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arxiv: 2606.19112 · v1 · pith:VEVOK4MUnew · submitted 2026-06-17 · ✦ hep-th · math-ph· math.MP

Post-Carroll Algebra, Conformal Extensions, and Field Theories

Mojtaba Najafizade This is my paper

Pith reviewed 2026-06-26 20:01 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MP
keywords post-Carroll algebraCarroll-Bargmann algebraCarroll-Schrödinger algebraconformal extensionspost-Carrollian CFTtwo-point functionscentral chargehigher-dimensional theories
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The pith

Incorporating c-dependent corrections to Carroll transformations produces the Carroll-Bargmann algebra whose conformal extension matches the symmetry algebra of higher-dimensional Carroll-Schrödinger theories.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces post-Carroll transformations that include leading corrections proportional to the speed of light. These transformations generate the post-Carroll algebra, which unlike the standard Carroll algebra admits a central charge in higher dimensions. This extended algebra is called the Carroll-Bargmann algebra. The work constructs its conformal extension, called the Carroll-Schrödinger algebra, and shows it coincides with the symmetry algebra of the higher-dimensional Carroll-Schrödinger theory. It also derives the general two-point functions for post-Carrollian conformal field theories, finding distinct sectors in one plus one dimensions versus higher dimensions.

Core claim

The conformal extension of the Carroll-Bargmann algebra precisely matches the symmetry algebra of the higher-dimensional Carroll-Schrödinger theory. This is obtained by first defining post-Carroll transformations consistent with post-Carrollian mechanics, forming the post-Carroll algebra, extending it to include a central charge, and then building the conformal version which reproduces the known symmetries.

What carries the argument

The Carroll-Bargmann algebra, formed by adding a central charge to the post-Carroll algebra in higher dimensions, which carries the argument by allowing conformal extensions that align with Carroll-Schrödinger symmetries.

Load-bearing premise

The leading c-dependent corrections to the Carroll transformations remain consistent with the post-Carrollian mechanics from prior work.

What would settle it

A direct computation showing that the generators of the Carroll-Schrödinger algebra do not satisfy the commutation relations of the conformal extension of the Carroll-Bargmann algebra would falsify the central claim.

Figures

Figures reproduced from arXiv: 2606.19112 by Mojtaba Najafizade.

Figure 1
Figure 1. Figure 1: Extensions of the post-Carroll algebra with the generators M, D, K, Ki . We subsequently construct field theories invariant under each of these algebras. We find that the corresponding theories must involve complex fields. In other words, the post-Carroll algebra, and its extensions, do not constitute a symmetry for field theories of real fields. As we will see, this is due to the existence of the “radial … view at source ↗
read the original abstract

By incorporating leading $c\,$-dependent corrections to the Carroll transformations, we introduce the ``post-Carroll transformations''. We demonstrate that these transformations are consistent with post-Carrollian mechanics \cite{Najafizadeh:2025ksm}; furthermore, they give rise to the so-called ``post-Carroll algebra''. We show that, unlike the Carroll algebra, this new structure allows for a central charge in higher dimensions; we refer to it as the ``Carroll-Bargmann algebra''. To construct conformal extensions, we first build the conformal extension of the post-Carroll algebra and study field theories invariant under this symmetry. We then construct the conformal extension of the Carroll-Bargmann algebra, referred to as the ``Carroll-Schr\"odinger algebra'', and demonstrate that it precisely matches the symmetry algebra of the higher-dimensional Carroll-Schr\"odinger theory \cite{Najafizadeh:2024imn}. Finally, we derive the general form of two-point functions in a post-Carrollian CFT, which in $1+1$ dimensions exhibits both electric and magnetic sectors, while in higher dimensions only the magnetic sector survives.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes post-Carroll transformations by incorporating leading c-dependent corrections to the Carroll transformations. These are shown to be consistent with post-Carrollian mechanics and lead to the post-Carroll algebra. This algebra admits a central charge in higher dimensions, termed the Carroll-Bargmann algebra. The conformal extension of the post-Carroll algebra is constructed, along with field theories invariant under it. The conformal extension of the Carroll-Bargmann algebra, called the Carroll-Schrödinger algebra, is shown to match the symmetry algebra of the higher-dimensional Carroll-Schrödinger theory. Finally, the general form of two-point functions in post-Carrollian CFTs is derived, exhibiting electric and magnetic sectors in 1+1 dimensions but only the magnetic sector in higher dimensions.

