# Anti-de Sitter Momentum Space in 3D and 4D Quantum Gravity

Giovanni Amelino-Camelia<EMAIL_ADDRESS>Dipartimento di Fisica Ettore
Pancini, Università di Napoli “Federico II”, and INFN, Sezione di Napoli,
Complesso Univ. Monte S. Angelo, I-80126 Napoli, Italy  Iarley P. Lobo
<EMAIL_ADDRESS>Dipartimento di Fisica, Università di Roma “La Sapienza",
P.le A. Moro 2, 00185 Roma, Italy  Department of Chemistry and Physics,
Federal University of Paraíba, Rodovia BR 079 - km 12, 58397-000 Areia-PB,
Brazil  Physics Department, Federal University of Lavras, Caixa Postal 3037,
37200-900 Lavras-MG, Brazil  Giovanni Palmisano<EMAIL_ADDRESS>Dipartimento di Fisica, Università di Roma “La Sapienza", P.le A. Moro 2,
00185 Roma, Italy  INFN, Sez. Roma1, P.le A. Moro 2, 00185 Roma, Italy

###### Abstract

There has been strong interest in the possibility that in the quantum-gravity
realm momentum space might be curved, mainly focusing, especially for what
concerns phenomenological implications, on the case of a de Sitter momentum
space. We here take as starting point the known fact that quantum gravity
coupled to matter in $2+1$ spacetime dimensions gives rise to an effective
picture characterized by a momentum space with anti-de Sitter geometry, and we
point out some key properties of $2+1$-dimensional anti-de Sitter momentum
space. We observe that it is impossible to implement all of these properties
in theories with a $3+1$-dimensional anti-de Sitter momentum space, and we
then investigate, with the aim of providing guidance to the relevant
phenomenology focusing on possible modified laws of conservation of momenta,
the implications of giving up, in the $3+1$-dimensional case, some of the
properties of the $2+1$-dimensional case.

##  I Introduction

Over the last few years a growing number of quantum-gravity studies has
focused on the possibility of having a Planck-scale-curved momentum space,
with the associated emergence of DSR-relativistic Amelinodsr1  spacetime
symmetries (“DSR” stands for “doubly-special”, or occasionally “deformed-
special”, relativity). This concerns the possibility that in the quantum-
gravity realm relativistic symmetries might be deformed in such a way that one
would not only have a speed scale $c$ as non-trivial relativistic invariant,
but also a length/inverse-momentum scale.

Besides its intrinsic conceptual interest, this scenario has also been studied
as a rare example of quantum-gravity scenario which provides some viable
phenomenological avenues Amelinodsr2 ; Amelino-Camelia2013 ; Addazi:2021xuf .
Additional elements of motivation for such studies are provided by the fact
that $2+1$-dimensional quantum gravity, the only quantum-gravity we have
managed to solve freidellivineeffective ; MatschullWelling , does indeed lead
to DSR-relativistic symmetries and curvature of momentum space. In $2+1$
spacetime dimensions the gravitational degrees of freedom are topological and
many results can be obtained more easily then in the $3+1$-dimensional case.
In particular, by describing particles as topological defects (“punctures”) of
the spacetime manifold, it is possible to integrate out the gravitational
degrees of freedom and the resulting effective model is characterized
freidellivineeffective  by a curved momentum space within a DSR-relativistic
picture RellocinAdS .

The 4D (3+1-dimensional) quantum gravity we care about surely is very
different from 3D quantum gravity, but, since the two theories have in common
some of their aspects of conceptual complexity, it may well be the case that
they also have some features in common. Perhaps the most appealing argument
for the emergence of DSR-relativistic symmetries in the 3+1D context is found
in some recent studies Amelino-Camelia:2016gfx ; Bojowald:2012ux ;
Brahma:2016tsq ; Cianfrani:2016ogm , where models inspired by the loop-
quantum-gravity approach were shown to lead to modifications of the
hypersurface deformation algebra: in turn those deformations of the
hypersurface deformation algebra were found to give rise to DSR-relativistic
symmetries in the appropriate limiting regime Amelino-Camelia:2016gfx ;
Bojowald:2012ux . It is however still difficult to guess from these
preliminary 4D results and from what is known about 3D quantum gravity what is
the type of DSR-relativistic symmetries that should be favoured for 4D quantum
gravity. Most related studies assumed a de Sitter geometry of momentum space
(see, e.g., Refs.GiuliaFlavio ; Carmona:2019fwf  and references therein). To
our knowledge the only study considering the anti-de Sitter momentum-space
possibility is the one reported in Ref.Arzano:2014jua , but it focused
exclusively on issues relevant for single-particle phenomenology (mainly
correction terms for the laws of propagation and a mechanism for dimensional
reduction in the ultra-violet regime).

We find it to be particularly noteworthy that the DSR-relativistic symmetries
of 3D quantum gravity are codified in a momentum space with anti-de Sitter
geometry freidellivineeffective ; MatschullWelling , in a way that is fully
compatible with the so-called “Relative Locality framework” RellocinAdS ;
principle , and admits formulation in terms of a Hopf algebra of relativistic
symmetries Schroers2011 . We here here take as working assumption that also in
4D quantum gravity it should be possible to describe curvature of momentum
space within the Relative Locality framework, and we investigate whether or
not one gets a sensible picture by adopting also for 4D quantum gravity a
momentum space with anti-de Sitter geometry. In fact, quantum gravity effects
being manifest in the form of curved momentum space and Relative Locality has
been recently explored in different frameworks Freidel:2013zga ;
Freidel:2015pka ; Freidel:2016pls ; Freidel:2018apz ; Freidel:2022ryr ;
Berglund:2023vrm . Of particular significance for phenomenology are our
results concerning possible laws of composition of momenta on an anti-de
Sitter momentum space, since they can affect particle reactions (see, e.g.,
Refs.Carmona:2014lqa  and references therein).

A key aspect of our analysis concerns the identification of some properties
that characterize the 3D law of composition of momenta, reflecting properties
of the anti-de Sitter geometry of momentum space: some interesting
implications arise investigating whether the same properties can be imposed in
the 4D setting. Empowered by the geometric tools introduced by the relative-
locality framework principle , we find appropriate to focus on four properties
of the law of composition of momenta, which we here list but will be more
carefully discussed later in the manuscript:

$R1$) Existence of identity;

$R2$) Associativity;

$R3$) Isometric action of the composition law on the geodesic distance in
momentum space;

$R4$) Relativistic covariance of the theory.

It is noteworthy that these 4 properties not only are present in 3D quantum
gravity, but are also present in a particular formulation of de Sitter
momentum space, the one amenable to interpretation GiuliaFlavio ;
arzano2011anatomy  in terms of the $\kappa$-Poincaré Hopf algebra
lukierski1991q ; Lukierski1992344 ; MajidRUEGG ; Amelinodsr1 . We here
conjecture that the availability of a Hopf-algebra interpretation is connected
to the validity of the properties $R1$, $R2$, $R3$ and $R4$.

Section II is devoted to a summary of relevant 3D-quantum-gravity results and
results on the relative-locality momentum space which admits $\kappa$-Poincaré
interpretation. In section III we show that adopting an anti-de Sitter
momentum space for 4D quantum gravity one is unable to enforce the
requirements $R1$, $R2$, $R3$ and $R4$: essentially there is an
incompatibility between $R3$ and $R4$ for 4D anti-de Sitter momentum space (as
implemented in the relative-locality framework). This is consistent with the
findings of previous studies (see, in particular, the appendix of
Ref.Carmona:2019fwf ), which however focused on scenarios with spatial
isotropy, whereas we here extend the result to the anisotropic case. Our main
objective is to provide guidance to the relevant phenomenology: having found
that $R1$, $R2$, $R3$ and $R4$ cannot be simultaneously implemented within a
$3+1$-dimensional anti-de Sitter setting, in section IV we highlight some
examples of viable physical pictures which are obtained by relaxing one or
another of these requirements. We provide further guidance to the
phenomenology of anti-de Sitter momentum space in Section V, where we consider
a much-studied DSR scenario, first introduced by Magueijo and Smolin
2002PhRvL..88s0403M , and show that it is an example of 4D setting adopting
anti-de Sitter momentum space (whereas previous studies had erroneously linked
it to de Sitter momentum space), and that it is a scenario such that only
$R1$, $R2$, $R4$ are enforced ($R3$ does not hold in the scenario of
Ref.2002PhRvL..88s0403M ).

##  II $2+1$ Gravity and $\kappa$-Poincaré momentum spaces

In this section we offer a brief summary of previous results which are
relevant for our analysis. We start by offering a sketchy description of the
relative-locality framework, the framework which provides us the tools for
analyzing the properties of relativistic kinematics in terms of geometric
structures. Then we discuss the results on 3D quantum gravity which are most
relevant for our analysis, and similarly we discuss some properties of the de
Sitter momentum space which admits description in terms of the
$\kappa$-Poincaré Hopf algebra. As we go along we shall characterize the
requirements $R1$, $R2$, $R3$ and $R4$ in greater detail then already done in
our introductory remarks.

###  II.1 The Relative Locality framework

We fix notation by describing the on-shell relation and the law of composition
of momenta, in general, in the following form:

$\begin{cases}m^{2}=d_{\kappa}^{2}(k),\\\
q=p\oplus_{\kappa}p^{\prime},\end{cases}$ (1)

where $q$ is the momentum obtained composing momenta $p$ and $p^{\prime}$,
while $m$ is the mass of a particle of momentum $k$. The functions
$d_{\kappa}^{2}$ and $\oplus_{\kappa}$ depend on momenta and (in general) a
deformation parameter $\kappa$, a momentum scale often identified with the
Planck scale (we work in units with the speed-of-light scale and Planck’s
constant set to 1).

Within the relative-locality framework, introduced in principle , one gives a
geometrical picture of relativistic kinematics, by associating the on-shell
relation to a metric on momentum space and the composition law to a connection
on momentum space, thereby characterizing momentum space as a Cartan manifold:

$\begin{array}[]{ccc}d_{\kappa}^{2}&\leftrightarrow&g^{\mu\nu},\\\
\oplus_{\kappa}&\leftrightarrow&\Gamma_{\lambda}^{\mu\nu}.\end{array}$ (2)

The on-shell relation is described as the geodesic distance of a momentum $p$
from the origin of momentum space, where the geodesic is evaluated according
to the metric $g^{\mu\nu}$ on momentum space. This means that:

$m^{2}=d_{\kappa}^{2}(p,0)=\int
dt\sqrt{g^{\mu\nu}\dot{\gamma}_{\mu}\dot{\gamma}_{\nu}}\,.$ (3)

For the association between composition law and connection on momentum space
the original proposal was principle ; balenascience :

$\Gamma_{\lambda}^{\mu\nu}(p)=-\frac{\partial}{\partial
q_{\mu}}\frac{\partial}{\partial
k_{\nu}}(q\oplus_{\kappa}^{[p]}k)_{\lambda}|_{q=k=p},$ (4)

where $\oplus^{[p]}$ is such that
$q\oplus_{\kappa}^{[p]}k=p\oplus_{\kappa}\left(\ominus_{\kappa}p\oplus_{\kappa}q\right)\oplus_{\kappa}\left(\ominus_{\kappa}p\oplus_{\kappa}k\right)$
and $\ominus_{\kappa}$ is the antipode operation, such that
$p\oplus_{\kappa}\left(\ominus_{\kappa}p\right)=0$.

