Trace formula for PSDOs

Question

In Getzler's famous paper "Pseudodifferential Operators on Supermanifolds and the Atiyah-Singer Index Theorem", he states that for a (trace-class) pseudo-differential operator $P$ on a Riemannian manifold $M$, one has the formula $$ \mathrm{Tr} P = \int_{T^*M} \sigma(P), $$ where $\sigma(P)$ is the full symbol of $P$, which is calculated using "the exponential map of $M$ to pull back the pseudodifferential operator near $x \in M$ to $T_xM$ and calculate its symbol on $T^*_xM$ using the Euclidean structure" (quote of p.164).

However, as far as I know, the full symbol of a $\Psi$DO can be defined using the Levi-Civita connection, but only up to a smoothing symbol. And of course in applications, $P$ is usually a smoothing operator.Also, I could not get my hands on a full copy of the paper of Widom ("A complete symbolic calculus for pseudodifferential operators") cited there, and the preview of it on google books does not seem to help me.

So, my question is: How does one define $\sigma(P)$ to make the above formula true?

Answer 1

Consider the modified question: How does one define the operator $P=\mathrm{Op}(p)$ so that the trace formula holds with $\sigma(P)=p$ as the full symbol? The following quantization rule answers this with the help of the exponential map of the compact Riemannian manifold $M$, $n=\dim M$: $$Pu(x)=(2\pi)^{-n}\int_{T_x^* M} \int_{T_x M} e^{-i\eta v}\chi(x,v)p(x,\eta)u(\exp_x v)\,dv\,d\eta.$$ The quantization depends on the choice of $\chi\in C_c^\infty(TM)$ which equals unity in a neighbourhood of the zero section and localizes to an open subset mapped diffeomorphically by $\exp$ to an open neighbourhood of the diagonal in $X\times X$. Using normal geodesic coordinates one proves that this quantization rule does indeed give the standard classes $\Psi^m(M)$ of pseudo-differential operators. In a neighbourhood of the diagonal, the Schwartz kernel $K$ is given as follows: $$K(x,y)dy=((2\pi)^{-n}\int_{T_x^* M} e^{-i\eta v} p(x,\eta)d\eta) dv, \quad y=\exp_x v.$$ In particular, a different choice of cutoff $\chi$ modifies $P$ only by adding a smoothing operator. Moreover, we get, if the order of $P$ is strictly less than $-n$, the trace formula: $$\mathrm{Tr}P=\int_M K(x,x) dx= (2\pi)^{-n}\int_{T^*M} p(x,\xi)dx d\xi.$$ In fact, the first equality is known, and the second follows from the above.

In can now answer the question you posed. Multiply the Schwartz kernel of the operator with a fixed cutoff function which localizes to a small neighbourhood of the diagonal, pull this product back under $\exp$ to a distribution on $TM$, and (up to obvious adjustments) define the symbol by Fourier transformation in the fiber variable.

You can find details of the full symbol calculus, but not the trace formula, for operators acting between bundles in a semiclassical setup, in the appendix of my paper (also available at arXiv). Full symbol calculi were developed by Bokobza (1969), Widom (1980), Pflaum (1998), Sharafutdinov (2004), and Safarov.