Quantization of Carrollian fermions

Emre Onur Kahya; Ertu\u{g}rul Ekiz; Utku Zorba

REVIEW 3 major objections 3 minor 3 cited by

Quantization of Carrollian fermions

T0 review · 3 major / 3 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Taking the electric Carrollian limit of Yukawa theory yields quantized Dirac fermions whose interaction is a time-dependent Dirac-delta potential, and one-loop renormalization flow that is either unstable or Gaussian.

desk verdict The quantization half is solid and worth a referee; the one-loop fixed-point claim is withdrawn by the paper's own Appendix A and should not survive in current form. read the letter →

arxiv 2502.05645 v3 pith:B5TRSFXR submitted 2025-02-08 hep-th gr-qc

Ertuğrul Ekiz , Emre Onur Kahya , Utku Zorba This is my paper · ORCID

classification hep-thgr-qc

keywords CarrolliansymmetryultralocalfieldtheoryDiracfermionquantizationYukawaWilsonianrenormalizationgroupfixedpointsCPTtransformationsflat-bandsystems

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Carrollian field theories are the $c \to 0$ contraction of relativistic theories: the light cone closes, particles cannot move through space, and only time evolution remains in the electric limit. This paper claims to provide the first quantized interacting example with Dirac fermions, obtained as the electric Carrollian limit of Yukawa theory. The fermions are canonically quantized, their charge-conjugation, parity, and time-reversal properties are worked out, and the tree-level Yukawa potential is exactly a spatial Dirac delta with a time-dependent factor, $V(t,x) = - i g^2 e^{-iM|t|}\delta^3(x)/(2M)$. The paper then applies one-loop Wilsonian renormalization to the same model and finds that the fixed-point structure is inverted relative to the relativistic theory: with a $\phi^4$ term the Carrollian theory has an unstable non-Gaussian fixed point, and without it only the Gaussian fixed point survives. The broader interest is that this connects Carrollian physics to well-studied point-like interactions in quantum optics and condensed matter and provides a field-theoretic origin for ultralocal interactions.

What carries the argument

The load-bearing object is the electric Carrollian limit, implemented by the rescalings $\tilde{\Psi} = \sqrt{c}\,\Psi$ and $\tilde{m} = m/c$, which sends the Dirac equation to $i\gamma^0\dot{\Psi} = m\Psi$ and the Klein-Gordon equation to $\ddot{\phi} + M^2\phi = 0$. Because no spatial derivatives survive, all propagators depend only on the frequency $w$, and the interaction vertex is local in space; this is what turns the Yukawa potential into a time-dependent Dirac delta. The Wilsonian machinery then scales only the frequency, $w' = bw$, leaving the spatial momenta untouched, which produces the unusual scaling dimensions $d_\phi = -1/2$ and $d_\psi = 0$ and the linear terms in the Carrollian $\beta$ functions. The divergent spatial momentum integral $K_3 = \int d^3k$ is regularized as $K_3 = a\Lambda$, so every loop correction carries this factor and the fixed-point values depend on it.

What would settle it

Compute the one-loop Carrollian $\beta$ functions keeping the field-strength renormalization terms and without imposing $M^2 \ll \Lambda^2$; if the non-Gaussian fixed point at $M_*^2 = -\Lambda^2/3$ shifts or disappears, the reported fixed-point structure is an artifact of the regulator. A complementary check is to repeat the Wilsonian analysis with a dimensionally regularized spatial integral instead of $K_3 = a\Lambda$; genuine regulator independence would keep the fixed points fixed, while scheme dependence would expose the assumption.

