\tau} f, \varphi \rangle = \int_{\mathbb{R}^3} f \cdot \overline{\left( -i c \alpha \cdot \nabla \varphi + m c^2 \beta \varphi \right)} \text{d} x + \int_\Sigma (f_+|_\Sigma + f_-|_\Sigma) \cdot \frac{1}{2} (\eta I_4 + \tau \beta) \overline{\varphi|_\Sigma} \text{d} \sigma,$$ where the notation $\alpha \cdot x = \alpha_1 x_1 + \alpha_2 x_2 + \alpha_3 x_3$ for a vector $x = (x_1,x_2,x_3)$ was used. On the other hand, one would expect that the operator $A_{\eta, \tau}$ acts for $x \notin \Sigma$ as $$\label{action_A_eta_formal}
  A_{\eta, \tau} f(x) = -i c \alpha \cdot \nabla f(x) + m c^2 \beta f(x),$$ which leads via integration by parts in $\Omega_\pm$ to the observation that $$\begin{split}
    \langle A_{\eta, \tau} f, \varphi \rangle &= \int_{\Omega_+ \cup \Omega_-} \big( -i c \alpha \cdot \nabla f + m c^2 \beta f \big) \cdot \overline{\varphi} \text{d} x\\
    &=\int_{\mathbb{R}^3} f \cdot \overline{\left( -i c \alpha \cdot \nabla \varphi + m c^2 \beta \varphi \right)} \text{d} x - \int_\Sigma i c \alpha \cdot \nu (f_+|_\Sigma - f_-|_\Sigma) \cdot \overline{\varphi|_\Sigma} \text{d} \sigma
  \end{split}$$ should hold for $f \in {\mathrm{dom}\,}A_{\eta, \tau}$. Comparing the two expressions for $\langle A_{\eta, \tau} f, \varphi \rangle$ we conclude that a function $f \in {\mathrm{dom}\,}A_{\eta, \tau}$ should satisfy the jump condition 