The unitary representations
of the Poincaré group
in any spacetime dimension

Xavier Bekaerta,111E-mail address<EMAIL_ADDRESS>. and
Nicolas Boulangerb,222E-mail address<EMAIL_ADDRESS>; Chargé de
Recherches FNRS (Belgium).

a Laboratoire de Mathématiques et Physique Théorique,

Unité Mixte de Recherche $6083$ du CNRS, Fédération Denis Poisson,

Université François Rabelais,

Parc de Grandmount, 37200 Tours (France)

b Service de Mécanique et Gravitation,

Université de Mons-Hainaut, Académie Wallonie-Bruxelles,

6 avenue du Champ de Mars, 7000 Mons (Belgium)

An extensive group-theoretical treatment of linear relativistic wave equations
on Minkowski spacetime of arbitrary dimension $D\geqslant 3$ is presented in
these lecture notes. To start with, the one-to-one correspondence between
linear relativistic wave equations and unitary representations of the isometry
group is reviewed. In turn, the method of induced representations reduces the
problem of classifying the representations of the Poincaré group
$ISO(D-1,1)^{\uparrow}$ to the classication of the representations of the
stability subgroups only. Therefore, an exhaustive treatment of the two most
important classes of unitary irreducible representations, corresponding to
massive and massless particles (the latter class decomposing in turn into the
“helicity” and the “infinite-spin” representations) may be performed via the
well-known representation theory of the orthogonal groups $O(n)$ (with
$D-3\leqslant n\leqslant D-1$). Finally, covariant wave equations are given
for each unitary irreducible representation of the Poincaré group with non-
negative mass-squared. Tachyonic representations are also examined. All these
steps are covered in many details and with examples. The present notes also
include a self-contained review of the representation theory of the general
linear and (in)homogeneous orthogonal groups in terms of Young diagrams.

Expanded version of the lectures presented at the second Modave summer school
in mathematical physics (Belgium, August 2006).

##  1 Group-theoretical preliminaries

Elementary knowledge of the theory of Lie groups and their representations is
assumed (see e.g. the textbooks [1, 2] or the lecture notes [3]). The basic
definitions of the Lorentz and Poincaré groups together with some general
facts on the theory of unitary representations are reviewed in order to fix
the notation and settle down the prerequisites.

###  1.1 Universal covering of the Lorentz group

The group of linear homogeneous transformations
$x^{\prime\mu}=\Lambda^{\mu}_{~{}\nu}x^{\nu}$ ($\mu,\nu=0,1,\ldots,D-1$)
preserving the Minkowski metric $\eta_{\mu\nu}$ of “mostly plus” signature
$(-,+,\ldots,+)\,$,

$\Lambda^{T}\eta\Lambda=\eta\,,$

where $\Lambda^{T}$ denotes the matrix transpose of $\Lambda\,$, is called the
Lorentz group $O(D-1,1)$.

A massless particle propagates on the light-cone $x^{2}=0\,$. Without loss of
generality, one may consider that its momentum points along the $(D-1)$th
spatial direction. Then it turns out to be convenient to make use of the
light-cone coordinates

$x^{\pm}=\frac{1}{\sqrt{2}}\,(\,x^{D-1}\,\pm\,x^{0}\,)\quad\mbox{and}\quad
x^{m}\quad(m=1,\ldots,D-2)\,,$

where the Minkowski metric reads $\eta_{++}=0=\eta_{--}\,$,
$\eta_{+-}=1\,=\eta_{-+}\,$ and $\eta_{mn}=\delta_{mn}$ ($m,n=1,\ldots,D-2$).

On physical grounds, one will mainly be interested in the matrices $\Lambda$’s
with determinant $+1\,$ and such that $\Lambda^{0}_{~{}0}\geqslant 0\,$. It
can be shown that such matrices $\Lambda$’s also form a group that one calls
the proper orthochronous Lorentz group denoted by $SO(D-1,1)^{\uparrow}\,$. It
is connected to the identity, but not simply connected, that is to say, there
exist loops in the group manifold $SO(D-1,1)^{\uparrow}$ which are not
continuously contractible to a point. In order to study the representations
(reps) of $SO(D-1,1)^{\uparrow}\,$, one may first determine its universal
covering group, i.e. the Lie group which is simply connected and whose Lie
algebra is isomorphic to $\mathfrak{so}(D-1,1)\,$, the Lie algebra of
$SO(D-1,1)^{\uparrow}\,$. For $D\geqslant 4\,$, the universal covering group,
denoted $Spin(D-1,1)\,$, is the double cover of $SO(D-1,1)^{\uparrow}\,$. The
spin groups $Spin(D-1,1)$ have no intrinsically projective representations.
Therefore, a single (or double) valued “representation” of
$SO(D-1,1)^{\uparrow}$ is meant to be a genuine representation of
$Spin(D-1,1)\,$.

Warning: The double cover of $SO(2,1)^{\uparrow}$ is the group $SU(1,1)$, in
close analogy to the fact that the double cover of $SO(3)$ is $SU(2)\,$. The
group $SU(2)$ is also the universal covering group of $SO(3)\,$, but beware
that the universal cover of $SO(2,1)^{\uparrow}$ is actually
${\mathbb{R}}^{3}\,$, covering $SO(2,1)^{\uparrow}$ infinitely often. Thus one
may not speak of the spin group for the “degenerate case” of the proper
orthochronous Lorentz group in spacetime dimension three. The analogue is that
the universal cover of $SO(2)\cong U(1)$ is $\mathbb{R}$ covering it
infinitely often, so that one may not speak of the spin group for the
“degenerate case” of the rotation group in two spatial dimensions.

###  1.2 The Poincaré group and algebra

The transformations

$x^{\prime\mu}=\Lambda^{\mu}_{~{}\nu}x^{\mu}+a^{\mu}$

where $a$ is a spacetime translation vector, form the group of all
inhomogeneous Lorentz transformations. If one denotes a general transformation
by $(\Lambda,a)\,$, the multiplication law in the Poincaré group is given by

$(\Lambda_{2},a_{2})\cdot(\Lambda_{1},a_{1})=(\Lambda_{2}\Lambda_{1},a_{2}+\Lambda_{2}a_{1})\,,$

so that the inhomogeneous Lorentz group is the semi-direct product denoted by

$IO(D-1,1)={\mathbb{R}}^{D}\rtimes O(D-1,1)\,.$

The subgroup $ISO(D-1,1)^{\uparrow}$ of inhomogeneous proper orthochronous
Lorentz transformations is called the Poincaré group. The Lie algebra
$\mathfrak{iso}(D-1,1)$ of the Poincaré group is presented by the generators
$\\{\,P_{\mu}\,,\,M_{\nu\rho}\,\\}$ and by the commutation relations

$\displaystyle i\,[M_{\mu\nu},M_{\rho\sigma}]$ $\displaystyle=$
$\displaystyle\eta_{\nu\rho}M_{\mu\sigma}-\eta_{\mu\rho}M_{\nu\sigma}-\eta_{\sigma\mu}M_{\rho\nu}+\eta_{\sigma\nu}M_{\rho\mu}\,$
(1) $\displaystyle i\,{[}P_{\mu},M_{\rho\sigma}{]}$ $\displaystyle=$
$\displaystyle\eta_{\mu\rho}P_{\sigma}-\eta_{\mu\sigma}P_{\rho}\,,$ (2)
$\displaystyle i\,{[}P_{\mu},P_{\rho}{]}$ $\displaystyle=$ $\displaystyle
0\,.$ (3)

Two subalgebras must be distinguished: the Lie algebra $\mathfrak{so}(D-1,1)$
of the Lorentz group presented by the generators $\\{\,M_{\nu\rho}\,\\}$ and
by the commutation relations (1), and the Lie algebra ${\mathbb{R}}^{D}$ of
the Abelian translation group presented by the generators $\\{\,P_{\mu}\,\\}$
and by the commutation relations (3). The latter algebra ${\mathbb{R}}^{D}$ is
an ideal of the Poincaré algebra, as can be seen from (2). Altogether, this
implies that the Lie algebra of the Poincaré group is the semi-direct sum
$\mathfrak{iso}(D-1,1)={\mathbb{R}}^{D}\niplus\mathfrak{so}(D-1,1)\,$.

The Casimir elements of a Lie algebra $\mathfrak{g}$ are homogeneous
polynomials in the generators of $\mathfrak{g}$ providing a distinguished
basis of the center ${\cal Z}\Big{(}{\cal U}(\mathfrak{g})\Big{)}$ of the
universal enveloping algebra ${\cal U}(\mathfrak{g})$ (see e.g. the part V of
the lecture notes [3]). The quadratic Casimir operator of the Lorentz algebra
$\mathfrak{so}(D-1,1)$ is the square of the generators $M_{\mu\nu}$:

${\cal
C}_{2}\Big{(}\mathfrak{so}(D-1,1)\Big{)}\,=\,\frac{1}{2}\,M^{\mu\nu}M_{\mu\nu}\,.$
(4)

The quadratic Casimir operator of the Poincaré algebra $\mathfrak{iso}(D-1,1)$
is the square of the momentum

${\cal C}_{2}\Big{(}\mathfrak{iso}(D-1,1)\Big{)}\,=\,-P^{\mu}P_{\mu}\,,$ (5)

while the quartic Casimir operator is

${\cal C}_{4}\Big{(}\mathfrak{iso}(D-1,1)\Big{)}\,=\,-{1\over
2}P^{2}M_{\mu\nu}M^{\mu\nu}+M_{\mu\rho}P^{\rho}M^{\mu\sigma}P_{\sigma}\,,$

which, for $D=4$, is the square of the Pauli-Lubanski vector $W^{\mu}$,

$W^{\mu}:=\frac{1}{2}\,\varepsilon^{\mu\nu\rho\sigma}M_{\nu\rho}P_{\sigma}\,.$

###  1.3 ABC of unitary representations

The mathematical property that all non-trivial unitary reps of a non-compact
simple Lie group must be infinite-dimensional has some physical significance,
as will be reviewed later.

Finite-dimensional unitary reps of non-compact simple Lie groups: Let
$\textsc{U}:G\rightarrow U(n)$ be a unitary representation of a Lie group $G$
acting on a (real or complex) Hilbert space $\cal H$ of finite dimension
$n\in\mathbb{N}$. Then U is completely reducible. Moreover, if U is
irreducible and if $G$ is a connected simple non-compact Lie group, then U is
the trivial representation.

Proof: For the property that U is completely reducible, we refer e.g. to the
proof of the proposition 5.15 in [1]. The image $\textsc{U}(G)$ for any
unitary representation U defines a subgroup of $U(n)\,$. Moreover, the kernel
of U is a normal subgroup of the simple group $G$. Therefore, either the
representation is trivial and $\ker\textsc{U}=G\,$, or it is faithfull and
$\ker\textsc{U}=\\{e\\}\,$. In the latter case, U is invertible and its image
is isomorphic to its domain, $\textsc{U}(G)\cong G$. Actually, the image
$\textsc{U}(G)$ is a non-compact subgroup of $U(n)$ because if $\textsc{U}(G)$
was compact, then $\textsc{U}^{-1}\big{(}\textsc{U}(G)\big{)}=G$ would be
compact since $\textsc{U}^{-1}$ is a continuous map. But the group $U(n)$ is
compact, thus it cannot contain a non-compact subgroup. Therefore the
representation cannot be faithful, so that it is trivial. (A different proof
of the second part of the theorem may be found in the section 8.1.B of [2].) ∎

Another mathematical result which is of physical significance is the following
theorem on unitary irreducible representations (UIRs) of compact Lie groups.

Unitary reps of compact Lie groups: Let U be a UIR of a compact Lie group $G$,
acting on a (real or complex) Hilbert space $\cal H$. Then $\cal H$ is finite-
dimensional. Moreover, every unitary representation of $G$ is a direct sum of
UIRs (the sum may be infinite).

Proof: The proofs are somewhat lengthy and technical so we refer to the
section 7.1 of [2] for complete details. ∎

Examples of (not so) simple groups:

$\bullet$ On the one hand, all (pseudo)-orthogonal groups $O(p,q)$ are either
Abelian ($p+q=2$) or simple ($p+q>2$). Moreover, orthogonal groups ($p\,q=0$)
are compact, while pseudo-orthogonal groups ($p\,q\neq 0$) are non-compact.

$\bullet$ On the other hand, the inhomogeneous Lorentz group $IO(D-1,1)$ is
non-compact (both ${\mathbb{R}}^{D}$ and $O(D-1,1)$ are non-compact) but it is
not semi-simple (because its normal subgroup ${\mathbb{R}}^{D}$ is Abelian).

##  2 Elementary particles as unitary irreducible representations of the
isometry group

Except for the final remarks, this section is based almost ad verbatim on the
introduction of the illuminating work of Bargmann and Wigner [4], modulo some
changes of notation and terminology in order to follow the modern conventions.

The wave functions $\mid\psi\,\,\rangle$ describing the possible states of a
quantum-mechanical system form a linear vector space $\cal H$ which, in
general, is infinite-dimensional and on which a positive-definite inner
product $\langle\,\,\phi\mid\psi\,\,\rangle$ is defined for any two wave
functions $\mid\phi\,\,\rangle$ and $\mid\psi\,\,\rangle$ (i.e. they form a
Hilbert space). The inner product usually involves an integration over the
whole configuration or momentum space and, for particles of non-vanishing
spin, a summation over the spin indices.

If the wave functions in question refer to a free particle and satisfy
relativistic wave equations, there exists a correspondence between the wave
functions describing the same state in different Lorentz frames. The
transformations considered here form the group of all inhomogeneous Lorentz
transformations (including translations of the origin in space and time). Let
$\mid\psi\,\,\rangle$ and $\mid\psi\,\,\rangle^{\prime}$ be the wave functions
of the same state in two Lorentz frames $L$ and $L^{\prime}$, respectively.
Then $\mid\psi\,\,\rangle^{\prime}=U(\Lambda,a)\mid\psi\,\,\rangle$, where
$U(\Lambda,a)$ is a linear unitary operator which depends on the
transformation $(\Lambda,a)$ leading from $L$ to $L^{\prime}\,$. By a proper
normalization, $U$ is determined by $\Lambda$ up to a factor $\pm 1\,$.
Moreover, the operators $U$ form a single- or double-valued representation of
the inhomogeneous Lorentz group, i.e., for a succession of two transformations
$(\Lambda_{1},a_{1})$ and $(\Lambda_{2},a_{2})$, we have

$U(\Lambda_{2},a_{2})U(\Lambda_{1},a_{1})=\pm
U(\Lambda_{2}\Lambda_{1},a_{2}+\Lambda_{2}a_{1})\,.$ (6)

Since all Lorentz frames are equivalent for the description of our system, it
follows that, together with $\mid\psi\,\,\rangle\,$,
$U(\Lambda,a)\mid\psi\,\,\rangle$ is also a possible state viewed from the
original Lorentz frame $L\,$. Thus, the vector space $\cal H$ contains, with
every $\mid\psi\,\,\rangle\,$, all transforms
$U(\Lambda,a)\mid\psi\,\,\rangle\,$, where $(\Lambda,a)$ is any inhomogenous
Lorentz transformation.

The operators $U$ may also replace the wave equation of the system. In our
discussion, we use the wave functions in the “Heisenberg” representation, so
that a given $\mid\psi\,\,\rangle$ represents the system for all times, and
may be chosen as the “Schrödinger” wave function at time $t=0$ in a given
Lorentz frame $L$. To find $\mid\psi\,\,\rangle_{t_{0}}\,$, the Schrödinger
function at time $t_{0}\,$, one must therefore transform to a frame
$L^{\prime}$ for which $t^{\prime}=t-t_{0}\,$, while all other coordinates
remain unchanged. Then
$\mid\psi\,\,\rangle_{t_{0}}=U(\Lambda,a)\mid\psi\,\,\rangle\,$, where
$(\Lambda,a)$ is the transformation leading from $L$ to $L^{\prime}\,$.

A classification of all unitary representations of the inhomogeneous Lorentz
group, i.e. of all solution of (6), amounts, therefore, to a classification of
all possible relativistic wave equations. Two reps $U$ and
$\widetilde{U}=VUV^{-1}\,$, where $V$ is a fixed unitary operator, are
equivalent. If the system is described by wave functions
$\mid\psi\,\,\rangle\,$, the description by

$\widetilde{\mid\psi\,\,\rangle}=V\mid\psi\,\,\rangle$ (7)

is isomorphic with respect to linear superposition, with respect to forming
the inner product of two wave functions, and also with respect to the
transition from one Lorentz frame to another. In fact, if
$\mid\psi\,\,\rangle^{\prime}=U(\Lambda,a)\mid\psi\,\,\rangle\,$, then

$\widetilde{\mid\psi\,\,\rangle}^{\prime}=V\mid\psi\,\,\rangle^{\prime}=VU(\Lambda,a)V^{-1}\widetilde{\mid\psi\,\,\rangle}=\widetilde{U}(\Lambda,a)\widetilde{\mid\psi\,\,\rangle}\,.$

Thus, one obtains classes of equivalent wave equations. Finally, it is
sufficient to determine the irreducible representations (irreps) since any
other may be built from them.

Two descriptions which are equivalent according to (7) may be quite different
in appearance. The best known example is the description of the
electromagnetic field by the fieldstrength and the vector potential,
respectively. It cannot be claimed either that equivalence in the sense of (7)
implies equivalence in every physical aspect. It should be emphasized that any
selection of one among the equivalent systems involves an explicit or implicit
assumption as to possible interactions, etc. Our analysis is necessarily
restricted to free particles and does not lead to any assertion about possible
interactions.

The present discussion is not based on any hypothesis about the structure of
the wave equations provided that they be covariant. In particular, it is not
necessary to assume differential equations in configuration space. But it is a
result of the group-theoretical analysis that every irreducible wave equation
is equivalent, in the sense of (7), to a system of differential equations for
fields on Minkowski spacetime.

Remarks:

$\bullet$ An important theorem proved by Wigner is that any symmetry
transformation that is continuously related to the identity must be
represented by a linear unitary operator (see e.g. the appendix A of [5]).
Strictly speaking, physical states are represented by rays in a Hilbert space.
Therefore the unitary representations of the symmetry group are actually
understood to be projective representations. In spacetime dimensions
$D\geqslant 4\,$, this subtlety reduces to the standard distinction between
single and double valued representations of the Poincaré group, as was taken
for granted in the text.

