%TEX, All definitions and formats are included. \magnification 1200 %def. matem. \def\DAlamb{{\vcenter{\vbox{\hrule height.4pt \hbox{\vrule width.4pt height 5pt \kern5pt \vrule width.4pt} \hrule height.4pt}}}} \def\reali{\hbox{\rm I\hskip-2pt\bf R}} \def\realipunto{\hbox{\rm I}\hskip-2pt \dot {\bf R}} \def\complessi{\hbox{\rm I\hskip-5.9pt\bf C}} \def\nearrowro{\hbox{\bf T\hskip-7.4pt\rm I}} \def\razionali{\hbox{\rm I\hskip-5.7pt\bf Q}} \def\interi{\hbox{\rm{Z\hskip-8.2pt Z}}} \def\naturali{\hbox{\rm I\hskip-5.5pt\bf N}} \def\dsec#1#2{{{\partial^2 #1 } \over {\partial #2^2}}} \def\dpar#1#2{{{\partial #1 } \over {\partial #2}}} \def\modulo#1{\left| #1 \right|} \def\norma#1{{\left\| #1 \right\|}} %spaziature \def\bsk{\bigskip} \def\msk{\medskip} \def\ssk{\smallskip} \def\ni{\noindent} % Fonti di caratteri \font\titfnt=cmbx10 at 14.40 truept \font\tentex=cmtex10 \font\ninerm=cmr9 \font\eightrm=cmr8 \font\sixrm=cmr6 % Definizione formato della pagina \newdimen\pagewidth \newdimen\pageheight \newdimen\ruleht \hsize=31pc \vsize=45pc \maxdepth=2.2pt \parindent=19pt \pagewidth=\hsize \pageheight=\vsize \ruleht=.5pt \abovedisplayskip=10pt minus 2pt \belowdisplayskip=11pt minus 2pt \abovedisplayshortskip=8pt minus 2pt \belowdisplayshortskip=9pt minus 2pt \baselineskip=15pt minus 0.5pt \lineskip=2pt minus 0.5pt \lineskiplimit= 2pt \mathsurround=1pt % Definizioni per l'output %(numeri in alto alternati) \nopagenumbers \def\testos{\null} \def\testod{\null} \headline={\if T\tpage{\gdef\tpage{F}{\hfil}} \else{\ifodd\pageno\rightheadline\else\leftheadline\fi} \fi} \gdef\tpage{T} \def\rightheadline{\hfil{\tensl\testod}\hfil{\tenrm\folio}} \def\leftheadline{{\tenrm\folio}\hfil{\tensl\testos}\hfil} %(numeri in alto al centro) %\nopagenumbers %\headline={\hss\tenrm\folio\hss} %\voffset=2\baselineskip % Registrazione referenze su file esterno e macro di conteggio \newcount\numref \global\numref=1 \newwrite\fileref \immediate\openout\fileref=ref.tmp \immediate\write\fileref{\parindent 30pt} \def\citaref#1{${[\the\numref]}$\immediate\write\fileref {\par\noexpand\item{{[\the\numref]\enspace}}}\ignorespaces \immediate\write\fileref{{#1}}\ignorespaces \global\advance\numref by 1\ignorespaces} \def\references{\immediate\closeout\fileref \par\vfill\eject \null\msk \centerline{\bf References} \bsk \input ref.tmp} %Fine delle def. generali \def \H {{\cal H}} \def \Hpi {{\cal H_\pi}} \def \vp {{\varphi}} \def \eps {{\varepsilon}} \def \A {{\cal A}} \def \D {{\cal D}} \def \F {{\cal F}} \def \G {{\cal G}} \def \Z {{\cal Z}} \def \O {{\cal O}} \def \L {{\cal L}} \def \S {{\cal S}} \def \bg {{\beta_g}} \def \ap {{\alpha_p}} \def \oz {{\omega_0}} \def \og {{\omega_g}} \def \aal {{\alpha_{a,\Lambda}}} \def \undp {{1\over 2 \pi}} \def \unm {{1\over 2}} \def \cs {C^{*}} \def \dm {d\mu} \def \dmz {d\mu^0} \def \piz {{\pi_0}} \def \psz {{\psi_0}} \def \Pz {{{\cal P}_0}} \def \Aobs {{\cal A}_{obs}} \def \endproof {$q.e.d.$} \def \intdp {{\int_{0}^{2 \pi}}} \def \intinf {{\int_{-\infty}^{\infty}}} \def \Fpi {{\F^0_\pi }} \def \pinv {{\pi_0^{inv}}} \def \Fmn {{F^{\mu \nu}}} \def \glm {{\gamma^{\lambda \mu}}} %Fine delle definizioni \hbox{} \bsk\bsk \hfill IFUP--TH 20/95 \bsk\bsk\bsk \centerline{\titfnt Charged fields, Higgs Phenomenon and Confinement.} \msk \centerline{\titfnt Lesson from soluble models $ ^{*}$} \bsk\bsk\bsk \centerline{G. Morchio} \msk \centerline{\it Dipartimento di Fisica dell'Universit\`a and INFN, Pisa, Italy} \bsk\bsk\bsk\bsk\bsk \ni {\bf Abstract}. Higgs phenomenon and confinement are discussed on the basis of a general framework for gauge and Poincar\'e transformations on field algebras arising in gauge theories, in positive gauges. The resulting phenomena and structures are explicitly controlled for a class of soluble models. \bsk\bsk\bsk\bsk\bsk \ni $ ^{*} $ Invited talk at the \lq\lq Colloquium on New Problems in the General Theory of Fields and Particles\rq\rq , Paris, July 1994. \vfill \eject \ni {\bf 1. Motivations} \bsk The r\^ole of gauge theories has evolved, in the last 30 years, from that of a special case, Quantum Electrodynamics (QED), to that of a common tool for the description of all the fundamental interactions. The fact that QED, and in general gauge theories, are usually described in terms of field algebras which do not completely fit into the Wightman framework \citaref{R.F.Streater and A.S.Wightman, {\it PCT, Spin and Statistics and All That\/}, Ben\-ja\-min, New York, 1964; Addison--Wesley, 1989; R.Jost, {\it The General Theory of Quantized Fields\/}, Am. Math. Society, 1965.}, poses therefore a general problem, and requires the identification and investigation of characteristic structures. In fact, locality and covariance of gauge fields hold only in \lq\lq local gauges\rq\rq , where positivity fails; on the other hand, the status of gauge fields in positive gauges is problematic, especially when there is confinement (i.e. absence of charged sectors with finite energy), or screening (\lq\lq Higgs phenomenon\rq\rq ). The space--time symmetries present problems already in the QED case, where the Lorentz group is spontaneously broken in the charged sectors \citaref{J.Fr\"ohlich, G.Morchio, F.Strocchi, Ann. Phys. {\bf 119}, 241 (1979); Phys. Lett. {\bf B 89}, 61 (1979). } \citaref{D.Buchholz, Phys. Lett. {\bf B 174}, 331 (1986).