remain irreducible with respect to SO${}_{0}(1,4)$, then the SO$(4)$ $\supset$
SO$(3)$ basis of SO${}_{0}(1,4)$ is already a complete basis of the
SO${}_{0}(2,4)$ irreducible representation. To examine this very point, we
first recall that the UIR’s of the SO$(4)$ group, denoted here by
$\textbf{D}^{(j_{l},j_{r})}$, are determined by two numbers
$j_{l},j_{r}\in\mathbb{N}/2$, which are related to the values of the SO$(4)$
Casimir operators by:

$\displaystyle\frac{1}{2}L_{\texttt{A}\texttt{B}}L^{\texttt{A}\texttt{B}}=2\big{(}j_{l}(j_{l}+1)+j_{r}(j_{r}+1)\big{)}{\mathbbm{1}}\,,$
(H.7) $\displaystyle\frac{1}{8}{\cal
E}^{\texttt{A}\texttt{B}\texttt{C}\texttt{D}}L_{\texttt{A}\texttt{B}}L_{\texttt{C}\texttt{D}}=\big{(}j_{l}(j_{l}+1)-j_{r}(j_{r}+1)\big{)}{\mathbbm{1}}\,.$
(H.8)

From Eqs. (H.6) and (H.7), we directly obtain:

$\displaystyle\mbox{spectrum}\;(\Gamma_{0})^{2}=\lambda+2\big{(}j_{l}(j_{l}+1)+j_{r}(j_{r}+1)\big{)}\,.$
(H.9)

This implies that, besides the SO${}_{0}(2,4)$ generators lying inside
SO${}_{0}(1,4)$, the spectrum of $\Gamma_{0}$ (up to a sign) is also
characterized by that of SO$(4)$ in the irreducible representation of
SO${}_{0}(2,4)$. Therefore, the SO$(4)$ $\supset$ SO$(3)$ basis of
SO${}_{0}(1,4)$ is indeed a complete basis of the SO${}_{0}(2,4)$ irreducible
representation.

We are now ready to go through a complete classification of UIR’s of
SO${}_{0}(2,4)$ characterized by the additional representation relation (H.2).
Considering all the above, while we compare our UIR’s characterized by (H.4)
and (H.5) with the complete list of the SO${}_{0}(1,4)$ UIR’s given by Dixmier
(see section XII), possible values of $\lambda\in\mathbb{R}$ are:

$\displaystyle\lambda=1-s^{2}\,,\;\;\;\;\;\;\;s=0,\frac{1}{2},1,\frac{3}{2},2,\;...\,,$
(H.10)

where:

  * •

The case $s=0$ corresponds to the dS4 UIR
${\underline{U}}^{\mbox{\small{cs}}}_{0,\frac{1}{2}}$, which in turn with
respect to $\mathrm{SO}_{0}(1,4)\supset\mathrm{SO}(4)$ reduces as
${\underline{U}}^{\mbox{\small{cs}}}_{0,\frac{1}{2}}\;\big{|}_{\mathrm{SO}(4)}={\textbf{D}}^{(0,0)}$
(see section XII).

  * •

Other values of $s$ correspond to the dS UIR’s $\Pi^{\pm}_{s,s}$, for which we
have $\Pi^{-}_{s,s}\;\big{|}_{\mathrm{SO}(4)}={\textbf{D}}^{(s,0)}$ and
$\Pi^{+}_{s,s}\;\big{|}_{\mathrm{SO}(4)}={\textbf{D}}^{(0,s)}$ (see section
XII).

The above SO${}_{0}(1,4)$ UIR’s not only extend to SO${}_{0}(2,4)$, but they
also precisely extend to two inequivalent UIR’s of SO${}_{0}(2,4)$. This
additional doubling comes to fore when we take into account the sign of
$\Gamma_{0}$. Actually, if we substitute the possible values of $\lambda$
given in Eq. (H.10) (and correspondingly, the respective values of
$(j_{l},j_{r})$ given in the above itemization) into (H.9), we find:

$\displaystyle\mbox{spectrum}\;\Gamma_{0}\equiv E_{0}=\pm(s+1)\,.$ (H.11)

Note that no operator in SO${}_{0}(2,4)$ changes the sign of $(s+1)$ and
consequently the sign of conformal energy $E_{0}$. Therefore, the sign of
$E_{0}$ is another invariant of the SO${}_{0}(2,4)$ UIR’s. Accordingly, in
this paper, we denote the SO${}_{0}(2,4)$ UIR’s by
${\cal{C}}^{\gtrless}_{E_{0},j_{l},j_{r}}$, in which the superscripts
‘$\gtrless$’ refer to the positive/negative sign of conformal energy,
respectively.

