Topics in quantum dynamics (original) (raw)
Galilei general relativistic quantum mechanics
to appear, 1994
We present a general relativistic approach to quantum mechanics of a spinless charged particle, subject to external classical gravitational and electromagnetic fields in a curved space-time with absolute time. The scheme is also extended in order to treat the n-body quantum mechanics.
Galilei general relativistic quantum mechanics revisited
… e Outros Ensaios, AS Alves, FJ …, 1998
We review the recent advances in the generally covariant and geometrically intrinsic formulation of Galilei relativistic quantum mechanics. The main concepts used are Galilei-Newton space-time, Newtonian gravity and electromagnetism, space-time connection and cosymplectic form, quantum line bundle and quantum connection, Schrodinger equation and Hilbert bundle, quantisable functions and quantum operators. The paper contains a number of improvements and simplifications with respect to the already published or announced results.
Quantum Mechanics and General Relativity
Fundamental subjects of quantum mechanics and general relativity are presented in a unitary framework. A quantum particle is described by wave packets in the two conjugate spaces of the coordinates and momentum. With the time dependent phases proportional to the Lagrangian, the group velocities of these wave packets are in agreement with the fundamental Hamilton equations. When the relativistic Lagrangian, as a function of the metric tensor and the matter velocity field, is considered, the wave velocities are equal to the matter velocity. This means that these waves describe the matter propagation, and that the equality of the integrals of the matter densities over the spatial and the momentum spaces, with the mass in the Lagrangian of the time dependent phases, which describes the particle dynamics, represent a mass quantization rule. Describing the interaction of a quantum particle with the electromagnetic field by a modification of the particle dynamics, induced by additional terms in the time dependent phases, with an electric potential conjugated to time, and a vector potential conjugated to the coordinates, Lorentz’s force and Maxwell’s equations are obtained. With Dirac’s Hamiltonian, and operators satisfying the Clifford algebra, dynamic equations similar to those used in the quantum field theory and particle physics are obtained, but with an additional relativistic function, depending on the velocity, and the matter-field momentum. For particles and antiparticles, wavefunctions for finite matter distributions are obtained. The particle transitions, and Fermi’s golden rule, are described by the Lagrangian matrix elements over the Lagrangian eigenstates and densities of these states. Transition rates are obtained for the two possible processes, with the spin conservation or with the spin inversion. Dirac’s formalism of general relativity, with basic concepts of Christoffel symbols, covariant derivative, scalar density and matter conservation, the geodesic dynamics, curvature tensor, Bianci equations, Ricci tensor, Einstein’s gravitation law and the Schwarzschild matric elements, are presented in detail. From the action integrals for the gravitational field, matter, electromagnetic field, and electric charge, Lorentz’s force and Maxwell’s equations in the general relativity are obtained. It is also shown that the gravitational field is not modified by the electromagnetic field. For a black hole, the velocity and the acceleration of a particle are obtained. It is shown that, in the perfect spherical symmetry hypothesis, an outside particle is attracted only up to three times the Schwarzschild radius, between this distance and the Schwarzschild radius the particle being repelled, so that it reaches this boundary only in an infinite time, with null velocity and null acceleration. At the formation of a black hole, as a perfectly spherical object of matter gravitationally concentrated inside the Schwarzschild boundary, the central matter explodes, the inside matter being carried out towards this boundary, but reaching there only in an infinite time, with null velocity and null acceleration. In this way, our universe is conceived as a huge black hole. Based on this model, the essential properties, as big bang, inflation, the low large-scale density, the quasi-inertial behavior of the distant bodies, redshift, the dark matter and the dark energy, are unitarily explained. From the description of a gravitational wave by harmonically oscillating coordinates, the wave equation for the metric tensor is obtained, the propagation direction of such a wave being taken for reference. For a quantum particle as a distribution of matter interacting with a gravitational field, according to the proposed model, it is obtained that this field rotates with the angular momentum 2, called the graviton spin, as a rotation of the metric tensor which is correlated to the matter velocity, as the particle matter rotates with a half-integer spin for Fermions, and an integer spin for Bosons.
Classical and quantum study of the motion of a particle in a gravitational field
Journal of Mathematical Chemistry, 2005
The classical and the quantum mechanical description of a one-dimensional motion of a particle in the presence of a gravitational field is thoroughly discussed. The attention is centered on the evolution of classical and quantum mechanical position probability distribution function. The classical case has been compared with three different quantum cases: (a) a quantum stationary case, (b) a quantum non-stationary zero approximation case, where the wave packet has the shape of the first eigenfunction, and (c) a quantum non-stationary general case, where the wave packet is a superposition of stationary states.
2010
The literature on quantum-gravity-inspired scenarios for the quantization of space–time has so far focused on particle-physics-like studies. This is partly justified by the present limitations of our understanding of quantum gravity theories, but we here argue that valuable insight can be gained through semi-heuristic analyses of the implications for gravitational phenomena of some results obtained in the quantum space–time literature.