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\begin{titlepage}
\begin{flushright}
UWThPh-1995-12\\
ESI-226-1995\\
May 19, 1995
\end{flushright}
\vspace{2cm}
\begin{center}
{\Large \bf A Fermi Field Algebra as Crossed Product}\\[50pt]
Nevena Ilieva$^{*,\, \sharp}$, Heide Narnhofer \\
\vspace{0.2cm}
Institut f\"ur Theoretische Physik \\
Universit\"at Wien \\
and\\
Erwin Schr\"odinger International Institute \\
for Mathematical Physics \\


\vspace{2cm}
{\bf Abstract} \\
\end{center}
On the example of Luttinger model and Schwinger model we consider the 
observable algebra of interacting fermi systems in two--dimensional 
space--time and construct field algebra related to it as a crossed product
with some automorphism group. Fermi statistics results for conveniently
chosen automorphisms. The extension of time evolution to the field algebra
and its asymptotic behaviour are treated. For the Luttinger model time 
evolution is asymptotically anticommutative, while for the Schwinger model
we find a reformulation of confinement.
%\end{center}

\vfill
\noindent $^{\star)}$ Work supported in part by ``Fonds zur F\"orderung der
wissenschaftlichen Forschung in \"Osterreich'' under grant 
M083--PHY--1994/95 
\vspace{0.1cm}

\noindent $^{*)}$ Lise--Meitner Fellow
\vspace{0.1cm}

\noindent $^{\sharp)}$ permanent address: Bulgarian Academy of Sciences, 
%\hskip3.5truecm
Institute for Nuclear Research and Nuclear Energy, Sofia 
\end{titlepage}

\section{Introduction}
The Bose--Fermi duality in one space dimension has been successfully
used for solving various problems in $(1+1)$--dimensional field theories
and in 1--dimensional models in solid state physics. Starting from the
pioneering works by Jordan \cite{J}, Born \cite{BN}, Mattis--Lieb 
\cite{ML}, Klaiber \cite{K},
different aspects of this phenomenon and different approaches, both to
its technical realization and to its physical meaning have been
considered.
%(for a recent review on the literature, see, for example \cite{*}).
However, consistent expressions exist so far only for fermion bilinears 
(directly connected to the observables of the theory) while the 
explicit reconstruction of fermions themselves back from the bosonic
variables is more subtle. This problem traces back to a
question of principal importance: whether and in which cases conclusions
about the time evolution of charged fields can be drawn from the
knowledge of the time evolution of the observables.

First rigorous results on the possible fermi behaviour of operators acting 
on a bose field can be found in \cite{SW} on the example of the massless Skyrme
model. 
An important contribution to the solution of the problem is done
in \cite{CR} where local fermi fields are constructed as strong
limits on a dense set of states of specific bosonic models. In a series
of papers \cite{AMSa} further progress is achieved since these local fermi 
fields are constructed as ultrastrong limits of bosonic variables in all
representations that are locally Fock with respect to the ground state
of the massless scalar field. There, an appropriate framework for the
construction of anticommuting variables out of commuting ones is found
to be provided by the nonregular representations of (canonical
extensions of) CCR algebras \cite{AMSb}. However, the question about relation 
(if any) between charged and non--charged field evolution still remains
open.

In the present paper we propose a solution to this problem that makes
use of the construction of the field algebra as a crossed product of the
observable one by the $\alpha$--action of {\bf Z}, $\alpha$ being a
not--inner automorphism of the latter.

The relevance of the crossed product C*--algebra extensions for the relation
of the field algebra to the observable algebra is first pointed out in
\cite{DHR} where the problem of constructing field groups is reformulated as
a problem of constructing extensions of the observable algebra by a group
dual. Also, they are discussed in the context of C*-- and W*--dynamical
systems in \cite{H}. Explicitly,
crossed products of C*--algebras by semigroups of endomorphisms are
introduced when proving the existence of a compact global gauge group
in particle physics given only the local observables \cite{DR}. The problem
of extension of automorphisms from a unital C*--algebra to its
crossed product by the action of a compact group dual becomes important
in the structural analysis of symmetries in the algebraic setting of
Quantum Field Theory \cite{H}, where in the case of a broken symmetry this 
allows for concrete conclusions about the vacuum degeneracy \cite{BDLR}.
\nopagebreak{

We restrict ourselves to the more simple case of a
crossed product generated by the action of a not inner automorphism of the
observable algebra with a discrete group and identify the resulting
object with a charged field algebra. The conditions under which space
translations can be extended from the observable to the field algebra
make the algebra extension essentially unique.
This (noncanonical in the sense of \cite{AMSb}) extension of the observable
algebra yields an extension of its states whose properties are 
discussed. The question about the fermion evolution finds here a natural
answer, the crucial point being a compatibility relation between the
automorphism used in the crossed product and the time evolution, i.e.
a property taking place on the observable algebra. For its realization
the structure of the energy spectrum of the model under consideration
is essential. The gauge group and its action for a crossed product
field algebra are also defined.}

In our approach we stay state independent and do not consider strong limits. 
Thus we cannot get the CAR relations in the renormalized form 
$\{\Psi^\dg(x), \Psi(y)\} = z \delta (x-y)$, where the renormalization 
constant $z$ goes to zero in some limit. We rather take as characteristic of 
fermi fields their asymptotic anticommutativity.

Another advantage of envisaging the fermionization as a crossed product
is the fact that the field algebra inherits in a natural way the net
structure of the observable algebra. Therefore it is evident that
global properties do not affect the construction of fermions in
accordance with the observation in \cite{AMSa}.

On the other hand, the crossed product construction is not restrictive
enough to guarantee a statistic theorem. On the contrary, an interesting
feature of the algebra so obtained is the possibility, depending on the
particular choice of functions that determine the automorphism $\alpha$,
to endow this algebra with a ``zone'' structure, where also fields
with fractional statistics are present. The specific conditions under
which such fields could be provided with a stable time evolution will
be considered elsewhere.

\section{The crossed product algebra}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\setcounter{equation}{0}

We start with the CCR (Weyl) algebra $\A(\V_0,\sigma)$ over the real
symplectic space $\V_0$ with symplectic form $\sigma$, generated by
unitaries $W(\Phi)$, with $\Phi := (f,g) \in \V_0$, which satisfy
\beqa
W(\Phi_1)\; W(\Phi_2) &=& e^{i \sigma(\Phi_1,\Phi_2)/2}\; W(\Phi_1 +
\Phi_2) \no \\
W(\Phi)^* &=& W(-\Phi) = W(\Phi)^{-1}
\eeqa

The elements of $\A$ are of the form
\beq
A = \sum_i c^{(i)} \; W(\Phi_i) := \sum_i c^{(i)} \; W_i, \qquad
c^{(i)} \in {\bf C}, \qquad 
\sum_i |c^{(i)}| < \infty.
\eeq
We can also consider its closure as C*--algebra. This algebra can be
enlarged to another CCR algebra by enlarging the space $\V_0$ to a space
$\V$, in a way that $\sigma$ in (2.1) appears to be the restriction on
$\V_0$ of the symplectic form of $\V$. This view point was taken in
\cite{AMSa}. There, with appropriate
ultrastrong limits it was shown that fermi field operators can be
obtained, so that the fermi algebra belongs to the weak closure of
CCR$(\V,\sigma)$ in appropriately taken representations \cite{AMSa,CR}.
Instead of doing this we will construct a new algebra $\F$ such that
$$
\mbox{CCR}(\V_0) \subset \F \subset \mbox{CCR}(\V)
$$
without referring to representations and then
show that $\F$ can be considered as fermi field algebra.

