# New representations of Poincaré group
for consistent relativistic particle theories

Giuseppe Nisticò
Dipartimento di Matematica e Informatica, Università della Calabria, Italy
and
INFN – gruppo collegato di Cosenza, Italy
email<EMAIL_ADDRESS>
###### Abstract

Though the irreducible representations of the Poincaré group form the
groundwork for the formulation of relativistic quantum theories of a particle,
robust classes of such representations are missed in current formulations of
these theories. In this work the extended class of irreducible representations
with positive “mass” parameter is explicitly determined. Several new
representations in such extension, so far excluded, give rise to consistent
theories for Klein-Gordon particles and also to new species of particle
theories.

##  1 Introduction

The identification of the irreducible representations of the Poincaré group
$\mathcal{P}$ lays the groundwork for the formulation of the relativistic
quantum theories of one elementary free particle. Indeed, each such a theory
must contain [1] an irreducible representation $g\to U_{g}$ of $\mathcal{P}$
that realizes the quantum transformation of every quantum observable according
to $A\to S_{g}[A]=U_{g}AU_{g}^{-1}$. Unfortunately, the literature about
relativistic quantum theories of a single particle does not take into account
all possible irreducible representations of the Poincaré group $\mathcal{P}$.
One of the classes discarded is that of the irreducible representations with
anti-unitary space inversion operator [2], [3], [4]; in fact, not only quantum
theories of a particle characterized by anti-unitary space inversion operator
can be consistently developed [5], but even anti-unitary space inversion
operators turn out to be indispensable for formulating complete quantum
theories of Klein-Gordon particles without the inconsistencies that plagued
the early theory [6].

These arguments point out that the following tasks should be accomplished in
order to effectively identify the possible quantum theories of a free
particle.

  * T.1.

To single out the possible irreducible representations of $\mathcal{P}$
without a priori preclusions, such as the preclusion against representations
with anti-unitary space inversion operator.

  * T.2.

Then, the explicit determination of the theories for an elementary free
particle can be addressed by selecting which of the representations identified
by T.1 satisfy the further constraints imposed by the peculiar features
characterizing this specific physical system. Only the representations
inconsistent with these further constraints have to be excluded.

In this article we address the first task. In order to confer linearity to the
presentation, we carry out a general classification and identification by
means of a systematic self-contained derivation.

Task T.2 is outside the scope of the present article. However, in order to
ascertain that our work is not meaningless from a theoretical physics point of
view, in the final section 6 we show that consistent relativistic quantum
theories of a particle can be formulated, which are based on irreducible
representations singled out by the present work and not considered in the
literature. Section 2 introduces the notation and basic mathematical
prerequisites relative to Poincaré groups and their representations. In
section 3 we show that all irreducible representations of $\mathcal{P}$ can be
classified according to the following three criteria. “Mass” and “spin”
parameters $(\mu,s)$. Each irreducible representation of $\mathcal{P}$ must be
characterized by a unique pair $(\mu,s)$ ,
$\mu\in\hbox{{{I}}\kern-4.70001pt\hbox{C}}$,
$s\in\frac{1}{2}\hbox{I\kern-1.99997pt\hbox{N}}$, called mass and spin
parameters, respectively. In this work we restrict to the class of positive
mass irreducible representations. Spectrum of $P_{0}$. In every irreducible
representation there are only three mutually exclusive possibilities for the
spectrum $\sigma(P_{0})$ of the Hamiltonian operator $P_{0}$: Either
$\sigma(P_{0})=[\mu,\infty)\equiv{\mathcal{I}}_{\mu}^{+}$; or
$\sigma(P_{0})=(-\infty,-\mu]\equiv{\mathcal{I}}_{\mu}^{-}$; or
$\sigma(P_{0})={\mathcal{I}}_{\mu}^{-}\cup{\mathcal{I}}_{\mu}^{+}$. It is
shown how these possibilities are related to the unitary or anti-unitary
character of the space inversion operator
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ and of the time reversal operator
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$. Reducibility of $U^{\pm}$. Given
an irreducible representation $U$ of $\mathcal{P}$ characterized by a pair
$(\mu,s)$ and by one of the possible spectra of $P_{0}$, it turns out that two
particular sub-representations, $U^{+}$ or $U^{-}$, can be reducible or not.
The literature takes into account only irreducible representations with
$U^{\pm}$ irreducible. Our redetermination does not overlook the irreducible
representations with $U^{+}$ or $U^{-}$ reducible. In section 4 we explicitly
identify all irreducible representations of $\mathcal{P}$ with $U^{\pm}$
irreducible. The representations with $\sigma(P_{0})=I_{\mu}^{+}$ or
$\sigma(P_{0})=I_{\mu}^{-}$ are already well known. For
$\sigma(P_{0})=I_{\mu}^{+}\cup I_{\mu}^{-}$ we identify, besides the well
known representations with both space inversion operator
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ and time reversal operator
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ unitary, also all irreducible
representations with ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ anti-unitary
and ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ unitary, and with both
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$,
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ anti-unitary, neglected in the
literature. Section 5 deals with the class of the so far “ignored”
representations of $\mathcal{P}$ with $U^{+}$ or $U^{-}$ reducible. We
explicitly identify irreducible representations $U$ of $\mathcal{P}$ with
$U^{\pm}$ reducible in all three possible cases with
$\sigma(P_{0})=I_{\mu}^{+}$, $\sigma(P_{0})=I_{\mu}^{-}$ and
$\sigma(P_{0})=I_{\mu}^{+}\cup I_{\mu}^{-}$. They open to the possibility of
yet unknown particle theories. Section 6 shows that this possibility is
absolutely concrete.

##  2 Notation and mathematical prerequisites

###  2.1 Poincaré group.

Given any vector $\underline{x}=(x_{0},x_{1},x_{2},x_{3})\equiv(x_{0},{\bf
x})\in\mathop{{\rm I}\kern-1.99997pt{\rm R}}\nolimits^{4}$, we call $x_{0}$
the time component of $\underline{x}$ and ${\bf x}=(x_{1},x_{2},x_{3})$ the
spatial component of $\underline{x}$. The proper orthochronous Poincaré group
${\mathcal{P}}_{+}^{\uparrow}$ is the separable locally compact group of all
transformations of $\mathop{{\rm I}\kern-1.99997pt{\rm R}}\nolimits^{4}$
generated by the ten one-parameter sub-groups ${\mathcal{T}}_{0}$,
${\mathcal{T}}_{j},{\mathcal{R}}_{j}$, ${\mathcal{B}}_{j}$, $j=1,2,3$, of time
translations, spatial translation, proper spatial rotations and Lorentz
boosts, respectively. The Euclidean group $\mathcal{E}$ is the sub-group
generated by all ${\mathcal{T}}_{j}$ and ${\mathcal{R}}_{j}$. The sub-group
generated by ${\mathcal{R}}_{j}$, ${\mathcal{B}}_{j}$ is the proper
orthochronous Lorentz group ${\mathcal{L}}_{+}^{\uparrow}$ [7]; it does not
include time reversal ${{}^{\triangleleft}\kern-3.2pt\texttt{t}}$ and space
inversion ${{}_{\triangleleft}\kern-2.2pt\texttt{s}}$. Time reversal
${{}^{\triangleleft}\kern-3.2pt\texttt{t}}$ transforms
$\underline{x}=(x_{0},{\bf x})$ into $(-x_{0},{\bf x})$; space inversion
${{}_{\triangleleft}\kern-2.2pt\texttt{s}}$ transforms
$\underline{x}=(x_{0},{\bf x})$ into $(x_{0},-{\bf x})$. The group generated
by
$\\{{\mathcal{P}}_{+}^{\uparrow},{{}^{\triangleleft}\kern-3.2pt\texttt{t}},{{}_{\triangleleft}\kern-2.2pt\texttt{s}}\\}$
is the separable and locally compact Poincaré group $\mathcal{P}$. By
${\mathcal{L}}_{+}$ we denote the subgroup generated by
${\mathcal{L}}_{+}^{\uparrow}$ and
${{}^{\triangleleft}\kern-3.2pt\texttt{t}}$, while ${\mathcal{L}}^{\uparrow}$
denotes the subgroup generated by ${\mathcal{L}}_{+}^{\uparrow}$ and
${{}_{\triangleleft}\kern-2.2pt\texttt{s}}$; analogously, ${\mathcal{P}}_{+}$
denotes the subgroup generated by ${\mathcal{P}}_{+}^{\uparrow}$ and
${{}^{\triangleleft}\kern-3.2pt\texttt{t}}$, while ${\mathcal{P}}^{\uparrow}$
is the subgroup generated by ${\mathcal{P}}_{+}^{\uparrow}$ and
${{}_{\triangleleft}\kern-2.2pt\texttt{s}}$.

###  2.2 Mathematical structures.

The following mathematical structures, based on a complex and separable
Hilbert space $\mathcal{H}$, are of general interest in quantum theory.

-

The set $\Omega({\mathcal{H}})$ of all self-adjoint operators of
$\mathcal{H}$; in a quantum theory these operators represent quantum
observables.

-

The lattice $\Pi({\mathcal{H}})$ of all projections operators of
$\mathcal{H}$; in a quantum theory they represent observables with spectrum
$\\{0,1\\}$.

-

The set $\Pi_{1}({\mathcal{H}})$ of all rank one orthogonal projections of
$\mathcal{H}$.

-

The set ${\mathcal{S}}({\mathcal{H}})$ of all density operators of
$\mathcal{H}$; in a quantum theory these operators represent quantum states.

-

The set ${\mathcal{V}}({\mathcal{H}})$ of all unitary or anti-unitary
operators of the Hilbert space $\mathcal{H}$.

-

The set ${\mathcal{U}}({\mathcal{H}})$ of all unitary operators of
$\mathcal{H}$; trivially,
${\mathcal{U}}({\mathcal{H}})\subseteq{\mathcal{V}}({\mathcal{H}})$ holds.

###  2.3 Generalized representations of groups.

The following definition introduces generalized notions of group
representation. Definition 2.1. Let $G$ be a separable, locally compact group
with identity element $e$. A correspondence
$U:G\to{\mathcal{V}}({\mathcal{H}})$, $g\to U_{g}$, with $U_{e}=\hbox{\sl
1\kern-2.5pt\hbox{I}}$, is a generalized projective representation of $G$ if
the following conditions are satisfied.

i)

A complex function
$\sigma:{G}\times{G}\to{\hbox{{{I}}\kern-4.70001pt\hbox{C}}}$, called
multiplier, exists such that
$U_{g_{1}g_{2}}=\sigma(g_{1},g_{2})U_{g_{1}}U_{g_{2}}$; the modulus
$|\sigma(g_{1},g_{2})|$ is always 1, of course;

ii)

for all $\phi,\psi\in\mathcal{H}$, the mapping $g\to\langle
U_{g}\phi\mid\psi\rangle$ is a Borel function in $g$.

If $U_{g}$ is unitary for all $g\in G$, then $U$ is called projective
representation.

A generalized projective representation is said to be continuous if for any
fixed $\psi\in\mathcal{H}$ the mapping $g\to U_{g}\psi$ from $G$ to
$\mathcal{H}$ is continuous with respect to $g$.  If $g\to U_{g}$ is a
generalized projective representation of $\mathcal{P}$ and
$\theta(g)\in\mathop{{\rm I}\kern-1.99997pt{\rm R}}\nolimits$, then
$g\to\tilde{U}_{g}=e^{i\theta(g)}U_{g}$ is a generalized projective
representation, said equivalent [8] to $g\to U_{g}$. In [9] we have proved
that the following statement holds. Proposition 2.1. If $G$ is a connected
group, then every continuous generalized projective representation of $G$ is a
projective representation, i.e. $U_{g}\in{\mathcal{U}}({\mathcal{H}})$, for
all $g\in G$.

###  2.4 Generalized representations of the Poincaré group $\mathcal{P}$

All sub-groups ${\mathcal{T}}_{0}$, ${\mathcal{T}}_{j},{\mathcal{R}}_{j}$,
${\mathcal{B}}_{j}$ of ${\mathcal{P}}_{+}^{\uparrow}$ are additive; in fact,
${\mathcal{B}}_{j}$ is not additive with respect to the parameter relative
velocity $u$, but it is additive with respect to the parameter
$\varphi(u)=\frac{1}{2}\ln\frac{1+u}{1-u}$. Then, according to Stone’s theorem
[10], for every continuous projective representation of
${\mathcal{P}}_{+}^{\uparrow}$, an equivalent projective representation $U$
exists for which there are ten self-adjoint generators $P_{0}$, $P_{j}$,
$J_{j}$, $K_{j}$, $j=1,2,3$, of the ten one-parameter unitary subgroups
$\\{e^{iP_{0}t}\\}$, $\\{e^{-i{P}_{j}a_{j}},\,a\in{\mathop{{\rm
I}\kern-1.99997pt{\rm R}}\nolimits}\\}$,
$\\{e^{-i{J}_{j}\theta_{j}},\,\theta_{j}\in{\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits}\\}$, $\\{e^{i{K}_{j}\varphi(u_{j})},\,u_{j}\in{\mathop{{\rm
I}\kern-1.99997pt{\rm R}}\nolimits}\\}$ of ${\mathcal{U}}({\mathcal{H}})$ that
represent the one-parameter sub-groups ${\mathcal{T}}_{0}$,
${\mathcal{T}}_{j},{\mathcal{R}}_{j}$, ${\mathcal{B}}_{j}$ according to $U$.

####  2.4.1 Commutation relations.

The structural properties of ${\mathcal{P}}_{+}^{\uparrow}$ as a Lie group
imply that every continuous projective representation of
${\mathcal{P}}_{+}^{\uparrow}$ admits an equivalent projective representation
$U$ such that the following commutation relations [11] hold for its
generators. (i) $[{P}_{j},{P}_{k}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$, (ii)
$[{J}_{j},{P}_{k}]=i{\hat{\epsilon}}_{jkl}{P}_{l}$, (iii)
$[{J}_{j},{J}_{k}]=i{\hat{\epsilon}}_{jkl}{J}_{l}$,

(iv) $[{J}_{j},{K}_{k}]=i{\hat{\epsilon}}_{jkl}{K}_{l}$, (v)
$[{K}_{j},{K}_{k}]=-i\delta_{j,k}J_{l}$, (vi)
$[{K}_{j},{P}_{k}]=i\delta_{jk}P_{0}$, (1)

(vii) $[P_{j},P_{0}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$, (viii)
$[J_{j},P_{0}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$, (ix)
$[K_{j},P_{0}]=iP_{0}$, where ${\hat{\epsilon}}_{jkl}$ is the Levi-Civita
symbol ${\epsilon}_{jkl}$ restricted by the condition $j\neq l\neq k$.

