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.
Anne, I don’t think it’s quite right to say “decoherence occurs when the wave functions of two possible states of a system become sufficiently different.” A quantum system that is in a pure state, unentangled with anything else, is described by a single wave function. When an electron goes through the two slits, there is a single wave function that goes through both slits, not two different wave functions for the two different alternatives. To get decoherence, you necessarily need some other system to become involved. If there is a sensor at the slits, the wave function of the electron becomes entangled with the wave function of the sensor. Instead of the wave function of the electron being “went through slit 1” + “went through slit two,” we have a wave function of the entangled electron+sensor system, of the form “went through slit 1, sensed slit 1” + “went through slit 2, sensed slit 2.”
If we then threw away our knowledge of the wave function of the sensor, there ceases to be any such thing as “the wave function of the electron.” Instead, there is a statistical (not quantum-mechanical) set of two different wave functions, which can no longer interfere. That’s decoherence — the electron is no longer described by a wave function all its own.
In case of the sensor measuring the “which part information”, the amplitude of the interference term is given by the inner product of the two states of the sensor corresponding to the sensor deflecting the electron and the sensor not deflecting the electron.
In case of a perfect detector these two states should be orthogonal (because they should be eigenstates of some observable corresponding to different eigenvalues that tell you through which slit the electron went) and then the interference pattern will vanish.
The reason why you can’t use a sensor which is heavy has actually a lot to do with decoherence. When the sensor deflects the electron, it absorbs part of the momentum of the electron. If the changed state is to be oerthogonal to the original state (or at least has very small overlap), then it must be very sharply peaked in momentum space.
But by the uncertainty principle that means that in ordinary space it has to have a large spread (in this real space picture you can see that scattering off the sensor would erase phase information if the wavefunction is wide enough).
Now, decoherence leads wavefunctions of objects to collapse in the position basis. The coherence length becomes of the order of the thermal de Broglie wavelength which is of the order:
lambda = hbar/sqrt[m k T]
where m is the mass and T the temperature.
This means that in momentum space the coherence length is of the order
sqrt[m k T]
So, for large enough m, the coherence length in momentum space would be much larger than the absorbed electron momentum and it becomes impossible to detect the recoils.
Sean, I think I see what you’re getting at, but let me put my question another way. Let’s take the two-slit experiment with “sensor” (including the sensor in our wavefunction) and pick the position of a null in the interference pattern. Now, as we adjust the mass of the sensor, the probability changes from zero to the classical value. It seems to me that in some sense when you start getting the classical value you have “decoherence”: your system is not behaving quantum-mechanically any more, even though it has not interacted with the outside world at all. Or rather, it is behaving as if “slit A or slit B” were a probabilistic choice rather than a quantum-mechanical goes-both-ways option. Certainly the wave function represents both possibilities, but the superposition has no detectable effects. Does that make sense?
What I’m getting at is that the weirdness of QM’s “both possibilities happen” is not that they both happen (though I suppose that bothers philosophers) but rather that the fact that they both happen has measurable consequences: in the double-slit experiment, path integrals along the two paths give probability amplitudes that you add, rather than probabilities that you add. If the phases between the two paths are correlated, you see an interference pattern, direct physical evidence of this weird philosophical idea. And so what I’m trying to understand about decoherence is, in a closed system not interacting with the rest of the universe (until I observe it at the end), when do those observable effects of the superposition of states disappear? I don’t mind that Schrodinger’s cat is in a superposition of alive and dead states, but will there be measurable interference effects between the two? Will I be able to measure some quantity which will *show* that the cat was in both states?
Anne, see this article:
http://arxiv.org/abs/quant-ph/0210001
Double Slit Experiment
Start with Slit A and Slit B
We assume the particle emitter is centered between the two slits at some distance.
We assume that there finite number of particles N during the observation time.
N(A) = number of particles passing through slit A
N(B) = number of particles passing through slit B
N(AnB) = number of particles passing through both slit A and B
N(AuB) = number of particles passing through either A or B
We can write the following equation:
N = N(A) + N(B) – N(AnB) + (N – N(AuB))
divide through by N and we find:
1 = P(A) + P(B) – P(AnB) + (1 – P(AuB))
we can then write:
P(AuB) = P(A) + P(B) – P(AnB)
This equation is equivalent to:
sqrt[P(AuB)]^2 = sqrt[P(A)]^2 + sqrt[P(B)]^2 – sqrt[P(AnB)]^2
This describes the diagonal of a rectangular parallelepiped, where length along the respective axes are:
x= sqrt[P(A) – P(AnB)]
y= sqrt[P(B) – P(AnB)]
z= sqrt[P(AnB)]
thus:
P(AuB) = x^2 + y^2 + z^2
We can also define conditional probabilities as angles where:
P(AnB) / P(A) = P(B|A) = [sin (theta1)]^2
P(AnB) / P(B) = P(A|B) = [sin (theta2)]^2
Since the emitter is centered; then when:
theta1 = theta2 = pi/2 = 90 degrees
P(A) = P(B) = P(AnB)
and
x= sqrt[0] = 0
y= sqrt[0] = 0
and
P(AuB) = 0 + 0 + z^2
P(AuB) = P(AnB)
This means that P(A) and P(B) are perfectly correlated, which occurs when we have no knowledge of which slit the particles travel through.
