Axiomatization of quantum mechanics and physics in general ?

[microsansfil]

Hello

Every physical theory is formulated in terms of mathematical objects. It is thus necessary to establish a set of rules to map physical concepts and objects into mathematical objects that we use to represent them.

http://ocw.mit.edu/courses/nuclear-...s-fall-2012/lecture-notes/MIT22_51F12_Ch3.pdf

Is there a relationship between this so-called axiomatic formulation and mathematical formulation in formal system ?

In mathematics :

Formal proofs are sequences of well-formed formulas. For a wff to qualify as part of a proof, it might either be an axiom or be the product of applying an inference rule on previous wffs in the proof sequence. The last wff in the sequence is recognized as a theorem.

Once a formal system is given, one can define the set of theorems which can be proved inside the formal system. This set consists of all wffs for which there is a proof.

In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems

So far we have only talked about syntax

To give sense you need mathematical interpretation. In mathematics, model theory is the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic. A theory is a set of sentences in a formal language, and model of a theory a structure (e.g. an interpretation) that satisfies the sentences of that theory.

For example let say F is the closed formula. F = ∀x ∀y (p(x,y) → ∃ z ( p(x,z) ∧ p(z,y) ) p is a predicate.
Noting G the implication F :∀x ∀y ( G )

1/ Interpretation I1 for domain D1 which is the set of real. The binary predicate p is the order <

if p (x, y) is false, G is true
if p (x, y) is true, then x <y. Let z = (x + y) / 2. Then, p (x, z) is true, and p (z, y) also

In conclusion, the formula F is satisfiable and I1 is a model for F.

2/ Interpretation I2 for domain D2 which is the set of natural numbers. The binary predicate p is still the order <

if p (x, y) is false, G is true
if p (x, y) is true, then x <y. There is no integer between x and y when x and y are consecutive. G may be false.

In conclusion F therefore is not valid as there are interpretations which do not satisfy F, but F is not unsatisfiable.In other words about this example, the same syntatic formula is undecidable in the theory of total order, then it is a theorem in the theory of dense orders.

A mathematical statement is written in a certain language, and we can say it is true or false in a structure that interprets all the language elements.

Patrick

 

[atyy]

In mathematics, a rigourous, non-formal proof (eg. Fermat's last theorem) is generally thought to also have a formal proof. Generally the non-formal proof is much easier to read than the formal proof, and the existence of the formal proof is not doubted, so people don't usually check. However, some people have tried to check some non-formal proofs by formal proofs, for example http://www.newscientist.com/article/dn26041-proof-confirmed-of-400yearold-fruitstacking-problem.html.

In mathematical physics, the proofs (eg. Noether's theorem, Stinespring dilation theorem) are generally thought to be on the same level as the rigourous, non-formal proofs, with the existence of formal proofs assumed.

The difficulty is of course how the symbols are interpreted as corresponding to experiments and observations. Here generally we need to have some intuitive notions. This is not so bad, since even Goedel's incompleteness theorem usually uses the intuitive natural numbers. If one doesn't want to use the intuitive natural numbers and prefers to define the natural numbers via ZFC, one still needs a metalanguage to define ZFC.

 

[bhobba]

A formal mathematical system becomes applied when you map the formal objects of the theory to things out there.

In physics, and applied math in general, that mapping is intermixed with the formal objects in ways that's often not made explicit but assumed. Its what is known as a mathematical model - and you can find the detail of this stuff in books on mathematical modelling. Indeed sometimes its even left up in the air the exact correspondence because it doesn't really matter eg in probability you can either do it frequentest or Bayesian - most times it makes no difference - although sometimes it does make a difference to how you view the problem in which case it can be made explicit.

The most basic example is good old Euclidean geometry you learned about at school. It speaks of points of no size that only has location - such of course do not exist. It speaks of lines of length but no width. Such of course do not exist. When you apply it you usually draw diagrams that its obvious what you take as points and lines and that's how you apply it. Or you can do what Hilbert did and totally axiomatise Euclidean geometry and everything is abstract - that's the modern mathematical method.

Usually in physics what's done is the Euclid method rather than the formal method.

As an example, since this is a forum for QM, we will examine that.

