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Axiomatization of quantum mechanics and physics in general ?

  1. Sep 5, 2014 #1
    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
     
    Last edited: Sep 5, 2014
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  3. Sep 5, 2014 #2

    atyy

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    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.
     
  4. Sep 5, 2014 #3

    bhobba

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    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
     
    Last edited: Sep 5, 2014
  5. Sep 5, 2014 #4

    bhobba

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    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
     
  6. Sep 6, 2014 #5
    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
     
  7. Sep 6, 2014 #6

    atyy

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    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.
     
  8. Sep 6, 2014 #7

    bhobba

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    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/Geometry-Quantum-Theory-V-S-Varadarajan/dp/0387493859

    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
     
    Last edited by a moderator: May 6, 2017
  9. Sep 6, 2014 #8

    bhobba

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    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
     
  10. Sep 6, 2014 #9

    atyy

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    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
     
    Last edited: Sep 6, 2014
  11. Sep 7, 2014 #10
    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
     
    Last edited: Sep 7, 2014
  12. Sep 7, 2014 #11

    bhobba

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    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
     
  13. Sep 7, 2014 #12
    I don't read this in the articles give by atyy.

    I read

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

    Patrick
     
  14. Sep 7, 2014 #13
    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

    This line of research was recently revived by the influence of Quantum Information Theory, like this one "Informational axioms for quantum theory"

    Patrick
     
    Last edited: Sep 7, 2014
  15. Sep 7, 2014 #14

    bhobba

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  16. Sep 7, 2014 #15

    bhobba

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    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
     
    Last edited: Sep 8, 2014
  17. Sep 7, 2014 #16

    Fredrik

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    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
     
  18. Sep 7, 2014 #17

    bhobba

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    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 cant 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
     
    Last edited: Sep 7, 2014
  19. Sep 8, 2014 #18

    bhobba

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    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
     
  20. Sep 8, 2014 #19
    What is the point I want to make ?

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

    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
     
  21. Sep 8, 2014 #20
    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
     
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