Symmetric Sampling and the Discrete Laws of Physics (3): The Mathematics of Discrete Physics

The Nyquist-Shannon Sampling Theorem was refuted in part-1 of this series due to its mathematical flaws, including its circular proof. The theorem implicitly assumes the input signals to be already known deterministically, which makes the signals fundamentally not measurable. In part-2, the problems with the early sampling ideas are explained in verbal detail, while minimizing on the mathematics. This part-3 focuses primarily on the mathematical side of the measurement problem. This problem affects not only sampling, but all Laws of Physics including: Classical Mechanics; Quantum Mechanics; and General Relativity; since these are all based on determinism. Resolving this problem, requires the Laws of Physics to be rephrased in discrete form. The logical information flow in Physics must start with measurements, from which to derive discrete operators to replace the deterministic operators of Classical Physics (including Quantum Mechanics). Symmetric-sampling provides the rules for conducting discrete measurements on a physical system. From a large set of noisy physical measurements, we can derive the best-fit discrete operator and its main eigenvectors which best cover the physical experiment. Such discrete operators, can predict future experimental outcomes. New results can be appended to earlier datasets to further improve the discrete operator, taking advantage of the increased size of the “learning-set”. This “measurement-first” approach unifies Physics under a small set of postulates. The hypothetical “pre-collapse” QM wavefunction Ψ(r,t) in the Schrödinger equation gives way to a five-dimensional “post-collapse” mutual-current-density J(x,y;x',y',t) wavefunction (MCD), a successor to Zernike’s 5D mutual-intensity function: I(x,y;x',y',t). The Schrödinger equation itself will be replaced by a new partial differential equation operating on the MCD.