New representations of Poincar\'e group for consistent Relativistic Particle Theories

TL;DR

This paper extends the class of irreducible Poincaré group representations, enabling the development of consistent relativistic particle theories, including new particle species and improved formulations for Klein-Gordon particles.

Contribution

It explicitly determines an extended class of irreducible Poincaré group representations with positive mass, previously excluded, facilitating new relativistic particle theories.

Findings

New representations lead to consistent Klein-Gordon particle theories

Extended class includes previously missed irreducible representations

Enables formulation of new particle species

Abstract

Though the irreducible representations of the Poincare' 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.

Equations

$$[P_0^2-(P_1^2+P_2^2+P_3^2),U_g]=IO,$$

$$[W_0^2-(W_1^2+W_2^2+W_3^2),U_g]=IO,$$

$$P_0=\int \lambda d{E}_{\lambda }^{(0)},P_j=\int \lambda d{E}_{\lambda }^{(j)},j=1,2,3,$$

$$\underline{P}=\int \underline{p}dE\underline{p},withd{E}_{\underline{p}}=d{E}_{p_0}^{(0)}d{E}_{p_1}^{(1)}d{E}_{p_2}^{(2)}d{E}_{p_3}^{(3)}.$$

$$\sigma (\underline{P})=\{\underline{p}=(p_0,\mathbf{p})\in {\mathrm{IR}}^4\mid E({\Delta }_{\underline{p}})\ne IO\; for\; every\; neighborough{\Delta }_{\underline{p}}of\underline{p}\}.$$

$$U_g\underline{P}{U}_g^{-1}=\mathrm{g}(\underline{P}),$$

$$U_{g}E(\Delta )U_g^{-1}=E({\text{g}}^{-1}(\Delta ))holds\; for\; everyg\in {\mathcal{L}}_{+}^{\uparrow }.$$

$$(i)P_0^2-{\mathbf{P}}^2=\eta \text{sl}1Iand(ii)W^2\equiv W_0^2-(W_1^2+W_2^2+W_3^2)=\varpi \text{sl}1I.$$

$$S_{\mu }^{+}=\{\underline{p}\mid p_0^2-{\mathbf{p}}^2={\mu }^2,p_0>0\},(positive\; hyperboloid),$$

$$S_{\mu }^{-}=\{\underline{p}\mid p_0^2-{\mathbf{p}}^2={\mu }^2,p_0<0\},(negative\; hyperboloid).$$

$$\sigma (\underline{P})\subseteq \{\underline{p}\mid p_0^2-{\mathbf{p}}^2-{\mu }^2=0\}\equiv S_{\mu }^{+}\cup S_{\mu }^{-}.$$

$$\sigma (\underline{P})\cap S_{\mu }^{+}\ne \varnothing \; implies{S}_{\mu }^{+}\subseteq \sigma (\underline{P});\sigma (\underline{P})\cap S_{\mu }^{-}\ne \varnothing \; implies{S}_{\mu ,\varpi }^{-}\subseteq \sigma (\underline{P}).$$

$$\sigma (P_0)=-\sigma (P_0).$$

$${}_{◃}\text{S}={\rm Y},and{}^{◃}\text{T}=\tau \mathcal{K}{\rm Y},where$$

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

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

$${{}^{\triangleleft}\kern-1.5pt\texttt{T}}=\left[\begin{array}[]{cc}0&1\cr 1&0\end{array}\right]\,;\qquad{{}_{\triangleleft}\kern-2.0pt\texttt{S}}=\Upsilon\left[\begin{array}[]{cc}1&0\cr 0&1\end{array}\right]\quad\hbox{or}\quad{{}_{\triangleleft}\kern-2.0pt\texttt{S}}=\Upsilon\left[\begin{array}[]{cc}1&0\cr 0&-1\end{array}\right]\,.$$

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

$${{}_{◃}\text{S}}_{mn}{p}_0=p_0{{}_{◃}\text{S}}_{mn},{{}_{◃}\text{S}}_{mn}{p}_j=-p_j{{}_{◃}\text{S}}_{mn},{{}_{◃}\text{S}}_{mn}{\text{j}}_k={\text{j}}_k{{}_{◃}\text{S}}_{mn},{{}_{◃}\text{S}}_{mn}{\text{k}}_j=-{\text{k}}_j{{}_{◃}\text{S}}_{mn}.$$

$${\rm Y}{p}_j=-p_j{\rm Y},{\rm Y}\frac{\partial }{\partial p_j}=-\frac{\partial }{\partial p_j}{\rm Y},[{\rm Y},S_j]=IO,{{\rm Y}}^2=\text{sl}1I.$$

$${{}_{\triangleleft}\kern-2.0pt\texttt{S}}=\Upsilon\,{\bf n}\cdot{\vec{\sigma}},\;\hbox{ where }{\vec{\sigma}}=\left(\left[\begin{array}[]{cc}0&1\cr 1&0\end{array}\right],\left[\begin{array}[]{cc}0&-i\cr i&0\end{array}\right],\left[\begin{array}[]{cc}1&0\cr 0&-1\end{array}\right]\right).$$

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

$${{}^{◃}\text{T}}_{mn}{p}_0=p_0{{}^{◃}\text{T}}_{mn},{{}^{◃}\text{T}}_{mn}{p}_j=-p_j{{}^{◃}\text{T}}_{mn},{{}^{◃}\text{T}}_{mn}{\text{j}}_k=-{\text{j}}_k{{}^{◃}\text{T}}_{mn},{{}^{◃}\text{T}}_{mn}{\text{k}}_j={\text{k}}_j{{}^{◃}\text{T}}_{mn}.$$

$$\mathcal{K}{p}_j=p_j\mathcal{K},\mathcal{K}\frac{\partial }{\partial p_j}=\frac{\partial }{\partial p_j}\mathcal{K},\mathcal{K}{\rm Y}={\rm Y}\mathcal{K},{\mathcal{K}}^2=\text{sl}1I.$$

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

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

$$P_{0}=\left[\begin{array}[]{cccc}p_{0}&0&0&0\cr 0&-p_{0}&0&0\cr 0&0&p_{0}&0\cr 0&0&0&-p_{0}\end{array}\right],\quad P_{j}=\left[\begin{array}[]{cccc}p_{j}&0&0&0\cr 0&p_{j}&0&0\cr 0&0&p_{j}&0\cr 0&0&0&p_{j}\end{array}\right],$$

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

Full text

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 ${{}_{\triangleleft}\kern-2.2pt\texttt{s}}$.

[TABLE]

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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).

[TABLE]

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.

[TABLE]

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

[TABLE]

[TABLE]

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.

[TABLE]

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.

[TABLE]

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.

[TABLE]

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

[TABLE]

• - $\Upsilon$ 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

[TABLE]

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.

[TABLE]

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].

[TABLE]

(In the matrices (21) “$1$” and “[math]” 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 $\mathcal{H}$.

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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,

[TABLE]

and

[TABLE]

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

[TABLE]

[TABLE]

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.