Abstract Einstein’s Equivalence Principle implies that the Lorentz force equation can be derived from a geodesic equation by imposing a certain (necessary) condition on the electromagnetic potential (Trzetrzelewski, EPL 120:4, 2018). We analyze the quantization of that constraint and find the corresponding differential equations for the phase of the wave function. We investigate these equations in the case of Coulomb potential and show that physically acceptable solutions do not exist. This result signals an inconsistency between Einstein’s Equivalence Principle and Relativistic Quantum Mechanics at an atomic level.
Abstract Einstein’s Equivalence Principle implies that the Lorentz force equation can be derived from a geodesic equation by imposing a certain (necessary) condition on the electromagnetic potential (Trzetrzelewski, EPL 120:4, 2018). We analyze the quantization of that constraint and find the corresponding differential equations for the phase of the wave function. We investigate these equations in the case of Coulomb potential and show that physically acceptable solutions do not exist. This result signals an inconsistency between Einstein’s Equivalence Principle and Relativistic Quantum Mechanics at an atomic level.
The covariance of the Schr\"odinger equation under Galilei boosts and the compatibility of nonrelativistic quantum mechanics with Einstein's equivalence principle have been constrained for so long to the existence of a superselection rule which would prevent a quantum particle from being found in superposition states of different masses. In an effort to avoid this expedient, and thus allow nonrelativistic quantum mechanics to account for unstable particles, recent works have suggested that the usual Galilean transformations are inconsistent with the nonrelativistic limit implied by the Lorentz transformation. Here we approach the issue in a fundamentally different way. Using a formalism of unitary transformations and employing quantum reference frames rather than immaterial coordinate systems, we show that the Schr\"odinger equation, although form variant, is fully compatible with the aforementioned principles of relativity.
The removal of the peculiar degeneration arising in the classical concepts of rest frame and time parametrization is at the heart of the recently formulated equivalence principle (EP). The latter, stating that all physical systems can be connected by a coordinate transformation to the free one with vanishing energy, univocally leads to the quantum stationary HJ equation (QSHJE). This is a third order nonlinear differential equation which provides a trajectory representation of quantum mechanics (QM). The trajectories depend on the Planck length through hidden variables which arise as initial conditions. The formulation has manifest p-q duality, a consequence of the involutive nature of the Legendre transformation and of its recently observed relation with second order linear differential equations. This reflects in an intrinsic ψ D -ψ duality between linearly independent solutions of the Schrödinger equation. Unlike Bohm's theory, there is a nontrivial action even for bound states and no pilot waveguide is present. A basic property of the formulation is that no use of any axiomatic interpretation of the wave function is made. For example, tunneling is a direct consequence of the quantum potential which differs from the Bohmian one and plays the role of particle's self-energy. Furthermore, the QSHJE is defined only if the ratio ψ D /ψ is a local homeomorphism of the extended real line into itself. This is an important feature as the L 2 (ℝ) condition, which in the Copenhagen formulation is a consequence of the axiomatic interpretation of the wave function, directly follows as a basic theorem which only uses the geometrical gluing conditions of ψ D /ψ at q=±∞ as implied by the EP. As a result, the EP itself implies a dynamical equation that does not require any further assumption and reproduces both tunneling and energy quantization. Several features of the formulation show how the Copenhagen interpretation hides the underlying nature of QM. Finally, the nonstationary higher dimensional quantum HJ equation and the relativistic extension are derived.
