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# Two interacting conformal Carroll particles

Roberto Casalbuonia<EMAIL_ADDRESS>Daniele Dominicia)
<EMAIL_ADDRESS>Joaquim Gomisb<EMAIL_ADDRESS>a)Department of
Physics and Astronomy, University of Florence and INFN, 50019 Florence, Italy
b)Departament de Física Quàntica i Astrofísica
and Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona, Martí i
Franquès 1, 08028 Barcelona, Spain

###### Abstract

In this note we analyse two different models of two interacting conformal
Carroll particles that can be obtained as the Carrollian limit of two
relativistic conformal particles. The first model describes particles with
zero velocity and exhibits infinite dimensional symmetries which are
reminiscent of the BMS symmetries. A second model of interaction of Carrollian
particles is proposed, where the particles have non zero velocity and
therefore, as a consequence of the limit $c\to 0$, are tachyons. Infinite
dimensional symmetries are present also in this model.

###### pacs:

02.20.-a; 02.20.Tw; 03.30.+p

##  I Introduction

Carroll symmetry, which was introduced by LevyLeblond:1965 ; gupta1966analogue
as the limit of the Poincaré symmetry where the velocity of light is going to
zero, $c\to 0$, has recently received a lot of attention, mainly in connection
with its relation with the BMS group Bondi:1962px ; Sachs:1962zza . Some
applications of Carrollian physics allow, for example, to understand the
symmetries of null hypersurfaces, such as black-hole horizons Donnay:2019jiz ,
boundaries of asymptotically flat spacetimes Duval:2014uva , Carroll fluids
Ciambelli:2018wre . For an introductory review of certain aspects of Non-
Lorentzian theories, see Bergshoeff:2022eog .

In particular, in this paper we are interested to have a dynamical
understanding of the relation between the conformal Carroll structures
Duval:2014uva  and the symmetries of Carrollian particles.

A (massive) non-conformal Carroll particle was introduced as Carroll limit of
relativistic massive particle Bergshoeff:2014jla ; it can also be obtained
from a coadjoint orbit of the Carroll group Duval:2014uoa . The (massless)
Carroll particle was also considered in Bergshoeff:2014jla . A model of two
non-conformal interacting particles was also considered where the individual
particles move but the center of mass does not move.

The symmetry of massive and massless free Carroll particles is infinite
dimensional. In the massless case it contains the finite conformal Carroll
symmetry introduced in Bagchi:2016bcd , see also Chen:2021xkw . In the case of
the two non-conformal particles the symmetry algebra is finite dimensional
Bergshoeff:2014jla .

The causal structure of the Carroll space allows space-like intervals and
therefore the existence of Carroll tachyons. In deBoer:2021jej  a model is
constructed where a tachyonic Carroll particle with zero energy moves.

In this paper we propose two models for two conformal Carroll particles as
Carrollian limits of two conformal relativistic particles introduced in
Casalbuoni:2014ofa .

The first one describes particles which do not move and exhibits an infinite
dimensional symmetry algebra. Any free conformal Carroll particle has
vanishing energy, therefore two free Carrollian particles have total energy
equal to zero. The nice feature of this model is that the system can have a
total energy different from zero and still be Carroll conformal invariant.
Clearly the energy, which turns out to be constant, depends on the
interaction. We also study some possible connections with the infinite
dimensional conformal symmetry of Duval:2014uva  and the BMS symmetry.

In the second model the particles move, their individual energy is zero and
consequently they are Carrollian tachyons. Also in this case the symmetry
turns out to be infinite dimensional.

##  II Conformal Carroll particle

The canonical form of the action of a massless relativistic particle is given
by (we assume the signature of the metric $(+,-,-,-)$)

$S=\int d\tau[-p\cdot\dot{x}-\frac{e}{2}p^{2}]\,,$ (1)

where the dot denotes the differentiation with respect to an invariant
parameter $\tau$. The action is also invariant under the conformal group that
contains dilatations and special conformal transformations (SCT). The
infinitesimal transformations are given by the ordinary Poincaré
transformations, plus dilatations

$\delta x^{\mu}=\epsilon_{D}x^{\mu},\,\,\,\,\delta
p^{\mu}=-\epsilon_{D}p^{\mu},\,\,\,\delta e=2e\epsilon_{D}.$ (2)

and SCT:

$\delta x^{\mu}=b^{\mu}x^{2}-2(b\cdot x)x^{\mu},\,\,\,\,\delta
p^{\mu}=-2b\cdot px^{\mu}+2b\cdot xp^{\mu}+2p\cdot xb^{\mu},$ (3) $\delta
e=-4eb\cdot x.$ (4)

We next consider the Carroll limit which is given by

$x^{0}=\frac{t}{\omega},\quad p_{0}=\omega E,$ (5)

with $\omega\to\infty$. It is understood that, before taking the limit, we
rescale the einbein variable like

$\displaystyle e\to\frac{e}{\omega^{2}}.$ (6)

Note that we use a dimensionless parameter $\omega$ instead of the velocity of
light. Indeed, if one considers the Carrollian counterpart of the non-
relativistic limit of a string, one needs to use a dimensionless parameter
Gomis:2000bd ; Gomis:2005pg ; Barducci:2018wuj . The contractions we consider
in this work correspond to the limit $c\to 0$ of a world probed by particles.
There are more general possible contractions that correspond to the non-
Lorentzian limits of extended objects such as strings and branes.

In the Carroll limit the action (1) becomes

$\displaystyle S_{C}$ $\displaystyle=$ $\displaystyle\int
d\tau\,\big{[}-E\dot{t}+{\bf p}\cdot\dot{{\bf x}}-\frac{e}{2}E^{2}\big{]}\,.$
(7)

From the symplectic form of the canonical action we deduce the following
Poisson brackets

$\\{E,\ t\\}=1,\quad\\{e,\ \pi\\}=1,\quad\\{x^{i},p^{j}\\}=\delta^{ij}\,\quad
i=1,2,3.$ (8)

This action can also be obtained by the method of non-linear realisations
Bergshoeff:2014jla  and coadjoint orbit method Duval:2014uoa . Since the
action (1) is relativistic conformal invariant, the action (7) is also
invariant under the Carroll conformal transformations. We have

$\displaystyle\delta t={\bm{\beta}}\cdot{\bf x}+a_{t}\,,\hskip 56.9055pt\delta
x^{i}=\epsilon^{ijk}\theta^{j}x^{k}+a^{i}\,,$ $\displaystyle\delta
p^{i}=\epsilon^{ijk}\theta^{j}p^{k}+{\beta^{i}}E\,,\hskip 82.51282pt\delta
E=0\,,$ (9)

$\delta t=\epsilon_{D}t,\,\,\,\,\delta{\bf x}=\epsilon_{D}{\bf x},\,\,\,\delta
E=-\epsilon_{D}E,\delta{\bf p}=-\epsilon_{D}{\bf p},\,\,\,\delta
e=2\epsilon_{D}e\,,$ (10) $\delta t=-b^{0}{\bf x}^{2}+2{\bf b}\cdot{\bf
x}\,t,\,\,\,\,\delta{\bf x}=-{\bf b}\,{\bf x}^{2}+2({\bf b}\cdot{\bf x})\,{\bf
x}\,,$ (11) $\delta E=-2{\bf b}\cdot{\bf x}E,\,\,\,\delta{\bf
p}=-2(b^{0}E-{\bf b}\cdot{\bf p}){\bf x}-2{\bf b}\cdot{\bf x}\,{\bf
p}+2(Et-{\bf p}\cdot{\bf x}){\bf b}\,,$ (12) $\delta e=4{\bf b}\cdot{\bf
x}e\,,$ (13)

where $\beta^{i},a_{t},a^{i},\epsilon_{D},b^{0},b^{i}$ are the infinitesimal
parameters associated to Carrollian boost, space-time translations, dilatation
and special Carrollian conformal transformations.

The phase space realization of the conformal Carroll generators is the
following

$H={E}\,,\hskip 28.45274pt{\bf P}={\bf p}\hskip 28.45274pt{\bf G}=E{\bf
x}\,,\hskip 28.45274pt{\bf J}={\bf x}\times{\bf p}\,,$ (14) $D=-Et+{\bf
p}\cdot{\bf x},\quad K^{0}=-E{\bf x}^{2},\quad{\bf K}=2D{\bf x}-{\bf
x}^{2}{{\bf p}}\,.$ (15)

The corresponding transformations are obtained as $\delta=\\{G,\,\,\\}$.

