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\centerline{\bf References}
\bsk
\input ref.tmp}
%Fine delle def. generali
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\def \endproof {$q.e.d.$}
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%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