# Infraparticles with superselected direction of motion in two-dimensional
conformal field theory

Wojciech Dybalski111Supported by the DFG grant SP181/25.
Zentrum Mathematik, Technische Universität München,
D-85747 Garching, Germany
E-mail<EMAIL_ADDRESS>Yoh Tanimoto222Supported in part by the ERC
Advanced Grant 227458 OACFT “Operator Algebras and Conformal Field Theory”.
Dipartimento di Matematica, Università di Roma “Tor Vergata”
Via della Ricerca Scientifica, 1 - I–00133 Roma, Italy.
E-mail<EMAIL_ADDRESS>
###### Abstract

Particle aspects of two-dimensional conformal field theories are investigated,
using methods from algebraic quantum field theory. The results include
asymptotic completeness in terms of (counterparts of) Wigner particles in any
vacuum representation and the existence of (counterparts of) infraparticles in
any charged irreducible product representation of a given chiral conformal
field theory. Moreover, an interesting interplay between the infraparticle’s
direction of motion and the superselection structure is demonstrated in a
large class of examples. This phenomenon resembles the electron’s momentum
superselection expected in quantum electrodynamics.

##  1 Introduction

Particle aspects and superselection structure of quantum electrodynamics are
plagued by the infrared problem, which has been a subject of study in
mathematical physics for more than four decades [46, 28, 29, 39, 21, 40, 18,
33, 19, 30, 6, 7, 8, 10, 16, 41, 42, 22, 23, 24, 48, 34, 44]. The origin of
this difficulty, inherited from classical electrodynamics, is the emission of
photons which accompanies any change of the electron’s momentum. It has two
important consequences which are closely related: Firstly, the electron is not
a particle in the sense of Wigner [50], but rather an _infraparticle_ [46]
i.e. it does not have a precise mass. Secondly, the electron’s plane wave
configurations of different momenta cannot be superposed into normalizable
wavepackets. In fact, such configurations have different spacelike asymptotic
flux of the electric field, which imposes a superselection rule [7]. The
evidence for this phenomenon of the electron’s _momentum superselection_ comes
from two sources: On the one hand, it appears in models of non-relativistic
QED in the representation structure of the asymptotic electromagnetic field
algebra [18]. On the other hand, it is suggested by structural results in the
general framework of algebraic quantum field theory [7, 8, 16, 41, 42].
However, no examples of local, relativistic theories, describing
infraparticles with superselected momentum, have been given to date. Thus the
logical consistency of this property with the basic postulates of quantum
field theory remains to be settled. As a step in this direction, we
demonstrate in the present paper that a simple variant of this phenomenon -
superselection of direction of motion - occurs in a large class of two-
dimensional conformal field theories.

Conformal field theory has been a subject of intensive research over the last
two decades, both from physical and mathematical viewpoints, motivated, in
particular, by the search for non-trivial quantum field theories. (See e.g.
[3] and references therein). It exhibits particularly interesting properties
in two dimensions, where the symmetry group is infinite dimensional. Since the
seminal work of Buchholz, Mack and Todorov [15] the superselection structure
of these theories has been investigated [31] and deep classification results
have been obtained [36, 37]. It has remained unnoticed, however, that two-
dimensional conformal field theories have also a rich and interesting particle
structure: The concepts of Wigner particles and infraparticles have their
natural counterparts in this setting and both types of excitations appear in
abundance: Any chiral conformal field theory in a vacuum representation has a
complete particle interpretation in terms of Wigner particles. Although such
theories are non-interacting, their (Grosse-Lechner) deformations [14] exhibit
non-trivial scattering and inherit the property of asymptotic completeness as
we show in a companion paper [25]. It is verified in the present work that any
charged irreducible product representation of a chiral conformal field theory
admits infraparticles. In a large class of examples these infraparticles have
superselected direction of motion i.e. their plane wave configurations with
opposite directions of momentum cannot be superposed. Thus subtle particle
phenomena, which are not under control in physical spacetime, can be
investigated in these two-dimensional models.

To keep our analysis general, we rely on the setting of algebraic QFT [32]. We
base our discussion on the concept of a local net of $C^{*}$-algebras on
$\mathbb{R}^{2}$, defined precisely in Subsection 2.1: To any open, bounded
region $\mathcal{O}\subset\mathbb{R}^{2}$ we attach a $C^{*}$-algebra
${\mathfrak{A}}(\mathcal{O})$, acting on a Hilbert space $\mathcal{H}$ of
physical states. This algebra is generated by observables which can be
measured with an experimental device localized in $\mathcal{O}$. It is
contained in the quasilocal algebra ${\mathfrak{A}}$, which is the inductive
limit of the net $\mathcal{O}\to{\mathfrak{A}}(\mathcal{O})$. Moreover, there
acts a unitary representation of translations $\mathbb{R}^{2}\ni x\to U(x)$ on
$\mathcal{H}$, whose adjoint action
$\alpha_{x}(\,\cdot\,)=U(x)\,\cdot\,U(x)^{*}$ shifts the observables in
spacetime. The infinitesimal generators of $U$ are interpreted as the
Hamiltonian $H$ and the momentum operator $\boldsymbol{P}$. Their joint
spectrum is contained in the closed forward lightcone $V_{+}$, to ensure the
positivity of energy. If there exists a cyclic, unit vector
$\Omega\in\mathcal{H}$ which is the unique (up to a phase) joint eigenvector
of $H$ and $\boldsymbol{P}$ with eigenvalue zero, then we say that the theory
is in a vacuum representation. If each of the subspaces
$\mathcal{H}_{\pm}=\ker(H\mp\boldsymbol{P})$ includes some vectors orthogonal
to $\Omega$, then we say that the theory contains Wigner particles. Since we
do not assume that these particles are described by vectors in some
irreducible representation space of the Poincaré group, the present definition
is less restrictive than the conventional one. However, it is better suited
for a description of the dispersionless kinematics of two-dimensional massless
excitations. In particular, it allows us to apply the natural scattering
theory, developed by Buchholz in [5], which we outline in Subsection 2.2
below. We recall that in [5] these excitations are called ‘waves’, to stress
their composite character.

Due to this compound structure of Wigner particles (or ‘waves’), asymptotic
completeness in a vacuum representation is not in conflict with the existence
of charged representations with a non-trivial particle content. However, in
charged representations of massless two-dimensional theories Wigner particles
may be absent, as noticed in [12]. In this case scattering theory from [5]
does not apply and an appropriate framework for the analysis of particle
aspects is the theory of _particle weights_ [10, 47, 16, 35, 41, 42, 23, 24],
developed by Buchholz, Porrmann and Stein, which we revisit in Subsections 2.3
and 2.4. This theory is based on the concept of the asymptotic functional,
given by

$\sigma_{\Psi}^{\mathrm{out}}(C)=\lim_{t\to\infty}\int
d\boldsymbol{x}\,(\Psi|\alpha_{(t,\boldsymbol{x})}(C)\Psi),$ (1.1)

for any vector $\Psi\in\mathcal{H}$ of bounded energy, and suitable
observables $C\in{\mathfrak{A}}$. (In general, some time averaging and
restriction to a subnet may be needed before taking the limit). We remark for
future reference that this functional induces a sesquilinear form
$\psi_{\Psi}^{\mathrm{out}}$ on a certain left ideal of ${\mathfrak{A}}$. We
show in Theorem 2.11 below, that asymptotic functionals are non-zero in
theories of Wigner particles. If non-trivial asymptotic functionals arise in
the absence of Wigner particles, then we say that the theory describes
infraparticles333The conventional definition of infraparticles requires that
both $\mathcal{H}_{+}$ and $\mathcal{H}_{-}$ contain at most multiples of the
vacuum vector. Our (less restrictive) definition imposes this requirement on
one of these subspaces only. Thus theories containing ‘waves’ running to the
right but no ‘waves’ running to the left (or vice versa) describe
infraparticles according to our terminology. Such nomenclature turns out to be
more convenient in the context of two-dimensional, massless theories.. Using
standard decomposition theory, the GNS representation $\pi$ induced by the
sesquilinear form $\psi_{\Psi}^{\mathrm{out}}$ can be decomposed into a direct
integral of irreducible representations

$\pi\simeq\int_{X}d\mu(\xi)\,\pi_{\xi},$ (1.2)

where $(X,d\mu)$ is a Borel space and $\simeq$ denotes unitary equivalence
[49, 42]. Results from [2, 47] suggest that the measurable field of
irreducible representations $\\{\pi_{\xi}\,\\}_{\xi\in X}$ carries information
about all the (infra-)particle types appearing in the theory. In particular,
there exists a field of vectors $\\{\,q_{\xi}\,\\}_{\xi\in X}$ which can be
interpreted as the energy and momentum of plane wave configurations
$\\{\psi_{\xi}\\}_{\xi\in X}$ of the respective (infra-)particles [42]. The
sesquilinear forms $\\{\psi_{\xi}\\}_{\xi\in X}$, called _pure particle
weights_ , induce the representations $\\{\pi_{\xi}\\}_{\xi\in X}$ and satisfy

$\psi_{\Psi}^{\mathrm{out}}=\int_{X}d\mu(\xi)\,\psi_{\xi}.$ (1.3)

The existence of such a decomposition was shown, under certain technical
restrictions, in [41, 42].

The theory of particle weights is sufficiently general to accommodate the
phenomenon of the infraparticle’s momentum superselection, discussed above: In
this case $q_{\xi}\neq q_{\xi^{\prime}}$ should imply that $\pi_{\xi}$ is not
unitarily equivalent to $\pi_{\xi^{\prime}}$ for almost all labels $\xi$,
$\xi^{\prime}$ corresponding to the infraparticle in question. Superselection
of direction of motion is a milder property: It only requires that plane waves
$\psi_{\xi},\psi_{\xi^{\prime}}$, travelling in opposite directions, give rise
to representations $\pi_{\xi},\pi_{\xi^{\prime}}$ which are not unitarily
equivalent. This latter interplay between the infraparticle’s kinematics and
the superselection structure occurs in some two-dimensional conformal field
theories, as we explain below. We state this property precisely in Definitions
2.7 and 2.12, where we restrict attention to representations $\pi$ of (Murray-
von Neumann) type I with atomic center. This is sufficient for our purposes
and allows us to separate our central concept from ambiguities involved in the
general decompositions (1.2), (1.3).

Our discussion of conformal field theory relies on the notion of a local net
of von Neumann algebras on $\mathbb{R}$ which we introduce in Subsection 3.1.
(Such nets arise e.g. by restricting the familiar Möbius covariant nets on the
circle to the real line). With any open bounded region ${\cal
I}\subset\mathbb{R}$ we associate a von Neumann algebra ${\cal A}({\cal I})$,
acting on a Hilbert space $\mathcal{K}$, and denote the quasilocal algebra of
this net by ${\cal A}$. Moreover, the Hilbert space $\mathcal{K}$ carries a
unitary representation of translations $\mathbb{R}\ni s\to V(s)$, whose
spectrum coincides with $\mathbb{R}_{+}$. If there exists a cyclic, unit
vector $\Omega_{0}\in\mathcal{K}$, which is the unique (up to a phase) non-
zero vector invariant under the action of $V$, then we say that the theory is
in a vacuum representation. Given such a net, covariant under the action of
some internal symmetry group, one can proceed to the fixed-point subnet which
has a non-trivial superselection structure. In the simple case, considered in
Subsection 3.4, the action of $\mathbb{Z}_{2}$ is implemented by a unitary
$W\neq I$ on $\mathcal{K}$ s.t. $W^{2}=I$. The fixed-point subnet ${\cal
A}_{\mathrm{ev}}$ consists of all the elements of ${\cal A}$, which commute
with $W$. The subspace $\mathcal{K}_{\mathrm{ev}}=\ker(W-I)$ (resp.
$\mathcal{K}_{\mathrm{odd}}=\ker(W+I)$) is invariant under the action of
${\cal A}_{\mathrm{ev}}$ and gives rise to a vacuum representation (resp. a
charged representation) of the fixed-point theory.

Given two nets of von Neumann algebras on the real line, ${\cal A}_{1}$ and
${\cal A}_{2}$, acting on Hilbert spaces $\mathcal{K}_{1}$ and
$\mathcal{K}_{2}$, one obtains the two-dimensional chiral net
${\mathfrak{A}}$, acting on
$\mathcal{H}=\mathcal{K}_{1}\otimes\mathcal{K}_{2}$, by the standard
construction, recalled in Subsection 3.1: The two real lines are identified
with the lightlines in $\mathbb{R}^{2}$ and for any double cone
$\mathcal{O}={\cal I}\times\mathfrak{J}$ one sets444In the main part of the
paper ${\mathfrak{A}}({\cal I}\times\mathfrak{J})$ denotes a suitable weakly
dense ‘regular subalgebra’ of ${\cal A}_{1}({\cal I})\otimes{\cal
A}_{2}(\mathfrak{J})$. This distinction is not essential for the present
introductory discussion. ${\mathfrak{A}}({\cal I}\times\mathfrak{J})={\cal
A}_{1}({\cal I})\otimes{\cal A}_{2}(\mathfrak{J})$. If the nets ${\cal
A}_{1}$, ${\cal A}_{2}$ are in vacuum representations, with the vacuum vectors
$\Omega_{1}\in\mathcal{K}_{1}$, $\Omega_{2}\in\mathcal{K}_{2}$, then
${\mathfrak{A}}$ is also in a vacuum representation, with the vacuum vector
$\Omega=\Omega_{1}\otimes\Omega_{2}$. In spite of their simple tensor product
structure, chiral nets play a prominent role in conformal field theory. In
fact, with any local conformal net on $\mathbb{R}^{2}$ one can associate a
chiral subnet by restricting the theory to the lightlines. In the important
case of central charge $c<1$ these subnets were instrumental for the
classification results, mentioned above, which clarified the superselection
structure of a large class of models [43, 36]. As we show in the present work,
chiral nets also offer a promising starting point for the analysis of particle
aspects of conformal field theories: Any chiral net in a vacuum representation
is an asymptotically complete theory of Wigner particles. Moreover, any
charged irreducible product representation of such a net contains
infraparticles. With this information at hand, we exhibit examples of
infraparticles with superselected direction of motion. This construction is
summarized briefly in the remaining part of this Introduction.

Let us consider two fixed-point nets ${\cal A}_{1,\mathrm{ev}}$, ${\cal
A}_{2,\mathrm{ev}}$, obtained from ${\cal A}_{1}$ and ${\cal A}_{2}$ with the
help of the unitaries $W_{1}$ and $W_{2}$, implementing the respective actions
of $\mathbb{Z}_{2}$. The resulting chiral net ${\mathfrak{A}}_{\mathrm{ev}}$
acts on the Hilbert space $\mathcal{H}=\mathcal{K}_{1}\otimes\mathcal{K}_{2}$,
which decomposes into four invariant subspaces with different particle
structure:

$\mathcal{H}=(\mathcal{K}_{1,\mathrm{ev}}\otimes\mathcal{K}_{2,\mathrm{ev}})\oplus(\mathcal{K}_{1,\mathrm{odd}}\otimes\mathcal{K}_{2,\mathrm{ev}})\oplus(\mathcal{K}_{1,\mathrm{ev}}\otimes\mathcal{K}_{2,\mathrm{odd}})\oplus(\mathcal{K}_{1,\mathrm{odd}}\otimes\mathcal{K}_{2,\mathrm{odd}}).$
(1.4)

${\mathfrak{A}}_{\mathrm{ev}}$ restricted to
$\mathcal{H}_{0}:=\mathcal{K}_{1,\mathrm{ev}}\otimes\mathcal{K}_{2,\mathrm{ev}}$
is a chiral theory in a vacuum representation. Thus it is an asymptotically
complete theory of Wigner particles, by the result mentioned above.
$\mathcal{H}_{{\rm{R}}}:=\mathcal{K}_{1,\mathrm{odd}}\otimes\mathcal{K}_{2,\mathrm{ev}}$
contains ‘waves’ travelling to the right, but no ‘waves’ travelling to the
left. In
$\mathcal{H}_{{\rm{L}}}:=\mathcal{K}_{1,\mathrm{ev}}\otimes\mathcal{K}_{2,\mathrm{odd}}$
the opposite situation occurs. Thus ${\mathfrak{A}}_{\mathrm{ev}}$ restricted
to $\mathcal{H}_{{\rm{R}}}$ or $\mathcal{H}_{{\rm{L}}}$ describes
infraparticles, according to our terminology. Finally,
${\mathfrak{A}}_{\mathrm{ev}}$ restricted to
$\hat{\mathcal{H}}:=\mathcal{K}_{1,\mathrm{odd}}\otimes\mathcal{K}_{2,\mathrm{odd}}$
is a theory of infraparticles which does not contain ‘waves’. In Theorem 3.10
below, which is our main result, we establish superselection of direction of
motion for infraparticles described by the net
$\hat{\mathfrak{A}}={\mathfrak{A}}_{\mathrm{ev}}|_{\hat{\mathcal{H}}}$. The
argument proceeds as follows: ${\mathfrak{A}}_{\mathrm{ev}}$ is contained in
${\mathfrak{A}}$, which is an asymptotically complete theory of Wigner
particles. Thus we can use the scattering theory from [5] to compute the
asymptotic functionals (1.1) and obtain the decompositions (1.2) of their GNS
representations. Interpreted as a state on ${\mathfrak{A}}$, any vector
$\Psi_{1}\otimes\Psi_{2}\in\hat{\mathcal{H}}$ consists of two ‘waves’ at
asymptotic times: $\Psi_{1}\otimes\Omega_{2}$ travelling to the right and
$\Omega_{1}\otimes\Psi_{2}$ travelling to the left. (Cf. Theorem 3.3 below).
However, these two vectors belong to different invariant subspaces of
${\mathfrak{A}}_{\mathrm{ev}}$, namely to $\mathcal{H}_{{\rm{R}}}$ and
$\mathcal{H}_{{\rm{L}}}$. The corresponding representations of
$\hat{\mathfrak{A}}$ are not unitarily equivalent, since they have different
structure of the energy-momentum spectrum.

Our paper is organized as follows: Section 2, which does not rely on conformal
symmetry, concerns two-dimensional, massless quantum field theories and their
particle aspects: Preliminary Subsection 2.1 introduces the main concepts. In
Subsection 2.2 we recall the scattering theory of two-dimensional, massless
Wigner particles developed in [5]. Subsection 2.3 gives a brief exposition of
the theory of particle weights and introduces our main concept: superselection
of direction of motion. Subsection 2.4 presents our main technical result,
stated in Theorem 2.11, which clarifies the structure of asymptotic
functionals in theories of Wigner particles. Its proof is given in Appendix A.
In Section 3 we apply the concepts and tools presented in Section 2 to chiral
conformal field theories. Our setting, which is slightly more general than the
usual framework of conformal field theory, is presented in Subsection 3.1. In
Subsection 3.2 we show that any chiral theory in a vacuum representation has a
complete particle interpretation in terms of Wigner particles. In Subsection
3.3 we demonstrate that charged irreducible product representations of any
chiral theory describe infraparticles. Subsection 3.4 presents our main
result, that is superselection of the infraparticle’s direction of motion in
chiral theories arising from fixed-point nets of $\mathbb{Z}_{2}$ actions.
Proofs of some auxiliary lemmas are postponed to Appendix B. In Section 4 we
summarize our work and discuss future directions.

##  2 Particle aspects of two-dimensional massless theories

###  2.1 Preliminaries

In this section, which does not rely on conformal symmetry, we present some
general results on particle aspects of massless quantum field theories in two-
dimensional spacetime. We rely on the following variant of the Haag-Kastler
axioms [32]:

######  Definition 2.1.

A local net of $C^{*}$-algebras on $\mathbb{R}^{2}$ is a pair
$({\mathfrak{A}},U)$ consisting of a map
$\mathcal{O}\to{\mathfrak{A}}(\mathcal{O})$ from the family of open, bounded
regions of $\mathbb{R}^{2}$ to the family of $C^{*}$-algebras on a Hilbert
space $\mathcal{H}$, and a strongly continuous unitary representation of
translations $\mathbb{R}^{2}\ni x\to U(x)$ acting on $\mathcal{H}$, which are
subject to the following conditions:

  1. 1.

(isotony) If $\mathcal{O}_{1}\subset\mathcal{O}_{2}$, then
${\mathfrak{A}}(\mathcal{O}_{1})\subset{\mathfrak{A}}(\mathcal{O}_{2})$.

  2. 2.

(locality) If $\mathcal{O}_{1}\perp\mathcal{O}_{2}$, then
$[{\mathfrak{A}}(\mathcal{O}_{1}),{\mathfrak{A}}(\mathcal{O}_{2})]=0$, where
$\perp$ denotes spacelike separation.

  3. 3.

(covariance)
$U(x){\mathfrak{A}}(\mathcal{O})U(x)^{*}={\mathfrak{A}}(\mathcal{O}+x)$ for
any $x\in\mathbb{R}^{2}$.

