Euclidean Quantum Vector Fields

We study quantum vector fields in Euclidean space-time. These fields can be identified with generalized random vector fields, which we study in terms of their covariance. We prove that the conditions of translational invariance, covariance with respect to some representation τ=⊕ τj of the orthogonal group O(n), where none of the irreducible components τj is trivial, and the condition of reflection positivity cannot be fulfilled at the same time unless the test function space is restricted by some gauge condition. However, if the representation τ is trivial, i.e., if every matrix in O(n) is mapped to the identity, we can explicitly write down covariance matrices which lead to Gaussian fields which fulfill all conditions in the axiomatic framework.