# Renormalization on the DFR Quantum Spacetime

Juan F. López† and Andrés F. Reyes-Lega‡<EMAIL_ADDRESS>
<EMAIL_ADDRESS>Physics Department, Universidad de los Andes, Cra 1
N°18A-12,
Bogotá D.C., 111711,Colombia

((Day Month Year); (Day Month Year))

###### Abstract

An approach to renormalization of scalar fields on the Doplicher-Fredenhagen-
Roberts (DFR) quantum spacetime is presented. The effective non-local theory
obtained through the use of states of optimal localization for the quantum
spacetime is reformulated in the language of (perturbative) Algebraic Quantum
Field Theory. The structure of the singularities associated to the non-local
kernel that codifies the effects of noncommutativity is analyzed using the
tools of microlocal analysis.

###### keywords:

Quantum Spacetime; Renormalization; Algebraic Quantum Field Theory

Mathematics Subject Classification: 81T15, 81T18, 81T75

##  1 Introduction

One of the most difficult problems in theoretical physics has been, for many
decades, the formulation of a consistent quantum theory of gravity. Apart from
the conceptual and mathematical difficulties, the lack of observational
evidence for the quantum nature of spacetime at the Planck scale has been a
major obstacle. However, this state of affairs is likely to change in the
forthcoming years, as the emergence of the multi-messenger paradigm seems to
indicate [1, 2].

Well established approaches to the problem (like, e.g., loop quantum gravity
[3], asymptotic safety [4] or causal sets [5]) aim for a description of
quantum gravity at a fundamental level. In a more phenomenological vein, there
are other approaches that follow a “bottom-up” approach. That is the case with
several models of quantum spacetimes which incorporate the idea of a
fundamental minimal length scale, but still can be regarded as “deformations”
of, say, Minkowski spacetime. The idea of introducing a set of noncommutative
coordinate operators describing a quantum spacetime goes back to Heisenberg
and Snyder [6]. Much later, similar models were found as low-energy limits of
string theories [7]. The study of quantum field theory on such quantum
spacetimes has been a subject of notorious interest during the last two
decades. Since –by construction– such models still bear resemblance to the
physics of lower energy scales, they offer an interesting possibility for the
exploration of quantum gravity signatures.

In 1994, Doplicher, Fredehagen and Roberts (DFR) proposed a model for a
quantum spacetime that retains all the symmetries of Minkwoski spacetime, but
that also includes a parameter $\lambda_{P}$ (of the order of magnitude of
Planck’s length) which reflects the quantum nature of spacetime.
Heuristically, the idea is to promote the coordinates of events in Minkowski
spacetime to operators satisfying commutation relations that lead to
uncertainty relations compatible with the _Principle of Gravitational
Stability against localisation of events_ [8].

This principle basically says that events, regarded as points in a spacetime
$\mathcal{M}$ that is modeled as a smooth manifold, do not have a physically
coherent interpretation at small length scales (of the order of magnitude of
Planck’s length). The physical existence of a point $x\in\mathcal{M}$, in the
setting of Algebraic QFT (AQFT), would require to define a field localized in
a region of $\mathcal{M}$ arbitrarily close to $x$. From the Heisenberg
uncertainty principle $\Delta E\Delta t\gtrsim\hbar/2$, localization in time
increases uncertainty in energy density, giving place to creation and
annihilation of a lot of particles (of the theory). If this energy density is
localized in a region smaller than its Schwarszchild radius, it would generate
a trapped surface (e.g. a black hole) hiding the event at all (for an
interesting discussion of this issue in the context of QFT on curved
spacetimes with back reaction, see [9]). The principle solves this problem by
imposing certain _uncertainty relations_ on the spacetime coordinates that
forbid such extreme localization of events.

Mathematically, the model is formulated in terms of a $C^{*}$-algebra, which
is to be interpreted as the “algebra of functions” on this quantum spacetime,
much in the spirit of noncommutative geometry [10]. The possibility of
constructing quantum fields over this quantum spacetime was already discussed
in the original works [8, 11].

The theory of quantum fields over the DFR quantum spacetime has been the
subject of intense study during the last two decades (see, e.g., [12, 13], or
[14] for a review). More recently, techniques from perturbative Algebraic
Quantum Field Theory (pAQFT) have been applied to models of quantum fields on
the DFR quantum spacetime [15]. One of the main issues with standard
interacting QFT is the occurrence of (ultraviolet) divergent integrals order
by order in the loop expansion in perturbation theory [16]. Mathematically, it
is possible to trace the origin of these divergences in Fourier space back to
ill-defined products of distributions in coordinate space. Epstein-Glaser
renormalization [17] is the scheme which solves the problem of divergences via
the extension of distributions satisfying certain conditions in a completely
finite way. pAQFT [18, 19] has been recently developed in order to combine the
formal understanding of Algebraic (or axiomatic) QFT with the perturbative
methods. Its application to the study of quantum fields on the DFR quantum
spacetime is an important addition to the already extensive body of work where
other approaches to QFT have been extended to this non-commutative Minkowski
spacetime. Just to name a few, there is one approach inspired in the normal
order by point-splitting [9], another one based on the Yang-Feldman equation
[20]. The point-splitting version has been written in the pAQFT framework[15].
This model has been proved to be free of UV divergences and the existence of
its adiabatic limit has been established.

The different approaches to implement QFT over the noncommutative quantum
spacetime, focused on defining the interacting normally ordered action
functional, lead to inequivalent non-local effective models with very
different singularity structures. Nevertheless, when the commutative limit is
taken (by taking the limit $\lambda_{P}\rightarrow 0$), correlation functions
computed using those different approaches give the standard results for QFT on
Minkowski space.

Therefore, even in the UV finite approach by point-splitting, some
renormalization procedure will be necessary in order to compare with the
commutative, low-energy regime. Deviations from the commutative approximation,
in a bottom-up approach, could give information about the (still unknown)
quantum gravity theory, or its effective low-energy theory.

In the present paper, we have proposed a map that assigns to any _local_
interaction for a scalar field on the DFR Quantum Spacetime a _non-local_
effective interacting quantum field theory in the framework of pAQFT [18],
based on the original Moyal type product of fields by Doplicher, Fredenhagen
and Roberts [11]. Moreover, the singularity structure of this new effective
theory is described in terms of the wavefront set of some distribution [21].

