General proof of gauge independence of the sum of the dia- and paramagnetic magnetic moments

Full text: In his seminal 1932 book 'The Theory of Electric and Magnetic Susceptibilities' J.H. Van Vleck claimed that it could be shown that the sum of the orbital paramagnetic and diamagnetic susceptibilities was independent of the origin of the vector potential used to describe the magnetic field. In classical mechanics the force on a charged particle, such as an electron in a conductor, is best described by its coupling to the electric and magnetic fields. In contrast, in quantum mechanics charged particles are coupled to the electric and magnetic potentials. It is the space and time derivatives of these potentials that give the fields. Consequently many different potentials can give rise to the same fields. This is called gauge freedom and the arbitrariness of the origin of the magnetic vector potential is one example of it. Van Vleck never published a complete proof of the result he claimed. In the meantime some partial proofs were constructed but they were all restricted by being limited to perturbation theory and to magnetic fields that were uniform in space and constant in time. In this talk a proof will be presented that overcomes all these restrictions. It applies to the non-linear magnetic response and not just the perturbative part and is valid for magnetic fields that are non-uniform in space and time. The proof uses a set of potentials called the multipolar gauge, which although used in relativistic electrodynamics, has found hitherto little use in condensed matter physics. In it the potentials are expressed in terms of the instantaneous but non-local values of the electromagnetic fields