What theories address the fundamental questions about quantum mechanics?

[Fredrik]

It is assumed all the time. Every experimenter who experiments with a quantum system confined to his experimentation desk believes that the system's position is well-defined enough to be able to say that it is on his desk.

Right, but we were talking about the quote

"...gives the probability for the electron, arriving from the z-direction, to be thrown out into the direction designated by the angles alpha, beta, gamma, with the phase change delta"​

It sounds like he's talking about Coulomb scattering. In that case, the electron can be "thrown out" in any direction, and that means that it's going to be in a superposition of going in all directions. It won't be approximately localized until it has interacted with something else, like a bunch of air molecules.

That would be to assume that a particle in a superposition of states with approximately well-defined positions actually is in one of those locations at all times.

Only to someone already spoilt by the mutilated Born rule. For those who understand the Heisenberg uncertainty relation as what it is, a bound on uncertainties of the mean, it only says that the position is not better defined than the standard deviation, and that being in a location determined by a single real number is an impossibility.

I know what the uncertainty relation is. What you're saying here doesn't address what I said. What I said is that if the wavefunction has two (or more) peaks that are pretty far apart, it would be a mistake to think that the particle is approximately located at only one of the two peaks.

I know that you don't believe that to be the case, but Born's statement of his rule strongly suggests that he did. That's why I say that it's been improved, not mutilated.

Position is very different from spin.

Not so different that a superposition of localized wavefunctions has the interpretation "the partice is either here or there", while a superposition of spin up and spin down has an interpretation that's very different from "the spin is either up or down". (This claim about spin states is proved by Bell inequality violations). This would be the implication of what you're suggesting, but I see now that we are once again talking about very different things.

I think the claim that undetected particles have positions that just happen to be unknown is provably false, even though I don't know how to prove it myself.

I just proved the opposite.

No, you just made a case for something entirely different. Maybe I wasn't clear enough. To prove the opposite would be to (at least) prove that if the wavefunction has two separate peaks, the particle is really at one of those locations the whole time.

You argued that a particle is always approximately localized the whole time, but I think my first comment in this post pokes a hole in that. An even simpler counterargument is to just ask you to consider a double slit experiment. The particle will at best be approximately localized at two locations until it interacts with the screen.

By the way, I read about a double slit experiment with C70 molecules where they ran the experiment many times with different densities of the surrounding air, and found that the higher the density, the more the interference pattern looked like what you'd get when only one slit is open at a time. This beautifully illustrates that it's the particle's interactions with other things that localizes it.

Born's statement would have been OK if he, instead of "probability [...] to be thrown out into the direction...", had said "probability [...] to be approximately located in the direction [...], after it has interacted with its environment". The modern version has corrected the original, not mutilated it.

Haag means with position 'classical position with infinite precision. This infinite precision - and _only_ this leads to paradoxa.

I don't think that's what he meant, but I also don't want to spend more time than necessary analyzing quotes, so I'll just drop this one.

 

[A. Neumaier]

Right, but we were talking about the quote

"...gives the probability for the electron, arriving from the z-direction, to be thrown out into the direction designated by the angles alpha, beta, gamma, with the phase change delta"​

It sounds like he's talking about Coulomb scattering. In that case, the electron can be "thrown out" in any direction, and that means that it's going to be in a superposition of going in all directions. It won't be approximately localized until it has interacted with something else, like a bunch of air molecules.

Interacted, yes, but interactions are not yet measurements, according to the conventional terminology. As long as one is still talking about probabilities, one hasn't measured yet - each outcome is a possibility. (This is discussed in Mott's famous paper on particle tracks.)
The measurement happens when (and only when) one of the possibilities was actually realized.

I know what the uncertainty relation is. What you're saying here doesn't address what I said. What I said is that if the wavefunction has two (or more) peaks that are pretty far apart, it would be a mistake to think that the particle is approximately located at only one of the two peaks.

I know that you don't believe that to be the case, but Born's statement of his rule strongly suggests that he did. That's why I say that it's been improved, not mutilated.

The ''improved'' Wikipedia formulation of Born's rule simply doesn't do justice to the situation mentioned by Born, even when it is interpreted as measuring the angle of an electron. it is a continuous variable. Thus, no matter which angle beta is measured, it has probability zero (take M={beta} in Wikipedia's Born rule). So the improvement is not good enough.

On the other hand, I don't want to argue his formulation further since my Axiom 5 together with MI captures what really happens in much more precision and generality. I just found it interesting that he didn't refer to measurement. And

Not so different that a superposition of localized wavefunctions has the interpretation "the partice is either here or there", while a superposition of spin up and spin down has an interpretation that's very different from "the spin is either up or down". (This claim about spin states is proved by Bell inequality violations). This would be the implication of what you're suggesting, but I see now that we are once again talking about very different things.

Yes. I am talking about particles in the sense of Weinberg's essay, ''What is Quantum Field Theory, and What Did We Think It Is?'' http://arxiv.org/pdf/hep-th/9702027v1:
''In its mature form, the idea of quantum field theory is that quantum fields are the basic ingredients of the universe, and particles are just bundles of energy and momentum of the fields.'', not about the popular weird view.

No, you just made a case for something entirely different.

I made the case for my statement

Claiming that particles do not have positions unless they are measured was one of the worst disservices the mutilated Born rule has done to the understanding of quantum physics.

(which is a statement about a much more general claim than what Born discussed) upon which you responded with the above. Particles have approximate positions and momenta when one can meaningfully talk about them. In the remaining cases, there are no particles but only a field. The field has approximate mass densities that describe the situation, no matter whether an additional particle interpretation is or isn't meaningful.

You argued that a particle is always approximately localized the whole time, but I think my first comment in this post pokes a hole in that.

I probably wasn't clear enough. I meant to argue that a field may sensibly be interpreted as a particle only at the times where it is approximately localized. Thus a single photon passing through a half-silvered mirror becomes bilocal, and the particle picture (if retained) leads to the well-known paradoxes. The field picture, though, has no problems at all.

 

[Fredrik]

So you're just redefining the term "particle" to make your claim correct? Weinberg defines one-particle states as the members of a Hilbert space associated with an irreducible representation. Such states do not have to be localized at all times, and they're still called one-particle states.

