The central mystery of quantum mechanics (according to Feynman)

[Dadface]

Observing which slit a photon or particle goes through in the two slit experiment results in the formation of a diffraction pattern instead of an interference pattern. Were Feynmanns remarks about this made by referring to the classical experiments where there is high illumination or to the later experiments such as where there is one photon or one particle at a time or to all variations of the experiment.
By searching I have found it easy to find out what he said but have so far not been able to find out what exactly he was referring to.
Thank you

 

[atyy]

http://www.feynmanlectures.caltech.edu/III_01.html#Ch1-S7
"In this chapter we shall tackle immediately the basic element of the mysterious behavior in its most strange form. We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery. We cannot make the mystery go away by “explaining” how it works. We will just tell you how it works. In telling you how it works we will have told you about the basic peculiarities of all quantum mechanics."

Nowadays we know that Feynman's argument was inaccurate. Non-relativistic quantum phenomena can be explained in a somewhat "classical way" with Bohmian mechanics. It is still being researched whether a Bohmian viewpoint is satisfactory for relativistic quantum mechanics. However, Bohmian trajectories are not the same as classical Hamiltonian trajectories. More importantly, Bohmian mechanics is nonlocal, and is "non-classical" or "mysterious" in that sense. So it is not so much that Bohmian mechanics is an entirely "classical way" of explaining quantum mechanics, since it is nonlocal. Rather, the double slit experiment doesn't demonstrate a violation of the Bell inequality, and so it doesn't force nonlocality on us.

The violation of the Bell inequality by quantum mechanics requires two ingredients: non-commuting observables and entanglement.

Feynman's point was probably about non-commuting observables, ie. position and momentum cannot be simultaneously well-defined. So Feynman got one of the ingredients needed for encapsulating the mystery. He didn't get the second ingredient of entanglement, so most people nowadays would say that the double slit is not the "only" mystery.

These notes by Scarani http://arxiv.org/abs/1303.3081 explain why the violation of the Bell inequality cannot be simulated classically. The comments by Fuchs and Schack on the first 3 pages of http://arxiv.org/abs/1301.3274 explain why in some sense the double slit experiment is not mysterious, and simply illustrates that different experiments produce different results (I think you can also find this point in Ballentine's book; I don't recommend the rest of the article by Fuchs and Schack or Ballentine's book, because they are rather idiosyncratic, unless one has already learned QM the usual way, eg. Landau & Lifshitz.)

There is, however, one proposed interpretation of quantum mechanics called "Consistent Histories" in which it is said that the double slit is the only mystery.

 

[my2cts]

Keeping things a lot simpler, I think he referred to the fact that a single particle, such as an electron, appears to go through both slits. If the setup is modified so that it can be determined through which slit the electron goes no interference occurs.

 

[atyy]

Keeping things a lot simpler, I think he referred to the fact that a single particle, such as an electron, appears to go through both slits. If the setup is modified so that it can be determined through which slit the electron goes no interference occurs.

Yes, Feynman writes in the link given above "Is it true, or is it not true, that the electron either goes through hole 1 or it goes through hole 2?" The only answer that can be given is that we have found from experiment that there is a certain special way that we have to think in order that we do not get into inconsistencies.". So maybe he was talking about was what was later formalized as the Kochen-Specker theorem.

http://en.wikipedia.org/wiki/Kochen–Specker_theorem
"Hence, the KS theorem does only exclude noncontextual hidden variable theories."

Scarani also mentions that the Bell tests exclude more types of hidden variable theories than the Kochen-Specker tests.
http://arxiv.org/abs/1303.3081 (footnote 8): "Tests like "contextuality" a la Kochen Specker, or sequential measurements a la Feynman or Leggett-Garg, need a minimal amount of assumptions to prove that the outcomes do not come from a pre-established agreement. Indeed, no detailed knowledge of the system and the measurement is needed, but one must assume that the measurement device does not write in, nor reads from, other degrees of freedom than the relevant one."