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Extended reading notes

Core claim

The core claim is that an interacting quantum field theory of Carrollian Dirac fermions exists and is described by the electric Carrollian limit of Yukawa theory. The authors start from the Carrollian Dirac Lagrangian $\mathcal{L}_{CD} = \bar{\Psi} i\gamma^0 \dot{\Psi} - m\bar{\Psi}\Psi$, obtain the Feynman propagator $S_F(w) = i(w\gamma^0 + m)/(w^2 - m^2 + i\epsilon)$, and couple the fermions to a Carrollian scalar through $g\phi\bar{\Psi}\Psi$ plus a $\lambda\phi^4$ self-interaction. Computing the tree-level scattering amplitude gives $V(t,x) = - i g^2 e^{-iM|t|}\delta^3(x)/(2M)$, an ultralocal, time-dependent point interaction. At one loop, Wilsonian renormalization with only the frequency rescaled gives $\beta$ functions whose fixed points are summarized by Table II: in the Carrollian theory with $\lambda \neq 0$ there is a non-Gaussian fixed point $(M^2_* = -\Lambda^2/3, m_* = 0, \lambda_* = 128\pi^4\Lambda^3/(9K_3), g_* = 0)$ with a relevant direction, while deleting $\lambda\phi^4$ leaves only Gaussian fixed points; the relativistic theory shows the opposite pattern. The paper also shows that the allowed CPT-invariant Lagrangian terms are only the kinetic and mass terms, because $\Psi^\dagger\Psi$ is odd under charge conjugation and $i\bar{\Psi}\gamma^5\Psi$ is odd under time reversal.

Load-bearing premise

The load-bearing premise is that the divergent spatial momentum integral $K_3 = \int d^3k$ can be treated as a finite constant times the cutoff, $K_3 = a\Lambda$, and that the resulting cutoff-dependent $\beta$ functions still describe the physics; this assumption is strained by the theory's own fixed point $M_*^2 = -\Lambda^2/3$, which violates the $M^2 \ll \Lambda^2$ condition used in the calculation.

Editorial extensions

If this is right

The tree-level potential $V(t,x) = -i g^2 e^{-iM|t|}\delta^3(x)/(2M)$ gives Carrollian Yukawa theory as a field-theoretic origin for the time-dependent Dirac-delta potentials used in quantum tunnelling, quantum defects, and quantum optics.
Because the Carrollian beta functions contain linear terms in $g$ and $\lambda$, the deformed ($\lambda=0$) Carrollian theory has only irrelevant couplings and flows to the Gaussian fixed point, so it remains perturbatively trivial at all scales.
With $\lambda\phi^4$ included, the non-Gaussian Carrollian fixed point has stability eigenvalues $-3/2$, $-1$, $1-\sqrt{10}$, and $1+\sqrt{10}$, so exactly one direction is relevant and the fixed point is unstable.
In the relativistic theory the roles are reversed: the $\lambda \neq 0$ theory has only Gaussian fixed points, while the deformed theory has non-Gaussian points $m_* = \pm\Lambda$, $g_* = \pm4\pi$ with oscillatory unstable flow, as collected in Table II.
The Carrollian limit systematically converts marginal relativistic couplings into irrelevant couplings, which suggests that interacting Carrollian theories do not generate new universality classes at one loop.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: if the spatial-momentum integral $K_3$ factorizes at all loop orders, the ultralocal interaction may be exact rather than a tree-level accident, and a lattice regularization of the spatial directions would provide a direct numerical test.
Editorial inference: the negative scalar mass-squared at the non-Gaussian fixed point hints at a Carrollian analogue of spontaneous symmetry breaking; adding a symmetry-breaking term could lift the instability and turn the fixed point into a physical critical point.
Editorial inference: because the free Carrollian Dirac propagator carries no spatial momentum, the model may be exactly solvable in the ultralocal sector, and the same Wilsonian treatment applied to magnetic Carrollian fermions (which keep spatial derivatives) could restore a richer fixed-point structure.
Editorial inference: since $\Psi^\dagger\Psi$ is odd under charge conjugation, a Carrollian theory coupled to an electromagnetic-like field through this bilinear would break C and could produce distinct transport signatures in flat-band systems.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First: the free Carrollian Dirac quantization and the tree-level ultralocal potential are solid enough to be useful. The paper's real new content is the electric-limit Dirac field, its CPT table, and the Yukawa coupling whose tree-level amplitude becomes -i g^2/(2M) e^{-iM|t|} delta^3(x). That is a legitimate extension and the derivation is clean. For someone working in Carrollian QFT or ultralocal toy models, that half is worth reading.

