Defining Probability in Quantum Physics

[madhatter106]

I've noticed the use of a fairly innocuous word 'probability' being used in quantum physics and wonder if the actual meaning of the word is being implied vs the intent as it pertains to quantum physics, when used by students and professors ?

This may seem pedantic however it's easier to relate to the normal use of 'probability' then the mind twister of QM's version.

The correct intent and meaning behind that word draws a line in the sand many are not willing to cross, why?

 

[Feldoh]

What do you mean. What is your understanding of the "actual" meaning and "QM" meaning?

 

[madhatter106]

What do you mean. What is your understanding of the "actual" meaning and "QM" meaning?

Sure,
I take the std math definition as the possible set outcomes divided by the number of choices. I.e random

For QM probability is not the same as the possible outcomes are not random.

 

[Fredrik]

The predictions of QM come in the form of numbers in the closed interval [0,1]. The theory associates one such number p_R with each logically possible result R of a measurement, and claims that this p_R will be the relative frequency of that particular result in a long series of experiments on identically prepared systems. If you have performed an experiment N times, every result R will have been obtained a specific number of times N_R. The relative frequency of the result R is defined as f_R=N_R/N.

The theory is considered a good theory if |f_R-p_R| is found to be small for a wide range of results.

 

[madhatter106]

As I understand it the probability is that where you 'measure' is where you find it, it was never there but in all places before the measurement. The measurement is the probability, not the same as where is the marble between 3 cups, as the probability in that scenario changes when you look under a cup that the marble is not in.

I just think that the use of the word probability conveys to some a deterministic view when there isn't one.

 

[Feldoh]

As I understand it the probability is that where you 'measure' is where you find it, it was never there but in all places before the measurement. The measurement is the probability, not the same as where is the marble between 3 cups, as the probability in that scenario changes when you look under a cup that the marble is not in.

I just think that the use of the word probability conveys to some a deterministic view when there isn't one.

I still don't see a distinction.

In the case where a marble is (randomly) under one of three cups when you lift a cup you change the probability.

In a QM experiment you have a system and when you measure the system you change the probability.

In terms of probability both cases ask in the same way.


I see what you're trying to say but it has nothing to do with the definition of probability. What you're trying to describe with the marble case has an analogy in quantum mechanics. In quantum mechanics we call it a hidden-variable theory.

A hidden-variable theory essentially says that there is some way that nature determines the exact state of a system that we have just not found.

 

[Truecrimson]

Sure,
I take the std math definition as the possible set outcomes divided by the number of choices. I.e random

For QM probability is not the same as the possible outcomes are not random.

A uniform (random) distribution is only one special kind of probability distribution.
http://en.wikipedia.org/wiki/List_of_probability_distributions

 

[skippy1729]

Sure,
I take the std math definition as the possible set outcomes divided by the number of choices. I.e random

For QM probability is not the same as the possible outcomes are not random.

Another consistent rational definition is the subjective Bayesian: Probability ~ Degree of Belief. Its application to QM resolves many of the "paradoxes". qv http://arxiv.org/abs/1003.5209 or http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.77.8442&rep=rep1&type=pdf.

Skippy

 

[A. Neumaier]

I've noticed the use of a fairly innocuous word 'probability' being used in quantum physics and wonder if the actual meaning of the word is being implied vs the intent as it pertains to quantum physics

See Chapter A3: ''Probability'' of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html#A3

 

[Fredrik]

Another consistent rational definition is the subjective Bayesian: Probability ~ Degree of Belief. Its application to QM resolves many of the "paradoxes". qv http://arxiv.org/abs/1003.5209 or http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.77.8442&rep=rep1&type=pdf.

Skippy

I'm going to nitpick your choice of words a bit, and say that what you're describing is an interpretation of probability, not a definition.

I also have to say that I think the philosophy of probabilities is a bunch of nonsense. When someone makes a statement about what probabilities really are (or what anything "really is"), they're making a statement about reality, not about pure mathematics. This means that it's either science or pseudo-science. It's science if and only if it's a part of a scientific theory, and scientific theories must be at least statistically falsifiable, i.e. everything I said about QM in post #4 must hold for such a theory, and I think it's obvious that those things can't hold for a theory about what probabilities really are.

