May 2019
Pierre Martin-Dussaud111Aix Marseille Univ, Université de Toulon, CNRS, CPT,
Marseille, France
<EMAIL_ADDRESS>

A primer of group theory
for Loop Quantum Gravity
and Spin-foams

The aim of a theory really is, to a great extent, that of systematically
organizing past experience in

such a way that the next generation, our students and their students and so
on, will be able to

absorb the essential aspects in as painless a way as possible, and this is the
only way in which you

can go on cumulatively building up any kind of scientific activity without
eventually coming to a dead end.

M. F. Atiyah, [Ati74].

## Foreword

Calculations in Loop Quantum Gravity (LQG) and spin-foams theory rely heavily
on group theory of $SU(2)$ and $SL_{2}(\mathbb{C})$. Even though many
monographs exist devoted to this theory, the different tools needed (e.g.
representation theory, harmonic analysis, recoupling theory…) are often
dispersed in different books, with different conventions and notations. This
was the initial motivation for the compilation of the present document.
Generally, these notes can serve three main purposes:

  1. 1.

A concise introduction for students to the essential mathematical tools of
LQG. It bridges a gap between the level of students at the end of a Master
programme, and the minimum level required to start doing research in LQG. In
case the pace is too fast, paragraphs have been inserted for a quick
refresher. They are written in small font size, and introduced by ‘$\clubsuit$
Reminder’. Instead of introducing new formulae out of nowhere, we insist on
motivations for doing it. The proofs are done, if helpful for understanding,
but are sometimes only sketched . They are written in small font size
introduced by ‘$\blacktriangleright$ Proof’, so that they can also be skipped
easily.

  2. 2.

A convenient compendium for researchers. Instead of having each formula each
in a different heavy, old book, the most useful ones are now gathered in a
short toolbox.

  3. 3.

A translational hub between the conventions of the main references. For many
notions, no standard notations are universally used. Each author tends to use
their own notations, which makes it difficult to switch easily from one
reference to another. We have made some choices ourselves, but we show
explicitly how they relate to various major references: such discussions are
done in small font size introduced by ‘$\star$ Nota Bene’. We see this attempt
as a step towards a more widely accepted use of common notations. In
particular, we give the conventions of the Wolfram Language, which are helpful
for implementing numerical computations.

Although most of the content is not new, we also offer some new derivations of
results, simpler than what can be found elsewhere, or sometimes not written
anywhere else. A commented bibliography is provided at the end to give a
panorama of the existing literature and to help readers looking for more
details.

These notes are aimed both at physicists, caring about their tools being
mathematically well grounded, and at mathematicians, curious about how some of
their familiar abstract structures can reveal the beauty of quantum gravity.
For the latter, we have introduced specific short paragraphs in small font
size, introduced by ‘$\varheart$ Physics’, that provide general ideas and
references on how the mathematics has been exploited by theoretical
physicists, especially in quantum gravity.

The first chapter introduces $SU(2)$ and $SL_{2}(\mathbb{C})$ the two Lie
groups of main interest for quantum gravity. It is aimed at gathering the main
ingredients which will be extensively used later, fixing notations and
refreshing memories of the reader. Chapter 2 deals with representations of
$SU(2)$, and present various possible realisations that are used in the
literature. Chapter 3 condenses the main results of recoupling theory of
$SU(2)$. Chapter 4 deals with the representations of $SL_{2}(\mathbb{C})$.
Finally chapter 5 wraps everything up into a mathematical presentation of
quantum gravity (with loops and spin-foams).

In writing these notes, I received much help from Simone Speziale, Giorgio
Sarno, Fabio D’Ambrosio, Alexander Thomas, Daniel Martinez and Carlo Rovelli,
who I warmly thank for this.

###### Contents

  1. 1 Warmup
    1. 1.1 Basics of $SL_{2}(\mathbb{C})$
    2. 1.2 The restricted Lorentz group $SO^{+}(3,1)$
    3. 1.3 The sub-groups of $SL_{2}(\mathbb{C})$
    4. 1.4 Decomposing $SL_{2}(\mathbb{C})$
    5. 1.5 Basics of $SU(2)$
    6. 1.6 The rotations $SO(3)$
    7. 1.7 Integrable functions over $SU(2)$
    8. 1.8 Integrable functions over $SL_{2}(\mathbb{C})$
  2. 2 Representation theory of $SU(2)$
    1. 2.1 Irreps of $SU(2)$
    2. 2.2 Angular momentum realisation
    3. 2.3 Homogeneous realisation
    4. 2.4 Projective realisation
    5. 2.5 Spinorial realisation
  3. 3 Recoupling theory of $SU(2)$
    1. 3.1 Clebsch-Gordan coefficients
    2. 3.2 Invariant subspace
    3. 3.3 Wigner’s $3jm$-symbol
    4. 3.4 Wigner’s $4jm$-symbol
    5. 3.5 Wigner’s $6j$-symbol
    6. 3.6 Graphical calculus
  4. 4 Representation theory of $SL_{2}(\mathbb{C})$
    1. 4.1 Finite irreps
    2. 4.2 Infinite irreps
    3. 4.3 Principal unitary series
      1. 4.3.1 Homogeneous realisation
      2. 4.3.2 Projective realisation
      3. 4.3.3 $SU(2)$-realisation
      4. 4.3.4 Canonical basis
      5. 4.3.5 Action of the generators
      6. 4.3.6 $SL_{2}(\mathbb{C})$ Wigner’s matrix
    4. 4.4 Recoupling of $SL_{2}(\mathbb{C})$
  5. 5 Loops and Foams in a nutshell
    1. 5.1 Spin-network
    2. 5.2 Spin-foam
  6. A Representations and intertwiners
  7. B Induced representation
  8. C Commented bibliography

##  Chapter 1 Warmup

The central role of group theory in physics has been largely revealed in the
modern theories of the XXth century. Quantum gravity makes no exception. The
two main groups of interest for quantum gravity are $SL_{2}(\mathbb{C})$ and
its subgroup $SU(2)$. This may seem natural since $SL_{2}(\mathbb{C})$ is, in
some sense, the ‘quantum version’ (more precisely the universal cover) of the
restricted Lorentz group $SO^{+}(3,1)$, which is the symmetry group of
Minkowski spacetime, and $SU(2)$ is the subgroup obtained when a preferred
time-slice is chosen; but the reason why these groups come out in quantum
gravity is actually more subtle (see section 5).

This first chapter is a quick and dense summary of the very basic tools and
facts about $SL_{2}(\mathbb{C})$ and $SU(2)$, which will be later used
extensively. It also fixes many of the notations. If you already feel warmed
up, you would do well skipping this chapter. If you have never seen these
notions in your life, you would do better to first learn them with an
introductory book (see the references in the commented bibliography, appendix
C).

###  1.1 Basics of $SL_{2}(\mathbb{C})$

The algebra of $2\times 2$ complex matrices is denoted
$\mathcal{M}_{2}(\mathbb{C})$. The linear group

$GL_{2}(\mathbb{C})\overset{\rm
def}{=}\left\\{M\in\mathcal{M}_{2}(\mathbb{C})\mid\det M\neq 0\right\\}$ (1.1)

is a Lie group. Its associated Lie algebra $\mathfrak{gl}_{2}(\mathbb{C})$ is
actually isomorphic to $\mathcal{M}_{2}(\mathbb{C})$ endowed with the Lie
product $\left[M,N\right]=MN-NM$.

> $\clubsuit$ Reminder. The set $\mathcal{M}_{2}(\mathbb{C})$ is an algebra
> because it is a vector space (with addition of matrices) endowed with a
> bilinear product (the usual matrix product). $GL_{2}(\mathbb{C})$ is not a
> linear subspace, but it is a complex differential manifold of dimension $4$.
> What’s more, the inversion and the multiplication are both analytical maps,
> which makes $GL_{2}(\mathbb{C})$ a complex Lie group. A complex Lie group of
> dimension $n$ is also a real Lie group of dimension $2n$. Therefore
> $GL_{2}(\mathbb{C})$ is a real Lie group of dimension $8$. The associated
> Lie algebra is the tangent space over the neutral element of the Lie group,
> usually denoted with Gothic letters. It is indeed a Lie algebra, that is to
> say an algebra where the bilinear product, called the Lie bracket, is
> antisymmetric and satisfies the Jacobi identity.

The subset of invertible uni-modular matrices,

$SL_{2}(\mathbb{C})\overset{\rm def}{=}\left\\{M\in GL_{2}(\mathbb{C})\mid\det
M=1\right\\},$ (1.2)

is a complex Lie sub-group of dimension $3$, called the special linear group
$SL_{2}(\mathbb{C})$. Therefore, it is also a real Lie group of dimension $6$.
Topologically, $SL_{2}(\mathbb{C})$ is not compact, but it is simply
connected.

The Lie algebra of $SL_{2}(\mathbb{C})$ is

$\mathfrak{sl}_{2}(\mathbb{C})=\left\\{M\in\mathcal{M}_{2}(\mathbb{C})\mid{\rm
Tr\ }M=0\right\\}.$ (1.3)

It is a $3$-dimensional complex Lie sub-algebra of
$\mathfrak{gl}_{2}(\mathbb{C})$. A basis is given by the three Pauli matrices:

$\displaystyle\sigma_{1}\overset{\rm def}{=}\begin{pmatrix}0&1\\\
1&0\end{pmatrix},$ $\displaystyle\sigma_{2}\overset{\rm
def}{=}\begin{pmatrix}0&-i\\\ i&0\end{pmatrix},$
$\displaystyle\sigma_{3}\overset{\rm def}{=}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}.$ (1.4)

Interestingly, they satisfy111Here and everywhere else, Einstein notation is
understood over repeated indices.

$[\sigma_{i},\sigma_{j}]=2i\,\epsilon_{ijk}\,\sigma_{k}.$ (1.5)

With the identity matrix $\mathbb{1}$, the Pauli matrices also provide a basis
for $\mathcal{M}_{2}(\mathbb{C})$: any $a\in\mathcal{M}_{2}(\mathbb{C})$ can
be written uniquely

$a=a_{0}\mathbb{1}+\sum_{k=1}^{3}a_{k}\sigma_{k}\quad\text{with}\quad
a_{0},a_{1},a_{2},a_{3}\in\mathbb{C}.$ (1.6)

Note that in this basis, the determinant reads $\det
a=a_{0}^{2}-a_{1}^{2}-a_{2}^{2}-a_{3}^{2}$. Moreover, the identity matrix and
the Pauli matrices also provide a basis to the real vector space of $2\times
2$ hermitian matrices, defined by

$H_{2}(\mathbb{C})\overset{\rm def}{=}\\{M\in\mathcal{M}_{2}(\mathbb{C})\mid
M^{\dagger}=M\\}.$ (1.7)

Any $h\in H_{2}(\mathbb{C})$ can be written uniquely as

$h=h_{0}\mathbb{1}+\sum_{k=1}^{3}h_{k}\sigma_{k}\quad\text{with}\quad
h_{0},h_{1},h_{2},h_{3}\in\mathbb{R}.$ (1.8)

###  1.2 The restricted Lorentz group $SO^{+}(3,1)$

The spacetime of special relativity is Minkowski spacetime $\mathbb{M}$.
Mathematically, it is the vector space $\mathbb{R}^{4}$, endowed with a
Lorentzian inner product, whose signature is either $(-,+,+,+)$ (general
relativists convention) or $(+,-,-,-)$ (particle physicists convention). The
group of all isometries (distance-preserving transformations) of Minkoswki
spacetime is called the Poincaré group (sometimes the inhomogeneous Lorentz
group). The isometries that leave the origin fixed form a linear subgroup,
called the Lorentz group (sometimes the homogeneous Lorentz group), and
denoted $O(3,1)$ (or $O(1,3)$). It is composed of four connected components
related to each other by the operators of parity (space inversion) and time-
reversal. The identity component forms a subgroup of $O(3,1)$, made of
transformations that preserves orientation and the direction of time. It is
called the proper orthochronous Lorentz group, or the restricted Lorentz
group, denoted $SO^{+}(3,1)$.

As a real vector space, Minkowski spacetime $\mathbb{M}$ is isomorphic to
$H_{2}(\mathbb{C})$, with the map

$X=(t,x,y,z)\mapsto
h=t\mathbb{1}+x\sigma_{1}+y\sigma_{2}+z\sigma_{3}=\begin{pmatrix}t+z&x-iy\\\
x+iy&t-z\end{pmatrix}.$ (1.9)

The inverse map is given by

$h\mapsto X=\frac{1}{2}(\Tr h,\Tr h\sigma_{1},\Tr h\sigma_{2},\Tr
h\sigma_{3}),$ (1.10)

and the pseudo-scalar product (with convention $(+,-,-,-)$)

$X\cdot
X^{\prime}=tt^{\prime}-xx^{\prime}-yy^{\prime}-zz^{\prime}=-\frac{1}{4}\Tr\left(hh^{\prime}-h\sigma_{1}h^{\prime}\sigma_{1}-h\sigma_{2}h^{\prime}\sigma_{2}-h\sigma_{3}h^{\prime}\sigma_{3}\right)$
(1.11)

Note that the pseudo-norm of $\mathbb{M}$ is mapped to the determinant over
$H_{2}(\mathbb{C})$:

$X\cdot X=\det h.$ (1.12)

From the latter property, we see that the action of $a\in SL_{2}(\mathbb{C})$
upon $h\in H_{2}(\mathbb{C})$, given by

$h\mapsto aha^{\dagger},$ (1.13)

defines a linear isometry on $\mathbb{M}$. Thus, it defines an homomorphism
between $SO^{+}(3,1)$ and $SL_{2}(\mathbb{C})$, and it is easy to show the
following isomorphism of groups

$SL_{2}(\mathbb{C})/\\{\mathbb{1},-\mathbb{1}\\}\cong SO^{+}(3,1).$ (1.14)

$SL_{2}(\mathbb{C})$ is said to be the double cover, or the universal cover,
of $SO^{+}(3,1)$, or sometimes also the Lorentz spin group. This gives a first
glimpse on the role of $SL_{2}(\mathbb{C})$ in fundamental physics.

###  1.3 The sub-groups of $SL_{2}(\mathbb{C})$

There are many sub-groups of $SL_{2}(\mathbb{C})$. We describe below the main
ones. The figure 1 shows the relations of inclusion between them.

Figure 1: This graph represents the relations of inclusions between the
subgroups of $SL_{2}(\mathbb{C})$.

$\blacksquare$ $\boldsymbol{SU(2)}$, the unitary special group, is defined by:

$SU(2)\overset{\rm def}{=}\left\\{u\in SL_{2}(\mathbb{C})\mid
u^{\dagger}u=\mathbb{1}\right\\}.$ (1.15)

Any $u\in SU(2)$ can be uniquely written as

$u=u_{0}e+i\sum_{k=1}^{3}u_{k}\sigma_{k}\quad\text{with}\quad
u_{0},u_{1},u_{2},u_{3}\in\mathbb{R}\quad{\rm
and}\quad\sum_{k=0}^{3}u_{k}^{2}=1.$ (1.16)

> $\varheart$ Physics. Through the isomorphism 1.9 and the action 1.13, the
> definition 1.15 enables to see $SU(2)$ as the stabiliser of the unit time
> vector $(1,0,0,0)$. Sometimes, the stabiliser is also called the little
> group or the isotropy group. Physically, it means that $SU(2)$ only acts
> over the space, and not in the time direction. Choosing another time
> direction, related to $(1,0,0,0)$ by a boost $\Lambda$, would have defined
> another stabilizer, isomorphic to $SU(2)$, which makes physicists sometimes
> talk of a $SU(2)$, as if there were several.

$\blacksquare$ $\boldsymbol{SU(1,1)}$ is defined by:

$SU(1,1)\overset{\rm def}{=}\left\\{v\in SL_{2}(\mathbb{C})\mid
v^{\dagger}\sigma_{3}v=\sigma_{3}\right\\}.$ (1.17)

Any $v\in SU(1,1)$ can be uniquely written as:

$v=v_{0}e+v_{1}\sigma_{1}+v_{2}\sigma_{2}+iv_{3}\sigma_{3}\quad\text{with}\quad
v_{0},v_{1},v_{2},v_{3}\in\mathbb{R}\quad{\rm and}\quad
v_{0}^{2}-v_{1}^{2}-v_{2}^{2}+v_{3}^{2}=1.$ (1.18)

> $\varheart$ Physics. Similarly to the $SU(2)$ case, $SU(1,1)$ can be
> understood by its action in Minkoswki spacetime as the stabiliser of
> $(0,0,0,1)$.

$\blacksquare$ $\boldsymbol{SL_{2}(\mathbb{R})}$, the real linear special
group, is defined by

$SL_{2}(\mathbb{R})\overset{\rm
def}{=}\left\\{a\in\mathcal{M}_{2}(\mathbb{R})\mid\det a=1\right\\},$ (1.19)

and interestingly it is also

$SL_{2}(\mathbb{R})=\left\\{a\in SL_{2}(\mathbb{C})\mid
a^{\dagger}\sigma_{2}a=\sigma_{2}\right\\}.$ (1.20)

Any $a\in SU(1,1)$ can be uniquely written as:

$a=a_{0}e+a_{1}\sigma_{1}+ia_{2}\sigma_{2}+a_{3}\sigma_{3}\quad\text{with}\quad
a_{0},a_{1},a_{2},a_{3}\in\mathbb{R}\quad{\rm and}\quad
a_{0}^{2}-a_{1}^{2}+a_{2}^{2}-a_{3}^{2}=1.$ (1.21)

> $\varheart$ Physics. Similarly to the $SU(2)$ case, $SL_{2}(\mathbb{R})$ can
> be understood by its action in Minkoswki spacetime as the stabiliser of
> $(0,0,1,0)$.

> $\bigstar$ Nota Bene. Following the previous sequence, it would be fair to
> expect next sub-group to be the one defined by
>
> $\left\\{b\in SL(2,\mathbb{C})\mid
> b^{\dagger}\sigma_{1}=\sigma_{1}b^{-1}\right\\}.$ (1.22)
>
> To the knowledge of the author, this group has no name and is not much
> studied in the literature.

$\blacksquare$ The upper $\boldsymbol{K_{+}}$ and lower $\boldsymbol{K_{-}}$
triangular group are defined by:

$\displaystyle K_{+}\overset{\rm
def}{=}\left\\{\begin{pmatrix}\lambda^{-1}&\mu\\\
0&\lambda\end{pmatrix}\mid\lambda\in\mathbb{C}^{*}\quad{\rm
and}\quad\mu\in\mathbb{C}\right\\}$ $\displaystyle K_{-}\overset{\rm
def}{=}\left\\{\begin{pmatrix}\lambda^{-1}&0\\\
\mu&\lambda\end{pmatrix}\mid\lambda\in\mathbb{C}^{*}\quad{\rm
and}\quad\mu\in\mathbb{C}\right\\}.$

They are also called the Borel sub-groups or the parabolic sub-groups.

$\blacksquare$ The subgroups $\boldsymbol{Z_{+}}$ and $\boldsymbol{Z_{-}}$ are
defined by:

$\displaystyle Z_{+}\overset{\rm def}{=}\left\\{\begin{pmatrix}1&z\\\
0&1\end{pmatrix}\mid z\in\mathbb{C}\right\\}$ $\displaystyle Z_{-}\overset{\rm
def}{=}\left\\{\begin{pmatrix}1&0\\\ z&1\end{pmatrix}\mid
z\in\mathbb{C}\right\\}.$

$\blacksquare$ The diagonal group $\boldsymbol{D}$ is defined by:

$D\overset{\rm def}{=}\left\\{\begin{pmatrix}\delta&0\\\
0&\delta^{-1}\end{pmatrix}\mid\delta\in\mathbb{C}^{*}\right\\}.$ (1.23)

$\blacksquare$ $\boldsymbol{U(1)}$, the uni-dimensional unitary group, seen as
a subgroup of $SL_{2}(\mathbb{C})$ through the group isomorphism

$U(1)\overset{\rm
def}{=}\\{e^{i\theta}\mid\theta\in\mathbb{R}\\}\cong\left\\{\begin{pmatrix}e^{i\theta}&0\\\
0&e^{-i\theta}\end{pmatrix}\mid\theta\in\mathbb{R}\right\\}.$ (1.24)

$U(1)$ is also the maximal torus (biggest compact, connected, abelian Lie
subgroup) of $SU(2)$.

$\blacksquare$ $\boldsymbol{Z_{2}}$, the center, defined as the subset of
matrices which commute with all $SL_{2}(\mathbb{C})$, is shown to be

$Z_{2}=\left\\{\mathbb{1},-\mathbb{1}\right\\}.$ (1.25)

Since it is a normal subgroup (as any center of any group), the quotient
$SL_{2}(\mathbb{C})/Z_{2}$ is also a group, which can be shown to be
isomorphic to the restricted Lorentz group $SO^{+}(3,1)$, as was already said
in 1.14.

###  1.4 Decomposing $SL_{2}(\mathbb{C})$

The structural properties of a matrix group can be grasped through the study
of its decompositions. We are going to present four different decompositions
of $SL_{2}(\mathbb{C})$. First of all, let us introduce the notation $Sp(M)$
for the set of the eigenvalues of $M$, and

$H_{2}^{++}(\mathbb{C})\overset{\rm def}{=}\left\\{M\in
H_{2}(\mathbb{C})\mid\forall\lambda\in Sp(H),\quad\lambda>0\right\\},$ (1.26)

for the set of $2\times 2$ hermitian positive-definite matrices.

> $\clubsuit$ Reminder. Recall that any hermitian matrix is diagonalisable. It
> is the content of the spectral theorem that says that $H$ is an hermitian
> matrix if and only if there exists a unitary matrix $U$ and a diagonal
> matrix $D$ with real coefficients such that
>
> $\displaystyle H=U^{\dagger}DU.$
>
> The coefficients of $D$ are the eigenvalues of $H$.