 

[my2cts]

In fact if an experiment is arranged such that it can be determined if the electron goes left or right, this implies that the two possibilities are described by mutually orthogonal wavefunctions. So it is the absence of physical distinguishability that leads to the interference phenopmenon. I guess that if a system could be set up such that the wavefunctions describing the two possibility are partially but not completely overlapping, such that it would be possible to determine with say 95% certainty through which slit the electron passed, then a strongly reduced interference pattern would be observed.

 

[Dadface]

Thank you atyy and my2cts. My question was enquiring about the history of interference and I wanted to know what variation(s) of the two slit experiment Feynman was referring to.
I intended to look at the theory in greater detail but was thrown a bit by the above responses. To me the suggestion seems to be that Bohmian mechanics can explain the results of those interference experiments where which way information can be obtained. Is that so, can Bohmian mechanics even explain the results of quantum eraser type experiments of the type where which way information is not actually obtained but where there is the potential to do so?
Clarification would be greatly appreciated.
Thank you

 

[stevendaryl]

...Rather, the double slit experiment doesn't demonstrate a violation of the Bell inequality, and so it doesn't force nonlocality on us.

The violation of the Bell inequality by quantum mechanics requires two ingredients: non-commuting observables and entanglement.

I have a slightly different perspective on Bell's inequality. I don't think that its violation in experiments is a mystery about quantum theory itself. Instead it dashes the hopes for replacing quantum theory by a future, less mysterious theory. So to me, it's not so much that Bell introduced any new mysteries of quantum theory, he just made it clear that we are pretty much stuck with all the old mysteries.

 

[DevilsAvocado]

Were Feynmanns remarks about this made by referring to the classical experiments where there is high illumination or to the later experiments such as where there is one photon or one particle at a time or to all variations of the experiment.

Maybe this video could help you.

Richard Feynman on the Double Slit Paradox: Particle or Wave?
https://www.youtube.com/watch?v=hUJfjRoxCbk
http://www.youtube.com/embed/hUJfjRoxCbk

To me the suggestion seems to be that Bohmian mechanics can explain the results of those interference experiments where which way information can be obtained.

Ask atyy about initial conditions and the Born rule.

 

[DevilsAvocado]

Instead it dashes the hopes for replacing quantum theory by a future, less mysterious theory. So to me, it's not so much that Bell introduced any new mysteries of quantum theory, he just made it clear that we are pretty much stuck with all the old mysteries.

Agree 100%. :thumbs:

Kepler had discovered that the planets move along ellipses. Galileo had discovered that falling objects move along parabolas. Each was expressed by a simple mathematical curve and each was a partial decoding of the secret of motion. Separately they were profound discoveries; together they were the seeds of the Scientific Revolution, which was about to flower.

This is not unlike the present juncture in theoretical physics. We have two great discoveries, quantum theory and general relativity, whose unification we seek. Having worked on this problem for most of my life, I’m impressed by the progress we’ve made. At the same time, I’m certain that some simple idea lies hidden in plain sight that will be the key to its resolution. Admitting that progress can be held up as we await the invention of nothing more substantial than an idea is humbling, but it’s happened before.

 

[atyy]

I intended to look at the theory in greater detail but was thrown a bit by the above responses. To me the suggestion seems to be that Bohmian mechanics can explain the results of those interference experiments where which way information can be obtained. Is that so, can Bohmian mechanics even explain the results of quantum eraser type experiments of the type where which way information is not actually obtained but where there is the potential to do so?

I haven't worked through BM for a quantum eraser experiment, but my understanding is that the consensus is that BM is the only established interpretation of non-relativistic quantum mechanics, apart from Copenhagen (other leading approaches to interpretation are many-worlds and consistent histories, but it is not universally acknowledged that they solve all problems). So BM should be able to handle the quantum eraser experiment.

I have a slightly different perspective on Bell's inequality. I don't think that its violation in experiments is a mystery about quantum theory itself. Instead it dashes the hopes for replacing quantum theory by a future, less mysterious theory. So to me, it's not so much that Bell introduced any new mysteries of quantum theory, he just made it clear that we are pretty much stuck with all the old mysteries.

I agree, and that's what I intend to say: the violation of the Bell inequality encapsulates the mysteries of quantum mechanics.

 

[Dadface]

Maybe this video could help you.

