The classical sampling theorem was refuted in part-1 of this series due to its circular proof: the proof is always true, whether the input signal is under-sampled or not. The symmetric-sampling theory of part-1, is here presented, while minimizing on the mathematics. The classical sampling theorem [Shannon 1949] is based on sampling deterministic signals and produces a finite number of measurements at specific sampling points. At each sampling point, however, the measurement must reach infinite precision. That creates a measurement problem in sampling theory, which is mirrored by an equally fundamental measurement problem in Quantum Mechanics and in all of Physics. Deterministic Classical Physics does not include measurements; it only includes predicting infinitely precise results postulated by “Laws of Physics”. Those postulated results serve only to confirm theoretical predictions. Physics must rather be based on collecting information, that is, on learning from the real physical experiments. Classical Physics is thus constructed with “inverted logic”. Solving this measurement problem calls for a reformulation of the Laws of Physics in discrete form. The mathematical details of such a Discrete Physics are covered in part-3 of this series. Discrete measurements (the act of information collection) on a physical system can never lead to deterministic results since measuring anything to infinite precision requires an infinite amount of energy. Representing a deterministic function in a digital computer to infinite precision, is equally impossible. Symmetric-sampling allows one to assess the physical world by measurements and calculations, without ever deranging into singularities or infinities: not at the small-scale (high-frequency) limit, and not at the large-scale (low-frequency) limit. Symmetric-sampling, combined with a coherent information harvesting strategy, resolves the measurement problem in Physics, including Classical Sampling; Quantum Mechanics; General Relativity, etc. Jumping ahead to the overall conclusions of this series: the 1920s Classical Quantum Mechanics is not only “incomplete”, but it is also superfluous. That conclusion will allow us to leapfrog straight into the realm of a discrete version of Quantum Field Theory (QFT) in future parts of this series.
The classical sampling theorem was refuted in part-1 of this series due to its circular proof: the proof is always true, whether the input signal is under-sampled or not. The symmetric-sampling theory of part-1, is here presented, while minimizing on the mathematics. The classical sampling theorem [Shannon 1949] is based on sampling deterministic signals and produces a finite number of measurements at specific sampling points. At each sampling point, however, the measurement must reach infinite precision. That creates a measurement problem in sampling theory, which is mirrored by an equally fundamental measurement problem in Quantum Mechanics and in all of Physics. Deterministic Classical Physics does not include measurements; it only includes predicting infinitely precise results postulated by “Laws of Physics”. Those postulated results serve only to confirm theoretical predictions. Physics must rather be based on collecting information, that is, on learning from the real physical experiments. Classical Physics is thus constructed with “inverted logic”. Solving this measurement problem calls for a reformulation of the Laws of Physics in discrete form. The mathematical details of such a Discrete Physics are covered in part-3 of this series. Discrete measurements (the act of information collection) on a physical system can never lead to deterministic results since measuring anything to infinite precision requires an infinite amount of energy. Representing a deterministic function in a digital computer to infinite precision, is equally impossible. Symmetric-sampling allows one to assess the physical world by measurements and calculations, without ever deranging into singularities or infinities: not at the small-scale (high-frequency) limit, and not at the large-scale (low-frequency) limit. Symmetric-sampling, combined with a coherent information harvesting strategy, resolves the measurement problem in Physics, including Classical Sampling; Quantum Mechanics; General Relativity, etc. Jumping ahead to the overall conclusions of this series: the 1920s Classical Quantum Mechanics is not only “incomplete”, but it is also superfluous. That conclusion will allow us to leapfrog straight into the realm of a discrete version of Quantum Field Theory (QFT) in future parts of this series.
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.
