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# Hall effects in Carroll dynamics

Loïc Marsot, Pengming Zhang, Maxim N. Chernodub, P.A. Horvathy

► To cite this version:

Loïc Marsot, Pengming Zhang, Maxim N. Chernodub, P.A. Horvathy. Hall effects in Carroll dynamics. Physics Reports, 2023, 1028, pp.1-60. 10.1016/j.physrep.2023.07.007. hal-03896861

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# Hall effects in Carroll dynamics

Loïc Marsot, Pengming Zhang, Maxim N. Chernodub, P.A. Horvathy

► To cite this version:

Loïc Marsot, Pengming Zhang, Maxim N. Chernodub, P.A. Horvathy. Hall effects in Carroll dynamics. Physics Reports, 2023, 1028, pp.1-60. 10.1016/j.physrep.2023.07.007. hal-03896861v2

HAL Id: hal-03896861

https://hal.science/hal-03896861v2

Submitted on 27 May 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

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# Hall effects in Carroll dynamics*

L.Marsot<sup>1,2,3†</sup>, P.-M. Zhang<sup>4‡</sup>, M. Chernodub<sup>5 §</sup> and P. A. Horvathy<sup>5¶</sup>

<sup>1</sup>School of Science, Sun Yat-sen University,
Shenzhen 518107, (China)

<sup>2</sup>Aix Marseille Univ,
Université de Toulon
CNRS, CPT, Marseille, (France)

<sup>3</sup>Department of Physics,
Babeş-Bolyai University,
Kogalniceanu Street,
400084 Cluj-Napoca, (Romania)

<sup>4</sup>School of Physics and Astronomy,
Sun Yat-sen University,
Zhuhai 519082, (China)

<sup>5</sup>Institut Denis Poisson,
Tours University – Orléans University, UMR 7013
Tours 37200 (France)

(Dated: December 13, 2022)

<footnote>* Research initiated by Christian Duval [1] and dedicated to his memory.</footnote>

<footnote>† marsot@cpt.univ-mrs.fr</footnote>

<footnote>‡ corresponding author. zhangpm5@mail.sysu.edu.cn</footnote>

<footnote>§ mailto:maxim.chernodub@univ-tours.fr</footnote>

<footnote>¶ horvathy@lmpt.univ-tours.fr</footnote>
# Hall motions in Carroll dynamics*

L. Marsot<sup>1,2,3†</sup>, P.-M. Zhang<sup>4‡</sup>, M. Chernodub<sup>5§</sup> and P. A. Horvathy<sup>5¶</sup>

<sup>1</sup>*School of Science,*
*Shenzhen campus of Sun Yat-sen University,*
*Shenzhen, Guangdong 518107, P.R. China*

<sup>2</sup>*Aix Marseille Univ,*
*Université de Toulon*
*CNRS, CPT, Marseille, France*

<sup>3</sup>*Department of Physics,*
*Babeş-Bolyai University,*
*Kogalniceanu Street,*
*400084 Cluj-Napoca, (Romania)*

<sup>4</sup>*School of Physics and Astronomy,*
*Sun Yat-sen University,*
*Zhuhai 519082, (China)*

<sup>5</sup>*Institut Denis Poisson CNRS/UMR 7013 - Université de Tours - Université d'Orléans Parc de Grandmont,*
*37200, Tours, (France)*

(Dated: May 20, 2025)

<footnote>* Research initiated by Christian Duval [1] and dedicated to his memory.</footnote>

<footnote>† marsot3@mail.sysu.edu.cn</footnote>

<footnote>‡ corresponding author. zhangpm5@mail.sysu.edu.cn</footnote>

<footnote>§ mailto:maxim.chernodub@univ-tours.fr</footnote>

<footnote>¶ horvathy@lmpt.univ-tours.fr</footnote>
Abstract

"Do Carroll particles move?" The answer depends on the characteristics of the particle such as its mass, spin, electric charge, and magnetic moment. A massive Carroll particle (closely related to fractons) does not move; its immobility follows from the Carroll boost symmetry. A strictly massless Carroll particle may propagate by following the Hall law consistently with partial breaking of their Carroll boost symmetry. In $d = 2$ space dimensions, the Carroll group has a two-fold central extension which allows us to generalize the dynamics to massive and massless particles, including anyons. The anyonic spin and magnetic moment combine with the doubly-extended structure parameterized by two Casimir invariants interpreted as intrinsic magnetization and non-commutativity parameter. The extended Carroll particle subjected to an electromagnetic background field moves following a generalized (anomalous) Hall law which includes a Zeeman force. Our theory is illustrated by massless, uncharged anyons with doubly-centrally extended structure (we call "exotic photons") which move on the horizon of a Kerr-Newman Black Hole, giving rise to an anyonic spin-Hall Effect.

Key words: Carroll symmetry, Hall motions, fractons, anyonic spin-Hall effects, motion on Black Hole horizon.

PACS numbers: 04.20.-q Classical general relativity;
Abstract

"Do Carroll particles move?" The answer depends on the characteristics of the particle such as its mass, spin, electric charge, and magnetic moment. A massive Carroll particle (closely related to fractons) does not move; its immobility follows from Carroll boost symmetry which implies dipole conservation, but not conversely. A massless Carroll particle may propagate by following the Hall law, consistently with the partial breaking of the Carroll boost symmetry. The framework is extended to Carroll field theory. In $d = 2$ space dimensions, the Carroll group has a two-fold central extension which allows us to generalize the dynamics to massive and massless particles, including anyons. The anyonic spin and magnetic moment combine with the doubly-extended structure parameterized by two Casimir invariants interpreted as intrinsic magnetization and non-commutativity parameter. The extended Carroll particle subjected to an electromagnetic background field moves following a generalized Hall law which includes a Zeeman force. This theory is illustrated by massless, uncharged anyons with doubly-centrally extended structure we call exotic photons, which move on the horizon of a Black Hole, giving rise to an anyonic spin-Hall Effect.

Key words: Carroll symmetry, Hall motions, fractons, anyonic spin-Hall effects, motion on Black Hole horizon.

PACS numbers: 04.20.-q Classical general relativity;
# Contents

<table><tr><td>I. Introduction</td><td>4</td></tr><tr><td>II. Galilean dynamics</td><td>7</td></tr><tr><td>A. The massive Galilean model</td><td>9</td></tr><tr><td>B. Exotic Galilean dynamics in the plane</td><td>10</td></tr><tr><td>C. Galilean anyons with spin-field coupling</td><td>12</td></tr><tr><td>III. Carroll dynamics</td><td>13</td></tr><tr><td>A. Massive Carroll particles</td><td>15</td></tr><tr><td>B. Fractons</td><td>18</td></tr><tr><td>C. Massless Carroll particles</td><td>22</td></tr><tr><td>D. Immobility and Carroll boost symmetry</td><td>27</td></tr><tr><td>E. “No-particle” motion</td><td>28</td></tr><tr><td>F. Carroll “no motion”: interpretation through General Relativity</td><td>30</td></tr><tr><td>G. On the role of the mass</td><td>32</td></tr><tr><td>IV. Doubly-extended Carroll dynamics</td><td>32</td></tr><tr><td>A. Free massive doubly-extended particle</td><td>32</td></tr><tr><td>B. Massless doubly-extended particles</td><td>37</td></tr><tr><td>C. Coupling to a gauge field: eppur si muove</td><td>39</td></tr><tr><td>D. Extension to vanishing Carroll mass, m = 0</td><td>41</td></tr><tr><td>E. Carrollian anyons</td><td>43</td></tr><tr><td>F. A toy model</td><td>44</td></tr><tr><td>V. Contraction from relativistic anyons</td><td>48</td></tr><tr><td>VI. Hall on the Hole (motion on the Horizon)</td><td>49</td></tr><tr><td>A. Schwarzschild Horizons</td><td>50</td></tr><tr><td>B. Motion on the Kerr-Newman horizon</td><td>52</td></tr><tr><td>C. Dynamics on the Kerr-Newman black hole horizon</td><td>55</td></tr><tr><td>D. Partially broken Carroll and BMS on the Kerr-Newman horizon</td><td>57</td></tr><tr><td>VII. Conclusion</td><td>60</td></tr></table>
# Contents

<table><tr><td>I. Introduction</td><td>4</td></tr><tr><td>II. From Euclidean to Carroll: a guided tour in the Symmetry Zoo</td><td>9</td></tr><tr><td>A. Galilean, Carrollian, and Aristotle geometries</td><td>9</td></tr><tr><td>B. The (pre)symplectic framework and notations</td><td>16</td></tr><tr><td>III. Galilean particle dynamics</td><td>28</td></tr><tr><td>A. Galilean anyons with spin-field coupling</td><td>32</td></tr><tr><td>B. Galilean/Bargmann symmetry in field theory</td><td>33</td></tr><tr><td>IV. Carroll dynamics</td><td>35</td></tr><tr><td>A. Massive Carroll particles</td><td>37</td></tr><tr><td>B. Multiparticle Carrollian systems</td><td>43</td></tr><tr><td>C. Massless Carroll particles</td><td>46</td></tr><tr><td>D. “No-particle” motion</td><td>50</td></tr><tr><td>E. Carroll “no motion”: interpretation through General Relativity</td><td>53</td></tr><tr><td>F. Carroll field theories</td><td>56</td></tr><tr><td>V. Fractons and Carroll particles</td><td>59</td></tr><tr><td>A. The fracton model</td><td>59</td></tr><tr><td>B. Dipole vs Carroll boost symmetry</td><td>62</td></tr><tr><td>VI. Doubly-extended Carroll dynamics</td><td>68</td></tr><tr><td>A. Free massive doubly-extended particle</td><td>68</td></tr><tr><td>B. Massless doubly-extended particles</td><td>73</td></tr><tr><td>C. Coupling to a gauge field: eppur si muove</td><td>74</td></tr><tr><td>D. Extension to vanishing Carroll mass, m=0</td><td>77</td></tr><tr><td>E. Carrollian anyons</td><td>79</td></tr><tr><td>F. A toy model</td><td>81</td></tr><tr><td>VII. Contraction from relativistic anyons</td><td>84</td></tr><tr><td>VIII. Motion in a plane gravitational wave</td><td>86</td></tr></table>
# I. INTRODUCTION

The Carroll algebra is a contraction of the Poincaré algebra in which the speed of light is sent to zero [2–4]. In this limit particles do not move, because they would otherwise violate the causality constraint imposed by special relativity [2–7].

Interest in Carrollian dynamics has long been delayed, precisely, due to this “no-go” statement. The situation started to change after physical applications were found [8–10]. The celebrated Bondi-Metzner-Sachs (BMS) group of General Relativity [11] is, for example, a conformal extension of the Carroll group [12–16]. Carrollian hydrodynamics has attracted considerable recent attention [17–19]. Further applications include plane gravitational waves and the “memory effect” [20–23, 25]. The geometric approach is nicely complemented from the Lie algebra point of view [26].

A recent development which overturns the negative assessment mentioned above comes from condensed matter physics. As it will be explained in sec.III B, the limited mobility of quasiparticles called fractons [24, 27] follows from that they are indeed Carroll particles [30]. Incredibly, the “original sin” of Carrollian physics has turned into a virtue!

A sprain to the “no-motion-for-Carroll” tenet comes from recent study of an “exotic photon” i.e., a massless Carroll particle with spin, magnetic moment but with no electric charge on the horizon of a Kerr-Newman black hole (see sec. IV F). In [31, 32] we found that an “exotic photon” can move, namely by following a Hall-type law.

The particular interest of our “exotic photon” is underlined by that they can in principle be imitated using appropriate meta-materials and thus studied in laboratories [25, 33].

From the theoretical point of view, the celebrated move-or-not-to-move-dilemma is remarkably related to the behavior under Carroll boosts,

$$x \to x, \quad s \to s - \mathbf{b} \cdot x, \qquad (I.1)$$

where $x$ is the position and $s$ is Carrollian time [5, 30, 32].

This paper is devoted to a comprehensive study of Carroll dynamics and its applications to general relativity and condensed matter physics.
<table><tr><td><b>IX. Hall on the Hole: motion on the Horizon</b></td><td>91</td></tr><tr><td>A. Schwarzschild Horizons</td><td>92</td></tr><tr><td>B. Motion on the Kerr-Newman horizon</td><td>94</td></tr><tr><td>C. Partially broken Carroll and BMS symmetry on the horizon</td><td>99</td></tr><tr><td><b>X. Conclusion</b></td><td>102</td></tr><tr><td><b>Acknowledgments</b></td><td>104</td></tr><tr><td><b>References</b></td><td>104</td></tr><tr><td>A. Appendix</td><td>119</td></tr></table>

# I. INTRODUCTION

The Carroll algebra $\mathcal{C}$, as first introduced by Lévy-Leblond and Sen Gupta [2, 3], is a contraction of the Poincaré algebra $\mathfrak{P}$ in which the velocity of light is sent to zero instead of going to infinity as in the Galilean limit [2–6]. In the Carroll limit massive particles do not move, because they would otherwise violate the causality constraint imposed by special relativity [2–4, 6–8, 10].

Interest in Carrollian dynamics has long been limited, precisely, due to the lack of motion.

The situation started to change after physical applications were found [11–13]. The celebrated Bondi-Metzner-Sachs (BMS) group of General Relativity [14] is, for example, a conformal extension of the Carroll group [15–19]. Carrollian hydrodynamics has attracted considerable recent attention [20–22]. Further applications include the memory effect in a plane gravitational wave [23–27].

The geometric approach is nicely complemented from the Lie algebra point of view [28–30]. Non elativistic (including Carrollian) gravitation has become a popular topic [31–33].

A sprain to the “no-motion-for-Carroll” tenet comes from recent study of an “exotic photon” i.e., a massless Carroll particle with spin, magnetic moment but with no electric charge on the *horizon of a Kerr-Newman black hole* (see sec. VIF). In [34, 35] we found that an “exotic photon” *can move*, namely by following a *Hall-type law*.

Further developments which overturn the negative assessment comes also from *condensed*
Historically, the archetyp e of Hall-type motions arises when a charged particle is put into combined perpendicular electric field and magnetic fields in such a way that the electric and the Lorentz forces compensate each other. This requires the particle to move forcelessly with the Hall velocity [34–37],

$$
\left(\frac{dx^i}{dt}\right)^{\text{Hall}} = \epsilon_{ij} \frac{E_j}{B}, \quad i, j = 1, 2, \quad B \equiv B_3, \qquad (\text{I.2})
$$

where the motion is considered in a $d = 2$ plane normal to the direction of magnetic field **B** = (0, 0, $B_3$). To fix our terminology, we shall agree that a Hall motion is one which satisfies relation (I.2).

Many years after Hall’s original discovery, an anomalous Hall Effect (i.e. one with no magnetic field) was observed in certain ferromagnetic crystals and explained by an anomalous current [38]. A similar effect was proposed, even later, for massive particles with spin [39] and then further extended to light [40–42] and to chiral fermions [43–45]. Spin-Hall effects in curved space were considered in [46–49].

Reviewing 150 years of research of Hall-type Effects goes beyond our scope here. The interested reader is advised the literature, see, e.g. [35–37]. Here we just mention that a semi-classical explanation based on the two-parameter “exotic” *central extension* of the planar Galilei group [50–54] was proposed using a Berry phase–extended framework [55–57].

The rôle of *central extensions* for physics has been first recognized by Bargmann in his seminal paper [58]: for a massive non-relativistic system it is not the Galilei group itself but its 1-parameter *central extension* by the mass (called the Bargmann group [59]) which is physically relevant.

Unexpectedly, the dynamics is indeed even richer in the plane: Galilean systems admit a second (“exotic”) central extension [1, 50, 60, 61] which yields *non-commutating position coordinates* [1, 51, 52, 60–62],

$$
\{x, y\} = \frac{\kappa}{m^2} = \theta, \qquad (I.3)
$$

where $\kappa$ is the extension parameter.

Returning to Carroll, a curious fact is that while in $d \ge 3$ space dimensions the group has no nontrivial central extension, in $d = 2$ it does admit one with *two central parameters*, $\kappa_{\text{exo}}$ and $\kappa_{\text{mag}}$ [6, 31, 63]$^1$. The doubly extended Carroll group will be denoted by $\tilde{C}$. The

<footnote n="1">¹ We changed our notation with respect to Ref. [31]: $q_1 \rightarrow -\frac{1}{2}\kappa_{\text{mag}}$ and $q_2 \rightarrow +\frac{1}{2}\kappa_{\text{exo}}$. In this paper we</footnote>
matter physics: the limited mobility of the quasiparticles called fractons [36–45]<fnref n="1" /> follows from that they are indeed Carroll particles [41, 44]: the original defect of Carrollian physics has become a virtue!

A link between these notions appear from the principal property of the fracton: it is a (quasi)particle which, if set in isolation, is unable to move in response to any applied force. However, fractons can move a collective excitations by forming combinations – that distantly resemble bound states – which that can propagate through the medium [37, 38]. More details on fractions can be found in Sec. V.

From the theoretical point of view, the *move-or-not-to-move-dilemma* is related to the symmetry under shifting Carroll time we denote by $s$,

$$x \to x, \quad s \to s - b \cdot x, \qquad (I.1)$$

where $x$ is the position. what we shall call (see sec.IV) a Carroll or a *C-boost* [7, 35, 41].

This paper is devoted to a comprehensive study of the Carroll dynamics and its applications to condensed matter physics and to general relativity.

C-boosts are related to the *dipole symmetry* of fractons [40–45]. Close similarities can indeed be established between the Carroll algebra and the space-time symmetry group of fracton theories once the generator of C-boosts is identified with the dipole generator, whereas the Carroll-time translation operator (Hamiltonian) becomes the electric charge generator [48–50].

Historically, Hall-type motion arises when a charged particle is put into combined perpendicular electric and magnetic fields in such a way that the forces compensate and the particle to moves forcelessly with the Hall velocity [51–54],

$$\left(\frac{dx^i}{dt}\right)^{\text{Hall}} = \epsilon_{ij} \frac{E_j}{B}, \quad i, j = 1, 2, \quad B \equiv B_3. \qquad (I.2)$$

(The motion is considered in a $d=2$ plane normal to the direction of magnetic field $B = (0, 0, B_3)$). To fix our terminology, we shall agree that a Hall motion is one which satisfies relation (I.2).

<footnote n="1">¹ The term “fractons”, which is now widely associated with immobile excitations in condensed matter systems, has been used earlier to denote particles that carry fractional charges [46] and collective quantized phonon-like excitations on a substrate with a fractal structure [47]. These two objects should not be confused with the immobile excitations which are also called “fractons” [36–44].</footnote>
associated dynamics has been worked out in [31].

In our investigations we are led by the remarkable analogy (called in [5] a “duality”) between extended planar Carrollian dynamics outlined in Sec. IV below with its “exotic” Galilean counterpart [50–53], highlighted by that the central charges combine, for both cases, with the external magnetic field, B, into an effective mass:

$$m^* = \begin{cases} m(1 - \theta eB) & \text{Galilei} \\ m\left(1 - \frac{\kappa_{\text{exo}}}{m^2} B^*\right) & \text{Carroll} \end{cases} \qquad (I.4)$$

where

$$B^* = eB + \kappa_{\text{mag}}. \qquad (I.5)$$

The comparison of these equations allow us to clarify the physical sense of the double-fold central extension: the parameter $\kappa_{\text{exo}}$ in the Carrollian dynamics corresponds to the noncommutativity parameter $\theta$ of its Galilean counterpart (with the identification $\theta \leftrightarrow \kappa_{\text{exo}}/m^2$) while the parameter $\kappa_{\text{mag}}$ generates a magnetic-field background by shifting the physical background field$^2 eB \to B^* = eB + \kappa_{\text{mag}}$.

The relationship between $B^*$ and B in Eq. (I.5) is reminiscent the relationship between the magnetic fields **B** and **H**, and the magnetization **M** inside a media: **B** = **H** + 4$\pi$$M$, where we used Gaussian units. Within this interpretation, the parameter $\kappa_{\text{mag}}$ corresponds to the z-component of the intrinsic magnetization of the material, $\kappa_{\text{mag}} = 4\pi M_z$.

The double-extension of the Carroll system has a peculiar relation with the anomalous Hall Effect (AHE) which emerges in the absence of the background magnetic field **B** [38]. One could naively think that AHE is caused by the intrinsic magnetization $\kappa_{\text{mag}}$ which could play the role of a background magnetic **B** inside the system even when the external magnetic field vanishes, $B = 0$.

However, the Casimir invariant $\kappa_{\text{mag}}$ is not the origin of the anomalous Hall effect. It is the other invariant, $\kappa_{\text{exo}}$, which is responsible for the AHE: it generates the non-commutativity in the phase space, (I.3) with $\theta \neq 0$, which induces an anomalous velocity/momentum relation (II.17a) and leads, in turn to, Hall-like motion even in the absence of the background magnetic field [see [43, 44, 51–53, 56] and also Eqs. (IV.28) below].

<footnote>provide also a “dictionary” from Souriau’s notations to a more common language [64].</footnote>

<footnote>$^2$Notice that the field $B^*$ in Eq. (I.4) is defined without the standard electric charge factor $e$. This intentional omission keeps the field $B^*$ finite in the neutral-charge limit $e \to 0$.</footnote>
Many years after Hall's original discovery, an anomalous Hall Effect (i.e. one with no magnetic field) was observed in certain ferromagnetic crystals and explained by an anomalous current [55]. A similar effect was proposed, later, for massive particles with spin [56] and then further extended to light [57–59] and to chiral fermions [60–63]. Spin-Hall effects in curved space were considered in [64–67].

Reviewing 150 years of research of Hall-type effects goes beyond our scope here. The interested reader is advised to consult the literature, see, e.g. [52–54]. Here we just mention that a semi-classical explanation based on the two-parameter “exotic” central extension of the planar Galilei group [68–72] was proposed using a Berry phase-extended framework [73–75].

The rôle of central extensions for physics has been first recognized by Bargmann in his seminal paper [76]: for a massive non-relativistic system it is not the Galilei group $\mathfrak{G}$ itself, but its 1-parameter central extension by the mass (called the Bargmann group [77] and denoted by $\mathfrak{B}$), which is physically relevant.

Unexpectedly, the dynamics is indeed even richer in the plane: Galilean systems admit a second (“exotic”) central extension [1, 68, 78, 79] which yields non-commuting position coordinates [1, 69, 70, 78–80],

$$ \{x, y\} = \frac{\kappa}{m^2} = \theta, \qquad (I.3) $$

where $\kappa$ is the extension parameter.

Returning to Carroll, a curious fact is that while in $d \ge 3$ space dimensions the group has no nontrivial central extension, in $d = 2$ it does admit one with two central parameters, $\kappa_{\text{exo}}$ and $\kappa_{\text{mag}}$ [8, 34, 81]<fnref n="2" /> which escaped attention until recently. The doubly extended planar Carroll group will be denoted by $\hat{\mathfrak{C}}$. The associated dynamics has been worked out in [34].

In our investigations we are led by the remarkable analogy (called in [7] a “duality”) between extended planar Carrollian dynamics outlined in Sec. VI with its “exotic” Galilean counterpart [1, 68–71], highlighted by that the central charges combine, in both cases, with

<footnote n="2">² We changed our notation with respect to Ref. [34]: $q_1 \to -\frac{1}{2}\kappa_{\text{mag}}$ and $q_2 \to +\frac{1}{2}\kappa_{\text{exo}}$. We provide also a “dictionary” from Souriau’s notations to a more common language [82, 83].</footnote>
In the solid-state language, the AHE is produced by the Berry curvature of occupied elec-
tronic bands which appears due to the topology of Fermi surfaces in particular ferromagnets
when spin-orbit coupling is included [65].

When the effective mass vanishes, $m^* = 0$, the system becomes singular and requires Hamiltonian (alias “Faddeev-Jackiw”) reduction [51–53, 66]. In the Galilean case the only allowed motions then obey the Hall law (I.2). The Carroll case is more subtle, as it is explained in sections IV and VI.

Our paper is organized as follows. After a short reminder of the Souriau framework, we
recall how “exotic” (i.e., doubly extended) Galilean particles [50–52, 60, 61] are constructed
by the Kirillov-Kostant-Souriau (KKS) orbit method [59, 67, 68] applied to the Bargmann
group and conveniently studied in the “Eisenhart-Duval” framework [69–74].

Sec.III is devoted to non-centrally-extended Carroll particles [2–9]. The “no-motion-for-
fractons” statement [24, 30] follows at once from their relation to our massive Carrollian
model III B. The wave equation of free fractons, Eqn. (III.26) below, is indeed identical to
the Carrollian one studied in [31].

Doubly-extended Carroll particles [6, 31, 32, 63] are introduced in sec IV and then ex-
tended to massless particles. Theorems III.1-IV.7 address the recurrent “moving or not”
question with answers listed in Tables I and II, and illustrated in sec.IV F. The Carrollian
model is related to relativistic anyons in sec.V. Sect.VI illustrates the general results by a
comprehensive study of motion on the *horizon of a black hole* [16, 31, 32].

II. GALILEAN DYNAMICS

We start with a brief outline of the Souriau approach [59], spelled out in pedestrian terms.
The fundamental object is a “presymplectic” form i.e., a closed 2-form of constant rank $\sigma$
we call the *Souriau form*<fnref n="3" />, which is defined on a manifold $\mathcal{E}$ called the “evolution space”.
The classical motions are determined by the kernel of $\sigma$. The closedness property $d\sigma = 0$
implies that $\sigma = d\bar{\omega}$ locally, and the “Cartan 1-form” $\bar{\omega}$ may be used for a variational
interpretation [59, 76].

<footnote n="3">³ Souriau calls his 2-form $\sigma$ the “Lagrange form”. He claims indeed to have found his framework hidden in sect. V. of the second (1811) edition of “Mécanique Analytique” of Lagrange [75]. A physically relevant example with a 3-dimensional kernel is given by *massless relativistic particles with spin* [43, 46, 59].</footnote>
the external magnetic field, B, into an effective mass:

$$m^* = \begin{cases} m(1 - \theta eB) & \text{Galilei } \mathfrak{G} \\ m\left(1 - \frac{\kappa_{\text{exo}}}{m^2} B^*\right) & \text{Carroll } \mathfrak{C} \end{cases} \quad \text{where} \quad B^* = eB + \kappa_{\text{mag}}. \qquad (I.4)$$

The comparison of these equations allows us to clarify the physical sense of the double central extension: the parameter $\kappa_{\text{exo}}$ in the Carrollian dynamics corresponds to the noncommutativity parameter $\theta$ of its Galilean counterpart (with the identification $\theta \leftrightarrow \kappa_{\text{exo}}/m^2$) while the parameter $\kappa_{\text{mag}}$ generates a magnetic-field background by shifting the physical background field<fnref n="3" /> as in (I.4).

The relationship between $B^*$ and B is reminiscent the relationship between the magnetic fields **B** and **H**, and the magnetization **M** inside a medium: $B = H + 4\pi M$, respectively, where we used Gaussian units. Within this interpretation, the parameter $\kappa_{\text{mag}}$ corresponds to the z-component of the intrinsic magnetization of the material, $\kappa_{\text{mag}} = 4\pi M_z$.

The double-extension of the Carroll system has a peculiar relation with the anomalous Hall Effect (AHE) which emerges in the absence of the background magnetic field B [55].