  4. 4.

(positivity of energy) The spectrum of $U$ is contained in the closed forward
lightcone
$V_{+}:=\\{\,(\omega,\boldsymbol{p})\in\mathbb{R}^{2}\,|\,\omega\geq|\boldsymbol{p}|\,\\}$.

  5. 5.

(regularity) The group of translation automorphisms
$\alpha_{x}(\,\cdot\,)=U(x)\,\cdot\,U(x)^{*}$ satisfies $\lim_{x\to
0}\|\alpha_{x}(A)-A\|=0$ for any $A\in{\mathfrak{A}}$.

We also introduce the quasilocal $C^{*}$-algebra of this net
${\mathfrak{A}}=\overline{\bigcup_{\mathcal{O}\subset\mathbb{R}^{2}}{\mathfrak{A}}(\mathcal{O})}$.

For any given net $({\mathfrak{A}},U)$ there exists exactly one unitary
representation of translations $U^{\mathrm{can}}$ s.t. $U^{\mathrm{can}}$
implements $\alpha$, all the operators $U^{\mathrm{can}}(x)$,
$x\in\mathbb{R}^{2}$ are contained in ${\mathfrak{A}}^{\prime\prime}$, the
spectrum of $U^{\mathrm{can}}$ is contained in $V_{+}$ and has Lorentz
invariant lower boundary upon restriction to any subspace of $\mathcal{H}$
invariant under the action of ${\mathfrak{A}}^{\prime\prime}$ [4]. We assume
that this _canonical_ representation of translations has been selected above,
i.e. $U=U^{\mathrm{can}}$. We denote by $(H,\boldsymbol{P})$ the corresponding
energy-momentum operators i.e. $U(x)=e^{iHt-i\boldsymbol{P}\boldsymbol{x}}$,
$x=(t,\boldsymbol{x})$. As we are interested in scattering of massless
particles, we introduce the single-particle subspaces ${\cal
H}_{\pm}:=\ker(H\mp\boldsymbol{P})$ and denote the corresponding projections
by $P_{\pm}$. The intersection $\mathcal{H}_{+}\cap\mathcal{H}_{-}$ contains
only translationally invariant vectors. If
$\mathcal{H}_{+}\neq\mathcal{H}_{+}\cap\mathcal{H}_{-}$ and
$\mathcal{H}_{-}\neq\mathcal{H}_{+}\cap\mathcal{H}_{-}$ then we say that the
theory describes Wigner particles. If $U$ has a unique (up to a phase)
invariant unit vector $\Omega\in\mathcal{H}$ and $\Omega$ is cyclic under the
action of ${\mathfrak{A}}$ then we say that the net $({\mathfrak{A}},U)$ is in
a vacuum representation. In this case ${\mathfrak{A}}$ acts irreducibly on
$\mathcal{H}$. (Cf. Theorem 4.6 of [1]). Scattering theory for Wigner
particles in a vacuum representation, developed in [5], will be recalled in
Subsection 2.2.

In the absence of Wigner particles we will apply the theory of particle
weights [10, 16, 41, 42], outlined in Subsection 2.3, to extract the
(infra-)particle content of a given theory. In this context it is necessary to
consider various representations of the net $({\mathfrak{A}},U)$. A
representation of the net $({\mathfrak{A}},U)$ is, by definition, a family of
representations $\\{\pi_{\cal O}\\}$ of local algebras which are consistent in
the sense that if ${\cal O}_{1}\subset{\cal O}_{2}$ then it holds that
$\pi_{{\cal O}_{2}}|_{{\mathfrak{A}}({\cal O}_{1})}=\pi_{{\cal O}_{1}}$. Since
the family of open bounded regions in ${\mathbb{R}}^{2}$ is directed, this
representation uniquely extends to a representation $\pi$ of the quasilocal
$C^{*}$-algebra ${\mathfrak{A}}$. Conversely, a representation of
${\mathfrak{A}}$ induces a consistent family of representations of local
algebras. In the following $\pi$ may refer to a representation of
${\mathfrak{A}}$ or a family of representations. We say that a representation
$\pi:{\mathfrak{A}}\to B(\mathcal{H}_{\pi})$ is covariant, if there exists a
strongly continuous group of unitaries $U_{\pi}$ on $\mathcal{H}_{\pi}$, s.t.

$\pi(\alpha_{x}(A))=U_{\pi}(x)\pi(A)U_{\pi}(x)^{*},\quad
A\in{\mathfrak{A}},\,\,x\in\mathbb{R}^{2}.$ (2.1)

Moreover, we say that this representation has positive energy, if the joint
spectrum of the generators of $U_{\pi}$ is contained in $V_{+}+q$ for some
$q\in\mathbb{R}^{2}$. We denote the corresponding canonical representation of
translations by $U_{\pi}^{\mathrm{can}}$ and note that
$(\pi({\mathfrak{A}}),U^{\mathrm{can}}_{\pi})$ is again a local net of
$C^{*}$-algebras in the sense of Definition 2.1. We say that the net
$(\pi({\mathfrak{A}}),U^{\mathrm{can}}_{\pi})$ is in a charged irreducible
representation, if $\pi({\mathfrak{A}})$ acts irreducibly on a non-trivial
Hilbert space $\mathcal{H}_{\pi}$ which does not contain non-zero invariant
vectors of $U^{\mathrm{can}}_{\pi}$.

We call two representations $(\pi_{1},\mathcal{H}_{\pi_{1}})$ and
$(\pi_{2},\mathcal{H}_{\pi_{2}})$ of $({\mathfrak{A}},U)$ unitarily
equivalent, (in short
$(\pi_{1},\mathcal{H}_{\pi_{1}})\simeq(\pi_{2},\mathcal{H}_{\pi_{2}})$), if
there exists a unitary $W:\mathcal{H}_{\pi_{1}}\to\mathcal{H}_{\pi_{2}}$ s.t.

$\displaystyle W\pi_{1}(A)=\pi_{2}(A)W,\quad\phantom{444}A\in{\mathfrak{A}}.$
(2.2)

If $\pi_{1}$ is a covariant, positive energy representation then so is
$\pi_{2}$ and it is easy to see that

$WU_{\pi_{1}}^{\mathrm{can}}(x)=U_{\pi_{2}}^{\mathrm{can}}(x)W,\quad
x\in\mathbb{R}^{2}.$ (2.3)

######  Remark 2.2.

We note that our (non-standard) Definition 2.1 of the local net neither
imposes the Poincaré covariance nor the existence of the vacuum vector. Thus
it applies both to vacuum representations and charged representations, which
facilitates our discussion. Apart from the physically motivated assumptions,
we adopt the regularity property 5, which can always be assured at the cost of
proceeding to a weakly dense subnet. This property seems indispensable in the
general theory of particle weights [41] e.g. in the proof of Proposition 2.10
stated below. For consistency of the presentation, we proceed to regular
subnets also in our discussion of conformal field theories in Section 3. We
stress, however, that this property is not needed there at the technical
level.

###  2.2 Scattering states

Scattering theory for Wigner particles in a vacuum representation of a two-
dimensional massless theory $({\mathfrak{A}},U)$ was developed in [5]. For the
reader’s convenience we recall here the main steps of this construction.
Following [5], for any $F\in{\mathfrak{A}}$ and $T\geq 1$ we introduce the
approximants:

$\displaystyle F_{\pm}(h_{T})=\int h_{T}(t)F(t,\pm t)dt,$ (2.4)

where $F(x):=\alpha_{x}(F)$,
$h_{T}(t)=|T|^{-\varepsilon}h(|T|^{-\varepsilon}(t-T))$, $0<\varepsilon<1$ and
$h\in C_{0}^{\infty}(\mathbb{R})$ is a non-negative function s.t. $\int
dt\,h(t)=1$. By applying the mean ergodic theorem, one obtains

$\lim_{T\to\infty}F_{\pm}(h_{T})\Omega=P_{\pm}F\Omega.$ (2.5)

Moreover, for $F\in{\mathfrak{A}}(\mathcal{O})$ and sufficiently large $T$ the
operator $F_{+}(h_{T})$ (resp. $F_{-}(h_{T})$) commutes with any observable
localized in the left (resp. right) component of the spacelike complement of
$\mathcal{O}$. Exploiting these two facts, the following result was
established in [5]:

######  Proposition 2.3 ([5]).

Let $F,G\in{\mathfrak{A}}$. Then the limits

$\displaystyle\Phi_{\pm}^{\mathrm{out}}(F):=\underset{T\to\infty}{\mathrm{s}\textrm{-}\lim}\;F_{\pm}(h_{T})\quad$
(2.6)

exist and are called the (outgoing) asymptotic fields. They depend only on the
respective vectors $\Phi_{\pm}^{\mathrm{out}}(F)\Omega=P_{\pm}F\Omega$ and
satisfy:

  1. (a)

$\Phi_{+}^{\mathrm{out}}(F)\mathcal{H}_{+}\subset\mathcal{H}_{+},\quad\Phi_{-}^{\mathrm{out}}(G)\mathcal{H}_{-}\subset\mathcal{H}_{-}$.

  2. (b)

$\alpha_{x}(\Phi_{+}^{\mathrm{out}}(F))=\Phi_{+}^{\mathrm{out}}(\alpha_{x}(F)),\quad\alpha_{x}(\Phi_{-}^{\mathrm{out}}(G))=\Phi_{-}^{\mathrm{out}}(\alpha_{x}(G))$
for $x\in{\mathbb{R}}^{2}$.

  3. (c)

$[\Phi_{+}^{\mathrm{out}}(F),\Phi_{-}^{\mathrm{out}}(G)]=0$.

The incoming asymptotic fields $\Phi_{\pm}^{\mathrm{in}}(F)$ are constructed
analogously, by taking the limit $T\to-\infty$.

With the help of the asymptotic fields one defines the scattering states as
follows: Since ${\mathfrak{A}}$ acts irreducibly on $\mathcal{H}$, for any
$\Psi_{\pm}\in\mathcal{H}_{\pm}$ we can find $F_{\pm}\in{\mathfrak{A}}$ s.t.
$\Psi_{\pm}=F_{\pm}\Omega$ [45]. The vectors

$\displaystyle\Psi_{+}\overset{\mathrm{out}}{\times}\Psi_{-}=\Phi_{+}^{\mathrm{out}}(F_{+})\Phi_{-}^{\mathrm{out}}(F_{-})\Omega$
(2.7)

are called the (outgoing) scattering states. By Proposition 2.3 they do not
depend on the choice of $F_{\pm}$ within the above restrictions. The incoming
scattering states $\Psi_{+}\overset{\mathrm{in}}{\times}\Psi_{-}$ are defined
analogously. The physical interpretation of these vectors, as two independent
excitations travelling in opposite directions at asymptotic times, relies on
the following proposition from [5]:

######  Proposition 2.4 ([5]).

Let $\Psi_{\pm},\Psi_{\pm}^{\prime}\in\mathcal{H}_{\pm}$. Then:

  1. (a)

$(\Psi_{+}\overset{\mathrm{out}}{\times}\Psi_{-}|\Psi^{\prime}_{+}\overset{\mathrm{out}}{\times}\Psi^{\prime}_{-})=(\Psi_{+}|\Psi^{\prime}_{+})(\Psi_{-}|\Psi^{\prime}_{-})$,

  2. (b)

$U(x)(\Psi_{+}\overset{\mathrm{out}}{\times}\Psi_{-})=(U(x)\Psi_{+})\overset{\mathrm{out}}{\times}(U(x)\Psi_{-})$,
for $x\in\mathbb{R}^{2}$.

Analogous relations hold for the incoming scattering states.

Following [5], we define the subspaces spanned by the respective scattering
states:

$\mathcal{H}^{\mathrm{in}}=\mathcal{H}_{+}\overset{\mathrm{in}}{\times}\mathcal{H}_{-}\,\,\textrm{
and
}\,\,\mathcal{H}^{\mathrm{out}}=\mathcal{H}_{+}\overset{\mathrm{out}}{\times}\mathcal{H}_{-}.$
(2.8)

Next, we introduce the wave operators
$\Omega^{\mathrm{out}}:\mathcal{H}_{+}\otimes\mathcal{H}_{-}\to\mathcal{H}^{\mathrm{out}}$
and
$\Omega^{\mathrm{in}}:\mathcal{H}_{+}\otimes\mathcal{H}_{-}\to\mathcal{H}^{\mathrm{in}}$,
extending by linearity the relations

$\Omega^{\mathrm{out}}(\Psi_{+}\otimes\Psi_{-})=\Psi_{+}\overset{\mathrm{out}}{\times}\Psi_{-}\,\,\textrm{
and
}\,\,\Omega^{\mathrm{in}}(\Psi_{+}\otimes\Psi_{-})=\Psi_{+}\overset{\mathrm{in}}{\times}\Psi_{-}.$
(2.9)

These operators are isometric in view of Proposition 2.4 (a). The scattering
operator $S:\mathcal{H}^{\mathrm{out}}\to\mathcal{H}^{\mathrm{in}}$, given by

$S=\Omega^{\mathrm{in}}(\Omega^{\mathrm{out}})^{*},$ (2.10)

is also an isometry. Now we are ready to introduce two important concepts:

######  Definition 2.5.

(a) If $S=I$ on $\mathcal{H}^{\mathrm{out}}$, then we say that the theory is
non-interacting.
(b) If $\mathcal{H}^{\mathrm{in}}=\mathcal{H}^{\mathrm{out}}=\mathcal{H}$ then
we say that the theory is asymptotically complete (in terms of ‘waves’).

We show in Theorem 3.3 below that any chiral conformal field theory in a
vacuum representation is both non-interacting and asymptotically complete. (We
demonstrated these facts already in [25] in a different context).

To conclude this subsection, we introduce some other useful concepts which are
needed in Theorem 2.11 below: Let us choose some closed subspaces ${\cal
K}_{\pm}\subset\mathcal{H}_{\pm}$, invariant under the action of $U$, and
denote by ${\cal K}_{+}\overset{\mathrm{out}}{\times}{\cal K}_{-}$ the linear
span of the respective scattering states. For any $\Psi\in{\cal
K}_{+}\overset{\mathrm{out}}{\times}{\cal
K}_{-}\subset\mathcal{H}^{\mathrm{out}}$ we introduce the positive functionals
$\rho_{\pm,\Psi}$, given by the relations

$\displaystyle\rho_{+,\Psi}(A)$ $\displaystyle=$
$\displaystyle((\Omega^{\mathrm{out}})^{-1}\Psi|(A\otimes
I)(\Omega^{\mathrm{out}})^{-1}\Psi),$ (2.11) $\displaystyle\rho_{-,\Psi}(A)$
$\displaystyle=$ $\displaystyle((\Omega^{\mathrm{out}})^{-1}\Psi|(I\otimes
A)(\Omega^{\mathrm{out}})^{-1}\Psi),$ (2.12)

where $A\in B(\mathcal{H})$ and the embedding ${\cal K}_{+}\otimes{\cal
K}_{-}\subset\mathcal{H}\otimes\mathcal{H}$ is understood. These functionals
can be expressed as follows

$\rho_{\pm,\Psi}(\,\cdot\,)=\sum_{n\in\mathbb{N}}(\Psi_{\pm,n}|\,\cdot\,\Psi_{\pm,n}),$
(2.13)

where $\Psi_{\pm,n}\in{\cal K}_{\pm}$ and
$\sum_{n\in\mathbb{N}}\|\Psi_{\pm,n}\|^{2}=\|\Psi\|^{2}$. It follows easily
from Lemma A.2, that for $\Psi\in P_{E}({\cal
K}_{+}\overset{\mathrm{out}}{\times}{\cal K}_{-})$, where $P_{E}$ is the
spectral projection on vectors of energy not larger than $E$, one can choose
$\Psi_{\pm,n}\in P_{E}{\cal K}_{\pm}$. We note that for $\|\Psi\|=1$ the
functionals $\rho_{\pm,\Psi}$ are just the familiar reduced density matrices.

###  2.3 Particle weights

Similarly as in the previous subsection we consider a local net of
$C^{*}$-algebras $({\mathfrak{A}},U)$ acting on a Hilbert space $\mathcal{H}$.
However, we do not assume that $\mathcal{H}$ contains the vacuum vector or
non-trivial single-particle subspaces $\mathcal{H}_{\pm}$. To study particle
aspects in this general situation we use the theory of particle weights [10,
16, 41, 42] which we recall in this and the next subsection. With the help of
this theory we formulate in Definitions 2.7 and 2.12 below the central notion
of this paper: superselection of direction of motion.

First, we recall two useful concepts: almost locality and the energy
decreasing property. An observable $B\in{\mathfrak{A}}$ is called almost
local, if there exists a net of operators
$\\{\,B_{r}\in{\mathfrak{A}}(\mathcal{O}_{r})\,|\,r>0\,\\}$, s.t. for any
$k\in\mathbb{N}_{0}$

$\lim_{r\to\infty}r^{k}\|B-B_{r}\|=0,$ (2.14)

where
$\mathcal{O}_{r}=\\{(t,\boldsymbol{x})\in\mathbb{R}^{2}\,|\,|t|+|\boldsymbol{x}|<r\,\\}$.
We say that an operator $B\in{\mathfrak{A}}$ is energy decreasing, if its
energy-momentum transfer is a compact set which does not intersect with the
closed forward lightcone $V_{+}$. We recall that the energy-momentum transfer
(or the Arveson spectrum w.r.t. $\alpha$) of an observable
$B\in{\mathfrak{A}}$ is the closure of the union of supports of the
distributions

$(\Psi_{1}|\tilde{B}(p)\Psi_{2})=(2\pi)^{-1}\int
d^{2}x\,e^{-ipx}(\Psi_{1}|B(x)\Psi_{2})$ (2.15)

over all $\Psi_{1},\Psi_{2}\in\mathcal{H}$, where $p=(\omega,\boldsymbol{p})$,
$x=(t,\boldsymbol{x})$ and $px=\omega t-\boldsymbol{p}\boldsymbol{x}$.

Following [16, 41], we introduce the subspace
$\mathcal{L}_{0}\subset{\mathfrak{A}}$, spanned by operators which are both
almost local and energy decreasing, and the corresponding left ideal in
${\mathfrak{A}}$:

$\mathcal{L}:=\\{\,AB\,|\,A\in{\mathfrak{A}},B\in\mathcal{L}_{0}\,\\}.$ (2.16)

Particle weights form a specific class of sesquilinear forms on $\mathcal{L}$:

######  Definition 2.6.

A particle weight is a non-zero, positive sesquilinear form $\psi$ on the left
ideal $\mathcal{L}$, satisfying the following conditions:

  1. 1.

For any $L_{1},L_{2}\in\mathcal{L}$ and $A\in{\mathfrak{A}}$ the relation
$\psi(AL_{1},L_{2})=\psi(L_{1},A^{*}L_{2})$ holds.

  2. 2.

For any $L_{1},L_{2}\in\mathcal{L}$ and $x\in\mathbb{R}^{2}$ the relation
$\psi(\alpha_{x}(L_{1}),\alpha_{x}(L_{2}))=\psi(L_{1},L_{2})$ holds.

  3. 3.

For any $L_{1},L_{2}\in\mathcal{L}$ the map $\mathbb{R}^{2}\ni
x\to\psi(L_{1},\alpha_{x}(L_{2}))$ is continuous. Its Fourier transform is
supported in a shifted lightcone $V_{+}-q$, where $q\in V_{+}$ does not depend
on $L_{1},L_{2}$.

