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

Technical Summary: Generalized Fock–Lorentz Transformations from Projective Conformal Coordinates

Problem Statement
The paper addresses the need for a unified and mathematically consistent formulation of the generalized Fock–Lorentz (FL) transformations. While standard FL transformations provide a nonlinear, projective realization of Lorentz symmetry using a single time-like conformal factor ($1 + ct/R$), it remains unclear how this construction generalizes when the conformal factor is replaced by an arbitrary linear form $1 + a_\mu x^\mu / R$. The authors aim to derive these generalized transformations to distinguish between time-like, space-like, and null projective sectors, clarifying the structure of invariant intervals, the role of the conformal factor, and the operational meaning of coordinate-dependent physical quantities.

Methodology
The construction relies on a family of auxiliary Minkowski coordinates, $X^\mu$, defined via a projective conformal map:
$$X^\mu = \frac{x^\mu}{1 + a_\nu x^\nu / R}$$
where $R$ is a deformation length scale and $a_\mu$ is a constant deformation vector. The methodology proceeds as follows:

1. Linear Action on Auxiliary Coordinates: Ordinary linear Lorentz transformations ($\Lambda^\mu_\nu$) are applied to the auxiliary coordinates $X^\mu$.
2. Induced Nonlinearity: By inverting the map, the authors derive the induced nonlinear transformations for the physical coordinates $x^\mu$. The denominator of the transformation is shown to be entirely determined by the conformal factor, eliminating arbitrary choices.
3. Sector Classification: The causal character of the deformation vector $a_\mu$ (time-like, space-like, or null) is used to categorize the transformations. The metric convention is $\eta_{\mu\nu} = \text{diag}(-1, 1, 1, 1)$.
4. Momentum-Space Dual: The authors apply the momentum-space dual of the FL conformal factor to the mass shell relation, defining a position-dependent "apparent mass" $m_{\text{app}}(x)$.
5. Quantum Application: Within the time-like sector, the authors construct symmetrized one-dimensional Klein–Gordon and Dirac oscillators by replacing the constant mass with the time-dependent apparent mass $m_{\text{app}}(t)$. They analyze these systems in the adiabatic regime.

Key Contributions and Results

• Generalized Transformations: The central result is the induced nonlinear transformation of physical coordinates:
  $$x'^\mu = \frac{\Lambda^\mu_\nu x^\nu}{D_a(x; \Lambda)}, \quad D_a(x; \Lambda) = 1 + \frac{a_\rho x^\rho - a_\rho \Lambda^\rho_\sigma x^\sigma}{R}$$
  This formulation renders the structure of the invariant interval transparent:
  $$\frac{c^2 t^2 - r^2}{(1 + a_\rho x^\rho / R)^2} = \text{invariant}$$

• Sector Distinctions:

  • Time-like ($a_\mu = (-1, 0, 0, 0)$): Recovers the standard FL transformations where the conformal factor depends only on time ($1 + ct/R$). The deformation is spatially homogeneous.
  • Space-like ($a_\mu = (0, \mathbf{n})$): The conformal factor depends on the spatial projection ($1 + \mathbf{n} \cdot \mathbf{r}/R$). This sector introduces anisotropy, singling out a preferred spatial axis.
  • Null ($a_\mu = (-1, \mathbf{n})$): The conformal factor depends on the light-front coordinate ($1 + (ct + \mathbf{n} \cdot \mathbf{r})/R$).

• Coordinate Velocity of Light: The paper clarifies that an explicit formula for the coordinate velocity of light is available immediately only in the time-like sector. In space-like and null sectors, the velocity must be obtained by solving an implicit linear vector equation, correcting the misconception that one can simply substitute spatial terms into the time-like velocity formula.

• Apparent Mass and Wave Equations: The deformation leads to an apparent mass $m_{\text{app}}(x) = m_0 / (1 + a_\mu x^\mu / R)$.

  • In the time-like sector, this yields a time-dependent mass $m_{\text{app}}(t)$. The authors construct a symmetrized Klein–Gordon oscillator and a Dirac oscillator. The resulting instantaneous spectra acquire explicit FL corrections:
    $$E^2_{n, \text{KG}}(t) = m^2_{\text{app}}(t)c^4 + 2m_{\text{app}}(t)c^2\hbar\omega\left(n + \frac{1}{2}\right)$$
    $$E^2_{n, \pm}(t) = m^2_{\text{app}}(t)c^4 + 2m_{\text{app}}(t)c^2\hbar\omega n_\pm$$
    As $R \to \infty$, these reduce continuously to standard relativistic oscillator spectra.
  • In the space-like sector, the spatial dependence of the mass introduces a weak-gradient anharmonicity. Specifically, a cubic term ($x^3$) breaks the parity symmetry of the ordinary oscillator. The authors note that a full quantum treatment here requires careful analysis of operator ordering and self-adjointness, which is beyond the scope of the exact spectra derived for the time-like case.

Significance and Claims
The paper claims to provide a "compact and systematic formulation" that removes ambiguities regarding the denominator in FL transformations, showing it is strictly determined by the conformal factor. The significance lies in:

1. Unification: Demonstrating that time-like, space-like, and null sectors are distinct projective structures rather than mere coordinate variations of a single transformation.
2. Operational Clarity: Distinguishing between coordinate-dependent velocities and locally measured invariant velocities, noting that local measurements of light speed and inertia remain $c$ and $m_0$ respectively due to projective rigidity.
3. Controlled Application: Providing explicit analytic spectra for relativistic oscillators in the time-like sector, serving as a testbed for how projective conformal factors modify quantum systems in the adiabatic regime.
4. Phenomenological Constraints: The authors modestly state that for cosmological scales of $R$, local variations are negligible. Observable effects would likely require nonlocal comparisons (e.g., high-redshift spectral ratios) or directional tests in anisotropic sectors.

The work concludes that the generalized FL construction is a deformation of special relativity controlled by the dimensionless ratio $a_\mu x^\mu / R$, offering a cleaner basis for future applications to deformed dispersion relations and relativistic kinematics.