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20191102 https://www.quora.com/How-is-the-Ehrenfest-paradox-resolved/answer/Alan-Cooper-5
20191208 How-does-special-relativity-explain-the-twin-paradox-since-acceleration-is-required?
20200929 How-do-you-explain-Einstein-s-twin-paradox-in-an-easy-to-understand-summary?
20210515 Why-does-time-dilation-slow-only-one-twin-in-the-twin-paradox?
20220109 How-does-the-twin-clock-paradox-in-SR-really-work?-Please-see-the-comment-for-the-specific-case
Calabi-Yau manifolds (there’s not just one) are a type of mathematical concept, but they’re not “just” that as they do have applications in certain attempts to describe physics.
The role they usually play in physics is to help us formalize the relationships that we postulate between the various internal variables that describe what particles are likely to show up at a point in space time. As such the theory often combines the six dimensions of a Calabi-Yau manifold with the four dimensions of the space-time that we are ‘inside’ to get a total of ten dimensions. But the extra dimensions are often either considered to be very small in some sense, or to have the part that contributes to the physics we see be just a slice through the whole thing. In the first case it makes more sense to say (as in James Bridgeman’s answer) that there’s a C-Y manifold at every point inside us (rather than vice versa), and in the second case that the entire space-time we live in is just a (4d) slice through the extended C-Y manifold (with other slices or “branes” perhaps corresponding to “alternate universes” of some kind). But neither of these cases is in any sense known to be true. So far it’s all just speculative construction of mathematical models that might eventually be shown to describe our actual physics.
Physical measurements are generally considered to produce real number valued results, but these can always be re-expressed as collections of Yes or No answers (to questions about whether or not the observed value is within various narrower and narrower intervals for example).
And a physical theory is a procedure for predicting the probabilities of positive answers to some such questions from knowledge of others – typically in the context of some particular experimental setup or system.
For example the system might consist of an electron emitted from a cathode or electron gun at a particular location into an evacuated environment which includes a barrier with two slits between the cathode and a phosphorescent screen. In this case questions with known answers include the position and time of emission and what we want to predict are probabilities of seeing a flash in various regions on the screen (which might be combined into the form of a probability density function).
In a classical theory, any range of values of momentum, position and time determines an experimentally testable question and in principle the electron could have arbitrarily precise position and momentum at any particular time, and the lattice of all such questions is just like that of all propositions in classical logic. The most complete specification of the state at each time is just a point in phase space (ie a precise value of position and momentum) and incompletely specified states can be identified as statistical mixtures of these pure states corresponding to probability density functions on the phase space.
But in quantum theory the position and momentum cannot both be measured with arbitrary precision at the same time and the lattice of all experimentally testable propositions does not have all the completeness properties of a classical logic. The most complete specification of the state in this theory corresponds to a ray (or unit vector) in a Hilbert Space (of “wave functions”) but such a pure state may still not have precise values for all observables and a pure state with precise values for one observable (often called an eigenstate for that observable) may correspond to a linear superposition of eigenstates with different eigenvalues for complementary observables. Even less completely specified states can be identified as statistical mixtures of these pure states corresponding to weighted sums or integrals of pure state projection operators to yield trace class operators called density matrices.
When QM is used to predict probabilities for outcomes at the end of an experiment where the system under study is essentially destroyed (as with the electron being absorbed into the phosphor of the screen) the question of what happens during and after the measurement does not arise. But if we place a non-destructive detector within the experiment then we need to ask how to analyse and predict what happens to the combined system. It appears in that case that if we ignore the quantum nature of the detector then the result for the electron looks as if, at the time when the intermediate measurement is made, the electron’s state suddenly jumps to the eigenstate for the value that happens to be observed. And explaining how this is consistent with a quantum description of the combined system of electron and detector is what is known as the “measurement problem”
See Quora questions:
What is the measurement problem of quantum mechanics?