$\blacksquare$ Polar decomposition. For all $M\in GL_{2}(\mathbb{C})$, there
exists a unique unitary matrix $U\in U(2)$ and a unique positive-definite
hermitian matrix $H\in H_{2}^{++}(\mathbb{C})$ such that:

$M=HU.$ (1.27)

> $\blacktriangleright$ Proof. The polar decomposition actually works for the
> set $H_{2}^{++}(\mathbb{C})$ of $n\times n$ positive-definite hermitian
> matrices. Recall that if $H\in H_{n}^{++}(\mathbb{C})$, then there exists a
> unique matrix $S\in H_{n}^{++}(\mathbb{C})$, called the square root of $H$,
> such that $H=S^{2}$.
>
> Existence. Let $M\in GL_{n}(\mathbb{C})$ be the matrix to be decomposed,
> then $M^{\dagger}M$ is hermitian and positive. Thus we have a unique matrix
> $S\in H_{n}^{++}(\mathbb{C})$ such that $M^{\dagger}M=S^{2}$. We check
> finally that $MS^{-1}$ is unitary.
>
> Uniqueness. If $M=QS$ with $Q$ unitary and $S$ definite-positive hermitian,
> then $M^{\dagger}=SQ^{-1}$, so $M^{\dagger}M=S^{2}$. But $M^{\dagger}M$ is
> positive hermitian and has therefore a unique positive hermitian square
> root, so that $S$ is this square root and $Q$ is equal to $MS^{-1}$.
>
> For $n=1$, it is the decomposition $z=re^{i\theta}$ of a non-zero complex
> number. It is the reason why we call it polar decomposition (kind of
> generalisation of polar coordinates). $\Box$

Remarks:

  1. 1.

The order does not matter, and the theorem would also be true with $M=UH$.

  2. 2.

If $M=HU\in SL_{2}(\mathbb{C})$, then $U\in SU(2)$ and $\det H=1$.

> $\varheart$ Physics. The polar decomposition has been used notably by
> Thiemann and Winkler in their analysis of the coherent states of quantum
> gravity (see [Thi01, TW01b, TW01c, TW01a]).

$\blacksquare$ Cartan decomposition. For all $g\in SL_{2}(\mathbb{C})$, there
exists $u,v\in SU(2)$ and $r\in\mathbb{R}_{+}$ such that:

$g=ue^{r\sigma_{3}/2}v^{-1}.$ (1.28)

> $\blacktriangleright$ Proof. The proof is essentially the same as
> previously. Existence. Let $g\in SL_{2}(\mathbb{C})$ be decomposed.
> $g^{\dagger}g$ is positive-definite hermitian. With the spectral theorem, we
> have $v\in U(2)$ and $d$ a real diagonal matrix with strictly positive
> coefficients such that $g^{\dagger}g=v^{\dagger}dv$. If $\det
> v=e^{i\theta}$, then $u=e^{-i\theta}v\in SU(2)$ and
> $g^{\dagger}g=u^{\dagger}du$. Since $\det d=1$, one can write
> $d=e^{r\sigma_{3}}$. Note that
>
> $e^{r\sigma_{3}/2}=\begin{pmatrix}e^{r/2}&0\\\ 0&e^{-r/2}\end{pmatrix},$
> (1.29)
>
> Then we show $gue^{r\sigma_{3}/2}\in SU(2)$. $\Box$

Remark:

  1. 1.

The number $r$ is called the rapidity of the boost along the axis $z$.

  2. 2.

This theorem can be generalized to the case $SL_{n}(\mathbb{C})$.

  3. 3.

Given the decomposition $g=ue^{r\sigma_{3}/2}v^{-1}$, $r$ is uniquely
determined, but $u$ and $v$ are not. The other possible choices are obtained
by $(u,v)\mapsto(ue^{i\theta\sigma_{3}},ve^{i\theta\sigma_{3}})$, with
$\theta\in\mathbb{R}$.

  4. 4.

The polar decomposition of $SL_{2}(\mathbb{C})$ is a particular case of the
Cartan decomposition where $v^{-1}=\mathbb{1}$.

> $\varheart$ Physics. This decomposition has been used notably for the
> asymptotics of spin-foams amplitude[Spe17], and also for twisted geometries
> in [LS16].

$\blacksquare$ Gauss decomposition. Let $g\in SL_{2}(\mathbb{C})$ such that
$g_{22}\neq 0$. There exists a unique triplet $(z_{+},d,z_{-})\in Z_{+}\times
D\times Z_{-}$ such that

$g=z_{+}dz_{-}$ (1.30)

> $\blacktriangleright$ Proof. Explicit computation. If
> $g=\begin{pmatrix}\alpha&\beta\\\ \gamma&\delta\end{pmatrix}\in
> SL(2,\mathbb{C})$, then one can write:
>
> $g=\begin{pmatrix}1&\beta\delta^{-1}\\\
> 0&1\end{pmatrix}\begin{pmatrix}\delta^{-1}&0\\\
> 0&\delta\end{pmatrix}\begin{pmatrix}1&0\\\ \gamma\delta^{-1}&1\end{pmatrix}$
> (1.31)
>
> $\Box$

$\blacksquare$ Iwasawa decomposition. According to the Japanese mathematician
(1917 - 1998), for any matrix $M\in SL_{2}(\mathbb{C})$, there exists a unique
triplet $(Z,D,U)\in Z_{+}\times D_{\mathbb{R}_{+}}\times SU(2)$ that
decomposes $M$:

$M=ZDU=\begin{pmatrix}1&z\\\ 0&1\end{pmatrix}\begin{pmatrix}\lambda^{-1}&0\\\
0&\lambda\end{pmatrix}\begin{pmatrix}\alpha&-\beta^{*}\\\
\beta&\alpha^{*}\end{pmatrix}$ (1.32)

with
$(z,\lambda,\alpha,\beta)\in\mathbb{C}\times\mathbb{R}^{*}_{+}\times\mathbb{C}^{2}$.

> $\varheart$ Physics. This decomposition has been used notably for the study
> of covariant twisted geometries [LST12].

###  1.5 Basics of $SU(2)$

Let us now focus on the special unitary subgroup

$SU(2)\overset{\rm def}{=}\left\\{u\in SL_{2}(\mathbb{C})\mid
u^{\dagger}u=\mathbb{1}\right\\}.$ (1.33)

It is a $3$-dimensional real Lie subgroup of the $6$-dimensional real Lie
group $SL_{2}(\mathbb{C})$. Any $u\in SU(2)$ can be uniquely written as

$u=\begin{pmatrix}\alpha&-\beta^{*}\\\ \beta&\alpha^{*}\end{pmatrix}\quad{\rm
with}\quad(\alpha,\beta)\in\mathbb{C}^{2},|\alpha|^{2}+|\beta|^{2}=1,$ (1.34)

or equivalently

$u=\begin{pmatrix}a+ib&-c+id\\\ c+id&a-ib\end{pmatrix}\quad{\rm
with}\quad(a,b,c,d)\in\mathbb{R}^{4},a^{2}+b^{2}+c^{2}+d^{2}=1.$ (1.35)

The latter expression shows that $SU(2)$ is diffeomorphic to $S^{3}$ (the unit
sphere of $\mathbb{R}^{4}$). Therefore it is connected, simply connected and
compact. The center of $SU(2)$ is $Z_{2}=\\{\mathbb{1},-\mathbb{1}\\}$, and
the quotient $SU(2)/Z_{2}$ is also a group, which happens to be isomorphic to
$SO(3)$ (see section 1.6).

> $\varheart$ Physics. The group $SU(2)$ is central in quantum physics. First,
> it appears for the theory of the angular momentum (spin). Historically, it
> was also used as an approximate symmetry group for the isospin that relates
> protons and neutrons. Then it reappeared to describe the electro-weak
> interaction. In LQG, $SU(2)$ comes with the holonomies, which are obtained
> by exponentiation of the Ashtekar variables, used for the quantization (see
> section 5).

The real Lie algebra of $SU(2)$ is

$\mathfrak{su}(2)=\left\\{M\in\mathcal{M}_{2}(\mathbb{C})\mid M^{\dagger}=-M\
{\rm and}\ {\rm Tr\ }M=0\right\\}.$ (1.36)

It is a real vector space, of which a basis is given by
$(i\sigma_{1},i\sigma_{2},i\sigma_{3})$. Since $SU(2)$ is a compact Lie group,
any element of $SU(2)$ can be written (non uniquely) as the exponential of an
element of the associated Lie algebra $\mathfrak{su}(2)$ (it is a general
theorem for compact Lie groups).

$\blacksquare$ Exponential decomposition. If $u\in{\rm SU}(2)$, there exists a
(non-unique) $\vec{\alpha}\in\mathbb{R}^{3}$ such that

$u=e^{i\vec{\alpha}\cdot\vec{\sigma}}=\cos\|\vec{\alpha}\|I+i\sin\|\vec{\alpha}\|\frac{\vec{\alpha}}{\|\vec{\alpha}\|}\cdot\vec{\sigma}.$
(1.37)

> $\blacktriangleright$ Proof. Let $M=\begin{pmatrix}a&-b^{*}\\\
> b&a^{*}\end{pmatrix}\in SU(2)$. The equality is easy to check for
> $r=\arccos(\real a)$ and $\alpha_{1}=\frac{r}{\sin r}\imaginary b$,
> $\alpha_{2}=-\frac{r}{\sin r}\real b$, $\alpha_{3}=\frac{r}{\sin
> r}\imaginary a$. $\Box$

$\blacksquare$ Euler angles decomposition. For all $u\in SU(2)$, there exists
$\alpha,\beta,\gamma\in\mathbb{R}$ (called Euler angles) such that:

$u=e^{-\frac{i\alpha}{2}\sigma_{3}}e^{-\frac{i\beta}{2}\sigma_{2}}e^{-\frac{i\gamma}{2}\sigma_{3}}$
(1.38)

The choice can be made unique by restricting the domain of definition of the
angles, with for instance $\alpha\in]-2\pi,2\pi[,\beta\in[0,\pi]\ \text{and}\
\gamma\in[|\alpha|,4\pi-|\alpha|[$.

> $\blacktriangleright$ Proof. Explicit computation. The right hand side (RHS)
> gives
>
>
> $\begin{pmatrix}e^{-\frac{i(\alpha+\gamma)}{2}}\cos\beta/2&-e^{\frac{i(\gamma-\alpha)}{2}}\sin\beta/2\\\
> e^{-\frac{i(\gamma-\alpha)}{2}}\sin\beta/2&e^{\frac{i(\alpha+\gamma)}{2}}\cos\beta/2\end{pmatrix}$
> (1.39)
>
> and for any $u\in SU(2)$, it is clearly possible to find
> $\alpha,\beta,\gamma\in\mathbb{R}$ to write $u$ in this form. Note that
> there are other conventions for the definition of Euler angles. The
> definition we have chosen is the one of Varshalovich ([VMK87] p. 27) and
> Sakurai ([SN11] p. 177). Instead, Rühl ([Rüh70] p. 43) and the Wolfram
> Language have chosen the convention
> $u=e^{\frac{i\alpha}{2}\sigma_{3}}e^{\frac{i\beta}{2}\sigma_{2}}e^{\frac{i\gamma}{2}\sigma_{3}}$.
> $\Box$

###  1.6 The rotations $SO(3)$

As we have said in section 1.3, the action of $SU(2)$ over Minkowski space,
given by 1.13, preserves the time direction. Then, $SU(2)$ acts on the spatial
dimensions as the group of rotations over the Euclidean space
$\mathbb{R}^{3}$. The group of rotations of $\mathbb{R}^{3}$ is

$SO(3)\overset{\rm def}{=}\left\\{M\in\mathcal{M}_{3}(\mathbb{R})\mid
M^{T}M=\mathbb{1}\ {\rm and}\ \det M=1\right\\}.$ (1.40)

It is a Lie group whose Lie algebra is

$\mathfrak{so}(3)=\left\\{M\in\mathcal{M}_{3}(\mathbb{R})\mid M^{T}+M=0\ {\rm
and}\ \Tr M=0\right\\}.$ (1.41)

We can show the following isomorphism of Lie algebra

$\mathfrak{so}(3)\cong\mathfrak{su}(2).$ (1.42)

Besides we have the following group isomorphism:

$SO(3)\cong SU(2)/Z_{2}.$ (1.43)

This can be seen with the map

$g:\begin{pmatrix}\alpha&-\beta^{*}\\\
\beta&\alpha^{*}\end{pmatrix}\mapsto\begin{pmatrix}\frac{1}{2}(\alpha^{2}+\alpha^{*2}-\beta^{2}-\beta^{*2})&\frac{i}{2}(\alpha^{2}-\alpha^{*2}-\beta^{2}+\beta^{*2})&\alpha\beta^{*}+\alpha^{*}\beta\\\
\frac{i}{2}(-\alpha^{2}+\alpha^{*2}-\beta^{2}+\beta^{*2})&\frac{1}{2}(\alpha^{2}+\alpha^{*2}+\beta^{2}+\beta^{*2})&i(-\alpha\beta^{*}+\alpha^{*}\beta)\\\
-\alpha\beta-\alpha^{*}\beta^{*}&i(-\alpha\beta+\alpha^{*}\beta^{*})&\alpha\alpha^{*}-\beta\beta^{*}\end{pmatrix}$
(1.44)

which is a $2$-to-$1$ onto homomorphism from $SU(2)$ to $SO(3)$. It satisfies
notably $g(u)=g(-u)$. Topologically, $SO(3)$ is homeomorphic to the sphere
$S^{3}$ where the antipodal points have been identified. It is notably
connected, but not simply connected.

The action of the homomorphism 1.44 over the Euler decomposition 1.38, shows
that any rotation $r\in SO(3)$, can be decomposed as

$r=r_{z}(\alpha)r_{y}(\beta)r_{z}(\gamma)\\\ \text{with}\
r_{z}(\phi)=\begin{pmatrix}\cos\phi&\sin\phi&0\\\ -\sin\phi&\cos\phi&0\\\
0&0&1\end{pmatrix}\quad\text{and}\quad
r_{y}(\beta)=\begin{pmatrix}\cos\beta&0&\sin\beta\\\ 0&1&0\\\
-\sin\beta&0&\cos\beta\end{pmatrix}.$ (1.45)

where $(\alpha,\beta,\gamma)$ are (any choice of) Euler angles of (any choice
of) one of the two antecedents of $r$ by $g$. The unicity of the decomposition
can be obtained for instance with the restriction
$\alpha\in]-\pi,\pi[,\beta\in[0,\pi]\ \text{and}\
\gamma\in[|\alpha|,2\pi-|\alpha|[$.

###  1.7 Integrable functions over $SU(2)$

##### Haar measure.

It can be shown that there exists a unique quasi-regular Borel measure $\mu$
over $SU(2)$ which satisfies

  1. 1.

Invariant: $\mu(u)=\mu(gu)=\mu(ug)$;

  2. 2.

Normalised: $\mu(SU(2))=1$.

It is called the (two-sided normalised) Haar measure of $SU(2)$.

> $\clubsuit$ Reminder. A Borel set in $SU(2)$ is any subset of $SU(2)$
> obtained from open sets through countable union, countable intersection, or
> taking the complement. All Borel sets form an algebra called the Borel
> algebra $\mathcal{B}(SU(2))$. A Borel $SU(2)$-measure $\mu$ is a non-
> negative function over $\mathcal{B}(SU(2))$ for which $\mu(\emptyset)=0$,
> and which is countable additive (the measure of a disjoint union is the sum
> of the disjoint sets). A Borel measure is said to be quasi-regular if it is
> outer regular ($\mu(S)=\inf\\{\mu(U)\mid S\subseteq U,U\ \text{open}\\}$)
> and inner regular: $\mu(S)=\sup\\{\mu(K)\mid K\subseteq S,K\
> \text{compact}$).

The Haar measure enables definition of integrals of functions $f$ over
$SU(2)$:

$\int_{SU(2)}f(u)\,\differential\mu(u)\quad\text{also
denoted}\quad\int_{SU(2)}f(u)\,\differential u$ (1.46)

##### The Hilbert space $L^{2}(SU(2))$.

The space of complex functions over $SU(2)$ satisfying

$\int_{SU(2)}|f(u)|^{2}du<\infty,$ (1.47)

is denoted $L^{2}(SU(2))$. It is an infinite-dimensional Hilbert space with
the scalar product

$(f_{1},f_{2})\overset{\rm def}{=}\int_{SU(2)}f_{1}^{*}(u)f_{2}(u)du.$ (1.48)

###  1.8 Integrable functions over $SL_{2}(\mathbb{C})$

In Vilenkin ([GGV66] pp. 214-215), an invariant measure $da$ over
$SL_{2}(\mathbb{C})$ is defined, so that for any $g\in SL_{2}(\mathbb{C})$, we
have

$da=d(ga)=d(ag)=d(a^{-1}).$ (1.49)

It is given explicitly by

$da=\left(\frac{i}{2}\right)^{3}|a_{12}|^{-2}\,da_{11}d\overline{a_{11}}\,da_{12}d\overline{a_{12}}\,da_{22}d\overline{a_{22}}.$
(1.50)

In Rühl [Rüh70], the invariant measure is given in terms of the coefficients
in the decomposition 1.6, by

$da=\pi^{-4}\delta\left(a_{0}-\sum_{k=1}^{3}a_{k}^{2}-1\right)da_{0}d\overline{a_{0}}\,da_{1}d\overline{a_{1}}\,da_{2}d\overline{a_{2}}\,da_{3}d\overline{a_{3}}.$
(1.51)

It is normalised so that the induced measure over $SU(2)$ is the same Haar
measure defined in the previous section. In Rühl ([Rüh70] p. 285), it is shown
that using the Cartan decomposition $a=ue^{r\sigma_{3}/2}v^{-1}$, we have:

$d\mu(a)=\frac{1}{4\pi}\sinh^{2}rdrdudv.$ (1.52)

##  Chapter 2 Representation theory of $SU(2)$

Motivated by their omnipresence in quantum physics, we are going to study the
representations of $SU(2)$ over finite-dimensional Hilbert spaces111Hilbert
spaces are of main interest for quantum physics. But many physicists may not
be aware of the real representations of $SU(2)$, which are much less studied
(see [IRS91] for more about them).. Now, due to Peter-Weyl’s theorem (see
appendix A), a complex finite representation of a compact group can be
decomposed into a direct sum of irreps. We thus focus on irreps of $SU(2)$.

###  2.1 Irreps of $SU(2)$

To start with, we make use of the following correspondence, which is a
particular case of the so-called Weyl’s unitary trick. For finite dimensional
representations, we can show that the following sets of representations are in
one-to-one correspondence:

  1. 1.

Holomorphic representations222‘Holomorphic representation’ means that the map
defined by the representation over the vector space is holomorphic. of
$SL_{2}(\mathbb{C})$,

  2. 2.

Representations of $SU(2)$,

  3. 3.

Representations of $\mathfrak{su}(2)$,

  4. 4.

$\mathbb{C}$-linear representations333As we will see later,
$\mathfrak{sl}_{2}(\mathbb{C})$ can be seen both as a complex or as a real
vector space. ‘$\mathbb{C}$-linear representations’ means that we care about
the complex structure of $\mathfrak{sl}_{2}(\mathbb{C})$. We will care about
the $\mathbb{R}$-linear representations of $\mathfrak{sl}_{2}(\mathbb{C})$ in
section 4.1. of $\mathfrak{sl}_{2}(\mathbb{C})$.

Moreover, this correspondence preserves invariant subspaces and equivalences
of representations.

> $\blacktriangleright$ Proof. We can show the following bijections [Kna86]:
>
>   * $(1)\Rightarrow(2)$
>
> Restriction of the action of $SL_{2}(\mathbb{C})$ to that of its subgroup
> $SU(2)$.
>
>   * $(2)\Rightarrow(3)$
>
> Differentiation as shown in appendix A.
>
>   * $(3)\Rightarrow(4)$
>
> Use the fact that $\mathfrak{sl}_{2}(\mathbb{C})\cong\mathfrak{su}(2)\oplus
> i\mathfrak{su}(2)$.
>
>   * $(4)\Rightarrow(1)$
>
> If $G$ and $H$ are two analytical groups with $G$ simply connected, and if
> $\phi:\mathfrak{g}\rightarrow\mathfrak{h}$ is an homomorphism between there
> Lie algebras, then there exists a smooth homomorphism $\Phi:G\rightarrow H$
> whose differential at the identity is $\phi$. In our case,
> $SL_{2}(\mathbb{C})$ is simply connected, and since $\phi$ is assumed
> $\mathbb{C}$-linear, $\Phi$ is holomorphic.
>
>

>
> $\Box$

We can now describe all the $\mathbb{C}$-linear irreps of
$\mathfrak{sl}_{2}(\mathbb{C})$.

##### Theorem.

For all $n\in\mathbb{N}$, there exists an $n$-dimensional $\mathbb{C}$-linear
irreducible representation of $\mathfrak{sl}_{2}(\mathbb{C})$, unique up to
equivalence.

> $\blacktriangleright$ Proof. The proof is constructive. Analysis: assume the
> existence of an $(n+1)$-dimensional representation over a vector space
> $\mathcal{H}$. One shows then that there exists a basis $(v_{i})$ with
> $i=0..n$ such that the action of the elements
> $h,e,f\in\mathfrak{sl}_{2}(\mathbb{C})$, defined by
>
> $\displaystyle h\overset{\rm def}{=}\sigma_{3}=\begin{pmatrix}1&0\\\
> 0&-1\end{pmatrix}$ $\displaystyle e\overset{\rm
> def}{=}(\sigma_{1}+i\sigma_{2})/2=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}$
> $\displaystyle f\overset{\rm
> def}{=}(\sigma_{1}-i\sigma_{2})/2=\begin{pmatrix}0&0\\\ 1&0\end{pmatrix},$
>
> and which satisfy the commuting relations
>
> $\displaystyle[h,e]=2e$ $\displaystyle[h,f]=-2f$ $\displaystyle[e,f]=h,$
>
> is given by:
>
> $\displaystyle h\cdot v_{k}=2(j-k)v_{k},$ $\displaystyle e\cdot
> v_{k}=k(n-k+1)v_{k-1},$ $\displaystyle f\cdot v_{k}=v_{k+1}.$
>
> Synthesis: one choose any basis $(v_{i})$ and defines the action of $h$,
> $e$, and $f$ by the previous formulas. One checks easily that the obtained
> representation is irreducible. $\Box$

The $3$-dimensional complex vector space $\mathfrak{sl}_{2}(\mathbb{C})$ can
also be seen as a $6$-dimensional real vector space, which has
$\mathfrak{su}(2)$ as its subspace. Thus, by restriction of the action of
$\mathfrak{sl}_{2}(\mathbb{C})$ to $\mathfrak{su}(2)$, the previously found
$\mathbb{C}$-linear irreps of $\mathfrak{sl}_{2}(\mathbb{C})$, define also
irreps of $\mathfrak{su}(2)$. Finally, by exponentiating with 1.37, we find
all irreps of $SU(2)$ over complex vector spaces.

