What theories address the fundamental questions about quantum mechanics?

[Fredrik]

So, after this clarifying dispute, please look again at my axioms in post #5 and find out that I give 5 short axioms that specify the basic concepts in a concise and complete way (both much more specific and much more realistic than what I found in the literature), not referring at all to measurement, and then a single interpretation rule that specifies the minimal consensus among all interpretations of quantum mechanics I know of.

Born's rule is not among the axioms or interpretation rules but follows under the assumptions under which it can be expected to hold in practice.

What you're saying in #5 looks OK-ish to me, but (as you know) I wouldn't think of it as defining a theory of physics. A1 and A2 makes a connection with reality by saying (A1) that each physical system is associated with a Hilbert space, and (A2) that the properties of a system are represented by a state operator on its Hilbert space. But then you're essentially just defining a few more mathematical terms. I guess that's OK, since you have made it clear that your intentions are to define the mathematical aspects of the theory.

So let's look at the individual axioms:

A1. Most people include the requirement that the Hilbert space be separable. Why didn't you? Note btw that it can be a little bit weird to say that each system is associated with a separable Hilbert space, since all infinite-dimensional separable Hilbert spaces are isomorphic. I understand that an appropriate choice of Hilbert space will make it easier to define the operators we're interested in, but you might want to add something like that to the comments.

A2. You haven't yet imposed the "isolated from its environment" requirement. Did you intend the state operator (=density matrix) to be a "reduced density matrix" in those cases when it's not? (I think the only justification for using reduced density matrices is the Born rule, so I'm concerned about circularity). I also think that the assumption that the state operator represents the properties of the system, instead of the properties of an ensemble, is too strong. If it represents the properties of a system, it also represents the properties of an ensemble, and the claim that it represents the properties of an ensemble is strong enough. I wouldn't use an axiom that's stronger than it needs to be, and I think this one in particular will get us into philosophical difficulties. I think it might even be provably wrong (although I haven't proved it) since these operators can represent mixed states as well as pure states.

A3. This just defines a term, so there's not much to say.

A4. I would describe this as a comment, not an axiom. It might as well be moved into the discussion following the axioms.

A5. When I said that the axioms need to include a version of the Born rule, you protested and said that your axioms contain something that it can be derived from instead. I would say that this is a version of the Born rule, or at least that it will be turned into one when it's explained that the expectation corresponds to the average value of a long series of measurements.

A6. This also sounds like a comment, not an axiom.

 

[Fredrik]

It is not unreasonable to have such a book. But it _is_ unreasonable to have a book-sized axiom system.

Foundations should be concise, unambiguous, and simple.

I agree, and I think the appropriate way to deal with it is to

1) State the mathematical axioms and definitions. (This will be concise, unambiguous, and simple. Maybe even unnecessary, since these things should already appear in math books).

2) State the physical axioms, i.e. the statements that specify how to interpret the mathematics as predictions about results of experiments, given an identification between specific mathematical observables and specific measuring devices. (This will be concise, simple, and not ambiguous enough to cause any problems as far as I can tell).

3) State the identification between specific mathematical observables and specific measuring devices. (This one will be anything but concise and simple, and I don't know the best way to do it yet, but it's an essential part of physics, whether we like it or not).

The difference to assuming functional analysis is that measurement theory assume the very theory it is supposed to found, according to your view.

This problem is no less severe in your approach. You have just chosen to ignore the problem. As I said before, I don't know to what extent it can be resolved. I don't think anyone has even tried to think this through to the end, probably because philosophers don't understand physics, and physicists think philosophy is useless. (I think that's just a slight exaggeration).

You can criticize me all you want for explicitly mentioning something (step 3 above) that is obviously problematic, but the only reason you see a problem with what I'm saying and not with what everyone else is saying, is that this is an elephant in the room that everyone else is ignoring.

