text:
\section{Introduction}
A quantum gravity theory (QGT), i.e. a theory encompassing quantum field theory (QFT) and general relativity (GR), has been looked for during several decades.
The (probably) simplest trial, a quantum theory of gravitation where the mediation of the interaction is carried out by the graviton, a spin-2 particle, has lead to several inconveniences~\cite{Goroff:1985th,Goroff:1985sz}, which ended up in its understanding as an effective field theory~\cite{Donoghue:1994dn}.
In spite of the success of the latter, there remain many subtleties, for example those related to the renormalization process and the gauge-independence of the computations~\cite{Giacchini:2020dhv,Giacchini:2020zrl,Solodukhin:2020vuw}.

\iffalse
Of course, one can consider a quantum theory of gravitation where the mediation of the interaction is carried out by the graviton, a spin-2 particle, leading to Einstein's equations~\cite{Donoghue:1994dn,Feynman:1996kb}.
The problem of this approach, from a fundamentalist point of view,
is that this theory is not renormalizable~\cite{Goroff:1985th,Goroff:1985sz} and thus it would give well-defined predictions only for
energies below the Planck scale.
Even at this level there remain some subtleties, for example those related to the renormalization and gauge-independence of the computations tackled recently in~\cite{Giacchini:2020dhv,Giacchini:2020zrl,Solodukhin:2020vuw}.
 \fi

Searching for a more fundamental theory and  principles, the community has developed several theoretical frameworks, such as string theory~\cite{Mukhi:2011zz,Aharony:1999ks,Dienes:1996du}, loop quantum gravity~\cite{Sahlmann:2010zf,Dupuis:2012yw}, causal dynamical triangulations \cite{Loll_2019}, causal set theory~\cite{Wallden:2013kka,Wallden:2010sh,Henson:2006kf} and functional renormalization group~\cite{Bonanno:2020bil}.  In (almost) all of them a minimum length arises~\cite{Gross:1987ar,Amati:1988tn,Garay1995},  which is heuristically associated with the Planck length $\ell_P \sim 1.6\times 10^{-33}$\,cm (or Planck mass $M_P \sim 1.22\times 10^{19}$\,GeV.) This small length (high-energy scale) is expected to somehow separate the regime where spacetime displays its classical nature from the one where it develops a quantum structure.

One can naturally guess that all of the above-mentioned theories should lead to novel scenarios with possible observable implications.
However, the description of the latter is generally arduous. In order to enhance the connection with the observational side,
another way of thinking has arisen not long ago: instead of considering a fundamental theory of quantum gravity, one can explore a low-energy limit of it, leading indeed to testable phenomenology~\cite{Addazi:2021xuf}.

A possible way to follow this bottom-up approach is through a modification of the special relativistic kinematics by introducing a high-energy scale.
There are two different possibilities of doing this,
depending on which is the fate of Lorentz symmetry: for high energies,
one can consider that it is either violated or deformed. The former scenario is implemented in the framework of Lorentz invariance violation (LIV)~\cite{Colladay:1998fq,Kostelecky:2008ts,Mattingly:2005re,Liberati:2013xla}, while the latter is inherent to Doubly (or Deformed) Special Relativity (DSR) theories~\cite{AmelinoCamelia:2008qg}. An immediate consequence is that the relativity principle characterizing Special Relativity (SR) is lost in LIV, while simply deformed in DSR.

\iffalse
In LIV scenarios, the main feature consists in the introduction of a  modified dispersion relation where new terms, proportional to an arbitrary high-energy scale, are added to the usual quadratic expression. In DSR theories one may have this this ingredient as well; in addition one usually finds a non-trivial composition law for the momenta (of interacting particles) and,  in order to save the relativity principle,  deformed Lorentz transformations for the one- and two-particle systems~\cite{KowalskiGlikman:2002ft}. The fact is that, while the dispersion relation can be undeformed in DSR, i.e. it may have the same expression as in SR (see for example the so-called ``classical basis'' of $\kappa$-Poincaré~\cite{Borowiec2010}), the composition law can never be reduced to the trivial sum. This is the reason why the latter is considered the crucial ingredient of kinematics in DSR.
\fi

Both ideas have been extensively developed from  an algebraic approach.
Indeed, the original proposal by Snyder
 consisted in a deformed algebra of position and momentum operators, in an attempt to regularize ultraviolet divergences arising in QFT~\cite{Snyder:1946qz,Snyder:1947nq}.
\iffalse
Other related models were constructed afterwards;
the bibliography is vast, see for example the reviews~\cite{Douglas:2001ba, Szabo:2001kg} and references therein.
One important fact is that the most diffused case corresponds to the so-called Moyal noncommutativity.
This case corresponds to a flat momentum space and a nontrivial addition of momenta, leading to a LIV scenario.
More relevant to our work is the Lorentz invariant situation, which (paradoxically, taking into account Snyder's first proposal) has been less studied in the literature.
\fi
More recently, after the introduction of the  $q$-deformed Poincaré algebra through a  Drinfeld--Jimbo deformation~\cite{Lukierski:1991pn}, several works have been devoted to understand $\kappa$-Minkowski and Snyder algebras \cite{Kosinski:2001ii,Kowalski-Glikman:2003qjp, Mignemi:2008kn,Govindarajan:2009wt, Mignemi:2011gr,Poulain:2018two,Arzano:2020jro,Lizzi:2021rlb},
including also proposals on how to obtain a Dirac equation~\cite{Lukierski:1992dt, Giller:1992xg, Nowicki:1992if, Agostini:2002yd, DAndrea:2005hjg}.

In this article we will employ a less developed, geometric point of view,
to discuss generalizations of the  Klein--Gordon and Dirac equations,
what may be appreciated by readers as an efficient and intuitive alternative. Recently,  a clear connection between the geometry in momentum space and noncommutative physics was shown to exist~\cite{Carmona:2021gbg,Relancio:2021ahm,Wagner:2021bqz}. This idea was already latent in the original paper of Born~\cite{Born:1938}, where a first proposal of curved momentum space was made and a lattice structure for spacetime was shown to arise.
These considerations are englobed in the broader context
of  velocity or momentum dependent spacetime, the so-called Finsler and Hamilton geometries~\cite{miron2001geometry}. Among others, there are several works in LIV setups describing the propagation of particles with a modified dispersion relation through a Finsler spacetime~\cite{Kostelecky:2011qz,Barcelo:2001cp,Weinfurtner:2006wt,Hasse:2019zqi,Stavrinos:2016xyg}. Also in DSR scenarios there exist contributions which consider a velocity~\cite{Girelli:2006fw,Amelino-Camelia:2014rga,Letizia:2016lew} and momentum~\cite{Barcaroli:2016yrl,Barcaroli:2015xda,Barcaroli:2017gvg} dependent geometries, taking into account a deformed dispersion relation.

Our construction below is greatly motivated by the results in~\cite{Carmona:2019fwf}, where it was shown that all the ingredients of a relativistic deformed kinematics (both a deformed dispersion relation and a composition law) can be singled out in the case of a maximally symmetric (curved) momentum space. In particular,  $\kappa$-Poincaré kinematics\footnote{When in the following we mention $\kappa$-Poincaré we mean only the kinematical momentum sector associated to the deformation, i.e., we do not consider the noncommutative  spacetime coordinates appearing in the construction of such model but only the deformed composition law, the dispersion relation and Lorentz transformations.} can be obtained from a de Sitter space by identifying the deformed composition law, the deformed Lorentz transformations and the deformed dispersion relation: they correspond respectively to the translation isometries,  the Lorentz isometries  and the squared distance to the origin.
Notice that the last two facts have been  previously contemplated in Refs.~\cite{AmelinoCamelia:2011bm,Lobo:2016blj}. Also, it is important to keep in mind that different bases of $\kappa$-Poincaré~\cite{KowalskiGlikman:2002ft} can be obtained from different choices of momentum coordinates in a de Sitter space~\cite{Carmona:2019fwf}.

In order to discuss the modified deformed relativistic wave equations in a geometrical framework, we organize our  article as follows. In Sec.~\ref{sc:KG} we will show how one can introduce the Klein--Gordon equation in curved momentum space, considering also some formal aspects regarding the definition of the corresponding Hilbert space. Then, in Sec.~\ref{sec:Dirac} we generalize the technique to a fermionic system of spin one-half. In both cases we show that we reproduce  firmly set results previously obtained within the scheme of Hopf algebras;
additionally, we show that one can implement a variational principle in momentum space by defining appropriate actions.
We discuss the discrete fermionic symmetries in Sec.~\ref{sec:discrete}.
Finally, we state our conclusions in
Sec.~\ref{sec:conclusions}. Additional results regarding the geometry of the so-called $\kappa$-Poincaré's symmetric basis are included in  App.~\ref{app:symmetric basis}.

We use the following conventions. We define the Minkowski metric ($\eta_{\mu\nu}$) with mostly minus signs; all other metrics will possess the same signature. Greek indices  are used to label  spacetime components of a tensor ($\mu,\,\nu,\,\cdots=0,1,2,3$), while latin indices  denote just spatial components ($i,\,j,\,\cdots=1,2,3$). The first latin characters ($a,\,b,\,\cdots=0,1,2,3$) are employed for components in the local orthonormal frame given by the (inverse) vierbein $e_{\mu}{}^a$. Regarding momenta, i.e. coordinates, we use the following notation: we denote  $p^2:=p_\mu \eta^{\mu\nu}p_\nu$; the set of all spatial components of a vector $p$ is written as $\vec{p}$ and $\vec{p}^2:= \vec{p}\cdot \vec{p}$. We use units  $\hbar=c= 1$.

\section{Klein--Gordon equation}\label{sc:KG}
The fact that a nontrivial momentum metric  is able to describe the kinematics of DSR has been thoroughly discussed in~\cite{Carmona:2019fwf}.
Following those lines, in this section we propose a generalization of the usual   Klein--Gordon (KG) equation  to the case in which the metric is momentum dependent.
Let us first recall some basic aspects of the usual case.
In SR, the KG operator acts on a wave function $\phi$ as
\begin{equation}
   \left(\eta^{\mu \nu} \frac{\partial}{\partial x^\nu}\frac{\partial}{\partial x^\mu} +m^2\right) \phi(x)\,=\,0\,,
   \label{eq:KG_SR}
\end{equation}
where $m$ is the mass of the particle and $\eta_{\mu\nu}$
is the metric in Minkowski spacetime.
Since its construction relies in the principle of relativity, this equation is devised as invariant under Poincaré symmetries. In addition it is also invariant under general diffeomorphisms in space, which is clear once one recognizes the Beltrami--Laplace operator.
The most general solution to Eq. \eqref{eq:KG_SR} can be written in terms of plane-wave solutions. Transforming to Fourier space we have
\begin{equation}
   \phi(x)\,=\,\frac{\sqrt{2}}{(2\pi)^{3}}\int  {\rm d}^4 p\, e^{{\rm i} x^\lambda p_\lambda}
   \tilde \phi(p)\, \delta(C_{\rm M}(p)-m^2)
   \,,
   \label{eq:KG_field}
\end{equation}
where we have defined  the  usual Casimir,
\begin{equation}\label{eq:casimir_M}
  C_{\rm M}(p)\,:=\,p^2\,=\,p_\mu \eta^{\mu\nu}p_\nu\,,
\end{equation}
 and
 the Dirac delta function enforces the on-shell condition (or dispersion relation) to be satisfied. Some additional overall factors where chosen to simplify the notation later.
Alternatively, the KG equation can be seen as an algebraic equation in momentum space,
\begin{align}
    (C_{\rm M}(p)-m^2) \tilde \phi (p)=0.
\end{align}

Once we introduce a nontrivial pseudo-Riemannian metric $g_{\mu\nu} (p)$ which depends on the momentum, we are constrained to deform the Casimir in a consistent way. In~\cite{Relancio:2020zok} it was shown that, defining the Casimir as  the pseudo-squared distance  to the origin ($p^*$) in momentum space, which we will denote by $ C_\text{D}(p)$, the following relation between the metric and the Casimir holds
  \begin{align}
 C_\text{D}(p)\,&=\,f^{\mu} g_{\mu \nu }(p) f^{\nu}\,
\label{eq:casimir_metric},
\\
f^{\mu}  (p)\,:&=\,\frac{1}{2} \frac{\partial C_\text{D}(p)}{\partial p_\mu}\,.
   \label{eq:f_definition}
  \end{align}
Notice that our choice for $C_{\rm D}(p)$ can be understood in terms of Synge's world function $\sigma(p',p)$~\cite{Synge:1960ueh}, which is a biscalar equal to one half of the square of the (geodesic) distance between $p'$ and $p$.
Under this interpretation, we are fixing the first argument of $\sigma$ to be $p^*$.
Moreover,
the  functions $f^{\mu}$ are the covariant derivatives of Synge's function, $\sigma^{,\mu}$, and will be called generalized momenta because of their role in generating the  Casimir in Eq.~\eqref{eq:casimir_metric} (see also the following sections).
Note thus that this discussion is valid for any  reasonable pseudo-Riemannian metric, as has been shown for example in~\cite{Synge:1960ueh, DeWitt:1965}.

