Axiomatization of quantum mechanics and physics in general ?

[bhobba]

As I don't see any reference here I thought I would mention this is no. 6 of Hilbert's problems.

That's far from complete eg it is generally thought virtual particles are merely an artefact of the perturbation methods used rather than actually being part of the theory.

Since Hilberts time things have got a LOT more difficult with no end in sight.

Thanks
Bill

 

[bhobba]

Do you have a more specific reference, like a page number? I don't understand what you mean that the problem is. I also don't understand the "part of a statistical theory" comment.

I would like to see a reference as well because I can't follow it either.

Thanks
Bill

 

[microsansfil]

Do you have a more specific reference, like a page number? I don't understand what you mean that the problem is. I also don't understand the "part of a statistical theory" comment.

atyy had already given the reference "Axiomatique quantique" by C. Piron. It is in French (Page 6 or 443 on pdf)

Patrick

 

[microsansfil]

As I don't see any reference here I thought I would mention this is no. 6 of Hilbert's problems.

Thank

The solution of Hilbert's sixth problem thus remains open.

The aim is therefore to express physics in mathematical logic of the first order.

So Godel's theorems should apply.

Patrick

 

[bhobba]

The aim is therefore to express physics in mathematical logic of the first order.

I would say most definitely NOT. Even the aim of expressing it in the language of pure math, which is weaker than mathematical logic, is only being pursued by a very small number.

So Godel's theorems should apply.

Of course. But so? To see why its not really that big a deal in practice, it is in fact logically equivalent to the halting problem. That a computer program can't be written to determine if a program will halt or not is hardly an Earth shattering issue in practice.

Thanks
Bill

 

[bhobba]

atyy had already given the reference "Axiomatique quantique" by C. Piron. It is in French (Page 6 or 443 on pdf)

I can't read French but it simply looks like a paper on Pirons Axioms.

Here is his axioms:
file:///C:/Users/Administrator/Downloads/hpa-001_1968_41_1_a_004_d%20(1).pdf

Cant find probability mentioned anywhere there.

In the idea of these authors, a measure that always has some uncertainty, only those proposals that can be defined as part of a statistical theory are physically valid.

It would be helpful if, in your own words, you can explain how those uncertainties come about?

Thanks
Bill

 

[microsansfil]

I would say most definitely NOT. Even the aim of expressing it in the language of pure math, which is weaker than mathematical logic, is only being pursued by a very small number.

Then, you have to define what a mathematical axiom in the context of physics because in mathematic I know only one definition.

In mathematics the objects are clearly defined : http://en.wikipedia.org/wiki/Axiom#Mathematical_logic

Patrick
PS
Hilbert's sixth problem

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.

---

On the Logical Foundations of the Jauch-Piron Approach to Quantum Physics

We make a critical analysis of the basic concepts of the Jauch-Piron (JP) approach to quantum physics. Then, we exhibit a formalized presentation of the mathematical structure of the JP theory by introducing it as a completely formalized syntactic system, i.e., we construct a formalized language L e and formally state the logical-deductive structure of the JP theory by means of L e.
Finally, we show that the JP syntactic system can be endowed with an intended interpretation, which yields a physical model of the system. A mathematical model endowed with a physical interpretation is given which establishes (in the usual sense of the model theory) the coherence of the JP syntactic system.

Patrick

 

[microsansfil]

It would be helpful if, in your own words, you can explain how those uncertainties come about?

What i understand. They reject the axiom of atomicity, because we must admit the existence of atoms, that is to say, accurate measurements of physical quantities.


Patrick

 

[bhobba]

Then, you have to define what a mathematical axiom in the context of physics because in mathematic I know only one definition.

Then you need to broaden your experience. See post 2 by Atty.

Patrick quoted
'We make a critical analysis of the basic concepts of the Jauch-Piron (JP) approach to quantum physics. Then, we exhibit a formalized presentation of the mathematical structure of the JP theory by introducing it as a completely formalized syntactic system, i.e., we construct a formalized language L e and formally state the logical-deductive structure of the JP theory by means of L e.
Finally, we show that the JP syntactic system can be endowed with an intended interpretation, which yields a physical model of the system. A mathematical model endowed with a physical interpretation is given which establishes (in the usual sense of the model theory) the coherence of the JP syntactic system.'