Significance. Should the algebraic identifications and derivations hold, this work extends the Carroll algebra to include central extensions in higher dimensions and provides a conformal version that aligns with known Carroll-Schrödinger theories. The explicit construction of two-point functions offers a testable prediction for such CFTs and could contribute to the study of non-relativistic or Carrollian limits in quantum field theory.

major comments (2)
  1. [Abstract] The assertion that the Carroll-Schrödinger algebra 'precisely matches' the symmetry algebra of the higher-dimensional Carroll-Schrödinger theory is central but is supported primarily by reference to the cited work Najafizadeh:2024imn rather than an explicit re-derivation or mapping of generators within this manuscript.
  2. [Abstract] The introduction of the post-Carroll algebra and its allowance for a central charge in higher dimensions is presented as a key result, yet the explicit commutation relations defining this algebra and the demonstration of the central charge are not provided in the abstract and appear to depend on the consistency check with the prior mechanics paper.
minor comments (2)
  1. The abstract refers to 'we demonstrate that these transformations are consistent with post-Carrollian mechanics'; including a short summary of this demonstration would aid readers unfamiliar with the cited work.
  2. Notation for the algebras (post-Carroll, Carroll-Bargmann, Carroll-Schrödinger) should be clearly defined upon first use in the main text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their report and for highlighting points that can improve the clarity of our presentation. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] The assertion that the Carroll-Schrödinger algebra 'precisely matches' the symmetry algebra of the higher-dimensional Carroll-Schrödinger theory is central but is supported primarily by reference to the cited work Najafizadeh:2024imn rather than an explicit re-derivation or mapping of generators within this manuscript.

    Authors: We agree that an explicit mapping of generators would strengthen the manuscript. While the body constructs the Carroll-Schrödinger algebra from the conformal extension of the Carroll-Bargmann algebra and states the matching, we will add a short table or paragraph in the revised version that explicitly identifies each generator with the corresponding symmetry of the higher-dimensional Carroll-Schrödinger theory. revision: yes

  2. Referee: [Abstract] The introduction of the post-Carroll algebra and its allowance for a central charge in higher dimensions is presented as a key result, yet the explicit commutation relations defining this algebra and the demonstration of the central charge are not provided in the abstract and appear to depend on the consistency check with the prior mechanics paper.

    Authors: Abstracts are summaries and do not contain full commutation relations. The post-Carroll algebra, including the central extension in higher dimensions, is derived directly in the main text from the post-Carroll transformations; the reference to Najafizadeh:2025ksm provides an independent consistency check with mechanics but is not required for the algebraic derivation itself. No revision to the abstract is needed, though we can add a clarifying sentence in the introduction if the referee prefers. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained algebraic constructions

full rationale

The paper introduces post-Carroll transformations from c-dependent corrections, derives the post-Carroll algebra and its Carroll-Bargmann extension with central charge, builds conformal extensions explicitly, and shows by direct computation that the Carroll-Schrödinger algebra matches the symmetry of the higher-dimensional theory. The two self-citations supply the definition of post-Carrollian mechanics and the target Carroll-Schrödinger theory but are not load-bearing for the new derivations or the matching step performed here; no step reduces by construction to a fit, ansatz, or prior result of the same authors. The central claims remain independent algebraic identities.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; all structures are introduced by modifying prior Carroll transformations.

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