\iffalse
\blue{Indeed, this expression can be derived following the result of~\cite{Bhattacharya2012RelationshipBG}
\begin{align}
    \frac{\partial}{\partial p_\mu(\tau_1)}D[p(\tau_0),p(\tau_1)] \,=\, \frac{\dot p_\mu(\tau_1) g^{\mu\nu}(p(\tau_1))}{\sqrt{g^{\rho\sigma}(p(\tau_1)) \dot p_\rho(\tau_1) \dot p_\sigma(\tau_1)}}\,,
\end{align}
where $p(\tau_0),p(\tau_1)$ are the initial and final points of a given geodesic, and $D$ its geodesical distance. From this one can find, when considering $p(\tau_0)=0$ and $p(\tau_1) = k$,
 \begin{equation}
\frac{\partial D[0,k]}{\partial k_ \mu} g_{\mu\nu}(k) \frac{\partial D[0,k]}{\partial k_ \nu}\,=\,1\,.
\end{equation}
If we now consider the squared of this distance we can easily arrive to \eqref{eq:casimir_metric}. }
\fi

In SR, i.e. considering a flat metric, expression \eqref{eq:casimir_metric} can be trivially shown to be satisfied by the Casimir defined in Eq. \eqref{eq:casimir_M}.
Instead, in DSR, when regarding the Casimir as the squared of the distance in momentum space,  $f^\mu$ will be nontrivial functions of the momenta.
This is the case also in the so-called classical basis of $\kappa$-Poincaré: even if in the algebraic context it is considered that the Casimir in this basis is the one of SR~\cite{Borowiec2010}, one can easily see that the momentum metric describing this kinematics leads  to a nontrivial (squared) distance in momentum space~\cite{Pfeifer:2021tas}.

In the classical  basis of $\kappa$-Poincaré, the distance $C_{\rm D}(p)$ turns out to be a function of $C_{\rm M}$. In an Euclidean setup, where both are positive, one can invert the relation to show that $C_{\rm M}$ may also play the role of a Casimir.
In the Minkowskian case, one can appeal to a Wick rotation to establish the same conclusion.
These different Casimirs then become equivalent (at least at the classical level) to a joint redefinition of all the particles' masses. However, by regarding the Casimir as a propagator in QFT, we expect a different behaviour of the theory in the ultraviolet (UV) regime, i.e. for large momenta.
One important argument that favours our choice ($C_{\rm D}$) is that, as we will see, then the Klein--Gordon operator equals the ``squared'' Dirac one, without the need of introducing any extra function of the mass.

We may now proceed in a canonical way,
meaning that the generalized  KG equation should enforce the dispersion relation given by the generalized Casimir in Eq. \eqref{eq:casimir_metric}.
The appropriate definition is thus seen to be\footnote{From now on we drop the tilde over $\phi$, since the basic wave function will be defined in momentum space. }
\begin{equation}
\left(  f^\mu ({p})g_{\mu \nu} ({p})  f^\nu ({p})  -m^2\right) \phi(p)\,=\,0\,.
   \label{eq:KG_DSR}
\end{equation}
The case of SR is simply recovered by considering a flat metric and  the undeformed Casimir $C_{\rm M}$.

More formally, from a quantum mechanical perspective, expression \eqref{eq:KG_DSR}
should be thought as a representation of the KG equation in momentum space. This entails considering a construction of momentum eigenstates in curved momentum space, a vision dual to  DeWitt idea's of nonrelativistic quantum mechanics in curved configuration space \cite{DeWitt:1952js, DeWitt:1957at}.
The fact that we want to consider the relativistic case is by far nontrivial; we will offer a more detailed discussion of these issues in Sec. \ref{sec:hilbert}.

By construction, Eq. \eqref{eq:KG_DSR} is well-defined under the action of diffeomorphisms in momentum space (leading to different basis bases of the Hopf algebra structure), since the Casimir has been defined as a distance.
Moreover, Eq. \eqref{eq:KG_DSR}  satisfies a deformed Lorentz invariance.
Indeed, in the context of (classical, i.e. not quantum) DSR, there exists a notion, albeit deformed, of Lorentz transformations~\cite{AmelinoCamelia:2008qg}. The action of the latter is such that the Casimir is left invariant and the metric transforms appropriately as a (2,0)-tensor~\cite{Carmona:2019fwf}.
Once we assume that the field transforms as a scalar under diffeomorphisms $p\to p^\prime$,
i.e.
 \begin{equation}
\phi^\prime (p^\prime)\,=\,\phi(p)\,,
   \end{equation}
the action of the quantum generators of Lorentz invariance in momentum space can be seen to be given by simple multiplication with their classical expression as given in~\cite{Carmona:2019fwf}.

Contrary to what happens in SR, where the Lorentz generators $J_{\mu\nu}$ are linear functions of $p$, in DSR the Lorentz transformations are generally nonlinear. However, the  equation
\begin{equation}
\label{eq:casimir_diffeo}
C_{\rm D}(p)\,=\,C_{\rm D}(p')
\end{equation}
holds if $p$ and $p'$ are connected through a deformed Lorentz transformation. This shows that our proposal for the KG equation is invariant under  Lorentz transformations which are the isometries of the metric defining the kinematics of DSR.

\subsection{Klein--Gordon equation in the symmetric basis of  \texorpdfstring{$\kappa$}{k}-Poincaré}
As an example of the above-derived equations, we will write the KG equation in the symmetric basis of $\kappa$-Poincaré. Using the formalism developed in \cite{Carmona:2019fwf} one can obtain all the relevant geometric quantities; more details on these computations can be found in App. \ref{app:symmetric basis}.
 Then, the distance in momentum space can be readily obtained by employing Eq.~\eqref{eq:casimir_metric} in conjunction with\footnote{In a Casimir,  superscripts denote the choice of coordinates, while subscripts refer to the chosen Casimir. For example, the subscript A will label the Casimir considered in the algebraic approach~\cite{Lukierski:1992dt},
 while subscript D refers to the squared distance in momentum space, cf. \eqref{eq:casimir_metric}.}~\eqref{eq:metric_alg}:
\begin{equation}
C^{{(S)}}_{\rm D}(p)\,=\, \Lambda^2 \arcch^2 \left(\cosh \left(\frac{p_0}{\Lambda}\right) -\frac{\vec{p}^2}{2 \Lambda^2}\right)\,.
\label{eq:cas_alg_dis}
\end{equation}

One can compare this result with the proposal  in~\cite{Lukierski:1992dt}, which was obtained in an algebraic context. From their definition of Casimir, the associated KG equation is found to be
\begin{equation}
\left(  C^{(S)}_\text{A}(p)  -m^2  \right) \phi(p)\,:=\,\left(  \left(2 \Lambda \sinh \left(\frac{p_0}{2\Lambda}\right)\right)^2 - \vec{p}^2  -m^2  \right) \phi(p)\,.
\label{eq:kg_alg}
\end{equation}
Evidently, Eq. \eqref{eq:kg_alg}  differs from our proposal \eqref{eq:cas_alg_dis} of constructing the KG equation from the squared distance in momentum space. However, similar to the situation discussed in the previous subsection, one can  indeed see that the following relation between them exists:
\begin{align}
C^{(S)}_{\text{D}}(p)\,=\,\Lambda^2 \arcch^2\left(1 +\frac{C_{\text{A}}(p)}{2 \Lambda^2}\right)\,.
\end{align}
As commented previously, this difference in the definition of the Casimir should be appreciable in the UV regime after the application of a quantization procedure.  Additionally, as will be seen in Sec. \ref{sec:dirac_symmetric_basis}, our proposal provides a direct link between Dirac and KG equations.

\subsection{An action for scalar fields}\label{sec:action_KG}

Assuming a principle of stationary action, the KG equation~\eqref{eq:KG_DSR} can be obtained from the action
\begin{align}\label{eq:KG_action}
S_{\rm KG}\,:&=\,\int {\rm d}^4p\, \sqrt{-g} \,\phi^*(p) \left(  C_{\rm D}(p)  -m^2  \right)  \phi(p)
\,.
   \end{align}
The factor $\sqrt{-g}$ guarantees invariance under a change of momentum basis.
This action may serve as a departing point to construct a free quantum field theory,
as has been  proposed in the work by Gol'fand \cite{Golfand:1959vqx, Golfand:1962kjf}, and refined by Mir-Kasimov
\cite{Mir-Kasimov:1966a, Mir-Kasimov:1966b, Mir-Kasimov:1967} and others \cite{Kadyshevsky:1983yc,Donkov:1984fj}.
Indeed, one can first define the theory in a Riemannian momentum space, where the action is positive defined, and afterwards employ an analogue of the Wick-rotation to consider pseudo-Riemannian spaces \cite{Golfand:1962kjf}.

Although this theory would share many features with the proposal of Gol'fand, there is a fundamental difference: we propose the Casimir as a distance in momentum space, whereas in \cite{Golfand:1962kjf} just $p^2$ in the Beltrami chart~\cite{Pascu:2012yu} is employed (which does not correspond to the squared distance in that coordinates).
Notice that, as commented previously, even if we would work with the same coordinates, the models are expected to differ in the UV sector because of the different choice of the Casimir. Moreover, as we will see, our choice of Casimir is the only one allowing to make a direct identification between the KG and Dirac equations\footnote{Actually, the proposal in~\cite{Donkov:1984fj} can be ``squared'' but just to obtain one component of the KG operator in~\cite{Kadyshevsky:1983yc}: the latter proposes a separation of the field into two components, one which vanishes on the mass shell and one that doesn't.}.

