Construction of quantum fields from Euclidean tensor fields

Abstract

We define Euclidean tensor fields over 𝒮 (R4), from which we construct quantum tensor fields satisfying all the Wightman axioms except the uniqueness of the vacuum. By a process of reduction, it is possible to obtain, from some suitably chosen Euclidean tensor field, a quantum field satisfying all the Wightman axioms except the uniqueness of the vacuum and transforming according to any arbitrarily chosen one-valued finite-dimensional irreducible representation of the restricted Lorentz group L↑+. We give a Euclidean vector field and a Euclidean tensor field of rank two as examples, leading respectively to the real Proca Wightman field and the free electromagnetic Wightman field.