Some of us advocated balenascience  an alternative association between
composition law and connection on momentum space such that to any pair of
momenta $q$ and $k$ one associates the autoparallel111An autoparallel curve
$\gamma$ is such that
$\frac{d^{2}\gamma_{\lambda}}{dt}+\Gamma_{\lambda}^{\mu\nu}\left(\gamma\right)\frac{d\gamma_{\mu}}{dt}\frac{d\gamma_{\nu}}{dt}=0$.
curves $\gamma^{(q)}$ and $\gamma^{(k)}$ which connect $q$ and $k$ to the
origin of momentum space. A parametric surface
$\gamma\left(s,t\right):[0,1]\times[0,1]\rightarrow M$ function of the
parameters $s$ and $t$ is then defined on the momentum space $M$ by:

$\begin{cases}\frac{d}{dt}\frac{d}{ds}\gamma_{\lambda}+\Gamma_{\lambda}^{\mu\nu}\left(\gamma\right)\frac{d\gamma_{\mu}}{dt}\frac{d\gamma_{\nu}}{ds}=0,\\\
\gamma_{\lambda}\left(s,0\right)=\gamma_{\lambda}^{(k)}\left(s\right),\\\
\gamma_{\lambda}\left(0,t\right)=\gamma_{\lambda}^{(q)}\left(t\right).\end{cases}$
(5)

The momentum composed of $q$ and $k$ is the point of momentum space given by:

$q\oplus_{\kappa}k=\gamma\left(1,1\right).$ (6)

In most cases of interest, and in particular when the composition law is
associative, the two ways to link a composition law with a connection on
momentum space are equivalent.

###  II.2 $2+1$ Gravity momentum space

As we mentioned in the introduction, in $2+1$ dimensions the description of
the interactions between particles and the gravitational field is such that
freidellivineeffective ; MatschullWelling  the gravitational degrees of
freedom can be integrated out, thereby providing an effective description of
the particles in which particle momenta live on a curved space, which turns
out to be the Lie group SL$\left(2,\mathbb{R}\right)$.

Any element of SL$\left(2,\mathbb{R}\right)$ can be written as
2014SIGMA..10..079A :

$\hat{p}=\frac{p_{4}}{\kappa}\hat{I}-2\frac{p_{\mu}}{\kappa}\hat{X}^{\mu},$
(7)

where the hat denotes matrix elements, $p_{4}$ and $p_{\mu}$ $(\mu=0,1,2)$ are
coefficients, $\hat{I}$ is the identity matrix and the $\hat{X}^{\mu}$ belongs
to the Lie algebra of SL$\left(2,\mathbb{R}\right)$, for example they can be
chosen as:

$\begin{array}[]{ccc}\hat{X}^{0}=\frac{1}{2}\left(\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right),&\hat{X}^{1}=\frac{1}{2}\left(\begin{array}[]{cc}0&-1\\\
-1&0\end{array}\right),&\hat{X}^{2}=\frac{1}{2}\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right).\end{array}$ (8)

The fact that $\hat{p}$ belongs to SL$\left(2,\mathbb{R}\right)$ means that
its determinant has to be $1$ and this translates into the condition:

$p_{4}^{2}+p_{\mu}p^{\mu}=\kappa^{2}.$ (9)

The coefficients $p_{4}$ and $p_{\mu}$ can then be interpreted as the
coordinates of a curved momentum space with an anti-de Sitter structure.

The on-shell relation can be constructed by taking the Killing form of the Lie
algebra of SL$\left(2,\mathbb{R}\right)$ RellocinAdS  and takes the form:

$d_{AdS}\left(p,0\right)=\begin{cases}\kappa\arccos\left(p_{4}/\kappa\right)\
;&p_{4}<\kappa\,,\\\ 0\ ;&p_{4}=\kappa\,,\\\
\kappa\operatorname{arccosh}\left(p_{4}/\kappa\right)\
;&p_{4}>\kappa\,,\end{cases}$ (10)

where $\kappa$ is usually identified with the Planck mass. The composition law
$p\oplus_{\kappa}q$ can be introduced in terms of MatschullWelling ;
RellocinAdS

$\hat{p}\cdot\hat{q}=\kappa^{-1}\left(p\oplus_{\kappa}q\right)_{4}\hat{I}-2\kappa^{-1}\left(p\oplus_{\kappa}q\right)_{\mu}\hat{X}^{\mu},$
(11)

where the dot stands for matrix multiplication.

From (7) and (11) we get:

$\begin{cases}\left(q\oplus_{\kappa}p\right)_{\lambda}=\frac{1}{\kappa}\left(q_{\lambda}p_{4}+q_{4}p_{\lambda}-\epsilon_{\lambda}^{\
\alpha\beta}q_{\alpha}p_{\beta}\right),\\\
\left(q\oplus_{\kappa}p\right)_{4}=\frac{1}{\kappa}\left(q_{4}p_{4}-q_{\alpha}p^{\alpha}\right).\end{cases}$
(12)

We adopt the Cartesian parameterization of anti-de Sitter, such that

$p_{4}=\begin{cases}\sqrt{\kappa^{2}-p^{2}}\ ;\ &p_{4}>0\,,\\\
-\sqrt{\kappa^{2}-p^{2}}\ ;\ &p_{4}<0\,,\end{cases}$ (13)

where $p^{2}=p_{\mu}p^{\mu}$ . Then the induced metric reads

$g^{\mu\nu}\left(p\right)=\eta^{\mu\nu}-\frac{p^{\mu}p^{\nu}}{p^{2}-\kappa^{2}}\,,$
(14)

and the associated geodesic distance is given by:

$d_{AdS}\left(q,k\right)=\begin{cases}\kappa\arccos\left[\left(q_{4}k_{4}+q_{\mu}k^{\mu}\right)/\kappa^{2}\right]\
;&q_{4}k_{4}+q_{\mu}k^{\mu}<\kappa^{2}\,,\\\ 0\
;&q_{4}k_{4}+q_{\mu}k^{\mu}=\kappa^{2}\,,\\\
\kappa\operatorname{arccosh}\left[\left(q_{4}k_{4}+q_{\mu}k^{\mu}\right)/\kappa^{2}\right]\
;&q_{4}k_{4}+q_{\mu}k^{\mu}>\kappa^{2}\,.\end{cases}$ (15)

By picking as origin the point $p_{4}=\kappa$, $p_{\mu}=0$, the on-shell
relation can be obtained from (3) and reduces to (10).

For what concerns the connection, since the composition law is associative one
can rely equivalently on (4) or (5), obtaining

$\Gamma_{\lambda}^{\mu\nu}\left(0\right)=\frac{1}{\kappa}\epsilon_{\lambda}^{\
\mu\nu}\,,$ (16)

It is easy to see that the on-shell relation and the composition law are both
invariant under the (undeformed) group of Lorentz transformations,
$p\rightarrow\Lambda(p)$:

$\begin{cases}d_{\kappa}\left(\Lambda\left(p\right),0\right)=d_{\kappa}\left(p,0\right),\\\
\Lambda\left(q\right)\oplus_{\kappa}\Lambda\left(k\right)=\Lambda\left(q\oplus_{\kappa}k\right).\end{cases}$
(17)

###  II.3 $\kappa$-Poincaré momentum space

The $\kappa$-Poincaré momentum space is the most studied scenario for deformed
relativistic kinematics. It can be characterized by the following on-shell
relation and law of composition of momenta KowalskiGlikman:2004tz :

$\begin{cases}d_{\kappa}\left(\tilde{p}\right)=\kappa\operatorname{arccosh}\left[\cosh\left(\tilde{p}_{0}/\kappa\right)-\frac{1}{2\kappa^{2}}\tilde{p}_{1}^{2}e^{p_{0}/\kappa}\right],\\\
\left(\tilde{q}\oplus_{\kappa}\tilde{k}\right)_{0}=\tilde{q}_{0}+\tilde{k}_{0}\,,\\\
\left(\tilde{q}\oplus_{\kappa}\tilde{k}\right)_{1}=\tilde{q}_{1}+\tilde{k}_{1}e^{-\tilde{q}_{0}/\kappa}\,,\end{cases}$
(18)

where for simplicity we are giving explicit formulas for the case of $1+1$
spacetime dimensions. The on-shell relation can be associated GiuliaFlavio  to
a de Sitter metric in the so called comoving coordinates (see appendix A):

$g^{\mu\nu}=\left(\begin{array}[]{cc}1&0\\\
0&-e^{-\tilde{p}_{0}/\kappa}\end{array}\right),$ (19)

and the composition law can be associated to a connection of the form:

$\Gamma_{\lambda}^{\mu\nu}\left(\tilde{p}\right)=\frac{1}{\kappa}\delta_{\lambda}^{1}\delta_{0}^{\mu}\delta_{1}^{\nu}\,.$
(20)

This connection is torsionful and its curvature vanishes.

The $\kappa$-Poincaré on-shell relation is invariant under the following
deformation of Lorentz transformations:

$\displaystyle\tilde{\Lambda}_{\kappa}\left(\xi,\tilde{p}\right)=\begin{cases}\tilde{\Lambda}{}_{0}\left(\tilde{p}\right)=&\kappa\ln\left\\{\frac{1}{\kappa}\left[\cosh\left(\xi\right)\left[\kappa\sinh\left(\tilde{p}_{0}/\kappa\right)+\frac{1}{2\kappa}\tilde{p}_{1}^{2}e^{\tilde{p}_{0}/\kappa}\right]+\sinh\left(\xi\right)e^{\tilde{p}_{0}/\kappa}\tilde{p}_{1}\right.\right.\\\
&\left.\left.+\kappa\cosh\left(\tilde{p}_{0}/\kappa\right)-\frac{1}{2\kappa}\tilde{p}_{1}^{2}e^{\tilde{p}_{0}/\kappa}\right]\right\\}\,,\\\
\tilde{\Lambda}{}_{1}\left(\tilde{p}\right)=&\kappa\frac{\cosh\left(\xi\right)e^{\tilde{p}_{0}/\kappa}\tilde{p}_{1}+\sinh\left(\xi\right)\left[\kappa\sinh\left(\tilde{p}_{0}/\kappa\right)+\frac{1}{2\kappa}\tilde{p}_{1}^{2}e^{\tilde{p}_{0}/\kappa}\right]}{\sinh\left(\xi\right)e^{\tilde{p}_{0}/\kappa}\tilde{p}_{1}+\cosh\left(\xi\right)\left[\kappa\sinh\left(\tilde{p}_{0}/\kappa\right)+\frac{1}{2\kappa}\tilde{p}_{1}^{2}e^{\tilde{p}_{0}/\kappa}\right]+\kappa\cosh\left(\tilde{p}_{0}/\kappa\right)-\frac{1}{2\kappa}\tilde{p}_{1}^{2}e^{\tilde{p}_{0}/\kappa}}\,,\end{cases}$
(21)

where $\xi$ is the rapidity parameter.

The composition law is covariant in a non-standard way: the action of the
deformed boosts (21) on the composition law is such that:

$\tilde{\Lambda}_{\kappa}\left(q\oplus_{\kappa}k,\xi\right)=\tilde{\Lambda}_{\kappa}\left(q,\xi\right)\oplus_{\kappa}\tilde{\Lambda}_{\kappa}\left(k,\xi^{\prime}\left(q\right)\right),$
(22)

where $\xi^{\prime}\left(q\right)$ is a function of $q$ of the form

$\xi^{\prime}=2\operatorname{arcsinh}\left(\frac{e^{-\tilde{p}_{0}/\kappa}\sinh\left(\xi/2\right)}{\sqrt{\left(\cosh\left(\xi/2\right)+\frac{1}{\kappa}\tilde{p}_{1}\sinh\left(\xi/2\right)\right)^{2}-e^{-2\tilde{p}_{0}/\kappa}\sinh^{2}\left(\xi/2\right)}}\right).$
(23)

We shall refer in the following to such kind of behavior as a “non-homomorphic
action” of the Lorentz boost on the composition law. Notice that the
covariance of the $\kappa$-Poincaré composition law is non-standard in two
ways, some of which are encoded in the form of (21), while others are encoded
in the fact that boosts act with a non-homomorphic action on the composition
law. Still, as shown in Refs.balenascience ; tesipalmisano , it is a fully
legitimate (logically consistent) way of enforcing relativistic covariance.

