Locally covariant Wick polynomials and tadpole renormalisation

Abstract: In this talk, I will show how the Wick polynomial construction can be regarded as a trivialization of a certain algebra bundle over the space of parametrices of a differential operator. When local functionals are included in the algebra, the construction corresponds exactly to the renormalisation of diagrams with short loops (tadpoles). This renormalisation can, of course, be performed using the widely employed point-splitting procedure with the Hadamard parametrix. I will demonstrate how some classical microlocal results of Guillemin, employing analytic regularisation, can be used to achieve the same end. As a byproduct, I will show that the Wick polynomials constructed via the microlocal tadpole renormalisation satisfy a smoothness condition which is essential to the classification of the finite renormalisation freedom.