Feynman on the use of imaginary numbers
Enter Feynman: imaginary exponents
Whenever I want to get to deeper and more natural insights, I turn to Feynman. Feynman has a characteristic treatment of the imaginary exponents in his books (Lectures: chapter 22). Feynman carefully re-derives what he calls "the most remarkable formula in mathematics"
(22.9) e(it)=cos(t) + i sin(t).
The treatment is thorough and bears the aura of 'naturalness' but really involves 2 hand tricks that the untrained eye will miss (I missed it on first read).
a/ the derivation starts with a clumsy differential definition of the exponential, after 22.6 he just parachutes the linear development (10^t=1+2.3025t). b/ between 22.5 and 22.6 there is a misdirection where he simply writes that the conjugate of e(it) is e(-it). Both statements are simply parachuted in his case.
Just a phase
Just as characteristically Feynman shines in a few spots. For example he develops his expose by using 10 as the power basis for imaginary numbers, not the usual e. This underscores the arbitrary nature of the base. We have become so accustomed to using e that we never really question why it is there, why e, why not pi?. He shows that whatever you use as a power base, say P, P^(it)x^P(-it) = P^(0)= 1. So that the norm of a complex exponent is always 1.
So the complex exponents are all on the complex circle of norm one no matter what the base. Using P or using e, is just a rescaling of the t factor since if e=P^s then it = it(s/s) and P^(it)=P^(its/s)=(P^(s))(it/s) = e^(it/s) by simple algebra of exponents. The exponent can be identified with a phase on the unit complex circle and when we are in e base then the phase length is 2pi, yielding an immediate mapping between the phase and with the arc-length of a circle of radius 1.
The purely algebraic treatment of the properties of imaginary exponentials is followed by the chapter on resonance which is really the application of said numbers to differential equations. Feynman clearly says one should always take the real part of the equations and that the imaginary part is non sensical. He insists on the fact that this is possible only with linear equations so we are never multiplying imaginary numbers and making them real, thereby mixing them (23-1).
The intellectual honesty of Feynman is remarkable, no other textbook or teacher I have ever had goes to so much length to establish the validity of the basic constructs. In the case of linear differential equations, the solving can be done by inspection by replacing the differentiation by a iw factor. The formula for resonance just pops out. The use of imaginary number finds a natural justification as a powerful tool to solve basic differential equations.
More gripes re: QM
Feynman is rigorous up to that level. In this sense imaginary numbers are a mathematical trick to deal with linear differential equations. Yet in Quantum Mechanics (QM), which he himself contributed so much to via Quantum Electro-Dynamics (QED), one takes the square of amplitude mixing real and imaginary in order to find probabilities and non linear equations are the cat's meow in solid state physics anyways, where we will have powers of the imaginary numbers, again mixing real and imaginary components. As a matter of fact interference is modeled by this mixing. The use of imaginary numbers reverts to axiomatic status in QM (hypothesis), but epistemologically, the fact that it describes nature so precisely points to the fact that these imaginary numbers are a representation for something 'real', what that 'something' is (a phase, a twistor, whatever) is never really explained in most text book. Practitioners seldom question their practices.
Thermodynamics and grand ensemble statistics
More importantly and far from textbook land, one of the open frontiers of QM is why the path integral formalism looks so much like that of thermodynamics (average over large ensembles) but with (ixt) replaced by 1/T, where t is time and T is temperature. One treatment is dynamic (time) the other static and comprises a classical temperature. I asked that very question once to a Nobel worthy professor, he smiled and said "just replace it by 1/T and you are done" I pointed out it wasn't physical, that the dimensions didn't even match. When I repeated the same question 6 month and claimed I could just replace it by 1/T he would say "aaah, but you can't do that, you need to explain why there is a "i" there, that is the mystery"... it made me smile...
This is the reason I pay so much attention to the presence and justifications for the presence of i in these equations. I have no doubt they should be there, after all the formalism works. But if QM is indeed a statistical ensemble (non-deterministic) of things existing at the planck level, then something brings about the 'i' behavior, things that twist in a funky way, things that have a phase. I remind myself that planck constructs exist at 10^-35m length and that the standard model operates at 10^-15m, there are 20 orders of magnitude in between. In 3D that is a volume with 60 orders of magnitude small planck volume. In other words, an electron is hardly a particle but seems to contain 10^60 things in it. There are more planck things in a electron, than electrons on earth, it would be surprising if the electron was not a statistical beast. The standing of i in our statistical ensembles is then epistemologically important.