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Non legge i files di bibliografia. %% %% Come si usa. Strutturare il file come segue: %% \def\REFERENCES{\input } %% OPZIONALE %% \input macrobib %% \noref %% < testo del file > %% \end %% \def\noref{\def\rfr(##1){##1} \def\references{}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Il comando \allref elenca tutte le referenze definite nei files %% indicati dal comando \REFERENCES e le cita tramite le sigle. %% %% Come si usa. Creare un file ausiliario strutturato come segue: %% \def\REFERENCES{\input } %% \input macrobib %% \allref %% \end %% \def\allref{\def\REF(##1)##2\endREF{\refno{##1}{##2}} \begref{References}{99}\REFERENCES\endref} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Il comando \checkref controlla che nei files indicati dal %% comando \REFERENCES non ci siano referenze con la stessa sigla. %% %% Come si usa. 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(??different files with same name??}\fi} \REFERENCES \def\REF(##1)##2\endREF{ \ifundef{w##1} \else\item{[##1]}##2\fi} \def\title##1{\medskip\noindent{\bf ##1}} \REFERENCES} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Il comando \createref crea un file di bibliografia %% estraendo le referenze citate nel file di riferimento da un %% (o una serie di) file(s) generali di bibliografia. %% Come si usa: strutturare il file di riferimento in questo modo: %% \def\REFERENCES{\input } %% \input macrobib %% \createref{} %% %% \references %% \end %% La prima volta che .tex viene 'teccato' si ottiene un typeset %% analogo a quello ottenuto col comando \alphanumref, e viene creato un file %% .bib che contiene tutte le entrate bibliografiche contenute nel %% file di bibliografia citate in .tex. La seconda volta si ottiene %% un typeset analogo a quello ottenuto col comando \autonumref, usando come %% sorgente il file .bib creato precedentemente. %% Attenzione: se i files bibliografici sono piu' d'uno, le entrate %% bibliografiche del file .bib non saranno in ordine alfabetico!! \def\createbibfile{ \def\REF(##1)##2\endREF{\wdch{r}{##1}{defined}}\REFERENCES \def\rfr(##1){\wdef{q##1}{quoted}\ifundef{r##1}$\diamondsuit$##1 \write15{\string \REF(##1) [##1] to be inserted \csname endREF\endcsname} \write16{!!ref ##1 was never defined!!}\else##1\fi} \def\references{ \def\REF(####1)####2\endREF{ \ifundef{q####1} \else\refno{####1}{####2} \write15{\string \REF(####1)####2\csname endREF\endcsname}\fi} \begref{References}{99}\REFERENCES\endref}} \def\createref#1{\openin14=#1.bib \ifeof14 \relax \createbibfile\openout15=#1.bib \else\def\REFERENCES{\input #1.bib}\autonumref\fi} %%%% Reference macroes %%%%%%%%%%% \def\REFERENCES{ \REF(BiWi1) Bisognano J., Wichmann E., ``{\it On the duality condition for a Hermitian scalar field}'', J. Math. Phys. {\bf 16} (1975), 985-1007. \endREF \REF(Borc1) Borchers H.J., ``{\it The CPT theorem in two-dimensional theories of local observables}'', Commun. Math. Phys. {\bf 143} (1992), 315. \endREF \REF(BrRo1) Bratteli O., Robinson D.W., `` Operator Algebras and Quantum Statistical Mechanics, I.'' Sprin\-ger-Verlag, Berlin-Heidelberg-New York, 1979. \endREF \REF(BGLo1) Brunetti R., Guido D., Longo R., ``{\it Modular structure and duality in conformal quantum field theory}'', Commun. Math. Phys., {\bf 156} (1993), 201-219. \endREF \REF(BuFr1) Buchholz D., Fredenhagen K.,``{\it Locality and the structure of particle states}'' Commun. Math. Phys. {\bf 84} (1982), 1-54 \endREF \REF(BuMT1) Buchholz D., Mach G., Todorov I.T.,``{\it The current algebra on the circle as a germ of local field theories}'' Nucl. Phys. B (Proc. Suppl.) {\bf 5B}, (1998) 20. \endREF \REF(DHRo1) Doplicher S., Haag R., Roberts J.E., ``{\it Local observables and particle statistics I}'', Commun. Math. Phys. {\bf 23} (1971), 199-230. \endREF \REF(DHRo2) Doplicher S., Haag R., Roberts J.E., ``{\it Local observables and particle statistics II}'', Commun. Math. Phys. {\bf 35} (1974), 49-85. \endREF \REF(Fred1) Fredenhagen K., ``{\it Generalization of the theory of superselection sectors}'', in `The algebraic Theory of Superselection Sectors', D. Kastler ed., World Scientific, Singapore 1990. \endREF \REF(FrJo1) Fredenhagen K., J\"or\ss\ M. ``{\it Conformal Haag-Kastler nets, pointlike localized fields and the existence of operator product expansion}'', DESY preprint, october 1994. \endREF \REF(FrRS1) Fredenhagen K., Rehren K.-H., Schroer B., ``{\it Superselection sectors with braid group statistics and exchange algebras. I}'' Commum. Math. Phys. {\bf 125} (1989) 201-226 \endREF \REF(FrRS2) Fredenhagen K., Rehren K.-H., Schroer B., ``{\it Superselection sectors with braid group statistics and exchange algebras. II}'' Rev. Math. Phys. {\bf Special Issue} (1992) 113-157 \endREF \REF(FrGM1) Fr\"ohlich J., Gabbiani F., Marchetti P., ``{\it Braid statistics in three-dimensional local quantum theory}'' in `The algebraic Theory of Superselection Sectors', D. Kastler ed., World Scientific, Singapore 1990. \endREF \REF(GaFr1) Fr\"ohlich J., Gabbiani F., ``{\it Operator algebras and Conformal Field Theory}'', Commun. Math. Phys. {\bf 155} (1993), 569-640. \endREF \REF(GuLo1) Guido D., Longo R., ``{\it Relativistic invariance and charge conjugation in quantum field theory}'', Commun. Math. Phys. {\bf 148} (1992), 521-551. \endREF \REF(GuLo2) Guido D., Longo R., ``{\it An algebraic spin and statistics theorem}'', to appear in Commun. Math. Phys. \endREF \REF(Haag1) Haag R. , ``Local Quantum Physics'', Springer- Verlag, Berlin-Heidelberg-New York 1992. \endREF \REF(HiLo1) Hislop P., Longo R., ``{\it Modular structure of the local algebras associated with the free massless scalar field theory}'', Commun. Math. Phys. {\bf 84} (1982), 84. \endREF \REF(Jone1) Jones V., ``{\it Index for subfactors}'' Invent. Math. {\bf 2} (1983) 1, 25. \endREF \REF(Long2) Longo R., ``{\it Index of subfactors and statistics of quantum fields. I}'', Commun. Math. Phys. {\bf 126} (1989), 217-247. \endREF \REF(Long3) Longo R., ``{\it Index of subfactors and statistics of quantum fields. II Correspondences, braid group statistics and Jones polynomial }'', Commun. Math. Phys. {\bf 130} (1990), 285-309. \endREF \REF(Long4) Longo R., ``{\it Minimal index and braided subfactors}'' J. Funct. Anal. {\bf 109} (1992), 98-112. \endREF \REF(Long5) Longo R., ``{\it Von~Neumann algebras and quantum field theory}'' Proceedings of the International Congress of Mathematicians, Zurich 1994, Birkh\"auser (to appear). \endREF \REF(LoRo1) Longo R., Roberts J.E., ``{\it A theory of dimension}'', preprint. \endREF \REF(PiPo1) Pimsner M., Popa S. {\it Entropy and index for subfactors} Ann. Sci. Ecole Norm. Sup. {\bf 19} (1986), 57-106. \endREF \REF(PrSe1) Pressley A., Segal G., `` Loop Groups'', Oxford Science publ., Oxford 1986 \endREF \REF(Robe1) Roberts J.E., ``{\it Local cohomology and superselection structure}'', Commun. Math. Phys {\bf 51}, (1976) 107-119. \endREF \REF(StWi1) Streater R.F., Wightman A.S., ``PCT, Spin and Statistics, and all that'', Addison Wesley, Reading (MA), 1989. \endREF \REF(Take1) Takesaki M., `` Tomita theory of modular Hilbert algebras '', Lecture Notes in Math. {\bf 128}, Springer Verlag, New York--Heidelberg--Berlin 1970. \endREF \REF(Take2) Takesaki M., ``Theory of Operator Algebras, I'', Springer, New York, 1979. \endREF \REF(WWWi1) Wick G.C., Wightman A.S., Wigner E.P. ``{\it The intrinsic parity of elementary particles}'' Phys. Rev. {\bf 88} (1952), 101-105. \endREF \REF(Zimm1) Zimmer R. ``Ergodic Theory of Semisimple Groups'', Birkh\"auser, Boston-Basel-Stuttgart, 1984. \endREF } \autonumref %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %%% TITLE %%% \topskip3.cm \font\ftitle=cmbx12 scaled\magstep1 \vskip2truecm \centerline{\ftitle The Conformal Spin and Statistics Theorem} \bigskip\bigskip \centerline{Daniele Guido$^1$\footnote{$^*$} { Supported in part by MURST and CNR-GNAFA.} and Roberto Longo$^{1,2*}$} \footnote{}{E-mail:\ guido@mat.utovrm.it, longo@mat.utovrm.it } \vskip1.truecm \item{$(^1)$} Dipartimento di Matematica, Universit\`a di Roma ``Tor Vergata'' \par via della Ricerca Scientifica, I--00133 Roma, Italia. \item{$(^2)$} Centro Linceo Interdisciplinare, Accademia Nazionale dei Lincei \par via della Lungara 10, I--00165 Roma, Italia \vfill\eject \topskip0.cm \titlea {Introduction} During the recent years Conformal Quantum Field Theory has become a widely studied topic, especially on a low dimensional space-time, because of physical motivations such as the desire of a better understanding of two-dimensional critical phenomena, and also for its rich mathematical structure providing remarkable connections with different areas such as Hopf algebras, low dimensional topology, knot invariants, subfactors among many others. The Operator Algebra approach furnishes a powerful tool of investigation in this context, not only because it naturally leads to a model independent and intrinsic analysis, focusing on essential aspects such as the relative position of the local von Neumann algebras, but also because it makes visible otherwise hidden natural structures bringing to results inaccessible by different methods. Two examples of this kind, the geometric description of the Tomita-Takesaki modular structure of the local von~Neumann algebras [\rfr(BiWi1),\rfr(HiLo1),\rfr(BGLo1)], and the connection of the statistics of a superselection sector with the Jones index theory of subfactors [\rfr(Long2)], will play a fundamental role in this paper. These methods are present and important in general Quantum Field Theory, but provide an even richer structure in the low-dimensional case, conformal theories on $S^1$ in particular. In the early seventies Doplicher, Haag and Roberts [\rfr(DHRo1),\rfr(DHRo2)] developed a theory of superselection sectors, in the sense of [\rfr(WWWi1)], in the algebraic framework proposed by Haag and Kastler [\rfr(Haag1)] starting from first principles. They described a superselection sector by a localized endomorphism $\r$ of the $C^*$-algebra generated by the local observable von Neumann algebras on the usual Minkowski space. In particular they showed that the statistics