Questions of gauge and basis sets for the representation of electromagnetic interactions in quantum mechanics

Recent concerns about the gauge dependence of transition amplitudes describing the changes of state of quantized systems interacting with electromagnetic radiations have led to the development of invariant schemes for the expression of the time-dependent quantities involved. It is shown in this work that such schemes are not uniquely specified by the requirement for gauge invariance. Moreover, they simply exchange a dependence upon the gauge of the electromagnetic potentials for a previously unrecognized dependence upon the representation of basis states describing initial and final states of the process that is equivalent to a dependence upon gauge. However, general expressions are developed in this work which show that resonant transitions which correspond to physically observable processes involving the pure or mixed absorption and emission of $N$ photons are gauge invariant regardless of the multipolarity of the transition moments and independent of the representation of the basis states of the system. A preferred representation is introduced which is shown to accommodate the physical effects generally encountered.