# Quantum phase transition of infrared radiation

Bartosz Biadasiewicz and Wojciech Dybalski

Faculty of Mathematics and Computer Science
Adam Mickiewicz University in Poznań
ul. Uniwersytetu Poznańskiego 4, 61–614 Poznań, Poland.
E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

We describe a phase transition of infrared radiation, driven by quantum
fluctuations, which takes place at the boundary of (the conformal diagram of)
Minkowski spacetime. Specifically, we consider a family of states
interpolating between the vacuum and the Kraus-Polley-Reents infravacuum. A
state from this family can be imagined as a static source emitting flashes of
infrared radiation in distant past. The flashes are in suitable squeezed
states and the time intervals between them are controlled by a certain
parameter $r$. For $r<0$ the states are lightcone normal, thus physically
indistinguishable from local excitations of the vacuum. They suffer from the
usual infrared problems such as disintegration of the Bloch-Nordsieck
$S$-matrix and rotational symmetry breaking by soft photon clouds. However,
for $r>0$ lightcone normality breaks down, the $S$-matrix is stabilized by the
Kraus-Polley-Reents mechanism and the rotational symmetry is restored. We
interpret these two situations as ordered ($r<0$) and disordered ($r>0$) phase
of infrared radiation, and show that they can be distinguished by asymptotic
fluctuations of the fields. We also determine the singular behaviour of some
$S$-matrix elements near the critical point $r=0$.

Keywords: Phase transition, infrared problems, symmetry breaking.

###### Contents

  1. 1 Introduction
    1. 1.1 Phase transition on Minkowski spacetime
    2. 1.2 Spontaneous symmetry breaking
    3. 1.3 Correlation length
    4. 1.4 Phase transition in the future lightcone
    5. 1.5 Organization of the paper
  2. 2 Interpolating states $\omega_{r}$
  3. 3 The existence of the $S$-matrix in the disordered phase
  4. 4 Non-existence of the $S$-matrix in the ordered phase and at the critical point
  5. 5 The $S$-matrix near the critical point
  6. 6 The absence of lightcone normality in the disordered phase and at the critical point
  7. 7 Lightcone normality in the ordered phase
  8. 8 Order parameters in the ordered phase
  9. 9 Conclusion and outlook
  10. A Outline of the proof of estimates (7.14)
  11. B Omitting finite number of modes from $\omega_{r}$

##  1 Introduction

Phase transitions are abundant in Nature and belong to the most studied
phenomena in Physics [So11, Zi]. In spite of tremendous theoretical progress,
especially pertaining to renormalization and universality, they seem to escape
any definite classification. Without any intention of reviewing the subject,
let us give some examples illustrating this diversity: On the one hand, in the
textbook case of the ferromagnetic phase transition the relevant states are in
thermal equilibrium and the transition is driven by thermal fluctuations. On
the other hand, quantum phase transitions typically occur at zero temperature,
the relevant states are ground states, and the transitions are caused by
quantum fluctuations [Vo03]. In the same time, the concept of phase
transitions is not reserved for ground states and thermal states - there is a
rich field of non-equilibrium phase transitions with an important example of
directed percolation [Hi06]. To broaden the perspective, it should also be
noted that the bulk and the boundary of a physical system may have distinct
phase diagrams [Di97]. In particular the boundary may undergo a phase
transition even if the bulk is far from criticality [KSMLSC22].

The boundary degrees of freedom of massless QFT enjoyed a lot of attention
over the last decade. In fact, the Strominger’s ‘infrared triangle’ [HMPS14,
HPS16, St17, Pa17] linking the Weinberg’s soft photon theorem, asymptotic
charges and memory effects is located at the boundary of the (conformal
diagram of) Minkowski spacetime. All the vertices of this ‘infrared triangle’
have a long history [We65, AS81, Br77, Sta81, He95, BG13] summarized in
[He16], but the emergence of the triangle clearly intensified efforts to
understand the asymptotic structure of electrodynamics and gravity, see e.g.
[CL15, GS16, CE17, HIW16, KPRS17, DH19, DW19, RS20, PSW22, MRS22]. In spite of
such activity, to our knowledge, the phase diagram of boundary degrees of
freedom in massless QFT has not been studied to date. In the present paper we
demonstrate that phase transitions actually occur for these degrees of
freedom, even in massless scalar free field theory.

(a) Ordered phase($r<0$)

(b) Critical point($r=0$)

(c) Disordered phase($r>0$)

Fig. 1. Schematic illustration of the phase transition in terms of the double
cones from Fig. 2. The future lightcone with tip at the origin is indicated in
red.

###  1.1 Phase transition on Minkowski spacetime

Let us explain in non-technical terms how the transition we found comes about,
leaving the detailed construction of the states to Section 2. Let us imagine a
stationary source emitting flashes of infrared radiation in distant past.
Specifically, suppose that the corresponding wave packets have widths
$\varepsilon_{i}>0$ in momentum space, where the sequence
$\\{\varepsilon_{i}\\}_{i\in\mathbb{N}}$ converges to zero. One such emission
event is depicted in Fig. 2. At the moment of emission the wave packet is
essentially localized in space

tx Fig. 2. Radiation emitted from a base of a double cone.

in a ball of radius $1/\varepsilon_{i}$. This ball is a base of a double
cone111A double cone of radius $a$ centered at zero is the open set
$\mathcal{O}_{a}:=\\{\,(t,\boldsymbol{x})\in^{4}\,|\,|t|+|\boldsymbol{x}|<a\,\\}$.
in Fig. 2 which serves to determine the region of spacetime to which the
radiation will travel. The $i$-th flash occurs at time $-\tau^{(r)}_{i}$,
where

$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\tau^{(r)}_{i}:=\left(\frac{1}{\varepsilon_{i}}\right)\varepsilon_{i}^{r}},$
(1.1)

and $r\in$ is our control parameter. Thus for $r=0$ this time-shift is equal
to the radius of the double cone, as illustrated in Fig. 1 (b). In Fig. 1 (a)
we presented a family of such double cones for $\tau^{(r)}_{i}$ given by (1.1)
and $r$ negative, slightly smaller than zero. Here the time-shifts are larger
than the radii of the double cones. Consequently, the double cones are located
one below the other and the infrared radiation evades large parts of
spacetime, including the future lightcone. On the other hand, in the case of
positive $r$, slightly larger than zero, we obtain a family of double cones
inscribed within each other, as indicated in Fig. 1 (c). Consulting again Fig.
2, we see that in this case the infrared radiation fills the entire spacetime.
The qualitative difference between these two situations (a) and (c) suggests
the presence of a phase transition at $r=0$. We will show that this is,
indeed, the case.

To exhibit a phase transition it is necessary to specify order parameters. For
this purpose, let us denote by $\phi$ the massless scalar free field on Fock
space $\mathcal{F}$ and by $\omega_{r}$ the physical state of radiation as
discussed above. For a smooth, real valued function $f$, we define, cf.
[Bu86],

$\displaystyle\phi_{R}(f):=\frac{1}{R^{3}}\int\phi(x)f(x/R)\,d^{4}x$ (1.2)

and choose it as order parameter in the limit $R\to\infty$. We will be
interested in two cases - $f$ supported in the spacelike complement of zero
and in the future lightcone. We will refer to the respective quantities
$\phi_{R}(f)$ as the _spacelike_ and _timelike_ order parameter. Denoting by
$\Omega$ the Fock space vacuum, we obviously have
$\langle\Omega,\phi_{R}(f)\Omega\rangle=0$ and222We use the following
conventions for the Fourier transforms in space, time and spacetime:
$\tilde{f}(\boldsymbol{k})=\frac{1}{(2\pi)^{3/2}}\int{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}d^{3}\boldsymbol{x}}\,\mathrm{e}^{-\mathrm{i}\boldsymbol{k}\cdot\boldsymbol{x}}f(\boldsymbol{x})$,
$\tilde{f}(k^{0})=\int dt\,\mathrm{e}^{\mathrm{i}k^{0}t}f(t)$,
$\tilde{f}(k)=\frac{1}{(2\pi)^{3/2}}\int
d^{4}x\,\mathrm{e}^{\mathrm{i}kx}f(x)$.

$\displaystyle\langle\Omega,\phi_{R}(f)^{2}\Omega\rangle=\int\frac{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}d^{3}\boldsymbol{k}}}{2|\boldsymbol{k}|}|\tilde{f}(|\boldsymbol{k}|,\boldsymbol{k})|^{2},$
(1.3)

which is the rationale for the scaling chosen in (1.2). Since $\omega_{r}$ is
a quasi-free state, we still have $\omega_{r}(\phi_{R}(f))=0$ for all $r\in$
but the fluctuations distinguish the two phases. Namely,

$\displaystyle\lim_{R\to\infty}\omega_{r}(\phi_{R}(f)^{2})=\left\\{\begin{array}[]{ll}\int\frac{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}d^{3}\boldsymbol{k}}}{2|\boldsymbol{k}|}|\tilde{f}(|\boldsymbol{k}|,\boldsymbol{k})|^{2}&\textrm{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}for
$r<0$,}}\\\
\infty&\textrm{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}for
$r\geq 0$,}}\end{array}\right.$ (1.6)

where the divergence to infinity is possibly only along a subsequence and in
both cases we will impose some restrictions on $f$. In line with standard
terminology we will call the region of parameters with bounded fluctuations
the ordered phase and the region with unbounded fluctuations the disordered
phase. Since the fluctuations have quantum character and change in a
discontinuous manner with $r$, we interpret this phenomenon as a quantum phase
transition. As the considered states are neither thermal nor vacua, the
transition has a non-equilibrium character. It consists in a qualitative
change of a certain dynamics, namely the Bloch-Nordsieck process, which we
describe in the next subsection.

###  1.2 Spontaneous symmetry breaking

The transition from the disordered to the ordered phase is typically
accompanied by spontaneous symmetry breaking. This phenomenon can be concisely
described in the language of algebraic QFT. This exact mathematical apparatus,
proposed long time ago by Haag and Kastler [HK64, Ha], is currently gaining
popularity in theoretical high-energy physics [Wi18, CLPW22, So23]. We find it
very helpful for describing subtle properties of infrared radiation which are
the topic of this paper. Denote by $\mathfrak{A}$ the $C^{*}$-algebra
generated by operators $\exp(\mathrm{i}\phi(f))$, where $f\in D(^{4};)$, i.e.,
a smooth, compactly supported, real valued function on 4. Now $\omega_{r}$ is
a state on $\mathfrak{A}$, that is a positive, normalized, linear functional.
We denote its GNS representation by $\pi_{r}$. Symmetries of our physical
system are described by automorphisms $\alpha$ of $\mathfrak{A}$. We say that
a symmetry $\alpha$ is unitarily implemented in representation $\pi_{r}$ if
there exists a unitary $U$ such that
$\pi_{r}\circ\alpha(A)=U\pi_{r}(A)U^{*},A\in\mathfrak{A}$. In the usual
shorthand notation

$\displaystyle\pi_{r}\circ\alpha\simeq\pi_{r}.$ (1.7)

On the other hand, if such a unitary does not exist, we say that the symmetry
$\alpha$ is broken in $\pi_{r}$.

