Quantum fields obtained from convoluted generalized white noise never have positive metric

TL;DR

This paper proves that relativistic quantum fields derived from convoluted generalized white noise only have a positive metric if the noise is Gaussian, highlighting a fundamental limitation in non-Gaussian cases.

Contribution

It establishes a necessary and sufficient condition for positive metric in quantum fields from convoluted generalized noise, based on a criterion by Baumann and the Dell'Antonio-Robinson-Greenberg theorem.

Findings

Positive metric fields are Gaussian

Non-Gaussian convoluted noise fields lack positive metric

Connection between noise type and quantum field positivity

Abstract

It is proven that the relativistic quantum fields obtained from analytic continuation of convoluted generalized (L\'evy type) noise fields have positive metric, if and only if the noise is Gaussian. This follows as an easy observation from a criterion by K. Baumann, based on the Dell'Antonio-Robinson-Greenberg theorem, for a relativistic quantum field in positive metric to be a free field.