$\bullet$ Notice that the previous discussion remains entirely valid if the
Minkowski spacetime ${\mathbb{R}}^{D-1,1}$ is replaced everywhere by any other
maximally symmetric spacetime (i.e. de Sitter spacetime $dS_{D}$, or anti de
Sitter spacetime $AdS_{D}$) under the condition that the inhomogeneous Lorentz
group $IO(D-1,1)$ be also replaced everywhere by the corresponding group of
isometries (respectively, $\,O(D,\,1)$ or, $O(D-1,\,2)\,$).

$\bullet$ In first-quantization, particles are described by fields on the
spacetime and isometries are represented by unitary operators. A particle is
said to be “elementary” if the representation is irreducible, and “composite”
if the representation is made of a product of irreps. In second-quantization,
a unitary representation of the isometry group describes the one-particle
Hilbert space of states.

Another point of view: On the one hand, the rules of quantum mechanics imply
that quantum symmetries correspond to unitary representations of the symmetry
group carried by the Hilbert space of physical states. Furthermore, if time
translations are a one-parameter subgroup of the symmetry group, then the
Schrödinger equation is essentially a unitary representation of this subgroup.
On the other hand, the principle of relativity dictates that the isometries of
spacetime be symmetries of the physical system. All together, this implies
that linear relativistic wave equations may be identified with unitary reps of
the isometry group.

##  3 Classification of the unitary representations

###  3.1 Induced representations

The method of induced reps was introduced by Wigner in his seminal paper [6]
on the unitary representations of the inhomogeneous Lorentz group $IO(3,1)$ in
four spacetime dimensions, which admits a straightforward generalization to
any spacetime dimension $D$, as reviewed now. The content of this subsection
finds its origin in the section 2.5 of the comprehensive textbook [5].

From (3) one sees that all the translation generators commute with each other,
so it is natural to express physical states $\mid\psi\,\,\rangle$ in terms of
eigenvectors of the translation generators $P^{\mu}\,$. Introducing a label
$\sigma$ to denote all other degrees of freedom, one thus considers states
$\Psi_{q,\sigma}$ with ${P}_{\mu}\Psi_{q,\sigma}=q_{\mu}\Psi_{q,\sigma}\,$.
From the infinitesimal translation $U=\hbox{
1\hskip-3.0ptl}-iP^{\mu}\epsilon_{\mu}$ and repeated applications of it, one
finds that finite translations are represented on $\cal H$ by $U(\hbox{
1\hskip-3.0ptl},a)=\exp(-i\,P^{\mu}a_{\mu})\,$, so one has

$U(\hbox{ 1\hskip-3.0ptl},a)\,\Psi_{q,\sigma}\,=\,e^{-i\,q\cdot
a}\,\Psi_{q,\sigma}\,.$

Using (2), one sees that the effect of operating on $\Psi_{p,\sigma}$ with a
quantum homogeneous transformation $U(\Lambda,0)\equiv U(\Lambda)$ is to
produce an eigenvector of the translation generators with eigenvalue $\Lambda
p$ :

$\displaystyle P^{\mu}U(\Lambda)\Psi_{p,\sigma}$ $\displaystyle=$
$\displaystyle
U(\Lambda)[U^{-1}(\Lambda)P^{\mu}U(\Lambda)]\Psi_{p,\sigma}=U(\Lambda)({\Lambda^{-1}}_{\rho}^{~{}\mu}P^{\rho})\Psi_{p,\sigma}$
$\displaystyle=$
$\displaystyle\Lambda^{\mu}_{~{}\rho}p^{\rho}U(\Lambda)\Psi_{p,\sigma}\,,$

since ${(\Lambda^{-1})}^{\rho}_{~{}\nu}=\Lambda_{\nu}^{~{}\rho}\,$. Hence
$U(\Lambda)\Psi_{p,\sigma}$ must be a linear combination of the states
$\Psi_{\Lambda p,\sigma}$ :

$U(\Lambda)\Psi_{p,\sigma}=\sum_{\sigma^{\prime}}C_{\sigma^{\prime}\sigma}(\Lambda,p)\Psi_{\Lambda
p,\sigma^{\prime}}\,.$ (8)

In general, it is possible by using suitable linear combinations of the
$\Psi_{p,\sigma}$ to choose the $\sigma$ labels in such a way that the matrix
$C_{\sigma^{\prime}\sigma}(\Lambda,p)$ is block-diagonal; in other words, so
that the $\Psi_{p,\sigma}$ with $\sigma$ within any one block by themselves
furnish a representation of the Poincaré group. It is natural to identify the
states of a specific particle type with the components of a representation of
the Poincaré group which is irreducible, in the sense that it cannot be
further decomposed in this way. It is clear from (8) that all states
$\Psi_{p,\sigma}$ in an irrep of the Poincaré group have momenta $p$ belonging
to the orbit of a single fixed momentum, say $q^{\mu}$.

One has to work out the structure of the coefficients
$C_{\sigma^{\prime}\sigma}(\Lambda,p)$ in irreducible representations of the
Poincaré group. In order to do that, note that the only functions of $p^{\mu}$
that are left invariant by all transformations $\Lambda^{\mu}_{~{}\nu}\in
SO(D-1,1)^{\uparrow}$ are, of course, $p^{2}=\eta_{\mu\nu}p^{\mu}p^{\nu}$ and,
for $p^{2}\leqslant 0\,$, also the sign of $p^{0}\,$. Hence, for each value of
$p^{2}\,$, and (for $p^{2}\leqslant 0$) each sign of $p^{0}\,$, one can choose
a standard four-momentum, say $q^{\mu}\,$, and express any $p^{\mu}$ of this
class as

$p^{\mu}=L^{\mu}_{~{}\nu}(p)q^{\nu}\,,$

where $L^{\mu}_{~{}\nu}$ is some standard proper orthochronous Lorentz
transformation that depends on $p^{\mu}\,$, and also implicitly on our choice
of $q^{\mu}\,$. One can define the states $\Psi_{p,\sigma}$ of momentum
$p^{\mu}$ by

$\Psi_{p,\sigma}\equiv N(p)\,U\Big{(}L(p)\Big{)}\,\Psi_{q,\sigma}\,,$ (9)

where $N(p)$ is a numerical normalization factor. Operating on (9) with an
arbitrary homogeneous Lorentz transformation $U(\Lambda)\,$, one now finds

$\displaystyle U(\Lambda)\Psi_{p,\sigma}$ $\displaystyle=$ $\displaystyle
N(p)\,U\Big{(}\Lambda L(p)\Big{)}\,\Psi_{q,\sigma}$ (10) $\displaystyle=$
$\displaystyle N(p)\,U\Big{(}L(\Lambda p)\Big{)}\,U\Big{(}L^{-1}(\Lambda
p)\Lambda L(p)\Big{)}\Psi_{q,\sigma}\,.$

The point of this last step is that the Lorentz transformation $L^{-1}(\Lambda
p)\Lambda L(p)$ takes $q$ to $L(p)q=p\,$, then to $\Lambda p\,$, and finally
back to $q\,$, so it belongs to the subgroup of the Lorentz group consisting
of Lorentz transformations $W^{\mu}_{~{}\nu}$ that leave $q^{\mu}$ invariant :
$W^{\mu}_{~{}\nu}q^{\nu}=q^{\mu}\,$. This stability subgroup is called the
little group corresponding to $q\,$. For any $W,\bar{W}$ in the little group,
one has

$U(W)\Psi_{q,\sigma}=\sum_{\sigma^{\prime}}D^{q}_{\sigma^{\prime}\sigma}(W)\Psi_{q,\sigma^{\prime}}$
(11)

and

$D^{q}_{\sigma^{\prime}\sigma}(\bar{W}W)=\sum_{\sigma^{\prime\prime}}D^{q}_{\sigma^{\prime}\sigma^{\prime\prime}}(\bar{W})D^{q}_{\sigma^{\prime\prime}\sigma}(W)\,,$

that is to say, the coefficients $D^{q}(W)$ furnish a representation of the
little group. In particular, for $W(\Lambda,p)\equiv L^{-1}(\Lambda p)\Lambda
L(p)\,$, (10) becomes

$U(\Lambda)\Psi_{p,\sigma}=N(p)\sum_{\sigma^{\prime}}D_{\sigma^{\prime}\sigma}(W(\Lambda,p))U\Big{(}L(\Lambda
p)\Big{)}\Psi_{q,\sigma^{\prime}}$

or, recalling the definition (9),

$U(\Lambda)\Psi_{p,\sigma}=\frac{N(p)}{N(\Lambda
p)}\sum_{\sigma^{\prime}}D_{\sigma^{\prime}\sigma}\big{(}W(\Lambda,p)\big{)}\Psi_{\Lambda
p,\sigma^{\prime}}\,.$ (12)

Apart from the question of normalization, the problem of determining the
coefficients $C_{\sigma^{\prime}\sigma}$ in the transformation rule (8) has
been reduced to the problem of determining the coefficients
$D_{\sigma^{\prime}\sigma}$. In other words, the problem of determining all
possible irreps of the Poincaré group has been reduced to the problem of
finding all possible irreps of the little group, depending on the class of
momentum to which $q^{\mu}$ belongs. This approach, of deriving
representations of a semi-direct product like the inhomogeneous Lorentz group
from the representations of the stability subgroup, is called the method of
induced representations.

The wave function $\Psi_{p,\sigma}$ depends on the momentum, therefore its
Fourier transform $\Psi_{x,\sigma}$ depends on the spacetime coordinate, so
that the wave function is called a (complex) field on Minkowski spacetime
${\mathbb{R}}^{D-1,1}$ and the quantities $\Psi_{x,\sigma}$ at fixed $x$ and
for varying $\sigma$ are referred to as its physical components at $x\,$.

###  3.2 Orbits and stability subgroups

The orbit of a given non-vanishing vector $q^{\mu}$ of Minkowski spacetime
${\mathbb{R}}^{D-1,1}$ under Lorentz transformations is, by definition, the
hypersurface of constant momentum square $p^{2}\,$. Geometrically speaking, it
is a quadric of curvature radius $m>0$. More precisely, the hypersurface

  * •

$p^{2}=-m^{2}$ is a two-sheeted hyperboloid, each sheet of which is called a
mass-shell. The corresponding UIR is said to be massive.

  * •

$p^{2}=0$ is a cone, each half of which is called a light-cone. The
corresponding UIR is said to be massless ($m=0$).

  * •

$p^{2}=+m^{2}$ is a one-sheeted hyperboloid. The corresponding UIR is said to
be tachyonic.

Orthochronous Lorentz transformations preserve the sign of the time component
of vectors of non-positive momentum-squared, thus the orbit of a time-like
(light-like) vector is the mass-shell (respectively, light-cone) to which the
corresponding vector belongs.

Remarks:

$\bullet$ Notice that the Hilbert space carrying the irrep is indeed an
eigenspace of the quadratic Casimir operator (5), the eigenvalue of which is
${\cal C}_{2}=\pm\,m^{2}\,$ (the eigenvalue is real because the representation
is unitary), as it should according to Schur’s lemma. Moreover, the quadratic
Casimir classifies the UIRs but does not entirely characterize them.

$\bullet$ As quoted in Section 2, it is not necessary to assume differential
equations in position space, because the group-theoretical analysis leads
directly to a wave function which is a function of the momenta on the orbit,
the Fourier transform of which is a function in position space obeying the
Klein–Gordon equation
$\Box\,\Psi_{x,\sigma}\,=\,\pm\,m^{2}\,\Psi_{x,\sigma}\,$.

By going to a rest-frame, it is easy to show that the stabilizer of a time-
like vector $q^{\mu}=(m,\overrightarrow{0})\neq 0$ is the rotation subgroup
$SO(D-1)\subset SO(D-1,1)^{\uparrow}$. For a space-like vector, one may choose
a frame such that the non-vanishing momentum is along the $(D-1)$th spatial
axis: $q^{\mu}=(0,0,\ldots,0,m)\neq 0$. Thus its stabilizer is the subgroup
$SO(D-2,1)^{\uparrow}\subset SO(D-1,1)^{\uparrow}$. In the case of a light-
like vector, the little group “is not quite so obvious” to determine, as was
stressed by Wigner himself [7]. It clearly contains the rotation group
$SO(D-2)$ in the space-like hyperplane ${\mathbb{R}}^{D-2}$ transverse to the
light-ray along the momentum. Now, we will provide an algebraic proof that the
stabilizer of a light-like vector is the Euclidean group $ISO(D-2)\,$.
According to Wigner, reviewing his $D=4$ analysis, “no simple argument is
known (…) to show directly that the group of Lorentz transformations which
leave a null vector invariant is isomorphic to the two-dimensional Euclidean
group, desirable as it would be to have such an argument. Clearly, there is no
plane in the four-space of momenta in which these transformations could be
interpreted directly as displacements (…) because all transformations
considered here are homogeneous” [7]. Even though there is no simple geometric
way to understand this fact, the algebraic proof reviewed here is rather
straightforward.

Proof: By going in a light-cone frame (see Section 1.1), it is possible to
express the components of a momentum $p^{\mu}$ obeying $p^{2}=0$ as
$p_{\mu}=(p_{-},0,0,\ldots,0)\,$. In words, one can set the component $p_{+}$
to zero, as well as all the transverse components $p_{m}\,$
($m=1,\ldots,D-2$). The condition that the component $p_{-}$ be unaffected by
a Lorentz transformation translates as
$0\stackrel{{\scriptstyle!}}{{=}}i[p_{-},M_{\nu\rho}]=\eta_{-\nu}\,p_{\rho}-\eta_{-\rho}\,p_{\nu}$
due to (2). Obviously, the transformation generated by $M_{+-}$ does modify
$p_{-}$, hence it cannot be part of the little group for $p\,$. The other
Lorentz generators preserve $p_{-}\,$, but they should also satisfy the
equations $[p_{m},M_{\mu\nu}]=0=[p_{+},M_{\mu\nu}]\,$. It is readily seen that
$i[p_{m},M_{n-}]=\delta_{mn}p_{-}\neq 0$ (for $m=n$), therefore $M_{n-}$ does
not belong to the little group of $p_{\mu}$ either. We are left with the
generators $\\{M_{mn},M_{+n}\\}$ which preserve the (vanishing) value of
$p_{+}\,$. It turns out to be more convenient for later purpose to work with
the generators $\pi_{n}:=p_{-}M_{+n}=p^{\mu}M_{\mu\,n}$ instead. This
redefinition does not modify the algebra since $p_{-}$ commutes with all the
generators of the little group. From the Poincaré algebra (1)–(3) one finds,
in the light-cone frame,

$\displaystyle i\,[M_{mn},M_{pq}]$ $\displaystyle=$
$\displaystyle\delta_{np}M_{mq}-\delta_{mp}M_{nq}-\delta_{qm}M_{pn}+\delta_{qn}M_{pm}\,,$
(13) $\displaystyle i\,[\pi_{m},M_{np}]$ $\displaystyle=$
$\displaystyle\delta_{mn}\pi_{p}-\delta_{mp}\pi_{n}\,,$ (14) $\displaystyle
i\,{[}\pi_{m},\pi_{n}{]}$ $\displaystyle=$ $\displaystyle 0\,.$ (15)

As can be seen, the generators $\\{M_{mn},\pi_{m}\\}$ span the Lie algebra of
the inhomogeneous orthogonal group $ISO(D-2)\,$. ∎

For later purpose, notice that the quadratic Casimir operator of the Euclidean
algebra $\mathfrak{iso}(D-2)$ presented by the generators
$\\{M_{mn},\pi_{m}\\}$ and the relations (13)-(15) is the square of the
“translation” generators

${\cal C}_{2}\Big{(}\mathfrak{iso}(D-2)\Big{)}\,=\,\pi^{m}\pi_{m}\,.$ (16)

To end up this discussion, one should consider the case of a vanishing
momentum. Of course, the orbit of a vanishing vector under linear
transformations is itself while its stabilizer is the whole linear subgroup.
Therefore, the subgroup of $SO(D-1,1)^{\uparrow}$ leaving invariant the zero-
momentum vector $p^{\mu}=0$ is the whole group itself. This ends up the
determination of the orbit and stabilizer of any possible vector
$\in{\mathbb{R}}^{D-1,1}\,$.

Remark: The zero-momentum ($q^{\mu}=0$) representations are essentially UIRs
of the little group $SO(D-1,1)^{\uparrow}$ because the translation group acts
trivially. The proper orthochronous Lorentz group may be identified with the
isometry group of the de Sitter spacetime $dS_{D-1}$. In other words, the wave
function of the zero-momentum representation may be interpreted as a wave
function on a lower-dimensional de Sitter spacetime, and conversely. Even
though their physical meaning may differ, both UIRs may be mathematically
identified.

###  3.3 Classification

To summarize the previous subsection, the UIRs of the Poincaré group
$ISO(D-1,1)^{\uparrow}$ have been divided into four classes according to the
four possible orbits of the momentum, summarized in the following table (where
$m^{2}>0\,$):

Gender | Orbit | Stability subgroup | UIR
---|---|---|---
$p^{2}=-m^{2}$ | Mass-shell | $SO(D-1)$ | Massive
$p^{2}=0$ | Light-cone | $ISO(D-2)$ | Massless
$p^{2}=+m^{2}$ | Hyperboloid | $SO(D-2,1)^{\uparrow}$ | Tachyonic
$p_{\mu}=0$ | Origin | $SO(D-1,1)^{\uparrow}$ | Zero-momentum

The problem of classifying the UIRs of the Poincaré group
$ISO(D-1,1)^{\uparrow}$ has been reduced to the classication of the UIRs of
the stability subgroup of the momentum, which are either a unimodular
orthogonal group, an Euclidean group or a proper orthochronous Lorentz group.

