Alain Connes (pdf)

where A is a concrete algebra of coordinates represented on a Hilbert space H and the
operator D is the inverse of the line element.
ds = 1/D. (64)
This definition is entirely spectral; the elements of the algebra are operators, the points,
if they exist, come from the joint spectrum of operators and the line element is an
operator.
The basic properties of such spectral triples are easy to formulate and do not make any
reference to the commutativity of the algebra A. They are
[D, a] is bounded for any a ∈ A , (65)
D = D ∗ and (D + λ) −1 is a compact operator ∀ λ ∉ C . (66)
(Of course D is an unbounded operator).
There is no difficulty to adapt the above formula for the distance in the general noncommutative
case, one uses the same, the points x and y being replaced by arbitrary
states ϕ and ψ on the algebra A. Recall that a state is a normalized positive linear
form on A such that ϕ(1) = 1,
ϕ : Ā → C , ϕ(a ∗ a) ≥ 0 , ∀ a ∈ Ā , ϕ(1) = 1 . (67)
The distance between two states is given by,
d(ϕ, ψ) = Sup {|ϕ(a) − ψ(a)| ; a ∈ A , ‖[D, a]‖ ≤ 1} . (68)
The significance of D is two-fold. On the one hand it defines the metric by the above
equation, on the other hand its homotopy class represents the K-homology fundamental
class of the space under consideration.
There is an equally simple formula for the Yang-Mills action in general The analogue of
the Yang-Mills action functional and the classification of Yang-Mills connections on the
noncommutative tori were developped in [20], with the primary goal of finding a ”manifold
shadow” for these noncommutative spaces. These moduli spaces turned out indeed
to fit this purpose perfectly, allowing for instance to find the usual Riemannian space of
gauge equivalence classes of Yang-Mills connections as an invariant of the noncommutative
metric. We refer to [12] for the construction of the metrics on noncommutative tori
from the conceptual point of view. All natural axioms of noncommutative geometry
are fulfilled in that case.
These constructions were greatly generalised to isospectral deformations of Riemannian
geometries of rank > 1 in joint work with G. Landi and M. Dubois-Violette .
In essence the transition from the Riemannian paradigm of geometry to the above spectral
one parallels the evolution of the unit of length in physics. The ”meter” for instance
was defined in the eighteen century as a small fraction of the earth circonference, and
concretely represented by a platinum bar localized near Paris. The actual standard of
length is given nowadays in a quite different way: it is the wave-length corresponding to
specific transitions in the cesium atom. The inverse frequencies are computed from the
corresponding Dirac Hamiltonian in exact analogy with the above spectral definition
of ”ds”. This unit of length is no longer localized, owing to its lack of commutativity
with the space-time coordinates.
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