Divergent integrals in quantum field theory (QFT) can be given well defined existence as Lorentz covariant complex measures, which may be analyzed by means of a spectral calculus. The case of the photon self energy is considered and the spectral vacuum polarization function is shown to have very close agreement with the vacuum polarization function obtained using dimensional regularization / renormalization in the timelike domain. Using the spectral vacuum polarization function a potential function defined in the timelike domain is derived. The Uehling potential function, from which the Uehling contribution to the Lamb shift may be computed, is derived from an analytic continuation into the spacelike domain of this potential function. The spectral running coupling for QED is computed from this analytically continued potential function. The integral defining the spectral running coupling constant is shown to converge for all non-zero energies while that for the running coupling constant computed using dimensional regularization / renormalization is shown to diverge for all non-zero energies. It is seen that the spectral running coupling does not have a Landau pole and agrees both qualitatively and quantitatively with the results of scattering experiments at all energies.

量子场论 (QFT) 中的发散积分可以作为洛伦兹协变复测度给出明确定义的存在，这可以通过谱微积分进行分析。考虑了光子自能的情况，并且光谱真空极化函数与在类时域中使用维度正则化/重整化获得的真空极化函数非常一致。使用光谱真空极化函数，可以导出在类时域中定义的势函数。可以计算 Uehling 对兰姆位移的贡献的 Uehling 势函数是从该势函数的类空间域的解析延拓导出的。QED 的频谱运行耦合是从这个解析连续的势函数计算出来的。定义频谱运行耦合常数的积分显示为对所有非零能量收敛，而使用维度正则化/重整化计算的运行耦合常数的积分显示为对所有非零能量发散。可以看出，光谱运行耦合没有朗道极点，并且在定性和定量上与所有能量的散射实验结果一致。