## Good references for Rigged Hilbert spaces?

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Every now and then I attempt to understand better quantum mechanics and quantum field theory, but for a variety of possible reasons, I find it very difficult to read any kind of physicist account, even when the physicist is trying to be mathematically respectable. (I am not trying to be disrespectful or controversial here; take this as a confession of stupidity if it helps.) I am generally interested in finding online mathematical accounts which ideally would come close to being of "Bourbaki standard": definition-theorem-proof and written for mathematicians who prefer conceptual explanations, and ideally with tidy or economical notation (e.g., eschewing thickets of subscripts and superscripts).

More specifically, right now I would like a (mathematically trustworthy) online account of rigged Hilbert spaces, if one exists.

Maybe I am wrong, but the Wikipedia account looks a little bit suspect to me: they describe a rigged Hilbert space as consisting of a pair of inclusions $i: S \to H$, $j: H \to S^\ast$ of topological vector space inclusions, where $S^\ast$ is the strong dual of $S$, $H$ is a (separable) Hilbert space, $i$ is dense, and $j$ is the conjugate linear isomorphism $H \simeq H^\ast$ followed by the adjoint $i^\ast: H^\ast \to S^\ast$. This seems a little vague to me; should $S$ be more specifically a nuclear space or something? My guess is that a typical application would be where $S$ is Schwartz space on $\mathbb{R}^4$, with its standard dense inclusion in $L^2(\mathbb{R}^4)$, so $S^\ast$ consists of tempered distributions.

I also hear talk of a nuclear spectral theorem (due to Gelfand and Vilenkin) used to help justify the rigged Hilbert space technology, but I don't see precise details easily available online.

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## 5 Answers

Some time ago I was interested in rigged Hilbert space to get a better understanding of quantum physics. On that occasion I collected some references on this subject, see below. It's quite comprehensive. A good starting point for an overview could be the works of Madrid and Gadella. Note that there are different versions of "rigged Hilbert space" (in context of quantum physics) in literature.

J.-P. Antoine. Dirac formalism and symmetry problems in quantum mechanics. i. general dirac formalism. Journal of Mathematical Physics, 10(1):53--69, 1969.

N.Bogoliubov, A.Logunov, and I.Todorov. Introduction to Axiomatic Quantum Field Theory, chapter 1 Some Basic Concepts of Functional Analysis 4 The Space of States, pages 12--43, 113--128. Benjamin, Reading, Massachusetts, 1975.

R.de la Madrid. Quantum Mechanics in Rigged Hilbert Space Language. PhD thesis, Depertamento de Fisica Teorica Facultad de Ciencias. Universidad de Valladolid, 2001. (available here)

M.Gadella and F.Gómez. A unified mathematical formalism for the dirac formulation of quantum mechanics. Foundations of Physics, 32:815--869, 2002. (available here)

M.Gadella and F.Gómez. On the mathematical basis of the dirac formulation of quantum mechanics. International Journal of Theoretical Physics, 42:2225--2254, 2003.

M.Gadella and F.Gómez. Dirac formulation of quantum mechanics: Recent and new results. Reports on Mathematical Physics, 59:127--143, 2007.

I.M. Gelfand and N.J. Vilenkin. Generalized Functions, vol. 4: Some Applications of Harmonic Analysis, volume4, chapter 2-4, pages 26--133. Academic Press, New York, 1964.

A.R. Marlow. Unified dirac-von neumann formulation of quantum mechanics. i. mathematical theory. Journal of Mathematical Physics, 6:919--927, 1965.

E.Prugovecki. The bra and ket formalism in extended hilbert space. J. Math. Phys., 14:1410--1422, 1973.

J.E. Roberts. The dirac bra and ket formalism. Journal of Mathematical Physics, 7(6):1097--1104, 1966.

J.E. Roberts. Rigged hilbert spaces in quantum mechanics. Commun. math. Phys., 3:98--119, 1966. (available here)

Tjøstheim. A note on the unified dirac-von neumann formulation of quantum mechanics. Journal of Mathematical Physics, 16(4):766--767, 4 1975.

Edit I remember that there is also a discussion about Gelfand triples in physics in the Funktionalanalysis books by Siegfried Großmann but I don't have a copy handy the moment. Though it is in german it might be interesting for you, too.

