Pith. sign in

REVIEW 3 major objections 4 minor 41 references

In teleparallel Mielke–Baekler spacetimes, spin splits particle dynamics into regular and critical sectors, and for massless particles it reduces Noether symmetries from conformal transformations to isometries.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-01 13:48 UTC pith:D7LP4IBC

load-bearing objection Careful coadjoint-orbit actions for spinning particles in teleparallel MB/AdS3 with a genuinely new sector split; the massless Noether claim is real but proven only under a restricted ansatz, so the abstract overreaches until that is fixed. the 3 major comments →

arxiv 2607.26532 v1 pith:D7LP4IBC submitted 2026-07-29 hep-th

Massive and massless particles in Mielke--Baekler geometries

Carles Batlle , Roberto Casalbuoni , Daniele Dominici , Jos\'{e} Figueroa-O'Farrill , Joaquim Gomis This is my paper
classification hep-th PACS 04.20.-q04.50.Kd11.30.Cp
keywords Mielke–Baekler spacetimesteleparallel geometrynonlinear realisationscoadjoint orbitsconstrained systemsinvariant connectionsconformal algebraPapapetrou equation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper builds worldline actions for massive and massless spinning particles in three-dimensional Mielke–Baekler geometries—homogeneous spacetimes whose invariant metric connection is allowed to carry torsion. Working in the teleparallel branch, where the invariant connection is flat but torsionful (realized by anti-de Sitter spacetime), it shows that the Wess–Zumino term responsible for spin splits each theory into a generic 'regular' sector and an exceptional 'critical' sector with fewer physical degrees of freedom. In the massive critical sector (m + qs = 0), momentum transport relative to the Levi-Civita connection acquires a Papapetrou-type spin–curvature forcing term; in the regular sector the spin–curvature force vanishes and trajectories are ordinary geodesics. For massless particles, conformal transformations are Noether symmetries only when spin vanishes; for nonzero spin only the Killing fields survive in the regular sector, and any conformal Killing contribution can be removed by gauge transformations in the critical sector. The paper thus identifies spin as the control parameter for both the sector structure and the surviving spacetime symmetries.

Core claim

The central claim is that adding a Wess–Zumino spin term to a particle moving in a teleparallel Mielke–Baekler spacetime qualitatively changes the dynamics and the symmetry content. The spin term causes a dichotomy between a regular sector (generic coadjoint orbit, m+qs≠0) and a critical sector (exceptional two-dimensional orbit, m+qs=0), with the critical sector having two extra first-class constraints and hence two fewer physical degrees of freedom. In the massive case, momentum is always parallel-transported by the transposed Weitzenböck connection—the flat torsionful connection with reversed torsion—but in the critical sector the equivalent Levi-Civita description is a Papapetrou equatio

What carries the argument

The load-bearing object is the Mielke–Baekler algebra with brackets [JA,JB]=εABC JC, [JA,PB]=εABC PC, [PA,PB]=εABC(pJC+qPC); its canonical connection has invariant torsion and curvature, and the teleparallel branch p=0 is a flat Weitzenböck connection with torsion. The accompanying coframe θA satisfies the Maurer–Cartan equation dθA = −(q/2)εABC θB∧θC, which encodes the torsion and drives all transport equations. The spin term is a Wess–Zumino term s(cosh v−1)φ̇ in the massive action; it is what creates the regular/critical sector split through the combination m+qs, and through the constraint matrix it determines which variables are second-class, first-class, or gauge. The transposed Weitzen

Load-bearing premise

The claim that nonzero spin eliminates conformal Noether symmetries depends on restricting spacetime point transformations to δxM = αM(x) independent of the internal variables u and φ; allowing more general transformations could bring conformal symmetries back.

