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Linear relationship between frequency and energy of a wave train

Peifeng Wang

Published May 3, 2019• 0 citations
Publisher

Abstract

We demonstrate that, under Lorentz transformation, the frequency and the energy of an electromagnetic wave train form a linear relationship, emulating the equation of photon energy E = ℏ ω . This shows that wave-particle duality of light is compatible under space time transformation.
DOI
10.22541/au.155686535.57688463

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This letter proposes an explanation for the century-old puzzle of the wave-particle-duality of light, which thousands of physicists, including Einstein, Plank, Feynman …, have been trying to resolve. Since a ‘particle ’ is localized in space, it is mathematically characterized here as an impulse-function in space. Then it is argued that if this ‘particle’ has anything to do with waves then we can know it by taking Fourier-transform of the impulse-function in space. When Fourier-transformed into wave-number-domain; we find that a ‘particle ’ should contain a wide ‘set ’ of waves, and not just a single frequency. Then we show that in the experiments performed so far [1] the red lasers had significantly wide line-width, means the sources have been producing a wide set of waves, and not just a pure single frequency. Similarly, in the single-particle interference-experiments incandescent filament-lamps were used with green filters inserted to isolate single photons; but it is obvious that at the frequencies of light very narrow-band-filters are not yet technically feasible, so the green filters used allowed significantly wide band of waves. So in the experiments performed so far we got ‘particles ’ localized in space. And in the double-slit-interference-experiments this wide band of waves passed from both the slits, interfered like waves, and whenever and wherever they got coherently added, a ‘particle ’ called ‘photon ’ got detected. Atoms emit short-duration pulses of radiation, so their Fourier transform contains a wide band of frequencies. And at the time of detection, an atom needs only a small piece of continuous ‘wave’, of a suitable duration, amplitude and frequency, for ejection of an electron. So it is concluded here that: emission and detection of light is in the form of ‘photons’; whereas its propagation in space is in the form of a wide band of ‘waves’. 2

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A simple covariant derivation of the de Broglie relations breaks down when one considers propagation of light in an anisotropic medium. This breakdown results in two distasteful conclusions. The first is that the de Broglie relations are not completely universal and their covariance is restricted. The second is that wave-particle duality is not universal. In this paper a resolution of these problems is suggested, a resolution which is an extrapolation of ideas of de Broglie and Andrade e Silva. The relations are generalized for propagation in anisotropic media by defining a new phase wave with wave vector l in the direction of energy transport and distinct from the conventional electromagnetic wave. The introduction of l follows the method by which de Broglie associated a wave with an electron. The magnitude of l is omega u/c-squared where omega is the angular frequency and u the velocity of energy transport. The physical meaning of l and the covariance of physical laws in the presence of matter are discussed. An experiment is described which would test the existence of the phase wave associated with the new wave vector l. This experiment is essentially a Young two-slit experiment carried out within an anisotropic crystal. Keywords: Scattering; Anisotropic materials.

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In the framework of Special Relativity and in agreement with the electromagnetic approach used to connect Quantum and Relativistic effects in Bridge Theory, a first order wave-equation able to emulate the wave-particle duality in terms of local electromagnetic interactions is proposed. The equation is able to describe the propagation of a fermion as a de Broglie's wave and simultaneously to characterise the wave in motion with a mass energy as a particle. The solution of the wave-equation proposed is compatible with the de Broglie's idea of an empty pilot wave, carrying information about the dynamical properties of a particle materialising during each local interaction with the observers.

