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Is there a way to motivate, retrospectively, that observables must be representable by

  1. linear operators
  2. on a Hilbert space?

Specifically, there seems to be a hint to something in the accepted answer to this question. There, the author writes that

"[...] and since these commutators satisfy the Jacobi identity, they can be represented by linear operators on a Hilbert space."

  • Is this true? If an observable can be written as a commutator (like the three coordinates of angular momentum), does it automatically follow that it corresponds to some linear operator? If yes, how / is this a theorem with a name?

Thanks for further hints, and for all hints so far. Quantum mechanics is still really strange to me, additional motivation for one of the postulates would be edit: is nice.

If you want me to clarify further, please say so.

This question: "How did the operators come about" seems related, but the answer doesn't help since it starts from the postulates. My question is about possible motivations one could give for one of the postulates.

Revision history part 1: The question was quite general before, it also asked "where did the operators come from" which aready has a very good answer here.

For completeness / Revision history part 2: Another approach might be via Wigner's theorem, that any symmetry operator on a Hilbert space is either linear unitary or antilinear antiunitary (and then one could maybe go from symmetry operators to their generators, and find out that they correspond to observables). Originally the question was also whether this line of argumentation works as well. But probably, this is covered in the historical sources, given here and in the comments.

dasWesen
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    The person who developed the mathematical formalism of QM was Von Neumann (Mathematische Grundlagen der Quantenmechanik). I might be wrong but I think he also gave its name to the "Hilbert space". Before him, the ideas about this were quite confused – John Donne Dec 04 '17 at 20:40
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    von Neumann formalized Dirac's 1930 QM textbook, freely available. Dirac worked off of finite dimensional analogies that Heisenberg's matrix mechanics suggested. The question duplicates How did quantum mechanics operators come into being? question on hsm SE and How did the operators come about? on this SE. – Conifold Dec 04 '17 at 21:00
  • @Conifold The second of your reference I mentioned in my question, stating why it is no answer. The first one (I'm still reading) has good references for the historical developement. What I would have liked though are the actual motivations one could think of (I will modify the question title.) Please do not comment after only reading the title. – dasWesen Dec 04 '17 at 21:49
  • I'll have a look at the von Neumann book again for sure, then. However the second part of my question is very specific. I hope someone knows what the person was referencing to. (And since the preface of the von Neumann book describes it as 'important historically, but outdated, might even contain mistakes' - it seems likely it might not contain the answer.) – dasWesen Dec 04 '17 at 21:56
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    What do you mean by "the actual motivations one could think of"? Dirac's book is freely available online, you can find there what he thought. Extending from matrices seems pretty motivated though, so you'll have to be much more specific as to what you mean. As for the second question, most SE's have one question per question policies to permit answers of reasonable focus and length. You should split it off into a separate question. – Conifold Dec 04 '17 at 22:18
  • @Conifold You're right, and the 'what did lead to it' part of my question you answered already in the other post. Really interesting, would upvote if I could. So, the best is probably I reduce the question to the second part. – dasWesen Dec 04 '17 at 22:31
  • Judging by this question you may want to register on hsm in any case. There is a big blue Join This Community button in the upper right corner, it should automatically link it to your other SE accounts when you press it. – Conifold Dec 04 '17 at 22:59
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    If one considers general probabilistic theories, e.g. in a quantum logic approach (for a basic introduction see e.g. plato.stanford.edu/entries/qt-quantlog/), which means formalizing the general logical structure of probabilistic statements about physical systems, one finds a number of hypothetically plausible possibilities, among them the one equivalent to the Hilbert space formalism. Afaik (consistent with above link) no elementary explanation is known singling out this particular structure. There are other approaches similar to "Quantum Theory From Five Reasonable Axioms" by L. Hardy – Adomas Baliuka Dec 04 '17 at 23:40
  • Please don't make your post look like a revision history – Kyle Kanos Dec 05 '17 at 11:00
  • @AdomasBaliuka So this one is just one of multiple ways to 'write down' quantum mechanics? Maybe the others are easier to make sense of, if this one fails for me :) Thanks! – dasWesen Dec 07 '17 at 21:45
  • @KyleKanos Better? – dasWesen Dec 07 '17 at 21:45
  • @dasWesen no, you still have all sorts of ridiculous remarks about edits made to the post. Get rid of them entirely by making a cohesive set of statements, then I will be happy. – Kyle Kanos Dec 08 '17 at 00:35
  • @dasWesen anything you can think of is just one way of writing it down. The possibilities I was allduding to do not describe the statistics of quantum experiments. One of the possible logical structures is real quantum mechanics, based on real hilbert spaces. It has complicated entanglement properties and is completely ruled out by experiment as a probabilistic theory of experimental outcomes. It is only of interest to mathematicians. Experiments can and do single out the Standard quantum formalism. It's just that we don't have a theoretical justification for this (which need not be a problem) – Adomas Baliuka Dec 15 '17 at 14:32

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From the physical standpoint, it is natural to suppose that it is possible to manipulate observables mathematically in a suitable way. It should be possible to sum them, multiply them, and scale them in order to obtain new observables. In addition, it is often convenient to extend the concept of observable to complex objects for which it is also possible to do "complex conjugation" (abstractly, called involution).

Complex numbers, or complex valued functions, allow the above manipulations, and could be taken as observables (and the latter are in classical theories). To describe quantum systems, however, such observables should also be non-commutative, since effects due to non-commutativity are observed experimentally.

Mathematically, objects with the properties above form an involutive algebra, abelian if they commute, non-abelian if not. There is a theorem, of Gelfand and Naimark, based on a construction of Gelfand, Naimark, and Segal, that says the following.

Every *-algebra is isomorphic to an algebra of linear operators acting on a Hilbert space

It is therefore natural to represent quantum observables as linear operators. For classical (abelian) observables, the theorem shows that the operators in that case are multiplication operators, and that abelian algebras could also be represented as an algebra of complex-valued functions acting on a suitable topological space. The latter representation is the one usually chosen for classical theories.

yuggib
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