}, with substantial implications on the transformation properties of charged fields. The absence of charged sectors with well--defined (unitarily implemented) time evolution in Quantum Cromodynamics (QCD) is not a very distant phenomenon from the breaking of the Lorentz group in QED, the main difference being produced by the smallness of the coupling constant in QED, which makes the problem relevant only at very large distances. We sketch below a general framework for these features and problems in terms of characteristic structures of gauge field algebras in positive gauges. As we will see, \ssk\ni i) unbroken gauge groups cannot commute with those Poincar\'e transformations which do not leave the corresponding charged sectors invariant; the gauge group does not therefore commute with the Lorentz transformations in QED, and with the time translations in QCD. \ssk\ni ii) In the case of a (partially) broken gauge group, the algebra of observables is strongly dense in the field subalgebra which is pointwise invariant under the residual unbroken gauge subgroup, and there are no charged states corresponding to the broken charges. \ssk\ni iii) In case ii), Poincar\'e automorphisms commuting with the gauge group can be defined only on an extension of the field algebra with non--trivial centre; the appearance of central variables in the time evolution of the field algebra is at the basis of non--trivial mass spectra associated to the breaking of the gauge group. These phenomena can be seen as the relativistic counterpart of general features of non--relativistic systems with Coulomb interactions, where variables at infinity appear in the time evolution of local variables as a consequence of the removal of infrared or volume cutoffs. \msk Before introducing {\it field\/} algebras, we recall the point of view and the results of general Algebraic Local Quantum Theory, which always apply to the {\it observable} algebras. \bsk\bsk\goodbreak \ni {\bf 2. Observables and fields} \bsk The {\it observable} algebra of a gauge QFT can be assumed to be given by a net $\A(\O)$ of local (Von Neumann) algebras, defined in the vacuum representation. The whole physical content of the theory can in principle be extracted from the observable algebra, in terms of the classification of its representations, under suitable criteria of \lq\lq physical relevance\rq\rq . We do not consider for the moment the problem of which properties distinguish the observable nets which arise in gauge theories from those which arise in standard Quantum Field Theory (QFT), and therefore we consider any local net $\A(\O)$, with the standard assumptions \citaref{R.Haag, Local Quantum Physics, Springer, Berlin, 1992.}. The following possibilities arise for relevant representations: \ssk\ni 1. Representations labelled by \lq\lq localizable\rq\rq\ charges, obtained from the vacuum representation through morphisms which are localizable in finite space--time regions, in the sense that they leave invariant all observables in the causal complement of a double cone. This has been thoroughly investigated by Doplicher, Haag and Roberts (DHR). Space--time covariance of the charged sectors follows, together with the classification of statistics and the construction of a field algebra and a compact \lq\lq gauge group\rq\rq which classifies the representations of $\A$ \citaref{S.Doplicher, R.Haag, J.E.Roberts, Comm. Math. Phys. {\bf 23}, 199 (1971); J.E.Roberts, Lectures on Algebraic Q.F.T., in {\it The Algebraic Theory of Superselection Sectors\/}, D.Kastler ed., World Scientific, 1990, and refs. therein.}. \ssk\ni 2. Representations stable under space--time translations, defined by (particle) states with ener\-gy--momentum spectrum in an isolated iperboloid with positive mass. By the results of ref. \citaref{D.Buchholz, K.Fredenhagen, Comm. Math. Phys. {\bf 84}, 1 (1982).} they can all be obtained by morphisms which are either localized in the sense of DHR, or can be localized in {\it any} space--like cone, i.e. the morphism can be chosen so that all observables localized in the causal complement of any given spacelike cone are left invariant. \ssk\ni 3. Representations stable under space--time translations, with relativistic spectral condition, labelled by charges which obey a Gauss' law (which forbids [3] eigenstates of the mass operator). The localization region of the corresponding morphisms includes at least a spacelike cone \citaref{D.Buchholz, Comm. Math. Phys. {\bf 85}, 49 (1982).}. \ssk QED, being characterized by a charge which obeys a Gauss' law, is a candidate for case 3. However, the generalization of the DHR analysis to this case, and the full characterization of the properties of the charged sectors (including particle statistics) seems to require further information which, in our opinion, can be obtained by studying the properties of the charged fields in the (standard) positive gauge formulations. QCD seems to be characterized by the absence of charged sectors stable under space--time translations and with positive energy. It is not clear how such a theory can be characterized as a gauge theory along the lines of 1--3. Perhaps, again some insight can be obtained by studying the algebra of charged fields in positive gauges, which should define sectors not covariant under (space)--time translations. The Higgs--Kibble model, and more generally gauge theories exhibiting the Higgs phenomenon, do not have sectors labelled by a Gauss' charge, because of screening. The spontaneous breaking of gauge symmetry implies that charged fields in positive gauges cannot give rise to charged sectors, (see Proposition 3 below). Again, even if for different reasons, it is not clear how such a kind of theories can be characterized by a gauge group, and, more generally, whether they can be recognized in terms of properties of the observable algebra. Here too, the study of the field algebra in the standard approach should give relevant hints. \ssk The difficulty in the characterization of the phenomena exhibited by gauge theories on the basis of the classification of the representations of the observable algebra can in general be traced back in the idea that the \lq\lq gauge group\rq\rq\ must be identified in terms of the \lq\lq particle representations\rq\rq\ of $\A$; these exist only in the \lq\lq QED case\rq\rq , and even in this case do not have the localization properties which appear as most natural from the point of view of the local structure of the observables. \ssk In conclusion, in order to i) investigate a possible algebraic characterization of gauge QFT, ii) discuss possible algebraic characterizations of confinement and of screening, iii) get information on the charged sectors of QED, we propose to study the general properties of the algebra of fields in positive gauges. This strategy is very similar to that used by DHR in their first paper \citaref{S.Doplicher, R.Haag, J.E.Roberts, Comm. Math. Phys. {\bf 13}, 1 (1965).