Mathematically what causes wavefunction collapse?

[bhobba]

I was actually referring to a human conscious observer, but that isn't to say that the conscious observer is the only entity that can make a record of a quantum observation. They must interact as part of a classical system with the quantum system on the terms of quantum mechanics.

Then I don't understand why you wish to introduce consciousness at all.

One of the first books on QM I studied was Von Neumann's classic because my background is math and not physics - I mathematically couldn't quite grasp Dirac's treatment, however being grounded in the Hilbert space formalism I learned in my undergrad studies I cottoned onto Von Neumann fairly well. I know why he introduced consciousness - the Von Neumann cut could be placed anywhere and if you trace it back the only place different was consciousness. But things have moved on considerably since then and we now understand decoherence a lot better - and that looks the best place to put the cut - in fact it gives the APPEARANCE of collapse. Von Neumann didn't live long enough for this development, but the other high priest of it, Wigner, did. When he learned of some early work on decoherence by Zurek he did a complete 180% turn and believed collapse was an actual physical process that occurred out there.

Now I don't necessarily agree with that because for me the appearance of collapse is good enough - and decoherence explains that - but for the life of me I can't see why anyone these days wants to introduce consciousnesses into it at all.

Thanks
Bill

 

[craigi]

Then I don't understand why you wish to introduce consciousness at all.

One of the first books on QM I studied was Von Neumann's classic because my background is math and not physics - I mathematically couldn't quite grasp Dirac's treatment, however being grounded in the Hilbert space formalism I learned in my undergrad studies I cottoned onto Von Neumann fairly well. I know why he introduced consciousness - the Von Neumann cut could be placed anywhere and if you trace it back the only place different was consciousness. But things have moved on considerably since then and we now understand decoherence a lot better - and that looks the best place to put the cut - in fact it gives the APPEARANCE of collapse. Von Neumann didn't live long enough for this development, but the other high priest of it, Wigner, did. When he learned of some early work on decoherence by Zurek he did a complete 180% turn and believed collapse was an actual physical process that occurred out there.

Now I don't necessarily agree with that because for me the appearance of collapse is good enough - and decoherence explains that - but for the life of me I can't see why anyone these days wants to introduce consciousnesses into it at all.

Thanks
Bill

The reason that the topic of consciousness arose was really with respect to the experimentalist's inherent inability to isolate themselves from the quantum system under observation and how this seems to lead to a more uncomfortable understanding than emerges from other experiments. Whenever such isolation does exist it must be brought to an end in order to take a result.

I don't subscribe to "consciousness causes collapse" arguments. Though I do think that when we search for an ontological description of the universe we should be careful not to discount the role of consciousness.

It's relevant to both definitions of realism from physics and psychology, to which forms of ontological descriptions we find most appealing, to our preconceptions of time and causality and to anthropic bias.

For a functional description of quantum mechanics, I'd agree that it's unlikley to play a role.

 

[bhobba]

The reason that the topic of consciousness arose was really with respect to the experimentalist's inherent inability to isolate themselves from the quantum system under observation and how this seems to lead to a more uncomfortable understanding than emerges from other experiments. Whenever isolation does exist it must be brought to an end in order to take a result.

Cant quite follow that one.

My understanding, for example, is in the hunt for particles like the Higgs experimentalists used computers to sieve through the mountains of data - the experimentalist didn't seem too involved with it at all. An when a candidate was found it was only then they looked at it - way after the experiment was done.

From my perspective this poses a lot of problems with involving the experimentalist, and consciousness, in it at all.

I want to add none of this violates consciousness being involved - its like sophism in that nothing can really disprove it - but, especially with modern technology such as computers, leads to an ever increasingly weird view of the world.

Thanks
Bill

 

[Sugdub]

Well it is shared by Ballentine and the many others that hold to the Ensemble interpretation such as myself, and certain variants of Copenhagen as well... Its like when you throw a dice - its state has gone from a state vector with 1/6 in each entry to one where its 1 in a entry - ...

Obviously you didn't catch my point. As I've explained, the “measurement problem” could not arise if physicists would remain factual, I mean defining the state vector as a mathematical representation for a property of the experiment (NOT a property of a “system” in the world) and reporting that this property (and therefore the associated state vector) evolves in a continuous (respectively discontinuous) way in response to a continuous (respectively discontinuous) change of the experimental set-up (NOT in response to a change of the so-called “state of the system”). Should physicists adopt “simply a codification of the results of possible observations”, there would be no such thing as the “state of the system” and the state vector would not evolve across space and neither across time.

However, all interpretations of the quantum theory (including the Copenhagen interpretation) add a first postulate on top of the experimental facts reported above (on top of the simple “codification of the results of possible observations”) whereby the state vector represents equally a property of “something” of the world, namely a property of the “system” being observed or measured by the experiment. It is this double definition of the state vector (a property of an experiment and a property of a physical “system” involved in the experiment and therefore localised in space and time) which makes the “measurement problem” to arise. Because the evolution of the property of the experiment, which according to the first definition is real but does take place in a configuration space which should not be confused with space-time, is then assumed to also trace an evolution of the property of the “system” according to the second definition, whereas the latter can only occur somewhere inside the experimental set-up, during the experiment i.e. in space-time. So the redundant definition of the state vector contained in the first postulate leads to a contradiction concerning the nature of the manifold in which the state vector evolves.

