#  Twisted tensor products of field algebras

Ezio Vasselli
Dipartimento di Matematica, Università di Roma “Tor Vergata”,
Via della Ricerca Scientifica, I-00133 Roma, Italy.
<EMAIL_ADDRESS>

###### Abstract

Let $\mathcal{A}$ be a C*-algebra, ${\boldsymbol{h}}$ a Hilbert space and
$\mathcal{C}_{\boldsymbol{h}}$ the CAR algebra over ${\boldsymbol{h}}$. We
construct a twisted tensor product of $\mathcal{A}$ by
$\mathcal{C}_{\boldsymbol{h}}$ such that the two factors are not necessarily
one in the relative commutant of the other. The resulting C*-algebra may be
regarded as a generalized CAR algebra constructed over a suitable Hilbert
$\mathcal{A}$-bimodule. As an application, we exhibit a class of fixed-time
models where a free Dirac field (giving rise to the
$\mathcal{C}_{\boldsymbol{h}}$ factor) in general is not relatively local to a
free scalar field (which yields the $\mathcal{A}$ factor). In some of the
models, gauge-invariant combinations of the two (not relatively local) fields
form a local observable net.

Mathematics Subject Classification. 46L06, 46L08, 81T05
Keywords. C*-algebras, Hilbert modules, Fock spaces, quantum field theory

##  1 Introduction

In quantum field theory, a common way to construct a field system is to
perform the tensor product of Fock spaces and define for each factor the
corresponding free field. This method is used in both exactly solvable models
and perturbative theory, as a preliminary step to perform Wick or time-ordered
products and get the corresponding Wightman or Green functions. At the
algebraic level the typical construction is given by the spatial tensor
product $\mathcal{F}\doteq\mathcal{C}_{\boldsymbol{h}}\otimes\mathcal{A}$,
where $\mathcal{C}_{\boldsymbol{h}}$ is the CAR algebra over a Hilbert space
${\boldsymbol{h}}$ (generalized free Fermi field) and $\mathcal{A}$ is a CCR
algebra (generalized free Bosonic field).

The starting point of the present paper is the remark that we may regard
$\mathcal{F}$ as the C*-algebra generated by a Fermi field $\psi$ intended in
a broader sense, defined in terms of creation and annihilation operators
living in a fermionic Fock bimodule $\EuFrak{F}_{-}(\EuFrak h)$. Here,
$\EuFrak h\doteq{\boldsymbol{h}}\otimes\mathcal{A}$ is the free Hilbert
bimodule carrying the _trivial left action_ uniquely determined by the
relations $wA=Aw$, $w\in{\boldsymbol{h}}$, $A\in\mathcal{A}$. Adopting this
point of view, $\mathcal{A}$ appears as part of the target space for the ”non-
commutative anticommutator function” of $\psi$.

This suggests a strategy to escape from tensor products and possibly produce
interesting models. The idea is that of considering Hilbert bimodules carrying
a non-trivial left action, with the aim of constructing field systems where
the Bosonic components do not necessarily commute with the fermionic ones.
This is what happens, for example, in QED, where (in positive gauges) charged
fields cannot be relatively local to the electromagnetic field [14].

It is aim of the present paper to provide an algebraic machinery to construct
field C*-algebras having the above mentioned property. We start by reviewing
generic Hilbert bimodules, their Fock bimodules and the corresponding GNS
Hilbert spaces obtained by applying states of $\mathcal{A}$ §2. On these
spaces there are well-defined annihilation and creation operators, and the
”zero particle space” is the GNS space of $\mathcal{A}$ instead of
${\mathbb{C}}$ as in usual Fock space. Annihilation operators –while they
perform the usual operation of annihilating states defined by $\EuFrak h$–
modify (without annihilating) states in the GNS space of $\mathcal{A}$. Thus
in the present paper the term _annihilation operator_ should be intended in
this broader sense.

To define fermionic spaces, one should introduce a permutation symmetry on
Fock bimodules. Since in general this is an impossible task, we focus on free
Hilbert bimodules for which it is possible to define a permutation symmetry in
the obvious way §3. This does not necessarily lead to the trivial construction
$\mathcal{F}=\mathcal{C}_{\boldsymbol{h}}\otimes\mathcal{A}$, because the
crucial property to avoid it is non-triviality of the left
$\mathcal{A}$-action. Having defined our fermionic Fock space, we face the
fact that antisymmetry

$f\otimes_{-}g\ =\ -g\otimes_{-}f\ \ \ ,\ \ \ f,g\in\EuFrak h\,,$ (1.1)

does not hold in full generality. The reason relies on commutation properties
of $\mathcal{A}$: namely, there are suitable support C*-subalgebras
$\mathcal{A}(f),\mathcal{A}(g)$ of $\mathcal{A}$, and a sufficient condition
for (1.1) holding true is that

$[\mathcal{A}(f)\,,\,\mathcal{A}(g)]\ =\ 0$ (1.2)

together with the fact that the left $\mathcal{A}(g)$-action on $f$ is trivial
and analogously for $\mathcal{A}(f)$ on $g$, Remark 3.3. In this case we say
that $f$ and $g$ are _mutually free_.

An other consequence of the fact that we are dealing with Hilbert modules
rather than Hilbert spaces is given by non-stability of the fermionic Fock
module both under the action of creation operators and the left
$\mathcal{A}$-action. Both these problems are solved by considering a
particular class of left $\mathcal{A}$-actions that we call _$\mathcal{G}$
-twists_, where $\mathcal{G}$ is a group of unitary generators of
$\mathcal{A}$ (Lemma 3.4). $\mathcal{G}$-twists are given by group morphisms
$u:\mathcal{G}\to\mathcal{U}({\boldsymbol{h}})$ that are used to perturb the
trivial left $\mathcal{A}$-action, Def.3.2. We exhibit two classes of
C*-algebras that naturally admit $\mathcal{G}$-twists: the first is given by
Weyl algebras (giving rise to generalized free Bosonic fields), Example 3.1;
the second, Example 3.2, is the universal C*-algebra of the electromagnetic
field [8].

Finally in §4 we construct our generalized CAR algebra having as input the
free module $\EuFrak h$ and the twist $u$, Theorem 4.3. Adding as a third
ingredient a suitable conjugation $\kappa$ acting on ${\boldsymbol{h}}$ we
obtain what we call a _Dirac triple over $\mathcal{A}$_, from which we
construct a Dirac field (4.16) and the corresponding field algebra (4.18). It
is this field algebra that defines the desired twisted tensor product of
$\mathcal{A}$ by $\mathcal{C}_{\boldsymbol{h}}$.

As an application we exhibit a family of models in a fixed-time formulation,
depending on a tempered distribution
$\sigma\in\mathcal{S}^{\prime}({\mathbb{R}}^{3})$ that defines the twist. In
our models, a free Dirac field (generating the C*-algebra
$\mathcal{C}_{\boldsymbol{h}}$) and a free scalar field (generating a Weyl
C*-algebra $\mathcal{W}$) give rise to the following situations: (1)
$\mathcal{C}_{\boldsymbol{h}}$ and $\mathcal{W}$ do not commute, but are
relatively local (when $\sigma$ is the Dirac delta); (2)
$\mathcal{C}_{\boldsymbol{h}}$ and $\mathcal{W}$ are not relatively local (for
$\sigma$ having support with non-empty interior); (3) $\mathcal{W}$ is not
relatively local to $\mathcal{C}_{\boldsymbol{h}}$, but is in the commutant of
the fixed-point algebra of $\mathcal{C}_{\boldsymbol{h}}$ under the gauge
action (when $\sigma$ is the Lebesgue measure).

A discussion of our results, work in progress and perspectives are given in
the final section §5.

In the following points we fix some conventions:

  * •

Throughout the present paper, $\mathcal{A}$ will denote a C*-algebra with unit
${\bf 1}$ and $\mathcal{S}(\mathcal{A})$ its state space. Unless otherwise
stated, Hilbert space representations and *-morphisms are assumed to be non-
degenerate (hence unital).

  * •

We reserve Euler Fraktur fonts for Hilbert modules, related objects ($\EuFrak
h$, $\EuFrak{B}(\EuFrak h)$, $\EuFrak{F}(\EuFrak h)$ and so on), and objects
defined starting from Hilbert modules (for example, the Hilbert spaces
$\EuFrak h^{\omega}$ for $\omega\in\mathcal{S}(\mathcal{A})$). Instead, unless
otherwise stated the calligrafic font is used for Hilbert spaces
($\mathcal{H}$) and C*-algebras ($\mathcal{A}$, $\mathcal{B}(\mathcal{H})$,
…). As an exception to this rule, we adopt a lower case bold font for ”one-
particle Hilbert spaces”, that therefore are denoted by ${\boldsymbol{h}}$.

  * •

Free Hilbert modules usually appear in literature with the notation $\EuFrak
h={\boldsymbol{h}}\otimes\mathcal{A}$, where ${\boldsymbol{h}}$ is a Hilbert
space. To use the symbol $\otimes$ without ambiguities, in the sequel up to
rare exceptions we will write
${\boldsymbol{h}}\otimes\mathcal{A}\equiv{\boldsymbol{h}}\mathcal{A}$ and
$v\otimes A\equiv vA$ for $v\in{\boldsymbol{h}}$ and $A\in\mathcal{A}$. The
symbol $\otimes$ shall be used for the internal tensor product of Hilbert
bimodules or the Hilbert space tensor product.

  * •

For generic elements of $\EuFrak h={\boldsymbol{h}}\mathcal{A}$ we will often
use the notation of implicit sum for repeated indices,
$\sum_{i}v_{i}A_{i}\equiv v_{i}A_{i}$. The sum may be infinite, and in this
case our notation should be understood in terms of norm convergence in
$\EuFrak h$.

##  2 Fock spaces over C*-algebras

In the present section we collect some standard facts on Hilbert bimodules
that we present in a form suitable for our purposes. References are
Blackadar’s book [5], and the papers [15],[2, 16, 17] for GNS-representations
and Fock bimodules respectively. For the use of Hilbert bimodules in
mathematical physics see [1, 17, 18], [10, 11], [3, 4], and the recent review
[2].

### Hilbert modules.

A _(right Hilbert) $\mathcal{A}$-module_ is a complex vector space $\EuFrak h$
carrying a right $\mathcal{A}$-module action and endowed with a
$\mathcal{A}$-valued, strictly positive and right $\mathcal{A}$-linear scalar
product ${\left\langle\right.}\cdot,\cdot{\left.\right\rangle}$, which induces
the norm $\|{\left\langle\right.}\cdot,\cdot{\left.\right\rangle}\|$ under
which $\EuFrak h$ is a Banach space. It is worth noting that then

$A{\left\langle\right.}f,g{\left.\right\rangle}\,=\,({\left\langle\right.}g,f{\left.\right\rangle}A^{*})^{*}\,=\,({\left\langle\right.}g,fA^{*}{\left.\right\rangle})^{*}\,=\,{\left\langle\right.}fA^{*},g{\left.\right\rangle}\,,$
(2.1)

for all $f,g\in\EuFrak h$, $A\in\mathcal{A}$. An $\mathcal{A}$-module $\EuFrak
h$ defines the C*-algebra $\EuFrak{B}(\EuFrak h)$ of linear operators
$T:\EuFrak h\to\EuFrak h$ such that there is $T^{*}\in\EuFrak{B}(\EuFrak h)$
with
${\left\langle\right.}T^{*}f,g{\left.\right\rangle}={\left\langle\right.}f,Tg{\left.\right\rangle}$,
$f,g\in\EuFrak h$. This implies that $T\in\EuFrak{B}(\EuFrak h)$ is right
$\mathcal{A}$-linear, _i.e._ $T(fA)=(Tf)A$, and bounded. The C*-algebra of
compact operators is given by the closed ideal $\EuFrak{K}(\EuFrak
h)\subseteq\EuFrak{B}(\EuFrak h)$ generated by the elementary operators
$|f{\left.\right\rangle}{\left\langle\right.}g|h\doteq
f{\left\langle\right.}g,h{\left.\right\rangle}$, $h\in\EuFrak h$. We say that
$\EuFrak h$ is a _Hilbert $\mathcal{A}$-bimodule_ whenever there is a
*-morphism $\lambda:\mathcal{A}\to\EuFrak{B}(\EuFrak h)$, called the left
action (Hilbert bimodules are also known as _C*-correspondences_ [2]). In the
present paper we shall assume that $\lambda$ is faithful and non-degenerate,
and following a standard notation we will write

$Af\,\equiv\,\lambda(A)f\ \ \ ,\ \ \ \forall f\in\EuFrak
h\,,\,A\in\mathcal{A}\,.$

Let $\omega\in\mathcal{S}(\mathcal{A})$ be a state. Then
${\left\langle\right.}f,g{\left.\right\rangle}_{\omega}\doteq\omega({\left\langle\right.}f,g{\left.\right\rangle})$
is a scalar product and defines the Hilbert space $\EuFrak h^{\omega}$ whose
elements $f^{\omega}\in\EuFrak h^{\omega}$ are defined in correspondence with
$f\in\EuFrak h$. We have the representation

$\pi^{\omega}:\EuFrak{B}(\EuFrak h)\to\mathcal{B}(\EuFrak h^{\omega})\ \ \ ,\
\ \ T^{\omega}f^{\omega}\,\doteq\,(Tf)^{\omega}\,.$ (2.2)

The argument to prove that (2.2) is well-defined is standard, so we give just
a sketch. Let $T\in\EuFrak{B}(\EuFrak h)$ and $v\in\EuFrak h$ such that
$v^{\omega}=0$, i.e.
${\left\langle\right.}v,v{\left.\right\rangle}_{\omega}=0$; then
$\|T\|^{2}-T^{*}T$ is a positive element of $\EuFrak{B}(\EuFrak h)$ and this
implies that
${\left\langle\right.}v,(\|T\|^{2}-T^{*}T)v{\left.\right\rangle}\in\mathcal{A}$
is positive for any $v\in\EuFrak h$. As a consequence

$0\ =\ \|T\|^{2}{\left\langle\right.}v,v{\left.\right\rangle}_{\omega}\ \geq\
{\left\langle\right.}v,T^{*}Tv{\left.\right\rangle}_{\omega}\ =\
{\left\langle\right.}Tv,Tv{\left.\right\rangle}_{\omega}\ \geq\ 0\,,$

implying $(Tv)^{\omega}=0$, and $T^{\omega}$ is a well-defined linear operator
on $\mathcal{B}(\EuFrak h^{\omega})$ with norm $\leq\|T\|$. Note that (2.2)
restricts to the representation of the left $\mathcal{A}$-action

$\pi^{\omega}:\mathcal{A}\to\mathcal{B}(\EuFrak h^{\omega})\ \ \ ,\ \ \
A^{\omega}f^{\omega}\,\doteq\,(Af)^{\omega}\,.$ (2.3)

We call (2.2) and (2.3) the GNS-representations induced by $\omega$.

######  Example 2.1.

We set $\EuFrak h\doteq\mathcal{A}$ with right (left) $\mathcal{A}$-module
action given by right (left) multiplication and scalar product
${\left\langle\right.}f,g{\left.\right\rangle}\doteq f^{*}g$, $f,g\in\EuFrak
h$. If $\omega\in\mathcal{S}(\mathcal{A})$, then it is readily seen that (2.3)
is the GNS representation in the usual sense. For future use, we write
$v_{A}^{\omega}$ to indicate the GNS vector corresponding to
$A\in\mathcal{A}$, so that $\|v_{A}^{\omega}\|^{2}=\omega(A^{*}A)$.

######  Example 2.2.

Let $\EuFrak h\doteq{\boldsymbol{h}}\mathcal{A}$ denote the free right Hilbert
$\mathcal{A}$-module, with right action $vB\doteq w\otimes AB$ and scalar
product ${\left\langle\right.}v,v^{\prime}{\left.\right\rangle}\doteq
A^{*}A^{\prime}{\left\langle\right.}w,w^{\prime}{\left.\right\rangle}$, where
$v\doteq w\otimes A$, $v^{\prime}\doteq w^{\prime}\otimes A^{\prime}$,
$w,w^{\prime}\in{\boldsymbol{h}}$, $A,A^{\prime},B\in\mathcal{A}$. If
$\lambda:\mathcal{A}\to\EuFrak{B}(\EuFrak
h)\simeq\mathcal{B}({\boldsymbol{h}})\otimes\mathcal{A}$ is a *-morphism, then
we have the left $\mathcal{A}$-action $Af\doteq\lambda(A)f$. Given
$\omega\in\mathcal{S}(\mathcal{A})$, the Hilbert space $\EuFrak h^{\omega}$ is
isomorphic to ${\boldsymbol{h}}\otimes{\boldsymbol{h}}^{\omega}$, where
${\boldsymbol{h}}^{\omega}$ is the GNS space of $\omega$.