In this paper, the quantum dynamics was obtained in the framework of the general theory of relativity, where a quantum particle is described by a distribution of matter, with amplitude functions of the matter density, in the two conjugate spaces of the spatial coordinates and of the momentum, called wave functions. For a free particle, these wave functions are conjugate wave packets in the coordinate and momentum spaces, with time-dependent phases proportional to the relativistic Lagrangian, as the wave velocities in the coordinate space are equal to the distribution velocity described by the wave packet in this space. From the wave velocities of the particle wave functions, Lorentz’s force and the Maxwell equations were obtained. From the wave/group equation in the momentum space describing the Lorentz force, the expressions of the electric and magnetic fields as functions of the electric potential conjugated to time and of the vector potential conjugated to the coordinates in the particle-field Lagrangian were obtained. With these expressions, the electric and magnetic fields that satisfy the Faraday-Maxwell law of electromagnetic induction and the two Gauss-Maxwell laws of these fields were obtained. The Ampère-Maxwell law is obtained only by taking into account the physical consistency of the matter-field interaction of the equality of the propagation field velocity with the maximum relativistic velocity c. For a quantum particle in the electromagnetic field, dynamic equations in the coordinate and momentum spaces and the particle and antiparticle wave functions were obtained. It was shown that the electromagnetic potentials as functions of the coordinates describing the matter distribution of the quantum particle do not alter this distribution – under the action of an electromagnetic a quantum particle moves as a whole. The scattering or tunneling rate in an electromagnetic field, for the two possible cases, with the spin conservation, or inversion, were obtained. This description of a quantum particle as a distribution of matter with a density amplitude/wavefunction of the form of a wave packet, with the time-dependent phase proportional to the relativistic Lagrangian as a function of the metric tensor including also the gravitational field, enables the application of this theory in quantum gravity and quantum field theory in agreement with general relativity.
Abstract We present an exact quantum observable analog of the weak equivalence principle for a ‘relativistic’ quantum particle. The quantum geodesic equations are obtained from Heisenberg equations of motion as an exact analog of a fully covariant classical Hamiltonian evolution picture, with the proper identification of the canonical momentum variables as p µ , rather than p µ . We discuss the meaning of the equations in relation to projective measurements as well as equations with solution curves as ones in the noncommutative geometric picture of spacetime, and a plausible approach to quantum gravity as a theory about quantum observables as physical quantities including the notion of quantum coordinate transformation.
This is the second of the two related papers analyzing the origins and possible explanations of a paradoxical phenomenon of the quantum potential (QP). It arises in quantum mechanics (QM) of a particle in a Riemannian n-dimensional configurational space obtained by various procedures of quantization of non-relativistic natural Hamilton systems. Now, two questions are investigated: 1) Does a QP appear in non-relativistic QM generated by the quantum theory of a scalar field (QFT) non-minimally coupled to the space-time metric? 2) To which extent is it in accord with quantization of natural systems? To this end, the asymptotic non-relativistic equation for particle-interpretable wave functions and operators of canonical observables are obtained from the primary QFT objects. It is shown that, in globally static space-time, the Hamilton operators coincide at the origin of the quasi-Euclidean space coordinates in both alternative approaches for any non-minimality constant ξ~ , but a certain requirement of the Equivalence Principle to the quantum field propagator distinguishes the unique value ξ~=1/6 = 1/6. Just the same value had the constant ζ in the quantum Hamiltonians arising from the traditional quantizations of natural systems: the DeWitt canonical, Pauli-DeWitt quasiclassical, geometric and Feynman ones, as well as in the revised Schrödinger variational quantization. Thus, the QP generated by mechanics is tightly related to non-minimality of the quantum scalar field. Meanwhile, an essential discrepancy exists between the non-relativistic QMs derived from the two alternative approaches: QFT generates a scalar QP, whereas various quantizations of natural mechanics lead to QPs depending on the choice of space coordinates as physical observables and nonvanishing even in flat space if the coordinates are curvilinear.
A derivation is presented of the quantummechanical wave equations based upon the Equity Principle of Einstein's General Relativity Theory. This is believed to be more generic than the common derivations based upon Einstein's energy relationship for moving particles. It is shown that Schrodinger's Equation, if properly formulated, is relativisticly covariant. This makes the critisized Klein-Gordon Equation for spinless massparticles obsolete. Therefore Dirac's Equation is presented from a different viewpoint and it is shown that the relativistic covariance of Schrodinger's Equation gives a natural explanation for the dual energy outcome of Dirac's derivation and for the nature of antiparticles. The propagation of wave functions in an energy field is studied in terms of propagation along geodesic lines in curved space-time, resulting in an equivalent formulation as with Feynman's path integral. It is shown that Maxwell's wave equation fits in the developed framework as the massless limit of moving particles. Finally the physical appearance of electrons is discussed including a quantitative calculation of the jitter phenomenon of a free moving electron.