qA is called the potential momentum, and is part of the canonical momentum. The line integral of A over a closed loop, Γ, is equal to the magnetic flux, Φ, through a surface, S, that it encloses: Therefore, the units of A are also equivalent to Weber per metre. The above equation is useful in the flux quantization of superconducting loops. Although the magnetic field B is a pseudovector (also called axial vector), the vector potential A is a polar vector. This means that if the right-hand rule for cross products were replaced with a left-hand rule, but without changing any other equations or definitions, then B would switch signs, but A would not change. This is an example of a general theorem: The curl of a polar vector is a pseudovector, and vice versa. Gauge choices The above definition does not define the magnetic vector potential uniquely because, by definition, we can arbitrarily add curl-free components to the magnetic potential without changing the observed magnetic field. Thus, there is a degree of freedom available when choosing A. This condition is known as gauge invariance. Maxwell's equations in terms of vector potential Using the above definition of the potentials and applying it to the other two Maxwell's equations (the ones that are not automatically satisfied) results in a complicated differential equation that can be simplified using the Lorenz gauge where A is chosen to satisfy: Using the Lorenz gauge, Maxwell's equations can be written compactly in terms of the magnetic vector potential A and the electric scalar potential ϕ: In other gauges, the equations are different. A different notation to write these same equations (using four-vectors) is shown below. Calculation of potentials from source distributions The solutions of Maxwell's equations in the Lorenz gauge (see Feynman and Jackson) with the boundary condition that both potentials go to zero sufficiently fast as they approach infinity are called the retarded potentials, which are the magnetic vector potential and the electric scalar potential due to a current distribution of current density , charge density , and volume Ω, within which ρ and J are non-zero at least sometimes and some places): where the fields at position vector r and time t are calculated from sources at distant position r′ at an earlier time t′. The location r′ is a source point in the charge or current distribution (also the integration variable, within volume