### The Fock bimodule.

Let $n\in{\mathbb{N}}$ and $\EuFrak h$ denote a Hilbert bimodule. We consider
the tensor product of complex vector spaces $\EuFrak h^{\odot n}$, that we
endow with the scalar product

${\left\langle\right.}v,w{\left.\right\rangle}\,\doteq\,{\left\langle\right.}v_{n}\,,\,{\left\langle\right.}v_{n-1}\,,\,{\left\langle\right.}\ldots{\left\langle\right.}v_{1},w_{1}{\left.\right\rangle}\ldots{\left.\right\rangle}w_{n-1}{\left.\right\rangle}\,w_{n}{\left.\right\rangle}\,,$
(2.4)

where $v\doteq v_{1}\otimes\ldots\otimes v_{n}$ and $w$ are in $\EuFrak
h^{\odot n}$. Note that the term ${\left\langle\right.}v_{n-1},\ldots
w_{n-1}{\left.\right\rangle}$ appearing in the previous expression belongs to
$\mathcal{A}$, so it makes sense to consider its left action on $w_{n}$, and
analogously for the nested terms ${\left\langle\right.}v_{n-k},\ldots
w_{n-k}{\left.\right\rangle}$, $k=2,\ldots,n-1$. The completion obtained by
(2.4) is denoted by $\EuFrak h^{n}$ and is called the _internal $n$-fold
tensor product_: it is endowed with the $\mathcal{A}$-module actions

$Av\doteq Av_{1}\otimes\ldots\otimes v_{n}\ \ \ ,\ \ \ vA\doteq
v_{1}\otimes\ldots\otimes v_{n}A\,,$

and therefore it forms a Hilbert $\mathcal{A}$-bimodule. By construction, it
turns out

$\ldots\otimes v_{k}\otimes Av_{k+1}\otimes\ldots\,=\,\ldots\otimes
v_{k}A\otimes v_{k+1}\otimes\ldots\ \ \ ,\ \ \ \forall A\in\mathcal{A}\,.$
(2.5)

######  Remark 2.1.

A delicate point of the tensor product of Hilbert bimodules is that an
elementary tensor $v\in\EuFrak h^{\odot n}$ may have norm zero even when the
all the factors $v_{1},\ldots,v_{n}$ do not. For example, if
$\mathcal{A}=C(X)$ is commutative, $\EuFrak h$ is the module of sections of a
vector bundle $E\to X$, and $Af=fA$ is defined by pointwise multiplication for
all $A\in C(X)$ and $f\in\EuFrak h$, then
${\left\langle\right.}v,v{\left.\right\rangle}={\left\langle\right.}v_{1},v_{1}{\left.\right\rangle}{\left\langle\right.}v_{2},v_{2}{\left.\right\rangle}=0$
for $v\doteq v_{1}\otimes v_{2}$ and
${\mathrm{supp}}\,(v_{1})\cap{\mathrm{supp}}\,(v_{2})=\emptyset$.

The direct sum of Hilbert $\mathcal{A}$-bimodules $\EuFrak h,\EuFrak
h^{\prime}$ is defined in the obvious way, and yields the
$\mathcal{A}$-bimodule $\EuFrak h\oplus\EuFrak h^{\prime}$. Thus we define the
_Fock $\mathcal{A}$-bimodule_

$\EuFrak{F}(\EuFrak h)\,\doteq\,\bigoplus_{n=0}^{\infty}\EuFrak h^{n}\,,$
(2.6)

with $\EuFrak h^{0}\doteq\mathcal{A}$ and $\EuFrak h^{1}\doteq\EuFrak h$; any
$v\in\EuFrak{F}(\EuFrak h)$ is a sequence $v=\\{v^{n}\in\EuFrak h^{n}\\}$. Let
now $f\in\EuFrak h$, $n\geq 1$, and $v\doteq v_{1}\otimes\ldots\otimes
v_{n}\in\EuFrak h^{n}$. We set

$\left\langle
f\right|v\,\doteq\,{\left\langle\right.}f,v_{1}{\left.\right\rangle}v_{2}\otimes\ldots\otimes
v_{n}\,,$ (2.7)

obtaining a right $\mathcal{A}$-linear operator $\left\langle f\right|:\EuFrak
h^{n}\to\EuFrak h^{n-1}$. We define the annihilation operator

$(a(f)v)^{n-1}\,\doteq\,\sqrt{n}\,\,\left\langle f\right|v^{n}\ \ \ ,\ \ \
n\geq 1\,,$ (2.8)

with domain $\,{\mathrm{Dom}}\,(a(f))\doteq\EuFrak{F}^{\\#}(\EuFrak
h)\doteq\left\\{v\in\EuFrak{F}(\EuFrak
h)\,:\,\exists\sum_{n}n{\left\langle\right.}v^{n},v^{n}{\left.\right\rangle}\in\mathcal{A}\right\\}$.
Next we define the creation operator

$\left\\{\begin{array}[]{l}(a^{*}(f)v)^{n+1}\,\doteq\,\sqrt{n+1}\,\,f\otimes
v^{n}\ \ \ ,\ \ n\geq 1\,,\\\ (a^{*}(f)v)^{1}\doteq fv^{0}\ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \,,\ \ n=0\,,\end{array}\right.$ (2.9)

for $v\in\,{\mathrm{Dom}}\,(a^{*}(f))\doteq\EuFrak{F}^{\\#}(\EuFrak h)$. Note
that $v^{0}\in\mathcal{A}$, so the expression $fv^{0}\in\EuFrak h$ makes
sense. Moreover, if $v^{n}=0$ for all $n\geq 1$ then $a(f)v=0$, so that
$\mathcal{A}=\EuFrak h^{0}$ is in the kernel of $a(f)$. We have

$a(f)a^{*}(g)\,=\,(n+1){\left\langle\right.}f,g{\left.\right\rangle}\ \ \ ,\ \
\ a^{*}(g)a(f)\,=\,ng\otimes\left\langle f\right|\ \ \ ,\ \ \ f,g\in\EuFrak
h\,.$ (2.10)

The above notation suggests that $a^{*}(f)$ is the adjoint of $a(f)$ and,
actually, for elementary tensors $v\in\EuFrak h^{n}$, $w\in\EuFrak h^{n-1}$,
we have

$\begin{array}[]{l}{\left\langle\right.}v,a^{*}(f)w{\left.\right\rangle}\ =\
\sqrt{n}\,{\left\langle\right.}v,f\otimes w{\left.\right\rangle}\ =\\\
\sqrt{n}\,{\left\langle\right.}v_{2}\otimes\ldots\otimes
v_{n}\,,\,{\left\langle\right.}v_{1},f{\left.\right\rangle}w{\left.\right\rangle}\
=\
\sqrt{n}\,{\left\langle\right.}{\left\langle\right.}f,v_{1}{\left.\right\rangle}v_{2}\otimes\ldots\otimes
v_{n}\,,\,w{\left.\right\rangle}\ =\\\
{\left\langle\right.}a(f)v\,,\,w{\left.\right\rangle}\,.\end{array}$ (2.11)

Thus $a^{*}(f)$ behaves as the adjoint of $a(f)$ on the common domain
$\EuFrak{F}^{\\#}(\EuFrak h)$. Note that $a^{*}(f)$ and $a(f)$ are unbounded,
thus we should be careful when we use the term _adjoint_. Yet, since we are
interested in the fermionic case where it will be shown that the creation and
annihilation operators are bounded, we prefer to not discuss this point here.

### Hilbert spaces in Hilbert modules.

A _Hilbert space in $\EuFrak h$_ is given by a closed vector subspace
${\boldsymbol{h}}\subset\EuFrak h$ such that
${\left\langle\right.}w,w^{\prime}{\left.\right\rangle}\in{\mathbb{C}}{\bf
1}$, $\forall w,w^{\prime}\in{\boldsymbol{h}}$. The proof of the following
result is trivial, therefore it is omitted.

######  Lemma 2.2.

Let ${\boldsymbol{h}}\subset\EuFrak h$ be a Hilbert space. Then there is an
injective linear mapping $\mathcal{F}({\boldsymbol{h}})\to\EuFrak{F}(\EuFrak
h)$ preserving the scalar product, where $\mathcal{F}({\boldsymbol{h}})$ is
the Fock space in the usual sense.

### GNS-Hilbert spaces of the Fock bimodule.

Let now $\omega\in\mathcal{S}(\mathcal{A})$ and $\EuFrak h^{n,\omega}$ denote
the Hilbert space obtained by completion of $\EuFrak h^{n}$ under the scalar
product ${\left\langle\right.}\cdot,\cdot{\left.\right\rangle}_{\omega}$.
Moreover, let $\EuFrak{F}^{\omega}(\EuFrak h)$ denote the Hilbert space
obtained by the analogous completion of $\EuFrak{F}(\EuFrak h)$. Basic
properties of this Hilbert space are resumed in the following result.

######  Proposition 2.3.

Let $\EuFrak h$ be a Hilbert $\mathcal{A}$-bimodule and
$\omega\in\mathcal{S}(\mathcal{A})$. With the above notation, there is a
decomposition

$\EuFrak{F}^{\omega}(\EuFrak h)\,\simeq\,\bigoplus_{n=0}^{\infty}\EuFrak
h^{n,\omega}\,.$ (2.12)

The component $\EuFrak h^{0,\omega}$ is the usual GNS Hilbert space of
$\omega$, and there is a representation

$\hat{\pi}^{\omega}:\mathcal{A}\to\mathcal{B}(\EuFrak{F}^{\omega}(\EuFrak h))\
\ ,\ \ \hat{\pi}^{\omega}\,=\,\oplus_{n}\pi^{n,\omega}\,,$ (2.13)

where each $\pi^{n,\omega}$ is defined as in (2.3) and
$\pi^{0,\omega}:\mathcal{A}\to\mathcal{B}(\EuFrak h^{0,\omega})$ is the usual
GNS representation. Given a Hilbert space ${\boldsymbol{h}}\subset\EuFrak h$,
the closed vector subspace of $\EuFrak{F}^{\omega}(\EuFrak h)$ spanned by
elements of the type $(wA)^{\omega}$, $w\in{\boldsymbol{h}}^{n}$,
$A\in\mathcal{A}$, $n\in{\mathbb{N}}$, is a Hilbert space isomorphic to
$\mathcal{F}({\boldsymbol{h}})\otimes\EuFrak h^{0,\omega}$, where
$\mathcal{F}({\boldsymbol{h}})$ is the Fock space in the usual sense, so that
there is an embedding

$\iota:\mathcal{F}({\boldsymbol{h}})\otimes\EuFrak
h^{0,\omega}\to\EuFrak{F}^{\omega}(\EuFrak h)\ \ ,\ \ \iota(w\otimes
v_{A}^{\omega})\,\doteq\,(wA)^{\omega}\,.$ (2.14)

###### Proof.

The direct sum decomposition (2.12) is obvious. About $\EuFrak h^{0,\omega}$,
by definition it is given by the completion of $\EuFrak h^{0}=\mathcal{A}$
under the scalar product $\omega(A^{*}A^{\prime})$,
$A,A^{\prime}\in\mathcal{A}$, so it is the GNS-space of $\omega$. The
decomposition (2.13) trivially follows by the previous points, as well as the
fact that $\pi^{0,\omega}$ is the GNS representation of $\omega$. Finally, the
embedding (2.14) follows by the fact that, given elementary tensors
$w,w^{\prime}\in{\boldsymbol{h}}^{n}$, $A,A^{\prime}\in\mathcal{A}$,

${\left\langle\right.}wA,w^{\prime}A^{\prime}{\left.\right\rangle}\ =\
A^{*}\,\prod_{k}{\left\langle\right.}w_{k},w^{\prime}_{k}{\left.\right\rangle}\,A^{\prime}\,,$

having used the fact that

${\left\langle\right.}w_{n},{\left\langle\right.}w_{n-1},{\left\langle\right.}\ldots{\left\langle\right.}w_{1},w^{\prime}_{1}{\left.\right\rangle}\ldots{\left.\right\rangle}w^{\prime}_{n-1}{\left.\right\rangle}w^{\prime}_{n}{\left.\right\rangle}\
=\ \prod_{k}{\left\langle\right.}w_{k},w^{\prime}_{k}{\left.\right\rangle}$

(recall that
${\left\langle\right.}w_{k},w^{\prime}_{k}{\left.\right\rangle}\in{\mathbb{C}}$
for all $k$). ∎

######  Remark 2.4 (Non-Fock nature of $\EuFrak{F}^{\omega}(\EuFrak h)$).

We emphasize that in general $\EuFrak h^{n,\omega}$ is not the $n$-fold tensor
product $(\EuFrak h^{\omega})^{\otimes n}$ in the sense of Hilbert spaces, and
as a consequence $\EuFrak{F}^{\omega}(\EuFrak h)$ is not the Fock space of
$\EuFrak h^{\omega}$. This is readily seen by comparing on elementary tensors
$v,w\in\EuFrak h^{n}$ the scalar product obtained by (2.4) ,

${\left\langle\right.}v^{\omega},w^{\omega}{\left.\right\rangle}\,\doteq\,{\left\langle\right.}v_{n}\,,\,{\left\langle\right.}v_{n-1}\,,\,{\left\langle\right.}\ldots{\left\langle\right.}v_{1},w_{1}{\left.\right\rangle}\ldots{\left.\right\rangle}w_{n-1}{\left.\right\rangle}\,w_{n}{\left.\right\rangle}_{\omega}$
(2.15)

with the scalar product of $(\EuFrak h^{\omega})^{\otimes n}$,

${\left\langle\right.}v^{\omega},w^{\omega}{\left.\right\rangle}^{\prime}\
\doteq\
\prod_{k}{\left\langle\right.}v_{k},w_{k}{\left.\right\rangle}_{\omega}\,,$
(2.16)

the latter corresponding to the Fock space of $\EuFrak h^{\omega}$.

######  Remark 2.5.

Let ${\boldsymbol{h}}\subset\EuFrak h$ be a Hilbert space. Then the embedding
of $\mathcal{F}({\boldsymbol{h}})$ into $\EuFrak{F}(\EuFrak h)$ (Lemma 2.2)
factorizes through the isometric mapping

$\mathcal{F}({\boldsymbol{h}})\simeq\mathcal{F}({\boldsymbol{h}})\otimes\Omega\to\EuFrak{F}^{\omega}(\EuFrak
h)\ \ \ ,\ \ \ w\otimes\Omega\mapsto w^{\omega}\,,$

where $\Omega\in\EuFrak h^{0,\omega}$ is the GNS-vector (”vacuum”) obtained by
(2.14) for $A\equiv{\bf 1}$. Thus $\mathcal{F}({\boldsymbol{h}})$ embeds in
any GNS space $\EuFrak{F}^{\omega}(\EuFrak h)$.