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Thanks, student. I am not particularly near a good library where I can access these to see which would suit my purposes. I remember seeing something by Madrid on the arXiv and it wasn't quite what I was looking for, but I'll look at his thesis. Which of these do you think are in definition-theorem-proof format at a level of rigor that would satisfy mathematicians?  – Todd Trimble Oct 23 2010 at 20:18
When I collected the references I was interested to have  as much rigor as possible. If I remember correctly all of those are written in a usual mathematical style. In Madrid's thesis there are many examples concerning quantum mechanics. For a more general approach I would look at   M.Gadella and F.Gómez. A unified mathematical formalism for the dirac formulation of quantum mechanics. Foundations of Physics, 32:815--869, 2002  In this paper they tried to unify some (perhaps most) versions of rigorous frameworks for rigged Hilbert space in view of quantum physics.  – student Oct 23 2010 at 21:20
I think the problem with this subject is, that there are many different attempts to give a rigorous framework for rigged Hilbert spaces in physics. I don't know if there is a generally accepted useful version. Hence it's not surprising that there are now good free online resources about this subject.  – student Oct 23 2010 at 21:27
@student: thanks again for all your references, but regarding your last comment, you've so far given me one online resource (Madrid), which gives no proofs. So this is not in a style that I was asking for above.  – Todd Trimble Oct 23 2010 at 22:10
I added links to those articles which are currently open access (only two more, sorry) – student Oct 23 2010 at 23:27
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"Generalized functions volume 4" by Gelʹfand, Vilenkin, (Math review number 0146653) has a long an detailed discussion of rigged Hilbert spaces and nuclear spaces. The book by Glimm and Jaffe has a brief summary of the theory.

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Thank you, Richard. At physicsforums.com/showthread.php?t=294488 I read some hearsay about some alleged flaws in their arguments; do you know what they're talking about and whether a mathematician should be worried?  – Todd Trimble Oct 23 2010 at 17:48

The Springer online Encyclopedia of Mathematics' entry on RHS looks more rigorous albeit also more succinct than Wikipedia; for another online intro see the nlab entry. In addition to the references listed there, a rigorous discussion of the RHS can be found (as far as I recall -- I do not have a copy handy) e.g. in the two-volume book Principles of Advanced Mathematical Physics by Robert D. Richtmyer. Also, it appears that, unlike the physics community, the name Gelfand triple (rather than RHS) is more commonly used by the mathematicians.

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Thanks for the tips, mathphysicist. The first sentence of that Springer Encyclopedia reference gives the same definition as wikipedia (so maybe that definition is perfectly adequate after all), but then a little later it says, "The most interesting case is that in which is $\Phi$ [my $S$] is a nuclear space." Then they cite a spectral theorem, but I can't tell if they mean to include the nuclear hypothesis in the theorem or not.  – Todd Trimble Oct 23 2010 at 17:27
Um, mathphysicist, I know you don't realize it, but I was the one who wrote (most of) that nLab article!  – Todd Trimble Oct 23 2010 at 20:09
Well, Todd, unfortunately I didn't, sorry :( Should have looked at the edits history :) – mathphysicist Oct 24 2010 at 2:24
Please feel free to remove the nLab reference from my answer if you find this necessary and/or appropriate.  – mathphysicist Oct 24 2010 at 2:25

This is not precisely related to your question, but a certain notion of rigged Hilbert space occurs in the theory of C*-algebras. Particularly, one should look at the work of Marc Rieffel, e.g. http://math.berkeley.edu/~rieffel/papers/morita_equivalence.pdf. I figured I'd mention this because it is decidedly mathematical, and a useful idea.

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Thanks, Jon. Interesting blast from the not-too-distant past.    – Todd Trimble Oct 25 2010 at 12:50
You're welcome, Todd! – Jon Bannon Oct 25 2010 at 14:21
Correct me if I'm wrong, but Rieffel's rigged spaces are what we call Hilbert C*-modules, right? Is that related to the rigged spaces everyone here is talking about? I'm asking because I really don't understand...  Anyway, if so, a nice book would also be Lance's "Hilbert C*-Modules: A Toolkit for Operator Algebraists", since its a toolkit and all, very concise and extremely pedagogic. – Yul Otani Nov 14 at 19:00
@Yul: I was wondering the same thing way back when I asked this (hence my first sentence). I don't see a direct relation, come to think of it...except via the Gelfand triple bit...but that is not a precise relation. If you happen to find out, please let me know. – Jon Bannon Nov 14 at 23:00

I would highly recommend looking at the chapter on Sobolev Towers in the book by Engel and Nagel One-Parameter Semigroups for Linear Evolution Equations or the "baby edition" A Short Course on Operator Semigroups.

It provides a really nice example of rigged Hilbert spaces. For example, if $A:D(A) \subset L^2 \to L^2$ is the (Dirichlet) Laplacian, then one can identify $D(A^n)$, $n=1,2,\ldots$ with Sobolev spaces and $D(A^{-n})$ with the negative Sobolev spaces (i.e. extrapolation spaces of $A$).

This concept can be taken further if one considers analytic semigroups and fractional powers of operators and also into the Banach space setting (see Amann's book Linear and Quasilinear Parabolic Problems: Abstract linear theory).

Basically, the concept of rigged Hilbert spaces becomes really natural if one keeps PDEs and Sobolev spaces in mind.

Finally, the book by Reed and Simon Methods of Modern Mathematical Physics - Vol 1: Functional analysis provides a number of references for rigged Hilbert spaces at the end of Section VII (page 244).

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