An imaging system is proposed for matter-wave functions, which is based on producing a quadratic phase modulation on the wavefunction of a charged particle, analogous to that produced by a space or time lens. The modulation is produced by co-propagating the wavepacket within an extremum of the harmonic vector and scalar potentials associated with a slow-wave electromagnetic structure. By preceding and following this interaction with appropriate dispersion, which is characteristic of a solution to the time-dependent Schrödinger equation, a system results that is capable of magnifying (i.e., stretching or compressing the space- and time-scales) and time-reversing an arbitrary quantum wavefunction.
An imaging system is proposed for matter-wave functions, which is based on producing a quadratic phase modulation on the wavefunction of a charged particle, analogous to that produced by a space or time lens. The modulation is produced by co-propagating the wavepacket within an extremum of the harmonic vector and scalar potentials associated with a slow-wave electromagnetic structure. By preceding and following this interaction with appropriate dispersion, which is characteristic of a solution to the time-dependent Schrödinger equation, a system results that is capable of magnifying (i.e., stretching or compressing the space- and time-scales) and time-reversing an arbitrary quantum wavefunction.
Optical temporal imaging is a technique of stretching or compressing an optical waveform in time, while fully preserving its temporal structure [1]. The key element of a temporal imaging system is a time lens, introducing a quadratic in time phase modulation into the signal field, which may be based, for example, on the sum-frequency generation (SFG). A quantum treatment of the time lens becomes important in some special cases, in particular when the information on some physical process is conveyed by single photons or by squeezed light [2,3].
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.
This paper aims to point out that a wave function is a description of a quantum spatiotemporal entanglement and the transformation between time and space. In any quantum mechanical representation, the real part of the wave function represents the space wave and the imaginary part represents the time wave, and they are the space-time itself. Time wave is not limited by space and dominates the nonlocality and integrity of a quantum. Matter wave is a four-dimensional space-time wave, and the basic unit of vacuum is a four-dimensional space-time element stationary relative to the observer. The essence of quantum measurement or interaction is that a conjugate condensation equivalent to that determined by inner product operation occurs between Space-time waves. Particle property is only the localization effect of quantum global collapse when quantum position measurement or equivalent interaction is made.
Wave mechanics, geometrical optics, classical Hamilton--Jacobi dynamics, electron optics, and gauge-coupled quantum mechanics are usually introduced as conceptually distinct frameworks. Yet they share a common mathematical structure: an environment modifies a local phase relation, the phase defines a local propagation momentum, and the resulting momentum field determines characteristic curves along which propagation occurs. This paper develops a propagation-first synthesis of these structures for particles with nonzero rest mass. For a nonrelativistic spinless particle with wave functionΨ(r,t)=R(r,t)eiS(r,t)/ℏ,the Schrödinger equation in scalar and electromagnetic potentials is exactly equivalent, wherever the amplitude is sufficiently regular, to a continuity equation and a quantum Hamilton--Jacobi equation,∂t∂R2+∇⋅[R2m∇S−qA]=0,∂t∂S+2m(∇S−qA)2+qϕ+U+Q=0,whereQ=−2mℏ2R∇2R. The gauge-invariant kinetic propagation momentum is thereforeΠ=∇S−qA,and the associated local propagation velocity isv=mΠ. We call this coupled amplitude--phase representation the \textbf{Unified Propagation System}. The name denotes an organizational framework rather than a new fundamental equation. Within this framework, external interactions can be interpreted as modifications of a local phase--dispersion structure. For static scalar potentials, spatial variation of the local propagation momentum defines an effective matter-wave refractive index and reproduces the optical--mechanical analogy familiar from Hamiltonian and electron optics. Electromagnetic vector potentials generalize this picture: they enter directly into the gauge-invariant phase gradient and therefore modify propagation direction in a manner that cannot in general be represented by a scalar refractive index. This motivates a generalized definition of refraction: \textbf{refraction is the bending of propagation characteristics caused by spatial or temporal variation of the local phase--dispersion relation}. In the semiclassical limit this definition reduces directly to geometrical ray bending. Beyond that limit, the full amplitude-dependent term Q provides an additional modification of the propagation structure. The same equations can be written in a second-order form,mdtdv=q(E+v×B)−∇U−∇Q,showing that force-based and propagation-based descriptions can represent the same dynamics at different levels. We therefore propose that electromagnetic deflection, potential scattering, and matter-wave refraction may be regarded as different manifestations of a common propagation principle rather than as fundamentally unrelated mechanisms. The paper deliberately avoids a deeper ontological claim. It does not assert that the wave function, electromagnetic field, and particle arise from a common physical substrate. It instead identifies the propagation structure that any such deeper account would have to reproduce. A relativistic extension based on the gauge-covariant Hamilton--Jacobi equation suggests that the same architecture survives beyond the nonrelativistic Schrödinger regime.
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.