Construction of interacting local relativistic quantum fields in four spacetime dimensions

Abstract

We announce the solution of the old problem of relativistic quantum field theory, namely the (nonperturbative) construction of interacting local relativistic quantum fields over four-dimensional spacetime satisfying all axioms. The fields are obtained starting from euclidean random fields satisfying systems of coupled stochastic partial differential equations. Using their Markov property with respect to three-dimensional hyperplanes and gauge invariance, we construct from these fields new euclidean random fields which are Osterwalder-Schrader reflection positive, yielding a symmetric contraction semigroup generated by a positive hamiltonian on a L 2-space of gauge invariant functions. The relativistic fields are then obtained by analytic continuation. The constructed fields are interacting, asymptotically free and are the continuum limits of lattice fields. An expression for their hamiltonian is provided.