Abstract The transformation from a time-dependent random walk to quantum mechanics converts a modified Bessel function into an ordinary one together with a phase factor e,ir/2 for each time the electron flips both direction and handedness. Causality requires the argument to be greater than the order of the Bessel function. Assuming equal probabilities for jumps ± 1 , the normalized modified Bessel function of an imaginary argument is the solution of the finite difference differential Schrödinger equation whereas the same function of a real argument satisfies the diffusion equation. In the nonrelativistic limit, the stability condition of the difference scheme contains the mass whereas in the ultrarelativistic limit only the velocity of light appears. Particle waves in the nonrelativistic limit become elastic waves in the ultrarelativistic limit with a phase shift in the frequency and wave number of 7r/2. The ordinary Bessel function satisfies a second order recurrence relation which is a finite difference differential wave equation, using non-nearest neighbors, whose solutions are the chirality components of a free-particle in the zero fermion mass limit. Reintroducing the mass by a phase transformation transforms the wave equation into the Klein-Gordon equation but does not admit a solution in terms of ordinary Bessel functions. However, a sign change of the mass term permits a solution in terms of a modified Bessel function whose recurrence formulas produce all the results of special relativity. The Lorentz transformation maximizes the integral of the modified Bessel function and determines the paths of steepest descent in the classical limit. If the definitions of frequency and wave number in terms of the phase were used in special relativity, the condition that the frame be inertial would equate the superluminal phase velocity with the particle velocity in violation of causality. In order to get surfaces of constant phase to move at the group velocity, an integrating factor is required which determines how the intensity decays in time. The phase correlation between neighboring sites in quantum mechanics is given by the phase factor for the electron to reverse its direction, whereas, in special relativity, it is given by the Doppler shift.
Abstract The transformation from a time-dependent random walk to quantum mechanics converts a modified Bessel function into an ordinary one together with a phase factor e,ir/2 for each time the electron flips both direction and handedness. Causality requires the argument to be greater than the order of the Bessel function. Assuming equal probabilities for jumps ± 1 , the normalized modified Bessel function of an imaginary argument is the solution of the finite difference differential Schrödinger equation whereas the same function of a real argument satisfies the diffusion equation. In the nonrelativistic limit, the stability condition of the difference scheme contains the mass whereas in the ultrarelativistic limit only the velocity of light appears. Particle waves in the nonrelativistic limit become elastic waves in the ultrarelativistic limit with a phase shift in the frequency and wave number of 7r/2. The ordinary Bessel function satisfies a second order recurrence relation which is a finite difference differential wave equation, using non-nearest neighbors, whose solutions are the chirality components of a free-particle in the zero fermion mass limit. Reintroducing the mass by a phase transformation transforms the wave equation into the Klein-Gordon equation but does not admit a solution in terms of ordinary Bessel functions. However, a sign change of the mass term permits a solution in terms of a modified Bessel function whose recurrence formulas produce all the results of special relativity. The Lorentz transformation maximizes the integral of the modified Bessel function and determines the paths of steepest descent in the classical limit. If the definitions of frequency and wave number in terms of the phase were used in special relativity, the condition that the frame be inertial would equate the superluminal phase velocity with the particle velocity in violation of causality. In order to get surfaces of constant phase to move at the group velocity, an integrating factor is required which determines how the intensity decays in time. The phase correlation between neighboring sites in quantum mechanics is given by the phase factor for the electron to reverse its direction, whereas, in special relativity, it is given by the Doppler shift.
