On the production of $e^+e^-$-pairs from the vacuum under the influence of an electric field

A study is made of the problem of pair production of particies with spin $1/2$ in an external homogeneous electric field $\mathbf E(t)$. The Heisenberg equations of motion are found for the operators of creation and annihilation, a solution for which has the form of a linear canonical transformation. A system of differential equations that determine the coefficients $U(t)$ and $V(t)$ of this transformation are obtained. The probabilities of pair creation are expressed in terms of the eigenvalues of the matrix $VV^+$. The ease of particles with anomalous magnetic moment is also considered.