### Creation and annihilation operators in $\EuFrak{F}^{\omega}(\EuFrak h)$.

Let $f\in\EuFrak h$ and $n\in{\mathbb{N}}$. Then the mapping $w\mapsto
f\otimes w$ defines a right $\mathcal{A}$-linear operator $T(f):\EuFrak
h^{n}\to\EuFrak h^{n+1}$, and since

${\left\langle\right.}(T(f)w\,,\,v{\left.\right\rangle}\ =\
{\left\langle\right.}f\otimes w\,,\,v{\left.\right\rangle}\ =\
{\left\langle\right.}w\,,\,{\left\langle\right.}f,v_{1}{\left.\right\rangle}(v_{2}\otimes\ldots\otimes
v_{n}){\left.\right\rangle}\ =\
{\left\langle\right.}w\,,\,{\left\langle\right.}f|v{\left.\right\rangle}\,,$

we find that $T(f)$ has adjoint $T(f)^{*}={\left\langle\right.}f|$ and, as a
consequence, is bounded [5, Prop.13.2.2] (this also implies that
${\left\langle\right.}f|$ is bounded and adjointable). Therefore, the argument
used to construct the representation (2.3) allows to define the linear
operators

$T(f)^{\omega}:\EuFrak h^{n,\omega}\to\EuFrak h^{n+1,\omega}\ \ \ ,\ \ \
{\left\langle\right.}f|^{\omega}:\EuFrak h^{n+1,\omega}\to\EuFrak
h^{n,\omega}\,.$

Rescaling these operators by the $\sqrt{n}$ factors in dependence of the order
of the tensor powers we obtain the creation and annihilation operators

$(a^{\omega}(f)v^{\omega})^{n-1}\ \doteq\
\sqrt{n}\,{\left\langle\right.}f|^{\omega}v^{\omega}\ =\
\sqrt{n}\,({\left\langle\right.}f|v)^{\omega}\,,$ (2.17)

$\left\\{\begin{array}[]{l}(a^{*,\omega}(f)v^{\omega})^{n+1}\,\doteq\,\sqrt{n+1}\,\,(f\otimes
v^{n})^{\omega}\ \ \ ,\ \ n\geq 1\,,\\\ \\\
(a^{*,\omega}(f)v)^{1}\,\doteq\,(fv^{0})^{\omega}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \,\,,\ \ n=0\,,\end{array}\right.$ (2.18)

where $v^{\omega}=\\{v^{n,\omega}\\}$ is defined for
$v\in\EuFrak{F}^{\\#}(\EuFrak h)$. Clearly $a^{*,\omega}(f)$ is (contained in)
the adjoint of $a^{\omega}(f)$ in the sense of Hilbert spaces,

$\begin{array}[]{lcl}{\left\langle\right.}w^{\omega,n-1}\,,\,(a^{\omega}(f)v^{\omega})^{n-1}{\left.\right\rangle}_{\omega}&=&\sqrt{n}\,{\left\langle\right.}w^{n-1}\,,\,{\left\langle\right.}f,v_{1}{\left.\right\rangle}(v_{2}\otimes\ldots\otimes
v_{n}){\left.\right\rangle}_{\omega}\ =\\\
&=&\sqrt{n}\,{\left\langle\right.}f\otimes
w^{n-1}\,,\,v_{1}\otimes\ldots\otimes v_{n}{\left.\right\rangle}_{\omega}\
=\\\
&=&{\left\langle\right.}(a^{*,\omega}(f)w^{\omega})^{n}\,,\,v^{\omega}{\left.\right\rangle}_{\omega}\,.\end{array}$

Note that since $a^{*}(f)$ may have non-trivial kernel (Remark 2.1), the same
is true for $a^{*,\omega}(f)$.

##  3 Twisted module actions and permutation symmetry

In the present section we study the permutation symmetry on tensor powers of
Hilbert bimodules or, to be honest, the factors that in general prevent the
realization of this property. These factors are easily identified and are non-
commutativity of $\mathcal{A}$ and the fact that in general $Af\neq fA$,
$A\in\mathcal{A}$, $f\in\EuFrak h$. Anyway even with the above limitations we
shall give a version of the fermionic Fock space for free Hilbert bimodules,
and define the corresponding restrictions of the creation and annihilation
operators. The tool used to perform these constructions is the one of _twist_
, namely a ”perturbation” of the trivial left $\mathcal{A}$-module action
defined using a group of unitary generators of $\mathcal{A}$. We remark that
thanks to the Kasparov stabilization theorem [5] the choice of considering
only free bimodules is not a severe requirement.

Let us now consider an elementary tensor $v=v_{1}\otimes\ldots\otimes
v_{n}\in\EuFrak h^{n}$. Given a permutation $\varrho\in{\mathbb{P}}(n)$, we
would like to define the operator

$U_{\varrho}v\ \doteq\ v_{\varrho(1)}\otimes\cdots\otimes v_{\varrho(n)}\,.$
(3.1)

It is obvious that the above operator is well-defined on the algebraic
elementary tensors $v_{1}\odot\ldots\odot v_{n}$, thus the question is what
happens when one applies the norm induced by $\mathcal{A}$. Here we
immediately encounter a problem, illustrated by the following example.
Consider a Hilbert bimodule $\EuFrak h$ of the type Example 2.1 with $\EuFrak
h=\mathcal{A}=\mathcal{O}_{2}$, the Cuntz algebra generated by mutually
orthogonal isometries $\psi_{1},\psi_{2}$; we take $v_{1}\doteq\psi_{1}$,
$v_{2}\doteq\psi_{2}^{*}$ and evaluate the scalar products

${\left\langle\right.}v_{1}\otimes v_{2}\,,\,v_{1}\otimes
v_{2}{\left.\right\rangle}\ =\
{\left\langle\right.}v_{2},{\left\langle\right.}v_{1},v_{1}{\left.\right\rangle}v_{2}{\left.\right\rangle}\
=\ \psi_{2}\psi_{1}^{*}\psi_{1}\psi_{2}^{*}\ =\ \psi_{2}\psi_{2}^{*}\,,$
${\left\langle\right.}v_{2}\otimes v_{1}\,,\,v_{2}\otimes
v_{1}{\left.\right\rangle}\ =\
{\left\langle\right.}v_{1},{\left\langle\right.}v_{2},v_{2}{\left.\right\rangle}v_{1}{\left.\right\rangle}\
=\ \psi_{1}^{*}\psi_{2}\psi_{2}^{*}\psi_{1}\ =\ 0\,.$

This example shows that the above operators $U_{\varrho}$ in general are ill-
defined, because they map zero norm algebraic elementary tensors into tensors
with norm $\neq 0$. This also spoils the preservation of compositions of
permutations in (3.1). For example, given the flip $\varrho(1)=2$,
$\varrho(2)=1$, in the situation of the previous example we find

$\|U_{\varrho}(v_{2}\otimes v_{1})\|=\|v_{1}\otimes v_{2}\|\neq 0\ \ \ ,\ \ \
U_{\varrho}U_{\varrho}(v_{1}\otimes v_{2})=0\ \ \ ,\ \ \
U_{\varrho\varrho}(v_{1}\otimes v_{2})=v_{1}\otimes v_{2}\,.$

### Free modules.

In the language adopted in the present paper, a Hilbert bimodule $\EuFrak h$
is free if, and only if, there is a Hilbert space
${\boldsymbol{h}}\subset\EuFrak h$ such that

$\EuFrak h\ =\ {\boldsymbol{h}}\mathcal{A}\ \doteq\ {\mathrm{closed\
span}}\\{wA\,:\,w\in{\boldsymbol{h}}\,,\,A\in\mathcal{A}\\}\,.$ (3.2)

######  Lemma 3.1.

Let $\EuFrak h$ be a free Hilbert bimodule. Then there is an isomorphism of
right Hilbert modules $\EuFrak{F}(\EuFrak
h)\simeq\mathcal{F}({\boldsymbol{h}})\mathcal{A}$ and, given a state
$\omega\in\mathcal{S}(\mathcal{A})$, there is an isomorphism of Hilbert spaces
$\EuFrak{F}^{\omega}(\EuFrak
h)\simeq\mathcal{F}({\boldsymbol{h}})\otimes\EuFrak h^{0,\omega}$ where
$\EuFrak h^{0,\omega}$ is the usual GNS Hilbert space. The left
$\mathcal{A}$-module action $\lambda$ defines by evaluation the representation

$\lambda^{\omega}:\mathcal{A}\to\mathcal{B}(\mathcal{F}({\boldsymbol{h}})\otimes\EuFrak
h^{0,\omega})\,.$ (3.3)

###### Proof.

Let $n\in{\mathbb{N}}$ and $v=v_{1}\otimes\ldots\otimes v_{n}\in\EuFrak
h^{n}$; then by (3.2) we may write $v_{i}=w_{k}A_{k;i}$, where $\\{w_{k}\\}$
is an orthonormal basis of ${\boldsymbol{h}}$,
$A_{k;i}\doteq{\left\langle\right.}w_{k},v_{i}{\left.\right\rangle}\in\mathcal{A}$,
and the (possibly infinite) sum on the repeated index $k$ is performed. Using
the above decomposition, it is easily seen that
$v\in{\boldsymbol{h}}^{n}\mathcal{A}$ and we conclude that $\EuFrak
h^{n}\subseteq{\boldsymbol{h}}^{n}\mathcal{A}$. Since the opposite inclusion
is obvious, we have $\EuFrak h^{n}={\boldsymbol{h}}^{n}\mathcal{A}$, and this
proves that $\EuFrak h^{n}\simeq{\boldsymbol{h}}^{n}\mathcal{A}$ implying
$\EuFrak{F}(\EuFrak h)\simeq\mathcal{F}({\boldsymbol{h}})\mathcal{A}$. Thus
$\EuFrak{F}^{\omega}(\EuFrak
h)\simeq\mathcal{F}({\boldsymbol{h}})\otimes\EuFrak h^{0,\omega}$ by
Proposition 2.3. ∎

We shall see in the following sections that in general
$\lambda^{\omega}(\mathcal{A})$ is not in the commutant of
$\mathcal{B}(\mathcal{F}({\boldsymbol{h}}))$ in
$\mathcal{B}(\mathcal{F}({\boldsymbol{h}})\otimes\EuFrak h^{0,\omega})$;
moreover, operators of the type $\lambda^{\omega}(A)$, $A\in\mathcal{A}$, mix
vectors of $\mathcal{F}({\boldsymbol{h}})$ and $\EuFrak h^{0,\omega}$ in the
sense that, given the GNS-vector $\Omega\doteq v_{\bf 1}^{\omega}\in\EuFrak
h^{0,\omega}$, typically we will find
$\lambda^{\omega}(A)(v\otimes\Omega)=\sum_{k}v_{k}\otimes v^{0}_{k}$ with
$v_{k}\neq v$ and $v^{0}_{k}\neq\Omega$. In contrast, let us define the left
$\mathcal{A}$-module action

$A(wB)\ \doteq\ wAB\ \ \ ,\ \ \ w\in{\boldsymbol{h}}\,,\,A,B\in\mathcal{A}$
(3.4)

(in standard notation, $A(w\otimes B)\doteq w\otimes AB$). We call (3.4) the
_trivial left action_. At the level of Fock space, it does not induce a mixing
because $\lambda(A)=1\otimes A$, and clearly $\lambda^{\omega}(\mathcal{A})$
and $\mathcal{B}(\mathcal{F}({\boldsymbol{h}}))$ commute for any
$\omega\in\mathcal{S}(\mathcal{A})$.

### Twists and left actions.

The following notion concerns a class of left actions well-behaved with
respect to the permutation symmetry, obtained by twisting the trivial one by
means of a group action.

######  Definition 3.2.

Let $\mathcal{G}\subseteq\mathcal{U}\mathcal{A}$ be a group generating
$\mathcal{A}$ as C*-algebra and $\EuFrak h={\boldsymbol{h}}\mathcal{A}$ a free
bimodule. Then the left $\mathcal{A}$-action $\lambda$ is said to be
$\mathcal{G}$-twisted whenever there is a group morphism
$u:\mathcal{G}\to\mathcal{U}({\boldsymbol{h}})$, that we call the
$\mathcal{G}$-twist, such that

$\lambda(\gamma)w\equiv\gamma w\,=\,(u_{\gamma}w)\gamma\ \ ,\ \
\forall\gamma\in\mathcal{G}\,,\,w\in{\boldsymbol{h}}\,.$ (3.5)

We say that the $\mathcal{G}$-twist is trivial whenever $u$ is the trivial
representation.

For reader’s convenience we check the consistence of the previous definition
by verifying that the r.h.s. of (3.5) defines an adjointable (and as a
consequence $\mathcal{A}$-linear and bounded) operator. We write
$wA,w^{\prime}A^{\prime}\in\EuFrak h$ for $w,w^{\prime}\in{\boldsymbol{h}}$,
$A,A^{\prime}\in\mathcal{A}$, and compute

${\left\langle\right.}wA,\gamma w^{\prime}A^{\prime}{\left.\right\rangle}\ =\
A^{*}{\left\langle\right.}w,u_{\gamma}w^{\prime}{\left.\right\rangle}\gamma
A^{\prime}\ =\
(\gamma^{*}A)^{*}{\left\langle\right.}u_{\gamma}^{*}w,w^{\prime}{\left.\right\rangle}A^{\prime}\
=\
{\left\langle\right.}\gamma^{*}wA,w^{\prime}A^{\prime}{\left.\right\rangle}$

(note that
${\left\langle\right.}w,u_{\gamma}w^{\prime}{\left.\right\rangle}\in{\mathbb{C}}$);
thus (3.5) has adjoint $\gamma^{*}$ as expected. We remark that since
$\mathcal{G}$ generates $\mathcal{A}$ the left action $\lambda$ is determined
by $u$. Despite that, $u$ may have a kernel even when $\lambda$ does not (as
shall be evident in the following example, where $\mathcal{A}$ can be simple).
Finally we note that in the previous definition we did not assume strong
continuity of $u$, so in general it is not a unitary representation; therefore
we say that $u$ is a _unitary morphism_.

######  Example 3.1 (Weyl algebras).

Let $\mathcal{S}$ be a real vector space with a symplectic form $\eta$ and
$\mathcal{W}$ denote the associated Weyl C*-algebra generated by unitary
symbols $W_{s}$, $s\in\mathcal{S}$. Let
$\mathcal{G}\subseteq\mathcal{U}\mathcal{W}$ denote the group generated by
$W_{s}e^{1/2i\theta}$ for $s\in\mathcal{S}$ and $\theta\in{\mathbb{R}}$. The
Weyl relations imply that an Abelian quotient of $\mathcal{G}$ is given by
$\mathcal{S}$ as an additive group. Thus any unitary morphism
$u_{ab}:\mathcal{S}\to\mathcal{U}({\boldsymbol{h}})$ lifts to a morphism
$u:\mathcal{G}\to\mathcal{U}({\boldsymbol{h}})$ such that
$u(W_{s})u(W_{s^{\prime}})=u(W_{s+s^{\prime}})$ 111 As a matter of fact, such
unitary morphisms can be easily obtained, for example by exponentials of
linear functionals $\rho:\mathcal{S}\to{\mathbb{R}}$. . We then consider the
free Hilbert module $\EuFrak h\doteq{\boldsymbol{h}}\mathcal{W}$, and set
$W_{s}^{\lambda}(wB)\doteq(u(W_{s})w)\,W_{s}B$, $s\in\mathcal{S}$,
$w\in{\boldsymbol{h}}$, $B\in\mathcal{W}$, obtaining unitary operators
$W_{s}^{\lambda}\in\EuFrak{B}(\EuFrak h)$. Since

$\begin{array}[]{lcl}W_{s}^{\lambda}W_{s^{\prime}}^{\lambda}wB&=&(u(W_{s})u(W_{s^{\prime}})w)\,W_{s}W_{s^{\prime}}B\
=\\\ &=&e^{1/2i\eta(s,s^{\prime})}\,(u(W_{s+s^{\prime}})w)\,W_{s+s^{\prime}}B\
=\\\
&=&e^{1/2i\eta(s,s^{\prime})}\,W_{s+s^{\prime}}^{\lambda}\,wB\,,\end{array}$

we have that $W_{s}^{\lambda}$ fulfil the Weyl relations, so they define a
*-morphism $\lambda:\mathcal{W}\to\EuFrak{B}(\EuFrak h)$ that by construction
is a left $\mathcal{W}$-action twisted by $u$.

