  In the sixties, DeWitt discovered that the advanced and retarded Green
functions of the wave operator on metric perturbations in the de Donder gauge
make it possible to define classical Poisson brackets on the space of
functionals that are invariant under the action of the full diffeomorphism
group of spacetime. He therefore tried to exploit this property to define
invariant commutators for the quantized gravitational field, but the operator
counterpart of such classical Poisson brackets turned out to be a hard task. On
the other hand, the mathematical literature studies often an approximate
inverse, the parametrix, which is, strictly, a distribution. We here suggest
that such a construction might be exploited in canonical quantum gravity. We
begin with the simplest case, i.e. fundamental solution and parametrix for the
linear, scalar wave operator; the next step are tensor wave equations, again
for linear theory, e.g. Maxwell theory in curved spacetime. Last, the nonlinear
Einstein equations are studied, relying upon the well-established
Choquet-Bruhat construction, according to which the fifth derivatives of
solutions of a nonlinear hyperbolic system solve a linear hyperbolic system.
The latter is solved by means of Kirchhoff-type formulas, while the former
fifth-order equations can be solved by means of well-established parametrix
techniques for elliptic operators. But then the metric components that solve
the vacuum Einstein equations can be obtained by convolution of such a
parametrix with Kirchhoff-type formulas. Some basic functional equations for
the parametrix are also obtained, that help in studying classical and quantum
version of the Jacobi identity.
