Four lectures on renormalisation; analytic aspects Sylvie PAYCHA

1 Regularised and renormalised evaluators at zero “Evaluating ” meromorphic functions in one variable (resp. in several variables) at zero with poles at that point requires regularising (resp. renormalising). We shall describe and compare various regularisation (resp. renormalisation) methods one can use to extend ordinary evaluators at zero to certain algebras of meromorphic functions. The Mellin transform proves to be a useful tool in that context. We shall also investigate extended homogeneous distributions, comparing Hadamard finite part methods with Riesz regularisation, as a prototype for the more sophisticated integrals and sums we will be studying in the coming lectures. 2 Translation invariant linear forms on symbols A basic analytic tool in the context of regularisation is pseudodifferential calculus. In particular, the zeta function generalises to a linear form on the algebra of classical pseudodifferential symbols on IR, the one loop Feynman integral generalises to a linear form on the algebra of classical pseudodiffer-ential symbols on IR4. But neither of these are translation invariant. We show that the only translation invariant linear form on the algebra of classical pseudodifferentia