Transition Probability and Preferential Gauge

This paper is concerned with whether or not the preferential gauge can ensure the uniqueness and correctness of results obtained from the standard time-dependent perturbation theory, in which the transition probability is formulated in terms of matrix elements of Hamiltonian. For a dynamical quantum process, the major objective of the existing perturbation theory is to calculate the transition probability between different eigenstates. The theory, which was proposed by Dirac at the very early stage of quantum theory[1] and has been serving as an important part of quantum mechanics in textbooks[2], gives an analytical expression of transition probability in terms of matrix elements of the Hamiltonian representing perturbations. It was noticed that the transition probability given by the Dirac theory is not gauge-invariant[3], which suggests that the uniqueness of the theory, thus the correctness of the theory, is, at least in principle, questionable. A debate concerning the gauge uncertainty of the theory occurred in the last several decades and the debate was " finally " ended up with the concept of the preferential gauge[3-5], which implies a vanishing vector potential A(t) whenever the electromagnetic perturbing field becomes zero. With the introduction of the concept, the gauge difficulty of the Dirac theory is regarded by many of the community as being resolved. However, as we believe and this paper will argue, the issue is far from closed. Many related questions, of which some are quite essential and fundamental , can be raised. Firstly, the preferential gauge does not seem to be a basic concept in the practical realm: Except imposing an additional constraint upon the perturbation theory, it has not found any way into other physical theories. Secondly, the relationship between the preferential gauge and the perturbation theory is kind of peculiar: The preferential gauge is assumed vital for the theory to hold, but one is not able to build it into the derivation of the theory. (In standard textbooks, the perturbation theory is derived from the Schrödinger equation without recourse to a special gauge.) Thirdly, one notices that invoking a special gauge to save a theory is never a good exercise whereas effort associated with pursuing a gauge-invariant theory usually boosts up physics development. (At this point, it is quite rel