Quantum theory based on real numbers can be experimentally falsified
Note that it is trivial to split the real and imaginary parts into two separate real-numbers and write quantum mechanics that way with only real numbers. Instead of i you get a 90 degree rotation matrix, instead of individual numbers you get a 2-element vector, etc.Lacking "numbers" with the right arithmetic properties for other things in quantum mechanics, we indeed use matrices and vectors for other stuff all the time. Dirac figured he needed some 4×4 matrices in order to be able to take the square root of some operator at some point, which is how his equation predicted the existence of antimatter (because it implied the wavefunction had to be a vector with more components - some of the other components turned out to be antimatter).But complex numbers happen to have the right properties that at least we can do without matrices and vectors for the lowest-level quantities in quantum mechanics: the state amplitudes or the values of wavefunctions of spinless, non-relativistic particles. This article is saying that you can't do away with these properties at the lowest level of quantum mechanics, whether or not you actually use complex numbers to represent them.