Generalized Fock--Lorentz Transformations from Projective Conformal Coordinates and Their Application to One-Dimensional Relativistic Oscillators

We present a compact and systematic formulation of the generalized Fock--Lorentz (FL) transformations. The construction is based on a family of auxiliary Minkowski coordinates defined through a projective conformal map, $X^{\mu}=x^{\mu}/[1+a_{\nu}x^{\nu}/R]$, where $R$ denotes a deformation length and $a^{\mu}$ a constant deformation vector. Ordinary Lorentz transformations, acting linearly on $X^{\mu}$, thereby induce nonlinear transformations of the physical coordinates $x^{\mu}$. This formulation renders transparent the structure of the invariant interval, the role of the conformal factor, and the distinct limits associated with the time-like, space-like, and null (light-like) choices of $a^{\mu}$. We further clarify the operational meaning of the coordinate-dependent speed of light, as well as its momentum-space dual, which yields the apparent mass $\mapp(x)=m_{0}/[1+a_{\mu}x^{\mu}/R]$. As a controlled application, we construct a symmetrized one-dimensional Klein--Gordon oscillator and a one-dimensional Dirac oscillator in the time-like sector, for which $\mapp(t)=m_{0}/(1+ct/R)$. Within the adiabatic regime, the associated instantaneous spectra acquire explicit FL corrections and reduce continuously to the standard relativistic oscillator spectra in the limit $R\to\infty$. Finally, we determine the leading weak-gradient anharmonicity induced by a one-dimensional space-like deformation and identify the conditions required for a consistent quantum treatment.