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A Boost For Quantum Reality
Eponymous Hero sends this excerpt from Nature:
"The philosophical status of the wavefunction — the entity that determines the probability of different outcomes of measurements on quantum-mechanical particles — would seem to be an unlikely subject for emotional debate. Yet online discussion of a paper claiming to show mathematically that the wavefunction is real has ranged from ardently star-struck to downright vitriolic since the article was first released as a preprint in November 2011. ... [The authors] say that the mathematics leaves no doubt that the wavefunction is not just a statistical tool, but rather, a real, objective state of a quantum system."
No. Well, maybe for the cat, we're not able to do the experiment to tell.
But in the equivalent test using a photon in place of a cat and orthogonal polarization states in place of dead or alive, the photon most certainly does not "know" what state it is in.
This is the essence of Bell's inequality and the fact that there is no local hidden variable theorem compatible with the results of QM.
Tim.
God said, "div D = rho, div B = 0, curl E = -@B/@t, curl H = J + @D/@t," and there was light.
http://arxiv.org/abs/1111.3328v2
http://arxiv.org/pdf/1111.3328v2.pdf
http://stephan.sugarmotor.org
If the wave function has an effect then it what way is it not real? Maybe its the mathematician in me but if reality can only be understood mathematically then I have no problem with that, thats just a problem with our imagination. I have always thought the divided universes interpretation of quantum physics multiple states was reading too much into things, a bit like during the steam age everybody wanted to interpreted things in terms of steam engines, thats useful, but the model implies things which the pure maths itself doesn't.
Think of probability distributions. If you throw a die and don't look at the result, you don't know which of the possible results happened. However you know that if you throw that die often enough, you know that each result happens approximately the same number of time. Therefore you can assign the same probability to each result, i.e. 1/6 each. But the probability distribution does not describe the current state of the die; the current state of the die is that it shows one of the numbers 1 to 6. It just tells you about your knowledge of that state; the equal probability just means "I have no idea which result happened, and there's no reason to favour either one."
Now assume that a trusted friend looks at the cube and tells you that it is not a 6. Now suddenly the probability distribution you assign to the cube changes: You'll assign probability 0 to the 6, and probability 1/5 to all other results. However the physical state of the cube does not change at all. Only your knowledge about it changes.
Finally you look at the die, and find e.g. it shows the 3. At that point the probability distribution "collapses" to the distribution which assigns 1 to the result 3, and 0 to all other results.
Now the idea of non-real wave functions is exactly like that. For those interpretations the wave function doesn't tell you what state of the system is, but only which results you get how often when you measure certain properties. When you measure, your knowledge changes, and therefore the wave function "collapses" just the same way the probability distribution "collapses" when you look at the die.
The Tao of math: The numbers you can count are not the real numbers.
The article confused me greatly so I read some of the arxiv preprint linked above. Here's the idea and context as I understand it. I've included some basic quantum background since most people here don't have it.
* Intro to wavefunctions via an example. Electrons have a property called "spin" which has two states, "up" or "down". These can be measured in, for instance, the Stern-Gerlach experiment where those electrons with spin up are deflected up by a magnetic field and those with spin down go down. The wavefunction corresponds to a list of the probability of each outcome occurring. The probabilities evolve through time via the Schrodinger equation which allows predictions to be made. One might prepare an electron where its spin wavefunction corresponds to the list [1/3, 2/3], so 1/3 of the particles go up and 2/3rds go down. [I've oversimplified; wavefunctions are actually elements of an abstract Hilbert space and complex-number amplitudes are used instead of real-number probabilities. I love Hilbert space but it's too much to explain here.]
* Spin is not a classical property. One can measure spin "left" and "right" in addition to "up" and "down" by rotating the Stern-Gerlach (SG) device mentioned above and measuring left/right deflection. Suppose you run a stream of electrons through an up/down SG device which gives 80% of them "up". You then run those "up" electrons through a left/right SG device--it will always come out with 50% "left" and 50% "right". Even more strangely, if you then run the "left" electrons through another up/down SG device, the probabilities will now be 50%/50%, even though you selected only spin up electrons at the first stage so you'd expect 100%/0%. The act of going through the left/right device altered the spin up/down state somehow.
* Hidden variables. Perhaps the electrons above have definite "spin vertical" and "spin horizontal" properties before the experiment starts. The act of going through a device must change the other property, though everything might be deterministic if there is some further hidden property controlling which electrons have their spin up/down states altered in which ways by passing through the "left" SG device. The alternative is that there are no definite properties which determine the wavefunction; the wavefunction is all there is, reality is somehow fundamentally probabilistic, and the wavefunction is "real" instead of a statistical construct.
