The Weil's integrality condition of prequantization is generalized to two-dimensional phase space with boundaries. It is shown that in the prequantization condition a term related to the symplectic potential on th...The Weil's integrality condition of prequantization is generalized to two-dimensional phase space with boundaries. It is shown that in the prequantization condition a term related to the symplectic potential on the boundary appears. The necessity of the generalized condition is proved by analyzing the isolated singularities of the Hermitian bundle while the sufficiency of the condition is proved via geometric construction on the space of equivalence class.展开更多
A new theorem on random walks suggest some possible revisions of the foundations of Quantum Mechanics. This is presented below in the simplified framework of the description of the evolution of a material point in spa...A new theorem on random walks suggest some possible revisions of the foundations of Quantum Mechanics. This is presented below in the simplified framework of the description of the evolution of a material point in space. Grossly speaking, it is shown that the probabilities generated by normalizing the square modulus of a sum of probability amplitudes, in the setup of Quantum Mechanics, becomes asymptotically close (under the appropriate limiting conditions) to the probabilities generated by the usual causal processes of Classical Mechanics. This limiting coincidence has a series of interesting potential applications. In particular it allows us to reintroduce the concept of causality within the core of Quantum Mechanics. Moreover, it suggests, among other consequences, that gravitational interaction may not even exist. Even though the interpretations of Quantum Mechanics which follow from this mathematical result may seem to bring some unexpected innovations in the context of theoretical physics, there is an obvious necessity to study its theoretical impact on Quantum Mechanics. The first steps toward this aim are taken in the present article.展开更多
文摘The Weil's integrality condition of prequantization is generalized to two-dimensional phase space with boundaries. It is shown that in the prequantization condition a term related to the symplectic potential on the boundary appears. The necessity of the generalized condition is proved by analyzing the isolated singularities of the Hermitian bundle while the sufficiency of the condition is proved via geometric construction on the space of equivalence class.
文摘A new theorem on random walks suggest some possible revisions of the foundations of Quantum Mechanics. This is presented below in the simplified framework of the description of the evolution of a material point in space. Grossly speaking, it is shown that the probabilities generated by normalizing the square modulus of a sum of probability amplitudes, in the setup of Quantum Mechanics, becomes asymptotically close (under the appropriate limiting conditions) to the probabilities generated by the usual causal processes of Classical Mechanics. This limiting coincidence has a series of interesting potential applications. In particular it allows us to reintroduce the concept of causality within the core of Quantum Mechanics. Moreover, it suggests, among other consequences, that gravitational interaction may not even exist. Even though the interpretations of Quantum Mechanics which follow from this mathematical result may seem to bring some unexpected innovations in the context of theoretical physics, there is an obvious necessity to study its theoretical impact on Quantum Mechanics. The first steps toward this aim are taken in the present article.