In order to find the relevant automorphisms $\alpha$, let us recapitulate the
Bloch-Nordsieck discussion of infrared problems [BN37]: Let
$\phi_{\mathrm{int}}$ be the massless scalar quantum field interacting with an
external source $j$ of the form

$\displaystyle
j(t,\boldsymbol{x})=\left\\{\begin{array}[]{ll}j_{0}(\boldsymbol{x}-\boldsymbol{v}^{\mathrm{out}}t)&\textrm{for
$t\geq 0$},\\\ j_{0}(\boldsymbol{x}-\boldsymbol{v}^{\mathrm{in}}t)&\textrm{for
$t<0$},\\\ \end{array}\right.$ (1.10)

where $j_{0}\in D(^{3};)$ is spherically symmetric and satisfies
$\int{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}d^{3}\boldsymbol{x}}\,j_{0}(\boldsymbol{x})=:q\neq
0$. That is, the source corresponds to a charged particle with initial
velocity $\boldsymbol{v}^{\mathrm{in}}$ and final velocity
$\boldsymbol{v}^{\mathrm{out}}$, both strictly smaller than the speed of
light. The system is governed by the field equation

$\displaystyle\Box\phi_{\mathrm{int}}(x)=-j(x)$ (1.11)

with an initial condition
$\phi_{\mathrm{int}}(0,\boldsymbol{x})=\phi_{0}(0,\boldsymbol{x})$ which
differs from $\phi$ at most by some real valued function. (We cannot set
$\phi_{\mathrm{int}}(0,\boldsymbol{x})=\phi(0,\boldsymbol{x})$ here, since we
will identify $\phi$ with the incoming field below). Setting
$V(t):=\int{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}d^{3}\boldsymbol{x}}\,\phi(t,\boldsymbol{x})j(t,\boldsymbol{x})$
we write the textbook solution for $t\geq 0$

$\displaystyle\phi_{\mathrm{int}}(t,\boldsymbol{x})=W_{t}\phi_{0}(t,\boldsymbol{x})W_{t}^{*},\quad
W_{t}:=\mathrm{\bar{T}exp}\big{(}-\int_{0}^{t}d\tau\,V(\tau)\big{)},$ (1.12)

where $\bar{T}$ orders the times $\tau$ ascendingly from left to right. We
thus identify the interacting dynamics as
$U(t):=W_{t}\mathrm{e}^{\mathrm{i}tH_{0}}$, where $H_{0}$ is the free
Hamiltonian, and compute the outgoing asymptotic field according to the LSZ
prescription:

$\displaystyle\phi_{\mathrm{int}}^{\mathrm{out}}(f)=\lim_{\tau\to\infty}U(\tau)\mathrm{e}^{-\mathrm{i}\tau
H_{0}}\phi_{0}(f)\mathrm{e}^{\mathrm{i}\tau
H_{0}}U(\tau)^{*}=\phi_{0}(f)-\mathrm{i}\mathrm{Im}\langle
g^{\mathrm{out}},F\rangle,$ (1.13)

where

$\displaystyle
g^{\mathrm{out}}(\boldsymbol{k}):=\mathrm{i}\tilde{j}_{0}(\boldsymbol{k})\,\frac{1}{|\boldsymbol{k}|^{3/2}(1-\hat{\boldsymbol{k}}\cdot\boldsymbol{v}^{\mathrm{out}})},\quad\,\hat{\boldsymbol{k}}:=\boldsymbol{k}/|\boldsymbol{k}|,\quad
F(\boldsymbol{k}):=|\boldsymbol{k}|^{-1/2}\tilde{f}(|\boldsymbol{k}|,\boldsymbol{k}).$
(1.14)

The latter definition is motivated by the relation
$\phi(f)=\frac{1}{\sqrt{2}}(a^{*}(F)+a(F))$, where $a^{(*)}$ are the
creation/annihilation operators333Our convention is
$[a(F_{1}),a^{*}(F_{2})]=\langle F_{1},F_{2}\rangle$, where $\langle
F_{1},F_{2}\rangle:=\int
d^{3}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\boldsymbol{k}}\,\bar{F}_{1}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\boldsymbol{k}})F_{2}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\boldsymbol{k}})$
is the scalar product and $\|F\|_{2}=\langle F,F\rangle^{1/2}$ the norm in
$L^{2}(^{3})$. on the Fock space $\mathcal{F}$. We repeat the computation for
negative times which amounts to changing the direction of time ordering in
(1.12) and replacing ‘out’ with ‘in’ in (1.13) and (1.14). Now we identify
$\phi$ with the incoming field $\phi_{\mathrm{int}}^{\mathrm{in}}$ and define
the automorphism $\alpha$ by the relation

$\displaystyle\mathrm{e}^{\mathrm{i}\phi_{\mathrm{int}}^{\mathrm{out}}(f)}=\alpha(\mathrm{e}^{\mathrm{i}\phi_{\mathrm{int}}^{\mathrm{in}}(f)})=\mathrm{e}^{-\mathrm{i}\mathrm{Im}\langle
g^{\mathrm{out}}-g^{\mathrm{in}},F\rangle}\mathrm{e}^{\mathrm{i}\phi_{\mathrm{int}}^{\mathrm{in}}(f)}.$
(1.15)

Comparing this relation with (1.7) it is clear that the implementing unitary
$U$ (if it exists) is simply the $S$-matrix for the Bloch-Nordsieck process.
We will denote this unitary $S_{r}$.

In fact, the $S$-matrix should intertwine the incoming and outgoing asymptotic
fields. Its absence in the vacuum representation $\pi_{\mathrm{vac}}$ is the
essence of the infrared problem. Since
$\lim_{r\to{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-\infty}}\omega_{r}=\omega_{\mathrm{vac}}$,
i.e., our states tend weakly to the vacuum state for large negative $r$, it is
not a surprise that the $S$-matrix does not exist in the ordered phase of our
system (see Section 4). As a matter of fact, it is also not a surprise that
the Bloch-Nordsieck $S$-matrix
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}S_{r}}$
does exist in the disordered phase. In fact, we construct our states in such a
way that
$\lim_{r\to{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\infty}}\omega_{r}=\omega_{\mathrm{KPR}}$,
where $\omega_{\mathrm{KPR}}$ is the Kraus-Polley-Reents (KPR) infravacuum
[Re74, KPR77, Kr82, Ku98, CD19, BD22]. The latter state is known to have the
implementation property (1.7) for the automorphisms in question as we recall
in Section 3. It is thus clear from the fact that $\\{\omega_{r}\\}_{r\in}$
interpolates between $\omega_{\mathrm{vac}}$ and $\omega_{\mathrm{KPR}}$ that
the disintegration of $S_{r}$ must occur for some intermediate value of $r$.
We will show that it happens at $r=0$ as a symmetry breaking phenomenon
accompanying the phase transition described in Subsection 1.1.

As the symmetry transformation $\alpha$, discussed above, may appear somewhat
abstract, let us have a brief look at the familiar case of rotations. It turns
out that our states $\omega_{r}$ are invariant under rotations for all $r\in$,
thus initially there is no rotational symmetry breaking. The situation becomes
more interesting if instead of $\omega_{r}$ we consider the family
$\omega_{r}\circ\alpha$. In view of (1.15), we simply transport our states
from the algebra of incoming fields to the algebra of outgoing fields. Thus we
can interpret the resulting states as the background radiation from Fig. 1 and
Fig. 2 accompanied by a soft photon cloud emitted by the external source. In
this case we have

$\displaystyle\\!\\!\\!\pi_{r}\circ\alpha\circ\alpha_{\mathcal{R}}\not\simeq\pi_{r}\circ\alpha\,\quad\textrm{for}\quad{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}r\leq
0},$ (1.16)
$\displaystyle\\!\\!\\!\pi_{r}\circ\alpha\circ\alpha_{\mathcal{R}}\simeq\pi_{r}\circ\alpha\,\quad\textrm{for}\quad{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}r>0},$
(1.17)

where $\alpha_{\mathcal{R}}$, $\mathcal{R}\in SO(3)$, are the usual rotation
automorphisms of scalar free field theory. Again, the extreme cases
$r\to\pm\infty$ can be extracted from the literature [Ro70, KPR77]. For the
interpolating family of states $\omega_{r}\circ\alpha$ we establish (1.17) and
(1.16) in Sections 3 and 4, respectively. We conclude that the rotational
symmetry, unitarily implemented in the disordered phase, is broken in the
ordered phase, similar to the familiar case of the ferromagnetic phase
transition in the $O(3)$ Heisenberg model.

###  1.3 Correlation length

Apart from constructing the Bloch-Nordsieck $S$-matrix $S_{r}$ in the
disordered phase, we also obtain detailed information about the behaviour of
its matrix elements near the critical point. Specifically, we obtain

$\displaystyle\langle\Omega,S_{r}\Omega\rangle\sim\left\\{\begin{array}[]{ll}0&\textrm{for\quad\quad\
\\!\\!\\!
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$r\leq
0$},}\\\
\mathrm{e}^{-\frac{c}{r^{3/2}}}&\textrm{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}for
$0<r\leq 1$.}\end{array}\right.$ (1.20)

To explain the first line of (1.20) let us first consider the vacuum case
$r={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-\infty}$.
Here the standard computation with an infrared cut-off $\sigma>0$ gives

$\displaystyle\langle\Omega,S_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}r=-\infty}}^{\sigma}\Omega\rangle\sim\frac{\sigma}{\Lambda},$
(1.21)

where $\Lambda$ is a fixed UV scale introduced by $j_{0}$. This expression
vanishes as $\sigma\to 0$ and the same is true in the ordered phase ($r<0$),
as we show in Theorem 4.1 below. The second line of (1.20), which we verify in
Section 5, means, more precisely, that
$C_{1}\mathrm{e}^{-\frac{c_{1}}{r^{3/2}}}\leq\langle\Omega,S_{r}\Omega\rangle\leq
C_{2}\mathrm{e}^{-\frac{c_{2}}{r^{3/2}}}$, for suitable positive constants
independent of $0<r\leq 1$. Consequently
$\lim_{r\to{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0^{+}}}\langle\Omega,S_{r}\Omega\rangle=0$,
thus the dependence $\ni r\mapsto\langle\Omega,S_{r}\Omega\rangle$ is
continuous, unlike the behaviour of the order parameter fluctuations in (1.6).
Relation (1.21) suggests an interpretation of
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\xi:=}\langle\Omega,S_{r}\Omega\rangle^{-1}$
as the correlation length of our phase transition: All photons of larger
wavelengths, emitted by the external source $j$, are effectively screened by
the background radiation. This definition is adapted to our particular
situation, in which even in the disordered phase the theory is genuinely
massless and exponential decay of correlations is not expected. We observe
from (1.20), that our correlation length has a similar dependence on the
control parameter $r$ as in the case of the Berezinskii–Kosterlitz–Thouless
transition.

###  1.4 Phase transition in the future lightcone

Even if $\\{\omega_{r}\\}_{r\in}$ tends to the vacuum as $r\to-\infty$, one
can ask how exotic these states are for small negative $r$, close to the phase
transition under study. As emphasized in [BR14], the observers have access
only to their future lightcones, so it suffices to consider this question only
inside the open future lightcone $V_{+}$ with tip at zero. Let us denote by
$\mathfrak{A}(V_{+})$ the $C^{*}$-algebra generated by operators
$\exp(\mathrm{i}\phi(f))$, where $f\in D(V_{+};)$. As we see from Fig. 1 (a),
in the ordered phase ($r<0$) the radiation does not enter the future
lightcone. It is therefore not a surprise that in this case the representation
$\pi_{r}$ is quasi-equivalent to the vacuum representation
$\pi_{\mathrm{vac}}$, i.e, unitarily equivalent up to multiplicity [BR,
Section 2.4.4.]. In a shorthand notation, we prove that

$\displaystyle\pi_{r}|_{\mathfrak{A}(V_{+})}\approx\pi_{\mathrm{vac}}|_{\mathfrak{A}(V_{+})}\quad\textrm{for}\quad
r<0.$ (1.22)

This property of _lightcone normality_ means physically that the observer
cannot decide by experiments inside the lightcone if the radiation is in a
state from the familiar vacuum sector, or in the state $\omega_{r}$, on a
verge of the phase transition. In fact, by Definition 6.2 below, the observer
can describe the radiation by a density matrix from the vacuum sector.

As shown in [CD20], the relevant automorphisms $\alpha$ satisfy

$\displaystyle\pi_{\mathrm{vac}}\circ\alpha|_{\mathfrak{A}(V_{+})}\simeq\pi_{\mathrm{vac}}|_{\mathfrak{A}(V_{+})}.$
(1.23)

By combining this with (1.22), we note that these automorphisms are unitarily
implemented on $\mathfrak{A}(V_{+})$ also in the ordered phase, perhaps up to
multiplicity. Thus the phenomenon of symmetry breaking, visible on the full
Minkowski spacetime, essentially disappears in the future lightcone. So one
may ask, if the phase transition can actually be observed in $V_{+}$. The
answer is yes, because for $f$ supported in $V_{+}$ the order parameters
$\phi_{R}(f)$ are measurable in $V_{+}$ for all $R>0$. The distinction (1.6)
remains valid, and actually implies

$\displaystyle\pi_{r}|_{\mathfrak{A}(V_{+})}\not\approx\pi_{\mathrm{vac}}|_{\mathfrak{A}(V_{+})}\quad\textrm{for}\quad{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}r\geq
0}$ (1.24)

as we show in Section 6. Thus by (1.22), (1.24) the property of lightcone
normality is another feature that distinguishes the two phases. Given Fig. 1
(b) one may be surprised that lightcone normality fails at $r=0$. In this
schematic figure the radiation seems to avoid the future lightcone also at the
critical point. The reason for the failure is that the flashes of radiation
only approximately fit into the double cones at the moment of emission. At the
critical point their tails matter as we explain in Section 7 below estimate
(7.14).

###  1.5 Organization of the paper

| Ordered phase $(r<0)$ | Critical point $(r=0)$ | Disordered phase $r>0$
---|---|---|---
$S$-matrix | ✗ | ✗ | ✓
Lightcone normality | ✓ | ✗ | ✗
Table 1: Summary of main results.

Our paper is organized as follows: In Section 2 we define the states
$\omega_{r}$ and reformulate them in terms of certain symplectic maps. In
Section 3 we recall the KPR mechanism for stabilizing the Bloch-Nordsieck
$S$-matrix and show that it is at work in the entire disordered phase ($r>0$).
Section 4 confirms the absence of the $S$-matrix in the ordered phase ($r<0$)
and at the critical point ($r=0$). In Section 5 we have a closer look at the
disintegration of the $S$-matrix near the critical point and derive formula
(1.20) for the correlation length. In Section 6 we verify the infinite
fluctuations of the timelike order parameter in the disordered phase and at
the critical point. We conclude the absence of lightcone normality. We also
demonstrate infinite fluctuations of the spacelike order parameter in the
disordered phase and at the critical point. We point out that infinite
fluctuations of the spacelike order parameter are a necessary condition for
the existence of the $S$-matrix. Section 7 is devoted to lightcone normality
in the ordered phase, which is our main technical result. In Section 8 we
indicate that lightcone normality implies vacuum fluctuations of the timelike
order parameter, as stated in the first line of (1.6). More surprisingly,
certain technical ingredients from the proof of lightcone normality also
ensure vacuum fluctuations of the spacelike order parameter. Recalling from
above that finite fluctuations of the spacelike order parameter are in
conflict with the KPR mechanism, this sheds some light on the intriguing
incompatibility between the existence of the Bloch-Nordsieck $S$-matrix and
lightcone normality, see Table 1.