of $\r$, a representation of the permutation group, is intrinsically encoded in $\r$ and classified it by an associated statistical parameter $\l_\r$. It was more recently realized that in the low dimensional case the statistics becomes a representation of Artin braid group. By applying generalized DHR methods, a first analysis in this case was given in [\rfr(Long2),\rfr(FrRS1)]. In the simplest cases (small index or few channels) the statistics parameter classifies the braid group statistics by the Jones polynomial invariant for knots and links and its generalizations, see [\rfr(Long3),\rfr(Long5)]. A key point in the analysis of superselection sectors is the index-statistics theorem [\rfr(Long2)] showing that, in any space-time dimension, $$ {\rm Ind}(\r)=d(\r)^2 $$ where ${\rm Ind}(\r)$ is the minimal index of $\r$, an extension of the Jones index [\rfr(Jone1)], and $d(\r):=|\l_\r|^{-1}$ is the DHR statistical dimension of $\r$. We refer to [\rfr(Long5)] for a survey and for references on the index theory for infinite factors, but we recall that the square root of the minimal index of an endomorphism of a factor has the meaning of a dimension, that find an identification in this context by the above equation. On the other hand important informations on the statistics are also contained in the statistics phase $\k_\r:=\l_\r/|\l_\r|$ of $\r$: on the 4-dimensional space-time $\k_\r = \pm 1$, a sign labeling the fundamental Fermi-Bose alternative. Therefore it is natural to look at a counterpart of the index-statistics relation for the statistics phase. Based on the classical spin-statistics connection (see [\rfr(StWi1)]), one may easily conjecture that in a conformal theory on $S^1$ the statistics phase has to agree with the univalence of the sector $\r$ $$ s_\r=\k_\r $$ where $s_\r:=e^{2\pi i L_\r}$ ($L_\r$ the conformal spin, the lowest eigenvalue of the conformal Hamiltonian) is a label for the central extension associated with the occurring projective representation of the M\"obius group $PSL(2,\Re)$. Attempts to prove this relation have been made in particular by Fredenhagen, Rehren and Schroer [\rfr(FrRS2)] and, in the related $2+1$-dimensional context, by Fr\"ohlich, Gabbiani and Marchetti [\rfr(FrGM1)]. Starting with assumptions on the existence of a global conjugate charge and of complete reducibility, they obtained a spin summation rule, which implies the equality up to a sign $s_\r=\pm\k_\r$. But the conformal spin-statistics theorem remained unproven unless adding ad hoc undesirable assumptions. Based on different ideas, this paper will show how the full strength of Operator Algebras provides the general and intrinsic spin and statistics relation, namely the equality $s_\r=\k_\r$. We deal with conformal theories on $S^1$ (one-dimensional components of two-dimensional chiral conformal theories) and base our analysis only on first principles: isotony of the local von Neumann algebras, locality, conformal invariance with positive energy, existence of the vacuum. We thus obtain the complete relation $$ {\rm statistics\ parameter}= {{\rm univalence}\over{\sqrt{\rm minimal\ index}}}\ . $$ Note that $\k_\r$ has a local nature while $s_\r$ is a global invariant. This is reminiscent of familiar situations in Geometry and suggests that extensions of our result to more general (curved) space-times should reveal further geometrical aspects. Our theorem is not only a prototype for further generalizations, but it already provides a number of immediate extensions or variants, like for the case of topological charges on a $2+1$-dimensional space-time [\rfr(BuFr1)]. This is due to the fact that we shall use the conformal invariance only indirectly, not in an essential way. For convenience we shall discuss these aspects together with related points and examples in a separate paper. Our paper follows a previous work [\rfr(GuLo1)] where we reconsidered the classical spin and statistics theorem in Quantum Field Theory [\rfr(StWi1)] and derived it in the algebraic setting assuming the ``modular covariance property", namely the geometric meaning of the modular groups of the von~Neumann algebras associated with wedge regions, consistently to the Bisognano-Wichmann theorem. That work, not directly extendible to the lower dimensional case due to the occurrence of the braid group statistics, focused however on the role played by the modular covariance property. The latter was shown to hold in conformal field theory on general grounds [\rfr(BGLo1),\rfr(GaFr1)], and set thus the basis for the present analysis. Ultimalety only the geometric description of the modular conjugations is essential in our analysis. We now pass to the description of the more specific content of this paper. In Section 1 we recall the basic properties shared by the local von Neumann algebras $\A(I)$ associated with intervals $I$ of $S^1$. Like in the classical case, the spin-statistics relation is strictly tied up with the PCT symmetry. Section 2 is indeed devoted to the construction of a global conjugate charge for a superselection sector $\r$ with finite statistics, a key point relevant in itself, previously an assumption in the related literature. As shown in [\rfr(GuLo1)], the sector $$\bar\r:=j\cdot\r\cdot j$$ is locally a conjugate of $\r$ in the sense that if $\r$ is an endomorphism localized in an interval $I_0$ and $j$ is the adjoint (geometric) action given by the modular conjugation of an interval, one has the identity $$ \bar\r|_{\A(I)}=\ov {\r|_{\A(I)}} $$ where $I$ is any interval containing $I_0$ and its reflection by $j$; the bar on the right hand side denotes the conjugate endomorphism in the sense of the sectors of the factor $\A(I)$ [\rfr(Long3)], a framework equivalent to the setting of the correspondences of Connes. In the irreducible case $\bar\r$ is characterized by by the existence of an isometry $V_I\in\A(I)$ that intertwines the identity and $\bar\r\r|_{\A(I)}$. But the problem remained whether there is a global intertwiner $V$ independent of $I$. We solve this problem positively by using an argument inspired by the ``vanishing of the matrix coefficient theorem'' for connected simple Lie groups, see the Appendix B. We prove in fact the equivalence between the local and the global intertwiners for superselection sectors with finite index, namely the embedding into the sectors (endomorphisms modulo inners) of the factor $M:=\A(I)$ determined, via the restriction map, $$ {\rm Superselection\, sectors}\rightarrow{\rm Sect}(M) $$ corresponds by the index-statistics theorem to a faithful functor of tensor $C^*$-categories with conjugates which is full (no new intertwiner arises in the range). This implies that the fusion rules of the superselection sectors are entirely described by the theory of subfactors. As a first consequence we shall see in Section 3 that the (internal) intertwiner property of the above isometry $V$ is equivalent to the (spatial) property of being the standard implementation of $\r$, according to Araki, Connes and Haagerup, see Appendix A, with respect to the vacuum vector. To extract information from this fact we localize $\r$ in the upper-right quarter-circle and consider the standard implementations $V_1$ and $V_2$ of $\r$ as an endomorphism of the upper and of the right semicircle von Neumann algebra respectively and observe that $$ \mu_\r:=V_1^*V_2^*V_1V_2 \eqno(3.1) $$ is a scalar invariant for $\r$ that reflects by both analytic-algebraic and geometric aspects. It is indeed natural to look at $\mu_\r$ as a generalized multiplicative commutator of local intertwiners, in the spirit of the statistics, and identify it with the statistics parameter $\l_\r$, or as an invariant obtained by reversing the orientation, in the spirit of the spin, and identify $\mu_\r$ with ${\rm Ind}(\r)^{-{1\over 2}}$--times the univalence of $\r$. In more detail we shall obtain the spin-statistics relation by ``squaring'' a more primitive identity between operators (see eq. $(3.8)$) where a further invariant $c_\r$ enters. Our result is then completed by showing that $c_\r$ is a conjugate-invariant character on the semi-ring of the superselection sectors, so that it takes only the values $\pm 1$. This reaches the goal of our paper, but leaves out the full understanding of the invariant $c_\r$, in particular whether the value $c_\r=-1$ might actually occur. We think this is the case, that reflects a cohomological obstruction, and plan to return to this point somewhere else. Our work has been announced in [\rfr(Long5)]. \titlea{1. General properties of conformal precosheaves on S$^1$} In this section we recall the basic properties enjoyed by the family of the von~Neumann algebras associated with a conformal Quantum Field Theory on $S^1$. By an {\it interval} we shall always mean an open connected subset $I$ of $S^1$ such that $I$ and the interior $I'$ of its complement are non-empty. We shall denote by $\I$ the set of intervals in $S^1$. A precosheaf $\A$ of von~Neumann algebras on the intervals of $S^1$ is a map $$ I\to\A(I) $$ from $\I$ to the von~Neumann algebras on a Hilbert space $\H$ that verifies the following property: \begassumption{A. Isotony} If $I_1$, $I_2$ are intervals and $I_1\subset I_2$, then $$ \A(I_1)\subset\A(I_2)\ . $$ \endassumption $\A$ is a {\it conformal precosheaf} of von Neumann algebras if the following properties B--E hold too. \begassumption{B. Conformal invariance} There is a unitary representation $U$ of $\bG$ (the universal covering group of $PSL(2,\Re)$) on $\H$ such that $$ U(g)\A(I)U(g)^*=\A(gI)\ , \qquad g\in\bG,\ I\in\I. $$ \endassumption The group $PSL(2,\Re)$ is identified with the M\"obius group of $S^1$, i.e. the group of conformal transformations on the complex plane that preserve the orientation and leave the unit circle globally invariant. Therefore $\bG$ has a natural action on $S^1$. \begassumption{C. Positivity of the energy} The generator of the rotation subgroup $U(R(\cdot))$ is positive. \endassumption Here $R(\th)$ denotes the (lifting to $\bG$ of the) rotation by an angle $\th$. In the following we shall often write $U(\th)$ instead of $U(R(\th))$. We may associate two one-parameter groups with any interval $I$. Let $I_1$ be the upper semi-circle, i.e. the interval $\{e^{i\th},\th\in (0,\pi)\}$. We identify this interval with the positive real line $\Re_+$ via the Cayley transform $C:S^1\to\Re\cup\{\infty\}$ given by $z\to -i(z-1)(z+1)^{-1}.