Let us finish this introduction with a description of the remaining sections
of the paper. In section 2, the DFR Quantum Spacetime is presented following
the main results of [11]. Section 3 is devoted to a quick review of the basics
of interacting pAQFT. The next two sections contain the main results of this
paper. In section 4 we obtain an effective non-local interacting Quantum Field
model and express it in the language of pAQFT. The modified Feynman diagrams
that are obtained for this model and their corresponding rules are also
presented. In section 5 we present an analysis of the microlocal properties of
the non-local kernel previously obtained. We finish with some final remarks in
section 6

##  2 Quantum Space-Time

A Quantum Space-Time (QST) shall be understood as a noncommutative space in
the context of noncommutative geometry [10, 22]. By this, functions over the
QST are elements of a C*-algebra $\mathcal{E}$ generated by the unbounded
affiliated selfadjoint elements $\\{q^{\mu}\\}_{\mu=0}^{3}$, the coordinate
operators; which satisfy following the Space-Time Uncertainty Relations (STUR)
[8]

$\displaystyle\Delta q^{0}\sum_{j=1}^{3}\Delta q^{i}$
$\displaystyle\geq\frac{\lambda_{P}^{2}}{2},$ (1) $\displaystyle\sum_{1\leq
j<k\leq 3}\Delta q^{i}\Delta q^{j}$
$\displaystyle\geq\frac{\lambda_{P}^{2}}{2},$ (2)

where $\lambda_{P}$ is some constant with units of the order of the Planck
length. Here, an affiliated element $q$ affiliated to $\mathcal{E}$ is taken
in the sense of Woronowicz [23] as the C*-homomorphism

$\begin{matrix}C_{0}^{\infty}(\mathbb{R})&\rightarrow&M(\mathcal{E})\\\
f&\mapsto&f(q),\end{matrix}$ (3)

where $M$ stands for the multiplier algebra [24]. A state
$\omega\in\mathcal{S}(\mathcal{E})$ is in the domain of $q$ if

$sup\left\\{\omega(f(q))|f\in\ C_{0}(\mathbb{R}),\ f(x)<x^{2}\ \forall
x\in\mathbb{R}\right\\}<\infty,$ (4)

and the Space-Time Uncertainty Relations are assumed to hold in any state in
the domain of all $q^{\mu}$, where
$\Delta(q)=\sqrt{\omega(q^{2})-\omega(q)^{2}}$. Doplicher, Fredenhagen and
Roberts proposed [11] an implementation of the (1) and (2) relations imposing
commutation relations for the coordinate operators (this approach will be
followed for the remaining of this section)

$\left[q^{\mu},q^{\nu}\right]=i\lambda_{P}^{2}Q^{\mu\nu},$ (5)

with $Q^{\mu\nu}$ selfadjoint elements which are assumed to be central
elements (commuting with the coordinate generators)

$\left[q^{\mu},Q^{\sigma\rho}\right]=0,$ (6)

and satisfying the Quantum Conditions

$\displaystyle Q^{\mu\nu}Q_{\mu\nu}$ $\displaystyle=0,$ (7)
$\displaystyle\frac{1}{4}\left(\epsilon_{\mu\nu\sigma\rho}Q^{\mu\nu}Q^{\sigma\rho}\right)^{2}$
$\displaystyle=1.$ (8)

The C*-algebra generated by these operators is isomorphic to the trivial
bundle of C*-algebras

$\mathcal{E}\cong C_{0}(\Sigma,\mathcal{K}),$ (9)

where $\mathcal{K}$ denotes the space of compact operators on a separable
Hilbert space, and $\Sigma$ is the joint spectrum of the central operators
$Q^{\mu\nu}$. Topologically, $\Sigma$ is the product of the tangent bundle of
a sphere and a 2-point set

$\Sigma\cong TS^{2}\times\\{\pm\\}\cong SL_{2}(\mathbb{C})/D.$ (10)

The second equality in (10) is obtained by considering the constant values
$\sigma\in\Sigma$ of $Q$ on irreducible representations of $\mathcal{E}$ (the
spectral values) as antisymmetric $4\times 4$ matrices with electric and
magnetic components orthogonal and equal in norm. This is a homogeneous space
with action of $SL_{2}(\mathbb{C})$ (the double covering of the proper Lorentz
group) as a second rank tensor, and isotropy $D$ group the space of $2\times
2$ diagonal matrices. The only representations we shall be concerned with are
those for which the Weyl-like operators $e^{ik_{\mu}q^{\mu}}$ are well defined
and are strongly continuous in each $k^{\mu}$. Those representations are
called regular realizations. A faithful representation can be obtained via the
Fourier transform. Using the central elements $Q$ as Casimir operators to
label irreducible representations, a faithful representation has the direct
integral form

$\mathcal{E}\cong\int_{\oplus}\mathcal{\pi}_{\sigma}(\mathcal{E}),$ (11)

such that $\pi_{\sigma}(Q^{\mu\nu})=\sigma^{\mu\nu}$. For $F\in
C_{0}(\Sigma,L^{1}(\mathbb{R}^{4}))$ such that
$F(\sigma)(k)=\hat{f}(\sigma,k)$, it is possible to define the operator

$\pi_{\sigma}(f(q))=\int\frac{dk}{(2\pi)^{4}}\hat{f}(\sigma,k)e^{ik_{\mu}\pi_{\sigma}(q^{\mu})};$
(12)

or in short notation

$f(q)=\int\frac{dk}{(2\pi)^{4}}\hat{f}(Q,k)e^{ik_{\mu}q^{\mu}}.$ (13)

When restricted to a representation $\pi_{\sigma}$, the commutation relations
(5), (6) lead to the Moyal product

$e^{ik^{1}_{\mu}q^{\mu}}*e^{ik^{2}_{\mu}q^{\mu}}=e^{i\frac{\lambda_{P}^{2}}{2}k^{1}_{\mu}\sigma^{\mu\nu}k^{2}_{\nu}}e^{i(k^{1}+k^{2})_{\mu}q^{\mu}},$
(14)

which completely determines the algebra $\mathcal{E}$. Let us denote with
$\mathcal{S}(\mathcal{E})$ the space of states of $\mathcal{E}$. Following, we
will work with states $\omega\in\mathcal{S}(\mathcal{E})$ of the form

$\omega(f(q))=\int_{\Sigma}d\mu(\sigma)\int\frac{dk}{(2\pi)^{4}}\hat{f}(\sigma,k)\omega_{\sigma}(e^{ik_{\nu}\pi_{\sigma}(q^{\nu})}),$
(15)

where $\mu$ is a probability measure over $\Sigma$ and $\omega_{\sigma}$ is a
continuous collection of states in each irreducible representation
$\pi_{\sigma}$. An interesting example of a such a state in $\mathcal{E}$ is
given by the optimally localized state around $x\in\mathbb{R}^{4}$. This state
minimizes the uncertainty $\sum_{\nu}\Delta(q^{\mu})^{2}$ and it is given by a
measure $\mu$ supported only on $\Sigma_{1}$ -the zero section of $\Sigma$\-
and

$\omega_{\sigma}(e^{ik_{\nu}\pi_{\sigma}(q^{\nu})})=exp\left[-\frac{\lambda_{P}^{2}}{2}\sum_{\nu}\left(k_{\nu}\right)^{2}\right]e^{ik_{\nu}x^{\nu}}.$
(16)

Clearly, this state is not Lorentz invariant, neither any state with a measure
$\mu$ compactly supported in $\Sigma$. Moreover, measures with non-compact
support allow arbitrary delocalized states (with localization measured by
$\sum_{\mu}\Delta_{\omega}(q^{\mu})^{2}$). The simplest choice, which we will
follow in this work, is to pick in each sphere the unique measure induced by
rotational symmetry. Another state derived from (15), takes into account the
integral of the operator $f(q)$ in the whole euclidean space at a fixed time
$t$. With the invariant measure at $\Sigma_{1}$ (16)

$\int_{q^{0}=t}f(q)d^{3}q:=\int_{\Sigma_{1}}d\mu(\sigma)\int\frac{dk_{0}}{2\pi}\hat{f}(\sigma;k_{0},\vec{0})e^{ik_{0}t}.$
(17)

These type of expressions appear in the definition of the Hamiltonian at time
$t$ in terms of some density (e.g. interactions), as it is presented below.