###  II.4 Some key properties of $2+1$ gravity and $\kappa$-Poincaré momentum
space

It is useful for our purposes to highlight a few key properties that are found
both for the composition law of $2+1$ gravity and for the composition law of
the $\kappa$-Poincaré momentum space. We start from property $R1$, the
composition laws admit an identity element, such that

$\begin{array}[]{cc}R1:&\begin{cases}q\oplus_{\kappa}p|_{p=0}=q\,,\\\
q\oplus_{\kappa}p|_{q=0}=p\,.\end{cases}\end{array}$ (24)

The composition laws of $2+1$ gravity and of the $\kappa$-Poincaré momentum
space also have properties $R2$ and $R3$, i.e. they preserve the geodesic
distance in momentum space,

$\begin{array}[]{cc}R2:&d_{\kappa}\left(p\oplus_{\kappa}q,p\oplus_{\kappa}k\right)=d_{\kappa}\left(q,k\right)\,;\end{array}$
(25)

and they are associative,

$\begin{array}[]{cc}R3:&\left(p\oplus_{\kappa}\left(q\oplus_{\kappa}k\right)\right)_{\lambda}=\left(\left(p\oplus_{\kappa}q\right)\oplus_{\kappa}k\right)_{\lambda}\,.\end{array}$
(26)

For what concerns property $R4$ (relativistic covariance) we have seen in
subsections II.2 and II.3 that the compatibility with a relativistic
description is encoded in the following covariance properties:

$\begin{array}[]{cc}&\begin{cases}2+1\
\text{Gravity}&\Lambda\left(q\oplus_{\kappa}k\right)=\Lambda\left(q\right)\oplus_{\kappa}\Lambda\left(k\right)\,,\\\
\mbox{$\kappa$}\text{-Poincar\'{e}}&\tilde{\Lambda}_{\kappa}\left(q\oplus_{\kappa}k,\xi\right)=\tilde{\Lambda}_{\kappa}\left(q,\xi\right)\oplus_{\kappa}\tilde{\Lambda}_{\kappa}\left(k,\xi^{\prime}\left(q\right)\right)\,.\end{cases}\end{array}$
(27)

In general, in a relativistic momentum space there will be a deformation
$\tilde{\Lambda}\left(p,\xi\right)$ of the Lorentz transformations, and
suitable functions $\xi^{\prime}\left(\xi,p\right)$ and
$\xi^{\prime\prime}\left(\xi,p\right)$ such that under the action of
$\tilde{\Lambda}$ the on-shell relation is invariant

$d_{\kappa}\left(\tilde{\Lambda}\left(p\right)\right)=d_{\kappa}\left(p\right)\,,$
(28)

and the composition law is covariant, meaning that

$\begin{array}[]{cc}R4:&\tilde{\Lambda}_{\kappa}\left(q\oplus_{\kappa}k,\xi\right)=\tilde{\Lambda}_{\kappa}\left(q,\xi^{\prime}\left(\xi,k\right)\right)\oplus_{\kappa}\tilde{\Lambda}_{\kappa}\left(k,\xi^{\prime\prime}\left(\xi,q\right)\right)\,.\end{array}$
(29)

The triple $\left(\tilde{\Lambda},\xi^{\prime},\xi^{\prime\prime}\right)$
should be global, meaning that for any momentum $p$ and any given finite $\xi$
we have that both $\tilde{\Lambda}\left(\xi,q\right)$, and
$\tilde{\Lambda}\left(q,\xi^{\prime}\left(\xi,k\right)\right)\oplus\tilde{\Lambda}\left(k,\xi^{\prime\prime}\left(\xi,q\right)\right)$
are finite.

In the following, a composition law satisfying the properties (24)-(29) with
respect to some metric will be called a “Hopf composition law” associated to
the given metric. We label as “Hopf” a composition law satisfying the
properties (24)-(29), since we shall conjecture a link between momentum spaces
governed by (24)-(29), and the mathematics of Hopf algebras.

One last point we want to remark here concerns the requirement of covariance
and particularly the possible distinction between the general realization of
covariance expressed in (29) and the requirement of homomorphic covariance,
i.e.

$\begin{array}[]{cc}R4b:&\tilde{\Lambda}\left(q\oplus_{\kappa}k,\xi\right)=\tilde{\Lambda}\left(q,\xi\right)\oplus_{\kappa}\tilde{\Lambda}\left(k,\xi\right)\,,\end{array}$
(30)

where we can see that the parameters identifying the Lorentz transformations
are all identical on both sides of the equation.

Although the distinction between the homomorphic and the non-homomorphic
realizations of covariance are useful in classifying the possible structures
of momentum spaces, we stress that both cases constitute a legitimate
implementation of the relativity principle, as discussed in balenascience ;
tesipalmisano .

We notice that for maximally symmetric momentum spaces parametrized in
Cartesian coordinates, the on-shell relation is preserved by the standard
undeformed Lorentz transformations $\Lambda$ and the requirement of
homomorphic covariance is equivalent to the condition that the composition law
can be expressed in the explicitly covariant form:

$\left(q\oplus_{\kappa}k\right)_{\lambda}=A\left(q,k\right)q_{\lambda}+B\left(q,k\right)k_{\lambda}\,,$
(31)

where $A$ and $B$ are scalar functions of $q$ and $k$ .

On the other hand, the requirement of covariance in the more general form
(29), is equivalent to the condition that:

$\left(q\oplus_{\kappa}k\right)_{\lambda}=A\left(q,k\right)\chi_{\lambda}^{\
\alpha}q_{\alpha}+B\left(q,k\right)\Omega_{\lambda}^{\ \alpha}k_{\alpha}\,,$
(32)

where $\chi$ and $\Omega$ are now arbitrary Lorentz transformations, functions
of $q,$ $k$ and any other fixed arbitrary momenta. For example, introducing a
single external vector $v$ we can consider Lorentz transformations of the
form:

$\Omega_{\lambda}^{\
\alpha}=\omega_{1}\left(q,v\right)\delta_{\lambda}^{\alpha}+\omega_{2}\left(q,v\right)q_{\lambda}q^{\alpha}+\omega_{3}\left(q,v\right)q_{\lambda}v^{\alpha}+\omega_{4}\left(q,v\right)q_{\lambda}q^{\alpha}+\omega_{5}\left(q,v\right)\epsilon_{\lambda}^{\
\alpha\beta\gamma}q_{\beta}v_{\gamma}\,,$ (33)

where $\omega_{n}$ ($n=1,...,5$) are scalar functions of $q$ and $v$.222Of
course the scalars $\omega_{n}$ are not independent; they have to be
compatible with the (pseudo) orthogonality conditions for $\Omega$.

The compatibility between the introduction of an external vector and the
covariance of the composition law will be discussed in subsection III.4.

###  II.5 “Proper de Sitter” and “Snyder-de Sitter” momentum spaces

While most of the work in the context of deformed relativistic kinematic has
been developed in ways that establish a consistency with the formalism of Hopf
algebras, it is important for our purposes to stress that alternatives to this
“Hopf-algebra paradigm” can also be legitimately considered, and in fact they
have been already considered in the literature, most notably in
Refs.balenascience ; 2014PhRvD..90g6010B . These are two cases of momentum
spaces with de Sitter metric not admitting a Hopf-algebraic description, and
therefore can serve as reference models for the investigations we shall also
report here on momentum spaces with anti-de Sitter metric not admitting a
Hopf-algebraic description.

We describe both the model of Ref.balenascience  and the model of
Ref.2014PhRvD..90g6010B  adopting Cartesian coordinates on de Sitter momentum
space.333In balenascience  Proper de Sitter momentum space has been introduced
and developed in comoving coordinates (see the appendix A). These can be
derived from the embedding coordinates (see the appendix A) by imposing:

$p_{4}=\begin{cases}\sqrt{\kappa^{2}+p^{2}}\ ;\ &p_{4}>0\,,\\\
-\sqrt{\kappa^{2}+p^{2}}\ ;\ &p_{4}<0\,.\end{cases}$ (34)

In particular, both in Ref.balenascience  and in Ref.2014PhRvD..90g6010B  work
within the $p_{4}>0$ sector of de Sitter momentum space. The induced metric is

$g^{\mu\nu}\left(p\right)=\eta^{\mu\nu}-\frac{p^{\mu}p^{\nu}}{p^{2}+\kappa^{2}}\,,$
(35)

and the associated on-shell relation is

$d_{\kappa}^{2}\left(p\right)=\kappa^{2}\operatorname{arccosh}^{2}\left(\sqrt{p^{2}+\kappa^{2}}/\kappa\right)\,.$
(36)

The “Proper de Sitter” composition law of Ref.balenascience  is defined as the
composition law (5) associated to the Levi Civita de Sitter connection that in
Cartesian coordinates is given by
$\Gamma_{\lambda}^{\mu\nu}\left(p\right)=\frac{1}{\kappa}p_{\lambda}g^{\mu\nu}\left(p\right)$.

At second order in the Planck length $\kappa^{-1}$ it is possible
balenascience  to give a general solution to the equations (5) in the form

$\displaystyle\left(q\oplus_{\kappa}p\right)_{\lambda}=q_{\lambda}+k_{\lambda}-\frac{1}{\kappa}\Gamma_{\lambda}^{\alpha\beta}\left(0\right)q_{\alpha}k_{\beta}-\frac{1}{\kappa^{2}}\left\\{\left(\partial^{\rho}\Gamma_{\lambda}^{\alpha\beta}\left(0\right)\right)q_{\alpha}k_{\beta}\left(\frac{q_{\rho}+k_{\rho}}{2}\right)\right.$
$\displaystyle\left.-\frac{1}{2}\left(\Gamma_{\lambda}^{\alpha\beta}\Gamma_{\alpha}^{\gamma\delta}\right)\left(0\right)q_{\gamma}k_{\delta}k_{\beta}-\frac{1}{2}\left(\Gamma_{\lambda}^{\alpha\beta}\Gamma_{\beta}^{\gamma\delta}\right)\left(0\right)q_{\alpha}q_{\gamma}k_{\delta}\right\\}\,.$
(37)

Substituting the Levi Civita connection into (37) one gets

$\left(q\oplus_{\kappa}p\right){}_{\lambda}=\left(q_{\lambda}+k_{\lambda}\right)\left(1-\frac{1}{2\kappa^{2}}q\cdot
k\right)\,.$ (38)

We notice that the composition law (38) satisfies $R1$ and is commutative but
not associative. Furthermore it satisfies $R4b$ with respect to the standard
Lorentz boosts:

$\Lambda\left(q\oplus_{\kappa}k\right)=\Lambda\left(q\right)\oplus_{\kappa}\Lambda\left(k\right)\,.$
(39)

The “Snyder” composition law of Ref.2014PhRvD..90g6010B  is

$\begin{cases}(p\oplus
q)_{\mu}=\frac{1}{\kappa}p_{\mu}q_{4}+\left[\delta_{\mu}^{\alpha}+\frac{p_{\mu}p^{\alpha}}{\kappa(\kappa+p_{4})}\right]q_{\alpha}\,,\\\
(p\oplus q)_{4}=\frac{1}{\kappa}\left(p_{4}q_{4}+p_{\mu}q^{\mu}\right)\
\end{cases}$ (40)

It satisfies the requirements $R1$, $R2$ and $R4b$ (again, with respect to the
standard Lorentz boost) and is non-associative and non-commutative.

