Here, we present an algebraic and kinematical analysis of non-commutative
$kappa$-Minkowski spaces within Galilean (non-relativistic) and Carrollian
(ultra-relativistic) regimes. Utilizing the theory of Wigner-In”{o}nu
contractions, we begin with a brief review of how one can apply these
contractions to the well-known Poincar'{e} algebra, yielding the corresponding
Galilean (both massive and mass-less) and Carrollian algebras as $c to infty$
and $cto 0$, respectively. Subsequently, we methodically apply these
contractions to non-commutative $kappa$-deformed spaces, revealing compelling
insights into the interplay among the non-commutative parameters $a^mu$ (with
$|a^nu|$ being of the order of Planck length scale) and the speed of light $c$
as it approaches both infinity and zero. Our exploration predicts a sort of
“branching” of the non-commutative parameters $a^mu$, leading to the emergence
of a novel length scale and time scale in either limit. Furthermore, our
investigation extends to the examination of curved momentum spaces and their
geodesic distances in appropriate subspaces of the $kappa$-deformed Newtonian
and Carrollian space-times. We finally delve into the study of their deformed
dispersion relations, arising from these deformed geodesic distances, providing
a comprehensive understanding of the nature of these space-times.
Here, we present an analysis of non-commutative $kappa$-Minkowski spaces within Galilean and Carrollian regimes. We use the theory of Wigner-In”{o}nu contractions to apply these contractions to the well-known Poincar'{e} algebra, obtaining the corresponding Galilean and Carrollian algebras as $c$ approaches infinity and zero, respectively.
Next, we apply these contractions to non-commutative $kappa$-deformed spaces, exploring the interplay between the non-commutative parameters $a^mu$ (of the order of Planck length scale) and the speed of light $c$ as it approaches both infinity and zero. Interestingly, our analysis predicts a “branching” of the non-commutative parameters $a^mu$, leading to the emergence of a novel length scale and time scale in each limit.
We also investigate curved momentum spaces and their geodesic distances in appropriate subspaces of the $kappa$-deformed Newtonian and Carrollian space-times. This exploration allows us to study deformed dispersion relations arising from these deformed geodesic distances, providing a comprehensive understanding of the nature of these space-times.
Future Roadmap
Our analysis opens up several avenues for future research in the field of non-commutative $kappa$-Minkiwoski spaces and their applications. Here is a suggested roadmap for readers interested in further exploration:
1. Experimental Tests
One potential challenge is to design and perform experimental tests to validate the predictions made by our analysis. Investigating the effects of non-commutativity at high-energy regimes or in extreme gravitational fields could provide valuable insights into the validity of these theoretical concepts.
2. Mathematical Refinements
There is still room for further mathematical refinements in the study of non-commutative $kappa$-Minkowski spaces. Analyzing the algebraic properties and symmetry transformations of these spaces in more detail could lead to a deeper understanding of their structures.
3. Cosmological Implications
It would be interesting to explore the cosmological implications of non-commutative $kappa$-Minkowski spaces. Investigating their effects on inflationary models or the early universe could provide valuable insights into the fundamental nature of space and time.
4. Quantum Field Theory on Non-Commutative $kappa$-Minkowski Spaces
Extending the study to quantum field theory on non-commutative $kappa$-Minkowski spaces could shed light on the behavior of fundamental particles in these exotic space-time backgrounds. Understanding their effects on particle interactions and scattering processes could have significant implications for particle physics.
5. Generalizations to Other Non-Relativistic and Ultra-Relativistic Regimes
Exploring the applicability of our analysis to other non-relativistic and ultra-relativistic regimes beyond Galilean and Carrollian algebras could unveil new insights and possibilities. Investigating the behavior of non-commutative $kappa$-Minkowski spaces in different physical contexts could lead to unexpected phenomena.
6. Gravitational Aspects
An intriguing avenue for future research is to incorporate gravitational aspects into the study of non-commutative $kappa$-Minkowski spaces. Analyzing the interplay between gravity and non-commutativity could uncover novel gravitational effects and potentially reconcile quantum mechanics with general relativity.
In summary, our analysis of non-commutative $kappa$-Minkowski spaces opens up a wide range of future research directions. While there are challenges in experimental validation and mathematical refinement, the opportunities for exploring cosmological implications, quantum field theory, generalizations to other regimes, and gravitational aspects are promising.