Existence and Uniqueness of the Method of the 2+2 Split

Can an arbitrary vector potential be written in terms of four suitably chosen scalars $\psi^{T.E.},\psi^{T.M.},\psi$ and $\phi$ so as to satisfy Eqs.(10), (12), and (15)? If the answer is ``yes'' then any e.m. field can be expressed in terms of these scalars, and one can claim that these scalars give a complete and equivalent description of the e.m. field. It turns out that this is indeed the case. In fact, the description is also unique. Indeed, given the vector potential, Eq.(7), there exist four unique scalars which are determined by this vector potential so as to satisfy Eqs.(10), (12), and (15). The determining equations, obtained by taking suitable derivatives, are

$$\frac{\partial^2 \psi^{T.E.}}{\partial x^2}+ \frac{\partial^2 \psi^{T.E.}}{\partial y^2} = \partial_y A_x -\partial_x A_y$$ (19)

$$-\frac{\partial^2 \psi^{T.M.}}{\partial t^2}+ \frac{\partial^2 \psi^{T.M.}}{\partial z^2} = \partial_z A_t -\partial_t A_z$$ (20)

$$\frac{\partial^2 \psi}{\partial x^2}+ \frac{\partial^2 \psi}{\partial y^2} = \partial_x A_x +\partial_y A_y$$ (21)

$$-\frac{\partial^2 \phi}{\partial t^2}+ \frac{\partial^2 \phi}{\partial z^2} = -\partial_t A_t +\partial_z A_z$$ (22)

These equations guarantee the existence of the sought after scalar functions $\psi^{T.E.},\psi^{T.M.},\psi$ and $\phi$ . Their uniqueness follows from their boundary conditions in the Euclidean $(x,y)$-plane and their initial conditions in the Lorentzian $(t,z)$-plane. Consequently, Eqs.(19)-(22) together with Eqs.(10), (12), and (15) establish a one-to-one linear correspondence between the space of vector potentials and the space of four ordered scalars,

$$(\psi^{T.E.},\psi^{T.M.},\psi,\phi)\leftarrow \!\!\!\rightarrow (A_t,A_z,A_x,A_y)$$

Of the four scalars, three are gauge invariants, namely $\psi^{T.E.}$, $\psi^{T.M.}$, and the difference $\psi -\phi$, a result made obvious by inspecting Eqs.(19)-(22).