 

[vanhees71]

Of course, the state vector (or better said the state, i.e., the statistical operator $\hat{R}$) used to represent the state is a property of the system, namely the way to describe our knowledge about the system based on an established preparation procedure the system has undergone. The only knowledge we have about the system is probabilistic, according to Born's rule. I.e., if we measure an observable $A$ exactly the possible outcome of the measurement is a value in the spectrum of the associated self-adjoint operator $\hat{A}$. Let's denote $|a,\beta \rangle$ an arbitrary orthonormal (generalized) basis of the corresponding (generalized) eigenspace, where $\beta$ is a label (consisting of one or more further real parameters). Then the probability to find the value $a$ when measuring the observable $A$ is
$$P(a|\hat{R})=\sum_{\beta} \langle a,\beta|\hat{R}|a,\beta \rangle.$$
Given the Hamiltonian you can evaluate how the description of the system changes with time in terms of the Statistical operator $\hat{R}(t)$ and observable operators $\hat{A}(t)$. The corresponding time dependences of these objects are determined up to a unitary time-dependent transformation of state and observable operators, which can be chosen arbitrarily without changing the outcome of physical properties (probabilities, expectation values, etc.).

Nowhere have I made the assumption that the state operators are more than a description of our knowledge about the system, given the (equivalence class of) preparation procedures on the system. Indeed, using this strictly physical meaning of the abstract formalism of quantum theory there is no necessity for a state collapse or a measurement problem.

The only thing you must assume is that there are measurement devices for the observable you want to measure, which allow to determine values of observables and store them irreversibly for a sufficient time so that I can read off these values. Experience shows that such devices exist in practice, e.g., to measure the position, momentum, angular momentum, etc. of single particles or other (even sometimes macroscopic) systems showing quantum behavior. There's nothing mysterious with quantum theory in this point of view.

For some people, among them famous scientists like Einstein, Planck and Schrödinger, this view is unacceptable, because they insist on what they call "realism", i.e., that the abstract elements of the theory are in one-to-one correspondence with physical properties of the system (e.g., the position and momentum vectors of a small "pointlike" body in classical mechanics, denoting a deterministic reality of the location and velocity of this body). Within quantum theory such a view is hard to maintain, as Bell's theorem shows (except one gives up locality, which attempts to my knowledge however so far has not lead to consistent theories about the physical world).

 

[bhobba]

I mean defining the state vector as a mathematical representation for a property of the experiment (NOT a property of a “system” in the world) and reporting that this property (and therefore the associated state vector) evolves in a continuous (respectively discontinuous) way in response to a continuous (respectively discontinuous) change of the experimental set-up (NOT in response to a change of the so-called “state of the system”). Should physicists adopt “simply a codification of the results of possible observations”, there would be no such thing as the “state of the system” and the state vector would not evolve across space and neither across time.

I don't think you quite grasp just how much this is not a 'definitional' thing but to a large extent is forced on us, not just by experiment, which of course is the ultimate justification for any theory, but by considerations of a pretty fundamental and basic nature.

I posted this before - but will post it again:
http://arxiv.org/pdf/quant-ph/0101012.pdf

The evolution thing, while not usually presented this way, but Ballentine is one source that that breaks the mould, is in fact forced on us by symmetries and a very fundamental theorem called Wigners theorem:
http://en.wikipedia.org/wiki/Wigner's_theorem

However, all interpretations of the quantum theory (including the Copenhagen interpretation) add a first postulate on top of the experimental facts reported above (on top of the simple “codification of the results of possible observations”) whereby the state vector represents equally a property of “something” of the world, namely a property of the “system” being observed or measured by the experiment.

I have zero idea where you got that from but its simply not true.

What the state is is very interpretation dependent. Some like the Ensemble interpretation (as espoused by Ballentine in his standard textbook mentioned previously) use it to describe the statistical properties of a conceptual ensemble of systems and observational apparatus while for others like many worlds its very real indeed.

Of course in the Ensemble interpretation its a 'property' just like the probabilities assigned to the faces of a dice is a property of the dice - but it doesn't exist out there in a real sense like say an electric field does.

If you disagree then simply get a copy of Ballentine.

Thanks
Bill

 

[craigi]

I'm not convinced that we can avoid interpretational issues by sticking to the mathematical formalism, since the formalism deals with probability and probability itself is open to interpetation.

See frequentist and Bayesian interpretations of probability.

There's a nice brainteaser here that illustrates how different interpretations of probability give different results.
http://www.behind-the-enemy-lines.com/2008/01/are-you-bayesian-or-frequentist-or.html

 

[Sugdub]

...Nowhere have I made the assumption that the state operators are more than a description of our knowledge about the system, ...