See post 137:
https://www.physicsforums.com/showthread.php?t=763139&page=8

The fundamental axiom is:
An observation/measurement with possible outcomes i = 1, 2, 3 ... is described by a POVM Ei such that the probability of outcome i is determined by Ei, and only by Ei, in particular it does not depend on what POVM it is part of.

Notice the use of observation/measurement. That's something that exists out there and understanding what it's saying leads us immediately into the difficult interpretational problems of QM - without detailing here exactly what they are.

It also leaves up in the air exactly what probability is. That's common in applied math with it not really making much difference most of the time if its either of the two main interpretations - namely frequentest or Bayesian. But most applied mathematicians choose frequentest because it's much more concrete - a notable exception being Bayesian Inference which is more naturally Bayesian. But QM is (mostly) not an expectation, I would say most applied math types would chose a frequentest type view and hold to a interpretation like the Ensemble. An exception would be many worlds where Bayesian is the best view.

Philosophers however all have their own take which is different from the utility view of applied maths.

Thanks
Bill

 

[bhobba]

In mathematics, a rigourous, non-formal proof (eg. Fermat's last theorem) is generally thought to also have a formal proof. Generally the non-formal proof is much easier to read than the formal proof, and the existence of the formal proof is not doubted, so people don't usually check. However, some people have tried to check some non-formal proofs by formal proofs, for example http://www.newscientist.com/article/dn26041-proof-confirmed-of-400yearold-fruitstacking-problem.html.

In mathematical physics, the proofs (eg. Noether's theorem, Stinespring dilation theorem) are generally thought to be on the same level as the rigourous, non-formal proofs, with the existence of formal proofs assumed.

The difficulty is of course how the symbols are interpreted as corresponding to experiments and observations. Here generally we need to have some intuitive notions. This is not so bad, since even Goedel's incompleteness theorem usually uses the intuitive natural numbers. If one doesn't want to use the intuitive natural numbers and prefers to define the natural numbers via ZFC, one still needs a metalanguage to define ZFC.

That is true.

But applied math takes it further by semantically intermixing the objects its being applied to and the mathematical objects of the theory in an informal and often not explicitly stated way.

Thanks
Bill

 

[microsansfil]

The difficulty is of course how the symbols are interpreted as corresponding to experiments and observations.

It can exist also other axiomatic. From Constantin Piron quantum mechanics can be reduced to three axiom, competitor postulates of quantum mechanics (Dixit Wiki).

As you can also see special relativity in a different mathematical langage.

In all case you need semantic to build model to make experiment. The physical's purpose is the experimentation.

Patrick

 

[atyy]

It can exist also other axiomatic. From Constantin Piron quantum mechanics can be reduced to three axiom, competitor postulates of quantum mechanics (Dixit Wiki).

As you can also see special relativity in a different mathematical langage.

In all case you need semantic to build model to make experiment. The physical's purpose is the experimentation.

Piron's work is cited in Hardy's http://arxiv.org/abs/quant-ph/0101012 and http://arxiv.org/abs/1303.1538, and Chiribella, D'Ariano and Perinotti's http://arxiv.org/abs/1011.6451 which are other axiomatizations of quantum mechanics (but I think only for the finite dimensional case). Also interesting is Leifer and Spekkens's http://arxiv.org/abs/1107.5849.

 

[bhobba]

[URL]http://en.wikipedia.org/wiki/Constantin_Piron

[URL]http://en.wikipedia.org/wiki/Constantin_Piron

Sure.

That's part of geometric QM. There are some texts on it about eg I have a copy of Varadarajan - Geometry of Quantum Theroery:
https://www.amazon.com/dp/0387493859/?tag=pfamazon01-20

It is in fact our most powerful formalism.

The issue however is its mathematically non trivial - which is a codeword for HARD. It stretches my mathematics to its limit.

The reason its generally not done that way is there is simply no gain. The issue it penetrates most deeply isn't a particularly worrying one in practice - namely the most mathematically elegant way to view the correspondence between classical and quantum.

Also there is a gulf in the language used by both approaches. To see this have look at the following thread:
https://www.physicsforums.com/showthread.php?t=758125

Frederick used the language of function analysis, which Geometric QM uses (it's more used by pure mathematicians than applied - although most applied mathematicians are aware of it as I am from studies into things like Hilbert spaces etc) - I used a more informal approach that avoided it. Most physicists are more informal.

