So, oddly enough, quantum mechanics is entirely local in the common meaning of the word. When physicists say that it is non-local, they mean that particles which have a common origin but then were separated can be stronger correlated than particles without quantum properties could ever be. I know this sounds somewhat lame, but that’s what quantum non-locality really means.This is correct. A particle has quantum properties if it obeys the Heisenberg uncertainty principle. A particle without quantum properties obeys a classical theory of local hidden variables.

The theory is local. The only claim to non-locality is just a statement about correlations.

Dear Roger,

ReplyDeleteHow do you square it off (the assertion of locality, made in reference to correlations in measurements) with the following two postulates of QM? (Numbering of the postulates is in an arbitrary order.)

P1. The state of a system is completely specified by its complex-valued wavefunction, and

P2. The wavefunction evolves in time as per the TDSE.

P1 implies that evolution in the probability cloud---a real-valued measure---is insufficient to completely specify a state. Correlations are even more restricted in scope. They refer not to probabilities, but only to the concretely measured values---specific real-valued numbers merely sampled out during experiments out of all possible values of a real-valued probability distribution. If correlations are so restricted of scope, how can they be used to completely specify a system state? And, if a system state cannot be completely specified with them, then how can they be used to characterize the entire theory? At the most, you can say that the concrete outcomes of the observables show locality. But not the theory.

P2 implies that the method of solving the TDSE will have an essential say in deciding the character of the theory, because the nature of the integration procedure too will shape the nature of the physical evolution---if the theory and the solution procedure have at all been validated. Well, here, the theory involved in the solution procedure is the Fourier analysis. It directly implies non-locality. (It did so even for the original purpose for which it was invented, viz. the classical diffusion. But in classical diffusion, this theory is taken as a mere mathematical approximation. In contrast, in QM, \Psi evolution cannot be regarded as an approximation. Else, you violate P1.

OK. You are trapped on (at least) these two, interrelated, counts. How do you propose to break out of it? Just asking ;-)

Best,

--Ajit

My response is that those are not really the postulates of quantum mechanics. P1 is just untested speculation. P2 is violated by measurements.

DeleteIf locality is what you want to save, then you don't have to jettison P1, and for that matter not even P2. (In fact, you don't have to remove any one of the standard QM postulates.)

DeleteTo save locality, you have to jettison only the Fourier theory. That is to say, TDSE remains, but the solution procedure involves not Fourier theory, but some other procedures or ansatz's that *in principle* remain local throughout.

Instead, if you focus on correlations, a few important insights regarding the nature of QM could sure be had. However, mere correlations would never be powerful enough in order to build a theory of all QM phenomena. Correlations are real valued; that's why.

In the first version of this comment, I wrote something showing how to have locality in solving TDSE, but it made the comment too big. So I dropped posting it, and wrote this new reply afresh.

... Actually, what I had written about was only the diffusion equation, but the passage to the Helmholtz and TDSE is very obvious. ...

The idea, in brief, is this: (i) Begin with the usual finite difference method (FDM), and (ii) ensure locality in this discrete description, as the first thing. Ensuring locality requires sticking to the explicit time-marching method; the implicit time- marching is inherently global, and so, useless for our purpose.

Then, through suitable limiting processes---which is where the core idea lies---you can combine both (i) locality and (ii) continuity in solutions.

I will post my idea via an entry at my blog, once I get some free time from the current bout of Python simulations (involving particle(s) in a box). To the best of my knowledge, none else has put forward this idea. Even if someone did (which I very much doubt), chances are bright that he didn't make any connection to the local vs. global issue in QM. Else, the locality issue being so hot (Einstein!), someone or the other would have said something or the other about it (apart from in the basic lit. on diffusion, PDEs, or computational physics.)

So, I could write a paper too. ... But again, only after the current trials end, and then, after I get a job in data science.

In the meanwhile, if anyone wants to have a look at the first draft (containing the idea), send me an email, and I will email it back to you.

Best,

--Ajit

So when a single photon or electron interferes with itself by passing through two slits at once in the double-slit experiment, you would argue that the wavefunction has only local properties?

ReplyDeleteYes. The term "local" does not mean confined to a point. The photon (or electron) is really a wave that occupies a non-zero volume of space. Some of that wave goes thru each slit. Effects have to propagate at some speed not greater than the speed of light.

