text:
\section{Introduction}
	While the most natural setting of a physical theory in spacetime is a $(3+1)$-dimensional manifold, there is no obstruction, in principle, to formulating self-consistent theories in a higher- or lower-dimensional manifold~\cite{ehrenfest,hardy}.
	From a strictly physical point of view, these theories can provide a description of phenomena occurring in presence of an effective dimensional reduction, or, on the other hand, encode for convenience a theory in $3+1$ dimensions into a larger spacetime.

	Many partial differential equations of physics can be generalized to arbitrary spatial dimensions: this is notably the case for all problems where the Laplacian is the relevant differential operator acting on the spatial variables. In these cases, one can investigate how the solutions of equations appearing in the same form are affected by dimensionality, and possibly explore those properties that are peculiar to specific dimensions, $3+1$ in particular. A paramount example is the wave equation: around 1900, J.~Hadamard found out qualitative differences for propagation of a localized perturbation in odd and larger than 1 spatial dimensions, where perturbation concentrates around the wavefront, and even dimensions, characterized by the presence of a trailing edge following the wavefront~\cite{hadamard,ehrenulen,balasz,courant,evans}.

	Apart from fundamental reasons, the interest in low-dimensional theories is twofold. On one hand, they are of tantamount importance in the formulation of quantized field theories \cite{qed1,qed2,qed3,qed4,qed5,qed6}, on the other hand, the technological developments of the last few years have enabled us to engineer and control truly low-dimensional systems, yielding some fascinating dimension-dependent features~\cite{wheelerdirac,qsim_book,qsim1,qsim2,qsim3,qsim4,qsim5,qsim6,roy,wqed1,wqed2,graphene}. It is thus worth studying whether self-consistent low-dimensional theories can be obtained starting from a more familiar $(3+1)$-dimensional one.

	There are two possible approaches to dimensional reduction. The first and most common one consists in formulating a lower dimensional version of the theory in $3+1$ dimensions, characterized by the same \textit{ab initio} properties of the starting point, as long as they are allowed by the new dimensionality \cite{lapidus,moreno,wheelerem,mcdonald,boito,GBB}.
	An alternative approach is represented by the \textit{method of descent}. This tool was originally formulated by Hadamard to solve several evolution problems of classical physics in a generic number of spatial dimensions: according to his own words, ``it consists in noticing that he who can do more can do less''~\cite{hadamard}. The descent method represented the key to identify the aforementioned difference in the behavior of wave equation solutions for even and odd spatial dimensions.

	While the descent method was conceived as a tool for solving equations, the underlying idea can be used in the reverse direction to perform the dimensional reduction of a given physical theory. Actually, a low-dimensional model can be considered as an instance of the theory where all the relevant quantities are uniform along one or more spatial directions. The low-dimensional equations of motion then follow directly from those in $3+1$ dimensions, by imposing such invariance. Reasonably enough, dimensional reduction by descent should yield a reduced version of the original $3+1$ theory among its products. However, the application of the descent method to electromagnetism~\cite{descent} shows us that this is not the whole story: besides the expected theory, other independent theories in the considered reduced dimensionality can emerge, sometimes with strikingly different physical properties.

	Following this line of research, in this work we shall apply the descent method to the equations of motion of a charged spin-\textonehalf\@ particle. We will start from the free Dirac theory, and then extend the results to the case of the minimal coupling with the electromagnetic field. After recalling, in Section~\ref{sec:prelim}, relevant notions about the Dirac equation in $3+1$ dimensions, we shall tackle in Section~\ref{sec:freedirac} the problem of dimensional reduction of the free Dirac equation, comparing the results with the natural generalization to an arbitrary dimension of the Dirac operator as a ``square root'' of the Klein-Gordon operator.

	We will show that the method of descent provides two independent sectors, governed by non-interacting Dirac equation in $2+1$ dimensions.
	Then, in Section~\ref{sec:diracem}, after briefly recalling the results obtained for the free electromagnetic field, we shall apply the same procedure to the coupled Dirac-Maxwell equations. We will observe that it is not trivial, in the case of interacting fields, to obtain a decoupled $2+1$ version of the Dirac theory by simply requiring invariance along one direction. The usual formulation of QED in $2+1$ dimensions emerges only by considering specific classes of solutions after dimensional reduction. An interesting spinoff of our analysis will be a nontrivial extension of Hadamard's descent method, with the introduction of the covariant derivative.

	\section{Preliminaries}\label{sec:prelim}

	The starting manifold of our analysis is the familiar $(3+1)$-dimensional Minkowski spacetime with metric tensor $(\eta^{\mu\nu})=\diag(+1,-1,-1,-1)$. Unless otherwise specified, Einstein summation convention is understood. In order to clearly distinguish between a $(3+1)$- and a $(2+1)$-dimensional context, three different kinds of indices will often be used: as common practice, Greek indices $\mu,\nu,\rho$, taking values in $\{0,1,2,3\}$, will refer to the $(3+1)$-dimensional coordinates, and  Latin indices $i, j, k$, with values in $\{1, 2, 3\}$, to its spatial coordinates. In addition,  Latin indices $a,b,c$, with values in $\{0,1,2\}$, will be reserved for the $(2+1)$-dimensional coordinates. This convention applies to free and contracted indices. Natural units will be adopted, and EM units will be rationalized.

	\subsection{Dirac equation in \texorpdfstring{$3+1$}{3+1} dimensions}\label{sec:normdirac}

	In a $(3+1)$-dimensional spacetime, the Dirac Lagrangian, describing a free spin-\textonehalf{} particle of mass $m$, reads
	\begin{equation}\label{eq:lagrDirac}
		\mathcal L=
		\overline \Psi\qty(\iu\gamma^\mu\partial_\mu-m \id_4)\Psi\, .
	\end{equation}
	Here, $\Psi=\Psi(x)$ is a four-component wavefunction, often called \emph{Dirac spinor} or \emph{bispinor}, $\id_4$ is the $4\times 4$ identity matrix (that will be kept implicit in most of the equations henceforth), $(\gamma^\mu)_{\mu=0,1,2,3}$ is a quadruple of $4\times 4$ matrices, usually referred to as the \emph{gamma matrices}, satisfying the anticommutation property
	\begin{equation}\label{eq:cliff}
		\acomm{\gamma^\mu}{\gamma^\nu}=2\eta^{\mu\nu}\id_4\,,
	\end{equation}
	and $\overline\Psi=\Psi^\dagger\gamma^0$ is the \emph{Dirac adjoint spinor}.
	By requiring that
	\begin{equation}\label{eq:hermigamma}
		{\gamma^\mu}^{\dagger}=\gamma^0\gamma^\mu\gamma^0\,,
	\end{equation}
	one gets that $\gamma^0$ is Hermitian (hence $\overline\Psi=(\gamma^0\Psi)^\dagger$), and the $\gamma^i$'s  are anti-Hermitian, and the action associated to the Lagrangian $\mathcal L$ is real-valued.
	It is also useful to introduce a fifth matrix
	\begin{equation}
		\gamma^5=\iu\gamma^0\gamma^1\gamma^2\gamma^3 ,
	\end{equation}
	which is Hermitian, involutive, and anticommuting with all the $\gamma^\mu$'s. The matrices $\id_4$, $(\gamma^\mu)$, $(\gamma^\mu \gamma^\nu)_{\mu<\nu}$, $(\gamma^5 \gamma^\mu )$, and $\gamma^5$ form a basis of the space of $4\times 4$ complex matrices~\cite{Go55}. The anticommutation relations~\eqref{eq:cliff} have the following immediate consequences:  the gamma matrices are traceless, $\gamma^0$ being involutive, and each $\gamma^i$ anti-involutive (therefore, the condition~\eqref{eq:hermigamma} is equivalent to assuming the gamma matrices to be unitary).

	The stationary points of the action corresponding to the Dirac Lagrangian~\eqref{eq:lagrDirac} are determined by the Dirac equation~\cite{dirac,thaller}
	\begin{equation}\label{eq:Dirac}
		\qty(\iu\gamma^\mu\partial_\mu-m\id_4)\Psi=0\,.
	\end{equation}
	The four-component differential operator $(\iu\gamma^\mu\partial_\mu-m\id_4)$ applied to the Dirac spinor squares to the Klein-Gordon operator, in the following sense:
	\begin{equation}
		(\iu\gamma^\mu\partial_\mu-m\id_4)^\dagger\,(\iu\gamma^\mu\partial_\mu-m\id_4)=\left(\partial_\mu\partial^\mu+m^2 \right)\id_4 .
	\end{equation}
	The matrices $S^{\mu\nu}=\tfrac{\iu}{4}[\gamma^\mu,\gamma^\nu]$ generate the $(\tfrac{1}{2},0)\oplus(0,\tfrac{1}{2})$ representation of the restricted Lorentz group.
	In particular, by introducing the six Hermitian matrices
	\begin{equation}\label{eq:alphabeta}
		\Sigma^i=\tfrac{\iu}{2}\epsilon^{ijk}\gamma^j\gamma^k, \qquad \alpha^i=\gamma^0\gamma^i \,,
	\end{equation}
	a proper rotation is represented by $\exp(-\tfrac{\iu}{2}\vb*\theta\cdot\vb*\Sigma)$, and a boost by $\exp(-\tfrac{1}{2}\vb*\eta\cdot\vb*\alpha)$. The Noether current
	\begin{equation}\label{eq:jmuvett}
		(j^\mu)=\qty(\overline{\Psi}\gamma^\mu\Psi)=\qty(\Psi^\dagger\Psi,
		\Psi^\dagger\vb*\alpha\Psi)\,,
	\end{equation}
	related to the $\mathrm{U}(1)$ symmetry $\Psi\to\mathrm{e}^{\iu\theta}\Psi$ of the Lagrangian~\eqref{eq:lagrDirac}, is manifestly covariant, behaving as a vector under restricted Lorentz transformations, and can be interpreted as a probability density four-current.

	Distinct families of matrices satisfying Eq.~\eqref{eq:cliff} correspond to different \textit{representations} of the Dirac spinors and the Dirac algebra, all of them being linked via unitary transformations $U\in\mathrm{U}(4)$,
	\begin{equation}
		\Psi\to \Psi'=U\Psi, \qquad \gamma^{\mu}\to {\gamma'}^{\mu}=U\gamma^{\mu}U^{\dagger} ,
	\end{equation}
	preserving both the anticommutation relations~\eqref{eq:cliff} (as a more general similarity transformation would do) and the conditions~\eqref{eq:hermigamma}~\cite{Go55}. In the following, different representations will be labelled by specific labels on the spinor and the gamma matrices. Dirac's original choice,
	\begin{equation}\label{eq:alphabetaDirac}
		\gamma_{\mathrm{D}}^0
		=\mqty(\id_2& 0\\ 0&-\id_2)	, \qquad
		\gamma_{\mathrm{D}}^i
		=\mqty( 0&\sigma^{i}\\ -\sigma^{i}&0 ) , \qquad
		\Psi_{\mathrm{D}}=\mqty(\phi\\		\chi) ,
	\end{equation}
	where
	\begin{equation}
		\sigma^1 = \mqty( 0 & 1 \\ 1 & 0), \qquad \sigma^2 = \mqty( 0 & -\iu \\ \iu & 0), \qquad \sigma^3 = \mqty( 1 & 0 \\ 0 & -1)
	\end{equation}
	are the Pauli matrices,
	is known as the \emph{standard} or \emph{Dirac} \emph{representation}. In this representation, the generators~\eqref{eq:alphabeta}
	take the simple forms 	\begin{equation}\label{eq:alphasigmaSR}
		\vb*\Sigma_{\mathrm {D}}
		=\mqty( 		\vb*\sigma&0\\ 0 &		\vb*\sigma), \qquad
		\vb*\alpha_{\mathrm {D}}
		=\mqty(  0 &		\vb*\sigma\\ 		\vb*\sigma&0).
	\end{equation}
	In particular, it is worth noticing that the spin operator along the $z$ direction, $\Sigma_{\mathrm{D}}^3$, is diagonal in the Dirac representation.

	\subsection{Dirac equation in arbitrary spatial dimensions}

	The anticommutation relations~\eqref{eq:cliff} can be imposed in a Min\-kowski spacetime of arbitrary spatial dimensionality $n$, starting from  $n+1$ square matrices $(\Gamma^A)_{0\le A\le n}$, of order $N=2^{\lfloor(n+1)/2\rfloor}$ (with $\lfloor x\rfloor$ denoting the integer part of $x$), satisfying the anticommutation algebra $\acomm{\Gamma^A}{\Gamma^B}=2\eta^{AB}\id_{N}$~\cite{Go55,BrauerWeyl}, with
	$(\eta^{AB})=\diag(+1,-1,\dots,-1)$, and $\id_N$ being the $N\times N$ identity matrix.
	Therefore, the free Dirac equation can be generalized to an arbitrary number of spatial dimensions, a feature essentially noticed from the beginning by Dirac himself \cite{history}. A ``natural'' notion of an $(n+1)$-dimensional Dirac theory can be achieved by following the same pattern as in Subsection~\ref{sec:normdirac}. Such a theory will be formulated in terms of $N$-dimensional objects, with $(\iu \Gamma^A\partial_A-m I_N)^{\dagger}(\iu \Gamma^A\partial_A-m I_N)$ equaling the Klein-Gordon operator in $n+1$ dimensions.

