Magnetic fields gauge dependence

Here (r - Rc) (r - Rq) is the dot product times a unit matrix (i.e. (r — Rg) (r — Rg)I) and (r - RG)(r — Rg) is a 3x3 matrix containing the products of the x,y,z components, analogous to the quadrupole moment, eq. (10.4). Note that both the L and P operators are gauge dependent. When field-independent basis functions are used the first-order property, the HF magnetic dipole moment, is given as the expectation value over the unperturbed wave funetion (for a singlet state) eqs. (10.18)/(10.23). 

A more recent implementation, which completely eliminates the gauge dependence, is to make the basis functions explicitly dependent on the magnetic field by inclusion of a complex phase factor refening to the position of the basis function (usually the nucleus). 

The Aharonov-Bohm effect is self-inconsistent in U(l) electrodynamics because [44] the effect depends on the interaction of a vector potential A with an electron, but the magnetic field defined by = V x A is zero at the point of interaction [44]. This argument can always be used in U(l) electrodynamics to counter the view that the classical potential A is physical, and adherents of the received view can always assert in U(l) electrodynamics that the potential must be unphysical by gauge freedom. If, however, the Aharonov-Bohm effect is seen as an effect of 0(3) electrodynamics, or of SU(2) electrodynamics [44], it is easily demonstrated that the effect is due to the physical inhomogeneous term appearing in Eq. (25). This argument is developed further in Section VI. 

This means that a magnetic field is always a quantity that depends on motion, or a current. If there is no magnetic field, there is no electric current, that is, no motion of charge. The use of Cartesian indices for the internal 0(3) gauge space therefore corresponds to an electrostatic situation where there is no movement of charge. The use of complex circular indices corresponds to electrodynamics. 

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