######  Example 3.2 (The universal C*-algebra of the electromagnetic field).

Let $k\in{\mathbb{N}}$ and $\mathcal{D}_{k}({\mathbb{R}}^{4})$ denote the
vector space of smooth, compactly supported $k$–forms on ${\mathbb{R}}^{4}$
(with Minkowski metric). Let
$\mathcal{C}_{1}({\mathbb{R}}^{4})\subset\mathcal{D}_{1}({\mathbb{R}}^{4})$ be
the subspace of 1–forms $g=(g^{\mu})\in\mathcal{D}_{1}({\mathbb{R}}^{4})$ such
that $\delta g\doteq\partial_{\mu}g^{\mu}=0$. Then $\delta
f\doteq-2\partial_{\nu}f^{\mu\nu}$ defines a 1–form $\delta
f\in\mathcal{C}_{1}({\mathbb{R}}^{4})$ for any
$f=(f^{\mu\nu})\in\mathcal{D}_{2}({\mathbb{R}}^{4})$. With this notation, we
define the C*-algebra $\mathcal{A}$ generated by the group $\mathcal{G}$ of
unitary symbols $V(g)$, $g\in\mathcal{C}_{1}({\mathbb{R}}^{4})$, with
relations

$V(a_{1}g)V(a_{2}g)=V((a_{1}+a_{2})g)\ \ ,\ \ V(g)^{*}=V(-g)\ \ ,\ \
V(0)=1\,,$ (3.6) $V(\delta f_{1})V(\delta f_{2})=V(\delta f_{1}+\delta f_{2})\
\ ,\ \ {\mathrm{supp}}\,f_{1}\perp{\mathrm{supp}}\,f_{2}\,,$ (3.7)
$\left[V(g_{1}),V(g_{2})\right]_{\bullet}\,\in\mathcal{A}\cap\mathcal{A}^{\prime}\
\ ,\ \ {\mathrm{supp}}\,g_{1}\perp{\mathrm{supp}}\,g_{2}\,,$ (3.8)

where $a_{1},a_{2}\in{\mathbb{R}}$. In the above expressions we used the
symbol $\perp$ to indicate spacelike separation, whilst
$\left[U,V\right]_{\bullet}\doteq UVU^{*}V^{*}$ is the group commutator and
$\mathcal{A}\cap\mathcal{A}^{\prime}$ is the centre of $\mathcal{A}$. Any
Wightman field $F(f)$, $f\in\mathcal{D}_{2}({\mathbb{R}}^{4})$, fulfilling the
Maxwell equations, defines a representation $\pi$ of $\mathcal{A}$ such that
$\pi(V(\delta f))=e^{iF(f)}$ [8]. Note that since
$\mathcal{C}_{1}({\mathbb{R}}^{4})$, as an additive group, is a quotient of
$\mathcal{G}$ under the map $V(g)\mapsto g$, any unitary morphism of
$\mathcal{C}_{1}({\mathbb{R}}^{4})$ induces a unitary morphism of
$\mathcal{G}$. Let now
$u:\mathcal{C}_{1}({\mathbb{R}}^{4})\to\mathcal{U}({\boldsymbol{h}})$ be a
unitary morphism; we set $\EuFrak h\doteq{\boldsymbol{h}}\mathcal{A}$ and
define $V^{\lambda}(g)\,wA\doteq(u_{g}w)V(g)A$, $w\in{\boldsymbol{h}}$,
$A\in\mathcal{A}$. A straightforward check shows that the operators
$V^{\lambda}(g)$ are in $\EuFrak{B}(\EuFrak h)$ and fulfil (3.6-3.8): for
example, about (3.7) we have

$\begin{array}[]{lcl}V^{\lambda}(\delta f_{1})V^{\lambda}(\delta
f_{2})\,wA&=&(u_{\delta f_{1}}u_{\delta f_{2}}w)\,V(\delta f_{1})V(\delta
f_{2})A\ =\\\ &=&(u_{\delta f_{1}+\delta f_{2}}w)\,V(\delta f_{1}+\delta
f_{2})A\ =\\\ &=&V^{\lambda}(\delta f_{1}+\delta f_{2})\,wA\,.\end{array}$

Thus we have the left action $\lambda:\mathcal{A}\to\EuFrak{B}(\EuFrak h)$,
$\lambda(V(g))\equiv V^{\lambda}(g)$, that by construction is
$\mathcal{G}$-twisted.

### Permutation symmetry.

Lemma 3.1 allows to define a permutation symmetry in the obvious way, by
extending the one defined on $\mathcal{F}({\boldsymbol{h}})$:

$U_{\varrho}(wA)\,\doteq\,w_{\varrho}A\ \ \ ,\ \ \
\varrho\in{\mathbb{P}}(n)\,,\,w\in{\boldsymbol{h}}^{n}\,,\,n\in{\mathbb{N}}\,,$
(3.9)

where $w_{\varrho}\in{\boldsymbol{h}}^{n}$ is the vector transformed under the
usual permutation symmetry in Fock space. Of course, to apply the previous
definition we must express a tensor of the type $v_{1}\otimes\ldots\otimes
v_{n}$, $v_{1},\ldots,v_{n}\in\EuFrak h$, in terms of tensors of the type
$(w_{1}\otimes\ldots\otimes w_{n})A$, $w_{1},\ldots,w_{n}\in{\boldsymbol{h}}$,
$A\in\mathcal{A}$. By definition $U_{\varrho}$ is unitary on
$\EuFrak{F}(\EuFrak h)$ in the sense of right Hilbert modules, thus we get the
spectral projection
$P_{-}\doteq\oplus_{n}P_{-}^{n}\in\EuFrak{B}(\EuFrak{F}(\EuFrak h))$,
$P_{-}^{n}\doteq(n!)^{-1}\sum_{\varrho}\epsilon_{\varrho}U_{\varrho}$, where
$\epsilon_{\varrho}$ is the sign of $\varrho\in{\mathbb{P}}(n)$. We write
$\EuFrak{F}_{-}(\EuFrak h)\doteq P_{-}\EuFrak{F}(\EuFrak h)$; it is obvious
that $\EuFrak{F}_{-}(\EuFrak h)$ is a right $\mathcal{A}$-module, and that it
is free with $\EuFrak{F}_{-}(\EuFrak
h)\simeq\mathcal{F}_{-}({\boldsymbol{h}})\otimes\mathcal{A}$ where
$\mathcal{F}_{-}({\boldsymbol{h}})$ is the ordinary fermionic Fock space. We
have the decomposition

$\EuFrak{F}_{-}(\EuFrak h)\ \doteq\ \bigoplus_{n\geq 0}\EuFrak h^{n}_{-}\
\simeq\ \bigoplus_{n\geq 0}\,({\boldsymbol{h}}^{n}_{-}\mathcal{A})\,,$ (3.10)

where $\EuFrak h^{n}_{-}\doteq P_{-}^{n}\EuFrak h^{n}$,
${\boldsymbol{h}}^{n}_{-}\doteq P_{-}^{n}{\boldsymbol{h}}^{n}$. Of course,
$\EuFrak h^{0}_{-}=\mathcal{A}$ and $\EuFrak h^{1}_{-}=\EuFrak
h\simeq{\boldsymbol{h}}\mathcal{A}$.

######  Remark 3.3 (The Pauli principle).

In spite of the simplicity of our definition some care is needed to handle
antisymmetric tensors. For example, if $f,g\in\EuFrak h$ then it is not
ensured that $P_{-}(f\otimes g)=-P_{-}(g\otimes f)$, unless $f$ and $g$ belong
to ${\boldsymbol{h}}$. In particular, the Pauli principle

$P_{-}(f\otimes f)\ =\ 0\ \ \ ,\ \ \ f\in\EuFrak h\,,$ (3.11)

does not hold in general, and a priori its validity is ensured only for
$f\in{\boldsymbol{h}}$. A more general class of examples for which the Pauli
principle holds is the following. Given $f\in\EuFrak h$, we define the support

$\mathcal{A}(f)\,\doteq\,C^{*}\\{{\left\langle\right.}v,f{\left.\right\rangle}\in\mathcal{A}\,:\,v\in{\boldsymbol{h}}\\}\,\subseteq\mathcal{A}\,.$
(3.12)

The support has the property that if $f=f_{h}A_{h}$, where
$\\{f_{h}\\}\subset{\boldsymbol{h}}$ is an orthonormal set and
$A_{h}\in\mathcal{A}$, then
$A_{h}={\left\langle\right.}f_{h},f{\left.\right\rangle}\in\mathcal{A}(f)$ for
all $h\in{\mathbb{N}}$. Given $f,g\in\EuFrak h$, we write

$f\bowtie g\ \ \stackrel{{\scriptstyle.}}{{\Leftrightarrow}}\ \
fB^{\prime}=B^{\prime}f\ \ ,\ \ gB=Bg\ \ ,\ \ \left[B,B^{\prime}\right]\ =\
0\,,$ (3.13)

for all $B\in\mathcal{A}(f)\,,\,B^{\prime}\in\mathcal{A}(g)$. In this case we
say that $f$ and $g$ are mutually free, and we find

$\begin{array}[]{lcl}P_{-}(f\otimes g)&=&P_{-}(f_{h}\otimes A_{h}g)\ =\
P_{-}(f_{h}\otimes g)A_{h}\ =\\\ &=&P_{-}(f_{h}\otimes
g_{k})\,A^{\prime}_{k}A_{h}\ =\ -P_{-}(g_{k}\otimes
f_{h})\,A^{\prime}_{k}A_{h}\ =\\\ &=&-P_{-}(g_{k}\otimes
f_{h})\,A_{h}A^{\prime}_{k}\ =\ -P_{-}(g_{k}\otimes f)\,A^{\prime}_{k}\ =\\\
&=&-P_{-}(g\otimes f)\,,\end{array}$

so that

$P_{-}(f\otimes g)\ =\ -P_{-}(g\otimes f)\ \ ,\ \ f\bowtie g\,.$ (3.14)

We conclude that validity of the Pauli principle is related to commutation
properties of $\mathcal{A}$ (locality, when $\mathcal{A}$ is the C*-algebra of
a Haag-Kastler net), and properties of the left $\mathcal{A}$-action, that is,
the eventuality that it is trivial on the involved vectors of $\EuFrak h$ and
elements of $\mathcal{A}$. Note for $f\in{\boldsymbol{h}}$ we have
$\mathcal{A}(f)={\mathbb{C}}$ so that $f,g\in{\boldsymbol{h}}$ implies
$f\bowtie g$.

### Fermionic creation and annihilation operators.

To define and handle fermionic creation and annihilation operators we make
some remarks on elementary tensors, using the attention needed in the case of
Hilbert bimodules.

We start by noting that we may arrange order $n$ permutations by considering
for any $k=1,\ldots,n$ the set of those permutations that bring the $k$-th
object at first position. Thus for any elementary tensor $v=wA\in\EuFrak
h^{n}$, $w\in{\boldsymbol{h}}^{n}$, $A\in\mathcal{A}$, we may write

$v_{-}\ \doteq\ P_{-}^{n}v\ =\
\frac{1}{n}\sum_{k=1}^{n}(-1)^{k-1}\,w_{k}\otimes w^{(k)}_{-}A\,\in\EuFrak
h^{n}_{-}\ ,$ (3.15) $\displaystyle
w^{(k)}_{-}\,\doteq\,\ldots\otimes_{-}\hat{w}_{k}\otimes_{-}\ldots=\frac{1}{(n-1)!}\sum_{\varrho\in{\mathbb{P}}_{n-1,k}}\epsilon_{\varrho}\,w_{\varrho(1)}\otimes\ldots\otimes
w_{\varrho(n)}\,\in{\boldsymbol{h}}^{n-1}_{-}\,,$

where the notation $\hat{w}_{k}$ indicates that $w_{k}$ does not appear in the
tensor and ${\mathbb{P}}_{n-1,k}$ is understood as the permutation group of
the set $\\{1,\ldots,n\\}\setminus\\{k\\}$; the term $(-1)^{k-1}$ in (3.15) is
the sign of the transposition bringing $w_{k}$ at first position. The above
expression makes manifest that in general $\EuFrak{F}_{-}(\EuFrak h)$ is not
stable under the left $\mathcal{A}$-action, because the terms
$Bw_{k}=\sum_{i}w_{i}B_{ik}$,
$B_{ik}\doteq{\left\langle\right.}w_{i},Bw_{k}{\left.\right\rangle}\in\mathcal{A}$,
typically belong to $\EuFrak h$ and induce a mixing in the tensor product. The
point is that in general $Bw_{k}$ does not belong to ${\boldsymbol{h}}$, thus
we must perform the operations of the proof of Lemma 3.1 to get a tensor of
the form $Bv_{-}=\sum_{i}w^{\prime}_{i}A^{\prime}_{i}$ with
$w^{\prime}_{i}\in{\boldsymbol{h}}^{n}$. It is after this operation that we
can apply the projection $P_{-}^{n}$ and get
$P_{-}^{n}Bv_{-}=\sum_{i}w^{\prime}_{i,-}A_{i}$.

In the sequel, we shall write $\EuFrak h_{\circ}\subseteq\EuFrak h$ for the
vector space spanned by (finite linear combinations of) vectors of the type
$w\gamma$, $w\in{\boldsymbol{h}}$, $\gamma\in\mathcal{G}$. By construction
$\EuFrak h_{\circ}$ is dense in $\EuFrak h$.

We now establish some properties of antisymmetric tensors in case of a
$\mathcal{G}$-twist. The most important is that, in spite of the previous
remark, the fermionic space is stable both under the $\mathcal{G}$-action and
antisymmetric tensor products by vectors in $\EuFrak h_{\circ}$.

######  Lemma 3.4 (Twists and permutation symmetry).

Let $\EuFrak h={\boldsymbol{h}}\mathcal{A}$ be a free Hilbert bimodule with
$\mathcal{G}$-twist $u:\mathcal{G}\to\mathcal{U}({\boldsymbol{h}})$. Then the
following properties hold:

  1. 1.

For any $\gamma\in\mathcal{G}\subseteq\mathcal{U}\mathcal{A}$ it turns out
$\gamma\EuFrak h^{n}_{-}\in\EuFrak h^{n}_{-}$, so that $\EuFrak{F}_{-}(\EuFrak
h)$ is stable under the left module action by $\mathcal{G}$.

  2. 2.

Let $g=g_{i}\gamma_{i}\in\EuFrak h_{\circ}$ with $g_{i}\in{\boldsymbol{h}}$,
$\gamma_{i}\in\mathcal{G}$. Then for any tensor $w_{-}A\in\EuFrak h^{n}_{-}$
with $w_{-}\doteq
w_{1}\otimes_{-}\ldots\otimes_{-}w_{n}\in{\boldsymbol{h}}_{-}^{n}$,
$A\in\mathcal{A}$, we have

$P_{-}^{n+1}(g\otimes w_{-}A)\ =\
g_{i}\otimes_{-}(u_{\gamma_{i}}w_{1})\otimes_{-}\ldots\otimes_{-}(u_{\gamma_{i}}w_{n})\cdot\gamma_{i}A\
\in\EuFrak h^{n}_{-}\,,$ (3.16)

so $\EuFrak{F}_{-}(\EuFrak h)$ is stable under antisymmetric tensor product by
elements of $\EuFrak h_{\circ}$;

  3. 3.