$ Then we consider the one-parameter groups $\L_{I_1}(s)$ and $T_{I_1}(t)$ of diffeomorphisms of $S^1$ (cf. Appendix~B) such that $$ C\L_{I_1}(s)C^{-1}x=e^{s}x,\qquad C T_{I_1}(t)C^{-1}x=x+t, \quad t,s,x\in\Re. $$ We also associate with $I_1$ the reflection $r_{I_1}$ given by $$ r_{I_1}z=\bar z $$ where $\bar z$ is the complex conjugate of $z$. We remark that $\L_{I_1}$ restricts to an orientation preserving diffeomorphisms of $I_1$, $r_{I_1}$ restricts to an orientation reversing diffeomorphism of $I_1$ onto $I'_1$ and $T_{I_1}(t)$ is an orientation preserving diffeomorphism of $I_1$ into itself if $t\geq0$. Then, if $I$ is an interval and we chose $g\in\bG$ such that $I=gI_1$ we may set (see also the Appendix B) $$ \L_I=g \L_{I_1} g^{-1},\qquad r_I=g r_{I_1} g^{-1},\qquad T_I=g T_{I_1} g^{-1}. $$ The elements $\L_I(s)$, $s\in\Re$ and $r_I$ are well defined, while the one parameter group $T_I$ is defined up to a scaling of the parameter. However, such a scaling plays no role in this paper. We note also that $T_{I'}(t)$ is an orientation preserving diffeomorphism of $I$ into itself if $t\leq0$. Lemma B.4 in Appendix~B states the equivalence between the positivity of the conformal Hamiltonian, i.e. the generator of the rotation group $U(R(\cdot))$, and the positivity of the usual Hamiltonian, i.e. the generator of the translations on the real line in the above specified identification of $S^1$ with $\Re\cup\{\infty\}$. \begassumption{D. Locality} If $I_0$, $I$ are disjoint intervals then $\A(I_0)$ and $A(I)$ commute. \endassumption The lattice symbol $\vee$ will denote `the von Neumann algebra generated by'. \begassumption{E. Existence of the vacuum} There exists a unit vector $\Q$ (vacuum vector) which is $U(\bG)$-invariant and cyclic for $\vee_{I\in\I}\A(I)$. \endassumption Let $r$ be an orientation reversing isometry of $S^1$ with $r^2=1$ (e.g. $r_{I_1}$). The action of $r$ on $PSL(2,\Re)$ by conjugation lifts to an action $\s_r$ on $\bG$, therefore we may consider the semidirect product of $\bG\times_{\s_r}\Ze_2$. Any involutive orientation reversing isometry has the form $R(\th)r_{I_1}R(-\th)$, thus $\bG\times_{\s_r}\Ze_2$ does not depend on the particular choice of the isometry $r$. Since $\bG\times_{\s_r}\Ze_2$ is a covering of the group generated by $PSL(2,\Re)$ and $r$, $\bG\times_{\s_r}\Ze_2$ acts on $S^1$. We call (anti-)unitary a representation $U$ of $\bG\times_{\s_r}\Ze_2$ by operators on $\H$ such that $U(g)$ is unitary, resp. antiunitary, when $g$ is orientation preserving, resp. orientation reversing. \begproposition{1.1} Let $\A$ be a conformal precosheaf. The following hold: \item{$(a)$} {\it Reeh-Schlieder theorem} [\rfr(FrJo1)]: $\Q$ is cyclic and separating for each von Neumann algebra $\A(I)$, $I\in\I$. \item{$(b)$} {\it Bisognano-Wichmann property} [\rfr(BGLo1),\rfr(GaFr1)]: $U$ extends to an (anti-)unitary representation of $\bG\times_{\s_r}\Ze_2$ such that, for any $I\in\I$, $$ \eqalignno{U(\L_I(2\pi t))&=\D_I^{it}\ ,&(1.1)\cr U(r_I)&=J_I\ ,&(1.2)\cr} $$ where $\D_I$, $J_I$ are the modular operator and the modular conjugation associated with $(\A(I),\Q)$ [\rfr(Take1)]. For each $g\in\bG\times_{\s_r}\Ze_2$ $$ U(g)\A(I)U(g)^*=\A(gI)\ . $$ \item{$(c)$} {\it Additivity} [\rfr(FrJo1)]: if a family of intervals $I_i$ covers the interval $I$, then $$ \A(I)\subset\bigvee_i\A(I_i)\ . $$ \item{$(d)$} {\it Spin and statistics for the vacuum sector} [\rfr(GuLo2)]: $U$ is indeed a representation of $PSL(2,\Re )$, i.e. $U(2\pi)=1$. \item{$(e)$} {\it Haag duality} [\rfr(BGLo1),\rfr(GaFr1)]: $$\A(I')=\A(I)'\ ,\quad I\in\I.$$ \endproposition \begProof We sketch here only the proof of $(d)$ and refer to the original literature for the rest. Note however that: the usual Reeh-Schlieder argument shows that $(c)$ implies $(a)$; $(b)$ is proved by using a theorem of Borchers [\rfr(Borc1)]; $(e)$ is an immediate consequence of $(b)$. To get $(d)$ let $I_1$ and $I_2$ be the upper and the right semicircle respectively, then $J_{I_1}$ fixes $\Omega$ and implements an anti-automorphism of $\A(I_{I_2})$, thus it commutes with $J_{I_2}$. By property $(b)$ $J_{I_1}J_{I_2}=U(\pi)$, thus $U(\pi)$ is an involution. \endProof \begassumption{F. Uniqueness of the vacuum (or irreducibility)} The only $U(\bG)$-invariant vectors are the scalar multiples of $\Q$. \endassumption The term irreducibility is due to the following. \begproposition{1.2} The following are equivalent: \item{$(i)$} $\Co\Q$ are the only $U(\bG)$-invariant vectors. \item{$(ii)$} The algebras $\A(I)$, $I\in\I$, are factors. In this case they are type III$_1$ factors. \item{$(iii)$} If a family of intervals $I_i$ intersects at only one point $\z$, then $\cap_i\A(I_i)=\Co$. \item{$(iv)$} The von~Neumann algebra $\vee\A(I)$ generated by the local algebras coincides with $\B(\H)$ ($\A$ is irreducible). \endproposition \begProof $(i)\imply(ii)$. Indeed $(i)$ implies (c) of Corollary B.2 in the Appendix B, hence the modular group of $\A(I)$ with respect to $\Q$ is ergodic, showing that $\A(I)$ is a type III$_1$ factor. \np $(ii)\imply(iii)$. If $\z$ is a boundary point of an interval $I$, then by additivity and duality $\cap_i\A(I_i)$ commutes both with $\A(I)$ and $\A(I')$, and is therefore trivial. \np $(iii)\imply(iv)$. We have $\vee_{I\in\I}\A(I)\supset\vee_{\z\notin I}\A(I)=\B(\H)$. \np $(iv)\imply(i)$. Let $I$ be an interval and $x\in\A(I)$ such that $U(g)x\Omega=x\Omega$ for all $g\in\bG$. since $\Omega$ is locally separating, we have $x=U(g)xU(g)^{-1}$. Since $\bG$ acts transitively on the intervals, $x$ is in the commutant of $\cup_{I\in\I}\A(I)$, and is therefore a scalar. Since $\A(I)\Q$ is dense in $\H$, by the Ergodic Theorem $\Q$ is the only $U(\bG)$-invariant vector. \endProof By Corollary~B.2 the irreducibility of $\A$ is also equivalent to $\Q$ being unique invariant for any of the one-parameter subgroups of $U$ corresponding to $T_I$,$\Lambda_I$ or $R$. Now any conformal precosheaf decomposes uniquely into a direct integral of irreducible conformal precosheaves. This can be seen as in Proposition 3.1 of [\rfr(GuLo2)]. We will therefore always assume that our precosheaves are irreducible. \titlea {2. Superselection structure. Constructing the global conjugate charge} \titleb {2.1 Generalities on superselection sectors with finite index} In this section $\A$ is an irreducible conformal precosheaf of von Neumann algebras as defined in Section 1. A covariant {\it representation} $\pi$ of $\A$ is a family of representations $\pi_I$ of the von Neumann algebras $\A(I)$, $I\in\I$, on a Hilbert space $\H_\pi$ and a unitary representation $U_\pi$ of the covering group $\bG$ of $PSL(2,\Re)$, with {\it positive energy}, i.e. the generator of the rotation unitary subgroup has positive generator, such that the following properties hold: $$ \eqalign{ I\subset {\tilde I}\imply\pi_{\tilde I}\big|_{\A(I)}=\pi_I\quad &(isotony)\cr \ad U_\pi(g)\cdot\pi_I=\pi_{gI}\cdot\ad U(g) \quad &(covariance).\cr} \eqno(2.1) $$ A unitary equivalence class of representations of $\A$ is called {\it superselection sector}. Assuming $\H_\pi$ to be separable, the representations $\pi_I$ are normal because the $\A(I)$'s are factors [\rfr(Take2)]. Therefore for any given $I_0$, $\pi_{I'_0}$ is unitarily equivalent $id_{\A(I'_0)}$ because $\A(I'_0)$ is a type III factor. By identifying $\H_\pi$ and $\H$, we can thus assume that $\pi$ is localized in a given interval $I_0\in\I$, i.e. $\pi_{I'_0}=id_{\A(I'_0)}$ (cf. [\rfr(BuMT1)]). By Haag duality we then have $\pi_I(\A(I))\subset\A(I)$ if $I\supset I_0$. In other words, given $I_0\in\I$ we can choose in the same sector of $\p$ a {\it localized endomorphism} with localization support in $I_0$, namely a representation $\r$ equivalent to $\p$ such that $$ I\in\I,\,I\supset I_0\Rightarrow\r_I\in{\rm End}\A(I), \qquad\r_{I_0'}=id_{I_0'}. $$ In the following (with the exception of subsection 2.4) representations or endomorphisms are always assumed to be covariant with positive energy\fonote{Assuming strong additivity (i.e. Haag duality on the real line) the covariance property with positive energy follows automatically in the finite index case; in fact the weaker assumption of {\it 3-regularity} is sufficient (cf. [\rfr(GuLo1)]). $\A$ is said to be $n$-regular if, after removing $n$ points from $S^1$, the $C^*$-algebra generated by the local operators is irreducible. By Haag duality and factoriality any conformal precosheaf is 2-regular, but the validity of 3-regularity is not known in general. An example violating 4-regularity has been pointed out to us by H. Wiesbrock.}. To capture the global point of view we may consider the {\it universal algebra} $\ua$. Recall that $\ua$ is a C$^*$-algebra canonically associated with the precosheaf $\A$ (see [\rfr(Fred1),\rfr(GuLo1)]). There are injective embeddings $\iota_I:\A(I)\to\ua$ so that the local von Neumann algebras $\A(I),\ I\in\I,$ are identified with subalgebras of $\ua$ and generate all together a dense $^*$-subalgebra of $\ua$, and every representation of the precosheaf $\A$ factors through a representation of $\ua$. Conversely any representation of $\ua$ restricts to a representation of $\A$. The vacuum representation $\pi_0$ of $\ua$ corresponds to the identity representation of $\A$ on $\H$, thus $\pi_0$ acts identically on the local von Neumann algebras. We shall often drop the symbols $\iota_I$ and $\p_0$ when no confusion arises. By the universality property, for each $g\in PSL(2,\Re)$ the isomorphism $\ad U(g):\A(I)\to\A(gI)$, $I\in\I$ lifts to an automorphism $\a_g$ of $\ua$. It will be convenient to lift the map $g\to\a_g$ to a representation, still denoted by $\a$, of the universal covering group $\bG$ of $PSL(2,\Re)$ by automorphisms of $\ua$. The covariance property for an endomorphism $\r$ of $\ua$ localized in $I_0$ means that $\a_g\cdot\r\cdot\a_{g^{-1}}$ is equivalent to $\r$ for any $g\in\bG$, i.e. $$ \ad z_\r(g)^*\cdot\r=\a_g\cdot\r\cdot\a_{g^{-1}}\, \qquad g\in\bG.