One could naively think that the AHE is caused by the intrinsic magnetization $\kappa_{\text{mag}}$ which could play the role of a background magnetic B inside the system even when the external magnetic field vanishes, $B = 0$. However, the Casimir invariant $\kappa_{\text{mag}}$ is not the origin of the anomalous Hall effect. It is the other invariant, $\kappa_{\text{exo}}$, which is responsible for the AHE: it generates the non-commutativity in the phase space, (I.3) with $\theta \neq 0$, which induces an anomalous velocity/momentum relation (III.13a) and leads in turn to Hall-type motion even in the absence of the background magnetic field [60–62, 69–73]. See also sec. VI B.

In solid-state language, the AHE is produced by the Berry curvature of occupied electronic bands which appears due to the topology of Fermi surfaces in particular ferromagnets when spin-orbit coupling is included [84].

When the effective mass vanishes, $m^* = 0$, the system becomes singular and requires Hamiltonian (alias “Faddeev-Jackiw”) reduction [69–71, 85]. In the Galilean case the only allowed motions then obey the Hall law (I.2). The Carroll case is more subtle, as it is explained in sections VI and IX.

<footnote n="3">³ Notice that the field $B^*$ in Eq. (I.4) is defined without the standard electric charge factor $e$. This intentional omission keeps the field $B^*$ finite in the neutral-charge limit $e \to 0$.</footnote>
For a non-relativistic system, for example, ker σ can be 1-dimensional, parametrized by non-relativistic time, t. The restriction to t = t₀ = const. of the evolution space, M₀ ≡ Mₜ₀, is the “phase space at t₀” and the restriction of σ to M₀ is then a 2-form Ω₀ = σ|M₀ which we call the “symplectic form at t₀”. In a somewhat sloppy way, the choice of t₀ is neglected and one speaks of “the” phase space M endowed “the” symplectic form Ω.<fnref n="4" />

After the tacit choice of t₀ (say t₀ = 0), the Souriau 2-form can be split as

$$ \sigma = \Omega - d\mathcal{H} \wedge dt, \qquad (II.1) $$

where H is the Hamiltonian. Note that while this decomposition does depend on the choice of t₀, different choices generate equivalent Hamiltonian structures.

The next step is to introduce minimal coupling to an electromagnetic field represented by a closed 2-form on space-time, F = 1/2 F_μν dx^μ ∧ dx^ν = Eᵢdxᵢ ∧ dt + 1/2 Fᵢⱼdxᵢ ∧ dxⱼ, where the spatial components of the field strength tensor encode the magnetic field, Fᵢⱼ = ϵᵢⱼₖBₖ. Then Souriau’s rule is to add F to the free form σ₀ = dω₀,

$$ \sigma_0 \rightarrow \sigma = \sigma_0 + eF, \qquad (II.2) $$

where e is the electric charge. For a free non-relativistic particle with mass m, for example, σ₀ = Ω₀ − dH₀ ∧ dt and H₀ = p²/2m. In the static case, adding eF to σ₀ as in (II.2) modifies both the symplectic form and the Hamiltonian, respectively,

$$ \Omega_0 \rightarrow \Omega = \Omega_0 + \frac{e}{2} F_{ij} dx^i \wedge dx^j \quad \text{and} \quad \mathcal{H}_0 \rightarrow \mathcal{H}_{em} = \frac{p^2}{2m} + eV. \qquad (II.3) $$

The vector gauge field A and the electrostatic potential V determine the magnetic and electric fields,<fnref n="5" />

$$ \nabla \times A = B, \qquad \nabla V = -E, \qquad (II.4) $$

respectively.

Poisson brackets can be introduced when Ω = 1/2 Ω_αβ dz^α ∧ dz^β is regular (where the z_α are coordinates on “the” phase space, M). They involve the inverse matrix Ω^αβ, Ω_αβΩ^βγ = δ_α^γ,

$$ \{f, g\} = \Omega^{\alpha\beta} \frac{\partial f}{\partial z^{\alpha}} \frac{\partial g}{\partial z^{\beta}}. \qquad (II.5) $$

<footnote n="4">⁴ Notice that σ = π*Ω, where π is the projection from the evolution space E onto E/ker σ called the “space of motions” [59].</footnote>

<footnote n="5">⁵ Space and time are treated symmetrically in (II.2) but not in (II.3), which is in general different from the naive “minimal coupling” rule p_μ → p_μ − eA_μ with A^μ = (A^0 ≡ V, A). A counter-example is provided, e.g., by an “exotic” Galilean particle [51, 52] see Sec. II B.</footnote>
Our fundamental tool to find a classical system with a given symmetry group is to apply “(coadjoint) orbit method” called, with a slight abuse, the Kirillov, Kostant, and Souriau (KKS) method [77, 86, 87].

Our review is organized as follows. After a short review of the relevant Lie algebras and a pedestrian introduction to Souriau’s presymplectic framework in section II, we recall how “exotic” (i.e., doubly extended) Galilean particles [1, 68–70, 78, 79] are constructed by the KKS algorithm [88–93].

Sec.IV discusses Carroll particles with no central extension [2–4, 7, 8, 10–12]. The unusual form of the C-boost momentum $D^{Gal}$ in (IV.8c), whose conservation implies immobility, comes from absence of the conventional kinetic term in (IV.11) - (IV.12).

Massless Carroll particles have a richer dynamics, though IV C. The relation of immobility and Carroll boost symmetry is highlighted in subsec.IV A.

Fractons, to which section V is devoted, attract much current attention in condensed matter physics [36–44]. Their much-discussed immobility follows from the conservation of their dipole momentum [41, 42, 44, 45],

$$D = mx \qquad (I.5)$$

which follows in turn from the relation of fractons with the Carrollian model, highlighted in sec.IV F. The wave equation of free fractons, eqn. (V.5), is indeed identical to the Carrollian one studied in [34]. The intimate relation of dipole and Carroll symmetries is explained in sec.V B. The absence of spatial derivative terms from the Carroll wave equations (IV.61) or (IV.66) is the field-theory equivalent of absence of the usual kinetic term.

Doubly-extended Carroll particles [8, 34, 35, 81] are studied in sec. VI and then extended to massless particles. Theorems VI.1-VI.7 address the recurrent “moving or not?” question with answers listed in Tables I and II, and illustrated in sec.VI F.

The Carrollian model is related to relativistic anyons in sec. VII, which are particles in two spatial dimensions with statistics described by a continuous parameter. The standard paradigm implies that, in three spatial dimensions, local particle fields can possess only two types of statistics: integer spins are associated with bosonic particles characterized by commuting fields, while half-integer spins describe fermionic particles described by anti-commuting fields, as follows from the spin-statistics theorem [94]. In two spatial dimensions, the spin-statistics theorem does not work anymore because the group of rotations becomes
A. The massive Galilean model

For the sake of comparison and also for later use, we spell out how the massive Galilean model is obtained. Instead of Souriau’s original approach [59], we start with the Bargmann group, i.e. the 1-parameter (non-trivial) central extension of the Galilei group by the mass [58] denoted by $\tilde{G}$, represented by the matrices,

$$ \begin{pmatrix} A_{ij} & \mathbf{b} & 0 & \mathbf{c} \\ 0 & 1 & 0 & h \\ -b_i A_{ji} & -\mathbf{b}^2/2 & 1 & f \\ 0 & 0 & 0 & 1 \end{pmatrix} \qquad (II.6) $$

with $(A_{ij}) \in \text{SO}(d)$, $\mathbf{b}, \mathbf{c} \in \mathbb{R}^d$, $h, f \in \mathbb{R}$ representing rotations, boosts, space translations, time translations, and “vertical” translations along the central extension. Then the *coadjoint orbit method* of Kirillov, Kostant, and of Souriau (KKS) can be used to build the evolution space $\mathcal{E}$ [59, 67, 68].

In the Galilean case a non-relativistic spinless massive particle with internal energy $E_0$ and mass $m$ is obtained by contracting the Maurer-Cartan form $g^{-1}dg$ with the basepoint $\mu_0 = (0, 0, 0, E_0, m)$ chosen in the dual Lie algebra. The KKS method then yields the Cartan resp. Souriau forms,

$$ \varpi_0^{\text{Gal}} = \mu_0 \cdot \Theta = -E_0 dt - m ds + \mathbf{p} \cdot d\mathbf{x} - \frac{\mathbf{p}^2}{2m} dt, \qquad (II.7a) $$

$$ \sigma_0^{\text{Gal}} = d\mathbf{p} \wedge \left( d\mathbf{x} - \frac{\mathbf{p}}{m} dt \right). \qquad (II.7b) $$

These forms are defined on the extended evolution space with coordinates $(\mathbf{x}, \mathbf{p}, t, s)$, where $t$ is non-relativistic time and $s$ is the “vertical coordinate” along the central extension.

The equations of motion are given by the kernel of the Souriau form [59]; for (II.7b) one of these equations requires, in particular, the *velocity relation*

$$ \frac{d\mathbf{x}}{dt} = \frac{\mathbf{p}}{m} \qquad (II.8) $$

to be satisfied. The construction is completed by applying the coupling rule (II.2). Massive nonrelativistic systems are conveniently studied in the “Bargmann” framework [69–71, 73, 74].
Abelian, and the value of particle spin can take any value [95]. The latter property supports the existence of the third kind of particles, which was coined in Ref. [96] “the anyons”. Similarly to bosons and fermions, statistics plays a principal role in the physics of many-particle anyonic systems affecting their thermodynamical properties (for a recent account, see Ref. [97]) as well as transport phenomena that include electric (super)conductivity [98] and heat flow transport [99] as well as Hall-type effects [100], not mentioning a vital role of anyons in topological quantum computations essential for the construction of a fault-tolerant quantum computer [101].

In sec. VIII the role of Carroll symmetry is highlighted by the geodesic motion in a plane gravitational wave [23, 25]. Sect. IX illustrates the general results by a comprehensive study of motion of an hypothetical massless particle with anyonic magnetic moment on the horizon of a black hole [19, 34, 35].

Our notations are: $\mathcal{C}$ (for Carroll), $\mathcal{B}$ (for Bargmann), etc are space-time manifolds whose isometry groups $\mathcal{C}, \mathcal{B}$, etc act through the matrices denoted by C, B, etc.

## II. FROM EUCLIDE TO CARROLL: A GUIDED TOUR IN THE SYMMETRY ZOO

In this section, we review the different geometries and symmetries which will appear through the paper. The presymplectic framework used to derive classical models from symmetries is also recalled.

### A. Galilean, Carrollian, and Aristotle geometries

A relativistic structure is given by a couple $(\mathcal{M}, g)$ : the manifold $\mathcal{M}$ is endowed with a Lorentzian metric $g$. Geometries that give rise to Galilean or Carrollian symmetries are less straightforward, though. Before proceeding we recall, for future reference, a few definitions.

**Definition II.1.** A Galilean structure on a $(d+1)$-dimensional manifold $\mathcal{M}$ is [102–106] given by a triple $(\mathcal{M}, \gamma^{ab}, \theta)$ where,

1. $\gamma^{ab}$ $(a, b = 0, 1, \dots, d)$ is a twice-contravariant symmetric tensor field of signature $(0, +, \dots, +)$;
The well-known conserved quantities associated with Galilean symmetry will not be re-produced here with the exception of the boost momentum,

$$g^{\text{Gal}} = m\left(\boldsymbol{x} - \frac{\boldsymbol{p}}{m}t\right), \qquad (\text{II.9})$$

we record for later comparison with (III.10b), (III.46) and (IV.3b).

We mention for completeness that by changing the basepoint $\mu_0$ the Souriau model can be extended to: (i) spinning particles [40, 59, 76] and (ii) to massless particles, in both the Relativistic and Galilean frameworks [59] including curved space. Light which propagates in an optical medium can, in particular, be viewed as a massless particle which moves in a curved manifold [41, 42, 47, 77]. The optical index of the medium, $n(\boldsymbol{x})$, allows us to define a metric

$$g_{ij} = n^2(\boldsymbol{x}) \delta_{ij} \qquad (\text{II.10})$$

on space such that the gradient, $\nabla(1/n)$, plays the rôle of an effective “electric” force. When $n = \text{const.}$ the motion is free and instantaneous (as seen above). For light with spin $s$, the non-uniformoptical index $n = n(\boldsymbol{x})$ yields an anomalous velocity term with approximate equations of motion Eqn. (14) of [41],

$$\dot{\boldsymbol{r}} \approx \boldsymbol{p} - \frac{s}{k} \boldsymbol{\nabla} \left( \frac{1}{n} \right) \times \boldsymbol{p}, \qquad \dot{\boldsymbol{p}} \approx -n^3 k^2 \boldsymbol{\nabla} \left( \frac{1}{n} \right), \qquad (\text{II.11})$$

consistently with [40].

## B. Exotic Galilean dynamics in the plane

In the plane the Galilei group has been known to admit a second, central extension [50] which allows for an extended “exotic” dynamics [51–54, 60, 61]. Now we spell out the Souriau framework for the “exotic” (i.e. doubly-centrally extended) Galilean particle with mass $m$ subjected to the background electromagnetic field. Setting $\theta = \kappa/m^2$ as in (I.3), the “exotic” Souriau 2-form

$$\sigma_{\text{exo}}^{\text{Gal}} = \underbrace{(dp_i - eE_i dt) \wedge (dx^i - \frac{p_i}{m} dt)}_{\sigma_{em}^{\text{Gal}}} + \underbrace{\frac{1}{2} eB \epsilon_{ij} dx^i \wedge dx^j}_{\text{exotic}} + \underbrace{\theta \epsilon_{ij} dp^i \wedge dp^j}_{\text{exotic}} \qquad (\text{II.12})$$
2. closed “clock” one-form $\tau = \tau_a dx^a$ which spans the 1-dimensional kernel of $\gamma^{ab}$.

The coordinates on $\mathcal{M}$ are typically denoted by $(x^i, t)$ and are thought of as position and
Galilean time. The flat structure is given by the clock $\tau = dt$ and the “metric” $\gamma = \delta^{ij}\partial_i\partial_j$.

**Definition II.2.** A Newton-Cartan structure [102–107] on a (d+1)-dimensional manifold $\mathcal{M}$ with coordinates $x^i, t$ is given by a quadruple $(\mathcal{M}, \gamma^{ab}, \theta, \Gamma)$, where

1. $(\mathcal{M}, \gamma^{ab}, \theta)$ is a Galilean structure as in definition II.1 above;

2. $\Gamma$ is a Galilean connection, i.e., a symmetric linear connection compatible with $\gamma$ and $\tau$.

Note that the connection $\Gamma$ is *not* uniquely determined by the Galilei structure $(M, \gamma, \theta)$,
one really has an equivalence class of connections.

**Definition II.3.** A weak Carroll structure [12] is a triple $(\mathcal{C}, g, \xi)$ composed of a manifold $\mathcal{C}$ of dimension $d+1$, endowed with:

1. a degenerate twice symmetric covariant tensor $(g_{\mu\nu})$ such that $\dim\ker(g_{\mu\nu}) = 1$;

2. a nowhere vanishing vector field $(\xi^\mu)$ in the kernel, $g_{\mu\nu}\xi^\nu = 0$.

The coordinates are typically denoted by $(x^i, s)$ with $s$ referred to as “Carrollian time”. $$ \xi = \partial_s. $$

**Definition II.4.** A strong Carroll structure [7, 11, 16, 17, 90] is a quadruple $(\mathcal{C}, g, \xi, \Gamma)$ composed of a manifold $\mathcal{C}$ of dimension $d+1$, where

1. $(\mathcal{C}, g, \xi)$ is a weak Carroll structure as in def. II.3;

2. $\Gamma$ is a Carrollian connection, i.e., a symmetric linear connection compatible with $g$ and $\xi$ (much like for Newton-Cartan structures in definition (II.2)).

**Definition II.5.** An Aristotelian structure [108, 109] is given by a $(d+1)$-dimensional manifold $\mathcal{M}$ carrying

1. a “clock” 1-form $\tau = (\tau_\mu)$
is defined on the evolution space $\mathcal{E} = \mathbb{R}^2 \times \mathbb{R}^2 \times \mathbb{R} = \{\mathbf{x}, \mathbf{p}, t\}$. It can be split as $\sigma_{\text{exo}}^{\text{Gal}} = \Omega_{\text{exo}}^{\text{Gal}} - d\mathcal{H} \wedge dt$, where

$$ \Omega_{\text{exo}}^{\text{Gal}} = \underbrace{d\mathbf{p} \wedge d\mathbf{x} + \frac{e}{2} F_{ij} dx^i \wedge dx^j}_{\Omega_{em}} + \underbrace{\frac{\kappa}{2m^2} \epsilon_{ij} dp^i \wedge dp^j}_{\text{exotic}}, \quad \mathcal{H}_{em} = \frac{\mathbf{p}^2}{2m} + eV. \quad (II.13) $$

The system is regular when the determinant of the symplectic form does not vanish,

$$ \det(\Omega_{ij}^{\text{Gal}}) = \left(\frac{m^*}{m}\right)^2 \neq 0, \quad \text{where} \quad m^* \equiv m_G^* = m(1 - \theta eB). \quad (II.14) $$

Here $m^*$ is an *effective mass* [51, 52]. Dropping the suffix G, the Poisson brackets associated with (II.12) are, for $m^* \neq 0$,

$$ \{x_i, x_j\} = \frac{m}{m^*} \theta \epsilon_{ij}, \quad \{x_i, p_j\} = \frac{m}{m^*} \delta_{ij}, \quad \{p_i, p_j\} = \frac{m}{m^*} eB \epsilon_{ij}, \quad (II.15) $$

or equivalently,

$$ \{x_i, x_j\} = \left(\frac{1}{1 - \theta eB}\right) \theta \epsilon_{ij}, \quad \{x_i, p_j\} = \left(\frac{1}{1 - \theta eB}\right) \delta_{ij}, \quad \{p_i, p_j\} = \left(\frac{1}{1 - \theta eB}\right) eB \epsilon_{ij}. \quad (II.16) $$

In particular, the coordinates do not commute when $\theta \neq 0$. The associated Hamilton equations are

$$ \frac{m^*}{m} \dot{x}_i = \frac{p_i}{m} - \theta \epsilon_{ij} E_j, \quad (II.17a) $$

$$ \dot{p}_i = eE_i + eB\epsilon_{ij}\dot{x}^j. \quad (II.17b) $$

Note the velocity and the momentum cease to be parallel when $\theta \mathbf{E} \neq 0$. For $\theta = 0$ we recover the usual expressions for a charged particle in a constant magnetic field and for $B = 0$ we get the exotic expression $(2.3)$ in [52]. The system becomes singular when the effective mass vanishes, $m^* = 0$, and Hamiltonian (alias “Faddeev-Jackiw”) reduction [52, 66] yields [51–53]. The behavior when the effective mass changes sign is studied conveniently using chiral decomposition [78, 79]. We have:

**Theorem II.1.** Eqn. (II.17) implies that when its effective mass (II.14) vanishes,

$$ m^* = 0, \quad (II.18) $$

then an “exotic” Galilean particle follows the Hall law (I.2).
2. a symmetric tensor $g_{\mu\nu}$ of euclidean signature such that

3. the 1-dimensional kernel of $g_{\mu\nu}$ is spanned by a vector $\xi^\mu$ dual to the clock, $\tau_\mu\xi^\nu = \delta_\mu^\nu$, $g_{\mu\nu}\xi^\mu = 0$.

An Aristotelian structure thus combines the Galilean and the Carrollian definitions.

**Definition II.6.** A Bargmann structure [88–91, 109] is a triple $(\mathcal{B}, g, \xi)$ composed of a $(d+2)$ dimensional manifold $\mathcal{B}$ carrying

1. a Lorentzian metric $g$

2. a nowhere vanishing covariantly constant null vector field $\xi$

The key features of a Bargmann structure are that:

1. A Newton-Cartan structure (definition (II.2)) is obtained by projection $\mathcal{M} = \mathcal{B}/\mathbb{R}\xi$ [89, 107]

2. The $t$ = const. hypersurfaces $C_t$ of a Bargmann manifold $\mathcal{B}$ are Carroll manifolds [7]. More generally, any null-submanifold of a Lorentzian spacetime is a Carroll manifold [92, 93].

Most of our subsequent investigations focus at spacetimes in two spatial dimensions, therefore we collect the various “non-Einsteinian” symmetry groups in $d = 2$. However these groups (except central extensions in general) can be defined at higher dimensions, so we keep the notation $d$. We only list the nonzero Lie brackets. Non-trivial central extensions are also considered as they play an important role to build models for elementary particles [77, 86, 87, 111]. With some mild confusion between Lie groups and their generating algebras, we follow [77, 108].

1. The Euclidean group $\mathcal{E} = \text{SE}(d) \simeq \text{SO}(d) \times \mathbb{R}^d$, whose action on space is,

$$x^i \mapsto R_j^i x^j + c^i. \qquad (II.1)$$
The Souriau form (II.12) is locally $\sigma_{\text{exo}}^{\text{Gal}} = d\varpi_{\text{exo}}^{\text{Gal}}$, where the Cartan 1-form $\varpi_{\text{exo}}^{\text{Gal}}$ corresponds to the “exotic” phase space Lagrangian [52]$^6$,

$$ \mathcal{L}_{\text{exo}}^{\text{Gal}} = \underbrace{(\mathbf{p} - e\mathbf{A}) \cdot \frac{d\mathbf{x}}{dt} - \mathcal{H}_{\text{em}} dt}_{\mathcal{L}_{\text{em}}} + \underbrace{\frac{\theta}{2} \mathbf{p} \times \frac{d\mathbf{p}}{dt}}_{\text{exotic}}. \qquad (\text{II.19}) $$

We record for further reference that in a constant B-field the *conserved angular momentum* is [52]

$$ \ell = \mathbf{x} \times \mathbf{p} + \frac{1}{2} eB \mathbf{x}^2 + \frac{1}{2} \theta \mathbf{p}^2 + \chi, \qquad (\text{II.20}) $$

where the real constant $\chi$ is identified as the *anyonic spin* [80–82]. The relation to planar vortex dynamics [83] will be studied elsewhere.

### C. Galilean anyons with spin-field coupling

The planar model can be further extended. In $d=2$ spatial dimensions, the anyonic spin is just a constant $\chi$, and a term involving the spin-field Hamiltonian$^7$,

$$ \mathcal{H}_{\text{any}} = -\mu\chi B, \qquad (\text{II.21}) $$

where $\mu$ is a magnetic momentum can be added to the spinless expression (II.12). Then we end up with the Souriau form of a massive, exotic anyon with spin in an em field,

$$ \sigma^{\text{Gal}} = \underbrace{(dp_i - eE_i dt) \wedge (dx^i - \frac{p^i}{m} dt)}_{\text{electric}} + \underbrace{\frac{1}{2} eB \epsilon_{ij} dx^i \wedge dx^j}_{\text{magnetic}} + \underbrace{\frac{\theta}{2} \epsilon_{ij} dp^i \wedge dp^j}_{\text{exotic}} + \underbrace{\mu\chi dB \wedge dt}_{\text{spin-field}}. \qquad (\text{II.22}) $$

The new term merely shifts the force,

$$ eE_i \rightarrow E_i^* = eE_i + \mu\chi\partial_i B, \qquad (\text{II.23}) $$

and the equations of motion become again (II.17) up to this shift.

Note the important role which is played by the magnetic field B in Eq. (II.23). The gradient of B behaves as an effective electric field which adds up to the background electric

<footnote>$^6$ In $d=2$ spatial dimensions, the vector product is a pseudoscalar, $\mathbf{a} \times \mathbf{b} = \epsilon_{ij}a^i b^j$.</footnote>

<footnote>$^7$ For an “exotic photon”, the spin-field term (II.21) plays a particular role on the horizon, see Sec. VI. Another example is studied in [84].</footnote>
For $d = 2$, $\mathfrak{E}$ is generated by $(J, P_i)$ s.t.<fnref n="4" />

$$[J, P_i] = \epsilon_{ij} P_j; \qquad (II.2)$$

2. For $d = 2$, $\mathfrak{E}$ admits a 1-parameter “magnetic” central extension $\hat{\mathfrak{E}} \simeq \mathfrak{E} \times \mathbb{R}$ [83] with generators $(J, P_i, A_{\text{mag}})$ which satisfy <fnref n="5" />:

$$[J, P_i] = \epsilon_{ij} P_j, \quad [P_i, P_j] = \epsilon_{ij} A_{\text{mag}}; \qquad (II.3)$$

3. The *Aristotle* group [77, 108] $\mathfrak{A} \simeq \text{SE}(d) \times \mathbb{R}$ is obtained by trivially extending the Euclidean group $\mathfrak{E}$ by time translations, which commute with the $\mathfrak{E}$ subgroup. This yields an action on spacetime,

$$\begin{pmatrix} x^i \\ t \end{pmatrix} \mapsto \begin{pmatrix} R_j^i x^j + c^i \\ t + h \end{pmatrix} \qquad (II.4)$$

$\mathfrak{A}$, generated by $(J, P_i, P_0)$,

$$[J, P_i] = \epsilon_{ij} P_j, \qquad (II.5)$$

is the group of automorphism of flat Aristotle structures (definition (II.5)).

4. The *magnetic Aristotle* group is the central extension $\hat{\mathfrak{A}} \simeq \mathfrak{A} \times \mathbb{R}$, for $d = 2$. It has generators $(J, P_i, P_0, A_{\text{mag}})$ s.t.

$$[J, P_i] = \epsilon_{ij} P_j, \quad [P_i, P_j] = \epsilon_{ij} A_{\text{mag}}. \qquad (II.6)$$

Particles with planar Aristotle symmetry were constructed in [9].

5. The *Galilei group* $\mathfrak{G} \simeq (\text{SO}(d) \times \mathbb{R}) \times (\mathbb{R}^d \times \mathbb{R}^d)$, is the group of automorphisms of the flat Newton-Cartan structure (see the definition (II.2)) and acts as,

$$\begin{pmatrix} x^i \\ t \end{pmatrix} \mapsto \begin{pmatrix} R_j^i x^j + b^i t + c^i \\ t + h \end{pmatrix}. \qquad (II.7)$$

$$\begin{align*} [M_{ij}, M_{kl}] &= \delta_{jk}M_{il} - \delta_{ik}M_{jl} - \delta_{jl}M_{ik} + \delta_{il}M_{jk}, \\ [M_{ij}, P_k] &= \delta_{ik}P_j - \delta_{jk}P_i \end{align*}$$

However, since the present review is mainly about $d = 2$ physics, we will prefer the notation valid in only a specific dimension with $J = M_{12}$, here (II.2).

<footnote n="4">⁴ In arbitrary spatial dimension $d$, one can write the generators as $(M_{ij}, P_i)$ where $M$ is a skewsymmetric $d \times d$ matrix for rotation and $P$ for translation such that</footnote>

<footnote n="5">⁵ $A_{\text{mag}}$, etc are elements of the extended Lie algebra while $\kappa_{\text{mag}}$ etc belong to the dual algebra.</footnote>
field **E** in Eq. (II.23). The effective electric field appears from the interaction of the anyonic spin with the background magnetic field described by the Hamiltonian (II.21). The spin-field interaction plays the same role as the electrostatic potential *eV* in the Hamiltonian (II.3) for a usual electrically charged particle. This analogy identifies the effective electrostatic potential for the anyons, *eV*<sub>eff</sub> = −μχB. The inhomogeneities in the latter produce the force on the anyon which can be described, according to Eq. (II.4), via an effective electric field **E**<sub>eff</sub> = (μχ/e)∇**B**.