} for theories with localizable charges, namely to abstract general structural properties from concrete field algebras, and use them as a basis for a general algebraic approach. \bsk\bsk \ni {\bf 3. Charged field algebra as an extension of the observable algebra} \bsk The charged fields in gauge theories are characterized by being charged with respect to a Gauss' charge, and therefore they cannot be local with respect to the observables in the abelian case, and cannot be relatively local in the non--abelian case. Furthermore, the experience with QED suggests that charged fields cannot in general be covariant under the Lorentz group. Locality and covariance are obtained in renormalizable gauges, but at the price of giving up positivity; the physical interpretation is then obtained through a subsidiary condition on the states, and the solutions of such a condition require, for the charged sectors, a non--local construction (see ref. \citaref{G.Morchio, F.Strocchi, Nucl. Phys. {\bf B 211}, 471 (1983); {\bf B 232}, 547 (1984); Infrared problem, Higgs phenomenon and long range interactions, in {\it Fundamental problems of gauge field theory\/}, G.Velo and A.S.Wightman ed., Plenum Press 1986.}. This is at the basis of the absence of locality and covariance of the states and of the morphisms which may define them in terms of representations of the observable algebra. In a spirit similar to that of ref.[8], we consider a $\cs$ algebra $\F$, as the {\it algebra of fields} in a positive gauge. The {\it gauge group} $\G$ is defined as the group of ${*}$ automorphisms $\bg$, $ g \in \G$, of $\F$, which leave an {\it observable subalgebra} $\A \subset \F$ pointwise invariant. Clearly, given $\F$ and $\G$, $\A$ can be defined as the subalgebra of $\F$ which is pointwise invariant under $\G$, or, alternatively, one may consider $\A$ as a given {\it observable} algebra, $\F$ an extension of $\A$, and $\G$ {\it defined} by the above relation. For the observable algebra $\A$ the standard general assumptions can be made, i.e.: \ssk\ni i) {\it $\A$ is the $\cs$ completion of a {\it local net} $$ \O \mapsto \A(\O) $$ defined for all double cones $\O$ in Minkowski space, $\A = \overline{\cup_\O \A(\O)}^{|| \ ||}$, and the Poincar\'e group is assumed to act as a group of automorphisms $\aal $ of $\A$}. \ssk A vacuum state is assumed to exist as a pure state on $\A$, with unique (pure) extension to $\F$: \ssk\ni ii) {\it there exists a pure state $\oz$ on $\F$, such that its restriction to $\A$ is pure and Poincar\'e invariant}. \ssk The first issue is the space--time covariance properties of the field algebra and the relation between the space--time translations and the gauge group. In the standard approach to gauge QFTs in positive gauges, the gauge group is believed to be a \lq\lq global\rq\rq\ one, the local gauge group having been broken by fixing the gauge, and the folklore seems to take for granted that such (\lq\lq residual\rq\rq ) gauge group commutes with the space--time translations. It is worthwhile to see whether this property can indeed be assumed and what is its origin in the present framework. Assume therefore that \ssk\ni iii) {\it a subgroup $\Pz$ of the Poincar\'e group defines a group of automorphisms $\alpha_p$, $p \in \Pz$ of $\F$, which extend the Poincar\'e automorphisms defined on $\A$}. \ssk We will also use later the assumptions \ssk\ni iv) {\it As a state on $\F$, $\oz$ is left invariant by the automorphisms $\ap$}: $\oz(B) = \oz (\alpha_p (B))$, $\forall B \in \F , \, p \in \Pz$. \ssk\ni v) {\it in the GNS representation $\piz$ defined by $\oz$ and $\F$, $\oz$ is the only state invariant under all $\alpha_p$, $p \in \Pz$}. \msk Since $\oz$ is pure, $\F$ is irreducible in $\piz$; all the gauge automorphisms $\bg$ which leave $\oz$ invariant have unitary implementers in $\piz$, which commute with $\A$, and it is reasonable to assume that all unitary operators in $\piz$ which commute with $\A$ define automorphisms of $\F$; in fact, by irreducibility of $\F$, this can always be achieved, by enlarging $\F$, if necessary, with strong limits in $\piz$. The representation $\piz$ of $\F$ will be our primary object of interest; but an important r\^ole is played in the following by {\it gauge invariant\/} strong topologies on $\F$, defined by {\it gauge invariant\/} representations $\pi$ of $\F$, i.e. by representations which are stable under the action of the gauge group, $\pi \circ \bg = \pi$, $\forall g \in \G$; the gauge automorphisms are automatically continuos with respect to any gauge invariant strong topology, as a consequence of the invariance under $\bg^{*}$ of the folium of states associated to $\pi$. Assuming iv) and v), the representation $\piz$ will turn out to be gauge invariant if and only if $\oz$ is invariant under $\bg$; in any case, a gauge invariant representation is obtained by taking the direct sum over the gauge group of the GNS representations defined by the states $\og \equiv \oz \circ \bg$. This representation will be denoted by $\pi_0^{inv}$. The following Propositions show the implications of the assumption that a group of automorphisms of $\F$, in particular $\alpha_p$, $p \in \Pz$, have \lq\lq gauge invariant generators\rq\rq . \msk\ni {\bf Proposition 1}. {\it Let $\pi$ be a gauge invariant representation of $\F$, and $\gamma$ an automorphism of $\F$}. \ni i) {\it if $\gamma$ $\F$ is the strong limit in $\pi$ of automorphisms $\gamma_L$ which commute with $\bg$, then} $ [ \gamma, \bg ] = 0 $. \ni ii) {\it in