arXiv subjects
Publications and source records attributed to E. Atzmon.
We show that the probabilistic distribution over the space in the spectator world, can be associated via noncommutative geometry (with some modifications) to a metric in which the particle lives. According to this geometrical view, the metric in the particle world is ``contracted'' or ``stretched'' in an inverse proportion to the probability distribution.
In a one-dimensional lattice, the induced metric (from a noncommutative geometry calculation) breaks translation invariance. This leads to some inconsistencies among different spectator frames, in the observation of the hoppings of a test particle between lattice sites. To resolve the inconsistencies between the different spectator frames, we replace the test particle's bare mass by an effective locally dependent mass. This effective mass also depends on the lattice constant - i.e. it is a scale dependent variable (a "running" mass). We also develop an alternative approach based on a compensating potential. The induced potential between a spectator frame and the test particle is attractive on the average. We then show that the entire formalism holds for a quantum particle represented by a wave function, just as it applies to the mechanics of a classical point particle.
Measuring distances on a lattice in noncommutative geometry involves square, symmetric and real ``three-diagonal'' matrices, with the sum of their elements obeying a supremum condition, together with a constraint forcing the absolute value of the maximal eigenvalue to be equal to 1. In even dimensions, these matrices are unipotent of order two, while in odd dimensions only their squares are Markovian. We suggest that these bi-graded Markovian matrices (i.e. consisting in the square roots of Markovian matrices) can be thought of as non-local Dirac operators. The eigenvectors of these matrices are spinors. Treating these matrices as determining the stochastic time evolution of states might explain why one observes only left handed neutrinos. Some other physical interpretations are suggested. We end by presenting a mathematical conjecture applying to q-graded Markovian matrices.
Using a simple analysis based on the measurement procedure for a quantized area we explain the 1/4 factor in the Bekenstein-Hawking black hole formula A/4 for the entropy.
In the following paper we continue the work of Bimonte-Lizzi-Sparano on distances on a one dimensional lattice. We succeed in proving analytically the exact formulae for such distances. We find that the distance to an even point on the lattice is the geometrical average of the ``predecessor'' and ``successor'' distances to the neighbouring odd points.