In this paper, problems of relativistic kinematics are studied, and a formula of infinite string oscillations U(x, t) = cosФ (1) (x, t) cos S(x, t) with the phase motion Ф (1) =0 in accordance with the arbitrarily given law x = X(t) (|v| < c, where v = X; c is the sound velocity) is obtained. The obtained formula allows us to trace the emergence of one-dimensional relativistic dynamics laws for a material point. It is shown that S(x, t) is a solution of Hamilton Jacobi equation and can be considered as the action of the «particle» with Ф (1) = 0 phase. The phase motion is likely to be influenced by the potential V(x, t) = p(t)(X(t) x) (where p is a momentum of the velocity v), and obeys the Newton equation and the Hamilton equations. The function ψ = exp (iS) is a solution of the Schrodinger equation with a relativistic Hamiltonian operator with the potential V(x,t), in which the operator √-c 2V 2 + m 2c 4 is expanded into series. In a non-relativistic case with the particle speed v ≪ c, this equation coincides with the normal Schrodinger equation. It is demonstrated that the relativistic equation is linked with the one-dimensional Dirac equation in the Foldy Wouthuysen representation under condition of magnetic field absence. Further, the possibility of complex structure objects introduction in the framework of a linear wave model is investigated.
In this paper, problems of relativistic kinematics are studied, and a formula of infinite string oscillations U(x, t) = cosФ (1) (x, t) cos S(x, t) with the phase motion Ф (1) =0 in accordance with the arbitrarily given law x = X(t) (|v| < c, where v = X; c is the sound velocity) is obtained. The obtained formula allows us to trace the emergence of one-dimensional relativistic dynamics laws for a material point. It is shown that S(x, t) is a solution of Hamilton Jacobi equation and can be considered as the action of the «particle» with Ф (1) = 0 phase. The phase motion is likely to be influenced by the potential V(x, t) = p(t)(X(t) x) (where p is a momentum of the velocity v), and obeys the Newton equation and the Hamilton equations. The function ψ = exp (iS) is a solution of the Schrodinger equation with a relativistic Hamiltonian operator with the potential V(x,t), in which the operator √-c 2V 2 + m 2c 4 is expanded into series. In a non-relativistic case with the particle speed v ≪ c, this equation coincides with the normal Schrodinger equation. It is demonstrated that the relativistic equation is linked with the one-dimensional Dirac equation in the Foldy Wouthuysen representation under condition of magnetic field absence. Further, the possibility of complex structure objects introduction in the framework of a linear wave model is investigated.
In a previous paper, we showed that the Hamilton equations of motion of a quantum particle are obtained as group velocities of the wave packet describing this particle only if the time dependent phase of a particle wave is proportional to the Lagrangian, not to the Hamiltonian as in a solution of the conventional Schrodinger equation. When a Lagrangian of relativistic form is considered, the wave packet of a quantum particle takes a physical form, with a finite spectrum of a cut off velocity (c). Based on a relativistic quantum principle, asserting the invariance of the time dependent phase for an arbitrary change of coordinates, we obtained the relativistic kinematics and dynamics, the electromagnetic field equations, the spin and the electromagnetic and gravitational interactions. When the Lagrangian is considered as a function of the Hamiltonian, we obtain a Schrodinger type equation with an additional term depending on the velocity and the momentum operator. Based on this equation, we investigate the dynamics of a relativistic quantum particle. In this framework, such a particle is described as a continuous distribution of conservative matter, according to the general theory of relativity. In an electromagnetic field, any time dependent phase variation is modified with a term proportional to a vector potential conjugated to the spatial coordinates and a scalar potential conjugated to time. However, this classical description tells us nothing about the structure of the physical world. Only Quantum Mechanics tells us something about the structure of this world. Namely, that this world is composed of species of identical quantum particles. On one hand, experimentally, it has been found that these particles are of a wavy nature. On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with momenta conjugated to coordinates and energy conjugated to time Consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrodinger equation. We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves. We consider a quantum particle as a distribution of conservative matter propagating according to the General Theory of Relativity. We obtain the dynamics of this matter in a gravitational field, the propagation in plane waves perpendicular to geodesic tracks, and equations of conservation. We describe the matter dynamics as a positively defined density and show that, according to the general theory of relativity, such a distribution can be conceive only as of a fragment of matter with a finite mass equal to a mass , as a characteristic of the matter dynamics, – the matter quantization. The group velocities of the Fourier conjugate representations in the coordinate and momentum spaces describe the dynamics of a quantum particle in agreement with the Hamiltonian equations. Under the action of an external (non-gravitational) field, the acceleration of the quantum matter has two components: 1) A component perpendicular to the velocity, given by the relativistic mechanical component of the time dependent phase, and 2) A component given by the additional field terms of this phase. A free quantum particle is described by a non-dispersing wave function , contrary to the solution of the Schrodinger equation. A coherent electromagnetic field, in resonance with a system of active quantum particles in a Fabry-Perot cavity, has a wave vector approximately proportional to the metric elements, as the resonance frequency is approximately constant – a gravitational wave can be detected by the transmission characteristics of an active Fabry-Perot cavity. In a constant gravitational field, a quantum particle undertakes a velocity and an acceleration, which, at the boundary of a black hole are null – absorption and evaporation processes at the boundary of a black hole arise only by gravitational perturbations. Generally, a quantum particle is described by a time-space volume, called graviton, with a spin 2, and a distribution of a specific matter in this volume, with a half-integer spin for Fermions and an integer spin for Bosons. A graviton Lagrangian is obtained as a curvature integral on a graviton volume, and a Hamiltonian tensor is obtained for the gravitational coordinates and velocities. In a gravitational field, the time space coordinates are deformed. In such a field, any plane wave remains perpendicular geodesic, while an additional acceleration is possible in the wave plane.