 

[A. Neumaier]

What you're saying in #5 looks OK-ish to me, but (as you know) I wouldn't think of it as defining a theory of physics.

So, if you combine the six axioms with the interpretation rule MI also stated in #5, what is missing that you require as essential for defining a theory of physics?

I don't think that _anything_ beyond what I wrote there is uncontroversial about quantum mechanics, whereas what I stated is used virtually everywhere in actual applications of quantum physics to real physical systems.

A1 and A2 makes a connection with reality by saying (A1) that each physical system is associated with a Hilbert space, and (A2) that the properties of a system are represented by a state operator on its Hilbert space. But then you're essentially just defining a few more mathematical terms. I guess that's OK, since you have made it clear that your intentions are to define the mathematical aspects of the theory.

Actually, the axioms make as much contact with reality as my former axioms for projective planes. Using the word ''system'' is no more contact to reality than using ''point'' or ''line'' when discussing projective planes - unless one pretends to know already what these terms mean. But the purpose of foundations is just that - to specify what the concepts should mean.

Only the explanations in parentheses starting with ''e.g.'' give some interpretative aid in how one should intuitively think of the formal terms. Strictly speaking, these are not part of the axiom system but serve as commentary by means of examples.

A1. Most people include the requirement that the Hilbert space be separable. Why didn't you?

I don't include this restrictive assumption for two reasons:
1. It is not needed for much of the development, and
2. The modelling of QED seems to require nonseparable Hilbert spaces to accommodate the infrared behavior (work by Kibble).

Note btw that it can be a little bit weird to say that each system is associated with a separable Hilbert space, since all infinite-dimensional separable Hilbert spaces are isomorphic. I understand that an appropriate choice of Hilbert space will make it easier to define the operators we're interested in, but you might want to add something like that to the comments.

It doesn't matter which of the Hilbert spaces one chooses, since they are all isomorphic.
The extra structure needed to do physics is actually encoded in my Axiom A4, which you had dismissed as a mere comment.

A2. You haven't yet imposed the "isolated from its environment" requirement.

The reason is that there is only a single isolated physical system that contains any of the things we are interested in - namely the whole universe (including the parts unobservable by us). All other physical systems are not isolated from the environment. Since my axioms shall be a foundation - I don't want to be more restrictive than necessary. Moreover, ''environment'' is an undefined, problematic term that I want to keep out of the foundations - it should figure only in the interpretation.

Did you intend the state operator (=density matrix) to be a "reduced density matrix" in those cases when it's not? (I think the only justification for using reduced density matrices is the Born rule, so I'm concerned about circularity).

I stated precisely what I intended to state - the axioms are the fruit of a long sequence of improvements. They do not contain a circularity, only requirements on how the various terms that I am using are related.

Note that real physical states are _always_ reduced density matrices since all physical systems
- with exception of only the universe as a whole - are part of a bigger system.

I also think that the assumption that the state operator represents the properties of the system, instead of the properties of an ensemble, is too strong.

I define in this axiom the meaning of the term ''particular system''. People today apply routinely quantum mechanics to a single 'ion in the ion trap on this particular desk', and describe its evolution by a density matrix. I don't want to exclude such standard usages from being covered.

But whether this ion is or isn't interpreted as an ensemble is again a matter of interpretation. Loading the axiom system with such interpretive issues would make it too vague (and too controversial) to serve as foundation. The term ''ensemble'', and what precisely constitutes one is too vague and controversial, hence should be not part of the axioms but of the interpretation. A system in the formal sense defined here may or may not be an ensemble in a conventional sense, depending on the precise meaning of the conventions followed.

I think this one in particular will get us into philosophical difficulties.

Since none of the axioms refer to reality but are on the same level as the axioms for projective planes, how can there be philosophical difficulties? There can only be logical consistency or a logical contradiction. But the axioms are consistent if set theory is consistent, since it is easy to give mathematical realizations of the axioms.