I have no issue with those things.

I simply want to know how uncertainties enter into Pirons axioms.

Specifically I would like to hear your explanation of how the Born Rule is deduced.

Thanks
Bill

 

[bhobba]

What i understand. They reject the axiom of atomicity, because we must admit the existence of atoms, that is to say, accurate measurements of physical quantities.

Cant follow that. Could you be saying its an experimental fact that atomic measurements are statistical?

If so exactly how does this follow from Pirons 3 axioms?

Thanks
Bill

 

[Fredrik]

I can't read French but it simply looks like a paper on Pirons Axioms.

Here is his axioms:

I think we would need to be on your computer to be able to use the URL you posted.

This is a google translation of the first half of that page (the end of section 3).

G. Birkhoff and J. von Neumann criticized the view (discussed here) that any subset of the phase space is a proposal. It seems they (we translate) artificial to consider a proposal statement such as the angular velocity of the Earth around the sun is a rational number (in radians per second), and it seems their best, at least statistically, to consider classes that are measurable modulo a subset of measure zero, which correspond to propositions subsets. Such considerations lead them to abandon the axiom of atomicity. So the idea of these authors, a measure that always has some uncertainty, only those proposals that can be defined as part of a statistical theory are physically valid. But if, without restricting us to a particular theory, we consider all proposals for a conventional system, we must admit the existence of atoms, that is to say, accurate measurements of quantities such as position or momentum, although its existence is not defined in act but in power (not in actu sed in potentia). This at least suggests that the fact that it is always possible practice ment to improve the result of a measurement. A satisfactory abstract lattice axioms OTC, and distributive properties of atomicity is called a complete Boolean lattice (J. von Neumann as continuous) and Atomic. However, such a mat can still be considered as the lattice of the subsets of the set of its atoms. This provides a characterization of the lattice proposals for a conventional system.​

Yes, it does sound artificial to assign a probability to the proposition that the angular velocity is a rational number. That's a set of measure 0, so the probability would be 0, and the proposition would be impossible to test in experiments anyway. But how is it more artificial and more of a problem than other idealizations that are made in theories of physics?

We always use 4-tuples of real numbers to identify events in spacetime, even though that description is almost certainly inadequate on small scales. In QM, we allow unbounded operators to represent measurements, even though a set of possible results of a measurement is always finite in the real world, due to our inability to measure with infinite precision.

The artificial thing that the text is talking about shows up when we reinterpret classical mechanics as a probability theory. So it's already present in classical mechanics stated in the usual way. It's just better hidden there. Classical mechanics seems to work quite well in spite of this.

 

[Fredrik]

Cant follow that. Could you be saying its an experimental fact that atomic measurements are statistical?

If so exactly how does this follow from Pirons 3 axioms?

Thanks
Bill

These are the definitions of "atom" and "atomic" from Piron's book:

(1.16): DEFINITION If b ≠ C and b < c, one says that C covers b
when b < x < C $\Rightarrow$ x = b or x = c. An element which covers 0
is called an atom (or point). A lattice is said to be atomic if for every
b≠0 there exists at least one atom p smaller than b (i.e. p < b).​

O denotes the minimal element, i.e. $\varnothing$ in the case of $\sigma$-algebras, $\{0\}$ in the case of a lattice of closed subspaces of a Hilbert space, and the 0 operator in the case of a lattice of projection operators.

I prefer the notation ≤ over <, and I like to denote the minimal element by 0. So I'd say that an atom is an element $a$ such that $a\neq 0$, $0\leq a$, and for all $c$ such that $0\leq c\leq a$, we have $c=0$ or $c=a$. This is a singleton subset in the case of $\sigma$-algebras, a 1-dimensional subspace in the case of a lattice of closed subspaces, and a projection operator for a 1-dimensional subspace in the case of a lattice of projections.

So a $\sigma$-algebra is atomic since every non-empty subset contains a point, and a lattice of closed subspaces is atomic since every subspace that isn't 0-dimensional contains a 1-dimensional subspace.

