Pith. sign in

REVIEW 4 major objections 5 minor 39 references

Carrollian Quantum Mechanics: Time-like, Space-like and Hybrid Sectors

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read By contracting the Klein-Gordon equation as the speed of light approaches zero, this paper derives three Carroll-invariant quantum theories and shows that the time-like one exhibits temporal tunneling with the Bogoliubov relation $\lvert…

desk verdict The time-like Carroll sector is solid, but the claimed three-sector derivation from the c→0 contraction is not; the hybrid limit gives a massless wave equation and the space-like equation is an Ansatz. read the letter →

arxiv 2608.08384 v1 pith:7HTFP44L submitted 2026-08-09 hep-th

Mehdi Ahmadi-Jahmani This is my paper · ORCID
classification hep-th
keywords CarrollsymmetryKlein-Gordonequationultra-relativisticlimittemporaltunnelingBogoliubovtransformationfractondualitytachyonicmodesnullstates
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

Taking the speed of light to zero is not a single limit: the paper shows that a systematic contraction $c\to0$ of the Klein-Gordon equation splits into three Carroll-invariant quantum theories—time-like, space-like, and hybrid—each with its own continuity equation, probability density, current, and exact solutions. The time-like sector is immobile in space but supports temporal tunneling: a time-dependent barrier mixes positive and negative frequencies, yielding the Bogoliubov relation $\lvert T\rvert^2=1+\lvert R\rvert^2$ and emergent antiparticle-like modes. The space-like sector requires a non-dynamical compensating field to restore Carroll boosts; its propagating energy eigenstates are null states with vanishing density, their physical content carried by the spatial flux. The hybrid sector interpolates between the two, giving a damped or amplified wave equation whose box solutions divide into underdamped, critically damped, and overdamped regimes. The framework matters because Carroll symmetry is the ultra-relativistic counterpart of Galilean symmetry and controls the dynamics of null surfaces and immobile ('fractonic') excitations.

What carries the argument

The machinery is the controlled Carrollian contraction: replacing $c$ by $\epsilon c$ and sending $\epsilon\to0$ under three different scalings of the coordinates, mass, and wavefunction. The space-like and hybrid sectors are carried by a non-dynamical Stückelberg-like compensating field $\chi$ (an auxiliary field with no kinetic term) whose Carroll boost variation is a differential operator, e.g. $\delta_C\chi=\beta^2\partial_t+2\beta\cdot\partial_x$; it cancels the non-invariance of the spatial Laplacian. The time-like sector is carried by the indefinite Klein-Gordon norm, which turns temporal scattering into a Bogoliubov transformation preserving $\lvert E\rvert^2-\lvert F\rvert^2=1$, i.e. $\lvert T\rvert^2=1+\lvert R\rvert^2$. These two objects—the contraction scalings and the compensating field—fix which sectors exist and which states are physical.

What would settle it

A direct check is to apply a non-zero Carroll boost to the space-like equation, keep all terms, and verify that the $\chi$-dependent terms cancel the Laplacian's extra terms for every smooth $\psi$; if no local choice of $\chi$ closes the algebra, or if the canonical equal-time commutator of the zero-norm states yields no measurable flux, the central claim fails.

Watch

Extended reading notes

Core claim

The central claim is that the ultra-relativistic limit of relativistic spin-0 quantum mechanics is not a single equation but a three-way branching. Depending on how coordinates, mass, and wavefunction are rescaled before $c\to0$, the Klein-Gordon equation contracts to (i) the time-like Carroll equation $(\hbar^2\partial_t^2+m_0^2)\psi=0$, which is second-order in time and has no spatial derivatives; (ii) a space-like sector, first-order in time with a compensating field $\chi$ whose boost variation is $\delta_C\chi=\beta^2\partial_t+2\beta\cdot\partial_x$, giving purely imaginary energies and zero-norm propagating states; and (iii) a hybrid sector containing both $\partial_t^2$ and $\partial_x^2$ plus a first-order $\chi\partial_t$ term, which is a damped/amplified oscillator for real wavefunctions and tachyonic for complex ones. For each sector the paper constructs the continuity equation, the probability density and current, and exact particle-in-a-box and rectangular-barrier solutions. The signature result is temporal tunneling: a rectangular potential step in time converts part of an incoming positive-frequency mode into a negative-frequency mode, with amplitudes satisfying $\lvert T\rvert^2=1+\lvert R\rvert^2$, the indefinite-norm analogue of $\lvert T\rvert^2+\lvert R\rvert^2=1$.

