The principle of invariance(also called phase harmonic law) in classical wave dynamics as a basic physical principle is discussed and an underlying intuitive reason which lies behind it——the order invariance of physical quantities is proposed.Such invariance reflects the principle of relativity.From phase harmonic law we deduced the transformation laws for wave vector,frequency,and group velocity of wave packets while space-time coordinates satisfying Galileo and Lorentz transformation,respectively.Furthermore we discussed the problem of transformation for Schrdinger equation and Klein-Gordon equation.For Schrdinger equation,we believe that the complex nature of wave function releases the wave function from invariance which is inevitable for classical waves(where only the real part of wave function have physical meaning),thus expanded the physical region.It becomes the basic quantum equation of nonrelativistic particles.Klein-Gordon equation,however,satisfies phase harmonic law,we believe that there should be some kind of order invariance for physical quantities behind it.We discussed one possibility from professor Guangjiong Ni's two-component(particle and antiparticle) notion to identify such an order.
The principle of invariance(also called phase harmonic law) in classical wave dynamics as a basic physical principle is discussed and an underlying intuitive reason which lies behind it——the order invariance of physical quantities is proposed.Such invariance reflects the principle of relativity.From phase harmonic law we deduced the transformation laws for wave vector,frequency,and group velocity of wave packets while space-time coordinates satisfying Galileo and Lorentz transformation,respectively.Furthermore we discussed the problem of transformation for Schrdinger equation and Klein-Gordon equation.For Schrdinger equation,we believe that the complex nature of wave function releases the wave function from invariance which is inevitable for classical waves(where only the real part of wave function have physical meaning),thus expanded the physical region.It becomes the basic quantum equation of nonrelativistic particles.Klein-Gordon equation,however,satisfies phase harmonic law,we believe that there should be some kind of order invariance for physical quantities behind it.We discussed one possibility from professor Guangjiong Ni's two-component(particle and antiparticle) notion to identify such an order.
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.
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.
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 work develops a phase-based formulation of relativistic wave dynamics, demonstrating that the Klein--Gordon equation emerges naturally from the foundational assumption of intrinsic phase periodicity in material fields. Mapping the phase directly onto the classical action, we postulate that localized excitations possess an invariant rest-frame oscillation governed by a proper frequency ω0. This physical condition establishes an operational mass-frequency relation, m=ℏω0/c2, without requiring rest energy as an independent, axiomatic input. We show that the Klein--Gordon equation arises as the minimal local, linear, Lorentz-invariant field equation compatible with this internal phase structure. Within this framework, mass acts as an intrinsic frequency scale governing wave propagation, and relativistic kinematics is fully recovered as a structural consequence of phase coherence. This approach provides a unified wave-mechanical interpretation where particle dynamics maps onto the group velocity of dispersive wave packets, offering an intuitive account of free propagation, dispersion, and tunneling across potential barriers.
Ψ∂ (4.1) The harmonic trial solution of this equation with respect to a fixed inertial frame ( is given by )tx, ( ) ( ) ( )ctxikectxktx −Ψ=Ψ−Ψ=Ψ 00 orcos, In another inertial frame( ),, tx ′′ considered to be in uniform translational motion with velocity as described by the Galilean transformation (2.13), with respect to the fixed frame, the wave is represented by a wave function ,v ( ) ( ), related to ,x t x′ ′Ψ Ψ t as 2 2i.e., x t x x x t x x x x ′ ′∂Ψ ∂Ψ ∂ ∂Ψ ∂ ∂Ψ ∂ Ψ ∂ Ψ= + = =′ ′ ′ ′∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ and 2 2i.e., 2 x t v v v t x t t t x t x tt x ′ ′∂Ψ ∂Ψ ∂ ∂Ψ ∂ ∂Ψ ∂Ψ ∂ Ψ ∂ Ψ ∂ Ψ ∂ Ψ= + = − + = − +′ ′ ′ ′ ′ ′ ′∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 2t∂ ∂ ∂ where use has been made of known properties of partial derivatives. It follows that obeys the following wave equation, with respect to the moving inertial frame ( tx ′′Ψ , ) ( ) tx v tx vc ′∂′∂ Ψ∂−′∂ Ψ∂=′∂ Ψ∂− 2 22 2 (4.2) which is obviously different from the wave equation (4.1) that holds in the fixed inertial frame. In other words the electromagnetic wave equation is not invariant with respect to a uniform translation of the inertial frame, using a Galilean transformation. Inserting into Eq.