##  III $3+1$-dimensional anti-de Sitter momentum spaces

In the previous section we have highlighted four properties for the laws of
composition of momenta in $2+1$ gravity and in the $\kappa$-Poincaré momentum
space: $R1$ (existence of identity), $R2$ (isometricity), $R3$
(associativity), and $R4$ (relativistic covariance, with variant $R4b$ for
homomorphic covariance).

In this section we will show that these four properties $R1$-$R4$ enjoyed by
the laws of composition of momenta in $2+1$ gravity and in the
$\kappa$-Poincaré momentum space, cannot be enforced in the case of the
$3+1$-dimensional anti-de Sitter momentum space.

###  III.1 Geodesic distance and on-shell relation

We start by identifying the form of the geodesic distance in $3+1$ dimensional
anti-de Sitter momentum space. The anti-de Sitter manifold can be
characterized as a hypersurface, embedded in a $3+2$ dimensional flat space,
characterized by:

$p_{0}^{2}-p_{1}^{2}-p_{2}^{2}-p_{3}^{2}+p_{4}^{2}=\kappa^{2}\,.$ (41)

The geodesic distance between any pair of momenta $q$ and $k$ is:

$d_{AdS}\left(q,k\right)=\begin{cases}\kappa\arccos\left[\left(q_{4}k_{4}+q_{\mu}k^{\mu}\right)/\kappa^{2}\right]\
;\ &q_{4}k_{4}+q_{\mu}k^{\mu}<\kappa^{2}\,,\\\ 0\ ;\
&q_{4}k_{4}+q_{\mu}k^{\mu}=\kappa^{2}\,,\\\
\kappa\operatorname{arccosh}\left[\left(q_{4}k_{4}+q_{\mu}k^{\mu}\right)/\kappa^{2}\right]\
;\ &q_{4}k_{4}+q_{\mu}k^{\mu}>\kappa^{2}\,.\end{cases}$ (42)

As specified by the relative-locality framework principle , the on-shell
relation can be inferred from (42) by choosing as origin of our momentum space
the point $p_{O}=\left(0,0,0,0,\kappa\right)$, obtaining Arzano:2014jua :

$m_{AdS}\left(p\right)=d_{AdS}\left(p,p_{O}\right)=\begin{cases}\kappa\arccos\left(p_{4}/\kappa\right)\
;\ &p_{4}<\kappa\,,\\\ 0\ ;\ &p_{4}=\kappa\,,\\\
\kappa\operatorname{arccosh}\left(p_{4}/\kappa\right)\ ;\
&p_{4}>\kappa\,.\end{cases}$ (43)

###  III.2 General isometric composition laws

Our next task is to identify what are the isometric composition laws, i.e. the
composition laws which satisfy the first three properties $R1$, $R2$, $R3$.

The requirement of invariance of the geodesic distance under the action of the
composition of momenta, characterizes the composition law as an element of
$\text{SO}\left(2,3\right)/SO\left(1,3\right)$ which is the isometry group of
anti-de Sitter quotiented by the subgroup of isometries that preserve the
origin we picked for the momentum space. In particular it can be written as a
linear action:

$\left(p\oplus_{H}q\right)=T\left(p\right)\cdot q\,,$ (44)

where $\oplus_{H}$ stands for “Hopf composition law”, $q$ is a five
dimensional vector and $T$ is a $5\times 5$ matrix of
$\text{SO}\left(2,3\right)$ and a suitable function of $p$.444A similar
approach has been recently explored in Carmona:2019vsh ; Carmona:2019fwf ;
Relancio:2022mia ; Carmona:2021pxw ; Relancio:2021ahm ; Carmona:2021gbg .

We can write $T$ in all generality as:

$T\left(p\right)=\bar{T}\left(p\right)\Omega\left(p\right),$ (45)

where $\Omega\left(p\right)$ is an arbitrary Lorentz transformation and
$\bar{T}$ is such that $R1$ and $R2$ are satisfied for any choice of
$\Omega\left(p\right)$.

It is easy to see that the anti-de Sitter geodesic distance (42) is preserved
by the action of a composition law of the form:

$\left(p\oplus_{H}q\right)_{I}=\left[\frac{1}{\kappa}p_{I}\delta_{4}^{J}-\frac{1}{\kappa}p^{\alpha}\delta_{I}^{4}\delta_{\alpha}^{J}+\delta_{I}^{\lambda}\left(\delta_{\lambda}^{\alpha}-\frac{p_{\lambda}p^{\alpha}}{\kappa(\kappa+p_{4})}\right)\delta_{\alpha}^{J}\right]q_{J}\,,$
(46)

or, in a more explicit form:

$\begin{cases}\left(p\oplus_{H}q\right)_{\lambda}=\frac{1}{\kappa}p_{\lambda}q_{4}+\left(\delta_{\lambda}^{\alpha}-\frac{p_{\lambda}p^{\alpha}}{\kappa(\kappa+p_{4})}\right)q_{\alpha}\,,\\\
\left(p\oplus_{H}q\right)_{4}=\frac{1}{\kappa}\left(p_{4}q_{4}-p_{\mu}q^{\mu}\right)\,.\end{cases}$
(47)

A more general possibility, provided by the possible choices of $\Omega$ in
(45), has the form:

$\begin{cases}\left(p\oplus_{H}q\right)_{\lambda}=\frac{1}{\kappa}p_{\lambda}q_{4}+\left(\delta_{\lambda}^{\alpha}-\frac{p_{\lambda}p^{\alpha}}{\kappa(\kappa+p_{4})}\right)\Omega\left(p\right)_{\alpha}^{\
\beta}q_{\beta}\,,\\\
\left(p\oplus_{H}q\right)_{4}=\frac{1}{\kappa}\left(p_{4}q_{4}-p^{\alpha}\Omega\left(p\right)_{\alpha}^{\
\beta}q_{\beta}\right)\,.\end{cases}$ (48)

This is the general form of an anti-de Sitter isometric composition law.

###  III.3 Explicitly covariant isometric composition laws

Our next step is to look for a form of Lorentz transformations $\Omega$ such
that the composition law is associative. We first look for Lorentz
transformations $\Omega$ with an explicitly covariant form. In $2+1$
dimensions such kind of transformations can be written in all generality as
follows:

$\Omega_{\alpha}^{\
\beta}=\omega_{1}\left(p\right)\delta_{\alpha}^{\beta}+\frac{1}{\kappa^{2}}\omega_{2}\left(p\right)p_{\alpha}p^{\beta}+\frac{1}{\kappa}\omega_{3}\left(p\right)\epsilon_{\alpha}^{\
\beta\gamma}p_{\gamma}\,,$ (49)

where $\omega_{1}$, $\omega_{2}$ and $\omega_{3}$ are scalars that are
constrained by the orthonormality conditions:

$\begin{cases}\left(\omega_{3}\right)^{2}=\frac{\kappa^{2}}{p^{2}}\left[1-\left(\omega_{1}\right)^{2}\right]\,,\\\
2\left(\omega_{1}\right)\left(\omega_{2}\right)+\left(\omega_{2}\right)^{2}\frac{p^{2}}{\kappa^{2}}-\frac{\kappa^{2}}{p^{2}}\left[1-\left(\omega_{1}\right)^{2}\right]=0\,.\end{cases}$
(50)

Evidently one of the scalars $\omega_{1}$, $\omega_{2}$ or $\omega_{3}$ can be
arbitrarily chosen, and in particular the choice

$\begin{array}[]{ccc}\omega_{1}=\frac{p_{4},}{\kappa},&\omega_{2}=\frac{\kappa}{\kappa+p_{4}},&\omega_{3}=-1\end{array}$
(51)

leads to the associative composition law (12) arising in $2+1$ gravity, as one
can easily check.

In the four dimensional case, on the other hand, the most general explicitly
covariant Lorentz transformation has the form:

$\Omega_{\alpha}^{\
\beta}=\omega_{1}\left(p\right)\delta_{\alpha}^{\beta}+\frac{1}{\kappa^{2}}\omega_{2}\left(p\right)p_{\alpha}p^{\beta}\,,$
(52)

where the scalar functions $\omega_{1}$ and $\omega_{2}$ are fixed by the
orthonormality conditions to be $\omega_{1}=1$ and $\omega_{2}$ either zero or
equal to $-\frac{2\kappa^{2}}{p^{2}}$. The arising composition laws would have
then the two possible forms:

$\begin{cases}1)&\begin{cases}\left(p\oplus
q\right)_{\lambda}=\frac{1}{\kappa}p_{\lambda}q_{4}+\left(\delta_{\lambda}^{\alpha}-\frac{p_{\lambda}p^{\alpha}}{\kappa(\kappa+p_{4})}\right)q_{\alpha}\,,\\\
\left(p\oplus
q\right)_{4}=\frac{1}{\kappa}\left(p_{4}q_{4}-p^{\alpha}q_{\alpha}\right)\,.\end{cases}\\\
2)&\begin{cases}\left(p\oplus
q\right)_{\lambda}=\frac{1}{\kappa}p_{\lambda}q_{4}+\left(\delta_{\lambda}^{\alpha}-\frac{p_{\lambda}p^{\alpha}}{\kappa(\kappa-
p_{4})}\right)q_{\alpha}\,,\\\ \left(p\oplus
q\right)_{4}=\frac{1}{\kappa}\left(p_{4}q_{4}+p^{\alpha}q_{\alpha}\right)\,,\end{cases}\end{cases}$
(53)

which remind us of the Snyder composition law discussed in subsection II.5.

We notice that both the composition laws in (53) are not associative.
Therefore we conclude that in $3+1$ dimensions it is not possible to find a
composition law satisfying $R1$, $R2$, $R3$ and $R4b$. This is consistent with
and generalizes the result reported in the appendix of Carmona:2019fwf , which
showed the incompatibility of the conditions $R1$-$R4$. Our observation is
stronger since we also contemplated replacing $R4$ with the weaker $R4b$, and
still found incompatibility. From the phenomenology perspective it is
important to notice that the main difference between $R4$ and $R4b$ is that
$R4$ assumes spatial isotropy, not assumed in $R4b$. In the appendix of
Carmona:2019fwf , it was it is shown that, assuming isotropy, it is not
possible to find a Lie subalgebra of translation generators in four
dimensional anti-de Sitter space, which in turn indeed implies that there is
no isotropic, associative, covariant composition law defined from the
translation generators in four-dimensional anti-de Sitter space, i.e. the
conditions $R1$-$R4$.

The main goal of the study we are here reporting, to which we devote the
remaining subsections of this section, is to provide guidance to the
phenomenology of 4D anti-de Sitter momentum space, taking as starting point
the incompatibility between the requirements $R1$, $R2$, $R3$ and $R4$ (or
$R4b$). If there had been compatibility between these requirements then that
scenario would have been a natural first target for the phenomenology. That
not being the case (and the option of exploring in full generality the
phenomenology of anti-de Sitter momentum space not being an available option,
considering the smallness of the Planck length) it is valuable to identify
some target scenarios which might deserve priority in phenomenology, and of
course renounce to one or another of the conditions $R1$, $R2$, $R3$ and
$R4$(/$R4b$).

