Constructive \(\phi^{4}\) field theory without tears.

*(English)*Zbl 1141.81022Both authors are known for their magnificent contributions to constructive quantum field theory for more than three decades. Their work together with the efforts of a small and dedicated group of physicists concerned the mathematically complete construction of Boson fields with polynomial selfinteraction and Fermi fields with Yukawa interaction. While fermionic theories may be rather easily constructed by a convergent resummation of the perturbation series of the Schwinger functions, Bosonic constructive field theory in four Euclidean dimensions remains awfully difficult. Many renormalization group steps are needed. Shedding tears one is forced to discritize space from the beginning, thus loosing Euclidean invariance.

The present paper explains how to avoid this. The authors identify an infinite family of graphs giving rise to a convergent loop expansion for the connected Schwinger functions of the \(\phi^4\) theory in four dimensions without lattice, cluster or Mayer expansions. The loop expansion also seems suited to treat matrix models. In fact, this was the goal when Rivasseau began his study one year ago.

The present paper explains how to avoid this. The authors identify an infinite family of graphs giving rise to a convergent loop expansion for the connected Schwinger functions of the \(\phi^4\) theory in four dimensions without lattice, cluster or Mayer expansions. The loop expansion also seems suited to treat matrix models. In fact, this was the goal when Rivasseau began his study one year ago.

Reviewer: Gert Roepstorff (Aachen)