 

[A. Neumaier]

So you're just redefining the term "particle" to make your claim correct?

I am differentiating things in order to be able to clearly describe what is going on. I am looking at the actual usage of the terms in experimental situations. The textbook explanations are usually far too idealized to describe actual practice.

Weinberg defines one-particle states as the members of a Hilbert space associated with an irreducible representation. Such states do not have to be localized at all times, and they're still called one-particle states.

There is a difference between particles and 1-particle states. The former is a semiclassical concept without a fully precise meaning, the latter a precisely defined field-theoretic term. Or, in your terms: The former is an interpretation in the real world, the latter a mathematical concept.

A field is in a 1-particle state if the state of the field is an eigenstate of the particle number operator with eigenvalue 1, but this doesn't make the field a particle. As Weinberg says, a particle is a bundle of energy and momentum (with particle number 1 if it is an elementary particle) .

But a plane wave 1-particle state doesn't bundle in the way required for a particle.

In an N-particle state of a field theory with N>1, one cannot even represent a _single_ property of the constituting particles anymore (see the current discussion in the thread ''Difference between 'Quantum theories'''). The formal terminology sticks, although nothing about it is observable anymore.

Independent of whether it is or isn't in a 1-particle state, a field may be viewed as a particle at any time where its state is fairly well localized, provided it remains localized for a time sufficiently long to make it detectable. In this case, it has a reasonably well-defined position and momentum, which can be experimentally checked. (Below this time, it would be a resonance, only indirectly detectable.)

 

[A. Neumaier]

There is a difference between particles and 1-particle states. The former is a semiclassical concept without a fully precise meaning, the latter a precisely defined field-theoretic term. Or, in your terms: The former is an interpretation in the real world, the latter a mathematical concept.

http://en.wikipedia.org/wiki/Particle :
''In the physical sciences, a particle is a small localized object to which can be ascribed several physical properties such as volume or mass.''

 

[A. Neumaier]

A full definition of a specific theory includes all of the following:

1. Definitions of mathematical terms.
   
2. A set of statements that tells us how to use a piece of mathematics, and a given set of definitions of terms for measuring devices, to make predictions about results of experiments.
   
3. Definitions of terms for measuring devices.

Step 1 will appear to be short, but only because we choose not to write down definitions that we expect mathematically sophisticated readers to already be familiar with. A full specification of step 1 would include definitions of terms like "function" and "integral", and even definitions of terms like "proof", from mathematical logic.

The usual convention in mathematics is that when giving axioms, one only states what is beyond the stuff already defined earlier. This is what makes an axiom system concise.
You only need to say which concept you give physical names. Thus, given all current mathematics, this step _is_ short (and doesn't only appear so).

Step 2 will actually be short. The statement "a clock measures the proper time of the curve in spacetime that represents its motion" is a good example of the sort of thing we will see in step 2. This particular statement tells us (when combined with the rest of steps 1-2) how to use clocks to find out how accurate some of the theory's predictions are, but it doesn't tell us what a clock is. That's why steps 1-2 only defines a class of theories. A theory must be falsifiable, but we need step 3 to get falsifiability. Each definition of the term "clock" would give us a different theory in the class of theories defined by the list of step 2 statements.

Step 2 is what I call interpretation rules. There is little point calling them axioms, since - unlike axioms, which must be self-explaining given what has been defined before - they are nothing precise but a guide to relate the formal terms to stuff considered known according to the current social conventions.

Step 3 is anything but short. It tells us e.g. what measuring devices we should call "clocks". An instruction manual that describes how to build a cesium clock would of course be very long. To understand step 3, one must understand the refinement process I talked about in #97.

This cannot be part of the foundation of a subject since in order to be precise, it needs the full-blown theory, developed on the basis of the foundation. If you require step 3 to be part of the foundations, everything becomes circular.

 

[Fredrik]

Step 2 is what I call interpretation rules. There is little point calling them axioms, since - unlike axioms, which must be self-explaining given what has been defined before - they are nothing precise but a guide to relate the formal terms to stuff considered known according to the current social conventions.

I like the term "axiom" because these rules are an essential part of the definition of a theory, and because they are postulated, not derived. The only reason I can think of to choose another term is that some people feel very strongly that the term shouldn't be used outside of pure mathematics. I like the term "interpretation rule", but it has problems too. I think it suggests too strongly that the interpretation rules for QM define an "interpretation of QM", when in fact they (together with the definitions of mathematical terms and terms for measuring devices) define the theory itself. An interpretation of the theory is defined by additional assumptions.

This cannot be part of the foundation of a subject since in order to be precise, it needs the full-blown theory, developed on the basis of the foundation. If you require step 3 to be part of the foundations, everything becomes circular.

The definitions of terms for measuring devices can't not be part of the definition of a specific theory, because scientific theories need to be falsifiable. The refinement process described in #97 isn't circular, it's just annoying.

(It's not circular because when you refine the definition of "clock" for example, you use theories based on the old definition).

 

[A. Neumaier]

I like the term "axiom" because these rules are an essential part of the definition of a theory, and because they are postulated, not derived. The only reason I can think of to choose another term is that some people feel very strongly that the term shouldn't be used outside of pure mathematics.

The main reason they shouldn't be called axioms is that axioms must be self-explaining (in earlier times one said ''self-evident''). If the axioms contain concepts that are more complex than what the axiom is supposed to explain, it is neither self-explaining nor self-evident.

I like the term "interpretation rule", but it has problems too. I think it suggests too strongly that the interpretation rules for QM define an "interpretation of QM", when in fact they (together with the definitions of mathematical terms and terms for measuring devices) define the theory itself. An interpretation of the theory is defined by additional assumptions.

One wouldn't need additional assumptions if the interpretation rules were clear in the first place. If the rules from step 2 would specify how the theory relates to reality, what would be the use of additional interpretations? Interpretation problems appear only in as far the interpretation rules are fuzzy or incomplete.

The definitions of terms for measuring devices can't not be part of the definition of a specific theory, because scientific theories need to be falsifiable.