When the effective mass vanishes, $m^* = 0$, the system becomes once again singular.
However then the magnetic field, $B = B_{crit} = m^2/e\kappa$, is necessarily *constant* and therefore
the new, $\mu$-term drops out: the anyonic spin has *no impact* on the (Hall) motions.

Massless Galilean particle models can also be constructed by the KKS method by choosing
appropriately the base point $\mu_0$ in the dual Lie algebra. They can be spinless (“Fermat
particle”– see sec.III C), or can also carry spin [41, 42, 59]. The same method works also in
the relativistic case. Such a particle can be coupled to a background metric [46, 47, 49].

The “exotic” model introduced in [51, 52] has anomalous gyromagnetic factor *g* = 0.
More general cases are studied in [85]. An optical medium can be viewed also as a special
metric [42]; then for massless particles we get a *Spin-Hall effect for light* [40–43].

### III. CARROLL DYNAMICS

After our brief review of the Galilean case, henceforth we focus our attention at Carroll
particles which are indeed our main objects of our interest. The label {.}Carr analogous to
{.}Gal will be omitted; all our formulae refer to the Carroll case in what follows.

Let us first recall that a *Carroll structure* [5, 8, 9, 13, 14, 71] is a triple (*M*, *g*, *ξ*) composed
of:

1. a manifold $\mathcal{M}$ of dimension $d + 1$;

2. a degenerate twice symmetric covariant tensor ($g_{\mu\nu}$) such that $\dim\ker(g_{\mu\nu}) = 1$;

3. a nowhere vanishing vector field ($\xi^\mu$) in the kernel, $g_{\mu\nu}\xi^\nu = 0$.

Its “isometries” – i.e., transformations which preserve both ($g_{\mu\nu}$) and $\xi$ – span an infinite-
dimensional group. A finite dimensional group is obtained if, mimicking the Galilean case
The generators $(J, P_i, P_0, K_i)$ satisfy,

$$[J, K_i] = \epsilon_{ij} K_j, \quad [J, P_i] = \epsilon_{ij} P_j, \quad [K_i, P_0] = P_i. \qquad (II.8)$$

6. The Galilei group has, in arbitrary dimension, a 1-parameter central extension by the mass, $M$, [76] called the *Bargmann group*. $\mathfrak{B} \simeq \mathfrak{G} \times \mathbb{R}$ is generated by $(J_i, P_i, P_0, K_i, M)$,

$$\begin{align}
[J, K_i] &= \epsilon_{ij} K_j, \quad [J, P_i] = \epsilon_{ij} P_j, \tag{II.9} \\
[K_i, P_j] &= \delta_{ij} M, \quad [K_i, P_0] = P_i, \nonumber
\end{align}$$

The elements of the Bargmann group can be conveniently represented by the $(d+3) \times (d+3)$ matrices [110],

$$B = \begin{pmatrix} 1 & -b_k R^k_j & -\mathbf{b}^2/2 & b_0 \\ 0 & R^i_j & \mathbf{b} & \mathbf{c} \\ 0 & 0 & 1 & h \\ 0 & 0 & 0 & 1 \end{pmatrix} \in \mathfrak{B} \qquad (II.10)$$

with $(R_{ij}) \in \text{SO}(d)$, $\mathbf{b}, \mathbf{c} \in \mathbb{R}^d$, $h, b_0 \in \mathbb{R}$ representing rotations, Galilean boosts, space translations, Galilean time translations and “vertical” translations along the central extension. A $\mathfrak{B}$-matrix acts affinely on Bargmann spacetime represented by the column vectors $\begin{pmatrix} s \\ x^j \\ t \\ 1 \end{pmatrix}$ by matrix multiplication, yielding,

$$\begin{pmatrix} s \\ x^i \\ t \end{pmatrix} \mapsto \begin{pmatrix} s - b_i R^i_j x^j + b_0 \\ R^i_j x^j + b^i t + c^i \\ t + h \end{pmatrix}. \qquad (II.11)$$

7. The *planar Galilei* group has a 2-parameter “exotic” central extension $\hat{\mathfrak{B}} \simeq \mathfrak{G} \times (\mathbb{R} \times \mathbb{R})$ [2, 70], with generators $(J, P_i, P_0, K_i, M, A_{\text{exo}})$:

$$\begin{align}
[J, K_i] &= \epsilon_{ij} K_j, \quad [K_i, K_j] = \epsilon_{ij} A_{\text{exo}}, \quad [J, P_i] = \epsilon_{ij} P_j, \tag{II.12} \\
[K_i, P_j] &= \delta_{ij} M, \quad [K_i, P_0] = P_i, \nonumber
\end{align}$$

which includes both the mass-extension ($M$) above and also the “exotic” extension specific for $d=2$, discussed below in sec.III.
[86], a connection is chosen. A strong Carroll structure is then obtained by completing the above definition to a quadruple $(\mathcal{M}, g_{\mu\nu}, \xi^{\mu}, \nabla)$ with:

4. a (non unique) connection $\nabla$ compatible with both the “metric” ($g_{\mu\nu}$) and the vector field $\xi^{\mu}$, i.e. $\nabla(g_{\mu\nu}) = 0$ and $\nabla_{\mu}\xi^{\nu} = 0$.

A Carroll manifold is obtained, for example, by constraining a Bargmann space [69–72] to a $t = t_0 = \text{const. submanifold}$ [5]. More generally, any null-submanifold is a Carroll manifold [73, 74]. The isometries of the flat strong Carroll structure are conveniently represented by $(d+2) \times (d+2)$ matrices,

$$ \begin{pmatrix} A^i_j & 0 & c^i \\ -b_k A^k_j & 1 & f \\ 0 & 0 & 1 \end{pmatrix}, \qquad (\text{III.1}) $$

where $(A_{ij}) \in \text{SO}(d)$ is a rotation, $\mathbf{c}$ is a space translation, $\mathbf{b}$ is a boost, and $f$ is a “Carrollian time translation”. The matrix acts on the Carrollian coordinates by matrix multiplication,

$$ \begin{pmatrix} \mathbf{x} \\ s \\ 1 \end{pmatrix} \rightarrow \begin{pmatrix} A\mathbf{x} + \mathbf{c} \\ s - \mathbf{b} \cdot A\mathbf{x} + f \\ 1 \end{pmatrix}, \qquad (\text{III.2}) $$

where we denoted Carrollian time by $s$ to distinguish it from Galilean time, $t$.<fnref n="8" /> The implementation (III.2) highlights the characteristic feature of Carroll structures: unlike in the Galilean case, boosts leave the position invariant and shift instead Carrollian time, $s$, cf. (I.1).

The dynamics of a Carroll particle in an electromagnetic field can be derived in two steps as outlined in section II: first a free model is constructed by the KKS method II A applied to the Carroll group [5, 6, 14, 31, 59], which is then coupled to the em field by the rule (II.2).

An element $g = g(A, \mathbf{b}, \mathbf{c}, f)$ of the Carroll group is parametrized by $\frac{d(d+3)}{2} + 1$ numbers. Dual to them are the “moments” [59], which are the physical parameters that describe the state of a particle at a point. A moment is $\mu = \mu(\ell, \mathbf{g}, \boldsymbol{\pi}, m)$ where $\ell \in \mathfrak{so}(d)^*$ is the angular momentum, $\mathbf{g}$ represents the boost momentum [center of mass], $\boldsymbol{\pi}$ is the conserved momentum, and $m$ is the mass of the particle. Notice that the mass is not an externally given quantity but a *moment* and a *Casimir invariant* associated now *not* with a central

<footnote n="8">⁸ Our notation is consistent with the Bargmann framework [69–72].</footnote>
8. The *Carroll group* $\mathcal{C} \simeq \text{SE}(d) \times (\mathbb{R}^d \times \mathbb{R})$ is the group of isometries of a flat strong Carroll structure (definition (II.4)), and has generators $(J, P_i, P_0, K_i)$ with non-zero Lie brackets

$$[J, K_i] = \epsilon_{ij} K_j, \quad [J, P_i] = \epsilon_{ij} P_j, \quad [K_i, P_j] = \delta_{ij} P_0; \qquad (II.13)$$

A representation of $\mathcal{C}$ is readily obtained from that of the Bargmann group, (II.10). The Carroll group is a subgroup of the latter, obtained by reducing the dimension of the representation by erasing the line and column which correspond to $t$ and providing us with $(d+2) \times (d+2)$ matrices,

$$C = \begin{pmatrix} 1 & -b^k R_{kj} & b_0 \\ 0 & R_j^i & c^i \\ 0 & 0 & 1 \end{pmatrix} \in \mathcal{C}, \qquad (II.14)$$

where $(R_{ij}) \in \text{SO}(d)$ is a rotation, $\mathbf{c}$ is a space translation, $\mathbf{b}$ is a (Carroll or) C-boost, and $b_0 = \text{const.} \in \mathbb{R}$ is a “Carrollian time translation” (In order to distinguish it from Galilean time, $t$, we denoted the Carroll time by $s$). The affine action of $C$ on $\begin{pmatrix} s \\ 1 \end{pmatrix}$ induces one on Carroll space-time $\mathcal{C} = \{(s, \mathbf{x})\}$,

$$\begin{pmatrix} s \\ \mathbf{x} \end{pmatrix} \mapsto \begin{pmatrix} s + b_0 - \mathbf{b} \cdot R\mathbf{x} \\ R\mathbf{x} + \mathbf{c} \end{pmatrix}. \qquad (II.15)$$

This is the implementation we used for plane gravitational waves [25], see sec.VIII.

We mention for completeness that the Carroll group as introduced by Lévy-Leblond [2] acts instead on *usual spacetime*,

$$\begin{pmatrix} \mathbf{x} \\ t \end{pmatrix} \rightarrow \begin{pmatrix} R\mathbf{x} + \mathbf{c} \\ t + \mathbf{b} \cdot \mathbf{x} + h \end{pmatrix}, \qquad (II.16)$$

which can be obtained from the affine action of the matrix

$$\tilde{\mathcal{C}} = \begin{pmatrix} R_j^i & 0 & c^i \\ b_k R_j^k & 1 & h \\ 0 & 0 & 1 \end{pmatrix} \in \tilde{\mathcal{C}} \qquad (II.17)$$

which should to be compared with (II.15). It is this, “t-based” implementation that is considered in the BMS context [16, 20–22], and it is also the one that the authors of
extension as for Galilei/Bargmann theory, but with (Carrollian) “time” translations along Carrollian time s. It is a sort of “Carrollian energy” cf. (III.3).

The word “momentum” has indeed three related but somewhat different meanings here. Firstly, one has **p**, which we define to be always *mv*. Secondly there is the *canonical* or *generalized* momentum, defined as $\mathbf{P} = \partial\mathcal{L}/\partial\mathbf{x}'$. For a non relativistic particle in a magnetic field, for instance, $\mathbf{P} = mv - e\mathbf{A}$. At last, “momentum” is also the *conserved quantity* dual to space translation symmetry (a “moment” in Souriau’s terminology [59]), which we shall call here “impulsion” and denote by $\pi$. The three definitions coincide in some simple cases but not always, see, e.g., (IV.3).

The dynamical description for Carroll particles is readily found by embedding the Carroll group $C$ (III.1) as a subgroup of the Bargmann group $\tilde{G}$ (II.6) by “freezing out” (Galilean) time translations [5, 22],

$$h = 0. \qquad (III.3)$$

The moments allow us, e.g., to define Casimir invariants. As said above, the first one is *m*, the mass of the particle.

Carroll particles can also carry spin, defined as an additional Casimir invariant. However the planar case discussed in Sec. IV will be sufficient for our purposes here.

## A. Massive Carroll particles

We consider first the free massive case $m \neq 0$ [with no spin]. The $dt$-terms in (II.7) are switched off and we end up (consistently with # (A.9) and (A.10) in [5]), with,

$$\varpi_0 = \mathbf{p} \cdot d\mathbf{x} - mds, \qquad (III.4a)$$

$$\sigma_0 = d\varpi_0 = dp \wedge dx. \qquad (III.4b)$$

Note that Carrollian time, $s$, does not appear in the Souriau form. These forms are defined on “Carroll evolution space”

$$\mathcal{E} = \mathbb{R}^d \times \mathbb{R}^d \times \mathbb{R} = \{\mathbf{x}, \mathbf{p}, s\}. \qquad (III.5)$$

Compared to the Galilean theory of sec. II A, not only $E_0$, the internal energy is lost but, even more importantly, the usual kinetic term ($\mathbf{p}^2/2m$) $dt$ in (II.3) of the Galilean theory
[41] call the Carroll action. Eqns. (II.14)-(II.15) correspond in turn to the “internal” or “dipole symmetry” of condensed matter physicists, see sec.V.

The “t-based” Aristotle group $\tilde{\mathfrak{A}}$ involves t-translations by $h$ and is a subgroup of both of the t-based Carroll ($\tilde{\mathfrak{C}}$) and Galilei groups. It is obtained by switching boosts off in (II.16). Likewise, an “s-based” Aristotle subgroup of $\mathfrak{C}$, obtained by trading $t$ for $s$, i.e., by eliminating C-boosts from (II.14), involves s-translations by $b_0$.

Both implementations (II.15) and (II.16) highlight the characteristic feature of Carroll structures: C-boosts leave the position invariant and shift instead time — either $s$ or $t$, cf. (I.1) and (II.16).

While (II.14) is, properly speaking, “the” Carroll group, we will call “(generalized) Carroll group” any (conformal) isometry group of a chosen Carroll structure.

9. The group of isometries of a flat weak Carroll structure defined in (II.3) is infinite dimensional, and is denoted by $\mathfrak{A}^\infty \simeq \text{SE}(d) \times C^\infty(\mathbb{R}^d, \mathbb{R})^6$. It acts on spacetime as

$$ \begin{pmatrix} \boldsymbol{x} \\ s \end{pmatrix} \mapsto \begin{pmatrix} R\boldsymbol{x} + \boldsymbol{c} \\ s + T(\boldsymbol{x}) \end{pmatrix}. \qquad (II.18) $$

Compared to the Carroll group, we see that the s-translation and Carroll boosts are promoted to shifts by an arbitrary function $T(\boldsymbol{x})$ and yields an infinite dimensional version of the Aristotle group. While we may call this group “super Aristotle” or “infinitely extended Aristotle”, do note that it spans the isometries of a flat weak Carroll structure, and not of an Aristotle structure.

The shift in (II.18) is analogous to the celebrated Bondi - van der Burg - Metzner - Sachs (BMS) group [14], which is indeed the conformal isometry group of a Carroll structure with topology $\mathbb{S}^2 \times \mathbb{R}$ [16]. Hence, following the BMS literature, $s \to s+T(\boldsymbol{x})$ will be called a *supertransl*. Its zeroth-order term can be thought of as an s-translation, and the first-order term as a C-boost (I.1).

From the algebraic point of view, the Carroll group lies, “at halfway” between the Aristotle and the super-Aristotle” groups,

$$ \mathfrak{A} \subset \mathfrak{C} \subset \mathfrak{A}^\infty. \qquad (II.19) $$

<footnote n="6">⁶ The Galilean counterpart of this group (not used in this paper) is the Coriolis group $C^\infty(\mathbb{R}, \text{SE}(d)) \times \mathbb{R}$.</footnote>
is missing from the Cartan form (III.4a) (or equivalently, from the Hamiltonian $\mathcal{H}$ (III.6) and/or from the $\mathcal{L}$ Lagrangian (III.11)).

Note that the usual $-(\mathbf{p}/m)ds$ is “missing” from behind the $dx$ in $\sigma_0$, leaving $\mathbf{p}$ and $dx/ds$ unrelated: the Galilean velocity relation (II.8) is lost. This difference between Galilei/Bargmann and Carroll systems would not be altered by adding further (electromagnetic or central extension) terms.

Splitting the Souriau form into symplectic form and Hamiltonian as in (II.1),

$$ \sigma_0 = \Omega_0 - d\mathcal{H}_0 \wedge ds \quad \text{with} \quad \Omega_0 = d\mathbf{p} \wedge d\mathbf{x} \quad \text{and} \quad \mathcal{H}_0 \equiv 0, \qquad (III.6) $$

shows that a free massive Carroll particle has an identically zero Hamiltonian. The corresponding Hamilton equations read as follows:

$$ (x^i)' = \{x^i, \mathcal{H}_0\} = 0, \quad p_i' = \{p_i, \mathcal{H}_0\} = 0, \qquad (III.7) $$

where the prime,

$$ \{ \cdot \}' = d/ds, \qquad (III.8) $$

denotes a derivative with respect to Carrollian time. Equations (III.7) imply therefore:

**Theorem III.1.** A massive free Carroll particle does not move,

$$ \mathbf{x}(s) = \mathbf{x}_0, \quad \mathbf{p}(s) = \mathbf{p}(0), \qquad (III.9) $$

cf. (A.11) in [5].

A free massive particle is a Carroll-symmetric by construction. The associated conserved quantities (Souriau’s “moment” [59]) in $d=2$ spatial dimensions are as follows:

$$ \ell = \mathbf{x} \times \mathbf{p} \qquad \text{angular momentum,} \qquad (III.10a) $$

$$ \mathbf{g} = m\mathbf{x} \qquad \text{boost momentum,} \qquad (III.10b) $$

$$ \pi = \mathbf{p} \qquad \text{linear momentum.} \qquad (III.10c) $$

Comparison with the Galilean boosts in (II.9) shows that the difference comes, once again, from the decoupling of position and momentum. These expressions will later be compared with (IV.3). See also [30].

**Corollary 1.** The conservation of the boost momentum $\mathbf{g}$ in (III.10b) implies, consistently with Theorem III.1, that a massive particle with Carroll boost symmetry can not move.
The Bargmann group $\mathfrak{B}$ contains two different copies of the *Aristotle group* $\mathfrak{A}$: one projecting on Galilean space, and one acting on “vertical” Carroll space. Indeed, we readily see from the Bargmann action on spacetime (II.11) that the first copy is the subgroup of $\mathfrak{B}$ without boosts and Carroll time translations, $b_i = 0$ and $b_0 = 0$, and projects on Galilei spacetime as in (II.4), while the second one is the subgroup of $\mathfrak{B}$ without boosts and Galilean time translations, yielding the action (II.15) on Carroll spacetime with $b_i = 0$ and $h = 0$,

10. In the plane, for $d = 2$, we have a doubly extended Carroll group $\hat{\mathfrak{C}} \simeq \mathfrak{C} \ltimes (\mathbb{R} \times \mathbb{R})$, which is a 2-parameter central extension with generators $(J, P_i, P_0, K_i, A_{\text{mag}}, A_{\text{exo}})$:

$$ \begin{gathered} [J, K_i] = \epsilon_{ij} K_j, \quad [K_i, K_j] = \epsilon_{ij} A_{\text{exo}}, \qquad (II.20) \\ [J, P_i] = \epsilon_{ij} P_j, \quad [K_i, P_j] = \delta_{ij} P_0, \quad [P_i, P_j] = \epsilon_{ij} A_{\text{mag}}. \end{gathered} $$

At this point we mention a curious fact. The planar Euclidean ($\mathfrak{E}$), *Aristotle* ($\mathfrak{A}$), and Carroll ($\mathfrak{C}$), groups all admit a one-parameter “magnetic” central extension, defined by the commutator of space translations, cf. (II.3)-(II.6)-(II.20), which could be interpreted as an additional, “internal” magnetic field. But this central extension does not appear in the Galilei algebra $\mathfrak{G}$ in (II.12), obtained by further extension by Galilean boosts: the magnetic extension is broken when passing from $\mathfrak{A}$ to $\mathfrak{G}$ (or from $\mathfrak{C}$ to $\mathfrak{B}$). One may wonder where the difference comes from. Let us recall that the bracket of a Lie algebra should respect the Jacobi identity, which imposes strong constraints. For the Galilei algebra the Jacobi identity for a time translation, a boost in one, and a spatial translation in the other direction,

$$ 0 = [P_0, [K_1, P_2]] + [K_1, [P_2, P_0]] + [P_2, [P_0, K_1]] = [P_2, \underbrace{[P_0, K_1]}_{K_1}] = [P_2, P_1] $$

requires the magnetic extension to vanish. For the Carroll algebra instead, the identity is trivially satisfied because Carroll time translations commute with boosts, $[P_0, K_1] = 0$.

## B. The (pre)symplectic framework and notations

### From Lagrange mechanics to presymplectic geometry

Classical mechanics is usually taught through Lagrangian or Hamiltonian mechanics. While these tools are perfectly fine to study simple enough systems, the results shown
## Lagrangian formulation

The Carrollian dynamics can also be described by a Lagrangian. In the free massive case, viewing **p** and **x**' as independent variables on the evolution space $\mathcal{E}$, the Euler-Lagrange equation of the system

$$ \mathcal{L}_0 = \mathbf{p} \cdot \mathbf{x}', \qquad (III.11) $$

confirm the “no motion” conclusion (III.9).

Coupling our Carroll particle to an electromagnetic (EM) field by the rule (II.2) amounts to

$$ \mathbf{p} \rightarrow \mathbf{p} - e\mathbf{A} \quad \text{and} \quad 0 \equiv \mathcal{H}_0 \rightarrow \mathcal{H}_{em} = eV \qquad (III.12) $$

which yields, in Lagrangian form,

$$ \mathcal{L}_{em} = (\mathbf{p} - e\mathbf{A}) \cdot \mathbf{x}' - eV. \qquad (III.13) $$

Variation w.r.t. **p** and **x** viewed as independent variables implies the equations of motion # (3.21a-b) of Ref. [31],

$$ \mathbf{x}' = 0, \quad \mathbf{p}' = e\mathbf{E}. \qquad (III.14) $$

Note that the magnetic field does not play any role in the derivation. To understand how this comes about, note that the variational equations are,

$$ (x^i)' = 0 \quad \text{and} \quad p_i' = eE_i + eB\epsilon_{ij}(x^j)' . \qquad (III.15) $$

The first equation here switches off the Lorentz-force in the second one. In words: “no motion” implies “no Lorentz force”, allowing us to state the “no-motion theorem” which generalises Theorem III.1:

**Theorem III.2.** A massive charged Carroll particle does not move in the electromagnetic background field :

$$ \mathbf{x} = \mathbf{x}_0 = \text{const.} \qquad (III.16) $$

We notice that while the presence of the electric field $\mathbf{E} \neq 0$ produces the change in the momentum $\mathbf{p}$ of the particle, it does not affect its position, $\mathbf{x}$, in the course of the evolution of the system with Carrollian time $s$.
in this review call for a more advanced framework, which is (pre)symplectic geometry [77]. While one may still find it possible to work in terms of Lagrangians, and most of the time we will write the corresponding Lagrangian, deriving them, through e.g. the Kirillov-Kostant-Souriau theorem, or computing their symmetries, and so on, is a lot more convenient in the (pre)symplectic framework.

The following is a quick outline of [77, §7] with more standard notations. For more details, especially around subtle mathematical points, see [112] (in French).

Lagrangians are functions $L \in C^\infty(R \times TM)$, where $M$ is the space manifold, with $\dim M = n$, which are used to compute an action between $t_a$ and $t_b$,

$$A = \int_{t_a}^{t_b} L(t, \boldsymbol{x}(t), \boldsymbol{v}(t))dt, \qquad (II.21)$$

where bold symbols are for vectors on $M$ (i.e. spatial vectors). While $\boldsymbol{v}$ is, at this point, set to be $\boldsymbol{v}(t) = d\boldsymbol{x}(t)/dt$, this condition will be relaxed later on, and $\boldsymbol{v}(t)$ will be considered as $n$ additional degrees of freedom. The variables describing the evolution of a dynamical system can thus be denoted $(t, \boldsymbol{x}, \boldsymbol{v}) \in \mathcal{E}$, where $\mathcal{E}$ will be called the “evolution space”.

By introducing a new parameter $s$ to describe the curve

$$s \mapsto (t(s), \boldsymbol{x}(s), \boldsymbol{v}(s)), \qquad (II.22)$$

and generalized coordinates,

$$q = \begin{pmatrix} \boldsymbol{x} \\ t \end{pmatrix} \in Q, \qquad \dot{q} = \begin{pmatrix} \boldsymbol{v} \\ \dot{t} \end{pmatrix} \in T_q Q, \qquad (II.23)$$

one can reparametrize the Lagrangian as$^7$,

$$l(q, \dot{q}) = L(t, \boldsymbol{x}, \frac{\boldsymbol{v}}{\dot{t}})\dot{t} \qquad (II.24)$$

where the dot is the derivative with respect to $s$.

As is well-known, the Euler-Lagrange equations of motion are obtained by asking that the action is stationary,

$$\delta A = 0 \Leftrightarrow \left[ \frac{\partial l}{\partial \dot{q}^\mu} \delta q^\mu \right]_{s_a}^{s_b} + \int_{s_a}^{s_b} \left( \frac{\partial l}{\partial q^\mu} - \frac{d}{ds} \frac{\partial l}{\partial \dot{q}^\mu} \right) \delta q^\mu ds = 0, \qquad (II.25)$$

<footnote>$^7$ Note that it naively looks like the Lagrangian is now defined as $l \in C^\infty(TQ)$, where $\dim TQ = 2n + 2 = \dim(R \times TM) + 1$. This seemingly additional degree of freedom is however not relevant. One can always choose to write $\dot{t} = 1$, the other gauges (other than the forbidden $\dot{t} = 0$) being equivalent, see [112] for details.</footnote>
For completeness we record also the Souriau form obtained by the rule (II.2),

$$ \sigma_{em} = \sigma_0 + e\mathbb{F} = dp \wedge dx + \frac{e}{2} F_{ij} dx^i \wedge dx^j + eE_i dx^i \wedge ds, \qquad (III.17) $$

whose kernel yields again (III.14)<fnref n="9" />.

## B. Fractons

The Carrollian particles are closely related to fractons which, associated with immobile (or partially mobile) quasiparticles in condensed matter systems, attract much current attention [24, 27, 30]. The fracton phase of matter can be described by complex scalar field theories endowed with dipole symmetry [24].<fnref n="10" /> The simplest Lagrangian which describes the interacting fractons has the following form:

$$ \mathcal{L} = \underbrace{|\partial_s \Phi|^2 - m^2 |\Phi|^2}_{\mathcal{L}_0} - \lambda \underbrace{(\partial_i \Phi \partial_j \Phi - \Phi \partial_i \partial_j \Phi)(\partial_i \Phi^* \partial_j \Phi^* - \Phi^* \partial_i \partial_j \Phi^*)}_{\text{self-interaction } \mathcal{L}_{\text{int}}}, \qquad (III.18) $$

where $\Phi(x, s)$ is a complex scalar field [30]. We denote, for reasons which will be clear later, the time variable by $s$. The model (III.18) possess the Aristotelian spacetime symmetries since it is invariant under spacetime translations and spatial rotations but not under the boosts.

In the absence of interactions, $\lambda = 0$, the model (III.18) possess the trivial energy spectrum:

$$ \varepsilon_k = m, \qquad (III.19) $$

implying that the group velocity of the excitations is zero:

$$ \mathbf{v}_k = \frac{\partial \varepsilon_k}{\partial \mathbf{k}} = 0. \qquad (III.20) $$

Similarly to the massive Carrollian particles, the quasiparticles described by the free Lagrangian $\mathcal{L}_0$ in Eq. (III.18) do not move [24, 27, 30].