###  2.2 Angular momentum realisation

In physics textbooks, the representations of $\mathfrak{sl}_{2}(\mathbb{C})$
are indexed by a half-integer, called a spin. To each spin $j\in\mathbb{N}/2$
is associated a Hilbert space $\mathcal{Q}_{j}$ of dimension $2j+1$. The
canonical basis, also called magnetic basis, is composed of the vectors (or
‘kets’ in the Dirac language) denoted $\ket{j,m}$ with
$m\in\\{-j,...,0,...,j\\}$. It is made orthonormal by choosing the scalar
product that satisfies the property (in Dirac notations):

$\innerproduct{j,m}{j,n}=\delta_{mn}.$ (2.1)

We now define the angular momentum observables $J_{i}\overset{\rm
def}{=}\frac{1}{2}\sigma_{i}$, sometimes called simplygenerators of $SU(2)$ or
generators of rotations. They satisfy:

$\left[J_{i},J_{j}\right]=i\epsilon_{ijk}J_{k}.$ (2.2)

We then define their linear action over $\mathcal{Q}_{j}$ by

$\displaystyle
J_{1}\ket{j,m}=\frac{1}{2}\sqrt{(j-m)(j+m+1)}\ket{j,m+1}+\frac{1}{2}\sqrt{(j+m)(j-m+1)}\ket{j,m-1},$
(2.3) $\displaystyle
J_{2}\ket{j,m}=\frac{1}{2i}\sqrt{(j-m)(j+m+1)}\ket{j,m+1}-\frac{1}{2i}\sqrt{(j+m)(j-m+1)}\ket{j,m-1},$
$\displaystyle J_{3}\ket{j,m}=m\ket{j,m}.$

> $\varheart$ Physics. In some textbooks, the generators are defined as
> $J_{i}\overset{\rm def}{=}\frac{\hbar}{2}\sigma_{i}$ where $\hbar$ is the
> reduced Planck constant, which has the dimension of an action. Indeed, this
> realisation originally comes from atomic physics, where the $J_{i}$
> represent ‘observables’ of angular momentum. For simplicity, we are working
> in the units where $\hbar=1$, keeping in mind the possibility to restore
> $\hbar$ explicitly at any moment by dimensional analysis. By the way, notice
> also that since ‘observables’ are required to be hermitian operators, the
> $J_{i}$ are elements of $i\mathfrak{su}(2)$ and not of $\mathfrak{su}(2)$.

The action of the generators over $\mathcal{Q}_{j}$ defines a
$(2j+1)$-dimensional irrep of $SU(2)$, called the spin-$j$ representation.
This is shown by exhibiting the following equivalence444Two representations
are equivalent if there exists a bijective intertwiner, i.e. a bijective map
$T$ between the two vector spaces, so that it commutes with the action of the
group (see appendix A). It would be too heavy to write explicitly
$\ket{jm}=T(v_{j-m})$, so that we choose to write rather
$\ket{jm}=T(v_{j-m})$. Similarly, for the operators we write $J_{+}\cong e$,
rather than $J_{+}=T\circ e\circ T^{-1}$.  with the irreps defined in the
previous section: $\ket{jm}\cong v_{j-m}$. The generators are related by
$J_{3}\cong h/2$, $J_{+}\cong e$ and $J_{-}\cong f$, where we define the
ladder operators $J_{+}\overset{\rm def}{=}J_{1}+iJ_{2}$ (up) and
$J_{-}\overset{\rm def}{=}J_{1}-iJ_{2}$ (down). Their action is

$\displaystyle J_{+}\ket{j,m}=\sqrt{(j-m)(j+m+1)}\ket{j,m+1},$ (2.4)
$\displaystyle J_{-}\ket{j,m}=\sqrt{(j+m)(j-m+1)}\ket{j,m-1}.$

Sometimes the action of $J_{+}$ over $\ket{jm}$ is written with a constant
phase $e^{i\delta}$. This choice defines an equivalent representation, but the
choice of a null phase (called the Condon-Shortley convention from [CS59]) is
the most widespread.

Finally notice that $\ket{jm}$ is also an eigenvector of the total angular
momentum $\vec{J}^{2}\overset{\rm def}{=}J_{1}^{2}+J_{2}^{2}+J_{3}^{2}$:

$\vec{J}^{2}\ket{jm}=j(j+1)\ket{jm}.$ (2.5)

In fact, the $\ket{jm}$ form the unique orthonormal basis that diagonalises
simultaneously the commuting operators $J_{3}$ and $\vec{J}^{2}$. We say that
$J_{3}$ and $\vec{J}^{2}$ form a complete set of commuting operators (CSCO).
From a mathematical perspective, notice also that $\vec{J}^{2}$ is not a
element of the algebra $i\mathfrak{su}(2)$, but an element of the universal
enveloping algebra $\mathcal{U}(i\mathfrak{su}(2))$ whose action can be easily
computed by successive action of $\mathfrak{su}(2)$. Since $\vec{J}^{2}$ has
the property to be a quadratic element that commutes with all of
$\mathcal{U}(i\mathfrak{su}(2))$, it is called the Casimir operator of
$\mathcal{U}(i\mathfrak{su}(2))$.

##### Wigner matrix.

The action of the generators of $SU(2)$ over $\mathcal{Q}_{j}$ defines a
linear action of the group $SU(2)$ by exponentiation (see appendix A). The
Wigner matrix $D^{j}(g)$, is the matrix that represents the action of $g\in
SU(2)$ in the $\ket{j,m}$ basis. It is thus a square matrix of size $2j+1$. By
definition, the coefficients of the Wigner matrix are the functions

$D^{j}_{mn}(g)\overset{\rm def}{=}\matrixelement{j,m}{g}{j,n}.$ (2.6)

> $\bigstar$ Nota Bene. One should be aware of a small ambiguity in the
> notation ‘$\matrixelement{j,m}{g}{j,n}$’ that arises when $g$ is a matrix
> that belongs simultaneously to $SU(2)$ and to $\mathfrak{su}(2)$. Then it
> should be said explicitly if one considers the group action or the algebra
> action when computing $\matrixelement{j,m}{g}{j,n}$, because it gives a
> different result. This ambiguity comes from the fact that physicists do not
> usually write explicitly if they consider the group representation $\rho$,
> or its differential $D\rho$. Mathematicians would write
> $\matrixelement{j,m}{\rho(g)}{j,n}$, or
> $\matrixelement{j,m}{D\rho(g)}{j,n}$. From equation 1.37, if $g=e^{a}\in
> SU(2)\cap\mathfrak{su}(2)$, with $a\in\mathfrak{su}(2)$, then we know from
> A, that $\rho(g)=e^{D\rho(a)}$, but $\rho(g)\neq{D\rho(e^{a})}=D\rho(g)$. In
> the definition of the Wigner matrix above, it is the group action which is
> considered.

The functions $D^{j}_{mn}$ form an orthogonal family of $L^{2}(SU(2))$:

$\int_{SU(2)}dg\
\overline{D^{j^{\prime}}_{m^{\prime}n^{\prime}}(g)}D^{j}_{mn}(g)=\frac{1}{2j+1}\delta_{jj^{\prime}}\delta_{mm^{\prime}}\delta_{nn^{\prime}}.$
(2.7)

> $\blacktriangleright$ Proof. The left hand side (LHS) is the coefficient
> $\matrixelement{j^{\prime}n^{\prime}}{A}{jn}$ of the operator
> $A:\mathcal{Q}_{j}\to\mathcal{Q}_{j^{\prime}}$ defined by
>
> $A\overset{\rm def}{=}\int du\
> u^{\dagger}\outerproduct{j^{\prime}m^{\prime}}{jm}u.$ (2.8)
>
> We first show that $A$ is an intertwiner. If $j\neq j^{\prime}$, then, by
> Schur’s lemmas (cf. section A), $A=0$. Otherwise, $j^{\prime}=j$, and $A$ is
> bijective, and there exists $\lambda\in\mathbb{C}$ so that
> $A=\lambda\mathbb{1}$. Taking the trace on both sides, we see that
> $\lambda=\frac{\delta_{mm^{\prime}}}{2j+1}$. $\Box$

In fact, the Peter-Weyl theorem (see A) even asserts that the functions
$D^{j}_{mn}$ form a basis of $L^{2}(SU(2))$: any function $f\in L^{2}(SU(2))$
can be written

$f(g)=\sum_{j\in\mathbb{N}/2}\sum_{m=-j}^{j}\sum_{n=-j}^{j}f^{j}_{mn}D^{j}_{mn}(g),$
(2.9)

with coefficients $f^{j}_{mn}\in\mathbb{C}$. It implies notably an equivalence
between the following Hilbert spaces

$L^{2}(SU(2))\cong\bigoplus_{j\in\mathbb{N}/2}(\mathcal{Q}_{j}\otimes\mathcal{Q}_{j}^{*}).$
(2.10)

We are going to derive explicit expressions for computing $D^{j}_{mn}(g)$, but
we first need to introduce another realisation of the spin-$j$ irreps.

###  2.3 Homogeneous realisation

Let $\mathbb{C}_{2j}[z_{0},z_{1}]$ be the vector space of polynomials of two
complex variables, homogeneous of degree $2j\in\mathbb{N}$. If
$P(z_{0},z_{1})\in\mathbb{C}_{2j}[z_{0},z_{1}]$, it can be written as

$P(z_{0},z_{1})=\sum_{k=0}^{2j}a_{k}z_{0}^{k}z_{1}^{2j-k},$ (2.11)

with coefficients $a_{0},...,a_{2j}\in\mathbb{C}$. The action of $SU(2)$ given
by

$g\cdot P(\mathbf{z})=P(g^{T}\mathbf{z})$ (2.12)

defines a $(2j+1)$-dimensional group representation.

> $\blacktriangleright$ Proof. The action satisfies
>
> $\displaystyle(g_{1}g_{2})\cdot P=g_{1}\cdot(g_{2}\cdot P)$
> $\displaystyle{\rm and}$ $\displaystyle e\cdot P=P,$
>
> which defines a group action over the vector space
> $\mathbb{C}_{2j}[z_{0},z_{1}]$. $\Box$

> $\bigstar$ Nota Bene. We could also have defined the action by
> $P(a^{-1}\boldsymbol{z})$, $P(a^{\dagger}\boldsymbol{z})$ or
> $P(\boldsymbol{z}a)$. In fact $P(\boldsymbol{z}a)=P(a^{T}\boldsymbol{z})$,
> defines the same action. The convention that we have chosen here is the one
> of Rovelli ([RV14] p. 173). Moreover our convention is consistent with the
> choice we have made later for the representations of the principal series
> (cf. 4.3.1). In Bernard ([BLR12] p. 128) the convention
> $P(a^{-1}\boldsymbol{z})$ is used.

This group action induces the following action of the generators555See the
footnote of page 4 for the notation $\cong$.

$J_{+}\cong z_{0}\frac{\partial}{\partial z_{1}}\qquad J_{-}\cong
z_{1}\frac{\partial}{\partial z_{0}}\qquad
J_{3}\cong\frac{1}{2}\left(z_{0}\frac{\partial}{\partial
z_{0}}-z_{1}\frac{\partial}{\partial z_{1}}\right).$ (2.13)

> $\blacktriangleright$ Proof. Let $g(t)=e^{tM}$. The differential of the
> representation is given by (see A):
>
> $\displaystyle M\cdot P(z_{0},z_{1})$ $\displaystyle=\frac{{\rm d}}{{\rm
> d}t}\left(g(t)\cdot P(z_{0},z_{1})\right)\big{|}_{t=0}$
> $\displaystyle=\frac{{\rm d}}{{\rm
> d}t}\left(P(g_{11}(t)z_{0}+g_{21}(t)z_{1},g_{12}(t)z_{0}+g_{22}(t)z_{1})\right)\big{|}_{t=0}$
> $\displaystyle=(M_{11}z_{0}+M_{21}z_{1})\frac{\partial P}{\partial
> z_{0}}(z_{0},z_{1})+(M_{12}z_{0}+M_{22}z_{1})\frac{\partial P}{\partial
> z_{1}}(z_{0},z_{1}).$
>
> Then it suffices to apply to $J_{3},J_{+},J_{-}$. $\Box$

This representation is equivalent to the spin-$j$ irrep through the
correspondence:

$\ket{j,m}\cong\left(\frac{(2j)!}{(j+m)!(j-m)!}\right)^{1/2}z_{0}^{j+m}z_{1}^{j-m},$
(2.14)

The RHS is sometimes denoted with Dirac notations
$\innerproduct{z_{0}z_{1}}{jm}$.

> $\blacktriangleright$ Proof. One looks for an intertwiner
> $\Phi:\mathcal{Q}_{j}\to\mathbb{C}_{2j}[z_{0},z_{1}]$, such that for all
> $i$, $J_{i}\cdot\Phi(\ket{j,m})=\Phi(J_{i}\ket{j,m})$. We show easily the
> necessary condition:
>
>
> $\Phi(\ket{j,m})=c\left(\frac{(2j)!}{(j+m)!(j-m)!}\right)^{1/2}z_{0}^{j+m}z_{1}^{j-m},$
> (2.15)
>
> We choose $c=1$. $\Box$

The homogeneous realisation is very convenient to derive an explicit
expression for the Wigner matrix coefficients, as we show now.

##### Wigner matrix formula.

$D^{j}_{mn}(g)=\left(\frac{(j+m)!(j-m)!}{(j+n)!(j-n)!}\right)^{1/2}\sum_{k}\binom{j+n}{k}\binom{j-n}{j+m-k}g_{11}^{k}g_{21}^{j+n-k}g_{12}^{j+m-k}g_{22}^{k-m-n}.$
(2.16)

The sum is done over the integers $k\in\\{\max(0,m+n),...,\min(j+m,j+n)\\}$.

> $\blacktriangleright$ Proof. First of all remark that the action 2.12 is
> well defined for any $g=\begin{pmatrix}g_{11}&g_{12}\\\
> g_{21}&g_{22}\end{pmatrix}\in GL_{2}(\mathbb{C})$. Explicitly, it acts over
> the canonical basis of $\mathbb{C}_{2j}[z_{0},z_{1}]$ like
>
> $\displaystyle g\cdot z_{0}^{k}z_{1}^{2j-k}$
> $\displaystyle=P\left(\begin{pmatrix}g_{11}&g_{21}\\\
> g_{12}&g_{22}\end{pmatrix}\binom{z_{0}}{z_{1}}\right)=P\left(\binom{g_{11}z_{0}+g_{21}z_{1}}{g_{12}z_{0}+g_{22}z_{1}}\right)$
> $\displaystyle=(g_{11}z_{0}+g_{21}z_{1})^{k}(g_{12}z_{0}+g_{22}z_{1})^{2j-k}$
> $\displaystyle=\left(\sum_{i=0}^{k}\binom{k}{i}(g_{11}z_{0})^{i}(g_{21}z_{1})^{k-i}\right)\left(\sum_{l=0}^{2j-k}\binom{2j-k}{l}(g_{12}z_{0})^{l}(g_{22}z_{1})^{2j-k-l}\right)$
> [binomial theorem]
> $\displaystyle=\sum_{i=0}^{k}\sum_{l=0}^{2j-k}\binom{k}{i}\binom{2j-k}{l}g_{11}^{i}g_{21}^{k-i}g_{12}^{l}g_{22}^{2j-k-l}z_{0}^{i+l}z_{1}^{2j-i-l}$
> $\displaystyle=\sum_{i=0}^{k}\sum_{n=i}^{2j-k+i}\left(\binom{k}{i}\binom{2j-k}{n-i}g_{11}^{i}g_{21}^{k-i}g_{12}^{n-i}g_{22}^{2j-k-n+i}\right)z_{0}^{n}z_{1}^{2j-n}$
> $\displaystyle[n:=i+l]$
> $\displaystyle=\sum_{n=0}^{2j}\left(\sum_{i=\max(0,n+k-2j)}^{\min(k,n)}\binom{k}{i}\binom{2j-k}{n-i}g_{11}^{i}g_{21}^{k-i}g_{12}^{n-i}g_{22}^{2j-k-n+i}\right)z_{0}^{n}z_{1}^{2j-n}$
>
> Up to there, nothing but simple computation. Now, a subtle change of
> variables:
>
> $m:=n-j\qquad q:=k-j.$ (2.17)
>
> (The subtlety is that $j$ is a spin, and we generalise the notation $\sum$
> for half-integers bounds with still a step of $1$. A general polynomial is
> written
>
> $P(\mathbf{z})=\sum_{m=-j}^{j}a_{j+m}z_{0}^{j+m}z_{1}^{j-m}.$ (2.18)
>
> The action of $GL_{2}(\mathbb{C})$ is:
>
> $\displaystyle g\cdot z_{0}^{j+q}z_{1}^{j-q}$
> $\displaystyle=\sum_{m=-j}^{j}\left(\sum_{i=\max(0,m+q)}^{\min(j+m,j+q)}\binom{q+j}{i}\binom{j-q}{m+j-i}g_{11}^{i}g_{21}^{q+j-i}g_{12}^{m+j-i}g_{22}^{-q-m+i}\right)z_{0}^{j+m}z_{1}^{j-m}.$
>
> $\Box$

This explicit formula is useful to show some symmetry properties like

$\overline{D^{j}_{mn}(u)}=(-1)^{m-n}D^{j}_{-m,-n}(u).$ (2.19)

> $\varheart$ Physics. This formula is much used for numerical computations
> with spin-networks and spin-foams, like in [BDF+10].

##### Euler angles expression.

Wigner proposed also another explicit expression for its matrix, in terms of
the so-called Euler angles. If $u\in SU(2)$, and
$\alpha,\beta,\gamma\in\mathbb{R}^{3}$ are the Euler angles of $u$, such that
$u=e^{-\frac{i\alpha}{2}\sigma_{3}}e^{-\frac{i\beta}{2}\sigma_{2}}e^{-\frac{i\gamma}{2}\sigma_{3}}$,
then

$\displaystyle D^{j}_{m^{\prime}m}(u)=e^{-i(\alpha m^{\prime}+\gamma
m)}d^{j}_{m^{\prime}m}(\beta)$

with the reduced Wigner matrix

$d^{j}_{m^{\prime}m}(\beta)=\left(\frac{(j+m^{\prime})!(j-m^{\prime})!}{(j+m)!(j-m)!}\right)^{\frac{1}{2}}\sum_{k=\max(0,m^{\prime}+m)}^{\min(j+m^{\prime},j+m)}(-1)^{m^{\prime}+j-k}\binom{j+m}{k}\binom{j-m}{j-k+m^{\prime}}\\\
\times\left(\cos\frac{\beta}{2}\right)^{2k-m-m^{\prime}}\left(\sin\frac{\beta}{2}\right)^{m+m^{\prime}+2j-2k}.$
(2.20)

> $\blacktriangleright$ Proof. The proof is very easy once given formula 2.16.
> It only consists in applying it to the matrix
>
> $\displaystyle
> e^{-\frac{i\alpha}{2}\sigma_{3}}e^{-\frac{i\beta}{2}\sigma_{2}}e^{-\frac{i\gamma}{2}\sigma_{3}}=\begin{pmatrix}e^{-\frac{i(\alpha+\gamma)}{2}}\cos\beta/2&-e^{\frac{i(\gamma-\alpha)}{2}}\sin\beta/2\\\
> e^{-\frac{i(\gamma-\alpha)}{2}}\sin\beta/2&e^{\frac{i(\alpha+\gamma)}{2}}\cos\beta/2\end{pmatrix}.$
>
> This is what Rühl does ([Rüh70] p. 43), except that in his convention, the
> Euler angles are defined with no minus sign in front, so
> $\alpha\mapsto-\alpha$, $\beta\mapsto-\beta$, $\gamma\mapsto-\gamma$, and so
> $d^{j}_{m^{\prime}m}(\beta)|_{\text{R\"{u}hl}}=d^{j}_{m^{\prime}m}(-\beta)$,
> which finally also equals the RHO of 2.20 provided the coefficient
> $(-1)^{m^{\prime}+j-k}$ is changed into $(-1)^{m+j-k}$. It is also the
> convention chosen by the Wolfram Language to define its function WignerD,
> hence:
>
> $\texttt{WignerD}[\\{j,m,n\\},\alpha,\beta,\gamma]=e^{i(\alpha m+\gamma
> n)}d^{j}_{mn}(-\beta).$ (2.21)
>
> A proof that does not presuppose formula 2.16 can be found in Sakurai
> ([SN11] p. 236-238), which has the same convention as ours for Euler angles.
> It uses the Schwinger’s oscillator model for angular momentum. It obtains
> the same formula as ours, but written in a slightly different way, changing
> the index of summation $k\rightarrow j+m-k$. Finally, Varshalovich ([VMK87]
> p. 76), which also has the same convention, gives a series of variations in
> the way of writing the above formula. $\Box$

###  2.4 Projective realisation

The spin-$j$ irrep can be realised over $\mathbb{C}_{2j}[z]$, the vector space
of complex polynomials of one variable $z$ of degree at most $2j$. This
realisation is obtained from the $\mathbb{C}_{2j}[z_{0},z_{1}]$ realisation by
the map:

$\left\\{\begin{array}[]{ll}\mathbb{C}_{2j}[z_{0},z_{1}]&\to\mathbb{C}_{2j}[z]\\\
P(z_{0},z_{1})&\mapsto P(z,1)\end{array}\right.$ (2.22)

This map is constructed from a projection from $\mathbb{C}^{2}$ to
$\mathbb{C}$, hence the name ‘projective’ that we give to this realisation.
Sometimes it is also named the ‘holomorphic’ realisation. From this we deduce
the action of $SU(2)$

$a\cdot
f(z)=(a_{12}z+a_{22})^{2j}f\left(\frac{a_{11}z+a_{21}}{a_{12}z+a_{22}}\right),$
(2.23)

and of the algebra $\mathfrak{su}(2)$

$J_{+}\cong-z^{2}\frac{d}{dz}+2jz\quad J_{3}\cong z\frac{d}{dz}-j\quad
J_{-}\cong\frac{d}{dz}.$ (2.24)

The canonical basis becomes

$\ket{j,m}\cong\sqrt{\frac{(2j)!}{(j+m)!(j-m)!}}z^{j+m}.$ (2.25)

We can give the following explicit expression for the scalar product that
makes the canonical basis orthonormal:

$\innerproduct{f}{g}\overset{\rm
def}{=}\frac{i}{2}\frac{2j+1}{\pi}\int_{\mathbb{C}}\overline{f(z)}g(z)\frac{dzd\overline{z}}{(1+|z|^{2})^{2j+2}}.$
(2.26)

###  2.5 Spinorial realisation

The following realisation of the spin-$j$ irreps relies on notations which
have been much developed by Penrose [PR84]. It was found useful for twistor
theory [PR86], and later in quantum gravity for the so-called twisted
geometries [FS10, LS16].