The one-loop renormalization section is the soft spot, and the stress-test note is right. The beta functions (101) are all multiplied by K3 = integral d^3k, the total spatial volume, not by a loop-momentum shell. Treating it as a*Lambda is an arbitrary choice; the fixed point values and even their existence depend on it. More seriously, the fixed point in Table II, M*^2 = -Lambda^2/3, violates the M^2 << Lambda^2 assumption used in the same derivation, and Appendix A says so explicitly: in that regime only Gaussian fixed points remain. That is an internal retraction of the central RG claim. I don't think this is fatal to the quantization part, but it is a load-bearing flaw for the abstract's claims about renormalization and fixed points.

There are smaller things: the Wilsonian procedure is only sketched, the distinction between shell integration and volume regularization is never confronted, and the comparison with relativistic beta functions in (83) is standard but not the issue. The citation pattern looks fine; the Carrollian references are appropriate, and the self-citations to the scalar RG work are used as benchmarks, not as inputs. The paper is honest about its own limitations, which I appreciate.

Who this is for: people doing Carrollian QFT and ultralocal models will get value from the quantization and the tree-level potential. The RG section should not be taken at face value.

Recommendation: I would not desk-reject it, because the quantization half deserves a referee. I'd tell the referees to focus on whether the RG section can be repaired or should be cut. If I were the editor, I'd send it to review and expect heavy revision.

Referee Report

3 major / 3 minor

Summary. The paper studies the electric Carrollian limit of Dirac fermions in 3+1 dimensions. It constructs the free Carrollian Dirac action, canonically quantizes the field, computes the Feynman propagator, and analyzes charge conjugation, parity, and time reversal for the relevant bilinears. It then couples the fermions to a Carrollian scalar through a Yukawa interaction and obtains the tree-level scattering amplitude, which yields an ultralocal, time-dependent delta-function potential, Eq. (66). The rest of the paper applies Wilsonian renormalization at one loop: the authors derive beta functions (101), identify a non-Gaussian fixed point (102), analyze its stability via the matrix (104), and summarize the fixed-point structure in Table II. An appendix presents a more complete one-loop calculation and reaches a different conclusion about the Carrollian fixed points.

Significance. The free-field quantization of the electric Carrollian Dirac fermion and the derivation of the tree-level ultralocal Yukawa potential are clean, self-contained, and potentially useful: they provide an explicit fermionic example in Carrollian field theory, connect the Carrollian limit to time-dependent point-like interactions relevant in condensed matter and quantum mechanics, and the discrete symmetry analysis is systematic. The paper is also honest in presenting limitations in Appendix A. However, the renormalization-group analysis, which is advertised in the abstract and Section V, is not supported as written: the non-Gaussian fixed point in the main text is contradicted by the paper's own Appendix A, and the one-loop beta functions depend on an ad hoc regularization of the divergent spatial momentum integral K3. The fixed-point claim should therefore be corrected or substantially qualified before the paper can be accepted.

major comments (3)