##  3 Quantum Field Theory

In the following sections, the framework of perturbative Algebraic Field
Theory [18, 25] will be used. In this setting, the configuration space of off-
shell fields for the real scalar field is
$\mathcal{C}:=C^{\infty}(\mathbb{R}^{4})$, i.e. the equations of motion are
not imposed at the very beginning. The fundamental fields are defined in the
following way. For fixed $x\in\mathbb{R}^{4}$, we define $\phi(x)$ to be the
evaluation map

$\begin{split}\phi(x):\mathcal{C}&\rightarrow\mathbb{R}\\\ h&\mapsto
h(x).\end{split}$ (18)

Analogously, for $a\in\mathbb{N}^{4}$ $\partial^{a}\phi(x)$ is defined by

$\begin{split}\partial^{a}\phi(x):\mathcal{C}&\rightarrow\mathbb{R}\\\
h&\mapsto\partial^{a}h(x)=\frac{\partial^{|a|}h(x)}{\partial^{a_{0}}x^{0}\ldots\partial^{a_{3}}x^{3}},\end{split}$
(19)

where $|a|=a_{0}+...+a_{3}$. The set of fields (i.e. the observables of the
theory), denoted by $\mathcal{F}$, are complex valued functions on the
configuration $\mathcal{C}$ with all its functional derivatives well defined,
and non-zero only up to finite order. That is, if $F\in\mathcal{F}$

$\begin{split}F:\mathcal{C}&\rightarrow\mathbb{C}\\\ \phi&\mapsto
F(\phi)=f_{0}+\sum_{n=0}^{N}f_{n}(\phi^{\otimes n}),\end{split}$ (20)

where $f_{0}\in\mathbb{C}$ and $f_{n}$ are complex valued, compactly supported
symmetric distributions in $\mathbb{R}^{4n}$; such that its wavefront set
satisfies

$WF(f_{n})\cap\left(\overline{V_{+}}\cup\overline{V_{-}}\right)=\emptyset,$
(21)

where $\overline{V_{\pm}}$ are the closures of the forward/backward cones for
a metric $\eta=diag(+,-,-,-)$ in Minkowski space-time given by

$V_{\pm}:=\\{x\in\mathbb{R}^{4}|x^{2}:=g_{\mu\nu}x^{\mu}x^{\nu}>0,\ \pm
x^{0}>0\\}.$ (22)

We will use the following short notation for products following [12]. For
$x_{1},\cdots x_{n}\in\mathbb{R}^{4}$, $F\in\mathcal{E}(\mathbb{R}^{4n})$ and
$G\in\mathcal{F}$

$\begin{split}d\underline{x}^{n}&:=\prod_{j=1}^{n}\prod_{\mu=0}^{4}dx_{i}^{\mu},\\\
F(\underline{x})&:=F(x_{1},\cdots x_{n}),\\\
\phi(\underline{x})^{n}&:=\phi(x_{1})\cdots\phi(x_{n}),\\\
\frac{\delta^{n}G}{\delta\phi(\underline{x})^{n}}&:=\frac{\delta^{n}G}{\delta\phi(x_{1})\cdots\delta\phi(x_{n})}.\end{split}$
(23)

Fields with pointwise multiplication and involution given by complex
conjugation, define the *-algebra of classical fields. Quantum fields are
described by $\mathcal{F}\llbracket\hbar\rrbracket$, formal power series in
the Planck constant with coefficients in $\mathcal{F}$; and with a non-
commutative product. For $F,\ G\in\mathcal{F}$, the quantum product is

$F*G=\sum_{n=0}^{+\infty}\frac{\hbar^{n}}{n!}\int
d\underline{x}^{n}d\underline{y}^{n}\frac{\delta^{n}F}{\delta\phi(\underline{x})^{n}}\left(\Delta^{+}(\underline{x}-\underline{y})\right)^{n}\frac{\delta^{n}G}{\delta\phi(\underline{y})^{n}},$
(24)

where

$\left(\Delta^{+}(\underline{x}-\underline{y})\right)^{n}:=\prod_{\ell=1}^{n}\Delta^{+}(x_{\ell}-y_{\ell}),$
(25)

with $\Delta^{+}$ the Wightman two-point function, which corresponds to the
choice of positive frequency part of the Green function for the Klein Gordon
operator $K=\Box+m^{2}$. The fields $\mathcal{F}\llbracket\hbar\rrbracket$
with this non-commutative product modulo the ideal
$\mathbb{S}K_{1}\mathcal{F}\llbracket\hbar\rrbracket$; where $K_{1}$ [26] is
the Klein-Gordon operator on the first variable and $\mathbb{S}$ is the
symmetrization operator, define the on-shell free scalar fields. Elements of
$\mathcal{F}\llbracket\hbar\rrbracket$ are normally ordered fields and the
quantum product is just the implementation of the Wick theorem.

Another important non-commutative product in quantum field theory is the
T-product (time ordered product). This product is fundamental on the
definition of a perturbative Field Theory defined by an interaction
Hamiltonian (e.g. causal S-matrix formalism of Bogoliubov-Epstein-Glaser
[27]). T-product defines elements which are formal power series in $\kappa$
(the coupling constant) and Laurent series in $\hbar$. From de Dyson series in
perturbation theory for quantum mechanics, the S-matrix operator is

$S=1+\sum_{n=1}^{+\infty}\frac{(-i\kappa)^{n}}{n!\hbar^{n}}\int dt_{1}\cdots
dt_{n}T_{n}\left(H_{int}(t_{1})\otimes\cdots\otimes H_{int}(t_{n})\right),$
(26)

where $T_{n}\left(H_{int}(t_{1})\otimes\cdots\otimes
H_{int}(t_{n})\right)=H_{int}(t_{\pi(1)})*\cdots*H_{int}(t_{\pi(n)})$ for
$\pi\in S_{n}$ such that $t_{\pi(1)}>...>t_{\pi(n)}$. In QFT the T-product is
exactly this time ordering for point $x_{i}\neq x_{j}$. For distributions with
support in coinciding points, it is necessary to implement some
renormalization procedure, i.e. extension of functionals from
$\mathbb{R}^{4n}\backslash\Delta^{n}$ ($\Delta^{n}$ the thin diagonal) to the
whole space.