\eqno(2.2) $$ for a suitable unitary $z_\r(g)\in\ua$. The map $g\to z_\r(g)$ can be chosen to be a localized $\a$-cocycle, i.e. $$ \eqalign{ z_\r(g)&\in\A(I_0\cup gI_0)\quad \forall g\in\bG:I_0\cup gI_0\in\I\cr z_\r(gh)&= z_\r(g)\a_g(z_\r(h)),\qquad g,h\in\bG.\cr} \eqno(2.3) $$ The relations between $(\pi,U_\pi)$ and $(\r,z_\r)$ are $$ \eqalign{ \pi&=\pi_0\cdot\r\cr \pi_0(z_\r(g))&=U_\pi(g)U(g)^*\ .\cr} \eqno(2.4) $$ As is known ([\rfr(Robe1)], see also [\rfr(GuLo1)]) that the localized cocycle $z_\r$ reconstructs the endomorphism $\r$ via $$ \r|_{\A(gI'_0)}=\ad z_\r(g)|_{\A(gI'_0)}\eqno(2.5) $$ A localized endomorphism of $\ua$ is said {\it irreducible} if the associated representation $\pi$ is irreducible. Note that the representations $\pi_0\cdot\r_1$ and $\pi_0\cdot\r_2$ associated with the endomorphisms $\r_1$, $\r_2$ of $\ua$ are unitarily equivalent if and only if $\r_1$ and $\r_2$ are equivalent endomorphisms of $\A$, i.e. $\r_2$ is a perturbation of $\r_1$ by an inner automorphism of $\A$. An endomorphism of $\ua$ localized in an interval $I_0$ is said to have {\it finite index} if $\r_I$ $(=\r|_{\A(I)})$ has finite index, $I_0\subset I$ (see [\rfr(Long2),\rfr(Long5)]). The index is indeed well defined due to the following. \begproposition{2.1} Let $\r$ be an endomorphism localized in the interval $I_0$. Then the index ${\rm Ind}(\r):= {\rm Ind}(\r_I)$, the minimal index of $\r_I$, does not depend on the interval $I\supset I_0$. \endproposition \begProof We show indeed that all the inclusions $\r(\A(I))\subset \A(I)$ are isomorphic if $I\supset I_0$ (they are isomorphic to the inclusion $\pi(\A(I))\subset\pi(\A(I'))'$ for all $I\in\I$ ). This follows because, if $g\in\bG$ and $z_\r(g)$ are chosen as in $(2.2),(2.3)$ with $I\supset I_0$ and $gI = I_0$, then $$ \eqalign{ \{\r(\A(I_0))\subset\A(I_0)\} &= \{U_\r(g)\r(\A(I))U_\r(g)^*\subset U(g)(\A(I))U(g)^*\}\cr &\simeq\{\r(\A(I))\subset z_\r(g^{-1})\A(I)z_\r(g^{-1})^*\}\cr} $$ and $z_\r(g)\in \A(I)$. \endProof \begproposition{2.2} Let $\r$ be a covariant (not necessarily irreducible) endomorphism with finite index. Then the representation $U_\r$ described before is unique. In particular, any irreducible component of $\r$ is a covariant endomorphism. \endproposition \begProof If $\r$ is localized in $I_0$ and has finite index the following inclusion shows that $\pi(\ua)'$ is finite-dimensional, $\pi:=\pi_0\cdot\r$: $$ \pi(\ua)'\subset (\pi(\A(I))\cup\pi(\A(I')))'=\pi(\A(I))'\cap\A(I)\ , \qquad I_0\subset I.\eqno(2.6) $$ Since $U_\pi$ implements automorphisms of $\pi(\A)$, it implements an action of $\bG$ by automorphisms of $\pi(\A)'$, that must be trivial because $\bG$ has no non-trivial action by automorphisms of a finite-dimensional $C^*$-algebra. Indeed such an action should be trivial on the center because $\bG$ is connected, thus it admits a faithful invariant trace that defines a scalar product unitarizing the representation, but the only finite-dimensional unitary representation of $\bG$ is the identity. Therefore we proved that $U_\pi\in\pi(\ua)''$, and this fact implies that any irreducible subsector of $\r$ is covariant. \np Let $U'_{\pi}$ be another representation of $\bG$ as in (2.1). Then, for each $x\in\A(I)$, $I\in\I$, $$ U'_{\pi}(g)U_{\pi}(g)^*{\pi}(x)=U'_{\pi}{\pi}(\a_g(x))U_{\pi}(g)^* ={\pi}(x)U'_{\pi}(g)U_{\pi}(g)^*, $$ which implies $U'_{\pi}(g)U_{\pi}(g)^*$ to belong to the center of $\pi(\ua)''$. Therefore $$ U_{\pi}(g)U'_{\pi}(g)^*U'_{\pi}(h)U_{\pi}(h)^*= U'_{\pi}(g)U'_{\pi}(h)U_{\pi}(h)^*U_{\pi}(g)^*= U'_{\pi}(gh)U_{\pi}(gh)^*\ , $$ i.e. $g\to U'_{\pi}(g)U_{\pi}(g)^*$ is a representation of $\bG$. Since $\bG$ is perfect, any abelian representation is trivial, i.e. $U_{\pi}=U'_{\pi}$ \endProof By the above proposition the {\it univalence} of an endomorphism $\r$ is well defined by $$ s_\r = U_{\r}(2\pi)\ . $$ By definition $s_\r$ belongs to $\pi(\ua)'$ therefore, when $\r$ is irreducible, $s_\r$ is a complex number of modulus one $$s_\r=e^{2\pi iL_\r}$$ with $L_\r$ the lowest weight of $U_\r$. In this case, since $U_{\r'}(g):=\pi_0(u)U_\r(g)\pi_0(u)^*$, where $\r'(\cdot):=u\r(\cdot)u^*$, $u\in\ua$, then $s_\r$ depends only on the superselection class of $\r$. Let $\r_1$, $\r_2$ be endomorphisms of an algebra $\cal B$. Their intertwiner space is defined by $$ (\r_1,\r_2)= \{T\in{\cal B}: \r_2(x)T=T\r_1(x),\ x\in{\cal B}\}.\eqno(2.7) $$ In case ${\cal B} =\ua$, $\r_i$ localized in the interval $I_i$ and $T\in (\r_1,\r_2)$, then $\pi_0(T)$ is an intertwiner between the representations $\pi_0\cdot\r_i$. If $I\supset I_1\cup I_2$, then by Haag duality its embedding $\iota_I\cdot \pi_0(T)$ is still an intertwiner in $(\r_1,\r_2)$ and a local operator. We shall denote by $(\r_1,\r_2)_I$ the space of such local intertwiners $$ (\r_1,\r_2)_I= (\r_1,\r_2)\cap\A(I). $$ If $I_1$ and $I_2$ are disjoint, we may cover $I_1\cup I_2$ by an interval $I$ in two ways: we adopt the convention that, unless otherwise specified, a {\it local intertwiner} is an element of $(\r_1,\r_2)_I$ where $I_2$ follows $I_1$ inside $I$ in the clockwise sense. We now define the statistics. Given the endomorphism $\r$ of $\A$ localized in $I\in\I$, choose an equivalent endomorphism $\r_0$ localized in an interval $I_0\in\I$ with $\bar I_0\cap\bar I =\emptyset$ and let $u$ be a local intertwiner in $(\r,\r_0)$ as above, namely $u\in (\r,\r_0)_{\tilde I}$ with $I_0$ following clockwise $I$ inside $\tilde I$. The {\it statistics operator} $\varepsilon := u^*\r(u) = u^*\r_{\tilde I}(u) $ belongs to $(\r^2_{\tilde I},\r^2_{\tilde I})$. An elementary computation shows that it gives rise to a presentation of the Artin braid group $$\varepsilon_i\varepsilon_{i+1}\varepsilon_i = \varepsilon_{i+1}\varepsilon_i\varepsilon_{i+1},\qquad \varepsilon_i\varepsilon_{i'} =\varepsilon_{i'}\varepsilon_i \,\quad{\rm if}\,\, |i-i'|\geq 2,$$ where $\varepsilon_i=\r^{i-1}(\varepsilon)$. The (unitary equivalence class of the) representation of the braid group thus obtained is the {\it statistics} of the superselection sector $\r$. Recall that if $\r$ is an endomorphism of a C$^*$-algebra $\B$, a {\it left inverse} of $\r$ is a completely positive map $\Phi$ from $\B$ to itself such that $\Phi\cdot\r=id$. We shall see in Corollary 2.12 that if $\r$ is irreducible there exists a unique left inverse $\Phi$ of $\r$ and that the {\it statistics parameter} $$ \l_\r:=\Phi(\varepsilon)\eqno(2.8) $$ depends only on the sector of $\r$. The {\it statistical dimension} $d(\r)$ and the {\it statistics phase} $\k_\r$ are then defined by $$ d(\r) = |\lambda_\r|^{-1}\ ,\qquad \k_\r = {\lambda_\r \over |\lambda_\r|}\ . $$ We shall indeed prove the equality between the statistics phase and the univalence while the statistical dimension equals the square root of the index [\rfr(Long2)] (see Corollary 3.7). \titleb {2.2 Equivalence between local and global intertwiners} If $\r,\s$ are endomorphisms of $\ua$ localized in the interval $I$, we may consider their intertwiner space $(\r_I,\s_I):=\{T\in \A(I):\s(x)T=T\r(x), \, \forall x\in \ua\}$. We always have $(\r,\s)_I\subset(\r_I,\s_I)$. \begtheorem{2.3} Let $\r$, $\s$ be endomorphisms with finite index localized in $I_0$. Then $$ (\r_I,\s_I)=(\r,\s)_I $$ for any $I\in\I$ that contains $I_0$. In other words if $T\in(\r_I,\s_I)$ then $\iota_I(T)$ intertwines $\r$ and $\s$ in $\ua$ \endtheorem The proof of this theorem will be carried on in a few steps. In the following $\r$ denotes an endomorphism of $\ua$ with finite index localized in an interval $I_0$. Let $\z\in I_0'$ and identify $S^1\backslash\z$ with $\Re$. Then $\r$ restricts to an endomorphism of each von Neumann algebra $\A(-\infty,\ell)$, for sufficiently large $\ell\in\Re$, hence it gives rise to an endomorphism $\r_\z$ the $C^*$-algebra $\A_\z$, the norm closure of $\cup_{\ell\in\Re}\A(-\infty,\ell)$. Let $\bP$ be the stabilizer of the point $\z$ for the $PSL(2,\Re)$ action, namely the semidirect product of the translations $T(t)$ and dilations $\L(s)$ on $\Re$: each $g\in\bP$ is written uniquely as a product $g=T(t)\L(s)$. Notice that $\bP$ is canonically embedded in $\bf G$ since $\bP$ is simply connected and its Lie algebra is a subalgebra of the Lie algebra of $PSL(2,\Re)$ that coincides with the Lie algebra of $\bf G$. It follows that $U_\r$ restricts to a representation of $\bP$ and we set $$ \b_g(x)=U_\r( g)xU_\r( g)^*=z_\r(g)U(g)xU(g)^*z_\r(g)^*,\quad x\in \A_\z,\,g\in \bP, $$ so that $\b$ is an action of $\bP$ by automorphisms of $\A_\z$, due to the fact that the cocycle $z_\r$ is a local operator. We consider now the semigroup $\bP_0$, the semidirect product of negative dilations with positive translations. $\bP_0$ is an amenable semigroup and we need an invariant mean $m$ constructed as follows: first we average (with an invariant mean) on positive translations and then over negative dilations. Observe that $f\to\int_{\bP_0}f(g)dm(g)$ gives an invariant mean on all $\bP$ vanishing on $f$ if, for any given $s\in\Re$, the map $t\to f(T(t)\L(s))$ has support in a left half line. Then we associate to $m$ the completely positive map $\Phi_\z$ of $\A_\z$ to $\B(\H)$ given by $$ \Phi_\z(x):=\int_{\bP_0} z_\r(g)^* xz_\r(g)dm(g), \quad x\in \A_\z. \eqno(2.9) $$ \beglemma{2.4} $\Phi_\z$ is a left inverse of $\r_\z$. Moreover $\Phi_\z$ is locally normal, i.e. has normal restriction to $\A(-\infty,\ell)$, $\ell\in\Re$, and $\bP$-invariant, namely $$ \Phi_\z=\a_g^{-1}\Phi_\z\b_g,\quad g\in\bP. $$ \endlemma \begProof Let $x$ belong to $\A(-\infty,\ell)$, $\ell\in\Re$. By formula $(2.2)$ $$ \Phi_\z(\r_\z(x))=\int_{\bP_0} \a_g(\r_\z(\a_{g^{-1}}(x)))dm(g) =x $$ because of the above property of $m$ since the integrand is constantly equal to $x$ on the set ${g\in\bP_0:g^{-1}(-\infty,a)\cap I_0=\emptyset}$. Then the localization of $\r_\z$ and Haag duality imply that the range of $\Phi_\z$ is contained in $\A_\z$. Setting $E=\r_\z\cdot\Phi_\z$ we have a conditional expectation of $\A_\z$ onto the range of $\r_\z$ that restricts to a conditional expectation $E_{(-\infty,\ell)}$ of $\A(-\infty,\ell)$ onto $\r(\A(-\infty,\ell))$ if $(-\infty,\ell)\supset I_0$. Since $\r_{(-\infty,\ell)}$ is assumed to have finite index, $E_{(-\infty,\ell)}$ is automatically normal [\rfr(Long3)]. Therefore $\Phi_\z|_{\A(-\infty,\ell)}= \r_{(-\infty,\ell)}^{-1}E_{(-\infty,\ell)}$ is normal for $\ell$ large enough, hence for any $\ell$. \np Concerning the $\bP$-invariance of $\Phi_\z$ we have, making use of the cocycle condition, $$ \eqalign{ \a_g^{-1}\Phi_\z\b_g(x) &=\a_g^{-1}( \int_{\bP_0} z_{\r}(h)^*\b_g(x)z_\r(h)dm(h))\cr &=\a_g^{-1}(\int_{\bP_0}z_\r(h)^*z_\r(g) \a_g(x)z_\r(g)^*z_\r(h)dm(h))\cr &=\int_{\bP_0} z_\r(hg^{-1})^*x z_\r(hg^{-1})dm(h)=\Phi_\z(x)\cr} $$ \endProof \begcorollary{2.5} $\f = \omega\Phi_\z$ is a locally normal $\b$-invariant state on $\A_\z$, where $\omega = (\ \cdot\ \Omega, ,\Omega)$. \endcorollary \begProof We have $\f\b_g=\omega\Phi_\z\b_g=\omega\a_g\Phi_\z=\omega\Phi_\z =\f$ and $\f$ is locally normal because both $\omega$ and $\Phi_\z$ are locally normal. \endProof Let $\{\p_\f,\x_\f,\H_\f\}$ be the GNS triple associated with the above state $\f$ and $V$ be the unitary representation of $\bP$ on $\H_\f$ given by $V_g x\x_\f=\b_g(x)\x_\f$ for $x\in\A_\z$. Notice that $V$ is strongly continuous because $\f$ is locally normal. We now need a variation of known results, see [\rfr(DHRo2),\rfr(BuFr1)]. \beglemma{2.6} If $\r_\z$ is irreducible then $$ \f(x) =\int_{\bP_0}\b_g(x)dm(g),\quad x\in\A_\z $$ \endlemma \begProof If $x\in\A(-\infty,\ell)$ and $y\in\A_\z$ is localized in a bounded interval, the commutator function $t\to[\b_{T(t)\L(s)}(x),\r(y)] =\b_{T(t)\L(s)}([x,\r(\a_{T(t)\L(s)}^{-1}(y)])$ vanishes on a right half line, hence $[\int_{\bP_0}\b_g(x),\r(y)dm(g)]=\int_{\bP_0}[\b_g(x),\r(y)]dm(g)=0$. Since $\r_\z$ is locally normal, $\int_{\bP_0}\b_{g}(x)dm(g)$ commutes with every $\r(\A(-\infty,\ell))$, thus with $\r_\z(\A_\z)$; since $\r_\z$ is irreducible it therefore a scalar equal to its vacuum expectation value: $$ \int_{\bP_0}\b_g(x)dm(g)= \int_{\bP_0}\omega(\b_g(x))dm(g) =\int_{\bP_0}\omega(z_g^*xz_g)dm(g)=\omega\Phi_\z(x)=\f(x), $$ due to the fact that $\omega$ is normal and $\a$-invariant. \endProof \begcorollary{2.7} If $\r_\z$ is irreducible, the one-parameter (translation) unitary group $V(T(t))$ has positive generator. \endcorollary \begProof If $f\in L^1(\Re)$ has Fourier transform $\hat f$ with support in $(-\infty,0)$, we have to show that $V_f:=\int_\Re f(t)V(T(t))dt=0$. Choose by Lemma B.4 a non zero $\psi\in\H$ such that ${\rm sp}_{U_\r}(\psi)+{\rm supp}\hat f\subset (-\infty,0)$, where ${\rm sp}_{U_\r}$ denotes the spectrum relative to $U_\r(T(\cdot))$. Setting $\b_f:=\int_{\Re}f(t)\b_{T(t)}dt$, for any $x\in\A_\z$ the vector $\b_g(\b_f(x))\psi=0$, for all $g\in \bP_0$, since it has negative spectrum relative to $U_\r(T(\cdot))$. By averaging over $\bP_0$ the vector $\b_g(\b_f(x^*)\b_f(x))\psi$, Lemma 2.6 implies $\|V_fx\x_\f\|^2=\f(\b_f(x)^*\b_f(x))=0$. \endProof \begcorollary{2.8} If $\r_\z$ is irreducible, $\f$ is faithful on $\cup\r(\A(-\infty,\ell))$. \endcorollary \begProof $\A_\z$ is a simple $C^*$-algebra since it is the inductive limit of type III factors (that are simple $C^*$-algebras). Therefore $\p_\f$ is one-to-one and the statement will follow if we show that $\x_\f$ if cyclic for $\B_\ell:=\r(\A(-\infty,\ell))',\,\ell>0$. To this end we may use a classical Reeh-Schlieder argument. If $\psi\in\H$ is orthogonal to $\B_\ell\x_\f$, and $\ell_0>\ell$, then for all $x\in\B_{\ell_0}$ we have $(x\x_\f,V(T(t))\psi)=0$ for $t$ in a neighborhood of $0$, thus for all $t\in\Re$ by positivity of the generator shown by Corollary 2.7. Hence, setting $\a_t\equiv\a_{T(t)}$ and $\b_t\equiv\b_{T(t)}$, $\psi$ is orthogonal to $(\cup_t \b_t(\B_{\ell_0}))\x_\f$, thus $\psi=0$ because $\cup_t \b_t(\B_{\ell_0})$ is irreducible since $$ \eqalign{ (\bigcup_t\b_t(\B_{\ell_0}))' &=\bigcap_t \b_t(\r(\A(-\infty,\ell_0))= \bigcap_t\r(\a_t(\A(-\infty,\ell_0)))\cr & = \r(\bigcap_t \a_t(\A(-\infty,\ell_0))) =\bigcap_\ell \A(-\infty,\ell)= \Co\cr} $$ by the local normality of $\r$. \endProof \begproposition{2.9} $(\r_I,\r_I)$ does not depend on the interval $I\supset I_0$. \endproposition \begProof We begin with the case in which $\r_\z$ is irreducible and assume for convenience that $\bar I_0\subset (-\infty,0)$. Notice then that $(\r_{(-\infty,0)},\r_{(-\infty,0)})$ is finite-dimensional and, by covariance, globally $\b_g$-invariant with $g$ in the subgroup of dilations because these transformations preserve $(-\infty,0)$. Therefore $(\r_{(-\infty,0)},\r_{(-\infty,0)})\x_\f$ is a finite-dimensional subspace of $\H_\f$ globally invariant for $V(\L(s))$, $s\in\Re$. By Proposition B.3 of the appendix B we thus have $V(T(t))x\x_\f=x\x_\f$ for every element $x\in (\r_{(-\infty,0)},\r_{(-\infty,0)})$, thus $\b_{T(t)}(x)=x$ because $\x_\f$ is separating. It follows that if $x\in(\r_{(-\infty,0)},\r_{(-\infty,0)})$ and $y\in \A(-\infty,0)$ $$ [x,\r(\a_g(y))]=\b_g([\b_g^{-1}(x),\r(y)])=\b_g([x,\r(y)])=0 $$ namely $$ x\in (\r_{(-\infty,0)},\r_{(-\infty,0)})\Rightarrow x\in (\r_\z,\r_\z)=\Co I $$ Since the converse implication is obvious by Haag duality we have the equality of the two intertwiner spaces. Now if $\r$ is any endomorphism with finite index, $(\r_\z,\r_\z)$ is finite-dimensional by the inclusion $(2.6)$, and $\r_\z$ decomposes into a direct sum of irreducible endomorphisms of $\A_\z$ which are covariant by Proposition~2.2, therefore the preceding analysis shows that also in this case $(\r_{(-\infty,0)},\r_{(-\infty,0)})=(\r_\z,\r_\z)$. Since $(\r_\z,\r_\z)$ is translation invariant, we get $(\r_{(-\infty,\ell)},\r_{(-\infty,\ell)})=(\r_\z,\r_\z)$ whenever $I_0\subset(-\infty,\ell)$ and, since $\z$ was arbitrary, we get the thesis. \endProof \begProofof{Theorem 2.3} The case $\s=\r$ follows immediately by Proposition 2.6: if $T\in (\r_I,\r_I)$ then $T$ also belongs to $(\r_{\tilde I},\r_{\tilde I})$ for any interval $\tilde I\supset I$ hence by additivity $T$ is a self-intertwiner of $\r$ on the whole algebra $\ua$. To handle the general case consider a direct sum endomorphism $\eta:=\r\oplus\s$ localized in $I$, then $$ {\rm dim}(\eta_I,\eta_I)={\rm dim}(\r_I,\r_I)+{\rm dim}(\s_I,\s_I) +2{\rm dim}(\r_I,\s_I) $$ while $$ {\rm dim}(\eta,\eta)_I={\rm dim}(\r,\r)_I+{\rm dim}(\s,\s)_I +2{\rm dim}(\r,\s)_I $$ therefore ${\rm dim}(\r_I,\s_I)={\rm dim}(\r,\s)_I$ and since we always have $(\r,\s)_I\subset (\r_I,\s_I)$ these two intertwiner spaces coincide. \endProof In particular we have proved the following. \begcorollary{2.10} Let $\r$ be an endomorphism of $\ua$ with finite index localized in $I_0$. The following are equivalent: \item{$(i)$} $\p_0\cdot\r$ is an irreducible representation of $\ua$ \item{$(ii)$} $\r(\A(I))'\cap\A(I)=\Co$ for some, hence for all, $I\supset I_0$ \item{$(iii)$} $\r_\z(\A_\z)'\cap \A_\z =\Co$ \item{$(iv)$} $\r_\z$ is an irreducible representation of $\A_\z$. \npindent Moreover any finite index representation $\pi$ of $\ua$ is the direct sum of irreducible representations. \endcorollary \titleb {2.3 The conjugate sector} Let $\r$ be an endomorphism of $\ua$ with finite index and localized in the interval $I_0$ as before. We shall say that the endomorphism $\bar\r$ is a conjugate of $\r$ if there exist isometries $V\in (id,\bar\r\r)$ and $\bar V\in (id,\r\bar\r)$ such that $$ \bar V^*\bar\r(V)={1 \over d},\quad V^*\r(\bar V)={1 \over d}\eqno(2.10) $$ where $d$ is a positive scalar. In this case one can in fact choose $V,\bar V$ so that $d$ is the square root of the minimal index of $\r$. Denote by $j_I$ the lifting to an anti-automorphism of $\ua$ of the adjoint action of the modular conjugation $J_I$ on the precosheaf $\A$. \begtheorem{2.11} Let $\r$ be a covariant endomorphism with finite index. There exists a conjugate endomorphism $\bar\r$, unique as superselection sector. $\bar\r$ is covariant with positive energy and is given by the formula $$ \bar\r=j\cdot\r\cdot j \eqno(2.11) $$ where $j=j_I$. If both $\r$ and $\bar\r$ are localized in the interval $I$, then there exist isometries $V\in (id,\bar\r\r)_I$ and $\bar V\in (id,\r\bar\r)_I$ such that the conjugate equations $(2.10)$ holds with $d=\sqrt{{\rm Ind}(\r)}$. \npindent If moreover $\r$ is irreducible, then $\bar\r$ is the unique irreducible endomorphism of $\ua$, up to inner automorphisms, such that $\r\bar\r$ contains the identity and in this case there exists a unique (up to a phase) isometry $V\in(id,\bar\r\r)_I$. \endtheorem \begProof As shown in [\rfr(GuLo1)], $\bar\r :=j_I\cdot\r\cdot j_I$ is an endomorphism of $\ua$ locally conjugate to $\r$, namely $\bar\r_{\tilde I}$ is a conjugate endomorphism of $\r_{\tilde I}$ according to [\rfr(Long3)], for any interval $\tilde I$ such that both $\r$ and $\bar\r$ are localized in $\tilde I$. Fixing such an interval $\tilde I$, since $\r_I$ has finite index, there exist isometries $V\in(id_{\tilde I},\bar \r_{\tilde I}\r_{\tilde I})$, $\bar V\in(id_{\tilde I},\r_{\tilde I}\bar \r_{\tilde I})$ such that $\bar V^*\bar\r(V)={1 \over d}$, $V^*\r(\bar V)={1 \over d}$, with $d=\sqrt{{\rm Ind}(\r_{\tilde I})}$ [\rfr(Long3)]. By Theorem 2.3 $V$ and $\bar V$ are global intertwiners, namely $\bar \r$ is a global conjugate. The uniqueness of $\bar\r$, the characterization of $\bar\r$ in the irreducible case and the uniqueness of $V$ follow again by the corresponding statements for sectors of factors [\rfr(Long3)] because of Theorem 2.3. The covariance of $\bar \r$ follows by the formula $\bar\r=j\cdot\r\cdot j$, see [\rfr(GuLo1)]. \endProof \begcorollary{2.12} If $\r$ is a an endomorphism of $\ua$ with finite index, there exists a (global) faithful left inverse $\Phi$ of $\r$ which is given by the formula $$ \Phi = V^*\bar\r(\cdot)V \eqno(2.12) $$ where $V\in(id,\bar\r\r)$ verifies the conjugate equations $(2.10)$ and all faithful left inverses have this form. If $\r$ is localized in $I$, also $\Phi$ is localized in $I$ and $\Phi|_{\A({\tilde I})}$ is normal if $\tilde I\supset I$. \npindent If $V$ $\bar V$ are chosen so that the constant $d$ in $(2.10)$ is equal to $\sqrt{{\rm Ind}(\r_I)}$, then $\Phi$ is uniquely determined. In particular if $\r$ is irreducible then $\Phi$ is the unique left inverse of $\r$. \endcorollary \begProof Only the uniqueness of $\Phi$ needs still to be proved. We assume that $\r$ is localized in $I$ and $V\in\A(I)$. By the same argument as in Corollary 5.7 of [\rfr(Long2)], essentially the push-down lemma in [\rfr(PiPo1)], every element $x\in\ua$ can be written as $$ x={\rm Ind}(\r)\r\Phi(x\bar V^*)\bar V.