>>11004706We very much know how it be defined. In fact, free relativistic QFTs a la Haag-Kastler are mathematically well-defined in terms of local nets of observables and representation theory thereof, or via Baez-Zhang in terms of symplectic nets of probability measures.
The entire problem is with scattering, in which particle interactions may lead to no amplitudes satisfying both the required physical (e.g. crossing symmetry) and mathematical (e.g. analyticity/meromorphicity) conditions.
There are several classes of mathematically well-defined interacting relativistic QFTs, however:
1. QFTs with fields satisfying the Strocchi-Swieca conditions, which have well-defined -matrices,
2. QFTs reconstructed from an interacting Euclidean FT, whose amplitudes have relaxed analytic conditions and the relativistic versions can be Wick rotated,
3. QFTs whose (quantum) dynamics can be described by SDEs, and whose renormalization can be captured via recently developed "regularity structures" of Hairer, or
4. QFTs obtained on hyperkahler manifolds that admit a Kostant-Souriau quantization of the jet bundle.
Not to mention relativistic QFTs with mostly topological/geometric data (i.e. very little dynamics), such as TQFTs and CFTs, can be mathematically defined, written down and classified completely categorically. This line of thinking has recently been pushed to study superconformals, SuGras and (SUSY) strings and sees application in homological mirror symmetry and even geometric Langlands.
Even standard texts like Weinberg covers some of the required foundations such as Dirac's quantization of classical holonomic constraints. So no, the problem is with you, not with relativistic QFT.
