space of order one which consists of functions $f \in L^2(\Omega)$ with $\nabla f \in L^2(\Omega)^3$, where $\nabla f$ is the distributional gradient of $f$. Similarly $H^1(\mathbb{R}^3)$ is introduced. Moreover, we define the trace space $$H^{1/2}(\Sigma) := \{ f |_\Sigma: f \in H^1(\Omega) \}$$ equipped with the norm $\| \varphi \|_{1/2} := \inf\{ \| f \|_{H^1(\Omega)}: f \in H^1(\Omega),~ f|_\Sigma = \varphi \}$. One verifies that the trace mapping $$\label{trace_theorem}
  H^1(\Omega) \ni f \mapsto f|_\Sigma \in H^{1/2}(\Sigma)$$ is a bounded, surjective linear map and one can further show that $H^{1/2}(\Sigma) \subset L^2(\Sigma)$, cf. [@HW08 Section 4.2 and Theorem 4.2.1].

Since we are not interested in the semiclassical limit, we choose units in  in such a way that $\hbar=1$. However, we keep the mass of the particle $m$ and the speed of light $c$ both as positive constants. The Dirac matrices $\alpha := (\alpha_1, \alpha_2, \alpha_3)$ and $\beta$ are defined for $j \in \{ 1,2,3\}$ by $$\label{def_Dirac_matrices}
  \alpha_j := \begin{pmatrix} 0 & \sigma_j \\ \sigma_j & 0 \end{pmatrix}
  \quad \text{and} \quad \beta := \begin{pmatrix} I_2 & 0 \\ 0 & -I_2 \end{pmatrix},$$ where $I_d$ denotes the $d\times d$-identity matrix and $\sigma_1, \sigma_2, \sigma_3$ are the Pauli spin matrices $$  \sigma_1 := 