The conformal Carroll algebra in phase space in four dimensions is given by
(see Bagchi:2016bcd , and also Chen:2021xkw )

$\displaystyle\\{E,P^{i}\\}=\\{E,G^{i}\\}=\\{E,J^{i}\\}=\\{E,K^{0}\\}=0\,,\\{E,D\\}=-E\,,\\{E,K^{i}\\}=-2G^{i}\,,$
(16)
$\displaystyle\\{P^{i},P^{j}\\}=0\,,\\{P^{i},G^{j}\\}=-\delta^{ij}E\,,\\{P^{i},J^{j}\\}=\epsilon^{ijk}P^{k}\,,$
$\displaystyle\\{P^{i},D\\}=-P^{i}\,,\\{P^{i},K^{0}\\}=2G^{i}\,,\\{P^{i},K^{j}\\}=2\epsilon^{ijk}J^{k}-2D\delta^{ij}\,\,,$
$\displaystyle\\{G^{i},G^{j}\\}=0\,,\\{G^{i},J^{j},\\}=\epsilon^{ijk}G^{k}\,,\\{G^{i},D\\}=0\,,\\{G^{i},K^{0}\\}=0\,,\\{G^{i},K^{j}\\}=\delta^{ij}K^{0}\,,$
$\displaystyle\\{J^{i},J^{j}\\}=\epsilon^{ijk}J^{k}\,,\\{J^{i},D\\}=0\,,\\{J^{i},K^{0}\\}=0\,,\\{J^{i},K^{j}\\}=\epsilon^{ijk}K^{k}\,,$
$\displaystyle\\{D,K^{0}\\}=-K^{0}\,,\\{D,K^{i}\\}=-K^{i}\,,$
$\displaystyle\\{K^{i},K^{0}\\}=0\,,\\{K^{i},K^{j}\\}=0\,.$

The conformal Carroll algebra can be obtained from the relativistic conformal
algebra with generators $M^{\mu\nu},P^{\mu},K^{\mu},D$ associated to Lorentz,
space translations, SCT and dilatations

$\displaystyle\left[\tilde{M}^{\mu\nu},\tilde{M}^{\rho\sigma}\right]$
$\displaystyle=$ $\displaystyle\eta^{\nu\rho}\;\tilde{M}^{\mu\sigma}+...,$
$\displaystyle\left[\tilde{P}^{\rho},\tilde{K}^{\sigma}\right]$
$\displaystyle=$ $\displaystyle
2\,(\eta^{\rho\sigma}\;\tilde{D}\;+\;\tilde{M}^{\rho\sigma}\;),$
$\displaystyle\left[\tilde{P}^{\mu},\tilde{M}^{\rho\sigma}\right]$
$\displaystyle=$ $\displaystyle\eta^{{\mu}[\rho}\;\tilde{P}^{\sigma]},$
$\displaystyle\left[\tilde{K}^{\mu},\tilde{M}^{\rho\sigma}\right]$
$\displaystyle=$ $\displaystyle\eta^{{\mu}[\rho}\;\tilde{K}^{\sigma]},$
$\displaystyle\left[\tilde{P}^{\rho},\tilde{D}\right]$ $\displaystyle=$
$\displaystyle-\;\tilde{P}^{\rho},$
$\displaystyle\left[\tilde{K}^{\rho},\tilde{D}\right]$ $\displaystyle=$
$\displaystyle\tilde{K}^{\rho},$
$\displaystyle\left[\tilde{P}^{\mu},\tilde{P}^{\nu}\right]$ $\displaystyle=$
$\displaystyle\left[\tilde{K}^{\mu},\tilde{K}^{\nu}\right]=\left[\tilde{D},\tilde{M}^{\mu\nu}\right]=0$
(17)

by the following contractions

$E=\frac{1}{\omega}\,\tilde{P}^{0},\,G^{i}=\frac{1}{\omega}\,\tilde{M}^{i0},\,K^{0}=\frac{1}{\omega}\tilde{K}^{0}\,$
(19)

and the identification

$J^{i}=\frac{1}{2}\epsilon^{ijk}\tilde{M}^{jk}$ (20)

###  II.1 Infinite-dimensional Symmetries

Now we want to analyse which is the most general point transformation of the
Carroll particle action (1) following the steps of Bergshoeff:2014jla .

In order to do that we want to write the Carollian Killing equations. Let us
first consider the equations of motion deduced from (7)

$\dot{t}=-eE,\quad\dot{\bf{x}}=0,\quad\dot{e}=\lambda(\tau),\quad\dot{E}=0,\quad\dot{\bf{p}}=0\,,\quad\dot{\pi}=-1/2E^{2}\,,$
(21)

where $\lambda(\tau)$ is an arbitrary function and $\pi$ is the momentum
associated to the einbein variable which is constrained by $\pi=0$.

Consider the following generator of the canonical transformations

$G=E\xi^{0}({\bf x},t)-{\bf p}\cdot{\bm{\xi}}({\bf x},t)+\gamma({\bf
x},t)\pi,$ (22)

with $\xi^{0}({\bf x},t)\,,\xi^{i}({\bf x},t)$ and $\gamma({\bf x},t)$
arbitrary functions of the space-time coordinates. The transformations
generated by this generator are given by

$\displaystyle\delta t$ $\displaystyle=\\{G,t\\}=$ $\displaystyle\xi^{0}({\bf
x},t),\quad\delta x^{i}=\\{G,x^{i}\\}=\xi^{i}({\bf x},t),\quad\quad\delta
e=\\{e,G\\}=\gamma({\bf x},t)\,,$ $\displaystyle\delta E$
$\displaystyle=\\{G,E\\}=$ $\displaystyle\partial_{t}\xi^{0}({\bf
x},t)E+\partial_{t}\xi^{i}({\bf x},t)p_{i}+\partial_{t}\gamma({\bf
x},t)\pi\,,$ $\displaystyle\delta p^{i}$ $\displaystyle=\\{G,p^{i}\\}=$
$\displaystyle\partial_{i}\xi^{0}({\bf x},t)E+\partial_{i}\xi^{j}({\bf
x},t)p_{j}+\partial_{i}\gamma({\bf x},t)\pi\,.$ (23)

These transformations are symmetries of the free Carroll particle, provided
that $G$ is a constant of motion, i.e., $dG/d\tau=0$.This leads to the
following restriction, after use of Eq. (21) and $\pi$ that is a primary
constraint:

$\displaystyle 0$ $\displaystyle=E\left[\dot{t}\partial_{t}\xi^{0}({\bf
x},t)+\dot{x_{j}}\partial^{j}\xi^{0}({\bf
x},t)\right]+p_{i}\left[\dot{t}\partial_{t}\xi^{i}({\bf
x},t)+\dot{x_{j}}\partial^{j}\xi^{i}({\bf x},t)\right]+\dot{\pi}\gamma({\bf
x},t)$ (24) $\displaystyle=-eE^{2}\partial_{t}\xi^{0}({\bf
x},t)-eEp_{i}\partial_{t}\xi^{i}({\bf x},t)-\frac{1}{2}\gamma({\bf
x},t)E^{2}\,.$

From this equation we deduce the following Killing equations corresponding to
the free Carroll particle Bergshoeff:2014jla :

$\partial_{t}\xi^{0}(t,{\bf
x})=-\frac{\gamma}{2e}\,,\quad\partial_{t}\xi^{i}(t,{\bf x})=0$ (25)

for arbitrary parameter $\gamma({\bf x},t)$. This leads to the following
generator of the conformal Killing transformations

$G=E\xi^{0}(t,{\bf x})+p_{i}\xi^{i}({\bf x})-2\pi e\partial_{t}\xi^{0}(t,{\bf
x})\,,$ (26)

which generates an infinite dimensional symmetry, which will be called ${\cal
G}_{1}$. These transformations include Carroll conformal transformations.
However they are more general. The dependence of $\xi^{0}(t,{\bf x})$ on the
parameter $t$ is arbitrary while in BMS is linear Duval:2014uva . This
arbitrary dependence on $t$ is reminiscent of the Newman-Unti group, which
contains BMS which is isomorphic to the infinite extension of the Carroll
conformal group Duval:2014uva .