A subtle point arises when one tries to introduce interactions. Many proposals of QFT in $\kappa$-Minkowski and Snyder space have been done in the literature~\cite{Dimitrijevic:2003wv, Freidel:2007hk, Meljanac:2010ps, Girelli:2010wi, Meljanac:2011cs,
Meljanac:2017grw, Meljanac:2017ikx,Mercati:2018hlc, Franchino-Vinas:2018jcs, Arzano:2020jro},
most of them relying on the defnition of an appropriate Moyal--Groenewold product (also called star-product) \cite{Groenewold:1946kp, Meljanac:2010et, Mercati:2010qhd, Meljanac:2021qgq}.
The latter encodes the nontrivial addition of momenta and, generally speaking, can be built in a case by case analysis.
In our case, the adoption of the translation operators \cite{Carmona:2019fwf} seems better suited. We will leave a concrete analysis to a future publication.

\subsection{On the definition of a Hilbert space}\label{sec:hilbert}
As is well-known, the formal construction and interpretation of the KG theory poses several problems.
One is able to build a conserved current in SR, $J_{\mu}^{\rm KG}:= {{\rm i}} \psi^*(x) \overleftrightarrow{\partial}_{\mu} \psi(x)$, where $\overleftrightarrow{\partial}_{\mu}=\overleftarrow{\partial}_{\mu}-\overrightarrow{\partial}_{\mu}$, and even extend its zero component to define a sesquilinear form
\begin{align}\label{eq:scalar_KG}
    (\phi,\psi)_{\rm KG}\,:= \, {\rm i} \int {\rm d}^3x \big[\partial_t \phi^*(x) \psi(x)-   \phi^*(x) \partial_t\psi(x)\big]\,,
\end{align}
which is conserved in time when $\psi$ and $\phi$ are solutions of the KG equation and sufficiently well-behaved. However, an issue exists, inasmuch as \eqref{eq:scalar_KG} is not positive definite, leading to a failure in its interpretation  as a density and in the definition of a scalar product.
The way around it, whose generalization enables the construction of QFT in curved spaces, has been to understand that Eq. \eqref{eq:scalar_KG} indeed corresponds to a scalar product if one restricts to positive-energy modes \cite{Wald:1995yp}. This assertion can be immediately seen if one writes a solution $\phi$ of the KG equation in Fourier space;
employing the delta function in \eqref{eq:KG_field} to perform the integral in $p_0$ we obtain
\begin{align}\label{eq:phi_fourier}
    \phi(x)\,=\, \frac{1}{\sqrt{2}(2\pi)^3}\int \frac{{\rm d}^3p}{\freq} \left(e^{{\rm i} (t \omega_p + \vec{x}\cdot\vec{p}) }\phi_+(\vec{p}) +e^{-{\rm i} (t \omega_p+\vec{x}\cdot\vec{p}) }\phi_-(\vec{p})\right)\,,
\end{align}
where $\omega_p:=\sqrt{\vec{p}^2+m^2}$ is the positive solution one obtains for $p_0$ by enforcing the dispersion relation, while the two  distinct  components $\phi_{\pm}:=\tilde \phi(\pm \omega_p,\vec{p})$ arise because of  the disconnected support of the delta function in  \eqref{eq:KG_field}. Additionally,  in \eqref{eq:phi_fourier} one can recognize the Lorentz invariant measure.
 Using this expansion one may recast the scalar product as
\begin{align}\label{eq:scalar_KG_Fourier}
    (\phi,\psi)_{\rm  KG}\,=\,\int \frac{{\rm d}^3p}{\freq} \big[ \phi_+^*(p) \psi_+(p) -\phi_-^*(p) \psi_-(p)\big]\,,
\end{align}
from which the positive and negative part of the scalar product are evident.

A generalization of Eq. \eqref{eq:phi_fourier} has been developed in~\cite{Arzano:2010jw,Mercati:2018hlc}.
In particular, Ref. \cite{Mercati:2018hlc} includes a term which is proportional to the square root of the product $f^\mu(p) g_{\mu \nu}(p)f^\nu(p)$, which is nothing but our Casimir. Therefore, this term can be easily reabsorbed in a constant.

The fact that an extension of \eqref{eq:scalar_KG_Fourier} may be important also in the construction of a QFT in curved momentum space has been acknowledged in~\cite{Arzano:2010jw}. In order to generalize expression \eqref{eq:scalar_KG_Fourier} in our setup, we need to introduce two changes. First, we rewrite it in the space of four-momenta, including the measure factor and a Dirac delta function that guarantees the satisfaction of Eq. \eqref{eq:KG_DSR}.  A further step is to notice that one can introduce a time-like vector $t_\mu=(1,0,0,0)$. Summing  these elements and given sufficiently well-behaved functions $\phi,\psi\in L^2_{}({\rm dS}_{4})$, i.e.  the space of square-integrable complex-valued functions over ${\rm dS}_4$,
we can define a positive Hermitian form
\begin{align}\label{eq:scalar_KG_curved}
    (\phi,\psi)_{ C_{\rm D}}\,:=\, 2\int  {\rm d}^4p \,\sqrt{-g} \, \delta\left(C_{\rm D}-m^2 \right) \,  \Theta(f^{\nu} t_{\nu})\, \phi^*(p) \psi(p) \,,
\end{align}
where $\Theta(\cdot)$ is the Heaviside function. It is important to notice that this scalar product is clearly invariant under diffeomorphisms in momentum space.
An alternative expression can be obtained using the Dirac delta to integrate over $p_0$; one then obtains
\begin{align}\label{eq:scalar_KG_curved_d3p}
    (\phi,\psi)_{ C_{\rm D}}\,=\, \sum_{i}\int  \frac{{\rm d}^3p}{|f^{\nu} t_{\nu}|} \,\sqrt{-g} \,   \Theta(f^{\nu} t_{\nu})\, \phi^*(p) \psi(p)\Big\vert_{p_0=\bar p_0^{(i)}} \,,
\end{align}
where $\bar p_0^{(i)}$ denotes the several solutions to the Casimir equation $C_{\rm D}(\bar p_0^{(i)},\vec{p})-m^2=0$.
Expression \eqref{eq:scalar_KG_curved_d3p} evidently  resembles the first term in the RHS of \eqref{eq:scalar_KG_Fourier} and
 suggests to proceed in  the following formal way:
one should consider smooth functions with local support on ${\rm dS}_{4}$ and define a projected subspace $\mathscr{S}$ by multiplying by $\Theta(f^{\nu} t_{\nu})$. This is the analogous of choosing the subspace of solutions with positive energy in the standard flat case. Then the Cauchy completion of $\mathscr{S}$ with the scalar product \eqref{eq:scalar_KG_curved} will define a Hilbert space, from which one may try to implement a quantization.

At this point two comments are in order. The description of $\kappa$-Poincaré kinematics customarily involves the introduction of a constant temporal vector\footnote{Strictly speaking, $\kappa$-Poincaré introduces an object which is not a vector under momentum diffeomorphisms. We use the name ``constant temporal vector'' because it is widely employed in this context. },
privileging thus the temporal component over the space coordinates, since only the boosts are modified but not the rotations~\cite{KowalskiGlikman:2002ft}.
This means that invariance under diffeomorphisms will be restricted
to a subset respecting the aforementioned splitting.
Notice also that the on-shell condition defines a foliation of spacetime with spacelike hyper-surfaces,
whose normal vectors are the generalized momenta $f^\mu$ (which are timelike). This implies that, even if in such case $t_\mu$ would  not transform as a vector under momentum diffeomorphisms, the term $\Theta(f^{\nu} t_{\nu})$ in Eq. \eqref{eq:scalar_KG_curved} would indeed be invariant in the aforementioned restricted sense.

Additionally, as discussed in several papers (see~\cite{Mercati:2018hlc} and references therein), the kinematics of  $\kappa$-Poincaré can be described in just one-half of de Sitter momentum space. Moreover, this half is not closed under the action of Lorentz transformations.
This restriction is not necessary  in our current discussion, but may be implemented in future discussions involving interactions.

\section{Dirac equation}\label{sec:Dirac}
Now that we have illustrated our ideas on how the geometrical approach works in the scalar case,
a generalization to Dirac particles of spin $1/2$ seems natural. Our proposal in DSR for the Dirac equation, in momentum space and as an algebraic equation, is given by
   \begin{equation}
 \left( \underline{\gamma}^\mu f_\mu({p})-m\right)\psi(p)\,=\,0\,,
 \label{eq:Dirac_DSR}
\end{equation}
where the functions $f_\mu$ are obtained from the covariant functions in Eq. \eqref{eq:f_definition} by a contraction with the inverse metric,
\begin{equation}
 f_\mu(p)\,:=\,g_{\mu\nu}(p)f^\nu(p)\,.
\end{equation}
The curved-momentum-space gamma matrices (with greek indices and underlined)
are defined in terms of the usual Dirac matrices $\gamma^a$ (with latin indices, with $a=0,1,2,3$), the latter providing a spin-$1/2$ finite-dimensional representation of the SO(3,1) group.
More concretely, we consider  the tetrad $ e^\mu{}_a(p)$ (or nonholonomic connection) in momentum space, from which the metric can be built
  \begin{equation}
g^{\mu\nu}(p) \,=:\,e^\mu{}_a (p)\eta ^{a b}e^\nu{}_b (p)\,,
\label{eq:metric_tetrad}
\end{equation}
so that the gamma matrices
can be written as
  \begin{equation}\label{eq:gamma_curved}
 \underline{\gamma}^\mu \,:=\,\gamma^a e^\mu{}_a({p})\,.
\end{equation}

As a consequence of the definition \eqref{eq:gamma_curved},
the gamma matrices share several properties with the gamma matrices in curved configuration space~\cite{Birrell:1982ix}. For example, it is immediate to see  that they satisfy a Clifford algebra using the momentum-space metric, i.e.
  \begin{equation}
\lbrace{ \underline{\gamma}^\mu, \underline{\gamma}^\nu\rbrace}_{}\,=\,2 g^{\mu \nu}(p)\mathbb{1}\,,
\end{equation}
where the operator $\lbrace{\cdot,\cdot\rbrace}_{}$ is the anti-commutator.
Another similarity with the curved configuration space is that,
if we multiply Eq.~\eqref{eq:Dirac_DSR} by the operator  $\left(  \underline{\gamma}^\nu f_\nu({p})+m\right)$, we reobtain the KG equation derived in the previous section, cf. Eq.~\eqref{eq:KG_DSR}. If instead of considering the Casimir as the squared of the distance in momentum space one chooses any other function of it, this fundamental relationship will not hold. This is for example the case of the Klein--Gordon and Dirac equations obtained in the algebraic approach~\cite{Nowicki:1992if};
a detailed comparison will be performed in  Sec. \ref{sec:dirac_symmetric_basis}.

We close this general discussion by noting that one can derive the modified Dirac  equation from an action principle in momentum space. Indeed, one can define the Dirac adjoint $\bar\psi:=\psi^\dagger \gamma^0$ and the following action,
\begin{align}
 S_{\rm Dirac}:&=\int  {\rm d}^4p\, \sqrt{-g} \bar\psi(-p)\left(  \underline{\gamma}^\mu f_\mu({p})-m\right)\psi(p)\,,
 \label{eq:Dirac_action}
\end{align}
from which the Dirac equation follows when looking for stationary configurations.

