# Construction of wedge-local nets of observables through Longo-Witten
endomorphisms. II

Marcel Bischoff 111Supported in part by the ERC Advanced Grant 227458 OACFT
“Operator Algebras and Conformal Field Theory”. and Yoh Tanimoto∗
e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
Dipartimento di Matematica, Università di Roma “Tor Vergata”
Via della Ricerca Scientifica, 1 - I–00133 Roma, Italy.

###### Abstract

In the first part, we have constructed several families of interacting wedge-
local nets of von Neumann algebras. In particular, there has been discovered a
family of models based on the endomorphisms of the ${\rm U(1)}$-current
algebra ${{\mathcal{A}}^{(0)}}$ of Longo-Witten.

In this second part, we further investigate endomorphisms and interacting
models. The key ingredient is the free massless fermionic net, which contains
the ${\rm U(1)}$-current net as the fixed point subnet with respect to the
${\rm U(1)}$ gauge action. Through the restriction to the subnet, we construct
a new family of Longo-Witten endomorphisms on ${{\mathcal{A}}^{(0)}}$ and
accordingly interacting wedge-local nets in two-dimensional spacetime. The
${\rm U(1)}$-current net admits the structure of particle numbers and the
S-matrices of the models constructed here do mix the spaces with different
particle numbers of the bosonic Fock space.

Dedicated to Roberto Longo on the occasion of his 60th birthday

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Fermi nets on $S^{1}$
    2. 2.2 Subnets and the character argument
    3. 2.3 Scattering theory of waves in ${\mathbb{R}}^{2}$ (revisited)
    4. 2.4 Restriction of wedge-local nets
  3. 3 Examples of fermi nets
    1. 3.1 ${\rm U(1)}$-current net ${{\mathcal{A}}^{(0)}}$
    2. 3.2 The free complex fermion net $\mathrm{Fer}_{\mathbb{C}}$
    3. 3.3 ${\rm U(1)}$-current net as a subnet of $\mathrm{Fer}_{\mathbb{C}}$
  4. 4 A new family of Longo-Witten endomorphisms on ${\rm U(1)}$-current net
  5. 5 Interacting wedge-local net with particle production
    1. 5.1 Construction of scattering operators
    2. 5.2 Action of the S-matrix on the 1+1 particle space
  6. 6 Conclusion and outlook

##  1 Introduction

As already explained in Part I [Tan11a], construction of interacting models of
Quantum Field Theory in (physical) four spacetime dimensions has been a long-
standing open problem, and recently the algebraic approach had several
progress [Lec08, GL07, GL08, BS08, BLS11, Lec11] and two dimensional cases
work particularly well: these works constructed models of QFT with weaker
localization property, and in some case such models turned out to be strictly
local and fully interacting [Lec08]. One should recall, however, that the
models in [Lec08] allow a complete interpretation in terms of particles
(asymptotic completeness) and the particle number is preserved under the
scattering operator. On the other hand, it is known that in four dimensions an
interacting model inevitably involves particle production [Aks65]. In the
present paper, we construct a further new family of interacting wedge-local
two-dimensional massless models and find that their S-matrices mix the spaces
with different particle numbers.

In fact, the requirement to involve particle production non-perturbatively is
already not simple. On the one hand, an asymptotically complete model must
behave like the free theory and hence must be compatible with the Fock space
structure at asymptotic time. On the other hand, a particle production process
properly means a violation of the Fock structure at physical time. To overcome
this difficulty, one would have to “deform” the free theory in a somewhat
involved way (cf. [Lec11]) or should rely on a nice trick. Here we take the
second way. Standard examples and techniques from Conformal Field Theory
provide such a trick.

Conformal Field Theory has been well studied particularly on the circle, which
can be seen as a chiral part of 1+1 dimensional theory. There are many
important examples of such models, or nets in operator-algebraic terms, and
both field-theoretic and operator-algebraic techniques allow one to analyze
their interrelationships. Our trick can be briefly summarized as follows: we
consider the free complex fermionic field $\psi$ on the circle; the field
$\psi$ admits a gauge group action by ${\rm U(1)}$, and the fixed point with
respect to this action is known to be isomorphic to the algebra of the ${\rm
U(1)}$-current $J$. Both fields are free fields acting naturally on the Fock
space (fermionic and bosonic, respectively) but the correspondence between the
spaces is quite involved. The passage to 1+1 dimensional models is simply the
tensor product of two such chiral parts. Now, we can easily “deform” the two-
dimensional Dirac field (built up from the chiral parts
$\psi\otimes{\mathbbm{1}}$ and ${\mathbbm{1}}\otimes\psi$) in such a way that
it commutes with the product action of the gauge group ${\rm U(1)}\times{\rm
U(1)}$. Hence the deformation restricts to the algebra of the conserved
current $J^{\mu}=(J^{0},J^{1})=(J\otimes{\mathbbm{1}}+{\mathbbm{1}}\otimes
J,{\mathbbm{1}}\otimes J-J\otimes{\mathbbm{1}})$, and this deformation is
sufficiently complicated so that the resulting S-matrix does not preserve the
bosonic Fock structure, thanks to the involved fermion-boson correspondence.

In Part I, we have constructed a family of two-dimensional massless models
based on the free current $J^{\mu}$ or more precisely its net
${{\mathcal{A}}^{(0)}}\otimes{{\mathcal{A}}^{(0)}}$ of von Neumann algebras of
observables. The main ingredient was endomorphisms of the algebra
${{\mathcal{A}}^{(0)}}({\mathbb{R}}_{+})$ of observables localized in the
positive half-line ${\mathbb{R}}_{+}$ commuting with the translations. A
family of such endomorphisms has been studied first by Longo and Witten [LW11]
in order to construct Quantum Field Theory with boundary. We used those
endomorphisms to construct two-dimensional models without boundary. In the
present article, we study the fermi net $\mathrm{Fer}_{\mathbb{C}}$ generated
by the free complex fermionic field $\psi$ and its Longo-Witten endomorphisms.
We construct endomorphisms of $\mathrm{Fer}_{\mathbb{C}}$ which commute with
the gauge action of ${\rm U(1)}$, hence restrict to the fixed point subnet of
${{\mathcal{A}}^{(0)}}$. It turns out that the restricted endomorphisms cannot
be implemented by second quantization operators, hence are different from the
ones considered in [LW11]. We again knit them up to construct S-matrices and
wedge-local nets.

Then the fixed point with respect to the action of ${\rm U(1)}\times{\rm
U(1)}$ is considered. We find that its asymptotic behaviour is the same as the
free (bosonic) current $J^{\mu}$ and the S-matrix does not preserve the space
of $1+1$ particles ($1$ left and $1$ right moving particle) in the sense of
the Fock space structure. We stress that the Fock space particle number has no
intrinsic meaning as particles, because we are in a massless case where just a
scattering between two waves is considered. One has to pass to the massive
case to talk about particle production. We will discuss in more detail the
implication of this phenomenon at the end of Section 5.

This paper is organized as follows. In Section 2 we recall the standard
notions of algebraic QFT and the scattering theory of two-dimensional massless
models [Buc75, DT11]. Some simple observations are given about subtheories and
inner symmetries. Main examples of nets, the free complex fermionic net
$\mathrm{Fer}_{\mathbb{C}}$ and the ${\rm U(1)}$-current net
${{\mathcal{A}}^{(0)}}$, are introduced in Section 3. Although it is well-
known [KR87, Kac98, Reh98] that the fixed point subnet
$\mathrm{Fer}_{\mathbb{C}}$ with respect to ${\rm U(1)}$ is
${{\mathcal{A}}^{(0)}}$ at the field-theoretic level, we prove it in the
framework of algebraic approach. Section 4 is devoted to the construction of
new Longo-Witten endomorphisms on ${{\mathcal{A}}^{(0)}}$. They are used in
Section 5 to construct new interacting wedge-local nets. Outlook and open
problems are summarized in Section 6.

##  2 Preliminaries

###  2.1 Fermi nets on $S^{1}$

Here we give a summary of one-dimensional nets, since they will be our
building blocks of the construction of two-dimensional interacting models. In
the first part, we considered local nets of von Neumann algebras on $S^{1}$.
Since we need to exploit the free fermionic field in this second part, a
generalized concept of nets is recalled.

We follow the definition in [CKL08] and denote by ${\rm
M\ddot{o}b}^{(2)}(\cong{\rm SL}(2,{\mathbb{R}})\cong{\rm SU}(1,1))$ the double
cover of the Möbius group ${\rm PSL}(2,{\mathbb{R}})$. We denote by
$\mathcal{I}$ the set of proper intervals $I\subset S^{1}$, where proper means
that $I$ is open and connected and neither dense nor the empty set. A (Möbius
covariant) fermi net is an assignment of von Neumann algebras
${\mathcal{F}}_{0}(I)$ on ${\mathcal{H}}_{{\mathcal{F}}_{0}}$ to intervals
$I\in\mathcal{I}$ on $S^{1}$ satisfying the following conditions:

  1. (1)

Isotony. If $I_{1}\subset I_{2}$, then
${\mathcal{F}}_{0}(I_{1})\subset{\mathcal{F}}_{0}(I_{2})$.

  2. (2)

Möbius covariance. There exists a strongly continuous unitary representation
$U_{0}$ of the group ${\rm M\ddot{o}b}^{(2)}$ such that for any interval
$I\in\mathcal{I}$ it holds that

$U_{0}(g){\mathcal{F}}_{0}(I)U_{0}(g)^{*}={\mathcal{F}}_{0}(gI),\mbox{ for
}g\in{\rm M\ddot{o}b}^{(2)},$

where the action of ${\rm M\ddot{o}b}^{(2)}\cong{\rm SU}(1,1)$ on $S^{1}$ is
defined through linear fractional transformation.

  3. (3)

Positivity of energy. The generator of the one-parameter subgroup of the lift
of rotations in $\mathrm{M\ddot{o}b}$ in the representation $U_{0}$ is
positive.

  4. (4)

Existence of the vacuum. There is a unique (up to a phase) unit vector
$\Omega_{0}$ in ${\mathcal{H}}_{{\mathcal{F}}_{0}}$ which is invariant under
the action of $U_{0}$, and cyclic for $\bigvee_{I\Subset
S^{1}}{\mathcal{F}}_{0}(I)$.

  5. (5)

${\mathbb{Z}}_{2}$-grading. There is a unitary operator $\Gamma_{0}$ with
$\Gamma_{0}^{2}={\mathbbm{1}}$ such that $\Gamma_{0}\Omega_{0}=\Omega_{0}$ and
${\hbox{\rm Ad\,}}\Gamma_{0}({\mathcal{F}}_{0}(I))={\mathcal{F}}_{0}(I)$.

  6. (6)

Graded locality. If $I_{1}\cap I_{2}=\emptyset$, then
$[{\mathcal{F}}_{0}(I_{1}),{\hbox{\rm
Ad\,}}Z_{0}({\mathcal{F}}_{0}(I_{2}))]=0$, where
$Z_{0}:=\frac{{\mathbbm{1}}-\mathrm{i}\Gamma_{0}}{1-\mathrm{i}}$.

If the grading operator is trivial: $Z_{0}={\mathbbm{1}}$, then the net
${\mathcal{F}}_{0}$ is said to be local.

Among the consequences of these conditions are (see [CKL08]):

  1. (7)

Reeh-Schlieder property. The vector $\Omega_{0}$ is cyclic and separating for
each ${\mathcal{F}}_{0}(I)$.

  2. (8)

Additivity. If $I=\bigcup_{i}I_{i}$, then
${\mathcal{F}}_{0}(I)=\bigvee_{i}{\mathcal{F}}_{0}(I_{i})$.

  3. (9)

Twisted Haag duality on $S^{1}$. For an interval $I\in\mathcal{I}$, it holds
that ${\mathcal{F}}_{0}(I)^{\prime}={\hbox{\rm
Ad\,}}Z_{0}({\mathcal{F}}_{0}(I^{\prime}))$, where $I^{\prime}$ is the
interior of the complement of $I$ in $S^{1}$.

  4. (10)

Bisognano-Wichmann property. The modular group
$\Delta_{{\mathcal{F}}_{0}({\mathbb{R}}_{+})}^{\mathrm{i}t}$ of
${\mathcal{F}}_{0}({\mathbb{R}}_{+})$ with respect to $\Omega_{0}$ is equal to
$U_{0}(\delta(-2\pi t))$, where $S^{1}$ is identified as the one-point
compactification of ${\mathbb{R}}$ as below and $\delta$ is the one-parameter
group of dilations.

  5. (11)

Irreducibility. It holds that
$\bigvee_{I\in\mathcal{I}}{\mathcal{F}}_{0}(I)=B({\mathcal{H}}_{{\mathcal{F}}_{0}})$.

Each algebra ${\mathcal{F}}_{0}(I)$ is referred to as a local algebra (even
for a fermi net). Note that if the grading operator $\Gamma_{0}$ is trivial,
then the definition of fermi net coincides with the one of local Möbius-
covariant net. We identify the circle $S^{1}$ and the compactified real line
${\mathbb{R}}\cup\\{\infty\\}$ through the Cayley transform

$t=-\mathrm{i}\frac{z-1}{z+1}\Longleftrightarrow
z=-\frac{t-\mathrm{i}}{t+\mathrm{i}},\phantom{...}t\in{\mathbb{R}},\phantom{..}z\in
S^{1}\subset{\mathbb{C}}$

and refer to the algebra ${\mathcal{F}}_{0}(I)$ for an interval
$I\subset{\mathbb{R}}$. The representation $U_{0}$ of ${\rm
M\ddot{o}b}^{(2)}\cong\mathrm{SL}(2,{\mathbb{R}})$ restricts indeed to a
projective unitary representation of ${\rm PSL}(2,{\mathbb{R}})$ [CKL08]. Let
$\rho$ be the ($4\pi$ periodic) lift of the rotations in $\mathrm{PSU}(1,1)$
(acting by $\rho(\theta)z=\mathrm{e}^{\mathrm{i}\theta}z$) to
$\mathrm{M\ddot{o}b}^{(2)}$ and let us denote
$R_{0}(\theta)=U_{0}(\rho(\theta))=\mathrm{e}^{\mathrm{i}\theta L_{0}}$. Under
the identification between $S^{1}$ and ${\mathbb{R}}\cup\\{\infty\\}$, one can
talk about the translations and dilations of ${\mathbb{R}}$, which are
included in $\mathrm{M\ddot{o}b}$. In particular, the representation of
translations (which we denote by $\tau$) plays a crucial role. Let us denote
$T_{0}(t)=U_{0}(\tau(t))$.

A Longo-Witten endomorphism of a fermi net ${\mathcal{F}}_{0}$ is an
endomorphism of the algebra ${\mathcal{F}}_{0}({\mathbb{R}}_{+})$ implemented
by a unitary $V_{0}$ which commutes with the translation $T_{0}(t)$. A family
of Longo-Witten endomorphisms has been found for the ${\rm U(1)}$-current net
and the real free fermion net [LW11]. The examples will be explained later in
detail.

Note that a Longo-Witten endomorphism is uniquely implemented up to scalar.
Indeed, since it commutes with translation, ${\hbox{\rm Ad\,}}V_{0}$ is an
endomorphism of ${\mathcal{F}}_{0}({\mathbb{R}}_{+}+t)$ for any
$t\in{\mathbb{R}}$. If there is another unitary $W_{0}$ which satisfies
${\hbox{\rm Ad\,}}W_{0}(x)={\hbox{\rm Ad\,}}V_{0}(x)$ for any
$x\in{\mathcal{F}}_{0}({\mathbb{R}}_{+}+t)$, $t\in{\mathbb{R}}$, then by the
irreducibility $W_{0}^{*}V_{0}$ must be scalar.

###  2.2 Subnets and the character argument

Let ${\mathcal{F}}_{0}$ be a fermi (or local) net on
${\mathcal{H}}_{{\mathcal{F}}_{0}}$. Another assignment ${\mathcal{A}}_{0}$ of
von Neumann algebras $\\{{\mathcal{A}}_{0}(I)\\}_{I\in\mathcal{I}}$ on
${\mathcal{H}}_{{\mathcal{F}}_{0}}$ is called a subnet of ${\mathcal{F}}_{0}$
if it satisfies isotony, Möbius covariance with respect to the same $U_{0}$
for ${\mathcal{F}}_{0}$ and it holds that
${\mathcal{A}}_{0}(I)\subset{\mathcal{F}}_{0}(I)$ for every interval
$I\in\mathcal{I}$. We simply write
${\mathcal{A}}_{0}\subset{\mathcal{F}}_{0}$. In this case, let us denote
${\mathcal{H}}_{{\mathcal{A}}_{0}}=\overline{\bigvee_{I\in\mathcal{I}}{\mathcal{A}}_{0}(I)\Omega_{0}}$.
Then it is immediate to see that ${\mathcal{A}}_{0}(I)$ and $U_{0}$ restrict
to ${\mathcal{H}}_{{\mathcal{A}}_{0}}$, and by this restriction
${\mathcal{A}}_{0}|_{{\mathcal{H}}_{{\mathcal{A}}_{0}}}$ becomes a fermi net
with the representation of covariance
$U_{0}|_{{\mathcal{H}}_{{\mathcal{A}}_{0}}}$. This restriction is also said to
be a subnet of ${\mathcal{F}}_{0}$ if no confusion arises.

For a fermi net ${\mathcal{F}}_{0}$ on $S^{1}$, a gauge automorphism
$\alpha_{0}$ is a family of automorphisms $\\{\alpha_{0,I}\\}$ of local
algebras which satisfies the consistency condition

$\alpha_{0,I_{2}}|_{{\mathcal{A}}_{0}(I_{1})}=\alpha_{0,I_{1}}\,\,\mbox{ for
}I_{1}\subset I_{2}\,.$

If a gauge automorphism $\alpha_{0}$ preserves the vacuum state
$\langle\Omega_{0},\cdot\,\Omega_{0}\rangle$, it is said to be an inner
symmetry. An inner symmetry $\alpha_{0}$ can be unitarily implemented by the
formula $V_{\alpha_{0}}x\Omega_{0}=\alpha_{0}(x)\Omega_{0}$, where $x$ is an
element of some local algebra ${\mathcal{F}}_{0}(I)$. We say that a compact
group $G$ acts on the net ${\mathcal{F}}_{0}$ when there is automorphisms
$\\{\alpha_{0,g}\\}_{g\in G}$ which satisfy the composition law when
restricted to local algebras. The fixed point subnet with respect to this
action of $G$ is the subnet defined by
${\mathcal{F}}_{0}^{G}(I):={\mathcal{F}}_{0}(I)^{G}$.

