Fractional Spacetime Measures in Quantum Field Theory: Non-Hermitian Dynamics, Lorentz Violation, and Neutrino Phenomenology

We investigate a class of modified quantum field theories based on a fractional-inspired deformation of the action, where nonlocality is introduced through a spacetime-dependent weight rather than fractional derivatives. Using a Riemann–Liouville–type kernel implemented directly in the measure, we derive modified Klein–Gordon and Dirac equations and show that the resulting dynamics acquires additional first-order derivative terms governed by gradients of the weight. These corrections lead to nonconservative evolution and non-Hermitian effective Hamiltonians in the conventional inner product, while preserving a conserved weighted current and a generalized norm. The framework explicitly breaks Lorentz invariance by introducing a preferred spacetime structure, with potential implications for discrete symmetries, including possible violations of parity, time reversal, and CPT. We analyze the structure of the modified Dirac equation, including its Hamiltonian formulation and charge-conjugation properties, and discuss the consistency of the theory in terms of a weighted Hilbert space. Phenomenological implications are explored with emphasis on neutrino propagation, where high-precision oscillation and time-of-flight measurements provide stringent constraints. We estimate that current and future experiments bound deviations from the standard theory, highlighting the challenge of observationally distinguishing the model while identifying potential signatures such as energy-dependent dispersion and CPT asymmetries.