Let $U:{\mathcal{P}}\to{\mathcal{V}}({\mathcal{H}})$ be a generalized
projective representation of $\mathcal{P}$, whose restriction to
${\mathcal{P}}_{+}^{\uparrow}$ is continuous. Since time reversal
${{}^{\triangleleft}\kern-3.2pt\texttt{t}}$ and space inversion
${{}_{\triangleleft}\kern-2.2pt\texttt{s}}$ are not connected with the
identity transformation $e\in\mathcal{P}$, the operators
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}=U_{{}^{\triangleleft}\kern-3.2pt\texttt{t}}$
and
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}=U_{{}_{\triangleleft}\kern-2.2pt\texttt{s}}$
can be unitary or anti-unitary. The phase factor $e^{i\theta(g)}$ can be
always chosen [11] in such a way that the following statements hold in the
equivalent generalized projective representation. If
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ is unitary,

then
$[{{}_{\triangleleft}\kern-2.0pt\texttt{S}},P_{0}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$,
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}P_{j}=-P_{j}{{}_{\triangleleft}\kern-2.0pt\texttt{S}}$,
$[{{}_{\triangleleft}\kern-2.0pt\texttt{S}},J_{k}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$,
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}K_{j}=-K_{j}{{}_{\triangleleft}\kern-2.0pt\texttt{S}}$;
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}^{2}=\hbox{\sl
1\kern-2.5pt\hbox{I}}$; (2) If ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ is
anti-unitary,

then
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}P_{0}=-P_{0}{{}_{\triangleleft}\kern-2.0pt\texttt{S}}$,
$[{{}_{\triangleleft}\kern-2.0pt\texttt{S}},P_{j}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$,
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}J_{k}=-J_{k}{{}_{\triangleleft}\kern-2.0pt\texttt{S}}$,
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}K_{j}=K_{j}{{}_{\triangleleft}\kern-2.0pt\texttt{S}}$,
(3)

and ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}^{2}=c\hbox{\sl
1\kern-2.5pt\hbox{I}}$, so that
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}^{-1}=c{{}_{\triangleleft}\kern-2.0pt\texttt{S}}$,
where $c=1$ or $c=-1$. If ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ is
unitary,

then
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}P_{0}=-P_{0}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}$,
$[{{}^{\triangleleft}\kern-1.5pt\texttt{T}},P_{j}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$,
$[{{}^{\triangleleft}\kern-1.5pt\texttt{T}},J_{k}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$,
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}K_{j}=-K_{j}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}$;
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}^{2}=\hbox{\sl
1\kern-2.5pt\hbox{I}}$ (4) If ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ is
anti-unitary,

then
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}P_{0}=P_{0}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}$,
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}P_{j}=-P_{j}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}$,
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}J_{k}=-J_{k}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}$,
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}K_{j}=K_{j}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}$,
(5)

and ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}^{2}=c\hbox{\sl
1\kern-2.5pt\hbox{I}}$, so that
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}^{-1}=c{{}^{\triangleleft}\kern-1.5pt\texttt{T}}$,
either $c=1$ or $c=-1$ must hold. The commutation condition for the pair
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$,${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$,
is

${{}_{\triangleleft}\kern-2.0pt\texttt{S}}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}=\omega{{}^{\triangleleft}\kern-1.5pt\texttt{T}}{{}_{\triangleleft}\kern-2.0pt\texttt{S}}$,
with $\omega\in\hbox{{{I}}\kern-4.70001pt\hbox{C}}$ and $|\omega|=1$. (6) From
now on the continuity hypothesis for $U\mid_{{\mathcal{P}}_{+}^{\uparrow}}$ is
implicitly assumed. A quantum theory based on a generalized projective
representation is indistinguishable in all respects from the theory based on
an equivalent $e^{i\theta}U$. For this reason we assume that a generalized
projective representation of $\mathcal{P}$ satisfies (1)-(6). Proposition 2.2.
If $U:{\mathcal{P}}\to{\mathcal{V}}({\mathcal{H}})$ is a generalized
projective representation, then the relations (1)-(6) imply that the following
equalities hold for all $g\in{{\mathcal{P}}}$, including
${{}^{\triangleleft}\kern-3.2pt\texttt{t}}$ and
${{}_{\triangleleft}\kern-2.2pt\texttt{s}}$.

$[P_{0}^{2}-(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}),U_{g}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}\,,$
$None$
$[W_{0}^{2}-(W_{1}^{2}+W_{2}^{2}+W_{3}^{2}),U_{g}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}\,,$
$None$

where $W_{0}={\bf P}\cdot{\bf J}$ and $W_{j}=P_{0}J_{j}-({\bf P}\times{\bf
K})_{j}$ define the Lubański four-operator $\underline{W}=(W_{0},{\bf W})$.

###  2.5 Spectral properties of the self-adjoint generators

Spectral properties of the self-adjoint generators are now derived. Relations
(1.i), (1.vii) establish that the generators $P_{0}$, $P_{1}$, $P_{2}$,
$P_{3}$ of a generalized projective representation $U$ of $\mathcal{P}$
commute with each other; therefore, according to spectral theory [12] a common
spectral measure $E:{\mathcal{B}}({\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits}^{4})\to\Pi({\mathcal{H}})$ exists such that

$P_{0}=\int\lambda dE^{(0)}_{\lambda}\,,\quad P_{j}=\int\lambda
dE^{(j)}_{\lambda}\,,\;j=1,2,3,$ $None$

where $E^{(0)}_{\lambda}=E((-\infty,\lambda]\times{\mathop{{\rm
I}\kern-1.99997pt{\rm R}}\nolimits}^{3})$, $E^{(1)}_{\lambda}=E(\mathop{{\rm
I}\kern-1.99997pt{\rm R}}\nolimits\times(-\infty,\lambda]\times{\mathop{{\rm
I}\kern-1.99997pt{\rm R}}\nolimits}^{2})$, $E^{(2)}_{\lambda}=E({\mathop{{\rm
I}\kern-1.99997pt{\rm
R}}\nolimits}^{2}\times(-\infty,\lambda]\times{\mathop{{\rm
I}\kern-1.99997pt{\rm R}}\nolimits})$, $E^{(3)}_{\lambda}=E({\mathop{{\rm
I}\kern-1.99997pt{\rm R}}\nolimits}^{3}\times(-\infty,\lambda])$ are the
resolutions of the identity of the individual operators $P_{0}$, $P_{1}$,
$P_{2}$, $P_{3}$.

Once introduced the four-operator
$\underline{P}=(P_{0},P_{1},P_{2},P_{3})\equiv(P_{0},{\bf P})$, the equalities
(9) can be rewritten in the more compact form

$\underline{P}=\int\underline{p}\,dE{\underline{p}}\,,\hbox{ with
}dE_{\underline{p}}=dE^{(0)}_{p_{0}}dE^{(1)}_{p_{1}}dE^{(2)}_{p_{2}}dE^{(3)}_{p_{3}}\,.$
$None$

The spectrum of $\underline{P}$ can be defined as the following closed subset
of $\mathop{{\rm I}\kern-1.99997pt{\rm R}}\nolimits^{4}$.

$\sigma({\underline{P}})=\\{\underline{p}=(p_{0},{\bf p})\in\mathop{{\rm
I}\kern-1.99997pt{\rm R}}\nolimits^{4}\mid
E(\Delta_{\underline{p}})\neq\hbox{{{I}}\kern-4.70001pt\hbox{O}}\hbox{ for
every neighborough }\Delta_{\underline{p}}\hbox{ of }\underline{p}\\}\,.$
$None$

By making use of (1), the following proposition can be proved. Proposition
2.3. Let $U:{\mathcal{P}}\to{\mathcal{U}}({\mathcal{H}})$ be a projective
representation of ${\mathcal{P}}_{+}^{\uparrow}$, Then for every Lorentz
transformation $g\in{\mathcal{L}}_{+}^{\uparrow}$ the following relation holds

$U_{g}\underline{P}U_{g}^{-1}=\hbox{\rm{g}}(\underline{P}),$ $None$

where ${\rm\textsf{g}}:\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{4}\to\mathop{{\rm I}\kern-1.99997pt{\rm R}}\nolimits^{4}$ is the
function that transforms any $\underline{p}\in\mathop{{\rm
I}\kern-1.99997pt{\rm R}}\nolimits^{4}$ as a four-vector according to $g$.
Moreover, the following statement is a straightforward implication of (12).

$U_{g}E(\Delta)U_{g}^{-1}=E({\textsf{g}}^{-1}(\Delta))\hbox{ holds for every
}g\in{\mathcal{L}}_{+}^{\uparrow}.$ $None$

##  3 Classification of positive “mass” irreducible representations of
$\mathcal{P}$

A generalized projective representation
$U:{\mathcal{P}}\to{\mathcal{V}}({\mathcal{H}})$ can be reducible or not; in
the case that it is reducible, however, it must be the direct sum or the
direct integral of irreducible ones [7]. Therefore, to determine all possible
generalized projective representations of ${\mathcal{P}}$ it is sufficient to
identify the irreducible ones. For this reason, from now on we specialize to
irreducible generalized projective representations of $\mathcal{P}$. Hence,
from Prop. 2.2 the following proposition follows. Proposition 3.1. If a
generalized projective representation of $\mathcal{P}$ is irreducible, then
two real numbers $\eta$, $\varpi$ exist such that the following equalities
hold.

$(i)\;\;P_{0}^{2}-{\bf P}^{2}=\eta\hbox{\sl
1\kern-2.5pt\hbox{I}}\quad\hbox{and}\quad(ii)\;\;W^{2}\equiv
W_{0}^{2}-(W_{1}^{2}+W_{2}^{2}+W_{3}^{2})=\varpi\hbox{\sl
1\kern-2.5pt\hbox{I}}\,.$ $None$

Therefore every irreducible generalized projective representation of
${\mathcal{P}}$ is characterized by the real constants $\eta,\varpi$. We
restrict our investigation to those irreducible generalized representations
for which $\eta>0$, so that $\eta=\mu^{2}$, with $\mu>0$; with this
restriction it can be proved that
$s\in\frac{1}{2}\hbox{I\kern-1.99997pt\hbox{N}}$ exists such that
$\varpi=-\mu^{2}s(s+1)$. The parameters $\mu$ and $s$ are called mass and spin
parameters, respectively.

###  3.1 Spectral characterization of positive “mass” irreducible
representations of $\mathcal{P}$

Now we show that for an irreducible generalized projective representation of
${\mathcal{P}}$, characterized by specific parameters $\mu>0$ and $s$, the
spectrum $\sigma(\underline{P})$ of the four-operator
$\underline{P}=(P_{0},{\bf P})$, must be one of three definite subsets
$S^{+}_{\mu}$, $S^{-}_{\mu}$, $S^{+}_{\mu}\cup S^{-}_{\mu}$ of $\mathop{{\rm
I}\kern-1.99997pt{\rm R}}\nolimits^{4}$, where

$S^{+}_{\mu}=\\{\underline{p}\mid p_{0}^{2}-{\bf
p}^{2}=\mu^{2},\,p_{0}>0\\}\,,\hbox{ (positive hyperboloid)}\,,$ $None$
$S^{-}_{\mu}=\\{\underline{p}\mid p_{0}^{2}-{\bf
p}^{2}=\mu^{2},\,p_{0}<0\\}\,,\hbox{ (negative hyperboloid)}\,.$

Proposition 3.2. If $U:{\mathcal{P}}\to{\mathcal{V}}({\mathcal{H}})$ is an
irreducible generalized projective representation, then there are only the
following mutually exclusive possibilities for the spectra
$\sigma(\underline{P})$ and $\sigma(P_{0})$. (u)
$\sigma(\underline{P})=S^{+}_{\mu}$ and
$\sigma(P_{0})=[\mu,\infty)\equiv{\mathcal{I}}_{\mu}^{+}$, “up” spectrum; (d)
$\sigma(\underline{P})=S^{-}_{\mu}$ and
$\sigma(P_{0})=(-\infty,-\mu]\equiv{\mathcal{I}}_{\mu}^{-}$, “down” spectrum;
(s) $\sigma(\underline{P})=S^{+}_{\mu}\cup S^{-}_{\mu}$ and
$\sigma(P_{0})={\mathcal{I}}_{\mu}^{+}\cup{\mathcal{I}}_{\mu}^{-}$,
“symmetrical” spectrum.  Proof. Since $P_{0}^{2}-{\bf
P}^{2}-\mu^{2}=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$, if
$\underline{p}\in\sigma(\underline{P})$ then $p_{0}^{2}-{\bf p}^{2}-\mu^{2}=0$
must hold, i.e.

$\sigma(\underline{P})\subseteq\\{\underline{p}\mid p_{0}^{2}-{\bf
p}^{2}-\mu^{2}=0\\}\equiv S^{+}_{\mu}\cup S^{-}_{\mu}\,.$ $None$

On the other hand, if $\underline{p}\in\sigma(\underline{P})$, then according
to spectral theory
$\textsf{g}(\underline{p})\in\sigma(\textsf{g}(\underline{P}))$ holds for all
$g\in{\mathcal{L}}_{+}^{\uparrow}$, of course; but
$\textsf{g}(\underline{P})=U_{g}\underline{P}U_{g}^{-1}$ by Prop. 2.3;
therefore, $\underline{p}\in\sigma(\underline{P})$ if and only if
$\textsf{g}(\underline{p})\in\sigma(\underline{P})$ because $\underline{P}$
and $U_{g}\underline{P}U_{g}^{-1}$ have the same spectra. Hence, the following
statements hold.

$\sigma(\underline{P})\cap S^{+}_{\mu}\neq\emptyset\hbox{ implies
}S^{+}_{\mu}\subseteq\sigma(\underline{P})\,;\;\;\sigma(\underline{P})\cap
S^{-}_{\mu}\neq\emptyset\hbox{ implies
}S^{-}_{\mu,\varpi}\subseteq\sigma(\underline{P})\,.$ $None$

Since $\sigma(\underline{P})\neq\emptyset$, (16) and (17) imply that only one
of the three cases (u), (d) or (s) can occur. $\bullet$ In the case (s) the
restriction $U:{\mathcal{P}}_{+}^{\uparrow}\to{\mathcal{U}}({\mathcal{H}})$ is
always reducible, namely $U\mid_{{\mathcal{P}}_{+}^{\uparrow}}$ is reduced by
the projection operators
$E^{+}=\int_{S^{+}_{\mu}}dE_{\underline{p}}\equiv\int_{\mu}^{\infty}p_{0}dE_{p_{0}}^{(0)}$
and
$E^{-}=\int_{S^{-}_{\mu}}dE_{\underline{p}}\equiv\int_{-\infty}^{-\mu}p_{0}dE_{p_{0}}^{(0)}$,
with ranges ${\mathcal{M}}^{+}=E^{+}{\mathcal{H}}$ and
${\mathcal{M}}^{-}=E^{-}{\mathcal{H}}$, respectively. We prove this statement
in the following Proposition. Proposition 3.3. In an irreducible generalized
projective representation $U:{\mathcal{P}}\to{\mathcal{V}}({\mathcal{H}})$ the
relation $[E^{\pm},U_{g}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$ holds for all
$g\in{\mathcal{P}}_{+}^{\uparrow}$. Hence, the following consequences can be
immediately implied.

  * i)

In the case of symmetrical spectrum $\sigma(\underline{P})=S_{\mu}^{+}\cup
S_{\mu}^{-}$, the restriction $U\mid_{{\mathcal{P}}_{+}^{\uparrow}}$ is
reduced by $E^{+}$ into $U^{+}=E^{+}U\mid_{{\mathcal{P}}_{+}^{\uparrow}}E^{+}$
and $U^{-}=E^{-}U\mid_{{\mathcal{P}}_{+}^{\uparrow}}E^{-}$;

  * ii)

if $\sigma(\underline{P})=S_{\mu}^{+}$ (resp.,
$\sigma(\underline{P})=S_{\mu}^{-}$), then
$U\mid_{{\mathcal{P}}_{+}^{\uparrow}}=U^{+}$ (resp.,
$U\mid_{{\mathcal{P}}_{+}^{\uparrow}}=U^{-}$).

