Generalized Random Vector Fields and Euclidean Quantum Vector Fields

Abstract

We study generalized random vector fields in the framework of Euclidean quantum field theory. A recent no-go theorem about the non-existence of covariant reflection-positive random vector fields with locally integrable covariance is discussed and some of its implications are pointed out. We study Euclidean quantum fields obtained as pullbacks by translation invariant, covariant and weakly local operators and present examples of Gaussian random vector fields that fulfil all axioms for Euclidean random fields and that have a covariance which is not locally integrable. Finally we point out that in space-time dimension 2 there exist interacting Euclidean quantum vector fields obtained as pullbacks of P(Φ)2-theories. However, the different components of these fields do not couple.KeywordsVector FieldRandom FieldAnalytic ContinuationPartial Differential OperatorReflection PositivityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.