In textbook derivations of the Lorentz transformations for electric and magnetic fields from relativistic kinematics and “rest-frame” E-M laws, either one of two assumptions is used: that charge is conserved in all frames of reference, or that charge is quantized. I see no intuitive reason to favour the former, and most texts I’ve flipped through have used the latter. As it turns out, the latter implies the former, because if charge is quantized, then the total charge of any system must be an integer multiple of the smallest unit of charge, and integers are scalar invariants under Lorentz transformations, so the total charge must also be a Lorentz invariant. In other words, quantization of charge + Lorentz invariance => conservation of charge. Using conservation of charge, you get , which, together with Lorentz contraction of space together and , implies that is the product of a scalar invariant and the four-velocity, and hence is a four-vector itself. That allows one to go on to prove that is a four-vector, blah blah.

What bothered me about this was that it seems counterintuitive that special relativistic electrodynamics is somehow dependent on the quantization of charge. Other than this mathematical requirement for justifying how the Lorentz transformations are a ‘derivation’ from ‘normal’ electrodynamics + relativistic kinematics, there doesn’t seem to be anything in electrodynamics itself that suggests it would work better with discrete rather than continuous charges. If anything, those damn differential equations cry out for lovely continuous flowing ribbons of charge rather than discrete blobs. A more plausible dependency scenario to me would be that of electrodynamics being directly dependent on the Lorentz invariance of charge. In which case, why don’t they just derive the Lorentz transformations using that assumption, rather than go through the more onerous* and (in my eyes) somewhat physically artificial route of “quantization, therefore charge conservation”? Could it be that like me, said textbook authors think that taking charge as a Lorentz invariant quantity as a fundamental assumption is not a sufficiently justifed move?

Another problem with that could, I suppose be dismissed as a “mere” historical problem. The quantization of electric charge was demonstrated only in 1913, eight years after Einstein had put forward special relativity. Presumably, special relativity had been integrated into electrodynamics before that (or, to be precise, people had worked out the details of how electromagnetism was inherently a relativistic theory). I suppose they used the (now judged to be unsatisfactory) Lorentz invariance of total charge as an assumption for that? And this new reason they give in modern textbooks is one of those things massaged in for pedagogical purposes, even though it was not the original reasoning through which the Lorentz transformations were derived?

I know nothing about particle theory, so I expect that someone is going to come along anytime now and say something like “you dumb shit, the relationship between charge quantization and special relativity is a simple consequence of electroweak unification”.

*Wangsness points out in his text that the inference from quantization to charge conservation depends not just on integers being Lorentz invariants, but also on the basic unit of charge being a Lorentz invariant.

**Update**: I am told that the textbook derivations are not, in fact, derivations, but justifications on the side for something we think should be true for other reasons. The reply M. O. gave was somewhat unclear, but he agreed when I paraphrased it as follows: we must make an assumption of some sort (in this case, charge invariance), in order to get into the logical circle of showing that the relevant quantities are four-vectors. My vague idea is that we can’t even empirically prove that charge is Lorentz invariant unless we already have the formulae for the Lorentz transformations of E and/or B. But we need the invariance of charge to ‘derive’ those transformations. So we have to assume something, and invariance of charge, I suppose, is as good a choice as any, and certainly better (more intuitively immediate) than assuming the specific four-vectors or the E and B tensor. (But, I maintain, quantization of charge is not — why do so many textbook authors use that? M. O. suggested that perhaps someone famous had used it.)

I wish there were “physics for philosophers” textbooks along the lines of “physics for poets” books. But nevermind the probable lack of market demand, there is a serious lack of people who are qualified to write those books. Most physicists certainly aren’t.