The relation between the T-product and the interacting theory is two-fold. In
experiments in high energy physics, the role of the S-matrix is to connect
asymptotic in and out states (which are assumed to be described by the free
theory, far away from the support of the interaction before taking the
adiabatic limit). On the other side, there are those retarded wave (Möller)
operators $r_{S_{1},S_{0}}$ [25], which map solutions of the free theory
described by the free action

$S_{0}=\frac{1}{2}\int
dx\left(\partial_{\nu}\phi\partial^{\nu}\phi-m^{2}\phi^{2}\right)$ (27)

to solutions of the interacting theory with action $S_{1}=S_{0}+\kappa
S_{int}$

$S_{int}[g]=-\int dx\ g(x)\mathcal{H}_{int}(x),$ (28)

where a function $g\in\mathcal{D}(\mathbb{R}^{4})$ introduced in order to
avoid IR divergences, and $\mathcal{H}_{int}$ is the interaction Hamiltonian
density. The retarded operator can be defined as a perturbative series in
$\kappa$ in terms of the R-product (retarded product). Let $F\in\mathcal{F}$

$F^{ret}:=r_{S_{1},S_{0}}(F)=\sum_{n=0}^{+\infty}\frac{1}{n!}R_{n,1}(S_{int}[g]\otimes...\otimes
S_{int}[g],F).$ (29)

The R-product can be calculated in terms of the T-product using the Bogoliubov
formula

$\displaystyle F^{ret}=R\left(e^{\kappa S_{int}/\hbar}_{\otimes},F\right)$
$\displaystyle=\frac{\hbar}{i}\frac{d}{d\lambda}\Big{|}_{\lambda=0}\textbf{S}(\kappa
S_{int}/\hbar)^{*-1}*\textbf{S}(\kappa S_{int}/\hbar+\lambda F),$ (30)
$\displaystyle R_{n}(S\otimes...\otimes S,F)$
$\displaystyle=i^{n}\sum_{I\subseteq\\{1,...,n\\}}(-1)^{|I|}\bar{T}_{|I|+1}(S_{I}\otimes
F)*T_{|I^{c}|}(S_{I^{c}})$ (31)

with S matrix

$\textbf{S}(F)=1+\sum_{n=1}^{+\infty}\frac{i^{n}}{n!\hbar^{n}}T_{n}(F^{\otimes
n})=T\left(e_{\otimes}^{iF/\hbar}\right),$ (32)

where the exponential with respect to the tensor product is defined by

$e_{\otimes}^{F}:=1+\sum_{n=1}^{+\infty}\frac{1}{n!}\underbrace{F\otimes...\otimes
F}_{n-times}=1+\sum_{n=1}^{+\infty}\frac{1}{n!}F^{\otimes n},$ (33)

and anti chronological ($\bar{T}$-) product given by the inverse order of the
T-product.

##  4 Quantum Fields in Quantum Spacetime

The aim of this section is to associate to certain class of fields in standard
pAQFT, new fields depending on the non-commutativity parameter $\lambda_{P}$,
using the properties discussed for the algebra of functions on the QST in the
definition. This new fields should be applied in perturbation theory (e.g.
Dyson series, retarded fields) for the associated interacting effective
Hamiltonians for interactions of the form $\phi^{n}$ [16].

Given that, from the second section, the available product over the QST is
defined for functions with the same coordinate generator $q$, the
aforementioned association in the last paragraph is expected to be a map from
local fields $\mathcal{F}_{loc}$ to fields $\mathcal{F}$, where the image
shall be understood as the effective field over the QST. As it is customary in
approaches to QFT from non-commutative geometry, e.g. the Moyal plane [28]; if
the departure point of the interaction is a local action, the effective action
taking into account the non-commutativity effects results non-local[29, 30,
31].

For this reason, during the following sections, the fields of interest are
assumed to be local, i.e. the support of the distributions which define the
fields shall be in the thin diagonal

$supp\ f_{n}\subseteq\Delta^{n}.$ (34)

A quite general set of examples of those fields are of the form

$\begin{split}F[\phi]&=\sum_{n=1}^{N}\sum_{\begin{subarray}{c}a=(a_{1},...,a_{n})\in\mathbb{N}^{d}\\\
|a|<\infty\end{subarray}}\int
d\underline{x}^{n}(-1)^{|a|}\partial^{a}\left[f_{a}(x_{n})\delta(x_{1}-x_{n})...\delta(x_{n-1}-x_{n})\right]\phi(\underline{x})^{n}\\\
&=\sum_{n=1}^{N}\sum_{\begin{subarray}{c}a=(a_{1},...,a_{n})\in\mathbb{N}^{d}\\\
|a|<\infty\end{subarray}}\int dx\
f_{a}(x)\underline{\partial^{a}\phi(x)},\end{split}$ (35)

where

$\begin{split}|a|&:=|a_{1}+...+a_{n}|,\\\
\partial^{a}&:=\partial_{x_{1}}^{a_{1}}...\partial_{x_{n}}^{a_{n}},\\\
\underline{\partial^{a}\phi(x)}&:=\partial^{a_{1}}\phi(x)\cdots\partial^{a_{n}}\phi(x),\end{split}$
(36)

and $f_{a_{1}...a_{n}}$ are compactly supported distributions in
$\mathbb{R}^{4}$. An example of this kind of local fields is the $\phi^{n}$
interaction

$S_{int}[g]=-\frac{1}{n!}\int dx\ g(x)\phi(x)^{n},$ (37)

where $g\in\mathcal{D}(\mathbb{R}^{4})$. With the definition of the derivative
of a function over the QST

$\frac{\partial}{\partial x^{\mu}}f(q)=\frac{\partial}{\partial
a^{\mu}}\Big{|}_{a^{\mu}=0}f(q+a^{\mu}I)=\int\frac{dk}{(2\pi)^{4}}\hat{f}(Q,k)(ik_{\mu})e^{ik_{\mu}q^{\mu}}.$
(38)

The respective fundamental quantum field over the quantum spacetime is defined
by

$\phi(q)=\int\frac{dk}{(2\pi)}\hat{\phi}(k)\otimes e^{ik_{\mu}q^{\mu}},$ (39)

where the Fourier transform of the field is the usual

$\hat{\phi}(k)=\int dx\ \phi(x)e^{-ik_{\mu}x^{\mu}}.$ (40)

This is an element of the tensor product of *-algebras
$\mathcal{F}\otimes\mathcal{E}$. In the following, the $\otimes$ symbol shall
be omitted. Making use of the product in $\mathcal{E}$, the monomial field
over the QST is