##  III Two conformal relativistic particles

In this section we briefly recall the most relevant features of a model of two
interacting conformal particles proposed in Casalbuoni:2014ofa . The canonical
action is formulated, in terms of the coordinates $x_{1},x_{2}$, the momenta
$p_{1},p_{2}$, the einbeins $e_{1},e_{2}$ and the associated momenta
$\pi_{1},\pi_{2}$ that are primary constraints, by

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d\tau\left(-p_{1}{{\dot{x}}}_{1}-p_{2}\cdot{\dot{x}}_{2}+\pi_{1}{\dot{e}_{1}}+\pi_{2}{\dot{e}_{2}}-H\right)\,,$
(27)

where the Hamiltonian $H$ is given by

$\displaystyle H$ $\displaystyle=$ $\displaystyle-
e_{1}\phi_{1}-e_{2}\phi_{2}-\pi_{1}\mu_{1}-\pi_{2}\mu_{2}\,,$ (28)

$\mu_{1},\mu_{2}$ are arbitrary functions of $\tau$ and the mass-shells
constraints are

$\displaystyle\phi_{1}=\frac{1}{2}\left(p_{1}^{2}-\frac{\alpha^{2}}{4}\sqrt{\frac{e_{2}}{e_{1}}}\frac{1}{r^{2}}\right),$
$\displaystyle\phi_{2}=\frac{1}{2}\left(p_{2}^{2}-\frac{\alpha^{2}}{4}\sqrt{\frac{e_{1}}{e_{2}}}\frac{1}{r^{2}}\right)\,$
(29)

where $r=x_{1}-x_{2}$ it is relative space-time coordinate. Note that when the
interaction, which is regulated by the parameter $\alpha$, vanishes the model
describes two massless free relativistic particles. Notice also that the model
is conformal invariant with effective masses different from zero, a property
that is not possible for a single conformal particle.

The equations of motion are given by

$\dot{x}_{a}^{\mu}=\\{x^{\mu}_{a},H\\}=e_{a}p_{a}^{\mu},\,\,\,a=1,2,$ (30)
$\dot{p}_{1}^{\mu}=\\{p^{\mu}_{1},H\\}=\frac{\alpha^{2}}{4}\sqrt{e_{1}e_{2}}\\{p^{\mu}_{1},\frac{1}{r^{2}}\\}=-\sqrt{e_{1}e_{2}}\frac{\alpha^{2}}{2r^{2}}r^{\mu}=-\dot{p}_{2}^{\mu},$
(31) $\dot{e}_{a}=\\{e_{a},H\\}=\mu_{a}\,\,\,a=1,2,$ (32)
$\dot{\pi}_{a}=\\{\pi_{a},H\\}=\phi_{a},\,\,\,a=1,2\,.$ (33)

The stability of the primary constraints $\pi_{a}$ implies the secondary
constraints $\phi_{a}=0$. Note the appearance of the einbeins in the
constraints $\phi_{a}$.

###  III.1  Killing equations and symmetries

Like in the free particle case we would like to find the most general point
transformations of the model. We want to write the corresponding Killing
equations for the two particle model. In order to do that we write the most
general generator

$G=\sum_{a=1}^{2}\left[\xi_{a\mu}(x_{1},x_{2})p_{a}^{\mu}+\gamma_{a}\pi_{a}\right].$
(34)

As shown in Casalbuoni:2014ofa  the requirement ${dG}/{d\tau}=0$ is satisfied
provided the following conditions are verified

$\xi_{a}(x_{1},x_{2})=\xi_{a}(x_{a}),\,\,\,a=1,2,$ (35)
$\frac{\partial\xi_{a\mu}}{\partial
x_{a}^{\nu}}+\frac{\partial\xi_{a\nu}}{\partial
x_{a}^{\mu}}-g_{\mu\nu}\tilde{\gamma}_{a}=0,\,\,\,a=1,2,$ (36)

where

$\tilde{\gamma}_{a}=\frac{\gamma_{a}}{e_{a}},\,\,\,a=1,2,$ (37)

and

$(\xi_{1}-\xi_{2})\cdot
r=\frac{r^{2}}{8}\sum_{a=1}^{2}\partial_{a\nu}\xi_{a}^{\nu}.$ (38)

From Eq. (36) we also obtain

$\tilde{\gamma}_{a}=\frac{1}{2}\partial_{a\nu}\xi_{a}^{\nu},\,\,\,a=1,2.$ (39)

We have verified that the model is invariant under the diagonal subgroup of
the two conformal groups $SO(4,2)_{1,2}$ acting on the two variables $x_{1}$
and $x_{2}$ respectively by checking that Eqs. (36) and (38) are satisfied for
a generic infinitesimal transformation of the diagonal conformal subgroup,

$\xi_{a}^{\mu}=\epsilon^{\mu}+\omega^{\mu}_{a\nu}x^{\nu}_{a}+\epsilon_{D}x_{a}^{\mu}+b^{\mu}x_{a}^{2}-2(b\cdot
x_{a})x^{\mu}_{a},\,\,\,a=1,2.$ (40)

We can now compute the transformation properties of the variables under
dilatations and SCT using

$\delta x_{a}^{\mu}=\\{G,x^{\mu}\\}=\xi_{a}^{\mu},\,\,\,\,\delta
p_{a}^{\mu}=\\{G,p_{a}^{\mu}\\}=-\frac{\partial\xi_{a}^{\nu}}{\partial
x_{a\mu}}{p_{a\nu}},\,\,a=1,2,$ (41) $\delta e_{a}=\\{e_{a},G\\},\,\,a=1,2.$
(42)

Under dilatations:

$\delta x_{a}^{\mu}=\epsilon_{D}x_{a}^{\mu},\,\,\,\,\delta
p_{a}^{\mu}=-\epsilon_{D}p_{a}^{\mu},\,\,\,\delta
e_{a}=2e_{a}\epsilon_{D},\,\,a=1,2.$ (43)

Under special conformal transformations:

$\delta x_{a}^{\mu}=b^{\mu}x_{a}^{2}-2(b\cdot x_{a})x^{\mu}_{a},\,\,\,\,\delta
p_{a}^{\mu}=-2b\cdot p_{a}x_{a}^{\mu}+2b\cdot x_{a}p_{a}^{\mu}+2p_{a}\cdot
x_{a}b^{\mu},\,\,a=1,2,$ (44) $\delta
e_{a}=e_{a}\tilde{\gamma}_{a}=-e_{a}4b\cdot x_{a},\,\,a=1,2.$ (45)

##  IV Two conformal Carroll particles

Here we consider the Carroll limit of the conformal relativistic canonical
action (27) by assuming

${p_{a}}^{0}=\omega E_{a},\,\,\,x_{a}^{0}=\frac{1}{\omega}t_{a},\,\,\,a=1,2$
(46)

From the mass-shell constraints given in Eq. (III) we get

$\displaystyle\phi_{1}=\frac{1}{2}\left(\omega^{2}E_{1}^{2}-{\bf
p}_{1}^{2}+\frac{\alpha^{2}}{4}\sqrt{\frac{e_{2}}{e_{1}}}\frac{1}{\frac{1}{\omega^{2}}r_{0}^{2}-{\bf
r}^{2}}\right),$
$\displaystyle\phi_{2}=\frac{1}{2}\left(\omega^{2}E_{2}^{2}-{\bf
p}_{2}^{2}+\frac{\alpha^{2}}{4}\sqrt{\frac{e_{1}}{e_{2}}}\frac{1}{\frac{1}{\omega^{2}}r_{0}^{2}-{\bf
r}^{2}}\right)\,.$ (47)

By defining

${\tilde{e}_{a}}=e_{a}\omega^{2},\,\,\,{\tilde{\alpha}}^{2}=\frac{\alpha^{2}}{\omega^{2}},\,\,\,a=1,2,$
(48)

we obtain, in the limit $\omega\to\infty$,

$\displaystyle e_{1}\phi_{1}$ $\displaystyle=$
$\displaystyle\frac{{\tilde{e}}_{1}}{2}\left(E_{1}^{2}-\frac{{\tilde{\alpha}}^{2}}{4}\sqrt{\frac{{\tilde{e}}_{2}}{{\tilde{e}}_{1}}}\frac{1}{{\bf
r}^{2}}\right),\quad$ (49)
$e_{2}\phi_{2}=\frac{{\tilde{e}}_{2}}{2}\left(E_{2}^{2}-\frac{{\tilde{\alpha}}^{2}}{4}\sqrt{\frac{{\tilde{e}}_{1}}{{\tilde{e}}_{2}}}\frac{1}{{\bf
r}^{2}}\right)\,,$ (50)

so that the Carroll canonical Lagrangian is given by

$\displaystyle L$ $\displaystyle=$ $\displaystyle-E_{1}\dot{t}_{1}+{\bf
p}_{1}\cdot{\dot{\bf x}}_{1}-E_{2}\dot{t}_{2}+{\bf p}_{2}\cdot{\dot{\bf
x}}_{2}$ (51)
$\displaystyle+\frac{{\tilde{e}}_{1}}{2}\left(E_{1}^{2}-\frac{{\tilde{\alpha}}^{2}}{4}\sqrt{\frac{{\tilde{e}}_{2}}{{\tilde{e}}_{1}}}\frac{1}{{\bf
r}^{2}}\right)+\frac{{\tilde{e}}_{2}}{2}\left(E_{2}^{2}-\frac{{\tilde{\alpha}}^{2}}{4}\sqrt{\frac{{\tilde{e}}_{1}}{{\tilde{e}}_{2}}}\frac{1}{{\bf
r}^{2}}\right)\,,$

From now on we neglect the tilde, by renaming $\tilde{e}\to
e,\,\,\,\tilde{\alpha}\to\alpha$.