Our methodological aim is to make easy-to-interpret one of the abstract symmetries symmetry with respect to local gauge transformations. An infinite homogeneous string in three-dimensional space is considered. We first assume that free oscillations of the string are described by the function u(x, t) = cos kx exp [-ikct iF(x, t)]. From the external observers viewpoint each point of the string rotates in the YZ plane with an additional phase F due to direction changes of axes Y and Z in space and time. Continuously performing active Poincare transformations on the standing wave u(x, t) and not affecting the function F we obtain the function for the forced oscillations of a special kind U(x, t) = Ψ(x, t)cosФ(x, t), where Ψ = exp(iS(x,t)). The phase Ф(x, t) = 0 is called “particle”. It is shown that S is the action of this particle. The particle total energy and generalized momentum that include the potential functions V, A are derived from S. Ψ reduces to an identity the Schrodinger equation with nonlocal Hamiltonian that contains the functions V, A. The identity remains valid when replacing the F with F f (x, t). This replacement is equivalent to a local gauge transformation in the form of simultaneous replacement of Ψ with exp (if (x,t)Ψ, of V with V dtf (x, t) and of A with A + dxf (x, t). Thus, the investigated model has local gauge symmetry.
We consider a quantum particle as a wave packet in the coordinate space. When the conjugate wave packet in the momentum space is considered, we find that the group velocities of these two wave packets, which describe the particle dynamics, are in agreement with the Hamilton equations only if in the time dependent phases one considers the Lagrangian instead of the Hamiltonian which leads to the conventional Schrodinger equation. We define a relativistic quantum principle asserting that a quantum particle has a finite frequency spectrum, with a cutoff propagation velocity c as a universal constant not depending on the coordinate system, and that any time dependent phase variation is the same in any system of coordinates. From the time dependent phase invariance, the relativistic kinematics is obtained. We consider two types of possible interactions: 1) An interaction with an external field, by a modification of the time dependent phase differential with the terms proportional to the differentials of the space-time coordinates multiplied with the components of this field four-potential, and 2) an interaction by a deformation of the space-time coordinates, due to a gravitational field. From the invariance of the time dependent phase with field components, we obtain a mechanical force of the form of Lorentz’s force, and three Maxwell equations: The Gauss-Maxwell equations for the electric and magnetic fluxes, and the Faraday-Maxwell equation for the electromagnetic induction. When the fourth equation, Ampere-Maxwell, is considered, the interaction field takes the form of the electromagnetic field. For a low propagation velocity of the particle waves, we get a packet of waves with the time dependent phases proportional to the relativistic Hamiltonian, as in Dirac’s famous theory of spin, and a slowly-varying amplitude with a phase proportional to the momentum and this velocity. In the framework of our theory, the spin is obtained as an all quantum effect, without any additional assumption to the quantum theory. When a space-time deformation is considered in the time dependent phase of a quantum particle, from the group velocity we get the particle dynamics according to the general theory of relativity. In this way, the relativistic dynamics, the electromagnetic field, and the spin of a quantum particle are obtained only from the invariance of the time dependent phases of the particle wave functions.