Let us now summarize the pertinent properties of particle weights established
in [41] (in a slightly different framework). As a consequence of Theorem 2.9,
stated below, particle weights satisfy the following clustering property [41]

$\int d\boldsymbol{x}\,|\psi(L_{1},\alpha_{\boldsymbol{x}}(L_{2}))|<\infty,$
(2.17)

valid for $L_{1}=B_{1}^{*}A_{1}B_{1}^{\prime}$,
$L_{2}=B_{2}^{*}A_{2}B_{2}^{\prime}$, where
$B_{1},B_{1}^{\prime},B_{2},B_{2}^{\prime}\in\mathcal{L}_{0}$ and
$A_{1},A_{2}\in{\mathfrak{A}}$ are almost local. In view of this bound, the
GNS representation $(\pi_{\psi},\mathcal{H}_{\pi_{\psi}})$ induced by a
particle weight $\psi$ is well suited for a description of physical systems
which are localized in space (e.g. configurations of particles). The Hilbert
space $\mathcal{H}_{\pi_{\psi}}$ is given by

$\mathcal{H}_{\pi_{\psi}}=(\,\mathcal{L}/\\{\,L\in\mathcal{L}\,|\,\psi(L,L)=0\\})^{\mathrm{cpl}}$
(2.18)

and the respective equivalence class of an element $L\in\mathcal{L}$ is
denoted by $|L\rangle\in\mathcal{H}_{\pi_{\psi}}$. The completion is taken
w.r.t. the scalar product $\langle L_{1}|L_{2}\rangle:=\psi(L_{1},L_{2})$. The
representation $\pi_{\psi}$ acts on $\mathcal{H}_{\pi_{\psi}}$ as follows

$\pi_{\psi}(A)|L\rangle=|AL\rangle,\quad A\in{\mathfrak{A}}.$ (2.19)

This representation is covariant and the translation automorphisms are
implemented by the strongly continuous group of unitaries $U_{\pi_{\psi}}$,
given by

$U_{\pi_{\psi}}(x)|L\rangle=|\alpha_{x}(L)\rangle,\quad
x\in\mathbb{R}^{2},\,L\in\mathcal{L}$ (2.20)

which is called the standard representation of translations in the
representation $\pi_{\psi}$. By property 3 in Definition 2.6 above, its
spectrum is contained in a shifted closed forward lightcone. The corresponding
canonical representation will be denoted by $U_{\pi_{\psi}}^{\mathrm{can}}$.
(Cf. the discussion below Definition 2.1). We also introduce operators
$(Q^{0},\boldsymbol{Q})$ of _characteristic energy-momentum_ of $\psi$ which
are the generators of the following group of unitaries on
$\mathcal{H}_{\pi_{\psi}}$

$U^{\mathrm{char}}_{\pi_{\psi}}(x)=U^{\mathrm{can}}_{\pi_{\psi}}(x)U_{\pi_{\psi}}(x)^{-1}\in\pi_{\psi}({\mathfrak{A}})^{\prime},$
(2.21)

i.e.
$U^{\mathrm{char}}_{\pi_{\psi}}(x)=e^{iQ^{0}t-i\boldsymbol{Q}\boldsymbol{x}}$.
We call a particle weight _pure_ , if its GNS representation is irreducible.
It follows from definition (2.21) that the operator of characteristic energy-
momentum of such a weight is a vector
$q=(q^{0},\boldsymbol{q})\in\mathbb{R}^{2}$. It can be interpreted as the
energy and momentum of the plane wave configuration of the particle described
by this weight [2, 41].

To extract properties of elementary subsystems (particles) of a physical
system described by a given (possibly non-pure) particle weight, it is natural
to study irreducible subrepresentations of its GNS representation. To ensure
that there are sufficiently many such subrepresentations, we restrict
attention to particle weights $\psi$ whose GNS representations $\pi_{\psi}$
are of type I with atomic center555i.e. whose center is a direct sum of one-
dimensional von Neumann algebras.. (In particular, $\pi_{\psi}$ appearing in
our examples in Subsection 3.4 below belong to this family). Then, by Theorem
1.31 from Chapter V of [49], there exists a unique family of Hilbert spaces
$(\mathfrak{H}_{\alpha},\mathfrak{K}_{\alpha})_{\alpha\in\mathbb{I}}$ and a
unitary
$W:\mathcal{H}_{\pi_{\psi}}\to\bigoplus_{\alpha\in\mathbb{I}}\\{\mathfrak{H}_{\alpha}\otimes\mathfrak{K}_{\alpha}\\}$
s.t.

$\displaystyle W\pi_{\psi}({\mathfrak{A}})^{\prime\prime}W^{-1}$
$\displaystyle=$
$\displaystyle\bigoplus_{\alpha\in\mathbb{I}}\\{B(\mathfrak{H}_{\alpha})\otimes{\mathbb{C}}I\\},$
(2.22) $\displaystyle W\pi_{\psi}({\mathfrak{A}})^{\prime}W^{-1}$
$\displaystyle=$
$\displaystyle\bigoplus_{\alpha\in\mathbb{I}}\\{{\mathbb{C}}I\otimes
B(\mathfrak{K}_{\alpha})\\}.$ (2.23)

We note that a subspace ${\cal K}_{\alpha,e}\subset\mathcal{H}_{\pi_{\psi}}$
carries an irreducible subrepresentation $\pi_{\alpha,e}$ of $\pi_{\psi}$, if
and only if $W{\cal K}_{\alpha,e}=\mathfrak{H}_{\alpha}\otimes{\mathbb{C}}e$
for some $\alpha\in\mathbb{I}$ and $e\in\mathfrak{K}_{\alpha}$. Clearly,
$\pi_{\alpha,e}$ and $\pi_{\alpha,e^{\prime}}$ are unitarily equivalent for
any fixed $\alpha$ and arbitrary vectors
$e,e^{\prime}\in\mathfrak{K}_{\alpha}$. Choosing in any
$\mathfrak{K}_{\alpha}$ an orthonormal basis $B_{\alpha}$, we obtain

$\pi_{\psi}=\bigoplus_{\begin{subarray}{c}\alpha\in\mathbb{I}\\\ e\in
B_{\alpha}\end{subarray}}\pi_{\alpha,e}.$ (2.24)

It is clear from the above discussion that any irreducible subrepresentation
of $\pi_{\psi}$ is unitarily equivalent to some $\pi_{\alpha,e}$ in the
decomposition above.

If all the representations in the decomposition (2.24) are unitarily
equivalent to some fixed vacuum representation, then we call the particle
weight $\psi$ neutral. Otherwise we call $\psi$ charged. In the case of
charged particle weights there may occur an interplay between the
translational and internal degrees of freedom of the system which we call
_superselection of direction of motion_. To introduce this concept, we need
some terminology: Let $\mathcal{H}_{\pi_{\psi},\mathrm{R}}$ (resp.
$\mathcal{H}_{\pi_{\psi},\mathrm{L}}$) be the spectral subspace of the
characteristic momentum operator $\boldsymbol{Q}$ of $\psi$, corresponding to
the interval $[0,\infty)$ (resp. $(-\infty,0)$). Let $\pi$ be an irreducible
subrepresentation of $\pi_{\psi}$, acting on a subspace ${\cal
K}\subset\mathcal{H}_{\pi_{\psi}}$. Then we say that $\pi$ is _right-moving_
(resp. _left-moving_), if ${\cal K}\neq\\{0\\}$ and ${\cal
K}\subset\mathcal{H}_{\pi_{\psi},\mathrm{R}}$ (resp. ${\cal
K}\subset\mathcal{H}_{\pi_{\psi},\mathrm{L}}$). By a suitable choice of the
bases $B_{\alpha}$ one can ensure that each representation $\pi_{\alpha,e}$,
appearing in decomposition (2.24), has one of these properties. (In fact,
exploiting relations (2.21), (2.23), one can choose such basis vectors
$e\in\mathfrak{K}_{\alpha}$ that
$W^{-1}(\mathfrak{H}_{\alpha}\otimes{\mathbb{C}}e)$ belong to
$\mathcal{H}_{\pi_{\psi},\mathrm{R}}$ or
$\mathcal{H}_{\pi_{\psi},\mathrm{L}}$). After this preparation we define the
central concept of the present paper:

######  Definition 2.7.

Let $\mathcal{W}$ be a family of particle weights and assume that their GNS
representations
$\\{\,(\pi_{\psi},\mathcal{H}_{\pi_{\psi}})\,|\,\psi\in\mathcal{W}\,\\}$ are
of type I with atomic centers. Suppose that for any
$\psi,\psi^{\prime}\in\mathcal{W}$ the following properties hold:

  1. 1.

$\pi_{\psi}$ has both left-moving and right-moving irreducible
subrepresentations.

  2. 2.

No right-moving, irreducible subrepresentation of $\pi_{\psi}$ is unitarily
equivalent to a left-moving irreducible subrepresentation of
$\pi_{\psi^{\prime}}$.

Then we say that this family of particle weights has superselected direction
of motion.

Let us now relate superselection of direction of motion in the above sense to
our discussion of this concept in the Introduction. For this purpose we
consider a particle weight $\psi$, whose GNS representation is of type I with
atomic center and acts on a _separable_ Hilbert space
$\mathcal{H}_{\pi_{\psi}}$. Making use of formula (2.24) and identifying
unitarily each $\pi_{\alpha,e}$, acting on
$W^{-1}(\mathfrak{H}_{\alpha}\otimes{\mathbb{C}}e)$, with
$\pi_{\alpha}:=\pi_{\alpha,e_{0}}$ acting on ${\cal
K}_{\alpha}:=W^{-1}(\mathfrak{H}_{\alpha}\otimes{\mathbb{C}}e_{0})$ for some
chosen $e_{0}\in B_{\alpha}$, we obtain

$\pi_{\psi}({\mathfrak{A}})\simeq\bigoplus_{\alpha\in\mathbb{I}}\\{\pi_{\alpha}({\mathfrak{A}})\otimes{\mathbb{C}}I\\},$
(2.25)

where the r.h.s. acts on $\bigoplus_{\alpha\in\mathbb{I}}\\{{\cal
K}_{\alpha}\otimes\mathfrak{K}_{\alpha}\\}$. In the sense of the same
identification

$\pi_{\psi}({\mathfrak{A}})^{\prime}\simeq\bigoplus_{\alpha\in\mathbb{I}}\\{{\mathbb{C}}I\otimes
B(\mathfrak{K}_{\alpha})\\}.$ (2.26)

Now, following [42], we choose a maximal abelian von Neumann algebra ${\cal
M}$ in $\pi_{\psi}({\mathfrak{A}})^{\prime}$, containing
$\\{\,U^{\mathrm{char}}_{\pi_{\psi}}(x)\,|\,x\in\mathbb{R}^{2}\,\\}$. As a
consequence of formula (2.26)

${\cal M}\simeq\bigoplus_{\alpha\in\mathbb{I}}\\{{\mathbb{C}}I\otimes{\cal
M}_{\alpha}\\},$ (2.27)

where ${\cal M}_{\alpha}\subset B(\mathfrak{K}_{\alpha})$ are maximal abelian
von Neumann subalgebras. For any such ${\cal M}_{\alpha}$ there exists a Borel
space $(Z_{\alpha},d\mu_{\alpha})$ s.t. $({\cal
M}_{\alpha},\mathfrak{K}_{\alpha})\simeq(L^{\infty}(Z_{\alpha},d\mu_{\alpha}),L^{2}(Z_{\alpha},d\mu_{\alpha}))$.
(This fact uses separability of the Hilbert space. See Theorem II.2.2 of
[20]). Adopting this identification in (2.25) and (2.26), we obtain

$U^{\mathrm{char}}_{\pi_{\psi}}\simeq\bigoplus_{\alpha\in\mathbb{I}}\\{I\otimes
U^{\mathrm{char}}_{\alpha}\\},$ (2.28)

where $U^{\mathrm{char}}_{\alpha}(x)\in L^{\infty}(Z_{\alpha},d\mu_{\alpha})$
is the operator of multiplication by (the equivalence class of) the function
$Z_{\alpha}\ni z\to e^{iq_{\alpha,z}x}$, where
$q_{\alpha,z}=(q_{\alpha,z}^{0},\boldsymbol{q}_{\alpha,z})\in\mathbb{R}^{2}$.
Introducing the field of representations
$(\pi_{\alpha,z},\mathfrak{H}_{\alpha,z})_{z\in Z_{\alpha}}$ s.t.
$\pi_{\alpha,z}=\pi_{\alpha}$ and $\mathfrak{H}_{\alpha,z}={\cal K}_{\alpha}$
for all $z\in Z_{\alpha}$, we obtain from relation (2.25) the existence of a
unitary
$\tilde{W}:\mathcal{H}_{\pi_{\psi}}\to\bigoplus_{\alpha\in\mathbb{I}}\int^{\oplus}d\mu_{\alpha}(z)\,\mathfrak{H}_{\alpha,z}$
s.t.

$\tilde{W}\pi_{\psi}(\,\cdot\,)\tilde{W}^{-1}=\bigoplus_{\alpha\in\mathbb{I}}\int^{\oplus}_{Z_{\alpha}}d\mu_{\alpha}(z)\,\pi_{\alpha,z}(\,\cdot\,).$
(2.29)

This is an example of decomposition (1.2), stated in the Introduction.
Moreover, as a consequence of (2.28),

$\tilde{W}U^{\mathrm{char}}_{\pi_{\psi}}(x)\tilde{W}^{-1}=\bigoplus_{\alpha\in\mathbb{I}}\int^{\oplus}_{Z_{\alpha}}d\mu_{\alpha}(z)\,e^{iq_{\alpha,z}x},$
(2.30)

where $\\{q_{\alpha,z}\\}_{z\in Z_{\alpha}}$ is the field of characteristic
energy-momentum vectors666This terminology is consistent with the discussion
after formula (2.21). In fact, under some technical restrictions each
$\pi_{\alpha,z}$ is induced by some pure particle weight $\psi_{\alpha,z}$,
whose characteristic energy-momentum vector is $q_{\alpha,z}$ [41, 42]. Cf.
also formula (1.3).  of the representations
$(\pi_{\alpha,z},\mathfrak{H}_{\alpha,z})_{z\in Z_{\alpha}}$. As we required
in the Introduction, for any particle weight with superselected direction of
motion, the relation
$\boldsymbol{q}_{\alpha,z}\cdot\boldsymbol{q}_{\alpha^{\prime},z^{\prime}}\leq
0$ should imply that $\pi_{\alpha,z}$ is not unitarily equivalent to
$\pi_{\alpha^{\prime},z^{\prime}}$ for almost all $z,z^{\prime}$. This is in
fact the case in view of the following proposition.

######  Proposition 2.8.

Suppose that $\psi$ belongs to a family of particle weights which has
superselected direction of motion in the sense of Definition 2.7 and s.t. its
GNS representation acts on a separable Hilbert space. Then
$\\{\pi_{\alpha,z}\\}_{z\in Z_{\alpha}}$, appearing in the decomposition
(2.29) of $\pi_{\psi}$, is a field of right-moving (resp. left-moving)
representations, if and only if $\boldsymbol{q}_{\alpha,z}\geq 0$ (resp.
$\boldsymbol{q}_{\alpha,z}<0$) for almost all $z\in Z_{\alpha}$.

Proof. Suppose that $\pi_{\alpha}$ is a right-moving subrepresentation of
$\pi_{\psi}$ i.e. ${\cal
K}_{\alpha}\subset\mathcal{H}_{\pi_{\psi},\mathrm{R}}$. We recall that
$\pi_{\alpha}$ coincides with $\pi_{\alpha,e_{0}}$ acting on ${\cal
K}_{\alpha,e_{0}}={\cal K}_{\alpha}$. Since every $\pi_{\alpha,e}$,
$e\in\mathfrak{K}$, is unitarily equivalent to $\pi_{\alpha,e_{0}}$, the
property of superselection of direction of motion implies that ${\cal
K}_{\alpha,e}\subset\mathcal{H}_{\pi_{\psi},\mathrm{R}}$ for all
$e\in\mathfrak{K}$. Consequently,
$\mathcal{H}_{\alpha}=W^{-1}(\mathfrak{H}_{\alpha}\otimes\mathfrak{K}_{\alpha})\subset\mathcal{H}_{\pi_{\psi},\mathrm{R}}$.
Since the projection $P_{\alpha}$ on $\mathcal{H}_{\alpha}$ is central, this
subspace is invariant under the action of $U^{\mathrm{char}}_{\pi_{\psi}}$.
Formula (2.28) gives

$U^{\mathrm{char}}_{\pi_{\psi}}(x)P_{\alpha}\simeq I\otimes
U^{\mathrm{char}}_{\alpha}(x),$ (2.31)

thus the spectra of the generators of $\mathbb{R}^{2}\ni x\to
U^{\mathrm{char}}_{\alpha}(x)$ and $\mathbb{R}^{2}\ni x\to
U^{\mathrm{char}}_{\pi_{\psi}}(x)P_{\alpha}$ coincide. In particular the
spectrum of the generator of space translations of
$U^{\mathrm{char}}_{\alpha}$ is contained in $[0,\infty)$. The opposite
implication follows immediately from relation (2.31). $\Box$

###  2.4 Asymptotic functionals

In this subsection we consider a concrete class of particle weights,
introduced in [9, 10, 41], which have applications in scattering theory. Their
construction relies on the following result due to Buchholz (which remains
valid in higher dimensions).

######  Theorem 2.9 ([10]).

Let $({\mathfrak{A}},U)$ be a local net of $C^{*}$-algebras on
$\mathbb{R}^{2}$. Then, for any $E\geq 0$, $L\in\mathcal{L}$,

$\left\|P_{E}\int_{K}d\boldsymbol{x}\,(L^{*}L)(\boldsymbol{x})P_{E}\right\|\leq
c,$ (2.32)

where $P_{E}$ is the spectral projection on vectors of energy not larger than
$E$, $K\subset\mathbb{R}$ is a compact interval, and $c$ is a constant
independent of $K$.

Following [41], we introduce the algebra of detectors
$\mathcal{C}=\mathrm{span}\\{L_{1}^{*}L_{2}:L_{1},L_{2}\in\mathcal{L}\\}$ and
equip it with a locally convex topology, given by the family of seminorms

$p_{E}(C)=\sup\left\\{\int
d\boldsymbol{x}\,|(\Psi|C(\boldsymbol{x})\Psi)|\,\,|\,\,\Psi\in
P_{E}\mathcal{H},\,\|\Psi\|\leq 1\,\right\\},\,\quad C\in\mathcal{C},$ (2.33)

labelled by $E\geq 0$, which are finite by Theorem 2.9. Next, for any
$\Psi\in\mathcal{H}$ of bounded energy, (i.e. belonging to $P_{E}\mathcal{H}$
for some $E\geq 0$), we define a sequence of functionals
$\\{\sigma_{\Psi}^{(T)}\\}_{T\in\mathbb{R}}$ from the topological dual of
$\mathcal{C}$:

$\sigma_{\Psi}^{(T)}(C):=\int dt\,h_{T}(t)\int
d\boldsymbol{x}\,(\Psi|C(t,\boldsymbol{x})\Psi),\quad C\in\mathcal{C}.$ (2.34)

As this sequence is uniformly bounded in $T$ w.r.t. any seminorm $p_{E}$, the
Alaoglu-Bourbaki theorem gives limit points
$\sigma_{\Psi}^{\mathrm{out}}\in\mathcal{C}^{*}$ as $T\to\infty$, which are
called the asymptotic functionals. The following fact was shown in [41]:

######  Proposition 2.10 ([41]).

If $\sigma_{\Psi}^{\mathrm{out}}\neq 0$, then the sesquilinear forms on
$\mathcal{L}$, given by

$\psi_{\Psi}^{\mathrm{out}}(L_{1},L_{2}):=\sigma_{\Psi}^{\mathrm{out}}(L_{1}^{*}L_{2}),$
(2.35)

are particle weights, in the sense of Definition 2.6.

Fundamental results from [2] suggest a physical interpretation of the particle
weights $\psi_{\Psi}^{\mathrm{out}}$ as mixtures of plane wave configurations
of all the particle types described by the theory. (Cf. formulas (1.2),
(1.3)). Accordingly, we say that a given theory has a non-trivial _particle
content_ , if it admits some non-zero asymptotic functionals
$\sigma_{\Psi}^{\mathrm{out}}$. This is the case in any massless two-
dimensional theory of Wigner particles (in a vacuum representation) as a
consequence of the following theorem. A proof of this statement, which is our
main technical result, is given in Appendix A.

######  Theorem 2.11.

Let $({\mathfrak{A}},U)$ be a local net of $C^{*}$-algebras on
$\mathbb{R}^{2}$ in a vacuum representation, acting on a Hilbert space
$\mathcal{H}$. Then, for any $\Psi\in P_{E}\mathcal{H}^{\mathrm{out}}$, $E\geq
0$,

$\displaystyle\psi_{\Psi}^{\mathrm{out}}(L_{1},L_{2})$ $\displaystyle=$
$\displaystyle\lim_{T\to\infty}\int dt\,h_{T}(t)\int
d\boldsymbol{x}\,(\Psi|(L_{1}^{*}L_{2})(t,\boldsymbol{x})\Psi)$ (2.36)
$\displaystyle=$ $\displaystyle\int
d\boldsymbol{x}\,(\rho_{+,\Psi}+\rho_{-,\Psi})\big{(}(L_{1}^{*}L_{2})(\boldsymbol{x})\big{)},$

where the functionals $\rho_{\pm,\Psi}$ are defined by (2.11), (2.12). In
particular, $\psi_{\Psi}^{\mathrm{out}}=0$, if and only if
$\Psi\in{\mathbb{C}}\Omega$.