##### Abstract indices.

We are going to use the clever conventions of abstract indices of Penrose
([PR84] pp. 68-115). To start with we need a set of ‘abstract indices’
$\mathcal{L}$, that is to say a countable set of symbols. We use for instance
capital letters:

$\mathcal{L}\overset{\rm def}{=}\\{A,B,...,Z,A_{0},...,Z_{0},A_{1},...\\}.$
(2.27)

Then we denote $\mathfrak{S}^{\bullet}\overset{\rm def}{=}\mathbb{C}^{2}$, and
for any abstract index $A\in\mathcal{L}$, $\mathfrak{S}^{A}\overset{\rm
def}{=}\mathfrak{S}^{\bullet}\times\\{A\\}$. Obviously $\mathfrak{S}^{A}$ is
isomorphic to $\mathbb{C}^{2}$ as a complex vector space. An element of
$\mathfrak{S}^{A}$ will be typically denoted $z^{A}=(z,A)\in\mathfrak{S}^{A}$.
The abstract index $A$ serves as a marker to ‘type’ the vector
$z\in\mathbb{C}^{2}$ (thus $z^{A}\neq z^{B}$). This notation is very efficient
to deal with several copies of the same space (here $\mathbb{C}^{2}$), like in
tensor theory.

The vector space of linear forms from $\mathbb{C}^{2}$ to $\mathbb{C}$, is
called the dual space, and denoted $\mathfrak{S}_{\bullet}$. Similarly, we
denote $\mathfrak{S}_{A}\overset{\rm
def}{=}\mathfrak{S}_{\bullet}\times\\{A\\}$, which is trivially isomorphic to
the dual space of $\mathfrak{S}^{A}$. Its elements, called covectors, are
denoted with an abstract lower capital index, $z_{A}$. Then the evaluation of
a covector $y_{A}=(y,A)$ on a vector $z^{A}=(z,A)$ (called a ‘contraction’ or
a ‘scalar product’) is denoted $y_{A}z^{A}=y(z)\in\mathbb{C}$ (the order does
not matter $y_{A}z^{A}=z^{A}y_{A}$).

##### Spinors.

Consider the space of formal (commutative and associative) finite sums of
formal (commutative and associative) products of elements, one from each
$\mathfrak{S}^{A_{1}},...,\mathfrak{S}^{A_{p}},\mathfrak{S}_{B_{1}},...,\mathfrak{S}_{B_{q}}$.
A typical element can be written:

$t^{A_{1}...A_{p}}_{\hskip 30.0ptB_{1}...B_{q}}=\sum_{i=1}^{m}z_{1,i}^{\hskip
10.0ptA_{1}}\,...\,z_{p,i}^{\hskip 10.0ptA_{p}}\ y^{1,i}_{\hskip
10.0ptB_{1}}\,...\,y^{q,i}_{\hskip 10.0ptB_{q}}.$ (2.28)

Then impose the rules

  1. 1.

(Homogeneity) $\forall\alpha\in\mathbb{C},\quad(\alpha z_{1}^{\
A_{1}})\,z_{2}^{\ A_{2}}\,...\,z_{p}^{\ A_{p}}=z_{1}^{\ A_{1}}\,(\alpha
z_{2}^{\ A_{2}})\,...\,z_{p}^{\ A_{p}}$

  2. 2.

(Distributivity) $(z_{1}^{\ A_{1}}+z_{2}^{\
A_{2}})\,z_{3}^{A_{3}}\,...\,z_{p}^{\ A_{p}}=z_{1}^{\ A_{1}}\,z_{3}^{\
A_{3}}\,...\,z_{p}^{\ A_{p}}+z_{2}^{\ A_{2}}\,z_{3}^{\ A_{3}}\,...\,z_{p}^{\
A_{p}}$.

The resulting space is a vector space denoted
$\mathfrak{S}^{A_{1}...A_{p}}_{B_{1}...B_{q}}$. Its elements are called
spinors of type $(p,q)$, and its dimension is $2^{p+q}$.

> $\bigstar$ Nota Bene. The spinor space
> $\mathfrak{S}^{A_{1}...A_{p}}_{B_{1}...B_{q}}$ is isomorphic, but not equal,
> to
> $\mathfrak{S}^{A_{1}}\otimes...\otimes\mathfrak{S}^{A_{p}}\otimes\mathfrak{S}_{B_{1}}\otimes...\otimes\mathfrak{S}_{B_{q}}$.
> The difference is the commutativity of the product. For instance the formal
> product of $\mathfrak{S}^{AB}$ is commutative (by assumption) in the sense
> that, for $z^{A}\in\mathfrak{S}^{A}$ and $y^{B}\in\mathfrak{S}^{B}$,
> $z^{A}y^{B}=y^{B}z^{A}$, whereas the tensor product is not, $z^{A}\otimes
> y^{B}\neq y^{B}\otimes z^{A}$, simply because $z^{A}\otimes
> y^{B}\in\mathfrak{S}^{A}\otimes\mathfrak{S}^{B}$ and $y^{B}\otimes
> z^{A}\in\mathfrak{S}^{B}\otimes\mathfrak{S}^{A}$ do not belong to the same
> set. Heuristically it can be said that the abstract indices keep track of
> the position in the tensor product.

The spinor space is endowed with a bunch a basic operations defined by a set
of rules. It would be utterly unpedagogical to state these rules in the most
general case. On the contrary they are very intuitive for simple examples, and
generalise without ambiguities for higher order spinors.

  1. 1.

(Index substitution) If $z^{A}=(z,A)\in\mathfrak{S}^{A}$, we denote
$z^{B}=(z,B)\in\mathfrak{S}^{B}$. Thus $z^{A}\neq z^{B}$.

  2. 2.

(Index permutation) If
$t^{AB}=\sum_{i}z_{i}^{A}y_{i}^{B}\in\mathfrak{S}^{AB}$, we denote
$t^{BA}=\sum_{i}z_{i}^{B}y_{i}^{A}\in\mathfrak{S}^{AB}$.

  3. 3.

(Symmetrisation) $t^{(AB)}\overset{\rm def}{=}\frac{1}{2}(t^{AB}+t^{BA})$ or
generally $z^{(A_{1}...A_{n})}\overset{\rm def}{=}\frac{1}{n!}\sum_{\sigma\in
S_{n}}z^{A_{\sigma(1)}...A_{\sigma(n)}}$.

  4. 4.

(Anti-symmetrisation) $t^{[AB]}\overset{\rm def}{=}\frac{1}{2}(t^{AB}-t^{BA})$
or generally $z^{[A_{1}...A_{n}]}\overset{\rm
def}{=}\frac{1}{n!}\sum_{\sigma\in
S_{n}}\epsilon_{\sigma}z^{A_{\sigma(1)}...A_{\sigma(n)}}$, with
$\epsilon_{\sigma}$ the signature of the permutation $\sigma$.

  5. 5.

(Contraction) If $\sum_{i}z_{i}^{A}y^{i}_{B}=t^{A}_{\ B}$, then $t^{A}_{\
A}=\sum_{i}z_{i}^{A}y^{i}_{A}\in\mathbb{C}$.

##### Index dualisation

We denote the canonical basis of $\mathbb{C}^{2}$:

$e_{0}\overset{\rm def}{=}\binom{1}{0}\hskip 85.35826pte_{1}\overset{\rm
def}{=}\binom{0}{1}.$ (2.29)

It is easy to show that there exists a unique normalised skew-symmetric spinor
of type $(0,2)$. It is denoted $\epsilon_{AB}$, and satisfies by definition:

$\epsilon_{AB}=-\epsilon_{BA},\hskip
85.35826pt\epsilon_{AB}e_{0}^{A}e_{1}^{B}=1.$ (2.30)

> $\bigstar$ Nota Bene. $\epsilon_{AB}$ corresponds over $\mathbb{C}^{2}$ to
> the unique $2$-form $\epsilon$ normalised by the condition
> $\epsilon(e_{0},e_{1})=1$, which is nothing but the determinant over
> $\mathbb{C}^{2}$.

For two vectors $z=(z_{0},z_{1})$ and $y=(y_{0},y_{1})$, we show easily that:

$\epsilon_{AB}z^{A}y^{B}=z_{0}y_{1}-z_{1}y_{0}.$ (2.31)

Interestingly, $\epsilon_{AB}$ defines a canonical mapping between
$\mathfrak{S}^{A}$ and $\mathfrak{S}_{A}$, given by

$z^{A}\mapsto z_{A}=z^{B}\epsilon_{BA}.$ (2.32)

It is called index dualisation.

> $\bigstar$ Nota Bene. Actually the mapping could also have been defined by
> $z^{A}\mapsto z_{A}=z^{B}\epsilon_{AB}$. It gives the same mapping ‘up to a
> sign’, since $\epsilon_{AB}$ is skew-symmetric. This other convention is
> chosen by Rovelli ([RV14] p. 23). Our convention is the one of Penrose
> ([PR84] p. 104).

The covectors of the dual space $\mathfrak{S}_{\bullet}$ can also be described
by a pair of components in the dual basis. The index dualisation can then be
expressed in components as $((z_{0},z_{1}),A)\mapsto((-z_{1},z_{0}),A)$.
Similarly to the usual Dirac notation $\ket{z}=(z_{0},z_{1})$, a notation is
sometimes introduced for the dual $[z|=(-z_{1},z_{0})$. With this choice, the
RHS of 2.31 reads $[z|y\rangle$.

##### Conjugation.

The conjugation of $z\in\mathbb{C}$ is denoted $\bar{z}$ or $z^{*}$. We define
the conjugation over $\mathfrak{S}^{A}$ by
$\overline{z^{A}}=\overline{(z,A)}\overset{\rm
def}{=}(\overline{z},\dot{A})=\bar{z}^{\dot{A}}\in\mathfrak{S}^{\dot{A}}$.
Thus we have introduced a new set of abstract indices, the dotted indices:

$\dot{\mathcal{L}}\overset{\rm
def}{=}\\{\dot{A},\dot{B},...,\dot{Z},\dot{A}_{0},...,\dot{Z}_{0},\dot{A}_{1},...\\}.$
(2.33)

We impose moreover that $\overline{\bar{z}^{\dot{A}}}=z^{A}$, i.e.
$\ddot{A}=A$, so that the conjugation is an involution. Importantly, we regard
the set $\mathcal{L}$ and $\dot{\mathcal{L}}$ as incompatible classes of
abstract indices, meaning that we forbid index substitution between them two.
In other words, dotted and undotted indices commute: for any
$t^{A\dot{B}}\in\mathfrak{S}^{A\dot{B}}$, we have $t^{A\dot{B}}=t^{\dot{B}A}$.

> $\bigstar$ Nota Bene. One way to formalise this ‘incompatibility’ between
> dotted and undotted indices would be to define rather $z^{A}=(z,A,0)$ and
> $z^{\dot{A}}=(z,A,1)$. Thus the index substitution $z^{A}=(z,A,0)\mapsto
> z^{A}=(z,B,0)=z^{B}$ clearly does not enable to translate from a dotted to
> an undotted index. Only the complex conjugation can through
> $z^{A}=(z,A,0)\mapsto(\overline{z},A,1)=z^{\dot{A}}$.

##### Inner product.

We define the map $J$ by:

$J\binom{z_{0}}{z_{1}}=\binom{-\overline{z_{1}}}{\overline{z_{0}}}.$ (2.34)

Using the previously introduced generalised Dirac notation, we read
$J\ket{z}=|z]$. Since $J^{2}=-1$, the map $J$ behaves over $\mathbb{C}^{2}$
very much as the imaginary number $i$ behaves over $\mathbb{C}$. For this
reason the map $J$ is said to define a complex structure over
$\mathbb{C}^{2}$. A combination of $\epsilon_{AB}$ and $J$ defines an inner
product over $\mathfrak{S}^{A}$:

$-\epsilon_{AB}(Jz)^{A}y^{B}=\overline{z_{0}}y_{0}+\overline{z_{1}}y_{1}.$
(2.35)

In generalised Dirac notations, we read $-[Jz|y\rangle=\innerproduct{z}{y}$,
which is consistent with the usual Dirac notation for the scalar product.

> $\bigstar$ Nota Bene. With the matrix action over spinors, defined just
> below, equation 2.36, we can see that the inner product is invariant under
> the action of $SU(2)$: $\innerproduct{u\cdot z}{u\cdot
> y}=\innerproduct{z}{y}$. There is no surprise since it is actually one way
> of defining $SU(2)$. However the inner product is not invariant under
> $SL_{2}(\mathbb{C})$ (contrary to the determinant). So, to ‘another choice
> of $SU(2)$’, in the sense of a stabilizer of a time direction (see 1.3),
> would correspond another invariant inner product, and thus another complex
> structure $J$. For instance, in [LST12], they choose rather $(-J)$ for the
> complex structure. The choice we have made here is the one of Rovelli
> ([RV14] p. 24).

##### Representation.

The vector space $\mathcal{M}_{2}(\mathbb{C})$ is isomorphic to
$\mathfrak{S}^{A}_{B}$, through the isomorphism that associates to any
$t\in\mathcal{M}_{2}(\mathbb{C})$ the unique spinor $t^{A}_{\ \ B}$ such that:

$\forall z\in\mathbb{C}^{2},\quad(tz)^{A}=t^{A}_{\ \ B}\,z^{B}.$ (2.36)

Then the groups $SL_{2}(\mathbb{C})$ and $SU(2)$ can be represented over
$\mathfrak{S}^{A_{1}...A_{p}}$ such as:

$u\cdot z^{A_{1}...A_{p}}=u^{A_{1}}_{\ \ B_{1}}...u^{A_{p}}_{\ \
B_{p}}\,z^{B_{1}...B_{p}}.$ (2.37)

Yet this representation is not irreducible, since it is stable over the
subspace of completely symmetric spinors $\mathfrak{S}^{(A_{1}...A_{p})}$.

> $\blacktriangleright$ Proof. Let’s see that the action is stable over this
> subspace. Suppose $z^{A_{1}...A_{p}}$ is completely symmetric, i.e.
> $z^{(A_{1}...A_{p})}=z^{A_{1}...A_{p}}$. Then (sketching the proof, the
> details are left to the reader):
>
> $\displaystyle u\cdot z^{(A_{1}...A_{p})}$ $\displaystyle=u^{(A_{1}}_{\ \
> B_{1}}...u^{A_{p})}_{\ \ B_{p}}\,z^{B_{1}...B_{p}}$
> $\displaystyle=u^{A_{1}}_{\ \ (B_{1}}...u^{A_{p}}_{\ \
> B_{p})}\,z^{B_{1}...B_{p}}$ $\displaystyle=u^{A_{1}}_{\ \
> B_{1}}...u^{A_{p}}_{\ \ B_{p}}\,z^{(B_{1}...B_{p})}$
> $\displaystyle=u^{A_{1}}_{\ \ B_{1}}...u^{A_{p}}_{\ \
> B_{p}}\,z^{B_{1}...B_{p}}$ $\displaystyle=u\cdot z^{A_{1}...A_{p}}$
>
> thus $u\cdot z^{A_{1}...A_{p}}$ is also completely symmetric. $\Box$

Thus $SL_{2}(\mathbb{C})$ and $SU(2)$ can be represented of the vector space
$\mathfrak{S}^{(A_{1}...A_{p})}$ of dimension $p+1$. A basis is given by

$\\{e^{(A_{1}}_{i_{1}}...e^{A_{p})}_{i_{p}}\mid
i_{1},...,i_{p}\in\\{0,1\\}\\}=\\{e^{(A_{1}}_{0}...e^{A_{m}}_{0}e^{A_{m+1}}_{1}...e^{A_{p})}_{1}\mid
m\in\\{0,...,p\\}\\}.$ (2.38)

This representation is irreducible and equivalent to the spin $p/2$
representation through the intertwiner:

$e^{(A_{1}}_{0}...e^{A_{m}}_{0}e^{A_{m+1}}_{1}...e^{A_{p})}_{1}\cong
z_{0}^{m}z_{1}^{p-m}.$ (2.39)

> $\blacktriangleright$ Proof. To see this, it is simpler to write the spinors
> in the canonical basis of $\mathfrak{S}^{A_{1}...A_{p}}$:
>
>
> $\xi^{A_{1}...A_{p}}=\sum_{i_{1},i_{2},...,i_{p}=0}^{1}c^{i_{1},...,i_{p}}\,e^{A_{1}}_{i_{1}}...e^{A_{p}}_{i_{p}}$
> (2.40)
>
> The total symmetry of $\xi^{A_{1}...A_{p}}$ imposes a total symmetry of the
> coefficients $c^{i_{1},...,i_{p}}$. Then we define the following bijection
> between $\mathfrak{S}^{(A_{1}...A_{p})}$ and $\mathbb{C}_{p}[z_{0},z_{1}]$:
>
>
> $\xi^{A_{1}...A_{p}}\cong\sum_{p_{1},...p_{n}=0}^{1}c^{p_{1}...p_{n}}z_{p_{1}}...z_{p_{2}}$
> (2.41)
>
> This defines an intertwiner as can be checked by looking at the action of a
> group element. $\Box$

##  Chapter 3 Recoupling theory of $SU(2)$

The $SU(2)$ irreps provide the fundamental building blocks of quantum space-
time. From a mathematical perspective, irreps are the fundamental bricks from
which other representations are built. Indeed any finite representation of
$SU(2)$ is completely reducible (i.e. it can be written as a direct sum of
irreps). In particular, a tensor product of irreps can be decomposed into a
direct sum of irreps: i.e. there exists a bijective intertwiner that maps the
tensor product to a direct sum of irreps. Such an intertwiner is sometimes
called a ‘coupling tensor’ (see Moussouris [Mou83] pp. 10-11). This naming
comes from quantum physics: when two systems couple (i.e. interact), the total
system is described by states of the tensor product of the Hilbert spaces of
the subsystems. Notice that there may exist several coupling tensors between a
tensor product and its corresponding sum of irreps. It is precisely the goal
of ‘recoupling theory’ to describe these coupling tensors and to understand
how one can translate from one decomposition to another.

###  3.1 Clebsch-Gordan coefficients

Given $\mathcal{Q}_{j_{1}}$ and $\mathcal{Q}_{j_{2}}$, two irreps of $SU(2)$,
the tensor representation is defined over
$\mathcal{Q}_{j_{1}}\otimes\mathcal{Q}_{j_{2}}$. The canonical basis of
$\mathcal{Q}_{j_{1}}\otimes\mathcal{Q}_{j_{2}}$ is given by the elements

$\ket{j_{1}m_{1};j_{2}m_{2}}\overset{\rm
def}{=}\ket{j_{1},m_{1}}\otimes\ket{j_{2},m_{2}}$ (3.1)

where $m_{1}$ and $m_{2}$ belong to the usual range of magnetic indices.
Interestingly, this basis is the unique orthonormal basis that diagonalises
simultaneously the commuting operators:

$J_{3}\otimes\mathbb{1},\quad\mathbb{1}\otimes
J_{3},\quad\vec{J}^{2}\otimes\mathbb{1},\quad\mathbb{1}\otimes\vec{J}^{2}.$
(3.2)

Another complete set of commuting operators is given by

$J_{3}\otimes\mathbb{1}+\mathbb{1}\otimes
J_{3},\quad\left(\vec{J}\otimes\mathbb{1}+\mathbb{1}\otimes\vec{J}\right)^{2},\quad\vec{J}^{2}\otimes\mathbb{1},\quad\mathbb{1}\otimes\vec{J}^{2}.$
(3.3)

Therefore, there exists an orthonormal basis that diagonalises them
simultaneously. It is given by

$\ket{j_{1}j_{2};k;n}\quad{\rm with}\
k\in\\{|j_{1}-j_{2}|,...,j_{1}+j_{2}\\}\quad{\rm and}\ n\in\\{-k,...,k\\},$
(3.4)

and characterised by the action of the operators:

$\displaystyle(J_{3}\otimes\mathbb{1}+\mathbb{1}\otimes
J_{3})\ket{j_{1}j_{2};k;n}=n\ket{j_{1}j_{2};k;n}$ (3.5)
$\displaystyle\left(\vec{J}\otimes\mathbb{1}+\mathbb{1}\otimes\vec{J}\right)^{2}\ket{j_{1}j_{2};k;n}=k(k+1)\ket{j_{1}j_{2};k;n}$
(3.6)
$\displaystyle\vec{J}^{2}\otimes\mathbb{1}\ket{j_{1}j_{2};k;n}=j_{1}(j_{1}+1)\ket{j_{1}j_{2};k;n}$
(3.7)
$\displaystyle\mathbb{1}\otimes\vec{J}^{2}\ket{j_{1}j_{2};k;n}=j_{2}(j_{2}+1)\ket{j_{1}j_{2};k;n}.$
(3.8)

A proof can be found in Sakurai ([SN11] pp. 217-231). This result proves that
$\mathcal{Q}_{j_{1}}\otimes\mathcal{Q}_{j_{2}}$ can be decomposed into a
direct sum of irreps, namely we have the following equivalence of
representations

$\mathcal{Q}_{j_{1}}\otimes\mathcal{Q}_{j_{2}}\cong\bigoplus_{k=|j_{1}-j_{2}|}^{j_{1}+j_{2}}\mathcal{Q}_{k}.$
(3.9)

The equivalence is given by the bijective intertwiner
$\iota:\mathcal{Q}_{j_{1}}\otimes\mathcal{Q}_{j_{2}}\to\bigoplus_{k=|j_{1}-j_{2}|}^{j_{1}+j_{2}}\mathcal{Q}_{k}$,
which satisfies

$\iota\ket{j_{1},j_{2};km}=\ket{km}.$ (3.10)

We define the Clebsch-Gordan coefficients by the scalar product

$C^{jm}_{j_{1}m_{1}j_{2}m_{2}}\overset{\rm
def}{=}\innerproduct{j_{1}m_{1}j_{2}m_{2}}{j_{1}j_{2};jm}.$ (3.11)

Said differently, we have

$\ket{j_{1}j_{2};j,m}=\sum_{m_{1}=-j_{1}}^{j_{1}}\sum_{m_{2}=-j_{2}}^{j_{2}}C^{jm}_{j_{1}m_{1}j_{2}m_{2}}\ket{j_{1}m_{1}j_{2}m_{2}}.$
(3.12)

The Clebsh-Gordan coefficients can be seen as the matrix coefficients of the
intertwiner in the canonical bases.

