Mathematically what causes wavefunction collapse?

[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.

 

[Nugatory]

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.

Nope - each element of the ensemble includes its own measuring apparatus. You could think of it as an ensemble of laboratories, all prepared through the same procedure to conduct the same experiment.

 

[bhobba]

Nope - each element of the ensemble includes its own measuring apparatus. You could think of it as an ensemble of laboratories, all prepared throught the same procedure to conduct the same experiment.

Exactly. But its a slight blemish you have to do this.

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.

Yea - that's an issue with that interpretation - not much of an issue IMHO (I agree with Nugatory) - but an issue.

That's one reason (there are others - but it is one - if it was only this issue I probably wouldn't worry about it) why I hold to the ignorance ensemble interpretation with decoherence.

Not that Ballentine agrees that decoherence is of any value interpretatively - he doesn't - but I respectfully disagree with him on that point.

Thanks
Bill

 

[OCR]

(as entering "random walks" in Google has not proved particularly fruitful!).

That's odd...

https://www.google.com/#q=Random+walk

It seems to be the first "hit", then merely a jaunt to...

http://en.wikipedia.org/wiki/Random_walk

Then to...

http://en.wikipedia.org/wiki/Category:Variants_of_random_walks

We then arrive to view the... you guessed it...

http://en.wikipedia.org/wiki/Wiener_process

And, look at all the processes at the bottom of the page ... wow!



OCR...

 

[Sugdub]

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...

May be it is wise to also relate this to my last input as #88 in the other thread related to Ballentine's ensemble-interpretation:

(in a slightly different context, ...)... I would agree with your statement if the word “systems” was replaced with a proper concept. Paraphrasing one of your previous inputs I would say: "It is undeniable that the state vector can and should be thought of as a representation of the statistical property of an iterative run of a uniquely prepared experiment which delivers, at each run, one amongst a set of possible outcomes". Stating that the statistical property relates to the flow of qualitative pieces of information produced by an experiment is the only true minimal position that cannot be challenged. Stating that the distribution relates to some physical system or stating that each individual piece of information relates to an individual physical system, that already goes beyond the bare minimum since it cannot be proven experimentally...
Thanks

 

[Superposed_Cat]

imaginary time?

 

[Jilang]

That's odd...

https://www.google.com/#q=Random+walk

It seems to be the first "hit", then merely a jaunt to...

http://en.wikipedia.org/wiki/Random_walk

Then to...

http://en.wikipedia.org/wiki/Category:Variants_of_random_walks

We then arrive to view the... you guessed it...

http://en.wikipedia.org/wiki/Wiener_process

And, look at all the processes at the bottom of the page ... wow!



OCR...

Ha Ha, that's really funny - that Wiki Page is one of my bestest favourites!
What I should have said "not particulary fruitful when looking for published articles pertaining to QM"
Thanks for pointing all out the processes though - I'd forgotten just how much it applies to! Interesting that they don't seem to mention Quantum Mechanics though...even the Wiener part only talk about fluid dynamics.

 

[Jilang]

imaginary time?

Cat, It sounds like science fiction doesn't it? Hawking and physicist James B. Hartle have applied the concept of imaginary time in their research on the origin of the universe, including their efforts to develop a unified theory derived from Einstein’s theory of relativity and from Richard Feynman’s concept of multiple possible histories of the universe.

 

[stevendaryl]

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.

This is just fooling around with symbols, but...

The probability amplitude to go from state $| A\rangle$ at time $t$ to state $|B\rangle$ at time $t + \delta t$ is given by:

$\langle A | e^{-i H \delta t/\hbar} | B \rangle$

If we assume that this formula works when $\delta t < 0$, then the probability amplitude for going from state $| B \rangle$ at time $t + \delta t$ to state $|A\rangle$ at time $t$ is given by:

$\langle B | e^{+i H \delta t/\hbar} | A \rangle$

So the amplitude for going from $| A\rangle$ to $|B\rangle$ and back in time to $| A\rangle$ would be the product:

$\langle A | e^{-i H \delta t/\hbar} | B \rangle\langle B | e^{+i H \delta t/\hbar} | A \rangle = |\langle A | e^{-i H \delta t/\hbar} | B \rangle|^2$

which is the Born expression for the probability of going from $| A\rangle$ to $|B\rangle$.

So, mathematically, the probability of going from $| A\rangle$ to $|B\rangle$ is the probability amplitude of making a "round-trip" back to the starting point (and starting time).

 

[Jilang]

So the amplitude for going from $| A\rangle$ to $|B\rangle$ and back in time to $| A\rangle$ would be the product:

$\langle A | e^{-i H \delta t/\hbar} | B \rangle\langle B | e^{+i H \delta t/\hbar} | A \rangle = |\langle A | e^{-i H \delta t/\hbar} | B \rangle|^2$

which is the Born expression for the probability of going from $| A\rangle$ to $|B\rangle$.

So, mathematically, the probability of going from $| A\rangle$ to $|B\rangle$ is the probability amplitude of making a "round-trip" back to the starting point (and starting time).

Thanks for this. I can see what you mean. It's given me a lot to think about. I think Swinger introduced circles in time in 1960. I wonder what determines whether a particle travels clockwise or anti-clockwise. Logically it would need to go both ways so phase factors would cancel I think.

 

[zonde]

Nope - each element of the ensemble includes its own measuring apparatus. You could think of it as an ensemble of laboratories, all prepared through the same procedure to conduct the same experiment.

Can you elaborate what is your statement?
It's your belief? Or do you mean that it's Ballentine's interpretation? Or maybe you think it's experimentally verified fact?