	Here, we shall focus on the case of $2$ spatial dimensions, so the triple of $2\times 2$ gamma matrices $(\Gamma^a)$ will satisfy
	\begin{align}\label{eq:cliffLD}
		\acomm{\Gamma^a}{\Gamma^b}=2\eta^{ab}\id_2, \qquad {\Gamma^a}^{\dagger} = \Gamma^0 \Gamma^a \Gamma^0 .
	\end{align}
	The corresponding Dirac Lagrangian $\overline{\psi}\qty(\iu\Gamma^a\partial_a-m\id_2)\psi$, with $\psi$ a two-component function of three variables and $\overline{\psi}=\psi^{\dagger}\Gamma^0$, yields the Euler-Lagrange equation
	\begin{equation}\label{eq:DiracLow}
		\qty(\iu\Gamma^a\partial_a-m\id_2)\psi=0\,,
	\end{equation}
	which will be referred to as the $(2+1)$-dimensional Dirac equation. One of the possible choices of $(2+1)$-dimensional gamma matrices in $2+1$ dimensions is represented by $(\sigma^3, \iu\sigma^2,-\iu\sigma^1)$~\cite{thaller}.

	\section{Dimensional reduction of the free Dirac theory}\label{sec:freedirac}
	We will perform the descent on the Dirac equation from $3$ to $2$ spatial dimensions along the $z$ coordinate, but any other direction would lead to analogous results. We expect a sensible $(2+1)$-dimensional free Dirac theory
	to be covariant with respect to the $(2+1)$-dimensional Lorentz group $\mathrm{O}(1,2)$ of the $z$-preserving transformations
	\begin{align}\label{eq:lambda}
		\varLambda
		=\left(\begin{BMAT}{c1c}{c1c}
			\begin{BMAT}[5pt]{c}{c} L \end{BMAT} & \\
			&\,Q\,
		\end{BMAT}\right)\,,
	\end{align}
	with $L \in\mathrm{O}(1,2)$, and $Q \in\mathrm{O}(1)=\{+1,-1\}$. In a $(2+1)$-dimensional context, attributes such as ``covariant'', ``vector'', ``scalar'' will always be referred to this group or one of its subgroups.

	Therefore, we specialize the Dirac differential problem into a $z$-in\-de\-pen\-dent one by imposing the \emph{descent condition}
	\begin{equation}
		\partial_3\Psi=0\,,
	\end{equation}
	which is both re\-pre\-sen\-ta\-tion-in\-de\-pen\-dent, and covariant.
	We end up with the ``reduced Dirac equation''
	\begin{equation}\label{eq:descent}
		\qty(\iu \gamma^a \partial_a-m\id_4)	\Psi=0\,,
	\end{equation}
	that can be regarded as the Euler-Lagrange equation of the ``reduced Lagrangian''
	\begin{equation}\label{eq:lagrDesc}
		\tilde{\mathcal L}=
		\overline \Psi\qty(\iu\gamma^a\partial_a-m\id_4)\Psi\,.
	\end{equation}

	As we will show in the following, Eq.~\eqref{eq:descent} is equivalent to the pair of  uncoupled equations:
	\begin{align}
		\qty(\iu\gamma_{+}^{a}\partial_a-m\id_4)\Psi_{+}=0\,,\label{eq:Gammap}\\
		\qty(\iu\gamma_{-}^{a}\partial_a-m\id_4)\Psi_{-}=0\,,\label{eq:Gammam}
	\end{align}
	where $(\gamma_{+}^{a})_{0\le a\le 2}$ and $(\gamma_{-}^{a})_{0\le a\le 2}$ are families of $4\times 4$ matrices satisfying
	\begin{equation}\label{cliffa}
		\acomm{\gamma_{\pm}^a}{\gamma_{\pm}^b}=2\eta^{ab}P_\pm\,,
		\qquad
		\gamma^a_\pm P_\pm=\gamma^a_\pm=P_\pm\gamma^a_\pm\,,
	\end{equation}
	with $P_+$ and $P_-$ two complementary orthogonal projections of rank 2, thus satisfying
	\begin{equation}\label{eq:projIdem}
		P_+ P_-=0, \qquad P_{\pm}^2 = P_{\pm}^{\dagger} = P_{\pm}  , \qquad P_++P_-=\id_4 \,.
	\end{equation}

	Moreover, Eqs.~\eqref{eq:Gammap}--\eqref{eq:Gammam} turn out to be $(2+1)$-dimensional Dirac equations, that can be explicitly put  in the form of Eq.~\eqref{eq:DiracLow} in a suitable basis, such that $P_+=\id_2 \oplus\, 0_2$ and $P_-= 0_2 \oplus \id_2$.
	Notice that, despite the formal analogy between the equations, dimensional reduction generates \textit{two independent Dirac theories} in (2+1) dimensions. The derivation of two independent dimensionally-reduced models by descent was already found in the electromagnetic case, as we shall outline in the following. There, however, the theories are also formally different.

	In Section~\ref{sec:sr}, we shall first discuss the problem in the standard representation (which, quite by chance, turns out to be particularly advantageous to perform the descent along $z$), and then formulate in Subsection~\ref{sec:alg} a general proof of the above statement that is independent of the representation.
	A discussion of these results is finally given in Section~\ref{sec:remarks}.

	\subsection{Descent in the standard representation}\label{sec:sr}
	By conveniently labelling the spinor components in the standard representation as
	\begin{align}\label{eq:spinoSR}
		\Psi_{\mathrm{D}}=\mqty(\phi\\		\chi)
		=\mqty( \phi_\uparrow\\	\phi_\downarrow\\ \chi_\downarrow\\ \chi_\uparrow)
		\,,
	\end{align}
	the Dirac equation~\eqref{eq:Dirac} has the matrix form
	\begin{align}
		\begin{pmatrix}
			\iu\partial_0-m&0&\iu\partial_3&\iu\partial_1+\partial_2\\
			0&\iu\partial_0-m&\iu\partial_1-\partial_2&-\iu\partial_3\\
			-\iu\partial_3&-\iu\partial_1-\partial_2&-\iu\partial_0-m&0\\
			-\iu\partial_1+\partial_2&\iu\partial_3&0&-\iu\partial_0-m
		\end{pmatrix}
		\begin{pmatrix}
			\phi_\uparrow\\	\phi_\downarrow\\	\chi_\downarrow\\	\chi_\uparrow
		\end{pmatrix}=0\,,\label{eq:diracDirac}
	\end{align}
	equivalent to the system of four equations
	\begin{equation}
		\begin{cases}
			(\iu\partial_0-m)	\phi_\uparrow+\iu\partial_3\chi_\downarrow+ (\iu\partial_1+\partial_2)\chi_\uparrow=0\,,
			\\
			(\iu\partial_0-m)\phi_\downarrow+(\iu\partial_1-\partial_2)\chi_\downarrow-\iu\partial_3\chi_\uparrow=0\,,
			\\
			(\iu\partial_0+m)\chi_\downarrow	+\iu\partial_3\phi_\uparrow+(\iu\partial_1+\partial_2)\phi_\downarrow
			=0\,,	\label{eq:FreeIII}\\
			(\iu\partial_0+m)\chi_\uparrow+
			(\iu\partial_1-\partial_2)\phi_\uparrow-\iu\partial_3\phi_\downarrow=0\,.

		\end{cases}
	\end{equation}
	Notice that all the coupling terms between $\uparrow$ and $\downarrow$ components are in the form of a derivative $\partial_3$ with respect to $z$.

	Consequently, the condition $\partial_3\Psi_\mathrm{D}=0$ breaks all and only the couplings between the $\uparrow$ and $\downarrow$ components, so that the Dirac equation takes the matrix form
	\begin{equation}\label{eq:diracDiracz}
		\left(\begin{array}{@{}c|cc|c@{}}
			\iu\partial_0-m&0&0&\iu\partial_1+\partial_2\\
			\hline
			0& \iu\partial_0-m& \iu\partial_1-\partial_2&0\\
			0&-\iu\partial_1-\partial_2&-\iu\partial_0-m&0\\
			\hline
			-\iu\partial_1+\partial_2&0&0&\-\iu\partial_0-m
		\end{array}\right)
		\left(\begin{array}{@{}c@{}}
			\phi_\uparrow\\
			\hline
			\phi_\downarrow\\
			\chi_\downarrow\\
			\hline
			\chi_\uparrow
		\end{array}\right)=0\,,
	\end{equation}
	and the wavefunction is partitioned into two independently evolving sectors, respectively spanned by the $\uparrow$ components, whose dynamics is determined by
	\begin{equation}
		\begin{cases}
			(+\iu\partial_0-m)	\phi_\uparrow+(+\iu\partial_1+\partial_2)\chi_\uparrow=0\,,
			\\
			(-\iu\partial_0-m)\chi_\uparrow+(-\iu\partial_1+\partial_2)\phi_\uparrow=0\,,\label{eq:descFreeII}
		\end{cases}
	\end{equation}
	and the $\downarrow$ components, obeying the equations
	\begin{equation}
		\begin{cases}
			(+\iu\partial_0-m)\phi_\downarrow+(+\iu\partial_1-\partial_2)\chi_\downarrow=0\,
			\\
			(-\iu\partial_0-m)\chi_\downarrow+(-\iu\partial_1-\partial_2)\phi_\downarrow=0\,.\label{eq:descFreeIV}
		\end{cases}
	\end{equation}
	The only difference between the $\uparrow$ and $\downarrow$ equations consists in the sign before the $\partial_2$ terms, which entails that the subsystems~\eqref{eq:descFreeII} and~\eqref{eq:descFreeIV} are transformed into each other by a reflection of the $y$-axis $(t,x,y,z)\to(t,x,-y,z)$.

	At this point, by observing that both the triples of $2 \times 2$ matrices
	\begin{align}\label{GammaD}
		\qty(\Gamma_{\uparrow}^a)=(\sigma^3,\iu\sigma^2,-\iu\sigma^1)	\,,
		&&\qty(\Gamma_{\downarrow}^a)=(\sigma^3,\iu\sigma^2,+\iu\sigma^1)\,,
	\end{align}
	satisfy the anticommutation relations~\eqref{eq:cliffLD},
	and by introducing the two-components wavefunctions
	\begin{equation}\label{eq:Dspin}
		\psi_{\uparrow}
		=\begin{pmatrix}
			\phi_\uparrow\\		\chi_\uparrow
		\end{pmatrix}\,,
		\qquad
		\psi_{\downarrow}
		=\begin{pmatrix}
			\phi_\downarrow\\\chi_\downarrow
		\end{pmatrix}\,,
	\end{equation}
	the subsystems~\eqref{eq:descFreeII} and~\eqref{eq:descFreeIV} can be rewritten as
	\begin{align}
		\qty(\iu\Gamma_{\uparrow}^{a}\partial_a-m\id_2)\psi_{\uparrow}=0\,,\label{eq:Gammaup}\\
		\qty(\iu\Gamma_{\downarrow}^{a}\partial_a-m\id_2)\psi_{\downarrow}=0\,,\label{eq:Gammadown}
	\end{align}
	namely, as a pair of $(2+1)$-dimensional Dirac equations.

	The partition of the spinor in Eq.~\eqref{eq:diracDiracz} can be easily interpreted by looking at the generators of $\mathrm{SO^+(1,2)}$, which are proportional to the matrices
	\begin{align}
		\Sigma^3_{\mathrm D}=
		\left(\begin{BMAT}[1pt]{c1cc1c}{c1cc1c}
			1&&\phantom{-1}&\\
			\phantom{-1}&-1&&\\
			&&1&\\
			&&&-1
		\end{BMAT}\right),
		&&
		\alpha^1_{\mathrm D}=
		\left(\begin{BMAT}[1pt]{c1cc1c}{c1cc1c}
			&&\phantom{-1}&\iu\\
			&\phantom{-1}&\iu&\\
			\phantom{-1}&\iu&&\\
			\iu&&&\phantom{-1}
		\end{BMAT}\right),
		&&
		\alpha^2_{\mathrm D}
		=\left(\begin{BMAT}[1pt]{c1cc1c}{c1cc1c}
			&&&1\\
			&\phantom{-1}&-1&\\
			&1&&\phantom{-1}\\
			-1&&&
		\end{BMAT}\right).
	\end{align}
	Actually, the transformation generated by $(\Sigma_{\mathrm{D}}^3,\alpha_{\mathrm{D}}^1,\alpha_{\mathrm{D}}^2)$ does not mix the $\uparrow$ and $\downarrow$ components: our partition thus coincides with the decomposition of the spinor with respect to this group, satisfying the expected $(2+1)$-dimensional covariance.

	It is worth remarking that the pair~\eqref{eq:Gammaup}--\eqref{eq:Gammadown} of uncoupled Dirac equations emerges naturally in the  low-energy  description  of graphene  (see  e.g.\  Eq.~(19)  of~\cite{graphene}),  which  represents  a prototypical example of ($2+1$)-dimensional \textit{Dirac matter}.

