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 -> infinity a countably-infinite basis of the primitive correlators. The second argument is that any conceivable exact analytic calculation of the primitive correlators involves, in the continuum limit, a linear system of coupled partial differential equations on an infinite number of unknown functions, namely the primitive correlators, evolving with respect to an infinite number of independent variables.