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Physics of Wave Processes and Radio Systems

Feynman integrals as connecting component between radio еngineering, fractal paradigm and quantum mechanics

Alex Potapov, Потапов А. А, Alexander E. Rassadin, А. Э. Рассадин

Published Sep 15, 2015• 1 citation
OpenAlex

Abstract

Analysing Feynman integral for one-dimensional harmonic oscillator we have found that it is possible to introduce quasiparticle of radio systems. This quasiparticle we have called radion. One can understand radion as object moving along fractal trajectories in phase space of oscillator. The same arguments have led us to the conclusion about natural fractality of all radio systems working with spectrum of signal. Under radiation of electromagnetic wave into medium with arbitrary dielectric permeability radion continues its fractal movement. Interrelations of radion and photon also have been discussed.

Related papers

The latest achievements in the development of the concept of radio- and optical systems quasiparticle based on Feynman integrals

Alexander A. Potapov, Alexander E. Rassadin

In a number of previous papers authors have introduced quasiparticle of radio- and optical systems. We have called this quasiparticle by ‘radion’. The basis for this is the representation of Green’s function of equation of quasioptics by Feynman integral. It means that radion has quantum mechanical properties. In particular in approximation of quasioptics one can interprete amplitude of electromagnetic field as amplitude of probability of radion. In the paper presented we describe new features of radion which we have established recently namely we have found some exact solutions of evolutionary equation for field of quantum mechanical averages of momenta of ensemble of radions possessing by average of momentum in initial state depending on average of coordinate in it. We demonstrate three the simplest solutions from this set of exact solutions. These solutions in the form of power series on dimensionless time are applicable up to the moment of gradient catastrophe. For representation of solutions of parabolic equation in approximation of quasioptics Remizov method of quasi-Feynman formulae has been used. All these results due to suggested by us physical analogy are valid for free one-dimensional nonrelativistic quantum mechanical particle too. Also in the framework of Remizov method a number of new formulae for statistical radiophysics and optics has been obtained. ‘Natural’ fractality of radio- and optical systems has been discussed.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Apr 6, 2017•2 citations

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Quantum phenomena are discussed using extended measures. The complex measurable processes introduced earlier are studied in greater depth in the context of harmonic oscillators. Some of the fundamental problems of radiation and interaction of radiation with matter are analysed in the general frame of extended measure theory. The connection to Feynman path integrals is also discussed and it is shown how the present approach circumvents the divergence difficulties particularly in the estimation of Lamb shift.

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Inconsistencies of some standard quantum mechanical models (Madelung’s, de Broglie’s models) are eliminated as- suming the micro particle movements on continuous, but non-differentiable curves (fractal curves). This hypothesis, named by us the fractal approximation of motion, will allow an unitary approach of the phenomena in quantum me-chanics (separation of the physical motion of objects in wave and particle components depending on the scale of resolution, correlated motions of the wave and particle, i.e. wave-particle duality, the mechanisms of duality, by means of both phase wave-particle coherence and wave-particle incoherence, the particle as a clock, particle incorporation into the wave and the implications of such a process). Moreover, correspondences with standard gravitational models (Einstein’s model, string theory) can be also distinguished.

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Quantum nanomechanics: a new perspective on the quantum harmonic oscillator (Invited Paper)

E. K. Twyeffort Irish, Keith Schwab

The standard mathematical formalism of cavity QED leads us to consider photons as excitations of a quantum harmonic oscillator. Although it is one of the most familiar problems of quantum mechanics, some aspects of the quantum harmonic oscillator remain difficult to visualize, particularly in the rather abstract context of an electromagnetic field. Recently, modern microfabrication and refrigeration techniques have begun to allow the creation of nanoscale mechanical oscillators which can be cooled close to the quantum regime. Despite the extreme physical differences between an electromagnetic cavity and a nanomechanical resonator, both systems may be approximated by the same quantum harmonic oscillator model. However, the conceptual consequences of quantum behavior, and the challenges to physical intuition, are quite different in the two cases. Taking a mechanical point of view therefore allows fresh insight into the quantum harmonic oscillator problem. To illustrate the connection and how it may aid our understanding of light, the mathematical parallelism between an electromagnetic cavity and a mechanical resonator is demonstrated. Current nanomechanics experiments are discussed, and some possible quantum measurements are introduced. Finally, the discrepancies between the predictions of quantum mechanics and our experience of classical vibrating beams are considered, with an emphasis on how nanomechanics may be able to offer a new perspective on the nature of photons.