	\subsection{Descent in a representation-independent approach}\label{sec:alg}
	The calculations in the Dirac representation are facilitated by the fact that the generator $\Sigma_3$, representing the spin component along the descent direction $z$, is diagonal. Had we chosen a different representation, or had we performed the descent in the same representation but along a different direction, the resulting decoupling would have been less evident: of course we would have ended with two independently evolving 2-dimensional subspaces, but in general their vectors would have been written in terms of linear combinations of the spinor components. Therefore, it is worth reviewing the decoupling mechanism from a re\-pre\-sen\-ta\-tion-in\-de\-pen\-dent point of view.

	We can say that a proper linear
	decoupling consists in the existence of a pair of projections on nontrivial complementary subspaces, namely two operators $P_\pm$ satisfying the properties~\eqref{eq:projIdem}, with the possible exception of Hermiticity,
	acting exclusively on the spinorial degrees of freedom of $\Psi$, such that the projected wavefunctions $P_+\Psi$ and $P_-\Psi$ are mutually decoupled in the \textit{reduced} Dirac equation~\eqref{eq:descent}. The latter can be decomposed, by means of the identity resolution provided by $P_{\pm}$, into the pair
	\begin{align}
		P_+(\iu\gamma^a\partial_a-m)P_+\Psi+\iu P_+\gamma^a P_-\partial_a\Psi&=0\,,\\
		P_-(\iu\gamma^a\partial_a-m)P_-\Psi+\iu P_-\gamma^a P_+\partial_a\Psi&=0\,.
	\end{align}

	A necessary and sufficient condition for $P_+\Psi$ and $P_-\Psi$ to be decoupled is the vanishing of the coupling terms $\iu P_+\gamma^a P_-\partial_a\Psi$ and $\iu P_-\gamma^a P_+\partial_a\Psi$. Considering the property $P_+P_-=P_-P_+=0$ and the arbitrariness of the spinor derivatives, these conditions are equivalent to
	\begin{equation}\label{eq:projCommuto}
		\left[P_+,\gamma^a\right]=0 \qquad \text{for } a=0,1,2.
	\end{equation}

	In order to determine the $4\times 4$ complex matrices $P_{\pm}$, we decompose them in the form
	\begin{equation}\label{eq:decomposition}
		P_{\pm}=a_0\id+\sum_\mu b_\mu\gamma^\mu+ \sum_{\mu<\nu}c_{\mu\nu}\gamma^\mu\gamma^\nu+\sum_\mu d_\mu\gamma^\mu\gamma^5 + a_5 \gamma^5\,,
	\end{equation}
	where we have temporarily dropped Einstein's convention, yielding the commutators
	\begin{equation}
		\tfrac{1}{2}[P_{\pm},\gamma^a]=\sum_{\mu\ne a} b_\mu\gamma^\mu\gamma^a+\eta^{aa}\biggl(- \sum_{a<\nu}c_{a\nu}\gamma^\nu
		+\sum_{\mu<a}c_{\mu a}\gamma^\mu- d_a\gamma^5\biggr)- a_5 \gamma^a\gamma^5\,.
	\end{equation}
	By imposing the commutation conditions~\eqref{eq:projCommuto}, while letting the $(2+1)$-dimensional index $a$ take the values $0,1,2$, all the coefficients but $d_3$ must vanish, so $P_{\pm}=a_0\id+d_3\gamma^3\gamma^5$. Finally, by imposing the idempotence condition~\eqref{eq:projIdem},
	\begin{equation}
		a_0\id+d_3\gamma^3\gamma^5=\qty(a_0^2+{d_3}^2)\id + 2 a_0 d_3 \gamma^3\gamma^5\,,
	\end{equation}
	we get $a_0=1/2$ and $d_3=\pm1/2 $, and the projections are finally obtained:
	\begin{equation}\label{eq:Ppm}
		P_\pm=\frac{\id \pm \gamma^3\gamma^5}{2}\,.
	\end{equation}
	We are thus led to consider the matrix
	\begin{equation}
		\kappa^3= P_+ - P_- =\gamma^3\gamma^5=\iu\gamma^0\gamma^1\gamma^2
		=\gamma^0\Sigma^3\,,
	\end{equation}
	which is traceless, because $\mathrm{tr}\qty(\gamma^3\gamma^5)=-\mathrm{tr}\qty(\gamma^5\gamma^3)$, as well as Hermitian and involutive, since it is the product of the commuting Hermitian and involutive matrices $\gamma^0$ and $\Sigma^3$. Therefore, its eigenvalues are $1$ and $-1$, each with multiplicity $2$, corresponding to the \emph{orthogonal} projections $P_+$ and $P_-$, respectively. Moreover, by construction we have $[\gamma^a,\kappa^3]=0$.

	Now, by setting
	\begin{equation}\label{eq:Decomp}
		\Psi_\pm=P_\pm\Psi, \qquad
		\gamma^a_\pm=P_\pm\gamma^a P_\pm,
	\end{equation}
	the following decompositions hold:
	\begin{align}
		\Psi&=\Psi_+ +\Psi_-\,,\label{eq:PsiDecomp}\\
		\gamma^a&=\gamma^a_+ +\gamma^a _-\,,\label{eq:gammaADecomp}
	\end{align}
	and the reduced equation~\eqref{eq:descent} can be written as the pair of uncoupled equations
	\begin{align}
		\qty(\iu\gamma_{+}^{a}\partial_a-m\id_4)\Psi_{+}=0\,,\label{eq:gammap}\\
		\qty(\iu\gamma_{-}^{a}\partial_a-m\id_4)\Psi_{-}=0\,,\label{eq:gammam}
	\end{align}
	where $(\gamma_{\pm}^a)$ are two sets  matrices satisfying
	\begin{equation}\label{eq:clifflike}
		\acomm{\gamma^a_\pm}{\gamma^b_\pm}=\acomm{P_\pm\gamma^a P_\pm}{P_\pm\gamma^b P_\pm}=2\eta^{ab}P_\pm\,.
	\end{equation}

	At this point, we can show that that Eqs.~\eqref{eq:gammap}--\eqref{eq:gammam} are $(2+1)$-dimensional Dirac equations, in the following sense.
	First, we can always choose a representation where the projections are diagonalized in the form
	\begin{equation}\label{eq:projrepr}
		\hat P_{+}=\mqty(\id_2& 0\\0& 0 )\,,
		\qquad
		\hat P_{-}=\mqty(0& 0\\0&\id_2)\,.
	\end{equation}
	According to the decompositions~\eqref{eq:PsiDecomp}--\eqref{eq:gammaADecomp}, $\hat\gamma^a$ and $\hat\Psi$, the expressions of the $a$-th gamma matrix and of the wavefunction in the chosen representation, have the block structure
	\begin{equation}\label{eq:diagonal_decomp}
		\hat\gamma^a=\mqty(\hat\Gamma^a_+ & 0\\0&\hat\Gamma^a_-)\,, \qquad \hat\Psi=\mqty(\hat\psi_+\\ \hat\psi_-)\,.
	\end{equation}
	In this regard, notice that the existence of a projection commuting with each $\gamma^a$, in turn a necessary and sufficient condition for the decoupling of the reduced Dirac equation, is also equivalent to the simultaneous block-diagonalizability of the $(\gamma^a)$.

	Then the sets $\bigl(\hat\Gamma^a_\pm\bigr)$ inherit their anticommutation relations from those of $\bigl(\hat\gamma^a_\pm\bigr)$ in Eq.~\eqref{eq:clifflike}, and turn out to be $(2+1)$-dimensional gamma matrices. The decoupled equations~\eqref{eq:gammap}--\eqref{eq:gammam} are equivalent to
	\begin{align}
		\bigl(\iu\hat\Gamma_{+}^{a}\partial_a-m\id_2\bigr)\hat\psi_{+}=0\,,\label{eq:hatGammap}\\
		\bigl(\iu\hat\Gamma_{-}^{a}\partial_a-m\id_2\bigr)\hat\psi_{-}=0\,,\label{eq:hatGammam}
	\end{align}
	which are in the form~\eqref{eq:DiracLow}, hence $(2+1)$-dimensional Dirac equations, defined on the two-dimensional images of $P_+$ and $P_-$, respectively.

	\subsection{Discussion of the results}\label{sec:remarks}

	\subsubsection{Reconnecting the pictures.}

	In order to check the consistency between the results of the last two subsections, we write $\kappa^3$ in the standard representation. Since
	\begin{equation}
		\kappa^3_\mathrm{D} = \gamma_{\mathrm{D}}^3 \gamma_{\mathrm{D}}^5 =\diag(1,-1,-1,1) ,
	\end{equation}
	its eigenspaces corresponding to $1$ or $-1$ are exactly the subspaces spanned by the $\uparrow$ and $\downarrow$ spinor components, respectively. Accordingly, Eqs.~\eqref{eq:Gammaup}--\eqref{eq:Gammadown} and Eqs.~\eqref{eq:gammap}--\eqref{eq:gammam} are just different expressions (in terms of algebraic objects of different orders, $2$ and $4$, respectively) of the same couple of $(2+1)$-dimensional Dirac equations.

	\subsubsection{Properties of the decoupling.}

	First of all, we check that the decomposition underlying our decoupling, as well as our reduced equations are covariant with respect to the $(2+1)$-dimensional  Lorentz group~\eqref{eq:lambda}. Indeed, let us recall that $\gamma^5$ anticommutes with all the gamma matrices, hence commuting with all the generators of the restricted Lorentz group. As a consequence, we observe that $\kappa^3$ is left invariant by the restricted Lorentz transformations of the form~\eqref{eq:lambda}:
	\begin{align}
		D(\varLambda^{-1})\kappa^3 D(\varLambda)=
		D(\varLambda^{-1})\gamma^3	D(\varLambda)
		D(\varLambda^{-1})\gamma^5	D(\varLambda)
		={\varLambda^3}_\mu\gamma^\mu\gamma^5
		=Q\gamma^3\gamma^5
		=\kappa^3\,,
	\end{align}
	for $Q=+1$, with $D$ denoting the $(\tfrac{1}{2},0)\oplus(0,\tfrac{1}{2})$ representation of the restricted Lorentz group.

	The reduction of the original equation in a pair of decoupled $(2+1)$-dimensional Dirac equations can also be interpreted at the level of Lagrangians as follows: the reduced Lagrangian~\eqref{eq:lagrDesc} can be written, without any further assumption, as the sum of two independent Lagrangians,
	\begin{equation}\label{eq:lagrDescDecomp}
		\tilde{\mathcal L}
		=\overline \Psi_+\qty(\iu\gamma_+^a\partial_a-m)\Psi_+
		+ \overline \Psi_-\qty(\iu\gamma_-^a\partial_a-m)\Psi_-
		=\tilde{\mathcal L}_+ +\tilde{\mathcal L}_-\,,
	\end{equation}
	with Eqs.~\eqref{eq:gammap}--\eqref{eq:gammam} being their respective Euler-Lagrange equations.

	\subsubsection{``Block-diagonal'' representations.}

	Let us try to decompose $\gamma^3$ in a similar way as Eq.~\eqref{eq:gammaADecomp}.
	Its behaviour with respect to the projections $P_\pm$,
	\begin{align}
		\gamma^3 P_\pm=P_\mp\gamma^3\,,
	\end{align}
	yields the decomposition
	\begin{align}\label{eq:gamma3Decomp}
		\gamma^3=\gamma^3_{+,-}+\gamma^3_{-,+}\,,
	\end{align}
	where we have set
	\begin{align}\label{eq:gamma3PM}
		\gamma^3_{\pm,\mp}=P_\pm\gamma^3 P_\mp=\gamma^3 P_\mp=P_\pm\gamma^3\,.
	\end{align}
	The decompositions~\eqref{eq:gammaADecomp} and~\eqref{eq:gamma3Decomp} of the gamma matrices extend to the vector current~\eqref{eq:jmuvett}, since by setting
	\begin{align}\label{eq:japm}
		j_\pm^a&=\Psi^\dagger_\pm\gamma^0_\pm\gamma^a_\pm\Psi_\pm\,,\\
		j_{\pm,\mp}^3&=\Psi^\dagger_\pm\gamma^0_\pm\gamma^3_\mp\Psi_\mp\,.
	\end{align}
	we can accordingly write
	\begin{align}
		\overline{\Psi}\gamma^a\Psi&=j_+^a+j_-^a\,,\label{eq:currdecompa}\\
		\overline{\Psi}\gamma^3\Psi&=j_{+,-}^3+j_{-,+}^3\,.\label{eq:currdecomp3}
	\end{align}
	In matrix terms, Eq.~\eqref{eq:gamma3Decomp} tells us that, in the representation~\eqref{eq:projrepr}, where each $\gamma^a$ is block-diagonal, $\gamma^3$ has to be off-block-diagonal, specifically:
	\begin{align}
		\hat\gamma^3=\mqty(0 & \hat\Gamma^3_{+,-}\\\hat\Gamma^3_{-,+}&0 )\,.
	\end{align}

	The decomposition, determined by the descent, of the free Dirac theory into two independent models is thus clearly visualized in any representation such that the first three Dirac matrices are block diagonal. The Dirac equation takes the form:
	\begin{align}
		\left(\begin{BMAT}[4pt]{cc}{cc}
			\iu\hat\Gamma_{+}^{a}\partial_a-m\id_2&\iu\hat\Gamma_{+,-}^{3}\partial_3\\
			\iu\hat\Gamma_{-,+}^{3}\partial_3&\iu\hat\Gamma_{-}^{a}\partial_a-m\id_2
		\end{BMAT}\right)
		\left(\begin{BMAT}[4pt]{c}{cc}\hat\psi_+\\\hat\psi_-\end{BMAT}\right)=0\,.
	\end{align}
	In this way, since terms proportional to $\partial_3$ are cancelled by the descent condition, the reduced equation~\eqref{eq:descent} can be recast in the block-diagonal form
	\begin{align}
		\left(\begin{BMAT}[4pt]{c1c}{c1c}
			\iu\hat\Gamma_{+}^{a}\partial_a-m\id_2 & 0\\
			0 & \iu\hat\Gamma_{-}^{a}\partial_a-m\id_2
		\end{BMAT}\right)
		\left(\begin{BMAT}[4pt]{c}{c1c}
			\hat\psi_+\\
			\hat\psi_-\end{BMAT}\right)=0\,.
	\end{align}
	and the decoupling into two $(2+1)$-dimensional Dirac equation manifestly emerges.