If your argument were correct then my foundations (defined by Axioms A1-A6 and the interpretation rule MI) would not be falsifiable, since it has no definition of terms for measurement devices. But if MI were not satisfied in practice, my foundations would be falsified by what was already known in 1930.

The refinement process described in #97 isn't circular, it's just annoying.
(It's not circular because when you refine the definition of "clock" for example, you use theories based on the old definition).

This is an illusion. A theory based on the definition of a clock based on the rotation of the Earth will prove that a cesium clock is working irregularly, hence it cannot be used to define a cesium clock. If such a definition were used, it would have to be part of the theory, and hence be in conflict with the intended improved definition.

 

[Fredrik]

The main reason they shouldn't be called axioms is that axioms must be self-explaining (in earlier times one said ''self-evident''). If the axioms contain concepts that are more complex than what the axiom is supposed to explain, it is neither self-explaining nor self-evident.

I don't know what "self-explaining" means, but the idea that axioms should be "self-evident" is, as you say, from an earlier time. The modern view is of course that all axioms are part of a definition of something, not some sort of "obvious truths". (Yes, I know you didn't say that axioms are obvious truths, but you did use the term "self-evident", which means the same thing to me, even though it seems to mean something different to you).

If your complaint is that I'm using terms like "clock" in step 2, and not defining them until step 3, that problem is solved by a trivial reordering of the steps.

One wouldn't need additional assumptions if the interpretation rules were clear in the first place. If the rules from step 2 would specify how the theory relates to reality, what would be the use of additional interpretations? Interpretation problems appear only in as far the interpretation rules are fuzzy or incomplete.

The interpretation rules are meant to tell us (together with the definitions of mathematical terms and terms for measuring devices) how to interpret the mathematics as predictions about results of experiments. The interpretation rules shouldn't include assumptions that have no effect on the predictions, because then we can delete those assumptions and get a simpler theory that makes identical predictions.

An interpretation of QM is meant to tell us what "actually happens" to the system at all times, even at times between state preparation and measurement. This is a much more ambitious goal, and it clearly isn't accomplished by a minimal set of interpretation rules. The only thing that can define an interpretation of QM is an additional set of assumptions. These assumptions aren't supposed to change the theory's predictions, because if they do, they give us a new theory, not an interpretation of the one we already have. This is why interpretations of QM aren't science. (I'm not saying that interpretations are useless. I think of them in the same way I think about Venn diagrams. They are tools that can help us think about difficult things in a more intuitive way).

If your argument were correct then my foundations (defined by Axioms A1-A6 and the interpretation rule MI) would not be falsifiable, since it has no definition of terms for measurement devices. But if MI were not satisfied in practice, my foundations would be falsified by what was already known in 1930.

It's "satisfied in practice" precisely because physicists have used the refinement procedure described in #97 (and clarified below) to get a better and better idea about how to define a correspondence between measuring devices and mathematical observables. (OK, they haven't used exactly that procedure, but what they actually did is close enough. What I'm describing is, as it should, an idealization. The way it was actually done worked because the refinement process isn't too sensitive to the exact details). That's the "practice" part. Without it, your foundations wouldn't be falsifiable. But you haven't actually omitted the refinement procedure from your foundations. You have just hidden it in terms like "traditional cultural setting".

This is an illusion. A theory based on the definition of a clock based on the rotation of the Earth will prove that a cesium clock is working irregularly, hence it cannot be used to define a cesium clock. If such a definition were used, it would have to be part of the theory, and hence be in conflict with the intended improved definition.

That's not how the refinement process works. To go from the nth level in the hierarchy to the (n+1)th, you just write down a set of instructions on how to built a (n+1)th level measuring device that can be understood and followed by someone who understands the nth level theories and has access to nth level measuring devices. There's nothing circular about this.

 

[strangerep]

I
The interpretation rules are meant to tell us (together with the definitions of mathematical terms and terms for measuring devices) how to interpret the mathematics as predictions about results of experiments. The interpretation rules shouldn't include assumptions that have no effect on the predictions, because then we can delete those assumptions and get a simpler theory that makes identical predictions.

An interpretation of QM is meant to tell us what "actually happens" to the system at all times, even at times between state preparation and measurement. [...]

Your first paragraph above is an essential element of the scientific method.
But the second paragraph describes fiction and thus seems quite at odds
with the first. We can "delete" those parts of the interpretation that tell us
what "actually happens" between preparation and measurement and get a
"simpler theory that makes identical predictions".

 

[Fredrik]

Your first paragraph above is an essential element of the scientific method.
But the second paragraph describes fiction and thus seems quite at odds
with the first. We can "delete" those parts of the interpretation that tell us
what "actually happens" between preparation and measurement and get a
"simpler theory that makes identical predictions".

I agree of course, as you can tell from the comments I made immediately after the text you quoted.

The funny thing is that in classical mechanics, both kinds of interpretations are defined by the same statements. A statement like "x(t) is the position of the particle at time t" can actually be interpreted in three different ways: 1) as the definition of the mathematical term "position", 2) as a claim about what's "actually happening", and 3) as a prediction about results of experiments. In fact, I think most people would think of the third interpretation as a logical consequence of the second, because if a particle is at position coordinates x₀, the result of a position measurement should be x₀.

This has spoiled us into thinking that a good theory must tell us what "actually happens". In the early days of QM, a theory that does that was referred to as "a complete theory", and it was said that QM "can't be a complete theory". Now that we're more familiar with QM, I think it's more natural to say that there's a difference between making predictions about reality and describing reality, and that a theory only needs to do the former.

 

[A. Neumaier]

A statement like "x(t) is the position of the particle at time t" can actually be interpreted in three different ways: 1) as the definition of the mathematical term "position", 2) as a claim about what's "actually happening", and 3) as a prediction about results of experiments. In fact, I think most people would think of the third interpretation as a logical consequence of the second, because if a particle is at position coordinates x₀, the result of a position measurement should be x₀.

One expects the result of a position measurement of a classical point at x_0 to be only approximately equal to x_0...

Now that we're more familiar with QM, I think it's more natural to say that there's a difference between making predictions about reality and describing reality, and that a theory only needs to do the former.