For $\alpha$ a free \cite{T} (so not inner) automorphism of CCR$(\V_0,\sigma)
= \A$ we can consider the crossed product
$$
\F = \A \st{\alpha}{\times} {\bf Z}.
$$
This corresponds (compare \cite{A}) to adding a unitary operator $U$
with all its powers, so that one can formally write
$$
\F =  \sum_n \A \; U^n,
$$
with $U$ implementing the automorphism $\alpha$ in $\A$:
$$
U \; A \; U^{-1} = \alpha \; A.
$$
$U$ should be thought of as charge creating operator and $\F$ is a
minimal extension.

The multiplication law in $\F$ is
\beq
\sum_n A_n \; U^n \sum_k B_k \; U^k = \sum_{n,k} A_n \; \alpha^n \; B_k \;
U^{n+k}
\eeq
and we take $\alpha$ to be
$$
\alpha \; W(\Phi) = e^{i\sigma(\bar \Phi,\Phi)} \; W(\Phi), \qquad
\alpha \equiv \alpha_{\bar \Phi}
$$
$$
\bar \Phi := (\bar f,\bar g) \in \V\setminus\V_0, \qquad \V_0
\subset \V.
$$

Crossed products are unitarily equivalent, i.e.
$$
\A \st{\alpha}{\times} {\bf Z} \approx \A \st{\wh \alpha}{\times}
{\bf Z}
$$
if $\alpha \circ \wh \alpha^{-1}$ is an inner automorphism of $\A$. 
Therefore our algebra $\F$ depends only on the equivalence class
$\{ \bar \Phi\} \in \V/\V_0$, though for the explicit calculations we
will specify $\bar \Phi \in \V\setminus\V_0$. The automorphism 
$\alpha$ has to be free, so $\F$ has trivial center like $\A$ . 
Since $\alpha$ is
implemented by $W(\bar \Phi)$ in CCR$(\V)$, $\F$ in a natural way
(identifying $U = W(\bar \Phi))$ is a subalgebra of CCR$(\V)$.

By writing an element $F \in \F$ as $F = \sum_n \; A_n \; U^n, A_n
\in \A$, we see that it is convenient to consider $\F$ as (infinite)
vector space with $U^n$ as basic unit vectors and $A_n =: (F)_n$ the
components of $F =: \{A_n\}$. The algebraic structure of $\F$ is such
that multiplication is not componentwise but (2.3) says that
$$ 
(F \cdot G)_m = \sum_{n} F_n \; \alpha^n \; G_{m-n}
$$
The algebra $\A$ can be identified with the zero component in $\F$,
$\F$ is actualy a left $\A$--module. In components we have
\beqan
(U^k)_n &=& \delta_{kn} \\
(U^*)_n &=& \delta_{-1,n}
\eeqan
so that the left action of $U$ is the shift.

Further, we can write
\beq
F = \sum_n  A_n \; U^n = \sum_{n,i} c_n^{(i)}W_i U^n 
:= \bigoplus_n  \sum_i  c_n^{(i)}  F_i^{(n)}.
\eeq

The (non normalized) operator
$$
F_i = \sum_n W_i \; U^n
$$
defines the $U$--orbit through $W_i$, so the last of eqs. (2.4) gives
an orbit decomposition of the elements of $\F$. 

There are two questions that naturally arise. Given an automorphism on $\A$,
can it be extended to $\F$ and how unique is this extension. In the 
physical applications we are especially concerned with space translations and
time evolution. A similar question concerns the extension and its uniqueness 
for given states on $\A$.

We concentrate first on the extension $\wt \rho$ of an automorphism $\rho$ 
of $\A$. Since
all elements can be written as sums and product of $A \in \A$ and $U$, i.e.
$\{ \delta_{1n}\}$ and the action of $\wt \rho$ on $A$ must coincide
with $\rho$, we make the ansatz
$$
\wt \rho \{ \delta_{1n}\} = \{ V_{\rho_n}^{(1)}\}, \qquad 
$$
$\wt \rho \in \F $ requires $V_{\rho_n}^{(1)} \in \A$.
Then $\{ V_{\rho_n}^{(1)}\}$ will fix $\wt \rho$.
The consistency of $\rho$ and $\wt \rho$ on the subalgebra $\A$ of $\F$ 
requires
$$
1 = U \cdot U^* = \wt \rho U \cdot \wt \rho U^* = \left\{ \sum_n 
V_{\rho_n}^{(1)} \alpha^n
V^{-1)}_{\rho_{k-n}} \right\} = \{ \delta_{0k}\}.
$$

This equation is satisfied by
\beq
\{ V_{\rho_n}^{(1)}\} = \{ V_{\rho}^{(1)} \; \delta_{1n}\}, \qquad
V_{\rho}^{(1)} \in \A
\eeq
We refer to \cite{HN} for a discussion on the uniqueness of this choice.

Further, for $W \in \A$, we have
$$
\wt \rho(U \cdot W) = \wt \rho U \cdot \rho W = \wt \rho(\alpha W \cdot U)
$$
and from (2.3), (2.5) follows
$$
\{ V_{\rho}^{(1)} \delta_{1n}\} \{\rho W \delta_{0n}\} = 
\{ \rho \alpha W \delta_{0n}\} \{ V_{\rho}^{(1)} \delta_{1n}\}
$$
so that
$$
V_{\rho}^{(1)} \alpha \rho W = \rho \alpha(W) V_{\rho}^{(1)}
$$
or, equivalently,
$$
V_{\rho}^{(1)} \alpha \rho W  V^{*(1)}_{\rho} 
=: \gamma_\rho \alpha \rho W = \rho \alpha W.
$$

This can only be satisfied for some $V_\rho^{(1)} \in \A$,
if the automorphism $\gamma_\rho$,
\beq
\gamma_\rho = \rho \alpha \rho^{-1} \alpha^{-1} 
\eeq
is an inner automorphism of $\A$. For example, 
$\alpha$ is easily seen to be extendible to an automorphism also of $\F$, 
since the 
corresponding $\gamma_{\alpha}$ is the identity transformation, so that
$\wt \alpha U = U$.
Since the CCR$(\V_0)$ algebra $\A$ has trivial center, the unitary operator
that implements an automorphism is unique up to a phase factor. 

Apart from the condition that $\rho \alpha \rho^{-1} \alpha^{-1}$ is
inner no other conditions have to be satisfied.

We should mention that the question if an automorphism of the observable
algebra can be extended to the field algebra is also treated in \cite{BDLR} in
the context of the theory of \cite{DR} where the field algebra is obtained
as crossed product over a specially directed symmetric monoidal subcategory 
End $\A$ of unital endomorphisms of $\A$ as generalization
of our automorphism group $\alpha$. There two conditions enter, one is the 
appropriate replacement of our demand that $\rho \alpha \rho^{-1} \alpha^{-1}$
has to be inner, the other is a compatibility condition
with the net structure. We do not have any counterpart to this condition.
It will turn out that in our case the net structure of the field algebra is a
consequence of the net structure of the observable algebra and of the
compatibility relation for $\alpha$. 

