Analytic Classification of Hidden-Variable Extensions in Relativistic Quantum Field Theory

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.