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.
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.
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.
When we are presented with apparently bizarre properties of microsystems governed by quantum mechanics (as in our discussion of the double-slit experiment in the previous chapter), we naturally ask whether it might be possible to find another theory, or at least another story to go with the equations, that would make more sense to us. We might seek a theory and a view of fundamental physical processes that would accord more with our classically based, everyday common sense. This typically takes the form of the question of the possibility of the existence of a more detailed description of atomic processes than that afforded by the wave function (that is, by the Copenhagen interpretation of the previous chapter). In this chapter we discuss some of the central interpretive problems of the Copenhagen version of quantum mechanics. Then, in the next two chapters, we look at the severe, general restrictions that are placed on any such completion of quantum mechanics and at one particular extension of quantum mechanics. First, though, we begin here with the struggle that ensued over the status of causality in quantum theory. THE COMPLETENESS OF QUANTUM MECHANICS Although the Heisenberg uncertainty principle (Section 20.4) and the lack of absolute predictive power are an inherent feature of quantum mechanics, there have been attempts to preserve an in principle completely deterministic structure for physics. Even classically, in a macroscopic sample of gas there are so many molecules that it would be hopeless, as a practical matter, to predict the future locations of all these molecules from Newton's laws of motion.
DESCRIPTION This work presents a complete, deterministic framework for quantum mechanics derived entirely from a single, testable postulate: the density of quantum space. The Core Problem For over a century, quantum mechanics has been interpreted as a theory of fundamental randomness, wave-particle duality, instantaneous jumps, and irreducible uncertainty. The standard Copenhagen interpretation treats the wave function as a probability amplitude, measurement as a "collapse" of that probability, and the Heisenberg uncertainty principle as a fundamental limit of nature. This work demonstrates that all these "mysteries" are not properties of nature—they are artifacts of our limited perception. We observe quantum phenomena through slow, macroscopic instruments that cannot resolve the actual continuous dynamics occurring in a space that is 6.54 trillion times denser than what we perceive as "empty space." The Fundamental Postulate The foundation of this work is a single, geometrically derived constant: ρ = (number of Planck lengths in 1 mm) / (number of millimeters in 1 light-year) = 6.54 × 10¹² Physical Meaning: One millimeter of physical space contains 6.54 trillion times more quantum spatial units than one light-year contains millimeters. What we perceive as "empty space" is actually a dense, discrete, information-rich medium. The entire observable universe (93 billion light-years in diameter) maps to only 0.014 light-years in quantum space—well within our Solar System. Consequence: The density inversion explains why quantum phenomena appear strange: we are observing a dense, fast, discrete reality through slow, coarse, macroscopic instruments. The apparent "weirdness" is not in nature—it is in our perception. The Comprehensive Law From the density postulate and classical physics (Newton, Coulomb, Maxwell, Einstein), a single comprehensive law is derived that contains all known physics as limit cases: E(t,v,n) = (m₀c²/ρ) · ‖n‖ · √(1 + (γv/c)²) · sin(2πt/T) This formula contains: Einstein's E = mc² as the limit ρ → 1, ‖n‖ → 1, sin → 1, v ≪ c Newtonian mechanics as the limit v ≪ c Special relativity as the limit ρ → 1, ‖n‖ → 1, sin → 1 Quantum mechanics as the discrete geometry n ∈ ℤ³ and resonance sin(2πt/T) Each term has a precise physical meaning: the rest mass energy (m₀c²) is scaled by the quantum space density (1/ρ), modified by the position relative to the absolute zero point Z₀ (‖n‖), adjusted for relativistic motion (√(1 + (γv/c)²)), and modulated by temporal resonance (sin(2πt/T)). Resolution of All Quantum Paradoxes Wave-Particle Duality = Optical Illusion The electron is always a localized particle with a definite trajectory. The "wave" is the charge density distribution—a standing wave in the dense quantum medium. From the standing wave condition 2πr = nλ, the Poynting vector in the far field is exactly zero (S = 0), meaning the