<footnote n="9">⁹ We also get $\mathbf{E} \cdot \mathbf{x}' = 0$ i.e., that the velocity is perpendicular to the electric field. However this is reduced to an identity because of the first relation in (III.14). The spinning Carroll equations in $d = 3$ spatial dimensions were first presented in [31].</footnote>

<footnote n="10">¹⁰ The term “fractons”, which is now widely accepted to be associated with immobile excitations in condensed matter systems, has been earlier used to denote particles that carry fractional charges [28] and collective quantized phonon-like excitations on a substrate with a fractal structure [29]. These two objects should not be confused with the immobile excitations which are also called “fractons” [24, 27, 30].</footnote>
where $\mu, \nu$ are indices running over the space $Q$.

The (canonical) momentum is defined as,

$$p_{\mu} = \frac{\partial l}{\partial \dot{q}^{\mu}} \qquad (II.26)$$

Since Lagrangians of the form (II.24) are homogeneous of degree one in velocity per their definition, i.e. they obey $l(q, \lambda \dot{q}) = \lambda l(q, \dot{q})$, they can always be written $l(q, \dot{q}) = p_{\mu} \dot{q}^{\mu}$. Differentiating this latter expression leads to

$$\frac{\partial l}{\partial q^{\mu}} dq^{\mu} = dp_{\mu} \dot{q}^{\mu} \qquad (II.27)$$

Using (II.26) and (II.27), the variation of the action (II.25) can be written,

$$\delta A = 0 \Leftrightarrow [p_{\mu} \delta q^{\mu}]_{s_a}^{s_b} + \int_{s_a}^{s_b} (\delta p_{\mu} d\dot{q}^{\mu} - dp_{\mu} \delta q^{\mu}) = 0 \qquad (II.28)$$

Now, if one defines, with some abuse of notation, $(y^a) = \begin{pmatrix} q^{\mu} \\ p_{\mu} \end{pmatrix} \in Y \simeq \mathcal{E}^8$, where $a, b$ are indices running over the evolution space $\mathcal{E}$, and the 1-form on $\mathcal{E}^9$,

$$\varpi = p_{\mu} dq^{\mu} \in T_{y}^{*} \mathcal{E} \qquad (II.29)$$

then, contracting $\varpi$ with a variation $\delta y = (\delta q^{\mu}, \delta p_{\mu})$ yields $\varpi_a \delta y^a = p_{\mu} \delta q^{\mu}$. This 1-form will be called “Cartan 1-form”. In fact, since $l = p_{\mu} \dot{q}^{\mu}$, the Cartan 1-form is merely the Lagrangian “up to $ds$” (or $dt$ once one sets $\dot{t} = 1$) [116],

$$\int \varpi = \int l \, ds. \qquad (II.30)$$

The variation of the action (II.28) can finally be written (again with some slight abuse of notation),

$$\delta A = 0 \Leftrightarrow [\varpi_a \delta y^a]_{s_a}^{s_b} + \int_{s_a}^{s_b} \frac{1}{2} (d\varpi)_{ab} \delta y^a \frac{dy^b}{ds} ds = 0, \qquad (II.31)$$

where $d\varpi$ is the exterior derivative of the 1-form $\varpi$, i.e. $d\varpi = dp_{\mu} \wedge dq^{\mu} = dp_{\mu} \otimes dq^{\mu} - dq^{\mu} \otimes dp_{\mu}$. This 2-form will be denoted $\sigma = d\varpi$, and is called the *presymplectic* 2-form, or *Souriau form*.<fnref n="10" />

<footnote n="8">⁸ Recall that, given the above notations, $\dim \mathcal{E} = 2n + 1$, and coordinates on $\mathcal{E}$ are $(x, v, t)$.</footnote>

<footnote n="9">⁹ The basis of the space $T_y^*\mathcal{E}$ is $dy^a = (dq^\mu, dp_\mu)$.</footnote>

<footnote n="10">¹⁰ Souriau calls his 2-form $\sigma$ the “Lagrange form”. He claims indeed to have found his framework hidden in sect. V. of the second (1811) edition of *Mécanique Analytique* of Lagrange [113]. Opticians [114, 115] call it the Lagrange integral invariant or Lagrange bracket.</footnote>
The independence of the particle energy level on its momentum is the characteristic of the so-called flat bands, the subject which is under staring interest in condensed matter physics. The flat bands appear in the magic-angle bilayer graphene [104], which hosts unconventional superconductivity at low temperatures [105]. The fermionic models and the relation to Carroll physics of flat bands has been very recently discussed Ref. [103].

The full Lagrangian (III.18) is invariant under rigid and linear-in-the-position rotations of the phase of $\Phi$,

$$ \Phi(\mathbf{x}, s) \rightarrow \hat{\Phi}(\mathbf{x}, s) = e^{i(\beta_0 - \vec{\beta}_1 \cdot \vec{x})} \Phi(\mathbf{x}, s), \qquad (III.21) $$

or infinitesimally,

$$ \delta\Phi(\mathbf{x}, s) = i\beta_0 \Phi(\mathbf{x}, s), \qquad (III.22a) $$

$$ \delta\Phi(\mathbf{x}, s) = -i(\beta_1 \cdot \mathbf{x}) \Phi(\mathbf{x}, s) \qquad (III.22b) $$

where we separated the transformations parameterized by the coordinate-independent pa- rameters $\beta_0$ and $\beta_1$. We have $|\hat{\Phi}|^2 = |\Phi|^2$ and $|\partial_s \hat{\Phi}|^2 = |\partial_s \Phi|^2$ from which we infer that the free Lagrangian $\mathcal{L}_0$ is left invariant.

Let us now recall the Noether theorem for a general scalar field theory: if the action $\mathcal{S} = \int \mathcal{L}(\phi) \sqrt{g} d^{D+1}x$ (where $g$ is the determinant of the metric on the manifold upon which $\phi$ is defined) changes under a transformation $\phi \rightarrow \phi + \delta\phi$ by a surface term, $\delta\mathcal{S} = \int \partial_\alpha K^\alpha d^{D+1}x$, then we have the conserved current

$$ J^\alpha = - \frac{\delta(\sqrt{g}\mathcal{L})}{\delta(\partial_\alpha \phi)} \delta\phi + K^\alpha, \qquad \partial_\alpha J^\alpha = 0, \qquad (III.23) $$

and thus a conserved charge,

$$ Q = \int \left( - \frac{\delta(\sqrt{g}\mathcal{L})}{\delta(\partial_0 \phi)} \delta\phi + K^0 \right) d^3\vec{x}. \qquad (III.24) $$

Applied to the free Lagrangian $\mathcal{L}_0$ (III.18) yields the conserved quantities which correspond to the symmetry transformations (III.22a) and (III.22b), respectively,

$$ q = \frac{i}{2} \int (\Phi^* \partial_s \Phi - \Phi \partial_s \Phi^*) d^3 \mathbf{x}, \qquad (III.25a) $$

$$ g = -\frac{i}{2} \int (\partial_s \Phi^* \mathbf{x} \Phi - \partial_s \Phi \mathbf{x} \Phi^*) d^3 \mathbf{x}, \qquad (III.25b) $$

which are interpreted as the electric charge and the dipole moment carried by the field $\Phi$.
Now, for the above equation to be true for all variations $\delta y$,

$$ \sigma_{ab} \frac{dy^b}{ds} = 0 \qquad (II.32) $$

must hold. In other words, $\frac{dy^b}{ds}$ must be in the kernel of $\sigma$ $^{11}$. The $\frac{dy^b}{ds}$ then yield the (Hamilton) equations of motion of the variational problem$^{12}$.

As the above shows, the description of a dynamical system by a Lagrangian implies that it is possible to describe the system in terms of a 2-form $\sigma$ (the converse is not necessarily true). While the former description yields equations of motion after writing down the Euler-Lagrange equations, the latter description yields equations of motion after studying the kernel of the 2-form. The main selling point of the presymplectic description is that classical mechanics is recast in geometrical terms, which allows for a precise and powerful framework to prove theorems, such as the Kirillov-Kostant-Souriau theorem, which we will recall later on.

While the above derivation of presymplectic mechanics was done with Galilean mechanics in mind, i.e. on a spacetime with a clear separation between space and time, it is more general. Any space $\mathcal{E}$ endowed with a closed 2-form (of constant rank) is called a presymplectic manifold [77].

The last notion of presymplectic geometry we need to introduce for the understanding of this review is its version of Noether's theorem, which says, [77, §11],

**Theorem II.1.** An (infinitesimal) symmetry is given by a vector field $X$ on the evolution space $\mathcal{E}$ which leaves the Souriau form (II.43) invariant, $L_X\sigma = 0$, where $L_X = d \circ i_X + i_X \circ d$ is the Lie derivative along the direction defined by $X$. Then we have the conserved quantity $f_X$ $^{13}$

$$ i_X\sigma = -df_X. \qquad (II.33) $$

In terms of the Cartan form $\varpi$, the symmetry condition and conserved quantity are

$$ L_X\varpi = dK_X, \qquad f_X = i_X\varpi - K_X. \qquad (II.34) $$

<footnote>$^{11}$ The dimension of the kernel must be at least 1. While in standard classical mechanics the kernel has usually dimension 1, a physical example with a 3-dimensional kernel is given by massless relativistic particles with spin [61, 64, 77].</footnote>

<footnote>$^{12}$ Note that since we can always work with $\dot{t} = 1$, having the equations for $\frac{dy^b}{ds}$ means having the equations for $\frac{d\tilde{x}}{dt}$ and $\frac{d\tilde{v}}{dt}$.</footnote>

<footnote>$^{13}$ Provided that $i_X\sigma$ is an exact form. Also, in fact, $f_X = \mu \cdot \tilde{X}$ where $\mu : \mathcal{E} \to \mathfrak{g}^*$ is called the moment map and $\tilde{X} \in \mathfrak{g}$ is the Lie algebra element corresponding to its action $X \in T_y\mathcal{E}$ on evolution space.</footnote>
To get more insight, we note that variation of $\mathcal{L}_0$ yields the truncated Klein-Gordon equation

$$ \partial_s^2 \Phi = -m^2 \Phi \qquad (III.26) $$

from which the kinetic term $|\nabla \Phi|^2$ is missing$^{11}$. Here we recognize the free quantum equation of a Carroll particle [31] mentioned in the Introduction.

The surprising forms (III.18) and (III.26) can actually be derived from the relativistic theory by a contraction familiar from the Carroll theory [5, 31]. Let us indeed start with the Klein-Gordon equation for a free relativistic particle of mass $m$ in $1+d$ dimensions with coordinates $x^0, x^1 \dots x^d$ and metric $(-1, \underbrace{1, \dots, 1}_{d})$,

$$ \partial_{\mu} \partial^{\mu} \Phi - m^2 C^2 \Phi = 0, \qquad (III.27) $$

where the parameter $C$ has a dimension of velocity and is, in general, different from the physical speed of light $c$. Introducing Carroll time, $s$, by putting $x^0 = s/C$ [5, 31] (III.27) becomes,

$$ \left[ \frac{1}{C^2} \Delta - \partial_s^2 - m^2 \right] \Phi = 0. \qquad (III.28) $$

Then letting $C \to \infty$ we end up with (III.26). Alternatively, we rewrite the Klein-Gordon Lagrangian in terms of Carroll time $s$,

$$ \mathcal{L}_{KG} = C^2 \left( \partial_s \Phi \partial_s \Phi^* + \frac{1}{C^2} |\nabla \Phi|^2 - m^2 \Phi \Phi^* \right). \qquad (III.29) $$

The overall factor $C^2$ can be dropped and the $C \to \infty$ limit of the bracketed quantity is $\mathcal{L}_0$ in (III.18).

We now relate fractons to Carroll particles through the relativistic Klein-Gordon theory by passing through the Bargmann space. We start with the free case. Eqn. (III.26) can indeed be solved as$^{12}$,

$$ \Phi(\boldsymbol{x}, s) = e^{ims} \Psi(\boldsymbol{x}), \qquad (III.30) $$

where $\Psi(\boldsymbol{x})$ is an arbitrary function of $\boldsymbol{x}$ only. We then note that (III.30) is defined on the Carroll slice $C = \{(\boldsymbol{x}, t = 0, s)\}$ (naturally extended to Bargmann space) as an equivariant function w.r.t. Carroll time, $\partial_s \Phi = im\Phi$.

<footnote>$^{11}$ The absence of $|\nabla \Phi|^2$ from the Lagrangian (III.18) and/or space derivatives in (III.26) is reminiscent of the absence of the kinetic term $p^2/2m$ in the classical Carroll dynamics and anticipates the analogy to be discussed below.</footnote>

<footnote>$^{12}$ We choose positive mass solutions, $m > 0$.</footnote>
In conventional terms, this requires that the Lagrangian changes by a total derivative. Applications of the presymplectic framework are presented in the Appendix.

For later use we recall also the field theory version of Noether’s theorem using Lagrangian mechanics:

**Theorem II.2.** If the action $\mathcal{S} = \int \mathcal{L}(\phi)\sqrt{g} d^{d+1}\mathbf{x}$, where $g$ is the determinant of the metric, changes under a transformation $\phi \to \phi + \delta\phi$ by a surface term, $\delta\mathcal{S} = \int \partial_{\alpha}K^{\alpha}d^{d+1}\mathbf{x}$, then we have the conserved current

$$J^{\alpha} = - \frac{\delta(\sqrt{g}\mathcal{L})}{\delta(\partial_{\alpha}\phi)} \delta\phi + K^{\alpha}, \quad \partial_{\alpha}J^{\alpha} = 0, \qquad (II.35)$$

and thus a conserved charge,

$$Q = \int \left( - \frac{\delta(\sqrt{g}\mathcal{L})}{\delta(\partial_0 \phi)} \delta\phi + K^0 \right) d^3\vec{x}. \qquad (II.36)$$

## A Galilean example

As an example of the above presymplectic construction, consider standard Galilean mechanics with its well-known Lagrangian, $L(t, \mathbf{x}, \mathbf{v}) = \frac{1}{2}mv^2 - U(\mathbf{x})$, which may also be written $l(q, \dot{q}) = \frac{1}{2}mv^2/\dot{t} - Ut$.

One finds for the canonical momentum,

$$p = \frac{\partial l}{\partial \dot{q}} = \begin{pmatrix} mv/\dot{t} \\ -\frac{1}{2}m(\dot{v}/\dot{t})^2 - U \end{pmatrix} =: \begin{pmatrix} \mathbf{p} \\ -E \end{pmatrix}, \qquad (II.37)$$

and we easily verify that $p_i \dot{q}^i = m\dot{x}^i/\dot{t} - \frac{1}{2}m(\dot{x}/\dot{t})^2\dot{t} - Ut = l(q, \dot{q})$.

The 1-form $\varpi = p_{\mu}dq^{\mu}$ is then,

$$\varpi = \mathbf{p} \cdot d\mathbf{x} - \frac{\mathbf{p}^2}{2m} dt - U dt, \qquad (II.38)$$

where the coefficient in front of the $dt$ is usually denoted $\mathcal{H} = \frac{\mathbf{p}^2}{2m}dt + U$, it is the Hamiltonian of the system. The 2-form $\sigma = d\varpi$ is then,

$$\sigma = d\mathbf{p} \wedge d\mathbf{x} - \frac{\mathbf{p}}{m} \cdot d\mathbf{p} \wedge dt - \frac{\partial U}{\partial x^i} dx^i \wedge dt. \qquad (II.39)$$
The subgroup of the Bargmann group which leaves the slice $\mathcal{C}$ invariant is the Carroll group, (III.2). Viewing (III.30) as defined on Bargmann space, the action of the Carroll group on $\mathcal{C}$ induces,

$$ \Phi(\boldsymbol{x}, s) \rightarrow e^{-im(\boldsymbol{b} \cdot (\boldsymbol{x}-\boldsymbol{c})+f)} \Phi(A^{-1}(\boldsymbol{x}-\boldsymbol{c}), s). \qquad (\text{III.31}) $$

For $A = \text{Id}$, $\boldsymbol{c} = 0$, $f = 0$, the infinitesimal version of (III.31) is the Carroll boosts (III.22b). Dipole moment conservation thus follows from Carroll boost invariance of the massive Carroll model [2, 5, 30]. Therefore the “fractons do not move” statement is a rephrasing of our Theorem III.1, itself consistent with the conservation of $\boldsymbol{g}$ in (III.10b).

The free Lagrangian (III.18) is indeed invariant under BMS-like symmetries [30]. The transformation

$$ \Phi(\boldsymbol{x}, s) \rightarrow \hat{\Phi}(\boldsymbol{x}, s) = e^{iT(\boldsymbol{x})} \Phi(\boldsymbol{x}, s) \qquad (\text{III.32}) $$

leaves indeed $\mathcal{L}_0$ manifestly invariant for any supertranslation $T(\boldsymbol{x})$.

We proceed by adding yet another remark about the celebrated “no-motion” property. Let us observe in fact that for any solution (III.30) of the wave equation (III.26) the particle density is independent of Carrollian time,

$$ |\Phi(\boldsymbol{x}, s)|^2 = |\Psi(\boldsymbol{x})|^2, \qquad (\text{III.33}) $$

which is indeed the field-theoretical analog of immobility.

This same conclusion can also be obtained by looking at the conserved dipole moment alias Carroll boost moment $\boldsymbol{g}$ in (III.25b). Evaluated on a solution (III.30), we get the field-theoretical analog of (III.10b),

$$ \boldsymbol{g} = m \int |\Psi(\boldsymbol{x})|^2 \boldsymbol{x} \, d^2\boldsymbol{x}, \qquad (\text{III.34}) $$

which does not depend on $s$. For completeness, we record also the conserved charge (III.25a),

$$ q = -m \int |\Phi|^2 d\boldsymbol{x} = -m \int |\Psi(\boldsymbol{x})|^2 d\boldsymbol{x}. \qquad (\text{III.35}) $$

Turning to the full Lagrangian (III.18), we note that the self-interaction term $\mathcal{L}_{\text{int}}$ is invariant under both transformations in (III.21),

$$ \mathcal{L}_{\text{int}}(\Phi) = \mathcal{L}_{\text{int}}(\hat{\Phi}). \qquad (\text{III.36}) $$
We are interested in the equations of motion $\frac{dy^a}{ds} = \xi^a$ which are in the kernel of $\sigma$, where we will denote $(\xi^a) = (\xi_x^i, \xi_v^i, \xi_t)$. We want,

$$i_{\xi} \sigma = \sigma(\xi, \cdot) = \left( \xi_i^v + \xi_t \frac{\partial U}{\partial x^i} \right) dx^i + \left( \xi_t \frac{p^i}{m} - \xi_x^i \right) dp_i - \left( \frac{p^i \xi_i^v}{m} + \xi_x^i \frac{\partial U}{\partial x^i} \right) dt \quad (II.40)$$

to vanish, which yields,

$$\xi_i^v + \xi_t \frac{\partial U}{\partial x^i} = 0, \qquad (II.41a)$$

$$\xi_t \frac{p^i}{m} - \xi_x^i = 0, \qquad (II.41b)$$

$$\frac{p^i \xi_i^v}{m} + \xi_x^i \frac{\partial U}{\partial x^i} = 0. \qquad (II.41c)$$

The first equation gives $\frac{dp_i}{ds} = -\frac{dt}{ds} \frac{\partial U}{\partial x^i}$, while the second is $\frac{dt}{ds} \frac{p^i}{m} = \frac{dx^i}{ds}$. The third equation is automatically satisfied. These two equations yield the usual

$$\frac{dp_i}{dt} = -\frac{\partial U}{\partial x^i} = F_i, \qquad (II.42a)$$

$$\frac{dx^i}{dt} = \frac{p^i}{m}, \qquad (II.42b)$$

when accounting for the fact that we can work with $\dot{t} = 1$.

## Phase space, Poissons brackets, symplectic geometry

Now, there is one more step in this framework which is to define the “phase space” $\mathcal{M}$ of the system, or space of motions according to the terminology in [77]. It is obtained as the quotient $\mathcal{M} = \mathcal{E}/\ker \sigma$, and it is endowed with a symplectic form $\Omega$ such that $\sigma = \pi^*\Omega$ where $\pi$ is the projection from $\mathcal{E}$ to $\mathcal{M}$. A symplectic form is a regular closed 2-form, in the sense that its kernel has dimension 0. A symplectic manifold is then a manifold $\mathcal{M}$ endowed with a symplectic 2-form.

The evolution space $\mathcal{E}$ contains the variables that describe the system along its worldline. For instance with the previous Galilean mechanics example, coordinates on $\mathcal{E}$ are $(\mathbf{x}, \mathbf{v}, t)$, and following the trajectory of a system means following a line (assuming dim $\sigma = 1$) $(\mathbf{x}(s), \mathbf{v}(s), t(s))$ in $\mathcal{E}$, just like in Lagrangian mechanics. Hence, the evolution space is foliated by trajectories. Computing the quotient of $\mathcal{E}$ by $\ker \sigma$ (this kernel representing the equations of motion) means “reducing” each trajectories down to a point. The space of motions $\mathcal{M}$ thus holds its name, in that each point in $\mathcal{M}$ represents a complete trajectory.
Thus (III.22) are symmetries for (III.18), as said in Ref. [30]. Then the Noether theorem shows that the free expressions (III.25) are indeed correct for the full dynamics.

Let us now consider how the free fractons described by the Lagrangian $\mathcal{L}_0$, Eq. (III.18) with $\lambda = 0$, move when coupled to the background electromagnetic field. One immediately recognizes that due to the absence of the spatial derivative $\nabla$ in the free Lagrangian $\mathcal{L}_0$, the vector part of the gauge potential, $A$, does not enter the Lagrangian picture. The electrostatic potential $V = V(x)$ does appear via the formal extension of the derivative, $\partial_s \rightarrow \partial_s - ieV(x)$, but it only modifies the phase of the solution (III.30):

$$ \Phi(x, s) = e^{i(m+eV(x))s} \Psi(x), \qquad (III.37) $$

without affecting the particle density (III.33). Therefore, in the simplest Lagrangian picture (III.18), free fractons do not perform a Hall-like motion neither in the massless nor in the massive case. This is the field-theoretical version of the classical “no-motion”.

Then one can wonder if the BMS-type higher symmetry survives the addition of the $\lambda \neq 0$-term in (III.18). Plugging the Ansatz (III.32) into (III.18), we readily see that

$$ \mathcal{L} \rightarrow \widehat{\mathcal{L}} = \mathcal{L} + \lambda \Phi^4 (\partial_i \partial_j T) (\partial_i \partial_j T). \qquad (III.38) $$

Not being a total derivative, the shift breaks the symmetry, unless

$$ \partial_i \partial_j T = 0 \qquad (III.39) $$

for all $i, j$. Thus only terms up to linear order in $x$, $T(x) = \beta_0 - \vec{\beta}_1 \cdot \vec{x}$ survive in general, as in (III.21). Similarly, a dilation, $\Phi \rightarrow \hat{\Phi} = \Lambda\Phi$, $\Lambda = \text{const.}$ breaks the scale invariance of the free Lagrangian,

$$ \mathcal{L}_0 \rightarrow \widehat{\mathcal{L}}_0 = \Lambda^2 \mathcal{L}_0 \quad \text{but} \quad \mathcal{L}_{\text{int}} \Rightarrow \Lambda^4 \mathcal{L}_{\text{int}}. \qquad (III.40) $$

### C. Massless Carroll particles

Massless Carroll particles can also be constructed along the KKS lines [14]. There are in fact, two classes. The conserved norm of their momentum $|\pi|$ is promoted to a Casimir invariant. However massless particles with zero or with non-zero momentum behave differently, providing us with two different classes of massless Carroll particles as will be seen below.
In a less rigorous, but more explicit way, consider a Galilean system, where dim ker $\sigma = 1$, parametrized by non-relativistic time $t$. The restriction to $t = t_0 = \text{const.}$ of the evolution space, $\mathcal{E}_0 \equiv \mathcal{E}_{t_0} = \{\mathbf{x}, \mathbf{p}, t_0\}$, is (a chart of) the “phase space $\mathcal{M}_0 = \{\mathbf{x}, \mathbf{p}\}$ at $t_0$” and the restriction of $\sigma$ to $\mathcal{M}_0$ is then a 2-form $\Omega_0 = \sigma|_{\mathcal{M}_0}$ which we call the “symplectic form at $t_0$”. In a somewhat sloppy way, the choice of $t_0$ is neglected and one speaks of “the” phase space $\mathcal{M}$ endowed “the” symplectic form $\Omega$. After the tacit choice of $t_0$ (say $t_0 = 0$), the Souriau 2-form can be split as

$$ \sigma = \Omega - d\mathcal{H} \wedge dt, \qquad (II.43) $$

where $\mathcal{H}$ is the Hamiltonian. Compare with the example given in (II.39). Note that while this decomposition does depend on the choice of $t_0$, different choices generate equivalent Hamiltonian structures.

Poisson brackets can now be introduced. Since the symplectic 2-form $\Omega = \frac{1}{2}\Omega_{\alpha\beta}dz^\alpha dz^\beta$ (where the $z_\alpha$ are coordinates on $\mathcal{M}$) is regular, $\Omega_{\alpha\beta}$ is invertible, $\Omega_{\alpha\beta}\Omega^{\beta\gamma} = \delta^\gamma_\alpha$. The Poisson bracket between two functions $f, g \in C^\infty(\mathcal{M})$ is,

$$ \{f, g\} = \Omega^{\alpha\beta} \frac{\partial f}{\partial z^\alpha} \frac{\partial g}{\partial z^\beta}. \qquad (II.44) $$

### The orbit method, central extensions, minimal coupling

A powerful result of (pre)symplectic geometry is the Kirillov-Kostan-Souriau (KKS) theorem [77, 86, 87], which is the cornerstone of the present review. This theorem states that

**Theorem II.3.** Any coadjoint orbit of a Lie group is a symplectic manifold. Conversely, any $G$-homogeneous symplectic manifold is (locally) symplectomorph to a coadjoint orbit of $G$ or of a central extension of $G$.

As we have seen, the space of motions $\mathcal{M}$ of a dynamical system is a symplectic manifold. A symplectic manifold is said to be $G$-homogeneous when a Lie group $G$ acts on it such that $\forall x, y \in \mathcal{M}, \exists g \in G, y = gx$. In other words, it is $G$-homogeneous when the transformations in the group $G$ allow us to visit the whole manifold $\mathcal{M}$ from any starting point. The physical intuition for homogeneity of the space of motions is that we can always relate two trajectories by a transformation, e.g. we can always send one (free) Galilean trajectory to another Galilean trajectory by a suitable Galilean transformation, provided the Galilei group is a symmetry of the system.
Consider first massless particles with non-zero conserved momentum,

$$ \pi \neq 0. \qquad (III.41) $$

Then the KKS algorithm yields the spinless “Fermat particles” of geometrical optics — which are indeed both Galilean and relativistic [14, 41, 42, 59]. This is because the coadjoint orbits of interest for us of the Galilei, Poincaré and Carroll groups are in fact those of their common Euclidean subgroup, obtained when boosts and time translations are dropped [42]. Geometric Optics is Euclidean — both in direct and historical sense [87].