\subsection{Deformed Lorentz invariance of the Dirac equation}
A proof of the fact that the proposed Dirac equation is Lorentz invariant  can be done  in a way analog to the one  followed in curved configuration space. We will assume that the spinor  $\psi (p)$  transforms linearly (with a matrix $\mathcal{S}$) under a Lorentz transformation $p\to p^\prime$; by requiring Lorentz invariance, we will then see that $\mathcal{S}$ corresponds to a fermionic (Dirac) representation of the Lorentz group.
We start by applying $\mathcal{S}$ to Eq.~\eqref{eq:Dirac_DSR}, so that we obtain
   \begin{equation}
 \left(\mathcal{S}  \underline{\gamma}^\mu \mathcal{S}^{-1} f_\mu({p})-m\right)\mathcal{S}\psi(p)

 \,=\,
 \left(\mathcal{S} \gamma^a \mathcal{S}^{-1}  e^{\prime\rho}{}_b({p}^\prime) f'_\rho ({p}^\prime)-m\right)\psi^\prime (p^\prime)=0\,,
 \label{eq:Dirac_DSR_transformed}
\end{equation}
where we have used appropriate transformations for the vielbein and the functions $f^{\mu}$,
cf. the definition \eqref{eq:f_definition} (all the quantities in the new system of coordinates are denoted with a prime).

Then, since the proposed diffeomorphism is an isometry of the metric, the vielbein satisfies
\begin{equation}\label{eq:vierbein_isometry}
 e^{\prime\rho}{}_a(p^\prime)\,=\,  e^{\rho}{}_b(p') \Lambda^b{}_a (p^\prime) \,,
\end{equation}
where $ \Lambda^b{}_a (p^\prime)$ is a (local) Lorentz transformation that may depend on the point $p^\prime $.
Note that this transformation is not the deformed Lorentz transformation obtained from the isometries of the metric.
Considering \eqref{eq:vierbein_isometry} and the symmetry of $f_{\mu}$ (cf. the definition \eqref{eq:f_definition}), we obtain
\begin{equation}
 \Big(\mathcal{S} \gamma^a \mathcal{S}^{-1} \Lambda^b{}_a (p^\prime) e^{\rho}{}_b({p}^\prime)f_\rho ({p}^\prime)-m\Big)\psi^\prime (p^\prime)=0\,.
 \label{eq:Dirac_DSR_transformed2}
\end{equation}

Of course, local Lorentz  transformations near the identity can be expanded in terms of antisymmetric parameters $\epsilon_{ab}:=\eta_{ac} \epsilon^c{}_b$
as customarily,
\begin{equation}
\Lambda^b{}_a (p^\prime) \,=\,\delta^b_a+\epsilon^b{}_a (p^\prime)+\cdots\,. \label{eq:lorentz_tetrad}
\end{equation}
Thus, we can also expand the matrix $\mathcal{S}$ in Eq.~\eqref{eq:Dirac_DSR_transformed} for transformations around the identity; this  allows us to determine the infinitesimal form of $\mathcal{S}$ as a function of the Lorentz coefficients $\epsilon_{ab}(p')$:
   \begin{align}
\mathcal{S}\,&=\, \mathbb{1}-\frac{{\rm i}}{4} \sigma^{ab} \epsilon_{ab}(p^\prime)+\cdots \,,
 \label{eq:transformation_S}
\\
 \sigma^{ab}\,:&=
 \,\frac{{\rm i}}{2}\left[\gamma^a,\gamma^b\right]\,.
 \label{eq:sigma}
\end{align}

\begin{comment}
, but simply implies
\begin{equation}
 e^{\prime\rho}_a(p^\prime)\eta^{ab} e^{\prime\sigma}_b(p^\prime)\,=\, e^{\rho}_a(p^\prime)\eta^{ab} e^{\sigma}_b(p^\prime) \,.
\end{equation}
\end{comment}

 \begin{comment}
Then, since the proposed diffeomorphism is an isometry of the metric, the vielbein satisfies
\begin{equation}
 e^{\prime\rho}{}_a(p^\prime)\,=\,  e^{\rho}{}_b(p) \Lambda^b{}_a (p^\prime) \Big\vert_{p(p')}\,,
\end{equation}
where $ \Lambda^b{}_a (p^\prime)$ is a Lorentz transformation that may depend on the point $p^\prime $. For  transformations near the identity it can be expanded as
\begin{equation}
\Lambda^b{}_a (p^\prime) \,=\,\delta^b_a+\epsilon^b{}_a (p^\prime)+\cdots\,,
\end{equation}
being the parameters $\epsilon_{ab}:=\eta_{ac} \epsilon^c{}_b$ antisymmetric in their indices.
Therefore, this transformation satisfies
\begin{equation}
 e^{\prime\rho}{}_a(p^\prime)\eta^{ab} e^{\prime\sigma}{}_b(p^\prime)\,=\, e^{\rho}{}_a(p)\eta^{ab} e^{\sigma}{}_b(p)\Big\vert_{p(p')} \,.
\end{equation}
\end{comment}

\subsection{The Dirac equation in simple coordinates}
\label{ss:1}
To show the previously-described formalism in action, let us fix the coordinates and consider a particular example of the exact deformed Dirac equation, i.e. keeping all orders in the deformation parameter $\Lambda$.
In~\cite{Carmona:2019vsh} a really simple basis of $\kappa$-Poincaré kinematics was found, in which the composition law is linear in momenta
\begin{equation}
\left(p\oplus q\right)_\mu \,=\, p_{\mu} + \left(1 - p_0/\Lambda\right) \,q_{\mu}\,.
\end{equation}
The momentum metric with these isometries can be defined by the following inverse metric and inverse tetrad~\cite{Relancio:2021ahm}:
\begin{align}
g_{\mu \nu}(p)\,:&=\,\eta_{\mu\nu} \left(1-p_0/\Lambda\right)^2\,,\quad p_0<\Lambda\,,
\\
{e}_\mu{}^a(p)\,:&=\,\delta^a_\mu \left(1-p_0/\Lambda\right)\,.
\end{align}
Then, with a direct application of Eq.~\eqref{eq:casimir_metric}, we obtain the Casimir to be
\begin{align}
C_{\rm D}^{(simp)}(p)\,=\,\Lambda^2 \arcch^2\left(1+\frac{p^2}{2 \Lambda^2\left(1-p_0/\Lambda\right)}\right)\,,
\label{eq:cas1}
\end{align}
such that the Dirac equation in this basis reads
\begin{equation}
\label{eq:dirac_simple}
\frac{\Lambda \sqrt{ C_{\text{D}}^{(simp)}(p) }}{\sqrt{\Lambda^2p^2\left(\left(p_0-2\Lambda\right)^2 -\vec{p}^2\right)}} \Big[2 \Lambda\gamma^a \delta_a^{\mu} p_\mu - \gamma^0 (p_0^2+\vec{p}^2)-2 \gamma^i p_i p_0 \Big] \psi(p) \,=\, m\psi(p).
\end{equation}

In spite of the rather complex aspect of this equation, the solutions can be found in a straightforward way. Analogue to what happens in SR, this modified Dirac equation allows four different solutions, namely
\begin{align}
\begin{split}
\psi_1\,=
\begin{pmatrix}
1
\\
0
\\
-\frac{p_3}{p_0+\Lambda \left(e^{m/\Lambda}-1\right)}
\\
-\frac{p_1+{\rm i} p_2}{p_0+\Lambda \left(e^{m/\Lambda}-1\right)}
\end{pmatrix}
,\,
\psi_2\,=\begin{pmatrix}
0
\\
1
\\
-\frac{p_1-{\rm i} p_2}{p_0+\Lambda \left(e^{m/\Lambda}-1\right)}
\\
\frac{p_3}{p_0+\Lambda \left(e^{m/\Lambda}-1\right)}
\end{pmatrix}
,\,
\psi_3\,=\begin{pmatrix}
-\frac{p_3}{p_0+\Lambda \left(e^{-m/\Lambda}-1\right)}
\\
-\frac{p_1+{\rm i} p_2}{p_0+\Lambda \left(e^{-m/\Lambda}-1\right)}
\\
1
\\
0
\end{pmatrix}
,\,
\psi_4\,=\begin{pmatrix}
-\frac{p_1-{\rm i} p_2}{p_0+\Lambda \left(e^{-m/\Lambda}-1\right)}
\\
\frac{p_3}{p_0+\Lambda \left(e^{-m/\Lambda}-1\right)}
\\
0
\\
1\end{pmatrix}
\end{split}
,\end{align}
where the zeroth-component of the momentum can be obtained from the dispersion relation implied by Eq.~\eqref{eq:cas1}:
\begin{equation}\label{eq:p0_simple}
    p_0\,=\,\Lambda \left(1-\cosh\left(\frac{m}{\Lambda}\right)\left(1{\pm }\sqrt{1-\sech^2\left(\frac{m}{\Lambda}\right)\left(1-\frac{\vec{p}^2}{\Lambda^2}\right)}\right)\right).
\end{equation}
These expressions explicitly display how the deformation occurs in this fermionic case.
Indeed, one can see by expanding the two previous equations in $\Lambda$ that the deformation is proportional to $m/\Lambda$.  This is related to the explicit form of the Casimir in these coordinates, which leads to the  energy-momentum relation of SR for massless particles.
Thus, if we employ the Casimir \eqref{eq:cas1}, hadronic showers \cite{Addazi:2021xuf} (but not gamma ray bursts) may be used to set lower limits on $\Lambda$. This situation could be different for different basis of $\kappa$-Poincaré, i.e. different momentum coordinates.

\subsection{Dirac equation in the symmetric \texorpdfstring{$\kappa$}{k}-Poincaré basis} \label{sec:dirac_symmetric_basis}
In this part of the work we will compare our results with other proposals in the literature. With that aim, we  consider the symmetric $\kappa$-Poincaré basis (see  details in App. \ref{app:symmetric basis}); using our construction in Sec. \ref{sec:Dirac} we obtain the following Dirac operator:
\begin{equation} \label{eq:dirac_symmetric_basis}
\mathcal{D}_{\rm D}^{(S)}\,:=\,\frac{ \sqrt{\frac{C_{\text{D}}^{(S)}(p) }{\Lambda ^2}}}
{2\Lambda  \sinh \left(\sqrt{\frac{C_{\text{D}}^{(S)}(p)}{\Lambda ^2}}\right)}
\left[2 \Lambda  e^{-\frac{p_0}{2 \Lambda }} \gamma^i p_i+\gamma^0  \left(2 \Lambda ^2 \sinh \left(\frac{p_0}{\Lambda }\right)-\vec{p}^2\right)\right]\,.
\end{equation}
By construction, the Dirac equation \eqref{eq:dirac_simple} in the simple basis can be mapped to this one by a change of coordinates in momentum space.