* Bell's theorem. Suppose spin up/down and spin left/right are definite properties and some hidden variables explain the above results. Using entanglement (which I'll leave undefined) and the assumption that information cannot travel faster than light, one can measure both the spin left/right and spin up/down values of a particle before the hidden variables have a chance to act (note: they might act in a very bizarre, perhaps even non-deterministic, manner, but we get to measure things before they have that chance). This gives a testable prediction which differs from quantum mechanics. If the experiment is performed, the "definite property" theory does not predict reality while the use of wavefunctions does predict reality. This is strong evidence for the reality of wavefunctions, though it's not completely conclusive.
* The paper. It derives Bell's fundamental contradiction from fewer assumptions. In its own words,
The result is in the same spirit as Bell's theorem, which states that no local theory [i.e. one without faster-than-light communication] can reproduce the predictions of quantum theory. Both theorems need to assume that a system has a objective physical state L such that probabilities for measurement outcomes depend only on L. But our theorem only assumes this for systems prepared in isolation from the rest of the universe in a quantum pure state [e.g. a particle measured as spin "up" right after the SG experiment above]. This is unlike
As I said I love Hilbert space, so your comment is enough motivation for me to write up a brief explanation.
The n-dimensional Hilbert space is the collection of length-n lists of complex numbers. One can add these lists and scale them, so for instance [1, i] + [2, 1] = [3, i+1] and 2*[i, -1] = [2i, -2]. Physically, each component of the list corresponds to a possible experimental outcome. More specifically, the probability of the outcome corresponding to the ith component is the square of the magnitude of the ith component. For the electron spin up/down experiment I talked about the wavefunction [1, 0] gives a |1|^2 = 100% chance of measuring spin up (and 0% chance of measuring spin down; this is called a pure state). [sqrt(1/3), sqrt(2/3)] corresponds to a 1/3 chance to measure spin up and 2/3rds to measure spin down. You may wonder why the magnitude-squared business is used at all (why not just keep track of the probabilities?) which is where the complex numbers come in to play. The state [sqrt(1/3), i * sqrt(2/3)] has the same experimental outcomes given this single measurement as the previous state, [sqrt(1/3), sqrt(2/3)] but it is fundamentally different from it since the two components are "out of phase". More elaborate experiments can detect the difference. In this case it turns out the result of the spin left/right experiment is encoded in the phase difference between the two components.
Hilbert space comes with an important operation called an inner product, which I'll denote by the term "dot". It can "single out" the entry at a particular position in a list. For instance, by definition [1, i] dot [0, 1] = i, singling out the second component. The operation is extended to more general lists on the right-hand-side by rules I won't discuss, and it has a physical interpretation in terms of probabilities--the magnitude of (A dot B) squared is the probability of measuring a particle with wavefunction A in the state described by wavefunction B, which fits what I said above in light of the computation |[sqrt(1/3), sqrt(2/3)] dot [1, 0]|^2 = |sqrt(1/3)|^2 = 1/3. Note that the sum of the squares of the magnitudes of the entries in the list must be 1 since the experiment will have some outcome with 100% certainty.
One can have infinite dimensional Hilbert space where the lists are allowed to have infinite length. Sequence space is a popular example: it contains [1/1, i/2, 1/3, i/4, 1/5, ...] and [0, 1, 0, 0, 0, ...]. We often restrict ourselves to lists where the sum of the magnitudes squared are 1 since these are the only physically meaningful wavefunctions, giving the so-called projective Hilbert space. [1, 1, 1, ...] is certainly not in that space since it has infinite sum-of-squares. Actually, [1/1, i/2, 1/3, i/4, 1/5, ...] doesn't work here either, but sqrt(6)/pi * [1/1, i/2, 1/3, i/4, 1/5, ...] does work. (There's a beautiful proof using Parseval's theorem.) [1, 1, 1, ...] fails particularly badly since it cannot be scaled to an element of projective Hilbert space as we were able to do with the other list, so we don't allow it in regular Hilbert space at all. Any other lists that have infinite sum-of-squares are similarly excluded. The inner product is extended in a natural way to infinite lists. That's all the structure one requires.
I should note that Hilbert space is more often defined as an abstract vector space over the complex numbers equipped with a positive-definite sesquilinear inner product which is moreover Cauchy complete with respect to the induced norm. Projective Hilbert space is usually defined as projective equivalence classes over a Hilbert space with semi-canonical norm-1 representatives. My definitions are equivalent, assuming the axiom of choice (everybody does), and they're obviously more accessible (though it's much less pretty IMO). I should also mention that wavefunctions and elements of Hilbert space are usually written with the bra-ket notation and as sums of pure states (as the paper does); my notation is from Python and was chosen considering the audience.