# Physical implication of $\textbf{E}\rightarrow\textbf{B},~~\textbf{B}\rightarrow -\textbf{E}$ invariance of the Maxwell's equations

An interesting observation to consider about the Maxwell's equation is that in absence of the sources, the equations are symmetric under the interchange $$\textbf{E}\rightarrow\textbf{B},~~\textbf{B}\rightarrow -\textbf{E}\tag{1}$$ apart from some numerical factors of $\mu_0$ and $\epsilon_0$. In terms of the electromagnetic field strength tensor $F^{\mu\nu}$ and its dual $\tilde{F}^{\mu\nu}$ the transformation (1) is equivalent to $$F^{\mu\nu}\to \tilde{F}^{\mu\nu},~~\tilde{F}^{\mu\nu}\to -F^{\mu\nu}.\tag{2}$$ It also turns out that if both electric and magnetic sources$^1$, $j^\mu$ and $k^\mu$, are included the duality can still be preserved if Eq.(2) is supplemented with $$j^\mu\to k^\mu,~~k^\mu\to -j^\mu.\tag{3}$$

Does this invariance have any physical significance/consequence in classical electrodynamics itself, in particular, if magnetic sources were there?

$^1$ Inclusion of magnetic sources modify the homogeneous Maxwell's equations to $\partial_\mu\tilde{F}^{\mu\nu}=k^\nu\neq 0$. This implies that the four-potential $A^\mu$ cannot be defined.

One approach is to consider the Noether charge associated to this symmetry. In the absence of sources, it turns out that the Noether current associate to the electric-magnetic duality symmetry takes the form \begin{align} J^\mu=(h,\vec{s}), \qquad h=E\cdot C-B\cdot A, \quad \vec s=E\times A+B\times C \end{align} where $$F=dA, \star F=dC$$. The physical meaning of $$h$$ is the helicity density, while $$\vec s$$ is the spin angular momentum density. The conservation of the current implies that the time variation of total helicity is equal to the flux of spin angular momentum. Another important result is that the spin and orbital part of the angular momentum of are separately conserved (see Mandel and Wolf section 10.6). The interaction of matter and light can transfer spin angular momentum to orbital angular momentum and vice versa. This has been observed in the lab.
Another observation is that if we define $\mathbf{F}:=\mathbf{E}+ic\mathbf{B}$ the duality achieves $\mathbf{F}\mapsto -i\mathbf{F}$, suggesting complex exponential waves will play a natural role in describing electromagnetism.