[

B]A3.[/B] This just defines a term, so there's not much to say.

This is a precise formal substitute for the vague ''isolated from the environment'' that you wanted to see in Axiom A2.

A4. I would describe this as a comment, not an axiom. It might as well be moved into the discussion following the axioms.

As already mentioned, this it what fills Hilbert space with life. It is an axiom since it gives the requirements on the usage of the term ''observable''. I do not require that all selfadjoint operators are observables, but that certain particular ones are. Which ones, and which properties (commutation rules) are assigned depends on the system, whence I gave a long
list of examples.

A5. When I said that the axioms need to include a version of the Born rule, you protested and said that your axioms contain something that it can be derived from instead. I would say that this is a version of the Born rule, or at least that it will be turned into one when it's explained that the expectation corresponds to the average value of a long series of measurements.

What you refer to is not commonly called Born's rule! http://en.wikipedia.org/wiki/Born_rule

Born's rule is applicable only to very idealized measurments: instantaneous, perfect (projective) measurements of a single observable with a discrete, fully known spectrum (see my discussion in post #1 of the thread https://www.physicsforums.com/showthread.php?t=470982 ).

But foundations of quantum mechanics should be applicable to the real world, hence should not depend on idealizations in their axioms. Moreover, the notion of a measurement is very vague, hence must be avoided in foundations that aim to be clear.

A6. This also sounds like a comment, not an axiom.

This is a nontrivial axiom, analogous to the induction axiom in Peano's axiom system for the natural numbers. It defines the meaning of the term ''quantum mechanical prediction'', and says more or less that only what can be concluded from Axioms A1-A5 without the use of additional assumptions is to be regarded as quantum mechanics.

You forgot to comment on the interpretation rule MI. It states the common ensemble interpretation in as unambiguous terms as possible, and is followed by a discussion of why it would be harmful to consider it as an axiom.

 

[A. Neumaier]

I
3) State the identification between specific mathematical observables and specific measuring devices. (This one will be anything but concise and simple, and I don't know the best way to do it yet, but it's an essential part of physics, whether we like it or not). [...]

You can criticize me all you want for explicitly mentioning something (step 3 above) that is obviously problematic, but the only reason you see a problem with what I'm saying and not with what everyone else is saying, is that this is an elephant in the room that everyone else is ignoring.

But the same elephant is in the room of general relativity. Nevertheless, there it causes hardly any problems because the separation of theory and interpretation is there much more thorough than in the quantum mechanical tradition. Once things are separated there, too, the controversies will also recede into th background.

 

[A. Neumaier]

A5. When I said that the axioms need to include a version of the Born rule, you protested and said that your axioms contain something that it can be derived from instead. I would say that this is a version of the Born rule, or at least that it will be turned into one when it's explained that the expectation corresponds to the average value of a long series of measurements.

What you refer to is not commonly called Born's rule! http://en.wikipedia.org/wiki/Born_rule

Born's rule is applicable only to very idealized measurments: instantaneous, perfect (projective) measurements of a single observable with a discrete, fully known spectrum (see my discussion in post #1 of the thread https://www.physicsforums.com/showthread.php?t=470982 ).

I was just rereading Born's 1926 paper (reprinted in English translation in pp.52-55 of the reprint volume ''Quantum Theory and Measurement'' by Wheeler and Zurek) - which introduced the probabilistic interpretation that earned him a Nobel prize.

To my surprise, his whole paper does nowhere refer to measurements or something equivalent! This implies that the most common form in which Born's rule is stated (namely the one given in the above wikipedia link) is not by Born but a later mutilated version that mixes Born's clear analysis with the muddy waters of the measurement problem.

In place of the wikipedia form of the rule, ''the probability of measuring a given eigenvalue lambda_i will equal <psi|P_i|psi>, where P_i is the projection onto the eigenspace of A corresponding to lambda_i'', Born has the following, which doesn't depend on anything being measured (let alone to be assigned a precise numerical measurement value):
''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''.

