$\displaystyle\;\sin\psi^{\prime}\sin\theta^{\prime}\cos\phi^{\prime}=\sin\psi\sin\theta\cos\phi\,,$
$\displaystyle{z}^{\prime 2}={z}^{3}\sin t+{z}^{2}\cos t\;$
$\displaystyle\Rightarrow$
$\displaystyle\;\sin\psi^{\prime}\sin\theta^{\prime}\sin\phi^{\prime}=(\sin\psi\cos\theta)\sin
t+(\sin\psi\sin\theta\sin\phi)\cos t\,,$ $\displaystyle{z}^{\prime
3}={z}^{3}\cos t-{z}^{2}\sin t\;$ $\displaystyle\Rightarrow$
$\displaystyle\;\sin\psi^{\prime}\cos\theta^{\prime}=(\sin\psi\cos\theta)\cos
t-(\sin\psi\sin\theta\sin\phi)\sin t\,.$

These equations allow to calculate the derivatives of
$\psi^{\prime}(t),\theta^{\prime}(t),\phi^{\prime}(t)$, and
$N(\underline{g}_{k},\textbf{z})$ with respect to $t$:

$\displaystyle\frac{\partial\psi^{\prime}}{\partial t}\Big{|}_{t=0}$
$\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle\frac{\partial\theta^{\prime}}{\partial t}\Big{|}_{t=0}$
$\displaystyle=$ $\displaystyle\sin\phi\,,$
$\displaystyle\frac{\partial\phi^{\prime}}{\partial t}\Big{|}_{t=0}$
$\displaystyle=$ $\displaystyle\cot\theta\cos\phi\,,$
$\displaystyle\frac{\partial N(\underline{g}_{k=1},\textbf{z})}{\partial
t}\Big{|}_{t=0}$ $\displaystyle=$ $\displaystyle 0\,.$

The associated infinitesimal operator then takes the form:

$\displaystyle\underline{\hat{Y}}_{1}=\mathrm{i}\Big{(}\sin\phi\;\frac{\partial}{\partial\theta}+\cot\theta\cos\phi\;\frac{\partial}{\partial\phi}\Big{)}\,.$
(XIII.8)

Similarly, for $k=2$ and $k=3$, we respectively obtain:

$\displaystyle\underline{\hat{Y}}_{2}$ $\displaystyle=$
$\displaystyle\mathrm{i}\Big{(}-\cos\phi\;\frac{\partial}{\partial\theta}+\cot\theta\sin\phi\;\frac{\partial}{\partial\phi}\Big{)}\,,$
(XIII.9) $\displaystyle\underline{\hat{Y}}_{3}$ $\displaystyle=$
$\displaystyle\mathrm{i}\Big{(}-\frac{\partial}{\partial\phi}\Big{)}\,.$
(XIII.10)

The other infinitesimal operators associated with the scalar representations
$\underline{U}^{\mbox{\small{ps}}}_{0,\tau}$ are obtained in the same way.
They respectively read Rabeie :

  * •

The infinitesimal operator of time translations:

$\displaystyle\underline{\hat{X}}_{0}=\mathrm{i}\Big{(}-\tau\cos\psi+\sin\psi\;\frac{\partial}{\partial\psi}\Big{)}\,.$
(XIII.11)

  * •

The infinitesimal operators of space translations:

$\displaystyle\underline{\hat{X}}_{1}$ $\displaystyle=$
$\displaystyle\mathrm{i}\Big{(}-\sin\theta\cos\phi\;\frac{\partial}{\partial\psi}-\cot\psi\cos\theta\cos\phi\;\frac{\partial}{\partial\theta}+\frac{\cot\psi\sin\phi}{\sin\theta}\;\frac{\partial}{\partial\phi}\Big{)}\,,$
(XIII.12) $\displaystyle\underline{\hat{X}}_{2}$ $\displaystyle=$
$\displaystyle\mathrm{i}\Big{(}-\sin\theta\sin\phi\;\frac{\partial}{\partial\psi}-\cot\psi\cos\theta\sin\phi\;\frac{\partial}{\partial\theta}-\frac{\cot\psi\cos\phi}{\sin\theta}\;\frac{\partial}{\partial\phi}\Big{)}\,,$
(XIII.13) $\displaystyle\underline{\hat{X}}_{3}$ $\displaystyle=$
$\displaystyle\mathrm{i}\Big{(}-\cos\theta\;\frac{\partial}{\partial\psi}+\cot\psi\sin\theta\;\frac{\partial}{\partial\theta}\Big{)}\,.$
(XIII.14)

  * •

The infinitesimal operators of boosts:

$\displaystyle\underline{\hat{Z}}_{1}$ $\displaystyle=$
$\displaystyle\mathrm{i}\Big{(}-\tau\sin\psi\sin\theta\cos\phi-\cos\psi\sin\theta\cos\phi\;\frac{\partial}{\partial\psi}-\frac{\cos\theta\cos\phi}{\sin\psi}\;\frac{\partial}{\partial\theta}+\frac{\sin\phi}{\sin\psi\sin\theta}\;\frac{\partial}{\partial\phi}\Big{)}\,,$
(XIII.15) $\displaystyle\underline{\hat{Z}}_{2}$ $\displaystyle=$
$\displaystyle\mathrm{i}\Big{(}-\tau\sin\psi\sin\theta\sin\phi-\cos\psi\sin\theta\sin\phi\;\frac{\partial}{\partial\psi}-\frac{\cos\theta\sin\phi}{\sin\psi}\;\frac{\partial}{\partial\theta}-\frac{\cos\phi}{\sin\psi\sin\theta}\;\frac{\partial}{\partial\phi}\Big{)}\,,$
(XIII.16) $\displaystyle\underline{\hat{Z}}_{3}$ $\displaystyle=$
$\displaystyle\mathrm{i}\Big{(}-\tau\sin\psi\cos\theta-\cos\psi\cos\theta\;\frac{\partial}{\partial\psi}+\frac{\sin\theta}{\sin\psi}\;\frac{\partial}{\partial\theta}\Big{)}\,.$
(XIII.17)

One can check that the given infinitesimal operators verify the following
commutation relations:

$\displaystyle\big{[}\underline{\hat{Y}}_{i},\underline{\hat{Y}}_{j}\big{]}$
$\displaystyle=$
$\displaystyle\mathrm{i}{{\cal{E}}_{ij}}^{k}\;\underline{\hat{Y}}_{k}\,,$
$\displaystyle\big{[}\underline{\hat{Y}}_{i},\underline{\hat{X}}_{j}\big{]}$
$\displaystyle=$
$\displaystyle\mathrm{i}{{\cal{E}}_{ij}}^{k}\;\underline{\hat{X}}_{k}\,,$
$\displaystyle\big{[}\underline{\hat{X}}_{i},\underline{\hat{X}}_{j}\big{]}$
$\displaystyle=$
$\displaystyle\mathrm{i}{{\cal{E}}_{ij}}^{k}\;\underline{\hat{Y}}_{k}\,,$
$\displaystyle\big{[}\underline{\hat{Y}}_{i},\underline{\hat{Z}}_{j}\big{]}$
$\displaystyle=$
$\displaystyle\mathrm{i}{{\cal{E}}_{ij}}^{k}\;\underline{\hat{Z}}_{k}\,,$
$\displaystyle\big{[}\underline{\hat{X}}_{i},\underline{\hat{Z}}_{j}\big{]}$
$\displaystyle=$
$\displaystyle-\mathrm{i}\delta_{ij}\;\underline{\hat{X}}_{0}\,,$
$\displaystyle\big{[}\underline{\hat{Z}}_{i},\underline{\hat{Z}}_{j}\big{]}$
$\displaystyle=$
$\displaystyle-\mathrm{i}{{\cal{E}}_{ij}}^{k}\;\underline{\hat{Y}}_{k}\,,$
$\displaystyle\big{[}\underline{\hat{X}}_{0},\underline{\hat{X}}_{i}\big{]}$
$\displaystyle=$ $\displaystyle-\mathrm{i}\underline{\hat{Z}}_{i}\,,$
$\displaystyle\big{[}\underline{\hat{X}}_{0},\underline{\hat{Z}}_{i}\big{]}$
$\displaystyle=$ $\displaystyle-\mathrm{i}\underline{\hat{X}}_{i}\,,$
$\displaystyle\big{[}\underline{\hat{Y}}_{i},\underline{\hat{X}}_{0}\big{]}$
$\displaystyle=$ $\displaystyle 0\,,$ (XIII.18)

where, again, $i,j,k=1,2,3$ and ${{\cal{E}}_{ij}}^{k}$ is the three-
dimensional totally antisymmetric Levi-Civita symbol. The above infinitesimal
operators demonstrate the orbital part of the $\mathfrak{sp}(2,2)$ algebra.
Actually, by defining:515151An interesting point to be noticed here is that
usually when we speak about the orbital part of the $\mathfrak{sp}(2,2)$
algebra, we are thinking about actions on the functions of the ambient space
coordinates of the dS4 hyperboloid, but now we present the orbital part in
terms of functions on $\mathbb{S}^{3}$. The link between these two
realizations, let us say, the spacetime and the $\mathbb{S}^{3}$ realizations,
will be given in part III.

$\displaystyle
M_{4k}\equiv\underline{\hat{X}}_{k}\,,\;\;\;\;\;\;\;M_{04}\equiv\underline{\hat{X}}_{0}\,,\;\;\;\;\;\;\;M_{ki}\equiv{{\cal{E}}_{ki}}^{j}\;\underline{\hat{Y}}_{j}\,,\;\;\;\;\;\;\;M_{0k}\equiv\underline{\hat{Z}}_{k}\,,$
(XIII.19)

we explicitly get:

$\displaystyle[M_{AB},M_{CD}]=-\mathrm{i}\big{(}\eta_{AC}{M_{BD}}+\eta_{BD}{M_{AC}}-\eta_{AD}{M_{BC}}-\eta_{BC}{M_{AD}}\big{)}\,,\;\;\;\;\;\;\;A,B=0,1,2,3,4\,.$
(XIII.20)

Note that $M_{AB}=-M_{BA}$.

Quite analogous to the situation in the $1+1$-dimensional case, the above
operators cannot be defined on the whole Hilbert space
$L^{2}_{\mathbb{C}}(\mathbb{S}^{3})$, since they are unbounded. Below, we will
show that they are indeed essentially self-adjoint operators on the common
dense invariant subspace $\underline{\Delta}\;\big{(}\subset
L^{2}_{\mathbb{C}}(\mathbb{S}^{3})\big{)}$ made of all finite linear
combinations of elements of the orthonormal basis
$\big{\\{}|{Llm}\rangle\big{\\}}\equiv\big{\\{}{\cal{Y}}_{Llm}(\textbf{z})\big{\\}}$,
where ${\cal{Y}}_{Llm}(\textbf{z})$’s, with
$(L,l,m)\in\mathbb{N}\times\mathbb{N}\times\mathbb{Z}$, $0\leqslant l\leqslant
L$ and $-l\leqslant m\leqslant l$, are the hyperspherical harmonics on
$\mathbb{S}^{3}$ (for the explicit form of ${\cal{Y}}_{Llm}(\textbf{z})$’s,
see appendix D). Then, to check the irreducibility of the corresponding
representations $\underline{U}^{\mbox{\small{ps}}}_{0,\tau}$, it is sufficient
to show that $\underline{\Delta}$ does not contain any nontrivial subspace
invariant for the given infinitesimal operators. Considering the relations
given in appendix D, the actions of the above infinitesimal operators on the
basis $\big{\\{}|{Llm}\rangle\big{\\}}$’s respectively read Rabeie :

  * •

The actions of the space-rotation operators:

$\displaystyle\underline{\hat{Y}}_{1}\;|{Llm}\rangle$ $\displaystyle=$
$\displaystyle\mathrm{i}\Big{(}\sin\phi\;\frac{\partial}{\partial\theta}+\cot\theta\cos\phi\;\frac{\partial}{\partial\phi}\Big{)}\;|{Llm}\rangle$
(XIII.21) $\displaystyle=$
$\displaystyle-\frac{1}{2}\sqrt{(l+m)(l-m+1)}\;|{Ll,m-1}\rangle-\frac{1}{2}\sqrt{(l-m)(l+m+1)}\;|{Ll,m+1}\rangle\,,$
$\displaystyle\underline{\hat{Y}}_{2}\;|{Llm}\rangle$ $\displaystyle=$
$\displaystyle\mathrm{i}\Big{(}-\cos\phi\;\frac{\partial}{\partial\theta}+\cot\theta\sin\phi\;\frac{\partial}{\partial\phi}\Big{)}\;|{Llm}\rangle$
(XIII.22) $\displaystyle=$
$\displaystyle-\frac{\mathrm{i}}{2}\sqrt{(l+m)(l-m+1)}\;|{Ll,m-1}\rangle+\frac{\mathrm{i}}{2}\sqrt{(l-m)(l+m+1)}\;|{Ll,m+1}\rangle\,,$
$\displaystyle\underline{\hat{Y}}_{3}\;|{Llm}\rangle=\mathrm{i}\Big{(}-\frac{\partial}{\partial\phi}\Big{)}\;|{Llm}\rangle=m\;|{Llm}\rangle\,.$
(XIII.23)

  * •

The action of the time-translation operator:

$\displaystyle\underline{\hat{X}}_{0}\;|{Llm}\rangle$ $\displaystyle=$
$\displaystyle\mathrm{i}\Big{(}-\tau\cos\psi+\sin\psi\;\frac{\partial}{\partial\psi}\Big{)}\;|{Llm}\rangle$
(XIII.24) $\displaystyle=$
$\displaystyle\frac{\mathrm{i}}{2}\sqrt{\frac{(L-l+1)(L+l+2)}{(L+1)(L+2)}}\;(-\tau+L)\;|{L+1,lm}\rangle$
$\displaystyle-\frac{\mathrm{i}}{2}\sqrt{\frac{(L+l+1)(L-l)}{L(L+1)}}\;(\tau+L+2)\;|{L-1,lm}\rangle\,.$

  * •

The actions of the space-translation operators:

$\displaystyle\underline{\hat{X}}_{3}\;|{Llm}\rangle$ $\displaystyle=$
$\displaystyle\mathrm{i}\Big{(}-\cos\theta\;\frac{\partial}{\partial\psi}+\cot\psi\sin\theta\;\frac{\partial}{\partial\theta}\Big{)}\;|{Llm}\rangle$
(XIII.25) $\displaystyle=$
$\displaystyle\mathrm{i}\sqrt{\frac{(l-m+1)(L+l+2)(L-l)(l+m+1)}{(2l+1)(2l+3)}}\;|{L,l+1,m}\rangle$
$\displaystyle-\mathrm{i}\sqrt{\frac{(l-m)(L+l+1)(L-l+1)(l+m)}{(2l+1)(2l-1)}}\;|{L,l-1,m}\rangle\,,$
$\displaystyle\underline{\hat{X}}_{2}\;|{Llm}\rangle=-\mathrm{i}[\underline{\hat{X}}_{3},\underline{\hat{Y}}_{1}]\;|{Llm}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{2}\Bigg{(}\sqrt{\frac{(l-m+1)(l-m+2)(L+l+2)(L-l)}{(2l+1)(2l+3)}}\;|{L,l+1,m-1}\rangle$
(XIII.26)
$\displaystyle+\sqrt{\frac{(l+m-1)(l+m)(L+l+1)(L-l+1)}{(2l+1)(2l-1)}}\;|{L,l-1,m-1}\rangle$
$\displaystyle+\sqrt{\frac{(l+m+1)(l+m+2)(L+l+2)(L-l)}{(2l+1)(2l+3)}}\;|{L,l+1,m+1}\rangle$
$\displaystyle+\sqrt{\frac{(l-m)(l-m-1)(L+l+1)(L-l+1)}{(2l+1)(2l-1)}}\;|{L,l-1,m+1}\rangle\Bigg{)}\,,\;\;\;\;\;$
$\displaystyle\underline{\hat{X}}_{1}\;|{Llm}\rangle=-\mathrm{i}[\underline{\hat{X}}_{2},\underline{\hat{Y}}_{3}]\;|{Llm}\rangle$
$\displaystyle=$
$\displaystyle-\frac{\mathrm{i}}{2}\Bigg{(}\sqrt{\frac{(l-m+1)(l-m+2)(L+l+2)(L-l)}{(2l+1)(2l+3)}}\;|{L,l+1,m-1}\rangle$
(XIII.27)
$\displaystyle+\sqrt{\frac{(l+m-1)(l+m)(L+l+1)(L-l+1)}{(2l+1)(2l-1)}}\;|{L,l-1,m-1}\rangle$
$\displaystyle-\sqrt{\frac{(l+m+1)(l+m+2)(L+l+2)(L-l)}{(2l+1)(2l+3)}}\;|{L,l+1,m+1}\rangle$
$\displaystyle-\sqrt{\frac{(l-m)(l-m-1)(L+l+1)(L-l+1)}{(2l+1)(2l-1)}}\;|{L,l-1,m+1}\rangle\Bigg{)}\,.\;\;\;\;\;$

  * •

The actions of the boost operators:

$\displaystyle\underline{\hat{Z}}_{3}\;|{Llm}\rangle=\mathrm{i}[\underline{\hat{X}}_{0},\underline{\hat{X}}_{3}]\;|{Llm}\rangle$
$\displaystyle=$
$\displaystyle\frac{\mathrm{i}(-\tau+L)}{2}\Bigg{(}-\sqrt{\frac{(l-m)(l+m)(L-l+1)(L-l+2)}{(L+1)(L+2)(2l+1)(2l-1)}}\;|{L+1,l-1,m}\rangle$
(XIII.28)
$\displaystyle+\sqrt{\frac{(l-m+1)(l+m+1)(L+l+2)(L+l+3)}{(L+1)(L+2)(2l+1)(2l+3)}}\;|{L+1,l+1,m}\rangle\Bigg{)}$
$\displaystyle-\frac{\mathrm{i}(\tau+L+2)}{2}\Bigg{(}\sqrt{\frac{(l-m)(l+m)(L+l+1)(L+l)}{L(L+1)(2l+1)(2l-1)}}\;|{L-1,l-1,m}\rangle$
$\displaystyle-\sqrt{\frac{(l-m+1)(l+m+1)(L-l)(L-l-1)}{L(L+1)(2l+1)(2l+3)}}\;|{L-1,l+1,m}\rangle\Bigg{)}\,,$
$\displaystyle\underline{\hat{Z}}_{2}\;|{Llm}\rangle=\mathrm{i}[\underline{\hat{X}}_{0},\underline{\hat{X}}_{2}]\;|{Llm}\rangle$
$\displaystyle=$
$\displaystyle\frac{-\tau+L}{4}\Bigg{(}\sqrt{\frac{(l+m-1)(l+m)(L-l+1)(L-l+2)}{(L+1)(L+2)(2l+1)(2l-1)}}\;|{L+1,l-1,m-1}\rangle$
(XIII.29)
$\displaystyle+\sqrt{\frac{(l-m+1)(l-m+2)(L+l+2)(L+l+3)}{(L+1)(L+2)(2l+1)(2l+3)}}\;|{L+1,l+1,m-1}\rangle$
$\displaystyle+\sqrt{\frac{(l+m+1)(l+m+2)(L+l+2)(L+l+3)}{(L+1)(L+2)(2l+1)(2l+3)}}\;|{L+1,l+1,m+1}\rangle$
$\displaystyle+\sqrt{\frac{(l-m)(l-m-1)(L-l+1)(L-l+2)}{(L+1)(L+2)(2l+1)(2l-1)}}\;|{L+1,l-1,m+1}\rangle\Bigg{)}$
$\displaystyle+\frac{\tau+L+2}{4}\Bigg{(}\sqrt{\frac{(l+m-1)(l+m)(L+l+1)(L+l)}{L(L+1)(2l+1)(2l-1)}}\;|{L-1,l-1,m-1}\rangle$
$\displaystyle+\sqrt{\frac{(l-m)(l-m-1)(L+l+1)(L+l)}{L(L+1)(2l+1)(2l-1)}}\;|{L-1,l-1,m+1}\rangle$
$\displaystyle+\sqrt{\frac{(l+m+1)(l+m+2)(L-l)(L-l-1)}{L(L+1)(2l+1)(2l+3)}}\;|{L-1,l+1,m+1}\rangle$
$\displaystyle+\sqrt{\frac{(l-m+1)(l-m+2)(L-l)(L-l-1)}{L(L+1)(2l+1)(2l+3)}}\;|{L-1,l+1,m-1}\rangle\Bigg{)}\,,$
$\displaystyle\underline{\hat{Z}}_{1}\;|{Llm}\rangle=\mathrm{i}[\underline{\hat{X}}_{0},\underline{\hat{X}}_{1}]\;|{Llm}\rangle$
$\displaystyle=$
$\displaystyle\frac{\mathrm{i}(-\tau+L)}{4}\Bigg{(}-\sqrt{\frac{(l+m-1)(l+m)(L-l+1)(L-l+2)}{(L+1)(L+2)(2l+1)(2l-1)}}\;|{L+1,l-1,m-1}\rangle$
(XIII.30)
$\displaystyle-\sqrt{\frac{(l-m+1)(l-m+2)(L+l+2)(L+l+3)}{(L+1)(L+2)(2l+1)(2l+3)}}\;|{L+1,l+1,m-1}\rangle$
$\displaystyle+\sqrt{\frac{(l+m+1)(l+m+2)(L+l+2)(L+l+3)}{(L+1)(L+2)(2l+1)(2l+3)}}\;|{L+1,l+1,m+1}\rangle$
$\displaystyle+\sqrt{\frac{(l-m)(l-m-1)(L-l+1)(L-l+2)}{(L+1)(L+2)(2l+1)(2l-1)}}\;|{L+1,l-1,m+1}\rangle\Bigg{)}$
$\displaystyle-\frac{\mathrm{i}(\tau+L+2)}{4}\Bigg{(}\sqrt{\frac{(l+m-1)(l+m)(L+l+1)(L+l)}{L(L+1)(2l+1)(2l-1)}}\;|{L-1,l-1,m-1}\rangle$
$\displaystyle-\sqrt{\frac{(l-m)(l-m-1)(L+l+1)(L+l)}{L(L+1)(2l+1)(2l-1)}}\;|{L-1,l-1,m+1}\rangle$
$\displaystyle-\sqrt{\frac{(l+m+1)(l+m+2)(L-l)(L-l-1)}{L(L+1)(2l+1)(2l+3)}}\;|{L-1,l+1,m+1}\rangle$
$\displaystyle+\sqrt{\frac{(l-m+1)(l-m+2)(L-l)(L-l-1)}{L(L+1)(2l+1)(2l+3)}}\;|{L-1,l+1,m-1}\rangle\Bigg{)}\,.$

The above relations reveal that, on $\underline{\Delta}\subset
L^{2}_{\mathbb{C}}(\mathbb{S}^{3})$, the given infinitesimal operators of the
scalar representations $\underline{U}^{\mbox{\small{ps}}}_{0,\tau}$ are well
defined (in the allowed ranges of parameters), and also that the common dense
$\underline{\Delta}$ is invariant, with no nontrivial subspace invariant,
under the actions of the aforementioned operators. The irreducibility of the
given representations then is proved.

