stranger than we can imagine Bell's theorem Limits on hidden variable accounts
De Broglie s hypothesis (Lunn, 1921) De Broglie proposed that every particle has an associated wave (called a pilot wave), and every wave has an associated particle. The relationship between the two is always the same: E = hf and p = h/l (or vector version, p=(h/2y)k This doesn't yet explain atoms, but there's a suggestive relation: if there were an integer number of DeBroglie wavelengths around a circular orbit, Bohr quantization would result! (Again, this is NOT the way it really works- almost everything about the Bohr model was wrong.) The full solution requires understanding what persistent wave patterns can exist in the atom, which requires finding the wave equation. The waves will be genuine 3-D waves, not waves on an imaginary 1-D orbit. The electron is described by a wave function, y(r,t), which obeys a differential equation. The non-relativistic version is called Schrödinger s equation. (also first due to Lunn) First term, (squared momentum), depends on how y wiggles in space. Second term, (potential energy), due to various neighbors (whose positions are presumed fixed in our reference frame). Third term (total energy) is how fast y changes in time.
What is y? Although that equation is deterministic, the resulting behavior isn t. Where those individual blips show up on the screen looks random. Those interference patterns only show up after many blips are accumulated. How does random behavior come from the deterministic equation?
Born s probability interpretation Recall that the intensity (energy density) of a wave goes as the square of the amplitude (for light the magnitude of the electric field, for water ripples the height change). Quantum mechanics says that if we consider an ensemble (collection) of identically prepared electrons, each described by similar wave functions, y(x,t), (obviously with starting t shifted) y (x,t) 2 DV is the probability that an electron would be found in the little volume DV near point x at time t, if an experiment is done that could locate it that accurately. y 2 gives a probability density, so for a large ensemble it tells us the rate at which electrons arrive at the spot of interest on the screen. In the places where the two waves interfere destructively, the probability is less than the sum of the two individual probabilities, and may even be zero. There will be a fundamental loss of determinism unless there's something else beyond the wave function (i.e. not in the theory) to guide the outcome. This recipe does not claim to tell us what y "is".
Einstein-Podolsky-Rosen Einstein and collaborators (EPR) proposed that by using the conservation laws, one could show that QM was missing something. E.g. consider a particle that flies apart into two particles, each detected somewhere on a sphere of detectors. The blue pair or the red pair might occur, but not a mixture, which would violate conservation of momentum Conservation of momentum says the particles have to go opposite directions. QM says they don't know which way they're going. Possible resolutions: The particles don't have to be detected in opposite directions, the conservation laws only hold on the average. (Bohr thought this at one time, but it's completely wrong experimentally.) The particles are always found in opposite directions, because there is some hidden variable which allows them to know which way they are going. QM is incomplete! Even though it is predetermined that the particles go opposite directions, what those directions are is not determined until one is (randomly) detected. The other somehow knows which way to go, faster than the speed of light! (Einstein called this "spooky correlations at a distance") Einstein believed that this argument showed the incompleteness of QM. Was he right?
A Science Fiction Story Say that you want to find out why people like pepperoni, mushrooms, and olives on pizza. You ask many people, and they give Yes/No answers when asked about each. (Say 50% yes for each.) But you don't notice any distinguishing properties of the people who say Y. Does that mean you can conclude that the answer is random, some sort of momentary glitch that occurs in a person's response? Of course not- you may just have missed the "hidden variables" needed to understand taste. There MAY be something in each person's mind ahead of time that determines their answer, or maybe not.
Ask Couples Each couple gives the same answer to the one question you ask them, so you know there was already something in their heads that determined the answer. Otherwise how could they all get their stories straight? So there WAS some hidden variable determining the outcome. A slight complication- all these folks seem to get confused after one question. If you ask each member of the couples a second question, the perfect correlation between them is lost. So you're only allowed to ask one of each person. Now you try asking one person about pepperoni, and the other about mushrooms. You find that 85% of the time, they give the same answer (YY or NN). So you conclude that whatever makes people like mushrooms usually makes them like pepperoni. You try the same thing asking about mushrooms and olives. Again 85% of the time, they agree. (YY or NN) Now you ask about pepperoni and olives. What extent of agreement do you expect? The 15% who had different opinions on M-P and the 15% with different opinions on M-O might be the same people. Then there would be 0% disagreement on P-O. Or maybe they're all different people. Then there would be 15%+15% =30% disagreement on P-O. Or it could come out anywhere in between.