When we set up a detector at slit B to count how many particles actually go through (and we’ll assume that our detector is perfect), then:
P(AuB) = P(A) + P(B)
because our perfect knowledge of particles through B means:
P(AnB) = 0
which occurs when:
theta1 = theta2 = 0 degrees
This makes P(A) and P(B) completely independent!
This means that when we count the particles going through B (or A if we wanted), the number of particles going through slit A and slit B are independent of each other, and we see 2 lines instead of a diffraction pattern!
p.s.
This also implies that it is our knowledge of past events (or rather the environment’s retention of data) which leads to decoherence and the observed classical independence of objects
Lawrence @ 25,
Your explanation is unclear to me. Aren’t you conflating two different things together-the effects of the photon on the wavefunction (actually density matrix) of the object and its “classical dynamics”? The ehrenfest equations of motion that govern the time evolution of the expectation values of position and momentum will have classically chaotic solutions, however, the scattering of the photons affects the quantum density matrix of the system. It seems to me that there is no inconsistency in classically chaotic trajectories being unaffected by quantum fluctuations (since said fluctuations average out into the expectation values).
Also why should the chaotic behavior in the classical limit be preserved upon adding quantum corrections, except in the sense that it is retained in the evolution of momentum and position expectation values?
Please cite this comment.
Under the mathematical construction I provide in comment #55, it becomes apparent that one can in fact cycle the detector at B off and on.
The Zeeman effect is a splitting of spectral lines in the presence of a static magnetic field.
Our situation in the double slit experiment is analogous (and no this has nothing to do with metaphysics as suggested in the Gravity is an Important Force post, comment 17 by ObsessiveMathsFreak)
We are in fact generating a measurable field effect when we introduce the detector. In the Zeeman effect, we see a splitting of spectral lines, in the Double Slit experiment we see a “collapse”.
The field that is generated is linked to the amount of information we possess. We should be able to cycle the detector on and off, and observe the effects of the cycling. By changing the proportion of on and off time, we are changing the conditional probabilities (angles theta1 and theta2 as discussed in comment #55)
Presumably the propagation of the change in the observed diffraction pattern should occur at the speed of light.
Neil B. ¤ on Oct 24th, 2008 at 7:00 pm
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.
BTW, how many of you saw the very interesting and poignant Nova show about Hugh Everett and his musician son Mark?
—————-
The modulus squares of the probability amplitudes determine the probabilities. With the reduction of the off diagonal terms c*_0c_1, which is complex valued, the density matrix is reduced to a diagonal matrix of classical-like probabilities. So the quantum dynamics involves a linear summation of probability amplitudes A_i, while the classical-like outcomes are due to a linear summation of probabilities, which are the P_i = |A_i|^2. There are some subtle issues going on here, in particular this is a case of what might be called a transversal problem.
In classical or macroscopic physics we are able dublicate things, but with QM one can’t do this. One can’t in a unitary manner duplicate a quantum state. Yet if the world is ultimately quantum mechanical then how is it that we can duplicate macroscopic information? Well we can’t perfectly. In TCPI/IP protocal one has to perform parity bit corrections using Hamming distances, or just with photocopiers we have all seen how copies of copies of … , degrade with each iteration. It appears that classical cloning is an approximation, and the noise which results might betray an underlying quantum substrate.
The MWI received some promotion by NOVA. I am honestly somewhat agnostic about this idea. There are problems with getting MWI consistent with the Born rule.
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Otis on Oct 24th, 2008 at 7:46 pm wrote
My question is: Why do we have classical mechanics at all in our universe?
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That really is a big question. In general how the macroscopic world, which include finite temperature physics or statistical mechanics, fits into quantum physics is an open question. Bohr of course proposed the Copenhagen interpretation, which is really just a “first order” approximation that works well enough for most problems.
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Brody Facoum wrote: 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)?