\eqno(2.13) $$ If $\Psi$ is a left inverse of $\r$ and satisfies the conjugate equations with $d=\sqrt{{\rm Ind}(\r)}$, then $\Psi$ and $\Phi$ have the same restriction to $\A(I)$ because the corresponding statement is true for endomorphisms of factors [\rfr(Long3)] and, by Corollary~2.10, $\Psi(\bar V)=\Phi(\bar V)$. Thus, by formula $(2.13)$, $$ \Psi(x)={\rm Ind}(\r)\Phi(x\bar V^*)\Psi(\bar V)= {\rm Ind}(\r)\Phi(x\bar V^*)\Phi(\bar V)=\Phi(x). $$ \endProof If $\r$ is a finite index endomorphism of $\ua$, we define $\l_\r=\Phi(\varepsilon)$ where $\Phi$ is the unique ``minimal" left inverse provided by Corollary 2.12. As shown in [\rfr(Long2)], $\Phi$ is a standard left inverse in the sense of [\rfr(DHRo2)], namely $\l_\r$ is a positive scalar multiple of a unitary $\k_\r\in(\r,\r)_I$ and the statistical dimension is then defined by $d(\r)=\|\l_\r\|^{-1}$. By the index-statistics theorem (see Corollary 3.7) if $\r$ has finite index, then also $d(\r)$ is finite. \begcorollary{2.13} If $\r$ is irreducible with finite index, the statistics parameter $\l_\r$ in formula $(2.8)$ is a non-zero scalar. \endcorollary \begProof $\l_\r=\Phi(\varepsilon)=\Phi_\z(\varepsilon)$ belongs to $(\r_I,\r_I)$ thus is a scalar by Corollary 2.10. $\l_\r$ does not vanish as mentioned above. \endProof \titleb{2.4 Equivalence between finite index and finite statistics.} If a covariant, positive energy superselection sector $\r$ has finite index, then also the statistical dimension is finite. In fact Corollary 3.7 will relate the two quantities in the general reducible case. For completeness, in this subsection we will outline an argument showing a converse of this assertion. We shall say that a localized endomorphism $\r$ of $\ua$ has {\it finite statistics} if there exists a left inverse $\Phi$ of $\r$ such that the statistical parameter $\l_\r:=\Phi(\varepsilon)$ is an invertible operator; even in the irreducible case we do not know a priori that $\l_\r$ is a scalar since Corollary 2.10 has not been proved. In the following proposition $\r$ is a covariant endomorphism of $\ua$, but positive energy is not assumed. \begproposition{2.14} If $\r$ is covariant with finite statistics, then $\r$ has finite index and positive energy. \endproposition \begProof Let $\r$ be localized in $I_0$, $\z\in I'_0$ and $\Phi_\z:=\Phi|_{\A_\z}$. Because of finite statistics the DHR inequality holds: $$ \|\Phi_\z(x)\|\geq c\|x\|,\quad x\in\A_\z^+\eqno(2.14) $$ where $c=\|\l_\r^{-1}\|^{-2}>0$, by reasoning as in [\rfr(DHRo2)]. Indeed if $x=x^*\in\A(-\infty,\ell)$ with $I_0$ $\subset$ $(-\infty,\ell)$ and $u$ is a unitary such that $u\r(\cdot)u^*$ is localized in $(\ell,\infty)$, so that $\r(x)=u^*xu$ and $\varepsilon$ $=$ $u^*\r(u)$, we have $\Phi(u^*x)$ $=$ $\Phi(\r(x)u^*)$ $=$ $\Phi(\r(x)\varepsilon\r(u^*))$ $=$ $x\l_\r u^*$ and therefore $\|\Phi(x^2)\|$ $\geq$ $\|\Phi(xu)\Phi(u^*x)\|$ $=$ $\|\l_\r^*x^2\l_\r\|$ $=$ $\|x\l_\r^*\l_\r x\|$ $=$ $\geq c\|x^2\|$. As $\r_\z$ is isometric, the inequality (2.14) is clearly equivalent to the Pimsner-Popa inequality [\rfr(PiPo1)] $$ \|E(x)\|\geq c\|x\|,\quad x\in\A_\z^+, $$ with $E=\r_\z\cdot\Phi_\z$ the associated conditional expectation onto the range of $\r_\z$, and it is also equivalent to $$ E(x)\geq cx,\quad x\in\A_\z^+,\eqno(2.15) $$ (see [\rfr(Long2)] for the version of these inequalities on infinite factors). In particular $E|_{\A(I)}$ is normal and $\r_I$ has finite index $I\supset I_0$. We can now replace $\Phi_\z$ by its average $\Phi'_\z$ over $\bP$ with respect to an invariant mean, e.g. the $m$ in the previous section, $\Phi'_\z:=\int_{{\bP}_0}\a_g^{-1}\Phi_\z\b_gdm(g)$. Since $\r$ is locally normal $\r\Phi'_\z$ still satisfies the inequality $(2.15)$ and hence $\Phi'_\z$ the inequality $(2.14)$. At this point the state $\varphi=\omega\Phi'_\z$ in Corollary 2.5 is again locally normal and faithful, thus Proposition 2.9 applies and provides the global conjugate in Theorem 2.11. The usual additivity of the spectrum argument then shows that $\r$ is a positive energy representation. \endProof \titlea {3. The conformal spin-statistics theorem} \titleb {3.1 A first relation between spin and statistics} In this subsection we prove a first relation between spin and statistics. We shall not use the full conformal invariance, but only the covariance with respect to the rotation subgroup and the geometric interpretation of the modular conjugations. In the following $I_1$ and $I_2$ always denote the upper semicircle $\{e^{i\theta},\th\in (0,\pi)\}$ and the right semicircle $\{e^{i\th},\th\in (-{\pi\over 2},{\pi\over 2})\}$ respectively and $\r$ is an irreducible covariant with positive energy endomorphism of $\ua$ with finite index localized in an interval whose closure is contained in $I_1\cap I_2$. Then $\r_{I_i}=\r |_{\A(I_i)}$ is an irreducible finite-index endomorphism of $\A(I_i)$ and we denote by $V_i$ the standard implementation of $\r_{I_i}$, $i=1,2$, with respect to the vacuum vector, see Appendix A. We also shorten the notations: $J_i$ stands for the modular conjugation $J_{I_i}$, $\ov\r_i$ for the conjugate $j_i\r j_i$ of $\r$ where $j_i$ is the promotion to an anti-automorphism of $\ua$ of the precosheaf anti-automorphism $J_i \cdot J_i$. The symbol $\ad U$ denotes the automorphism of $\ua$ corresponding to a unitary $U$ (e.g. $\ad U_\r(g):=\ad z_\r(g)\cdot\alpha_g$). \beglemma{3.1} We have $V_1\in\A(I_2)$, $V_2\in\A(I_1)$ and $V_i$ is the unique isometry (up to a phase) with this localization support that intertwines the identity and $\r\ov\r_i$, $i=1,2$. \endlemma \begProof By the geometric meaning of $J_1$, both $\r$ and $\ov\r_1$ are localized in $I_2$, thus by Theorem 2.11 we can take an isometry $v\in (id,\ov\r_1\r)_{I_2}$, in fact $v$ belongs to $\A(I)$ if $I$ is any subinterval of $I_2$ that contains both the localization support of $\r$ and of $\ov\r_1$. Since $\r_{I_2}$ is irreducible, $v$ is uniquely determined (up to a phase) by such properties. Therefore we may choose $v$ so that $j_1(v)=v$. By additivity $v$ implements $\r_{I_1}$ and since it also commutes with $J_1$ we have $V_1=\pm v$ by Lemma A.3. The argument for $V_2$ is similar. \endProof Since $\r,\bar\r_1,\bar\r_2$ are localized in disjoint intervals, they pairwise commute, thus $V_1V_2$ and $V_2V_1$ both belong to $(id,\r^2\ov\r_1\ov\r_2)_{I_1\cup I_2}$ hence $$ \mu_\r=V_1^*V_2^*V_1V_2 $$ is a scalar. It is an invariant for $\r$ that, by construction, reflects algebraic, analytical and geometric aspects. By looking at $\mu_\r$ from these different point of view we shall identify it, with different arguments, with the statistics parameter and with the univalence of $\r$ times $d(\r)^{-1}$, proving the conformal spin-statistics theorem. \beglemma{3.2} The following identities between endomorphisms of $\ua$ hold: \item{$(a)$} $\r\ov\r_1=\ad U_\r(\pi)\r\ov\r_2\ad U(\pi),$ \item{$(b)$} $\r\ov\r_1j_2\r\ov\r_1j_2=\r\ov\r_2j_1\r\ov\r_2j_1.$ \endlemma \begProof By formula $(1.2)$ we have $J_1J_2=U(\pi)$, hence $j_1j_2=j_2j_1=\ad U(\pi)$, therefore $$ \ov\r_1=\ad U(\pi)\ov\r_2\ad U(\pi). $$ Thus by covariance $$ \r\ov\r_1=\r\ad U(\pi)\ov\r_2\ad U(\pi)= \ad U_\r(\pi)\r\ov\r_2\ad U(\pi) $$ and, since $\ov\r_1$ and $\ov\r_2$ are localized in disjoint intervals and thus commute, $$ \r\ov\r_1j_2\r\ov\r_1j_2=\r\ov\r_1\ov\r_2\ad U(\pi)\r\ad U(\pi) =\r\ov\r_2\ov\r_1\ad U(\pi)\r\ad U(\pi)=\r\ov\r_2j_1\r\ov\r_2j_1 $$ \endProof \beglemma{3.3} We have $$ U_\r(\pi)V_2U(\pi)=c_\r V_1\eqno(3.2) $$ where $c_\r$ is a complex number of modulus one. \endlemma \begProof By Lemma 3.1 $V_1$ is the unique isometry (up to a phase) in $(id,\r\ov\r_1)_{I_2}$. By Lemma 3.2 $(a)$, also $U_\r(\pi)V_2U(\pi)$ belongs to $(id,\r\ov\r_1)$. Moreover, if $x\in \A(I'_2)$, then ad$U(\pi)(x)\in\A(I_2)$ hence $$ {\ad}U_\r(\pi)V_2U(\pi)(x)={\ad}U_\r(\pi)\r{\ad}U(\pi)(x) ={\ad}U_\r(2\p)\r(x)=\r(x)=x $$ showing that $U_\r(\pi)V_2U(\pi)$ belongs to $\A(I_2)$ too, thus it coincides with $V_1$ up to a phase. \endProof \beglemma{3.4} $\b_\r:=(V_1J_2V_1J_2)^*V_2J_1V_2J_1 $ belongs to $(0,1]$. \endlemma \begProof According to Lemma~3.2 $ (ii)$, $V_1J_2V_1J_2$ and $V_2J_1V_2J_1$ are both isometries in $(id, \r\ov\r_1j_2\r\ov\r_1j_2)$ and both belongs to the same local von Neumann algebra $\A(I)$ where $\r\ov\r_1j_2\r\ov\r_1j_2$ is localized in $I\in\I$ therefore $\b_\r$ is a complex scalar. Setting $e_i:=V_iV_i^*$ we deduce that $$ \b_\r V_1J_2V_1J_2=e_1J_2e_1J_2\ V_2J_1V_2J_1.\eqno(3.3) $$ Since $V_i$ is the standard implementation of $\r_{I_i}$, $V_i$ preserves the positive cone ${\cal P^{\natural}}(\A(I_i),\Q)$. Moreover $J_1$ preserves ${\cal P^{\natural}}(\A(I_2),\Q)$ because it implements an antiautomorphism of $\A(I_2)$ and fixes $\Omega$, and $J_2$ preserves ${\cal P^{\natural}}(\A(I_1),\Q)$ analogously. By the definition of the natural positive cones and the relations $V_1,e_1\in\A(I_2)$, $V_2\in\A(I_1)$, we have that $V_2J_1V_2J_1\Q$ and $V_1J_2V_1J_2\Q$ belong to ${\cal P^{\natural}}(\A(I_1),\Q)\cap {\cal P^{\natural}}(\A(I_2),\Q)$ and $e_1J_2e_1J_2\Q\in{\cal P^{\natural}}(\A(I_2),\Q)$. Since the scalar product of non-zero vectors in a natural cone is non-negative, and furthermore positive if one of the vectors is cyclic (equivalently separating), and since $(e_2J_1e_2J_1\Q,\Q)$ $=\|\D_{I_1}^{1\over 4}e_2\Q\|^2\neq 0$ we have $$(V_1J_2V_1J_2\Q,\Q)>0,\quad (e_2J_1e_2J_1V_2J_1V_2J_1\Q,\Q)=(V_2J_1V_2J_1\Q,e_2J_1e_2J_1\Q)>0$$ that entails $\b_\r>0$ by comparing with (3.3), provided we show that $V_2J_1V_2J_1\Q$ is separating for $\A(I_1)$. But this is true because if $x\in\A(I_1)$ and $xV_2J_1V_2J_1\Q=0$ then $$(j_1\Phi j_1\Phi(x^*x)\Q,\Q)= (J_1V^*_2J_1V^*_2x^*xV_2J_1V_2J_1\Q,\Q)=0$$ and this implies $x=0$ because the left inverse $\Phi$ of $\r$ is faithful. The rest is clear since by definition $||\b_\r||\leq 1$. \endProof \beglemma{3.5}$ \l_\r=V_1^*V_2^*V_1V_2\ . $ \endlemma \begProof As in [\rfr(DHRo1)] we get $\l_\r=\r(V_1^*)V_1;$ indeed if $\r'$ is localized in $I_1\cap I'_2$ and $u$ is a unitary in $(\r,\r')_{I_1}$, then $\ad u^*|_{\A(I_2)}=\r_{I_2}$, thus $\r(V_1^*)V_1 = u^*V_1^*uV_1 = u^*\Phi(u) = \Phi (\varepsilon_\r) = \l_\r$. Since $V_1\in\A(I_2)$ and $V_2$ implements $\r$ on $\A(I_2)$ we thus have $$ V_1^*V_2^*V_1V_2=V_1^*\Phi(V_1)=\Phi(\r(V_1^*)V_1)=\Phi(\l_\r)= \l_\r.