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 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 M±(Λ) where Λ = l 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 Λ in a four-dimensional spacetime is near the electroweak length scale, i.e. Λmax ∼ lelectroweak ∼ 10−18 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 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 (Λ → 0), the results of this paper are compatible with the results of the usual Proca model.