Generally there is a gulf between the methods acceptable to[pure mathematicians and those of applied. An example would be the functional derivative:
http://en.wikipedia.org/wiki/Functional_derivative

See the section on 'Using the delta function as a test function'

Doing that you quite frequently end up with polynomials in the Dirac Delta function - but higher orders of that function (its not even really a function) are not even defined. But you generally don't run into problems - even though what you are doing is WRONG. Its applied math vs pure.

Thanks
Bill

 

[bhobba]

Piron's work

He uses the Geometric approach which based on the Symplectic Geometric view of Classical Mechanics:
http://research.microsoft.com/en-us/um/people/cohn/thoughts/symplectic.html

Its mathematically very elegant - which is why mathematicians love it.

But practical applications is not its strong point - which is why its not generally used by applied mathematicians and physicists.

As one whit said - you have these applied theories, but when mathematicians get a hold of it and express it in their most elegant formalism, such as Symplectic Geometry and classical mechanics, it becomes unrecognisable.

Thanks
Bill

 

[atyy]

It's available free!

http://retro.seals.ch/digbib/view?pid=hpa-001:1964:37::443
Piron, C.
Axiomatique Quantique
Helvetica Physica Acta, 37: 439–468, 1964

http://retro.seals.ch/digbib/view?pid=hpa-001:1969:42::844
Jauch, J.M., and Piron, C.
On the Structure of Quantal Propositional Systems
Helvetica Physica Acta, 42: 842–848, 1969

Interesting also is that Jauch and Piron tried to strengthen the von Neumann theorem excluding hidden variables. Their proof is presumably wrong, as discussed by Bohm and Bub.

http://retro.seals.ch/digbib/view?pid=hpa-001:1963:36::833
Jauch, J.M., and Piron, C.
Can hidden variables be excluded in quantum mechanics?
Helvetica Physica Acta, 36: 827–848, 1969

http://www.physics.nmsu.edu/~bkiefer/HISTORY/BOHM_BUB_1966.pdf
A refutation of the proof by Jauch and Piron that hidden variables can be excluded in quantum mechanics
Rev Mod Phys, 1966

 

[microsansfil]

It's available free!
...

http://www.physics.nmsu.edu/~bkiefer/HISTORY/BOHM_BUB_1966.pdf
A refutation of the proof by Jauch and Piron that hidden variables can be excluded in quantum mechanics
Rev Mod Phys, 1966

Thank for all this articles. This shows that the debate is on semantics. Indeed in the last article the assumption seem to be "the impossibility of proposition that describe simultaneously the results of measurements of two non-commutatif observables in an "empirical fact

...
"

"ie, that the current linguistic structure of quantum mechanics is the only one that can be used correctly to describe the empirical facts underlying the theory"

One could almost make the similarity to the model theory (the semantic approach) in mathematics. Distinguish the views semantic and syntactic. The first is model theory, while the second characterizes the proof theory.

The proof theory defines formal theories, and model theory gives interpretations. This for first-order theories, that is to say, the theories expressed in first-order languages​​. These are formal languages​​. That is to say, the languages ​​that are first defined by the syntax, without any reference to the meaning of their expression.

You can see a kind of division of labor.

Patrick

 

[bhobba]

"ie, that the current linguistic structure of quantum mechanics is the only one that can be used correctly to describe the empirical facts underlying the theory"

Its well known that many different 'theories' equally well describe QM.

Since you raise Pirons proof, that it is invalid has nothing to do with linguistics or semantics, its got to to with the assumption he made being false, namely the expectation assumption ie < A > + < B >=< (A + B) >. Its exactly the same assumption Von Neumann made except Piron only assumed it for commuting observables. Its false for hidden variable theories.

Thanks
Bill

 

[microsansfil]

its got to to with the assumption he made being false, namely the expectation assumption ie < A > + < B >=< (A + B) >. Its exactly the same assumption Von Neumann made except Piron only assumed it for commuting observables. Its false for hidden variable theories.

I don't read this in the articles give by atyy.

