The variable transformation of the potentials is called a gauge transformation. To obtain a unique solution, choose the gauge—that is, put constraints on Ψ that make the solution unique. Another way of expressing this additional condition is to put a constraint on ∇ · A. A vector field is uniquely defined up to a constant if both ∇ · A and ∇ × A are given. This is called Helmholtz’s theorem.

One particular gauge is the Coulomb gauge given by the constraint: ∇ ⋅ A = 0.