(4.2) harmonic solution written as we obtain 2 2 2 2 2 2( ) ( ) 2 ( ) i.e., orc v k k w v k w w v c w c− − = − − = − ± = ± v (4.3) Hence Galileo's transformation predicts that the velocity of light should be different from c, if measured with respect to a moving inertial frame. We therefore have a choice to make before we can fully understand the electromagnetic phenomena. We can either accept Galileo's principle of relativity which states that all inertial frames are equivalent or the Galilean transformation (2.13). We can sacrifice Galileo's principle of relativity by saying that there is a privileged inertial frame, called the ether, in which it is supposed that the electromagnetic wave velocity is c as given by Maxwell's equations, although they made no essential reference to the ether. A long series of investigations has been carried out, in order to detect some effect of the motion of the earth through the ether, always with negative results. All of the experiments (the best known being that of Michelson-Morley) have indicated that the speed of light is always the same in all directions and is independent of the relative motion of observer, source and transmitting medium. Einstein has reaffirmed Galileo's principle of relativity on the equivalence of all the inertial frames and, in addition, has assumed as a basic postulate the experimental fact that the speed of light is always constant. As a consequence, the Galilean transformation had to be replaced by another, in order to preserve the same speed of light in all systems. Einstein's basic assumptions are known as the postulates of special relativity: 1. The laws of physics take the same form in all inertial systems 2. The speed of light in empty space is a universal constant c and is the same for all observers. It is independent of the state of motion of the emitting body Special relativity is only concerned with transformations between inertial frames, as indicated by the restriction contained in the first postulate. General relativity covers reference frames in arbitrary relative motion, and is based on the principle of equivalence, confirmed experimentally, which unifies gravitational and inertial forces. The first of Einstein's postulates implies that it is only possible, by means of physical measurements, to demonstrate that two gravity-free reference frames are moving relative to each other. There is no physical basis to infer that any such frame is intrinsically at rest or in uniform motion. The notion of absolute rest is not consistent with the first postulate. The second postulate is in contradiction with the Galilean transformation which would preserve the form of Newton's laws of motion, as illustrated by Eq.(4.3). Therefore, we must find a transformation between two frames moving with uniform relative motion, which is consistent with the second postulate. We must then generalize the laws of Newtonian mechanics in a form which is invariant under that transformation, as required by the first postulate. A fundamental requirement of special relativity is that the laws of Newtonian mechanics are valid in the low speed limit and must be obtained as limiting cases of relativistic formulae for .cv << This assumption plays, for special relativity, a role analogous to that of Bohr's correspondence principle in quantum mechanics (see Chapter 28). 4.2. THE LORENTZ TRANSFORMATION Einstein's procedure is to locate an event in a reference frame by its spatial Cartesian coordinates x, y and z and a temporal coordinate t. Consider two frames S and S' moving with uniform relative motion, such that the Cartesian axis system in coincides with that in S, Oxyz, at time ,S ′ zyxO ′′′′ .0==′ tt As illustrated in Figure 4.1, the x and axes are parallel to the direction of relative motion: in moves in the x direction with velocity v, and in x′ , the frameS ′S S, the frameS ′ moves with velocity in the direction. A pulse of light emitted from O at time v− x′ 0=t spreads as a spherical wave travelling with speed c, so that the equation of the wave front in S, at a time t will be (4.4) 22222 tczyx =++ 0 0' z' y' x,x' z y Figure 4.1. Cartesian axes of two inertial frames in uniform relative motion The second postulate requires that the wave front in S ′ can be represented by (4.5) 22222 tczyx ′=′+′+′ In other words an observer in also sees a spherical wave travelling from its origin if not, any change of the wave front could be used to infer that the system is moving uniformly instead of being at rest. The invariance of the expression S ′ ,O′ S ′ (4.6) 2222222222 tczyxtczyx ′−′+′+′=−++ is a basic requirement that the transformation between the two systems must satisfy. The Galilean transformation (2.13), which in our case reads ttzzyyvtxx =′=′=′−=′ ,,, is not consistent with the invariance property (4.6), as it gives xvttvtczyxtczyx 2222222222222 −+−++=′−′+′+′ The first postulate brings some simplification to the possible form of the desired transformation. Since a uniform rectilinear motion in S must go over into a uniform rectilinear motion in S', the transformation must be linear. A nonlinear transformation would predict acceleration in S' even if the velocity were constant in S. The choice of the origins in space and time has avoided constant terms, and the transformation will be homogenous. The directions perpendicular to the relative motion, which are effectively at rest, must be left unchanged by the transformation. We then take the general form BtAxx +=′ yy =′ (4.7) zz =′ FtDxt +=′ Further simplification comes from the argument that the origin O', which has the coordinate moves along the x axis with velocity ,0=′x ,/ vdtdx = that is ABvBtAx /or0 −=+= while the origin O, with coordinate ,0=x moves along x-axis with velocity Recalling that when , we can write .'/' vdtdx −= 0=x 0=t 00 ort D B Bx A B t v F F F ′ − ⋅′ ′= ⋅ + = − = so that Inserting now Eqs.(4.7) into Eq.