The use of complex fields in electromagnetism Virtually all treatments of electromagnetic wave propagation, and in particular of monochromatic plane waves, use basic complex analysis to simplify calculations. I am comfortable with these manipulations, and with the basic math of linear algebra in terms of understanding $\Bbb{R}^3$ and $\Bbb{C}^3$ as abstract vector spaces over $\Bbb{R}$ and $\Bbb{C}$ respectively. Usually the manipulations in the complex domain are justified by appealing to the linearity of Maxwell's equations so that we can just "keep track" of the real part of the complex vector in question. This is where I don't quite follow. It seems that textbooks implicitly use some map (which I will refer to as $Re$) from $\Bbb{C}^3$ to $\Bbb{R}^3$ in order to "get back to the real field" such that, for $(c_1,c_2,c_3) \in \Bbb{C}^3$,
$$Re(c_1,c_2,c_3)=(Re(c_1),Re(c_2),Re(c_3))\in \Bbb{R}^3.$$ Is there any resource which you are aware of which is very careful about this correspondence, and which makes clear at every step of the way why a given operation on $\Bbb{C}^3$ does not break this correspondence effected by $Re$? I suppose that ultimately this question is about phasors more generally, but I was hoping for a resource specifically for electromagnetism since here we have complex vectors, as opposed to just complex numbers in, say, AC circuit theory.
 A: Suppose we have some equation involving complex variables, such as $a = b + c$ or $a = bc$. You simply want to know when the equation remains true if we replace all the complex variables with their real parts. (Of course, here we are defining the real part of a complex vector to be a vector whose entries are the real parts of the entries of the complex vector, the real part of a complex vector field to be the real vector field whose value at each point is the real part of the complex vector field, and so on. In all cases we extend the definition of the phrase "the real part" naturally.)
The general rule is that it works whenever we only perform linear operations, which means multiplication by scalars and adding complex variables together, which follows from high school algebra. Thus, $a = b + c$ implies $\text{Re}(a) = \text{Re}(b) + \text{Re}(c)$, as well as $\text{Re}(\alpha a) = \text{Re}(\alpha b) + \text{Re}(\alpha c)$ for any complex number $\alpha$, while $a = bc$ does not imply $\text{Re}(a) = \text{Re}(b) \, \text{Re}(c)$ in general.
Is it really so simple? Don't the textbooks do all kinds of operations, like time derivatives, divergences, curls, gradients, and Laplacians as well? If you recall the definition of a derivative,
$$f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$$
you'll see that a derivative is merely a particular example of subtracting two complex variables and then dividing by a scalar. That's true for all kinds of derivatives, so you're allowed to take any kind of derivative of both sides while preserving the desired property. There's really not much to it, which is why textbooks don't keep careful track of it.
Of course, the fact that you can't multiply variables together means you need to be careful when dealing with quantities that are quadratic in the field, such as the power, energy, or momentum.
A: I agree with you that I've never really seen a rigorous discussion of this procedure of "promoting" real-valued waves to complex-valued ones, and I've always slightly shared your concern that some subtle issues don't slip in here involving multiplying complex numbers together - especially when we deal with things like impedances and complex permittivities. Nor have I ever seen a discussion about whether these complex physical quantities depend on your choice of exactly how to promote the real functions to complex ones.
I think that the procedure of adding a complex part to a real value function is at least uniquely defined, via the concept of the analytic representation. Given an arbitrary (sufficiently nice) real-valued function, there is an natural corresponding purely imaginary function to add to it to make it "nicer", which is ($i$ times) its Hilbert transform. This gives the analytic representation of the original function, which is guaranteed to have no negative-frequency components. So at least there's no ambiguity as to which pure imaginary function to add (although maybe it usually doesn't really matter, since you drop it at the end).
I think that the key to doing this rigorously is to mentally divide all of your formally complex quantities into two sets:

*

*the quantities that get made real at the end of the day, like promoted E&M fields. These are the "vectors" (in the linear algebra sense, not the geometric sense) that you're taking complex linear combinations of.

*The quantities that don't get made real, but that multiply functions that do (like impedances, complex permittivities, etc.). These are the complex scalars in this linear-algebra conceptualization.

And as knzhou pointed out, you need to be very careful whenever you multiply together functions that will be made real. As long as you don't do that, everything always works out. Multiplying two such functions together breaks the linear algebra structure.
I agree with knzhou that nothing conceptually new comes up with vector-valued functions, and the subtleties are all already there with scalar functions.