\eqno(3.5) $$\endProof \begproposition{3.6} The following relations hold: $$ \eqalignno{ \b_\r&=d(\r)^{-1} &(3.6)\cr s_\r&=c_\r^2 \k_\r&(3.7)\cr} $$ where $\k_\r$ is the phase of the statistical parameter. \endproposition \begProof Taking adjoints in (3.2), we have $U(\pi)V^*_1U_\r(\pi)=c_\r V_2^*$, and multiplying side by side this expression with formula (3.2) we have $$c_\r^2 V_2^*V_1= s_\r U(\pi)V_1^*V_2U(\pi)\eqno(3.8) $$ because $s_\r:=U_\r(2\pi)$. Since $J_i$ commutes with $V_i$ and $J_1J_2=J_2J_1=U(\pi)$ we have $$ \b_\r=J_1\b_\r J_1=J_1(V_1^*J_2V_1^*J_2V_2J_1V_2J_1)J_1 =V_1^*U(\pi)V_1^*V_2U(\pi)V_2\eqno(3.9) $$ therefore, by inserting formula (3.8) in the expression for $\l_\r$ given by Lemma~3.5 and comparing with (3.9) we obtain $$ \l_\r=V_1^*V_2^*V_1V_2=c_\r^{-2} s_\r V_1^*U(\pi)V_1^*V_2U(\pi)V_2=c_\r^{-2} s_\r\b_\r\eqno(3.10) $$ and the thesis easily follows. \endProof \begcorollary{3.7} (Index-statistics theorem) For every covariant endomorphism $\r$ of $\ua$ we have ${\rm Ind}(\r)=d(\r)^2.$ \endcorollary \begProof If $\r$ is irreducible we have $\bar V_1^*\r_{I_2}(V_1)$ $=$ ${1\over d}$ with $d=\sqrt{{\rm Ind}(\r)}$ by Corollary 2.12 and comparing with formula $(3.5)$ we have the thesis since $\bar V_1$ and $ V_1$ are equal up to a phase. The general case follows by additivity of both the statistical dimension and the square root of the minimal index (or by a direct argument). The case of infinite index is treated in subsection 2.4.\endProof \titleb{3.2 The spin-statistics theorem.} We prove now that $c^2_\r =1$, completing our result. In this step the role of the conformal invariance is to fix uniquely the representation of the rotation group $U_\r(\th)$, otherwise defined up to a one-dimensional representation, as the restriction of the unique representation of $\bG$. We could nevertheless fix $U_\r(\th)$ by using the positivity of the conformal Hamiltonian. It is convenient to extend the definition of $c_\r$ to the case of a reducible finite index $\r$. To this end notice that, as in the proof Lemma 3.3, both $U_\r(\pi)V_2U(\pi)$ and $c_\r V_1$ belong to $(id,\r\bar\r_1)_{I_2}$, thus there exists $c_\r \in(\r\bar\r_1,\r\bar\r_1)_{I_2}$ such that formula (3.2) holds. Replacing $c_\r$ by its push-down if necessary, we may further assume that $c_\r \in(\r,\r)$ and this condition define it uniquely, see [\rfr(LoRo1)]. In the following $\r$, $\s$ are finite index endomorphisms of $\ua$. \beglemma{3.8} Let $\r$ and $\s$ be localized in $I_1\cap I_2$, with $\r$ an irreducible subsector of $\s$ and $p_\r\in\A(I_1\cap I_2)$ is the minimal idempotent in $(\s,\s)_{I_1\cap I_2}$ corresponding to $\r$, then $c_\s p_\r=c_\r p_\r$. In particular, if $c_\s$ is a scalar, then $c_\r=c_\s.$ \endlemma \begProof With $w\in (\r,\s)$ an isometry in $\A(I_1\cap I_2)$, we have by Lemma A.4 of the appendix $$ \sqrt{d(\r)}V_i^\r=\sqrt{d(\s)}w^*Jw^*JV_i^\s.\eqno(3.11) $$ The projection $p_\r=w^*w\in\A(I_1\cap I_2)$ commutes with the range of $\s$, hence it commutes with $U_\s$ (see the proof of Proposition 2.2), therefore $$(w^*U_\s(g)w)(w^*U_\s(h)w)=w^*U_\s(gh)w,\quad g,h\in{\bf G},\eqno(3.12)$$ namely $g\to w^*U_\s(g)w$ is a unitary representation. Since for every $x\in\ua$ we have $$\eqalign{ (w^*U_\s(g)w)\r(x)(w^*U_\s(g)^*w) &=w^*U_\s(g)\s(x)U_\r(g)^*w\cr &=w^*\s(U(g)xU(g)^*)w=\r(U(g)xU(g)^*),\cr} $$ we get by the uniqueness of the representation in Proposition 2.2 $$U_\r(g)=w^*U_\s(g)w.\eqno(3.13)$$ Since $c_\s$ leaves in a finite-dimensional algebra, we may assume that $p_\r$ is an eigen-projection of $c_\s$ namely $c_\s p_\r =\ell p_\r$ with $\ell\in\Co$. Making substitutions in the formula $(3.2)$ according to the equations $(3.12)$, $(3.13)$, we then get $$\eqalign{\sqrt{d(\r)}c_\r V^\r_1&= \sqrt{d(\s)} w^*U_\s(\p)ww^*J_2w^*J_2V_2^\s U(\p)\cr &=\sqrt{d(\s)}w^*U_\s(\p)J_2w^*J_2V_2^\s U(\p)\cr &= \sqrt{d(\s)}w^* U_\s(\p)J_2w^*J_2U_\s(-\p)c_\s V_1^\s \cr &= \sqrt{d(\s)} w^*U(\p)z_\s(\p)J_2w^*J_2z^*_\s(\pi)U(\p)c_\s V_1^\s\cr &= \sqrt{d(\s)} w^*U(\p)J_2w^*J_2U(\p)c_\s V_1^\s\cr &=\sqrt {d(\s)} w^*J_1w^*J_1 c_\s V_1^\s\cr &=\sqrt{d(\s)} w^* c_\s J_1w^*J_1V_1^\s =\sqrt{d(\r)} \ell V^\r_1 \cr} $$ where we have used that $[J_2w^*J_2,z_\s(\p)]=0$ due to the localization in disjoint intervals of $w^*$ and $z_\s((\pi))$ and again of the identity $J_1J_2=U(\pi)$, and this concludes the proof. \np\endProof Our choice of the intervals $I_1$ and $I_2$ is, of course, conventional. If we replace them by their rotates $R(\th)I_1$, $R(\th)I_2$, we would get a priori another invariant $c_\r(\th)$ for a $\r$ localized in their intersection. But this is soon seen to be equal to $c_{\r_\th}$, the old invariant for $\r_\th := {\rm ad}U(-\th)\r{\rm ad}U(\th)={\rm ad}z_\r(-\th)\r$ (because $U(\th)$ establishes an isomorphism between the old and the rotated structures). Next lemma implies that $c_{\r_\th}=c_\r$ if also $\r_\th$ is localized in $I_1\cap I_2$. \beglemma{3.9} $c_\r$ depends only on the superselection class of $ \r$ and not on its representative $\r$ nor on the choice of $I_1$ and $I_2$ as above. \endlemma \begProof If $\r$ is localized in $I_1\cap I_2$ and $\s={\rm ad}W^*\cdot\r$ for some unitary $W\in\A(I_1\cap I_2)$ then $V_i^\r=W^*JW^*JV_i^\s$ and by a computation similar to the one in the Lemma 3.7 we see that $c_\s=c_\r$. By the comment preceding the this lemma it thus follows that $c_\r$ remains unchanged if we rotate the $I_i$'s provided $\r$ stays localized in the intersection of the intervals. Thus, in finitely many steps, replacing $\r$ by an equivalent endomorphism and making small rotations of the intervals, we see that $c_\r$ does not vary in its superselection class. \endProof \beglemma{3.10} $c_\r=c_{\bar\r}$. \endlemma \begProof By Lemma 3.9 we may choose $\bar\r=\bar\r_1=j_1\r j_1$. Thus $\bar\r$ is localized in $I_1'\cap I_2$ and $c_{\bar\r}$ is definable with respects to the intervals $I_2=R(-{\pi\over 2})I_1$,$\,I_1'=R(-{\pi\over 2})I_2$. The standard implementations of $\bar\r$ relative to these intervals are respectively given by $J_1V_2^\r J_1$ and $J_1V_1^\r J_1=V_1^\r$, moreover $U_{\bar\r}(\th)=J_1U_\r(-\th)J_1$, see [\rfr(GuLo1)]. Inserting these identities in the defining expression $(3.2)$ for $c_{\bar\r}$ we thus have $J_1U_\r(-\pi)J_1V_1^\r U(\pi)=c_{\bar\r}J_1V_2^\r J_1$ and after cancellations this gives the stated equality. \endProof \beglemma{3.11} Let $\r$, $\s$ be irreducible and localized in $I_1\cap I_2$. Then $c_{\r\s}=c_\r c_\s$. \endlemma \begProof By the cocycle equation $z_{\r\s}(g) =z_{\r}(g)\r(z_{\s}(g))$ and the multiplicativity of the standard implementations $V_i^{\r\s}=V_i^{\r}V_i^{\s}$ the equation $(3.2)$ for $\r\s$ gives $$\eqalign{ c_\r c_\s U(\p)V_1^\r V_1^\s U(\p) &=z_\r(\p)V_2^\r z_\s(\p) V_2^\s =z_\r(\p)\r(z_\s(\pi))V_2^\r V_2^\s\cr &=z_{\r\s}(\p)V_2^{\r\s} =c_{\r\s}U(\p)V_1^{\r\s}U(\p)\cr}\eqno(3.14)$$ where we used that $z_\s(\p)\in\A(I_2\cup I_1')$ and that $V_2^\r$ implements $\r$ on $\A(I_2\cup I_1')$. Since $V_1^{\r\s}=V_1^\r V_1^\s$ we have the thesis. \endProof \begcorollary{3.12} $c_\r ^2=1$. \endcorollary \begProof If $\r$ is irreducible, then by the Lemmas 3.8 and 3.9 we have $c_\r^2=c_\r c_{\bar \r}= c_{\r{\bar \r}}= 1$. The general case follows by Lemma 3.8. \endProof Now the spin and statistics relation immediately follows immediately by Proposition 3.6. \begtheorem{3.13} (Spin and Statistics) Let $\r$ be a superselection sector with finite statistics. Then $ \k_\r=s_\r.$ \endtheorem \titlea{Appendix A. Standard implementation of left inverses} We will deal here with the the notion of standard implementation (see e.g. [\rfr(BrRo1)]) in the endomorphism case. Let $M$ be a von~Neumann algebra on a Hilbert space $\H$ and $\r$ a unital injective endomorphism of $M$. The left inverses $\Phi$ of $\r$ correspond bijectively to the conditional expectations $E$ of $M$ onto $\r(M)$: $$ \eqalign{ \Phi&\to E=\r\cdot\Phi\cr E&\to\Phi=\r^{-1}\cdot E\,.\cr} \eqno(A.1) $$ We shall say that an isometry $V\in\B(\H)$ {\it implements the left inverse} $\Phi$ if $$ V^*xV=\Phi(x)\,,\qquad x\in M\eqno(A.2) $$ \beglemma{A.1} Let the isometry $V$ implement $\Phi$. Then \item{$(a)$} $Vx=\r(x)V$, \quad $x\in M,$ \item{$(b)$} $exe=E(x)e$, \quad $x\in M,$ \np where $e=VV^*$ and $E=\r\Phi$. Conversely if $(a)$ and $(b)$ hold then $V$ implements $\Phi$. \endlemma \begProof If we set $x\equiv\r(y)$ in $(A.2)$ we have $V^*\r(y)V=y$ for all $y\in M$ hence $$ e\r(y)V=Vy.