###  III.4 Aside on the possibility to describe general implementations of the
relativity principle

The results of the previous subsection on the impossibility of finding a
composition law satisfying $R1$, $R2$, $R3$ and $R4b$, mean that in $3+1$
dimensions, with anti-de Sitter momentum space, one cannot reproduce exactly
the structure encountered in 2+1-dimensional quantum gravity. Our next task is
to establish whether or not it is at least possible to trade $R4{b}$ for the
requirement of relativistic covariance $R4$. This is inspired by the fact that
in the de Sitter case by enforcing $R4$ (but not $R4b$) one finds the
interesting case of the $\kappa$-Poincaré momentum space.

Once we drop the request of explicit covariance, we can choose the Lorentz
transformation $\Omega$ with a more general form than the one we considered in
(52). The possibility that we will consider consists in the introduction of an
external vector $v$, that we call the stabilizer, in terms of which $\Omega$
can be written, in all generality as:

$\displaystyle\Omega_{\alpha}^{\
\beta}=\omega_{1}\left(p^{2},v\right)\delta_{\alpha}^{\beta}+\frac{1}{\kappa^{2}}\omega_{2}\left(p^{2},v\right)p_{\alpha}p^{\beta}+\frac{1}{\kappa}\omega_{3}\left(p^{2},v\right)\epsilon_{\alpha}^{\
\beta\gamma\delta}p_{\gamma}v_{\delta}$
$\displaystyle+\frac{1}{\kappa}\omega_{4}\left(p^{2},v\right)v_{\alpha}p^{\beta}+\frac{1}{\kappa}\omega_{5}\left(p^{2},v\right)p_{\alpha}v^{\beta}\,,$
(54)

where $\epsilon_{\alpha}^{\ \beta\gamma\delta}$ is the Levi Civita symbol and
the scalars $\omega_{n}$ are constrained by the orthonormality conditions:

$\begin{cases}\left(\omega_{1}\right)^{2}+\frac{1}{\kappa^{2}}\left(\omega_{3}\right)^{2}\left(p^{2}v^{2}-\left(p\cdot
v\right)^{2}\right)=0\,,\\\
\left(\omega_{5}\right)^{2}=\left(\omega_{3}\right)^{2}\,,\\\
\frac{1}{\kappa}\left(p\cdot
v\right)\left(\omega_{3}\right)^{2}+\omega_{1}\omega_{4}+\omega_{1}\omega_{5}+\frac{1}{\kappa^{2}}\omega_{2}\omega_{5}p^{2}+\frac{1}{\kappa}\omega_{4}\omega_{5}\left(v\cdot
p\right)=0\,,\\\
2\omega_{1}\omega_{2}+\frac{1}{\kappa^{2}}\left(\omega_{2}\right)^{2}p^{2}+\left(\omega_{4}\right)^{2}v^{2}+\frac{1}{\kappa}2\omega_{4}\omega_{2}\left(v\cdot
p\right)-\left(\omega_{3}\right)^{2}v^{2}=0\,.\end{cases}$ (55)

In the next subsection we will discuss whether it is possible to choose the
scalars $\omega_{n}$ ($n=1,...,5$) compatibly with (55), and in such a way
that the composition law is associative. First here we discuss how the
introduction of an external vector $v$ can be compatible with the overall DSR
relativistic compatibility of the composition law. As it has been shown in
tesipalmisano , the key point is given by the fact that $v$ enters the
composition law just inside the Lorentz transformations $\Omega$. Standard
properties of the Lorentz transformations ensure that for any such
transformation $\Lambda_{\xi}$ associated to a rapidity parameter $\xi$, we
have:

$\Omega\left(\Lambda_{\xi}q;v\right)=\left(\Lambda_{\xi}\Omega\left(q;v\right)\bar{\Lambda}_{\xi}\right)\bar{\Lambda}\left(\xi^{\prime}\right)\,,$
(56)

where $\bar{\Lambda}$ is the inverse of $\Lambda$ and $\xi^{\prime}$ is a set
of parameters, function of $q$ and $v$, defined by:555The possibility to find
the explicit solution for $\xi^{\prime}$ is ensured by the fact that the term
$\Lambda_{\xi}\bar{\Omega}\left(q;v\right)\Omega\left(q;\bar{\Lambda}_{\xi}v\right)\bar{\Lambda}_{\xi}$
is a combination of Lorentz transformations and it is then also a Lorentz
transformation.

$\bar{\Lambda}\left(\xi^{\prime}\right)=\Lambda_{\xi}\bar{\Omega}\left(p;v\right)\Omega\left(p;\bar{\Lambda}_{\xi}v\right)\bar{\Lambda}_{\xi}\,.$
(57)

By using (56) it is then easy to verify that the composition law (48)
satisfies the generalized form of covariance:

$\Lambda_{\xi}\left(q\oplus_{H}k\right)=\Lambda_{\xi}\left(q\right)\oplus_{H}\Lambda_{\xi^{\prime}}\left(k\right)\,.$
(58)

We refer to balenascience ; tesipalmisano  for a discussion on the possibility
to characterize (58) as a legitimate implementation of the relativity
principle.

###  III.5 Unavailability of Hopf composition laws in the anti-de Sitter case

Our next task is to find the scalars $\omega_{n}$ in (54), compatible with
(55) and such that the overall composition law is associative. A solution can
be found for $v_{\lambda}=\delta_{\lambda}^{1}$ in the form:

$\left(q\oplus_{H}k\right)_{\lambda}=\frac{1}{\kappa}\left(k\cdot
v+k_{4}\right)q_{\lambda}+k_{\lambda}+v_{\lambda}\frac{1}{\left(q\cdot
v+q_{4}\right)}\left(-q\cdot k+\left(\kappa-q_{4}\right)k\cdot v\right)$ (59)

and making $v$ explicit, we have:

$\left(q\oplus_{H}k\right)_{\lambda}=\frac{1}{\kappa}\left(k_{1}+k_{4}\right)q_{\lambda}+k_{\lambda}+\delta_{\lambda}^{1}\frac{1}{\left(q_{1}+q_{4}\right)}\left(-q\cdot
k+\left(\kappa-q_{4}\right)k_{1}\right)\,.$ (60)

In the appendix B we provide the proof of the uniqueness of the solution (60)
up to second order in the Planck length. Assuming that also in the exact case
the solution is unique, let us analyze its structure. The point of interest
for us is the fact that the above composition law is not defined for
$q_{1}+q_{4}=0$. A way out of this problem could be found by performing a
change of coordinates such that the point $q_{1}+q_{4}$ is sent to infinity.
An example of this kind are the so called Horospherical coordinates defined in
the sector $p_{1}+p_{4}>0$ (for simplicity we work in $1+1$ dimensions, but
the result is valid for any dimensions) by:

$\begin{cases}\tilde{p}_{1}=\kappa\ln\left[\frac{1}{\kappa}\left(p_{1}+p_{4}\right)\right]=f_{1}\left(p\right),\\\
\tilde{p}_{0}=\frac{\kappa p_{0}}{p_{1}+p_{4}}=f_{0}\left(p\right),\\\
p_{1}=\kappa\sinh\left(\frac{1}{\kappa}\tilde{p}_{1}\right)+\frac{1}{2\kappa}\tilde{p}_{0}^{2}e^{\tilde{p}_{1}/\kappa}=\bar{f}_{1}\left(\tilde{p}\right),\\\
p_{0}=e^{\tilde{p}_{1}/\kappa}\tilde{p}_{0}=\bar{f}_{0}\left(\tilde{p}\right),\\\
p_{4}=\kappa\cosh\left(\frac{1}{\kappa}\tilde{p}_{1}\right)+\frac{1}{2\kappa}\tilde{p}_{0}^{2}e^{\tilde{p}_{1}/\kappa}=\digamma\left(\tilde{p}\right).\end{cases}$
(61)

In the Horospherical coordinates the composition laws indeed are free from
singularities and take the form:666We use here the same symbol $\oplus_{H}$
for the composition law both in embedding and in comoving coordinates in order
to simplify the notation.

$\begin{cases}\left(\tilde{q}\oplus_{H}\tilde{k}\right)_{0}=\tilde{q}_{0}+\tilde{k}_{0}e^{-\tilde{q}_{1}/\kappa}\,,\\\
\left(\tilde{q}\oplus_{H}\tilde{k}\right)_{1}=\tilde{q}_{1}+\tilde{k}_{1}\,.\end{cases}$
(62)

The problem here arises when we map the (standard) Lorentz transformations
which preserve the on-shell relation (43) to horospherical coordinates. We
find in fact that:

$\displaystyle\tilde{\Lambda}\left(\xi,\tilde{p}\right)=\begin{cases}\tilde{\Lambda}{}_{0}\left(\tilde{p}\right)=&\kappa\frac{\cosh\left(\xi\right)e^{\tilde{p}_{1}/\kappa}\tilde{p}_{0}+\sinh\left(\xi\right)\left[\kappa\sinh\left(\tilde{p}_{1}/\kappa\right)+\frac{1}{2\kappa}\tilde{p}_{0}^{2}e^{\tilde{p}_{1}/\kappa}\right]}{\sinh\left(\xi\right)e^{\tilde{p}_{1}/\kappa}\tilde{p}_{0}+\cosh\left(\xi\right)\left[\kappa\sinh\left(\tilde{p}_{1}/\kappa\right)+\frac{1}{2\kappa}\tilde{p}_{0}^{2}e^{\tilde{p}_{1}/\kappa}\right]+\kappa\cosh\left(\tilde{p}_{1}/\kappa\right)-\frac{1}{2\kappa}\tilde{p}_{0}^{2}e^{\tilde{p}_{1}/\kappa}}\,,\\\
\tilde{\Lambda}{}_{1}\left(\tilde{p}\right)=&\kappa\ln\left\\{\frac{1}{\kappa}\left[\cosh\left(\xi\right)\left[\kappa\sinh\left(\tilde{p}_{1}/\kappa\right)+\frac{1}{2\kappa}\tilde{p}_{0}^{2}e^{\tilde{p}_{1}/\kappa}\right]+\sinh\left(\xi\right)e^{\tilde{p}_{1}/\kappa}\tilde{p}_{0}+\right.\right.\\\
&\left.\left.\kappa\cosh\left(\tilde{p}_{1}/\kappa\right)-\frac{1}{2\kappa}\tilde{p}_{0}^{2}e^{\tilde{p}_{1}/\kappa}\right]\right\\}\,,\end{cases}$
(63)

and these are ill-defined for:

$\cosh\left(\xi\right)\bar{f}_{1}\left(\tilde{p}\right)+\sinh\left(\xi\right)\bar{f}_{0}\left(\tilde{p}\right)+F\left(\tilde{p}\right)=0\,,$
(64)

which is always solvable.

The main point is that the map (61) is ill-defined for $p_{1}+p_{4}=0$ and
that this constraint breaks evidently the covariance of the theory. A similar
observation was also done in Arzano:2014jua . This is a general result in that
in order to avoid the singularity in $p_{1}+p_{4}$ we have to use a map that
is singular in $p_{1}+p_{4}$.

We therefore conclude that it is not possible, when adopting the
$3+1$-dimensional anti-de Sitter metric, to find a composition law satisfying
the Hopf properties (24)-(29).