#####  XIII.1.4 Quantum Casimir operator

Having in mind the identities given in Eq. (XIII.19), the corresponding
(scalar) quadratic Casimir operator reads:525252Note that the quartic Casimir
operator vanishes in this case.

$\displaystyle
Q_{0}^{(1)}=-\frac{1}{2}M_{AB}M^{AB}=\underline{\hat{X}}_{0}^{2}+\sum_{i=1}^{3}\underline{\hat{Z}}_{i}^{2}-\sum_{i=1}^{3}\underline{\hat{X}}_{i}^{2}-\sum_{i=1}^{3}\underline{\hat{Y}}_{i}^{2}=-\tau(\tau+3)\mathbbm{1}\,,$
(XIII.31)

with $\tau=-\frac{3}{2}-\mathrm{i}\nu$ ($\nu\in\mathbb{R}$). Considering the
above, and quite analogous to the discussion (subsequent to Eq. (VI.36)) given
in the previous part, we here would like to draw attention to the fact that
the functions $f(\textbf{z})\in\underline{\Delta}$, which identify the (true)
quantum states carrying the dS4 scalar principal representations, are indeed
eigenfunctions of the quadratic Casimir operator $Q_{0}^{(1)}$ for the
eigenvalues $(\frac{9}{4}+\nu^{2})$, namely,
$\big{(}Q_{0}^{(1)}-(\frac{9}{4}+\nu^{2})\big{)}f(\textbf{z})=0$. Again, this
point (extended to the whole three series of the dS4 UIR’s) will be employed
in part III, when the spacetime realization of the dS4 UIR’s is considered, to
present the “wave equations” of dS4 elementary systems.

####  XIII.2 Principal series: general case

In a general case possessing spin $s$, the Hilbert space carrying the
principal series UIR’s of Sp$(2,2)$ is characterized by
$L^{2}_{{\mathbb{C}}^{2s+1}}(\mathbb{S}^{3})$. The action of the
representation operators $\underline{U}^{\mbox{\small{ps}}}_{s,\nu}$ on the
functions $f(\textbf{z})\in L^{2}_{{\mathbb{C}}^{2s+1}}(\mathbb{S}^{3})$
reads:

$\displaystyle\mathrm{Sp}(2,2)\ni\underline{g}=\begin{pmatrix}\textbf{a}&\textbf{b}\\\
\textbf{c}&\textbf{d}\end{pmatrix}\;:\;f(\textbf{z})\;\mapsto\;\Big{(}\underline{U}^{\mbox{\small{ps}}}_{s,\nu}(\underline{g})f\Big{)}(\textbf{z})=N(\underline{g},\textbf{z})\;{\cal{D}}^{s}\Big{(}\frac{-\textbf{z}^{\scriptscriptstyle\bigstar}\textbf{b}+\textbf{d}}{|-\textbf{z}^{\scriptscriptstyle\bigstar}\textbf{b}+\textbf{d}|}\Big{)}\;f(\underline{g}^{-1}\diamond\textbf{z})\,,$
(XIII.32)

where the definitions of $N(\underline{g},\textbf{z})$ and
$\underline{g}^{-1}\diamond\textbf{z}$ are precisely the same as those given
in Eq. (XIII.2), and ${\cal{D}}^{s}$’s stand for the $2s+1$-dimensional UIR’s
of $\mathrm{SU}(2)$ (to see more on the $\mathrm{SU}(2)$ UIR’s, one can refer
to appendix E). Clearly, by adjusting $s=0$, the UIR’s (XIII.32) coincide with
the scalar ones (XIII.2).

Taking steps parallel to those pointed out in subsubsection XIII.1.3, one can
check that the infinitesimal operators $L_{AB}$ of the UIR’s
$\underline{U}^{\mbox{\small{ps}}}_{s,\nu}$ are constituted by an orbital part
$M_{AB}$, which is exactly the one already given in Eq. (XIII.19), and a
spinorial part $S_{AB}$ ($L_{AB}=M_{AB}+S_{AB}$). These infinitesimal
operators explicitly read:

$\displaystyle L_{4k}$ $\displaystyle=$
$\displaystyle\big{(}M_{4k}\big{)}\mathbbm{1}_{2s+1}+\hat{J}_{k}\,,$
$\displaystyle L_{04}$ $\displaystyle=$
$\displaystyle\big{(}M_{04}\big{)}\mathbbm{1}_{2s+1}-\sum_{k=1}^{3}z^{k}\hat{J}_{k}\,,$
$\displaystyle L_{ki}$ $\displaystyle=$
$\displaystyle\big{(}M_{ki}\big{)}\mathbbm{1}_{2s+1}-{{\cal{E}}_{ki}}^{j}\;\hat{J}_{j}\,,$
$\displaystyle L_{0k}$ $\displaystyle=$
$\displaystyle\big{(}M_{0k}\big{)}\mathbbm{1}_{2s+1}+z^{4}\hat{J}_{k}+{{\cal{E}}_{ki}}^{j}\;z^{i}\hat{J}_{j}\,,$
(XIII.33)

where $i,j,k=1,2,3$ and $z^{1},\;...\;,z^{4}$ are given by the identities in
Eq. (XIII.1.3), while the matrix elements of the $(2s+1)\times(2s+1)$ matrices
$\hat{J}_{k}$ are given by:

$\displaystyle\left(\hat{J}_{k=1}\right)_{mm^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{(s+m)(s-m+1)}\;\delta_{m,m^{\prime}+1}+\frac{1}{2}\sqrt{(s-m)(s+m+1)}\;\delta_{m,m^{\prime}-1}\,,$
$\displaystyle\left(\hat{J}_{k=2}\right)_{mm^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\mathrm{i}}\sqrt{(s+m)(s-m+1)}\;\delta_{m,m^{\prime}+1}-\frac{1}{2\mathrm{i}}\sqrt{(s-m)(s+m+1)}\;\delta_{m,m^{\prime}-1}\,,$
$\displaystyle\left(\hat{J}_{k=3}\right)_{mm^{\prime}}$ $\displaystyle=$
$\displaystyle m\;\delta_{m,m^{\prime}}\,,$ (XIII.34)

where $m$ and $m^{\prime}$ are such that $-s\leqslant m,m^{\prime}\leqslant s$
and $s\pm m,\;s\pm m^{\prime}$ are integers. Note that the $\hat{J}_{k}$’s set
exhibits the matrix realization of the spin $s$ representation of the
$\mathfrak{su}(2)$ Lie algebra:

$\displaystyle[\hat{J}_{k},\hat{J}_{i}]=-\mathrm{i}{{\cal{E}}_{ki}}^{j}\;\hat{J}_{j}\,.$
(XIII.35)

Considering the above, one can also check that the infinitesimal operators
$L_{AB}$ verify the commutation relations (XII.2), and that the two Casimir
operators (XII.3) and (XII.4) respectively take the following forms:

$\displaystyle Q^{(1)}=\Big{(}-s(s+1)-\tau(\tau+3)\Big{)}\mathbbm{1}\,,$
(XIII.36) $\displaystyle
Q^{(2)}=\Big{(}-s(s+1)(\tau+1)(\tau+2)\Big{)}\mathbbm{1}\,,$ (XIII.37)

again, with $\tau=-3/2-\mathrm{i}\nu$ ($\nu\in\mathbb{R}$).

####  XIII.3 Principal series: restriction to the maximal compact subgroup
$\mathrm{SU}(2)\times\mathrm{SU}(2)$

In this subsection, we explicitly present the given action of
$\underline{U}^{\mbox{\small{ps}}}_{s,\nu}({\underline{g}})$’s in the Hilbert
space $L^{2}_{{\mathbb{C}}^{2s+1}}(\mathbb{S}^{3})$ (see Eq. (XIII.32)), when
it is restricted to the elements ${\underline{g}}$ belonging to the maximal
compact subgroup of Sp$(2,2)$, that is,
$\underline{\cal{K}}\sim\mathrm{SU}(2)\times\mathrm{SU}(2)$. Such elements are
determined by ${\underline{g}}=\begin{pmatrix}\textbf{v}&\textbf{0}\\\
\textbf{0}&\textbf{w}\end{pmatrix}$, with
$\textbf{v},\textbf{w}\in\mathrm{SU}(2)$ (see subsection IX.2). On this basis,
the action (XIII.32) takes the form:

$\displaystyle\Big{(}\underline{U}^{\mbox{\small{ps}}}_{s,\nu}(\underline{k})f\Big{)}(\textbf{z})={\cal{D}}^{s}(\textbf{w})f(\textbf{v}^{\scriptscriptstyle\bigstar}\textbf{z}\textbf{w})\,.$
(XIII.38)

Now, let ${\textbf{D}}^{(j_{l},j_{r})}$ denote the
$(2j_{l}+1)\times(2j_{r}+1)$-dimensional UIR of
$\mathrm{SU}(2)\times\mathrm{SU}(2)$ on the vectors $f\in
L^{2}_{{\mathbb{C}}^{2s+1}}(\mathbb{S}^{3})$; note that, again, the subscripts
‘$l$’ and ‘$r$’ refer to the left and right regular UIR’s of $\mathrm{SU}(2)$,
respectively, and that ${\cal{D}}^{s}\sim{\textbf{D}}^{(0,s)}$. Then, the
representations $\underline{U}^{\mbox{\small{ps}}}_{s,\nu}(\underline{g})$,
with $\underline{g}\in\mathrm{SU}(2)\times\mathrm{SU}(2)$, decompose into the
infinite direct sum:

$\displaystyle\bigoplus_{j\in\mathbb{N}/2}{\textbf{D}}^{(0,s)}\otimes{\textbf{D}}^{(j,j)}=\bigoplus_{j\in\mathbb{N}/2}\;\bigoplus_{|j-s|\leqslant
i\leqslant j+s}{\textbf{D}}^{(j,i)}\,.$ (XIII.39)

In the above decomposition, the UIR ${\textbf{D}}^{(j,i)}$ appears once and
only once in the reduction of the restriction
$\underline{U}^{\mbox{\small{ps}}}_{s,\nu}\;\big{|}_{\mathrm{SU}(2)\times\mathrm{SU}(2)}$.

####  XIII.4 Complementary series

The complementary series of the Sp$(2,2)$ UIR’s, like its principal
counterpart, is issued from the Iwasawa factorization of Sp$(2,2)$. The
carrier Hilbert space of the complementary series UIR’s, in a general case
with spin $s=0,1,2,\;...\;$, is
$L^{2}_{{\mathbb{C}}^{2s+1}}(\mathbb{S}^{3}\times\mathbb{S}^{3})$, while the
measure can be found through a reproducing kernel (see the process pointed out
in the dS2 case). For instance, in the scalar case $s=0$, we have Takahashi' :

$\displaystyle\langle
f_{1},f_{2}\rangle_{-\tau-3}=\frac{\Gamma(-\tau-1)\;\Gamma(\tau+3)}{2\pi^{2}\;\Gamma(-2\tau-3)}\iint_{\mathbb{S}^{3}\times\mathbb{S}^{3}}f^{\ast}_{1}(\textbf{z}_{1})f_{2}(\textbf{z}_{2})\;|\textbf{z}_{1}-\textbf{z}_{2}|^{-2\tau-6}\;\mathrm{d}\mu(\textbf{z}_{1})\mathrm{d}\mu(\textbf{z}_{2})\,,$
(XIII.40)

where $\tau$ is real and bounded, $-3<\tau<-3/2$, and again
$\mathrm{d}\mu(\textbf{z})$ is the $O(4)$-invariant measure on
$\mathbb{S}^{3}$. Note that by expanding the kernel in terms of the
hyperspherical harmonics on $\mathbb{S}^{3}$ (see appendix F):535353Note that
the value $\tau=-3$, for which the kernel reduces to $1$, represents the
critical value separating the scalar complementary series UIR’s from the
discrete ones. We will discuss this matter in the following subsection.

$\displaystyle\frac{\Gamma(-\tau-1)\;\Gamma(\tau+3)}{2\pi^{2}\;\Gamma(-2\tau-3)}\;|\textbf{z}-\textbf{z}_{2}|^{-2\tau-6}=\sum_{L,l,m}\frac{\Gamma(L+\tau+3)}{\Gamma(L-\tau)}{\cal{Y}}_{Llm}(\textbf{z}_{1}){\cal{Y}}^{\ast}_{Llm}(\textbf{z}_{2})\,,$
(XIII.41)

we achieve the following orthonormal basis of the corresponding Hilbert space
$L^{2}_{\mathbb{C}}(\mathbb{S}^{3}\times\mathbb{S}^{3})$:

$\displaystyle\Bigg{\\{}\widetilde{{\cal{Y}}}_{Llm}^{\tau}(\textbf{z})\equiv\sqrt{\frac{\Gamma(L-\tau)}{\Gamma(L+\tau+3)}}\;{\cal{Y}}_{Llm}(\textbf{z}),\;\;L\in\mathbb{N},\;0\leqslant
l\leqslant L,\;-l\leqslant m\leqslant l\Bigg{\\}}\,.$ (XIII.42)

The action of the associated representation operators
$\underline{U}^{\mbox{\small{cs}}}_{0,\tau}$ on the functions
$f(\textbf{z})\in L^{2}_{\mathbb{C}}(\mathbb{S}^{3}\times\mathbb{S}^{3})$
reads:

$\displaystyle\mathrm{Sp}(2,2)\ni\underline{g}=\begin{pmatrix}\textbf{a}&\textbf{b}\\\
\textbf{c}&\textbf{d}\end{pmatrix}\;:\;f(\textbf{z})\;\mapsto\;\Big{(}\underline{U}^{\mbox{\small{cs}}}_{0,\tau}(\underline{g})f\Big{)}(\textbf{z})$
$\displaystyle=$
$\displaystyle\big{|}-\textbf{b}^{\scriptscriptstyle\bigstar}\textbf{z}+\textbf{d}^{\scriptscriptstyle\bigstar}\big{|}^{2\tau}f\Big{(}\frac{\textbf{a}^{\scriptscriptstyle\bigstar}\textbf{z}-\textbf{c}^{\scriptscriptstyle\bigstar}}{-\textbf{b}^{\scriptscriptstyle\bigstar}\textbf{z}+\textbf{d}^{\scriptscriptstyle\bigstar}}\Big{)}$
(XIII.43) $\displaystyle\equiv$ $\displaystyle
N(\underline{g},\textbf{z})\;f(\underline{g}^{-1}\diamond\textbf{z})\,.$

In a spinorial case, with spin $s=1,2,3,\;...\;$, this action extends to:

$\displaystyle\mathrm{Sp}(2,2)\ni\underline{g}=\begin{pmatrix}\textbf{a}&\textbf{b}\\\
\textbf{c}&\textbf{d}\end{pmatrix}\;:\;f(\textbf{z})\;\mapsto\;\Big{(}\underline{U}^{\mbox{\small{cs}}}_{s,\tau}(\underline{g})f\Big{)}(\textbf{z})$
$\displaystyle=$
$\displaystyle\big{|}-\textbf{b}^{\scriptscriptstyle\bigstar}\textbf{z}+\textbf{d}^{\scriptscriptstyle\bigstar}\big{|}^{2\tau}\;{\cal{D}}^{s}\Big{(}\frac{-\textbf{z}^{\scriptscriptstyle\bigstar}\textbf{b}+\textbf{d}}{|-\textbf{z}^{\scriptscriptstyle\bigstar}\textbf{b}+\textbf{d}|}\Big{)}\;f\Big{(}\frac{\textbf{a}^{\scriptscriptstyle\bigstar}\textbf{z}-\textbf{c}^{\scriptscriptstyle\bigstar}}{-\textbf{b}^{\scriptscriptstyle\bigstar}\textbf{z}+\textbf{d}^{\scriptscriptstyle\bigstar}}\Big{)}$
(XIII.44) $\displaystyle\equiv$ $\displaystyle
N(\underline{g},\textbf{z})\;{\cal{D}}^{s}\Big{(}\frac{-\textbf{z}^{\scriptscriptstyle\bigstar}\textbf{b}+\textbf{d}}{|-\textbf{z}^{\scriptscriptstyle\bigstar}\textbf{b}+\textbf{d}|}\Big{)}\;f(\underline{g}^{-1}\diamond\textbf{z})\,,$

where, in this case, the functions $f(\textbf{z})\in
L^{2}_{{\mathbb{C}}^{2s+1}}(\mathbb{S}^{3}\times\mathbb{S}^{3})$ and, again,
${\cal{D}}^{s}$’s are the $2s+1$-dimensional UIR’s of $\mathrm{SU}(2)$, while
$-2<\tau<-3/2$. Apart from the the values of $\tau$, the above actions are
quite similar to those given in the principal case (see Eqs. (XIII.2) and
(XIII.32), respectively). Clearly, the infinitesimal operators and
correspondingly the two Casimir operators in this series of the UIR’s also
take the same forms as those appeared in the principal one (again, with the
exception of the values of $\tau$).

####  XIII.5 Discrete series

The discrete series of the Sp$(2,2)$ UIR’s is issued from the Cartan
decomposition of the group (see subsection IX.2). The corresponding UIR’s are
specified by two parameters $p$ and $q$, namely,
$\underline{U}^{\mbox{\small{ds}}}\equiv\Pi^{\pm}_{p,q}$, such that
$p,q\in\mathbb{N}/2$, $1\leqslant q\leqslant p$, and $p-q\in\mathbb{N}$
Takahashi' . In a general case with given $p$ and $q$ in their respective
allowed ranges, these UIR’s act in the Hilbert space
$L^{2}_{\mathbb{C}^{2p+1}}(B)$ as the space of vector-valued functions
$f(\textbf{z})$, analytic inside the open unit-ball $B$, with values in
$\mathbb{C}^{2p+1}$; the latter determines the carrier space of either UIR’s
${\cal{D}}^{p}\otimes\mathbbm{1}$ or $\mathbbm{1}\otimes{\cal{D}}^{p}$ of the
maximal compact subgroup
$\underline{{\cal{K}}}\sim\mathrm{SU}(2)\times\mathrm{SU}(2)$ of
Sp$(2,2)$,545454Note that ${\cal{D}}^{p}$’s refer to the $2p+1$-dimensional
UIR’s of $\mathrm{SU}(2)$ (see appendix E). which are respectively referred to
by the superscripts ‘$-$’ and ‘$+$’ in the representation operators (see Eq.
(XIII.47)). The functions $f(\textbf{z})$ are also supposed to be square
integrable with respect to the scalar product Takahashi' :

$\displaystyle\langle f_{1},f_{2}\rangle=c_{p,q}\int_{B}\langle
f_{1}(\textbf{z}),f_{2}(\textbf{z})\rangle_{\mathbb{C}^{2p+1}}\;\big{(}1-|\textbf{z}|^{2}\big{)}^{2q-2}\mathrm{d}\mu(\textbf{z})\,,$
(XIII.45)

where:

$\displaystyle c_{p,q}=\frac{(2q-1)(p+q)(p-q+1)}{\pi^{2}}\,,$ (XIII.46)

while $\langle\cdot,\cdot\rangle_{\mathbb{C}^{2p+1}}$ stands for the scalar
product in $\mathbb{C}^{2p+1}$, and
$\big{(}1-|\textbf{z}|^{2}\big{)}^{2q-2}\mathrm{d}\mu(\textbf{z})$ for the
invariant measure. [Note that for the limit cases $(p,q=1/2)$ and $(p,q=0)$, a
specific treatment is needed. We will study the case $(p,q=0)$ in the
following subsubsection.] The action of the representation operators
$\Pi^{\pm}_{p,q}$ in the Hilbert space $L^{2}_{\mathbb{C}^{2p+1}}(B)$ reads:

$\displaystyle\mathrm{Sp}(2,2)\ni\underline{g}=\begin{pmatrix}\textbf{a}&\textbf{b}\\\
\textbf{c}&\textbf{d}\end{pmatrix}\;:\;f(\textbf{z})\;\mapsto\;\Big{(}\Pi^{\pm}_{p,q}(\underline{g})f\Big{)}(\textbf{z})$
$\displaystyle=$
$\displaystyle\big{|}-\textbf{b}^{\scriptscriptstyle\bigstar}\textbf{z}+\textbf{d}^{\scriptscriptstyle\bigstar}\big{|}^{2\tau}\;{\cal{D}}^{p}\Big{(}\big{(}\eta^{\pm}(\underline{g},\textbf{z})\big{)}^{-1}\Big{)}\;f\Big{(}\frac{\textbf{a}^{\scriptscriptstyle\bigstar}\textbf{z}-\textbf{c}^{\scriptscriptstyle\bigstar}}{-\textbf{b}^{\scriptscriptstyle\bigstar}\textbf{z}+\textbf{d}^{\scriptscriptstyle\bigstar}}\Big{)}$
(XIII.47) $\displaystyle\equiv$ $\displaystyle
N(\underline{g},\textbf{z})\;{\cal{D}}^{p}\Big{(}\big{(}\eta^{\pm}(\underline{g},\textbf{z})\big{)}^{-1}\Big{)}f(\underline{g}^{-1}\diamond\textbf{z})\,,$

where $\tau=-q-1$ and:

