The paper by W. Heisenberg and H. Euler discusses the implications of Dirac's theory of the positron on electromagnetic fields. It shows that every electromagnetic field tends to produce pairs, leading to modifications in Maxwell's equations for the vacuum. These modifications are calculated for a specific case where no real electrons or positrons exist and the field changes only slightly over Compton wavelengths. The resulting Lagrange function for the field is derived, and its development terms describe scattering processes of light by light. For large fields, the field equations derived differ significantly from Maxwell's equations, which are compared with Born's suggestions. The ability of matter and radiation to transform into each other introduces new features in quantum electrodynamics. This transformation implies that Maxwell's equations must be replaced by more complex equations even in empty space. The authors argue that it is not possible to separate processes in empty space from material processes, as fields can generate matter if they have sufficient energy. Even in regions where energy is insufficient for matter generation, the virtual possibility of polarization of the vacuum leads to changes in Maxwell's equations. The paper focuses on the polarization of the vacuum, which will lead to a distinction between the vectors \(\mathfrak{B}\), \(\mathfrak{E}\), \(\mathfrak{D}\), and \(\mathfrak{H}\). The polarization functions \(\mathfrak{P}\) and \(\mathfrak{M}\) can be complex functions of the field strengths and their derivatives. For small field strengths, these functions can be approximated as linear functions of \(\mathfrak{E}\) and \(\mathfrak{B}\), and the changes in Maxwell's theory have been determined by Uehling and Serber. For slowly varying field strengths, \(\mathfrak{P}\) and \(\mathfrak{M}\) are functions of \(\mathfrak{E}\) and \(\mathfrak{B}\) at the same point, and the development of \(\mathfrak{P}\) and \(\mathfrak{M}\) follows odd powers of \(\mathfrak{E}\) and \(\mathfrak{B}\). The goal of the paper is to determine the functions \(\mathfrak{P}(\mathfrak{E}, \mathfrak{B})\) and \(\mathfrak{M}(\mathfrak{E}, \mathfrak{B})\) for the case of slowly varying field strengths. This involves calculating the energy density of the field \(U(\mathfrak{E}, \mathfrak{B})\) as a function of \(\mathfrak{E}\) and \(\mathfrak{B}\), which can be derived from the Hamiltonian formalism. The Lagrange function must be Lorentz invariant and can only depend onThe paper by W. Heisenberg and H. Euler discusses the implications of Dirac's theory of the positron on electromagnetic fields. It shows that every electromagnetic field tends to produce pairs, leading to modifications in Maxwell's equations for the vacuum. These modifications are calculated for a specific case where no real electrons or positrons exist and the field changes only slightly over Compton wavelengths. The resulting Lagrange function for the field is derived, and its development terms describe scattering processes of light by light. For large fields, the field equations derived differ significantly from Maxwell's equations, which are compared with Born's suggestions. The ability of matter and radiation to transform into each other introduces new features in quantum electrodynamics. This transformation implies that Maxwell's equations must be replaced by more complex equations even in empty space. The authors argue that it is not possible to separate processes in empty space from material processes, as fields can generate matter if they have sufficient energy. Even in regions where energy is insufficient for matter generation, the virtual possibility of polarization of the vacuum leads to changes in Maxwell's equations. The paper focuses on the polarization of the vacuum, which will lead to a distinction between the vectors \(\mathfrak{B}\), \(\mathfrak{E}\), \(\mathfrak{D}\), and \(\mathfrak{H}\). The polarization functions \(\mathfrak{P}\) and \(\mathfrak{M}\) can be complex functions of the field strengths and their derivatives. For small field strengths, these functions can be approximated as linear functions of \(\mathfrak{E}\) and \(\mathfrak{B}\), and the changes in Maxwell's theory have been determined by Uehling and Serber. For slowly varying field strengths, \(\mathfrak{P}\) and \(\mathfrak{M}\) are functions of \(\mathfrak{E}\) and \(\mathfrak{B}\) at the same point, and the development of \(\mathfrak{P}\) and \(\mathfrak{M}\) follows odd powers of \(\mathfrak{E}\) and \(\mathfrak{B}\). The goal of the paper is to determine the functions \(\mathfrak{P}(\mathfrak{E}, \mathfrak{B})\) and \(\mathfrak{M}(\mathfrak{E}, \mathfrak{B})\) for the case of slowly varying field strengths. This involves calculating the energy density of the field \(U(\mathfrak{E}, \mathfrak{B})\) as a function of \(\mathfrak{E}\) and \(\mathfrak{B}\), which can be derived from the Hamiltonian formalism. The Lagrange function must be Lorentz invariant and can only depend on