###  III.6 Hopf composition laws for the de Sitter case

In relation to the obstruction we uncovered for the anti-de Sitter case, one
might gain some insight by looking at the composition law that is found for
the de Sitter case by enforcing isometricity and associativity:

$\left(q\oplus_{H}k\right)_{\lambda}=\frac{1}{\kappa}\left(k_{0}+k_{4}\right)q_{\lambda}+k_{\lambda}+\delta_{\lambda}^{0}\frac{1}{\left(q_{0}+q_{4}\right)}\left(-q\cdot
k+\left(\kappa-q_{4}\right)k_{0}\right).$ (65)

This composition law is evidently ill defined for $q_{0}+q_{4}=0$, but in the
de Sitter case it is possible to enforce the requirement $q_{0}+q_{4}>0$ as a
relativistic feature, since $q_{0}+q_{4}>0$ is invariant under the action of
the Lorentz boosts on the de Sitter momentum space and is also preserved by
the composition law, meaning that if $q_{0}+q_{4}>0$ and $k_{0}+k_{4}>0$, than
we will also have
$\left(q\oplus_{H}k\right)_{0}+\left(q\oplus_{H}k\right)_{4}>0$. Moreover,
there is no obstruction in setting $q_{4}>0$ and $q_{0}>0$, so that
$q_{0}+q_{4}>0$ is guaranteed. On the contrary, in the anti-de Sitter case the
condition $q_{1}>0$ would introduce, severe interpretational challenges
($q_{0}>0$ is for positive energies, which is physically desireable, while
$q_{1}>0$ would suggest positive spatial momentum in the $1$ direction, the
pathological requirement of motion only in the positive direction of a spatial
axis).

We therefore have a good picture ($q_{4}>0$ and $q_{0}>0$) such that in the de
Sitter case the composition law is nonsingular. Still it is interesting to
notice that in the de Sitter case the removal of the divergence for
$q_{0}+q_{4}=0$ has different outcome with respect to the anti-de Sitter case.
By adopting comoving coordinates defined (in the portion of de Sitter
$p_{0}+p_{4}>0$) by:

$\begin{cases}\tilde{p}_{0}=\kappa\ln\left[\frac{1}{\kappa}\left(p_{0}+p_{4}\right)\right]=f_{0}\left(p\right),\\\
\tilde{p}_{1}=\frac{\kappa p_{1}}{p_{0}+p_{4}}=f_{1}\left(p\right),\\\
p_{0}=\kappa\sinh\left[\frac{1}{\kappa}\tilde{p}_{0}\right]+\frac{1}{2\kappa}\tilde{p}_{1}^{2}e^{\tilde{p}_{0}/\kappa}=\bar{f}_{0}\left(\tilde{p}\right),\\\
p_{1}=e^{\tilde{p}_{0}/\kappa}p_{1}=\bar{f}_{1}\left(\tilde{p}\right),\\\
p_{4}=\kappa\cosh\left(\frac{1}{\kappa}\tilde{p}_{0}\right)+\frac{1}{2\kappa}\tilde{p}_{1}^{2}e^{\tilde{p}_{0}/\kappa}=\digamma\left(\tilde{p}\right),\end{cases}$
(66)

the composition law (65) takes the form:

$\begin{cases}\left(\tilde{q}\oplus_{H}\tilde{k}\right)_{0}=\tilde{q}_{0}+\tilde{k}_{0}\,,\\\
\left(\tilde{q}\oplus_{H}\tilde{k}\right)_{1}=\tilde{q}_{1}+\tilde{k}_{1}e^{-\tilde{q}_{0}/\kappa}\,,\end{cases}$
(67)

which is clearly well-defined and corresponds to the $\kappa$-Poincaré
composition law discussed in subsection II.3. In particular the Lorentz boost
in comoving coordinates takes the form (21) that here we report again:

$\displaystyle\tilde{\Lambda}\left(\xi,\tilde{p}\right)=\begin{cases}\tilde{\Lambda}{}_{0}\left(\tilde{p}\right)=&\kappa\ln\left\\{\frac{1}{\kappa}\left[\cosh\left(\xi\right)\left[\kappa\sinh\left(\tilde{p}_{0}/\kappa\right)+\frac{1}{2\kappa}\tilde{p}_{1}^{2}e^{\tilde{p}_{0}/\kappa}\right]+\sinh\left(\xi\right)e^{\tilde{p}_{0}/\kappa}\tilde{p}_{1}\right.\right.\\\
&\left.\left.+\kappa\cosh\left(\tilde{p}_{0}/\kappa\right)-\frac{1}{2\kappa}\tilde{p}_{1}^{2}e^{\tilde{p}_{0}/\kappa}\right]\right\\}\,,\\\
\tilde{\Lambda}{}_{1}\left(\tilde{p}\right)=&\kappa\frac{\cosh\left(\xi\right)e^{\tilde{p}_{0}/\kappa}\tilde{p}_{1}+\sinh\left(\xi\right)\left[\kappa\sinh\left(\tilde{p}_{0}/\kappa\right)+\frac{1}{2\kappa}\tilde{p}_{1}^{2}e^{\tilde{p}_{0}/\kappa}\right]}{\sinh\left(\xi\right)e^{\tilde{p}_{0}/\kappa}\tilde{p}_{1}+\cosh\left(\xi\right)\left[\kappa\sinh\left(\tilde{p}_{0}/\kappa\right)+\frac{1}{2\kappa}\tilde{p}_{1}^{2}e^{\tilde{p}_{0}/\kappa}\right]+\kappa\cosh\left(\tilde{p}_{0}/\kappa\right)-\frac{1}{2\kappa}\tilde{p}_{1}^{2}e^{\tilde{p}_{0}/\kappa}}\,,\end{cases}$
(68)

It is easy to check that, unlike the analogous anti-de Sitter case, the
transformation (68) is well-defined in the sector $p_{0}>0$.777The boost (68)
would be ill defined for
$\cosh\left(\xi\right)\bar{f}_{0}\left(\tilde{p}\right)+\sinh\left(\xi\right)\bar{f}_{1}\left(\tilde{p}\right)+\digamma\left(\tilde{p}\right)=0$,
but this condition is not solvable for a positive $\tilde{p}_{0}$ (or, which
is the same, a positive $p_{0}$).

##  IV Alternative Scenarios with anti-de Sitter momentum space

In the previous section we have seen that in the realm of DSR-compatible
(relativistic) deformations it is not possible to require the properties (24),
(25), (26), (29) (i.e. $R_{1}$, $R_{2}$, $R_{3}$, $R_{4}$) for composition
laws implemented in $3+1$-dimensional anti-de Sitter momentum space. Our next
task is consider the possibility of composition laws which are explicitly
relativistically covariant but are such that either the condition of
associativity or the condition of isometricity are lost.

###  IV.1 Explicit covariance and isometricity: anti-Snyder momentum space

As we have seen in section III.3, the two possible composition laws that
satisfy both the conditions of explicit covariance and isometricity, at the
expense of the associativity condition, are those of Eq. (53), that we here
report:

$\begin{cases}1)&\begin{cases}\left(p\oplus_{(1)}q\right)_{\lambda}=\frac{1}{\kappa}p_{\lambda}q_{4}+\left(\delta_{\lambda}^{\alpha}-\frac{p_{\lambda}p^{\alpha}}{\kappa(\kappa+p_{4})}\right)q_{\alpha}\,,\\\
\left(p\oplus_{(1)}q\right)_{4}=\frac{1}{\kappa}\left(p_{4}q_{4}-p^{\alpha}q_{\alpha}\right).\end{cases}\\\
2)&\begin{cases}\left(p\oplus_{(2)}q\right)_{\lambda}=\frac{1}{\kappa}p_{\lambda}q_{4}+\left(\delta_{\lambda}^{\alpha}-\frac{p_{\lambda}p^{\alpha}}{\kappa(\kappa-
p_{4})}\right)q_{\alpha}\,,\\\
\left(p\oplus_{(2)}q\right)_{4}=\frac{1}{\kappa}\left(p_{4}q_{4}+p^{\alpha}q_{\alpha}\right).\end{cases}\end{cases}$
(69)

We can see that both the composition laws are divergent respectively for
$\kappa+p_{4}=0$ and $\kappa-p_{4}=0$. Furthermore, it is not possible to
choose a portion of anti-de Sitter such that $\kappa+p_{4}>0$ and close under
the action of $\oplus_{(1)}$ and the Lorentz boosts, nor any portion
characterized by $\kappa-p_{4}=0$ and close under the action of $\oplus_{(2)}$
and the Lorentz boosts.888Considering for example the first composition law
$\oplus_{(1)}$, this can be seen by fixing, for any couple of momenta $q$ and
$p$, the values of $q_{4}$ and $p_{4}$ , and of the on-shell relations
$m_{q}^{2}=q_{0}^{2}-\vec{q}^{2}$ and $m_{p}^{2}=p_{0}^{2}-\vec{p}^{2}$ . By
picking the important case in which the particle $q$ is at rest, then the
condition $p_{4}q_{4}-p^{\alpha}q_{\alpha}$ translates in the constraint
$\left(\kappa^{2}-m_{p}^{2}\right)\frac{q_{4}^{2}}{m_{q}^{2}}-m_{p}^{2}>p_{1}^{2}$,
which is not admissible because is not preserved by the action of the Lorentz
group.

Although the two composition laws alone seem to have some pathologies, still
it is possible to construct a well-defined composition law from (69) with this
prescription:

$\left(p\oplus_{\kappa}q\right)=\begin{cases}\left(p\oplus_{(1)}q\right)\ ;\
&p_{4}\geq 0\,,\\\ \left(p\oplus_{(2)}q\right)\ ;\ &p_{4}<0\,.\end{cases}$
(70)

Because of the resemblance of this composition law with the Snyder one
discussed in subsection II.5, we will refer in the following to (70) as the
anti-Snyder composition law.

###  IV.2 Explicit covariance and associativity

We consider now the case of an explicitly covariant and associative (but non-
isometric) composition law. The general explicitly covariant composition law
can be written as:

$\left(q\oplus_{\kappa}k\right)_{\lambda}=A\left(q,k\right)q_{\lambda}+B\left(q,k\right)k_{\lambda}\,,$
(71)

where $A$ and $B$ are scalars, i. e. arbitrary functions of $q^{2}$, $k^{2}$
and $q\cdot k$.