Let ${\mathcal{F}}_{0}$ be a fermi net and ${\mathcal{A}}_{0}$ be a subnet.
Recall that, for a Möbius covariant fermi net, the Bisognano-Wichmann property
is automatic. As a consequence, for each interval there is a conditional
expectation $E_{0,I}:{\mathcal{F}}_{0}(I)\to{\mathcal{A}}_{0}(I)$ which
preserves the vacuum state $\langle\Omega_{0},\cdot\,\Omega_{0}\rangle$ and
implemented by the projection $P_{{\mathcal{A}}_{0}}$ onto
${\mathcal{H}}_{{\mathcal{A}}_{0}}$ (see [Tak03, Theorem IX.4.2]). This
projection $P_{{\mathcal{A}}_{0}}$ contains much information of
${\mathcal{A}}_{0}$.

Consider the case where ${\mathcal{A}}_{0}={\mathcal{F}}_{0}^{G}$ is the fixed
point subnet with respect to an action $\alpha_{0}$ of a compact group $G$ by
inner symmetry. Then we have a unitary representation $V_{\alpha_{0}}$ of $G$
on ${\mathcal{H}}_{{\mathcal{F}}_{0}}$. If we write the set of invariant
vectors with respect to $V_{\alpha_{0}}$ by
${\mathcal{H}}_{{\mathcal{F}}_{0}}^{G}$, it holds that
${\mathcal{H}}_{{\mathcal{F}}_{0}}^{G}={\mathcal{H}}_{{\mathcal{A}}_{0}}$.
Indeed, the inclusion
${\mathcal{H}}_{{\mathcal{A}}_{0}}\subset{\mathcal{H}}_{{\mathcal{F}}_{0}}^{G}$
is obvious. On the other hand, for $x\in{\mathcal{F}}_{0}(I)$, we have

$\left(\int_{G}\alpha_{0}(x)\,\mathrm{d}g\right)\Omega_{0}=\int_{G}\left(V_{\alpha_{0}}(g)x\Omega_{0}\right)\,\mathrm{d}g,$

which implies that any vector in ${\mathcal{H}}_{{\mathcal{F}}_{0}}^{G}$ can
be approximated from ${\mathcal{H}}_{{\mathcal{A}}_{0}}$ by the Reeh-Schlieder
property.

For the later use, we put here a simple observation.

######  Proposition 2.1.

In the situation above, if a Longo-Witten endomorphism is implemented by
$W_{0}$ and $W_{0}$ commutes with $V_{\alpha_{0}}$, then ${\hbox{\rm
Ad\,}}W_{0}$ restricts to a Longo-Witten endomorphism of the fixed point
subnet ${\mathcal{A}}_{0}$.

###### Proof.

The unitary $W_{0}$ commutes with the projection $P_{{\mathcal{A}}_{0}}$,
hence also with the conditional expectation $E_{0}$ onto ${\mathcal{A}}_{0}$.
∎

Let ${\mathcal{F}}_{0}$ be fermi (or local) net on
${\mathcal{H}}_{{\mathcal{F}}_{0}}$. The Hilbert space
${\mathcal{H}}_{{\mathcal{F}}_{0}}$ is graded by the action of the rotation
subgroup $R_{0}(\theta)=\mathrm{e}^{\mathrm{i}\theta L_{0}}$:

${\mathcal{H}}_{{\mathcal{F}}_{0}}={\mathbb{C}}\Omega_{0}\oplus\bigoplus_{r\in\frac{1}{2}{\mathbb{N}}}{\mathcal{H}}_{r}=\bigoplus_{r\in\frac{1}{2}{\mathbb{N}}_{0}}{\mathcal{H}}_{r}$

with
${\mathcal{H}}_{r}=\\{\xi\in{\mathcal{H}}_{{\mathcal{F}}_{0}}:R_{0}(\theta)\xi=\mathrm{e}^{\mathrm{i}r\theta}\xi\\}$
and the sum only going over ${\mathbb{N}}_{0}$ for a local net. The conformal
character of the net ${\mathcal{F}}_{0}$ is given as a formal power series of
$t=\mathrm{e}^{-\beta}$:

$\operatorname{tr}_{{\mathcal{H}}_{{\mathcal{F}}_{0}}}(\mathrm{e}^{-\beta
L_{0}})=\sum_{r\in\frac{1}{2}{\mathbb{N}}_{0}}^{\infty}{\hbox{dim}\,}{\mathcal{H}}_{r}\cdot
t^{r}\,.$

Let us assume that there is an action of $G={\rm U(1)}$ by inner symmetry. We
denote by $V_{0}(\theta)$ the implementing unitary. Then $V_{0}$ and $U_{0}$
commute and ${\mathcal{H}}_{F_{0}}$ is graded also by the gauge action
$V_{0}(\theta)=\mathrm{e}^{\mathrm{i}\theta Q_{0}}$:

${\mathcal{H}}_{{\mathcal{F}}_{0}}={\mathbb{C}}\Omega_{0}\oplus\bigoplus_{r\in\frac{1}{2}{\mathbb{N}},q\in{\mathbb{Z}}}{\mathcal{H}}_{r,q}=\bigoplus_{q\in{\mathbb{Z}}}{\mathcal{H}}_{\,\cdot\,,q},\qquad\text{with}\qquad{\mathcal{H}}_{\,\cdot\,,q}:=\bigoplus_{r\in\frac{1}{2}{\mathbb{N}}_{0}}{\mathcal{H}}_{r,q}$

and the character is given as a formal power series in $t=\mathrm{e}^{-\beta}$
and $z=\mathrm{e}^{-E}$:

$\operatorname{tr}_{{\mathcal{H}}_{{\mathcal{F}}_{0}}}(\mathrm{e}^{-\beta
L_{0}-EQ_{0}})=\sum_{r\in\frac{1}{2}{\mathbb{N}}_{0},q\in{\mathbb{Z}}}{\hbox{dim}\,}{\mathcal{H}}_{r,q}\cdot
t^{r}z^{q}\,.$

Recall that it holds that
${\mathcal{H}}_{{\mathcal{F}}_{0}}^{G}={\mathcal{H}}_{{\mathcal{A}}_{0}}$. The
operator $Q_{0}$ acts by $0$ on ${\mathcal{H}}_{{\mathcal{F}}_{0}}^{G}$, hence
we can obtain the conformal character of ${\mathcal{A}}_{0}$ just by taking
the coefficient of $z^{0}$ in
$\operatorname{tr}_{{\mathcal{H}}_{{\mathcal{F}}_{0}}}(\mathrm{e}^{-\beta
L_{0}-EQ_{0}})$.

Later in this paper we need to compare the size of two subnets. Let
${\mathcal{A}}_{0}\subset{\mathcal{B}}_{0}\subset{\mathcal{F}}_{0}$ be an
inclusion of three fermi nets. If the conformal characters of
${\mathcal{A}}_{0}$ and ${\mathcal{B}}_{0}$ coincide, then this means that the
subspaces ${\mathcal{H}}_{{\mathcal{A}}_{0}}$ and
${\mathcal{H}}_{{\mathcal{B}}_{0}}$ coincide, since we have already an
inclusion
${\mathcal{H}}_{{\mathcal{A}}_{0}}\subset{\mathcal{H}}_{{\mathcal{B}}_{0}}$
and the coefficients of the conformal character are the dimensions of
eigenspaces of $L_{0}$. This in turn implies that two subnets
${\mathcal{A}}_{0}$ and ${\mathcal{B}}_{0}$ are the same since the conditional
expectations which are implemented by
$P_{{\mathcal{A}}_{0}},P_{{\mathcal{B}}_{0}}$ are the same. We will see such
an argument in an example.

###  2.3 Scattering theory of waves in ${\mathbb{R}}^{2}$ (revisited)

Here we just collect some basic notions regarding scattering theory of two-
dimensional massless models. As recalled in Part I [Tan11a], this theory has
been established by Buchholz [Buc75] and extended to the wedge-local case
[DT11]. A Borchers triple on a Hilbert space ${\mathcal{H}}$ is a triple
$({\mathcal{M}},T,\Omega)$ of a von Neumann algebra ${\mathcal{M}}\subset
B({\mathcal{H}})$, a unitary representation $T$ of ${\mathbb{R}}^{2}$ on
${\mathcal{H}}$ and a vector $\Omega\in{\mathcal{H}}$ such that

  * •

${\hbox{\rm Ad\,}}T(t_{0},t_{1})({\mathcal{M}})\subset{\mathcal{M}}$ for
$(t_{0},t_{1})\in
W_{\mathrm{R}}=\\{(x_{0},x_{1})\in{\mathbb{R}}^{2}:x_{1}>|x_{0}|\\}$, the
standard right wedge.

  * •

The joint spectrum ${\rm sp}\,T$ is contained in the closed forward lightcone
$\overline{V}_{+}=\\{(p_{0},p_{1})\in{\mathbb{R}}_{2}:p_{0}\geq|p_{1}|\\}$.

  * •

$\Omega$ is a unique (up to scalar) invariant vector under $T$, and cyclic and
separating for ${\mathcal{M}}$.

We recall that one interprets ${\mathcal{M}}$ as the algebra assigned to the
wedge $W_{\mathrm{R}}$. Let ${\mathcal{W}}$ be the set of wedges, i.e. the set
of all $W=gW_{\mathrm{R}}$ where $g$ is a Poincaré transformation, then we
define the wedge-local net ${\mathcal{W}}\ni W\mapsto{\mathcal{M}}(W)$
associated with the Borchers triple $({\mathcal{M}},T,\Omega)$ by
${\mathcal{M}}(W_{R}+a)=T(a){\mathcal{M}}T(a)^{\ast}$ and
${\mathcal{M}}(W_{R}^{\prime}+a)=T(a){\mathcal{M}}^{\prime}T(a)^{\ast}$. With
the help of the modular objects one can define a representation of the
Poincaré group extending the one of translations $T$ [Bor92]. For details we
refer to the first part.

Take a Borchers triple $({\mathcal{M}},T,\Omega)$ and $x\in B({\mathcal{H}})$.
We write $x(a)={\hbox{\rm Ad\,}}T(a)(x)$ for $a\in{\mathbb{R}}^{2}$ and
consider observables sent to lightlike directions with parameter
${\mathcal{T}}$:

$x_{\pm}(h_{\mathcal{T}}):=\int h_{\mathcal{T}}(t)x(t,\pm t)\,\mathrm{d}t,$

where
$h_{\mathcal{T}}(t)=|{\mathcal{T}}|^{-\varepsilon}h(|{\mathcal{T}}|^{-\varepsilon}(t-{\mathcal{T}}))$,
$0<\varepsilon<1$ is a constant, ${\mathcal{T}}\in{\mathbb{R}}$ and $h$ is a
nonnegative symmetric smooth function on ${\mathbb{R}}$ such that $\int
h(t)\,\mathrm{d}t=1$. Then for $x\in{\mathcal{M}}$, the limits
$\Phi^{\mathrm{out}}_{+}(x):=\underset{{\mathcal{T}}\to+\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,x_{+}(h_{\mathcal{T}})$
and
$\Phi^{\mathrm{in}}_{-}(x):=\underset{{\mathcal{T}}\to-\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,x_{-}(h_{\mathcal{T}})$
exist. Furthermore we set
$\Phi_{+}^{\mathrm{in}}(y^{\prime}):=J_{\mathcal{M}}\Phi_{+}^{\mathrm{out}}(J_{\mathcal{M}}y^{\prime}J_{\mathcal{M}})J_{\mathcal{M}},\,\,\Phi_{-}^{\mathrm{out}}(y^{\prime}):=J_{\mathcal{M}}\Phi_{-}^{\mathrm{in}}(J_{\mathcal{M}}y^{\prime}J_{\mathcal{M}})J_{\mathcal{M}}$
for $y^{\prime}\in{\mathcal{M}}^{\prime}$, where $J_{\mathcal{M}}$ is the
modular conjugation of ${\mathcal{M}}$ with respect to $\Omega$. The
properties of these asymptotic fields are summarized in [DT11, Tan11a]. For
example, it holds that
$\Phi_{+}^{\mathrm{in}}(y^{\prime})=\underset{{\mathcal{T}}\to-\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,y^{\prime}_{+}(h_{\mathcal{T}})$
and
$\Phi_{-}^{\mathrm{out}}(y^{\prime})=\underset{{\mathcal{T}}\to+\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,y^{\prime}_{-}(h_{\mathcal{T}})$.

Let ${\mathcal{H}}_{+}$ (respectively by ${\mathcal{H}}_{-}$) be the space of
the single excitations with positive momentum, (respectively with negative
momentum), i.e. ${\mathcal{H}}_{+}=\\{\xi\in{\mathcal{H}}:T(t,t)\xi=\xi\mbox{
for }t\in{\mathbb{R}}\\}$ (respectively
${\mathcal{H}}_{-}=\\{\xi\in{\mathcal{H}}:T(t,-t)\xi=\xi\mbox{ for
}t\in{\mathbb{R}}\\}$). For $\xi_{+}\in{\mathcal{H}}_{+}$,
$\xi_{-}\in{\mathcal{H}}_{-}$, there are sequences of local operators
$\\{x_{n}\\},\\{y_{n}\\}\subset{\mathcal{M}}$ and
$\\{x^{\prime}_{n}\\},\\{y^{\prime}_{n}\\}\subset{\mathcal{M}}^{\prime}$ such
that
$\underset{n\to\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,P_{+}x_{n}\Omega=\underset{n\to\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,P_{+}x^{\prime}_{n}\Omega=\xi_{+}$
and
$\underset{n\to\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,P_{-}y_{n}\Omega=\underset{n\to\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,P_{+}y^{\prime}_{n}\Omega=\xi_{-}$.
We define collision states as in [DT11]:

$\xi_{+}{\overset{{\mathrm{in}}}{\times}}\xi_{-}=\underset{n\to\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,\Phi^{\mathrm{in}}_{+}(x^{\prime}_{n})\Phi^{\mathrm{in}}_{-}(y_{n})\Omega,\quad\xi_{+}{\overset{{\mathrm{out}}}{\times}}\xi_{-}=\underset{n\to\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,\Phi^{\mathrm{out}}_{+}(x_{n})\Phi^{\mathrm{out}}_{-}(y^{\prime}_{n})\Omega\,.$

We denote by ${\mathcal{H}}^{\mathrm{in}}$ (respectively
${\mathcal{H}}^{\mathrm{out}}$) the subspace generated by
$\xi_{+}{\overset{{\mathrm{in}}}{\times}}\xi_{-}$ (respectively
$\xi_{+}{\overset{{\mathrm{out}}}{\times}}\xi_{-}$). The isometry

$S:{\mathcal{H}}^{\mathrm{out}}\ni\xi_{+}{\overset{{\mathrm{out}}}{\times}}\xi_{-}\longmapsto\xi_{+}{\overset{{\mathrm{in}}}{\times}}\xi_{-}\in{\mathcal{H}}^{\mathrm{in}}$

is called the scattering operator or the S-matrix of the Borchers triple
$({\mathcal{M}},T,\Omega)$. We say that the Borchers triple
$({\mathcal{M}},T,\Omega)$ is interacting if $S$ is not equal to the identity
operator on ${\mathcal{H}}^{\mathrm{out}}$ and asymptotically complete (with
respect to waves) if it holds that
${\mathcal{H}}^{\mathrm{in}}={\mathcal{H}}^{\mathrm{out}}={\mathcal{H}}$.

We have studied the general structure of asymptotically complete local and
wedge-local nets (using Borchers triple) in [Tan11a, Section 3]. The point was
that for a given (strictly local) $({\mathcal{M}},T,\Omega)$ we can construct
the chiral net, and the original object ${\mathcal{M}}$ can be recovered from
the chiral net and a single operator $S$. Here we rephrase this observation
from the point of view of constructing examples based on chiral components.
See also the general structure of asymptotically complete strictly local nets
[Tan11a, Section 3]

$x$$t$$W_{\mathrm{R}}$$\scriptstyle\mathrm{Ad\,}S({\mathbbm{1}}\otimes{\mathcal{F}}_{-}({\mathbb{R}}_{+}))$$\scriptstyle{\mathcal{F}}_{+}({\mathbb{R}}_{-})\otimes{\mathbbm{1}}$$W_{\mathrm{R}}+(t_{1},x_{1})$$\scriptstyle\mathrm{Ad\,}S({\mathbbm{1}}\otimes{\mathcal{F}}_{-}({\mathbb{R}}_{+}+\frac{t_{1}+x_{1}}{\sqrt{2}}))$$\scriptstyle{\mathcal{F}}_{+}({\mathbb{R}}_{-}+\frac{t_{1}-x_{1}}{\sqrt{2}})\otimes{\mathbbm{1}}$$x$$t$
Figure 1: On the definition of the wedge-local net

######  Proposition 2.2.

Let ${\mathcal{F}}_{\pm}$ be two fermi nets on $S^{1}$ defined on
${\mathcal{H}}_{\pm}$ and assume that there is a unitary operator $S$ on
${\mathcal{H}}_{+}\otimes{\mathcal{H}}_{-}$ commuting with $T_{+}\otimes
T_{-}$, leaving ${\mathcal{H}}_{+}\otimes\Omega_{-}$ and
$\Omega_{+}\otimes{\mathcal{H}}_{-}$ pointwise invariant, such that
$x\otimes{\mathbbm{1}}$ commutes with ${\hbox{\rm
Ad\,}}S(x^{\prime}\otimes{\mathbbm{1}})$ where
$x\in{\mathcal{F}}_{+}({\mathbb{R}}_{-})$ and $x^{\prime}\in{\hbox{\rm
Ad\,}}Z_{+}({\mathcal{F}}_{+}({\mathbb{R}}_{+}))$, and ${\hbox{\rm
Ad\,}}S({\mathbbm{1}}\otimes y)$ commutes with ${\mathbbm{1}}\otimes
y^{\prime}$ where $y\in{\mathcal{F}}_{-}({\mathbb{R}}_{+})$ and
$y^{\prime}\in{\hbox{\rm Ad\,}}Z_{-}({\mathcal{F}}_{-}({\mathbb{R}}_{-}))$.
Then the triple

  * •

${\mathcal{M}}_{S}:=\\{x\otimes{\mathbbm{1}},{\hbox{\rm
Ad\,}}S({\mathbbm{1}}\otimes
y):x\in{\mathcal{F}}_{+}({\mathbb{R}}_{-}),y\in{\mathcal{F}}_{-}({\mathbb{R}}_{+})\\}^{\prime\prime}$,

  * •

$T(t,x):=T_{+}(\frac{t-x}{\sqrt{2}})\otimes T_{-}(\frac{t+x}{\sqrt{2}})$,

  * •

$\Omega:=\Omega_{+}\otimes\Omega_{-}$

is an asymptotically complete Borchers triple with the S-matrix $S$.

