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Anatomy of Null Contractions
High Energy Physics - Theory · Submitted 21 Jun 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We find, similar to Susskind's original observation of the non-relativistic case, that the Poincare algebra, written in the lightcone coordinates, naturally contains Carrollian sub-algebras in one lower dimension.
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The abstract, sentence by sentence
We introduce null contractions of the Poincare and relativistic conformal algebras. The longitudinal null contraction involves writing the algebra in lightcone coordinates and contracting one of the null directions. For the Poincare algebra, this yields two non-overlapping co-dimension one Carroll algebras. The transverse contraction is a limit on the spatial dimensions and yields two non-overlapping co-dimension one Galilean algebras. We find, similar to Susskind's original observation of the non-relativistic case, that the Poincare algebra, written in the lightcone coordinates, naturally contains Carrollian sub-algebras in one lower dimension. The effect of the longitudinal contraction, which essentially focusses on the null direction, is to disentangle the two Carroll algebras that now correspond to the symmetries of the two null boundaries. The transverse contraction similarly separates the overlapping Galilean sub-algebras of the original Poincare algebra. We discuss aspects of the conformal case, where we get lower dimensional Carroll Conformal algebras and Schrodinger algebras.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We introduce null contractions of the Poincare and relativistic conformal algebras. | contribution 1.00 | 0.94 | 0.12 | asserted 0.99 |
| s1 | The longitudinal null contraction involves writing the algebra in lightcone coordinates and contracting one of the null directions. | method 0.95 | 0.58 | 0.05 | asserted 1.00 |
| s2 | For the Poincare algebra, this yields two non-overlapping co-dimension one Carroll algebras. | result 1.00 | 0.88 | 0.19 | asserted 1.00 |
| s3 | The transverse contraction is a limit on the spatial dimensions and yields two non-overlapping co-dimension one Galilean algebras. | result 0.77 | 0.82 | 0.12 | asserted 1.00 |
| s4 | We find, similar to Susskind's original observation of the non-relativistic case, that the Poincare algebra, written in the lightcone coordinates, naturally contains Carrollian sub-algebras in one lower dimension. | result 0.84 | 0.69 | 0.30 | asserted 1.00 |
| s5 | The effect of the longitudinal contraction, which essentially focusses on the null direction, is to disentangle the two Carroll algebras that now correspond to the symmetries of the two null boundaries. | interpretation 0.76 | 0.78 | 0.18 | asserted 1.00 |
| s6 | The transverse contraction similarly separates the overlapping Galilean sub-algebras of the original Poincare algebra. | result 0.67 | 0.79 | 0.10 | asserted 1.00 |
| s7 | We discuss aspects of the conformal case, where we get lower dimensional Carroll Conformal algebras and Schrodinger algebras. | contribution 0.59 | 0.86 | 0.10 | asserted 0.99 |
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Original abstract
We introduce null contractions of the Poincare and relativistic conformal algebras. The longitudinal null contraction involves writing the algebra in lightcone coordinates and contracting one of the null directions. For the Poincare algebra, this yields two non-overlapping co-dimension one Carroll algebras. The transverse contraction is a limit on the spatial dimensions and yields two non-overlapping co-dimension one Galilean algebras. We find, similar to Susskind's original observation of the non-relativistic case, that the Poincare algebra, written in the lightcone coordinates, naturally contains Carrollian sub-algebras in one lower dimension. The effect of the longitudinal contraction, which essentially focusses on the null direction, is to disentangle the two Carroll algebras that now correspond to the symmetries of the two null boundaries. The transverse contraction similarly separates the overlapping Galilean sub-algebras of the original Poincare algebra. We discuss aspects of the conformal case, where we get lower dimensional Carroll Conformal algebras and Schrodinger algebras.
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References
How the literature uses it
6 arXiv papers cite this in 11 sentences
Papers citing it · 9 sentences without a confident label
Compares with it 1 paper
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In contrast to foundational and follow-up approaches (e.g., [ref] ) based on inertial worldsheets and assuming eternal ($𝕋 →∞$) strings, where this parameter was treated as an abstract contraction tool to access tensionless limits at $ϵ→0$, our construction demonstrates that $ϵ$ admits a concrete geometric realization.
New-born strings are tensionless · Concluding remarks and outlook · 0.95
Cites it as support 1 paper
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The reduction of the Lorentz algebra induced by the light-cone decomposition mirrors the symmetry reduction inherent to Carrollian spacetimes, as shown in [ref] .
Carroll fermions, expansions and the lightcone · Oddity in odd dimensions and lessons from the lightcone · 0.99
What it cites · 18 sentences without a confident label
Uses the method or tool of 1 paper
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* Carroll expansions: One of the important recent tools employed in Carrollian theories is the method of expansion [ref] .
Carroll symmetry, dark energy and inflation · Future directions · 0.82
Cites as background 10 papers
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Carrollian manifolds [ref] are defined by a pair $(τ^μ, h_μν)$, where $τ^μ$ is a nowhere vanishing vector field and $h_μν$ is a degenerate covariant metric with $τ^μh_μν=0$.
Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time · Usual contractions of the conformal algebra · 0.95
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Carrollian manifolds [ref] are defined by a pair $(τ^μ, h_μν)$, where $τ^μ$ is a nowhere vanishing vector field and $h_μν$ is a degenerate covariant metric with $τ^μh_μν=0$.
Conformal Carroll groups and BMS symmetry · Usual contractions of the conformal algebra · 0.95
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This formed the basis of what has come to be known as the infinite momentum frame (earlier introduced by Weinberg in [ref] ) and also is central to the ideas of Discrete Light Cone Quantization (DLCQ), especially in the context of Matrix theory [ref] (the BFSS matrix model [ref] ) and related to more recent ideas of null reductions.
M Theory As A Matrix Model: A Conjecture · Discussions · 0.95
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