For $g=g_{i}\gamma_{i}\in\EuFrak h_{\circ}$ it turns out

$P_{-}^{n+1}(g\otimes w_{-}A)\ =\ \frac{1}{n}\,\left(g\otimes
w_{-}A-\sum_{k=1}^{n}(-1)^{k-1}w_{k}\otimes P_{-}^{n}(g\otimes
w^{(k)}_{-}A)\right)\,.$ (3.17)

###### Proof.

Point 1. Let $w_{-}A\in\EuFrak h^{n}_{-}$ with
$w=w_{1}\otimes_{-}\ldots\otimes_{-}w_{n}$ an elementary antisymmetric tensor.
Then for any permutation $\varrho$ it turns out

$\begin{array}[]{lcl}\gamma w_{\varrho(1)}\otimes\ldots\otimes
w_{\varrho(n)}&=&(u_{\gamma}w_{\varrho(1)})\otimes\ldots\otimes(u_{\gamma}w_{\varrho(n)})\gamma\,.\end{array}$

Thus, defining $u_{\gamma}^{-}\doteq
P_{-}^{n}\cdot\otimes^{n}u_{\gamma}\in\mathcal{U}({\boldsymbol{h}}_{-}^{n})$
we find that $\gamma w_{-}A=(u_{\gamma}^{-}w_{-})\gamma A$ belongs to $\EuFrak
h^{n}_{-}$ as claimed. Point 2. With the notation of the previous point, with
$g=g_{i}\gamma_{i}$ we have

$\begin{array}[]{lcl}g\otimes
w_{-}A&=&\sum_{\varrho}\epsilon_{\varrho}\,g_{i}\otimes\gamma_{i}w_{\varrho(1)}\otimes\ldots\otimes
w_{\varrho(n)}A\,=\\\
&=&\sum_{\varrho}\epsilon_{\varrho}\,g_{i}\otimes(u_{\gamma_{i}}w_{\varrho(1)})\otimes\ldots\otimes(u_{\gamma_{i}}w_{\varrho(n)})\gamma_{i}A\,=\\\
&=&g_{i}\otimes(u_{\gamma_{i}}w_{1})\otimes_{-}\ldots\otimes_{-}(u_{\gamma_{i}}w_{n})\cdot\gamma_{i}A\,.\end{array}$

Thus, using the fact that $P_{-}^{n}$ is a right $\mathcal{A}$-linear
operator, we conclude that

$\begin{array}[]{lcl}P_{-}^{n}(g\otimes
w_{-}A)&=&P_{-}^{n}(g_{i}\otimes((u_{\gamma_{i}}w_{1})\otimes_{-}\ldots\otimes_{-}(u_{\gamma_{i}}w_{n}))\,\gamma_{i}A\,=\\\
&=&g_{i}\otimes_{-}(u_{\gamma_{i}}w_{1})\otimes_{-}\ldots\otimes_{-}(u_{\gamma_{i}}w_{n})\,\gamma_{i}A\,.\end{array}$

Point 3. We have, by applying (3.15) and (3.16),

$\begin{array}[]{lcl}P_{-}^{n+1}(g\otimes
w_{-}A)&=&(g_{i}\otimes_{-}u_{\gamma_{i}}^{-}w_{-})\gamma_{i}A\,=\\\
&=&1/n\left(g_{i}\otimes
u_{\gamma_{i}}^{-}w_{-}-\sum_{k=1}^{n}(-1)^{k-1}\,w_{k}\otimes(g_{i}\otimes_{-}u_{\gamma_{i}}^{-}w^{(k)}_{-})\right)\gamma_{i}A\,=\\\
&=&1/n\left(g_{i}\gamma_{i}\otimes
w_{-}-\sum_{k=1}^{n}(-1)^{k-1}\,w_{k}\otimes(g_{i}\gamma_{i}\otimes_{-}w^{(k)}_{-})\right)A\,=\\\
&=&1/n\left(g\otimes
w_{-}-\sum_{k=1}^{n}(-1)^{k-1}\,w_{k}\otimes(g\otimes_{-}w^{(k)}_{-})\right)A\,.\end{array}$

∎

Now, by (3.15), given $f\in\EuFrak h$ we have

${\left\langle\right.}f|v_{-}\ =\ \frac{1}{n}\sum_{k}(\pm
1)^{k-1}{\left\langle\right.}f,w_{k}{\left.\right\rangle}\,w^{(k)}_{-}A\,\in\EuFrak
h^{n-1}\,,$ (3.18)

thus in general we cannot say that ${\left\langle\right.}f|v_{-}\in\EuFrak
h^{n-1}_{-}$. Yet we have the following property:

######  Lemma 3.5.

Assume that there is a $\mathcal{G}$-twist on $\EuFrak h$ and let $f\in\EuFrak
h_{\circ}$. Then ${\left\langle\right.}f|v_{-}\in\EuFrak h^{n-1}_{-}$ for all
$v_{-}=w_{-}A\in\EuFrak h^{n}_{-}$, and $\EuFrak{F}_{-}(\EuFrak h)$ is stable
under the action of the operator ${\left\langle\right.}f|$.

###### Proof.

Starting from (3.18) and writing $f=f_{i}\gamma_{i}$,
$f_{i}\in{\boldsymbol{h}}$, $\gamma_{i}\in\mathcal{G}$, we find

$\begin{array}[]{lcl}{\left\langle\right.}f|v_{-}&=&1/n\sum_{k}(\pm
1)^{k-1}{\left\langle\right.}f_{i},w_{k}{\left.\right\rangle}\,\gamma_{i}^{*}w^{(k)}_{-}A\,=\\\
&=&1/n\sum_{k}(\pm
1)^{k-1}{\left\langle\right.}f_{i},w_{k}{\left.\right\rangle}\,(u_{\gamma_{i}}^{*}w_{1})\otimes_{-}\ldots\otimes_{-}(u_{\gamma_{i}}^{*}w_{n})\,\gamma_{i}^{*}A\,.\end{array}$

Since ${\left\langle\right.}f_{i},w_{k}{\left.\right\rangle}\in{\mathbb{C}}$,
the last term belongs to $\EuFrak h^{n-1}_{-}$ as claimed. ∎

The next computations will allow to evaluate the anti-commutation relations.
By (3.17), for $f\in\EuFrak h$ and $g=g_{i}\gamma_{i}\in\EuFrak h_{\circ}$ we
get

${\left\langle\right.}f|P_{-}^{n+1}(g\otimes w_{-}A)\ =\
\frac{1}{n}\,\left({\left\langle\right.}f,g{\left.\right\rangle}w_{-}A-\sum_{k=1}^{n}(-1)^{k-1}{\left\langle\right.}f,w_{k}{\left.\right\rangle}P_{-}^{n}(g\otimes
w^{(k)}_{-})A\right)$ (3.19)

and

$P_{-}^{n+1}(g\otimes{\left\langle\right.}f|w_{-}A)\ =\
\frac{1}{n}\sum_{k}(-1)^{k-1}P_{-}^{n}(g\otimes{\left\langle\right.}f,w_{k}{\left.\right\rangle}\,w^{(k)}_{-})A\,.$
(3.20)

About the last equality, we note that due to the antisymmetrization operator
$P_{-}^{n}$ the factor
${\left\langle\right.}f,w_{k}{\left.\right\rangle}\in\mathcal{A}$ appears in
$i$-th position of the involved elementary tensor for any $i=1,\ldots,n$. This
implies that (3.20) may differ from the sum at r.h.s. of (3.19), because in
general ${\left\langle\right.}f,w_{k}{\left.\right\rangle}$ cannot freely
shift on the left of the involved elementary tensor.

Let now $P_{-}\doteq\oplus_{n}P_{-}^{n}$, where $P_{-}^{0}$ and $P_{-}^{1}$
are the identity. We introduce the notation $\EuFrak{F}_{-}^{\\#}(\EuFrak
h)\doteq\EuFrak{F}_{-}(\EuFrak h)\cap\EuFrak{F}^{\\#}(\EuFrak h)$, and for any
$f\in\EuFrak h_{\circ}$ define the fermionic annihilation and creation
operators

$\left\\{\begin{array}[]{l}{\boldsymbol{a}}_{-}(f):\EuFrak{F}_{-}^{\\#}(\EuFrak
h)\to\EuFrak{F}_{-}^{\\#}(\EuFrak h)\ \ \ ,\ \ \
{\boldsymbol{a}}_{-}(f)\,\doteq\,a(f)\restriction\EuFrak{F}_{-}^{\\#}(\EuFrak
h)\,,\\\ \\\ {\boldsymbol{a}}_{-}^{*}(f):\EuFrak{F}_{-}^{\\#}(\EuFrak
h)\to\EuFrak{F}_{-}^{\\#}(\EuFrak h)\ \ ,\ \
{\boldsymbol{a}}_{-}^{*}(f)\,\doteq\,P_{-}a^{*}(f)\restriction\EuFrak{F}_{-}^{\\#}(\EuFrak
h)\,.\end{array}\right.$ (3.21)

Note that the property

${\boldsymbol{a}}_{-}(f)\EuFrak{F}_{-}^{\\#}(\EuFrak
h)\,\subseteq\,\EuFrak{F}_{-}^{\\#}(\EuFrak h)\ \ ,\ \
{\boldsymbol{a}}_{-}(f)\,=\,P_{-}{\boldsymbol{a}}_{-}(f)\ \ ,\ \ \forall
f\in\EuFrak h_{\circ}\,,$ (3.22)

tacitly understood in (3.21) is a consequence of the hypothesis that there is
a $\mathcal{G}$-twist and Lemma 3.5. Also note that for the moment we do not
know whether the fermionic creation and annihilation operators are bounded,
thus in (3.21) we make use of the domain $\EuFrak{F}_{-}^{\\#}(\EuFrak h)$. In
the following result we give an interpretation of the twist
$u:\mathcal{G}\to\mathcal{U}({\boldsymbol{h}})$ as an obstacle to make the
creation and annihilation operators commute with elements of $\mathcal{A}$.

######  Lemma 3.6.

Let $v\in{\boldsymbol{h}}$ and $\gamma\in\mathcal{G}$. Then
$\gamma{\boldsymbol{a}}_{-}^{*}(v)={\boldsymbol{a}}_{-}^{*}(u_{\gamma}v)\gamma$
and
$\gamma{\boldsymbol{a}}_{-}(u_{\gamma}^{*}v)={\boldsymbol{a}}_{-}(v)\gamma$.

###### Proof.

Let $w_{-}A\in\EuFrak h_{-}^{n}$, $w_{-}\in{\boldsymbol{h}}_{-}^{n}$,
$A\in\mathcal{A}$. By the argument of the proof of Lemma 3.4, Point 1, we find

$\begin{array}[]{lcl}\gamma{\boldsymbol{a}}_{-}^{*}(v)w_{-}A&=&\gamma
P_{-}(v\otimes w_{-})A\ =\ (u_{\gamma}v\otimes_{-}u_{\gamma}^{-}w_{-})\gamma
A\ =\\\ &=&{\boldsymbol{a}}_{-}^{*}(u_{\gamma}v)(u_{\gamma}^{-}w_{-})\gamma A\
=\ {\boldsymbol{a}}_{-}^{*}(u_{\gamma}v)\gamma w_{-}A\,.\end{array}$

With an analogous argument the claim about ${\boldsymbol{a}}_{-}(v)$ is
proved. ∎

######  Lemma 3.7.

Let $f\in\EuFrak h_{\circ}$. Then ${\boldsymbol{a}}_{-}^{*}(f)$ is the adjoint
of ${\boldsymbol{a}}_{-}(f)$ over the domain $\EuFrak{F}_{-}^{\\#}(\EuFrak
h)$.

###### Proof.

By applying (3.21) and (3.22) we find

${\left\langle\right.}v,{\boldsymbol{a}}_{-}(f)v^{\prime}{\left.\right\rangle}\
=\ {\left\langle\right.}v,a(f)P_{-}v^{\prime}{\left.\right\rangle}\ =\
{\left\langle\right.}P_{-}a^{*}(f)v,v^{\prime}{\left.\right\rangle}\ =\
{\left\langle\right.}{\boldsymbol{a}}_{-}^{*}(f)v,v^{\prime}{\left.\right\rangle}\,,$

for all $v,v^{\prime}\in\EuFrak{F}_{-}^{\\#}(\EuFrak h)$. This proves the
Lemma. ∎

Let $\omega\in\mathcal{S}(\mathcal{A})$. We define
$\EuFrak{F}_{-}^{\omega}(\EuFrak h)$ as the Hilbert space obtained by
evaluation of $\EuFrak{F}_{-}(\EuFrak h)$ over $\omega$. In the previous lines
we proved that ${\boldsymbol{a}}_{-}(f)$ and ${\boldsymbol{a}}_{-}^{*}(f)$ are
well-defined, $\mathcal{A}$-linear and one the adjoint of the other in the
sense of Hilbert modules (on $\EuFrak{F}_{-}^{\\#}(\EuFrak h)$); thus, by the
argument used to construct $a^{\omega}(f)$ and $a^{*,\omega}(f)$, we define

${\boldsymbol{a}}_{-}^{\omega}(f)\ \ \ ,\ \ \
{\boldsymbol{a}}_{-}^{*,\omega}(f)\ \ \ ,\ \ \ f\in\EuFrak h_{\circ}\,,$

as the evaluations of ${\boldsymbol{a}}_{-}(f)$ and
${\boldsymbol{a}}_{-}^{*}(f)$ over $\omega$. The domain of these operators is
clearly given by $\EuFrak{F}_{-}^{\\#,\omega}(\EuFrak h)$, defined by
evaluation of vectors in $\EuFrak{F}_{-}^{\\#}(\EuFrak h)$.

##  4 CARs and fermionic fields

In the present section we study the anti-commutation relations that our
fermionic creation and annihilation operators fulfill, and define the
corresponding Dirac field assuming the presence of a suitable conjugation. We
bring on the light some features that in general prevent anticommutators to be
expressed only in terms of the given $\mathcal{A}$-valued scalar product and,
primarily, make anticommutators non-local in the sense that they do not vanish
even when the involved ”spinors” are orthogonal. In fact, not surprisingly, to
get anticommutators of the usual form besides orthogonality we must require
mutual freeness Rem.3.3.

We proceed by maintaining the assumptions of the previous section, so that
$\EuFrak h={\boldsymbol{h}}\mathcal{A}$ is free and there is a
$\mathcal{G}$-twist $u:\mathcal{G}\to\mathcal{U}({\boldsymbol{h}})$,
$\mathcal{G}\subseteq\mathcal{U}\mathcal{A}$. Before computing our anti-
commutation relations, for convenience we give a notion of mutual freeness
explicitly designed to handle vectors in $\EuFrak h_{\circ}$. Let
$f=f_{i}\gamma_{i}$ and $g=g_{h}\gamma_{h}\in\EuFrak h_{\circ}$, with
$f_{i},g_{h}\in{\boldsymbol{h}}$,
$\gamma_{i},\gamma^{\prime}_{h}\in\mathcal{G}$. We write $f\bowtie_{\circ}g$
whenever for all $i,h$ it turns out

$[\gamma_{i}\,,\,\gamma^{\prime}_{h}]\,=\,0\ \ ,\ \
u_{\gamma_{i}}g_{h}\,=\,g_{h}\ \ ,\ \
u_{\gamma^{\prime}_{h}}f_{i}\,=\,f_{i}\,.$ (4.1)

######  Lemma 4.1.

If $f,g\in\EuFrak h_{\circ}$ and $f\bowtie_{\circ}g$, then $f\bowtie g$.

###### Proof.

As a preliminary step we note that any generator
${\left\langle\right.}w,f{\left.\right\rangle}$ of $\mathcal{A}(f)$, defined
for $w\in{\boldsymbol{h}}$, is a linear combination in $\\{\gamma_{i}\\}$, so
$\mathcal{A}(f)$ is contained in the C*-algebra generated by
$\\{\gamma_{i}\\}$. Thus the hypothesis $f\bowtie_{\circ}g$ and (4.1) imply
$[A,B]=0$ for all $A\in\mathcal{A}(f)$, $B\in\mathcal{A}(g)$; moreover,
$\gamma_{i}g=(u_{\gamma_{i}}g_{h})\gamma_{i}\gamma^{\prime}_{h}=(u_{\gamma_{i}}g_{h})\gamma^{\prime}_{h}\gamma_{i}=g\gamma_{i}$,
and analogously for $f$ and $\gamma^{\prime}_{h}$. This implies $f\bowtie g$,
as claimed. ∎

### Anticommutators of creation operators.

Let $f\bowtie_{\circ}g\in\EuFrak h_{\circ}$ and $v_{-}=w_{-}A\in\EuFrak
h_{-}^{n}$, $w_{-}\in{\boldsymbol{h}}_{-}^{n}$, $A\in\mathcal{A}$. Writing
$f=f_{i}\gamma_{i}$, $g=g_{h}\gamma^{\prime}_{h}$ we find

$\begin{array}[]{lcl}{\boldsymbol{a}}_{-}^{*}(f){\boldsymbol{a}}_{-}^{*}(g)w_{-}A&=&P_{-}^{n+2}(f\otimes
P_{-}^{n+1}(g\otimes w_{-}))A\ =\\\
&=&P_{-}^{n+2}(f_{i}\otimes\gamma_{i}P_{-}^{n+1}(g_{h}\otimes
u^{\prime}_{h}w_{-}))\gamma^{\prime}_{h}A\ =\\\ &=&P_{-}^{n+2}(f_{i}\otimes
P_{-}^{n+1}(g_{h}\otimes
u_{i}u^{\prime}_{h}w_{-}))\gamma_{i}\gamma^{\prime}_{h}A\ =\\\
&=&-P_{-}^{n+2}(g_{h}\otimes P_{-}^{n+1}(f_{i}\otimes
u_{i}u^{\prime}_{h}w_{-}))\gamma_{i}\gamma^{\prime}_{h}A\ =\\\
&=&-P_{-}^{n+2}(g_{h}\otimes
P_{-}^{n+1}(f_{i}\otimes\gamma_{i}\gamma^{\prime}_{h}w_{-}))A\ =\\\
&=&-P_{-}^{n+2}(g_{h}\otimes P_{-}^{n+1}(f\otimes\gamma^{\prime}_{h}w_{-}))A\
=\\\ &=&-P_{-}^{n+2}(g\otimes P_{-}^{n+1}(f\otimes w_{-}))A\ =\\\
&=&-{\boldsymbol{a}}_{-}^{*}(g){\boldsymbol{a}}_{-}^{*}(f)w_{-}A\,.\end{array}$

In the previous computation, for any $n\in{\mathbb{N}}$ we defined the
antisymmetric tensor powers $u_{i}\doteq u(\gamma_{i})^{-}$,
$u^{\prime}_{h}\doteq
u(\gamma^{\prime}_{h})^{-}\in\mathcal{U}({\boldsymbol{h}}_{-}^{n})$; moreover,
we used (3.16) and the fact that expressions of the type
$P_{-}^{n+2}(f_{i}\otimes P_{-}^{n+1}(g_{h}\otimes u_{i}u^{\prime}_{h}w_{-}))$
are ordinary antisymmetric tensor powers on Hilbert space, so the Pauli
principle applies and the minus sign appears when we exchange $f_{i}$ and
$g_{h}$. We conclude that

$[{\boldsymbol{a}}_{-}^{*}(f)\,,\,{\boldsymbol{a}}_{-}^{*}(g)]_{+}\,=\,0\ \ ,\
\ f\bowtie_{\circ}g\in\EuFrak h_{\circ}\,.$ (4.2)

Note that the need to assume $f\bowtie_{\circ}g$ to make the above
anticommutators vanish is an aspect of non-validity of the Pauli principle in
full generality, Remark 3.3.