Load-bearing premise

The space-like and hybrid sectors survive only if adding a non-dynamical auxiliary field that shifts under Carroll boosts lets a single scalar field carry spatial flux, in contrast to the known result the paper cites that a bare Carrollian scalar cannot propagate.

Editorial extensions

If this is right

  • Temporal barriers act as tunable beam splitters in the time-like sector: a pure positive-frequency state emerges as a superposition of positive- and negative-frequency parts with probabilities set by the barrier height and duration.
  • A particle confined in a temporal box has quantized energy levels; for a zero interior potential the rest mass itself is quantized in units of $\pi\hbar/T$.
  • Propagating space-like Carroll states are null in the sense of zero probability density, so measurements in that sector should be organized around the spatial flux rather than local density.
  • Hybrid Carrollian particles tunnel through a static spatial barrier with the same transmission and reflection probabilities as an ordinary non-relativistic particle, while the overall amplitude is multiplied by a global damping or amplification factor.
  • Time-like Carroll quantum mechanics coincides with the 0+1-dimensional reduction of a dipole-symmetric scalar field theory, providing a quantum-mechanical realization of isolated fractonic monopoles.

Reading between the lines

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

  • If the null states of the space-like sector are taken literally, an immediate test is to construct interference experiments where two such fluxes meet; the density would remain zero everywhere while the current pattern should still show interference fringes—something impossible in standard non-relativistic quantum mechanics.
  • The same contraction recipe should produce an exactly parallel trio of Galilean sectors in the $c\to\infty$ limit, with a 'spatial tunneling' analogue of temporal tunneling; the paper only sketches this direction.
  • The relation $\lvert T\rvert^2=1+\lvert R\rvert^2$ is identical in form to particle creation by time-dependent backgrounds in quantum field theory; if the analogy holds, the temporal barrier should also produce a calculable entanglement entropy between the positive- and negative-frequency sectors.
  • One could look for an experimental realization in ultracold atom or photonic simulators where an engineered time-dependent dispersion mimics the Carrollian slow light, checking whether the predicted zero-density propagating mode appears as a dark state that still carries flux.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper claims to develop Carrollian quantum mechanics by taking c→0 contractions of the Klein-Gordon equation (2.1). Three sectors are announced: time-like, space-like, and hybrid. For the time-like sector the paper derives the Marsot equation, a continuity equation with vanishing current, the temporal box spectrum, and temporal barrier scattering with the Bogoliubov relation |T|²=1+|R|². The space-like sector is obtained by adding a compensating field χ (or an additive χ̃) and is claimed to describe tachyonic null states with spatial current. The hybrid sector combines temporal and spatial second derivatives with a first-order χ∂t term and is analyzed in damped/amplified regimes. The paper also claims a fracton/time-like Carroll duality and presents a classical hybrid particle action in Appendix A.

Significance. If the three-sector derivation were valid, the paper would provide a useful catalogue of Carrollian quantum-mechanical models with exact box and barrier solutions. The time-like part is genuinely valuable: the temporal tunneling computation is algebraically consistent and the identity |E|²−|F|²=1 follows correctly from the indefinite Klein-Gordon norm. However, the central claim is not established: the space-like and hybrid equations are not obtained from the announced contractions, and the space-like probabilistic interpretation has an inconsistency between the current and the physical flux. The no-propagation obstruction cited as ref. [40] is not engaged. These issues affect the core of the paper rather than presentation.