I read

The argument by which Jauch and Piron attemp to prove that the structure of quantum theory is incompatible with the assumptionof hidden variables is based on an analysis of the type of experimental question that can be asked in the theory. Thus, they consider those observables of a physical system which are associated with only two alternative or possibilities, which may be designed by 1 or 0, yes or no, true or false.

And in the article about "Quantum Axiomatics" C. Piron explain his theory base on propositional calculus; The conclusion of the article

We developed for the propositional calculus a very general formalism valid both in classical physics that quantum physics. Thus it includes not only the mechanics of Newton n particles but also the phenomenological theory of fluid, electromagnetism and gravity (in the classical sense).

Patrick

 

[microsansfil]

Hello

It seem that there are several axiomatic approach to the foundations of quantum mechanics available in the physical and mathematical literature.

http://www2.latech.edu/~greechie/1973 Quantum Logics.pdf

in particular Günther Ludwig (1918–2007) which was a German physicist mainly known for his work on the foundations of quantum theory. In Ludwig (1970, 1985, 1987), he published an axiomatic account of quantum mechanics, which was based on the statistical interpretation of quantum theory.

An axiomatic Basis for Quantum Machanics ; Structuralism in Physics G. Ludwig

The mathematical theory MT used in a physical theory PT contains as its core a “species of structure” Σ. This is a meta-mathematical concept of Bourbaki which Ludwig introduced into the structuralistic approach.

This line of research was recently revived by the influence of Quantum Information Theory, like this one "http://www.qubit.it/research/publications/cdp-vaxjo.pdf"

Patrick

 

[bhobba]

I don't read this in the articles give by atyy.

Its well known though - Bell and others sorted it out long ago:
http://fy.chalmers.se/~delsing/QI/Bell-RMP-66.pdf

Thanks
Bill

 

[bhobba]

It seem that there are several axiomatic approach to the foundations of quantum mechanics available in the physical and mathematical literature.

I don't quite follow the point you want to make.

There are several axiomatic foundations to many areas of math and physics. They are either generally assumed to be logically equivalent or formal proofs exist showing they are.

The above just seems to be a variant of Hardy's in that it also is based on information ideas - except Hardy seems more elegant
http://arxiv.org/pdf/quantph/0101012.pdf

Thanks
Bill

 

[Fredrik]

I clicked the link to Piron's Wikipedia entry, and found a link to Varadarajan's review of Piron's book. Varadarajan mentioned a book called "The logico-algebraic approach to quantum mechanics", so I googled it. There's an article in it called "A survey of axiomatic quantum mechanics" by Stanley P Gudder that I find interesting. Unfortunately I have only found it at google books, and some of the pages don't show up in the preview. Here's the link anyway (mainly so I can find it myself in the future :tongue:) http://books.google.com/books?id=HN... mechanics&hl=sv&pg=PA323#v=onepage&q&f=false

 

[bhobba]

Varadarajan mentioned a book called "The logico-algebraic approach to quantum mechanics"

I think its exactly the same as, or at least incorporated in, the Geometric approach to QM as for example found in Varadarajan - Geometry Of Quantum Theory.

It's discussed in Chapter 3 - The Logic Of A Quantum Mechanical System.

The notes at the end of that chapter mention the works of Piron. Evidently the book to get on it, from those notes, is 'The Logic Of Quantum Mechanics - Volume 15 - Encyclopaedia Of Mathematics And Its Applications'

All this is tied up with an open question in the geometric approach to QM - namely exactly how much does the logic of QM imply the Hilbert Space struture of QM.

Piron did a famous theorem that almost, but not quite, proved it:
http://plato.stanford.edu/entries/qt-quantlog/#5

This maybe is what the OP is referring to with regard to Piron - not his famous proof about hidden variables.

If that's the case Pirons Theorem has recently been superseded by Solèr’s Theorem:
http://golem.ph.utexas.edu/category/2010/12/solers_theorem.html

It gets us very close to that elusive goal - but again isn't quite there yet. Although there is something in the back of my mind it does accomplish it with an extra reasonable additional assumption - but I can't recall what it is.

There is zero doubt as far as the foundations of QM is concerned this is our most penetrating formalism. But most physicists don't use it because it's notoriously hard, and doesn't really penetrate the issue most physicists are concerned with - how to apply it.