Unless the lambda_i are (as for polarization, spin or angular momentum in a particular direction - the common subjects of experiments involving Alice and Bob) system-independent, discrete, and known a priori - in which case one can label each measurement record with these numbers -, the wikipedia form of Born's rule is highly unrealistic.

 

[Fredrik]

Using the word ''system'' is no more contact to reality than using ''point'' or ''line'' when discussing projective planes - unless one pretends to know already what these terms mean.

I assumed that "system" referred to something in the real world, not something mathematical, but I see what you meant now. I think this is an unusual way to define the term, but I kind of like it actually. I think I would have chosen to be more formal about it, e.g. by saying that a system is a pair (K,H) where K and H are...what you said. That way A1 would look more like what it really is, a definition of a mathematical term.

Since none of the axioms refer to reality but are on the same level as the axioms for projective planes, how can there be philosophical difficulties?

If A2 just defines a term, then there obviously aren't any difficulties. I thought you were saying that a mathematical state represents all the properties of the real-world counterpart of what you call a "particular system". That would at least have been a controversial statement.

You forgot to comment on the interpretation rule MI. It states the common ensemble interpretation in as unambiguous terms as possible, and is followed by a discussion of why it would be harmful to consider it as an axiom.

To be honest, I had not read that far when I wrote my comments. I have read it now. My only objection is that I wouldn't have understood the phrase "statistically consistent with independent realizations of a random vector X with measure as defined in axiom A5" if I didn't already understand QM.

To my surprise, his whole paper does nowhere refer to measurements or something equivalent! This implies that the most common form in which Born's rule is stated (namely the one given in the above wikipedia link) is not by Born but a later mutilated version that mixes Born's clear analysis with the muddy waters of the measurement problem.

In place of the wikipedia form of the rule, ''the probability of measuring a given eigenvalue lambda_i will equal <psi|P_i|psi>, where P_i is the projection onto the eigenspace of A corresponding to lambda_i'', Born has the following, which doesn't depend on anything being measured (let alone to be assigned a precise numerical measurement value):
''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''.

That statement strongly suggests that particles have well-defined positions at all times. That's a very controversial suggestion given our current understanding of QM. I think his original idea has been improved, not mutilated.

 

[Fredrik]

1) State the mathematical axioms and definitions. (This will be concise, unambiguous, and simple. Maybe even unnecessary, since these things should already appear in math books).

2) State the physical axioms, i.e. the statements that specify how to interpret the mathematics as predictions about results of experiments, given an identification between specific mathematical observables and specific measuring devices. (This will be concise, simple, and not ambiguous enough to cause any problems as far as I can tell).

3) State the identification between specific mathematical observables and specific measuring devices. (This one will be anything but concise and simple, and I don't know the best way to do it yet, but it's an essential part of physics, whether we like it or not).

I've been thinking about step 3. I think the only way to describe this process is in terms of a hierarchy of theories. (Keep in mind that by my definitions, step 2 above is the main part of the definition of a specific theory, but step 3 is a part of it too...I will however have more to say about that at the end). You start with the definitions of the purely mathematical parts of a collection of theories (say pre-relativistic classical mechanics, SR and QM). Postulate a correspondence between mathematical observables and measuring devices in any way you can. You can e.g. define the term "clock" by a describing an hourglass or something, and define a "second" by saying that it's the time it takes a certain amount of sand to run through. A few such definitions is enough to define "version 1" of pre-relativistic classical mechanics and start using it to make predictions.

Experiments will show you that you're on the right track. So now you have a reason to believe that the theory says something useful. One of the things it tells you is that the swings of a pendulum take roughly the same time. So you redefine a "second" to be the time it takes a specific pendulum to swing away and back, and you define "version 2" of the theory with the term "clock" defined by a description of how to build a pendulum clock. This way you can continue to define new versions of the theory, each one more accurate than the previous version.

