyperion,” you really mean the state we would get by averaging over all the possible states of the photons we didn’t keep track of. And that averaging process — considering the state of a certain quantum system when we haven’t kept track of the states of the many other systems with which it is entangled — leads to decoherence. Roughly speaking, the photons bouncing off of Hyperion act like a series of many little “observations of the wavefunction,” collapsing it into a state of definite orientation.
So, in the real world, not only does this particular moon (of Saturn) exist when we’re not looking, it’s also in a pretty well-defined orientation — even if, in a simple model that excludes the rest of the universe, its wave function would be all spread out after only 20 years of evolution. As Zurek and Paz conclude, “Decoherence caused by the environment … is not a subterfuge of a theorist, but a fact of life.” (As if one could sensibly distinguish between the two.)
Update: Scientific American has been nice enough to publicly post a feature by Martin Gutzwiller on quantum chaos. Thanks due to George Musser.
Michael Berry, in his paper on this subject in 2001, argued that a single photon would be enough to induce decoherence, since Hyperion’s angular-momentum levels are so closely spaced.
Incidentally, Berry made much the same point as I did in #14 about the timescale of classical chaos, but goes onto say that decoherence renders this point moot.
George
James, yes, you can choose whatever basis is convenient for you in pracice. But we are now considering the philosphical question: “Does Hyperion have a definite orientation before we observe it?”
So, we should consider the slightly less reduced density matrix in which you also keep your own mental state. I cannot observe myself in a superposition of two different mental states, I’m always one part of such a superposition. So, I’m going to assume that there exists a preferred physical basis for the mental states.
Then, instead of tracing out the mental states, we should consider the reduced density matrices:
< m|rho|m>
where rho is the density matrix in which everything except Hyperion and the observer’s degrees of freedom have been traced out, and m denotes a particular preferred mental basis state. The question is then if this reduced density matrix describes a pure state for general m.
Only if it is a pure state can we say that before we observe it, Hyperion already was in a definite state. So, paradoxically, a pure state in this case corresponds to a collapsed state and a mixed state corresponds to the system being in a superposition. 🙂
As I argued above you should expect that the state will be a mixed state which then means that Hyperion doesn’t have a definite orientation given our mental state before we measure it.
My non-scientific mind understood a scant 50% of what you said in your article yet I still found it wildly fascinating.
Cheers to that.
Typo:
instead of tracing out the mental states we project out particular mental basis states, so the reduced density matrix is:
< m|rho|m>
Then I was wrong to say that one should look for arbitrary m in the preffered basis. Instead one should try to find out if this can describe a pure state if the brain and thus the mental states m are perturbed by photons scattering off Hyperion and then hitting the observer on Earth. That doesn’t look plausible to me, as any such effects will be swamped by other local effects.
Count Iblis,
You say:
“…I cannot observe myself in a superposition of two different mental states, I’m always one part of such a superposition. So, I’m going to assume that there exists a preferred physical basis for the mental states.”
Have you never felt “in two minds” about something? Research seems to have shown that the brain is massiveely parallel, just from a classical point of view, and there are those who would argue that it exploits QM parallelism in its working (personally I’m not convinced, but I can’t rule it out).
The density matrix – or the reduced one for tht mater – which I remember fondly (not) from work in quantum optics, is a statistical mechanical tool, and sheds no light on fundamental questions – such as what Hyperion is doing when nobody’s looking.
-James
Topics like this are why I stick with this blog. If there’s a more fascinating topic in modern science I have yet to encounter it.
You roll one die, you can’t predict what number you’re going to get.
You roll 100000000000000000000000000000000000000000000000 dice, you can predict with accuracy refined to a similar number of decimal places that 1/6th of them will be 1, a 1/6th 2 etc etc. The more you roll, the greater the accuracy of your prediction.
I’m a total and utter layman when it comes to the fine details of quantum mechanics, but surely the “decohernace” everyone is discussing is just the increasing predictability of increasing numbers of “dice rolls” (or particle interactions) writ large across the universe?
On our macro scale, we’re used the the ‘dice’ being rigidly defined quantities of discrete numbers between one and six that we can point to and identify. When we observe a single die that is in the process of rolling, we can’t call it 1, 2, 3, 4, 5 or 6, we must define it as something that has the potential of coming up 1, 2, 3, 4, 5 or 6 but is no single one of those things until it stops rolling, and all of those things together all the time.
When you describe dice like that, they sound just as fantastical as an object behaving in a quantum manner, but they’re still dice. Obviously this is horrifically simplified, but if you remove the image of a comfortably familiar object and remember that we’re dealing with particles, I think the point still stands?
