I Quantum mechanics is random in nature?

[entropy1]

Where have you been dude. There are tons of them, and many are simply philosophical arguments about the meaning of probability:
http://math.ucr.edu/home/baez/bayes.html

That is interesting. Unfortunately, the format of the link is a series of emails and the formulas in it are skewed.

 

[bhobba]

That is interesting. Unfortunately, the format of the link is a series of emails and the formulas in it are skewed.

The key bit is at the top before the emails and will not take long to read.

And Feller is definitely a must read of the early chapters.

Thanks
Bill

 

[Zafa Pi]

To the contrary, randomness is a very mathematical concept, called probability theory and the theory of stochastic processes. The latter are, of course, very much motivated by physics (starting with kinetic theory in the 19th century by Maxwell and Boltzmann with some preliminary work by Bernoulli).

bhobba, in posts #49, #50, and #52 refers to Feller's classic two volume set. I agree that Feller is a worthy reference. On page 20 of volume 1 Feller says, "The word "random" is not well defined," and then he goes on to mention the example I gave in post #43. In fact I know of no serious book on probability that defines the word "random" or "randomness". If you think to the contrary, please cite a reference.

Probability theory is a mathematical theory that attempts to model intuitive concepts around randomness, just as QM is a mathematical theory that attempts to model the behavior of light (among other things). But QM does not define light, nor can you derive the definition of light from the axioms of QM. It is up to the physicist to assign states and observables that model the behavior he wishes to observe, or predict about light in the lab. It is up to the statistician to assign probabilities to the "random" phenomena she wishes to model with the theory. To say that "randomness is a very mathematical concept" is to confuse the map with the territory.

 

[houlahound]

Would it be accurate to say randomness in the classical physics sense is just the lack of knowledge of the details, ie the trajectory of a single particle in a gas is random but the macroscopic behavior of kajillions of particles is mundanely predictable via thermodynamics.

 

[Zafa Pi]

No, but there's nothing special about QM in that regard.

Consider what happens with classical gravitation. We start by investigating such diverse phenomena as falling objects, thrown stones, the atmospheric pressure, the motion of the planets. We don't understand any of them, we follow many false paths, and eventually Isaac Newton discovers the law of gravity. One equation, $F=Gm_1m_2/r^2$, explains everything and we understand gravity in all of its varied manifestations...
Except that then some clever high school kid who should be doing her exercises and calculating the speed of a dropped object interrupts her teacher to ask "What makes the objects want to move together? What's going on that makes $F=Gm_1m_2/r^2$ work so well? What's the reason the math works?". The answer is going to be some variant of of "It works. Experiments prove it. Shut up and calculate, you still haven't finished your exercises".

What's different here is that classical gravitation fits in well enough with our common sense that once we see how well it works we tend to accept it without digging deeper. QM, on the other hand, is counterintuitive enough to provoke that "yes, but why?" question, and a feeling of deep dissatisfaction when no answer is forthcoming.

When my kid was 4 and took a fall, he asked me why the ground kept pulling him down. I said, "I don't know why." The poor kid had lousy classical intuition.

 

[Stephen Tashi]

Randomness is not a mathematical concept. There are random variables (functions) and probabilities, but no definition of random.

I agree if we are talking about the common non-mathematical concept of "randomness" (or "probablility" or "stochastic process") as something that involves a state of "potential" or "tendency" or "degree of possibility" which then becomes an "actuality" or a "realization". When the mathematical theory of probability is applied to a specific situation, people think about randomness in that way (e.g. a fair coin has the "tendency" to land heads half the time and when thrown it "actually" lands heads or doesn't.) However, the formal development of probability based on measure theory there is no axiomatization of the concept of a "probability" transforming to an "actuality".

In the formal theory, events have probabilities, but there is no mathematical definition for an event with probability becoming an "actual" event. The closest one gets to the concept of a transition from "probable" to "actual" is in the definition of conditional probability. However, that definition is quite abstract and it merely defines one probability measure in terms of other probability measures. The fact that the conditional probability "given event E" can be defined does not entail any axiom that event E can exist in two states - a "probable" state and an "actual" state.