I have, in fits and starts, delved into it. Mathematical beauty of the first order - but one is left with the question, so? Aside of course from the appreciation of beauty of this sort by those of mathematical bent which IMHO is very worthwhile - but opinions on such things will vary.

If anyone wants the gory detail on it:
http://arxiv.org/pdf/math/9504224v1.pdf
http://arxiv.org/pdf/quant-ph/0105107v1.pdf

Like I say - very penetrating - but mathematically non trivial.

Thanks
Bill

 

[bhobba]

quantum mechanics can be reduced to three axiom

I have no reason to doubt its true. I even reduced it to one, Ballentine to two.

But care is required on what is reasonable and what is an axiom. I suspect strongly, as is the case with my and Ballentine's, one makes reasonable assumptions along the way such as the continuity assumption for filtering type observations.

Thanks
Bill

 

[microsansfil]

I don't quite follow the point you want to make.

What is the point I want to make ?

There are several axiomatic foundations to many areas of math and physics. They are either generally assumed to be logically equivalent or formal proofs exist showing they are.

In the article write by R.l.GREECHIE ANDS.P.GUDDER we can read

Models present different approaches to what appears to be essentially the same underlying theory. In fact there havec been studies made comparing these different approches [27, 35, 63].
Now it may seem, at first sight, to be wasteful and redundant to proliferate the literature with different approaches
to the same subject.

What concerns me is the question : These axioms are they useful for the experimental physicist ? How do they build their models, their experiments from these axioms ? For example in this experiment http://arxiv.org/abs/1401.4318

Patrick

 

[microsansfil]

I have no reason to doubt its true. I even reduced it to one, Ballentine to two.

If you reduced it to one this mean that other axiom are theorem in your axiomatics theoric.
It can not be independent since otherwise it would form another Theory.

Have you demonstrated this ?

Patrick

 

[bhobba]

What concerns me is the question : These axioms are they useful for the experimental physicist ?

No. And I think its pretty certain that's true.

How do they build their models,

Well I can't speak for others but the way I came up with my current view and the axioms I use is I read a lot and picked the eyes out of it based on my particular view of elegance. And one thing I can assure anyone is elegance is a very personal thing. The Geometric view is elegance incarnate - but its difficulty level is not my cup of tea. So I chose a different route - the one in Ballentine which had a strong effect on me. It just uses two axioms and the development is just right for the level of rigour I enjoy.

their experiments from these axioms ? For example in this experiment http://arxiv.org/abs/1401.4318

Its like any applied area you may learn axioms at the start (and I have to say most QM books I have read are not axiomatically presented - its sort of a historical mish mash - eg Griffiths - which is still a good book BTW) but once you build up an intuition that's what is used.

In my degree I got caught up in the rigour thing. I well remember the teacher that cured me. He said I can show you books that do that - but you wouldn't read them. I got one - and he was right.

Me and a couple of friends heard of Russell's Principia Mathematica and we thought its so important we really should go through it. I got about 1/4 way through and gave it away - the best was about 1/3.

Thanks
Bill

 

[bhobba]

If you reduced it to one this mean that other axiom are theorem in your axiomatics theoric. Have you demonstrated this ?

No.

The reason is there are hidden axioms in all treatments eg the continuity one I mentioned.

I can virtually guarantee that three axiom treatment you mention doesn't explicitly state a number of important things. Since its by Piron I can say it will be via the Geometrical treatment that makes heavy use of Gleason's theorem - there are a number of hidden assumptions if you do that eg non-contextuality and the strong superposition principle. That's two axioms right there.

However there is only one way to find out - post the three axioms. I am pretty sure I can spot some it left out.

Added Later:
I found them:
http://arxiv.org/pdf/quant-ph/0008019.pdf

'Gleason’s theorem, together with the spectral theorem, the classical results of Stone, Wigner, Weyl and von Neumann, and Mackey’s own work on induced unitary representations, allow one essentially to derive the entire apparatus of non-relativistic quantum mechanics (including its unitary dynamics, the CCRs, etc.), from the premise that the logic of experimental propositions is represented by the projection lattice P(H).'

Its the geometric view. Notice that word - essentially. Other assumptions like the previous axioms I mentioned are required.

What my single axiom does is make the subsequent ones so reasonable you may not even notice it.

