I have been playing around with fourier transforms on Wolfram Alpha. I noticed that F{cos kt, w} = sqrt(pi/2)*diracDelta(k+/-w) where +/- means either + or - and w is the frequency transform variable. I discovered to my amazement that the fourier transform F{cos(kt+phi),w} = sqrt(pi/2)*exp(+/-i *phi)*diracDelta(k+/-w). That is the imaginary terms pop out because the wave undergoes a phase shift phi. I am now wondering about all the physicists who are jumping up and down about the ontology of imaginary numbers in QM are just simply insisting on a preferred notation to represent a phase shift when the phase shift would go away simply by choosing a suitable galilean or lorentz reference frame.

Whilst no one seems to be able to assign a physical meaning to psi, most are quick to point out the meaning of psi^2 as the probability density function. However it has no units so its fairly useless to apply dimensional analysis to it.

So why do we insist psi is complex when we can represent it with a real valued phase shift?

Whilst no one seems to be able to assign a physical meaning to psi, most are quick to point out the meaning of psi^2 as the probability density function. However it has no units so its fairly useless to apply dimensional analysis to it.

So why do we insist psi is complex when we can represent it with a real valued phase shift?

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