It would be hard to develop a concept of "actual occurrence" that is consistent with both intuitive idea of "actual" and the theory of probability. For example, we can't say an event with probability 0 will not "actually" occur if we admit the procedure of taking a random sample form a normally distributed random variable. Any specific value we realize from such a distribution has probability zero of occurring. One may side step paradox in practical applications by saying that we cannot "actually" take a random sample from a normal distribution, we can only obtain a value that has finite precision. However, this leaves the theoretical problem of whether some "actual" value was realized (with zero probability) and then was measured with finite precision.

Mathematical probability theory avoids any axiomatic treatment of how events that have probabilities become "actual" or don't. The theory uses suggestive terminology like "almost surely" to suggest how probability can be applied, but there is nothing in the measure theoretic definition of probability that asserts an event can transform from a state of having a probability to a state of being "actual'.

 

[Zafa Pi]

Would it be accurate to say randomness in the classical physics sense is just the lack of knowledge of the details, ie the trajectory of a single particle in a gas is random but the macroscopic behavior of kajillions of particles is mundanely predictable via thermodynamics.

The 1st half of your sentence, up to the comma, is generally considered correct.
The ie (sic) in your sentence is inappropriate. The 2nd half of your sentence is not a retelling or refinement of the 1st half.

 

[bhobba]

If you think to the contrary, please cite a reference.

Sorry if I wasn't clear but that was my point. I was trying to get people to read it rather than simply regurgitate what it says and in doing so wasn't 100% clear in what I was trying to get across.

In speaking of Feller I was talking of the introduction - The Nature Of Probability Theory.

The whole chapter needs to be read. If I was to post about it would simply repeat that chapter and I urge anyone interested in applying probability theory, and indeed applied math in general, to read it.

However a quote from page 3 sums it up:
'We shall no more attempt to explain the true meaning of probability than the modern physicist dwells on the real meaning of mass and energy or the geometer discusses the nature of a point. Indeed we shall prove theroms and see how they are applied'

In particular the Kolmogorov axioms leave the notion of event undefined. Applying it requires the mapping of that concept to 'things' out there which only comes from practice and experience. Its similar to the concept of point in geometry. In Euclid's axioms a point has position and no size. Such of course do not exist, but from drawing diagrams and seeing how theorems are proved you get an intuitive feel how to map it to things about there such pegs (for points) and strings (for lines) surveyors use. It's usually so obvious no one explicitly states it. The same in probability - by seeing how events are mapped to things like coin tosses etc etc we gradually build up an understanding.

In QM the primitive is observation. The usual formalism no more attempts to define precisely what that is than probability does event. It can be done - see the Geometry of Quantum Theory by Varadarajan - but like other highly abstract approaches to physics such as Symplectic geometry and mechanics its mathematical beauty is sublime - applying it however is another matter - which is why physicists in general take a different route. Its like Hilbert's axioms of Euclidean geometry. Mathematically it defines exactly what Euclidean geometry is, but from an applied viewpoint Euclid's axioms are used because how to apply it is much clearer - but from the viewpoint of pure math has issues - which is why Hilbert came up with his axioms.

An interesting thing about probability is the same Kolomogorov axioms are used for both pure and applied approaches.

Thanks
Bill

 

[bhobba]

Would it be accurate to say randomness in the classical physics sense is just the lack of knowledge of the details, ie the trajectory of a single particle in a gas is random but the macroscopic behavior of kajillions of particles is mundanely predictable via thermodynamics.

The truth is in the Kolmogorov axioms.

What it means is left up in the air - how you map its undefined concept of event is its content. As I posted above its similar to point in geometry. No one worries about exactly what that is, you simply see how its applied.

Of course philosophers can and do argue about such things, but they generally get nowhere in the sense no one agrees on anything. Because of that mathematicians and physicts avoid it and we by forum rules don't generally discuss it here. There was a very famous example of this with the great philosopher Kant and perhaps even greater mathematician Gauss. Kant thought he knew what Euclidean geometry was thinking it was a priori. Gauss however knew differently, having discovered non euclidean geometry that was just as consistent as euclidean geometry but due to Kant's prestige didn't publish it. It was a lesson well learned and nowadays mathematicians and physicists generally don't worry about such things. Axioms are freely chosen, its meaning is purely in how the undefined concepts of those axioms are mapped to whatever you apply them to. Or you can go the pure math route and don't actually do that mapping and simply prove the consequences of the axioms.

