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An investigation of the validity of the semiclassical approximation to quantum electrodynamics in 1 + 1 dimensions is given. The criterion for validity used here involves the impact of quantum fluctuations introduced through a two-point function which emerges naturally when considering the stability of the backreaction equation to linear order perturbations, resulting in the linear response equation. Consideration is given to the case of a spatially homogeneous electric field generated by a classical source, coupled to a quantized massive spin ½ field. Solutions to the linear response equation as well as the impact of quantum fluctuations introduced through the current density two-point correlation function are presented for two relevant electric field-to-mass parameter values qE=m^2, indicative of the strength of the backreaction process. Previous efforts utilized approximate solutions to the linear response equation that were expected to be valid for early times. A comparative analysis is given between the exact and approximate solutions in order to validate this conjecture. 

In a four-dimensional Schwarzschild–de Sitter background, the spherically symmetric (l = 0) contribution to the Hadamard two-point correlation function is computed for a massless minimally coupled scalar field in the Unruh state. Consideration is given to spacetime points located between the black hole and cosmological horizons. Previously it was found in two dimensions at late times for spatially separated points that the Hadamard function exhibits unbounded linear growth in time, with a rate of growth proportional to the sum of the black hole and cosmological surface gravities. Here it is shown numerically that this instability persists in four dimensions, but with a modification of the two-dimensional result due to scattering effects associated with the scalar field modes. An analytic approximation is derived for the growth rate in four dimensions and, in the limit that the black hole vanishes, is found to be equivalent to the rate of growth for the Hadamard function found previously for de Sitter space in cosmological coordinates.