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Haven't read it, but it's obviously wrong. To expand: any time you read "...magical complex numbers" just mentally replace "complex" with "negative" and then examine how odd the original text now reads. There's nothing fundamentally different between the concept of negative numbers, and complex numbers.
I have always felt like "imaginary" was a poorly-chosen name. After all, I can plot, in two dimensions, a function that has "imaginary" roots, and yet I can see those roots in the graph. There is no discontinuity.
The intuition I developed with them while studying physics was that, unlike "real" numbers which interact by stacking, complex numbers interact by stacking and rotating. This is bizarre to think about with single numbers in a 1D world, but we don't live in a 1D world. In higher dimensions they rotate and sheer rather than just scale.
And indeed, QM (at least as it was thought to me) would fall apart without complex numbers. Whether a physical theory can be consistent without them is an interesting question, but not because a physical theory with them creates some kind of metaphysical paradox.
“Complex numbers” are rather poorly named. They are more naturally understood as simply a vector which has a magnitude, can be rotated and scaled. As geometric objects they are much more intuitive. The subject geometry algebra takes a great approach of generalizing this idea and augmenting basic linear algebra to unify complex numbers and beyond (quaternions, ect) with geometric objects and operations. This also fits…
One of the first things we were taught in physics was "don't think that imaginary or complex numbers have physical significance. just do the math." 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.
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…
I have always felt like "imaginary" was a poorly-chosen name. After all, I can plot, in two dimensions, a function that has "imaginary" roots, and yet I can see those roots in the graph. There is no discontinuity.
The name "imaginary" was due to Descartes and it absolutely was intended as a pejorative, even though they're necessary to algebraically close the reals. Some ancient Greeks, IIRC, were similarly hostile to negative numbers. Of course the "real" numbers have never been controversial despite the whole concept being a lot weirder (and uncomputable), probably because their informal aspects just so happen to line up with…
This is your regularly scheduled reminder that complex numbers have (real) matrix representations, and what matters in any model is the properties it has not its identity as an object.
Did someone claim otherwise?
>Marco made a curious face, so Toni posed the question: “Can standard quantum theory work without imaginary numbers?”