Undergrad Quantum mechanics is random in nature?

[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.