particular, if $\ap(A)$, $p \in \Pz$, is the strong limit of $U_L(p) A U_L(p)^{*}$, $U_L \in \A$, then} $$ [ \ap , \bg ] = 0 \ \ \ \forall g \in \G \eqno(1) $$ \ni iii) {\it if $\gamma$ $\F$ is implemented in $\pi$ by (unitary) operators $U$ in the Von Neumann algebra generated by $\A$ in $\pi$, then} $$ [ \gamma , \bg ] = 0 \ \ \ \forall g \in \G $$ \ssk\ni {\bf Proof}. i): The stability of $\pi$ under $\bg$ implies that $\bg$ is strongly continuous, and therefore, $\forall A \in \F$, $$ \bg \gamma (A) = \bg \, s-\lim \gamma_L (A) = s-\lim \bg \gamma_L (A) = s-\lim \gamma_L \bg (A) = \gamma \bg (A) $$ ii): All $\bg$ are strongly continuous and have therefore a unique strongly continuous extension to the Von Neumann algebra generated by $\F$ in $\pi$, which leaves the Von Neumann algebra generated by $\A$ pointwise invariant. \msk We conclude that if Poincar\'e transformations of $\F$ can be constructed from observable local implementers, or are implemented by strong limits of observables operators, then they commute with the gauge group. The delicate points are here: \ni a) The use of a gauge invariant strong topology, which is essential for the argument; we will see below the non--trivial implication of this fact for broken gauge groups. \ni b) The Poincar\'e group is not always implemented by observable operators in gauge theories, nor do local implemeters always converge; this cannot in fact be the case for the Lorentz boosts in QED, since they do not leave the charged sectors stable [2],[3],[7],[9], and for the time translations in confined models, if confinement corresponds to the instability of charged sectors under time translations, as also suggested by the models discussed below. \msk >From Proposition 1 it also follows: \ssk\ni {\bf Proposition 2}. {\it If $ [ \ap , \bg ] \neq 0 $ for some $p \in \Pz$ and $g \in \G$, then, in any representation $\pi$, either \ni $1$) $\bg$ is broken, or \ni $2$) $\ap$ is not implemented by operators in the strong closure of the observable algebra}. \ssk In the case of time translations $\alpha_t$, Proposition 2 says that if $[\alpha_t , \bg] \neq 0$ for some $g \in \G$, then one has for $\bg$ either the Higgs phenomenon or the confinement; a non--zero mass spectrum is in fact associated in general to the breaking of $\bg$, as a consequence of the lack of commutativity with $\alpha_t$, see ref.\citaref{G.Morchio, F.Strocchi, Comm. Math. Phys. {\bf 99}, 153 (1985); J. Math. Phys. {\bf 28}, 622 (1987).}, and the second alternative exclude the existence of energy as a (non local) observable. The alternative $2$) can be replaced, for the group of space--time translations $\alpha_x$, by \ni $2^\prime$) {\it $ \alpha_x $ is not implemented by a unitary group satisfying the relativistic spectral condition}. \ni In fact, by Borchers' theorem \citaref{H.J.Borchers, Comm. Math. Phys. {\bf 2}, 49 (1966).} the implementers could then be chosen in the strong closure of $\A$. \ssk A converse of Proposition 2, i.e. the fact that if a Poincar\'e transformation commutes with the gauge group, then it is implemented by observables, requires to consider the possibility of broken gauge transformations, and will be given below (Propositions 5 and 6). \ssk Now we discuss the possibility that the gauge group is broken in the vacuum representation $\piz$ of $\F$. This point has sharp implications on the relation between $\A$ and $\F$, and it is convenient to remark first that {\it in a gauge invariant representation} $\pi$ $\A$ {\it cannot be strongly dense in} $\F$, if the gauge group is non--trivial. This follows immediately from the strong continuity of $\bg$, which forbids the existence of a strongly dense pointwise invariant subalgebra. Moreover, for a GNS representation over a pure state $\omega$ invariant under the gauge group, in particular for $\piz$ whenever it is stable under gauge transformations, the GNS subspace generated by $\A$ is never dense, if the gauge group is not trivial: in fact, the invariance of $\omega$ implies the existence of unitary implementers of the gauge group, which reduce to the identity on the GNS subspace generated by $\A$. Given a representation $\pi$ of $\F$, the unbroken subgroup $\G^0_\pi$ of the gauge group is given by the gauge automorphisms $\bg$ which leave $\pi$ invariant, $\pi \circ \bg = \pi$. For the vacuum representation $\piz$, if iv), v) hold, and $\bg$ commutes $\ap$, $\forall g \in \G , \, p \in \Pz$, the unbroken subgroup $\G_0$ is given by the gauge automorphisms $\bg$ satisfying $\bg^{*} (\oz) = \oz$. We call $\F^0_\pi$ ($\F_0$ when $\pi = \piz$) the subalgebra of $\F$ pointwise invariant under $\G^0_\pi$. From the definition of the gauge group it follows that for a gauge invariant representation $\F^0_\pi = \A$, whereas, for representation with gauge group broken to the identity, $\F^0_\pi = \F$. \msk\ni {\bf Proposition 3}. {\it In the GNS representation $\piz$ of $\F$ defined by $\oz$, the observable algebra $\A$ is strongly dense in $\F_0$. The GNS subspace generated by $\A$ is therefore dense in the GNS subspace generated by $\F_0$. The strong closure $\overline\A = \overline{\F_0}$ coincides with the Von Neumann algebra of the operators, in the representation space of $\piz$, which are invariant under (the unique continuous extension of) $\bg$, $g \in \G_0$}. \ssk\ni {\bf Proof}. Assume that the strong closure of $\A$ is contained properly in the strong closure of $\F_0 $; the commutant of $\F_0$ is then contained properly in the commutant of $\A$, i.e. there exists an operator in the representation space $\Hpi$ which commutes with $\A$ but not with $\F_0$; by taking the hermitean (or antihermitean) part, and using the spectral theorem, a unitary operator is constructed with the same properties. This defines an automorphism of $\F$ which leaves $\A$ pointwise invariant, and therefore a gauge automorphism of $\F$ implemented in $\piz$, which does not act trivially