What else could describing reality mean than making predictions about it?

We just have become content with demanding no unreasonable accuracy of our descriptions,
and we know better what ''unreasonable'' means.

 

[A. Neumaier]

I don't know what "self-explaining" means, but the idea that axioms should be "self-evident" is, as you say, from an earlier time. The modern view is of course that all axioms are part of a definition of something, not some sort of "obvious truths". (Yes, I know you didn't say that axioms are obvious truths, but you did use the term "self-evident", which means the same thing to me, even though it seems to mean something different to you).

''self-explaining'' (which, as I had said, replaces the now obsolete older ''self-evident'' = ''obvious'') means that the only terms not explained through the axioms themselves are terms that were already defined in prior theories that are considered more fundamental - such as logic for set theory, set theory for calculus, differential geometry for general relativity, and functional analysis for quantum mechanics.

Thus an axiom containing the term ''measured'' (or ''clock'') is self-explaining only if the full relational content telling the legal ways of using the term is defined through the axioms themselves, rather than through an interpretation rule. The latter only tells how the term thus defined is used in real life.

If your complaint is that I'm using terms like "clock" in step 2, and not defining them until step 3, that problem is solved by a trivial reordering of the steps.

My complaint is not about step 2 (which, as we agree, is short and consists of the interpretation rules - though you prefer to call them differently). My complaint is that you make step 3 an integral part of the ''theory''. This is not needed for falsifiability, and indeed, it is not the common view; cf. http://en.wikipedia.org/wiki/Scientific_theory

The interpretation rules are meant to tell us (together with the definitions of mathematical terms and terms for measuring devices) how to interpret the mathematics as predictions about results of experiments. The interpretation rules shouldn't include assumptions that have no effect on the predictions, because then we can delete those assumptions and get a simpler theory that makes identical predictions.

I'd call the latter 'illustrations'' or ''fantasies'', while you apparently call them collectively an ''interpretation of QM''.

An interpretation of QM is meant to tell us what "actually happens" to the system at all times, even at times between state preparation and measurement.

I wouldn't make any distinction between ''what actually happens'' and ''what is in principle testable''. The former is meaningless without the latter.

It's "satisfied in practice" precisely because physicists have used the refinement procedure described in #97 (and clarified below) to get a better and better idea about how to define a correspondence between measuring devices and mathematical observables.

That my MI is satisfied although it is conceivable that it couldn't be satisfied means that
MI is falsifiable but not falsified, hence is an excellent interpretation rule. This disproves your claim that in order to be falsifiable, a theory must both contain step 2 and 3 in its foundation.

But you haven't actually omitted the refinement procedure from your foundations. You have just hidden it in terms like "traditional cultural setting".

It cannot be otherwise - you can never completely describe the cultural setting; so its description must be reduced to the bare minimum needed to inform the mature participant in the culture. This is the only way to make it both short and universally agreeable - properties a foundation _must_ have to be intelligible. (According to _your_ requirements, there are no foundations for quantum mechanics - since we don't have steps 1, 2, and 3 assembled anywhere.)

 

[Fredrik]

One expects the result of a position measurement of a classical point at x_0 to be only approximately equal to x_0...

Obivously. I should perhaps have thrown in an "approximately" in there somewhere, but you know that I know that all position measurements are approximate, so there was no need to even mention this. This is going to be very frustrating if you're going to complain every time you can instead of when it adds something to the discussion.

What else could describing reality mean than making predictions about it?

In that sentence, "description of reality" refers to a collection of statements that (attempts to) tell you what "actually happens" to the system at all times. But you knew that already.

Thus an axiom containing the term ''measured'' (or ''clock'') is self-explaining only if the full relational content telling the legal ways of using the term is defined through the axioms themselves, rather than through an interpretation rule.

I don't understand your objection. Take "a clock measures the proper time of the curve in spacetime that represents its motion" as an example. The only terms that aren't defined by the other steps are "measures" and "represents". The meaning of "measure" is part of what we already know. We don't explain it for the same reason that we don't explain what a function is. I would however consider it appropriate to explain the concept further in a text that describe features that all theories have in common. The same thing goes for "represents". The idea that mathematical concepts can represent real-world concepts is the most fundamental idea in all of physics.

If you have a point here that I still don't see, it would still only be an argument against the term "axiom", to be weighed against similar arguments against the alternatives. Ultimately it comes down to a matter of taste.

My complaint is that you make step 3 an integral part of the ''theory''. This is not needed for falsifiability, and indeed, it is not the common view; cf. http://en.wikipedia.org/wiki/Scientific_theory

It's obviously impossible to test the accuracy of a prediction about say, the speed of an object in free fall, without a specification of what sort of device measures velocity (or lengths and times separately). If Wikipedia says otherwise, they're wrong.

I'd call the latter 'illustrations'' or ''fantasies'', while you apparently call them collectively an ''interpretation of QM''.

I would be fine with those terms too, but I wonder why you think an interpretation of QM is something different? Don't you see e.g. that David Mermin's suggestion that reality is described by correlations between subsystems is neither derived from QM nor a part of its definition? How about the idea that different terms (in an expression for the the state vector in terms of an orthonormal basis, or a state operator in terms of pure states) represent actual, different universes? These are the sort of claims that are made by advocates of different "interpretations of QM". They go far beyond what's needed to make predictions.

That my MI is satisfied although it is conceivable that it couldn't be satisfied means that
MI is falsifiable but not falsified, hence is an excellent interpretation rule. This disproves your claim that in order to be falsifiable, a theory must both contain step 2 and 3 in its foundation.

You have just hidden step 3, not omitted it.

you can never completely describe the cultural setting

You can describe an idealized process that if it had been carried out to the letter, would have given us results no worse than the process that was actually carried out to give us the current "cultural setting". It's pointless to be concerned about deviations from the idealized process as long as the predictions produced using the current cultural setting agree with experiments.

 

[A. Neumaier]

Obivously. I should perhaps have thrown in an "approximately" in there somewhere, but you know that I know that all position measurements are approximate, so there was no need to even mention this. This is going to be very frustrating if you're going to complain every time you can instead of when it adds something to the discussion.