I think that atomicity is one of Piron's axioms. It seems that some people, including Birkhoff and von Neumann, reject it. My interpretation of the "modulo" comment in the translation from French is that they would like to do something like this: Define two subsets A,B to be equivalent if $(A-(A\cap B))\cup(B-(A\cap B)$ has measure 0, and then consider the set of equivalence classes of subsets instead of the set of all subsets.

 

[microsansfil]

Then you need to broaden your experience. See post 2 by Atty.

He speak about non-formal proof, C. Piron and other peoples speak about Axiom defined in formal logic (It is just non-classical but it is a formal logic) and its application to quantum mechanics.

This can not be more clear,it's written in black on white.

In this context (of quantum logic) "Logic" is a mathematical model for deductive thought. A logical system is dened by a formal
structure for constructing sentences, called syntax, and for attributing meaning to these sentences,
termed semantics.

Patrick

 

[bhobba]

He speak about non-formal proof, C. Piron and other peoples speak about Axiom defined in formal logic and its application to quantum mechanics.

Don't know why my link can't be downloaded. Hopefully this will work:
https://www.google.com.au/url?sa=t&...FaVqd-uDnX8Xx1E0A&sig2=0DfTvHfVEwb_ln7SAzJ6NA

The paper did not use formal logic (see the proof of Lemma 4.2 for example) - it used what Atty suggested - rigorous informal language - as every book on math (aside from my experience with the Principa) uses.

No modern mathematician or physicist uses formal logic. The only tome I know that does is Russell's famous Principa - and that was three volumes for just arithmetic.

Thanks
Bill

 

[bhobba]

These are the definitions of "atom" and "atomic" from Piron's book:

Ok got it - standard Quantum Logic stuff.

The point I was trying to get across though, and to see if Patrick had noticed it, was in that approach one introduces probabilities and the Born rule by invoking Gleason. While I couldn't read the paper he linked to I did a search - and sure enough - it mentions Gleason.

Now if you do that you subsume Kolmogorov's axioms by interpreting the measure defined via Gleason as a probability. There is nothing I can see in those axioms that mentions probability.

It's what I have been pointing out all through this thread - axiomitsations in physics are rarely if every complete - they always assume other stuff. Pirons axioms are no different - and having studied Geometric Quantum Theory a bit I am pretty sure it doesn't fit the bill - it's simply not that complete - this was the intent of 'essentially' I highlighted in one of my quotes.

But going even further than that, no modern mathematical of physical tome uses formal logic.

What Patrick is after doesn't exist, nor, to the best of my knowledge is anyone interested in doing such.

Thanks
Bill

 

[microsansfil]

rigorous informal language

This is unknown from mathematics. A door must be open or closed.

C. Piron Write in his article

Certains auteurs ont voulu voir dans les axiomes précédents les règles d'une nouvelle logique. En fait, ces axiomes ne sont que des règles de calcul et la logique habituelle s'applique sans avoir besoin d'être modifiée.

Some authors have wanted to see in the previous axioms, the rules of a new logic. In fact, these axioms are only rules of calculation and the usual logic applies without needing to be changed.

Patrick

 

[bhobba]

On the Logical Foundations of the Jauch-Piron Approach to Quantum Physics

OK - I managed to downloded that document.

It does not use formal logic - it's what Atty mentioned previously - informal rigorous language intermixed with formal logic.

From page 1338:
Hence, this representation yields a many-to-one representation of questions onto suitable operators which is embodied in our mathematical model; indeed, according to the latter, every variable of Le which is bound to range over questions in the physical intended interpretation is made to range over operators representing questions, according to the aforesaid representation, in our model. It must be stressed that the spectral values of the representative operators must not be interpreted as possible outcomes of measurements of the corresponding questions, but as probabilities of the yes outcome (Garola and Solombrino, 1983); therefore, a question must not be confused with the observable represented by the same operator according to the usual Hilbert representation. We also remark that every variable of Le which is bound to range over states in the physical intended interpretation is made to range over operators representing pure states (according to the aforesaid representation) in our model.