“Spinless light” [alias Fermat particle] does move: it propagates instantaneously along the light rays of geometrical optics. In detail, the geometrical model of Ref. [59] describes a light ray by a pair $(x, u)$, where $x$ is an arbitrary point on the ray and $u$ is a unit vector such that $\pi = ku$ is oriented along the ray. The “evolution space” $\mathcal{E} = \{(x, u)\}$, is thus 5-dimensional.

### Coadjoint action of the euclidean group.

An element $(A, c)$ of the euclidean group acts on $\mu_0 = (\ell, \pi)$ in the dual euclidean algebra $\mathfrak{e}^*$ labeled as,

$$ \ell \rightarrow A\ell - c \times A\pi, \qquad \pi \rightarrow A\pi. \qquad (III.42) $$

The Casimir invariants are,

$$ |\pi| = k, \qquad \ell \cdot \frac{\pi}{k} = j, \qquad (III.43) $$

where the non-zero constant $k$ is Souriau’s “color” [59], which corresponds to the absolute value of the conserved momentum (our “impulsion”). $j$ is the spin of the particle. Choosing the basepoint $\mu_0 = (0, \pi) \in \mathfrak{e}^*$, the KKS algorithm [59, 67, 68] endows $\mathcal{E}$ with the Cartan resp. Souriau forms<fnref n="13" /> [42, 59],

$$ \varpi_0 = k u \cdot dx, \qquad \sigma_0 = k du \wedge dx. \qquad (III.44) $$

A light ray is determined by the characteristic foliation of $\sigma_0$ in (III.44), $\delta u = 0$ and $\delta x \propto u$, which is indeed a straight line oriented along $u$. The description by $(u, x)$ is however redundant: $x_1$ and $x_2$ lie on same ray if they differ by a multiple of $u$. A light ray

<footnote n="13">¹³ Equivalently, with the Lagrangian $\mathcal{L}_0 = \pi \cdot x' = ku \cdot x'$, cf. (III.11).</footnote>
A coadjoint orbit is an orbit of the coadjoint action of a group $G$ on the dual of its Lie algebra $\mathfrak{g}^*$. Let us first emphasize the physical relevance of $\mathfrak{g}^*$. There is a natural pairing between each symmetry, i.e. between each generator of the Lie algebra $\mathfrak{g}$, and an element $\mu$ in the dual, which is called a *moment* [77]. This can particularly be seen in Noether's theorem II.1. As such, the dual of the Lie algebra is parametrized by physical quantities which describe the system. The coadjoint action of a group $G$ will then transform a moment, i.e. a set of quantities describing a physical state, into another.

For instance for the Galilei group, elements $Z$ in the algebra are parametrized as $Z = (\omega, \beta, \gamma, \epsilon)$, for $\omega$ infinitesimal rotations, $\beta$ boosts, $\gamma$ spatial translations and $\epsilon$ time translation. Consequently, elements in the dual of the Galilei algebra are parametrized as $\mu = (\ell, \mathfrak{g}, \mathbf{p}, E)$, with $\ell$ the angular momentum, $\mathfrak{g}$ the center of mass (or boost momentum), $\mathbf{p}$ the linear momentum, and $E$ the energy, with the pairing $\mu \cdot Z = \ell \cdot \omega - \mathfrak{g} \cdot \beta + \mathbf{p} \cdot \gamma - E\epsilon \in \mathbb{R}$<fnref n="14" />. The coadjoint action of the Galilei group then tells us how a moment $\mu$, meaning a set of physical quantities $(\ell, \mathfrak{g}, \mathbf{p}, E)$, transforms to another, i.e. how the momentum, the energy, etc transform when we change reference frame.

In order to define the coadjoint representation, one first needs to define the adjoint representation. The *adjoint representation* $\mathrm{Ad}(g) : \mathfrak{g} \to \mathfrak{g}$ is such that

$$ \mathrm{Ad}(g)Z = gZg^{-1}, \quad \forall g \in G, Z \in \mathfrak{g}. $$

Then, the *Coadjoint representation* (or coadjoint action) $\mathrm{Coad}(g) : \mathfrak{g}^* \to \mathfrak{g}^*$, is such that,

$$ (\mathrm{Coad}(g)\mu) \cdot Z = \mu \cdot (\mathrm{Ad}(g^{-1})Z), \quad \forall g \in G, \mu \in \mathfrak{g}^*, \forall Z \in \mathfrak{g}, $$

where the dot $\cdot : \mathfrak{g}^* \times \mathfrak{g} \to \mathbb{R}$ is the pairing between the algebra and its dual.

Consider, again, the Galilei group. Knowing the group law, which can be immediately found from the action of the group on spacetime (II.7), we can compute the coadjoint

<footnote n="14">¹⁴ The signs are chosen so that the moments have the expected sign later on. For instance, we want the energy $E$ to be positive.</footnote>
is thus labeled ultimately by the two orthogonal vectors,

$$ u \quad \text{and} \quad q = x - (u \cdot x)u, \quad u \cdot q = 0. \qquad (\text{III.45}) $$

The “space of motions” of a Fermat particle is thus $\mathcal{M} = TS^2$, the tangent bundle of the 2-sphere, depicted in Fig. 1. In conclusion:

<figure><img src="image_2.png" /><figcaption>FIG. 1: (a) A Fermat particle in flat space moves along oriented straight lines. The unit vector **u** gives its direction and the vector **q** points to its closest point to a (chosen) origin. **x** is any point on the ray. (b) The space of motions is thus the tangent bundle of the unit 2-sphere, endowed with k-times its canonical symplectic form.</figcaption></figure>

**Theorem III.3.** A free, mass, spin and charge-less Carroll particle is equivalent<fnref n="14" /> to the previously considered euclidean “Fermat particle” and moves, instantaneously, along the oriented straight lines of geometrical optics.

An intuitive explanation is that there is no canonical quantity with the dimension of velocity in that theory, so either they do not move at all, or they move instantaneously. Since we know that they do move (the trajectory is a ray), it has to be instantaneous.

Another way to see how such “spinless light” may escape the “no-motion-for-Carroll” tenet is to recall that Carroll boosts, (I.1), leave the free Souriau-form $\sigma_0$ in (III.44) obviously

<footnote n="14">¹⁴ Up to interchanging Galilean and Carrollian times, $t \leftrightarrow s$.</footnote>
representation of the Galilei group, Coad(g)μ = (ℓ', g', p', E'), with

$$ \ell' = R\ell - \mathbf{b} \times (R\mathbf{g}) + \mathbf{c} \times (R\mathbf{p}), \qquad (II.45a) $$

$$ \mathbf{g}' = R(\mathbf{g} - \mathbf{p}e), \qquad (II.45b) $$

$$ \mathbf{p}' = R\mathbf{p}, \qquad (II.45c) $$

$$ E' = E + \langle \mathbf{b}, R\mathbf{p} \rangle. \qquad (II.45d) $$

There are two important conclusions to draw from this example. The first one is that clearly, unlike a symplectic manifold, the dual g is not (in general) homogeneous under the coadjoint action. For instance here, the norm of the momentum p is invariant under the coadjoint action. So, starting from a moment with a given p, we cannot reach another physical state with a different norm of the momentum by a coadjoint action. This means that while, on a symplectic level, we can map one trajectory to any other possible trajectory with a suitable transformation in G, it is not possible to reach all moments (i.e. physical states) from one moment. One then say that there are different orbits of the coadjoint representation.

The Coadjoint orbits of a moment μ₀ ∈ g* are defined as

$$ \mathcal{O}_{\mu_0} = \{\mu = \mathrm{Coad}(g)\mu_0 | g \in G\} \subset g^*. $$

In other words, an orbit Oμ₀ is the set of all physical states a system can reach starting from a state μ₀ by transformations in G. The orbits partition the dual g*, which is depicted on the figure 1.

Now, as we have also seen from the Galilean example, there are invariants under the coadjoint action. These invariants then help index each coadjoint orbit, they are called *Casimir invariants* <fnref n="15" />.

The Kirillov-Kostant-Souriau theorem then tells us that all coadjoint orbits Oμ₀ are symplectic manifolds. In fact, an orbit will be the symplectic manifold that describe the dynamical system (e.g. an elementary particle) corresponding to the Casimir invariants on this orbit.

<footnote n="15">¹⁵ In the literature, it is more common to see Casimir invariants in the context of Lie algebras, not so much in the context of dual of Lie algebras. However due to the (linear) pairing between the two, these definitions are equivalent.</footnote>
invariant; the associated conserved quantity is [14],

$$g \equiv 0 \qquad (III.46)$$

cf. eqn. # (VI.17) of [14], consistently with putting $m = 0$ into the conserved quantities (III.10). Thus, unlike in the massive case, we can not now infer from the conservation of the boost momentum that $\boldsymbol{x}$ is fixed [30] — consistently with what we had found above.

Extension to spin in $d \ge 3$ space dimensions (which allows us to study the “spin-Hall effect for light”, see [40–42]) is not considered in this paper.

Anyonic spin in $d = 2$ will be studied in sec. IV. The description of Carroll particles can be extended to curved space [7].

## Coupling to a background electromagnetic field.

A free particle constructed by the KKS method can be coupled to an electromagnetic field by the rule (II.2). The Carrollian version can be obtained by taking the Carrollian limit [88]. One starts with the relativistic electromagnetic tensor

$$F = F_{ij}dx^i \wedge dx^j + \frac{E_i}{c}dx^i \wedge dx^0. \qquad (III.47)$$

However, rescaling the electric field as $E_i \mapsto \tilde{E}_i = E_i/(cC)$ one gets, after introducing Carrollian time by $x^0 = s/C$ (so that $C \to \infty$ is the Carrollian limit),

$$F = F_{ij}dx^i \wedge dx^j + \tilde{E}_i dx^i \wedge ds \qquad (III.48)$$

where Carroll time, $s$, was replaced the usual time coordinate<fnref n="15" />.

The electric and magnetic fields defined above satisfy the Carroll-Maxwell laws [5], either in their electric version,

$$\left\{ \begin{array}{ll} \epsilon_{ijk} \partial_j E^k + \partial_s B^i = 0, & \partial_i B^i = 0 \\ \partial_s E^i = 0, & \partial_i E^i = 0 \end{array} \right. \qquad \text{electric} \qquad (III.49)$$

or in their magnetic version,

$$\left\{ \begin{array}{ll} \partial_s B^i = 0, & \partial_i B^i = 0 \\ \epsilon_{ijk} \partial_j B^k - \partial_s E^i = 0, & \partial_i E^i = 0 \end{array} \right. \qquad \text{magnetic} \qquad (III.50)$$

respectively<fnref n="16" />.

<footnote n="15">¹⁵ The same result is obtained [5, 73] by switching to light-cone coordinates $u$ and $v$. Then the restriction to the $u = \text{const.}$ null hypersurface is again (III.48) with $\tilde{E}_i = E_i - F_{iz}$.</footnote>

<footnote n="16">¹⁶ When compared to Galilean electromagnetism [5, 88] the words “electric” and “magnetic” are inter-</footnote>
On may wonder then what would happen when we couple (hypothetical) massless Carroll particle with non-zero “color” $k$ in (III.43), and endowed also with an electric charge to electromagnetism. Such particles would be described by the Souriau 2-form

$$ \sigma = k du_i \wedge dx^i + \frac{e}{2} F_{ij} dx^i \wedge dx^j + e E_i dx^i \wedge ds, \qquad (III.51) $$

whose integral leaves $(\delta\mathbf{x}, \delta\mathbf{u}, \delta s) \in \ker \sigma$ are given by,

$$ \left\{ \begin{array}{l} e E_i \delta x^i = 0, \\ \delta \mathbf{x} \propto \mathbf{u}, \\ k \delta \mathbf{u} - e \mathbf{E} \delta s + e \mathbf{B} \times \delta \mathbf{x} = 0, \end{array} \right. \qquad (III.52) $$

augmented with the constraint $|\mathbf{u}| = 1$.

Such a model looks similar to a massive, non relativistic charged particle in a background electromagnetic field, — but with the color $k$ playing the role of a mass, and $\mathbf{u}$ playing that of its momentum. The only formal difference is that here $\mathbf{u}$ is a unit vector.

The upper two relations in (III.52) imply that the propagation is orthogonal to the electric field,

$$ E_i u^i = 0. \qquad (III.53) $$

The system (III.52) has several kinds of motions. First, one should note that such particles cannot exist in a pure electric field, as the equations of motion become inconsistent for $k = \text{const.}$. Then, as for standard Newtonian mechanics of charged particles subject to the Lorentz force in a pure magnetic field, we get helical motion. The motion becomes straight if the initial velocity is parallel to the magnetic field while the helical motion degenerates into a circle if the initial velocity is orthogonal to it.

While pure electric fields are forbidden, the system (III.52) admits Hall-type solutions. To see this let us assume, for example, that the fields $\mathbf{E} = \text{const.}$ and $\mathbf{B} = \text{const.}$ are as in the usual Hall scenario where $\mathbf{E}$ is in the plane and $\mathbf{B}$ is the third component of the magentic field perpendicular to the plane. Let us assume, moreover, that the ray’s direction is fixed,

$$ \mathbf{u} = \text{const.} \neq 0. \qquad (III.54) $$

<footnote>changed, underlining the dual nature of Carrollian vs Galilean physics. These equations are written in $d = 3$ spatial dimensions.</footnote>
commutes with all $A_i$<fnref n="16" />. If the resulting algebra is isomorphic to the original algebra $\mathfrak{g}$ then the extension is said to be trivial (and in group terms, one would then have $H = G \times E$), however if the resulting algebra is not isomorphic to $\mathfrak{g}$ then the central extension is said to be non-trivial.

Trivial central extensions have no physical meaning. However, non-trivial central extensions have an important impact on the physical parameters used to describe a system. Non-trivial central extensions display an accounting problem: a dynamical system may require more physical parameters to be described than the number of symmetries in the group. Also, such extensions display, in a sense, an obstruction of the possibility to represent a given group on certain spaces, typically on phase space and wavefunctions [76, 111]. This obstruction is then parametrized by a constant (or $n$ constants for an $n$ dimensional extension), which enters the list of physical parameters used to describe a system.

In particular, the Galilei group always features at least one<fnref n="17" /> non-trivial central extension, which yields the Bargmann group [76], whose definition we recalled in (II.10). It features an additional generator, whose dual we denote $m$: it corresponds to physical mass<fnref n="18" />. The coadjoint action of the Bargmann group is then $\mathrm{Coad}(g)\mu = (l', g', p', E', m')$, with

$$ \ell' = R\ell - \mathbf{b} \times (R\mathfrak{g}) + \mathbf{c} \times (R\mathbf{p}) + mc \times \mathbf{b}, $$

$$ \mathfrak{g}' = R(\mathfrak{g} - \mathbf{p}e) + m(\mathbf{c} - \mathbf{b}e), $$

$$ \mathbf{p}' = R\mathbf{p} + mb, $$

$$ E' = E + \langle \mathbf{b}, R\mathbf{p} \rangle + mb^2/2, $$

$$ m' = m, $$

<footnote n="16">¹⁶ And check that the resulting set of generators and their commutation relations do form indeed an algebra, i.e. obey the Jacobi identity.</footnote>

<footnote n="17">¹⁷ Non-trivial central extensions, which are related to the second cohomology group of $G$, $H^2(G)$, are a rather complicated topic and, when considering groups acting on spacetime, their number (the dimension of $H^2(G)$) typically depends on the dimension of spacetime.</footnote>

<footnote n="18">¹⁸ One can think of this central extension situation for the Galilei group as follows. The Poincaré group has 10 generators (in 3 + 1 dimensions), and dual to time and spatial translation generators are the energy and momentum. Now, in relativistic mechanics, the mass of the system is not a degree of freedom as it is computed from the energy and momentum. However, when taking the contraction $c \to \infty$ of the Poincaré group to obtain the Galilei group, the number of generators stays the same, but the relation between momentum, energy and mass is broken. The Galilean mass then appears as an additional degree of freedom, which can only be described by an extension of the Galilei group.</footnote>
Then from the third equation in (III.52) we deduce that

$$ \frac{\delta x^i}{\delta s} = \epsilon^{ij} \frac{E_j}{B}, \qquad (III.55) $$

which means Hall motion. We notice also that for vanishing electric field, $E = 0$ but $B \neq 0$, we get the circular motion.

### D. Immobility and Carroll boost symmetry

(Im)mobility and Carroll symmetry are closely related [2, 5, 30], as said before. We first recall the description by a $2d + 1$ dimensional evolution space $\mathcal{E} = \{(x, v, s)\}$, which is invariant under Carroll boosts. The Cartan 1-form is $^{17}$,

$$ \varpi = A_i(x, v)dx^i + B_i(x, v)dv^i - \mathcal{H}(x, v, s)ds, \qquad (III.56) $$

which is the most general 1-form on $\mathcal{E}$ with $A$ and $B$ do not depend explicitly on time. Then Carroll boost-invariance requires the Cartan 1-form to obey $L_X \varpi = df$ for some $f$ and $X = \beta_i x^i \partial_s$, implying $dL_X \varpi = 0$. Then

$$ dL_X \varpi = \beta_j (\partial_{x^i} \mathcal{H} dx^j \wedge dx^i - \partial_{v^i} \mathcal{H} dx^j \wedge dv^i) + \beta \cdot x (\partial_s \partial_{x^i} \mathcal{H} ds \wedge dx^i + \partial_s \partial_{v^i} \mathcal{H} ds \wedge dv^i) \qquad (III.57) $$

which vanishes if $^{18}$

$$ \partial_{x^i} \mathcal{H} = 0 \quad \text{and} \quad \partial_{v^i} \mathcal{H} = 0. \qquad (III.58) $$

Thus $\mathcal{H}(s)ds$ is exact and can be dropped (i.e., $\mathcal{H}$ is gauge-equivalent to $\mathcal{H}_0 = 0$). The system can thus be described by a Carroll-boosts-invariant Cartan 1-form. Thus assuming that the system is regular, the equations of motion are,

$$ \frac{d\boldsymbol{x}}{ds} = 0 \quad \text{and} \quad \frac{d\boldsymbol{v}}{ds} = 0. $$

In conclusion, the invariance under Carroll boosts prevents motion [2, 5, 30].

Non-trivial motion requires to introduce an external electromagnetic field, which breaks the Carroll boost invariance.

<footnote>$^{17}$ In a more standard language, $\mathcal{H}$ here is in fact the Hamiltonian.</footnote>

<footnote>$^{18}$ Note that the second condition is the one which forbids the kinetic $mv^2/2$ term in the Lagrangian of Carroll systems.</footnote>
where now **p** and the other physical quantities have the expected transformations under a group action. This coadjoint action has three Casimir invariants, provided $m \neq 0$,

$$C_1 = S^2, \quad S = \ell - \mathbf{g} \times \mathbf{p}/m,$$

$$C_2 = E - \frac{\mathbf{p}^2}{2m},$$

$$C_3 = m$$

As the Galilean example shows, considering whether a group has non-trivial central extensions is crucially important in order not to miss complete classes of physical states (massive Galilean systems here).

Finally, after all the considerations explained above, one can start to build the symplectic structure on the orbits, in order to derive physical models for dynamical systems.

The Maurer-Cartan form (with a slight abuse of notation) $\Theta = g^{-1}dg$ is a one form on $G$ which takes value in the Lie algebra $\mathfrak{g}$. We can thus consider its pairing with an element $\mu_0$ in the dual $\mathfrak{g}^*$, which yields the Cartan 1-form $\varpi$ of the dynamical system which is able to reach the physical state $\mu_0$,

$$\varpi = \mu_0 \cdot \Theta. \qquad (II.46)$$

Note that the choice of basepoint $\mu_0$ within an orbit $O_{\mu_0}$ does not matter, as the (seemingly) different presymplectic structures obtained are equivalent. Hence, in practice, one chooses the simplest $\mu_0$ in the orbit of the system considered.

Let us give the example of a non-relativistic spinless massive particle with internal energy $E_0$ and mass $m$. Such a system is described “at rest” by the basepoint moment $\mu_0 = (0, 0, 0, E_0, m)$ in the dual of the Bargmann algebra. The contraction of $\mu_0$ with the Maurer-Cartan form $\Theta = g^{-1}dg$ of the Bargmann group $\mathfrak{B}$ displayed in (II.10) yields<fnref n="19" /> the Cartan resp. Souriau forms,

$$\varpi_0^{\text{Gal}} = \mu_0 \cdot \Theta = -E_0 dt - m ds + \mathbf{p} \cdot d\mathbf{x} - \frac{\mathbf{p}^2}{2m} dt, \qquad (II.47a)$$

$$\sigma_0^{\text{Gal}} = d\varpi_0^{\text{Gal}} = d\mathbf{p} \wedge \left( d\mathbf{x} - \frac{\mathbf{p}}{m} dt \right). \qquad (II.47b)$$

These forms are defined on the evolution space $\mathcal{E} \simeq \mathfrak{B}/\text{SO}(3)$ with coordinates $(\mathbf{x}, \mathbf{p}, t, s)$, where $t$ is non-relativistic time and $s$ is the “vertical” coordinate along the central extension.

<footnote n="19">¹⁹ After the reparametrization $h \to t$, $c \to \mathbf{x}$, $b \to \mathbf{p}/m$, $b_0 \to s$ on the group.</footnote>
Remarkably, the limited mobility of Carroll particles is related to partial breaking of the Carroll symmetry. Indeed, if one considers a planar system with a broken Carroll boost symmetry in (say the $x$) direction, then we see from (III.57) that $\mathcal{H}$ must not depend on the orthogonal (i.e. $y$) variable (as well as satisfy some restrictions on the time dependence of $\mathcal{H}$). There is no restriction on $\mathcal{H}$ in the remaining (i.e. $x$) direction, leaving the possibility for motion in that direction.

The allowed resp. forbidden directions can also be identified using the conserved quantities. For the example considered above, the conserved quantity associated to the remaining boost symmetry is $i_X \varpi = \beta_x x \mathcal{H}(x)$ which forbids motion in the unbroken $x$-boost direction. Motion is instead allowed in turn along the broken Carroll boost in the $y$-direction. In conclusion,

**Theorem III.4.** A regular dynamical system which is invariant under Carroll boosts in at least 2 directions cannot move. If all but one Carroll boosts are broken, motion is forbidden in the unbroken, but is possible in the broken boosts' directions.

<table><caption>TABLE I: Motion of an unextended Carroll particle with vanishing spin, $s = 0$.</caption><thead><tr><th></th><th>free Carroll no charge e = 0</th><th>charged e ≠ 0 in EM field</th><th>in curved metric</th></tr></thead><tbody><tr><th>m ≠ 0</th><td>no motion</td><td>no motion</td><td>no motion</td></tr><tr><th>m = 0, k = 0</th><td>no motion</td><td>Hall motions</td><td>no motion</td></tr><tr><th>m = 0, k ≠ 0</th><td>instantaneous motion</td><td>helical/Hall motions</td><td>geometric optics</td></tr><tr><th>Fermat particle</th><td>straight lines</td><td></td><td></td></tr></tbody></table>

### E. “No-particle” motion

So far we assumed that the momentum $\pi$ does not vanish. However a curious degenerate case arises for

$$ \pi = 0, \qquad (III.59) $$

for which the KKS algorithm would yield identically zero Cartan (and thus Souriau) forms,

$$ \varpi_{m=0} \equiv 0 \quad \text{and} \quad \sigma_{m=0} \equiv 0, \qquad (III.60) $$
We see that we recover the forms (II.38) (up to irrelevant total differentials) and (II.39), for free particles.

The KKS construction is thus a powerful method to canonically obtain the dynamics of a system invariant under any Lie group $G$. However, it only yields models without interactions. There is no canonical way to obtain the model of a system with interactions, only heuristics.

In particular, we can introduce minimal coupling to an electromagnetic field represented by a closed 2-form on space-time, $F = \frac{1}{2}F_{\mu\nu}dx^\mu \wedge dx^\nu = E_idx^i \wedge dt + \frac{1}{2}F_{ij}dx^i \wedge dx^j$, where the spatial components of the field strength tensor encode the magnetic field, for instance $F_{ij} = \epsilon_{ijk}B_k$ in 3 + 1 dimensions. Then Souriau's rule is to add $F$ to the free form $\sigma_0 = d\varpi_0$<fnref n="20" />,

$$ \sigma_0 \rightarrow \sigma = \sigma_0 + eF, \qquad (II.48) $$

where $e$ is the electric charge.

### III. GALILEAN PARTICLE DYNAMICS

While the main focus of this review is about Carroll dynamics, there are strong parallels with Galilean dynamics, and hence some review of the latter is warranted for future comparison, which is what this section will be used for.

As computed in the previous section (II.47), the Souriau 2-form for a non-relativistic (Galilean) spinless particle with mass $m$ is

$$ \sigma_0^{\text{Gal}} = d\boldsymbol{p} \wedge \left(d\boldsymbol{x} - \frac{\boldsymbol{p}}{m}dt\right). \qquad (\text{III.1}) $$

Now, one of the equation of motion obtained when studying the kernel of this 2-form is the velocity/momentum relation,

$$ \frac{d\boldsymbol{x}}{dt} = \frac{\boldsymbol{p}}{m}. \qquad (\text{III.2}) $$

While we usually take for granted that the momentum on phase space is related to the velocity of the particle, we will see with Carroll models that this is not always the case.

<footnote n="20">²⁰ Note that this minimal coupling rule is in general different from the naive “minimal coupling” rule $p_\mu \rightarrow p_\mu - eA_\mu$ with $A^\mu = (A^0 \equiv V, A)$. An example where the two heuristics are different is, e.g. an “exotic” Galilean particle [69, 70] see Sec. III.</footnote>
respectively, — and thus no dynamics at all. However coupling “it” naively to electro-magnetism by the rule (II.2) yields ... a Carroll dynamics with a “purely electromagnetic Souriau form,

$$ \sigma^{(0)} = \frac{e}{2} F_{ij} dx^i \wedge dx^j + e E_i dx^i \wedge ds, \qquad (III.61) $$

whose characteristic foliation, in 2+1 dimensions, yields nevertheless non-trivial equations of motion. Remarkably, the electric charge drops out (as long as it does not vanish), leaving us with

**Corollary 2.** Coupling minimally a charged degenerate “no-particle” with vanishing conserved momentum, $\pi = 0$ (III.59) and therefore no free dynamics, (III.60), the system will move by following the Hall law (I.2):

$$ (x^i)' = \epsilon_{ij} \frac{E^j}{B}. \qquad (III.62) $$

How can a “no-particle” have non-trivial motion? The mystery is clarified by observing that belonging to the kernel of $\sigma^{(0)}$ means simply that the combined forces vanish along the trajectories and perpendicularly to the “electric” direction the motion is thus “free”.

Alternatively, from the symmetry point of view, a Carroll boost $X = \beta \cdot x \partial_s$ with $\beta = (\beta_x, \beta_y)$ is implemented on the Souriau 2-form (III.61) as,

$$ L_X \sigma^{(0)} = (\mathbf{E} \times \boldsymbol{\beta}) dx \wedge dy, \qquad (III.63) $$

whose vanishing is consistent with Thm III.4 above.