We return to our explicitly chosen $\alpha$. We consider $\rho$ that
are quasifree automorphisms on CCR$(\V_0)$, which means that they are 
of the form $\rho W(\Phi) = W(\Phi_{\rho})$. The inverse of the map 
$\Phi \rightarrow \Phi_{\rho}$ we denote by $\Phi \rightarrow 
\Phi_{-\rho}$ and $\rho$ has to preserve the symplectic structure in 
$\V_0$, so that $\sigma (\Psi, \Phi_{-\rho}) = \sigma (\Psi_{\rho}, 
\Phi)$. To start, this bijection is defined on $\V_0$. We have
$$
\gamma_\rho W(\Phi) = e^{i\sigma(\bar \Phi,\Phi_{-\rho}) - i \sigma(\bar
\Phi,\Phi)} \; W(\Phi).
$$
Assume that
$$
\sigma(\bar \Phi,\Phi_{-\rho}) - \sigma(\bar \Phi,\Phi) = \sigma(\Psi,
\Phi)
$$
for some $\Psi \in \V_0$. Then, on one hand, we have enlarged $\rho$ to
a quasifree automorphism on CCR$(\V)$ with $\bar \Phi_\rho = \bar \Phi
+ \Psi$, on the other hand, $\gamma_\rho$ satisfies our requirement with
\beq
V_\rho = W(\Psi) = W(\bar \Phi_\rho - \bar \Phi) \in \A.
\eeq
That the condition is satisfied for appropriately chosen $\bar \Phi$
if we consider space translation and with some restriction on time 
evolution will be discussed in Section 3. How this restriction is 
satisfied in physical models and what are the physical consequences 
will be discussed in Section 4.

The second principal question concerns construction of states over $\F$. Let
$\omega(\cdot)$ be a state over the algebra $\A$ and $\pi_\omega$ the
cyclic representation of $\A$ associated with it through the GNS
construction
$$
\omega(W(\Phi)) = \langle \omega |\pi_\omega(W(\Phi)) | \omega\rangle =
\langle \omega | \Phi\rangle_\omega,
$$
where $| \omega \rangle$ denotes the vacuum. Then the vectors
$|\Phi\rangle_\omega =  \pi_\omega(W(\Phi))|
\omega\rangle$ generate $\Ha_0$, the representation space of $\pi_\omega$.

The representation $\pi_\omega$ itself is given by
$$
\pi_\omega(W(\chi))|\Phi\rangle_\omega = e^{i \sigma(\chi,\Phi)/2}|
\Phi + \chi\rangle_\omega
$$
and the scalar product is
$$
{}_\omega\langle \chi|\Phi\rangle_\omega = e^{-i\sigma(\chi,\Phi)/2}
\omega(W(\Phi - \chi)).
$$
The crossed product algebra acts in a larger Hilbert space $\Ha$ which
may be considered as a direct sum of charge--$n$ subspaces (the
justification for this terminology will be given in Section 3), each of
them being a representation space corresponding to the state $\omega \circ
\alpha^{-n}$. We can imbed $\Ha_0$ into $\Ha$ and denote now the vacuum
by $|\Omega\rangle$, thus expressing the fact that we consider it as a 
vector in $\Ha$.
Then $U^k|\Omega\rangle$ can be denoted as
$$
U^k |\Omega \rangle = |\Omega_k\rangle
$$
and the vector space structure of $\F$ suggests that
$\langle \Omega_k|\Omega_n\rangle = \delta_{kn}$
with the identification $|\Omega\rangle = | \Omega_0\rangle$.

Then
\beqa
|F_i^{(k)}\rangle &=& W(\Phi_i)|\Omega_k\rangle = 
U^k \alpha^{-k} W(\Phi_i)|\Omega\rangle = \no\\
&=& e^{-ik\sigma(\bar \Phi,\Phi_i)} \; U^k|\Phi_i\rangle =
e^{-ik\sigma(\bar \Phi,\Phi_i)} | \Phi_i^{(k)}\rangle
\eeqa
so that $|F_i^{(k)}\rangle$, varying over $\Phi_i$ generate the charge--$k$
space $\Ha^{(k)}$ and varying over $k$ we get the complete Hilbert space $\Ha$.

%Vectors in subspaces $\Ha^{(n)}$ have the form
%\beq
%|F^{(n)}\rangle = \sum_i c_n^{(i)} e^{-in\sigma(\bar \Phi,\Phi_i)}
%| \Phi_i^{(n)}\rangle.
%\eeq
Arbitrary linear functionals built by vectors in $\Ha$ considered as states 
over $\A$ read
\beq
\langle F_i^{(n)}|W(\chi)|F_i^{(k)}\rangle =
\delta_{nk} e^{-i\sigma(\Phi_i,\chi)}\omega(\alpha^{-n} W(\chi)) =
\delta_{nk} e^{-i \sigma(\Phi_i + n\bar \Phi,\chi)} \omega(W(\chi)).
\eeq
On the other hand, given two states
on $\A$, $\;\omega_1$ and $\omega_2$, a quantum mechanical superposition of 
them to a state on $\F$ is only possible if the same representation $\pi$
is associated with both  $\omega_1$ and 
$\omega_2 \circ \alpha^k$  for some $k$ so that the new state 
is constructed with a vector 
$$
c_0|\Phi_1^{(0)}\rangle + c_k|\Phi_2^{(k)}\rangle, \qquad c_0, c_k \in {\bf C}
$$
If we take into account that $\omega$ is irreducible, $\alpha$ free and
$\omega \circ \alpha^n$ not normal with respect to $\omega$ we
can conclude that the extension of the state over $\A$ to a state over
$\F$ is uniquely given by the expectation value with $|\Omega_0\rangle$
in this representation. With
$$
F = \sum_{i,n} c_n^{(i)} \; W(\Phi_i) U^n
$$
we get
$$
\omega(F^*F) = \sum_{i,j,n} \bar c_n^{(j)} c_n^{(i)} e^{i\sigma(\Phi_i,
\Phi_j)/2} \; e^{-in\sigma(\bar\Phi, \Phi_i-\Phi_j)} \;
\omega(W(\Phi_i - \Phi_j)).
$$

Therefore, the states over $\F$ inherit in a natural way the whole
structure and symmetry properties from the states over $\A$. 

\section{The crossed product as field algebra}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\setcounter{equation}{0}

The important result in \cite{DR} is the theorem that the observable algebra
$\A$ together with the set of particle states (that form a DR--category) 
can be enlarged to a field algebra on which a gauge group acts, that
leaves the observable algebra elementwise invariant.

In the case of (free) fermions in one dimension the algebra is built by
creation and annihilation operators $a(f)$, $a^\dg(g)$, $f,g \in 
L^2({\bf R})$. The observable algebra is built by monomials with the
same number of creation and annihilation operators
\beq
\prod_{i=1}^n a^\dg(f_i) \prod_{j=1}^n a(g_j).
\eeq
They are invariant under the automorphism group
\beq
\gamma_\nu \; a(f) = e^{2 \pi i \nu} \; a(f) , \qquad
\nu \in [0,1) = S_1.
\eeq
The observable algebra contains the current algebra built by $a^\dg(x)a(x)$, 
invariant under the local gauge group $a(f) = a(e^{i\nu(x)}f(x)$. One still 
has to check whether this algebra is well defined.
>From e.g. \cite{ML} we know that the current algebra leads to
CCR$(\V_0)$ in appropriate representations.