### Anticommutators of annihilation operators.

It should not be a surprise that passing to the adjoint of (4.2) the
anticommutators of annihilation operators vanish. Anyway it is instructive to
perform the explicit computations, both to understand the interplay of
elements of $\mathcal{G}$ with the relation $\bowtie_{\circ}$ and to keep in
evidence details that may be useful to approach (unbounded) bosonic
annihilation operators.

We maintain the hypothesis $f\bowtie_{\circ}g\in\EuFrak h_{\circ}$ and the
notation for the orthonormal decompositions $f=f_{i}\gamma_{i}$,
$g=g_{h}\gamma^{\prime}_{h}$. As a first step, we further apply (3.15) and get

$w_{-}\,=\,\frac{1}{n(n-1)}\left(\sum_{k<h}(-1)^{k+h-1}w_{k}\otimes
w_{h}\otimes w^{(h,k)}+\sum_{k>h}(-1)^{k+h}w_{h}\otimes w_{k}\otimes
w^{(h,k)}\right)$ (4.3)
$=\,\frac{2}{n(n-1)}\sum_{k<h}(-1)^{k+h-1}(w_{k}\otimes_{-}w_{h})\otimes
w^{(h,k)}\,,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\,\,\,\,$ (4.4)

where

$w^{(h,k)}\,\doteq\,\left\\{\begin{array}[]{l}\ldots\hat{w}_{k}\otimes_{-}\ldots\otimes_{-}\hat{w}_{h}\ldots\
\ ,\ \ k<h\,,\\\
\ldots\hat{w}_{h}\otimes_{-}\ldots\otimes_{-}\hat{w}_{k}\ldots\ \ ,\ \
k>h\,;\end{array}\right.$ (4.5)

in (4.3) we used the fact that for $k<h$ the term $w_{h}$ needs a cyclic
permutation with order $h-2$ to shift on the left of the tensor
$w_{1}\otimes\ldots\hat{w}_{k}\ldots\otimes w_{n}$, whilst for $k>h$ the order
is $h-1$. Using elementary properties of symmetric tensors, we find
$w^{(h,k)}=-w^{(k,h)}$, $h\neq k$. After these preparations, we get

$\displaystyle{\left\langle\right.}g|{\left\langle\right.}f|v_{-}$
$\displaystyle=\,\frac{2}{n(n-1)}\sum_{k<h}(-1)^{k+h-1}{\left\langle\right.}f\otimes
g\,,\,w_{k}\otimes_{-}w_{h}{\left.\right\rangle}\,w^{(h,k)}A$
$\displaystyle=\,\frac{2}{n(n-1)}\sum_{k<h}(-1)^{k+h-1}{\left\langle\right.}f\otimes
g\,,\,w_{k}\otimes_{-}w_{h}{\left.\right\rangle}\,w^{(h,k)}A\,.$

We analyze in details the terms

$\begin{array}[]{l}{\left\langle\right.}f\otimes
g\,,\,w_{h}\otimes_{-}w_{k}{\left.\right\rangle}\ =\\\
{\left\langle\right.}g_{h}\gamma^{\prime}_{h}\,,\,{\left\langle\right.}f_{i}\gamma_{i},w_{h}{\left.\right\rangle}\,w_{k}{\left.\right\rangle}-{\left\langle\right.}g_{h}\gamma^{\prime}_{h}\,,\,{\left\langle\right.}f_{i}\gamma_{i},w_{k}{\left.\right\rangle}\,w_{h}{\left.\right\rangle}\,=\\\
{\gamma^{\prime}_{h}}^{*}{\left\langle\right.}g_{h}\gamma_{i}\,,\,{\left\langle\right.}f_{i},w_{h}{\left.\right\rangle}\,w_{k}{\left.\right\rangle}-{\gamma^{\prime}_{h}}^{*}{\left\langle\right.}g_{h}\gamma_{i}\,,\,{\left\langle\right.}f_{i},w_{k}{\left.\right\rangle}\,w_{h}{\left.\right\rangle}\,=\\\
{\gamma^{\prime}_{h}}^{*}\gamma_{i}^{*}({\left\langle\right.}f_{i},w_{h}{\left.\right\rangle}{\left\langle\right.}g_{h},w_{k}{\left.\right\rangle}-{\left\langle\right.}f_{i},w_{k}{\left.\right\rangle}{\left\langle\right.}g_{h},w_{h}{\left.\right\rangle})\,=\\\
{\gamma^{\prime}_{h}}^{*}\gamma_{i}^{*}({\left\langle\right.}f_{i}\,,\,{\left\langle\right.}g_{h},w_{k}{\left.\right\rangle}w_{h}{\left.\right\rangle}-{\left\langle\right.}f_{i}\,,\,{\left\langle\right.}g_{h},w_{h}{\left.\right\rangle}w_{k}{\left.\right\rangle})\,=\\\
{\left\langle\right.}f_{i}\gamma_{i}\gamma^{\prime}_{h}\,,\,{\left\langle\right.}g_{h},w_{k}{\left.\right\rangle}w_{h}{\left.\right\rangle}-{\left\langle\right.}f_{i}\gamma_{i}\gamma^{\prime}_{h}\,,\,{\left\langle\right.}g_{h},w_{h}{\left.\right\rangle}w_{k}{\left.\right\rangle}\,=\\\
{\left\langle\right.}\gamma^{\prime}_{h}f\,,\,{\left\langle\right.}g_{h},w_{k}{\left.\right\rangle}w_{h}{\left.\right\rangle}-{\left\langle\right.}\gamma^{\prime}_{h}f\,,\,{\left\langle\right.}g_{h},w_{h}{\left.\right\rangle}w_{k}{\left.\right\rangle}\,=\\\
{\left\langle\right.}f\,,\,{\left\langle\right.}g,w_{k}{\left.\right\rangle}w_{h}{\left.\right\rangle}-{\left\langle\right.}f\,,\,{\left\langle\right.}g,w_{h}{\left.\right\rangle}w_{k}{\left.\right\rangle}\,=\\\
{\left\langle\right.}g\otimes f\,,\,w_{k}\otimes
w_{h}{\left.\right\rangle}-{\left\langle\right.}g\otimes f\,,\,w_{h}\otimes
w_{k}{\left.\right\rangle}\,=\\\ -{\left\langle\right.}g\otimes
f\,,\,w_{h}\otimes_{-}w_{k}{\left.\right\rangle}\,,\end{array}$ (4.6)

and conclude that
$({\left\langle\right.}g|{\left\langle\right.}f|+{\left\langle\right.}f|{\left\langle\right.}g|)v_{-}=0$,
obtaining

$[{\boldsymbol{a}}_{-}(f)\,,\,{\boldsymbol{a}}_{-}(g)]_{+}\,=\,0\ \ ,\ \
f\bowtie_{\circ}g\in\EuFrak h_{\circ}\,.$ (4.7)

### Mixed Anticommutators.

Finally we analyze the anticommutators
$[{\boldsymbol{a}}_{-}(f)\,,\,{\boldsymbol{a}}_{-}^{*}(g)]_{+}$. At a first
stage we consider $f,g\in\EuFrak h_{\circ}$ without further hypothesis, and
vectors of the type $v_{-}=w_{-}A\in\EuFrak h^{n}_{-}$, with
$w_{-}=w_{1}\otimes_{-}\ldots\otimes_{-}w_{n}\in{\boldsymbol{h}}_{-}^{n}$,
$A\in\mathcal{A}$; then using (3.19) and (3.20) respectively, we compute

${\boldsymbol{a}}_{-}(f){\boldsymbol{a}}_{-}^{*}(g)v_{-}\ =\
{\left\langle\right.}f,g{\left.\right\rangle}v_{-}-\sum_{k}(-1)^{k-1}{\left\langle\right.}f,w_{k}{\left.\right\rangle}\,P_{-}^{n}(g\otimes
w^{(k)}_{-})A$ (4.8)
${\boldsymbol{a}}_{-}^{*}(g){\boldsymbol{a}}_{-}(f)v_{-}\,=\,\sum_{k}(-1)^{k-1}P_{-}^{n}(g{\left\langle\right.}f,w_{k}{\left.\right\rangle}\otimes
w^{(k)}_{-})A\,.$ (4.9)

A quick look to the previous equalities is sufficient to realize that the
terms

${\left\langle\right.}f,w_{k}{\left.\right\rangle}\,P_{-}^{n}(g\otimes
w^{(k)}_{-})A\ \ \ ,\ \ \
P_{-}^{n}(g{\left\langle\right.}f,w_{k}{\left.\right\rangle}\otimes
w^{(k)}_{-})A$ (4.10)

may prevent the realization of the anti-commutation relations that one could
expect. In fact, whilst in the case $\mathcal{A}={\mathbb{C}}$ they eliminate
each other leaving only the term
${\left\langle\right.}f,g{\left.\right\rangle}v_{-}$ present, in general they
could differ because the scalar products
${\left\langle\right.}f,w_{k}{\left.\right\rangle}$ are not free to shift
through the elementary tensors. Thus we adopt the usual hypothesis
$f\bowtie_{\circ}g$ and analyze the terms in (4.10) more in details.

As a first step we write as usual $f=f_{i}\gamma_{i}$,
$g=g_{h}\gamma^{\prime}_{h}$ and, to be concise, we write $u_{i}\doteq
u_{\gamma_{i}}$, $u^{\prime}_{h}\doteq
u_{\gamma^{\prime}_{h}}\in\mathcal{U}({\boldsymbol{h}})$,
${\left\langle\right.}f,w_{k}{\left.\right\rangle}=\gamma_{i}^{*}z_{ik}\in\mathcal{A}(f)$
with
$z_{ik}\doteq{\left\langle\right.}f_{i},w_{k}{\left.\right\rangle}\in{\mathbb{C}}$.
With this notation, the hypothesis $f\bowtie_{\circ}g$ implies
$[\gamma_{i},\gamma^{\prime}_{h}]\ =\ [\gamma_{i}^{*},\gamma^{\prime}_{h}]=0$,
so that

$[u_{i}\,,\,u^{\prime}_{h}]\ =\ [u_{i}^{*}\,,\,u^{\prime}_{h}]\ =\ 0\,.$

For the first expression in (4.10) we compute

$\begin{array}[]{l}{\left\langle\right.}f,w_{k}{\left.\right\rangle}\,P_{-}^{n}(g\otimes
w^{(k)}_{-})A\ =\\\
(n-1)!^{-1}\sum_{\varrho\in{\mathbb{P}}_{n-1,k}}\epsilon_{\varrho}\,\gamma_{i}^{*}z_{ik}\,P_{-}^{n}(g_{h}\gamma^{\prime}_{h}\otimes
w_{\varrho(1)}\otimes\ldots\otimes w_{\varrho(n)})A\ =\\\
(n-1)!^{-1}\sum_{\varrho\in{\mathbb{P}}_{n-1,k}}\epsilon_{\varrho}\,\gamma_{i}^{*}z_{ik}\,P_{-}^{n}(g_{h}\otimes
u^{\prime}_{h}w_{\varrho(1)}\otimes\ldots\otimes
u^{\prime}_{h}w_{\varrho(n)})\gamma^{\prime}_{h}A\,.\end{array}$

The term $P_{-}^{n}(g_{h}\otimes
u^{\prime}_{h}w_{\varrho(1)}\otimes\ldots\otimes
u^{\prime}_{h}w_{\varrho(n)})$ is a linear combination of terms of the type

$\epsilon_{\pi}\,u^{\prime}_{h}w_{\pi\rho(l)}\otimes\ldots g_{h}\ldots\otimes
u^{\prime}_{h}w_{\pi\rho(m)}\,\gamma^{\prime}_{h}A\,,$

where $\pi$ is any permutation of the terms in the argument of $P_{-}^{n}$ and
$g_{h}$ appears in position $\pi(1)$. Applying the operators
${\left\langle\right.}f,w_{k}{\left.\right\rangle}=\gamma_{i}^{*}z_{ik}$, we
get terms of the type

$\begin{array}[]{l}\epsilon_{\pi}\,\gamma_{i}^{*}z_{ik}u^{\prime}_{h}w_{\pi\rho(l)}\otimes\ldots
g_{h}\ldots\otimes u^{\prime}_{h}w_{\pi\rho(m)}\,\gamma^{\prime}_{h}A\ =\\\
\epsilon_{\pi}\,u_{i}^{*}u^{\prime}_{h}w_{\pi\rho(l)}\otimes\ldots\gamma_{i}^{*}g_{h}\gamma^{\prime}_{h}\ldots\otimes
w_{\pi\rho(m)}z_{ik}A\ =\\\
\epsilon_{\pi}\,u_{i}^{*}u^{\prime}_{h}w_{\pi\rho(l)}\otimes\ldots\gamma_{i}^{*}g\ldots\otimes
w_{\pi\rho(m)}z_{ik}A\ =\\\
\epsilon_{\pi}\,u_{i}^{*}u^{\prime}_{h}w_{\pi\rho(l)}\otimes\ldots
g\gamma_{i}^{*}\ldots\otimes w_{\pi\rho(m)}z_{ik}A\ =\\\
\epsilon_{\pi}\,u_{i}^{*}u^{\prime}_{h}w_{\pi\rho(l)}\otimes\ldots
g_{h}\ldots\otimes
u^{\prime}_{h}u_{i}^{*}w_{\pi\rho(m)}z_{ik}\gamma_{h}\gamma_{i}^{*}A\ =\\\
\epsilon_{\pi}\,u_{i}^{*}u^{\prime}_{h}w_{\pi\rho(l)}\otimes\ldots
g_{h}\ldots\otimes
u_{i}^{*}u^{\prime}_{h}w_{\pi\rho(m)}z_{ik}\gamma_{i}^{*}\gamma_{h}A\,.\end{array}$

In conclusion,

$\begin{array}[]{l}{\left\langle\right.}f,w_{k}{\left.\right\rangle}\,P_{-}^{n}(g\otimes
w^{(k)}_{-})A\ =\\\
(n-1)!^{-1}\sum_{\varrho,\pi}\epsilon_{\varrho}\epsilon_{\pi}\,u_{i}^{*}u^{\prime}_{h}w_{\pi\rho(l)}\otimes\ldots
g_{h}\ldots\otimes
u_{i}^{*}u^{\prime}_{h}w_{\pi\rho(m)}z_{ik}\gamma_{i}^{*}\gamma_{h}A\,.\end{array}$