$\underline{\partial^{a}\phi(q)}=\int\frac{d\underline{k}^{n}}{(2\pi)^{4n}}exp\left\\{i\frac{\lambda_{P}^{2}}{2}\sum_{j<m}k^{j}_{\mu}Q^{\mu\nu}k^{m}_{\nu}\right\\}(i\underline{k})^{a}\hat{\phi}(\underline{k})^{n}e^{i\sum_{j}k^{j}_{\mu}q^{\mu}},$
(41)

where $(ik)^{a}=\prod_{\mu=0}^{3}(ik_{\mu})^{a^{\mu}}$ for
$a=(a^{0},...,a^{3})\in\mathbb{N}^{4}$. Elements
$\Phi\in\mathcal{F}\otimes\mathcal{E}$ can be seen as maps from
$\mathcal{E}^{\prime}$ (functionals over the QST, such as the states
$\mathcal{S}(\mathcal{E}))$ to fields in $\mathcal{F}$. For a general element
of the form

$\Phi(q)=\int\frac{dk}{(2\pi)^{4}}\hat{\Phi}(Q,k)e^{ik_{\mu}q^{\mu}},$ (42)

and a functional
$\omega=\int_{\Sigma}d\mu(\sigma)\omega_{\sigma}\in\mathcal{E}^{\prime}$, a
field is defined by the pairing

$\langle\Phi,\omega\rangle=\int_{\Sigma}d\mu(\sigma)\omega_{\sigma}\left(e^{ik_{\mu}\pi_{\sigma}(q^{\mu})}\right)\hat{\Phi}(\sigma,k).$
(43)

In particular, it is possible to define a generalization of the integral from
the calculus at the end of section II, to define the action of a compactly
supported distribution $f\in\mathcal{D}^{\prime}(\mathbb{R}^{4})$. Define the
linear functional $I(f)\in\mathcal{E}^{\prime}$ with the rotationally
invariant measure $\mu$ on $\Sigma_{1}$ and its action on irreducible
representations

$I(f)_{\sigma}\left(e^{ik_{\nu}\pi_{\sigma}(q^{\nu})}\right)=\hat{f}(-k).$
(44)

Then, given the local field $F[\phi]\in\mathcal{F}_{loc}$; the effective non-
local field $F^{(Q)}[\phi]\in\mathcal{F}$ is defined in the momentum space as

$F^{(Q)}[\phi]=\sum_{n=1}^{N}\sum_{\begin{subarray}{c}a=(a_{1},...,a_{n})\in\mathbb{N}^{d}\\\
|a|<\infty\end{subarray}}\langle\underline{\partial^{a}\phi(q)},I(f_{a})\rangle,$
(45)

where

$\begin{split}\langle\underline{\partial^{a}\phi(q)},I(f)\rangle&=\int_{\Sigma_{1}}d\mu(\sigma)\int\frac{d\underline{k}^{n}}{(2\pi)^{4n}}\hat{f}\left(-\sum_{j=1}^{n}k^{j}\right)(i\underline{k})^{a}exp\left\\{i\frac{\lambda_{P}^{2}}{2}\sum_{j<l}k^{j}_{\mu}\sigma^{\mu\nu}k^{l}_{\nu}\right\\}\hat{\phi}(\underline{k})^{n}\end{split}$
(46)

For example, the effective action for the $\phi^{n}$ interaction action
obtained by the previous map, in the momentum space, is

$S^{(Q)}_{int}[g]=-\frac{1}{n!}\int_{\Sigma_{1}}d\mu(\sigma)\int\frac{d\underline{k}^{n}}{(2\pi)^{4n}}\hat{g}\left(-\sum_{j}k^{j}\right)exp\left\\{i\frac{\lambda_{P}^{2}}{2}\sum_{j<m}k^{j}_{\mu}\sigma^{\mu\nu}k^{m}_{\nu}\right\\}\hat{\phi}(\underline{k})^{n}$
(47)

In order to obtain explicit formulas for the action, as well as an interacting
Hamiltonian, let us first calculate the integration term associated with the
non-commutativity parameter [11]. That term is defined by

$\begin{split}\Lambda_{n}(\underline{k})&=\int_{\Sigma_{1}}d\mu(\sigma)exp\left\\{i\frac{\lambda_{P}^{2}}{2}\sum_{j<m}k^{j}_{\mu}\sigma^{\mu\nu}k^{m}_{\nu}\right\\}\\\
&=\frac{1}{2}\left(\frac{sin(\beta_{+})}{\beta_{+}}+\frac{sin(\beta_{-})}{\beta_{-}}\right),\end{split}$
(48)

where

$\beta_{\pm}:=\frac{\lambda_{P}^{2}}{2}\left\lVert\sum_{j<l}k_{0}^{j}\vec{k}^{l}-k_{0}^{l}\vec{k}^{j}\pm\vec{k}^{j}\times\vec{k}^{l}\right\rVert.$
(49)

Its inverse Fourier transform is denoted by

$\tilde{\Gamma}_{n}(\underline{x})=\int\frac{d\underline{k}^{n}}{(2\pi)^{4n}}\Lambda_{n}(\underline{k})e^{i\sum_{j}k^{j}_{\mu}x_{j}^{\mu}};$
(50)

and the translation to the point $x\in\mathbb{R}^{4}$

$\Gamma_{n}(\underline{x};x)=\tilde{\Gamma}_{n}(\underline{x}-x),$ (51)

where

$\tilde{\Gamma}_{n}(\underline{x}-x)=\tilde{\Gamma}_{n}(x_{1}-x,\cdots
x_{n}-x).$ (52)

Completes the definition of the non-commutative action in terms of a non-local
interaction distribution $\Gamma$, as follows

$S^{(Q)}_{int}[g]=-\frac{1}{n!}\int dx\
d\underline{x}^{n}g(x)\Gamma(\underline{x};x)\phi(\underline{x})^{n}.$ (53)

For the general local field $F[\phi]$, the action of this map defines a non-
local effective field, which in the configuration space if given by

$\langle\underline{\partial^{a}\phi(q)},I(f)\rangle=\int
dxd\underline{x}^{n}\Gamma_{n}(\underline{x};x)\underline{\partial^{a}\phi(x)}$
(54)

It is straightforward that in the weak limit when $\lambda_{P}\rightarrow 0$,
the non-local kernel

$\lim_{\lambda_{P}\rightarrow
0}\Gamma(x_{1},\ldots,x_{n};x)=\delta(x_{1}-x)\cdots\delta(x_{n}-x),$ (55)

and the standard QFT is recovered. The effective action above $S^{(Q)}_{int}$
can be understood as the associated action of some perturbative QFT induced by
the non-local Hamiltonian density

$\mathcal{H}_{int}(x)=\frac{1}{n!}\int
d\underline{x}^{n}\Gamma_{n}(\underline{x};x)\phi(\underline{x})^{n}.$ (56)

