Question on the "probabilistic" nature of QM

yoda jedi

yes, I have read previously on zurech attempts, but i think the last solution it will come from a more wider theory, a nonlinear one like trace dynamics or an epistemic ontic model.


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ThomasT

That view is somewhat puzzling to me. Perhaps you might post in the What's Your Philosophy of Mathematics? thread?

yoda jedi

yes, but macroreality is no indefinite.
maybe modal quantum theory with definite values is the answer.
or a nonlinear quantum mechanics destroying the superposition.


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zonde

Hmm, I do not see why there should be any reluctance to accept such indefiniteness.

Well, this part is rather unclear. First, are you redefinig "probability" without giving new definition or what?
And second, how does it helps to resolve the mystery if you don't interpret the probabilities as ignorance of the system?

Hmm, maybe disagreement is actually about the mystery to be solved.
For example, as I see the mystery is not in indefiniteness but that this indefiniteness is carrying some amount of amorphous definiteness and about particular properties of this amorphous definiteness. And I don't see how your arguments are getting closer to resolution of this mystery.

ThomasT

It's not clear to me what you mean by indefiniteness. If you just mean that exactly what one perceives is relative to one's viewpoint, then ok. But physical science has to do with publicly discernible/countable phenomena. In what way might these phenomena of public historical record be considered indefinite?

Of course, in any probabilistic formulation prior to observation there are encoded multiple possibilities given specific probability values. But then an experiment is done and a qualitative result recorded. Is there any sense in which a qualitative result should be considered indefinite?

Jano L.

These are very good points, I almost agree with you. The only difficulty is that deterministic equation of evolution is not enough to call the theory deterministic. One also needs quantity that fully describes physical state of the system, with no reference to probability.

However, the function Psi describes the state in a probabilistic way. I do not think there is a way to understand Psi as a specification of a physical state. The only thing we know about it is that it gives probabilities.

It seems that if one wants to have deterministic theory, one also has to introduce additional quantitities.

Hurkyl

Consider first ordinary classical mechanics. We have a phase space X representing all possible states of some system under study, and an observable like the "x-coordinate of the third particle" is a function on X: to each element of the phase space it assigns a real number. An ordinary interpretation of ordinal classical mechanics would imply that the 'state of reality' corresponds to some point in X, and questions like "What is the x coordinate of the third particle?" make sense as questions about reality and have precise, definite answers.

Unfortunately, due to engineering concerns, we don't have sufficient knowledge and precision to actually answer such questions. So we layer a new theory on top of classical mechanics to try and model our ignorance. And if we look over all of the questions we say "Z happens with 75% probability", and it turns out we said that 100,000 times and roughly 75,000 of those were correct, we're content with our model -- both of reality and of our ignorance.


Now, consider a variation on classical mechanics where phase space is not X, but instead the class of open subsets of X. In our interpretation, we do not say that the 'state of reality' corresponds to a point of X, but instead to an open subset of X. Questions like "What is the x coordinate of the third particle" no longer have definite answers, because the state of reality is some open subset U of X, and the value of our observable varies on the domain U.

So now assertions like "the x coordinate of the third particle is 3" still make sense as assertions about reality, but they do not necessarily have definite true/false values. Instead, they can also take on some 'intermediate' values between true and false. It might make more sense to think of it as being partially true and partially false. In fact, while the question above can definitely false, it can never be definitely true. A question like "the x coordinate of the third particle is between 3 and 4" could be definitely true, though.


Another variation is rather than phase space being open subsets of X, they are probability distributions on X, in the sense of Kolmogorov. Now, the physical quantity "What is the x coordinate of the third particle?" again makes sense. But instead of being a (definite) real number, the answer to this question is a random variable (again, in the sense of Kolmogorov). Again, let me emphasize that, in the interpretation of this variant, the physical state of the system is a probability distribution, and physical quantities are random variables.

These last two variations are what I mean by indefiniteness.


We can, of course, still layer ignorance probabilities on top of this. So we could be expressing our ignorance about the state of reality as being a probability distribution across the points of phase space -- that is, a probability distribution of probability distributions.