International Journal of Applied Physics•Jun 25, 2020•3 citations
The quantum mechanical model of electromagnetic waves is considered, in which the photon, as a particle, propagates with the speed of light, while the wave function of the photon spreads out with an infinite speed. The Lorentz transformation for the energy flux of electromagnetic wave is obtained while assuming the Galilei transformation for the space and time. The energy of the electromagnetic wave also follows the Lorentz transformation with the volume restricting the wave train not following any transformation.
This letter proposes an explanation for the century-old puzzle of the wave-particle-duality of light, which thousands of physicists, including Einstein, Plank, Feynman …, have been trying to resolve. Since a ‘particle ’ is localized in space, it is mathematically characterized here as an impulse-function in space. Then it is argued that if this ‘particle’ has anything to do with waves then we can know it by taking Fourier-transform of the impulse-function in space. When Fourier-transformed into wave-number-domain; we find that a ‘particle ’ should contain a wide ‘set ’ of waves, and not just a single frequency. Then we show that in the experiments performed so far [1] the red lasers had significantly wide line-width, means the sources have been producing a wide set of waves, and not just a pure single frequency. Similarly, in the single-particle interference-experiments incandescent filament-lamps were used with green filters inserted to isolate single photons; but it is obvious that at the frequencies of light very narrow-band-filters are not yet technically feasible, so the green filters used allowed significantly wide band of waves. So in the experiments performed so far we got ‘particles ’ localized in space. And in the double-slit-interference-experiments this wide band of waves passed from both the slits, interfered like waves, and whenever and wherever they got coherently added, a ‘particle ’ called ‘photon ’ got detected. Atoms emit short-duration pulses of radiation, so their Fourier transform contains a wide band of frequencies. And at the time of detection, an atom needs only a small piece of continuous ‘wave’, of a suitable duration, amplitude and frequency, for ejection of an electron. So it is concluded here that: emission and detection of light is in the form of ‘photons’; whereas its propagation in space is in the form of a wide band of ‘waves’. 2
A simple covariant derivation of the de Broglie relations breaks down when one considers propagation of light in an anisotropic medium. This breakdown results in two distasteful conclusions. The first is that the de Broglie relations are not completely universal and their covariance is restricted. The second is that wave-particle duality is not universal. In this paper a resolution of these problems is suggested, a resolution which is an extrapolation of ideas of de Broglie and Andrade e Silva. The relations are generalized for propagation in anisotropic media by defining a new phase wave with wave vector l in the direction of energy transport and distinct from the conventional electromagnetic wave. The introduction of l follows the method by which de Broglie associated a wave with an electron. The magnitude of l is omega u/c-squared where omega is the angular frequency and u the velocity of energy transport. The physical meaning of l and the covariance of physical laws in the presence of matter are discussed. An experiment is described which would test the existence of the phase wave associated with the new wave vector l. This experiment is essentially a Young two-slit experiment carried out within an anisotropic crystal. Keywords: Scattering; Anisotropic materials.
This paper reviews humanity's understanding of the photon, the evolution of quantum mechanics and quantum electrodynamics (QED), and analyzes their physical interpretations and philosophical implications. It seeks to elucidate the true nature of the photon by examining the possibility of hidden variables in the double-slit experiment. The wave-particle duality of the photon, as posited by quantum mechanics, appears to me to be less than entirely convincing. By designing a thought experiment involving a photon double-slit setup, I aim to refute the notion that photons generate Coherence, or so-called wave-function superposition, within the light field between the slits and the screen. This endeavour aims to demonstrate the pure particle nature of photons. Furthermore, by examining the wavelength of electromagnetic waves, this paper explores the intrinsic relationship between electromagnetic waves and photons, positing that electromagnetic waves are fundamentally pure particle-waves. Analysis of QED elucidates the potential influence of photon momentum on photon interactions, an effect that may profoundly impact quantum information transmission. To comprehensively describe the mathematical model of photon queues, this paper introduces the concept of a photon train. This metaphor likens photons travelling through space to a train in motion. The distance between carriages is denoted by (d), representing the wavelength of the photon。 