In a theory of Wigner particles
${\mathbb{C}}\Omega\neq\mathcal{H}_{\pm}\subset\mathcal{H}^{\mathrm{out}}$,
thus the particle content is non-trivial by the above result. However, non-
zero asymptotic functionals may also appear in the absence of Wigner particles
i.e. when one or both of the subspaces $\mathcal{H}_{\pm}$ equal
$\mathcal{H}_{+}\cap\mathcal{H}_{-}$. If this is the case, then we say that
the net $({\mathfrak{A}},U)$ describes _infraparticles_. Theorem 3.6 below
provides a large class of such theories. In Theorem 3.10 we show that some of
these models describe excitations whose direction of motion is superselected
in the following sense:

######  Definition 2.12.

Let $({\mathfrak{A}},U)$ be a net describing infraparticles. We say that the
infraparticles of the net $({\mathfrak{A}},U)$ have superselected direction of
motion, if $\\{\,\psi_{\Psi}^{\mathrm{out}}\,|\,\Psi\neq 0,\,\Psi\in
P_{E}\mathcal{H},\,E\geq 0\,\\}$ is a family of particle weights with
superselected direction of motion in the sense of Definition 2.7.

##  3 Particle aspects of conformal field theories

###  3.1 Preliminaries

In this section we are interested in particle aspects of chiral conformal
field theories. To emphasize the relevant properties of these models, we base
our investigation on the concept of a local net of von Neumann algebras on
${\mathbb{R}}$, defined below. There are many examples of such nets. In
particular, they arise from Möbius covariant nets on $S^{1}$ by means of the
Cayley transform and the subsequent restriction to the real line. The simplest
example is the so-called $U(1)$-current net [15], whose subnets and extensions
are well-studied. For certain classes of nets on $S^{1}$ even classification
results have been obtained [36, 37].

######  Definition 3.1.

A local net of von Neumann algebras on $\mathbb{R}$ is a pair $({\cal A},V)$
consisting of a map ${\cal I}\to{\cal A}({\cal I})$ from the family of open,
bounded subsets of $\mathbb{R}$ to the family of von Neumann algebras on a
Hilbert space $\mathcal{K}$ and a strongly continuous unitary representation
of translations $\mathbb{R}\ni s\to V(s)$, acting on $\mathcal{K}$, which are
subject to the following conditions:

  1. 1.

(isotony) If ${\cal I}\subset\mathfrak{J}$, then ${\cal A}({\cal
I})\subset{\cal A}(\mathfrak{J})$.

  2. 2.

(locality) If ${\cal I}\cap\mathfrak{J}=\varnothing$, then $[{\cal A}({\cal
I}),{\cal A}(\mathfrak{J})]=0$.

  3. 3.

(covariance) $V(s){\cal A}({\cal I})V(s)^{*}={\cal A}({\cal I}+s)$ for any
$s\in\mathbb{R}$.

  4. 4.

(positivity of energy) The spectrum of $V$ coincides with $\mathbb{R}_{+}$.

We also denote by ${\cal A}$ the quasilocal $C^{*}$-algebra of this net i.e.
${\cal A}=\overline{\bigcup_{{\cal I}\subset\mathbb{R}}{\cal A}({\cal I})}$.

Since we assumed that ${\cal A}({\cal I})$ are von Neumann algebras, we cannot
demand norm continuity of the functions $s\to\beta_{s}(A)$, $A\in{\cal A}$,
where $\beta_{s}(\,\cdot\,)=V(s)\,\cdot\,V(s)^{*}$. This regularity property
holds, however, on the following weakly dense subnet of $C^{*}$-algebras

${\cal I}\to\bar{\cal A}({\cal I}):=\\{\,A\in{\cal A}({\cal I})\,|\,\lim_{s\to
0}\|\beta_{s}(A)-A\|=0\,\\}.$ (3.1)

The corresponding quasilocal algebra is denoted by $\bar{\cal A}$.

If $V$ has a unique (up to a phase) invariant, unit vector
$\Omega_{0}\in\mathcal{K}$ and $\Omega_{0}$ is cyclic under the action of any
${\cal A}({\cal I})$ (the Reeh-Schlieder property) then we say that the net
$({\cal A},V)$ is in a vacuum representation. In this case ${\cal A}$ acts
irreducibly on $\mathcal{K}$. In the course of our analysis we will also
consider other representations of $({\cal A},V)$. We say that a representation
$\pi:{\cal A}\to B(\mathcal{K}_{\pi})$ is covariant, if there exists a
strongly continuous group of unitaries $V_{\pi}$ on $\mathcal{K}_{\pi}$, s.t.

$\pi(\alpha_{s}(A))=V_{\pi}(s)\pi(A)V_{\pi}(s)^{*},\quad A\in{\cal
A},\,\,s\in\mathbb{R}.$ (3.2)

Moreover, we say that this representation has positive energy, if the spectrum
of $V_{\pi}$ coincides with $\mathbb{R}_{+}$. If $\pi$ is locally normal (i.e.
its restriction to any local algebra ${\cal A}({\cal I})$ is normal) then
$(\pi({\cal A}),V_{\pi})$ is again a net of von Neumann algebras in the sense
of Definition 3.1.

Let $({\cal A}_{1},V_{1})$ and $({\cal A}_{2},V_{2})$ be two nets of von
Neumann algebras on ${\mathbb{R}}$, acting on Hilbert spaces $\mathcal{K}_{1}$
and $\mathcal{K}_{2}$. To construct a local net $({\mathfrak{A}},U)$ on
${\mathbb{R}}^{2}$, acting on the tensor product space ${\cal H}={\cal
K}_{1}\otimes{\cal K}_{2}$, we identify the two real lines with the lightlines
$I_{\pm}=\\{\,(t,\boldsymbol{x})\in\mathbb{R}^{2}\,|\,\boldsymbol{x}\mp
t=0\,\\}$ in ${\mathbb{R}}^{2}$. We first specify the unitary representation
of translations

$U(t,\boldsymbol{x}):=V_{1}\left(\frac{1}{\sqrt{2}}(t-\boldsymbol{x})\right)\otimes
V_{2}\left(\frac{1}{\sqrt{2}}(t+\boldsymbol{x})\right),$ (3.3)

whose spectrum is easily seen to coincide with $V_{+}$ as a consequence of
property 4 from Definition 3.1. We mention for future reference that if
$\alpha_{(t,\boldsymbol{x})}(\,\cdot\,):=U(t,\boldsymbol{x})\,\,\cdot\,\,U(t,\boldsymbol{x})^{*}$
is the corresponding group of translation automorphisms and
$\beta^{(1/2)}_{s}(\,\cdot\,):=V_{1/2}(s)\,\,\cdot\,\,V_{1/2}(s)^{*}$, then

$\alpha_{(t,\boldsymbol{x})}(A_{1}\otimes
A_{2})=\beta^{(1)}_{\frac{1}{\sqrt{2}}(t-\boldsymbol{x})}(A_{1})\otimes\beta^{(2)}_{\frac{1}{\sqrt{2}}(t+\boldsymbol{x})}(A_{2}),\quad
A_{1}\in{\cal A}_{1},\,A_{2}\in{\cal A}_{2}.$ (3.4)

Any double cone $D\subset{\mathbb{R}}^{2}$ can be expressed as a product of
intervals on lightlines $D={\cal I}_{1}\times{\cal I}_{2}$. We define the
corresponding local von Neumann algebra by
${\mathfrak{A}}^{\textrm{vN}}(D):={\cal A}_{1}({\cal I}_{1})\otimes{\cal
A}_{2}({\cal I}_{2})$, and for a general open region $\mathcal{O}$ we put
${\mathfrak{A}}^{\textrm{vN}}(\mathcal{O})=\bigvee_{D\subset\mathcal{O}}{\mathfrak{A}}^{\textrm{vN}}(D)$.
The net of von Neumann algebras $({\mathfrak{A}}^{\textrm{vN}},U)$, which we
call the chiral net, satisfies all the properties from Definition 2.1 except
for the regularity property 5. Therefore, we introduce the following weakly
dense subnet of $C^{*}$-algebras

$\mathcal{O}\to{\mathfrak{A}}(\mathcal{O}):=\\{\,A\in{\mathfrak{A}}^{\textrm{vN}}(\mathcal{O})\,|\,\lim_{x\to
0}\|\alpha_{x}(A)-A\|=0\,\\},$ (3.5)

and denote the corresponding quasilocal algebra by ${\mathfrak{A}}$. Then
$({\mathfrak{A}},U)$ is a local net of $C^{*}$-algebras in the sense of
Definition 2.1. We will call it the regular chiral net and refer to $({\cal
A}_{1},V_{1})$, $({\cal A}_{2},V_{2})$ as its chiral components. We note for
future reference that if ${\mathfrak{A}}$ acts irreducibly on $\mathcal{H}$,
then $U$ is automatically the canonical representation of translations of this
net (cf. Subsection 2.1). Another useful fact is the obvious inclusion

$\bar{\cal A}_{1}\otimes_{\mathrm{alg}}\bar{\cal A}_{2}\subset{\mathfrak{A}},$
(3.6)

where $\otimes_{\mathrm{alg}}$ is the algebraic tensor product.

Let $({\mathfrak{A}}^{\textrm{vN}},U)$ be a chiral net, whose chiral
components are $({\cal A}_{1},V_{1})$ and $({\cal A}_{2},V_{2})$. Let
$\pi_{1}$, $\pi_{2}$ be locally normal, covariant, positive energy
representations of the respective nets on $\mathbb{R}$. Then the chiral net of
$(\pi_{1}({\cal A}_{1}),V_{\pi_{1}})$, $(\pi_{2}({\cal A}_{2}),V_{\pi_{2}})$
is a covariant, positive energy representation of
$({\mathfrak{A}}^{\textrm{vN}},U)$, which will be denoted by
$(\pi({\mathfrak{A}}^{\textrm{vN}}),U_{\pi})$, $\pi=\pi_{1}\otimes\pi_{2}$ and
$\pi$ is called the product representation of $\pi_{1}$ and $\pi_{2}$. We note
that $(\pi({\mathfrak{A}}),U_{\pi})$ is contained in the regular subnet of
$(\pi({\mathfrak{A}}^{\textrm{vN}}),U_{\pi})$. For faithful $\pi$ these two
nets coincide, due to Proposition 2.3.3 (2) of [17]. It is easily seen that
$\pi$ is faithful (resp. irreducible), if $\pi_{1}$ and $\pi_{2}$ are faithful
(resp. irreducible). (Cf. Theorems 5.2 and 5.9 from Chapter IV of [49]).

###  3.2 Vacuum representations and asymptotic completeness

A regular chiral net $({\mathfrak{A}},U)$ is in a vacuum representation, with
the vacuum vector $\Omega\in\mathcal{H}$, if and only if its chiral components
$({\cal A}_{1},V_{1})$, $({\cal A}_{2},V_{2})$ are in vacuum representations
with the respective vacuum vectors $\Omega_{1}\in{\cal K}_{1}$,
$\Omega_{2}\in{\cal K}_{2}$ s.t. $\Omega=\Omega_{1}\otimes\Omega_{2}$. (Cf.
Proposition 3.5 below). In this subsection we show that any such regular
chiral net has a complete particle interpretation in terms of non-interacting
Wigner particles. These facts follow from our results in [25], but the
argument below is more direct.

We start from the observation that the asymptotic fields have a particularly
simple form in chiral theories:

######  Proposition 3.2.

Let $({\cal A}_{1},V_{1})$, $({\cal A}_{2},V_{2})$ be two local nets of von
Neumann algebras in vacuum representations, with the respective vacuum vectors
$\Omega_{1}$, $\Omega_{2}$. Then, for any $A_{1}\in\bar{\cal A}_{1}$,
$A_{2}\in\bar{\cal A}_{2}$

$\displaystyle\Phi_{+}^{\mathrm{out}/\mathrm{in}}(A_{1}\otimes A_{2})$
$\displaystyle=$ $\displaystyle A_{1}\otimes(\Omega_{2}|A_{2}\Omega_{2})I,$
(3.7) $\displaystyle\Phi_{-}^{\mathrm{out}/\mathrm{in}}(A_{1}\otimes A_{2})$
$\displaystyle=$ $\displaystyle(\Omega_{1}|A_{1}\Omega_{1})I\otimes A_{2}.$
(3.8)

Proof. We consider only $\Phi_{+}^{\mathrm{out}}$, as the remaining cases are
analogous. From the defining relation (2.6) and formula (3.4), we obtain

$\Phi_{+}^{\mathrm{out}}(A_{1}\otimes
A_{2})=\underset{T\to\infty}{\mathrm{s}\textrm{-}\lim}\;A_{1}\otimes\int
dt\,h_{T}(t)\beta_{\sqrt{2}t}^{(2)}(A_{2}).$ (3.9)

We set $A_{2}(h_{T}):=\int dt\,h_{T}(t)\beta_{\sqrt{2}t}^{(2)}(A_{2})$. This
sequence has the following properties:

$\displaystyle\lim_{T\to\infty}A_{2}(h_{T})\Omega_{2}$ $\displaystyle=$
$\displaystyle(\Omega_{2}|A_{2}\Omega_{2})\Omega_{2},$ (3.10)
$\displaystyle\lim_{T\to\infty}\|[A_{2}(h_{T}),A]\|$ $\displaystyle=$
$\displaystyle 0,\textrm{ for any }A\in\bar{\cal A}_{2}.$ (3.11)

The first identity above follows from the mean ergodic theorem and the fact
that $\Omega_{2}$ is the only vector invariant under the action of $V_{2}$.
The second equality is a consequence of the locality assumption from
Definition 3.1. Since $\bar{\cal A}_{2}$ acts irreducibly, any
$\Psi\in\mathcal{K}_{2}$ has the form $\Psi=A\Omega_{2}$ for some
$A\in\bar{\cal A}_{2}$ [45]. Thus we obtain from (3.10), (3.11)

$\underset{T\to\infty}{\mathrm{s}\textrm{-}\lim}\;A_{2}(h_{T})=(\Omega_{2}|A_{2}\Omega_{2})I,$
(3.12)

which completes the proof. $\Box$
Now we can easily prove the main result of this subsection:

######  Theorem 3.3.

Any regular chiral net $({\mathfrak{A}},U)$ in a vacuum representation is
asymptotically complete. More precisely:

$\displaystyle\mathcal{H}_{+}=\mathcal{K}_{1}\otimes{\mathbb{C}}\Omega_{2},$
(3.13)
$\displaystyle\mathcal{H}_{-}={\mathbb{C}}\Omega_{1}\otimes\mathcal{K}_{2},$
(3.14)
$\displaystyle\mathcal{H}_{+}\overset{\mathrm{out}}{\times}\mathcal{H}_{-}=\mathcal{H}_{+}\overset{\mathrm{in}}{\times}\mathcal{H}_{-}=\mathcal{H}.$
(3.15)

Moreover, any such theory is non-interacting.

######  Remark 3.4.

This result and Theorem 2.11 imply the convergence of the asymptotic
functional approximants $\\{\sigma_{\Psi}^{(T)}\\}_{T\in\mathbb{R}_{+}}$ for
all $\Psi\in\mathcal{H}$ of bounded energy in any regular chiral net in a
vacuum representation.

Proof. Using formula (2.5) and the cyclicity of the vacuum $\Omega$ under the
action of ${\mathfrak{A}}$, we obtain

$\mathcal{H}_{\pm}=[\,\Phi_{\pm}^{\mathrm{out}}(F)\Omega\,|\,F\in{\mathfrak{A}}\,],$
(3.16)

where $[\,\cdot\,]$ denotes the norm closure. Applying Proposition 3.2 and
exploiting the cyclicity of $\Omega_{1/2}$ under the action of $\bar{\cal
A}_{1/2}$, we obtain (3.13) and (3.14). The asymptotic completeness relation
(3.15) also follows from Proposition 3.2: For any $A_{1}\in\bar{\cal A}_{1}$,
$A_{2}\in\bar{\cal A}_{2}$

$\displaystyle\Phi_{+}^{\mathrm{out}}(A_{1}\otimes
I)\Phi_{-}^{\mathrm{out}}(I\otimes
A_{2})\Omega=\Phi_{+}^{\mathrm{in}}(A_{1}\otimes
I)\Phi_{-}^{\mathrm{in}}(I\otimes A_{2})\Omega=A_{1}\Omega_{1}\otimes
A_{2}\Omega_{2}.$ (3.17)

Exploiting once again cyclicity of $\Omega_{1/2}$, we obtain that scattering
states are dense in the Hilbert space.

Now let us show the lack of interaction: Let $\Psi_{\pm}\in\mathcal{H}_{\pm}$.
Then, by (3.13), (3.14) and the irreducibility of the action of $\bar{\cal
A}_{1/2}$ on $\mathcal{K}_{1/2}$, there exist $A_{1}\in\bar{\cal A}_{1}$,
$A_{2}\in\bar{\cal A}_{2}$ s.t. $\Psi_{+}=A_{1}\Omega_{1}\otimes\Omega_{2}$
and $\Psi_{-}=\Omega_{1}\otimes A_{2}\Omega_{2}$. Then

$\displaystyle\Psi_{+}\overset{\mathrm{out}}{\times}\Psi_{-}=\Phi_{+}^{\mathrm{out}}(A_{1}\otimes
I)\Phi_{-}^{\mathrm{out}}(I\otimes A_{2})\Omega=A_{1}\Omega_{1}\otimes
A_{2}\Omega_{2}$ $\displaystyle=\Phi_{+}^{\mathrm{in}}(A_{1}\otimes
I)\Phi_{-}^{\mathrm{in}}(I\otimes
A_{2})\Omega=\Psi_{+}\overset{\mathrm{in}}{\times}\Psi_{-}.$ (3.18)

Hence the scattering operator, defined in (2.10), equals the identity on
$\mathcal{H}$. $\Box$

###  3.3 Charged representations and infraparticles

It is the goal of this subsection to clarify the particle content of chiral
conformal field theories in charged representations. More detailed particle
properties of such theories, e.g. superselection of direction of motion, will
be studied in the next subsection.

Let us first note the following simple relation between the single-particle
subspaces of a regular chiral net and the invariant vectors of its chiral
components.

######  Proposition 3.5.

Let $({\cal A}_{1},V_{1})$, $({\cal A}_{2},V_{2})$ be local nets of von
Neumann algebras on $\mathbb{R}$. Then $V_{1}$ (resp. $V_{2}$) has a non-
trivial invariant vector, if and only if the single-particle subspace
$\mathcal{H}_{-}$ (resp. $\mathcal{H}_{+}$) of the corresponding regular
chiral net $({\mathfrak{A}},U)$ is non-trivial.

Proof. Suppose there exists a non-zero $\Omega_{1}\in\mathcal{K}_{1}$,
invariant under the action of $V_{1}$. Then, for any
$\Psi_{2}\in\mathcal{K}_{2}$,

$\displaystyle U(t,-t)(\Omega_{1}\otimes\Psi_{2})$ $\displaystyle=$
$\displaystyle\Omega_{1}\otimes\Psi_{2},$ (3.19)

for $t\in\mathbb{R}$. Hence the subspace $\mathcal{H}_{-}$ is non-trivial.
Similarly, the existence of a non-zero $\Omega_{2}\in\mathcal{K}_{2}$,
invariant under the action of $V_{2}$, implies the non-triviality of
$\mathcal{H}_{+}$.

Now suppose $\Psi\in\mathcal{H}_{-}$ and $V_{1}$ has no non-trivial, invariant
vectors. Then, by the mean ergodic theorem,

$\displaystyle\Psi=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}dt\,U(t,-t)\Psi=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}dt\,\big{(}V_{1}(\sqrt{2}t)\otimes
I\big{)}\Psi=0.$ (3.20)

Thus we established that $\mathcal{H}_{-}=\\{0\\}$. Similarly, the absence of
non-trivial, invariant vectors of $V_{2}$ implies that
$\mathcal{H}_{+}=\\{0\\}$. $\Box$
Let $({\mathfrak{A}},U)$ be a regular chiral net in a charged irreducible
(product) representation. That is ${\mathfrak{A}}$ acts irreducibly on a non-
trivial Hilbert space, which has the tensor product structure, by our
definition of chiral nets, and does not contain non-zero invariant vectors of
$U$. The particle structure of such theories is described by the following
theorem.

######  Theorem 3.6.

Let $({\mathfrak{A}},U)$ be a regular chiral net in a charged irreducible
(product) representation acting on a Hilbert space $\mathcal{H}$. Then:

  1. (a)

$\mathcal{H}_{+}=\\{0\\}$ or $\mathcal{H}_{-}=\\{0\\}$ i.e. the theory does
not describe Wigner particles.

  2. (b)

For any non-zero vector $\Psi\in P_{E}\mathcal{H}$, $E\geq 0$, all the limit
points of the net $\\{\sigma_{\Psi}^{(T)}\\}_{T\in\mathbb{R}_{+}}$, given by
(2.34), are different from zero.

Hence $({\mathfrak{A}},U)$ describes infraparticles.