Actually, the method of induced representation may also be applied to the
classification of the UIRs of the Euclidean group $ISO(D-2)$, the little group
of a massless particle. The important thing to understand is that the light-
like momentum $p^{\mu}$ is fixed and that what should be examined is the
action of the little group on the physical components characterized by
$\sigma\,$. From (15) one sees that the $D-2$ “translation” generators
$\pi^{i}$ commute with each other, so it is natural to express physical states
$\Psi_{p,\sigma}$ in terms of eigenvectors $\xi^{m}$ of these generators
$\pi^{m}$. Introducing a label $\varsigma$ to denote all remaining physical
components, one thus considers states $\Psi_{p,\,\xi,\,\varsigma}$ with
$\pi_{m}\Psi_{p,\,\xi,\,\varsigma}=\xi_{m}\Psi_{p,\,\xi,\,\varsigma}\,$. The
discussion presented in Subsection 3.1 may be repeated almost identically, up
to the replacement of the momentum $p$ by the eigenvector $\xi$, the label
$\sigma$ by $\varsigma$, the Poincaré group $ISO(D-1,1)^{\uparrow}$ by the
Euclidean group $ISO(D-2)$ and the proper orthochronous Lorentz group
$SO(D-1,1)^{\uparrow}$ by the unimodular orthogonal group $SO(D-2)\,$. The
conclusion is therefore similar: the problem of determining all possible
irreps of the massless little group $ISO(D-2)$ has been reduced to the problem
of finding all possible irreps of the stability subgroup of the $(D-2)$-vector
$\xi\,$, called the short little group in the literature [8].

The massless representations induced by a non-trivial representation of the
little group may therefore be divided into distinct categories, depending on
the class of momentum to which $\xi^{m}$ belongs. The situation is simpler
here because there exist only two possible classes of orbits for a vector in
the Euclidean space ${\mathbb{R}}^{D-2}$: either the origin $\xi^{m}=0\,$, or
a $(D-3)$-sphere of radius $\mu>0\,$. In the first case, the action of the
elusive “translation” operators $\pi^{m}$ is trivial and, effectively, the
little group is identified with the short little group $SO(D-2)$. These
representations are most often referred to as helicity representations by
analogy with the $D=4$ case. In the second case, the corresponding
representations are most often referred to as continuous spin representations
[8], even though Wigner also used the name infinite spin [7]. The former name
originates from the fact that the transverse vector $\xi^{m}$ has a continuous
range of values. Nevertheless, the latter name is more adequate in some
respect, as will be argued later on. Roughly speaking the point is that, on
the orbit $\xi^{2}=\mu^{2}$, the components spanned by the internal vector
$\xi^{m}$ take values on the sphere $S^{D-3}\subset{\mathbb{R}}^{D-2}$ of
radius $\mu=|\xi|\,$. The “radius” $\mu$ of this internal sphere has actually
the dimension of a mass parameter (the reason is that the sphere $S^{D-3}$ is
somehow in internal “momentum” space). Indeed, for massless representations,
the parameter $\mu$ classifying the various irreps should be understood as the
analogue of the mass for massive irreps, while the angular coordinates on the
sphere $S^{D-3}$ are the genuine “spin” degrees of freedom, the Fourier
conjugates of which are discrete variables as is more usual for spin degrees
of freedom. This point of view motivates the name “infinite spin.”333Actually,
in Subsection 5.3 an explicit derivation of the continuous spin representation
from a proper “infinite spin” limit of a massive representation is reviewed.
All the former comments find their natural interpretation in this point of
view.

To summarize, the UIRs of the Euclidean group $ISO(D-2)$ are divided into two
classes according to the two possible orbits of the $(D-2)$-vector $\xi_{m}$,
summarized in the following table:

Gender | Orbit | Stability subgroup | Massless UIR
---|---|---|---
$\xi^{2}\,=\,\mu^{2}$ | Sphere | $SO(D-3)$ | Infinite spin
$\xi_{m}\,=\,0$ | Origin | $SO(D-2)$ | Helicity

As a consequence of the method of induced representations, the physical
components of a first-quantized elementary particle span a UIR of the little
group. The number of local degrees of freedom (or of physical components) of
the field theory is thus given by the dimension of the Hilbert space carrying
the UIR of the little group. In the light of the standard results of
representation theory (reviewed in Subsection 1.3) and using the method of
induced representation, the UIRs of the Poincaré group may alternatively be
divided into two distinct classes: the finite-component ones (the massive and
the helicity reps) for which the (short) little group is compact, and the
infinite-component ones (the infinite-spin, the tachyonic and the zero-
momentum reps) for which the little group is non-compact.

Remarks:

$\bullet$ More precisely, the lower-dimensional cases $D=2,3$ are degenerate
in the following sense (when $p^{\mu}\neq 0$). In $D=2\,$, all little groups
are trivial, thus all physical fields are scalars. In $D=3\,$, all little
groups are Abelian (massive: $SO(2)$, massless: $\mathbb{R}$, tachyonic:
$SO(1,1)^{\uparrow}\cong\mathbb{R}$) hence all their UIRs have (complex)
dimension one: generically, fields have one physical degrees of freedom.
Notice that the helicity reps may be assigned a “conformal spin” if they are
extended to irreps of the group $SO(D,2)\supset SO(D-1,1)^{\uparrow}\,$.
Notice also that the “spin” of a massive representation is not discretized in
$D=3$ but is an arbitrary real number444This peculiarity is related to the
existence of anyons in three spacetime dimensions. [9] because the universal
cover of $SO(2,1)^{\uparrow}$ covers it infinitely often.

$\bullet$ For massive and helicity representations, the number of local
physical degrees of freedom may be determined from the well known formulas for
the dimension of any UIR of the orthogonal groups (reviewed in Subsection 4.3
for the tensorial irreps).

$\bullet$ This group-theoretical analysis does not probe topological theories
(such as Chern-Simons theory) because such theories correspond to identically
vanishing representations of the little group since they have no local
physical degrees of freedom.

The following corollary provides a group-theoretical explanation of the fact
that combining the principle of relativity with the rules of quantum mechanics
necessarily leads to _field_ theory.

Corollary: Every non-trivial unitary irreducible representation of the
isometry group of any maximally-symmetric spacetime is infinite-dimensional.

Proof: The Hilbert space carrying a non-trivial unitary representation of the
Poincaré group is infinite-dimensional because (i) in the generic case,
$q_{\mu}\neq 0$, the orbit is either a hyperboloid ($p^{2}\neq 0$) or a cone
($p^{2}=0$) and the space of wave functions on the orbit is infinite-
dimensional, (ii) the zero-momentum representations of the Poincaré group are
unitary representations of the de Sitter isometry group. Thus, the proof is
ended by noticing that all non-trivial unitary representations of the isometry
group of (anti) de Sitter spacetimes $(A)dS_{D}$ also are infinite-
dimensional, because their isometry groups are _pseudo_ -orthogonal Lie
groups.∎

##  4 Tensorial representations and Young diagrams

Most of the material reviewed here may be found in textbooks such as [10].
Nevertheless, large parts of this section are either copied or adapted from
the reference [11] because altogether it provides an excellent summary, both
for its pedagogical and comprehensive values. The material collected in the
present section goes slightly beyond what is strictly necessary for these
lectures, but the reader may find it useful in specific applications.

###  4.1 Symmetric group

An (unlabeled) Young diagram, consisting of $n$ boxes arranged in $r$ (left
justified) rows, represents a partition of the integer $n$ into $r$ parts:

$\displaystyle
n=\sum_{a=1}^{r}\lambda_{a}\qquad(\lambda_{1}\geqslant\lambda_{2}\geqslant\ldots\geqslant\lambda_{r})\,.$

That is, $\lambda_{a}$ is the number of boxes in the $a$th row. Successive row
lengths are non-increasing from top to bottom. A simpler notation for the
partition is the list of its parts:
$\lambda=\\{\lambda_{1},\lambda_{2},\ldots,\lambda_{r}\\}\,$. For instance,

Examples: There are five partitions of 4:

$\displaystyle\\{4\\},\\{3,1\\},\\{2,2\\},\\{2,1,1\\},\\{1,1,1,1\\}\,.$ (17)

Partitions play a key role in the study of the symmetric group
$\mathfrak{S}_{n}\,$. This is the group of all permutations of $n$ objects. It
has $n!$ elements and _its inequivalent irreducible representations may be
labeled by the partitions of $n\,$_. [In the following, Greek letters
$\lambda$, $\mu$ and $\nu$ will be used to specify not only partitions and
Young diagrams but also irreducible representations of the symmetric group and
other groups.]

The connection between the symmetric group and tensors was initially developed
by H. Weyl. This connection can be approached in (at least) two equivalent
ways.

  1. A.

Let $T_{\mu_{1}\ldots\,\mu_{n}}$ be a ‘generic’ $n$-index tensor, without any
special symmetry property. [For the moment, ‘tensor’ just means a function of
$n$ indices, not necessarily with any geometrical realization. It must be
meaningful, however, to _add_ — and form linear combinations of — tensors of
the same rank.] A Young tableau, or labeled Young diagram, is an assignment of
the numbers $1,2,\ldots,n$ to the $n$ boxes of a Young diagram $\lambda\,$.
The tableau is standard if the numbers are increasing both along rows from
left to right and down columns from top to bottom. The entries $1,\ldots,n$ in
the tableau indicate the $n$ successive indices of
$T_{\mu_{1}\ldots\mu_{n}}\,$. The tableau defines a certain symmetrization
operation on these indices: _symmetrize_ on the set of indices indicated by
the entries in each row, then _antisymmetrize_ the result on the set of
indices indicated by the entries in each column. The resulting object is a
tensor, $\widetilde{T}$, with certain index symmetries. Now let each
permutation of $\mathfrak{S}_{n}$ act (separately) upon $\widetilde{T}\,$. The
$n!$ results are not linearly independent; they span a vector space
$V_{\lambda}^{\mathfrak{S}_{n}}$ which supports an irreducible representation
of $\mathfrak{S}_{n}\,$. Different tableaux corresponding to the same diagram
$\lambda$ yield equivalent (by not identical) representations.

Example: The partition $\\{2,2\\}$ of $4$ has two standard tableaux:

(18)

Let us construct the symmetrized tensor $\widetilde{T}_{abcd}:=R_{ab|cd}$
corresponding to the second of these:

(19)

First symmetrize over the first and third indices ($a$ and $c$), and over the
second and fourth ($b$ and $d$):

$\frac{1}{4}\,(T_{abcd}+T_{cbad}+T_{adcb}+T_{cdab})\,.$

Now antisymmetrize the result over the first and second indices ($a$ and $b$),
and over the third and fourth ($c$ and $d$);555Here we adopt the convention
that the second round of permutations interchanges indices with the same
_names_ , rather than indices in the same _positions_ in the various terms.
The opposite convention is tantamount to antisymmetrizing _first_ , which
leads to a different, but mathematically isomorphic, development of the
representation theory. The issue here is analogous to the distinction between
space-fixed and body-fixed axes in the study of the rotation group (active or
passive transformations). dropping the combinatorial factor $\frac{1}{16}$, we
get

$\displaystyle R_{ab|cd}$ $\displaystyle=$ $\displaystyle
T_{abcd}+T_{cbad}+T_{adcb}+T_{cdab}-T_{bacd}-T_{cabd}-T_{bdca}-T_{cdba}$
$\displaystyle-$ $\displaystyle
T_{abdc}-T_{dbac}-T_{acdb}-T_{dcab}+T_{badc}+T_{dabc}+T_{bcda}+T_{dcba}\,.$

It is easy to check that $R$ possesses the symmetries of the Riemann tensor.
There are two independent orders of its indices (e.g. $R_{ab|cd}$ and
$R_{ac|bd}$), and applying any permutation to the indices produces some linear
combination of those two basic objects. On the other hand, performing on $T$
the operations prescribed by the first tableau in (18) produces a different
expression $P_{ac|bd}\,$, which, however, generates a two-dimensional
representation of $\mathfrak{S}_{4}$ with the same abstract index structure as
that generated by $R\,$. A non-standard tableau would also yield such a
representation, but the tensors within it would be linear combinations of
those already found. One finds

$\displaystyle P_{ac|bd}$ $\displaystyle=$ $\displaystyle
T_{abcd}+T_{bacd}+T_{abdc}+T_{badc}-T_{cbad}-T_{bcad}-T_{cbda}-T_{bcda}$
$\displaystyle-$ $\displaystyle
T_{adcb}-T_{dacb}-T_{adbc}-T_{dabc}+T_{cdab}+T_{dcab}+T_{cdba}+T_{dcba}\,.$

As the reader may check, no linear combinations of $P$ can reproduce $R\,$.
The objects $P_{ab|cd}$, $P_{ac|bd}$, $R_{ab|cd}$ and $R_{ac|bd}$ are linearly
independent. Although $R$ and $P$ are characterized by the same Young
_diagram_ , they are associated with different standard Young _tableaux_ and
therefore span two _different_ irreducible representations of
$\mathfrak{S}_{n}$.

Example: Define a _symmetrized Riemann tensor_ (the Jacobi tensor) by
$J_{ad;bc}:=\frac{1}{2}\,(R_{ab|cd}+R_{ac|bd})\,$. It obeys
$J_{ab;cd}=J_{ba;cd}=J_{ab;dc}\,$. Then it is easy to show that
$R_{ab|cd}=\frac{2}{3}\,(J_{ad;bc}-J_{bd;ac})\,$. Thus the tensor $J$ has no
fewer independent components and contains no less information than the tensor
$R$, despite the extra symmetrization; $R$ is recovered from $J$ by an
antisymmetrization. The tensors $R$ and $J$ are really the same tensor
expressed with respect to different bases.

  2. B.

The regular representation of $\mathfrak{S}_{n}$ is the $n!$-dimensional
representation obtained by letting $\mathfrak{S}_{n}$ act by left
multiplication on the formal linear combinations of elements of
$\mathfrak{S}_{n}\,$. [That is, one labels the basis vectors of
$\mathbb{R}^{n!}$ by elements of $\mathfrak{S}_{n}$, defines that action of
each permutation on the basis vectors in the natural way, and extends this
definition to the whole space by linearity.] Equivalently, the vector space of
the regular representation is the space of real-valued functions defined on
$\mathfrak{S}_{n}\,$. [In general the regular representation is defined with
complex scalars, but for $\mathfrak{S}_{n}$ it is sufficient to work with real
coefficients.]

Regular representation: The regular representation contains every irreducible
representation with a multiplicity equal to its dimension. Each Young diagram
$\lambda$ corresponds to an irreducible representation of
$\mathfrak{S}_{n}\,$. Its dimension and multiplicity are equal to the number
of standard tableaux of diagram $\lambda\,$.

The symmetrization procedure described under A. can be transcribed to the more
abstract context B. to construct a projection operator onto the subspace of
$\mathbb{R}^{n!}$ supporting each representation. [The numerical coefficient
needed to normalize the tableau operation as a projection — an operator whose
square is itself — is not usually the same as that accumulated from the
individual symmetrization operations. For example, to make $R_{abcd}$ into a
projection of $T_{abcd}$, one needs to divide by $12$, not $16$.]

Example: In (17), the partition $\\{4\\}$ corresponds to the totally symmetric
four-index tensors, a one-dimensional space $V^{\mathfrak{S}_{4}}_{\\{4\\}}$.
Similarly, $\\{1,1,1,1\\}$ yields the totally antisymmetric tensors. A generic
rank-four tensor, $T_{abcd}$, can be decomposed into the sum of its symmetric
and antisymmetric parts, plus a remainder. The theory we are expounding here
tells how to decompose the remainder further. The partition $\\{2,2\\}$ yields
two independent two-dimensional subrepresentations of the regular
representation; in more concrete terms, there are two independent pieces of
$T_{abcd}$ ($\frac{1}{12}\,R_{ab|cd}$ and $\frac{1}{12}\,P_{ac|bd}$)
constructed as described in connection with (18). One of these ($R_{ab|cd}$)
has exactly the symmetries of the Riemann tensor; the other ($P_{ac|bd}$,
coming from the first tableau of (18)) has the same abstract symmetry as the
Riemann tensor, but with the indices ordered differently. Finally, each of the
remaining partitions in (17) can be made into a standard tableau in three
different ways. Therefore, each of these two representations has three
separate pieces of $T$ corresponding to it, and each piece is three-
dimensional (has three independent index orders after its symmetries are taken
into account). Thus the total number of independent tensors which can be
formed from the irreducible parts of $T_{abcd}$ by index permutations is

$1^{2}+1^{2}+2^{2}+3^{2}+3^{2}=24=4!$

which is simply the total number of permutations of the indices of $T$ itself,
as it must be.

To state a formula for the dimension of an irreducible representation
$V^{\mathfrak{S}_{n}}_{\lambda}$ of $\mathfrak{S}_{n}\,$, we need the concept
of the hook length of a given box in a Young diagram $\lambda\,$. The hook
length of a box in a Young diagram is the number of squares directly below or
directly to the right of the box, including the box once:

(0,0)(10.5,0)1(10,10) (10.5,0)(10.5,0)1(10,10) (21,0)(10.5,0)1(10,10)
(31.5,0)(10.5,0)1(10,10) (0,-10.5)(10.5,0)1(10,10)
(10.5,-10.5)(10.5,0)1(10,10) (21,-10.5)(10.5,0)1(10,10)
(0,-21)(10.5,0)1(10,10) $\downarrow$$\longrightarrow$$-$$|$.$\bullet$

Example: In the following diagram, each box is labeled by its hook length:

(0,0)(10.5,0)1(10,10)6 (10.5,0)(10.5,0)1(10,10)4 (21,0)(10.5,0)1(10,10)3
(31.5,0)(10.5,0)1(10,10)1 (0,-10.5)(10.5,0)1(10,10)4
(10.5,-10.5)(10.5,0)1(10,10)2 (21,-10.5)(10.5,0)1(10,10)1
(0,-21)(10.5,0)1(10,10)1 .