At the end, we must underline that the conformal extension of the above dS4
UIR’s is equivalent to the conformal extension of the Poincaré massless UIR’s.
[For the latter point, which is beyond the scope of this paper, readers are
referred to Ref. Mack1977 .] In this sense, the above dS4 UIR’s are recognized
as dS4 massless representations.

###  Appendix I DS4 infinitesimal generators in terms of the conformal
coordinates

In this appendix, we present the (orbital part of the) dS4 infinitesimal
generators
$M_{AB}=-\mathrm{i}\big{(}x_{A}\partial_{B}-x_{B}\partial_{A}\big{)}$, with
$A,B=0,1,2,3,4$, in terms of the conformal coordinates $x=x(\rho,\textbf{u})$,
where $-\pi/2<\rho<\pi/2$ and $\textbf{u}\in\mathbb{S}^{3}$ (see Eq.
(XVI.16)). In this context, the three infinitesimal generators of boosts take
the form:

$\displaystyle M_{01}$ $\displaystyle=$
$\displaystyle\mathrm{i}\Big{(}-\cos\rho\sin\psi\sin\theta\cos\phi\frac{\partial}{\partial\rho}-\sin\rho\cos\psi\sin\theta\cos\phi\frac{\partial}{\partial\psi}-\frac{\sin\rho\cos\theta\cos\phi}{\sin\psi}\frac{\partial}{\partial\theta}+\frac{\sin\rho\sin\phi}{\sin\psi\sin\theta}\frac{\partial}{\partial\phi}\Big{)}\,,$
(I.1) $\displaystyle M_{02}$ $\displaystyle=$
$\displaystyle\mathrm{i}\Big{(}-\cos\rho\sin\psi\sin\theta\sin\phi\frac{\partial}{\partial\rho}-\sin\rho\cos\psi\sin\theta\sin\phi\frac{\partial}{\partial\psi}-\frac{\sin\rho\cos\theta\sin\phi}{\sin\psi}\frac{\partial}{\partial\theta}-\frac{\sin\rho\cos\phi}{\sin\psi\sin\theta}\frac{\partial}{\partial\phi}\Big{)}\,,$
(I.2) $\displaystyle M_{03}$ $\displaystyle=$
$\displaystyle\mathrm{i}\Big{(}-\cos\rho\sin\psi\cos\theta\frac{\partial}{\partial\rho}-\sin\rho\cos\psi\cos\theta\frac{\partial}{\partial\psi}+\frac{\sin\rho\sin\theta}{\sin\psi}\frac{\partial}{\partial\theta}\Big{)}\,.$
(I.3)

On the other hand, the infinitesimal generator of time translation reads:

$\displaystyle
M_{04}=\mathrm{i}\Big{(}-\cos\rho\cos\psi\frac{\partial}{\partial\rho}+\sin\rho\sin\psi\frac{\partial}{\partial\psi}\Big{)}\,.$
(I.4)

The other six generators (of space rotations $M_{ki}$ and space translations
$M_{4k}$ ($k,i=1,2,3$)), associated with the compact $SO(4)$ subgroup, take
the same form as those already given in subsection XIII.1 (see Eq. (XIII.19)).

Note that the $O(1,4)$-invariant measure on the dS4 hyperboloid is:

$\displaystyle\mathrm{d}\mu=(\cos\rho)^{-4}\;\mathrm{d}\rho\mathrm{d}\mu(\textbf{u})\,,$
(I.5)

where $\mathrm{d}\mu(\textbf{u})$ refers to the $O(4)$-invariant measure on
$\mathbb{S}^{3}$ (see appendix E).

###  Appendix J Precision on Eq. (XX.15)

In section XX, we have shown that the AdS4 group SO${}_{0}(2,3)$ represents
the relativity group for an elementary system which is a deformation of both a
relativistic free particle (with the rest energy $mc^{2}$) and a harmonic
oscillator (with the rest energy $\frac{3}{2}\hbar\omega$,
$\omega=\frac{c}{R}$). In this appendix, following Ref. GazRe1993 , we aim to
make this argument more explicit through the contraction process on the
group/algebra level, as we have discussed in section X. Of course, in order to
avoid technical (but not conceptual) difficulties, we deal with
($1+1$)-dimensional spacetime (AdS2 case). The rest energy of the (relevant)
harmonic oscillator then would be $\frac{1}{2}\hbar\omega$ instead of
$\frac{3}{2}\hbar\omega$.

