Mathematically what causes wavefunction collapse?

[zonde]

I have zero idea why you say that. Its simply not true.

The logic is dead simple. By the law of large numbers we can find an ensemble associated with an observation where the proportion of outcomes is the probability. This follows from simply assuming the outcome can be described probabilistically. The state is not even introduced at this point. The Ensemble Interpretation associates the state not with individual systems but with the ensemble. Its that easy. If you still don't get it I will have to leave it to someone else because I simply can't explain it any better.

I understand that part perfectly well. The part I don't understand is what in that (Ballentine's) interpretation changes if you associte it with individual system. And as I see it nothing changes if you say it's applicable to individual systems.

 

[bhobba]

I understand that part perfectly well. The part I don't understand is what in that (Ballentine's) interpretation changes if you associte it with individual system. And as I see it nothing changes if you say it's applicable to individual systems.

Got it now.

You face the discontinuous collapse issue if you think the state applies to an individual system and is in some sense real - that's the key point both Einstein and Ballentine didn't make clear in their objection. If its simply a level of confidence like the Baysian view of probability it doesn't matter one whit.

Thanks
Bill

 

[stevendaryl]

Got it now.

You face the discontinuous collapse issue if you think the state applies to an individual system and is in some sense real - that's the key point both Einstein and Ballentine didn't make clear in their objection. If its simply a level of confidence like the Baysian view of probability it doesn't matter one whit.

Thanks
Bill

I don't understand how the ensemble approach avoids the discontinuous collapse issue. I'm not trying to be argumentative, but I just don't see it.

 

[bhobba]

I don't understand how the ensemble approach avoids the discontinuous collapse issue. I'm not trying to be argumentative, but I just don't see it.

Its dead simple.

The interpretation assumes an observation selects an element from the conceptual ensemble. This is the sole purpose of the state in that interpretation. Nothing physical changed - the state simply refers to a conceptualization that with the observable determines the proportion of the outcomes in the conceptual ensemble.

To spell it out in excruciating detail given an observable and a state you can calculate the probabilities of the possible outcomes of the observation. This determines an ensemble of outcomes where the proportion of each outcome is the probability of that outcome. The interpretation assumes the observation simply picks a random element of the ensemble and that's the result. Since it all refers to just a conceptualization nothing physical changed.

To be even clearer apply it to throwing a coin. Its state is the vector 1/2, 1/2. Throw the coin and it picks a random entry from the ensemble that is half heads and half tales. The new state is now 0,1 or 1,0 depending if a head or tale came up. The state discontinuously changed - but so what - its just a conceptualization - an aid to figuring out the likelihood of an observation outcome.

Thanks
Bill

 

[stevendaryl]

Its dead simple.

The interpretation assumes an observation selects an element from the conceptual ensemble.

That makes perfect sense for classical ensembles. You have a collection of systems that agree on the macroscopic variables (say, number of particles, or total energy, or something). But the details of how particles are moving differs from system to system. When you measure some quantity that varies from one system to another, nothing changes, you're just discovering which system (or sub-ensemble) is the "real" world.

You could try the same tactic with quantum nondeterminism: The quantity that you are measuring--angular momentum, for example--doesn't have a definite value before the measurement, simply because all you know is that the real world is one system out of an ensemble, and different members of the ensemble have different values for that observable. After the measurement, you haven't done anything other than identify which system (or sub-ensemble) is the real world.

But to assume that the system had a definite value for angular momentum before you measured it is a hidden-variables assumption, isn't it? Why don't Bell-type inequalities rule that out?

 

[stevendaryl]

But to assume that the system had a definite value for angular momentum before you measured it is a hidden-variables assumption, isn't it? Why don't Bell-type inequalities rule that out?

One could assume that a quantum system has definite values for all variables at all times, and the only reason for nondeterminism is classical ignorance. One way to frame the results of the various mathematical no-go theorem (Bell's theorem, the Kochen-Specker theorem, etc.) is that if observables have definite values, then our ignorance about those values cannot be described using measurable sets.

 

[Jilang]

The question, why Born's rule holds true and why the description of nature on a fundamental level is indeterministic is not asked in the realm of physics. You may wonder about it and try to find a simpler or more intuitive set of postulates defining quantum theory (e.g., Weinberg discusses at length, whether Born's postulate can be derived from the other postulates, i.e., the usual kinematical and dynamical postulates in terms of the Hilbert-space formulation with observable operators and state operators, coming to the conclusion that it cannot be derived), but as long as there is no empirical evidence against quantum theory, you better keep this theory.

