What theories address the fundamental questions about quantum mechanics?

[DarMM]

So you can't back up your statement with a proof. With all due respect, I'll just then disregard it.

Perhaps I didn't explain myself well. It isn't the type of statement which has a proof. For example take the theorem that every operator is bounded on a finite dimensional Hilbert space. The analogue of my statement is that not every operator is bounded in an infinite dimensional Hilbert space. You don't prove this, you just give examples, since it is just description of what happens when another theorem doesn't hold.

For an example see Appendix C of:
Abhay Ashtekar, Jerzy Lewandowski, Donald Marolf, José Mourão, and Thomas Thiemann;Quantization of Diffeomorphism Invariant Theories of Connection with Local Degrees of Freedom, J. Math. Phys. 36, 6456

 

[A. Neumaier]

I'm not satisfied with this answer. The question was about the description of states, not of observables. The states in your formulation are described by the density operator rho(t). So my question remains: how do you describe the states of that system using this operator ?

This is specified in Axioms A2 and A3. There are lots of Hermitian, positive semidefinite, linear trace class operators rho_0 on the 3-particle Hilbert space K of the helium atom satisfying trace rho_0=1. Solving the initial value problem for d/dt rho(t) = i/hbar [rho(t),H] with rho(0)=rho_0 produces as many states satisfying at all times the conditions trace rho(t) = 1.

Note that the state at time t _is_ the operator rho(t), by definition.

Hmmm...No answer provided to my 3rd question.

The same holds for your third question.

 

[A. Neumaier]

Perhaps I didn't explain myself well. It isn't the type of statement which has a proof.

Actually, any counterexample _is_ a proof. All that was missing was the reference.

 

[DarMM]

Actually, any counterexample _is_ a proof. All that was missing was the reference.

Oh yeah!

 

[Fredrik]

Please, do comment, if possible, both on my set and on Arnold's one.

I was planning to do this today, but this morning I decided that I wanted to write down my own set of axioms before I comment on someone else's. I started by writing down a few general thoughts about axioms (instead of the actual axioms), and it turned into a detailed examination of the concepts of "state", "observable", and "measurement" that I'm still not done with. So even though I spent a few hours on this today, I still haven't written down a single axiom, or read any of yours or A.N.'s.

The only thing that's perfectly clear to me right now is that what's been bothering me about the axioms of QM can't be fixed by choosing a different set of axioms. All the significant problems I have are with the concepts of "state", "observable" and "measurement", and those parts of the identification of mathematical and real-world concepts that aren't even mentioned in the axioms.

I will continue to think about this tomorrow, and I intend to post my thoughts here when they're coherent enough to sound like an actual argument. I hope that will be tomorrow, but I can't promise anything.

 

[bg032]

In addition to these formal axioms one needs a rudimentary
interpretation relating the formal part to experiments.
The following _minimal_interpretation_ seems to be universally
accepted.

MI. Upon measuring at times t_l (l=1,...,n) a vector X of observables
with commuting components, for a large collection of independent
identical
(particular) systems closed for times t<t_l, all in the same state
rho_0 = lim_{t to t_l from below} rho(t)
(one calls such systems _identically_prepared_), the measurement
results are statistically consistent with independent realizations
of a random vector X with measure as defined in axiom A5.


Note that MI is no longer a formal statement since it neither defines
what 'measuring' is, nor what 'measurement results' are and what
'statistically consistent' or 'independent identical system' means.
Thus Axiom MI has no mathematical meaning. That's why it is already
part of the interpretation of formal quantum mechanics.

However, the terms 'measuring', 'measurement results', 'statistically
consistent', and 'independent' already have informal meaning in the
reality as perceived by a physicist. Everything stated in Axiom MI is
understandable by every trained physicist. Thus statement MI is not
for formal logical reasoning but for informal reasoning in the
traditional cultural setting that defines what a trained physicist
understands by reality.

It seems to me that the MI collocates your system of axioms in the context of the Copehagen interpretation, where a macroscopic classical realm, including notions such as measuring apparatuses, is assumed to exist independently of the the quantum realm. For me this is unsatisfactory, because it implies that two different and independent theories, namely classical and quantum mechanics, are necessary in order to explain our empirical perceptions. I would like a formulation of QM (which is arguably more fundamental then CM) in which mathematical elements clearly corresponding to our empirical experience of a classical evolution (e.g., trajectories) were present.

