Formulas for the probability of double $\ensuremath{\beta}$ decay including energy distributions and angular correlations of the emitted electrons according to the theory of Feynman and Gell-Mann are given. A strong dependence on the change of angular momentum exists. The formulas also exhibit interference between different intermediate states. The half-life of $_{20}\mathrm{Ca}_{28}$ is calculated using matrix elements of $j\ensuremath{-}j$ shell-model configurations. It is found to be $t\ensuremath{\sim}{10}^{17}\ensuremath{\cdots}{10}^{20}$ years. Actually, $t$ will be much greater, since the matrix elements used are those of favored transitions.