The Simple Harmonic motion is ubiquitous in nature at every scale from macroscopic bodies to the very microscopic elementary particles like electrons and photons. In the usual macroscopic scales motion of particles obey Newtonian and Relativistic dynamics which is modeled very accurately by linear Mathematics developed over the last few centuries. But at the quantum scales we inherently have to deal with probabilities and the wave particle dual nature becomes apparent at microscopic scales. To describe the energy of photons in Quantum Field Theory we have to use the notion of frequency which is a Simple Harmonic Motion Notion which in turn requires Sines and Cosines. In this Paper I would like to formulate a Mathematical Model by generalizing some symmetry principles from Simple Harmonic motion and show how this Model reduces to linear Mathematics at high frequencies or longer time scales and how this model maybe useful in general for quantum physicists in particular for dealing with infinities.
The Simple Harmonic motion is ubiquitous in nature at every scale from macroscopic bodies to the very microscopic elementary particles like electrons and photons. In the usual macroscopic scales motion of particles obey Newtonian and Relativistic dynamics which is modeled very accurately by linear Mathematics developed over the last few centuries. But at the quantum scales we inherently have to deal with probabilities and the wave particle dual nature becomes apparent at microscopic scales. To describe the energy of photons in Quantum Field Theory we have to use the notion of frequency which is a Simple Harmonic Motion Notion which in turn requires Sines and Cosines. In this Paper I would like to formulate a Mathematical Model by generalizing some symmetry principles from Simple Harmonic motion and show how this Model reduces to linear Mathematics at high frequencies or longer time scales and how this model maybe useful in general for quantum physicists in particular for dealing with infinities.
Quantum mechanics is introduced here as a fundamental theory that must treat particles equally often as waves. Here, classical waves are introduced in some detail, with discussions of the important concepts of waves: medium, amplitude, frequency, wavelength, period, harmonics, interference, standing waves, etc. Waves can be oscillatory waves with a repeating pattern, or pulse waves, which are localized in space. All of the elements of waves are important in understanding how the Schrödinger equation describes particles. Classical standing waves must have wave lengths that fit between boundaries giving them discrete allowed frequencies, a harbinger of quantized energies in quantum mechanics. Complicated waves can be analyzed in terms of their component frequencies, the fundamental idea of Fourier analysis. This discussion shows that a classical wave has a limit in how accurately one can specify simultaneously the wave’s position and frequency, in analogy with Heisenberg uncertainty principle of quantum mechanics.
A solitary electron in free space is not a fixed particle at rest in a given inertial frame of reference. It is always oscillating in a simple harmonic motion (SHM) in its own electromagnetic inertia field, even at zero kelvin temperature. A Newtonian particle exhibits uniform motion in a straight line. A quantum particle like the electron exhibits uniform motion along a sine wave path. This explains the physical concept of the wave-particle nature of the electron. As the mass of a quantum particle increases, the wavelength of the particle decreases as per French physicist Louis de Broglie. In the limiting case the sine wave becomes a straight line. Also, a hypothetical electron without an electric charge is like the bob of a simple pendulum without a string. In an electron-spin qubit, the electron can oscillate either clockwise or anti-clockwise and cannot oscillate both clockwise and anti-clockwise at the same time. It cannot be both spin-up and spin-down at the same time.
The underlying rules for a natural system describing cellular automata are simple, but produce highly complex behavior. A mathematical basis for the spectra of discrete coherent and non-coherent electromagnetic (EM) frequencies was derived, in which the algorithm exhibits an information distribution according to ratios of 2:3 in 1:2 at a semi-harmonic manner. This generalized music (GM) model shows that energy both in elementary particles and animate systems is semi-harmonic, quantized and discrete. A support for an ontological basis of the Standard Model was found, and indicates that the GM-model underlies the quantum field theory of subatomic particles. The present theory combines quantum mechanics and classical periodic systems, obeys to locality and solves the “hidden variable theory of Bohm”. The discovered pattern of electromagnetic field eigenvalues, within a broad range of discrete frequencies, points at a de Broglie/Bohm type of causal interpretation of quantum mechanics, implying an integral resonant pilot-wave/particle modality. The model has been substantiated by a meta-analysis of measured discrete energies of: 37 different Elementary Particles, 45 different EPR-measurements, zero-point energies of elements and about 450 electromagnetic wave frequencies of cells with a mean accuracy of 0.58%. It has been shown that the GM-scale is frequency-locked with zero-point oscillations, and thereby evidently implies involvement of entanglement.
