#  Multiparticle states in braided lightlike
$\kappa$-Minkowski noncommutative QFT

(1)Giuseppe<EMAIL_ADDRESS>(2)Flavio
<EMAIL_ADDRESS>
(1)Dipartimento di Fisica Ettore Pancini, Università di Napoli “Federico II”;
and INFN, Sezione di Napoli, Complesso Univ. Monte S. Angelo, I-80126 Napoli,
Italy;
(2)Departamento de Física, Universidad de Burgos, 09001 Burgos, Spain.

###### Abstract

In this study, we construct a 1+1-dimensional, relativistic, free, complex
scalar Quantum Field Theory on the noncommutative spacetime known as lightlike
$\kappa$-Minkowski. The associated $\kappa$-Poincaré quantum group of
isometries is triangular, and its quantum R matrix enables the definition of a
braided algebra of N points that retains $\kappa$-Poincaré invariance.
Leveraging our recent findings, we can now represent the generators of the
deformed oscillator algebra as nonlinear redefinitions of undeformed
oscillators, which are nonlocal in momentum space. The deformations manifest
at the multiparticle level, as the one-particle states are identical to the
undeformed ones. We successfully introduce a covariant and involutive deformed
flip operator using the R matrix. The corresponding deformed
(anti-)symmetrization operators are covariant and idempotent, allowing for a
well-posed definition of multiparticle states, a result long sought in Quantum
Field Theory on $\kappa$-Minkowski. We find that P and T are not symmetries of
the theory, although PT (and hence CPT) is. We conclude by noticing that
identical particles appear distinguishable in the new theory, and discuss the
fate of the Pauli exclusion principle in this setting.

###### Contents

  1. 1 Introduction
  2. 2 Noncommutative geometry of lightlike $\kappa$-Minkowski
    1. 2.1 The lightlike $\kappa$-Minkowski spacetime and the $\kappa$-Poincaré quantum group
    2. 2.2 Braided N-point algebra and its representations
  3. 3 Braided lightlike $\kappa$-deformed QFT
    1. 3.1 Old and new-type noncommutative plane waves and momentum space
    2. 3.2 Covariant quantization and oscillator algebra
    3. 3.3 Representation of the deformed oscillator algebra
    4. 3.4 1-particle Fock state and C, P, T symmetries
    5. 3.5 Braided flip operator and multiparticle states
  4. 4 Physical interpretation of deformed multi-particle states
    1. 4.1 On the indistinguishability of identical particles
    2. 4.2 On the Pauli exclusion principle
  5. 5 Conclusions

##  1 Introduction

Quantum Field Theory (QFT) on noncommutative spacetimes has been studied for
decades [1, 2, 3]. The initial motivations came from the desire to regularize
the ultraviolet divergences of quantum electrodynamics [4], together with the
intuition that quantum theory might require spacetime itself to be quantized
in some sense. This proposal entails some serious interpretational challenges,
as the smooth topology of spacetime would have to be replaced by something
new. The discussion of these issues echoes the debate that followed the
introduction of quantization conditions on phase-space orbits by Bohr [5]. The
idea of describing quantum phase space as a non-smooth, “pointless” geometry
remained in the back burner until the development of the theory of von Neumann
algebras, which represented the birth of noncommutative geometry [6]. The
commutative $C^{*}$-algebras of functions on a topological manifold have been
shown to completely characterize the topology of the manifold [7, 8].
Replacing this algebra with a noncommutative algebra leads to the modern
notion of a noncommutative geometry [6]. The tools of classical differential
topology and Riemannian geometry are insufficient to describe objects that
lack the notion of infinitesimal points, so the study of the properties of
these noncommutative spaces requires a purely algebraic formulation. Connes,
Woronowicz and Drinfel’d generalized the notion of a differential structure to
the noncommutative setting [9, 10, 11], which led to the definition of gauge
theories on a large class of noncommutative spaces. The “Noncommutative
Standard Model” of Connes, Lott and Chamseddine [12, 13] allows to unify the
Standard Model fields with the Higgs boson as well as the gravitational field
as an effective description of a “quasi-commutative” geometry. This model has
been extensively studied for decades, and, although so far it has not been
possible to deduce unambiguous predictions of some Standard Model parameters
that would make the model falsifiable, it still represents today a possible
avenue towards unification.

The main motivation for a noncommutative structure of spacetime comes from
Quantum Gravity. Already at the effective level of QFT coupled to classical
general relativity, it can be shown that the Planck length,
$\ell_{P}=\sqrt{{G\hbar}/{c^{3}}}$, represents a limit to the localizability
of fields [14]. Rather than attacking frontally the full problem of Quantum
Gravity, one could incorporate these restrictions to locality in effective
models that feature uncertainty relations among some noncommutative
“coordinate” operators. A compelling evidence that such a “noncommutative QFT”
could be a realistic, intermediate level of description between commutative
QFT and the full quantum theory of gravity comes from the only model of
Quantum Gravity that is well-understood as a QFT: 2+1-dimensional General
Relativity. This theory lacks local propagating degrees of freedom (gravitons)
and can be therefore quantized with topological QFT methods. Coupling the
theory to matter and integrating away the gravitational degrees of freedom
gives rise to an effective theory of matter propagating on a noncommutative
spacetime [15, 16]. The Planck scale in this model ends up playing the role of
a scale of noncommutativity.

Further evidence for spacetime noncommutativity as an effective description
comes from String Theory, in which the intrinsic length scale of strings has
been conjectured to prevent probing arbitrarily small distances [17].
Moreover, a connection between String Theory and noncommutative geometry was
found by Witten in the context of interacting bosonic open strings [18], and
later with Seiberg, with the identification of a regime in which the string
dynamics is described by a gauge theory on a noncommutative space [19]. Models
of noncommutative QFT inspired by the Seiberg–Witten one (so-called
$\theta$-Moyal-type spacetimes) have been extensively studied in their own
right [1], and a 1+1-dimensional toy model of this kind, studied by Grosse and
Wulkenhaar [20], was proven to be finite at all energy. This is the only known
example of an interacting QFT that is well-defined at all scales, and realizes
Snyder’s original dream of keeping the divergences of QFT under control via
noncommutativity. Notice that the Seiberg–Witten model [19] relies on a tensor
B-field taking an expectation value in the vacuum and providing a Lorentz-
breaking preferred background frame with respect to which the noncommutativity
of coordinates is specified. This, of course, raises the question of the
destiny of Lorentz invariance in noncommutative QFT, which becomes
particularly pressing in view of the stringent constraints available today on
Lorentz invariance violations [21, 22]. Beyond the need to evade these
constraints, we would also like to establish whether spacetime
noncommutativity can be compatible with relativistic invariance, which is a
question of great interest in itself. That the minimal lengths appearing in
Quantum Gravity might not imply Lorentz invariance violation has been long
conjectured. In Loop Quantum Gravity, it has been argued that the discreteness
of the spectra of geometric operators like the area is analogue to what
happens in quantum mechanics with the angular momentum operator [23]: one can
have a rotationally-symmetric state of non-zero spin, which of course is
impossible in classical mechanics. In quantum mechanics, these rotationally-
invariant states appear the same to all observers, because, although upon
measurement they get nonzero spins along one direction, this happens with a
spherically-symmetric probability distribution, and the symmetry is only
broken upon choosing a direction in space and realizing a projective
measurement.

In this paper, we are interested in conjectured field theories that deform,
rather than break, Poincaré invariance. In a theory of this type, the scale of
noncommutativity would represent a second relativistic scale, on par with the
speed of light $c$, which appears the same to all observers. This would be a
concrete realization of the principle of Doubly Special Relativity conjectured
more than two decades ago by Amelino-Camelia [24]. In such a theory, the
transformation rules between inertial observers are changed in such a way that
not only the speed of light, but also a new length/energy scale appears the
same to all observers. The existence of deformed-Poincaré-invariant QFTs has
long been conjectured, and models that may qualify as such have been studied
extensively for decades, encountering severe conceptual and technical
difficulties. The main reason to believe that such theories exist comes again
from noncommutative geometry: there are many examples of noncommutative
homogeneous spaces, which are invariant under quantum group symmetries [25,
26]. These are generalizations of the notion of a Lie group, in which the
algebra of functions on the group is noncommutative, and the transformation
parameters become fuzzy, just like the coordinates of the space. Deforming the
Poincaré group into a quantum group depending on an invariant length scale
seems therefore to be a concrete realization of the Doubly Special Relativity
principle mentioned above [27]. We now have a few candidates for a
noncommutative Minkowski spacetime, invariant under a quantum Poincaré group
[28, 27, 29], however the debate on how to build a QFT on such spaces in such
a way that Poincaré invariance is preserved is still open. In the case of
$\theta$-Moyal-type noncommutative spacetimes, the standard techniques used to
derive QFT predictions from the theory is to introduce a star product, which
is an infinite-dimensional representation of the noncommutative product
between coordinates on a space of commutative functions. Then the
noncommutative action of a field theory can be written as a nonlocal action in
terms of commutative fields. The nonlocality is due to the fact that the
action depends on derivatives of all orders of the commutative fields.
Treating such an action as a commutative one, one gets correlation functions
that depend on the noncommutativity parameters in an apparently Lorentz-
violating way. This seems at odds with the existence of quantum group
deformations of the Poincaré group that leave the $\theta$-Moyal-type
noncommutative spacetime (and the corresponding QFT actions) invariant [30,
31, 32, 33]. Solutions have been proposed, that require a distinction between
active and passive transformations [34, 35, 36]. In [37, 38], a different
approach was proposed, that involves a more careful treatment of the concept
of multilocal functions. In the commutative case, these are simply functions
from several points on the spacetime manifold onto the complex or real
numbers, and in terms of the (commutative) algebra of functions on the
manifold, they can be simply formulated as elements of tensor products of
copies of the same algebra of functions. In the noncommutative setting, taking
simply the tensor product fails to produce a covariant structure: in other
words, assuming that the coordinates of different points commute with each
other is not covariant under the quantum group Poincaré transformations.
Fortunately, mathematicians found a generalization of the concept of tensor
product, called braiding [25], which allows one to identify a noncommutative
algebra of N-points that is invariant under the relevant quantum group. In
[37, 38], this structure was used to define the Wightman functions of a
noncommutative QFT in a consistent way. Interestingly, the difference between
noncommutative coordinates of different points belong to a commutative
subalgebra of the braided N-point algebra, so that the N-point functions of
the theory are commutative and admit a simple interpretation as correlation
functions in perfect analogy to commutative QFT. There have been various
attempts to produce testable predictions of QFT on the $\theta$-Moyal
spacetime, including some conjectured violations of the Pauli Exclusion
principle [39, 40, 41]. The original approach to $\theta$-Moyal braided QFT
suggests that the theory is equivalent to the commutative one, at least at the
perturbative level [42]. More recent approaches [43, 44, 45, 46, 47], based on
Oeckl’s algebraic definition of braided QFT [48, 49], find that N-point
functions for $N>3$ are indeed deformed [46], although the one-loop self-
energy of a $\lambda\,\phi^{4}$ theory is not. The approach of [43, 44, 45,
46, 47] works for any quantum group deformation of the Poincaré group that can
be expressed as a twist, and therefore applies, in principle, to the case
studied in the present paper. It would be interesting to compare that approach
with ours.

Recently, some of us developed a similar construction for QFT on a different
noncommutative spacetime [50, 51], finding a rich and previously unnoticed
level of complexity. The noncommutative geometry in question is the so-called
“lightlike” $\kappa$-Minkowski spacetime, symmetric under the “lightlike”
$\kappa$-Poincaré quantum group [52, 53, 54, 55]. This model has great
interest for physics, because its noncommutativity parameter has the
dimensions of a length (unlike the $\theta$-Moyal models whose parameter is an
area), which suggests that it might capture the first order in an expansion in
powers of the Planck length of an effective theory of fields on a quantum-
gravitational background. A closely related noncommutative spacetime (the so-
called “timelike” version of $\kappa$-Minkowski) has been studied for decades
[56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74,
75, 76, 77, 78, 79, 80, 81], and many efforts went into building a consistent
QFT on it [82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97]. A
lot of progress has been made on the problem, although much work remains to be
done, and a fully satisfactory QFT, both from the interpretational and the
technical/mathematical point of view, remains elusive.

In our first paper [50], the covariant braided N-point algebra for a general
parametrization of $\kappa$-Minkowski-like noncommutative spacetimes was
constructed, and it was proven that its associativity is only compatible with
the lightlike model. In fact, the structure constants of the coordinate
algebra of this class of spacetimes are usually expressed in terms of four
parameters forming a vector (conventionally called $v^{\mu}$), and the
braiding construction turned out to be possible only if said vector is
lightlike (or null), hence the name of the model. Furthermore, the coordinate
differences between different points (and therefore all N-point functions)
were shown to be commutative, just like in the work of Wess and Fiore [37,
38]. In [50], a proposal for a covariant Pauli–Jordan function was put
forward, however a technical obstacle prevented us from defining general
Lorentz-invariant N-point functions. Namely, the momentum space of the theory
was not closed under Lorentz transformations, which practically meant that
certain momentum space integrals would have a Lorentz-breaking upper bound
related to the deformation energy scale. Thanks to a recent observation [98],
this problem of the non-closure of momentum space under Lorentz
transformations can be solved by enlarging the basis of noncommutative
functions that are used in the Fourier expansion of fields, to plane waves
that include a constant complex contribution to the frequency. This allows one
to “double” momentum space into two halves that are connected to each other by
Lorentz transformations, and together, are globally Lorentz invariant. This
observation was used in the recent work [51] to define a free complex scalar
field theory consistently (using covariant quantization based on a
Pauli–Jordan function), and derive the associated deformed construction and
annihilation operator algebra. Unfortunately, this algebra turned out to be
extremely complicated, due to the presence of the additional region of
momentum space, which, together with the mass shell, splits the commutator of
two creation/annihilation operators into no less than twenty cases which need
to be listed separately.

In this paper, we build upon the results of [50] and [51], and find a
substantial simplification for the oscillator algebra. We are able to find a
simple representation for our deformed creation and annihilation operators
that can be expressed in one line, and is based on infinite nonlinear
combinations of standard creation and annihilation operators. Such
representation makes the unwieldy algebra of [51] treatable, and allows us to
begin drawing some physical conclusions from the theory. First, the one-
particle sector is completely undeformed and coincide with that of a
commutative free complex scalar QFT. Secondly, the charge conjugation operator
is undeformed and Poincaré covariant. This was not the case in other
approaches to QFT on $\kappa$-Minkowski (in the case of timelike $v^{\mu}$)
[98]. In particular, the recent [99] shows that the charge conjugation
operator sends a one-particle state into a one-antiparticle state with
different momentum. This phenomenon is not present in our model. Regarding P
and T symmetries, these are not symmetries of the commutation relations
between coordinates, and this fact manifests itself already at the level of
the one-particle sector: we are not able to introduce a P or a T operator that
acts on the oscillator algebra or on the Fock space in the desired way. This,
however, does not prevent PT symmetry from being realized: thanks to the
antilinearity of the T operator, both the coordinate commutation relations and
the free field theory can be shown to be PT-invariant. Having C and PT, the
CPT invariance of the model is also guaranteed.