on $\F_0$, contrary to its definition. By the same argument one proves the last statement. \msk It follows from Proposition 3 that, if the gauge group is broken to the identity, then $\F_\pi^0 = \F$, and the representation space of $\piz$ coincides with that of the GNS representation of the observables over $\oz$, i.e. all the states are obtained by applying observables to the vacuum. The Poincar\'e automorphisms of $\A$ are then implementeted by unitary operators $U(a,\Lambda)$ in this representation, which belong, by irreducibility, to the strong closure of $\A$. If the extension of the Poincar\'e automorphisms to $\F$ is done in $\piz$ by $U(a,\Lambda) B U^{*}(a,\Lambda)$, $B\in \F$, it does not in general commute with the gauge group; in fact, even if $U(a,\Lambda)$ are the strong limits in $\piz$ of $U_L(a,\Lambda) \in \A$, such limits are taken in a strong topology which is not gauge invariant, and the gauge automorphisms are not continuous with respect to it. A gauge invariant extension requires the use of a gauge invariant strong topology, given by a representation stable under $\bg$. In the representation $\pi_0^{inv}$, obtained as a direct sum of the GNS representations of $\F$ over $\oz \circ \bg$, the strong convergence of $\piz(U_L(a,\Lambda))$ implies the strong convergence of $\pinv(U_L(a,\Lambda))$, by definition of $\pinv$ and invariance of $U_L$ under $\bg$. Their limit is invariant under the unique strongly continuous extension of $\bg$, and defines an extension of the Poincar\'e group to the strong closure of $\F$ in $\pi_0^{inv}$, which commutes with the gauge group. It is immediate to see that the strong closure of the field algebra in the representation $\pinv$ has a centre, $\Z_F$, which is abelian, because $\pinv$ is a direct sum if irreducible representations of $\F$, and has a spectrum isomorphic to the gauge group (with the discrete topology). The Poincar\'e automorphisms do not in general leave $\F$ stable, and it follows from their construction that they leave invariant the algebra generated by $\F$ and $\Z_F$, which may be taken as a new field algebra, on which the Poincar\'e automorphisms always exist and commute with the gauge group. We have therefore proven \msk\ni {\bf Proposition 4}. {\it If the gauge group is broken in $\piz$ to the identity, then the Poincar\'e automorphisms extend to automorphisms of the algebra generated by $\F$ and the centre of the Von Neumann algebra generated by $\pinv(\F)$, and commute with the gauge group}. \ssk The centre can be essential for the gauge invariance of the Poincar\'e automorphisms, as we will see in the models below. The appearance of central variables in the dynamics of $\F$ allows for an evasion of the Goldstone theorem, and is at the basis of the (Higgs) phenomenon of mass generation accompanying the spontaneous breaking of the gauge group ({\it spontaneous} indicating the commutation between the gauge group and the time translations) [10]. The point is that in general central variables appear if the Poncar\'e automorphisms and the gauge group are formulated so that they commute; in the ordinary Goldstone theorem such central variables are excluded by the assumption that a symmetry is generated by a local current. If the action of local implementers converges strongly on $\F$ in $\piz$, then a very similar argument to that given above shows that they converge strongly in $\pinv$, and the limit may then involve central variables {\it as a consequence of the non local character of the charged fields}. The same structures are present in non relativistic models with long range (Coulomb) or mean field interactions [10],\citaref{G.Morchio, F.Bagarello, J. Stat. Phys. {\bf 66}, 849 (1992).}, a prototype being Haag's treatment of the BCS model \citaref{R.Haag, Nuovo Cimento {\bf 25}, 1078 (1962).}. We may also observe that Proposition 3 applies to any symmetry group, with $\A$ playing the r\^ole of the neutral subalgebra, but a mass gap is produced only if central variables appear in the dynamics of charged fields, and therefore only if the latter are sufficiently non--local with respect to the observables. \ssk We can now discuss a converse of Proposition 2, and its implications on confinement. \msk\goodbreak\ni {\bf Proposition 5}. {\it If $ [ \ap , \bg ] = 0 $ for all $p \in \Pz$ and for all $\bg$ which are not broken in $\piz$, then the automorphisms $\ap$ are implemented in $\piz$ by unitary operators belonging to the strong closure of $\piz(\A)$}. \ssk\ni {\bf Proof}. $\oz$ is invariant under $\bg$, as a consequence of v), and under $\ap$, because of iv); there exist therefore implementers $U(p)$ and $V(g)$, which by construction leave invariant $\psz$, the representative vector of $\oz$; hence $$ U(p) V(g) B \psz = \ap \bg (B) \psz = \bg \ap (B) \psz V(g) U(p) B \psz \ \ \ \ \forall B \in \F $$ and therefore $U$ and $V$ commute, by the ciclicity of $\psz$. Thus, by Prop.3, $U(p) \in \overline\piz(\A)$. \msk By applying the construction in the proof of Proposition 4, which only uses the fact that in $\piz$ $\ap$ is implemented by strong limits of observables, the assumptions of Proposition 5 imply that all $\ap$, $p \in \Pz$, extend to automorphisms of the algebra generated by $\F$ and $\Z_F$ in $\pinv$, which commute with $\bg$ for {\it all} $g \in \G$. Since the form of the automorphisms which commute with the {\it broken} gauge transformations is determined by the construction given in the proof of Proposition 4, a version of Proposition 5 also applies to a field algebra with Poincar\'e transformations invariant under all the gauge group, and implies the existence of implemeters invariant under all the gauge group: \msk\ni {\bf Proposition 6}. {\it If $ [ \ap , \bg ] = 0 $ for all $g \in \G$, then all the automorphisms $\ap$ are implemented in $\pinv$ by gauge invariant operators, i.e. operators in the strong closure of $\pinv(\A)$}. \msk It follows from Propositions 5 and 6 that the existence of Poincar\'e automorphisms commuting with the gauge group always leads to implementers which leave invariant the Hilbert sectors defined (in $\piz$ or in $\pinv$) by the representations of the observable algebra $\A$; Poincar\'e automorphisms are therefore in this case never broken in the observable sectors. Since the Lorentz boosts are broken in QED, and since the breaking of time translations is typical of confined models (see below), we conclude that Poincar\'e automorphism, if they exist, cannot commute with the gauge group in these cases; the lack of commutativity between gauge transformations and time translations may in fact characterize confinement \citaref{F.Acerbi, G.Morchio, F.Strocchi, J. Math. Phys. {\bf 34}, 899 (1993); Lett. Math. Phys. {\bf 27}, 1 (1993); Rep. Math. Phys. {\bf 33}, 7 (1993).