[Section V.A.2, Eq. (102) and Table II] The non-Gaussian fixed point M*^2 = -Λ^2/3, λ* = 128π^4Λ^3/(9K3) presented in the main text is internally contradicted by Appendix A. The beta functions (101) were derived assuming M^2 << Λ^2, but the fixed point violates this condition. Appendix A explicitly states that the non-Gaussian fixed points for M^2 are O(Λ^2), that this 'invalidates the condition M^2 << Λ^2', and that 'Consequently, we are left with only Gaussian fixed points.' Table II and the discussion around Eq. (102) therefore do not represent a consistent result of the paper as a whole; this contradiction must be resolved before the renormalization claim can be accepted.
[Section V.A.2, Eqs. (96)-(101) and Appendix A] The one-loop results depend on the divergent spatial volume K3 = ∫d^3k, which is treated as a finite constant. Since the Carrollian propagators depend only on frequency, K3 factors out of every loop integral and is not a Wilsonian momentum-shell contribution. The replacement K3 = aΛ used in Appendix A is an ad hoc regularization, and the fixed point value λ* in Eq. (102) changes with the choice of K3 (for example K3 ~ (4π/3)Λ^3 versus K3 = aΛ); the appendix's own conclusion is that only Gaussian fixed points survive in the regime of validity. The beta functions (101) and the stability analysis (104) are therefore regulator-dependent as stated, and a physical justification for K3, or a regulator-independent statement of the results, is required.
[Section V.A.2 and Appendix A] The main text derives the beta functions (101) after setting external momenta to zero and neglecting field-strength renormalization, referring to Appendix A for details. The full one-loop beta functions in Appendix A contain additional contributions from field-strength renormalization, and the appendix finds that the non-Gaussian Carrollian fixed points lie outside the M^2 << Λ^2 regime used in the derivation. The relationship between the truncated beta functions (101) and the more complete expressions (A9)-(A11) is not explained, and it is not shown that the fixed point (102) is robust under the inclusion of the omitted terms. The authors should either justify the truncation or present the fixed-point structure only in the regime where the approximations are controlled.

minor comments (3)

[Eq. (66)] The tree-level potential V(t,x) = -i g^2 e^{-iM|t|} δ^3(x)/(2M) is complex, while the interaction term -g φ Ψbar Ψ in the Lagrangian (62) is Hermitian. The authors should clarify whether this is an effective or non-Hermitian potential and how the imaginary coefficient is to be interpreted physically.
[Eq. (34)] The operator ordering in the expression for the Carroll boost charge C^i is not normal ordered; using the anticommutation relation (29) may introduce a divergent c-number term. The authors should specify the normal-ordering prescription or verify the commutator [P^i, C^j] = iδ^{ij}H with the ordering as written.
[References] Reference [7] appears to contain an incomplete author name ('J., E. Have' should presumably be 'J. Hartong, E. Have'), and a few equation numbers are missing or inconsistently referenced; a final proofreading pass would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Carrollian fermion quantization, tree-level ultralocal potential, and one-loop beta functions are derived self-containedly from the stated Lagrangian.

full rationale

The paper's central derivations are self-contained. The Carrollian Dirac action (20) is obtained by the stated electric limit and quantization conditions, and the Feynman rules (63) follow from that action. The tree-level Yukawa potential (66) is a direct Fourier transform of the amplitude (64), with no input equivalent to the output. The one-loop beta functions (101) are computed from the standard Wilsonian procedure applied to the diagrams of Figure 2; they contain no fitted parameter and no quantity defined in terms of the target fixed points. The spatial-momentum integral K3 is a regulator-dependent constant, and treating it as a*Lambda in Appendix A is a regularization assumption, not a circular definition or a fit. The fixed points (102) are then solved from beta=0 rather than assumed. Self-citations (e.g., [34,49,50]) appear only as background or as extensions, not as load-bearing justification for the present results; the key comparison [30] is independent prior work by different authors. The Appendix A limitation, which notes that the non-Gaussian fixed point has M*^2 = -Lambda^2/3 and thereby invalidates the M^2 << Lambda^2 approximation, is an internal consistency and physical-interpretation concern, not a circularity: the derivation chain does not reduce to its own inputs. All model-specific predictions (ultralocal interaction, beta functions, fixed-point structure) are consequences of the explicitly written Lagrangian and standard perturbative rules.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The central claims rest on the standard Carrollian limiting procedure and on an anisotropic Wilsonian RG. The only ad hoc element is the regularization of the spatial volume factor K3, which directly controls the beta functions and fixed points. No new particles or forces are introduced.

free parameters (1)