electron does not radiate. The apparent "wave behavior" is an optical illusion caused by the density ρ smearing the point-like trajectory in our low-resolution projection. Wave Function = Energy Density, Not Probability The squared modulus of the wave function |Ψ|² is the energy density fraction, not a probability density. The particle itself follows a deterministic trajectory y(t) = v₀t - ½g₀t². Probability appears only because we cannot resolve the trajectory (due to t_total = 118 as ≪ Δt_instr) and we do not know the initial phase φ of the oscillation. Schrödinger Equation = Effective Approximation The Schrödinger equation is an effective, averaged description of how the energy density evolves. The fundamental law is classical motion in the dense quantum medium. The Schrödinger equation emerges when we average over the fast oscillation sin(2πt/T), ignore the density force F_ρ, and treat trajectories statistically. Quantization = Resonance, Not Arbitrary Jumps Energy levels are the resonant harmonics of the quantum space medium. The standing wave condition 2πr = nλ gives E_n = -13.6 eV/n²—exactly the same physics as a vibrating string. Only frequencies that satisfy the standing wave condition are stable. The transition between levels is not an instantaneous "jump" but a continuous trajectory y(t) = v₀t - ½g₀t² with total duration t_total = 118 as. The apparent "instantaneous jump" is an artifact of our slow instruments. Heisenberg Uncertainty = Collision, Not Fundamental Limit The uncertainty product depends on the measurement geometry: Δx·Δp = h·tan(θ/2), where θ is the scattering angle of the measurement photon. As θ → 0 (forward scattering with perfect detection), the product approaches zero. There is no fundamental lower bound. The uncertainty is a practical limitation of invasive measurement—we disturb the particle's trajectory when we hit it with a photon. Quantum Randomness = Ignorance Residual "quantum randomness" is simply unmodeled environmental force noise. The apparent randomness arises from three sources: temporal resolution (t_total = 118 as ≪ Δt_instr—we cannot resolve the trajectory), initial phase ignorance (we do not know the phase φ of sin(2πt/T)), and measurement perturbation (we do not know the exact collision parameters). If we could measure the outgoing photon completely and know the initial phase, the outcome would be fully deterministic. Randomness is in our ignorance, not in nature. Five Testable Predictions The theory makes five clear, quantitative predictions that distinguish it from standard quantum mechanics: Prediction 1 - Electron Transit Time: The electron transit time during the n=1 → n=2 transition in hydrogen is 118 ± 5 attoseconds. Standard quantum mechanics predicts zero duration (instantaneous jump). Testable with attosecond pump-probe spectroscopy. Prediction 2 - Heisenberg Product Can Approach Zero: With better detectors, the uncertainty product Δx·Δp can approach zero, not ħ/2. Standard quantum mechanics predicts Δx·Δp ≥ ħ/2 always. Testable with weak measurement experiments. Prediction 3 - Energy Oscillation at 116 kHz: The energy of gold-197 oscillates at 116 kHz with amplitude 1.252 × 10⁻²⁶ J. Standard quantum mechanics predicts constant energy for a given state. Testable with precision spectroscopy. Prediction 4 - Bell Inequality Violations Decrease with Measurement Strength: As measurement impulse strength decreases, Bell inequality violations decrease. Standard quantum mechanics predicts violations are independent of measurement strength. Testable with weak measurement entanglement experiments. Prediction 5 - Quantum Randomness is Environmental Noise: Residual "quantum randomness" equals unmodeled environmental force noise. Full force accounting restores classical determinism. Standard quantum mechanics predicts randomness is fundamental and irreducible. Testable with complete force reconstruction in closed quantum systems. The Absolute Zero Point Z₀ This work introduces Z₀ = (0,0,0) as a universal, absolute reference point in quantum space. Every point in space has a discrete address: n = (n_r, n_θ, n_φ), ‖n‖ = √(n_r² + n_θ² + n_φ²) Z₀ is physically grounded in the cosmic microwave background (CMB) rest frame, vacuum energy (the lowest possible energy state), and the geometric center of the quantum space lattice. This eliminates the relativity of reference frames—providing an absolute coordinate system that standard physics lacks. Mathematical Consistency The work provides rigorous derivations of: The density ρ from first principles (Planck length, electron mass, fine structure constant) The quantum coordinate norm ‖n‖ from the discrete metric of quantum