Within the "particle train" model of the photon queue, the wavelength λ is defined as the distance d between adjacent photons. This yields a structural equivalence between wavelength and photon spacing: ; . The longer the wavelength (smaller f), the greater the distance d between photons (corresponding to radio waves, where photons are "sparsely" dispersed). The shorter the wavelength (larger f), the closer the distance d between photons (corresponding to X-rays or γ-rays, where photons are "focused"). f represents the frequency of light. In standard physics, frequency is typically defined as the number of oscillations of an electromagnetic field per unit time, expressed by the relationship: . This is directly linked to the energy of a photon. However, although mathematically and experimentally complete, this definition obscures the intuitive informational structure of frequency. In this paper, we introduce a longitudinally structured interpretation: viewing frequency as the manifestation of "front-to-back density" or "longitudinal spacing" of light quanta along their propagation direction. This understanding does not replace the standard physical definition but operates in parallel to reveal frequency's deeper significance in energy transmission and information organization. For simplicity, we denote this approach as the "photon queue/longitudinal serialization (PQ/LS) model". PQ/LS denotes the theoretical framework of the photon queue model. A thought experiment on the conversion rate of quantum information into biological information reveals that, under identical light intensity (equivalent total energy input), the response of living systems is not solely determined by the 𝐼, but strongly depends on the spatial arrangement of the photon queue (where λ approximates the inter-photon distance).Bacterial-level matching of visible-light queues achieves the highest information-transmission efficiency (maximum mutual information and signal-to-noise ratio), thereby enabling dual optimization of energy utilization and rhythmic synchronization. This constitutes the core prediction of the model (PQ/LS): frequency (queue spacing), rather than total power (𝐼) alone, determines the conversion efficiency of quantum information into biological information. The paper explores the non-wave-particle duality of photons through double-slit experiments and double-slit observer experiments. The essence of photons is purely particle-like. The thought experiment of the photon double-slit is pivotal in demonstrating that photons, like other particles such as electrons, atoms, and macromolecules, are purely particle-like. In space, photons can interfere with each other's paths based on their momentum. This can simultaneously alter the trajectory and even the velocity of photons. That is, a photon beam with higher momentum may carry a portion with lower momentum, allowing it to propagate over vast distances. The wavelength of a light wave is fundamentally the distance between consecutive photons as they propagate through space. The longer the wavelength of a light wave, the greater the distance between successive photons, causing them to diverge more widely in space, as observed with radio waves. Conversely, shorter wavelengths result in closer distances between photons, leading to denser emission and highly directional propagation, exemplified by gamma rays. Photons within the visible-light spectrum may occasionally transmit information that carries specific content. This information, conveyed through light quanta, can propagate over distances ranging from several metres to thousands of kilometres. The essence of light quanta: particle queues (PQ/LS) and the profound meaning of frequency: light is fundamentally a sequence of light quanta (photons) advancing in spatial order. Regardless of whether the propagation path is straight or curved, the propagation maintains this queued formation, akin to a continuous jet of water. Within this model (PQ/LS), the wavelength λ of light observed macroscopically fundamentally represents the average distance between successive light quanta. The frequency f (or ν) is inversely proportional to this distance: the closer two light quanta are, the higher the frequency f; the greater the distance, the lower the frequency f. It can be expressed through the classical wave velocity formula: c=fλ. Within the visible spectrum, λ≈0.5×10⁻⁶ metres (500 nanometres). Remarkably, this distance corresponds precisely to the typical dimensions of bacteria (approximately 0.5–5 micrometres). The evolutionary tree of life takes bacteria as its root, while the wavelength of visible light—the fundamental scale for human vision and quantum information interaction—aligns exactly with this root-level dimension. As the product of billions of years of bacterial evolution, our human perception, communication, and even consciousness may possess an intrinsic connection to this "bacterial-scale" queue distance of light quanta. Philosophically, this unification reveals a grander picture: the frequency-distance relationship of the light quantum queue (PQ/LS) not only determines the magnitude of physical energy but also bridges the quantum realm with the origins of life. The evolution of human vision, which aligns precisely with this "bacterial-scale" dimension, suggests that our perceptual systems are rooted in life's most primordial mode of information exchange. As an information carrier, light transforms energy (E = hf) into biological signals, genetic expression, and