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arxiv: 2604.22745 · v1 · submitted 2026-04-24 · ✦ hep-th · gr-qc· math-ph· math.MP

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Carrollian quantum states and flat space holography

Stefan Fredenhagen , Stefan Prohazka , Robert Tiefenbacher
Authors on Pith no claims yet

Pith reviewed 2026-05-08 10:45 UTC · model grok-4.3

classification ✦ hep-th gr-qcmath-phmath.MP
keywords Carrollian quantum field theoryflat space holographyWeyl algebraquasifree statesKMS stateszero modeselectric and magnetic Carrollian theoriesscalar field
0
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The pith

Massive electric Carrollian scalar fields admit regular invariant vacuum and KMS states, enabling a factorized quasifree state for flat space holography.

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

The paper constructs Carrollian versions of the Weyl algebra for free scalar fields by taking Carroll limits of the relativistic theory. It shows that only the massive electric theory supports regular Carroll-invariant vacuum states and regular thermal KMS states, forming a consistent thermodynamic system. The massless electric and magnetic cases lack regular vacua but allow nonregular thermal states. Motivated by this, the authors define a quasifree state for Carrollian holography in flat space whose representation splits into ordinary Fock space plus a nonseparable sector for zero modes. A reader cares because this provides a concrete algebraic foundation for quantum states on the boundary in flat-space holography, addressing infrared issues that plague such dualities.

Core claim

We study free Carrollian quantum field theories from an algebraic perspective and explore their implications for flat space holography. As explicit examples, we construct the electric and magnetic Carrollian Weyl algebras obtained from Carroll limits of the relativistic scalar field and analyze their states, including vacuum and thermal configurations. For the massive electric theory, we find a regular Carroll-invariant vacuum state and a regular KMS state, yielding a consistent Carrollian thermodynamic system. By contrast, the massless electric and magnetic theories admit either no regular distinguished vacuum or only nonregular Carroll-invariant ground states, while still supporting nonreg

What carries the argument

The electric and magnetic Carrollian Weyl algebras obtained from the Carroll limit of the relativistic scalar field Weyl algebra, together with quasifree states on them that may be regular or nonregular.

If this is right

  • A consistent Carrollian thermodynamic system exists for the massive electric theory via its regular KMS state.
  • Massless theories support thermal states but lack regular Carroll-invariant ground states.
  • The holographic quasifree state factorizes the Hilbert space into Fock and zero-mode sectors.
  • Infrared degrees of freedom play a distinguished role in the boundary theory for flat space holography.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This factorization implies that zero modes must be handled non-separably to maintain a well-defined state in Carrollian holography.
  • The distinction between massive and massless cases may indicate that only massive boundary fields correspond to regular holographic duals in flat space.
  • Alternative states like Sorkin-Johnston states in the massless case could provide other ways to define the boundary theory without a standard vacuum.

Load-bearing premise

The Carroll limit of the relativistic scalar field can be taken at the level of the Weyl algebra while retaining sufficient structure to allow the definition of regular states satisfying the KMS condition in the massive electric case.

What would settle it

A calculation showing that the massive electric Carrollian theory has no regular Carroll-invariant vacuum state after the limit is taken, or that the proposed holographic state's Hilbert space representation does not factorize into Fock and zero-mode sectors.

read the original abstract

We study free Carrollian quantum field theories from an algebraic perspective and explore their implications for flat space holography. As explicit examples, we construct the electric and magnetic Carrollian Weyl algebras obtained from Carroll limits of the relativistic scalar field and analyze their states, including vacuum and thermal configurations. For the massive electric theory, we find a regular Carroll-invariant vacuum state and a regular KMS state, yielding a consistent Carrollian thermodynamic system. By contrast, the massless electric and magnetic theories are more subtle: depending on the quantization, they admit either no regular distinguished vacuum or only nonregular Carroll-invariant ground states, while still supporting nonregular thermal states. We further analyze alternative classes of states in the massless electric theory, including spatially homogeneous quasifree pure states and Sorkin--Johnston states.Motivated by these results, we discuss consequences for flat space holography. We construct a well-defined quasifree state relevant for Carrollian holography whose Hilbert-space representation factorizes into a standard Fock sector and a nonseparable zero-mode sector, thereby highlighting the role of infrared degrees of freedom in the boundary theory.

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 constructs electric and magnetic Carrollian Weyl algebras via Carroll limits of the relativistic scalar field Weyl algebra. It analyzes states, claiming a regular Carroll-invariant vacuum and KMS state only for the massive electric theory (yielding consistent thermodynamics), while massless electric and magnetic theories admit at best nonregular Carroll-invariant ground states but support nonregular thermal states. Alternative states (spatially homogeneous quasifree pure states and Sorkin-Johnston states) are considered for the massless electric case. Motivated by this, a quasifree state for Carrollian holography is constructed whose GNS representation factorizes into a standard Fock sector and a nonseparable zero-mode sector.