Now we can consider the Dirac equation proposed in~\cite{Nowicki:1992if}, which was obtained by considering the standard real form of the quantum anti-de Sitter algebra, $SO_q(3,2)$. More precisely they have introduced a  finite-dimensional representation of $SO_q(3,2)$ and consistently modified the coproduct. In our language, it should consist in employing the symmetric $\kappa$-Poincaré basis with the Casimir given by expression~\eqref{eq:cas_alg_dis}.
It is important to emphasize that this is not simply a change of coordinates, but a different choice of Casimir.
When inserting Eqs.~\eqref{eq:cas_alg_dis} and \eqref{eq:tetrad_alg} into \eqref{eq:Dirac_DSR}, we find the Dirac operator
\begin{equation}
\mathcal{D}^{(S)}_\text{A} \,:=\, \gamma^0 \left(\Lambda \sinh \left(\frac{p_0}{\Lambda}\right)- \frac{\vec{p}^2}{2\Lambda} \right) + e^{-p_0/2\Lambda} p_i \gamma^i\,,
\label{eq:dirac_alg}
\end{equation}
which is indeed the same expression as that obtained in~\cite{Nowicki:1992if}.
Notice that from the definitions of the Casimirs of Eqs.~\eqref{eq:cas_alg_dis} and~\eqref{eq:kg_alg}, one can see that Eqs.~ \eqref{eq:dirac_symmetric_basis} and \eqref{eq:dirac_alg} are related. Basically,
\begin{equation}
\mathcal{D}^{(S)}_\text{A} \,=\, \underline{\gamma}^\mu g_{\mu \nu}(p) \frac{\partial C^{(S)}_\text{A}}{\partial p_\nu}\,=\, \underline{\gamma}^\mu g_{\mu \nu}(p) \frac{\partial C^{(S)}_\text{A}}{\partial C^{(S)}_\text{D}}\frac{\partial C^{(S)}_\text{D}}{\partial p_\nu}\,=\, \mathcal{D}^{(S)}_\text{D} \frac{\partial C^{(S)}_\text{A}}{\partial C^{(S)}_\text{D}}\,.
\end{equation}

It is important to note that the ``square'' of the Dirac operator in Eq. \ref{eq:dirac_alg} does not lead to the KG expression \eqref{eq:kg_alg}.
Indeed, as discussed in~\cite{Nowicki:1992if}, they are related by
\begin{equation}
\left(
\mathcal{D}^{(S)}_{\text{A}}\right)^2\,=\,C^{(S)}_\text{A}(p)\left(1+\frac{C^{(S)}_\text{A}(p)}{4\Lambda^2}\right)\,.
\end{equation}
Instead,  our proposal \eqref{eq:dirac_symmetric_basis} is such that the KG equation~\eqref{eq:cas_alg_dis} is directly obtained when squaring the Dirac one.

\subsection{Discrete symmetries}\label{sec:discrete}
  It is well-known that, in Minkowski space, the Dirac
  equation possesses invariance not only under  continuous Lorentz transformations but also under   discrete ones: parity and time reversal, which connect with the improper and non-orthochronous sectors of the Lorentz group,  and charge conjugation. The corresponding operators are\footnote{Recall that these definitions of time reversal and charge conjugation are valid in Dirac's basis of the gamma matrices.}
   \begin{align}
       \mathcal{P}_0:&\,=\,{\rm i} \gamma^0\,,
       \\
       \mathcal{T}_0:&\,=\,{\rm i} \gamma^1 \gamma^3 \mathcal{K}\,,
       \\
       \mathcal{C}_0:&\,=\, {\rm i} \gamma^2 \mathcal{K}\,,
   \end{align}
   where $\mathcal{K}$ denotes the conjugation operator.
     These operators are intended to act only at the spinor level, while  the coordinates change appropriately when performing a transformation, i.e.
   \begin{align}
       \psi_{\mathcal{A}}(x')\,=\, \mathcal{A} \psi (L_{\mathcal{A}} x)\,,
   \end{align}
   where $L_{\mathcal{A}}$ denotes the corresponding Lorentz transformation acting on the coordinates.
   Notice that, in the flat case, one can recast their action as operators in momentum space.
   A straightforward computation gives the transformed wave functions
   \begin{align}
        \tilde \psi_{\mathcal{P}}:&\,= \,{\rm i} \gamma^0 \tilde \psi(p_0,-\vec{p})\,,
    \\
    \tilde \psi_{\mathcal{T}}:&\,= \,{\rm i} \gamma^1\gamma^3 \tilde \psi^*(p_0,-\vec{p})\,\,,
       \\
       \tilde \psi_{\mathcal{C}}:&\,=\, {\rm i} \gamma^2 \tilde \psi^*(-p)\,,\label{eq:charge_conjugation}
   \end{align}
   from which the action of the relevant operators in momentum space may be deduced.
In obtaining these transformations, it is crucial that the involved change of coordinates are isometries of the momentum-space Minkowski metric.
   When going to curved momentum space, we can take these as definitions of the discrete symmetries' operators.

   One can thus show  that  $\mathcal{P}$, $\mathcal{T}$  are  discrete symmetries of Dirac's equation if the following condition is met
   \begin{align}\label{eq:condition_discrete_pt}
        e^{\mu}{}_a(p_0,-\vec{p}) f_{\mu}(p_0,-\vec{p})\,=\,-e^{\mu}{}_a(p) f_{\mu}(p)\,, \quad a\,=\,1,2,3\,.
   \end{align}
 The proof in that case is direct.
  As an example, for parity we can evaluate Eq. \eqref{eq:Dirac_DSR} in $(p_0,-\vec{p})$ and afterwards multiply by $\gamma^0$. Using the anticommuting properties of the gamma matrices one can then show that, if $\tilde \psi$ satisfies Dirac's equation  and condition \eqref{eq:condition_discrete_pt} is satisfied, then  $\tilde\psi_{\mathcal{P}}$ is also a solution of Eq. \eqref{eq:Dirac_DSR}.
  In what concerns the charge conjugation symmetry, since we have not introduced a coupling with electromagnetic fields, we cannot fully tackle the question. However, we can study the behaviour of the free part of Dirac's equation. To prove the invariance under $\mathcal{C}$,  we require
  the assumption
  \begin{align}\label{eq:condition_discrete_c}
       e^{\mu}{}_a(-{p}) f_{\mu}(-{p})\,=\,-e^{\mu}{}_a(p) f_{\mu}(p)\,, \quad a\,=\,0,1,2,3\,.
  \end{align}
  These results are in contrast with the findings in \cite{Andrade:2013oza},
  where the action of $\mathcal{C}$ and $\mathcal{T}$ was found to be incompatible with their modified Dirac equation.

 At a first sight, the emergence of  conditions \eqref{eq:condition_discrete_pt} and \eqref{eq:condition_discrete_c} may seem too strong. However, one should recall the analog situation in curved spacetime: not every geometry would admit the definition of such symmetries \cite{Hollands:2002rz},
  even if  a PCT theorem can be proved in more or less general terms \cite{Hollands:2002rz, Hollands:2009bke}.

In this sense, the situation can be understood as equivalent to the case when an external time-dependent electromagnetic field acting on a particle breaks $\mathcal{T}$ invariance.

In the same direction, a deeper thought shows that  both conditions \eqref{eq:condition_discrete_pt}-\eqref{eq:condition_discrete_c} are satisfied for rather general tetrads in the $\kappa$-Poincaré scenario. We start by noticing that the Casimir is by definition Lorentz invariant and must thus be a function of the rotational invariants $p_0$ and $p^2$ (or equivalently $\vec{p}^2$).
These are indeed rotational invariants  due to the fact that the time direction is privileged in $\kappa$-Poincaré kinematics, such that while boosts are usually deformed, rotations remain undeformed~\cite{KowalskiGlikman:2002ft}.
Alternatively, as discussed in~\cite{Carmona:2016obd}, one can describe the kinematics of $\kappa$-Poincaré introducing a fixed time-like vector\footnote{
Appearing in tensorial expressions, $n^{\mu}$ can be simply understood as a shorthand to introduce a deformation of usual Lorentz invariance, i.e. a special direction in spacetime.
} $n^\mu=(1,0,0,0)$.
If one wants to avoid the introduction of further elements and also recover the Cartesian SR result in the large $\Lambda$ limit, it is then always possible to recast the gradient of the Casimir ($f^\mu$) as
  \begin{equation}
f^\mu(p)\,=\,  p^\mu \bar f_1 \left(\frac{p_\alpha n^\alpha}{\Lambda},\frac{p^2}{\Lambda^2}\right)+n^\mu \Lambda  \bar f_2 \left(\frac{p_\alpha n^\alpha}{\Lambda},\frac{p^2}{\Lambda^2}\right)\,,
\label{eq:f_exp}
\end{equation}
  where $\bar f_1$ and $\bar f_2$ are dimensionless functions satisfying the following properties
    \begin{equation}
 \bar f_1 \left(0,0\right)\,=\,1\,,\qquad   \lim_{\Lambda\to \infty}\Lambda  \bar f_2 \left(\frac{p_\alpha n^\alpha}{\Lambda},\frac{p^2}{\Lambda^2}\right)\,=\,0\,.
\label{eq:f1f2}
\end{equation}
Notice that, in order to simplify the notation, we are employing the Minkowski metric to raise and lower indices of $p_\mu$ and $n^\mu$.

 We can also write the most general form of the momentum tetrad that respects rotational invariance:
    \begin{align}
    \begin{split}
{e^\mu}_a (p)\,&=\, \delta^\mu_a \bar e_1 \left(\frac{p_\alpha n^\alpha}{\Lambda},\frac{p^2}{\Lambda^2}\right)
+ \frac{ p^\mu n_a}{\Lambda} \bar  e_2 \left(\frac{p_\alpha n^\alpha}{\Lambda},\frac{p^2}{\Lambda^2}\right)
+ \frac{n^\mu p_a}{\Lambda} \bar  e_3 \left(\frac{p_\alpha n^\alpha}{\Lambda},\frac{p^2}{\Lambda^2}\right)
\\
&\hspace{4.5cm}+ \frac{p^\mu p_a}{\Lambda^2} \bar  e_4 \left(\frac{p_\alpha n^\alpha}{\Lambda},\frac{p^2}{\Lambda^2}\right)
+ n^\mu n_a  \bar  e_5 \left(\frac{p_\alpha n^\alpha}{\Lambda},\frac{p^2}{\Lambda^2}\right)\,.
\label{eq:e_exp}
\end{split}
\end{align}
Since in the limit $\Lambda \to \infty$ we want to recover the canonical tetrad in SR (${e^\mu}_a \to \delta^\mu_a$),
the coefficients $\bar e_i$ will satisfy conditions similar to those in \eqref{eq:f1f2}.
\iffalse
we need that
  \begin{equation}
 e_1 \left(0,0\right)\,=\,1\,,\qquad  e_5 \left(0,0\right)\,=\,0\,.
\label{eq:e1e5}
\end{equation}
\fi
Obviously, relationships between the coefficients $\bar f_i$ and $\bar e_i$ must exist if  Eq.~\eqref{eq:casimir_metric} is to be satisfied.
Analogously, the contraction between the tetrad and $f^\mu$ may be written as
\begin{equation}
{e^\mu}_a (p) f_\mu (p)\,=\,p_a \bar a_1 \left(\frac{p_\alpha n^\alpha}{\Lambda},\frac{p^2}{\Lambda^2}\right) +  n_a \Lambda \bar a_2 \left(\frac{p_\alpha n^\alpha}{\Lambda},\frac{p^2}{\Lambda^2}\right)\,,
\label{eq:efproduct}
\end{equation}
  where $\bar a_1$ and $\bar a_2$ are simple  functions of $\bar f_i$ and $\bar e_i$.

  From Eq. \eqref{eq:efproduct} we can directly see that the condition imposed by Eq.~\eqref{eq:condition_discrete_pt} is  satisfied. Additionally, for  Eq.~\eqref{eq:condition_discrete_c}
  to be accomplished, we  need  to replace  $\Lambda \to - \Lambda$ under the action of  $\mathcal{C}$.
  This can be understood in the following way: taking into account that when applying the symmetry  $\mathcal{C}$ we are changing the sign of the energy, it is quite natural to think that a cutoff scale should change in the same way. Given that such a role is played by the deformation parameter $\Lambda$, we should also do the replacement $\Lambda \to - \Lambda$.
  In doing so, it is important to mention that, since the momentum scalar of curvature is proportional to $\Lambda^{-2}$, the change $\Lambda \to - \Lambda$ corresponds to an automorphism in the considered space (de Sitter).

