Quantum Mechanics without Measurement

[atyy]

This may be a subtIety but I don't agree. Contrary to von Neumann's quantum logic approach, Griffiths doesn't modify the laws of logic. He says that in order to talk consistently about probabilities, you must set up a sample space first. This defines the properties to which the probabilities are assigned.

It is not unusual that there are multiple ways to do this. Wikipedia uses the multiple properties of cards in a deck as an example. What is unusual is that these sample spaces can not always be combined in QM. This is because the Born rule tells us to use projectors to assign probabilities to properties. If we try to combine these properties, the probabilities may depend on the order we assign them because the projectors may be non-commuting. So combining properties doesn't always make sense. Independent of CH, I think this is a very simple and elegant way to see why naive realism isn't compatible with QM.

So what Griffiths says is that your statement A is necessarily of the form "in framework X, the particle has the property Y" and there's no problem to talk about A AND B.

Yes, in framework X, it's just classical stochastic processes. So in each framework, one history is real, so let's say that statement A is true in framework X. However, all frameworks "exist simultaneously". So we also have that statement B is true in framework Y. Normal logic would say that it is then true to say A is true in framework X and B is true in framework Y. But apparently this is forbidden.

 

[craigi]

Yes, in framework X, it's just classical stochastic processes. So in each framework, one history is real, so let's say that statement is A is true in framework X. However, all frameworks "exist simultaneously". So we also have that statement B is true in framework Y. Normal logic would say that it is then true to say A is true in framework X and B is true in framework Y. But apparently this is forbidden.

A : Property X in Framework Y
B : Property X' in Framework Y'

Now we can talk of A and B. It's not forbidden by any particular rule, but there is no framework where this has physical meaning.

We sort of knew this all along under CI as soon as we encountered spin, but we were told to shut up if we asked too many questions.

The problem was without the clarification of CH, CI could be easily misinterpreted as saying a particle can't have Sx and Sz both equal to 1/2 at the same time, for instance. I'm sure many of us have said it, but it's not true and it's not false either.We can create classical analogs, by considering photographic projections, for instance:

A : The object is facing X in photograph Y
B : The object is facing X' in photograph Y'

 

[kith]

Yes, in framework X, it's just classical stochastic processes. So in each framework, one history is real, so let's say that statement A is true in framework X. However, all frameworks "exist simultaneously". So we also have that statement B is true in framework Y. Normal logic would say that it is then true to say A is true in framework X and B is true in framework Y. But apparently this is forbidden.

No, this should be a perfectly valid statement.

 

[atyy]

No, this should be a perfectly valid statement.

Then why is it said that the frameworks are in compatible and statements from them cannot be combined. What you are saying here is that they can be combined. To me if this is possible, aren't they just like different reference frames?

 

[bhobba]

Agreed, but help me out here. How is what I said wrong?

The environment is NOT just the classical world. Its entirely quantum.

Thanks
Bill

 

[kith]

Then why is it said that the frameworks are in compatible and statements from them cannot be combined.

They refer to statements from within frameworks and not to the kind of meta-statements we have made.

What you are saying here is that they can be combined. To me if this is possible, aren't they just like different reference frames?

In a way, yes.

 

[craigi]

The environment is NOT just the classical world. Its entirely quantum.

Thanks
Bill

I'm probably using non-standard definitions, because I see them as equivalent.

Is it right for me to consider the classical world as a degenerate case of the quantum world and the environment to exhibit exactly the same lack of coherence?

 

[atyy]

What you are saying here is that they can be combined. To me if this is possible, aren't they just like different reference frames?

In a way, yes.

So if in some sense, the frameworks are compatible, shouldn't there be an "invariant object" that is the true reality underlying them? In the analogy with reference frames, we cannot combine coordinate statements from different frames, but we can speak of the geometrical object which is coordinate-independent. Is there none here? Or could there be one that hasn't been discovered? (Actually, doesn't the Gell-mann and Hartle paper http://arxiv.org/abs/1106.0767 try to do that?)

 

[craigi]

So if in some sense, the frameworks are compatible, shouldn't there be an "invariant object" that is the true reality underlying them? In the analogy with reference frames, we cannot combine coordinate statements from different frames, but we can speak of the geometrical object which is coordinate-independent. Is there none here? Or could there be one that hasn't been discovered? (Actually, doesn't the Gell-mann and Hartle paper http://arxiv.org/abs/1106.0767 try to do that?)