 

[Fredrik]

It seems to me that the MI collocates your system of axioms in the context of the Copehagen interpretation, where a macroscopic classical realm, including notions such as measuring apparatuses, is assumed to exist independently of the the quantum realm. For me this is unsatisfactory, because it implies that two different and independent theories, namely classical and quantum mechanics, are necessary in order to explain our empirical perceptions. I would like a formulation of QM (which is arguably more fundamental then CM) in which mathematical elements clearly corresponding to our empirical experience of a classical evolution (e.g., trajectories) were present.

People always read too much into the fact that the state of a measuring device at the end of a measurement can for all practical purposes be described classically. It doesn't mean that measuring devices follow a different set of rules than microscopic systems. It just acknowledges that we wouldn't consider a device that's in a superposition of quantum states to have measured something.

 

[A. Neumaier]

The only thing that's perfectly clear to me right now is that what's been bothering me about the axioms of QM can't be fixed by choosing a different set of axioms. All the significant problems I have are with the concepts of "state", "observable" and "measurement", and those parts of the identification of mathematical and real-world concepts that aren't even mentioned in the axioms.

All this cannot be part of the axioms at all. To understand what axioms are, consider the axioms for projective planes:

The points form a set P.
The lines form a set L.
There is an incidence relation I subset P x L.
Say x in l. or l contains x if (x,l) in I.
Any two distinct points are in a unique line.
Any two distinct lines contain a unique point.

That's all. The axioms say everything needed to work with projective planes.

Although no explanation is given of the meaning of the concepts of ''point'', ''line'', ''incidence''. This is not part of the _axioms_ but part of their _interpretation_ in real life. And indeed, here all the philosophical problems appear...

The purpose of an axiom system is precisely to separate the stuff that is problematic but peripheral from the stuff that is essential and allows rational deductions.

My axioms in the section ''Postulates for the formal core of quantum mechanics'' of Chapter A1 of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html#postulates give such a clear separation (and also explain some things about their interpretation).

 

[A. Neumaier]

It seems to me that the MI collocates your system of axioms in the context of the Copehagen interpretation, where a macroscopic classical realm, including notions such as measuring apparatuses, is assumed to exist independently of the the quantum realm.

Far from that.

MI is what _every_ interpretation I know of assumes (and has to assume) at least implicitly in order to make contact with experiments. It relates the axioms not to a hypothetical classical realm but to a nonphysical entity: the social conventions of the community of physicists.

Indeed, all interpretations I know of assume much more, but they differ a lot in what they assume beyond MI.

 

[Fra]

Although no explanation is given of the meaning of the concepts of ''point'', ''line'', ''incidence''. This is not part of the _axioms_ but part of their _interpretation_ in real life. And indeed, here all the philosophical problems appear...

In this decomposition, also the PHYSICAL problems appear there. After as long as it's purely axiomatic, it's pure mathematics, not only do you shave off the philosophy, but also the physical content.

The purpose of an axiom system is precisely to separate the stuff that is problematic but peripheral from the stuff that is essential and allows rational deductions.

AFAIK most real life problems and physics, are not something where deductive reasoning is used. Deductive reasoning is within pure mathematics.

I would like to claim that actually most relevant (non-idealized) problems in the real world required reasoning and decision making based upon incompelte information. Ie. it's some form of inference, but not deductive logic. Most some evolving inductive evolving logic.

Deductive logic is extremely efficient and precise, and useful, but it's also somewhat "sterile" and inflexible, lacking traits that are needed in most real situations.

/Fredrik

 

[A. Neumaier]

In this decomposition, also the PHYSICAL problems appear there. After as long as it's purely axiomatic, it's pure mathematics, not only do you shave off the philosophy, but also the physical content.

AFAIK most real life problems and physics, are not something where deductive reasoning is used. Deductive reasoning is within pure mathematics.

This amounts to claiming that most of theoretical physics has no physical content.
A very strange position.

 

[Fra]

This amounts to claiming that most of theoretical physics has no physical content.
A very strange position.

*IF* you insist on the hard decompsition, and see a PURE axiomatic picture, and then argue that the work of theoretical physicsists is to construct and prove theorems, then it's not physics. It's merely mathematical elaboration and extension of the language and tools of physics.

In that picture, I'd say the physics part lies there in selecting and constructing the axiom system. And this process is not deductive.

Indeed a lot of theoretical physics DO borderline to mathematics. It's more of mathematical elaboration of physical models, than DEVELOPING physical models. At least from my perspective.

Personally I do not draw a clear line. The message and the language develops hand by hand. So in MY view the philosophical, physical and mathematical developing does and should intermix.

/Fredrik

 

[Fra]

This amounts to claiming that most of theoretical physics has no physical content.
A very strange position.