As in the previous section, the S matrix is defined in terms of the
T-products, linear maps

$T_{n}:\mathcal{F}^{\otimes
n}\rightarrow\mathcal{F}\llbracket\kappa,\hbar,\hbar^{-1}\rrbracket,$ (57)

which, roughly speaking, order the fields $\phi$ with respect to the time
coordinate; but as in the definition from Section III for standard QFT, the
time ordering is performed with respect to the time coordinate $x$ of the
Hamiltonian density, in this case ignoring the dummy coordinates integrated
with the non-local kernel $\Gamma$. For this reason, the $x$ coordinate is
referred as time stamp . For example, at order $N=2$ in the interaction
coupling, the S matrix is

$\displaystyle S[g]=$ $\displaystyle 1-\frac{i\kappa}{n!\hbar}\int\
dxd\underline{x}g(x)\Gamma_{n}(\underline{x};x)\phi(\underline{x})^{n}+$
$\displaystyle-\frac{\kappa^{2}}{2(n!)^{2}\hbar^{2}}\int
dx_{1}dx_{2}d\underline{x_{1}}^{n}d\underline{x_{2}}^{n}g(x_{1})g(x_{2})T_{2}\big{(}\Gamma_{n}(\underline{x_{1}};x_{1})\phi(\underline{x_{1}})^{n}\otimes$
$\displaystyle\otimes\Gamma_{n}(\underline{x_{2}};x_{2})\phi(\underline{x_{2}})^{n}\big{)}+\mathcal{O}(\kappa^{3}),$

for $x_{1}^{0}\neq x_{2}^{0}$, $\underline{x_{i}}=(x_{i,1},\cdots,x_{i,n})$
for $i=1,2$ and

$\displaystyle
T_{2}\big{(}\Gamma_{n}(\underline{x_{1}};x_{1})\phi(\underline{x_{1}})^{n}\otimes$
$\displaystyle\Gamma_{n}(\underline{x_{2}};x_{2})\phi(\underline{x_{2}})^{n}\big{)}=\Gamma_{n}(\underline{x_{1}};x_{1})\Gamma_{n}(\underline{x_{2}};x_{2})\times$
$\displaystyle\times\big{(}\theta(x_{1}^{0}-x_{2}^{0}))\phi(\underline{x_{1}})^{n}*\phi(\underline{x_{2}})^{n}+\theta(x_{2}^{0}-x_{1}^{0}))\phi(\underline{x_{2}})^{n}*\phi(\underline{x_{1}})^{n}\big{)}.$

For both T- and R- products, it is possible to obtain a set of Feynman rules
modified by the non-local effects of the effective interacting QFT, the latter
one introducing the T-product into the Bogoliubov formula.

As an example of the previous formulas, let us calculate the interacting field
$\phi_{S}(x)$, defined as a formal power series in $\kappa$ and $\hbar$ for
$n=4$ order by order as

$\begin{split}\phi_{S}(x):=R\left(e_{\otimes}^{\kappa
S_{I}[g]/\hbar},\phi(x)\right)=\phi(x)+\frac{\kappa}{\hbar}R_{1,1}\left(S_{I}[g],\phi(x)\right)+\\\
+\frac{\kappa^{2}}{2\hbar^{2}}R_{2,1}\left(S_{I}[g]\otimes
S_{I}[g],\phi(x)\right)+\mathcal{O}(\kappa^{3}),\end{split}$ (58)

keeping concern to the 2nd order approximation in $\kappa$. The zeroth order
is simply $\phi_{S}^{(0)}(x)=\phi(x)$, the free field. The first order term is

$\frac{\kappa}{\hbar}R_{1,1}\left(S_{I}[g],\phi(y)\right)=\frac{\kappa}{4!}\int
dxd\underline{x}^{4}g(x)\Gamma_{4}(\underline{x};x)\theta(y^{0}-x^{0})\sum_{j=1}^{4}\Delta_{m}(y-x_{j})\prod_{\begin{subarray}{c}k=1\\\
k\neq j\end{subarray}}^{4}\phi(x_{k}),$ (59)

using from the Bogoliubov formula that

$R_{1,1}(S,F)=i\left[T_{2}(S,F)-S\star_{m}F\right]$ (60)

As it was mentioned above, the time order (in the $\theta$ functions) is
related only between the time stamps $y^{0}$ and $x^{0}$, but not the dummy
variables $x_{1},\ldots,x_{n}$; and
$\Delta_{m}(z)=\Delta_{m}^{+}(z)-\Delta_{m}^{+}(-z)$ is the commutator (Pauli-
Jordan) function. The latter term can be represented as the Feynman diagram at
Fig. 1.

Figure 1: A one-vertex tree level Feynman diagram for a
$\phi^{4}$-interaction.

Assuming that $\Gamma_{4}$ is symmetrized in $x_{1},\ldots,x_{4}$; it
corresponds to a normally orderded quantum field of the form

$\int dx\,dx_{1}\cdots
dx_{3}f^{(3)}(x_{1},\ldots,x_{3};y)\phi(x_{1})\cdots\phi(x_{3})$ (61)

with the distribution in $\mathcal{D}^{\prime}(\mathbb{R}^{4\times 3})$

$f^{(3)}(x_{1},...,x_{3};y)=\kappa\int
dxdx_{4}g(x)\Gamma_{n}(\underline{x};x)\theta(y^{0}-x^{0})\Delta_{m}(y-x_{4})$
(62)

This corresponds to a tree diagram, then this term is due to classical
interacting field contribution over non-commutative spacetime. The second
order contribution results more involved because it requires to calculate

$\begin{split}R_{2,1}(S_{1}\otimes S_{2},F)=T_{3}(S_{1}\otimes S_{2}\otimes
F)-S_{1}\star T_{2}(S_{2}\otimes F)+\\\ -S_{2}\star T_{2}(S_{1}\otimes
F)-T_{2}(S_{1}\otimes S_{2})\star F+S_{1}\star S_{2}\star F+S_{2}\star
S_{1}\star F\end{split}$ (63)

For higher order contributions, the calculation of several terms leads to the
following Feynman rules for the interacting field $\phi_{S}(x)$ at order $k$
in $\kappa$ are as follows:

  * •

Draw every diagram with one vertex $x$ (with only one leg) and $k$ vertices
labeled $x_{1},\ldots,x_{k}$ which are the time stamps. Also for each of the
latter labels there will be required $x_{i.j}$ non-local variables (which are
not time-ordered) for $i\in\\{1,\ldots,k\\}$ and $j\in\\{1,\ldots,n\\}$. Also,
pick up a temporal order $x^{0}_{\pi(1)}<\cdots<x^{0}_{\pi(k)}$ for $\pi\in
S_{k}$.