Acknowledgement: We would like to thank Detlev Buchholz for useful discussions
and pointing out the reference [DFG84]. We also thank Daniela Cadamuro and
Henning Bostelmann for helpful discussions. Financial support from the grant
‘Preludium’ 2021/41/N/ST1/02755 of the National Science Centre, Poland, is
gratefully acknowledged.

##  2 Interpolating states $\omega_{r}$

In this section we give a definition of the states $\omega_{r}$ which
interpolate between the vacuum at $r=-\infty$ and the KPR infravacuum at
$r=\infty$. We start from the following family of modes of radiation,
introduced first in [Re74, KPR77]:

  * •

We introduce sequences $\varepsilon_{i}:=2^{-(i-1)}$ and $b_{i}:=\frac{1}{i}$
for $i=1,2,3\ldots$

  * •

We define normalized functions
$\zeta^{\circ}_{i}(|\boldsymbol{k}|):=c_{0}\frac{\chi_{[\varepsilon_{i+1},\varepsilon_{i}]}(|\boldsymbol{k}|)}{|\boldsymbol{k}|^{3/2}}\in
L^{2}(_{+},|\boldsymbol{k}|^{2}d|\boldsymbol{k}|)$, where
$\chi_{[\varepsilon_{i+1},\varepsilon_{i}]}$ is the characteristic function of
$[\varepsilon_{i+1},\varepsilon_{i}]$ and $c_{0}:=(\log 2)^{-1/2}$ is the
normalization constant.

  * •

We define the sets

$\displaystyle
I_{n}:=\\{\,\beta=(i,\ell,m)\,|\,i={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}1},2,3,4\ldots
n,\ 0\leq\ell\leq i,\ -\ell\leq m\leq\ell\ \\}$ (2.1)

and let
$\zeta_{\beta}(\boldsymbol{k}):=\zeta^{\circ}_{i}(|\boldsymbol{k}|)Y_{\ell
m}(\hat{\boldsymbol{k}})$ for $\beta\in I_{n}$, where $Y_{\ell m}$ are the
real valued spherical harmonics.

As explained in Subsection 1.1, we want to create flashes of radiation at
times $-\tau^{(r)}_{\beta}:=-\tau^{(r)}_{i}$ defined in (1.1). Thus we
introduce the time-translated KPR-modes

$\displaystyle\zeta^{(r)}_{\beta}(\boldsymbol{k}):=\mathrm{e}^{-\mathrm{i}|\boldsymbol{k}|\tau^{(r)}_{\beta}}\zeta_{\beta}(\boldsymbol{k}).$
(2.2)

A family of $n$ flashes of radiation, as considered in Subsection 1.1, is
described by the following squeezed state of radiation

$\displaystyle\Psi^{(r)}_{n}=\prod_{\beta\in
I_{n}}\exp\bigg{(}-\frac{1}{\hat{b}_{\beta}}a^{*}(\zeta^{(r)}_{\beta})^{2}\bigg{)}\Omega,$
(2.3)

where $\hat{b}_{\beta}:=\frac{b_{i}^{2}+1}{b_{i}^{2}-1}$ and we omitted
normalization444For $i=1$ we have $\hat{b}_{\beta}=\infty$ thus
$1/\hat{b}_{\beta}=0$ in (2.3). So we could have omitted $i=1$ mode from the
beginning. Actually, omitting any finite number of modes does not change our
conclusions, cf. Appendix B.. Now the states $\omega_{r}$, consisting of
infinitely many flashes, are given by

$\displaystyle\omega_{r}(A)=\lim_{n\to\infty}\frac{\langle\Psi^{(r)}_{n},A\Psi^{(r)}_{n}\rangle}{\langle\Psi^{(r)}_{n},\Psi^{(r)}_{n}\rangle},\quad
A\in\mathfrak{A}.$ (2.4)

The above representation of the states $\omega_{r}$ is closest in spirit to
the discussion from Subsection 1.1, but is not always convenient for
computations. Therefore, we rearrange it as follows: Let us note that by
Gaussian integration

$\displaystyle\frac{\Psi^{(r)}_{n}}{\|\Psi^{(r)}_{n}\|}=\mathcal{U}_{r}(I_{n})\Omega:=\prod_{\beta\in
I_{n}}\bigg{(}\frac{z_{\beta}}{2\pi}\bigg{)}^{1/2}\int
ds_{\beta}\,\mathrm{e}^{-{\frac{1}{4}}(\hat{b}_{\beta}-1)s_{\beta}^{2}}W(\mathrm{i}s_{\beta}\zeta^{(r)}_{\beta})\Omega,$
(2.5)

where $z_{\beta}:=\frac{1}{2}\sqrt{\hat{b}_{\beta}^{2}-1}$ and
$W(\zeta):=\exp(\mathrm{i}\frac{1}{\sqrt{2}}(a^{*}(\zeta)+a(\zeta)))$,
$\zeta\in L^{2}(^{3})$, are the Weyl operators. Inserting this identity to
(2.4), using the basic relations

$\displaystyle
W(\zeta)W(\zeta^{\prime})=\mathrm{e}^{-\frac{1}{2}\mathrm{i}\mathrm{Im}\langle\zeta,\zeta^{\prime}\rangle}W(\zeta+\zeta^{\prime}),\quad\langle\Omega,W(\zeta)\Omega\rangle=\mathrm{e}^{-\frac{1}{4}\|\zeta\|_{2}^{2}},$
(2.6)

and performing the resulting Gaussian integrals, we conclude that

$\displaystyle\omega_{r}(W(F))=\langle\Omega,W(T_{r}{}F)\Omega\rangle.$ (2.7)

Here $T_{r}{}$ is a certain symplectic map on the space
$\mathcal{L}:=\\{\,F=\phi(f)\Omega\,|\,f\in D(^{4};)\\}$ equipped with the
symplectic form $\mathrm{Im}\langle\,\cdot\,,\cdot\,\rangle$. Then the GNS
representation of $\omega_{r}$ acts naturally on $\mathcal{F}$ by

$\displaystyle\pi_{r}(W(F))=W(T_{r}{}F).$ (2.8)

The symplectic map has the form

$\displaystyle
T_{r}{}=1+\sum_{i=1}^{\infty}(b_{i}^{-1}-1)u_{-\tau^{(r)}_{i}}Q_{i}\mathrm{Re}\,u_{\tau^{(r)}_{i}}+\mathrm{i}\sum_{i=1}^{\infty}(b_{i}-1)u_{-\tau^{(r)}_{i}}Q_{i}\mathrm{Im}\,u_{\tau^{(r)}_{i}},$
(2.9)

where $u_{\tau}:=\mathrm{e}^{\mathrm{i}\mu\tau}$, with
$\mu(\boldsymbol{k}):=|\boldsymbol{k}|$, are time translations and $Q_{i}$ are
orthogonal projections given by

$Q_{i}=|\zeta^{\circ}_{i}\rangle\langle\zeta^{\circ}_{i}|\otimes\tilde{Q}_{i}\quad\text{with}\quad\tilde{Q}_{i}:=\sum_{0\leq\ell\leq
i}\sum_{m=-\ell}^{\ell}|Y_{\ell m}\rangle\langle Y_{\ell m}|.$ (2.10)

We note in passing that above reasoning is an instance of the general theory
of implementation of symplectic maps [Sh62, SS65, Ru78, AY82]. In the
representation (2.7), (2.9) it is easy to see that
$\lim_{r\to-\infty}\omega_{r}(W(F))=\omega_{\mathrm{vac}}(W(F))$. In fact, due
to large oscillations only $1$ from (2.9) survives this limit as a consequence
of the Riemann-Lebesgue lemma and dominated convergence. It is also clear that
the limit $r\to\infty$ amounts to replacing $u_{\pm\tau^{(r)}_{i}}$ with $1$
in (2.9). Then we recover the usual KPR maps $T$ from [KPR77].

From formulas (2.7)–(2.10) it is also manifest that states $\omega_{r}$ are
invariant under rotations. Let
$(u_{\mathcal{R}}F)(\boldsymbol{k}):=F_{\mathcal{R}}(\boldsymbol{k}):=F(\mathcal{R}^{-1}\boldsymbol{k})$,
$\mathcal{R}\in SO(3)$, be the standard representation of rotations on
$L^{2}(^{3})$. It gives rise to automorphisms of $\mathfrak{A}$ via
$\alpha_{\mathcal{R}}(W(F))=W(F_{\mathcal{R}})$. Since $\tilde{Q}_{i}$ are
spectral projections of the total angular momentum operator $\mathrm{L}^{2}$,
we easily see that $T_{r}{}\circ u_{\mathcal{R}}=u_{\mathcal{R}}\circ
T_{r}{}$. This immediately gives
$\omega_{r}\circ\alpha_{\mathcal{R}}=\omega_{r}$, hence

$\displaystyle\pi_{r}\circ\alpha_{\mathcal{R}}(\,\cdot\,)=U_{\mathcal{R}}\pi_{r}(\,\cdot\,)U_{\mathcal{R}}^{*}$
(2.11)

for a unitary representation of rotations $\mathcal{R}\mapsto U_{\mathcal{R}}$
on $\mathcal{F}$.

##  3 The existence of the $S$-matrix in the disordered phase

The main technical result of this section is the following:

######  Theorem 3.1.

$T_{r}{}g\in L^{2}(^{3})$ for all $g$ as in (1.14) and $r>0$.

Here we wrote $g:=g^{\mathrm{out}}$ for brevity, and to stress that the
general form of these functions is the same in the ‘in’ and ‘out’ case. Since
$g$ is not in $\mathcal{L}$ and not even in $L^{2}(^{3})$ due to a singularity
at $\boldsymbol{k}=0$, the operation $T_{r}{}g$ requires clarification: We
introduce an intermediate infrared cut-off $\sigma>0$ and set
$g_{\sigma}(\boldsymbol{k}):=\chi_{[\sigma,\infty)}(|\boldsymbol{k}|)g(\boldsymbol{k})$,
where $\chi$ is the characteristic function. Then, more precisely, Theorem 3.1
requires that the limit $T_{r}{}g:=\lim_{\sigma\to 0}T_{r}{}\,g_{\sigma}$
exists in $L^{2}(^{3})$.

Before we prove Theorem 3.1, let us indicate its relevance to the problem of
the existence of the Bloch-Nordsieck $S$-matrix discussed in Subsection 1.2.
By considering relation (1.15) in representation $\pi_{r}$ we obtain

$\displaystyle\pi_{r}\circ\alpha(W(F))$ $\displaystyle\\!\\!\\!=$
$\displaystyle\\!\\!\\!\mathrm{e}^{-\mathrm{i}\mathrm{Im}\langle
g^{\mathrm{out}}-g^{\mathrm{in}},F\rangle}\pi_{r}(W(F))$ (3.1)
$\displaystyle\\!\\!\\!=$
$\displaystyle\\!\\!\\!\mathrm{e}^{-\mathrm{i}\mathrm{Im}\langle
T_{r}{}(g^{\mathrm{out}}-g^{\mathrm{in}}),T_{r}{}F\rangle}\pi_{r}(W(F))$
$\displaystyle\\!\\!\\!=$
$\displaystyle\\!\\!\\!W(T_{r}{}(g^{\mathrm{out}}-g^{\mathrm{in}}))\pi_{r}(W(F))W(T_{r}{}(g^{\mathrm{out}}-g^{\mathrm{in}}))^{*},$

where in the second step we used that $T_{r}{}$ is symplectic and in the last
step we exploited the Weyl relations (2.6). Thus, by Theorem 3.1, the
$S$-matrix $S_{r}:=W(T_{r}{}(g^{\mathrm{out}}-g^{\mathrm{in}}))$ is well
defined. This is how the $S$-matrix is stabilized by the KPR mechanism.
Combining (3.1) with (2.11) we also obtain

$\displaystyle\pi_{r}\circ\alpha\circ\alpha_{\mathcal{R}}(W(F))=S_{r}U_{\mathcal{R}}S_{r}^{*}\,\pi_{r}\circ\alpha(W(F))\,S_{r}U_{\mathcal{R}}^{*}S_{r}^{*},$
(3.2)

which confirms (1.17).