One could also say that it has a "frozen" physics content. Which is why it's sterile.

Physics is a living science, and I don't see how the progression of physics as a science is a deductive process.

/Fredrik

 

[Fredrik]

All this cannot be part of the axioms at all. To understand what axioms are, consider the axioms for projective planes:

It seems that you're the one who needs to be told what axioms are. Axioms of theories of physics are clearly not the same thing as axioms for mathematical structures. For example, the axioms of special relativity are not the axioms for Minkowski spacetime. SR is defined by a set of statements that tells us how to how to interpret the mathematics of Minkowski spacetime as predictions about results of experiments. An example of such a statement is "a clock measures the proper time of the curve in spacetime that represents its motion".

You won't find a list of axioms that define a theory of physics this way in any physics book. That doesn't mean I'm wrong. It only means that physicists are really sloppy with these details.

The axioms of QM will include some version of the Born rule. It is sometimes stated in terms of probabilities, and sometimes in terms of expectation values. Let's consider the second option. The rule would associate an expectation value E(s,A) with each pair (s,A) where s is a state and A is an observable. Such a rule doesn't actually say anything unless we also specify how mathematical states and observables correspond to things in the real world (preparation procedures and measuring devices).

 

[A. Neumaier]

It seems that you're the one who needs to be told what axioms are. Axioms of theories of physics are clearly not the same thing as axioms for mathematical structures.

Where do you take this assertion from?

I am told what axioms are by David Hilbert, who wrote in 1924 the first (and very influential) textbook on mathematical physics http://en.wikipedia.org/wiki/Methods_of_Mathematical_Physics

The axiomatic tradition started with Hilbert. He defined in his famous 1900 address in the context of the sixth problem what an axiomatization of physics should mean: http://en.wikipedia.org/wiki/Hilbert's_sixth_problem

''6. Mathematical Treatment of the Axioms of Physics. The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which already today mathematics plays an important part; in the first rank are the theory of probabilities and mechanics.''

This is a quest for giving axioms for physics that are of the same clarity and interpretation independence as those for geometry.

For example, the axioms of special relativity are not the axioms for Minkowski spacetime. SR is defined by a set of statements that tells us how to how to interpret the mathematics of Minkowski spacetime as predictions about results of experiments.

You mix axioms and their interpretation, which Hilbert took so great pains to separate.

The _same_ quantum mechanics has multiple interpretations, according to the different schools. But the axioms tell precisely what you are allowed to do in quantum mechanics, and are independent of such interpretations.

An example of such a statement is "a clock measures the proper time of the curve in spacetime that represents its motion".

How can your example be an axiom if it contains terms such as ''proper time'' which already presupposes the theory it is supposed to 'found?

Also, a complete set of axioms must be such that it allows one to deduce everything about the objects talked about in the axioms. How can your example be an axiom if it contains vague terms such as ''measures'' and ''motion'' without well-specified properties?

The axioms of QM will include some version of the Born rule.

Not necessarily. It only needs to be derivable from the axioms. This is the case in my axiom system. Born's rule is derivable from Axiom A5 if the assumptions hold under which Born's rule is valid.

And this is good so, since Born's rule is not a universal law, but a property of very special measurements.

 

[Fredrik]

Where do you take this assertion from?

It follows from the definition of science. A theory needs to make predictions about results of experiments to have some kind of falsifiability.

You mix axioms and their interpretation,

If you mean that I'm mixing mathematical axioms with physical axioms, that's simply not true. I keep them as separate as possible, but I also understand that what we're talking about isn't physics until we have made the connection between mathematics and experiments.

How can your example be an axiom if it contains terms such as ''proper time'' which already presupposes the theory it is supposed to 'found?

Are you seriously suggesting that the choice of what to call the result of the integration matters? If the term "proper time" bothers you, we can call it "flurpy" instead, but there's clearly no need to do this. The theory isn't made circular by a choice of terms that's inspired by the fact that we already know what we're going to use them for.

Also, a complete set of axioms must be such that it allows one to deduce everything about the objects talked about in the axioms. How can your example be an axiom if it contains vague terms such as ''measures'' and ''motion'' without well-specified properties?

I agree that axioms of a theory of physics are never completely well-defined, for the reasons you have correctly identified. Yes, that's annoying, but it's impossible to do better. If you think this is a good enough reason to not use the word "axioms", fine, let's call them "schmaxioms" instead. Then a theory of physics is defined by set of "schmaxioms" that tells us how to interpret the mathematics (defined by axioms) as predictions about results of experiments.