Whoops You now ask the couples do you like pepperoni/olives? they give the same answer 50% of the time. What gives? Maybe there was a statistical fluke. You try the same thing again with 100,000 couples- and get the same result. Maybe the couples were secretly signaling each other, getting their stories coordinated (on some questions) AFTER the questions were asked. Ask the questions in sealed boxes. Same result. Ask the questions SIMULTANEOUSLY (in Earth frame). Same result. Maybe the couples were getting their stories straight only on the questions which they knew they would be asked. You draw the questions out of hats, in the sealed boxes, simultaneously. And get the same result.
Science Fiction? Conclusion: this story is obviously fiction. The identical results on any ONE question tell you that (whether or not you believe that people in general are spontaneous and random) on these particular issues there had to be some prior content in their heads determining the answers. But there is NO WAY of assigning answers to the 3 questions (M,O,P) which gives these results for the disagreements: 15% M 15% P 50% O
tell the same story about particles "couples" are pairs emitted together in a decay process. "Like pepperoni?" becomes vertical polarization?" "Like olives?" becomes 45 degree polarization?" "Like mushrooms?" becomes polarized at 22.5 degrees between Experimentally the results are as we have described. Obviously when we concluded that this story is science fiction we assumed something that's false. What did we assume about reality? we assumed nothing whatever about QM-» we didn't even mention it.» It happens that QM precisely predicts the results, but we are now finding how the world differs from our assumptions, not how quantum theory differs from our assumptions.
WE assumed Realism. Not that the world must be deterministic, but only that those aspects of it which can be predicted perfectly are determined by some element of reality. You can predict the result on any particle by making the corresponding measurement on its partner. "If, without in any way disturbing a system, we can predict with certainty (i.e., probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity." (Einstein, Podolsky, Rosen, 1935) Local causality. The value of a property possessed by an isolated system cannot be affected by any operations carried out at a sufficient (i.e., spacelike) separation from it. Induction. (no conspiracies) The unmeasured values (chosen by random quantum processes) are not statistically different from the measured values. The properties of an ensemble are defined completely by the preparation conditions. In particular, the distribution of possessed values of a variable for the subensemble which we actually measure is identical to the distribution for the complete ensemble. (random sampling) Which one are you willing to give up?
Loopholes? 1. The initial passage of the particles through the angular momentum selectors are space-like separated. In the first generation of experiments, the conversion of those microevents to some large-scale device setting was slow enough for it to be conceivable that a time-like signal could propagate between the detectors before "measurement" is complete. This loophole is now closed in some experiments, e.g. with satellites many km apart. 2. Detection efficiency. There s a complication in that many particles are missed, requiring some extrapolation. This is now closed in some experiments using atoms rather than photons and now in photon experiments here! No experiment yet reported quite closes all the loopholes simultaneously. Still, it s a long-shot to think that these loopholes offer any escape.
For How Long Could the Detectors Trade Info? Long enough for a computer bit to change? But why not have an ambiguous bit setting until the message is received from the other end? Why do we think the definite result ever happens?
On Induction or Counter-factual definiteness or no conspiracy In a completely deterministic world, how can you argue about what would have happened if? We claim that if the world is anything like what we think, whatever the photons are doing (which polarization features they have) should not depend on whatever random remote events are setting the polarizers. So if they have real polarizations when the measured ones are (A,A) etc. then they should have them even if the choices are (A,B) etc. Barring some universal conspiracy
Does the problem lie with probabilities? We concentrated on probabilities and correlations. This may give the incorrect impression that the fundamental issue is the probabilities. It is possible to devise a 3-particle spin measurement in which the distinction between QM and LR can be seen in every single measurement. Prepare a 3-particle (each spin 1/2) state with the z-components of the spins described by (uuu - ddd)/2 1/2. Measure the x-components of the spins. QM predicts s 1 s 2 s 3 = -1, always, LR predicts s 1 s 2 s 3 = +1, always. A single measurement of the 3 spins could do the job. (See article by N. D. Mermin in Physics Today, June 1990.) Here's some old notes, left in for historical interest: "Although it is conceivable that this hypothetical experiment, if it is done, will agree with LR rather than QM, don't bet on it. It would be very strange for QM to do a perfect job of predicting the violations of LR so far, then break down and have LR work in the nonprobabilistic context. There's no theory that predicts that, and there was only a vague hunch that somehow the application of probabilistic reasoning might have something to do with the failure of LR."