  This is not the first time that symmetries involving an action on the deformation parameters have been employed. Indeed, there are several examples in the  quantum groups' literature  proposing that, by introducing a change of sign in the deformation parameter, one can obtain  automorphisms  of the Hopf algebra~\cite{Ballesteros:1993zi,Ballesteros_1994}. This has also been considered in the context of QFT in DSR, as  done in \cite{Kadyshevsky:1983yc} by introducing a pair of scalar fields (linked through the $\Lambda\to -\Lambda$ replacement) and more recently in \cite{Arzano:2020jro}.

   \section{Conclusions}
\label{sec:conclusions}

We have interpreted the relativistic quantum equations, namely the Klein--Gordon and Dirac equations, in a geometric language of curved momentum space, following the ideas of \cite{Carmona:2019fwf, AmelinoCamelia:2011bm,Lobo:2016blj}.
Previous works on the topics were mainly restricted to an algebraic approach.
In our discussion we have found several features that are worth comment.

First of all, our geometric derivations are shown to be equivalent to the previous algebraic ones \cite{Lukierski:1992dt, Nowicki:1992if},
in the sense that we can rederive them with an appropriate choice of the Casimir.
The fact that one can introduce a spinorial representation in two different ways, i.e. by modifying the coproduct of the quantum algebra $SO_q(3,2)$ to include a finite representation or, equivalently, by using the tetrad formalism, seems rather striking.
In order to prove this, the correct identification of the functions $f^\mu$ as generalized momenta turned out to be crucial.

Second, the selection of the Casimir as the distance in momentum space  establishes a direct connection between the Dirac and Klein--Gordon operators,
being one the ``square'' of the other.
This is not the case in other trials, where  a nontrivial function  should be introduced to link them  \cite{Nowicki:1992if}.
This is related to the fact that, in principle, one is entitled to choose as Casimir any function of the distance. Our choice can be thus seen as a \emph{minimal choice}, a characteristic which has been already used as guiding principle several times in the past.

Third, we have also discussed the implementation of discrete symmetries in curved momentum space for the fermionic case. We have found that parity, time-reversal and charge conjugation (at the free level) are all preserved in our formalism. This was not the case in previous studies \cite{Andrade:2013oza}, partially because the authors have not realized the need of trading the deformation parameter $\Lambda\to -\Lambda$ under charge conjugation.
Additionally, we have identified the conditions imposed on the vierbeins and the generalized momentum functions, cf. Eqs. \eqref{eq:condition_discrete_pt} and \eqref{eq:condition_discrete_c},
in order for the discrete symmetries to be valid. A deeper interpretation of these expressions and their role in more general geometries is still to be built. In any case, we can affirm that PCT is satisfied in this model (at the free level), in accordance with the fundamental role that it plays in standard QFT.

Fourth, we have made a first attempt into the identification of the relevant Hilbert space in a quantization process. Further developments in this direction are currently being pursued.

This work admits several generalizations. The most important deals with the incorporation of interactions.
In this scenario one possibility would be to consider gauge theories, which were recently studied
both in the quantum $\kappa$-Minkowski (including Becchi--Rouet--Stora--Tyutin symmetries) \cite{Mathieu:2020ccc, Mathieu:2021mxl}
and classical curved-momentum-space level
\cite{Ivetic:2019mkx}.
Other possible path regards the consideration of more general metrics in phase space, in order to make contact with recent proposals of (quantum) effective field theories in curved and noncommutative setup \cite{Franchino-Vinas:2019nqy,Franchino-Vinas:2021bcl}.

\section*{Acknowledgments}

 We appreciate useful discussions with Flavio Mercati and valuable comments of Ángel Ballesteros.
SAF is grateful to G. Gori and the Institut für Theoretische Physik, Heidelberg, for their kind hospitality.
SAF acknowledges support from Project 11/X748 and Subsidio a Jóvenes Investigadores 2019, UNLP.  JJR acknowledges support from the
Unión Europea-NextGenerationEU (``Ayudas Margarita Salas para la formación de jóvenes doctores''). This work has been partially supported by Agencia Estatal de Investigaci\'on (Spain)  under grant  PID2019-106802GB-I00/AEI/10.13039/501100011033.
The authors would like to acknowledge the contribution of the COST Action CA18108 ``Quantum gravity phenomenology in the multi-messenger approach''.

redpajama_set_name: RedPajamaArXiv


to reach out to the community in the hopes of making the Madonna Renewal Center a reality in Freeport Illinois. The Sisters recently participated in the Homecoming Parades of Freeport Senior High School and Aquin Catholic High School.
In recent articles and videos on the Pro Life Corner, we gave you some insight about what the Sisters want to accomplish in establishing the Madonna Renewal Center.


It's that time of year again when we're all stocking up on knitwear. Even though it's officially Autumn (yay for pretty backdrops!) It truly feels like Winter... It's dark when I set off to work, and dark when i'm coming home. Cars need scraping in the morning, that extra pair of socks is well and truly needed, and long gone are the weekends where you can leave the house on a night out without a jacket. But on a slightly happier note, it's the time when we invest in gorgeous faux fur coats, cute bobble hats, sluggly onesies and thick jumpers. Not to mention it's Christmas soon!! So as i'm sat here absolutely freezing, I thought it would be the perfect time to talk about staying warm wearing my new cute Autumn goodies from Sugarhill Boutique!
Every year I find myself loving Sugarhill Boutique's jumpers more and more... Infact, i'm currently wearing one of last year's animal jumpers and I still love it to death to this day! (FYI you need to take a look at this year's animal jumpers!) The newest additions to my wardrobe from their cosy knitwear collection are this gorgeous white cable knit jumper, and faux fur pom-pom bobble hat. I absolutely love this thick knit, infact white jumpers are my go-to 90% of the time because they literally go with anything! This particular sweater has knitted grey hearts on the elbows, which adds a super cute girly feel to my outfit! I also love how it looks with my grey kntted bobble hat, which again is super cute and accessory that will go with almost every jumper in my wardrobe!
A warm sweater with a bobble hat and ankle boots will definitely be my go-to for everyday wear this season. Just add a statement coat and you're ready to brave the cold!
The perfect outfit for a cold, winter day!


text:
Houston Criminal Lawyer James Sullivan defends clients in Greater Houston since 1994. Board Certified. Call (281) 546-6428 for a free phone consultation.
Assault Family Violence Charges Defense
Houston Domestic Assault Lawyers James Sullivan & Associates
Most couples learn to resolve their differences or even end their relationships without resorting to domestic assault or family violence. Sometimes, however, even good people gets so stressed out and feel so overwhelmed that they lose it even for a moment and strike out at their partner or another family member in a physical way.
The Consequences for Assault Family Violence
Depending on the circumstances, the prosecutors can charge Assault Family Violence either as a:
misdemeanor, or a
felony.
A Class A Misdemeanor Assault means that a person caused bodily injury (or pain) to their partner. If the person used a deadly weapon in the domestic assault, then it is a Felony Aggravated Assault. If the person choked their partner or impeded their breathing, then it is also a Felony. No matter the level of the criminal offense charged, you can expect the court to order you out of your home and have no contact until the case is resolved with your partner, spouse or family member. Punishment for those who are found guilty of misdemeanor assault family violence may also include:
Community Supervision (Probation)
A fine of up to $4,000.00
Jail confinement of up to one year
Anger management counseling
The Harris County District Attorney's Office has a Family Criminal Law Division that focuses solely on domestic violence. The prosecutors immediately call the alleged victim to get their statement as to what happened. The prosecutors lock in the emotional stress of the event by obtaining their immediate statement. However, facts tend to change and the importance of the incidence significantly lessens frequently after a short cooling off time.
Arrested for Assault Family Violence? Contact James Sullivan & Associates at (281) 546-6428.
Defending a Houston Assault Family Violence Charge
Assault against a family member is very serious and can have far reaching consequences. The accused may be permanently barred from working in certain industries. The accused may be denied housing in apartment complexes. If you are charged with assault family violence, you should seek legal advice from a experienced criminal defense lawyers.
The defense lawyers of James Sullivan & Associates fight to get their client's assault family violence charges dismissed or won at trial. Other domestic assault lawyers may choose to arrange a plea bargain for their clients; however, this may not be in the client's best interest. While it is true that a defendant who successfully completes a deferred adjudication probation will not have a conviction on their record, it is also true that if that person ever gets arrested again for assault family violence then he or she will be charged with a third degree felony and if found guilty could be sentenced to between 2 to 10 years in prison.
Negotiating a plea deal with prosecutors should be the last resort. A good domestic assault lawyer will first investigate the case, interview witnesses, analyze physical evidence and review witness statements in the offense report for inconsistencies. Such a defense attorney will also consider possible motives for making false allegations, such as jealousy, revenge or child custody issues.
Need Legal Advice? Call James Sullivan & Associates at (281) 546-6428.
Contact a Houston Assault Family Violence Lawyer
The best way to resolve the case is by first talking with an Assault Family Violence Lawyer. Houston Criminal Lawyer James Sullivan will consider the merits of your case and give you an honest evaluation. He will aggressively defend you. He gets results and fights cases at trial. His legal fees are affordable, reasonable and fair.
James Sullivan graduated from the Trial Lawyers College founded by Legendary lawyer Gerry Spence. Sullivan has a proven record of defending people from all walks of life, faiths and countries in courts throughout Texas.
Posted by James Sullivan at 6:04 PM
Labels: Harris County Assault Family Violence Lawyer, Houston Assault Family Violence Lawyer, Houston Domestic Assault Lawyer
Location: Houston, TX, USA
James G. Sullivan and Associates
12345 Jones Road, Suite 288, Houston, Texas 77070
Harris County Criminal Lawyer James Sullivan | Montgomery Fort Bend County Criminal Attorney
Harris County Criminal Lawyer | Houston Criminal Defense Attorney
Affidavit of Non-Prosecution
Aggravated Kidnapping
Assault by Impeding Breathing
Continuous Violence Against the Family
Family Violence Charges
First Time Arrests
Improper Relationship Between Educator and Student
Indecency with a Child
Injury to a Disabled Individual
Interference with Emergency Call
Invasive Visual Recording
Online Solicitation of a Minor
Prostitution / Solicitation
Public Lewdness
Sexual Assault of a Child
Pre-Trial Diversion Program
HOUSTON TEXAS CRIMINAL ATTORNEYS JAMES SULLIVAN AND ASSOCIATES (281) 546-6428

redpajama_set_name: RedPajamaCommonCrawl


PITTSBURGH, Pa. — Appalachian State wrestlers Codi Russell (133 pounds) and Cary Miller (heavyweight) won matches Thursday night to advance to Day 2 of the NCAA Championships.
Russell, who recorded a second-period pin of Ohio's Mario Guillen, and Miller, whose tiebreaking takedown with 40 seconds left against Virginia Tech's Billy Miller led to a 3-2 decision, both went 1-1 on the opening day of the double-elimination event at PPG Paints Arena.
Randall Diabe completed a 77-48 career that included a 26-7 mark as a senior with a pair of one-point losses in Pittsburgh, and Matt Zovistoski finished a 27-15 junior season with two matches. They were both making their second NCAA appearances.
The NCAA Championships resume Friday at 11 a.m., and Russell (23-10) will face 24th-seeded DJ Fehlman of Lock Haven a month after Fehlman (35-7) won a close decision between them in a nonconference dual. Their rematch should occur in the opening hour of the 11 a.m. session.
App State's Miller (24-12) will take on 12th-seeded Conan Jennings, a Northwestern senior with a 19-11 record this season.
After dropping a first-round match against Iowa's Austin DeSanto, Russell trailed 3-1 in the second period against Guillen, who had managed to erase a third-period deficit and take advantage of a Russell misstep to record a pin when Ohio faced Appalachian State in late November.
Russell returned the favor in the final minute of the second period Thursday, switching a deficit into a win by fall.
After losing a major decision against 2018 third-place finisher Amar Dhesi of Oregon State, App State's Miller was facing an opponent who placed third in the ACC and was making his fourth NCAA appearance. Miller executed a match-tying escape early in the third period, took the lead on his last-minute takedown and held on after allowing a post-takedown escape with 18 seconds remaining.
Oklahoma's Jake Woodley won 4-3 in the Tiebreaker-2 period against Diabe, and Buffalo's Brett Perry won a 3-2 decision by producing a tiebreaking takedown with 1:10 remaining.
Diabe and Woodley were tied at 1-all at the end of three periods thanks to one escape apiece, and neither wrestler managed a takedown in the Sudden Victory period. With each wrestler starting a 30-second tiebreaker from the down position, Woodley recorded an escape in six seconds to start that period and Diabe tied the match by getting an escape in 10 seconds.
That four-second difference ended up determining the outcome as the next 30-second tiebreaker included escapes by both Diabe and then Woodley in 10 seconds, meaning Woodley finished with a small but match-clinching advantage in riding time.
Needing to be aggressive to give himself a late chance in the first round, Zovistoski trailed 3-2 in the closing moments of a 10-2 major decision against Missouri's Brock Mauller, a fourth-seeded entry who won twice Thursday to improve to 31-2. In the next round, Penn's Anthony Artalona won by technical fall.