Thanks
Bill

 

[Fredrik]

I think its exactly the same as, or at least incorporated in, the Geometric approach to QM as for example found in Varadarajan - Geometry Of Quantum Theory.

It's discussed in Chapter 3 - The Logic Of A Quantum Mechanical System.

I haven't tried to figure out what exactly the book is about, but the specific chapter I linked to describes 4 different approaches to QM. Varadarajan's approach is number 2. The chapter describes the pros and cons of these approaches, and I think that could be interesting. Unfortunately I haven't found it online, so I would have to go to a library to read the pages that don't show up in the preview.

The notes at the end of that chapter mention the works of Piron. Evidently the book to get on it, from those notes, is 'The Logic Of Quantum Mechanics - Volume 15 - Encyclopaedia Of Mathematics And Its Applications'

That book is often referenced, even though it's out of print. I have digital copy though.

Piron did a famous theorem that almost, but not quite, proved it:
http://plato.stanford.edu/entries/qt-quantlog/#5
[...]
...Pirons Theorem has recently been superseded by Solèr’s Theorem:
http://golem.ph.utexas.edu/category/2010/12/solers_theorem.html

Interesting. According to these two articles, Piron showed that a simple set of assumptions about the lattice imply that it's isomorphic to the lattice of $\perp$-closed subspaces ($M^{\perp\perp}=M$) of some inner product space V over an involutive division ring D, and then Solér showed that if such an inner product space is orthomodular ($M+M^\perp=V$) and contains an infinite sequence, then D must be one of the three amigos $\mathbb R$, $\mathbb C$, $\mathbb H$, and V is a Hilbert space.

So when we combine these two theorems, we find that Piron's assumptions take us to what Varadarajan calls "the standard logics". I thought that I had read somewhere that this is what Piron proved, but maybe I misinterpreted something.

There is zero doubt as far as the foundations of QM is concerned this is our most penetrating formalism. But most physicists don't use it because it's notoriously hard, and doesn't really penetrate the issue most physicists are concerned with - how to apply it.

Well said.

I think that to some extent, the problem is that no one has written a good book about these things. Varadrajan's book contains a lot of fantastic stuff that's you won't find anywhere else, but it's insanely hard to read.

I have, in fits and starts, delved into it. Mathematical beauty of the first order - but one is left with the question, so? Aside of course from the appreciation of beauty of this sort by those of mathematical bent which IMHO is very worthwhile - but opinions on such things will vary.

People who are only concerned with how to use QM and other theories of physics will ask "so?"...and they probably don't even realize that they're doing essentially the same thing as the non-scientist who asks that question about things like the discovery of the Higgs particle.

I think the fact that we should start with a lattice (or at least a partially ordered set) can be derived from the assumption of falsifiability alone. Then falsifiable theories can be classified by the additional assumptions we make about their lattices. Piron's assumptions simply define a class of falsifiable theories.

In my opinion, the real beauty of the quantum logic approach is that once we have decided to develop a theory in which yes-no experiments are represented by the closed subspaces of a complex Hilbert space, the definitions of "state" and "observable" are very natural, even obvious, instead of being pulled out of a hat, as in the traditional Hilbert space approach.

 

[bhobba]

I think that to some extent, the problem is that no one has written a good book about these things. Varadrajan's book contains a lot of fantastic stuff that's you won't find anywhere else, but it's insanely hard to read.

Yes - well said

In my opinion, the real beauty of the quantum logic approach is that once we have decided to develop a theory in which yes-no experiments are represented by the closed subspaces of a complex Hilbert space, the definitions of "state" and "observable" are very natural, even obvious, instead of being pulled out of a hat, as in the traditional Hilbert space approach.

Everything about it is just so beautiful and natural. The mathematician in me loves it - but its just so damn hard.

Also I don't think it quite achieves what the OP would want an axiomatic approach to achieve eg Gleason only woks for dimensions 3 or greater, plus other stuff I mentioned. It essentially achieves its aims - even by the standards of mathematicians - but there are some blemishes.

Thanks
Bill

 

[microsansfil]

In my opinion, the real beauty of the quantum logic approach is that once we have decided to develop a theory in which yes-no experiments are represented by the closed subspaces of a complex Hilbert space, the definitions of "state" and "observable" are very natural, even obvious, instead of being pulled out of a hat, as in the traditional Hilbert space approach.