Proof. Part (a) follows immediately from Proposition 3.5 and the absence of
non-zero invariant vectors of $U$ in $\mathcal{H}$. As for part (b), since
${\mathfrak{A}}$ acts irreducibly on $\mathcal{H}$, its chiral components
$({\cal A}_{1/2},V_{1/2})$ act irreducibly on their respective Hilbert spaces
${\cal K}_{1/2}$. We note that for any non-zero vector $\Psi\in
P_{E}\mathcal{H}$ we can find a sequence of vectors
$\\{\Psi_{n}\\}_{n\in\mathbb{N}}$ from ${\cal K}_{1}$ s.t. $\Psi_{1}\neq 0$
and

$(\Psi|(C\otimes I)\Psi)=\sum_{n\in\mathbb{N}}(\Psi_{n}|C\Psi_{n})$ (3.21)

for all $C\in B({\cal K}_{1})$. Moreover, we can assume without loss of
generality that ${\cal K}_{1}$ does not contain non-trivial invariant vectors
of $V_{1}$. Then we obtain from Lemma A.1 (b) the existence of a local
operator $A\in{\cal A}_{1}$ and $f\in S(\mathbb{R})$ s.t.
$\mathrm{supp}\,\tilde{f}\cap\mathbb{R}_{+}=\varnothing$, which satisfy
$A(f)\Psi_{1}\neq 0$. We note that any $B:=A(f)\otimes I$ is a non-zero
element of ${\mathfrak{A}}$ which is almost local and energy decreasing.
Consequently, $B^{*}B$ belongs to the algebra of detectors $\mathcal{C}$ of
the net $({\mathfrak{A}},U)$. We consider the corresponding asymptotic
functional approximants

$\displaystyle\sigma_{\Psi}^{(T)}(B^{*}B)$ $\displaystyle=$ $\displaystyle\int
dt\,h_{T}(t)\int
d\boldsymbol{x}\,(\Psi|\alpha_{(t,\boldsymbol{x})}(B^{*}B)\Psi)$ (3.22)
$\displaystyle=$ $\displaystyle\int dt\,h_{T}(t)\int
d\boldsymbol{x}\,(\Psi|\beta_{(\sqrt{2})^{-1}(t-\boldsymbol{x})}^{(1)}(A(f)^{*}A(f))\otimes
I)\Psi)$ $\displaystyle\geq$ $\displaystyle\int
d\boldsymbol{x}\,(\Psi_{1}|(\beta^{(1)}_{(\sqrt{2})^{-1}\boldsymbol{x}}(A(f)^{*}A(f)))\Psi_{1})\neq
0,$

where in the last step we made use of (3.21). As the last expression is
independent of $T$, all the limit points of
$\\{\sigma_{\Psi}^{(T)}\\}_{T\in\mathbb{R}_{+}}$ are different from zero.
$\Box$

###  3.4 Infraparticles with superselected direction of motion

In Theorem 3.6 above we have shown that any charged irreducible (product)
representation of a chiral conformal field theory contains infraparticles. In
this subsection we demonstrate that in a large class of examples these
infraparticles have superselected direction of motion in the sense of
Definition 2.12.

Let $({\cal A},V)$ be a local net of von Neumann algebras on $\mathbb{R}$,
acting on a Hilbert space $\mathcal{K}$. We assume that this net is in a
vacuum representation, with the vacuum vector $\Omega_{0}\in\mathcal{K}$. Let
$W$ be a unitary operator on $\mathcal{K}$ which implements a symmetry of this
net i.e.

$\displaystyle W{\cal A}({\cal I})W^{*}\subset{\cal A}({\cal I}),$ (3.23)
$\displaystyle WV(t)W^{*}=V(t),$ (3.24) $\displaystyle
W\Omega_{0}=\Omega_{0},$ (3.25)

for any open, bounded interval ${\cal I}\subset\mathbb{R}$ and any
$t\in\mathbb{R}$. We assume that $W$ gives rise to a non-trivial
representation of the group $\mathbb{Z}_{2}$ i.e. ${\hbox{\rm Ad}}W\neq{\rm
id}$ and $W^{2}=I$. We define the subspaces

$\displaystyle{\cal A}_{\mathrm{ev}}({\cal I})$ $\displaystyle=$
$\displaystyle\\{\,A\in{\cal A}({\cal I})\,|\,WAW^{*}=A\,\\},$ (3.26)
$\displaystyle{\cal A}_{\mathrm{odd}}({\cal I})$ $\displaystyle=$
$\displaystyle\\{\,A\in{\cal A}({\cal I})\,|\,WAW^{*}=-A\,\\}.$ (3.27)

Let ${\cal A}_{\mathrm{ev}}$ (resp. ${\cal A}_{\mathrm{odd}}$) be the norm-
closed linear span of all operators from some ${\cal A}_{\mathrm{ev}}({\cal
I})$ (resp. ${\cal A}_{\mathrm{odd}}({\cal I})$), ${\cal
I}\subset{\mathbb{R}}$. Clearly, $({\cal A}_{\mathrm{ev}},V)$ is again a local
net of von Neumann algebras on the real line acting on $\mathcal{K}$. We
introduce the subspaces $\mathcal{K}_{\mathrm{ev}}=[{\cal
A}_{\mathrm{ev}}\Omega_{0}]$, $\mathcal{K}_{\mathrm{odd}}=[{\cal
A}_{\mathrm{odd}}\Omega_{0}]$, where $[\,\cdot\,]$ denotes the closure, which
are invariant under the action of ${\cal A}_{\mathrm{ev}}$ and $V$, and
satisfy
$\mathcal{K}=\mathcal{K}_{\mathrm{ev}}\oplus\mathcal{K}_{\mathrm{odd}}$.
$\mathcal{K}_{\mathrm{odd}}$ gives rise to the representation

$\displaystyle\pi_{\mathrm{odd}}(A)$ $\displaystyle=$ $\displaystyle
A|_{\mathcal{K}_{\mathrm{odd}}},\quad A\in{\cal A}_{\mathrm{ev}},$ (3.28)
$\displaystyle V_{\mathrm{odd}}(t)$ $\displaystyle=$ $\displaystyle
V(t)|_{\mathcal{K}_{\mathrm{odd}}},\quad t\in\mathbb{R}.$ (3.29)

Its relevant properties are summarized in the following lemma, which we prove
in Appendix B.

######  Lemma 3.7.

$(\pi_{\mathrm{odd}},\mathcal{K}_{\mathrm{odd}})$ is a covariant, positive
energy representation of $({\cal A}_{\mathrm{ev}},V)$, in which the
translation automorphisms are implemented by $V_{\mathrm{odd}}$. Moreover:

  1. (a)

$\pi_{\mathrm{odd}}$ is a locally normal, faithful and irreducible
representation of ${\cal A}_{\mathrm{ev}}$.

  2. (b)

$V_{\mathrm{odd}}$ does not admit non-trivial invariant vectors.

We set $\hat{\cal A}:=\pi_{\mathrm{odd}}({\cal A}_{\mathrm{ev}})$,
$\hat{V}(t):=V_{\mathrm{odd}}(t)$. By the above lemma $(\hat{\cal A},\hat{V})$
is again a local net of von Neumann algebras on the real line. We define its
representation on $\mathcal{K}_{\mathrm{ev}}$

$\displaystyle\pi_{\mathrm{ev}}(\hat{A})$ $\displaystyle=$
$\displaystyle\pi_{\mathrm{odd}}^{-1}(\hat{A})|_{\mathcal{K}_{\mathrm{ev}}},\quad\hat{A}\in\hat{\cal
A},$ (3.30) $\displaystyle V_{\mathrm{ev}}(t)$ $\displaystyle=$ $\displaystyle
V(t)|_{\mathcal{K}_{\mathrm{ev}}},\quad t\in\mathbb{R}$ (3.31)

and state the following fact, whose proof is given in Appendix B.

######  Lemma 3.8.

$(\pi_{\mathrm{ev}},\mathcal{K}_{\mathrm{ev}})$ is a covariant, positive
energy representation of $(\hat{\cal A},\hat{V})$, in which the translation
automorphisms are implemented by $V_{\mathrm{ev}}$. Moreover

  1. (a)

$\pi_{\mathrm{ev}}$ is a locally normal, faithful and irreducible
representation of $\hat{\cal A}$.

  2. (b)

$V_{\mathrm{ev}}$ admits a unique (up to a phase) invariant vector, which is
cyclic for any $\pi_{\mathrm{ev}}(\hat{\cal A}({\cal I}))$.

We conclude that $(\pi_{\mathrm{ev}}(\hat{\cal A}),V_{\mathrm{ev}})$ is a
local net of von Neumann algebras in a vacuum representation with the vacuum
vector $\Omega_{0}\in\mathcal{K}_{\mathrm{ev}}$.

We remark that the above abstract construction can be performed in a number of
concrete cases. If a Möbius covariant net ${\cal I}\to{\cal A}({\cal I})$ on
$S^{1}$, in a vacuum representation, admits an automorphism777An automorphism
$\gamma$ of a net ${\cal A}$ is an automorphism of the quasilocal algebra
${\cal A}$ which preserves each local algebra ${\cal A}({\cal I})$. $\gamma$
of order 2 which preserves the vacuum state, then one can define $W$ by

$WA\Omega_{0}=\gamma(A)\Omega_{0},\quad A\in{\cal A}({\cal I}).$ (3.32)

This does not depend on the choice of the interval ${\cal I}$ and defines a
unitary operator thanks to the invariance of the vacuum state. This $W$
automatically commutes with the action of the Möbius group (in particular with
the action of translations) as a consequence of the Bisognano-Wichmann
property [31]. Thus, upon restriction to the real line, we obtain a local net
equipped with a unitary $W$ which satisfies (3.23)-(3.25). Non-trivial
automorphisms $\gamma$ appear, in particular, in the $U(1)$-current net
($\gamma:J(z)\to-J(z)$) [15], in loop group nets of a compact group $G$ with a
${\mathbb{Z}}_{2}$-subgroup in $G$ [51] and in the tensor product net ${\cal
A}\otimes{\cal A}$ for an arbitrary Möbius covariant net ${\cal A}$, where
$\gamma$ is the flip symmetry.

Coming back to the abstract setting, we introduce the class of two-dimensional
theories, we are interested in: Let $({\cal A}_{1},V_{1})$, $({\cal
A}_{2},V_{2})$ be two local nets of von Neumann algebras on $\mathbb{R}$, in
vacuum representations, acting on Hilbert spaces $\mathcal{K}_{1}$,
$\mathcal{K}_{2}$. We denote the respective vacuum vectors by $\Omega_{1}$,
$\Omega_{2}$ and introduce the corresponding regular chiral net
$({\mathfrak{A}},U)$. We assume the existence of unitaries $W_{1}$, $W_{2}$,
which give rise to non-trivial representations of $\mathbb{Z}_{2}$ and
implement symmetries of the respective nets on $\mathbb{R}$ as defined in
(3.23)-(3.25). By the construction described above we obtain the nets
$(\hat{\cal A}_{1},\hat{V}_{1})$, $(\hat{\cal A}_{2},\hat{V}_{2})$, acting on
$\mathcal{K}_{1,\mathrm{odd}}$, $\mathcal{K}_{2,\mathrm{odd}}$. We denote by
$(\hat{\mathfrak{A}}^{\textrm{vN}},\hat{U})$ the corresponding chiral net
acting on
$\hat{\mathcal{H}}=\mathcal{K}_{1,\mathrm{odd}}\otimes\mathcal{K}_{2,\mathrm{odd}}$
and by $(\hat{\mathfrak{A}},\hat{U})$ its regular subnet. Let us summarize its
properties.

######  Proposition 3.9.

The regular chiral net $(\hat{\mathfrak{A}},\hat{U})$, whose chiral components
are $(\hat{\cal A}_{1},\hat{V}_{1})$, $(\hat{\cal A}_{2},\hat{V}_{2})$, has
the following properties:

  1. (a)

$\hat{\mathfrak{A}}$ acts irreducibly on $\hat{\mathcal{H}}$.

  2. (b)

$(\hat{\mathfrak{A}},\hat{U})$ does not admit Wigner particles
($\hat{\mathcal{H}}_{\pm}=\\{0\\}$), but all the asymptotic functionals of the
form $\\{\,\psi_{\Psi}^{\mathrm{out}}\,|\,\Psi\neq 0,\,\Psi\in
P_{E}\hat{\mathcal{H}},\,E\geq 0\,\\}$ are non-zero.

  3. (c)

$\pi_{{\rm{R}}}=\iota_{1}\otimes\pi_{2,\mathrm{ev}}$ and
$\pi_{{\rm{L}}}=\pi_{1,\mathrm{ev}}\otimes\iota_{2}$ are irreducible,
faithful, covariant representations of $(\hat{\mathfrak{A}},\hat{U})$, acting
on
$\mathcal{H}_{\pi_{{\rm{R}}}}:=\mathcal{K}_{1,\mathrm{odd}}\otimes\mathcal{K}_{2,\mathrm{ev}}$
and
$\mathcal{H}_{\pi_{{\rm{L}}}}:=\mathcal{K}_{1,\mathrm{ev}}\otimes\mathcal{K}_{2,\mathrm{odd}}$,
respectively. The respective (canonical) unitary representations of
translations are given by
$U_{\pi_{\rm{R}}}(x):=U(x)|_{\mathcal{H}_{\pi_{{\rm{R}}}}}$ and
$U_{\pi_{\rm{L}}}(x):=U(x)|_{\mathcal{H}_{\pi_{{\rm{L}}}}}$.

  4. (d)

$\mathcal{H}_{\pi_{{\rm{R}}},-}=\\{0\\}$ and
$\mathcal{H}_{\pi_{{\rm{R}}},+}\neq\\{0\\}$ while
$\mathcal{H}_{\pi_{{\rm{L}}},-}\neq\\{0\\}$ and
$\mathcal{H}_{\pi_{{\rm{L}}},+}=\\{0\\}$. Consequently, $\pi_{{\rm{R}}}$ is
not unitarily equivalent to $\pi_{{\rm{L}}}$.

In part (c) $\iota_{1/2}$ are the defining representations of $\hat{\cal
A}_{1/2}$. Representations $\pi_{1/2,\mathrm{ev}}$ are defined as in (3.30),
(3.31).

Proof. Part (a) follows from the irreducibility of $\pi_{1/2,\mathrm{odd}}$,
shown in Lemma 3.7. As for part (b), we obtain from Lemma 3.7 (b) and
Proposition 3.5 that $\hat{\mathcal{H}}_{\pm}=\\{0\\}$. On the other hand,
Theorem 3.6 ensures that the relevant asymptotic functionals are non-zero.
Irreducibility and faithfulness of $\pi_{{\rm{R}}/{\rm{L}}}$ in part (c)
follow from Lemma 3.8 (a) and Lemma 3.7 (a). Proceeding to part (d), we note
that, by faithfulness of $\pi_{{\rm{R}}}$, the net
$(\pi_{{\rm{R}}}(\hat{\mathfrak{A}}),U_{\pi_{{\rm{R}}}})$ coincides with the
regular chiral subnet of
$(\pi_{{\rm{R}}}(\hat{\mathfrak{A}}^{\textrm{vN}}),U_{\pi_{{\rm{R}}}})$, whose
chiral components are $(\hat{\cal A}_{1},\hat{V}_{1})$ and
$(\pi_{2,\mathrm{ev}}(\hat{\cal A}_{2}),V_{2,\mathrm{ev}})$. From Lemma 3.7
(b), Lemma 3.8 (b) and Proposition 3.5 we obtain that
$\mathcal{H}_{\pi_{{\rm{R}}},-}=\\{0\\}$ and
$\mathcal{H}_{\pi_{{\rm{R}}},+}\neq\\{0\\}$. An analogous reasoning, applied
to $\pi_{{\rm{L}}}$, shows that $\mathcal{H}_{\pi_{{\rm{L}}},-}\neq\\{0\\}$
and $\mathcal{H}_{\pi_{{\rm{L}}},+}=\\{0\\}$. Hence, due to relation (2.3),
the two nets are not unitarily equivalent. $\Box$
In view of part (b) of the above proposition, the theory
$(\hat{\mathfrak{A}},\hat{U})$ describes infraparticles. In the following
theorem, which is our main result, we show that these infraparticles have
superselected direction of motion, in the sense of Definition 2.12.

######  Theorem 3.10.

Consider the regular chiral net $(\hat{\mathfrak{A}},\hat{U})$, constructed
above. Let $\psi\in\\{\,\psi_{\Psi}^{\mathrm{out}}\,|\,\Psi\neq 0,\,\Psi\in
P_{E}\hat{\mathcal{H}},\,E\geq 0\,\\}$ and let $\pi_{\psi}$ be its GNS
representation. Then $\pi_{\psi}$ is a type I representation with atomic
center. It contains both right-moving and left-moving irreducible
subrepresentations which are unitarily equivalent to $\pi_{{\rm{R}}}$ and
$\pi_{{\rm{L}}}$, respectively. Hence the theory describes infraparticles with
superselected direction of motion.

######  Remark 3.11.

Let us consider the regular chiral net $({\mathfrak{A}},U)$ in the vacuum
representation. Then, similarly as in the theorem above, the GNS
representation $\pi_{\psi}$ induced by any particle weight
$\psi\in\\{\,\psi_{\Psi}^{\mathrm{out}}\,|\,\Psi\notin{\mathbb{C}}\Omega,\Psi\in
P_{E}\mathcal{H},E\geq 0\,\\}$ is of type I with atomic center. However, any
non-trivial irreducible subrepresentation of $\pi_{\psi}$ is unitarily
equivalent to the defining vacuum representation i.e. $\psi$ is neutral. These
facts are easily verified by modifying the proof below.

Proof. Let us first consider the regular chiral net $({\mathfrak{A}},U)$
acting on $\mathcal{H}$. By Theorem 3.3, ${\cal K}_{+}:={\cal
K}_{1,\mathrm{odd}}\otimes{\mathbb{C}}\Omega_{2}\subset\mathcal{H}_{+}$ and
${\cal K}_{-}:={\mathbb{C}}\Omega_{1}\otimes{\cal
K}_{2,\mathrm{odd}}\subset\mathcal{H}_{-}$. Any vector $\Psi\in P_{E}({\cal
K}_{+}\overset{\mathrm{out}}{\times}{\cal K}_{-})$, $E\geq 0$, gives rise to
functionals $\rho_{\Psi,\pm}$, defined by (2.11), (2.12). They have the form

$\rho_{\pm,\Psi}(\,\cdot\,)=\sum_{n\in\mathbb{N}}(\Psi_{\pm,n}|\,\cdot\,\Psi_{\pm,n}),$
(3.33)

where $\Psi_{\pm,n}\in P_{E}{\cal K}_{\pm}$. (Cf. formula (2.13) and the
subsequent discussion). Since $\Psi\neq 0$, we can assume that $\Psi_{+,1}\neq
0$ and $\Psi_{-,1}\neq 0$. We also note for future reference that ${\cal
K}_{+}\subset\mathcal{H}_{\pi_{{\rm{R}}}}$ and ${\cal
K}_{-}\subset\mathcal{H}_{\pi_{{\rm{L}}}}$.