However, if we consider as observable algebra the C*--algebra obtained as a 
norm closure from (3.1) the passage to the CAR--algebra as crossed
product is not possible as it was shown in \cite{B}. Therefore the closure
has to be taken with respect to some other topology.
Consideration of CCR$(\V_0)$ as a von
Neumann algebra would solve the problem, but we do not favour it because 
we want to stay representation independent as
much as possible. On the other hand, we cannot ignore the representation
completely, because in the C*--norm CCR$(\V_0)$ is not separable
whereas the fermi observable algebra is. This is not really a problem:
we are only interested in states that are locally normal with respect to 
the vacuum.
Therefore we take the following view point: 
we consider CCR$(\V_0)$ as a net of von Neumann algebras closed
locally in some representations, so that there is no contradiction with 
\cite{B}. To be more precise, we consider the local net
% The fact whether the
% field algebra is a subalgebra of CCR$(\V)$ is by far less evident.
$$
\A_\Lambda \subset  \{\prod a^\dg(f) a(g), \mbox{ supp }f,g \in 
\Lambda\}'',
$$
$$
\A = \ol{\bigvee_\Lambda \A_\Lambda}
$$
The union is taken in norm and this algebra does coincide with
$$
\A = \ol{\bigvee_\Lambda \mbox{CCR}(\V_0,\Lambda)''}
$$
where we have some freedom in choosing CCR$(\V_0,\Lambda)$, e.g. 
$(f,g) \in \V_0 := (\C_0^{\infty} \times \C_0^{\infty})$, 
supp~$f, g \subset
\Lambda$, where $\C_0^{\infty}$ is the space of test functions that are
infinitely differentiable and with compact support.

Our first step in the identification of $\F$ with a Fermi type algebra
$$
\ol{\bigvee_\Lambda \{ b(f),b^\dg(g),\mbox{ supp }f, g \in
\Lambda\}''}
$$
is to find the gauge automorphism, but this is trivial in the context of
crossed products. Defining
\beq
\gamma_\nu \; U^n = e^{2\pi i \nu n}\; U^n \qquad
 \gamma_\nu \; W_i = W_i
\eeq
and with (2.4) taken into account, we get
\beq
\gamma_\nu \;F_i = \sum_n e^{2\pi i \nu n}\; W_i U^n = 
\bigoplus_n e^{2\pi i \nu n}\; F_i^{(n)}
\eeq
Evidently, $F_i^{(0)}$, the elements of $\A$ as subalgebra of $\F$,
are invariant under the gauge automorphism (3.3)
$$
\gamma_\nu \; F_i^{(0)} = \gamma_\nu \; \{W_i \; \delta_{0k}\} = 
\{ W_i \; \delta_{0k}\} = F_i^{(0)}
$$
Also, $\gamma_\nu$ commutes with the structural automorphism $\alpha$,
$\;\alpha \circ \gamma_\nu = \gamma_\nu \circ \alpha$. 

For the representation $\pi_\Omega$ discussed in Section 2 we observe
(see (2.8))
\beq
\gamma_\nu ( W(\Phi_i) | \Omega_k\rangle) =
e^{2\pi i \nu k} \{W(\Phi_i)\delta_{kn}\} |\Omega\rangle = \\
e^{2\pi i \nu k} \; W(\Phi_i)|\Omega_k \rangle
\eeq
so that we really can interpret vectors $|F_i^{(k)}\rangle$ as belonging to the
charge--$k$ subspace. Thus, the (gauge invariant) state over $\A$,
$\omega(A)$, is extended to a gauge invariant state over $\F$
$$
\gamma_\nu \circ \Omega\;(\{F_i^{(n)}\}) = \Omega\;(\gamma_\nu \{F_i^{(n)}\}) = 
\delta_{n0} \; \Omega\;(\{e^{2\pi i \nu n}\;F_i^{(n)}\}) = \Omega 
\;(F_i^{(n)}).
$$

The next task is to reconstruct the net character of the field algebra.
This means that we want to find subalgebras $\F_{\,\Lambda}$ for which
the following relations take place
\beqan
\F_{\,\Lambda} \subset \F_{\,\bar \Lambda}, & & \mbox{ if } \;
\Lambda \subset \bar \Lambda \\
\sigma_x \F_{\,\Lambda} &=& \F_{\,\Lambda + x} \\
\F &=& \ol{\bigvee_\Lambda \; \F_{\,\Lambda}}
\eeqan

To show that this is really the case, we shall make use of two
important features of the crossed product algebras in question:
first, the extendibility of space translations to automorphisms of
the field algebra, and second, the unitary equivalence of crossed
products with structural automorphisms which differ by an inner
automorphism of the observable algebra $\A$.

Let us consider the observable algebra for a given region $\Lambda$
and choose $\bar \Phi \in \V\setminus\V_0$,
$\bar \Phi_x - \bar \Phi \in \V_0$ such that
$$
\left. \alpha_{\bar \Phi}\right|_{\A_{\, \wh \Lambda}} = id, \qquad
\wh \Lambda \subset \Lambda^c,
$$
where $\Lambda^c$ is the causal complement of $\Lambda$. Then we define
$$
\F_{\, \Lambda} := \A_{\, \Lambda} \st{\alpha_{\bar \Phi}}{\times} {\bf Z}
\subset \F.
$$
Space translations act in $\F_{\, \Lambda}$ as
$$
\sigma_x \{A_n\} = \{ \sigma_x A_n \cdot U_x^{(n)}\},
$$
with $U_x^{(n)}$ implementing the (inner) automorphism $\sigma_x
\alpha_{\bar \Phi}^n \sigma_x^{-1} \alpha_{\bar \Phi}^{-n}$. We then
get, in accordance with (2.3),
$$
\{ \sigma_x A_n U_x^{(n)}\} \{ \sigma_x B_m U_x^{(m)}\} =
\left\{ \sum_k \sigma_x A_k \; \alpha_{\bar \Phi_x}^k \,
B_{n-k} U_x^{(n)}\right\},
$$
which is exactly the multiplication law for the crossed product
algebra 
$$
\sigma_x \A_{\, \Lambda} \st{\alpha_{\bar \Phi_x}}{\times} {\bf Z}
$$
with
\beqan
\alpha_{\bar \Phi_x} &=& \sigma_x \alpha_{\bar \Phi}\sigma_{-x} \\
\left.\alpha_{\bar \Phi_x}\right|_{ A_{\, \wh\Lambda}} &=& id, \qquad
\wh \Lambda \subset (\Lambda + x)^c
\eeqan

Therefore, we have
$$
\sigma_x \F_{\,\Lambda} = \sigma_x \A_{\,\Lambda} \st{
\alpha_{\bar \Phi_x}}{\times} {\bf Z} = \F_{\,\Lambda + x}
$$

The net structure of $\F$ appears as a consequence of the extendibility
of space translations to an automorphism in $\F$ which requires for a
choice of the structural automorphism $\alpha_{\bar \Phi}$ that is consistent
with the net structure of $\A$.

Finally, we have to verify compatibility of anticommutation relations
with the structure of the crossed product algebra $\F$ in order to
ensure the existence of odd elements in it. This will complete the
identification of $\F$ with a fermi field algebra corresponding to
the observable algebra $\A$. 

