[AntonioLao]

Maxwell’s 1873 treatise on electricity and magnetism definitely avoided the use of imaginary exponentials for describing the space-time variations of the electric and magnetic fields. For this reason he invented real hyperbolic functions. The hyperbolic sine and cosine are given as sinh(x)=(exp(x)-exp(-x))/2 and cosh(x)=(exp(x)+exp(-x))/2. It appears that the argument x is most likely rational. In contrast the imaginary exponentials are given by sin(q)=(exp(iq)-exp(-iq))/2i and cos(q)=(exp(iq)+exp(-iq))/2. It appears that the argument q is most likely a multiple of irrational number, p.

Modern conscientious authors on classical electromagnetism cautiously pointed out these differences and strongly emphasized the use of only the real parts in every application. However, the physical descriptions of the imaginary parts started the old and new quantum revolution: old – Bohr, Sommerfeld, and Schrödinger; new – Heisenberg, Dirac, Born, and Jordan. Starting with Schrödinger complex imaginary wave equation, even though its solutions are imaginary wavefunctions, quantum physicists were capable of extracting eigenvalues of real energy quanta. Furthermore, the square absolute of the conjugate product becomes the probability of a particular location for the manifestation of elementary particles. Nonetheless, those fields of infinite degrees of freedom remain imaginary with brief moment’s appearances of virtual particles in high energy experiments. True dynamic physical balance between real and imaginary exponential pairs can be restored only by the use of square symmetric Hadamard matrices as space-time quanta of energy squares.