Let us now proceed to the net $(\hat{\mathfrak{A}},\hat{U})$, acting on
$\hat{\mathcal{H}}={\cal K}_{+}\overset{\mathrm{out}}{\times}{\cal
K}_{-}\subset\mathcal{H}$, and let $\hat{\mathcal{L}}$ be the left ideal of
$\hat{\mathfrak{A}}$, given by definition (2.16). For any
${\hat{L}}\in\hat{\mathcal{L}}$ we define

$L=(\pi_{1,\mathrm{odd}}^{-1}\otimes\pi_{2,\mathrm{odd}}^{-1})({\hat{L}})\in\mathcal{L},$
(3.34)

where $\mathcal{L}$ is the corresponding left ideal of ${\mathfrak{A}}$. We
note that such $L$ leaves the subspaces $\mathcal{H}_{\pi_{{\rm{R}}}}$ and
$\mathcal{H}_{\pi_{{\rm{L}}}}$ invariant. Exploiting Theorem 2.11 and formula
(2.13), we obtain

$\displaystyle\psi_{\Psi}^{\mathrm{out}}({\hat{L}}_{1},{\hat{L}}_{2})=\sum_{n\in\mathbb{N}}\int
d\boldsymbol{x}\,\\{(\Psi_{+,n}|(L_{1}^{*}L_{2})(\boldsymbol{x})\Psi_{+,n})+(\Psi_{-,n}|(L_{1}^{*}L_{2})(\boldsymbol{x})\Psi_{-,n})\\}.$
(3.35)

It follows from Theorem 2.9 that for any $L$ given by (3.34) the Fourier
transforms

$\displaystyle L\tilde{\Psi}_{+,n}(\boldsymbol{p})$ $\displaystyle:=$
$\displaystyle(2\pi)^{-1/2}\int
d\boldsymbol{x}\,e^{-i\boldsymbol{p}\boldsymbol{x}}LU_{\pi_{\rm{R}}}(\boldsymbol{x})^{*}\Psi_{+,n},$
(3.36) $\displaystyle L\tilde{\Psi}_{-,n}(\boldsymbol{p})$ $\displaystyle:=$
$\displaystyle(2\pi)^{-1/2}\int
d\boldsymbol{x}\,e^{i\boldsymbol{p}\boldsymbol{x}}LU_{\pi_{\rm{L}}}(\boldsymbol{x})^{*}\Psi_{-,n}$
(3.37)

belong to $\mathcal{H}_{\pi_{{\rm{R}}}}\otimes
L^{2}(\mathbb{R}_{+},d\boldsymbol{p})$ and
$\mathcal{H}_{\pi_{{\rm{L}}}}\otimes L^{2}(\mathbb{R}_{+},d\boldsymbol{p})$
respectively. Since $\pi_{{\rm{R}}}(\hat{\mathfrak{A}})$ acts irreducibly on
$\mathcal{H}_{\pi_{{\rm{R}}}}$ and $U_{\pi_{\rm{R}}}$ does not have non-zero
invariant vectors, we obtain from Lemma A.1 (a) the existence of
${\hat{L}}_{+}\in\hat{\mathcal{L}}$ s.t. $L_{+}\Psi_{+,1}\neq 0$. Since
$L_{+}U_{\pi_{\rm{R}}}(\boldsymbol{x})^{*}\Psi_{+,1}$ is a continuous function
of $\boldsymbol{x}$, it is nonzero as a square-integrable function, hence
$\\{L_{+}\tilde{\Psi}_{+,1}(\boldsymbol{p})\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\neq
0$. Analogously, we can find ${\hat{L}}_{-}\in\hat{\mathcal{L}}$ s.t.
$\\{L_{-}\tilde{\Psi}_{-,1}(\boldsymbol{p})\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\neq
0$. For future reference, we note the equalities

$\displaystyle\alpha_{x}(L)\tilde{\Psi}_{+,n}(\boldsymbol{p})$
$\displaystyle=$ $\displaystyle
e^{-i(\boldsymbol{p},\boldsymbol{p})x}U_{\pi_{\rm{R}}}(x)L\tilde{\Psi}_{+,n}(\boldsymbol{p}),$
(3.38) $\displaystyle\alpha_{x}(L)\tilde{\Psi}_{-,n}(\boldsymbol{p})$
$\displaystyle=$ $\displaystyle
e^{-i(\boldsymbol{p},-\boldsymbol{p})x}U_{\pi_{\rm{L}}}(x)L\tilde{\Psi}_{-,n}(\boldsymbol{p}),$
(3.39)

which hold in the sense of $\mathcal{H}_{\pi_{{\rm{R}}}}\otimes
L^{2}(\mathbb{R}_{+},d\boldsymbol{p})$ and
$\mathcal{H}_{\pi_{{\rm{L}}}}\otimes L^{2}(\mathbb{R}_{+},d\boldsymbol{p})$
respectively. These relations are easily verified for such $\Psi_{\pm,n}\in
P_{E}{\cal K}_{\pm}$ that $\mathbb{R}\ni\boldsymbol{x}\to
LU_{\pi_{{\rm{R}}/{\rm{L}}}}(\boldsymbol{x})^{*}\Psi_{\pm,n}$ decay rapidly in
norm as $|\boldsymbol{x}|\to\infty$, since in this case the Fourier transform
is pointwise defined. The general case follows from the fact that such vectors
form a dense subspace in $P_{E}{\cal K}_{\pm}$ (cf. formula (3.53) below) and
that the maps $P_{E}{\cal
K}_{\pm}\ni\Psi\to\\{L\tilde{\Psi}(\boldsymbol{p})\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\in\mathcal{H}_{\pi_{{\rm{R}}/{\rm{L}}}}\otimes
L^{2}(\mathbb{R}_{+},d\boldsymbol{p})$ are norm-continuous. This latter fact
is a consequence of Theorem 2.9 and the (Hilbert space valued) Plancherel
theorem.

After this preparation we study the structure of the GNS representation
induced by $\psi$. Let us first consider the following auxiliary
representation of $(\hat{\mathfrak{A}},\hat{U})$

$\pi_{1}(\,\cdot\,):=\bigoplus_{n\in\mathbb{N}}\big{(}\\{\pi_{{\rm{R}}}(\,\cdot\,)\otimes
I\\}\oplus\\{\pi_{{\rm{L}}}(\,\cdot\,)\otimes I\\}\big{)},$ (3.40)

acting on
$\mathcal{H}_{\pi_{1}}:=\bigoplus_{n\in\mathbb{N}}\big{(}\\{\mathcal{H}_{\pi_{{\rm{R}}}}\otimes
L^{2}(\mathbb{R}_{+},d\boldsymbol{p})\\}\oplus\\{\mathcal{H}_{\pi_{{\rm{L}}}}\otimes
L^{2}(\mathbb{R}_{+},d\boldsymbol{p})\\}\big{)}$. From definition (3.40) and
relation (2.22) we conclude that $\pi_{1}$ and its subrepresentations are of
type I with atomic center. Moreover, $\pi_{1}$ is covariant and it is easily
seen that the canonical representation of translations is given by

$U^{\mathrm{can}}_{\pi_{1}}(x)=\bigoplus_{n\in\mathbb{N}}\big{(}\big{\\{}U_{\pi_{\rm{R}}}(x)\otimes
I\big{\\}}\oplus\big{\\{}U_{\pi_{\rm{L}}}(x)\otimes I\big{\\}}).$ (3.41)

We note that $\pi_{\psi}$ is unitarily equivalent to a subrepresentation of
$\pi_{1}$. In fact, the map
$W_{1}:\mathcal{H}_{\pi_{\psi}}\to\mathcal{H}_{\pi_{1}}$, given by

$W_{1}:|{\hat{L}}\rangle\to\bigoplus_{n\in\mathbb{N}}\big{(}\\{\,L\tilde{\Psi}_{+,n}(\boldsymbol{p})\,\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\oplus\\{\,L\tilde{\Psi}_{-,n}(\boldsymbol{p})\,\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\big{)},$
(3.42)

intertwines the two representations and is an isometry by formula (3.35) and
the (Hilbert space valued) Plancherel theorem. It is easily checked that the
canonical representation of translations $U_{\pi_{\psi}}^{\mathrm{can}}$ in
the representation $\pi_{\psi}$ is given by the relation

$W_{1}U^{\mathrm{can}}_{\pi_{\psi}}(x)=U^{\mathrm{can}}_{\pi_{1}}(x)W_{1}.$
(3.43)

Recalling that
$U^{\mathrm{char}}_{\pi_{\psi}}(x)=U^{\mathrm{can}}_{\pi_{\psi}}(x)U_{\pi_{\psi}}^{-1}(x)$,
where $U_{\pi_{\psi}}$ is given by (2.20), we obtain

$\displaystyle W_{1}U^{\mathrm{char}}_{\pi_{\psi}}(x)|{\hat{L}}\rangle$
$\displaystyle=$ $\displaystyle
W_{1}U^{\mathrm{can}}_{\pi_{\psi}}(x)|\alpha_{-x}({\hat{L}})\rangle=U^{\mathrm{can}}_{\pi_{1}}(x)W_{1}|\alpha_{-x}({\hat{L}})\rangle$
(3.44) $\displaystyle=$
$\displaystyle\bigoplus_{n\in\mathbb{N}}\big{(}\\{\,U_{\pi_{\rm{R}}}(x)\alpha_{-x}(L)\tilde{\Psi}_{+,n}(\boldsymbol{p})\,\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\oplus\\{\,U_{\pi_{\rm{L}}}(x)\alpha_{-x}(L)\tilde{\Psi}_{-,n}(\boldsymbol{p})\,\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\big{)}$
$\displaystyle=$ $\displaystyle\bigoplus_{n\in\mathbb{N}}\big{(}\\{\,I\otimes
e^{i(\boldsymbol{p},\boldsymbol{p})x}\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\oplus\\{I\otimes
e^{i(\boldsymbol{p},-\boldsymbol{p})x}\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\big{)}W_{1}|{\hat{L}}\rangle,$

where in the last step we made use of relations (3.38), (3.39). Now let
$\boldsymbol{Q}$ be the generator of space translations of
$U^{\mathrm{char}}_{\pi_{\psi}}$ and let $\mathcal{H}_{\pi_{\psi},\mathrm{R}}$
(resp. $\mathcal{H}_{\pi_{\psi},\mathrm{L}}$) be its spectral subspace
corresponding to the interval $[0,\infty)$ (resp. $(-\infty,0)$). Then, by
formula (3.44),

$\displaystyle
W_{1}\mathcal{H}_{\pi_{\psi},\mathrm{R}}=P_{{\rm{R}}}W_{1}\mathcal{H}_{\pi_{\psi}},$
(3.45) $\displaystyle
W_{1}\mathcal{H}_{\pi_{\psi},\mathrm{L}}=P_{{\rm{L}}}W_{1}\mathcal{H}_{\pi_{\psi}},$
(3.46)

where $P_{{\rm{R}}/{\rm{L}}}$ are the projections on the subspaces
$\bigoplus_{n\in\mathbb{N}}\\{\mathcal{H}_{\pi_{{\rm{R}}/{\rm{L}}}}\otimes
L^{2}(\mathbb{R}_{+},d\boldsymbol{p})\\}$ in $\mathcal{H}_{\pi_{1}}$. From
definition (3.42) and the remarks after formula (3.37) we conclude that
$P_{{\rm{R}}}W_{1}\mathcal{H}_{\pi_{\psi}}\neq\\{0\\}$ and
$P_{{\rm{L}}}W_{1}\mathcal{H}_{\pi_{\psi}}\neq\\{0\\}$. Consequently
$\pi_{\psi}$ has both right-moving and left-moving irreducible
subrepresentations.

Let $\pi$ be an irreducible subrepresentation of $\pi_{\psi}$, acting on a
non-trivial subspace ${\cal K}\subset\mathcal{H}_{\pi_{\psi},\mathrm{R}}$
(i.e. a right-moving subrepresentation). Then $W_{1}\pi(\,\cdot\,)W_{1}^{*}$
is an irreducible subrepresentation of $\pi_{\rm{R}}(\,\cdot\,)\otimes I$
acting on
$\mathcal{H}_{\pi_{{\rm{R}}}}\otimes\big{(}\bigoplus_{n\in\mathbb{N}}L^{2}(\mathbb{R}_{+},d\boldsymbol{p})\big{)}$.
By irreducibility, we conclude that $\pi$ is unitarily equivalent to
$(\pi_{{\rm{R}}}(\,\cdot\,)\otimes
I)|_{\mathcal{H}_{\pi_{{\rm{R}}}}\otimes{\mathbb{C}}e}$ for some non-zero
$e\in\bigoplus_{n\in\mathbb{N}}L^{2}(\mathbb{R}_{+},d\boldsymbol{p})$. This
latter representation can be identified with $\pi_{{\rm{R}}}$. An analogous
argument shows that any left-moving irreducible subrepresentation of
$\pi_{\psi}$ is unitarily equivalent to $\pi_{{\rm{L}}}$. Hence, by
Proposition 3.9 (d), $(\hat{\mathfrak{A}},\hat{U})$ describes infraparticles
with superselected direction of motion. $\Box$
Let us assume for a moment that $\hat{\mathcal{H}}$ is separable. Then we
obtain from the above theorem and formula (2.29) that the GNS representation
of any particle weight $\psi^{\mathrm{out}}_{\Psi}$ of the net
$(\hat{\mathfrak{A}},\hat{U})$, where $\Psi\neq 0$ is a vector of bounded
energy, has the form

$\displaystyle\pi_{\psi^{\mathrm{out}}_{\Psi}}\simeq\int^{\oplus}_{Z_{{\rm{R}}}}d\mu_{{\rm{R}}}(z)\,\pi_{{\rm{R}},z}\oplus\int^{\oplus}_{Z_{{\rm{L}}}}d\mu_{{\rm{L}}}(z)\,\pi_{{\rm{L}},z}.$
(3.47)

Here $(Z_{{\rm{R}}/{\rm{L}}},d\mu_{{\rm{R}}/{\rm{L}}})$ are some Borel spaces
and $\pi_{{\rm{R}}/{\rm{L}},z}=\pi_{{\rm{R}}/{\rm{L}}}$ for all $z\in
Z_{{\rm{R}}/{\rm{L}}}$. A decomposition of $\psi^{\mathrm{out}}_{\Psi}$ into
pure particle weights, which induce the irreducible representations appearing
in the decomposition of $\pi_{\psi^{\mathrm{out}}_{\Psi}}$, was obtained by
Porrmann in [41, 42] (cf. formula (1.3) above). However, to apply Porrmann’s
abstract argument, one has to restrict attention to countable (resp.
separable) subsets of all the relevant objects and it is not guaranteed that
the resulting (restricted) pure particle weights extend to the original
domains. It is therefore worth pointing out that the theory
$(\hat{\mathfrak{A}},\hat{U})$ admits a large class of particle weights, whose
decomposition can be performed in the original framework. To our knowledge
this is the first such decomposition in the presence of infraparticles. (See
however [35] for some partial results on the Schroer model). These particle
weights belong to the set
$\\{\,\psi^{\mathrm{out}}_{\Psi}\,|\,\Psi\in\mathcal{D}\,\\}$, where
$\mathcal{D}\subset\hat{\mathcal{H}}$ is a dense domain spanned by vectors of
the form

$\Psi=F_{1}\Omega_{1}\otimes F_{2}\Omega_{2},$ (3.48)

where $F_{1}\in{\cal A}_{1,\mathrm{odd}},F_{2}\in{\cal A}_{2,\mathrm{odd}}$
are s.t. $F_{1}\otimes I,I\otimes F_{2}\in{\mathfrak{A}}$ are almost local and
have compact energy-momentum transfer (see formula (2.15)). The proof of the
following proposition exploits some ideas from [26].

######  Proposition 3.12.

Consider the regular chiral net $(\hat{\mathfrak{A}},\hat{U})$ constructed
above. Denote by $\hat{\mathcal{L}}$ its left ideal, given by definition
(2.16). Then, for any non-zero vector $\Psi\in\mathcal{D}$, there exist
continuous fields of pure particle weights
$\Delta_{{\rm{R}},n}\ni\boldsymbol{p}\to\psi_{{\rm{R}},n,\boldsymbol{p}}(\,\cdot\,,\,\cdot\,)$
and
$\Delta_{{\rm{L}},m}\ni\boldsymbol{p}\to\psi_{{\rm{L}},m,\boldsymbol{p}}(\,\cdot\,,\,\cdot\,)$
s.t. for any ${\hat{L}}_{1},{\hat{L}}_{2}\in\hat{\mathcal{L}}$

$\displaystyle\psi_{\Psi}^{\mathrm{out}}({\hat{L}}_{1},{\hat{L}}_{2})=\sum_{n\in
C_{{\rm{R}}}}\int_{\Delta_{{\rm{R}},n}}d\boldsymbol{p}\,\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}}_{1},{\hat{L}}_{2})+\sum_{m\in
C_{{\rm{L}}}}\int_{\Delta_{{\rm{L}},m}}d\boldsymbol{p}\,\psi_{{\rm{L}},m,\boldsymbol{p}}({\hat{L}}_{1},{\hat{L}}_{2}),$
(3.49)

where $C_{{\rm{R}}},C_{{\rm{L}}}\subset\mathbb{N}$ are non-empty finite
subsets and $\Delta_{{\rm{R}},n},\Delta_{{\rm{L}},m}\subset\mathbb{R}_{+}$ are
non-empty, open subsets for any $n\in C_{{\rm{R}}}$, $m\in C_{{\rm{L}}}$.
Moreover:

  1. (a)

The characteristic energy-momentum vectors of the weights
$\psi_{{\rm{R}},n,\boldsymbol{p}}$ (resp. $\psi_{{\rm{L}},m,\boldsymbol{p}}$)
are equal to $q_{{\rm{R}},n,\boldsymbol{p}}=(\boldsymbol{p},\boldsymbol{p})$
(resp. $q_{{\rm{L}},m,\boldsymbol{p}}=(\boldsymbol{p},-\boldsymbol{p})$).

  2. (b)

The GNS representation induced by any $\psi_{{\rm{R}},n,\boldsymbol{p}}$
(resp. $\psi_{{\rm{L}},m,\boldsymbol{p}}$) is unitarily equivalent to
$\pi_{{\rm{R}}}$ (resp. $\pi_{{\rm{L}}}$).

The representations $\pi_{{\rm{R}}/{\rm{L}}}$ appeared in Proposition 3.9.

######  Remark 3.13.

Parts (b) and (d) of Proposition 3.9 show that spectral properties of the
energy-momentum operators in the representations induced by the pure particle
weights $\psi_{{\rm{R}}/{\rm{L}},n,\boldsymbol{p}}$ are different from those
in the original representation: In the case of $U_{\pi_{{\rm{R}}}}$ the right
branch of the lightcone contains the singularities characteristic for Wigner
particles, while in the left branch such singularities are absent. (For
$U_{\pi_{{\rm{L}}}}$ the opposite situation occurs). For infraparticles in
physical spacetime a more radical version of this phenomenon may occur: There
one expects isolated singularities at the characteristic energy-momentum
values of the respective pure particle weights. (Cf. Section 2 (iii) of [16]).

Proof. Any vector $\Psi\in\mathcal{D}$ has the form

$\Psi=\sum_{k,l}c_{k,l}F_{{\rm{R}},k}\Omega_{1}\otimes
F_{{\rm{L}},l}\Omega_{2},$ (3.50)

where the sum is finite and $F_{{\rm{R}},k}$, $F_{{\rm{L}},l}$ have properties
specified below formula (3.48). Applying the Gram-Schmidt procedure, we can
ensure that the systems of vectors $\\{F_{{\rm{R}},k}\Omega_{1}\\}_{k=0}^{M}$,
$\\{F_{{\rm{L}},l}\Omega_{2}\\}_{l=0}^{N}$ are orthonormal. Since
$\hat{\mathcal{H}}=\mathcal{K}_{1,\mathrm{odd}}\otimes\mathcal{K}_{2,\mathrm{odd}}\subset\mathcal{K}_{1}\otimes\mathcal{K}_{2}=\mathcal{H}$,
we can write

$\Psi=\sum_{k,l}c_{k,l}\Phi^{\mathrm{out}}_{+}(F_{{\rm{R}},k}\otimes
I)\Phi^{\mathrm{out}}_{-}(I\otimes F_{{\rm{L}},l})\Omega,$ (3.51)

where we made use of Proposition 3.2 applied to the net $({\mathfrak{A}},U)$.
For any ${\hat{L}}\in\hat{\mathcal{L}}$ we define
$L=(\pi_{1,\mathrm{odd}}^{-1}\otimes\pi_{2,\mathrm{odd}}^{-1})({\hat{L}})\in\mathcal{L}$,
where $\mathcal{L}$ is the left ideal of ${\mathfrak{A}}$, given by definition
(2.16). In view of Theorem 2.11, we get

$\displaystyle\psi_{\Psi}^{\mathrm{out}}({\hat{L}}_{1},{\hat{L}}_{2})$
$\displaystyle=$ $\displaystyle\sum_{n\in C_{{\rm{R}}}}\int
d\boldsymbol{x}\,((G_{{\rm{R}},n}\otimes
I)\Omega|(L_{1}^{*}L_{2})(\boldsymbol{x})(G_{{\rm{R}},n}\otimes I)\Omega)$
(3.52) $\displaystyle+$ $\displaystyle\sum_{m\in C_{{\rm{L}}}}\int
d\boldsymbol{x}\,((I\otimes{G}_{{\rm{L}},m})\Omega|(L_{1}^{*}L_{2})(\boldsymbol{x})(I\otimes{G}_{{\rm{L}},m})\Omega),$

where $G_{{\rm{R}},n}=\sum_{k}c_{k,n}F_{{\rm{R}},k}$,
$G_{{\rm{L}},m}=\sum_{l}c_{m,l}F_{{\rm{L}},l}$ and the sets $C_{{\rm{R}}}$ and
$C_{{\rm{L}}}$ are chosen so that $\Psi_{{\rm{R}},n}:=(G_{{\rm{R}},n}\otimes
I)\Omega\neq 0$ and $\Psi_{{\rm{L}},m}:=(I\otimes G_{{\rm{L}},m})\Omega\neq 0$
for $n\in C_{{\rm{R}}}$ and $m\in C_{{\rm{L}}}$. We note that both sets are
non-empty, if $\Psi\neq 0$. (Cf. formula (2.13) and the subsequent remarks).