$\{\delta_{1n}\}$, the element in $\F$, that implements $\alpha_{\bar \Phi}$,
is an odd element in $\F_{\,\Lambda}$ if
\beq
\sigma_x \{\delta_{1n}\} \cdot \{\delta_{1n}\} + \{\delta_{1n}\} \cdot
\sigma_x \{\delta_{1n}\} = 0 \qquad \forall \; |x| > |\Lambda|.
\eeq
With (2.3), (2.5) this means
\beqan
\lefteqn{\{ W(\bar \Phi_x - \bar \Phi) \delta_{1n}\} \cdot \{ \delta_{1n}\}
+ \{\delta_{1n}\} \cdot \{W(\bar \Phi_x - \Phi)\delta_{1n}\}= }  \\
&=& \{ W(\bar \Phi_x - \bar \Phi)\delta_{2n}\} + 
\{ \alpha_{\bar \Phi} W(\bar \Phi_x - \bar \Phi)\delta_{2n}\} = \\
&=& \{(1 + e^{i \sigma(\bar \Phi,\bar \Phi_x - \bar \Phi)}) W(\bar \Phi_x - 
\bar \Phi) \delta_{2n}\} = 0
\eeqan

Thus, eq.(3.6) is satisfied if for $|x| > |\Lambda|$   
$$
e^{i \sigma(\bar \Phi,\bar \Phi_x - \bar \Phi)} = -1 \qquad 
$$
that is if the following relation holds
\beq
\sigma (\bar \Phi, \bar \Phi_x) = (2k + 1)\pi, \qquad k \mbox{ integer. }
\eeq
This is exactly the normalization condition used in \cite{AMSa} to choose an
approrpiate $\bar \Phi$ for the construction of the fermion creation operator.
It reads
$$
\lim_{x \ra \pm \infty} \sigma(\bar\Phi,\bar\Phi_x) = 
\pm (\bar g(\infty) - \bar g(-\infty))
\int \bar f(x) dx = (2k + 1)\pi,
$$
but  the limit is already attained for
$|x| > |\Lambda|$ because supp~$f,g \subset \Lambda$.

Then, for the $r$--th class elements in $\F$ it follows:
\beq
\sigma_x \{\delta_{rn}\} \{\delta_{rn}\} =
W_r (\bar \Phi_x - \bar \Phi) \delta_{2r,k} =
(-1)^{r^2} \{\delta_{rn}\} \sigma_x \{ \delta_{rn}\}
\eeq
so that for $r = 2n$ the elements commute and for $r = 2n + 1$ they
anticommute, thus providing a graded structure of $\F$.

Here, the sensitivity of the crossed product to the particular choice
of $\bar \Phi$ shows up. If the pair $\bar \Phi$ is scaled to
$\lambda \bar \Phi$, so that $\sigma$ is scaled to $\lambda^2 \sigma$, 
condition (3.6) in general fails. Instead, we get
\beq
\sigma_x \{ \delta_{1n}\} \{\delta_{1n}\} =
e^{-i \lambda^2(2k +1)\pi} \{\delta_{1n}\} \sigma_x \{\delta_{1n}\},
\eeq
which can be interpreted as a fractional statistics and therefore describes
an essentially different physical system.

However, it might happen that elements obeying fractional statistics, are
naturally present in the field algebra $\F$. In fact, this is exactly 
the situation, if the first odd element of $\F$ is not $\{\delta_{1n}\}$
but some $\{\delta_{\bar nn}\}$, i.e. if
\beqan
\lefteqn{
\sigma_x \{\delta_{\bar nn}\} \{\delta_{\bar nn}\} + \{\delta_{\bar nn}\}
\sigma_x \{\delta_{\bar nn}\} } \\
&=& \{ (W_{\bar n}(\bar \Phi_x - \bar \Phi) + \alpha^{\bar n} W_{\bar n} 
(\bar \Phi_x - \bar \Phi)) \delta_{2\bar n,k}\} = \\
&=& \{ (1 + e^{i \bar n^2 \sigma(\bar \Phi,\bar \Phi_x - \bar \Phi)})
W_{\bar n} (\bar \Phi_x - \Phi) \delta_{2 \bar n,k}\} = 0
\eeqan
so, instead of (3.6) we have the relation
$$
\sigma(\bar \Phi, \bar \Phi_x) = \frac{2k + 1}{\bar n^2} \pi.
$$
The graded structure is still present, with $2k \bar n$--classes being
commuting and $(2k+1)\bar n$--classes anticommuting ones. The elements
in the classes with numbers $m \in {\bf Z}/{\bf Z}_{\bar n}$ are
characterized by fractional statistics, satisfying a relation in formal
analogy to (3.9):
$$
\sigma_x \{ \delta_{mn}\} \{\delta_{mn}\} = e^{-i (m/\bar n)^2 (2k+1)\pi}
\{ \delta_{mn}\} \sigma_x \{\delta_{mn}\}.
$$
This offers an alternative approach to construct models with fractional
statistics.

Finally we note that $\A$ is a subalgebra of $\F$ for the gauge group
$\T = [0,1)$ while it is a subalgebra of CAR for the gauge group 
$\T \otimes {\bf R}$. Thus the crossed product algebra $\F$ being really
a Fermi algebra, does not coincide with CAR but is only contained in it.

%Finally we know that the fermi algebra allows also a local gauge group.
%Here (3.3) is replaced by
%$$
%\gamma \; a(f(x)) = a (e^{2\pi i \nu(x)} \; f(x))
%$$
%so that the gauge group is only felt in a local region. We want to
%rediscover a similar behaviour in our present fermi net. $\alpha_{\bar
%\Phi}$ corresponds to the fact that in
%$$
%W(\Phi_i) U^n
%$$
%we have created $n$ charges in the region $\Lambda$ together with pairs
%of fermions and antifermions corresponding to $\Phi_i$. Therefore $\Phi$
%can destroy the fermions in the region $\Lambda$ and can recreate them
%in another region $\bar \Lambda$. Correspondingly we can use a local
%gauge automorphism
%$$
%\gamma_s^{\chi} W(\Phi_i) U^n = e^{is(n - \langle \chi|\Phi_i\rangle)}
%W(\Phi_i)U^n
%$$
%with $\langle \chi|\bar \Phi\rangle = 1$ so that it acts trivially if the
%charge created by $\alpha_{\bar \Phi}$ is completely annihilated in the
%region $\Lambda$ by $\Phi$.

\section{Examples}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\setcounter{equation}{1}

We are now going to discuss two typical examples which demonstrate the
sensitivity of the construction described above to the physical content 
of the models. These two examples are Luttinger model \cite{L} and Schwinger
model \cite{S}. It is not our aim here to give an overview on the
enormous literature on these models or to enter in detail the far going 
conclusions drawn on their basis. What is important from the point of
view of the crossed product algebra construction is the essential
difference between the interactions they describe. The Luttinger model is
an example of a one--dimensional interacting fermionic system which is
nevertheless realistic enough (recently it has become even more popular
in connection with the ``Luttinger liquid'' behaviour of normal metals
\cite{And}). The Schwinger model gives an example of confinement, being
equivalent to a free massive scalar field theory in $(1+1)$--dimensional
space--time.