Thanks
Bill

 

[zonde]

No, but there's nothing special about QM in that regard.

Consider what happens with classical gravitation. We start by investigating such diverse phenomena as falling objects, thrown stones, the atmospheric pressure, the motion of the planets. We don't understand any of them, we follow many false paths, and eventually Isaac Newton discovers the law of gravity. One equation, $F=Gm_1m_2/r^2$, explains everything and we understand gravity in all of its varied manifestations...
Except that then some clever high school kid who should be doing her exercises and calculating the speed of a dropped object interrupts her teacher to ask "What makes the objects want to move together? What's going on that makes $F=Gm_1m_2/r^2$ work so well? What's the reason the math works?". The answer is going to be some variant of of "It works. Experiments prove it. Shut up and calculate, you still haven't finished your exercises".

What's different here is that classical gravitation fits in well enough with our common sense that once we see how well it works we tend to accept it without digging deeper. QM, on the other hand, is counterintuitive enough to provoke that "yes, but why?" question, and a feeling of deep dissatisfaction when no answer is forthcoming.

I would say that in classical gravitation the concept of "field" tries to answer the "why?" question.

 

[Nugatory]

I would say that in classical gravitation the concept of "field" tries to answer the "why?" question.

Sure, but that just pushes the "Why?" question down one level. Why do masses create gravitational fields and why do these fields behave the way they do?

 

[zonde]

Sure, but that just pushes the "Why?" question down one level. Why do masses create gravitational fields and why do these fields behave the way they do?

Yes, but gravitational field is metaphysics. And this shows [assuming you agree that gravitational field is metaphysics] how important it is to push the "Why?" question down just one level.

 

[Zafa Pi]

I would say that in classical gravitation the concept of "field" tries to answer the "why?" question.

How does mass manufacture the field?

 

[zonde]

How does mass manufacture the field?

Mass does not manufacture the field. Field just "sits" there, but it can have different potentials and mass changes this potential of field where it is.

 

[Zafa Pi]

Yes, but gravitational field is metaphysics. And this shows [assuming you agree that gravitational field is metaphysics] how important it is to push the "Why?" question down just one level.

The gravitational field is physics. It is as much so as the electric field. It appears in a zillion physics texts.

 

[Zafa Pi]

Mass does not manufacture the field. Field just "sits" there, but it can have different potentials and mass changes this potential of field where it is.

OK, how does mass make the changes?

 

[Zafa Pi]

Sorry if I wasn't clear but that was my point. I was trying to get people to read it rather than simply regurgitate what it says and in doing so wasn't 100% clear in what I was trying to get across.

I was quoting Feller to make a point about post #47 by vanhees71. Do you call that reguritation?

In speaking of Feller I was talking of the introduction - The Nature Of Probability Theory.

In post #49 you mention the early chapters.

The whole chapter needs to be read. If I was to post about it would simply repeat that chapter and I urge anyone interested in applying probability theory, and indeed applied math in general, to read it.

However a quote from page 3 sums it up:
'We shall no more attempt to explain the true meaning of probability than the modern physicist dwells on the real meaning of mass and energy or the geometer discusses the nature of a point. Indeed we shall prove theroms and see how they are applied'

In particular the Kolmogorov axioms leave the notion of event undefined. Applying it requires the mapping of that concept to 'things' out there which only comes from practice and experience. Its similar to the concept of point in geometry. In Euclid's axioms a point has position and no size. Such of course do not exist, but from drawing diagrams and seeing how theorems are proved you get an intuitive feel how to map it to things about there such pegs (for points) and strings (for lines) surveyors use. It's usually so obvious no one explicitly states it. The same in probability - by seeing how events are mapped to things like coin tosses etc etc we gradually build up an understanding.

An event is defined in probability theory as a subset of the probability space.

In QM the primitive is observation. The usual formalism no more attempts to define precisely what that is than probability does event. It can be done - see the Geometry of Quantum Theory by Varadarajan - but like other highly abstract approaches to physics such as Symplectic geometry and mechanics its mathematical beauty is sublime - applying it however is another matter - which is why physicists in general take a different route. Its like Hilbert's axioms of Euclidean geometry. Mathematically it defines exactly what Euclidean geometry is, but from an applied viewpoint Euclid's axioms are used because how to apply it is much clearer - but from the viewpoint of pure math has issues - which is why Hilbert came up with his axioms.