Richard Feynman on the Double Slit Paradox: Particle or Wave?
https://www.youtube.com/watch?v=hUJfjRoxCbk
http://www.youtube.com/embed/hUJfjRoxCbk



Ask atyy about initial conditions and the Born rule.

Thank you DevilsAvocado. When I clicked on the link I remembered having seen that video in the past. I probably just scanned through it. I will take another look

 

[Dadface]

I haven't worked through BM for a quantum eraser experiment, but my understanding is that the consensus is that BM is the only established interpretation of non-relativistic quantum mechanics, apart from Copenhagen (other leading approaches to interpretation are many-worlds and consistent histories, but it is not universally acknowledged that they solve all problems). So BM should be able to handle the quantum eraser experiment.



I agree, and that's what I intend to say: the violation of the Bell inequality encapsulates the mysteries of quantum mechanics.

Hello atty.My feeling at present is that all interpretatiuons of QM have some or even the same conceptual difficulties, for example in terms of anti intuitive results, when dealing with quantum eraser type experiments. If at any time you take a look at the experiments I would be interested to hear of any opinions you may form. Thank you again.

 

[bhobba]

for example in terms of anti intuitive results,

The interesting thing I have found about QM is to start with it seems very anti intuitive, but after a while, when you understand some of the more modern ways of viewing it, such as the most reasonable probability theory that allows continuous transformations, things seem a lot more reasonable.

The argument goes something like this. Suppose we have a system in 2 states represented by the vectors [0,1] and [1,0]. These states are called pure. These can be randomly presented for observation and you get the vector [p1, p2] where p1 and p2 give the probabilities of observing the pure state. Such states are called mixed. Now consider the matrix A that say after 1 second transforms one pure state to another with rows [0, 1] and [1, 0]. But what happens when A is applied for half a second. Well that would be a matrix U^2 = A. You can work this out and low and behold U is complex. Apply it to a pure state and you get a complex vector. This is something new. Its not a mixed state - but you are forced to it if you want continuous transformations between pure states.

QM is basically the theory that makes sense out of pure states that are complex numbers. There is really only one reasonable way to do it - by the Born rule (you make the assumption of non contextuality - ie the probability is not basis dependant, plus a few other things no need to go into here) - as shown by Gleason's theorem. But it can also be done without such high powered mathematical machinery:
http://arxiv.org/pdf/quant-ph/0101012.pdf

In my opinion intuition has a lot to do with understanding, familiarity and the way you view things.

Thanks
Bill

 

[Dadface]

The interesting thing I have found about QM is to start with it seems very anti intuitive, but after a while, when you understand some of the more modern ways of viewing it, such as the most reasonable probability theory that allows continuous transformations, things seem a lot more reasonable.

The argument goes something like this. Suppose we have a system in 2 states represented by the vectors [0,1] and [1,0]. These states are called pure. These can be randomly presented for observation and you get the vector [p1, p2] where p1 and p2 give the probabilities of observing the pure state. Such states are called mixed. Now consider the matrix A that say after 1 second transforms one pure state to another with rows [0, 1] and [1, 0]. But what happens when A is applied for half a second. Well that would be a matrix U^2 = A. You can work this out and low and behold U is complex. Apply it to a pure state and you get a complex vector. This is something new. Its not a mixed state - but you are forced to it if you want continuous transformations between pure states.

QM is basically the theory that makes sense out of pure states that are complex numbers. There is really only one reasonable way to do it - by the Born rule (you make the assumption of non contextuality - ie the probability is not basis dependant, plus a few other things no need to go into here) - as shown by Gleason's theorem. But it can also be done without such high powered mathematical machinery:
http://arxiv.org/pdf/quant-ph/0101012.pdf

In my opinion intuition has a lot to do with understanding, familiarity and the way you view things.

Thanks
Bill

Thank you Bill,
What I'm particularly interested in at the moment are the quantum eraser type experiments. I can see that theory can succesfully predict the variety of interference/diffraction patterns that can be observed but can theory explain the circumstances that are necessary to observe opposite type of patterns? In other words why does the potential to get which way information result in diffraction only even when, in some circumstances, the paths are not actually marked?