We adapt the notion of elementary systems to the one-dimensional harmonic
oscillator, while the Newton group BacryLevi  is considered as the relativity
group. The Newton Lie algebra is generated by:

$\displaystyle
H=\frac{p^{2}}{2m}+\frac{1}{2}m\omega^{2}q^{2}\,,\;\;\;\;\;\;\;P\equiv
p\,,\;\;\;\;\;\;\;L\equiv q\,,$ (J.1)

with Poisson brackets:

$\displaystyle\big{\\{}H,P\big{\\}}=m\omega^{2}L\,,\;\;\;\;\;\;\;\big{\\{}H,L\big{\\}}=-\frac{P}{m}\,,\;\;\;\;\;\;\;\big{\\{}P,L\big{\\}}=1\,.$
(J.2)

We also consider the Lie algebra of the Poincaré group generated by:

$\displaystyle
H_{\mbox{\tiny{tot}}}=\sqrt{p^{2}c^{2}+m^{2}c^{4}}\,,\;\;\;\;\;\;\;P\equiv
p\,,\;\;\;\;\;\;\;L\equiv\frac{qH_{\mbox{\tiny{tot}}}}{mc^{2}}\,,$ (J.3)

where $H_{\mbox{\tiny{tot}}}$ is the energy, $P$ the momentum, and $L$ the
observable associated with the (properly re-scaled) Lorentz boost. The
corresponding Poisson brackets are given by:

$\displaystyle\big{\\{}H_{\mbox{\tiny{tot}}},P\big{\\}}=0\,,\;\;\;\;\;\;\;\big{\\{}H_{\mbox{\tiny{tot}}},L\big{\\}}=-\frac{P}{m}\,,\;\;\;\;\;\;\;\big{\\{}P,L\big{\\}}=-\frac{H_{\mbox{\tiny{tot}}}}{mc^{2}}\,.$
(J.4)

If we restrict our attention to the kinetic energy instead of the total one,
$H=H_{\mbox{\tiny{tot}}}-mc^{2}$, we get:

$\displaystyle\big{\\{}H,P\big{\\}}=0\,,\;\;\;\;\;\;\;\big{\\{}H,L\big{\\}}=-\frac{P}{m}\,,\;\;\;\;\;\;\;\big{\\{}P,L\big{\\}}=1-\frac{H}{mc^{2}}\,.$
(J.5)

One can simply show that the Lie algebras (J.2) and (J.5) contract,
respectively, for $\omega\rightarrow 0$ and for $c\rightarrow\infty$, towards
the symmetry Lie algebra for the Galilean free particle:

$\displaystyle\big{\\{}H,P\big{\\}}=0\,,\;\;\;\;\;\;\;\big{\\{}H,L\big{\\}}=-\frac{P}{m}\,,\;\;\;\;\;\;\;\big{\\{}P,L\big{\\}}=1\,.$
(J.6)

Now, we take into account the Lie algebra for the AdS2 elementary system,
namely, for a free particle with mass $\mathfrak{M}_{\mbox{\tiny{AdS}}_{2}}$
living in AdS2 spacetime:

$\displaystyle\big{\\{}H,P\big{\\}}=\frac{\mathfrak{M}_{\mbox{\tiny{AdS}}_{2}}c^{2}}{R^{2}}L\,,\;\;\;\;\;\;\;\big{\\{}H,L\big{\\}}=-\frac{P}{\mathfrak{M}_{\mbox{\tiny{AdS}}_{2}}}\,,\;\;\;\;\;\;\;\big{\\{}P,L\big{\\}}=1-\frac{H}{\mathfrak{M}_{\mbox{\tiny{AdS}}_{2}}c^{2}}\,.$
(J.7)

This algebra can be identified to the AdS2 Lie algebra $\mathrm{SO}_{0}(2,1)$.
The Poincaré and Newton relativities then can be simply achieved from the AdS2
relativity through the contraction process; the limit $R\rightarrow\infty$
results in the Poincaré algebra
($\mathfrak{M}_{\mbox{\tiny{AdS}}_{2}}\longrightarrow m$), whereas the limit
$c,R\rightarrow\infty$ ($c/R=\omega$ being fixed) results in the Newton
algebra. Actually, quite similar to the dS4 case (see section X), we have:

$\displaystyle\begin{array}[]{ccccccc}{\mbox{AdS${}_{2}$ group
(algebra)}}&\longrightarrow&{\mbox{Poincar\'{e} group (algebra)}}&\\\ \\\
\big{\downarrow}&&\big{\downarrow}&\\\ \\\ {\mbox{Newton group
(algebra)}}&\longrightarrow&{\mbox{Galilei group (algebra)}}&\end{array}$
(J.13)

where, again, the arrows ‘$\longrightarrow$’ stand for the group (algebra)
contractions.

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