Dirac Equation and Representation Dependent Scattering Phenomena

TL;DR

The paper investigates whether the 1D Dirac equation with scalar potentials exhibits representation-dependent scattering, finding spin-flip and interference effects in representations other than the standard Dirac form, despite the same dispersion $E^2=p^2+m^2$. It compares the standard Dirac representation with Ajaib's nilpotent $\eta$ representation and a non-unitary, pseudo-Hermitian representation based on $\xi_1,\xi_2$, showing that transmission and reflection coefficients across a step potential depend on the chosen representation. In the Ajaib representation, spin-flip occurs in both relativistic and non-relativistic limits and the paper provides explicit expressions for $T_\uparrow$, $T_\downarrow$, $R_\uparrow$, $R_\downarrow$, highlighting how the boundary conditions couple spin channels via off-diagonal terms. In the non-unitary representation, scattering exhibits multi-channel quantum interference with a decomposition $T_{tot}=T_1+T_2+T_{int}$ and a large negative interference term that preserves probability, alongside persistent Klein tunneling at high energies, implying hidden physics in spinor-barrier coupling with potential experimental tests.

Abstract

We show that spin-flip probabilities emerge in the relativistic regime for scalar potentials, absent in the standard Dirac representation. We examine 1D scattering for the Dirac equation employing an alternate matrix representation introduced by the Author in an earlier study. We demonstrate that the transmission (T) and reflection (R) coefficients can depend on the chosen representation of the Clifford algebra despite the two representations being related by a unitary or non-unitary transformation. We also show that for the non-unitary case quantum interference may arise in scattering phenomena, a testable experimental signature. This representation dependence hints at hidden physics in how spinor components couple to external steps/barriers, even when the relativistic dispersion relation (E^2=p^2+m^2) is the same.

Figures (3)

• Figure 1: Particle incident on a step potential.
• Figure 2: Comparison of transmission and reflection coefficients for an electron incident on a potential barrier with $V_0= 1 \ keV$. We can see that even in the relativistic case there is a non-zero probability for the electron to flip its spin.
• Figure 3: Energy dependence of transmission and reflection coefficients for a spin-up incident electron encountering a step potential in the $\{\xi_1, \xi_2\}$ representation for the $E > V_0 + m$ case. (a) Decomposition of total transmission into amplitude components: $T_1$ (red, diagonal contribution $|t_{\uparrow\uparrow}|^2$), $T_2$ (blue, off-diagonal contribution $|t_{\uparrow\downarrow}|^2$), and $T_{\text{int}}$ (green, quantum interference term $2\text{Re}(t^*_{\uparrow\uparrow} t_{\uparrow\downarrow})$). The large individual magnitudes and substantial negative interference produce the physical total transmission $T_{\text{tot}} = T_1 + T_2 + T_{\text{int}} \le 1$, reflecting strong quantum correlations between spin channels. (b) Total transmission $T_{\text{tot}}$ (red) and reflection $R_{\text{tot}}$ (blue) demonstrating probability conservation $T_{\text{tot}} + R_{\text{tot}} = 1$ (black dashed line). The transition from reflection-dominated to transmission-dominated behavior exhibits Klein tunneling, with near-perfect transmission at high energies.