$\displaystyle\big{(}\eta^{+}(\underline{g},\textbf{z})\big{)}^{-1}=\frac{-\textbf{z}^{\scriptscriptstyle\bigstar}\textbf{b}+\textbf{d}}{|-\textbf{z}^{\scriptscriptstyle\bigstar}\textbf{b}+\textbf{d}|}\,,\;\;\;\;\;\;\;\big{(}\eta^{-}(\underline{g},\textbf{z})\big{)}^{-1}=\frac{-\textbf{z}\textbf{c}+\textbf{a}}{|-\textbf{z}^{\scriptscriptstyle\bigstar}\textbf{b}+\textbf{d}|}\,.$
(XIII.48)

Note that
$|-\textbf{z}^{\scriptscriptstyle\bigstar}\textbf{b}+\textbf{d}|=|-\textbf{z}\textbf{c}+\textbf{a}|$.
Moreover, considering the Cartan decomposition of Sp$(2,2)$ (see subsection
IX.2), we have:

$\displaystyle
B\ni\textbf{z}\;\mapsto\;\textbf{z}^{\prime}\equiv\underline{g}^{-1}\diamond\textbf{z}=(\textbf{a}^{\scriptscriptstyle\bigstar}\textbf{z}-\textbf{c}^{\scriptscriptstyle\bigstar})(-\textbf{b}^{\scriptscriptstyle\bigstar}\textbf{z}+\textbf{d}^{\scriptscriptstyle\bigstar})^{-1}\in
B\,,$ (XIII.49)

Proceeding as before, it is a matter of straightforward calculations to show
that the infinitesimal operators $L_{AB}$ of the UIR’s $\Pi^{\pm}_{p,q}$,
constituted by an orbital part $M_{AB}$ and a spinorial part $S_{AB}$
($L_{AB}=M_{AB}+S_{AB}$), are given respectively by:

$\displaystyle L_{4k}$ $\displaystyle=$
$\displaystyle\mathrm{i}\Big{(}z_{k}\frac{\partial}{\partial
z^{4}}-z_{4}\frac{\partial}{\partial
z^{k}}\Big{)}\mathbbm{1}_{2s+1}\pm\hat{J}_{k}\,,$ $\displaystyle L_{04}$
$\displaystyle=$ $\displaystyle\mathrm{i}\Big{(}-\tau
z_{4}+z_{4}\sum_{\sigma=1}^{4}z^{\sigma}\frac{\partial}{\partial
z^{\sigma}}-\frac{|\textbf{z}|^{2}+1}{2}\frac{\partial}{\partial
z^{4}}\Big{)}\mathbbm{1}_{2s+1}\mp\sum_{k=1}^{3}\frac{z^{k}}{|\textbf{z}|}\hat{J}_{k}\,,$
$\displaystyle L_{ki}$ $\displaystyle=$
$\displaystyle-\mathrm{i}\Big{(}z_{k}\frac{\partial}{\partial
z^{i}}-z_{i}\frac{\partial}{\partial
z^{k}}\Big{)}\mathbbm{1}_{2s+1}-{{\cal{E}}_{ki}}^{j}\;\hat{J}_{j}\,,$
$\displaystyle L_{0k}$ $\displaystyle=$ $\displaystyle\mathrm{i}\Big{(}-\tau
z_{k}+z_{k}\sum_{\sigma=1}^{4}z^{\sigma}\frac{\partial}{\partial
z^{\sigma}}-\frac{|\textbf{z}|^{2}+1}{2}\frac{\partial}{\partial
z^{k}}\Big{)}\mathbbm{1}_{2s+1}\pm\frac{z^{4}}{|\textbf{z}|}\hat{J}_{k}+{{\cal{E}}_{ki}}^{j}\;\frac{z^{i}}{|\textbf{z}|}\hat{J}_{j}\,,$
(XIII.50)

where, again, $i,j,k=1,2,3$ and the $(2s+1)\times(2s+1)$-matrices
$\hat{J}_{k}$ stand for the matrix realization (XIII.2) of the spin $s$
representation of the $\mathfrak{su}(2)$ Lie algebra. One can easily show that
the above infinitesimal operators obey the commutation relations (XII.2), and
that the two Casimir operators (XII.3) and (XII.4) explicitly read:

$\displaystyle Q^{(1)}=\Big{(}-p(p+1)-\tau(\tau+3)\Big{)}\mathbbm{1}\,,$
(XIII.51) $\displaystyle
Q^{(2)}=\Big{(}-p(p+1)(\tau+1)(\tau+2)\Big{)}\mathbbm{1}\,,$ (XIII.52)

again, with $\tau=-q-1$.

Here, we underline again that the above representations form the square-
integrable part of the discrete series UIR’s of Sp$(2,2)$. Below, we will
study the scalar discrete series representations, lying at the “lowest limit”
of this series, which are not square integrable, and as we will point out
below, they rather deserve the appellation “degenerate scalar complementary
series”.

#####  XIII.5.1 Discrete series: scalar case ($\Pi_{p\geqslant 1,0}$)

Let ${\cal{H}}_{p-1}\equiv L^{2}_{\mathbb{C}}(\mathbb{S}^{3})$ denote the
carrier Hilbert space of $\Pi_{p,0}$ ($p=1,2,...$), that is, the closure of
the linear span of all square-integrable functions
$\mathbb{S}^{3}\ni\textbf{z}\mapsto f(\textbf{z})\in\mathbb{C}$, according to
the inner product Takahashi' :

$\displaystyle\langle
f_{1},f_{2}\rangle_{p-1}=\frac{(-1)^{p+1}}{4\pi^{2}(2p-1)!}\iint_{\mathbb{S}^{3}\times\mathbb{S}^{3}}f_{1}^{\ast}(\textbf{z}_{1})f_{2}(\textbf{z}_{2})\;|\textbf{z}_{1}-\textbf{z}_{2}|^{2(p-1)}\log|\textbf{z}_{1}-\textbf{z}_{2}|^{-2}\;\mathrm{d}\mu(\textbf{z}_{1})\mathrm{d}\mu(\textbf{z}_{2})\,,$
(XIII.53)

where $\mathrm{d}\mu(\textbf{z})$ is the $O(4)$-invariant measure on
$\mathbb{S}^{3}$. The functions $f(\textbf{z})$ are also supposed to verify
the following orthogonality condition:

$\displaystyle\int_{\mathbb{S}^{3}}{\cal{Y}}_{Llm}^{\ast}(\textbf{z})f(\textbf{z})\;\mathrm{d}\mu(\textbf{z})=0\,,$
(XIII.54)

for all triplets $(Llm)$, with $0\leqslant L\leqslant p-1$. Therefore, if the
function $f(\textbf{z})$ belongs to $L^{2}_{\mathbb{C}}(\mathbb{S}^{3})$, it
lies in the subspace orthogonal to the finite-dimensional subspace $V_{p-1}$
with the orthonormal basis $\big{\\{}{\cal{Y}}_{Llm}\;;\;0\leqslant L\leqslant
p-1,\;0\leqslant l\leqslant L,\;-l\leqslant m\leqslant l\big{\\}}$. The
subspace $V_{p-1}$, considering the allowed ranges of $L$, $l$, and $m$, is of
$p(p+1)(2p+1)/6$ dimension, and carries the irreducible (nonunitary!) dS4
finite-dimensional representations, which, with respect to the notations given
in appendix G, are determined by $(n_{1}=0,n_{2}=p-1)$. These representations
are _Weyl equivalent_ 555555If two representations are Weyl equivalent, then
they share same Casimir eigenvalue. to the UIR’s $\Pi_{p,0}$. [We will revisit
the above mathematical structure in detail later in subsection XVI.3, when the
spacetime realization of the representations is taken into account.]

An orthonormal basis for the Hilbert space ${\cal{H}}_{p-1}$, with respect to
the identities given in appendix F, is:

$\displaystyle\Bigg{\\{}\widetilde{{\cal{Y}}}_{Llm}^{\tau=-p-2}(\textbf{z})\equiv\sqrt{\frac{(L+p+1)!}{(L-p)!}}\;{\cal{Y}}_{Llm}(\textbf{z}),\;\;L\geqslant
p,\;0\leqslant l\leqslant L,\;-l\leqslant m\leqslant l\Bigg{\\}}\,.$ (XIII.55)

The action of the UIR’s $\Pi_{p,0}$ in ${\cal{H}}_{p-1}$ is realized by:

$\displaystyle\mathrm{Sp}(2,2)\ni\underline{g}=\begin{pmatrix}\textbf{a}&\textbf{b}\\\
\textbf{c}&\textbf{d}\end{pmatrix}\;:\;f(\textbf{z})\;\mapsto\;\Big{(}\Pi_{p,0}(\underline{g})f\Big{)}(\textbf{z})$
$\displaystyle=$
$\displaystyle\big{|}-\textbf{b}^{\scriptscriptstyle\bigstar}\textbf{z}+\textbf{d}^{\scriptscriptstyle\bigstar}\big{|}^{2\tau}f\Big{(}\frac{\textbf{a}^{\scriptscriptstyle\bigstar}\textbf{z}-\textbf{c}^{\scriptscriptstyle\bigstar}}{-\textbf{b}^{\scriptscriptstyle\bigstar}\textbf{z}+\textbf{d}^{\scriptscriptstyle\bigstar}}\Big{)}$
(XIII.56) $\displaystyle\equiv$ $\displaystyle
N(\underline{g},\textbf{z})\;f(\underline{g}^{-1}\diamond\textbf{z})\,,$

with $\tau=-p-2$. Note that the infinitesimal operators and correspondingly
the two Casimir operators in the scalar discrete representations again take
the same forms as those appeared in the scalar principal series, except the
values of $\tau$ which here, as already mentioned, are given by $\tau=-p-2$.

The fact that the above representations are the UIR’s of Sp$(2,2)$
corresponding to what we could call the “complementary degenerate” series
rests on proofs which necessitate expansion formulas of kernels of the type
$|\textbf{z}_{1}-\textbf{z}_{2}|^{2p-2}\log|\textbf{z}_{1}-\textbf{z}_{2}|^{-2}$,
given in appendix F.

###  XIV “Massive”/“massless” dS4 UIR’s and the Poincaré contraction

At this stage, it is critical to understand the physical content of the dS4
UIR’s in terms of their Poincaré contraction limit
($R\rightarrow\infty$).565656Here, it is perhaps worthwhile mentioning that
the contraction of the dS UIR’s to the Poincaré ones was first put forward in
Ref. Phillips  in $1+1$ dimension and then in Ref. Mickelsson  for the
representations of the dS group in $1+n$ dimension. On this basis, naturally,
three categories of the dS4 UIR’s come to fore: those which contract to the
Poincaré massive UIR’s; those which possess a massless content; and finally
those which either have nonphysical Poincaré contraction limit or do not have
Poincaré contraction limit at all. Below, we will briefly study the first two
categories. But, before that, it would be convenient to take a look at the
notion of group contraction on the representation level. In this regard, we
again follow the lines sketched in Ref. Garidi Thesis , and present such a
notion in a way that is more suited to the needs of this paper.

Note that, in this section, the parameters $c$ (the speed of light) and
$\hbar$ (the Planck constant) are no longer normalized to unity. Together with
$R$, the radius of curvature of the dS4 hyperboloid $\underline{M}_{R}$, they
represent dimensionally independent quantities, which are employed to fix the
natural unit of “mass” $\hbar/cR$ in dS4 relativity.

####  XIV.1 Group contraction (the representation level): a brief introduction

Let $U^{R}$ and $U$ respectively denote a family of representations of a group
$G$ acting in a Hilbert space ${\cal{H}}_{R}$ and a family of representations
of a group $G^{\prime}$ acting in a Hilbert space ${\cal{H}}$. The two given
groups $G$ and $G^{\prime}$ are supposed to be close enough to be put into
bijection $i\;:\;G\rightarrow G^{\prime}$ (which is not a homomorphism). We
also need a topological space $E$ (see FIG. 4), in which we can write the
representations of $G$ and $G^{\prime}$. This is achieved by having an
injective map $A_{R}\;:\;{\cal{H}}_{R}\mapsto
E\;\big{(}\supset{\cal{H}}\big{)}$ such that, for all $f\in{\cal{H}}_{R}$, we
get:

$\displaystyle\lim_{R\rightarrow\infty}A_{R}f=h\,,\;\;\;\;\;\;\;h\in{\cal{H}}\,.$
(XIV.1)

Considering the above, the representation $U$ is said to be the contraction of
the representation $U^{R}$, symbolized here by $U^{R}\longrightarrow U$, if:

$\displaystyle\mbox{for
all}\;f\in{\cal{H}}_{R}\,,\;\;\;\;\;\;\;\lim_{R\rightarrow\infty}A_{R}U^{R}(g)f=U(g^{\prime})h=U(g^{\prime})\lim_{R\rightarrow\infty}A_{R}f\,,$
(XIV.2)

where $g^{\prime}\in G^{\prime}$ corresponds to $g\in G$ by the bijection $i$.

Figure 4: Topological space $E$, in which the representations of $G$ and
$G^{\prime}$ can be written Garidi Thesis .

####  XIV.2 DS4 massive UIR’s

We now turn to the aforementioned categories of the dS4 UIR’s, given in terms
of their null-curvature limit. For the first category, i.e., those dS4 UIR’s
which contract to the Poincaré massive ones, the dS4 principal series UIR’s
$\underline{U}^{\mbox{\small{ps}}}_{s,\nu}$, with $s\in\mathbb{N}/2$ and
$\nu\in\mathbb{R}$, are merely involved. In this sense, they are usually
called dS4 massive representations. Technically, for these representations, by
introducing a relation between the dS4 representation parameter $\nu$ and the
Poincaré-Minkowski mass $m$ as:

$\displaystyle m=\frac{\hbar\nu}{cR}\,,$ (XIV.3)

the Poincaré contraction explicitly reads Mickelsson ; Garidi zero curvature
limit :

$\displaystyle\underline{U}^{\mbox{\small{ps}}}_{s,\nu}\;\;\underset{\underset{\hbar\nu/cR=m}{R\rightarrow\infty,\;\nu\rightarrow\infty}}{\longrightarrow}\;\;c_{>}{\cal{P}}^{>}_{s,m}\;\oplus\;c_{<}{\cal{P}}^{<}_{s,m}\,,$
(XIV.4)

where ${\cal{P}}^{\gtrless}_{s,m}$ respectively stand for the
positive/negative energy Wigner UIR’s of the Poincaré group, possessing mass
$m$ and spin $s$. Among the ‘coefficients’ $c_{>}$ and $c_{<}$, one can be set
$1$, while the other is zero. [We will go through the mathematical details of
this contraction limit later in subsections XVI.4 and XVII.2, when the
spacetime realization of the representations is taken into account.]

Here, having in mind the lines sketched in the previous sections, we would
like to once again underline that during the Poincaré contraction procedure,
symbolized by Eq. (XIV.4), the Lorentz subgroup remains intact. In other
words, the contraction procedure is carried out with respect to the Lorentz
subgroup in the dS4 and Poincaré groups. This is actually the point that gives
sense to the notion of spin in the context of dS4 massive elementary systems
(associated with the principal series UIR’s), since it arises from the same
$\mathrm{SU}(2)$ (that is, the rotations subgroup of the Lorentz group) that
the notion of spin in Poincaré relativity does.

Another point to be noticed here is the possible breaking of the
irreducibility of the dS4 principal (massive) UIR’s, through the Poincaré
contraction limit, into a direct sum of two Poincaré massive UIR’s with
positive and negative energies. Actually, from this evidence of the possible
breaking of irreducibility, the inquiry into the concept of “rest energy” in
dS4 (generally, dS) relativity leads to ambiguity, when one follows the
procedure of the Poincaré contraction. We will come back to this important
point in subsection XIV.4.

####  XIV.3 DS4 massless UIR’s

In comparison with the dS4 massive UIR’s, the massless ones are more subtle.
As a matter of fact, the dS4 group has no UIR analogous to the so-called
massless infinite-spin UIR’s of the Poincaré group. Then, the dS4 massless
UIR’s are distinguished as those UIR’s with a unique extension to the UIR’s of
the conformal group $\mathrm{SO}_{0}(2,4)$, while that extension is equivalent
to the conformal extension of the Poincaré massless UIR’s Barut ; Mack1977 .
Accordingly, two different categories of the dS4 massless UIR’s come to fore
(see appendix H):

  * •

The massless scalar case, which involves the only physical representation (in
the sense of contraction/extension to the Poincaré/conformal UIR’s) of the dS4
complementary series, labelled in our notations by
$\underline{U}^{\mbox{\small{cs}}}_{s=0,\nu=\frac{1}{2}}$:

$\displaystyle\left.\begin{array}[]{ccccccc}&&{\cal{C}}^{>}_{1,0,0}&&{\cal{C}}^{>}_{1,0,0}&\hookleftarrow&{\cal{P}}^{>}_{0,0}\\\
\underline{U}^{\mbox{\small{cs}}}_{0,\frac{1}{2}}&\hookrightarrow&\oplus&\underset{R\rightarrow\infty}{\longrightarrow}&\oplus&&\oplus\\\
&&{\cal{C}}^{<}_{-1,0,0}&&{\cal{C}}^{<}_{-1,0,0}&\hookleftarrow&{\cal{P}}^{<}_{0,0}\,.\end{array}\right.$
(XIV.8)

  * •

The “helicity $=\pm s$” cases, which involve all representations
$\Pi^{\pm}_{p=s,q=s}$,575757Here, the superscript ‘$\pm$’ stands for the
helicities $\pm s$. with $s>0$, lying at the lower limit of the dS4 discrete
series:

$\displaystyle\left.\begin{array}[]{ccccccc}&&{\cal{C}}^{>}_{s+1,0,s}&&{\cal{C}}^{>}_{s+1,0,s}&\hookleftarrow&{\cal{P}}^{>}_{-s,0}\\\
\Pi^{+}_{s,s}&\hookrightarrow&\oplus&\underset{R\rightarrow\infty}{\longrightarrow}&\oplus&&\oplus\\\
&&{\cal{C}}^{<}_{-s-1,0,s}&&{\cal{C}}^{<}_{-s-1,0,s}&\hookleftarrow&{\cal{P}}^{<}_{-s,0}\,.\end{array}\right.$
(XIV.12)
$\displaystyle\left.\begin{array}[]{ccccccc}&&{\cal{C}}^{>}_{s+1,s,0}&&{\cal{C}}^{>}_{s+1,s,0}&\hookleftarrow&{\cal{P}}^{>}_{s,0}\\\
\Pi^{-}_{s,s}&\hookrightarrow&\oplus&\underset{R\rightarrow\infty}{\longrightarrow}&\oplus&&\oplus\\\
&&{\cal{C}}^{<}_{-s-1,s,0}&&{\cal{C}}^{<}_{-s-1,s,0}&\hookleftarrow&{\cal{P}}^{<}_{s,0}\,,\end{array}\right.$
(XIV.16)

Note that: (i) Above, we have denoted by the arrows ‘$\hookrightarrow$’ unique
extension and by ${\mathcal{P}}^{\gtrless}_{s,0}$, respectively, the
positive/negative energy Poincaré massless representations, possessing
helicity $s$. (ii) Conformal invariance involves the discrete series
representations (and their lower end) of the (universal covering of the)
conformal group or its double covering group $\mathrm{SO}_{0}(2,4)$ or its
fourth covering group $\mathrm{SU}(2,2)$. Above, the associated conformal
UIR’s are denoted by ${\mathcal{C}}^{\gtrless}_{E_{0},j_{l},j_{r}}$, where the
parameters $(j_{l},j_{r})\in\mathbb{N}/2\times\mathbb{N}/2$ label the UIR’s of
$\mathrm{SU}(2)\times\mathrm{SU}(2)$, while $E_{0}$ refers to the
positive/negative conformal energy.

####  XIV.4 Discussion: rehabilitating the dS4 physics from the point of view
of a local Minkowskian observer

We here draw attention to Eq. (XIV.4), in particular, to the possible breaking
of the irreducibility of the dS4 principal UIR’s, during the Poincaré
contraction limit, into a direct sum of two Poincaré UIR’s possessing positive
and negative energies. This phenomenon can be discussed on two levels.

On one hand, it refers to the energy ambiguity in dS4 (generally, dS)
relativity, which actually originates from the existence of the discrete
symmetry (VIII.16) in the dS4 group sending any point
$(x^{0},\textbf{x})\in\underline{M}_{R}$ to its mirror image with respect to
the $x^{0}$-axis, that is, $(x^{0},-\textbf{x})\in\underline{M}_{R}$.585858To
see the latter point, one can also consider Eq. (IX.22), when $\psi=0$ and
${\textbf{w}}^{2}\;\big{(}=(\cos{\theta},\sin{\theta}\vec{w})\big{)}=-1$ (that
is, $\theta=\pi$). Considering this discrete symmetry, the dS4 infinitesimal
generators $L_{\texttt{A}0}$ (see Eq. (XII.1)), with $\texttt{A}=1,2,3,4$,
transform into their respective opposites with possibly different signs of the
corresponding conserved charges, depending on the sign of x. This, for
instance, implies that whether the generator $L_{40}$, which contracts to the
Poincaré energy operator, moves us forwards or backwards in time (towards
increasing or decreasing $x^{0}$) depends on the sign of x, and hence, cannot
be precisely determined. In this sense, this is the best we can do: there is
no positive conserved energy in dS4 (generally, dS) spacetime.