Mathematically, of course, we can simplify and collapse it all down into one giant probability distribution, and it looks the same as if we just had the original, definite classical mechanics with ignorance probabilities on top.

And "forgetting" about the physical distribution works out well because the dynamics of classical mechanics works "pointwise" on X, without influence from the physical probability distribution, so if we pretend the physical distribution is just ignorance probabilities, we never run into a paradox where the dynamics of the system appear to be influenced by our 'information' about it.



Before continuing further, the reader needs to understand that the third variant of classical mechanics I mentioned above, the physical probability distribution really is part of the physical state space of the theory. It is not a combined "classical mechanics + ignorance" amalgamation: it is a theory that posits the state of reality really is a probability distribution across X.



Now, if we turn to quantum mechanics, and decoherence in particular. The promising lead of decoherence is that if we apply unitary evolution to a large system and restrict our attention to the behavior of a subsystem, the state of the subsystem decoheres into something that can FAPP be described as a probability distribution across outcomes.

But the important thing to notice is that this probability distribution is an aspsect of the physical state space. It is not ignorance probability, it is part of the physical state of the system as posited by "Hilbert space + unitary evolution" (or similar notion).

But unlike the classical case, the dynamics do depend on the full state of the system. And we really do observe this in physical experiments.

The classic "Alice and Bob have an entangled pair of qubits" thought experiment, for example. Because of the entanglement, Alice's particle has decohered into a fully mixed state: mixture of 50% spin up and 50% spin down around whichever axis she chooses to measure. Any experiment performed entirely in her laboratory will respect this mixture. But when Alice and Bob compare their measurements, the full state of the system reasserts itself in showing a correlation between their measurements.


In the third version of classical mechanics I described above, we can layer ignorance probabilities on top, then forget the difference between physical and ignorance probability -- in other words, replacing the decohered state with a collapsed state + ignorance about which state it collapsed to.

But forgetting the difference fails badly for quantum mechanics, because the dynamics do depend on the full state, and so if we're being forgetful, we do run into all sorts of issues where the physical evolution of a system appears to depend on our knowledge of the system.

the_pulp

Thanks for replying! I was really looking for if I was totally wrong or if you see some light. Let me tell you how I see it. For me, Psi is something that is useful to label states. Then, by knowing the symetries that the states should follow under space time rotations, you have what is the Hilbert operator representation of P, X, H and so on (and as a consequence its eigenstates).
Until here there are no probabilities, no Born Rule, nothing. Then we can interact with a system by the use of an instrument in a way that perhaps we can deterministically make it leave its, for example, P eigenstate and go to a predefined, for example, X eigenstate. Now again, there are no probabilities.
Finally we can make a game. We are going to arrange a complex interaction with an instrument, that makes the system go to an eigenstate of X and that has zillons of particles and we dont know the initial state of it. There is randomness, and probabilities, from the point of view of the scientist (but if theres a god, he would know the exact state of the instrument and he would know where the system will finnish). This is called "experiment" because it looks as if the scientist is discovering a property of the system (while he is not, the system does not have this "X" property. He is the one that makes it go to an eigenstate of it). Gleasons theorems (or Saunders) assures that if there are going to be probabilities and this probabilities depend only on the state of the system (here it is like this from the point of view of the scientist), then these should be calculated by the Born Rule.
So, in this interpretation, the Born Rule is not an axiom, just a theorem.

Finally I just want to say that Im obviously not sure of what Im saying. It is just the way I like things to be in order to make sense to me. Ive been reading these stuff for the last 4 years and only now I found an interpretation that suits with my senses! Again, I have not found it in Wikipedia or anything, so I really appreciate your answers, views etc.

So... What do you think?

Thanks!

Naty1

I liked post #67 regarding the OP's question....

".... our great models have been like shadows, that fit some projection of reality but are later found to not be the reality..."

we ARE dealing with a mathematical model. It's as far as we have gotten to date.

The original question involved:

I just posted these explanations on uncertainty in another thread and they may afford the OP a perspective regarding the representation [model] of what 'is actually happening' regarding 'probability'. These are a bit more basic than the Hilbert spaces and associated mathematical representations of the last half dozen or so posts.