What would settle it

Relax the ansatz and allow δxM = αM(x,u,φ) in the Noether symmetry equations; finding a conformal Killing solution with s≠0 would refute the Killing-only conclusion. A second check: in the flat limit q=0, s≠0, the critical condition m+qs=0 becomes m=0; verify that the reduced phase-space brackets reproduce the known massless anyon brackets—any mismatch would indicate a flaw in the sector analysis.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • In the regular sector of both massive and massless particles, the unparametrised spacetime trajectories are geodesics of the metric; spin does not add a propagating degree of freedom but deforms the reduced phase-space geometry, making the coordinate Dirac brackets spin-dependent.
  • At the critical locus, the physical phase space collapses from four to two dimensions and two new gauge symmetries appear; the momentum–velocity relation becomes degenerate, so the theory describes fewer degrees of freedom than the generic orbit.
  • For massive particles in the critical sector, Levi-Civita transport is not geodesic: a Papapetrou spin–curvature forcing term −½ R̃_MN AB ẋ^N S^AB appears, which vanishes in the regular sector.
  • For massless particles, conformal transformations are Noether symmetries only when s=0; for s≠0 the symmetry algebra of the spinning action is reduced to the six-dimensional Killing (isometry) algebra, and in the critical sector the conformal Killing pieces are gauge-trivial.
  • The teleparallel MB translations differ from ordinary translations in conformally flat coordinates; they are given by PA = ΠA − (q/2)JA − (q²/8)KA, so the same conformal algebra looks q-dependent in the teleparallel basis but remains isomorphic to so(3,2).

Where Pith is reading between the lines

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

  • Inference: because the massless Noether analysis restricts transformations to δxM = αM(x), a more general ansatz allowing dependence on the internal variables u and φ could in principle restore conformal symmetries for s≠0; the paper explicitly notes this restriction.
  • Inference: the critical sector's degenerate momentum–velocity map and enlarged gauge freedom resemble the phase-space behavior of anyons and of massless representations with continuous spin; quantizing the reduced critical phase space could yield spin-statistics relations in AdS3 with torsion.
  • Inference: the Papapetrou-type forcing term gives a concrete classical probe of torsion: a spinning test particle in the critical sector will deviate from the Levi-Civita geodesic in a way controlled by the constant torsion parameter q, offering a sharp observable difference between teleparallel and purely metric 3D gravity.
  • Inference: the q→0 limit of the Dirac brackets reduces to the well-known anyonic noncommutativity, so the MB geometry can be viewed as a torsionful deformation of anyon dynamics; exploring non-teleparallel branches could reveal whether the sector structure persists beyond AdS3.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The manuscript constructs coadjoint-orbit worldline actions, via nonlinear realisations, for massive and massless spinning particles on three-dimensional Mielke--Baekler homogeneous spacetimes, concentrating on the teleparallel p=0 branch (Weitzenb\"ock connection). It derives a two-sector structure: the constraint matrices (3.89) and (4.41) have determinants (3.90) and (4.42), which change rank at m+qs=0 and \mu+qs=0 respectively. In the massive critical sector the Levi-Civita description acquires a Papapetrou-type spin-curvature force (3.71); in the regular sector the force vanishes and momentum is parallel transported by the transposed Weitzenb\"ock connection. For massless particles the paper studies Noether symmetries under the point-transformation ansatz \delta x^M=\alpha^M(x), \delta u=\beta, \delta\phi=\gamma and concludes that conformal symmetries survive only for zero spin; for s\neq0 the regular sector admits only Killing fields, while in the critical sector conformal contributions are gauge-equivalent to Killing. Appendices provide the Lie-algebra deformation interpretation, teleparallel geometry review, gauge-fixing details, and the null-triad identity.

Significance. If the claims hold, this is a useful systematic treatment. The explicit constraint matrices, determinants, and Dirac brackets are detailed and reproducible; the sector dichotomy is derived from Poisson-bracket ranks rather than imported. The Papapetrou identification in the massive critical sector is explicitly and cleanly derived, and the relation between teleparallel MB translations and the conformal algebra in \S4.6 is a nice observation. The paper also names and builds on prior AdS3 sector work [19,20]. The main caveat is that the massless Noether conclusion is presented with more generality than the derivation supports; that issue must be fixed before the central claim is reliable.