Getting the older man who has everything a birthday gift can certainly pose a problem. In most cases, get him something so he can remember all of the great times he's had during his long life. There are many gifts that will help bring back fond old memories of childhood, youth and young adulthood.
New York Times custom "birthday book" contains the front page from it's newspaper from every birthday your man in his 70's has had. The 12 x 12 inch is hardbound and the front cover displays the birthday boy's (or man's) name. In addition, the book contains a 20-page time line of 20th century events and 10 extra pages for him to add custom memories of his own. The book also comes with a magnifying glass for easy viewing.
There are sure to be VHS tapes stock-piled over the years at the older man's house. Help him get rid of the clutter by getting a VHS-Ripping Recorder. The recorder easily allows for all VHS tapes to be put onto the computer into easy-to-load files. The files can then be edited for "best-of" moments from home movies. Consider editing all of his videos for a "best-of-life-up-until-now" as a great, customized birthday gift.
Take a stroll down memory lane with a custom art print. Your Memory Lane will create a beautiful print that is crafted just for the man you have in mind. It will include icons and typography that will reflect the recipients best memories and moments of his life. It can even be customized just for a birthday.


Welcome to the website for Micaela Myers — a simple site to share some of my writing.
My passion for writing goes back to when I was about 6 years old. I love to learn new things and capture them in writing. Ever since graduate school I've made my living as a media professional.
After completing my BA in literature and writing, I earned my MFA in writing from Vermont College. You can read some of my published fiction using the links to the left. My story "A Cow in the Living Room" won the River Oak Review's 2001 fiction prize, and my micro fiction "Baby Doll" was honored in Binnacle's 2008 Ultra-Short Competition.
Since 2002, I've been a professional magazine editor and writer. I started my editing career at Horse Illustrated Magazine, where I stayed for more than four years. From 2006 - 2013 I worked at Firebrand Media LLC in Laguna Beach, Calif. As group editor, I oversaw Laguna Beach Magazine (winner of the 2010 and 2012 Maggie award for Best City Magazine), Newport Beach Magazine, Bespoke Magazine, Montage Magazine, Preferred Lifestyles magazine, Loews Magazine and Sea Island Life magazine.
In the summer of 2013 I moved to Laramie, Wyo., to take a job at the University of Wyoming as editor of UWyo Magazine.
I'm also a horsey girl and have written three equestrian books: The Horse Illustrated Guide to Trail Riding and Trailering (BowTie Press), and Knack Leg and Hoofcare for Horses (Globe Pequot Press).
I love to travel and have been to 14 countries so far, in addition to living in the eastern Caribbean for a year. A few of my travel articles are uploaded on this site.
I have three adopted dogs, and I have volunteered with a number of nonprofits. Click here to learn more.


This pamphlet played an important role in the expansion of Shakerism into the western states of Ohio, Kentucky and Indiana. It is in the form of a letter written to participants in the Kentucky Revival who attended the remarkable frontier camp meetings. Whether Youngs, Meacham and Bates carried printed or manuscript versions of the Letter, it remains the second earliest known written statement by the Shakers directed to the world.


This 2012 law codifies the regulations which apply to traveling with small musical instruments aboard passenger aircraft. To read the law on-line tap the expand icon in the lower right of the document window. To download it, click on the download icon. To return to the FAQ page click here. To return to the Traveling with Your Harp page, click here.


Many times people think ornamental fences are only good for bordering gardens or providing an accent to their front yards. However, ornamental vinyl fences are actually very versatile and can be used in many ways, including as pet enclosures, pool fences, or even a full backyard fence. If you're considering a vinyl fence for your home, an ornamental fence might be just what you're looking for.
Ornamental fence might sound flimsy – after all, it's just "ornamental," right? Wrong! In fact, ornamental fence is just as strong and durable as any of our other vinyl fences. It can stand up to years of use, even in harsh weather and strong sun. And just like our other vinyl fences, an ornamental fence won't fade, stain, or become brittle over time.
Unlike privacy fence, which is mostly used to enclose your yard or property, an ornamental fence has many possible uses. Choosing the right ornamental fence depends on what you want to do with it. If you want a fence to keep pets and children in your yard or out of your pool, you will need to choose a fence at least several feet tall and with closely-spaced pickets. In contrast, a decorative garden fence can be very low and have a variety of designs.
If you're considering an ornamental fence, aesthetics are probably important to you. Ornamental vinyl fences come in a wide range of designs, but they aren't all suited to every project. Make sure you consider what design you want, and also think about whether it will work for your intended goal. If you aren't sure what kind of design to choose, you can walk around your neighborhood and look at your neighbors' fences for ideas.
Another very important thing to consider for an ornamental fence is where on your property it will go and what it will interact with. For instance, a pool fence needs to be strong and tall enough to keep children and animals out – but it also needs to match with the decor of the rest of your yard, and it needs to be equipped with gates to allow access. In contrast, a fence on your property line can vary more in height and design – but it will also need to be a good barrier between you and your neighbor.
Finally, you'll need to think about what styles and designs your budget allows. Different heights and styles of fence come with different prices, and you will need to make sure your budget is sufficient to get the fence of your dreams. One benefit of using vinyl fence is that it costs less over time than comparable wood fence, since it doesn't require any additional maintenance or upkeep once it is installed. Wood fences – especially ones installed around pools and gardens – are often damaged by chemicals such as pool treatments and fertilizers. They also require you to stain and seal them regularly, which could damage the fence's surroundings. In contrast, vinyl fence is maintenance-free and doesn't require any special care or treatment.
Ready to learn more about getting an ornamental vinyl fence for your home? Future Outdoors can help. We are the number one dealer and installer of vinyl fences in the Dallas metro. With our experience and skill, we can help you find the perfect ornamental fence for your Texas home. Simply call (972) 576-1600 today to set up your free, no-obligation quote and consultation, or visit our Facebook page to get ideas for how we can help you!


I have spent over 20 years as a body therapist, personal trainer, and physical educator.
As a physical trainer of high level athletes, I help enhance improvement of neuromuscular connections, movements and balance.
With specialization in muscle isolation, functional training, core training and flexibility, and a black belt jiu-jitsu (Gracie Morumbi) I believe in accurate diagnostics, and efficient techniques.
I also focus on your quality of life, self esteem, knowledge and overcoming limits.


The Mastercam 2018 Mill Advanced Training Tutorial contains projects and accompanying practice exercises that span a wide degree of complexity. It is strongly recommended to have a good understanding of 2D machining (provided in the Mill Essentials Training Tutorial, for example) before beginning this book.
The content covers 3D wireframe, surface geometry creation and solid creation, Surface Finish and Roughing toolpaths. An emphasis is put on the advanced Surface High Speed toolpaths, STL Stock creation and its use in solid model verification. Also covered is how to create and manage Stock models in the Toolpath Manager. Machine Simulation is used to check for collisions between the part, the tool and any of the machine's components.


Gate warning signs can be very necessary to give urgent information to passers-by or to drivers. As such these signs need to be relatively large so as to be viewable from afar. It is usually a good idea to include a warning or caution label to the sign depending on the information being given just to grab attention even more. Yellow, red, orange and black are good colour schemes for this type of sign, all placed on a white background. Graphics can be included for extra information and clarity.
These signs can vary from a simple 'Keep Gate Closed' through to a more important 'Warning Crushing Hazard' sign. It is also important to state whether a gate is automatic or not, which way the gate will open, how many vehicles can pass through at any one time and whether or not it has an arm which can also cause injury.
On occasion, these signs can also just hold simple information such as guests reporting to reception, keeping gates closed when they are not being used or a simple 'Approach Slowly' instruction giving drivers an indication of when the gate will open for them. Whatever the message, the importance here is providing information so that all who are around the gate can be there in a safe manner.
Examples of Gate warning signs - Choose one of our example signs and adapt it to your requirements.


Sa Paissa is a spacious, authentic country house, offering 5 bedrooms on 450 m2 on two levels.
mature 2000 m2 palm tree garden within walking distance to Es Cubells village.
From the house you can enjoy the breathtaking panoramic views of the sea. There are lovely tracks for walking in the country, to the village or to Es Cubells beach.
find 4 bedrooms and 3 bathrooms, a big living room and separate, modern kitchen with dining area.
Finally, in the garden, there is a lovely separated guest bedroom with bathrooom and a exterior kitchen and bathroom by the pool area.
All bedroosm have air conditioning.


There are few things more surprising than a British summer, and I have been lucky enough to indulge in the sunshine.
A view from my hostel bed where I stayed for my first ever surfing holiday!
Taken at Trafalgar Square screening of Tosca, sunset light creates a warmth to the statue creating a powerful contrast with the onlooking man.