On the other hand and besides the global considerations concerning the notion
of energy in dS4 (generally, dS) relativity, it is yet critical to understand
the physical content of dS4 relativity with respect to its null-curvature
limit at a given point $x\in\underline{M}_{R}$, namely, from the point of view
of a local (tangent) Minkowskian observer, for whom the fundamental physical
conservation laws are understood from the principles of Einstein-Poincaré
relativity. Considering Eq. (XIV.4), it seems that, even at a specific point
(say $x\in\underline{M}_{R}$), one cannot give a precise meaning to the dS4
“rest energy” in terms of the Poincaré contraction of the representations. In
part III, we will come back to this significant point. We will show that there
is a proper choice of dS4 (global) modes, which, at the zero-curvature limit,
tend to the usual plane waves with exclusively positive frequencies, as far as
their analyticity domain has been chosen properly. Respecting the analyticity
prerequisite of these modes, the Poincaré contraction of the dS4 UIR’s can be
controlled in such a way that they contract merely to the Poincaré UIR’s with
positive energy.595959Note that, besides the analyticity prerequisite which is
of particular interest in this paper, there is also another possible way out
to lift the ambiguity of the dS4 “rest energy” in terms of the Poincaré
contraction of the representations which is based on a causality de Sitterian
semi-group. For this approach, readers are referred to Ref. Mizony1984 . This
rehabilitates dS4 (generally, dS) relativity from the point of view of the
interpretation that can be made by considering the Poincaré contraction limit.
Of course, this argument by no means implies that the energy concept can be
defined globally in dS4 (generally, dS) spacetime. As a matter of fact, under
Bogoliubov transformations, the given modes at a point $x\in\underline{M}_{R}$
may turn into modes at some point $x^{\prime}\in\underline{M}_{R}$ which their
flat limit is of negative energy. See the details in part III.

####  XIV.5 For comparison: AdS4 UIR’s and the Poincaré contraction

AdS4 spacetime is most easily described as embedded in $\mathbb{R}^{5}$
provided with the metric
$\eta_{A^{\prime}B^{\prime}}=\mbox{diag}(1,-1,-1,-1,1)$, where the indices
$A^{\prime}$ and $B^{\prime}$ take the values $0,1,2,3,5$. [Note that the
missing number $4$ is left apart for a possible extension to conformal
theories.] Points in $\mathbb{R}^{5}$, therefore, are denoted by
$x=(x^{0},x^{1},x^{2},x^{3},x^{5})$. In this context, AdS4 spacetime can be
visualized as (the covering space of) the connected hyperboloid
$(x)^{2}=\eta_{A^{\prime}B^{\prime}}x^{A^{\prime}}x^{B^{\prime}}=(x^{0})^{2}-(x^{1})^{2}-(x^{2})^{2}-(x^{3})^{2}+(x^{5})^{2}=R^{2}$,
where $R$ is the radius of curvature.

The AdS4 relativity group is SO${}_{0}(2,3)$, or its double covering
$\mathrm{Sp}(4,\mathbb{R})$, or even its universal covering
$\widetilde{\mathrm{SO}_{0}(2,3)}$.606060The time in this spacetime has a
periodic nature, and is proportional to the rotation parameter relevant to the
subgroup $SO(2)$. This periodicity of time can be circumvented by considering
the covering space of group of motions $\widetilde{\mathrm{SO}_{0}(2,3)}$.
Then, the time is not bounded. A realization of the associated Lie algebra,
quite similar to the dS4 case, is achieved by the linear span of the following
ten Killing vectors:

$\displaystyle
K_{A^{\prime}B^{\prime}}=x_{A^{\prime}}\partial_{B^{\prime}}-x_{B^{\prime}}\partial_{A^{\prime}}\,,\;\;\;\;\;\;\;K_{A^{\prime}B^{\prime}}=-K_{B^{\prime}A^{\prime}}\,.$
(XIV.17)

Here, however, contrary to the dS4 case, there exists one globally timelike
Killing vector, i.e., $K_{50}$.

On the quantum level, the above Killing vectors are represented by
(essentially) self-adjoint operators in the Hilbert space of (spinor-)tensor
valued functions, square integrable according to some invariant inner product
of Klein-Gordon type (or else) on the AdS4 manifold (or on respective phase
spaces). In the former case, these representations read as:
$K_{A^{\prime}B^{\prime}}\mapsto
L_{A^{\prime}B^{\prime}}=M_{A^{\prime}B^{\prime}}+S_{A^{\prime}B^{\prime}}$,
where the definitions of $M_{A^{\prime}B^{\prime}}$ and
$S_{A^{\prime}B^{\prime}}$ are the same as the dS4 case (see Eq. (XII.1) and
its subsequent discussions). In this context, and on the physical level, three
classes of the AdS4 UIR’s (besides the trivial representations) appear:

  * •

The first class is constituted by those AdS4 UIR’s, which belong to the
_holomorphic_ discrete series and its lower limit. In this class, the spectrum
of the “energy” operator $L_{50}$ has a lower (positive) bound, say
$\varsigma>0$, while the spin operator $S_{12}$ admits eigenvalues
$-s,\;...\;,s$, with $s\in\mathbb{N}/2$.

  * •

The second class, constituted by the AdS4 _anti-holomorphic_ discrete series
UIR’s, possesses the same features as the first one, but with an upper bound
for the spectrum of the “energy” operator, i.e., $-\varsigma>0$.

  * •

The third class includes those AdS4 UIR’s with unbounded energy and spin.

In the first case (i.e., the physically meaningful AdS4 UIR’s), the
representations are usually denoted by $D_{s,\varsigma}$, with
$s\in\mathbb{N}/2$ and $\varsigma\geqslant s+1$ (with the exception of a few
of them),616161To be more precise, the parameter $\varsigma$ takes the values
$\varsigma\in\mathbb{N}/2$, for the UIR’s in the strict sense of the
$\mathrm{Sp}(4,\mathbb{R})$ discrete series, and the values
$\varsigma\in\mathbb{R}$, while $\varsigma\geqslant s+1$, for the “discrete”
series UIR’s of $\widetilde{\mathrm{SO}_{0}(2,3)}$. where the parameters $s$
and $\varsigma$ carry the physical meanings of spin and the lowest “energy”,
respectively. [Note that, to some extent, the parameter $\varsigma$ plays the
role of the parameter $\nu$ in the classification of the dS4 UIR’s.] Among the
AdS4 UIR’s $D_{s,\varsigma}$, one must distinguish between the UIR’s with
$\varsigma>s+1$ and the two significant limit cases:

  * •

The limit scalar cases $D_{0,1}$ and $D_{0,\frac{1}{2}}$ (the latter is named
the “Rac” Fronsdal ).

  * •

The limit spinorial or tensorial cases $D_{s,s+1}$ and $D_{\frac{1}{2},1}$
(the latter is named the “Di” Fronsdal ).

Here, quite similar to the dS4 case, there exist two Casimir operators and
their eigenvalues entirely characterize the AdS4 UIR’s. These Casimir
operators explicitly read:

$\displaystyle Q^{(1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}L_{A^{\prime}B^{\prime}}L^{A^{\prime}B^{\prime}}\,,$
(XIV.18) $\displaystyle Q^{(2)}$ $\displaystyle=$ $\displaystyle-
W_{A^{\prime}}W^{A^{\prime}}\,,\;\;\;\;\;\;\;W_{{}^{\prime}}=-\frac{1}{8}{\cal{E}}_{\tiny{A^{\prime}B^{\prime}C^{\prime}D^{\prime}E^{\prime}}}L^{B^{\prime}C^{\prime}}L^{D^{\prime}E^{\prime}}\,,$
(XIV.19)

and, according to the parameters introduced above, their respective
eigenvalues are:

$\displaystyle\langle Q^{(1)}\rangle_{\text{\tiny{AdS${}_{4}$}}}$
$\displaystyle=$ $\displaystyle s(s+1)+\varsigma(\varsigma-3)\,,$ (XIV.20)
$\displaystyle\langle Q^{(2)}\rangle_{\text{\tiny{AdS${}_{4}$}}}$
$\displaystyle=$ $\displaystyle-s(s+1)(\varsigma-1)(\varsigma-2)\,.$ (XIV.21)

Proceeding as the previous subsections, we now point out those AdS4 UIR’s
which contract to the Poincaré massive UIR’s (the so-called AdS4 massive
representations) and those AdS4 UIR’s which have a massless content (the so-
called AdS4 massless representations). For the AdS4 massive cases, the
(holomorphic) AdS4 discrete series representations $D_{s,\varsigma}$, with
$s\in\mathbb{N}/2$ and $\varsigma>s+1$, are only involved Evans67 . For them,
defining the Poincaré-Minkowski mass $m$ as:

$\displaystyle m=\frac{\hbar\varsigma}{cR}\,,$ (XIV.22)

the Poincaré contraction yields Evans67 :

$\displaystyle
D_{s,\varsigma}\underset{{\underset{\hbar\varsigma/cR=m}{R\rightarrow\infty,\;\varsigma\rightarrow\infty}}}{\longrightarrow}{\cal
P}^{>}_{s,m}\,.$ (XIV.23)

Evidently, for the AdS4 (generally, AdS) massive cases, quite contrary to the
dS4 (generally, dS) relativity (see Eq. (XIV.4)), the Poincaré contraction
entails no energy ambiguity; the AdS4 (generally, AdS) massive UIR’s contract
merely to the positive energy Poincaré massive UIR’s (in their respective
dimensions). It is also interesting to note that the Poincaré massive UIR’s
with negative energy can be obtained when we choose the _anti-holomorphic_
discrete series representations $D_{s,-\varsigma}$:

$\displaystyle
D_{s,-\varsigma}\underset{R\rightarrow\infty,\;\varsigma\rightarrow\infty}{\longrightarrow}{\cal
P}^{<}_{s,m}\,.$ (XIV.24)

On the other hand, for the AdS4 massless (conformal) representations, two
different categories appear Barut ; Angelopoulos :

  * •

The massless scalar case, involving the UIR $D_{0,1}$.

  * •

The spinorial or tensorial cases, involving all UIR’s $D_{s,s+1}$, with $s>0$,
lying at the lower end of the holomorphic AdS4 discrete series.

For the above massless representations, the following extensions hold Barut ;
Angelopoulos :

$\displaystyle D_{0,1}\;\;\hookrightarrow\;\;{\cal
C}^{>}_{1,0,0}\;\;\underset{R\rightarrow\infty}{\longrightarrow}\;\;{\cal
C}^{>}_{1,0,0}\;\;\hookleftarrow\;\;{\cal P}^{>}_{0,0}\,,$ (XIV.25)
$\displaystyle\left.\begin{array}[]{ccccccc}&&{\cal{C}}^{>}_{s+1,s,0}&&{\cal{C}}^{>}_{s+1,s,0}&\hookleftarrow&{\cal{P}}^{>}_{s,0}\\\
D_{s,s+1}&\hookrightarrow&\oplus&\underset{R\rightarrow\infty}{\longrightarrow}&\oplus&&\oplus\\\
&&{\cal{C}}^{>}_{s+1,0,s}&&{\cal{C}}^{>}_{s+1,0,s}&\hookleftarrow&{\cal{P}}^{>}_{-s,0}\,.\end{array}\right.$
(XIV.29)

The arrows ‘$\hookrightarrow$’, as before, denote unique extension. As is
obvious from the above, for the AdS4 massless cases, contrary to dS4
relativity, the extensions entail no energy ambiguity. There is, however, an
ambiguity concerning helicity. As a matter of fact, the notion of helicity is
not defined in AdS4 relativity at all. At the end, we must underline that all
other AdS4 representations have either nonphysical Poincaré contraction limit
or do not have Poincaré contraction limit.

##  Part III $1+3$-dimensional dS (dS4) geometry and relativity (QFT)

In the previous part, we have employed the dS4 relativity group Sp$(2,2)$ and
its UIR’s (in the Wigner sense) to provide a robust mathematical structure
describing (free) elementary systems in dS4 spacetime on the classical and
quantum mechanics levels. In the present part, utilizing the above
mathematical materials, on one hand and on the other hand, adopting the
Wightman-Gärding axioms along with analyticity requirements in the
complexified pseudo-Riemanian manifold, we proceed with a consistent QFT
description of dS4 elementary systems. Technically, to manage the group
representations, the whole process is performed in terms of coordinate-
independent (global) dS4 plane waves. The latter, defined in their relevant
tube domains of the complex dS4 manifold, are the formal analogue of the usual
plane waves in Minkowski spacetime. As a matter of fact, letting the curvature
go to zero, the dS4 plane waves (for massive cases that such a limit exists;
see section XIV) precisely reduce to their Minkowskian counterparts in such a
way that, as far as the analyticity domain has been chosen properly, no
negative frequency mode appears in this limiting process. These waves also
provide us with a remarkable (global) dS4 Fourier transform which turns into
the ordinary Fourier transform at the null-curvature limit.626262The dS4
(generally, dSd) plane waves are also of great significance in constructing
possible models of dS4/CFT3 (generally, dSd/CFTd-1) correspondence. For this
case, which is beyond the scope of this paper, readers are referred to Refs.
CFT Hollands ; CFT Tanhayi ; CFT Wrochna .

This approach to QFT reading of elementary systems in dS4 spacetime is
justified by the fact that the resulting theory has all the properties which
one might require from a (free) quantum field on a spacetime with high
symmetry. Indeed, a quantum field in this context is, roughly speaking, a
distribution $\hat{\Psi}$ on dS4 spacetime $\underline{M}_{R}$, solution to
the field equation, with values in a set of symmetric operators in some inner
product (Fock) space ${\mathscr{H}}$, and fulfilling the following physical
requirements:

  * •

Covariance, in the usual strong sense; there is a unitary representation
$\underline{\mathscr{U}}$ of the dS4 group Sp$(2,2)$ on the Fock space of
states ${\mathscr{H}}$, such that
$\underline{\mathscr{U}}(\underline{g})\hat{\Psi}(x)\underline{\mathscr{U}}^{-1}(\underline{g})=\hat{\Psi}(\underline{g}\diamond
x)$ for any $\underline{g}\in\mathrm{Sp}(2,2)$ and $x\in\underline{M}_{R}$.
[Borrowing the notations used in the previous part, let $\underline{U}$ be the
natural representation of the dS4 group in a Hilbert space ${\cal{H}}$. Then,
$\underline{\mathscr{U}}$ denotes the extension of $\underline{U}$ to the Fock
space ${\mathscr{H}}$ built on ${\cal{H}}$.]

  * •

Existence of a distinguished state $\Omega\in{\mathscr{H}}$, called vacuum,
which is invariant under the representation
$\underline{\mathscr{U}}(\underline{g})$, for all
$\underline{g}\in\mathrm{Sp}(2,2)$.

  * •

Locality/(anti-)commutativity, with respect to the dS4 causal structure (see
section VII).

  * •

Positive definiteness of all physical states.

Moreover, this construction allows to control the thermal properties and, as
pointed out above, the zero-curvature limit of the fields (if exists!).

Below, mainly considering dS4 scalar fields, we will make this QFT
construction explicit. Technically, dS4 scalar fields, in spite of their
simplicity, provide us with a complete framework to discuss in detail all
essential ingredients of this QFT formulation of dS4 elementary systems.
Moreover, it has been already shown that all other (spinor-)tensor fields in
dS4 spacetime can be given in terms of a copy of dS4 scalar fields: for the
massive and massless spin-$\frac{1}{2}$ fields, see Ref. Massive/Massless 1/2
, for the massive and massless spin-$1$ fields, respectively, Refs. Massive 1
and Massless 1 , for the massive and massless spin-$\frac{3}{2}$ fields, Ref.
Massive 3/2 , for the massive spin-$2$ field, Ref. Massive 2 , and finally for
the massless spin-$2$ field (the dS4 linear quantum gravity), Refs. Massless 2
; Massless 2' ; Massless 2'' ; Bamba 1 ; dS gravity 1 ; dS gravity 2 ;
DehghaniTakook . Then, in this sense also, the choice of dS4 scalar fields
seems quite natural for our discussions.

###  XV DS4 wave equations

The whole QFT construction that is meant to be presented here, as pointed out
above, is carried out in terms of dS4 plane waves, defined globally on the dS4
hyperboloid $\underline{M}_{R}$. For a detailed explanation of these global
waves, we first need to introduce the corresponding “wave equations”. This is
our task in this section.

Technically, the dS4 plane waves are considered as _eigendistributions_
636363See the next section. of the quadratic Casimir operator $Q^{(1)}$ for
the eigenvalues associated with the principal, complementary, and discrete
series of the dS4 UIR’s.646464Here, it is perhaps worthwhile to recall the
arguments subsequent to Eq. (XIII.31). Accordingly, with respect to the
Dixmier notations (see section XII), the wave equations read:

$\displaystyle\Big{(}Q^{(1)}+p(p+1)+(q+1)(q-2)\Big{)}\Psi(x)=0\,,\;\;\;\;\;\;\;x\in\underline{M}_{R}\,,$
(XV.1)

with the specific allowed ranges of values assumed by the parameters $p$ and
$q$ for the three series of the dS4 UIR’s (see subsections XII.1, XII.2, and
XII.3).

For the scalar waves (denoted here by $\phi(x)$), which are of particular
interest in our study, two different cases appear: those associated with the
scalar principal and complementary series which are determined by $p=0$, and
the ones associated with the scalar discrete series which are given by $q=0$
(see section XII). In both cases, the associated plane waves are characterized
by solutions to the following wave equations:

$\displaystyle\Big{(}Q^{(1)}+\tau(\tau+3)\Big{)}\phi(x)=0\,,\;\;\;\;\;\;\;x\in\underline{M}_{R}\,,$
(XV.2)

where the unifying complex parameter $\tau$, for the scalar principal series,
takes the values $\tau=-q-1=-3/2-\mathrm{i}\nu$, with $\nu\in\mathbb{R}$, for
the scalar complementary series, the values $\tau=-q-1=-3/2-\nu$, with
$\nu\in\mathbb{R}$ and $0<|\nu|<3/2$, and for the scalar discrete series, the
values $\tau=p-1$ or $\tau=-p-2$, with $p=1,2,...$ . In a shortcut, we assert
that the solutions to the wave equations (XV.2) are well defined for all
values of $\tau$ with $\mbox{Re}(\tau)<0$ (see subsection XVI.3). Accordingly,
for the scalar discrete series, the values $\tau=p-1$ need to be dropped.
Indeed, this series begins with $\tau=-3$, which is precisely where the
complementary series ends on its left.

####  XV.1 Computation in ambient notations

In ambient space notations, dS4 fields are represented by symmetric
(spinor-)tensor fields on the hyperboloid:

$\displaystyle\underline{M}_{R}\ni
x\;\mapsto\;\Psi(x)\equiv\Psi^{(r)}_{A_{1}\;...\;A_{n}}(x)\,,$ (XV.3)

where $r=n+1/2$ ($n$ being the tensorial rank) and
$A_{1},\;...\;,\;A_{n}=0,1,2,3,4$. Note that: (i) For the sake of simplicity,
the spinorial index, characterizing the four spinor components, has been
omitted. (ii) From now on, whenever is possible, we also omit the tensorial
indices.

The (spinor-)tensor fields $\Psi^{(r)}(x)$ are supposed to be homogeneous
functions in the $\mathbb{R}^{5}$-variables $x^{A}$ with some arbitrarily
chosen homogeneity degree $\ell$:

$\displaystyle x\cdot\partial\Psi^{(r)}(x)\;\Big{(}\equiv
x^{A}\frac{\partial}{\partial x^{A}}\Big{)}=\ell\Psi^{(r)}(x)\,.$ (XV.4)

Trivially, every homogeneous (spinor-)tensor field $\Psi^{(r)}(x)$ of the
$\mathbb{R}^{5}$ variables does not represent a dS4 physical entity. Actually,
to make sure that the field $\Psi^{(r)}(x)$ lies in the dS4 tangent spacetime,
it also must verify the transversality requirement for all indices
$A_{1},\;...\;,\;A_{n}$:

$\displaystyle
x^{A_{i}}\Psi^{(r)}_{A_{1}\;...\;A_{i}\;...\;A_{n}}(x)=(x\cdot\Psi)^{(r-1)}_{A_{1}\;...\;\breve{A}_{i}\;...\;A_{n}}(x)=0\,,$
(XV.5)

or more concisely $x\cdot\Psi^{(r)}(x)=0$, where $\breve{A}_{i}$ means that
this index is omitted.

We define here the symmetric, “transverse projector”
$\theta_{AB}=\eta_{AB}+R^{-2}x_{A}x_{B}$, verifying
$\theta_{AB}x^{A}=\theta_{AB}x^{B}=0$. [This projector is indeed the
transverse form of the dS4 metric in ambient space notations; this point will
be clarified in the next subsection.] We employ $\theta_{AB}$ to construct
transverse entities, such as
$\bar{\partial}_{A}=\theta_{AB}\partial^{B}=\partial_{A}+R^{-2}x_{A}x\cdot\partial$,
called transverse derivative, for which we have:

$\displaystyle\bar{\partial}_{A}x_{B}=\theta_{AB}\,,\;\;\;\;\;\;\;\mbox{and}\;\;\;\;\;\;\;\bar{\partial}_{A}(x)^{2}=0\,.$
(XV.6)

The latter identity shows that the differential operator $\bar{\partial}$
commutates with $(x)^{2}$, which means that $\bar{\partial}$ is intrinsically
defined on the dS4 hyperboloid $(x)^{2}=-R^{2}$. Considering the above, for a
general (spinor-)tensor field $\Psi^{(r)}_{A_{1}\;...\;A_{n}}(x)$, the
operator $\mathfrak{T}$ with the following definition:

$\displaystyle(\mathfrak{T}\Psi)^{(r)}_{A_{1}\;...\;A_{n}}(x)=\Bigg{(}\prod_{i=1}^{n}\theta^{B_{i}}_{A_{i}}\Bigg{)}\Psi^{(r)}_{B_{1}\;...\;B_{n}}(x)\,,$
(XV.7)

guarantees the transversality in each tensorial index; since the degree of
homogeneity of $\theta_{AB}$ is zero, the above definition does not change the
degree of homogeneity of the given (spinor-)tensor field.