###### Proof.

As in Part I [Tan11a], the conditions on $T$ and $\Omega$ are automatic
because they are just tensor products of objects for fermi nets. Similarly,
the condition that ${\hbox{\rm
Ad\,}}T(a){\mathcal{M}}_{S}\subset{\mathcal{M}}_{S}$ for $a\in W_{\mathrm{R}}$
is easily seen from the assumption that $T$ commutes with $S$ and the
covariance of fermi nets.

What remains is the cyclicity and separating property of $\Omega$ for
${\mathcal{M}}_{S}$. Cyclicity is immediate because we have

$\displaystyle{\mathcal{M}}_{S}\Omega$ $\displaystyle\supset$
$\displaystyle\\{x\otimes{\mathbbm{1}}\cdot S({\mathbbm{1}}\otimes
y)S^{*}\cdot\Omega:x\in{\mathcal{F}}_{+}({\mathbb{R}}_{-}),y\in{\mathcal{F}}_{-}({\mathbb{R}}_{+})\\}$
$\displaystyle=$ $\displaystyle\\{x\otimes
y\cdot\Omega:x\in{\mathcal{F}}_{+}({\mathbb{R}}_{-}),y\in{\mathcal{F}}_{-}({\mathbb{R}}_{+})\\}$

by the assumed property of $S$, and the latter set is total in
${\mathcal{H}}_{+}\otimes{\mathcal{H}}_{-}$ by the Reeh-Schlieder property for
fermi nets. As for the separating property, we define:

${\mathcal{M}}_{S}^{1}:=\\{{\hbox{\rm
Ad\,}}S(x^{\prime}\otimes{\mathbbm{1}}),{\mathbbm{1}}\otimes
y^{\prime}:x^{\prime}\in{\hbox{\rm
Ad\,}}Z_{+}({\mathcal{F}}_{+}({\mathbb{R}}_{+})),y^{\prime}\in{\hbox{\rm
Ad\,}}Z_{-}({\mathcal{F}}_{-}({\mathbb{R}}_{-}))\\}^{\prime\prime}.$

By an analogous proof, one sees that $\Omega$ is cyclic for
${\mathcal{M}}_{S}^{1}$. Furthermore, ${\mathcal{M}}_{S}$ and
${\mathcal{M}}_{S}^{1}$ commute by assumption. Hence $\Omega$ is separating
for ${\mathcal{M}}_{S}$. In other words, $({\mathcal{M}}_{S},T,\Omega)$ is a
Borchers-triple.

It is immediate that
$\Phi^{\mathrm{out}}_{+}(x\otimes{\mathbbm{1}})=x\otimes{\mathbbm{1}}$ and
$\Phi^{\mathrm{in}}_{-}({\hbox{\rm Ad\,}}S({\mathbbm{1}}\otimes y))={\hbox{\rm
Ad\,}}S({\mathbbm{1}}\otimes y)$ (the latter follows since $S$ commutes with
$T$). Similarly, we have $\Phi^{\mathrm{in}}_{+}({\hbox{\rm
Ad\,}}S(x^{\prime}\otimes{\mathbbm{1}}))={\hbox{\rm
Ad\,}}S(x^{\prime}\otimes{\mathbbm{1}})$ and
$\Phi^{\mathrm{out}}_{-}({\mathbbm{1}}\otimes y^{\prime})={\mathbbm{1}}\otimes
y^{\prime}$. From this, one concludes that the Borchers triple
$({\mathcal{M}}_{S},T,\Omega)$ is asymptotically complete and its S-matrix is
$S$. ∎

We remark that we see $({\mathcal{M}}_{S},T,\Omega)$ as a fermi (i.e. twisted
local) net defined by ${\mathcal{M}}(W_{\mathrm{R}}^{\prime})={\hbox{\rm
Ad\,}}Z_{+}\otimes Z_{-}({\mathcal{M}})$ and that the scattering theory of
waves [Buc75] is considered to be an analogue of the Haag-Ruelle scattering
theory and it is not intended to be applied to fermionic nets. But we will not
pay much attention to this restriction, since our result is a construction of
wedge-local nets with a free massless bosonic net as the asymptotic net, and
fermionic nets appear as auxiliary objects.

###  2.4 Restriction of wedge-local nets

We consider a Borchers triple $({\mathcal{M}},T,\Omega)$. It is in some cases
interesting to consider a subalgebra ${\mathcal{N}}$ of ${\mathcal{M}}$. Let
us denote ${\mathcal{H}}_{\mathcal{N}}:=\overline{{\mathcal{N}}\Omega}$.

######  Proposition 2.3.

If the subspace ${\mathcal{H}}_{\mathcal{N}}$ is invariant under $T$ and
${\hbox{\rm Ad\,}}T(a)({\mathcal{N}})\subset{\mathcal{N}}$ for $a\in
W_{\mathrm{R}}$. Then
$({\mathcal{N}}|_{{\mathcal{H}}_{\mathcal{N}}},T|_{{\mathcal{H}}_{\mathcal{N}}},\Omega)$
is a Borchers triple on ${\mathcal{H}}_{\mathcal{N}}$.

###### Proof.

The components ${\mathcal{N}},T$ and $\Omega$ naturally restricts to
${\mathcal{H}}_{\mathcal{N}}$. The conditions on $T$ and $\Omega$ are trivial,
even restricted to ${\mathcal{H}}_{\mathcal{N}}$. The cyclicity of $\Omega$ is
immediate from the definition of ${\mathcal{H}}_{\mathcal{N}}$. Since $\Omega$
is already separating for ${\mathcal{M}}$, so is also for ${\mathcal{N}}$.
Endomorphic action of $T$ on ${\mathcal{N}}$ is in the hypothesis. ∎

We call a triple $({\mathcal{N}},T,\Omega)$ a (Borchers) subtriple of
$({\mathcal{M}},T,\Omega)$ if ${\mathcal{N}}$ is a subalgebra of
${\mathcal{M}}$, ${\mathcal{H}}_{\mathcal{N}}$ is invariant under $T(a)$,
${\hbox{\rm Ad\,}}T(a)({\mathcal{N}})\subset{\mathcal{N}}$ for $a\in
W_{\mathrm{R}}$, and ${\mathcal{N}}$ is invariant under ${\hbox{\rm
Ad\,}}\Delta_{\mathcal{M}}^{\mathrm{i}t}$, where $\Delta_{\mathcal{M}}$ is the
modular operator of ${\mathcal{M}}$ with respect to $\Omega$.

Recall that a Borchers triple $({\mathcal{M}},T,\Omega)$ gives rise to a
strictly local net if $\Omega$ is cyclic for ${\mathcal{M}}\cap{\hbox{\rm
Ad\,}}T(a)({\mathcal{M}})^{\prime}$ for any $a\in W_{\mathrm{R}}$. We call
such a triple therefore strictly local. The following proposition shows that
the concept of Borchers subtriple corresponds to the one of a local subnet.

######  Proposition 2.4.

If a Borchers triple $({\mathcal{M}},T,\Omega)$ is strictly local, any
subtriple $({\mathcal{N}},T,\Omega)$ is again strictly local when restricted
on $\overline{{\mathcal{N}}\Omega}$.

###### Proof.

Since ${\mathcal{N}}$ is invariant under the modular automorphism ${\hbox{\rm
Ad\,}}\Delta_{\mathcal{M}}^{\mathrm{i}t}$, there is a conditional expectation
$E$ from ${\mathcal{M}}$ onto ${\mathcal{N}}$ which preserves the state
$\langle\Omega,\,\cdot\,\Omega\rangle$ and is implemented by the projection
$P_{\mathcal{N}}$ (see [Tak03, Theorem IX.4.2] for the original reference and
[Tan11b, Appendix A] for an application to nets).

We have to show that $\Omega$ is cyclic for the relative commutant
${\mathcal{N}}\cap{\hbox{\rm Ad\,}}T(a)({\mathcal{N}})^{\prime}$ on the
subspace ${\mathcal{H}}_{\mathcal{N}}$. Let us denote
${\mathcal{M}}_{0,a}:={\mathcal{M}}\cap{\hbox{\rm
Ad\,}}T(a)({\mathcal{M}})^{\prime}$. We claim that $E({\mathcal{M}}_{0,a})$ is
contained in ${\mathcal{N}}\cap{\hbox{\rm Ad\,}}T(a)({\mathcal{N}})^{\prime}$.
Indeed, by the definition of $E$, the image of $E$ is contained in
${\mathcal{N}}$. Furthermore, if $x\in{\mathcal{M}}_{0,a}$, $y\in{\hbox{\rm
Ad\,}}T(a)({\mathcal{N}})\subset{\hbox{\rm Ad\,}}T(a)({\mathcal{M}})$, then

$E(x)y=E(xy)=E(yx)=yE(x),$

hence they commute and the image $E({\mathcal{M}}_{0,a})$ lies in the relative
commutant. Now we have

$\overline{\left({\mathcal{N}}\cap{\hbox{\rm
Ad\,}}T(a)({\mathcal{N}})^{\prime}\right)\Omega}\supset\overline{E({\mathcal{M}}_{0,a})\Omega}\supset\overline{P_{\mathcal{N}}{\mathcal{M}}_{0,a}\Omega}={\mathcal{H}}_{\mathcal{N}}$

by the assumed strict locality of $({\mathcal{M}},T,\Omega)$. ∎

Let $({\mathcal{B}},U,\Omega)$ be an asymptotically complete local Poincaré
covariant net on ${\mathbb{R}}^{2}$ fulfilling the Bisognano-Wichmann property
(see [Tan11a] for related definitions). We recall that one can define the
(out-) asymptotic algebras ${\mathcal{B}}_{+}\otimes{\mathcal{B}}_{-}$ and the
scattering operator $S$ which is a unitary operator, and that it is possible
to recover the original net by the formula

${\mathcal{B}}(W_{\mathrm{R}})=\\{x\otimes{\mathbbm{1}},{\hbox{\rm
Ad\,}}S({\mathbbm{1}}\otimes
y):x\in{\mathcal{B}}_{+}({\mathbb{R}}_{-}),y\in{\mathcal{B}}_{-}({\mathbb{R}}_{+})\\}^{\prime\prime}.$

Note that $({\mathcal{B}}(W_{\mathrm{R}}),U|_{{\mathbb{R}}^{2}},\Omega)$ is an
asymptotically complete, strictly local Borchers triple. Here we exhibit a
simple way to construct subtriples. Let ${\mathcal{A}}_{+},{\mathcal{A}}_{-}$
be (Möbius covariant) subnets of ${\mathcal{B}}_{+},{\mathcal{B}}_{-}$,
respectively. If we define

${\mathcal{N}}=\\{x\otimes{\mathbbm{1}},{\hbox{\rm
Ad\,}}S({\mathbbm{1}}\otimes
y):x\in{\mathcal{A}}_{+}({\mathbb{R}}_{-}),y\in{\mathcal{A}}_{-}({\mathbb{R}}_{+})\\}^{\prime\prime},$

then $({\mathcal{N}},U|_{{\mathbb{R}}^{2}},\Omega)$ is a Borchers subtriple of
$({\mathcal{B}}(W_{\mathrm{R}}),U|_{{\mathbb{R}}^{2}},\Omega)$. Indeed,
conditions regarding ${\mathcal{N}},U|_{{\mathbb{R}}^{2}},\Omega$ are
immediate. As for the invariance of ${\mathcal{N}}$ under ${\hbox{\rm
Ad\,}}\Delta_{\mathcal{M}}^{\mathrm{i}t}$, it suffices to note that $S$ and
$\Delta_{\mathcal{M}}^{\mathrm{i}t}$ commute ([Tan11a, Lemma 2.4], cf.
[Buc75]) and that ${\mathcal{A}}_{+}({\mathbb{R}}_{-})$ and
${\mathcal{A}}_{-}({\mathbb{R}}_{+})$ are preserved by ${\hbox{\rm
Ad\,}}\Delta_{\mathcal{M}}^{\mathrm{i}t}$ because of Bisognano-Wichmann
property.

The trouble is, however, that such Borchers triples constructed as above are
not necessarily asymptotically complete in general. Indeed, the out-asymptotic
states span the subspace
$\overline{{\mathcal{A}}_{+}({\mathbb{R}}_{-})\Omega}\otimes\overline{{\mathcal{A}}_{-}({\mathbb{R}}_{+})\Omega}$.
It is easy to see that this coincides with the full space
$\overline{{\mathcal{N}}\Omega}$ if and only if it is invariant under $S$.

Since a clear-cut scattering theory is so far available only for
asymptotically complete nets, it is worthwhile to give a general condition to
assure that subnets are asymptotically complete. For simplicity, we consider
the following situation: let ${\mathcal{A}}_{0}$ be a fermi net on
${\mathcal{H}}_{0}$ with an action of a compact group $G$ by inner symmetry
implemented by $V_{g}$. Suppose that there is a unitary operator $S$ on
${\mathcal{H}}_{0}\otimes{\mathcal{H}}_{0}$ such that
$({\mathcal{M}}_{S},T,\Omega)$ is a Borchers triple where

  * •

${\mathcal{M}}_{S}:=\\{x\otimes{\mathbbm{1}},{\hbox{\rm
Ad\,}}S({\mathbbm{1}}\otimes
y):x\in{\mathcal{A}}_{0}({\mathbb{R}}_{-}),y\in{\mathcal{A}}_{0}({\mathbb{R}}_{+})\\}^{\prime\prime}$,

  * •

$T(t,x):=T_{0}(\frac{t-x}{\sqrt{2}})\otimes T_{0}(\frac{t+x}{\sqrt{2}})$,

  * •

$\Omega:=\Omega_{0}\otimes\Omega_{0}$,

as in Proposition 2.2.

######  Proposition 2.5.

If $S$ commutes with $V_{g}\otimes V_{g^{\prime}}$, $g,g^{\prime}\in G$, then
the triple

  * •

${\mathcal{N}}_{S}:=\\{x\otimes{\mathbbm{1}},{\hbox{\rm
Ad\,}}S({\mathbbm{1}}\otimes
y):x\in{\mathcal{A}}_{0}^{G}({\mathbb{R}}_{-}),y\in{\mathcal{A}}_{0}^{G}({\mathbb{R}}_{+})\\}^{\prime\prime}$
(restricted to $\overline{{\mathcal{N}}_{S}\Omega}$),

  * •

$T(t,x):=T_{0}(\frac{t-x}{\sqrt{2}})\otimes T_{0}(\frac{t+x}{\sqrt{2}})$,
(restricted to $\overline{{\mathcal{N}}_{S}\Omega}$)

  * •

$\Omega:=\Omega_{0}\otimes\Omega_{0}$

is an asymptotically complete Borchers triple with asymptotic algebra
${\mathcal{A}}_{0}^{G}\otimes{\mathcal{A}}_{0}^{G}$ and scattering operator
$S|_{\overline{{\mathcal{N}}_{s}\Omega}}$.

###### Proof.

As remarked above, $({\mathcal{N}}_{S},T,\Omega)$ is a Borchers triple on
${\mathcal{H}}_{{\mathcal{N}}_{S}}$, hence the only thing to be proven is
asymptotic completeness. We show that the subspace
$\overline{{\mathcal{A}}_{0}^{G}({\mathbb{R}}_{-})\otimes{\mathcal{A}}_{0}^{G}({\mathbb{R}}_{+})\Omega}$
is invariant under $S$.

We claim that $\overline{{\mathcal{A}}_{0}^{G}({\mathbb{R}}_{-})\Omega_{0}}$
coincides with the subspace ${\mathcal{H}}_{0}^{G}$ of invariant vectors under
$\\{V_{g}\\}_{g\in G}$. Indeed, for any $x\in{\mathcal{A}}_{0}$, the averaging
$\int_{g}V_{g}x\Omega_{0}\,\mathrm{d}g=\left(\int_{g}\alpha_{g}(x)\,\mathrm{d}g\right)\Omega_{0}$
gives a projection onto ${\mathcal{H}}_{0}^{G}$. By the Reeh-Schlieder
property, any vector in ${\mathcal{H}}_{0}^{G}$ can be approximated by vectors
in ${\mathcal{A}}_{0}^{G}({\mathbb{R}}_{-})\Omega_{0}$. The converse inclusion
is obvious.

Now it is easy to see that
$\overline{{\mathcal{A}}_{0}^{G}({\mathbb{R}}_{-})\otimes{\mathcal{A}}_{0}^{G}({\mathbb{R}}_{+})\Omega}={\mathcal{H}}_{0}^{G}\otimes{\mathcal{H}}_{0}^{G}$.
This is the space of invariant vectors under the action $\\{V_{g}\otimes
V_{g^{\prime}}:g,g^{\prime}\in G\\}$. Since $S$ commutes with $V_{g}\otimes
V_{g^{\prime}}$ by assumption, this subspace is preserved under $S$. Then, as
remarked before, $\overline{{\mathcal{N}}_{S}\Omega}$ coincides with
$\overline{{\mathcal{A}}_{0}^{G}({\mathbb{R}}_{-})\otimes{\mathcal{A}}_{0}^{G}({\mathbb{R}}_{+})\Omega}$
and we obtain the asymptotic completeness.

