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.
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.
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 1948, Richard Feynman published the first paper on his new approach to non-relativistic quantum mechanics. Before Feynman's work there were two mathematical formulations of quantum mechanics. Schrödinger's formulation was based on PDE (the Schrödinger equation) and states representation by wave functions, so it was in the framework of analysis and differential equations. The other formulation was Heisenberg's matrix algebra. Initially, they were thought to be competing. The proponents of one claimed that the other was “ wrong. ” Within a couple of years, John von Neumann had proved that they are equivalent.\nAlthough Feynman's theory was not fundamentally new, it nonetheless offered an entirely fresh and different perspective: via a precise formulation of Bohr's correspondence principle, it made quantum mechanics similar to classical mechanics in a precise sense. In addition, Feynman's approach made it possible to explain physical experiments, and, via diagrams, link them directly to computations. What resulted was a very powerful device for computing energies and scattering amplitudes - the famous Feynman's diagrams.\nIn his formulation, Feynman aimed at representing the solution to the non-relativistic Schrödinger equation in the form of an “ average ” over histories or paths of a particle. This solution is commonly known as the Feynman path integral. It plays an important role in the theory but appears as a postulate based on intuition coming from physics rather than a justified mathematical object. This is why Feynman's vision has caught the attention of many mathematicians as well as physicists.\nThe papers of Gelfand, Cameron, and Nelson are among the first, and more substantial, attempts to supply Feynman's theory with a rigorous mathematical foundation. These attempts were followed by many others, but unfortunately all of them were not quite satisfactory. The difficulty comes from a need to define a measure on an infinite-dimensional space of continuous functions that represent all possible paths of a particle. This Feynman's measure has to produce an integral with the properties requested by Feynman. In particular, the expression for the Feynman measure has to involve the non-absolutely integrable Fresnel integrands. The non-absolute integrability of the Fresnel integrands makes the measure fail to be positive and to have the countably additive property. Thus, a well-defined measure in the case of the Feynman path integral does not exist.\nExtensive research has been done on the methods of relating the Feynman path integral to the integral with respect to the Wiener measure. The method of analytic continuation in mass defines the Feynman path integral as a certain limit of the Wiener integrals. Unfortunately, this method can be used as definition for only almost all values of the mass parameter in the Schrödinger equation. For physicists, this is not a satisfactory result and needs to be improved.\nIn this work we examine those questions which originally led to the Feynman path integral. By now we know that Feynman's “ dream ” cannot be realized as a positive and countably additive measure on the path-space. Here, we offer a new way out by modifying Feynman's question, and thereby achieving a solution to the Schrödinger equation via a different kind of averages in the path-space.\nWe give our version of the question that Feynman “ should have asked ” in order to realize the elusive path integral. In our formulation, we get a Feynman path integral as a limit of linear functionals, as opposed to the more familiar inductive limits of positive measures, traditionally used for constructing the Wiener measure, and related Gaussian families. We adapt here an approach pioneered by Patrick Muldowney. In it, Muldowney suggested a Henstock integration technique in order to deal with the non-absolute integrability of the kind of Fresnel integrals which we need in our solution to Feynman's question. By applying Henstock's theory to Fresnel integrals, we construct a complex-valued “ probability distribution functions ” on the path-space. Then we use this “ probability ” distribution function to define the Feynman path integral as an inductive limit. This establishes a mathematically rigorous Feynman limit, and at the same time, preserves Feynman's intuitive idea in resulting functional. In addition, our definition, and our solution, do not place any restrictions on any of the parameters in the Schrödinger equation, and have a potential to offer useful computational experiments, and other theoretical insights.
This paper presents new examples of the functional-integral description of nonrelativistic quantum processes. The main point is the development and application of a functional-integral version which gives an adequate representation of the coherent dynamics of short impulses of light in media. An exact correspondence is received between the physical picture and functional-integral trajectories as the solutions of Euler-Lagrange equations with given initial and final conditions. The observed examples demonstrate the powerfulness of the path-integral method in the functional treatment and will promote the non-operator representation of quantum mechanics and quantum statistics.