major comments (4)
  1. [Section 3.2, Eqs. (3.5)–(3.7)] The announced contractions do not produce the equations used in Sections 5 and 6. For the hybrid scaling (3.7), substitute c=ϵc̄, x=ϵX, m0=M/ϵ into Eq. (2.1). After multiplying by ϵ², the leading-order equation is (1/c̄²)∂t²Φ−∂X²Φ=0, a massless wave equation, not Eq. (6.1); no ℏ²χ∂t term is generated. If c is instead kept fixed, (3.7) leads to −∂X²Φ+M²Φ=0 at leading order, still without the χ∂t term. The same structural objection applies to the space-like sector: no rescaling of the second-order Klein-Gordon equation can produce the first-order derivative in Eq. (5.4). Equations (5.4) and (6.1) are therefore introduced by hand after the limit, and the abstract's claim of a systematic three-sector derivation from c→0 contractions is unsupported.
  2. [Section 5.3, Eqs. (5.14)–(5.16)] The continuity and probability interpretation in the space-like sector are inconsistent. The continuity equation is derived for J=ℏ²(ψ*∂xψ−ψ∂xψ*), which is purely imaginary for plane-wave modes. The paper then defines a 'physical flux' J≡−iJ, but with this substitution the conserved equation becomes ∂tρ+i∂xJ=0, not ∂tρ+∂xJ=0. Thus the real flux is not the current appearing in the conservation law, and the claimed probabilistic interpretation with a real current is not demonstrated.
  3. [Section 5.2 and ref. [40]] The paper cites ref. [40], which states that single Carrollian scalars cannot propagate, but it does not explain why the space-like and hybrid sectors circumvent this obstruction. The space-like sector is explicitly said to retain spatial propagation, yet its energy eigenstates have purely imaginary energies and hence real exponential time dependence, so the solutions do not exhibit oscillatory propagation. If the χ coupling is meant to evade the no-go result, an explicit argument is required; without it, the propagation claim in Section 5.1 is unsupported.
  4. [Section 4.5, Eq. (4.46)] The claimed fracton/time-like Carroll duality contains a sign error. The Lagrangian L=|∂tϕ|²+m0²|ϕ|² given in Eq. (4.45) yields the Euler-Lagrange equation ℏ²∂t²ϕ−m0²ϕ=0, not ℏ²∂t²ϕ+m0²ϕ=0 as written in Eq. (4.46). The equation with the plus sign is not the oscillator equation of this Lagrangian, so the duality as stated is incorrect unless the Lagrangian is also changed.
minor comments (5)
  1. [Section 3.1 vs. Section 3.2] Section 3.1 specifies c→ϵc, but the scalings (3.5)–(3.7) never list the c-scaling. This ambiguity is directly related to the contraction failure identified above and should be clarified.
  2. [Section 5.1, Eq. (5.4)] The equation is described as Hermitian for complex ψ, but with χ=iχ′ the operator −ℏ²χ′∂t is anti-Hermitian. The terminology should be corrected or the Hermiticity claim justified with respect to a specific inner product.
  3. [Appendix A] The classical compensating field χ of Eq. (A.5) transforms as δχ=−e p·β, whereas the quantum compensating field in Eq. (5.3) transforms as δχ=β²∂t+2β·∂x. The paper does not connect these two mechanisms or explain why the same symbol χ is used for both.
  4. [References] References [18] and [19] are identical duplicates and should be merged.
  5. [Throughout] There are numerous typos and stylistic slips (e.g., 'factories' instead of 'factorizes' in Section 4.4.1, 'thistemporal' in Section 4.4). A careful proofreading pass is needed.

Circularity Check

4 steps flagged · score 6.0 of 10

Two of the three 'contracted' sectors are constructed, not derived: the χ-dependent equations are built to enforce Carroll invariance, Approach 2 is an identity, and the hybrid contraction (3.7) does not actually yield Eq. (6.1). The time-like sector is independent and not circular.

  1. self definitional [Sec. 5.1, Approach 2, Eqs. (5.6)-(5.8)]
    "An alternative derivation yields the constraint equation ℏ2˜χ + (−ℏ2∂2x + m20)ψ = 0. ... The field ˜χ is not an independent field but a functional of ψ, encoding its spatial structure: ˜χ = ∂2xψ − m20 ℏ2 ψ."

    With this definition, Eq. (5.6) reduces to 0=0. The 'constraint equation' is simply the definition of χ̃; the 'alternative derivation' adds no equation beyond relabeling the spatial Klein-Gordon operator as a source. The Carroll invariance of the formulation is then just a statement about the transformation chosen for χ̃ to cancel that same operator, so this sector is an identity rather than a contraction of (2.1).

  2. self definitional [Sec. 6.1, Approach 2, Eq. (6.7)]
    "The field ˜χ is determined entirely by ψ and its derivatives via ℏ2˜χ = −ℏ2∂2t ψ + ℏ2∂2x ψ − m20 ψ."