Proof. In the case (u) and (d), the statement is trivial because
$E^{\pm}=\int_{S_{\mu}^{\pm}}dE_{\underline{p}}\equiv\int_{\sigma(\underline{P})}dE_{\underline{p}}=\hbox{\sl
1\kern-2.5pt\hbox{I}}$. Then we suppose that
$\sigma(\underline{P})=S_{\mu}^{+}\cup S_{\mu}^{-}$. Since
$E^{+}=\chi_{S^{+}_{\mu}}(\underline{P})$, where $\chi_{S^{+}_{\mu}}$ is the
characteristic functional of $S^{+}_{\mu}$, the relations (1.i),(1.vii) imply
that $E^{+}$ commutes with $P_{0}$ and with all $P_{j}$. Therefore it remains
to show that ${\mathcal{M}}^{+}$ is left invariant by $U_{g}$, for every
$g\in{\mathcal{L}}_{+}^{\uparrow}$. If $\psi\in{\mathcal{M}}^{+}$, then for
every $g\in{\mathcal{L}}_{+}^{\uparrow}$ we have
$U_{g}\psi=U_{g}\int_{S^{+}_{\mu}}dE_{\underline{p}}\psi=\int_{{S^{+}_{\mu}}}dU_{g}E_{\underline{p}}{U_{g}}^{-1}(U_{g}\psi)=\int_{S^{+}_{\mu}}dE_{{\textsf{g}}^{-1}({\underline{p}})}(U_{g}\psi)$,
by Prop. 2.3. The last integral is a vector of
${\mathcal{M}}^{+}=E^{+}\mathcal{H}$, because
${\textsf{g}}^{-1}(\underline{p})\in S_{\mu}^{+}$ if $\underline{p}\in
S_{\mu}^{+}$ for $g\in{\mathcal{L}}_{+}^{\uparrow}$.

The same argument, suitably adapted, proves that ${\mathcal{M}}^{-}$ is left
invariant by $U_{g}$, for every $g\in{\mathcal{P}}_{+}^{\uparrow}$. The
consequences (i) and (ii) are straightforward. $\bullet$

####  3.1.1 Spectral implications of the unitary or anti-unitary character of
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ and
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$

Now we show how each of the possibilities for $\sigma(\underline{P})$
established by Prop.3.2 is characterizable according to the unitary or anti-
unitary character of the time reversal and the space inversion operators
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ and
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$. Lemma 3.1. Let $T$ be a unitary
or anti-unitary operator, and let $A$ be a self-adjoint operator with spectral
measure $E^{A}:{\mathcal{B}}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits)\to\Pi({\mathcal{H}})$. If $TAT^{-1}=f(A)$, where $f$ is a
continuous bijection of $\mathop{{\rm I}\kern-1.99997pt{\rm R}}\nolimits$,
then $TE^{A}(\Delta)T^{-1}=E^{A}(f^{-1}(\Delta))$, for all
$\Delta\in{\mathcal{B}}(\mathop{{\rm I}\kern-1.99997pt{\rm R}}\nolimits)$.
Proof. We recall that if $T$ is unitary or anti-unitary, then an operator $D$
is a projection operator if and only if $TDT^{-1}$ is a projection operator;
moreover, if $\Delta\to E^{A}(\Delta)$ is the spectral measure of $A$, then
$\Delta\to F(\Delta)=TE_{A}(\Delta)T^{-1}$ is the spectral measure of $f(A)$.
Now, let us define $\tilde{\Delta}=f^{-1}(\Delta)$. If $\pi(-a,a)$ is a
partition of the interval $[-a,a]$ formed by sub-intervals
$\tilde{\Delta}_{j}$ with $\tilde{\lambda}_{j}\in\tilde{\Delta}_{j}$, then
according to spectral theory we can write

$f(A)=\lim_{{\|\pi(-a,a)\|\to
0}\atop{a\to\infty}}\sum_{j}f(\tilde{\lambda}_{j})E(\tilde{\Delta}_{j})=\lim_{{\|\pi(-a,a)\|\to
0}\atop{a\to\infty}}\sum_{j}\lambda_{j}E(f^{-1}(\Delta_{j}))$

$=\lim_{{\|\pi(-a,a)\|\to
0}\atop{a\to\infty}}\sum_{j}\lambda_{j}F(\Delta_{j})$, where
$\lambda_{j}=f(\tilde{\lambda}_{j})\in\Delta_{j}$.

Therefore, for the uniqueness of the spectral measure $F$ of $f(A)$ we have

$F(\Delta)=E(f^{-1}(\Delta))=TE(\Delta)T^{-1}$. $\bullet$ Proposition 3.4. Let
$U:{\mathcal{P}}\to{\mathcal{U}}({\mathcal{H}})$ be an irreducible generalized
projective representation. If ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ is
anti-unitary and ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ is unitary, then
either $\sigma(\underline{P})=S_{\mu}^{+}$ or
$\sigma(\underline{P})=S_{\mu}^{-}$, and hence
$\sigma(\underline{P})=S_{\mu}^{+}\cup S_{\mu}^{-}$ cannot occur. Proof. First
we show that the hypotheses imply that ${\mathcal{M}}^{+}$ and
${\mathcal{M}}^{-}$ are invariant under both
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ and
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$. According to (5) the relation
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}P_{0}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}^{-1}=P_{0}$
holds when ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ is anti-unitary;
therefore Lemma 3.1 applies with $A=P_{0}$,
$T={{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ and $f$ the identity function,
so that
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}E^{(0)}(\Delta){{}^{\triangleleft}\kern-1.5pt\texttt{T}}^{-1}=E^{(0)}(\Delta)$
holds. This implies that if $\psi$ is any non vanishing vector in
${\mathcal{M}}^{+}$, then
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}\psi=\int_{\mu}^{\infty}d({{}^{\triangleleft}\kern-1.5pt\texttt{T}}E_{p_{0}}^{(0)}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}^{-1}){{}^{\triangleleft}\kern-1.5pt\texttt{T}}\psi=\int_{\mu}^{\infty}d\,E^{(0)}_{p_{0}}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}\psi\equiv
E^{+}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}\psi$. Thus
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}\psi$ is a vector in
${\mathcal{M}}^{+}$. This argument can be repeated with
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ instead of
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$, to deduce, by making use of (2),
that ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}\psi\in{\mathcal{M}}^{+}$ for
all $\psi\in{\mathcal{M}}^{+}$. The invariance of ${\mathcal{M}}^{-}$ is
proved quite similarly.

Now, since ${\mathcal{M}}^{+}$ and ${\mathcal{M}}^{-}$ are invariant under the
restriction $U\mid_{{\mathcal{P}}_{+}^{\uparrow}}$ according to Prop. 3.2,
they are invariant under the whole $U$. If
$\sigma(\underline{P})=S_{\mu}^{+}\cup S_{\mu}^{-}$ held, then
${\mathcal{M}}^{+}$ would be a proper subspace of $\mathcal{H}$, so that $U$
would be reducible, in contradiction with the hypothesis of irreducibility.
$\bullet$ Proposition 3.5. If ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ is
unitary then $\sigma(\underline{P})=S_{\mu}^{+}\cup S_{\mu}^{-}$ holds,
independently of ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$. Proof. If
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ is unitary, then
$\sigma({{}^{\triangleleft}\kern-1.5pt\texttt{T}}P_{0}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}^{-1})=\sigma(P_{0})$.
But
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}P_{0}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}^{-1}=-P_{0}$
holds by (4); therefore
$-\sigma(P_{0})\equiv\sigma(-P_{0})=\sigma({{}^{\triangleleft}\kern-1.5pt\texttt{T}}P_{0}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}^{-1})=\sigma(P_{0})$,
i.e.

$\sigma(P_{0})=-\sigma(P_{0}).$ $None$

Now, in general we have $\sigma(\underline{P})=S_{\mu}^{+}$ if and only if
$\sigma(P_{0})=[\mu,\infty)$, $\sigma(\underline{P})=S_{\mu}^{-}$ if and only
if $\sigma(P_{0})=(-\infty,-\mu]$, and $\sigma(\underline{P})=S_{\mu}^{+}\cup
S_{\mu}^{-}$ if and only if $\sigma(P_{0})=(-\infty,-\mu]\cup[\mu,\infty)$;
equation (18) holds only if $\sigma(P_{0})=(-\infty,-\mu]\cup[\mu,\infty)$;
thus $\sigma(\underline{P})=S_{\mu}^{+}\cup S_{\mu}^{-}$. $\bullet$
Proposition 3.6. If ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ is anti
unitary then $\sigma(\underline{P})=S_{\mu}^{+}\cup S_{\mu}^{-}$,
independently of ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$. Proof. Since
$\sigma(P_{0})$ is not empty, at least one of the projection operators
$E^{+}=E^{(0)}([\mu,\infty))$ or $E^{-}=E^{(0)}((-\infty,-\mu])$ must be
different from the null operator IO. Let us suppose that
$E^{(0)}([\mu,\infty))\neq\hbox{{{I}}\kern-4.70001pt\hbox{O}}$, so that
$S_{\mu}^{+}\subseteq\sigma(\underline{P})$. Since
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}P_{0}{{}_{\triangleleft}\kern-2.0pt\texttt{S}}^{-1}=-P_{0}$
holds by (3), Lemma 3.1 can be applied to deduce
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}E^{(0)}([\mu,\infty)){{}_{\triangleleft}\kern-2.0pt\texttt{S}}^{-1}=E^{(0)}((-\infty,-\mu])$;
hence $E^{(0)}((-\infty,-\mu])$ is a non null projection operator because
$E^{(0)}([\mu,\infty))$ is non-null and
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ is anti-unitary. This means that
$\sigma(P_{0})\cap(-\infty,-\mu]$ is not empty, that is to say that
$\sigma(\underline{P})\cap S_{\mu}^{-}\neq\emptyset$; therefore, according to
Prop. 3.2, $\sigma(\underline{P})=S_{\mu}\cup S_{\mu}^{-}$.

In the case $E^{(0)}((-\infty,-\mu])\neq\hbox{{{I}}\kern-4.70001pt\hbox{O}}$
the argument is easily adapted to reach the same conclusion. $\bullet$ The
following proposition is an easy corollary of these results Proposition 3.7.
If $\sigma(\underline{P})=S^{+}_{\mu}\cup S^{-}_{\mu}$ every vector
$\psi\in\mathcal{H}$ can be represented as a column vector
$\psi\equiv\left[\begin{array}[]{c}\psi^{+}\cr\psi^{-}\end{array}\right]$,
where $\psi^{+}=E^{+}\psi$ and $\psi^{-}=E^{-}\psi$. Coherently with such a
representation, any linear or anti-linear operator $A$ is represented by a
matrix $A\equiv\left[\begin{array}[]{cc}A_{11}&A_{12}\cr
A_{21}&A_{22}\end{array}\right]$, where $A_{11}=E^{+}AE^{+}$,
$A_{1}=E^{+}AE^{-}$, $A_{21}=E^{-}AE^{+}$ and $A_{22}=E^{-}AE^{-}$, in such a
way that $A\psi=\left[\begin{array}[]{cc}A_{11}&A_{12}\cr
A_{21}&A_{22}\end{array}\right]\left[\begin{array}[]{c}\psi^{+}\cr\psi^{-}\end{array}\right]$.
Since
$[E^{\pm},P_{0}]=[E^{\pm},P_{j}]=[E^{\pm},J_{j}]=[E^{\pm},K_{j}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$,
the generators $P_{0}$, $P_{j}$, $J_{k}$, $K_{j}$ have a diagonal form.

The operators ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ and
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ have diagonal representation only
if are unitary and anti-unitary, respectively.

###  3.2 General classification

An effective help, in explicitly identifying the possible structures of the
irreducible generalized projective representations of the Poincaré group, will
be provided just by the investigation of the reductions $U^{+}$ or $U^{-}$
singled out by Prop. 3.3. In general, even if the “mother” irreducible
generalized projective representation $U$ is irreducible, the reductions
$U^{+}$ or $U^{-}$ can be reducible or not. Let us denote the class of all
irreducible generalized projective representations of $\mathcal{P}$ by
${\mathcal{I}}_{\mathcal{P}}$ (unitarily equivalent representations are
identified in ${\mathcal{I}}_{\mathcal{P}}$). In virtue of Prop. 3.1, we can
operate a classification of the representations in
${\mathcal{I}}_{\mathcal{P}}$ according to the characterizing parameters $\mu$
and $s$: C.1.
${\mathcal{I}}_{\mathcal{P}}=\cup_{\mu>0,s\in\frac{1}{2}\hbox{I\kern-1.39998pt\hbox{N}}}\;{\mathcal{I}}_{\mathcal{P}}(\mu,s)$,
where ${\mathcal{I}}_{\mathcal{P}}(\mu,s)$ is the class of all representations
in ${\mathcal{I}}_{\mathcal{P}}$ such that $P_{0}^{2}-{\bf
P}^{2}=\mu^{2}\hbox{\sl 1\kern-2.5pt\hbox{I}}$ and
$W^{2}=-\mu^{2}s(s+1)\hbox{\sl 1\kern-2.5pt\hbox{I}}$. In its turn, by Prop.
3.2, each class ${\mathcal{I}}_{\mathcal{P}}(\mu,s)$ in C.1, can be decomposed
as C.2.
${\mathcal{I}}_{\mathcal{P}}(\mu,s)={\mathcal{I}}_{\mathcal{P}}(S_{\mu}^{+},s)\cup{\mathcal{I}}_{\mathcal{P}}(S_{\mu}^{-},s)\cup{\mathcal{I}}_{\mathcal{P}}(S_{\mu}^{+}\cup
S_{\mu}^{-},s)$, where ${\mathcal{I}}_{\mathcal{P}}(S_{\mu}^{\pm},s)$ is the
class of all representations in ${\mathcal{I}}_{\mathcal{P}}(\mu,s)$ such that
$\sigma(\underline{P})=S_{\mu}^{\pm}$, and
${\mathcal{I}}_{\mathcal{P}}(S_{\mu}^{+}\cup S_{\mu}^{-},s)$ is the class of
all representations in ${\mathcal{I}}_{\mathcal{P}}(\mu,s)$ such that
$\sigma(\underline{P})=S_{\mu}^{+}\cup S_{\mu}^{-}$. Each component of
${\mathcal{I}}_{\mathcal{P}}(\mu,s)$ in C.2 can be further decomposed into two
sub-classes according to the reducibility of $U^{+}$ or $U^{-}$: C.3.u.
${\mathcal{I}}_{\mathcal{P}}(S_{\mu}^{+},s)={\mathcal{I}}_{\mathcal{P}}(S_{\mu}^{+},s,U^{+}{\rm
irred.})\cup{\mathcal{I}}_{\mathcal{P}}(S_{\mu}^{+},s,U^{+}{\rm red.})$,
C.3.d.
${\mathcal{I}}_{\mathcal{P}}(S_{\mu}^{-},s)={\mathcal{I}}_{\mathcal{P}}(S_{\mu}^{-},s,U^{-}{\rm
irred.})\cup{\mathcal{I}}_{\mathcal{P}}(S_{\mu}^{-},s,U^{-}{\rm red.})$,
C.3.s. ${\mathcal{I}}_{\mathcal{P}}(S_{\mu}^{+}\cup
S_{\mu}^{-},s)={\mathcal{I}}_{\mathcal{P}}(S_{\mu}^{+}\cup
S_{\mu}^{-},s,U^{\pm}{\rm
irred.})\cup{\mathcal{I}}_{\mathcal{P}}(S_{\mu}^{+}\cup
S_{\mu}^{-},s,U^{+}\;{\rm or}\;U^{-}\,{\rm red.})$, with obvious meaning of
the notation. In section 4 we completely identify the possible irreducible
generalized projective representations $U$ of $\mathcal{P}$ for which $U^{+}$
and $U^{-}$ are irreducible, i.e. the components
${\mathcal{I}}_{\mathcal{P}}(S_{\mu}^{+},s,U^{+}{\rm irred.})$,
${\mathcal{I}}_{\mathcal{P}}(S_{\mu}^{-},s,U^{-}{\rm irred.})$ and
${\mathcal{I}}_{\mathcal{P}}(S_{\mu}^{+}\cup S_{\mu}^{-},s,U^{\pm}{\rm
irred.})$ of the decompositions C.3.u., C.3.d. and C.3.s.. In doing so we
shall identify, besides the well known irreducible generalized projective
representations $U$ of $\mathcal{P}$ with
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ anti-unitary and
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ unitary, or with
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ unitary and
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ unitary, also those with
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ anti-unitary and
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ anti-unitary, or with
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ unitary and
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ anti-unitary, which are not taken
into account in the literature about elementary particles theory. The class
${\mathcal{I}}_{\mathcal{P}}(S_{\mu}^{+}\cup S_{\mu}^{-},s,U^{+}\;{\rm
or}\;U^{-}\,{\rm red.})$ is investigated in section 5.