The nontriviality of the model manifests itself all in the multi-particle
sector. Already at the level of two particles one sees that the total momentum
depends nonlinearly on the momenta of the two particles, and the action of
Lorentz transformations on two momenta becomes nonlinear and mixes the
components of the momenta of the two particles (something dubbed
“backreaction” in previous works [76, 100]). Finally, we are able to introduce
a “braided flip operator” that exchanges the momenta of two particles in a
nonlinear way, which possesses all the properties that such an operator
should: it is Lorentz covariant and is an involution (its square is the
identity operator). This operator can be used to define symmetric and
antisymmetric states, which are necessary to define the Fock space of bosonic
and fermionic fields. Recent work by another group [101] showed that, in the
case of the “timelike” $\kappa$-Minkowski spacetime, such a flip operator does
not exist. The next best things are either non-Lorentz-covariant at all
orders, or are not involutive [102, 103, 104, 105], which means that one can
build an infinite tower of two-particle states that all share the same total
momentum. The conclusion of the authors of [101] is that the very notion of
identical particles, and (anti-)symmetrized multiparticle states loses
meaning. These results do not apply to the model considered in the present
paper, as our flip operator is both involutive and Lorentz-covariant. This
allows us to introduce a well-defined notion of multi-particle states, which
is something that has eluded studies of QFT on $\kappa$-Minkowski for decades.
The deformed multi-particle states allow for a revision of the classical
concepts of indistinguishability of identical particles and of the Pauli
exclusion principle. We find that, given enough precision, particles of the
same species which are described by a deformed (anti)-symmetric state can be
distinguished by an experiment measuring their momenta. Moreover, the class of
states prohibited by the Pauli Exclusion Principle is instead allowed in this
setting, while another class of states not excluded by the standard Principle
is instead prohibited.

##  2 Noncommutative geometry of lightlike $\kappa$-Minkowski

###  2.1 The lightlike $\kappa$-Minkowski spacetime and the $\kappa$-Poincaré
quantum group

The $d+1$-dimensional $\kappa$-Minkowski noncommutative space-time is defined
by commutations relations among coordinates of the form

$[x^{\mu},x^{\nu}]=\frac{i}{\kappa}(v^{\mu}x^{\nu}-v^{\nu}x^{\mu})\,,\quad\mu=0,1,\dots,d\,,$
(1)

where $\kappa$ is a deformation parameter with the dimensions of energy (in
natural units $c=\hbar=1$), and $v^{\mu}$ is a set of four real parameters.
The algebra of functions on Minkowski space-time is hence deformed into a non-
commutative algebra $\mathcal{A}$, generated by $x^{\mu}$ and the identity
$1$, and equipped with a non-commutative product defined by (1). One can
introduce a (commutative) arbitrary constant metric tensor $g_{\mu\nu}$, and
require that it is preserved by a quantum group of symmetries, which also
leaves leaves the commutation relations (1) invariant. One obtains different
quantum groups depending on the relationship between the parameters $v^{\mu}$
and the metric $g_{\mu\nu}$. If the parameters form a lightlike/null vector,
i.e. $v^{\mu}v^{\nu}g_{\mu\nu}=0$, one obtains a triangular Hopf algebra [25],
which is the best-behaved case (see below). This quantum group has been
discovered in [52, 53, 54, 55]. The spacelike case has been discussed in [106,
107], while the timelike one, first introduced in [58, 56, 108], is by far the
most-studied one. The appeal of the timelike case is that, superficially, the
algebra (1) appears spatially isotropic, and indeed it is invariant under
commutative/undeformed spatial rotations. At an early time of investigation of
the physics of quantum groups, when some phenomenological consequences were
being conjectured, undeformed spatial isotropy seemed compelling, because
before clarifying the difference between symmetry breaking and symmetry
deformations, a non-isotropic model could be feared to be incompatible with
very basic observations of the isotropy of empty space [109]. At the present
stage of understanding of the model, these worries result unfounded. Of course
(1) cannot be invariant under the full Lorentz (or Poincaré) group, unless one
replaces the group with a quantum group, as we will show momentarily. In this
case, whether the commutators (1) appear spatially isotropic or not is an
irrelevant point: the only way this affects the theory is that there is a
basis for the quantum Poincaré algebra of invariance of (1) in which the
rotation generators appear “more commutative/undeformed” (specifically: their
coproducts are primitive). This does not have any real consequences, because,
as we are about to show, there is a sense in which, for any choice of
$v^{\mu}$, the $\kappa$-Poincaré group has a Lorentz subgroup that is
commutative/undeformed, and all the noncommutativity is relegated to the
translations, which act on the Lorentz group in a nontrivial way. For these
reasons, we do not find any valid reason to prefer a particular choice of
$v^{\mu}$ vector, at this stage.

In developing field theories on the non-commutative space-time (1), the notion
of $N$-point functions is essential. These can be defined starting from the
braided tensor product algebra $\mathcal{A}^{\tilde{\otimes}N}$, which deforms
the standard tensor product of $N$ copies of $\mathcal{A}$ by introducing
nontrivial commutation relations between the coordinates of different points,
like $[x^{\mu}_{a},x^{\nu}_{b}]\neq 0$, with $a,b$ referring to different
copies of $\mathcal{A}$. Details of this construction can be found in [50,
51], where it was also shown that the $\kappa$-Poincaré invariance of the
$\mathcal{A}^{\tilde{\otimes}N}$ commutation relations, together with the
imposition of the Jacobi rule, selects only $v^{\mu}$ such that
$g_{\mu\nu}v^{\mu}v^{\nu}=0$.

Therefore, from now on (and as done in [50, 51]), we will restrict our
attention to the lightlike $\kappa$-Minkowski non-commutative space-time,
which in $1+1$ dimensions is characterized by the commutation relations

$[x^{+},x^{-}]=\frac{2i}{\kappa}\,x^{-}\,,\qquad x^{\pm}=x^{0}\pm x^{1}\,.$
(2)

In the following, we will work in units in which $\kappa=1$.

The symmetries of (2) are expressed in terms of the $\kappa$-Poincaré quantum
group, denoted by $\mathbb{C}_{\kappa}[ISO({\,1,1})]$. This notation
characterizes $\kappa$-Poincaré as a noncommutative deformation of the Hopf
algebra $\mathbb{C}_{\kappa}[ISO({\,1,1})]$ of complex functions on the
Poincaré group $ISO({\,1,1})$. The algebra sector reads [110] (all greek
indices run in the set $\\{+,-\\}$)

$\displaystyle[\Lambda^{\mu}_{\\!\ \\!\ \nu},\Lambda^{\rho}_{\\!\ \\!\
\sigma}]=0,\quad[a^{\mu},a^{\nu}]=i(v^{\mu}a^{\nu}-v^{\nu}a^{\mu})$ (3)
$\displaystyle[a^{\gamma},\Lambda^{\mu}_{\\!\ \\!\
\nu}]=i[(\Lambda^{\mu}_{\\!\ \\!\
\alpha}v^{\alpha}-v^{\mu})\Lambda^{\gamma}_{\\!\ \\!\
\nu}+(\Lambda^{\alpha}_{\\!\ \\!\
\nu}g_{\alpha\beta}-g_{\nu\beta})v^{\beta}g^{\mu\gamma}]$
$\displaystyle\Lambda^{\mu}_{\\!\ \\!\ \alpha}\Lambda^{\nu}_{\\!\ \\!\
\beta}g^{\alpha\beta}=g^{\mu\nu},\quad\Lambda^{\rho}_{\\!\ \\!\
\mu}\Lambda^{\sigma}_{\\!\ \\!\ \nu}g_{\rho\sigma}=g_{\mu\nu},$

where

$v^{\mu}=(2,0)\,,\qquad g_{\mu\nu}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)\qquad\Rightarrow\qquad g_{\mu\nu}v^{\mu}v^{\nu}=0\,.$
(4)

The coproduct $\Delta$, antipode $S$ and counit $\epsilon$, which codify
information on the quantum group product, inversion and identity are un-
deformed, and their expressions are given by

$\displaystyle\Delta[\Lambda^{\mu}_{\\!\ \\!\ \nu}]=\Lambda^{\mu}_{\\!\ \\!\
\alpha}\otimes\Lambda^{\alpha}_{\\!\ \\!\
\nu},\quad\Delta[a_{\mu}]=\Lambda^{\mu}_{\\!\ \\!\ \nu}\otimes
a^{\nu}+a^{\mu}\otimes 1$ (5) $\displaystyle S[\Lambda^{\mu}_{\\!\ \\!\
\nu}]=(\Lambda^{-1})^{\mu}_{\\!\ \\!\ \nu},\quad
S[a^{\mu}]=-(\Lambda^{-1})^{\mu}_{\\!\ \\!\
\nu}a^{\nu},\quad\epsilon[{\Lambda^{\mu}_{\\!\ \\!\
\nu}}]=\delta^{\mu}_{\nu},\quad\epsilon[a^{\mu}]=0.$

The Poincaré transformations of spacetime coordinates can be understood in
terms of a left co-action operator
$\cdot\,{}^{\prime}:\mathcal{A}\rightarrow\mathbb{C}_{\kappa}[ISO({\,1,1})]\otimes\mathcal{A}$.
We will write this co-action in the following compact way:

$x^{\prime}{}^{\mu}=\Lambda^{\mu}_{\\!\ \\!\ \nu}x^{\nu}+a^{\mu}\,,$ (6)

where the product on the right-hand side is understood as the tensor product
$\Lambda^{\mu}_{\\!\ \\!\ \nu}\otimes x^{\nu}+a^{\mu}\otimes 1$. In this
notation, it is understood that $[\Lambda^{\mu}_{\\!\ \\!\
\nu},x^{\rho}]=[a^{\mu},x^{\nu}]=0$. It is easy to check that, given the
coordinate transformation (6) and the commutation rules (3), the commutator
(2) is left invariant, in the sense that

$[x^{\prime+},x^{\prime-}]=2i\,x^{\prime-}\,,$ (7)

and the commutation relations appear identical to all inertial observers. The
symmetries can also be described in terms of the dual Hopf Algebra
$\mathcal{U}_{\kappa}[\mathfrak{iso}(1,1)]$, which can be thought of as a non-
commutative deformation of the universal enveloping algebra
$\mathcal{U}[\mathfrak{iso}(1,1)]$ of the Poincaré Lie algebra
$\mathfrak{iso}(1,1)$. To extract the relevant structures of
$\mathcal{U}_{\kappa}[\mathfrak{iso}(1,1)]$, we apply a finite transformation
on non-commutative plane waves, with a given ordering, and extract the action
of the generators of the algebra by evaluating the first order of the
transformation rules of plane waves. In this calculation and throughout the
manuscript, we choose to work with the $x^{+}$ to-the-right ordering, and a
transformed plane wave can be written as

$e^{ik_{-}{x^{\prime}}^{-}}e^{ik_{+}{x^{\prime}}^{+}}\,,$ (8)

where ${x^{\prime}}^{-},{x^{\prime}}^{+}$ can be read off from (6) and
$k_{\mu}\in\mathbb{C}$.333Here and in the following, we consider ordered
exponentials of the non-commutative coordinates with both real and complex
parameters. The properties of the exponentials do not depend whether the
parameters are real of complex, in general. For the sake of simplicity, we
will refer to these functions as plane waves, even if the parameter has an
imaginary component (see section 3.1). From the last line of (3), specified by
(4), the Lorentz part of the transformation can be parametrized by a single
operator $\tau$, as follows:

$\Lambda^{\mu}{}_{\nu}=\left(\begin{array}[]{cc}e^{\tau}&0\\\
0&e^{-\tau}\end{array}\right)\,.$ (9)

From commutators (3), it is possible to show that

$[a^{+},\tau]=2i(e^{\tau}-1)\,,\qquad[a^{-},\tau]=0\,.$ (10)

Using relations (10) and techniques developed in [50], we can write (8) as

$e^{ik_{-}e^{-\tau}x^{-}}e^{\frac{i}{2}\log[1+e^{\tau}(e^{2k_{+}}-1)]x^{+}}e^{ik_{-}a^{-}}e^{ik_{+}a^{+}}\,.$
(11)

Focusing on the Lorentz sector of the transformation, we want to write

$e^{ik_{-}e^{-\tau}x^{-}}e^{\frac{i}{2}\log[1+e^{\tau}(e^{2k_{+}}-1)]x^{+}}\approx\quantity(1+i\tau
N\triangleright)e^{ik_{-}x^{-}}e^{ik_{+}x^{+}}+\mathcal{O}(\tau^{2})\,,$ (12)

where $N$ is the boost operator in $\mathcal{U}_{\kappa}[\mathfrak{iso}(1,1)]$
and $\triangleright$ is a left action
$\triangleright:\mathcal{U}_{\kappa}[\mathfrak{iso}(1,1)]\otimes\mathcal{A}\to\mathcal{A}$.
Expanding the left hand-side at first order in $\tau$, one finds

$e^{ik_{-}e^{-\tau}x^{-}}e^{\frac{i}{2}\log[1+e^{\tau}(e^{2k_{+}}-1)]x^{+}}\approx\quantity(1-i\tau
x^{-}k_{-})e^{ik_{-}x^{-}}\quantity(1+i\tau
x^{+}\quantity(\frac{1-e^{-2k_{+}}}{2}))e^{ik_{+}x^{+}}\,.$ (13)

This can be understood as a non-linear deformation of the action of the
standard boost operator on a commutative plane-wave which is then mapped to a
non-commutative one with a given ordering (in this case $x^{+}$ to the right),
by means of a Weyl map $\Omega:C[\mathbb{R}^{2}]\rightarrow\mathcal{A}$ [111],
defined as ($kx$ is a shorthand for $k_{\mu}x^{\mu}$):

$\Omega(e^{ikx})=e^{ik_{-}x^{-}}e^{ik_{+}x^{+}}\,.$ (14)

The action of the boost operator $N$ can thus be written as

$N\triangleright\Omega(e^{ikx})=\Omega\quantity[\quantity((ix^{-}\partial_{-})+x^{+}\quantity(\frac{1-e^{2i\partial_{+}}}{2}))e^{ikx}]\,.$
(15)

By inspecting the translation sector of the transformation, the action of the
$\tilde{P_{\pm}}$ generators can be defined as

$\tilde{P}_{\pm}\triangleright\Omega(e^{ikx})=\Omega(-i\partial_{\pm}e^{ikx})=k_{\pm}\Omega(e^{ikx})$
(16)

Using expressions (15)-(16) and applying the generators in succession on a
single plane waves, one finds the commutators:

$[N,\tilde{P}_{+}]=i\quantity(\frac{1-e^{-2\tilde{P}_{+}}}{2})\qquad[N,\tilde{P}_{-}]=-i\tilde{P}_{-}\,,$
(17)

which can be easily shown to satisfy the Jacobi identities. The coproducts
encode the deviation from the Leibniz rule, and are found by applying the
generators on products of plane waves. The antipode is obtained by acting on
“inverse” plane waves, i.e. plane waves which multiplied by their standard
counterpart give the identity. The counit codifies the action of the
generators on plane waves with $k=0$.

$\displaystyle\Delta[\tilde{P}_{+}]=\tilde{P}_{+}\otimes
1+1\otimes\tilde{P}_{+},\quad\Delta[\tilde{P}_{-}]=\tilde{P}_{-}\otimes
1+e^{-2\tilde{P}_{+}}\otimes\tilde{P}_{-}$ (18)
$\displaystyle\Delta[N]=N\otimes 1+e^{-2\tilde{P}_{+}}\otimes N,\quad
S[N]=-Ne^{2\tilde{P}_{+}},\quad
S[\tilde{P}_{-}]=-\tilde{P}_{-}e^{2\tilde{P}_{+}}$ $\displaystyle
S[\tilde{P}_{+}]=-\tilde{P}_{+},\quad\epsilon[N]=\epsilon[\tilde{P}_{+}]=\epsilon[\tilde{P}_{-}]=0\,.$

The procedures outlined above define a Hopf Algebra: all its axioms are
satisfied, including the compatibility rules with the commutators (17) (i.e.,
the homomorphism property of $\Delta$, $S$ and $\epsilon$). The structures
thus obtained define the lightlike $\kappa$-Poincaré Hopf algebra in the so-
called bicrossproduct basis (characterized by momenta which close a Hopf
subalgebra [59, 55]). In particular, expressions (15),(16) define the
infinite-dimensional representation of
$\mathcal{U}_{\kappa}[\mathfrak{iso}(1,1)]$ in the bicrossproduct basis.

