On the Unsolvability of Bosonic Quantum Fields

TL;DR

This paper presents two broad arguments demonstrating the fundamental unsolvability of interacting bosonic quantum field theories, based on lattice Dyson-Schwinger equations and cardinality considerations, highlighting inherent computational limitations.

Contribution

It introduces novel unsolvability arguments for bosonic quantum fields using lattice and continuum approaches, emphasizing the complexity of primitive correlators.

Findings

Primitive correlators grow exponentially with lattice size.

Continuum limit yields a countably-infinite basis of correlators.

Exact solutions involve infinite coupled PDEs for primitive correlators.

Abstract

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.