Clear, but you have made the assumption that the knowledge gained is about a "system", about "something in the world", and this is precisely what I challenge. Until you catch the distinction between the knowledge gained about the properties of an experiment and the knowledge gained about the properties of a so-called "system" hypothetically involved in the experiment, I can't see how we could understand each other. My statement is that the measurement problem is a direct consequence of your belief whereby the knowledge formalised in the state vector is about a "system" in the world. Thanks.

 

[bhobba]

I'm not convinced that we can avoid interpretational issues by sticking to the mathematical formalism

I don't think that's the point Vanhees is making - I think the point is interpretations beyond the bare minimum are not required.

And indeed one of the fundamental differences between the Ensemble and most of the versions of Copenhagen is the Ensemble views the state as describing a conceptual ensemble of systems and observational apparatus (ie is a variant of the frequentest view of probably) and Copenhagen views the state as describing a single system, but it represents a subjective level of confidence about the results of observations - ie is related to the Baysean view.

There could even be a third view, but I have never seen it presented, that its an even more abstract thing with the probabilities of the Born Rule being interpreted via the Kolmogorov axioms - that would include both views - but more work such as connecting it to frequencies of outcomes via the law of large number would be required. From a theoretical viewpoint it may have advantages in showing Copenhagen and the Ensemble interpretation are really the same thing.

My background is in applied math and most applied mathematicians tend to favor looking at probabilities from the frequentest perspective, but to avoid circularity its based on the Kolmogerov axioms and connected by the law of large numbers.

Thanks
Bill

 

[bhobba]

There's a nice brainteaser here that illustrates how different interpretations of probability give different results.

I am neither - I am a Kolmogorovian :rofl::rofl::rofl::rofl:

Seriously though, at its foundations, and when considering fundamental issues regarding probability, its best to view probability as something very abstract defined via the Kolmogorov axioms. That way both the frequentest and Baysean view are seen as really variants of the same thing. When viewed in that light Copenhagen and the Ensemble interpretation are not necessarily that different.

Thanks
Bill

 

[bhobba]

My statement is that the measurement problem is a direct consequence of your belief whereby the knowledge formalised in the state vector is about a "system" in the world.

I can't quite grasp the point you are making.

Maybe you can detail what you think the measurement problem is.

My view is as detailed by Schlosshauer in the reference given previously. It has 3 parts:

(1) The problem of the preferred basis.
(2) The problem of the non-observability of interference.
(3) The problem of outcomes - ie why do we get any outcomes at all.

These days it is known that decoherence solves (2) for sure, quite likely solves (1) but more work needs to be done - the real issue is (3) - why we get any outcomes at all - that is very interpretation dependent.

The Ensemble interpretation addresses it by simply assuming a measurement selects an outcome from a conceptual ensemble. It is this conceptualization a state describes - its not describing anything real out there - but rather a conceptualization to aid in describing the outcome of measurement, observations etc etc. While a property of the system, it is not applicable to a single system which is one of the key characteristics of that interpretation.

Thanks
Bill

 

[zonde]

Nowhere have I made the assumption that the state operators are more than a description of our knowledge about the system, given the (equivalence class of) preparation procedures on the system.

Don't know but surely this part looks very strange:

Given the Hamiltonian you can evaluate how the description of the system changes with time in terms of the Statistical operator $\hat{R}(t)$ and observable operators $\hat{A}(t)$.

Our knowledge about the system changes with time?

 

[Sugdub]

I can't quite grasp the point you are making. Maybe you can detail what you think the measurement problem is.

I think there is a very large gap between statements made by physicists and what is actually backed-up by their experiments.
It is experimentally true that the information flow produced by some so-called “quantum experiments” can be qualified, statistically, by a measured distribution which results from counting discrete events, and formalised mathematically into the orientation of a unit vector in a multi-dimensions manifold (the state vector). Whether the state vector established through running the experiment in an iterative way can be projected as a property of each iteration taken individually is a dogma, not an experimental fact. Whether the property assigned to one single iteration (the state vector) can in turn be projected as the property of a subset of the experimental device (the so-called “preparation”) and then “measured” by the remaining of the device (the so-called “measurement apparatus”) is a second dogma, not an experimental fact. Whether the state vector assigned as a property of the preparation of a unique iteration can in turn be projected as a property of a so-called physical “system” presumably lying or moving through the device during the experiment is a third dogma, not an experimental fact. Finally, the assumption whereby the production of a qualitative information (e.g. that a particular “detector” has been activated) constitutes the outcome of a “measurement” reveals a misconception of what a measurement delivers: a quantitative information.
What I have explained earlier in this thread is that the only way to properly eliminate any form of the “measurement problem” is to reject all dogmas and misconception pointed above and to stick to experimental facts. Then, the continuous and the non-continuous evolutions of the state vector, defined as a property of a quantum experiment, won't cause any trouble. The consequence of this approach is that the quantum theory does not deal any longer with what happens in the world.

 

[zonde]

There is such a thing as Heisenberg picture and in this picture you still have connection with reality but it might be much closer to Sugdub's viewpoint (and have the benefits of that viewpoint).