One then has the following hook length formula for the dimension of the
representation $V^{\mathfrak{S}_{n}}_{\lambda}$ of $\mathfrak{S}_{n}$
corresponding to the Young diagram $\lambda\,$:

$\displaystyle\dim V^{\mathfrak{S}_{n}}_{\lambda}=\frac{n!}{\prod(\mbox{ hook
lengths})}\,.$ (20)

Remark: Note carefully that the “dimension” we have been discussing up to now
is the number of independent _index orders_ of a tensor, not the number of
independent _components_ when the tensor is realized geometrically with
respect to a particular underlying vector space or manifold. The latter number
depends on the dimension (say $D$) of that underlying space, while the former
is independent of $D$ (so long as $D$ is sufficiently large, as we tacitly
assume in generic discussions). For example, the number of components of an
antisymmetric two-index tensor is $\frac{D(D-1)}{2}\,$, but the number of its
index orders is always $1$, except in dimension $D=1$ where no non-zero
antisymmetric tensors exist at all.

###  4.2 General linear group

We now turn to the representation theory of the general linear and orthogonal
groups, where the (spacetime) dimension $D$ plays a key role. The theory of
partitions and of the representations of the permutation groups is the
foundation on which this topic is built.

Let $\\{v_{a}\\}$ represent a generic element of $\mathbb{R}^{D}$ (or of the
cotangent space at a point of a $D$-dimensional manifold). The action of non-
singular linear operators on this space gives a $D$-dimensional irreducible
representation $V$ of the general linear group $GL(D)\,$; indeed, this
representation defines the group itself. The rank-two tensors, $\\{T_{ab}\\}$,
carry a larger representation of $GL(D)$ ($V\otimes V$, of dimension $D^{2}$),
where the group elements act on the two indices simultaneously. The latter
representation is reducible: it decomposes into the subspace of symmetric and
antisymmetric rank-two tensors $V\otimes V\cong(V\odot V)\oplus(V\wedge V)$,
of respective dimensions $\frac{D(D+1)}{2}\,$ and $\frac{D(D-1)}{2}\,$.
Similarly, the tensor representation of rank $n$, $V^{\otimes n}$, decomposes
into irreducible representations of $GL(D)$ which are associated with the
irreducible representations of $\mathfrak{S}_{n}$ acting on the indices, which
in turn are labeled by the partitions of $n\,$, hence by Young diagrams. Young
diagrams with more than $D$ rows do not contribute [if $\lambda$ is a
partition of $n$ into more than $D$ parts, then the associated index
symmetrization of a $D$-dimensional rank-$n$ tensor yields an expression that
vanishes identically; in particular, there are no non-zero totally
antisymmetric rank-$n$ tensors if $n>D\,$].

More precisely, let $\lambda$ be a Young _tableau_. The Schur module
$V_{\lambda}^{GL(D)}$ is the vector space of all rank-$n$ tensors $\tilde{T}$
in $V^{\otimes n}$ such that:

> $(i)$ the tensor $\tilde{T}$ is completely antisymmetric in the entries of
> each column of $\lambda\,$,
>
> $(ii)$ complete antisymmetrization of $\tilde{T}$ in the entries of a column
> of $\lambda$ and another entry of $\lambda$ that is on the right-hand side
> of the column vanishes.

This construction is equivalent to the construction A.

Example: Associated with the Young tableau (19), the tensor $R_{ab|cd}$
introduced in the subsection 4.1 obeys to the conditions (i) and (ii):
$R_{ab|cd}=-R_{ba|cd}=-R_{ab|dc}$ and $R_{ab|cd}+R_{bc|ad}+R_{ca|bd}=0\,$.

As explained in the footnote 5, if one interchanges everywhere in the previous
constructions the words “symmetric” and “antisymmetric,” then the (reducible)
representation spaces characterized by the same Young _diagram_ [but not by
the same Young _tableau_] are isomorphic and the conditions (i)-(ii) must be
replaced with:

> $(a)$ the tensor is completely symmetric in the entries of each column of
> $\lambda$,
>
> $(b)$ complete symmetrization of the tensor in the entries of a row of
> $\lambda$ and another entry of $\lambda$ that sits in a lower row vanishes.

Example: Taking the standard Young tableau (19) and constructing, following
the “manifestly symmetric convention”, the irreducible tensor associated with
it, one obtains a tensor ${\cal{R}}$ with the same abstract index symmetries
as $J$ [i.e. obeying the constraints $(a)$ and $(b)$] but which is however
linearly independent from $J\,$, thence linearly independent from $R$ alone.
The tensor ${\cal{R}}$ can be expressed as a linear combination of _both_ $R$
_and_ $P\,$. Similarly, taking the first standard Young tableau in (18) and
following the manifestly symmetric convention, one obtains a tensor $\cal P$
obeying $(a)$ and $(b)$. This tensor is linearly independent from $P$ alone as
it is a linear combination of _both_ $P$ and $R\,$. Summarizing, associated
with the Young _diagram_ $\\{2,2\\}$ we have the (reducible) representation
space spanned by either $\\{R,P\\}$ in the manifestly antisymmetric convention
or by $\\{\cal{R},\cal{P}\\}$ in the manifestly symmetric convention.

Remarks:

$\bullet$ An important point to note is that, by the previous construction
featuring irreducible tensors with definite symmetry properties, we have got
_all_ the finite-dimensional irreducible representations of
$GL(D,\mathbb{R})\,$.

$\bullet$ In order to make contact with an alternative road to the
representation theory of $GL(D)$, one says that the irreducible representation
$\Gamma_{\lambda^{1}\,\ldots\,\lambda^{D-1}}$ of
${\mathfrak{sl}}(D,\mathbb{C})\equiv A_{D-1}$ with highest weight
$\Lambda=\lambda^{1}\Lambda_{(1)}+\lambda^{2}\Lambda_{(2)}+\ldots+\lambda^{D-1}\Lambda_{(D-1)}$
[see e.g. the Part II of the lecture notes [3] for definitions and notations]
is obtained by applying the Schur functor ${\mathbb{S}}_{\lambda}$ [i.e. the
construction presented above] to the standard representation $V$, where the
Young diagram is

$\displaystyle\lambda=\\{\lambda^{1}+\ldots+\lambda^{D-1}\,,\,\lambda^{2}+\ldots+\lambda^{D-1}\,,\,\ldots\,,\lambda^{D-1}\,,\,0\\}\,.$

In terms of the Young diagram for $\lambda\,$, the Dynkin labels $\lambda^{a}$
($1\leqslant a\leqslant D-1$) are the differences of lengths of rows:
$\lambda^{a}=\lambda_{a}-\lambda_{a+1}\,$.

Example: If $D=6$, then

(0,10)(10.5,0)1(10,10) (10.5,10)(10.5,0)1(10,10) (21,10)(10.5,0)1(10,10)
(31.5,10)(10.5,0)1(10,10) (42,10)(10.5,0)1(10,10) (52.5,10)(10.5,0)1(10,10)
(63,10)(10.5,0)1(10,10) (73.5,10)(10.5,0)1(10,10) (84,10)(10.5,0)1(10,10)
(0,-0.5)(10.5,0)1(10,10) (10.5,-0.5)(10.5,0)1(10,10) (21,-0.5)(10.5,0)1(10,10)
(31.5,-0.5)(10.5,0)1(10,10) (42,-0.5)(10.5,0)1(10,10)
(52.5,-0.5)(10.5,0)1(10,10) (0,-11)(10.5,0)1(10,10) (10.5,-11)(10.5,0)1(10,10)
(21,-11)(10.5,0)1(10,10) (31.5,-11)(10.5,0)1(10,10) (0,-21.5)(10.5,0)1(10,10)
(10.5,-21.5)(10.5,0)1(10,10) (21,-21.5)(10.5,0)1(10,10)
(31.5,-21.5)(10.5,0)1(10,10) (0,-32)(10.5,0)1(10,10) $\underbrace{\hskip
30.0pt}_{\lambda^{1}}$$\underbrace{\hskip
20.0pt}_{\lambda^{2}}$$\underbrace{\hskip
30.0pt}_{\lambda^{4}}$${}_{\lambda^{5}}$

is the Young diagram corresponding to the irrep $\Gamma_{3,2,0,3,1}$ of
$A_{5}\equiv{\mathfrak{sl}}(6,\mathbb{C})\,$.

The dimension of the representation $V^{GL(D)}_{\lambda}$ of $GL(D)$
corresponding to the Young diagram $\lambda\,$ is:

$\displaystyle\dim
V^{GL(D)}_{\lambda}=\,\prod\,\frac{D-\mbox{row}+\mbox{column}}{\mbox{ hook
length}}\,,$ (21)

where the product is over the $n$ boxes while “row” and “column” respectively
give the place of the corresponding box. As was underlined before, the formula
(21) is distinct from the hook length formula (20).

Examples:

$\bullet$ In the following diagram

(0,0)(10.5,0)1(10,10)5 (10.5,0)(10.5,0)1(10,10)6 (21,0)(10.5,0)1(10,10)7
(31.5,0)(10.5,0)1(10,10)8 (0,-10.5)(10.5,0)1(10,10)4
(10.5,-10.5)(10.5,0)1(10,10)5 (21,-10.5)(10.5,0)1(10,10)6
(0,-21)(10.5,0)1(10,10)3 ,

each box is labeled by its value in the numerator of (21) for $D=5$. Observe
that, for the corresponding diagram $\lambda\,$, $\dim
V^{GL(5)}_{\lambda}=1050\neq 70=\dim V^{\mathfrak{S}_{8}}_{\lambda}\,$.

$\bullet$ The space of (anti)symmetric tensors of $V$ of rank $n$ are denoted
by $\odot^{n}(V)$ (respectively, $\wedge^{n}(V)$). It carries an irreducible
representation of $GL(D)$ labeled by a Young diagram made of one row
(respectively, column) of length $n\,$. The dimensions

$\dim\odot^{n}(V)=\Big{(}\begin{array}[]{c}D+n-1\\\
n\end{array}\Big{)}\,,\quad\quad\dim\wedge^{n}(V)=\Big{(}\begin{array}[]{c}D\\\
n\end{array}\Big{)}\,,$ (22)

are easily computed from the formula (21) and reproduce the standard results
obtained from combinatorial arguments.

If $T_{1}$ and $T_{2}$ are tensors of ranks $n_{1}$ and $n_{2}$, then their
product is a tensor of rank $n_{1}+n_{2}\,$. Each factor $T_{j}$ transforms
under index permutation according to some representation of
$\mathfrak{S}_{n_{j}}\,$, and under linear transformation by the corresponding
representation of $GL(D)\,$. It follows immediately that the product tensor
$T_{1}\otimes T_{2}$ transforms as some representation of
$\mathfrak{S}_{n_{1}}\times\mathfrak{S}_{n_{2}}\,$. This _induces_ a
representation of the full permutation group $\mathfrak{S}_{n_{1}+n_{2}}$
which is associated with a corresponding representation of $GL(D)\,$, called
Kronecker product. It is possible to reduce these last two representations
into a sum of irreducible ones. We may assume that the factor representations
are irreducible, since the original tensors $T_{j}$ could have been broken
into irreducible parts at the outset.

Littlewood–Richardson rule: The decomposition of an “outer product”
$\mu\cdot\nu$ of irreducible representations $\mu$ and $\nu$ of
$\mathfrak{S}_{n_{1}}$ and $\mathfrak{S}_{n_{2}}$, respectively, into
irreducible representations of $\mathfrak{S}_{n_{1}+n_{2}}$ can be determined
by means of the following algorithm involving Young diagrams. The product is
commutative, so it does not matter which factor is regarded as the “right-
hand” one. [In practice, on should choose the simpler Young diagram for that
role.]

  * (I)

Label each box in the top row of the right-hand diagram, $\nu$, by “$a$”, each
box in the second row by “$b$”, etc.

  * (II)

Add the labeled boxes of $\nu$ to the left-hand diagram $\mu$, one at a time,
first the $a$s, then the $b$s, …, subject to these constraints:

    1. (A)

No two boxes in the same column are labeled with the same letter;

    2. (B)

At all stages the result is a legitimate Young diagram;

    3. (C)

At each stage, if the letters are read right-to-left along the rows, from top
to bottom, one never encounters more $b$s than $a$s, more $c$s than $b$s, etc.

  * (III)

Each of the distinct diagrams constructed in this way specifies an irreducible
subrepresentation $\lambda$, appearing in the decomposition of the outer
product. The same labeled Young diagram may arise in more than one way; the
multiplicity of that representation must be counted accordingly.

Remarks:

$\bullet$ This rule enables products of _distinct_ tensors to be decomposed.
When the factors are the same tensor, the list is further restricted by the
requirement of symmetry under interchange of the factors. This is the problem
of plethysm, whose solution requires more complicated techniques than the
Littlewood–Richardson rule.

$\bullet$ Representations with too many parts (columns of length greater than
$D$) must be deleted from the list of subrepresentations of the $GL(D)$. [If
irreducible representations of the special linear group $SL(D)$ are considered
instead, every column of length $D$ must be removed from the corresponding
Young diagram.]

###  4.3 Orthogonal group

It remains to consider index contractions. Up to now we considered only
covariant tensors, because in the intended application there is a metric
tensor which serves to relate contravariant and covariant tensors.
Contractions are mediated by this metric. Implicitly, therefore, one is
restricting the symmetry group of the problem from the general linear group to
the subgroup that leaves the metric tensor invariant, the orthogonal group
$O(D)\,$. [If the metric has indefinite signature, the true symmetry group is
a non-compact analogue of the orthogonal group, such as the Lorentz group.
This does not affect the relevant aspects of the finite-dimensional
representation theory.] Each irreducible $GL(D)$ representation
$V^{GL(D)}_{\lambda}$ decomposes into irreducible $O(D)$ representations
$V_{\nu}^{O(D)}$, labeled by Young diagrams $\nu$ obtained by removing an even
number of boxes from $\lambda\,$. The branching rule for this process involves
a sort of inverse of the Littlewood–Richardson rule:

Restriction from $GL(D)$ to $O(D)$: The irreps of $GL(D)$ may be reduced to
direct sums of irreps of $O(D)$ by extracting all possible trace terms formed
by contraction with products of the metric tensor and its inverse.

The reduction is given by the branching rule for $GL(D)\downarrow O(D)$:

$V_{\lambda}^{GL(D)}=V_{\lambda/\Delta}^{O(D)}\equiv V_{\lambda}^{O(D)}\oplus
V_{\lambda/\\{2\\}}^{O(D)}\oplus V_{\lambda/\\{4\\}}^{O(D)}\oplus
V_{\lambda/\\{2,2\\}}^{O(D)}\oplus\,\ldots$ (23)

where $\Delta$ is the formal infinite sum [12]

$\Delta=\,1\,+\begin{picture}(30.0,15.0)(-5.0,2.0)\multiframe(0,0)(10.5,0){2}(10,10){}{}
\end{picture}+\begin{picture}(55.0,15.0)(-5.0,2.0)\multiframe(0,0)(10.5,0){4}(10,10){}{}{}{}
\end{picture}+\begin{picture}(30.0,15.0)(-5.0,2.0)\multiframe(0,5)(10.5,0){2}(10,10){}{}
\multiframe(0,-5)(10.5,0){2}(10,10){}{} \end{picture}+\ldots$

corresponding to the sum of all possible plethysms of the metric tensor, and
where $\lambda/\mu$ means the sum of the Young diagrams $\nu$ such that
$\nu\cdot\mu$ contains $\lambda$ according to the Littlewood–Richardson rule
(with the corresponding multiplicity).

Examples:

$\bullet$ The $GL(D)$ irreducible representation labeled by the Young diagram
$\\{2,2\\}$ decomposes with respect to $O(D)$ according to the direct sum
$\\{2,2\\}/\Delta=\\{2,2\\}+\\{2,0\\}+\\{0,0\\}$ which corresponds to the
decomposition of the Riemann tensor into the Weyl tensor, the traceless part
of the Ricci tensor and the scalar curvature, respectively.

$\bullet$ The $GL(D)$ irreducible representation labeled by the Young diagram
$\\{n\\}$ decomposes with respect to $O(D)$ according to the direct sum
$\\{n\\}/\Delta=\\{n\\}+\\{n-2\\}+\\{n-4\\}+\,\ldots\,$, corresponding to the
decomposition of a completely symmetric tensor or rank $n$ into its traceless
part, the traceless part of its trace, etc. This provides an alternative proof
of the obvious fact that the number of independent components of a traceless
symmetric tensor of rank $n$ is equal to the number of independent components
of a symmetric tensor of rank $n$ minus the number of independent components
of a symmetric tensor of rank $n-2$ (its trace): $\dim
V_{\\{n\\}}^{O(D-2)}=\dim V_{\\{n\\}}^{GL(D)}-\dim V_{\\{n-2\\}}^{GL(D)}\,$.
Using the formula (22) allows to show that

$\dim V_{\\{n\\}}^{O(D)}=\frac{(D+2n-2)(D+n-3)!}{n!(D-2)!}\,.$ (24)

The very useful formula (24) contains as a particular case the well-known fact
that all the traceless symmetric tensorial representations of $O(2)$ are two-
dimensional (indeed, any UIR of an Abelian group is of complex dimension one).
Moreover, the traceless symmetric tensorial representations of rank $n$ of the
rotation group $O(3)$ are the well-known integer spin representations of
dimension equal to $2n+1\,$.

The following theorem is very important (see e.g. the first reference of
[10]):

Vanishing irreps for (pseudo-)orthogonal groups: Whenever the sum of the
lengths of the first two columns of a Young diagram $\lambda$ is greater than
$D$, then the irreducible representation of $O(D)$ labeled by $\lambda$ is
identically zero.

Young diagrams such that the sum of the lengths of the first two columns does
not exceed $D$ are said to be allowed.

Finite-dimensional irreps of (pseudo-)orthogonal groups: Each non-zero finite-
dimensional irreducible representation of $O(p,q)$ is isomorphic to a
completely traceless tensorial representation, the symmetry properties of
which are labeled by an allowed Young diagram $\lambda\,$.

The dimension of the tensorial irrep is determined by the following rule due
to King [13]:

  * ($\alpha$)

The numbers $D-1$, $D-3$, $D-5$, $\ldots$, $D-2r+1$are place in the end boxes
of the $1$st, $2$nd, $3$rd, $\ldots$, $r$th rows of the diagram $\lambda\,$. A
labeled Young diagram of $n$ numbers is then constructed by inserting in the
remaining boxes of the diagram, numbers which increase by one in passing from
one box to its left-hand neighbor.