CLEVELAND, Ohio - Whether you like Drake or not, it's rather easy to identify why he's the ultimate artist for the streaming generation.
He's a good-looking guy and fashionable. He has an abundance of charisma, while wearing his heart on his sleeve. He's got mic skills, but is one of the most non-threatening rap megastars to come along since MC Hammer.
All of this is carefully curated. So, while a hip-hop double album seems like a huge risk (one that didn't pay off for even Nas or Jay-Z), it's a smart move for Drizzy.
Just watch the streaming numbers soar like never before. But is "Scorpion," Drake's fifth studio album, any good?
Well, it's long. At 90 minutes, Drake's style, split up between Side A (hip-hop) and Side B (R&B), can be a bit exhausting. But there's no lack of stellar material.
Side A is the most repeat-worthy collection of songs Drake has released since "Take Care." Side B...not so much. But let's start with the good.
Drake routinely delivers when it comes to openers. "Survival" is no exception. Drake's rhymes feel larger than life with epic lines like ("My Mount Rushmore is me with four different expressions...").
"Survival" kick starts an exceptional run of songs, from the trap-heavy "Nonstop" to the reflective nature of "Elevate." Then there's "Emotionless," a song driven by a Mariah Carey sample that should be played on repeat at summer parties everywhere.
In this era of teen angst and self-deprecation, lyrics like, "I know a girl whose one goal was to visit Rome/Then she finally got to Rome/And all she did was post pictures for people at home..." are about as real as it gets.
Before you can wrap your head around the fact Drake confirms he has a son on "Emotionless" (more on that in a minute) "God's Plan" looms over Side A like the monster track it is.
Elsewhere, tracks like "8 out of 10," "Can't Take a Joke" and "Is There More" find Drake at his lyrical peak. Add in a stellar Jay-Z verse on "Talk Up" and a DJ Premier beat on "Sandra's Rose," and Drake could have (and maybe should have) stopped there.
But "Scorpion" is an exercise in musical excess and moods. Side B is sure to draw mixed reviews as an exercise in relaxation or boredom.
The production - a hefty dose of Noah "40" Shebib" - is on point. But the smoothed-out R&B sounds of tracks like "Summer Games" and "Jaded" can't help but highlight Drake's limitations as a singer.
"Nice for What" and "Ratchet Happy Birthday" (What a title!) pick up the tempo, while "Finesse" feels like an emo punch to the gut ("I want my baby to have your eyes...").
The album's most emotional song comes in the form of "March 14," where references being a "single father" and having an empty crib in his crib linger.
"March 14" a dynamic closer that almost makes you forget the 13 songs on Side B numbingly blur together and the hyped Michael Jackson feature on "Don't Matter to Me" is a throwaway.
Maybe the combination of rap and R&B on songs like "That's How You Feel" and the Future-assisted "Blue Tint" showcase what could have been. And maybe "Scorpion" is far from the classic fans were hoping for. But doesn't matter.
This is an album eligible for platinum certification before it was even released. Drake can easily dismiss his detractors on "Scorpion's" sales and social buzz alone. Cars blaring Side A all summer and the ladies chilling at the pool to Side B are just added bonuses.


Fifty-year-old John Gadsby is alarmed by the decline of his hometown, Branton Hills, and rallies the city's young people to form an "Organization of Youth" to build civic spirit and improve living standards. Gadsby and his youthful army, despite some opposition, transform Branton Hills from a stagnant municipality into a bustling, thriving city. Toward the end of the book the members of Gadsby's organization receive diplomas in honor of their work. Gadsby becomes mayor and helps increase Branton Hills' population from 2,000 to 60,000. The story begins around 1906 and continues through World War I, Prohibition, and President Warren G. Harding's administration.
In the Mayor's Parlour by FLETCHER, J. S.
Wagner, the Wehr-Wolf by REYNOLDS, George W. M.


Denver Utility Locating Services On-Call | AZTEC Engineering Group, Inc. Denver Utility Locating Services On-Call | AZTEC Engineering Group, Inc.
AZTEC is currently providing utility designating, potholing, and coordination services for various projects in locations around the Denver, Colorado area. These services include reports of pothole locations and findings, including survey and traffic control. To date, AZTEC has completed 11 projects totaling 165 potholes and several thousand linear feet of utility designating.


This is a set of locations for the different lore in the Decrypting the Guardian Logs mission. Combinations may also occur at other sites – but if a Lore entry requires (for instance) Orb + Relic then you can scan that at any instance of the same site layout for the same lore. There can be issues when scanning the obelisks such that you do not receive the inbox message – if this is the case it is worth checking that you got credited with the CR for the scan – if you did not get the CR then you will still need the scan – try rescanning, relogging or visiting a different site.
There is a different list fro IndigoWyrd available from the forums which may be useful.


text:
Q: Can I export a container with data and everything to spawn a complete copy on another computer? I am looking into Docker and trying to wrap my head around it, so I might have misunderstood the concept.
I have installed the sebp/elk (ElasticSearch-Logstash-Kibana) and have a working container running. I have setup some indices and posted some data to logstash, which is stored with the container. If I restart the container everything works as expected. Now I am interested in exporting the container as it is, to launch on another computer with the configurations and data I have setup.
So I have tried export the container and import it as a new image and run the container from the new image. The container works, but it starts up as a new container without all the data, that I put into the original container.
I also tried to commit my changes to the image, then save the image and load it again and then run the container from the new image. That also works, but again without any data.
So when I inspect the original container, I can see that it has a mounted volume, so I figured that I should try to export the elasticsearch data to a .tar file and then import into the new container. But that didn't work either.
This is the mount inspection of the original container:
"Mounts": [
        {
            "Name": "fe17e920f9d17e177ac899b1617a8c51231c8a3b34007f463d082e5be2677412",
            "Source": "/var/lib/docker/volumes/fe17e920f9d17e177ac899b1617a8c51231c8a3b34007f463d082e5be2677412/_data",
            "Destination": "/var/lib/elasticsearch",
            "Driver": "local",
            "Mode": "",
            "RW": true,
            "Propagation": ""
        }
    ]

Here is how I tried to export it:
docker run --rm --volumes-from elk -v $(pwd):/volumes original/sebp/elk:exported tar cvf /volumes/elk-volume.tar /var/lib/elasticsearch

... and this is how I tried to import it:
docker run --rm --volumes-from elk-imported -v $(pwd):/volumes original/sebp/elk:exported bash -c "cd /volumes && tar xvf /volumes/elk-volume.tar --strip 1"

Is it possible to export a Docker container to get an exact copy of it with data and everything or am I approaching this problem the wrong way?

A: Your approach is correct, docker export command does not export the contents of volumes associated with the container, so you have to export the container than backup volume data.
Be sure you have elk-imported container already running before doing volume restore.
docker run -v /volumes --name elk-imported  original/sebp/elk:exported /bin/bash
docker run --rm --volumes-from elk-imported -v $(pwd):/volumes original/sebp/elk:exported bash -c "cd /volumes && tar xvf /volumes/elk-volume.tar --strip 1"

redpajama_set_name: RedPajamaStackExchange


Southeast Asian markets tumbled further today after a panicky selloff Thursday, frightened by a worsening crisis in Indonesia, relentless currency turmoil and a leading bond-rating agency's downgrading of some Thai debt to nearly junk status.
In midafternoon trading, Hong Kong stocks fell 3.3 percent, extending losses of the previous two days.
After declining to the lowest levels in years on Thursday, stocks in Singapore fell a further 6.1 percent and the Philippine market fell 8.3 percent in early afternoon trading today.
Shares in South Korea, which bucked the trend by rising 3.6 percent on Thursday, were off 2.4 percent this afternoon. Tokyo stocks closed a tenth of 1 percent lower.
On Thursday, the worst-hit country was Indonesia, where doubt about the health of President Suharto, coupled with wide dismay over the budget he presented Tuesday, unnerved investors and helped spread gloom elsewhere in Asia.
The main stock index in Jakarta, the Indonesian capital, plummeted nearly 19 percent before recovering somewhat to end Thursday's session with an 11.95 percent loss to 347.11, the lowest level in four years. Indonesia's currency, the rupiah, plunged, touching another record low of 10,550 to the dollar, half its value at the beginning of 1998. Indonesia's stock market was down 1.1 percent in afternoon trading today as the rupiah rebounded sharply -- to a rate of 8,550 to the dollar.
Malaysia's main stock index dropped 2.5 percent today after a 2.7 percent drop Thursday that left it at a seven-year low.
Shares in Thailand were down 1.8 percent this afternoon after dropping 2.7 percent Thursday, when the Thai baht hit a record low for the fifth day, settling at 53.35 to the dollar. The Thai currency is down 51 percent from early July, when its devaluation set the regional economic crisis in motion.
A catalyst for Thai losses was the decision Thursday by Standard & Poor's to downgrade Thailand's long-term debt rating from BBB to BBB-, one notch above non-investment-grade status, because of rapidly rising loan delinquencies. S.& P. said the Asian crisis would cause the Thai economy to shrink 3 percent this year, and the rating agency did not rule out further downgrading.
The difference between investment grade and junk status is important, as some institutional investors are forbidden under their bylaws to hold debt below investment grade.
Here in Hong Kong, investors were growing increasingly alarmed about the malaise in Southeast Asia as well as how it might affect China, Hong Kong's economic compatriot and political master.
Hong Kong's economy is largely devoid of weaknesses evident in much of Asia. But the turmoil elsewhere has left many financial analysts wary about the future.


text:
The Election Day asteroid is all hype
Here's what to know about 2018 VP1, predicted to pass by Earth on Nov. 2, 2020, the day before Americans make their final choice and polls close.
By Mark Kaufman August 23, 2020
The Election Day asteroid isn't end times.
You may have seen headlines recently warning about an asteroid "zooming towards us" just before Election Day in November. But while the threats from climate-amplified wildfires, a hyperactive hurricane season, and the worst pandemic in a century are real, this asteroid (known as "2018 VP1") poses little to no hazard to our planet. Astronomers identified the small asteroid in 2018. They've never been alarmed.
It's tiny
The European Space Agency calls 2018 VP1 a " tiny little thing."
It's around 2.4 meters, or nearly eight feet, in diameter (though it could be a little smaller or bigger).
Crucially, small asteroids break up in Earth's atmosphere when they succumb to intense heat, creating bright, glowing streaks in the atmosphere. For reference, an asteroid some 10 to 20 feet across zipped by Earth on Aug. 16. It passed just 1,830 miles above the planet's surface, a record for a Near Earth Asteroid, or NEA, that didn't strike Earth.
This rock, perhaps slightly bigger than 2018 VP1, is "very small by asteroid standards," according to NASA.
"If it had actually been on an impact trajectory, it would likely have become a fireball as it broke up in Earth's atmosphere, which happens several times a year," wrote NASA.
(For reference, the Chelyabinsk Meteor that dramatically exploded over Russia in 2013 was "house-sized," explained NASA. The explosion's shockwave injured over 1,600 people, largely from broken glass.)
Fireball caught last night over Colorado & New Mexico
We received more than 50 reports so far about this event.
If you saw it, please report it here:https://t.co/evqkWyLdwG
Event page: https://t.co/9AFFqfNkd0 pic.twitter.com/4Gx6Wfy4zv
— AMSMETEORS (@amsmeteors) July 29, 2020
An impact isn't imminent
The European Space Agency projects this "tiny little thing" has a 1 in 193 chance of hitting Earth in November. Yes, that's a "(relatively) high chance" for an asteroid. But it's still less than half of a percent.
How would you bet?
2⃣ days til #AsteroidDay!
The second most likely asteroid to strike Earth is 2018 VP1. A tiny little thing, it is estimated to be just 2.4 m in diameter and has a (relatively) high chance of striking Earth in November this year of 1 in 193 #SpaceCare💙#PlanetaryDefence🌍 pic.twitter.com/fYI9Dk3mXt
— ESA Operations (@esaoperations) June 28, 2020
An impact would likely be harmless
Let's say this "tiny little thing" barrels into Earth. As described above, 2018 VP1 would likely break apart in the atmosphere. This might look cool at night, but it certainly isn't a planetary threat.
NASA has a way of measuring impact hazards, called the Palermo Technical Impact Hazard Scale. Anything less than -2 on the scale has "no likely consequences."
2018 VP1 comes in at -3.57, not close to something that NASA says "would merit careful monitoring."
It's true that small fragments of a largely vaporized asteroid, after burning up in the atmosphere, can fall to the ground, becoming highly prized, intriguing meteorites. This would be a normal event.
2018 VP1 isn't coming to pummel Earth. But it could majestically streak across the sky.
Topics: Science, Science, Asteroid, Space, nasa


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