	The computations in Section~\ref{sec:sr} can be used to single out a particular ``block-diagonal representation'', that can be defined through the unitary transformation relating it to the standard representation. By permuting the standard components as
	\begin{equation}\label{eq:perm}
		\hat\Psi=U\Psi_{ \mathrm{D}}
		=\left(\begin{BMAT}[1pt]{c}{cccc}
			\phi_\uparrow\\ \chi_\uparrow\\
			\phi_\downarrow\\ \chi_\downarrow	\end{BMAT}\right)\,, \qquad
		U=		\left(\begin{BMAT}[1pt]{cccc}{cccc}
			\vphantom{\phi_\uparrow}\mathmakebox[\widthof{$\phi_\uparrow$}]{1}&0&0&0\\ 	0&0&0&\vphantom{\chi_\uparrow}\mathmakebox[\widthof{$\chi_\uparrow$}]{1}\\ 	0&\vphantom{\phi_\downarrow}\mathmakebox[\widthof{$\phi_\downarrow$}]{1}&0&0\\ 	0&0&\vphantom{\chi_\downarrow}\mathmakebox[\widthof{$\chi_\downarrow$}]{1}&0
		\end{BMAT}\right)\,,
	\end{equation}
	the corresponding gamma matrices, $\hat\gamma^\mu =U\gamma_{\mathrm{D}}^\mu U^{\dagger}$, explicitly read
	\begin{align}\label{eq:Diracfreeunci}
		\hat\gamma^0=\mqty( \sigma^3&0 \\ 0&\sigma^3)\,,
		&&
		\hat\gamma^1=\mqty( \iu\sigma^2&0 \\ 0&\iu\sigma^2)\,,
		&&
		\hat\gamma^2=\mqty( -\iu\sigma^1&0 \\ 0&\iu\sigma^1)\,,
		&&
		\hat\gamma^3
		=\mqty( 0&\sigma^1 \\ -\sigma^1&0)\,,
	\end{align}
	and the resulting $(2+1)$-dimensional equations coincide with Eqs.~\eqref{eq:Gammaup}--\eqref{eq:Gammadown}.

	The algebraic background of this discussion  as well as its generalization to arbitrary dimensions will be the topic of future reserach.

	\subsubsection{Conserved \texorpdfstring{$\kappa_3$}{k3}-charge.}

	An interesting consequence of the descent is that the transformation
	\begin{equation}\label{eq:sym}
		\Psi(x)\to \mathrm{e}^{\iu\theta \kappa^3}\Psi(x)
	\end{equation}
	becomes a symmetry of the reduced Lagrangian~\eqref{eq:lagrDesc}, with Noether current
	\begin{equation}
		j_{\kappa^3}^\mu=\overline\Psi\gamma^\mu \kappa^3\Psi\,.
	\end{equation}
	We stress that the transformation~\eqref{eq:sym} is not a symmetry of the Dirac Lagrangian~\eqref{eq:lagrDirac}, as $\kappa^3$ does not commute (in fact, it anticommutes) with $\gamma^3$:
	\begin{equation}\label{eq:Pgamma3}
		\comm{\gamma^3}{\kappa^3}=-2\gamma^5\,, \qquad \acomm{\gamma^3}{\kappa^3}=0\,.
	\end{equation}
	To better understand the physical meaning of the symmetry~\eqref{eq:sym}, it is useful to evaluate the Noether charge
	\begin{equation}
		Q_{\kappa^3}=\int j^0_{\kappa^3}\, \dd^2{\vb*x}=\int\left(\abs{\Psi_+}^2-\abs{\Psi_-}^2\right) \dd^2{\vb*x},
	\end{equation}
	whose conservation implies that, after the descent, the current $I_+=\partial_0 \int \abs{\Psi_+}^2 \dd^2{\vb*x}$ is always balanced by $I_-=\partial_0 \int \abs{\Psi_-}^2 \dd^2{\vb*x}$. The theory is split in two sectors with  superselection charge $\kappa_3=P_+-P_-$.

	\subsubsection{Analogy with the massless theory.}

	We can draw a somewhat broad parallel between the reduced Dirac theory and the massless case. When $m=0$, the Dirac equation splits into two uncoupled Weyl equations, a left- and a right-handed one, in terms of the left- and the right-handed part of the wavefunction, which are eigenvectors of $\gamma^5$ corresponding to the eigenvalues $1$ and $-1$ respectively. This decomposition has a restricted covariance, and is manifest in the Weyl representation. Moreover the chiral tran\-sfor\-ma\-tion $\Psi\to \mathrm{e}^{\iu\theta \gamma^5}\Psi$ is a symmetry of the massless Dirac Lagrangian~\eqref{eq:lagrDirac}, associated to the axial current $\overline{\Psi}\gamma^\mu\gamma^5\Psi$. The roles of mass, chirality, and the Weyl representation for the massless case are played by the $z$-component of the momentum, the superselection charge $\kappa^3$, and any block-diagonal representation in the reduced theory, respectively.

	\section{Dimensional reduction of the minimally coupled Dirac theory}\label{sec:diracem}

	\subsection{Descent of the Maxwell equations}\label{sec:freeem}
	Let us recall the basics of electromagnetic (EM) theory in presence of non-dynamical sources. In the field formulation, the Maxwell equations,
	\begin{numcases}{}\label{eq:MaxwellHom}
		\epsilon^{\mu\nu\rho\sigma}	\partial_{\nu}F_{\rho\sigma}=0\,,\\
		\partial_{\nu}F^{\nu\mu}=j^{\mu}\,,            \label{eq:MaxwellInhom}
	\end{numcases}
	are a system of eight first-order differential equations for the components of the EM field tensor $(F^{\mu\nu})$, with the source terms corresponding to the components of the four-current $(j^{\mu})$:
	\begin{equation}
		\label{eq:Fj}
		(F^{\mu\nu})=
		\left(\begin{array}{@{}cccc@{}}
			0 & -E_x & -E_y & -E_z\\
			E_x & 0 & -B_z & B_y\\
			E_y & B_z & 0 &-B_x\\
			E_z & -B_y&B_x  & 0
		\end{array}\right),
		\qquad
		(j^\mu)=	\left(\begin{array}{@{}c@{}}
			\rho\\
			J_x\\
			J_y\\
			J_z
		\end{array}\right).
	\end{equation}
	In the potential formulation, the homogeneous equations~\eqref{eq:MaxwellHom} are equivalent to the existence of a four-potential $(A^\mu)=(\varPhi, \vb*A )$, determining the field components via
	\begin{equation}\label{eq:MaxIden}
		F^{\mu\nu}=\partial^\mu A^\nu-\partial^\nu A^\mu\,,
	\end{equation}
	and the inhomogeneous equations~\eqref{eq:MaxwellInhom} are rewritten as a system of four second-order differential equations in the $4$-potential components,
	\begin{equation}\label{eq:poten}
		\partial_\rho(\partial^\rho A^\mu-\partial^\mu A^\rho)=j^\mu\,,
	\end{equation}
	that can be obtained as the Euler-Lagrange equations of the EM Lagrangian
	\begin{align}\label{eq:lagrEM}
		\mathcal{L}_{\mathrm{em}}
		&=-\tfrac{1}{4}F_{\mu\nu}F^{\mu\nu}-J_\mu A^\mu
		=\tfrac{1}{2}\qty(\vb*{E}^2-\vb*{B}^2)-\rho\,\varPhi+\vb*J\cdot\vb*A\,,
	\end{align}
	which is invariant with respect to gauge transformations
	\begin{align}
		A^\mu(x)\to A^\mu(x) -\partial^\mu\chi(x)\,.
	\end{align}

	The dimensional reduction of EM has been extensively discussed in~\cite{descent}. The descent from $3$ to $2$ spatial dimensions of the Maxwell equations~\eqref{eq:MaxwellHom}--\eqref{eq:MaxwellInhom} is quite similar to the one of the free Dirac equation of Section~\ref{sec:freedirac}: chosen $z$ as the descent direction, Maxwell's differential problem is specialized into a $z$-in\-de\-pen\-dent one by imposing the conditions
	\begin{equation}\label{eq:de3EM}
		\partial_3F^{\mu\nu}=0\,, \qquad \partial_3 j^{\mu}=0\,,
	\end{equation}
	which are covariant with respect to the $\mathrm{O}(1,2)$ group~\eqref{eq:lambda}.
	As a consequence, the original system splits up into two decoupled subsystems of four equations:
	\begin{align}
		&	\begin{cases}
			\epsilon^{b c d} \partial_{b} F_{c d}=0\,,
			\\
			\partial_{b}F^{ba}=j^{a},								\label{eq:Fab}
		\end{cases}\\
		&\begin{cases}
			\epsilon^{a b c} \partial_{b} {F_{c}}^3=0,
			\\
			\partial_{b}F^{b 3}=j^3\,,							\label{eq:Fa3}
		\end{cases}
	\end{align}
	and both the EM field tensor and the $4$-current are partitioned into two mutually independent sectors, according to the $\mathrm{O}(1,2)$ decomposition:
	\begin{align}
		\label{eq: 2+1decomposition}
		(F^{\mu\nu})=
		\left(\begin{array}{@{}ccc|c@{}}
			0 & -E_x & -E_y & -E_z\\
			E_x & 0 & -B_z & B_y \\
			E_y & B_z & 0 & -B_x\\
			\hline
			E_z & -B_y& B_x  & 0		\end{array}\right),
		&&
		(j^\mu)=	\left(\begin{array}{@{}c@{}}
			\rho\\ J_x\\ J_y\\ \hline J_z
		\end{array}\right).
	\end{align}
	As the two Dirac equations, these two subsystems describe independent EM worlds, each evolving on its own. We label them according to the number of electric and magnetic components involved:
	\begin{itemize}
		\item the ``EEB'' equations~\eqref{eq:Fab} are written in terms of the field components $(E_x,E_y,B_z)$, determining the tensor block $(F^{ab})$, and of the sources $(\rho,J_x,J_y)$, determining the vector block $(j^a)$;
		\item the ``BBE'' equations~\eqref{eq:Fa3} involve the components $(B_x,B_y,E_z)$, determining the vector block $(F^{a3})$, and the source $j^3=J_z$.
	\end{itemize}
	This splitting is analogue to the one occurring in the description of the stationary EM phenomena, obeying the two independent theories of electrostatics and magnetostatics: while the descent~\eqref{eq:de3EM} consists in assuming all the EM quantities to be uniform along a spatial direction, the stationary equations are obtained by requiring uniformity along the time direction.

	Since the descent is performed on the equations of motion of the fields, the decoupling at the level of the potentials is not as manifest as in the field formulation. The descent conditions~\eqref{eq:de3EM}
	actually do \textit{not} imply $\partial_3A^{\mu}=0$, and the relations in Eq.~\eqref{eq:poten} specialize into
	\begin{numcases}{}
		\partial_{a}\partial^{a}A^{b}-\partial^{b}\partial_{a}A^{a}= j^{b}\,,   \label{eq:pota}\\
		\partial_{a}\partial^{a}A^3-\partial_3\partial_{a}A^{a}=j^3\,.   \label{eq:pot3}
	\end{numcases}
	The EEB subsystem translates into Eq.~\eqref{eq:pota}, which involve no $z$ derivative term, and determine a $(2+1)$-dimensional problem, in terms of the $(2+1)$-dimensional vectors $(A^a)$ and $(j^a)$ only, that is, of the components of the four-potential and of the four-current ``transversal to the descent direction''. The BBE subsystem translates into Eq.~\eqref{eq:pot3}, involving not only the components $A^3$ and $j^3$ ``along the descent direction'', but also $(A^a)$, via the term $\partial_3 \partial_b  A^b$, which is also the only surviving $z$-derivative.
	Therefore, while the components $(A^a)$  evolve on their own, the evolution of $A^3$ is affected by the value of $(A^a)$, as opposed to what happens at the level of the fields, where $(F^{ab})$ and $(F^{a3})$ evolve irrespectively of each other.

