series expansion of the function $f(X)$\]. In particular, we will consider the case where the functions in the exponents are linear either in $X^{\mu}$ or in $P^{\mu}$: $$\begin{aligned}
\label{Trans_X}
\exp(i(Xq))g(P)\exp(-i(Xq))&=g(P+q),\\
\label{Trans_P}
\exp(i(Py))f(X)\exp(-i(Py))&=f(X-y),\end{aligned}$$ where $q^{\mu}$ and $y^{\mu}$ are constant four-vectors.

In addition, the commutation relations $[\phi,P_{\phi}]=[T,P_T]=i$ imply in particular the identities $$\begin{aligned}
\label{Trans_phi}
\exp(ia\phi)\tilde{g}(P_{\phi})\exp(-ia\phi)&=\tilde{g}(P_{\phi}-a),\\
\label{Trans_PT}
\exp(ibP_T)\tilde{f}(T)\exp(-ibP_T)&=\tilde{f}(T+b),\end{aligned}$$ with $a$ and $b$ being two constants and $\tilde{f}(T)$ and $\tilde{g}(P_{\phi})$ being two arbitrary functions, which we will use below.

Note that if $|x\rangle$ ($|p\rangle$) is the eigenstate of the four-position (four-momentum) operator $X^{\mu}$ ($P^{\mu}=i\partial^{\mu}$) with eigenvalue $x^{\mu}$ ($p^{\mu}$), i.e., $X^{\mu}|x\rangle=x^{\mu}|x\rangle$ ($P^{\mu}|p\rangle=p^{\mu}|p\rangle$), then, by normalizing the eigenstates $|x\rangle$ ($|p\rangle$) such that $\langle x|y\rangle=\delta^{(4)}(x-y)$ \[$\langle p|q\rangle=(2\pi)^4\delta^{(4)}(p-q)$\], it is $\langle x|p\rangle=\exp(-i(px))=\exp[-i(p_+\phi+p_-T-\bm{p}_{\perp}\cdot\bm{x}_{\perp})]$ and $P_{\phi}|p\rangle=-p_+|p\rangle$, $P_T|p\rangle=-p_-|p\rangle$, and $\bm{P}_{\perp}|p\rangle=\bm{p}_{\perp}|p\rangle$. Also, the operator completeness relations hold $$\begin{aligned}
\label{C_x}
\int d^4x\,|x\rangle\langle x|&=1,\\
\int \frac{d^4p}{(2\pi)^4}\,|p\rangle\langle p|&=1.\end{aligned}$$ The Volkov states are the exact, analytical solutions of the Dirac equation in a plane wave [@Volkov_1935; @Landau_b_4_1982]. The positive-energy Volkov states $U_s(p,x)$ can be classified by means of the asymptotic momentum quantum numbers $\bm{p}$ (and then the energy $\varepsilon =\sqrt{m^2+\bm{p}^2}$) and of the asymptotic spin quantum number $s=1,2$ in the remote past, i.e. for $t\to-\infty$ (for notational simplicity, we have indicated the functional dependence on the 