major comments (3)
  1. [\S4.5, Eqs. (4.57)--(4.58)] The abstract claims that for nonzero spin only the Killing subset survives as genuine Noether transformations, and \S5 claims that the NSE admit conformal solutions only when s=0. These claims are proved only under the ansatz \delta x^M=\alpha^M(x) with \alpha^M independent of u,\phi. The text itself notes that this is 'not the most general point transformations' (p. 32), but the abstract and conclusions do not carry this qualification. If \alpha^M is allowed to depend on u and \phi, then \alpha^A=\iota_\alpha \theta^A and hence K^A_B in (4.66) and the NSE (4.68)--(4.70) acquire new terms involving \partial_u\alpha and \partial_\phi\alpha; the argument leading to \beta=\chi_A n^A in (4.75) and the Killing reduction (4.81)--(4.86) no longer applies. No argument in the text excludes such transformations. This is load-bearing for the massless symmetry-breaking claim. I do not assert the con
  2. [\S4.5, Eqs. (4.87)--(4.91)] The critical-sector statement that 'any conformal Killing contribution can be set to zero by a gauge transformation' is established only for the particular decomposition of \Delta^A obtained within the same restricted ansatz, using the specific gauge choices \rho_u=-qc, \rho_\phi=-qb, \varepsilon=a/(\mu e^\phi). The text does not demonstrate that every conformal Killing vector, or every possible \alpha^M in an enlarged class, admits this decomposition and removal. Since this is part of the same unqualified conclusion, it should either be proved under the same assumptions or stated with its limitations.
  3. [\S4.5, Eqs. (4.103)--(4.110)] For s\neq0 the analysis establishes a necessity statement: any solution of the NSE within the ansatz has \alpha Killing. However, the paper explicitly declines to solve the overdetermined first-order system for \beta and \gamma for general Killing vectors. Thus the phrase 'only the Killing subset survives' also requires an existence argument: one must show that every Killing vector extends to a Noether transformation of the full action. The G-invariant coadjoint-orbit construction makes this very plausible, but the manuscript should either verify it or downgrade the statement.
minor comments (4)
  1. [\S4.5, after (4.77)] The phrase 'A little analysis' is used at a key step. The derivation of (4.75) from the quadratic dependence on u in n^A should be shown or at least outlined; it is short but important.
  2. [\S4.6] The conformal algebra commutators are given in a particular sign convention. A brief consistency check with (4.157) and (4.168), or a statement of convention, would help avoid confusion.
  3. [References] Please check the identifiers for Refs. [8] and [20]; the arXiv prefixes '2604' and '2509' look possibly anachronistic or erroneous.
  4. [General] The notation with hats on the right-invariant generators in \S4.6.1 is fine, but it would be helpful to state explicitly that they generate the left action of the translation subgroup, not the right action, to match the conventional naming.

Circularity Check

0 steps flagged

No significant circularity: all load-bearing results are re-derived from stated inputs and explicit computations.

full rationale

The paper's central derivations are internally constructed from first principles and explicit algebra. The sector structure for massive particles is not imported: the determinant of the constraint matrix is computed directly as det M = m^2 sinh^4(v)(m+qs)^2 (eq. 3.90), and the massless analogue det M = e^{4φ} μ^2(μ+qs)^2 (eq. 4.42) is derived from the Poisson brackets of the constraints. The regular/critical dichotomy therefore follows from the model's own equations, and the citations to Refs. [19,20] only note a prior observation of a similar sector phenomenon; they are not load-bearing for the derivation. Similarly, the momentum transport equation (3.47), the transposed-Weitzenböck rewriting (3.50), and the Papapetrou formula (3.71) are obtained by substituting the definition p_M = -m u^A θ^A_M and using the Maurer–Cartan relation (3.40), not by assuming the result. The massless Noether analysis is derived from the Noether symmetry equations (4.68)–(4.70) and the identity in Appendix D. The only caveat is that the analysis is restricted to point transformations with α^M(x) independent of the internal variables u, φ, as the paper explicitly acknowledges; this is a stated assumption that could limit the strength of the negative claim for s≠0, but it is not a circular reduction of the result to its inputs. No fitted constants are renamed as predictions, and no uniqueness or ansatz is smuggled in via self-citation. Hence the paper's derivations are self-contained and not circular.

Axiom & Free-Parameter Ledger

6 free parameters · 7 axioms · 0 invented entities

The central claim rests on the standard coadjoint-orbit/nonlinear-realization formalism, the MB algebra ansatz (2.14), and the teleparallel p=0 restriction. No constants are fitted to data; q, m, s, μ are model inputs. No new physical entities are postulated.