The new parable You can ask one question each of Anne,(A) Belinda (B), or Cary (C). The questions can be "olives?" (X) or "Onions?" (Y). The answers can be "yes" (+1) or "no" (-1). You always find that if you ask one "olives?" and two "onions?" the product of the answers is +1. So it sounds as if each answer is determined by an "element of reality" since it is predictable by asking two OTHER questions. X 1 Y 2 Y 3 = Y 1 X 2 Y 3 = Y 1 Y 2 X 3 =1 So if those are just numbers we're talking about: X 1 Y 2 Y 3 Y 1 X 2 Y 3 Y 1 Y 2 X 3 = X 1 X 2 X 3 Y 1 2 Y 2 2 Y 3 2 = X 1 X 2 X 3 = 1 ALWAYS BUT QM says something strange: X 1 X 2 X 3 = -1 ALWAYS Essentially this experiment (using photons rather than spin=1/2) has been done.
Three-particle Results Nature 403, 515-519 (3 February 2000) doi:10.1038/35000514 Experimental test of quantum nonlocality in three-photon Greenberger Horne Zeilinger entanglement Jian-Wei Pan 1, Dik Bouwmeester 2, Matthew Daniell 1, Harald Weinfurter 3 & Anton Zeilinger 1 More than 85% of the time X 1 X 2 X 3 = -1 was found, which is impossible classically. The remaining 15% or so is attributed to imperfections in the analyzers, etc.
A quantum description of "measurement The macroscopic set-up creates a situation describable by y (which describes the quantum system) and j (which describes the macroscopic apparatus). Initially, these are independent, so if y has two possible values, y 1 and y 2, the overall wavefunction of the whole thing would be y changes in time, as described by the Schrödinger equation. When the micro-system (say a single particle) encounters a measurement apparatus, the wave functions describing the particle and the apparatus become "entangled", i.e. they are no longer independent. Either y goes into state 1, and all the needles, etc. represented by j go to read "1", or each goes to "2. So far, we have just described how the wave-function obeys the equation. Interference between possibilities (1) and (2) now disappears, because there are zillions of particles in different positions in, and there is no chance whatever that the waves representing these two possibilities will overlap.
Loss of Interference Here's the key point: if you have just one particle, going through two slits, the two paths show interference only if they get to the same place. The (x,y,z,t) coordinates must all be the same. The wave function representing MANY particles is a function of ALL their coordinates, so if there are two lumps of this wave function evolving in time, they show interference only if ALL the coordinates of ALL the particles can get to the same places by each path, at the same time. This simply never happens once many particles are involved in a complicated system. Thus we now have two distinct possibilities, represented by : We now have gotten rid of the interference, while postulating nothing different from the linear wave equation, and without worrying about "duality" or any such philosophy. The "projection postulate" turns out to follow naturally: obeying the wave eq, represents a situation in which, if the apparatus measures y again, it will get the same result. That is, no piece of the wavefunction represents a solution with the successive measurements of the same thing giving opposite results. So why is there any philosophical problem about QM, other than the usual shedding old ideas?
The Output State At this point the solution to the equation gives us: Both distinct possibilities are still there, even though they don't interfere! Why should you be troubled that both possibilities remain? Schrödinger's cat: Say that the micro-variable is a quantum spin, and the measurement apparatus is set up to kill a cat if the spin is up, and give it some food and water if the spin is down. This is not a science-fiction idea, but a relatively trivial thing to set up in an ordinary lab. The result of the solution of the linear wave equation is that the cat is both alive and dead, in a superposition. This does not mean "in a coma" or "almost dead" but BOTH fully alive and purring or thoroughly dead and decomposing. Furthermore, once you look, your wave function becomes entangled with those of the cat, etc. The solution of the linear wave equation now describes a superposition of a you who has seen the dead cat and a you who has seen the live cat! Which of youse guys is for real? If the linear wave equation by itself describes the world of our experience- it must describe many such worlds!
For How Long Could the detectors trade info? Long enough for a computer bit to change? But why not have an ambiguous bit setting until the message is received? Why do we think the definite result ever happens? We like to think that there s just one version of us. How long does it take to be a distinct you? Maybe a few milliseconds per thought? That s a lot of kilometers.