##  4 Irreducible $U$ with $U^{\pm}$ irreducible

###  4.1 The case $\sigma(\underline{P})=S_{\mu}^{\pm}$ with $U^{\pm}$
irreducible

The irreducible generalized projective representation of ${\mathcal{P}}$ with
$\sigma(\underline{P})=S_{\mu}^{+}$ (resp.,
$\sigma(\underline{P})=S_{\mu}^{+}$) and with $U^{+}$ (resp., $U^{-}$)
irreducible are well known [2],[11]. For each allowed pair $\mu>0$ and
$s\in\frac{1}{2}\hbox{I\kern-1.99997pt\hbox{N}}$ of the parameters
characterizing the representation, modulo unitary isomorphisms there is only
one irreducible projective representation of ${\mathcal{P}}_{+}^{\uparrow}$
with $\sigma(\underline{P})=S_{\mu}^{+}$ and only one with
$\sigma(\underline{P})=S_{\mu}^{-}$, that we report. The Hilbert space of the
projective representation is the space $L_{2}(\mathop{{\rm
I}\kern-1.99997pt{\rm
R}}\nolimits^{3},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu)$ of all
functions $\psi:\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3}\to\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1}$, ${\bf
p}\to\psi({\bf p})$, square integrable with respect to the invariant measure
$d\nu({\bf p})=\frac{dp_{1}dp_{2}dp_{3}}{\sqrt{\mu^{2}+{\bf p}^{2}}}$. The
irreducible generalized representations of $\mathcal{P}$ are obtained by
adding ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ and
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$, accorind the next sections 4.1.2
and 4.1.3.

####  4.1.1 The case $\sigma(\underline{P})=S_{\mu}^{+}$

Fixed $\mu$ and $s$, for the irreducible generalized projective representation
with $\sigma(\underline{P})=S_{\mu}^{+}$ the following statements hold.

  * –

The generators $P_{j}$ are the multiplication operators defined by
$(P_{j}\psi)({\bf p})=p_{j}\psi({\bf p})$; as consequence

  * –

$(P_{0}\psi)({\bf p})=p_{0}\psi({\bf p})$ where $p_{0}=+\sqrt{\mu^{2}+{\bf
p}^{2}}$, because $P_{0}$ has a positive spectrum;

  * –

the generators $J_{j}$ are given by
$J_{j}=i\left(p_{k}\frac{\partial}{\partial
p_{l}}-p_{l}\frac{\partial}{\partial p_{k}}\right)+S_{j}$, $(j,k,l)$ being a
cyclic permutation of $(1,2,3)$, where $S_{1},S_{2},S_{3}$ are the self-
adjoint generators of an irreducible projective representation
$L:SO(3)\to\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1}$ such that
$S_{1}^{2}+S_{2}^{2}+S_{3}^{2}=s(s+1)\hbox{\sl 1\kern-2.5pt\hbox{I}}$; hence,
they can be fixed to be the three spin operators of
$\;\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1}$;

  * –

the generators $K_{j}$ are given by $K_{j}=ip_{0}\frac{\partial}{\partial
p_{j}}-\frac{({\bf S}\land{\bf p})_{j}}{\mu+p_{0}}$;

  * –

the space inversion – unitary – and the time reversal – anti-unitary –
operators are

${{}_{\triangleleft}\kern-2.0pt\texttt{S}}=\Upsilon,\quad\hbox{and}\quad{{}^{\triangleleft}\kern-1.5pt\texttt{T}}=\tau{\mathcal{K}}\Upsilon\,,\quad\hbox{where}$
$None$

  * \- $\Upsilon$ is the unitary operator defined by $(\Upsilon\psi)({\bf p})=\psi(-{\bf p})$,

  * \- $\tau$ is a unitary matrix of $\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1}$ such that $\tau{\overline{S}_{j}}\tau^{-1}=-{S_{j}}$, for all $j$; such a matrix always exists and it is unique up a complex factor of modulus 1; moreover, if $s\in\hbox{I\kern-1.99997pt\hbox{N}}$ then $\tau$ is symmetric and $\tau\overline{\tau}=1$, while if $s\in\left(\hbox{I\kern-1.99997pt\hbox{N}}+\frac{1}{2}\right)$ then $\tau$ is anti-symmetric and $\tau\overline{\tau}=-1$ [11];

  * \- $\mathcal{K}$ is the anti-unitary complex conjugation operator defined by ${\mathcal{K}}\psi({\bf p})=\overline{\psi({\bf p})}$.

####  4.1.2 The case $\sigma(\underline{P})=S_{\mu}^{-}$.

For the irreducible projective representation with characterizing parameters
$\mu$, $s$ and $\sigma(\underline{P})=S_{\mu}^{-}$, the following symmetrical
statements hold.

  * –

The generators $P_{j}$ are the multiplication operators defined by
$(P_{j}\psi)({\bf p})=p_{j}\psi({\bf p})$; as consequence

  * –

$(P_{0}\psi)({\bf p})=-p_{0}\psi({\bf p})$, because $P_{0}$ has a negative
spectrum;

  * –

the generators $J_{j}$ are given by
$J_{j}=i\left(p_{k}\frac{\partial}{\partial
p_{l}}-p_{l}\frac{\partial}{\partial p_{k}}\right)+S_{j}$, $(j,k,l)$ being a
cyclic permutation of $(1,2,3)$;

  * –

the generators $K_{j}$ are given by $K_{j}=-ip_{0}\frac{\partial}{\partial
p_{j}}+\frac{({\bf S}\land{\bf P})_{j}}{\mu+p_{0}}$;

  * –

the space inversion and time reversal operators are
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}=\Upsilon$ and
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}=\tau{\mathcal{K}}\Upsilon$.

###  4.2 The case $\sigma(\underline{P})=S_{\mu}^{+}\cup S_{\mu}^{-}$ with
$U^{+}$ irreducible

Now we establish results that allow us to identify all the irreducible
generalized projective representations with
$\sigma(\underline{P})=S_{\mu}^{+}\cup S_{\mu}^{-}$ and $U^{\pm}$ irreducible.
Prop.s 3.4-3.6 imply that ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ is
unitary or ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ is anti-unitary.
Moreover, according to Prop. 3.3, $U\mid_{{\mathcal{P}}_{+}^{\uparrow}}$ is
always reduced by $E^{+}=E(S_{\mu}^{+})\equiv
E^{(0)}[\mu,\infty)=\chi_{[\mu,\infty)}(P_{0})$ and
$E^{-}=E(S_{\mu}^{-})\equiv
E^{(0)}(-\infty,-\mu]=\chi_{(-\infty,-\mu]}(P_{0})$, so that for all
$g\in{\mathcal{P}}_{+}^{\uparrow}$ we can write $U_{g}=U^{+}_{g}+U^{-}_{g}$,
where $U^{+}_{g}=E^{+}U_{g}E^{+}$ and $U^{-}_{g}=E^{-}U_{g}E^{-}$.

Each of these two components $U^{+}$ and $U^{-}$ can be reducible or not, in
its turn. The following proposition entails that the reducibility of $U^{+}$
is equivalent to the reducibility of $U^{-}$. Proposition 4.1. Let $U$ belong
to ${\mathcal{I}}_{\mathcal{P}}(S_{\mu}^{+}\cup S_{\mu}^{-}$. If $F_{+}$ is a
projection operator that reduces $U^{+}$, then the following statements hold.

  * (i)

In the case that ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ is unitary, the
projection operator
$F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}}={{}^{\triangleleft}\kern-1.5pt\texttt{T}}F_{+}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}$
reduces $U^{-}$, and
$F^{{}^{\triangleleft}\kern-3.2pt\texttt{t}}=F_{+}+F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}}$
reduces $U\mid_{{\mathcal{P}}_{+}}$;

  * (ii)

in the case that ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ is anti-unitary,
the projection operator
$F_{-}^{{}_{\triangleleft}\kern-2.2pt\texttt{s}}={{}_{\triangleleft}\kern-2.0pt\texttt{S}}F_{+}{{}_{\triangleleft}\kern-2.0pt\texttt{S}}$
reduces $U^{-}$, and
$F^{{}_{\triangleleft}\kern-2.2pt\texttt{s}}=F_{+}+F_{-}^{{}_{\triangleleft}\kern-2.2pt\texttt{s}}$
reduces $U\mid_{{\mathcal{P}}^{\uparrow}}$.

Proof. If ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ is unitary, then
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}^{-1}={{}^{\triangleleft}\kern-1.5pt\texttt{T}}$
and
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}P_{0}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}=-P_{0}$
follow from (7); this implies
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}E^{+}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}={{}^{\triangleleft}\kern-1.5pt\texttt{T}}\chi_{[\mu,\infty)}(P_{0}){{}^{\triangleleft}\kern-1.5pt\texttt{T}}=\chi_{(-\infty,-\mu]}=E^{-}$
by Lemma 3.1. If $F_{+}$ is a projection operator that reduces $U^{+}$, and
hence $\hbox{{{I}}\kern-4.70001pt\hbox{O}}<F_{+}<E^{+}$, then
$F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}}E^{-}=({{}^{\triangleleft}\kern-1.5pt\texttt{T}}F_{+}{{}^{\triangleleft}\kern-1.5pt\texttt{T}})E^{-}=({{}^{\triangleleft}\kern-1.5pt\texttt{T}}F_{+}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}){{}^{\triangleleft}\kern-1.5pt\texttt{T}}E^{+}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}={{}^{\triangleleft}\kern-1.5pt\texttt{T}}F_{+}E^{+}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}={{}^{\triangleleft}\kern-1.5pt\texttt{T}}F_{+}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}$
since $F_{+}<E^{+}$. Therefore,
$\hbox{{{I}}\kern-4.70001pt\hbox{O}}<F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}}<E_{-}$
is satisfied. Now we show that
$[F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}},P_{0}^{-}]=[F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}},P_{j}^{-}]=[F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}},K_{j}^{-}]=[F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}},J_{k}^{-}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$,
i.e. that $F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}}$ reduces $U^{-}$.
Since $P_{0}^{-}=E^{-}P_{0}E^{-}$ and
$[F_{+},P_{0}]=[F_{+},P_{0}^{+}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$, we have

$\begin{array}[]{ll}P_{0}^{-}F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}}&=P_{0}^{-}F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}}E^{-}=E^{-}P_{0}E^{-}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}F_{+}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}E^{-}=E^{-}P_{0}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}E^{+}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}F_{+}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}E^{-}\cr&=-E^{-}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}P_{0}E^{+}F_{+}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}E^{-}=-E^{-}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}E^{+}P_{0}F_{+}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}E^{-}=-E^{-}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}E^{+}F_{+}P_{0}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}E^{-}\cr&=-E^{-}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}F_{+}P_{0}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}E^{-}=E^{-}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}F_{+}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}P_{0}E^{-}=E^{-}F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}}P_{0}E^{-}=F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}}E^{-}P_{0}E^{-}=F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}}P_{0}^{-}.\end{array}$

A similar derivation shows that
$[F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}},P_{j}]=[F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}},K_{j}^{-}]=[F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}},J_{k}^{-}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$;
therefore $F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}}$ reduces $U^{-}$.
Now we see that
$F^{{}^{\triangleleft}\kern-3.2pt\texttt{t}}=F_{+}+F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}}$
reduces $U\mid_{{\mathcal{P}}_{+}}$. The equalities
$F^{{}^{\triangleleft}\kern-3.2pt\texttt{t}}P_{0}=(F_{+}+F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}})P_{0}=P_{0}(F_{+}+F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}})=P_{0}F^{{}^{\triangleleft}\kern-3.2pt\texttt{t}}$
immediately follow from $P_{0}=E^{+}P_{0}E^{+}+E^{-}P_{0}E^{-}$ and
$F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}}E^{-}=F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}}$,
$F_{+}E^{+}=F_{+}$,
$F_{+}E^{-}=F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}}E^{+}=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$;
similarly,
$[F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}},P_{j}]=[F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}},J_{k}]=[F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}},K_{j}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$
hold. Hence, $F^{{}^{\triangleleft}\kern-3.2pt\texttt{t}}$ reduces
$U\mid_{{\mathcal{P}}_{+}^{\uparrow}}$. Moreover,
$F^{{}^{\triangleleft}\kern-3.2pt\texttt{t}}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}=F_{+}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}+F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}={{}^{\triangleleft}\kern-1.5pt\texttt{T}}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}F_{+}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}+{{}^{\triangleleft}\kern-1.5pt\texttt{T}}F_{+}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}={{}^{\triangleleft}\kern-1.5pt\texttt{T}}F_{-}^{{}^{\triangleleft}\kern-3.2pt\texttt{t}}+{{}^{\triangleleft}\kern-1.5pt\texttt{T}}F_{+}={{}^{\triangleleft}\kern-1.5pt\texttt{T}}F^{{}^{\triangleleft}\kern-3.2pt\texttt{t}}$.
Therefore, $F^{{}^{\triangleleft}\kern-3.2pt\texttt{t}}$ reduces also
$U\mid_{{\mathcal{P}}_{+}}$. A quite similar argument proves statement (ii).
$\bullet$ Corollary. Under the hypotheses of Prop. 3.1, $U^{+}$ is reducible
if and only if $U^{-}$ is reducible. Prop. 3.1 and its corollary indicate that
the irreducible generalized projective representations of $\mathcal{P}$ can be
classified according to the reducibility of $U^{+}$.

