Spin and pseudospin symmetries and the equivalent spectra of relativistic spin-1/2 and spin-0 particles

Abstract

We show that the conditions which originate the spin and pseudospin symmetries in the Dirac equation are the same that produce equivalent energy spectra of relativistic spin-1/2 and spin-0 particles in the presence of vector and scalar potentials. The conclusions do not depend on the particular shapes of the potentials and can be important in different fields of physics. When both scalar and vector potentials are spherical, these conditions for isospectrality imply that the spin-orbit and Darwin terms of either the upper component or the lower component of the Dirac spinor vanish, making it equivalent, as far as energy is concerned, to a spin-0 state. In this case, besides energy, a scalar particle will also have the same orbital angular momentum as the (conserved) orbital angular momentum of either the upper or lower component of the corresponding spin-1/2 particle. We point out a few possible applications of this result. When describing some strong interacting systems it is often useful, because of simplicity, to approximate the behavior of relativistic spin-1/2 particles by scalar spin-0 particles obeying the Klein-Gordon equation. An example is the case of relativistic quark models used for studying quark-hadron duality because of the added complexity of structure functions of Dirac particles as compared to scalar ones. It turns out that some results (e.g., the onset of scaling in some structure functions) almost do not depend on the spin structure of the particle [1]. In this work we will give another example of an observable, the energy, whose value may not depend on the spinor structure of the particle, i.e., whether one has a spin-1/2 or a spin-0 particle. We will show that when a Dirac particle is subjected to scalar and vector potentials of equal magnitude, it will have exactly the same energy spectrum as a scalar particle of the same mass under the same potentials. As we will see, this happens because the spin-orbit and Darwin terms in the second-order equation for either the upper or lower spinor component vanish when the scalar and vector potentials have equal magnitude. It is not uncommon to find physical systems in which strong interacting relativistic particles are subject to Lorentz scalar potentials (or position-dependent effective masses) that are of the same order of magnitude of potentials which couple to the energy (time components of Lorentz four-vectors). For instance, the scalar and vector (hereafter meaning time-component of a four-vector potential) nuclear mean-field potentials have opposite signs but similar magnitudes, whereas relativistic models of mesons with a heavy and a light quark, like D-or B-mesons, explain the observed small spin-orbit splitting by having vector and scalar potentials with the same sign and similar strengths [2].It is well-known that all the components of the free Dirac spinor, i.e., the solution of the free Dirac equation, satisfy the free Klein-Gordon equation. Indeed, from the free Dirac equationone getswhere use has been made of the relation γ µ γ ν ∂ µ ∂...

Fermions in the background of mixed vector–scalar–pseudoscalar square potentials

“…This equivalence confirm the results obtained by P. Alberto and collaborators [35], which show that the spin and pseudospin symmetries in Dirac equation produce an equivalent energy espectra for relativistic spin-1/2 and spin-0 particles in the presence of vector and scalar potentials.…”

Dynamical symmetries of the Klein–Gordon equation

“…These symmetries, which have been observed in the hadron and nuclear spectroscopies for a long time [13,14], are derived from the investigation of the dynamics between a quark and an antiquark [15,16]. The very lately studies [9,10] have revealed that, the motion of a spin-1/2 particle with SVPEM satisfies the same differential equation and has the same energy spectrum as a scalar particle. When both scalar and vector potentials are spherical, Alberto et.…”

“…Recently, many works about the Dirac or the Klein-Gordon (KG) equation with scalar and vector potentials of equal magnitude (SVPEM) are reported [1,2,3,4,5,6,7,8,9,10,11,12] .When the potentials are spherical, the Dirac equation is said to have the spin or pseudospin symmetry corresponding to the same or opposite sign. These symmetries, which have been observed in the hadron and nuclear spectroscopies for a long time [13,14], are derived from the investigation of the dynamics between a quark and an antiquark [15,16].…”

General spin and pseudospin symmetries of the Dirac equation

“…Since those terms arise from the coupling of those spinor components in their first-order Dirac equations, they have a non-trivial, i.e., different from identity, matrix structure. Therefore, their suppression amounts to have the upper (spin symmetry) or lower (pseudospin symmetry) spinors satisfying second-order equations of Shrödinger type, i.e, with no matrix structure (see [6][7][8][9][10], and [11] for a brief review). This is possible because we have both scalar and vector potentials S and V .…”