 

[bhobba]

but can theory explain the circumstances that are necessary to observe opposite type of patterns? In other words why does the potential to get which way information result in diffraction only even when, in some circumstances, the paths are not actually marked?

Well if the eraser experiments were not explainable by QM that would be BIG news. They are of course so really your worry is what's going on.

Its not difficult really. In principle you can unscramble dehoherence but its impossible at a practical level when a large number degrees of freedom are involved. All the eraser experiments show its possible to do that in simple cases when there is a small number.

See:
https://www.physicsforums.com/showthread.php?t=623648

Thanks
Bill

 

[Dadface]

Well if the eraser experiments were not explainable by QM that would be BIG news. They are of course so really your worry is what's going on.

Its not difficult really. In principle you can unscramble dehoherence but its impossible at a practical level when a large number degrees of freedom are involved. All the eraser experiments show its possible to do that in simple cases when there is a small number.

See:
https://www.physicsforums.com/showthread.php?t=623648

Thanks
Bill

Thank you but does the explanation really conform to intuition? To quote professor Jim Al Khalili:

"If you can explain this using common sense and logic do let me know because there is a Nobel Prize waiting for you."

The remark was probably a bit tongue in cheek but it referred to the conceptual difficulties of the phenomenom.

(See the mini U tube presentation "Double slit explained by! Jim Al Khalili")

The professor seemed to be referring to just the basic one photon at a time experiments and not to, what I think is the more challenging, one photon at a time quantum eraser experiments.

 

[bhobba]

Thank you but does the explanation really conform to intuition?

Stand on a platform that can rotate with a spinning bicycle wheel in your hand. Rotate the wheel and you start to spin around. When most people first see it they are dumbfounded - gyroscopic effects are actually quite puzzling at first. But once you understand it its pretty ho hum really.

The same with QM once you understand its conceptual core.

"If you can explain this using common sense and logic do let me know because there is a Nobel Prize waiting for you."

I can and have given you some links that do it. But I doubt if I could ever collect because part of it is what is common-sensical changes as you understand more. Gyroscopic behaviour isn't common-sensical either until you understand it - but when you do - it's quite easy.

The issue with QM is it's not understandable in terms of everyday experience. Extend that experience a bit and the barriers fall away - the same with many areas really.

We have met the enemy and he is us - Pogo

Thanks
Bill

 

[Dadface]

Stand on a platform that can rotate with a spinning bicycle wheel in your hand. Rotate the wheel and you start to spin around. When most people first see it they are dumbfounded - gyroscopic effects are actually quite puzzling at first. But once you understand it its pretty ho hum really.

The same with QM once you understand its conceptual core.



I can and have given you some links that do it. But I doubt if I could ever collect because part of it is what is common-sensical changes as you understand more. Gyroscopic behaviour isn't common-sensical either until you understand it - but when you do - it's quite easy.

The issue with QM is it's not understandable in terms of everyday experience. Extend that experience a bit and the barriers fall away - the same with many areas really.

We have met the enemy and he is us - Pogo

Thanks
Bill

I think I get the point you're making. Something that might seem difficult and weird loses its difficulty when you understand it. It can lose its weirdness also.
I don't think,however, that an understanding of the presently available knowledge and theories of QM necessarily results in a loss of the apparent weirdness of some parts of the subject. In his day Feynman knew more about QM than the majority but still pointed out a "mystery". Now we have people like Khalili still pointing out the logical and common sense problems with the subject even though their knowledge is far broader than was available to Feynman.

 

[bhobba]

I don't think,however, that an understanding of the presently available knowledge and theories of QM necessarily results in a loss of the apparent weirdness of some parts of the subject. In his day Feynman knew more about QM than the majority but still pointed out a "mystery". Now we have people like Khalili still pointing out the logical and common sense problems with the subject even though their knowledge is far broader than was available to Feynman.

Like I said:

We have met the enemy and he is us

I personally feel very comfortable with it - maybe with time you will to.

Thanks
Bill

 

[Dadface]

Like I said:


I personally feel very comfortable with it - maybe with time you will to.

