Seminarium KMMF "Teoria Dwoistości"

The construction of the $\Phi^4$ model in 3 dimensions in the 1970s has been one of the major achievements of mathematical physics. In the 1980s Parisi and Wu suggested an alternative way of constructing Euclidean quantum field theories by viewing the correlation functions as moments of invariant measures of certain stochastic PDEs. However, the highly singular nature of these equations prevented their application in rigorous constructions until the breakthroughs in the area of singular stochastic PDEs in the past decade. After explaining the basic idea behind stochastic quantization proposed by Parisi and Wu I will show how to apply this technique to construct the measure of a certain quantum field theory model generalizing the $\Phi^4_3$ model called the fractional $\Phi^4$ model. The measure of this model is obtained as a perturbation of the Gaussian measure with covariance given by the inverse of a fractional Laplacian. Since the Gaussian measure is supported in the space of Schwartz distributions and the quartic interaction potential of the model involves pointwise products, to construct the measure it is necessary to solve the so-called renormalization problem. The talk is based on joint work with M. Gubinelli and P. Rinaldi.