The mass Casimir element of this algebra is

$C=\frac{1}{2}\tilde{P}_{-}(e^{2\tilde{P}_{+}}-1)\,.$ (19)

The action of the Weyl map is also useful to define generic non-commutative
functions in $\mathcal{A}$, by means of a non-commutative Fourier transform

$f(x)=\int\differential^{2}k\,\tilde{f}(k)\Omega(e^{ikx})\,.$ (20)

For such generic functions, a $\kappa$-Poincaré transformation can be written
as ($id$ is the identity map, and the dots indicate all higher order monomials
in the transformation parameters, with a given, specified ordering: in this
case, $\tau$ is chosen to be to the right of $a^{+}$, which is, in turn, to
the right of $a^{-}$):

$f(x^{\prime})=e^{i\,a^{-}\otimes\tilde{P_{-}}}e^{i\,a^{+}\otimes\tilde{P_{+}}}e^{i\,\tau\otimes\,N}(id\otimes\triangleright)f(x)=1\otimes
f(x)+i\,a^{\mu}\otimes\tilde{P}_{\mu}\triangleright f(x)+i\,\tau\otimes
N\triangleright f(x)+\dots$ (21)

with $\tilde{P}_{\mu},N\in\mathcal{U}_{\kappa}[\mathfrak{iso}(1,1)]$ and the
left action on coordinates is easily read from (15),(16),

$\tilde{P}_{\mu}\triangleright x^{\nu}=-i\,\delta^{\mu}{}_{\nu}\,,\qquad
N\triangleright x^{\pm}=\pm ix^{\pm}\,.$ (22)

A peculiarity of the $\kappa$-lightlike Hopf algebra, which will prove to be
useful in characterizing the physical results of this work is the fact it is
quasi-triangular,444In our specific case, a stronger condition holds: the
R-matrix is triangular, meaning that
$R^{BA}_{DC}R^{CD}_{EF}=\delta^{A}{}_{E}\,\delta^{B}{}_{F}$ [112]. i.e., it
admits a quantum $R$-matrix. It has been derived in [112, 113, 55], by
exploiting an isomorphism between $\mathcal{C}_{\kappa}(ISO(d,1))$ and
$\mathcal{U}_{\kappa}(\mathfrak{iso}(d,1))$, where $d=1,2,3$. In $1+1$
dimensions, the expression of the R-matrix is given by

$R=e^{-2i\tilde{P}_{+}\otimes N}e^{2iN\otimes\tilde{P}_{+}}\,,$ (23)

and in terms of it, relations (3), specified by (4) can be written in a
compact way as “RTT” relations, often used in the quantum group literature
[114]. Relations (22) define a three-dimensional representation
$\rho^{A}_{B}$, with $A,B=\\{+,-,2\\}$ for $\tilde{P}_{\mu},N$ acting on
vectors of the form $X^{A}\equiv(x^{\mu},1)$:

$\rho(\tilde{P}_{+})^{A}{}_{B}=\left(\begin{array}[]{ccc}0&0&-i\\\ 0&0&0\\\
0&0&0\end{array}\right)\,,\qquad\rho(\tilde{P}_{-})^{A}{}_{B}=\left(\begin{array}[]{ccc}0&0&0\\\
0&0&-i\\\
0&0&0\end{array}\right)\,,\qquad\rho(N)^{A}{}_{B}=\left(\begin{array}[]{ccc}-i&0&0\\\
0&+i&0\\\ 0&0&0\end{array}\right)\,.$ (24)

By noticing that the $\rho(\tilde{P}_{\pm})^{A}_{B}$ matrices are nilpotent,
(23) reduces to:

$R=1\otimes 1-2i\tilde{P}_{+}\otimes N+2iN\otimes\tilde{P}_{+}\,,$ (25)

and by realizing the tensor product as the standard Kronecker product, in
components we find

$\begin{aligned}
R^{AB}_{CD}&=\delta^{A}_{C}\delta^{B}_{D}+2\,i\,\left(\delta^{A}_{+}\delta^{B}_{+}\delta^{2}_{C}\delta^{+}_{D}-\delta^{A}_{+}\delta^{B}_{-}\delta^{2}_{C}\delta^{-}_{D}-\delta^{A}_{+}\delta^{B}_{+}\delta^{+}_{C}\delta^{2}_{D}+\delta^{A}_{-}\delta^{B}_{+}\delta^{-}_{C}\delta^{2}_{D}\right)\end{aligned}\,.$
(26)

By defining

$T^{A}{}_{B}=\left(\begin{array}[]{ccc}\Lambda^{+}{}_{+}&\Lambda^{+}{}_{-}&a^{+}\\\
\Lambda^{-}{}_{+}&\Lambda^{-}{}_{-}&a^{-}\\\
0&0&1\end{array}\right)=\left(\begin{array}[]{ccc}e^{\tau}&0&a^{+}\\\
0&e^{-\tau}&a^{-}\\\ 0&0&1\end{array}\right)\,,$ (27)

one can explicitly verify that the expression

$T^{A}{}_{C}\,T^{B}{}_{D}\,R^{DC}_{EF}=R^{BA}_{CD}\,T^{C}{}_{E}\,T^{D}{}_{F}\,,$
(28)

reproduces the commutation relations (3). Moreover, the commutation relations
between coordinates can be written in a compact way as

$X^{A}\,X^{B}=R^{BA}_{CD}\,X^{C}\,X^{D}\,.$ (29)

Equivalently, these “RXX” relations can also be verified using the infinite-
dimensional representation of $\mathcal{U}_{\kappa}(\mathfrak{iso}(d,1))$ and
relations (22). For instance:

$x^{+}x^{-}=\mu\circ R\triangleright(x^{-}\otimes x^{+})=x^{-}x^{+}+2i\,,$
(30)

where $\mu:\mathcal{A}\otimes\mathcal{A}\to\mathcal{A}$ is the noncommutative
multiplication of $\mathcal{A}$. This deformed flip property can also be
extended to products of plane waves. Indeed, one can verify that

$\mu\circ
R\triangleright\left[\Omega(e^{iqx})\otimes\Omega(e^{ikx})\right]=\Omega(e^{ikx})\Omega(e^{iqx})\,,$
(31)

where $k,q\in\mathbb{C}^{2}$. As we will see in the subsequent sections, the
R-matrix proves to be a valuable instrument in switching plane waves even in
the braided tensor product algebra introduced in [50, 51], so that the task of
covariant quantization of the non-commutative scalar field becomes more
feasible.

In what follows, we will work in a different basis for
$\mathcal{U}_{\kappa}[\mathfrak{iso}(1,1)]$, which is connected to the
bicrossproduct one by a redefinition of the $+$ momentum, given by

$P_{+}=\frac{1}{2}\left(e^{2\tilde{P}_{+}}-1\right)\,.$ (32)

The $\mathcal{U}_{\kappa}[\mathfrak{iso}(1,1)]$ commutators are, in this
basis, the undeformed ones of the Poincaré algebra,

$[N,P_{+}]=iP_{+}\qquad[N,P_{-}]=-iP_{-}\,.$ (33)

All the non-linearity is moved to the coproducts and the antipodes, which now
take the form

$\displaystyle\Delta[P_{+}]=P_{+}\otimes 1+1\otimes P_{+}+2P_{+}\otimes
P_{+},\qquad S(P_{+})=-\frac{P_{+}}{1+2P_{+}},$ (34)
$\displaystyle\Delta[P_{-}]=P_{-}\otimes 1+1\otimes
P_{-}-\frac{2P_{+}}{1+2P_{+}}\otimes P_{-},\qquad S(P_{-})=-P_{-}(1+2P_{+}),$
$\displaystyle\Delta[N]=N\otimes 1+1\otimes N-\frac{2P_{+}}{1+2P_{+}}\otimes
N,\qquad S(N)=-N(1+2P_{+})\,,$

while the counits remain all zero. The Casimir element in these variables can
be obtained by substituting (32) in (19) and is undeformed:

$C=P_{+}P_{-}\,,$ (35)

in agreement with the linearity of commutators (33).

Nonlinear transformations of the translation generators lead to different
bases for the $\mathcal{U}_{\kappa}[\mathfrak{iso}(1,1)]$ Hopf algebra, which,
as we will see in the following, correspond to different coordinate systems on
momentum space.555Notice that Hopf algebras are structures that are invariant
under general nonlinear changes of basis [26]. This is in stark contrast with
Lie algebras, which can only be meaningfully said to be invariant under linear
transformations of the basis. This enlarged invariance is guaranteed by the
additional structures like coproduct, counit and antipode. One can usually
change the commutators of a Hopf algebra to make them take almost any desired
form, but only at the cost of changing the coproducts, etc. The theory we are
presenting in this paper has the aspiration of being invariant under general
coordinate transformations on momentum space. This would imply that the
physical observables do not depend on the momentum coordinate systems used in
their prediction (see the discussion in [115], Sec. II). The presence of such
an invariance in our theory is supported by the preliminary results in [51]
(sec. 2.3), which show that the two-point functions of the theory are the same
regardless of the coordinate system on momentum space that was used to
calculate them.

Even in a generally-covariant theory, certain situations are better described
by certain choices of coordinates, e.g. Cartesian coordinates in Minkowski
space are preferred because they transform covariantly under Lorentz
transformations. In our model, the choice of coordinates $P_{\pm}$ has the
same advantage: they transform in an undeformed fashion under boosts. For this
reason, we find it convenient to work with them. As it turns out, plane waves
are eigenfunctions of the momenta, and the $P_{\pm}$ basis has the following
eigenvalues:

$P_{+}\Omega(e^{ikx})=\frac{1}{2}(e^{2k_{+}}-1)\Omega(e^{ikx})\qquad
P_{-}\Omega(e^{ikx})=k_{-}\Omega(e^{ikx})\,.$ (36)

These relations inspire a redefinition of the momenta appearing in the plane
waves:

$\xi_{-}=k_{-}\,,\qquad\xi_{+}=\frac{1}{2}(e^{2k_{+}}-1)\,,\qquad\Rightarrow\qquad\Omega(e^{i\,kx})=e^{i\xi_{-}x^{-}}e^{\frac{i}{2}\ln(1+2\xi_{+})x^{+}}\,,$
(37)

so that

$P_{\pm}\Omega(e^{ikx})=P_{\pm}\triangleright
e^{i\xi_{-}x^{-}}e^{\frac{i}{2}\ln(1+2\xi_{+})x^{+}}=\xi_{\pm}E[\xi]\,,$ (38)

where we defined $E[\xi]\equiv
e^{i\xi_{-}x^{-}}e^{\frac{i}{2}\ln(1+2\xi_{+})x^{+}}$. The algebraic
properties of non-commutative plane waves reflect the non-linear structure of
momentum space. First of all, from redefinition (37), we notice that the
$\xi_{+}$ component of the momentum is bounded from below, so that plane waves
$E[\xi]$ only cover half of momentum space [50, 51]. Products of plane waves
define the deformed composition law for momentum (denoted by $\Delta)$ and the
deformed inverse momenta (denoted with $S$), which mimick the structures of
coproduct and antipode, respectively:

$\begin{aligned}
&\Delta(\xi,\eta)=\left(\xi_{-}+\frac{\eta_{-}}{1+2\xi_{+}},\xi_{+}+\eta_{+}+2\xi_{+}\eta_{+}\right)\\\
&S(\xi)=\left(-\xi_{-}(1+2\xi_{+}),-\frac{\xi_{+}}{1+2\xi_{+}}\right)\end{aligned}\,,$
(39)

for $\xi,\eta\in\mathbb{C}^{2}$. The Hopf algebra properties then imply the
following consistency relations between composition law and antipode, which
can be checked explicitly using (39):

$\displaystyle\Delta[\xi,\Delta(\eta,\chi)]=\Delta[\Delta(\xi,\eta),\chi]=\Delta[\xi,\eta,\chi]\,,$
(40) $\displaystyle\Delta[\xi,S(\xi)]=\Delta[S(\xi),\xi]=0\,,$ $\displaystyle
S[\Delta(\xi,\eta)]=\Delta[S(\eta),S(\xi)]\,,$

for $\xi,\eta,\chi\in\mathbb{C}^{2}$. The first relation implies the
associativity of the composition law, the second the existence of a momentum
inverse, and the third codifies the anti-homomorphism property of the
antipode. The Casimir element (35) defines mass-shells in momentum-space
through the constraint

$m^{2}=\xi_{+}\xi_{-}\,,$ (41)

just as in the ordinary theory.

Later on, we will see that the set of operators $P_{+},P_{-},N$ can also be
represented in terms of creation-annihilation operators of standard quantum
field theory, and their action on well defined (multi)-particle states follows
the Hopf algebraic structures displayed above. Before diving into such
considerations, we review some of the basics elements needed to construct a
consistent quantum field theory on the $1+1$D lightlike $\kappa$-Minkowski
quantum space-time, following [50, 51].

###  2.2 Braided N-point algebra and its representations

The construction of the $1+1$D braided lightlike $\kappa$-Minkowski algebra
$\mathcal{A}^{\tilde{\otimes}N}$ was developed in [50]. The defining
commutation relations are given by

$[x^{+}_{a},x^{+}_{b}]=2i(x^{+}_{a}-x^{+}_{b}),\quad[x^{+}_{a},x^{-}_{b}]=2ix^{-}_{b},\quad[x^{-}_{a},x^{-}_{b}]=0\,,$
(42)

with $a,b=1,...,N$. These relations can equivalently be written in terms of
center of mass and relative coordinates:

$x^{\mu}_{cm}=\frac{1}{N}\sum_{a=1}^{N}x^{\mu}_{a}\qquad
y^{\mu}_{a}=x^{\mu}_{a}-x^{\mu}_{cm}\,,$ (43)

so that (42) becomes

$[x^{+}_{cm},x^{-}_{cm}]=2ix^{-}_{cm},\quad[x^{+}_{cm},y_{a}^{\pm}]=\mp
2iy_{a}^{\pm},\quad[x^{-}_{cm},y_{a}^{\pm}]=0.$ (44)

It is easy to check that the coordinate differences $\Delta
x^{\mu}_{ab}\coloneqq x^{\mu}_{a}-x^{\nu}_{b}$ are commutative (a feature also
shared by the braided tensor product of the $\theta$-Moyal non-commutative
space-time [37]):

$[\Delta x^{\mu}_{ab},\Delta x^{\nu}_{cd}]=0,\quad
a,b,c,d=1,\dots,N\,\,\,\text{ and }\mu,\nu=+,-.$ (45)

This, combined with the fact that $\kappa$-Poincaré invariant $N$-point
functions depend solely on coordinate differences (proved in [50]),
immediately tells us that $\kappa$-Poincaré invariant $N$-point functions are
commutative themselves. This greatly simplifies the interpretation of the
theory, given that all physical information should be encoded in $N$-point
functions. Once again, commutation relations (42) can be written in terms of
an $R$-matrix [114] as

$X^{A}_{a}\,X^{B}_{b}=R^{BA}_{CD}\,X^{C}_{b}\,X^{D}_{a}\,,$ (46)

where $X^{A}_{a}=(x^{\mu}_{a},1)$, and the operators appearing in the
$R$-matrix act in the same way on the $x^{\mu}_{a}$ coordinates whatever the
value of $a$.

