On generally covariant mathematical formulation of Feynman integral in Lorentz signature

It is widely accepted that the Feynman integral one of most promising methodologies for defining a generally covariant formulation nonperturbative interacting quantum field theories (QFTs) without fixed prearranged causal background. Recent literature suggests if spacetime metric not fixed, e.g. because it to be quantized along with other fields, may able avoid considering in original Lorentz signature, Wick rotation. Several mathematical phenomena are known, however, which at some point showstoppers mathematically sound definition signature. The formulation, known have differential reformulation, called master Dyson--Schwinger (MDS) equation correlators. In this paper shown particular presentation MDS can cast into rigorously defined form: involved function spaces and operators strictly their properties established. Therefore, serve as substitute integral, constructive QFT, arbitrary background structure. also even such setting, there canonical way define Wilsonian regularization equation. main result necessary sufficient condition regularized solution space nonempty, conformally invariant Lagrangians. This theorem provides an iterative approximation algorithm obtaining solutions, guaranteed convergent whenever nonempty. could eventually method putting signature QFTs onto lattice,