I guess I should clarify one thing. It's not the fact that people are thinking about these things that bother me. If they would think of these "interpretations" the same way we think of Venn diagrams, i.e. as tools that enable us to develop an intuition about a piece of mathematics, I wouldn't have a problem with it. What bothers me is that people seem to think that mathematicians or philosophers should try harder to find the correct answer to the question of "what probabilities really are". In my opinion, this indicates a fundamental misunderstanding. There's no way that this question can ever be answered.

 

[Fra]

I didn't get exactly which objection to OP had but I think calling the probability debate as nonsense is too strong and risk oversimplifying a difficult problem simply due to an in adequate classification system.

I agree this discussion is not a mathematical problem. There is nothing ambigous in the formal definition of probability. Mathematically probability in QM is no different than in classical physics. What's different is the logic and scheme by which the probabilites are computed. This is a different question, and I'm not sure if that was the original point or not. But once computed, the probability is what is always was.

I don't think this is what the deeper question is about.

The question is just wether and how these mathematical abstractions (or probability spaces, ensembles, computations etc) are the best mathematical tools for learning. And how to attach these abstractions to something physical or at hand. If you call this interpretation or something I'm not sure. But regardless how you classify it it's an important question that concerns also the science.

However I certainly agree that this question really doesn't belong in the QM section. I think it belongs to the BTSM section.

This is best exemplifed by looking at this statement

>and scientific theories must be at least statistically falsifiable

This implicitly makes use of the probability concept in order to define what a scientific theory is. Thus, if the probability concept is not well defined, operationally, neither is science.

I think everyone understand exactly how this still makes sense in practice, where there is no problem to repeat experiment, or where sufficiently large "samples" ARE realizable as approximations to the imaginary ensembles. But the point is that first there is no hard proof that any realizable approximation is good, and second sometimes the system we study are evolving so fast that not repetitive representative sampling of something well defined is possible.

Does this mean that such situations does not need to be understood, simply because our classification system doesn't view it as science?

I'm not answering this here, I'm just raising the question to illustrate the point:
For example: Does this mean that cosmology is not science? this certainly can be repeated. No anywhere sensible sampling of different universes is ever possible, in any way. yet there is NO poblem to picture this mathematically!

But does it make sense, if so how?

There are plenty of non-trivial questions here I just want to add that although these discussions tend to always contain plenty of confusion that may be nonsense, the general discussion isn't nonsense as there are some hard problems in here are are intellectually worthy just as much as anything that falls withing a simplistic view of "science" as a statistical description.

/Fredrik

 

[Fredrik]

>and scientific theories must be at least statistically falsifiable

This implicitly makes use of the probability concept in order to define what a scientific theory is. Thus, if the probability concept is not well defined, operationally, neither is science.

I described how probabilities are to be interpreted in this context in post #4. This should be considered part of the definition of "science". As you can see, I'm talking about relative frequencies. Philosophers who talk about "the relative frequency interpretation of probability" say that it only makes sense if the limit of N_R/N exists as N→∞. I say that this limit is completely irrelevant, mainly because there are much bigger obstacles to making science well-defined. The biggest problem by far is how to specify what sort of measuring devices should be used in the experiment. There's no way to make that well-defined, because the specification would have to consist of statements about real-world objects, not just mathematics, and only mathematical statements can be perfectly well-defined.

Does this mean that such situations does not need to be understood, simply because our classification system doesn't view it as science?

I don't have an answer that covers every type of situation that you can imagine, so I'll just say something really obvious: We don't need to understand things that can't be understood.

For example: Does this mean that cosmology is not science?

I don't think I want to get into a lengthy discussion about such things. I would probably have issues with some aspects of quantum cosmology, but I haven't thought it through, so I don't really know. For now, I'll just say that if a theory associates probability 1 with the possible result that very distant galaxies will be observed as redshifted, we clearly don't have to create a new universe to repeat the measurement. Just look again in a different telescope, or even the same one.

 

[skippy1729]

I'm going to nitpick your choice of words a bit, and say that what you're describing is an interpretation of probability, not a definition.

I also have to say that I think the philosophy of probabilities is a bunch of nonsense. When someone makes a statement about what probabilities really are (or what anything "really is"), they're making a statement about reality, not about pure mathematics. This means that it's either science or pseudo-science. It's science if and only if it's a part of a scientific theory, and scientific theories must be at least statistically falsifiable, i.e. everything I said about QM in post #4 must hold for such a theory, and I think it's obvious that those things can't hold for a theory about what probabilities really are.