  * •

For the line connecting $x$ with $x_{i}$, put a factor

$\theta(x^{0}-x^{0}_{i})\sum_{j=1}^{4}\Delta_{m}(x-x_{i.j}).$ (64)

Inside the sum $x_{i.j}$ is an occupied label.

  * •

For every $t$ lines connecting $x_{i}$ with $x_{j}$ with
$x^{0}_{i}>x^{0}_{j}$, put a factor inside the previous sums put a factor

$\theta(x^{0}_{i}-x^{0}_{j})\sum_{m_{1},\ldots,m_{t}}\sum_{n_{1},\ldots,n_{t}}\left(\prod_{\ell=1}^{t}\Delta_{m}^{+}\left(x_{i.m_{\ell}}-x_{j.n_{\ell}}\right)-\prod_{\ell=1}^{t}\Delta_{m}^{+}\left(x_{j.n_{\ell}}-x_{i.m_{\ell}}\right)\right),$
(65)

where $x_{i.m_{\ell}}$ and $x_{j.n_{\ell}}$ are taken over every label which
is not occupied. The labels inside $\Delta_{m}^{+}$ in the sum are now also
occupied.

  * •

Once the steps above are completed, for each free leg in vertex $x_{i}$, a
field $\phi(x_{i,j})$ is multiplied, with $j$ taking values over each label
which is not occupied.

  * •

If $L$ is the number of lines, the last expression is, up to some symmetry
numerical factor, the integrand of

$\kappa^{k}\hbar^{L}\int d\underline{x}^{k}d\underline{x_{1}}^{n}\cdots
d\underline{x_{k}}^{n}g(\underline{x})^{k}\Gamma_{n}(\underline{x_{1}};x)\cdots\Gamma_{n}(\underline{x_{k}};x_{k})$
(66)

  * •

Sum over all orders $\pi\in S_{k}$ and over all diagrams. This is $R_{k,1}$

Example: Let us consider the contributions of all the diagrams of the form
represented in Fig. 2 for $n=4$.

Figure 2: A two-vertex one loop Feynman diagram for a $\phi^{4}$-interaction.

This is a $k=2$ and $L=3$ contribution to the interacting field. The field
associated with this diagram for $x_{1}^{0}>x_{2}^{0}$ is

$\displaystyle\frac{\kappa^{2}\hbar}{4(4!)^{2}}\int
d\underline{x}^{2}d\underline{x_{1}}^{4}d\underline{x_{2}}^{4}g(\underline{x})^{2}\Gamma_{4}(\underline{x_{1}};x_{1})\Gamma_{4}(\underline{x_{2}};x_{2})\times$
$\displaystyle\times\Bigg{\\{}\theta(x^{0}-x_{1}^{0})\theta(x_{1}^{0}-x_{2}^{0})\sum_{j=1}^{4}\Delta_{m}(y-x_{1.j})\sum_{\begin{subarray}{c}j_{1}=1\\\
j_{1}\neq j\end{subarray}}^{4}\sum_{\begin{subarray}{c}j_{2}=1\\\ j_{2}\neq
j,j_{1}\end{subarray}}^{4}\sum_{k_{1}=1}^{4}\sum_{\begin{subarray}{c}k_{2}=1\\\
k_{1}\neq
k_{1}\end{subarray}}^{4}\Bigg{(}\prod_{t=1}^{2}\Delta_{m}^{+}(x_{1.j_{t}}-x_{2.k_{t}})$
$\displaystyle-\prod_{t=1}^{2}\Delta_{m}^{+}(x_{2.k_{t}}-x_{1.j_{t}})\Bigg{)}\prod_{\begin{subarray}{c}m=1\\\
m\neq
j,j_{1},j_{2}\end{subarray}}^{4}\phi(x_{1.m})\prod_{\begin{subarray}{c}n=1\\\
n\neq k_{1},k_{2}\end{subarray}}^{4}\phi(x_{2.n})\Bigg{\\}}.$

In any case, the fields and the propagators are mixed, due to the non-local
kernels $\Gamma_{4}$, at different points of the spacetime.

##  5 Singularity structure of the non-local kernel $\Gamma_{n}$

In standard perturbation theory for QFT, when the expansion in terms of
$\kappa$ and $\hbar$ is performed, divergences in Fourier transform space
appear commonly, and are addressed with regularization techniques like
analytic regularization [32] (e.g. dimensional regularization [33]) or
momentum space cut-off [16]. With closer attention to the obtained
distributions in Minkowski (physical) space-time from the Feynman diagrams,
the resulting divergences can be traced back to ill-defined products of
distributions. Bogoliubov-Epstein-Glaser renormalization [17, 27] deals with
the problem of defining those products in two steps. First, for functions
supported where the product of distributions is well defined, i.e. where
Hörmander criterion is satisfied [21] (as it will be explained below), the
usual product is taken. Then, the restricted distribution is extended (in a
non necessarily unique way) to a distribution to the whole spacetime, with
suitable scaling properties, known as the scaling and mass expansion axiom
[18]. In this section, the wavefront set for the distributions involved in our
modified Feynman rules will be presented. This set will allow us to describe
on which points the products of distributions are well defined.

There are simple cases when the product of distributions can be defined [21].
Given a distribution $\Gamma\in\mathcal{D}^{\prime}(\mathbb{R}^{n})$; we say
that $\Gamma$ is a smooth function restricted to the open set
$X\subseteq\mathbb{R}^{n}$ if there is a smooth function $\psi_{\Gamma}$ such
that for any $\phi\in\mathcal{D}(X)$

$\Gamma(\phi)=\int dx\psi_{\Gamma}(x)\phi(x).$ (67)

The complement of the biggest $X$ where this is possible, the set where
$\Gamma$ fails to be a function, is called the singular support of $\Gamma$
and is denoted by $sing\ supp(\Gamma)$. It is clear when the singular supports
of $\Gamma_{1}$ and $\Gamma_{2}$ are disjoint, the product is well defined:
given a smooth function $\chi$ which is 1 at $sing\ supp(\Gamma_{1})$ and 0 at
$sing\ supp(\Gamma_{2})$, for any $\phi\in\mathcal{D}$

$(\Gamma_{1}\cdot\Gamma_{2})(\phi)=\Gamma_{1}(\psi_{\Gamma_{2}}\chi\psi)+\Gamma_{2}(\psi_{\Gamma_{1}}(1-\chi)\phi).$
(68)

The Fourier transform of a compactly supported distribution
$\Gamma\in\mathcal{E}^{\prime}(\mathbb{R}^{n})$ is

$\hat{\Gamma}(k)=\Gamma(e^{-ik_{\mu}x^{\mu}}).$ (69)

From the Paley-Wiener theorem [21], another characterization when a
distribution is a function is given in terms of the decay properties of its
Fourier transform. $\Gamma$ is a distribution if and only if $\hat{\Gamma}$
decays faster than any inverse power of $k$. The set that measure the failure
of a distribution to be a smooth function in the Fourier space, in analogy to
the singular support, is the cone.