 

[stevendaryl]

I'm not convinced that we can avoid interpretational issues by sticking to the mathematical formalism, since the formalism deals with probability and probability itself is open to interpetation.

See frequentist and Bayesian interpretations of probability.

There's a nice brainteaser here that illustrates how different interpretations of probability give different results.
http://www.behind-the-enemy-lines.com/2008/01/are-you-bayesian-or-frequentist-or.html

The computation there seems like a very complicated way to get to the point. It seems to me that the same point is made much simpler with a smaller number of coin-flips:

What if you flip a coin twice, and get "heads" both times? What's the probability that the next flip will result in "heads"?

It seems that the author of that article would say the answer is 100%, if you are a frequentist, because you estimate that the probability of an event is equal to the relative frequency of that event's occurrence so far.

In contrast, the Bayesian probability is more complicated to compute. It's something like, letting $p$ be the unknown probability of "heads":

$P(H | data) = \int dp P(H|p) P(p|data) = \int dp P(H| p) P(data|p) P(p)/P(data)$

where $P(p)$ is the prior probability distribution for $p$, and
$P(data) = \int dp P(data|p) P(p)$, and where $data$ means the fact that the first two flips resulted in "heads". If we use a completely uninformative flat distribution for $p$, then $P(p) = 1$, $P(data | p) = p^2$, $P(data) = \int dp P(data|p) = \int p^2 dp = 1/3$. So

$P(H|data) = \int dp p \cdot p^2 \cdot 1/\frac{1}{3} = 3 \int p^3 dp = 3/4$

So the Bayesian probability is 3/4, not 1.

With a very small number of flips, it's clearer that nobody would believe the frequentist prediction; just because a coin produced heads-up twice in a row doesn't mean it'll produce heads-up three times in a row. When the number of flips gets very large, the frequentist predictions gets more sensible, but also, the difference between frequentist and Bayesian predictions diminishes.

 

[Jilang]

There are several contenders that "explain" wave function collapse, but the one I lean towards is the Many Worlds Interpretation. That said the usual version leaves a lot to be desired in that it requires infinite dimensions (in state space) and I am far happier with the introduction of another time dimension. The way to think about his is that there are many universes all at a different angle to each other. See if this makes any sense to you. Not my idea, but it's a good one!
http://arxiv.org/pdf/quant-ph/9902037v3.pdf

 

[craigi]

With a very small number of flips, it's clearer that nobody would believe the frequentist prediction; just because a coin produced heads-up twice in a row doesn't mean it'll produce heads-up three times in a row. When the number of flips gets very large, the frequentist predictions gets more sensible, but also, the difference between frequentist and Bayesian predictions diminishes.

This encapsulates the reason that I posted it pretty well.

Until we can be clear about whether a probability represents a property of an object or if it represents a subject's knowledge of a system and we have an explanation for what the hypothetical, or even real, infinite population that we're sampling actually is, then we can't hope to avoid other inerpretational issues when applying the formalism to the real world.

 

[bhobba]

Whether the state vector established through running the experiment in an iterative way can be projected as a property of each iteration taken individually is a dogma, not an experimental fact.

I would point out the same could be said about flipping a coin and assigning probabilities to it. In modern times probabilities is defined by the Kolmogorov axioms which is an abstract property assigned to an event (in your terminology iteration).

One then shows, via the law of large numbers, that is mathematically provable as a theorem from those axioms (plus a few reasonableness assumptions of the sort used in applied math all the time, but no need to go into that here) that for all practical purposes, if its done enough times the proportion of an event will equal the probability. This is the view taken by the Ensemble interpretation and what the state applies to - a conceptualization of a large number of iterations, events etc such that the proportion is the probability predicted by the Borne rule. When one makes an observation, in that interpretation, its selecting an element from that ensemble and wave-function collapse, in applying only to this conceptual ensemble, and nothing in any sense real, is of no concern at all.

It is a fundamental assumption of the theory that such is possible, but like heaps of stuff in physics usually not explicitly stated - it is assumed by merely mentioning probabilities in the Born rule such is understood. Its like when one defines acceleration as the derivative of velocity you are implicitly assuming the second derivative of position exists.

There is another view of probability that associates this abstract thing, probability, as defined in the Kolmogorov axioms, with a subjective confidence in something. This is the Bayesian view and is usually expressed via the so called Cox axioms - which are equivalent to the Kolmogorov axioms. This view leads to an interpretation along the lines of Copenhagen which takes the state as a fundamental property of an individual system, but gives a subjective confidence instead.

But we also have a very interesting theorem called Gleason's theorem. What this theorem shows, is if you want to associate a number between 0 and 1 on elements of a Hilbert space, and do it in a mathematically consistent way that respects the basis independence of those elements, then the only way to do it is via the Born rule. The reason this theorem is not usually used to justify the Born rule is the physical significance of that mathematical assumption is an issue - its tied up with what's called contextuality - but no need to go into that here - the point is there is quite a strong reason to believe the only reasonable way to assign probabilities to quantum events is via the Born rule. Oh and I forgot to mention it can be shown the Born Rule obeys the Kolmogorov axioms - that proof is not usually given because its assumed when you say gives the probability in an axiom you are assuming it does, but Ballentine, for example, is careful enough to show it.