The first step of the proof of Theorem 3.1 is to check the case $r=\infty$.
Then the time translations disappear in (2.9) and the maps $T_{r}{}$ become
simply the KPR maps $T$. Noting that $g$ is purely imaginary, we have

$\displaystyle Tg=(1+\sum_{i=1}^{\infty}(b_{i}-1)Q_{i})g.$ (3.3)

The somewhat intricate definition of the KPR states and maps from [KPR77] is
motivated precisely by the requirement that the above vector is in
$L^{2}(^{3})$. As there is a detailed modern account in [CD19, Proposition
4.2], we can be brief here. We can replace $g$ from (1.14) with

$\displaystyle
g^{\prime}(\boldsymbol{k}):=\mathrm{i}q\chi_{[0,1]}(|\boldsymbol{k}|)\,\frac{1}{|\boldsymbol{k}|^{3/2}(1-\hat{\boldsymbol{k}}\cdot\boldsymbol{v})},$
(3.4)

where $q$ appeared below (1.10), since the difference is an infrared regular
vector from $L^{2}(^{3})$. We have

$\displaystyle
Tg^{\prime}=\sum_{i=1}^{\infty}b_{i}Q_{i}g^{\prime}+(g^{\prime}-\sum_{i=1}^{\infty}Q_{i}g^{\prime}).$
(3.5)

One can show that the first term on the r.h.s. is in $L^{2}(^{3})$ using the
fact that $b_{i}\to 0$. As for the second term, we exploit that
$g^{\prime}=\sum_{i=1}^{\infty}(|\zeta^{\circ}_{i}\rangle\langle\zeta^{\circ}_{i}|\otimes
1)g^{\prime}$ thanks to $\chi_{[0,1]}$ in (3.4). Thus we obtain

$\displaystyle(g^{\prime}-\sum_{i=1}^{\infty}Q_{i}g^{\prime})=\sum_{i=1}^{\infty}\sum_{\ell>i}\sum_{m=-\ell}^{\ell}(|\zeta^{\circ}_{i}\rangle\langle\zeta^{\circ}_{i}|\otimes|Y_{\ell
m}\rangle\langle Y_{\ell m}|)g^{\prime}.$ (3.6)

Now using $\langle Y_{\ell
m},g^{\prime}\rangle=\frac{1}{\sqrt{\ell(\ell+1)}}\langle Y_{\ell
m},\mathrm{L}^{2}g^{\prime}\rangle$ and $\ell>i$ we get sufficiently strong
convergence of the sum to conclude that (3.6) is also in $L^{2}(^{3})$. This
gives Theorem 3.1 for $r=\infty$.

In order to prove Theorem 3.1 for arbitrary $r>0$, we come back to definition
(2.9) of $T_{r}{}$. Recalling that $g$ as in (1.14) is purely imaginary, we
have

$\displaystyle T_{r}{}g$ $\displaystyle\\!\\!\\!=$
$\displaystyle\\!\\!\\!g+\sum_{i=1}^{\infty}(b_{i}^{-1}-1)u_{-\tau^{(r)}_{i}}Q_{i}(\mathrm{i}\,\sin(\mu\tau^{(r)}_{i})g)+\mathrm{i}\sum_{i=1}^{\infty}(b_{i}-1)u_{-\tau^{(r)}_{i}}Q_{i}(-\mathrm{i}\cos(\mu\tau^{(r)}_{i})g).$
(3.7)

Exploiting (3.3), we can write

$\displaystyle T_{r}{}g$ $\displaystyle\\!\\!\\!=$
$\displaystyle\\!\\!\\!(T_{r}{}-T)g+Tg$ (3.9) $\displaystyle\\!\\!\\!=$
$\displaystyle\\!\\!\\!\sum_{i=1}^{\infty}(b_{i}^{-1}-1)u_{-\tau^{(r)}_{i}}Q_{i}(\mathrm{i}\,\sin(\mu\tau^{(r)}_{i})g)+\sum_{i=1}^{\infty}(b_{i}-1)[u_{-\tau^{(r)}_{i}}Q_{i}(\cos(\mu\tau^{(r)}_{i})-1)g]$
$\displaystyle\\!\\!\\!\phantom{444444444444444444444444444444}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+}\sum_{i=1}^{\infty}(b_{i}-1)(u_{-\tau^{(r)}_{i}}-1)Q_{i}g+Tg.$

We know already that $Tg\in L^{2}(^{3})$, so it suffices to consider the
remaining terms. In the case of (3.9) it is clear that
$\sin(\mu\tau^{(r)}_{i})\sim\mu\tau^{(r)}_{i}$ and
$\cos(\mu\tau^{(r)}_{i})-1\sim(\mu\tau^{(r)}_{i})^{2}$ have a regularizing
effect on the $\mu^{-3/2}$ singularity of $g$. We note the estimates on the
$L^{2}$-norms for $l=1,2$

$\displaystyle\|Q_{i}(\mu\tau^{(r)}_{i})^{l}g\|_{2}\leq(\tau^{(r)}_{i})^{l}\|\chi_{[\varepsilon_{i+1},\varepsilon_{i}]}(\mu)\mu^{l}g\|_{2}\leq
c(\tau^{(r)}_{i}\varepsilon_{i})^{l}=c2^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-r(i-1)l}}$
(3.10)

for a constant $c$ independent of $i$. Using this, we immediately obtain the
convergence in $L^{2}(^{2})$ of the sums in (3.9) for $r>0$.

The regularizing effect of $(u_{-\tau^{(r)}_{i}}-1)\sim\mu\tau^{(r)}_{i}$ in
(3.9) is slightly less obvious as this operator does not act directly on $g$.
We argue as follows: On the one hand,

$\displaystyle\|(u_{-\tau^{(r)}_{i}}-1)Q_{i}\|\leq\tau^{(r)}_{i}\|\mu
Q_{i}\|\leq\tau^{(r)}_{i}\|\mu\zeta^{\circ}_{i}\|_{2}\leq
c\tau^{(r)}_{i}\varepsilon_{i}=c2^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-r(i-1)}},$
(3.11)

where $\|\,\cdot\,\|$ above is the operator norm. On the other hand

$\displaystyle\|Q_{i}g\|_{2}\leq\|\chi_{[\varepsilon_{i+1},\varepsilon_{i}]}(\mu)g\|_{2}\leq
c.$ (3.12)

Using (3.11) and (3.12), the $L^{2}$-convergence of the sum in (3.9) follows
for $r>0$. This completes the proof of Theorem 3.1.

##  4 Non-existence of the $S$-matrix in the ordered phase and at the critical
point

The main technical result of this section is the following theorem, which
relies on the assumption $0\neq q:=\int
d^{3}\boldsymbol{x}\,j_{0}(\boldsymbol{x})$.

######  Theorem 4.1.

Let $r\leq 0$. Then $\lim_{\sigma\to
0}\|T_{r}{}(g^{\mathrm{out}}_{\sigma}-g^{\mathrm{in}}_{\sigma})\|_{2}=\infty$
for all $g^{\mathrm{in}/\mathrm{out}}$ as in (1.14) and
$\boldsymbol{v}^{\mathrm{in}}\neq\boldsymbol{v}^{\mathrm{out}}$.

Let us first indicate, following [Ro70], how the property from Theorem 4.1
prevents the existence of the $S$-matrix. Coming back to computation (3.1),
suppose by contradiction that there exists a unitary operator $S_{r}$ such
that

$\displaystyle\mathrm{e}^{-\mathrm{i}\mathrm{Im}\langle
T_{r}{}(g^{\mathrm{out}}-g^{\mathrm{in}}),T_{r}{}F\rangle}\pi_{r}(W(F))=S_{r}\pi_{r}(W(F))S_{r}^{*}.$
(4.1)

By evaluating both sides on $\langle\Omega,\,\cdot\,\,\Omega\rangle$ and using
(2.6), (2.8), we obtain

$\displaystyle\mathrm{e}^{-\mathrm{i}\mathrm{Im}\langle
T_{r}{}(g^{\mathrm{out}}-g^{\mathrm{in}}),T_{r}{}F\rangle}=\mathrm{e}^{\frac{1}{4}\|T_{r}{}F\|_{2}}\langle\Omega,S_{r}W(T_{r}{}F)S_{r}^{*}\Omega\rangle.$
(4.2)

Treating both sides as functions of $T_{r}{}F$ we note that the l.h.s. is
discontinuous in the topology of $L^{2}(^{3})$ by Theorem 4.1 and the Schur
lemma. However, the r.h.s. is continuous by the continuity of the maps
$L^{2}(^{3})\ni\zeta\mapsto W(\zeta)\Psi$, $\Psi\in\mathcal{F}$, between
Hilbert spaces. Using that $T_{r}{}:\mathcal{L}\to L^{2}(^{3})$ has a dense
range (cf. [Ku98, Lemma 3.3]) we obtain a contradiction.

By a similar token we obtain the rotational symmetry breaking in the ordered
phase. Let $\alpha_{\boldsymbol{v}}$ be the automorphism $\alpha$
corresponding to $g$ given by (1.14) with
$\boldsymbol{v}:=\boldsymbol{v}^{\mathrm{out}}$. Let us assume for simplicity
that $\boldsymbol{v}^{\mathrm{in}}=0$. Now suppose that there is a unitary
$V_{\mathcal{R}}$ such that

$\displaystyle\pi_{r}\circ\alpha_{\boldsymbol{v}}\circ\alpha_{0}^{-1}\circ\alpha_{\mathcal{R}}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A})=V_{\mathcal{R}}\pi_{r}\circ\alpha_{\boldsymbol{v}}\circ\alpha_{0}^{-1}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A})V_{\mathcal{R}}^{*},\quad{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(4.3)

Then, by exploiting (2.11), noting
$\alpha_{\mathcal{R}^{-1}}\circ\alpha_{\boldsymbol{v}}\circ\alpha_{\mathcal{R}}=\alpha_{\mathcal{R}\boldsymbol{v}}$
and redefining $\alpha_{0}^{-1}(A)\to A$, we obtain

$\displaystyle\pi_{r}\circ\alpha_{\mathcal{R}\boldsymbol{v}}\circ\alpha_{\boldsymbol{v}}^{-1}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A})=U_{\mathcal{R}}^{*}V_{\mathcal{R}}\pi_{r}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A})V_{\mathcal{R}}^{*}U_{\mathcal{R}},\quad{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(4.4)

By evaluating this relation on a Weyl operator we obtain an identity which is
analogous to (4.1). Thus we arrive at a contradiction which confirms (1.16).

Let us now move on to the proof of Theorem 4.1. As in the proof of Theorem
3.1, we can replace $g^{\mathrm{in}/\mathrm{out}}$ with
$g^{\prime}{}^{\mathrm{in}/\mathrm{out}}$ which have the factor
$\tilde{j}_{0}(\boldsymbol{k})$ replaced with $q\chi_{[0,1]}(\boldsymbol{k})$
as the differences are infrared regular vectors in $L^{2}(^{3})$. We set
$g=g^{\prime}{}^{\mathrm{out}}-g^{\prime}{}^{\mathrm{in}}$ and compute
$\|T_{r}{}g_{\sigma}\|_{2}$ using (3.7)

$\displaystyle\|T_{r}{}g_{\sigma}\|_{2}=\bigg{\|}\sum_{i=1}^{\infty}u_{-\tau^{(r)}_{i}}\bigg{\\{}u_{\tau^{(r)}_{i}}\chi_{\Delta_{i}}g_{\sigma}+(b_{i}^{-1}-1)Q_{i}(\mathrm{i}\,\sin(\mu\tau^{(r)}_{i})g_{\sigma})+\mathrm{i}(b_{i}-1)Q_{i}(-\mathrm{i}\cos(\mu\tau^{(r)}_{i})g_{\sigma})\bigg{\\}}\bigg{\|}_{2},\,\,$
(4.5)

where $\chi_{\Delta_{i}}$ are characteristic functions of the concentric
shells
$\Delta_{i}:=\\{\boldsymbol{k}\in^{3}\,|\,\varepsilon_{i+1}\leq|\boldsymbol{k}|\leq\varepsilon_{i}\,\\}$.
Since the vectors in curly brackets are supported in distinct shells, we can
drop the action of $u_{-\tau^{(r)}_{i}}$ under the norm. Decomposing these
vectors into real and imaginary parts, we obtain a lower bound

$\displaystyle\|T_{r}{}g_{\sigma}\|_{2}$ $\displaystyle\\!\\!\\!\geq$
$\displaystyle\\!\\!\\!\bigg{\|}\sum_{i=1}^{\infty}\bigg{\\{}\mathrm{i}\sin(\mu\tau^{(r)}_{i})\chi_{\Delta_{i}}g_{\sigma}+(b_{i}^{-1}-1)Q_{i}(\mathrm{i}\,\sin(\mu\tau^{(r)}_{i})g_{\sigma})\bigg{\\}}\bigg{\|}_{2}.$
(4.6)

Hence, noting that the cut-off $\sigma$ truncates the sum to some finite
$n(\sigma)$, such that $\lim_{\sigma\to 0}n(\sigma)\to\infty$, we have

$\displaystyle\|T_{r}{}g_{\sigma}\|^{2}_{2}$ $\displaystyle\\!\\!\\!\geq$
$\displaystyle\\!\\!\\!\sum_{i=1}^{n(\sigma)}\langle
g_{\sigma},\chi_{\Delta_{i}}\sin^{2}(\mu\tau^{(r)}_{i})g_{\sigma}\rangle+\sum_{i=1}^{n(\sigma)}(b_{i}^{-2}-1)\langle
g_{\sigma},\sin(\mu\tau^{(r)}_{i})Q_{i}\sin(\mu\tau^{(r)}_{i})g_{\sigma}\rangle.$
(4.7)

Since both sums above are positive, it suffices to show that the first of them
blows up in the limit $\sigma\to 0$. For $r<0$ we have

$\displaystyle\langle
g_{\sigma},\chi_{\Delta_{i}}\sin^{2}(\mu\tau^{(r)}_{i})g_{\sigma}\rangle$
$\displaystyle\\!\\!\\!=$
$\displaystyle\\!\\!\\!c\int_{\varepsilon_{i+1}}^{\varepsilon_{i}}\frac{d|\boldsymbol{k}|}{|\boldsymbol{k}|}\sin^{2}(2^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-r}(i-1)}|\boldsymbol{k}|/\varepsilon_{i})$
(4.8) $\displaystyle\\!\\!\\!=$
$\displaystyle\\!\\!\\!c\int_{1/2}^{1}\frac{d|\boldsymbol{k}|}{|\boldsymbol{k}|}\sin^{2}(2^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-r}(i-1)}|\boldsymbol{k}|)\underset{i\to\infty}{\to}c\int_{1/2}^{1}\frac{d|\boldsymbol{k}|}{2|\boldsymbol{k}|}.$

Here $c>0$ is an inessential factor coming from angular integration and the
last step follows from $\sin^{2}(x)=\frac{1}{2}(1-\cos(2x))$ and the Riemann-
Lebesgue lemma. For $r=0$ the expression on the l.h.s. of (4.8) is independent
of $i$ and non-zero. Thus the sum in (4.7) diverges as $\sigma\to 0$ for
$r\leq 0$. This concludes the proof of Theorem 4.1.