}, since it is equivalent (Prop.2 and Prop.5) to the non--existence of the energy as an observable, in the charged sectors. However, such characterization does not cover the case of time translations implemented by an (observable) energy unbounded from below, a mechanism which seems to occur in QED(2+1), if the \lq\lq photons\rq\rq\ do not acquire a mass \citaref{G.Morchio, F.Strocchi, Ann. Phys. {\bf 172}, 267 (1986).}. \bsk\bsk \ni {\bf 4. Models} \bsk\ni The general structures outlined above can be seen and explicitly controlled in soluble models. We discuss in the following the St\"uckelberg--Kibble and the Schwinger model; the first is a prototype of the Higgs phenomenon, the second of confinement. As we shall see, however, confinement takes place also in the S--K model, for low space dimensions, and this phenomenon is explicitely seen to depend in a very direct way upon the general alternative discussed above for the field algebra and the gauge group. \ssk The S--K model is defined by a linearization of the (abelian) Higgs--Kibble model, corresponding to fixing the modulus of the Higgs field and treating the phase as a scalar field; the Lagrangean density is $$ \L_{SK} = - {1 \over 4} F_{\mu \nu}^2 - {1 \over 2} (\partial_\mu \chi - e A_\mu)^2 $$ with $F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu $, and $\chi$ a scalar (Higgs) field. It will be considered for space--time dimensions $d + 1$, $d = 3,\, 2,\, 1 \,$. \ssk The Schwinger model is given, in bosonized form, by the Lagrangean density $$ \L_S = - {1 \over 4} F_{\mu \nu}^2 - {1 \over 2} (\partial_\mu \vp )^2 + e \, \eps^{\mu \nu} \partial_\mu \vp A_\nu $$ with $\vp$ the (pseudo--) scalar field in terms of which the fermion field is expressed, in two space--time dimensions. \bsk\ni {\it Observable algebras} \ssk We will first discuss the models in terms of observable algebras. These are defined by the correlation functions of gauge invariant fields on the vacuum, which are by definition independent of the gauge fixing. The r\^ole of the gauge fixing is that of giving (non local) relations between the fields which appear in the Lagrangean and the observables, allowing for the construction of the corresponding field algebras. The construction follows therefore a different logic, compared to that of ref. \citaref{G.Morchio, F.Strocchi, J. Math. Phys. {\bf 28}, 1912 (1987); Comm. Math. Phys. {\bf 111}, 593 (1987).}, where the field algebra at a fixed time was first defined in terms of canonical variables, and then the time evolution was constructed, meeting problem and features very close to those of non--relativistic Coulomb systems \citaref{G.Morchio, F.Strocchi, Ann. Phys. {\bf 170}, 2 (1986).}. \ssk To obtain the observable algebras, we start from the equations of motion for the observable fields given by the above Lagrangeans and assume local commutativity for the corresponding quantum fields; it is then easy to characterize (all) the Poincar\'e invariant correlation functions of such fields satisfying the relativistic spectral condition, and obtain the complete algebraic structure of the observable fields, which will correspond to a canonical (Weyl) algebra. \ssk For both models, the equation of motion for $F_{\mu \nu}$ are the Maxwell equations, $$ \partial_\mu \Fmn - j^\nu = 0 \eqno(2) $$ with $j^\nu$ given, in the S--K model, by $$ j^\nu = e \, (\partial^\nu \chi - e A^\nu) \eqno(3) $$ and, in the Schwinger model, by $$ j^\nu = e \, \eps^{\mu \nu} \partial_\mu \vp \eqno(4) $$ \ssk For the S--K model, from eq.(4) it follows $$ e^2 \, \Fmn = - (\partial^\mu j^\nu - \partial^\nu j^\mu) \eqno(5) $$ and, from eqs. (2) and (5), $$ \DAlamb \, j^\mu + e^2 j^\mu = 0 \ \ \ \ \ \ \ \ \ \ \ \ \partial_\mu j^\mu = 0 \eqno(6) $$ The current fields $j^\mu$ are therefore, in the S--K model, free massive fields of mass $m^2 = e^2$. Their two point function on a Poincar\'e invariant state satisfying the relativistic spectral condition is determined (up to a constant factor) by eq.(6), $$ = J^{\mu \nu} (x-y) $$ $$ \tilde J^{\mu \nu} (k) = (g^{\mu \nu} - { k^\mu k^\nu \over m^2 } ) \, \delta(k^2 - m^2) \, \theta(k^0) \eqno(7) $$ By local commutativity and the the Jost--Schroer theorem, all the truncated correlation funtions of $j^\mu$ vanish, and the commutator $ [j^\mu (x), j^\nu (y)] $ is a c--number valued distribution, determined by eq.(7). It follows that, in the Hilbert space $\H$ given by the Wightman reconstruction, the exponentials of the smeared fields $W(f) \equiv \exp(j^\mu (f_\mu))$, $f_\mu$ real in the Schwartz space $\S(\reali^{d+1})$, generate a Weyl algebra $\A$, defined by the symplectic form $$ = \int d^4 k \; (g^{\mu \nu} - { k^\mu k^\nu \over m^2 } ) \, \delta(k^2 - m^2) \, \eps(k^0) \, \tilde f_\mu (-k) \, \tilde g_\nu (k) \eqno(8) $$ The Poincar\'e invariant state on the Wightman fields $j^\mu$ defines a state $\oz$ on $\A$, given by $$ \oz (W(f)) = e^{ - [f,f] /4} \eqno(9) $$ $$ [f,f] = \int d^4 k \; (g^{\mu \nu} - { k^\mu k^\nu \over m^2 } ) \, \delta(k^2 - m^2) \, \tilde f_\mu (-k) \, \tilde g_\nu (k) \eqno(10) $$ and $\H$ is the GNS space of $\A$ on the state $\oz$. We have therefore obtained the observable algebra and its vacuum representation. \ssk For the Schwinger model, the equation of motion for $\vp$ is $$ \partial_\mu (- \partial^\mu \vp + e \, \eps^{\mu \nu} A_\nu) = 0 \eqno(11) $$ Eq. (11) can also be written $$ \DAlamb \, \vp - e F_{01} = 0 \eqno(12) $$ Equations (2) and (4) give the relations $$ \partial_\mu (e \vp - F_{01}) = 0 \ \ \ \ \ \ \ \ \ \ \mu = 0,1 \eqno(13) $$ and therefore $$ e \, \vp = F_{01} + \sigma \eqno(14) $$ with $ \sigma $ a field invariant under space--time translations. Eqs.