The framework of classical physics encompasses the well-established and complete Helmholtz wave equation. Capable of accurately describing the propagation, interference and diffraction of steady-state harmonic waves such as acoustic and electromagnetic waves at the macroscopic scale, this equation has been validated by centuries of experiments and is widely recognized as a definitive formula in fundamental physics. Nevertheless, conventional physics has long imposed a rigid scope of application. The academic community has always adopted one-dimensional real scalar coordinates for the Helmholtz equation, limiting its use to macroscopic material wave fields, while never attempting to apply it to the microscopic quantum realm or the underlying ontological structure of the universe. Meanwhile, starting from the microscopic mathematical characteristics, Heisenberg’s matrix mechanics derives the uncertainty principle based on the non-commutation relation of matrices. It directly proves that microscopic mechanical quantities cannot be fully represented by real scalars, and that the microscopic universe is inherently governed by matrix-dimensional structures. This is the fundamental reason why numerous paradoxes persist within quantum mechanics under the framework of real coordinates. Addressing the dual limitations of existing physical theories, this paper presents original research. While fully retaining the differential operators, wave constraints and steady-state solution rules of the classical Helmholtz equation, and introducing no new physical constants or artificial auxiliary hypotheses, this work merely replaces the real spatial coordinates X and Y in the equation with second-order two-dimensional linear matrix coordinates. On this basis, a novel theoretical system of two-dimensional linear waves tailored for the microscopic ontology of the universe is independently established. To clarify: classical physics only adopts the Helmholtz wave with real coordinates to interpret macroscopic wave motions. By contrast, the Helmholtz wave model built upon two-dimensional linear matrix coordinates is an exclusive original theory proposed in this paper. It forms a brand-new independent classical system for depicting the true microscopic essence of the universe. Employing rigorously verified classical mathematical and physical tools, including the classical Helmholtz wave principle, Heisenberg’s matrix mechanics, the uncertainty principle, the linear projection theorem and the wave field superposition principle, this paper systematically elaborates the mathematical structure, dimensional properties and evolutionary laws of two-dimensional linear waves. It further reveals the ultimate physical nature of quantum measurement collapse. All substances in the microscopic world exist inherently as two-dimensional linear Helmholtz waves. Human macroscopic perception and detection devices can only process one-dimensional real-valued information. Essentially, the act of observation is not the random collapse of the wave function as defined in standard quantum mechanics. Instead, the observing system actively extracts one degree of freedom from the two-dimensional matrix wave. The remaining single dimension undergoes dimensionality reduction and projection onto the one-dimensional real physical space, giving rise to the apparent phenomenon of localized particles. This original theory explains two core experimental puzzles that have perplexed quantum physics for a century, without logical contradictions or ad hoc supplements. First, continuous observation of the interference pattern in the double-slit experiment collects global steady-state information of the superimposed two-dimensional waves, and no dimensional extraction occurs, so no collapse takes place. Second, when a single photon reaches the human retina or a single-point detector conducts fixed-position detection, the observing system imposes localized boundary conditions to forcibly extract dimensions from the wave field. This projects the two-dimensional wave onto the one-dimensional physical space, which manifests macroscopically as wave function collapse. This research breaks down the barrier between macroscopic classical waves and microscopic quantum phenomena. The proposed original classical model provides a unified interpretation of microscopic effects including wave-particle duality and quantum measurement collapse, offering a new fundamental perspective for the research on the grand unification of fundamental physics.
These notes offer a basic introduction to the primary mathematical concepts of quantum physics, and their physical significance, from the operator and Hilbert space point of view, highlighting more what are essentially the abstract algebraic aspects of quantisation in contrast to more standard treatments of such issues, while also bridging towards the path integral formulation of quantisation. A discussion of the (first) Noether theorem and Lie symmetries is also included to complement the presentation. Emphasis is put throughout, as illustrative examples threading the presentation, on the quantum harmonic oscillator and the dynamics of a charged particle coupled to the electromagnetic field, with the ambition to bring the reader onto the threshold of relativistic quantum field theories with their local gauge invariances as a natural framework for describing relativistic quantum particles in interaction and carrying specific conserved charges.