I thought someone who works hard towards understanding a real proof of the spectral theorem rather than be content with the hand-waving derivation of a typical theoretical physics course would appreciate a discussion in precise terms, especially when it costs little compared to what is needed for rigorous proofs - only a few words of diligence here and there.

In that sentence, "description of reality" refers to a collection of statements that (attempts to) tell you what "actually happens" to the system at all times. But you knew that already.

But what actually happens is, in my mind, identically to what can be predicted given what can be said unambiguously about a system. How can you say something ''actually'' happens if you have no unambiguous way of expressing it? But if you can express it, you can predict it.
So the ''description of reality'' is synonymous.with ''what can be predicted from a complete knowledge of the state of a system''.

I don't understand your objection. Take "a clock measures the proper time of the curve in spacetime that represents its motion" as an example. The only terms that aren't defined by the other steps are "measures" and "represents".

The term ''clock'' is not defined either.

The meaning of "measure" is part of what we already know.

In place of a definition you refer to a social convention. So you do precisely the same as what I do in MI. But you criticize me for having only an incomplete definition of the theory.

We don't explain it for the same reason that we don't explain what a function is.

No. The reasons are very different.

We don't explain what a function is because we have already defined it in axiomatic set theory. Thus we have a very clear notion of a function, and don't need a further explanation.

Whereas our notion of measuring is very fuzzy; we don't explain it because we cannot. (We can't tell precisely what counts as a measurement device, when a measurement begins or ends, what should be the value of a measurement if two different people get slightly different pointer readings - should the result of the more skilled person count, or an average taken? In the latter case, the geometric or the arithmetic mean? Etc.. It is impossible to reach agreement in the community, except under a dictatorship or under pressure to agree on something definite - a pressure not present in our current social environment.)

It's obviously impossible to test the accuracy of a prediction about say, the speed of an object in free fall, without a specification of what sort of device measures velocity (or lengths and times separately).

I don't care about the speed of an object in free fall - we are discussing in the thread ''Axioms of quantum mechanics''. And we agreed already that it _is_ possible to test the accuracy of my description of Quanrtum mechanics based on Axions A1-A6, the single interpretation rule MI,
and an informal understanding of the practices mentioned in MI. This informal understanding is of the same kind you assumed above when telling me that there is no need to define the meaning of "measures" and "represents". With the same argument, there is no need to define the meaning of the terms I used in MI.

I would be fine with those terms too, but I wonder why you think an interpretation of QM is something different?

It is something very different. I think an interpretation of QM is an attempt to solve certain real puzzles that must be solved in the axioms (your step 1) and interpretation rules (your step 2) rather than in illustrations and fantasies, as your remark would suggest. Once a solution is found that satisfies the community (and not before that), the interpretation problem will subside and interest will be so small as current interest in the foundations of classical mechanics -which is the desirable, healthy state.

I spent years to figure out how quantum mechanics should be interpreted to make sense rationally, and studied all the interpretations in detail to find out what they contribute to understanding QM. And some of them did, though my own resulting interpretation (the thermal interpretation of thermodynamics, see the entry ''Foundations independent of measurements'' of Chapter A4 of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html#found0 ) leaves hardly any trace of how I reached my insights.

You have just hidden step 3, not omitted it.

In my interpretation rule MI, I haven't hidden more than you swept under the carpet by saying that there is no need to define the meaning of "measures" and "represents".

 

[Fredrik]

I think we have come as far as we can. To discuss this further will not bring us to an agreement about what the best way to handle foundational issues is. But I appreciate that you forced me to clarify some of my points, in particular step 3, because now I understand it better than I did before. So this has definitely been a useful discussion, even though we didn't reach an agreement.

I'm curious about one thing though. What "real issues" would you say that QM has? If QM is a theory only in the sense that it makes predictions about probabilities of possible results of experiments, and those predictions agree with experiments, how can it have issues? I don't think it has any issues other than those that are caused by additional assumptions of the sort that you described as "fantasies".

 

[A. Neumaier]

I'm curious about one thing though. What "real issues" would you say that QM has? If QM is a theory only in the sense that it makes predictions about probabilities of possible results of experiments, and those predictions agree with experiments, how can it have issues? I don't think it has any issues other than those that are caused by additional assumptions of the sort that you described as "fantasies".

The real issue is that the foundations are far from being precise enough to get a consensus about the meaning of QM. My interpretation rule MI is the minimal consensus that _every_ interpretation respects, and it is enough to make many predictions that agree with experiments. But only a small number of experiments that are actually performed fall under that minimal consensus.

The real issue therefore is to augment MI in such a way that it is comprehensive and yet acceptable to everyone.

For currently, beyond the minimal consensus, there are only a number of ad hoc rules employed by experimenters and theorists that force the experiments somehow (and intuitively) into the framework of the theory, and there is where the muddy waters begin. For example, in reality, many quantum observations violate the MI assumption of being independently prepared. But people close the eyes and pretend that they have an ensemble to which the standard Born rule and its consequences can be applied. But this no longer follows from the axioms as stated anywhere I know. And there are many more such issues.

It took me a long time before I began to understand what is going on there, and I am still perfecting my views on this - until I'll be able to write a book that is so clear that people will say: yes, of course, this way QM makes sense.

The discussions here are part of this clarification process: They force me to express myself more clearly than I'd do without the corrective coming from the responses. They indicate to me that I wasn't yet clear enough.

In my mind, QM is far from being the weird theory it is often pictured as. On the contrary, it is a very orderly, intuitive theory in which one can think almost classically if one uses the right visualization. My axioms and interpretation are already far stronger and far less idealizing than those I found in the literature. And I see that some of what I say is already persuasive. As long as the persuasive power of my arguments is still increasing, it makes sense for me to continue such discussions. In the end, quantum mechanics and its interpretation will be as crystal clear as classical Hamiltonian mechanics is.

 

[dextercioby]

[...]It took me a long time before I began to understand what is going on there, and I am still perfecting my views on this - until I'll be able to write a book that is so clear that people will say: yes, of course, this way QM makes sense.
[...]

