Abstract In previous work, the Hamilton-Jacobi equation has been associated with the metrics of general relativity and shown to be a generalized Dirac equation for quantum mechanics. This lends itself to a natural definition of wave-particle duality in quantum mechanics. This theory is now further developed to show that a free spinless quantum particle moving with velocity v obeys the standard wave equation of electro-magnetism. We also discuss the implications for the zitterbewegung problem and its relationship to isotropy. Moreover, it is also shown that for the theory to be consistent, the momentum defined by the Hamilton-.!acobi function presupposes the existence of a universal parameter internal to the system. In the case of particles with mass this invariant can be defined by dX = dt/m(t) where t has the units of time and m = m(t) has the units of mass.
Abstract In previous work, the Hamilton-Jacobi equation has been associated with the metrics of general relativity and shown to be a generalized Dirac equation for quantum mechanics. This lends itself to a natural definition of wave-particle duality in quantum mechanics. This theory is now further developed to show that a free spinless quantum particle moving with velocity v obeys the standard wave equation of electro-magnetism. We also discuss the implications for the zitterbewegung problem and its relationship to isotropy. Moreover, it is also shown that for the theory to be consistent, the momentum defined by the Hamilton-.!acobi function presupposes the existence of a universal parameter internal to the system. In the case of particles with mass this invariant can be defined by dX = dt/m(t) where t has the units of time and m = m(t) has the units of mass.
The proposed paper presents the unobserved inadequacies in de Broglie's concepts of wave-particle duality and matter waves in the year 1923. The commonly admitted quantum energy or frequency expression hν=γmc2 is shown to be inappropriate for matter waves and is acceptable only for photons. The superluminal phase velocity expression c2/\u{psion}, for matter waves, is investigated in detail and is also reported to be inadequate. The rectifications in the inadequate concepts of de Broglie's theory and refinements in the analogy implementation between light waves and matter waves are presented, which provides the modified frequency and phase velocity expression for matter waves. Mathematical proofs for the proposed modified frequency and phase velocity expression are also presented. In accordance with the proposed concepts, a wave-particle duality picture is presented which elucidates the questions coupled with the wave-particle duality, existing in the literature. Consequently, particle type nature is shown to be a characteristic of waves only, independent from the presence of matter. The modifications introduced in the frequency expression for matter waves leads to variation in the wave function expression for a freely moving particle and its energy operators, with appropriate justifications. A new relation between the Kinetic energy and Momentum of the moving body is proposed and is subsequently applied to introduce novel General and Relativistic Quantum Mechanical Wave Equations. Applications of these equations in bound state quantum mechanical systems provide the information regarding particle's general and relativistic behavior in such systems. Moreover, the proposed wave equations can also be transformed into Schrödinger's and Dirac's equations. The interrelation of Schrödinger's, Dirac's and proposed equations with the universal wave equation is also presented.
A quantum particle is often described as exhibiting dual particle-wave nature. A particle seems to be an object which obeys Newton’s laws/special relativity with a wave also being a phenomenon existing in Newtonian mechanics (special relativity). In other words the particle and wave are governed by x-t external variables which have infinite resolution. It was originally thought that x and t in addition to being external were absolute, but special relativity changed this idea. We have suggested in previous notes that a free quantum particle with energy E and momentum p creates its own space time i.e time intervals are proportional to 1/E and space to 1/p. This means no external clock is needed and different internal rulers and clocks exist for different E,p. It also suggests finite time and space intervals. In fact, an absolute is introduced namely A=0 = -Et+px with t and x being independent. This suggests that t→b/E and x→b/p. A is the free particle Lagrangian (both relativistic and nonrelativistic) with x and t being considered independent. Thus we suggest that the external x ruler and t clock of Newtonian mechanics represent an average of the internal picture and are also described by A, but with x/t=v and not x and t being treated as independent. In (1) we showed that having discrete x spacing with a running time (i.e. Netwon’s time variable) and insisting on spatial invariance leads to dA/dx partial = p. Thus instead of considering particle wave duality we suggest an underlying wave structure which creates its own spacetime intervals (1/E, 1/p), but may be mapped in certain situations to the external running x and t of Newton’s theory. For example, consider an electron (a quantum particle) with a fixed p moving over a distance much greater than its wavelength. It may be mapped to Newton’s external running x,t values with x=vt, but this is an average mapping into continuous x and t. In reality the situation is more complicated because there may be a Bohm-Aharonov effect occurring. Thus the underlying wave picture is in fact physically important and linked to interactions. Thus a self-ruler is not simply a measuring device - interactions