The symmetrical but infinitely varied growth of snowflakes can be modeled by assuming that the rate of growth at corners relative to sides varies with atmospheric conditions that are essentially uniform over the size of one flake but may vary over time while the crystal is growing. But the details of how those growth rates depend on conditions are quite complicated and have long been an area of serious research by Caltech physicist Kenneth Libbrecht. Veritasium has produced a new video which includes some recent advances in Liebrecht’s work (in particular a more detailed explanation of why different temperature regimes favour columnar over planar growth or vice versa)
Someone on Quora recently made reference to an article by Ivars Peterson which reported an interview with the distinguished physicist David Mermin where he drew attention to the distinction between comparison of averages and case-by-case correlation – by making an analogy between the independence of a baseball game’s outcome on the actions of a TV viewer and that of an experiment on one particle on the possible set up of a remote experiment on a related particle.
But when it comes to the key point of the matter, the article states it without justification:
The colors in the hypothetical A1-B4 experiment should agree 15 percent of the time (or disagree 85 percent of the time). But that’s impossible because the colors flashed at both detectors in the hypothetical A3-B4 experiment disagree 15 percent of the time, the results of the real A3-B2 experiment disagree 15 percent of the time and the results of the hypothetical A1-B2 experiment disagree 15 percent of the time. If the Strong Baseball Principle were correct, the colors in the hypothetical A1-B4 experiment could therefore differ at most about 45 percent of the time.
The article concludes with reference to an exchange that occurred in the pages of ‘Physics Today’ between Mermin and a couple of older distinguished physicists Paul Feshbach and Victor Weisskopf; but the background to that goes back a long way – all the way to Einstein vs Born&Bohr really, with focus on the famous Einstein-Podolsky-Rosen(EPR) gedanken-experiment of 1935.
The original EPR discussion emphasized the apparent possibility of simultaneously determining both the positions and momenta of a pair of particles by measuring one observable on each particle and using their fixed relationship to determine the same observable on the other particle. Because of complications related to keeping track of the different positions of the separated particles, it later became more popular to use spin components as the non-comensurable variables with position as an essentially irrelevant extra factor.
The paradox would then be that we could measure the spin of the two particles in perpendicular directions at the same time, and so (because of the two spins being equal) we have determined both spins at the same time for each particle.
[Of course we could measure both spins in succession on one particle, but after the second measurement the spin in the first direction would again be unknown, whereas for the two spins measured simultaneously it is not clear which (if any) is subsequently uncertain.]Another way of looking at this is that whichever spin is measured first, the measurement of that spin immediately changes the situation of both particles (and since they may by then be far apart, this appears to involve faster than light communication).
Fifty years after the EPR paper, David Mermin wrote a fancifully titled article in ‘Physics Today’ where he reviewed subsequent developments, including Bell’s inequality and the then recent experimental test of that by Alain Aspect.
As Pascual Jordan put it: “Observations not only disturb what has to be measured, they produce it. … We compel [the electron] to assume a definite position. … We ourselves produce the results of measurement.”
Faced with spooky actions at a distance, Einstein preferred to believe
that things one cannot know anything about (such as the momentum of a particle with a definite position) do exist all the same.
A bit (well 3 years) later, Feshbach and Weisskopf wrote a foolishly titled opinion piece suggesting that that the role of indeterminacy in quantum mechanics has been “greatly exaggerated”. Although probability plays an important role, they say, it doesn’t follow that the predictions of quantum mechanics are necessarily uncertain. Furthermore, they contend that there’s nothing particularly mysterious about photon indeterminacy and that the sense of mystery comes from asking “inappropriate” questions (Do they acknowledge though the appropriateness of questions about the actual correlations being in conflict with what is possible for particles with well defined properties and no instantaneous action at a distance?)
Although not explicitly named as an exaggerator, Mermin replied, suggesting that
I would rather celebrate the strangeness of quantum theory than deny it, because I believe it still has interesting things to teach us about how certain powerful but flawed verbal and mental tools we once took for granted continue to infect our thinking in subtly hidden ways.
and provide us with
the stimulus for exploring some very intriguing questions about the limitations in how we think and how we are capable of apprehending the world.
and
to be resolved by refraining from asking foolish questions is the puzzlement engendered in some by contemplating the Einstein-Podolsky- Rosen experiment.