  * ($\beta$)

This labeled Young diagram is extended to the limit of the triangular Young
diagram $\tau$ of $r$ rows. This produces a Young diagram
$\widetilde{\lambda}$ the $a$th row of which has length equal the maximum
between the two integers $\tau_{a}=r-a+1$ and $\lambda_{a}$.

  * ($\gamma$)

The series of numbers in any row of the Young diagram $\widetilde{\lambda}$ is
then extended by inserting in the remaining boxes of the diagram, numbers
which decrease by one in passing from one box to its right-hand neighbor. The
resulting numbers will be called the “King length.”

  * ($\delta$)

The row lengths $\lambda_{1}$, $\lambda_{2}$, $\ldots$, $\lambda_{r}$ are then
added to all of the numbers of the Young diagram $\widetilde{\lambda}$ which
lie on lines of unit slope passing through the first box of the $1$st, $2$nd,
$\ldots$, $r$th rows, respectively, of the Young diagram $\lambda\,$.

The dimension is equal to the product of the integers in the resulting labeled
Young diagram $\widetilde{\lambda}$ divided by the product of

  * -

the hook length of each box of $\lambda$, and of

  * -

the King length of each box of $\widetilde{\lambda}$ outside $\lambda\,$.

Examples:

$\bullet$ In the following diagram, allowed for $D=5$,

(0,0)(10.5,0)1(10,10)7 (10.5,0)(10.5,0)1(10,10)6 (21,0)(10.5,0)1(10,10)5
(31.5,0)(10.5,0)1(10,10)4 (0,-10.5)(10.5,0)1(10,10)4
(10.5,-10.5)(10.5,0)1(10,10)3 (21,-10.5)(10.5,0)1(10,10)2
(0,-21)(10.5,0)1(10,10)0 ,

each box is labeled by its King length, while in the diagram

(0,0)(10.5,0)1(10,10)11 (10.5,0)(10.5,0)1(10,10)9 (21,0)(10.5,0)1(10,10)6
(31.5,0)(10.5,0)1(10,10)4 (0,-10.5)(10.5,0)1(10,10)7
(10.5,-10.5)(10.5,0)1(10,10)4 (21,-10.5)(10.5,0)1(10,10)2
(0,-21)(10.5,0)1(10,10)1 ,

each box is labeled by the number obtained at the very end of King’s rule.
Observe that, for the corresponding diagram $\lambda\,$, it was not necessary
to perform the steps ($\beta$)-($\gamma$) and that, $\dim
V^{O(5)}_{\lambda}=231<1050=\dim V^{GL(5)}_{\lambda}\,$.

$\bullet$ In the following Young diagram $\lambda=\\{2,2,1\\}\,$, allowed for
$D=5$,

(0,0)(10.5,0)1(10,10)5 (10.5,0)(10.5,0)1(10,10)4 (0,-10.5)(10.5,0)1(10,10)3
(10.5,-10.5)(10.5,0)1(10,10)2 (0,-21)(10.5,0)1(10,10)0 ,

each box is labeled by the number obtained after step $(\alpha)\,$. The step
$(\beta)$ is now necessary and gives the Young diagram
$\widetilde{\lambda}=\\{3,2,1\\}\,$. At the end of steps $(\gamma)$ and
$(\delta)$, respectively, the result is

$\stackrel{{\scriptstyle(\gamma)}}{{\longrightarrow}}$(0,0)(10.5,0)1(10,10)5
(10.5,0)(10.5,0)1(10,10)4 (21,0)(10.5,0)1(10,10)3 (0,-10.5)(10.5,0)1(10,10)3
(10.5,-10.5)(10.5,0)1(10,10)2 (0,-21)(10.5,0)1(10,10)0
$\stackrel{{\scriptstyle(\delta)}}{{\longrightarrow}}$(70,0)(10.5,0)1(10,10)7
(80.5,0)(10.5,0)1(10,10)6 (91,0)(10.5,0)1(10,10)4 (70,-10.5)(10.5,0)1(10,10)5
(80.5,-10.5)(10.5,0)1(10,10)3 (70,-21)(10.5,0)1(10,10)1 ,

so that $\dim V^{O(5)}_{\lambda}=$ $\frac{7\cdot 6\cdot 5\cdot 4\cdot
3}{(4\cdot 3\cdot 2)\cdot(3)}\,=35<75$ $=\dim V^{GL(5)}_{\lambda}\,$.

$\bullet$ The space of traceless symmetric tensors of $V$ of rank $n$ carries
an irreducible representation of $O(D)$ labeled by a Young diagram made of one
row of length $n\,$ for which the dimension (24) is easily reproduced from the
King rule, since the rules ($\beta$)-($\gamma$) may be omitted

$\bullet$ Computing the number of components of the Weyl tensor and of a
symmetric, traceless, rank-two tensor in $D=4$ dimensions, enables one to give
the decomposition $\\{2,2\\}/\Delta=\\{2,2\\}+\\{2,0\\}+\\{0,0\\}$ of the
Riemann tensor into the Weyl tensor, the traceless part of the Ricci tensor
and the scalar curvature, respectively, in terms of the corresponding
dimensions. This gives the well-known result $20=10+9+1\,$.

Unitary irreps of orthogonal groups: Each non-zero inequivalent UIR of $O(D)$
corresponds to an allowed Young diagram $\lambda\,$, and conversely.

Proof: The orthogonal group is compact, thence any UIR is finite-dimensional
(see Subsection 1.3). Furthermore, any finite-dimensional irrep of the
orthogonal group is labeled by an allowed Young diagram. Moreover, an
important result is that any finite-dimensional representation may be endowed
with a sesquilinear form which makes it unitary.∎

The quadratic Casimir operator of the orthogonal algebra $\mathfrak{so}(D)$
presented by its generators and its commutation relations

$i\,[M_{\mu\nu},M_{\rho\sigma}]=\delta_{\nu\rho}M_{\mu\sigma}-\delta_{\mu\rho}M_{\nu\sigma}-\delta_{\sigma\mu}M_{\rho\nu}+\delta_{\sigma\nu}M_{\rho\mu}\,$
(25)

is the sum of square of the generators (similarly to the definition (4) for
$\mathfrak{so}(D-1,1)$ since these two complex algebras are isomorphic). Its
eigenvalue on a finite-dimensional irrep labeled by an allowed Young diagram
$\lambda=\\{\lambda_{1},\lambda_{2},\ldots,\lambda_{r}\\}$ is given in the
subsection 9.4.C of [2]:

$\Big{[}\,\,{\cal
C}_{2}\Big{(}\mathfrak{so}(D)\Big{)}\,-\,\sum_{a=1}^{r}\lambda_{a}(\lambda_{a}+D-2a)\,\Big{]}\,V_{\lambda}^{O(D)}=0\,.$
(26)

Examples:

$\bullet$ The UIRs of the Abelian $O(2)\cong U(1)$ are labeled by one integer
only, which is the eigenvalue of the single generator on the irrep, say
$h\in\mathbb{Z}\,$. The only allowed Young diagrams are made of a single row
of length equal to the non-negative integer $s=|h|\,$. The traceless symmetric
tensorial representations of $O(2)$ are two-dimensional, the sum of the two
irreps labeled by $h=\pm s\,$. The formula (26) with $D=2\,$, $r=1$ and
$\lambda_{1}=s$ gives the obvious eigenvalue $s^{2}\,$, since the quadratic
Casimir operator of the rotation group $O(2)$ is equal to the square of the
single generator.

$\bullet$ The quadratic Casimir operator of the rotation group $O(3)$ is the
square of the angular momentum. The irrep of $O(3)$ with spin $s\in\mathbb{N}$
is labeled by the allowed Young diagram made of a single row of length equal
to the integer $s\,$. The formula (26) with $D=3\,$, $r=1$ and $\lambda_{1}=s$
gives the celebrated eigenvalue $s(s+1)\,$.

$\bullet$ The irrep of $O(D)$ carried by the space of traceless symmetric
tensors of rank $n$ is labeled by the allowed Young diagram $\\{n\\}$ made of
a single row of length equal to an integer $n\,$. The formula (26) with $r=1$
and $\lambda_{1}=n$ gives the eigenvalue $n(n+D-2)$ for the quadratic Casimir
operator.

The following branching rule is extremely useful in the process of dimensional
reduction.

Restriction from $GL(D)$ to $GL(D-1)$: The restriction to the subgroup
$GL(D-1)\subset GL(D)$ of a finite-dimensional irrep of $GL(D)$ determined by
the Young diagram $\lambda$ contains each irrep of $GL(D-1)$ labeled by Young
diagrams $\mu$ such that

$\lambda_{1}\geqslant\mu_{1}\geqslant\lambda_{2}\geqslant\mu_{2}\geqslant\ldots\geqslant\mu_{r-1}\geqslant\lambda_{r}\geqslant\mu_{r}\geqslant
0\,,$

with multiplicity one. The same theorem holds for the restriction
$O(D)\downarrow O(D-1)$ where $\lambda$ is an allowed Young diagram.

These rules are discussed in the section 8.8.A of [2]. They may be summarized
in the following branching rule for $GL(D)\downarrow GL(D-1)$,

$V_{\lambda}^{GL(D)}=V_{\lambda/\Sigma}^{GL(D-1)}\equiv
V_{\lambda}^{GL(D-1)}\oplus V_{\lambda/\\{1\\}}^{GL(D-1)}\oplus
V_{\lambda/\\{2\\}}^{GL(D-1)}\oplus
V_{\lambda/\\{3\\}}^{GL(D-1)}\oplus\,\ldots$ (27)

where $\Sigma$ is the formal infinite sum of all Young diagrams made of a
single row.

Example: The branching rule applied to symmetric irrep labeled by a Young
diagram $\\{n\\}$ made of one row of length $n$ gives as a result:

$\\{n\\}/\Sigma\,=\,\\{\,n\,\,\\}+\\{n-1\\}+\\{n-2\\}+\ldots+\\{1\\}+\\{0\\}\,.$

This implies the obvious fact that a completely symmetric tensor of rank $n$
whose indices run over $D$ values may be decomposed as a sum of completely
symmetric tensors of rank $n$, $n-1$, $\ldots\,$, $1$, $0$ whose indices run
over $D-1$ values. A non-trivial instance of the branching rule for
$O(D)\downarrow O(D-1)$ is that the same result is true for traceless
symmetric tensors as well.

###  4.4 Auxiliary variables

Let $\lambda$ be a Young diagram with $s$ columns and $r$ rows.

The Schur module $V_{\lambda}^{GL(D)}$ in the “manifestly antisymmetric
convention” can be built via a convenient construction in terms of polynomials
in $s\times D$ graded variables satisfying appropriate conditions. More
precisely, the vector space $V_{\lambda}^{GL(D)}$ is isomorphic to a subspace
of the algebra

$\displaystyle\odot^{s}\big{(}\wedge(V)\,\big{)}\equiv\underbrace{\wedge(V)\odot\ldots\odot\wedge(V)}_{\mbox{$s$
factors}}\,.$ (28)

of $s$ symmetric tensor products of antisymmetric forms $\in\wedge(V)$. The
elements of $\odot^{s}\big{(}\wedge(V)\,\big{)}$ are usually called
multiforms. The $D$ generators of the $I$th factor $\wedge(V)$ are written
$d_{{}_{I}}x^{\mu}$ ($\,I=1,2,\ldots,s\,$). By definition, the multiform
algebra is presented by the graded commutation relations

$\displaystyle
d_{{}_{I}}x^{\mu}\,d_{{}_{J}}x^{\nu}\,=\,(-)^{\delta_{{}_{IJ}}}\,d_{{}_{J}}x^{\nu}\,d_{{}_{I}}x^{\mu}\,,$
(29)

where the wedge and symmetric products are not written explicitly. The
condition (i) of Subsection 4.2 is automatically verified for any element
$\Phi\in\odot^{s}\big{(}\wedge(V)\,\big{)}$ due to the fact that the variables
are anticommuting in a fixed column ($I=J$). The $GL(D)$-irreducibility
condition (ii) of Subsection 4.2 is implemented by the conditions

$\Big{(}\,d_{{}_{I}}x\cdot\frac{\partial^{L}}{\partial(d_{{}_{J}}x)}\,-\,\delta_{{}_{IJ}}\,\ell_{{}_{I}}\,\Big{)}\Phi\,=\,0\,,\quad(I\leqslant
J)$ (30)

where the dot stands for the contraction of the indices, $\ell_{I}$ for the
length of the $I$th column in the Young diagram $\lambda$ and $\partial^{L}$
stands for “left” derivative. By the Weyl construction, an element
$\Phi\in\odot^{s}\big{(}\wedge(V)\,\big{)}$ satisfying (30) belongs to the
Schur module $V_{\lambda}^{GL(D)}\,$. Following the discussion of Subsection
4.3, if $\lambda$ denotes an allowed Young diagram, such an element $\Phi\in
V_{\lambda}^{GL(D)}$ is irreducible under the (pseudo)-orthogonal group
$O(p,q)$ ($p+q=D$) if it is traceless, that is

$\Big{(}\,\frac{\partial^{L}}{\partial(d_{{}_{I}}x)}\cdot\frac{\partial^{L}}{\partial(d_{{}_{J}}x)}\,\Big{)}\Phi\,=\,0\,,\quad(\forall\;I,J)$
(31)

where the dot stands now for the contraction of indices via the use of the
metric preserved by $O(p,q)$. An element
$\Phi\in\odot^{s}\big{(}\wedge(V)\,\big{)}$ such that (30)-(31) are fulfilled
belongs to the Schur module $V_{\lambda}^{O(p,q)}$ labeled by the Young
diagram $\lambda\,$.

The Schur module $V_{\lambda}^{GL(D)}$ admits another convenient realization
in terms of polynomials in $r\times D$ commuting variables. In other words,
the vector space $V_{\lambda}^{GL(D)}$ is isomorphic to a subspace of the
polynomial algebra in the variables $u^{\mu}_{a}$ ($a=1,2,\ldots,r$) where the
index $a$ corresponds to each row. The condition (a) of Subsection 4.2 is
automatically verified for any such polynomial due to the fact that the
variables are commuting in a fixed row. The $GL(D)$-irreducibility condition
(b) of Subsection 4.2 is implemented by the conditions

$\Big{(}\,u_{a}\cdot\frac{\partial}{\partial
u_{b}}\,-\,\delta_{{}_{ab}}\,\lambda_{a}\,\Big{)}\Phi\,=\,0\,,\quad(a\leqslant
b)$ (32)

where the dot still stands for the contraction of the indices. The degree of
homogeneity of the polynomial $\Phi$ in the variables $u^{\mu}_{a}$ (for fixed
$a$) is $\lambda_{a}\,$. The corresponding coefficients are tensors
irreducible under the general linear group. By the Weyl construction, a
polynomial $\Phi(u_{a})$ satisfying (32) belongs to the Schur module
$V_{\lambda}^{GL(D)}\,$. Again, such an element $\Phi\in V_{\lambda}^{GL(D)}$
is irreducible under the (pseudo)-orthogonal group $O(p,q)$ ($p+q=D$) iff it
is traceless, that is

$\Big{(}\,\frac{\partial}{\partial u_{a}}\cdot\frac{\partial}{\partial
u_{b}}\,\Big{)}\Phi\,=\,0\,,\quad(\forall\;a,b)$ (33)

where the dot stands for the contraction of indices via the use of the metric
preserved by $O(p,q)$. A polynomial $\Phi(u_{a})$ such that (32)-(33) are
fulfilled belongs to the Schur module $V_{\lambda}^{O(p,q)}$ labeled by an
allowed Young diagram $\lambda\,$.

Example: Consider an irreducible representation of the orthogonal group $O(D)$
labeled by the Young diagram $\\{n\\}$ made of a single row of length equal to
an integer $n\,$. The polynomial $\Phi(u)\in V_{\\{n\\}}^{O(D)}$ obeys to the
irreducibility conditions

$\Big{(}\,u\cdot\frac{\partial}{\partial
u}\,-\,n\,\Big{)}\Phi\,=\,0\,,\quad\quad\Big{(}\,\frac{\partial}{\partial
u}\cdot\frac{\partial}{\partial u}\,\Big{)}\Phi\,=\,0\,.$ (34)

They mean that the polynomial is homogeneous (of degree equal to $n$) and
harmonic, so that its components correspond to a symmetric traceless tensor of
rank $n\,$:

$\Phi(u)\,=\,\frac{1}{n!}\Phi_{\mu_{1}\ldots\mu_{n}}\,u^{\mu_{1}}\ldots
u^{\mu_{n}}\,,\quad\quad\delta^{\mu_{1}\mu_{2}}\Phi_{\mu_{1}\mu_{2}\mu_{3}\ldots\mu_{n}}=0\,.$

Of course the integral of the square of such a polynomial over
${\mathbb{R}}^{D}$ is, in general, infinite. But the restriction of an
harmonic polynomial on the unit sphere $\overrightarrow{u}^{2}=1$ is square
integrable on $S^{D-1}$. This restriction is called a spherical harmonic of
degree $n\,$. Therefore the space of spherical harmonics of degree $n$
provides an equivalent realization of the Schur module $V_{\\{n\\}}^{O(D)}\,$.
For $D=3\,$, the space $V_{\\{n\\}}^{O(3)}\,$ is spanned by the usual
spherical harmonics $Y^{m}_{n}(\theta,\phi)$ on the two-sphere with
$|m|\leqslant n\,$.

Remarks:

$\bullet$ The infinitesimal generators of the pseudo-orthogonal group $O(p,q)$
are represented by the operators

$M_{\mu\nu}=i\sum\limits_{a=1}^{r}u_{a}^{\rho}\Big{(}g_{\rho\mu}\frac{\partial}{\partial
u^{\nu}_{a}}-g_{\rho\nu}\frac{\partial}{\partial u^{\mu}_{a}}\Big{)}\,.$

Reordering the factors and making use of (32)-(33) allows to reproduce the
formula (26) for the eigenvalues of the quadratic Casimir operator.