The starting point of this research is a representation of a quantum particle according to the Schrodinger equation of the conventional quantum mechanics. In this representation, a quantum particle is described by a wave packet in the coordinate space and the conjugated wave packet in the momentum space. The problem is that while in the coordinate space the group velocity is in agreement with one of the Hamilton equations, the group velocity in the momentum space is in contradiction with the other Hamilton equation - a minus sign is missing. Group velocities in agreement with the Hamilton equations are obtained only when in the time dependent phase of a quantum particle wave packet, instead of the Hamiltonian coming from the conventional Schrodinger equation, the Lagrangian is considered. This suggests us to consider the relativistic Lagrangian in the time dependent phase. In this way, the conventional relativistic principle of invariance of the time-space interval gets the more physically understandable form of the invariance of the time dependent phase of a quantum particle - the time dependent phase of a quantum particle is the same in any system of coordinates. Based on the relativistic time dependent phase invariance of a quantum particle, from the group velocities of this particle the relativistic kinematics and dynamics are obtained. The interaction with an electromagnetic field is described by a modification of the time dependent phase with a scalar potential conjugated to time and a vector potential conjugated to the space coordinates. In this paper, we find that, according to the formalism of the general relativity, any matter element in a field of forces is accelerated only perpendicularly to its velocity. This means that the matter propagation of a quantum particle can be conceived in planes perpendicular to velocity, while the mater distribution can be considered in a Fourier representation - quantum waves. We study the quantum particle dynamics in gravitational waves and derive the graviton spin. We consider a quantum particle in electromagnetic field and obtain a Schrödinger-Dirac-type equation, with additional terms describing the velocity dependence of the particle dynamics.
A wave equation already exists in classical physics (for example for a vibrating string) with a solution of the form exp(ikx-iwt) where k=2*3.14/wavelength and w=2*3.14 frequency. K and w are not simply linked to momentum and energy, but the wave is physically visible. The wavelike function exp(ikx-iEt) also appears in solutions for the photon, the relativistic Klein Gordon equation (and the Dirac too) as well as in the Schrodinger equation. This solution separates x and k and E and t so -id/dx exp(ikx-iEt) = k exp(ikx-iEt) and id/dt exp(ikx-iEt) = E exp(ikx-iEt). If a classical equation exists linking energy and momentum (like EE=pp+momo or E=.5mvv), one may convert it into a differential equation with independent fluctuations in x and t, but linked through the energy relationship. This retains the classical information, but introduces a wave solution. For the Schrodinger equation with a potential, exp(-iEt + i Integral (0,x) dx1 v(x1)m is a solution which is equivalent to exp(i Integral (0,t) L dt1) where L is the Lagrangian. In this case, -id/dx exp(i action) = p(x) exp(i action) and id/dt exp(i action) = E exp(i action). This also holds for the relativistic Lagrangian -mo sqrt(1-vv). In the case of an electromagnetic potential v dot A, one may write Integral dt1 v dot A = Integral (0,x) dx1 dot A and so A behaves exactly like p(x). For the case of light, exp(ikx-iEt) follows as a solution of a wave equation for either the electric or magnetic fields (which follows from Maxwell’s equations). The equation: d/dt (.5eoEl El + .5/uo BB) + grad dot El x B = 0 (for no source) also follows from Maxwell’s equations. It is an equation relating energy density to momentum flux and also allows for an exp(ikx-iEt) solution if El and B have the same magnitude (c=1) and are perpendicular to each other and the motion of the photon.
Essentially, we solve the particle-field equations obtained in the matter distribution quantum mechanics. In the framework of this theory, a particle is described by a density amplitude function, we call wave function, as a packet of waves with the time dependent phases proportional to the relativistic Lagrangian. The mass quantization of such a particle, as a distribution of matter, is obtained from the equality of the spatial integral of the density with the mass as a characteristic of the relativistic dynamics of this matter. The particle dynamics, as a motion of a mass spatial distribution, is obtained from the equality of the wave propagation velocities with the distribution coordinate velocities. This description is a physical one, compared to the Schrödinger-Heisenberg description leading to unphysical results as a null velocity for the kinetic energy as a constant of motion, or Zitterbewegung. The description of a particle interaction with an electromagnetic field by additional Lagrangian terms, with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and an electric potential conjugated to time, leads to the Lorentz force, and the Maxwell equations. In the two conjugated spaces, of the coordinates and of the momentum, the dynamics of a quantum particle is described by propagation operators, depending on the coordinate-momentum product, applied to time dependent wave functions satisfying matter-field equations. Unlike the standard Schrödinger or Schrödinger-Dirac equations, depending only on the Hamiltonian operator, instead of the Hamiltonian these equations contain the Lagrangian, which, besides the Hamiltonian, include additional terms with an explicit dependence on velocity, as it is expected in a relativistic theory. We obtain the wave function of a particle-antiparticle system interacting with an electromagnetic field, as a wave packet of spinors propagating in space with finite dimensions, given by the finite definition domain of the density function, in the space of the matter-field momentum, as a constant of motion. We show that an electrically charged quantum particle, rotating in a central electric field, generates a vector potential with a magnetic field perpendicular to the rotational plane and an electric field in this plane. By a quantum transition, a photon is emitted in the particle rotational plane, with the magnetic field perpendicular to this plane, and the electric field contained in this plane. In this framework, the vector potential rotation is correlated to a particle momentum rotation, for the canonical matter-field momentum as a constant of motion, without any electromagnetic field radiated outside.