Finally we evaluate the second term in (4.10),

$\begin{array}[]{l}P_{-}^{n}(g{\left\langle\right.}f,w_{k}{\left.\right\rangle}\otimes
w^{(k)}_{-})A\ =\\\
(n-1)!^{-1}\sum_{\varrho\in{\mathbb{P}}_{n-1,k}}\epsilon_{\varrho}\,P_{-}^{n}(g_{h}\gamma^{\prime}_{h}\gamma_{i}^{*}z_{ik}\otimes
w_{\varrho(1)}\otimes\ldots\otimes w_{\varrho(n)})A\ =\\\
(n-1)!^{-1}\sum_{\varrho\in{\mathbb{P}}_{n-1,k}}\epsilon_{\varrho}\,P_{-}^{n}(g_{h}\gamma_{i}^{*}\gamma^{\prime}_{h}z_{ik}\otimes
w_{\varrho(1)}\otimes\ldots\otimes w_{\varrho(n)})A\ =\\\
(n-1)!^{-1}\sum_{\varrho\in{\mathbb{P}}_{n-1,k}}\epsilon_{\varrho}\,P_{-}^{n}(g_{h}\otimes
u_{i}^{*}u^{\prime}_{h}w_{\varrho(1)}\otimes\ldots\otimes
u_{i}^{*}u^{\prime}_{h}w_{\varrho(n)})z_{ik}\gamma_{i}^{*}\gamma^{\prime}_{h}A\,=\\\
(n-1)!^{-1}\sum_{\varrho,\pi}\epsilon_{\varrho}\epsilon_{\pi}\,u_{i}^{*}u^{\prime}_{h}w_{\varrho(1)}\otimes\ldots
g_{h}\ldots\otimes
u_{i}^{*}u^{\prime}_{h}w_{\varrho(n)}z_{ik}\gamma_{i}^{*}\gamma^{\prime}_{h}A\,,\end{array}$

concluding that

${\left\langle\right.}f,w_{k}{\left.\right\rangle}\,P_{-}^{n}(g\otimes
w^{(k)}_{-})A\ =\
P_{-}^{n}(g{\left\langle\right.}f,w_{k}{\left.\right\rangle}\otimes
w^{(k)}_{-})A$

actually conspire to obtain, starting from (4.8) and (4.9), the anti-
commutation relations

$[{\boldsymbol{a}}_{-}(f)\,,\,{\boldsymbol{a}}_{-}^{*}(g)]_{+}\ =\
{\left\langle\right.}f,g{\left.\right\rangle}v_{-}\ \ \ ,\ \ \
f\bowtie_{\circ}g\,\in\EuFrak h_{\circ}\,.$ (4.11)

### Norm and generalized CARs.

As a particular case we now consider $f,g\in{\boldsymbol{h}}$ that clearly
implies $f\bowtie_{\circ}g$; moreover, we have
${\left\langle\right.}f,w{\left.\right\rangle}\in{\mathbb{C}}$ for all
$w\in{\boldsymbol{h}}$, thus the undesirable terms in (4.11) vanish. As a
consequence we find

$[{\boldsymbol{a}}_{-}(f)\,,\,{\boldsymbol{a}}_{-}^{*}(g)]_{+}\,=\,{\left\langle\right.}f,g{\left.\right\rangle}\in{\mathbb{C}}\
\ ,\ \
[{\boldsymbol{a}}_{-}^{*}(f)\,,\,{\boldsymbol{a}}_{-}^{*}(g)]_{+}\,=\,0\ \ ,\
\ f,g\in{\boldsymbol{h}}\,.$ (4.12)

Now, ${\boldsymbol{a}}_{-}(f)$ and ${\boldsymbol{a}}_{-}^{*}(g)$ act like the
usual annihilation and creation operators when restricted to the Fock space
$\mathcal{F}_{-}({\boldsymbol{h}})\subset\EuFrak{F}_{-}(\EuFrak h)$: we denote
the corresponding restrictions by $a_{-}(f)$, $a_{-}^{*}(g)$ (without bold
font), and note that $\|a_{-}(f)\|=\|a_{-}^{*}(f)\|=\|f\|$ [7, Vol.2,
Prop.5.2.2]. Given $w_{-}\in\mathcal{F}_{-}({\boldsymbol{h}})$,
$A\in\mathcal{A}$, by right $\mathcal{A}$-linearity we have

${\boldsymbol{a}}_{-}(f)(w_{-}A)\ =\ (a_{-}(f)w_{-})A\ \ \ ,\ \ \
{\boldsymbol{a}}_{-}^{*}(f)(w_{-}A)\ =\ (a_{-}^{*}(f)w_{-})A\ \ \ ,\ \ \
f\in{\boldsymbol{h}}\,,$

so that

$\begin{array}[]{lcl}\|{\left\langle\right.}{\boldsymbol{a}}_{-}(f)(w_{-}A)\,,\,{\boldsymbol{a}}_{-}(f)(w_{-}A){\left.\right\rangle}\|&=&\|A^{*}{\left\langle\right.}w_{-},a_{-}^{*}(f)a_{-}(f)w_{-}{\left.\right\rangle}A\|\leq\\\
&\leq&\|f\|^{2}\|A^{*}{\left\langle\right.}w_{-},w_{-}{\left.\right\rangle}A\|=\\\
&=&\|f\|^{2}\|w_{-}A\|^{2}\,,\end{array}$

having used the fact that $\|f\|^{2}-a_{-}^{*}(f)a_{-}(f)$ is a positive
operator on $\mathcal{F}_{-}({\boldsymbol{h}})$. Thus
$\|{\boldsymbol{a}}_{-}(f)\|=\|{\boldsymbol{a}}_{-}^{*}(f)\|=\|f\|$ for all
$f\in{\boldsymbol{h}}$. We use this property to prove the following, more
general, result:

######  Lemma 4.2.

Let $f\in\EuFrak h_{\circ}$. Then ${\boldsymbol{a}}_{-}(f)$ and
${\boldsymbol{a}}_{-}^{*}(f)$ are bounded.

###### Proof.

We start proving our assertion for the annihilation operator. Given
$f=f_{i}\gamma_{i}$, $f_{i}\in{\boldsymbol{h}}$, $\gamma_{i}\in\mathcal{G}$
(finite sum), for the usual elementary tensors $w_{-}A\in\EuFrak h_{-}^{n}$ we
compute

$\begin{array}[]{lcl}{\boldsymbol{a}}_{-}(f)w_{-}A&=&{\boldsymbol{a}}_{-}(f_{i}\gamma_{i})w_{-}A\,=\\\
&=&\frac{1}{\sqrt{n}}\sum_{k}(-1)^{k-1}\gamma_{i}^{*}{\left\langle\right.}f_{i},w_{k}{\left.\right\rangle}\,w^{(k)}_{-}A\,=\\\
&=&\sum_{i}\gamma_{i}^{*}\,{\boldsymbol{a}}_{-}(f_{i})w_{-}A\,.\end{array}$

The previous relations say that ${\boldsymbol{a}}_{-}(f)$ is sum of the
operators $\gamma_{i}^{*}{\boldsymbol{a}}_{-}(f_{i})$, where $\gamma_{i}^{*}$
are regarded as unitary operators on $\EuFrak h$ and
${\boldsymbol{a}}_{-}(f_{i})$ are, by the previous remarks, bounded. Thus we
conclude that ${\boldsymbol{a}}_{-}(f)$ is bounded for $f$ (finite) linear
combination in $\EuFrak h_{\circ}$. ∎

In the following result we give a synthesis of (4.2), (4.7), (4.11), and Lemma
4.2.

######  Theorem 4.3.

Let $\EuFrak h={\boldsymbol{h}}\mathcal{A}$ be a free Hilbert
$\mathcal{A}$-bimodule with twist
$u:\mathcal{G}\to\mathcal{U}({\boldsymbol{h}})$,
$\mathcal{G}\subseteq\mathcal{U}\mathcal{A}$. Then for any $f,g\in\EuFrak
h_{\circ}$ the creation and annihilation operators are bounded right
$\mathcal{A}$-module operators on $\EuFrak{F}_{-}(\EuFrak h)$, and the
following properties hold: (1) If $f\bowtie_{\circ}g$, then

$[{\boldsymbol{a}}_{-}(f)\,,\,{\boldsymbol{a}}_{-}(g)]_{+}\ =\
[{\boldsymbol{a}}_{-}^{*}(f)\,,\,{\boldsymbol{a}}_{-}^{*}(g)]_{+}\ =\ 0\,,$
(4.13)

and

$[{\boldsymbol{a}}_{-}(f)\,,\,{\boldsymbol{a}}_{-}^{*}(g)]_{+}\,=\,{\left\langle\right.}f,g{\left.\right\rangle}\in\mathcal{A}\,.$
(4.14)

(2) If $f,g\in{\boldsymbol{h}}$, then the previous anti-commutation relations
hold with ${\left\langle\right.}f,g{\left.\right\rangle}\in{\mathbb{C}}$.

### Dirac fields.

Let $\mathcal{U}_{*}({\boldsymbol{h}})$ denote the set of anti-unitary
operators on ${\boldsymbol{h}}$, and
$\kappa=\kappa^{*}\in\mathcal{U}_{*}({\boldsymbol{h}})$ a conjugation such
that

$[\kappa\,,\,u_{\gamma}]\,=\,0\ \ \ ,\ \ \
\forall\gamma\in\mathcal{G}\subseteq\mathcal{U}\mathcal{A}\,.$ (4.15)

Setting $\kappa(vA)\doteq(\kappa v)A^{*}$, $v\in{\boldsymbol{h}}$,
$A\in\mathcal{A}$, we extend $\kappa$ to the vector space spanned by
elementary tensors in $\EuFrak h$, obtaining a densely defined antilinear map.
Note that in particular $\kappa(v\gamma)=(\kappa v)\gamma^{*}$, so $\kappa$ is
defined on $\EuFrak h_{\circ}$. In the following result we check the
compatibility of $\kappa$ with the mutual freeness relation (4.1):

######  Lemma 4.4.

The following properties hold: (1) $\mathcal{A}(f)=\mathcal{A}(\kappa f)$ for
all $f\in\EuFrak h_{\circ}$; (2) If $f,g\in\EuFrak h_{\circ}$ and $f\bowtie
g$, then ${\left\langle\right.}f,\kappa
g{\left.\right\rangle}={\left\langle\right.}g,\kappa f{\left.\right\rangle}$;
(3) Let $f\bowtie_{\circ}g$; then $f\bowtie_{\circ}\kappa g$, $\kappa
g\bowtie_{\circ}f$ and $\kappa f\bowtie_{\circ}\kappa g$.

###### Proof.

(1) We can write $f=w_{i}A_{i}$ with $\\{w_{i}\\}$ an orthogonal base in
${\boldsymbol{h}}$ and
$A_{i}\doteq{\left\langle\right.}w_{i},f{\left.\right\rangle}\in\mathcal{A}$:
this implies that $\mathcal{A}(f)$ is generated by the set $\\{A_{i}\\}$. On
the other hand, $\kappa f=\kappa(w_{i})A_{i}^{*}$ where also $\\{\kappa
w_{i}\\}$ is a base of ${\boldsymbol{h}}$, implying that $\mathcal{A}(\kappa
f)$ is generated by $\\{A_{i}^{*}\\}$. Thus $\mathcal{A}(f)=\mathcal{A}(\kappa
f)$ as claimed. (2) Writing $g=v_{h}B_{h}$, $v_{h}\in{\boldsymbol{h}}$,
$B_{h}\in\mathcal{A}(g)$, we get

${\left\langle\right.}f,\kappa g{\left.\right\rangle}\ =\
A_{i}^{*}B_{h}^{*}{\left\langle\right.}w_{i},\kappa
v_{h}{\left.\right\rangle}\ =\
B_{h}^{*}A_{i}^{*}{\left\langle\right.}v_{h},\kappa
w_{i}{\left.\right\rangle}\ =\ {\left\langle\right.}v_{h}B_{h},(\kappa
w_{i})A_{i}^{*}{\left.\right\rangle}\ =\ {\left\langle\right.}g,\kappa
f{\left.\right\rangle}\,,$

having used the fact that $f\bowtie g$ implies $[A_{i},B_{h}]=0$. (3) We write
$f=f_{i}\gamma_{i}$, $g=g_{h}\gamma^{\prime}_{h}$ and check that
$f\bowtie_{\circ}\kappa g$:

$\begin{array}[]{lcl}(\kappa
g)\gamma_{i}&=&\kappa(g_{h}\gamma^{\prime}_{h})\gamma_{i}\ =\ (\kappa
g_{h})\gamma_{i}{\gamma^{\prime}_{h}}^{*}\ =\
\gamma_{i}(u_{\gamma_{i}}^{*}\kappa g_{h}){\gamma^{\prime}_{h}}^{*}\\\
&=&\gamma_{i}(\kappa u_{\gamma_{i}}^{*}g_{h}){\gamma^{\prime}_{h}}^{*}\ =\
\gamma_{i}(\kappa g_{h}){\gamma^{\prime}_{h}}^{*}\ =\ \gamma_{i}(\kappa
g)\,.\end{array}$

The other cases are verified in an analogous way, so the Lemma is proved. ∎

A _Dirac triple over $\mathcal{A}$_, written $(\EuFrak h,u,\kappa)$, is given
by a free Hilbert $\mathcal{A}$-bimodule $\EuFrak
h={\boldsymbol{h}}\mathcal{A}$ with $\mathcal{G}$-twist $u$, and a conjugation
$\kappa\in\mathcal{U}_{*}({\boldsymbol{h}})$ fulfilling (4.15). The (self-
dual) _Dirac field_ associated with $(\EuFrak h,u,\kappa)$ is defined by

$\hat{\psi}(f)\
\doteq\,\frac{1}{\sqrt{2}}\,({\boldsymbol{a}}_{-}^{*}(f)+{\boldsymbol{a}}_{-}(\kappa
f))\ \ ,\ \ f\in\EuFrak h_{\circ}\subseteq\EuFrak h\,.$ (4.16)

It yields operators $\hat{\psi}(f)\in\EuFrak{B}(\EuFrak{F}_{-}(\EuFrak h))$,
and by Lemma 3.7 we have

$\hat{\psi}^{*}(f)\ =\ \hat{\psi}(\kappa f)\,.$ (4.17)

By applying Lemma 3.6, Theorem 4.3 and Lemma 4.4 we obtain

$\left\\{\begin{array}[]{lll}[\hat{\psi}(f)\,,\,\hat{\psi}(g)]_{+}\ =\
{\left\langle\right.}\kappa
f,g{\left.\right\rangle}\in\mathcal{A}&,&f\bowtie_{\circ}g\in\EuFrak
h_{\circ}\,,\\\ \\\ \gamma\hat{\psi}(w)\ =\
\hat{\psi}(u_{\gamma}w)\gamma&,&w\in{\boldsymbol{h}}\,,\,\gamma\in\mathcal{G}\subseteq\mathcal{U}\mathcal{A}\,.\end{array}\right.$
(4.18)

We denote the C*-algebra generated by the operators
$\hat{\psi}(f),\gamma\in\EuFrak{B}(\EuFrak{F}_{-}(\EuFrak h))$, $f\in\EuFrak
h_{\circ}$, $\gamma\in\mathcal{G}$, by $\mathcal{F}_{\EuFrak h,u,\kappa}$, and
call it the _field C*-algebra_ of $(\EuFrak h,u,\kappa)$. By construction
$\mathcal{F}_{\EuFrak h,u,\kappa}$ contains $\lambda(\mathcal{A})$ and the CAR
algebra $\mathcal{C}_{\boldsymbol{h}}$, and fulfils the relations (4.17) and
(4.18) 222 Relations similar to (4.18) appeared in [12, 13] in the special
case where $\mathcal{A}$ is a Weyl algebra describing suitable asymptotic
configurations of the electromagnetic field. Whilst we use the C*-norm induced
by our Fock bimodule, the C*-algebra of the above reference is endowed with a
maximal C*-norm, that is non-trivial because a (non-separable) representation
is exhibited. . Any state $\omega\in\mathcal{S}(\mathcal{A})$ induces a
Hilbert space representation of $\mathcal{F}_{\EuFrak h,u,\kappa}$, defined as
in (2.13). We remark that at the abstract level at which we worked no topology
has been defined on $\EuFrak h_{\circ}$, thus no continuity property is
required for $\hat{\psi}(f)$ at varying of $f\in\EuFrak h_{\circ}$.

