Universal Emergence of Bosonic and Fermionic Algebras in a Deterministic Proper-Time Framework

Abstract

This work develops the deterministic, discrete proper-time framework introduced in our previous work, Eur.\ Phys.\ J.\ C \textbf{86} (2026) 829, in which quantum field theory emerges as an effective infrared description characterized by a running Planck constant. There, the effective quantization scale was inferred from the microscopic multiplicity unresolved by coarse-graining. Here, we provide its dynamical and operatorial realization and extend the construction to fermionic degrees of freedom. First, consistency under changes of macroscopic resolution leads to the structure of the Renormalization Group Equation, while the running Planck constant governs the crossover toward the deterministic regime. Second, we represent the reversible microscopic dynamics through finite-difference translations in field-configuration space. After coarse-graining, the resulting operator-valued canonical commutation relations reproduce the same quantization scale previously obtained from statistical microstate counting, thereby linking microscopic evolution to the emergent canonical algebra. Third, representing the same update within a Grassmann algebra yields the corresponding canonical anticommutation relations. The bosonic and fermionic sectors thus inherit a common effective Planck constant without introducing an independent fermionic update or quantization scale. Finally, we discuss possible implications for high-energy loop amplitudes and effective Hawking radiation.