You do the same to your other theories, including QM. At some point, you will see that to go from version n to version n+1 of pre-relativistic classical mechanics, you will have to use a version of QM(!), because it's the predictions of (some version of) QM that justify the new definition of a second that we're going to use in version n+1 (a statement about radiation emitted from a cesium-137 atom). At this point we define the term "clock" by a description of how to build an cesium clock, and we won't be able to do that without using earlier versions of several theories, including pre-relativistic classical mechanics and QM.

So the process of refining the correspondence between mathematical observables and measuring devices involves a large number of steps, and it's also clear that theories aren't refined in isolation from each other.

Because of this, I'm going to retreat from my position that step 3 should be considered part of the definition of each specific theory. I will not retreat all the way back to step 1 however, because I don't think it's a good idea, or even consistent with the standard usage of the concept of falsifiability, to define a theory to be a collection of definitions of mathematical terms. The way I see it now, a theory of physics is defined by a set of statements (which I will continue to refer to as "axioms") that tells us how to use a given correspondence between mathematical observables and measuring devices to interpret some piece of mathematics as predictions about results of experiments.

 

[A. Neumaier]

So the process of refining the correspondence between mathematical observables and measuring devices involves a large number of steps, and it's also clear that theories aren't refined in isolation from each other.

Yes, the book on measurement I was referring to.

Because of this, I'm going to retreat from my position that step 3 should be considered part of the definition of each specific theory.

Ok - a step in the right direction.

I will not retreat all the way back to step 1 however, because I don't think it's a good idea, or even consistent with the standard usage of the concept of falsifiability, to define a theory to be a collection of definitions of mathematical terms. The way I see it now, a theory of physics is defined by a set of statements (which I will continue to refer to as "axioms") that tells us how to use a given correspondence between mathematical observables and measuring devices to interpret some piece of mathematics as predictions about results of experiments.

Now please analyze what is needed to do step 2 in an unambiguous way. Note that there are hundreds of measurement devices already for a distance, and none of them defines it to 1000 decimals of accuracy.

My take on this (as you can see from my axioms) is that I make the theory contain mathematically precise definitions of (Platonic, idealized) concepts with the same names as their less idealized, imprecise but familiar cousins in the real world. This makes step 2 part of step 1. In this way, theory can make fully precise statements without claiming in the axioms the least about correspondence to the real world. This correspondence is, however, strongly but informally suggested by the names of the concepts. But in case of doubt, it is the theory version of the concept that dictates its meaning.

Indeed - precisely this sort of definition and reasoning allows us to be able to discuss the quality of clocks - and to choose one particular that makes reality fit closest to the theory.
It is just what is done in practice: Defining clocks via the rotation of the Earth was found to lead to smaller deviations from physical laws than that of a town major church clock, and redefining them via cesium atoms was a further qualitative improvement. If the clock were part of the definition of the theory, the latter would change each time the standard for a clock were changes. But in physics, the latter is not considered a correction of the theory.

I'll reply to your other post separately (later).

 

[A. Neumaier]

I have read it now. My only objection is that I wouldn't have understood the phrase "statistically consistent with independent realizations of a random vector X with measure as defined in axiom A5" if I didn't already understand QM.

One needs an understanding of this already for interpreting classical measurements.
Measuring the position and momentum of a Newtonian particle (like Jupiter, one of the particles for which Newton created his theory) at various times gives random vectors z(t) with six components, and, these must be statistically consistent with the predictions of classical mechanics (that the initial data determine the whole curve) in a precise sense, requiring a little knowledge about means, standard errors, statistical independence, and the method of least squares - which I summarized in the some vague term ''statistically consistent''.

That statement strongly suggests that particles have well-defined positions at all times. That's a very controversial suggestion given our current understanding of QM. I think his original idea has been improved, not mutilated.