	In order to decouple the potential components into two independent theories, part of the gauge freedom must be used to \emph{impose} $\partial_3 A^{\mu}=0$. Then, $(F^{a3})$ is determined only by $A^3$, and Eq.~\eqref{eq:pot3} reduces to $\partial_a\partial^a A^3=j^3$, namely a $(2+1)$-dimensional wave equation for $A_3$. This choice is analogous, in the context of stationary EM, to describing fields in terms of a time-independent four-potential, decomposing into a scalar and purely electrostatic component, and a vector and purely magnetostatic one.

	\subsection{Descent of the coupled Dirac-Maxwell equations}
	The Lagrangian associated with a Dirac fermion, of mass $m$ and electric charge $q$, interacting with an EM field, is the sum of the free EM Lagrangian~\eqref{eq:lagrEM}, and a modified Dirac Lagrangian~\eqref{eq:lagrDirac}, where, according to the minimal coupling prescription, $p^\mu\to p^\mu-q A^\mu$, the ordinary derivatives are replaced by the covariant ones:
	\begin{equation}
		\partial_\mu\to D_\mu=\partial_\mu+\iu qA_\mu .
	\end{equation}
	This requirement promotes the $\mathrm{U}(1)$ invariance of the free Dirac theory to a gauge symmetry:
	\begin{align}
		\Psi(x)\to\mathrm{e}^{\iu q\chi(x)}\Psi(x) , \qquad A_\mu(x)\to A_\mu(x) -\partial_\mu\chi(x) .
	\end{align}
	Therefore, the resulting Lagrangian is the sum of the free Lagrangians~\eqref{eq:lagrDirac} and~\eqref{eq:lagrEM}, with an additional interaction term coupling the EM four-potential and the fermionic current~\eqref{eq:jmuvett}:
	\begin{align}
		\mathcal L_{\mathrm{mc}}
		&=-\tfrac{1}{4}F_{\mu\nu}F^{\mu\nu}+
		\overline \Psi\qty(\iu\gamma^\mu \partial_\mu-m)\Psi
		-qA_\mu\overline \Psi \gamma^\mu \Psi\,.
	\end{align}
	The corresponding Euler-Lagrange equations, which will be referred to as the Dirac-Maxwell equations, read
\begin{eqnarray}
		(\iu\gamma^\mu \partial_\mu-m)\Psi
		&=&qA_\mu\gamma^\mu \Psi\,,
		\label{eq:dirMax} \\
		\partial_\mu\partial^\mu A^\nu-\partial^\nu \partial_\mu A^\mu
		&=& q\overline \Psi \gamma^\nu \Psi\,.
		\label{eq:maxDir}
\end{eqnarray}
	This is a system of four first-order and four second-order partial differential equations in the four complex components of $\Psi$ and the four real components of $A^\mu$, coupled to each other. Each equation has a nonlinear term of order zero, which has been written on the right-hand side, to suggest its interpretation as a dynamical source. Equation~\eqref{eq:dirMax} is a Dirac equation minimally coupled to a dynamical EM field, whereas Eq.~\eqref{eq:maxDir} represents the dynamics of an EM field with a dynamical spinor current.

	We are now going to discuss the dimensional reduction of the Dirac-Maxwell equations. We require that the reduced theory retains gauge-covariance (at the level of the equations of motion). In this regard, while the descent condition for the free EM theory, $\partial_3 F^{\mu\nu}=0$, is covariant, the one for the free Dirac theory, $\partial_3\Psi=0$, is not. We overcome this problem by generalizing the latter to the gauge-covariant condition $D_3\Psi=0$. As a consequence, the spinor is constrained to the form
	\begin{equation}
		\Psi(t,x,y,z)=\exp\left(\iu q\int^z \!A^3(t,x,y,\zeta)\,\dd{\zeta} \right) \Phi(t,x,y)\,,
	\end{equation}
	with $\Phi$ any four-component $z$-independent function. Notice that a dependence of $\Psi$ on $z$ is still allowed, as long as it is limited to a phase factor. This implies that any bilinear quantity $\Psi^\dagger M\Psi$, with $M$ a $4\times 4$ matrix, is $z$-independent:
	\begin{equation}
		\partial_3\left(\Psi^\dag M\Psi\right)=0 .
		\label{eq:physconstr}
	\end{equation}
	In particular, the spinor current is $z$-in\-de\-pen\-dent, consistently with the descent condition~\eqref{eq:de3EM} for an external four-current. In fact, the converse holds too:	it is easy to prove that the constraint~\eqref{eq:physconstr} on physical bilinear observables implies the gauge-covariant constraint on the spinor $D_3\Psi=0$.

	Summing up, we specialize the Dirac-Maxwell differential problem to a $z$-in\-de\-pen\-dent one by imposing the re\-pre\-sen\-ta\-tion-inde\-pen\-dent, Lo\-rentz-co\-va\-riant, and gauge-co\-va\-riant descent conditions:
	\begin{equation}\label{eq:D3Psi}
		D_3\Psi=0 , \qquad
		\partial_3\qty(\partial^\mu A^\nu-\partial^\nu A^\mu)=0 .
	\end{equation}
	Observe that the first equation represents a generalization of Hadamard's method. Each condition pertains to just one of the Dirac-Maxwell equations~\eqref{eq:dirMax}--\eqref{eq:maxDir}, hence their respective descents will be discussed one at a time.

	The Dirac equation~\eqref{eq:dirMax} is reduced to the equation
	\begin{equation}\label{eq:DirRedCova}
		\qty(\iu\gamma^a D_a-m\id_4)\Psi=0\,,
	\end{equation}
	which differs from the free Eq.~\eqref{eq:descent} only by the replacement of the partial derivatives with the covariant ones: the fermion is now minimally coupled to the dynamical vector potential $(A^a)$, whose components appear in the covariant derivatives $D_a$, namely to the $(2+1)$-dimensional EEB model. Using the resolution of identity provided by the projections~\eqref{eq:Ppm}, Eq.~\eqref{eq:DirRedCova} can be rewritten as
	\begin{align}
		\qty(\iu\gamma_{+}^{a}D_a-m\id_4)\Psi_{+}&=0\,,			\label{eq:DirPcoup}\\
		\qty(\iu\gamma_{-}^{a}D_a-m\id_4)\Psi_{-}&=0\,,			\label{eq:DirMcoup}
	\end{align}
	which generalizes the pair~\eqref{eq:gammap}--\eqref{eq:gammam} of $(2+1)$-dimensional free Dirac equations: the two fermionic sectors in $2+1$ dimensions are now minimally coupled to the same low-dimensional EM field.

	The Maxwell equations~\eqref{eq:maxDir} are reduced to
	\begin{align}
		\partial_a F^{ab}
		&= q \overline \Psi\gamma^{b} \Psi\,,										\label{eq:DescMaxDir012}\\
		\partial_a F^{a3}
		&=q \overline \Psi\gamma^3\Psi\,,														\label{eq:DescMaxDir3}
	\end{align}
	which are respectively equal to Eq.~\eqref{eq:pota}, for the EEB model, and Eq.~\eqref{eq:pot3}, for the BBE model, except that the external $(2+1)$-dimensional vector and scalar currents $(j^{ a})$ and $j^3$ have been replaced respectively by the $(2+1)$-dimensional blocks $(q\overline \Psi\gamma^{ a} \Psi)$ and $q\overline \Psi\gamma^3\Psi$ of the dynamical spinor current.

	Summarizing, the reduced Dirac-Maxwell equations can be finally written as
	\begin{equation}
		\begin{cases}
			(\iu\gamma^a \partial_a-m\id_4)\Psi=qA_a\gamma^a \Psi\,,

			\\
			\partial_{a}(\partial^{a}A^{b}-\partial^{b}A^{a})
			= q\overline \Psi \gamma^b \Psi\,,

			\\
			\partial_{a}(\partial^{a}A^{3}-\partial_{3}A^{a})
			= q\overline \Psi \gamma^3 \Psi\,.
			\label{eq:maxDir3}
		\end{cases}
	\end{equation}
	in terms of $\Psi$, $(A^a)$ and $A^3$, or, equivalently, by using the decompositions~\eqref{eq:Decomp},
	\begin{equation}
		\begin{cases}
			\qty(\iu\gamma_{+}^{a}\partial_a-m\id_4)\Psi_{+}=qA_a\gamma_+^a \Psi_+\,,
			\\
			\qty(\iu\gamma_{-}^{a}\partial_a-m\id_4)\Psi_{-}=qA_a\gamma_-^a \Psi_-\,,
			\\
			\partial_{a}(\partial^{a}A^{b}-\partial^{b}A^{a})
			=j^{b}_++j^{b}_-\,,
			\\
			\partial_{a}(\partial^{a}A^{3}-\partial_{3}A^{a})
			=j^3_{+,-}+j^3_{-,+}\,.
		\end{cases}
	\end{equation}
	Furthermore, by a block-diagonal choice~\eqref{eq:diagonal_decomp} for the gamma matrices, the Dirac-Maxwell equations can be recast in the form
	\begin{numcases}{}
		\bigl(\iu\hat\Gamma_{+}^{a}\partial_a-m\id_2\bigr)\hat\psi_{+}=qA_a\hat\Gamma_{+}^{a} \hat\psi_+\,,			\label{eq:dirMaxP}\\
		\bigl(\iu\hat\Gamma_{-}^{a}\partial_a-m\id_2\bigr)\hat\psi_{-}=qA_a\hat\Gamma_{-}^{a} \hat\psi_-\,,			\label{eq:dirMaxM}\\
		\partial_{a}\partial^{a}A^{b}-\partial^{b}\partial_{a}A^{a}
		=j^{b}_++j^{b}_-\,,							\label{eq:EMa} \\
		\partial_{a}\partial^{a}A^{3}+\partial_{3}\partial_{a}A^{a}
		=j^3_{+,-}+j^3_{-,+}\,.	\label{eq:EM3}
	\end{numcases}
	in which the number of independent components is explicit.

	\begin{figure}[tp]
		\centering
		\includegraphics[width=0.8\textwidth]{diagram.png}
		\caption{Relationship between descent and minimal coupling. The solid straight lines represent the descent from a $3+1$ theory to a $2+1$ one, the solid wavy lines represent the minimal coupling, whereas the dotted line represents the restriction of a theory to a particular class of solutions.}
		\label{fig-schema}
	\end{figure}

	\subsection{Discussion of the results}

	As opposed to what happens in the free theories, the Dirac-Maxwell equations do \textit{not} split by descent into two separate, non-interacting sectors, as variables are directly or indirectly coupled to each other. In particular, $\Psi_{+}$ and $\Psi_{-}$ are coupled between them through the EM field. Decoupling is only partial, with Eqs.~\eqref{eq:dirMaxP}--\eqref{eq:EMa} forming a subsystem in which only $\Psi$ and $(A^a)$ explicitly appear, although $A^3$ is still coupled to $(A^a)$ via the term $\partial_{3}\partial_{a}A^{a}$ (which retains a residual $z$-dependence allowed by gauge invariance), and to $\Psi$ via $j^3$.

	However, one can identify two classes of solutions in which decoupling effectively occurs throughout the evolution:
	\begin{enumerate}
		\item The solutions in which $\Psi_{-}$ identically vanishes, and the remaining fields consistently evolve according to
		\begin{equation}
			\begin{cases}
				\qty(\iu\gamma_{+}^{a}D_a-m \id_4)\Psi_{+}=0\,
				\\
				\partial_{a}F^{ab}=j^{b}_
				\\
				\partial_{a}F^{a3}=0\,.
			\end{cases}
		\end{equation}
		\item The solutions in which $\Psi_{+}$ identically vanishes, and the remaining fields consistently evolve according to
		\begin{equation}
			\begin{cases}
				\qty(\iu\gamma_{-}^{a}D_a-m \id_4)\Psi_{-}=0\,
				\\
				\partial_{a}F^{ab}=j^{b}_
				\\
				\partial_{a}F^{a3}=0\,.
			\end{cases}
		\end{equation}
	\end{enumerate}
	Notice that, in order to obtain the above classes of solutions, it is sufficient to impose the initial conditions $\Psi_-(t=0)=0$ and $\Psi_+(t=0)=0$, respectively. We remind that the above set of equations can be used to describe the physics of graphene interacting with an electromagnetic field \cite{graphene} [see comments on Eqs.~\eqref{eq:Gammaup}--\eqref{eq:Gammadown} in Section~\ref{sec:sr}].

	Formally, both these theories describe a two-com\-po\-nent spinor in $2+1$ dimensions interacting with an EEB theory, plus a non-interacting BBE sector which evolves freely. We remark that this result provides a justification \textit{a posteriori} of the traditional formulation of QED in $2+1$ dimensions, in term of two-component spinors.