It would seem reasonable that there could be such an invariant object. In the photographic projection analog that I offered earlier, this would be the 3D world, where we can relate the different orientations.

I'm pretty sure that if we try to define such an invariant object for QM, we get what we refer to as the multiverse in certain interpretations, but CH doesn't go that far.

 

[kith]

So if in some sense, the frameworks are compatible, shouldn't there be an "invariant object" that is the true reality underlying them? In the analogy with reference frames, we cannot combine coordinate statements from different frames, but we can speak of the geometrical object which is coordinate-independent. Is there none here? Or could there be one that hasn't been discovered? (Actually, doesn't the Gell-mann and Hartle paper http://arxiv.org/abs/1106.0767 try to do that?)

The underlying object is the quantum system. The fact that we cannot describe "the real thing" in an unambigous way doesn't imply that it doesn't exist. The CH seems to be still kind of a minimal interpretation and the search for an unambigous description goes beyond it. In order to get it, I think you have to assume something more than CH does. Gell-Mann and Hartle for example talk about negative probabilities in the abstract of their paper (I haven't read it).

 

[atyy]

It would seem reasonable that there could be such an invariant object. In the photographic projection analog that I offered earlier, this would be the 3D world, where we can relate the different orientations.

I'm pretty sure that if we try to define such an invariant object for QM, we get what we refer to as the multiverse in certain interpretations, but CH doesn't go that far.

The underlying object is the quantum system. The fact that we cannot describe "the real thing" in an unambigous way doesn't imply that it doesn't exist. The CH seems to be still kind of a minimal interpretation and the search for an unambigous description goes beyond it. In order to get it, I think you have to assume something more than CH does. Gell-Mann and Hartle for example talk about negative probabilities in the abstract of their paper (I haven't read it).

Or could one say that the fact the the multiple frameworks are compatible in the sense that A is true in framework X and B is true in framework Y, imply the existence of this unknown object? What Gell-Mann and Hartle seem to be searching for does seem to match something like common sense reality, especially since if I include myself in the quantum system, I will be in all frameworks, and there is nothing outside to say in framework X or framework Y.

 

[craigi]

Or could one say that the fact the the multiple frameworks are compatible in the sense that A is true in framework X and B is true in framework Y, imply the existence of this unknown object?

That is a matter of taste.

We natually try apply the principle that the simplest explanation is the best, but some believe it's simpler to say it doesn't exist and some believe it's simpler to say that it does.

There is a new development though, which weakens the former argument somewhat. It's still controversial, but cosmologists increasingly believe that there may sufficient space, if not infinite space, beyond our cosmic horizon, for many duplicate universes, which realize all possibilities of the wavefunction. This field isn't purely theoretical either. There has been experimental verification of some predictions of this theory, though direct observation of another universe is impossible.

Possibly the most compelling thing about these interpretations is that we recover determinism is a very natural way.

CH isn't concerned with any of this though.

 

[stevendaryl]

A : Property X in Framework Y
B : Property X' in Framework Y'

Now we can talk of A and B. It's not forbidden by any particular rule, but there is no framework where this has physical meaning.

We sort of knew this all along under CI as soon as we encountered spin, but we were told to shut up if we asked too many questions.

The problem was without the clarification of CH, CI could be easily misinterpreted as saying a particle can't have Sx and Sz both equal to 1/2 at the same time, for instance. I'm sure many of us have said it, but it's not true and it's not false either.


We can create classical analogs, by considering photographic projections, for instance:

A : The object is facing X in photograph Y
B : The object is facing X' in photograph Y'

There is an easy resolution to that classical analogue, which is to say that both can be true simultaneously. The property is not "facing X", but "facing X in photograph Y". The same object can satisfy "facing X in photograph Y" and "facing X' in photograph Y'" at the same time.

The incompatibility of frameworks seems more extreme than that. It seems that you can't say: "X is true in framework F, and X' is true in framework F'" if F and F' are incompatible.

 

[craigi]

There is an easy resolution to that classical analogue, which is to say that both can be true simultaneously. The property is not "facing X", but "facing X in photograph Y". The same object can satisfy "facing X in photograph Y" and "facing X' in photograph Y'" at the same time.