I guess I should clarify one thing. It's not the fact that people are thinking about these things that bother me. If they would think of these "interpretations" the same way we think of Venn diagrams, i.e. as tools that enable us to develop an intuition about a piece of mathematics, I wouldn't have a problem with it. What bothers me is that people seem to think that mathematicians or philosophers should try harder to find the correct answer to the question of "what probabilities really are". In my opinion, this indicates a fundamental misunderstanding. There's no way that this question can ever be answered.

So, are you saying that a physical theory where probabilities defined as subjective degrees of belief are not statistically falsifiable and therefore not scientific theories?

 

[madhatter106]

I'll try and clarify. The use of probability is fine until it meets up with QM, because unlike all other uses of probabilities the understanding of a probability in QM is not the exact same thing.

I'm not getting into the philosophical because that is not answerable nor fitting here. I was wondering if the use of that specific word has caused a misunderstanding or confusion in the true nature of duality in QM? I'm not referring to just the lay person but also those who've studied physics and either starting into QM or didn't at all.

I hope that makes some sense.

 

[skippy1729]

I'll try and clarify. The use of probability is fine until it meets up with QM, because unlike all other uses of probabilities the understanding of a probability in QM is not the exact same thing.

I'm not getting into the philosophical because that is not answerable nor fitting here. I was wondering if the use of that specific word has caused a misunderstanding or confusion in the true nature of duality in QM? I'm not referring to just the lay person but also those who've studied physics and either starting into QM or didn't at all.

I hope that makes some sense.

What do you mean by duality in this context?

 

[madhatter106]

Superposition

 

[netheril96]

Sure,
I take the std math definition as the possible set outcomes divided by the number of choices. I.e random

For QM probability is not the same as the possible outcomes are not random.

No, the standard definition of probability is a special measure confined to a σ-algebra.

 

[Fredrik]

So, are you saying that a physical theory where probabilities defined as subjective degrees of belief are not statistically falsifiable and therefore not scientific theories?

Probabilities are numbers assigned by probability measures. They are never defined as degrees of belief. You can choose to think of them that way, but we will still test each theory by comparing the predicted probabilities with the observed relative frequencies. We certainly can't test them by comparing the predicted probabilities with our beliefs.

 

[Fra]

I understand what Fredrik said, and I sort of agree, but there are still some points that I would like to add.

Probabilities are numbers assigned by probability measures. They are never defined as degrees of belief.

Probability theory can be defined in different ways.

The axiomatic way of Kolmogorov, or as rational reasoning - an extension of deductive logic, for example as lined out by ET Jaynes and Cox.

ET Jaynes, describes a rational system for reasoning. And then DEFINES a notion of "degree of belief" that is REPRESENTED by a real number [0,1].

Then, using several plausability postulates, that you require from any rational inference system, the same mathematical formalism as the kolmogorov axioms are infered. This is a MUCH deeper view that the axiomatic way. So you have two formal systems, one with probability measures, one with degrees of belief, but the mathematics is the same. But the way the axioms are inferred is different.

We certainly can't test them by comparing the predicted probabilities with our beliefs.

If you take this view seriously, probabilites are not predicted nor falsified like you suggest - I agree. Instead the probabilites are understood not as statements of nature, but as interaction tools, in a learning system. The IMPORTANT part is not to falsify a probability, the importatnt part is how the probability are updated in the event of new evidence; thus conforming to rational belief.

If you discuss this, I agree that one can argue that relative frequences can be verified by experiments, but subjective probabilities can't. But that isn't the rigt way to think about the subjective probabilities.

The focus shifts from descriptive to a inferencial. The focus is not to corroborate or falsify a prior; that is meaningless. The focus is to improve and revise it rationally.

This is analogous to the deeper view of what a theory is; a description of nature that is either wrong or corroborated - or an interaction tool for LEARNING about nature.

One can certainly ask which focus is more fundamental - eternal descriptions of nature in terms of eternal laws, or optimal inferencees about nature. Also note that degree of belief is just a methapor and doesn't necessarily refer to human beleif!

On the contrary can it be defined in terms of counting historical evidence, but the counting and representation of counters states are inherently observer dependent. I personally think that this is a much more sensible view than resorting to the fictive ensembles or limits.

