Abstract
Standard relativistic quantum mechanics often falters due to the non-local square-root operator in the kinetic term. Traditional approaches rely on non-relativistic limits or numerical approximations, limiting analytical precision. This research overcomes these constraints by shifting the dynamical formulation entirely to momentum space, effectively isolating the non-locality.
By invoking a hyperbolic kinematic transformation, we introduce the rapidity variable, which linearizes the non-linear square-root operator into a tractable first-order framework. This methodology is applied to two critical systems:
Linear Potentials: In the rapidity domain, the wave equation simplifies to a first-order system. Utilizing the Jacobi-Anger expansion, we project the continuous phase onto a discrete superposition of Bessel functions, yielding an exact series solution. This framework also provides a deterministic, semiclassical derivation of the Schwinger pair-production probability, recovering the canonical tunneling exponent via complex rapidity integration.
Relativistic Coulomb Interaction: The system manifests as a three-dimensional integral equation. We employ spherical multipole expansion to reduce this to a one-dimensional radial equation. Finally, applying scale-invariant Mellin transforms allows us to map complex momentum transfer kernels directly into Beta function scaling exponents.
This approach establishes a rigorous, exact analytical paradigm for relativistic potentials, bypassing numerical basis truncations and providing a powerful tool for exploring fundamental quantum dynamics.