Here are some explanations I saved [and edited] from a very, very long discussion in these forums on Heisenberg uncertainty:

Course Lecture Notes, Dr. Donald Luttermoser,East Tennessee State University:


[QUOTE] The HUP strikes at the heart of classical physics: the trajectory. Obviously, if we cannot know the position and momentum of a particle at t[0] we cannot specify the initial conditions of the particle and hence cannot calculate the trajectory.......Due to quantum mechanics probabilistic nature, only statistical information about aggregates of identical systems can be obtained. QM can tell us nothing about the behavior of individual systems. ….QUOTE]

what the quotes means: unlike classical physics, quantum physics means future predications of state [like position, momentum] are NOT precise.]

But why does this situation pop up in quantum mechanics? because of our mathematical representations.

[scattering, mentioned below, happens to be a convenient example of our limited ability to make measurements of arbitrary precision. ]


The basic ideas are these: HUP [Heinsenberg uncertainty principle] is a result of nature, not of experimental based uncertainties. From the axioms of QM and the math that is used to build observables and states of systems, it turns out that position and velocity (and also momentum) are examples of what are called "canonical conjugates" [a function and its Fourier transform]; They cannot be both be "sharply localized". That is, they cannot be measured to an arbitrary level of precision. It is a mathematical fact that any function and its Fourier transform cannot both be made sharp.

The wave function describes not a single scattering particle but an ensemble of similarly accelerated particles. Physical systems [like particles] which have been subjected to the same state preparation will be similar in some of their properties but not all of them. The physical implication of the uncertainty principle is that NO STATE PREPARATION PROCEDURE IS POSSIBLE WHICH WOULD YIELD AN ENSEMBLE OF SYSTEMS IDENTICAL IN ALL OF THEIR OBSERVABLE PROPERTIES.

A few cornerstone mathematical underpinnings:
The wave function describes an ensemble of similarly prepared particles rather than a single scattering particle. A wave function with a well defined wavelength must have a large special extension, and conversely a wave function which is localized in a small region of space must be a Fourier synthesis of components with a wide range of wavelengths. We cannot measure them both to an arbitrary level of precision. A function and its Fourier transform cannot both be made sharp. This is a purely a mathematical fact and so has nothing to do with our ability to do experiments or our present-day technology. As long as QM is based on the present mathematical theory an arbitrary level of precision cannot be achieved.


The HUP isn't about the knowledge of the conjugate observables of a single particle in a single measurement. that the uncertainty theorem is about the statistical distribution of the results of future measurements. The theorem doesn't say anything about whether you can measure both at the same time. That is a separate issue.


A single scattering experiment consists of shooting a single particle at a target and measuring its angle of scatter. Quantum theory does not deal with such an experiment but rather with the statistical distribution of the results of an ensemble of similar results. Quantum theory predicts the statistical frequencies of the various angles through which a an ensemble of similarly prepared particles may be scattered.

What we can't do is to prepare a state such that we would be able to make an accurate prediction about what the result of a position measurement would be, and an accurate prediction about what the result of a momentum measurement would be.

Physical systems which have been subjected to the same state preparation will be similar in some of their properties but not all of them. The physical implication of the uncertainty principle is that NO STATE PREPARATION PROCEDURE IS POSSIBLE WHICH WOULD YIELD AN ENSEMBLE OF SYSTEMS IDENTICAL IN ALL OF THEIR OBSERVABLE PROPERTIES.

Ken G

Well, thank you bhobba and ThomasT, and I certainly agree with the implication that the value is not as much in the answer each of us arrives at, as it is in the tensions between the possible answers, around just what is this amazing relationship between us, our environment, and our mathematics that tries to connect the two. I suspect answers like this will continue to be moving targets, as much as what physics is will itself continue to be a moving target. It is not just physics theories that change, but physics itself. Although we may fall into thinking that the modern version is what physics "is", that doesn't seem to do justice to the remarkable evolutionary resilience of this animal. To me, its most remarkable attribute is the way the accuracy and generality of its predictions converges steadily, yet the ingenious ontologies it invokes to achieve this convergence of accuracy do not converge at all.

Jazzdude

Sorry for warming up this thread again.

