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.
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.
The derivation of the Heisenberg Uncertainty Principle (HUP) from the Uncertainty Theorem of Fourier Transform theory demonstrates that the HUP arises from the dependency of momentum on a wave number that exists at the quantum level. It also establishes that the HUP is purely a relationship between the effective widths of Fourier transform pairs of variables (i.e. conjugate variables). We note that the HUP is not a quantum mechanical measurement principle per se. We introduce the Quantum Mechanical equivalent of the Nyquist-Shannon Sampling Theorem of Fourier Transform theory, and show that it is a better principle to describe the measurement limitations of Quantum Mechanics. We show that Brillouin zones in Solid State Physics are a manifestation of the Nyquist-Shannon Sampling Theorem at the quantum level. By comparison with other fields where Fourier Transform theory is used, we propose that we need todiscern between measurement limitations and inherent limitations when interpreting the impact of the HUP on the nature of the quantum level. We further propose that while measurement limitations result in our perception of indeterminism at the quantum level, there is no evidence that there are any inherent limitations at the quantum level, based on the Nyquist-Shannon Sampling Theorem
The derivation of the Heisenberg Uncertainty Principle (HUP) from the Uncertainty Theorem of Fourier Transform theory demonstrates that the HUP arises from the dependency of momentum on wave number that exists at the quantum level. It also establishes that the HUP is purely a relationship between the effective widths of Fourier transform pairs of variables (i.e. conjugate variables). We note that the HUP is not a quantum mechanical measurement principle per se. We introduce the Quantum Mechanical equivalent of the Nyquist-Shannon Sampling Theorem of Fourier Transform theory, and show that it is a better principle to describe the measurement limitations of Quantum Mechanics. We show that Brillouin zones in Solid State physics are a manifestation of the Nyquist-Shannon Sampling Theorem at the quantum level. By comparison with other fields where Fourier Transform theory is used, we propose that we need to discern between measurement limitations and inherent limitations when interpreting the impact of the HUP on the nature of the quantum level. We further propose that while measurement limitations result in our perception of indeterminism at the quantum level, there is no evidence that there are any inherent limitations at the quantum level, based on the Nyquist-Shannon Sampling Theorem.
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.
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.
Abstract Fourier transform is an integral part of harmonic analysis, information theory and signal processing with wide applications. The transformation between the Fourier pair is governed by an uncertainty relation, most generally captured by the Bandwidth theorem, . The usual approaches to the uncertainty principle employ sophisticated methods which lack physical interpretation of the origin and description of the uncertainty for real systems. In this paper, we have proposed a physically inspired model of the Fourier transform with an aim to describe its underlying dynamics and how it leads to the fundamental uncertainty principle. A signal is modeled as helical spring in 3D space, which is wrapped around a circle with controllable wrapping frequency. This process is formulated to reproduce the Fourier transform. The model is used to describe the origin of uncertainty, and to formulate a new generator-based approach to the Bandwidth theorem.
Since the publication of Heisenberg’s paper1 on the “anschaulichen Inhalt” of quantum mechanics, discussions of the fundamental limitations on the accuracy of physical measurements have been much in the foreground. According to Heisenberg, the quantum mechanics implies that it is impossible to measure simultaneous values of a coordinate and its conjugate momentum with unlimited precision. Instead, if Δp be the estimated error or uncertainty in a momentum and Δq that in the associated coordinate one must have the inequality,
ΔpΔq>2πh
(1)
This inequality has come to be known quite generally as Heisenberg’s uncertainty relation.