Lagrangian formulation of an infinite derivative real scalar field theory in the framework of the covariant Kempf–Mangano algebra in a (D+1)-dimensional Minkowski space–time
In 2017, G. P. de Brito and co-workers suggested a covariant generalization of the Kempf–Mangano algebra in a (D+1)-dimensional Minkowski space–time (Kempf and Mangano, 1997; de Brito et al., 2017). It is shown that reformulation of a real scalar field theory from the viewpoint of the covariant Kempf–Mangano algebra leads to an infinite derivative Klein–Gordon wave equation which describes two bosonic particles in the free space (a usual particle and a ghostlike particle). We show that in the low-energy (large-distance) limit our infinite derivative scalar field theory behaves like a Pais–Uhlenbeck oscillator for a spatially homogeneous field configuration ϕ(t, x-vector )=ϕ(t). Our calculations show that there is a characteristic length scale δ in our model whose upper limit in a four-dimensional Minkowski space–time is close to the nuclear scale, i.e., δ{sub max}∼δ{sub nuclear scale}∼10{sup −15} m. Finally, we show that there is an equivalence between a non-local real scalar field theory with a non-local form factor K(x−y)=−((□{sub x})/((1−((δ{sup 2})/2)□{sub x}){sup 2})) δ{sup (D+1)}(x−y) and an infinite derivative real scalar field theory from the viewpoint of the covariant Kempf–Mangano algebra.