I'm going to quibble here. A primitive of a theory is an undefined term, such as mass, length, time and force in Newtonian Mechanics. BTW energy is not a primitive, but is define in terms of the others. Observation = measurement is well define in QM by an axiom, as is state.

An interesting thing about probability is the same Kolomogorov axioms are used for both pure and applied approaches.

Thanks
Bill

Since you liked the the vanhees71 post you must have issues with my posts #43, #45, #53. I am curious as to what they may be. It seems to me that Feller supports my position.

 

[atyy]

Why does nobody in the physics world care about the random/non-random question?

Many do, including the greatest physicists like Bohr, Schroedinger, Dirac and Weinberg. However, a better and clearer way of stating the problem is: how can a theory that requires an observer be fundamental? Shouldn't there be laws of nature governing the universe before and after the existence of observers? If such laws exist, what are they?

If you carefully read what Dirac says when he advocates what we might call "shut up and calculate", you will find that he does not dismiss the measurement problem as a non-existent problem. He says the problem is hard, and that we can make progress without solving it. But he does imagine that the problem will be solved by a theory beyond quantum theory.

"Of course there will not be a return to the determinism of classical physical theory. Evolution does not go backward. It will have to go forward. There will have to be some new development that is quite unexpected, that we cannot make a guess about, which will take us still further from Classical ideas but which will alter completely the discussion of uncertainty relations. And when this new development occurs, people will find it all rather futile to have had so much of a discussion on the role of observation in the theory, because they will have then a much better point of view from which to look at things. So I shall say that if we can find a way to describe the uncertainty relations and the indeterminacy of present quantum mechanics that is satisfying to our philosophical ideas, we can count ourselves lucky. But if we cannot find such a way, it is nothing to be really disturbed about. We simply have to take into account that we are at a transitional stage and that perhaps it is quite impossible to get a satisfactory picture for this stage."

http://blogs.scientificamerican.com/guest-blog/the-evolution-of-the-physicists-picture-of-nature/

 

[Zafa Pi]

Many do, including the greatest physicists like Bohr, Schroedinger, Dirac and Weinberg. However, a better and clearer way of stating the problem is: how can a theory that requires an observer be fundamental? Shouldn't there be laws of nature governing the universe before and after the existence of observers? If such laws exist, what are they?

I don't think that most physicists think that QM requires an observer (as opposed von Neumann). The usual axioms don't require it. What do you think?

 

[atyy]

I don't think that most physicists think that QM requires an observer (as opposed von Neumann). The usual axioms don't require it. What do you think?

The usual axioms require it. It may be stated in slightly different language (eg. a classical measuring apparatus), but it is required.

See also bhobba's post #58 for yet another way of stating this.

 

[bhobba]

Yes, but gravitational field is metaphysics.

Come again - space time curvature is as real as you can get.

Thanks
Bill

 

[bhobba]

I don't think that most physicists think that QM requires an observer (as opposed von Neumann). The usual axioms don't require it. What do you think?

It doesn't, as interpretations such as MW and BM that do not have it prove.

Thanks
Bill

 

[bhobba]

The usual axioms require it. It may be stated in slightly different language (eg. a classical measuring apparatus), but it is required. See also bhobba's post #58 for yet another way of stating this.

I said observation was a primitive of the theory. What it means, and if it requires an observer, or even an observation in a more limited sense of being the outcome of an interaction, is very interpretation dependent. For example in BM the mixed state after decoherence is actually in that state independent of an observer or observation (ie decoherence outcome - it has actual values at all times).

Thanks
Bill

 

[bhobba]

OK, how does mass make the changes?

This is not the place to discuss it, the relativity forum is, but GR is very very elegant. Pretty much the assumption of no prior geometry all by itself leads to the Einstein Field equations. Mathematically this means the geometry is a dynamical variable ie obeys a least action principle - see for example section 4 of the following:
http://www.if.nu.ac.th/sites/default/files/bin/BS_chakkrit.pdf

Of course it solves nothing in a fundamental sense because you have simply changed the question to - why is there no prior geometry. Its just no prior geometry seems pretty intuitive - why should nature single out one geometry over another.

