Multipolar gauge

In the Hamiltonian conventionally used for derivations of molecular magnetic properties, the applied fields are represented by electromagnetic vector and scalar potentials [1,20] and if desired, canonical transformations are invoked to change the magnetic gauge origin and/or to introduce electric and magnetic fields explicitly into the Hamiltonian, see e.g. refs. [1,20,21]. Here we take as our point of departure the multipolar Hamiltonian derived in ref. [22] without recourse to vector and scalar potentials. 

The identity of the Kramers-Heisenberg formula obtained in the two gauges is a celebrated example [22], It should be noted, however, that the popular multipolar Hamiltonian ... 

The particular merit of multipolar gauge is that is allows one to express the scalar and vector potentials directly in terms of the fields E and B, thus facilitating the identification of electric and magnetic multipoles for generally time-dependent fields. We will follow the three-vector derivation given by Bloch [68]. We will furthermore in this section make extensive use of the Einstein summation convention for coordinate indices. Consider a Taylor expansion of the scalar potential... 

In a different gauge, it is possible to construct the multipolar Hamiltonian which is obtained by applying a unitary transformation to the minimal coupling Hamiltonian [75-77,106]. In the multipolar Hamiltonian, it is the transverse electric field, and the magnetic field, B(r) (satisfying Maxwell s equation, V x Et = - f), that appear, rather than the vector potential. Now, the interaction is written as... 

This system in which the longitudinal and transverse motions of the emitter are separated to a good approximation, provides a convenient example for the consideration of the motional effects discussed above. As shown by Healy", the canonical transfoimiation from the minimal-coupling to the multipolar Hamiltonian has the same form for any convenient reference-point R (not necessarily the center-of-mass) relative to which the polarizations are defined. This arbitrariness, which amounts to a gauge... 

The dressed photon is theoretically described by assuming a multipolar quantum electro-dynamic Hamiltonian in Coulomb gauge and single-particle states in a finite... 

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