/Fredrik

 

[Fra]

So, are you saying that a physical theory where probabilities defined as subjective degrees of belief are not statistically falsifiable and therefore not scientific theories?

The resolution of asking this questions has I think two outcomes.

1) conclude that anything that can't be statistically falsifiable (read: repeated identically an infinity of times, and handling of all this data) isn't worthy science.

2) conclude that this is unreasonable and that instead the mentioned characterisation of the scientific process is inadequate.

I subscribe to the second option. 1 is still a special case, since the statistical perspective is still recovered for sufficiently small subsystems. This is the same discussion as that of statistics and cosmology.

/Fredrik

 

[Fredrik]

ET Jaynes, describes a rational system for reasoning. And then DEFINES a notion of "degree of belief" that is REPRESENTED by a real number [0,1].

What is that definition? Can you post it, or a link to it?

2) conclude that this is unreasonable and that instead the mentioned characterisation of the scientific process is inadequate.

I subscribe to the second option.

Why? There are no experiments that indicate that we need to change the definition of science. Perhaps the answer is that we don't know if we need to change it, but if we ever will, this will be the logical way to do it. In that case, what are the arguments for this?

 

[Fra]

What is that definition? Can you post it, or a link to it?

First I'd just want to say that I do not fully share the reasoning of Jaynes and Cox, but in the process of arguing with you here, I am on their side. The points of disagreement I have with their views are even more subtle. I believe this idea traces back to Cox.

E.T Jaynes wrote a book called "Probability theory - the logic of science". The printed book is a completed version of what was only partially complete when he passed away. The incomplete book is available at http://bayes.wustl.edu/etj/prob/book.pdf. But there are more, theorems of Cox etc, but these are referred to in this work.

One needs to appreciate the CONTEXT to see where this fits it. But in short he introduces a concept called "degree of belief", this is then postulated or axiomatically decided to be represented by a real number.

In short, just let me cite part of the preface that illustrates a little bit.

"However, Polya demonstrated this qualitative agreement in such complete, exhaustive detail as to suggest that there must be more to it. Fortunately, the consistency theorems of R. T. Cox were enough to clinch matters; when one added Polya's qualitative conditions to them the result was a proof that, if degrees of plausibility are represented by real numbers, then there is a uniquely determined set of quantitative rules for conducting inference. That is, any other rules whose results conict with them will necessarily violate an elementary|and nearly inescapable|desideratum of rationality or consistency.

But the final result was just the standard rules of probability theory, given already by Bernoulli and Laplace; so why all the fuss? The important new feature was that these rules were now seen as uniquely valid principles of logic in general, making no reference to \chance" or \random variables"; so their range of application is vastly greater than had been supposed in the conventional probability"

IMHO, one can say that we take the probability concept from a descriptive problem to a decision problem.

The modern introduction of probability is set theoretic; ie it is descpriptive, concerning properties of sets or ensembles and how they combine.

This view is regarding the problem of rational reasoning; trying to CONSTRUCT it using rational requiredments of how to rationally manipulated degrees of belief and merely postulating that degrees of believe are represented by real numbers; the only consistent system for "manipulating degrees of beliefs" RATIONALLY is identical mathematically to probability theory.

Ie. the degrees of believe are the same as probabilities, but understood differently, without set theoretic justifications.

/Fredrik

 

[Fra]

Why? There are no experiments that indicate that we need to change the definition of science. Perhaps the answer is that we don't know if we need to change it, but if we ever will, this will be the logical way to do it. In that case, what are the arguments for this?

I've given reasons in several posts, but for some reason the point are hard to convey.

One example is what we discussed before, how to do science in say cosmology. Here there is no way to realized the ensembles. You then talked about making several measurements on the same object. But then we are closing in on the degree of belief notion. The argument is that it's a more general understanding of probability.

In cosmology, we do not consider ensembles of universes. We do not make man-made "preparations of the universe". Instead we collect and count evidence, collected from the earht based perspective. This way we can form rational degrees of beleif of the cosmological scence.

This is just one example, the other area is when it comes to understanding a generalisation of RG, or how theories actually SCALE with thte observer. For example what laws would a baryon "see".

At it's extreme all this things have impacts to measurement problems in cosmological models, to unification of actions, and to scaling of theories (Renormalization group). Current RG is in fact only a special case. All this is open issue ans of debate so all I do is try to convey that there are deep and important things to understand here, and it's far from nonsense.

/Fredrik