Thus it is assuming more than the stated axioms - it is assuming probability axioms.

In fact it requires Gleason as I have mentioned a number of times.

Thanks
Bill

 

[bhobba]

This is unknown from mathematics. A door must be open or closed.

That's wrong - simple as that.

You have read somewhere that mathematics is formal logic and for some reason don't seem to understand math in practice is not done that way.

Here is a book on rigorous math (analysis):
http://math.univ-lyon1.fr/~okra/2011-MathIV/Zorich1.pdf

It's not done by formal logic - some axioms are stated that way - but the treatment is not by formal logic.

In Pirons stuff you linked to the same is done - axioms are stated that way - but the development is not formal - nor can it be. To do it that way for even arithmetic took Russell three volumes of the most dry boring unilluminating math you can imagine.

Can you point me to any math textbook on any advanced area such as topology, analysis, linear algebra, functional analysis that uses formal logic?

Thanks
Bill

 

[microsansfil]

That's wrong - simple as that.

You have read somewhere that mathematics is formal logic and for some reason don't seem to understand math in practice is not done that way.

Nawak.

Anyone who studied model theory, Set theory, Proof theory, Computability, Axiomatic System (show that all mathematical theory could be reduced to some collection of axioms) ... knows what I mean.

I move.

Patrick
PS
Another interesting paper Quantum Logic as Classical Logic : http://arxiv.org/pdf/1406.3526v2.pdf

And This http://indigo.uic.edu/bitstream/handle/10027/10195/DeJonghe_Richard.pdf?sequence=2which is the inverse : Rebuilding Mathematics on a Quantum Logical Foundation

 

[atyy]

These are the definitions of "atom" and "atomic" from Piron's book:

(1.16): DEFINITION If b ≠ C and b < c, one says that C covers b
when b < x < C $\Rightarrow$ x = b or x = c. An element which covers 0
is called an atom (or point). A lattice is said to be atomic if for every
b≠0 there exists at least one atom p smaller than b (i.e. p < b).​

O denotes the minimal element, i.e. $\varnothing$ in the case of $\sigma$-algebras, $\{0\}$ in the case of a lattice of closed subspaces of a Hilbert space, and the 0 operator in the case of a lattice of projection operators.

I prefer the notation ≤ over <, and I like to denote the minimal element by 0. So I'd say that an atom is an element $a$ such that $a\neq 0$, $0\leq a$, and for all $c$ such that $0\leq c\leq a$, we have $c=0$ or $c=a$. This is a singleton subset in the case of $\sigma$-algebras, a 1-dimensional subspace in the case of a lattice of closed subspaces, and a projection operator for a 1-dimensional subspace in the case of a lattice of projections.

So a $\sigma$-algebra is atomic since every non-empty subset contains a point, and a lattice of closed subspaces is atomic since every subspace that isn't 0-dimensional contains a 1-dimensional subspace.

I think that atomicity is one of Piron's axioms. It seems that some people, including Birkhoff and von Neumann, reject it. My interpretation of the "modulo" comment in the translation from French is that they would like to do something like this: Define two subsets A,B to be equivalent if $(A-(A\cap B))\cup(B-(A\cap B)$ has measure 0, and then consider the set of equivalence classes of subsets instead of the set of all subsets.

Yes, it seems that atomicity is one of Piron's axioms. It seems that one gets it "for free" when one uses Kolmogorov's axioms for probability.

If we do what seems to be Birkhoff and von Neumann's "modulo" thing, does anything change? Or is it equivalent to not conditioning on sets of measure zero? If I understand correctly, standard probability based on Kolmogorov's axioms does not allow conditioning on sets of measure zero, eg. http://jmanton.wordpress.com/2012/06/28/sets-of-measure-zero-in-probability/. In fact that link explains the inadmissibility of conditioning on sets of measure zero by explaining that the conditional probability is an equivalence class, which souds very similar to the "modulo" idea of Birkhoff and von Neumann.

 

[atyy]

This is unknown from mathematics. A door must be open or closed.

That's wrong - simple as that.

You have read somewhere that mathematics is formal logic and for some reason don't seem to understand math in practice is not done that way.

