Rigged Hilbert Space (Gel'fand Triples)

Dirac's $∣ x ⟩$ and $∣ p ⟩$ kets, and the continuous-spectrum expansions of quantum mechanics, are not literally elements of a Hilbert space — a delta function and a plane wave are not square-integrable. The rigged Hilbert space (Gel'fand triple) is the framework that makes them rigorous, unifying the spectral theorem for continuous spectra with the distribution theory of the previous page. This is the precise home of the improper states used in QFT/fock-space-inventory.md.

The problem

The spectral theorem handles continuous spectrum via a projection-valued measure, but physicists prefer the eigenket language: expand $∣ ψ ⟩ = \int d x ψ (x) ∣ x ⟩$ with $\overset{x}{^} ∣ x ⟩ = x ∣ x ⟩$ and $⟨ x ∣ x^{'} ⟩ = δ (x - x^{'})$ . Yet $∣ x ⟩ \in / L^{2}$ (its "norm" is $δ (0) = \infty$ ) and $∣ p ⟩ = e^{i p x /ℏ} \in / L^{2}$ either. The kets are convenient fictions — until the rigged space makes them honest.

The Gel'fand triple

Introduce a dense subspace of good states $Φ$ (e.g. the Schwartz space, or the domain of all powers of the Hamiltonian) with a finer topology, and take its dual $Φ^{'}$ of continuous functionals. Since $Φ \subseteq H ≅ H^{*} \subseteq Φ^{'}$ , one obtains the Gel'fand triple (rigged Hilbert space) $Φ \subseteq H \subseteq Φ^{'}$ The middle is the ordinary Hilbert space of normalizable states; $Φ$ is the well-behaved test states (rapidly decaying, infinitely differentiable); and $Φ^{'}$ is large enough to contain the improper kets. A "bra" $⟨ x ∣$ is a continuous functional on $Φ$ — an element of $Φ^{'}$ — exactly as $δ$ is a distribution on test functions in distributions.md. The triple simply applies that distribution theory to the abstract Hilbert space.

The nuclear spectral theorem

The payoff legitimizes Dirac's formalism:

Gel'fand–Maurin (nuclear spectral) theorem. For a self-adjoint operator $A$ on a rigged Hilbert space with $Φ$ nuclear, there is a complete set of generalized eigenvectors in $Φ^{'}$ : functionals $∣ λ ⟩ \in Φ^{'}$ with $A ∣ λ ⟩ = λ ∣ λ ⟩$ (in the dual sense) and the expansion $∣ ψ ⟩ = \int d μ (λ) ∣ λ ⟩ ⟨ λ ∣ ψ ⟩ > (ψ \in Φ) .$

So the "eigenket expansion" is a theorem, not a heuristic: $\overset{x}{^}$ has generalized eigenkets $∣ x ⟩ \in Φ^{'}$ , $\overset{p}{^}$ has $∣ p ⟩$ , and the resolution of the identity $1 = \int d x ∣ x ⟩ ⟨ x ∣$ holds when sandwiched between good states. This is exactly the machinery underlying the momentum-space Fock construction and the continuous-spectrum manipulations of QM/preliminaries.md.

What the triple buys

Object	Naive status	Rigged status
$∣ x ⟩$ , $∣ p ⟩$	"not in $H$ "	genuine elements of $Φ^{'}$
$⟨ x ∣ x^{'} ⟩ = δ (x - x^{'})$	ill-defined	distributional pairing
$1 = \int ∣ x ⟩ ⟨ x ∣ d x$	formal	valid on $Φ$
continuous-spectrum "eigenstate"	none in $H$	generalized eigenvector in $Φ^{'}$

The rigged Hilbert space is thus the rigorous reconciliation of the physicist's Dirac calculus with the mathematician's spectral theorem — the two describing the same operator, one with generalized kets, the other with a spectral measure.

References

Gel'fand & Shilov, Generalized Functions, Vol. 4.
Bohm & Gadella, Dirac Kets, Gamow Vectors and Gel'fand Triplets.
de la Madrid, "The role of the rigged Hilbert space in quantum mechanics", Eur. J. Phys. 26, 287 (2005).