	On the other hand, a solution with $F^{ab}$ identically vanishing would also entail, for $q\neq 0$, an identically vanishing spinor, since
	\begin{equation}
		\partial_a F^{a3} = 0 \quad \Rightarrow \quad \Psi^{\dagger}\Psi = 0 ,
	\end{equation}
	leaving the free BBE sector as the only nontrivial theory.

	All the results discussed in this work are schematically summarized in  Fig.~\ref{fig-schema}, where the sectors D\textsubscript{+} and D\textsubscript{--} respectively refer to the free $(2+1)$-dimensional Dirac equations~\eqref{eq:Gammap} and~\eqref{eq:Gammam}, whereas the  EEB+D\textsubscript{+} and EEB+D\textsubscript{--} denote the above decoupled theories obtained when $\Psi_{-}=0$ and $\Psi_{+}=0$, respectively.

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

	We have shown that dimensional reduction through the descent method entails a splitting of the Dirac theory for a free spin-\textonehalf{} particle in a $(3+1)$-dimensional flat spacetime into two noninteracting sectors in a $(2+1)$-dimensional spacetime. Each sector obeys an equation that could also be obtained \textit{ab initio} as a Dirac equation in a $(2+1)$-dimensional spacetime, when defined as a formal ``square root'' of the Klein-Gordon equation. In this sense, Dirac's approach and dimensional reduction prove to be mutually compatible.

	We have subsequently applied the reduction procedure to a spin-\textonehalf{} particle minimally coupled to an electromagnetic field: while a complete splitting of the theory does not occur in such a case at the level of equations, two particular classes of solutions exist in which some sectors evolve trivially. In other words, the splitting occurs at the level of solutions. We emphasize that Hadamard's method has been extended by using the covariant derivative.

	Interestingly, the existence of \emph{two} $(2+1)$-dimensional Dirac equations after dimensional reduction is not a mere academic curiosity, since an analogous result is obtained, although under different physical conditions, in the low-energy description of the electronic properties of graphene close to the Dirac points~\cite{graphene}.

	We have shown that the dimensional reduction implies the emergence of a superselection charge, which is the analogous of chirality in the massless case. As it is well-known, the chiral symmetry of the $(3+1)$-dimensional massless Dirac Lagrangian is actually broken at the level of quantum field theory, a phenomenon which is usually referred to as chiral anomaly~\cite{Adler,RPBG21}. Therefore, an interesting open question is whether our newly found symmetry~\eqref{eq:sym} is still conserved in a corresponding $(2+1)$-dimensional quantum field theory.

	\section*{Acknowledgments}

	This research was funded by MIUR via PRIN 2017 (Progetto di Ricerca di Interesse Nazionale), project QUSHIP (2017SRNBRK), by the Italian National Group of Mathematical Physics (GNFM-INdAM), by Istituto Nazionale di Fisica Nucleare (INFN) through the project ``QUANTUM'', and by Regione Puglia and QuantERA ERA-NET Cofund in Quantum Technologies (Grant No. 731473), project QuantHEP.

	\section*{References}

redpajama_set_name: RedPajamaArXiv


What if Anakin Had KILLED Palpatine and Ruled the Republic?
Posted on April 11, 2018 November 16, 2018 Author Diego Comments Off on What if Anakin Had KILLED Palpatine and Ruled the Republic?
What if instead of falling for Palpatine's deceit Anakin had killed Palpatine and taken control of the Republic? We go through that scenario in this episode.


The evocative title of this book, The Alchemy of Wolves and Sheep, hints at a journey to be taken by readers. Intrigued, I entered this book with curious anticipation in hope of finding a text that integrates our working knowledge of complex trauma alongside the phenomenon of dissociation, while advancing a culturally reliable framework. While this text falls short of delivering an integrated cultural perspective, there are gems scattered across chapters that facilitate in the reader that sense of taking a voyage into unchartered territories. Admittedly, I read the book twice as it was not clear until the last chapter that Schwartz navigated trauma literature through Jungian and archetypal theoretical territories. Having grasped this, in my second read, the theses underlying the earlier chapters made more sense. Regardless of orientation, this book topic is essential to our ongoing debate about the relationship between dissociation and trauma within the fields of counselling and psychotherapy.


The wheel of cheese on the Raclette machine is heated to the right temperature and scraped onto a plate, to be enjoyed with condiments such as bread, raw vegetables, fruit, meat, potatoes and many more.
If the only melted cheese you've tried is on pizza or a toastie, then you have a lot to experience in the incredibly satisfying world of molten cheese. But when it comes to melting cheese not all cheeses are equal – some melt better than others, allowing unlocked textures, flavours and taste to ooze out. Being a cheese lover, I sure you will like watching the cheese melt and its flavour.
Chef Pierre Gay, a cheese master, cheese connoisseur and expert from France, shared his extensive knowledge on European cheese through a Raclette party jointly organized by the French Interbranch Dairy Organization and European Union, to promote the pre-eminence and finesse of European Cheese.
The beautiful assortment of condiments such as sliced capsicum, carrots, mushrooms, meat as well as pickles, grapes and pineapple, to be enjoyed with the various types of European Cheese.
"Raclette is a fairly new concept in Malaysia and for a country that loves cheese so much, this is an experience that Malaysians do not want to miss," explains Chef Pierre Gay. "Raclette is a cheese made of cow milk that has been broiled or melted under direct heat until oozy and a bit crispy and usually scraped over potatoes, crunchy vegetables or cured meats" says Chef Pierre Gay.
Chef Pierre Gay, the cheese master, cheese connoisseur and expert from France explaining what Raclette means and demonstrating a Raclette machine at the European Cheese event on 22 March at Cocott' KL.
He went on to explain that watching the cheese melt and being scraped over the food is part of the sensory experience of a Raclette. Chef Pierre Gay is the owner Pierre Gay Fromagerie – one of the oldest cheese houses in Annecy, in the Rhone-Alps in eastern France. The fromagerie known in the region for its fine cheeses was started in 1935, and Pierre took over the company in 1989 from his father. Pierre's dedication and efforts paid off when he was crowned Meilleur Ouvrier de France (MOF) year 2011, a recognition of being the top in his trade.
With a mission to educate the public in Malaysia of the different varieties of European cheeses, and how best to enjoy them, Chef Pierre Gay gave an insight into the world of European cheese through a Raclette Party recently at Cocott' KL.
Members of the media and guests also had a chance to melt and enjoy Brie, Reblochon, Fourme D'Ambert, Comte, Saint Nectaire, Gouda and Scamorza over table top Raclette machines, enjoyed with various cooked and raw vegetable such as Sliced Fresh Pepper, Button Mushroom, Tomatoes, Corn Kernels, Sliced Green Onions, Pineapple, Baked Potatoes, Cooked Carrots & Peas, Sliced Conference Pears, Sliced Green Apples and Seedless Grapes along with different meats including lamb, beef bacon, smoked chicken and sausages. The tantalizing spread also featured baguette bread, trimmings such as silverskin onions, gherkins/cornichons and spices such as pepper, salt and chilli flakes.
(from left) Chef Pierre Gay, the cheese master from France, Richard Krystkowiak representing SOPEXA and Chef Jean Michel Fraisse posing in front of various types of European cheeses demonstrated at the Raclette fete on 22 March 2018 at Cocott' KL.
The Chef used Raclette cheese which he melted in a large Raclette machine, then scraped over the cured meats and condiments to the delight of the guests. "The Raclette cheese is an excellent melting cheese and the main cheese used in raclettes and fondues. Pungent yet savoury, the Raclette's pH level gives it the perfect melting point resulting in a molten liquid that is stringy and holds its shape at the same time," he added.
Considered to be one of the most versatile foods in the world, cheese can be eaten as it is, paired with a classy wine or even incorporated into any dishes – from cold to hot, sweet to spicy, from appetisers to desserts – most cheese recipes only need a few additional items added to the star ingredient to make them fantastic. With stringent quality control and regulations in Europe, European cheeses are always of the highest quality, and the best cheese products are readily available to the consumers in Malaysia.
A Raclette which is a new concept in Malaysia is a great way to enjoy European cheeses and can be recreated easily at home to be enjoyed with friends and family, giving the traditional cheese platter a noveler and enjoyable upgrade. Organise your own Raclette Party and get ready to melt into an epicurean journey with the world's best European cheeses.


text:
A CLASSIC REMATCH
New York and Baltimore head for another titanic national championship showdown
Tex Maule
The best football game ever played—last year's championship epic between the Baltimore Colts and the New York Giants—seems likely to return for another showing this season, the casts even stronger.
The New York Giants, who grow mightier from week to week as the pro football year wanes, clinched the Eastern Conference title by demolishing Cleveland 48-7 last Sunday; Baltimore, after a shaky spell in midseason, took over the Western Conference lead by outrunning the San Francisco 49ers through the last handful of minutes in their game 34-14. Baltimore need only tie the feckless Los Angeles Rams on Saturday to ensure its second conference title.
San Francisco, which plays the Green Bay Packers, and the Chicago Bears, who meet Detroit, retain mathematical chances to tie for the Western title, but in the light of the recent form of the Colts, theirs must be faint hopes.
The Colt and Giant victories were very similar performances, although the New York triumph was the more impressive. Both these teams are superbly equipped with very large, very strong football players who have been in the National Football League long enough to qualify as experts in their arduous trade. Both have really fine quarterbacks, both have good running backs, cagey, sure-handed receivers, and both have a certain serene confidence in their ability which allows them to retain complete poise and assurance in almost any circumstance.
The Colts flew to San Francisco Thursday, facing a jinx which had seen them lose 10 games in a row in Kezar Stadium. They were a relaxed, unperturbed group, however; they laughed off the inevitable predictions of defeat they were exposed to in the Clift Hotel as they waited for Saturday afternoon. A bootblack, not recognizing a member of the Colt party in a barber chair, bet a shoe-polish salesman a year's free shines against 75 cans of shoe polish, and the Colt grinned. The bishop of the Catholic diocese in San Francisco granted special dispensation for the Colt Catholics to eat meat on Friday evening. "I shouldn't, after what you did to us in Baltimore," he said gruffly. "But I will because I don't want your boys to collapse after our boys whip them."
Surprisingly, San Francisco fans seemed immensely confident that the 49ers could reverse the 45-14 shellacking visited upon them by the Colts in Baltimore only two weeks before. Tickets were impossible to find at $50. apiece. "Sell?" said one fan. "Gosh, it'd cost you five bucks just to look at my ticket." The chef who prepared the team meals for the Colts at the hotel advised Coach Weeb Ewbank, Owner Carroll Rosenbloom and General Manager Don Kellett that the 49ers would whip the Colts; he was unceremoniously ushered from the dining room as a result. Ewbank worried for a while that the chef might retaliate by serving the team a case of food poisoning, but the meals were good and wholesome.
Riding to the game, the Colts were relaxed and confident. When the bus passed University of San Francisco, Bill Pellington, a linebacker, pointed to the campus. "Hey, Gino," he said to Marchetti, Baltimore's all-pro defensive end. "They tell me you were so famous there that they retired your grades." Marchetti, who may have attended four classes during his career at USF, nodded agreement.
In their dressing room, the Colts were still relaxed, although they were no longer joking. In the 49er dressing room, the atmosphere was grim and still. "It's our turn now," someone said into the silence.
But it wasn't. The 49ers played a much better game than they had in the massacre in Baltimore, but they simply could not match Colt manpower or finesse. Raymond Berry, one of the finest ends of the last decade, caught two touchdown passes, one in the end zone on a prodigious leap for an unbelievable grab. ("It'll work, Coach," he told Ewbank earlier in the game, of the pattern he ran to get free. "I'm aware of it, Raymond," said Ewbank. "I just wanted you to know," Berry replied.) Lenny Moore ran 64 yards to a touchdown with a Unitas pass after changing his planned route in mid-course; he was supposed to slant in over the middle. "I saw that cat come in too far," he said, speaking of the defensive halfback covering him. "So I just broke back out to the outside and hoped Johnny would read me." Johnny read, him, and the pass found Moore in the clear behind the 49er defender.
Milt Davis, thin and scholarly, who plays defensive halfback for the Colts (and off season studies dentistry), had one of the six interceptions which crushed San Francisco hopes. He dived to pick off the pass, fell to his knees and ran 57 yards for a touchdown. "Somebody was hollering 'Lateral, lateral!' " he said. "I got about 11 blocks though and I figured I'd go for the six." Davis later pulled a hamstring muscle and may miss the Ram game. Alan Ameche, the powerful Colt fullback, has bruised ribs but should play. Both would be available for a championship game.
ORGANIZED THIEVERY
The six interceptions—four off John Brodie, who started the game, and two off Tittle—were the result of a closely coordinated pass defense. "The line, the linebackers and the secondary worked together very well," Ewbank said. "We put enough pressure on their passer, and the linebackers, particularly, did a fine job." The Colt linebackers have intercepted 13 passes this season, four of them in this game. Don Shinnick alone has six interceptions, which must be close to a record for linebackers.
The Giants hit an offensive peak against the Browns, behind Charlie Conerly, a quarterback who ages as gracefully as wine. The two Giant lines—offense and defense—established astonishing mastery over Cleveland. A pro scout, watching from the press box, was awed.
"I've never seen anything like it," he said. "They don't have to red dog on defense. The four big men put on enough pressure. And no Brown has gotten through the Giant offensive line. I've never seen a line take a whipping like that before. The Giants are on them like mad dogs."
The Giants, who like beating Cleveland more than any other team in the league, completely dominated the game. Cleveland had injuries: Ray Renfro, one of its top receivers, was out with a pulled muscle; Walt Michaels, a key linebacker, was hurt and some of the other Cleveland linemen were hobbled by various sprains and contusions. Jim Brown, Cleveland's phenomenal fullback, was knocked silly by a cracking collision with the Giant line, abetted by Linebacker Sam Huff, early in the game. He played in a daze for most of the first period before he was taken out.
But it would have made little difference had all the Browns been healthy. Conerly called a beautifully intelligent game, sending his receivers scooting into gaps in the Cleveland secondary time and again' and throwing with marvelous accuracy. The Giants, after running Washington out of Yankee Stadium last week, passed the Browns out of contention in this game. Conerly threw 21 times, completed 14 for 271 yards and three touchdowns; Don Heinrich, the much-booed replacement for the Giants' old pro, had a respectable 8 of 16 for 92 yards and one touchdown.
The supposedly sophisticated New York fans poured out of the stands about two minutes before the game ended and made a determined effort to tear down the goal posts while the two teams were going through the motions of finishing the game. Police were powerless to stop them; then the public-address announcer informed the unruly that unless they cleared the field, the game could be declared forfeit to the Browns. In the face of so grievous a contingency, the mob reluctantly squeezed itself behind the side and end lines and the game was played out.
"I wouldn't have asked for a forfeit even if the last two minutes of the game hadn't been played," Cleveland Coach Paul Brown said afterward. He had discreetly led his warriors to the dressing room when the New York fans invaded the field, returning when order was restored. "We didn't even belong on the same field with them today," he added.
Brown's sentiments were not echoed by Coach Red Hickey of San Francisco. "We gave it all we had," he said dolefully after his loss to Baltimore. "We played a far better game than we played against them in Baltimore. But we just couldn't cope with them. This Baltimore team hasn't any flaws. And they've got a great bench. Ameche gets hurt and they send in Pricer and it doesn't slow them down at all. And Davis goes out and they've got another defensive halfback just about as good. We had some boys crippled up, but that didn't matter. You're not fit to play in this league if you can't disregard minor injuries. That's the mark of a pro and our boys were pros today."
Then, in answer to a question, he sounded an unwitting warning to the Giants.
"How do I compare the Colts with Cleveland?" he said. "There is no comparison. This is a truly great team—one of the finest pro teams of modern football. Cleveland doesn't compare with them at all."
[originallink:10495411:42535]
MARVIN NEWMAN
DISASTER LOOMS here over Cleveland's Quarterback Milt Plum in the person of giant Giant Rosey Grier. New York defense crushed Plum and the Brown team.
GIANT HALFBACK ALEX WEBSTER STARTS THROUGH YAWNING HOLE IN CLEVELAND LINE TO SET UP ANOTHER NEW YORK TOUCHDOWN
DISHEVELED FANS CRADLE PIECES OF GOAL POST RIPPED DOWN BY HAPPY CROWD
COLOR OF THE WEEK: COLT ON THE LOOSE
Raymond Berry, Baltimore's league-leading pass catcher, clutches a Johnny Unitas pass to his stomach in the end zone of the San Francisco 49ers for Colts' first touchdown in 34-14 rout. The official signals a touchdown as 49ers' Abe Woodson (40) and home town fans watch helplessly.