 

[atyy]

The axioms of QM will include some version of the Born rule. It is sometimes stated in terms of probabilities, and sometimes in terms of expectation values. Let's consider the second option. The rule would associate an expectation value E(s,A) with each pair (s,A) where s is a state and A is an observable. Such a rule doesn't actually say anything unless we also specify how mathematical states and observables correspond to things in the real world (preparation procedures and measuring devices).

Yes, definitely. So I tend to take the "axioms" of QM more like Newton's laws are "axioms" for classical mechanics, which need to be supplemented by particular force laws like the law of gravitation or the law of friction.

The difference between the "axioms" of QM and classical mechanics is that those of QM are already self-contradictory even without eg. the Lagrangian of the standard model and the LHC, since the unitary evolution and wave function collapse are in tension - unless one accepts MWI.

 

[Fredrik]

The difference between the "axioms" of QM and classical mechanics is that those of QM are already self-contradictory even without eg. the Lagrangian of the standard model and the LHC, since the unitary evolution and wave function collapse are in tension - unless one accepts MWI.

In my opinion, there is no contradiction in the axioms of QM. The measurement problem appears when we make two additional assumptions: 1. A state vector represents all the properties of the system. 2. There's only one world.

The assumption "1 and not 2" defines a MWI that makes a lot more sense than Everett's (because we haven't crippled the theory by dropping the Born rule too). The assumption "2 and not 1" defines an ensemble interpretation.

 

[Fredrik]

I decided that I wanted to write down my own set of axioms before I comment on someone else's. I started by writing down a few general thoughts about axioms (instead of the actual axioms), and it turned into a detailed examination of the concepts of "state", "observable", and "measurement" that I'm still not done with.
...
I will continue to think about this tomorrow, and I intend to post my thoughts here when they're coherent enough to sound like an actual argument. I hope that will be tomorrow, but I can't promise anything.

I'm still working on this. It has helped me get some of my thoughts in order, so I won't consider my time wasted even if I would choose not to post my conclusions. I'm hesitating because I wonder if anyone would even be interested in reading two really long posts (one with my general comments and one about states and observables) and then a third post, with the actual axioms. The first two are 95% finished, but I haven't begun writing the third yet. Maybe I should put this stuff in a new thread if I do post it.

 

[A. Neumaier]

It follows from the definition of science. A theory needs to make predictions about results of experiments to have some kind of falsifiability.

Please quote or cite the definition of science which implies that

Axioms of theories of physics are clearly not the same thing as axioms for mathematical structures.

Certainly your single-line argument does not prove this.

If you mean that I'm mixing mathematical axioms with physical axioms, that's simply not true. I keep them as separate as possible, but I also understand that what we're talking about isn't physics until we have made the connection between mathematics and experiments.

No I mean that axioms specify in unambiguous terms all properties that are ascribed to the concepts used, while interpretation rules tell how these concepts are applied as models of the real world.

For example, the axioms of projective geometry are just those I had given, and can be stated in precise terms, whereas the interpretation rules are ambiguous and approximate, of the kind:
-- A point is what has no parts.
-- A point is an object without extension.
-- A point is a mark on paper.
These are already three different, mutually incompatible but common interpretation rules for the projective point (and doesn't yet incorporate the interpretation of the points at infinity). Writing interpretation rules for a projective line is much more complicated and controversial.

This sort of observations prompted Hilbert to promote the axiomatization of theories as a means for making the content of a theory as precise as possible, separating the objective substance from the controversial philosophy.

Hilbert was a very good physicist - co-discoverer of the laws of general relativity, creator of the Hilbert space on which all quantum mechanics today is based, and very productive
in using the equations of physics to extract information tat can be compared with experiment. Deviating from the exiomatic tradition that he promoted in a way that changed mathematics and science requires very strong reasons.

It is no accident that today's quantum mechanics is based on Hilbert spaces rather than wave functions and Born's rule!

Are you seriously suggesting that the choice of what to call the result of the integration matters? If the term "proper time" bothers you, we can call it "flurpy" instead, but there's clearly no need to do this.

The names don't matter. The point is that ''proper time'' (or if you rename it, ''flurpy'')
is not even defined before you have the theory in place. One cannot formulate the interpretation rules in a clear way unless one first has the axioms that define the concepts.

In anything more complex than 19th century science, the concepts (the main ingredient that makes physics differ from Nature) are _defined_ by the axioms and the subsequent formal theory. They are then _interpreted_ by rules that usually assume both the concepts and some social conventions about how experiments are done.

I agree that axioms of a theory of physics are never completely well-defined, for the reasons you have correctly identified. Yes, that's annoying, but it's impossible to do better.

