Supersymmetry, Foldy-Wouthuysen Transformation and Stability of the Dirac Sea

Abstract

It is shown that for a large class of potential problems in the Dirac equation the positive and negative energy solutions do not mix even in the strong coupling limit We prove that this property, which implies a stability of the Dirac sea, is connected to the presence of superalgebra operators in the Dirac equation. The exact and closed form for the Foldy-Wouthuysen Hamiltonian which is used to prove this property are given. The potentials include the Dirac oscillator, the uniform time-independent magnetic field and the odd potentials and its nonabelian generalizations.

Self-sustained exothermic reaction of anti-Stokes gamma transitions in long-lived isomeric nuclei. I

“…For the ordinary mass term, the SUSY-QM structure of the Dirac equation in a magnetic field is broken softly in the sense that one can still solve the Landau problem for massive Dirac fermions making use of the supercharge operator by redefining the energy eigenvalue in the form E → E = √ E 2 − m 2 [12,13]. Thus, if we consider an ordinary mass gap for graphinos which is the same for both the sublattices, we can retain the mapping of the dynamics between sublattices.…”

“…It is known that the ordinary Dirac equation possesses a supersymmetric structure in the quantum mechanical sense [12,13]. The corresponding Dirac equation for a graphino exhibits features which are intrinsic to theories formulated in odd spacetime dimensions, namely two inequivalent, irreducible representations of the Dirac matrices, which for fermions moving in a plane correspond to pseudo-spin or flavor labels, and the generalized definition of parity transformation [14,15].…”

On connection between the two-body Dirac oscillator and Kemmer oscillators

“…The representation analogous to (22) has already been recognized for the Dirac Hamiltonian (either for a free [17] or for an interacting [18] particle). Such a representation is useful for searching for parasupersymmetries of the approximate nonrelativistic Hamiltonians and for constructing of the Foldy-Wouthuysen transformation [9,10].…”

Parasupersymmetries and Non-Lie Constants of Motion for Two-Particle Equations

“…(2.1 ). Such a representation has already been recognized for the Dirac equation (either for free 33 or interacting 34 particle, the corresponding β 0 is the Dirac matrix andQ 1 is a supercharge), and for the KDP equation. 18 An important property of the representation (5.3) is that it enables to construct the FW transformation using relations (5.2), refer to Sec.…”