Extending the Schrödinger quantum field to parametrized quantum field theory: A variational approach using the Pavšič–Barut action
This paper extends the Schrödinger quantum field—a wave function treated as a fundamental quantum field describing particle states and probabilities—to a fully relativistic framework within Parametrized Relativistic Quantum Theory (PRQT). A hallmark of parametrized theories is the use of an invariant evolution parameter, which addresses key conceptual challenges in standard quantum field theory (QFT). Feynman's path integral formulation is reviewed to illustrate the historical role of such parameters in early QFT developments. We then present a variational approach, based on the Pavšič–Barut action principle with a parametrized Lagrangian density, to derive field equations that naturally incorporate the invariant evolution parameter and yield a mass operator. The consistency of this formalism with physical measurements is demonstrated through applications to the Stueckelberg equation, including its implications for particle trajectories and mass states. The potential for resolving Standard Model anomalies is discussed.