##  5 The $S$-matrix near the critical point

Recall that $S_{r}$ denotes the Bloch-Norsieck $S$-matrix in the disordered
phase
$({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}r>0})$.
In this section we describe the disintegration of this $S$-matrix as we
approach the critical point, that is as
$r\to{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0^{+}}$.
Specifically, we will establish the second line in (1.20).

For this purpose we obtain from (3.1), (2.6)

$\displaystyle\langle\Omega,S_{r}\Omega\rangle=\mathrm{e}^{-\frac{1}{4}\|T_{r}{}g\|^{2}_{2}}$
(5.1)

and recall that the quantity $\|T_{r}{}g\|^{2}_{2}$ was the subject of our
interest in Sections 3, 4. (We allow a modification of $g$ as in (3.4), as it
cannot affect the singular behaviour of $S_{r}$ near the critical point). In
particular, formula (3.9)–(3.9) and estimates (3.10)-(3.12) immediately give
the following upper bound for $1\geq r>0$

$\displaystyle\|T_{r}{}g\|_{2}$
$\displaystyle\\!\\!\\!{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\leq}$
$\displaystyle\\!\\!\\!{\|\sum_{i=1}^{\infty}(b_{i}^{-1}-1)Q_{i}(\mathrm{i}\,\sin(\mu\tau^{(r)}_{i})g)\|_{2}+\sum_{i=1}^{\infty}|b_{i}-1|\|Q_{i}(\cos(\mu\tau^{(r)}_{i})-1)g\|_{2}}$
$\displaystyle\\!\\!\\!\phantom{4444444444444444444444444444}{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+}\sum_{i=1}^{\infty}|b_{i}-1|\|(u_{-\tau^{(r)}_{i}}-1)Q_{i}g\|_{2}+\|Tg\|_{2}}$
$\displaystyle\\!\\!\\!\leq$
$\displaystyle\\!\\!\\!\bigg{(}c\sum_{i=1}^{\infty}(b_{i}^{-1}-1)^{2}2^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-2r}(i-1)}\bigg{)}^{1/2}+c\sum_{i=1}^{\infty}2^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-2r}(i-1)}+c\sum_{i=1}^{\infty}2^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-r}(i-1)}+c\leq\frac{c^{\prime}}{r^{3/2}},$
(5.3)

where the constants $c,c^{\prime}$ are independent of $r$.

Now we are looking for a lower bound for this quantity. We observe, that in
the reasoning (4.5)–(4.8) we used the assumption $r\leq 0$ only in the very
last step, while the preceding steps hold for arbitrary $r\in$. As the first
sum on the r.h.s. of (4.7) leads to a weak upper bound, which does not match
the dependance (5.3), we look at the second sum in (4.7)

$\displaystyle\|T_{r}{}g\|^{2}_{2}$ $\displaystyle\\!\\!\\!\geq$
$\displaystyle\\!\\!\\!\sum_{i=1}^{\infty}(b_{i}^{-2}-1)\langle
g,\sin(\mu\tau^{(r)}_{i})Q_{i}\sin(\mu\tau^{(r)}_{i})g\rangle.$ (5.4)

We used here the fact that we are in the disordered phase and the bounds
(3.10)-(3.12) allow us to take the limit $\sigma\to 0$. Next, we compute

$\displaystyle\langle
g,\sin(\mu\tau^{(r)}_{i})Q_{i}\sin(\mu\tau^{(r)}_{i})g\rangle$
$\displaystyle\\!\\!\\!\geq$
$\displaystyle\\!\\!\\!c_{i}(2^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-2r}})^{i-1}\bigg{|}\int_{\frac{1}{2}}^{1}d|\boldsymbol{k}|\frac{\sin(|\boldsymbol{k}|(2^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-r}})^{i-1})}{|\boldsymbol{k}|(2^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-r}})^{i-1}}\bigg{|}^{2}$
(5.5) $\displaystyle\\!\\!\\!\geq$
$\displaystyle\\!\\!\\!c_{i}(2^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-2r}})^{i-1}\frac{1}{16},$

where in the last step we used $\frac{\sin x}{x}\geq 1/2$ for $|x|\leq 1$. The
constants $c_{i}$ arise from angular integration. As they have the form
$\langle g_{\mathrm{ang}},\tilde{Q}_{i}g_{\mathrm{ang}}\rangle$, where
$g_{\mathrm{ang}}\neq 0$ is the angular part of $g$, and
$\tilde{Q}_{i}\nearrow 1_{L^{2}(S^{2})}$ is an increasing family of
projections, we know that $c_{i}\geq c_{i_{0}}>0$ for some finite $i_{0}$. On
the other hand, first few $c_{i}$ may vanish for certain choices of
$\boldsymbol{v}^{\mathrm{in}}\neq\boldsymbol{v}^{\mathrm{out}}$. Coming back
to (5.4), we can write

$\displaystyle\|T_{r}{}g\|^{2}_{2}$ $\displaystyle\\!\\!\\!\geq$
$\displaystyle\\!\\!\\!\frac{c_{i_{0}}}{16}\sum_{i=i_{0}}^{\infty}(i^{2}-1)(2^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-2r}})^{i-1}\geq\frac{c_{i_{0}}}{16}\frac{2\mathrm{e}^{-2a}}{(1-\mathrm{e}^{-a})^{3}}\mathrm{e}^{-(i_{0}-1)a}|_{a=2|r|\log(2)}\geq\frac{c}{r^{3}}.$
(5.6)

Thus we have verified (1.20).

##  6 The absence of lightcone normality in the disordered phase and at the
critical point

The main result of this section is the following:

######  Theorem 6.1.

The representations $\pi_{r}$ are not lightcone normal for $r\geq 0$.

The breakdown of lightcone normality in the disordered phase and at the
critical point, we alluded to in formula (1.24), is driven by infinite
fluctuations of the timelike order parameter $\phi_{R}(f)$, which we stated in
(1.6). For future convenience we note the relation
$\mathrm{e}^{\mathrm{i}\phi_{R}(f)}=W(F_{R})$, where

$\displaystyle F_{R}(\boldsymbol{k}):=R^{3/2}F(\boldsymbol{k}R),\quad
F(\boldsymbol{k})=(2\pi)^{-3/2}|\boldsymbol{k}|^{-1/2}\int\mathrm{e}^{\mathrm{i}|\boldsymbol{k}|t-\mathrm{i}\boldsymbol{k}\cdot\boldsymbol{x}}f(t,\boldsymbol{x})dt{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}d^{3}\boldsymbol{x}}$
(6.1)

and $f$ is supported in the future lightcone $V_{+}$. Using this notation, we
can write

$\displaystyle\langle\Omega,\pi_{r}(W(F_{R}))\Omega\rangle=\mathrm{e}^{-\frac{1}{4}\|T_{r}{}F_{R}\|^{2}_{2}}=\mathrm{e}^{-\frac{1}{4}\omega_{r}(\phi_{R}(f)\phi_{R}(f))}$
(6.2)

and the last expression tends to zero if timelike fluctuations of the order
parameter tend to infinity with $R$. By Lemma 6.3 below, this property
prevents lightcone normality of the representation.

In Subsection 1.4 we defined the property of lightcone normality of a given
representation $\pi$ as quasi-equivalence of $\pi$ and $\pi_{\mathrm{vac}}$ on
$\mathfrak{A}(V_{+})$ that is unitary equivalence up to multiplicity. In this
section it is more convenient to use a different (but equivalent) definition
[BR, Section 2.4.4.]:

######  Definition 6.2.

If $\pi$ is a representation of a $C^{*}$-algebra $\mathfrak{A}$ then a state
$\omega$ of $\mathfrak{A}$ is said to be $\pi$-normal if there exists a
density matrix $\rho$ such that

$\displaystyle\omega(A)=\mathrm{Tr}(\rho\pi(A))$ (6.3)

for all $A\in\mathfrak{A}$. Two representations $\pi_{1}$ and $\pi_{2}$ of
$\mathfrak{A}$ are said to be quasi-equivalent, written
$\pi_{1}\approx\pi_{2}$, if each $\pi_{1}$-normal state is $\pi_{2}$-normal
and conversely.

The following lemma, which is an instance of the method of central sequences,
gives a useful criterion to exclude lightcone normality. The irreducibility of
$\pi_{r}$, which is stated as an assumption, is a consequence of the dense
range of $T_{r}{}$. For the latter property we refer to [Ku98, Lemma 3.3].

######  Lemma 6.3.

Let $\pi:\mathfrak{A}\to B(\mathcal{F})$ be an irreducible representation.
Suppose that the following limit exists (possibly along a subsequence) and

$\displaystyle\lim_{R\to\infty}\langle\Omega,\pi(W(F_{R}))\Omega\rangle\neq\langle\Omega,W(F)\Omega\rangle.$
(6.4)

Then $\pi$ is not lightcone normal.

Proof. Suppose, by contradiction, that $\pi$ is lightcone normal. Then
$\omega_{\mathrm{vac}}$ is $\pi$-normal, hence there exists a density matrix
$\rho$ such that

$\displaystyle\langle\Omega,W(F)\Omega\rangle=\langle\Omega,W(F_{R})\Omega\rangle=\mathrm{Tr}\big{(}\rho\,\pi(W(F_{R}))\big{)}=\sum_{j=1}^{\infty}p_{j}\langle\Psi_{j},\pi(W(F_{R}))\Psi_{j}\rangle,$
(6.5)

where we expressed
$\rho=\sum_{j=1}^{\infty}p_{j}|\Psi_{j}\rangle\langle\Psi_{j}|$ via an
orthonormal system $\\{\Psi_{j}\\}_{j\in\mathbb{N}}$ in $\mathcal{F}$ and
$\\{p_{j}\\}_{j\in\mathbb{N}}$ is a sequence of positive numbers, summable to
one. Now we can write $\Psi_{j}=U_{j}\Omega$ for some unitaries
$U_{j}\in\pi(\mathfrak{A})$ by the irreducibility assumption and the Kadison
transitivity theorem [KR, Theorem 10.2.1]. We observe that the localization
region of $W(F_{R})=\mathrm{e}^{\mathrm{i}\phi_{R}(f)}$ is shifted to timelike
infinity as $R\to\infty$. Consequently, by the Huyghens principle, i.e.
commutation of the massless scalar free field at timelike separation of
arguments, $\lim_{R\to\infty}[W(F_{R}),A]=0$ in norm for any
$A\in\mathfrak{A}$. This gives

$\displaystyle\lim_{R\to\infty}\langle\Psi_{j},\pi(W(F_{R}))\Psi_{j}\rangle=\lim_{R\to\infty}\langle\Omega,\pi(W(F_{R}))\Omega\rangle.$
(6.6)

Now by taking the limit $R\to\infty$ in (6.5) and using (6.4), we conclude the
proof. $\Box$

Let us now prove Theorem 6.1 using Lemma 6.3. Due to (6.2) it suffices to show
that $\lim_{R\to\infty}\|T_{r}{}F_{R}\|_{2}=\infty$ along a subsequence.
Coming back to representation (2.9), we can write

$\displaystyle\|T_{r}{}F_{R}\|_{2}\geq\|\sum_{i=1}^{\infty}(b_{i}^{-1}-1)u_{-\tau^{(r)}_{i}}Q_{i}\mathrm{Re}\,u_{\tau^{(r)}_{i}}F_{R}\|_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}2}}+O(1).$
(6.7)