    Substituting Eq. (6.7) into Eq. (6.5) gives 0=0. The 'inhomogeneous Klein-Gordon equation with a source term' is the original Klein-Gordon operator renamed as χ̃; there is no new hybrid wave equation. Presenting Approach 2 as a third-sector formulation is a relabeling of the input equation, not a derivation from the contraction.

2 more flagged steps
  1. other [Sec. 6.1, Approach 1, Eq. (6.1), from Eq. (3.7)]
    "Applying the hybrid contraction limit (3.7) to the Klein-Gordon equation (2.1) and requiring Carroll boost invariance yields the hybrid Carrollian wave equation (ℏ2∂2t −ℏ2∂2x +ℏ2χ∂t +m20)ψ = 0."

    The stated scaling (3.7) leaves t unscaled and introduces no χ; direct substitution into (2.1) gives a rescaled Klein-Gordon equation without any χ∂t term. The first-order temporal derivative is inserted afterward to satisfy the boost requirement, with χ's transformation (6.4) chosen to cancel the boost variation of the Laplacian. The 'derived' hybrid equation is therefore fixed by the symmetry requirement, not produced by the contraction; the claimed derivation is an ansatz presented as a limit.

  2. other [Sec. 5.1, Approach 1, Eq. (5.2), from Eq. (3.6)]
    "Applying the space-like Carroll limit (3.6) to the Klein-Gordon equation (2.1) for a real wavefunction gives (−ℏ2∂2x +m20)ψ=0, which is not Carroll boost invariant. To restore invariance, a compensating field χ is introduced, leading to the modified wave equation (ℏ2χ∂t −ℏ2∂2x +m20)ψ=0."

    The paper's own text separates the contraction result (5.1) from the final equation (5.2): (5.2) is obtained by adding χ∂t, with the transformation (5.3) designed to cancel the boost variation of ∂x². Thus the 'space-like Carroll equation' is not a limit of the Klein-Gordon equation; its Carroll invariance is guaranteed by the definition of χ. The sector exists by construction, not by the announced contraction.

full rationale

The time-like sector is not circular: applying the scaling (3.5) to the Klein-Gordon equation does produce the Marsot equation (4.1), and the Bogoliubov relation |T|²=1+|R|² follows from the standard indefinite Klein-Gordon norm via explicit matching. The central circularity burden is in the space-like and hybrid sectors. For both, the paper itself shows that the naive contraction gives an equation without the χ∂t term, and the compensating field χ is then introduced with a transformation law chosen to cancel the non-invariance of the Laplacian. The resulting 'Carroll-invariant' equation is therefore true by construction of χ, not by contraction of (2.1). The hybrid case is stronger still: the stated scaling (3.7) contains no time rescaling and no χ, and direct substitution does not yield Eq. (6.1). In addition, Approach 2 in both sectors defines χ̃ as the negative of the operator acting on ψ, making the 'constraint equation' or 'inhomogeneous Klein-Gordon equation' an identity. These are not independent predictions; they are relabelings or ansätze. The paper's self-citation [13] is present but not the main source of circularity; the main issue is that two of the three claimed sectors reduce to definitions. The time-like sector and its tunneling analysis retain independent content, so the overall score is a 6 rather than higher.

Assumptions & free parameters 1 free parameters · 5 assumptions · 2 invented entities

The framework leans on two hand-inserted objects: the contraction scalings, which are chosen to deliver the desired equations, and the compensating fields χ and χ~, which are added to secure Carroll invariance. Neither is derived from an action or from a unique mathematical criterion. The time-like sector avoids these additions and is the cleanest part of the paper.