> $\varheart$ Physics. The Clebsh-Gordan coefficients appear largely in the
> quantum theory of angular momentum. Two spin-systems are described as one
> single spin-system with a larger total angular momentum.

##### Remarks

  1. 1.

Due to the Condon-Shortley convention for the $SU(2)$-action, we have
$C^{jm}_{j_{1}m_{1}j_{2}m_{2}}\in\mathbb{R}$.

  2. 2.

The coefficients $C^{jm}_{j_{1}m_{1}j_{2}m_{2}}$ are well-defined and non-
zero, only if the following Clebsch-Gordan inequality (aka triangle
inequality) is satisfied

$|j_{1}-j_{2}|\leq j\leq j_{1}+j_{2}.$ (3.13)

Otherwise, we choose by convention, that $C^{jm}_{j_{1}m_{1}j_{2}m_{2}}=0$.

  3. 3.

If $m\neq m_{1}+m_{1}$, then $C^{jm}_{j_{1}m_{1}j_{2}m_{2}}=0$.

  4. 4.

Since the $\ket{jm}$ form an orthonormal basis, we have the following
‘orthogonality relations’

$\sum_{m_{1}=-j_{1}}^{j_{1}}\sum_{m_{2}=-j_{2}}^{j_{2}}C^{jm}_{j_{1}m_{1}j_{2}m_{2}}C^{j^{\prime}m^{\prime}}_{j_{1}m_{1}j_{2}m_{2}}=\delta_{j,j^{\prime}}\delta_{m,m^{\prime}}.$
(3.14)

The Clebsch-Gordan coefficients are numbers, but their definition is quite
implicit. Hopefully, there are also explicit formulas to compute them!

##### Explicit formula

$C^{jm}_{j_{1}m_{1}j_{2}m_{2}}=\delta_{m,m_{1}+m_{2}}\sqrt{2j+1}\sqrt{\frac{(j+m)!(j-m)!(-j+j_{1}+j_{2})!(j-j_{1}+j_{2})!(j+j_{1}-j_{2})!}{(j+j_{1}+j_{2}+1)!(j_{1}+m_{1})!(j_{1}-m_{1})!(j_{2}+m_{2})!(j_{2}-m_{2})!}}\\\
\times\sum_{k}\frac{(-1)^{k+j_{2}+m_{2}}(j+j_{2}+m_{1}-k)!(j_{1}-m_{1}+k)!}{(j-j_{1}+j_{2}-k)!(j+m-k)!k!(k+j_{1}-j_{2}-m)!}$
(3.15)

> $\blacktriangleright$ Proof. Other similar expressions can be found in
> Varshalovich ([VMK87] p. 238), with references to various proofs. It can be
> notably convenient to recognise in the sum above the so-called hyper-
> geometrical function ${}_{3}F_{2}$:
>
>
> $C^{jm}_{j_{1}m_{1}j_{2}m_{2}}=\delta_{m,m_{1}+m_{2}}\sqrt{2j+1}\frac{\sqrt{(j+j_{1}-j_{2})!(j-j_{1}+j_{2})!}}{\sqrt{(-j+j_{1}+j_{2})!(j+j_{1}+j_{2}+1)!}}\frac{\sqrt{(j+m)!(j-m)!(j_{1}+m_{1})!(j_{2}-m_{2})!}}{\sqrt{(j_{1}-m_{1})!(j_{2}+m_{2})!}}\\\
> \times{}_{3}F_{2}(j-j_{1}-j_{2},m_{1}-j_{1},-j_{2}-m_{2};j-j_{2}+m_{1}+1,j+j_{1}-m_{2}+1;1).$
> (3.16)
>
> $\Box$

In the Wolfram Language, they are implemented as

$C^{j_{3}m_{3}}_{j_{1}m_{1}j_{2}m_{2}}=\texttt{ClebschGordan}[\\{j_{1},m_{1}\\},\\{j_{2},m_{2}\\},\\{j_{3},m_{3}\\}].$
(3.17)

.

##### Exercise.

Show that

$D^{j_{1}}_{m_{1}n_{1}}(g)D^{j_{2}}_{m_{2}n_{2}}(g)=\sum_{j\in\mathbb{N}/2}\sum_{m=-j}^{j}\sum_{m^{\prime}=-j}^{j}C^{jm}_{j_{1}m_{1}j_{2}m_{2}}C^{jm^{\prime}}_{j_{1}n_{1}j_{2}n_{2}}D^{j}_{mm^{\prime}}(g).$
(3.18)

> $\blacktriangleright$ Proof.
>
> $\displaystyle D^{j_{1}}_{m_{1}n_{1}}(g)D^{j_{2}}_{m_{2}n_{2}}(g)$
> $\displaystyle=\bra{j_{1}m_{1}}g\ket{j_{1}n_{1}}\bra{j_{2}m_{2}}g\ket{j_{2}n_{2}}$
> $\displaystyle=\bra{j_{1}m_{1};j_{2}m_{2}}g\ket{j_{1}n_{1};j_{2}n_{2}}$
> $\displaystyle=\sum_{j\in\mathbb{N}/2}\sum_{m=-j}^{j}\sum_{j^{\prime}\in\mathbb{N}/2}\sum_{m^{\prime}=-j^{\prime}}^{j^{\prime}}\innerproduct{j_{1}m_{1}j_{2}m_{2}}{jm}\bra{jm}g\ket{j^{\prime}m^{\prime}}\innerproduct{j^{\prime}m^{\prime}}{j_{1}n_{1}j_{2}n_{2}}$
> $\displaystyle=\sum_{j\in\mathbb{N}/2}\sum_{m=-j}^{j}\sum_{m^{\prime}=-j}^{j}C^{jm}_{j_{1}m_{1}j_{2}m_{2}}C^{jm^{\prime}}_{j_{1}n_{1}j_{2}n_{2}}D^{j}_{mm^{\prime}}(g)$
>
> $\Box$

###  3.2 Invariant subspace

A general tensor product of $n$ irreps can be decomposed into a direct sum

$\bigotimes_{i=1}^{n}\mathcal{Q}_{j_{i}}\cong\bigoplus_{k=0}^{J}\left(\underbrace{\mathcal{Q}_{k}\oplus...\oplus\mathcal{Q}_{k}}_{d_{k}\
{\rm times}}\right),$ (3.19)

where $J=\sum_{i}j_{i}$ and $d_{k}$ is the degeneracy of the irrep
$\mathcal{Q}_{k}$. Here, ‘decomposing’ means ‘finding a bijective intertwiner
between the two spaces’. Concretely, such a decomposition is obtained by
applying successively the decomposition of only two, given by 3.9. The
operator $\mathbb{1}\otimes...\otimes J_{i}\otimes...\otimes\mathbb{1}$
corresponding to $J_{i}$ acting on the $k^{th}$ Hilbert space of the product
is denoted $(J_{i})_{k}$. The three components
$(J_{1})_{k},(J_{2})_{k},(J_{3})_{k}$ form the vectorial operator
$\vec{J}_{k}$.

> $\varheart$ Physics. In quantum gravity, such tensor spaces appear in the
> kinematical Hilbert space $\mathcal{H}$. The description of the dynamics
> requires to impose constraints that select subspaces of $\mathcal{H}$. One
> important constraint is the Gauss constraint which reduces $\mathcal{H}$ to
> its $SU(2)$-invariant subspace ${\rm Inv}_{SU(2)}\mathcal{H}$ that we define
> below.

We define the $SU(2)$-invariant subspace as

${\rm
Inv}_{SU(2)}\left(\bigotimes_{i=1}^{n}\mathcal{Q}_{j_{i}}\right)\overset{\rm
def}{=}\left\\{\psi\in\bigotimes_{i=1}^{n}\mathcal{Q}_{j_{i}}\mid\forall g\in
SU(2),\ g\cdot\psi=\psi\right\\}.$ (3.20)

It can also be characterized rather by the action of the algebra:

${\rm
Inv}_{SU(2)}\left(\bigotimes_{i=1}^{n}\mathcal{Q}_{j_{i}}\right)=\left\\{\psi\in\bigotimes_{i=1}^{n}\mathcal{Q}_{j_{i}}\mid\forall
s\in\mathfrak{su}(2),\ s\cdot\psi=0\right\\}.$ (3.21)

From this, it is easy to see that

${\rm
Inv}_{SU(2)}\left(\bigotimes_{i=1}^{n}\mathcal{Q}_{j_{i}}\right)\cong\underbrace{\mathcal{Q}_{0}\oplus...\oplus\mathcal{Q}_{0}}_{d_{0}\
{\rm times}},$ (3.22)

where $\mathcal{Q}_{0}\cong\mathbb{C}$ is the trivial representation.
Interestingly, we also have the following isomorphism:

${\rm
Inv}_{SU(2)}\left(\bigotimes_{i=1}^{n}\mathcal{Q}_{j_{i}}\right)\cong{\rm
Hom}_{SU(2)}\left(\bigotimes_{i=1}^{n}\mathcal{Q}_{j_{i}},\mathcal{Q}_{0}\right),$
(3.23)

where the RHS is the vector space of $SU(2)$-intertwiners between
$\bigotimes_{i=1}^{n}\mathcal{Q}_{j_{i}}$ and $\mathcal{Q}_{0}$.

> $\blacktriangleright$ Proof. Let
> $T:\bigotimes_{k=1}^{n}\mathcal{Q}_{j_{k}}\rightarrow\mathcal{Q}_{0}$ be an
> intertwiner. Since $T$ is a linear form, there exists
> $\psi_{T}\in\bigotimes_{k=1}^{n}\mathcal{Q}_{j_{k}}$ such that
> $T(\phi)=\innerproduct{\phi}{\psi_{T}}$. Since $T$ is also an intertwiner,
> we have for all $\psi\in\bigotimes_{k=1}^{n}\mathcal{Q}_{j_{k}}$ and $u\in
> SU(2)$,
> $\innerproduct{\phi}{u\cdot\psi_{T}}=\innerproduct{u^{\dagger}\cdot\phi}{\psi_{T}}=T(u^{\dagger}\cdot\phi)=u^{\dagger}\cdot
> T(\phi)=T(\phi)=\innerproduct{\phi}{\psi_{T}}$. So
> $u\cdot\psi_{T}=\psi_{T}$. We can check that the map $T\mapsto\psi_{T}$ is
> linear and bijective. QED. $\Box$

##### Orthogonal projector.

By definition, the orthogonal projector $P$ over ${\rm
Inv}_{SU(2)}\left(\bigotimes_{i=1}^{n}\mathcal{Q}_{j_{i}}\right)$ satisfies

$P^{2}=P\quad\text{and}\quad P^{\dagger}=P.$ (3.24)

It is easy to show that

$P=\int_{SU(2)}dg\bigotimes_{k=1}^{n}D^{j_{k}}(g).$ (3.25)

Remember also that if $\ket{j}$ is an orthonormal basis of ${\rm
Inv}_{SU(2)}\left(\bigotimes_{i=1}^{n}\mathcal{Q}_{j_{i}}\right)$, then $P$
can also be written as

$P=\sum_{j}\outerproduct{j}{j}.$ (3.26)

###  3.3 Wigner’s $3jm$-symbol

We can decompose
$\mathcal{Q}_{j_{1}}\otimes\mathcal{Q}_{j_{2}}\otimes\mathcal{Q}_{j_{3}}$ into
a direct sum by applying 3.9 first on the left tensor product:

$\mathcal{Q}_{j_{1}}\otimes\mathcal{Q}_{j_{2}}\otimes\mathcal{Q}_{j_{3}}\rightarrow\left(\bigoplus_{j_{12}}\mathcal{Q}_{j_{12}}\right)\otimes\mathcal{Q}_{j_{3}}\rightarrow\bigoplus_{j_{12}=|j_{1}-j_{2}|}^{j_{1}+j_{2}}\bigoplus_{k=|j_{12}-j_{3}|}^{j_{12}+j_{3}}\mathcal{Q}_{k}$
(3.27)

Thus we construct an orthonormal basis of
$\mathcal{Q}_{j_{1}}\otimes\mathcal{Q}_{j_{2}}\otimes\mathcal{Q}_{j_{3}}$
given by the states

$\ket{(j_{1}j_{2})j_{3};j_{12}kn}=\sum_{m_{1},m_{2},m_{3},m_{12}}C^{j_{12}m_{12}}_{j_{1}m_{1}j_{2}m_{2}}C^{kn}_{j_{12}m_{12}j_{3}m_{3}}\bigotimes_{i=1}^{3}\ket{j_{i},m_{i}},\\\
{\rm with}\ j_{12}\in\\{|j_{1}-j_{2}|,...,j_{1}+j_{2}\\}\quad{\rm and}\
k\in\\{|j_{12}-j_{3}|,...,j_{12}+j_{3}\\}\quad{\rm and}\ n\in\\{-k,...,k\\}.$
(3.28)

> $\blacktriangleright$ Proof. This orthonormal basis is obtained by twice
> applying the relation 3.28, first to the left tensor product. Notice that
> applying it first on the right would instead build the states:
>
>
> $\ket{j_{1}(j_{2}j_{3});j_{23}kn}=\sum_{m_{1},m_{2},m_{3},m_{23}}C^{j_{23}m_{23}}_{j_{2}m_{2}j_{3}m_{3}}C^{kn}_{j_{1}m_{1}j_{23}m_{23}}\bigotimes_{i=1}^{3}\ket{j_{i},m_{i}}.$
> (3.29)
>
> $\Box$

Then we show

${\rm
Inv}_{SU(2)}\left(\mathcal{Q}_{j_{1}}\otimes\mathcal{Q}_{j_{2}}\otimes\mathcal{Q}_{j_{3}}\right)={\rm
Span}\left\\{\ket{(j_{1}j_{2})j_{3};j_{3}00}\right\\}.$ (3.30)

> $\blacktriangleright$ Proof. First we show the equivalence:
>
>
> $\displaystyle\vec{J}^{2}\ket{(j_{1}j_{2})j_{3};j_{12}kn}=0\quad\Leftrightarrow\quad
> k=0.$
>
> We conclude using the characterisation 3.21. $\Box$

Thus if the Clebsch-Gordan condition is satisfied ($|j_{1}-j_{2}|\leq
j_{3}\leq j_{1}+j_{2}$), ${\rm
Inv}_{SU(2)}(\mathcal{Q}_{j_{1}}\otimes\mathcal{Q}_{j_{2}}\otimes\mathcal{Q}_{j_{3}})$
is one dimensional. Otherwise ${\rm
Inv}_{SU(2)}(\mathcal{Q}_{j_{1}}\otimes\mathcal{Q}_{j_{2}}\otimes\mathcal{Q}_{j_{3}})=\\{0\\}$.
Now supposing that the condition is satisfied, there exists a unique unit
vector in ${\rm
Inv}_{SU(2)}(\mathcal{Q}_{j_{1}}\otimes\mathcal{Q}_{j_{2}}\otimes\mathcal{Q}_{j_{3}})$,

$\ket{0}=\sum_{m_{1},m_{2},m_{3}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\\
m_{1}&m_{2}&m_{3}\end{pmatrix}\bigotimes_{k=1}^{3}\ket{j_{k},m_{k}},$ (3.31)

such that the coefficients $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\\
m_{1}&m_{2}&m_{3}\end{pmatrix}$ in the canonical basis are real and satisfy
the symmetry properties:

$\begin{pmatrix}j_{1}&j_{2}&j_{3}\\\
m_{1}&m_{2}&m_{3}\end{pmatrix}=\begin{pmatrix}j_{3}&j_{1}&j_{2}\\\
m_{3}&m_{1}&m_{2}\end{pmatrix}=\begin{pmatrix}j_{2}&j_{3}&j_{1}\\\
m_{2}&m_{3}&m_{1}\end{pmatrix}.$ (3.32)

The coefficients $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\\
m_{1}&m_{2}&m_{3}\end{pmatrix}$ are called the Wigner’s $3jm$-symbol and are
related to the Clebsch-Gordan coefficients by

$\begin{pmatrix}j_{1}&j_{2}&j_{3}\\\
m_{1}&m_{2}&m_{3}\end{pmatrix}=\frac{(-1)^{j_{1}-j_{2}-m_{3}}}{\sqrt{2j_{3}+1}}C^{j_{3},-m_{3}}_{j_{1}m_{1}j_{2}m_{2}}.$
(3.33)

In Mathematica, they are given by
ThreeJSymbol$[\\{j_{1},m_{1}\\},\\{j_{2},m_{2}\\},\\{j_{3},m_{3}\\}]$.

> $\blacktriangleright$ Proof. In ${\rm
> Inv}_{SU(2)}\left(\mathcal{Q}_{j_{1}}\otimes\mathcal{Q}_{j_{2}}\otimes\mathcal{Q}_{j_{3}}\right)$,
> all vectors are proportional to
>
> $\displaystyle\ket{(j_{1}j_{2})j_{3};j_{3}00}$
> $\displaystyle=\sum_{m_{1},m_{2},m_{3},m}C^{00}_{jmj_{3}m_{3}}C^{jm}_{j_{1}m_{1};j_{2}m_{2}}\ket{j_{1}m_{1};j_{2}m_{2};j_{3}m_{3}}$
> [applying 3.28]
> $\displaystyle=\sum_{m_{1},m_{2},m_{3},m}\delta_{m,-m_{3}}\delta_{j,j_{3}}\frac{(-1)^{j_{3}+m_{3}}}{\sqrt{2j_{3}+1}}C^{jm}_{j_{1}m_{1};j_{2}m_{2}}\ket{j_{1}m_{1};j_{2}m_{2};j_{3}m_{3}}$
> [computing 3.15]
> $\displaystyle=\sum_{m_{1},m_{2},m_{3}}\frac{(-1)^{j_{3}+m_{3}}}{\sqrt{2j_{3}+1}}C^{j_{3},-m_{3}}_{j_{1}m_{1};j_{2}m_{2}}\ket{j_{1}m_{1};j_{2}m_{2};j_{3}m_{3}}$
> $\displaystyle\text{ [simplifying]}.$
>
> The proportionality factor is chosen to be $(-1)^{j_{1}-j_{2}+j_{3}}$ to
> match the reality and the symmetry requirements. $\Box$

##### Remarks.

  1. 1.

These symbols satisfy nice symmetry properties such as:

$\begin{pmatrix}j_{1}&j_{2}&j_{3}\\\
m_{1}&m_{2}&m_{3}\end{pmatrix}=(-1)^{j_{1}+j_{2}+j_{3}}\begin{pmatrix}j_{2}&j_{1}&j_{3}\\\
m_{2}&m_{1}&m_{3}\end{pmatrix}=(-1)^{j_{1}+j_{2}+j_{3}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\\
-m_{1}&-m_{2}&-m_{3}\end{pmatrix}.$ (3.34)

  2. 2.

The orthogonality relations 3.14 become

$\sum_{jm}(2j+1)\begin{pmatrix}j_{1}&j_{2}&j\\\
m_{1}&m_{2}&m\end{pmatrix}\begin{pmatrix}j_{1}&j_{2}&j\\\
m_{1}^{\prime}&m_{2}^{\prime}&m\end{pmatrix}=\delta_{m_{1}m_{1}^{\prime}}\delta_{m_{2}m_{2}^{\prime}}$
(3.35) $\sum_{m_{1}m_{2}}(2j+1)\begin{pmatrix}j_{1}&j_{2}&j\\\
m_{1}&m_{2}&m\end{pmatrix}\begin{pmatrix}j_{1}&j_{2}&j^{\prime}\\\
m_{1}&m_{2}&m^{\prime}\end{pmatrix}=\delta_{jj^{\prime}}\delta_{mm^{\prime}}$
(3.36)

##### Exercise.

Show that

$\int_{SU(2)}D^{j_{1}}_{m_{1}n_{1}}(u)D^{j_{2}}_{m_{2}n_{2}}(u)D^{j_{3}}_{m_{3}n_{3}}(u)du=\begin{pmatrix}j_{1}&j_{2}&j_{3}\\\
m_{1}&m_{2}&m_{3}\end{pmatrix}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\\
n_{1}&n_{2}&n_{3}\end{pmatrix}$ (3.37)

> $\blacktriangleright$ Proof. The result can be directly obtained by equating
> 3.25 and 3.26, and expressing the equality in the magnetic basis. $\Box$

###  3.4 Wigner’s $4jm$-symbol

Similarly to the previous section we can decompose
$\mathcal{Q}_{j_{1}}\otimes\mathcal{Q}_{j_{2}}\otimes\mathcal{Q}_{j_{3}}\otimes\mathcal{Q}_{j_{4}}$
into a direct sum by applying 3.9 successively:

$\mathcal{Q}_{j_{1}}\otimes\mathcal{Q}_{j_{2}}\otimes\mathcal{Q}_{j_{3}}\otimes\mathcal{Q}_{j_{4}}\cong\bigoplus_{j_{12}=|j_{1}-j_{2}|}^{j_{1}+j_{2}}\bigoplus_{k=|j_{12}-j_{3}|}^{j_{12}+j_{3}}\bigoplus_{l=|k-j_{4}|}^{k+j_{4}}\mathcal{Q}_{l}$
(3.38)

In particular, we can see that

${\rm
Inv}_{SU(2)}\left(\bigotimes_{i=1}^{4}\mathcal{Q}_{j_{i}}\right)\cong\underbrace{\mathcal{Q}_{0}\oplus...\oplus\mathcal{Q}_{0}}_{d_{0}\
\text{times}}\\\ \text{with}\
d_{0}=\max(|j_{1}-j_{2}|,|j_{3}-j_{4}|)-\min(j_{1}+j_{2},j_{3}+j_{4}).$ (3.39)

An orthonormal basis of ${\rm
Inv}_{SU(2)}\left(\bigotimes_{i=1}^{4}\mathcal{Q}_{j_{i}}\right)$ is given by

$\ket{j}_{12}=\sum_{m_{1},m_{2},m_{3},m_{4}}\sqrt{2j+1}\begin{pmatrix}j_{1}&j_{2}&j_{3}&j_{4}\\\
m_{1}&m_{2}&m_{3}&m_{4}\end{pmatrix}^{(j)}\bigotimes_{k=1}^{4}\ket{j_{k},m_{k}},\\\
{\rm with}\
j\in\\{\max(|j_{1}-j_{2}|,|j_{3}-j_{4}|),...,\min(j_{1}+j_{2},j_{3}+j_{4})\\}$
(3.40)

and

$\begin{pmatrix}j_{1}&j_{2}&j_{3}&j_{4}\\\
m_{1}&m_{2}&m_{3}&m_{4}\end{pmatrix}^{(j)}\overset{\rm
def}{=}\sum_{m}(-1)^{j-m}\begin{pmatrix}j_{1}&j_{2}&j\\\
m_{1}&m_{2}&m\end{pmatrix}\begin{pmatrix}j&j_{3}&j_{4}\\\
-m&m_{3}&m_{4}\end{pmatrix}$ (3.41)