free parameters (6)
  • q = non-zero (undetermined)
    Torsion/AdS scale in MB bracket (2.14); central to the teleparallel geometry and to the critical loci m+qs=0 and μ+qs=0. Not fitted; input parameter.
  • p = 0 (teleparallel branch)
    Curvature parameter; set to zero in §§3-4 because the invariant connection is flat. This is the main modeling restriction.
  • m
    Mass label in the coadjoint orbit α=-mπ0+sλ0 (§3.2).
  • s (massive)
    Spin/WZ coupling in massive action (3.19); critical locus m+qs=0.
  • μ
    Momentum scale in massless orbit α=μ(π0-π1)-s(λ0-λ1) (§4.1).
  • s (massless)
    Spin parameter in massless WZ action (4.21); critical locus μ+qs=0.
axioms (7)
  • standard math Three-dimensional lorentzian space forms have exactly a one-parameter family of invariant metric connections, given by Nomizu maps N(P_A,P_B)=t ε_ABC P^C.
    §2.1, taken from [26] classification.
  • domain assumption There is a Lie algebra bracket (2.14) whose canonical connection reproduces the MB torsion/curvature; this MB algebra is the kinematical algebra of the spacetime.
    §2.2; motivated by [9], not derived from experiments.
  • standard math Particle actions are obtained from coadjoint orbits by S=∫⟨α,(g∘γ)^*ϑ⟩; no central extension exists, so coadjoint orbits classify homogeneous symplectic manifolds.
    §3.1; H^2=0 is proved in §2.3.
  • domain assumption Spin is introduced by a Wess–Zumino term s ϖ^J (massive) or s ϖ^- (massless), with α as momentum.
    §3.2, §4.1; physical model choice.
  • domain assumption The Noether symmetry analysis restricts to point transformations δx^M=α^M(x) with α independent of internal variables (eq. 4.57-4.58).
    §4.5; the strongest massless conclusion is relative to this ansatz.
  • standard math For massless particles, the momentum can be taken proportional to π0-π1 without loss of generality.
    §4.1, using the Lorentz orbit of null momenta; standard.
  • domain assumption The parametrization of Goldstone boosts has v≠0 and sinh v≠0 when solving equations and computing det M (eq. 3.33, 3.90).
    v=0 and sinh v=0 loci are excluded; this is a technical regularity assumption.

pith-pipeline@v1.3.0-daily-deepseek · 37123 in / 15273 out tokens · 137367 ms · 2026-08-01T13:48:59.265006+00:00 · methodology

0 comments
read the original abstract

The Mielke--Baekler geometries are three-dimensional reductive homogeneous spacetimes together with a choice of invariant connection which is compatible with a lorentzian metric. The spacetimes generalise Minkowski and (anti)de~Sitter spacetimes in that the invariant metric connection can have torsion, a peculiarity of three dimensions. Using coadjoint orbits and the techniques of nonlinear realisations, we construct worldline actions for massive and massless spinning particles moving in these spacetimes. We pay particular attention to the so-called teleparallel branch, in which the curvature of the invariant connection vanishes. Apart from the trivial Minkowski case, this singles out anti-de~Sitter spacetime, as the only of these lorentzian manifolds admitting an invariant Weitzenb\"ock connection; that is, a flat connection with torsion. The introduction of a Wess--Zumino term describing spin has, as a main consequence, the appearance of dynamical sectors (denoted ``regular'' and ``critical''), with a different number of physical degrees of freedom. In particular, in the massive case, we discuss the formulation of the dynamics in terms of either the Weitzenb\"ock or the Levi-Civita connection, and the emergence of a Papapetrou-type forcing term in the critical sector. For the massless particle we study the Noether symmetries of the action, which, for the spinless case, include the conformal transformations. For nonzero spin, only the Killing subset survives as genuine Noether transformations in the regular sector, while in the critical sector any conformal Killing contribution can be set to zero by a gauge transformation.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

41 extracted references · 1 canonical work pages

  1. [1]

    Ach´ ucarro and P.K

    A. Ach´ ucarro and P.K. Townsend,A Chern-Simons Action for Three-Dimensional Anti-de Sitter Supergravity Theories,Phys. Lett. B180(1986) 89

    1986

  2. [2]

    Witten,(2+1)-Dimensional Gravity as an Exactly Soluble System,Nucl

    E. Witten,(2+1)-Dimensional Gravity as an Exactly Soluble System,Nucl. Phys. B311(1988) 46

    1988

  3. [3]

    Mielke and P

    E.W. Mielke and P. Baekler,Topological gauge model of gravity with torsion,Phys. Lett. A156 (1991) 399

    1991

  4. [4]

    Blagojevi´ c,Gravitation and gauge symmetries, Series in High Energy Physics, Cosmology and Gravitation, IOP Publishing, Bristol (2002), 10.1887/0750307676

    M. Blagojevi´ c,Gravitation and gauge symmetries, Series in High Energy Physics, Cosmology and Gravitation, IOP Publishing, Bristol (2002), 10.1887/0750307676

    work page doi:10.1887/0750307676 2002

  5. [5]