2 Usually what is offered in support of this claim is the observation that there is nothing mysterious in the measurement of the spin of one particle being correlated with the probability distribution of the spin of the other, even if the two are far apart, since the two particles originate from a common source. Nobody would quarrel with that, but what many people find mysterious is not the existence of such correlations but their particular character, which turns out to be utterly inconsistent with some extremely simple and apparently very
reasonable ideas about the kinds of correlations it is possible to have between far-apart noninteracting systems exclusively as the result of their having once been together in the same place. It may well be that to ask for any explanation of this “unreasonable” character of the correlations is to ask a foolish question. But the question cannot fairly be dismissed as foolish without saying
what it is and making explicit the simple and apparently reasonable ideas that have to be thrown out with it.
And even if the question is foolish, he adds, in conclusion:
There is, however, an interesting nonfoolish question: Why do many knowledgeable and thoughtful people feel so strongly impelled to ask the foolish one?
My current version of the answer, not very well developed, is that it has
something to do with certain deterministic presuppositions that are built into our thought and language at some deep and not very accessible level, and that have somehow infected even the way we think about probability distributions. Being of this frame of mind, I am therefore unwilling to be told both that the importance of indeterminism in quantum mechanics has been grossly exaggerated and that there is nothing peculiar about the EPR correlations.Einstein once wrote to Schrodinger that “the Heisenberg-Bohr tranquilizing philosophy—or religion?—is so delicately contrived that, for the time being, it
provides a gentle pillow for the true believer.” When I rest my head on a quantum pillow I would like it to be fat and firm; the recently available
pillows have been a little too lumpy to soothe me back to sleep.
My understanding of the reasons for Mermin’s focus on the baseball game analogy was increased by reading his discussion of the Einstein-Born exchanges where it appears that Born persisted in thinking Einstein’s objection was to the inherent uncertainty in QM rather than to the relationship between different uncertainties – or as Mermin emphasizes the case-by-case lesser uncertainty than should be required in a classical situation.
But this leaves me wondering where the conflict was (between M and F&W). For here he seems to be agreeing that those who celebrate the weirdness of quantum mechanics focus too much on uncertainty when it’s perhaps often the lack of sufficient uncertainty that we should be surprised by.
The original EPR discussion (in the famous Einstein-Podolsky-Rosen paper of 1935) emphasized the apparent possibility of simultaneously determining both the positions and momenta of a pair of identical particles emitted in opposite directions from the same point by measuring one observable on each particle and using their fixed relationship to determine the same observable on the other particle. In order not to violate the uncertainty principle EPR conclude that whichever measurement is done first must immediately randomize the complemetary variable for the distant partner. But this seems to involve instant effects at a distance, which some would find unacceptable even if it didn’t conflict with special relativity. So without actually doing the (fairly complicated) experiment it seems that whichever way it goes something is rotten in the state of Denmark.
Because of complications related to keeping track of the different positions of the separated moving particles, it later became more popular to use spin components as the non-comensurable variables, with position as an essentially irrelevant extra factor.
The apparent “paradox” would then be that we could measure the spins of the two particles in perpendicular directions, and so (because of the two particles having spins that are always opposite (or in some versions equal)) we have determined spin components in both directions for each particle – which appears to conflict with the uncertainty principle of quantum mechanics (unless the first measurement immediately randomizes the spins of both particles in the other direction). [Of course we could measure both spins in succession on one particle; but after the second measurement the spin in the first direction would again be unknown.]
So if the uncertainty principle does hold, then the measurement of either spin in one direction seems to somehow instantly randomize that of both spins in the other direction no matter how far apart they may be.
So the two main questions are:
Regarding 1. The prediction of quantum mechanics (confirmed by experiment) is that when each observer measures a component of spin on one of the particles: if both choose the same direction to measure, then their spin values will always be opposite (ie perfect correlation); but if they measure in perpendicular directions, then for whatever result one gets the other will be equally likely to get positive or negative value (ie no correlation at all); and if they measure at other angles to one another there is a specific formula for the extent to which they will be correlated – namely, if the angle is [math]\theta[/math] then when one observer records spin up the other will record spin down with probability [math]cos^2(\theta/2)[/math].