We now turn to the explicit description of the dS4 quadratic Casimir operator
in ambient space notations. As already mentioned (see section XII), this
operator is written in terms of the self-adjoint operators $L_{AB}$ associated
with each of the ten Killing vectors (VIII.1). In the Hilbert space of
symmetric (spinor-)tensors $\Psi^{(r)}_{A_{1}...}(x)$ on $\underline{M}_{R}$,
square integrable according to some invariant inner (Klein-Gordon type)
product, the generator representatives $L_{AB}^{(r)}$ are defined by Moylan :

$\displaystyle L_{AB}^{(r)}=M_{AB}+S_{AB}^{(n)}+S^{(\frac{1}{2})}_{AB}\,,$
(XV.8)

where the orbital part is given by
$M_{AB}=-\mathrm{i}(x_{A}\partial_{B}-x_{B}\partial_{A})=-\mathrm{i}(x_{A}\bar{\partial}_{B}-x_{B}\bar{\partial}_{A})$,
while the spinorial parts $S_{AB}^{(n)}$ and $S^{(\frac{1}{2})}_{AB}$
respectively act on the tensorial indices as:

$\displaystyle
S_{AB}^{(n)}\Psi^{(r)}_{A_{1}\;...\;A_{n}}=-\mathrm{i}\sum_{i=1}^{n}\Big{(}\eta_{AA_{i}}\Psi^{(r)}_{A_{1}\;...\;(A_{i}\rightarrow
B)\;...\;A_{n}}-(A\rightleftharpoons B)\Big{)}\,,$

and on the spinorial indices by
$S^{(\frac{1}{2})}_{AB}=-\frac{\mathrm{i}}{4}[\gamma_{A},\gamma_{B}]$ (recall
from section VIII that the five $4\times 4$-matrices $\gamma_{A}$ generate the
Clifford algebra). The action of the quadratic Casimir operator on a given
spinor-tensor field $\Psi^{(r)}_{A_{1}\;...\;A_{n}}(x)$ then reads:

$\displaystyle
Q_{r}^{(1)}\;\Psi^{(r)}(x)=-\frac{1}{2}L^{(r)}_{AB}L^{(r)AB}\;\Psi^{(r)}(x)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\Big{(}M_{AB}+S_{AB}^{(n)}+S^{(\frac{1}{2})}_{AB}\Big{)}\Big{(}M^{AB}+S^{(n)AB}+S^{(\frac{1}{2})AB}\Big{)}\Psi^{(r)}(x)$
(XV.9) $\displaystyle=$
$\displaystyle-\frac{1}{2}\Big{(}M_{AB}M^{AB}+S_{AB}^{(n)}S^{(n)AB}+S^{(\frac{1}{2})}_{AB}S^{(\frac{1}{2})AB}\Big{)}\Psi^{(r)}(x)$
$\displaystyle-\Big{(}M_{AB}S^{(n)AB}+M_{AB}S^{(\frac{1}{2})AB}+S^{(\frac{1}{2})}_{AB}S^{(n)AB}\Big{)}\Psi^{(r)}(x)\,,$

with:

$\displaystyle\frac{1}{2}M_{AB}M^{AB}\;\Psi^{(r)}\;\Big{(}=-Q_{0}^{(1)}\;\Psi^{(r)}\Big{)}$
$\displaystyle=$ $\displaystyle R^{2}\bar{\partial}^{2}\,,$
$\displaystyle\frac{1}{2}S_{AB}^{(n)}S^{(n)AB}\;\Psi^{(r)}$ $\displaystyle=$
$\displaystyle n(n+3)\Psi^{(r)}-2\Sigma_{2}\eta\Psi^{(r)\prime}\,,$
$\displaystyle\frac{1}{2}S_{AB}^{(\frac{1}{2})}S^{(\frac{1}{2})AB}\;\Psi^{(r)}$
$\displaystyle=$ $\displaystyle\frac{5}{2}\Psi^{(r)}\,,$ $\displaystyle
M_{AB}S^{(n)AB}\;\Psi^{(r)}$ $\displaystyle=$ $\displaystyle
2\Sigma_{1}\partial
x\cdot\Psi^{(r)}-2\Sigma_{1}x\partial\cdot\Psi^{(r)}-2n\Psi^{(r)}\,,$
$\displaystyle M_{AB}S^{(\frac{1}{2})AB}\;\Psi^{(r)}$ $\displaystyle=$
$\displaystyle-\frac{\mathrm{i}}{2}\gamma_{A}\gamma_{B}M^{AB}\Psi^{(r)}\,,$
$\displaystyle S^{(\frac{1}{2})}_{AB}S^{(n)AB}\;\Psi^{(r)}$ $\displaystyle=$
$\displaystyle
n\Psi^{(r)}-\Sigma_{1}\gamma\big{(}\gamma\cdot\Psi^{(r)}\big{)}\,.$ (XV.10)

Note that: (i) The subscript ‘$r$’ in $Q_{r}^{(1)}$ refers to the fact that
the carrier space is constituted by rank $r$ spinor-tensors. (ii) $\Sigma_{m}$
designates the symmetrizer of the tensor product of two symmetric
(spinor-)tensors $\Phi$ and $\Psi$ of, respectively, tensorial rank $m$ and
$n-m$, where $m\leqslant\lfloor n/2\rfloor$, based upon which the components
of the symmetrized tensor product are given by:

$\displaystyle\big{(}\Sigma_{m}\Phi\Psi\big{)}_{A_{1}\;...\;A_{n}}=\sum_{i_{1}<i_{2}<\;...\;<i_{m}}\big{(}\Phi_{A_{i_{1}}A_{i_{2}}\;...\;A_{i_{m}}}\big{)}\big{(}\Psi_{A_{1},\;...\;\breve{A}_{i_{1}}\;...\;\breve{A}_{i_{2}}\;...\;\breve{A}_{i_{m}}\;...\;A_{n}}\big{)}\,,$
(XV.11)

where $\breve{A}_{i_{1}},\breve{A}_{i_{2}}$, and $\breve{A}_{i_{m}}$ mean that
these terms are omitted. (iii) The trace, associated with the tensorial part,
of the spinor-tensor field $\Psi^{(r)}$ of tensorial rank $n$ is denoted by
$\Psi^{(r)\prime}$, which is a symmetric spinor-tensor field of tensorial rank
$n-2$, given by:

$\displaystyle\Psi^{(r)\prime}_{A_{1}\;...\;A_{n-2}}=\eta^{A_{n-1}A_{n}}\Psi^{(r)}_{A_{1}\;...\;A_{n-2}A_{n-1}A_{n}}\,.$
(XV.12)

(iv) The operators $Q_{r}^{(1)}$ and $L^{(r)}_{AB}$ commutate with $(x)^{2}$.
Therefore, they are intrinsically defined on the dS4 hyperboloid
$(x)^{2}=-R^{2}$.

Finally, taking all the above identities into account, we can explicitly
rewrite Eq. (XV.9) as:

$\displaystyle Q_{r}^{(1)}\;\Psi^{(r)}(x)$ $\displaystyle=$
$\displaystyle\Big{(}Q_{0}^{(1)}+\frac{\mathrm{i}}{2}\gamma_{A}\gamma_{B}M^{AB}-n(n+2)-\frac{5}{2}\Big{)}\Psi^{(r)}$
(XV.13) $\displaystyle-2\Sigma_{1}\partial
x\cdot\Psi^{(r)}+2\Sigma_{1}x\partial\cdot\Psi^{(r)}+\Sigma_{1}\gamma\big{(}\gamma\cdot\Psi^{(r)}\big{)}+2\Sigma_{2}\eta\Psi^{(r)\prime}\,.$

Now, substituting Eq. (XV.13) into (XV.1), we get the explicit form of the
wave equations (XV.1) in terms of ambient notations. The very point that must
be noticed here is that clearly due to the form of $Q_{r}^{(1)}$ in ambient
notations (see Eq. (XV.13)), for a given (spinor-)tensor field, the space of
solutions to the relevant wave equation contains some invariant subspaces
which must be eliminated if one wishes to be left with the space that solely
carries the corresponding dS4 UIR’s. In this sense, the aforementioned list of
requirements (the homogeneity and transversality) for a given (spinor-)tensor
field $\Psi^{(r)}_{A_{1}\;...\;A_{n}}(x)$ must be supplemented as follows:

$\displaystyle x\cdot\partial\Psi^{(r)}$ $\displaystyle=$ $\displaystyle
0\,,\;\;\;\;\;\;\;\mbox{homogeneity ,}$ $\displaystyle x\cdot\Psi^{(r)}$
$\displaystyle=$ $\displaystyle 0\,,\;\;\;\;\;\;\;\mbox{transversality ,}$
$\displaystyle\partial\cdot\Psi^{(r)}$ $\displaystyle=$ $\displaystyle
0\,,\;\;\;\;\;\;\;\mbox{divergencelessness ,}$
$\displaystyle\gamma\cdot\Psi^{(r)}$ $\displaystyle=$ $\displaystyle
0\,,\;\;\;\;\;\;\;\mbox{tracelessness (associated with the spinorial part)
conditions}\,.$ (XV.14)

Note that: (i) Here, for the sake of simplicity, we set the degree of
homogeneity $\ell=0$. Accordingly, for instance, regarding formulas that will
be given in the next subsection, one can easily show that the d’Alembertian
operator $\square_{R}\equiv\nabla_{\mu}\nabla^{\mu}$ on dS4 spacetime
($\nabla_{\mu}$, with $\mu=0,1,2,3$, being the covariant derivative given in
local (intrinsic) coordinates) coincides with its counterpart
$\square_{5}\equiv\partial^{2}$ on $\mathbb{R}^{5}$. (ii) The transversality
and divergencelessness conditions together yield $\Psi^{(r)\prime}=0$.

Considering the above conditions along with the relations leading to Eq.
(XV.13), the dS4 wave equations for spinor-tensor fields
$\Psi^{(r=n+1/2)}_{A_{1}\;...\;A_{n}}(x)$ (see Eq. (XV.1)), in ambient
notations, take the form:

$\displaystyle\Big{(}Q_{0}^{(1)}+\frac{\mathrm{i}}{2}\gamma_{A}\gamma_{B}M^{AB}-n(n+2)-\frac{5}{2}+[p(p+1)+(q+1)(q-2)]\Big{)}\Psi^{(r)}_{A_{1}\;...\;A_{n}}(x)=0\,,$
(XV.15)

for tensor fields $\Psi^{(r=n)}_{A_{1}\;...\;A_{n}}(x)$:

$\displaystyle\Big{(}Q_{0}^{(1)}-n(n+1)+[p(p+1)+(q+1)(q-2)]\Big{)}\Psi^{(n)}_{A_{1}\;...\;A_{n}}(x)=0\,,$
(XV.16)

and particulary, for scalar fields $\Psi^{(r=0)}(x)\equiv\phi(x)$:

$\displaystyle\Big{(}Q_{0}^{(1)}+\tau(\tau+3)\Big{)}\phi(x)=0\,,$ (XV.17)

where the allowed ranges of $\tau$, corresponding to the three series of the
dS4 scalar UIR’s, have been already listed below Eq. (XV.2).

####  XV.2 Link to intrinsic coordinates

Since, often in the literature concerning dS4 (generally, dS) QFT, the fields
are presented in terms of local (intrinsic) coordinates, it would be
convenient here to demonstrate the link between the intrinsic and ambient
coordinates.

The intrinsic field $\Psi^{(r)}_{{\mu_{1}}\;...\;{\mu_{n}}}(X)$ is locally
characterized by $\Psi^{(r)}_{A_{1}\;...\;A_{n}}(x)$ through the following
identity:

$\displaystyle\Psi^{(r)}_{{\mu_{1}}\;...\;{\mu_{n}}}(X)=x^{A_{1}}_{\,\,,\,\mu_{1}}\;...\;x^{A_{n}}_{\,\,,\,\mu_{n}}\;\Psi^{(r)}_{A_{1}\;...\;A_{n}}\big{(}x(X)\big{)}\,,$
(XV.18)

where $x^{A_{i}}_{\,\,,\,\mu_{i}}=\partial x^{A_{i}}/\partial X^{\mu_{i}}$ and
$X^{\mu}$’s, with $\mu=0,1,2,3$, refer to the four local spacetime coordinates
for the dS4 hyperboloid $\underline{M}_{R}$. The corresponding dS4 metric
$g_{\mu\nu}$ is determined by inducing the natural metric of $\mathbb{R}^{5}$
on $\underline{M}_{R}$:

$\displaystyle\mathrm{d}s^{2}=\eta_{AB}\mathrm{d}x^{A}\mathrm{d}x^{B}\Big{|}_{(x)^{2}=-R^{2}}=g_{\mu\nu}\mathrm{d}X^{\mu}\mathrm{d}X^{\nu}\,.$
(XV.19)

Considering Eq. (XV.18), it follows that $\theta_{AB}$ is the only symmetric
and transverse tensor which is connected to the dS4 metric;
$g_{\mu\nu}=x^{A}_{\,\,,\,\mu}x^{B}_{\,\,,\,\nu}\theta_{AB}$. In this context,
the covariant derivatives are transformed as:

$\displaystyle\nabla_{\mu}\nabla_{\nu}\cdot\cdot\cdot\nabla_{\rho}\;\Psi^{(r)}_{\lambda_{1}\;...\;\lambda_{n}}=x^{A}_{\,\,,\,\mu}x^{B}_{\,\,,\,\nu}\;...\;x^{C}_{\,\,,\,\rho}\;\;x^{D_{1}}_{\,\,,\,\lambda_{1}}\;...\;x^{D_{n}}_{\,\,,\,\lambda_{n}}\;\mathfrak{T}\bar{\partial}_{A}\mathfrak{T}\bar{\partial}_{B}\;...\;\mathfrak{T}\bar{\partial}_{C}\;\Psi^{(r)}_{D_{1}\;...\;D_{n}}\,.$
(XV.20)

For scalar fields, which are of particular interest in this paper, the
d’Alembertian operator reads:

$\displaystyle\square_{R}\phi=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi$
$\displaystyle=$ $\displaystyle
g^{\mu\nu}x^{A}_{\,\,,\,\mu}x^{B}_{\,\,,\,\nu}\Big{(}\bar{\partial}_{A}\bar{\partial}_{B}-R^{-2}x_{B}\bar{\partial}_{A}\Big{)}\phi$
(XV.21) $\displaystyle=$
$\displaystyle\theta^{AB}\Big{(}\bar{\partial}_{A}\bar{\partial}_{B}-R^{-2}x_{B}\bar{\partial}_{A}\Big{)}\phi=\bar{\partial}^{2}\;\phi\,.$

Considering this equation along with the first identity given in (XV.1), we
get $Q^{(1)}_{0}=-R^{2}\square_{R}$. Then, the scalar-wave equations (XV.17),
by adjusting $-\tau(\tau+3)=R^{2}m^{2}_{R}$ ($m_{R}$ being a “mass”656565The
name “mass” is purely formal here, being in general not related to at rest
energy in the Poincaré symmetry sense. term ($\hbar=c=1$)), yield the ordinary
dS4 Klein-Gordon-like field equations
$\big{(}\Box_{R}+m^{2}_{R}\big{)}\phi(x)=0$.

###  XVI Plane-wave type solutions

From now on, as already mentioned, to avoid technical (but not conceptual)
complications, we merely stick to dS4 scalar fields. According to Refs.
GazeauPRL ; Bros 2point func , there is a continuous family of simple
solutions, called dS4 plane waves, to the scalar-wave equations (XV.17)
defined by:

$\displaystyle\phi_{\tau,\xi}(x)=\Big{(}\frac{x\cdot\xi}{R}\Big{)}^{\tau},$
(XVI.1)

where $x\in\underline{M}_{R}$ and $\xi$ lies in the null-cone $C$ in
$\mathbb{R}^{5}$:

$\displaystyle C=\Big{\\{}\xi\in\mathbb{R}^{5}\;;\;(\xi)^{2}=0\Big{\\}}\,.$
(XVI.2)

The plane waves (XVI.1), as functions of $\xi$ in the null-cone $C$, are
homogeneous with degree of homogeneity $\tau$. In this sense, they can be
completely characterized by specifying their values on a well-chosen curve
(the orbital basis) $\daleth$ of $C$. [Here, it is worthwhile noting that: (i)
On the dS4 submanifold $\underline{M}_{R}$ ($\subset\mathbb{R}^{5}$) defined
by $x\cdot x=-R^{2}$, with $R$ being constant, these waves are also
homogeneous with degree of homogeneity $\tau$. (ii) As functions of
$\mathbb{R}^{5}$, they are homogeneous, but with degree of homogeneity zero,
since in this case $R$ should be viewed as a function of $x$:
$R(x)=-\sqrt{-x\cdot x}$.]

Here, in a shortcut and without getting involved in mathematical detail, we
would like to highlight two critical features of the dS4 plane waves (XVI.1).
First, these waves, as _functions_ on the dS4 manifold $\underline{M}_{R}$,
are only locally defined on connected open subsets of $\underline{M}_{R}$,
because they are singular on some specific lower dimensional subsets of the
manifold, for instance, on the spatial boundaries given by $x^{0}=\pm
x^{4}\Leftrightarrow(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}=R^{2}$; to see the
latter point, it is sufficient to associate with these boundaries,
respectively, $\xi=(\xi^{0}=\pm\xi^{4},0,0,0,\xi^{4})\in C$, for which we get
$x\cdot\xi=0$ (recall that $\tau$ is generally a complex number, with
$\mbox{Re}(\tau)<0$). Moreover, these waves, as _functions_ on
$\underline{M}_{R}$, are multi-valued, since $x\cdot\xi$ can take negative
values, as well. In order to get a single-valued, global definition of these
waves, they need to be considered as _distributions_ , strictly speaking, as
the boundary values of analytic continuations of the solutions (XVI.1) to
suitable domains in the complexified dS4 manifold
$\underline{M}_{R}^{(\mathbb{C})}$. It turns out that the minimal domains of
analyticity, which lead to such a single-valued, global definition of the dS4
plane waves, are the _forward_ and _backward tubes_ of
$\underline{M}_{R}^{(\mathbb{C})}$.

Second, the plane waves (XVI.1) are not square integrable with respect to the
Klein-Gordon inner product. However, they can be considered as generating
functions for physically meaningful dS4 entities, like square-integrable
eigenfunctions of the dS4 quadratic Casimir operator. [Such square-integrable
eigenfunctions, being constructed through continuous superpositions of these
waves (by varying $\xi$ in $C$), generate (projective) Hilbert spaces carrying
the dS4 UIR’s. They actually generate the spacetime realization of those
Hilbert spaces that have been already presented in the previous part in the
context of the $\mathbb{S}^{3}$ realization of the representations.] In this
sense, in dS4 (generally, dS) relativity, the above plane waves behave quite
analogue to the usual ones in Minkowskian or Galilean quantum mechanics,
where, by superposition of nonsquare-integrable plane waves, one can build up
physical wave functions (wave packets).

In the following subsections, strongly inspired by Refs. GazeauPRL ; Bros
2point func ; sigma , we will elaborate these points in details, respectively.
Then, following Ref. Garidi zero curvature limit , we will end our discussions
in this section by investigating the behavior of the dS4 plane waves in the
flat (Minkowskian) limit. We will show that, as far as the analyticity domain
has been chosen properly, no negative frequency plane wave appears in this
limiting process, whatever the point around which the flat limit is
calculated.

####  XVI.1 A global definition: dS4 plane waves in their tube domains

Before getting involved with the definition of dS4 plane waves in their tube
domains, we need to elaborate some geometrical notions relevant to the dS4
complex hyperboloid:

$\displaystyle\underline{M}_{R}^{(\mathbb{C})}\equiv\Big{\\{}z=x+\mathrm{i}y\in{\mathbb{C}}^{5}\;;\;(z)^{2}=\eta_{AB}z^{A}z^{B}=(z^{0})^{2}-(z^{1})^{2}-(z^{2})^{2}-(z^{3})^{2}-(z^{4})^{2}=-R^{2}\Big{\\}}\,,$
(XVI.3)

where ${\mathbb{C}}^{5}$ stands for the ambient complex Minkowski spacetime.
We begin with the dS4 complex hyperboloid $\underline{M}_{R}^{(\mathbb{C})}$
itself, which equivalently can also be realized by the following set:

$\displaystyle\underline{M}_{R}^{(\mathbb{C})}\equiv\Big{\\{}(x,y)\in{\mathbb{R}}^{5}\times{\mathbb{R}}^{5}\;;\;(x)^{2}-(y)^{2}=-R^{2},\;x\cdot
y=0\Big{\\}}\,.$ (XVI.4)

In order to provide a clear visualization of
$\underline{M}_{R}^{(\mathbb{C})}$, we first point out that the identity
$(x)^{2}-(y)^{2}=-R^{2}$ designates the following distinguished sets of points
$(x,y)$ in ${\mathbb{R}}^{5}\times{\mathbb{R}}^{5}$:

  * •

$(x)^{2}<0$, or equivalently $(y)^{2}<R^{2}$,

  * •

$x=0$ (that is, $x=(0,0,0,0,0)$), or equivalently $(y)^{2}=R^{2}$, and finally

  * •

$(x)^{2}\geqslant 0$ ($x\neq 0$), or equivalently $(y)^{2}\geqslant R^{2}$.

