Critique of a Proposed Dynamical Group for Relativistic Quantum Mechanics

The dynamical group ${\tilde{G}}_5$ for relativistic quantum mechanics phenomenologically suggested by Aghassi, Roman, and Santilli is derived from the analysis of symmetry properties of Lagrangians and corresponding equations of motion for a free relativistic particle. All physical observables such as position, momentum, angular momentum, and mass squared are represented by well-defined operators which close the algebra of the dynamical group ${\tilde{G}}_5$. The unitary irreducible representations of this group, which are possible states of the physical system, are found. The particles accommodated in the single unitary irreducible representations have various spins starting from the lowest spin value and going up to infinity in integral steps. The mass-squared operator $P_{\mu }{P}^{\mu }$ lies in the enveloping algebra of ${\tilde{G}}_5$, and its eigenvalues are not necessarily quantized and can have any positive or negative values. It is pointed out that this group has several failures and thus it cannot be accepted as the reliable dynamical group for particles within relativistic quantum mechanics.