Thanks
Bill

In my case I can get to understand certain things, according to currently available knowledge, and I can get to feel comfortable with certain things. However, when I get to think more deeply about some of those things I often realize how little I really understand. That can make it more interesting.

Thank you for your comments Bill. They were much appreciated.

 

[jk22]

I read somewhere that Feynman considers the superposition principle as the mystery of QM. In some sense we could say that this principle is a way of expressing ignorance more precisely : if we don't know which result it is, then it is a superposition of the possible result with given probabilities. This way of expressing ignorance seems to have very accurate prediction abilities.

In some sense QM is going from the classical local-causal laws to a non-causal law from the superposition to the effective results.

On this other mystery, I find Bell's theorem strange, since it computes : <AB-AB'+A'B+A'B'>=<A(B-B')>+<A'(B+B')>.

But we know that the measurement results of the sum is not the sum of the measurement results, since $$\nu(B-B')=\pm\sqrt{2}\neq\nu(B)-\nu(B')$$ hence the quantity above becomes :

$$(\underbrace{\pm 1}_A)( \underbrace{\pm\sqrt{2}}_{B-B'})+(\underbrace{\pm 1}_{A'})( \underbrace{\pm\sqrt{2}}_{B+B'})\leq 2\sqrt{2}$$.

As we see this is a sum of results that are product of result in A place and B place, which gives greater than two. So what does it mean to add the measurement results of B and B' ? Even if it is mathematically correct, it does not correspond to any measurement of B+B'...which could be seens as the superposition of the measurement operators ?

 

[bhobba]

But we know that the measurement results of the sum is not the sum of the measurement results

I have zero idea what you are trying to get at.

But in QM one of the central mysteries is that for operators the sum of the measurement results of any two operators is linear in the sense of expectation values ie if A and B are observables then E(aA+bB) = aE(A) + bE(B). Its pretty intuitive. This in fact implies the Born rule. A = ∑yi |bi><bi| E(A) = ∑ yi E(|bi><bi|). Let P = ∑ E(|bi><bi|) |bi><bi|. E(A) = Trace (PA) - which is the Born Rule. Since from the definition of observables E(|bi><bi|) must be positive and sum to one P is a positive operator of unit trace. For pure states this immediately implies the principle of superposition because they form a vector space, and all the rest of the QM formalism.

The key is in fact linearity. It took physicists a little while to see this very intuitive assumption doesn't always apply for hidden variable theories because they can be contextual. The above argument is essentially Von Neumans famous proof against hidden variable theories - but turned on its head to deduce QM. Gleason's Theorem is basically a stronger version of it based on much weaker assumptions.

Thanks
Bill

 

[jk22]

I wondered if there exist a matrix operation let's name it &, such that A & B has as eigenvalues the sum of the eigenvalues of A and B, in particular if A and B are 2 nxn matrices, then, A & B should be a n^2 x n^2 matrix.

If this exist then this special "sum" of operator would correspond to make the sum of the eigenvalues, and hence there would be a structure in the comparison inside Bell's theorem. But I fear there exist no such operation &.

 

[jk22]

Add :

why by the way the density probability of the hidden variable is the same for AB and AB' ?

 

[Jilang]

I read somewhere that Feynman considers the superposition principle as the mystery of QM. In some sense we could say that this principle is a way of expressing ignorance more precisely : if we don't know which result it is, then it is a superposition of the possible result with given probabilities. This way of expressing ignorance seems to have very accurate prediction abilities.

In some sense QM is going from the classical local-causal laws to a non-causal law from the superposition to the effective results.

On this other mystery, I find Bell's theorem strange, since it computes : <AB-AB'+A'B+A'B'>=<A(B-B')>+<A'(B+B')>.

But we know that the measurement results of the sum is not the sum of the measurement results, since $$\nu(B-B')=\pm\sqrt{2}\neq\nu(B)-\nu(B')$$ hence the quantity above becomes :

$$(\underbrace{\pm 1}_A)( \underbrace{\pm\sqrt{2}}_{B-B'})+(\underbrace{\pm 1}_{A'})( \underbrace{\pm\sqrt{2}}_{B+B'})\leq 2\sqrt{2}$$.