### A class of fixed-time models.

We briefly present a family of models for the notion of Dirac triple and the
associated Dirac field. A more detailed exposition of these and other models
is postponed to a future publication.

We start by considering the symplectic space $\mathcal{S}$ given by pairs of
compactly supported test functions
$s=(s_{0},s_{1})\in\mathcal{S}({\mathbb{R}}^{3})\oplus\mathcal{S}({\mathbb{R}}^{3})$,
with symplectic form

$\eta(s,s^{\prime})\ \doteq\ \int(s_{1}s^{\prime}_{0}-s_{0}s^{\prime}_{1})$

(Lebesgue measure) and the associated Weyl C*-algebra $\mathcal{W}$. It is
readily seen that $\mathcal{W}$ is the C*-algebra associated to the
restriction at a fixed time of the free scalar field, with

$W(s)\ =\ e^{i(\phi(s_{0})+\dot{\phi}(s_{1}))}\,.$

Here, the field $\phi(s_{0})$ and its conjugate $\dot{\phi}(s_{1})$,
$s_{0},s_{1}\in\mathcal{S}({\mathbb{R}}^{3})$, are the initial conditions at
time $t_{0}$ of the free scalar field [6, §8.4.A]. As explained in Example
3.1, any unitary morphism of $\mathcal{S}$ as an additive group yields a
unitary morphism of the group $\mathcal{G}$ generated by Weyl unitaries and
phases.

We then consider the Hilbert spaces ${\boldsymbol{h}}_{+}\doteq
L^{2}({\mathbb{R}}^{3},{\mathbb{C}}^{4})$, ${\boldsymbol{h}}_{-}\doteq
L^{2}({\mathbb{R}}^{3},{\mathbb{C}}^{4,*})$, where ${\mathbb{C}}^{4,*}$ is the
conjugate space, and define
${\boldsymbol{h}}\doteq{\boldsymbol{h}}_{+}\oplus{\boldsymbol{h}}_{-}$ with
conjugation

$\kappa\in\mathcal{U}_{*}({\boldsymbol{h}})\ \ \ ,\ \ \
\kappa(w_{+}\oplus\bar{w}_{-})\,\doteq\,w_{-}\oplus\bar{w}_{+}$

(here $\bar{w}$ is the conjugate map
$\bar{w}(w^{\prime})\doteq{\left\langle\right.}w,w^{\prime}{\left.\right\rangle}$
defined by $w\in{\boldsymbol{h}}_{+}$). Given a tempered distribution
$\sigma\in\mathcal{S}^{\prime}({\mathbb{R}}^{3})$, we consider the unitary
morphism

$u_{\sigma}:\mathcal{S}\to\mathcal{U}({\boldsymbol{h}})\ \ ,\ \
u_{\sigma,s}(w_{+}\oplus\bar{w}_{-})\,\doteq\,e^{-i\sigma\star
s_{0}}w_{+}\oplus e^{i\sigma\star s_{0}}\bar{w}_{-}\ \ ,\ \
w\in{\boldsymbol{h}}\,,$

where $\sigma\star s_{0}\in C^{\infty}({\mathbb{R}}^{3})$ is the convolution
(note that $s_{1}$ does not come into play). It is then clear that $\kappa$
fulfils (4.15).

We are now in condition to form the free Hilbert bimodule $\EuFrak
h={\boldsymbol{h}}\mathcal{W}$ carrying the twisting defined by $u_{\sigma}$;
we have $\EuFrak h=\EuFrak h_{+}\oplus\EuFrak h_{-}$ with obvious meaning of
the symbols, and any $(\EuFrak h,u_{\sigma},\kappa)$ is a Dirac triple over
$\mathcal{W}$. With this input, we have the self dual Dirac field
$\hat{\psi}(h)$, $h\in\EuFrak h_{\circ}$, from which for convenience we
extract the electron field $\psi(f)\doteq\hat{\psi}(f\oplus 0)$, $f\in\EuFrak
h_{+,\circ}$, fulfilling the relations

$\left\\{\begin{array}[]{lll}\left[\psi^{*}(f)\,,\,\psi(g)\right]_{+}\ =\
{\left\langle\right.}f,g{\left.\right\rangle}\in\mathcal{W}&,&f\bowtie_{\circ}g\in\EuFrak
h_{+,\circ}\,,\\\ \\\ W(s)\psi(w)\ =\ \psi(e^{-i\sigma\star
s_{0}}w)W(s)&,&w\in{\boldsymbol{h}}_{+}\,,\,s\in\mathcal{S}\,.\end{array}\right.$
(4.19)

We denote the associated field C*-algebra by $\mathcal{F}_{\sigma}$. It is
endowed with the gauge action

$\beta:{\mathbb{U}}(1)\to{\bf aut\,}\mathcal{F}_{\sigma}\ \ \ ,\ \ \
\beta_{z}(\psi(f))\,\doteq\,\bar{z}\psi(f)\,.$

Now, given $f=f_{i}W(s_{i})\in\EuFrak h_{+,\circ}$,
$f_{i}\in{\boldsymbol{h}}_{+}$, $s_{i}\in\mathcal{S}$, we define the support
${\mathrm{supp}}\,(f)\subset{\mathbb{R}}^{3}$ as the union of the ”fermionic”
and ”bosonic” supports

${\mathrm{supp}}\,_{\psi}(f)\doteq\cup_{i}{\mathrm{supp}}\,(w_{i})\ \ \ ,\ \ \
{\mathrm{supp}}\,_{\mathcal{W}}(f)\doteq\cup_{i}{\mathrm{supp}}\,(s_{i})\,,$

and introduce the C*-algebras $\mathcal{F}_{\sigma}(A)$ generated by those
$\psi(f)$ having support in the open set $A\subset{\mathbb{R}}^{3}$. There are
two subnets $\mathcal{C}_{\boldsymbol{h}}$ and $\mathcal{W}$ of
$\mathcal{F}_{\sigma}$, the first defined by the operators $\psi(w)$,
$w\in{\boldsymbol{h}}_{+}$, and the second given by the unitaries $W(s)$,
$s\in\mathcal{S}$: the two subnets are defined by the free Dirac field and the
free scalar field respectively. In particular,

$[\psi(w),\psi(w^{\prime})]_{+}\ =\ [\psi^{*}(w_{1}),\psi(w_{2})]_{+}\ =\
0\,,$ (4.20)

for all $w,w^{\prime}\in{\boldsymbol{h}}_{+}$ and
$w_{1},w_{2}\in{\boldsymbol{h}}_{+}$ such that
${\mathrm{supp}}\,(w_{1})\cap{\mathrm{supp}}\,(w_{2})=\emptyset$. We discuss
the field net $\mathcal{F}_{\sigma}$ for several choices of
$\sigma\in\mathcal{S}^{\prime}({\mathbb{R}}^{3})$.

  * 1.

_$\sigma$ is the Dirac delta at the origin_. In this case $\sigma\star
s_{0}=s_{0}$ and the unitaries $W(s)$ induce by adjoint action the local gauge
transformations

$\psi(w)\,\to\,\psi(e^{-is_{0}}w)\,.$

If ${\mathrm{supp}}\,(w)\cap{\mathrm{supp}}\,(s_{0})=\emptyset$ then
$u_{\sigma,s}w=w$ and $[\psi(w),W(s)]=0$. Therefore the subnets
$\mathcal{C}_{\boldsymbol{h}}$ and $\mathcal{W}$ are relatively local, that
is, $\mathcal{C}_{\boldsymbol{h}}(A)\subset\mathcal{W}(B)^{\prime}$ for $A\cap
B=\emptyset$, and $\mathcal{F}_{\sigma}$ is local in the sense that it fulfils
normal commutation relations. For $A\cap B\neq\emptyset$ the second of (4.19)
in general holds with $u_{\sigma,s}w\neq w$ and $[W(s),\psi(w)]\neq 0$. A
sufficient condition to having $f\bowtie_{\circ}g$ is
${\mathrm{supp}}\,_{a}(f)\cap{\mathrm{supp}}\,_{b}(g)=\emptyset$ for all
combinations in $a,b=\psi,\mathcal{W}$ different from $a=b=\psi$: in this case

$[\psi^{*}(f),\psi(g)]_{+}\,=\,\sum_{ij}{\left\langle\right.}f_{i},g_{j}{\left.\right\rangle}W(s^{\prime}_{j}-s_{i})\,\in\mathcal{W}\,,$
(4.21)

having written $g=g_{j}W(s^{\prime}_{j})$. If $f,g$ are not mutually free,
then terms of the type (4.8-4.9), that are not in $\mathcal{W}$, appear in the
corresponding anticommutator.

  * 2.

_$\sigma$ has support with a non-empty interior and contained in the 3–ball
$B_{r}$, $r\in(0,\infty]$_. In this case ${\mathrm{supp}}\,(\sigma\star
s_{0})\subset{\mathrm{supp}}\,(s_{0})+B_{r}$. We may have
$u_{\sigma,s}w=e^{-i\sigma\star s_{0}}w\neq w$ even for
${\mathrm{supp}}\,(s)\cap{\mathrm{supp}}\,(w)=\emptyset$, and

$W(s)\psi(w)\ =\ \psi(e^{-i\sigma\star s_{0}}w)W(s)$

implying that in general $\mathcal{C}_{\boldsymbol{h}}$ and $\mathcal{W}$ are
not relatively local. If $f,g\in\EuFrak h_{+,\circ}$, then a sufficient
condition to having $f\bowtie_{\circ}g$ is that
$({\mathrm{supp}}\,_{a}(f)+B_{r})\cap({\mathrm{supp}}\,_{b}(g)+B_{r})=\emptyset$
for $(a,b)\neq(\psi,\psi)$; in that case (4.21) holds. Again, terms of the
type (4.8-4.9) appear for $f,g$ not mutually free.

  * 2.1

_$\sigma$ is the fundamental solution of the Poisson equation_ [9, §9.4]. In
this case

$(\sigma\star
s_{0})({\boldsymbol{x}})\,=\,\frac{1}{4\pi}\int\frac{1}{|{\boldsymbol{x}}-{\boldsymbol{y}}|}s_{0}({\boldsymbol{y}})\,d^{3}{\boldsymbol{y}}$

in general has non-compact support and is non-constant (for example, take
$s_{0}$ a non-negative bump function supported around the origin).
$\mathcal{W}$ is not relatively local both to $\mathcal{C}_{\boldsymbol{h}}$
and the fixed-point subnet $\mathcal{C}_{\boldsymbol{h}}^{\beta}$; in
particular,

$W(s)\psi(w_{1})\psi^{*}(w_{2})\ =\ \psi(e^{-i\sigma\star
s_{0}}w_{1})\,\psi^{*}(e^{i\sigma\star s_{0}}w_{2})W(s)$ (4.22)

even when ${\mathrm{supp}}\,(s)$ is disjoint from the supports of
$w_{1},w_{2}$.

  * 2.2.

_$\sigma$ is the Lebesgue measure_. In this case

$(\sigma\star s_{0})({\boldsymbol{x}})\,=\,\int
s_{0}({\boldsymbol{x}}-{\boldsymbol{y}})\,d^{3}{\boldsymbol{y}}\,=\,-{\left\langle\right.}s_{0}{\left.\right\rangle}\,\doteq\,-\int
s_{0}$

is constant and

$W(s)\psi(w)\ =\
e^{i{\left\langle\right.}s_{0}{\left.\right\rangle}}\psi(w)W(s)$ (4.23)

for all $s\in\mathcal{S}$ and $w\in{\boldsymbol{h}}_{+}$. The above equality
has a two-fold interpretation. The first is that the Weyl unitaries $W(s)$
induce the global gauge transformations
$e^{i{\left\langle\right.}s_{0}{\left.\right\rangle}}$ on the charged fields
$\psi(w)$. The second is that the operators $\psi(w)$ intertwine the identity
and the automorphism $\alpha\in{\bf aut\,}\mathcal{W}$, $\alpha(W(s))\doteq
e^{-i{\left\langle\right.}s_{0}{\left.\right\rangle}}W(s)$, so that (4.23)
becomes

$\alpha(W(s))\psi(w)\ =\ \psi(w)W(s)\,.$ (4.24)

The net $\mathcal{F}_{\sigma}$ is not local, in fact
$\mathcal{C}_{\boldsymbol{h}}(A)$ and $\mathcal{W}(B)$ are never one in the
commutant of the other. Yet by (4.23) we have that $\mathcal{W}$ is in the
commutant of $\mathcal{C}_{\boldsymbol{h}}^{\beta}$ and
$\mathcal{C}_{\boldsymbol{h}}$ is in the commutant of the subalgebra
$\mathcal{W}^{0}$ generated by test functions $s$ with
${\left\langle\right.}s_{0}{\left.\right\rangle}=0$. A local, gauge-invariant
subnet $\mathcal{A}$ of $\mathcal{F}_{\sigma}$ is the one generated by
operators of the type $A_{s,w_{1},w_{2}}\doteq\psi(w_{1})W(s)\psi^{*}(w_{2})$,
$w_{1},w_{2}\in{\boldsymbol{h}}_{+},s\in\mathcal{S}$, in fact

$[\,A_{s,w_{1},w_{2}}\,\,,\,A_{s^{\prime},w^{\prime}_{1},w^{\prime}_{2}}\,]\,=\,0$

for
$({\mathrm{supp}}\,(s)\cup{\mathrm{supp}}\,(w_{1})\cup{\mathrm{supp}}\,(w_{2}))\cap({\mathrm{supp}}\,(s^{\prime})\cup{\mathrm{supp}}\,(w^{\prime}_{1})\cup{\mathrm{supp}}\,(w^{\prime}_{2}))=\emptyset$,
having used (4.20), (4.23) and $[W(s),W(s^{\prime})]=0$. Note that
$\mathcal{A}$ is local both to $\mathcal{C}_{\boldsymbol{h}}^{\beta}$ and
$\mathcal{W}$.

##  5 Conclusions

In the present paper we presented a construction based on Hilbert bimodules,
in which the spatial tensor product of a CAR algebra by a C*-algebra is
replaced by a twisted product. This allows to construct field systems with
non-trivial commutation relations, as in (4.19) and (4.24). The technical
obstacles concerning tensor products and (the absence of) permutation symmetry
in Hilbert bimodules have been overcome by introducing the notion of twist,
which yields a class of left actions for which these drawbacks are under
control 333 We remark that the same technique may be used to construct bosonic
Fock bimodules and the corresponding fields: in such a scenario, the
construction in [18] would be analogous to a bosonic Fock
$\mathcal{A}$-bimodule, with $\mathcal{A}$ the finite $d$-dimensional Weyl
algebra, ${\boldsymbol{h}}=L^{2}({\mathbb{R}}^{d})$ and $\EuFrak
h={\boldsymbol{h}}\mathcal{A}$ endowed with the trivial left action in the
sense of the present paper..

The models presented in the previous section are elementary, yet they pose
questions that in our opinion deserve to be discussed. For example, the model
1 exhibits Weyl unitaries that induce local gauge transformations, thus in
regular representations of $\mathcal{W}$ we expect to find bosonic fields
assuming the role usually played by the zero components of the Dirac current
[19, §4.6.1] or the ”longitudinal photon field” in Gupta-Bleuer gauge [19,
§7.3.2]. In the model 2.1, $\sigma$ is related to electrostatic potentials and
not surprisingly it poses the problem of extracting a _local_ observable
subnet from which the initial (non-local) field net should be reconstructed.
In this regard, the model 2.2 provides a simple illustration of the fact that
this problem can be successfully solved in specific situations.

As a final remark, we point out that the physical understanding of the notion
of twist is a topic that has not been discussed in the present paper, in which
we used this object as a mathematical input. The correct interpretation should
be obtained by a deeper discussion of our models, especially in regular
representations of $\mathcal{W}$ [20]. In this regard, working in a fixed-time
_régime_ allows to avoid complications and easily produce examples, yet our
aim is to construct and discuss models in Minkowski space, entering in this
way in an explicitly relativistic scenario [21].

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