For creating a measurable Physics, we must first revisit its theoretical foundations. Deterministic Physics cannot be measured and therefore cannot be proven experimentally since an infinite amount of energy is required to confirm a Law of Physics to infinite precision. This “measurement problem” existed ever since Physics – centuries ago – slowly separated from its deterministic theological roots. Given the experimental successes of deterministic Newtonian Mechanics, deterministic Physics nevertheless became the law-of-the-land. The introduction of Quantum Mechanics in the 1920s threw a spanner in the works and re-emphasised the importance of that underlying measurement problem. Rather than removing Classical Physics' deterministic inheritance, however, the science community reacted by introducing deterministic postulates to make Quantum Mechanics “fit” the prevailing classical framework. Unphysical postulates like the existence of a wavefunction, assumed to be a “complete” description of a physical system, and its square was assumed to be “measurable”, created a series of problems at the core of Physics, which remained unresolved for more than a century. This (part-4) paper promotes a discrete uncertainty principle as the foundation of a Discrete Physics. Symmetric-sampling incorporates this discrete uncertainty principle from the onset and provides comprehensive rules for handling measurements and discrete computer representations of signals and functions. This principle applies to all of Physics, including classical Quantum Mechanics and General Relativity. The red line of this paper is to complete the original “X-ray-microscope” model experiment proposed by Heisenberg in 1927. Retracing Heisenberg’s original “Gedanken” experiment helps us pinpoint the deterministic flaws at the core of Classical Physics and Quantum Mechanics, while assessing a fresh comprehensive Discrete Physics.
Donoho and Stark have shown that a precise deterministic recovery of missing information contained in a time interval shorter than the time-frequency uncertainty limit is possible. We analyze this signal recovery mechanism from a physics point of view and show that the well-known Shannon-Nyquist sampling theorem, which is fundamental in signal processing, also uses essentially the same mechanism. The uncertainty relation in the context of information theory, which is based on Fourier analysis, provides a criterion to distinguish Shannon-Nyquist sampling from compressed sensing. A new signal recovery formula, which is analogous to Donoho-Stark formula, is given using the idea of Shannon-Nyquist sampling; in this formulation, the smearing of information below the uncertainty limit as well as the recovery of information with specified bandwidth take place. We also discuss the recovery of states from the domain below the uncertainty limit of coordinate and momentum in quantum mechanics and show that in principle the state recovery works by assuming ideal measurement procedures. The recovery of the lost information in the sub-uncertainty domain means that the loss of information in such a small domain is not fatal, which is in accord with our common understanding of the uncertainty principle, although its precise recovery is something we are not used to in quantum mechanics. The uncertainty principle provides a universal sampling criterion covering both the classical Shannon-Nyquist sampling theorem and the quantum mechanical measurement.
In this section we introduce the framework of quantum mechanics as it pertains to the types of systems we will consider for quantum computing. Here we also introduce the notion of a quantum bit or ‘qubit’, which is a fundamental concept for quantum computing. At the beginning of the twentieth century, it was believed by most that the laws of Newton and Maxwell were the correct laws of physics. By the 1930s, however, it had become apparent that these classical theories faced serious problems in trying to account for the observed results of certain experiments. As a result, a new mathematical framework for physics called quantum mechanics was formulated, and new theories of physics called quantum physics were developed in this framework. Quantum physics includes the physical theories of quantum electrodynamics and quantum field theory, but we do not need to know these physical theories in order to learn about quantum information. Quantum information is the result of reformulating information theory in this quantum framework. We saw in Section 1.6 an example of a two-state quantum system: a photon that is constrained to follow one of two distinguishable paths. We identified the two distinguishable paths with the 2-dimensional basis vectors and then noted that a general ‘path state’ of the photon can be described by a complex vector with |α0|2 +|α1|2 = 1. This simple example captures the essence of the first postulate, which tells us how physical states are represented in quantum mechanics. Depending on the degree of freedom (i.e. the type of state) of the system being considered, H may be infinite-dimensional. For example, if the state refers to the position of a particle that is free to occupy any point in some region of space, the associated Hilbert space is usually taken to be a continuous (and thus infinite-dimensional) space. It is worth noting that in practice, with finite resources, we cannot distinguish a continuous state space from one with a discrete state space having a sufficiently small minimum spacing between adjacent locations. For describing realistic models of quantum computation, we will typically only be interested in degrees of freedom for which the state is described by a vector in a finite-dimensional (complex) Hilbert space.