DeleteConsider a double slit whose slits are 1,000,000 miles apart. A photon is emitted and the wavefunction approaches both slits. Just before the photon arrives at the upper slit, a detector is placed just behind it. If the detector clicks, it means the the photon went through the upper slit. If the detector doesn't click, it means that the photon went through the lower slit.

DeleteHow did placing a detector at the upper slit immediately affect what happened a million miles away?

Dear Roger,

DeleteYou write, "The term "local" does not mean confined to a point. The photon (or electron) is really a wave that occupies a non-zero volume of space. Some of that wave goes thru each slit. Effects have to propagate at some speed not greater than the speed of light."

Consider a double slit whose slits are 1,000,000 miles apart. A photon is emitted and the wavefunction approaches both slits. Just before the photon arrives at the upper slit, a detector is placed just behind it. If the detector clicks, it means the the photon went through the upper slit. If the detector doesn't click, it means that the photon went through the lower slit.

How did placing a detector at the upper slit immediately affect what happened a million miles away?

Even if the photon is detected at the upper slit, I would still say that the wave went thru both slits. If removing the detectors results in an interference pattern, then we have to have the wave going thru both slits.

DeleteAs a practical matter, I don't know how you send one wave thru slits that are 1M miles apart, but let's assume that is possible.

If one photon is emitted, and the wave goes thru both slits, your next question is going to be why we never detect the photon behind BOTH slits. I say we don't because that would violate conservation of energy. Your next response is going to be that the explanation is unsatisfactory, because it seems as if each detector would need to know what is going on at the other detector.

If you assume that the photon must be going thru one slit or the other, then I would agree that it seems to contradict locality. But without that assumption, I don't see how to get such a contradiction.

Dear Roger,

ReplyDeleteYou write, "Even if the photon is detected at the upper slit, I would still say that the wave went thru both slits. If removing the detectors results in an interference pattern, then we have to have the wave going thru both slits."

Please feel free to place the detector just in front of the slit, instead of behind the slit, by moving it a centimeter. If the detector goes off and detects the photon, would you still argue that the photon went through both slits?

You write, "As a practical matter, I don't know how you send one wave thru slits that are 1M miles apart, but let's assume that is possible." If you are uncomfortable with the distance, please adjust the distance to a number that you are comfortable with. What distance are you comfortable with? What is the cutoff point for your comfort?

You write, "If one photon is emitted, and the wave goes thru both slits, your next question is going to be why we never detect the photon behind BOTH slits." No, that is not my ext question.

You write, "I say we don't because that would violate conservation of energy. Your next response is going to be that the explanation is unsatisfactory, because it seems as if each detector would need to know what is going on at the other detector. If you assume that the photon must be going thru one slit or the other, then I would agree that it seems to contradict locality. But without that assumption, I don't see how to get such a contradiction."

By choosing to place the detector in front of the first slit just as the photon arrives, and by doing it a million years after the photon was emitted, we can show that we affected what happened at the second slit in an instantaneous manner, or at least in a manner faster than light.

If you detect the photon before the slit, then there is no longer any reason to say it goes thru both slits.

ReplyDeleteSome of what you say would be just as paradoxical with classical physics. If I shoot a bullet into a cliff, and then I go find the bullet, I instantly eliminate the other possible locations for it. Nobody ever says that there is anything nonlocal about that.

So you fire a photon, detect it in one location, and immediately conclude that it is not in other locations. What's the mystery? Isn't that exactly what you would expect in any local theory?

Dear Roger,

ReplyDeleteThere exists no classical theory which could reproduce these results.

We're not shooting bullets at cliffs. We're dealing with photon and double slits.

You write, "So you fire a photon, detect it in one location, and immediately conclude that it is not in other locations. What's the mystery?"

Why are you ignoring the double slit and the interference? If you ignore the double slit and interference, then you are fighting your own straw men.

Again, imagine a stream of a million photons emitted from 100,000,000 miles away, which approaches a double slit whose slits are separated by 1,000,000 miles.

If there are just the slits, and no detectors, we see an interference pattern. The photons all went through both slits.

Now, if we place a detector before the first slit, one trillionth of a second before the million photons arrive at the screen, the interference pattern will disappear, as every photon will go through one slit or the other.

So it is that the wave function must collapse at a rate faster than the velocity of light, and thus quantum mechanics is indeed nonlocal.