K3 (spatial momentum volume) = K3 = a Lambda
The loop integrals over spatial momenta are independent of k and produce the divergent volume K3 = integral d^3k. The paper regularizes it as K3 = a Lambda with an arbitrary constant a. All beta functions (101) and the fixed point (102) depend on this constant.

assumptions (3)

domain assumption The electric Carrollian limit is obtained by the rescalings phi = c phi_tilde, M = M_tilde/c, Psi = sqrt(c) Psi_tilde, m = m_tilde/c, g = g_tilde/c^2, lambda = lambda_tilde/c^4 followed by c to 0.
This is the standard Carrollian limiting procedure used throughout the paper (Sections III and V), and it defines the theory under study.
domain assumption The Wilsonian renormalization group can be applied to Carrollian theories by scaling only the energy component w' = b w, leaving spatial momenta k unchanged.
Used in Section V.A.2; the validity of this anisotropic RG procedure for Carrollian field theories is assumed from reference [30].
ad hoc to paper The spatial momentum integral K3 = integral d^3k can be treated as a finite constant a Lambda.
Introduced in Appendix A to regularize the divergent spatial volume in loop integrals; the beta functions depend on this ad hoc regulator.

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Cite this review

Pith. "Pith review of Quantization of Carrollian fermions." pith.science (2026). https://pith.science/paper/B5TRSFXR

@misc{pith2026250205645,
  author       = {Pith},
  title        = {Pith review of: Quantization of Carrollian fermions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B5TRSFXR}},
  note         = {Machine review of arXiv:2502.05645}
}

read the original abstract

We provide the first example of interacting quantized Carrollian Dirac fermions and investigate their discrete symmetries, including charge conjugation (C), parity (P), and time reversal (T) transformations. As a toy model, we couple these fermions to a Carrollian scalar field using Carrollian Yukawa theory and compute the tree-level diagram, revealing an ultralocal interaction between the Carrollian fermions and the scalar field. This interaction, widely known as a Dirac delta interaction with time-dependent factor, frequently appears in quantum physics. We then address the renormalization of the theory by employing the Wilsonian procedure at one-loop order. Furthermore, we analyze the fixed points and stability properties of Carrollian Yukawa theory, comparing them with their relativistic counterparts. Beyond the specific Yukawa model studied here, we expect that our framework will have broader applications in Carrollian physics, particularly in understanding ultralocal interactions and their role in condensed matter systems, where similar phenomena arise in strongly correlated and non-relativistic regimes.

Figures

Figures reproduced from arXiv: 2502.05645 by the authors.

**Figure 2.** FIG. 2. Here diagrams (a) and (b) illustrate the scalar’s self-energy diagrams resulting from the couplings [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

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Forward citations

Cited by 3 Pith papers

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

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Reference graph

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

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Charge conjugation defines the symmetry between particles and antiparticles and is implemented by a unitary operator C

Charge conjugation We begin our discussion with the charge conjugation transformation. Charge conjugation defines the symmetry between particles and antiparticles and is implemented by a unitary operator C. This operator transforms a fermion 4 Note that CPT transformations in two and three dimensions are discussed in [86, 87] in different context. 8 into ...

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Parity Now, we proceed by investigating the parity transformation of the Carrollian Dirac field. The parity operator transforms the state as† ⃗k |0⟩ into as† −⃗k |0⟩ as we know from the Lorentzian case. Consequently, we may interpret the operator as follows: P as ⃗kP = ηaas −⃗k & P bs ⃗kP = ηbbs −⃗k , (53) 9 where ηa and ηb represents possible phase facto...

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− 1 M 2 (Λ2 + M 2) + log 1 + M 2/Λ2 M 4 # βg = g3 8π2 Λ2 Λ2 + M 2 + g3 8π2 + g3Λ4 8π2

Carrollian theory A similar analysis can be carried out for the Carrollian regime. However, it is crucial to note that in the Carrollian regime, as seen from propagators and other quantities, fields strictly depend on the energy component but not on the spatial momentum component. Since the fields do not explicitly depend on spatial momentum, any rescalin...

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