Imposing associativity we get the system of equations:

$\begin{cases}A\left(\left(q\oplus
k\right),p\right)A\left(q,k\right)=A\left(q,\left(k\oplus p\right)\right),\\\
A\left(\left(q\oplus k\right),p\right)B\left(q,k\right)=B\left(q,\left(k\oplus
p\right)\right)A\left(k,p\right),\\\ B\left(\left(q\oplus
k\right),p\right)=B\left(q,\left(k\oplus
p\right)\right)B\left(k,p\right).\end{cases}$ (72)

We search for a perturbative solution of (72) at second order in
$\kappa^{-1}$. We then expand the scalars $A$ and $B$ as follows:

$\begin{cases}A\left(q,k\right)=1+\frac{1}{\kappa^{2}}\left(bk^{2}+cq\cdot
k\right),\\\ B\left(q,k\right)=1+\frac{1}{\kappa^{2}}\left(dq^{2}+eq\cdot
k\right),\end{cases}$ (73)

where $b$, $c$, $d$ and $e$ are constants. Inserting the expansion above in
(72) we get:

$\begin{cases}\left[1+\frac{1}{\kappa^{2}}\left(bp^{2}+c\left(q+k\right)\cdot
p\right)\right]\left[1+\frac{1}{\kappa^{2}}\left(bk^{2}+cq\cdot
k\right)\right]=\left[1+\frac{1}{\kappa^{2}}\left(b\left(k+p\right)^{2}+cq\cdot\left(k+p\right)\right)\right]\,,\\\
\left[1+\frac{1}{\kappa^{2}}\left(bp^{2}+c\left(q+k\right)\cdot
p\right)\right]\left[1+\frac{1}{\kappa^{2}}\left(dq^{2}+eq\cdot
k\right)\right]=\left[1+\frac{1}{\kappa^{2}}\left(dq^{2}+eq\cdot\left(k+p\right)\right)\right]\left[1+\frac{1}{\kappa^{2}}\left(bp^{2}+cp\cdot
k\right)\right]\,,\\\
\left[1+\frac{1}{\kappa^{2}}\left(d\left(q+k\right)^{2}+e\left(q+k\right)\cdot
p\right)\right]=\left[1+\frac{1}{\kappa^{2}}\left(dq^{2}+eq\cdot\left(k+p\right)\right)\right]\left[1+\frac{1}{\kappa^{2}}\left(dk^{2}+ep\cdot
k\right)\right]\,,\end{cases}$ (74)

which is solved by:

$\begin{cases}c=2b,\\\ c=e,\\\ 2d=e,\end{cases}$ (75)

and we get the unique solution:

$\left(q\oplus_{\kappa}k\right)_{\lambda}=q_{\lambda}+k_{\lambda}+\frac{1}{\kappa^{2}}\left[b\left(k^{2}+2q\cdot
k\right)q_{\lambda}+b\left(q^{2}+2q\cdot k\right)k_{\lambda}\right]\,.$ (76)

###  IV.3 Explicit covariance and commutativity: Proper anti-de Sitter

The last possibility that we will consider is an example of a composition law
which is neither associative nor isometric, but is commutative. We proceed in
analogy with the proper de Sitter composition law discussed in subsection
II.5, so we shall label this option as the “Proper anti-de Sitter” composition
law.

Adopting Cartesian parameterization (see the appendix A) and substituting in
(37) the anti-de Sitter Levi-Civita connection
$\Gamma_{\lambda}^{\mu\nu}=\frac{1}{\kappa^{2}}p_{\lambda}g^{\mu\nu}\left(p\right)$
we then get:

$\left(q\oplus_{\kappa}p\right){}_{\lambda}=\left(q_{\lambda}+k_{\lambda}\right)\left(1-\frac{1}{2\kappa^{2}}q\cdot
k\right).$ (77)

We notice that the Proper anti-de Sitter composition law is non-associative
but commutative and covariant under the action of the standard Lorentz
transformations.

##  V Aside on the Magueijo-Smolin proposal

The notion of DSR-relativistic theory was first introduced by one of us in the
studies Amelinodsr1 ; Amelinodsr2 , using as illustrative model a description
of relativistic symmetries which within the relative-locality framework should
be formulated in terms of a de Sitter momentum space. A second DSR-
relativistic model was then introduced in a follow-up study by Magueijo and
Smolin, and we shall here show that this Magueijo-Smolin model can be
formulated in terms of anti-de Sitter momentum space. This is interesting
since in a previous study Granik:2002wd  the Magueijo-Smolin model was
erroneously described in terms of the bicrossproduct basis of
$\kappa$-Poincaré, which would imply a description in terms of a de Sitter
momentum space. We shall explain the origin of that confusion and also analyze
the Magueijo-Smolin model in light of our general results concerning theories
with anti-de Sitter momentum space.

The Magueijo-Smolin model was inspired by some work by Fock fock1969theory
for spacetime, basically adopting a momentum-space picture which reproduces
the Fock spacetime picture. Central to the Magueijo-Smolin proposal is the map
$U(p_{0},\kappa)$, such that

$\displaystyle p_{\mu}\mapsto U(p_{0},\kappa)\triangleright p_{\mu},$ (78)
$\displaystyle U(p_{0},\kappa)\triangleright
p_{\mu}=\frac{p_{\mu}}{1-p_{0}/\kappa}.$

This map induces a deformation of boost transformations and a dispersion
relation given by

$\|p\|^{2}=\frac{\eta^{\mu\nu}p_{\mu}p_{\nu}}{(1-p_{0}/\kappa)^{2}}=\eta^{\mu\nu}(U(p_{0},\kappa)\triangleright
p_{\mu})(U(p_{0},\kappa)\triangleright p_{\nu}).$ (79)

In order to provide a geometric characterization of the Magueijo-Smolin
proposal let us choose the following set of coordinates for anti-de Sitter
momentum space:

$\displaystyle P_{\mu}=\frac{\kappa p_{\mu}}{\sqrt{p^{2}+(\kappa-
p_{0})^{2}}},$ (80) $\displaystyle P_{4}=\frac{\kappa(\kappa-
p_{0})}{\sqrt{p^{2}+(\kappa-p_{0})^{2}}},$ (81)

where $p^{2}\equiv p_{0}^{2}-p_{1}^{2}-p_{2}^{2}-p_{3}^{2}$.

We rely again on the relative-locality framework to describe the on-shell
relation in terms of the geodesic distance from the origin of momentum space,
intended as $(0,0,0,0,\kappa)$, to an arbitrary point in the manifold. This
gives $P_{4}^{2}=\kappa^{2}\cos^{2}(d/\kappa)$, which can be rewritten as
follows:

$P_{4}^{2}=\kappa^{2}\cos^{2}(d/\kappa)=\frac{\kappa^{2}(\kappa-
p_{0})^{2}}{p^{2}+(\kappa-
p_{0})^{2}}\Rightarrow\|p\|^{2}:=\frac{p^{2}}{(1-p_{0}/\kappa)^{2}}=\kappa^{2}\tan^{2}(d/\kappa).$
(82)

Therefore the Magueijo-Smolin on-shellness relation is a function of the
geodesic distance in anti-de Sitter momentum space. For a de Sitter momentum
space, which we can obtain by performing a Wick rotation, one finds a
functional dependence on a hyperbolic tangent function, rather than a tangent
function. The hyperbolic tangent is a bounded function, leading to a
description of the mass of particles which has a maximum allowed value,
whereas Magueijo and Smolin specified very explicitly 2002PhRvL..88s0403M
that they were proposing a picture with unbounded mass. From this one must
inevitably conclude that the geometric picture of the Magueijo-Smolin proposal
must be given in terms of a momentum space with anti-de Sitter geometry.

The work of Magueijo and Smolin on DSR-relativistic scenarios also gives us
another structure that deserves to be analyzed from our perspective. This is
the proposal for the law of composition of momenta which they put forward in
Ref.2003PhRvD..67d4017M , a proposal that was intended to ensure that the
deformation of relativistic laws would not affect significantly macroscopic
systems (composed of many fundamental particles). In that proposal the total
momentum, $p_{\mu}^{(1,...,N)}\equiv p^{(1)}\oplus...\oplus p^{(N)}$, obtained
composing $N$ fundamental-particle momenta, is described implicitly through

$U(p_{0}^{(1,...,N)},N\kappa)\triangleright(p^{(1)}\oplus...\oplus
p^{(N)})=U(p_{0}^{(1)},\kappa)\triangleright
p^{(1)}+...+U(p_{0}^{(N)},\kappa)\triangleright p^{(N)}\,.$ (83)

When written in terms of the coordinates that we are using, this definition
takes the form:

$\frac{p_{\mu}^{(1,...,N)}}{1-p_{0}^{(1,...,N)}/N\kappa}=\sum_{i}^{N}\frac{p_{\mu}^{(i)}}{1-p_{0}^{(i)}/\kappa},$
(84)

For this composed momentum $p_{\mu}^{(1,...,N)}$ one finds that it is governed
by Lorentz transformations of the same form as the fundamental-particle
Lorentz transformations, but with deformation parameter rescaled from $\kappa$
to $N\kappa$. Of course this also implies that the on-shelleness condition for
the composed momentum has the same form as the on-shelleness condition for
fundamental particles, but with deformation parameter rescaled from $\kappa$
to $N\kappa$.

We can explicitly give a geometrical interpretation for this kind of
composition law. Let $\mathbb{M}^{5}$ be the $5$-dimensional Minkowski space
with signature $(+---+)$, labeled by Cartesian coordinates $\\{P_{A}\\}$,
$A=0,...,4$, let us denote with $adS_{\kappa}$, the anti-de Sitter space with
curvature $\kappa^{-1}$, the subset of $\mathbb{M}^{5}$ satisfying the
constraint
$(P_{0})^{2}-(P_{1})^{2}-(P_{2})^{2}-(P_{3})^{2}+(P_{4})^{2}=\kappa^{2}$, and
let ${\cal P}\subset adS_{\kappa}=\\{P^{(i)}\\}=\\{P_{A}^{(i)}\\}$,
$i=1,...,N$, be the set of $N$ points defined in $adS_{\kappa}$. We can then
introduce a composition law $\bigoplus_{i=1}^{N}$ as the map

$\displaystyle\bigoplus_{i=1}^{N}:{\cal P}\subset
adS_{\kappa}\subset\mathbb{M}^{5}\longrightarrow
adS_{N\kappa}\subset\mathbb{M}^{5},$ $\displaystyle
P_{A}^{(i)}\mapsto\bigoplus_{i=1}^{N}P_{A}^{(i)}=P_{A}^{(1,...,N)},{}A=0,...,4,$
(85)

where

$\displaystyle
P_{\mu}^{(1,...,N)}=\kappa\left(\sum_{i=1}^{N}\frac{P_{\mu}^{(i)}}{P_{4}^{(i)}}\right)\left[1+\frac{1}{N^{2}}\eta^{\alpha\beta}\sum_{j,k=1}^{N}\left(\frac{P_{\alpha}^{(j)}}{P_{4}^{(j)}}\right)\left(\frac{P_{\beta}^{(k)}}{P_{4}^{(k)}}\right)\right]^{-1/2}{},{}\mu=0,...,3,$
(86) $\displaystyle
P_{4}^{(1,...,N)}=N\kappa\left[1+\frac{1}{N^{2}}\eta^{\mu\nu}\sum_{i,j=1}^{N}\left(\frac{P_{\mu}^{(i)}}{P_{4}^{(i)}}\right)\left(\frac{P_{\nu}^{(j)}}{P_{4}^{(j)}}\right)\right]^{-1/2}.$
(87)

It is straightforward to verify that
$\eta^{\mu\nu}P_{\mu}^{(1,...,N)}P_{\nu}^{(1,...,N)}+\left(P_{4}^{(1,...,N)}\right)^{2}=(N\kappa)^{2}$.
For the proposal of Magueijo and Smolin 2002PhRvL..88s0403M  we have

$\frac{p_{\mu}^{(1,...,N)}}{1-p_{0}^{(1,...,N)}/N\kappa}=\sum_{i=1}^{N}\frac{p_{\mu}^{(i)}}{1-p_{0}^{(i)}/\kappa}.$
(88)

Form our perspective it noteworthy that this Magueijo-Smolin proposal for the
law of composition of momenta is commutative but not associative, and is
geometrically described by a map between anti-de Sitter spaces (i.e., from a
anti-de Sitter space with curvature given by the scale $\kappa^{-1}$ to
another anti-de Sitter space with curvature $(N\kappa)^{-1}$).

##  VI Concluding remarks

Our analysis intends to affect the future development of research inspired by
recent results supporting the possibility that momentum space is curved in
several approaches to the quantum-gravity problem. While in the 4D context
different approaches provide different expectations, in 3D quantum gravity all
approaches agree on the fact that momentum space has anti-de Sitter geometry,
with certain characterizing properties which we here highlighted, $R1$, $R2$,
$R3$, $R4b$. It would be natural then to explore the possibility that also 4D
quantum gravity might be characterized by an anti-de Sitter momentum space
with those properties. To our surprise our investigations showed that this is
not possible: in a 4D context with anti-de Sitter momentum space the
properties $R1$, $R2$, $R3$ are incompatible with the type of relativistic
covariance coded in $R4$ and are also incompatible with the weaker notion of
relativistic covariance coded in our requirement $R4b$.