The bottom line here is that physicists didn't pull this stuff out of a hat - its more or less forced on them by the Hilbert space formalism.

Thanks
Bill

 

[bhobba]

With a very small number of flips, it's clearer that nobody would believe the frequentist prediction; just because a coin produced heads-up twice in a row doesn't mean it'll produce heads-up three times in a row. When the number of flips gets very large, the frequentist predictions gets more sensible, but also, the difference between frequentist and Bayesian predictions diminishes.

I think if you go even further back to the Kolmogorov axioms you would not fall into any of this in the first place.

The frequentest view requires a very large number for the law of large numbers to apply - the exact number depending on what value in the convergence in probability you want to accept as for all practical purposes being zero eg you could use the Chebyshev inequality to figure out a suitable number to give a sufficiently low probability. Still it's is a very bad view for carrying out experiments to estimate probabilities. The Bayesian view is much better for that because you update your confidence as you go - you simply keep doing it until you have a confidence you are happy with. However for other things the frequentest view is better - you choose whatever view suits the circumstances knowing they both derive from its real justification - the Kolmogorov axioms.

I think its Ross in his book on probability models that points out regardless of what view you subscribe to its very important to learn how to think probabilistically, and that usually entails thinking in terms of what applies best to a particular situation.

But its good to know the real basis for both is the Kolmogorov axioms and Baysean and frequentest are really just different realizations of those axioms.

Thanks
Bill

 

[bhobba]

Until we can be clear about whether a probability represents a property of an object or if it represents a subject's knowledge of a system and we have an explanation for what the hypothetical, or even real, infinite population that we're sampling actually is, then we can't hope to avoid other inerpretational issues when applying the formalism to the real world.

I say it represents neither - it represents a number that obeys the Kolmogorov axioms. Both the Baysian and frequentest approaches are simply different realizations of those axioms. You choose the view that suits the circumstances.

If you want to use the frequentest view in QM then you are led to something like the Ensemble interpretation.

If you want the Bayesisan view you are led to Copenhagen.

In the MWI the Bayesian view seems to work best because the 'probability' represents a confidence you will find yourself in a particular world - viewing it in a random way like throwing a dice doesn't sit well with a deterministic theory.

I think Consistent Histories views it Bayesian

Thanks
Bill

 

[bhobba]

In contrast, the Bayesian probability is more complicated to compute. It's something like, letting $p$ be the unknown probability of "heads":

If I remember correctly, and its ages since I studied Baysian statistics, what you usually do is assign it some resonsonable starting probability such as for a coin 1/2 and a 1/2 then you carry out experiments to update this probability until you get it at a confidence level you are happy with.

There is something in the back of my mind from my mathematical statistics classes attended 30 years ago now that this converges quicker than using stuff like the Chebychev inequality to estimate the number of trials to get a reasonable confidence level - but don't hold me to it.

But in QM we have this wonderful Gleason's Theorem that if you want a probability that respects the formalism of vector spaces whose properties are not dependent on a particular basis then the Born Rule is the only way to do it.

Of course that assumption may not be true - but you really have to ask yourself why use a Hilbert space formalism in the first place if it isn't.

Thanks
Bill

 

[zonde]

Whether the state vector established through running the experiment in an iterative way can be projected as a property of each iteration taken individually is a dogma, not an experimental fact.

This is the view taken by the Ensemble interpretation and what the state applies to - a conceptualization of a large number of iterations, events etc such that the proportion is the probability predicted by the Borne rule. When one makes an observation, in that interpretation, its selecting an element from that ensemble and wave-function collapse, in applying only to this conceptual ensemble, and nothing in any sense real, is of no concern at all.

Hmm, it seems there is more than one Ensemble interpretation out there:
Einstein said: "The attempt to conceive the quantum-theoretical description as the complete description of the individual systems leads to unnatural theoretical interpretations, which become immediately unnecessary if one accepts the interpretation that the description refers to ensembles of systems and not to individual systems."

But we also have a very interesting theorem called Gleason's theorem. What this theorem shows, is if you want to associate a number between 0 and 1 on elements of a Hilbert space, and do it in a mathematically consistent way that respects the basis independence of those elements, then the only way to do it is via the Born rule.

Gleason's theorem does not say what these numbers mean physically, right? But Born rule says that these numbers are probabilities.

 

[bhobba]

Hmm, it seems there is more than one Ensemble interpretation out there:
Einstein said: "The attempt to conceive the quantum-theoretical description as the complete description of the individual systems leads to unnatural theoretical interpretations, which become immediately unnecessary if one accepts the interpretation that the description refers to ensembles of systems and not to individual systems."

Like most interpretations there are a number of variants. The one Einstein adhered to is the one presented by Ballentine in his book and the usual one people mean when they talk about it. And indeed it refers to an ensemble of systems exactly as I have been saying in this tread about the state referring to an ensemble of similarly prepared systems - its the one more or less implied if you want to look on probability the frequentest way.