Classical mechanics of a free particle is based on Newtonian mechanics and Lagrangian/Hamilton theories. A single frame of reference is chosen (the lab) and particles with the same mass mo (for example) moving at different velocities are considered in the free particle scenario (i.e in the absence of force (potentials)). We argue that the above is identical to considering a single particle with mass mo at time=to and x=0 viewed by various frames moving at different constant velocities -v. From the point of view of each frame, the particle moves at a velocity v. Thus the Newtonian picture of an mo particle moving at different speeds may be considered as a problem of special relativity. We argue (as in (1)) that (x,t)=(0,t) transforms to (x’,t’) such that x’/t’=v. If one uses a metric (1 0) (first row) (0 -1) second row, one may derive special relativistic relationships. In particular, one may obtain the Lorentz invariant -Et + px if (p,E) and (x,t) transform in the same manner. We argue that the relationships obtained from special relativity pertain to classical mechanics and so one may mathematically obtain a function L(v) such that p=dL/dv and E=pv - L for a free particle. In other words, special relativity governs Lagrangian results in the free particle case and is not really needed (in this special case). More importantly, it is special relativity which links E with t and p with x. As noted above, if one considers Netwonian mechanics in a single frame, the notion of special relativity does not arise, but the scaling of time by E and of x by p does i.e. A= -Et+px is the classical action both relativistically and nonrelativistically. It is just not written in this way usually (i.e. A = -mo t sqrt(1-vv) (c=1) relatiivstically and A =t .5mo vv nonrelativistically) We argued in (2) that scaling must manifest itself physically and that it accounts for the physical wavelength in quantum mechanics. We argue that the quantum mechanics of a free particle ultimately follows from special relativity. As a result, ideas of quantum mechanics would not have appeared in Newtonian mechanics nor Lagrangian or Hamiltonian theories. When special relativity was developed it focused on transformations related to different frames, but these are equivalent to describing one frame with many different velocities (same mo). Thus the physics in one frame is governed by special relativity even if it is nonrelativistic. We argue that the notion of the wavelength which appears even in the Schrodinger formalism follows from scaling of x by p which ultimately originates from special relativity. In other words, both the Lagrangian formalism and quantum mechanics follow from special relativity in the free particle case. We argue that free particle quantum mechanics is the starting point of overall quantum mechanics.
In (1) a Galilean transformation x’ = x-vt and t’=t is applied to the Schrodinger equation id/dt (partial) W(x,t) = -1/2m d/dx d/dx W(x,t) for a free particle, where W(x,t) is the wavefunction, and it is argued that W’(x’,t’) does not equal W(x(x’t,t’),t(x’,t’)). Instead it is shown that W’(x’,t’) = W(x,t) exp( -i/bar [mvx’+.5mvvt’] with x=x’+vt’ and t=t’. The author’s of (1) suggest this is due to remnants of special relativity creeping into the nonrelativistic situation. They then argue that this issue is already found in nonrelativistic classical mechanics, in particular in the Hamiltonian-Jacobi equation H(x, dS/dx) + dS/dt = 0 where S is the free particle action= t *Lagrangian and H is the Hamiltonian i.e 1/2m dS/dx dS/dx. They show that S’(x’,t’) = S(x,t) - (mvx’ +.5mvvt’) and also argue that this is due to a remnant of special relativity creeping into nonrelativistic classical mechanics. In this note, we consider the free particle action S= -Et+px nonrelativistically. (This form also holds for the relativistic free particle). It may be seen that such a solution satisfies the Hamilton-Jacobi equation, but the catch seems to be that one must transform both (x,t) and (p,E) to find a solution in a moving frame. The Galilean transformation leads to two different transformations, one for (x,t) and another for (p,E). A Lorentz transformation on the other hand treats (x,t) and (p,E) in the same way and -Et+px behaves as a dot product with a (1,-1) type metric. We argue that given the physical example of a particle with rest mass mo at x=0 at t=to, a moving frame transforms this into a new energy (including mo and kinetic energy), a p and x’,t’ such that x’/t’=v where -v is the velocity of the frame. One may suggest a more general transformation than the Galilean i.e. replace t’=t with t’=g(v) t. This then leads to -Et+px written in terms of E’,t’, p’,x’ equalling -E’t’ + p’x’ if g(v) = 1/sqrt(1-vv/bb) where b is a constant to make v/b dimensionless. Applying the transformation, which we assume holds for both (x,t) and (p,E) to light yields b=c. One may then show that -Et+px, written in terms of E’,t’,p’,x’, yields -E’t’ + p’x’,
In special relativity, time and space are transformed according to the Lorentz transformation. Relativistic quantum mechanics is quantum mechanics that is consistent with special relativity. Nevertheless, relativistic quantum mechanics is only an approximation to a fully self-consistent relativistic theory of known particle interactions because it does not describe cases where the number of particles changes. In special relativity, mass can emerge from energy, which is not possible in quantum mechanics based on the conservation of probability. Dirac proposed the equation named after him to solve the difficulty encountered in the Klein-Gordon equation stemmed from the fact that it is a second-order differential equation in time.