When $\alpha=0$ the Lagrangian is invariant under two independent Carroll
transformations acting on the corresponding particle coordinates. When
$\alpha\neq 0$, the Lagrangian is invariant under the following diagonal
Carroll transformations for each particle

$\delta_{C}x^{i}_{a}=\epsilon^{i}-\Lambda^{ij}x^{j}_{a},\,\,\,\delta_{C}p^{i}_{a}=-\Lambda^{ij}p^{j}_{a}+\beta^{j}E_{a}\,,$
(52) $\delta_{C}t_{a}=h+{\bm{\beta}}\cdot{\bf
x}_{a},\,\,\,\delta_{C}E_{a}=0,\,\,\,\delta e_{a}=0,\,\,\,a=1,2.$ (53)

Lagrangian equations of motion are

$\displaystyle\dot{\bf p}_{1}$ $\displaystyle=$
$\displaystyle\frac{\alpha^{2}}{2}\frac{\sqrt{e_{1}e_{2}}}{{\bf r}^{4}}{\bf
r},$ $\displaystyle\dot{\bf p}_{2}$ $\displaystyle=$
$\displaystyle-\frac{\alpha^{2}}{2}\frac{\sqrt{e_{1}e_{2}}}{{\bf r}^{4}}{\bf
r},$ $\displaystyle\dot{\bf x}_{a}$ $\displaystyle=$ $\displaystyle
0,\,\,\,\dot{E}_{a}=0,\,\,\,\dot{t}_{a}=e_{a}E_{a},$ $\displaystyle 0$
$\displaystyle=$ $\displaystyle\dot{\pi}_{1}=\frac{\partial L}{\partial
e_{1}}=\frac{1}{2}\left(E_{1}^{2}-\frac{\alpha^{2}}{4{\bf
r}^{2}}\sqrt{\frac{e_{2}}{e_{1}}}\right)$ $\displaystyle 0$ $\displaystyle=$
$\displaystyle\dot{\pi}_{2}=\frac{\partial L}{\partial
e_{2}}=\frac{1}{2}\left(E_{2}^{2}-\frac{\alpha^{2}}{4{\bf
r}^{2}}\sqrt{\frac{e_{1}}{e_{2}}}\right)\,.$ (54)

This model describes interacting Carroll particles that have zero velocity.
Since the distance among the particles is constant, it is only fixed by the
initial conditions.

###  IV.1 Eliminating the einbeins

The einbeins are non-dynamical variables and can be eliminated through their
equations of motion Pons:2009ch . In fact from the equations of motion of
$e_{1}$ we obtain

$\sqrt{\frac{e_{1}}{e_{2}}}=\frac{\alpha^{2}}{4E_{1}^{2}{\bf r}^{2}}$ (55)

and by squaring

${e_{1}}={e_{2}}\left(\frac{\alpha^{2}}{4E_{1}^{2}{\bf r}^{2}}\right)^{2}\,.$
(56)

Then the canonical Carroll Lagrangian can be rewritten as

$L=-E_{a}\dot{t}_{a}+\sum_{a=1,2}{\bf p}_{a}\cdot\dot{\bf
x}_{a}+\frac{e_{2}}{2}\left(E_{2}^{2}-(\frac{\alpha^{2}}{4{\bf
r}^{2}})^{2}\frac{1}{E_{1}^{2}}\right)\,.$ (57)

The equation of motion for $E_{2}$ gives

$\dot{t}_{2}=e_{2}E_{2}$ (58)

and substituting in Eq. (57) we obtain

$L=-E_{1}\dot{t}_{1}-\frac{1}{2}\frac{\dot{t}_{2}^{2}}{e_{2}}+\sum_{a=1,2}{\bf
p}_{a}\cdot\dot{\bf x}_{a}-\frac{e_{2}}{2}(\frac{\alpha^{2}}{4{\bf
r}^{2}})^{2}\frac{1}{E_{1}^{2}}\,.$ (59)

By using the equation for $e_{2}$

$e_{2}=\pm\frac{4{\bf
r}^{2}}{\alpha^{2}}\left(E_{1}^{2}t_{2}^{2}\right)^{1/2}\,.$ (60)

and substituting in Eq. (59), the Lagrangian can be rewritten as

$L=-E_{1}\dot{t}_{1}+\sum_{a=1,2}{\bf p}_{a}\cdot\dot{\bf
x}_{a}\mp\frac{\alpha^{2}}{4{\bf
r}^{2}}\left(\frac{\dot{t}_{2}^{2}}{E_{1}^{2}}\right)^{1/2}\,,$ (61)

For the equation of motion of $E_{1}$ we get

$E_{1}=\pm\frac{\alpha}{2}(\dot{t}_{2}^{2})^{1/4}\frac{1}{({{\bf
r}^{2}\dot{t}_{1}})^{1/2}}\,.$ (62)

By substituting $E_{1}$ in Eq. 61 the final Lagrangian is (with a suitable
choice of $\pm$ sign)

$\displaystyle L$ $\displaystyle=$ $\displaystyle\sum_{a=1,2}{\bf
p}_{a}\cdot\dot{\bf
x}_{a}-\alpha\left[\frac{{\dot{t}}_{1}^{2}{\dot{t}}_{2}^{2}}{{\bf
r}^{4}}\right]^{1/4}$ (63) $\displaystyle=$ $\displaystyle\sum_{a=1,2}{\bf
p}_{a}\cdot\dot{\bf x}_{a}-\frac{\alpha}{|{\bf
r}|}\left[{\dot{t}}_{1}{\dot{t}}_{2}\right]^{1/2}\,.$

We assume ${\dot{t}}_{1}{\dot{t}}_{2}>0$.

Since the Lagrangian is homogeneous of degree one in the velocities, it is
invariant under diffeomorphisms and gives rise to the primary constraint

$E_{1}E_{2}-\frac{\alpha^{2}}{4{\bf r}^{2}}=0\,.$ (64)

In the case of two conformal free particles their individual energies are
zero. Instead in this model, when the interaction is turned on ($\alpha\neq
0$), the energies are different from zero and constant. The particles are
interacting, indeed, as shown in Eq. (64), the energy of each particle depend
on the energy and the position of the other one. In conclusion the model is
Carroll conformal even if the two particles have non-vanishing energies.

Since the energies $E_{1}$ and $E_{2}$ are constants of motion, the meaning of
the Eq. (64) is that each particle stays inside its own light cone,
corresponding to the lines with constant values of ${\bf x}_{1}$ and ${\bf
x}_{2}$ (the velocities are zero, see eqs. (54)) and of ${\bf r}$. Introducing
the total and the relative energies of the two particles:

$E_{T}=E_{1}+E_{2},~{}~{}~{}E_{r}=E_{1}-E_{2}\,,$ (65)

we get

$E_{T}^{2}=E_{r}^{2}+\frac{\alpha^{2}}{{\bf r}^{2}}\,.$ (66)

This equation shows that the minimum of the total energy is for
$E_{1}=E_{2}=0$ and for ${\bf r}\to\infty$.