I hold to a slight variant however - called the ignorance ensemble interpretation that incorporates decoherence - check out the following for the detail:
http://philsci-archive.pitt.edu/5439/1/Decoherence_Essay_arXiv_version.pdf

Gleason's theorem does not say what these numbers mean physically, right? But Born rule says that these numbers are probabilities.

No it doesn't. But if you want to define a probability on the vector space and you want it not to depend on your choice of basis (this is the assumption of non-contextuality which in the Hilbert space formalism seems almost trivial - it actually took physicists like Bell to sort out exactly what was going on) it proves there is only one way to do it.

The assumption you make if you accept Gleason's theorem would go something like this - I don't know what outcome will occur but it seems reasonable I can associate some kind of probability to them. And if you do that then what the theorem shows is there is only one way to do it, namely via the Born Rule, and moreover that way obeys the Kolmogorov axioms. That is in fact a very innocuous assumption because all you are really doing is saying I can assume some kind reasonable confidence level can be associated with each outcome such as the Cox axioms. Or you believe if you do the observation enough times it will tend to a steady limit. But strictly speaking - yes its an assumption - however its so innocuous most would probably not grant it that status - I personally wouldn't.

Thanks
Bill

 

[stevendaryl]

If I remember correctly, and its ages since I studied Bayesian statistics, what you usually do is assign it some reasonable starting probability such as for a coin 1/2 and a 1/2 then you carry out experiments to update this probability until you get it at a confidence level you are happy with.

The way I have used Bayesian probability in the past (and I'm uncertain about the relationship between Bayesian probability and Bayesian statistics), what you are trying to do is to describe your situation in terms of parameters, and then use whatever data is available (including none!) to estimate the likelihood of the various possible values of those parameters.

So relative frequency only very indirectly comes into play. The probabilities are degrees of belief in the values of something, that something may not be a "random variable" at all--it might be a constant such as the mass of some new particle. Actually, that's usually the case, the parameters that you are dealing with are usually one of a kind things, not repeatable events. As for confidence intervals, I don't think those are as important in Bayesian probability as in frequentist. A probability is your confidence in the truth of some claim.

In general, you have some parametrized theory, and you're trying to figure out the values for the parameters.

The way that I would handle the problem of coin tosses would be to parameterize by a parameter $p$ (the probability of heads) that ranges from $0$ to $1$. This parameter, like any other unknown parameter, has a probability distribution for its possible values. Then you use the available data to refine that probability distribution.

So initially, you guess a flat distribution:

$P(p) = 1$ for the range $0 \leq p \leq 1$

According to this flat distribution for $p$, you can compute your prior estimate of the likelihood of heads:

$P(H) = \int dp P(p) \cdot P(H | p) = \int dp \ 1 \cdot p = 1/2$

So I come to the same conclusion, that the likelihood of getting "heads" based on no data at all, is 1/2. But it's not that I guessed that--that's computed based on the guess that the parameter $p$ has a flat distribution in the range $[0,1]$.

 

[Sugdub]

I would point out the same could be said about flipping a coin and assigning probabilities to it. In modern times probabilities is defined by the Kolmogorov axioms which is an abstract property assigned to an event (in your terminology iteration).

There are two aspects which require some attention.
First, one must clarify the rationale for assigning a probability (which is a form of property) to a discrete occurrence of an event-type, better than assigning this probability to the event-type representing one category of events that may be observed when running the experiment. In the first case the probability is a property of the unique iteration of the experiment which produced the discrete information, but in the second case it is a property of the iterative implementation of the experiment. What I said in my previous input is that the second case formalises what is experimentally true, whereas the first one stems from a dogma which can be accepted or rejected. I do think that the second approach, which is minimal because it endeavours relying exclusively on experimental truth and what can logically be derived from it, should be used as a reference whenever other approaches based on non-verifiable hypotheses lead to paradoxes.

Second, assuming the minimal approach is followed, there might be no compelling need for referring to “probabilities”. The “state vector”, more exactly the orientation of a unit vector, represents an objective property of a quantum experiment run in an iterative way (i.e. the distribution of discrete events over a set of event-types). The quantum formalism transforms the orientation of a unit vector into another orientation of the same unit vector. The new orientation computed by the quantum formalism relates to the objective property of a modified experiment (the distribution pattern remaining over the same set of event-types) or a combination of such experiments, still assuming an iterative run of that set-up. It should be noted that in a manifold the orientation of a unit vector (i.e. a list of cosines) is the canonical representation for a distribution. Hence the choice of a vectorial representation for the quantum theory implies that the formalism will manipulate/transform a set of cosines (the so-called "amplitudes of probability") instead of their squared values which account for relative frequencies. (I'm not aware of any alternative / simple explanation for this peculiar feature of the quantum formalism often presented as a mystery, but I'd be keen to learn about them). Eventually references to the “probability” concept, and more significantly to the "amplitude of probability" mis-concept can be dropped since the former only stands for "relative frequency observed in the iterative mode" whereas the latter has lost any physical significance according to the proposed approach.