The statement on S-matrix is immediate from the definition and by Proposition
2.2. ∎

##  3 Examples of fermi nets

###  3.1 ${\rm U(1)}$-current net ${{\mathcal{A}}^{(0)}}$

Let $U_{1}$ be the irreducible unitary positive-energy representation of
$\operatorname{M\ddot{o}b}$ with lowest weight $1$ on a Hilbert space denoted
by ${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$, which can be
identified with the one-particle space of the ${\rm U(1)}$-current. This has
the following concrete realization: consider $C^{\infty}(S^{1},{\mathbb{R}})$,
where we write the periodic function $f\in C^{\infty}(S^{1},{\mathbb{R}})$ as
a Fourier series

$f(\theta)=\sum_{k\in{\mathbb{Z}}}\hat{f}_{k}\mathrm{e}^{\mathrm{i}k\theta},\quad\hat{f}_{k}=\int_{0}^{2\pi}\mathrm{e}^{-\mathrm{i}k\theta}f(\theta)\frac{\,\mathrm{d}\theta}{2\pi}=\overline{\hat{f}_{-k}}\,.$

We introduce a semi-norm

$\|f\|^{2}=\sum_{k=1}^{\infty}k\cdot|\hat{f}_{k}|^{2}$

and a complex structure, i.e. an isometry $\mathcal{J}$ w.r.t. $\|\,\cdot\,\|$
satisfying $\mathcal{J}^{2}=-1$, by
$\mathcal{J}:\hat{f}_{k}\mapsto-\mathrm{i}\operatorname{sign}(k)\hat{f}_{k}$
and finally we get the Hilbert space
${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}=\overline{C^{\infty}(S^{1},{\mathbb{R}})/{\mathbb{R}}}^{\|\,\cdot\,\|}$
by completion with respect to the norm $\|\,\cdot\,\|$, where ${\mathbb{R}}$
is identified with the constant functions. By abuse of notation we denote also
the image of $f\in C^{\infty}(S^{1},{\mathbb{R}})$ in
${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$ by $f$. The scalar
product (linear in the second component) and the sesquilinear form
$\omega(\,\cdot\,,\,\cdot\,)\equiv\Im\langle\,\cdot\,,\,\cdot\,\rangle$ are
given by

$\displaystyle\langle f,g\rangle$
$\displaystyle=\sum_{k=1}^{\infty}k\hat{f}_{k}\hat{g}_{-k}\,,$
$\displaystyle\omega(f,g)$
$\displaystyle=\frac{-\mathrm{i}}{2}\sum_{k\in{\mathbb{Z}}}k\hat{f}_{k}\hat{g}_{-k}=\frac{1}{2}\int_{0}^{2\pi}f(\theta)g^{\prime}(\theta)\frac{\,\mathrm{d}\theta}{2\pi}=\frac{1}{4\pi}\int
f\,\mathrm{d}g\,,$

respectively. The unitary action $U_{1}$ of $\operatorname{M\ddot{o}b}$ on
${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$ is induced by the action
on $C^{\infty}(S^{1},{\mathbb{R}})$
$(U_{1}(g)f)=(g_{\ast}f)(\theta):=f(g^{-1}(\theta))$.

For $I\in\mathcal{I}$ we denote by $H(I)$ the closure of the subspace of real
functions with support in $I$. This space is standard (i.e.
$H(I)+\mathrm{i}H(I)$ is dense in
${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$ and
$H(I)\cap\mathrm{i}H(I)=\\{0\\}$) and the family
$\\{H(I)\\}_{I\in\mathcal{I}}$ is a local Möbius covariant net of standard
subspaces [Lon08, LW11].

We explain briefly the bosonic second quantization procedure in general. Let
${\mathcal{H}}^{1}$ be a separable Hilbert space, the one-particle space, and
$\omega(\,\cdot\,,\,\cdot\,)=\Im\langle\,\cdot\,,\,\cdot\,\rangle$ the
sesquilinear form. There are unitaries $W(f)$ for $f\in{\mathcal{H}}^{1}$
fulfilling

$W(f)W(g)=\mathrm{e}^{-\mathrm{i}\omega(f,g)}W(f+g)=\mathrm{e}^{-2\mathrm{i}\omega(f,g)}W(g)W(f)$

and acting naturally on the bosonic Fock space
$\mathrm{e}^{{\mathcal{H}}^{1}}$ over ${\mathcal{H}}^{1}$. This space is given
by
$\mathrm{e}^{{\mathcal{H}}^{1}}=\oplus_{n=0}^{\infty}P_{n}({\mathcal{H}}^{1})^{\otimes
n}$, where $P_{n}$ is the projection
$P_{n}(\xi_{1}\otimes\cdots\otimes\xi_{n})=1/n!\sum_{\sigma}\xi_{\sigma(1)}\otimes\cdots\otimes\xi_{\sigma(n)}$
and the sum goes over all permutations. The set of coherent vectors
$\mathrm{e}^{h}:=\oplus_{n=0}^{\infty}h^{\otimes n}/\sqrt{n!}$ with
$h\in{\mathcal{H}}^{1}$ is total in $\mathrm{e}^{{\mathcal{H}}^{1}}$ and it
holds $\langle\mathrm{e}^{f},\mathrm{e}^{h}\rangle=\mathrm{e}^{\langle
f,h\rangle}$. The vacuum is given by $\Omega=\mathrm{e}^{0}$ and the action of
$W(f)$ is given by
$W(f)\mathrm{e}^{0}=\mathrm{e}^{-\frac{1}{2}\|f\|^{2}}\mathrm{e}^{f}$, in
other words the vacuum representation is characterized by
$\phi(W(f))=\mathrm{e}^{-\frac{1}{2}\|f\|^{2}}$, where
$\phi(\,\cdot\,)=\langle\Omega,\,\cdot\,\Omega\rangle$. For a real subspace
$H\subset{\mathcal{H}}^{1}$, we define the von Neumann algebra

$\displaystyle R(H)=\\{W(f):f\in H\\}^{\prime\prime}\subset
B(\mathrm{e}^{{\mathcal{H}}^{1}})\,.$

Let $U$ be a unitary on the one-particle space ${\mathcal{H}}^{1}$ then
$\mathrm{e}^{U}:=\oplus_{n=0}^{\infty}U^{\otimes n}$ acts on coherent states
by $\mathrm{e}^{U}\mathrm{e}^{h}=\mathrm{e}^{Uh}$ and is therefore a unitary
on $\mathrm{e}^{{\mathcal{H}}^{1}}$, the second quantization unitary.

We obtain the ${\rm U(1)}$-current net ${{\mathcal{A}}^{(0)}}$ on
${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}:=\mathrm{e}^{{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}}$
with $\Omega_{0}=\mathrm{e}^{0}$ by defining
${{\mathcal{A}}^{(0)}}(I):=R(H(I))$ which is covariant with respect to
$U(g):=\mathrm{e}^{U_{1}(g)}$. For $f\in C^{\infty}(S^{1},{\mathbb{R}})$ we
consider a self-adjoint operator $J(f)$ given by the generator of the unitary
one-parameter group $W(t\cdot f)=\mathrm{e}^{\mathrm{i}t\cdot J(f)}$ with
$t\in{\mathbb{R}}$. This defines the usual current (field operator) smeared
with the real test function $f$, which fulfills
$J(f)\Omega_{0}=f\in{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$ and

$\displaystyle[J(f),J(g)]$
$\displaystyle=2\mathrm{i}\omega(f,g)=\sum_{k}k\hat{f}_{k}\hat{g}_{-k}=\frac{\mathrm{i}}{2\pi}\int
f\,\mathrm{d}g\,.$

It can be extended to complex test functions via
$J(f+\mathrm{i}g)=J(f)+\mathrm{i}J(g)$, and one obtains the usual operator
valued ($z$-picture) distribution $J(z)$ with the relations

$\displaystyle J(f)$
$\displaystyle=\sum_{n\in{\mathbb{Z}}}\hat{f}_{n}J_{n}=\oint_{S^{1}}f(z)J(z)\frac{\,\mathrm{d}z}{2\pi\mathrm{i}},$
$\displaystyle J(z)=\sum_{n}J_{n}z^{-n-1}$ $\displaystyle[J_{m},J_{n}]$
$\displaystyle=m\delta_{m+n,0}\,,$

where the modes $J_{n}=J(e_{n})$ with
$e_{n}(\theta)=\mathrm{e}^{\mathrm{i}n\theta}$ satisfy $J_{n}\Omega_{0}=0$ for
$n\geq 0$.

The space ${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}$ is spanned by
vectors of the form $\xi=J_{-n_{1}}\cdots J_{-n_{k}}\Omega_{0}$ with
$0<n_{1}\leq\cdots\leq n_{k}$ with “energy” $N=\sum_{m}n_{m}$, i.e.
$R(\theta)\xi=\mathrm{e}^{\mathrm{i}N\theta}\xi$. Therefore it is graded with
respect to the rotations

$\displaystyle{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}$
$\displaystyle={\mathbb{C}}\Omega_{0}\oplus\bigoplus_{n\in{\mathbb{N}}}{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}},n}$
$\displaystyle{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}},n}$
$\displaystyle=\bigoplus_{k=1}^{n}\bigoplus_{\begin{subarray}{c}0<n_{1}\leq\cdots\leq
n_{k}\\\ n_{1}+\cdots+n_{k}=n\end{subarray}}{\mathbb{C}}J_{-n_{1}}\cdots
J_{-n_{k}}\Omega_{0}$

and ${\hbox{dim}\,}{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}},n}$ is the
number of partitions of $n$ elements, whose generating function $p(t)$ is the
inverse of Euler’s function $\phi(t)=\prod_{k=1}^{\infty}(1-t^{k})$ and
therefore the conformal character of the ${\rm U(1)}$-current net is given by
($t=\mathrm{e}^{-\beta}$):

$\operatorname{tr}_{{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}}(\mathrm{e}^{-\beta
L_{0}})=\sum_{n=0}^{\infty}{\hbox{dim}\,}{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}},n}\cdot
t^{n}=\prod_{n\in{\mathbb{N}}}(1-t^{n})^{-1}$

(a conformal character is defined as a formal power series, but it is often
convergent for $|t|<1$ and here we used the formula
$(1-z)^{-1}=1+z+z^{2}\cdots$). It will be convenient to use the real
parametrization $x\in{\mathbb{R}}\cong S^{1}\setminus\\{-1\\}$ of the cut
circle and use the conventions

$\displaystyle
f(s)=\int_{\mathbb{R}}\mathrm{e}^{-\mathrm{i}sp}\hat{f}(p)\,\mathrm{d}p.$

By writing $f(s)=f_{0}(\theta(s))$ for $f_{0}\in
C^{\infty}(S^{1},{\mathbb{R}})$ where $\theta(s)=2\arctan(s)$, the space
${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$ above can be identified
with the space $L^{2}({\mathbb{R}}_{+},p\,\mathrm{d}p)$ in which the space
${\mathcal{S}}({\mathbb{R}},{\mathbb{R}})$ embeds by restriction of the
Fourier transformation to ${\mathbb{R}}_{+}$. In other words
${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$ can be seen as the
closure of the space ${\mathcal{S}}({\mathbb{R}},{\mathbb{R}})$ with complex
structure $\mathcal{J}\hat{f}(p)=\mathrm{i}\operatorname{sign}(p)\hat{f}(p)$
and the scalar product and sesquilinear form given by:

$\displaystyle\langle f,g\rangle$
$\displaystyle=\int_{{\mathbb{R}}_{+}}\hat{f}(-p)\hat{g}(p)p\,\mathrm{d}p\,,$
$\displaystyle\omega(f,g)$
$\displaystyle=\frac{-\mathrm{i}}{2}\int_{{\mathbb{R}}}\hat{f}(-p)\hat{g}(p)p\,\mathrm{d}p=\frac{1}{4\pi}\int_{\mathbb{R}}f(x)g^{\prime}(x)\,\mathrm{d}x\,.$

Using the above identification we denote for
$f\in{\mathcal{S}}({\mathbb{R}},{\mathbb{R}})$ by $J(f)$ the smeared current
with $J(f)\Omega_{0}=f\in{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$.
In this parametrization commutation relations read:

$\displaystyle[J(f),J(g)]$
$\displaystyle=\frac{\mathrm{i}}{2\pi}\int_{\mathbb{R}}f(x)g^{\prime}(x)\,\mathrm{d}x=\int_{{\mathbb{R}}}\hat{f}(-p)\hat{g}(p)p\,\mathrm{d}p\,.$

###  3.2 The free complex fermion net $\mathrm{Fer}_{\mathbb{C}}$

We construct the net of the free complex fermion on the circle, which can be
seen as the chiral part of the net of the free massless Dirac (or complex)
fermion on two dimensional Minkowski space. The notations of this section are
basically in accordance with [Was98], but we use a different convention of
positive-energy, which leads to the conjugated complex structure. For giving a
simple description of the one-particle space, we consider first the Hilbert
space $L^{2}(S^{1})$ and the Hardy space $H^{2}(S^{1})$, namely

$H^{2}(S^{1}):=\left\\{f:\mbox{ analytic on the unit disk }D,\sup_{0\leq
r<1}\int_{0}^{2\pi}|f(r\mathrm{e}^{\mathrm{i}\theta})|^{2}\,\mathrm{d}\theta<\infty\right\\}\,.$

Any function in $H^{2}(S^{1})$ has a $L^{2}$-boundary value and can be
considered as an element of $L^{2}(S^{1})$. In this sense, $H^{2}(S^{1})$ is a
subspace of $L^{2}(S^{1})$. Furthermore, it holds that

$H^{2}(S^{1})=\\{f\in L^{2}(S^{1}):\hat{f}_{n}=0\mbox{ for }n<0\\},$

where $\hat{f}_{n}$ is the $n$-th Fourier component of $f$. We denote the
orthogonal projection onto $H^{2}(S^{1})$ by $P$.

The group

$\mathrm{SU}(1,1):=\left\\{\left(\begin{matrix}\alpha&\beta\\\
\overline{\beta}&\overline{\alpha}\end{matrix}\right)\in
M_{2}({\mathbb{C}}):|\alpha|^{2}-|\beta|^{2}=1\right\\}$

acts on the circle $S^{1}$ by $g\cdot z=\frac{\alpha
z+\beta}{\overline{\beta}z+\overline{\alpha}}$ and there is a unitary action
of $\mathrm{SU}(1,1)$ on $L^{2}(S^{1})$ by

$(U(g)f)(z):=(V_{g}f)(z)=\frac{1}{-\overline{\beta}z+\alpha}f(g^{-1}\cdot
z)\,.$

One sees that the projection $P$ commutes with $V_{g}$ , since $V_{g}f$ is
still an analytic function for $|\alpha|>|\beta|$.

Then one defines a new Hilbert space
${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{1}=\overline{PL^{2}(S^{1})}\oplus({\mathbbm{1}}-P)L^{2}(S^{1})$:
namely, ${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{1}$ is identical with
$L^{2}(S^{1})$ as a real linear space and the multiplication by $\mathrm{i}$
is given by $-\mathrm{i}(2P-{\mathbbm{1}})$, or in other words, by
$-\mathrm{i}$ on $PL^{2}(S^{1})$ and $\mathrm{i}$ on
$({\mathbbm{1}}-P)L^{2}(S^{1})$. Because $P$ and $U(g)$ commute, the action of
$\mathrm{SU}(1,1)$ remains unitary on
${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{1}$.

Then for $I\in\mathcal{I}$ one takes real Hilbert subspaces $K(I):=L^{2}(I)$
of ${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{1}$. This subspaces turn out
to be standard [Was98, Theorem (p. 497)]. If $I_{1}$ and $I_{2}$ are disjoint
intervals, $K(I_{1})$ are real orthogonal to $K(I_{2})$, in other words
$K(I_{1})\subset K(I_{2})^{\perp}$, where
$K^{\perp}=\\{\xi\in{\mathcal{H}}:\Re\langle\xi,K\rangle=0\\}$. It turns out
that $I\mapsto K(I)$ is a twisted-local Möbius covariant net of standard
subspaces.

We briefly explain the fermionic second quantization in general. Let
${\mathcal{H}}^{1}$ be a complex Hilbert space and
${\mathcal{H}}=\Lambda({\mathcal{H}}^{1})$ the antisymmetric (fermionic) Fock
space obtained by completing the exterior algebra with the inner product. For
$A\in B({\mathcal{H}}^{1})$ with $\|A\|\leq 1$ we define $\Lambda(A)$ to be
$A^{\otimes k}$ on
${\mathcal{H}}^{k}:=\Lambda^{k}({\mathcal{H}}^{1})\subset({\mathcal{H}}^{1})^{\otimes
k}$. The space is ${\mathbb{Z}}_{2}$ graded by
$\Gamma:=\Lambda(-{\mathbbm{1}})$. We define
$Z=\frac{{\mathbbm{1}}-\mathrm{i}\Gamma}{1-\mathrm{i}}$ and note that
$Z^{2}=\Gamma$. For $f\in{\mathcal{H}}^{1}$ let $a(f)$ be the bounded operator
obtained by continuing the exterior multiplication $f\wedge\cdot$. The
operators fulfill the complex Clifford relations
$a(f)^{\ast}a(g)+a(g)a(f)^{\ast}=\langle f,g\rangle$ and
$\\{a(f),a(g)\\}=\\{a(f)^{\ast},a(g)^{\ast}\\}=0$ for all
$f,g\in{\mathcal{H}}^{1}$. For a standard subspace $K\subset{\mathcal{H}}^{1}$
we define the von Neumann algebra

$C(K)=\\{c(f):f\in K\\}^{\prime\prime}\subset B(\Lambda{\mathcal{H}}^{1})$

where $c(f)=a(f)+a(f)^{\ast}$, which fulfills the real Clifford relations
$c(f)c(g)+c(g)c(f)=2\Re\langle f,g\rangle$. By $\Omega=1\in\Lambda^{0}$ we
denote the vacuum which is cyclic and separating for $C(K)$ for every standard
subspace $K\subset{\mathcal{H}}^{1}$. Further it holds Haag-Araki duality,
i.e. $C(K^{\perp})$ equals $C(K)^{\perp}:=ZC(K)^{\prime}Z^{\ast}$, the twisted
commutant of $C(K)$. For a unitary $U$ on ${\mathcal{H}}^{1}$ it holds
$\Lambda(U)c(f)\Lambda(U^{\ast})=c(Uf)$, which implies that $C$ is covariant
with respect to the unitaries $U({\mathcal{H}}^{1})$, i.e.
$\Lambda(U)C(K)\Lambda(U)^{\ast}=C(UK)$.

We note that in the case like the complex fermion the one-particle space is
obtained from a Hilbert space ${\mathcal{H}}^{1}$ (the space of test
functions) and a projection $P$ by
${\mathcal{H}}^{1}_{P}=P{\mathcal{H}}^{1}\oplus\overline{P^{\perp}{\mathcal{H}}^{1}}$
and one gets a new representation of the complex Clifford algebra on
$\Lambda({\mathcal{H}}^{1}_{P})$ by
$a_{P}(f)=a(Pf)+a(\overline{P^{\perp}f})^{\ast}$ where $a(f)$ is the creation
operator. For a standard subspace $K\subset{\mathcal{H}}^{1}_{P}$ which is
invariant under the multiplication of $\mathrm{i}_{{\mathcal{H}}^{1}}$ in
${\mathcal{H}}^{1}$, the von Neumann algebra $C(K)$ on
$\Lambda({\mathcal{H}}_{P}^{1})$ coincides with the von Neumann algebra
$\\{a_{P}(f),a_{P}(f)^{\ast}:f\in K\\}^{\prime\prime}$. Indeed, the one
inclusion follows from $c(f)=a_{P}(f)+a_{P}(f)^{\ast}$ and the other follows
from Araki-Haag duality and $\\{a_{P}(f),c(g)\\}=\langle
g,f\rangle_{{\mathcal{H}}^{1}}=\Re\langle
g,f\rangle_{{\mathcal{H}}^{1}_{P}}-\mathrm{i}\Re\langle
g,\mathrm{i}_{{\mathcal{H}}^{1}}f\rangle_{{\mathcal{H}}^{1}_{P}}=0$ for $f\in
K$ and $g\in K^{\perp}$. We further note that the space
$\Lambda({\mathcal{H}}^{1}_{P})$ is as a real Hilbert space the same as
$\Lambda({\mathcal{H}}^{1})$ and can be identified canonically with
$\Lambda(P{\mathcal{H}}^{1})\otimes\Lambda(\overline{P^{\perp}{\mathcal{H}}^{1}})$.