    Blagojevic and B

    M. Blagojevic and B. Cvetkovic,Canonical structure of 3-D gravity with torsion, gr-qc/0412134

    Pith/arXiv arXiv

  6. [6]

    Giacomini, R

    A. Giacomini, R. Troncoso and S. Willison,Three-dimensional supergravity reloaded,Class. Quant. Grav.24(2007) 2845 [hep-th/0610077]

    Pith/arXiv arXiv 2007

  7. [7]

    Cvetkovic and M

    B. Cvetkovic and M. Blagojevic,Supersymmetric 3D gravity with torsion: Asymptotic symmetries,Class. Quant. Grav.24(2007) 3933 [gr-qc/0702121]

    Pith/arXiv arXiv 2007

  8. [8]

    Barriga, P

    F. Barriga, P. Concha, N. Merino and E. Rodr ´ ıguez,Non-Relativistic Chern–Simons Supergravity with Torsion,2604.24615

    Pith/arXiv arXiv

  9. [9]

    Geiller, C

    M. Geiller, C. Goeller and N. Merino,Most general theory of 3d gravity: Covariant phase space, dual diffeomorphisms, and more,JHEP02(2021) 120 [2011.09873]

    Pith/arXiv arXiv 2021

  10. [10]

    F.W. Hehl, P. von der Heyde, G.D. Kerlick and J.M. Nester,General Relativity with Spin and Torsion: Foundations and Prospects,Rev. Mod. Phys.48(1976) 393

    1976

  11. [11]

    Shapiro,Physical Aspects of the Space-Time Torsion,Phys

    I.L. Shapiro,Physical Aspects of the Space-Time Torsion,Phys. Rept.357(2002) 113 [hep-th/0103093]

    Pith/arXiv arXiv 2002

  12. [12]

    Coleman, J

    S.R. Coleman, J. Wess and B. Zumino,Structure of phenomenological lagrangians. i,Phys. Rev. 177(1969) 2239

    1969

  13. [13]

    Callan, S.R

    C.G. Callan, S.R. Coleman, J. Wess and B. Zumino,Structure of phenomenological lagrangians. ii,Phys. Rev.177(1969) 2247

    1969

  14. [14]

    Volkov,Phenomenological Lagrangians,Sov

    D.V. Volkov,Phenomenological Lagrangians,Sov. J. Part. Nucl.4(1973) 3

    1973

  15. [15]

    Gomis, K

    J. Gomis, K. Kamimura and P.C. West,The Construction of brane and superbrane actions using non-linear realisations,Class. Quant. Grav.23(2006) 7369 [hep-th/0607057]

    Pith/arXiv arXiv 2006

  16. [16]

    Bergshoeff, J

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

    Pith/arXiv arXiv 2023

  17. [17]

    Aldrovandi and J.G

    R. Aldrovandi and J.G. Pereira,Teleparallel Gravity: An Introduction, Springer (2013), 10.1007/978-94-007-5143-9

    doi:10.1007/978-94-007-5143-9 2013

  18. [18]

    Pereira and Y.N

    J.G. Pereira and Y.N. Obukhov,Gauge Structure of Teleparallel Gravity,Universe5(2019) 139

    2019

  19. [19]

    Batlle, J

    C. Batlle, J. Gomis, K. Kamimura and J. Zanelli,Dynamical sectors for a spinning particle in AdS3,Phys. Rev. D90(2014) 065017 [1407.2355]

    Pith/arXiv arXiv 2014

  20. [20]

    Batlle and J

    C. Batlle and J. Gomis,A general action for a particle in 2 + 1 anti-de Sitter space,Geom. Mech.02(2025) 309 [2509.18959]. – 54 –

    arXiv 2025

  21. [21]

    Mathisson,Neue Mechanik materieller Systeme,Acta Phys

    M. Mathisson,Neue Mechanik materieller Systeme,Acta Phys. Polon.6(1937) 163

    1937

  22. [22]

    Papapetrou,Spinning test-particles in general relativity

    A. Papapetrou,Spinning test-particles in general relativity. I,Proc. Roy. Soc. London Ser. A209 (1951) 248

    1951

  23. [23]

    Tulczyjew,Motion of multipole particles in general relativity theory,Acta Phys

    W. Tulczyjew,Motion of multipole particles in general relativity theory,Acta Phys. Polon.18 (1959) 393

    1959

  24. [24]

