In this paper, we find a class of Carrollian and Galilean contractions of
(extended) BMS algebra in 3+1 and 2+1 dimensions. To this end, we investigate
possible embeddings of 3D/4D Poincar'{e} into the BMS${}_3$ and BMS${}_4$
algebras, respectively. The contraction limits in the 2+1-dimensional case are
then enforced by appropriate contractions of their Poincar'{e} subalgebra. In
3+1 dimensions, we have to apply instead the analogy between the structures of
Poincar'{e} and BMS algebra. In the case of non-vanishing cosmological
constant in 2+1 dimensions, we consider the contractions of $Lambda$-BMS${}_3$
algebras in an analogous manner.

Examining Conclusions

This paper explores the concept of Carrollian and Galilean contractions of the (extended) BMS algebra in both 3+1 and 2+1 dimensions. The study focuses on investigating the potential embeddings of 3D/4D Poincaré into the BMS₃ and BMS₄ algebras, respectively. By enforcing appropriate contractions on the Poincaré subalgebra, the contraction limits in the 2+1-dimensional case are achieved. On the other hand, in the 3+1-dimensional scenario, a comparison between the structures of Poincaré and BMS algebra is necessary. The analysis also considers the contractions of Λ-BMS₃ algebras when dealing with a non-zero cosmological constant in 2+1 dimensions.

Future Roadmap

Looking ahead, several challenges and opportunities arise in the field of Carrollian and Galilean contractions of the extended BMS algebra.


  • Mathematical Complexity: Further exploration is required to fully understand the mathematical intricacies of these contractions. Researchers will face challenges in developing rigorous mathematical models and proofs to support their findings.
  • Data Validation: Empirical validation of these contractions using experimental data and observations can be a challenging task. It will require collaborative efforts between theoretical physicists and experimental scientists to verify the theoretical predictions.
  • Generalization: The current study focuses on specific dimensions (3+1 and 2+1). Generalizing these contractions to higher dimensions poses additional challenges that need to be addressed.


  • New Insights into Fundamental Physics: Carrollian and Galilean contractions offer a deeper understanding of the connections between different algebraic structures and their relevance to fundamental physics. This research opens up opportunities to uncover novel insights and potentially revise existing theories.
  • Application in Cosmology: The study of contractions of Λ-BMS₃ algebras in the presence of a non-zero cosmological constant holds promise for advancing our understanding of the universe’s evolution. It may provide valuable insights into phenomena such as cosmic inflation and dark energy.
  • Advanced Quantum Field Theory: The findings in this paper lay the groundwork for further exploration of advanced quantum field theories. Researchers can build upon these contractions to develop new frameworks that incorporate both classical and quantum effects.


The examination of Carrollian and Galilean contractions of the extended BMS algebra in different dimensions yields valuable insights into the structure of Poincaré and BMS algebras. While challenges in terms of mathematical complexity, data validation, and generalization exist, the opportunities for advancing our understanding of fundamental physics, cosmology, and quantum field theory are substantial. Further research in this area promises to provide a roadmap towards uncovering new discoveries and enhancing our knowledge of the universe.

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