In [50], a representation for the center of mass and relative coordinates has
been found, and reads

$\displaystyle\hat{x}^{+}_{cm}=2ix^{-}_{cm}\frac{\partial}{\partial{x^{-}_{cm}}}+i+2i\sum_{a=1}^{N-1}\bigg{(}y_{a}^{+}\frac{\partial}{\partial
y_{a}^{+}}-y_{a}^{-}\frac{\partial}{\partial y_{a}^{-}}\bigg{)},\quad$
$\displaystyle\hat{x}^{-}_{cm}=x^{-}_{cm},\quad\hat{y}_{a}^{+}=y_{a}^{+},\quad\hat{y}_{a}^{-}=y_{a}^{-},$
(47)

and is such that $x_{cm}^{\pm},y_{a}^{\pm}$ are Hermitian. For purposes which
shall be clear once we discuss non-commutative plane waves in more detail, we
will consider a more general, one-parameter class of representations, given by

$\displaystyle\hat{x}^{+}_{cm}=2i\left(x^{-}_{cm}\frac{\partial}{\partial{x^{-}_{cm}}}+s\right)+2i\sum_{a=1}^{N-1}\bigg{(}y_{a}^{+}\frac{\partial}{\partial
y_{a}^{+}}-y_{a}^{-}\frac{\partial}{\partial y_{a}^{-}}\bigg{)},\quad$
$\displaystyle\hat{x}^{-}_{cm}=x^{-}_{cm},\quad\hat{y}_{a}^{+}=y_{a}^{+},\quad\hat{y}_{a}^{-}=y_{a}^{-},$
(48)

where $0<s<1$, and the Hermitian representation is regained with $s=1/2$. When
analyzing plane waves, in the subsequent sections, it is useful to study the
action of operators of the type $e^{itx_{cm}^{+}}$ on functions of the braided
tensor product algebra. Using (48), it is easy to check that, for any complex
$t$,

$e^{it\hat{x}^{+}_{cm}}f(x^{-}_{cm},y_{a}^{+},y_{a}^{-})=e^{-t\,a}f(e^{-2t}x^{-}_{cm},e^{2t}y_{a}^{+},e^{-2t}y_{a}^{-})\,.$
(49)

Then, exploiting the fact that

$e^{itx_{a}^{+}}=e^{it(x^{+}_{cm}+y_{a}^{+})}=e^{i\left(\frac{e^{2t}-1}{2}\right)y_{a}^{+}}e^{itx^{+}_{cm}}\,,$
(50)

we obtain

$e^{-\frac{\pi}{2}{x}^{+}_{a}}=e^{i\left(\frac{e^{\pi\,i}-1}{2}\right)y_{a}^{+}}e^{-\frac{\pi}{2}x^{+}_{cm}}=e^{-i\,y_{a}^{+}}e^{-\frac{\pi}{2}{x}^{+}_{cm}}\,,$
(51)

so that

$e^{-\frac{\pi}{2}\hat{x}^{+}_{a}}f(x^{-}_{cm},y_{a}^{+},y_{a}^{-})=e^{-iy_{a}^{+}}e^{-i\pi\,s}f(e^{-i\pi}x^{-}_{cm},e^{i\pi}y_{a}^{+},e^{-i\pi}y_{a}^{-})=e^{-i\pi
s}e^{-iy_{a}^{+}}f(-x^{-}_{cm},-y_{a}^{+},-y_{a}^{-})\,.$ (52)

The square of this operator is then simply

$e^{-\pi\,\hat{x}^{+}_{a}}f(x^{-}_{cm},y_{a}^{+},y_{a}^{-})=e^{-2i\pi\,s}f(x^{-}_{cm},y_{a}^{+},y_{a}^{-})\,.$
(53)

and thus

$e^{-n\pi{x}^{+}_{a}}f(x^{-}_{cm},y_{a}^{+},y_{a}^{-})=e^{-2i\pi\,s\,n}f(x^{-}_{cm},y_{a}^{+},y_{a}^{-})\,.$
(54)

Having introduced the one-parameter family of representations (48), we would
like to find a condition that fixes the parameter $s$. This will be identified
in the next Section, in order to eliminate a sign ambiguity that emerges when
introducing a certain type of noncommutative plane waves (first introduced in
[98, 51]) that are necessary to ensure the covariance of the theory. In the
meantime, we need to briefly discuss the Hermiticity/self-adjointness
properties of the N-point coordinate operators $x_{a}^{\mu}$. This will be
necessary, as later we will need to introduce an involution that sends a
noncommutative plane wave into its inverse, which is necessary in order to
discuss field theory. What we would like is an involutive, anti-linear anti-
homomorphism which sends $E[\xi]$ (from Eq. (38)) into $E^{\dagger}[\xi]$ such
that $E^{\dagger}[\xi]E[\xi]=E[\xi]E^{\dagger}[\xi]=1$. We start by defining a
putative operator $*$ as the ”naive” Hermitian conjugation on operators, such
that its action on $\hat{x}^{+}_{cm}$ is given by

$(\hat{x}_{cm}^{+})^{*}=\hat{x}_{cm}^{+}+2i(2s-1)$ (55)

where, as expected, we obtain that $(\hat{x}_{cm}^{+})^{*}=\hat{x}_{cm}^{+}$
only when $s=1/2$, which corresponds to the symmetric ordering for the
representation (48). We can now define $\dagger$ as the operator that leaves
$\hat{x}_{cm}^{+}$ invariant for any choice of $s$,
$(\hat{x}_{cm}^{+})^{\dagger}=\hat{x}_{cm}^{+}$, so that its relation with $*$
is simply given by

$(\hat{x}_{cm}^{+})^{\dagger}=(\hat{x}_{cm}^{+})^{*}-2i(2s-1)$ (56)

The $(\cdot)^{*}$ operator is the Hermitian conjugate with respect to the
standard inner product of $L^{2}(\mathbbm{R}^{2N-1})$:

$\int_{\mathbbm{R}^{2N-1}}\bar{\psi}~{}\varphi~{}dx_{cm}^{-}dy_{1}^{-}\dots
dy_{N-1}^{-}dy_{1}^{+}\dots dy_{N-1}^{+}\,,$ (57)

where $\psi,\phi$ are square-integrable functions on $\mathbbm{R}^{2N-1}$. The
$\dagger$ operation is the Hermitian conjugate with respect to a different
inner product:

$\int_{\mathbbm{R}^{2N-1}}(x_{cm}^{-})^{2s-1}\,\bar{\psi}\,\varphi\,dx_{cm}^{-}dy_{1}^{-}\dots
dy_{N-1}^{-}dy_{1}^{+}\dots dy_{N-1}^{+}\,,$ (58)

where, in this case, the space of functions that have a finite norm is
different from $L^{2}(\mathbbm{R}^{2N-1})$. For $s>1/2$, it includes
$L^{2}(\mathbbm{R}^{2N-1})$, and also functions that diverge sufficiently
slowly in $x_{cm}^{-}\to 0$. For $s<1/2$, the space is smaller than
$L^{2}(\mathbbm{R}^{2N-1})$, as the functions need to go to zero sufficiently
fast at $x_{cm}^{-}\to 0$. The fact that the representations of $\mathcal{A}$
and the related braided algebras require different inner products for the
self-adjointness of the generators has been already noticed in [116, 62, 63].
From now on, we will use the $\dagger$ operator to conjugate plane waves.

##  3 Braided lightlike $\kappa$-deformed QFT

###  3.1 Old and new-type noncommutative plane waves and momentum space

We have introduced plane waves for a single copy of the $\mathcal{A}$ in
section 2.1 using the linear momentum parametrization. As it will be relevant
for what follows, we add a label indicating in which copy of the braided
tensor product algebra $\mathcal{A}^{\tilde{\otimes}N}$ the plane wave is
defined:

$E_{a}[\xi]=e^{i\xi_{-}x_{a}^{-}}e^{\frac{i}{2}\ln(1+2\xi_{+})x_{a}^{+}}\,.$
(59)

Under the involution that leaves $x^{+}$ invariant, the above transformes as

$E^{\dagger}_{a}[\xi]=E_{a}[S(\xi)]\,,$ (60)

where $S(\xi)$ is the antipode defined in (39), while the product of two plane
waves gives a composition law compatible with the coproducts in
(LABEL:eq:lincop):

$E_{a}[\xi]E_{a}[\eta]=E_{a}[\Delta(\xi,\eta)]\,.$ (61)

The space-time coordinates $x^{\pm}$ close the Lie algebra (2) of the affine
group of the line, $\text{aff}(1)$. Plane waves (59) span the connected
component of the identity of the corresponding Lie group, $\text{Aff}(1)$.
This is just a semiplane of 1+1-dimensional Minkowski space, bounded by a
straigth line [50, 51]. The boundary is given by the reality constraint for
the logarithm term, $\xi_{+}>-1/2$. This implies that such plane waves only
cover half of the Minkowski momentum space, an issue already pointed out in
[50]. There, it was shown that a field theory built from plane waves (59)
spoils $\kappa$-Poincaré invariance. This can easily be seen by considering a
$\kappa$-Poincaré transformation of (59):

$E_{a}^{\prime}[\xi]=e^{ie^{-\tau}x^{-}}e^{\frac{i}{2}\ln(1+2e^{\tau}\xi_{+})x^{+}}\,e^{i\xi_{-}a^{-}}e^{\frac{i}{2}\ln(1+2\xi_{+})a^{+}}\,.$
(62)

Notice that in this linear parametrization, the boost simply acts as a
dilation on $\xi_{-},\xi_{+}$, given the linear structure of the commutators
(33). For any value of $\tau$, a positive value of $\xi_{+}$ remains positive,
and we obtain a different group element connected to the identity. When
$\xi_{+}$ is negative, an excessively large boost may result in
$e^{\tau}\xi_{+}<-1/2$, so that the argument of the logarithm in (59) becomes
negative and we obtain a group element not connected to the identity, which we
can think of as a plane wave of the form (59) with a complex argument. In
[51], these “new type” plane waves were identified as the missing piece of the
puzzle needed to construct a consistent $\kappa$-Poincaré invariant field
theory.

Suppose we boost a plane wave of the form (59), such that
$e^{\tau}\xi_{+}<-1/2$; then, the logarithm term can be written as

$\ln[-|1+2e^{\tau}\xi_{+}|]=i\,\pi+\ln|1+2e^{\tau}\xi_{+}|+2\,n\,\pi\,i\,.$
(63)

Focusing on the Lorentz transformation sector of (62):

$e^{i\,e^{-\tau}\xi_{-}x^{-}_{a}}e^{\frac{i}{2}\ln[-|1+2e^{\tau}\xi_{+}|]\,x^{+}_{a}}=e^{i\,e^{-\tau}\xi_{-}x^{-}_{a}}e^{\frac{i}{2}\ln|1+2e^{\tau}\xi_{+}|\,x^{+}_{a}}e^{-\frac{\pi}{2}\,x^{+}_{a}}e^{-n\,\pi\,x^{+}_{a}}.$
(64)

We now come to an issue not discussed in [51]. There, using representation
(47), a sign ambiguity emerges in (64), due to the fact that $e^{-n\pi
x_{a}^{+}}\equiv(-1)^{n}$. In our novel parametric representation (48), using
the identification $e^{-n\pi x_{a}^{+}}\equiv e^{-2i\pi sn}$ from (54), (64)
becomes

$e^{i\,e^{-\tau}\xi_{-}x^{-}_{a}}e^{\frac{i}{2}\ln[-|1+2e^{\tau}\xi_{+}|]\,x^{+}_{a}}=e^{i\,e^{-\tau}\xi_{-}x^{-}_{a}}e^{\frac{i}{2}\ln|1+2e^{\tau}\xi_{+}|\,x^{+}_{a}}e^{-\frac{\pi}{2}\,x^{+}_{a}}e^{-2i\,\pi
s\,n}.$ (65)

To avoid the aforementioned sign ambiguity, we may choose $s=1$. Notice that
this implies that the coordinates are only Hermitian with respect to the inner
product (58). Nevertheless, the physical quantities characterizing our quantum
field theory (two point functions) are not affected by this choice. From now
on, whenever we refer to the (braided or not) $\kappa$-Minkowski coordinate
algebra we mean representation (48) with the choice $s=1$, and the Hermitian
conjugate operator $\dagger$ defined in Eq. (56). Having solved the sign
ambiguity, we have singled out one new type plane wave, among the infinite
possibilities arising from crossing the momentum space boundary with a too
large boost:

$E_{a}[\xi]\rightarrow\mathcal{E}[e^{-\tau}\xi_{-},\frac{1}{2}\ln|1+2e^{\tau}\xi_{+}|]\,,$
(66)

where, as in [51], we have defined

$\mathcal{E}_{a}[\xi]\coloneqq E_{a}[\xi]e^{-\frac{\pi}{2}x_{a}^{+}}$ (67)

The properties of plane waves of the type (67) have been discussed in detail
in [51], in particular all the rules to multiply these plane waves among each
other and with old-type plane waves, which are necessary for the discussion of
QFT. For what follows, we recall that the on-shell relation for this other
half of momentum space is still given by (41).

###  3.2 Covariant quantization and oscillator algebra

We expand a scalar field $\phi(x_{a})$ in terms of old type and new type plane
waves, as outlined in [51].

$\phi(x_{a})=\int
d^{2}\xi\,\delta(\xi_{+}\xi_{-}-m^{2})\tilde{\phi_{1}}(\xi)E_{a}[\xi]+\int
d^{2}\eta\,\delta(\eta_{+}\eta_{-}-m^{2})\tilde{\phi_{2}}(\eta)\mathcal{E}_{a}[\eta]\,.$
(68)

Enforcing the on-shell constraints and recalling the ranges of validity of
expressions for both types of plane waves, the field can be expressed as

$\displaystyle\phi(x_{a})=$
$\displaystyle-\int_{-1/2}^{0}\frac{d\xi_{+}}{2\xi_{+}}\tilde{\phi_{1}}(\xi_{+})E_{a}\left[\frac{m^{2}}{\xi_{+}},\frac{1}{2}\ln(1+2\xi_{+})\right]+\int_{0}^{\infty}\frac{d\xi_{+}}{2\xi_{+}}\tilde{\phi_{1}}(\xi_{+})E_{a}\left[\frac{m^{2}}{\xi_{+}},\frac{1}{2}\ln(1+2\xi_{+})\right]+$
(69) $\displaystyle-$
$\displaystyle\int_{-\infty}^{-1/2}\frac{d\xi_{+}}{2\xi_{+}}\tilde{\phi_{2}}(\xi_{+})\mathcal{E}_{a}\left[\frac{m^{2}}{\xi_{+}},\frac{1}{2}\ln\left(1+2\xi_{+}\right)\right]$

For convenience, let us introduce the shorthand notation

$\displaystyle
e_{a}(\xi_{+})\coloneqq\exp\left[i\frac{m^{2}}{\xi_{+}}x_{a}^{-}\right]\exp\left[\frac{i}{2}\ln(1+2\xi_{+})x_{a}^{+}\right]\qquad\xi_{+}>-\frac{1}{2}$
(70)
$\displaystyle\epsilon_{a}(\xi_{+})\coloneqq\exp\left[i\frac{m^{2}}{\xi_{+}}x_{a}^{-}\right]\exp\left[\frac{i}{2}\ln(1+2\xi_{+})x_{a}^{+}\right]\qquad\xi_{+}<-\frac{1}{2}$

for on-shell plane waves. Given expression (70), we have
$e_{a}^{\dagger}(\xi_{+})=e_{a}(S(\xi_{+}))$ and
$\epsilon_{a}^{\dagger}(\xi_{+})=\epsilon_{a}(S(\xi_{+}))$. Hereafter, we will
only focus on on-shell plane waves. Therefore, to further simplify the
notation, we remove the $+$ subscript from linear momentum and implicitly
refer to the $+$ component of momenta unless otherwise stated. We now make a
key observation regarding the antipode function $S(\xi)$. We notice that $S$
maps the $\xi>0$ region into $-1/2<\xi<0$ region and vice versa. The region
$\xi<-1/2$ is mapped onto itself via the application of $S$. This suggests
that, while the integral containing old type plane waves may be customarily
expanded in terms of both old type plane waves _and_ their Hermitian
conjugates, for the integral containing new type plane waves, only one between
$\mathcal{E}$ and $\mathcal{E}^{\dagger}$ is needed, otherwise one would be
overcounting Fourier modes. As a result, the field expansion (69) now reads

$\displaystyle\phi(x_{a})=$
$\displaystyle\int_{0}^{\infty}\frac{d\xi}{2\xi}\left[\frac{1}{2\xi+1}\tilde{\phi}_{1}\left(S(\xi)\right)e_{a}^{\dagger}(\xi)+\tilde{\phi_{1}}(\xi)e_{a}(\xi)\right]+$
(71) $\displaystyle+$
$\displaystyle\int_{-\infty}^{-\frac{1}{2}}\frac{d\xi}{2\xi(1+2\xi)}\tilde{\phi}_{2}\left(S(\xi)\right)\epsilon_{a}^{\dagger}(\xi)$