This got me thinking... If such a question is not asked in the realm of physics in what realm should it be asked? I would not have thought that the philosophers would have the maths, the mathematicians probably not the inclination...

I didn't realize there was any mystery about Born's postulate. Isn't it just the joint probability of something coming one way meeting something coming the other way in imaginary time?

 

[zonde]

Got it now.

You face the discontinuous collapse issue if you think the state applies to an individual system and is in some sense real - that's the key point both Einstein and Ballentine didn't make clear in their objection.

I would say that this quote clarifies Einstein's point:
"For if the statistical quantum theory does not pretend to describe the individual system (and its development in time) completely, it appears unavoidable to look elsewhere for a complete description of the individual system; in doing so it would be clear from the very beginning that the elements of such a description are not contained within the conceptual scheme of the statistical quantum theory." - http://www.marxists.org/reference/archive/einstein/works/1940s/reply.htm

I would say that basically the point is that details (or interpretation) of collapse is outside the scope of QM and in statistical interpretation we speak only about relative frequencies without going into details.

Well apart from that it looks very much like non-contextual (or intrinsic to particle) LHV approach as he speaks about complete description of the individual system as a "complete" version of quantum theory.

 

[bhobba]

But to assume that the system had a definite value for angular momentum before you measured it is a hidden-variables assumption, isn't it? Why don't Bell-type inequalities rule that out?

This is the Achilles Heel of the ensemble interpretation - its an ensemble of system and observational apparatus combined. Nothing is assumed about the value of any observable prior to observation.

Ballentine in his 1970 paper on it more or less stated he was assuming some kind of hidden variable so it was an ensemble of outcomes - but his book moved away from that.

This is the reason I hold to the ignorance ensemble interpretation with decoherence - you don't need this unnatural assumption.

Thanks
Bill

 

[bhobba]

One could assume that a quantum system has definite values for all variables at all times,

You run into problems with Kochen-Specker. The only way to do it is hidden variables.

You can also assume it after decoherence - which is the essence of the ignorance ensemble interpretation with decoherence.

Thanks
Bill

 

[bhobba]

This got me thinking... If such a question is not asked in the realm of physics in what realm should it be asked? I would not have thought that the philosophers would have the maths, the mathematicians probably not the inclination...

It can be asked in physics - the problem is exactly how meaningful is it without some experiment to decide on it. Vanhees obviously thinks its not a particularly meaningful thing because of it - but opinions vary. Personally I agree with him - but opinions are like bums - everyone has one - it doesn't make it correct.

There are philosophers around like David Wallice with the necessary background, having both a Phd in physics and philosophy, to address such issues, and they do. For example see his book the Emergent Multiverse I have a copy of:
http://www.amazon.com/dp/0199546967/?tag=pfamazon01-20

Of course that is the exception rather than the rule - to be blunt many philosophers comments about QM leave a lot to be desired.

Thanks
Bill

 

[bhobba]

I didn't realize there was any mystery about Born's postulate. Isn't it just the joint probability of something coming one way meeting something coming the other way in imaginary time?

I don't know what you mean by this.

There is no controversy about it per-se - its part of the formalism and just about all physicists/mathematicians accept it.

The issue is just how much does it depend on the other assumptions. We have Gleason's theorem and its variants that actually derive it. If there was no other assumption involved hidden variable theories would be kaput. But careful analysis shows there is an assumption - non contextuality - ie the probability doesn't depend on the basis. That's an almost trivial requirement mathematically in a theory with vector spaces - but physically its not quite so clear.

Thanks
Bill

 

[Nugatory]

I guess it's meant to be that way with all interpretations - you must decide which confusion is less confusing for the worldview you hold.

Even that might be too strong of a commitment to an interpretation. I find myself sometimes choosing an interpretation that "works" for the problem at hand, and dropping it just as quickly when another problem comes along.

 

[Maui]

Even that might be too strong of a commitment to an interpretation. I find myself sometimes choosing an interpretation that "works" for the problem at hand, and dropping it just as quickly when another problem comes along.

I deleted the original comment as I intended to write a more detailed post(so as not to be misunderstood) but have to attend to other things in the meantime and will get back to it.