We turn to the concrete case where ${\mathcal{H}}^{1}=L^{2}(S^{1})$ and define
the net
$\mathrm{Fer}_{\mathbb{C}}(I):=C(K(I))=\\{a_{P}(f),a_{P}(f)^{\ast}:f\in
L^{2}(I)\\}^{\prime\prime}$ (where here
$a_{P}(f):=a(\overline{Pf})+a(P^{\perp}f)^{\ast}$) on
${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}=\Lambda({\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{1})\cong\Lambda(\overline{PL^{2}(S^{1})})\otimes\Lambda(P^{\perp}L^{2}(S^{1}))$
which is isotonic by definition and fulfills twisted duality, namely by Haag-
Araki duality
$\mathrm{Fer}_{\mathbb{C}}(I^{\prime})=C(K(I)^{\perp})=C(K(I))^{\perp}=\mathrm{Fer}_{\mathbb{C}}(I)^{\perp}$.
In addition, the net $\mathrm{Fer}_{\mathbb{C}}$ is Möbius covariant. Indeed,
we can take the representation $\Lambda U(\cdot)$ by promoting the one-
particle representation $U$ to the second quantization operator. It is easy to
see that the covariance of this net $\mathrm{Fer}_{\mathbb{C}}$ follows from
the covariance of the net of standard spaces $K$. The representation $\Lambda
U$ has positive energy since so does the representation $U$, and leaves
invariant the vacuum vector $\Omega_{0}$ of the Fock space. Summing up, the
net $\mathrm{Fer}_{\mathbb{C}}$ is a fermi net (cf. [Was98]). This net is
referred to as the free complex fermi net on $S^{1}$. The scalar
multiplication by a constant phase $\mathrm{e}^{-\mathrm{i}\vartheta}$ in the
original structure of the one-particle space is still a unitary operator in
the new structure. Its promotion by the second quantization $V(\theta)$
implements an action of ${\rm U(1)}$ on $\mathrm{Fer}_{\mathbb{C}}$ by inner
symmetry. This will be referred to as the ${\rm U(1)}$-gauge action.

For $r\in\frac{1}{2}+{\mathbb{Z}}$ let $\psi_{r}=a_{P}(e_{-r-\frac{1}{2}})$
and $\bar{\psi}_{r}=a_{P}(e_{r-\frac{1}{2}})^{\ast}$ where $e_{r}\in
L^{2}(S^{1})$ with $e_{r}(\theta)=\mathrm{e}^{\mathrm{i}\theta r}$. The
$\psi_{r},\bar{\psi}_{r}$ are the modes of the free complex fermion, namely

$\displaystyle\\{\psi_{n},\psi_{m}\\}$
$\displaystyle=\\{\bar{\psi}_{m},\bar{\psi}_{n}\\}=0$
$\displaystyle\\{\bar{\psi}_{n},\psi_{m}\\}$ $\displaystyle=\delta_{m+n,0}$
$\displaystyle\psi_{n}^{\ast}$ $\displaystyle=\bar{\psi}_{-n}$

and it holds that $\psi_{r}\Omega_{0}=\bar{\psi}_{r}\Omega_{0}=0$ for
$r\in\frac{1}{2}+{\mathbb{N}}_{0}$. Each of $\psi_{r}$ or $\bar{\psi}_{r}$ has
norm $1$ following from the commutation relation. We can introduce the usual
fields ($f,g\in L^{2}(S^{1})$) and operator valued distributions in the
$z$-picture:

$\displaystyle\Psi(f)$
$\displaystyle=\sum_{r\in\frac{1}{2}+{\mathbb{Z}}}\hat{f}_{r}\Psi_{r}=\oint_{S^{1}}f(z)z^{-\frac{1}{2}}\Psi(z)\frac{\,\mathrm{d}z}{2\pi\mathrm{i}}\,,$
$\displaystyle\Psi(z)$
$\displaystyle=\sum_{r\in\frac{1}{2}+{\mathbb{Z}}}\Psi_{r}z^{-r-\frac{1}{2}}\,,$
$\displaystyle\bar{\psi}(f)$
$\displaystyle=\psi(\overline{f})^{\ast}=a_{P}(e_{-\frac{1}{2}}f)^{\ast}\,,$
$\displaystyle\\{\bar{\psi}(f),\psi(g)\\}$ $\displaystyle=\oint
f(z)g(z)\frac{\,\mathrm{d}z}{2\pi\mathrm{i}z}\,,$

where $\Psi$ is either $\psi$ or $\bar{\psi}$. The fields $\psi,\bar{\psi}$
are covariant, e.g. $U(g)\psi(f)U(g)^{\ast}=\psi(f_{g})$ with
$f_{g}(z)=\frac{1}{|\alpha-\overline{\beta}z|}f(g^{-1}z)$ for
$g=\left(\begin{matrix}\alpha&\beta\\\
\overline{\beta}&\overline{\alpha}\end{matrix}\right)\in\mathrm{SU(1,1)}$.

We note that vectors of the form

$\xi=\psi_{-r_{1}}\cdots\psi_{-r_{k}}\bar{\psi}_{-s_{1}}\cdots\bar{\psi}_{-s_{\ell}}\Omega_{0}$

with $0<r_{1}<\cdots<r_{k}$ and $0<s_{1}<\cdots<s_{\ell}$ form a basis of
${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}=\Lambda({\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{1})$
and that such a $\xi$ is an eigenvector for the rotations,
$R(\theta)\xi\equiv\mathrm{e}^{\mathrm{i}\theta
L_{0}}\xi=\mathrm{e}^{\mathrm{i}N\theta}\xi$ with
$N=\sum_{j=1}^{k}r_{j}+\sum_{j=1}^{\ell}s_{j}$ and of the gauge action
$V(\theta)\xi\equiv\mathrm{e}^{\mathrm{i}\theta
Q}\xi=\mathrm{e}^{\mathrm{i}(k-\ell)\theta}\xi$. In each vector of this basis
the $r$-th energy level can either be empty, be occupied by $\psi_{-r}$ or
$\bar{\psi}_{-r}$ or occupied by both. The contribution of this level to the
character
$\operatorname{tr}_{{\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}}(\mathrm{e}^{-\beta
L_{0}-EQ})$ is then $1$, $zt^{r}$, $z^{-1}t^{r}$ or $t^{2r}$, respectively,
where $t=\mathrm{e}^{-\beta}$ and $z=\mathrm{e}^{-E}$. By summing over all
possibilities one gets that the character of $\mathrm{Fer}_{\mathbb{C}}$ is
given by (cf. [Kac98, Reh98]):

$\displaystyle\operatorname{tr}_{{\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}}(\mathrm{e}^{-\beta
L_{0}-EQ})=\operatorname{tr}_{{\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}}(t^{L_{0}}z^{Q})$
$\displaystyle=\prod_{r\in{\mathbb{N}}_{0}+\frac{1}{2}}(1+zt^{r}+z^{-1}t^{r}+t^{2r})$
$\displaystyle=\prod_{r\in{\mathbb{N}}_{0}+\frac{1}{2}}(1+zt^{r})(1+z^{-1}t^{r})$
$\displaystyle=p(t)\sum_{q\in{\mathbb{Z}}}z^{q}t^{\frac{q^{2}}{2}}\,,$

where the last equality follows directly from the Jacobi triple product
formula (see [Apo76, Theorem 14.6])

$\displaystyle\prod_{r\in{\mathbb{N}}}(1+zw^{2r-1})(1+z^{-1}w^{2r-1})(1-w^{2r})=\sum_{q\in{\mathbb{Z}}}z^{q}w^{q}$

by setting $2r-1=2n$ and $t=w^{2}$. In particular, for the local net
$\mathrm{Fer}_{\mathbb{C}}^{{\rm U(1)}}$ the character is given by
$\operatorname{tr}_{{\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{{\rm
U(1)}}}(\mathrm{e}^{-\beta L_{0}})=p(t)$, since it is the fixed point with
respect to the ${\rm U(1)}$-gauge action and the conformal character is the
coefficient of $z^{0}$.

###  3.3 ${\rm U(1)}$-current net as a subnet of $\mathrm{Fer}_{\mathbb{C}}$

In this section we use the well-known fact that the Wick product
$:\\!\bar{\psi}\psi\\!:$ of the complex fermion $\psi$ equals the ${\rm
U(1)}$-current and give an analogue of the boson-fermion correspondence (see
e.g. [Kac98, 5.2]) in the operator algebraic setting. Let us denote by
${\mathcal{D}}_{0}$ the subspace of $\Lambda({\mathcal{H}}_{P}^{1})$ of
vectors with finite energy:

${\mathcal{D}}_{0}:=\mathrm{span}\left\\{\psi_{-r_{1}}\cdots\psi_{-r_{k}}\bar{\psi}_{-s_{1}}\cdots\bar{\psi}_{-s_{l}}\Omega_{0}:k,l\in{\mathbb{N}}_{0},r_{i},s_{j}\in{\mathbb{N}}+\frac{1}{2}\right\\}\,.$

Then we define the unbounded operators on the domain ${\mathcal{D}}_{0}$:

$\displaystyle J_{n}=\sum_{r+s=n}{:\\!\bar{\psi}_{r}\psi_{s}\\!:}$
$\displaystyle=$
$\displaystyle\sum_{r<0}\bar{\psi}_{r}\psi_{n-r}-\sum_{r>0}\psi_{n-r}\bar{\psi}_{r}$
$\displaystyle=$
$\displaystyle\sum_{r}\left(\bar{\psi}_{r}\psi_{n-r}-\langle\Omega_{0},\bar{\psi}_{r}\psi_{n-r}\Omega_{0}\rangle\right)$

with $r,s\in\frac{1}{2}+{\mathbb{Z}}$. Note that any vector in
${\mathcal{D}}_{0}$ is annihilated by $\psi_{r}$ for sufficiently large $r$,
thus the action of $J_{n}$ on such a vector can be defined and remains in
${\mathcal{D}}_{0}$. In particular, we have $J_{n}\Omega_{0}=0$ for
$n\in{\mathbb{N}}_{0}$.

######  Lemma 3.1.

On ${\mathcal{D}}_{0}$ it holds that

  1. 1.

$[J_{n},\psi_{k}]=-\psi_{n+k}$ and $[J_{n},\bar{\psi}_{k}]=\bar{\psi}_{n+k}$

  2. 2.

$[J_{m},J_{n}]=m\delta_{m+n,0}$

###### Proof.

Using $[ab,c]=a\\{b,c\\}-\\{a,c\\}b$, one obtains
$[\bar{\psi}_{r}\psi_{n},\psi_{k}]=-\delta_{r+k,0}\psi_{n}$ and
$[\psi_{n}\bar{\psi}_{r},\psi_{k}]=\delta_{r+k,0}\psi_{n}$ from which directly
follows
$[J_{n},\psi_{k}]=\sum_{r<0}[\bar{\psi}_{r}\psi_{n-r},\psi_{k}]-\sum_{r>0}[\psi_{n-m}\bar{\psi}_{r},\psi_{k}]=-\psi_{n+k}$.
Analogously one shows $[J_{n},\bar{\psi}_{k}]=\bar{\psi}_{n+k}$.

From the Jacobi identity, it follows immediately that $[J_{n},J_{m}]$ commutes
with all $\psi_{k}$ and $\bar{\psi}_{k}$ and hence $[J_{n},J_{m}]$ is a
multiple of the identity, therefore
$[J_{n},J_{m}]=\langle\Omega_{0},[J_{n},J_{m}]\Omega_{0}\rangle{\mathbbm{1}}$.
It is

$\displaystyle[J_{n},J_{p}]$
$\displaystyle=\sum_{r<0}[J_{n},\bar{\psi}_{r}\psi_{p-r}]-\sum_{r>0}[J_{n},\psi_{p-r}\bar{\psi}_{r}]$
$\displaystyle=-\sum_{r<0}\left(\bar{\psi}_{r}\psi_{p-r+n}-\bar{\psi}_{r+n}\psi_{p-r}\right)-\sum_{r>0}\left(\psi_{p-r}\bar{\psi}_{r+n}-\psi_{p-r+n}\bar{\psi}_{r}\right)$

and in the case $p\neq-n$ we get
$\langle\Omega_{0},[J_{n},J_{p}]\Omega_{0}\rangle=0$, and otherwise

$\displaystyle\langle\Omega_{0},[J_{n},J_{-n}]\Omega_{0}\rangle$
$\displaystyle=\begin{cases}\sum_{r<0}\langle\Omega_{0},\bar{\psi}_{r+n}\psi_{-r-n}\Omega_{0}\rangle=\sum_{r=\frac{1}{2}}^{n-\frac{1}{2}}\langle\Omega_{0},\\{\bar{\psi}_{r},\psi_{-r}\\}\Omega_{0}\rangle&n>0\\\
-\sum_{r>0}\langle\Omega_{0},\psi_{-r-n}\bar{\psi}_{r+n}\Omega_{0}\rangle=-\sum_{r=\frac{1}{2}}^{-n-\frac{1}{2}}\langle\Omega_{0},\\{\psi_{r},\bar{\psi}_{-r}\\}\Omega_{0}\rangle&n<0\end{cases}$
$\displaystyle=n\,,$

which completes the proof. ∎

Let $L_{0}$ be the generator of the rotation:
$R(\theta)=\mathrm{e}^{\mathrm{i}\theta L_{0}}$. From its action (see the end
of Section 3.2), one verifies that ${\mathcal{D}}_{0}$ is a core for $L_{0}$.

######  Lemma 3.2 (Linear energy bounds).

It holds that $[L_{0},J_{n}]=-nJ_{n}$ on ${\mathcal{D}}_{0}$. For a
trigonometric polynomial $f=\sum_{n}\hat{f}_{n}e_{n}$ where the sum is finite
and $\xi\in{\mathcal{D}}_{0}$, we have

$\displaystyle\|J(f)\xi\|$ $\displaystyle\leq c_{f}\|(L_{0}+1)\xi\|$
$\displaystyle\|[L_{0},J(f)]\xi\|$ $\displaystyle\leq
c_{\partial_{\theta}f}\|(L_{0}+1)\xi\|,$

where $c_{f}$ depends only on $f$.

###### Proof.

For the commutation relation, it is enough to choose an energy eigenvector
$\xi\in{\mathcal{D}}_{0}$, i.e. $L_{0}\xi=N\xi$. It is
$J_{n}L_{0}\xi=NJ_{n}\xi$ and

$L_{0}J_{n}\xi=L_{0}\left(\sum_{r<0}\bar{\psi}_{r}\psi_{n-r}\xi-\sum_{r>0}\psi_{n-r}\bar{\psi}_{r}\xi\right)=(N-n)J_{n}\xi,$

and the first statement follows.

We have seen that $\psi_{r}$ and $\bar{\psi}_{r}$ have norm $1$ in Section
3.2. First we claim that $\|J_{n}\xi\|\leq\|(2(L_{0}+1)+|n|)\xi\|$. Let $\xi$
be again an eigenvector of $L_{0}$, i.e. $L_{0}\xi=N\xi$. From the defining
sum of $J_{n}$, one sees that only $2N+|n|+2$ terms contribute to $J_{n}\xi$.
Hence we have $\|J_{n}\xi\|\leq(2N+|n|+2)\|\xi\|=\|2(L_{0}+1)+|n|\xi\|$. If
the inequality holds for eigenvectors, then for $\\{\xi_{r}\\}$ with different
eigenvalues, we have $\xi_{r}\perp\xi_{s}$ and $J_{n}\xi_{r}\perp
J_{n}\xi_{s}$, and hence

$\displaystyle\left\|J_{n}\sum_{r}\xi_{r}\right\|^{2}$ $\displaystyle=$
$\displaystyle\sum_{r}\|J_{n}\xi_{r}\|^{2}$ $\displaystyle\leq$
$\displaystyle\sum_{r}\|(2(L_{0}+1)+|n|)\xi_{r}\|^{2}$ $\displaystyle=$
$\displaystyle\left\|(2(L_{0}+1)+|n|)\sum_{r}\xi_{r}\right\|^{2}$

and the general case follows.

For a smeared field, we have

$\|J(f)\xi\|=\left\|\sum_{n}\hat{f}_{n}J_{n}\xi\right\|\leq
2\tilde{c}_{f}\|(L_{0}+1)\xi\|+\tilde{c}_{\partial_{\theta}f}\|\xi\|\leq(2\tilde{c}_{f}+\tilde{c}_{\partial_{\theta}f})\|(L_{0}+1)\xi\|,$

where $\tilde{c}_{f}=\sum_{n}|\hat{f}_{n}|$. By defining
$c_{f}=2\tilde{c}_{f}+\tilde{c}_{\partial_{\theta}f}$, we obtain the first
inequality of the statement. The rest follows by noting that
$[L_{0},J(f)]=J(\mathrm{i}\partial_{\theta}f)$. ∎

For a smooth function $f=\sum_{n\in{\mathbb{Z}}}\hat{f}_{n}e_{n}\in
C^{\infty}(S^{1})$, its Fourier coefficients $\hat{f}_{n}$ are strongly
decreasing and, in particular, it is summable:
$\sum_{n}|\hat{f}_{n}|=\tilde{c}_{f}<\infty$. Hence we can naturally extend
the definition of the smeared current to smooth functions using the above
estimate by

$J(f)=\sum_{n\in{\mathbb{Z}}}f_{n}J_{n}=\sum_{r,s\in\frac{1}{2}+{\mathbb{Z}}}f_{r+s}:\\!\psi_{r}\bar{\psi}_{s}\\!:,$

and the same inequality in Lemma 3.2 holds. The operator is closable since we
have $J(f)\subset J(\overline{f})^{*}$ and we still denote the closure by
$J(f)$. We note that from the above definition it follows that $J(f)$ is
obtained by a limit $\sum_{n}:\\!\psi(h_{n})\bar{\psi}(k_{n})\\!:$ with
suitable functions such that $\sum_{n}h_{n}(\theta)k_{n}(\vartheta)\to 2\pi
f(\theta)\delta(\theta-\vartheta)$. This implies covariance of the “field”,
i.e. $U(g)J(f)U(g)^{\ast}=J(f\circ g^{-1})$.