Taking inspiration from the undeformed QFT, we define

$\begin{cases}\tilde{\phi}_{1}(S(\xi))=a(\xi)\qquad\xi>0\\\
\tilde{\phi}_{1}(\xi)=\bar{b}(\xi)\qquad\xi>0\\\
\tilde{\phi}_{2}(\xi)=\alpha(\xi)\qquad\xi<-\frac{1}{2}\,,\end{cases}$ (72)

where the bar indicates complex conjugation. Upon quantization, $a(\xi)$ will
play the role of a particle annihilation operator while $b^{\dagger}(\xi)$
will play the role of an anti-particle construction operator. The newly
introduced operator $\alpha(\xi)$ is defined ”across” the momentum space
border and introduces a relation between operators $a(\xi)$ and
$b^{\dagger}(\xi)$ when the momenta involved are also across the border. The
expression for our scalar field is thus

$\displaystyle\phi(x_{a})=$
$\displaystyle\int_{0}^{\infty}\frac{d\xi}{2\xi}\left[\frac{1}{2\xi+1}a(\xi)e_{a}^{\dagger}(\xi)+\bar{b}(\xi)e_{a}(\xi)\right]+$
(73) $\displaystyle+$
$\displaystyle\int_{-\infty}^{-\frac{1}{2}}\frac{d\xi}{2\xi(1+2\xi)}\alpha(\xi)\epsilon_{a}^{\dagger}(\xi)$

We now promote the Fourier coefficients $a(\xi),b(\xi),\alpha(\xi)$ and their
complex conjugates to operators. The latter will be indicated by the $\dagger$
symbol rather than the bar one 666Although we will indicate the Hermitian
conjugates of these operators with the usual $\dagger$ symbol, as is also the
case with plane waves, it is important to keep in mind that they act on
different Hilbert spaces.. We adopt a covariant quantization approach, using
the Pauli-Jordan function $\Delta_{PJ}(x_{1}-x_{2})$, that is found to be
equal to the one employed in the commutative case [51] : one can obtain it
starting from the one found in [51] and performing a change of variables to
linear momentum. It reads:

$\displaystyle\Delta_{\text{PJ}}(x_{1}-x_{2})$
$\displaystyle=-\int_{0}^{+\infty}\frac{d\xi}{2\xi}e_{1}(\xi)e_{2}^{\dagger}(\xi)+\int_{0}^{+\infty}\frac{d\xi}{2\xi}\frac{1}{2\xi+1}e_{1}^{\dagger}(\xi)e_{2}(\xi)+$
(74)
$\displaystyle-\int_{-\infty}^{-\frac{1}{2}}\frac{d\eta}{2\eta}\epsilon_{1}(\eta)\epsilon_{2}^{\dagger}(\eta)\,.$

We are now equipped with all the ingredients needed to quantize the scalar
field (71), which we now denote as $\hat{\phi}$ and treat as an element of
$\mathcal{A}\otimes\mathcal{O}(\mathcal{H})$, where $\mathcal{O}(\mathcal{H})$
is the set of operators acting on the (anti-)particle Hilbert space. The
expression for $\hat{\phi}$ is

$\displaystyle\hat{\phi}(x_{a})=$
$\displaystyle\int_{0}^{\infty}\frac{d\xi}{2\xi}\left[\frac{1}{2\xi+1}a(\xi)e_{a}^{\dagger}(\xi)+b^{\dagger}(\xi)e_{a}(\xi)\right]+$
(75) $\displaystyle+$
$\displaystyle\int_{-\infty}^{-\frac{1}{2}}\frac{d\xi}{2\xi(1+2\xi)}\alpha(\xi)\epsilon_{a}^{\dagger}(\xi)\,,$

and the covariant quantization rules are

$[\hat{\phi}(x_{1}),\hat{\phi}^{\dagger}(x_{2})]=\Delta_{\text{PJ}}(x_{1}-x_{2}),\quad[\hat{\phi}(x_{1}),\hat{\phi}(x_{2})]=[\hat{\phi}^{\dagger}(x_{1}),\hat{\phi}^{\dagger}(x_{2})]=0.$
(76)

Commutators (76) then involve products of creation and annihilation operators
as well as products of non-commutative plane waves. In performing these
computations, we must choose a specific ordering for the coordinate functions
belonging to different copies of $\mathcal{A}$ in the braided tensor product
algebra. Following [50, 51], we order the non-commutative plane waves products
with the $x_{2}$ variables to the right. The general strategy to perform this
calculation is based on the fact that all plane waves, regardless of their
type, can be formally written as in (70), and they can be exchanged by making
use of the $R$-matrix. For any real values of $\xi,\eta$, we have:

$e_{2}(\eta)e_{1}(\xi)=\mu\circ R\triangleright e_{1}(\xi)\otimes
e_{2}(\eta)=e_{1}(\xi+2\xi\eta)e_{2}\quantity(\frac{\eta}{1+2\xi+4\xi\eta})\,,$
(77)

where $R$ is defined in (23), and in linear momentum variables reads

$R=e^{-i\ln(1+2P_{+})\otimes N}e^{iN\otimes\ln(1+2P_{+})}\,.$ (78)

Whether the waves above are of old or new type depends on the specific values
of $\xi,\eta$ considered. This leads to a division of the commutation
relations between creation and annihilation in various regions of momentum
space. The resulting list of commutation relations is still rather involved,
but the overall picture is much simpler than the one presented in [51], thanks
to the linear momentum redefinition. From the first commutator in (76), we
obtain:

$\bullet$ In the region $\xi\in]0;+\infty[$, $\eta\in]0;\frac{1}{4\xi}[$

$b^{\dagger}(\xi)b(\eta)-\frac{1}{1-4\xi\eta}b\left(\frac{\eta+2\xi\eta}{1-4\xi\eta}\right)b^{\dagger}\left(\frac{\xi+2\xi\eta}{1-4\xi\eta}\right)=-2\xi\delta(\xi-\eta)$
(79)

$\bullet$ In the region $\xi\in]0;+\infty[$, $\eta\in]\frac{1}{4\xi};\infty[$

$b^{\dagger}(\xi)b(\eta)+\frac{1}{1-4\xi\eta}\alpha^{\dagger}\left(-\frac{\eta+2\xi\eta}{1+2\eta}\right)\alpha\left(-\frac{\xi+2\xi\eta}{1+2\xi}\right)=-2\xi\delta(\xi-\eta)$
(80)

$\bullet$ In the region $\eta\in]0;+\infty[$, $\xi\in]0;\infty[$

$b^{\dagger}(\xi)a^{\dagger}(\eta)=\frac{1+2\eta}{1+2\eta+2\xi}a^{\dagger}\quantity(\eta+2\xi\eta)b^{\dagger}\quantity(\frac{\xi}{1+2\xi+4\xi\eta})$
(81)
$a(\xi)b(\eta)=\frac{1+2\xi}{1+2\xi+4\xi\eta}b\left(\frac{\eta}{1+2\xi+4\xi\eta}\right)a\left(\xi+2\xi\eta\right)$
(82)
$a(\xi)a^{\dagger}(\eta)-\frac{(1+2\xi)(1+2\eta)}{1+2\xi+2\eta}a^{\dagger}\left(\frac{\eta}{1+2\xi}\right)a\left(\frac{\xi}{1+2\eta}\right)=2\eta(1+2\eta)\delta(\xi-\eta)$
(83)

$\bullet$ In the region $\eta\in]-\infty;-\frac{1}{2}[$,
$\xi\in]-\infty;-\frac{1}{2}[$

$\alpha(\xi)\alpha^{\dagger}(\eta)+\frac{(1+2\xi)(1+2\eta)}{1+2\xi+2\eta}a^{\dagger}\left(\frac{\eta}{1+2\xi}\right)a\left(\frac{\xi}{1+2\eta}\right)=2\eta(1+2\eta)\delta(\xi-\eta)$
(84)

$\bullet$ In the region $\eta\in]-\infty;-\frac{1}{2}[$, $\xi\in]0;\infty[$

$b^{\dagger}(\xi)\alpha^{\dagger}(\eta)=\frac{1+2\eta}{1+2\eta+4\eta\xi}\alpha^{\dagger}(\eta+2\xi\eta)a\left(-\frac{\xi}{1+2\eta+2\xi+4\xi\eta}\right)$
(85)

$\bullet$ In the region $\eta\in]0;+\infty[$, $\xi\in]-\infty;-\frac{1}{2}[$

$\alpha(\xi)b(\eta)=\frac{1+2\xi}{1+2\xi+4\xi\eta}a^{\dagger}\left(-\frac{\eta}{1+2\eta+2\xi+4\xi\eta}\right)\alpha(\xi+2\eta\xi)$
(86)

$\bullet$ In the region $\eta\in]0;-\frac{1}{2}-\xi[$,
$\xi\in]-\infty;-\frac{1}{2}[$

$\alpha(\xi)a^{\dagger}(\eta)=\frac{(1+2\eta)(1+2\xi)}{1+2\eta+2\xi}b\left(-\frac{\eta}{1+2\eta+2\xi}\right)\alpha\left(\frac{\xi}{1+2\eta}\right)$
(87)

$\bullet$ In the region $\eta\in]-\frac{1}{2}-\xi;\infty[$,
$\xi\in]-\infty;-\frac{1}{2}[$

$\alpha(\xi)a^{\dagger}(\eta)=-\frac{(1+2\eta)(1+2\xi)}{1+2\eta+2\xi}\alpha^{\dagger}\left(\frac{\eta}{1+2\xi}\right)b^{\dagger}\left(-\frac{\xi}{1+2\xi+2\eta}\right)$
(88)

$\bullet$ In the region $\eta\in]-\frac{1}{2}-\xi;-\frac{1}{2}[$,
$\xi\in]0;+\infty[$

$a(\xi)\alpha^{\dagger}(\eta)=-\frac{(1+2\eta)(1+2\xi)}{1+2\eta+2\xi}b\left(-\frac{\eta}{1+2\eta+2\xi}\right)\alpha\left(\frac{\xi}{1+2\eta}\right)$
(89)

$\bullet$ In the region $\eta\in]-\infty;-\frac{1}{2}-\xi[$,
$\xi\in]0;+\infty[$

$a(\xi)\alpha^{\dagger}(\eta)=\frac{(1+2\eta)(1+2\xi)}{1+2\eta+2\xi}\alpha^{\dagger}\left(\frac{\eta}{1+2\xi}\right)b^{\dagger}\left(-\frac{\xi}{1+2\xi+2\eta}\right)$
(90)

From the $[\hat{\phi}(x_{1}),\hat{\phi}(x_{2})]=0$ commutator, the resulting
relations are:

$\bullet$ In the region $\eta\in]0;+\infty[$, $\xi\in]0;\infty[$

$b^{\dagger}(\xi)b^{\dagger}(\eta)=b^{\dagger}\left(\eta+2\xi\eta\right)b^{\dagger}\left(\frac{\xi}{1+2\eta+4\xi\eta}\right)$
(91)
$a(\xi)b^{\dagger}(\eta)=b^{\dagger}\left(\frac{\eta}{1+2\xi}\right)a\left(\frac{\xi}{1+2\eta}\right)$
(92)
$a(\xi)a(\eta)=a\left(\frac{\eta}{1+2\xi+4\xi\eta}\right)a\left(\xi+2\xi\eta\right)$
(93)

$\bullet$ In the region $\xi\in]0;+\infty[$, $\eta\in]0;\frac{1}{4\xi}[$

$b^{\dagger}(\xi)a(\eta)=a\left(\frac{\eta+2\xi\eta}{1-4\xi\eta}\right)b^{\dagger}\left(\frac{\xi+2\xi\eta}{1-4\xi\eta}\right)$
(94)

$\bullet$ In the region $\xi\in]0;+\infty[$, $\eta\in]\frac{1}{4\xi};\infty[$

$b^{\dagger}(\xi)a(\eta)=\alpha\left(\frac{\eta+2\xi\eta}{1-4\xi\eta}\right)\alpha\left(-\frac{\xi+2\xi\eta}{1+2\xi}\right)$
(95)

$\bullet$ In the region $\xi\in]0;+\infty[$, $\eta\in]-\infty;-\frac{1}{2}[$

$b^{\dagger}(\xi)\alpha(\eta)=\alpha\left(\frac{\eta+2\xi\eta}{1-4\xi\eta}\right)a\left(-\frac{\xi+2\xi\eta}{1+2\xi}\right)$
(96)

$\bullet$ In the region $\xi\in]0;+\infty[$,
$\eta\in]-\frac{1+2\xi}{4\xi};-\frac{1}{2}[$

$a(\xi)\alpha(\eta)=\alpha\left(\frac{\eta}{1+2\xi+4\xi\eta}\right)b^{\dagger}\left(-\frac{\xi+2\xi\eta}{1+2\xi+4\xi\eta}\right)$
(97)

$\bullet$ In the region $\xi\in]0;+\infty[$,
$\eta\in]-\infty;-\frac{1+2\xi}{4\xi}[$

$a(\xi)\alpha(\eta)=a\left(\frac{\eta}{1+2\xi+4\xi\eta}\right)\alpha\left(\xi+2\eta\xi\right)$
(98)

$\bullet$ In the region $\xi\in]-\infty;-\frac{1}{2}[$,
$\eta\in]0;-\frac{1}{2}-\xi[$

$\alpha(\xi)b^{\dagger}(\eta)=a\left(-\frac{\eta}{1+2\xi+2\eta}\right)\alpha\left(\frac{\xi}{1+2\eta}\right)$
(99)

$\bullet$ In the region $\xi\in]-\infty;-\frac{1}{2}[$,
$\eta\in]-\frac{1}{2}-\xi;+\infty[$

$\alpha(\xi)b^{\dagger}(\eta)=\alpha\left(-\frac{\eta}{1+2\xi+2\eta}\right)b^{\dagger}\left(-\frac{\xi}{1+2\eta+2\xi}\right)$
(100)

$\bullet$ In the region $\xi\in]-\infty;-\frac{1}{2}[$, $\eta\in]0;+\infty[$

$\alpha(\xi)a(\eta)=b^{\dagger}\left(-\frac{\eta}{1+2\eta+2\xi+4\xi\eta}\right)\alpha(\xi+2\xi\eta)$
(101)

$\bullet$ In the region $\xi\in]-\infty;-\frac{1}{2}[$,
$\eta\in]-\infty;-\frac{1}{2}[$

$\alpha(\xi)\alpha(\eta)=b^{\dagger}\left(-\frac{\eta}{1+2\xi+2\eta+4\eta\xi}\right)a\left(\xi+2\eta\xi\right)$
(102)

Commutation relations for the
$[\hat{\phi}^{\dagger}(x_{1}),\hat{\phi}^{\dagger}(x_{2})]=0$ commutator can
be obtained by taking the Hermitian conjugate of the commutators stemming from
$[\hat{\phi}(x_{1}),\hat{\phi}(x_{2})]=0$.

###  3.3 Representation of the deformed oscillator algebra

Figure 1: The mass-shells (in red) in light-cone coordinates are all confined
in the $\xi_{+}>0$, $\xi_{-}>0$ region.

