First-order condition for a real spectral triple

Let $(\mathcal A,H,D,J)$ satisfy the order-zero condition and write $b^\circ=Jb^*J^{-1}$ . The first-order condition is

\bigl[\, [D,a],b^\circ\,\bigr]=0 \qquad\text{for every }a,b\in\mathcal A,

where each $[D,a]$ , initially defined on $\operatorname{Dom}(D)$ , is replaced by its bounded extension. Equivalently, the derivation

a\longmapsto [D,a]

takes values in bounded operators that are right $\mathcal A$ -linear for the opposite action. The axiom is strictly stronger than order zero: order zero concerns $a$ itself, while first order concerns its differential $[D,a]$ .

Differential meaning

For a first-order differential operator, commuting once with multiplication by a function removes the derivative and leaves an order-zero operator. Such an operator commutes with multiplication by a second function. The double commutator is the operator-algebraic form of this property.

More generally, the axiom ensures that represented one-forms

\sum_j a_j[D,b_j]

are compatible with the right $\mathcal A$ -module structure. This is the reason the condition is separated from the bimodule-producing order-zero axiom.

Example and near miss

For the Dirac operator $\not D$ on a closed spin manifold,

[\not D,f]=c(df),

Clifford multiplication by $df$ . This is a bundle endomorphism and therefore commutes with right multiplication by every function $g$ , so first order holds.

The Laplace operator is a decisive near miss: although functions commute with the opposite action, $\bigl[\, [\Delta,f],g\,\bigr]$ is generally multiplication by a nonzero multiple of the metric pairing of $df$ and $dg$ . It therefore fails the first-order axiom.

References

A. Connes, “Noncommutative Geometry and Reality,” Journal of Mathematical Physics 36 (1995), 6194–6231. DOI record. Relevant: §2 on the order-one relation.
A. Connes and M. Marcolli, Noncommutative Geometry, Quantum Fields and Motives, American Mathematical Society, 2008. DOI record. Relevant: Definition 1.124, equation (1.473).
S. Lord, A. Rennie, and J. C. Várilly, “Riemannian Manifolds in Noncommutative Geometry,” Journal of Geometry and Physics 62 (2012), 1611–1638. DOI record. Relevant: §§2–3 on bimodules and first-order conditions.