Recall that $\|\psi_{r}\|=1$, hence the smeared field is still bounded:
$\|\psi(g)\|\leq\tilde{c}_{g}$. We claim that, for $f,g\in C^{\infty}(S^{1})$
and $\xi\in{\mathcal{D}}_{0}$, $\psi(g)\xi$ is in the domain of $J(f)$.
Indeed, for a trigonometric polynomial $g$, we have the estimate

$\displaystyle\|J(f)\psi(g)\xi\|$ $\displaystyle\leq$ $\displaystyle
c_{f}\|(L_{0}+1)\psi(g)\xi\|$ $\displaystyle\leq$ $\displaystyle
c_{f}(\tilde{c}_{g}\|\xi\|+\|[L_{0},\psi(g)]\xi+\psi(g)L_{0}\xi\|)$
$\displaystyle\leq$ $\displaystyle
c_{f}(\tilde{c}_{g}(\|\xi\|+\|L_{0}\xi\|)+\tilde{c}_{\partial_{\theta}g}\|\xi\|)\,.$

Then if we have a sequence of trigonometric polynomial $g_{n}$ converging to a
smooth function $g\in C^{\infty}(S^{1})$, the sequence
$\\{J(f)\psi(g_{n})\xi\\}$ is also converging.

######  Lemma 3.3.

For $\xi,\eta\in{\mathcal{D}}_{0}$, it holds that

$\displaystyle[J(f),\psi(g)]\xi$ $\displaystyle=-\psi(f\cdot g)\xi$
$\displaystyle[J(f),\bar{\psi}(g)]\xi$ $\displaystyle=\bar{\psi}(f\cdot g)\xi$
$\displaystyle\langle J(\bar{f})\xi,J(g)\eta\rangle$ $\displaystyle=\langle
J(\bar{g})\xi,J(f)\eta\rangle+2\mathrm{i}\omega(f,g)\langle\xi,\eta\rangle.$

###### Proof.

For trigonometric polynomials $f,g$, the statements can be proved easily from
Lemma 3.1. The general case is shown by approximating first $f$ by
polynomials, then $g$, according to the convergence considered above (as for
the third statement, obviously the order of limits does not matter). ∎

We need the following well-known result [DF77, Theorem 3.1]:

######  Theorem 3.4 (The commutator theorem).

Let $H$ be a positive self-adjoint operator and $A,B$ symmetric operators
defined on a core ${\mathcal{D}}_{0}$ for $(H+{\mathbbm{1}})^{2}$. Assume that
there is a constant $C$ such that

$\displaystyle\|A\xi\|\leq C\|(H+{\mathbbm{1}})\xi\|,\,\,\,\|B\xi\|\leq
C\|(H+{\mathbbm{1}})\xi\|,$ $\displaystyle\|[H,A]\xi\|\leq
C\|(H+{\mathbbm{1}})\xi\|,\,\,\,\|[H,B]\xi\|\leq C\|(H+{\mathbbm{1}})\xi\|,$
$\displaystyle\langle A\xi,B\eta\rangle=\langle B\xi,A\eta\rangle\mbox{ for
any }\xi,\eta\in{\mathcal{D}}_{0}.$

Then $A$ and $B$ are essentially self-adjoint on any core of $H$ and any
bounded functional calculus of $A$ and $B$ commute.

######  Remark 3.5.

In the original literature [DF77], this Theorem is proved under the assumption
of certain operator inequalities. In fact, what is really used in the proof of
commutativity of bounded functions is the norm estimates
$\|A(H+{\mathbbm{1}})^{-1}\|<C,\|[H,A](H+{\mathbbm{1}})^{-1}\|<C$ etc. and
they follow from the assumptions here. The essential self-adjointness of $A$
and $B$ can be proved by [RS75, Theorem X.37]. An analogous application of
this theorem with norm estimates can be found in [BSM90].

By the commutator theorem, we get that $J(f)$ is self-adjoint for $f\in
C^{\infty}(S^{1},{\mathbb{R}})$ and that all bounded functions of $J(f)$
commute with all bounded functions of $J(g)$ for $f,g\in
C^{\infty}(S^{1},{\mathbb{R}})$ with disjoint support.

Let $I$ be a proper interval and let us define the von Neumann algebra

${\mathcal{B}}(I)=\\{\mathrm{e}^{\mathrm{i}J(f)}:{\rm supp}f\subset
I\\}^{\prime\prime}.$

The local net ${\mathcal{B}}(I)$ restricted to
$\overline{{\mathcal{B}}(I)\Omega_{0}}$ can be identified with the ${\rm
U(1)}$-current net ${{\mathcal{A}}^{(0)}}$ on
${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}$, in particular we can
identify
$\overline{{\mathcal{B}}(I)\Omega_{0}}\cong{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}$.

######  Proposition 3.6.

Let $I$ be a proper interval, then
${\mathcal{B}}(I)\subset\mathrm{Fer}_{\mathbb{C}}^{{\rm U(1)}}(I)$.

###### Proof.

We see that ${\mathcal{B}}(I)$ commutes with
$\mathrm{Fer}_{\mathbb{C}}(I^{\prime})=\\{c(g):g\in
L^{2}(I^{\prime})\\}^{\prime\prime}$ because, for $f,g$ with disjoint
supports, $c(g)$ commutes with $J(f)$ on a core by Lemma 3.3 and therefore any
spectral projection of $c(g)$ commutes with $J(f)$, and hence with any bounded
functions of $J(f)$.

Further because $J(f)$ commutes by construction with the gauge action $V(t)$
and is in particular even because $V(\pi)=\Gamma$, it follows that
${\mathcal{B}}(I)$ lies in the twisted commutant
$\mathrm{Fer}_{\mathbb{C}}(I^{\prime})^{\perp}$. By twisted Haag duality it is
${\mathcal{B}}(I)\subset\mathrm{Fer}_{\mathbb{C}}(I^{\prime})^{\perp}=\mathrm{Fer}_{\mathbb{C}}(I)$
and therefore ${\mathcal{B}}(I)={\mathcal{B}}(I)^{{\rm
U(1)}}\subset\mathrm{Fer}_{\mathbb{C}}^{{\rm U(1)}}(I)$. ∎

Since the covariance has been seen, we have the following.

######  Corollary 3.7.

${\mathcal{B}}$ is a subnet of $\mathrm{Fer}_{\mathbb{C}}^{{\rm U(1)}}$.

Now the following is straightforward.

######  Proposition 3.8.

The ${\rm U(1)}$-fixed point subnet of the complex free fermion net
$\mathrm{Fer}_{\mathbb{C}}$ is the ${\rm U(1)}$-current net, i.e.
$\mathrm{Fer}_{\mathbb{C}}^{{\rm
U(1)}}={\mathcal{B}}\cong{{\mathcal{A}}^{(0)}}$.

###### Proof.

Let us see ${\mathcal{B}}$ as a subnet of the fermi net
$\mathrm{Fer}_{\mathbb{C}}^{{\rm U(1)}}$ on
${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{{\rm
U(1)}}\equiv{\mathcal{H}}_{\,\cdot\,,0}$. Further
$\overline{{\mathcal{B}}(I)\Omega}$ does not depend on $I$ by the same proof
of the Reeh-Schlieder property and is clearly a subspace of
${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{{\rm
U(1)}}\equiv{\mathcal{H}}_{\,\cdot\,,0}$.

In fact they coincide, since we have confirmed that
$\operatorname{tr}_{{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}}(\mathrm{e}^{-\beta
L_{0}})=\operatorname{tr}_{{\mathcal{H}}_{\,\cdot\,,0}}(\mathrm{e}^{-\beta
L_{0}})=p(t)$, where $\mathrm{e}^{-\beta}=t$, namely, their conformal
characters coincide (see also Section 2.2). ∎

We finish this section by giving the parametrization in $x$-picture, where the
action of the translation is more natural. With

$f(x)=\frac{1}{\sqrt{2\pi}}\sqrt{\left|\frac{\partial\theta(x)}{\partial
x}\right|}\mathrm{e}^{\mathrm{i}\theta(x)/2}f_{0}(\theta(x))$

we identify $L^{2}({\mathbb{R}})=L^{2}({\mathbb{R}},\,\mathrm{d}x)$ with
$L^{2}(S^{1})=L^{2}([0,2\pi],\,\mathrm{d}\theta/(2\pi))$ and therefore the
space ${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{1}$ is given by
$PL^{2}({\mathbb{R}})\oplus\overline{P^{\perp}L^{2}({\mathbb{R}})}$ with
$P:\hat{f}(p)\mapsto\Theta(p)\hat{f}(p)$ and it can be identified in “momentum
space” with $L^{2}({\mathbb{R}}_{+},2\pi\,\mathrm{d}p)\oplus
L^{2}({\mathbb{R}}_{+},2\pi\,\mathrm{d}q)$ by

$\displaystyle f(x)$
$\displaystyle\longmapsto\widehat{Pf}(p)\oplus\overline{\widehat{P^{\perp}f}(-q)}$
$\displaystyle p,q>0\,.$

The field operators are defined for $f\in L^{2}({\mathbb{R}})$ by
$\psi(f)=a_{P}(f)$ and $\bar{\psi}(f)=a_{P}(\overline{f})^{\ast}$. For
$\Psi\in{\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}$ we write its components

$\Psi_{m,n}\in{\mathcal{H}}_{m,n}:=L^{2}({\mathbb{R}}_{+}^{m+n},(2\pi)^{m+n}\,\mathrm{d}p_{1}\cdots\,\mathrm{d}p_{m}\,\mathrm{d}q_{1}\cdots
q_{n})_{-},$ (1)

where $-$ means the antisymmetrization within $p_{1},\ldots,p_{m}$ and
$q_{1},\ldots,q_{n}$. By this notation
$(\psi(f)\Omega_{0})_{1,0}(p)=\hat{f}(p)$ and
$(\bar{\psi}(f)\Omega_{0})_{0,1}(q)=\hat{f}(q)$. Further the bi-field
${:\\!\bar{\psi}(f)\psi(g)\\!:}=\bar{\psi}(f)\psi(g)-\langle\Omega_{0},\bar{\psi}(f)\psi(g)\Omega_{0}\rangle{\mathbbm{1}}$
creates from the vacuum $\Omega_{0}$ a fermionic 1+1 particle state
$\Psi_{f,g}:={:\\!\bar{\psi}(f)\psi(g)\\!:}\Omega_{0}$ with
$(\Psi_{f,g})_{1,1}(p,q)=-\hat{f}(q)\hat{g}(p)$ and it follows for $h\in
C^{\infty}({\mathbb{R}},{\mathbb{R}})$ that for the ${\rm U(1)}$-current $J$,
it holds $(J(h)\Omega_{0})_{1,1}(p,q)=-\frac{1}{2\pi}\hat{h}(p+q)$ which is
obtained by taking a limit $\sum_{n}\Psi_{f_{n},g_{n}}$ with test functions
$\sum_{n}f_{n}(x)g_{n}(y)\to h(x)\delta(x-y)$. We make the important
observation that the $J(f)\Omega_{0}$ generate the one-particle space which we
can identify with ${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$ and
this is obviously a proper subspace of the fermionic 1+1-particle space
${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{1,1}$.

##  4 A new family of Longo-Witten endomorphisms on ${\rm U(1)}$-current net

We use the description of
${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{1}=\overline{PL^{2}(S^{1})}+P^{\perp}L^{2}(S^{1})$
which equals $L^{2}(S^{1})$ as a real Hilbert space and is described in the
beginning of Section 3.2. First we decompose
${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{1}$ into irreducible
representations of $\mathrm{SU}(1,1)$ in a compatible way with $K(I)$. Let us
define

$\displaystyle{\mathcal{H}}_{\Re}$ $\displaystyle:=$
$\displaystyle\\{f\in{\mathcal{H}}^{1}_{\mathrm{Fer}_{\mathbb{C}}}:z^{\frac{1}{2}}f(z)\mbox{
is real}\\},$ $\displaystyle{\mathcal{H}}_{\Im}$ $\displaystyle:=$
$\displaystyle\\{f\in{\mathcal{H}}^{1}_{\mathrm{Fer}_{\mathbb{C}}}:z^{\frac{1}{2}}f(z)\mbox{
is pure imaginary}\\}.$

By their definition, it is clear that ${\mathcal{H}}_{\Re}$ and
${\mathcal{H}}_{\Im}$ are real Hilbert subspaces of $L^{2}(S^{1})$. In fact,
they are complex subspaces with respect to the new complex structure. To see
this, we take another description of ${\mathcal{H}}_{\Re}$: in terms of
Fourier components, it holds that $f\in{\mathcal{H}}_{\Re}$ if and only if
$f_{n}=\overline{f_{-n-1}}$. Recall that, on $L^{2}(S^{1})$, the new scalar
multiplication by $\mathrm{i}$ is given by $\mathrm{i}\cdot
f_{n}=-\mathrm{i}f_{n}$, $\mathrm{i}\cdot f_{-n-1}=\mathrm{i}f_{-n-1}$ for
$n\geq 0$. Hence this condition is preserved under the multiplication by
$\mathrm{i}$ and ${\mathcal{H}}_{\Re}$ is a complex subspace. An analogous
argument holds for ${\mathcal{H}}_{\Im}$. Next we see that
${\mathcal{H}}_{\Re}$ and ${\mathcal{H}}_{\Im}$ are orthogonal. Note that
because of the change of the complex structure, for $f(z)=\sum_{n}f_{n}z^{n}$
and $h(z)=\sum_{n}h_{n}z^{n}$ the inner product is written as follows:

$\langle f,h\rangle=\sum_{n\geq
0}f_{n}\overline{h_{n}}+\sum_{n<0}\overline{f_{n}}h_{n}.$

Now $f\in{\mathcal{H}}_{\Re}$ implies $f_{n}=\overline{f_{-n-1}}$ and
$h\in{\mathcal{H}}_{\Im}$ implies $h_{n}=-\overline{h_{-n-1}}$ for non-
negative $n$, hence it is easy to see that

$\langle f,h\rangle=\sum_{n\geq
0}f_{n}\overline{h_{n}}+\sum_{n<0}\overline{f_{n}}h_{n}=-\sum_{n\geq
0}\overline{f_{-n-1}}{h_{-n-1}}+\sum_{n<0}\overline{f_{n}}{h_{n}}=0.$

In other words, these two complex subspaces are mutually orthogonal.

Furthermore, ${\mathcal{H}}_{\Re}$ and ${\mathcal{H}}_{\Im}$ are invariant
under the action of $\mathrm{SU}(1,1)$. We recall that the action is given by
$(V_{g}f)(z)=\frac{1}{-\overline{\beta}z+\alpha}f\left(\frac{\overline{\alpha}z-\beta}{-\overline{\beta}z+\alpha}\right)$.
Then if $z^{\frac{1}{2}}f(z)$ is real then it holds that

$\displaystyle z^{\frac{1}{2}}(V_{g}f)(z)$ $\displaystyle=$
$\displaystyle\frac{1}{{\bar{z}}^{\frac{1}{2}}(-\overline{\beta}z+\alpha)}\left(\frac{\overline{\alpha}z-\beta}{-\overline{\beta}z+\alpha}\right)^{-\frac{1}{2}}\cdot\left(\frac{\overline{\alpha}z-\beta}{-\overline{\beta}z+\alpha}\right)^{\frac{1}{2}}f\left(\frac{\overline{\alpha}z-\beta}{-\overline{\beta}z+\alpha}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{(-\overline{\beta}+\alpha\bar{z})^{\frac{1}{2}}(\overline{\alpha}z-\beta)^{\frac{1}{2}}}\cdot\left(\frac{\overline{\alpha}z-\beta}{-\overline{\beta}z+\alpha}\right)^{\frac{1}{2}}f\left(\frac{\overline{\alpha}z-\beta}{-\overline{\beta}z+\alpha}\right)$

and both factors are real. Similarly one shows that ${\mathcal{H}}_{\Im}$ is
preserved under $V_{g}$. It is obvious that these two representations are
intertwined by the multiplication by $\mathrm{i}$ in the old complex
structure. This is still a unitary map, thus they are unitarily equivalent.
One can see that each representation is indeed irreducible, and when
restricted to $\mathrm{PSU}(1,1)={\rm PSL}(2,{\mathbb{R}})$, it is the
projective positive energy representation with lowest weight $\frac{1}{2}$.

It is easy to see that $e^{\Re}_{n}:=\\{e_{n}+e_{-n+1},n\geq 0\\}$ and
$e^{\Im}_{n}:=\\{\mathrm{i}(e_{n}+e_{-n+1}),n\geq 0\\}$ form bases of
${\mathcal{H}}_{\Re}$ and ${\mathcal{H}}_{\Im}$, respectively, where
$e_{n}(z)=z^{n}$ and the multiplication by $\mathrm{i}$ is given in the old
structure. Now we describe the gauge action in terms of this basis. By the
definition, for a given complex number $\alpha$ with modulus $1$, the action
is given by the multiplication in the old structure. Hence if
$\alpha=\cos\theta+\mathrm{i}\sin\theta$, we have
$U_{\alpha}e^{\Re}_{n}=\cos\theta e^{\Re}_{n}+\sin\theta e^{\Im}_{n}$ and
$U_{\alpha}e^{\Im}_{n}=-\sin\theta e^{\Re}_{n}+\cos\theta e^{\Im}_{n}$. This
means that $U_{\alpha}$ acts as the real rotation by $\theta$ in this basis.

#### Construction of endomorphisms

We construct Longo-Witten endomorphisms on the free fermion net
$\mathrm{Fer}_{\mathbb{C}}$ commuting with the gauge action. The key is the
following theorem. We remind that a standard pair $(\tilde{H},\tilde{T})$ is a
standard subspace $\tilde{H}\subset\tilde{\mathcal{H}}$ of a Hilbert space
$\tilde{\mathcal{H}}$ and a positive energy representation $\tilde{T}$ of
${\mathbb{R}}$ on $\tilde{\mathcal{H}}$, such that
$\tilde{T}(a)\tilde{H}\subset\tilde{H}$ for $a\geq 0$. If $\tilde{T}$ is
maximally abelian, the standard pair is said to be irreducible and there is a
(up to unitary equivalence) unique irreducible standard pair.