————–
I tend to think more according the (b), though if it turns out that the computing resources required to circumvent this infeasibility are greater then what is available in the universe (an infinitely large computer) then this problem segues into (a) as well.
Ultimately quantum physics is perfectly deterministic. What can be more deterministic than a linear wave equation? However, any large and local region has a myriad of states and making an accounting of them is not possible. So while the universe might from a fine grained perspective be completely quantum mechanical and deterministic from a coarse grained perspective things appears indeterministic and information about states is smeared out or “buried.” In the end the classical world might just be an illusion of sorts, where temperature and time are also illusions of inexact accounting of states. After all a Euclideanized time is t = hbar/kT, for T = temperature, and these both appear to reflect our inexactitude in measuring the world and incapcity in accounting for all possible states.
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kris on Oct 25th, 2008 at 5:50 pm wrote
Lawrence @ 25,
Your explanation is unclear to me. Aren’t you conflating two different things together-the effects of the photon on the wavefunction (actually density matrix) of the object and its “classical dynamics”?
————–
I guess you are referring to the estimate on the angle of orientation by using momentum across a moment arm or principal axis of the moon. This is meant only as an order of magnitude sort of argument.
Lawrence B. Crowell
I have a question regarding Sean’s comment number 4.
I see that if one prepared a classical chaotic system and a corresponding quantum chaotic system into a corresponding state, and then observed the evolution of both states in time, they would rapidly diverge from each other, because the quantum system would not be in exactly the same initial condition as the classical one.
Now suppose we prepare an ensemble of such corresponding pairs, prepared around different points in the phase space, and observed the evolution of all of these pairs. Would we find that the quantum system and the classical system had the same attractor? Would the value of the classical Lyapunov exponent tell us how long it would take for the pair to diverge?
My understanding is that the answer to both of those questions is yes.
I always wondered why when asked that Koan: “does a tree in the woods without an observer make a sound?” why don’t people respond: “does a guy dying an agonizing death in the woods with no observers make a sound?” no it’s a tree.
where do these philosophers get off. thinking inanimate matter computing.
The higher your philosopher’s D&D level is the more astounding feats of alternate reality you can. cut off your smell to spite your mind. that’s for level 30 philosophers with the liberal arts feat.
today is being weird day.
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Sean: “And those photons have their own quantum states, and when they bounce off Hyperion the states become entangled.”
I was under the impression that entanglement was something that happens to the pairs of particles that are created as a result of the collision and subsequent destruction of other particles. If so, then Hyperion’s entanglement with the universe at large would be a function of how many entangled halves of the particle pairs are retained as a part of its mass while the other half goes on its merry way. Are we assuming that half of the collision-generated (and therefore entangled) particles stay behind? If so, what justifies this assumption?
Am I missing something?
After a lashing about diction I just want to make this perfectly clear:
Zeeman effect – turn contraption on, lines split
Double slit experiment – turn contraption on, lines converge
The first case is a magnetic field effect, the second is a unified field effect.
I don’t need to be lectured about complex amplitudes and summing and squaring and quantum probabilities, I’m quite comfortable with those thank you. If people can’t see what’s before there eyes, I can’t help them.
Thank you.
To contribute to the discussion, I asked our web editor whether we could make an article on quantum chaos available for free, and he agreed. Written by Martin Gutzwiller, it appeared in our January 1992 issue, but I think it remains relevant. See http://www.sciam.com/article.cfm?id=quantum-chaos-subatomic-worlds
George
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A Spacetime View of Quantum Physics
Terry Bollinger, Oct 29, 2008
The issue of decoherence resulting from non-isolation was addressed a number of times by Richard Feynman in his writings, although he did not use that particular term. A good example is this quote from his Lectures on Physics (source: http://en.wikipedia.org/wiki/Symmetry#Quantum_objects):
“… if there is a physical situation in which it is impossible to tell which way it happened, it always interferes; it never fails.”
The word “interferes” in this context is a quick way of saying that such objects fall under the rules of quantum mechanics, in which they behave more like waves that interfere than like everyday large objects.
In other words, determining whether an event is quantum can be done simply by asking this question: Does there exist anywhere in the universe information about how the event occurred? If not, the event will follow the rules of quantum mechanics. If such information does exist, that part of the event that is addressed by information will necessarily be classical.
This has an interesting consequence. It means that quantum events are necessarily “ahistorical,” by which I mean that no single history can be assigned to them. Such quantum events become, as is vividly portrayed by Feynman path integrals, a confluence of all possible histories. Viewed from our information-rich classical perspective, this infinity of slight variations of possible particle paths and histories looks and behaves very much like a wave, provided only that we don’t “poke it” it too hard or in a way that precipitates a historical outcome.