(12) and (14) imply that $F_{01}$ is a free massive fields, of mass $m^2 = e^2$. It is important to remark that the (Wick) exponentials of $\vp$ are observable, since they correspond to the bilinears of the fermion field, so that the observable algebra must include (as unbounded operators affiliated to the local Von Neumann algebras) $F_{01}$ {\it and} $\vp$. The one point function of $F_{01}$ on a state invariant under the proper Poincar\'e group vanishes by eq.(12), and the two point function is that of a massive field. Assuming as before local commutativity, which implies the vanishing of all higher order truncated correlation functions, the observable algebra must then be identified as the algebra $\A_S$ generated by the Weyl exponentials of the massive free field $F_{01}(f)$, $f \in \S(\reali^2)$, and by the variable $ \exp i \alpha \sigma$, which, by local commutativity and space--time invariance, is in the centre of $\A_S$. The appearence of central variables in the observable algebra is related to chiral symmetry, which is here well defined as an automorphism of $\A$, commuting with the (proper) Poincar\'e group: $$\beta^\lambda \, e^{ i \vp(f)} = e^{i \lambda \tilde f (0)} \, e^{ i \vp(f)} \ \ \ \ \ \ \beta^\lambda \, e^{i F_{01}(f)} = e^{ i F_{01}(f)} \eqno(15) $$ The presence of the central variable $\sigma$ is essential for the validity of eq.(15), i.e. for the existence of chiral automorphisms commuting with the (proper) Poincar\'e group. In fact, if a factorial, in particular irreducible, representation $\pi$ of $\A$ is considered, then $\pi (\sigma)$ is a number, and any automorphism of $\pi(\A)$ which shifts $\vp$ must also shift the massive field $F_{01}$, and cannot therefore commute with the space--time translations. One recovers in this way the alternative, typical for symmetries in systems with long range forces, between \ni i) a symmetric algebraic dynamics (which naturally arises as the thermodynamic limit of a symmetric finite volume dynamics), which involves central variables [10] \ni ii) the use of a simple algebra, with the consequence of a non--symmetric dynamics; this is obtained as the infinite volume limit of finite volume dynamics generated by Hamiltonians with non--symmetric boundary terms \citaref{G.Morchio, F.Strocchi, Ann. Phys. {\bf 185}, 241 (1988).}. Moreover, the mass spectrum of the Schwinger model can be seen as the spectrum associated to the spontaneous breaking of the chiral transformations [16]. \bsk\ni {\it Field algebras} \ssk The field algebras defined by the lagrangean variables in the Coulomb gauge can now be constructed as extended Weyl algebras. For the details, see refs. [14],[16]. We start from the Coulomb gauge relation $$ - \triangle A^0 = j^0 \eqno(16)$$ in order to construct $A^0$. Once the variable $A^0$ has been constructed, the Higgs field $\chi$ and the $A^i$ fields in the S--K model follow immediately from eq.(3), which gives $ \partial^0 \chi$ in terms of $A^0$, and $A^i$ in terms of $j^i$ and $\partial^i \chi$. We look therefore for an operator valued distribution solution of eq.(16), defined in a Hilbert space $\H$, with a cyclic vector $\psz$ invariant under space--time translations. The one point function of $A^0$ is then constant, and we will fix it to 0 for the moment; the two point function $W(x-y)$ is of positive type, and its Fourier transform is therefore a measure, $\tilde W (k)$, satisfying $$ |{\bf k}|^4 \, \tilde W(k) = J^{00} (k) \eqno(17) $$ The solution of eq.(17) is unique up to $\delta({\bf k}) a(k^0)$, and this term is excluded if in $\H$ there is only one vector invariant under space translations. Moreover, the solution exists if and only if $$ { \tilde J^{00} (k) \over |{\bf k}|^4 } = {1 \over m^2 {\bf k}^2} \, \delta(k^2 - m^2) \, \theta(k^0) \eqno(18) $$ is a measure (see eqs.(9),(10)). This is true for the S--K model in space dimensions $d=3$, but not for $d=2,1$, nor for the Schwinger model. \msk\ni 1. St\"uckelberg--Kibble model in $3+1$ dimensions. \ssk The solution of eq.(16) is in this case uniquely determined by the one and two point functions, assuming that all the higher order truncated correlation functions vanish. In order to express the solution in terms of the observable Weyl algebra $\A$, it is enough to notice that the form $[g,g]$, which defines the state $\oz$ on $\A$, remains finite on $\triangle^{-1} \S$, defined in Fourier space (on the support of $\delta(k^2 - m^2)$) by $\{ \tilde f / {\bf k^2} \}$, $f \in \S(\reali^4)$; the Weyl operators can then be extended by strong continuity to $\triangle^{-1} \S$, since sequences $W(f_n)$ of Weyl operators converge strongly, in a GNS representation over a quasi--free state $\omega$, if and only if $f_n$ converge strongly in the scalar product $[f,f]$ which defines $\omega$ (eq.(9)). The solution of eq.(16) exists therefore in the strong closure of the observable algebra, in the vacuum representation. Given $A^0$, the Higgs field $\chi$ is determined by eq.