Thanks
Bill

 

[atyy]

I said observation was a primitive of the theory. What it means, and if it requires an observer, or even an observation in a more limited sense of being the outcome of an interaction, is very interpretation dependent. For example in BM the mixed state after decoherence is actually in that state independent of an observer or observation (ie decoherence outcome - it has actual values at all times).

Thanks
Bill

Observation is not a primitive in BM.

 

[zonde]

Come again - space time curvature is as real as you can get.

GR took gravity to a new level of course.
But I would change the wording of your statement: space time deformation is physical. I understand that "curvature" is the standard word but as I see it implies 5th dimension and that is metaphysics.

 

[Mentz114]

GR took gravity to a new level of course.
But I would change the wording of your statement: space time deformation is physical. I understand that "curvature" is the standard word but as I see it implies 5th dimension and that is metaphysics.

Spacetime curvature does not require a fifth dimension. It is intrinsic in the metric.

 

[Stephen Tashi]

What began as discussion of randomness has turned into debates about things that sound (to me) rather deterministic. Is there a line of reasoning that leads down this path? - or is it just a digression?

 

[atyy]

What began as discussion of randomness has turned into debates about things that sound (to me) rather deterministic. Is there a line of reasoning that leads down this path? - or is it just a digression?

Probability according to Kolmogorov can always be interpreted as ignorance of a deterministic process. In this sense, Kolmogorov probability is not truly random. In contrast, QM does not obey the pure Kolmogorov axioms. In contrast to Kolmogorovian probability, the QM state space is not a simplex, and may be "truly random" because it is not due to ignorance of a deterministic process.

The question is whether QM can be embedded in a larger theory whose state space is a simplex.

 

[Zafa Pi]

Probability according to Kolmogorov can always be interpreted as ignorance of a deterministic process. In this sense, Kolmogorov probability is not truly random. In contrast, QM does not obey the pure Kolmogorov axioms. In contrast to Kolmogorovian probability, the QM state space is not a simplex, and may be "truly random" because it is not due to ignorance of a deterministic process.

The question is whether QM can be embedded in a larger theory whose state space is a simplex.

But it doesn't have to be interpreted as ignorance of a deterministic process, though the BM interpretation does. The Kolmogorov axioms (the usual ones we now think of) are silent on such issues. And as I have pointed out in previous posts "random" or "truly random" are not defined in probability theory. They are intuitive ideas, usually connoting, "having no cause", what ever that means.

Could you please give a concrete where probabilistic notions in QM don't obey the Kolmogorov axioms.

 

[atyy]

But it doesn't have to be interpreted as ignorance of a deterministic process, though the BM interpretation does. The Kolmogorov axioms (the usual ones we now think of) are silent on such issues. And as I have pointed out in previous posts "random" or "truly random" are not defined in probability theory. They are intuitive ideas, usually connoting, "having no cause", what ever that means.

Could you please give a concrete where probabilistic notions in QM don't obey the Kolmogorov axioms.

Yes, however, Kolmogorov probability does allow notions such as a particle having a trajectory at all times.

So fundamentally, classical probability has a state space that is a simplex. That is the difference between QM and Kolmogorov.

 

[Zafa Pi]

Yes, however, Kolmogorov probability does allow notions such as a particle having a trajectory at all times.

So fundamentally, classical probability has a state space that is a simplex. That is the difference between QM and Kolmogorov.

Kolmogorov probability, which I now merely call probability theory, as given in chapter 1 of Feller also allows notions such as red eyed dragons. It doesn't say anything about dragons or particle trajectories. Within probability theory there is a probability space or sample space, but I can't find a state space within the theory. I know what a state space is in QM.

 

[bhobba]

Observation is not a primitive in BM.

Exactly.

Observation is a primitive in the standard formalism with no or very minimal interpretation. In interpretations that use decoherence it has morphed to why do we get outcomes at all which is its status in my ignorance ensemble. Although it could be reasonably argued that since decherence applies to any interpretation even just the formalism it has morphed in every interpretation, but I would not argue that since its really, IMHO just semantics on what interpretation means. In still others it is explained by other things. In the very formal mathematical approach such as the found in Varadarajan it is reduced to the logic of QM - here logic means in the formal mathematical sense of a lattice. But like Hilbert's axioms of Euclidean geometry is generally not used in applications where a slightly looser development is better.

