From the Dirac theory of the positron, it follows that electromagnetic fields tend to produce particle pairs, leading to modifications of Maxwell's equations for the vacuum. These modifications are calculated for the special case where no real electrons or positrons are present, and the field changes only slightly over Compton wavelengths. The resulting Lagrange function for the field is given by a complex expression involving integrals and trigonometric functions. The critical field strength $ |\mathfrak{E}_k| $ is defined as $ \frac{m^2 c^3}{e \hbar} $, which represents the threshold for pair production. For small fields, the equations describe light scattering processes, while for large fields, they differ significantly from Maxwell's equations. These equations are compared to those proposed by Born. The ability to convert matter into radiation and vice versa leads to new aspects of quantum electrodynamics. This includes the necessity to replace Maxwell's equations with more complex ones for vacuum processes, as fields can generate matter even in empty space. This leads to a polarization of the vacuum, which results in the distinction between vectors B, C and D, H. The polarization P and M can be complex functions of field strengths and their derivatives. For small fields, P and M can be approximated as linear functions of E and B. In this approximation, Uehling and Serber have determined the modifications of Maxwell's theory. For slowly varying fields, P and M are functions of E and B at the same point, with derivatives of E and B no longer appearing. The expansion of P and M in terms of E and B proceeds in odd powers, with third-order terms responsible for light scattering processes. The goal of this work is to fully determine the functions P(E, B) and M(E, B) for the case of slowly varying fields. This can be achieved by calculating the energy density of the field U(E, B), which can be derived from the Lagrange function using Hamilton's method. The Lagrange function must be a function of the two invariants $ \mathfrak{E}^2 - \mathfrak{B}^2 $ and $ (\mathbf{E}\mathbf{B})^2 $. The calculation of U(E, B) reduces to determining the energy density of the material field associated with constant fields E and B. Before addressing this problem, the mathematical framework of positron theory is briefly recapitulated.From the Dirac theory of the positron, it follows that electromagnetic fields tend to produce particle pairs, leading to modifications of Maxwell's equations for the vacuum. These modifications are calculated for the special case where no real electrons or positrons are present, and the field changes only slightly over Compton wavelengths. The resulting Lagrange function for the field is given by a complex expression involving integrals and trigonometric functions. The critical field strength $ |\mathfrak{E}_k| $ is defined as $ \frac{m^2 c^3}{e \hbar} $, which represents the threshold for pair production. For small fields, the equations describe light scattering processes, while for large fields, they differ significantly from Maxwell's equations. These equations are compared to those proposed by Born. The ability to convert matter into radiation and vice versa leads to new aspects of quantum electrodynamics. This includes the necessity to replace Maxwell's equations with more complex ones for vacuum processes, as fields can generate matter even in empty space. This leads to a polarization of the vacuum, which results in the distinction between vectors B, C and D, H. The polarization P and M can be complex functions of field strengths and their derivatives. For small fields, P and M can be approximated as linear functions of E and B. In this approximation, Uehling and Serber have determined the modifications of Maxwell's theory. For slowly varying fields, P and M are functions of E and B at the same point, with derivatives of E and B no longer appearing. The expansion of P and M in terms of E and B proceeds in odd powers, with third-order terms responsible for light scattering processes. The goal of this work is to fully determine the functions P(E, B) and M(E, B) for the case of slowly varying fields. This can be achieved by calculating the energy density of the field U(E, B), which can be derived from the Lagrange function using Hamilton's method. The Lagrange function must be a function of the two invariants $ \mathfrak{E}^2 - \mathfrak{B}^2 $ and $ (\mathbf{E}\mathbf{B})^2 $. The calculation of U(E, B) reduces to determining the energy density of the material field associated with constant fields E and B. Before addressing this problem, the mathematical framework of positron theory is briefly recapitulated.