It is shown that the alternative Klein-Gordon equation with positive definite probability density proposed in a letter by M.D. Kostin does not meet the requirements of relativistic (quantum) field theory and therefore does not allow for a meaningful physical interpretation. PACS code: 11.10.-z Keywords: Klein-Gordon equation, relativistic field theory. 2 The alternative formulation of the Klein-Gordon equation [1, 2] proposed by M.D. Kostin reads [3] i¯h ∂φ ∂t = +mc2φ + c ( ˆ⃗p ⃗ ψ) (1) i¯h ∂ ⃗ ψ ∂t = −mc2 ⃗ ψ + c ˆ ⃗pφ, (2) where φ(⃗r,t) and ⃗ ψ(⃗r,t) are ’scalar ’ and ’vector ’ probability amplitudes, respectively, and ˆ ⃗p = −i¯h ⃗ ∇. Defining the probability density P = φ ∗ φ + ( ⃗ ψ ∗ ⃗ ψ) (3) and the probability current density one readily derives the probability conservation equation ⃗S = c(φ ∗ ⃗ ψ + φ ⃗ ψ
We investigate the
d-dimensional Dunkl--Klein--Gordon equation for a scalar particle within an algebraic framework. By employing Schrödinger factorization, we construct the generators of the
su(1,1) algebra and establish the associated symmetry of the radial sector. The corresponding energy spectra are derived from unitary irreducible representations, and the Sturmian radial bases are obtained analytically. We analyze the
d-dimensional Dunkl--Klein--Gordon oscillator and the bound-state sector of the
d-dimensional Dunkl--Klein--Gordon equation with a Dunkl--Coulomb-like potential. Furthermore,
SU(1,1) coherent states are constructed and their time evolution is analyzed, revealing a characteristic radial oscillation behavior. The results show that the Dunkl deformation modifies the spatial and spectral structure of the system and, in the oscillator sector, introduces explicit parity-dependent energy shifts while preserving the underlying radial algebraic dynamics.