The cone of a distribution, denoted as $\Sigma(\Gamma)$, is the complement of
largest open conical neighborhood where $\hat{\Gamma}$ decays faster than any
inverse polynomial function. With these definitions, the main idea of
microlocal analysis is to define locally, the information of the singular
behavior described by $\Sigma(\Gamma)$. For $x\in\mathbb{R}^{n}$, the cone at
$x$ is

$\Sigma_{x}(\Gamma):=\bigcap_{\begin{subarray}{c}\phi\in\mathcal{D}(\mathbb{R}^{n})\\\
x\in supp(\phi)\end{subarray}}\Sigma(\phi\Gamma).$ (70)

The wavefront set $WF(\Gamma)$ encodes the information of both $sing\ supp$
and $\Sigma$. It is defined by

$WF(\Gamma)=\left\\{(x,k)\in\mathbb{R}^{n}\times\mathbb{R}^{n}|x\in sing\
supp(\Gamma),\ k\in\Sigma_{x}(\Gamma)\right\\},$ (71)

and for distributions over a manifold $M$, it can be defined as a subbundle of
$T^{*}M$. The Hörmander’s criterion establishes that the product of two
distributions $\Gamma_{1}\cdot\Gamma_{2}$ is a well defined distribution if
the pointwise sum $WF(\Gamma_{1})+WF(\Gamma_{2})$ does not intersect the zero
section, i.e.

$\left\\{(x,k_{1}+k_{2})|(x,k_{1})\in WF(\Gamma_{1})\ and\ (x,k_{2})\in
WF(\Gamma_{2})\right\\}\cap\mathbb{R}^{n}\times\\{0\\}=\emptyset.$ (72)

In this case, the product is defined using the convolution formula.

Some useful properties of the wavefront set are presented here. For $f,\ g$
distributions and $F=f\otimes g$ the tensor product, in the sense
$F(x,y)=f(x)g(y)$; then

$WF(F)\subseteq\left\\{(x,y;k_{1},k_{2})|(x;k_{1})\in WF(f),\ (y;k_{2})\in
WF(g)\right\\},$ (73)

in particular, if the distribution is independent of some set of variables
$G(x,y)=g(x)$

$WF(G)=\left\\{(x,y;k_{1},k_{2})|(x;k_{1})\in WF(g)\right\\}.$ (74)

From translational invariance, a distribution
$F(\underline{x},x)=f\underline{x}-x)$ has a wavefront set

$WF(F)=\left\\{(\underline{x},x;\underline{k},k)|(\underline{x}-x;\underline{k})\in
WF(f),\ \sum_{j=1}^{n}k_{j}+k=0\right\\}.$ (75)

Furthermore, the following wavefront sets for the delta, step and Wightman
two-point distributions turn to be useful for our problem [34]

$\displaystyle WF(\delta(x))$ $\displaystyle=\\{(x;k)|x=0,\ k\neq 0\\}$ (76)
$\displaystyle WF(\theta(x))$ $\displaystyle=\\{(x;k)|x=0,\ k\neq 0\\}$ (77)
$\displaystyle WF(\Delta_{m}^{+}(x))$
$\displaystyle=\\{(x;k)|k_{0}=|\vec{k}|,\ x^{0}=\lambda k_{0},\
\vec{x}=-\lambda\vec{k},\ \lambda\in\mathbb{R}\\}.$ (78)

Finally, for the definitions (48) and (49) for the Fourier transform of the
non-local kernel $\Gamma_{n}$ which defines the interaction (53), it is
possible to extract information about the structure of the singularities. Let
us define the algebraic projective varieties

$K_{\pm}=\left\\{(k^{1},\ldots,k^{n})\in\mathbb{R}^{4n}|\sum_{1\leq j<l\leq
n}\left(k_{0}^{j}k_{i}^{l}-k_{0}^{l}k_{i}^{j}\pm\epsilon^{imn}k^{j}_{m}k^{l}_{n}\right)=0,\
i=1,2,3\right\\},$ (79)

both of codimension 3 in $\mathbb{R}^{4n}$. The Fourier transform
$\Lambda_{n}$ in (48) is a non-zero constant for any point in
$K_{0}:=K_{+}\cup K_{-}$ and decays at least as $k^{-2}$ for any other
direction outside $K_{0}$. These later directions correspond to a behavior of
a continuous function, and can be avoided, for example by

$\begin{split}\Lambda_{n}&=\Lambda_{n}^{(c)}+\Lambda_{n}^{(\delta)}\\\
\Lambda_{n}^{(\delta)}&(\underline{k})=e^{-\beta_{+}^{2}-\beta_{-}^{2}}\\\
\Lambda_{n}^{(c)}&=\Lambda_{n}-\Lambda_{n}^{(\delta)}.\end{split}$ (80)

$\Lambda_{n}^{(\delta)}$ has a $\delta$-like singularity at directions in
Fourier space in $K_{0}$. The distribution $\Gamma^{(\delta)}_{n}$, whose
Fourier transform is $\Lambda_{n}^{(\delta)}$, has a wavefront set

$WF(\Gamma_{n}^{(\delta)})\subseteq\left\\{(\underline{x};\underline{k})|(\underline{x})\in
K_{0},\ (\underline{x})-\lambda(\underline{k})\in K_{0}\
\forall\lambda\in\mathbb{R}\right\\}$ (81)

##  6 Conclusion and perspectives

The $\phi^{n}$ interacting scalar field presented in [11] has been described
in the framework of pAQFT as a non-local effective theory and the set of
modified Feynman rules has been obtained. The effects of noncommutativity of
the spacetime are encoded in the non-local distributions $\Gamma_{n}$. The
structure of their singularities has been described using techniques of
microlocal analysis. Each distribution $\Gamma_{n}$ is expressed as the sum of
a continuous function plus a term that carries a $\delta$-type singularity,
whose wavefront set can be described explicitly, up to the solution of a
system of second order algebraic equations (geometrically given by a
projective variety).

In contrast to the result of [15], where the effective non-local QFT turns up
to be finite at any order in perturbation thery; our approach to pAQFT based
on the original work of Doplicher et. al. [11] keeps some non-trivial
singularities which have to be treated using some renormalization scheme (e.g.
BPHZ or Epstein-Glaser). These issues will be explored in a forthcoming
publication. A treatment at finite temperature or even a generalization to
curved backgrounds appears to be accessible in the presented framework.

## Acknowledgments

A.F. Reyes-Lega acknowledges financial support from the Faculty of Sciences of
Universidad de los Andes through project No. INV-2021-128-2324. J. F. López
acknowledges financial support from the Faculty of Sciences of Universidad de
los Andes through project No. INV-2022-144-2493.
Preprint of an article submitted for consideration in Reviewsin Mathematical
Physics © 2023 [copyright World Scientific Publishing Company]
https://www.worldscientific.com/worldscinet/rmp.

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