Concerning the Shape of the QTE Fourier Transform Wavenumber Spectrum, an Analysis of the Evolution of Our Ideas on the Matter, Starting with the Toth-Maatian Review Papers, 1990-93

Abstract

We start with the rectangular spectrum. What you get, from putting the spectral area = constant, is a relationship E(v). The question becomes, why is this relationship not reflected in the spectrum itself – instead, apparently, we just have a “flat top” spectrum, where 
the amplitude E does not vary with the wavenumber k(k_S, v). All the way back, right in the beginning, 1988, we consulted a book on Fourier Transforms and deduced a modification of the rectangular spectrum to a “dual triangular” spectrum. And that, without a lot of mathematical or physical justification, just “common sense”, was the form of our spectrum in the Toth-Maatian Review papers, 1990-93. It only became evident very recently that the spectrum has an electronic component and a positronic component, each contributes one half of the spectrum, the one fermion contributes a line of positive slope in the E-k profile, the other, a line of negative slope. You start out with the Toth-Maatian Review profile, dual triangular, see Figure 4 below. The electron and positron in phase, the photonic wavepacket you accelerate onto contributes both the fermion and the anti-fermionic ghost, in-phase, in the calculation. Then you displace the two spectral lines such that the two lines of wavenumber dependency “overlap”. This is now an out-of-phase association, (figures 5, 6 below), this time you are comparing the fermion with the anti-fermion, not the anti-fermionic ghost. And this time, the two wavenumber profiles, add together to get that constant spectral height that we started with! So comparing entities between two individual photons, the positronic one and the negative one, these are out-of-phase, in accordance with the Maxwellian wave, ψ = ψ₀
 × exp j(kx - ωt) = ψ = ψ₀ × [cos(kx - ωt) + jsin(kx - ωt)]. Astoundingly, the E(k) algebra is the same, whether we are considering fermion + anti-fermionic ghost within an individual photon, or considering the fermion versus anti-fermion content between two separate photons – not concerned with the ghosts, in this equivalent description.