What theories address the fundamental questions about quantum mechanics?

[dextercioby]

And by the way, we can view the wave function, or effects thereof: http://www.dailymail.co.uk/sciencet...-mechanics-shown-work-visible-world-time.html

Experimentalists observe phenomena, they measure physical quantities. We can never view, nor measure wave functions/density operators as they are simply mathematical tools to describe reality. I think of a quantum states as being a part of reality.

We will always measure (or determine from numerical analysis of experimental tests of quantum mechanics) probabilities or spectral values of self-adjoint operators, because, as it follows from the axiomatization I proposed in post 1 of the thread, they are the only items assuring the connection between mathematics (functional analysis) and experiment.

 

[QuantumClue]

Experimentalists observe phenomena, they measure physical quantities. We can never view, nor measure wave functions/density operators as they are simply mathematical tools to describe reality. I think of a quantum states as being a part of reality.

We will always measure (or determine from numerical analysis of experimental tests of quantum mechanics) probabilities or spectral values of self-adjoint operators, because, as it follows from the axiomatization I proposed in post 1 of the thread, they are the only items assuring the connection between mathematics (functional analysis) and experiment.

There are scientists who take the superpositioning principle of wave mechanics as a physical phenomenon quite seriously, and not merely a mathematical artefact as you are applying it soley to. And this is what they observe in the link I showed you. Then it is evident we can view a wave function [of] matter.

 

[dextercioby]

Here is an axiom system fully covering current mainstream quantum mechanics and quantum field theory (but not various speculations beyond the standard model). It covers both the nonrelativistic case and the relativistic case.

There are six basic axioms:

A1. A generic system (e.g., a 'hydrogen molecule')
is defined by specifying a Hilbert space K whose elements
are called state vectors and a (densely defined, self-adjoint)
Hermitian linear operator H called the _Hamiltonian_ or the _energy_.

A2. A particular system (e.g., 'the ion in the ion trap on this
particular desk') is characterized by its _state_ rho(t)
at every time t in R (the set of real numbers). Here rho(t) is a
Hermitian, positive semidefinite (trace class) linear operator on K
satisfying at all times the conditions
trace rho(t) = 1. (normalization)
[/url]

1. Is it apparent to me, or you introduce two different description of states, one through <state vectors> and the other through <states \rho (t)> ? Are they both necessary, thus independent of each other ?
2. How are these two these axioms used to describe the physical states of a helium atom ?
3. How does A2 apply to the simplest possible system, the nonrelativistic free massive particle ?

 

[DarMM]

This is exactly what I was thinking. Separable spaces are easier to work with. That's why we try using a separable space first.

You might find this interesting, but there is a somewhat justifiable reason not to use them. For a non-separable Hilbert space the fact that a Lie group has a representation on the Hilbert space doesn't imply that the Lie Algebra has a representation. So even though rotations might be represented by a group of unitary operators, angular momentum wouldn't be a well defined operator.

 

[dextercioby]

You might find this interesting, but there is a somewhat justifiable reason not to use them. For a non-separable Hilbert space the fact that a Lie group has a representation on the Hilbert space doesn't imply that the Lie Algebra has a representation. So even though rotations might be represented by a group of unitary operators, angular momentum wouldn't be a well defined operator.

Can you post or send a reference to a mathematical proof for that ? Thanks!

 

[A. Neumaier]

1. Is it apparent to me, or you introduce two different description of states, one through <state vectors> and the other through <states \rho (t)> ? Are they both necessary, thus independent of each other ?
2. How are these two these axioms used to describe the physical states of a helium atom ?
3. How does A2 apply to the simplest possible system, the nonrelativistic free massive particle ?

1. The state vectors are called so conventionally, without having to be states - they are just calculational tools. The physicall state is rho(t) and carries the information about experimental behavior. To remove the confusion, just replace Axiom A1 by the following improved version.

A1. A generic system (e.g., a 'hydrogen molecule') is defined by
specifying a Hilbert space K and a (densely defined, self-adjoint)
Hermitian linear operator H called the _Hamiltonian_ or the _energy_.

2. Here you need also Axiom A4. with three particles (alpha, e, e'). But e and e' are indistinguishable, so only symmetric functions of the labels e and e' are observable. H is given by the standard atomic Hamiltonian one can find in any textbook.

3. H= p^2/2m.

 

[DarMM]

Can you post or send a reference to a mathematical proof for that ? Thanks!

In the theory of representations of Lie groups on Hilbert spaces, the separability property allows you prove the existence of a representation of the Lie algebra. However without separability you cannot complete the proof, so there is no guarantee that the Lie algebra has a representation. There are several example theories where this is the case.

(In fact it was an issue in Loop Quantum Gravity at one point I believe, but I don't know much about that subject.)

There isn't really a proof, since it is a description of what occurs in a case where another proof (representations on separable Hilbert spaces) fails.

 

[A. Neumaier]

In the theory of representations of Lie groups on Hilbert spaces, the separability property allows you prove the existence of a representation of the Lie algebra. However without separability you cannot complete the proof, so there is no guarantee that the Lie algebra has a representation.

a) Is there a simple explicit example of this situation?

b) What about the converse: Does a unitary representation of a Lie algebra by self-adjoint operators always generate unitary representation of a Lie group?

 

[dextercioby]

[...]just replace Axiom A1 by the following improved version.

A1. A generic system (e.g., a 'hydrogen molecule') is defined by
specifying a Hilbert space K and a (densely defined, self-adjoint)
Hermitian linear operator H called the _Hamiltonian_ or the _energy_.

Alright, agreed.

2. Here you need also Axiom A4. with three particles (alpha, e, e'). But e and e' are indistinguishable, so only symmetric functions of the labels e and e' are observable. H is given by the standard atomic Hamiltonian one can find in any textbook.

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 the states of that system using this operator ?

3. H= p^2/2m.

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

 

[dextercioby]

In the theory of representations of Lie groups on Hilbert spaces, the separability property allows you prove the existence of a representation of the Lie algebra. However without separability you cannot complete the proof, so there is no guarantee that the Lie algebra has a representation. There are several example theories where this is the case. [...] There isn't really a proof, since it is a description of what occurs in a case where another proof (representations on separable Hilbert spaces) fails.

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

 

[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.

 

[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.