On the Unsolvability of Bosonic Quantum Fields

Two general unsolvability arguments for interacting bosonic quantum field theories are presented, based on Dyson-Schwinger equations on the lattice and cardinality considerations. The first argument is related to the fact that, on a lattice of size N, the system of lattice Dyson-Schwinger equations closes on a basis of "primitive correlators" which is finite, but grows exponentially with N. By properly defining the continuum limit, one finds for N to infinity a countably-infinite basis of the primitive correlators. The second argument in favor of the transcendency of exact solutions, is that any conceivable exact analytic calculation of the primitive correlators involves, in the continuum limit, a linear system of coupled partial differential equations with variable coefficients on an infinite number of unknown functions, namely the primitive correlators, evolving with respect to an infinite number of independent variables.