Charge Conservation, Klein’s Paradox and the Concept of Paulions in the Dirac Electron Theory

Abstract

An algebraic block-diagonalization of the Dirac Hamiltonian in a timeindependent external field reveals a charge-index conservation law which forbids the physical phenomena of the Klein paradox type and guarantees a single-particle nature of the Dirac equation in strong external fields. Simultaneously, the method defines simpler quantum-mechanical objects-paulions and antipaulions, whose 2-component wave functions determine the Dirac electron states through exact operator relations. Based on algebraic symmetry, the presented theory leads to a new understanding of the Dirac equation physics, including new insight into the Dirac measurements and a consistent scheme of relativistic quantum mechanics of electron in the paulion representation. Along with analysis of the mathematical anatomy of the Klein paradox falsity, a complete set of paradox-free eigenfunctions for the Klein problem is obtained and investigated via stationary solutions of the Pauli-like equations with respective paulion Hamiltonians. It is shown that the physically correct Dirac states in the Klein zone are characterized by the total particle reflection from the potential step and satisfy the fundamental charge-index conservation law.

Relativistic time-dependent quantum dynamics across supercritical barriers for Klein-Gordon and Dirac particles

“…In the fermionic case there is no superradiance (although some authors suggest differently, see e.g. [2,13]), and supercritical steps have been deemed to have an acceptable interpretation only within a Quantum Field Theory (QFT) approach [21,22], a point that appears to be supported by the wide variety of conflicting interpretations of Klein tunneling that have been proposed within the first quantized framework [5][6][7][8][9][10][11][12][13][14][15].…”

“…The main issue now is whether these results lend to a consistent interpretation of the Klein paradox within the first quantized formalism. This is known to be problematic when basing considerations on stationary plane waves, and indeed different, sometimes conflicting interpretations of superradiance and supercritical tunneling in a first quantized framework have been given [5][6][7][8][9][10][11][12][13][14][15]. Klein tunneling has even been denied to exist in the bosonic case [14] or for fermions [10].…”

“…This is known to be problematic when basing considerations on stationary plane waves, and indeed different, sometimes conflicting interpretations of superradiance and supercritical tunneling in a first quantized framework have been given [5][6][7][8][9][10][11][12][13][14][15]. Klein tunneling has even been denied to exist in the bosonic case [14] or for fermions [10]. Of course since particle creation is induced by supercritical potentials, a proper approach requires a Quantum Field Theory (QFT) treatment.…”

“…[1][2][3][4]) give different, often conflicting accounts of the Klein paradox. This situation is reflected even in recent works [5][6][7][8][9][10][11][12][13][14][15], that reach different conclusions generally based on time independent considerations.…”

The Stationary Dirac Equation as a Generalized Pauli Equation for Two Quasiparticles

“…Perhaps, the most paradoxical implications of the Dirac equation are the Klein tunneling [3][4][5][6][7][8][9][10] and the so-called 'Zitterbewegung' phenomenon [3,11,12]. Both are often mentioned in the current literature on this equation and both give rise to controversy among researchers.…”