The wave equation associated to the Eq. (64) is

$[\partial_{t_{1}}\partial_{t_{2}}+\frac{\alpha^{2}}{4{\bf
r}^{2}}]\Phi(t_{1},t_{2},\vec{x}_{1},\vec{x}_{2})=0$ (67)

###  IV.2 Invariance under dilatations and special conformal transformations

The Lagrangian (51) is invariant also under the dilatations and special
conformal transformations. Under dilatations we have:

$\delta t_{a}=\epsilon_{D}t_{a},\,\,\,\,\delta{\bf x}_{a}=\epsilon_{D}{\bf
x}_{a},\,\,\,\delta E_{a}=-\epsilon_{D}E_{a},\delta{\bf
p}_{a}=-\epsilon_{D}{\bf p}_{a},\,\,\,\delta
e_{a}=2\epsilon_{D}e_{a},\,\,\,a=1,2.$ (68)

Under SCT:

$\delta t_{a}=-b^{0}{\bf x}_{a}^{2}+2{\bf b}\cdot{\bf
x}_{a}\,t_{a},\,\,\,\,\delta{\bf x}_{a}=-{\bf b}\,{\bf x}_{a}^{2}+2{\bf
b}\cdot{\bf x}_{a}\,{\bf x}_{a},\,\,\,a=1,2.$ (69)

$\delta E_{a}=-2{\bf b}\cdot{\bf x}_{a}E_{a},\,\,\,\delta{\bf
p}_{a}=-2(b^{0}E_{a}-{\bf b}\cdot{\bf p}_{a}){\bf x}_{a}-2{\bf b}\cdot{\bf
x}_{a}\,{\bf p}_{a}+2(E_{a}t_{a}-{\bf p}_{a}\cdot{\bf x}_{a}){\bf b},\,\,$
(70) $\delta e_{a}=4{\bf b}\cdot{\bf x}_{a}e_{a},\,\,,a=1,2.$ (71)

These SCT are obtained from the covariant ones given in Eqs. (44) and (45),
considering

$x_{a}^{0}=\frac{1}{\omega}t_{a},\,\,\,p^{0}_{a}=\omega
E_{a},\,\,\,b^{0}=\frac{b^{0}}{\omega},\,\,\,a=1,2.$ (72)

###  IV.3 Infinite dimensional symmetries

Like in the previous sections, here we would like to find the most general
point symmetries of the model.

Let consider the generic Killing vector

$G=\sum_{a=1,2}\left[\xi^{0}_{a}(t_{1},t_{2},{\bf x}_{1},{\bf
x}_{2})E_{a}-{\bm{\xi}}_{a}(t_{1},t_{2},{\bf x}_{1},{\bf x}_{2})\cdot{\bf
p}_{a}+\gamma_{a}(t_{1},t_{2},{\bf x}_{1},{\bf x}_{2})\pi_{a}\right]\,.$ (73)

By considering the $\tau$ derivative we obtain

$\displaystyle\frac{dG}{d\tau}$ $\displaystyle=$
$\displaystyle\sum_{a,c=1,2}\Big{[}\frac{\partial\xi^{0}_{a}(t_{1},t_{2},{\bf
x}_{1},{\bf x}_{2})}{\partial
t_{c}}\dot{t}_{c}E_{a}-\frac{\partial\xi_{a}^{j}(t_{1},t_{2},{\bf x}_{1},{\bf
x}_{2})}{\partial t_{c}}\dot{t}_{c}p_{a}^{j}-{\bm{\xi}}_{a}(t_{1},t_{2},{\bf
x}_{1},{\bf x}_{2})\cdot\dot{\bf p}_{a}$ (74)
$\displaystyle+\gamma_{a}(t_{1},t_{2},{\bf x}_{1},{\bf
x}_{2})\dot{\pi}_{a}\Big{]}\,,$

where we have used the equations of motion

$\dot{E}_{a}=0,\,\,\,\dot{\bf x}_{a}=0\quad$ (75)

and the primary constraints $\pi_{a}=0$. Using Eqs. (54) we get

$\displaystyle\frac{dG}{d\tau}$ $\displaystyle=$
$\displaystyle\sum_{a,c=1,2}\Big{[}\frac{\partial\xi^{0}_{a}(t_{1},t_{2},{\bf
x}_{1},{\bf x}_{2})}{\partial
t_{c}}e_{c}E_{c}E_{a}-\frac{\partial\xi_{a}^{j}(t_{1},t_{2},{\bf x}_{1},{\bf
x}_{2})}{\partial
t_{c}}e_{c}E_{c}p_{a}^{j}+\frac{1}{2}\gamma_{a}(t_{1},t_{2},{\bf x}_{1},{\bf
x}_{2})E^{2}_{a}\Big{]}$ (76) $\displaystyle-\frac{\alpha^{2}}{2{\bf
r}^{4}}\sqrt{e_{1}e_{2}}({\bm{\xi}}_{1}-{\bm{\xi}}_{2})\cdot{\bf
r}-\frac{\alpha^{2}}{8{\bf r}^{2}}\left[\gamma_{1}(t_{1},t_{2},{\bf
x}_{1},{\bf x}_{2})\sqrt{\frac{e_{2}}{e_{1}}}+\gamma_{2}(t_{1},t_{2},{\bf
x}_{1},{\bf x}_{2})\sqrt{\frac{e_{1}}{e_{2}}}\right]\,.$

Therefore the Killing condition ${dG}/{d\tau}=0$ implies

$\frac{\partial\xi_{a}^{j}(t_{1},t_{2},{\bf x}_{1},{\bf x}_{2})}{\partial
t_{c}}=0,\,\,\,a,c=1,2,$ (77)

or $\xi_{a}^{j}=\xi_{a}^{j}({\bf x}_{1},{\bf x}_{2})$. We see that $\delta{\bf
x}_{a}\equiv{\bm{\xi}}_{a}$ do no depend on the times $t_{1},t_{2}$.

By defining

$\gamma_{a}=e_{a}\tilde{\gamma}_{a}$ (78)

from ${dG}/{d\tau}=0$ we must have also

$\frac{\partial\xi^{0}_{a}(t_{1},t_{2},{\bf x}_{1},{\bf x}_{2})}{\partial
t_{c}}=-\frac{1}{2}\delta_{ac}\tilde{\gamma}_{a},,\,\,\,a,b,c=1,2,$ (79)

and

$\frac{\alpha^{2}}{{\bf r}^{2}}({\bm{\xi}}_{1}-{\bm{\xi}}_{2})\cdot{\bf
r}=\frac{\alpha^{2}}{2}\frac{\delta{{\bf r}^{2}}}{{\bf
r}^{2}}=-\frac{\alpha^{2}}{4}(\tilde{\gamma}_{1}+\tilde{\gamma}_{2}),\,\,\,a=1,2.$
(80)

In the case $\alpha=0$, that is when we turn off the interaction, we see that
we have two sets of independent Killing equations, one set for each particle.
Therefore the infinite group of symmetry transformations is ${\cal
G}_{1}\times{\cal G}_{1}$. When the interaction is on and $\alpha\not=0$, we
get:

$\frac{1}{{\bf r}^{2}}({\bm{\xi}}_{1}-{\bm{\xi}}_{2})\cdot{\bf
r}=\frac{1}{2}\frac{\delta{{\bf r}^{2}}}{{\bf
r}^{2}}=-\frac{1}{4}(\tilde{\gamma}_{1}+\tilde{\gamma}_{2}),\,\,\,a=1,2.$ (81)

From Eq. (79) we have

$\frac{\partial\xi^{0}_{1}}{\partial
t_{2}}=0=\frac{\partial\xi^{0}_{2}}{\partial t_{1}}\,.$ (82)

Furthermore from Eq. (81) we note that since ${\bm{\xi}}_{a}$ are only
functions of ${\bf x}_{1},{\bf x}_{2}$, $\tilde{\gamma}_{a}$ depend only on
${\bf x}_{1},{\bf x}_{2}$. In conclusion we obtain

$\xi^{0}_{a}\equiv\delta t_{a}=-\frac{1}{2}\tilde{\gamma}_{a}({\bf x}_{1},{\bf
x}_{2})t_{a}+h_{a}({\bf x}_{1},{\bf x}_{2}),\quad\delta
e_{a}=\gamma_{a}={e_{a}}\tilde{\gamma}_{a}({\bf x}_{1},{\bf x}_{2})\,,$ (83)

with the extra condition given by Eq. (81).

Summarising the Killing generator is given by

$G=\sum_{a=1,2}\left[\big{(}-\frac{1}{2}\tilde{\gamma}_{a}({\bf x}_{1},{\bf
x}_{2})t_{a}+h_{a}({\bf x}_{1},{\bf x}_{2})\big{)}E_{a}-{\bm{\xi}}_{a}({\bf
x}_{1},{\bf x}_{2})\cdot{\bf p}_{a}+{e_{a}}\tilde{\gamma}_{a}({\bf x}_{1},{\bf
x}_{2})\pi_{a}\right]\,,$ (84)

where $h_{a},{\bm{\xi}}_{a},\tilde{\gamma}_{a}$ are arbitrary functions of
${\bf x}_{1},{\bf x}_{2}$ that must satisfy the condition (79). Therefore the
original ten arbitrary functions reduce to nine independent arbitrary
functions of ${\bf x}_{1},{\bf x}_{2}$.