######  Theorem 4.1 ([LW11, Theorem 2.6]).

Let $(\tilde{H},\tilde{T})$ be a standard pair with multiplicity $n$, i.e. it
decomposes into $n$-fold direct sum of irreducible standard pairs, each
unitarily equivalent to the unique standard pair $(H,T)$ and
$T(t)=\mathrm{e}^{\mathrm{i}tP}$. Then a unitary $\tilde{V}$ commuting with
the translation $\tilde{T}$ preserves $\tilde{H}$ if and only if $\tilde{V}$
is a $n\times n$ matrix $(V_{hk})$ (with respect to the decomposition of
$\tilde{\mathcal{H}}$ into $n$ direct sum as above) such that
$V_{hk}=\varphi_{hk}(P)$, where $\varphi_{hk}:{\mathbb{R}}\to{\mathbb{C}}$ are
complex Borel functions such that $(\varphi_{hk})$ is a unitary matrix for
almost every $p>0$, each $\varphi_{hk}$ is the boundary value of a function in
${\mathbb{H}}(\SS_{\infty})$ and is symmetric, i.e.
$\varphi_{hk}(-p)=\overline{\varphi_{hk}(p)}$.

Consider the one-particle space
${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{1}$ for
$\mathrm{Fer}_{\mathbb{C}}$. The pair of the standard space
$K({\mathbb{R}}_{+})=L^{2}({\mathbb{R}}_{+})$ defined in Section 3.2 (under
the identification of $S^{1}$ and ${\mathbb{R}}\cup\\{\infty\\})$ and the
natural translation has multiplicity 2. If we take a matrix-valued function
$(\varphi_{hk})$ as above and take the second quantization operator
$\Lambda(V)$ of the (matrix-valued) operator $(V_{kh})=(\varphi_{hk}(P))$,
then it implements a Longo-Witten endomorphism of $\mathrm{Fer}_{\mathbb{C}}$
(see [LW11]).

As the gauge group acts by real rotation
$\left(\begin{matrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{matrix}\right)$, any matrix-valued function of $p$
which commute with them must have the form
$\left(\begin{matrix}a(p)&\mathrm{i}b(p)\\\
-\mathrm{i}b(p)&a(p)\end{matrix}\right)$. If each component is symmetric, then
$a$ is symmetric and $b$ is antisymmetric. Such a matrix-valued function can
be diagonalized by the matrix $\left(\begin{matrix}1&\mathrm{i}\\\
\mathrm{i}&1\end{matrix}\right)$ and becomes
$\left(\begin{matrix}a(p)+b(p)&0\\\ 0&a(p)-b(p)\end{matrix}\right)$. We claim
that such $a$ and $b$ exist. Indeed, let $\varphi$ be a inner function (not
necessarily symmetric), namely the boundary value with modulus $1$ of a
bounded analytic function on the upper half-plane ${\mathbb{H}}$, and define
$a(p)=\frac{1}{2}(\varphi(p)+\overline{\varphi(-p)})$,
$b(p)=\frac{1}{2}(\varphi(p)-\overline{\varphi(-p)})$. Then it is obvious that
$a$ is symmetric and $b$ is antisymmetric. In addition, $a(p)+b(p)=\varphi(p)$
and $a(p)-b(p)=\overline{\varphi(-p)}$, hence the diagonalized matrix is
unitary for almost every $p$. By the theorem of Longo-Witten, the operator
$\varphi(P_{1}):=\left(\begin{matrix}a(P)&\mathrm{i}b(P)\\\
-\mathrm{i}b(P)&a(P)\end{matrix}\right)$ preserves the real Hilbert space
$\widetilde{H}:=K({\mathbb{R}}_{+})$, where $P_{1}$ is the generator of the
translation in ${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1}$ which has
multiplicity $2$ and $P$ is the generator of $T$ of the irreducible standard
pair $(H,T)$.

It is easy to see that the above diagonalization is given exactly by the
decomposition
${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{1}=\overline{PL^{2}(S^{1})}\oplus({\mathbbm{1}}-P)L^{2}(S^{1})$.

We remind that ${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{1}$ can be
identified with $L^{2}({\mathbb{R}})$ as a real space. In
$L^{2}({\mathbb{R}})$ the function $\varphi(P_{1})f$ is the function with
Fourier transform $\varphi(p)\hat{f}(p)$ and we remark that it also follows
directly from the Paley-Wiener theorem that $\varphi(P_{1})$ leaves
$L^{2}({\mathbb{R}}_{+})\subset L^{2}({\mathbb{R}})$ invariant for $\varphi$
inner. Further using that the space
${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}$ decomposes in
${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}=\bigoplus_{m,n\in{\mathbb{N}}_{0}}{\mathcal{H}}_{m,n}$
like in (1) with the gauge action given by
$V(\theta)\Psi_{m,n}=\mathrm{e}^{\mathrm{i}(m-n)\theta}\Psi_{m,n}$, the action
of the Longo-Witten unitary $V_{\varphi}=\Lambda(\varphi(P_{1}))$ is given by

$\displaystyle(V_{\varphi}\Psi)_{m,n}(p_{1},\cdots,p_{m},q_{1},\cdots,q_{n})$
$\displaystyle\quad=\varphi(p_{1})\cdots\varphi(p_{m})\overline{\varphi(-q_{1})}\cdots\overline{\varphi(-q_{n})}\Psi_{m,n}(p_{1},\cdots,p_{m},q_{1},\cdots,q_{n})\,.$

######  Lemma 4.2.

Let $\iota:\Psi\in
L^{2}({\mathbb{R}}_{+},p\,\mathrm{d}p)\equiv{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}\hookrightarrow{\mathcal{H}}_{1,1}\subset{\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}$
be the embedding given by $\iota(\Psi)_{1,1}(p,q)=-\frac{1}{2\pi}\Psi(p+q)$. A
second quantization Longo-Witten unitary $V_{\varphi}$ commuting with the
gauge action $V(\,\cdot\,)$ satisfies
$V_{\varphi}\iota{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}\subset\iota{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$
if and only if $V_{\varphi}=V(\theta)T(t)$ with $t\geq 0$.

###### Proof.

The translations commute with the gauge action and it follows immediately that
they leave ${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$ invariant. We
note that $\varphi(p)\overline{\varphi(-q)}\Psi(p+q)$ belongs to
${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$ only if it can be written
as a function of $g(p+q)$. This means that
$\varphi(p)\overline{\varphi(-q)}=\widetilde{\varphi}(p+q)$ for $p,q\geq 0$,
where $\widetilde{\varphi}$ is another function. Then, putting $q=0$ and $p=0$
respectively, we see that
$\varphi(p)\overline{\varphi(0)}=\widetilde{\varphi}(p)$ for $p\geq 0$ and
$\varphi(0)\overline{\varphi(-q)}=\widetilde{\varphi}(q)$ for $q\geq 0$, in
particular $\widetilde{\varphi}(0)=1$. Multiplying the each side of these
equations, one sees that
$\widetilde{\varphi}(p+q)=\widetilde{\varphi}(p)\widetilde{\varphi}(q)$
because $|\varphi(0)|=1$. Then it follows that
$\widetilde{\varphi}(p)=\mathrm{e}^{\mathrm{i}\kappa p}$ for some $\kappa\geq
0$, and $\varphi(p)=\mathrm{e}^{\mathrm{i}(\kappa p+\theta)}$ for some
$\theta\in{\mathbb{R}}$ (in fact, the arguments here should be treated with
care because the relation is given only almost everywhere, but both $\varphi$
and $\check{\varphi}$ analytically continue and in the domain of analyticity
it holds everywhere).

Such a $\varphi$ is a Longo-Witten unitary only for $\kappa\geq 0$. The
constant factor $\mathrm{e}^{\mathrm{i}\theta}$ corresponds to the factor
$V(\theta)$. ∎

######  Theorem 4.3.

Let $\varphi$ be an inner function as above. The endomorphism implemented by
the second quantization $V_{\varphi}$ of the operator constructed above
restricts to the ${\rm U(1)}$-current subnet. The restriction cannot be
implemented by any second quantization operator if
$\varphi(p)\neq\mathrm{e}^{\mathrm{i}(\kappa p+\theta)}$.

###### Proof.

The operator $V_{\varphi}$ restricts to the subnet ${{\mathcal{A}}^{(0)}}$ by
the general argument in Proposition 2.1. It cannot be implemented by a second
quantization operator, since any second quantization operator preserves the
particle number, while $V_{\varphi}$ does not for non-exponential $\varphi$ as
we saw above, and a Longo-Witten endomorphism is uniquely implemented up to
scalar (see Section 2.1). ∎

######  Remark 4.4.

By the construction in [LW11], each unitary
$V=V_{\varphi}|_{{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}}$ related to
an inner function $\varphi$ from above gives rise to a local, time-translation
covariant net of von Neumann algebras on the Minkowski half-space
$M_{+}=\\{(t,x)\in{\mathbb{R}}^{2}:x>0\\}$. This net is associated with the
${\rm U(1)}$-current net ${{\mathcal{A}}^{(0)}}$ and defined by
${{\mathcal{A}}^{(0)}}_{V}(O)={{\mathcal{A}}^{(0)}}(I_{1})\vee
V{{\mathcal{A}}^{(0)}}(I_{2})V^{\ast}$, where $O=I_{1}\times
I_{2}=\\{(t,x)\in{\mathbb{R}}^{2}:t-x\in I_{1},t+x\in I_{2}\\}$ is a double
cone with $\overline{O}\subset M_{+}$ corresponding uniquely to the two
intervals $I_{1}$ and $I_{2}$ with disjoint closures. In the case where
$\varphi$ is not exponential $V_{\varphi}$ does not come from second
quantization—in contrast to the unitaries constructed by Longo and Witten in
[LW11]—and therefore gives new examples.

##  5 Interacting wedge-local net with particle production

###  5.1 Construction of scattering operators

In the previous section we saw that, in the basis
$\\{e_{n}+e_{-n},e_{n}-e_{-n}\\}$ the matrix operator
$\left(\begin{matrix}a(P)&\mathrm{i}b(P)\\\
-\mathrm{i}b(P)&a(P)\end{matrix}\right)$ implements a Longo-Witten
endomorphism if $a$ is symmetric and $b$ is antisymmetric, and after the
simultaneous diagonalization it becomes $\left(\begin{matrix}\varphi(P)&0\\\
0&\check{\varphi}(P)\end{matrix}\right)$ where $\varphi$ is an inner function
and $\check{\varphi}(p)=\overline{\varphi(-p)}$ (note that if $\varphi$
extends to an analytic function $\varphi(z)$ on ${\mathbb{H}}$, then
$\check{\varphi}(z)=\overline{\varphi(-\overline{z})}$ also extends to
${\mathbb{H}}$, hence $\check{\varphi}$ is again an inner function). By the
same argument one sees that $\left(\begin{matrix}\check{\varphi}(P)&0\\\
0&\varphi(P)\end{matrix}\right)$ implements an endomorphism since
$\check{\check{\varphi}}=\varphi$.

With respect to the basis after diagonalization, we split the Hilbert space
${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1}=:{\mathcal{H}}_{+}\oplus{\mathcal{H}}_{-}$
and the generator of translation $P_{1}=:P_{+}\oplus P_{-}$. Then the tensor
product space can be written as follows:

${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1}\otimes{\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1}=\left({\mathcal{H}}_{+}\otimes{\mathcal{H}}_{+}\right)\oplus\left({\mathcal{H}}_{+}\otimes{\mathcal{H}}_{-}\right)\oplus\left({\mathcal{H}}_{-}\otimes{\mathcal{H}}_{+}\right)\oplus\left({\mathcal{H}}_{-}\otimes{\mathcal{H}}_{-}\right)\,.$

According to this decomposition into a direct sum of four subspaces, we define
an operator

$M_{\varphi}:=\varphi(P_{+}\otimes P_{+})\oplus\check{\varphi}(P_{+}\otimes
P_{-})\oplus\check{\varphi}(P_{-}\otimes P_{+})\oplus\varphi(P_{-}\otimes
P_{-})\,.$

Then this restricts to the subspace
${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1}\otimes{\mathcal{H}}_{+}=\left({\mathcal{H}}_{+}\oplus{\mathcal{H}}_{-}\right)\otimes{\mathcal{H}}_{+}$
and it is $\varphi(P_{+}\otimes P_{+})\oplus\check{\varphi}(P_{-}\otimes
P_{+})$, or we can decompose it with respect to the spectral measure of
$P_{+}$:

$\int_{{\mathbb{R}}_{+}}\left(\begin{matrix}\varphi(pP_{+})&0\\\
0&\check{\varphi}(pP_{-})\end{matrix}\right)\otimes\,\mathrm{d}E_{+}(p).$

Similarly, the restriction to
${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1}\otimes{\mathcal{H}}_{-}$ is
written as

$\int_{{\mathbb{R}}_{+}}\left(\begin{matrix}\check{\varphi}(pP_{+})&0\\\
0&\varphi(pP_{-})\end{matrix}\right)\otimes\,\mathrm{d}E_{-}(p).$

Using the two-point set ${\mathbb{Z}}_{2}=\\{+,-\\}$ we define

$\varphi_{+}(p,+):=\varphi(p),\,\,\,\,\varphi_{+}(p,-)=\check{\varphi}(p),\,\,\,\,\varphi_{-}(p,+)=\check{\varphi}(p),\,\,\,\,\varphi_{-}(p,-)=\varphi(p).$

By defining the spectral measure $E_{1}=E_{+}\oplus E_{-}$ on
${\mathcal{H}}^{1}$, $M_{\varphi}$ can be simply written as

$M_{\varphi}=\int_{{\mathbb{R}}_{+}\times{\mathbb{Z}}_{2}}\left(\begin{matrix}\varphi_{+}(pP_{+},\iota)&0\\\
0&\varphi_{-}(pP_{-},\iota)\end{matrix}\right)\otimes\,\mathrm{d}E_{1}(p,\iota),$

where $\iota=\pm$.

As in [Tan11a], we construct the scattering matrix first on the unsymmetrized
Fock space, then restrict it to the antisymmetric space. For an operator $A$
on
${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1}\otimes{\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1}$,
we denote by $A^{m,n}_{i,j}$ on
$({\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1})^{\otimes
m}\otimes({\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1})^{\otimes n}$ the
operator which acts only on the $i$-th factor in
$({\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1})^{\otimes m}$ and $j$-th
factor in $({\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1})^{\otimes n}$ as
$A$. As a convention, $A^{m,n}_{i,j}$ equals to the identity operator if $m$
or $n$ is $0$. Let us denote simply
$\widetilde{\varphi}(p,\iota):=\left(\begin{matrix}\varphi_{+}(p,\iota)&0\\\
0&\varphi_{-}(p,\iota)\end{matrix}\right)$ and
$\widetilde{\varphi}(P_{1},\iota):=\left(\begin{matrix}\varphi_{+}(P_{+},\iota)&0\\\
0&\varphi_{-}(P_{-},\iota)\end{matrix}\right)$. From the observation above, it
is straightforward to see that

$(M_{\varphi})^{m,n}_{i,j}=\int\left({\mathbbm{1}}\otimes\cdots\otimes\underset{i\mbox{-th}}{\widetilde{\varphi}(p_{j}P_{1},\iota_{j})}\otimes\cdots\otimes{\mathbbm{1}}\right)\otimes\,\mathrm{d}E_{1}(p_{1},\iota_{1})\otimes\cdots\otimes\,\mathrm{d}E_{1}(p_{n},\iota_{n})$

(the case where $m$ or $n$ is $0$ is treated separately). Then we define, as
in [Tan11a],

$\displaystyle S_{\varphi}^{m,n}$ $\displaystyle:=$
$\displaystyle\prod_{i,j}(M_{\varphi})^{m,n}_{i,j}$ $\displaystyle
S_{\varphi}$ $\displaystyle:=$
$\displaystyle\bigoplus_{m,n}S_{\varphi}^{m,n}\,.$

Let ${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{\Sigma}$ be the unsymmetrized
Fock space based on ${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1}$. Note that
$S_{\varphi}$ is defined on
${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{\Sigma}\otimes{\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{\Sigma}$,
and it naturally restricts to
${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}\otimes{\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{\Sigma}$,
${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{\Sigma}\otimes{\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}$
and
${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}\otimes{\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}$.
This $S_{\varphi}$ will be interpreted as the scattering matrix. In order to
confirm this, we have to take the spectral decomposition of $S_{\varphi}$ only
with respect to the right or left component. Namely,

$\displaystyle S_{\varphi}$ $\displaystyle:=$
$\displaystyle\bigoplus_{m,n}\prod_{i,j}(M_{\varphi})^{m,n}_{i,j}$
$\displaystyle=$
$\displaystyle\bigoplus_{m,n}\prod_{i,j}\int\left({\mathbbm{1}}\otimes\cdots\otimes\underset{i\mbox{-th}}{\widetilde{\varphi}(p_{j}P_{1},\iota_{j})}\otimes\cdots\otimes{\mathbbm{1}}\right)\otimes\,\mathrm{d}E_{1}(p_{1},\iota_{1})\otimes\cdots\otimes\,\mathrm{d}E_{1}(p_{n},\iota_{n})$
$\displaystyle=$
$\displaystyle\bigoplus_{m,n}\int\prod_{i,j}\left({\mathbbm{1}}\otimes\cdots\otimes\underset{i\mbox{-th}}{\widetilde{\varphi}(p_{j}P_{1},\iota_{j})}\otimes\cdots\otimes{\mathbbm{1}}\right)\otimes\,\mathrm{d}E_{1}(p_{1},\iota_{1})\otimes\cdots\otimes\,\mathrm{d}E_{1}(p_{n},\iota_{n})$
$\displaystyle=$
$\displaystyle\bigoplus_{n}\int\bigoplus_{m}\prod_{j}\left(\widetilde{\varphi}(p_{j}P_{1},\iota_{j})\right)^{\otimes
m}\otimes\,\mathrm{d}E_{1}(p_{1},\iota_{1})\otimes\cdots\otimes\,\mathrm{d}E_{1}(p_{n},\iota_{n})$
$\displaystyle=$
$\displaystyle\bigoplus_{n}\int\prod_{j}\bigoplus_{m}\left(\widetilde{\varphi}(p_{j}P_{1},\iota_{j})\right)^{\otimes
m}\otimes\,\mathrm{d}E_{1}(p_{1},\iota_{1})\otimes\cdots\otimes\,\mathrm{d}E_{1}(p_{n},\iota_{n})$
$\displaystyle=$
$\displaystyle\bigoplus_{n}\int\prod_{j}\Lambda(\widetilde{\varphi}(p_{j}P_{1},\iota_{j}))\otimes\,\mathrm{d}E_{1}(p_{1},\iota_{1})\otimes\cdots\otimes\,\mathrm{d}E_{1}(p_{n},\iota_{n})\,,$

where the integral and the product commute in the third equality since the
spectral measure is disjoint for different values of $p$’s and $\iota$’s, and
the sum and the product commute in the fifth equality since the operators in
the integrand act on mutually disjoint spaces, namely on
$({\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1})^{\otimes
m}\otimes{\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{\Sigma}$ for different
$m$. In the final expression, all operators appearing in the integrand are the
second quantization operators, thus this formula naturally restricts to the
partially antisymmetrized space
${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}\otimes{\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{\Sigma}$.