free parameters (1)
  • Carroll contraction scaling powers in (3.5)-(3.7) = hand-chosen: m~1/eps^2, psi~eps^2; for space-like/hybrid also t~1/eps, x~eps, m~1/eps
    The scalings are not derived from an independent principle; they are selected so that each sector keeps the desired derivative structure. This is the main hand-chosen input to the central claim.
assumptions (5)
  • domain assumption The Klein-Gordon probability density and current (2.9)-(2.10) are the correct objects to take the c→0 limit for probability interpretation.
    All sector continuity equations are built by manipulating the Klein-Gordon density and current; the paper assumes this probabilistic structure survives the contraction.
  • ad hoc to paper A non-dynamical background field χ with operator-valued boost variation (5.3) can restore Carroll invariance without changing the physics.
    Introduced in Sec. 5.1; it is not derived from the Klein-Gordon equation or from an action. This is the load-bearing device for the space-like and hybrid sectors.
  • domain assumption The physical Hilbert space in the time-like sector is the positive-frequency subspace.
    Sec. 4.2 excludes negative-frequency states to obtain a positive density; this truncation is an interpretive choice, not a consequence of the Carroll group.
  • domain assumption Zero-norm states in the space-like sector are physically meaningful because the underlying geometry is null.
    Sec. 5.3 asserts that ρ=0 states are physical due to Carrollian null geometry; this is an interpretation of the equations, not a derived consequence.
  • standard math Separation of variables and continuity matching at potential steps are valid for these equations.
    Used throughout for the box and scattering solutions; standard for linear second-order equations.
invented entities (2)
  • Compensating field χ
    purpose: Restores Carroll boost invariance of the space-like and hybrid wave equations by canceling non-invariant terms from the spatial Laplacian.
    A non-dynamical background field with no kinetic term and no falsifiable handle outside the paper; introduced specifically to enforce the target symmetry (Secs. 5.1-5.2, 6.1).
  • Compensating field χ~
    purpose: Additive variant in Approach 2; a functional of ψ rather than an independent field, encoding spatial structure in the constraint equations.
    Appears in Eqs. (5.6) and (6.5); it is determined entirely by ψ and its derivatives, and has no independent dynamics or observable prediction.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Carrollian Quantum Mechanics: Time-like, Space-like and Hybrid Sectors." pith.science (2026). https://pith.science/paper/7HTFP44L

@misc{pith2026260808384,
  author       = {Pith},
  title        = {Pith review of: Carrollian Quantum Mechanics: Time-like, Space-like and Hybrid Sectors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7HTFP44L}},
  note         = {Machine review of arXiv:2608.08384}
}
abstract

We develop a comprehensive theoretical framework for Carrollian quantum mechanics by performing systematic ultra-relativistic contractions of the Klein--Gordon equation in the limit $c \to 0$. This limiting process uncovers three distinct sectors---time-like, space-like, and hybrid---each governed by a Carroll-invariant wave equation and accompanied by a consistent probabilistic interpretation. The time-like sector exhibits a novel temporal tunneling phenomenon, characterized by the relation $|\mathcal{T}|^2 = 1 + |\mathcal{R}|^2$, which reflects the indefinite character of the Klein--Gordon norm. In the space-like sector, the presence of spatial propagation necessitates a tachyonic dispersion relation, which forces the density to vanish for energy eigenstates, yielding zero-norm (null) states that gain physical relevance within Carrollian physics due to the underlying null geometric structure. The hybrid sector combines features of both sectors and admits two distinct formulations, which may be either tachyonic or non-tachyonic, and can be interpreted as an inhomogeneous Klein--Gordon equation. For all three sectors, we derive the corresponding continuity equations, probability densities, and currents, and investigate canonical quantum systems---including the particle in a box and tunneling phenomena---within the Carrollian regime.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

39 extracted references · 6 canonical work pages

  1. [15]

    Quantum carroll/fracton particles,

    J. Figueroa-O’Farrill, A. Pérez, and S. Prohazka, “Quantum carroll/fracton particles,”JHEP10 (2023) 041,2307.05674

    arXiv 2023

  2. [40]

    Frozen Motion: Why Single Carrollian Scalars Cannot Propagate,

    A. J. Bruce, “Frozen Motion: Why Single Carrollian Scalars Cannot Propagate,”2603.07081. – 38 –

    arXiv

  3. [1]

    Une nouvelle limite non-relativiste du groupe de poincaré,

    J.-M. Lévy-Leblond, “Une nouvelle limite non-relativiste du groupe de poincaré,” inAnnales de l’IHP Physique théorique, vol. 3, pp. 1–12. 1965

    work page 1965

  4. [2]

    On an analogue of the galilei group,

    N. D. Sen Gupta, “On an analogue of the galilei group,”Il Nuovo Cimento A44(1966) 512–517

    work page 1966

  5. [3]

    Non-lorentzian theories with and without constraints,

    E. A. Bergshoeff, J. Gomis, and A. Kleinschmidt, “Non-lorentzian theories with and without constraints,”JHEP01(2023) 167,2210.14848

    arXiv 2023

  6. [4]