Apologies if I’m explaining something incredibly rudimentary that everybody is already talking beyond as pre-assumed ^^;
Given that the action for the rotation of this body is probably around 10^50 hbar the uncertainty in the angular position is ~ 10^{-50}. So a single photon which impacts the body across is primary axis d with momentum p = hbar-k can change the angular momentum by L ~ hbar*k*d, with a change in the angular position ~ 1/kd = lambda/d ~ 10^{-6}/10^5 (ball parking the size of the moon here) which is more than enough to change the angular position of the moon far greater than the HUP uncertainty.
As for density matrices of brain states and the like, one of the whole points of this is to show that there is no need to invoke any mental state of an observer.
Lawrence B. Crowell
I’m not so sure the world doesn’t appear quantum mechanical.
I’m going to watch the baseball game.
Is the outcome:
1)Pretermined from events that happened billions of years ago? (Newtonian determinism)
2)The reult of certain propensities of nature mixed with chance and the free choices of the conscious participants?(orthodox Von Nuemann QM)
I think the universe I live in is very quantum mechanical.
Paul on Oct 24th, 2008 at 4:19 pm wrote: surely the “decohernace” everyone is discussing is just the increasing predictability of increasing numbers of “dice rolls” (or particle interactions) writ large across the universe?
No it is not that. Suppose you have a two state system. There are two possible states it can be in |0) and |1). The dual to these states are written as (0| and (1|, which satisy some conditions
(0|0) = (1|1) = 1
(0|1) = (1|0) = 0,
or these states in the state space are perpendicular or as we more often say orthogonal. A general state vector for this system is usually written as
|Y) = c_0|0) + c_1|1)
the dual of this vector is then
(Y| = c*_0(0| + c*_1(1|
where the star * represents complex conjugation for i = sqrt(-1) changes sign to -i where ever it appears. The terms c_i are complex valued probability amplitudes. What physicists often do is to look at the density matrix which is written as rho = |Y)(Y| so it has components
rho_{00} = c*_0c_0 = |c_0|^2
rho_{11} = c*_1c_1
rho_{01} = c*_0c_1
rho_{10} = c*_1c_0
where rho_{01} and rho_{10} are complex conjugations of each other.
The rho_{01} and rho_{10} off diagonal terms are complex valued and represent phases for the superposition of these states. The diagonal terms just give the probabilities for the two states. If this system is coupled to some complex environment or a shower of photons these phases can become coupled to these external factors and removed. The density matrix is then reduce to the diagonal terms. The systems has decoherently entered into a “collapsed state.”
Lawrence B. Crowell
Well, let me reformulate my point without invoking density matrices. If you observe a system it will be in one of the eigenstates of the observable you are measuring. Now suppose that the system has already decohered before you make the measurement due to interactions with the environment. We perform a measurement and find that the system is in some state.
The question is if the system was in that state before you made that measurement. I claim that this is not the case. We can be sure that after the mesurement, the system has collapsed into some definite eigenstate. So, it is then in a pure state. If before measurement the system were in the same state it would thus already have to be in that same pure state.
The fact that in the density matrix formalism you find a mixed state is then explained by the fact that had you included your mental state, the density matrix would be diagonal with only 1 nonzero diagonal element. It would then be the tracing out over the possible mental states which yields the mixed state.
But this would imply that we could in principle have psychic powers to “feel” what the orientation of Hyperion is without observing it. That’s surely not possible, therefore one has to assume that the density matrix that includes your mental states would not describe a pure state (i.e. it would not have a single nonzero diagonal element, or a singe dirac delta in the continuous case).
Or formulated without referring to density matrices: In the full entangled state of system and rest of the universe, the terms corresponding to different orientations would not necessarily contain mental states that are orthogonal. I.e. if you group the terms for each mental state, it would contain states referring to different orientations.
What does decoherence mean in the context of the many worlds interpretation?
That it’s likely that the position of a macroscopic object like the moon is rather definitive in a given branch?
Lawrence, you forget that figures giving “probability” for a given state or outcome are based on collapses and specific events already happening, then fed into the decoherence pretended “explanation” of what Wikipedia calls “appearance” of collapse – it’s a circular argument, fallacious in the familiar way. None of you going on about that here or elsewhere really explain why the waves spreading around, interacting or not with each other, don’t just stay patterns of waves as they do in both classical physics and in the pure mathematics of wave evolution (as in the “deterministic evolution of the wave under the Schrodinger equation” etc.) Waves only get connected to “probability” because something weird about the universe gets them to express like that, it still doesn’t make any sense at the fundamental level – you are just putting out double talk that reminds me of the sophistry put out by Wittgenstein’s supporters. As for the density matrix: that’s just a way of talking about waves that we don’t know the particulars of, isn’t it? Like for example, a photon gun that may produce linear x or linear y, we don’t know which. It isn’t really a “state” that anything has per se, in any coherent (I mean, general meaning as a pun) sense.