A useful technique employed in studies of quantum field theories on non-
commutative space-time is to represent the creation and annihilation operators
of the deformed theory in terms of the ones of the standard theory [39, 117].
We introduce operators $c$ and $c^{\dagger}$ which satisfy the standard
bosonic commutation relations (in lightcone coordinates [118]):

$[c(\xi),c^{\dagger}(\eta)]=2\xi\delta(\xi-\eta)\qquad[c(\xi),c(\eta)]=[c^{\dagger}(\xi),c^{\dagger}(\eta)]=0\,,$
(103)

for any real value of $\xi,\eta$. These operators act on the usual Fock space
employed in quantum field theory. The vacuum state $\ket{0}$ is annihilated by
$c(\xi)$ and $c^{\dagger}(-\xi)$, for $\xi>0$. Then, single particle states
are defined as excitations of the vacuum state as

$c^{\dagger}(\xi)\ket{0}=\ket{\xi}_{P}\qquad\xi>0\,,$ (104)

while single anti-particle states are instead defined by

$c(-\xi)\ket{0}=\ket{\xi}_{AP}\qquad\xi>0\,.$ (105)

This parametrization of the standard oscillator algebra might not be familiar
to the reader: it is a compact way of expressing the bosonic algebra of a
complex scalar field in terms of a single infinite one-parameter set of
operators. This is possible because, when expressed in lightcone coordinates,
the creation and annihilation operators for particles and antiparticles depend
on a single positive parameter, $\xi>0$, which is the lightcone momentum (fig.
1). Instead of having different symbols for the particle and antiparticle
operators, we define $c(\xi)$ on negative values of $\xi$ too, and identify
$c(\xi)$ for negative $\xi$ with the creation operator for the antiparticles.
The corresponding annihilation operators will be the Hermitian conjugates of
those. This choice just amounts to a relabeling of the Fourier coefficients of
our scalar fields, and makes the notation more compact.

We recall the expressions for the Poincaré charges in terms of these
operators. The boost operator reads

$N=-i\int_{0}^{\infty}\frac{d\xi}{2\xi}\xi\left(\frac{dc^{\dagger}(\xi)}{d\xi}c(\xi)+\frac{dc(-\xi)}{d\xi}c^{\dagger}(-\xi)\right)\,,$
(106)

while the translations generators are given by

$P_{+}=\int_{0}^{\infty}\frac{d\xi}{2}(c^{\dagger}(\xi)c(\xi)+c(-\xi)c^{\dagger}(-\xi)),\qquad
P_{-}=\int_{0}^{\infty}m^{2}\frac{d\xi}{2\xi^{2}}(c^{\dagger}(\xi)c(\xi)+c(-\xi)c^{\dagger}(-\xi))\,.$
(107)

These generators close the standard Poincaré algebra given that they are
undeformed. Using (103) and (106), it is also easy to show that

$[N,c(\xi)]=-i\xi\frac{dc(\xi)}{d\xi}\qquad[N,c^{\dagger}(\xi)]=-i\xi\frac{dc^{\dagger}(\xi)}{d\xi}\,,$
(108)

for every $\xi$, and hence

$\displaystyle e^{ixN}c(\xi)e^{-ixN}=c(e^{x}\xi),$ (109) $\displaystyle
e^{ixN}c^{\dagger}(\xi)e^{-ixN}=c^{\dagger}(e^{x}\xi)\,.$

For what follows, it is convenient to introduce the shorthand notation for the
following finite boost transformation with momentum-dependent rapidity:

$e^{i\ln(1+2\xi)N}\coloneqq B_{\xi}\,.$ (110)

It allows to write a representation for $a(\xi),b^{\dagger}(\xi)$ in a compact
way, as follows (for $\xi>0$)

$\displaystyle
a(\xi)=\frac{1}{\sqrt{1+2S(\xi)}}B_{S(\xi)}c(\xi)=\frac{1}{\sqrt{1+2S(\xi)}}c(-S(\xi))B_{S(\xi)}$
(111) $\displaystyle
b^{\dagger}(\xi)=\frac{1}{\sqrt{1+2\xi}}c(-\xi)B_{\xi}=\frac{1}{\sqrt{1+2\xi}}B_{\xi}c(S(\xi))$
$\displaystyle
a^{\dagger}(\xi)=\frac{1}{\sqrt{1+2S(\xi)}}c^{\dagger}(\xi)B_{\xi}=\frac{1}{\sqrt{1+2S(\xi)}}B_{\xi}c^{\dagger}(-S(\xi))$
$\displaystyle
b(\xi)=\frac{1}{\sqrt{1+2\xi}}B_{S(\xi)}c^{\dagger}(-\xi)=\frac{1}{\sqrt{1+2\xi}}c^{\dagger}(S(\xi))B_{S(\xi)}\,.$

When $\xi<-1/2$, the commutators involving $\alpha(\xi),\alpha^{\dagger}(\xi)$
impose the constraints

$\displaystyle\alpha(\xi)=\frac{1}{\sqrt{1+2S(\xi)}}B_{S(\xi)}c(\xi)=a(\xi)=b^{\dagger}(S(\xi))$
(112)
$\displaystyle\alpha^{\dagger}(\xi)=\frac{1}{\sqrt{1+2S(\xi)}}c^{\dagger}(\xi)B_{\xi}=a^{\dagger}(\xi)=b(S(\xi))$

which can be obtained by extending the definitions in
(LABEL:eq:representations1) to negative $\xi$. Notice that these constraints
also identify $a(\xi)$ with $b^{\dagger}(S(\xi))$, which is consistent with
(LABEL:eq:representations1) and with the commutative limit. Indeed, when
$\kappa\rightarrow\infty$, the momentum space boundary $\xi=-\kappa/2$
vanishes, so that $a$ and $b^{\dagger}$ are not constrained anymore, as it
should be in the commutative quantum field theory of the complex scalar field.

The presence of the particular operator $B_{\xi}$ in these representations is
by no means incidental. It can be traced back to the plane wave flip governed
by the $R$-matrix (77), which is explicitly dependent on $N$. When changing
variables in the integrals appearing in the covariant quantization procedure,
the braiding of the momenta in plane waves is then reflected in the arguments
of the creation and annihilation operators. The deformed harmonic oscillator
algebra (112) is quite different from the one found in [119] for
$\theta$-Moyal non-commutative quantum field theory. There, the arguments of
the creation and annihilation operators are left untouched, but the
commutation relations are deformed by multiplication of a phase, dependent on
$p_{\mu}\theta^{\mu\nu}q_{\nu}$, with $p,q$ being the momenta involved and
$\theta^{\mu\nu}$ the antisymmetric matrix controlling the noncommutativity
between coordinates.

Having represented the deformed creation and annihilation operators in terms
of ordinary ones, we can now define (anti)-particle states of the deformed
theory making use of the ordinary operators $c$, $c^{\dagger}$ on the standard
Fock space.

###  3.4 1-particle Fock state and C, P, T symmetries

We start by exploring the $1$-particle states of the deformed theory. They are
elements of the $1$-particle Hilbert space $\mathcal{H}$. From representations
(LABEL:eq:representations1), it is immediate to see the the vacuum of the
ordinary theory, $\ket{0}$ is also annihilated by the annihilation operators
of the deformed theory

$a(\xi)\ket{0}=b(\xi)\ket{0}=0\,.$ (113)

Single particle states are then defined as excitations of the vacuum

$\frac{a^{\dagger}(\xi)}{\sqrt{1+2\xi}}\ket{0}=c^{\dagger}(\xi)B_{\xi}\ket{0}=c^{\dagger}(\xi)\ket{0}=\ket{\xi}_{P}\,,$
(114)

where we used the fact that $B_{\xi}\ket{0}=\ket{0}$ for every $\xi$ and the
square root factor in the denominator guarantees normalization. Single anti-
particle states are instead given by

$\frac{b^{\dagger}(\xi)}{\sqrt{1+2S(\xi)}}\ket{0}=c(-\xi)B_{\xi}\ket{0}=c(-\xi)\ket{0}=\ket{\xi}_{AP}\,.$
(115)

The single (anti)-particle states are thus identical to the ones defined in
the commutative quantum field theory. As a consequence of this, the action of
the momentum operators $P_{\pm}$ defined in (107) give the standard results

$\displaystyle P_{+}\ket{\xi}_{P}=\xi\ket{\xi}_{P}\,,$ $\displaystyle
P_{+}\ket{\xi}_{AP}=\xi\ket{\xi}_{AP}\,,$ (116) $\displaystyle
P_{-}\ket{\xi}_{P}=\frac{m^{2}}{\xi}\ket{\xi}_{P}\,,$ $\displaystyle
P_{-}\ket{\xi}_{AP}=\frac{m^{2}}{\xi}\ket{\xi}_{AP}\,.$

What about the $\alpha(\xi)$ operator? By letting it act on the vacuum, it is
easy to see that $\alpha(\xi)\ket{0}=\alpha^{\dagger}(\xi)\ket{0}=0$. So we
see that on the one-particle states, the $\alpha(\xi)$ leave no observable
trace.

We attempt to define the charge conjugation operator as is ordinarily done in
standard quantum field theory. We require that

$\mathcal{C}\hat{\phi}(x^{-},x^{+})\mathcal{C}^{-1}=\hat{\phi}^{\dagger}(x^{-},x^{+})\,.$
(117)

Recalling the expression for the scalar field (73), the above constraint
yields, for $\xi>0$

$\mathcal{C}\frac{a(\xi)}{1+2\xi}\mathcal{C}^{-1}=b(\xi)\qquad\mathcal{C}\,b^{\dagger}(\xi)\,\mathcal{C}^{-1}=\frac{a^{\dagger}(\xi)}{1+2\xi}\,,$
(118)

while for $\xi<-1/2$, we have

$\mathcal{C}\frac{\alpha(\xi)}{1+2\xi}\mathcal{C}^{-1}=-\alpha^{\dagger}(S(\xi))\,.$
(119)

For single particle states, (118) yields simply

$\mathcal{C}\ket{\xi}_{AP}=\ket{\xi}_{P}\,,$ (120)

as is the case in the undeformed quantum field theory. As a result, we can
write the charge conjugation operator as

$\mathcal{C}=\int_{0}^{\infty}d\xi\quantity[c^{\dagger}(\xi)c^{\dagger}(-\xi)+c(\xi)c(-\xi)]\,,$
(121)

which is just the usual expression one obtains also in commutative quantum
field theory. Using the above and representations (LABEL:eq:representations1),
(112) for the creation and annihilation operators, properties (118) and (119)
can be explicitly verified.

A remark on the consequences of Eq. (121): as can be seen from Eq. (116), the
one-particle state and the one-antiparticle state associated to it through the
charge conjugation operator carry the same momentum. This departs from what
was recently found in the timelike $\kappa$-Minkowski case in [99], where it
appears that the charge conjugation operator sends a one-particle state into a
one-antiparticle state with different momentum. This led to an interesting
phenomenology and the possibility of putting bounds to the noncommutativity
parameters only a few order of magnitude lower than the Planck energy [120,
121]. Unfortunately, these experimental bounds are irrelevant for the model
considered in this paper.

Regarding parity (P) and time revesal (T), in the commutative case, in
lightcone coordinates, these operators are introduced as, respectively:

$P:\,x^{\pm}\to x^{\mp}\,,\qquad T:\,x^{\pm}\to-x^{\mp}\,,$ (122)

which are mapped to two involutive operators $\mathcal{P}$ and $\mathcal{T}$
acting on the creation and annihilation operators, defined by

$\mathcal{P}\hat{\phi}(x^{-},x^{+})\mathcal{P}^{-1}=\hat{\phi}(x^{+},x^{-})\,,\qquad\mathcal{T}\hat{\phi}(x^{-},x^{+})\mathcal{T}^{-1}=\overline{\hat{\phi}(-x^{+},-x^{-})}\,,$
(123)

where the operator over the quantum field on the right hand side of the action
of the $\mathcal{T}$ operator is a complex conjugate, as opposed to a
Hermitian conjugate, as it acts only on the plane waves in the Fourier
expansion of the fields, and leaves the construction and annihilation
operators unchanged. It is necessary to compose the naïve time reversal
operator with this complex conjugate, thereby making the operator antilinear,
in order to have a well-behaved transformation on the Fock space (the naïve
operator $\hat{\phi}(x^{-},x^{+})\to\hat{\phi}(-x^{+},-x^{-})$ would be
unacceptable, as it would end up annihilating all one-particle states [122]).
Replacing in the above the expansion of an on-shell quantum field in creation
and annihilation operators [i.e. the commutative equivalent of Eq. (73)], one
gets the following action of $\mathcal{P}$ and $\mathcal{T}$:

$\displaystyle\mathcal{P}a(\xi)\mathcal{P}^{-1}=\pm
a\quantity(\frac{m^{2}}{\xi})\,,$
$\displaystyle\mathcal{P}b(\xi)\mathcal{P}^{-1}=\pm
b\quantity(\frac{m^{2}}{\xi})\,,$ (124)
$\displaystyle\mathcal{P}a^{\dagger}(\xi)\mathcal{P}^{-1}=\pm
a^{\dagger}\quantity(\frac{m^{2}}{\xi})\,,$
$\displaystyle\mathcal{P}b^{\dagger}(\xi_{+})\mathcal{P}^{-1}=\pm
b^{\dagger}\quantity(\frac{m^{2}}{\xi})\,,$

and

$\displaystyle\mathcal{T}a(\xi)\mathcal{T}^{-1}=a\quantity(\frac{m^{2}}{\xi})\,,$
$\displaystyle\mathcal{T}b(\xi)\mathcal{T}^{-1}=b\quantity(\frac{m^{2}}{\xi})\,,$
(125)
$\displaystyle\mathcal{T}a^{\dagger}(\xi)\mathcal{T}^{-1}=a^{\dagger}\quantity(\frac{m^{2}}{\xi})\,,$
$\displaystyle\mathcal{T}b^{\dagger}(\xi)\mathcal{T}^{-1}=b^{\dagger}\quantity(\frac{m^{2}}{\xi})\,.$

Acting on the vacuum with the left- and right-hand sides of the equations
above, one gets:

$\displaystyle\mathcal{P}\ket{\xi}_{P}=\pm\ket{\frac{m^{2}}{\xi}}_{P}\,,$
$\displaystyle\mathcal{P}\ket{\xi}_{AP}=\pm\ket{\frac{m^{2}}{\xi}}_{AP}$
$\displaystyle\,,$ (126)
$\displaystyle\mathcal{T}\ket{\xi}_{P}=\ket{\frac{m^{2}}{\xi}}_{P}\,,$
$\displaystyle\mathcal{T}\ket{\xi}_{AP}=\ket{\frac{m^{2}}{\xi}}_{AP}\,,$

where the $\pm$ sign depends on the parity of the particle and the overall
phase for the time-reversal was omitted since it has no effect on our
discussion.

One could imagine to extend this analysis to the noncommutative case, exactly
like what we did in Subsec. 3.4 for the charge conjugation operator. However,
an obstacle immediately manifests itself: there is no sense in which the
coordinate commutation relations (2) can be invariant under parity and time-
reversal transformations. In the noncommutative case, we have to choose what
these operators do to the noncommutative product between coordinates: they may
leave it unchanged, meaning that they are homomorphisms for this product, or
they may exchange the product order, in which case they are anti-
homomorphisms. This distinction is absent in the commutative case, precisely
because the products are commutative. So, for consistency with the commutative
limit, we need a linear P operator and an antilinear T operator, however we
are free to choose either of them as homomorphisms or anti-homomorphisms.
Regardless of what we choose, since the commutation relations (2) have $x^{-}$
on the right-hand side, an operator that sends $x^{-}$ to $x^{+}$ can never
leave them invariant.

If we insist on introducing a P and a T operator as in (123), acting on our
on-shell noncommutative quantum fields (73), then our on-shell plane waves are
sent to off-shell ones. For example, choosing P to be a homomorphism one gets
the following transformation rule for a noncommutative plane wave:

$e[\xi]\to
e^{i\frac{m^{2}}{\xi}x^{+}}e^{\frac{i}{2}\ln(1+2\xi)x^{-}}=e^{\frac{i}{2}\ln(1+2\xi)e^{-2\,\frac{m^{2}}{\xi}}x^{-}}e^{i\frac{m^{2}}{\xi}x^{+}}\,,$
(127)

and the pair of momentum components that appear on the right hand side:

$\left(\frac{1}{2}\ln(1+2\xi)e^{-2\,\frac{m^{2}}{\xi}},\frac{m^{2}}{\xi}\right)\,,$
(128)

does not satisfy the on-shell relation anymore. The same happens for the other
on-shell waves in the field expansion, including those of “new type”. If we
chose P to be an anti-homomorphism:

$e[\xi]\to e^{\frac{i}{2}\ln(1+2\xi)x^{-}}e^{i\frac{m^{2}}{\xi}x^{+}}\,,$
(129)

we end up with the following a pair of momentum components:

$\left(\frac{1}{2}\ln(1+2\xi),\frac{m^{2}}{\xi}\right)\,,$ (130)

which again does not satisfy the on-shell relation (the $\xi_{-}$ and
$\xi_{+}$ components are in the wrong order). Analogous calculations can also
be done for the time reversal operator $\mathcal{T}$, and still result in an
off-shell plane wave.

