Wodzicki residue

non-commutative 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].

References

[a1]	A. Connes, "Noncommutative geometry" , Acad. Press (1994) MR1303779 Zbl 0818.46076
[a2]	E. Elizalde, "Complete determination of the singularity structure of zeta functions" J. Phys. , A30 (1997) pp. 2735 MR1450345 Zbl 0919.58065
[a3]	M. Wodzicki, "Noncommutative residue I" Yu.I. Manin (ed.) , K-Theory, Arithmetic and Geometry , Lecture Notes in Mathematics , 1289 , Springer (1987) pp. 320–399 MR0923140 Zbl 0649.58033