    Carroll structures, null geometry and conformal isometries,

    L. Ciambelli, R. G. Leigh, C. Marteau, and P. M. Petropoulos, “Carroll structures, null geometry and conformal isometries,”Phys. Rev. D100(2019), no. 4, 046010,1905.02221

    arXiv 2019

  7. [5]

    Carrollian physics at the black hole horizon,

    L. Donnay and C. Marteau, “Carrollian physics at the black hole horizon,”Class. Quant. Grav. 36(2019), no. 16, 165002,1903.09654

    arXiv 2019

  8. [6]

    Carroll black holes,

    F. Ecker, D. Grumiller, J. Hartong, A. Pérez, S. Prohazka, and R. Troncoso, “Carroll black holes,”SciPost Phys.15(2023), no. 6, 245,2308.10947. – 36 –

    arXiv 2023

Show all 39 references
  1. [7]

    Carrollian manifolds and null infinity: a view from Cartan geometry,

    Y. Herfray, “Carrollian manifolds and null infinity: a view from Cartan geometry,”Class. Quant. Grav.39(2022), no. 21, 215005,2112.09048

    2022 arXiv

  2. [8]

    Carrollian structure of the null boundary solution space,

    H. Adami, A. Parvizi, M. M. Sheikh-Jabbari, V. Taghiloo, and H. Yavartanoo, “Carrollian structure of the null boundary solution space,”JHEP02(2024) 073,2311.03515

    2024 arXiv

  3. [9]

    Correspondence between asymptotically flat spacetimes and nonrelativistic conformal field theories,

    A. Bagchi, “Correspondence between asymptotically flat spacetimes and nonrelativistic conformal field theories,”Phys. Rev. Lett.105(2010) 171601,1006.3354

    2010 arXiv

  4. [10]

    Flat holography and Carrollian fluids,

    L. Ciambelli, C. Marteau, A. C. Petkou, P. M. Petropoulos, and K. Siampos, “Flat holography and Carrollian fluids,”JHEP07(2018) 165,1802.06809

    2018 arXiv

  5. [11]

    Carrollian conformal fields and flat holography,

    K. Nguyen and P. C. West, “Carrollian conformal fields and flat holography,”Universe9(2023), no. 7, 336,2305.02884

    2023 arXiv

  6. [12]

    Holography in flat spacetimes: the case for Carroll,

    A. Bagchi, P. Dhivakar, and S. Dutta, “Holography in flat spacetimes: the case for Carroll,” JHEP08(2024) 144,2311.11246

    2024 arXiv

  7. [13]

    Fracton and non-lorentzian particle duality: gauge field couplings and geometric implications,

    M. M. Ahmadi-Jahmani and A. Parvizi, “Fracton and non-lorentzian particle duality: gauge field couplings and geometric implications,”JHEP08(2025) 157,2503.21660

    2025 arXiv

  8. [14]

    Carroll/fracton particles and their correspondence,

    J. Figueroa-O’Farrill, A. Pérez, and S. Prohazka, “Carroll/fracton particles and their correspondence,”JHEP06(2023) 207,2305.06730

    2023 arXiv

  9. [16]

    Fractons, dipole symmetries and curved spacetime,

    L. Bidussi, J. Hartong, E. Have, J. Musaeus, and S. Prohazka, “Fractons, dipole symmetries and curved spacetime,”SciPost Phys.12(2022), no. 6, 205,2111.03668

    2022 arXiv

  10. [17]

    Dynamics of carroll strings,

    B. Cardona, J. Gomis, and J. M. Pons, “Dynamics of carroll strings,”JHEP2016(2016), no. 7, 1–11

    2016

  11. [19]

    Tensionless path from closed to open strings,

    A. Bagchi, A. Banerjee, and P. Parekh, “Tensionless path from closed to open strings,”Phys. Rev. Lett.123(2019), no. 11, 111601,1905.11732

    2019 arXiv

  12. [20]

    Scaling Symmetry and Carrollian Gravity,

    H. Afshar and M. Ahmadi-Jahmani, “Scaling Symmetry and Carrollian Gravity,”2512.20736

    arXiv

  13. [21]

