Deformed Galilean Relativity & Extended Kinematics

• Deformed Galilean Relativity is a theoretical framework that generalizes classical symmetry by incorporating invariant parameters c and R to modify inertial transformations.
• It replaces standard Galilean boosts with Lorentz-type and Fock–Lorentz inverted translations, linking Minkowski spacetime with anti–de Sitter structures.
• The framework adapts particle kinematics and action functionals, yielding novel dispersion relations relevant to high-energy, cosmological, and quantum gravity regimes.

Deformed Galilean Relativity denotes a class of theoretical frameworks and mathematical constructions in which the conventional Galilean symmetry of nonrelativistic physics is generalized or algebraically deformed. Such deformations typically introduce new invariant parameters or structural corrections, yielding nonstandard representations of inertial motion, modified kinematics, and alternative realizations of otherwise familiar symmetry groups, such as the Poincaré or anti-de Sitter groups. These generalizations form an important bridge between classical, nonrelativistic kinematics and the highly structured symmetries encountered in quantum gravity, high-energy phenomena, and cosmological-scale considerations.

1. Extended Galilei Algebra and Its Deformation

The extended Galilei algebra—composed of generators for time translations ($H$), spatial translations ($P_i$), Galilean boosts ($K_i$), and rotations ($J_i$)—underpins the symmetries of Newtonian inertial motion. A salient result (Manida, 2011) is that this algebra admits at least two distinct nontrivial deformations:

• Standard “Relativization” with invariant velocity $c$:

The customary deformation replaces Galilean boosts ($K_i = t\partial_i$) by Lorentz boosts,

$L_i = t\partial_i + \frac{1}{c^2} r_i \partial_t,$

thereby introducing $c$ as an invariant speed and yielding the familiar Poincaré group structure, with Minkowski spacetime metric $ds^2 = c^2dt^2 - d\mathbf{r}^2$.

• "Inverted" Translations with invariant length $R$:

A distinct deformation replaces spatial translations ($P_i$) by

$$F_i = -\partial_i + \frac{1}{R^2}\bigl(t\, r_k \partial_k + r_i r_k \partial_k\bigr),$$

where $R$ is an invariant length parameter. The finite transformations generated by $F_i$ are Fock–Lorentz (linear fractional) transformations:

$$t' = \frac{t - a^0}{1+\frac{a \cdot r}{R^2}},\quad r'_i = \frac{r_i - a_i}{1+\frac{a \cdot r}{R^2}}.$$

These two deformations are mathematically and physically distinct: the boost deformation introduces an invariant velocity, while the translation deformation introduces an invariant length, mixing coordinates in a non-affine manner and leading to nonstandard Poincaré symmetry realization.

2. Algebraic Structure: Combination and Anti-de Sitter Realization

Combining the two deformations—the Lorentz boosts (parametrized by $c$) and the “inverted” translations (parametrized by $R$)—produces a symmetry algebra isomorphic to that of Anti-de Sitter (AdS) space in Beltrami coordinates (Manida, 2011). Introducing rescaled generators $\tilde{L}_i = c L_i$, $\tilde{F}_i = R F_i$, and an additional generator $B = \frac{R}{c}H - \frac{c}{R}A$, the algebra is expressed via dimensionless structure constants. The fundamental commutators are of the form

$$[J_i, J_j] = \epsilon_{ijk}J_k,\quad [J_i, \tilde{L}_j] = \epsilon_{ijk}\tilde{L}_k,\quad [J_i, \tilde{F}_j] = \epsilon_{ijk}\tilde{F}_k,$$

and

$$[\tilde{F}_i, \tilde{F}_j] = -\frac{1}{R^2} \epsilon_{ijk} J_k,$$

where $A$ corresponds to an “inverse” time translation and the full algebra supports the AdS(3,2) group structure.

This combined structure encodes both $R$ and $c$, and, crucially, interpolates between Minkowski spacetime (the $R\to\infty$ limit) and “R-space” ($c\to\infty$), where Fock–Lorentz transformations dominate. The unification clarifies the algebraic flexibility of the extended Galilei algebra and frames spacetime symmetries in terms of a cosmological length scale and the speed of light.