The models in question are described by the following Lagrangians:
$$
\cL_S = - \frac{1}{4} F_{\mu\nu} F^{\mu\nu} + i \bar \psi \gamma^\mu
(\partial_\mu - i e A_\mu)\psi \eqno(4.1\mbox{a})
$$
$$
\cL_L = i \bar \psi \gamma^\mu \partial_\mu \psi - \int j^\mu(x) V(x-y)
j_\mu(y) dy \eqno(4.1\mbox{b})
$$
where $V(x-y)$ is an even smooth function, $\psi(x)$ is a two--component 
spinor, satisfying
\beq
\{ \psi_i^\dg(x), \psi_j(y)\} = \delta_{ij} \delta(x - y)
\eeq
all other anticommutators vanishing, $A_\mu(x)$ is the vector potential,
$$
F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu, \qquad
\mu,\nu = 0,1
$$
and currents $j_\mu(x)$ are defined as
\beqa
j_\mu(x) &=& \bar \psi(x) \gamma_\mu \psi(x) \no \\
j_{5\mu}(x) &=& \bar \psi(x) \gamma_\mu \gamma_5 \psi(x) \\
j_{R,L}(x) &=& \frac{1}{2} (j_0(x) \pm j_1(x)) \no
\eeqa
with the two--dimensional $\gamma$--matrices
$$
\gamma_0 = \sigma_1 , \qquad \gamma_1 = - i \sigma_2 , \qquad
\gamma_5 = \gamma_0 \gamma_1 = \sigma_3,
$$
$\sigma_k$ being the Pauli matrices.

Hamiltonian densities have the form
$$
\Ha_S(x) = i \bar \psi(x) \gamma_1 \partial_1 \psi(x) - \frac{e^2}{2}
\int (j_L(x) + j_R(x))|x-y|(j_L(y) + j_R(y)) dy \eqno(4.4\mbox{a})
$$
$$
\Ha_L(x) = i \bar \psi(x) \gamma_1 \partial_1 \psi(x) + 4 \int j_R(x)
V(x-y) j_L(y) dy \eqno(4.4\mbox{b})
$$
with $j_L(x)$ and $j_R(x)$ --- left and right current respectively.
Therefore, for a direct comparison of the results it is convenient
to generalize (4.1b) (hence, (4.4b)) to the more realistic type of
two--body interaction
\addtocounter{equation}{1}
\beq 
\int (j_R(x) + j_L(x)) V(x-y)(j_R(y) + j_L(y))dy
\eeq
which does not affect solvability of the model but only causes minor
changes in the spectrum of excitations.

Now, in the momentum space Hamiltonians are
\beqa
H_S &=& H_0 + \frac{e^2}{2\pi} \int_0^\infty [\rho_1(p) + \rho_2(p)]
\frac{1}{p^2} [\rho_1(-p) + \rho_2(-p)] dp \no \\
H_L &=& H_0 + \int_0^\infty \wt V(p) [\rho_1(p) + \rho_2(p)][\rho_1(-p)
+ \rho_2(-p)] dp
\eeqa
where $H_0$ is the free Hamiltonian and the following notation is used:
\beqa
\rho_i(p) &=& \int dk \; a^\dg_i(k+p) a_i(k), \qquad p > 0, \no \\
\rho_i(-p) &=& \int dk \; a^\dg_i(k) a_i(k+p), \qquad p> 0
\eeqa
with $a_i(k)$, $a^\dg_i(k)$ being the Fourier transformed of $\psi_i(x)$,
$\psi^\dg_i(x)$:
$$
\psi_i(x) = \frac{1}{\sqrt{2\pi}} \int e^{ipx} \; a_i(p) \; dp
$$
$$
\{ a^\dg_i(q),a_j(p)\} = \delta_{ij} \; \delta(p-q).
$$

The semiboundedness of the free Hamiltonian $H_0$ is achieved after a 
Bogolyubov transformation of $a$'s and $a^\dg$'s, which effectively
describes the negative energy states filling (filling of the Dirac sea),
\beqa
a_1(k) &=& b(k) \theta(k) + c^\dg(k) \theta(-k) \no \\
a_2(k) &=& b(k) \theta(-k) + c^\dg(k) \theta(k).
\eeqa

The new creation and annihilation operators $b,b^\dg,c,c^\dg$ satisfy 
canonical anticommutation relations, but the vacuum is already defined
as
\beq
b(k) |0\rangle = c(k) |0\rangle = 0 .
\eeq

This procedure results in the appearance of an anomalous term in the
commutator of currents (4.7), which otherwise commuted
\beqa
[\rho_1(p),\rho_1(p')] &=& \;p \delta(p-p') \no \\{}
[\rho_2(p),\rho_2(p')] &=& -p \delta(p-p')
\eeqa

It is eqs. (4.10) that justify the so--called bosonization of the
two--dimensional models with fermions.
The calculation of the anomalies is done usually in this new vacuum,
but this is not essential. In \cite{GMR} the same result was also
obtained in temperature states. This is not surprising. It is an
algebraic relation, so it has to be state independent, provided the
densities are well defined, i.e. smearing over $p$ gives an (unbounded)
operator. This only works in the new, Dirac vacuum and in all states that are
locally normal with respect to it. It is one of the
achievements of our approach that we can find the observable algebra
as local net, so restricting the permitted states only on a local
basis. From this local basis we come back to the field algebra and
there is no need to check the anomalies in every state (that has not to be
globally normal, i.e. permits temperature). In this sense we interpret the
appearance of anomalies as a local effect.

For the corresponding spectra we get 
\beqa
\omega_L(p) &=& |p| (1 - \wt V(p))^{1/2} \\
\omega_S(p) &=& |p| \left(1 + \frac{m^2}{p^2}\right)^{1/2},
\qquad m = \frac{e}{\sqrt{\pi}}.
\eeqa
The CCR algebra $\A(\V_0,\sigma)$ in both cases is generated by the
unitaries
$$
W(\Phi) := W(f,g) = \exp \{i \int [f(x) \rho_A(x) + g(x) \rho_V(x)]dx\}
$$
$$
(f,g) \in \V_0 = (\C_0^\infty \times \C_0^\infty),
$$
$$
\rho_A(x) = \rho_1(x) - \rho_2(x), \qquad
\rho_V(x) = \rho_1(x) + \rho_2(x)
$$
which satisfy
\beq
W(\Phi_1) \; W(\Phi_2) = e^{i \sigma(\Phi_1,\Phi_2)/2} \;
W(\Phi_1 +  \Phi_2),
\eeq
$$
\sigma(\Phi_1,\Phi_2) \equiv \sigma((f_1 g_1),(f_2g_2)) = \int (f'_1 g_2 - 
f'_2 g_1) dx.
$$

The field algebra $\F$ may be constructed, following the procedure 
described in Sections 1 -- 3, with the help of an automorphism
$\alpha_{\bar \Phi}$
$$
\alpha_{\bar \Phi} : W(\Phi) \ra e^{i\sigma(\bar \Phi,\Phi)} \; W(\Phi),
\qquad \Phi \in \V_0,
$$
$$
\bar \Phi := (\bar f,\bar g) \in \V = (\partial^{-1} \C_o^\infty \times
\partial^{-1} \C_0^\infty)
$$
(a function $f(x)$ belongs to $\partial^{-1} \C_0^\infty$, $f(x) \in 
\partial^{-1} \C_o^\infty$ if $\partial f(x) \in \C_0^\infty$). 
Therefore, the functions $f,g$ have bounded Fourier components at $p = 0$
\beq
\int f(x) dx \sim \wt f(0) < \infty \qquad \forall \; f(x) \in \C_o^\infty
\eeq
while for functions $\bar f,\bar g$, as well as for their space-- and
time--translated this components might diverge and only their boundary
values at $\pm \infty$ are related through
\beq
\int \partial \bar f(x) dx = \bar f(\infty) - \bar f(-\infty) = 
M_{\bar f} < \infty.
\eeq
The space translations as automorphism can be extended from the observable
algebra $\A$ (4.13) to the field algebra $\F$
\beq
\F = \A \st{\alpha_{\bar \Phi}}{\times} {\bf Z}
\eeq
if, according to (2.7), condition (4.14) is satisfied for the functions
$$
\bar f_x - \bar f = \wh f_x, \qquad \bar g_x - \bar g = \wh g_x
$$
i.e. if $\;\wh \Phi_x := (\wh f_x, \wh g_x) \in \C_0^\infty \;$ for finite $x$.