Let us consider the first sum in (3.52) above: Any $L\in\mathcal{L}$ is a
finite linear combination of operators of the form $AB$, where
$A,B\in{\mathfrak{A}}$ and $B$ is almost local and energy decreasing. Since we
assumed that $F_{{\rm{R}},k}\otimes I$ are almost local, the functions

$\displaystyle\mathbb{R}\ni\boldsymbol{x}\to
ABU_{\pi_{{\rm{R}}}}(\boldsymbol{x})^{*}(G_{{\rm{R}},n}\otimes
I)\Omega=A[B,(G_{{\rm{R}},n}\otimes I)(-\boldsymbol{x})]\Omega$ (3.53)

decrease in norm faster than any inverse power of $|\boldsymbol{x}|$.
Consequently, the Fourier transform

$\displaystyle L\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p}):=(2\pi)^{-1/2}\int
d\boldsymbol{x}\,e^{-i\boldsymbol{p}\boldsymbol{x}}LU_{\pi_{{\rm{R}}}}(\boldsymbol{x})^{*}(G_{{\rm{R}},n}\otimes
I)\Omega$ (3.54)

is a norm-continuous function. It is compactly supported in $\mathbb{R}_{+}$
due to the spectrum condition and the fact that the energy-momentum transfer
of each $G_{{\rm{R}},n}\otimes I$ is bounded. By the (Hilbert space valued)
Plancherel theorem, we can write

$\int d\boldsymbol{x}\,((G_{{\rm{R}},n}\otimes
I)\Omega|(L_{1}^{*}L_{2})(\boldsymbol{x})(G_{{\rm{R}},n}\otimes I)\Omega)=\int
d\boldsymbol{p}\,(L_{1}\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p})|L_{2}\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p})).$
(3.55)

We define

$\displaystyle\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}}_{1},{\hat{L}}_{2}):=(L_{1}\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p})|L_{2}\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p})).$
(3.56)

It is easy to see that non-zero $\psi_{{\rm{R}},n,\boldsymbol{p}}$ are
particle weights in the sense of Definition 2.6: Positivity and property 1 are
obvious. The continuity requirement in property 3 follows from the equality

$\displaystyle\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}}_{1},{\hat{L}}_{2}(y)-{\hat{L}}_{2})$
$\displaystyle=(2\pi)^{-1/2}\int
d\boldsymbol{x}\,e^{-i\boldsymbol{p}\boldsymbol{x}}(L_{1}\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p})|[(L_{2}(y)-L_{2}),(G_{{\rm{R}},n}\otimes
I)(-\boldsymbol{x})]\Omega)$ (3.57)

and from the dominated convergence theorem. Invariance under translations
(property 2) is a straightforward consequence of the formula

$\displaystyle\alpha_{x}(L)\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p})=e^{-i(\boldsymbol{p},\boldsymbol{p})x}U_{\pi_{{\rm{R}}}}(x)L\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p}),\quad
x\in\mathbb{R}^{2}.$ (3.58)

Making use of the above relation and the spectrum condition, it is easy to see
that the distribution

$\displaystyle\mathbb{R}^{2}\ni q\to(2\pi)^{-1}\int
d^{2}x\,e^{-iqx}\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}}_{1},\alpha_{x}({\hat{L}}_{2}))$
(3.59)

is supported in $V_{+}-(\boldsymbol{p},\boldsymbol{p})$.

Now let us show that any function
$\boldsymbol{p}\to\psi_{{\rm{R}},n,\boldsymbol{p}}(\,\cdot\,,\,\cdot\,)$,
$n\in C_{{\rm{R}}}$, is non-zero on a non-empty open set
$\Delta_{{\rm{R}},n}$: Since $(G_{{\rm{R}},n}\otimes
I)\Omega\in\mathcal{H}_{\pi_{{\rm{R}}}}$ is different from zero,
$U_{\pi_{{\rm{R}}}}$ does not have non-zero invariant vectors and
$\pi_{{\rm{R}}}(\hat{\mathfrak{A}})$ acts irreducibly on
$\mathcal{H}_{\pi_{{\rm{R}}}}$, Lemma A.1 ensures the existence of
${\hat{L}}\in\hat{\mathcal{L}}$ s.t. $L(G_{{\rm{R}},n}\otimes I)\Omega\neq 0$.
Consequently, $\mathbb{R}\ni\boldsymbol{x}\to
LU_{\pi_{{\rm{R}}}}(\boldsymbol{x})^{*}(G_{{\rm{R}},n}\otimes I)\Omega$ is a
non-zero function and so is the norm of its Fourier transform
$\mathbb{R}_{+}\ni\boldsymbol{p}\to\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}},{\hat{L}})$.
Since the functions
$\mathbb{R}_{+}\ni\boldsymbol{p}\to\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}}_{1},{\hat{L}}_{2})$
are continuous for all ${\hat{L}}_{1},{\hat{L}}_{2}\in\hat{\mathcal{L}}$, as
we have shown above, the sets

$\Delta_{{\rm{R}},n}:=\bigcup_{{\hat{L}}_{1},{\hat{L}}_{2}\in\hat{\mathcal{L}}}\\{\,\boldsymbol{p}\in\mathbb{R}_{+}\,|\,\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}}_{1},{\hat{L}}_{2})\neq
0\,\\}$ (3.60)

are open and non-empty.

Let us now fix some $n\in C_{{\rm{R}}}$,
$\boldsymbol{p}\in\Delta_{{\rm{R}},n}$ and consider the GNS representation
$\pi$ induced by $\psi_{{\rm{R}},n,\boldsymbol{p}}$, acting on the Hilbert
space
$\mathcal{H}_{\pi}:=(\hat{\mathcal{L}}/\\{\,{\hat{L}}\in\hat{\mathcal{L}}\,|\,\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}},{\hat{L}})=0\,\\})^{\mathrm{cpl}}$.
The equivalence class of ${\hat{L}}\in\hat{\mathcal{L}}$ is denoted by
$|{\hat{L}}\rangle$ and the scalar product is given by
$\langle{\hat{L}}_{1}|{\hat{L}}_{2}\rangle=\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}}_{1},{\hat{L}}_{2})$.
This GNS representation has the form

$\displaystyle\pi(\hat{A})|{\hat{L}}\rangle$ $\displaystyle=$
$\displaystyle|\hat{A}{\hat{L}}\rangle,\quad\,\,\,\,\,\,{\hat{L}}\in\hat{\mathcal{L}},\,\hat{A}\in\hat{\mathfrak{A}},$
(3.61) $\displaystyle U_{\pi}(x)|{\hat{L}}\rangle$ $\displaystyle=$
$\displaystyle|\alpha_{x}({\hat{L}})\rangle,\quad{\hat{L}}\in\hat{\mathcal{L}},\,x\in\mathbb{R}^{2},$
(3.62)

where $U_{\pi}$ is the standard representation of translations. We will show
that $(\pi(\hat{\mathfrak{A}}),U_{\pi})$ is unitarily equivalent to
$(\pi_{{\rm{R}}}(\hat{\mathfrak{A}}),U_{\pi_{{\rm{R}}}})$. To this end, we
introduce the map
$W_{{\rm{R}}}:\mathcal{H}_{\pi}\to\mathcal{H}_{\pi_{{\rm{R}}}}=\mathcal{K}_{1,\mathrm{odd}}\otimes\mathcal{K}_{2,\mathrm{ev}}$
given by

$W_{{\rm{R}}}|{\hat{L}}\rangle=L\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p}),\quad{\hat{L}}\in\hat{\mathcal{L}}.$
(3.63)

This map is clearly an isometry. Since $\pi_{{\rm{R}}}$ acts irreducibly on
$\mathcal{H}_{\pi_{{\rm{R}}}}$, we obtain that $W_{{\rm{R}}}$ has a dense
range and hence it is a unitary operator. From the relation

$W_{{\rm{R}}}\pi(\hat{A})|{\hat{L}}\rangle=\pi_{{\rm{R}}}(\hat{A})L\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p})=\pi_{{\rm{R}}}(\hat{A})W_{{\rm{R}}}|{\hat{L}}\rangle,\quad{\hat{L}}\in\hat{\mathcal{L}},\,\,\hat{A}\in\hat{\mathfrak{A}}$
(3.64)

we conclude that $\pi$ and $\pi_{{\rm{R}}}$ are unitarily equivalent. Next, we
obtain for any ${\hat{L}}\in\hat{\mathcal{L}}$ and $x\in\mathbb{R}^{2}$

$\displaystyle
U_{\pi_{{\rm{R}}}}(x)W_{{\rm{R}}}|{\hat{L}}\rangle=e^{i(\boldsymbol{p},\boldsymbol{p})x}\alpha_{x}(L)\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p})=e^{i(\boldsymbol{p},\boldsymbol{p})x}W_{{\rm{R}}}U_{\pi}(x)|{\hat{L}}\rangle,$
(3.65)

where in the first step we made use of relation (3.58). We recall that the
spectrum of $U_{\pi_{{\rm{R}}}}$ coincides with $V_{+}$ and note that
$\pi(\hat{\mathfrak{A}})$ acts irreducibly on $\mathcal{H}_{\pi}$, by relation
(3.64) and Proposition 3.9 (c). Thus, in view of equality (3.65),

$U^{\mathrm{can}}_{\pi}(x)=e^{i(\boldsymbol{p},\boldsymbol{p})x}U_{\pi}(x),\quad
x\in\mathbb{R}^{2}$ (3.66)

is the canonical representation of translations in the GNS representation of
$\psi_{{\rm{R}},n,\boldsymbol{p}}$. Relation (3.66) shows that
$q_{{\rm{R}},n,\boldsymbol{p}}=(\boldsymbol{p},\boldsymbol{p})$.

The analysis of the second term on the r.h.s. of (3.52) proceeds similarly:
For any $m\in C_{{\rm{L}}}$ and ${\hat{L}}\in\hat{\mathcal{L}}$ one introduces
vectors

$L\tilde{\Psi}_{{\rm{L}},m}(\boldsymbol{p}):=(2\pi)^{-1/2}\int
d\boldsymbol{x}\,e^{i\boldsymbol{p}\boldsymbol{x}}LU(\boldsymbol{x})^{*}(I\otimes
G_{{\rm{L}},m})\Omega$ (3.67)

and functionals
$\psi_{{\rm{L}},m,\boldsymbol{p}}({\hat{L}}_{1},{\hat{L}}_{2})=(L_{1}\tilde{\Psi}_{{\rm{L}},m}(\boldsymbol{p})|L_{2}\tilde{\Psi}_{{\rm{L}},m}(\boldsymbol{p}))$.
By an analogous reasoning as above one shows that for $\boldsymbol{p}$ in some
non-empty, open set $\Delta_{{\rm{L}},m}\subset\mathbb{R}_{+}$ these
functionals are particle weights with characteristic energy-momentum vectors
$q_{{\rm{L}},m,\boldsymbol{p}}=(\boldsymbol{p},-\boldsymbol{p})$. Their GNS
representations are unitarily equivalent to $\pi_{{\rm{L}}}$. $\Box$

##  4 Conclusions and outlook

In this work we carried out a systematic study of particle aspects of two-
dimensional conformal field theories both in vacuum representations and in
charged representations. In the former case we established a complete particle
interpretation in terms of Wigner particles (or ‘waves’ in the terminology of
[5]). In the latter case we proved the existence of infraparticles and
verified superselection of their direction of motion in a large class of
examples. We conclude that conformal field theories provide a valuable testing
ground for fundamental concepts of scattering theory.

An important question which remained outside of the scope of the present work
is the problem of asymptotic completeness in the case of infraparticles. We
remark that the theory of particle weights offers natural formulations of this
property [9, 11] which can be adapted to the case of massless, two-dimensional
theories. We conjecture that any charged representation of a chiral conformal
field theory has a complete particle interpretation in terms of
infraparticles.

A more technical circle of problems concerns the decomposition of particle
weights and their representations stated in formulas (1.2), (1.3). We recall
that the general procedure of [41, 42] is not canonical: Firstly, it involves
a choice of a maximal abelian subalgebra, acting on the representation space
of the original weight. Secondly, it relies on a selection of countable
subsets of all the objects involved. In view of these ambiguities it is not
yet possible to associate a unique family of (infra-)particle types with any
given quantum field theory. We feel that a satisfactory solution of these
problems requires a systematic study of examples. A useful criterion for their
classification is the type of representations induced by particle weights.
Thus in the present paper we focused on representations of type I (with atomic
center) which have a simple decomposition theory. Already in this elementary
case we found a physically interesting phenomenon: superselection of direction
of motion. It is a natural direction of further research to look for theories,
whose asymptotic functionals induce representations which are not of type I
with atomic center. We conjecture that such models exist and some of them
describe infraparticles with superselected momentum, similar to the electron
in QED.

Acknowledgements. The authors would like to thank Prof. D. Buchholz and Prof.
R. Longo for interesting discussions.

##  Appendix A Proof of Theorem 2.11

######  Lemma A.1.

(a) Let $({\mathfrak{A}},U)$ be a local net of $C^{*}$-algebras on
$\mathbb{R}^{2}$ in the sense of Definition 2.1, acting irreducibly on a
Hilbert space $\mathcal{H}$ and let $U=U^{\mathrm{can}}$. Let
$\Psi\in\mathcal{H}$ be s.t.

$A(f)\Psi:=\int d^{2}x\,\alpha_{x}(A)f(x)\Psi=0$ (A.1)

for all local operators $A\in{\mathfrak{A}}$ and all $f\in S(\mathbb{R}^{2})$
s.t. $\mathrm{supp}\,\tilde{f}$ is compact and $\mathrm{supp}\,\tilde{f}\cap
V_{+}=\varnothing$. Then $\Psi$ is invariant under the action of $U$. (Here
$\tilde{f}(p):=(2\pi)^{-1}\int d^{2}x\,e^{ipx}f(x)$ ).
(b) Let $({\cal A},V)$ be a local net of von Neumann algebras on $\mathbb{R}$
in the sense of Definition 3.1, acting irreducibly on a Hilbert space ${\cal
K}$. Let $\Psi\in{\cal K}$ be s.t.

$A(f)\Psi:=\int ds\,\beta_{s}(A)f(s)\Psi=0$ (A.2)

for all local operators $A\in{\cal A}$ and all $f\in S(\mathbb{R})$ s.t.
$\mathrm{supp}\,\tilde{f}$ is compact and
$\mathrm{supp}\,\tilde{f}\cap\mathbb{R}_{+}=\varnothing$. Then $\Psi$ is
invariant under the action of $V$. (Here
$\tilde{f}(\omega):=(2\pi)^{-\frac{1}{2}}\int ds\,e^{i\omega s}f(s)$ ).

Proof. The argument exploits some ideas from the proof of Proposition 2.1 of
[13]. As for part (a), suppose that $\Psi$ is not invariant under the action
of $U$. Since the map $B(\mathcal{H})\ni A\to A(f)$ is $\sigma$-weakly
continuous (cf. Lemma 5.3 of [41]) and ${\mathfrak{A}}$ acts irreducibly on
$\mathcal{H}$, condition (A.1) implies that

$P(\Delta_{1})A(f)P(\Delta_{2})\Psi=0,$ (A.3)

where $P(\,\cdot\,)$ is the spectral measure of $U$ and
$\Delta_{1},\Delta_{2}\subset\mathbb{R}^{2}$ are open bounded sets. Since the
spectrum of $U$ has Lorentz invariant lower boundary and $\Psi$ is not
invariant under translations, we can choose $\Delta_{1},\Delta_{2}$ s.t.
$P(\Delta_{1})\neq 0$, $P(\Delta_{2})\Psi\neq 0$ and the closure of
$(\Delta_{1}-\Delta_{2})$ does not intersect with $V_{+}$. Choosing $f\in
S(\mathbb{R}^{2})$ s.t. $\mathrm{supp}\,\tilde{f}\cap V_{+}=\varnothing$ and
$\tilde{f}(p)=1$ for $p$ in the closure of $(\Delta_{1}-\Delta_{2})$, we
obtain that

$P(\Delta_{1})AP(\Delta_{2})\Psi=0$ (A.4)

for any $A\in{\mathfrak{A}}$. Exploiting irreducibility again, we obtain
$P(\Delta_{1})=0$, which is a contradiction. The proof of part (b) is
analogous. $\Box$

######  Lemma A.2.

Let ${\cal K}_{\pm}\subset\mathcal{H}_{\pm}$ be closed subspaces, invariant
under the action of $U$. Let $\\{e_{+,m}\\}_{m\in\mathbb{I}}$ be a complete
orthonormal basis in $(P_{E}{\cal K}_{+})$ and let
$\\{e_{-,n}\\}_{n\in\mathbb{J}}$ be a complete orthonormal basis in
$(P_{E}{\cal K}_{-})$ for some $E\geq 0$. Then any $\Psi\in P_{E}({\cal
K}_{+}\overset{\mathrm{out}}{\times}{\cal K}_{-})$ can be expressed as

$\Psi=\sum_{m,n}c_{m,n}e_{+,m}\overset{\mathrm{out}}{\times}e_{-,n},$ (A.5)

where $\sum_{m,n}|c_{m,n}|^{2}<\infty$.

Proof. First, we define a strongly continuous unitary representation of
translations

$U_{0}(x)(\Psi_{+}\otimes\Psi_{-})=(U(x)\Psi_{+})\otimes(U(x)\Psi_{-}),\quad\Psi_{\pm}\in{\cal
K}_{\pm}$ (A.6)

on ${\cal K}_{+}\otimes{\cal K}_{-}$. Then we obtain from Proposition 2.4

$\Omega^{\mathrm{out}}U_{0}(x)=U(x)\Omega^{\mathrm{out}}.$ (A.7)

For $\Psi^{\prime}=(\Omega^{\mathrm{out}})^{-1}\Psi$ the above relation gives
$P_{0,E}\Psi^{\prime}=\Psi^{\prime}$, where $P_{0,E}$ is the spectral
projection of $U_{0}$ corresponding to the set
$\\{\,(\omega,\boldsymbol{p})\in\mathbb{R}^{2}\,|\,\omega\leq E\,\\}$. By the
functional calculus, we get $P_{0,E}=P_{0,E}(P_{E}\otimes P_{E})$. Hence

$\Psi^{\prime}=(P_{E}\otimes
P_{E})\Psi^{\prime}=\sum_{m,n}c_{m,n}e_{+,m}\otimes e_{-,n}.$ (A.8)

By applying $\Omega^{\mathrm{out}}$, we obtain relation (A.5). $\Box$

######  Lemma A.3.

Let $\Psi^{\prime}\in\mathcal{H}$ be a vector of bounded energy, let
$F_{1},F_{2}\in{\mathfrak{A}}$ be almost local and of compact energy-momentum
transfer, and let $L=\sum_{k=1}^{n}A_{k}B_{k}$, where
$A_{k},B_{k}\in{\mathfrak{A}}$ are almost local and $B_{k}$ are, in addition,
energy decreasing. Then

$\lim_{T\to\infty}(\Psi^{\prime}|[[Q_{T},\Phi^{\mathrm{out}}_{+}(F_{1})],\Phi^{\mathrm{out}}_{-}(F_{2})]\Omega)=0,$
(A.9)

where $Q_{T}=\int dt\,h_{T}(t)\int
d\boldsymbol{x}\,(L^{*}L)(t,\boldsymbol{x})$. (The above sequence is well
defined by Theorem 2.9).

Proof. First, we note that by Proposition 2.3 and Theorem 2.9

$\lim_{T\to\infty}(\Psi^{\prime}|[[Q_{T},\Phi^{\mathrm{out}}_{+}(F_{1})],\Phi^{\mathrm{out}}_{-}(F_{2})]\Omega)=\lim_{T\to\infty}(\Psi^{\prime}|[[Q_{T},F_{1,+}(h_{T})],F_{2,-}(h_{T})]\Omega),$
(A.10)

if the limit on the r.h.s. exists. We introduce the auxiliary operators:

$Q_{\pm,T}:=\int
dt\,h_{T}(t)\int_{\mathbb{R}_{\pm}}d\boldsymbol{x}\,(L^{*}L)(t,\boldsymbol{x}).$
(A.11)

As we show below, they satisfy

$\lim_{T\to\infty}\|P_{E}[Q_{\pm,T},F_{\mp}(h_{T})]P_{E}\|=0$ (A.12)

for any $E\geq 0$ and any $F\in{\mathfrak{A}}$ which is almost local and of
compact energy-momentum transfer. Making use of this relation and the fact
that $Q_{T}=Q_{+,T}+Q_{-,T}$, the proof is completed with the help of the
Jacobi identity and Proposition 2.3 (c).

Let us now verify (A.12). As the two cases are analogous, we focus on one of
them and estimate the corresponding expression as follows.

$\|P_{E}[Q_{-,T},F_{+}(h_{T})]P_{E}\|\leq\int
dtdt_{1}\,h_{T}(t)h_{T}(t_{1})\int_{\mathbb{R}_{-}}d\boldsymbol{x}\|[L^{*}L(t,\boldsymbol{x}),F(t_{1},t_{1})]\|.$
(A.13)

Since $L^{*}L$ and $F$ are almost local, we can find sequences
$C_{r},F_{r}\in{\mathfrak{A}}(\mathcal{O}_{r})$, s.t. for any $n\in\mathbb{N}$
there exist $C_{n},C^{\prime}_{n}$ s.t.