####  4.2.1 Hilbert space and self-adjoint generators.

In the case that $U^{+}$ is irreducible, with
$\sigma(\underline{P})=S_{\mu}^{+}\cup S_{\mu}^{-}$, according to Prop. 3.3
the restriction
$U:{\mathcal{P}}_{+}^{\uparrow}\to{\mathcal{U}}({\mathcal{H}})$ must be the
direct sum of
$U^{+}:{\mathcal{P}}_{+}^{\uparrow}\to{\mathcal{U}}({\mathcal{H}}^{+})$ and
$U^{-}:{\mathcal{P}}_{+}^{\uparrow}\to{\mathcal{U}}({\mathcal{H}}^{-})$, where
${\mathcal{H}}^{+}=E^{+}({\mathcal{H}})$,
${\mathcal{H}}^{-}=E^{-}({\mathcal{H}})$ and
${\mathcal{H}}^{+}\oplus{\mathcal{H}}^{-}={\mathcal{H}}$; according to Prop.
4.1, both $U^{+}$ and $U^{-}$ are irreducible projective representations of
${\mathcal{P}}_{+}^{\uparrow}$. Since $\sigma(P_{0}^{+})=[\mu,\infty)$ (resp.,
$\sigma(P_{0}^{-})=(-\infty,-\mu]$), the reduced projective representation
$U^{+}$ (resp., $U^{-}$) is unitarily isomorphic to the projective
representation
$U:{\mathcal{P}}_{+}^{\uparrow}\to{\mathcal{U}}(L_{2}(\mathop{{\rm
I}\kern-1.99997pt{\rm
R}}\nolimits^{3},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu))$ with
$\sigma(\underline{P})=S_{\mu}^{+}$ (resp., with
$\sigma(\underline{P})=S_{\mu}^{-}$) described in sect. 4.1, with the same
characterizing parameters $\mu$ and $s$ of the unrestricted irreducible
generalized projective representation $U$. Accordingly, there are two unitary
mappings $W^{+}:{\mathcal{H}}^{+}\to L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu)$, and
$W^{-}:{\mathcal{H}}^{-}\to L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu)$ such that
the reduced generators in the Hilbert space $W^{+}({\mathcal{H}}^{+})\equiv
L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu)$ are the
following.

$P_{0}^{+}=W^{+}(E^{+}P_{0}E^{+})$, so that $(P_{0}^{+}\phi)({\bf
p})=p_{0}({\bf p})\phi({\bf p})=\sqrt{\mu^{2}+{\bf p}^{2}}\phi({\bf p})$;

$P_{j}^{+}=p_{j}$;

$J_{k}^{+}={\textsf{j}}_{k}=i\left(p_{l}\frac{\partial}{\partial
p_{j}}-p_{j}\frac{\partial}{\partial p_{l}}\right)+S_{k}$;

$K_{j}^{+}={\textsf{k}}_{j}=ip_{0}\frac{\partial}{\partial p_{j}}-\frac{({\bf
S}\land{\bf P})_{j}}{\mu+p_{0}}$.

Symmetrically, the reduced generators in $W^{-}({\mathcal{H}}^{-})\equiv
L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu)$ are

$P_{0}^{-}=W^{-}(E^{-}P_{0}E^{-})=-p_{0}$; $P_{j}^{-}=p_{j}$;
$J_{k}^{-}={\textsf{j}}_{k}=i\left(p_{l}\frac{\partial}{\partial
p_{j}}-p_{j}\frac{\partial}{\partial p_{l}}\right)+S_{k}$;
$K_{j}^{-}=-{\textsf{k}}_{j}=-ip_{0}\frac{\partial}{\partial
p_{j}}+\frac{({\bf S}\land{\bf p})_{j}}{\mu+p_{0}}$.

Hence we have proved that, modulo unitary isomorphisms, the Hilbert space of
the representation is $L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu)\oplus
L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu)$. It is
convenient to represent each vector $\psi\in{\mathcal{H}}=E^{+}\psi+E^{-}\psi$
as a column vector
$\psi=\left[\begin{array}[]{c}\psi^{+}\cr\psi^{-}\end{array}\right]$, where
$\psi^{+}=W^{+}(E^{+}\psi)$ and $\psi^{-}=W^{-}(E^{-}\psi)$; in such a
representation the generators of $U\mid_{{\mathcal{P}}_{+}^{\uparrow}}$ take
the following form, known as the canonical form.

$P_{j}=\left[\begin{array}[]{cc}p_{j}&0\cr
0&-p_{j}\end{array}\right],\;\;P_{0}=\left[\begin{array}[]{cc}p_{0}&0\cr
0&-p_{0}\end{array}\right],\;\;J_{k}=\left[\begin{array}[]{cc}{\textsf{j}}_{k}&0\cr
0&{\textsf{j}}_{k}\end{array}\right],\;\;K_{j}=\left[\begin{array}[]{cc}{\textsf{k}}_{j}&0\cr
0&-{\textsf{k}}_{j}\end{array}\right].$ $None$

####  4.2.2 Time reversal and space inversion operators.

The condition $\sigma(\underline{P})=S_{\mu}^{+}\cup S_{\mu}^{-}$ implies that
the time reversal operator ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ must be
unitary or the space inversion operator
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ must be anti-unitary, according to
Prop.s 3.4-3.6. In the case in which both
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ and
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ are unitary their explicit form is
well known, up a complex factor of modulus 1 [11].

${{}^{\triangleleft}\kern-1.5pt\texttt{T}}=\left[\begin{array}[]{cc}0&1\cr
1&0\end{array}\right]\,;\qquad{{}_{\triangleleft}\kern-2.0pt\texttt{S}}=\Upsilon\left[\begin{array}[]{cc}1&0\cr
0&1\end{array}\right]\quad\hbox{or}\quad{{}_{\triangleleft}\kern-2.0pt\texttt{S}}=\Upsilon\left[\begin{array}[]{cc}1&0\cr
0&-1\end{array}\right]\,.$ $None$

(In the matrices (21) “$1$” and “$0$” denote the identity and null operators
of $\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1}$. This notation is adopted
throughout the paper, whenever it does not cause confusions) However,
irreducible generalized projective representations with
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ anti-unitary or
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ anti-unitary do exist, as we show
after the following Prop.4.2. Proposition 4.2. Let
$U:{\mathcal{P}}\to{\mathcal{V}}({\mathcal{H}})$ be an irreducible generalized
projective representation of $\mathcal{P}$, with $U^{+}$ irreducible. The
following statements hold.

  * i)

If ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ is anti-unitary then

  * ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}=\tau{\mathcal{K}}\Upsilon\left[\begin{array}[]{cc}1&0\cr 0&e^{i\theta}\end{array}\right]$; hence, ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ can be taken to be ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}=\tau{\mathcal{K}}\Upsilon\left[\begin{array}[]{cc}1&0\cr 0&1\end{array}\right]$ up a complex factor of modulus 1;

  * in particular, $s\in\hbox{I\kern-1.99997pt\hbox{N}}$ implies ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}^{2}=\hbox{\sl 1\kern-2.5pt\hbox{I}}$ and $s\in(\hbox{I\kern-1.99997pt\hbox{N}}+\frac{1}{2})$ implies ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}^{2}=-\hbox{\sl 1\kern-2.5pt\hbox{I}}$;

  * ii)

if ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ is anti-unitary then

  * ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}=\left[\begin{array}[]{cc}0&\tau\cr\tau&0\end{array}\right]{\mathcal{K}}$ when ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}^{2}=\hbox{\sl 1\kern-2.5pt\hbox{I}}$ and $s\in\hbox{I\kern-1.99997pt\hbox{N}}$, or when ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}^{2}=-\hbox{\sl 1\kern-2.5pt\hbox{I}}$ and $s\in(\hbox{I\kern-1.99997pt\hbox{N}}+\frac{1}{2})$,

  * ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}=\left[\begin{array}[]{cc}0&\tau\cr-\tau&0\end{array}\right]{\mathcal{K}}$ when ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}^{2}=-\hbox{\sl 1\kern-2.5pt\hbox{I}}$ and $s\in\hbox{I\kern-1.99997pt\hbox{N}}$, or when ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}^{2}=\hbox{\sl 1\kern-2.5pt\hbox{I}}$ and $s\in(\hbox{I\kern-1.99997pt\hbox{N}}+\frac{1}{2})$.

Proof. Since ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ is anti-unitary, the
operator
$\hat{T}=\tau{\mathcal{K}}\Upsilon{{}^{\triangleleft}\kern-1.5pt\texttt{T}}\equiv\left[\begin{array}[]{cc}T_{11}&T_{12}\cr
T_{21}&T_{22}\end{array}\right]$ is unitary, and
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}={\mathcal{K}}\Upsilon\tau^{-1}\hat{T}$.
Now, (5), $\tau^{-1}\overline{S_{k}}\tau=-S_{k}$ and the properties

$\Upsilon p_{j}=-p_{j}\Upsilon$, $\Upsilon\frac{\partial}{\partial
p_{j}}=-\frac{\partial}{\partial p_{j}}\Upsilon$, $\Upsilon^{2}=\hbox{\sl
1\kern-2.5pt\hbox{I}}$, ${\mathcal{K}}p_{j}=p_{j}{\mathcal{K}}$,
${\mathcal{K}}\frac{\partial}{\partial p_{j}}=\frac{\partial}{\partial
p_{j}}{\mathcal{K}}$, ${\mathcal{K}}\Upsilon=\Upsilon{\mathcal{K}}$,
${\mathcal{K}}^{2}=\hbox{\sl 1\kern-2.5pt\hbox{I}}$

imply $[\hat{T},P_{j}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$,
$[\hat{T},P_{0}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$,
$[\hat{T},J_{k}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$. The first two of these
last three equalities imply that $\hat{T}=\left[\begin{array}[]{cc}T_{1}({\bf
p})&0\cr 0&T_{2}({\bf p})\end{array}\right]$, where $T_{m}({\bf p})$ is a
$(2s+1)\times(2s+1)$ matrix for every ${\bf p}\in\mathop{{\rm
I}\kern-1.99997pt{\rm R}}\nolimits^{3}$, so that $[T_{m}({\bf
p}),p_{j}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$; the third equality implies
$[T_{m}({\bf p}),{\textsf{j}}_{k}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$. Then,
since $p_{1},p_{2},p_{3},{\textsf{j}}_{1},{\textsf{j}}_{2},{\textsf{j}}_{3}$
are the generators of an irreducible projective representation of
$\mathcal{E}$ in the Hilbert space $L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu)$, the
relations $[T_{m}({\bf p}),p_{j}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$ and
$[T_{m}({\bf p}),{\textsf{j}}_{k}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$ imply
$T_{m}({\bf p})=e^{i\theta_{m}}\hbox{\sl 1\kern-2.5pt\hbox{I}}$, i.e.
$\hat{T}=\left[\begin{array}[]{cc}e^{i\theta_{1}}&0\cr
0&e^{i\theta_{2}}2\end{array}\right]$, with $\theta_{1,2}$ constant. Hence we
have
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}={\mathcal{K}}\Upsilon\tau^{-1}\left[\begin{array}[]{cc}1&0\cr
0&e^{i\theta}\end{array}\right]=\pm\tau{\mathcal{K}}\Upsilon\left[\begin{array}[]{cc}e^{i\theta_{1}}&0\cr
0&e^{i\theta_{2}}\end{array}\right]$; the free phase can be chosen so that
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}=\tau{\mathcal{K}}\Upsilon\left[\begin{array}[]{cc}1&0\cr
0&e^{i\theta}\end{array}\right]$. By transforming each operator $B$ into
$WBW^{-1}$, where $W=\left[\begin{array}[]{cc}1&0\cr
0&e^{i\frac{\theta}{2}}\end{array}\right]$,
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ turns out to be transformed into
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}=\tau{\mathcal{K}}\Upsilon\left[\begin{array}[]{cc}1&0\cr
0&1\end{array}\right]$, while all generators $P_{j}$, $P_{0}$, $J_{k}$,
$K_{j}$ remain unchanged. Accordingly,
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}^{2}=\left[\begin{array}[]{cc}\tau\overline{\tau}&0\cr
0&\tau\overline{\tau}\end{array}\right]$. If
$s\in\hbox{I\kern-1.99997pt\hbox{N}}$, then $\tau\overline{\tau}=1$, so that
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}^{2}=\hbox{\sl
1\kern-2.5pt\hbox{I}}$; if
$s\in(\hbox{I\kern-1.99997pt\hbox{N}}+\frac{1}{2})$, then
$\tau\overline{\tau}=-1$, so that
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}^{2}=-\hbox{\sl
1\kern-2.5pt\hbox{I}}$. This proves (i).

The proof of (ii) is carried out along quite similar lines. $\bullet$ In an
irreducible generalized projective representation of $\mathcal{P}$ with
$\sigma(\underline{P})=S_{\mu}^{+}\cup S_{\mu}^{-}$ the combination
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ anti-unitary and
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ unitary is excluded by Prop.3.4.
However, all other combinations can occur. The combination
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ unitary and
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ unitary is already settled
according to (21). Then we check the consistency of the remaining
combinations; it is sufficient to verify that (6) is satisfied, since all
other conditions (1)-(5) for $P_{J}$, $P_{0}$, $J_{k}$, $K_{j}$,
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ and
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ are valid by construction.

  * a)

If ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ is anti-unitary and
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ is anti-unitary, then they have
the form shown by Prop. 4.2. By a straightforward calculation it is verified
that condition (6) is always satisfied.

  * b)

If ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ is unitary and
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ is anti-unitary, then they have
the form given by (21) and (ii) in Prop. 4.2. We see that (6) is always
satisfied.

Thus, besides the usually considered irreducible generalized projective
representations with the combination
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ unitary,
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ unitary, also the combinations
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ unitary,
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ anti-unitary and
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ anti-unitary,
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ anti-unitary can occur. They are
completely determined according to the following scheme.

###  4.3 Resulting scheme

For every $\mu>0$ and every $s\in\frac{1}{2}\hbox{I\kern-1.99997pt\hbox{N}}$
there are height irreducible generalized projective representations of
$\mathcal{P}$:

  * u.

the representation with up spectrum $\sigma(\underline{P})=S_{\mu}^{+}$,
identified in section 4.1.1;

  * d.

the representation with down spectrum $\sigma(\underline{P})=S_{\mu}^{-}$,
identified in section 4.1.2;

  * s.

six inequivalent representations $U^{(1)}$, $U^{(2)}$,…,$U^{(6)}$ with
symmetrical spectrum $\sigma(\underline{P})=S_{\mu}^{+}\cup S_{\mu}^{-}$,
identified in section 4.2, all with the same Hilbert space
${\mathcal{H}}=L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu)\oplus
L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu)$ and the
same self-adjoint generators (20); they differ just for the different
combinations of time reversal and space inversion operators.

  * $U^{(1)}$ has unitary ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}=\left[\begin{array}[]{cc}0&1\cr 1&0\end{array}\right]$ and unitary ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}=\Upsilon\left[\begin{array}[]{cc}1&0\cr 0&1\end{array}\right]$;

  * $U^{(2)}$ has unitary ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}=\left[\begin{array}[]{cc}0&1\cr 1&0\end{array}\right]$ and unitary ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}=\Upsilon\left[\begin{array}[]{cc}1&0\cr 0&-1\end{array}\right]$;

  * $U^{(3)}$ has unitary ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}=\left[\begin{array}[]{cc}0&1\cr 1&0\end{array}\right]$ and anti-unitary ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}=\left[\begin{array}[]{cc}0&\tau\cr\tau&0\end{array}\right]{\mathcal{K}}$;

  * $U^{(4)}$ has unitary ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}=\left[\begin{array}[]{cc}0&1\cr 1&0\end{array}\right]$ and anti-unitary ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}=\left[\begin{array}[]{cc}0&\tau\cr-\tau&0\end{array}\right]{\mathcal{K}}$;

  * $U^{(5)}$ has anti-unitary ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}=\tau{\mathcal{K}}\Upsilon\left[\begin{array}[]{cc}0&1\cr 1&0\end{array}\right]$ and anti-unitary${{}_{\triangleleft}\kern-2.0pt\texttt{S}}=\left[\begin{array}[]{cc}0&\tau\cr\tau&0\end{array}\right]{\mathcal{K}}$;

  * $U^{(6)}$ has anti-unitary ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}=\tau{\mathcal{K}}\Upsilon\left[\begin{array}[]{cc}0&1\cr 1&0\end{array}\right]$ and anti-unitary${{}_{\triangleleft}\kern-2.0pt\texttt{S}}=\left[\begin{array}[]{cc}0&\tau\cr-\tau&0\end{array}\right]{\mathcal{K}}$.

