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Transactions of the IRE Professional Group on Information Theory

Communication theory and physics

Деннис Габор

Published Feb 1, 1953• 107 citations
Publisher

Abstract

The electromagnetic signals used in communication are subject to the general laws of radiation. One obtains a complete representation of a signal by dividing the time-frequency plane into cells of unit area and associating with every cell a "ladder" of distinguishable steps in signal intensity. The steps are determined by Einstein's law of energy fluctuation, involving both waves and photons. This representation, however, gives only one datum per cell, viz. the energy, while in the classical description one has two data; an amplitude and a phase. It is shown in the second part of the paper that both descriptions are practically equivalent in the long-wave region, or for strong signals, as they contain approximately the same number of independent, distinguishable data, but the classical description is always a little less complete than the quantum description. In the best possible experimental analysis the number of distinguishable steps in the measurement of amplitude and phase is only the fourth root of the number of photons. Thus it takes a hundred million photons per cell in order to define amplitude and phase to one percent each.

DOI
10.1109/tit.1953.1188558

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Summary The electromagnetic signals used in communication are subject to the general laws of radiation. One obtains a complete representation of a signal by dividing the time-frequency plane into cells of unit area and associating with every cell a “ladder” of distinguishable steps in signal intensity. The steps are determined by Einstein's law of energy fluctuation, involving both waves and photons.

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Hitherto communication theory was based on two alternative methods of signal analysis. One is the description of the signal as a function of time; the other is Fourier analysis. Both are idealizations, as the first method operates with sharply defined instants of time, the second with infinite wave-trains of rigorously defined frequencies. But our everyday experiences—especially our auditory sensations—insist on a description in terms of both time and frequency. In the present paper this point of view is developed in quantitative language. Signals are represented in two dimensions, with time and frequency as co-ordinates. Such two-dimensional representations can be called “information diagrams,” as areas in them are proportional to the number of independent data which they can convey. This is a consequence of the fact that the frequency of a signal which is not of infinite duration can be defined only with a certain inaccuracy, which is inversely proportional to the duration, and vice versa. This “uncertainty relation” suggests a new method of description, intermediate between the two extremes of time analysis and spectral analysis. There are certain “elementary signals” which occupy the smallest possible area in the information diagram. They are harmonic oscillations modulated by a “probability pulse.” Each elementary signal can be considered as conveying exactly one datum, or one “quantum of information.” Any signal can be expanded in terms of these by a process which includes time analysis and Fourier analysis as extreme cases.These new methods of analysis, which involve some of the mathematical apparatus of quantum theory, are illustrated by application to some problems of transmission theory, such as direct generation of single sidebands, signals transmitted in minimum time through limited frequency channels, frequency modulation and time-division multiplex telephony.

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The electromagnetic signals used in communication are subject to the general laws of radiation. One obtains a complete representation of a signal by dividing the time-frequency plane into cells of unit area and associating with every cell a "ladder" of distinguishable steps in signal intensity. The steps are determined by Einstein's law of energy fluctuation, involving both waves and photons. This representation, however, gives only one datum per cell, viz. the energy, while in the classical description one has two data; an amplitude and a phase. It is shown in the second part of the paper that both descriptions are practically equivalent in the long-wave region, or for strong signals, as they contain approximately the same number of independent, distinguishable data, but the classical description is always a little less complete than the quantum description. In the best possible experimental analysis the number of distinguishable steps in the measurement of amplitude and phase is only the fourth root of the number of photons. Thus it takes a hundred million photons per cell in order to define amplitude and phase to one percent each.
Hitherto communication theory was based on two alternative methods of signal analysis. One is the description of the signal as a function of time; the other is Fourier analysis. Both are idealizations, as the first method operates with sharply defined instants of time, the second with infinite wave-trains of rigorously defined frequencies. But our everyday experiences—especially our auditory sensations—insist on a description in terms of both time and frequency. In the present paper this point of view is developed in quantitative language. Signals are represented in two dimensions, with time and frequency as co-ordinates. Such two-dimensional representations can be called “information diagrams,” as areas in them are proportional to the number of independent data which they can convey. This is a consequence of the fact that the frequency of a signal which is not of infinite duration can be defined only with a certain inaccuracy, which is inversely proportional to the duration, and vice versa. This “uncertainty relation” suggests a new method of description, intermediate between the two extremes of time analysis and spectral analysis. There are certain “elementary signals” which occupy the smallest possible area in the information diagram. They are harmonic oscillations modulated by a “probability pulse.” Each elementary signal can be considered as conveying exactly one datum, or one “quantum of information.” Any signal can be expanded in terms of these by a process which includes time analysis and Fourier analysis as extreme cases.These new methods of analysis, which involve some of the mathematical apparatus of quantum theory, are illustrated by application to some problems of transmission theory, such as direct generation of single sidebands, signals transmitted in minimum time through limited frequency channels, frequency modulation and time-division multiplex telephony.
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. 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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. 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Atoms and ions containing residual bound electrons do not quite resemble the simple harmonic oscillator model used so successfully in the classical theory of radiation. However, replacing the atoms or ions with sets of harmonic oscillators of a great number of discrete resonance frequencies and having various amplitudes, together with the results of classical radiation theory, go a long way toward a quantitative description of emission or absorption spectra. The set of resonance frequencies is obtained from measured or calculated energy levels using Ritz's combination principle. The amplitudes are associated with matrix elements of appropriate quantum mechanical operators between wave functions of the two energy eigenstates involved at a given frequency. In other words, quantities of the emitters, absorbers, or scatterers are described quantum-mechanically, whereas the electromagnetic field is treated classically. Such semi-classical description of matter-electromagnetic field interactions became unnecessary very early in the development of quantum theory. It will therefore not be discussed in any detail. Instead, we will begin immediately with the combined theory of matter and radiation (Heitler 1954, Dirac 1958, Loudon 1983). Quantum theory of particles and fields There are various ways to also quantize the electromagnetic fields (Cohen-Tannoudji, DuPont-Roc and Grynberg 1989), of which that performed on the combined Hamiltonian equations of motion for the field-matter system is followed here.
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A classical signal which agrees with the ideas of mathematical analysis on infinitesimal quantities possesses an infinite information capacity, which contradicts the concept of information that is finite by definition. A real signal should have some “threshold” for determining a finite step between its distinguishable states and the “limit” for the possible number of states. This paper considers the influence of the level of required energy, performance, and noise of the receiver on the capability of distinguishing the set of states in the signal. It is shown that the generalized threshold is the constraint on the spectral density of the signal energy and the limit is the constraint on the energy density with respect to time or spatial coordinate. In the microcosm, a signal is a gradient of potential; the constraint of its perception speed in the case of motion results, within the action threshold, in the fact that it changes in kinetic energy with changing potential cannot be determined. As a result, the particle motion turns out defined only in a finite number of “reference” points separated by intervals determined by the information threshold. Between reference points the motion has an uncertain, random, i.e., noisy character. The threshold equation for finding random deviations of motion parameters which should generalize the known Schrödinger and Klein-Gordon equations is discussed. Information transmission capacity of space for electromagnetic signals is estimated based on the analysis of the densities of the threshold and limiting energy.