BTW, how many of you saw the very interesting and poignant Nova show about Hugh Everett and his musician son Mark? The idea of constantly splitting parallel worlds is cool as a weird idea but I don’t buy it. One thing to consider: in the Schrodinger’s cat situation, there is an unstable nucleus that may or not have decayed after a given time. That means that the cat is superposed alive/dead, etc. But that means that not only is the wave representing say, an emitted beta (electron) spread out over all angles and not just a narrow ray (presumably – given ordinary angular uncertainty) but it can’t even be a nice crisp shell. IOW, emitted particles have to keep “leaking out” over time, and that makes everything even more difficult to sort out. REM that by contrast, usually we see the example put as, the shell expands to reach a spherical screen and must collapse (but still at a given moment!) somewhere on the shell, with impact time being a bit uncertain but not a major issue. Well?
Pingback: Gravity is an Important Force | Cosmic Variance
My question is: Why do we have classical mechanics at all in our universe?
In the beginning, at the Planck scale, there were no classical localized states. So when and why did decoherence produce the universe that we can live in?
I have heard it said that our approximately classical world is not fundamental. Instead, it is the result of special cosmic initial conditions, the result of a special quantum state, a relic of how the Big Bang came about.
So what does physics tell us about the mechanism for Big Bang emergence of localized states such that we can predict when the sun will come up tomorrow?
This is a fascinating topic. Thanks to Sean for the post and to others for the discussions.
Another issue I have with decoherence: Consider the classic case of the photon split by a beamsplitter. The waves travel at right angles towards distant detectors, separated from the BS and each other by empty space, and perhaps many kilometers away. We can have the photon coherence length much less than distance to detectors. The photon must absorb in one or the other detector, not both. Consider the split wave function as it reaches the detectors. One or the other detector will ping, and then the other one is barred from also pinging. There is no way for any interaction or interference of any actual waves, to reach from the pinged detector to the other one for collapsing the wave that “was there” as we imagine it while it was just propagating. Nothing actually crosses the spatial separation, and the forbidding of the double ping happens immediately despite the distance. (Sure, there’s some “connection” in entanglement but that isn’t the actual influence of one wave on another, in the environmental “decoherence” sense.) There’s no way to make that work out rationally. In any case, I get suspicious when apologists start talking of how something makes X “appear” to happen, that is a danger sign of BS (not a beam splitter!) at work.
Neil, you can easily work out exacly what happens using the usual formalism of quantum mechanics. What you get is a state like:
|ping, no ping) + |no ping, ping)
and then this quickly decoheres.
Pingback: Ars Mathematica » Blog Archive
A handful of questions:
9: Robert: Why do we have to introduce our brains? If we treat the standard model as real, what happens at the surface of an arbitrary lens surrounded by the practical vacuum of interplanetary space?
Whatever happens at that surface can be used to create a partial map of Hyperion; these maps can be constructed (mechanically or logically) at any time the lens is “allowed to look at [Hyperion]”, and the maps are in principle predictable from one moment to the next, even if we hide and unhide Hyperion from this arbitrary lens (and/or any/all other(s)). “Some eigenstate of the orientation operator” really means an infall of particles with which we can construct a *statistically normal* map.
However, surely a *normal* map differs from a representation of a subset of the *real* state of Hyperion at any point in the recent past not least because of the differences in accelerations of the various parts of Hyperion? Moreover, if our lens is not a massless pointlike event, it will also have its own differences in accelerations at the surface that will influence its interactions with infalling particles from the direction of Hyperion.
15: Lawrence B Crowell: does your “can’t be predicted” mean (a) “cannot be decided at all”, or (b) “is infeasible to compute”? Do we introduce uncertainty intervals to cope with (a) or with (b)?
That is, can we confidently do more — or less — than produce a phase space for Hyperion recursively considering the phase space of each of the quantized elements of Hyperion, and admit that the phase space for Hyperion itself may be incomplete because we cannot practically isolate it from external events?
IOW, is the analysis of the state of Hyperion properly in the domain of statmech, which is how I read Thomas Larsson’s question at 18.? I think you sorta say so in 25 and 33.
It just seems to me that trying to treat an object on the scale of Hyperion as a full set of quantum events is unnecessarily hard. (It also seems to get harder the more I think about how one would actually go about doing that; how does one account for particles it radiates that might interact only gravitationally for long long long periods of time?)