It is the ahistorical properties of such regions that make them baffling when they are viewed from a classical perspective. Jon Bell showed through his brilliantly insightful thought experiment that led to surmise that from a classical perspective, certain classes of events necessarily require the concept of “instantaneous” action at a distance.
Intriguingly, saying that such events require “instantaneous” action seriously understates the problem. What really occurs is when a Feynman integral is poked hard is that one particular instance of the infinity of possible paths in the Feynman integral suddenly decides to become real. That sounds innocuous enough until you begin to ponder this point: A Feynman path extends not just across space, but across time.
That is, forcing a quantum region of spacetime to become “historical” – that is, to have a classically defined outcome that can be observed and recorded – requires not just an “instantaneous” updating of events that could be distant in space, but also of events that could be distant in time. More specifically, you must in effect update the way the event originated in the past, even if that past is very distant. An example would be an in the case of an interstellar photon whose path integral extends several billion years back to the time of its launch from an ancient quasar. An example would be measuring the polarization of an intergalactic photon that started its journey to earth several billion years ago. Such a measurement results in the instantiation of a specific Feynman path that is the equivalent to saying that the photon “always” had a particular polarization since the time it was first emitted billions of years ago.
A reflex reaction to such scenarios is that they violate causality and so cannot possibly be correct interpretations of quantum mechanics. The very nature of quantum mechanical regions prevents any conflict, however, since a quantum region of spacetime can exist only if there is no historical information about what took place within it. A violation of causality cannot occur because no information has ever emerged from the region to become the cause of something! The very definition of what makes a region quantum simultaneously safeguards such regions from generating causality violations.
From such issues I would argue that there is a very deep, even tautological relationship between three issues: classical physics, history, and information. If no information about an event exists, it has no history, and so behaves under the rules of quantum mechanics in which classical causality is replaced by the wavelike sum of all possible histories. If information about what happened exists anywhere in the external universe (an observer is emphatically not required, only a bit of data here or there), the event becomes both historical and classical, subject to the standard laws of causality. Notably, Schrödinger’s Cat is actually a very poor example of quantum behavior: The cat simply lives or dies in a fully classical fashion, since the box in which it is hidden from view is nonetheless exuding enormous quantities of information about what is going on inside. There is heat from the cat, sound from breathing and motion, air molecules that are continually bumping and jostling the cat, and even gravity and particle effects. Only if the experiment can be updated to prevent all these other forms of information leakage does it become a viable thought experiment for understanding the nature of quantum physics.
So, can classical objects become quantum, as described in the article? Of course! But where is one detail that makes the creation of quantum regions for classical objects extraordinarily difficult to do: You have to keep such objects totally, completely, and absolutely incommunicado with the rest of the universe for a significant length of time.
It is this difficulty of creating information-isolated regions of spacetime for large time that is at the root of why quantum mechanics generally applies only to very small objects. It is easy to isolate an electron in a way that prevents it from exchanging information with the rest of the universe most of the time, while it is very nearly impossible to do the same thing with a spaceship flying through interstellar space. A single photon emitted from the ship and registered classically somewhere else in the universe is sufficient to establish a location and pull the ship out of the world of quantum physics. Clever physics and clever engineering can sometimes create ahistorical regions of space large enough for us to observe, such as superfluids, Bose condensates, and for that matter the light-reflecting conduction bands of metals, but for the most part information binds and bonds us together into a vibrantly complex world of history, information, and classical physics.
Lawrence B Crowell and Peter Erwin, thank you.
Arjen Dijksman: what can you really say about a single detected photon other than its measured quantum state? Surely you can’t be certain that the source of the photon was the event you’re interested in, you can only be mostly certain, if it has a particular frequency for example. You can improve this in laboratories, but what about in astronomy or observational cosmology? A single photon might arrive from the “right” direction with the “right” momentum, but how confident can you be that it’s not an environmental photon which evolves into the state you’re looking for? More importantly when thinking about large classical objects like Hyperion and Earth, what about single photons which are emitted by the event you’re interested in, but which exit the state you’re looking for because of accelerations, scattering or absorption? Did the event “not happen”?
To bring it back to your comment and this discussion topic,
can you really say that your single detected photon tells you anything certain about an object in uncontrolled real space, and that the problem with saying anything about such an object is only that you don’t know what other information it is presently sending into the environment that you are not in a position to detect?