$\bullet$ Instead of polynomial functions in the commuting variables, one may
equivalently consider distributions obeying to the same conditions. The space
of solutions would carry an equivalent irrep, as follows from the highest-
weight construction of the representation. However, it does not make sense any
more of talking about the “coefficients” of the homogeneous distribution so
that the link with the equivalent tensorial representation is more intricate.

The example of the spherical harmonics suggests that it might be convenient to
realize any unitary module of the orthogonal group $O(D)$ as a space of
functions on the unit hypersphere $S^{D-1}$ satisfying some linear
differential equations. Better, the symmetry under the orthogonal group would
be made manifest by working with homogeneous harmonic functions on the ambient
space ${\mathbb{R}}^{D}\,$, evaluated on any hypersphere
$S^{D-1}\subset{\mathbb{R}}^{D}\,$.

Spherical harmonics: To any UIR of the isometry group $O(D)$ of a hypersphere
$S^{D-1}$, one may associate manifestly covariant differential equations for
functions on $S^{D-1}$ embedded in ${\mathbb{R}}^{D}$ whose space of solutions
carry the corresponding UIR.

Proof: Any UIR of the isometry group $O(D)$ corresponds to a Schur module
$V_{\lambda}^{O(D)}$ which may be realized as the space of polynomials
$\Phi(\overrightarrow{u}_{a})$ such that (32)-(33) are obeyed. Let us
introduce the notation: $\overrightarrow{x}:=\overrightarrow{u}_{1}$ and
$\overrightarrow{t}_{a-1}:=\overrightarrow{u}_{a}$ for $a=2,\ldots,r\,$. One
interprets the polynomial
$\Phi(\overrightarrow{x},\overrightarrow{t}_{\underline{a}})$ (where the index
$\underline{a}$ runs from $1$ to $r-1$) as a tensor field on the Euclidean
space ${\mathbb{R}}^{D}$ parametrized by the Cartesian coordinates
$\overrightarrow{x}\,$, with some auxiliary variables
$\overrightarrow{t}_{\underline{a}}$ implementing the tensor components. The
conditions (32)-(33) for $a$ and $b$ strictly greater than $1$ imply that

$\Big{(}\,t_{\underline{a}}\cdot\frac{\partial}{\partial
t_{\underline{b}}}\,-\,\delta_{{}_{\underline{a}\underline{b}}}\,\lambda_{\underline{a}}\,\Big{)}\Phi\,=\,0\,,\quad(\underline{a}\leqslant\underline{b})\quad\quad\Big{(}\,\frac{\partial}{\partial
t_{\underline{a}}}\cdot\frac{\partial}{\partial
t_{\underline{b}}}\,\Big{)}\Phi\,=\,0\,,$ (35)

where $\underline{\lambda}=\\{\lambda_{2},\ldots,\lambda_{r}\\}$ is the Young
diagram obtained from $\lambda$ by removing its first row. Thus the components
of the “tensor field”
$\Phi(\overrightarrow{x},\overrightarrow{t}_{\underline{a}})$ carry an
irreducible representation of $O(D)$ labeled by $\underline{\lambda}$. The
conditions (32) for $a=b=1$ imply that

$(x\cdot\frac{\partial}{\partial x}-\lambda_{1})\Phi=0\,,$

so the polynomial
$\Phi(\overrightarrow{x},\overrightarrow{t}_{\underline{a}})$ is homogeneous
of degree $\lambda_{1}$ in the radial coordinate $|\overrightarrow{x}|\,$. The
condition (33) for $a=b=1$ is interpreted as the Laplace equation

$\Big{(}\,\frac{\partial}{\partial x}\cdot\frac{\partial}{\partial
x}\,\Big{)}\Phi\,=\,0$ (36)

on the ambient space ${\mathbb{R}}^{D}$, it imples that the tensor field
$\Phi$ is harmonic in ambient space. The condition (32) for $b>a=1$ states
that the radial components vanish,

$\Big{(}\,x\cdot\frac{\partial}{\partial
t_{\underline{a}}}\,\Big{)}\Phi\,=\,0\,,$ (37)

so the tensor components are longitudinal to the hyperspheres $S^{D-1}\,$.
Therefore the evaluation of the non-vanishing components of
$\Phi(\overrightarrow{x},\overrightarrow{t}_{\underline{a}})$ on the unit
hypersphere $|\overrightarrow{x}|=1$ is an intrinsic tensor field living on
the hypersphere $S^{D-1}$ and whose tensor components carry an irrep of the
stability subgroup $O(D-1)$ labeled by $\underline{\lambda}\,$. These tensor
fields generalize the spherical harmonics to the generic case $r\geqslant
1\,$. Finally, the condition (33) for $b>a=1$ states that the tensor field is
divergenceless in ambient space,

$\Big{(}\,\frac{\partial}{\partial x}\cdot\frac{\partial}{\partial
t_{\underline{a}}}\,\Big{)}\Phi\,=\,0\,.$ (38)

The differential equations (36) and (38) are written in ambient space but they
may be reformulated in intrinsic terms on the hypersphere, at the price of
losing the manifest covariance under the full isometry group $O(D)\,$.∎

###  4.5 Euclidean group

The method of induced representations was introduced in Subsection 3.1 for the
Poincaré group $ISO(D-1,1)^{\uparrow}$ and applied to the Euclidean group
$ISO(D-2)$ in Subsection 3.3. Focusing on the genuine (i.e. with a non-trivial
action of the translation generators) irreps of the inhomogeneous orthogonal
group, all of them are induced from an UIR of the stability subgroup. Using
the results of the previous section 4.3, one may summarize the final result
into the following classification.

Unitary irreps of the inhomogeneous orthogonal groups: Each inequivalent UIR
of the group $IO(D)$ with a non-trivial action of its Abelian normal subgroup
is associated with a positive real number $\mu$ and an allowed Young diagram
of the subgroup $O(D-1)\,$, and conversely.

The orbits of the linear action of the orthogonal group $O(D)$ on the
Euclidean space ${\mathbb{R}}^{D}$ are the hyperspheres $S^{D-1}$ of radius
$R\,$. The isometry group of any such hypersphere $S^{D-1}$ is precisely
$O(D)\,$. Considering a region of fixed size on these hyperspheres, in the
limit $R\rightarrow\infty$ the sphere becomes a hyperplane
${\mathbb{R}}^{D-1}\,$. Therefore the homogeneous and inhomogeneous orthogonal
groups are related by some infinite radius limit: $O(D)\rightarrow IO({D-1})$.
Such a process is frequently referred to as an Inönü-Wigner contraction in the
physics literature [14]. This is better seen at the level of the Lie algebra.
Specializing the $D$th directions, the commutation relations (25) take the
form

$\displaystyle i\,[M_{mn},M_{pq}]$ $\displaystyle=$
$\displaystyle\delta_{np}M_{mq}-\delta_{mp}M_{nq}-\delta_{qm}M_{pn}+\delta_{qn}M_{pm}\,,$
(39) $\displaystyle i\,[M_{mD},M_{pq}]$ $\displaystyle=$
$\displaystyle\delta_{mn}M_{pD}-\delta_{mp}M_{nD}\,,$ (40) $\displaystyle
i\,{[}M_{mD},M_{pD}{]}$ $\displaystyle=$ $\displaystyle M_{pm}\,.$ (41)

where the latin letters take $D-1$ values. Defining $M_{mD}\,=\,R\,P_{m}$ and
taking the limit $R\rightarrow\infty$ (with $P_{m}$ fixed) in the relations
(39)-(41) lead to

$\displaystyle i\,[M_{mn},M_{pq}]$ $\displaystyle=$
$\displaystyle\delta_{np}M_{mq}-\delta_{mp}M_{nq}-\delta_{qm}M_{pn}+\delta_{qn}M_{pm}\,,$
(42) $\displaystyle i\,[P_{m},M_{pq}]$ $\displaystyle=$
$\displaystyle\delta_{mn}P_{p}-\delta_{mp}P_{n}\,,$ (43) $\displaystyle
i\,{[}P_{m},P_{p}{]}$ $\displaystyle=$ $\displaystyle 0\,.$ (44)

As can be seen, the generators $\\{M_{mn},P_{m}\\}$ span the Lie algebra of
the inhomogeneous orthogonal group $IO(D-1)\,$. The former argument proves the
contraction $\mathfrak{so}(D)\rightarrow\mathfrak{iso}({D-1})\,$.

The limit of a sequence of irreps of the homogeneous orthogonal group $O(D)$,
in which one performs an Inönü-Wigner contraction, is automatically a
representation of the inhomogeneous orthogonal group $IO(D-1)$ (if the limit
is not singular). An interesting issue is the inverse problem: which irreps of
$IO(D-1)$ may be obtained as the limit of such a sequence of irreps of
$O(D)\,$? The problem is non-trivial because, generically, the limit of a
sequence of irreps is a reducible representation.

Contraction of UIRs of the homogeneous orthogonal groups: Each inequivalent
UIR of the group $IO(D-1)$ with a non-trivial action of its Abelian normal
subgroup may be obtained as the contraction of a sequence of UIRs of the group
$O(D)$.

More precisely, the Inönü-Wigner contraction $R\rightarrow\infty$ of a
sequence of UIRs of $O(D)\,$, labeled by allowed Young diagrams
$\nu=\\{s,\lambda_{1},\ldots,\lambda_{r}\\}$ such that the limit of the
quotient $s/R$ is a fixed positive real number $\mu$, is the UIR of $IO(D-1)$
labeled by the parameter $\mu$ and the Young diagram
$\lambda=\\{\lambda_{1},\ldots,\lambda_{r}\\}\,$.

Proof: The use of the spherical harmonics construction discussed at the end of
Subsection 4.4 is very convenient here. The main idea is to solve the
homogeneity condition in a neighborhood of $x^{D}\neq 0$ as follows:

$\Phi(x^{m},x^{D},t_{\underline{a}})\,=\,z^{s}\,\phi\left(\frac{x^{m}}{z},t_{\underline{a}}\right)\,,$
(45)

where $\overrightarrow{x}=(x^{m},x^{D})$ and
$\phi(y^{m},t_{\underline{a}}):=\Phi(y^{m},\frac{s}{\mu},t_{\underline{a}})\,$.
In other words, one may perform a convenient change of coordinates from the
homogenous coordinates $(x^{m},x^{D})$ to the set $(y^{m},z)$ where

$y^{m}=\frac{x^{m}}{z}$

are the inhomogenous coordinates (on the projective space
${\mathbb{P}}{\mathbb{R}}^{D-1}$ minus the point at infinity $z=0\,$) and

$z=\frac{\mu\,x^{D}}{s}$

is a scale variable.. The magic is that the equations for the generalized
spherical harmonics have a well-behaved limit $x^{D}\rightarrow\infty$ in
terms of $\phi(y^{m},t_{\underline{a}})$ when $x^{D}/s$ is fixed to be equal
to the ratio $z/\mu\,$, where $z$ and $\mu$ are finite [15]. To see that, one
should use the relations

$\displaystyle\frac{\partial}{\partial x^{m}}$ $\displaystyle=$
$\displaystyle\frac{1}{z}\,\frac{\partial}{\partial y^{m}}\,,$
$\displaystyle\frac{\partial}{\partial x^{D}}$ $\displaystyle=$
$\displaystyle\frac{\mu}{s}\left(\frac{\partial}{\partial
z}-\frac{1}{z}\,y^{m}\frac{\partial}{\partial y^{m}}\right)\,.$ (46)

Moreover, the equation in tis limit may be identified with equations for the
proper UIR of the inhomogeneous orthogonal group $IO(D-1)$ realized
homogeneously in terms of the inhomogenous coordinates. ∎

Example: The simplest instance is when $\lambda=\\{0\\}$ because one considers
the sequence of harmonic functions $\Phi(x^{m},x^{D})$ of homogeneity degree
$s\,$. The Laplace operator acting on $\Phi(x^{m},x^{D})$ reads in terms of
$\phi(y^{m})$ as follows

$\Delta_{{\mathbb{R}}^{D}}\Phi=z^{s-2}\left[{\partial\over\partial
y}\cdot{\partial\over\partial y}+{\mu^{2}\over
s^{2}}\left(s(s-1)-(2s-1)\left(y\cdot{\partial\over\partial
y}\right)+\left(y\cdot{\partial\over\partial
y}\right)^{2}\right)\right]\phi\,,$

due to the homogeneity condition (45) and the relations (46). The Laplace
equation $\Delta_{{\mathbb{R}}^{D}}\Phi=0$ is thus equivalent to the equation

$\left[{\partial\over\partial y}\cdot{\partial\over\partial y}+{\mu^{2}\over
s^{2}}\left(s(s-1)-(2s-1)\left(y\cdot{\partial\over\partial
y}\right)+\left(y\cdot{\partial\over\partial
y}\right)^{2}\right)\right]\phi=0\,,$

whose limit for $s\rightarrow\infty$ is the Helmholtz equation
$[\,\Delta_{{\mathbb{R}}^{D-1}}+\mu^{2}\,]\,\phi=0\,,$ where
$\Delta_{{\mathbb{R}}^{D-1}}={\partial\over\partial
y}\cdot{\partial\over\partial y}\,$. The space of solutions of the Helmholtz
equation carries an UIR of $IO(D-1)$ induced from a trivial representation of
the stability subgroup $O(D-2)\,$.

##  5 Relativistic wave equations

The Bargmann-Wigner programme amounts to associating, with any given UIR of
the Poincaré group, a manifestly covariant differential equation whose
positive-energy solutions transform according to the corresponding UIR.
Physically, it might be natural to restrict this programme to the two most
important classes of UIRs: the massive and massless representations.
Mathematically, this restriction is convenient because the group-theoretical
analysis is simpler since any of these UIRs is induced from an UIR of a
unimodular orthogonal group $SO(n)$ (with $D-3\leqslant n\leqslant D-1$), as
can be checked easily on the tables of Subsection 3.3.

In 1948, this restricted programme was completed by Bargmann and Wigner in
four dimensions when, for each such UIR of $ISO(3,1)^{\uparrow}\,$, a
relativistic wave equation was written whose positive-energy solutions
transform according to the corresponding UIR [4]. But this case ($D=4$) will
not be reviewed here in details because it may cast shadow on the generic
case. Indeed, it is rather peculiar in many respects:

  * •

The quadratic and quartic Casimir operators essentially classify the UIRs, but
this is no more true in higher dimensions where more Casimir operators are
necessary and the classification quickly becomes technically cumbersome in
this way. Moreover, one should stress that the eigenvalues of the Casimir
operators do not characterize uniquely an irreducible representation (for
instance, the quadratic and quartic Casimir operators vanish for all helicity
representations).

  * •

The (complex) Lorentz algebra $\mathfrak{so}(3,1)$ is isomorphic to the direct
sum of two (complex) rotation algebras
$\mathfrak{so}(3)\cong\mathfrak{sp}(2)\,$. These isomorphisms allow the use of
the convenient “dotted-undotted” formalism for the finite-dimensional (non-
unitary) irreps of the spin group $Spin(3,1)$.

  * •

The symmetric tensor-spinor fields are sufficient to cover all inequivalent
cases.

  * •

The helicity short little group $SO(2)$ is Abelian, therefore its irreps are
one-dimensional, for fixed helicity. Notice that the helicity is discretized
because the representation of the “little group” $SO(2)$ is a restriction of
the representation of the group $Spin(3)\cong SU(2)$ which has no
intrinsically projective representations.

  * •

The infinite-spin short little group $SO(1)$ is trivial, thus there are only
two inequivalent infinite-spin representations (single- or double-valued) [6].

  * •

etc.

Moreover, there exists an extensive literature on the subject of UIRs of
$ISO(3,1)^{\uparrow}$ and we refer to the numerous pedagogical reviews
available for more details on the four-dimensional case (see e.g. the
inspiring presentations of [5] and [16]).

It is standard to require time reversal and parity symmetry of the field
theory. More precisely, the wave equations we will consider are covariant
under the two previous transformations. As a consequence of the time reversal
symmetry, the representation is irreducible under the group $ISO(D-1,1)$ but
reducible under the Poincaré group $ISO(D-1,1)^{\uparrow}\,$: the Hilbert
space of solutions contain both positive and negative energy solutions.
Furthermore, the parity symmetry implies that the representation is
irreducible under the inhomogeneous Lorentz group $IO(D-1,1)$ but reducible
under the group $ISO(D-1,1)$ (for instance, both chiralities are present in
the massless case for $D$ even). To conclude, the Bargmann-Wigner programme is
actually understood as associating, with any given UIR of the inhomogeneous
Lorentz group, a manifestly covariant differential equation whose solutions
transform according to the corresponding UIR.

###  5.1 General procedure

The lesson on induced representations that we learned from Wigner implies the
following strategy:

  1. 1.

Pick a unitary representation of the (short) little group.

  2. 2.

Introduce a wave function on ${\mathbb{R}}^{D-1,1}$ taking values in some
(possibly non-unitary) representation of the Lorentz group $O(D-1,1)$ the
restriction of which to the (short) little group contains the representation
of step 1.

  3. 3.

Write a system of linear covariant equations, differential in position space
$x^{\mu}$ thus algebraic in momentum space $p_{\nu}\,$, for the wave function
of step 2. These equations may not be independent.

  4. 4.

Fix the momentum and check in convenient coordinates that the wave equations
of step 3 put to zero all “unphysical” components of the wave function. More
precisely, verify that its non-vanishing components carry the unitary
representation of step 1.

Proof: The fact that the set of linear differential equations is taken to be
manifestly covariant ensures that the Hilbert space of their solutions carries
a (infinite-dimensional) representation of $IO(D-1,1)\,$. The fourth step
determines the representation of the little group by which it is induced. ∎

In the physics literature, the fourth step is referred to as “looking at the
physical degrees of freedom.” If the (possibly reducible) representation is
proven to be unitary, then this property is summarized in a “no-ghost
theorem.”

The Klein-Gordon equation $(p^{2}\pm\,m^{2})\Psi=0$ is always, either present
in the system of covariant equations or a consequence thereof. Consequently,
the Klein-Gordon equation will be assumed implicitly from now on in the step
3. Therefore, the step 4 will be immediately performed in a proper Lorentz
frame. (We refer the reader to the Subsection 3.2 for more details.)

