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Axioms

Parametrized Relativistic Quantum Theory: Framework and Application to Scanning Tunneling Microscopy

John R. Fanchi

Published Oct 12, 2026• 0 citations
Publisher

Abstract

Parametrized Relativistic Quantum Theory (PRQT) extends conventional relativistic quantum mechanics by introducing an invariant evolution parameter (s) and a broader symmetry group. This paper presents the axiomatic and probabilistic foundations of PRQT. The resulting Stueckelberg equation is applied to the study of barrier penetration, with explicit calculations for scanning tunneling microscopy (STM). The STM application demonstrates high sensitivity to barrier width and quantitative agreement between exact and thick-barrier approximations.

DOI
10.3390/axioms15100760

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Parametrized Relativistic Quantum Theory (PRQT) extends conventional relativistic quantum mechanics by introducing an invariant evolution parameter (s) and a broader symmetry group. This paper presents the axiomatic and probabilistic foundations of PRQT. The resulting Stueckelberg equation is applied to the study of barrier penetration, with explicit calculations for scanning tunneling microscopy (STM). The STM application demonstrates high sensitivity to barrier width and quantitative agreement between exact and thick-barrier approximations.
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.
This research paper explores the theoretical foundations of quantum tunneling, explaining how particles can pass through potential barriers despite lacking the classical energy to do so. It develops the mathematical framework using the Schrödinger equation, derives the transmission coefficient for a rectangular potential barrier, introduces the WKB approximation for more realistic barriers, and examines key applications such as alpha decay, scanning tunneling microscopy (STM), and the Josephson effect. The paper also reviews the historical development, experimental verification, and limitations of quantum tunneling, highlighting its significance as one of the most fundamental and experimentally validated predictions of quantum mechanics.
Quantum tunneling is conventionally described by the Schrödinger wave equation with a prescribed static potential barrier, providing accurate transmission probabilities but offering limited insight into the microscopic dynamics of particle–barrier interactions. In this work, we develop a dynamical second-quantization framework in which the barrier is modeled as an ensemble of quantized internal modes rather than as an externally imposed classical potential. The tunneling particle interacts directly with these microscopic barrier excitations through a coupled particle–barrier Hamiltonian, from which the Heisenberg equations of motion are derived. Collective coherent excitations of the barrier modes give rise to an emergent effective barrier that naturally recovers the conventional rectangular barrier and the WKB transmission limit under appropriate conditions. Unlike standard treatments, the present formulation explicitly incorporates microscopic barrier dynamics and provides a unified description of particle–barrier coupling within a second-quantized formalism. The framework further suggests that repeated tunneling events may experience different microscopic interaction histories, motivating a statistical interpretation of tunneling times. Because both the particle and barrier are treated within the same operator formalism, the theory provides a natural foundation for extension to relativistic quantum transport, photonic barriers, cavity quantum electrodynamics, and other structured quantum media.
The `evolving constants' method of defining the quantum dynamics of time-reparametrization-invariant theories is investigated for a particular implementation of parametrized non-relativistic quantum mechanics (PNRQM). The wide range of time functions that are available to define evolving constants raises issues of interpretation, consistency, and the degree to which the resulting quantum theory coincides with, or generalizes, the usual non-relativistic theory. The allowed time functions must be restricted for the predictions of PNRQM to coincide with those of usual quantum theory. They must be restricted to have a notion of quantum evolution in a time parameter connected to spacetime geometry. They must be restricted to prevent the theory from making inconsistent predictions for the probabilities of histories. Suitable restrictions can be introduced in PNRQM but these seem unlikely to apply to a reparametrization-invariant theory like general relativity.
At the turn of the twentieth century, the establishment of quantum theory propelled rapid advancements, particularly in the understanding of quantum tunnelling—a fundamental phenomenon in quantum mechanics crucial for various physical processes. The quantum phenomenon of particle passes through potential barriers is of great importance. In classical physics, when the energy of a particle is less than the height of a double barrier structure, it is impossible for it to pass through. However, quantum mechanics allows a particle to penetrate the barrier and emerge on the other side. This paper explores the quantum tunnelling effect, focusing on the single potential barrier model in one dimension and subsequently extending to the double potential barrier model. The Schrödinger equation provides the foundational framework for elucidating the motion of microscopic particles, emphasizing wave-particle duality inherent in quantum mechanics. The analysis of the single potential barrier model involves solving the Schrödinger equation in different regions, determining wave functions and coefficients through boundary conditions. The transmission coefficient is derived, representing the probability of a particle passing through a barrier. In the case of a “thick” barrier, an approximate form for transmission coefficient is provided, demonstrating the exponential decrease in transmission probability with increasing barrier thickness.