For Born's statistical statement to be interpretable, it is enough to assume that particles have approximately well-defined positions at all times. Indeed, traffic monitoring services routinely make statistics on the position and number of classical cars although the latter do not have well-defined positions at all times.

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.

 

[Fredrik]

*** Deleted ***

I'm not satisfied with what I said here, and I'm too tired to fix it now. I'll take care of it tomorrow.

 

[Fredrik]

After some additional thought, I think I have to return to my previous position, or rather a refined version of it. If I define specific theories by "steps 1-2" stuff, they won't be falsifiable, and that's unacceptable to me. I think the only good way out of this is to let terms like "special relativity" and "quantum mechanics" refer to classes of theories instead of specific theories. The members of each class are what I previously (in #97) referred to as "versions" of a specific theory.

A full definition of a specific theory includes all of the following: (Forget my previous steps 1-3. This list replaces the old one).

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.

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 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.

In spite of what I just said, I would still find it more than OK to call special relativity a "theory". This is to be understood as a sloppy way of referring to the specific member (of the class of theories) that's singled out by the best definitions of measuring devices that we have at the moment. This terminology isn't any more sloppy than e.g. what we're doing when we define a group as a pair (G,*) and then start referring to G as a "group".

 

[Fredrik]

For Born's statistical statement to be interpretable, it is enough to assume that particles have approximately well-defined positions at all times.

Yes, but why would we assume that? 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. If don't know if there are Bell inequalities for position, as there are Bell inequalities for spin, but since the violation of the latter completely rules out the possibility that the entangled state |↑>|↓>+|↓>|↑> represents "particle 1 is either in the ↑ state or the ↓ state and particle 2 is in the opposite state", I expect something similar to hold for position.

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.

I strongly disagree of course, because of what I said above. 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.

Better educated people than myself seem to agree:

Take the example of a position measurement on an electron. It woud lead to a host of paradoxa if one wanted to assume that the electron has some position at a given time. "Position" is just not an attribute of an electron, it is an attribute of the "event" i.e. of the interaction process between the electron and an appropriately chosen measuring instrument (for instance a screen), not of the electron alone. The uncertainty about the position of the electron prior to the measurement is not due to our subjective ignorance. It arises from improperly attributing the concept of position to the electron instead of reserving it for the event.​

(Rudolf Haag, Local quantum physics, page 2).

 

[dextercioby]

Better educated people than myself seem to agree:

Take the example of a position measurement on an electron. It woud lead to a host of paradoxa if one wanted to assume that the electron has some position at a given time. "Position" is just not an attribute of an electron, it is an attribute of the "event" i.e. of the interaction process between the electron and an appropriately chosen measuring instrument (for instance a screen), not of the electron alone. The uncertainty about the position of the electron prior to the measurement is not due to our subjective ignorance. It arises from improperly attributing the concept of position to the electron instead of reserving it for the event.​

(Rudolf Haag, Local quantum physics, page 2).

I would argue that this essentially means that the electron has no other observable, simply because any <observable> would be "an attribute of the "event" i.e. of the interaction process between the electron and an appropriately chosen measuring instrument (for instance a screen), not of the electron alone".

I don't see why Haag's claim can't be extended to any possible observable.

 

[Fredrik]

I would argue that this essentially means that the electron has no other observable, simply because any <observable> would be "an attribute of the "event" i.e. of the interaction process between the electron and an appropriately chosen measuring instrument (for instance a screen), not of the electron alone".

My view is that what a measuring device does is to first prepare a correlation between eigenstates of the observable to be measured and states with a reasonably sharp position, and then make a "position measurement". The latter is done by a component that when it interacts with a particle of the type it's designed to detect, produces a signal that tells us that the interaction has taken place.

The position of the interaction is at least as well-defined as the position of the "detector" (the component that the particle interacted with to produce the signal).

I don't see why Haag's claim can't be extended to any possible observable.