The latter case, however, has no intersection with $x\cdot y=0$, and
therefore, must be dropped. To make the latter point apparent, utilizing the
quaternionic notations introduced in part II, we denote the points $x$ and $y$
belonging to the latter case by $(x^{0},\textbf{x})$ and $(y^{0},\textbf{y})$,
respectively. On this basis, we have:666666Note that the dot product of two
quaternions is exactly their usual vector dot product.

$\displaystyle\mbox{for}\;\;x^{0}y^{0}>0\;;\;\;x\cdot
y=|x^{0}||y^{0}|-\textbf{x}\cdot\textbf{y}\,,\;\;\;\;\;\;\;\mbox{and}\;\;\;\;\;\;\;\mbox{for}\;\;x^{0}y^{0}<0\;;\;\;x\cdot
y=-|x^{0}||y^{0}|-\textbf{x}\cdot\textbf{y}\,.$ (XVI.5)

Then, according to the Cauchy-Schwarz inequality (which implies that
$-|\textbf{x}||\textbf{y}|\leqslant\textbf{x}\cdot\textbf{y}\leqslant|\textbf{x}||\textbf{y}|$)
and the facts that $(x)^{2}\geqslant 0$ (i.e., $|x^{0}|\geqslant|\textbf{x}|$)
and $(y)^{2}\geqslant R^{2}>0$ (i.e., $|y^{0}|>|\textbf{y}|$), we have:

$\displaystyle\mbox{for}\;\;x^{0}y^{0}>0\;;\;\;x\cdot
y\geqslant|x^{0}||y^{0}|-|\textbf{x}||\textbf{y}|>0\,,\;\;\;\;\;\;\;\mbox{and}\;\;\;\;\;\;\;\mbox{for}\;\;x^{0}y^{0}<0\;;\;\;x\cdot
y\leqslant-|x^{0}||y^{0}|+|\textbf{x}||\textbf{y}|<0\,,$ (XVI.6)

which both inequalities explicitly reveal that $x\cdot y\neq 0$. Accordingly,
dropping the third case (with $(x)^{2}\geqslant 0$, $x\neq 0$),
$\underline{M}_{R}^{(\mathbb{C})}$ can be visualized as the set of points
$(x,y)\in{\mathbb{R}}^{5}\times{\mathbb{R}}^{5}$ with $(x)^{2}<0$ or $x=0$
(equivalently $(y)^{2}\leqslant R^{2}$).

We also need to remind the notions of forward and backward tubes in
${\mathbb{C}}^{5}$, respectively, denoted by $T^{+}$ and $T^{-}$. By
definition, $T^{\pm}=\mathbb{R}^{5}+\mathrm{i}\mathring{\underline{V}}^{\pm}$,
where the domains
$\mathring{\underline{V}}^{\pm}\equiv\big{\\{}y\in\mathbb{R}^{5}\;;\;(y)^{2}>0,\;y^{0}\gtrless
0\big{\\}}$ stem from the causal structure on $\underline{M}_{R}$ (see section
(VII)). Indeed, $T^{+}$ and $T^{-}$ can be visualized as the set of points
$z=x+\mathrm{i}y\in\mathbb{C}^{5}$
($(x,y)\in{\mathbb{R}}^{5}\times{\mathbb{R}}^{5}$), such that $(y)^{2}>0$,
with $y^{0}>0$ and $y^{0}<0$, respectively. [Here, in a shortcut, it is
convenient to point out that the tubes $T^{\pm}$ are the (standard)
analyticity domains of $1+4$-dimensional Minkowskian quantum fields, which
verify the positivity of the spectrum of the energy operator (see section
XVII).]

We finally define the following open subsets of
$\underline{M}_{R}^{(\mathbb{C})}$:

$\displaystyle{\cal{T}}^{+}=T^{+}\cap\underline{M}_{R}^{(\mathbb{C})}\,,\;\;\;\;\;\;\;{\cal{T}}^{-}=T^{-}\cap\underline{M}_{R}^{(\mathbb{C})}\,,$
(XVI.7)

where ${\cal{T}}^{+}$ and ${\cal{T}}^{-}$ are respectively called forward and
backward tubes of $\underline{M}_{R}^{(\mathbb{C})}$. Considering the above,
they can be viewed as the set of points $z=x+\mathrm{i}y\in{\mathbb{C}}^{5}$,
with $-R^{2}<(x)^{2}<0$ or $x=0$ (equivalently $0<(y)^{2}\leqslant R^{2}$),
while $y^{0}>0$ and $y^{0}<0$, respectively. Here, one must notice that the
tubes ${\cal{T}}^{\pm}$ are indeed domains and _tuboids_ 676767A tuboid is a
domain bordered by a set with real coordinates. above $\underline{M}_{R}$ in
$\underline{M}_{R}^{(\mathbb{C})}$, from which, in a well-defined way, one can
take _the boundary value on $\underline{M}_{R}$ of analytic functions in the
distribution sense_686868Let $\phi(z=x+\mathrm{i}y)$ be an analytic function
in a given local tube $\Theta=\Delta+\mathrm{i}\Xi$. Then, for all $f(x)$ in
$\mathfrak{D}(\Delta)$ (the latter being the space of infinitely
differentiable functions with _compact support_ on the $\mu$-measure space
$\Delta$), the sequence of distributions
$\big{\\{}\phi_{y}\;;\;y\in\Xi\big{\\}}$ defined in
$\mathfrak{D}^{\prime}(\Delta)$ by:
$\displaystyle\phi_{y}(f)=\int_{\Delta}\phi(x+\mathrm{i}y)f(x)\;\mathrm{d}\mu(x)\,,$
is weakly convergent when $y$ goes to zero in $\Xi$; this limit
($\phi_{y}|_{y\in\Xi,\;y\rightarrow 0}$) gives a distribution in
$\mathfrak{D}^{\prime}(\Delta)$, called boundary value of $\phi(z)$ on
$\Delta$ from the local tube $\Theta$. To see more detailed discussions on
this topic, one can refer to Refs. Bros 2point func ; Neeb . Moreover, to get
acquainted with the notion of _distribution_ and its relevant objects, readers
are referred to Ref. Streater . [As a final remark in this footnote, we would
like to point out that, by definition, the support of a function $f$, denoted
in the sequel by $\mbox{supp}(f)$, is the subset of the domain containing
those elements which are not mapped to zero. If the domain of $f$ is a
topological space, the support of $f$ is defined as the smallest closed set
containing all points that are not mapped to zero. Functions with compact
support on a topological space are those whose closed support is a compact
subset of the space.] Bros 2point func .

Now, we are in a position to encounter our main task in this subsection, that
is, presenting a single-valued, global definition of the dS4 plane waves. By
analytic continuation of the plane waves (XVI.1) to the dS4 complex
hyperboloid $\underline{M}_{R}^{(\mathbb{C})}$, the obtained complexified
waves $(z\cdot\xi/R)^{\tau}$ are defined globally and single valued, provided
that $z$ varies in ${\cal{T}}^{+}$ or ${\cal{T}}^{-}$ and $\xi$ lies in the
future null-cone
$C^{+}\equiv\big{\\{}\xi\in\mathbb{R}^{5}\;;\;(\xi)^{2}=0,\;\xi^{0}>0\big{\\}}$,
because then the imaginary part of $(z\cdot\xi/R)$ has a fixed sign and
moreover $z\cdot\xi\neq 0$. To see the point, let
$z=x+\mathrm{i}y\in{\cal{T}}^{+}$ or ${\cal{T}}^{-}$ (that is,
$y=(y^{0},\textbf{y})=\mbox{Im}(z)$ verify $0<(y)^{2}\leqslant R^{2}$, with
$y^{0}>0$ and $y^{0}<0$, respectively) and, again using the quaternionic
notations, $\xi=(\xi^{0},\bm{\xi})\in C^{+}$ (that is, $\xi^{0}>0$ and
$|\bm{\xi}|=\xi^{0}$). Taking parallel steps to the ones that led to Eqs.
(XVI.5) and (XVI.6), one can show that:

$\displaystyle\left\\{\begin{array}[]{rl}y\cdot\xi>0\;\;\;\;\;\;\;\mbox{for}\;\;\;z\in{\cal{T}}^{+}\,,\vspace{1mm}\\\
\vspace{1mm}y\cdot\xi<0\;\;\;\;\;\;\;\mbox{for}\;\;\;z\in{\cal{T}}^{-}\,.\end{array}\right.$
(XVI.10)

Therefore, for all $z=x+\mathrm{i}y$ in the domains ${\cal{T}}^{\pm}$ and
$\xi\in C^{+}$, we have $z\cdot\xi\neq 0$; no matter $x\cdot\xi\neq 0$ or not,
the term $y\cdot\xi$ is always nonzero. On the other hand, for an arbitrary
value of $\tau$ and a fixed sign of the imaginary part of $(z\cdot\xi/R)$, a
single-valued determination of $(z\cdot\xi/R)^{\tau}$ is given by:

$\displaystyle\Big{(}\frac{z\cdot\xi}{R}\Big{)}^{\tau}=\exp\Big(\tau\Big{[}\mathrm{i}\;\mbox{arg}\Big{(}\frac{z\cdot\xi}{R}\Big{)}+\log|\frac{z\cdot\xi}{R}\Big{|}\Big{]}\Big{missing})\,,\;\;\;\;\;\;\;\mbox{arg}\Big{(}\frac{z\cdot\xi}{R}\Big{)}\in\;]-\pi,\pi[\,.$
(XVI.11)

Accordingly, one can define the single-valued, global dS4 plane waves as the
boundary values, in the sense of distributions, of analytic continuation to
the forward (${\cal{T}}^{+}$) or backward (${\cal{T}}^{-}$) tube of the waves
(XVI.1):

$\displaystyle\phi^{\pm}_{\tau,\xi}(f)$ $\displaystyle=$
$\displaystyle\int_{\underline{M}_{R}}c\;\Big{(}\frac{(x+\mathrm{i}y)\cdot\xi}{R}\Big{)}^{\tau}\Big{|}_{\xi\in
C^{+},\;y\in\mathring{\underline{V}}^{\pm},\;y\rightarrow
0}\;f(x)\;\mathrm{d}\mu(x)$ (XVI.12) $\displaystyle=$
$\displaystyle\int_{\underline{M}_{R}}\underbrace{\Bigg{(}c\;\Big{[}\vartheta\Big{(}\frac{x\cdot\xi}{R}\Big{)}+\vartheta\Big{(}-\frac{x\cdot\xi}{R}\Big{)}e^{\pm\mathrm{i}\pi\tau}\Big{]}\;\Big{|}\frac{x\cdot\xi}{R}\Big{|}^{\tau}\Bigg{)}}_{\equiv\phi^{\pm}_{\tau,\xi}(x)}\;f(x)\;\mathrm{d}\mu(x)\,,$

where, in the above “Fourier transform”,
$f(x)\in\mathfrak{D}(\underline{M}_{R})$ ($\mathfrak{D}(\underline{M}_{R})$
being the space of infinitely differentiable functions with compact support on
$\underline{M}_{R}$), while $\mathrm{d}\mu(x)$ stands for the invariant
measure on $\underline{M}_{R}$, $\vartheta$ for the Heaviside function, and
$c$ for a real-valued constant (this constant, as we will discuss in section
XVII, is fixed by applying the local Hadamard condition on the corresponding
two-point function).

####  XVI.2 Precision on orbital basis of the cone

We now elaborate on the notion of orbital basis $\daleth$ of the future null-
cone $C^{+}$. Let $x_{e}$ and $\underline{{\cal{S}}}(x_{e})$ respectively
denote a unit vector (in the sense $|(x_{e})^{2}|=1$) in $\mathbb{R}^{5}$ and
its stabilizer subgroup in Sp$(2,2)$. In the present context, two types of
orbits are of interest:

  * •

If $x_{e}\in\mathring{\underline{V}}^{+}$, then $\daleth$ would be the section
of $C^{+}$ characterized by a hyperplane of the form $x_{e}\cdot\xi=a$
($a>0$), i.e., an orbit (of spherical type) of the stabilizer subgroup
$\underline{{\cal{S}}}(x_{e})\sim SO(4)$.696969We recall that under the action
of Sp$(2,2)$ in $\mathbb{R}^{5}$, the region $\mathring{\underline{V}}^{+}$ is
divided into a union of mutually disjoint orbits of different radii, for
which, according to the Cartan decomposition of Sp$(2,2)$, the subgroup
$SO(4)\sim\mathrm{SU}(2)\times\mathrm{SU}(2)$, isomorphic to the maximal
compact subgroup of Sp$(2,2)$, plays the role of stabilizer subgroup (see
subsection IX.2). To see the point, let us set $x_{e}=(1,0,0,0,0)$. With this
choice of the unit vector $x_{e}\in\mathring{\underline{V}}^{+}$, the orbit
$\daleth\equiv\daleth_{0}$ reads:

$\displaystyle\daleth_{0}=\Big{\\{}\xi\in
C^{+}\;;\;\xi^{0}=a\Big{\\}}=\Big{\\{}\xi\in
C^{+}\;;\;(\xi^{1})^{2}+(\xi^{2})^{2}+(\xi^{3})^{2}+(\xi^{4})^{2}=a^{2}\Big{\\}}\,.$
(XVI.13)

  * •

If $(x_{e})^{2}=-1$, $\daleth$ represents the union of the sections of $C^{+}$
characterized by two hyperplanes of the forms $x_{e}\cdot\xi=\pm b$ ($b>0$),
i.e., the union of two hyperboloid sheets, which are orbits of the stabilizer
subgroup $\underline{{\cal{S}}}(x_{e})\sim\mathrm{SO}_{0}(1,3)$.707070We
recall that under the action of Sp$(2,2)$ in $\mathbb{R}^{5}$, the exterior of
the cone $(x)^{2}=0$, which here $x_{e}$ belongs to, is divided into a union
of mutually disjoint orbits of different radii, for which, according to the
space-time-Lorentz decomposition of Sp$(2,2)$, the subgroup
$\mathrm{SO}_{0}(1,3)$, isomorphic to the Lorentz subgroup, plays the role of
stabilizer subgroup (see subsection IX.1). To see the point, we choose
$x_{e}=(0,0,0,0,1)$. With this choice, the orbit $\daleth\equiv\daleth_{4}$
takes the form:

$\displaystyle\daleth_{4}=\daleth^{+}_{4}\cup\daleth^{-}_{4}$ $\displaystyle=$
$\displaystyle\Big{\\{}\xi\in
C^{+}\;;\;\xi^{4}=+b\Big{\\}}\bigcup\Big{\\{}\xi\in
C^{+}\;;\;\xi^{4}=-b\Big{\\}}$ (XVI.14) $\displaystyle=$
$\displaystyle\Big{\\{}\xi\in
C^{+}\;;\;(\xi^{0})^{2}-(\xi^{1})^{2}-(\xi^{2})^{2}-(\xi^{3})^{2}=b^{2}\Big{\\}}\,.$

Here, in a shortcut, we assert that the latter parametrization is the most
proper choice when one deals with the zero-curvature limit of dS4 fields. In
this context, the null-vector $\xi\in C^{+}$ is considered in terms of the
four-momentum $(k^{0},\vec{k})$ of a Minkowskian particle with mass $m$ as
follows:

$\displaystyle\xi^{\pm}\;\big{(}\in\daleth^{\pm}_{4}\big{)}=\Bigg{(}\frac{k^{0}}{m}=\sqrt{\frac{\vec{k}\cdot\vec{k}}{m^{2}}+1},\frac{\vec{k}}{m},\pm
1\Bigg{)}\,,$ (XVI.15)

based upon which, we have $(k^{0})^{2}-\vec{k}\cdot\vec{k}=m^{2}$. We will
come back to this important point later.

####  XVI.3 DS4 plane waves as generating functions for square-integrable
eigenfunctions

For the sake of reasoning, in this subsection, we invoke a system of bounded
global coordinates appropriate for describing a bounded version of dS4
spacetime, namely $\mathbb{S}^{3}\times(-\pi/2,\pi/2)$. This system of
intrinsic coordinates, known as conformal coordinates, is given by:

$\displaystyle
x=(x^{0},\textbf{x})=\big{(}R\tan\rho,R(\cos\rho)^{-1}\textbf{u}\big{)}\,,$
(XVI.16)

where $-\pi/2<\rho<\pi/2$ and $\textbf{u}\in\mathbb{S}^{3}$ (for an explicit
form of u, see Eq. (XIII.1.3)). Note that the coordinate $\rho$ is actually
timelike, and plays the role of a conformal time; the closure of the
$\rho$-interval is taken into account, when one studies compactified spacetime
under conformal action.

#####  XVI.3.1 DS4 plane waves as generating functions

Again, employing the quaternionic notations, let $\xi=(\xi^{0},\bm{\xi})\in
C^{+}$ (that is, $\xi^{0}>0$ and $\xi^{0}=|\bm{\xi}|$). According to the
identities given in appendices B and E, one can demonstrate
$\bm{\xi}\in\mathbb{H}\sim\mathbb{R}^{4}$ as $\bm{\xi}=|\bm{\xi}|\textbf{v}$,
with $\textbf{v}\in\mathrm{SU}(2)\sim\mathbb{S}^{3}$. The dot product
$x\cdot\xi/R$ then can be given, in terms of the conformal coordinates
(XVI.16), as follows:

$\displaystyle\frac{x\cdot\xi}{R}=(\tan\rho)\xi^{0}-\frac{1}{\cos\rho}\textbf{u}\cdot\bm{\xi}=\frac{\xi^{0}e^{\mathrm{i}\rho}}{2\mathrm{i}\cos\rho}\big{(}1+r^{2}-2r(\textbf{u}\cdot\textbf{v})\big{)}\,,\;\;\;\;\;\;\;r=\mathrm{i}e^{-\mathrm{i}\rho}\,.$
(XVI.17)

Considering the above identity along with the generating function for the
Gegenbauer polynomials $C_{n}^{-\tau}(x)$ (see Eq. (D.8)), it is
straightforward to show that:

$\displaystyle\Big{(}\frac{x\cdot\xi}{R}\Big{)}^{\tau}$ $\displaystyle=$
$\displaystyle\Bigg{(}\frac{\xi^{0}e^{\mathrm{i}\rho}}{2\mathrm{i}\cos\rho}\Bigg{)}^{\tau}\big{(}1+r^{2}-2r(\textbf{u}\cdot\textbf{v})\big{)}^{\tau}$
(XVI.18) $\displaystyle=$
$\displaystyle\Bigg{(}\frac{\xi^{0}e^{\mathrm{i}\rho}}{2\mathrm{i}\cos\rho}\Bigg{)}^{\tau}\sum_{n=0}^{\infty}r^{n}\;C_{n}^{-\tau}(\textbf{u}\cdot\textbf{v})\,,\;\;\;\;\;\;\;\mbox{Re}(\tau)<\frac{1}{2}\,.$

Note that the above expansion is not valid in the function sense, because the
generating function for the Gegenbauer polynomials (as is manifest above) is
merely convergent for $|r|<1$, while, in our case, we have
$|r|=|\mathrm{i}e^{-\mathrm{i}\rho}|=1$. This failure, however, is
circumvented here by giving a negative imaginary part to the angle
$\rho\rightarrow\rho-\mathrm{i}\epsilon$ ($\epsilon>0$). This process,
ensuring the convergence of the expansion, indeed amounts to extend the
ambient coordinates to the backward tube ${\cal{T}}^{-}$ (see subsection
XVI.1).

