  Recently a one-parameter extension of the covariant Heisenberg algebra with
the extension parameter $l$ ($l$ is a non-negative constant parameter which has
a dimension of $[momentum]^{-1}$) in a $(D+1)$-dimensional Minkowski space-time
has been presented [G. P. de Brito, P. I. C. Caneda, Y. M. P. Gomes, J. T.
Guaitolini Junior and V. Nikoofard, Effective models of quantum gravity induced
by Planck scale modifications in the covariant quantum algebra, Adv. High
Energy Phys. 2017 (2017) 4768341]. 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 satisfy the modified dispersion relation
$\frac{\textbf{p}^2}{\left(1+\frac{\Lambda^2}{2\hbar^2}\textbf{p}^2\right)^2}=m^2c^2$
where $\Lambda=\hbar l$ is the characteristic length scale in our model. This
modified dispersion relation describes two massive vector particles with the
effective masses ${\cal
M}_{_\pm}(\Lambda)=\frac{2m}{1\mp\sqrt{1-2\left(\frac{mc\Lambda}{\hbar}\right)^2}}$.
Numerical estimations show that the maximum value of $\Lambda$ in a
four-dimensional space-time is near to the electroweak length scale, i.e.,
$\Lambda_{_{max}}\sim l_{_{electroweak}}\sim10^{-18}\; m$. 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
[A. Accioly, J. Helayel-Neto, G. Correia, G. Brito, J. de
  Almeida and W. Herdy, Interparticle potential energy for D-dimensional
electromagnetic models from the corresponding scalar ones, Phys. Rev. D 93
(2016) 105042].

High Energy Physics - Theory;General Relativity and Quantum Cosmology