I’ve been working on the Fourier transform. The idea was that the conventional probability should be transformed into a new domain (wave probability). To clarify, in the proj geom, the flat z=1 would refer to the observation of the outcome of some experiment and the slopes refer to N (the number of times the experiment is performed). So as N → inf, the slope tends to pi/2. Then after the transformation, those slopes would be expressed as n in sigma e^(2*pi*n*i).
After some wrestling, I realized this couldn’t be developed further. So instead sine waves in Fourier series were used. However, the idea was similar to the initial approach. In the formula of the Fourier series a new parameter, t, shows up in coefficients. This is an auxiliary parameter in Integral [from 0 to 2*pi] ( f(t)e^(i*n*t) dt). Since the integral is zero around those circles, all coefficients will be zero except when n=0 (and since f(t) is a complex function, at each iteration a new n is sent to zero by multiplying it by f(t)). This is not the main point and so I won’t explain it in detail here. The point I’m trying to make is that t in this formula is not a real parameter. As I said it’s more of an auxiliary parameter aimed to cancel out all coefficients except one.
However, I wanted a new parameter that would be similar to the change of domain from time to frequency in signal processing. Here, in the Fourier series, the domain is not changed but a new parameter, t, is included. Then I saw this video ( I can’t view many websites with my laptop; this is the video on YouTube: “But what is a Fourier series? From heat flow to drawing with circles” ). In this video, t turns into time. And so, over time, A*e^(i*2*pi*t) would generate rotation around the circle. There are inf many circles in a Fourier series, and so any complicated shape can be produced (as explained in the video).
I will explain that turning t to time, in fact, adds two new parameters: time (or memory) and observation. And these two parameters distinguish the conventional probability from the wave probability. Both parameters can be easily added by the new auxiliary parameter, t, in the Fourier series.
The shapes that have been drawn in the video are contained in the complex plane. Dividing the complex plane into the real and imaginary parts, those shapes are decomposed into two movements along the real and the imaginary axes. Those movements refer to variable probabilities of some event for some experiment as N gets larger (the imaginary part was discussed before; see 2 Aug 2024). So the drawings (on the complex plain) refer to the memories of past incidents and the real parts of those drawings refer to the time-dependent probabilities.
The other issue is that as N → inf, the circles shrink more and more and eventually at inf, it turns into a point. This suggests one thing: length contraction can be used here. This will add special relativity (SR) and, so observation, to the structure. In a different article, I tried to develop SR based on proj geom where length contraction would be produced by the contraction of the unit length (i.e. z=1 would turn into z=v’/c and then v’/c would shrink according to SR). By incorporating that construction and using it in the wave probability, observation will also be included.
SR is based on Lorentz invariance. This symmetry refers to the flat spacetime, Minkowski. In the wave probability, however, the “spacetime” is not necessarily flat.