\eqno(A.3) $$ In particular, if $y$ is unitary, $\|e\r(y)V\x\|$ $=$ $\|Vy\x\|$ $=$ $\|\x\|$ $=$ $\|\r(y)V\x\|$, $\x\in\H$, showing that $e\r(y)V\x=\r(y)V\x$, hence $e\r(y)e=\r(y)e$, so we have $$ e\r(y)=(\r(y^*)e)^*=(e\r(y^*)e)^*=e\r(y)e=\r(y)e $$ which implies $e\in\r(M)'$ because $M$ is generated by its unitaries. Formula $(A.3)$ then entails $(a)$. To check $(b)$ notice that $$ exe=VV^*xVV^*=V\Phi(x)V^*= =\r(\Phi(x))VV^*=E(x)e. $$ Conversely, assuming $(a)$ and $(b)$, we have $$ V^*xV=V^*exeV=V^*E(x)V=V^*\r(\Phi(x))V=\Phi(x),\quad x\in M\ . $$ \endProof We shall say that an isometry $V$ {\it implements the endomorphism $\r$} and that the projection $e$ {\it implements the conditional expectation} $E$ if the equations $(a)$ and $(b)$ of Lemma A.1 are respectively satisfied. We now fix a unit cyclic and separating vector $\Q\in\H$ for $M$ and its corresponding natural cone ${\cal P}^{\natural}(M,\Q)$. If $\Phi$ is a normal left inverse of $\r$ let us consider the state $$ \f=\w\cdot\Phi $$ where $\w=(\cdot\Q,\Q)$ and the corresponding vector $\x\in {\cal P}^{\natural}(M,\Q)$ such that $\f=(\cdot\x,\x)$. Let $e:=[\r(M)\x]\in\r(M)'$ and let $V_\Phi$ be the isometry of $\H$ with final projection $e$ such that $V_\Phi :\H\to e\H$ is the Araki-Connes-Haagerup standard implementation of $\r$ as an isomorphism of $M$ with $\r(M)$ with respect to the positive cones ${\cal P}^{\natural}(M,\Q)$ and ${\cal P}^{\natural}(\r(M),\x)$. Then $V_\Phi$ is given by $$ V_{\Phi}x\Q=\r(x)\x\,,\qquad x\in M $$ We check that $V_\Phi$ implements $\Phi$. To this end note first that $E=\r\Phi$ is $\f$-invariant since $$ \f\cdot E=\w\cdot\Phi\cdot\r\cdot\Phi=\w\cdot\Phi=\f $$ Then $$ \eqalign{ (x\r(b)\x,\r(a)\x)&=\f(\r(a^*)x\r(b))=\f\cdot E(\r(a^*)x\r(b))=\cr &=\f(\r(a^*)E(x)\r(b))=(E(x)\r(b)\x,\r(a)\x)\quad a,b,x\in M\cr} $$ i.e. $eE(x)e=exe$, $x\in M$, but $e\in\r(M)'$, hence $e$ implements $E$; in particular, if $\Phi$ is faithful, $e$ is the Takesaki projection for $E$. Moreover $V_\Phi$ implements $\Phi$ because $$ \r(x)V_\Phi y\Q =\r(x)\r(y)\x =\r(xy)\x=V_\Phi xy\Q\, \quad x,y\in M. $$ The isometry $V_\Phi$ will be called the {\it standard implementation} of $\Phi$ with respect to $\Q$. In case $\r$ has finite index, namely $\r(M)$ is a finite index subfactor of $M$, and $\Phi = \Phi_{\rm min}$, the minimal left inverse of $\r$, we shall denote $V_{\Phi_{\rm min}}$ by $V_\r$ and call it the standard implementation of $\r$ with respect to $\Q$. We collect here some properties of the standard implementations. \begproposition{A.2} \item {$(a)$} $V_\Phi$ is the unique isometry that implements $\Phi$ and sends ${\cal P}^{\natural}(M,\Q)$ into itself. In particular $V_\Phi$ depends on ${\cal P}^{\natural}(M,\Q)$ but not on the particular vector $\Q$. \item {$(b)$} $V_\Phi$ is the unique isometry that implements $\Phi$ and verifies $V_\Phi\Q\in {\cal P}^{\natural}(M,\Q)$. \item {$(c)$} $V_{\Phi_1\Phi_2}=V_{\Phi_1}V_{\Phi_2}$, with $\Phi_1$, $\Phi_2$ normal left inverses of $\r_1$, $\r_2$. In particular, if $\r_1$, $\r_2$ have finite index, $V_{\r_1\r_2}$ $=$ $V_{\r_1}V_{\r_2}$. \item {$(d)$} $JV_\Phi J=V_\Phi$, where $J$ is the modular conjugation of $(M,\Q)$. \endproposition \begProof By construction $V_\Phi$ implements $\Phi$ and maps ${\cal P}^{\natural}(M,\Q)$ into itself, in particular $V_\Phi\Q\in {\cal P}^{\natural}(M,\Q)$. Now suppose that an isometry $V$ implements $\Phi$ and $V\Q\in {\cal P}^{\natural}(M,\Q)$. Then $$ (\ \cdot\ V\Q,V\Q)=(V^*\cdot V\Q,\Q)=\w\cdot\Phi=\f $$ thus $V\Q$ is the unique vector $\x\in{\cal P}^{\natural}(M,\Q)$ associated with $\f$ and $Vx\Q=\r(x)V\Q=\r(x)\x$, namely $V=V_\Phi$ . This proves $(a)$ and $(b)$. \np $(c)$ is consequences of $(a)$ and of the multiplicativity of the minimal index [\rfr(Long4)]. \np $(d)$ $J$ restricted to the range of $e=V_\Phi V_\Phi^*$ coincides with the modular conjugation of $M_e$ because $\varphi$ preserves the conditional expectation $E$, thus $V_\Phi^*JV_\Phi =J$ because $V_\Phi$ is the standard implementation of $\r$ as an isomorphism of $M$ with $\r(M)$. \endProof \beglemma{A.3} Let $M$ be a factor and $\r$ a finite index endomorphism. If $W$ is an isometry that implements $\r$ and commutes with $J$, then $W$ implements a left inverse $\Phi$ of $\r$ and $W=mV_\r$ for some $m\in (\r,\r)$, which is invertible iff $\Phi$ is faithful. In particular, if $\r$ is irreducible, then $W=\pm V_\r$ \endlemma \begProof The partial isometry $Z=WV^*_\r$ commutes with $J$ and belongs to $\r(M)'$, thus $Z\in N'\cap M_1$, where we set $N=\r(M)$ and $M_1=JN'J$ denotes the Jones basic extension of $N\subset M$. Clearly we have $W = ZV_\r$. Let $m$ be the Pimsner-Popa push-down of $Z$, namely the unique element $m\in M$ such that $mV_\r = ZV_\r$. We have $m={\rm Ind}(\r)E(Ze)$ with $E=\r\Phi_{\rm min}$, thus $m\in \r(M)'\cap M$ and $W=mV_\r$ showing in particular that $W$ implements a left inverse $\Phi$ of $\r$. Clearly $\Phi$ is faithful if $m$ is invertible. Conversely, if $\Phi$ is faithful, $p\in(\r,\r)$ is a projection and $pm=0$ then $\Phi(p)=V_\r^*m^*pmV_\r =0$, thus $p=0$ so $m$ is invertible. If moreover $\r$ is irreducible, then $m\in \Co$, thus $m=\pm 1$ because both $W$ and $V_\r$ are isometries commuting with $J$. \endProof Recall now that the {\it dimension} $d(\r)$ of $\r$ is defined as the square root of the minimal index of $\r$. \begproposition{A.4} Let $\s$ be a finite index endomorphism of the factor $M$ and $\r$ an irreducible subsector of $\s$. If $w$ is an isometry in $(\r,\s)$, then $$ V_\r={\sqrt{{d(\s)}\over {d(\r)}}}w^*Jw^*JV_\s. $$ If $\s=\oplus_{i=1}^N n_i\r_i$ is an irreducible decomposition of $\s$ and for each $i$ $\{w^{(i)}_k,\ k=1,\dots n_i\}$ is an orthonormal basis of isometries in $(\r_i,\s)$, then $$ V_\s=\sum_{i=1}^N\sum_{k=1}^{n_i}{\sqrt{{d(\r_i)}\over {d(\s)}}} w^{(i)}_kJw^{(i)}_kJV_{\r_i}.\eqno (A.4) $$ \endproposition \begProof We prove the second assertion that implies the first one. Set $W$ equal to the right hand side in $(A.4)$. The ranges of the $w^{(i)}_k$'s are pairwise orthogonal and the coefficients verifies $\sum_{i=1}^Nn_i{d(\r_i)\over d(\s)}=1$, thus $W$ is an isometry and a direct verification shows that it implements $\s$. Moreover $W$ commutes with $J$, thus Lemma A.3 shows that $W$ implements a left inverse $\Phi$ of $\s$. But $W$ also preserves the natural cone ${\cal P}^\natural(M,\Q)$, because this is true for each of its terms, thus $W$ is the standard implementation of $\Phi$ by Proposition A.2. It remains to show that $\Phi$ is the minimal left inverse. Now a left inverse is determined by the state obtained by restricting it to $(\s,\s)$. The value of $\Phi$ on the minimal projection $w^{(i)}_k w^{(i)*}_k$ is ${d(\r_i)}\over {d(\s)}$, hence it is the minimal left inverse. \endProof \titlea{Appendix B. Invariant vectors for representations of SL(2,$\Re$).} We start by recalling the ``vanishing of the matrix coefficient theorem'' for a connected simple Lie group $\bf G$ with finite center, see [\rfr(Zimm1)]. \begtheorem{B.1 }Let $U$ be a unitary representation of $\bf G$ on a Hilbert space $\H$. If $U$ does not contain the identity, then $(U(g)\x,\eta)\to 0$ as $g\to\infty$ for all $\x,\eta\in\H$. \endtheorem As a consequence, if $U$ is a unitary representation of $\bf G$ and $\x\in\H$ then the subgroup $\{g\in{\bf G},\,U(g)\x=\x\}$ is either compact or equals to $\bf G$. In the following $\bf G$ always denotes the universal covering group of $SL(2,\Re)$ and we state an explicit corollary in this case. Let us consider the one-parameter subgroups of $\bf G$ of the translations, dilations and rotations defined as the lifting to $\bf G$ of the one-parameter subgroups of $SL(2,\Re)$ $$ T(t)=\left( \matrix{1&t\cr0&1}\right),\quad \Lambda(s)=\left(\matrix{e^{s\over2}&0\cr 0&e^{-{s\over2}}}\right),\quad R(\th)=\left(\matrix{{\rm cos}{\th\over2} &{\rm sin}{\th\over2}\cr -{\rm sin}{\th\over2} &{\rm cos}{\th\over2}}\right),\eqno(B.1) $$ and we still denote them by the same symbols $T,\Lambda,R$ (cf. the definitions in Section~1). \begcorollary{B.2} Let $U$ be a unitary representation of $\bf G$ and $\Q$ a vector of the Hilbert space $\H$. The following are equivalent: \item{$(i)$} $\Co\Q$ are the only $U$ invariant vectors. \item{$(ii)$} $\Co\Q$ are the only $U(T(\cdot))$ invariant vectors. \item{$(iii)$} $\Co\Q$ are the only $U(\Lambda(\cdot))$ invariant vectors. \np If moreover the generator of $U(R(\cdot))$ is positive then the former are also equivalent to \item{$(iv)$} $\Co\Q$ are the only $U(R(\cdot))$ invariant vectors. \endcorollary \begProof Although the cardinality of the center $\bf Z$ of $\bf G$ is infinite, we check that Theorem B.1 still applies. By decomposing $U$ into a direct integral of irreducible representations, it is sufficient to consider the case in which $U$ is irreducible. Since $U$ is infinite-dimensional, the tensor product with its conjugate representation $U\otimes\bar U$ does not contain the identity. Now $U\otimes\bar U$ is trivial on the center $\bf Z$, hence defines a representation of $PSL(2,\Re)$. If $\x\in\H$ then by Theorem B.1 $$ |(U(g)\x,\x)|^2=(U(g)\otimes\bar U(g)\x\otimes\bar\x,\x\otimes\bar\x)\to 0\quad{\rm as}\quad g\to\infty. $$ Then the first set of equivalences is then clear. Furthermore $(i)$ is equivalent to $(iv)$ if the conformal Hamiltonian is positive because the identity is the only irreducible unitary representation of $\bf G$ with lowest weight $0$. \endProof In this paper we need a result in the spirit of Theorem B.1 concerning representations of the subgroup $\bP$ of the upper triangular matrices in $SL(2,\Re)$, namely the group generated by the translations and the dilations. \begproposition{B.3} Let $U$ be a unitary representation of $\bf P$ on a Hilbert space $\H$. If $F\subset\H$ is a finite-dimensional subspace which is globally $U(\Lambda(\cdot))$-invariant, then $F$ is left pointwise fixed by $U(T(\cdot))$. \endproposition \begProof Setting $u(t):=U(T(t))$ and $v(s):=V(\Lambda(s))$ we have two one-parameter unitary groups on $\H$ satisfying the commutation relations $$ v(s)u(t)v(-s)=u(e^{s}t)\ , \qquad t,s\in\Re. \eqno(B.2) $$ Since $F$ is finite dimensional, we need to show that $u(t)\x=\x$ if $\x$ is a $v$-eigenvector, i.e. there exists a character $\chi\in\widehat\Re$ such that $$ v(s)\x=\chi(s)\x\ ,\quad s\in \Re\ .\eqno(B.3) $$ Indeed in this case by the formula $(B.2)$ implies $$ u(e^st)\x=v(s)u(t)v(-s)\x=\ov{\chi(s)}v(s)u(t)\x $$ hence $$ (u(e^st)\x,\x)=(u(t)\x,\x),\quad t,s\in\Re. $$ As $s\to-\infty$ we thus have $$ (\x,\x)=(u(t)\x,\x) $$ that implies $u(t)\x=\x$ by the limit case of the Schwartz inequality. \endProof Before concluding this appendix, we recall a known fact needed in the text. \beglemma{B.4} Let $U$ be a unitary representation of $\bf G$. The following are equivalent: \item{$(i)$} The generator of $U(R(\cdot))$ is positive. \item{$(ii)$} The generator of $U(T(\cdot))$ is positive. \np In this case, if $U$ is non-trivial, the spectrum of the generator of $U(T(\cdot))$ is $[0,\infty)$.\endlemma \begProof For the equivalence $(i)\Leftrightarrow (ii)$ see e.g. [\rfr(PrSe1)]. The last statement follows because the spectrum of $U(T(\cdot))$ has to be dilation invariant because of the commutation relations $(B.2)$. \endProof \references \end