# Electromagnetism - role of the phase between fundamental invariants?

The two Lorentz invariants are $$\mathbf{E}^2-\mathbf{B}^2$$ and $$2\mathbf{E}\cdot \mathbf{B}$$.

It is common in the literature to construct a complex vector:

$$\mathbf{F}=\mathbf{E}+i\mathbf{B}$$

whose square produces the Lorentz invariants:

$$\mathbf{F}^2=\mathbf{E}^2-\mathbf{B}^2+2i\mathbf{E}\cdot\mathbf{B}$$

Since this is a complex number, one can certainly understand it using the usual Euler equation:

$$\mathbf{F}=R \exp (i \theta)$$

where

$$R=\sqrt{(\mathbf{E}^2-\mathbf{B}^2)^2+4(\mathbf{E}\cdot\mathbf{B})^2}\\ \theta=\arctan(2\mathbf{E}\cdot\mathbf{B}/ (\mathbf{E}^2-\mathbf{B}^2))$$

Is there a place for $$\mathbf{F}$$, expressed using a phase $$\theta$$ and a length $$R$$ in the formulation of electromagnetism? Can one formulate electromagnetism using a phase/magnitude combo?

The norm squared of $$\textbf{F}$$ is the energy density of the electromagnetic field, meaning that the hamiltonian formulation would use it instead of $$\text{Re}(\textbf{F}^2)$$. I suspect that the phase is not useful because the $$E. B$$ invariant is a total derivative.