$\displaystyle\|L^{*}L-C_{r}\|\leq\frac{C_{n}}{r^{n}},\quad\|F-F_{r}\|\leq\frac{C^{\prime}_{n}}{r^{n}}.$
(A.14)

We choose $r=(1+\frac{1}{4}|\boldsymbol{x}|)^{\varepsilon}+T^{\varepsilon}$,
where $0<\varepsilon<1$ appeared in the definition of $h_{T}$. We write

$\displaystyle[(L^{*}L)(t,\boldsymbol{x}),F(t_{1},t_{1})]$ $\displaystyle=$
$\displaystyle[(L^{*}L-C_{r})(t,\boldsymbol{x}),F(t_{1},t_{1})]$ (A.15)
$\displaystyle+$
$\displaystyle[C_{r}(t,\boldsymbol{x}),(F-F_{r})(t_{1},t_{1})]$
$\displaystyle+$ $\displaystyle[C_{r}(t,\boldsymbol{x}),F_{r}(t_{1},t_{1})].$

By estimates (A.14), the first two terms on the r.h.s. above give
contributions to (A.13) which tend to zero in the limit $T\to\infty$. The
contribution of the last term can be estimated as follows, exploiting
locality,

$\displaystyle\int
dtdt_{1}\,h_{T}(t)h_{T}(t_{1})\int_{\mathbb{R}_{-}}d\boldsymbol{x}\|[C_{r}(t,\boldsymbol{x}),F_{r}(t_{1},t_{1})]\|$
$\displaystyle\phantom{444444444444}\leq c\int
dtdt_{1}\,h_{T}(t)h_{T}(t_{1})\int_{\mathbb{R}_{-}}d\boldsymbol{x}\chi(|\boldsymbol{x}-t_{1}|\leq|t-t_{1}|+2r),$
(A.16)

where $\chi$ is the characteristic function of the respective set and $c$ is a
constant independent of $T$. Let us now derive some inequalities which hold on
the support of the integrand on the r.h.s. of (A.16). First, we note that
$t,t_{1}\in\mathrm{supp}\,h_{T}$, if and only if $t,t_{1}\in
T^{\varepsilon}\mathrm{supp}\,h+T$, in particular $|t-t_{1}|\leq
c_{1}T^{\varepsilon}$ for some $c_{1}\geq 0$. Exploiting this fact, the
inequality $|\boldsymbol{x}-t_{1}|\leq|t-t_{1}|+2r$ and the relation
$r=(1+\frac{1}{4}|\boldsymbol{x}|)^{\varepsilon}+T^{\varepsilon}$, we find
such $c_{2}\geq 0$ that $|\boldsymbol{x}|\leq c_{2}T$ and $r\leq
c_{2}T^{\varepsilon}$, in particular the r.h.s. of (A.16) is finite for any
$T\geq 1$. Making use of the inequalities $r\leq c_{2}T^{\varepsilon}$ and
$|\boldsymbol{x}-t_{1}|\leq|t-t_{1}|+2r$, and of the fact that
$t,t_{1}\in\mathrm{supp}\,h_{T}$, we obtain that $|\boldsymbol{x}-T|\leq
c_{3}T^{\varepsilon}$ for some $c_{3}\geq 0$ which implies that
$\boldsymbol{x}>0$ for sufficiently large $T$. As the region of integration in
the $\boldsymbol{x}$ variable is restricted to $\mathbb{R}_{-}$, we conclude
that the r.h.s. of (A.16) is zero for such $T$. $\Box$

Proof of Theorem 2.11: Let $Q_{T}=\int dt\,h_{T}(t)\int
d\boldsymbol{x}\,(L^{*}L)(t,\boldsymbol{x})$, where
$L=\sum_{k=1}^{n}A_{k}B_{k}$ is an element of the left ideal $\mathcal{L}$,
$A_{k}$ are almost local and $B_{k}\in\mathcal{L}_{0}$. Moreover, we choose
$\Psi=\Phi^{\mathrm{out}}_{+}(F_{1})\Phi^{\mathrm{out}}_{-}(F_{2})\Omega$ and
$\Psi^{\prime}=\Phi^{\mathrm{out}}_{+}(F_{1}^{\prime})\Phi^{\mathrm{out}}_{-}(F_{2}^{\prime})\Omega$,
where $F_{1/2},F_{1/2}^{\prime}\in{\mathfrak{A}}$ are almost local and have
compact energy-momentum transfer. Since $\Psi$ and $\Psi^{\prime}$ are vectors
of bounded energy, we can write

$\displaystyle(\Psi^{\prime}|Q_{T}\Phi^{\mathrm{out}}_{+}(F_{1})\Phi^{\mathrm{out}}_{-}(F_{2})\Omega)$
$\displaystyle=$
$\displaystyle(\Psi^{\prime}|[[Q_{T},\Phi^{\mathrm{out}}_{+}(F_{1})],\Phi^{\mathrm{out}}_{-}(F_{2})]\Omega)$
(A.17) $\displaystyle+$
$\displaystyle(\Psi^{\prime}|\Phi^{\mathrm{out}}_{-}(F_{2})Q_{T}\Phi^{\mathrm{out}}_{+}(F_{1})\Omega)$
$\displaystyle+$
$\displaystyle(\Psi^{\prime}|\Phi^{\mathrm{out}}_{+}(F_{1})Q_{T}\Phi^{\mathrm{out}}_{-}(F_{2})\Omega).$

The term with the double commutator above vanishes as $T\to\infty$ due to
Lemma A.3. The second term on the r.h.s. of relation (A.17) is treated as
follows:

$\displaystyle\lim_{T\to\infty}(\Psi^{\prime}|\Phi^{\mathrm{out}}_{-}(F_{2})Q_{T}\Phi^{\mathrm{out}}_{+}(F_{1})\Omega)$
$\displaystyle\phantom{44444}=\lim_{T\to\infty}(\Psi^{\prime}|\Phi^{\mathrm{out}}_{-}(F_{2})\int
h_{T}(t)e^{iHt}\int
d\boldsymbol{x}\,(L^{*}L)(\boldsymbol{x})e^{-i\boldsymbol{P}t}\Phi^{\mathrm{out}}_{+}(F_{1})\Omega)$
$\displaystyle\phantom{444444}=\lim_{T\to\infty}(\Psi^{\prime}|\Phi^{\mathrm{out}}_{-}(F_{2})\int
h_{T}(t)e^{i(H-\boldsymbol{P})t}\int
d\boldsymbol{x}\,(L^{*}L)(\boldsymbol{x})\Phi^{\mathrm{out}}_{+}(F_{1})\Omega)$
$\displaystyle\phantom{4444444444444444444444}=(\Psi^{\prime}|\Phi^{\mathrm{out}}_{-}(F_{2})P_{+}\int
d\boldsymbol{x}\,(L^{*}L)(\boldsymbol{x})\Phi^{\mathrm{out}}_{+}(F_{1})\Omega),$
(A.18)

where in the first step we made use of the fact that
$\Phi^{\mathrm{out}}_{+}(F_{1})\Omega=P_{+}\Phi^{\mathrm{out}}_{+}(F_{1})\Omega$,
in the second step we exploited the invariance of $\int
d\boldsymbol{x}\,(L^{*}L)(\boldsymbol{x})$ under translations in space and in
the last step we made use of the mean ergodic theorem as in the proof of Lemma
1 of [5]. Next, we obtain

$\displaystyle(\Phi^{\mathrm{out}}_{+}(F_{1}^{\prime})\Phi^{\mathrm{out}}_{-}(F_{2}^{\prime})\Omega|\Phi^{\mathrm{out}}_{-}(F_{2})P_{+}\int
d\boldsymbol{x}\,(L^{*}L)(\boldsymbol{x})\Phi^{\mathrm{out}}_{+}(F_{1})\Omega)$
$\displaystyle\phantom{44}=(\Phi^{\mathrm{out}}_{-}(F_{2})^{*}\Phi^{\mathrm{out}}_{-}(F_{2}^{\prime})\Omega|\Phi^{\mathrm{out}}_{+}(F_{1}^{\prime})^{*}P_{+}\int
d\boldsymbol{x}\,(L^{*}L)(\boldsymbol{x})\Phi^{\mathrm{out}}_{+}(F_{1})\Omega)$
$\displaystyle\phantom{44444}=(\Omega|\Phi^{\mathrm{out}}_{-}(F_{2}^{\prime})^{*}\Phi^{\mathrm{out}}_{-}(F_{2})\Omega)\,(\Omega|\Phi^{\mathrm{out}}_{+}(F_{1}^{\prime})^{*}\int
d\boldsymbol{x}\,(L^{*}L)(\boldsymbol{x})\Phi^{\mathrm{out}}_{+}(F_{1})\Omega),$
(A.19)

where we made use of the facts that
$[\Phi^{\mathrm{out}}_{+}(F_{1}),\Phi^{\mathrm{out}}_{-}(F_{2})]=0$ and that
$\mathcal{H}_{+}/{\mathbb{C}}\Omega$ is orthogonal to
$\mathcal{H}_{-}/{\mathbb{C}}\Omega$ (as in the proof of Lemma 4 (a) of [5]).
The last term on the r.h.s. of (A.17) is treated analogously.

We note that any $\Psi_{\pm}\in P_{E}\mathcal{H}_{\pm}$ can be approximated by
a sequence of vectors of the form $P_{\pm}F_{n}\Omega$, where
$F_{n}\in{\mathfrak{A}}$ are quasilocal and have energy-momentum transfers in
some fixed compact set. Hence, any
$\Psi=\Psi_{+}\overset{\mathrm{out}}{\times}\Psi_{-}$ has bounded energy. By
the above considerations and Theorem 2.9, we obtain for any
$\Psi=\Psi_{+}\overset{\mathrm{out}}{\times}\Psi_{-}$,
$\Psi^{\prime}=\Psi_{+}^{\prime}\overset{\mathrm{out}}{\times}\Psi_{-}^{\prime}$,
$\Psi_{\pm},\Psi_{\pm}^{\prime}\in P_{E}\mathcal{H}_{\pm}$,

$\displaystyle\lim_{T\to\infty}(\Psi^{\prime}|Q_{T}\Psi)$ $\displaystyle=$
$\displaystyle(\Psi_{+}^{\prime}|\Psi_{+})\int
d\boldsymbol{x}\,(\Psi_{-}^{\prime}|(L^{*}L)(\boldsymbol{x})\Psi_{-})$ (A.20)
$\displaystyle+$ $\displaystyle(\Psi_{-}^{\prime}|\Psi_{-})\int
d\boldsymbol{x}\,(\Psi_{+}^{\prime}|(L^{*}L)(\boldsymbol{x})\Psi_{+}).$

Now in view of Lemma A.2, any $\Psi\in P_{E}\mathcal{H}^{\mathrm{out}}$ has
the form

$\Psi=\sum_{m,n}c_{m,n}e_{+,m}\overset{\mathrm{out}}{\times}e_{-,n},$ (A.21)

where $\\{e_{\pm,m}\\}_{m=0}^{\infty}$ are orthonormal systems in
$\\{P_{E}\mathcal{H}_{\pm}\\}$, which we choose so that $e_{\pm,0}=\Omega$.
Defining

$\displaystyle\Psi_{+,n}=\sum_{m}c_{m,n}e_{+,m},\quad\Psi_{-,n}=\sum_{m}c_{n,m}e_{-,m},$
(A.22)

we obtain
$\rho_{\pm,\Psi}(\,\cdot\,)=\sum_{n}\,(\Psi_{\pm,n}|\,\cdot\,\Psi_{\pm,n})$.
Relation (A.20) gives

$\lim_{T\to\infty}(\Psi|Q_{T}\Psi)=\int
d\boldsymbol{x}\,(\rho_{+,\Psi}+\rho_{-,\Psi})\big{(}(L^{*}L)(\boldsymbol{x})\big{)}.$
(A.23)

Exploiting the Cauchy-Schwarz inequality and the following bounds, valid for
$L=AB$, $A\in\mathfrak{A}$, $B\in\mathcal{L}_{0}$,

$\displaystyle|(\Psi|Q_{T}\Psi)|$ $\displaystyle\leq$
$\displaystyle\|P_{E}\int
d\boldsymbol{x}\,(B^{*}B)(\boldsymbol{x})P_{E}\|\,\|A^{*}A\|,$ (A.24)
$\displaystyle\int
d\boldsymbol{x}\,(\rho_{+,\Psi}+\rho_{-,\Psi})\big{(}(L^{*}L)(\boldsymbol{x})\big{)}$
$\displaystyle\leq$ $\displaystyle 2\|P_{E}\int
d\boldsymbol{x}\,(B^{*}B)(\boldsymbol{x})P_{E}\|\,\|A^{*}A\|,\,\,\,\,$ (A.25)

one extends (A.23) to any $L\in\mathcal{L}$. Now formula (2.36) follows by a
polarization argument.

Let us now show that $\psi_{\Psi}^{\mathrm{out}}=0$ only if
$\Psi\in{\mathbb{C}}\Omega$. By the above considerations we obtain, for any
$B\in\mathcal{L}_{0}$,

$\psi_{\Psi}^{\mathrm{out}}(B,B)=\sum_{n}\int
d\boldsymbol{x}\,\big{\\{}(\Psi_{-,n}|(B^{*}B)(\boldsymbol{x})\Psi_{-,n})+(\Psi_{+,n}|(B^{*}B)(\boldsymbol{x})\Psi_{+,n})\big{\\}}.$
(A.26)

If $\psi_{\Psi}^{\mathrm{out}}=0$, then $B\Psi_{\pm,n}=0$ for each $n$ and any
such $B$. Thus, by Lemma A.1 (a), $\Psi_{\pm,n}$ are proportional to $\Omega$.
Using definitions (A.21), (A.22) and the convention $e_{\pm,0}=\Omega$, it is
easily seen that $\Psi$ is proportional to $\Omega$. $\Box$

##  Appendix B Proofs of Lemmas 3.7 and 3.8

Proof of Lemma 3.7: As for the main part of the lemma, it suffices to show
that the spectrum of $V_{\mathrm{odd}}$ coincides with $\mathbb{R}_{+}$. It
follows from the assumption ${\hbox{\rm Ad}}W\neq{\rm id}$ and the Reeh-
Schlieder property of the net $({\cal A},V)$ that ${\cal
A}_{\mathrm{odd}}({\cal I})\neq\\{0\\}$ and ${\cal K}_{\mathrm{odd}}=[{\cal
A}_{\mathrm{odd}}({\cal I})\Omega_{0}]$ for any open, bounded subset ${\cal
I}\subset\mathbb{R}$. Let $P(\,\cdot\,)$ be the spectral measure of $V$ and
suppose that $P(\Delta){\cal K}_{\mathrm{odd}}=\\{0\\}$ for some open subset
$\Delta\subset\mathbb{R}_{+}$. We fix a non-zero $A\in{\cal
A}_{\mathrm{odd}}({\cal I})$. Then, for any $B\in{\cal A}({\cal I})$ the
distribution

$(\Omega_{0}|[B,\widetilde{A}(\omega)]\Omega_{0})=\frac{1}{\sqrt{2\pi}}\int
dt\,e^{-i\omega t}(\Omega_{0}|[B,\beta_{t}(A)]\Omega_{0})$ (B.1)

is supported outside of $\Delta\cup-\Delta$. Since, by locality, this
distribution is a holomorphic function, it must be zero for all
$\omega\in\mathbb{R}$. Thus for any $f\in S(\mathbb{R})$ s.t. $\tilde{f}$ is
supported in the interior of $\mathbb{R}_{+}$ we obtain
$(\Omega_{0}|BA(f)\Omega_{0})=(\Omega_{0}|B\tilde{f}(T)A\Omega_{0})=0$. Here
$T\geq 0$ is the generator of $V$, $A(f)=\int dt\,\beta_{t}(A)f(t)$ and we
made use of the fact that $A(f)^{*}\Omega_{0}=0$, due to the support property
of $\tilde{f}$ and the spectrum condition. Approximating the characteristic
function of the interior of $\mathbb{R}_{+}$ with such $\tilde{f}$ and making
use of the fact that $(\Omega_{0}|A\Omega_{0})=0$, we conclude that
$A\Omega_{0}=0$ and hence, by the Reeh-Schlieder property $A=0$, which
contradicts our assumption. Consequently, $P(\Delta){\cal
K}_{\mathrm{odd}}\neq\\{0\\}$ for any open subset $\Delta$ of
$\mathbb{R}_{+}$, which means that the spectrum of $V_{\mathrm{odd}}$
coincides with $\mathbb{R}_{+}$.

This fact can also be proven as follows: The representation of translations
$V$ can be extended to a representation of the $ax+b$ group thanks to the
Borchers theorem [27]. There is only one non-trivial, irreducible
representation of this group which has positive energy [38] and its spectrum
of translations is ${\mathbb{R}}_{+}$. Since ${\cal K}_{\mathrm{odd}}$ does
not contain non-trivial invariant vectors of $V$, the spectrum of $V|_{{\cal
K}_{\mathrm{odd}}}$ coincides with $\mathbb{R}_{+}$.

Let us now proceed to part (a) of the lemma. To show the irreducibility of
$\pi_{\mathrm{odd}}$, it suffices to check that any vector
$\Psi\in\mathcal{K}_{\mathrm{odd}}$ is cyclic under the action of
$\pi_{\mathrm{odd}}({\cal A}_{\mathrm{ev}})$. By contradiction, we suppose
that there is $\Psi^{\prime}\in\mathcal{K}_{\mathrm{odd}}$ s.t.
$(\Psi^{\prime}|A\Psi)=0$ for any $A\in{\cal A}_{\mathrm{ev}}$. But this
implies that $(\Psi^{\prime}|B\Psi)=0$ for any $B\in{\cal A}$, which
contradicts the irreducibility of the action of ${\cal A}$ on $\mathcal{K}$.
Next we verify the faithfulness of $\pi_{\mathrm{odd}}$ restricted to a local
algebra. Let $A\in{\cal A}_{\mathrm{ev}}({\cal I})$ be a positive local
element which is zero upon restriction to $\mathcal{K}_{\mathrm{odd}}$. For
any local odd element $B\in{\cal A}_{\mathrm{odd}}(\mathfrak{J})$ and for
sufficiently large $s$ we obtain

$0=(\Omega|\beta_{s}(B^{*})A\beta_{s}(B)\Omega)=(\Omega|\beta_{s}(B^{*}B)A\Omega)\to(\Omega|B^{*}B\Omega)\cdot(\Omega|A\Omega),$
(B.2)

where in the last step we took the limit $s\to\infty$. By the Reeh-Schlieder
property it follows that $A=0$. This implies that $\pi_{\mathrm{odd}}$ is
faithful on ${\cal A}_{\mathrm{ev}}({\cal I})$ by Proposition 2.3.3 (3) of
[17]. Now the faithfulness of $\pi_{\mathrm{odd}}$ on the quasilocal algebra
${\cal A}_{\mathrm{ev}}$ follows from Proposition 2.3.3 (2) of [17], which
says that $\pi_{\mathrm{odd}}$ is faithful, if and only if
$\|\pi_{\mathrm{odd}}(A)\|=\|A\|$ for any $A\in{\cal A}_{\mathrm{ev}}$. Local
normality of $\pi_{\mathrm{odd}}$ is obvious, since $\pi_{\mathrm{odd}}$ acts
by the restriction to a subspace. Indeed, making use of Lemma 2.4.19 from [17]
and of the fact that $\pi_{\mathrm{odd}}$ preserves the norm, it is easy to
check that
$\mathrm{l.u.b.}\pi_{\mathrm{odd}}(A_{\alpha})=\pi_{\mathrm{odd}}(\mathrm{l.u.b.}\,A_{\alpha})$,
where l.u.b denotes the least upper bound and
$\\{A_{\alpha}\\}_{\alpha\in\mathbb{I}}$ is a uniformly bounded increasing net
of positive operators from some ${\cal A}_{\mathrm{ev}}({\cal I})$.

Part (b) of the lemma follows from the uniqueness of the invariant vector of
$V$. $\Box$

Proof of Lemma 3.8: We know from Lemma 3.7 that ${\cal
A}_{\mathrm{ev}}\neq{\mathbb{C}}I$, since it can be irreducibly represented on
the infinite dimensional Hilbert space ${\cal K}_{\mathrm{odd}}$.
Consequently, we can find a non-zero $A\in{\cal A}_{\mathrm{ev}}({\cal I})$,
for some open, bounded ${\cal I}$, s.t. $(\Omega_{0}|A\Omega_{0})=0$.
Proceeding identically as in the proof of the main part of Lemma 3.7, we
conclude that the spectrum of $V_{\mathrm{ev}}$ coincides with
$\mathbb{R}_{+}$. Part (b) follows trivially from the fact that the net
$({\cal A},V)$ is in a vacuum representation. Irreducibility in part (a)
follows from part (b). The remaining part of the statement is proven
analogously as the corresponding part of Lemma 3.7. $\Box$

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