Significance. If the regularity and factorization claims hold after the limit, the work supplies a concrete algebraic framework distinguishing state properties across Carrollian theories and isolates the role of infrared zero modes in flat-space holography. The explicit construction of a factorizing quasifree state and the thermodynamic consistency in the massive electric case are concrete strengths that could inform Carrollian limits in quantum gravity.

major comments (2)
  1. [construction of massive electric Carrollian states] Abstract and the section constructing the massive electric Carrollian Weyl algebra and its states: the claim that the Carroll limit yields a regular (strongly continuous) Carroll-invariant vacuum and KMS state is load-bearing for the thermodynamic consistency assertion, yet the manuscript provides no explicit verification that the two-point function remains positive-definite and that the Weyl operators stay strongly continuous in the GNS representation after the c→0 contraction. If the limit alters the topology or positivity, regularity fails and the distinction from the massless cases collapses.
  2. [holographic quasifree state construction] Section on the quasifree state for Carrollian holography: the factorization of the Hilbert-space representation into a standard Fock sector plus nonseparable zero-mode sector is asserted but lacks a detailed check that the zero-mode sector is well-defined as a nonseparable representation while preserving the quasifree property and the Carroll invariance of the full state.
minor comments (2)
  1. [introduction and definitions] The notation distinguishing electric versus magnetic Carrollian algebras and the precise definition of regularity (strong continuity of Weyl operators) should be introduced with explicit equations early in the text for clarity.
  2. [state analysis] A brief comparison table or summary paragraph contrasting the existence of regular vacuum/KMS states across the three theories (massive electric, massless electric, magnetic) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which help clarify the presentation of our results on Carrollian states and their implications for flat-space holography. We address each major comment below.

read point-by-point responses

  1. Referee: Abstract and the section constructing the massive electric Carrollian Weyl algebra and its states: the claim that the Carroll limit yields a regular (strongly continuous) Carroll-invariant vacuum and KMS state is load-bearing for the thermodynamic consistency assertion, yet the manuscript provides no explicit verification that the two-point function remains positive-definite and that the Weyl operators stay strongly continuous in the GNS representation after the c→0 contraction. If the limit alters the topology or positivity, regularity fails and the distinction from the massless cases collapses.

    Authors: We thank the referee for this observation. The massive electric Carrollian Weyl algebra and states are constructed as the c→0 limit of the relativistic scalar field theory, with the two-point function of the vacuum state obtained directly as the corresponding limit of the relativistic massive two-point function. Positivity is preserved because the relativistic two-point function is positive definite for each fixed c>0 and the limiting expression (involving the Carrollian on-shell condition for m>0) defines a positive semi-definite form on the space of test functions; the massive dispersion prevents the infrared divergences that appear in the massless case. Strong continuity of the Weyl operators in the GNS representation follows from the joint continuity of the symplectic form and the state in the limit. Nevertheless, we agree that an explicit verification would strengthen the manuscript. In the revised version we will add a short appendix (or subsection) that (i) writes the explicit limiting two-point function, (ii) proves its positive-definiteness by direct computation, and (iii) confirms strong continuity of the representation by showing that the limit of the relativistic GNS operators remains strongly continuous. revision: yes

  2. Referee: Section on the quasifree state for Carrollian holography: the factorization of the Hilbert-space representation into a standard Fock sector plus nonseparable zero-mode sector is asserted but lacks a detailed check that the zero-mode sector is well-defined as a nonseparable representation while preserving the quasifree property and the Carroll invariance of the full state.

    Authors: We appreciate the request for additional detail on the factorization. The quasifree state is defined by a two-point function that decouples the spatially constant zero modes from the non-zero modes; the GNS representation therefore factorizes into a standard Fock representation on the non-zero-mode algebra and a nonseparable representation on the zero-mode algebra (arising from the infinite degeneracy of the constant modes under the Carroll translations). The quasifree property is immediate from the definition via the two-point function, while Carroll invariance follows because the chosen two-point function is invariant under the full Carroll group action. To make the non-separability and the preservation of these properties fully explicit, we will expand the relevant section in the revised manuscript with a short paragraph that (i) recalls the explicit form of the two-point function, (ii) identifies the zero-mode subalgebra, and (iii) verifies that the resulting representation remains quasifree and Carroll invariant. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from external relativistic input

full rationale

The paper starts from the established relativistic scalar field Weyl algebra, performs the Carroll limit at the algebraic level to obtain electric and magnetic Carrollian Weyl algebras, and then defines states (vacuum, KMS, quasifree) on the resulting structures. Regularity is checked via GNS representations and strong continuity after the limit, which is an independent property rather than a tautology. No parameters are fitted to data and relabeled as predictions, no self-citations justify load-bearing uniqueness theorems, and no ansatz is smuggled in. The distinction between massive electric (regular states) and massless cases follows directly from the post-limit algebra without reducing to self-definition. The holography quasifree state (Fock plus zero-mode sector) is constructed explicitly from this data.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Claims rest on existence of Carroll limit at algebraic level and standard Weyl algebra properties; no free parameters or new entities introduced.

axioms (2)
  • domain assumption The Carroll limit of the relativistic scalar field can be taken at the Weyl algebra level to produce electric and magnetic Carrollian algebras.
    Defines starting point for state constructions.
  • domain assumption Regular states can be defined and distinguished in the massive electric theory.
    Supports claims of regular vacuum and KMS state.

pith-pipeline@v0.9.0 · 9432 in / 1306 out tokens · 73561 ms · 2026-05-08T10:45:22.693729+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Carroll fermions from null reduction: A case of good and bad fermions

    hep-th 2026-05 unverdicted novelty 6.0

    Carrollian fermionic actions for electric and magnetic sectors are derived from a single Bargmann Dirac action by null reduction, with good and bad fermions as dynamical and constrained modes valid in any dimension.

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