Klein--Gordon and Dirac Oscillators with an Apparent Mass Induced by the Momentum-Space Dual of the Fock--Lorentz Transformations

Abstract

We propose a controlled momentum-space dual of the Fock--Lorentz (FL) transformations and use it to derive a deformed relativistic mass shell. Restricting the FL conformal factor to the cosmological-frame world line $\vx=0$, the invariant relation takes the form $(E^{2}-\vp^{2}c^{2})(1+ct/R)^{2}=m_{0}^{2}c^{4}$, which is equivalent to the standard special-relativistic dispersion law with a time-dependent apparent mass $\mapp(t)=m_{0}/(1+ct/R)$. Canonical quantization then yields Klein--Gordon (KG) and Dirac equations containing a slowly varying mass scale. We show explicitly that squaring the Dirac equation reproduces the KG operator, modulo first-order corrections proportional to $\dot\mapp$ that are suppressed by the ratio of the Compton wavelength to the FL scale. The construction is not presented as a unique covariant phase-space theory; rather, it is a world-line ansatz designed to isolate the spectral consequences of the FL conformal factor. As applications, we study the one-dimensional KG and Dirac oscillators. In the adiabatic regime, governed by the small parameter $\epsilon=c/(R\omega)\ll1$, closed-form instantaneous spectra are obtained. The Dirac-oscillator calculation is carried out in component form and then reduced to the physical spinor spectrum, thereby avoiding the double counting of the upper and lower component ladders. Dimensionless plots illustrate the apparent-mass drift, the induced spectral evolution, and the domain of adiabatic validity. For cosmological values of $R$, non-adiabatic corrections are entirely negligible; in the formal limit $t\to\infty$ the apparent mass tends to zero and, for fixed quantum number, the instantaneous levels collapse toward $E=0$.