Hilbert showed how to do it better, by separating axioms from interpretation rules.
The axioms precisely define what the theory is about, and the interpretation rules
loosely define how the theory applies to reality.

If you think this is a good enough reason to not use the word "axioms", fine, let's call them "schmaxioms" instead. Then a theory of physics is defined by set of "schmaxioms" that tells us how to interpret the mathematics (defined by axioms) as predictions about results of experiments.

What you suggest to call the set of schmaxioms is conventionally called ''interpretation''.
It is no accident that one talks about the many different interpretations of quantum mechanics. Their goal is precisely to interpret quantum mechanics (the precise theory)
as predictions about results of experiments. But they all assume that a precise theroy called quantum mechanics exists already, which is to be interpreted by an ''interpretations of quantum mechanics''

The established tradition about what to call an axiom is that of Hilbert. His notion of axiom is the one established in the literature. Try entering the key words
axioms physics
into either of http://scholar.google.com/ or http://en.wikipedia.org/ !

A theory of physics is defined by axioms that tell us precisely how the concepts of physics relate in a consistent matter to each other, and by interpretation rules that
tell us how the theory thus defined applies to interpret experiments.

The axioms of physics in the published volume on Hilbert's problems,

Mathematical Developments Arising from Hilbert Problems,
Proc. Symp. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974,
Amer. Math. Soc., Providence, RI, 1976,

are taken to be the Wightman axioms, not as the Born rule!
(DarMM will be pleased to hear that if he didn't know it already!)

 

[Fra]

Hilbert showed how to do it better, by separating axioms from interpretation rules. The axioms precisely define what the theory is about, and the interpretation rules loosely define how the theory applies to reality.

The problem of such strict picture is that it is nice from a mature, stable theory.

But how do you consider an evolving and developing theory? Then axioms (and interpretations) has to come and go.

Maybe that's the confusion as to what's science (or natural science). Strict formalisations of physical theories in the way you envision, is good (and is work for qualified mathematicians and logicians, not average physicists), but this is usually done once the theory is reasonably matured unless you think that CHOOSING the set of axioms that (given some interpretation) is the most FIT theory of reality, can be done from pure logic as a deductive process.

The strict formalisation does not describe the scientific process itself, like evolution of a system of axioms. It's in this part that I think the physics lies. So I think a balance between formal development and gaming is the only way.

A wolf does not excuse a rabbit that responds with syntax error instead of activating flight mode. Those rabbits soon are depleted in nature. Interactions with the unknown and survival in that environment is what demands a balance between flexibility and specific skills.

/Fredrik

 

[A. Neumaier]

The problem of such strict picture is that it is nice from a mature, stable theory.

But how do you consider an evolving and developing theory? Then axioms (and interpretations) has to come and go.

Yes. But quantum mechanics is a very mature theory. It is not hard to separate there the axioms from the interpretation. Hence there should be no incentive to mix them up.

Development in the way how a theory is exploited to obtain predictions do not affect
the foundations.

On the foundational level, only the models for gravitation are evolving and developing a lot, which shows in the many conflicting approaches. No agreed theory - no agreed axioms.

And there are minor developments in the standard model, essentially changes in the details of the action.

Finally, there are unresolved issues in proving the existence of interacting quantum field theories in 4D; this are open research problems but not of a more severe nature than the unresolved issues in the mathematics of the Navier-Stokes equations (which like QFT gave rise to a Millenium prize to be won).

 

[Fra]

It sounds I agree with most of what you say after all.

I think the apparent disagreement is simply because I think you are a mathematician trying to formalize current QM - I'm not. I have a different quest, I am looking into understanding the open issues (and thus possible changing the theory). In that stage, time is not ripe for strict formalisations.

So the risk I see, is that you may do a nice formalisation of a theory, that later is revised. Then I'm not sure how easy the reconstruction of the formalized system would be.

This is similar to my objection to for example Poppers view on science. Popper did his best to try to make the scientific process look as deductive as possible. He didn't like the fuzzy induction. What he missed is the logic of hypothesis generation. HOW does a falsified theory, HELP find a better hypothesis, rather than just discarding and start from scratch. This is where induction is superior, although admittedly not deductive.

I think mathematicians serve physicists good though, to back them up in the maturation phase of theories.

/Fredrik

 

[A. Neumaier]

I think the apparent disagreement is simply because I think you are a mathematician trying to formalize current QM - I'm not. I have a different quest, I am looking into understanding the open issues (and thus possible changing the theory). In that stage, time is not ripe for strict formalisations.

