A Unified First-Order Spacetime Derivative Framework for Quantum Fields of All Spins: Emergent Spin-2 Dynamics and Gravitational Structure

We develop a unified first-order framework for relativistic fields of different spin, in which the dynamics are governed by a common operator-based equation. This formulation provides a coherent description of scalar, spinor, vector, and tensor fields within a single structure and reproduces the corresponding second-order wave equations in appropriate limits. A central result is the emergence of a consistent spin-2 sector from the same underlying dynamics. By constructing the tensor field as a bilinear combination of internal spacetime degrees of freedom, we obtain a symmetric rank-2 field with the correct number of independent components. In the massless limit, the resulting equation matches the structure of linearized gravity, while source-like terms arise naturally from quadratic combinations of field derivatives, providing an intrinsic origin for an effective energy–momentum tensor. The Lagrangian formulation yields conserved quantities via Noether’s theorem and reproduces derivative structures consistent with the weak-field Einstein–Hilbert action. These results suggest that gravitational dynamics may emerge from a more fundamental first-order field theory.