The incompatibility of frameworks seems more extreme than that. It seems that you can't say: "X is true in framework F, and X' is true in framework F'" if F and F' are incompatible.

I think you can say it, but no inferences can be drawn from it.

In the classical case we can construct a higher dimensional space where it does have meaning, but is this not also true in the quantum case?

To borrow terminology from other interpretations, are we not effectively saying, that the electron has Sx = 1/2 in one universe AND Sz = 1/2 in another universe, for example?

That all sounds fine, but obviously we're not going to be able to draw any conclusions from that.

 

[atyy]

The incompatibility of frameworks seems more extreme than that. It seems that you can't say: "X is true in framework F, and X' is true in framework F'" if F and F' are incompatible.

That's what I thought too, but craigi (#182) and kith (#183) indicate above that it is possible.

 

[stevendaryl]

I think you can say it, but it has no meaning and no inferences can be drawn from it.

In the classical case we can construct a higher dimensional space where it does have meaning, but is this not also true in the quantum case?

No, in the quantum case there is no higher dimensional framework that can make sense of all of the frameworks simultaneously.

A framework consists of a sequence of times, and a choice of an observable at each time. A history for a framework consists of an assignment of a value to each observable at each time in the framework.

Within a framework, classical logic and classical probability hold, which means that you can reason as if probability is just due to ignorance. So you can pretend that "The particle has spin-up along the x-axis at time t" is a meaningful statement, either true or false, but you don't know which.

So within each framework, you can reason as if there is a single "real" history, while the other histories aren't real. So if it's possible for each framework to have a "real" history, why isn't it possible to assume that there is a "master history" that chooses one history to be real out of each framework? If there were such a master history, it would allow one to say, for every possible observable and for every possible time, what the value of that variable is at that time.

This would be sort of like dBB on steroids. dBB has definite (though unknown) values for particle positions at every moment in time, but it does not treat other observables in an egalitarian manner.

What prevents us from assuming that there is a master history? Really, it's not logic, it's probability theory. If we assume that there is a definite, but unknown, master history, then it means that we can use ordinary probability theory to reason about this history. That is, we can just use probability to reflect our ignorance about which master history is the real one.

But then what would prevent us from asking a question along the lines of:
"What is the probability that the particle has spin-up along the x-axis and spin-up along the y-axis at time t=0?" One way of interpreting Bell's theorem is that there is no consistent probability that we can assign to conjunctions of statements involving incompatible observables.

One way out (described by the late mathematical physicist Pitowsky, which I read about in Stanley Gudder's book on quantum probability) is to assume nonmeasurable sets. It's a mathematical curiosity that it is possible to come up with a set of reals for which there is no consistent way to assign a probability that a random real is in that set. That doesn't mean that the set doesn't exist. It doesn't mean that it's impossible to pick a random real in that set, it just means that there is no way to compute a probability for such an event.

 

[stevendaryl]

That's what I thought too, but craigi (#182) and kith (#183) indicate above that it is possible.

I don't agree. Remember, a framework is nothing more than a choice of which observables and which moments in time to talk about. So suppose framework F has observable O1 at time t1, observable O2 at time t2, etc. Framework F' has observable O1' at time t1', O2' at time t2', etc.

Then saying "X is true in framework F" is a statement of the form
"O1 has value X1 at time t1, O2 has value X2 at time t2, ..."

Saying "X' is true in framework F'" is similarly a statement of the form
"O1' has value X1' at time t1', O2' has value X2' at time t2', ..."

So the conjunction "X is true in framework F and X' is true in framework F'" just amounts to saying:
"O1 has value X1 at time t1, O1' has value X1' at time t1', O2 has value X2 at time t2, O2' has value X2' at time t2' ..."

So I don't see the difference between the meta statement and the corresponding conjunction of ordinary statements.

 

[atyy]

Nonmeasurable sets are also a potential way to evade Bell's theorem. strangerep has mentioned that on PF before. But I don't know if there are any successful constructions using that potential out.

 

[kith]

Then saying "X is true in framework F" is a statement of the form
"O1 has value X1 at time t1, O2 has value X2 at time t2, ..."

A framework is a choice of observables at certain times. The kind of statements I had in mind is "If the observables of framework F are chosen to be real, O1 has value X1 at time t1, O2 has value X2 at time t2, ..." AND "If the observables of framework F' are chosen to be real, O1' has value X1' at time t1', O2' has value X2' at time t2', ...". Such meta-statements can always be made.

