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According to the paper Quantum Mechanics Beyond Hilbert Space by J.P. Antoine, several mathematical structures have been devised to make mathematical sense of Dirac's formulation of quantum mechanics (specifically, QM with operators having continuous spectra). I was wondering if there was a general set of axioms that could be specified that we would want any such structure to satisfy for it to be a sufficient mathematical image of quantum mechanics.

scott
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  • I believe that's exactly what's done on von neumann's book, try looking on it and see if that's is what you are looking for. – Hydro Guy Jul 18 '13 at 04:09
  • I always thought Von Neumann's and Dirac's approaches to QM were significantly different. For example, I don't think that the VN formalism deals with distributions (e.g. the delta function) at all. – scott Jul 18 '13 at 04:18
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    Von Neumann's aproach don't deal directly with the with things like position/momentum eigen-vectors because of that kind of technicality, although, I believe that they give the same physical content. To deal with this kind of thing it's necessary to resort to [Rigged Hilbert Space][https://en.wikipedia.org/wiki/Rigged_Hilbert_space]. I'm trying to find some reference that explains in detail who to use it to do QM, but the general idea you can find in Ballentine's book. – Hydro Guy Jul 18 '13 at 04:26
  • Comment to the question (v2): It would be good if OP (or somebody else?) could provide precise reference to the link, so that we can reconstruct it in case of future link rot. – Qmechanic Jul 18 '13 at 04:58
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    See also. http://physics.stackexchange.com/questions/43515/rigged-hilbert-space-and-qm – user1504 Jul 18 '13 at 11:46
  • If you are a more visual study type, then the following lecture (series) from the Perimenter Institute should be interesting for you: http://pirsa.org/displayFlash.php?id=13010066 – Tobias Diez Jul 18 '13 at 22:46

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The standard treatment of quantum mechanics proceeds by way of asserting that the physical states of systems correspond to vectors in Hilbert spaces. There is no problem dealing with continuous spectra in this standard setting, see e.g. the Wikipedia page on decomposition of spectra. In this treatment, however, there are parts of the spectrum of an operator that do not correspond to eigenvalues of certain eigenvectors in the Hilbert space. Therefore, there exist points in the spectrum that don't correspond to physical states. In addition, in this treatment, there are certain "kets" that do not correspond to elements of the Hilbert space.

Some people find this state of affairs unsatisfactory, and as a result came the development of the rigged Hilbert space formulation of quantum mechanics. In this formulation, all elements of the spectrum of an operator are eigenvalues corresponding to eigenvectors in the rigged Hilbert space. In addition, all objects that one would like to call "kets," are elements of some rigged Hilbert space.

For example, for a one-dimensional free particle moving on the real line $\mathbb R$, the position "eigenvector" $|x\rangle$ does not correspond to an element of the Hilbert space $L^2(\mathbb R)$; it does not correspond to a square-integrable function. However, rigged Hilbert spaces are "larger" than Hilbert spaces and the rigged Hilbert space for a free particle moving on the real line does contain such objects.

In this sense, the rigged Hilbert space formulation is perhaps a more natural way of making Dirac notation, for example, rigorous. However, the decision to use rigged Hilbert spaces is, as far as I am aware, a matter of taste. The conventional formulation via Hilbert spaces can handle any situation you might come across in quantum mechanics, but it requires one to keep a clear distinction between physical states in the Hilbert space, and unphysical states, which are represented by distributions.

joshphysics
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  • Let's suppose I wanted to use the conventional Dirac formalism (as taught in a typical QM class). Where can I find the most complete list of heuristics that are used in this formalism? – scott Jul 18 '13 at 05:12
  • @scott By a list of heuristics, do you mean formal rules for manipulating states etc. in Dirac notation that does not delve into their mathematical constructions via Hilbert and rigged Hilbert spaces? – joshphysics Jul 18 '13 at 05:25
  • Yes. I guess essentially I'd like a list of objects, relations among them, operators on them, etc. that we assume to exist mathematically. I'm guessing this is essentially a rewording of my original question. – scott Jul 18 '13 at 05:30
  • @scott Hmmm sorry but I'm still not sure what you mean, but I'm also not aware of where you can find any such list if I am interpreting your question correctly. Essentially the only objects you have to worry about are Hilbert spaces, kets (which are just elements of Hilbert or rigged Hilbert spaces), bras (which are linear functionals), and linear operators. Everything else, like specific Hilbert spaces like $L^2(\mathbb R)$ and linear operators on them etc., is just a special case, physical-system-dependent detail. – joshphysics Jul 18 '13 at 05:51
  • @scott Perhaps you will find this http://arxiv.org/pdf/quant-ph/0502053.pdf and references therein useful. – joshphysics Jul 18 '13 at 16:26