Thanks
Bill

 

[bhobba]

Yes, however, Kolmogorov probability does allow notions such as a particle having a trajectory at all times.

Yes of course. Just to elaborate, the use of probability anywhere can result from a number reasons. Its truly random being one, inherent lack of knowledge is another, and there are probably other ways although I can't think of them off the top of my head.

BM is an example of a deterministic theory where because of inherent lack of knowledge (to be specific as implied by the indeterminacy relations) you don't know enough, and the theory forbids you from finding it, to predict outcomes so you must use probability theory. Another is the use of decision theory in MW which, while not the same as the Kolmogorov axioms, does imply them as professions like actuarial science that also make use of it know quite well.

Thanks
Bill

 

[bhobba]

Within probability theory there is a probability space or sample space, but I can't find a state space within the theory. I know what a state space is in QM.

Of course you are correct.

But to elaborate a bit one can define a measure over the state space (eg via Gleason) and relate that measure to events - which of course is what Gleason also does. In that way the Kolmogorov axioms are recovered.

Thanks
Bill

 

[Zafa Pi]

Of course you are correct.

But to elaborate a bit one can define a measure over the state space (eg via Gleason) and relate that measure to events - which of course is what Gleason also does. In that way the Kolmogorov axioms are recovered.

Thanks
Bill

And you are correct. Given any set whatsoever one can define a countably additive probability measure on some sigma-algebra of subsets.

 

[mikeyork]

Sorry to join this thread so late, but I need to point out that you can't talk about randomness without specifying what variable is random. In QM, you can have a particle with an exactly deterministic momentum, but then the position is completely (uniformly, infinitely) random. In Hilbert space any state vector that is a superposition in one basis can be an eigenstate in another.

 

[vanhees71]

You can NOT have a particle with precisely determined momentum. An observable can never have a determined value that's in the continuous part of the corresponding operator. For momentum it's immediately clear, because the generalized eigenfunction in position representation is a plane wave, which is not square integrable. That's the true meaning of the famous uncertainty principle $\Delta x \Delta p \geq \hbar/2$, which tells you that neither position (which has only continuous eigenvalues) and momentum (which as well has only continuous eigenvalues) can have 0 variance in any pure or mixed state possible!

 

[Stephen Tashi]

That's the true meaning of the famous uncertainty principle $\Delta x \Delta p \geq \hbar/2$, which tells you that neither position (which has only continuous eigenvalues) and momentum (which as well has only continuous eigenvalues) can have 0 variance in any pure or mixed state possible!

Are you interpreting $\Delta x$ and $\Delta p$ as standard deviations of random variables ? Does the fact that a random variable has a non-zero standard deviation preclude us from doing an experiment where one specific value of the random variable is realized?

As I said in a previous post, the (Kolmogorov) theory of probability doesn't say anything (pro or con) about whether random variables can take on "actual" values or whether we can do experiments that cause these actual values to occur. So demonstrating that a physical quantity cannot be precisely measured can't be done by a purely mathematical argument. I'd like to understand what physical argument is used to reach your conclusion.

 

[vanhees71]

What do you mean by "realized". Quantum theory tells you that you cannot prepare a particle such that it has a precisely determined momentum. You can prepare it at any precision (standard deviation) $\Delta p>0$ but never make $\Delta p=0$. It doesn't say that you cannot measure momentum at any accuracy. This only depends on your measurement apparatus. In principle you can always construct a device that measures momentum more precise than the $\Delta p$ of the prepared state of the particle. Then you'll find fluctuations around the mean value given by the corresponding probability distribution according to this standard deviation, when measuring the momentum at this higher accuracy, on a large set of equally prepared partices (an ensemble).

The physical argument is the overwhelming accuracy of quantum theory in describing all observations we have collected about nature today. There's not a single reproducible contradiction between quantum theory and observations, and quantum theory has been tested to extreme accuracy in some cases. So we have good reason to believe that quantum theory describes nature very accurately. Of course, as with all scientific knowledge, it's always possible that one day one discovers a phenomenon that cannot be described by quantum theory. This would be a real progress, because then we'd have learned something completely new about nature and we would have to adjust our theories leading to an even more comprehensive view about nature. In some sense you can say that finding discrepancies between the theories we have today and experiment is the true goals of scientific research in order to find even better theories.