As we see this is a sum of results that are product of result in A place and B place, which gives greater than two. So what does it mean to add the measurement results of B and B' ? Even if it is mathematically correct, it does not correspond to any measurement of B+B'...which could be seens as the superposition of the measurement operators ?

I have seen this before on other sites that claim this is the correct equation. Can anyone illucidate?

 

[jk22]

What I know from the stuff is that in QM the observables are operators, the measurement results are the eigenvalues of those operators.
Here the eigenvalues of B+B' are not 2,0,-2, even if the eigenvalues of B and B' are 1 and -1.

However Bell's theorem study the average of a sum of operators, which is the sum of the averages, then it supposes the form of the average using hidden variables and it is shown that there is a discrepancy between this ansatz and the qm calculation.

 

[bhobba]

What I know from the stuff is that in QM the observables are operators, the measurement results are the eigenvalues of those operators.

That's true.

But what you are getting at with your other stuff I have zero idea - basically it looks like gibberish to me and I have read a LOT of books on QM.

I suspect though you should become acquainted with Gleason's theorem:
http://kof.physto.se/cond_mat_page/theses/helena-master.pdf

There is only one way to circumvent Gleason - contextuality. This is how Bohmian mechanics, for example, has hidden variables that do not obey the usual quantum rules. It was the mistake Von Neumann made in his proof hidden variables did not exist. He assumed the expectation value is the sum of expectations. That however may not apply to hidden variables that are explicitly contextual.

Thanks
Bill

 

[bhobba]

I have seen this before on other sites that claim this is the correct equation. Can anyone illucidate?

He needs to make clear quite a few things because I have zero idea what he is on about - specifically he needs to detail where he is getting his equations from and context. Its fairly obvious he is pulling them from some paper or text - we need to know that context.

But, as I have posted before, one of the key features of QM is expectation values are linear.

This is deeply tied to non-contextualty and Gleason's theorem.

Thanks
Bill

 

[bhobba]

But we know that the measurement results of the sum is not the sum of the measurement results

Yea - that's only true for commuting observables.

Your point being?

Thanks
Bill

 

[jk22]

He assumed the expectation value is the sum of expectations. That however may not apply to hidden variables that are explicitly contextual.

I'm sorry if my expression is not very clear.

Here you mean that the sum of the average could be different from the average of the sum. That's where I would like to come with Bell's theorem. (I suppose A(theta,lambda) is a contextual expression, since the result depends explicitly on the measurement apparatus configuration).

We have a sum of averages containing for example : $$\int A(\theta_A,\lambda)B(\theta_B,\lambda)\rho(\lambda)d\lambda-\int A(\theta_A,\lambda)B(\theta'_B,\lambda)\rho(\lambda)d\lambda$$

As you said we probably cannot put those 2 integral in one directly, because the sum does not correspond to measurement results, hence :

$$\int A(\theta_A,\lambda)B(\theta_B,\lambda)\rho(\lambda)d\lambda-\int A(\theta_A,\lambda)B(\theta'_B,\lambda)\rho(\lambda)d\lambda\neq \int A(\theta_A,\lambda)(B(\theta_B,\lambda)-B(\theta'_B,\lambda))\rho(\lambda)d\lambda$$

Is that what you mean that for contextual hidden variable the sum of average could be different than the average of the sum ?

 

[bhobba]

Is that what you mean that for contextual hidden variable the sum of average could be different than the average of the sum ?

No.

It's simple.

If you have a look at Gleason you will see its watertight if the measure defined on a projection operator depends only on the operator itself and not on other operators it may be part of in a resolution of the identity. The only out is if that's not the case. It's not the case for some hidden variable theories.

I have zero idea where you are getting your other equations from, what they mean, what the terms mean, context or anything.

Please detail exact context and meaning.

They look like some equations I vaguely remember from papers on hidden variables, but you need to detail exactly what the terms mean and their derivation. A concern I have is when you say 'I suppose such and such is a contextual expression' it indicates you do not know what the terms mean. It's very hard for someone to figure out what you are driving at if that is the case.

Thanks
Bill