The traditional mathematical approach to quantum mechanics, as developed by von Neumann, is based on Hilbert spaces and unbounded self-adjoint operators. Another approach, which more closely resembles Dirac's original ideas, uses rigged Hilbert spaces and continuous self-adjoint operators.
More precisely, a rigged Hilbert space is a pair (A,μ), where A is a complex complete nuclear locally convex topological vector space and the inner product μ: Ā⊗A→C is a (continuous) morphism of such spaces that is symmetric and strictly positive (μ(x⊗y)=μ(y⊗x) and μ(x⊗x)>0 for x≠0). Here Ā⊗A denotes the unique tensor product of nuclear spaces.
Completing A with respect to μ results in a Hilbert space H together with a continuous embedding i: A→H, which has dense image, hence the name rigged Hilbert space, i.e., the Hilbert space H is “rigged” with the additional data of A. Continuous self-adjoint (with respect to μ) operators on A extend to unbounded self-adjoint operators on H. This procedure allows one to pass from the formalism of rigged Hilbert spaces to the von Neumann formalism.
Vice versa, if we assume that H is the space of L^2-sections of some vector bundle on a smooth manifold equipped with an inner product μ, then we can set A to be the space of all smooth sections of this vector bundle and let μ be the restriction of μ to A.
The advantage of the formalism of rigged Hilbert spaces lies in the fact that unbounded operators on a Hilbert space are difficult to deal with (one cannot always add or multiply them and has to be very careful about domains), whereas continuous operators on a rigged Hilbert space form an algebra.
Spectral triples, also known as unbounded Fredholm modules, were introduced by Alain Connes to study noncommutative spaces.
An odd spectral triple is a quadruple (E,H,ρ,D), where E is a C*-algebra, H is a Hilbert space, ρ: E→B(H) is a monomorphism of C*-algebras, and D is a self-adjoint unbounded operator on H with compact resolvent such that for all e∈E the commutant [D,ρ(e)] is bounded. For even spectral triples the space H is Z/2-graded, ρ is even, and D is odd. The choice of the name is motivated by the fact that every even/odd spectral triple represents a class in even/odd K-homology of E.
A typical example of an even spectral triple is given by (E,H,ρ,D), where E is the C*-algebra of continuous functions on a compact smooth Riemannian oriented spin manifold M, H is the space of L^2-sections of the spinor bundle of M, ρ acts by multiplication, and D is the (closure of) Dirac operator.
Since unbounded operators in spectral triples give rise to the same problems (absence of everywhere defined algebraic operations) one is led naturally to the following question: Can we define spectral triples using the formalism of rigged Hilbert spaces?
One might expect that rigged spectral triples should be quadruples (E,X,ρ,D), where E is an algebra belonging to a certain class of “rigged C*-algebras” (we expect all algebras of the form C^∞(M) to belong to this class; one can try certain generalizations of C*-algebras such as locally C*-algebras, perhaps with additional conditions such as nuclearity; perhaps we also need to supply an embedding of E into some C*-algebra), X is a rigged Hilbert space, ρ is a continuous representation of E on X, and D is a self-adjoint continuous endomorphism of X, possibly satisfying some additional conditions. For ordinary spectral triples the operator D is required to have compact resolvent and for all e∈E the commutator [D,ρ(e)] must be bounded. What is the right analog of this condition for rigged spectral triples?
Among desirable properties of rigged spectral triples one can name the following two:
One should be able to introduce a notion of homotopy of rigged spectral triples in such a way that homotopy classes of rigged spectral triples give K-homology of E.
Moreover, one should be able at least in some cases to complete a rigged spectral triple to an ordinary spectral triple, preserving the K-homology class.
A typical example of a rigged spectral triple should be given by (E,X,ρ,D), where E is the algebra of smooth functions on a compact smooth Riemannian oriented spin manifold M, H is the space of smooth sections of the spinor bundle of M with the natural inner product, ρ acts by multiplication, and D is the Dirac operator.