In this paper, we try to give a new approach to the quantum mechanics(QM) on the framework of quantum field theory(QFT). Firstly, we make a detail study on the (non-relativistic) Schrödinger field theory, obtaining the Schrödinger equation as a field equation, after field quantization, the Heisenberg equations for the momentum and position operators of the particles excited from the (Schrödinger) field and the Feynman path integral formula of QM are also obtained. We then give the probability concepts of quantum mechanics in terms of a statistical ensemble, realizing the ensemble(or statistical) interpretation. With these, we make a series of conceptual modifications to the standard quantum mechanics, especially propose a new assumption about the quantum measurement theory which can solve the EPR paradox from the view of the QFT. Besides, a field theoretical description to the double-slit interference experiment is developed, obtaining the required particle number distribution. In the end, we extend all the above concepts to the relativistic case so that the ensemble interpretation is still proper. Two extra topics are added, in the first one, an operable experiment is proposed to distinguish the Copenhagen interpretation from the ensemble one via very different experimental results. While the second topic concerns with the extensions of the concept of coherent state to both the Bosonic and Fermionic field cases, to obtain the corresponding classical fields. And in the concluding section, we make some general comparisons between the standard QM and the one derived from the QFT, from which we claim that the QFT is the fundamental theory.
One of the most intriguing quantum phenomena of light is the wave–particle duality which is related to the inability of the classical wave and particle concepts to fully describe the behavior of quantum objects. Theoretically, the wave features of the electromagnetic field, such as interference, follow from the solution of Maxwell's equations while particle features are a consequence of the field quantization, in which the electromagnetic field is described to consist of discrete energy packets, photons. Due to the wave–particle duality, for example, simultaneous description of optical absorption, emission, and interference of propagating fields has conventionally not been considered feasible using local optical models but has required using Maxwell's equations with stochastic source terms accounting for the wave–particle duality. Also the description of the momentum of light in a medium has been problematic.
In the research described in this thesis, the quantized fluctuational electrodynamics (QFED) model is derived based on combining the electromagnetic Green's functions to the field quantization to describe the position dependence of the photon number and local thermal balance in general non-equilibrium conditions, and to separate the electromagnetic field into parts propagating in different directions in resonant structures, which has conventionally been problematic due to interference effects. The QFED method is shown to resolve the previously found anomalies in the canonical commutation relations of photon ladder operators. The QFED method is also used to derive quantum optical field–matter interaction parameters fully capturing the interference, emission, damping, and scattering of propagating photons in stratified media leading to an interference-exact radiative transfer equation (RTE) model. Therefore, the derived interaction parameters solve the problem of simultaneously describing interference and losses in the widely used RTE framework.
The particle aspects of light are also studied by directly applying the conservation laws of nature and the special theory of relativity to show that light propagating in a medium must be described as mass-polariton (MP) quasiparticles, covariant coupled states of the field and matter. These quasiparticles are shown to have momentum of the Minkowski form and a nonzero mass that is carried by a mass density wave associated with light. The field–matter interaction related to the MP quasiparticles also leads to the dissipation of photon energy. These particle model results are also derived using a wave-based optoelastic continuum dynamics (OCD) approach following from the electrodynamics of continuous media and the continuum mechanics. The obtained results verify the full agreement between the wave and particle models when they are correctly applied. The key finding that a light pulse propagating in a medium is inevitably associated with an experimentally measurable mass implies a fundamental change in the prevailing perceptions of light.; Eras kiehtovimmista valon…
The Dirac equation in one (space) dimension has a solution in the form of a path integral. If the equation includes the influence of an external potential, then this solution takes the form of a Feynman-Kac style representation. We consider the Feynman-Kac functional, where a Borel measure η replaces the Lebesgue measure l in the time integration. Let Q(t) be the evolution operator associated with this functional through path integration. We obtain a series representation for this operator, adapting a construction given by T. Zastawniak to this setting. Using a technique we call operator time-reordering, we see that this series can be interpreted as a perturbation series. Restricting ourselves to the case where the discrete part ν of η has finite support, we use the series to show that Q(t), considered as a function of time t, satisfies a certain Volterra-Stieltjes integral equation. We then deduce from the integral equation that Q(t) satisfies a differential equation associated with the continuous part μ of η. When η=μ=l this differential equation reduces to the Dirac equation with a potential. Thus we establish a "Feynman-Kac formula with a Lebesgue-Stieltjes measure η." Moreover, we show that the function Q(t) suffers a jump discontinuity at every point in the support of ν. Although the development here is quite different, the results obtained are similar those given by M. L. Lapidus for the nonrelativistic setting. Again, restricting ourselves to the case where the discrete part ν of η has finite support, we obtain a "generalized Dyson series" for Q(t) which is analogous to the series obtained by G. W. Johnson and Lapidus in the nonrelativistic setting. The theory of semigroups of linear operators, and the theory of strong product integration for bounded linear operators play an important role in this development. The various series representations obtained for Q(t), and the method of operator time-reordering provide a rigorous means of carrying out the "disentangling" which is a central element of Feynman's time-ordered operational calculus for noncommuting operators.