AbstractStandard quantum mechanics is formulated using complex Hilbert spaces, real-valued probabilities, continuous one-parameter dynamics, and—where particles and fields have continuous degrees of freedom —function spaces constructed over continua such as R3 . In that precise mathematical sense, the real and complex continua are deeply embedded in the standard theory. It does not follow, however, that experiments have demonstrated the physical existence of an uncountable set of exact real numbers, still less that individual quantum systems physically “store” infinitely many digits. Every preparation, control operation, and measurement has finite precision, and many quantum predictions can be approximated arbitrarily closely using countable or finite descriptions.This distinction matters because the continuum is sometimes blamed for quantum theory's conceptual strangeness. That diagnosis is mostly unsupported. Superposition, interference, uncertainty, noncommutativity, entanglement, contextuality, Bell-inequality violations, and no-cloning already occur in finite-dimensional quantum systems and do not depend in any straightforward way on continuous position space or arbitrary irrational numbers. Restricting a Hilbert space to rational, algebraic, computable, or finitely specified states therefore does not automatically produce classical physics. Indeed, dense restricted subsets can approximate ordinary quantum predictions to arbitrary finite accuracy, although turning such a subset into a complete physical theory requires much more: closed dynamics, a composition rule, a measurement theory, robustness, and new empirical predictions.A particularly developed contemporary alternative is Tim Palmer's Invariant Set Theory and its recent formulation as Rational Quantum Mechanics (RaQM). RaQM is not accurately described as simply “quantum mechanics with irrational numbers banned.” It restricts the bases in which states are physically defined through number-theoretic rationality conditions, represents admissible states using finite bit strings, invokes a fractal invariant state space equipped with a non-Euclidean, p-adic-like metric, and violates Bell measurement independence at the level of exact counterfactual settings. Palmer's 2026 proposal further predicts a finite qubit-information capacity and an eventual saturation of quantum computational advantage. These are genuine, potentially falsifiable proposals, but they remain speculative and have not displaced standard quantum theory.The strongest conclusion is consequently asymmetric. There is a serious foundational question about whether the continuum is an ontological feature of nature or an extraordinarily efficient mathematical completion. Finite precision prevents existing experiments from proving a literal continuum. But there is presently little evidence that continuum mathematics causes quantum phenomena. Any discrete replacement must still account quantitatively for interference, contextuality, Bell correlations, entanglement, spectroscopy, atomic structure, and quantum computation. Discreteness can change quantum physics only when it is accompanied by additional physical postulates—and those postulates, rather than discreteness alone, must ultimately face experiment.
I. The arena of quantum mechanics and quantum field theory is the abstract, unobserved and unobservable, M-dimensional formal Hilbert space ≠ spacetime. II. The arena of observations—and, more generally, of all events (i.e., everything) in the real physical world—is the classical four-dimensional physical spacetime. III. The “Born rule” is the random process “magically” transforming I into II. Wavefunctions are superposed and entangled only in the abstract space I, never in spacetime II. Attempted formulations of quantum theory directly in real physical spacetime actually constitute examples of “locally real” theories, as defined by Clauser and Horne, and are therefore already empirically refuted by the numerous tests of Bell’s theorem in real, controlled experiments in laboratories here on Earth. Observed quantum entities (i.e., events) are never superposed or entangled as they: (1) exclusively “live” (manifest) in real physical spacetime and (2) are not described by entangled wavefunctions after “measurement” effectuated by III. When separated and treated correctly in this way, a number of fundamental problems and “paradoxes” of quantum theory vs. relativity (i.e., spacetime) simply vanish, such as the black hole information paradox, the infinite zero-point energy of quantum field theory and the quantization of general relativity.