The class of all such octets, for all permitted values of $\mu$ and $s$, does
not exhaust ${\mathcal{I}}_{\mathcal{P}}$, because the components with $U^{+}$
or $U^{-}$ reducible in the decompositions C.3 are not empty, as we show in
section 5.

##  5 Irreducible $U:{\mathcal{P}}\to{\mathcal{V}}({\mathcal{H}})$ with
$U^{+}$ or $U^{-}$ reducible

The current relativistic quantum theories of a particle are developed only on
irreducible generalized projective representations
$U:{\mathcal{P}}\to{\mathcal{V}}({\mathcal{H}})$ with $U^{+}$ and $U^{-}$
irreducible. This would be a correct practice if the irreducibility of the
whole $U$ implied the irreducibility of the reductions
$U^{\pm}=E^{\pm}U\mid_{{\mathcal{P}}_{+}^{\uparrow}}E^{\pm}$. This is not the
case. In this section, in fact, we show that irreducible generalized
projective representations $U$ of $\mathcal{P}$ exist such that $U^{\pm}$ is
reducible in the case $\sigma(\underline{P})=S_{\mu}^{\pm}$, as well as in the
case $\sigma(\underline{P})=S_{\mu}^{+}\cup S_{\mu}^{-}$.

###  5.1 The cases $\sigma(\underline{P})=S_{\mu}^{\pm}$

Given an irreducible generalized projective representation of $\mathcal{P}$,
Prop. 3.3 implies that if the restriction
$U\mid_{{\mathcal{P}}_{+}^{\uparrow}}$ is irreducible too, then either
$\sigma(\underline{P})=S_{\mu}^{+}$ or $\sigma(\underline{P})=S_{\mu}^{-}$.
The converse is not true; in other words, the condition
$\sigma(\underline{P})=S_{\mu}^{+}$ implies
$U\mid_{{\mathcal{P}}_{+}^{\uparrow}}=U^{+}$, but does not imply that $U^{+}$
is irreducible. In fact, now we identify irreducible generalized projective
representations $U:{\mathcal{P}}\to{\mathcal{V}}({\mathcal{H}})$ for which
$U^{+}$ is reducible.

We deal with the case $\sigma(\underline{P})=S_{\mu}^{+}$; the alternative
case $\sigma(\underline{P})=S_{\mu}^{-}$ can be addressed along identical
lines. We show that for any $\mu>0$ and any
$s\in\frac{1}{2}\hbox{I\kern-1.99997pt\hbox{N}}$ there are irreducible
generalized projective representations $U$ of $\mathcal{P}$ such that $U^{+}$
is the direct sum $U^{(1)}\oplus U^{(2)}$ of two identical projective
representations
$U^{(1)}:{\mathcal{P}}_{+}^{\uparrow}\to{\mathcal{U}}({\mathcal{H}}^{(1)})$
and
$U^{(2)}:{\mathcal{P}}_{+}^{\uparrow}\to{\mathcal{U}}({\mathcal{H}}^{(2)})$.

Let us consider two irreducible projective representations
$U^{(1)}:{\mathcal{P}}_{+}^{\uparrow}\to{\mathcal{U}}({\mathcal{H}}^{(1)})$
and
$U^{(2)}:{\mathcal{P}}_{+}^{\uparrow}\to{\mathcal{U}}({\mathcal{H}}^{(2)})$ of
${\mathcal{P}}_{+}^{\uparrow}$ of the form described in sect. 4.1.1, with the
same pair $\mu$, $s$ of parameters that determine the representations up
unitary isomorphisms, and with
${\mathcal{H}}^{(1)}={\mathcal{H}}^{(2)}=L_{2}(\mathop{{\rm
I}\kern-1.99997pt{\rm
R}}\nolimits^{3},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu)$. The
Hilbert space of the direct sum $U^{(1)}\oplus U^{(2)}$ is
${\mathcal{H}}=L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu)\oplus
L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu)$.

It is convenient to represent every vector $\psi=\psi_{1}+\psi_{2}$ in
$\mathcal{H}$, with $\psi_{1}\in{\mathcal{H}}^{(1)}$ and
$\psi_{2}\in{\mathcal{H}}^{(2)}$, as the column vector
$\psi\equiv\left[\begin{array}[]{c}\psi_{1}\cr\psi_{2}\end{array}\right]$, so
that every linear (resp., anti-linear) operator $A$ of $\mathcal{H}$ can be
represented by a matrix $\left[\begin{array}[]{cc}A_{11}&A_{12}\cr
A_{21}&A_{22}\end{array}\right]$, where $A_{mn}$ is a linear (resp., anti-
linear) operator of $L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu)$, and
$A\psi=\left[\begin{array}[]{cc}A_{11}&A_{12}\cr
A_{21}&A_{22}\end{array}\right]\left[\begin{array}[]{c}\psi_{1}\cr\psi_{2}\end{array}\right]=\left[\begin{array}[]{c}A_{11}\psi_{1}+A_{12}\psi_{2}\cr
A_{21}\psi_{1}+A_{22}\psi_{2}\end{array}\right]$. Let us introduce the
following operators of $\mathcal{H}$.

$P_{0}=\left[\begin{array}[]{cc}p_{0}&0\cr 0&p_{0}\end{array}\right]\,,\quad
P_{j}=\left[\begin{array}[]{cc}p_{j}&0\cr 0&p_{j}\end{array}\right]\,,\quad
J_{k}=\left[\begin{array}[]{cc}{\textsf{j}}_{k}&0\cr
0&{\textsf{j}}_{k}\end{array}\right]\,,\quad
K_{j}=\left[\begin{array}[]{cc}{\textsf{k}}_{j}&0\cr
0&{\textsf{k}}_{j}\end{array}\right]\,,$ $None$

where ${\textsf{j}}_{k}=i\left(p_{l}\frac{\partial}{\partial
p_{j}}-p_{j}\frac{\partial}{\partial p_{l}}\right)+S_{k}$ and
${\textsf{k}}_{j}=ip_{0}\frac{\partial}{\partial p_{j}}-\frac{({\bf
S}\land{\bf p})_{j}}{\mu+p_{0}}$.

These operators are self-adjoint and satisfy relations (1). Then (22) are the
generators of a reducible projective representation
$U:{\mathcal{P}}_{+}^{\uparrow}\to L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu)\oplus
L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu)$. Since for
this representation $\sigma(\underline{P})=S_{\mu}^{+}$, the possible
extensions to the whole $\mathcal{P}$ are obtained by introducing a time
reversal operator ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ and a space
inversion operator ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ in such a way
to satisfy (2), (5), (6). In sections 5.1.1 and 5.1.2 we show that, while
fixed $\mu$ and $s$ there is a unique such possibility for
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ up to unitary equivalence, there
are inequivalent possibilities for
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$. In section 5.1.3 we prove that
some of these possibilities give rise to irreducible generalized projective
representations of of $\mathcal{P}$.

####  5.1.1 Space inversion operator
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$.

According to Prop. 3.4-3.6, the condition $\sigma(\underline{P})=S_{\mu}^{+}$
implies that
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}=\left[\begin{array}[]{cc}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{11}&{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{12}\cr{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{21}&{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{22}\end{array}\right]$
is anti-unitary and
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}=\left[\begin{array}[]{cc}{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{11}&{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{12}\cr{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{21}&{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{22}\end{array}\right]$
is unitary. We begin by determining
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$. Relations (2) imply

${{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{mn}p_{0}=p_{0}{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{mn},\;{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{mn}p_{j}=-p_{j}{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{mn},\;{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{mn}{\textsf{j}}_{k}={\textsf{j}}_{k}{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{mn},\;{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{mn}{\textsf{k}}_{j}=-{\textsf{k}}_{j}{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{mn}.$
$None$

The unitary operator $\Upsilon$ defined on $\mathcal{H}$ satisfies the
following relations.

$\Upsilon p_{j}=-p_{j}\Upsilon,\quad\Upsilon\frac{\partial}{\partial
p_{j}}=-\frac{\partial}{\partial
p_{j}}\Upsilon,\quad[\Upsilon,S_{j}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}},\quad\Upsilon^{2}=\hbox{\sl
1\kern-2.5pt\hbox{I}}\,.$ $None$

Once introduced the unitary operator
$\hat{S}=\Upsilon{{}_{\triangleleft}\kern-2.0pt\texttt{S}}$, from (2), (24)
and (23) we derive
${\hat{S}}_{mn}p_{0}\equiv({\hat{S}}P_{0})_{mn}=\Upsilon{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{mn}p_{0}=p_{0}\Upsilon{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{mn}=p_{0}\hat{S}_{mn}$,
(25.i)
${\hat{S}}_{mn}p_{j}\equiv({\hat{S}}P_{j})_{mn}=\Upsilon{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{mn}p_{j}=-\Upsilon
p_{j}{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{mn}=p_{j}\Upsilon{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{mn}=p_{j}\hat{S}_{mn}$,
(25.ii)
${\hat{S}}_{mn}{\textsf{j}}_{k}\equiv({\hat{S}}J_{k})_{mn}=\Upsilon{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{mn}{\textsf{j}}_{k}=\Upsilon{\textsf{j}}_{k}{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{mn}=\Upsilon\left(ip_{l}\frac{\partial}{\partial
p_{j}}-ip_{j}\frac{\partial}{\partial
p_{l}}+S_{k}\right){{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{mn}=$ (25.iii)
$\qquad\quad=\left(ip_{l}\frac{\partial}{\partial
p_{j}}-ip_{j}\frac{\partial}{\partial
p_{l}}+S_{k}\right)\Upsilon{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{mn}={\textsf{j}}_{k}{\hat{S}}_{mn}$,
${\hat{S}}_{mn}k_{j}\equiv({\hat{S}}K_{j})_{mn}=\Upsilon{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{mn}k_{j}=-\Upsilon
k_{j}{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{mn}=-\Upsilon\left(ip_{0}\frac{\partial}{\partial
p_{j}}-\frac{[{\bf S}\land{\bf
p}]_{j}}{\mu+p_{0}}\right){{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{mn}=$
(25.iv) $\qquad\quad=\left(ip_{0}\frac{\partial}{\partial p_{j}}-\frac{[{\bf
S}\land{\bf
p}]_{j}}{\mu+p_{0}}\right)\Upsilon{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{mn}={\textsf{k}}_{j}{\hat{S}}_{mn}$.
Now, since each component projective representation
$U^{(m)}:{\mathcal{P}}_{+}^{\uparrow}\to{\mathcal{H}}^{(m)})$ is irreducible,
(25) imply that each ${\hat{S}}_{mn}$ is a multiple of the identity, so that
${\hat{S}}=\left[\begin{array}[]{cc}c_{11}&c_{12}\cr
c_{21}&c_{22}\end{array}\right]$. According to (2), the further constraint
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}^{2}=\hbox{\sl
1\kern-2.5pt\hbox{I}}$ can be imposed; it is satisfied if and only if
${\hat{S}}^{2}=\hbox{\sl 1\kern-2.5pt\hbox{I}}$; this implies that
$\hat{S}={\hat{S}}^{-1}={\hat{S}}^{\ast}$ is a constant hermitean matrix with
eigenvalues $+1$ and $-1$, where ${\hat{S}}^{\ast}$ denotes the adjoint of
$\hat{S}$; therefore, a unit vector ${\bf n}\in\mathop{{\rm
I}\kern-1.99997pt{\rm R}}\nolimits^{3}$ exists such that

${{}_{\triangleleft}\kern-2.0pt\texttt{S}}=\Upsilon\,{\bf
n}\cdot{\vec{\sigma}},\;\hbox{ where
}{\vec{\sigma}}=\left(\left[\begin{array}[]{cc}0&1\cr
1&0\end{array}\right],\left[\begin{array}[]{cc}0&-i\cr
i&0\end{array}\right],\left[\begin{array}[]{cc}1&0\cr
0&-1\end{array}\right]\right).$ $None$

If $\hat{W}=\left[\begin{array}[]{cc}w_{11}&w_{12}\cr
w_{21}&w_{22}\end{array}\right]$ is any constant unitary $2\times 2$ matrix,
then
$[\hat{W},P_{0}]=[\hat{W},P_{j}]=[\hat{W},J_{k}]=[\hat{W},K_{j}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$.
Such a matrix always exists such that
$\hat{W}{{}_{\triangleleft}\kern-2.0pt\texttt{S}}\hat{W}^{-1}=\Upsilon\hat{W}{\bf
n}\cdot{\vec{\sigma}}\hat{W}^{-1}=\Upsilon\left[\begin{array}[]{cc}0&1\cr
1&0\end{array}\right]$; therefore, by converting every operator $B$ into
$\hat{W}B\hat{W}^{-1}$ we obtain a unitarily equivalent irreducible
representation of $\mathcal{P}$. In so doing the operators $P_{0}$, $P_{j}$,
$J_{k}$ and $K_{j}$ remain unaltered because each of them has the form
$\left[\begin{array}[]{cc}A&0\cr 0&A\end{array}\right]$. Thus, up to a unitary
isomorphism, ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ satisfies (2), (6) if
and only if

${{}_{\triangleleft}\kern-2.0pt\texttt{S}}=\Upsilon\left[\begin{array}[]{cc}0&1\cr
1&0\end{array}\right]\,.$ $None$

####  5.1.2 Time reversal operator
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$.

Now we identify the time reversal operator
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ that completes the generalized
projective representation of $\mathcal{P}$ that extends the reducible
projective representation $U=U^{(1)}\oplus U^{(2)}$ of
${\mathcal{P}}_{+}^{\uparrow}$ to $\mathcal{P}$. The conditions (5) imply

${{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}p_{0}=p_{0}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}\,,\;{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}p_{j}=-p_{j}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}\,,\;{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}{\textsf{j}}_{k}=-{\textsf{j}}_{k}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}\,,\;{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}{\textsf{k}}_{j}={\textsf{k}}_{j}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}.$
$None$

The anti-unitary operator ${\mathcal{K}}$ satisfies the following relation

${\mathcal{K}}p_{j}=p_{j}{\mathcal{K}},\quad{\mathcal{K}}\frac{\partial}{\partial
p_{j}}=\frac{\partial}{\partial
p_{j}}{\mathcal{K}},\quad{\mathcal{K}}\Upsilon=\Upsilon{\mathcal{K}},\quad{\mathcal{K}}^{2}=\hbox{\sl
1\kern-2.5pt\hbox{I}}\,.$ $None$