This is easily seen to be the case. Note, however, that due to (4.8) space 
translations are generated by $\;\hat P = exp \{i|p|x\}\;$ which restricts 
the Fourier transforms of all test functions to the subspace of even 
functions. Then the invariance of symplectic form $\sigma$ under shifts in $x$
determines the Fourier decompositions:
\beqan
f(x) &=& \frac{1}{\sqrt {2\pi}} \int e^{i|p|x}\,\wt f(p)\,dp \\
g(x) &=& \frac{1}{\sqrt {2\pi}} \int e^{-i|p|x}\,\wt g(p)\,dp.
\eeqan

This gives for the zero modes
\beqa
\int_{-\infty}^\infty \wh f_x(y)dy &=& 
2\,\int_0^\infty e^{ipy} \wt{\bar f}(p) (e^{-ipx} -1) dp dy 
\no \\
&=& -ix \left.(p \wt{\bar f}(p))\right|_{p=0} = \\
&=& -x \int \partial \bar f(y) dy = -x M_{\bar f} < \infty, \no
\eeqa
according to (4.15), similarly for $\wh g_x$.
Eq. (4.17) also means that the singularity of the zero mode of
$\bar f, \bar g$ is of the type $1/p$. For example, we can choose for
$(\bar f,\bar g)$ appropriately smeared $\theta$--functions:
\beqan
\bar f(x) &=& \int F(x-y) \theta(y) dy = \int_0^\infty F(x-y) dy \\
\bar g(x) &=& \int G(x-y) \theta(y) dy = \int_0^\infty G(x-y) dy
\eeqan
with $F,G$ being $\C_0^\infty$--functions, so that for the Fourier 
components we get
\beq
\wt{\bar f}(|p|) = \frac{1}{i} \; \frac{\wt F(p)}{p - i\ve}, \qquad
\wt F(0) \mbox{ finite.}
\eeq
Since the space translations can be extended to the field algebra, we
can discuss the asymptotic statistical behaviour of its elements. In
particular, the existence of anticommuting variables, due to eq. (3.6), 
imposes some restrictions on the functions, defining the structural
automorphism $\alpha_{\bar \Phi}$. For the algebra (4.16) this
requirements reads
\beq
\sigma(\bar \Phi,\bar \Phi_x) = 
\pm [\bar g(\infty) - \bar g(-\infty)] \int \bar f'(y) dy = (2k +
1)\pi, \qquad \mbox{for } |x| > \Lambda,
\eeq
or
$$
\pm M_{\bar g} \; M_{\bar f} = (2 k +1)\pi, \qquad k \mbox{ integer.}
$$
This is exactly the condition used in \cite{AMSa} to choose the appropriate
$\bar \Phi$ for the construction of the fermionic creation and
annihilation operators, apart from the difference in the symplectic
form, hence, in the choice of $\V = (\C_0^\infty \times \partial^{-1} 
\C_0^\infty)$
there. The finite quantities $M_{\bar f}, M_{\bar g}$ then may be given
a meaning of charges.

In momentum representation condition (4.19) reads
\beqa
\lefteqn{\lim_{x \ra \pm \infty} i \int |p| \wt{\bar f}(|p|) \wt{\bar g}(|p|)
(e^{i|p|x} - e^{-i|p|x}) dp = } \no \\
&=& \lim_{x \ra \pm \infty} \int_{-\infty}^\infty |p| \;
\frac{\wt F(p) \wt G(-p)}{(p + i\ve)(p - i\ve)} (e^{i|p|x} - e^{-i|p|x})dp
\no \\
&=& (2k + 1)\pi = \pm 2 \pi \wt F(0) \wt G(0).
\eeqa

\section{Extension of the time evolution to the field algebra}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\setcounter{equation}{0}

The success to consider the
observable algebra as Weyl algebra stems from the fact that the time
evolution of interacting fermi systems can be described as quasifree
evolution of the Weyl algebra. To see how the extension procedure works
we have to be more explicit:
$$
\tau_t \; W(f,g) = W(f_t,g_t) = W(e^{-iht}f,e^{iht}g)
$$
where $h$ is the one--particle Hamiltonian. In our models $e^{iht}$ maps
$\C_0^\infty$ into $\C_0^\infty$. 
%since the evolution has finite velocity. 
According to (2.7),
$$
V_t = W(\bar f_t - \bar f, \bar g_t - \bar g),
$$
and extendibility of time evolution asks that both $\bar f_t - \bar f$, 
$\bar g_t - \bar g$ must belong to 
the space $\C_0^\infty$. If time evolution commutes with space translation,
we better work in momentum space, so that we have to consider
\beq
(e^{-i\omega(|p|)t} - 1) \wt{\bar f}(p), \qquad
(e^{i\omega(|p|)t} - 1) \wt{\bar g}(p).
\eeq
According to (4.18) $\wt{\bar f}(p)$ does not stay bounded for $p \ra 0$
but has a singularity of the type $1/(p - i\ve)$.

Extendibility of the time evolution from the observable to the crossed
product field algebra then depends on the structure of the spectrum
$\omega(p)$. If relation
\beq
\lim_{p \ra 0} \frac{\omega(|p|)}{p} = M_\omega < \infty
\eeq
holds, this is enough to cure the singularities of $\bar f, \bar g$,
so that $V_t \in \A$.

Let us now look at the models described in Section 4. For reasonable 
potentials the spectrum of the 
Luttinger model (4.11) satisfies condition (5.2), while for the spectrum
(4.12) this is not the case: there, an additional singularity is present,
corresponding to the appearance of a massive scalar particle that does
not allow extension of the time evolution as an automorphism on $\F$.