This is the view taken by the Ensemble interpretation and what the state applies to - a conceptualization of a large number of iterations, events etc such that the proportion is the probability predicted by the Borne rule. When one makes an observation, in that interpretation, its selecting an element from that ensemble and wave-function collapse, in applying only to this conceptual ensemble, and nothing in any sense real, is of no concern at all.

I'm sorry I don't understand this last sentence, in particular what you say about the link between the occurrence of an event and the collapse of the wave function. What I said is that a non-continuous modification of the experimental device is likely to translate into a non-continuous evolution of the observed distribution for the new device as compared to the initial distribution. There is no such thing as a collapse of the wave-function triggered or induced by the occurrence of a discrete event. The so-called wave function is a property of an experiment, not a property of a “system” and neither a state of our knowledge or belief.

There is another view of probability that associates this abstract thing, probability, as defined in the Kolmogorov axioms, with a subjective confidence in something. This is the Bayesian view and is usually expressed via the so called Cox axioms - which are equivalent to the Kolmogorov axioms. This view leads to an interpretation along the lines of Copenhagen which takes the state as a fundamental property of an individual system, but gives a subjective confidence instead

I don't think the formalism (Kolmogorov, Bayes, ...) determines whether the probability should be interpreted as a belief, as some knowledge about what may happen or as an objective state. Only the correspondence you explicitly establish between what you are dealing with and the mathematical objects involved in the probability formalism defines what the probability you compute deals with.
In the minimal approach I recommend to follow, the “probability” refers to an objective property of a quantum experiment, and it actually means “relative frequency observed in the iterative mode”.
Thanks.

 

[bhobba]

There are two aspects which require some attention.
First, one must clarify the rationale for assigning a probability (which is a form of property) to a discrete occurrence of an event-type, better than assigning this probability to the event-type representing one category of events that may be observed when running the experiment.

I have zero idea what you are trying to say. Being able to assign probabilities to events is pretty basic and if it was in anyway not valid great swaths of applied mathematics from actuarial science to statistical mechanics would be in trouble - but they obviously arent.

I'm sorry I don't understand this last sentence, in particular what you say about the link between the occurrence of an event and the collapse of the wave function.

Its very simple:
http://en.wikipedia.org/wiki/Ensemble_interpretation

Thanks
Bill

 

[zonde]

Like most interpretations there are a number of variants. The one Einstein adhered to is the one presented by Ballentine in his book and the usual one people mean when they talk about it. And indeed it refers to an ensemble of systems exactly as I have been saying in this tread about the state referring to an ensemble of similarly prepared systems - its the one more or less implied if you want to look on probability the frequentest way.

If you say that measurement outcome is described by probability you say that the rule applies to individual event (relative frequencies emerge from statistical ensemble of idependent events). So you contradict what Einstein was saying.
You have to allow possibility that relative frequencies appear as certainty by deterministic physical process. And then it's Ensemble interpretation.

The assumption you make if you accept Gleason's theorem would go something like this - I don't know what outcome will occur but it seems reasonable I can associate some kind of probability to them.

I assume that assigning probability to outcome might lead to false predictions.

 

[audioloop]

Hi all, I was wondering mathematically ,what causes wave function collapse? and why does it exist in all it's Eigen states before measurement? Thanks for any help and please correct my question if I have anything wrong.

math is just description.


.

 

[craigi]

If you say that measurement outcome is described by probability you say that the rule applies to individual event (relative frequencies emerge from statistical ensemble of idependent events). So you contradict what Einstein was saying.
You have to allow possibility that relative frequencies appear as certainty by deterministic physical process. And then it's Ensemble interpretation.

Einstein wasn't saying that an ensemble is required, only that if we interpret QM as a desription of ensembles rather than individual events we avoid "unnatural" interpretations.

In my opinion, the term unnatural seems to have been used in order to make the statement correct, but also makes it completely subjective. For it to be objective he would've actually had to define what he means by unnatural and if I recall correctly this was effectively an expression of his frustration with indeterminism. He was asserting his own prejudices on nature. It would've been written from a faith position in local realist hidden variable theories. Which we now know to be invalid if we require counterfactual definiteness.

 

[Jilang]

You have to allow possibility that relative frequencies appear as certainty by deterministic physical process.

I disagree with this. If everything were determined by physics process how would you explain something like a decay rate for an atom or particle. These events have a probability but are inherently random or appear to be so.

 

[Sugdub]

I have zero idea what you are trying to say. Being able to assign probabilities to events is pretty basic and if it was in anyway not valid great swaths of applied mathematics from actuarial science to statistical mechanics would be in trouble - but they obviously arent.

I had a look to the Ensemble interpretation article you referred to and I must admit I found it anything but clear. The first section displays a quote by Einstein (reproduced in this thread in #57 by Zonde). I would be extremely surprised if in the original context Einstein used the word “system” in a different meaning than a “microscopic object”, I mean something less precise but in the same range as a “particle”. May be somebody could clarify this point.