Finally, from Sean’s initial posting: ‘So when you speak about “the quantum state of Hyperion,” you really mean the state we would get by averaging over all the possible states of the photons we didn’t keep track of’ — but the photons scattered off or radiated by Hyperion only reflect (pardon the word) the state of Hyperion’s surface, which is a useful boundary that describes its overall orientation, but which *many* microstates can equally describe, particularly when you start considering things below Hyperion’s surface. Right?
Finally, does knowing that there are lots of interactions happening between Hyperion and events across the universe — environmental decoherence — *really* improve our ability to predict Hyperion’s orientation in, say, 2029?
Suppose there is no photons and other particles hitting Hyperion. Wouldn’t gravity itself enough to exert decoherence ? The same gravity which cause it’s instability…
Alex @ 20:
I was just wondering why Hyperion specifically is so unpredictable, why it’s quantum fluctuations evolve into macroscopic ones while most bodies follow the correspondence principle.
This is because Hyperion is classically chaotic and other moons are not. It doesn’t matter whether the fluctuations or uncertainties in question are quantum-mechanical in scale/origin or not (e.g., does small asteroid X bounce off Hyperion or not, thus giving it a kick); the tumbling of Hyperion is chaotic because of the specific gravitation situation it is in (the overlapping influences of Saturn and Titan, the locations of resonances, Hyperion’s particular shape and moment of inertia, etc.).
Quantum mechanics doesn’t explain[*] why Hyperion itself is chaotic and other bodies are not; classical mechanics (and the particular conditions of the system) does.
The argument of Zurek & Paz is that Hyperion does indeed follow the correspondence principle, so that despite the underlying QM nature of reality, Hyperion still tumbles in a classical (and in its case chaotic) fashion. The issue is why — that is, how & why does the correspondence principle still apply in situations that are classically chaotic?
[*] By which I mean “it’s not necessary (or it doesn’t help) to use QM to explain”
Brody Facoum @ 44:
Finally, does knowing that there are lots of interactions happening between Hyperion and events across the universe — environmental decoherence — *really* improve our ability to predict Hyperion’s orientation in, say, 2029?
No. What environmental decoherence does is ensure that Hyperion’s orientation obeys classical dynamics — that is, that the correspondence principle holds for Hyperion as well as for Titan, Saturn, and other macroscopic objects. Since classical dynamics tells us that Hyperion’s orientation evolves in a chaotic fashion, we are unable to predict its orientation more than a few days in advance.
Thanks for the best post of the year, possibly the millenium (IMHO). I think it answers a question I had for the previous QM-talk post, although some of the commenters seem to disagree on that. I will never understand QM (unless that means I do!), but it gives me some comfort that smarter people find it is non-supernatural, albeit weird.
When I receive a photon from an object, it informs me about one of the possible locations of the object: the location of the point of emission of the photon. The location of the object as a whole is not determined precisely through one single measurement, because I don’t have any information about the other possible points of photon emission of the extended object. The problem with quantum measurements is that I receive this information bit by bit and not classically as a whole. If the world doesn’t seem to be quantum-mechanical, it is a matter of perception.
I am a bit puzzled by the business of needing environmental photons to produce decoherence.
As I understand it, decoherence occurs when the wave functions of two possible states of a system become sufficiently different (that is, sufficiently nearly orthogonal, right?) that no interference effects between the two states are observable. If that’s the case, then shouldn’t two wildly different histories leading to wildly different orientations produce very nearly orthogonal states? If Hyperion did not decohere, what interference effects would we see?
Or, let’s take the double-slit experiment, slightly modified: an electron passes through one slit or the other (or, really, both) and passes on to hit the screen. But near one slit let’s put a charged object whose momentum we can check after the experiment. If the charged object is really massive, then the electron will not move it appreciably no matter which path it takes, and the interference pattern will remain (though distorted by the electron’s deflection). If the charged object is light enough, we can measure its momentum after the fact and determine which slit the electron passed through, and the interference pattern must disappear. Thus we have a parameter (mass of the “sensor”) we can adjust from “no observation” to “observation”.
As I understand decoherence, what happens is that when the “sensor” is light enough to actually tell which path the electron took, it unavoidably entangles its phase with the electron’s wavefunction, producing a phase shift in the “passed through slit A” possibility. This phase shift is random, presumably because of the uncertainty principle as applied to the sensor, the interference pattern on the screen disappears. (Actually you always have a phase shift, but if the object is massive the random part of the phase shift is so small that the interference pattern is not affected.) Put another way, once you include a light detector, the two alternatives “passes through slit A” and “passes through slit B” have such different wave functions that there is no detectable interference effect. Does that sound about right?