This is just a manifestation of the non-invariance of the basic commutation
relations (2), which are the starting point of the whole model. This theory is
parity- and time-reversal-breaking. However, the theory can still be said to
preserve combined PT invariance: if both P and T are chosen to have the same
behaviour with respect to the noncommutative product, i.e. they are both
homomorphisms or anti-homomorphisms, and P is assumed linear while T is
assumed antilinear, then the coordinate commutation relations (2) turn our to
be invariant. Such a PT operator would leave also the on-shell plane waves
$e[\xi]$ invariant, however the new-type waves could, in principle, change:
looking at Eq. (67),
$\mathcal{E}_{a}[\xi]=E_{a}[\xi]e^{-\frac{\pi}{2}x_{a}^{+}}$, it is clear
that, an antilinear homomorphism like our PT operator would leave $E_{a}[\xi]$
invariant, while changing the $e^{-\frac{\pi}{2}x_{a}^{+}}$ term into
$e^{+\frac{\pi}{2}x_{a}^{+}}$. This, however, is harmless, as we can easily
prove that $e^{-\frac{\pi}{2}x_{a}^{+}}=e^{+\frac{\pi}{2}x_{a}^{+}}$ in our
representation, when acting on functions of a single variable. Thus, the new-
type plane waves are also left invariant by PT.

It appears that PT transformations are still a symmetry of our theory. In
particular, PT acts like the identity on the scalar field Fock space (this is
true in the commutative case too for spin-zero fields [122]). Finally, CPT is
preserved too.

###  3.5 Braided flip operator and multiparticle states

We now begin exploring the multi-particle sector of the theory. To get a
feeling of the novelties introduced by our non-commutative framework, let us
start by focusing on two particle states (the conclusions drawn will be
analogous for anti-particle states):

$\begin{aligned}
\ket{\xi}_{P}\otimes\ket{\eta}_{P}=\frac{a^{\dagger}(\xi)}{\sqrt{1+2\xi}}\ket{0}\otimes\frac{a^{\dagger}(\eta)}{\sqrt{1+2\eta}}\end{aligned}\ket{0}\,,$
(131)

which are elements of the tensor product of two copies of the 1-particle
Hilbert space $\mathcal{H}$. The total momentum for this two-particle state is
obtained by acting with the coproducts of the translation generators, as
dictated by Hopf Algebra axioms when acting on the tensor product of its
representation. To obtain the total $+$ component of the momentum, we apply
the coproduct (LABEL:eq:lincop) for $P_{+}$,

$\displaystyle\Delta[P_{+}]\quantity(\ket{\xi}_{P}\otimes\ket{\eta}_{P})=P_{+}\ket{\xi}_{P}\otimes\ket{\eta}_{P}+\ket{\xi}_{P}\otimes
P_{+}\ket{\eta}_{P}+2P_{+}\ket{\xi}_{P}\otimes P_{+}\ket{\eta}_{P}=$ (132)
$\displaystyle=$
$\displaystyle\quantity(\xi+\eta+2\xi\eta)\quantity(\ket{\xi}_{P}\otimes\ket{\eta}_{P})=\Delta[\xi,\eta]_{+}\quantity(\ket{\xi}_{P}\otimes\ket{\eta}_{P})\,,$

where the $\Delta[\xi,\eta]$ operation for liner momentum was defined in (39).
In a similar fashion, we can calculate the $-$ component for the total
momentum, yielding

$\displaystyle\Delta[P_{-}]\quantity(\ket{\xi}_{P}\otimes\ket{\eta}_{P})=P_{-}\ket{\xi}_{P}\otimes\ket{\eta}_{P}+\ket{\xi}_{P}\otimes
P_{-}\ket{\eta}_{P}-\frac{2P_{+}}{1+2P_{+}}\ket{\xi}_{P}\otimes
P_{-}\ket{\eta}_{P}=$ (133) $\displaystyle=$
$\displaystyle\quantity(\frac{m^{2}}{\xi}+\frac{m^{2}}{\eta}-\frac{2\xi}{1+2\xi}\frac{m^{2}}{\eta})(\ket{\xi}_{P}\otimes\ket{\eta}_{P})=\Delta[\xi,\eta]_{-}\quantity(\ket{\xi}_{P}\otimes\ket{\eta}_{P})\,,$

The same line of reasoning can be applied to anti-particle states, obtaining
the same results for the total momenta.

In ordinary quantum field theory, multi-particle states live in symmetrized or
anti-symmetrized tensor-products of single-particle states, which characterize
the notion of identical particles. The key ingredient is the ordinary flip
operator $\sigma$, which is an involutive operation on the tensor product of
Hilbert spaces of single particle states, where the multi-particle states are
defined. In general, an analogous construction of the multi-particle Fock
space is not so straightforward for quantum field theories on non-commutative
space-time [101]. The main reason for this is that the standard flip operation
applied to a two-particle state yields another two-particle state carrying
different total momentum, due to the non-commutative nature of the coproducts.
In our specific model, this simply follows from observing that

$\Delta[P_{-}]\quantity(\ket{\xi}_{P}\otimes\ket{\eta}_{P})\neq\Delta[P_{-}]\quantity(\ket{\eta}_{P}\otimes\ket{\xi}_{P})\,.$
(134)

The way out of this empasse is to define a deformed notion of particle
exchange. This is possible, for example, in non-commutative quantum field
theory on the $\theta$-Moyal non-commutative space-time [37, 39], due to the
properties of the twist operator, linked to the existence of an $R$-matrix
[123]. For the much-studied timelike $\kappa$-Minkowski case, several works
[102, 104, 101] have tried to identify a braiding of single-particle states in
order to construct a deformed notion of symmetric and anti-symmetric states.
These approaches all present some shortcomings: either the braiding is not
involutive, or it is not covariant when constructing the theory at all orders
in $\kappa$. The recent [101] finds that, accepting a non-involutive flip
operator as the physical one, the notion of identical particles has to be
abandoned. The lack of involutivity of the flip operator leads, in fact, to an
infinite tower of states characterized by the same total momentum. In the
present work, we find that the $\kappa$-lightlike framework, although
characterized by the same non-abelian momentum Lie-group structure as the
timelike case, admits a well-defined notion of identical particles, thanks to
the existence of the universal $R$-matrix. Consider, for instance, the two-
particle state defined as

$\tilde{R}\quantity(\ket{\xi}_{P}\otimes\ket{\eta}_{P})=R\circ\sigma~{}\quantity(\ket{\xi}_{P}\otimes\ket{\eta}_{P})=R\quantity(\ket{\eta}_{P}\otimes\ket{\xi}_{P})\,,$
(135)

i.e., we act with the flip operator $\sigma$, where
$\sigma~{}\quantity(\ket{\xi}_{P}\otimes\ket{\eta}_{P})=\ket{\eta}_{P}\otimes\ket{\xi}_{P}$,
and then with the $R$-matrix defined in (78). In detail, we have

$\displaystyle\tilde{R}\quantity(\ket{\xi}_{P}\otimes\ket{\eta}_{P})=$
$\displaystyle e^{-2i\ln(1+2P_{+})\otimes
N}e^{2iN\otimes\ln(1+2P_{+})}(\ket{\eta}_{P}\otimes\ket{\xi}_{P})=$ (136)
$\displaystyle=$ $\displaystyle e^{-2i\ln(1+2P_{+})\otimes
N}\left(B_{\xi}\ket{\eta}_{P}\otimes\ket{\xi}_{P}\right)=$ $\displaystyle=$
$\displaystyle\ket{\eta+2\xi\eta}_{P}\otimes
B_{S(\eta+2\xi\eta)}\ket{\xi}_{P}=$ $\displaystyle=$
$\displaystyle\ket{\eta+2\eta\xi}_{P}\otimes\ket{\frac{\xi}{1+2\eta+4\eta\xi}}_{P}\,.$

The structure of the new obtained two-particle states mimicks the structure of
the Hermitian conjugate of commutation relation (93), where the deformation
emerges from applying the $R$-matrix to exchange plane waves upon performing
covariant quantization, as discussed in section 3.2. By acting with the
momentum coproducts (LABEL:eq:lincop), it is now easy to check that

$\Delta[P_{\pm}]\quantity(\ket{\eta+2\eta\xi}_{P}\otimes\ket{\frac{\xi}{1+2\eta+4\eta\xi}}_{P})=\Delta(\xi,\eta)_{\pm}\quantity(\ket{\eta+2\eta\xi}_{P}\otimes\ket{\frac{\xi}{1+2\eta+4\eta\xi}}_{P})\,,$
(137)

so the deformed symmetric state (135) has the same total momentum as (131),
thus being a suitable candidate for our construction of deformed
(anti-)symmetric states. It is also easy to show that $\tilde{R}$ is an
involutive operator, i.e. $(\tilde{R}^{2}=1)$. Indeed, repeating the same
analysis as in (136), one can show that

$\tilde{R}\,\left(\ket{\eta+2\eta\xi}_{P}\otimes\ket{\frac{\xi}{1+2\eta+4\eta\xi}}_{P}\right)=\ket{\xi}_{P}\otimes\ket{\eta}_{P}\,.$
(138)

This last property makes $\tilde{R}$ and ideal candidate for constructing a
deformed symmetrization operator, useful in defining deformed symmetric states
in our field theory. We are now ready to define the deformed symmetrization
operator:

$\mathcal{S}^{+}\coloneqq\frac{1}{2}(1\otimes 1+\tilde{R})\,,$ (139)

which is such that $(\mathcal{S}^{+})^{2}=\mathcal{S}^{+}$, i.e.
$\mathcal{S}^{+}$ is idempotent. Then, we can define deformed symmetric two-
particle states simply as

$\sqrt{2}\,\mathcal{S}^{+}\left(\ket{\xi}_{P}\otimes\ket{\eta}_{P}\right)\,,$
(140)

where the $\sqrt{2}$ factor is introduced for normalization. In an analogous
way, we can define the antisymmetrization operator $\mathcal{S}^{-}$:

$\mathcal{S}^{-}\coloneqq\frac{1}{2}(1\otimes 1-\tilde{R})\,,$ (141)

which is also idempotent and can be used to define antisymmetric multi-
particle states.777Although so far we only worked out the quantization of a
scalar field, we can already say something about fermionic fields and their
deformed Fock space, just by analyzing the general properties of the R-matrix.

So far, we have shown that there exists an involutive braiding that suggests
the definition of deformed symmetric two-particle states in lightlike
$\kappa$-Minkowski quantum field theory. We now show the covariance of such
braiding, in order to complete the picture.

A single particle state transforms under a finite boost of parameter $\tau$ as

$\ket{\xi}_{P}\rightarrow e^{i\tau N}\ket{\xi}_{P}=\ket{e^{\tau}\xi}_{P}$
(142)

When acting on the tensor product of single-particle states, the boost
coproduct (LABEL:eq:lincop) needs to be taken into account. For a finite
transformation, using the commutation relations (33), we can prove that

$e^{i\tau\,\Delta[N]}=e^{i\tau N}\otimes
e^{i\ln(\frac{(1+2P_{+})e^{\tau}}{1+2e^{\tau}P_{+}})N}\,.$ (143)

For our two particle state (131), this yields

$\ket{\xi}_{P}\otimes\ket{\eta}_{P}\rightarrow e^{i\tau N}\ket{\xi}_{P}\otimes
e^{i\ln(\frac{(1+2\xi)e^{\tau}}{1+2\xi
e^{\tau}})N}\ket{\eta}_{P}=\ket{e^{\tau}\xi}_{P}\otimes\ket{\frac{(1+2\xi)}{1+2e^{\tau}\xi}e^{\tau}\eta}_{P}\,.$
(144)

The deformed flipped state (136) is instead mapped into

$\ket{\eta+2\eta\xi}_{P}\otimes\ket{\frac{\xi}{1+2\eta+4\eta\xi}}_{P}\rightarrow\ket{e^{\tau}\eta(1+2\xi)}_{P}\otimes\ket{\frac{e^{\tau}\xi}{1+2e^{\tau}\eta(1+2\xi)}}_{P}\,.$
(145)

Conversely, by first boosting the two-particle state (131) and then flipping
it with $\tilde{R}$, the result is

$\tilde{R}\quantity(\ket{e^{\tau}\xi}_{P}\otimes\ket{\frac{(1+2\xi)}{1+2e^{\tau}\xi}e^{\tau}\eta}_{P})=\ket{e^{\tau}\eta(1+2\xi)}_{P}\otimes\ket{\frac{e^{\tau}\xi}{1+2e^{\tau}\eta(1+2\xi)}}_{P}\,,$
(146)

which is identical to the right-hand side of (145). We have thus proved that

$\tilde{R}e^{i\tau\Delta[N]}=e^{i\tau\Delta[N]}\tilde{R}\,.$ (147)

Basically, our deformed flip operator $\tilde{R}$ commutes with all the Hopf
Algebra generators $P_{\pm},N$, also given its compatibility with the momenta
coproducts shown above. Therefore, relativistic covariance is guaranteed.

##  4 Physical interpretation of deformed multi-particle states

###  4.1 On the indistinguishability of identical particles

In quantum mechanics, two particles of the same species are described by a
symmetric or anti-symmetric state [124], defined as

$\sqrt{2}\quantity(\frac{1\pm\sigma}{2})\ket{p}\otimes\ket{q}=\frac{\ket{p}\otimes\ket{q}\pm\ket{q}\otimes\ket{p}}{\sqrt{2}}\,,$
(148)

where $\sigma$ is the standard flip operator and $p,q$ are the linear momenta
of the particles. Operationally, the indistinguishability of the two particles
may be understood as follows. Suppose we have a calorimeter that can measure
the energy of one particle at a time, from which we can deduce the
corresponding momentum (we are in 1+1 dimensions). According to state (148),
the calorimeter can measure either $p$ or $q$. If our calorimeter measures
momentum $p$, for example, we have no way of knowing if the measured particle
is the one in the first or second place of the tensor product. This
indistinguishability simply follows from the (anti)-symmetric property of the
quantum mechanical state describing the two-particle system. What happens then
if the two-particle state is instead defined by the deformed
(anti)-symmetrization operators $\mathcal{S}^{\pm}$ in (139)? We reintroduce
the dimensional parameter $\kappa$, for clarity. Consider a decay process of
an initial particle of mass $M$ with momentum $\Pi_{\mu}=(\Pi,M^{2}/\Pi)$ (in
light-cone coordinates). The particle decays into two identical particles of
mass $m$ and momenta $\xi_{\mu}=(\xi,m^{2}/\xi)$,
$\eta_{\mu}=(\eta,m^{2}/\eta)$. We will call $\xi$ the momentum of the
particle that enters the coproduct (39), in the deformed momentum conservation
law, from the left, while $\eta$ is the label of the momentum on the right-
hand side of the coproduct. Notice that this labeling choice has nothing to do
with the placement of the particle momenta in the tensor product, and has no
physical consequences: we could choose the opposite convention and nothing
would change in the calculations. The deformed momentum conservation law
dictates:

$\left\\{\begin{aligned} &\Pi=\xi+\eta+2\xi\,\eta\,,\\\
&\frac{M^{2}}{\Pi}=\frac{m^{2}}{\xi}+\left(1-\frac{2\,\xi}{\kappa+2\xi}\right)\frac{m^{2}}{\eta}\,,\end{aligned}\right.$
(149)

the above two equations can be solved with respect to $\xi$ and $\eta$, and
they have two solutions (recall that all the on-shell momenta, $\Pi$, $\xi$
and $\eta$ are positive-definite):