(4.5), we obtain the equation .FA = ( ) ( ) 02 22222222222222222 =−+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −−++−=′−′+′+′ DcBAxt c BAtczyxDcAtczyx The identity (4.6) requires that 2222 =−=−=− DcB c BADcA Since ,B vA= we obtain 2222 /1 /, /1 , /1 cv cvD cv vB cv A − −= − −= − = (4.8) so that Eqs.(4.7) now read 22 /1 cv vtxx − −=′ yy =′ (4.9) zz =′ /1 / cv cvxtt − −=′ The four equations (4.9) are known as the Lorentz transformation, which in this case applies to the relative motion of inertial frames along the x-axis. It is a straightforward matter to show that Eqs.(4.9) are consistent with the invariance relation (4.6), which is thus a sufficient as well as a necessary condition for the Lorentz transformation. With the usual abbreviations ( )1 /1 1, 222 ≥ − = − == γ β γβ cvc v (4.10) the Lorentz transformation simplifies to ( ) ( )x x ct x x ctγ β γ β′ ′= − = + ′ a) yyyy ′==′ or b) (4.11) zzzz ′==′ c) ⎟⎠ ⎞⎜⎝ ⎛ ′+′=⎟⎠ ⎞⎜⎝ ⎛ −=′ x c ttx c tt βγβγ d) The right hand side shows the inverse transformations where the primed and unprimed quantities are interchanged and the sign of the relative velocity is reversed, because its direction is the only difference between the systems S and S'. In the low velocity limit the Lorentz transformation reduces to the Galilean transformation since the only difference consists in the presence of the factor γ and the term ,1< that, in Eq.(4.11a), γ indicates a change of measuring stick length and in Eq.(4.11d) a change in the clock rate in the Lorentz transformation. The term ( )xc/β− in Eq.(4.11d) shows that the time is dependent on the position of a given event. Therefore, the immediate consequences of the Lorentz transformation require the revision of the usual concepts of simultaneity, time and length. Simultaneity Consider two events occuring at the same time ttt == 21 at two different positions and in the unprimed system S. According to Eqs.(4.11) the events are recorded in the primed system S' at different times 1x 2x / /, 1 1 t x c t xt t cβ ββ β − −′ ′= = − − so that the apparent time interval in S' is given by ( ) 0 1 / 2 21 12 ≠− −=′−′=′∆ β β cxx ttt (4.12) and hence, it is a function of the relative velocity v. The events will not appear as simultaneous to observers in S'. We are then forced to abandon the intuitive idea of absolute simultaneity, which is based on our experience with objects of low velocity, and accept the relativity of simultaneity, which depends on the coordinate system and applies in the domain of high velocities. Length contraction Consider a rod at rest in the primed system S' lying along the x-axis and having the length 12 xx ′−′=λ given by the distance between its ends 21 and xx ′′ measured at the same instant of time t'. The length in an inertial frame in which the rod is at rest is called its proper length. To define length in a frame S in which the rod is moving, an observer must locate the position of both end points, 21 and xx ′′ , at the same time From the inverse equations (4.11) we obtain 1 2 .t t t= = 1 1 2 1 x ct x ct x xx x x xβ ββ β β − −′ ′ ′ ′= = − = − − − − so that the apparent length 12 xxl −= is 21 βλ −=l (4.13) This equation is known as the Lorentz contraction and states that a rod parallel to the direction of its motion is shortened by the factor .1 2β− It results from the fact that simultaneous events in the rest frame of one rod are not simultaneous in the rest frame of the other. A rod transverse to its direction of motion is unchanged in length. The proper length λ is independent of the orientation of the rod, as a manifestation of the isotropy of inertial rest frame S'. If the orientation of a rod is neither longitudinal nor transverse to the direction of relative motion, both the length and the inclination are changed in the laboratory frame S, as an effect of a contraction in the direction of the relative motion with no change of transverse dimensions.
A physics-oriented review is first given of wave equations: examples, the Cauchy problem, solutions in various coordinates, their symmetries, how to build a wave packet. Fourier analysis and dispersion relations are then introduced, and here the key tool is the symbol of differential operators, which makes it possible to study them in terms of algebraic polynomials involving cotangent-bundle variables. Further basic material deals with geometrical optics from the wave equation, phase and group velocity (and their dual relationship in momentum space), the Helmholtz equation and the eikonal approximation for the scalar wave equation. The Schrödinger equation is then derived with emphasis on the wave packet and its relation to classical behaviour in the light of the Einstein–de Broglie relation. For this purpose, it is shown that it is possible to build a wave packet for which the wave-like properties manifest themselves for distances of the order of atomic dimensions. The Fourier transform with respect to time of the wave packet obeys the stationary Schrödinger equation, while the wave function is found to obey a partial differential equation which is of first order in time. The wave equation The wave equation occurs in several branches of classical physics, e.g. the theory of sound, electromagnetic phenomena in vacuum and in material media, and elastic vibrations of material bodies. We consider first the simplest situation, i.e. the propagation of waves in a homogeneous, isotropic and stationary medium. Propagation is characterized by the refractive index n , a quantity that is independent of the point in space-time.