The weird-looking “naked” electromagnetic Souriau form (III.61) can actually be derived from the massive 2-form (III.17). Replacing $\mathbf{p}$ by $mv$ we get,

$$ \sigma^{(m)} = m d\mathbf{v} \wedge d\mathbf{x} + \frac{e}{2} F_{ij} dx^i \wedge dx^j + e E_i dx^i \wedge ds. \qquad (III.64) $$

Then letting here $m \to 0$ switches off the first term, leaving us just with (III.61). Using a more conventional language, putting $\mathbf{p} = mv$ into the massive Carroll Lagrangian with EM coupling (III.13) to get

$$ \mathcal{L}_{em} = (mv - e\mathbf{A}) \cdot \mathbf{x}' - eV, \qquad (III.65) $$

whose variational equations are, instead of (III.15)

$$ m(x^i)' = 0 \quad \text{and} \quad mv_i' = eE_i + eB\epsilon_{ij}(x^j)' . \qquad (III.66) $$

Then letting here $m \to 0$ switches off the first equation leaving $(x^i)'$ undetermined, while the second one becomes the Hall law, (III.62).
Note that massive non-relativistic systems can conveniently be studied in the “Bargmann” framework [88–90, 92], where dynamical systems are recast to move on a “Bargmann” spacetime, which has one more dimension than the usual non-relativistic spacetime. In practice, one adds a coordinate s, which is the same s appearing in the 1-form (II.47a). The dynamics then project down to a non-relativistic spacetime, without s. Such considerations are particularly convenient since the isometries of a Bargmann spacetime span exactly the Bargmann group, which removes the need to consider central extensions.

The well-known conserved quantities associated with Galilean symmetry will not be reproduced here with the exception of the Galilean boost momentum,

$$D^{\text{Gal}} = m\boldsymbol{x} - t\boldsymbol{p}, \qquad (\text{III.3})$$

we record for later comparison with (III.29), (IV.8c), (IV.38), (V.12b) and (VI.4c) and (IV.76).

Now, for the above free particle, we can write the 2-form as $\sigma_0^{\text{Gal}} = \Omega_0 - d\mathcal{H}_0 \wedge dt$ with $\mathcal{H}_0 = \boldsymbol{p}^2/2m$. Considering electromagnetic (EM) interactions, in the static case, amounts to add $eF$ to $\sigma_0$ as in (II.48), which modifies both the symplectic form and the Hamiltonian, respectively,

$$\Omega_0 \rightarrow \Omega = \Omega_0 + \frac{e}{2} F_{ij} dx^i \wedge dx^j \quad \text{and} \quad \mathcal{H}_0 \rightarrow \mathcal{H}_{em} = \frac{\boldsymbol{p}^2}{2m} + eV. \qquad (\text{III.4})$$

The vector gauge field $A$ and the electrostatic potential $V$ determine the magnetic and electric fields,

$$\nabla \times A = B, \qquad \nabla V = -E, \qquad (\text{III.5})$$

respectively.

We mention for completeness that by changing the basepoint $\mu_0$ (with different Casimir invariants, i.e. considering another orbit $O_{\mu_0'} \neq O_{\mu_0}$), one can build models for: (i) spinning particles [57, 77, 116] and (ii) for massless particles, in both the Relativistic and Galilean frameworks [77] including curved space.

Light which propagates in an optical medium can, in particular, be viewed as a massless particle which moves in a curved manifold [58, 59, 65, 117]. The optical index of the medium, $n(\boldsymbol{x})$, allows us to define a metric

$$g_{ij} = n^2(\boldsymbol{x}) \delta_{ij} \qquad (\text{III.6})$$
## F. Carroll “no motion”: interpretation through General Relativity

Carroll structures implying Carroll dynamics are plentiful in General Relativity: in fact, any null hypersurface of a Lorentzian spacetime carries a Carroll structure [5, 73, 74].

Finding null hypersurfaces amounts to finding null geodesics; Carroll structures are thus characterized by the motions of massless particles. Let us look at such particles in flat spacetime. Their geodesic motion is given by the set of first order equations,

$$ \dot{X}^{\mu} = P^{\mu}, \qquad (III.67) $$

$$ \dot{P}^{\mu} = 0, \qquad (III.68) $$

where the dot denotes the derivative with respect to some affine parameter. Pick coordinates $(x, y, z, t)$ such that the momentum for outgoing massless particles be $P = (0, 0, 1, 1)$, yielding the spacetime diagram in figure 2.

<figure><img src="image_3.png" /><figcaption>FIG. 2: Spacetime diagram. The light-cone is in **blue**. Photon trajectories are either **outgoing** ($\dot{z} > 0$) depicted in **green** or **ingoing** depicted in **red**.</figcaption></figure>

In- and outgoing null geodesics define new coordinates that are well suited for the study
on space such that the gradient, $\nabla(1/n)$, plays the rôle of an effective “electric” force. When $n = \text{const.}$ the motion is free and instantaneous (as seen above). For light with spin $s$, the non-uniform optical index $n = n(\mathbf{x})$ yields an anomalous velocity term with approximate equations of motion eqn. (14) of [58],

$$ \dot{\mathbf{r}} \approx \mathbf{p} - \frac{s}{k} \nabla \left(\frac{1}{n}\right) \times \mathbf{p}, \qquad \dot{\mathbf{p}} \approx -n^3 k^2 \nabla \left(\frac{1}{n}\right), \qquad (\text{III.7}) $$

consistently with [57].

## Exotic Galilean dynamics in the plane

In the plane the Galilei group has been known to admit a second, central extension [68] which allows for an extended “exotic” dynamics [69–72, 78, 79]. Now we spell out the Souriau framework for the “exotic” (i.e. doubly-centrally extended) Galilean particle with mass $m$ subjected to the background electromagnetic field. Setting $\theta = \kappa/m^2$ as in (I.3), the “exotic” Souriau 2-form

$$ \sigma_{\text{exo}}^{\text{Gal}} = \underbrace{(dp_i - eE_i dt) \wedge (dx^i - \frac{p_i}{m} dt)}_{\sigma_{\text{em}}^{\text{Gal}}} + \underbrace{\frac{1}{2} eB \epsilon_{ij} dx^i \wedge dx^j}_{\text{exotic}} + \underbrace{\theta \epsilon_{ij} dp^i \wedge dp^j}_{\text{exotic}} \qquad (\text{III.8}) $$

is defined on the evolution space $\mathcal{E} = \mathbb{R}^2 \times \mathbb{R}^2 \times \mathbb{R} = \{\mathbf{x}, \mathbf{p}, t\}$. It can be split as $\sigma_{\text{exo}}^{\text{Gal}} = \Omega_{\text{exo}}^{\text{Gal}} - d\mathcal{H} \wedge dt$, where

$$ \Omega_{\text{exo}}^{\text{Gal}} = \underbrace{d\mathbf{p} \wedge d\mathbf{x} + \frac{e}{2} F_{ij} dx^i \wedge dx^j}_{\Omega_{\text{em}}} + \underbrace{\frac{\kappa}{2m^2} \epsilon_{ij} dp^i \wedge dp^j}_{\text{exotic}}, \qquad \mathcal{H}_{\text{em}} = \frac{\mathbf{p}^2}{2m} + eV. \qquad (\text{III.9}) $$

The system is regular when the determinant of the symplectic form does not vanish,

$$ \det(\Omega_{ij}^{\text{Gal}}) = \left(\frac{m^*}{m}\right)^2 \neq 0, \quad \text{where} \quad m^* \equiv m_G^* = m(1 - \theta eB). \qquad (\text{III.10}) $$

Here $m^*$ is an *effective mass* [69, 70]. Dropping the suffix $G$, the Poisson brackets associated with (III.8) are, for $m^* \neq 0$,

$$ \{x_i, x_j\} = \frac{m}{m^*} \theta \epsilon_{ij}, \quad \{x_i, p_j\} = \frac{m}{m^*} \delta_{ij}, \quad \{p_i, p_j\} = \frac{m}{m^*} eB \epsilon_{ij}, \qquad (\text{III.11}) $$

or equivalently,

$$ \{x_i, x_j\} = \left(\frac{1}{1 - \theta eB}\right) \theta \epsilon_{ij}, \quad \{x_i, p_j\} = \left(\frac{1}{1 - \theta eB}\right) \delta_{ij}, \quad \{p_i, p_j\} = \left(\frac{1}{1 - \theta eB}\right) eB \epsilon_{ij}. \qquad (\text{III.12}) $$
of null trajectories. Let us then introduce null coordinates $u$ and $v$ such that

$$ u = \frac{ct + z}{2}, \quad v = \frac{ct - z}{2}. \qquad (III.69) $$

The next step is to compute the geodesic equation of motion (III.67) for a massless particle, for an “outgoing” particle with momentum $P$ chosen as above, for instance. We readily see from (III.69) that $\dot{u} = 1$ and $\dot{v} = 0$, meaning that outgoing massless particles have their trajectories along a $v = \text{const.}$ hypersurface, while $u$ is a natural evolution parameter. Conversely, an ingoing massless particle would follow $u = \text{const.}$ hypersurfaces, with $v$ the evolution parameter. The situation is depicted in figure 2. The other equations of an outgoing particle are, from (III.67), given by,

$$ \frac{dx}{du} = 0, \quad \frac{dy}{du} = 0, \qquad (III.70) $$

where we used $u$ as the evolution parameter.

What is said above allows us to understand why the “no-motion” statement for Carroll particles is so natural: the $v = \text{const.}$ hypersurfaces define the Carroll structure. Such a structure would be described by 2 spatial coordinates $x$ and $y$ and by another one “along” the null direction, $u$. Geometrically, $u$ would be what we called the “Carroll time”. The outgoing massless particles stay on these Carroll structures, and their equations of motion are thus induced from the ambient spacetime (III.70). Thus there is “no motion” in the $(x, y)$ plane, as the induced momentum vanishes in the spatial directions, as expected for Carroll dynamics.

For the Carroll structures that are found as null hypersurfaces in lorentzian spacetimes, the intrinsic statement that Carroll particles do not move mean simply that particles move in the null direction that defines the hypersurface, i.e., that it follows a null geodesic in the ambient spacetime. The massless particle has all its momentum directed toward the null direction, and has no spare momentum left to deviate in the other directions. In that sense, the statement of the Red Queen makes perfect sense for Carrollian physics.

So in a sense, we can think of Carroll structures/dynamics as an effective description of phenomena which follow null geodesics. This also implies that finding Carrollian motion means finding potential deviations from null geodesics.
In particular, the coordinates do not commute when $\theta \neq 0$. The associated Hamilton equations are

$$ \frac{m^*}{m} \dot{x}_i = \frac{p_i}{m} - e\theta\epsilon_{ij}E_j, \qquad (III.13a) $$

$$ \dot{p}_i = eE_i + eB\epsilon_{ij}\dot{x}^j. \qquad (III.13b) $$

Note the velocity and the momentum cease to be parallel when $\theta \mathbf{E} \neq 0$. For $\theta = 0$ we recover the usual expressions for a charged particle in a constant magnetic field and for $B = 0$ we get the exotic expression $(2.3)$ in [69]. The system becomes singular when the effective mass vanishes, $m^* = 0$, and Hamiltonian (alias “Faddeev-Jackiw”) reduction [69, 85] yields [69–71]:

**Theorem III.1.** eqn. (III.13) implies that when its effective mass (III.10) vanishes,

$$ m^* = 0, \qquad (III.14) $$

and when the electric field is constant then an “exotic” Galilean particle follows the Hall law (I.2).

Indeed, for $m^* = 0$, one has $e\theta = 1/B$, and thus the first equation (III.13a) yields for the momentum $p_i = m\epsilon_{ij}\frac{E_j}{B}$. Now, if the electric field is constant, then $\dot{p}_i = 0$, and the second equation (III.13b) reduces to the Hall law (I.2).

The behavior when the effective mass changes sign is studied conveniently using chiral decomposition [118, 119]. We have the Souriau form (III.8) is locally $\sigma_{\text{exo}}^{\text{Gal}} = d\varpi_{\text{exo}}^{\text{Gal}}$, where the Cartan 1-form $\varpi_{\text{exo}}^{\text{Gal}}$ corresponds to the “exotic” phase space Lagrangian [69]$^{21}$,

$$ \mathcal{L}_{\text{exo}}^{\text{Gal}} = \underbrace{(\mathbf{p} - e\mathbf{A}) \cdot \frac{d\mathbf{x}}{dt} - \mathcal{H}_{\text{em}} dt}_{\mathcal{L}_{\text{em}}} + \underbrace{\frac{\theta}{2} \mathbf{p} \times \frac{d\mathbf{p}}{dt}}_{\text{exotic}}. \qquad (III.15) $$

We record for further reference that in a constant $B$-field the *conserved angular momentum* is [69]

$$ \ell = \mathbf{x} \times \mathbf{p} + \frac{1}{2} eB \mathbf{x}^2 + \frac{1}{2} \theta \mathbf{p}^2 + \chi, \qquad (III.16) $$

where the real constant $\chi$ is identified as the *anyonic spin* [120–122]. The relation to planar vortex dynamics [123, 124] will be studied elsewhere.

<footnote>$^{21}$ In $d = 2$ spatial dimensions, the vector product is a pseudoscalar, $\mathbf{a} \times \mathbf{b} = \epsilon_{ij}a^i b^j$.</footnote>
## G. On the role of the mass

Comparison of the behaviors in the massive/massless cases yields further insight: Eqn. (III.14) implies that the mass $m \neq 0$ “glues” our particle to a fixed position. However in the massless limit the particle is “liberated” and moves as in (III.62). The system has obviously the half-broken Carrollian boost symmetry found earlier in Theorem III.4. This non-analytical behavior of the mass-induced spatial “confinement” of the Carroll particle has similar features with the conventional Hall effect (I.2) in two spatial dimensions.

The motion of a particle with a nonvanishing (even infinitesimally small, but nonzero) electric charge $e \neq 0$ is radically different from that of a particle with an exactly zero charge $e = 0$. In the first case, the electromagnetic field generates the Hall motion of the particle (I.2), while in the second case, the background field does not affect the motion at all. Moreover, despite the paramount requirement of the nonvanishing electric charge $e \neq 0$, the Hall motion does not depend explicitly on the actual value of the electric charge $e$ (I.2). In the Hall conditions (with perpendicular electric and magnetic fields), the motion is non-analytic in $e$ as $e = 0$ and $e \neq 0$ cases are qualitatively different.

The “no-particle” motion of the Carroll particle is a non-analytical phenomenon but now in the particle mass rather than in the electric charge: the motion is radically different for massive ($m \neq 0$) and massless ($m = 0$) cases. The second similarity appears because the actual value of the mass does not enter the solution of equations of motion for the Carroll particle, similarly to the electric charge in the Hall effect.

# IV. DOUBLY-EXTENDED CARROLL DYNAMICS

## A. Free massive doubly-extended particle

A fact which escaped attention until recently is that, in $d = 2$ space dimensions, the Carroll group admits a two-parameter central extension [6, 31, 63]. Its central charges shall be denoted by $\kappa_{\text{exo}}$ and $\kappa_{\text{mag}}$, respectively whose non-vanishing will henceforth be assumed. Such additional charges are allowed by the theory and they may play an important role for planar systems with Carroll symmetry.

The doubly-extended group, $\tilde{C}$, can be represented by $6 \times 6$ matrices (Eqn (2.9) in [31])
A. Galilean anyons with spin-field coupling

The planar model can be further extended. In $d = 2$ spatial dimensions, the anyonic spin is just a constant $\chi$, and a term involving the spin-field Hamiltonian$^{22}$,

$$ \mathcal{H}_{any} = -\mu\chi B, \qquad (III.17) $$

where $\mu$ is a magnetic momentum can be added to the spinless expression (III.8). Then we end up with the Souriau form of a massive, exotic anyon with spin in an electromagnetic (EM) field,

$$ \sigma^{\text{Gal}} = \underbrace{(dp_i - eE_i dt) \wedge (dx^i - \frac{p^i}{m} dt)}_{\text{electric}} + \underbrace{\frac{1}{2} eB \epsilon_{ij} dx^i \wedge dx^j}_{\text{magnetic}} + \underbrace{\frac{\theta}{2} \epsilon_{ij} dp^i \wedge dp^j}_{\text{exotic}} + \underbrace{\mu\chi dB \wedge dt}_{\text{spin-field}}. \quad (III.18) $$

The new term merely shifts the force,

$$ eE_i \rightarrow E_i^* = eE_i + \mu\chi\partial_i B, \qquad (III.19) $$

and the equations of motion become again (III.13) up to this shift.

Note the important role which is played by the magnetic field $B$ in Eq. (III.19). The gradient of $B$ behaves as an effective electric field which adds up to the background electric field $E$ in Eq. (III.19). The effective electric field appears from the interaction of the anyonic spin with the background magnetic field described by the Hamiltonian (III.17). The spin-field interaction plays the same role as the electrostatic potential $eV$ in the Hamiltonian (III.4) for a usual electrically charged particle. This analogy identifies the effective electrostatic potential for the anyons, $eV_{eff} = -\mu\chi B$. The inhomogeneities in the latter produce the force on the anyon which can be described, according to Eq. (III.5), via an effective electric field $E_{eff} = (\mu\chi/e)\nabla B$.

When the effective mass vanishes, $m^* = 0$, the system becomes once again singular. However then the magnetic field, $B = B_{crit} = m^2/e\kappa$, is necessarily constant and therefore the new, $\mu$-term drops out: the anyonic spin has no impact on the (Hall) motions.

<footnote>$^{22}$ For an “exotic photon”, the spin-field term (III.17) plays a particular role on the horizon, see Sec. IX. Another example is studied in [125].</footnote>
to be compared with (III.1),

$$ \left( \begin{array}{ccccc} A_j^i & 0 & c^i & 0 & \epsilon_k^i b^k \\ -b_k A_j^k & 1 & f & 0 & A_{\text{exo}} \\ 0 & 0 & 1 & 0 & 0 \\ c_k \epsilon_l^k A_j^l & 0 & A_{\text{mag}} & 1 & -(f + \mathbf{b} \cdot \mathbf{c}) \\ 0 & 0 & 0 & 0 & 1 \end{array} \right), \qquad (\text{IV.1}) $$

where $A_{\text{mag}}$ and $A_{\text{exo}}$ are new parameters associated with the double extension. The extended algebra has a 8-parameter moment map, $\mu(\ell, \mathbf{g}, \pi, m, \kappa_{\text{mag}}, \kappa_{\text{exo}})$ with commutators,

$$ \begin{gathered} [J_3, K_i] = \epsilon_{ij} K_j, \quad [K_i, K_j] = \epsilon_{ij} A_{\text{exo}}, \quad [J_3, P_i] = \epsilon_{ij} P_j, \quad [K_i, P_j] = \delta_{ij} P_0, \qquad (\text{IV.2}) \\ [J_3, P_0] = 0, \quad [K_i, P_0] = 0, \quad [P_i, P_j] = \epsilon_{ij} A_{\text{mag}}, \quad [P_i, P_0] = 0, \end{gathered} $$

where $J_3$ is the rotation, the $(K_i)$ are boosts, the $(P_i)$ spatial translations, $(P_0)$ is time translation, and $A_{\text{exo}}$ and $A_{\text{mag}}$ are the exotic and magnetic extensions, respectively.

More insight into the extended algebra structure is obtained by comparison with previously considered extensions. The one with $A_{\text{exo}}$ is plainly the same as the “exotic” extension considered for the planar Galilean algebra [50–52].

On the other hand, the “magnetic” extension which implies the non-commutativity of spatial translations, $[P_i, P_j] = \epsilon_{ij} A_{\text{mag}}$, does not appear in the Galilei algebra — while it can appear instead in the extension of the *planar Euclidean algebra*, which is a subalgebra of both of the Carroll and the Galilei algebras. Then one may wonder why does the magnetic extension appear for Carroll, but not for Galilei.

Let us recall that in order to be a Lie algebra, the bracket operation should respect the Jacobi identity, which imposes strong constraints. For the Galilei algebra, the Jacobi identity for a time translation, a boost in one and a spatial translation in the other direction,

$$ 0 = [P_0, [K_1, P_2]] + [K_1, [P_2, P_0]] + [P_2, [P_0, K_1]] = [P_2, [P_0, K_1]] = [P_2, P_1] $$

requires the magnetic extension to vanishes. For the Carroll algebra instead, this identity is trivially realized because Carroll time translation commutes with boosts.

The free doubly-extended model obtained by the KKS algorithm is, by construction, symmetric w.r.t. the doubly-extended Carroll group $\tilde{C}$, with associated conserved quantities,
Massless Galilean particle models can also be constructed by the KKS method by choosing appropriately the base point $\mu_0$ in the dual Lie algebra. They can be spinless (“Fermat particle” – see sec.IV C), or can also carry spin [58, 59, 77]. The same method works also in the relativistic case. Such a particle can be coupled to a background metric [64, 65, 67].

The “exotic” model introduced in [69, 70] has anomalous gyromagnetic factor $g = 0$. More general cases are studied in [126]. An optical medium can be viewed also as a special metric [59]; then for massless particles we get a Spin-Hall effect for light [57–59, 61].

## B. Galilean/Bargmann symmetry in field theory

So far we studied Galilean invariant classical models. Now we present a field-theoretical example to be compared later with the discussion of fractons in sect.V.

In [128] Lévy-Leblond proposed to describe a non-relativistic particle with spin 1/2 in an electromagnetic field by a counterpart of the Dirac equation. Although his equation can, of course, be studied in any dimension [129–133], here we restrict ourselves to $d = 2$. Then it can be obtained [134] by starting with the 4-component Dirac equation in 1+2 dimensions,

$$ (ic\gamma_{\pm}^{\alpha}D_{\alpha} - m)\psi_{\pm} = 0, \qquad (III.20) $$

where $D_\alpha = \partial_\alpha - ieA_\alpha$ is the gauge-covariant derivative, the two-component spinors $\psi_\pm$ are the chiral components and $\gamma_\pm^\alpha = (\pm(1/c)\sigma^3, i\sigma^1, i\sigma^2)$ are the 1+2 dimensional Dirac matrices. Setting

$$ \psi_{+} = e^{-imc^{2}t} \begin{pmatrix} \Psi_{+} \\ \tilde{\chi}_{+} \end{pmatrix} \quad \text{and} \quad \psi_{-} = e^{-imc^{2}t} \begin{pmatrix} \tilde{\chi}_{-} \\ \Psi_{-} \end{pmatrix}, \qquad (III.21) $$

dropping $iD_t\tilde{\chi} \ll mc^2\tilde{\chi}$, eqn. (III.20) becomes

$$ \begin{gathered} iD_t\Phi - (\boldsymbol{\sigma} \cdot \mathbf{D})\chi = 0, \qquad (III.22) \\ (\boldsymbol{\sigma} \cdot \mathbf{D})\Phi + 2mc\chi = 0, \end{gathered} $$

where $\Phi = \begin{pmatrix} \psi_+ \\ \psi_- \end{pmatrix}$ and $\chi = c \begin{pmatrix} \tilde{\chi}_- \\ \tilde{\chi}_+ \end{pmatrix}$. Using $(\mathbf{D} \cdot \boldsymbol{\sigma})^2 = \mathbf{D}^2 + eB\sigma_3$ allows us to deduce the 2nd-order equations

$$ \begin{aligned} iD_t\Phi &= -\frac{1}{2m}[\mathbf{D}^2 + eB\sigma_3]\Phi, \qquad (III.23) \\ iD_t\chi &= -\frac{1}{2m}[\mathbf{D}^2 + eB\sigma_3]\chi - \frac{e}{2m}(\boldsymbol{\sigma} \cdot \mathbf{E})\Phi. \end{aligned} $$
$$ \ell = \boldsymbol{x} \times \boldsymbol{p} + \frac{1}{2}\theta\boldsymbol{p}^2 + \frac{1}{2}\kappa_{\text{mag}}\boldsymbol{x}^2 + \chi, \qquad \text{angular momentum} \qquad (\text{IV.3a}) $$

$$ g^i = m(x^i + \theta \epsilon^{ij} p_j), \qquad \text{boost momentum} \qquad (\text{IV.3b}) $$

$$ \pi_i = p_i - \kappa_{\text{mag}} \epsilon_{ij} x^j, \qquad \text{"impulsion"} \qquad (\text{IV.3c}) $$

completed with $m, \kappa_{\text{exo}}, \kappa_{\text{mag}}$, cf. \# (3.18) in [31] and sec.III for the terminology.

These quantities are analogous to the “exotic” Galilean expressions in a constant magnetic field [51, 52]. The real constant $\chi$ in (IV.3a) is Carrollian/anyonic spin, defined in (IV.4). The boost momentum $g$ (IV.3b) plays a particularly important role, as it will be explained below.

The unusual form of the “impulsion” $\boldsymbol{\pi}$ in (IV.3c) deserves a comment. First we note that $A_i = -\frac{1}{2}(\kappa_{\text{mag}}/e)\epsilon_{ij}x^j$ is a vector potential for $\kappa_{\text{mag}}$ viewed as a (constant) “internal magnetic field”. Then $\boldsymbol{\pi}$ has the unexpected form $\pi_i = p_i + 2eA_i$, where $\boldsymbol{p}$ is just a coordinate. However in terms of the canonical momentum $P_i = p_i - \frac{1}{2}\kappa_{\text{mag}}\epsilon_{ij}x^j$ obtained from the Lagrangian (IV.12), $\pi_i = P_i - \frac{1}{2}\kappa_{\text{mag}}\epsilon_{ij}x^j = P_i + eA_i$.

**Casimirs.** When the mass does not vanish, $m \neq 0$, we have four Casimirs [31]: $\kappa_{\text{mag}}, \kappa_{\text{exo}}$, the mass, $m$, and the anyonic spin, $\chi$, defined by,

$$ m^*\chi = m^*\ell - \boldsymbol{g} \times \boldsymbol{\pi} + \frac{\kappa_{\text{exo}}}{2m}\boldsymbol{\pi}^2 + \frac{\kappa_{\text{mag}}}{2m}\boldsymbol{g}^2, \qquad (\text{IV.4}) $$

where

$$ m^* = m \left( 1 - \frac{\kappa_{\text{exo}}}{m^2} \kappa_{\text{mag}} \right) \qquad (\text{IV.5}) $$

is an effective mass, analogous to $m_G^*$ in (II.14) in the Galilean case.

Eqn. (IV.4) is valid also for $m^* = 0$ when both the spin $\chi$ and the total angular momentum $\ell$ drop out. In this case, realized for a special relation between the Casimir invariants $\kappa_{\text{mag}} \cdot \kappa_{\text{exo}} = m^2$, we get:

$$ \frac{1}{m} \boldsymbol{g} \times \boldsymbol{\pi} - \frac{1}{2\kappa_{\text{mag}}} \boldsymbol{\pi}^2 - \frac{1}{2\kappa_{\text{exo}}} \boldsymbol{g}^2 = 0. \qquad (\text{IV.6}) $$

Multiplying (IV.4) by $m^2 \neq 0$ and then letting $m \to 0$ yields in turn the expression valid in the massless case

$$ \chi = \ell - \frac{\boldsymbol{\pi}^2}{2\kappa_{\text{mag}}} - \frac{\boldsymbol{g}^2}{2\kappa_{\text{exo}}}, \qquad (\text{IV.7}) $$
The component $\Phi$ thus satisfies the two-component Pauli equation which further separates into two uncoupled scalar equations with opposite charges. The component $\chi$ loses in turn its proper dynamics and is determined by the large component, $\Phi$.

