Indeed they may be! However, that presents us with two very different possibilities: that imaginary numbers may be essential to our description of reality, or to reality itself. Epicycles were at one point essential to our description of reality; no longer.
It's neat how it produces wave functions that are familiar and useful. However, it strikes me that saying that the "are" orthogonal because it makes sense to graph them as such really is akin to saying that you can draw a picture of me (or you) and a unicorn at right angles. Then the picture of me/you and the picture of the unicorn are indeed at right angles, and they are equally real (as pictures). The difference is that one of them has a referent in the physical world, and the other doesn't; and the referents are not, therefore, equal. One of them is real -- indeed it is actual -- and the other is imaginary.
But the physicists and mathematicians are really saying something stronger than that, which is that 'the sense that it makes' to graph them this way implies that there is a rational relationship between the real and the impossible; and then, applying this equation to reality, that this relationship between the real and the impossible ends up giving rise to the actual. That's an extraordinary claim, which at least some of them really seem to believe.
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"Imaginary" is an unfortunate term. The square root of -1 is a bit of an abstraction, but so are the "real" numbers, and I've seen some videos by a fellow who is trying to resurrect the Greek approach of using ratios for everything (except construction). (Hey, why not? If it works, you learn something, and if it doesn't work, you learn something.) His approach is more sophisticated than anything recorded from antiquity.
Anyhow, to avoid invidious connotations, call them the posrad (real= positive radical) numbers and the negrad (imaginary) numbers. [You will not find these terms in the literature; I just made them up.]
As to "orthogonality": It does represent another dimension. To extend your example, you are a created thing and the unicorn is a legendary thing. I can define a set of created and legendary creatures, but I'd best not mix them up when I count--I need to count created and legendary separately. (#created, #legendary) = 2 dimensions that don't blend with each other. The count of things in my set is like an abstraction of 2 directions.
Similarly the posrad and negrad have to be treated separately.
Of course the complex plane isn't a simple vector space, since we have complex multiplication defined on it as well. So "orthogonality" isn't really quite appropriate, though the concept does help visualize things.
@ james "You will not find these terms in the literature, I just made them up." This is why I love you
Yes, the Greek system of ratios has some interesting features. You probably remember the discussion of Plato's Laws when it turns out that Plato wanted to build his model society around the number 5,040, which he says is especially convenient. This is because it is divisible by so many other numbers (some sixty!), so a ratio of x/12 or x/14 or x/15 or x/1008 will all be easily attained. I can definitely see that playing with the ancient math of ratios some more could be enlightening.
As to "orthogonality": It does represent another dimension.... The count of things in my set is like an abstraction of 2 directions.
Similarly the posrad and negrad have to be treated separately. Of course the complex plane isn't a simple vector space, since we have complex multiplication defined on it as well. So "orthogonality" isn't really quite appropriate, though the concept does help visualize things.
That's what caught my attention. They're abstracting to a dimension, which they then go on to talk about as productive of sine waves and so on. Those are then supposed to be not just an exercise in fun things to do with math, but predictive of actual physical reality -- that's what Schrodinger was really interested in, and why it's astonishing that he used imaginary numbers to get to a functional equation.
I understand it's highly simplified in presentation; the 'square'/orthogonal aspect really works best with i, and not so well for the square root of other negatives. Likewise "1" has some strange features that cease to apply with 2, 3, etc.
"To extend your example, you are a created thing and the unicorn is a legendary thing. I can define a set of created and legendary creatures, but I'd best not mix them up when I count--I need to count created and legendary separately. (#created, #legendary) = 2 dimensions that don't blend with each other."
Which is all well and good, but again: what we're doing with this is creating a predictive model for actuality. The legendary ends up having a necessary function in predicting the created, actual things.
Don't let the connotations of the word "imaginary" get a hold here.
The field of the complex plane is isomorphic to other fields, such as that of 2-d matrices spanned by
(1 0)
(0 1)
corresponding to "1" and
(0 -1)
(1 0)
corresponding to "i"
It would take a lot more ink to write out the model in this field, but it would work. Pairs of entries would correspond to physical quantities, with no complex numbers in sight--nothing "imaginary".
For a model representing physical reality to "work", aspects of the model should have physical significance--not because the map is the territory, but because it represents it. To say that the "imaginary part of the wave function" _is_ Y is merely sloppy writing.
Granted, in some models it's not easy to see the connection--as with gauge fields.
I've always wondered why, once we allow operations to be interpreted as valid in new, orthogonal, dimensions, why we can't allow the same concept to extend into other directions? What about the SqRt[i] such that
SqRt[i]*SqRt [i] = i
?