Here $O(1)$ denotes error terms which are manifestly bounded in $R$, because
the respective operators are bounded and $\|F_{R}\|_{2}$ is independent of
$R$. Thus it suffices to show that

$\displaystyle\lim_{n\to\infty}\sum_{i=1}^{\infty}(b_{i}^{-1}-1)^{2}\langle\mathrm{Re}(u_{\tau^{(r)}_{i}}F_{R_{n}}),Q_{i}\mathrm{Re}(u_{\tau^{(r)}_{i}}F_{R_{n}})\rangle=\infty,$
(6.8)

for some subsequence $\\{R_{n}\\}_{n\in\mathbb{N}}$, tending to infinity. As
we are interested in a bound from below, it suffices to consider the $n$-th
term in the sum over $i$ and the $\ell=0$ term in the sum over the spherical
harmonics hidden in $Q_{i}$. Let us pick in (6.1)
$f(t,\boldsymbol{x})=\delta(t-\tau)\eta(\boldsymbol{x})$, where $\tau>0$ is
sufficiently large depending on the support of $\eta$, so that $f$ is
supported inside the future lightcone. Furthermore, we choose $\eta$
spherically symmetric and such that $\tilde{\eta}$ is positive. The resulting
expression reads

$\displaystyle(b_{n}^{-1}-1)^{2}|\langle\zeta^{\circ}_{n},\mathrm{Re}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}u_{\tau^{(r)}_{i}}}F_{R_{n}})\rangle|^{2}=c_{0}^{2}(b_{n}^{-1}-1)^{2}\bigg{(}\int_{\varepsilon_{n+1}R_{n}}^{\varepsilon_{n}R_{n}}d|\tilde{\boldsymbol{k}}|\,\tilde{\eta}(|\tilde{\boldsymbol{k}}|)\cos(|\tilde{\boldsymbol{k}}|R_{n}^{-1}(2^{-{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(n-1)}|r|}\varepsilon^{-1}_{n}+\tau))\bigg{)}^{2}.\,\,$
(6.9)

Now we choose $R_{n}:=L/\varepsilon_{n}$ for some $L>0$. First, suppose that
$r>0$ and compute

$\displaystyle\lim_{n\to\infty}\int_{\varepsilon_{n+1}R_{n}}^{\varepsilon_{n}R_{n}}d|\tilde{\boldsymbol{k}}|\,\tilde{\eta}(\tilde{\boldsymbol{k}})\cos(|\tilde{\boldsymbol{k}}|R_{n}^{-1}(2^{-{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(n-1)}|r|}\varepsilon^{-1}_{n}+\tau))=\int_{L/2}^{L}d|\tilde{\boldsymbol{k}}|\,\tilde{\eta}(\tilde{\boldsymbol{k}}),$
(6.10)

which is strictly positive. Since $b_{n}^{-2}\to\infty$, we obtain (6.8).
Next, suppose $r=0$. Then the integral in (6.9) reads

$\displaystyle\int_{L/2}^{L}d|\tilde{\boldsymbol{k}}|\,\tilde{\eta}(\tilde{\boldsymbol{k}})\cos(|\tilde{\boldsymbol{k}}|/L),$
(6.11)

which is also strictly positive. Thus we conclude the proof of Theorem 6.1.

As we anticipated in (6.2), the timelike order parameter diverges to infinity
in the disordered phase and at the critical point. Actually the same is true
for the spacelike order parameter. In fact, the discussion (6.7)–(6.11) above
can be immediately adapted to the spacelike case by choosing $\tau=0$ and the
support of $\eta$ outside zero. Thus we have verified the second line in
(1.6).

Interestingly, for $r>0$ the divergence of the spacelike order parameter can
be seen directly from the existence of the $S$-matrix, using the method of
central sequences from the proof of Lemma 6.3. Coming back to (3.1), we have

$\displaystyle\mathrm{e}^{-\mathrm{i}\mathrm{Im}\langle
g^{\mathrm{out}}-g^{\mathrm{in}},F_{R}\rangle}\pi_{r}(W(F_{R}))$
$\displaystyle\\!\\!\\!=$
$\displaystyle\\!\\!\\!S_{r}\pi_{r}(W(F_{R}))S_{r}^{*},$ (6.12)

for a unitary $S_{r}$. We choose here $F_{R}$ as in (6.1), but for $f$
supported in the spacelike complement of zero. We compute

$\displaystyle\mathrm{Im}\langle
g^{\mathrm{out}}-g^{\mathrm{in}},F_{R}\rangle\underset{R\to\infty}{\to}\tilde{j}_{0}(0)\int{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}d^{3}\boldsymbol{k}}\,\frac{\mathrm{Re}(F(\boldsymbol{k}))}{|\boldsymbol{k}|^{3/2}}\bigg{\\{}\frac{1}{(1-\hat{\boldsymbol{k}}\cdot\boldsymbol{v}^{\mathrm{out}})}-\frac{1}{(1-\hat{\boldsymbol{k}}\cdot\boldsymbol{v}^{\mathrm{in}})}\bigg{\\}},$
(6.13)

which can easily be arranged to be different from zero (and multiples of
$2\pi$) for $\boldsymbol{v}^{\mathrm{out}}\neq\boldsymbol{v}^{\mathrm{in}}$.
Now arguing as in the proof of Lemma 6.3, we obtain

$\displaystyle\lim_{R\to\infty}(\mathrm{e}^{-\mathrm{i}\mathrm{Im}\langle
g^{\mathrm{out}}-g^{\mathrm{in}},F_{R}\rangle}-1)\langle\Omega,\pi_{r}(W(F_{R}))\Omega\rangle=0,$
(6.14)

which implies $\langle\Omega,\pi_{r}(W(F_{R}))\Omega\rangle\to 0$, hence
$\omega_{r}(\phi_{R}(f)\phi_{R}(f))\to\infty$, cf. (6.2). Thus diverging
fluctuations of the field at spacelike infinity are an unavoidable price for
stabilizing the Bloch-Nordsieck $S$-matrix, independent of the detailed
structure of the states $\omega_{r}$.

Unfortunately, such a general argument fails for fluctuations of the timelike
order parameter, as (6.13) is inevitably zero in this case [CD20]. (Unlike the
spacelike case, $F$ cannot be chosen real here). Thus for the divergence of
the fluctuations of the field in future timelike directions we have to rely on
computations presented earlier in this section which depended on the
particular structure of the states $\omega_{r}$. The same applies to the
absence of lightcone normality.

##  7 Lightcone normality in the ordered phase

In this section we prove the following theorem:

######  Theorem 7.1.

The representations $\pi_{r}$ are lightcone normal for $r<0$.

Our proof is inspired by [DFG84], but we consider a more complicated family of
states. We use a general fact that any state $\omega$ on $\mathfrak{A}$ such
that

$\displaystyle|\omega(A)-\omega_{\mathrm{vac}}(A)|<2\|A\|,\quad
A\in\mathfrak{A}(V_{+}),$ (7.1)

is lightcone normal, cf. [Ha, Theorem 2.2.16]. To check this criterion, we
will use representation (2.4) of states $\omega_{r}$. We recall the operators
$\mathcal{U}_{r}(I_{n})$ from (2.5) and generalize them to any subset
$I\subset I_{n}$. Now by the Weyl relations and the definition of the vacuum
(2.6) the approximating sequence of $\omega_{r}$ from (2.4) can be restated as
follows:

$\displaystyle\langle\mathcal{U}_{r}(I)\Omega,W(F)\mathcal{U}_{r}(I)\Omega\rangle$
$\displaystyle=\bigg{(}\prod_{\beta^{\prime}}\frac{z_{\beta^{\prime}}}{2\pi}\bigg{)}\int
d^{\\#I}s\int
d^{\\#I}t\,\mathrm{e}^{-{\frac{1}{4}}\sum_{\beta}(\hat{b}_{\beta}-1)s_{\beta}^{2}}\mathrm{e}^{-{\frac{1}{4}}\sum_{\gamma}(\hat{b}_{\gamma}-1)t_{\gamma}^{2}}\mathrm{e}^{-\frac{1}{4}\sum_{\beta}(s_{\beta}-t_{\beta})^{2}}$
(7.2)
$\displaystyle\times\mathrm{e}^{\frac{\mathrm{i}}{2}[t_{\beta}\mathrm{Im}\langle\tilde{f}_{\beta,\mu},F\rangle+s_{\gamma}\mathrm{Im}\langle\tilde{\bar{f}}_{\gamma,\mu},F\rangle+{\mathrm{Im}\langle
F,-s_{\gamma}\tilde{f}_{\gamma,\mu}-t_{\gamma}\tilde{\bar{f}}_{\gamma,\mu}\rangle}]}\mathrm{e}^{\frac{1}{4}2\mathrm{Re}\langle
F,(s-t)_{\beta}(\tilde{\bar{f}}_{\beta,\mu}-\tilde{f}_{\beta,\mu})\rangle}$
(7.3)
$\displaystyle\times\mathrm{e}^{\frac{1}{4}[s_{\beta}s_{\gamma}\mathrm{q}_{\beta,\gamma}+t_{\beta}t_{\gamma}\mathrm{p}_{\beta,\gamma}-2s_{\beta}t_{\gamma}\mathrm{r}_{\beta,\gamma}]}$
(7.4)
$\displaystyle\times\langle\Omega,W(-t_{\beta}d_{+,\beta}+s_{\beta}d_{-,\beta})W(F)W(s_{\gamma}d_{+,\gamma}-t_{\gamma}d_{-,\gamma})\Omega\rangle,$
(7.5)

where $F\in\mathcal{L}$, $f_{\beta}\in D(^{4};\mathbb{C})$,
$\tilde{f}_{\beta,\mu}(\boldsymbol{k}):=\mu^{-1/2}(\boldsymbol{k})\tilde{f}_{\beta}(|\boldsymbol{k}|,\boldsymbol{k})$
and summation over repeated indices $\beta,\gamma$ is understood.

Let us start the discussion of this lengthy formula from factor (7.3). Here,
by (7.1), it suffices to consider $F\in\mathcal{L}(V_{+})$, where
$\mathcal{L}(V_{+})=\\{\phi(f)\Omega\,|\,f\in D(^{4};),\mathrm{supp}(f)\subset
V_{+}\\}$. Restricting attention to $f_{\beta}\in D(V_{-};\mathbb{C})$, where
$V_{-}$ is the backward lightcone, this factor is equal to one by the Huyghens
principle. This is a significant step towards verifying (7.1), as now function
$F$ appears only in (7.5) and both sides of the equation extend to linear
functionals acting on $A\in\mathfrak{A}(V_{+})$.

Regarding factors (7.4), (7.5), the rule of the game is to choose $f_{\beta}$
in such a way that the quantities

$\displaystyle
d_{+,\beta}:=\mathrm{i}\zeta^{(r)}_{\beta}-\tilde{f}_{\beta,\mu},\ \
d_{-,\beta}:=\tilde{\bar{f}}_{\beta,\mu}$ (7.6)

are small. This will allow us, in particular, to control the factor (7.4),
since the quantities

$\displaystyle\mathrm{q}_{\beta,\gamma}:=w_{\beta,\gamma}-2\mathrm{i}u^{+,-}_{\beta,\gamma},\quad\mathrm{p}_{\beta,\gamma}:=w_{\beta,\gamma}+2\mathrm{i}u^{+,-}_{\beta,\gamma},\quad\mathrm{r}_{\beta,\gamma}:=\mathrm{Re}(w_{\beta,\gamma}){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-}\mathrm{i}(u^{+,+}_{\beta,\gamma}-u^{-,-}_{\beta,\gamma})$
(7.7)

depend on $d_{\pm}$ via

$\displaystyle\ \ w_{\beta,\gamma}:=\langle
d_{+,\beta}+d_{-,\beta},d_{+,\gamma}+d_{-,\gamma}\rangle,\ \
u^{\sigma_{1},\sigma_{2}}_{\beta,\gamma}:=\mathrm{Im}\langle
d_{\sigma_{1},\beta},d_{\sigma_{2},\gamma}\rangle,\quad\sigma_{1},\sigma_{2}\in\\{\pm\\}.$
(7.8)

Now if we find such $f_{\beta}$ that $d_{\pm,\beta}\to 0$ then the r.h.s. of
identity (7.2)–(7.5) approaches the vacuum state as needed to check criterion
(7.1).