That's a very ambitious plan you have. I hope you mean to write a book for the somewhat knowledgeables (i.e. for those one which already took a formal/superficial course with all the typical textbook examples which normally lack any mathematical rigorosity or abound in hand-waving arguments) and not for the beginners. That would interest me

In the end, quantum mechanics and its interpretation will be as crystal clear as classical Hamiltonian mechanics is.

That would be really interesting, if made true someday.

 

[A. Neumaier]

That's a very ambitious plan you have. I hope you mean to write a book for the somewhat knowledgeables (i.e. for those one which already took a formal/superficial course with all the typical textbook examples which normally lack any mathematical rigorosity or abound in hand-waving arguments) and not for the beginners. That would interest me

That would be really interesting, if made true someday.

You can get an idea of what I am aiming at if you look at

Arnold Neumaier and Dennis Westra,
Classical and Quantum Mechanics via Lie algebras,
2008.
http://lanl.arxiv.org/abs/0810.1019

It is the draft of an almost finished book (not yet 'the one' I envisioned in my previous mail, but one must work in stages to see what is feasible). Should you or anyone else here read it, I'd appreciate being informed by email (address at my home page) about inaccuracies and suggestions for improvements.

As you'll see, it starts off assuming a little familiarity with physics. But everything used (beyond elementary linear algebra and calculus up to partial derivatives) is actually defined; so a dedicated reader can use it for self-study. The requirements get high only late in the book. Indeed, I know of several 16 year olds who enjoyed reading large fractions of the book. On the other hand, the book already contains some new points of view even for experts.

Chapter 7 contains a first approximation to my thermal interpretation of quantum mechanics. It is much more realistic and down to Earth than anything else I have seen.

My goal for the future foundational book is to have something that nicely explains the world as wee see it, starting from basic axioms (like those presented in post #5 here),
and an exposition of QED (so no nuclear and subnuclear physics). Still missing (compared to the above draft) is most of field theory -- in particular, a rigorous version of QED (the hardest thing, since the literature does not even contain a rigorous definition of what QED should be) and nonequilibrium statistical mechanics (where I have lots of notes but not yet a coherent write-up), which is what will make everything realistic and close to macroscopic physics. In particular, it will be able to describe realistic measurements.

Unfortunately, since I have a full-time job as a math professor, working there on very different topics, work on this is slow. But I make steady progress each time I have a few weeks to concentrate on it.

 

[dextercioby]

I will try to make time to read on your work and will address any possible issues I may find in private, because the guidelines of this forum prohibit us from discussing WIP, but only either published books or published articles in peer-reviewed journals. Your referenced item doesn't currently classify to any of the 2 categories allowed for public debate.

For the record, I think your decision to join the forum made a remarkable increase of quality to the content written here. Also for the record, I remember your name from 2003-2005 when this forum was sharing posts published on http://groups.google.com/group/sci.physics.research/topics. Now the link has gone, but thankfully you joined PF.

On topic now, I remember one objection I made to your set written in post #5, namely not postulating the unique feature of systems of identical particles. If you describe the states by von Neumann density operators, how will this operator 'capture' the symmetrization/antisymmetrization of the tensor product of spaces ? I'm now referring to your statements in post #56 which do not contain a satisfactory answer for me to the questions I raised. Moreover, why shouln't the description of these particular systems be axiomatized ? Can it be then derived from another axioms ?

Thank you

Daniel

P.S. Another side note: if you're a colleague of Prof. Georg Teschl, and as I highly appreciate his work/book on quantum mechanics (it could be viewed as a complement of E. Prugovecki's 1970 book in the sense of providing the Hilbert space solution to the H atom in its simplest quantum mechanical description (I have't seen it in the literature in other place)), it would be nice or convenient for me, if the work you're preparing would have the same mathematical depth as his and as your mentioned book draft.

 

[strangerep]

I will try to make time to read (Arnold's) work and will address any possible issues I may find in private, because the guidelines of this forum prohibit us from discussing WIP, but only either published books or published articles in peer-reviewed journals. Your referenced item doesn't currently classify to any of the 2 categories allowed for public debate.

Although this is correct, under strict interpretation of PF guidelines, I'd be quite
disappointed to be locked out of such discussions as a consequence.

Arnold, it seems to me that such discussions of your public papers which are not
yet published in peer-reviewed places could "legally" take place over in
the Independent Research forum. Indeed, I'm sure such discussion would raise
the overall quality there, as your contributions here have done.

 

[A. Neumaier]

I will try to make time to read on your work and will address any possible issues I may find in private, because the guidelines of this forum prohibit us from discussing WIP, but only either published books or published articles in peer-reviewed journals. Your referenced item doesn't currently classify to any of the 2 categories allowed for public debate.

In view of the comment of strangerep, you might want to open a thread in the Independent Research forum. This must be justified, so you should include some background information such as the one given in your current post. Then that forum would discuss the content matter, while for things such as reporting misprints or making minor suggestions, you should use email.

On topic now, I remember one objection I made to your set written in post #5, namely not postulating the unique feature of systems of identical particles. If you describe the states by von Neumann density operators, how will this operator 'capture' the symmetrization/antisymmetrization of the tensor product of spaces ? I'm now referring to your statements in post #56 which do not contain a satisfactory answer for me to the questions I raised. Moreover, why shouln't the description of these particular systems be axiomatized ? Can it be then derived from another axioms ?

Instances satisfying the axioms are usually not part of the axiom system.
For example, we have the axioms for groups, and then we have - as part of the group theory built on top of that - special constructions such as the groups Sp(n,C), which are instances of a group but of course with additional structure.

Therefore, all parenthetical remarks in my axioms (including the one in A4 mentioning distinguishable particles) are not part of the axiom system but comments for the readers so that they associate the right intuition with the axioms. The list of examples given only has illustrative character and is far from being exhaustive.

I didn't mention indistinguishable particles in my examples for two reasons:
1. One cannot easily specify the set of relevant observables without introducing lots of additional notation or terminology - whereas the explanations of the axioms should be very short.
2. I think that the concept of indistinguishable particles is completely superseded by the concept of a quantum field. The latter gives much better intuition about the meaning of the formalism, and the former (which is difficult to justify and even more difficult to interpret intuitively) is then completely dispensable.