Let us introduce the operator
$\hat{T}=\left[\begin{array}[]{cc}{\hat{T}}_{11}&{\hat{T}}_{12}\cr{\hat{T}}_{21}&{\hat{T}}_{22}\end{array}\right]$,
with
${\hat{T}}_{mn}=\tau{\mathcal{K}}\Upsilon{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}$,
that is unitary, so that
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}=\Upsilon{\mathcal{K}}\tau^{-1}\hat{T}\equiv\tau{\mathcal{K}}\Upsilon\hat{T}$.
Relations (28), (24), (29) imply
$\hat{T}_{mn}p_{0}=\tau{\mathcal{K}}\Upsilon{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}p_{0}=\tau{\mathcal{K}}\Upsilon
p_{0}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}=\tau
p_{0}{\mathcal{K}}\Upsilon{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}=p_{0}\tau{\mathcal{K}}\Upsilon{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}=p_{0}\hat{T}_{mn}$,
(30.i)
$\hat{T}_{mn}p_{j}=\tau{\mathcal{K}}\Upsilon{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}p_{j}=-\tau{\mathcal{K}}\Upsilon
p_{j}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}=\tau
p_{j}{\mathcal{K}}\Upsilon{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}=p_{j}\tau{\mathcal{K}}\Upsilon{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}p_{j}\hat{T}_{mn}$,
(30.ii)
$\hat{T}_{mn}{\textsf{j}}_{k}=\tau{\mathcal{K}}\Upsilon{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}{\textsf{j}}_{k}=-\tau{\mathcal{K}}\Upsilon{\textsf{j}}_{k}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}=$
(30.iii)

$\qquad\quad=-\tau{\mathcal{K}}\Upsilon\left(ip_{l}\frac{\partial}{\partial
p_{j}}-ip_{j}\frac{\partial}{\partial
p_{l}}\right){{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}-\tau{\mathcal{K}}\Upsilon
S_{k}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}=$

$\qquad\quad=\tau\left(ip_{l}\frac{\partial}{\partial
p_{j}}-ip_{j}\frac{\partial}{\partial
p_{l}}\right){\mathcal{K}}\Upsilon{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}-\tau\overline{S_{k}}\tau^{-1}\tau{\mathcal{K}}\Upsilon{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}=$
$\qquad\quad=\left(ip_{l}\frac{\partial}{\partial
p_{j}}-ip_{j}\frac{\partial}{\partial
p_{l}}\right)\tau{\mathcal{K}}\Upsilon{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}+S_{k}\tau{\mathcal{K}}\Upsilon{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}={\textsf{j}}_{k}\hat{T}_{mn}$,
$\hat{T}_{mn}{\textsf{k}}_{j}=\tau{\mathcal{K}}\Upsilon{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}{\textsf{k}}_{j}=\tau{\mathcal{K}}\Upsilon{\textsf{k}}_{j}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}=\tau{\mathcal{K}}\Upsilon\left(ip_{0}\frac{\partial}{\partial
p_{j}}-\frac{[{\bf S}\land{\bf
p}]_{j}}{\mu+p_{0}}\right){{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}=$
(30.iv) $\qquad\quad=\tau\left(ip_{0}\frac{\partial}{\partial
p_{j}}+\frac{[\overline{\bf S}\land{\bf
p}]_{j}}{\mu+p_{0}}\right){\mathcal{K}}\Upsilon\hat{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}=\left(ip_{0}\frac{\partial}{\partial
p_{j}}+\frac{[\tau\overline{\bf S}\tau^{-1}\land{\bf
p}]_{j}}{\mu+p_{0}}\right)\tau{\mathcal{K}}\Upsilon\hat{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{mn}={\textsf{k}}_{j}\hat{T}_{mn}$.
The irreducibility of each component
$U^{(m)}:{\mathcal{P}}_{+}^{\uparrow}\to{\mathcal{U}}(L_{2}(\mathop{{\rm
I}\kern-1.99997pt{\rm
R}}\nolimits^{3},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu))$ implies
that ${\hat{T}}=\left[\begin{array}[]{cc}d_{11}&d_{12}\cr
d_{21}&d_{22}\end{array}\right]$, with $d_{mn}$ constant. Further constraints
are imposed by the condition
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}^{2}=c\hbox{\sl
1\kern-2.5pt\hbox{I}}$, with $c=\pm 1$. Now we have
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}^{2}=\tau{\mathcal{K}}\Upsilon\hat{T}\tau{\mathcal{K}}\Upsilon\hat{T}=\tau\overline{\tau}\overline{\hat{T}}\hat{T}$;
therefore
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}^{2}=\overline{\hat{T}}\hat{T}$ if
$s\in\hbox{I\kern-1.99997pt\hbox{N}}$ and
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}^{2}=-\overline{\hat{T}}\hat{T}$ if
$s\in(\hbox{I\kern-1.99997pt\hbox{N}}+\frac{1}{2})$. It is clear that there
are always unitary constant matrices
${\hat{T}}=\left[\begin{array}[]{cc}d_{11}&d_{12}\cr
d_{21}&d_{22}\end{array}\right]$ for which
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}=(\tau{\mathcal{K}}\Upsilon\hat{T})^{2}=\pm\hbox{\sl
1\kern-2.5pt\hbox{I}}$: it is sufficient that $\overline{\hat{T}}\hat{T}=\pm
1$; a trivial solution is ${\hat{T}}=\left[\begin{array}[]{cc}1&0\cr
0&1\end{array}\right]$, that satisfies also (6) with the operator
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ given by (27); but other less
trivial solutions can easily singled out, such as
$\hat{T}=\left[\begin{array}[]{cc}0&1\cr-1&0\end{array}\right]$.

So, extensions of $U^{(1)}\oplus U^{(2)}$ to generalized projective
representations of the whole $\mathcal{P}$ are easily realized. Example 5.1.
Let us study, for instance, the case $s=0$, where $\tau=1$. The condition
$\overline{\hat{T}}\hat{T}=\pm\hbox{\sl 1\kern-2.5pt\hbox{I}}$ entails
${\hat{T}}=c{\hat{T}}^{t}$, where $c=\pm 1$ ant $\hat{T}^{t}$ being the
transpose of $\hat{T}$.

If $c=1$, then ${\hat{T}}={\hat{T}}^{t}$, that implies
$\hat{T}=\left[\begin{array}[]{cc}d_{11}&d_{12}\cr
d_{21}&d_{22}\end{array}\right]=\left[\begin{array}[]{cc}d_{11}&d_{21}\cr
d_{12}&d_{22}\end{array}\right]$, i.e. $d_{12}=d_{21}$. Since
${\hat{T}}^{2}=\hbox{\sl 1\kern-2.5pt\hbox{I}}$ and $\hat{T}$ is unitary,
$\hat{T}={\hat{T}}^{-1}={\hat{T}}^{\ast}$, i.e. $\hat{T}$ is hermitean and has
two eigenvalues $+1$ and $-1$. Therefore $\hat{T}={\bf n}\cdot{\vec{\sigma}}$.
Condition (6) implies $\omega=\pm 1$, and if $\omega=1$ then
$\hat{T}=\left[\begin{array}[]{cc}0&1\cr 1&0\end{array}\right]$, whereas if
$\omega=-1$ then $\hat{T}=\left[\begin{array}[]{cc}1&0\cr
0&-1\end{array}\right]$.

If $c=-1$, then ${\hat{T}}=c{\hat{T}}^{t}$ implies
$\hat{T}=\left[\begin{array}[]{cc}0&d\cr-d&0\end{array}\right]$, with
$d\in\hbox{{{I}}\kern-4.70001pt\hbox{C}}$. In this case ${\hat{T}}^{2}=-1$, so
that $\left[\begin{array}[]{cc}-d^{2}&0\cr
0&-d^{2}\end{array}\right]=\left[\begin{array}[]{cc}-1&0\cr
0&-1\end{array}\right]$, i.e $d=\pm 1$ and we can take
$\hat{T}=\left[\begin{array}[]{cc}0&1\cr-1&0\end{array}\right]$.

The required commutation relation (6) between
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ and
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ is satisfied in case $c=-1$ with
$\omega=-1$. Indeed,

${{}_{\triangleleft}\kern-2.0pt\texttt{S}}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}=\Upsilon{\mathcal{K}}\Upsilon\left[\begin{array}[]{cc}0&1\cr
1&0\end{array}\right]\left[\begin{array}[]{cc}0&1\cr-1&0\end{array}\right]={\mathcal{K}}\left[\begin{array}[]{cc}-1&0\cr
0&1\end{array}\right],$

and

${{}^{\triangleleft}\kern-1.5pt\texttt{T}}{{}_{\triangleleft}\kern-2.0pt\texttt{S}}={\mathcal{K}}\left[\begin{array}[]{cc}0&1\cr-1&0\end{array}\right]\left[\begin{array}[]{cc}0&1\cr
1&0\end{array}\right]={\mathcal{K}}\left[\begin{array}[]{cc}1&0\cr
0&-1\end{array}\right]=-{{}_{\triangleleft}\kern-2.0pt\texttt{S}}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}.$

####  5.1.3 Irreducibility of the extension
$U:{\mathcal{P}}\to{\mathcal{U}}({\mathcal{H}})$.

Now we show that, for each possible value of $s$, there are irreducible
generalized projective representation
$U:{\mathcal{P}}\to{\mathcal{V}}({\mathcal{H}})$ that extend the reducible
projective representations
$U:{\mathcal{P}}_{+}^{\uparrow}\to{\mathcal{U}}({\mathcal{H}})$ of the kind we
are considering, that are irreducible.

Let $A=\left[\begin{array}[]{cc}A_{11}&A_{12}\cr
A_{21}&A_{22}\end{array}\right]$ be any self-adjoint operator of
${\mathcal{H}}=L_{2}(S_{\mu}^{+},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu)\oplus
L_{2}(S_{\mu}^{+},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu)$, such that
$[A,U_{g}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$ for all $g\in\mathcal{P}$, and
therefore $A$ commutes with all self-adjoint generators and with
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ and
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$. From
$[A,P_{j}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$ we imply that each $A_{mn}$
must be a function of $\bf p$: $A_{mn}=a_{mn}({\bf p})$, and in particular
$[A_{mn},p_{j}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$. Moreover,
$[A,J_{k}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$ implies
$[A_{mn},{\textsf{j}}_{k}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$. Then, since
$p_{1},p_{2},p_{3},{\textsf{j}}_{1},{\textsf{j}}_{2},{\textsf{j}}_{3}$ are the
generators of an irreducible projective representation of $\mathcal{E}$ in the
Hilbert space
$L_{2}(S_{\mu}^{+},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu)$, each
$A_{mn}$ is a multiple of the identity: $A_{mn}=a_{mn}\hbox{\sl
1\kern-2.5pt\hbox{I}}$. Now, the condition
$[A,{{}_{\triangleleft}\kern-2.0pt\texttt{S}}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$
implies $A=\left[\begin{array}[]{cc}a&b\cr b&a\end{array}\right]$. In the
generalized projective representation where
$\hat{T}=\left[\begin{array}[]{cc}0&1\cr-1&0\end{array}\right]$ we have
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}=\tau\Upsilon{\mathcal{K}}\left[\begin{array}[]{cc}0&1\cr-1&0\end{array}\right]$;
the condition
$[A,{{}^{\triangleleft}\kern-1.5pt\texttt{T}}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$
implies $b=0$. Therefore, a self-adjoint operator $A$ that commutes with all
$U_{g}$, $g\in{\mathcal{P}}$ must have the form
$A=\left[\begin{array}[]{cc}a&0\cr 0&a\end{array}\right]\equiv a\hbox{\sl
1\kern-2.5pt\hbox{I}}$, and therefore the generalized projective
representation $U$ is irreducible.

###  5.2 The case $\sigma(\underline{P})=S_{\mu}^{+}\cup S_{\mu}^{-}$

Now we determine irreducible representations $U$ of $\mathcal{P}$ with
$\sigma(\underline{P})=S_{\mu}^{+}\cup S_{\mu}^{-}$, such that $U^{+}$, and
hence $U^{-}$ by Prop. 4.1, is the direct sum of two irreducible projective
representations $U^{(1)}$ and $U^{(2)}$ of ${\mathcal{P}}_{+}^{\uparrow}$. Our
search will be successful for ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$
unitary and ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ anti-unitary.

The aimed irreducibility forces the characterizing parameters $\mu$ and $s$ of
$U$ to have the same values for the reduced components $U^{(1)}$ and
$U^{(2)}$; hence, $U^{(1)}$ and $U^{(2)}$ must be unitarily isomorphic, so
that they can be identified with two identical projective representations
according to section 4.2.1.

We consider the case where $s=0$, because its simplicity helps clearness. Each
of the Hilbert spaces ${\mathcal{M}}$ of $U^{(1)}$ and ${\mathcal{N}}$ of
$U^{(2)}$ can be identified with $L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3},d\nu)\oplus L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3},d\nu)$. According to Prop. 4.1.i, both subspaces
${\mathcal{M}}={\mathcal{M}}^{+}\oplus{\mathcal{M}}^{-}$ and
${\mathcal{N}}={\mathcal{N}}^{+}\oplus{\mathcal{N}}^{-}$, of
${\mathcal{H}}={\mathcal{M}}\oplus{\mathcal{N}}$, where
${\mathcal{M}}^{-}={{}^{\triangleleft}\kern-1.5pt\texttt{T}}{\mathcal{M}}^{+}$
and
${\mathcal{N}}^{-}={{}^{\triangleleft}\kern-1.5pt\texttt{T}}{\mathcal{N}}^{+}$
reduce $U\mid_{{\mathcal{P}}_{+}}$. Hence, every vector $\psi$ of the Hilbert
space $\mathcal{H}$ of the entire generalized projective representation of
$\mathcal{P}$ can be uniquely decomposed as
$\psi=\psi_{{\mathcal{M}}^{+}}+\psi_{{\mathcal{M}}^{-}}+\psi_{{\mathcal{N}}^{+}}+\psi_{{\mathcal{N}}^{-}}$,
with $\psi_{{\mathcal{M}}^{+}}\in{\mathcal{M}}^{+}$,
$\psi_{{\mathcal{M}}^{-}}\in{\mathcal{M}}^{-}$,
$\psi_{{\mathcal{N}}^{+}}\in{\mathcal{N}}^{+}$,
$\psi_{{\mathcal{N}}^{-}}\in{\mathcal{N}}^{-}$, so that $\psi$ can be
represented as a column vector
$\psi=\left[\begin{array}[]{c}\psi_{{\mathcal{M}}^{+}}\cr\psi_{{\mathcal{M}}^{-}}\cr\psi_{{\mathcal{N}}^{+}}\cr\psi_{{\mathcal{N}}^{-}}\end{array}\right]$.

In such a representation the self-adjoint generators of
${\mathcal{P}}_{+}^{\uparrow}$ satisfying (1) are

$P_{0}=\left[\begin{array}[]{cccc}p_{0}&0&0&0\cr 0&-p_{0}&0&0\cr
0&0&p_{0}&0\cr 0&0&0&-p_{0}\end{array}\right],\quad
P_{j}=\left[\begin{array}[]{cccc}p_{j}&0&0&0\cr 0&p_{j}&0&0\cr 0&0&p_{j}&0\cr
0&0&0&p_{j}\end{array}\right],$
$J_{k}=\left[\begin{array}[]{cccc}{\textsf{j}}_{k}&0&0&0\cr
0&{\textsf{j}}_{k}&0&0\cr 0&0&{\textsf{j}}_{k}&0\cr
0&0&0&{\textsf{j}}_{k}\end{array}\right],\quad
K_{j}=\left[\begin{array}[]{cccc}{\textsf{k}}_{j}&0&0&0\cr
0&-{\textsf{k}}_{j}&0&0\cr 0&0&{\textsf{k}}_{j}&0\cr
0&0&0&-{\textsf{k}}_{j}\end{array}\right]\,.$

According to Prop. 4.1, also the unitary operator
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ is reduced by $\mathcal{M}$ and
$\mathcal{N}$, where its irreducible components, by (21), are both
$\left[\begin{array}[]{cc}0&1\cr 1&0\end{array}\right]$. Then we have
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}=\left[\begin{array}[]{cccc}0&1&0&0\cr
1&0&0&0\cr 0&0&0&1\cr 0&0&1&0\end{array}\right]$. Now we seek for an anti-
unitary space inversion operator ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$.
In the case ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}^{2}=\hbox{\sl
1\kern-2.5pt\hbox{I}}$, since ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ is
anti-unitary, by imposing (3) we find
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}={\mathcal{K}}\left[\begin{array}[]{cccc}0&s_{1}&0&s_{2}\cr
s_{1}&0&s_{2}&0\cr 0&s_{2}&0&s_{3}\cr s_{2}&0&s_{3}&{}_{0}\end{array}\right]$,
with $s_{1},s_{2},s_{3}$ constant. So we have obtained a generalized
projective representation of $\mathcal{P}$, because (1)-(6) hold.