    Gauging the carroll algebra and ultra-relativistic gravity,

    J. Hartong, “Gauging the carroll algebra and ultra-relativistic gravity,”JHEP08(2015) 069, 1505.05011

    2015 arXiv

  14. [22]

    Carroll versus galilei gravity,

    E. Bergshoeff, J. Gomis, B. Rollier, J. Rosseel, and T. ter Veldhuis, “Carroll versus galilei gravity,”JHEP03(2017) 165,1701.06156

    2017 arXiv

  15. [23]

    Hall effects in carroll dynamics,

    L. Marsot, P.-M. Zhang, M. Chernodub, and P. A. Horvathy, “Hall effects in carroll dynamics,” 2212.02360

    arXiv

  16. [24]

    J. D. Bjorken and S. D. Drell,Relativistic Quantum Mechanics. McGraw-Hill, 1964

    1964

  17. [25]

    Greiner,Relativistic Quantum Mechanics: Wave Equations

    W. Greiner,Relativistic Quantum Mechanics: Wave Equations. Springer, 3 ed., 2000

    2000

  18. [26]

    J. J. Sakurai,Advanced Quantum Mechanics. Addison-Wesley, 1967

    1967

  19. [27]

    Dynamics of carroll particles,

    E. Bergshoeff, J. Gomis, and G. Longhi, “Dynamics of carroll particles,”Class. Quant. Grav.31 (2014), no. 20, 205009,1405.2264

    2014 arXiv

  20. [28]

    Classification of conformal carroll algebras,

    H. Afshar, X. Bekaert, and M. Najafizadeh, “Classification of conformal carroll algebras,”JHEP 12(2024) 148,2409.19953

    2024 arXiv

  21. [29]

    A non-lorentzian primer,

    E. Bergshoeff, J. Figueroa-O’Farrill, and J. Gomis, “A non-lorentzian primer,”SciPost Phys. Lect. Notes69(2023) 1,2206.12177. – 37 –

    2023 arXiv

  22. [30]

    Carroll symmetry, dark energy and inflation,

    J. de Boer, J. Hartong, N. A. Obers, W. Sybesma, and S. Vandoren, “Carroll symmetry, dark energy and inflation,”Front. in Phys.10(2022) 810405,2110.02319

    2022 arXiv

  23. [31]

    Carroll swiftons,

    F. Ecker, D. Grumiller, M. Henneaux, and P. Salgado-Rebolledo, “Carroll swiftons,”Phys. Rev. D110(2024), no. 4, L041901,2403.00544

    2024 arXiv

  24. [32]

    Planar carrollean dynamics, and the carroll quantum equation,

    L. Marsot, “Planar carrollean dynamics, and the carroll quantum equation,”J. Geom. Phys.179 (2022) 104574,2110.08489

    2022 arXiv

  25. [33]

    Quantum glassiness,

    C. Chamon, “Quantum glassiness,”Phys. Rev. Lett.94(2005), no. 4, 040402, cond-mat/0404182

    2005 arXiv

  26. [34]

    Fracton phases of matter,

    M. Pretko, X. Chen, and Y. You, “Fracton phases of matter,”Int. J. Mod. Phys. A35(2020), no. 06, 2030003,2001.01722

    2020 arXiv

  27. [35]

    Towards classification of fracton phases: the multipole algebra,

    A. Gromov, “Towards classification of fracton phases: the multipole algebra,”Phys. Rev. X9 (2019), no. 3, 031035,1812.05104

    2019 arXiv

  28. [36]

    Quantizing Carrollian field theories,

    J. Cotler, K. Jensen, S. Prohazka, A. Raz, M. Riegler, and J. Salzer, “Quantizing Carrollian field theories,”JHEP10(2024) 049,2407.11971

    2024 arXiv

  29. [37]

    Foundations of Carrollian geometry,

    L. Ciambelli and P. Jai-akson, “Foundations of Carrollian geometry,”Phys. Rept.1188(2026) 1–51,2510.21651

    2026 arXiv

  30. [38]

    Carroll stories,

    J. de Boer, J. Hartong, N. A. Obers, W. Sybesma, and S. Vandoren, “Carroll stories,” 2307.06827

    arXiv

  31. [39]

    Two interacting conformal carroll particles,

    R. Casalbuoni, D. Dominici, and J. Gomis, “Two interacting conformal carroll particles,” 2306.02614

    arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.