Let us indicate how to control the resulting error terms: First, we note the
straightforward estimate

$\displaystyle|t_{\beta}\,\mathrm{q}_{\beta,\gamma}s_{\gamma}|,|t_{\beta}\,\mathrm{p}_{\beta,\gamma}s_{\gamma}|,|t_{\beta}\,\mathrm{r}_{\beta,\gamma}s_{\gamma}|\leq\frac{3}{2}{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|d\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}^{2}\sum_{\beta}(|t_{\beta}|^{2}+|s_{\beta}|^{2})\|d_{\beta}\|_{2},$
(7.9)

where we set $\|d_{\beta}\|_{2}:=\|d_{+,\beta}\|_{2}+\|d_{-,\beta}\|_{2}$,
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|d\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}:=\sum_{\beta}\|d_{\beta}\|_{2}$
and applied the Cauchy-Schwarz inequality. Thus we obtain the following bound
for error terms originating from (7.4):

$\displaystyle|\mathrm{e}^{\frac{1}{4}[s_{\beta}s_{\gamma}\mathrm{q}_{\beta,\gamma}+t_{\beta}t_{\gamma}\mathrm{p}_{\beta,\gamma}-2s_{\beta}t_{\gamma}\mathrm{r}_{\beta,\gamma}]}-1|\leq{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|d\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}\mathrm{e}^{3\sum_{\beta}(|t_{\beta}|^{2}+|s_{\beta}|^{2})\|d_{\beta}\|_{2}},$
(7.10)

where we noted that $|\mathrm{e}^{x}-1|\leq y\mathrm{e}^{(1+y^{-1})|x|}$ for
$x\in,y>0$, and assumed that
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|d\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}\leq
1$. By functional calculus, the same bound can be established for quantities

$\displaystyle\|\big{(}W(-t_{\beta}d_{+,\beta}+s_{\beta}d_{-,\beta})^{*}-1\big{)}\Omega\|,\quad\|\big{(}W(s_{\gamma}d_{+,\gamma}-t_{\gamma}d_{-,\gamma})-1\big{)}\Omega\|,$
(7.11)

which appear in error terms originating from (7.5). Altogether, using (7.10)
and (7.2)–(7.5) we easily obtain the following bound

$\displaystyle\\!\\!\\!|\langle\mathcal{U}_{r}(I)\Omega,A\mathcal{U}_{r}(I)\Omega\rangle-\langle\Omega,A\Omega\rangle|$
$\displaystyle\\!\\!\\!\leq{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|d\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}\|A\|\bigg{(}\prod_{\beta}\frac{z_{\beta}}{2\pi}\bigg{)}\int
d^{\\#I}s\int
d^{\\#I}t\,\mathrm{e}^{-{\frac{1}{4}}\sum_{\beta}([\hat{b}_{\beta}-12\|d_{\beta}\|_{2}]-1)s_{\beta}^{2}}\mathrm{e}^{-{\frac{1}{4}}\sum_{\gamma}([\hat{b}_{\gamma}-12\|d_{\gamma}\|_{2}]-1)t_{\gamma}^{2}}\mathrm{e}^{-\frac{1}{4}\sum_{\beta}(s_{\beta}-t_{\beta})^{2}}$
$\displaystyle\\!\\!\\!={\left|\kern-1.07639pt\left|\kern-1.07639pt\left|d\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}\|A\|\bigg{(}\prod_{\beta}\frac{z_{\beta}}{\tilde{z}_{\beta}}\bigg{)},$
(7.12)

where $z_{\beta}:=\frac{1}{2}\sqrt{\hat{b}_{\beta}^{2}-1}$ as defined before,
$\tilde{z}_{\beta}:=\frac{1}{2}\sqrt{[\hat{b}_{\beta}-12\|d_{\beta}\|_{2}]^{2}-1}$
and in the last step we used Gaussian integration. Now by elementary estimates
we arrive at the following lemma:

######  Lemma 7.2.

Let
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|d\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}\leq
1$. Suppose that $\hat{b}_{\beta}-12\|d_{\beta}\|_{2}-1>0$ for $\beta\in I$.
Then the following estimate holds

$\displaystyle|\langle\mathcal{U}_{r}(I)\Omega,A\mathcal{U}_{r}(I)\Omega\rangle-\langle\Omega,A\Omega\rangle|\leq
c\|A\|{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|d\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}\exp\bigg{(}c^{\prime}\sup_{\gamma\in
I}\hat{b}_{\gamma}\sum_{\beta\in
I}\frac{\|d_{\beta}\|_{2}}{\hat{b}_{\beta}-12\|d_{\beta}\|_{2}-1}\bigg{)}\quad$
(7.13)

for $A\in\mathfrak{A}(V_{+})$ and some numerical constants $c$, $c^{\prime}$.

The next task is to find $f_{\beta}\in D(V_{-};\mathbb{C})$ such that
$d_{\pm,\beta}\to 0$ sufficiently fast, given our set of timelike shifts
$\\{\tau^{(r)}_{\beta}\\}_{\beta\in I_{n}}$ of (1.1). With such input, we
intend to estimate the r.h.s. of (7.13) so as to obtain (7.1). As a first
step, in Appendix A we find for any $\tilde{r}>0$ a family of functions
$f^{\circ}_{i}\in D(^{4};\mathbb{C})$ supported in double cones of radii
$2/\delta_{i}$,
$\delta_{i}:={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}2}\varepsilon_{i}^{1+\tilde{r}}$,
such that

$\displaystyle\|\zeta^{\circ}_{i}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathrm{i}Y_{\ell
m}}-\tilde{f}_{\mu,i}^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\circ}}\|_{2}\leq
c_{\tilde{r}}i^{2}\varepsilon_{i}^{\tilde{r}/4},\quad\|\tilde{\bar{f}}_{\mu}^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\circ}}\|_{2}\leq
c_{\tilde{r}}i^{2}\varepsilon_{i}^{\tilde{r}/4},$ (7.14)

where the constants $c_{\tilde{r}}$ are independent of $i$. Next, we assume
that $\tilde{r}\leq|r|$ and define the functions appearing in (7.6) as
$\tilde{f}_{\mu,i}:=\mathrm{e}^{-\mathrm{i}\mu{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\tau^{(r)}_{i}}}\tilde{f}_{\mu,i}^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\circ}}$.
Clearly, the fields $\phi(f_{i})$ are localized in the backward lightcone
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}V_{-}}$.
As a matter of fact, the bounds (7.14) are at the basis of the introductory
discussion in Subsection 1.1 and the double cones in Fig. 1 symbolize the
localization regions of the fields $\phi(f_{i})$ in the case
$0<\tilde{r}\ll|r|$. By invariance of the norm under the unitary time
evolution, the bounds (7.14) hold also for the quantities

$\displaystyle\|d_{+,\beta}\|_{2}=\|\mathrm{e}^{-\mathrm{i}\mu{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\tau_{i}}}\zeta^{\circ}_{i}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathrm{i}Y_{\ell
m}}-\mathrm{e}^{-\mathrm{i}\mu{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\tau_{i}}}\tilde{f}_{\mu,i}^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\circ}}\|_{2},\quad\|d_{-,\beta}\|_{2}=\|\mathrm{e}^{-\mathrm{i}\mu{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\tau_{i}}}\tilde{\bar{f}}_{\mu}^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(7.15)

After this preparation we are almost ready to prove Theorem 7.1. The last
remaining difficulty is to obtain from (7.13) the bound by $2\|A\|$ appearing
in (7.1). The problem is that $d_{\pm,\beta}\to 0$ does not mean that
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|d\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}:=\sum_{\beta}\|d_{\beta}\|_{2}$,
appearing on the r.h.s. of (7.13), is small. In fact, although the sequence
$\|d_{\beta}\|_{2}$ converges to zero, first few terms can be large. However,
the smallness of
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|d\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
can easily be ensured by dropping a finite number of modes from the states
$\omega_{r}$. (This amounts to dropping a finite number of terms in the sums
in (2.9)). It is intuitively clear that this leads to a unitarily equivalent
representation. After all, by the Stone-von Neumann uniqueness theorem, the
appearance of distinct sectors requires infinitely many degrees of freedom,
cf. Appendix B for details. Thus we conclude the proof of Theorem 7.1 using
Lemma 7.2 and estimate (7.1) given bounds (7.14) which hold also for (7.15).

##  8 Order parameters in the ordered phase

In this section we determine the timelike and spacelike order parameters in
the ordered phase. Thus we verify the first line in (1.6), which we left aside
so far. Regarding the timelike case, we can use the lightcone normality of
$\pi_{r}$: Since the representation is equivalent to the vacuum, it is not
surprising that the timelike asymptotic fluctuations of the field take the
vacuum value. At the technical level, this statement follows immediately from
Lemma 6.3. The spacelike case is more interesting as we cannot conclude the
vacuum form of the order parameter directly from lightcone normality. However,
some ingredients from the proof of lightcone normality will also prove
important here.

Since we are looking at the spacelike order parameter, we can choose such $f$
in (6.1) that the resulting $F$ is real-valued. Then we obtain from formula
(2.9)

$\displaystyle\omega_{r}(\phi_{R}(f)\phi_{R}(f))$ $\displaystyle\\!\\!\\!=$
$\displaystyle\\!\\!\\!\|T_{r}{}F_{R}\|^{2}_{2}$ (8.1)
$\displaystyle\\!\\!\\!=$ $\displaystyle\\!\\!\\!\|F\|_{2}^{2}+\sum_{\beta\in
I_{\infty}}\big{(}b_{\beta}^{-2}-1\big{)}|\mathrm{Re}\langle\zeta^{(r)}_{\beta},F_{R}\rangle|^{2}+\sum_{\beta\in
I_{\infty}}\big{(}b_{\beta}^{2}-1\big{)}|\mathrm{Im}\langle\zeta^{(r)}_{\beta},F_{R}\rangle|^{2}.$

Using functions $f_{\beta}$, which appeared in (7.6), we rewrite the above
scalar products as follows:

$\displaystyle\langle\zeta^{(r)}_{\beta},F_{R}\rangle=\langle\zeta^{(r)}_{\beta}-\tilde{f}_{\beta,\mu},F_{R}\rangle+\langle\tilde{f}_{\beta,\mu},F_{R}\rangle.$
(8.2)

Now recalling (7.14), (7.15), the first term satisfies

$\displaystyle|\langle\zeta^{(r)}_{\beta}-\tilde{f}_{\beta,\mu},F_{R}\rangle|\leq\|F\|_{2}c_{r}i^{2}\varepsilon_{i}^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}|r|}/4},$
(8.3)

where we chose $\tilde{r}=|r|$. As for the second term in (8.2), let us first
restrict attention to $\beta$, $R$ such that $\phi(f_{\beta})$ and
$\phi_{R}(f)$ commute. Then

$\displaystyle|\langle\tilde{f}_{\beta,\mu},F_{R}\rangle|=|\langle\phi(f_{\beta})\Omega,\phi_{R}(f)\rangle|{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}=}|\langle\phi_{R}(f)\Omega,\phi(\bar{f}_{\beta})\Omega\rangle|\leq\|F\|_{2}\|\tilde{\bar{f}}_{\beta,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mu}}\|_{2}\leq\|F\|_{2}c_{r}i^{2}\varepsilon_{i}^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}|r|}},$
(8.4)

where we used again (7.14), (7.15).

$\operatorname{supp}f$xt$\mathbf{z_{0}\hskip
1.42271pt\frac{2}{\delta_{\beta}}}$$c_{2}R$$c_{1}R$ Fig. 3. Schematic
illustration for relation (8.6).

Now we determine the set of $\beta$, $R$ for which the commutation of
$\phi(f_{\beta})$ and $\phi_{R}(f)$ may fail. We recall from the discussion
above (7.14), that the function $f^{\circ}_{\beta}$, for $\beta=(1,0,0)$, was
supported in the double cone of radius $1$ centered at zero. Since the support
of a function is compact and the double cone is an open set, this function is
also supported in a double cone of a slightly smaller radius $(1-z_{0})$. For
arbitrary $\beta$, functions $f^{\circ}_{\beta}$ are supported in double cones
of radii $\frac{2}{\delta_{\beta}}(1-z_{0})$, where $z_{0}$ is independent of
$\beta$. This is due to the fact that these supports simply arise by scaling
the supports of certain auxiliary functions $\tilde{\eta}$,
$\tilde{\eta}_{1}$, cf. Appendix A. Taking the timelike shifts into account,
the support of $f_{\beta}$ is contained in

$\displaystyle\mathcal{O}_{\frac{2}{\delta_{\beta}}(1-z_{0})}-e_{0}\tau^{(r)}_{\beta}=\mathcal{O}_{\frac{2}{\delta_{\beta}}(1-z_{0})}-e_{0}\frac{2}{\delta_{\beta}},$
(8.5)

where $e_{0}$ is the timelike unit vector and $\mathcal{O}_{a}$ is the double
cone of radius $a$ centered at zero. Now suppose that $f$ (entering into the
order parameter) is supported in the annulus $0<c_{1}<|x|<c_{2}$. Then
$\phi_{R}(f)$ and
$\phi(f_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta}})$
commute, unless

$\displaystyle
z_{0}\frac{2}{\delta_{\beta}}<c_{2}R\quad\Rightarrow\quad\frac{z_{0}}{2c_{2}}(2^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}|r|}})^{i-1}<\varepsilon_{i+1}R,$
(8.6)

where we recalled that
$\delta_{\beta}=2\varepsilon_{\beta}^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}1+|r|}}$.
This can be read off from Fig. 3, where $\tilde{r}=|r|$ was chosen (unlike
Fig. 1 drawn for $\tilde{r}\ll|r|$). For $\beta$ as in (8.6) we cannot argue
as in (8.4) but instead we get a similar bound on the quantity in (8.2) by an
explicit computation: Choosing $f$ depending only on $\boldsymbol{x}$ and
spherically symmetric, we obtain

$\displaystyle|\langle\zeta^{(r)}_{\beta},F_{R}\rangle|$
$\displaystyle\\!\\!\\!=$
$\displaystyle\\!\\!\\!(4\pi)\big{|}\int^{\varepsilon_{i}R}_{\varepsilon_{i+1}R}d|\boldsymbol{k}^{\prime}|\,\mathrm{e}^{\mathrm{i}|\boldsymbol{k}^{\prime}|\tau^{(r)}_{i}R^{-1}}\tilde{f}(|\boldsymbol{k}^{\prime}|)\big{|}$
(8.7) $\displaystyle\\!\\!\\!\leq$
$\displaystyle\\!\\!\\!(4\pi)\int_{\frac{z_{0}}{4c_{2}}(2^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}|r|}})^{i-1}}^{\infty}d|\boldsymbol{k}^{\prime}|\,|\tilde{f}(|\boldsymbol{k}^{\prime}|)|\leq
c_{N}[(2^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}|r|}})^{i-1}]^{-N},$

where in the last two steps we made use of (8.6) and of the rapid decay of
$f$. Now the discussion (8.2)–(8.7) gives

$\displaystyle|\langle\zeta^{(r)}_{\beta},F_{R}\rangle|\leq
c^{\prime}_{r}i^{2}\varepsilon_{i}^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}|r|}},$
(8.8)

for some constant $c^{\prime}_{r}$ independent of $R$,
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta}$.
There are no restrictions on $R$, $\beta$ anymore, as we treated all the
possibilities.