The completion [17] of the Bargmann-Wigner programme for finite-component
representations is reviewed in the subsections 5.2 and 5.3 for single-valued
UIRs of the Poincaré group.666Spinorial irreps may be adressed analogously by
supplementing the system of differential equations with Dirac-like equations
and gamma-trace constraints (see e.g. [15, 18] for more details). The
Bargmann-Wigner programme has been completed for anyonic representations in
three-dimensional Minkowski spacetime [19]. The tachyonic case777The
discussion presented in the section 5.4 was not published before, it directly
derives from private conversations between X.B. and J. Mourad. is more briefly
discussed in Subsection 5.4. The zero-momentum representation is not
considered here since it essentially is a unitary representation of the de
Sitter spacetime $dS_{D-1}$.

###  5.2 Massive representations

The Bargmann-Wigner programme is easy to complete for massive UIRs because the
massive stability subgroup is the orthogonal group $O(D-1)\subset O(D-1,1)\,$.
By going to a rest-frame, the time-like momentum vector takes the form
$p^{\mu}=(m,\overrightarrow{0})\neq 0\,$. The physical components of the field
are thus carrying a tensorial irrep of the group $O(D-1)$ of orthogonal
transformations in the spatial hyperplane ${\mathbb{R}}^{D-1}\,$. In other
words, the linear wave equations should remove all components including time-
like directions. These unphysical components are responsible for the fact that
the Fock space is not endowed with a positive-definite norm.

Step 1. From the sections 1.3 and 4, one knows that any unitary representation
of the orthogonal group $O(D-1)$ is a sum of UIRs which are finite-dimensional
and thus, equivalent to a tensorial representation. Let us consider the UIR of
$O(D-1)$ labeled by the allowed Young diagram
$\lambda=\\{\lambda_{1},\lambda_{2},\ldots,\lambda_{r}\\}$ (i.e. the sum of
the lengths of its first two columns does not exceed $D-1$).

Step 2. The simplest way to perform the Bargmann-Wigner programme in the
massive case is to choose a covariant wave function whose components carry the
(finite-dimensional and non-unitary) tensorial irrep of the Lorentz group
$O(D-1,1)$ labeled by the Young diagram $\lambda\,$. As explained in the
subsection 4.4, a convenient way of realizing this is in terms of a wave
function $\Phi(p,u_{a})$ polynomial in the auxiliary commuting variables
$u_{a}^{\mu}\,$ satisfying the irreducibility conditions (32)-(33).

Step 3. The massive Klein-Gordon equation has to be supplemented with the
transversality conditions

$\Big{(}\,p\cdot\frac{\partial}{\partial u_{a}}\,\Big{)}\Phi\,=\,0\,,$ (47)

of the wave function.

Step 4. Looking at a fixed-momentum mode in its corresponding rest-frame
$p^{\mu}=(m,\overrightarrow{0})$ leads to the fact that the components of the
wave function along the timelike momentum are set to zero by (47):
$\Phi=\Phi(p,\overrightarrow{u}_{a})\,$. In words, $\Phi$ does not depend on
the time components $u^{0}_{a}\,$, $\forall\;a\,$. In this case, the
conditions (32)-(33) read as irreducibility conditions under the orthogonal
group $O(D-1)\,$. ∎

Example: Massive symmetric representations with “spin” equal to $s$ correspond
to Young diagrams $\lambda=\\{s\\}$ made of one row of length equal to the
integer $s\,$. In four spacetime dimensions, this representation is precisely
what is usually called a “massive spin-$s$ field.”888To our knowledge, the
Bargmann-Wigner programme for the massive integer-spin representations in
four-dimensional Minkowski spacetime was adressed along the lines reviewed
here for the first time by Fierz in [20]. The covariant wave function
$\Phi(p,u)$ obeys to the irreducibility conditions (32)-(33) of the components

$\Big{(}\,u\cdot\frac{\partial}{\partial
u}\,-\,s\,\Big{)}\Phi\,=\,0\,,\quad\quad\Big{(}\,\frac{\partial}{\partial
u}\cdot\frac{\partial}{\partial u}\,\Big{)}\Phi\,=\,0\,.$ (48)

The wave function $\Phi$ is homogeneous of degree $s$ and harmonic in the
auxiliary variable $u\,$. If the wave function $\Phi(p,u)$ is polynomial in
the auxiliary variable $u\,$, then its components correspond to a symmetric
traceless tensor of rank $s$

$\Phi(p,u)\,=\,\frac{1}{s!}\Phi_{\mu_{1}\ldots\mu_{s}}(p)\,u^{\mu_{1}}\ldots
u^{\mu_{s}}\,,\quad\quad\eta^{\mu_{1}\mu_{2}}\Phi_{\mu_{1}\mu_{2}\mu_{3}\ldots\mu_{s}}(p)=0\,.$

The covariant wave equations are the massive Klein-Gordon equation together
with the transversality condition

$\Big{(}\,p\cdot\frac{\partial}{\partial u}\,\Big{)}\Phi\,=\,0\,,$ (49)

which reads in components as

$p^{\mu_{1}}\Phi_{\mu_{1}\mu_{2}\ldots\mu_{s}}(p)=0\,.$ (50)

The non-vanishing components of a solution of (50) must be along the spatial
directions, i.e. only $\Phi_{i_{1}\ldots i_{s}}(p)$ may be $\neq 0\,$. This
symmetric tensor field is traceless with respect to the spatial metric:
$\delta^{i_{1}i_{2}}\Phi_{i_{1}i_{2}i_{3}\ldots i_{s}}(p)=0\,$, thus the
physical components carry a symmetric irrep of the orthogonal group
$O(D-1)\,$, the dimension of which can be computed by making use of the
formula (24). The polynomial wave function $\Phi(p,u)$ evaluated on the
internal unit hypersphere $u^{i}u_{i}=1$ corresponds to a decomposition of the
physical components in terms of the spherical harmonics on the internal
hypersphere $S^{D-2}\,$, which is an equivalent, though rather unusual, way of
representing the physical components (usually, the use of spherical harmonics
is reserved to the “orbital” part of the wave function).

The quartic Casimir operator of the Poincaré algebra is easily evaluated in
components in the rest frame

$\displaystyle-{1\over 2}$ $\displaystyle
P^{2}M_{\mu\nu}M^{\mu\nu}\,\,+\,\,M_{\mu\rho}P^{\rho}\,M^{\mu\sigma}P_{\sigma}$
$\displaystyle=\,{1\over
2}\,\,m^{2}(M_{ij}M^{ij}+2M_{i0}M^{i0})\,-\,m^{2}M_{i0}M^{i0}\,=\,m^{2}\,{1\over
2}\,M_{ij}M^{ij}\,,$

giving as a final result for a massive representation associated with a Young
diagram $\lambda$

$\displaystyle{\cal C}_{4}\Big{(}\mathfrak{iso}(D-1,1)\Big{)}$
$\displaystyle=$ $\displaystyle{\cal
C}_{2}\Big{(}\mathfrak{iso}(D-1,1)\Big{)}\,{\cal
C}_{2}\Big{(}\mathfrak{so}(D-1)\Big{)}\,\,,$ (51) $\displaystyle=$
$\displaystyle m^{2}\,\sum_{a=1}^{r}\lambda_{a}(\lambda_{a}+D-2a-1)\,,$

where the eigenvalues of the quadratic Casimir operator of the rotation
algebra are given by the formula (26).

Example: In any dimension $D\,$, the eigenvalue of the quartic Casimir
operator for a massive symmetric representation of rank $s$ is equal to
$m^{2}\,s(s+D-3)$. In four spacetime dimensions, the square of the Pauli-
Lubanski vector acting on a massive field of spin-$s$ is indeed equal to
$m^{2}\,s(s+1)$.

Each massive representation in $D\geqslant 4$ dimensions may actually be
obtained as the first Kaluza–Klein mode in a dimensional reduction from $D+1$
down to $D$ dimensions. There is no loss of generality because the massive
little group $SO(D-1)$ in $D$ dimension is identified with the
$(D+1)$-dimensional helicity (short) little group. Such a Kaluza–Klein
mechanism leads to a Stückelberg formulation of the massive field.

The massless limit $m\rightarrow 0$ of a massive irrep with $\lambda$ fixed
is, in general, reducible because the irrep of the massive little group
$SO(D-1)$ is restricted to the helicity (short) little group $SO(D-2)\subset
SO(D-1)$. This argument combined with the known branching rule for
$O(D-1)\downarrow O(D-2)$ (reviewed in Subsection 4.3) allows to prove that
the massless limit of a massive irrep of the homogeneous Lorentz group labeled
by a fixed Young diagram $\lambda$ contains each helicity irrep labeled by
Young diagrams $\mu$ such that

$\lambda_{1}\geqslant\mu_{1}\geqslant\lambda_{2}\geqslant\mu_{2}\geqslant\ldots\geqslant\mu_{r-1}\geqslant\lambda_{r}\geqslant\mu_{r}\geqslant
0\,,$

with multiplicity one. The zero modes of a dimensional reduction from $D+1$
down to $D$ dimensions are determined by the same rule.

Example: The zero modes of the dimensional reduction of a massive symmetric
representations with “spin” equal to $s$ are all helicity symmetric
representations with integer “spins” not greater than the integer $s$, each
with multiplicity one. For the dimensional reduction of a gravitational theory
(i.e. a spin-two particle), one recovers the usual result that the massless
spectrum is made of one “graviton” (spin-$2$), one “photon” (spin-$1$) and one
“dilaton” (spin-$0$).

###  5.3 Massless representations

The quartic Casimir operator of the Poincaré algebra is evaluated easily in
components in the light-cone coordinates (see Subsection 3.2 for notations),

$\displaystyle-{1\over
2}\,P^{2}\,M_{\mu\nu}M^{\mu\nu}\,\,+\,\,M_{\mu\rho}P^{\rho}\,M^{\mu\sigma}P_{\sigma}\,=\,0\,+\,M_{m+}P^{+}\,M^{m-}P_{-}\,=\,\pi_{m}\pi^{m}\,,$

giving as a final result for a massless representation

${\cal C}_{4}\Big{(}\mathfrak{iso}(D-1,1)\Big{)}\,=\,{\cal
C}_{2}\Big{(}\mathfrak{iso}(D-2)\Big{)}\,=\,\mu^{2}$ (52)

where the quadratic Casimir operator of the massless little group is written
in (16).

####  5.3.1 Helicity representations

Helicity representations correspond to the case $\mu=0\,$, so that $\pi^{m}=0$
and in practice the representation is induced from a representation of the
orthogonal group $O(D-2)\,$.

Step 1. Again, any unitary representation of the orthogonal group $O(D-2)$ is
a sum of finite-dimensional UIRs. Let us consider the UIR of the helicity
short little group $O(D-2)$ labeled by the allowed Young diagram
$\lambda=\\{\lambda_{1},\lambda_{2},\ldots,\lambda_{r}\\}$ (that is, the sum
of the lengths of its first two columns does not exceed $D-2$):

$\begin{picture}(75.0,70.0)(-20.0,2.0)\put(-25.0,30.0){$\lambda=$}
\multiframe(0,0)(13.5,0){1}(15,10){}\put(20.0,0.0){$\lambda_{r}$}
\multiframe(0,10.5)(13.5,0){1}(20,10){}\put(25.0,10.5){$\lambda_{r-1}$}
\multiframe(0,21)(13.5,0){1}(40,20){\ldots}\put(45.0,25.0){$\vdots$}
\multiframe(0,41.5)(13.5,0){1}(55,10){} \put(60.0,41.5){$\lambda_{3}$}
\multiframe(0,52)(13.5,0){1}(70,10){} \put(75.0,52.0){$\lambda_{2}$}
\multiframe(0,62.5)(13.5,0){1}(110,10){} \put(115.0,62.5){$\lambda_{1}$}
\end{picture}\quad\quad.$ (53)

The step 2 is more subtle to perform than for massive representations because
the field equations must set to zero all components along the light-cone of
the covariant wave function, because they are unphysical. In other words, the
covariant wave equations should remove two directions, and not only one like
in the massive case. This fact implies that the transversality is not a
sufficient condition any more, it must be supplemented either by other
equations or by gauge symmetries asserting that one may quotient the solution
space by pure gauge fields. In these lecture notes, one focuses on two gauge-
invariant formulations which may be respectively referred to as “Bargmann-
Wigner formulation” in terms of the fieldstrength and “gauge-fixed
formulation” in terms of the potential.

Bargmann-Wigner equations

The so-called “Bargmann-Wigner equations” were actually first written by Dirac
[21] in four-dimensional Minkowski spacetime in spinorial form. Their name
originates from their decisive use in the completion of the Bargmann-Wigner
programme [4]. The generalization of the Bargmann-Wigner equations to any
dimension was presented in [17] for tensorial irreps (reviewed here) and in
[18] for spinorial irreps.

Step 2. Let
${\overline{\lambda}}=\\{\lambda_{1},\lambda_{1},\lambda_{2},\ldots,\lambda_{r}\\}$
be the Young diagram depicted as

$\begin{picture}(75.0,80.0)(-20.0,2.0)\put(-25.0,40.0){$\overline{\lambda}=$}
\multiframe(0,0)(13.5,0){1}(15,10){}\put(20.0,0.0){$\lambda_{r}$}
\multiframe(0,10.5)(13.5,0){1}(20,10){}\put(25.0,10.5){$\lambda_{r-1}$}
\multiframe(0,21)(13.5,0){1}(40,20){\ldots}\put(45.0,25.0){$\vdots$}
\multiframe(0,41.5)(13.5,0){1}(55,10){} \put(60.0,41.5){$\lambda_{3}$}
\multiframe(0,52)(13.5,0){1}(70,10){} \put(75.0,52.0){$\lambda_{2}$}
\multiframe(0,62.5)(13.5,0){1}(110,10){}\put(115.0,62.5){$\lambda_{1}$}
\multiframe(0,73)(13.5,0){1}(110,10){}\put(115.0,73.0){$\lambda_{1}$}
\end{picture}\quad\quad.$ (54)

It is obtained from the Young diagram $\lambda$ represented in (53) by adding
a row of equal length on top of the first row of $\lambda\,$. The Young
diagram ${\overline{\lambda}}$ has at least two rows of equal lengths and the
sum of the lengths of its first two columns does not exceed $D\,$. The
covariant wave function is chosen to take values in the Schur module
$V_{\overline{\lambda}}^{O(D-1,1)}$ realized in the manifestly antisymmetric
convention. Following Subsection 4.4, the wave function ${\cal
K}(p,d_{{}_{I}}x)$ is taken to be a polynomial in the graded variables
$d_{{}_{I}}x^{\mu}$ ($\,I=1,2,\ldots,\lambda_{1}\,$) obeying the commutation
relations (29). Moreover, the irreducibility conditions of the components
under the Lorentz group $O(D-1,1)$ are

$\Big{(}\,d_{{}_{I}}x^{\mu}\frac{\partial^{L}}{\partial(d_{{}_{J}}x^{\mu})}\,-\,\delta_{{}_{IJ}}\,\overline{\ell}_{{}_{I}}\,\Big{)}{\cal
K}\,=\,0\,,\quad(I\leqslant J)$ (55)

where $\overline{\ell}_{I}$ stands for the length of the $I$th column in the
Young diagram $\overline{\lambda}\,$, and

$\Big{(}\,\eta^{\mu\nu}\frac{\partial^{L}}{\partial(d_{{}_{I}}x^{\mu})}\frac{\partial^{L}}{\partial(d_{{}_{J}}x^{\nu})}\,\Big{)}{\cal
K}\,=\,0\,.$ (56)

Step 3. The covariant wave equations may be summarized in the assertion that
the wave function is a “harmonic” multiform in the sense that, $\forall\;I$,
it is “closed”

$\Big{(}\,p_{\mu}\,d_{{}_{I}}x^{\mu}\,\Big{)}{\cal K}\,=\,0\,,$ (57)

and “coclosed” (i.e. transverse)

$\Big{(}\,p^{\mu}\frac{\partial^{L}}{\partial(d_{{}_{I}}x^{\mu})}\,\Big{)}{\cal
K}\,=\,0\,.$ (58)

The operators $p\cdot d_{{}_{I}}x$ act as “exterior differentials” (or
“curls”), they are nilpotent and obey graded commutation relations. As one can
easily see, the wave equations (57) and (58), considered together, imply the
massless Klein-Gordon equation. Actually, the equations (57) may even be
imposed off-shell, whereas the equations (58) only hold on-shell [17].

Step 4. In the light-cone frame (see Section 1.1), the components of the
momentum may be taken to be $p_{\mu}=(p_{-},0,0,\ldots,0)$ with $p_{-}\neq
0\,$. On the one hand, the transversality condition (58) implies that the wave
function does not depend on the variables $d_{{}_{I}}x^{+}\,$. On the other
hand, the closure condition (57) reads $(p_{-}d_{{}_{I}}x^{-})\,{\cal K}=0\,$,
the general solution of which is ${\cal
K}\,=\,(\prod_{I}p_{-}d_{{}_{I}}x^{-})\,\phi\,$, where $\phi$ depends neither
on $d_{{}_{I}}x^{-}$ nor on $d_{{}_{I}}x^{+}$ (due to the transversality
condition). In other words, the directions along the light-cone have been
removed, since $\phi=\phi(p,d_{{}_{I}}x^{m})\,$ ($m=1,2,\ldots,D-2$). Focusing
on this field, one may show that the irreducibility conditions (55) become, in
terms of the function $\phi$,

$\Big{(}\,d_{{}_{I}}x^{m}\frac{\partial^{L}}{\partial(d_{{}_{J}}x^{m})}\,-\,\delta_{{}_{IJ}}\,\ell_{{}_{I}}\,\Big{)}\phi\,=\,0\,,\quad(I\leqslant
J)$ (59)

where $\ell_{{}_{I}}=\overline{\ell}_{{}_{I}}-1$, and the trace conditions
(56) implies

$\Big{(}\,\delta^{mn}\frac{\partial^{L}}{\partial(d_{{}_{I}}x^{m})}\frac{\partial^{L}}{\partial(d_{{}_{J}}x^{n})}\,\Big{)}\phi\,=\,0\,.$
(60)

Since $\ell_{{}_{I}}$ is the length of the $I$th column of the Young diagram
$\lambda\,$, the system of equations (59)-(60) states that the components of
the function $\phi$ carry a tensorial irrep of the orthogonal group
$O(D-2)\,$. Therefore, the same is true for the physical components of the
wave function ${\cal K}\,$.∎

This may be reformulated covariantly by saying that the closure (57) of the
wave function implies that

${\cal
K}\,=\,\Big{(}\,\prod_{I=1}^{\lambda_{1}}p_{\mu}d_{{}_{I}}x^{\mu}\,\Big{)}\,\phi\,.$
(61)

In components, this means that the tensor ${\cal K}$ is equal to $\lambda_{1}$
curls of the tensor $\phi\,$. This motivates the name “fieldstrength” for the
wave function ${\cal K}(p,d_{{}_{I}}x)\,$, the components of which are
irreducible under the Lorentz group (when evaluated on zero-mass shell) and
labeled by $\overline{\lambda}\,$, and the name “potential” or “gauge field”
for the wave function $\phi(p,d_{{}_{I}}x)\,$, the components of which may be
taken to be irreducible under the general linear group, with symmetries
labeled by the Young diagram $\lambda\,$.

