Relativistic supersymmetric quantum mechanics based on Klein–Gordon equation

Abstract

Witten's non-relativistic formalism of supersymmetric quantum mechanics was based on a factorization and partnership between Schrödinger equations. We show how it accommodates a transition to the partnership between relativistic Klein-Gordon equations.

Coupled-channel version of the PT-symmetric square well

“…We see here that the two operators R(σ) may be perceived as representing "reduced" This conclusion is compatible with formulae (36) and (22).…”

“…The frequent use of the coupling of channels in physics [21,22] attracted our attention to the partitioned…”

“…For this reason, the slightly modified 'brabra-ket' notation of ref. [22] is advocated and summarized here in Appendix B.…”

Non-Hermitian interaction representation and its use in relativistic quantum mechanics

“…It is instructive to verify that the latter operator is equal to the old untilded unobservable Hamiltonian of Eq. (37). One reveals that the demonstration of the identity H(t) = H(t) is easy due to the validity of Theorem 1.…”

Matrix Continued Fraction Solution to the Relativistic Spin-0 Feshbach–Villars Equations

“…They split the Klein-Gordon wave function into two components and for the components vector they arrived at a Schrödinger-like equation with first order in time derivative. Although the Feshbach-Villars formalism appear in some advanced quantum mechanics books [9,10,11,12,13,14,15], and they were utilized in gaining deeper insight into relativistic physics of Klein paradox pair production, [16,17,18,19,20,21,22,23], in exotic atoms [24,25,26], used in theoretical consideraions [27,28,29,31], study relativistic scattering [32,32] and optics [33] or demosntrate PT symmetry [34,35,36,37], they were hardly used as a computational tool. The equations look like ordinary coupled differential equations, but the components are coupled by the kinetic energy operator, which makes them very hard to solve.…”