Trace-Class Obstructions to Bell Factorization via Quadratic Forms in Relativistic Quantum Field Theory

Eduardo Gonzalez-Granda Fernandez

Published Apr 27, 2026• 0 citations

Abstract

We establish a functional-analytic obstruction to Bell-type factorizations of two-point distributions in relativisticquantum field theory. Working purely at the level of quadratic forms on test functions, we show that any positive sesquilinear form whoselocal trace diverges cannot admit a Bell factorization with uniformly L²-bounded response functions. Moreover, anyapproximating sequence of such factorizations necessarily exhibits unbounded L² norms — a continuous analogue of theexact cancellations characteristic of fine-tuning in classical causal models. The argument is fully rigorous and self-contained: it uses only finite orthonormal families, Bessel's inequality, andTonelli's theorem, avoiding any need for operator domains or spectral theory. As a physical motivation, we discusshow short-distance singularities in Wightman two-point functions heuristically imply infinite local trace, makingthe obstruction physically relevant. We also provide an explicit spectral computation for the free scalar fieldin four dimensions. The result provides a functional-analytic mechanism underlying Cavalcanti's theorem on fine-tuning in classicalcausal models, and complements our previous spectral no-go theorems for causally localized jamming in QFT. This paper belongs to a series on analytic obstructions to causal factorization in relativistic quantum field theory.

DOI: 10.5281/zenodo.19814939

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