Effect of Negative Energy Components in the Two-Nucleon System

A feature of the Lévy-Klein solution of the Bethe-Salpeter equation for the deuteron is the elimination of ${\phi }_{+-}$, ${\phi }_{-+}$, and ${\phi }_{--}$ in terms of the ${\phi }_{++}$ component of the wave function via a perturbation expansion. To investigate the validity of this procedure, the coupled equations for the various components in the first nonadiabatic approximation to the ${\Delta }_{+}$ interaction were examined. A set of first-order radial equations with multiplicative potentials (in the region $r>\mathrm{}\frac{1}{m}$) was obtained, involving ${\phi }_{\pm \pm }\mathrm{}\left(r\right)=\frac{1}{4}(1\mathrm{}\pm {\beta }_1)(1\mathrm{}\pm {\beta }_2)\phi \mathrm{}\left(r\right)\mathrm{}$. A rigorous elimination of ${\phi }_{+-}$ and ${\phi }_{-+}$ led to equations containing ${\phi }_{++}$ and ${\phi }_{--}$. By neglecting velocity-dependent terms, potentials of the type $\frac{{\mu }^2}{2m}\frac{g^2}{4\pi }{\tau }_1·{\tau }_2\frac{Y\left(4\right)\mathrm{}}{P_0+\frac{1}{2}\left(\frac{g^2}{4\pi }\right){\tau }_1·{\tau }_{2}Y\left(r\right)\mathrm{}}$ appear. Expanding the denominator yields the usual Yukawa second-order potential plus a term proportional to $g^4{\left({\tau }_1·{\tau }_2\right)}^2$. Such an expansion, however, is poor even for $r\lesssim \frac{1}{\mu }$, a pole actually existing near $r\sim \frac{0.7}{\mu }$ for the charge singlet state. (For $J=1\mathrm{}$, the structure of the lowest-order tensor interaction is greatly altered.) Thus, the perturbation expansion appears to radically alter the structure of the equations.