# On generally covariant mathematical formulation of Feynman integral in Lorentz signature

January 27, 2022

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

Keywords:

Feynman integral formulation, master Dyson-Schwinger equation, constructive field theory