Therefore the model of two conformal Carroll particles analysed in this note
has an infinite dimensional group of symmetries that we will call ${\cal
G}_{2}$. Notice that ${\cal G}_{2}\subset{\cal G}_{1}\times{\cal G}_{1}$. The
main difference between these two groups is the transformation laws of the
times $t_{a}$ that in the free case are arbitrary functions of the times and
the coordinates, while in the interacting case the Eq. (79) restricts the time
dependence of the transformations of $\xi^{0}_{a}$ to be linear in $t_{a}$.

It is convenient to make a comparison with Duval:2014uva . Let us generalise
to two particles the expression for the generators of the conformal Carrollian
algebra from Duval:2014uva . We consider the diagonal subgroup of the two
conformal Carroll groups.

For Carroll dilatations and SCT we have

$\xi^{i}_{a}=(\epsilon_{D}+2{\bf b}\cdot{\bf x}_{a})x^{i}_{a}-b^{i}{\bf
x}_{a}^{2}\,,$ (85) $\xi^{0}_{a}=(\epsilon_{D}+2{\bf b}\cdot{\bf
x}_{a})t_{a}+T_{a}({\bf x}_{1},{\bf x}_{2})\,.$ (86)

We have

$\frac{\partial\xi^{0}_{a}}{\partial t_{a}}=\epsilon_{D}+2{\bf b}\cdot{\bf
x}_{a}$ (87)

and

$\frac{1}{2}\left(\frac{\partial\xi^{0}_{1}}{\partial
t_{1}}+\frac{\partial\xi^{0}_{2}}{\partial t_{2}}\right)=\epsilon_{D}+{\bf
b}\cdot{\bf x}_{1}+{\bf b}\cdot{\bf x}_{2}$ (88)
$\displaystyle({\bm{\xi}}_{1}-{\bm{\xi}}_{2})\cdot{\bf r}$ $\displaystyle=$
$\displaystyle\epsilon_{D}{\bf r}^{2}+2({\bf b}\cdot{\bf x}_{1}){\bf
x}_{1}\cdot{\bf r}-{\bf b}\cdot{\bf r}({\bf x}_{1})^{2}-2({\bf b}\cdot{\bf
x}_{2}){\bf x}_{2}\cdot{\bf r}+{\bf b}\cdot{\bf r}({\bf x}_{2})^{2}$ (89)
$\displaystyle=$ $\displaystyle\epsilon_{D}{\bf r}^{2}+{\bf r}^{2}({\bf
b}\cdot{\bf x}_{1}+{\bf b}\cdot{\bf x}_{2})\,.$

In conclusion Eq. (81) is satisfied by Eq. (85) and Eq. (86) and the so-called
super-translations $T_{a}({\bf x}_{1},{\bf x}_{2})$ in our case are arbitrary
functions of both particle positions ${\bf x}_{1},{\bf x}_{2}$.

##  V Two conformal Carroll particles. A tachyonic model.

In this Section, following the approach of deBoer:2021jej  Gomis:2022spp , we
develop a model describing two interacting Carrollian tachyons. Let us
consider the Lagrangian

$\displaystyle L$ $\displaystyle=$
$\displaystyle\sum_{a=1,2}\left[-E_{a}\dot{t}_{a}+{\bf p}{\dot{\bf
x}}_{a}-{e_{a}}\phi_{a}-\chi_{a}E_{a}\right]$ (90) $\displaystyle=$
$\displaystyle-E_{1}\dot{t}_{1}+{\bf p}_{1}\cdot{\dot{\bf
x}}_{1}-E_{2}\dot{t}_{2}+{\bf p}_{2}\cdot{\dot{\bf x}}_{2}$
$\displaystyle-\frac{{e}_{1}}{2}\left({\bf
p}_{1}^{2}-\frac{\alpha^{2}}{4}\sqrt{\frac{e_{2}}{e_{1}}}\frac{1}{{\bf
r}^{2}}\right)-\frac{{e}_{2}}{2}\left({\bf
p}_{2}^{2}-\frac{{\alpha}^{2}}{4}\sqrt{\frac{{e}_{1}}{{e}_{2}}}\frac{1}{{\bf
r}^{2}}\right)$ $\displaystyle-\chi_{1}E_{1}-\chi_{2}E_{2}\,.$

Notice the presence of the Lagrangian multipliers $\chi_{1},\chi_{2}$, that
implement that the energies are zero; these terms are necessary in order to
have Carroll invariance. An analogous situation appears for single particle in
Gomis:2022spp  where the action is named the magnetic Carroll particle. In
fact this Lagrangian is invariant under the Carroll, Eqs. (52) and (53), scale
and SCT, Eqs. (68-71), by requiring the following transformations for the
einbein $\chi_{a}$

$\delta\chi_{a}=-e_{a}{\bf p}_{a}\cdot\bm{\beta},\quad a=1,2,\quad{\rm
Carroll~{}~{}transf.s}\,,$ (91)

$\delta\chi_{a}=\epsilon_{D}\chi_{a}\quad a=1,2,\quad{\rm
scale~{}~{}transf.s}\,,$ (92)

$\delta\chi_{a}=2\left[e_{a}(b^{0}{\bf p}_{a}\cdot{\bf x}_{a}-t_{a}{\bf
p}_{a}\cdot{\bf b})+\chi_{a}{\bf b}\cdot{\bf x}_{a}\right]\quad
a=1,2,\quad{\rm SCT~{}~{}transf.s}\,.$ (93)

The equations of motion derived from the Lagrangian (90) by varying with
respect to $t_{a},E_{a},{\bf p}_{a},{\bf x}_{a},\chi_{a},e_{a}$ are the
following

$\quad\dot{E}_{a}=0,\quad\dot{t}_{a}=-E_{a}\,,$ (94) $e_{a}{\bf
p}_{a}-\dot{\bf x}_{a}=0,\quad\dot{\bf p}_{1}=-\frac{\alpha^{2}}{2{\bf
r}^{4}}\sqrt{e_{1}e_{2}}{\bf r}=-\dot{\bf p}_{2}\,,$ (95) $E_{a}=0\,,$ (96)
$\phi_{a}=0\,.$ (97)

In this model the particles have non zero velocity and therefore, as a
consequence of the limit $c\to 0$, they are necessarily tachyons. Or
equivalently, since $E_{a}=0$, if we use relativistic mass-shell constraint
the two particle invariant squared masses are negative ($E_{a}^{2}-{\bf
p}_{a}^{2}<0$).

###  V.1 Eliminating the einbeins

Let us first study the equivalence of this model with the $c\to 0$ limit of
the model for two relativistic conformal particles Casalbuoni:2014ofa
described by the Lagrangian

$L=-\alpha\left(\frac{\dot{x}_{1}^{2}\dot{x}_{2}^{2}}{r^{4}}\right)^{1/4}\,.$
(98)

In order to do that, we start by eliminating the non-dynamical variable
$e_{1}$ in Eq. (90) through its own equation of motion. We obtain

$\sqrt{\frac{e_{1}}{e_{2}}}=\frac{\alpha^{2}}{4{\bf r}^{2}{\bf p}_{1}^{2}}$
(99)

and substituting in Eq. (90)

$\displaystyle L$ $\displaystyle=$ $\displaystyle-E_{1}\dot{t}_{1}+{\bf
p}_{1}\cdot{\dot{\bf x}}_{1}-E_{2}\dot{t}_{2}+{\bf p}_{2}\cdot{\dot{\bf
x}}_{2}$ (100) $\displaystyle-\frac{e_{2}}{2}\left[{\bf
p}_{2}^{2}-(\frac{\alpha^{2}}{4{\bf r}^{2}})^{2}\frac{1}{{\bf
p}_{1}^{2}}\right]$ $\displaystyle-\chi_{1}E_{1}-\chi_{2}E_{2}\,.$

The equations of motion of the non dynamical variables ${\bf p}_{2}$ are

${\bf p}_{2}=\frac{1}{e_{2}}\dot{\bf x}_{2}$ (101)

and the Lagrangian becomes

$\displaystyle L$ $\displaystyle=$ $\displaystyle-
E_{1}\dot{t}_{1}-E_{2}\dot{t}_{2}-\chi_{1}E_{1}-\chi_{2}E_{2}$ (102)
$\displaystyle+{\bf p}_{1}\cdot{\dot{\bf x}}_{1}+\frac{1}{2e_{2}}\dot{\bf
x}_{2}^{2}+\frac{e_{2}}{2}(\frac{\alpha^{2}}{4{\bf r}^{2}})^{2}\frac{1}{{\bf
p}_{1}^{2}}\,.$

Then we eliminate the non dynamical variable $e_{2}$ through its own equation
of motion. (choosing the $-$ sign)

