On a Geometrical Description of Quantum Mechanics

Quantum Mechanics on Curved Hypersurfaces

In this work, Schrödinger and Dirac equations will be examined in geometries that confine the particles to hypersurfaces. For this purpose, two methods will be considered. The first method is the thin layer method which relies on explicit use of geometrical relations and the squeezing of a certain coordinate of space (or spacetime). The second is Dirac's quantization procedure involving the modification of canonical quantization making use of the geometrical constraints. For the Dirac equation, only the first method will be considered. Lastly, the results of the two methods will be compared an…

Geometrical description of spin-2 fields

We show that the torsion of a Cartan geometry can be associated to two spin-2 fields. This structure allows a new approach to deal with the proposal of geometrization of spin-2 fields besides the traditional one dealt with in General Relativity. We use the associated Hilbert-Einstein Lagrangian $R$ for generating a dynamics for the fields.

On the geometrical hypotheses underlying wave functions and their emerging dynamics

Classical mechanics for individual physical systems and quantum mechanics of non-relativistic particles are shown to be exceptional cases of a generalized dynamics described in terms of maps between two manifolds, the source being configuration space. The target space is argued to be 2-dimensional and flat, and their orthogonal directions are physically interpreted. All terms in the map equation have a geometrical meaning in the target space, and the pull-back of its rotational Killing one-form allows the construction of a velocity field in configuration space. Identification of this velocity …

BTZ Black Hole Entropy: A spin foam model description

We present a microscopical explanation of the entropy of the BTZ black hole using discrete spin foam models of quantum gravity. The entropy of a black hole is given in geometrical terms which lead us to think that its statistical description must be given in terms of a quantum geometry. In this paper we present it in terms of spin foam geometrical observables at the horizon of the black hole.

The Chrono'Geometrical Structure of Special and General Relativity: Towards a Background-Independent Description of the Gravitational Field and Elementary Particles

Since the main open problem of contemporary physics is to find a unified description of the four interactions, we present a possible scenario which, till now only at the classical level, is able to englobe experiments ranging from experimental space gravitation to atomic and particle physics. After a reformulation of special relativistic physics in a form taking into account the non-dynamical chrono-geometrical structure of Minkowski space-time (parametrized Minkowski theories and rest-frame instant form) and in particular the conventionality of simultaneity (re-phrased as a gauge freedom), a …

On the Geometrical Character of Gravitation

The issue of whether some manifestations of gravitation in the quantum domain, are indicative or not of a non-geometrical aspect in gravitation is discussed. We examine some examples that have been considered in this context, providing a critical analysis of previous interpretations. The analysis of these examples is illustrative about certain details in the interpretation of quantum mechanics. We conclude that there are, at this time, no indications of such departure from the geometrical character of gravitation.

Quantum-mechanical description of Lense-Thirring effect for relativistic scalar particles

Exact expression for the Foldy-Wouthuysen Hamiltonian of scalar particles is used for a quantum-mechanical description of the relativistic Lense-Thirring effect. The exact evolution of the angular momentum operator in the Kerr field approximated by a spatially isotropic metric is found. The quantum-mechanical description of the full Lense-Thirring effect based on the Laplace-Runge-Lenz vector is given in the nonrelativistic and weak-field approximation. Relativistic quantum-mechanical equations for the velocity and acceleration operators are obtained. The equation for the acceleration defines …

Critique of an Experiment Regarding Quantum States in a Gravitational Field and Possible Geometrical Explanations

We discuss an experiment conducted by Nesvizhevsky et al. As it is the first experiment claimed to have observed gravitational quantum states, it is imperative to investigate all alternative explanations of the result. In a student project course in applied quantum mechanics, we consider the possibility of quantummechanical effects arising from the geometry of the experimental setup, due to the "cavity" formed. We try to reproduce the experimental result using geometrical arguments only. Due to the influence of several unknown parameters our result is still inconclusive.

Differential Structures - the Geometrization of Quantum Mechanics

The usual quantization of a classical space-time field does not touch the non-geometrical character of quantum mechanics. We believe that the deep problems of unification of general relativity and quantum mechanics are rooted in this poor understanding of the geometrical character of quantum mechanics. In Einstein's theory gravitation is expressed by geometry of space-time, and the solutions of the field equation are invariant w.r.t. a certain equivalence class of reference frames. This class can be characterized by the differential structure of space-time. We will show that matter is the tran…

Geometrical models for the study of astrophysical systems with spheroidal symmetry imbedded in a standard cosmology: The case of cosmic voids

We present a class of general prolate and oblate spheroidal spacetimes for the description of cosmic structures in the Universe. They are exact geometries which represent, in an appropriated way, the imbedding of spheroidal matter-energy distributions within a standard cosmological scenario, and therefore they allow for an improved description of a wider class of astrophysical systems from a more accurate point of view. These spacetimes can be used to describe overdensity or underdensity regions; in this work we consider the last case, that is, the description of cosmic voids. We introduce and…

利用连续介质力学类比探讨引力波的几何方面：现状与潜在影响

Some geometrical aspects of gravitational waves using continuum mechanics analogy: State of the art and potential consequences

本研究利用连续介质力学与广义相对论的类比，从弹性与晶体塑性角度分析LIGO/VIRGO测得的空间变形，将引力波极化与等效弹性介质变形张量关联，并引入几何扭转以重建连贯的3D环境。

Spinors in Quantum Geometrical Theory

Spinors have played an essential but enigmatic role in modern physics since their discovery. Now that quantum-gravitational theories have started to become available, the inclusion of a description of spin in the development is natural and may bring about a profound understanding of the mathematical structure of fundamental physics. A program to attempt this is laid out here. Concepts from a known quantum-geometrical theory are reviewed: (1) Classical physics is replaced by a suitable geometry as a fundamental starting point for quantum mechanics. (2) In this context, a resolution is found for…

Quantum-Mechanical Description of Spin-1/2 Particles and Nuclei Channeled in Bent Crystals

General quantum-mechanical description of relativistic particles and nuclei with spin 1/2 channeled in bent crystals is performed with the use of the cylindrical coordinate system. The previously derived Dirac equation in this system is added by terms characterizing anomalous magnetic and electric dipole moments. A transformation to the Foldy-Wouthuysen representation, a derivation of the quantum-mechanical equations of motion for particles and their spins, and a determination of classical limit of these equations are fulfilled in the general case. A physical nature of main peculiarities of de…