Utilizing Eqs. (D.8) and (D.9) along with (D.15), we get the following
auxiliary relation:

$\displaystyle\big{(}1+r^{2}-2r(\textbf{u}\cdot\textbf{v})\big{)}^{\tau}=2\pi^{2}\sum_{Llm}r^{L}\;{\cal
P}_{L}^{-\tau}(r^{2})\;{\cal Y}_{Llm}(\textbf{u}){\cal
Y}_{Llm}^{\ast}(\textbf{v})\,,$ (XVI.19)

where, again, ${\cal Y}_{Llm}$’s, with
$(L,l,m)\in\mathbb{N}\times\mathbb{N}\times\mathbb{Z}$, $0\leqslant l\leqslant
L$ and $-l\leqslant m\leqslant l$, are the hyperspherical harmonics on the
unit-sphere $\mathbb{S}^{3}$ (see appendix D), while the functions ${\cal
P}_{L}^{-\tau}(r^{2})$ are defined in terms of the hypergeometric functions
as:

$\displaystyle{\cal
P}_{L}^{-\tau}(r^{2})=\frac{1}{(L+1)!}\frac{\Gamma(L-\tau)}{\Gamma(-\tau)}\;_{2}F_{1}(L-\tau,-\tau-1;L+2;r^{2})\,,$
(XVI.20)

with the integral representation:

$\displaystyle r^{L}\;{\cal P}_{L}^{-\tau}(r^{2})\;{\cal
Y}_{Llm}(\textbf{u})=\frac{1}{2\pi^{2}}\int_{\mathbb{S}^{3}}\big{(}1+r^{2}-2r(\textbf{u}\cdot\textbf{v})\big{)}^{\tau}\;{\cal
Y}_{Llm}(\textbf{v})\;\mathrm{d}\mu(\textbf{v})\,.$ (XVI.21)

Now, combining the above auxiliary relation with Eq. (XVI.18), and
substituting $r=\mathrm{i}e^{-\mathrm{i}\rho}$, the expansion of the dS4 plane
waves takes the form:

$\displaystyle\Big{(}\frac{x\cdot\xi}{R}\Big{)}^{\tau}=2\pi^{2}\sum_{Llm}\Phi_{Llm}^{\tau}(x)\;(\xi^{0})^{\tau}\;{\cal
Y}_{Llm}^{\ast}(\textbf{v})\,,$ (XVI.22)

where we introduce three sets of functions
$\big{\\{}\Phi_{Llm}^{\tau}(x)\big{\\}}$, in the allowed ranges of the
unifying complex parameter $\tau$ corresponding to the three series of the dS4
scalar representations, on $\underline{M}_{R}$ as:

$\displaystyle\Phi_{Llm}^{\tau}(x)$ $\displaystyle=$
$\displaystyle\frac{\mathrm{i}^{L-\tau}\;e^{-\mathrm{i}(L-\tau)\rho}}{(2\cos\rho)^{\tau}}\;{\cal
P}_{L}^{-\tau}\big{(}-e^{-2\mathrm{i}\rho}\big{)}\;{\cal Y}_{Llm}(\textbf{u})$
(XVI.23) $\displaystyle=$
$\displaystyle\frac{\mathrm{i}^{L-\tau}\;e^{-\mathrm{i}(L-\tau)\rho}}{(2\cos\rho)^{\tau}}\;\frac{\Gamma(L-\tau)}{(L+1)!\;\Gamma(-\tau)}\;_{2}F_{1}(L-\tau\;,-\tau-1;L+2;-e^{-2\mathrm{i}\rho})\;{\cal
Y}_{Llm}(\textbf{u})\,.$

By using the Euler’s transformation (see Eq. (D.30)), one can also obtain the
following alternative form of Eq. (XVI.23):

$\displaystyle\Phi_{Llm}^{\tau}(x)=\mathrm{i}^{L-\tau}\;e^{-\mathrm{i}(L+\tau+3)\rho}\;(2\cos\rho)^{\tau+3}\;\frac{\Gamma(L-\tau)}{(L+1)!\;\Gamma(-\tau)}\;{}_{2}F_{1}\big{(}\tau+2,L+\tau+3;L+2;-e^{-2\mathrm{i}\rho}\big{)}\;{\cal
Y}_{Llm}(\textbf{u})\,.$ (XVI.24)

Here, we must underline that:

  * •

The introduced functions $\Phi_{Llm}^{\tau}(x)$ are well defined for all
values of $\tau$ with $\mbox{Re}(\tau)<0$,717171Note that this restriction on
$\tau$, namely, $\mbox{Re}(\tau)<0$, is issued from the domain of the Gamma
function $\Gamma(-\tau)$ appeared in the denominator of (XVI.23) (to see the
properties of the Gamma functions, we refer readers to Ref. Magnus ). and
therefore, are well defined for all the dS4 scalar UIR’s.

  * •

In the conformal coordinates $x=x(\rho,\textbf{u})$, these functions are
infinitely differentiable in their respective allowed ranges of parameters.

  * •

Due to the linear independence of ${\cal Y}_{Llm}$’s, the functions
$\Phi_{Llm}^{\tau}(x)$ play the role of the usual solutions to the scalar-wave
equations (XV.17) (again, in the respective allowed ranges of $\tau$), when
one proceeds with the proper separation of variables. [We will clarify this
point in subsubsection XVI.3.3.]

  * •

Considering the orthonormality of ${\cal Y}_{Llm}$’s on $\mathbb{S}^{3}$, the
integral representation of $\Phi_{Llm}^{\tau}(x)$’s, say the “Fourier
transform” on $\mathbb{S}^{3}$, is given by:

$\displaystyle\Phi_{Llm}^{\tau}(x)=\frac{1}{2\pi^{2}\big{(}\xi^{0}\big{)}^{\tau}}\int_{\mathbb{S}^{3}}\Big{(}\frac{x\cdot\xi}{R}\Big{)}^{\tau}{\cal
Y}_{Llm}(\textbf{v})\;\mathrm{d}\mu(\textbf{v})\,.$ (XVI.25)

[We will also come back to the meaning of this Fourier transform later in
subsubsection XVI.3.4.]

  * •

At the limit $\rho\rightarrow\pm\pi/2$, namely, the infinite dS4
“future”/“past”, the behavior of these functions, with respect to the form
(XVI.23), is determined by (see Eq. (D.31)):

$\displaystyle\Phi_{Llm}^{\tau}(x)\approx_{\rho\rightarrow\pm\frac{\pi}{2}}\;\;\frac{\mathrm{i}^{L-\tau}\;e^{-\mathrm{i}(L-\tau)\rho}}{(2\cos\rho)^{\tau}}\;\frac{\Gamma(L-\tau)\;\Gamma(2\tau+3)}{\Gamma(-\tau)\;\Gamma(\tau+2)\;\Gamma(L+\tau+3)}\;{\cal
Y}_{Llm}(\textbf{u})\,,$ (XVI.26)

for $-\frac{3}{2}<\mbox{Re}(\tau)<0$, and, with respect to the alternative
form (XVI.24), by (see Eq. (D.31)):

$\displaystyle\Phi_{Llm}^{\tau}(x)\approx_{\rho\rightarrow\pm\frac{\pi}{2}}\;\;\mathrm{i}^{L-\tau}\;e^{-\mathrm{i}(L+\tau+3)\rho}\;(2\cos\rho)^{\tau+3}\;\frac{\Gamma(-2\tau-3)}{\Gamma(-\tau)\;\Gamma(-\tau-1)}\;{\cal
Y}_{Llm}(\textbf{u})\,,$ (XVI.27)

for $\mbox{Re}(\tau)<-\frac{3}{2}$.727272Note that for
$\mbox{Re}(\tau)=-\frac{3}{2}$, which is the case for the scalar principal
series $\tau=-3/2-\mathrm{i}\nu$ ($\nu\in\mathbb{R}$), we need to give a small
imaginary part to $\nu$, then we can use the above asymptotic formulas (see
Ref. Bros 2002 ). Considering the asymptotic relations (XVI.26) and (XVI.27)
in their respective domains of the parameter $\tau$, it is clear that the
functions $\Phi_{Llm}^{\tau}(x)$ are singular at the limit
$\rho\rightarrow\pm\pi/2$ for all $\mbox{Re}(\tau)<-3$ corresponding to the
relation (XVI.27), where the asymptotic behavior is dominated by the factor
$(\cos\rho)^{\tau+3}$. Of course, one must notice that this singularity is
nothing but a direct result of the choice of coordinates, that we have already
made in order to express the dot product $x\cdot\xi$. Technically, this
singularity occurs for the scalar discrete series ($\tau=-p-2$) with
$p\geqslant 2$, which includes all the scalar discrete series UIR’s
$\Pi_{p,0}$, with the exception of $\Pi_{p=1,0}$ ($\tau=-3$) associated with
the so-called minimally coupled scalar field (this point will be used later in
subsection XVII.3).

#####  XVI.3.2 Normalized eigenfunctions

Here, we proceed with the examination of the introduced three sets of
functions $\big{\\{}\Phi_{Llm}^{\tau}(x)\big{\\}}$, in the respective allowed
ranges of $\tau$, as three sets of basis elements for the spacetime
realization of the (respective) carrier Hilbert spaces of the dS4 scalar
(principal, complementary, and discrete series) UIR’s. Such basis elements are
supposed to be scalar-valued functions on $\underline{M}_{R}$, infinitely
differentiable, and solutions to the scalar-wave equations
(XV.17),737373Again, for the latter point, it is convenient to recall the
arguments subsequent to Eq. (XIII.31). which, so far, all are well fulfilled
by $\Phi_{Llm}^{\tau}(x)$’s. These elements are also supposed to be square
integrable with respect to the so-called Klein-Gordon inner product. This is
actually the only criterion left here that needs to be examined concerning the
introduced sets of functions. Technically, for given solutions $\Phi_{1}$ and
$\Phi_{2}$ to the scalar-wave equations (XV.17), the Klein-Gordon inner
product is defined by:

$\displaystyle\langle\Phi_{1}(x),\Phi_{2}(x)\rangle=\mathrm{i}\int_{\Sigma}\Phi_{1}^{\ast}(x)\big{(}\overset{\rightarrow}{\partial}_{\nu}-\overset{\leftarrow}{\partial}_{\nu}\big{)}\Phi_{2}(x)\;\mathrm{d}\sigma^{\nu}\equiv\mathrm{i}\int_{\Sigma}\Phi_{1}^{\ast}(x)\overset{\leftrightarrow}{\partial}_{\nu}\Phi_{2}(x)\;\mathrm{d}\sigma^{\nu}\,,$
(XVI.28)

where $\Sigma$ and $\mathrm{d}\sigma^{\nu}$ respectively stand for a _Cauchy
surface_ 747474That is a spacelike surface in such a way that the Cauchy data
on it uniquely define a solution to the wave equations (XV.17). and the area
element vector on it. The Klein-Gordon inner product is dS4 invariant and of
course independent of the choice of $\Sigma$. Regarding the choice of global
coordinates $x=x(\rho,\textbf{u})$, that we have already made, this product
takes the form:

$\displaystyle\langle\Phi_{1}(x),\Phi_{2}(x)\rangle=\mathrm{i}R^{2}\int_{\rho=0}\Phi_{1}^{\ast}(\rho,\textbf{u})\;\overset{\leftrightarrow}{\partial}_{\rho}\;\Phi_{2}(\rho,\textbf{u})\;\mathrm{d}\mu(\textbf{u})\,,$
(XVI.29)

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

Considering the orthogonality relations of ${\cal Y}_{Llm}$’s, for the given
sets of functions $\big{\\{}\Phi_{Llm}^{\tau}(x)\big{\\}}$ (the normalizable
ones), we have :

$\displaystyle\langle\Phi_{L_{1}l_{1}m_{1}}^{\tau}(x),\Phi_{L_{2}l_{2}m_{2}}^{\tau}(x)\rangle=\delta_{L_{1}L_{2}}\delta_{l_{1}l_{2}}\delta_{m_{1}m_{2}}\|\Phi_{L_{1}l_{1}m_{1}}^{\tau}(x)\|^{2}\,,$
(XVI.30)

where, by admitting the form (XVI.23) and utilizing Eqs. (D.32), (D.33), and
(D.4), we obtain:

$\displaystyle\|\Phi_{Llm}^{\tau}(x)\|^{2}$ $\displaystyle=$ $\displaystyle\pi
R^{2}\;2^{2-2L}\;e^{-\pi\mbox{Im}(\tau)}\;\Big{|}\frac{\Gamma(L-\tau)}{\Gamma(-\tau)}\Big{|}^{2}$
(XVI.31) $\displaystyle\hskip
28.45274pt\times\Bigg{[}\mbox{Re}\Bigg{(}\Gamma^{\ast}\Big{(}\frac{L-\tau+1}{2}\Big{)}\;\Gamma\Big{(}\frac{L-\tau}{2}\Big{)}\;\Gamma^{\ast}\Big{(}\frac{L+\tau+4}{2}\Big{)}\;\Gamma\Big{(}\frac{L+\tau+3}{2}\Big{)}\Bigg{)}\Bigg{]}^{-1}\,.$

For real values of $\tau$, by employing the _Legendre duplication formula_
757575$\Gamma(z)\;\Gamma(z+1/2)=2^{1-2z}\sqrt{\pi}\;\Gamma(2z)$., Eq. (XVI.31)
simplifies to:

$\displaystyle\|\Phi_{Llm}^{\tau}(x)\|^{2}=2^{3}R^{2}\;\frac{\Gamma(L-\tau)}{\big{(}\Gamma(-\tau)\big{)}^{2}\;\Gamma(L+\tau+3)}\,.$
(XVI.32)

According to Eqs. (XVI.31) and (XVI.32), it is manifest that the sets of
functions $\big{\\{}\Phi_{Llm}^{\tau}(x)\big{\\}}$, with
$-3<\mbox{Re}(\tau)<0$, which is the case for the dS4 scalar principal and
complementary series, are normalizable. Hence, considering this fact along
with the properties listed below Eqs. (XVI.23) and (XVI.24), we argue that the
functions $\Phi_{Llm}^{\tau}(x)$, in the respective ranges of $\tau$
corresponding to the dS4 scalar principal and complementary series, are proper
candidates to generate the (respective) carrier Hilbert spaces of the
representations. For the scalar discrete series, characterized by
$\mbox{Re}(\tau)\leqslant-3$ ($\tau=-p-2$, with $p=1,2,...$), however, the
situation is more delicate. To make the point clear, we first need to consider
the alternative form (XVI.24) of the functions $\Phi_{Llm}^{\tau}(x)$, for
which, the range $\mbox{Re}(\tau)\leqslant-3$ is allowed. In this case, the
involved hypergeometric functions
${}_{2}F_{1}\big{(}-p,L-p+1;L+2;-e^{-2\mathrm{i}\rho}\big{)}$, with
$p=1,2,...$, reduce to polynomials of degree $p$ (see appendix D).
Accordingly, for the scalar discrete series ($\tau=-p-2$), the corresponding
set of functions $\big{\\{}\Phi_{Llm}^{\tau}(x)\big{\\}}$ is divided into two
parts:

  * •

The set of $\big{\\{}\Phi_{Llm}^{\tau}(x)\big{\\}}$, with $L<p$ (say
$\big{\\{}\Phi_{L_{<p}lm}^{\tau}(x)\big{\\}}$), which, with respect to the
Klein-Gordon inner product (XVI.29), is of null norm and clearly, considering
the allowed ranges of $L,l$, and $m$, is of $p(p+1)(2p+1)/6$ dimension.

  * •

The set of regular normalizable functions
$\big{\\{}\Phi_{Llm}^{\tau}(x)\big{\\}}$, with $L\geqslant p$ (say
$\big{\\{}\Phi_{L_{\geqslant p}lm}^{\tau}(x)\big{\\}}$).

Note that the invariant null-norm subspace
$\big{\\{}\Phi_{L_{<p}lm}^{\tau}(x)\big{\\}}$ can be interpreted as a space of
“gauge” states, which carries the irreducible (nonunitary) finite-dimensional
representations of the dS4 group. Regarding the notations introduced in
appendix G, these representations are characterized by ($n_{1}=0,n_{2}=p-1$).
They are indeed _Weyl equivalent_ 767676Again, if two representations are Weyl
equivalent, then they share same Casimir eigenvalue. to the discrete series
UIR’s $\Pi_{p,0}$.

Let us summarize the above results. In the allowed ranges of parameters, the
normalized eigenfunctions (denoted here by
$\widetilde{\Phi}_{Llm}^{\tau}(x)$’s), as basis elements of the carrier
Hilbert spaces of the dS4 scalar UIR’s, read Chernikov :

$\displaystyle\widetilde{\Phi}_{Llm}^{\tau}(x)=N_{L}^{\tau}\;\mathrm{i}^{L-\tau}\;e^{-\mathrm{i}(L+\tau+3)\rho}\;(\cos\rho)^{\tau+3}\;{}_{2}F_{1}\big{(}\tau+2,L+\tau+3;L+2;-e^{-2\mathrm{i}\rho}\big{)}\;{\cal
Y}_{Llm}(\textbf{u})\,,$ (XVI.33)

where, with respect to Eq. (XVI.31), the normalization factors $N_{L}^{\tau}$
are:

$\displaystyle N_{L}^{\tau}$ $\displaystyle=$
$\displaystyle\frac{1}{R\sqrt{\pi}}\;2^{L+\tau+2}\;e^{\frac{\pi}{2}\mbox{Im}(\tau)}\;\frac{\Gamma(L-\tau)}{|\Gamma(L-\tau)|}\frac{|\Gamma(-\tau)|}{\Gamma(-\tau)}\frac{1}{(L+1)!}$
(XVI.34)
$\displaystyle\times\sqrt{\mbox{Re}\Bigg{(}\Gamma^{\ast}\Big{(}\frac{L-\tau+1}{2}\Big{)}\;\Gamma\Big{(}\frac{L-\tau}{2}\Big{)}\;\Gamma^{\ast}\Big{(}\frac{L+\tau+4}{2}\Big{)}\;\Gamma\Big{(}\frac{L+\tau+3}{2}\Big{)}\Bigg{)}}\;.$

For real values of $\tau$, which is the case for the complementary and
discrete series (the normalizable ones), the normalization factors
$N_{L}^{\tau}$ simplify to:

$\displaystyle
N_{L}^{\tau}=\frac{2^{\tau+\frac{3}{2}}}{R}\;\frac{\sqrt{\Gamma(L-\tau)\;\Gamma(L+\tau+3)}}{(L+1)!}\,.$
(XVI.35)

In particular, for the scalar discrete series, determined by $\tau=-p-2$
($p=1,2,...$), the orthonormal system turns into ($L\geqslant p$):

$\displaystyle\widetilde{\Phi}_{L_{\geqslant p}lm}^{\tau}(x)=N_{L_{\geqslant
p}}^{\tau}\;\mathrm{i}^{L+p+2}\;e^{-\mathrm{i}(L-p+1)\rho}\;(\cos\rho)^{-p+1}\;{}_{2}F_{1}\big{(}-p,L-p+1;L+2;-e^{-2\mathrm{i}\rho}\big{)}\;{\cal
Y}_{Llm}(\textbf{u})\,,$ (XVI.36)

with:

$\displaystyle N_{L_{\geqslant
p}}^{\tau}=\frac{2^{-p-\frac{1}{2}}}{R}\;\frac{\sqrt{\Gamma(L+p+2)\;\Gamma(L-p+1)}}{(L+1)!}\,.$
(XVI.37)

As already mentioned, for the scalar discrete series with $p\geqslant 2$,
these functions are singular at the limits $\rho=\pm\pi/2$.

Now, for the scalar principal and complementary series, we can rewrite the
expansion formula (XVI.22), representing the dS4 plane waves as generating
functions, in terms of the orthonormal sets
$\big{\\{}\widetilde{\Phi}_{Llm}^{\tau}(x)\big{\\}}$ (in the respective ranges
of $\tau$) as:

$\displaystyle\Big{(}\frac{x\cdot\xi}{R}\Big{)}^{\tau}=2\pi^{2}\sum_{Llm}\|\Phi_{Llm}^{\tau}(x)\|\;\widetilde{\Phi}_{Llm}^{\tau}(x)\;(\xi^{0})^{\tau}\;{\cal
Y}_{Llm}^{\ast}(\textbf{v})\,,$ (XVI.38)

where $\|\Phi_{Llm}^{\tau}(x)\|$’s are determined by Eq. (XVI.31), for the
principal series, and by Eq. (XVI.32), for the complementary series. For the
scalar discrete series, the expansion needs to be split into two sectors:

$\displaystyle\Big{(}\frac{x\cdot\xi}{R}\Big{)}^{\tau=-p-2}$ $\displaystyle=$
$\displaystyle
2\pi^{2}\sum_{L=0}^{p-1}\sum_{lm}{\Phi}_{L_{<p}lm}^{\tau}(x)\;(\xi^{0})^{-p-2}\;{\cal
Y}_{Llm}^{\ast}(\textbf{v})$ (XVI.39)
$\displaystyle+2\pi^{2}\sum_{L=p}^{\infty}\sum_{lm}\|{\Phi}_{L_{\geqslant
p}lm}^{\tau}(x)\|\;\widetilde{\Phi}_{L_{\geqslant
p}lm}^{\tau}(x)\;(\xi^{0})^{-p-2}\;{\cal
Y}_{Llm}^{\ast}(\textbf{v})\,,\;\;\;\;\;\;\;p=1,2,...\;,$

where $\|{\Phi}_{L_{\geqslant p}lm}^{\tau}(x)\|$’s are determined by Eq.
(XVI.32), with $\tau=-p-2$. Accordingly, we have the “spherical” modes in dS4
spacetime in terms of the dS4 plane waves.