redpajama_set_name: RedPajamaCommonCrawl


With forecasts for many parts of the country to suffer more snowy and icy conditions the roads are likely to be treacherous for many drivers.
Avoiding driving if possible is often the best advice in such conditions but for many people that's not an option.
With that in mind here are a few things you should and shouldn't do to ensure you stay safe on the roads.
A few simple checks will ensure your car is in top condition in the worst weather.
It pays to be prepared for the worst so make sure you have a bag with essentials, in case you get stuck. Among the vitals are a torch, high-vis jacket or vest, sturdy footwear, shovel, a blanket or extra clothes, warning triangle, ice scraper, phone charger and food and drink.
Keep an eye on what the weather is predicted to do so you can plan your journey. Leave yourself more time to get to your destination and be prepared for delays and accidents that could hold you up.
The best way to stay safe is to stay off the road.
Startlingly, some people still think this is a good idea. The boiling water hitting frozen glass can be enough to crack or shatter the windscreen so just don't do it. Use de-icer, a scraper and some old-fashioned elbow grease.
Major roads are the ones most regularly cleared while smaller routes may not be treated at all, making them more treacherous. If you have an accident you're also less likely to be discovered quickly.
Failing to properly clear all the snow and ice from your car, including windows and roof, is massively dangerous and illegal. Not being able to see properly could land you with a £60 fine and three penalty points.


text:
Wundef TV
Home Tags Talkative pig chatham
Tag: talkative pig chatham
5 Ways to Turn Your Talkertive Nature into Strength
Charles Wundengba - December 1, 2019
Facebook Pay will allow you make payments on WhatsApp, Instagram, and Facebook
Charles Wundengba - November 13, 2019
Facebook is launching a new payments system today, appropriately named Facebook Pay. It will be available across Facebook, Messenger, Instagram, and WhatsApp, and it's...
Does Social Media Cause Depression?
Is using social media making people unhappy? Evidence is mounting that there is a link between social media and depression. In several recent studies, teenage and young adult users who spend the most time on Instagram, Facebook and other platforms were shown to have a substantially (from 13 to 66 percent) higher rate of reported depression than those who spent the least time.
What Marketers Need to Know About LinkedIn
Charles Wundengba - July 19, 2018
What if I told you that LinkedIn, a social media platform that is now 14 years old, is the next big opportunity for marketers...
How to Get Thousands of Views on Your LinkedIn Content
You spent four hours writing that LinkedIn article, and you were convinced that your network would love it. So, you hit the publish button and waited...
How To Get More Organic Reach on Facebook for Pages with Fewer Likes
Charles Wundengba - May 9, 2018
Facebook has almost 2 billion users, 500 million on Linkedin, 1 billion on Youtube and another 300 million on Twitter. Instagram also houses 500...
Are you in your 30s, Avoid these 3 Money Mistakes!
Charles Wundengba - January 12, 2020
The financial decisions you make in your 30s can have a big impact on your future. "Money is a tool. When we learn to use...
How Customers are Risk Rated to Prevent Crime
Richieson Gyeni-Boateng - January 6, 2020
As we begin the year 2020, institutions should put in place measures to determine the risk their customers pose to them. This is because,...
The Difference Between "Invention" and "Innovation"
Charles Wundengba - December 28, 2019
Two and a half years ago, I co-founded Stroome, a collaborative online video editing and publishing platform and 2010 Knight News Challenge winner. From its...
Forgiveness By Stacy M. Amewoyi
You must forgive to cleanse your heart. You can't change the past, nobody can go back in time and avoid situations that hurt us, I am sure that many of us would like to do it to fix what we did wrong and avoid the pain, but it is impossible!
Economic diversity shields African economies amid slowing global GDP growth
The report finds that, despite there being a gloomy backdrop created by slow growth in the United States, China and Europe, Africa remains a relative bright spot with many positive economic stories. Although the continent's two largest economies, Nigeria and South Africa, continue to struggle, the report finds that Africa will become home to half of the 10 fastest-growing economies on the planet over the next five years.
Former Child Soldier Now Turned Sales Motivational Speaker
Almost everything in this world runs on sales. You are selling either a service or a product. For Daniel Choudry, he believes everyone can sell as long as he can get your attitude changed. His an author and calls himself a "sales warrior from East Africa" Personally he has been selling since 2007 while at the university. With background in Business Administration, majoring in marketing from Makerere University Business Institute (MUBS) and a Masters in Marketing to horn his skills in sales indeed you can tell he has no regrets.
10 Things You Probably Didn't Know About the Mexican who is worth about 6% of his country's Gross Domestic Product (GDP), Carlos Slim Helú.
Charles Wundengba - September 15, 2019
Carlos Slim Helú, born 28 January 1940 in Mexico City, is a Mexican business magnate, engineer, investor and philanthropist. From 2010 to 2013,...
Meet Africa's Richest Man, Aliko Dangote
Aliko Dangote GCON (born 10 April 1957) is a Nigerian billionaire, who owns the Dangote Group, which has interests in commodities. The company operates...
He is championing the fight against Revenge Porn in Ghana. Meet the 'Chief Blogger', Barimah Amoaning Samuel.
Revenge porn is the revealing of sexually explicit images or videos of a person posted on the Internet, typically by a former sexual partner, without the consent of the subject and in order to cause them distress or embarrassment. It may refer to all sexually explicit videos and images of a person by former partner, old friend, hacker, blackmailer or whoever for whatever motive. One man has decided to stand in the gap and end the act and his name is Barimah Amoaning Samuel.
Meet the PR Student behind Bronya Artistry
Emmanuel Agyemfra Boateng - September 18, 2019
My visit to the Executive hostel of Odeneho Kwadaso based Christian Service University College early this year prior to their graduation, I...
How Is Digital Innovation Changing Marketing?
In 2013, Adobe conducted a comprehensive research study and found that 76% of marketers believe that marketing changed more in the past 2 years than in the previous 50. Digital technologies were changing rapidly, impacting how marketers build their strategies and market to their audiences. Today, fast digital innovation has been replaced with the strategic implementation of new technologies in marketing.
How to Get Started With Affiliate Marketing – A Beginner's Guide
Being a beginner in affiliate marketing can be an overwhelming place to be. It's been awhile for me, but I have a clear memory of feeling...
Ever thought about freelancing your skills? Read this!
Charles Wundengba - November 1, 2019
As companies scale back on expensive, benefit-heavy workforce, they're increasingly turning to outside--freelance--help. If you've got expertise in the right areas, there's a good chance you can parlay it into a freelance career by sharing your knowledge and skills with a variety of clients.
Six key benefits of DataOps for businesses
Charles Wundengba - September 3, 2019
Today companies need the latest data at their fingertips, without it they limit their agility and ability to maneuver potential competition, says Alasdair Parsonson, Technical Expert at Data Intelligence for Hitachi Vantara
10 Black Makeup Artists on Instagram You Should Follow
Dante Lee - January 2, 2019
For Black women, the makeup world can often be disappointing. Very few brands carry a shade range that complements darker skin tones, making it...
Wundef.com


