In the wave theory, each sample (for example to toss a coin for once) is considered an intrinsic rotation (or spin). Mathematically this can be explained by the real and imaginary images mentioned on the 30th of July.
In the article, I explain that each wave, phi, represents one particular outcome (for example head or tail, refer to the post on the 27th of July), so the final wave represents the probability linked to tossing a coin for N times (N-> inf) and consists of two waves, one for the head (phi_h) and another for tail (Phi_t). Then the Fourier transform of the wave linked to the head (phi_h) refers to all samples constituting phi_h and it expresses the probability of getting a head in that sample. As mentioned previously this probability is a fuzzy value and is expressed as a sine wave.
Each basic element in the Fourier transform consists of real and imaginary parts. So each of these elements can be expressed as a rotation around a complex circle of radius r ( r= S_(A^2 +B^2); where A and B are real and complex coefficients; however the radius is not important in this discussion and the radius is normalized in this discussion). The sine wave alone produces values in the range [0, 1]; r is normalized. But sine and cosine combined always deliver the value 1 (the normalized radius).
As mentioned before each of these basic elements represents one sample. So in this construction, the real part represents the possibility of one particular outcome (for example head or tail) and the imaginary part represents all other outcomes as one entity. That’s why the value of sine and cosine combined is always 1.
To clarify what it means it’s better to explain it with a social example. Consider the example of memory or peoples’ viewpoint about one social issue, for example, the US election. Harris supporters group all Trump supporters into one entity. People might support Trump for different reasons. For example, lowering taxes or immigration or racial issues, etc. but for a Harris supporter, all of them constitute one entity. All differences between those Trump supporters will be diminished and similarities will be emphasized so that all Trump supporters form on coherent entity. So a Harris supporter is on one side and the other entity (all Trump supporters combined) is on the other side.
That’s how wave probability is constructed. For example, one ball should be picked from a bag containing 3 balls of different colours (R, G, B). The probability wave psi consists of three waves: psi= A_R*phi_R + A_G*phi*G + A_B*phi_B (refer to 27 July 2024). Fourier transform of A_R*phi_R represents all samples of picking a ball for the particular outcome that the red ball is picked. Then the real part of this basic element of the Fourier transform represents the probability of picking the red ball and the imaginary part represents all other outcomes (blue and green combined).