Since the functions $\bar f,\bar g$ are defined with some additional
restrictions, following from the requirement for $\{ \delta_{1n}\}$
to be an odd element of $\F$, a question arises about the importance of
this additional condition (4.19) for the asymptotic statistical behaviour
of the time evolution. This means, we are interested in the limit
behaviour
$$
\lim_{t \ra \pm \infty} i \int |p| \wt{\bar f}(|p|) \wt{\bar g}(|p|)
(e^{i \omega(|p|)t} - e^{-i\omega(|p|)t}) dp
$$
with $\omega(p)$ satisfying (5.2) so that we have to consider 
integrals of the type
$$
\lim_{t \ra \infty} i \int_0^\infty \frac{\wt H(p)}{p^2 + \ve^2}
e^{i \omega(p)t} dp =
\lim_{t \ra \infty} i \int_0^\infty \frac{\wt H(\omega^{-1}(q))}
{[\omega^{-1}(q)^2 + \ve^2]} \frac{e^{iqt} dq}{\omega'(\omega^{-1}(q))}
$$
with $H(x)$ being an $\C_0^\infty$--function. These integrals have exactly the
same singularities (due to (5.2)) as those in (4.20). Then, together
with (4.19), an analogous relation takes place also for time--shifted
$\bar \Phi$--pair
$$
\lim_{t \ra \pm \infty} \sigma(\bar \Phi_t,\bar \Phi) = (2k +1)\pi,
\qquad k \mbox{ integer,}
$$
so that
$$
\{ \delta_{1n}\} \cdot \tau_t \{ \delta_{1n}\} + \tau_t \{\delta_{1n}\}
\cdot \{ \delta_{1n}\} = 0.
$$

Therefore, when an extension of the time evolution as automorphism from
observable to the field algebra is possible, the asymptotic
anticommutativity of space translations on the odd subalgebra of $\F$ 
provides asymptotic anti--Abelianess of the time evolution there 
(compare comments in \cite{H}, p.228).

We want to emphasize that it is by far not evident that
asymptotic behaviour of time and space translations is the same. For
example, in the XY--model new features appear \cite{A,HN}.

Finally, we wonder how time evolution can be interpreted if (5.2) is
violated, e.g. in the Schwinger model. In \cite{AMSb} the view point is
taken that the algebra is enlarged even more to CCR$(\V)$ where
$\V = \{ \bigcup_t \bar \Phi_t \cup \V_0\}$. $\V$ is always a linear space 
and we have already observed that $\lambda \bar \Phi$, $\lambda$ real, 
with varying $\lambda$ leads
to a larger algebra than the desired fermi field algebra (fractional
statistics). We prefer to take the view point that we do not want to
enlarge the algebra but are satisfied to have a well defined time
evolution of states since we have found the possibility to
extend any state on $\A$ to a gauge invariant state on $\F$ (Section 2).
Accordingly, any gauge invariant state on $\F$ has a well defined time
evolution. Especially, time invariant state on $\A$ induces a time 
invariant state on $\F$. Properties of such states also for finite
temperature are discussed in \cite{V}. 
%On the other hand, every state on $\F$ is normal with
%respect to a gauge invariant state.

We consider the states (2.10)
$$
\langle F^{(k)}|\pi(W(\psi))|F^{(k)}\rangle =
\langle \Phi|\pi(\alpha^{-k} W(\psi))| \Phi\rangle.
$$
They evolve in the course of time to
$$
\langle F^{(k)}_t|\pi(W(\psi))|F^{(k)}_t\rangle =
\langle \Phi|\pi(\alpha^k \tau_t W(\psi))| \Phi\rangle =
\langle \Phi| \pi \circ \tau_t \circ \tau_{-t} \alpha^k \tau_t W(\psi)
| \Phi\rangle.
$$

As we have already mentioned, two states, $\omega_1$ and $\omega_2$, 
over a gauge invariant
algebra can be combined in a not gauge invariant state iff the
representations $\pi_1$ and $\pi_2 \circ \alpha^k$ are equivalent for
some $k$. In the course of time $\omega_1$ and $\omega_2$ evolve to
states, corresponding to the representations $\pi_1 \circ \tau_t$ and
$\pi_2 \circ \tau_t$. Since $\pi_2$ by assumption is equivalent to 
$\pi_1 \circ \alpha^{-k}$, we demand that
\beqan
\pi_1 \circ \tau_t &\approx& \pi_2 \circ \tau_t \circ \alpha^{-k} \\
\pi_1 &\approx& \pi_2 \circ \alpha^{-k} \circ \alpha^k \tau_t \alpha
^{-k} \tau_{-t}
\eeqan
and this only holds if $\alpha^k \tau_t \alpha^{-k} \tau_{-t}$ is an
automorphism of $(\A)''$, which is not the case in the Schwinger
model. Therefore states
that are not gauge invariant have no well defined time evolution, so 
that they are not physically acceptable and we have screening of the
charge (confinement).

To summarize, the time evolution on the fermionic field algebra can be
obtained as a naturally extended automorphism from the time evolution
of the observable fields only in cases when short range interactions
determine the behaviour of the system. Existence of long range forces,
typical example being the Schwinger model, appears to be an obstacle to this.
Of course, the fermionic field algebra is still well defined but we can
only consider the time evolution of gauge--invariant, hence, not
charged fermionic structures (consisting of equal number fermions and
antifermions). This nonextensibility of the time evolution may be
viewed as another manifestation of the confinement, which takes place
in the Schwinger model.

\section*{Concluding remarks}
We have demonstrated on simple examples the possibility to construct
fermionic field algebra as a crossed product of the observable algebra
by a proper $\alpha$--action of the group of integer numbers {\bf Z}.
$\alpha$ has to be a free (not--inner) automorphism of the
observable algebra $\A$ which is a simplification for dimension 2 as
compared to the specially directed monoidal category of endomorphisms 
for 4 dimensions in \cite{BDLR} but still provides an analysis of various
models. The field algebra so obtained has a local net structure, the
ingredients being von Neumann algebras. The extension of automorphisms
from observable to the field algebra is shown to be possible under
a compatibility condition between the automorphism in question and the
structural one used in the crossed product. 
As a direct consequence of this compatibility relation for the special 
case of space translations and of the net structure of the observable 
algebra appears the net structure also of the field algebra so that
no further restrictions on $\alpha$ have to be imposed to guarantee the
latter. Also, the states are shown to be extendible to
the field algebra, inheriting the structure and properties of the states
over the algebra of observables.

In the two cases of automorphisms of particular interest --- space
translations and time evolution,  we have the following situation: the
conditions to be fulfilled in order to have space translations extended
to the field algebra and to have anticommuting fields present in it, 
are enough
to specify the automorphism $\alpha(\bar \Phi)$ by fixing $\bar \Phi$.
Then, time evolution appears to be extendible only in the case of short
range interactions but then it is also asymptotically anti--Abelian for 
the anticommuting fields, so that space translations and time evolution
have the same asymptotic statistical behaviour. In the cases when long
range forces prevent consistent extension of time evolution from
observable to the field algebra, a well defined time evolution is shown
to exist for gauge invariant states on the latter. In this context,
charge screening (or confinement) in the Schwinger model may be
understood as an absence of well defined time evolution for charged
states.

The crossed product field algebra allows also for fractional statistics
in one space dimension. This interesting possibility as well as
the description of vacuum degeneracy in gauge models in the crossed
product scheme will be considered separately.

\section*{Acknowledgements}
We want to thank Walter Thirring for his incouraging interest, valuable 
suggestions and critical remarks, Ivan Todorov for constructive discussions
and Thomas Hudetz for useful comments. 

One of us (N.I.)
thanks the Institute for Theoretical Physics of Vienna University and 
Erwin Schr\"odinger International Institute for Mathematical Physics for
their hospitality and ``Fonds zur F\"orderung der wissenschaftlichen
Forschung in \"Osterreich'' for the financial support under Lise Meitner
grant M083--PHY--1994/95.

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