Rigged Hilbert space

Template:Multiple issues Template:Short description In mathematics and physics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction which can enlarge a Hilbert space to a bigger space containing additional objects which are not in the Hilbert space but which one would like to think of alongside the Hilbert space. For example, in the quantum mechanical description of a non-relativistic particle using the Hilbert space of square-integrable functions on the real line, eigenstates of the position and momentum operators are not in the Hilbert space, but are in a suitably defined rigged Hilbert space. Informally, the term "rigged" means that the Hilbert space has been equipped to do more than it otherwise could, in analogy with rigging a boat.

This construction is designed to link the distribution and square-integrable aspects of functional analysis. Such spaces were introduced to study spectral theory. They bring together the 'bound state' (eigenvector) and 'continuous spectrum', in one place.

Using this notion, a version of the spectral theorem for unbounded operators on Hilbert space can be formulated. "Rigged Hilbert spaces are well known as the structure which provides a proper mathematical meaning to the Dirac formulation of quantum mechanics."

Motivation

A function such as $$x\mapsto e^{ix},$$ is an eigenfunction of the differential operator $$-i\frac{d}{dx}$$ on the real line Template:Math, but isn't square-integrable for the usual (Lebesgue) measure on Template:Math. To properly consider this function as an eigenfunction requires some way of stepping outside the strict confines of the Hilbert space theory. This was supplied by the apparatus of distributions, and a generalized eigenfunction theory was developed in the years after 1950.Template:Sfn

Definition

A rigged Hilbert space is a pair Template:Math with Template:Math a Hilbert space, Template:Math a dense subspace, such that Template:Math is given a topological vector space structure for which the inclusion map $$i:\mathrm{\Phi }\to H,$$ is continuous.Template:SfnTemplate:Sfn Identifying Template:Math with its dual space Template:Math, the adjoint to Template:Math is the map $$i^{\ast }:H=H^{\ast }\to {\mathrm{\Phi }}^{\ast }.$$

The duality pairing between Template:Math and Template:Math is then compatible with the inner product on Template:Math, in the sense that: $$⟨u,v{⟩}_{\mathrm{\Phi }\times {\mathrm{\Phi }}^{\ast }}=(u,v{)}_H$$ whenever $u\in \mathrm{\Phi }\subset H$ and $v\in H=H^{\ast }\subset {\mathrm{\Phi }}^{\ast }$. In the case of complex Hilbert spaces, we use a Hermitian inner product; it will be complex linear in Template:Math (math convention) or Template:Math (physics convention), and conjugate-linear (complex anti-linear) in the other variable.

The triple $(\mathrm{\Phi },H,{\mathrm{\Phi }}^{\ast })$ is often named the Gelfand triple (after Israel Gelfand). $H$ is referred to as a pivot space.

Note that even though Template:Math is isomorphic to Template:Math (via Riesz representation) if it happens that Template:Math is a Hilbert space in its own right, this isomorphism is not the same as the composition of the inclusion Template:Math with its adjoint Template:Math $$i^{\ast }i:\mathrm{\Phi }\subset H=H^{\ast }\to {\mathrm{\Phi }}^{\ast }.$$

Functional analysis approach

The concept of rigged Hilbert space places this idea in an abstract functional-analytic framework. Formally, a rigged Hilbert space consists of a Hilbert space Template:Math, together with a subspace Template:Math which carries a finer topology, that is one for which the natural inclusion $$\mathrm{\Phi }\subseteq H$$ is continuous. It is no loss to assume that Template:Math is dense in Template:Math for the Hilbert norm. We consider the inclusion of dual spaces Template:Math in Template:Math. The latter, dual to Template:Math in its 'test function' topology, is realised as a space of distributions or generalised functions of some sort, and the linear functionals on the subspace Template:Math of type $$\varphi \mapsto ⟨v,\varphi ⟩$$ for Template:Math in Template:Math are faithfully represented as distributions (because we assume Template:Math dense).

Now by applying the Riesz representation theorem we can identify Template:Math with Template:Math. Therefore, the definition of rigged Hilbert space is in terms of a sandwich: $$\mathrm{\Phi }\subseteq H\subseteq {\mathrm{\Phi }}^{\ast }.$$

The most significant examples are those for which Template:Math is a nuclear space; this comment is an abstract expression of the idea that Template:Math consists of test functions and Template:Math of the corresponding distributions.

An example of a nuclear countably Hilbert space $\mathrm{\Phi }$ and its dual ${\mathrm{\Phi }}^{\ast }$ is the Schwartz space $𝒮(ℝ)$ and the space of tempered distributions ${𝒮}^{\prime }(ℝ)$, respectively, rigging the Hilbert space of square-integrable functions. As such, the rigged Hilbert space is given byTemplate:Sfn $$𝒮(ℝ)\subset L^2(ℝ)\subset {𝒮}^{\prime }(ℝ).$$ Another example is given by Sobolev spaces: Here (in the simplest case of Sobolev spaces on ${ℝ}^n$) $$H=L^2({ℝ}^n),\mathrm{\Phi }=H^s({ℝ}^n),{\mathrm{\Phi }}^{\ast }=H^{-s}({ℝ}^n),$$ where $s>0$.

See also

• Fourier inversion theorem
• Fourier transform § Tempered distributions
• Self-adjoint operator § Spectral theorem

References

• J.-P. Antoine, Quantum Mechanics Beyond Hilbert Space (1996), appearing in Irreversibility and Causality, Semigroups and Rigged Hilbert Spaces, Arno Bohm, Heinz-Dietrich Doebner, Piotr Kielanowski, eds., Springer-Verlag, Template:Isbn. (Provides a survey overview.)
• J. Dieudonné, Éléments d'analyse VII (1978). (See paragraphs 23.8 and 23.32)
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• K. Maurin, Generalized Eigenfunction Expansions and Unitary Representations of Topological Groups, Polish Scientific Publishers, Warsaw, 1968.
• Template:Cite thesis
• de la Madrid Modino, R. "The role of the rigged Hilbert space in Quantum Mechanics," Eur. J. Phys. 26, 287 (2005); quant-ph/0502053.
• Template:Cite thesis