Regarding 2. We should note first that observation of one spin determining the other is not what seems to require instantaneous action at a distance. (Since they came from the same source, that is just as natural as the fact that if a pair of gloves is separated and someone sees that they have a right glove then they know immediately that whoever has the other glove has the left.)
The only surprising thing is the way the choice by one observer to measure in one direction randomizes what is seen in the other direction by a distant observer. So whichever spin is measured first, it seems that the measurement of that spin immediately changes the situation of both particles (and since they may by then be far apart, this appears to involve faster than light communication).
However, this situation for perpendicular measurement directions could be resolved by imagining that the electrons are drawn, apparently at random, from a pool of particles which are tagged in advance with variables which control the way they will respond to each kind of measurement.
In fact if each electron is any one of z+y+,z+y-,z-y+,z-y- with equal a priori probability, and if two such electrons are produced with opposite spin pairs (the first chosen at random and the second then fixed) with one sent off to each of two observers, say Alice and Bob, then if Alice measures z and gets + we can see that if Bob measures z the result will be – and that if he measures y the result has equal probability of being either + or – .
If that was all to see there would be no essential mystery. Even if there really were “hidden” variables corresponding to the spins we don’t measure (because for each particle our measurement the original spin in any one direction potentially randomizes all the other directions), the fact that they could exist would explain the observed statistics with no need to imagine some mysterious instantaneous action at a distance.
And if that was the correct explanation, then it might really make sense to say that the quantum mechanics we use is somehow “incomplete” as a description of how nature “really is”. [It has always puzzled me though that Einstein took this view, since he was the one who noted that if the aether is unobservable then the most “correct” version of physics is the one in which it is never mentioned (even though it would be perfectly possible to follow Lorentz and derive everything we see from a theory in which the aether does exist but where the laws of physics conspire to keep it always hidden from us).]
BUT THERE IS MORE! (though it wasn’t really pointed out ’til much later)
Bob and Alice can also measure spins in directions other than z and y. For example they could choose a direction halfway in between, ie at [math]45^o[/math] to both, which we’ll call w. [There is also the x direction in 3d of course, but a direction perpendicular to both y and z could be accounted for just with another two-valued hidden variable and it is only the intermediate directions that mess things up.]
It is clearly much more difficult to specify variables (attached to the individual particles) and rules (without involving instantaneous effects at a distance) which match the predictions of quantum mechanics for all of the relations between spin measurements in all possible directions. But it was not until 1964 that it was actually shown to be impossible.
The impossibility proof is due to John Bell who noted a relationship that must hold between the probabilities of different outcomes in any theory that allows the spin values in all directions to be set for each particle at the outset without any change at one particle as a result of what is done to the other. He noted that for any specification of each particle’s spin as either up(u), down(d) or unrestricted(*) in each of three directions, say y, w, and z, if we represent the corresponding probability by just listing the u or d values in the ywz order (so that ud*=udu+udd represents the probability of getting y up and w down regardless of the z value), then we must have ud*+*ud =udu+udd+uud+dud>=uud+udd=u*d. (The same relation is in fact true for any system of three yes or no questions regardless of where they come from.) So any prior assignment of spins in all directions must lead to the probabilities of y up with w down and of w up with z down adding up to more than the probability of y up with z down.
But for the case that w is at [math]45^o[/math] midway between z and y, the quantum theory predicts (and experiment confirms) that if Alice measures in the z direction and Bob in the w direction then their results will be opposite with probability [math]cos^2(22.5^o)\approx .85[/math] and the same with probability [math]cos^2(67.5^o)\approx .15[/math].
So in that case the quantum theory predicts probabilities of 15% for ud*+du* (ie for y opposite to w) and for *ud+*du (ie for w opposite to z) and 50% for u*d+d*u (ie for y opposite to z) – which is NOT consistent with Bell’s inequality.