I'm not sure what it would mean to extend it to other observables. My thoughts on "properties" of quantum systems in general, is that a system can only be said to have a property if it's been prepared in a state such that the probability of a positive result in an experiment designed to test if it has that property is 1. So a system can be said to have a specific value of an observable A if and only if it has been prepared in an eigenstate of A.

 

[A. Neumaier]

Yes, but why would we assume that? .

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. If he works with an electron beam, he beliefs that the electrons in the beam are very close to the center of the cross section of the beam, and he can check that at any time by putting something in the way to measure it. Except if the source is very weak, when he needs to wait until an electron arrives. Whereas if he measures outside the beam, he'll find no electron there.

The electron in a hydrogen atom has a position very close to the proton together with which it forms the atom - by the very definition of a hydrogen atom. Since an electron has a position operator, we can compute its mean position (which happens to agree with the position of the nucleus), and the root mean square deviation, which gives (as everywhere in statistics) the uncertainty in the mean. The fact that we cannot determine the position more accurately is because the electron is delocalized (i.e., thinned out, extended in space) and has no more accurate position.

It is not _very_ different to the uncertainty of the position of a car that I had used as an example. One can give some reference position (e.g., the position of the center of the ash tray in the car, or the center of the front car axis, and an uncertainty that tells you that it is meaningless to define the position of the car (or any other object whose length is more than 1m) with an accuracy of higher than 1m or so. One doesn't need Heisenberg's uncertainty relation for this.

As the car has a well-defined position up to some accuracy determined by its size (but no better), so the electron has a well-defined position up to some accuracy determined by its
size (but no better). The difference is only that a quantum particle behaves much more like a compressible fluid than a car does, and can (just like a cloud) change its size dependent on the
state it is in. (In case this needs some extra justification: Even nuclei, which are much more rigid than electrons, are often described as a classical fluid: http://en.wikipedia.org/wiki/Nuclear_structure )

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.

If don't know if there are Bell inequalities for position, as there are Bell inequalities for spin, but since the violation of the latter completely rules out the possibility that the entangled state |↑>|↓>+|↓>|↑> represents "particle 1 is either in the ↑ state or the ↓ state and particle 2 is in the opposite state", I expect something similar to hold for position.

Position is very different from spin. If we gave up on the existence of an objective 9but approximate) position of particles we'd completely lose control over any experiment, since we wouldn't know where particles are unless we happened to measure them (which we most often do not).

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. Perhaps you can exhibit at least the flaws in my argument?

Take the example of a position measurement on an electron. It woud lead to a host of paradoxa if one wanted to assume that the electron has some position at a given time. "Position" is just not an attribute of an electron, it is an attribute of the "event" i.e. of the interaction process between the electron and an appropriately chosen measuring instrument (for instance a screen), not of the electron alone. The uncertainty about the position of the electron prior to the measurement is not due to our subjective ignorance. It arises from improperly attributing the concept of position to the electron instead of reserving it for the event.[/indent]

(Rudolf Haag, Local quantum physics, page 2).

Haag means with position 'classical position with infinite precision. This infinite precision - and _only_ this leads to paradoxa. Note that he talks about uncertainty, which would be meaningless if the position were not approximately determined.

There are good reasons that we talk about Heisenberg's uncertainty relation for rather than his nonexistence relation. The uncertainty relations in the modern form http://en.wikipedia.org/wiki/Uncertainty_relation#Mathematical_derivations have nothing to do with measurement per se - they follow directly from the basics on Hilbert spaces without reference to measurement, and just state that (unless what happens for spin in a spin eigenstate), the expectation is not the precise value of position but has an intrinsic uncertainty.

And the conventional interpretation of the Ehrenfest theorem http://en.wikipedia.org/wiki/Ehrenfest_theorem as establishing the quantum-classical correspondence wouldn't make sense without the knowledge that expectations of positions are approximate positions - for any system, quantum or classical.

 

[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".