(3), namely $e \, \partial^0 \chi = - \partial_0^2 A^0$, and, as already discussed for $A^0$, we may construct $\chi$ as $$\chi = - {1 \over e} \, \partial_0 A^0 \eqno(19) $$ We have therefore constructed the field algebra, in the Coulomb gauge, as the Weyl algebra over the extended space $\triangle^{-1} \S$; moreover, since this algebra is regularly represented by $\oz$, we may include in the field algebra all the bounded functions of the fields. The observable algebra is {\it strongly dense} in the field algebra, in the vacuum representation $\piz$, and there are {\it no charged states\/}. {\it The gauge group\/} consists of the automorphisms $\gamma^{\lambda \mu}$ defined by $$ \gamma^{\lambda \mu} \, (e^{i A^0 (f)}) = e^{i \lambda \; Re \tilde f(m, {\bf0} )} \; e^{i \mu \; Im \tilde f(-m, {\bf 0} )} \; e^{i A^0(f)} \eqno(20) $$ corresponding to $$ A^0({\bf x},0) \mapsto A^0({\bf x},0) + \lambda \ \ \ \ \ \ \ \ \ \ \chi({\bf x},0) \mapsto \chi({\bf x},0) + \mu $$ and {\it it is broken\/} in $\piz$ to the identity (See Proposition 3). It follows imediately from eq.(20) that the space--time translations, defined (by construction) on the field algebra by the unitary group which implements the space--time translations for the observable algebra in the vacuum representation, do not commute with $\gamma^{\lambda \mu}$. A representation $\pinv$ of the field algebra can be immediately constructed as the direct sum of the GNS representations over the states $\gamma^{\lambda \mu \; {*}} \oz$, and space time translations commuting with all $\gamma^{\lambda \mu}$ can be easily constructed on the algebra generated by the Weyl algebra over $\triangle^{-1} \S$ and the centre of its strong closure in $\pinv$; they have gauge invariant implementers, given in each irreducible representation by the action of the gauge automorphisms (which are strongly continuos in $\pinv$) on the implementers in $\piz$. (See Prop.4). A field algebra with a non--trivial centre, and the same structure for the space--time translation automorphisms, is also obtained if the time evolution is constructed [16] as a strong limit of infrared cut--off dynamics defined by Hamiltonians invariant under the gauge group, with a strong topology invariant under the gauge automorphisms. The mass spectrum of the model is associated to the spontaneous breaking of the automorphisms $\glm$, through a generalized Goldstone theorem [10]. \msk\ni 2. St\"uckelberg--Kibble model in $2+1$ and $1+1$ dimensions; Schwinger model. \ssk For the S--K model in $2+1$ and $1+1$ dimensions, and for the Schwinger model, the above construction does not apply, since the quadratic form which defines the vacuum state on the observables is divergent on $\triangle^{-1} \S$, and cannot in fact be extended to a positive form on $\triangle^{-1} \S$ which still majorizes the extension of the symplectic form to $ < \triangle^{-1} f , g>$, $f,g \in \S$, a necessary condition for the positivity of the resulting state. The construction of the field algebras for these models can be done by an extension of the observable Weyl algebra to an algebra defined in an abstract way as the Weyl algebra over a space of $C^\infty$ functions (linearly bounded in the space variables) $\triangle^{-1} \S$, with a symplectic form $$ invariant under space--time translations and extending the symplectic form on $\S \times \S$ [14]. As a result of an analysis of the relation between the symplectic form $$, and the quadratic form $[g,g]$, $f \in \triangle^{-1} \S$, $g \in \S$, the extension of the vacuum state to the Weyl algebra of the fields is found to be unique and given by $$ \oz ( \exp i \alpha A^0 (f)) = 0 \ \ \ \ \ \ \ \ {\rm if} \ \ \tilde f (m,0) \neq 0 \eqno(21) $$ The vacuum representation of the field algebra is therefore {\it non regular}, i.e. not strongly continuous in the parameters of the Weyl groups; the variables $A^0$, and $\chi$ for the S--K model, do not exist as field variables, but only in the exponentiated (Weyl) form $\exp i \alpha A^0 (f)$, $\exp i \beta \chi (f)$, $f \in \S$. It follows from eq.(21) that the automorphisms $\glm$ are unbroken in the GNS representation of the field algebra defined by the unique extension of the vacuum state. The application to the vacuum of the charged field variables gives therefore rise to charged states, orthogonal to the vacuum representation of the observables, and the representations of the observable algebra obtained by the GNS construction over such states are easily seen to be inequivalent to the vacuum representation. The expectation value of the electric field gives rise to a non--trivial Gauss charge in the charged sectors. The space translation automorphisms are well defined on the (Coulomb gauge) field algebra and are implemented by strongly continuous unitary groups; the (space) momentum is therefore well defined also in the charged sectors. The time evolution automorphisms, which exist on the field algebra as a consequence of the invariance under time translations of the extended symplectic form, leave the (unique extension of) the vacuum state invariant; they are therefore unitarily implemented and define a time evolution of the charged states, which give rise to representations of the observable algebras which are inequivalent for different times. The same result is obtained, by invariance of the vacuum under time translations, if one considers the states obtained from a charged state by applying time translations automorphisms to the {\it observable} algebra. The implementers of the time translations are therefore not strongly continuous, and have no generator, i.e. the Hamiltonian does not exist in the charged sectors. (See Propositions 2 and 5). As in the general analysis given above, the reason is that the gauge automorphisms $\glm$ do not commute with the time evolution automorphisms, and therefore the time evolution of a charged state gives rise to states with different values of the (unbroken) charges, with an electric flux at infinity which oscillates in time. \bsk\bsk\ni {\bf Acknowledgements}. The scheme exposed here has been elaborated in collaboration with F.Strocchi. Thanks are also due to the organizers of the Colloquium for the opportunity given and for the very stimulating atmosphere. \vfill\eject\references \bye