$\displaystyle\xi=F_{1}(\Pi,M,m)\,,$
$\displaystyle\eta=G_{1}(\Pi,M,m,\kappa)\,,$ (150)
$\displaystyle\xi=F_{2}(\Pi,M,m)\,,$
$\displaystyle\eta=G_{2}(\Pi,M,m,\kappa)\,,$

where

$\begin{gathered}F_{1}(\Pi,M,m)=\frac{\Pi}{2}\left(1+\sqrt{1-\frac{4m^{2}}{M^{2}}}\right)\,,\qquad
F_{2}(\Pi,M,m)=\frac{\Pi}{2}\left(1-\sqrt{1-\frac{4m^{2}}{M^{2}}}\right)\,,\\\
G_{1}(\Pi,M,m,\kappa)=\frac{\kappa\Pi\left(M(\kappa+2\Pi)\sqrt{M^{2}-4m^{2}}-4m^{2}\Pi+M^{2}(\kappa+2\Pi)\right)}{8m^{2}\Pi^{2}+2\kappa
M^{2}(\kappa+2\Pi)}\,,\\\
G_{2}(\Pi,M,m,\kappa)=\frac{\kappa\Pi\left(-M(\kappa+2\Pi)\sqrt{M^{2}-4m^{2}}-4m^{2}\Pi+M^{2}(\kappa+2\Pi)\right)}{8m^{2}\Pi^{2}+2\kappa
M^{2}(\kappa+2\Pi)}\,.\end{gathered}$ (151)

If we choose the first solution, the final state will be the following:

$\ket{\psi_{1}}=\sqrt{2}\,\mathcal{S}^{+}\big{(}\ket{F_{1}}\otimes\ket{G_{1}}\big{)}=\frac{1}{\sqrt{2}}\bigg{(}\ket{F_{1}}\otimes\ket{G_{1}}+\tilde{R}\left[\ket{F_{1}}\otimes\ket{G_{1}}\right]\bigg{)}\,,$
(152)

while if we choose the second:

$\ket{\psi_{2}}=\sqrt{2}\,\mathcal{S}^{+}\big{(}\ket{F_{2}}\otimes\ket{G_{2}}\big{)}=\frac{1}{\sqrt{2}}\bigg{(}\ket{F_{2}}\otimes\ket{G_{2}}+\tilde{R}\left[\ket{F_{2}}\otimes\ket{G_{2}}\right]\bigg{)}\,.$
(153)

However, as it turns out, the two states are identical. In fact, it is
possible to show that

$\tilde{R}\left[\ket{F_{1}}\otimes\ket{G_{1}}\right]=\ket{F_{2}}\otimes\ket{G_{2}}\,,\qquad\tilde{R}\left[\ket{F_{2}}\otimes\ket{G_{2}}\right]=\ket{F_{1}}\otimes\ket{G_{1}}\,,$
(154)

implying $\ket{\psi_{1}}=\ket{\psi_{2}}$, just like in the undeformed theory.
This is a nontrivial rigidity of the theory, consequence of the Hopf-algebraic
constraints that entail its relativistic nature. The compatibility between the
deformed momentum composition law and the flip operator is what is behind it.
The final state, $\ket{\psi_{1}}=\ket{\psi_{2}}$ is proportional the sum of
the following two kets (at first order in $\kappa^{-1}$):

$\displaystyle\ket{F_{1}}\otimes\ket{G_{1}}$
$\displaystyle=\ket{F_{1}}\otimes\ket{F_{2}-\frac{2m^{2}\Pi^{2}}{\kappa\,M^{2}}+\mathcal{O}(\kappa^{-2})}\,,$
(155) $\displaystyle\tilde{R}\left[\ket{F_{1}}\otimes\ket{G_{1}}\right]$
$\displaystyle=\ket{F_{2}}\otimes\ket{F_{1}-\frac{2m^{2}\Pi^{2}}{\kappa\,M^{2}}+\mathcal{O}(\kappa^{-2})}\,.$

The result is consistent with the commutative limit $\kappa\rightarrow\infty$.
However, when the $\kappa$-deformation is switched on, the qualitative
features of this multi-particle state are completely different from their
undeformed counterpart. If the calorimeter measures the momentum of one of the
particles, we can obtain one of the following four results:

Unflipped, Left:
$\displaystyle\frac{\Pi}{2}\left(1+\sqrt{1-\frac{4m^{2}}{M^{2}}}\right)\,,$
(156) Flipped, Left:
$\displaystyle\frac{\Pi}{2}\left(1-\sqrt{1-\frac{4m^{2}}{M^{2}}}\right)\,,$
Flipped, Right:
$\displaystyle\frac{\Pi}{2}\left(1+\sqrt{1-\frac{4m^{2}}{M^{2}}}\right)-\frac{2m^{2}\Pi^{2}}{\kappa\,M^{2}}+\mathcal{O}(\kappa^{-2})\,,$
Unflipped, Right:
$\displaystyle\frac{\Pi}{2}\left(1-\sqrt{1-\frac{4m^{2}}{M^{2}}}\right)-\frac{2m^{2}\Pi^{2}}{\kappa\,M^{2}}+\mathcal{O}(\kappa^{-2})\,,$

according to whether we are measuring the left- or right-hand side of the
tensor product of the unflipped state, $\ket{F_{1}}\otimes\ket{G_{1}}$, or of
the flipped state, $\tilde{R}\left[\ket{F_{1}}\otimes\ket{G_{1}}\right]$. In
the noncommutative theory, the four momenta (156) are all different. Measuring
the momentum of one particle allows us to identify which side of the tensor
product it came from, and from which state (flipped or unflipped). Therefore,
there is a sense in which the indistinguishability of identical particles is
lost when the $\kappa$-deformation is taken into account. There is no avoiding
this if we want to construct a relativistic theory. As already stressed in
section 3.5, a state of the type (148) would not be covariant under the
$\kappa$-Poincaré transformations, which exhibit all their non-trivial
behaviour on multi-particle states, given that the coproduct is involved.

###  4.2 On the Pauli exclusion principle

We now explore the consequences of the deformed permutation symmetry on the
Pauli exclusion principle. In standard quantum theory, a state describing two
fermions with the same quantum numbers is annihilated by the undeformed anti-
symmetrizer:

$\quantity(\frac{1-\sigma}{2})\ket{\xi}\otimes\ket{\xi}=0\,.$ (157)

This is the essence of the Pauli exclusion principle, which has been confirmed
in a variety of experiments searching for classically prohibited transitions
to states of the form $\ket{\xi}\otimes\ket{\xi}$. It is then natural to ask
what is the fate of the Pauli exclusion principle in our $\kappa$-deformed
framework. Assuming that 2-fermion states are left invariant by
$\mathcal{S}^{-}$, what is the class of states annihilated by this operator?
We require that:

$\mathcal{S}^{-}\quantity(\ket{\xi}_{P}\otimes\ket{\eta}_{P})=0\,,$ (158)

which uniquely selects the two particle states

$\ket{\xi}_{P}\otimes\ket{-S(\xi)}_{P}=\ket{\xi}_{P}\otimes\ket{\xi/(1+2\xi)}_{P}\,.$
(159)

The same holds for $\ket{\eta/(1-2\eta)}\otimes\ket{\eta}$, which is the same
state as (159), just parametrized with respect to the momentum of the particle
on the right hand side of the tensor product. In the undeformed case, the
solution $\xi=\eta$ would have been selected, in agreement with (157). In
light of this reasoning, we can visualize the Pauli principle in a simple
manner. In the $(\xi,\eta)$ plane, which contains admissible pairs of momenta
that can be attributed to fermions, the Pauli principle excludes a one-
dimensional subset of the $(\xi,\eta)$ plane: the pairs lying on a curve
$\eta=f(\xi)$. In the commutative case, $f(\xi)=\xi$ (the bisector of the
plane), while the $\kappa$-deformed version is $f(\xi)=-S(\xi)$ (see fig. 2) .

Figure 2: The $+$-momentum space of momentum-pairs for two-particle states, in
units of $\kappa$. The dashed line represents the pairs excluded by the
undeformed Pauli exclusion principle. The thick curve represents the pairs
excluded by the deformed version of the exclusion principle when non-
commutativity is taken into account. For large $\xi$, the curve saturates at
$1/2\kappa$, which is the dotted asymptote in the plot.

Notice that our deformed identical-particles states are Lorentz-covariant,

$\Delta(e^{i\tau
N})\ket{\xi}_{P}\otimes\ket{-S(\xi)}_{P}=\ket{e^{\tau}\xi}_{P}\otimes\ket{-S(e^{\tau}\xi)}_{P}\,,$
(160)

in the sense that identical-particles states are sent to boosted identical-
particles states by a Lorentz transformation. This is due to the fact that the
curve $\eta=-S(\xi)$ lays on an orbit of the Lorentz group.

In light of the previous observations, we notice that the state
$\ket{\xi}_{P}\otimes\ket{\xi}_{P}$ is not annihilated by $\mathcal{S}^{-}$,
contrary to the commutative case. Notice, however, that this form of the state
is not preserved by Lorentz transformations: if an observer attributes the
same momentum $\xi$ to two particles, by the action of the finite boost
generator (143) on the state $\ket{\xi}_{P}\otimes\ket{\xi}_{P}$, a boosted
observer would attribute different momenta to them:

$\ket{\xi}_{P}\otimes\ket{\xi}_{P}\to\ket{e^{\tau}\xi}_{P}\otimes\ket{\frac{e^{\tau}\xi(1+2\xi)}{1+2e^{\tau}\xi}}_{P}\,,$
(161)

with $\tau$ being the boost parameter.

The discussion above highlights the fact that, in our model, states of the
form $\ket{\xi}_{P}\otimes\ket{\xi}_{P}$ are not excluded by the
antisymmetrization operator, because they are not the true “identical
particle” states of the theory. This could potentially lead to new physical
phenomena, which could be interpreted as departures from the Pauli Exclusion
Principle (PEP). Notice that the detection of a state of the form
$\ket{\xi}_{P}\otimes\ket{\xi}_{P}$ would not necessarily imply that the PEP,
and all of its physical consequences, is violated in our theory: for example,
it might well be the case that the theory keeps forbidding more than two
electrons to share the same atomic orbital, in view of the aforementioned
existence of classes of states that are excluded by the antisymmetrization
operator. At any rate, considering that there now are stringent bounds on PEP
violations [125, 126, 127], it would be interesting to study the basic
physical processes that are tested by these experiments, within the context of
our theory, and investigate possible observable consequences of
noncommutativity. Notice that such modelization would require significant
further development of the theory (at the very least, interacting QFTs with
Dirac fields). Results obtained through rudimentary/simplistic methodologies,
especially if they compromise Poincaré invariance, fall short of the necessary
rigor and hold no significance within the context of our theoretical
framework.

##  5 Conclusions

Building on our past results [50] and [51], we were able to define a QFT on
the $\kappa$-Minkowski noncommutative spacetime in the same spirit as [37,
38]: the coordinates of N different points cannot belong to the simple tensor
product algebra, otherwise it would not be $\kappa$-Poincaré covariant. One
needs to introduce a braiding, which requires a quantum R matrix for the
$\kappa$-Poincaré group. This exists only in the lightlike case, i.e. when the
commutators between the coordinates (1) are described by a vector $v^{\mu}$
that is lightlike, or null, with respect to the metric $g_{\mu\nu}$ that is
preserved by the $\kappa$-Poincaré group (3). Within this framework, one can
define consistently covariant N-point functions, which are the backbone of
QFT. The striking advantage of the approach of [37, 38], which is shared by
our model, as proven in [50], is that the translation-invariant combinations
of different coordinates (i.e. the coordinate differences) are commutative,
which implies that the N-point functions are all commutative. This hugely
simplifies the physical interpretation of the theory, as we do not have to
deal with noncommutative correlation functions, whose meaning would be rather
obscure. A similar conceptual simplification is achieved in many other
approaches to noncommutative QFT by using a star product, and defining a path
integral over commutative functions in which the action is turned into a
nonlocal, infinite-derivative functional of the fields. Then, the correlation
functions are commutative objects, simply obtained as expectation values or
functional variations of the partition function. However, there is no sense in
which these commutative N-point functions can be invariant under the quantum
group of isometries of the noncommutative spacetime they are supposed to live
in. In our approach, we have a way of writing commutative N-point functions
which are $\kappa$-Poincaré invariant, and we believe that this is a key
advantage of the approach based on braiding. As shown in [37, 38] in the case
of $\theta$-Moyal noncommutative spacetimes, with the braided structures one
can define a QFT built upon the Wightman axioms, and the quantization of a
free complex scalar field can be performed with the introduction of a
covariant Pauli–Jordan function. In the case of $\theta$-Moyal spacetimes, the
QFTs thus defined turned out to be completely indistinguishable from their
commutative counterparts, as all the N-point functions of the free theory, as
well as the perturbative expansion of the N-point functions of an interacting
theory, turn out to be undeformed. In our lightlike $\kappa$-Minkowski case,
we find that, although the Pauli-Jordan and two-point functions are
undeformed, a dependence on the deformation parameter appears at the level of
multiparticle states already in the free theory. The momentum, boost and
charge conjugation operators are undeformed, however the creation and
annihilation operators can be written, in a key advancement obtained in this
paper for the first time, as an infinite nonlinear combination of undeformed
creation and annihilation operators. The deformed creation operators act in a
trivial way on the vacuum, and the one-particle sector looks undeformed.
However, as soon as we create more than one particle we start seeing a
dependence on the noncommutativity parameter: the momentum of two particles is
a nonlinear combination of the two single-particle momenta. The way that two
particle momenta boost under Lorentz transformations is nonlinear and mixes
the momenta of the two particles. We can introduce a covariant and involutive
flip operator, which acts nonlinearly on the momenta of the two particles,
changing them in a more complicated way than simply exchanging them. This flip
is used to define two covariant and idempotent symmetrization and
antisimmetrization operators, whose image is the Fock space of bosonic and,
respectively, fermionic fields. The situation is substantially simpler
compared to the attempts at defining a QFT on the timelike $\kappa$-Minkowski
spacetime: in this case, the absence of a quantum R matrix makes it impossible
to define a flip operator that is both involutive and Lorentz-covariant [102,
103, 104, 105, 101], which implies that the notion of identical particles and
(anti-)symmetrized multiparticle states loses meaning [101]. We proved that
our theory is C-, PT- and CPT-invariant, however P and T symmetries do not
hold separately. This can already be seen at the level of the coordinate
commutation relations, which break P and T symmetry. The theory allows for the
existence of states which are excluded by the Pauli principle in the classical
setting. This opens up the interesting phenomenological opportunity of setting
bounds on the model by experimental results searching for evidence of
transitions into such states.

The noncommutative QFT defined in [50, 51] and completed here seems in healthy
shape, and motivates interest in several future research directions. The
simplest one is to write the N-point functions of the free theory for N larger
than two, to check whether they are undeformed too, or perhaps the
nontriviality of the multiparticle sector manifests into a dependence of
higher correlators on the noncommutativity parameter. A further issue to
consider is that the model studied so far is in 1+1 spacetime dimensions, and
its generalization to 3+1 dimensions seems straightforward, but it has not
been worked out explicitly and might yet hide some surprises. The next natural
step is to introduce an interaction, which is where the theory has the highest
chances of providing some predictions that depart from standard QFT on
commutative Minkowski space. Further down the road, gauge theories and
fermions might be explored, and perhaps a possible connection with CP
violation in the Standard Model.

## Acknowledgements

G.F. acknowledges financial support by the Programme STAR Plus, funded by
Federico II University and Compagnia di San Paolo, and by the MIUR, PRIN 2017
grant 20179ZF5KS. F.M. acknowledges support by the Q-CAYLE Proyect funded by
the Regional Government of Castilla y León (Spain) and by the Ministry of
Science and Innovation MICIN through NextGenerationEU (PRTR C17.I1).

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