If you are interested in how I think about indistinguishable particles, read Example 5.1.8(iii) on p.99 of the draft of my book, and the discussion of post #25-#41 in the thread https://www.physicsforums.com/showthread.php?t=471125 , as far as it concerns indistinguishable particles. If this doesn't explain enough, please start a new tread with a specific question.

P.S. Another side note: if you're a colleague of Prof. Georg Teschl,

I don't know Georg Teschl, but have a colleague called Gerald Teschl whose office is two doors from mine.

and as I highly appreciate his work/book on quantum mechanics (it could be viewed as a complement of E. Prugovecki's 1970 book in the sense of providing the Hilbert space solution to the H atom in its simplest quantum mechanical description (I have't seen it in the literature in other places),

This is only the tip of an iceberg. Read:
-- Chapter 21 of: BG Wybourne, Classical groups for physicists, Wiley 1974.
-- Section 4.1 and 4.2 of: Thirring, A course in mathematical physics, Vol. III.
-- B. Cordani, The Kepler Problem, Birkh"auser 2003.
-- Barut and Raczka, Theory of group representations and applications, Warszawa 1980.
(The last book has many group-based exercises on the hydrogen atom; probably in Chapter 12 or 13. But I don't have the book here, hence can't check.)

Section 17.5 of my draft book briefly summarizes what's going on. (We were running out of time. The finshed book will have a thorough treatment.)

it would be nice or convenient for me, if the work you're preparing would have the same mathematical depth as his and as your mentioned book draft.

I prefer to express physics in more elementary terms than he does, but the level of rigor should be the same.

 

[A. Neumaier]

the unique feature of systems of identical particles. [...] why shouln't the description of these particular systems be axiomatized ? Can it be then derived from another axioms ?

The most proper way to give an axiomatic approach to the whole of quantum physics is to give a formal definition of the state of the universe, and then to derive everything else from that - since everything we observe is part of the universe, hence must be encoded in its state. I am working towards this goal, but this requires quantum field theory, and as I said, this part of my foundations is far from finished.

 

[dextercioby]

I don't know Georg Teschl, but have a colleague called Gerald Teschl whose office is two doors from mine.

Yes, of course. Sorry, I didn't check his full name.

This is only the tip of an iceberg. Read:
-- Chapter 21 of: BG Wybourne, Classical groups for physicists, Wiley 1974.
-- Section 4.1 and 4.2 of: Thirring, A course in mathematical physics, Vol. III.
-- B. Cordani, The Kepler Problem, Birkh"auser 2003.
-- Barut and Raczka, Theory of group representations and applications, Warszawa 1980.
(The last book has many group-based exercises on the hydrogen atom; probably in Chapter 12 or 13. But I don't have the book here, hence can't check.)

Thank you for the references. The bolded one looks very interesting.

 

[A. Neumaier]

If you describe the states by von Neumann density operators, how will this operator 'capture' the symmetrization/antisymmetrization of the tensor product of spaces ?

The latter is already encoded in the Hilbert space. Any Hermitian, positive semidefinite, linear operator on the N-particle sector of Fock space with trace 1 automatically represents N correctly (anti)symmetrized indistinguishable particles.

 

[A. Neumaier]

I think we have come as far as we can. To discuss this further will not bring us to an agreement about what the best way to handle foundational issues is. But I appreciate that you forced me to clarify some of my points, in particular step 3, because now I understand it better than I did before.

Let me comment your step 3 with a quote from John Bell, taken from Mermin's paper http://arxiv.org/pdf/quant-ph/0612216 :

''Here are some words which . . . have no place in a formulation with any pretension to physical precision: system, apparatus, environment, microscopic, macroscopic, reversible, irreversible, observable, information, measurement. On this list of bad words the worst of all is “measurement”. . . . What exactly qualifies some physical systems to play the role of “measurer”? . . . The word has had such a damaging effect on the discussion, that I think it should now be banned altogether in quantum mechanics.''

But you want to have ''measurement'' figure very prominently in the foundations. You even accept it as self-evident, without needing the slightest explanation:

The only terms that aren't defined by the other steps are "measures" and "represents". The meaning of "measure" is part of what we already know. We don't explain it for the same reason that we don't explain what a function is.

 

[A. Neumaier]

In view of the comment of strangerep, you might want to open a thread in the Independent Research forum. This must be justified, so you should include some background information such as the one given in your current post. Then that forum would discuss the content matter, while for things such as reporting misprints or making minor suggestions, you should use email.

After reading the moderation procedure for the Independent research Forum, it seems better that I'd open this thread. Please let me know (here) whether you have already started to prepare something, and if yes please send me your draft (by email), so that I can build on it.

I didn't mention indistinguishable particles in my examples for two reasons:
1. One cannot easily specify the set of relevant observables without introducing lots of additional notation or terminology - whereas the explanations of the axioms should be very short.
2. I think that the concept of indistinguishable particles is completely superseded by the concept of a quantum field. The latter gives much better intuition about the meaning of the formalism, and the former (which is difficult to justify and even more difficult to interpret intuitively) is then completely dispensable.

If you are interested in how I think about indistinguishable particles, read Example 5.1.8(iii) on p.99 of the draft of my book, and the discussion of post #25-#41 in the thread https://www.physicsforums.com/showthread.php?t=471125 , as far as it concerns indistinguishable particles.

See also posts #54-#60 from https://www.physicsforums.com/showthread.php?t=473423

If this doesn't explain enough, please start a new tread with a specific question.

For the benefit of everyone, I'll start a new thread on indistinguishable particles putting everything together in one place. See
https://www.physicsforums.com/showthread.php?t=474321
https://www.physicsforums.com/showthread.php?t=474293

 

[Fredrik]

Let me comment your step 3 with a quote from John Bell, taken from Mermin's paper http://arxiv.org/pdf/quant-ph/0612216 :

''Here are some words which . . . have no place in a formulation with any pretension to physical precision: system, apparatus, environment, microscopic, macroscopic, reversible, irreversible, observable, information, measurement. On this list of bad words the worst of all is “measurement”. . . . What exactly qualifies some physical systems to play the role of “measurer”? . . . The word has had such a damaging effect on the discussion, that I think it should now be banned altogether in quantum mechanics.''