The free Lévy-Leblond equation<sup>23</sup>, (III.22) with $A_\mu = 0$, is, by construction, invariant w.r.t. the Bargmann group [76]— a non-trivial central extension of the Galilei group<sup>24</sup>, cf. eqn. (8) and (26) in ref. [128], If one works with projective representations, a Galilei boost by $\mathbf{b}$ is implemented as,

$$ \begin{pmatrix} \Phi(\mathbf{x}, t) \\ \chi(\mathbf{x}, t) \end{pmatrix} \rightarrow e^{im(\mathbf{b}\cdot\mathbf{x} + \frac{1}{2}\mathbf{b}^2t)} \begin{pmatrix} \mathbb{I}_2 & 0 \\ -\frac{1}{2}\boldsymbol{\sigma} \cdot \mathbf{b} & \mathbb{I}_2 \end{pmatrix} \begin{pmatrix} \Phi(\mathbf{x} + \mathbf{b}t, t) \\ \chi(\mathbf{x} + \mathbf{b}t, t) \end{pmatrix}. \qquad (\text{III.24}) $$

Thus boosts act on the component $\Phi$ as a usual Galilei boost does on a scalar,

$$ \Phi(\mathbf{x}, t) \rightarrow e^{im(\mathbf{b}\cdot\mathbf{x} + \frac{1}{2}\mathbf{b}^2t)} \Phi(\mathbf{x} + \mathbf{b}t, t), \qquad (\text{III.25}) $$

however it mixes the $\Phi$ and $\chi$ components. The well-known associated conserved quantities are listed in [128].

Studying Galilean field theories on a Newton-Cartan (N-C) spacetime (def. II.2) is not convenient: the central extension does not belong to the isometries of the N-C spacetime, therefore we have no true, only projective representations of the Galilei group.

The natural context to study Galilean field theories is provided by the Bargmann space [88–90], see def. II.6. Bargmann structures have an additional spacetime coordinate $s$; the isometries of a flat Bargmann structure are given by the Bargmann group, see (II.11). Bargmann wavefunctions are $\Psi(\mathbf{x}, t, s)$ and are equivariant with respect to the new coordinate,

$$ L_{\xi}\Psi = im\Psi \qquad (\text{III.26}) $$

($\xi = \partial_s$). This means that “ordinary” wavefunctions $\Phi$ are lifted to Bargmann space, where their phase freedom is incorporated into the $s$ coordinate.

$$ \Psi(\mathbf{x}, t, s) = e^{ims}\Phi(\mathbf{x}, t). \qquad (\text{III.27}) $$

It is this form which explains how wave functions (or scalar fields) behave under Bargmann/Galilei transformations, see (II.11): the projective factor in (III.24) and (III.25)

<footnote n="23"><sup>23</sup> The discussion also holds true for the Schrödinger equation.</footnote>

<footnote n="24"><sup>24</sup> The full group of invariance of the Lévy-Leblond equation (and of the Schrödinger equation) is the conformal extension of the Bargmann group, called the Schrödinger group [135].</footnote>
where the Casimir invariants $\kappa_{mag}$ and $\kappa_{exo}$ are assumed to be nonvanishing. Equation (IV.7) implies that the Carrollian anyonic spin $\chi$ is given by the angular momentum $\ell$ with contributions coming from the impulsion $\pi$ and the boost momentum $g$.

The KKS construction applied to $\tilde{\mathcal{C}}$ yields, for a particle with mass $m \neq 0$, the Cartan resp. Souriau-forms with two additional terms [31],

$$ \varpi_{\text{exo}} = \mathbf{p} \cdot d\mathbf{x} + \frac{\theta}{2} \mathbf{p} \times d\mathbf{p} + \frac{\kappa_{\text{mag}}}{2} \mathbf{x} \times d\mathbf{x}, \qquad (\text{IV.8a}) $$

$$ \sigma_{\text{exo}} = d\mathbf{p} \wedge d\mathbf{x} + \underbrace{\frac{1}{2}\theta \epsilon_{ij} dp^i \wedge dp^j}_{\text{exotic}} + \underbrace{\frac{\kappa_{\text{mag}}}{2} \epsilon_{ij} dx^i \wedge dx^j}_{\text{"internal magnetic"}}, \qquad (\text{IV.8b}) $$

defined on Carroll evolution space $\mathcal{E}$ (III.5), cf. # (3.10) in [31]. The constant

$$ \theta = \frac{\kappa_{\text{exo}}}{m^2} \qquad (\text{IV.9}) $$

here is interpreted as the non-commutativity parameter, see (IV.11) below.

The terms with $\theta \sim \kappa_{exo}$ in (IV.8) are those of exotic Galilean dynamics, (I.3), and are in fact present also for *relativistic anyons* from which the Galilean model can be deduced as it will be shown in sec. V. The terms with $\kappa_{mag}$ are reminiscent of a (constant) “internal magnetic” field, as mentioned above. Cf. also (II.12).

Eqn. (IV.8) is formally $\Omega_{exo} - d\mathcal{H}_0 \wedge ds$ with zero Hamiltonian,

$$ \Omega_{\text{exo}} = d\mathbf{p} \wedge d\mathbf{x} + \frac{\theta}{2} \epsilon_{ij} dp^i \wedge dp^j + \frac{\kappa_{\text{mag}}}{2} \epsilon_{ij} dx^i \wedge dx^j \quad \text{and} \quad \mathcal{H}_0 \equiv 0, \qquad (\text{IV.10}) $$

cf. (III.6). $\Omega_{exo}$ is a closed and regular (i.e. symplectic) 2-form on the $s = s_0 = \text{const.}$ submanifold $\mathbb{R}^2 \times \mathbb{R}^2 = \{\mathbf{x}, \mathbf{p}, s_0\}$ of the Carroll evolution space $\mathcal{E}$.

When the *effective mass* $m^*$ is non-zero, $m^* \neq 0$, i.e. $\kappa_{exo} \cdot \kappa_{mag} \neq m^2$, the associated Poisson brackets [6]

$$ \begin{aligned} \{x^i, x^j\} &= \left(\frac{1}{1-\theta\kappa_{\text{mag}}}\right) \theta\epsilon^{ij}, \\ \{x^j, p_i\} &= \left(\frac{1}{1-\theta\kappa_{\text{mag}}}\right) \delta_i^j, \\ \{p_i, p_j\} &= \left(\frac{1}{1-\theta\kappa_{\text{mag}}}\right) \kappa_{\text{mag}} \epsilon_{ij} \end{aligned} \qquad (\text{IV.11}) $$
follows, for example, from the Bargmann lift of the symmetry. Hence, these wavefunctions $\Psi(x, t, s)$ allow the definition of a true representation of the Bargmann group, without projective factors.

Lastly, the Bargmann structure is also natural for Galilean field theories in the sense of covariance. For instance, the Schrödinger equation on a Newton-Cartan spacetime is equivalent to a massless Klein-Gordon equation on a Bargmann structure, while the Lévy-Leblond equation is equivalent to a massless Dirac equation on Bargmann space which has 1 higher dimension [135].

These properties will play an important rôle for fractons, see sec. V.

Applying Noether's theorem II.2 to the spin 1/2 Lévy-Leblond system, we get [128, 135]

$$J^0 = |\Phi|^2, \quad J^i = \frac{1}{2im} (\Phi^\dagger \nabla \Phi - (\nabla \Phi)^\dagger \Phi) + \nabla \times \left( \frac{1}{2m} \Phi^\dagger \sigma_3 \Phi \right). \quad (\text{III.28})$$

The conserved quantities of a free particle, eqn. (5.6) of [135],

$$
\begin{align*}
\mathbf{P} &= \int \underbrace{\left\{ \frac{1}{2i} (\Phi^\dagger \nabla \Phi - (\nabla \Phi)^\dagger \Phi) \right\}}_{\mathcal{P}} d^2\mathbf{x} \quad \text{linear momentum} \\
\ell &= \int \mathbf{x} \times \mathcal{P} \, d^2\mathbf{x} + \int \frac{1}{2} \Phi^\dagger \sigma_3 \Phi \, d^2\mathbf{x} \quad \text{angular momentum} \\
\mathbf{D}^{\text{Gal}} &= m \int \Phi \Phi^\dagger \mathbf{x} \, d^2\mathbf{x} - t \mathbf{P} \quad \text{boost momentum} \tag{III.29} \\
H &= \int \left\{ \frac{1}{2m} \nabla \Phi (\nabla \Phi)^\dagger \right\} d^2\mathbf{x} \quad \text{energy} \\
M &= m \int \Phi \Phi^\dagger \, d^2\mathbf{x} \quad \text{mass}
\end{align*}
$$

span under Poisson bracket the Bargmann algebra. The angular momentum $\ell$ confirms that the components of $\Phi$ have spins $\pm \frac{1}{2}$. The conserved quantity associated with Galilean boosts, $\mathbf{D}$, should be compared with the boost momentum of a point particle, (III.3).

## IV. CARROLL DYNAMICS

After recalling the Galilean case, henceforth we focus our attention at Carroll particles which are indeed our main objects of interest. The label $\{\cdot\}^{\text{Car}}$ analogous to $\{\cdot\}^{\text{Gal}}$ will be understood and omitted.
are identical with Eqn. (II.16) of the “exotic” Galilei case under the replacement $\kappa_{\text{mag}} \sim eB$. The first relation implies that the coordinates do not commute when $\kappa_{\text{exo}} \neq 0$ and the last one implies noncommuting momenta for $\kappa_{\text{mag}} \neq 0$.

Alternatively (and equivalently when $m^* \neq 0$) we can consider the doubly-extended Carroll Lagrangian defined on the evolution space $\mathcal{E}$ in (III.5) which has, two more terms added to the free unextended Carroll Lagrangian $\mathcal{L}_0$ (III.11),

$$ \mathcal{L}_{\text{exo}} = \underbrace{\boldsymbol{p} \cdot \boldsymbol{x}'}_{\mathcal{L}_0} + \underbrace{\frac{1}{2} \theta \epsilon_{ij} p^i (p^j)'}_{\text{exotic}} + \underbrace{\frac{1}{2} \kappa_{\text{mag}} \epsilon_{ij} x^i (x^j)'}_{\text{magnetic}} . \qquad (\text{IV.12}) $$

The associated variational equations (which are valid without the assumption $m^* \neq 0$)

$$ (x^k)' + \theta \epsilon^{kl} (p_l)' = 0, \quad (p_i)' - \kappa_{\text{mag}} \epsilon_{ij} (x^j)' = 0, \qquad (\text{IV.13}) $$

are integrated at once to yield the conserved vectors

$$ x^k + \theta \epsilon^{kl} p_l = \text{const.} = Q^k, \qquad (\text{IV.14a}) $$

$$ p_i - \kappa_{\text{mag}} \epsilon_{ij} x^j = \text{const.} = \pi_i. \qquad (\text{IV.14b}) $$

In particular, the 2-vector $Q = (Q^k)$ in (IV.14a) is identified as the guiding center [37, 79].

The conservation of the boost momentum $g$ in (IV.3) alone does not now imply immobility for $x$ – however it does imply that of $Q$ in (IV.14a). Combining with the conservation of $\pi$ (IV.14b) then yields the pair of decoupled equations which involve the effective mass (IV.5),

$$ m^*(x_i)' = 0, \quad m^*(p_i)' = 0 \qquad (\text{IV.15}) $$

which allow us to infer:

**Theorem IV.1.** A free doubly-extended Carroll particle with non-vanishing effective mass, $m^* \neq 0$ in (IV.5), does not move.

However when the effective mass vanishes,

$$ m^* = 0 \quad \text{i.e.} \quad \theta \kappa_{\text{mag}} = 1 \quad \text{i.e.} \quad \kappa_{\text{exo}} \kappa_{\text{mag}} = m^2, \qquad (\text{IV.16}) $$

then no conclusion can be drawn from (IV.15). For $m^* = 0$ the determinant of the symplectic form<fnref n="19" />,

$$ \det (\Omega_{ij}) = \left( \frac{m^*}{m} \right)^2 = (1 - \theta \kappa_{\text{mag}})^2 \qquad (\text{IV.17}) $$

<footnote n="19">¹⁹ Compare with (II.14).</footnote>
The dynamics of a Carroll particle in an electromagnetic field can be derived in two steps as outlined in section III: first a free model is constructed by the KKS method applied to the Carroll group [7, 8, 17, 34, 77], which is then coupled to the em field by the rule (II.48).

An element $g = g(R, \mathbf{b}, \mathbf{c}, b_0)$ of the Carroll group $\mathfrak{C}$ is parametrized by $\frac{d(d+3)}{2} + 1$ numbers. Dual to them are the “moments” [77], which are the physical parameters that describe the state of a particle at a point. A moment is $\mu = \mu(\ell, \mathbf{D}, \pi, m) \in \mathfrak{ carr}^*(d)$, where $\ell \in \mathfrak{so}(d)^*$ is the angular momentum, $\mathbf{D}$ the boost momentum [center of mass], $\pi$ the conserved momentum, and $m$ is an arbitrary real number. The Carroll group acts on these moments with its coadjoint action,

$$ \ell' = \ell + \mathbf{b} \times \mathbf{R}\mathbf{D} - \mathbf{c} \times \mathbf{R}\pi + m\mathbf{b} \times \mathbf{c} \qquad (IV.1a) $$

$$ \mathbf{D}' = \mathbf{R}\mathbf{D} + m\mathbf{c} \qquad (IV.1b) $$

$$ \pi' = \mathbf{R}\pi + m\mathbf{b} \qquad (IV.1c) $$

$$ m' = m \qquad (IV.1d) $$

The invariants under the coadjoint action are the *Casimir invariants*. Notice that $m$ is not an externally given quantity but a *moment* and a *Casimir invariant* associated now not with a central extension as for Galilei/Bargmann theory, but with (Carrollian) “time” translations along Carrollian time $s$. It is a sort of “Carrollian energy” cf. (IV.2).

The KKS orbit calculation we follow here provides us with a mathematical framework but does not give a physical interpretation for the moments $\ell, \mathbf{D}, \pi, m$. In view of the physical applications which will come, for now we make the choice to interpret $m$ as the mass of a particle, due to the analogy of the Carroll and the Galilei groups. This interpretation is not fixed, though: in fracton theory reviewed in sec. V for instance, it can be viewed instead as an electric charge, which can plainly take both positive or negative values. The mathematical framework applies to both cases.

The word “momentum” has indeed three related but somewhat different meanings here. Firstly, one has $\mathbf{p}$, which we define to be always $mv$. Secondly there is the *canonical* or *generalized* momentum, defined as $\mathbf{P} = \partial\mathcal{L}/\partial\mathbf{x}'$. For a non relativistic particle in a magnetic field, for instance, $\mathbf{P} = mv - e\mathbf{A}$. At last, “momentum” is also the *conserved quantity* dual to space translation symmetry (a “moment” in Souriau’s terminology [77]), which we shall call here “impulsion” and denote by $\pi$. The three definitions coincide in some simple cases but not always, see, e.g., (VI.4).
becomes indeed zero, highlighting the fact that the variational system is singular, requiring “Faddeev-Jackiw” reduction [51, 52, 66]. The dimension of the phase space drops from 4 to 2. The variables $x$ and $p$ become redundant however their (fixed) combination $Q = (Q^k)$ in (IV.14a) is still a physical quantity. In its terms the reduced free symplectic form is simply

$$ \Omega_{red} = \kappa_{mag} dQ^1 \wedge dQ^2 = \frac{1}{\theta} dQ^1 \wedge dQ^2. \qquad (IV.18) $$

The reduced Poisson brackets and variational forms are thus

$$ \{Q_1, Q_2\}_{red} = -\theta, \qquad \bar{\omega}_{red} = \frac{1}{2\theta} \epsilon_{ij} Q^i dQ^j, \qquad \mathcal{L}_{red} = \frac{1}{2\theta} \mathbf{Q} \times \mathbf{Q}'. \qquad (IV.19) $$

The Hamiltonian $\mathcal{H}$ is identically zero and the equations of motion, $Q'_i = \{Q^i, \mathcal{H}\}$, are therefore,

$$ Q'_i = 0 \Rightarrow \mathbf{Q} = \mathbf{Q}_0 = \text{const.} \qquad (IV.20) $$

consistently with (IV.14a). In conclusion,

**Theorem IV.2.** When the effective mass of a massive system vanishes, $m^* = 0$, our free massive doubly-extended particle model becomes singular. The dynamics of $x$ and of $p$ can not be separately determined, however the guiding center $Q$ in (IV.14a), remains fixed. The conserved quantities in (IV.3) are expressed in terms of the guiding center alone,

$$ \ell = \frac{1}{2\theta} \mathbf{Q}^2 \qquad \text{angular momentum} \qquad (IV.21a) $$

$$ \mathbf{g} = m\mathbf{Q} \qquad \text{boost momentum (center of mass)} \qquad (IV.21b) $$

$$ \pi_i = -\frac{1}{\theta} \epsilon_{ik} Q^k \qquad \text{“impulsion”} \qquad (IV.21c) $$

consistently with the loss of two physical degrees of freedom.

For $m^* = 0$ the (no-)motion of the guiding center, (IV.20), follows directly from the conservation of $\mathbf{g}$ or of $\pi$, themselves proportional to $Q^k$.

We conclude that extended Carroll symmetry plays for the extended model a role analogous to that of Carroll in the unextended case, with the effective mass $m^*$ (IV.5) replacing the “naked” mass, $m$.

## B. Massless doubly-extended particles

Let us now study the double extension of the massless Carroll particles discussed in section III C. The coadjoint action of the Euclidean group (III.42) extends to one of the doubly
The dynamical description for Carroll particles is readily found by embedding the Carroll group $\mathcal{C}$ (II.14) into the Bargmann group $\mathfrak{B}$ (II.10) as a subgroup by “freezing out” (Galilean) time translations [7, 25] in $\mathfrak{B}$,

$$h = 0. \qquad (IV.2)$$

The moments allow us to define Casimir invariants. As said above, the first one is $m$, the mass of the particle. Carroll particles can also carry spin, defined as an additional Casimir invariant. However the planar case discussed in Sec. VI will be sufficient for our purposes here.

## A. Massive Carroll particles

We consider first the free massive case $m \neq 0$ with no spin. The $dt$-terms in (II.47) are switched off and we end up (consistently with $\# (A.9)$ and (A.10) in [7]), with,

$$\bar{\omega}_0 = \mathbf{p} \cdot d\mathbf{x} - mds, \qquad (IV.3a)$$

$$\sigma_0 = d\bar{\omega}_0 = d\mathbf{p} \wedge d\mathbf{x}. \qquad (IV.3b)$$

These forms are defined on “Carroll evolution space”

$$\mathcal{E} = \mathbb{R}^d \times \mathbb{R}^d \times \mathbb{R} = \{\mathbf{x}, \mathbf{p}, s\}. \qquad (IV.4)$$

Note that Carrollian time, $s$, does not appear in the Souriau 2-form.

Compared to the Galilean theory of sec. II B and sec. III, not only $E_0$, the internal energy is lost but, even more importantly, the usual kinetic term $(\mathbf{p}^2/2m) dt$ in (III.4) of the Galilean theory is missing from the Cartan form (IV.3a) (or equivalently, from the Hamiltonian $\mathcal{H}$ (IV.5) and thus also from the $\mathcal{L}$ Lagrangian (IV.9)).

The usual $-(\mathbf{p}/m)ds$ is missing from behind the $d\mathbf{x}$ in $\sigma_0$, leaving $\mathbf{p}$ and $d\mathbf{x}/ds$ unrelated: the usual Galilean velocity relation (III.2) is lost. This fundamental difference between Galilei and Carroll systems would not be altered by adding further (either electromagnetic or central extension) terms.

Splitting the Souriau form into symplectic form and Hamiltonian as in (II.43),

$$\sigma_0 = \Omega_0 - d\mathcal{H}_0 \wedge ds \quad \text{with} \quad \Omega_0 = d\mathbf{p} \wedge d\mathbf{x} \quad \text{and} \quad \mathcal{H}_0 \equiv 0, \qquad (IV.5)$$
extended Carroll group. The group element $(A, \mathbf{b}, \mathbf{c}, f, A_{\text{exo}}, A_{\text{mag}}) \in \tilde{C}$ is implemented on $\tilde{\mu}_0 = (\ell, \mathbf{g}, \boldsymbol{\pi}, m = 0, \kappa_{\text{exo}}, \kappa_{\text{mag}}) \in \tilde{\mathfrak{c}}^*$ as,

$$ \ell \rightarrow \ell - \mathbf{c} \times A\boldsymbol{\pi} + \mathbf{b} \times A\mathbf{g} - \frac{1}{2}\kappa_{\text{exo}}\mathbf{b}^2 - \frac{1}{2}\kappa_{\text{mag}}\mathbf{c}^2, \qquad (\text{IV.22a}) $$

$$ g^i \rightarrow (A\mathbf{g})^i + \kappa_{\text{exo}} \epsilon_{ij} b^j \qquad (\text{IV.22b}) $$

$$ \pi_i \rightarrow (A\boldsymbol{\pi})_i - \kappa_{\text{mag}} \epsilon_{ij} c^j, \qquad (\text{IV.22c}) $$

while $m = 0, \kappa_{\text{exo}}, \kappa_{\text{mag}}$ are left invariant, [31]. Having $\kappa_{\text{mag}} \neq 0$ in (IV.22b) then implies that the norm of the momentum is not a Casimir invariant anymore [31]. We thus have only one kind of massless doubly-extended particle — namely the trivial one associated with $\boldsymbol{\pi} = 0$. Put in another way, the “color” $k$ in a chosen basepoint can be eliminated by a translation which carries $\tilde{\mu}_0$ to the origin of the dual algebra, whose contribution to the KKS symplectic form is therefore obviously zero. Thus the first term in (IV.8a) disappears, leaving us with the “truncated” or “purely exotic” Cartan and Souriau forms,

$$ \varpi_{\text{exo}} = \frac{\kappa_{\text{exo}}}{2} \boldsymbol{v} \times d\boldsymbol{v} + \frac{1}{2} \kappa_{\text{mag}} \boldsymbol{x} \times d\boldsymbol{x}, \qquad (\text{IV.23a}) $$

$$ \sigma_{\text{exo}} = \underbrace{\frac{1}{2}\kappa_{\text{exo}}\epsilon_{ij}dv^i \wedge dv^j}_{\text{exotic}} + \underbrace{\frac{1}{2}\kappa_{\text{mag}}\epsilon_{ij}dx^i \wedge dx^j}_{\text{magnetic}}, \qquad (\text{IV.23b}) $$

where we reverted to the $\boldsymbol{v}$-notation since $\boldsymbol{p} = mv$ is not well-defined anymore<fnref n="20" /> which further extends the “no-particle” model discussed in subsec. III E. Had we set the extension parameters to zero we would find vanishing forms and thus no particle trajectories, as we had observed before in section III C. However keeping the terms induced by the central extension, we do find “trajectories”: the equations of motion become trivial and yield fixed positions<fnref n="21" />,

$$ (x^i)' = 0, \quad (p_i)' = 0 \quad \Rightarrow \quad \boldsymbol{x}(s) = \boldsymbol{x}_0 = \text{const.}, \quad \boldsymbol{p}(s) = \boldsymbol{p}_0 = \text{const.} \qquad (\text{IV.24}) $$

**Theorem IV.3.** A “purely exotic” free massless doubly-extended Carroll particle (IV.23) which has $\boldsymbol{\pi} = 0$ does not move: the double extension eliminates the free motions we studied in sec. III C.

This no-motion conclusion will be modified when the particle is coupled to external fields, see the section below.

<footnote n="20">²⁰ The same result is obtained alternatively by putting $\boldsymbol{p} = mv$ in (IV.8a) and then letting $m \rightarrow 0$.</footnote>

<footnote n="21">²¹ The vector $\boldsymbol{u}$ in the Fermat model of sec. III C is now ill-defined: null vectors have no direction.</footnote>
shows that a free massive Carroll particle has an identically zero Hamiltonian. The corresponding Hamilton equations read as follows:

$$ (x^i)' = \{x^i, \mathcal{H}_0\} = 0, \quad p_i' = \{p_i, \mathcal{H}_0\} = 0, \qquad (IV.6) $$

where the prime, {$\cdot$}' = d/ds, denotes a derivative with respect to Carrollian time. Equations (IV.6) imply therefore:

**Theorem IV.1.** *A free massive Carroll particle does not move,*

$$ \mathbf{x}(s) = \mathbf{x}_0, \quad \mathbf{p}(s) = \mathbf{p}_0, \qquad (IV.7) $$

cf. (A.11) in [7].

A free massive particle is Carroll-symmetric by construction and is also manifest by looking at (IV.3). The associated conserved quantities (Souriau’s “moment” [77]) in $d = 2$ spatial dimensions are as follows:

$$ \ell = \mathbf{x} \times \mathbf{p} \qquad \text{angular momentum,} \qquad (IV.8a) $$

$$ \pi = \mathbf{p} \qquad \text{linear momentum,} \qquad (IV.8b) $$

$$ \mathbf{D}^{\text{Car}} = m\mathbf{x} \qquad \text{C-boost momentum.} \qquad (IV.8c) $$

Comparison with the Galilean boosts in (III.3) shows that the difference comes, once again, from the decoupling of position and momentum. These expressions will later be generalized (VI.4). The superscript {$\cdot$}^{Car} will henceforth be dropped and we’ll write simply $\mathbf{D}^{\text{Car}} \equiv \mathbf{D}$.

**Corollary IV.1.** *The conservation of the C-boost momentum D in (IV.8c) implies, consistently with Theorem IV.1, that a massive particle with Carroll boost symmetry can not move.*

**Lagrangian formulation.** The Carrollian dynamics can also be described by a Lagrangian. In the free massive case, replacing $\mathbf{p}$ by $\mathbf{v} = \mathbf{p}/m$ and viewing $\mathbf{v}$ and $\mathbf{x}'$ as independent variables, the Euler-Lagrange equation of

$$ \mathcal{L}_0 = m\mathbf{v} \cdot \mathbf{x}', \qquad (IV.9) $$

confirm the “no motion” conclusion (IV.7): $\mathbf{v}$ is indeed a Lagrange multiplier, yielding the free equation of motion $\mathbf{x}' = 0$ in (IV.6), supplemented with $\mathbf{v} = \mathbf{v}_0 = \text{const.}$ We note also
### C. Coupling to a gauge field: eppur si muove

The free model can be coupled to an em field by the rule (II.2). The external and “internal” magnetic fields combine into the generalized magnetic field:

$$B^* = eB + \kappa_{\text{mag}}, \qquad (\text{IV.25})$$

and yield non-commuting coordinates and momenta [31],

$$\sigma_{\text{exo}} = dp_i \wedge dx_i + \frac{\theta}{2} \epsilon_{ij} dp^i \wedge dp^j + \frac{1}{2} B^* \epsilon_{ij} dx^i \wedge dx^j + eE_i dx_i \wedge ds, \qquad (\text{IV.26a})$$

$$\{x_i, x_j\} = \frac{m}{m^*} \theta \epsilon_{ij}, \qquad \{x_i, p_j\} = \frac{m}{m^*} \delta_{ij}, \qquad \{p_i, p_j\} = \frac{m}{m^*} B^* \epsilon_{ij}, \qquad (\text{IV.26b})$$

where the Carrollian effective mass which generalizes (IV.5) now involves the combined field (IV.25),

$$m^* = m(1 - \theta B^*), \qquad (\text{IV.27})$$

assumed not to vanish$^{22}$. For a static scalar potential $V = V(x)$, $E = -\nabla V$, for example, the Hamilton equations are [31]),

$$(x^i)' = - \frac{\theta}{1 - \theta B^*} \epsilon^{ij} eE_j, \qquad (\text{IV.28a})$$

$$(p_i)' = \frac{1}{1 - \theta B^*} eE_i. \qquad (\text{IV.28b})$$

The position $x$ splits off from $p$ which is coupled to $x$ through the electric field, — but its dynamics does not affect the motion of $x$.