[Aside: The original proposed experiment used three directions separated by angles of [math]120^o[/math] for which the QM prediction (and experimental results) give a probability for agreement between any two of [math]cos^2(60^o)= 0.25[/math] if they are different and 1 if they are the same. So if the measurements are made in randomly chosen directions (which will agree 1/3 of the time and disagree 2/3 of the time) the overall probability of agreement is [math]1/3(1)+2/3(.25)=1/3+1/6=1/2[/math]. But for any assignment of u or d to each of the possible measurement directions [math]a=0^o[/math],[math]b=120^o[/math],[math]c=240^o[/math], the probability of getting agreement between the measurements in randomly chosen directions is always more than 5/9 (because if both particles are either uuu or ddd then agreement is certain and in any other case like uud then of the nine possible choices of which direction is measured on which particle only four lead to disagreement so the probability of agreement is 5/9). So for any sequence of pairs with random measurement directions we should have at least a 5/9 chance of agreement.]So the upshot of all this is that there is no way of assigning in advance what each observer will see when detecting paired electrons in arbitrarily chosen directions. So it cannot be the case that before detection each electron has a separate identity with its own pattern of spin responses. Or as Bohr is famously reported to have said (or not said) “There is no quantum world. There is only an abstract quantum physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature.”.
So does this mean that collapse of the two particle wave function induced by the first measurement has an instantaneous effect at the position of the remote observer?
Well that depends on whether or not you are “foolish” enough to want to “interpret” Quantum Mechanics in terms of conceptual structures that evolved because they are useful for dealing with he universe at a very particular scale, and which may not be relevant at very much smaller (or larger) scales. Or, as David Mermin put it “it has something to do with certain deterministic presuppositions that are built into our thought and language at some deep and not very accessible level, and that have somehow infected even the way we think about probability distributions.”
Perhaps a more sensible approach is to treat QM like Popeye (it is what it is) and just consider what it actually says rather than (or at least before) trying to “interpret” it.
So what does it say? Perhaps the first thing to notice is that any application of quantum mechanics usually starts by identifying an isolated system – or at least one for which all outside interactions are precisely specified. And what it gives us are predictions about the probabilities of the various outcomes that we might see when the isolation is broken due to the system’s interaction with ourselves. The quantum state is thus a property of a system while it is isolated from the observer – but this may be different for different observers (think of Wigner’s friend for example), and so the state is not really defined in an absolute sense but only relative to particular observers.
A common way to start talking about the EPR “paradox” is with the parable of Bertelman’s socks.
One is green and the other red, and sadly for poor Bertelman, they have been separated and shipped to opposite ends of the universe in boxes to be opened by Alice and Bob.
No-one is surprised to hear that when Alice opens her box and sees a green sock then she knows immediately that Bob’s box contains a red sock. We all know that she could have been informed ahead of time about Bertelman’s sartorial oddity, and of course the socks both started out together. So no-one thinks that Alice’s ability to predict what Bob will see indicates some mysterious ability to instantaneously “turn his sock red”.
So there must be something more to the EPR issue than just knowing that if a pair is split into two boxes and if we find one in one box, then the other must be in the other box.
And indeed there is. The quantum problem is not with the values of one property but how those of different properties can be related. Bertelman’s boring socks were just too simple.
But Dr. Fahni’s funny socks socks are another matter. They are actually iridescent!
Depending on how they are illuminated, one is red and one green or one yellow and one blue.
When either Alice or Bob sees one as red then with the same lighting arrangement the other sees a green one and vice versa. And the same goes for yellow and Blue. And also for Orange and Turquoise. (Oh! I forgot to mention that. There is an intermediate lighting setting under which we see orange and turquoise. And there, as we shall see later, is the rub!)
If Alice sees red, then in the same light Bob sees green (and vice versa), but if he looks in the Yellow-Blue light he sees either with equal probability.
And if Alice sees yellow then in the same light Bob sees blue (and vice versa), but if he looks in the Red-Green light he sees either with equal probability.