Introduction of C. Piron in his book Quantum Axiomatics

Introduction

In quantum theory, an observable is represented by a linear operator (usually self-adjoint) acting in a Hilbert space.

But we know further that the existence of superselection rules implies the existence of self-adjoint which correspond to no observable linear operators.

In this work we will characterize the intrinsic structure of all observable physical system.

Then we can justify the use of the Hilbert space and linear operators with the peculiarities related to superselection rules.

We follow an axiomatic method whose starting point is due to G. Birkhoff and J. von Neumann1).But that is not the "logical" differences between the classical theory and quantum theory that concern us, our goal is to develop a general formalism valid in both cases.

That is why, after introduce all the variables of interest to us and discussed their relationship with logic, we will examine in detail the classical case and the usual quantum case (including the case of the oscillator).

Thus we shall be led to formulate our axioms.

Finally, we will define the notion of physical state as "generalized probability" according to an idea due to Mackey)

Propositional calculus

Given a family of physical systems, we consider among all possible measures those for which the measurement system results in yes or no.

As, indeed, any measure can be replaced by a yes-no experiments, the study of observable is reduced to the study of the structure of all the proposals.

We say that a proposition is true for a given system (ie prepared in a specific way), if the answer yes is undoubted.

...

The three sets of axioms 0, T and C define on the proposals a complete lattices orthocomplemented, because for any family of proposals, there is a lower bound, but also an upper bound.

...

In fact, these axioms are only rules of calculation and logic standard applies without the need to change.

...

I did not know it was possible to capture in the same axiomatic, classical mechanics and quantum mechanics.

Patrick

 

[atyy]

There is zero doubt as far as the foundations of QM is concerned this is our most penetrating formalism. But most physicists don't use it because it's notoriously hard, and doesn't really penetrate the issue most physicists are concerned with - how to apply it.

I think the other issue is - if QM is found to be experimentally false - what theoretical options do we have?

From this point of view, both of Hardy's derivations (http://arxiv.org/abs/quant-ph/0101012, http://arxiv.org/abs/1303.1538) are mystifying to me - they distinguish QM and classical probability by a single condition - so it would seem that to go beyond QM, we return to classical probability within his framework. If one likes Bohmian Mechanics, that is completely reasonable, since Bohmian Mechanics is classical probability and suggests "quantum non-equilibrium" as a simple way to go beyond QM.

But there are some approaches that are in Hardy's spirit, but by using slightly different sets of axioms seem suggest other ways beyond QM. For example, Popescu and Rohrlich http://arxiv.org/abs/quant-ph/9709026 show that there are theories more non-local than QM that are also consistent with relativity. Masanes and Mueller http://arxiv.org/abs/1004.1483 also talk about ways to go beyond QM.

Incidentally, I have not read much of the old work like Piron's and Ludwig's, but am more familiar with modern stuff like Hardy's and Chiribella, D'Ariano and Perinotti's (I put Hardy and Chiribella et al in the same class). Would someone who is familiar with Piron's, Ludwig's and Hardy's approaches care to say what the major conceptual differences are?

Actually, reading Hardy, he says that his approach is a development of Ludwig's:
http://arxiv.org/abs/1303.1538
"Many of these reconstruction attempts employ the so called “convex probabilities framework”. This goes back to originally to Mackey and has been worked on (and sometimes rediscovered) by many others since including Ludwig [32], Davies and Lewis [11], Gunson [21], Mielnik [36], Araki [2], Gudder et al. [20], Foulis and Randall [14], Fivel [12] as well as more recent incarnations [22, 3].

The circuit framework used here [27, 29] (see also [28, 25]) might be regarded as a marriage of the convex probabilities framework and the pictorial (or categorical) approach of Abramsky and Coecke [1, 9]. A similar framework has been developed by Chiribella, D’Ariano, and Perinotti [7]."

 

[Fredrik]

I did not know it was possible to capture in the same axiomatic, classical mechanics and quantum mechanics.

Classical mechanics can be viewed as a probability theory that assigns probabilities 0 or 1 to subsets of phase space. For each point s in phase space, there's a probability measure $P_s$ defined by
$$P_s(x)=\begin{cases}1 & \text{if }x\in E\\ 0 & \text{if }x\notin E.\end{cases}$$ The points in phase space are often called "states". The number $P_s(E)$ can be interpreted as the probability that the state of the system is in the set E, given that the state is specifically the point s.