$e_{2}=-\frac{4{\bf r}^{2}}{\alpha^{2}}\sqrt{\dot{\bf x}_{2}^{2}{\bf
p}_{1}^{2}}$ (103)

and substituting in $L$

$\displaystyle L$ $\displaystyle=$
$\displaystyle-\sum_{a=1,2}(E_{a}\dot{t}_{a}+\chi_{a}E_{a})$ (104)
$\displaystyle+{\bf p}_{1}\cdot{\dot{\bf x}}_{1}-\frac{\alpha^{2}}{4{\bf
r}^{2}}\sqrt{\frac{\dot{\bf x}_{2}^{2}}{{\bf p}_{1}^{2}}}\,.$

Finally, taking into account that

$\dot{\bf x}_{1}=-{\bf p}_{1}\frac{\alpha^{2}}{4{\bf r}^{2}}\sqrt{\dot{\bf
x}_{2}^{2}}({\bf p}_{1}^{2})^{-3/2}$ (105)

and

${\bf p}_{1}^{2}=\frac{\alpha^{2}}{4{\bf r}^{2}}\sqrt{\frac{\dot{\bf
x}_{2}^{2}}{\dot{\bf x}_{1}^{2}}}\,,$ (106)

we get the following expression for the Lagrangian

$\displaystyle L$ $\displaystyle=$
$\displaystyle-\sum_{a=1,2}(E_{a}\dot{t}_{a}+\chi_{a}E_{a})$ (107)
$\displaystyle-\alpha\left(\frac{\dot{\bf x}_{1}^{2}\dot{\bf x}_{2}^{2}}{{\bf
r}^{4}}\right)^{1/4}\,.$

By eliminating $E_{a}$ and $\chi_{a}$ we obtain

$\dot{t}_{a}=-\chi_{a},\quad E_{a}=0$ (108)

and the following expression for the Lagrangian

$\displaystyle L$ $\displaystyle=$ $\displaystyle-\alpha\left(\frac{\dot{\bf
x}_{1}^{2}\dot{\bf x}_{2}^{2}}{{\bf r}^{4}}\right)^{1/4}\,,$ (109)

which coincides with the $c\to 0$ limit of (98).

A primary constraint can be obtained by squaring ${\bf p}_{a}$

${\bf p}_{1}^{2}{\bf p}_{2}^{2}-\frac{\alpha^{2}}{16{\bf r}^{4}}=0\,.$ (110)

###  V.2  Killing equations and symmetries

In this Section we derive the Killing vector by working directly with the
Lagrangian given in Eq. (90). By taking the total derivative with respect to
$\tau$ of the Killing vector

$G=\sum_{a=1,2}(\xi_{a}^{0}E_{a}-{\bm{\xi}}_{a}\cdot{\bf
p}_{a}+\gamma_{a}\pi_{a}+\nu_{a}\pi_{a}^{\chi})\,,$ (111)

where

$\pi_{a}^{\chi}=\frac{\partial L}{\partial\dot{\chi}_{a}}\,,$ (112)

we get

$\displaystyle\frac{dG}{d\tau}$ $\displaystyle=$
$\displaystyle\sum_{a,b=1,2}\left(\frac{\partial\xi_{a}^{0}}{\partial
t_{b}}\dot{t}_{b}E_{a}+\frac{\partial\xi_{a}^{0}}{\partial{\bf x}_{b}}\dot{\bf
x}_{b}E_{a}-\frac{\partial{\bm{\xi}}_{a}}{\partial t_{b}}\dot{t}_{b}{\bf
p}_{a}-\frac{\partial\xi_{a}^{i}}{\partial
x_{b}^{j}}\dot{x}_{b}^{j}p_{a}^{i}\right)$ (113)
$\displaystyle-\sum_{a=1,2}({\bm{\xi}}_{a}\cdot\dot{\bf
p}_{a}+\gamma_{a}\dot{\pi}_{a}+\nu_{a}\dot{\pi}_{a}^{\chi})$ $\displaystyle=$
$\displaystyle\sum_{a,b=1,2}\left(-\frac{\partial\xi_{a}^{0}}{\partial
t_{b}}\chi_{b}E_{a}+\frac{\partial\xi_{a}^{0}}{\partial{\bf x}_{b}}e_{b}{\bf
p}_{b}E_{a}+\frac{\partial{\bm{\xi}}_{a}}{\partial t_{b}}\chi_{b}{\bf
p}_{a}-\frac{\partial\xi_{a}^{i}}{\partial
x_{b}^{j}}e_{b}p_{b}^{j}p_{a}^{i}\right)$
$\displaystyle-\sum_{a=1,2}({\bm{\xi}}_{a}\cdot\dot{\bf
p}_{a}+\gamma_{a}\phi_{a}+\nu_{a}E_{a})$ $\displaystyle=$
$\displaystyle\sum_{a,b=1,2}\Big{[}(-\frac{\partial\xi_{a}^{0}}{\partial
t_{b}}\chi_{b}+\frac{\partial\xi_{a}^{0}}{\partial{\bf x}_{b}}e_{b}{\bf
p}_{b}-\nu_{a})E_{a}-\frac{\partial{\bm{\xi}}_{a}}{\partial t_{b}}\chi_{b}{\bf
p}_{a}-\frac{\partial\xi_{a}^{i}}{\partial x_{b}^{j}}e_{b}p_{b}^{j}p_{a}^{i}$
$\displaystyle-\frac{1}{2}\gamma_{a}{\bf p}^{2}_{a}-\frac{\alpha^{2}}{2{\bf
r}^{4}}\sqrt{e_{1}e_{2}}({\bm{\xi}}_{1}-{\bm{\xi}}_{2})\cdot{\bf
r}-\frac{\alpha^{2}}{8r^{2}}\sqrt{e_{1}e_{2}}(\tilde{\gamma}_{1}+\tilde{\gamma}_{2}))\Big{]}\,,$

where use has been made of the primary constraints $\pi_{a}=\pi^{\chi}_{a}=0$,
of Eqs.(94),(95) and of

$\dot{\pi}_{a}=-\phi_{a}\,,$ (114) $\dot{\pi}^{\chi}_{a}=-E_{a}\,.$ (115)

The sum over latin indices is understood.

In order to get ${dG}/{d\tau}=0$, we must require

$\frac{\partial{\bm{\xi}}_{a}}{\partial t_{b}}=0\,,$ (116)

so that ${\bm{\xi}}_{a}$ is a function of only ${\bf x}_{1},{\bf x}_{2}$; we
require also the vanishing of the coefficient of $E_{a}$

$\sum_{b}(-\frac{\partial\xi_{a}^{0}}{\partial
t_{b}}\chi_{b}+\frac{\partial\xi_{a}^{0}}{\partial{\bf x}_{b}}e_{b}{\bf
p}_{b})-\nu_{a}=0\,.$ (117)

We also require, assuming ${\bm{\xi}}_{a}={\bm{\xi}}_{a}({\bf x}_{a})$, the
vanishing of the coefficient of $p_{a}^{i}p_{a}^{j}$

$\left(\frac{\partial\xi^{i}_{a}}{\partial
x_{a}^{j}}+\frac{\partial\xi^{j}_{a}}{\partial
x_{a}^{i}}\right)=-\delta^{ij}\tilde{\gamma}_{a}({\bf x}_{a})\,,$ (118)

so that ${dG}/{d\tau}$ implies, when $\alpha\neq 0$

$-\frac{1}{4}(\tilde{\gamma}_{1}+\tilde{\gamma}_{2})=\frac{1}{6}\left(\frac{\partial\xi_{1}^{i}}{\partial
x_{1}^{i}}+\frac{\partial\xi_{2}^{i}}{\partial
x_{2}^{i}}\right)=\frac{({\bm{\xi}}_{1}-{\bm{\xi}}_{2})\cdot{\bf r}}{{\bf
r}^{2}}\,.$ (119)

In conclusion the general solution for the Killing vector is

$G=\sum_{a=1,2}\left[\xi_{a}^{0}(t_{1},t_{2},{\bf x}_{1},{\bf
x}_{2})E_{a}-{\bm{\xi}}_{a}({\bf x}_{a})\cdot{\bf
p}_{a}+e_{a}\tilde{\gamma}_{a}\pi_{a}+\nu_{a}\pi_{a}^{\chi}\right]\,,$ (120)

where

$\tilde{\gamma}_{a}=-\frac{2}{3}\frac{\partial\xi^{i}_{a}}{\partial
x_{a}^{i}},\quad$ (121) $\nu_{a}=\sum_{b}(-\frac{\partial\xi_{a}^{0}}{\partial
t_{b}}\chi_{b}+\frac{\partial\xi_{a}^{0}}{\partial{\bf x}_{b}}e_{b}{\bf
p}_{b})\,,$ (122)

with $\xi_{a}^{0}$ arbitrary function of $t_{1},t_{2},{\bf x}_{1},{\bf x}_{2}$
and ${\bm{\xi}}_{a}({\bf x}_{a})$ satisfying Eq. (119). As expected from the
tachyonic Carroll particle case discussed in Section V, while
$\xi^{0}_{a},{\bm{\xi}}_{a},\tilde{\gamma}_{a}$ are arbitrary point functions
of $t_{a}$ and ${\bm{\xi}}_{a}$, in general $\nu_{a}$ depend also on the
momenta ${\bf p}_{a}$. Carroll, scale transformations and conformal
transformations satisfy Eq. (119).