Finally, using (8.8), we obtain bounds on the summands in (8.1) which are
independent of $R$ and summable. Thus, by the dominated convergence theorem,
we can enter with the limit $R\to\infty$ under the sums. Then it is clear from
the first step in (8.7) that the individual summands tend to zero and we are
left with the vacuum contribution $\|F\|_{2}^{2}$.

##  9 Conclusion and outlook

In this paper we described a family of states $\\{\omega_{r}\\}_{r\in}$
interpolating between the vacuum at $r\to-\infty$ and the Kraus-Polley-Reents
infravacuum at $r\to\infty$. We found a phase transition at $r=0$ separating
the ordered phase ($r<0$) from the disordered phase ($r>0$). Remarkably, in
the ordered phase the state is lightcone normal, that is, it cannot be
distinguished by any experiments in the future lightcone from a state in the
vacuum sector. Not surprisingly, this phase suffers from all the infrared
problems familiar from the vacuum sector: the Bloch-Nordsieck $S$-matrix
disintegrates and soft photon clouds break rotational symmetry. However, by an
infinitesimal change of the control parameter from
$r={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-\epsilon}$
to
$r={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+\epsilon}$
we enter the disordered phase, where the $S$-matrix is stabilized by the
Kraus-Polley-Reents mechanism and rotational symmetry is restored. The
transition can be observed in the future lightcone by measuring the timelike
asymptotic fluctuations of the field. Their abrupt increase prevents the
lightcone normality in the disordered phase.

Our phase transition is quite subtle and does not easily fit into standard
categories. As states $\omega_{r}$ are not ground states, it is a non-
equilibrium phase transition, which consists in a qualitative change of a
certain dynamics. This dynamics is, in our case, the Bloch-Nordsieck
scattering process and the qualitative change consists in the existence or
disintegration of its scattering matrix. This transition is also visible in
the asymptotic fluctuations of the order parameter stated in (1.6), which have
a discontinuity at $r=0$. On the one hand, such discontinuity suggests a first
order phase transition. On the other hand, the behaviour of the correlation
length $\langle\Omega,S_{r}\Omega\rangle^{-1}\sim\exp(c/r^{3/2})$, stated in
(1.20), points to a continuous transition. It would be interesting to study
the disintegration of other quantities near $r=0$, such as the angular
momentum in representations $\pi_{r}\circ\alpha$. It would also be desirable
to identify some universal features of the resulting dependencies (‘critical
exponents’) independent of the detailed construction of the states
$\omega_{r}$. Finally, the transition is clearly visible in the boundary
degrees of freedom, but it may be accompanied by some measurable effects in
the bulk. For example, the rate of decay of correlations
$\boldsymbol{x}\mapsto\omega_{r}(\phi(0)\phi(\boldsymbol{x}))$ might
distinguish the two phases. We leave these questions to future research.

The Kraus-Polley-Reents mechanism for curing infrared problems has obvious
advantages over the usual infraparticle (or Faddeev-Kulish) approach. Instead
of intricate, velocity dependent dressing transformations of the $S$-matrix,
it provides a regularizing background radiation which is independent of the
dynamics. After replacing the vacuum with a KPR infravacuum the usual Dyson
$S$-matrix is meaningful. This is clear in the external source situation, but
also thinkable in perturbation theory. Apart from the scalar field, the KPR
infravacua are available for the electromagnetic field [KPR77, CD19] and we
are confident that our findings could be generalized to this case. However, to
our knowledge the KPR mechanism has not yet been tested in the case of
massless higher spin particles. It would be interesting to fill this gap,
given the difficulties with applying the Faddeev-Kulish approach in the
presence of gravitons [PSW22]. In the next step one could revisit the approach
to the black hole information paradox from [HPS16] in the presence of KPR
radiation. As such radiation dramatically changes the superselection structure
of infrared degrees of freedom, this may lead to interesting new insights.

##  Appendix A Outline of the proof of estimates (7.14)

For brevity, we skip the index
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta}$
on $f$, $\zeta^{\circ}$, $\tau^{(r)}$ and write $Y:=Y_{\ell m}$. We consider
$f^{\circ}\in D(^{4};\mathbb{C})$ of the form
$f^{\circ}(t,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\boldsymbol{x}})=f_{0}(t)f_{1}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\boldsymbol{x}})$.
We require that it is supported in a double cone of radius $\frac{2}{\delta}$,
for some $0<\delta\leq 2$. We have

$\displaystyle\tilde{f}^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\circ}}_{\mu}(\boldsymbol{k})$
$\displaystyle\\!\\!\\!=$
$\displaystyle\\!\\!\\!|\boldsymbol{k}|^{-1/2}\frac{1}{(2\pi)^{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}3/2}}\int
dt{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}d^{3}\boldsymbol{x}}\,\mathrm{e}^{\mathrm{i}|\boldsymbol{k}|t}\mathrm{e}^{-\mathrm{i}\boldsymbol{k}\cdot\boldsymbol{x}}f_{0}(t)f_{1}(\boldsymbol{x})=|\boldsymbol{k}|^{-1/2}\tilde{f}_{0}(|\boldsymbol{k}|)\tilde{f}_{1}(\boldsymbol{k}),$
(A.1)
$\displaystyle\tilde{\bar{f}}^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\circ}}_{\mu}(\boldsymbol{k})$
$\displaystyle\\!\\!\\!=$
$\displaystyle\\!\\!\\!|\boldsymbol{k}|^{-1/2}\bar{\tilde{f}}_{0}(-|\boldsymbol{k}|)\bar{\tilde{f}}_{1}(-\boldsymbol{k}).$
(A.2)

We will choose $f_{0}(t)=\bar{f}_{0}(-t)$ so that $\tilde{f}_{0}$ is real. We
want to choose $\tilde{f}_{0}(k^{0})\simeq 1$ for $k^{0}>0$ and
$\tilde{f}_{0}(k^{0})\simeq 0$ for $k^{0}<0$. For this purpose, we set

$\displaystyle\tilde{f}_{0}(k^{0}):=(\eta_{0,\delta}*\chi_{[0,2]})(k^{0}),$
(A.3)

where
$\eta_{0,\delta}(\,\cdot\,):=\frac{1}{\delta}\eta_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0}(\frac{1}{\delta}(\,\cdot\,))$
and
$\eta_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0}\in
D(;)$ is positive, symmetric and such that $\int dk^{0}\,\eta_{0}(k^{0})=1$ so
that $\eta_{0,\delta}$ is a delta approximating sequence as $\delta\to 0$. We
also require that the Fourier transform of
$\eta_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0}$
is supported in the interior of a ball of radius one so that
$\tilde{\eta}_{\delta}$ is supported inside a ball of radius $1/\delta$. As
for $f_{1}$, we choose analogously a delta approximating sequence
$\eta_{1,\delta}(\,\cdot\,):=\frac{1}{\delta^{3}}\eta_{1}(\frac{1}{\delta}(\,\cdot\,))$
in momentum space, such that its Fourier transform is compactly supported in
the interior of a ball of radius $1/\delta$. We set

$\displaystyle\tilde{f}_{1}(\boldsymbol{k}):=(\eta_{1,\delta}*\mu^{1/2}{\zeta^{\circ}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathrm{i}Y})(\boldsymbol{k}).$
(A.4)

By slightly tedious, but straightforward estimates one checks that for any
$\tilde{r}>0$ and
$\delta:={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}2}\varepsilon_{i}^{1+\tilde{r}}$

$\displaystyle\|\zeta^{\circ}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathrm{i}Y}-\tilde{f}_{\mu}^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\circ}}\|_{2}\leq
c_{\tilde{r}}i^{2}\varepsilon_{i}^{\tilde{r}/4},\quad\|\tilde{\bar{f}}_{\mu}^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\circ}}\|_{2}\leq
c_{\tilde{r}}i^{2}\varepsilon_{i}^{\tilde{r}/4},$ (A.5)

where the constants $c_{\tilde{r}}$ are independent of $i$. The essential
ingredients here are the rapid decay of the functions $\eta_{0},\eta_{1}$, the
uniform bound $|\tilde{f}_{0}(k^{0})|\leq 1$ and basic properties of spherical
harmonics:

$\displaystyle|Y_{\ell m}(\hat{\boldsymbol{k}})|$ $\displaystyle\\!\\!\\!\leq$
$\displaystyle\\!\\!\\!c(\ell+1)^{1/2},$ (A.6) $\displaystyle|Y_{\ell
m}(\hat{\boldsymbol{k}})-Y_{\ell,m}(\hat{\boldsymbol{k}}^{\prime})|$
$\displaystyle\\!\\!\\!\leq$
$\displaystyle\\!\\!\\!\tilde{\alpha}\ell(\ell+1),$ (A.7)

where $\tilde{\alpha}$ is the angle between the unit vectors
$\hat{\boldsymbol{k}},\hat{\boldsymbol{k}}^{\prime}$. For (A.6) we refer to
[AH, formula (2.36)]. Estimate (A.7) follows by expressing the rotation from
$\hat{\boldsymbol{k}}$ to $\hat{\boldsymbol{k}^{\prime}}$ in terms of the
angular momentum operators. The factor $i^{2}$ on the r.h.s. of (A.5) comes
from estimates (A.6), (A.7) and the condition $\ell<i$ appearing in (2.10).

##  Appendix B Omitting finite number of modes from $\omega_{r}$

We provide here some details for the last step of the proof of Theorem 7.1. We
define $I^{n_{0}}_{n}:=\\{\,\beta=(i,\ell,m)\in I_{n}\,|\,i>n_{0}\\}$ for
$n_{0}<n$ and decompose $I_{n}=I_{n_{0}}\cup I^{n_{0}}_{n}$. By the arguments
which led to (2.7), (2.8) we can write

$\displaystyle\langle\mathcal{U}_{r}(I^{n_{0}}_{n})\Omega,W(F)\mathcal{U}_{r}(I^{n_{0}}_{n})\Omega{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\rangle}=\langle\Omega,W(T_{r,I^{n_{0}}_{n}}F)\Omega\rangle,$
(B.1)

where the symplectic map $T_{r,I^{n_{0}}_{n}}$ arises by keeping only
$n_{0}<i\leq n$ in (2.9). Denoting by $\pi_{r,I}$ the representation acting by
$\pi_{r,I}(W(F))=W(T_{r,I}F)$, we note the relation

$\displaystyle\langle\Omega,\pi_{r,I_{n}}(W(F))\Omega\rangle=\langle\mathcal{U}_{r}(I_{n})\Omega,W(F)\mathcal{U}_{r}(I_{n})\Omega\rangle=\langle\mathcal{U}_{r}(I_{n_{0}})\Omega,\pi_{r,I^{n_{0}}_{n}}(W(F))\mathcal{U}_{r}(I_{n_{0}})\Omega\rangle.$
(B.2)

By keeping $n_{0}$ fixed and taking the limit $n\to\infty$, we obtain

$\displaystyle\langle\Omega,\pi_{r}(W(F))\Omega\rangle=\langle\mathcal{U}_{r}(I_{n_{0}})\Omega,\pi_{r,I^{n_{0}}_{\infty}}(W(F))\mathcal{U}_{r}(I_{n_{0}})\Omega\rangle.$
(B.3)

Thus, by [CD19, Lemma A.1], $\pi_{r}$ is unitarily equivalent to
$\pi_{r,I^{n_{0}}_{\infty}}$ for any $n_{0}\in\mathbb{N}$.

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