Exact Results in Quantum Theory

Renormalization relies on the possibility of cutting off our ignorance of the physics at short scales by integrating out the degrees of freedom which are beyond direct observation. A literal implementation of this procedure, motivated by mathematical, rather than physical difficulties with the notion of measure on an infinite dimensional space, is the theory of cylinder measures. We will introduce generalized cylinder measures and show how the problem of renormalization can be cast in terms of them, instead of the Feynman diagrams of the $n$-point functions. Then, we will argue that the resulting problem can, under certain hypothesis, be reduced to that of the corresponding strongly coupled theory, and obtain a family of solutions for the case of scalar bosons whose interaction Lagrangian does not contain derivatives. We will finally show how the functional measure of a field with effectively pure quartic interaction at a given, fixed scale can be formally constructed, in arbitrary space-time dimension.