    Nomizu,Invariant affine connections on homogeneous spaces,Amer

    K. Nomizu,Invariant affine connections on homogeneous spaces,Amer. J. Math.76(1954) 33

    1954

  25. [25]

    Figueroa-O’Farrill and S

    J. Figueroa-O’Farrill and S. Prohazka,Spatially isotropic homogeneous spacetimes,JHEP01 (2019) 229 [1809.01224]

    Pith/arXiv arXiv 2019

  26. [26]

    Figueroa-O’Farrill, R

    J. Figueroa-O’Farrill, R. Grassie and S. Prohazka,Geometry and BMS Lie algebras of spatially isotropic homogeneous spacetimes,JHEP08(2019) 119 [1905.00034]

    Pith/arXiv arXiv 2019

  27. [27]

    Cartan and J.A

    ´E. Cartan and J.A. Schouten,On the geometry of the group manifold of simple and semisimple groups,Nederl. Akad. Wetensch. Proc. Ser. A29(1926) 803

    1926

  28. [28]

    Rossmann,Lie groups, vol

    W. Rossmann,Lie groups, vol. 5 ofOxford Graduate Texts in Mathematics, Oxford University Press, Oxford (2002)

    2002

  29. [29]

    Ivanov and V.I

    E.A. Ivanov and V.I. Ogievetskii,The Inverse Higgs Phenomenon in Nonlinear Realizations, Theor. Math. Phys.25(1975) 1050

    1975

  30. [30]

    McArthur,Nonlinear realizations of symmetries and unphysical Goldstone bosons,JHEP11 (2010) 140 [1009.3696]

    I.N. McArthur,Nonlinear realizations of symmetries and unphysical Goldstone bosons,JHEP11 (2010) 140 [1009.3696]

    Pith/arXiv arXiv 2010

  31. [31]

    Obukhov, F

    Y.N. Obukhov, F. Portales-Oliva, D. Puetzfeld and G.F. Rubilar,Invariant conserved currents in generalized gravity,1507.02191

    Pith/arXiv arXiv

  32. [32]

    Obukhov and G.F

    Y.N. Obukhov and G.F. Rubilar,Invariant conserved currents in gravity theories with local lorentz and diffeomorphism symmetry,Phys. Rev. D74(2006) 064002

    2006

  33. [33]

    Obukhov and G.F

    Y.N. Obukhov and G.F. Rubilar,Invariant conserved currents in gravity theories: Diffeomorphisms and local gauge symmetries,Phys. Rev. D76(2007) 124030

    2007

  34. [34]

    Puetzfeld and Y.N

    D. Puetzfeld and Y.N. Obukhov,Equations of motion in metric-affine gravity: A covariant unified framework,Phys. Rev. D90(2014) 084034 [1408.5669]

    Pith/arXiv arXiv 2014

  35. [35]

    Iosifidis and F.W

    D. Iosifidis and F.W. Hehl,Motion of test particles in spacetimes with torsion and nonmetricity, Phys. Lett. B850(2024) 138498

    2024

  36. [36]

    Pryce,The mass-centre in the restricted theory of relativity and its connexion with the quantum theory of elementary particles,Proc

    M.H.L. Pryce,The mass-centre in the restricted theory of relativity and its connexion with the quantum theory of elementary particles,Proc. Roy. Soc. Lond. A195(1948) 62

    1948

  37. [37]

    Chou, V.P

    C. Chou, V.P. Nair and A.P. Polychronakos,On the electromagnetic interactions of anyons,Phys. Lett. B304(1993) 105 [hep-th/9301037]

    Pith/arXiv arXiv 1993

  38. [38]

    Jackiw and V.P

    R. Jackiw and V.P. Nair,Relativistic wave equations for anyons,Phys. Rev. D43(1991) 1933

    1991

  39. [39]

    Horvathy and M.S

    P.A. Horvathy and M.S. Plyushchay,Non-relativistic anyons, exotic Galilean symmetry and noncommutative plane,JHEP06(2002) 033 [hep-th/0201228]

    Pith/arXiv arXiv 2002

  40. [40]

    Chevalley and S

    C. Chevalley and S. Eilenberg,Cohomology theory of Lie groups and Lie algebras,Trans. Amer. Math. Soc.63(1948) 85

    1948

  41. [41]

    Hochschild and J.-P

    G. Hochschild and J.-P. Serre,Cohomology of Lie algebras,Ann. of Math. (2)57(1953) 591. – 55 –

    1953