even the emergence of consciousness—from photons stimulating retinal cells to potential quantum biological processes (such as quantum coherence in photosynthesis or brain microtubules), light weaves an information network between the microscopic and macroscopic realms. Bacterial circadian rhythms, particularly the KaiABC system in cyanobacteria, function as living fossils: they document how visible light quantum queues (PQ/LS) were transformed into ordered biological information programmes during life's earliest stages. This experimental finding not only validates the functional significance of scale consistency but illuminates the earliest step in the evolutionary path from light to meaning—within the unicellular world, light ceased to be mere energy, becoming instead the language of time and the rhythm of information. The relatively poor correlation between archaea and most bacteria in circadian rhythms and responses to visible light, from another perspective, suggests that archaea and bacteria did not originate on Earth but rather within protoplanetary discs or planetesimals. These archaea and bacteria can survive in the harshest natural environments, unaffected by Earth's 24-hour circadian rhythms or visible light. It further substantiates my published theory that life did not originate on Earth. From the bacterial scale to the scale of consciousness: information density remains operative. Within the Photon Quantum Queue Model (PQ/LS), the wavelength of visible light (~0.5 μm interphoton spacing) aligns remarkably with the bacterial scale, suggesting that the informational "foundation" of life's origin is set by the interphoton spacing. As we ascend from single-celled bacteria (scale 0.5–5 μm) to multicellular organisms, nervous systems, and ultimately human consciousness (neuronal scale ~10–100 μm, brain as a whole ~0.1 m), a central question emerges: does this information density, originating from the photon quantum distance, continue to operate at higher scales, bridging microscopic origins with macroscopic awareness? From bacteria to conscious scales: Implications of information density for SETI and the search for extraterrestrial life. An intuitive interpretation of light's nature: Mainstream quantum electrodynamics (QED) views photons as quantum field excitations, where wavelength λ represents the phase oscillation scale associated with a single photon, not the physical distance between multiple photons. This theory deliberately adopts a semi-classical "particle queue" image, directly correlating λ with the average distance between adjacent photons. While not strictly valid in weak-light or single-photon experiments, the theory challenges the orthodoxy that "wavelength is solely a property of individual photons." The theory's stance can be summarized as one of maximizing continuity and wholeness within strict scientific boundaries. Our challenge lies in rejecting the compartmentalization of the quantum realm, the origins of life,
Though it has been widely viewed that the light consists of quantum particles known as photons, the light is not made of quantum particles or photons, and the light is always a wave. What is quantized in the light is the frequency. The electromagnetic frequency is always quantized due to the quantized nature of the electromagnetic energy; the quantum energy results in quantum frequency. The light is made up of many discrete electromagnetic wave bursts of constant durations; each burst consists of an electromagnetic wave of a single frequency determined by an integer multiple of the indivisible quantum energy units. The quantum energy and frequency are synonymous; they are one and the same. The duration of a wave burst is a universal constant, and it is the same for all the wave bursts irrespective of the frequency of the electromagnetic wave they contain. Since the quantum energy is in the frequency of the electromagnetic wave in a wave burst, the light can be propagated unlimited distances with the quantum energy intact, without which, the evolution of life would not have occurred. The quantum energy is not affected by any gradual path degradation even after the propagation of light for billions of light years since the quantum energy is the frequency; this would have been an impossible task by any other means such as carrying the quantum energy in a particle or photon on a wave. The light is always a wave, and never a particle; there is no wave-particle duality. There is absolutely no need for an unrealistic, artificial quantum light particle, or photon hypothesis. The hypothetical mass-less particles only exist in human imagination; they do not exist in the nature.
In the framework of Special Relativity and in agreement with the electromagnetic approach used to connect Quantum and Relativistic effects in Bridge Theory, a first order wave-equation able to emulate the wave-particle duality in terms of local electromagnetic interactions is proposed. The equation is able to describe the propagation of a fermion as a de Broglie's wave and simultaneously to characterise the wave in motion with a mass energy as a particle. The solution of the wave-equation proposed is compatible with the de Broglie's idea of an empty pilot wave, carrying information about the dynamical properties of a particle materialising during each local interaction with the observers.