 

[Jilang]

Mathematically what causes the collapse Is the application of a boundary condition in time. Prior to that you have an equation with lots of solutions.

 

[Jilang]

I don't know what you mean by this.

There is no controversy about it per-se - its part of the formalism and just about all physicists/mathematicians accept it.

....but physically its not quite so clear.

Thanks
Bill

The wave function evolves in imaginary time and is a probability distribution in imaginary time. It is just for historical reasons and perhaps unfortunate that we call "i" imaginary. (See Hawkins comments on this). I suppose it less of a mouthful than "something at right angles to". Consider an interaction between two particles described by wavefunctions a and b. the probability of the interaction is <a|b> Which is the joint probability of finding them at the same place at the same imaginary time. If there is a phase difference between any of the components they will be orthogonal and not at the time imaginary time and the result is zero for that component. You can think of it as all playing out on a circle which helps a bit. Real time spreading outwards, imaginary time around the circle.

 

[bhobba]

The wave function evolves in imaginary time and is a probability distribution in imaginary time.

If what you are talking about is Wick rotation then yes that's true ie its a Wiener process when you do that.

But its got nothing to do with Born's rule or the origin of probability.

The reason its true was sorted out by Feynman yonks ago - only by allowing complex numbers can phase cancellation occur on most paths leaving those of stationary action.

There is also another difference - a Wiener process gives the probably of a particular path - in QM all paths are taken simultaneously.

And yes it's mathematically well known so called imaginary numbers are no more imaginary or not imaginary than say real numbers.

Thanks
Bill

 

[kye]

Sorry to butt in but how does Ballentine Ensemble interpretation view superpositions? I'm asking because I'm wondering if Quantum Computer concept of Qbit still work if Ballentine Ensemble interpretation were true. Remember superposition in quantum computers work in real time (the particle is in all basis simultaneously and not separately like in Ensemble interpretation).

 

[bhobba]

Sorry to butt in but how does Ballentine Ensemble interpretation view superpositions?

Basically its the bog standard QM formalism with the frequentest interpretation of Born's rule stitched on.

The principle of superposition holds exactly the same - the state simply applies to ensembles for the purpose of observations - that's all. It only comes into play during observations.

Thanks
Bill

 

[stevendaryl]

If what you are talking about is Wick rotation then yes that's true ie its a Wiener process when you do that.

But its got nothing to do with Born's rule or the origin of probability.

The reason its true was sorted out by Feynman yonks ago - only by allowing complex numbers can phase cancellation occur on most paths leaving those of stationary action.

There is also another difference - a Wiener process gives the probably of a particular path - in QM all paths are taken simultaneously.

Mathematically, the Wiener path integral and the Feynman path integral seem very analogous: the first sums over all paths to get a probability, the other sums over all paths to get a probability amplitude. I don't see immediately why the second implies that "all paths are taken" more than the first.

I don't have a good intuition as to whether the similarity of the two indicates something profound, or is just a red herring. What's sort of interesting is that if you allow paths that go back and forth in time, then

The probability of going from A at time t_1 to B at time t_2 is equal (by the Born rule) to the probability amplitude of going from A to B and back to A.

 

[bhobba]

I don't have a good intuition as to whether the similarity of the two indicates something profound, or is just a red herring.

Its VERY VERY profound - at least I think it is anyway - but that doesn't mean its a mystery - we know very well what's going on.

Mathematically its very important because there are technical difficulties defining a Feynman integral rigorously. However there is a generalization of a Wiener process called a Hida distribution and by Wick rotation can be used to define the Feynman integral.

Thanks
Bill

 

[stevendaryl]

Its VERY VERY profound - at least I think it is anyway - but that doesn't mean its a mystery - we know very well what's going on.

Mathematically its very important because there are technical difficulties defining a Feynman integral rigorously. However there is a generalization of a Wiener process called a Hida distribution and by Wick rotation can be used to define the Feynman integral.

Thanks
Bill

I didn't just mean that the two are mathematically related--clearly they are. I was wondering whether the relationship between the Wiener integral (or Hida distribution--I never heard of that before) and the Feynman path integral is a clue about the nature of quantum mechanics. I don't know what kind of clue--maybe that we live in the analytic continuation of a classical world?

 

[Jilang]

If what you are talking about is Wick rotation then yes that's true ie its a Wiener process when you do that.

But its got nothing to do with Born's rule or the origin of probability.