We have checked that, by using the expression of $\xi_{a}^{0}$ for Carroll,
scale and SCT in Eq. (122), we recover the correct transformations for
$\chi_{a}$ given in Eqs. (91),(92),(93), by computing

$\delta\chi_{a}=\\{\chi_{a},G\\}$ (123)

and assuming

$\\{\chi_{a},\pi^{\chi}_{b}\\}=-\delta_{ab}$ (124)

Note also that the dependence of $\nu_{a}$ on ${\bf x}_{a},{\bf p}_{a}$ in
principle could give additional contributions to the transformations of ${\bf
p}_{a},{\bf x}_{a}$, which however are vanishing because proportional to the
primary constraints $\pi_{a}^{\chi}$.

Notice also that in the case of two non-conformal particles Bergshoeff:2014jla
the individual particles move with energies different from zero and they have
only a diagonal finite conformal symmetry.

##  VI Conclusions

Carroll symmetries have recently received a lot of attention because of
several applications in different domains of theoretical physics, from theory
of gravity and strings to studies of fractons in condensed matter.

In this paper, after reexamining the case of a single conformal Carroll
particle, we derive the conformal Carroll generators, their algebra and the
infinite extension of this algebra Bergshoeff:2014jla . Then we propose two
different models of Carroll interacting particles that provide dynamical
realisations of the Carroll conformal algebra. The models are obtained from
the relativistic model of interacting conformal particles proposed in
Casalbuoni:2014ofa  through suitable limits for $c\to 0$.

Concerning the dynamics, the first model describes particles with zero
velocity but with non vanishing energy as a consequence of the interaction.
Free conformal Carroll particles have zero energy; here, when the interaction
is on, the energy of one particle depends on the energy of the second one and
on the particle relative distance which is constant.

The second model is a tachyonic one: the particles have non zero velocity and
therefore, due to the limit $c\to 0$, they are necessarily tachyons.
Nevertheless their energy is vanishing.

Both models exhibit, after a complete analysis of the most general point
symmetry transformations, infinite dimensional symmetries which include super-
translations as in the case of the BMS group Bondi:1962px ; Sachs:1962zza ,
the symmetry that arises in asymptotically flat space-time at null infinity.
These infinite dimensional algebras contain the Carroll conformal one
Duval:2014uva , which are equivalent to BMS ones.

In the future several directions could be investigated: on the one hand it
would be interesting to extend the analysis of the two conformal Carroll
models to include corrections to the next order in $c\to 0$ expansion, see for
example Gomis:2022spp , on the other hand a deeper analysis of the infinite
symmetries could be performed to check similarities with the BMS group.

###### Acknowledgements.

We would like to thank interesting discussions with Carles Batlle, Josep Pons
and Axel Kleinschmidt for a careful reading of the first version version of
this paper. Also JG acknowledges the warm hospitality at the Max Planck Albert
Einstein Institute where this work was completed. The work of JG has been
supported in part by MINECO FPA2016-76005-C2-1-P and PID2019-105614GB-C21 and
from the State Agency for Research of the Spanish Ministry of Science and
Innovation through the Unit of Excellence Maria de Maeztu 2020-203 award to
the Institute of Cosmos Sciences (CEX2019-000918-M).

## References

  * (1) J.-M. Lévy-Leblond, “Une nouvelle limite non-relativiste du groupe de Poincaré,” Annales de l’I.H.P. Physique théorique 3 (1965) no. 1, 1–12.
  * (2) N. S. Gupta, “On an analogue of the Galilei group,” Nuovo Cim. A 44 (1966) no. 2, 512–517.
  * (3) H. Bondi, M. G. J. van der Burg and A. W. K. Metzner, Proc. Roy. Soc. Lond. A 269 (1962), 21-52 doi:10.1098/rspa.1962.0161
  * (4) R. Sachs, Phys. Rev. 128 (1962), 2851-2864 doi:10.1103/PhysRev.128.2851
  * (5) L. Donnay and C. Marteau, “Carrollian Physics at the Black Hole Horizon,” Class. Quant. Grav. 36 (2019) no.16, 165002 doi:10.1088/1361-6382/ab2fd5 [arXiv:1903.09654 [hep-th]].
  * (6) C. Duval, G. W. Gibbons and P. A. Horvathy, “Conformal Carroll groups and BMS symmetry,” Class. Quant. Grav. 31 (2014), 092001 doi:10.1088/0264-9381/31/9/092001 [arXiv:1402.5894 [gr-qc]].
  * (7) L. Ciambelli, C. Marteau, A. C. Petkou, P. M. Petropoulos and K. Siampos, JHEP 07 (2018), 165 doi:10.1007/JHEP07(2018)165 [arXiv:1802.06809 [hep-th]].
  * (8) E. Bergshoeff, J. Figueroa-O’Farrill and J. Gomis, “A non-lorentzian primer,” [arXiv:2206.12177 [hep-th]].
  * (9) E. Bergshoeff, J. Gomis and G. Longhi, “Dynamics of Carroll Particles,” Class. Quant. Grav. 31 (2014) no.20, 205009 doi:10.1088/0264-9381/31/20/205009 [arXiv:1405.2264 [hep-th]].
  * (10) C. Duval, G. W. Gibbons, P. A. Horvathy and P. M. Zhang, “Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time,” Class. Quant. Grav. 31 (2014), 085016 doi:10.1088/0264-9381/31/8/085016 [arXiv:1402.0657 [gr-qc]].
  * (11) A. Bagchi, R. Basu, A. Kakkar and A. Mehra, JHEP 12 (2016), 147 doi:10.1007/JHEP12(2016)147 [arXiv:1609.06203 [hep-th]].
  * (12) B. Chen, R. Liu and Y. f. Zheng, SciPost Phys. 14 (2023), 088 doi:10.21468/SciPostPhys.14.5.088 [arXiv:2112.10514 [hep-th]].
  * (13) J. de Boer, J. Hartong, N. A. Obers, W. Sybesma and S. Vandoren, “Carroll Symmetry, Dark Energy and Inflation,” Front. in Phys. 10 (2022), 810405 doi:10.3389/fphy.2022.810405 [arXiv:2110.02319 [hep-th]].
  * (14) R. Casalbuoni and J. Gomis, “Conformal symmetry for relativistic point particles,” Phys. Rev. D 90 (2014) no.2, 026001 doi:10.1103/PhysRevD.90.026001 [arXiv:1404.5766 [hep-th]].
  * (15) J. Gomis and H. Ooguri, “Nonrelativistic closed string theory,” J. Math. Phys. 42 (2001), 3127-3151 doi:10.1063/1.1372697 [arXiv:hep-th/0009181 [hep-th]].
  * (16) J. Gomis, J. Gomis and K. Kamimura, “Non-relativistic superstrings: A New soluble sector of AdS(5) x S**5,” JHEP 12 (2005), 024 doi:10.1088/1126-6708/2005/12/024 [arXiv:hep-th/0507036 [hep-th]].
  * (17) A. Barducci, R. Casalbuoni and J. Gomis, “Confined dynamical systems with Carroll and Galilei symmetries,” Phys. Rev. D 98 (2018) no.8, 085018 doi:10.1103/PhysRevD.98.085018 [arXiv:1804.10495 [hep-th]].
  * (18) J. M. Pons, “Substituting fields within the action: Consistency issues and some applications,” J. Math. Phys. 51 (2010), 122903 doi:10.1063/1.3525242 [arXiv:0909.4151 [hep-th]].
  * (19) J. Gomis and A. Kleinschmidt, “Infinite-Dimensional Algebras as Extensions of Kinematic Algebras,” Front. in Phys. 10 (2022), 892812 doi:10.3389/fphy.2022.892812 [arXiv:2202.05026 [hep-th]].