Simultaneity, Causality, and Spectral Representations

Recently Zangari and Censor discussed the non-uniqueness of the spatiotemporal world-view, and proposed a representative alternative based on the Fourier transform as a mathematical model. It was argued that this so called spectral representation, by virtue of the invertibility of the Fourier transform, is fully equivalent to our conventional spatiotemporal world-view, although in the two systems the information is ordered in a radically different manner. Criticism of the new conception can be traced back to the fundamental principles of simultaneity and causality, whose role in the spectral domain has not been sufficiently demonstrated. These questions are carefully investigated in the present study. Simple but concise examples are used to verbally and graphically clarify the mathematics involved in integral transforms, like the Fourier transform under consideration. The transition from the spatiotemporal domain to the spectral domain entails not only a different patterning of data points. What is involved here is that every point in one domain is affecting all points in the other domain, and to follow what happens to simultaneity and causality under such circumstances is not a trivial feat. Even for the general reader, the discussion based on the simple examples should suffice to critically follow the arguments as they unfold. For completeness, the general mathematical formulations are given too. In order to follow the footprints of the spatiotemporal simultaneity and causality concepts into the spectral domain, a special strategy is implemented here: Certain spatiotemporal situations are stated, and then their outcome in the spectral domain is examined. For example, it is shown that if a causal sequence of events is flipped over in time, thus reversing the order of cause and effect, in the spectral domain the