# Difference between revisions of "Wodzicki residue"

In algebraic quantum field theory, in order to write down an action in operator language one needs a functional that replaces integration [a1]. For the Yang–Mills theory (cf. Yang–Mills field) this is the Dixmier trace, which is the unique extension of the usual trace to the ideal $\mathcal{L}^{(1,\infty)}$ of the compact operators $T$ such that the partial sums of the spectrum diverge logarithmically as the number of terms in the sum. The Wodzicki (or non-commutative) residue [a3] is the only extension of the Dixmier trace to the class of pseudo-differential operators (ΨDOs) which are not in $\mathcal{L}^{(1,\infty)}$. It is the only trace one can define in the algebra of ΨDOs (up to a multiplicative constant), its definition being: $\mathrm{res} A = 2\mathrm{res}_{s=0} \tr(A\Delta^{-s})$, with $\Delta$ the Laplace operator. It satisfies the trace condition: $\mathrm{res}(AB) = \mathrm{res}(BA)$. A very important property is that it can be expressed as an integral (local form): $$\mathrm{res} A = \int_{S^*M} \tr a_{-n}(x,\xi) d\xi$$ with $S^*M \subset T^*M$ the co-sphere bundle on $M$ (some authors put a coefficient in front of the integral, this gives the Adler–Manin residue).
If $\dim M = n = -\ord A$ ($M$ a compact Riemannian manifold, $A$ an elliptic operator, $n \in \mathbf{N}$), it coincides with the Dixmier trace, and one has $$\mathrm{res}_{s=1} \zeta_A(s) = \frac{1}{n} \mathrm{res} A^{-1} \ .$$
The Wodzicki residue continues to make sense for ΨDOs of arbitrary order and, even if the symbols $a_j(x,\xi)$, $j < n$, are not invariant under coordinate choice, their integral is, and defines a trace. All residues at poles of the zeta-function of a ΨDO can be easily obtained from the Wodzicki residue [a2].