 

[Delta Kilo]

In QM, you can have a particle with an exactly deterministic momentum, but then the position is completely (uniformly, infinitely) random.

The particle does not "have" a random position. Instead, the result of a position measurement, should we choose to perform one, is random. Before the measurement the particle is in well defined state. Randomness only appears as a result of measurement process, which necessarily involves interaction with large number of particles in unknown state.

I don't get why so much fuss is made about randomness in QM. Everyone seems to be OK with classical Brownian motion of a speck of dust being random. But, they say, unlike QM, it is not a "true" randomness, they could predict it if it they knew the positions and velocities of all air molecules in a volume at time t0. But is it really so? Penrose gave this argument in "Road to Reality": moving 1kg by 1 meter somewhere in the vicinity of Alpha Centauri causes enough change in gravitation here on Earth to completely scramble the trajectories of molecules in 1m^3 volume of air within 1 minute. Given that gravitation travels at c and cannot be screened, we'd need to know the state of the entire universe all the way back to Big Bang. So for all intents and purposes the motion of individual molecules of air is truly random.

I feel it is exactly the same way with QM, I don't see much conceptual difference between QM spin measured 50/50 up or down and a classical problem of a ball on a knife's edge falling 50/50 left or right. In theory in the absence of external influences the ball stays on the edge forever and the detector remains in Schrodinger cat state of superposition. In real life the ball eventually falls down and the detector reports a single outcome.

 

[mikeyork]

Vanhees, you say "An observable can never have a determined value that's in the continuous part of the corresponding operator." in response to me. But I said deterministic (meaning in a definite eigenstate) not determined (i.e.observed).

 

[Stephen Tashi]

What do you mean by "realized".

It doesn't say that you cannot measure momentum at any accuracy.

I don't understand the distinction that you are apparently making between knowing a particles momentum was precisely measured to be a specific value at time t and concluding the particle "had" that precise momentum at time t. By a "realized" value , I mean that the value was measured and thus that the random physical quantity is thus know to have taken that specific value. Perhaps erroneously, I think of a theoretical measurement of momentum as a "realization" of a specific value of momentum.

I agree that any practical apparatus does not produce infinitely precise measurements. The conceptual question is whether the thing being measured "had" a exact value when an imperfect measuring apparatus measured it. is the argument that no particle ever had an exact momentum because all practical measuring equipment has limited precision ?[/QUOTE]

 

[Zafa Pi]

Vanhees, you say "An observable can never have a determined value that's in the continuous part of the corresponding operator." in response to me. But I said deterministic (meaning in a definite eigenstate) not determined (i.e.observed).

Confusion regularly results when people blur the lines between theory and reality. QM is a mathematical theory and in that theory a measurement by an obsversable (Hermitian operator) of a state (unit vector in a Hilbret space, or ray, or L² "wave" function) is a random variable (see Feller and Nielsen & Chuang). If the variance of that r.v. is 0 then the state is an eigenvector of the operator. What vanhees71 is saying (my interpretation) is that the momentum operator has no eigenvector in the state (Hilbert) space (same for the position operator). Thus Δp (of a state) is > 0 (Δp is the s.d. = sqrt of the variance)

Now in reality the experimental physicist selects a momentum measuring apparatus (hopefully modeled by QM) and prepares a large number of entities in the same state and procedes to measure them. The resulting measurements are not all the same in spite of the fact that each individual measurement is a single precise value. Thus the collection of all the measurements has a non-zero (statistical) variance. And as vanhees71 says QM and reality agree.

 

[DrChinese]

The particle does not "have" a random position. Instead, the result of a position measurement, should we choose to perform one, is random. Before the measurement the particle is in well defined state. Randomness only appears as a result of measurement process, which necessarily involves interaction with large number of particles in unknown state.

I don't get why so much fuss is made about randomness in QM. Everyone seems to be OK with classical Brownian motion of a speck of dust being random. But, they say, unlike QM, it is not a "true" randomness, they could predict it if it they knew the positions and velocities of all air molecules in a volume at time t0. But is it really so? Penrose gave this argument in "Road to Reality": moving 1kg by 1 meter somewhere in the vicinity of Alpha Centauri causes enough change in gravitation here on Earth to completely scramble the trajectories of molecules in 1m^3 volume of air within 1 minute. Given that gravitation travels at c and cannot be screened, we'd need to know the state of the entire universe all the way back to Big Bang. So for all intents and purposes the motion of individual molecules of air is truly random.

