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Analytic Obstructions to Classical Causal Factorization in Relativistic QFT: From Two-Point to Three-Point Functions

Eduardo Gonzalez-Granda Fernandez

Published Apr 26, 2026• 0 citations
Publisher

Abstract

This work investigates structural constraints imposed by analyticity in relativistic quantum field theory on classical causal explanations of quantum correlations. Building on previous results on analytic rigidity, we establish a conditional obstruction to the realization of classical causal factorizations through spacetime-localized modifications of two-point correlation functions. The argument derives analyticity directly from the spectral condition and identifies clustering as the key mechanism linking probabilistic factorization with analytic constraints. We then extend the analysis to three-point functions. A marginal spectral condition is introduced as a natural generalization compatible with dimensional reduction. Under this framework, we prove that any admissible modification must vanish upon reduction to all two-point marginals. The irreducible three-point sector is analyzed via connected correlation functions. A conditional obstruction is obtained under an extended analytic rigidity hypothesis in higher-dimensional tube domains, which is formulated explicitly as an open problem. The work provides a unified perspective connecting analytic properties of quantum field theory with causal modeling approaches to quantum correlations, and outlines a program for extending these results to higher-point functions.

DOI
10.5281/zenodo.19787039

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This work investigates structural constraints imposed by analyticity in relativistic quantum field theory on classical causal explanations of quantum correlations. Building on previous results on analytic rigidity, we establish a conditional obstruction to the realization of classical causal factorizations through spacetime-localized modifications of two-point correlation functions. The argument derives analyticity directly from the spectral condition and identifies clustering as the key mechanism linking probabilistic factorization with analytic constraints. We then extend the analysis to three-point functions. A marginal spectral condition is introduced as a natural generalization compatible with dimensional reduction. Under this framework, we prove that any admissible modification must vanish upon reduction to all two-point marginals. The irreducible three-point sector is analyzed via connected correlation functions. A conditional obstruction is obtained under an extended analytic rigidity hypothesis in higher-dimensional tube domains, which is formulated explicitly as an open problem. The work provides a unified perspective connecting analytic properties of quantum field theory with causal modeling approaches to quantum correlations, and outlines a program for extending these results to higher-point functions.
This work investigates the compatibility of hidden-variable modifications with the analytic structure of relativistic quantum field theory (QFT). Working within the framework of tempered distributions, we analyze three fundamental properties: the spectral condition, analyticity in the forward tube, and causal localization. We prove a rigidity result showing that any modification satisfying all three conditions must vanish identically. This establishes a structural obstruction linking analyticity and causal constraints in QFT. Building on this result, we provide a systematic classification of nontrivial hidden-variable extensions according to which analytic assumptions fail. We identify three distinct regimes: (i) delocalized modifications compatible with standard QFT, (ii) causally localized modifications that violate the spectral condition and introduce negative-frequency components, and (iii) localized modifications that necessarily break analyticity. These results show that nontrivial hidden-variable theories cannot be embedded within the standard analytic framework of relativistic QFT without violating at least one of its fundamental structural principles. The work provides a unified analytic perspective on the constraints governing hidden-variable extensions and clarifies the role of analyticity as a rigidity mechanism.
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We establish a structural obstruction to bounded Bell-type factorizations of two-point functions in relativistic quantum field theory. Working at the level of bilinear forms on Schwartz space, we show that any Bell-factorizable distribution with uniformly bounded L² response functions satisfies a universal quadratic bound. We then prove that Wightman two-point functions with unbounded spectral density (a standard ultraviolet property in interacting QFT) violate this bound. As a consequence, no such function admits a stable Bell factorization: any approximating sequence of Bell-factorizable distributions necessarily involves response functions whose norms diverge. This provides a functional-analytic formulation of the fine-tuning phenomenon identified by Cavalcanti in causal models, linking spectral properties of quantum field theory to the instability of classical causal representations. The results avoid assumptions of spacetime localization and instead rely solely on spectral structure and positivity, offering a new perspective on the incompatibility between relativistic QFT and classical causal factorizations.
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