Yes. My interest is in presenting the stuff that is ripe in the clearest possible way that I can manage. Formalizing current QM is not really difficult, as my axiom system shows.

 

[Fredrik]

Please quote or cite the definition of science which implies that

You're pulling a [citation needed] on me because I say that scientific theories need to be falsifiable. It's not exactly a controversial claim.

The point is that ''proper time'' (or if you rename it, ''flurpy'')
is not even defined before you have the theory in place. One cannot formulate the interpretation rules in a clear way unless one first has the axioms that define the concepts.

This is incorrect. It would be correct to say that you can't define "proper time" until you have defined "Minkowski spacetime" and "timelike curve", but the former is just a mathematical structure, and the latter is just a mathematical term for a particular kind of function from an interval of the real numbers to a subset of (the underlying set of) Minkowski spacetime. You can certainly define both (and proper time) without defining a theory of physics. So there is absolutely no circularity in making "a clock measures the proper time of the curve in spacetime that represents its motion" one of the "schmaxioms" that define the theory of physics that we call "special relativity".

In anything more complex than 19th century science, the concepts (the main ingredient that makes physics differ from Nature) are _defined_ by the axioms and the subsequent formal theory. They are then _interpreted_ by rules that usually assume both the concepts and some social conventions about how experiments are done.

I made it clear from the start that what you call an interpretation is what I call a theory of physics. It makes no sense to me to refer to a piece of mathematics as a theory of physics.

What you suggest to call the set of schmaxioms is conventionally called ''interpretation''.
It is no accident that one talks about the many different interpretations of quantum mechanics. Their goal is precisely to interpret quantum mechanics (the precise theory)
as predictions about results of experiments.

This is also incorrect. Their goal is, or at least should be, to turn a theory (defined by a piece of mathematics and an additional set of statements that describe how to interpret the mathematics as predictions about results of experiments) into a description of what's "actually happening" at all times. They do so by means of an additional set of statements that don't change the predictions. That's why interpretations of QM are not a part of science. (I'm not suggesting that they're useless. If an interpretation can improve your intuition about what QM will predict, I'm not going to suggest that you shouldn't use it).

In other words, there are two kinds of interpretations. You need to interpret a piece of mathematics to get a theory, and you need to interpret a theory to get a "description". Now, you can, and undoubtedly will, criticize my claims on the grounds that my definitions aren't standard definitions that everyone is using, but the fact is, there are no standard definitions. I would say that's exactly why the literature on "interpretations on QM" is such a mess.

By the way, Everett's MWI doesn't qualify as an interpretation of QM by my definitions, or even as a theory. When you just drop the Born rule, what you have left can't make predictions about results of experiments, so it doesn't qualify as a theory. To interpret a theory, you need to have a theory first. This doesn't mean that the idea of many worlds is dead. I believe (but haven't worked out the details) that the proper way to define a MWI is to keep the Born rule around, and use it to identify the interesting worlds. But now I'm starting to drift off topic. I don't want to turn this into a discussion of many-worlds interpretations.

 

[A. Neumaier]

You're pulling a [citation needed] on me because I say that scientific theories need to be falsifiable. It's not exactly a controversial claim.

No. I was accepting this for the sake of discussion, though I don't share the view (see the section ''Can good theories be falsified?'' of Chapter C1 of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html#falsified ).

I was complaining that falsifiability hasn't anything to do with your assertion ''Axioms of theories of physics are clearly not the same thing as axioms for mathematical structures'' which you claimed ''follows from the definition of science''.

This is incorrect. It would be correct to say that you can't define "proper time" until you have defined "Minkowski spacetime" and "timelike curve", but the former is just a mathematical structure, and the latter is just a mathematical term for a particular kind of function from an interval of the real numbers to a subset of (the underlying set of) Minkowski spacetime. You can certainly define both (and proper time) without defining a theory of physics.

Thus, for you, physics consists _only_ in the interpretation? Well, then most of what is done in theoretical physics is no physics. And 95% of what is in any common textbook on quantum mechanics is no physics.

Since there cannot be precise axioms for this impoverished version of physics, your view becomes understandable but irrelevant.

I made it clear from the start that what you call an interpretation is what I call a theory of physics. It makes no sense to me to refer to a piece of mathematics as a theory of physics.

Yours is a minority position. According to tradition, there is ''quantum mechanics''
(which is the formal, mathematical part that allows you to calculate predictions), and ''the interpretation of quantum mechanics'' which tells how the predictions relate to reality. Nobody calls the latter ''quantum mechanics'' and the former ''the mathematics of quantum mechanics''!