> $\blacktriangleright$ Proof. First, we construct an orthonormal basis of
> $\mathcal{Q}_{j_{1}}\otimes\mathcal{Q}_{j_{2}}\otimes\mathcal{Q}_{j_{3}}\otimes\mathcal{Q}_{j_{4}}$
> given by the states
>
>
> $\ket{((j_{1}j_{2})j_{3})j_{4};jklm}=\sum_{m_{1},m_{2},m_{3},m,n,m_{4}}C^{jm}_{j_{1}m_{1}j_{2}m_{2}}C^{kn}_{jmj_{3}m_{3}}C^{lm}_{knj_{4}m_{4}}\bigotimes_{i=1}^{4}\ket{j_{i},m_{i}},\\\
> \hfill{\rm with}\ j\in\\{|j_{1}-j_{2}|,...,j_{1}+j_{2}\\}\quad{\rm and}\
> k\in\\{|j-j_{3}|,...,j+j_{3}\\}\\\ {\rm and}\
> l\in\\{|k-j_{4}|,...,k+j_{4}\\}\quad{\rm and}\ n\in\\{-l,...,l\\}.$ (3.42)
>
> ${\rm Inv}_{SU(2)}\left(\bigotimes_{i=1}^{4}\mathcal{Q}_{j_{i}}\right)$ is
> spanned by the vectors with $l=0$. Similarly to the case $n=3$, we compute
>
> $\displaystyle\ket{((j_{1}j_{2})j_{3})j_{4};jj_{4}00}$
> $\displaystyle=(-1)^{j_{4}+j_{1}-j_{2}-j_{3}}\sum_{m_{1},m_{2},m_{3},m_{4}}\sqrt{2j+1}\begin{pmatrix}j_{1}&j_{2}&j_{3}&j_{4}\\\
> m_{1}&m_{2}&m_{3}&m_{4}\end{pmatrix}^{(j)}\bigotimes_{i=1}^{4}\ket{j_{i},m_{i}}.$
>
> $\Box$

This basis has the interesting property that it diagonalises
$(\vec{J}_{1}+\vec{J}_{2})^{2}$:

$(\vec{J}_{1}+\vec{J}_{2})^{2}\ket{j}_{12}=j(j+1)\ket{j}_{12}.$ (3.43)

The $4jm$-symbol also satisfy orthogonality relations:

$\sum_{m_{1},m_{2},m_{3}}\begin{pmatrix}j_{1}&j_{2}&j_{3}&j_{4}\\\
m_{1}&m_{2}&m_{3}&m_{4}\end{pmatrix}^{(j_{12})}\begin{pmatrix}j_{1}&j_{2}&j_{3}&l_{4}\\\
m_{1}&m_{2}&m_{3}&n_{4}\end{pmatrix}^{(l_{12})}=\frac{\delta_{j_{12}l_{12}}}{d_{j_{12}}}\frac{\delta_{j_{4}l_{4}}\delta_{m_{4}n_{4}}}{d_{j_{4}}}.$
(3.44)

Finally we can show, similarly to 3.37, that

$\int_{SU(2)}D^{j_{1}}_{m_{1}n_{1}}(u)D^{j_{2}}_{m_{2}n_{2}}(u)D^{j_{3}}_{m_{3}n_{3}}(u)D^{j_{4}}_{m_{4}n_{4}}(u)du\\\
=\sum_{j}(2j+1)\begin{pmatrix}j_{1}&j_{2}&j_{3}&j_{4}\\\
m_{1}&m_{2}&m_{3}&m_{4}\end{pmatrix}^{(j)}\begin{pmatrix}j_{1}&j_{2}&j_{3}&j_{4}\\\
n_{1}&n_{2}&n_{3}&n_{4}\end{pmatrix}^{(j)}.$ (3.45)

###  3.5 Wigner’s $6j$-symbol

In the previous section, we have exhibited an orthonormal basis for ${\rm
Inv}_{SU(2)}\left(\bigotimes_{i=1}^{4}\mathcal{Q}_{j_{i}}\right)$. It is built
from one possible decomposition of $\bigotimes_{i=1}^{4}\mathcal{Q}_{j_{i}}$
into irreps. Another possible decomposition leads to another basis

$\ket{j}_{23}=\sum_{m_{1},m_{2},m_{3},m_{4}}\sqrt{2j+1}\begin{pmatrix}j_{4}&j_{1}&j_{2}&j_{3}\\\
m_{4}&m_{1}&m_{2}&m_{3}\end{pmatrix}^{(j)}\bigotimes_{i=1}^{4}\ket{j_{i},m_{i}}$
(3.46)

> $\blacktriangleright$ Proof. From
>
>
> $\ket{(j_{1}(j_{2}j_{3}))j_{4};jklm}=\sum_{m_{1},m_{2},m_{3},m,n,m_{4}}C^{jm}_{j_{2}m_{2}j_{3}m_{3}}C^{kn}_{j_{1}m_{1}jm}C^{lm}_{knj_{4}m_{4}}\bigotimes_{i=1}^{4}\ket{j_{i},m_{i}},$
> (3.47)
>
> we show that
>
> $\displaystyle\ket{(j_{1}(j_{2}j_{3}))j_{4};jj_{4}00}$
> $\displaystyle=(-1)^{j_{1}+j_{2}-j_{3}+j_{4}}\sum_{m_{1},m_{2},m_{3},m_{4}}\sqrt{2j_{23}+1}\begin{pmatrix}j_{4}&j_{1}&j_{2}&j_{3}\\\
> m_{4}&m_{1}&m_{2}&m_{3}\end{pmatrix}^{(j_{23})}\bigotimes_{i=1}^{4}\ket{j_{i},m_{i}}.$
>
> $\Box$

The change of basis is given by

${}_{12}\innerproduct{j}{k}_{23}=\sqrt{2j+1}\sqrt{2k+1}(-1)^{j_{1}+j_{2}+j_{3}-j_{4}-2j-2k}\begin{Bmatrix}j_{1}&j_{2}&j\\\
j_{3}&j_{4}&k\end{Bmatrix}$ (3.48)

where we have defined a new symbol:

$\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\\
j_{4}&j_{5}&j_{6}\end{Bmatrix}\overset{\rm
def}{=}\sum_{m_{1},\dots,m_{6}}(-1)^{\sum_{i=1}^{6}(j_{i}-m_{i})}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\\
-m_{1}&-m_{2}&-m_{3}\end{pmatrix}\begin{pmatrix}j_{1}&j_{5}&j_{6}\\\
m_{1}&-m_{5}&m_{6}\end{pmatrix}\\\ \times\begin{pmatrix}j_{4}&j_{2}&j_{6}\\\
m_{4}&m_{2}&-m_{6}\end{pmatrix}\begin{pmatrix}j_{3}&j_{4}&j_{5}\\\
m_{3}&-m_{4}&m_{5}\end{pmatrix}$ (3.49)

In the Wolfram Language, it is returned by the function
SixJSymbol$[\\{j_{1},j_{2},j_{3}\\},\\{j_{4},j_{5},j_{6}\\}]$. These symbols
satisfy the symmetries

$\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\\
j_{4}&j_{5}&j_{6}\end{Bmatrix}=\begin{Bmatrix}j_{2}&j_{1}&j_{3}\\\
j_{5}&j_{4}&j_{6}\end{Bmatrix}=\begin{Bmatrix}j_{3}&j_{2}&j_{1}\\\
j_{6}&j_{5}&j_{4}\end{Bmatrix}=\begin{Bmatrix}j_{4}&j_{2}&j_{3}\\\
j_{1}&j_{5}&j_{6}\end{Bmatrix}.$ (3.50)

Similarly one can define the symbols $9j$ and $15j$.

> $\varheart$ Physics. The $\\{6j\\}$-symbol appeared in quantum gravity when
> Ponzano and Regge realised that the $\\{6j\\}$-symbol approximate the action
> of general relativity in the semi-classical limit [PR68]. More precisely
> they have shown that
>
> $\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\\
> j_{4}&j_{5}&j_{6}\end{Bmatrix}\underset{j_{i}\to\infty}{\sim}\frac{1}{\sqrt{12\pi
> V}}\cos(S(j_{i})+\pi/4)\,,$ (3.51)
>
> where $V$ is the volume of a tetrahedron whose edges have a length of
> $j_{i}+1/2$, and $S(j_{i})$ is the so-called Regge action, which is a
> discrete $3$-dimensional version of the Einstein-Hilbert action. This result
> was a important source of inspiration for later development of spin-foams.

###  3.6 Graphical calculus

The recoupling theory of $SU(2)$ can be nicely implemented graphically. The
underlying philosophy of it is to take advantage of the $2$-dimensional
surfaces offered by our sheets of paper and our black-boards to literally draw
our calculations, rather than restricting oneself to the usual one dimensional
lines of calculations. If done properly, the method can help to understand the
structure of analytical expressions, and make computations faster. Of course,
the first principle of graphical calculus is that there should be a one-to-one
correspondence between analytical expressions and diagrams. There exists many
conventions for this correspondence in the literature, so we have chosen one
that seems to be quite popular [SSS18], and which is described in details by
Varshalovich ([VMK87], Chap. 11).

##### Definitions.

The basic object of this graphical calculus is the $3$-valent node, that
represent the Wigner’s $3jm$ symbol:

$\begin{pmatrix}j_{1}&j_{2}&j_{3}\\\
m_{1}&m_{2}&m_{3}\end{pmatrix}=\begin{array}[]{c}\begin{overpic}[scale={0.6}]{3CG-
out.png}\put(5.0,45.0){$j_{1}$}\put(50.0,65.0){$j_{2}$}\put(90.0,55.0){$j_{3}$}\put(45.0,5.0){$-$}\end{overpic}\end{array}=\begin{array}[]{c}\begin{overpic}[scale={0.6}]{3CG-
out.png}\put(5.0,45.0){$j_{1}$}\put(50.0,65.0){$j_{3}$}\put(90.0,55.0){$j_{2}$}\put(45.0,5.0){$+$}\end{overpic}\end{array}.$
(3.52)

Remark that

  1. 1.

The signs $+/-$ on the nodes indicate the sense of rotation
(anticlockwise/clockwise) in which the spins must be read. To alleviate
notations we can decide not to write them by choosing conventionally that the
default sign of the nodes is minus, if not otherwise specified. The arrows on
the wires will be used below to define the operation of summation.

  2. 2.

Everywhere we implicitly assume that the Clebsch-Gordan inequalities are
satisfied.

  3. 3.

The magnetic indices are implicit in the diagram, which creates no ambiguity,
as long as we associate the $m_{i}$ to the spin $j_{i}$.

  4. 4.

The symmetry properties 3.32 are naturally implemented on the diagram, which
also guarantees the one-to-one correspondence between the analytical
expression and the diagram.

  5. 5.

Only the topology of the diagram matters, which means that all topological
deformations are allowed.

$\begin{array}[]{c}\begin{overpic}[scale={0.6}]{3CG-
out.png}\put(5.0,45.0){$j_{1}$}\put(50.0,65.0){$j_{2}$}\put(90.0,55.0){$j_{3}$}\end{overpic}\end{array}=\begin{array}[]{c}\begin{overpic}[scale={0.6}]{3CG-
star.png}\put(5.0,25.0){$j_{1}$}\put(50.0,70.0){$j_{2}$}\put(80.0,40.0){$j_{3}$}\end{overpic}\end{array}=\begin{array}[]{c}\begin{overpic}[scale={0.6}]{3CG-
octopus.png}\put(5.0,60.0){$j_{1}$}\put(50.0,75.0){$j_{2}$}\put(90.0,55.0){$j_{3}$}\end{overpic}\end{array}$
(3.53)

This principle of topological equivalence is a strong principle of graphical
calculus, that will hold for any other diagram constructed later.

Then we can define graphically the two basic operations of algebra:
multiplication and summation. Multiplication is implemented simply by
juxtaposition of two diagrams:

$\begin{array}[]{c}\begin{overpic}[scale={0.6}]{3CG-
out.png}\put(5.0,45.0){$j_{1}$}\put(50.0,65.0){$j_{2}$}\put(90.0,55.0){$j_{3}$}\end{overpic}\end{array}\begin{array}[]{c}\begin{overpic}[scale={0.6}]{3CG-
out.png}\put(5.0,45.0){$j_{4}$}\put(50.0,65.0){$j_{5}$}\put(90.0,55.0){$j_{6}$}\end{overpic}\end{array}=\begin{pmatrix}j_{1}&j_{2}&j_{3}\\\
m_{1}&m_{2}&m_{3}\end{pmatrix}\begin{pmatrix}j_{4}&j_{5}&j_{6}\\\
m_{4}&m_{5}&m_{6}\end{pmatrix}$ (3.54)

To define the summation, we shall first tell more about the orientation of
external wires. As you may have noticed, the arrows on the wires are all
outgoing. Now we define also the ingoing orientation with the general rule
that inverting the orientation111It can be seen as the contraction with the
‘metric tensor’ $\epsilon_{mm^{\prime}}=(-1)^{j-m}\delta_{m,-m^{\prime}}$, as
it is often said. of an external line $(jm)$ amounts analytically to
transforming $m$ to $-m$ and multiplying the overall expression by a factor
$(-1)^{j-m}$. For instance

$\begin{array}[]{c}\begin{overpic}[scale={0.6}]{3CG-
outin.png}\put(5.0,45.0){$j_{1}$}\put(55.0,65.0){$j_{2}$}\put(90.0,55.0){$j_{3}$}\end{overpic}\end{array}=(-1)^{j_{1}-m_{1}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\\
-m_{1}&m_{2}&m_{3}\end{pmatrix}\,.$ (3.55)

The summation over a magnetic index $m$ (from $-j$ to $j$) is now represented
by gluing two external wires with the same spin $j$ and magnetic index $m$,
but of opposite directions, like:

$\begin{array}[]{c}\begin{overpic}[scale={0.6}]{4CG.png}\put(-5.0,55.0){$j_{1}$}\put(25.0,55.0){$j_{2}$}\put(70.0,55.0){$j_{3}$}\put(95.0,55.0){$j_{4}$}\put(60.0,5.0){$j$}\end{overpic}\end{array}=\sum_{m=-j}^{j}\begin{array}[]{c}\begin{overpic}[scale={0.6}]{3CG-
out.png}\put(5.0,45.0){$j_{1}$}\put(55.0,65.0){$j_{2}$}\put(90.0,55.0){$j$}\end{overpic}\end{array}\begin{array}[]{c}\begin{overpic}[scale={0.6}]{3CG-
outin.png}\put(5.0,45.0){$j$}\put(55.0,65.0){$j_{3}$}\put(90.0,55.0){$j_{4}$}\end{overpic}\end{array}$
(3.56)

On the RHS, we recognise the definition of the $4jm$-symbol, so that

$\begin{array}[]{c}\begin{overpic}[scale={0.6}]{4CG.png}\put(-5.0,55.0){$j_{1}$}\put(25.0,55.0){$j_{2}$}\put(70.0,55.0){$j_{3}$}\put(95.0,55.0){$j_{4}$}\put(60.0,5.0){$j$}\end{overpic}\end{array}=\begin{pmatrix}j_{1}&j_{2}&j_{3}&j_{4}\\\
m_{1}&m_{2}&m_{3}&m_{4}\end{pmatrix}^{(j)}$ (3.57)

The line between two nodes, whose magnetic index is summed over, is called an
internal line, in opposition to external lines, which have a free hand.
Contrary to the previous rule of inversion for external lines, it is easy to
show that changing the orientation of an internal line gives a phase:

$\begin{array}[]{c}\begin{overpic}[scale={0.6}]{4CG.png}\put(-5.0,55.0){$j_{1}$}\put(25.0,55.0){$j_{2}$}\put(70.0,55.0){$j_{3}$}\put(95.0,55.0){$j_{4}$}\put(60.0,5.0){$j$}\end{overpic}\end{array}=(-1)^{2j}\begin{array}[]{c}\begin{overpic}[scale={0.6}]{4CG-
right.png}\put(0.0,55.0){$j_{1}$}\put(25.0,55.0){$j_{2}$}\put(60.0,55.0){$j_{3}$}\put(90.0,55.0){$j_{4}$}\put(60.0,5.0){$j$}\end{overpic}\end{array}.$

A powerful aspect of graphical calculus comes from the representation of the
Kronecker delta with a single line

$\begin{array}[]{c}\begin{overpic}[scale={0.6}]{up.png}\put(-30.0,85.0){$(j_{1},m_{1})$}\put(-25.0,15.0){$(j_{2},m_{2})$}\end{overpic}\end{array}=\delta_{j_{1}j_{2}}\delta_{m_{1}m_{2}}\qquad\text{or}\qquad\begin{array}[]{c}\begin{overpic}[scale={0.6}]{up.png}\put(10.0,60.0){$j$}\put(15.0,95.0){$m$}\put(20.0,5.0){$n$}\end{overpic}\end{array}=\delta_{mn}\,.$
(3.58)

The rule of summation applied to it enables to compute its trace:

$\begin{array}[]{c}\begin{overpic}[scale={0.6}]{circle.png}\put(5.0,60.0){$j$}\end{overpic}\end{array}=2j+1$
(3.59)

For instance, the orthogonality relation 3.36 now reads

$\begin{array}[]{c}\begin{overpic}[scale={0.6}]{loop.png}\put(30.0,90.0){$(j_{1},m_{1})$}\put(30.0,5.0){$(j_{2},m_{2})$}\put(5.0,60.0){$j_{3}$}\put(40.0,60.0){$j_{4}$}\end{overpic}\end{array}\begin{array}[]{c}\begin{overpic}[scale={0.6}]{circle.png}\put(5.0,70.0){$j_{1}$}\end{overpic}\end{array}=\begin{array}[]{c}\begin{overpic}[scale={0.6}]{down.png}\put(25.0,90.0){$(j_{1},m_{1})$}\put(25.0,5.0){$(j_{2},m_{2})$}\end{overpic}\end{array}$
(3.60)

##### Lemmas.

From all the rules described above, the following lemma can already be checked
as an exercise.

  1. 1.

Reversing all external lines has no effect:

$\begin{array}[]{c}\begin{overpic}[scale={0.6}]{3CG-
out.png}\put(5.0,45.0){$j_{1}$}\put(50.0,65.0){$j_{2}$}\put(90.0,55.0){$j_{3}$}\end{overpic}\end{array}=\begin{array}[]{c}\begin{overpic}[scale={0.6}]{3CG-
in.png}\put(5.0,45.0){$j_{1}$}\put(55.0,65.0){$j_{2}$}\put(90.0,55.0){$j_{3}$}\end{overpic}\end{array}\,.$
(3.61)

  2. 2.

Changing the sign of the node gives a phase

$\begin{array}[]{c}\begin{overpic}[scale={0.6}]{3CG-
out.png}\put(5.0,45.0){$j_{1}$}\put(50.0,65.0){$j_{2}$}\put(90.0,55.0){$j_{3}$}\put(45.0,5.0){$-$}\end{overpic}\end{array}=(-1)^{j_{1}+j_{2}+j_{3}}\begin{array}[]{c}\begin{overpic}[scale={0.6}]{3CG-
out.png}\put(5.0,45.0){$j_{1}$}\put(50.0,65.0){$j_{2}$}\put(90.0,55.0){$j_{3}$}\put(45.0,5.0){$+$}\end{overpic}\end{array}\,.$
(3.62)

  3. 3.

The evaluation of the so-called $\Theta$-graph:

$\begin{array}[]{c}\begin{overpic}[scale={0.6}]{theta.png}\put(20.0,70.0){$j_{1}$}\put(55.0,50.0){$j_{2}$}\put(75.0,10.0){$j_{3}$}\end{overpic}\end{array}=1.$
(3.63)

  4. 4.

