A second quantized approach to the Rabi problem

In the present work the Rabi Problem, involving the response of a spin 1/2 particle subjected to a magnetic field, is considered in a second quantized approach. In this concrete physical scenario, we show that the second quantization procedure can be applied directly in a non-covariant theory. The proposed development explicits not only the relation between the full quantum treatment of the problem and the semiclassical Rabi model, but also the connection of these approaches with the Jaynes-Cummings model. The consistency of the method is checked in the semiclassical limit. The treatment is then extended to the matter component of the Rabi problem so that the Schr\"odinger equation is directly quantized. Considering the spinorial field, the appearance of a negative energy sector implies a specific identification between Schr\"odinger's and Maxwell's theories. The generalized theory is consistent, strictly quantum and non-relativistic.