Can I suggest that both of you are right, and talking about different things? It is true that there is nowhere in the world written an axiomatization of quantum mechanics in formal language. I don't even know whether Kolmogorov's axioms for probability have been written in formal language. However, I don't think anyone doubts that if one wanted to, Kolmogorov has been precise enough that his axioms can be translated into a formal statements. Similarly, although the proof of Fermat's last theorem was certainly not formal, I don't think the experts doubt that it could be rewritten using Peano's axioms if they wanted to. If Peano's were for some strange reason not enough, I think everyone would be very surprised if they couldn't do it in ZFC.

So do we believe that there is an axiomatization of quantum mechanics that is precise enough that we believe a formalization of it exists in principle? My guess is that it should, after all it doesn't seem much more than linear algebra and Kolmogorov's axioms, both of which we do believe can be formalized if we wished. Or would others disagree? For simplicity, one could take finite dimensional quantum mechanics, and maybe Hardy's axioms for specificity - is there any doubt that Hardy's axioms can be formalized?

Of course the question above would not answer which physical operations we describe in natural language would correspond to the mathematical operations.

However, as far as I can tell, Piron was not that much interested in the formalization of quantum mechanics. He was more interested in reasonable axioms - very much as Hardy. After all, if one were just interested in formalization we can just postulate the Hilbert space and the Born rule straightaway. The point of Piron's derivation is to try to make the Hilbert space seem natural or reasonable. Similarly, the point of Gleason's is that if one considers non-contextuality natural, then the Born rule is implied.

 

[bhobba]

Can I suggest that both of you are right, and talking about different things? It is true that there is nowhere in the world written an axiomatization of quantum mechanics in formal language. I don't even know whether Kolmogorov's axioms for probability have been written in formal language. However, I don't think anyone doubts that if one wanted to, Kolmogorov has been precise enough that his axioms can be translated into a formal statements.

There may be a communication gap here.

No one doubts that the informal language used in practice will produce results in any way different to if formal logic such as found in Russell's Principa was used.

But that is not the sense I got - I got he was claiming its all done by formal logic - which isn't true.

OK let's move on from that.

Yes, it seems that atomicity is one of Piron's axioms. It seems that one gets it "for free" when one uses Kolmogorov's axioms for probability.

What's going on is this.

One takes Pirons axioms, and I assume he deliberately has them in a form he can apply his famous theorem, and shows the 'atoms' of his approach map to the yes-no projection operators on a Hilbert space.

One then invokes Gleason to show the only measure that can be defined on those projection operators is via the Born Rule.

Also note Gleason only works for dimension 3 or greater - dimension 2 is an issue if you are being rigorously exact.

But to proceed from that one needs to make further assumptions and introduce concepts like independent observations, show the measure defined by Gleason obeys the Kolmogorov axioms etc.

Thanks
Bill

 

[atyy]

Also note Gleason only works for dimension 3 or greater - dimension 2 is an issue if you are being rigorously exact.

Well, we could just use Busch's theorem. Anyway, intuitively I think that Busch's theorem is just saying that there's a Naimark extension for dimension 2, and we can apply Gleason's to the Naimark extension. And the Naimark extension is just the formalization of being able to place the Heisenberg cut in more than one place:)

Ok, that's silly, since it assumes Copenagen is intuitive (I confess it is:)

 

[atyy]

Similarly, although the proof of Fermat's last theorem was certainly not formal, I don't think the experts doubt that it could be rewritten using Peano's axioms if they wanted to. If Peano's were for some strange reason not enough, I think everyone would be very surprised if they couldn't do it in ZFC.

OK, googling suggests I was too hasty there. Here's a very interesting blog post, with interesting comments too: http://blog.computationalcomplexity.org/2014/01/fermats-last-theorem-and-large.html.

 

[atyy]

What's going on is this.

One takes Pirons axioms, and I assume he deliberately has them in a form he can apply his famous theorem, and shows the 'atoms' of his approach map to the yes-no projection operators on a Hilbert space.

One then invokes Gleason to show the only measure that can be defined on those projection operators is via the Born Rule.

