Confusion about the solution and inner product properties of Dirac equation
I'v just read the solutions of Dirac equation and haven't yet known anything about the quantization of the equation, so my confusion is about this level.The problem is trivial. The solutions of Dirac equation are as follows: $$\left|\psi ^1\right\rangle =\sqrt{\frac{m+e}{2 m}} e^{-\text{ipx}} \left( \begin{array}{c} 1 \ 0 \ \frac{p^3}{m+e} \ \frac{p^1+\text{ip}^2}{m+e} \ \end{array} \right)=u_1 e^{-\text{ipx}};$$ $$\left|\psi ^2\right\rangle =\sqrt{\frac{m+e}{2 m}} e^{-\text{ipx}} \left( \begin{array}{c} 0 \ 1 \ \frac{p^1-\text{ip}^2}{m+e} \ -\frac{p^3}{m+e} \ \end{array} \right)=u_2 e^{-\text{ipx}};$$ $$\left|\psi ^3\right\rangle =\sqrt{\frac{m+e}{2 m}} e^{\text{ipx}} \left( \begin{array}{c} \frac{p^3}{m+e} \ \frac{p^1+\text{ip}^2}{m+e} \ 1 \ 0 \ \end{array} \right)=v_2 e^{\text{ipx}};$$ $$\left|\psi ^4\right\rangle =\sqrt{\frac{m+e}{2 m}} e^{\text{ipx}} \left( \begin{array}{c} \frac{p^1-\text{ip}^2}{m+e} \ -\frac{p^3}{m+e} \ 0 \ 1 \ \end{array} \right)=v_1 e^{\text{ipx}}$$
The corresponding inner product relations are as follows: $$u_{s \overset{\rightharpoonup }{p}} u_{r \overset{\rightharpoonup }{p}}{}^{\dagger }=v_{s \overset{\rightharpoonup }{p}} v_{r \overset{\rightharpoonup }{p}}{}^{\dagger }=\frac{e \delta ^{\text{rs}}}{m};$$ $$v_{s \left(-\overset{\rightharpoonup }{p}\right)} u_{r \overset{\rightharpoonup }{p}}{}^{\dagger }=0;$$ My question is:
For two particle states $\left|\psi ^1\right\rangle$and $\left|\psi ^4\right\rangle$, if they are going to have a same momentum and opposite energy, then the condition $v_{s \left(-\overset{\rightharpoonup }{p}\right)} u_{r \overset{\rightharpoonup }{p}}{}^{\dagger }=0$ is satisfied. 402
This illustrates that the two particles cannot have their states related, therefore they do not interact. But, what if I choose to make these two particles have opposite momentum direction (i.e.$v_{s \overset{\rightharpoonup }{p}} u_{r \overset{\rightharpoonup }{p}}{}^{\dagger }\neq0$), then the inner product of the two states cannot vanish. Does this mean that under such circumstances the two particles can interact? If I am guessing right, what are the interactions? Annihilation? Sorry for being so lengthy and expatiatory. 298