On the construction of gauge independent transition amplitudes

Abstract

The conventional time dependent transition amplitude between a state ψ A ( t), in the presence of an external electromagnetic field described by the four-potential A μ, and an eigenfunction of the field-free hamiltonian H 0 has been shown to depend on the gauge used to describe the fields. This is a serious flaw, and may underlie the continuing difficulties with the interpretation of Lamb shift measurements recently discussed by Low and Lundeen et al. Aharonov and Au have demonstrated the connection between gauge changes and the corresponding changes in the representation of the field-free hamiltonian. We complete the explicit proof of the existence of gauge independent transition amplitudes by an explicit construction of the field-free hamiltonian as a functional of the four potential used to describe the fields. This construction clarifies the procedure suggested by Aharonov and Au, assures the gauge independence of the transition amplitudes, and avoids the introduction of a preferred gauge advocated by Yang and collaborators.