But you want to have ''measurement'' figure very prominently in the foundations. You even accept it as self-evident, without needing the slightest explanation:

Theories need to be falsifiable. To be falsifiable, they need to make predictions about results of measurements. I don't think the argument needs to be more complicated than that.

Bell is expressing his dissatisfaction with the fact that QM, as defined by a typical list of axioms, looks like a set of rules that tells us how to calculate probabilities of possibilities, instead of like a description of what actually happens (i.e. an interpretation/ontology/illustration/fantasy). When I read his statement, I see what kind of theory he was wishing for, but I don't see a reason to think that such a theory exists.

 

[A. Neumaier]

In view of the comment of strangerep, you might want to open a thread in the Independent Research forum.

After reading the moderation procedure for the Independent research Forum, it seems better that I'd open this thread.

To prepare for this, I decided to put a newer version of the book on the arXiv, but it turned out that to turn my current intermediate version into something reasonably coherent required more work on my part, and I am not yet finished with that. So it will take a bit longer before the (new version of the) book is on the arXiv, ready for discussion.

 

[A. Neumaier]

To prepare for this, I decided to put a newer version of the book on the arXiv, but it turned out that to turn my current intermediate version into something reasonably coherent required more work on my part, and I am not yet finished with that. So it will take a bit longer before the (new version of the) book is on the arXiv, ready for discussion.

A discussion forum for discussing the much expanded version 2 of the book has been approved: https://www.physicsforums.com/showthread.php?t=490492
Please post your comments there.

 

[andrebourbaki]

As you said in post 133,

Theories need to be falsifiable. To be falsifiable they must predict the results of
measurements. (Or predict the observed probabilities of obtaining various results of measurements.) Or, predict the results of experiments, and an experiment must be replicable, and these probabilities are indeed replicable although the individual results are not so replicable. We are, so far, in agreement, and so is John Bell.

But, you do not appreciate Bell's concern: a theory must also be a theory, i.e., precise and unambiguous. His complaint is that the theory, in particular the axioms, do not state what kind of system produces a measurement, and so the relation between 'system' or 'Hamiltonian' and measurement is not clear, but the practicioner is left free to choose
whether to treat a Geiger counter as a quantum system with a Hamiltonian, or as a measurement apparatus, and get two different answers: in the former, after the measurement, the electron is in an entangled state with the Geiger counter, but with the latter, it is in one definite separable state. These two predictions are hard to falsify, but
they are logically contradictory so the usual six axioms don't constitute a 'theory', let alone a falsifiable theory. A theory does not need good taste, skill, etc. to be employed...that is his point, in principle.

 

[Fredrik]

I don't see any of that as a problem. My view is that a theory only needs to assign probabilities to measurement results, given a preparation procedure and a measurement procedure. This means that a definition of a specific theory consists of a purely mathematical part, and a set of correspondence rules that tell us which measuring devices the probability assignments apply to. The usual axioms of QM only define the purely mathematical part of the theory.

 

[andrebourbaki]

If the theory does not tell you which set of rules to apply when, and this is Bell's point, then it is not the theory which is making the predictions, it is the user. That is, QM requires flair, savoir-faire, good taste, it is not a 'theory' in the precise sense of the word.
Bell agreed in print that for all practical purposes, this is not a problem. But from the logical point of view, it is a problem that there is no way to decide what is a measurement device. This problem may become practical very soon, and in two ways. What if one, with nanotechnology, produced a meso-scopic geiger counter? It would then be seen that the measurement axioms were only approximate. The second one is, the measurement part of the theory (observables, probability, reduction of the wave packet) have never been satisfactorily extended to the relativistic regime. What if the two observers attempted to
perform the same measurement at very different speeds and got different results...
Bell disagreed with your comment that he wanted a theory which gives one a picture of reality, he tried to emphasise that that was not his critique. It is this overlap in the applicability of the axioms that was his complaint, and Wigner's too. The theory itself ought to specify precisely when to apply the measurement axioms and reduction and when not to.

 

[Fredrik]

As I said, the way I see it, the theory consists of a purely mathematical part, and a set of correspondence rules. Only the purely mathematical part is covered comprehensively in QM books. The purely mathematical part can't possibly tell you what the measuring devices are. That's the sort of stuff that's covered by the correspondence rules, which unfortunately, aren't covered comprehensively anywhere. I guess that's what you're talking about, but using a slightly different terminology.

I don't think of this as a problem with QM. It's just an annoying but unavoidable feature of science.

 

[andrebourbaki]

We are using a slightly different terminology. We agree that both the math and the 'correspondence rules' are part of the theory. If the 'correspondence rules' are not covered comprehensively *anywhere*, then you are, with different terminology, conceding Bell's point 'the theory does not tell us ...' You also add that for you, this is not a problem. Bell agrees that *for you* this is not a problem, he also agrees that for all practical purposes it is not a problem.

But. Is there a principled, fundamental obstacle to the 'correspondence rules' *ever* being written down, are they, in principle, incapable of being written down? If so, then the theory cannot be written down.

Bell thought that of course the practice of Physics can never be cut and dry, with all procedures written down in advance, he explicitly allowed that the art of finding workable approximations which permit of making practical predictions requires flair, good taste, etc. But I hope you will be fair to his point of view: if even the fundamental theory (six axioms and a few correspondence rules, say for geiger counters and bubble chambers) cannot, even in principle, be stated clearly and correctly in language in a theory, then Fundamental Physics is not theoretical, the theory can never exist, it is therefore illogical not in the sense that it asserts a contradiction or falsity but in the sense that it cannot be expressed logically.

My experience is that half the physicists in the world are Aristotelians and agree with you, and would not be troubled in saying that even fundamental physics evades or transcends logic, or even that there can never be any such thing as fundamental physics, or an exact truth, or a final theory... (this list represents an increasingly radical degree, not everyone would go all the way down the list). But the other half are, like Weinberg and Dirac, Platonists and would agree that Bell's point, if valid, is a defect that hopefully will be fixed eventually.