Those attempting to use the results on 3D quantum gravity as guidance toward
4D quantum gravity should take our findings into account. The analogy between
3D quantum gravity and 4D quantum gravity cannot be too close, at least for
what concerns the aspects on which we here focused. Our perspective is that 3D
gravity and 4D gravity clearly have some important differences, already at the
classical level (3D gravity has no propagating gravitational degrees of
freedom), and the analogy, if at all fruitful, can only be partial. In the
final part of our study we investigated the possibility of 4D momentum spaces
with only some of the properties $R1$, $R2$, $R3$ and $R4$/$R4b$, and we feel
that a broader investigation of this possibility might be valuable.

### Acknowledgements

G.A.-C. acknowledges financial support by the Programme STAR Plus, funded by
Federico II University and Compagnia di San Paolo, and by the MIUR, PRIN 2017
grant 20179ZF5KS. IPL was partially supported by the National Council for
Scientific and Technological Development - CNPq grant 306414/2020-1 and by the
Grant 3197/2021, Paraiba State Research Foundation (FAPESQ). The authors would
like to acknowledge networking support by the COST Action QGMM (CA18108),
supported by COST (European Cooperation in Science and Technology).

##  Appendix A Cartesian, Comoving and Horospherical Coordinates

###  A.1 De Sitter space

A useful setting one can choose to analyse de Sitter space consists in
embedding it in a flat, higher-dimensional manifold 2016IJMPS..4160126L ;
moschella2006sitter . We use Greek letters to label coordinates for $0,...,3$.
One can describe a $4$-dimensional de Sitter manifold as a hypersurface
embedded in a $5$-dimensional Minkowski space with signature $(+----)$, with
an extra space-like coordinate. For a Minkowski metric

$ds^{2}=dp_{0}^{2}-dp_{1}^{2}-dp_{2}^{2}-dp_{3}^{2}-dp_{4}^{2},$ (89)

de Sitter space is defined as the hyperboloid satisfying

$p_{0}^{2}-p_{1}^{2}-p_{2}^{2}-p_{3}^{2}-p_{4}^{2}\doteq
p^{2}-p_{4}^{2}=-\kappa^{2}.$ (90)

####  A.1.1 Cartesian coordinates for de Sitter

Another possible choice of coordinate system for de Sitter space is simply to
use the coordinates of the embedding and pick half of de Sitter space, for
instance the one with

$p_{4}=\sqrt{p^{2}+\kappa^{2}}>0.$ (91)

This way, by calculating the differential $dp_{4}$ and substituting in 89, we
find the line element

$ds^{2}=g^{\mu\nu}dp_{\mu}dp_{\nu},$ (92)

for

$g^{\mu\nu}=\eta^{\mu\nu}-\frac{p^{\mu}p^{\nu}}{p^{2}+\kappa^{2}}\doteq\eta^{\mu\nu}-\frac{p^{\mu}p^{\nu}}{p_{4}^{2}}.$
(93)

####  A.1.2 Comoving coordinates

Another possible choice of coordinate system for de Sitter space is

$\displaystyle\left\\{\begin{array}[]{rcl}p_{0}&=&\kappa\sinh(\tilde{p}_{0}/\kappa)+e^{\tilde{p}_{0}/\kappa}(\tilde{p}_{1}^{2}+\tilde{p}_{2}^{2}+\tilde{p}_{3}^{2})/2\kappa,\\\
p_{i}&=&e^{\tilde{p}_{0}/\kappa}\tilde{p}_{i},\\\
p_{4}&=&\kappa\cosh(\tilde{p}_{0}/\kappa)-e^{\tilde{p}_{0}/\kappa}(\tilde{p}_{1}^{2}+\tilde{p}_{2}^{2}+\tilde{p}_{3}^{2})/2\kappa,\end{array}\right.$
(97)

These are called, comoving coordinates. The metric assumes the following form:

$ds^{2}=d\tilde{p}_{0}^{2}-e^{2\tilde{p}_{0}/\kappa}\left(d\tilde{p}_{1}^{2}+d\tilde{p}_{2}^{2}+d\tilde{p}_{3}^{2}\right),$
(98)

This is the coordinate system usually used to describe $\kappa$-Poincaré
momentum space GiuliaFlavio . It covers only half of de Sitter space, the
portion $p_{0}+p_{4}>0$.

###  A.2 Anti-de Sitter space

The anti-de Sitter metric can be found using a similar procedure of embedding
a hypersurface in a flat space 2016IJMPS..4160126L ; moschella2006sitter . Let
us do the example of embedding this $4$-dimensional manifold in a
$5$-dimensional one with signature $(+---+)$. For the Minkowski metric

$ds^{2}=dp_{0}^{2}-dp_{1}^{2}-dp_{2}^{2}-dp_{3}^{2}+dp_{4}^{2},$ (99)

anti-de Sitter space is defined as the hyperboloid satisfying

$p_{0}^{2}-p_{1}^{2}-p_{2}^{2}-p_{3}^{2}+P_{4}^{2}\doteq
p^{2}+p_{4}^{2}=\kappa^{2}.$ (100)

####  A.2.1 Cartesian coordinates for anti-de Sitter

Following a similar procedure of the last section, we might pick half of anti-
de Sitter space, for instance

$p_{4}=\sqrt{\kappa^{2}-p^{2}},$ (101)

to have

$g^{\mu\nu}=\eta^{\mu\nu}+\frac{p^{\mu}p^{\nu}}{\kappa^{2}-p^{2}}\doteq\eta^{\mu\nu}+\frac{p^{\mu}p^{\nu}}{p_{4}^{2}},$
(102)

which can be derived from the de Sitter case by a Wick rotation
$(p_{4},\kappa)\mapsto(ip_{4},i\kappa)$.

####  A.2.2 Horospherical coordinates

Another set of coordinates in similar to the comoving ones of de Sitter space
and are called horospherical coordinates. They are defined by the embedding

$\displaystyle\left\\{\begin{array}[]{rcl}p_{0}&=&\tilde{p}_{0}e^{\tilde{p}_{1}/\kappa},\\\
p_{1}&=&\kappa\sinh\left(\tilde{p}_{1}/\kappa\right)-e^{\tilde{p}_{1}/\kappa}(-\tilde{p}_{0}^{2}+\tilde{p}_{2}^{2}+\tilde{p}_{3}^{2})/2\kappa,\\\
p_{2}&=&\tilde{p}_{2}e^{\tilde{p}_{1}/\kappa},\\\
p_{3}&=&\tilde{p}_{3}e^{\tilde{p}_{1}/\kappa},\\\
p_{4}&=&\kappa\cosh\left(\tilde{p}_{1}/\kappa\right)+e^{\tilde{p}_{1}/\kappa}(-\tilde{p}_{0}^{2}+\tilde{p}_{2}^{2}+\tilde{p}_{3}^{2})/2\kappa,\end{array}\right.$
(108)

and the line element assumes the form

$ds^{2}=e^{2p_{1}/\kappa}\left(dp_{0}^{2}-dp_{2}^{2}-dp_{3}^{2}\right)-dp_{1}^{2}.$
(109)

##  Appendix B Uniqueness of Hopf Composition Law at Second Order in the
Planck Lenght

In order to see that the composition law (60) is the unique solution of
(24)-(29) up to an arbitrary rotation, we expand the composition law (48) and
the general boost $\Omega$ at second order in the Planck length:

$\begin{cases}\left(p\oplus_{H}q\right)_{\lambda}\approx
p_{\lambda}+q_{\lambda}+\frac{1}{\kappa}\omega_{\lambda}^{\
\beta\gamma}q_{\beta}p_{\gamma}-\frac{1}{2}\frac{1}{\kappa^{2}}p_{\lambda}\left(q^{2}+q\cdot
p\right)+\frac{1}{\kappa^{2}}X_{\lambda}^{\
\beta\gamma\lambda}p_{\gamma}p_{\lambda}q_{\beta}\,,\\\ \Omega_{\alpha}^{\
\beta}\approx\delta_{\alpha}^{\ \beta}+\frac{1}{\kappa}\omega_{\alpha}^{\
\beta\gamma}p_{\gamma}+\frac{1}{\kappa^{2}}X_{\alpha}^{\
\beta\gamma\lambda}p_{\gamma}p_{\lambda}\,,\end{cases}$ (110)

where the condition of (pseudo) orthogonality for $\Omega$ translates into the
constraints:

$\begin{cases}\omega^{\mu\nu\gamma}+\omega^{\nu\mu\gamma}=0,\\\
2X^{(\mu\nu)\alpha\beta}+\omega^{\theta\mu(\beta}\omega_{\theta}^{\
\nu\alpha)}=0.\end{cases}$ (111)

Imposing the condition of associativity for the composition law results in:

$0=\frac{1}{2}\left(\delta_{\lambda}^{\nu}\eta^{\alpha\mu}-\delta_{\lambda}^{\mu}\eta^{\alpha\nu}\right)+\omega_{\lambda}^{\
\alpha\theta}\omega_{\theta}^{\ \mu\nu}-\omega_{\lambda}^{\
\theta\nu}\omega_{\theta}^{\ \alpha\mu}+2X_{\lambda}^{\ \alpha\nu\mu}\,.$
(112)

Using the previous equations we find that $X$ is completely determined as a
function of $\omega$, which in turns has to satisfy the condition:

$0=\frac{1}{2}\left(\delta_{\lambda}^{\nu}\eta^{\alpha\mu}-\delta_{\lambda}^{\mu}\eta^{\alpha\nu}\right)+\omega_{\lambda}^{\
\alpha\theta}\omega_{\theta}^{\ [\mu\nu]}-\omega_{\lambda}^{\
\theta[\nu}\omega_{\theta}^{\ \alpha\mu]}\,.$ (113)

Parameterizing $\omega$ as:

$\omega_{\lambda}^{\ \mu\nu}=c\epsilon_{\lambda}^{\
\mu\nu\delta}v_{\delta}+d\left(v_{\lambda}\eta^{\mu\nu}-\eta_{\lambda}^{\mu}v^{\nu}\right),$
(114)

and inserting (114) into (113) we get:

$\delta_{\lambda}^{[\nu}\eta^{\alpha\mu]}+d^{2}v^{2}\delta_{\lambda}^{[\nu}\eta^{\alpha\mu]}=0,$
(115)

which tell us that $v$ has to be spacelike. In particular we will choose
$d^{2}=1=-v^{2}$ so that:

$\omega_{\lambda}^{\
\mu\nu}=v_{\lambda}\eta^{\mu\nu}-\delta_{\lambda}^{\mu}v^{\nu}\,,$ (116)

and the $X$ is given by (112):

$X_{\lambda}^{\
\alpha\nu\mu}=\frac{1}{2}\delta_{\lambda}^{(\nu}\eta^{\alpha\mu)}+v_{\lambda}v^{(\mu}\eta^{\alpha\nu)}-\frac{1}{2}v_{\lambda}\eta^{\mu\nu}v^{\alpha}\,,$
(117)

and is unique up to a spatial (three dimensional) rotation of $w$.

It is not hard to see that by expanding the composition law (60) at second
order in the Planck length we get indeed contribution of the form (110),
considering (116) and (117).

## References

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