Jackson's Electrodynamics (3rd ed, Sec. 6.11) argues that one can introduce magnetic charges and currents into the Maxwell equations without changing observable physics:
For this, he introduces the duality transformation (6.151) $$ \begin{align} \mathbf{E} & = \mathbf{E}' \cos\xi + Z_0\mathbf{H}' \sin\xi, & \mathbf{D} & = \mathbf{D}' \cos\xi + Z_0\mathbf{B}' \sin\xi, \\ Z_0\mathbf{H}' & = - \mathbf{E}'\sin\xi + Z_0 \mathbf{H}'\cos\xi & Z_0\mathbf{B}' & = - \mathbf{D}'\sin\xi + Z_0 \mathbf{B}'\cos\xi & \end{align} $$ as a rotation $\xi$ in ($\mathbf{E}, \mathbf{H}$) space. If electric and magnetic charges and currents are rotated in the same way, Eq. (6.152), $$ \begin{align} Z_0 \rho_e & = Z_0 \rho_e' \cos\xi + \rho_m' \sin\xi, & Z_0 \mathbf{J}_e & = Z_0 \mathbf{J}_e' \cos\xi + \mathbf{J}_m' \sin\xi, \\ \rho_m & = - Z_0 \rho_e'\sin\xi + \rho_m'\cos\xi & \mathbf{J}_m& = - Z_0 \mathbf{J}_m' \sin\xi + \mathbf{J}_m'\cos\xi & \end{align} $$ (the impedance $Z_0 = \sqrt{\mu_0/\epsilon_0}$ ensures consistency of prefactors), the extended Maxwell equations [Eq. (6.150)] $$ \begin{align} \mathbf{\nabla} \cdot \mathbf{D} & = \rho_e, & \mathbf{\nabla} \times \mathbf{H} & = \frac{\partial \mathbf{D}}{\partial t} + \mathbf{J}_e \\ \mathbf{\nabla} \cdot \mathbf{B} & = \rho_m, & -\mathbf{\nabla} \times \mathbf{E} & = \frac{\partial \mathbf{B}}{\partial t} + \mathbf{J}_m \end{align} $$ are invariant under the rotations (6.151) and (6.152) (if performed simultaneously), i.e., they describe the same observable physics. So we can always find an angle $\xi$ such that we have only electric charges. Jackson argues: "The invariance of the equations of electrodynamics under duality transformations shows that it is a matter of convention to speak of a particle possessing an electric charge, but not magnetic charge."
Later, starting from the field-strength tensor (11.137), $$F^{\alpha\beta} = \begin{pmatrix} 0 & -E_x & - E_y & -E_z\\ E_x & 0 & - B_z & B_y \\ E_y & B_z & 0 & - B_x \\ E_z & - B_y & B_x & 0 \end{pmatrix} $$ he introduces its dual (11.140), $$\mathcal{F}^{\alpha\beta} = \frac{1}{2}\epsilon^{\alpha\beta\gamma\delta} F_{\gamma\delta} = \begin{pmatrix} 0 & -B_x & -B_y & -B_z\\ B_x & 0 & E_z & -E_y \\ B_y & -E_z & 0 & E_x \\ B_z & E_y & -E_x & 0 \end{pmatrix} $$ which allows him to bring the homogeneous Maxwell equations into the covariant form (11.142) $$\partial_\alpha \mathcal{F}^{\alpha\beta} = 0.$$
My question is about Jackson's remark that the dual field-strength tensor (11.140) was a "special case of the duality transformation (6.151)." (page 556) What does that mean?
Of course, as Jackson says (p556): "The elements of the dual tensor $\mathcal{F}^{\alpha\beta}$ are obtained from $F^{\alpha\beta}$ by putting $\mathbf{E} \rightarrow \mathbf{B}$ and $\mathbf{B} \rightarrow -\mathbf{E}$ in (11.137)." And putting $\xi = \pi/2$ in (6.151) does the same thing. But my reading of the duality transformation (6.151) is that we should do the same rotation in (6.152). Then, the duality transformation expresses an invariance of observable physics [Eq. (6.150)] under $\xi$ rotations almost like a gauge transformation. All this is really a rather different story from what happens in (11.140). Then, using the same name for this seems, at best, misleading.
Or am I trying to dig too deep? Or is there something yet deeper that I am missing?
I know Sommerfeld's old geometric interpretation of the dual field-strength tensor. And I am also aware of question 37577. But it seems to me that these discuss very different questions. I am concerned about the physical interpretation of the duality transformation.