The reason its true was sorted out by Feynman yonks ago - only by allowing complex numbers can phase cancellation occur on most paths leaving those of stationary action.

There is also another difference - a Wiener process gives the probably of a particular path - in QM all paths are taken simultaneously.

And yes it's mathematically well known so called imaginary numbers are no more imaginary or not imaginary than say real numbers.

Thanks
Bill

Thanks very much for this. I had never heard of a Wiener process before today and it's exactly the word I needed (as entering "random walks" in Google has not proved particularly fruitful!). The Schroedinger equation looks very much a diffusion equation operating in imaginary time.

 

[Jilang]

Mathematically, the Wiener path integral and the Feynman path integral seem very analogous: the first sums over all paths to get a probability, the other sums over all paths to get a probability amplitude. I don't see immediately why the second implies that "all paths are taken" more than the first.

I don't have a good intuition as to whether the similarity of the two indicates something profound, or is just a red herring. What's sort of interesting is that if you allow paths that go back and forth in time, then

The probability of going from A at time t_1 to B at time t_2 is equal (by the Born rule) to the probability amplitude of going from A to B and back to A.

Thanks for this, it's really wonderful! I don't have such good maths, but I had a feeling this should be true. A Wick rotation of time would produce a space-type dimension (consider the metric) maybe that explains the similarity. So quantum mechanics could be described as random walks in imaginary time? If I ever win the lottery and get to write a book that's what I'll call it!

 

[Jilang]

If what you are talking about is Wick rotation then yes that's true ie its a Wiener process when you do that.
...
There is also another difference - a Wiener process gives the probably of a particular path - in QM all paths are taken simultaneously.

If the Wiener process was in imaginary time though all paths would be simultaneous (at the same radius on the circle of time) wouldn't they?

 

[Jilang]

The probability of going from A at time t_1 to B at time t_2 is equal (by the Born rule) to the probability amplitude of going from A to B and back to A.

If you have time could you expand on this a bit more. I'm very interested in the Born postulate and would love to have a better understanding of it. As it's defined it looks like a joint probability to me rather than a probability of a single entity. The similarity in its form to probability of transitions between the initial and final states and interactions has an implication that I'm trying to understand.

 

[bhobba]

I didn't just mean that the two are mathematically related--clearly they are. I was wondering whether the relationship between the Wiener integral (or Hida distribution--I never heard of that before) and the Feynman path integral is a clue about the nature of quantum mechanics. I don't know what kind of clue--maybe that we live in the analytic continuation of a classical world?

In that case I agree - what it tells us about the nature of QM is unclear.

Thanks
Bill

 

[bhobba]

If the Wiener process was in imaginary time though all paths would be simultaneous (at the same radius on the circle of time) wouldn't they?

You have totally lost me. You obviously have some kind of intuition about imaginary time beyond me.

Thanks
Bill

 

[bhobba]

I want to add with regard to the Ensemble interpretation the bible on it is Ballentine's superb book - QM - A Modern Development.

The CORRECT view of ensembles in that interpretation is found on page 46 (emphasis mine):

'However it is important to remember this ensemble is the CONCEPTUAL infinite set of all such systems that may potentially result from the state preparation procedure, and not a concrete set of systems that co-exist in space'

The only thing I will add is I do not view it as infinite, because my mathematics background has issues with such things, merely so large the law of large numbers applies giving an ensemble with proportion of outcomes the same as probability. And to avoid issues with the property being there prior to observation the ensemble is of system and observational apparatus combined - although in Ballentine's text its pretty obvious that's what he is talking about since it refers to the usual system preparation, transformation, then measurement one often finds in such discussions.

Thanks
Bill

 

[zonde]

The only thing I will add is I do not view it as infinite, because my mathematics background has issues with such things, merely so large the law of large numbers applies giving an ensemble with proportion of outcomes the same as probability. And to avoid issues with the property being there prior to observation the ensemble is of system and observational apparatus combined - although in Ballentine's text its pretty obvious that's what he is talking about since it refers to the usual system preparation, transformation, then measurement one often finds in such discussions.

I certainly agree that observational apparatus should be included into the system. But ...
then it would seem that you have to include preparation apparatus too ... and manipulation apparatus. And we end up at the same thing that Sugdub was saying earlier in discussion that the state is property of the whole experimental setup.

And yet another thing. If we include observational apparatus into the system then individual systems include the same observational apparatus (yet at different times and in different states) and are not really separate.