 

[atyy]

A framework is a choice of observables at certain times. The kind of statements I had in mind is "If the observables of framework F are chosen to be real, O1 has value X1 at time t1, O2 has value X2 at time t2, ..." AND "If the observables of framework F' are chosen to be real, O1' has value X1' at time t1', O2' has value X2' at time t2', ...". Such meta-statements can always be made.

Does this still make sense if the observer is included in the system, so that the observer is in all frameworks? Griffiths's version seems to make sense if the observer lies outside the system, but does it still make sense if the frameworks encompass everything? If the observer is in the system, who is doing the choosing of the framework that is real?

Could this be the reason for differences bewteen the Griffiths and Gell-Mann/Hartle versions of CH?

 

[stevendaryl]

A framework is a choice of observables at certain times. The kind of statements I had in mind is "If the observables of framework F are chosen to be real, O1 has value X1 at time t1, O2 has value X2 at time t2, ..." AND "If the observables of framework F' are chosen to be real, O1' has value X1' at time t1', O2' has value X2' at time t2', ...". Such meta-statements can always be made.

I don't see how it makes any difference whether you are talking statements or meta-statements. Take a very simple case: We have an electron prepared at time t=0 to have spin-up in the z-direction. Framework F1 consists of a single observable, the x-component of spin, at time t=1. Framework F2 consists of a different observable, the y-component of spin at time t=1. You can imagine frameworks for every possible orientation for spin.

Whatever difficulties we have with compound statement "s_x = +1/2 and s_y = +1/2", we'll have exactly the same difficulties with the compound statement: "If F1 is chosen as real, then s_x = +1/2 and if F2 is chosen as real, then s_y = +1/2". In either case, we're talking about a mathematical mapping from orientations to the two-element set \{ +1/2, -1/2 \}. What Bell's theorem shows is that there is no consistent assignment of probabilities to such mappings in a way that agrees with the predictions of quantum mechanics. Calling the mapping a "meta" fact doesn't change this. The same proof shows that there is no consistent assignment of probabilities to the set of all "meta" statements. So you haven't actually changed anything by letting it be "meta". You still have statements that seem to be meaningful in combination, but there is no consistent way to assign likelihoods of their being true.

You might as well have dropped the "meta", and just talked about spins themselves. It's perfectly meaningful to say "The electron has spin +1/2 in the x-direction, and spin +1/2 in the y-direction". There is no contradiction from making such a claim. But there is no consistent way to assess the probability that such a claim is true. Meta versus non-meta doesn't make any difference.

 

[DevilsAvocado]

What prevents us from assuming that there is a master history? Really, it's not logic, it's probability theory. If we assume that there is a definite, but unknown, master history, then it means that we can use ordinary probability theory to reason about this history. That is, we can just use probability to reflect our ignorance about which master history is the real one.

How about the double-slit? Check out Richard Feynman @49:45, in this video from Cornell University 1964. We're forbidden – even in theory – to in advance know about which slit, because if we did, the double-slit would stop working! This is not a matter of ignorance or 'bad tools'; it's an intrinsic property of QM.

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


P.S: I love nice and even numbers!

 

[stevendaryl]

How about the double-slit? Check out Richard Feynman @49:45, in this video from Cornell University 1964. We're forbidden – even in theory – to in advance know about which slit, because if we did, the double-slit would stop working! This is not a matter of ignorance or 'bad tools'; it's an intrinsic property of QM.

To say that we are forbidden to know which slit seems interpretation-dependent. For example, in the Bohm theory, the electron (or photon--I'm not sure if there is a Bohm theory for the photon) has a definite position at all times.

 

[stevendaryl]

Wow! That video plays the Cornell song at the beginning:

Far above Cayuga's waters,
With its waves of blue,
Stands our noble Alma Mater,
Glorious to view

I'm not a Cornell grad, but I do live in Ithaca.

That is a great video. I wish I could have learned physics from Feynman.

 

[craigi]

To say that we are forbidden to know which slit seems interpretation-dependent. For example, in the Bohm theory, the electron (or photon--I'm not sure if there is a Bohm theory for the photon) has a definite position at all times.

Agreed.

I think much of what we're talking about in latter section of this thread is really about level of abstraction.