I feel it is exactly the same way with QM, I don't see much conceptual difference between QM spin measured 50/50 up or down and a classical problem of a ball on a knife's edge falling 50/50 left or right. In theory in the absence of external influences the ball stays on the edge forever and the detector remains in Schrodinger cat state of superposition. In real life the ball eventually falls down and the detector reports a single outcome.

Clearly, there are tremendous differences between the classical examples you give and the quantum ones. Classical systems do not feature non-commuting observables. Non-commuting observables not only have specific limits in their precision, those limits can be seen in experiments on entangled pairs. So if you don't see the conceptual difference between these, you need to consider more experiments.

 

[David Lewis]

Given that gravitation travels at c and cannot be screened, we'd need to know the state of the entire universe all the way back to Big Bang. So for all intents and purposes the motion of individual molecules of air is truly random.

My understanding of classical mechanics there are no hypothetically random processes, even if we can't collect enough information to make a prediction.

 

[drschools]

Would not truly random fit the Aristotelian definition of God i.e. the "unmoved mover" ? We might therefore attribute all genuinely random or uncaused phenomenon to God. We would therefore be following a similar argument as the "God of the Gaps". It might be argued also that while "Newtonian phenomenon" increasingly shrinks the unexplained gaps and therefore evidence of God in the universe, "Quantum phenomenon" re-expands the mysterious unexplainable gaps of science and therefore evidence of God (of the gaps).

 

[DrChinese]

Would not truly random fit the Aristotelian definition of God i.e. the "unmoved mover" ? We might therefore attribute all genuinely random or uncaused phenomenon to God. We would therefore be following a similar argument as the "God of the Gaps". It might be argued also that while "Newtonian phenomenon" increasingly shrinks the unexplained gaps and therefore evidence of God in the universe, "Quantum phenomenon" re-expands the mysterious unexplainable gaps of science and therefore evidence of God (of the gaps).

Sure. Although it does beg the question: why are there any physical laws at all if god is reserving her efforts only to fill in those gaps?

 

[Zafa Pi]

The particle does not "have" a random position. Instead, the result of a position measurement, should we choose to perform one, is random. Before the measurement the particle is in well defined state. Randomness only appears as a result of measurement process, which necessarily involves interaction with large number of particles in unknown state.

I don't get why so much fuss is made about randomness in QM. Everyone seems to be OK with classical Brownian motion of a speck of dust being random. But, they say, unlike QM, it is not a "true" randomness, they could predict it if it they knew the positions and velocities of all air molecules in a volume at time t0. But is it really so? Penrose gave this argument in "Road to Reality": moving 1kg by 1 meter somewhere in the vicinity of Alpha Centauri causes enough change in gravitation here on Earth to completely scramble the trajectories of molecules in 1m^3 volume of air within 1 minute. Given that gravitation travels at c and cannot be screened, we'd need to know the state of the entire universe all the way back to Big Bang. So for all intents and purposes the motion of individual molecules of air is truly random.

I feel it is exactly the same way with QM, I don't see much conceptual difference between QM spin measured 50/50 up or down and a classical problem of a ball on a knife's edge falling 50/50 left or right. In theory in the absence of external influences the ball stays on the edge forever and the detector remains in Schrodinger cat state of superposition. In real life the ball eventually falls down and the detector reports a single outcome.

I believe yours' is the valid answer to the original question of this thread. The key is in the 2nd line of your 2nd paragraph : "if they knew the positions ... " But they don't know and cannot give any reasonable way to go about knowing. Those which want to make a distinction between random due to lack of knowledge and "pure random" will come down to events that have a cause and those that don't. A philosophical quagmire of the 1st order in which I have squandered my youth.

In knowing the state of the universe all the way back to the Big Bang is also the ultimate loop hole in the disproving of realism using the measurements of entangled entities (super-determinism). And nobody cares, nor should they.

 

[Zafa Pi]

Sure. Although it does beg the question: why are there any physical laws at all if god is reserving her efforts only to fill in those gaps?

God moves in mysterious ways just like Simone Biles.