3. Deformed Particle Kinematics and Action Functionals

The modifications in the symmetry algebra are reflected at the level of particle kinematics. The classical action for a free point particle in this deformed setting is

$$S = -\sum_n m_n c^2 R^2 \int \frac{\sqrt{1-\frac{1}{R^2}(\mathbf{r}_n - t\,\dot{\mathbf{r}}_n)^2}}{R^2 + c^2 t^2 - \mathbf{r}_n^2}\, dt,$$

explicitly incorporating both $R$ and $c$ (Manida, 2011). The resulting equations of motion and conserved quantities generally exhibit dependence on both invariants.

The dispersion relation becomes (see eq. (45) of (Manida, 2011)): $K^2 = H^2 - m^2,$ where energy $H$ and momentum $K$ are modified by $R$-dependent terms. In the “R-space” kinematics (the $c\to\infty$ limit with $R$ finite), the minimal energy occurs for motion satisfying $v = r/t$, i.e., “Hubble-like” velocity scaling. Thus, the relativistic kinematics are sensitive to both a limiting speed and an invariant length, with the standard relations recovered only as $R\to\infty$.

4. Physical Interpretation and Limit Transitions

The role of the fundamental invariant parameters is as follows:

• The invariant velocity $c$ arises from the deformation of boost generators and encapsulates the standard relativistic speed limit.
• The invariant length $R$ emerges from the deformation of translation generators and reflects a cosmological scale, above which the non-affine transformation properties become significant.

The framework encompasses two singular limit transitions:

Limit	Remaining structure	Physical regime
$R\to\infty$, $c$ finite	Ordinary (Minkowski) kinematics	Standard special relativity
$c\to\infty$, $R$ finite	“R-space” kinematics	Alternative relativity with cosmological length scale

These transitions allow interpolation between familiar relativistic kinematics and novel regimes where the cosmological scale $R$ fundamentally alters the transformation properties and particle dynamics.

5. Conceptual and Cosmological Implications

Deforming different subalgebras of the extended Galilei group—first by an invariant $c$ (boosts), then by an invariant $R$ (translations)—leads to distinct, and in combination, unified, nonstandard realizations of the Poincaré group (Manida, 2011). The introduction of $R$ suggests the possibility of a cosmologically motivated generalization of Minkowski space, relevant when lengths approach the observable universe size. This realization provides a route to incorporating cosmological scales directly into symmetry considerations, echoing the structure of anti–de Sitter space.

A plausible implication is that in regimes where $|r| \sim R$, the Fock–Lorentz nonlinearity becomes non-negligible, potentially leading to observable deviations from standard locality and relativity principles, particularly in early-universe or high-redshift cosmology where distances approach $R$.

6. Broader Frameworks and Limitations

The methodology outlined in (Manida, 2011) systematically replaces Galilean boosts with Lorentz-type generators and spatial translations with “inverted” generators, yielding a symmetry algebra admitting two limiting regimes and a combined phase described by AdS symmetry in Beltrami coordinates. The dynamics of free particles, as well as the realization of conserved charges and the form of dispersion relations, consistently incorporate both the velocity and length invariants.

However, practical applicability is bounded by the assumptions of homogeneity, isotropy, and the non-interacting particle limit. While the deformed structure is suggestive for cosmological extensions, its implications for field-theoretic or strongly interacting systems require a distinct analysis.

7. Summary Table: Deformation Parameters and Physical Regimes

Deformation parameter	Generator affected	Geometric interpretation	Limit ($\to\infty$)	Physical outcome
$c$	Boosts ($K_i \to L_i$)	Invariant velocity	$c\to\infty$	Nonstandard “R-space” kinematics
$R$	Translations ($P_i \to F_i$)	Invariant length (cosmology)	$R\to\infty$	Standard (Minkowski) kinematics

Retaining both $c$ and $R$ in the algebra gives the full anti–de Sitter group in Beltrami coordinates, constituting the most general deformed Galilean relativity covered in (Manida, 2011).

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By systematically deforming the Galilei algebra through the introduction of invariant velocity and length parameters, a variety of physically distinct and mathematically coherent generalizations of classical kinematics are constructed. These serve not only as testbeds for cosmological and quantum structure at large scales but also as prototypes for further quantum deformations and the paper of spacetime with nontrivial global properties.