#####  XVI.3.3 Usual solutions by separation of variables

Let us now show that the three sets of functions
$\big{\\{}\Phi_{Llm}^{\tau}\big{\\}}$ (or
$\big{\\{}\widetilde{\Phi}_{Llm}^{\tau}\big{\\}}$), corresponding to the three
series of the dS4 scalar representations, can be actually obtained by directly
solving the scalar-wave equations (XV.17). As already pointed out in
subsection XV.2, the scalar Casimir operator is proportional to the Laplace-
Beltrami operator in dS4 spacetime (i.e., $Q^{(1)}_{0}=-R^{2}\Box_{R}$),
which, with respect to the conformal coordinates $x=x(\rho,\textbf{u})$ (see
Eq. (XVI.16)), takes the form:

$\displaystyle\Box_{R}=\frac{1}{\sqrt{g}}\partial_{\nu}\sqrt{g}g^{\nu\mu}\partial_{\mu}=\frac{1}{R^{2}}\cos^{4}\rho\frac{\partial}{\partial\rho}\Big{(}\cos^{-2}\rho\frac{\partial}{\partial\rho}\Big{)}-\frac{1}{R^{2}}\cos^{2}\rho\;\Delta_{3}\,,$
(XVI.40)

with:

$\displaystyle\Delta_{3}=\frac{\partial^{2}}{\partial\psi^{2}}+2\cot\psi\;\frac{\partial}{\partial\psi}+\frac{1}{\sin^{2}\psi}\frac{\partial^{2}}{\partial\theta^{2}}+\cot\theta\;\frac{1}{\sin^{2}\psi}\frac{\partial}{\partial\theta}+\frac{1}{\sin^{2}\psi\sin^{2}\theta}\frac{\partial^{2}}{\partial\phi^{2}}\,,$
(XVI.41)

where the latter is the Laplace operator on $\mathbb{S}^{3}$ (note that here
we have used the parametrization given in Eq. (XIII.1.3), for which
$\mathbb{S}^{3}\ni\textbf{u}=\textbf{u}(\psi,\theta,\phi)$, with
$0\leqslant\psi,\theta\leqslant\pi$ and $0\leqslant\phi<2\pi$). Solutions to
the scalar-wave equations (XV.17), in the respective ranges of $\tau$, then
can be found by separation of variables,
$\Phi_{Llm}^{\tau}(x)=A(\rho)B(\textbf{u})$, based upon which we get Chernikov
; Kirsten :

$\displaystyle\big{(}\Delta_{3}+C\big{)}B(\textbf{u})=0\,,$ (XVI.42)
$\displaystyle\Big{(}\cos^{4}\rho\frac{\mathrm{d}}{\mathrm{d}\rho}\;\cos^{-2}\rho\frac{\mathrm{d}}{\mathrm{d}\rho}+C\cos^{2}\rho-\tau(\tau+3)\Big{)}A(\rho)=0\,.$
(XVI.43)

The relevant solutions to the angular part (XVI.42), with $C=L(L+2)$ and
$L\in\mathbb{N}$, are the usual hyperspherical harmonics on $\mathbb{S}^{3}$,
namely, $B(\textbf{u})={\cal{Y}}_{Llm}(\textbf{u})$. On the other hand, for
the $\rho$-dependent part and for a given $\tau$, the solutions (corresponding
to the Euclidean vacuum) are Kirsten :

$\displaystyle A(\rho)\equiv A_{\lambda
L}(\rho)=\chi_{L}\;(\cos\rho)^{\frac{3}{2}}\Big{(}P_{L+\frac{1}{2}}^{\lambda}(\sin\rho)-\frac{2\mathrm{i}}{\pi}Q_{L+\frac{1}{2}}^{\lambda}(\sin\rho)\Big{)}\,,$
(XVI.44)

where $P_{n}^{\lambda}$ and $Q_{n}^{\lambda}$ are the Legendre functions on
the cut, with $\lambda\equiv\lambda(\tau)=\pm(\tau+\frac{3}{2})$, and
$\chi_{L}$ is:

$\displaystyle\chi_{L}=\frac{\sqrt{\pi}}{2R}\Bigg{(}\frac{\Gamma(L-\lambda+\frac{3}{2})}{\Gamma(L+\lambda+\frac{3}{2})}\Bigg{)}^{\frac{1}{2}}.$
(XVI.45)

Here, it must be underlined that the above relations provide us with three
sets of solutions $\big{\\{}\Phi_{Llm}^{\tau}(x)=A_{\lambda
L}(\rho){\cal{Y}}_{Llm}(\textbf{u})\big{\\}}$ in the allowed ranges of $\tau$
associated with the three series of the dS4 scalar representations, except for
the case of discrete series ($\tau=-p-2$), with $L<p$, for which the above
formulas break down. These families of solutions obey the following
orthogonality prescription:

$\displaystyle\langle\Phi_{L^{\prime}l^{\prime}m^{\prime}}^{\tau},\Phi_{Llm}^{\tau}\rangle=\delta_{LL^{\prime}}\delta_{ll^{\prime}}\delta_{mm^{\prime}}\,,\;\;\;\;\;\;\;\langle\Phi_{L^{\prime}l^{\prime}m^{\prime}}^{\tau},\big{(}\Phi_{Llm}^{\tau}\big{)}^{\ast}\rangle=0\,.$
(XVI.46)

Now, to make apparent the link between the $\rho$-dependent part of the
solutions $A_{\lambda L}(\rho)$ and its counterpart in Eq. (XVI.23) or
equivalently in Eq. (XVI.24), given in terms of the hypergeometric functions,
one can either directly expand the Legendre functions in their arguments or
solve the differential Eq. (XVI.43), by changing the variables $\rho\mapsto
t\;\big{(}=r^{2}\big{)}=-e^{-2\mathrm{i}\rho}$:

$\displaystyle\Big{(}t^{2}(1-t)^{2}\;\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}+2t(1-t)\;\frac{\mathrm{d}}{\mathrm{d}t}-\frac{1}{4}L(L+2)(1-t)^{2}-t\tau(\tau+3)\Big{)}A(t)=0\,,\;\;\;\;\;\;\;A_{\lambda
L}(\rho)\equiv A(t)\,.$ (XVI.47)

Below, we proceed with the second approach.

_Frobenius solutions to Eq. ( XVI.47) in the neighborhood of $t=0$._777777At
first glance, it might be argued that, with respect to the definition
$t=-e^{-2\mathrm{i}\rho}$, the neighborhood of $t=0$ is not reachable at all
(since $|t|=1$). But, the very point to be noticed here is that the whole
above construction is performed when we gave a negative imaginary part to the
angle $\rho\rightarrow\rho-\mathrm{i}\epsilon$ (see the argument subsequent to
Eq. (XVI.18)). Therefore, the definition $t=-e^{-2\mathrm{i}\rho}$ should
actually be read as $t=-e^{-2\mathrm{i}\rho}e^{-2\epsilon}$. In this sense,
the neighborhood of $t=0$ is quite reachable by varying the value of
$\epsilon\;(>0)$. For the solutions of the form $A(t)=t^{c}\sum_{n\geqslant
0}a_{n}t^{n}$, the Frobenius indicial equation admits two sets of solutions,
respectively, associated with the case $c=c_{1}=L/2$ and the case
$c=c_{2}=-(L+2)/2$. The former set of solutions, in the neighborhood of $t=0$,
reads as:

$\displaystyle
A(t)=A_{c_{1}}(t)=t^{\frac{L}{2}}(1-t)^{\tau+3}\;_{2}F_{1}\big{(}\tau+2,L+\tau+3;L+2;t\big{)}\,,$
(XVI.48)

or alternatively, by making use of the Euler’s transformation (see Eq.
(D.30)), as:

$\displaystyle
A(t)=A_{c_{1}}(t)=t^{\frac{L}{2}}(1-t)^{-\tau}\;_{2}F_{1}\big{(}L-\tau,-\tau-1;L+2;t\big{)}\,.$
(XVI.49)

By substituting $t=-e^{-2\mathrm{i}\rho}$, one can easily check that the above
alternative forms of solutions respectively give the $\rho$-dependent part of
the solutions (XVI.24) and (XVI.23), up to constant factors.

On the other hand, since $c_{1}-c_{2}=L+1\in\mathbb{N}$, for the latter set of
solutions associated with $c=c_{2}=-(L+2)/2$, we face a degenerate case, for
which linearly independent solutions read:

$\displaystyle
A(t)=A_{c_{2}}(t)=\big{(}\log(t)\big{)}A_{c_{1}}(t)+\sum_{n=-L-1}^{\infty}b_{n}t^{n+\frac{L}{2}}\,,$
(XVI.50)

where the coefficients $b_{n}$ are recurrently specified by substituting the
above solutions into Eq. (XVI.47). Note that the second set of solutions is
singular at $t=0$, due to the logarithmic term. This set of solutions is
relevant to the scalar discrete series case, for which one has to deal with
the finite dimensional space of null-norm solutions (in this regard, we would
like to draw the attention of readers to subsection XVII.3, where we discuss
this matter for the simplest case ($p=1$)).

The above sets of solutions respectively possess the Klein-Gordon norms:

$\displaystyle\|A_{c_{1}}\|^{2}=2^{3}R^{2}\;\frac{\big{(}(L+1)!\big{)}^{2}}{\Gamma(L-\tau)\;\Gamma(L+\tau+3)}\,,$
(XVI.51)

which correspond to Eq. (XVI.31), and (for real values of $b_{n}$’s):

$\displaystyle\|A_{c_{2}}\|^{2}=\pi^{2}\|A_{c_{1}}\|^{2}+uR^{2}\Bigg{(}\frac{\sqrt{\pi}\;2^{2-L}(L+1)!}{\Gamma(\frac{L-\tau+1}{2})\;\Gamma(\frac{L+\tau+4}{2})}+4(-1)^{L}v\Bigg{)}\,,$
(XVI.52)

where:

$\displaystyle
u=\sum_{n=-L-1}^{\infty}(-1)^{n}b_{n}\,,\;\;\;\;\;\;\;v=\sum_{n=-L-1}^{\infty}(-1)^{n}(n+L/2)b_{n}\,.$
(XVI.53)

In the discrete series case ($\tau=-p-2$), with $L<p$, we conjecture that the
Klein-Gordon norm (XVI.52) vanishes, as the norm (XVI.51) does.

_Frobenius solutions to Eq. ( XVI.47) in the neighborhood of $t=1$, associated
with the discrete series_ ($\tau=-p-2$). For the solutions of the form
$A(t)=(1-t)^{d}\sum_{n\geqslant 0}c_{n}(1-t)^{n}$, the Frobenius indicial
equation has two sets of solutions, respectively, associated with
$d=d_{1}=-\tau=p+2$ and $d=d_{2}=\tau+3=1-p$. The second case represents the
$\rho$-dependent part ($t=-e^{-2\mathrm{i}\rho}$) of the (respective)
solutions (XVI.23) or (XVI.24). Solutions associated with the first case, in
the neighborhood of $t=1$, are given by:

$\displaystyle
A(t)=A_{d_{1}}(t)=t^{\frac{L}{2}}(1-t)^{p+2}\;_{2}F_{1}\big{(}L+p+2,p+1;L+2;1-t\big{)}\,,$
(XVI.54)

or equivalently, with respect to the Euler’s transformation (see Eq. (D.30)),
by:

$\displaystyle
A(t)=A_{d_{1}}(t)=t^{\frac{L}{2}-2p-1}(1-t)^{p+2}\;_{2}F_{1}\big{(}-p,L-p+1;L+2;1-t\big{)}\,.$
(XVI.55)

Again, because $d_{1}-d_{2}=2p+1\in\mathbb{N}$, one has to deal with a
degenerate case in the context of the second set of solutions. Linearly
independent solutions then are:

$\displaystyle
A(t)=A_{d_{2}}(t)=\big{(}\log(1-t)\big{)}A_{d_{1}}(t)+\sum_{n=-2p-1}^{\infty}e_{n}(1-t)^{n+p+2}\,,$
(XVI.56)

where the coefficients $e_{n}$ are recurrently specified by substituting the
above solutions into Eq. (XVI.47), while the latter admits the change of
variables $t\mapsto 1-t$. Here, again, the second set of solutions is
singular, due to singularity of the logarithmic term at $t=1$.

#####  XVI.3.4 DS4 UIR’s: spacetime realization versus $\mathbb{S}^{3}$
realization

This subsubsection is devoted to a more detailed discussion concerning the
Fourier transform (XVI.25). This transform actually intertwines two different
realizations of the dS4 scalar principal series UIR’s, namely the spacetime
and $\mathbb{S}^{3}$ realizations, while the dS4 plane waves
$(x\cdot\xi/R)^{\tau}$ serve as the (Fourier) kernel for passing from one
realization to the other. To see the point, while we have in mind the
arguments given in subsubsection XVI.3.2, it is sufficient to recall from
subsection XIII.1 that the space spanned by ${\cal Y}_{Llm}$’s on
$\mathbb{S}^{3}$ carries the ten essentially self-adjoint infinitesimal
operators associated with the scalar principal series UIR’s of the dS4 group
Sp$(2,2)$ (the $\mathbb{S}^{3}$ realization).

Here, for all three series of the dS4 scalar UIR’s, we introduce the following
kernel:

$\displaystyle\underline{M}_{R}\times\mathbb{S}^{3}\ni(x,\textbf{v})\;\mapsto\;K(x,\textbf{v})\equiv\frac{R^{-1}}{2\pi^{2}}\;2^{-\frac{3}{2}}\;e^{\frac{\pi}{2}\mbox{Im}(\tau)}\;|\Gamma(-\tau)|\;(\xi^{0})^{-\tau}\;\Big{(}\frac{x\cdot\xi}{R}\Big{)}^{\tau},$
(XVI.57)

where we remind that $\xi=(\xi^{0},\xi^{0}\textbf{v})\in C^{+}$, with
$\xi^{0}>0$ and $\textbf{v}\in\mathbb{S}^{3}$. This kernel for the scalar
principal series representations (with $\tau=-3/2-\mathrm{i}\nu$ and
$\nu\in\mathbb{R}$) expands as:

$\displaystyle
K^{\mbox{\small{ps}}}(x,\textbf{v})=\sum_{Llm}\widetilde{\Phi}_{Llm}^{\tau}(x)\;\Big{(}{\cal{Y}}_{Llm}^{\tau}(\textbf{v})\Big{)}^{\ast}.$
(XVI.58)

To get the above identity, we have used Eqs. (XVI.38) and (XVI.31), and then,
after substituting $\tau=-3/2-\mathrm{i}\nu$ and considering the fact that
$\Gamma^{\ast}(z)=\Gamma(z^{\ast})$, we have applied the Legendre duplication
formula. On the other hand, for the scalar complementary series
representations (with $\tau=-3/2-\nu$ and $\nu\in\mathbb{R}$, while
$0<|\nu|<3/2$), taking Eq. (XVI.32) into account, the expansion of this kernel
turns into:

$\displaystyle
K^{\mbox{\small{cs}}}(x,\textbf{v})=\sum_{Llm}\widetilde{\Phi}_{Llm}^{\tau}(x)\;\Big{(}\widetilde{{\cal{Y}}}_{Llm}^{\tau}(\textbf{v})\Big{)}^{\ast},$
(XVI.59)

where
$\widetilde{{\cal{Y}}}_{Llm}^{\tau}(\textbf{v})=\sqrt{\frac{\Gamma(L-\tau)}{\Gamma(L+\tau+3)}}\;{\cal{Y}}_{Llm}(\textbf{v})$.
For the scalar discrete series representations (with $\tau=-p-2$ and
$p=1,2,...$), one needs to consider Eqs. (XVI.39) and (XVI.32), based upon
which:

$\displaystyle
K^{\mbox{\small{ds}}}(x,\textbf{v})=R^{-1}2^{-\frac{3}{2}}(p+1)!\sum_{L=0}^{p-1}\sum_{lm}{\Phi}_{L_{<p}lm}^{\tau}(x)\;\Big{(}{\cal{Y}}_{Llm}^{\tau}(\textbf{v})\Big{)}^{\ast}+\sum_{L=p}^{\infty}\sum_{lm}\widetilde{\Phi}_{L_{\geqslant
p}lm}^{\tau}(x)\;\Big{(}\widetilde{{\cal{Y}}}_{Llm}^{\tau}(\textbf{v})\Big{)}^{\ast},$
(XVI.60)

where
$\widetilde{{\cal{Y}}}_{Llm}^{\tau=-p-2}(\textbf{v})=\sqrt{\frac{(L+p+1)!}{(L-p)!}}\;{\cal{Y}}_{Llm}(\textbf{v})$.
We recall from subsections XIII.4 and XIII.5 that
$\widetilde{{\cal{Y}}}_{Llm}^{\tau}(\textbf{v})$’s, respectively, with
$\tau=-3/2-\nu$ and $\tau=-p-2$ ($L\geqslant p$), generate the
$\mathbb{S}^{3}$ realization of the carrier Hilbert spaces of the dS4 scalar
complementary and discrete series representations.

Accordingly, for all three series of the dS4 scalar UIR’s, we have the two
transforms which, in the respective allowed ranges of $\tau$, connect the
“wave functions” $\widetilde{\Phi}_{Llm}^{\tau}(x)$ (square integrable with
respect to the Klein-Gordon inner product (XVI.29)) with the respective
elements $f(\textbf{v})$ in the Hilbert space
$L^{2}_{\mathbb{C}}(\mathbb{S}^{3})$, for the scalar principal series, in
$L^{2}_{\mathbb{C}}(\mathbb{S}^{3}\times\mathbb{S}^{3})$, for the scalar
complementary series, and again in $L^{2}_{\mathbb{C}}(\mathbb{S}^{3})$, for
the scalar discrete series. These two transforms, which are inverse to each
other, read:

$\displaystyle\widetilde{\Phi}_{Llm}^{\tau}(x)=\langle
K^{\ast}(x,\cdot),f\rangle_{L^{2}}\,,$ (XVI.61) $\displaystyle
f(\textbf{v})=\langle
K(\cdot,\textbf{v}),\widetilde{\Phi}_{Llm}^{\tau}\rangle_{KG}\,,$ (XVI.62)

where $\langle\cdot,\cdot\rangle_{L^{2}}$ and $\langle\cdot,\cdot\rangle_{KG}$
respectively refer to the $L^{2}(\mathbb{S}^{3})$ inner product (see Eq.
(XIII.1)) and the Klein-Gordon inner product (XVI.29). One can easily see
that, for the principal case, the Fourier transform (XVI.61) precisely
recovers the transform (XVI.25).

####  XVI.4 DS4 plane waves and the zero-curvature limit

In this subsection, following the lines sketched in Ref. Garidi zero curvature
limit , we aim to study the behavior of the dS4 scalar principal waves under
vanishing curvature.787878Recall that, among all the dS4 UIR’s, those
admitting a meaningful null-curvature limit merely involve the principal
series representations; they contract explicitly to the Poincaré massive
UIR’s. In this sense, they are also called dS4 massive representations (see
section XIV). We show that, at the flat (Minkowskian) limit, as far as the
analyticity domain is chosen properly, they precisely coincide with the usual
_positive-frequency_ , Minkowskian plane waves of a particle with mass $m$.

According to the discussions given in subsection XVI.1, the dS4 single-valued,
global plane waves can be obtained as the boundary values, in the distribution
sense, of analytic continuation to the forward (${\cal{T}}^{+}$) or backward
(${\cal{T}}^{-}$) tubes of the plane-wave solutions to the (relevant) wave
equations. Here, for the sake of reasoning, we stick to the waves analytic in
the backward tube ${\cal{T}}^{-}$. Then, for the dS4 scalar principal waves,
which are of interest in the current discussion, we have:

$\displaystyle\phi_{\nu,\xi}^{-}(x)=c\;\Big{[}\vartheta\Big{(}\frac{x\cdot\xi}{R}\Big{)}+\vartheta\Big{(}-\frac{x\cdot\xi}{R}\Big{)}e^{-\mathrm{i}\pi\tau}\Big{]}\;\Big{|}\frac{x\cdot\xi}{R}\Big{|}^{\tau},$
(XVI.63)

where $\xi\in C^{+}$, $\tau=-\frac{3}{2}-\mathrm{i}\nu$ ($\nu\in\mathbb{R}$),
and the real-valued constant $c\equiv c_{\nu}$ is:797979This point will be
explicitly discussed later (see Eq. (XVII.35) and its relevant discussions).

$\displaystyle
c_{\nu}=\sqrt{\frac{R^{-2}\;(\nu^{2}+1/4)}{2(2\pi)^{3}\;(1+e^{-2\pi\nu})\;m^{2}}}\;,$
(XVI.64)

in which $m$ stands for the Poincaré-Minkowski mass. Note that, in the sequel
(as in section XIV), we consider a relation between the representation
parameter $\nu$ and $m$ as $\nu=mR$ ($c=1=\hbar$).

Technically, to compute the flat limit ($R\rightarrow\infty$) of the waves
(XVI.63), one needs to consider an area around a given point
$x\in\underline{M}_{R}$, in which all the distances are negligible in
comparison with $R$. Regarding the homogeneity of the dS4 hyperboloid
$\underline{M}_{R}$ under the action of Sp$(2,2)$, one can take, for instance,
the point $x^{4}=R$ and $x^{\mu}=0$, with $\mu=0,1,2,3$.808080From physical
point of view, the only physical entity visible to a local observer on the dS4
hyperboloid $\underline{M}_{R}$ is the gravitational acceleration, naively
speaking, the radius of curvature $R$, which is the same all over the
hyperbolid. In this sense, the observer can never understand where is he/she
exactly located on $\underline{M}_{R}$. Therefore, for the sake of reasoning
and without losing the generality, we can consider any point on
$\underline{M}_{R}$. Letting $R$ go to infinity, dS4 spacetime in the
neighborhood of this point admits its tangent plane, that is, the
$1+3$-dimensional Minkowski spacetime. The coordinates $x^{\mu}$ in this
neighborhood are approximated as (see subsection IX.1, and the relations
(IX.24) and (IX.25)):

$\displaystyle
x^{\mu}=x_{\circ}^{\mu}+{\cal{O}}(R^{-1})\,,\;\;\;\;\;\;\;x^{4}=R+{\cal{O}}(1)\,.$
(XVI.65)

Now, we can deal with the flat limit of the waves (XVI.63). In the first step,
under the limit $R\rightarrow\infty$, for which $e^{-\mathrm{i}\pi\tau}\approx
0$ (recall that $\tau=-\frac{3}{2}-\mathrm{i}mR$), these waves reduce to:

$\displaystyle\lim_{R\rightarrow\infty}\phi_{\nu,\xi}^{-}(x)\approx\lim_{R\rightarrow\infty}c_{\nu}\Big{[}\vartheta\Big{(}\frac{x\cdot\xi}{R}\Big{)}\Big{]}\;\Big{|}\frac{x\cdot\xi}{R}\Big{|}^{\tau}.$
(XVI.66)

Subsequently, taking Eq. (XVI.64) and the parametrization (XVI.65) into
account, we obtain:

$\displaystyle\lim_{R\rightarrow\infty}\phi_{\nu,\xi}^{-}(x)\approx\frac{1}{\sqrt{2(2\pi)^{3}}}\;\vartheta(-\xi^{4})\;\Bigg{[}|\xi^{4}|^{\tau}\Big{(}1+\frac{\xi_{\mu}x_{\circ}^{\mu}}{R|\xi^{4}|}\Big{)}^{-\frac{3}{2}-\mathrm{i}mR}\Bigg{]}_{R\rightarrow\infty}.$
(XVI.67)

This limit solely exists for $\xi^{4}<0$, due to the Heaviside function
$\vartheta(-\xi^{4})$, and for $|\xi^{4}|=1$, due to the term
$|\xi^{4}|^{\tau}$. Therefore, we consider the orbital basis $\daleth^{-}_{4}$
on $C^{+}$ given in Eq. (XVI.14), based upon which we can write
$\xi^{-}\;\big{(}\in\daleth^{-}_{4}\big{)}=\Big{(}\frac{k^{0}}{m}=\sqrt{\frac{{\vec{k}}^{2}}{m^{2}}+1},\frac{\vec{k}}{m},-1\Big{)}$,