The short-distance behavior of an Abelian Proca model based on a one-parameter extension of the covariant Heisenberg algebra

Recently, a one-parameter extension of the covariant Heisenberg algebra with the extension parameter [Formula: see text] [Formula: see text] is a non-negative constant parameter which has a dimension of [Formula: see text] in a [Formula: see text]-dimensional globally flat spacetime has been presented which is a covariant generalization of the Kempf–Mangano algebra [see G. P. de Brito, P. I. C. Caneda, Y. M. P. Gomes, J. T. Guaitolini Junior and V. Nikoofard, Adv. High Energy Phys. 2017, 4768341 (2017) and A. Kempf and G. Mangano, Phys. Rev. D 55, 7909 (1997)]. The Abelian Proca model is reformulated from the viewpoint of the above one-parameter extension of the covariant Heisenberg algebra. It is shown that the free space solutions of the above modified Proca model describe two massive vector particles with different effective masses [Formula: see text] where [Formula: see text] is the characteristic length scale in our model. In addition, the Feynman propagator in momentum space for the modified Abelian Proca model is calculated analytically. Our numerical estimations show that the maximum value of [Formula: see text] in a four-dimensional spacetime is near the electroweak length scale, i.e. [Formula: see text]. We show that in the infrared/large-distance domain, the modified Proca model behaves like an Abelian massive Lee–Wick model which has been presented by Accioly and his co-workers in A. Accioly, J. Helayel-Neto, G. Correia, G. Brito, J. de Almeida and W. Herdy, Phys. Rev. D 93, 105042 (2016). The short-distance behavior of the modified Proca model is studied in the massless limit and the explicit forms of the inhomogeneous infinite derivative Maxwell equation and the infinite derivative Poisson equation are obtained. Finally, note that in the low-energy limit [Formula: see text], the results of this paper are compatible with the results of the usual Proca model.

COMPUTER SIMULATION FOR THE CROSSING TIME IN A DIPLOID, ASYMMETRIC, SHARPLY-PEAKED LANDSCAPE IN THE INFINITE POPULATION LIMIT

This study calculated the crossing time in the diploid mutation–selection model in an infinite population limit for various dominance parameters, h, and selective advantages, by switching on a diploid, asymmetric, sharply-peaked landscape, from an initial state which is the steady state in a diploid, sharply-peaked landscape. The crossing time for h < 1 was found to diverge at the critical fitness parameter, which increased with increasing selective advantage and decreased with increasing sequence length. When the sequence length was increased with a fixed extension parameter, there was no crossing time for h < 1 when the sequence length was longer than the critical sequence length, which increased with increasing selective advantage. The crossing time for h ≤ 1 was found to be an exponentially increasing function of the sequence length, and the crossing time for h > 1 became saturated at a long sequence length. The crossing time decreased with increasing selective advantage, mainly because the larger selective advantage caused the increase in relative density of the reversal allele to grow exponentially at an earlier time.

CROSSING TIME THROUGH THE FITNESS BARRIER IN AN ASYMMETRIC MULTIPLICATIVE OR ADDITIVE LANDSCAPE IN THE MUTATION-SELECTION MODEL

This study examines the dependence of crossing time on sequence length for a finite population in an asymmetric multiplicative, or additive, landscape with a positive asymmetric parameter and a fixed extension parameter in three types of mutation-selection model: the coupled discrete-time model, the coupled continuous-time model, and the decoupled continuous-time model. The crossing times for a finite population in the three types of mutation-selection model began to deviate from the crossing times for an infinite population at a critical sequence length. It then increased exponentially as a function of sequence length in the stochastic region where the sequence length was much longer than the critical sequence length. The exponentially increasing rates of the crossing times in the three types of mutation-selection model were similar to each other. These rates were decreased by increasing the asymmetric parameter. Once the asymmetric parameter reached a certain limit the crossing times for the three models in the stochastic region could not be decreased further by increasing the asymmetric parameter past this limit.

A FAMILY OF NONCOMMUTATIVE GEOMETRIES

It is shown that the noncommutativity in quantum Hall system may get modified. The self-adjoint extension of the corresponding Hamiltonian leads to a family of noncommutative geometry labeled by the self-adjoint extension parameters. We explicitly perform an exact calculation using a singular interaction and show that, when projected to a certain Landau level, the emergent noncommutative geometries of the projected coordinates belong to a one-parameter family. There is a possibility of obtaining the filling fraction of fractional quantum Hall effect by suitably choosing the value of the self-adjoint extension parameter.