Such a representation, however, is reducible. Indeed let
$A=\left[\begin{array}[]{cccc}A_{11}&A_{12}&A_{13}&A_{14}\cr
A_{21}&A_{22}&A_{23}&A_{24}\cr A_{31}&A_{32}&A_{33}&A_{34}\cr
A_{41}&A_{42}&A_{43}&A_{44}\end{array}\right]$ be any self-adjoint operator of
$\mathcal{H}$; the conditions
$[A,P_{0}]=[A,P_{j}]=[A,J_{k}]=[A,K_{j}]=[A,{{}^{\triangleleft}\kern-1.5pt\texttt{T}}]=[A,{{}_{\triangleleft}\kern-2.0pt\texttt{S}}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$
are satisfied if and only if $A=\left[\begin{array}[]{cccc}a&0&b&0\cr
0&a&0&b\cr\overline{b}&0&c&0\cr 0&\overline{b}&0&c\end{array}\right]$ where
$a,c\in\mathop{{\rm I}\kern-1.99997pt{\rm R}}\nolimits$ and
$b\in\hbox{{{I}}\kern-4.70001pt\hbox{C}}$, provided that $a+\overline{b}=b+c$.
Therefore, there are self-adjoint operators $A$ that commute with all
$U_{g}\in U({\mathcal{P}})$, different from a multiple of the identity. We
have to conclude that if ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}^{2}=1$
then $U:{\mathcal{P}}\to{\mathcal{V}}({\mathcal{H}})$ is reducible. Let us now
consider the case that
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}^{2}=-\hbox{\sl
1\kern-2.5pt\hbox{I}}$. We find that the conditions (3), (6) are satisfied if
and only if
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}={\mathcal{K}}\left[\begin{array}[]{cccc}0&0&0&1\cr
0&0&1&0\cr 0&-1&0&0\cr-1&0&0&0\end{array}\right]$. If $A$ is any self-adjoint
operator of $\mathcal{H}$, then this time the conditions
$[A,P_{0}]=[A,P_{j}]=[A,J_{k}]=[A,K_{j}]=[A,{{}^{\triangleleft}\kern-1.5pt\texttt{T}}]=[A,{{}_{\triangleleft}\kern-2.0pt\texttt{S}}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$
imply $A=\left[\begin{array}[]{cccc}a&0&0&0\cr 0&a&0&0\cr 0&0&a&0\cr
0&0&0&a\end{array}\right]\equiv a\hbox{\sl 1\kern-2.5pt\hbox{I}}$ with
$a\in\mathop{{\rm I}\kern-1.99997pt{\rm R}}\nolimits$. Thus $U$ is
irreducible. The results of sections 4 and 5 show that the whole class
${\mathcal{I}}_{\mathcal{P}}$ contains classes that are not considered in the
literature about relativistic quantum theories of single particles; for
instance, in [11] only the representations of sections 4.1.1, 4.1.2 and
$U^{(1)}$ and $U^{(2)}$ in section 4.3 are considered. Thus the present work
identifies two further (non-disjoint) robust classes of representations of
$\mathcal{P}$ that should be considered for the formulation of relativistic
quantum theories: ${\mathcal{I}}_{\mathcal{P}}({\rm
ant.}{{}_{\triangleleft}\kern-2.0pt\texttt{S}})$, i.e. the class that collects
all representation of the kind $U^{(3)}$-$U^{(6)}$;
${\mathcal{I}}_{\mathcal{P}}(U^{\pm}{\rm red.})$, i.e. the class of all
representations in ${\mathcal{I}}_{\mathcal{P}}$ with $U^{+}$ or $U^{-}$
reducible.

##  6 Consistent relativistic quantum theories of elementary particle

In the previous sections we have carried out a redetermination of the class of
the irreducible generalized projective representations of $\mathcal{P}$,
singling out classes of irreducible representations besides those currently
considered for the formulation of relativistic quantum theories of a particle.
Our work is meaningful, however, only if consistent theories based on these
further representations can be developed. This is the case, indeed; in this
section some consistent theories of localizable particle based on
representations in the new classes, derived in [5], are presented. By
localizable free particle, shortly free particle we mean an isolated system
whose quantum theory is endowed with a unique triple
$(Q_{1},Q_{2},Q_{3})\equiv{\bf Q}$ of quantum observables, called position
operator, such that

  * (Q.1)

$[Q_{j},Q_{k}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$, for all
$j,k\in\\{1,2,3\\}$. This condition requires that a measurement of position
yields all three values of the coordinates of the particle position.

  * (Q.2)

The triple $(Q_{1},Q_{2},Q_{3})\equiv{\bf Q}$ is characterized by the specific
properties of transformation of position with respect to the group
$\mathcal{P}$, expressed as relations for the transformed position observable
$S_{g}[{\bf Q}]=U_{g}{\bf Q}U_{g}^{-1}$. In particular,

  * (a) $S_{{}^{\triangleleft}\kern-3.2pt\texttt{t}}[{\bf Q}]={\bf Q}$ and $S_{{}_{\triangleleft}\kern-2.2pt\texttt{s}}[{\bf Q}]=-{\bf Q}$, equivalent to ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}{\bf Q}={\bf Q}{{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ and ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}{\bf Q}=-{\bf Q}{{}_{\triangleleft}\kern-2.0pt\texttt{S}}$.

  * (b) If $g\in\mathcal{E}$ then $S_{g}[{\bf Q}]=U_{g}{\bf Q}U_{g}^{-1}={\textsf{g}}({\bf Q})$, where ${\bf x}\to{\textsf{g}}({\bf x})$ is the function that realizes $g$.

A free particle is said elementary if the generalized projective
representation $U$ for which $S_{g}[A]=U_{g}AU_{g}^{-1}$ is irreducible.
Accordingly, by selecting the irreducible generalized projective
representations $U$ of $\mathcal{P}$, that admit such a triple $\bf Q$
satisfying (Q.1) and (Q.2) we identify the possible theories of elementary
free particles. For the projective representations with
$\sigma(\underline{P})=S_{\mu}^{\pm}$, $U^{\pm}$ irreducible and $s=0$,
identified in section 4.1, it turns out that conditions (Q.1) and (Q2.a,b) are
sufficient [5] to univocally determine $\bf Q$ as $Q_{j}=F_{j}$, where
$F_{j}=i\frac{\partial}{\partial p_{j}}-\frac{i}{2p_{0}^{2}}p_{j}$ are the
Newton and Wigner operators [13]. In this case, hence, we recover well known
theories [2],[3],[BM68].

###  6.1 Elementary particle theories with $s=0$ based on $U^{(3)}$ and
$U^{(5)}$

The explicit form of the tranformation properties with respect to
$\mathcal{P}$ is available only for the subgroup generated by the Euclidean
group $\mathcal{E}$ and
$\\{{{}_{\triangleleft}\kern-2.2pt\texttt{s}},{{}^{\triangleleft}\kern-3.2pt\texttt{t}}\\}$;
they are expressed by (Q.2,a,b). For the irreducible generalized projective
representations with $\sigma(\underline{P})=S_{\mu}^{+}\cup S_{\mu}^{-}$,
$U^{\pm}$ irreducible and $s=0$, identified in section 4.2, the known
transformation properties (Q.1) and (Q2.a,b) are sufficient [5] to completely
and univocally determine $\bf Q$ only for $U^{(3)}$ and $U^{(5)}$; the
position operator must be ${\bf Q}=\hat{\bf
F}=\left[\begin{array}[]{cc}F_{j}&0\cr 0&F_{j}\end{array}\right]$. Hence, we
have two complete theories based on the new representations $U^{(3)}$ and
$U^{(5)}$. Though in $U^{(3)}$ the space inversion operator is anti-unitary,
and in $U^{(5)}$ also the time reversal operator is anti-unitary, the theories
are perfectly consistent, in the sense that (Q.1) and (Q.2) are satisfied.
Thus, these new representations are indispensable to determine complete
theories with the nowadays available conditions. The early theory for such a
kind of particle is Klein-Gordon theory [14]-[16], that suffered serious
problems. A first problem is that the wave equation of Klein-Gordon theory is
second order in time, while according to the general laws of quantum theory it
should be first order.

Furthermore, Klein-Gordon theory interprets $\hat{\rho}(t,{\bf
x})=\frac{i}{2m}\left(\overline{\psi_{t}}\frac{\partial}{\partial
t}\psi_{t}-{\psi_{t}}\frac{\partial}{\partial t}\overline{\psi_{t}}\right)$ as
the probability of position density and $\hat{\bf j}(t,{\bf
x})=\frac{i}{2m}\left(\psi_{t}\nabla\overline{\psi_{t}}-\overline{\psi_{t}}\nabla\psi_{t}\right)$
as its current density. This interpretation is at the basis of the Dirac
concern that position probability density can be negative, due to the presence
of time derivatives of $\psi_{t}$ in $\hat{\rho}$. A way to overcome the
difficulty without making resort to quantum field theory [17] was proposed by
Feshbach and Villars [18]. They derive an equivalent form of Klein-Gordon
equation as a first order equation $i\frac{\partial}{\partial
t}\Psi_{t}=H\Psi_{t}$ for the state vector
$\Psi_{t}=\left[\begin{array}[]{c}\phi_{t}\cr\chi_{t}\end{array}\right]$,
where
$\phi_{t}=\frac{1}{\sqrt{2}}(\psi_{t}+\frac{1}{m}\frac{\partial}{\partial
t}\psi_{t})$,
$\chi_{t}=\frac{1}{\sqrt{2}}(\psi_{t}-\frac{1}{m}\frac{\partial}{\partial
t}\psi_{t})$, and $H=(\sigma_{3}+\sigma_{2})\frac{1}{2m}(\nabla+m\sigma_{3})$,
$\psi_{t}$ being the Klein-Gordon wave function; in this representation
$\hat{\rho}=|\phi_{t}|^{2}-|\chi_{t}|^{2}$, without time derivatives. The
minus sign in $\hat{\rho}$ forbids to interpret it as probability density of
position; Feshbach and Villars proposed to reinterpret it as density
probability of charge, so that negative values could be accepted.
Nevertheless, according to Barut and Malin [BM68], covariance with respect to
boosts should imply that $\hat{\rho}$ must be the time component of a four-
vector. Barut and Malin proved that is not the case. In order to check our
theories with respect to these problems, we reformulate the theories based on
$U^{(3)}$ and $U^{(5)}$ in equivalent forms, obtained by means of unitary
transformations operated by the unitary operator $Z=Z_{1}Z_{2}$, where
$Z_{2}=\frac{1}{\sqrt{p_{0}}}\hbox{\sl 1\kern-2.5pt\hbox{I}}$ and $Z_{1}$ is
the inverse of the Fourier-Plancherel operator, that transforms $\psi({\bf
p})$ into $(Z\psi)({\bf x})\equiv(\hat{\psi})({\bf x})$. In the so
reformulated theories the Hilbert space for both turns out to be
${\mathcal{H}}=Z\left(L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3},d\nu)\oplus L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3},d\nu)\right)\equiv L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3})\oplus L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3})$; the new self-adjoint generators are
${\hat{P}}_{j}=ZP_{j}Z^{-1}=\left[\begin{array}[]{cc}-i\frac{\partial}{\partial
x_{j}}&0\cr 0&-i\frac{\partial}{\partial x_{j}}\end{array}\right]$,
${\hat{P}}_{0}=\sqrt{\mu^{2}-\nabla^{2}}\left[\begin{array}[]{cc}1&0\cr
0&-1\end{array}\right]$, ${\hat{J}}_{k}=-i\left(x_{l}\frac{\partial}{\partial
x_{j}}-x_{j}\frac{\partial}{\partial
x_{l}}\right)\left[\begin{array}[]{cc}1&0\cr 0&1\end{array}\right]$;
${\hat{K}}_{j}=\frac{1}{2}\left(x_{j}\sqrt{\mu^{2}-\nabla^{2}}+\sqrt{\mu^{2}-\nabla^{2}}x_{j}\right)\left[\begin{array}[]{cc}1&0\cr
0&-1\end{array}\right]$.

The wave equation trivially is $i\frac{\partial}{\partial
t}\psi_{t}=P_{0}\psi_{t}$, that is first order.

The position operator turns out to be ${\hat{Q}}_{j}=Z\hat{\bf
F}Z^{-1}\equiv\left[\begin{array}[]{cc}x_{j}&0\cr 0&x_{j}\end{array}\right]$,
so that also the other problems disappear. Indeed, the position is represented
by the multiplication operator; therefore, the probability density of position
must necessarily be given by the non negative function $\rho(t,{\bf
x})=|\psi_{t}^{+}({\bf x})|^{2}+|\psi_{t}^{-}({\bf x})|^{2}$. On the other
hand, being $K_{j}$ and $\bf Q$ explicitly known, the covariance properties
with respect to boosts, according to (Q.2), are explicitly expressed in full
coherence by $S_{g}[{\bf Q}]=e^{iK_{j}\varphi(u)}{\bf
Q}e^{-iK_{j}\varphi(u)}$.

###  6.2 New species of particle theories

In the literature all irreducible representations taken as bases of elementary
particle theories are characterized by the irreducibility of $U^{\pm}$. Now,
in section 5.1 for each $\mu>0$ and every
$s\in\frac{1}{2}\hbox{I\kern-1.99997pt\hbox{N}}$ an irreducible representation
of $\mathcal{P}$ is identified characterized by
$\sigma(\underline{P})=S_{\mu}^{+}$ such that $U^{+}$ is reducible. It can be
shown [5] that conditions (Q.1), (Q.2.a,b) univocally determine the position
operator $\hat{\bf Q}$, and therefore gives rise to a consistent theory [5].
For these representations, where ${\mathcal{H}}=L_{2}(\mathop{{\rm
I}\kern-1.99997pt{\rm R}}\nolimits^{3},d\nu)\oplus L_{2}(\mathop{{\rm
I}\kern-1.99997pt{\rm R}}\nolimits^{3},d\nu)$, such position operator is
${\hat{Q}}_{j}=\left[\begin{array}[]{cc}F_{j}&0\cr 0&F_{j}\end{array}\right]$.
Therefore, complete consistent theories of an elementary free particle turn
out to be identified, which corresponds to none of the early theories.

Thus, the extension of the class of the irreducible representations of
$\mathcal{P}$ is meaningful, because it allows to identify consistent theories
and also new species of consistent theories.

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