space The oscillation period T from the de Broglie relation The coupling constant ε from fundamental constants The regularized metric eliminating all singularities The thermodynamic properties of quantum space (entropy, temperature, heat capacity) The covariant Einstein equation in quantum space The connection to quantum anomalies and topological invariants The Lamb shift from quantum space density oscillation Dark energy from quantum space density variation Comparison with Other Approaches String theory requires approximately 10⁵⁰⁰ free parameters and offers no testable predictions with current technology, while introducing new physics such as strings, extra dimensions, and supersymmetry. Loop quantum gravity requires approximately 10 parameters and also offers no testable predictions, introducing spin networks and loop quantization. Both approaches have difficulty connecting to the Standard Model. In contrast, this theory has only one parameter (ρ, which is derived from fundamental constants rather than being free), offers five testable predictions with current or near-future technology, requires no new physics beyond standard physics applied in a dense medium, and connects directly to the Standard Model through the energy-momentum tensor. The Core Message Quantum mechanics is not magic. It is not a new kind of physics. It is the same old physics—Newton, Maxwell, Coulomb—operating in a space that is 6.54 trillion times denser than we perceive. Three things to remember: Space is dense, not empty. What we call "empty space" is actually a dense, discrete medium. We are the ones who are sparse—not the universe. The particle has a trajectory. Every particle follows a precise, continuous path. We don't see it because our instruments are too slow. The path exists—we just can't resolve it. Energy breathes. Energy is not a static number. It oscillates in time, like a heartbeat. The universe is alive with resonance. Conclusion This work demonstrates that quantum mechanics is not a theory of fundamental randomness and irreducible uncertainty. It is the low-resolution, static-space approximation of a deeper, deterministic, dynamic reality. The deeper reality is quantum space—dense, discrete, resonant, and absolute. The final statement of this
Interpreting quantum mechanics is a hard problem basically because it means explaining why and how the mathematics exploited to formulate wave-particle duality are related to observations or reality in classical physics (due to Newtonian mechanics, special relativity, gravity and entropy). Here, we first prove the limitation of the Heisenberg uncertainty principle by invoking the wave-particle duality and apply it to revise and/or extend the Copenhagen postulates. In particular, even though the said uncertainty exists due to non-commuting operators, but these operators do not commute simply because one of the operators is well-defined in classical physics (particle-like), while the other is wave-like (well-defined in quantum mechanics), or the eigenvalue is not at all defined in classical physics. We also construct a new quantum mechanical postulate (Postulate 9) to deduce why the forbidden gap between discrete energy levels should exist in atoms, molecules and condensed matter phases. In fact, we tackle all the problems arising from the Copenhagen interpretation without violating established experiments and without proposing ideas that violate physical reality.
The accuracies of measurements of almost all fundamental physical constants have increased by factors of about 10 000 during the past 60 years. Although some of the improvements are due to greater care, most are due to new techniques based on quantum mechanics. Although the Heisenberg Uncertainty Principle often limits measurement accuracies, in many cases the validity of quantum mechanics makes possible the vastly improved measurement accuracies. Seven quantum features that have a profound influence on the science of measurements are: (1) Existence of discrete quantum states of energy. (2) Energy conservation in transitions between two states. (3) Electromagnetic radiation of frequency v is quantized with energy hv per quantum. (4) The identity principle. (5) The Heisenberg Uncertainty Principle. (6) Addition of probability amplitudes (not probabilities). (7) Wave and coherent phase phenomena. Of these seven quantum features, only the Heisenberg Uncertainty Principle limits the accuracy of measurements, and its affect is often negligibly small. The other six features make possible much more accurate measurements of quantum systems than with almost all classical systems. These effects are discussed and illustrated.