Quantum electrodynamics based on self-fields, without second quantization: A nonrelativistic calculation of g-2.

Abstract

Using a formulation of quantum electrodynamics that is not second quantized, but rather based on self-fields, we compute the anomalous magnetic moment of the electron to first order in the fine-structure constant \ensuremath{\alpha}. In the nonrelativistic (NR) case and in the dipole approximation, our result is ${a}_{e}$\ensuremath{\equiv}(g-2)/2=(4\ensuremath{\Lambda}/3m)(\ensuremath{\alpha}/2\ensuremath{\pi}), where \ensuremath{\Lambda} is a positive photon energy cutoff and m the electron mass. A reasonable choice of cutoff, \ensuremath{\Lambda}/m=(3/4, yields the correct sign and magnitude for g-2 namely, ${a}_{e}$=+\ensuremath{\alpha}/2\ensuremath{\pi}. In our formulation the sign of ${a}_{3}$ is correctly positive, independent of cutoff, and the demand that ${a}_{e}$=+\ensuremath{\alpha}/2\ensuremath{\pi} implies a unique value for \ensuremath{\Lambda}. This is in contradistinction to previous NR calculations of ${a}_{e}$ that employ electromagnetic vacuum fluctuations instead of self-fields; in the vacuum fluctuation case the sign of ${a}_{e}$ is cutoff dependent and the equation ${a}_{e}$=\ensuremath{\alpha}/2\ensuremath{\pi} does not have a unique solution in \ensuremath{\Lambda}.