Similarly, 3.44 implies

$\begin{array}[]{c}\begin{overpic}[scale={0.6}]{doubletheta.png}\put(60.0,90.0){$j_{1}$}\put(30.0,55.0){$j_{2}$}\put(55.0,40.0){$j_{3}$}\put(70.0,10.0){$j_{4}$}\put(5.0,40.0){$i$}\put(85.0,40.0){$k$}\end{overpic}\end{array}=\frac{\delta_{i,k}}{d_{i}}.$
(3.64)

##### Invariant functions.

One nice thing about this graphical calculus is that it makes easy to
represent and to remember the Wigner $6j$-symbols:

$\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\\
j_{4}&j_{5}&j_{6}\end{Bmatrix}=\begin{array}[]{c}\begin{overpic}[scale={0.6}]{tetrahedron.png}\put(25.0,63.0){$j_{1}$}\put(47.0,60.0){$j_{2}$}\put(75.0,60.0){$j_{3}$}\put(40.0,35.0){$j_{4}$}\put(25.0,20.0){$j_{5}$}\put(75.0,20.0){$j_{6}$}\end{overpic}\end{array}$
(3.65)

As we can see, the $6j$-symbol looks like a tetrahedron. All magnetic indices
are summed over, so that it is only a function of the spins $j_{i}$, what we
can call an invariant function. It gives us the idea to define other invariant
functions by finding other diagrams with no external links. For instance we
define the $9j$-symbol:

$\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\\ j_{4}&j_{5}&j_{6}\\\
j_{7}&j_{8}&j_{9}\end{Bmatrix}=\begin{array}[]{c}\begin{overpic}[scale={0.6}]{9CG.png}\put(25.0,80.0){$j_{1}$}\put(47.0,65.0){$j_{2}$}\put(73.0,73.0){$j_{3}$}\put(60.0,40.0){$j_{4}$}\put(80.0,40.0){$j_{5}$}\put(65.0,15.0){$j_{6}$}\put(25.0,15.0){$j_{7}$}\put(25.0,35.0){$j_{8}$}\put(5.0,45.0){$j_{9}$}\end{overpic}\end{array}$
(3.66)

Notice that we could have also defined the $9j$-symbol to be rather

$\begin{array}[]{c}\begin{overpic}[scale={0.6}]{9j-alter.png}\put(25.0,90.0){$j_{1}$}\put(45.0,75.0){$j_{2}$}\put(63.0,83.0){$j_{3}$}\put(50.0,55.0){$j_{4}$}\put(75.0,40.0){$j_{5}$}\put(55.0,5.0){$j_{6}$}\put(10.0,15.0){$j_{7}$}\put(25.0,35.0){$j_{8}$}\put(0.0,45.0){$j_{9}$}\end{overpic}\end{array}$
(3.67)

but this one can be actually rewritten as the product of two $6j$-symbol. Such
a decomposition cannot be done with the $9j$-symbol 3.66, so that it is said
‘irreducible’222For more details on the $9j$-symbol, see [Edm57] pp. 100-114..
We also have the $15j$-symbol:

$\begin{Bmatrix}j_{1}&j_{2}&j_{11}\\\ j_{4}&j_{5}&j_{15}\\\
j_{7}&j_{3}&j_{14}\\\ j_{9}&j_{6}&j_{13}\\\
j_{8}&j_{10}&j_{12}\end{Bmatrix}=\begin{array}[]{c}\begin{overpic}[scale={0.6}]{15j.png}\put(25.0,75.0){$j_{1}$}\put(33.0,47.0){$j_{2}$}\put(60.0,50.0){$j_{3}$}\put(80.0,70.0){$j_{4}$}\put(50.0,55.0){$j_{5}$}\put(55.0,35.0){$j_{6}$}\put(95.0,35.0){$j_{7}$}\put(37.0,37.0){$j_{8}$}\put(55.0,5.0){$j_{9}$}\put(0.0,35.0){$j_{10}$}\put(45.0,81.0){$j_{11}$}\put(0.0,60.0){$j_{12}$}\put(15.0,5.0){$j_{13}$}\put(80.0,10.0){$j_{14}$}\put(93.0,60.0){$j_{15}$}\end{overpic}\end{array}$
(3.68)

which is the definition used by [SSS18]. It is different from the convention
chosen in [Oog92], which is

$\begin{Bmatrix}l_{1}&l_{2}&l_{3}&l_{4}&l_{5}\\\
j_{1}&j_{2}&j_{3}&j_{4}&j_{5}\\\
l_{10}&l_{9}&l_{8}&l_{7}&l_{6}\end{Bmatrix}=\begin{array}[]{c}\begin{overpic}[scale={0.6}]{15j-ooguri.png}\put(25.0,75.0){$l_{2}$}\put(35.0,60.0){$j_{4}$}\put(50.0,45.0){$j_{3}$}\put(80.0,70.0){$l_{10}$}\put(60.0,57.0){$j_{1}$}\put(60.0,30.0){$j_{5}$}\put(95.0,35.0){$l_{8}$}\put(37.0,30.0){$j_{2}$}\put(55.0,5.0){$l_{6}$}\put(0.0,35.0){$l_{4}$}\put(45.0,81.0){$l_{1}$}\put(0.0,60.0){$l_{3}$}\put(15.0,5.0){$l_{5}$}\put(80.0,10.0){$l_{7}$}\put(93.0,60.0){$l_{9}$}\end{overpic}\end{array}$
(3.69)

Contrary to the $6j$-symbol, there is no consensus about what is called the
$15j$-symbol, but in all cases it corresponds to an invariant function
associated to a $3$-valent graph with 15 links. Actually, we can build $5$
topologically different $15j$-symbols333For more details on the $15j$-symbol,
see [YLV62] pp. 65-70.. Here we see the power of graphical calculus: imagine
if we had given the analytical formula for it… that is doable, but unreadable.

> $\varheart$ Physics. In the spirit of the result of Ponzano and Regge 3.51,
> Ooguri used the $15j$-symbol to provide a model of quantum gravity [Oog92].
> It still plays a major role in the EPRL model [Spe17].

##  Chapter 4 Representation theory of $SL_{2}(\mathbb{C})$

To put it in a nutshell, the kinematic aspects of loop quantum gravity deal
with representations of $SU(2)$, while the dynamics, in its spin-foam
formulation, lie in the representation theory of $SL_{2}(\mathbb{C})$. The
current models of spin-foams, like the EPRL one, use extensively the principal
series of $SL_{2}(\mathbb{C})$.

Fortunately all the irreps of $SL_{2}(\mathbb{C})$ are known. Whether or not
all representations, including non-reducible ones, have been classified is
unknown to the author. In section 4.1, we present the finite-dimensional ones.
In section 4.2, we summarize the infinite-dimensional ones. Finally, in
section 4.3, we focus on the principal series, which is the main interest for
quantum gravity.

###  4.1 Finite irreps

The finite irreps of $SL_{2}(\mathbb{C})$ are well-known. They can be obtained
from the irreps of $\mathfrak{sl}_{2}(\mathbb{C})$. In section 2.1, we have
already seen the finite irreps of its $3$-dimensional (complex) Lie algebra
$\mathfrak{sl}_{2}(\mathbb{C})$: they are indexed by a spin
$j\in\mathbb{N}/2$. It is also possible to see $\mathfrak{sl}_{2}(\mathbb{C})$
as a real Lie algebra of dimension $6$, in which case, we will rather denote
it $\mathfrak{sl}_{2}(\mathbb{C})_{\mathbb{R}}$. In this section, we will
describe the (real) linear representations of
$\mathfrak{sl}_{2}(\mathbb{C})_{\mathbb{R}}$. We have the following
isomorphism between real vector spaces:

$\mathfrak{sl}_{2}(\mathbb{C})_{\mathbb{R}}\cong\mathfrak{su}(2)\oplus\mathfrak{su}(2).$
(4.1)

The algebra $\mathfrak{su}(2)\oplus\mathfrak{su}(2)$ is the Lie algebra of
$SU(2)\times SU(2)$. A consequence of Peter-Weyl’s theorem is that the irreps
of a cartesian product are tensor products of the irreps of the factors
([Kna86] p. 32). Thus the irreps of
$\mathfrak{sl}_{2}(\mathbb{C})_{\mathbb{R}}$ are given by the usual tensor
representation over $\mathcal{Q}_{j_{1}}\otimes\mathcal{Q}_{j_{2}}$,
abbreviated by $(j_{1},j_{2})$. The action is given by:

$a\cdot(\ket{j_{1},m_{2}}\otimes\ket{j_{2},m_{2}})\overset{\rm
def}{=}(a\ket{j_{1},m_{1}})\otimes\ket{j_{2},m_{2}}+\ket{j_{1},m_{1}}\otimes(a\ket{j_{2},m_{2}}).$
(4.2)

The isomorphism 4.1 provides naturally a basis of
$\mathfrak{sl}_{2}(\mathbb{C})_{\mathbb{R}}$, given by the three Pauli
matrices $\sigma_{i}\in i\mathfrak{su}(2)$ and the three matrices
$i\sigma_{i}\in\mathfrak{su}(2)$. To match the earlier notations introduced in
section 2.2, we often denote the rotation generators $J_{i}\overset{\rm
def}{=}\frac{1}{2}\sigma_{i}$ and the boost generators $K_{i}\overset{\rm
def}{=}\frac{i}{2}\sigma_{i}$. These generators satisfy the commutation
relations:

$\displaystyle\left[J_{i},J_{j}\right]=i\varepsilon_{ijk}J_{k}$
$\displaystyle\left[J_{i},K_{j}\right]=i\varepsilon_{ijk}K_{k}$
$\displaystyle\left[K_{i},K_{j}\right]=-i\varepsilon_{ijk}J_{k}.$

> $\bigstar$ Nota Bene. Another basis is given by the complexified generators.
> Posing $A_{i}=\frac{1}{2}(J_{i}+iK_{i})$ and
> $B_{i}=\frac{1}{2}(J_{i}-iK_{i})$, the commutation relations become:
>
> $\displaystyle\left[A_{i},A_{j}\right]=i\varepsilon_{ijk}A_{k}$
> $\displaystyle\left[B_{i},B_{j}\right]=i\varepsilon_{ijk}B_{k}$
> $\displaystyle\left[A_{i},B_{j}\right]=0.$
>
> We can also define the scale operators. Posing $K_{\pm}\overset{\rm
> def}{=}K_{1}\pm iK_{2}$ and $J_{\pm}\overset{\rm def}{=}J_{1}\pm iJ_{2}$,
> the scale operators satisfy:
>
> $\displaystyle\left[J_{3},J_{\pm}\right]=\pm J_{\pm}$
> $\displaystyle\left[J_{+},J_{-}\right]=2J_{3}$
> $\displaystyle\left[K_{3},K_{\pm}\right]=\mp J_{\pm}$
> $\displaystyle\left[K_{+},K_{-}\right]=-2J_{3}$
> $\displaystyle\left[J_{+},K_{+}\right]=\left[J_{-},K_{-}\right]=\left[J_{3},K_{3}\right]=0$
> $\displaystyle\left[K_{3},J_{\pm}\right]=\pm K_{\pm}$
> $\displaystyle\left[J_{\pm},K_{\mp}\right]=\pm 2K_{3}$
> $\displaystyle\left[J_{3},K_{\pm}\right]=\pm K_{\pm}$

Then the various realisations, which where described in chapter 2 for the
action of $\mathfrak{su}(2)$, can be adapted to
$\mathfrak{sl}_{2}(\mathbb{C})_{\mathbb{R}}$.

##### Homogeneous realisation.

Let $m,n\geq 2$, and
$\mathbb{C}_{(m,n)}[z_{0},z_{1};\overline{z_{0}},\overline{z_{1}}]$ the vector
space of homogeneous polynomials of degree $m$ in $(z_{0},z_{1})$ and
homogeneous of degree $n$ in $(\overline{z_{0}},\overline{z_{1}})$. The action
of $SL_{2}(\mathbb{C})$ is given by

$\displaystyle g\cdot P(z_{0},z_{1})=P\left(g^{T}\binom{z_{0}}{z_{1}}\right).$

The associated action of the algebra
$\mathfrak{sl}_{2}(\mathbb{C})_{\mathbb{R}}$ is given by

$\displaystyle J_{+}\cong z_{0}\frac{\partial}{\partial
z_{1}}+\overline{z_{0}}\frac{\partial}{\partial\overline{z_{1}}}$
$\displaystyle J_{-}\cong z_{1}\frac{\partial}{\partial
z_{0}}+\overline{z_{1}}\frac{\partial}{\partial\overline{z_{0}}}$
$\displaystyle J_{3}\cong\frac{1}{2}\left(z_{0}\frac{\partial}{\partial
z_{0}}-z_{1}\frac{\partial}{\partial
z_{1}}+\overline{z_{0}}\frac{\partial}{\partial\overline{z_{0}}}-\overline{z_{1}}\frac{\partial}{\partial\overline{z_{1}}}\right).$

##### Projective realisation.

As seen in section 2.4, the restriction of $P(z_{0},z_{1})$ to $P(z,1)$
induces another realisation over the space of polynomials of degree at most
$m$ in $z$ and at most $n$ in $\overline{z}$. The action is given by

$\displaystyle
g\cdot\phi(\xi)=(g_{12}\xi+g_{22})^{m}\overline{(g_{12}\xi+g_{22})}^{n}\phi\left(\frac{g_{11}\xi+g_{21}}{g_{12}\xi+g_{22}}\right).$

##### Spinorial realisation.

Finally the $(m,n)$ representation of $SL_{2}(\mathbb{C})$ can be realised
over the space of totally symmetric spinors
$\mathfrak{S}^{(A_{1}...A_{m})(\dot{A}_{1}...\dot{A}_{n})}$ such as (see
Penrose [PR84] p. 142)

$u\cdot z^{A_{1}...A_{m}\dot{A}_{1}...\dot{A}_{n}}=u^{A_{1}}_{\ \
B_{1}}...u^{A_{m}}_{\ \ B_{m}}\,\overline{u}^{\dot{A}_{1}}_{\ \
\dot{B}_{1}}...\overline{u}^{\dot{A}_{n}}_{\ \
\dot{B}_{n}}\,z^{B_{1}...B_{m}\dot{B}_{1}...\dot{B}_{n}}.$ (4.3)

Finite representations of $SL_{2}(\mathbb{C})$ cannot be unitary (except the
trivial one), because it is a simply connected non-compact Lie group. If we
want unitary representations, we shall turn to infinite ones.

###  4.2 Infinite irreps

In this section, we describe all the infinite-dimensional irreps of
$SL_{2}(\mathbb{C})$.

> $\bigstar$ Nota Bene. All the unitary irreps of the Lorentz group have been
> found simultaneously in 1946 by Gel’fand and Naimark [GN47], by Harish-
> Chandra [HC47] and by Bargmann [Bar47]. It seems nevertheless that Gel’fand
> and Naimark were the first to publish (unfortunately their article is only
> in Russian). The question remained to find all the irreps, unitary or not,
> and this was solved also by Naimark in [Nai54]. In 1963, Gel’fand, Minlos
> and Shapiro published the first book (with english translation) that reviews
> all these results [GMS63]. In 1964, Naimark wrote a more detailed and well-
> written book that wraps up the subject for mathematically-orientated
> physicists [Nai64].

The infinite irreps of $SL_{2}(\mathbb{C})$ are parametrised by
$(m,\rho)\in\mathbb{Z}\times\mathbb{C}$ with $\imaginary\rho\geq 0$ and
$\rho^{2}\neq-(|m|+2n)^{2}\in\mathbb{N}^{*}$. A realisation is given over the
Hilbert space $L^{2}(\mathbb{C})$ endowed with the scalar product

$\innerproduct{\varphi}{\phi}=\frac{i}{2}\int_{\mathbb{C}}\overline{\varphi(\xi)}\phi(\xi)(1+|\xi|^{2})^{-\imaginary\rho}d\xi
d\overline{\xi},$ (4.4)

and the action

$a\cdot
f(z)=(a_{12}z+a_{22})^{\frac{m}{2}+\frac{i\rho}{2}-1}\overline{(a_{12}z+a_{22})}^{-\frac{m}{2}+\frac{i\rho}{2}-1}f\left(\frac{a_{11}z+a_{21}}{a_{12}z+a_{22}}\right).$
(4.5)

Remarks:

  1. 1.

If $\rho^{2}=-(|m|+2n)^{2}$ with $n\in\mathbb{N}^{*}$, the Hilbert space and
the action still defines a representation, but a reducible one. Then, if one
restricts the action to the subspace of polynomials of degree at most
$p=\frac{m}{2}+i\frac{\rho}{2}-1$ in $z$ and
$q=-\frac{m}{2}+i\frac{\rho}{2}-1$ in $\overline{z}$, the representation is
irreducible and equivalent to the finite-dimensional representation $(p,q)$.

  2. 2.

Not all the representations $(m,\rho)$ are unitary. They are unitary in only
two cases: when $\rho\in\mathbb{R}$ (principal series); when $m=0$ and
$i\rho\in]-2,0[$ (complementary series), provided another scalar product
chosen in the latter case (see below).

  3. 3.

Among these infinite irreps, only the principal representations $(\rho,k)$ and
$(-\rho,-k)$ are equivalent.

  4. 4.

A proof of the result above can be found in Naimark ([Nai64] pp. 294-295). A
sketch of it in the case of the principal series can be found in appendix B.

Principal series. When $\rho\in\mathbb{R}$, the scalar product over
$L^{2}(\mathbb{C})$ becomes the more usual

$(f_{1},f_{2})\overset{\rm
def}{=}\frac{i}{2}\int_{\mathbb{C}}\overline{f_{1}(z)}f_{2}(z)dzd\overline{z},$
(4.6)

and the representation $(\rho,m)\in\mathbb{R}\times\mathbb{Z}$ is unitary. It
is called the principal series. Moreover the representations $(\rho,m)$ and
$(-\rho,-m)$ are unitarily equivalent.

> $\bigstar$ Nota Bene. We have used below the convention of Naimark for
> indexing the representations of the principal unitary series,
> $(\rho,m)\in\mathbb{R}\times\mathbb{Z}$ (see [Nai64] p. 150). Vilenkin uses
> the same convention ([GGV66] p. 191). Rühl uses rather the convention
> $(\rho_{R},m_{R})=(\rho,-m)$ ([Rüh70] p. 54). Gel’fand uses the convention
> $(\rho_{G},m_{G})=(\rho/2,m)$ ([GMS63] p. 247). In LQG, it is common to use
> the conventions $(p,k)=(\rho/2,m/2)$, as Rovelli does ([RV14] p. 182) or
> Barrett [BDF+10]. Usually in LQG, the parameter $k$ is even restricted to be
> non-negative, which actually does not suppress any representation since
> $(p,k)$ and $(-p,-k)$ are equivalent.

Complementary series. When $m=0$ and $i\rho\in]-2,0[$, the action becomes

$a\cdot\phi(\xi)=|a_{12}\xi+a_{22}|^{i\rho-2}\phi\left(\frac{a_{11}\xi+a_{21}}{a_{12}\xi+a_{22}}\right).$
(4.7)

It also defines a unitary representation for the scalar product

$\innerproduct{\varphi}{\phi}=\left(\frac{i}{2}\right)^{2}\int_{\mathbb{C}^{2}}\frac{\overline{\varphi(\xi)}\phi(\eta)}{|\xi-\eta|^{2+i\rho}}d\xi
d\overline{\xi}d\eta d\overline{\eta}.$ (4.8)

###  4.3 Principal unitary series

In this section, we will present three realisations of the unitary principal
series. The construction of the principal series by Gel’fand and Naimark is
based on the induced representations method (see appendix B for a sketch of
the proof). Though rigorous from the mathematical point of view, it is not
very intuitive, especially for physicists. In 1962, Gel’fand, Graev and
Vilenkin published a book (referred to as ‘the Vilenkin’ hereafter) where they
build the principal series from a space of homogeneous functions, which may
seem more natural ([GGV66] pp. 139-201). A beautiful and concise exposure can
be found in the article of Dao and Nguyen ([DN67] pp. 18-21). We present the
method in the following subsection.

####  4.3.1 Homogeneous realisation

Consider $\mathcal{F}(\mathbb{C}^{2})$, the vector space of the complex
functions over $\mathbb{C}^{2}$. A function $F\in\mathcal{F}(\mathbb{C}^{2})$
is said to be homogeneous of degree $(\lambda,\mu)\in\mathbb{C}^{2}$ if it
satisfies for all $\alpha\in\mathbb{C}$ :

$F(\alpha\boldsymbol{z})=\alpha^{\lambda}\overline{\alpha}^{\mu}F(\boldsymbol{z}).$
(4.9)

To be consistently defined (when $\alpha=e^{2i\pi n}$), the degree should
satisfy the condition :

$\mu-\lambda\in\mathbb{Z}.$ (4.10)

Instead of $(\lambda,\mu)$, we will instead use in the following, the
parameters $(p=\frac{\mu+\lambda+2}{2i},k=\frac{\lambda-\mu}{2})$ (same choice
of parameters as Rovelli [RV14] p. 182). Define
$\mathcal{D}^{(p,k)}[z_{0},z_{1}]$ as the subspace of homogeneous functions of
degree $(\lambda,\mu)$ infinitely differentiable over
$\mathbb{C}^{2}\setminus\\{0\\}$ in the variables $z_{0},z_{1},\bar{z}_{0}$ et
$\bar{z}_{1}$ with a certain topology111The topology is defined by the
following property of convergence: a sequence $F_{n}(z_{0},z_{1})$ is said to
converge to $0$ if it converges to zero uniformly together with all its
derivatives on any compact set in the $(z_{0},z_{1})$-p1ane which does not
contain $(0,0)$ (see Vilenkin [GGV66] p. 142).. We define a continuous
representation $SL_{2}(\mathbb{C})$ over $\mathcal{D}^{(p,k)}[z_{0},z_{1}]$ by

$a\cdot F(\boldsymbol{z})\overset{\rm def}{=}F(a^{T}\boldsymbol{z}).$ (4.11)

> $\bigstar$ Nota Bene. We could also have defined the action by
> $F(a^{-1}\boldsymbol{z})$, $F(a^{\dagger}\boldsymbol{z})$ or
> $F(\boldsymbol{z}a)$. In fact $F(\boldsymbol{z}a)=F(a^{T}\boldsymbol{z})$,
> defines the same action. The action with $a^{\dagger}$ is obtained by the
> transformation $a_{ij}\mapsto\overline{a_{ij}}$. The action with $a^{-1}$ is
> obtained by the transformation
> $a_{ij}\mapsto(2\delta_{ij}-1)\sum_{kl}(1-\delta_{ik})(1-\delta_{jl})a_{kl}$.
> The convention that we have chosen here is the one of Vilenkin ([GGV66] p.
> 145), Dao and Nguyen ([DN67] p. 18), Rühl ([Rüh70] p. 53), Rovelli ([RV14]
> p. 182) and Barrett [BDF+10]. Knapp ([Kna86] p. 28) is using the convention
> $F(a^{-1}\boldsymbol{z})$.

Now define the following $2$-form over $\mathbb{C}^{2}$:

$\displaystyle\Omega(z_{0},z_{1})=\frac{i}{2}(z_{0}dz_{1}-z_{1}dz_{0})\wedge(\overline{z_{0}}d\overline{z_{1}}-\overline{z_{1}}d\overline{z_{0}}).$

Interestingly, it is invariant for the action of $SL_{2}(\mathbb{C})$:
$\Omega(a\boldsymbol{z})=\Omega(\boldsymbol{z})$. Let $\Gamma$ be a path in
$\mathbb{C}^{2}$ that intersects each projective line exactly once. Then
define the scalar product over $\mathcal{D}^{(p,k)}[z_{0},z_{1}]$:

$\displaystyle(F,G)=\int_{\Gamma}\overline{F(\boldsymbol{z})}G(\boldsymbol{z})\Omega(\boldsymbol{z}).$

Thus $\mathcal{D}^{(p,k)}[z_{0},z_{1}]$ is an Hilbert space. Interestingly,
the result does not depend on the path $\Gamma$ provided $p\in\mathbb{R}$,
which we consider to be the case in the following. This scalar product is
invariant for $SL_{2}(\mathbb{C})$: $(a\cdot F,a\cdot G)=(F,G)$. Thus the
representation is unitary. It could be also shown to be irreducible. In the
next subsection, we will see that the representation
$\mathcal{D}^{(p,k)}[z_{0},z_{1}]$ is equivalent to the representation
$(\rho=2p,m=2k)$ of the principal series described in section 4.2.