In the context of the question this thread I have been asked mutliple times in here, what my reasons are making certain statements about decoherence, many worlds or generally how quantum theory should be interpreted. I found that it is hard to answer all these questions here, but they motivated me enough to start a blog where I can describe what I think is the best approach to the problem of interpretation. Namely replacing interpretation with scientific deduction, that describes observation as an emergent phenomenon. And no, this is not many worlds. The content is partly based on a research paper that is currently in the publication pipeline.

Since the resulting discussion could become speculative at some points and I want to respect the forum rules, I have moved the blog elsewhere. If you have comments and would like to discuss you are invited to do it there. In any case I am looking forward to your feedback.

http://aquantumoftheory.wordpress.com

Cheers,

Jazz

audioloop

nice to know.

Zmunkz

But don't overlook the fact that Quantum mechanics, and any future theory of motion we devise, needs to reduce to Newtonian mechanics when applied to the same domain in which Newtonian physics was established (QM of course does).

My point is any future development we might discover that supersedes QM will itself need to reduce to a purely random formulation when addressing the domain of particle interactions and states in which we know QM randomness to be accurate. I have a hard time imagining how any deterministic formulation could, in any reduced case, produce the purely random results needed in the quantum domain (it's a contradiction of terms in fact). It therefore seem to me a fair conclusion that reality is, at bottom -- fundamentally if you prefer -- probabilistic. And I think we must concede that even while admitting QM is unlikely to be the final say in our understanding of particles.

(Sorry, I know I am quoting a post from early in this thread, but it caught my eye)

bhobba

Hi Jazz

Interesting stuff.

I have my own view based on the primacy of observables and their basis invariance - I must get around to writing it up one day. Although its bog standard QM it is an approach I haven't seen anyone else use. Its a more sophisticated version of Victor Stengers view:
http://www.colorado.edu/philosophy/v...g/SuperPos.htm

Thanks
Bill

bhobba

I to think that at rock bottom reality is fundamentally probabilistic. QM may indeed be superseded but I am not so sure that such is likely.

Thanks
Bill

Ken G

I would argue that reality cannot be probabilistic at rock bottom, but neither does it appear to be deterministic. The error is in thinking it has to be one or the other-- that's not the case, our models have to be one or the other, reality can just be whatever it is.

The reason it can't be probabilistic is that probabilistic theories are, almost by definition, not rock-bottom theories (probabilities reflect some process or information that is omitted on purpose, and probabilities are generated as placekeepers for what is omitted-- that's just what they are whenever we understand what we are actually doing). Hence, probability treatments are theories of what you are not treating, much more than they are theories of what you are treating. But if one rules out probabilistic theories from the status of "rock bottom" descriptions, one might imagine that all that is left is a deterministic description, but that's even worse-- there's no evidence that any physical description is exactly deterministic, determinism was always an effective concept in every application where it was ever used in practice. So probabilistic descriptions always have a "untreated underbelly", if you like, whereas deterministic descriptions are always effective at only an approximate level in all applications where they are used.

These are just factual statements about every example we can actually point at where we know what is going on, so why should we ever think they will not be true of some "final theory" that treats the "rock bottom" of reality? A much more natural conclusion seems to be that Bohr was right-- physics is what we can say about nature, and never was, nor ever should have been, a "rock bottom" description. We just don't get such a thing, we get determinations of probabilities, and that is all physics is intended to do.

As for "rock bottom" reality outside of what physics is intended to do, that is a fundamentally imprecise concept at best. Physics is all about creating a language that lets us talk about reality, so there is no such thing as a reality outside of physics that we could ever try to talk about intelligibly in a physics forum. Terms like "probabilistic" or "deterministic" are mathematical physics terms-- they have no meaning outside that context.

Zmunkz

You've outlined an interesting way of looking at this. Could you possibly elaborate on the above quotation? I'm trying to understand why by definition probabilistic theories cannot be foundational. I can see in the macro sense (something like flipping a coin for instance) probabilities are a stand-ins for actual non-probabilistic phenomenon... but I can't quite convince myself this analogy carries to everything. Could you maybe add a little on this?

This is the classic realist vs. instrumentalist debate. Looks like you fall on the instrumentalist side -- I am not sure if I can meet you there, although you make the case well.