Examples:

$\bullet$ The helicity vectorial representation corresponds to a Young diagram
$\lambda=\\{1\\}$ made of a single box. In four spacetime dimensions, this
representation is precisely what is usually called a “vector gauge field”. The
Young diagram $\overline{\lambda}=\\{1,1\\}$ is a single column made of two
boxes. The wave function in momentum space is given by

${\cal K}\,=\,\frac{1}{2}\,{\cal K}_{\mu\nu}(p)\,dx^{\mu}dx^{\nu}$

which carries an irrep of $GL(D,\mathbb{R})$: the antisymmetric rank-two
representation. As one can see, the wave function actually is a differential
two-form, the components of which transforming as an antisymmetric tensor of
rank two. The wave equations (57) and (58), respectively, read in components

$p_{\mu}{\cal K}_{\nu\rho}+p_{\nu}{\cal K}_{\rho\mu}+p_{\rho}{\cal
K}_{\mu\nu}=0\quad{\rm{(Bianchi~{}identities)}}$

and

$p^{\mu}{\cal K}_{\mu\nu}=0\qquad{\rm{(transversality~{}conditions)}}\,.$

The differential two-form $\cal K$ is indeed harmonic (closed and coclosed).
In physical terms, one says that the fieldstrength ${\cal K}_{\mu\nu}$ obeys
to the Maxwell equations. As usual, the Bianchi identities imply that the
fieldstrength derives from a potential: ${\cal
K}_{\mu\nu}=p_{\mu}\phi_{\nu}-p_{\nu}\phi_{\mu}\,$. In the light-cone
coordinates, the transversality implies that the components ${\cal K}_{+\nu}$
vanish, thus the only non-vanishing components are ${\cal
K}_{-n}=p_{-}\phi_{n}\,$. Therefore the only physical components correspond to
a $(D-2)$-vector in the hyperplane transverse to the light-cone.

$\bullet$ Helicity symmetric representations with “helicity” (or “spin”) equal
to $s$ correspond to Young diagrams $\lambda=\\{s\\}$ made of one row of
length equal to the integer $s\,$. In four spacetime dimensions, this
representation is precisely what is usually called a “massless spin-$s$
field.” The Young diagram $\overline{\lambda}=\\{s,s\\}$ is a rectangle made
of two row of length equal to the integer $s\,$. The wave function is thus a
polynomial in the auxiliary variables

${\cal K}\,=\,\frac{1}{2^{s}}\,{\cal
K}_{\mu_{1}\nu_{1}\,\mid\ldots\,\mid\,\mu_{s}\nu_{s}}\,d_{1}x^{\mu_{1}}d_{1}x^{\nu_{1}}\,\ldots\,d_{s}x^{\mu_{s}}d_{s}x^{\nu_{s}}$

satisfying the irreducibility equations (55)-(56) with $\ell_{I}=2$
($I=1,\ldots,s$). The fieldstrength tensor ${\cal
K}_{\mu_{1}\nu_{1}\,\mid\ldots\,\mid\,\mu_{s}\nu_{s}}$ was first introduced by
Weinberg in four spacetime dimensions [22]. The tensor ${\cal K}$ is, by
construction, antisymmetric in each of the $s$ sets of two indices

$\displaystyle{\cal
K}_{\mu_{1}\nu_{1}\,\mid\ldots\,\mid\,\mu_{s}\nu_{s}}\,=\,-\,{\cal
K}_{\nu_{1}\mu_{1}\,\mid\ldots\,\mid\,\mu_{s}\nu_{s}}=\,\ldots\,=\,-\,{\cal
K}_{\mu_{1}\nu_{1}\,\mid\ldots\,\mid\,\nu_{s}\mu_{s}}\,.$ (62)

Moreover, the complete antisymmetrization over any set of three indices gives
zero and all its traces are zero, so that the tensor indeed belongs to the
space irreducible under the Lorentz group $O(D-1,1)$ characterized by a two-
row rectangular Young diagram of length $s\,$. In four-dimensional Minkowski
spacetime, the irrep of the Lorentz group $O(3,1)$ carried by the (on-shell)
Weinberg tensor is usually denoted as $(s,0)\oplus(0,s)\,$. More precisely,
the symmetry properties of the tensor ${\cal
K}_{\mu_{1}\nu_{1}\,\mid\ldots\,\mid\,\mu_{s}\nu_{s}}$ are labeled by the
Young tableau

The equation (61) means that the components of the tensor ${\cal
K}_{\mu_{1}\nu_{1}\,\mid\ldots\,\mid\,\mu_{s}\nu_{s}}$ are essentially the
projection of $p_{\mu_{1}}\ldots p_{\mu_{s}}\phi_{\nu_{1}\ldots\nu_{s}}$ on
the tensor field irreducible under $GL(D,\mathbb{R})$ with symmetries labeled
by the above Young tableau. The physical components $\phi_{n_{1}\ldots n_{s}}$
of the symmetric tensor gauge field $\phi_{\nu_{1}\ldots\nu_{s}}$ are along
the $D-2$ directions transverse to the light-cone. The number of physical
degrees of freedom of a helicity symmetric field of rank $s$ can be computed
by making use of the formula (24).

$\bullet$ The helicity symmetric representation with “spin” equal to $2$
corresponds to the graviton. The fieldstrength has the symmetry properties of
the Riemann tensor. Its on-shell tracelessness indicates that it corresponds
to the Weyl tensor. The equations (57) are the Bianchi identities for the
linearized Riemann tensor on flat spacetime, whereas the equations (58) hold
as a consequence of the sourceless Einstein equations linearized around flat
spacetime.

Gauge-fixed equations

The following equations are somewhat unusual, but they proved to be
determinant in the completion of the Bargmann-Wigner programme for the
infinite spin representations [15].

Step 2. Let
$\widehat{\lambda}=\\{\lambda_{1}-1,\lambda_{2}-1,\ldots,\lambda_{r}-1\\}$ be
the Young diagram depicted as

$\begin{picture}(75.0,70.0)(-20.0,2.0)\put(-25.0,30.0){$\widehat{\lambda}=$}
\multiframe(0,0)(13.5,0){1}(9,10){}\put(14.0,0.0){$\lambda_{r}-1$}
\multiframe(0,10.5)(13.5,0){1}(14,10){}\put(19.0,10.5){$\lambda_{r-1}-1$}
\multiframe(0,21)(13.5,0){1}(30,20){\ldots}\put(35.0,25.0){$\vdots$}
\multiframe(0,41.5)(13.5,0){1}(45,10){} \put(50.0,41.5){$\lambda_{3}-1$}
\multiframe(0,52)(13.5,0){1}(60,10){} \put(65.0,52.0){$\lambda_{2}-1$}
\multiframe(0,62.5)(13.5,0){1}(100,10){} \put(105.0,62.5){$\lambda_{1}-1$}
\end{picture}\quad\quad,$ (63)

obtained from the Young diagram $\lambda$ represented in (53) by removing the
first column of $\lambda\,$. Therefore the sum of the length of the first two
columns of the Young diagram ${\widehat{\lambda}}$ does not exceed $D-2\,$.
The covariant wave function is chosen to take values in the Schur module
$V_{\widehat{\lambda}}^{O(D-1,1)}$ realized in the manifestly symmetric
convention. Actually, as anticipated in Subsection 4.4, it turns out to be
crucial to regard the wave function $\Phi(p,u_{a})$ as a distribution in the
commuting auxiliary variables $u^{\mu}_{a}\,$, obeying to

$\displaystyle\left[\left(u_{a}\cdot{\partial\over\partial
u_{b}}\right)-\widehat{\lambda}_{a}\,\delta_{ab}\right]\Phi$ $\displaystyle=$
$\displaystyle 0\,,\quad(a\leqslant b)\,.$ (64)
$\displaystyle\left({\partial\over\partial u_{a}}\cdot{\partial\over\partial
u_{b}}\right)\Phi$ $\displaystyle=$ $\displaystyle 0\,,$ (65)

Step 3. Proper wave equations are the transversality condition (47) combined
with the equation

$(p\cdot u_{a})\,\Phi=0\,.$ (66)

The equations (66) and (47) are the respective analogues of the closure and
coclosure conditions (57)-(58). A drastic difference is that the operators
$p\cdot u_{a}$ are not nilpotent (thus there is no underlying cohomology).
Actually, the equation (66) has no solution if $\Phi$ is assumed to be a
polynomial in all the variables.

Step 4. Equation (66) can be solved as

$\Phi=\delta(u_{a}\cdot p)\,\phi\,,$ (67)

where the distribution $\phi(p,u_{a})$ may actually be assumed to be a
function depending polynomially on the auxiliary variables $u_{a}$ for the
present purpose. The Dirac delta is a distribution of homogeneity degree equal
to minus one, hence the irreducibility conditions (64)-(65) imply that

$\displaystyle\left[\left(u_{a}\cdot{\partial\over\partial
u_{b}}\right)-\lambda_{a}\,\delta_{ab}\right]\phi$ $\displaystyle=$
$\displaystyle 0\qquad(a\leqslant b)\,,$ (68)
$\displaystyle\left({\partial\over\partial u_{a}}\cdot{\partial\over\partial
u_{b}}\right)\phi$ $\displaystyle=$ $\displaystyle 0\,.$ (69)

The function $\phi$ is defined from (67) modulo the equivalence relation

$\phi\sim\phi+\sum\limits_{a=1}^{r}\,(u_{a}\cdot p)\,\epsilon_{a}$ (70)

where $\epsilon_{a}$ are arbitrary functions. This means that (67) is
equivalent to the alternative road towards the Bargmann-Wigner programme: the
gauge symmetry principle with the irreducible components of $(u_{a}\cdot
p)\,\epsilon_{a}$ being pure gauge fields. As mentioned before, this path will
not be addressed here (see e.g. [17] and refs therein for more discussions on
the gauge-invariance issue). Therefore, one may say that the equation (66) is
the “remnant” of the gauge symmetries (70). In the light-cone coordinates, the
gauge symmetries (70) imply that one may choose a representative $\phi$ which
does not depend on the variables $u^{-}_{a}$ (the gauge is “fixed”). The
transversality condition (47) implies that $\phi$ is also transverse, implying
no dependence on $u^{+}_{a}$ (“gauge shoots twice”). Thus $\phi$ depends only
on the transverse auxiliary variables $u^{m}_{a}\,$, so one concludes by
observing that the physical components of $\phi$ carry a tensorial irrep of
$O(D-2)$ labeled by $\lambda\,$.∎

####  5.3.2 Infinite spin representations

Infinite spin representations correspond to the case $\mu\neq 0$ and, in
practice, the representation of the massless little group $IO(D-2)$ is induced
from a representation of the orthogonal group $O(D-3)\,$. The parameter $\mu$
is a real parameter with the dimension of a mass. Wigner proposed a set of
manifestly covariant equations to describe fields carrying these UIR in four
spacetime dimensions [23]. Recently, they were generalized to arbitrary
infinite-spin representations in any dimension [15].

Step 1. Again, any unitary representation of the orthogonal group $O(D-3)$ is
a sum of finite-dimensional UIRs. Let us consider the UIR of the helicity
short little group $O(D-3)$ labeled by the allowed Young diagram
$\lambda=\\{\lambda_{1},\lambda_{2},\ldots,\lambda_{r}\\}$ (that is, the sum
of the lengths of its first two columns does not exceed $D-3$).

Step 2. In order to have manifest covariance, it is necessary to lift the
eigenvalues $\xi^{m}$ of the generators $\pi^{m}$ in the massless little group
to a $D$-vector $\xi^{\mu}\,$. In practice, the covariant wave function is
taken to be a distribution $\Phi(p,\xi,u_{a})\,$ satisfying the conditions
(32)-(33). The tensorial components associated with the commuting variables
$u_{a}$ belong to the Schur module of the Lorentz group $O(D-1,1)$ labeled by
an allowed Young diagram $\lambda\,$.

Step 3. Relativistic equations describing a first-quantized particle with
infinite spin are

$\displaystyle(p\cdot\xi)\,\Phi$ $\displaystyle=$ $\displaystyle 0\,,$ (71)
$\displaystyle\left(p\cdot\frac{\partial}{\partial\xi}-i\right)\Phi$
$\displaystyle=$ $\displaystyle 0\,,$ (72)
$\displaystyle(\xi^{2}-\mu^{2})\,\Phi$ $\displaystyle=$ $\displaystyle 0\,,$
(73)

together with the transversality conditions

$\displaystyle(p\cdot u_{a})\,\Phi$ $\displaystyle=$ $\displaystyle 0\,,$ (74)
$\displaystyle\left(p\cdot{\partial\over\partial u_{a}}\right)\Phi$
$\displaystyle=$ $\displaystyle 0\,,$ (75)
$\displaystyle\left(\xi\cdot{\partial\over\partial u_{a}}\right)\Phi$
$\displaystyle=$ $\displaystyle 0\,.$ (76)

This system of equations is far from being independent. For instance,
compatibility condition of the systems (71)-(72) or (74)-(75) is the massless
Klein-Gordon equation.

Step 4. The equation (72) reflects the fact that the couples $(p\,,\xi)$ and
$(p\,,\xi+\alpha p)$ are physically equivalent for arbitrary
$\alpha\in\mathbb{R}$. Indeed, one gets

$\Phi(p\,,\xi+\alpha p)\,=\,e^{i\alpha}\,\Phi(p\,,\xi)$ (77)

from Equation (72). The equation (73) states that the internal vector $\xi$ is
a space-like vector while the mass-shell condition states that the momentum is
light-like. From the equation (71), one obtains that the internal vector is
transverse to the momentum. All together, one finds that $\xi$ may be taken to
live on the hypersphere $S^{D-3}$ of radius $\mu$ embedded in the transverse
hyperplane ${\mathbb{R}}^{D-2}\,$. In brief, the “continuous spin” degrees of
freedom essentially correspond to $D-3$ angular variables, whose Fourier
conjugates are discrete variables analogous to the usual spin degrees of
freedom. Finally, proceeding analogously to the “gauge-fixed” wave equations
of the helicity representations, one may show [15] that the conditions
(74)-(76) concretely remove three unphysical directions in the components, so
that the final result is a tensorial irrep of the short little group $O(D-3)$
fixing both the momentum $p$ and the internal vector $\xi\,$.

From the group theoretical point of view, the UIR of the homogeneous and
inhomogeneous orthogonal groups are related by an Inönü-Wigner contraction
$O(D-1)\rightarrow IO({D-2})$ (see Subsection 4.5). It follows that one can
obtain the continuous spin representations from the massive ones in a suitable
massless limit $m\rightarrow 0$ since their little group UIRs are related by a
contraction. The quartic Casimir operator of the Poincaré group for the
massive representation is related to its Young diagram $\nu$ labeling the UIR
of the little group $O(D-1)$ via the formula (51):

${\cal
C}_{4}\Big{(}\mathfrak{iso}(D-1,1)\Big{)}=m^{2}\,\sum_{a=1}^{r}\nu_{a}(\nu_{a}+D-2a-1)\,,$
(78)

In order to keep ${\cal C}_{4}$ non-vanishing, the massless limit must be such
that the product of the “spin” $\nu_{1}=s$ and the mass $m$ remains finite.
More precisely, one needs $sm\rightarrow\mu\,$ in order to reproduce (52), so
that the spin goes to infinity while the row lengths $\nu_{a}$ for $a\neq 1\,$
are kept equal to $\lambda_{a-1}\,$ [24, 15]. The Fourier transform (in the
internal space spanned by $\xi$) of the wave equations (71)-(76) may be
obtained in this way from the wave equations of a massive representation in
“gauge-fixed” form (see [15] for more details). This limit is very similar to
the contraction of Subsection 4.5.

###  5.4 Tachyonic representations

The tachyonic representations have some similarities with the massive
representations. The simpler one is the analogue of the Klein-Gordon equation,
up to a change of sign for the mass term. The other similarity is that the
linear equations should remove the components along the momentum. Of course,
the major difference is that the momentum is space-like. The quartic Casimir
operator of the Poincaré algebra is also evaluated easily in components,
giving as a final result for a tachyonic representation,

$\displaystyle{\cal C}_{4}\Big{(}\mathfrak{iso}(D-1,1)\Big{)}$
$\displaystyle=$ $\displaystyle{\cal
C}_{2}\Big{(}\mathfrak{iso}(D-1,1)\Big{)}\,{\cal
C}_{2}\Big{(}\mathfrak{so}(D-2,1)\Big{)}\,\,,$ (79)

where the eigenvalues of the quadratic Casimir operator of the rotation
algebra are given by the formula (26).

Step 1. The first step is more involved for the tachyonic case since it
requires the exhaustive knowledge of the UIR theory for the groups
$SO(D-2,1)^{\uparrow}\,$. Fortunately, complete results are available [25].
However, the steps 2-3 further require the completion of the Bargmann-Wigner
programme for the isometry group $SO(D-2,1)^{\uparrow}$ of the de Sitter
spacetime $dS_{D-2}\,$, which is still an open problem in full generality.

Let us assume that this programme has been performed through an ambient space