According to Schroedinger's ideas, classical dynamics of point particles should correspond to the " geometrical optics " limit of a linear wave equation, in the same way as ray optics is the limit of wave optics. It is shown that, using notions of modern wave theory, the " geometrical optics " analogy leads to the correspondence between a classical Hamiltonian H and a " quantum " wave equation in a natural and general way. In particular, the correspondence is unambiguous also in the case where H contains mixed terms involving momentum and position. In the line of Schroedinger's ideas, it is also attempted to justify the occurrence, in QM, of eigenvalues problems, not merely for energy, but also for momentum. It is shown that the wave functions of pure momentum states can be defined in a physically more satisfying way than by assuming plane waves. In the case of a spatially uniform force field, such momentum states have a singularity and move undeformed according to Newton's second law. The mentioned unambiguous correspondence allows to uniquely extend the Klein-Gordon relativistic wave equation to the case where a constant gravitational field is present. It is argued that Schroedinger's wave mechanics can be extended to the case with a variable gravitational field only if one accepts that the wave equation is a preferred-frame one. From this viewpoint, generally-covariant extensions of the wave equations of QM seem rather formal. Finally, it is conjectured that there is no need for a quantum gravity.
According to Schroedinger's ideas, classical dynamics of point particles should correspond to the " geometrical optics " limit of a linear wave equation, in the same way as ray optics is the limit of wave optics. It is shown that, using notions of modern wave theory, the " geometrical optics " analogy leads to the correspondence between a classical Hamiltonian H and a " quantum " wave equation in a natural and general way. In particular, the correspondence is unambiguous also in the case where H contains mixed terms involving momentum and position. In the line of Schroedinger's ideas, it is also attempted to justify the occurrence, in QM, of eigenvalues problems, not merely for energy, but also for momentum. It is shown that the wave functions of pure momentum states can be defined in a physically more satisfying way than by assuming plane waves. In the case of a spatially uniform force field, such momentum states have a singularity and move undeformed according to Newton's second law. The mentioned unambiguous correspondence allows to uniquely extend the Klein-Gordon relativistic wave equation to the case where a constant gravitational field is present. It is argued that Schroedinger's wave mechanics can be extended to the case with a variable gravitational field only if one accepts that the wave equation is a preferred-frame one. From this viewpoint, generally-covariant extensions of the wave equations of QM seem rather formal. Finally, it is conjectured that there is no need for a quantum gravity.
According to Schrödinger's ideas, classical dynamics of point particles should correspond to the « geometrical optics » limit of a linear wave equation, in just the same way as ray optics is the limit of wave optics. It is shown that the « geometrical optics » analogy leads to the correspondence between a classical Hamiltonian H and a linear wave equation in a natural and general way. In particular, the correspondence is unambiguous also in the case where H contains mixed terms involving momentum and position. This is obtained through a theory of the dispersion relations, which leads to properly define the group velocity and to enlighten its role. Using this latter notion, it is shown that, for a quite general class of potentials, « momentum states » can be defined in a physically more satisfying way than by assuming plane waves. These momentum states are solutions of the time-dependent Schrödinger equation and their amplitude functions move rigidly on a well-defined trajectory. In the case of a spatially uniform force field, such momentum states must have a singularity and the trajectory is defined by Newton's second law in the given force field.
According to Schrodinger's ideas, classical dynamics of point particles should correspond to the « geometrical optics » limit of a linear wave equation, in just the same way as ray optics is the limit of wave optics. It is shown that the « geometrical optics » analogy leads to the correspondence between a classical Hamiltonian H and a linear wave equation in a natural and general way. In particular, the correspondence is unambiguous also in the case where H contains mixed terms involving momentum and position. This is obtained through a theory of the dispersion relations, which leads to properly define the group velocity and to enlighten its role. Using this latter notion, it is shown that, for a quite general class of potentials, « momentum states » can be defined in a physically more satisfying way than by assuming plane waves. These momentum states are solutions of the time-dependent Schrodinger equation and their amplitude functions move rigidly on a well-defined trajectory. In the case of a spatially uniform force field, such momentum states must have a singularity and the trajectory is defined by Newton's second law in the given force field.
We examine the origin of the Newton–Schrödinger equations (NSEs) that play an important role in alternative quantum theories (AQT), macroscopic quantum mechanics and gravity-induced decoherence. We show that NSEs for individual particles do not follow from general relativity (GR) plus quantum field theory (QFT). Contrary to what is commonly assumed, the NSEs are not the weak-field (WF), non-relativistic (NR) limit of the semi-classical Einstein equation (SCE) (this nomenclature is preferred over the 'Moller–Rosenfeld equation') based on GR+QFT. The wave-function in the NSEs makes sense only as that for a mean field describing a system of N particles as , not that of a single or finite many particles. From GR+QFT the gravitational self-interaction leads to mass renormalization, not to a non-linear term in the evolution equations of some AQTs. The WF-NR limit of the gravitational interaction in GR+QFT involves no dynamics. To see the contrast, we give a derivation of the equation (i) governing the many-body wave function from GR+QFT and (ii) for the non-relativistic limit of quantum electrodynamics. They have the same structure, being linear, and very different from NSEs. Adding to this our earlier consideration that for gravitational decoherence the master equations based on GR+QFT lead to decoherence in the energy basis and not in the position basis, despite some AQTs desiring it for the 'collapse of the wave function', we conclude that the origins and consequences of NSEs are very different, and should be clearly demarcated from those of the SCE equation, the only legitimate representative of semiclassical gravity, based on GR+QFT.
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.
Abstract In the present work we use a point transformation defined in the extended configuration space to show that, in both classical and quantum formalisms, the evolution of a particle under the influence of a constant gravitational field can be related to the evolution of the harmonic oscillator. That is, in the classical mechanics regime, we show that this transformation allows us to establish the connection in the Newtonian, Lagrangian, Hamilton-Jacobi and Hamiltonian formalisms. In the quantum mechanics regime, the transformation determines a relationship between the wave functions, solutions to the corresponding Schrödinger equations, the propagators and the Wigner functions. Furthermore, by using the quantum potential approach to quantum mechanics, we show that the classical point transformation, in a natural manner, induces a quantum point transformation in the corresponding extended quantum configuration space, that determines the connection between the corresponding Madelung-Bohm trajectories. We find that the quantum transformation has the same functional form as the classical one even though the quantum or Madelung-Bohm trajectories are in general totally different from the classical ones. Then, we show that the general results can also be applied to the corresponding paraxial wave equations, and by using the stationary solutions of the quantum Harmonic oscillator and the Airy solution obtained by Berry and Balazs, as two applications, we determine the evolution of the Hermite-Gaussian and Airy beams in free space, in a linear and in a quadratic optical media. Finally, by using the point transformation, we construct two metrics whose geodesics correspond precisely to the classical trajectories for the harmonic oscillator and a particle in a uniform gravitational field.
According to De Broglie's idea of analogy, the relation between quantum mechanics and classical mechanics is similar to that between wave optics and geometric optics, we have given the quantum equation of the gravitational field intensity Eg(r,t) for matter particles, since the gravitational field intensity Eg(r,t) relatives to the particle position distribution function ψ(r,t), the quantum equation convert into the Schrodinger equation. In addition, we have further studied the essence and origin of quantum theory, and obtained some new results.