 

[DevilsAvocado]

To say that we are forbidden to know which slit seems interpretation-dependent. For example, in the Bohm theory, the electron (or photon--I'm not sure if there is a Bohm theory for the photon) has a definite position at all times.

Yes, but in dBB you have the 'magical' unknown initial conditions (of the universe?), that makes it impossible to make any predictions in advance. If it wasn't, dBB would be deterministic all they way through (and someone would get the Nobel Prize in Physics).

 

[DevilsAvocado]

Wow! That video plays the Cornell song at the beginning:

Enjoy!

 

[DevilsAvocado]

That is a great video. I wish I could have learned physics from Feynman.

Yes, he was truly brilliant. Have you seen the Sir Douglas Robb lectures at the University of Auckland (1979)? It's available on YouTube:

QED: The Strange Theory of Light and Matter

https://www.youtube.com/watch?v=LPDP_8X5Hug

The playlist consist of 30 videos between 10-15 min:
http://www.youtube.com/playlist?list=PL4C9818DC43C7E834

The original can be found here:
http://www.vega.org.uk/video/subseries/8

 

[stevendaryl]

Yes, but in dBB you have the 'magical' unknown initial conditions (of the universe?), that makes it impossible to make any predictions in advance. If it wasn't, dBB would be deterministic all they way through (and someone would get the Nobel Prize in Physics).

I'm just objecting to the claim that people are forbidden from knowing the positions of particles. If someone (God, maybe) whispered into your ear what the position of the electron was at time t=0, and told you what the "pilot wave" was, then you would know what the position was at all future times. If you don't have somebody omniscient whispering in your ear, then that calls for probabilities. You assume a probability distribution on initial positions, and on pilot waves, and then you end up getting a probabilistic prediction for future values of the position. That's the way ordinary classical probability works--anything you don't know, you can throw into the probability distributions to reflect your ignorance.

But when it comes to incompatible observables, such as the spins in the x-direction and the y-direction, it's not simply that you don't know enough. There is no consistent way to assign probabilities (and having the same predictions as quantum mechanics).

I don't think, though, that there is a proof that it is impossible to HAVE simultaneous values for incompatible observables. Only that there is no way to assign probabilities to sets of values for incompatible observables. I don't think the double-slit experiment proves otherwise. As I mentioned before, Pitowsky came up with a model for the EPR experiment which did assume that the spin in every direction was defined. His model escaped from the proof of Bell's theorem in that it did not assign probabilities for certain combinations of events (for the probability that the electron is spin-up in the x-direction and spin-down in the y-direction might be undefined). Bell's theorem amounts to a proof that there is no consistent way to assign probabilities to such events.

 

[DevilsAvocado]

I don't think the double-slit experiment proves otherwise.

In the case of the double-slit, we can be certain that knowledge of which slit will destroy interference. It's very hard (impossible) to get pass this simple fact, and the obvious reason is that you need "two sources" to get this kind of interference, whether this "source" is one particle 'splitting' to pass the two slits, or if it is only the wavefunction passing thru (like "water waves"), or if it is one particle guided by a pilot-wave passing thru, is still unknown.

Bell's theorem amounts to a proof that there is no consistent way to assign probabilities to such events.

Einstein was very skeptical about CFD, and maybe we are paying too much attention to this regarding EPR-Bell, I don’t know...

As you are saying, it only becomes a problem when we perform the measurement, i.e. suppose Bell required us to have a 'particle' with 6 incompatible values. Then we could build a model of a real spinning 'dice', that for some (unknown) reason will never let us see these 6 values simultaneously, and when we perform a measurement, we will only get one value, based solely on classical probabilities.

What's the problem!?

The problem is that this model works very fine for the 1935 version of EPR, where we theoretically could utilize a 'common influence' on the two 'twin dices', showing correlated behavior at measurement, i.e. if one shows even numbers, the other always shows odd, and vice versa.

This however is a dead parrot after 1964 and Bell's theorem, which mathematically proves that to have 'real twin dices' spinning at the source, the 'common influence' is not 'strong' enough to explain what happens in QM experiments (and predictions).

The only way to have spinning 'real dices' is to introduce a 'magical synchronization' that must be non-local, since the final parameters, setting the outcome probability for the 'twin dices', are set locally at the very last moment, and this 'new probability' is a combination of Alice local settings + Bob local settings, which makes any 'common source probability' faulty.

This is how it is.