I agree with you that you receive information bit by bit, photon by photon, however. 🙂
Finally, a general question: is there anything like a line separating observations in which the quantum states of individual particles is important and when they are only important probablistically? On the one hand we care about quantum states at some level when looking at the Lyman-Alpha forest or when doing galaxy surveys or studying the CMB or studying spectral lines generally; do we really care much when studying the orbit or rotation of a planet or moon, even one around a distant star?
Flipping this around, I think I am asking: is the discussion of decoherence a way of explaining that QM is still useful in the case of terrestrial observations of Hyperion? (I think I’m reading Peter Erwin’s comment 46&47 as a sort of “yes” to this question, on the grounds that the correspondence principle drives the selection of a QM theory that predicts classical-scale reality including classical limits on the predictability of dynamic systems). That is, is this whole discussion exploring the behaviour of QM rather than that of Hyperion? (I’ll accept a “duh! stupid question!” answer gladly 🙂 )
Brody:
Nothing I fear, we can only infer something. We aren’t even able to measure its quantum state. We are only able to measure some characteristics about the impact of the photon on the detecting electron which is at the origin of the detection cascade ending at our perceiving cells. The quantum state could well be a mix between |x> and |y>, but my detection process can filter only one of both “orthogonal states”. In that sense, I don’t see any difference between measurements on macroscopic and nanoscopic quantum objects apart from the fact that we receive way much more information about a macroscopic object. The huge quantity of information gathered about a single object allows us to square out the indeterminacy about direction, momentum, previous scattering, diffraction, absorption, non detection events of the photons.
The wave function or state is never directly measured. We only measure an obervable O which acts on a state O|n) = O_n|n). For all we know the state vector is just a mathematical construction we impose and nothing more.
Measurement is a bit of a strange topic, for a state vector |Y) = sum_n c_n|n) evolves by nice continuous unitary operators. Equivalently, one can consider the obervable operators as evolving by unitary operators in the Heisenberg picture. Yet a measurement is one where the classical “needle” of the detector pops into one of a number of discrete outcomes for a measured observable. So we never measure a system in a superposed state — Schrodinger’s cat is either alive or dead upon looking in the box. Decoherence which removes the entanglement phase of a system reduces the density matrix to a diagonal of probabilities. It does not tell us which of these outcomes are actually measured though.
We might of course ask why there is a classical world at all. For a system with a large action, given by a huge number of Planck units of action, the path integral describes a set of paths which are very close to each other. The issue which arises is how does the quantum world “know” to cluster around a certain path which we consider classical. This matter Zurek is investigating according to einselection processes. This is a topic which would take considerable writing to describe. Maybe it is a topic which one of the CV blog meisters will take up.
I will say that the decoherence and einselection approaches to these problems I find preferrable to the alternatives such as the many worlds interpretation.
Lawrence B. Crowell
Lawrence: “For all we know the state vector is just a mathematical construction we impose and nothing more”.
Isn’t the state vector imposed on us, rather than that we impose it as a mathematical construction? It fits so well as a representation of the physical system that I’m inclined to visualize it as the “hard reality” concerning the system. Line-shaped macroscopic objects, which we could also represent by mathematical vectors, follow the same rules:
– differential evolution d|vector>/dt = i . omega . |vector> (the vector difference is always perpendicular to the vector itself),
– indeterminacy of measurement result (measuring the location of the object may give any location on the length of the object),
– steering by a pilot-wave (in a cloud of line-shaped objects, a line-shaped object spins in phase with the other),
– square-law for probability of detection : . (the cross section between detecting and detector line-shaped objects evolves with the phase of the pilot-wave).
The wave function is not real — it is complex! In effect the state vector is really a sort of construction on our part. The only thing which is real are the eigenvalues of Hermitian operators. Those are the only thing which we actually detect, or which make a detector go “ping!” As for probabilities, it is of course the case those are computed by the amplitudes of the wave function. Yet we really don’t measure those directly, but only infer them with an ensemble of measurements.
Lawrence B. Crowell
Lawrence, I agree we don’t measure directly the amplitudes of the wave-function but only infer them from a set of measurements. Concerning the terminology real vs. complex, the distinction is mathematical. Complex values of the wavefunction refer to ‘real’ physics: the change of phase of a rotating vector that represents the particle. Of course measurement values take real values because a QM result is one single number, but this must not hide the fact that there is hard reality behind those measurements: particles that are described by a rotating vector (and not by a point).
The wave function in quantum mechanics only represents the information potentially available about a system. I suppose I don’t think of the wave function as real in the sense of “reification.” It really is just a field-like effect of sorts we use to make sense of things we measure.
Lawrence B. Crowell