In the second section of the same article, the “system” is defined as a single run of a quantum experiment, whereas an ensemble-system is defined as an iterative run of that experiment. That looks pretty similar to what I described in my previous inputs, although the use that is made of the word “system” makes the text quite harsh to digest. But then the key sentence according to which one should understand if and why the ensemble interpretation assumes that the wave-function is a property of one single iteration reads as follows:
“The ensemble interpretation may well be applied to a single system or particle, and predict what is the probability that that single system will have for a value of one of its properties, on repeated measurements”.
If “system” stands for “a single iteration of the experiment”, then the sentence actually assigns the “property” to the “repeated measurements” pattern, the ensemble-system, and not to a single run. If “systems” stands for a “microscopic system” (if not, the wording “system or particle” is irrational), then the sentence does not tell whether the property is assigned to a single run or not. The sentence does not include any justification anyway.
Further on an example is presented where a pair of dice, i.e. a physical object involved in the experimental device, plays the role of the so-called “system”. The ambiguity is maximal.

Let's make things simple. If one admits that the probabilistic property assigned to the iterative experiment reflects an underlying probabilistic property assigned to a more elementary level (the single iteration), then there is no reason why this second probabilistic property should not in turn reflect a third probabilistic property standing another level below, whatever the form it takes. This leads to a regression ad infinitum which can only stop when one specifies a level to which a deterministic property can be assigned. So the only realistic and credible alternative to stating that the property at the level of a single run is deterministic (which all physicists assume in the case of classical probabilities) is to accept that there is no property at all at this elementary level, so that the distribution pattern observed at the iterative level is a fundamental property which cannot be reduced to the appearance or synthesis of a more fundamental property.
I've explained in my previous input why and how the quantum formalism actually deals with transforming a distribution of relative frequencies into another distribution of the same nature, thanks to an appropriate mathematical representation using the orientation of a unit vector which makes the “amplitude of probability” an empty physical concept. The quantum formalism deals with a probabilistic property defined at the iterative level, reflecting the experimental truth.
Should there be a more fundamental property at a lower level, whichever level that means, then the quantum formalism would no longer be considered as the most fundamental theory dealing with quantum experiments. It would have to be replaced with a theory explicitly dealing with the lowest level property, and that property would necessarily be deterministic.

 

[bhobba]

If you say that measurement outcome is described by probability you say that the rule applies to individual event (relative frequencies emerge from statistical ensemble of idependent events). So you contradict what Einstein was saying.

That's simply not true.

It purely depends on your interpretation of probability. In the ensemble interpretation an observation selects an outcome from the conceptual ensemble and what that outcome is can only be described probabilistically.

In most versions of Copenhagen the state applies to an individual system, but is purely a representation of subjective knowledge about the outcome of observations.

Ballentine, correctly, in his book, points out, as Einstein did, the difficulty that arises if you consider it applies to something more definite that an ensemble (the collapse issue is the problem), but for some reason didn't consider the case where is was simply subjective knowledge, which is what most versions of Copenhagen think of the state as.

I assume that assigning probability to outcome might lead to false predictions.

But it doesn't.

Thanks
Bill

 

[bhobba]

Einstein wasn't saying that an ensemble is required, only that if we interpret QM as a desription of ensembles rather than individual events we avoid "unnatural" interpretations.

Exactly what Einstein was getting at is explained in Ballentine's book.

But basically its the collapse issue. The ensemble interpretation is one way out, considering it purely as a state of knowledge is another.

Also note, and it bears mentioning, Einstein did NOT disagree with QM as you will sometimes read - he considered it incomplete - not incorrect.

Thanks
Bill

 

[craigi]

I disagree with this. If everything were determined by physics process how would you explain something like a decay rate for an atom or particle. These events have a probability but are inherently random or appear to be so.

This would actually be pretty easy to construct a viable deterministic hidden variable theory for. Where they have problems, is when we consider separated entangled particles ans contexuality.

Classical systems that are considered fundamentally deterministic exhibit appararent randomness. In fact, a system that is fundamentally indeterministic can appear deterministic and vice versa.

Einstein believed that apparent indeterminism was fundamentally deterministic. I think that perhaps a better way to look at it, is how does determinism emerge so convincingly from indeterminism, in our experiences, that the human mind considers it to be so fundamental. There are indeterminstic processes taking place all around us on all scales, all the time, but we are much more atuned to the deterministic processes.

 

[bhobba]

I had a look to the Ensemble interpretation article you referred to and I must admit I found it anything but clear. The first section displays a quote by Einstein (reproduced in this thread in #57 by Zonde). I would be extremely surprised if in the original context Einstein used the word “system” in a different meaning than a “microscopic object”, I mean something less precise but in the same range as a “particle”. May be somebody could clarify this point.

In discussions about QM one often encounters an analysis of a typical measurement situation consisting of preparation, transformation, then measurement.

See figure 1 in the following for a discussion:
http://arxiv.org/pdf/quant-ph/0101012.pdf

Thanks
Bill