may occur unlike in the case of the external ruler. Given that p defines a kind of spatial invariance approximated by a periodic function exp(ipx) (eigenfunction of -id/dx) and that Newton’s running time is not needed in a quantum bound state equilibrium, one may wonder what one should do in the case of a particle bound in a V(x) potential. In Newtonian mechanics V(x) suggests non-spatial invariance and v(x) velocity changes with x as well in contrast with exp(ipx) which is periodically invariant. In fact Newtonian mechanics (and Lagrangian/Hamiltonian as well) focus on d/dt i.e x(t). We have argued above that time intervals 1/E are independent of 1/p spatial ones. It may be noticed, however, that different p values lead to different wavelengths and so even though there is a kind of spatial invariance for each, when compared to each other this invariance is broken and superpositions of exp(ipx) may create local x probability densities. At the same they create spatial momentum densities because -d/dx (d/dx) 1/2m yields a kind of average kinetic energy when normalized. This seems to describe the x-p uncertainty relationship. In order to create spatial resolution (other than the built-in one in the wavelength) one needs to superpose different momenta, each of which are spatially invariant according to their periodic scheme exp(ipx). Thus one may create an average 1/2m <pp> and obtain a vrms. This may be described in terms of Newtonian external running x and t variables which have infinite resolution, but this vrms does not map to a physical state i.e to exp(i (mvrms) x) localized at x. Rather the various exp(ipx)’s which create vrms are the physical entities as may be seen by knocking a particle out of a bound state. The physical exp(ipx) momenta states have spatial invariance and so cannot be associated with a specific position in space. Only their superposition creates the notion of specific position in space, but then there is a distribution of p. This seems to be the basis for the x-p uncertainty principle. Thus we suggest that the particle of particle-wave duality is created through a mapping of a physical wave scenario to a continuous (infinite resolution) x-t scheme which in reality is an abstraction because each p,E free particle state has it sown internal clock-ruler i.e. 1/E and 1/p time and space units. Given that 1/E and 1/p have finite values the idea of dt and dx tending to 0 only makes sense as an abstraction which only holds in certain situations.
Abstract An explanation of the quantum-mechanical particle-wave duality is given by the watt-less emission of gravitational waves from a particle described by the Dirac equation. This explanation is possible through the existence of negative energy, and hence negative mass solutions of Einstein’s gravitational field equations. They permit to understand the Dirac equation as the equation for a gravitationally bound positive–negative mass (pole–dipole particle) two-body configuration, with the mass of the Dirac particle equal to the positive mass of the gravitational field binding the positive with the negative mass particle, and with the mass particles making a luminal “Zitterbewegung” (quivering motion), emitting a watt-less oscillating positive–negative space curvature wave. It is shown that this thusly produced “Zitterbewegung” reproduces the quantum potential of the Madelung-transformed Schrödinger equation. The watt-less gravitational wave emitted by the quivering particles is conjectured to be de Broglie’s pilot wave. The hypothesised connection of the Dirac equation to gravitational wave physics could, with the failure to detect gravitational waves by the LIGO antennas and pulsar timing arrays, give a clue to extended theories of gravity, or a correction of astrophysical models for the generation of such waves.
A neo-classical relativistic mechanics theory is presented where the spin of an electron is a natural part of its space-time path as a point particle. The fourth-order equation of motion corresponds to the same Lagrangian function in proper time as in special relativity except for an additional spin energy term. The total motion can be decomposed into a sum of a local spin motion about a point and a global motion of this point, called the spin center. The global motion is sub-luminal and obeys Newton's second law in proper time, the time for a clock fixed at the spin center, while the total motion occurs at the speed of light c, consistent with the eigenvalues of Dirac's velocity operators having magnitude c. The local spin motion corresponds to Schrödinger's zitterbewegung and is a perpetual motion, which for a free electron has a circular path in the spin-center frame. In an electro-magnetic field, this spin motion generates magnetic and electric dipole energies through the Lorentz force on the electron's point charge. The corresponding electric dipole energy is consistent with the spin-orbit coupling term in the corrected Pauli non-relativistic Hamiltonian but the magnetic dipole energy is one half of that in Dirac's theory. By defining a spin tensor as the angular momentum of the electron's total motion about its spin center, the fundamental equations of motion can be re-written in an identical form to those of the Barut-Zanghi electron theory. These equations of motion can then be expressed using operators applied to a state function of proper time satisfying a Dirac-Schrödinger spinor equation. The operators produce dynamic variables without any probability implications. For the free electron, the state function satisfies Dirac's relativistic wave equation when the Lorentz transformation is applied to express proper time in terms of an observer's space-time coordinates.
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’,