Now, you can, and undoubtedly will, criticize my claims on the grounds that my definitions aren't standard definitions that everyone is using, but the fact is, there are no standard definitions.

At least there is a main stream view, well expressed by the introductory sentences in wikipedia:

''Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter.'' (http://en.wikipedia.org/wiki/Quantum_mechanics)

''An interpretation of quantum mechanics is a set of statements which attempt to explain how quantum mechanics informs our understanding of nature.'' (http://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics)

 

[A. Neumaier]

The point is that ''proper time'' (or if you rename it, ''flurpy'') is not even defined before you have the theory in place. One cannot formulate the interpretation rules in a clear way unless one first has the axioms that define the concepts.

This is incorrect. It would be correct to say that you can't define "proper time" until you have defined "Minkowski spacetime" and "timelike curve", but the former is just a mathematical structure, and the latter is just a mathematical term for a particular kind of function from an interval of the real numbers to a subset of (the underlying set of) Minkowski spacetime. You can certainly define both (and proper time) without defining a theory of physics. So there is absolutely no circularity in making "a clock measures the proper time of the curve in spacetime that represents its motion" one of the "schmaxioms" that define the theory of physics that we call "special relativity".

In anything more complex than 19th century science, the concepts (the main ingredient that makes physics differ from Nature) are _defined_ by the axioms and the subsequent formal theory. They are then _interpreted_ by rules that usually assume both the concepts and some social conventions about how experiments are done.

I made it clear from the start that what you call an interpretation is what I call a theory of physics. It makes no sense to me to refer to a piece of mathematics as a theory of physics.

You cannot separate the mathematics from the physics.A mathematical theory _is_ a theory of physics once its concepts agree with those of a branch of physics, and its assumptions and conclusions can be brought into correspondence with physical reality,
no matter how informal (or even unspoken) the interpretation rules are.

Let me give a more complex example. To define what it means to measure time, we cannot proceed without first having a definition of the unit of time in which to make the measurement. The official definition (found, e.g., at http://physics.nist.gov/cuu/Units/second.html ) is:
''The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.''

To be able to make sense of this interpretation rule (or ''axiom of physics'', as you would want to have it), one needs to assume both a lot about the theory of quantum mechanics (in the sense I stated: the formal mathematical theory of what can be deduced from axioms that make no reference to reality) and some additional informal rules that tell how to measure transitions between two levels, and how to prepare a cesium 133 atom so that the quantity described can be measured.

To understand the latter, one needs more results from quantum mechanics of the formal, mathematical kind, and more informal rules that tell how these results are interpreted in an experiment. Etc..

One ends up with a whole book on measurement theory instead of a simple axiom system.

This book would also have to tell how one recognizes a Cesium 133 atom. The correct answer is: By verifying that it behaves like the theoretical model of a Cesium 133 atom. This is the only criterion - if an atom does not behave like that, we conclude with certainty that it is not a Cesium 133 atom.

The situation is here not different from the thermodynamical situation characterized by H.B. Callen in his famous textbook
H.B. Callen.
Thermodynamics and an introduction to thermostatistics,
2nd. ed. Wiley, New York, 1985.
He writes on p.15: ''Operationally, a system is in an equilibrium state if its properties are consistently described by thermodynamic theory.'' (This quote can also be found at the end of Section 2 of the article http://www.polyphys.mat.ethz.ch/education/lec_thermo/callen_article.pdf )

Thus the only way to get sound foundations of a theory of physics is to give clear, fully precise axioms for the formal, mathematical part, then describe its consequences, and finally, with the conceptual apparatus created by the theory (of course with lots of hindsight, arrived at through prior, less rigorous stages) to specify the conditions when it applies to reality in a more informal way, but still attempting to preserve as much clarity as possible.

 

[Fredrik]

No. I was accepting this for the sake of discussion, though I don't share the view (see the section ''Can good theories be falsified?'' of Chapter C1 of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html#falsified ).

Your assertion that theories can't be falsified is obviously true if you define a theory as a piece of mathematics. The fact that your very non-standard view on falsifiability follows from your way of thinking about theories should tell you that your way of thinking about theories isn't exactly mainstream.

Anyway, it's becoming increasingly clear that most of our "disagreements" aren't disagreements at all. For example we both require proper definitions of all the mathematical terms that we're going to use, and we both understand that the mathematics need to be interpreted as predictions about results of experiments. We use different definitions of common terms, and as a result we can say the same thing and still appear to contradict each other.