Also note Gleason only works for dimension 3 or greater - dimension 2 is an issue if you are being rigorously exact.

But to proceed from that one needs to make further assumptions and introduce concepts like independent observations, show the measure defined by Gleason obeys the Kolmogorov axioms etc.

Thanks
Bill

OK, if I understand Piron tries to have some natural axioms from which one can get (close to) the Hilbert space. Then if one believes non-contextuality is natural, one can also get the Born Rule by Gleason's.

What is a bit opaque to me is - in Hardy's derivation, where is it that non-contextuality enters?

 

[bhobba]

Another interesting paper Quantum Logic as Classical Logic : http://arxiv.org/pdf/1406.3526v2.pdf

And This http://indigo.uic.edu/bitstream/handle/10027/10195/DeJonghe_Richard.pdf?sequence=2which is the inverse : Rebuilding Mathematics on a Quantum Logical Foundation

All that stuff is well known.

It simply completes a line of reasoning started by Von-Neumann that dates back to the early days of QM.

Its the basis of the Geometric approach, the reference I have being Varadarajan that I have been studying on and off for a while now.

Formally its what is called a logic - which is what Piron's axioms define. One can define observables, states, even probability measures on those states, all sorts of things in a logic - see Chapter 3 of Varadarajan.

The key idea is to show the particular logic is equivalent to the projection operators on a Hilbert space - and by choosing the axioms of your QM logic carefully one can invoke Pirons Theorem or similar to prove that equivalence.

Then one invokes Gleason to derive the Born rule for those probability measures.

All very beautiful and mathematically satisfying.

BUT - and here is the clanger - one must introduce other axioms to arrive at QM eg something to address the dimension limitation of Gleason. Another is for filtering type observation to show the resultant state is an eigenvector of the observable you need to assume continuity. There are undoubtedly others as well.

This approach is extremely beautiful and alluring - which is why I in fits and starts keep studying it - but falls short of the aim of fully axiomatising QM.

Thanks
Bill

 

[atyy]

What!? http://www.mth.kcl.ac.uk/~streater/piron.html

 

[bhobba]

What is a bit opaque to me is - in Hardy's derivation, where is it that non-contextuality enters?

Well spotted.

He takes probabilities as his fundamental thing and shows via his axioms, QM, as a probability model, is what results.

However we come to the nasty little issue of applying it. Are the observables defined in that probability model the only things in the theory? That's where non-contextuality comes into it - hidden variables can be contextual.

Its the same with Piron of course. But that isn't my main concern - one defines observables etc and assumes, just like if you apply Hardy, you run into the issue of exactly how good a model it is.

Personally I give both Hardy and Piron a pass on that. The treatment of both define the theory pretty clearly - its simply how good a model is it.

Again well spotted - it never even occurred to me.

Thanks
Bill

 

[atyy]

Personally I give both Hardy and Piron a pass on that. The treatment of both define the theory pretty clearly - its simply how good a model is it.

But maybe they don't need a pass? Piron only tried to derive the Hilbert space, not the Born rule, so he doesn't obviously need contextuality. And yes, to get from Piron to the Born rule via Gleason, we understand nowadays that we need non-contextuality.

But does Hardy fail to mention the assumption of non-contextuality? Or is it in there, and just in a more natural or "reasonable" way, as he intends?

 

[bhobba]

OK, if I understand Piron tries to have some natural axioms from which one can get (close to) the Hilbert space.

By careful choice of the axioms you get exactly a Hilbert space - but some bits are not as 'natural' as one would like.

Soler's Theorem is a bit more of an advance in the natural department:
http://golem.ph.utexas.edu/category/2010/12/solers_theorem.html
http://arxiv.org/pdf/math/9504224v1.pdf
http://arxiv.org/pdf/quant-ph/0105107v1.pdf

But, while so tantalisingly close, still isn't quite there yet.

John Baez discussed it in some of his finds articles - if I remember correctly that is. You can almost hear him weep - if only - it would be just so beautiful if it was. I think its the natural reaction of those with a mathematical bent to this stuff (and of course I am one).

Thanks
Bill