We could graph numbers with a factor of SrRt[i] along an axis perpendicular to both the real and imaginary axis. In math systems that swallow tesseracts without complaint, one might imagine still other axis still perpendicular to the existing ones, upon with other "roots" might be measured.
Soon one does arrive in territories like this one:
https://www.youtube.com/watch?v=BKorP55Aqvg
But intervals along a line --and the direction OF that line-- are hardly the only way to represent progress along a system. If a number represents a color ( red lines, drawn in transparent ink? ) or a shape or a speed then things can get interesting.
https://www.quantamagazine.org/behold-modular-forms-the-fifth-fundamental-operation-of-math-20230921/
Mathematically you are all way over my head, but as someone who works with spatial models regularly maybe I have a different take.
If the formula works in reality, than the "imaginary" numbers (coined by Decartes apparently, derogatorily) are in fact real- the question is how, or what the relationship is. Mathematically it's laid out as orthogonal (and then he describes yet a third axis in the video, e to the ix- I have no idea what that means). So the question for mathematicians becomes, I think, what those axes actually represent. Perhaps we should start by renaming the axes, as James suggests. "Imaginary" is clearly not correct, so Pos Rad and Neg Rad could work, or perhaps we replace "Imaginary" with "invisible" since there is some reason we disconnected that number system from reality. If I knew more about that reason why, I might have more insight into this problem.
This may all also be complete garbage from someone who admittedly is in way over his head, so forgive me if so.
“If the formula works in reality, than the "imaginary" numbers (coined by Decartes apparently, derogatorily) are in fact real…”
Well, no: epicycles worked to produce reliable navigation in reality, but they weren’t real. They were, well, an imagination designed to align an inaccurate system of physics to observations. It was perfectly functional, though, indeed better for a long time than heliocentric alternatives.
Imaginary numbers aren’t real either, but real is a term that is dangerously equivocal here. James is right to point out that “real numbers” aren’t real in the same way that you are; they’re a set of concepts that applies to all the counting numbers, their negatives, and their and their negatives’ divisions. But you’ll never meet a number in the street; they’re ideas.
So in a way, you can say that the imaginary numbers are ‘as real’ as the real numbers. But it’s still true that no real number can be multiplied by itself and yield -1. You can’t count to that number, and you can’t divide to it. They’re neither a whole number nor a ratio (unlike pi, which is irrational but a ratio).
If they’re fundamentally necessary to the best model of reality, then either the model is wrong— like epicycles— or the math that underpins reality isn’t anything like what we encounter in actuality. That implies that actuality, our reality, isn’t what is fundamental to reality. Either possibility is, well, possible.
FWIW, I suppose you could think of epicycles as analogous to a Taylor series but with circles instead of polynomials.
But think of something simpler. Have you ever met a -1 in the world, all by itself? Yet it is extremely useful for understanding the physical world, as well as more abstract things like cash flow.
I don't doubt that you can devise a formalism for quantum mechanics which does not involve complex numbers, substituting complicated rules that do the same thing. Physicists have a bias towards simplicity, though you wonder if that's still true if you look at some recent theory papers.
Agreed with all that "imaginary" is a poor term for the useful operations and sets of numbers in question. Just as poor as "real".
I won't argue either way but notions, arguments, and conceptions of the reality -- or mere utility -- of numbers start at least as far back as Socrates, the Pythagoreans, and the "irrational" square root of two. (I am of the ignorant opinion that Archimedes would not have argued for a "transcendental" value of Pi had not the "irrational" issue already been settled.)
Mathematicians apparently delight in confusing the rest of us. If you can spare 15 minutes to get an image in your head about "Normal Numbers" I think you'll be amused.
https://youtu.be/5TkIe60y2GI
"But think of something simpler. Have you ever met a -1 in the world, all by itself? Yet it is extremely useful for understanding the physical world, as well as more abstract things like cash flow."
Yes, it's really all numbers you don't meet: even the positive counting integers, like "one" and "two," are only abstracted from the things you actually meet. "Two oranges" is really just an orange, and another orange; the number is conceptual, but the oranges are actual.
What remains interesting is that i and the like are not like things that might apply to oranges. You can't add, subtract, multiply, or divide to get to these kinds of numbers. They are in a sense placeholders for ideas that ordinary mathematical operations can't produce. It really is another degree more abstract, because it's not applicable to anything in the world that we encounter as an actual thing. It's a way of, well, imagining what it might be like if there was a number that could do what the real numbers can't.
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