I was complaining that falsifiability hasn't anything to do with your assertion ''Axioms of theories of physics are clearly not the same thing as axioms for mathematical structures'' which you claimed ''follows from the definition of science''.

I have never heard anyone drop that the falsifiability requirement before. Everyone requires scientific theories to be falsifiable. A piece of mathematics can't be falsified, but statements about how to use a piece of mathematics to make predictions about results of experiments can be falsified. It's as simple as that.

Thus, for you, physics consists _only_ in the interpretation? Well, then most of what is done in theoretical physics is no physics. And 95% of what is in any common textbook on quantum mechanics is no physics.

I didn't say that. I just said that each theory (by my definition of "theory") is defined by what you call an interpretation. Physics isn't just about defining theories. Theoretical physics is also about how to find their predictions, and experimental physics is about finding out how accurate those predictions are.

Since there cannot be precise axioms for this impoverished version of physics, your view becomes understandable but irrelevant.

That would mean that everything that isn't pure mathematics is irrelevant.

A mathematical theory _is_ a theory of physics once its concepts agree with those of a branch of physics, and its assumptions and conclusions can be brought into correspondence with physical reality,
no matter how informal (or even unspoken) the interpretation rules are.

If we need to bring the assumptions and conclusions into correspondence with physical reality, then that's precisely what makes a theory of physics different from a piece of mathematics. The "bringing" isn't implied by the mathematics, so it must be postulated separately.

To define what it means to measure time, we cannot proceed without first having a definition of the unit of time in which to make the measurement.
...
To be able to make sense of this interpretation rule (or ''axiom of physics'', as you would want to have it), one needs to assume both a lot about the theory of quantum mechanics (in the sense I stated: the formal mathematical theory of what can be deduced from axioms that make no reference to reality) and some additional informal rules that tell how to measure transitions between two levels, and how to prepare a cesium 133 atom so that the quantity described can be measured.
...

You're not wrong here. My example axiom used the word "clock", and there doesn't seem to be a way to define that term other than to write down a set of assembly instructions and then say that the things you build using these instructions are called "clocks". But you won't be able to write down a really good set of instructions unless you already know what some theory that hasn't been properly defined yet (possibly the one you're trying to define) is going to predict. This is annoying as hell, but your approach doesn't avoid these issues, it just ignores them!

I don't know to what extent the problem of how to define specific measuring devices can be solved, but I'm sure it can't be solved in a way that leaves everyone satisfied. It also can't be avoided, and I refuse to ignore it. I'm going to spend some time thinking about it over the next few days.

One ends up with a whole book on measurement theory instead of a simple axiom system.

It doesn't sound unreasonable to have a book on measurement theory define the terms used in axioms of theories of physics, like "state", "observable" and "clock". If we can require that people study functional analysis or differential geometry before they study a list of axioms, we can certainly require that they study some measurement theory as well.

Thus the only way to get sound foundations of a theory of physics is to give clear, fully precise axioms for the formal, mathematical part, then describe its consequences, and finally, with the conceptual apparatus created by the theory (of course with lots of hindsight, arrived at through prior, less rigorous stages) to specify the conditions when it applies to reality in a more informal way, but still attempting to preserve as much clarity as possible.

The only difference between this and what I'm doing is that I want the correspondence between mathematics and reality to be spelled out explicitly rather than swept under the rug. That correspondence is what defines the difference between mathematics and physics, and it's far to important to be ignored.

 

[A. Neumaier]

Anyway, it's becoming increasingly clear that most of our "disagreements" aren't disagreements at all.

Everyone requires scientific theories to be falsifiable. A piece of mathematics can't be falsified, but statements about how to use a piece of mathematics to make predictions about results of experiments can be falsified. It's as simple as that.

But it still doesn't follow that one cannot clearly separate the axioms that define the formal concepts from the interpretation rules that relate them to reality, which was my main point. Mixing these causes confusion, as evidenced by 85 years of foundational problems for quantum mechanics

The only difference between this and what I'm doing is that I want the correspondence between mathematics and reality to be spelled out explicitly rather than swept under the rug. That correspondence is what defines the difference between mathematics and physics, and it's far to important to be ignored.

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. This is not shown in #5 but is discussed in the section ''Postulates for the formal core of quantum mechanics'' of Chapter A1 of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html#postulates

 

[A. Neumaier]

One ends up with a whole book on measurement theory instead of a simple axiom system.

It doesn't sound unreasonable to have a book on measurement theory define the terms used in axioms of theories of physics, like "state", "observable" and "clock". If we can require that people study functional analysis or differential geometry before they study a list of axioms, we can certainly require that they study some measurement theory as well.

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.

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