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.
A new relativistic quantum wave equation has been derived by applying the quantum prescription to the momentum and the kinetic energy rather than to the momentum and the total energy, since after all it is the kinetic energy that generates the momentum. The resulting equation reduces to the Schrodinger equation in the nonrelativistic limit and to the Klein-Gordon equation for “massless particles” in the relativistic limit, i.e., if the velocity of the particle approaches that of light, c. For massive particles in general, the new equation deviates from the Klein-Gordon equation. The same equation is shown to decouple according to the Dirac formalism, yielding a modified form of Dirac equation. When applied to a rest particle, the modified Dirac equation is shown to avoid a negative energy solution and instead include a constant solution. The other, the time-dependent particle solution of the modified Dirac equation, has the characteristic frequency Mc2/(ћ/2) , i.e., twice those of the Dirac solutions, Mc2/ћ.
The Klein–Gordon equation, written down in 1926 by Oskar Klein and Walter Gordon (and earlier in unpublished form by Schrödinger himself), is the natural relativistic extension of quantum mechanics for spin‑zero particles: (□ + m²c²/ℏ²) Ψ = 0 where □ = (1/c²) ∂ₜ² − ∇². The equation is second‑order in both time and space, treats them symmetrically, and reduces to the free‑particle Schrödinger equation in the non‑relativistic limit. Its plane‑wave solutions satisfy the relativistic mass‑shell condition E² = p²c² + m²c⁴. Why is a relativistic version needed at all? The Schrödinger equation describes how quantum systems evolve when they’re moving slowly compared to the speed of light — fine for atoms, molecules, and everyday chemistry, where electrons move at less than 1% of c. But the Schrödinger equation is first‑order in time and second‑order in space, treating the two on different footings. Special relativity says space and time are unified into a single four‑dimensional structure where Lorentz transformations mix them, and any equation that privileges time over space breaks Lorentz invariance. A truly fundamental quantum equation must respect this symmetry. Klein–Gordon is the minimal scalar‑field equation that does. The standard derivation runs by analogy: start with the relativistic energy–momentum relation E² = p²c² + m²c⁴, replace E by iℏ∂ₜ and p by −iℏ∇, apply both sides to a wavefunction. This produces the right equation but it’s not really a derivation — it’s a guess motivated by the Schrödinger operator‑substitution procedure. This walkthrough shows how the Event Density framework derives the Klein–Gordon equation from substrate primitives — the same primitives that gave the Born rule, the Schrödinger equation, and the Bell–Tsirelson bound — plus the commitment to Lorentz covariance. The participation measure becomes Lorentz‑covariant. The unique second‑order Lorentz‑scalar differential operator on a scalar field is the d’Alembertian; the minimum‑order Lorentz‑scalar linear equation is therefore (□ + M²) Ψ = 0. Dimensional analysis fixes M² = m²c²/ℏ². Local U(1) gauge invariance forces the covariant derivative and the conserved four‑current. The non‑relativistic limit, via rest‑energy factorization, recovers the free‑particle Schrödinger equation exactly. The structural payoff: the Klein–Gordon equation is what you get when the substrate ontology is pushed through a Lorentz‑covariant reformulation. No new primitives are required. Relativistic and non‑relativistic quantum mechanics are not two theories glued together by analogy; they are the same substrate ontology applied at different velocity regimes. The walkthrough is honest about the known caveats — negative‑energy solutions and the non‑positive‑definite j₀ current — both of which resolve at the Dirac level and at QFT.