####  4.3.2 Projective realisation

Consider the map

$\displaystyle\iota:\left\\{\begin{array}[]{lll}\mathbb{C}&\rightarrow&\mathbb{C}^{2}\\\
\zeta&\mapsto&(\zeta,1)\end{array}\right.$

$\iota$ is a diffeomorphism from $\mathbb{C}$ to its range. It parametrises a
horizontal straight line of the complex plane. The projective construction
consists in restricting the domain of definition of the homogeneous function
to this line. If $F\in\mathcal{F}(\mathbb{C}^{2})$, define
$\iota^{*}F\in\mathcal{F}(\mathbb{C})$ as

$\iota^{*}F(z)\overset{\rm def}{=}F\circ\iota(z)=F(z,1).$ (4.12)

The $2$-form $\Omega$ becomes similarly

$\displaystyle\iota^{*}\Omega(z)=\frac{i}{2}dz\wedge d\overline{z},$

which is nothing but the usual Lebesgue measure over $\mathbb{C}$. Thus we
define the Hilbert space $L^{2}(\mathbb{C})$ with the the scalar product

$\displaystyle(f,g)=\frac{i}{2}\int_{\mathbb{C}}\overline{f(z)}g(z)dz\wedge
d\overline{z}.$

Thus we have a map $\iota^{*}:\mathcal{D}^{(p,k)}[z_{0},z_{1}]\rightarrow
L^{2}(\mathbb{C})$. In fact $\iota^{*}$ is bijective: for all $f\in
L^{2}(\mathbb{C})$, there exists a unique
$F\in\mathcal{D}^{(p,k)}[z_{0},z_{1}]$ such that $f=\iota^{*}F$. $F$ is given
explicitly by

$F(z_{0},z_{1})=z_{1}^{-1+ip+k}\bar{z}_{1}^{-1+ip-k}f\left(\frac{z_{0}}{z_{1}}\right).$
(4.13)

Importantly, $\iota^{*}$ induces naturally an action of $SL_{2}(\mathbb{C})$
over $\mathcal{F}(\mathbb{C})$, such that $\iota^{*}$ becomes an intertwiner
between two equivalent representations. After computation, we obtain:

$a\cdot f(z)\overset{\rm
def}{=}(a_{12}z+a_{22})^{-1+ip+k}\overline{(a_{12}z+a_{22})}^{-1+ip-k}f\left(\frac{a_{11}z+a_{21}}{a_{12}z+a_{22}}\right).$
(4.14)

This formula is exactly the same formula as 4.5, with the indices
$(p,k)=(\rho/2,m/2)$. Thus we have constructed explicitly the representations
of the principal series, and we have shown the equivalence of the realisations
$\mathcal{D}^{(p,k)}[z_{0},z_{1}]$ and $L^{2}(\mathbb{C})$.

####  4.3.3 $SU(2)$-realisation

We are going to build another realisation of the unitary principal
representations. It is based on a space of ‘$U(1)$-covariant’ functions over
$SU(2)$. The general idea lies over the observation that
$SU(2)/U(1)\cong\mathbb{C}P^{1}$. To be precise, consider the map

$\displaystyle\kappa:\left\\{\begin{array}[]{lll}SU(2)&\rightarrow&\mathbb{C}^{2}\\\
u&\mapsto&(u_{21},u_{22})\end{array}\right.$

$\kappa$ is a diffeomorphism to its range. In some sense $SU(2)$ can be seen
as the ‘unit circle’ of $\mathbb{C}^{2}$, so that $\kappa$ can be seen as the
injection of the circle in the plane $\mathbb{C}^{2}$.

Then, define
$\kappa^{*}:\mathcal{F}(\mathbb{C}^{2})\rightarrow\mathcal{F}(SU(2))$ such
that

$\kappa^{*}F(u)\overset{\rm
def}{=}F\circ\kappa(u)=F\left(u_{21},u_{22}\right).$ (4.15)

If $F\in\mathcal{D}^{(p,k)}[z_{0},z_{1}]$, then we show easily that
$\kappa^{*}F$ satisfies the covariance property

$\kappa^{*}F(e^{i\theta\sigma_{3}}u)=e^{-2i\theta k}\kappa^{*}F(u).$ (4.16)

We denote $\mathcal{D}^{(p,k)}[u]\overset{\rm
def}{=}\kappa^{*}\mathcal{D}^{(p,k)}[z_{0},z_{1}]$. Thus
$\kappa^{*}:\mathcal{D}^{k}[z_{0},z_{1}]\to\mathcal{D}^{(p,k)}[u]$ is a
bijection. Its inverse is given explicitly by

$F(z_{0},z_{1})=(|z_{0}|^{2}+|z_{1}|^{2})^{-1+ip}\phi\left(\frac{1}{\sqrt{|z_{0}|^{2}+|z_{1}|^{2}}}\begin{pmatrix}z_{1}^{*}&-z_{0}^{*}\\\
z_{0}&z_{1}\end{pmatrix}\right).$ (4.17)

> $\blacktriangleright$ Proof. We have to check that 4.17 is the inverse
> expression of 4.15. Then we check the homogeneity property
> $(\kappa^{*})^{-1}\phi(\alpha(z_{0},z_{1}))=\alpha^{\lambda}\overline{\alpha}^{\mu}(\kappa^{*})^{-1}\phi(z_{0},z_{1})$.
> $\Box$

We could also translate the measure $\kappa^{*}\Omega$, and thus endow
$\mathcal{D}^{(p,k)}[u]$ with the structure of an Hilbert space.
Interestingly, it is a subspace of $L^{2}(SU(2))$. As previously, one can
translate the action of $SL_{2}(\mathbb{C})$ over $\mathcal{D}^{(p,k)}[u]$
such that $\kappa^{*}$ become a bijective intertwiner, and we obtain

$a\cdot\phi(u)=(|\beta_{a,u}|^{2}+|\alpha_{a,u}|^{2})^{-1+ip}\phi\left(\frac{1}{\sqrt{|\beta_{a,u}|^{2}+|\alpha_{a,u}|^{2}}}\begin{pmatrix}\alpha_{a,u}&-\beta_{a,u}^{*}\\\
\beta_{a,u}&\alpha_{a,u}^{*}\end{pmatrix}\right),$ (4.18)

with $\alpha_{a,u}\overset{\rm def}{=}(u_{21}a_{12}+u_{22}a_{22})^{*}$ and
$\beta_{a,u}\overset{\rm def}{=}u_{21}a_{11}+u_{22}a_{21}$. Thus
$\mathcal{D}^{(p,k)}[u]$ is a third equivalent realisation of the unitary
principal series. The equivalence with $L^{2}(\mathbb{C})$ is made through
$(\kappa\circ\iota^{-1})^{*}$ which gives explicitly

$\phi(u)=u_{22}^{-1+ip+k}\overline{u_{22}}^{-1+ip-k}f\left(\frac{u_{21}}{u_{22}}\right),$
(4.19)

and conversely

$f(z)=(1+|z|^{2})^{-1+ip}\phi\left(\frac{1}{\sqrt{1+|z|^{2}}}\begin{pmatrix}1&-z^{*}\\\
z&1\end{pmatrix}\right).$ (4.20)

####  4.3.4 Canonical basis

The advantage of the $SU(2)$-realisation is that we already know interesting
function over $SU(2)$, namely the coefficients of the Wigner matrix
$D^{j}_{pq}$. Are they elements of $\mathcal{D}^{(p,k)}[u]$? They are indeed,
provided they satisfy the covariance property 4.16. We compute easily

$\displaystyle
D^{j}_{pq}\left(e^{i\theta\sigma_{3}}u\right)=\sum_{k=-j}^{j}\bra{j,p}e^{i\theta\sigma_{3}}\ket{j,k}D^{j}_{kq}\left(u\right)=\sum_{k=-j}^{j}\delta_{pk}e^{2ik\theta}D^{j}_{kq}\left(u\right)=e^{2ip\theta}D^{j}_{pq}\left(u\right).$

Thus, the covariance property it is satisfied if $p=-k$, and so

$\forall j\in\\{|k|,|k|+1,...\\},\quad\forall q\in\\{-j,...,j\\},\quad
D^{j}_{-k,q}\in\mathcal{D}^{(p,k)}[u].$ (4.21)

Since the $D^{j}_{mn}(u)$ form a basis of $L^{2}(SU(2))$, we show easily that
the subset exhibited in 4.21 form a basis of $\mathcal{D}^{(p,k)}[u]$. Another
consequence is the following decomposition of $\mathcal{D}^{(p,k)}[u]$ into
irreps of $SU(2)$:

$\mathcal{D}^{(p,k)}[u]\cong\bigoplus_{j=|k|}^{\infty}\mathcal{Q}_{j}.$ (4.22)

We then call canonical basis of $\mathcal{D}^{(p,k)}[u]$ the set of functions:

$\phi^{(p,k)}_{jm}(u)\overset{\rm
def}{=}\sqrt{\frac{2j+1}{\pi}}D^{j}_{-k,m}(u),\quad\text{with}\
j=|k|,|k|+1,...\ \text{and}\ -j\leq m\leq j.$ (4.23)

From 2.7, we see that they satisfy the orthogonality relations

$\int_{SU(2)}\differential u\
\overline{\phi^{(p,k)}_{jm}(u)}\phi^{(p,k)}_{ln}(u)=\frac{1}{\pi}\delta_{jl}\delta_{mn}.$
(4.24)

> $\bigstar$ Nota Bene. In Rühl ([Rüh70] p. 59), the factor
> $\frac{1}{\sqrt{\pi}}$ is absent from the definition of the
> $\phi^{(p,k)}_{jm}$. Thus, the orthogonality relations do not show a factor
> $\frac{1}{\pi}$ on the RHS. We have chosen this factor so that the canonical
> basis $f^{(p,k)}_{jm}$ of $L^{2}(\mathbb{C})$ (see below 4.25) is
> orthonormal for the usual scalar product with the Lebesgue measure ${\rm
> d}z$ (for Rühl the measure is ${\rm d}z/\pi$).
>
> Moreover $\phi^{(p,k)}_{jm}$ could have been defined with a phase factor
> $e^{i\psi(p,j)}$. This is is set to zero in some literature including
> [Rüh70, BDF+10], and we follow that convention here. An alternative phase
> convention leading to real $SL_{2}(\mathbb{C})$-Clebsch-Gordan coefficients
> is obtained for the choice [KVM78, Spe17]
> $e^{i\psi(p,j)}=(-1)^{-\frac{j}{2}}\frac{\Gamma(j+i\rho+1)}{|\Gamma(j+i\rho+1)|}$.
> This phase convention is the one used in the approach of [Spe17] to compute
> the coefficients and vertex amplitudes and the numerical analysis [DS18,
> Don18, FMDS] that implements this approach. From the perspective of the
> large spin asymptotics this additional phase plays no role and can be taken
> into account independently, if so desired.
>
> An intermediate choice of phase is the one of [DN67, Ras03], which has the
> advantage of simplifying the recursion relations satisfied by the Clebsch-
> Gordan coefficients [ARRW70b, ARRW70a]. The latter are now either real or
> purely imaginary.

The intertwiner $\kappa^{*}$ enables to translate this basis in
$\mathcal{D}^{(p,k)}[z_{0},z_{1}]$, and we obtain the canonical basis:

$\displaystyle F^{(p,k)}_{jm}(z_{0},z_{1})$
$\displaystyle=\sqrt{\frac{2j+1}{\pi}}(|z_{0}|^{2}+|z_{1}|^{2})^{ip-1}D^{j}_{-k,m}\left(\frac{1}{\sqrt{|z_{0}|^{2}+|z_{1}|^{2}}}\begin{pmatrix}z_{1}^{*}&-z_{0}^{*}\\\
z_{0}&z_{1}\end{pmatrix}\right),$

where an explicit expression for $D^{j}_{-k,m}$ is given by equation 2.16. The
same is done with the intertwiner $\iota^{*}$ to $L^{2}(\mathbb{C})$, and we
obtain the canonical basis:

$f^{(p,k)}_{jm}(z)=\sqrt{\frac{2j+1}{\pi}}(1+|z|^{2})^{ip-1-j}D^{j}_{-k,m}\begin{pmatrix}1&-z^{*}\\\
z&1\end{pmatrix}$ (4.25)

> $\clubsuit$ Reminder. From 2.16 we have
>
> $D^{j}_{-k,m}\begin{pmatrix}1&-z^{*}\\\
> z&1\end{pmatrix}=\left(\frac{(j-k)!(j+k)!}{(j+m)!(j-m)!}\right)^{1/2}\sum_{i=\max(0,m-k)}^{\min(j-k,j+m)}\binom{j+m}{i}\binom{j-m}{j-k-i}z^{j+m-i}(-z^{*})^{j-k-i}.$
> (4.26)

The constant factors of 4.23 have been chosen so that

$\displaystyle\frac{i}{2}\int_{\mathbb{C}}\overline{f^{(p,k)}_{jm}(z)}f^{(p,k)}_{ln}(z)dzd\overline{z}=\delta_{jl}\delta_{mn}.$

Finally, in ket notations, the canonical basis is denoted $\ket{p,k,jm}$.

####  4.3.5 Action of the generators

Similarly to 2.13, the action of the $SL_{2}(\mathbb{C})$-generators can be
computed from the action of the group (see Naimark [Nai64] pp. 104-117). The
generators of the rotations ‘stay inside’ the same $SU(2)$-irreps:

$\left\\{\begin{array}[]{l}J_{3}\ket{p,k,j,m}=m\ket{p,k,j,m}\\\
J_{+}\ket{p,k,j,m}=\sqrt{(j+m+1)(j-m)}\ket{p,k,j,m+1}\\\
J_{-}\ket{p,k,j,m}=\sqrt{(j+m)(j-m+1)}\ket{p,k,j,m-1}\end{array}\right.$
(4.27)

The generators of the boost spread over the neighbouring subspaces:

$K_{3}\ket{p,k,j,m}=\alpha_{j}\sqrt{j^{2}-m^{2}}\ket{p,k,j-1,m}+\gamma_{j}m\ket{p,k,j,m}\\\
-\alpha_{j+1}\sqrt{(j+1)^{2}-m^{2}}\ket{p,k,j+1,m},$ (4.28)
$K_{+}\ket{p,k,j,m}=\alpha_{j}\sqrt{(j-m)(j-m-1)}\ket{p,k,j-1,m+1}\\\
+\gamma_{j}\sqrt{(j-m)(j+m+1)}\ket{p,k,j,m+1}\\\
+\alpha_{j+1}\sqrt{(j+m+1)(j+m+2)}\ket{p,k,j+1,m+1}$ (4.29)
$K_{-}\ket{p,k,j,m}=-\alpha_{j}\sqrt{(j+m)(j+m-1)}\ket{p,k,j-1,m-1}\\\
+\gamma_{j}\sqrt{(j+m)(j-m+1)}\ket{p,k,j,m-1}\\\
-\alpha_{j+1}\sqrt{(j-m+1)(j-m+2)}\ket{p,k,j+1,m-1}$ (4.30)

with $\gamma_{j}\overset{\rm def}{=}\frac{kp}{j(j+1)}$ and
$\alpha_{j}\overset{\rm
def}{=}i\sqrt{\frac{(j^{2}-k^{2})(j^{2}+p^{2})}{j^{2}(4j^{2}-1)}}$. From these
expressions, it is possible to compute the action of the two Casimir
operators:

$\displaystyle(\vec{K}^{2}-\vec{L}^{2})\ket{p,k,j,m}=(p^{2}-k^{2}+1)\ket{p,k,j,m}\
,$ (4.31) $\displaystyle\vec{K}\cdot\vec{L}\ket{p,k,j,m}=pk\ket{k,p,j,m}\ .$

####  4.3.6 $SL_{2}(\mathbb{C})$ Wigner’s matrix

We define the $SL_{2}(\mathbb{C})$ Wigner’s matrix by its coefficients

$D^{(p,k)}_{j_{1}q_{1}j_{2}q_{2}}(a)\overset{\rm
def}{=}\bra{p,k;j_{1}q_{1}}a\ket{p,k;j_{2}q_{2}}.$ (4.32)

These coefficients satisfy the orthogonality relations:

$\int_{SL_{2}(\mathbb{C})}dhD^{(p_{1},k_{1})}_{j_{1}m_{1}l_{1}n_{1}}(h)D^{(p_{2},k_{2})}_{j_{2}m_{2}l_{2}n_{2}}(h)=\frac{1}{4(p_{1}^{2}+k_{1}^{2})}\delta(p_{1}-p_{2})\delta_{k_{1}k_{2}}\delta_{j_{1}j_{2}}\delta_{l_{1}l_{2}}\delta_{m_{1}m_{2}}\delta_{n_{1}n_{2}}.$
(4.33)

##### Explicit expressions.

To compute explicitly, we use a concrete realisation, for instance

$D^{(p,k)}_{jplq}(g)=\frac{i}{2}\int_{\mathbb{C}}\differential\omega\differential\overline{\omega}\,\overline{f^{(p,k)}_{jp}(\omega)}(g_{12}\omega+g_{22})^{-1+ip+k}\overline{(g_{12}\omega+g_{22})}^{-1+ip-k}f^{(p,k)}_{lq}\left(\frac{g_{11}\omega+g_{21}}{g_{12}\omega+g_{22}}\right).$
(4.34)

Cartan decomposition states that any $g\in SL_{2}(\mathbb{C})$ can be written
(non uniquely) as $g=ue^{r\sigma_{3}/2}v^{-1}$, with $u,v\in SU(2)$ and
$r\in\mathbb{R}_{+}$ (see 1.4). Then, we show that

$D^{(p,k)}_{jmln}(g)=\sum_{q=-\min(j,l)}^{\min(j,l)}D^{j}_{mq}(u)d^{(p,k)}_{jlq}(r)D^{l}_{qn}(v^{-1}).$
(4.35)

with the reduced $SL_{2}(\mathbb{C})$ Wigner’s matrix defined as

$d^{(p,k)}_{jlm}(r)\overset{\rm def}{=}D^{(p,k)}_{jmlm}(e^{r\sigma_{3}/2}).$
(4.36)

> $\blacktriangleright$ Proof.
>
> $\displaystyle D^{(p,k)}_{jmln}(g)$
> $\displaystyle=\sum_{p\geq|k|}\sum_{q=-p}^{p}\sum_{p^{\prime}\geq|k|}\sum_{q^{\prime}=-p^{\prime}}^{p^{\prime}}D^{(p,k)}_{jmpq}(u)D^{(p,k)}_{pqp^{\prime}q^{\prime}}(e^{r\sigma_{3}/2})D^{(p,q)}_{p^{\prime}q^{\prime}ln}(v^{-1})$
> $\displaystyle=\sum_{p\geq|k|}\sum_{q=-p}^{p}\sum_{p^{\prime}\geq|k|}\sum_{q^{\prime}=-p^{\prime}}^{p^{\prime}}\delta_{jp}D^{j}_{mq}(u)D^{(p,k)}_{pqp^{\prime}q^{\prime}}(e^{r\sigma_{3}/2})\delta_{p^{\prime}l}D^{l}_{q^{\prime}n}(v^{-1})$
> $\displaystyle=\sum_{q=-j}^{j}\sum_{q^{\prime}=-l}^{l}D^{j}_{mq}(u)D^{(p,k)}_{jqlq^{\prime}}(e^{r\sigma_{3}/2})D^{l}_{q^{\prime}n}(v^{-1})$
> $\displaystyle=\sum_{q=-j}^{j}\sum_{q^{\prime}=-l}^{l}D^{j}_{mq}(u)\matrixelement{p,k,jq}{e^{r\sigma_{3}/2}}{p,k,lq^{\prime}}D^{l}_{q^{\prime}n}(v^{-1})$
> $\displaystyle=\sum_{q=-\min(j,l)}^{\min(j,l)}D^{j}_{mq}(u)D^{(p,k)}_{jqlq}(e^{r\sigma_{3}/2})D^{l}_{qn}(v^{-1})$
>
> $\Box$

We have the following symmetry properties:

$d^{(p,k)}_{jlm}(r)=d^{(-p,k)}_{ljm}(-r)=d^{(p,-k)}_{jl,-m}(r)=(-1)^{j-l}d^{(-p,-k)}_{ljm}(r)=\overline{d^{(p,k)}_{ljm}(-r)}.$
(4.37)

$\blacksquare$ Integral formula 1.

$d^{(p,k)}_{jlm}(r)=\sqrt{(2j+1)(2l+1)}\left(\frac{(j-k)!(j+k)!}{(j+p)!(j-p)!}\right)^{1/2}\left(\frac{(l-k)!(l+k)!}{(l+p)!(l-p)!}\right)^{1/2}\\\
\times\sum_{i=\max(0,p-k)}^{\min(j-k,j+p)}\sum_{i^{\prime}=\max(0,p-k)}^{\min(l-k,l+p)}\Bigg{[}(-1)^{j+l-2k-i-i^{\prime}}\binom{j+p}{i}\binom{j-p}{j-k-i}\binom{l+p}{i^{\prime}}\binom{l-p}{l-k-i^{\prime}}\\\
\times
e^{r(ip-1+p-k-2i^{\prime})}\int^{\infty}_{0}2|\omega|(1+|\omega|^{2})^{-ip-1-j}(e^{-2r}+|\omega|^{2})^{ip-1-l}|\omega|^{2(j+l-i^{\prime}-i+p-k)}d|\omega|\Bigg{]}$
(4.38)

> $\blacktriangleright$ Proof.
>
> $\displaystyle D^{(p,k)}_{jplp}(e^{r\sigma_{3}/2})$
> $\displaystyle=\int_{\mathbb{C}}d\omega\overline{f^{(p,k)}_{jp}(\omega)}e^{r(1-ip)}f^{(p,k)}_{lp}(e^{r}\omega)$
> $\displaystyle=\int_{\mathbb{C}}d\omega\sqrt{\frac{2j+1}{\pi}}(1+|\omega|^{2})^{-ip-1-j}\overline{D}^{j}_{-k,p}\begin{pmatrix}1&-\omega^{*}\\\
> \omega&1\end{pmatrix}e^{r(1-ip)}$
> $\displaystyle\quad\times\sqrt{\frac{2l+1}{\pi}}(1+e^{2r}|\omega|^{2})^{ip-1-l}D^{l}_{-k,p}\begin{pmatrix}1&-e^{r}\omega^{*}\\\
> e^{r}\omega&1\end{pmatrix}.$
>
> We conclude using the explicit expression of equation 4.26. $\Box$

$\blacksquare$ Integral formula 2.

$d^{(p,k)}_{jlm}(r)=\sqrt{(2j+1)(2l+1)}\sqrt{\frac{(j-k)!(j+k)!(l-k)!(l+k)!}{(j+m)!(j-m)!(l+m)!(l-m)!}}\\\
\times\sum_{n_{1},n_{2}}\Bigg{[}(-1)^{j+l+2m-n_{1}-n_{2}}\binom{j+m}{n_{1}}\binom{j-m}{j-k-
n_{1}}\binom{l+m}{n_{2}}\binom{l-m}{l-k-n_{2}}\\\ \times