This is one of the complicated steps in physics (I have a PhD in physics (and forgot everything since)).

First you have some math that goes along discovering physics. You split vectors, multiply mass by something and it's fine.

Then you have math that helps you with physics. Simple differentials equations that uncover while laws of nature (cooling down speed for instance). This is the golden time for many because you're at this sweet spot where it is exciting but not too hard.

Then comes the travel in desert of abstract things you have no idea about and winner why someone hates you by shoving Abel groups down your throat for no reason.

Finally comes that sight of relief when you can binding do some maths to end up with a real life solution without too much thinking because you have solid tools.

The last part is a bit morally complicated because you have the feeling that you are cheating. Renormalization, I am looking at you.

But then I forgot everything because I left academia and my memories may be faulty.

There really isn't anything weird or suspect about renormalization (except the name, perhaps). Read A. Zee's book on Quantum Field Theory.

> There really isn't anything weird or suspect about renormalization

This is the first time I've heard anyone say that. To me, renormalization is extremely weird, if anything because it's so unrigorous and ad-hoc that I find it hard to believe it even works. Sure, it does the job it's supposed to, and I understand how it does that (for the most part anyway), but that doesn't make it any less weird.

That's always been my take too.

Like, ok, we get testable answers and they match experiments but also this is _so_ hacky and I can't shake the feeling that one day someone will come along and show that there's some reason why these bad assumptions work out fine. You know, like how "to find the Schwarzschild radius for a black hole of known mass, calculate the radius at which the escape velocity is equal to the speed of light" gives the correct answer even though the theory implied by this method is naive and wrong.

Read "Quantum Field Theory in a Nutshell."

You are certainly right, like I said it was a long time ago.

But doing splits and advanced acrobatics to get rid of infinities always felt like a hack (Feynman felt the same do at least I am not alone :))

"shut up and calculate", also known as the Feynman approach to quantum mechanics.

It's not imprecise. It reproduces experimental results from theory, so it's in fact the most precise approach in existence.

> don't think that imaginary or complex numbers have physical significance.

Yeah I'd say that's the most common approach but I think it's misguided. Complex numbers aren't any less physical than any other number. It just turns out that for historical reasons, it makes sense to define observable quantities using self-adjoint operators (which have real eigenvalues, and the latter are used to measure things like energy). But that doesn't mean the rest is not physical. Just because we can't take a picture of an object in the dark, it doesn't mean the object isn't there when the lights are off.

I'd say that complex numbers are the only ones that have physical significance. They are what's actually happens in the real world until we disturb it with experiment.

"And as imprecise as that sounds, many of the formulas that take complex numbers as inputs multiply them with other complex numbers in such a way that the imaginary side cancels out. "

The only thing imprecise about this is "many". Really any formula for an observable of any kind (including probabilities) has to come out to a real number.

Not really! Often you get a complex solution and both the real and imaginary components are valid.

A complex solution can be valid but you never measure a complex number. I'd argue that in situations like using complex numbers to simulate time varying electrical activity the ontological status of the imaginary part of the solution is uncontroversial: the complex numbers in that situation have no ontological status at all and what is present is charges. In that case we're simply using the complex numbers as a convenient notation for a variable and its conjugate. In quantum mechanics one is more easily led to wonder about whether the ontological status of the complex numbers in that theory really can be settled so easily.

>A complex solution can be valid but you never measure a complex number.

I see where you are coming from, and I'm asking this as a genuine question rather than to argue, but what's stopping me from measuring the length and the mass of an object and saying the "length-mass" of it is length + i(mass)? I suppose it isn't useful since complex numbers are not ordered, but aren't "numbers" arbitrary? In measure theory, measures are defined as outputting positive real numbers and +infinity because those happen to align with our intuition about how measures work, but as far as I know, maths(and physics here I guess) does not care about the representation of my quantity which I'm measuring, but it only cares about it's properties.

Well, for one thing, for such quantity to make physical sense, both the real part and the imaginary part should be of the same dimension, e.g. "length." Also, the result of a measurement is supposed to come from (be an eigenvalue of) an observable - an operator, and, on the one hand, I think I'd have a hard time conjuring one up; on the other hand, the eigenvalues are "supposed to be" real anyway! So, no, that doesn't work.

Nothing stops you from representing the value that way, but when you go to a meter or lay a yardstick against something, you are measuring a real number (or, at the very least, a number which has no complex character to it). "This many ticks on a ruler" or "this many clicks on a clock."

Same holds for negative numbers. There are no negative quantities in physics, negative numbers as quantities only appear if you order your equations wrong. (And one can argue against the other appearances of negative numbers and minus signs.)

But, how would one handle positive and negative charge?

You just do the bookkeeping, taking into account that if charges are opposite then they attract, etc.