Now we define

  * •

${\mathcal{M}}_{\varphi}:=\\{x\otimes{\mathbbm{1}},{\hbox{\rm
Ad\,}}S_{\varphi}({\mathbbm{1}}\otimes
y):x\in\mathrm{Fer}_{\mathbb{C}}({\mathbb{R}}_{-}),y\in\mathrm{Fer}_{\mathbb{C}}({\mathbb{R}}_{+})\\}^{\prime\prime}$,

  * •

$T(t,x):=T_{0}(\frac{t-x}{\sqrt{2}})\otimes T_{0}(\frac{t+x}{\sqrt{2}})$,

  * •

$\Omega:=\Omega_{0}\otimes\Omega_{0}$.

As the net $\mathrm{Fer}_{\mathbb{C}}$ is fermionic by nature, the
interpretation of the scattering theory of [Buc75] is not clear. Nevertheless,
we can show the following by an almost same proof as in [Tan11a, Lemma 5.2,
Theorem 5.3].

######  Lemma 5.1.

The triple $({\mathcal{M}}_{\varphi},T,\Omega)$ is a Borchers triple.

###### Proof.

To apply Proposition 2.2, it is immediate that $S_{\varphi}$ commutes with
translation since it is defined through the spectral measure as above. It
preserves ${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}\otimes\Omega_{0}$ and
$\Omega_{0}\otimes{\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}$ pointwise, since
these subspaces correspond to the case where $m$ or $n$ is $0$ in the above
decomposition and $S_{\varphi}$ acts as the identity operator by definition.
What remains to show is the commutation property.

As we saw above, the operator $S_{\varphi}$ can be written as

$S_{\varphi}=\bigoplus_{n}\int\prod_{j}\Lambda(\widetilde{\varphi}(p_{j}P_{1},\iota_{j}))\otimes\,\mathrm{d}E_{1}(p_{1},\iota_{1})\otimes\cdots\otimes\,\mathrm{d}E_{1}(p_{n},\iota_{n})\,.$

The point is that the operators which appear in the integrand implement Longo-
Witten endomorphisms as we saw above since $p_{j}\geq 0$ in the support of the
integration.

Let $x^{\prime}\in\mathrm{Fer}_{\mathbb{C}}({\mathbb{R}}_{+})$ and consider
$x^{\prime}\otimes{\mathbbm{1}}$ as an operator on
${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}\otimes{\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{\Sigma}$.
We have

${\hbox{\rm
Ad\,}}S_{\varphi}(x^{\prime}\otimes{\mathbbm{1}})=\bigoplus_{n}\int{\hbox{\rm
Ad\,}}\left(\prod_{j}\Lambda(\widetilde{\varphi}(p_{j}P_{1},\iota_{j}))\right)(x^{\prime})\otimes\,\mathrm{d}E_{1}(p_{1},\iota_{1})\otimes\cdots\otimes\,\mathrm{d}E_{1}(p_{n},\iota_{n})\,.$

Although this formula is not closed on
${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}\otimes{\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}$,
the left hand side obviously restricts there. One sees that the integrand
remains in $\mathrm{Fer}_{\mathbb{C}}({\mathbb{R}}_{+})$.

Recall the operator $Z_{0}$ which gives the graded locality of
$\mathrm{Fer}_{\mathbb{C}}$. One has to remind that
$Z_{0}=\frac{{\mathbbm{1}}-\mathrm{i}\Gamma_{0}}{1-\mathrm{i}}$ where
$\Gamma_{0}=\Lambda(-{\mathbbm{1}})$, hence $Z_{0}$ commutes with any second
quantization operator. Then by the disintegration above (and the corresponding
disintegration with respect to the left component), it is easy to see that
$Z_{0}\otimes{\mathbbm{1}}$ commutes with $S_{\varphi}$.

Let us check the commutation property of the assumptions in Proposition 2.2.
Note that ${\hbox{\rm Ad\,}}Z_{0}(x)\otimes{\mathbbm{1}}$ and ${\hbox{\rm
Ad\,}}Z_{0}(x)\in\mathrm{Fer}_{\mathbb{C}}({\mathbb{R}}_{+})^{\prime}$ for
$x\in\mathrm{Fer}_{\mathbb{C}}({\mathbb{R}}_{-})$. Since
$Z_{0}\otimes{\mathbbm{1}}$ and $S_{\varphi}$ commute as we saw above, to
prove the first commutation relation, it is enough to show that $[{\hbox{\rm
Ad\,}}Z_{0}(x)\otimes{\mathbbm{1}},{\hbox{\rm
Ad\,}}S_{\varphi}(x^{\prime}\otimes{\mathbbm{1}})]=0$ for
$x\in\mathrm{Fer}_{\mathbb{C}}({\mathbb{R}}_{-})$ and
$x^{\prime}\in\mathrm{Fer}_{\mathbb{C}}({\mathbb{R}}_{+})$. As operators
acting on
${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}\otimes{\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{\Sigma}$,
this is done by the above disintegration of ${\hbox{\rm
Ad\,}}S_{\varphi}(x^{\prime}\otimes{\mathbbm{1}})$. Then both operators
naturally restrict to
${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}\otimes{\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}$,
and we obtain the claim (cf. [Tan11a, Lemma 5.2, Theorem 5.3]). The second
commutation relation for Proposition 2.2 can be proven analogously. ∎

Finally we arrive at a new family of interacting Borchers triples with
asymptotic algebra ${{\mathcal{A}}^{(0)}}\otimes{{\mathcal{A}}^{(0)}}$.

######  Theorem 5.2.

Let us define

  * •

${\mathcal{N}}_{\varphi}:=\\{x\otimes{\mathbbm{1}},{\hbox{\rm
Ad\,}}S_{\varphi}({\mathbbm{1}}\otimes y):x\in\mathrm{Fer}_{\mathbb{C}}^{{\rm
U(1)}}({\mathbb{R}}_{-}),y\in\mathrm{Fer}_{\mathbb{C}}^{{\rm
U(1)}}({\mathbb{R}}_{+})\\}^{\prime\prime}$,

  * •

$T(t,x):=T_{0}(\frac{t-x}{\sqrt{2}})\otimes T_{0}(\frac{t+x}{\sqrt{2}})$,

  * •

$\Omega:=\Omega_{0}\otimes\Omega_{0}$.

Then the triple $({\mathcal{N}}_{\varphi},T,\Omega)$, restricted to
$\overline{{\mathcal{N}}_{\varphi}\Omega}$, is an asymptotically complete,
interacting Borchers triple with the asymptotic algebra
${{\mathcal{A}}^{(0)}}\otimes{{\mathcal{A}}^{(0)}}$ and scattering operator
$S_{\varphi}|_{\overline{{\mathcal{N}}_{\varphi}\Omega}}$. It also holds that
$\overline{{\mathcal{N}}_{\varphi}\Omega}=\overline{{{\mathcal{A}}^{(0)}}(I_{+})\otimes{{\mathcal{A}}^{(0)}}(I_{-})\Omega}$
for arbitrary intervals $I_{+},I_{-}$.

###### Proof.

Substantial arguments are already done: In Lemma 5.1 we constructed Borchers
triples with $\mathrm{Fer}_{\mathbb{C}}\otimes\mathrm{Fer}_{\mathbb{C}}$ as
the asymptotic algebra. We have seen in Section 3.3 the ${\rm U(1)}$-current
net ${{\mathcal{A}}^{(0)}}$ is the fixed point subnet of
$\mathrm{Fer}_{\mathbb{C}}$ with respect to the action of ${\rm U(1)}$. From
the construction in Section 5.1 and Theorem 4.3, it is easy to see that
$S_{\varphi}$ commutes with the product action of the inner symmetries. Then
all the statements of the Theorem follow from the general consideration of
Proposition 2.5. ∎

###  5.2 Action of the S-matrix on the 1+1 particle space

In this Section we want to analyze the action of the S-matrix of the models
constructed in Section 5.1 on the 1+1 particle space
${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}\otimes{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$,
i.e. one left and one right moving particle, where we use the word particle in
the sense of Fock space excitations. We note that on the $n$+0 and 0+$n$
particle spaces ${\mathcal{H}}_{n}\otimes{\mathbb{C}}\Omega_{0}$ and
${\mathbb{C}}\Omega_{0}\otimes{\mathcal{H}}_{n}$, respectively, the S-matrix
$S$ acts trivially. A typical vector in
${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}\otimes{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$
is of the form $\Psi:=J(f)\Omega_{0}\otimes J(g)\Omega_{0}$ which we express
as the function $\Psi(p,\bar{p})=\hat{f}(p)\hat{g}(\bar{p})$. The embedding
$\iota:L^{2}({\mathbb{R}}_{+},p\,\mathrm{d}p)\otimes
L^{2}({\mathbb{R}}_{+},\bar{p}\,\mathrm{d}\bar{p})\cong{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}\otimes{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}\hookrightarrow{\mathcal{H}}_{1,1}\otimes{\mathcal{H}}_{1,1}\subset{\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}\otimes{\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}$
is given by
$\iota(\Psi)_{1,1;1,1}(p,q,\bar{p},\bar{q})=\frac{1}{(2\pi)^{2}}\Psi(p+q,\bar{p}+\bar{q})$.
We have an analogue of Lemma 4.2

######  Proposition 5.3.

Let $\varphi$ be some inner function. The unitary $S_{\varphi}$ satisfies
$S_{\varphi}({\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}\otimes{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1})\subset{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}\otimes{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$
if and only if $\varphi(p)=\mathrm{e}^{\mathrm{i}(kp+\theta)}$.

###### Proof.

The action of $S_{\varphi}$ on
$\Psi\in{\mathcal{H}}_{1}\otimes{\mathcal{H}}_{1}$ is given by

$S_{\varphi}\Psi(p+q,\bar{p}+\bar{q})=\varphi(p\cdot\bar{p})\check{\varphi}(q\cdot\bar{p})\check{\varphi}(p\cdot\bar{q})\varphi(q\cdot\bar{q})\Psi(p+q,\bar{p}+\bar{q})$

which is again in
${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}\otimes{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$
if it can be written as a function $\tilde{\Psi}(p+q,\bar{p}+\bar{q})$, in
particular if
$\varphi(p\cdot\bar{p})\check{\varphi}(q\cdot\bar{p})\check{\varphi}(p\cdot\bar{q})\varphi(q\cdot\bar{q})=\widetilde{\varphi}(p+q,\bar{p}+\bar{q})$.
Setting $\bar{p}=1$ and $\bar{q}=0$, we have
$\varphi(p)\check{\varphi}(q)=\widetilde{\varphi}(p+q,1)$. The rest follows as
Lemma 4.2.

∎

######  Remark 5.4.

In the case $\varphi(p)=\mathrm{e}^{\mathrm{i}\kappa p}$, one gets the models
obtained in [DT11] using warped convolution.

######  Proposition 5.5.

Let $e$ be the projection on
${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}\otimes{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$,
then $eS_{\varphi}e=\tilde{\varphi}(P\otimes P)$, where $\tilde{\varphi}$ is
boundary value of an analytic function in $\mathbb{H}$ with
$|\tilde{\varphi}(p)|\leq 1$ and $P$ is the generator of translation
restricted to the one-particle space (which gives rise the irreducible
standard pair).

###### Proof.

It can be checked that

$(e_{0}f)(p,q)=\frac{1}{p+q}\int_{0}^{p+q}f(p+q-x,x)\,\mathrm{d}x$

is the projection on
${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}\subset{\mathcal{H}}_{1,1}$.
Then the action of $eS_{\varphi}$ on a
$f\in{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}\otimes{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$
can be calculated to be $\varphi^{\prime}(P\otimes 1,1\otimes P)$ with

$\varphi^{\prime}(p,q)=\frac{1}{p\cdot
q}\int_{0}^{p}\int_{0}^{q}\varphi((p-x)\cdot(q-y))\varphi(x\cdot
y)\check{\varphi}((p-x)\cdot
y)\check{\varphi}(x\cdot(q-y))\,\mathrm{d}y\,\mathrm{d}x$

and it is easy to check that with $\tilde{\varphi}(p):=\varphi^{\prime}(p,1)$
it holds $\varphi^{\prime}(p,q)=\tilde{\varphi}(p\cdot q)$ for all $p,q>0$.
That $|\tilde{\varphi}(p)|\leq 1$ can be checked directly or follows from the
fact that $S_{\varphi}$ is unitary. ∎

######  Remark 5.6.

It is a general feature of asymptotically complete Borchers triples with
asymptotic algebra ${{\mathcal{A}}^{(0)}}\otimes{{\mathcal{A}}^{(0)}}$ that
the restriction of the scattering matrix $S$ to $eSe$ is a functional calculus
of $P\otimes P$. Indeed, both $e$ and $S$ commute with the translation $T$,
but $T$ is maximally abelian when restricted to
${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}\otimes{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$,
hence there is a function $\varphi_{S}$ such that
$eSe=\varphi_{S}(P\otimes{\mathbbm{1}},{\mathbbm{1}}\otimes P)$. Furthermore,
both $e$ and $S$ commute with boosts, so does $\varphi_{S}$ and one obtains
the form $eSe=\varphi_{S}^{\prime}(P\otimes P)$.

We note that the proof above shows that $|\tilde{\varphi}(M^{2}/2)|$ is the
probability that an improper state in
${\mathcal{H}}^{1}_{{\mathcal{A}}^{(0)}}\otimes{\mathcal{H}}^{1}_{{\mathcal{A}}^{(0)}}$
with mass $M^{2}$ is scattered elastically in the sense of Fock space
particles, where

$\tilde{\varphi}(p)=\frac{1}{p}\int_{0}^{p}\int_{0}^{1}\varphi((p-x)(1-y))\varphi(xy)\check{\varphi}((p-x)y)\check{\varphi}(x(1-y))\,\mathrm{d}y\,\mathrm{d}x\,.$

As we discussed in 3.1, the Hilbert space of the ${\rm U(1)}$-current net, and
hence the tensor product of two copies of it, admit the bosonic Fock space
structure, hence we can consider the particle number. Although we admit that
this concept does not have an intrinsic meaning, we claim that it is possible
to interpret this as the number of massless particles.

An evidence comes from the comparison with massive cases. In [Lec08] Lechner
has constructed a family of massive interacting models parametrized by so-
called scattering functions, and later he reinterpreted them as deformations
of the massive free field [Lec11]. If one applies the same deformation
procedure to the derivative of the massless free field whose net is
${{\mathcal{A}}^{(0)}}\otimes{{\mathcal{A}}^{(0)}}$ (with scattering functions
satisfying $S_{2}(0)=1$), he obtains the Borchers triples with
${{\mathcal{A}}^{(0)}}\otimes{{\mathcal{A}}^{(0)}}$ as the asymptotic net
constructed in [Tan11a] 222Private communication with Gandalf Lechner and Jan
Schlemmer. This will be presented elsewhere.. Hence the models in [Tan11a]
should be considered as the massless versions of the models in [Lec08].
Likewise, it can be said that the models constructed in the present paper are
the deformed (in an appropriate sense) version of the massless free field.

In massive case, there is a mass gap in the spectrum of the spacetime
translation and the one-particle space of the Fock space has an intrinsic
meaning. In massless case, such an intrinsic interpretation is lost but there
is still the Fock space structure. Thus we think that, if the two-particle
space in the Fock structure is not preserved by the S-matrix, as in the case
where $\varphi$ is not exponential (see Proposition 5.3), then it represents
massless particle production.

##  6 Conclusion and outlook

In this paper we have constructed a new family of Longo-Witten endomorphisms
on ${{\mathcal{A}}^{(0)}}$ through the inclusion
${{\mathcal{A}}^{(0)}}=\mathrm{Fer}_{\mathbb{C}}^{\rm
U(1)}\subset\mathrm{Fer}_{\mathbb{C}}$. We combined them to construct
interacting wedge-local nets with
${{\mathcal{A}}^{(0)}}\otimes{{\mathcal{A}}^{(0)}}$ as the asymptotic algebra
and showed that their S-matrices do not preserve the $n$-particle space of the
bosonic Fock space. Particle production is a necessary feature of interacting
models in higher dimensions [Aks65], thus this result opens up some hope for
algebraic construction of higher dimensional interacting models.

However, there are at least two shortcomings with the present method. The
first is that we proved only wedge-locality of the models. As already shown in
[Tan11a], a wedge-local net can be dilation-covariant and at the same time
interacting. On the other hand, a strictly local dilation-covariant
(asymptotically complete) net is necessarily not interacting [Tan11b]. Hence,
interaction of wedge-local nets could be just a false-positive and strict
locality is desired. The second is the fact that the concept of particle in
massless case is not intrinsically defined. Although the Fock space structure
is easily understood, its interpretations should be treated with care.

These issues could be overcome by considering massive cases. As for strict
locality, it has been shown that the deformation of the massive free field by
a suitably regular function is again strictly local [Lec08, Lec11]. On the
other hand, in massless situation, even the simplest case $\varphi(p)=-1$
(where $\varphi$ is an inner symmetric function used in [Tan11a] to deform
directly ${{\mathcal{A}}^{(0)}}\otimes{{\mathcal{A}}^{(0)}}$) is already not
strictly local [Tan11a]. Hence we believe that strict locality should be
addressed in massive models. Furthermore, for a massive asymptotically
complete model, the notion of particle production is intrinsic. Fortunately,
it is known that the construction in [Tan11a] coincides with the deformation
of the massive free field as we remarked in the last section, hence a further
correspondence between massive and massless cases are expected. We hope to
investigate this problem in a future publication.

Of course, interacting models in higher dimensions are always one of the most
important issues. Although conformal nets themselves are not interacting
[BF77], some new constructions based on CFT could be possible and ideas from
the present article could be useful.

#### Acknowledgment.

We thank our supervisor Roberto Longo for his constant support and useful
suggestions. Y. T. thanks Gandalf Lechner and Jan Schlemmer for discussions
on the relation between the present construction and the deformation of
[Lec11].

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