In fact we can get this randomness just by starting with a drawer full of pairs of socks in which each left sock is any one of RY,GY,RB,GB (with RY meaning that in Red-Green lighting it shows Red and in Yellow-Blue it shows Yellow) and the corresponding right sock is its opposite (ie GB,RB,GY,RY respectively). Now if a pair is drawn at random and sent to Alice and Bob, then if Alice sees Green she knows that in the Red-Green light Bob will see Red but she has no idea what he will see in the Yellow-Blue light. So it may be getting a little bit complicated to arrange, but there’s still no great mystery in either the randomness or the correlation.
But now let’s look at what happens when the Orange-Turquoise measurement is compared to the other two.
One possible set-up is to build a bigger sock drawer with pairs in which the left sock is of type RYT,GYT,RBT,GBT,RYO,GYO,RBO,GBO and the right always opposite (ie respectively GBO,RBO,GYO,RYO,GBT,RBT,GYT,RYT). If there are equal numbers of each type of pair then when Alice sees Red she still knows that if he looks in RGlight Bob will see Green but if he looks in YBlight he will see either Yellow or Blue with equal probability and if he looks in OTlight he will see Orange or Turquoise with equal probability (and similarly for all the other combinations).
BUT, since Orange is “closer” to Red and Yellow than to Green and Blue, and Turquoise is closer to Green and Blue than to Red and Yellow, Dr. Fahni wants to maintain the opposition by ensuring that whenever Alice sees Red, Bob is actually less likely to see Orange than Turquoise.
In fact when Alice sees Red, if Bob uses OTlight he sees Turquoise just 15% of the time and Orange 85%, and when she sees Green those figures are reversed. Also, if Alice looks in OTlight and sees Orange, then if Bob uses RGlight he will see Red 85% of the time and Green just 15%, and if he uses YBlight he will see Yellow 85% and Blue 15%.
So let’s see how Dr Fahni might have filled up his sock drawer to make this happen. The obvious strategy would be to have unequal numbers of the different pair types with RYT and GBO being least likely, RYO and GBT most likely, and the others somewhere in between.
A first try might be to have just GBT or RYO on the left sock (each matched with the other on the right sock) but this won’t work because we still want GY and GB to be equally likely (and add up to the total proportion green results in RGlight). And similarly for RY and RB.
So in fact we need GY=GB=RY=RB=25%.
But Bell noted that RB=RBT+RBO<=RBT+RBO+RYT+GBO=RBT+RYT+RBO+GBO=RT+BO
So if the situation is symmetric with RT and BO equal, they cannot be less than 12.5%
But with GYT+GBT=85%of50%=RYO+RBO and RYT+RBT=15%of50%=GYO+GBO, we get RT=BO=7.5%.
So the observed correlations cannot be achieved by just selecting at random from a set of pre-prepared pairs of socks.
What is, to me, particularly remarkable in all this is the fact that all quantum mechanics purports to predict are the relationships between
The key to all this is the fact that quantum mechanics only claims to predict observations on isolated systems. In order to have a well defined pure state, a system must be isolated; and any measurement process ends that isolation. We can, of course, consider the system of interest combined with its measuring apparatus as a larger system, and apply quantum mechanics to this extended system – and also to whatever else it interacts with up to any point before the chain of interactions reaches our brain to register a sense that something has happened. Some other observer might include our own brain as part of the system under study, but we ourself cannot. At the point where our brain registers an experience, the uncertainty of our own history gets resolved and our perspective on the state of the universe collapses to whichever of the possible outcomes is the one we actually experience. But from the perspective of a different observer, our state may still remain incompletely determined and not collapse until that observer senses something that tells them what we have seen. The point here is that the state is not a property just of the system itself, but of the perspective from which it is being studied. So rather than imagining many worlds, I prefer to think of many perspectives on a single world, in each of which there are many possible histories (or configurations in space-time) of that one world. In this view the result of an observation is not some ill-defined splitting of the universe but rather just a shrinking of the family of all histories that are compatible with our total experience to date.
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