The term "pure state" is used both for the point s, and for the associated probability measure $P_s$. If $\{s_1,\dots,s_n\}$ is a finite subset of phase space, and $c_1,\dots,c_n$ are numbers in the interval [0,1] such that $c_1+\cdots+c_n=1$, then $\sum_i c_i P_{s_i}$ is a probability measure. These measures are called "states". A state that isn't equal to any pure state is said to be "mixed". Unlike the pure states, the range of a mixed state is not just {0,1}. It's a larger subset of [0,1].

The numbers $c_i$ can be interpreted as the probability that the system is in the state $s_i$. This means that all non-trivial probabilities (i.e. probabilities that aren't 0 or 1) in classical mechanics are due to ignorance about what the state is.

There's a simple way to change this into a theory where even the pure states can assign non-trivial probabilities. Instead of taking the pure states to be the points of some set, and having them assign probabilities to subsets of that set, we take the pure states to be 1-dimensional subspaces of an inner product space, and have them assign probabilities to subspaces of that inner product space. In the classical case, the probability is 0 if the pure state isn't in the set, and 1 if it is. In these new theories, the probability is 0 if the pure state is orthogonal to the subspace, is 1 if it's a subset of the subspace, and is a number between 0 and 1 in all other cases. Dumbed down only a little, the probability is determined by the "angle" between the pure state and the subspace that is assigned a probability. If we're just trying to write down such a theory for fun, it makes sense to just use a finite-dimensional inner product space over $\mathbb C$, because they are the easiest ones to work with.

The set of all subsets of a set, and the set of all subspaces of a finite-dimensional vector space, both satisfy the definition of a lattice. Technically, in the classical case, we wouldn't use the set of all subsets. The standard choice is to use the Borel σ-algebra. This is the smallest σ-algebra that contains all the open sets. When we're trying to develop a useful quantum theory along these lines, we wouldn't use a finite-dimensional vector space. We would use a separable infinite-dimensional Hilbert space. We would also use the lattice of closed subspaces rather than the lattice of subspaces (i.e. we would require the subspaces to be closed sets and therefore Hilbert spaces, rather than just vector spaces). These technical details aren't very relevant to the main point, which is that we're dealing with a lattice in both the classical case and the quantum case. (All σ-algebras are lattices).

From this point of view, the key difference between classical and quantum mechanics is that the lattice of a classical theory is an especially simple type of lattice called a Boolean algebra, while the lattice of a quantum theory is a different type of lattice.

These ideas have been developed further into a fancy mathematical theory called quantum measure theory. I don't understand much of it yet, but I'll try to explain what I know. Given a von Neumann algebra W (a subset of the set of bounded linear operators on a Hilbert space that satisfies some conditions), the set of projections in W (linear operators P such that $P^2=P=P^*$), is a lattice. If you define a probability measure on such a lattice, you end up with a generalized probability theory (generalized because the standard definition requires the domain of a probability measure to be a σ-algebra), and somehow the commutative von Neumann algebras correspond to classical probability theories, while the non-commutative ones correspond to quantum theories.

 

[microsansfil]

Classical mechanics can be viewed as a probability theory that assigns probabilities 0 or 1 to subsets of phase space. For each point s in phase space, there's a probability measure $P_s$ defined by
$$P_s(x)=\begin{cases}1 & \text{if }x\in E\\ 0 & \text{if }x\notin E.\end{cases}$$ The points in phase space are often called "states". The number $P_s(E)$ can be interpreted as the probability that the state of the system is in the set E, given that the state is specifically the point s.

Yes

However (Dixit C. Piron in his book) C. G.Birkhoff and J. von Neumann criticized the view that any subset of the phase space is a proposal. To them, they seems it is artificial to consider, as a rational number (in radians per second), a proposal statement such as the angular velocity of the Earth around the sun.

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.

Patrick

 

[Fredrik]

However (Dixit C. Piron in his book) C. G.Birkhoff and J. von Neumann criticized the view that any subset of the phase space is a proposal.

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.

 

[pbuk]

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

 

[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