These equations are reminiscent of but different from their Galilean counterparts in (II.17): $p_i$ is missing from the “velocity” relation (IV.28a) which is purely anomalous, whereas the Lorentz force is missing from the purely electric $p$-equation (IV.28b). The external magnetic field is hidden in $m^*$ through $B^*$.

**Theorem IV.4.** The position coordinate $x$ of a massive doubly-extended Carroll particle in a static EM field with non-vanishing effective mass, $m^* \neq 0$, moves, namely by following the (anomalous) Hall law, (IV.28a).

However combining the equations (IV.28) implies,

<footnote>$^{22}$ Compare with (II.12), (II.14) and (II.16) in the Galilean case.</footnote>
en passant that eqn. (IV.9) confirms the C-boost symmetry: it is enough to consider the Hamiltonian action

$$ \mathcal{S} = \int \mathcal{L}_0 ds = \int m\mathbf{v} \cdot d\mathbf{x}, \qquad (IV.10) $$

which does not involve $s$ and is therefore trivially invariant under C-boosts (I.1). The Noether moment is $\mathbf{D}$ in (IV.8c).

Coupling a Carroll particle to an electromagnetic (EM) field by the rule (II.48) amounts to

$$ m\mathbf{v} \rightarrow m\mathbf{v} - e\mathbf{A} \quad \text{and} \quad 0 \equiv \mathcal{H}_0 \rightarrow \mathcal{H}_{em} = eV \qquad (IV.11) $$

which yields, in Lagrangian form,

$$ \mathcal{L}_{em} = (m\mathbf{v} - e\mathbf{A}) \cdot \mathbf{x}' - eV. \qquad (IV.12) $$

Variation w.r.t. $\mathbf{p}$ and $\mathbf{x}$ viewed as independent variables implies the equations of motion (3.21a-b) of Ref. [34],

$$ \mathbf{x}' = 0, \qquad \mathbf{v}' = \frac{e}{m}\mathbf{E}. \qquad (IV.13) $$

Note that the magnetic field is not involved here. To understand how this comes about, note that the variational equations are,

$$ (x^i)' = 0 \quad \text{and} \quad mv_i' = eE_i + eB\epsilon_{ij}(x^j)'. \qquad (IV.14) $$

The first equation here switches off the Lorentz-force in the second one: “no motion” implies “no Lorentz force”, allowing us to generalise Theorem IV.1:

**Theorem IV.2.** A massive charged Carroll particle in an electromagnetic background field does not move: $\mathbf{x} = \mathbf{x}_0 = \text{const.}$

This statement will be modified for centrally extended Carroll particles, see Theorem VI.4.

We notice that while the presence of the electric field $\mathbf{E} \neq 0$ changes $\mathbf{v}$, this does not affect the position, $\mathbf{x}$, in its evolution w.r.t. Carrollian time $s$.

For completeness we record also the Souriau form obtained by the rule (II.48),

$$ \sigma_{em} = \sigma_0 + e\mathbb{F} = m\mathbf{d}\mathbf{v} \wedge d\mathbf{x} + \frac{e}{2}F_{ij}dx^i \wedge dx^j + eE_i dx^i \wedge ds, \qquad (IV.15) $$

whose kernel yields again (IV.13)$^{25}$.

<footnote>$^{25}$ We also get $\mathbf{E} \cdot \mathbf{x}' = 0$ i.e., that the velocity is perpendicular to the electric field. However this is reduced to an identity because of the first relation in (IV.13). Spinning Carroll equations in $d = 3$ spatial dimensions were presented in [34].</footnote>
**Theorem IV.5.** The guiding center $Q^i = x^i + \theta \epsilon^{ij} p_j$ in (IV.14a) of a doubly-extended particle with $m^* \neq 0$, coupled to an electromagnetic field does not move,

$$Q'_i = 0 \Rightarrow \mathbf{Q} = \mathbf{Q}_0 = \text{const.} \qquad (\text{IV.29})$$

When $m^* \neq 0$ the real-space motion is recovered by solving the $p$ equation and using (IV.14a),

$$x^i(s) = Q_0^i - \theta \epsilon^{ij} p_j(s). \qquad (\text{IV.30})$$

For constant EM fields s.t. $m^* \neq 0$, for example, the motion is perpendicular to the electric field,

$$x^i(s) = Q_0^i - \epsilon^{ij} \left( \frac{e\theta}{1 - \theta B^*} \right) E_j s. \qquad (\text{IV.31})$$

For vanishing effective mass, $m^* = 0$ the $x$-motion can not be recovered as said before (and consistently with (IV.28) and (IV.31)).

Another example is obtained by a radial electric field $\mathbf{E} = f(|\mathbf{x}|) \hat{\mathbf{x}}$, where $f(|\mathbf{x}|)$ is a function of the radius in the $d=2$ plane. Multiplying (IV.28a) by $\hat{\mathbf{x}}$ shows that the motion which is circular. In terms of the complex variables $\zeta = x^1 + ix^2$ and $\Pi = p_1 + ip_2$,

$$\zeta(s) = |\mathbf{x}_0| \exp[-i\Omega s] \quad \text{where} \quad \Omega = \frac{e\theta}{1 - \theta B^* (|\mathbf{x}_0|)} \frac{f(|\mathbf{x}_0|)}{|\mathbf{x}_0|}, \qquad (\text{IV.32a})$$

$$\Pi(s) = i \frac{|\mathbf{x}_0|}{\theta} \exp[-i\Omega s] + \Pi_0 = \frac{i}{\theta} \zeta(s) + \Pi_0. \qquad (\text{IV.32b})$$

Thus the momentum rotates around $\Pi_0 = \text{const.}$ (which can be absorbed into the guiding center) with the same angular velocity $\Omega$ as $\zeta$, but with a phase advance by $\pi/2$.

For $m^* = 0$ but $\mathbf{E} \neq 0$ the eqns (IV.28) are ill-defined, however their guiding center, i.e. their combination $\mathbf{Q}$ in (IV.14a) remains fixed when $m^*$ changes sign by sweeping through zero,

$$\frac{m^*}{m} (Q^i)' = \frac{m^*}{m} ((x^i)' + \theta \epsilon^{ij} p_j') = 0 \qquad (\text{IV.33})$$

by (IV.29), including at $m^* = 0$. See Fig. 3 for an insight into the “phase transition”. The external forces vary the frequency $\Omega = \Omega(|\mathbf{x}|)$ but the trajectories remain circular with fixed radius $|\mathbf{x}_0|/\theta$ for all values of $m^*$, cf. [79].
Immobility and Carroll boost symmetry

The relation between immobility and Carroll boost symmetry emphasized in Corollary IV.1 can also be understood as follows. Let us consider a massive particle and assume that it is invariant under Carroll boosts and time translations only. Let us emphasize that since we do not consider rotation and spatial translation symmetries, the model considered here is more general than that of usual Carroll particles. This particle is described by a 2d + 1 dimensional evolution space $\mathcal{E} = \{(x, v, s)\}$. Its Cartan 1-form is,

$$ \varpi = A_i(x, v, s)dx^i + B_i(x, v, s)dv^i - \mathcal{H}(x, v, s)ds, \qquad (IV.16) $$

which is the most general 1-form on $\mathcal{E}$. Here $\mathcal{H}$ is the Hamiltonian.

The equations of motion for this most-general dynamical system are then given by,

$$ \left\{ \begin{aligned} \frac{dx^j}{ds}(\partial_j A_i - \partial_i A_j) + \frac{dv^j}{ds}(\partial_{v_j} A_i - \partial_i B_j) &= -\partial_i \mathcal{H} - \partial_s A_i \\ \frac{dx^j}{ds}(\partial_{v_i} A_j - \partial_j B_i) - \frac{dv^j}{ds}(\partial_{v_j} B_i - \partial_{v_i} B_j) &= \partial_{v_i} \mathcal{H} + \partial_s B_i \end{aligned} \right. \qquad (IV.17) $$

The next step is to require Carroll boost-invariance and time translation invariance. This requires the Cartan 1-form obeys $L_X \varpi = df$ (cf. II.1), for

$$ X = \beta_i \partial_{v_i} - \beta_i x^i \partial_s, \qquad (IV.18) $$

for some function $f$, implying $dL_X \varpi = 0$ (cf. II.1). Likewise for the time symmetry, $dL_T \varpi = 0$, with $T = \partial_s$.
<figure><img src="image_4.png" /><figcaption>FIG. 3: Trajectories for unit charge and exotic parameter in a radial electric field $\mathbf{E} = \mathbf{x}/|\mathbf{x}|^3$ in the plane for effective masses passing from negative to positive, unfolded in Carroll time s. The rotation around the fixed guiding center $\mathbf{Q}(s) = \mathbf{Q}_0$ speeds up when the effective mass increases from $m^* < 0$. The motion is instantaneous for $m^* = 0$ and changes orientation for $m^* > 0$.</figcaption></figure>

### D. Extension to vanishing Carroll mass, $m = 0$

The above results can be extended from massive to massless particles, i.e., such that their Carroll mass vanishes, $m = 0$. The clue is to rewrite (IV.28) for $m \neq 0$ as,

$$ (x^i)' = \frac{1}{B^* - m^2/\kappa_{\text{exo}}} \epsilon^{ij} eE_j, \quad p_i' = m^2 \frac{eE_i}{m^2 - \kappa_{\text{exo}}B^*}. \qquad (\text{IV.34}) $$

Then letting $m \to 0$ we get:

**Theorem IV.6.** A massless, charged doubly-extended Carroll particle in a combined electric and effective magnetic field $B^* = eB + \kappa_{\text{mag}}$ in (IV.25) moves according to the generalized Hall law,

$$ (x^i)' = \epsilon^{ij} \frac{eE_j}{B^*} \quad \text{and} \quad p_i' = 0. \qquad (\text{IV.35}) $$

We record for later use that the Hall law above can be derived from the first-order Lagrangian resp. of Souriau form,

$$ \mathcal{L}_{2+1} = \frac{1}{2} B^* \epsilon_{ij} x^i (x^j)' - eV(\mathbf{x}), \qquad (\text{IV.36a}) $$

$$ \sigma_{2+1} = \frac{1}{2} B^* \epsilon_{ij} dx^i \wedge dx^j + eE_i dx^i \wedge ds. \qquad (\text{IV.36b}) $$

Returning to the Souriau framework, coupling the free massless particle with “purely exotic dynamics”, (IV.23), to an external EM field by the rule (II.2) yields,

$$ \sigma_{m=0} = \underbrace{\frac{1}{2} \kappa_{\text{exo}} \epsilon_{ij} dv^i \wedge dv^j + \frac{1}{2} B^* \epsilon_{ij} dx^i \wedge dx^j}_{\Omega_{m=0}} + eE_i dx^i \wedge ds. \qquad (\text{IV.37}) $$
These symmetry considerations require the following conditions,

$$ \beta_k \partial_{v_k} (\partial_j A_i - \partial_i A_j) + \beta_i (\partial_s A_j + \partial_j \mathcal{H}) - \beta_j (\partial_s A_i + \partial_i \mathcal{H}) = 0 \qquad (IV.19a) $$

$$ \partial_s (\partial_j A_i - \partial_i A_j) = 0 \qquad (IV.19b) $$

$$ \beta_k \partial_{v_k} (\partial_{v_j} A_i - \partial_i B_j) + \beta_i (\partial_s B_j + \partial_{v_j} \mathcal{H}) = 0 \qquad (IV.19c) $$

$$ \partial_s (\partial_{v_j} A_i - \partial_i B_j) = 0 \qquad (IV.19d) $$

$$ \beta_k \partial_{v_k} (\partial_{v_j} B_i - \partial_i B_j) = 0 \qquad (IV.19e) $$

$$ \partial_s (\partial_{v_j} B_i - \partial_i B_j) = 0 \qquad (IV.19f) $$

$$ \beta_k \partial_{v_k} (\partial_s A_i + \partial_i \mathcal{H}) = 0 \qquad (IV.19g) $$

$$ \partial_s (\partial_s A_i + \partial_i \mathcal{H}) = 0 \qquad (IV.19h) $$

$$ \beta_k \partial_{v_k} (\partial_s B_i + \partial_i \mathcal{H}) = 0 \qquad (IV.19i) $$

$$ \partial_s (\partial_s B_i + \partial_{v_j} \mathcal{H}) = 0 \qquad (IV.19j) $$

At this point, let us consider two possibilities. First, we will consider Cartan 1-forms/Lagrangians which do not depend on acceleration (i.e. $dv^i$) which means that $B_i = 0$. In a latter case, we will consider general systems.

a. Acceleration-independent Lagrangians The first natural set of dynamical systems to look at are those whose Lagrangian/Cartan 1-form does not depend on the acceleration, i.e. $B_i = 0$. In that case, the equations of motion and the constraints obtained above simply considerably. In particular, the symmetries requirements imply that $\partial_s \partial_{v_j} A_i = 0$, which in turns means that $\partial_{v_k} \partial_i H = 0$. With the other equations, one finds that $\partial_{v_i} H$ is a constant.

Hence, looking at the second equation of motion (IV.17), for systems with accelerations-independent Lagrangians, non-trivial motions require the Hamiltonian to be of the form $H = c_i v^i + g(x^i)$, for some constants $c_i$ and a function $g$. The second equation of motion is thus of the form,

$$ \frac{dx^j}{ds} \partial_{v_i} A_j = c_n. \qquad (IV.20) $$

Note that since we want the system to be regular (in the sense that there must be a unique solution to the equations of motion), we have $\partial_{v_i} A_j \neq 0$. This term however, cannot depend on $s$, must be so that $\partial_i \partial_{v_k} A_j = \partial_j \partial_{v_k} A_i$, and must be so that $c_i = -\frac{1}{n} \partial_{v_j} \partial_{v_i} A_j$ per (IV.19c).

Solving all the above conditions, the most general acceleration independent Cartan 1-form invariant under Carroll boosts and time translations, with non-trivial motion is described
Viewed in another way, rewriting the Souriau form $\sigma_{\text{exo}}$ (IV.26a) in terms of $v = p/m$ and then letting $m \to 0$ suppresses the first term and yields once again (IV.37). The determinant of the symplectic form $\Omega_{m=0}$ in (IV.37) is,

$$ \det (\Omega_{m=0}) = (\kappa_{\text{exo}})^2 (B^*)^2. \qquad (\text{IV.38}) $$

Thus our previous conditions $\kappa_{\text{exo}} \neq 0$ and $B^* \neq 0$ guarantee that $\Omega_{m=0}$ is regular. However the dynamics trivially projects to Carroll spacetime $\{(x, s)\}$: the first term in (IV.37) is switched off, leaving us with (IV.36) and “forgetting” about the “irrelevant” dynamics of $p$.

What can be the physical realization of these particles? Rigorously speaking, there are no strictly massless charged fields in the Standard Model of fundamental interactions in its low-temperature phase, which is relevant to present-day phenomenology [97]. While genuinely massive constituents of the model can be treated as effectively massless particles at very high energies (such as leptons and quarks in heavy ion collisions [98]), they propagate in three spatial dimensions and, therefore, are not relevant for the peculiar features of two-dimensional physics considered in this section.

However, suitable examples do exist in the condensed matter context: massless charged particles in two spatial dimensions were realized in graphene, a two-dimensional crystal made of carbon atoms, discovered experimentally in 2004 [99]. In this material, the dynamics of the low-energy electronic excitations is described by the $2+1$ massless Dirac Hamiltonian,

$$ \mathcal{H} = \bar{\psi}i\boldsymbol{\alpha}(\boldsymbol{\nabla} - i\boldsymbol{e}\boldsymbol{A})\psi + eV, \qquad (\text{IV.39}) $$

where $\boldsymbol{\alpha} = (\gamma^0\gamma^1, \gamma^0\gamma^2)$ and $\gamma^\mu$ are spatial Dirac matrices in $2+1$ dimensions. Being charged, the massless electronic quasiparticles interact with the background electromagnetic field $A^\mu = (V, \boldsymbol{A})$, thus providing us with a system that exhibits equivalences with quantum electrodynamics [100].

In the absence of background electromagnetic fields, the Dirac particle described by the Hamiltonian (IV.39) possesses the 'relativistic' Dirac particle-like dispersion, $\varepsilon_p = v_F|p|$, where $p = (p_1, p_2)$ corresponds to a quasimomentum of the particle with the origin at the tip of a Dirac cone. The particles propagate relativistically with the Fermi velocity, $v_F$, which is typically hundreds of times smaller than the speed of light.

The massless nature of the excitations free-standing graphene is protected by the symmetries of the underlying carbon crystal lattice, thus making the massless nature ($m=0$)
by,

$$ \bar{\omega} = \left( \alpha v_i - c_k v^k v_i + \frac{1}{2} v^2 c_i + \sum_i f_i + \gamma_i s \right) dx^i - (c_k v^k + g(x^i)) ds \quad (IV.21) $$

for $\alpha, \gamma_i$ constants, and the functions $f_i = f_i(v_k - v_i, x_j)$, with $k \neq i$, e.g. for $n = 3$, $f_1 = f_1(v_2 - v_1, v_3 - v_1, x_j)$, satisfying $\partial_i \partial_{v_k} f_j = \partial_j \partial_{v_k} f_i$. At least one of the $c_i$ must be non-zero in order for the system to move.

Assuming that the functions $f_i$ vanish, the equations of motion are,

$$ \left\{ \begin{array}{l} \left( (\alpha - c_k v^k) \delta_{ij} + c_i v_j - c_j v_i \right) \frac{dv^j}{ds} = -\partial_i g(x^j) - \gamma_i \\ \\ \left( (\alpha - c_k v^k) \delta_{ij} + c_i v_j - c_j v_i \right) \frac{dx^j}{ds} = c_i \end{array} \right. \quad (IV.22) $$

For standard Carroll dynamics (IV.3a), which features no motion, $\alpha = 1$, $c_i = 0$, $\gamma_i = 0$, and $f_i = 0$.

If one asks at this point that the Lagrangian must not depend on terms higher than the square of the velocity, which is the case for most physical systems, then $c_i = 0$<fnref n="26" /> and the system does not move.

**Theorem IV.3.** A regular massive dynamical system, described by an acceleration-independent Lagrangian, containing no higher terms than velocity square ones, invariant under Carroll boosts and time translations do not move.

Note that the equations restricting drastically the form of the Hamiltonian, and forbidden motion for “physical” systems, were $\beta_k \partial_{v_k} \partial_i \mathcal{H} = 0$, and $\beta_k \partial_{v_k} \partial_{v_i} \mathcal{H} = 0$. If one of the boost symmetry were to be broken, for instance in $d = 2$, $\beta = (\beta_x, 0)$, the Hamiltonian is only constrained to be $H = c_x v_x + g(v_y, x, y)$. The arbitrary dependency on $v_y$ will then allow non-trivial motion. Hence, for “reasonable physical systems”, breaking the Carroll boost invariance, even only partially, is necessary in order to allow motion.

b. Acceleration dependent Lagrangians A more general class of dynamical systems are those whose Lagrangian depend on the acceleration, see e.g. [79, 141] for physical examples. This is actually the case of some systems in this review, due to the “exotic” terms of “extended” dynamical systems in $d = 2$, for instance (III.15) for Galilean systems, and later with (VI.9a) for Carroll systems.

<footnote n="26">²⁶ $v^2 c_i dx^i / ds$ being a third power of velocity term.</footnote>
of these charged excitations ($e \neq 0$) a robust feature of the material [101]. Therefore, the quasiparticles in graphene can be viewed as a charged, doubly-extended Carroll particle in the massless limit described in this section.

## E. Carrollian anyons

Carroll particles with anyonic spin, $\chi \neq 0$ in (IV.3), can also be studied along the same lines. The Souriau 2-form $\sigma_{\text{exo}}$ (IV.26a) is generalized by following the rule (II.2),

$$ \sigma_{\text{any}} = dp_i \wedge dx_i + \frac{\theta}{2} \epsilon_{ij} dp^i \wedge dp^j + \frac{1}{2} eB^* \epsilon_{ij} dx^i \wedge dx^j + E_i^* dx^i \wedge ds, \qquad (\text{IV.40}) $$

where the electric field and the gradient of the magnetic field are again combined (as in (II.23) in the Galilean theory), into an effective electric term $^{23}$,

$$ E_i^* = eE_i + \mu\chi\partial_i B. \qquad (\text{IV.41}) $$

The equations of motion take the form (IV.28) with $e\mathbf{E}$ replaced by $\mathbf{E}^*$.

The extension to zero mass is obtained along the same lines as above:

**Theorem IV.7.** Both massive and massless Carrollian anyons move by following the Hall law (I.2) but with the electric and magnetic fields replaced by the effective values $\mathbf{E}^*$ and $\mathbf{B}^*$, (IV.41) and (IV.25), respectively,

$$ (x^i)' = \epsilon^{ij} \left( \frac{eE_j + \mu\chi\partial_j B}{eB + \kappa_{\text{mag}}} \right) = \epsilon^{ij} \left( \frac{E_j^*}{B^*} \right) \quad \text{and} \quad p_i' = 0. \qquad (\text{IV.42}) $$

Note here the double role played by the magnetic field $B$.

The massless limit can also be discussed in the Souriau framework. We start with (IV.40) with $m \neq 0$ and eliminate $\mathbf{p}$ in favor of $\mathbf{v} = \mathbf{p}/m$. Then letting $m \to 0$ we get a regular Souriau form which, compared to (IV.23b), has an additional “effective electric” term,

$$ \sigma_{\text{any}} = \frac{1}{2}\kappa_{\text{exo}} \epsilon_{ij} dv^i \wedge dv^j + \frac{1}{2} B^* \epsilon_{ij} dx^i \wedge dx^j + E_i^* dx^i \wedge ds. \qquad (\text{IV.43}) $$

The kernel of $\sigma_{\text{any}}$ yields the equation of motion (IV.35) [with $p_i$ replaced by $v_i$].

<footnote>$^{23}$ A constant shift of the magnetic field, $B \to B + \text{const.}$, produces no effect since the spin coupling involves only the gradient of $B$.</footnote>
For systems with acceleration, i.e. when $B_i \neq 0$, the constraints (IV.19a) imposed by the Carroll boost symmetry and the time translation symmetry are not enough to restrict motion in a meaningful manner. We get that the right hand sides of the equations of motion (IV.17) do not depend on the velocity nor on time, but they may still depend on the position.

As a conclusion, we see that “physically reasonable” dynamical systems, which do not have acceleration terms, or very specific high order velocity terms, in their Lagrangian, and which are invariant under Carroll boosts (and time translation), cannot move. There are two ways to find motion for these systems:

* Include acceleration terms in their Lagrangian description, which is effectively what we will study in sections VI, IX.

* Or, consider systems with only partial Carroll boost symmetry, which is what we will study in some following sections, for instance section IV D, and also VI C.

Things are even more subtle for massless particles, as it will be explained in sec. IV C.

## B. Multiparticle Carrollian systems

Now we discuss briefly multiparticle systems, which are discussed in [10, 142, 143]. Consider first a collection of $N$ isolated free massive Carroll particles in $d$ dimensions which do not interact between themselves. We described them by the generalization of (IV.10),

$$ \mathcal{S}_0 = \int \left( \sum_{a=1}^{N} m_a \boldsymbol{v}_a \cdot \boldsymbol{x}'_a \right) ds = \int \sum_{a=1}^{N} m_a \boldsymbol{v}_a \cdot d\boldsymbol{x}_a. \qquad (IV.23) $$

Each $\boldsymbol{v}_a$ here can be viewed as an independent Lagrange multiplier, and therefore we get decoupled equations of motion

$$ m_a \boldsymbol{x}'_a = 0 \Rightarrow \boldsymbol{x}_a = (\boldsymbol{x}_a)_0 \quad \text{and} \quad \boldsymbol{v}'_a = 0 \Rightarrow \boldsymbol{v}_a = (\boldsymbol{v}_a)_0, \qquad (IV.24) $$

$a = 1, \dots, N$. Recalling that C-boosts (I.1) act on Carroll time $s$, but not on $\boldsymbol{x}$ implies the $s$-independence of the action (IV.23), from which we infer the C-boost symmetry of the system. The associated conserved boost momentum is the sum of the individual expressions,
<table><caption>TABLE II: Motion of doubly-extended Carroll particles in $d = 2$ spatial dimensions.</caption><thead><tr><th></th><th>free extended Carroll</th><th>in EM field</th><th>in curved metric</th></tr></thead><tbody><tr><td><i>m</i> ≠ 0</td><td>no motion</td><td>generalized Hall</td><td>no motion</td></tr><tr><td><i>m</i> = 0, <i>e</i> ≠ 0</td><td>no motion</td><td>generalized Hall</td><td>no motion</td></tr><tr><td><i>m</i> = 0, <i>e</i> = 0, μχ ≠ 0</td><td>no motion</td><td>generalized Hall</td><td>no motion</td></tr><tr><td><i>m</i>* = 0</td><td>Q fixed, <i>x</i> motion undetermined</td><td></td><td></td></tr></tbody></table>

The generalized Hall law (IV.42) contains an “anomalous” anyonic contribution activated by the inhomogeneities of the magnetic field (IV.41). For massive particles, the anyonic term can be interpreted as a conventional Zeeman force

$$ \mathbf{F} = g\mu_B s \nabla B, \qquad (IV.44) $$

exerted by the spatial gradient of the transverse magnetic field $B \equiv B_z$ on the particle propagating in the plane. Here $g$ is the Landé factor, and $\mu_B$ is the Bohr magneton. A comparison of Eqs. (IV.41) and (IV.44) provide us with the identification $e\mu\chi \equiv g\mu_B s$.

In spintronic devices, the electrons with different orientations of their spins $s \equiv s_z$ are split by the Zeeman force (IV.44). This phenomenon lies in the heart of a mesoscopic Stern-Gerlach effect which provides an efficient spin filter for a current traversing a specially fabricated device that incorporates an inhomogeneous magnetic field $B = B_z$ [102].

Thus, two-dimensional Carrollian anyons can be interpreted as spin-polarized charged particles with a nonzero Landé factor.

## F. A toy model

Let us recall that an “exotic photon” studied in Ref. [32] is defined by being uncharged, carry magnetic moment, anyonic spin, and have non-vanishing “exotic charges”,

$$ e = 0, \quad \mu \neq 0, \quad \chi \neq 0, \quad \kappa_{\text{exo}} \neq 0, \quad \kappa_{\text{mag}} \neq 0, \qquad (IV.45) $$

which corresponds to the Carroll Hamiltonian of a spinning anyon,

$$ \mathcal{H} = -\mu\chi B. \qquad (IV.46) $$

cf. (II.21). Its motion on the horizon of a black hole will be revisited in Sec. VI.