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Analysing Feynman integral for one-dimensional harmonic oscillator we have found that it is possible to introduce quasiparticle of radio systems. This quasiparticle we have called radion. One can understand radion as object moving along fractal trajectories in phase space of oscillator. The same arguments have led us to the conclusion about natural fractality of all radio systems working with spectrum of signal. Under radiation of electromagnetic wave into medium with arbitrary dielectric permeability radion continues its fractal movement. Interrelations of radion and photon also have been discussed.
In a number of previous papers authors have introduced quasiparticle of radio- and optical systems. We have called this quasiparticle by ‘radion’. The basis for this is the representation of Green’s function of equation of quasioptics by Feynman integral. It means that radion has quantum mechanical properties. In particular in approximation of quasioptics one can interprete amplitude of electromagnetic field as amplitude of probability of radion. In the paper presented we describe new features of radion which we have established recently namely we have found some exact solutions of evolutionary equation for field of quantum mechanical averages of momenta of ensemble of radions possessing by average of momentum in initial state depending on average of coordinate in it. We demonstrate three the simplest solutions from this set of exact solutions. These solutions in the form of power series on dimensionless time are applicable up to the moment of gradient catastrophe. For representation of solutions of parabolic equation in approximation of quasioptics Remizov method of quasi-Feynman formulae has been used. All these results due to suggested by us physical analogy are valid for free one-dimensional nonrelativistic quantum mechanical particle too. Also in the framework of Remizov method a number of new formulae for statistical radiophysics and optics has been obtained. ‘Natural’ fractality of radio- and optical systems has been discussed.
Quantum phenomena are discussed using extended measures. The complex measurable processes introduced earlier are studied in greater depth in the context of harmonic oscillators. Some of the fundamental problems of radiation and interaction of radiation with matter are analysed in the general frame of extended measure theory. The connection to Feynman path integrals is also discussed and it is shown how the present approach circumvents the divergence difficulties particularly in the estimation of Lamb shift.
The Feynman path-integral formulation of quantum mechanics is developed. The math- ematical framework is worked out from Schrödinger's equation, illustrating the compati- bility of the two formulations. We introduce the sum over all paths, known as the kernel K, and discuss its implications for events occurring in succession. We compute the kernel for quadratic Lagrangians. Then, making use of imaginary time, we consider the discrete version of the kernel, known as the Euclidean path integral, and use it to solve the (quan- tum) harmonic and anharmonic oscillators through numerical Monte Carlo simulations. Lastly, we use the framework to quantize the electromagnetic field, as a motivation to discuss perturbation theory via Feynman diagrams.
A new application of a fractal concept to quantum physics has been developed. The fractional path integrals over the paths of the Levy flights are defined. It is shown that if fractality of the Brownian trajectories leads to standard quantum mechanics, then the fractality of the Levy paths leads to fractional quantum mechanics. The fractional quantum mechanics has been developed via the new fractional path integrals approach. A fractional generalization of the Schrodinger equation has been discovered. The new relationship between the energy and the momentum of the nonrelativistic fractional quantum-mechanical particle has been established, and the Levy wave packet has been introduced into quantum mechanics. The equation for the fractional plane wave function has been found. We have derived a free particle quantum-mechanical kernel using Fox's H-function. A fractional generalization of the Heisenberg uncertainty relation has been found. As physical applications of the fractional quantum mechanics we have studied a free particle in a square infinite potential well, the fractional "Bohr atom" and have developed a new fractional approach to the QCD problem of quarkonium. We also discuss the relationships between fractional and the well-known Feynman path integral approaches to quantum mechanics. (c) 2000 American Institute of Physics.
Inconsistencies of some standard quantum mechanical models (Madelung’s, de Broglie’s models) are eliminated as- suming the micro particle movements on continuous, but non-differentiable curves (fractal curves). This hypothesis, named by us the fractal approximation of motion, will allow an unitary approach of the phenomena in quantum me-chanics (separation of the physical motion of objects in wave and particle components depending on the scale of resolution, correlated motions of the wave and particle, i.e. wave-particle duality, the mechanisms of duality, by means of both phase wave-particle coherence and wave-particle incoherence, the particle as a clock, particle incorporation into the wave and the implications of such a process). Moreover, correspondences with standard gravitational models (Einstein’s model, string theory) can be also distinguished.
The standard mathematical formalism of cavity QED leads us to consider photons as excitations of a quantum harmonic oscillator. Although it is one of the most familiar problems of quantum mechanics, some aspects of the quantum harmonic oscillator remain difficult to visualize, particularly in the rather abstract context of an electromagnetic field. Recently, modern microfabrication and refrigeration techniques have begun to allow the creation of nanoscale mechanical oscillators which can be cooled close to the quantum regime. Despite the extreme physical differences between an electromagnetic cavity and a nanomechanical resonator, both systems may be approximated by the same quantum harmonic oscillator model. However, the conceptual consequences of quantum behavior, and the challenges to physical intuition, are quite different in the two cases. Taking a mechanical point of view therefore allows fresh insight into the quantum harmonic oscillator problem. To illustrate the connection and how it may aid our understanding of light, the mathematical parallelism between an electromagnetic cavity and a mechanical resonator is demonstrated. Current nanomechanics experiments are discussed, and some possible quantum measurements are introduced. Finally, the discrepancies between the predictions of quantum mechanics and our experience of classical vibrating beams are considered, with an emphasis on how nanomechanics may be able to offer a new perspective on the nature of photons.