text:
RUSSIA / USSR
1. conventional submarines
1. DELFIN submarine (1904)
2. "Holland" type submarines (SOM) (1904 - 1907)
3. "Lake" type submarines (OSYOTR) (1904 - 1906)
4. KASATKA submarines (1905 - 1908)
5. KARP submarines (1907)
6. POCHTOVY submarine (1908)
7. MINOGA submarine (1909)
8. KAYMAN submarines (1909 - 1910)
9. AKULA submarine (1911)
10. MORZH submarines (1915)
11. KRAB minelaying submarine (1915)
12. SVYATOY GEORGIY submarine
13. NARVAL submarines (1915 - 1916)
14. BARS submarines (1915 - 1917), YORSH minelaying submarines (1917))
15. "AG" submarines (AG 11) (1916 - 1923)
16. SVYATOY GEORGIY submarine (1917)
17. B.1 submarines
18. V.1 submarines
19. Z.1 minelaying submarines
20. "Series I" submarines (DEKABRIST) (1930 - 1931)
21. L-55 submarine (1918 / 1931)
22. "Series II" minelaying submarines (LENINETS) (1933)
23. "Series III" submarines (SHCHUKA) (1933)
24. "Series V" submarines (Shch-11 KARAS`) (1933 - 1934)
25. "Series V-bis" submarines (STERLYAD`) (1934 - 1935)
26. "Series VI" submarines (M-1) (1934 - 1935)
27. "Series V-bis-2" submarines (Shch-121) (1935 - 1936)
28. "Series VI-bis" submarines (M-53) (1935 - 1936)
29. "Series IV" submarines (P-1) (1936)
30. "Series IX" submarines (N-1) (1936 - 1938)
31. "Series X" submarines (Shch-126) (1936 - 1939)
32. "Series XI" minelaying submarines (L-7) (1936 - 1938)
33. "Series XII" submarines (M-87) (project 40) (1937 - 1943)
34. "Series XIII" minelaying submarines (L-13) (1938 - 1939)
35. "Series IX-bis, IX-bis-2" submarines (S-4) (1939 - 1948)
36. "Series XIV" submarines (K-1) (1939 - 1944)
37. RONIS submarines (1927 / 1940)
38. KALEV submarines (1937 / 1940)
39. "Series X-bis" submarines (Shch-405) (1941 - 1946)
40. "Series XIII-1938" minelaying submarines (L-20) (1941 - 1943)
41. "Series XV" submarines (M-200) (project 96) (1943 - 1953)
42. "Series XVI" submarines (S-47) (project 97)
43. V-1 submarine (1937 / 1944)
44. V-2 submarines (1942 / 1944)
46. TS-1 submarine (1943 / 1944)
48. TS-3 submarine (1936 / ---)
49. M-401 submarine (project 95) (1945)
50. N-22 submarines (1943 - 1944 / 1946)
51. N-24 submarines (1944 / 1946)
52. N-26 submarine (1944 / 1946)
55. TS-16 submarine (1935 / ---)
56. I-41 submarine (1943 / 1949)
58. S-61 submarines (project 613) (1951 - 1958)
59. B-61 submarines (project 611) (1953 - 1958)
60. M-254 submarines (projects 615, A-615, 637) (1953 - 1958)
61. S-99 submarine (project 617) (1955)
63. S-350 submarines (project 633) (1959 - 1961)
64. B-443 submarines (project 641B) (1973 - 1982)
65. B-248 submarines (projects 877, 877V) (1980 - 1994)
66. MS-520 small submarines (project 865) (1988 - 1990)
67. B-340 submarines (projects 636M, 06363) (2005- )
68. B-90 SAROV experimental submarine (project 20120) (2008)
69. B-585 SANKT-PETERBURG submarines (project 677) (2010-)
2. conventional cruise missile submarines
1. K-156 cruise missile submarines (project 651) (1963 - 1968)
3. conventional ballistic missile submarines
1. B-41 ballistic missile submarines (project 629) (1959 - 1962)
4. nuclear powered attack submarines
1. K-3 nuclear powered attack submarines (projects 627, 627A) (1958 - 1964)
2. K-27 nuclear powered attack submarine (project 645) (1963)
3. K-38 nuclear powered attack submarines (projects 671, 671V) (1967 - 1974)
4. K-64 nuclear powered attack submarines (projects 705, 705K) (1971-1981)
5. K-387 nuclear powered attack submarines (project 671RT) (1972 - 1978)
6. K-524 nuclear powered attack submarines (projects 671RTM, 671RTMK) (1977 - 1992)
7. K-278 nuclear powered attack submarine (project 685) (1983)
8. K-239 nuclear powered attack submarines (project 945) (1984 - 1987)
9. K-284 nuclear powered attack submarines (projects 971, 09711, 971I (1984 - 2009)
10. K-534 nuclear powered attack submarines (project 945A) (1990 - 1993)
11. K-560 SEVERODVINSK nuclear powered multipurpose submarines (projects 08850, 08851) (2013- )
12. KS-329 BELGOROD special weapon submarine (project 09852)
13. KHABAROVSK special weapon submarine (project 09851)
5. nuclear powered cruise missile submarines
1. K-45 nuclear powered cruise missile submarines (project 659) (1961 - 1963)
2. K-166 nuclear powered cruise missile submarines (project 675) (1963 - 1968)
4. K-162 nuclear powered cruise missile submarine (project 661) (1969)
5. K-452 nuclear powered cruise missile submarines (project 670M) (1973 - 1980)
7. K-148 nuclear powered cruise missile submarines (project 949A) (1986 - 1996)
6. nuclear powered ballistic missile submarines
1. K-19 nuclear powered ballistic missile submarines (project 658) (1960 - 1962)
2. K-137 nuclear powered ballistic missile submarines (projects 667A, 667AU) (1967 - 1974)
3. K-279 nuclear powered ballistic missile submarines (project 667B) (1972 - 1977)
4. K-182 nuclear powered ballistic missile submarines (project 667BD) (1975)
5. K-424 nuclear powered ballistic missile submarines (project 667BDR) (1976 - 1982)
6. TK-208 nuclear powered ballistic missile submarines (project 941) (1981 - 1989)
7. K-51 IMENI XXVI S'YEZDA KPSS nuclear powered ballistic missile submarines (project 667BDRM) (1984 - 1990)
8. K-535 YURIY DOLGORUKIY nuclear powered ballistic missile submarines (projects 955, 955A) (2012- )
7. midget submarines
1. Drzewiecki type III midget submarines (1880-1881)
2. MATROS PYOTR KOSHKA midget submarine (1904)
3. PORT-ARTURETS midget submarine
4. FOREL midget submarine (1904)
5. "Holland" type midget submarines (NO1) (1914)
6. "APSS" type midget submarines (1935 - 1936)
7. APL midget submarine, PIGMEY midget submarine
8. TM-4 midget submarines (1941 / 1944)
9. "SH" type midget submarines (1948)


text:
CNN Reporter Alisyn Camerota Face-Palms While Listening to Donald Trump Supporters' Voter Fraud Claims
CNN reporter Alisyn Camerota couldn't hide her frustration with a group of Donald Trump supporters after they falsely claimed that there was widespread voting fraud in California. Watch her reaction in the clip above.
PHOTOS: Donald Trump's Most Offensive and Outrageous Quotes Ranked
In a segment that aired Thursday, December 1, Camerota met with a group of five Trump supporters who she had spoken with throughout the campaign to hear if they still agreed with several of the President-elect's controversial stances. One woman said, "Voting is a privilege in this country, and you need to be legal — not like in California, where 3 million illegals voted."
When Camerota, the host of CNN's New Day, questioned where she was getting this information, the female Trump supporter said that she read it in the media and on CNN. The woman also alleged that President Barack Obama was encouraging illegal immigrants to vote.
PHOTOS: Celebrities Supporting Donald Trump for President
Camerota, 50, then asked the whole group of five, "Did you hear President Obama say illegal people could vote?" They unanimously nodded their heads, so Camerota pushed back again and asked where they read it.
"Google it. You can find it on Facebook," another female Trump supporter replied. Camerota quickly pulled up an article on her phone about Fox Business Network deceptively editing a clip that took the POTUS' comments out of context.
PHOTOS: Stars Who Played the President
Camerota continued to challenge the guests on their voter fraud claims. "You, as you sit here today, think that millions of illegal people voted in this country and you believe that there was widespread voting abuse? In the millions of people?"
The first woman replied, "I think there was in some states. California allows it."
Camerota face-palmed, apparently in disbelief about what she was hearing. "They do not allow illegals — you mean voter fraud, California allows?" she asked again. The woman replied, "I believe there was voter fraud in this country."
The business mogul, 70, has frequently claimed that the election was "rigged," and earlier this week, he tweeted about unsubstantiated claims of voter fraud. "In addition to winning the Electoral College in a landslide, I won the popular vote if you deduct the millions of people who voted illegally," he wrote, adding in another tweet, "Serious voter fraud in Virginia, New Hampshire and California – so why isn't the media reporting on this? Serious bias – big problem!"


text:
You Are Here : Home / Council & the Shire/Permits, Applications & Forms
Find common application and permit forms in the one location. Includes livestock grazing and animal registrations, change of contact details, waste, building and planning forms, event and venue booking forms and forms for the Port of Apollo Bay.
Application forms for Planning, Building, Report and Consent, Permit/Plan Searches, Property Information Certificates
Information and nomination forms for the Colac Otway Shire 2023 Community Awards are available here
Dial Before You Dig is a nationwide referral service for information on locating underground utilities networks on public and on private land.
Colac Otway Shire offers a range of iconic locations for filming, including the Great Ocean Road. Find out about the guidelines and apply for a permit to film within the shire.
Forms relating to running an event or tourism related matters.
The object of the Freedom of Information Act 1982 is to extend as far as possible the right of the community to access information in the possession of the Government of Victoria.
The Legal Point of Discharge (LPoD) is a point specified by Council where stormwater from a property must be discharged.
Permits and applications for the Port of Apollo Bay are available here.
A range of local laws and compliance-related forms, including application forms for livestock grazing, permits for signage display and hoardings
Information and forms for temporary renaming a sports ground.
Council permits and requirements for conducting works on a Council road or within a Council road reserve.


text:
Douro es una denominación de origen controlada (DOC) portuguesa para vinos producidos en la región demarcada de Douro, que abarca las subregiones de Baixo Corgo, Cima Corgo y Douro Superior, situadas al norte del país, alrededor del río Duero. Esta región es la misma que se encuentra demarcada para la producción del oporto.

Variedades de uva
 Tintas: Bastardo, Mourisco Tinto, Tinta Amarela, Tinta Barroca, Tinto Cão, Tinta Roriz (Tempranillo), Touriga Franca y Touriga Nacional (entre otras).
 Blancas: Donzelinho Branco, Gouveio, Malvasia Fina, Rabigato y Viosinho (entre otras).

Véase también
Vinos de Portugal
Gastronomía de Portugal
Región vitícola del Alto Duero

Enlaces externos
 Web del Instituto da Vinha e do Vinho (en portugués)

Vinos de Portugal con denominación de origen
Río Duero

redpajama_set_name: RedPajamaWikipedia


text:
Het vaartijdenboek is een verplicht document voor elk binnenschip of ander vaartuig, met uitzondering van veerponten en veerboten, dat bedrijfsmatig en zelfstandig op het binnenwater van Nederland of België vaart. In het vaartijdenboek moeten onder andere de vaar- en rusttijden, het aantal bemanningsleden en hun functie worden genoteerd. Hiervoor is de gezagvoerder verantwoordelijk. Bij het vaartijdenboek hoort een verklaring van afgifte vaartijdenboek.

Geschiedenis
Het vaartijdenboek is ingesteld in het verlengde van de Rijtijdenwet 1936, die in Nederland voor het wegverkeer werd ingevoerd. In Europees verband werd geopperd om ook de vaar- en rusttijden voor het vervoer over water te regelen. Dit resulteerde in het eerste nationale vaar- en rusttijdenboek, blauw van kleur. In 1988 is een nieuw vaartijdenboek, rood van kleur, van kracht geworden.

Wettelijke basis
Nederland:
 Wet vaartijden en bemanningssterkte binnenvaart (WVBB), die per 30 december 2008 wordt vervangen door de Binnenvaartwet.
België: Met ingang van 1 juli 2007
KB van 9 maart 2007, Belgisch Staatsblad van 16 maart 2007: Voorschriften voor de bemanning van binnenschepen op de scheepvaartwegen van het Koninkrijk.
Internationaal:
 Reglement onderzoek schepen op de Rijn (ROSR). Het maakt niet uit welke nationaliteit het schip voert. Er zijn ontheffingen en vrijstellingen mogelijk.

Gebruik
Het gebruik van het boek is ingesteld om oververmoeidheid en concentratiegebrek van de bemanning tegen te gaan en is daarmee een sociale regeling voor de Rijn- en binnenvaart. Het dient ervoor de arbeidsbescherming aan boord van de schepen en de veiligheid van de vaart zo veel mogelijk te waarborgen. Daarom zijn er wettelijk eisen gesteld aan het aantal uren dat bemanningsleden maximaal mogen werken en minimaal moeten rusten. Deze zijn afhankelijk van de exploitatiewijze van het schip. In een aantal gevallen is een tachograaf voorgeschreven als ondersteuning, maar in beperktere mate dan de Rijtijdenwet 1936. Ook de samenstelling van de bemanning wordt hierin geregeld.

Controles
Met het vaartijdenboek kunnen de dienstboekjes worden gecontroleerd. Een dienstboekje is een op naam gesteld document voor bemanningsleden van een professioneel varend binnenschip. Buiten de gezagvoerder moet elk lid van de bemanning van zowel een Nederlands als tegenwoordig ook een Belgisch binnenschip in het bezit zijn van een op naam gesteld dienstboekje. De verplichting ging voor bemanningsleden van Belgische binnenschepen in per 1 juli 2008. Het dienstboekje dient voor personen die een patent of vaarbewijs willen verkrijgen ook voor het aantonen van vaartijd en scheepsreizen op de Rijn en op andere vaarwegen.

Verkrijgbaarheid
Naast een aanvraagformulier is een van de volgende documenten nodig:
 (voorlopig) Certificaat van Onderzoek schepen op de Rijn of
 (voorlopig) Communautair Certificaat voor Binnenschepen of
 (voorlopig) Certificaat van Deugdelijkheid of
 meetbrief indien het schip niet certificaatplichtig is.

Voor schepen met een certificaat van onderzoek schepen op de Rijn geldt dat het eerste vaartijdenboek en de verklaring van afgifte moeten worden aangevraagd bij de Commissie van Deskundigen voor de Rijnvaart die het certificaat van onderzoek aan het schip heeft uitgereikt. Het volgende vaartijdenboek kan door de nationale instanties worden verstrekt. Zij mogen echter alleen tegen overlegging van het voorgaande vaartijdenboek worden afgegeven. Het voorgaande vaartijdenboek moet van de onuitwisbare aantekening "ongeldig" worden voorzien en dient aan de schipper te worden teruggegeven. Bij een nieuw vervolg vaartijdenboek moet het oude volle en ongeldig verklaarde vaartijdenboek gedurende zes maanden na de laatste aantekening ter controle aan boord worden bewaard.

 Elk volgende vaartijdenboek is in Nederland te verkrijgen bij de Stichting Afvalstoffen & Vaardocumenten Binnenvaart (SAB) te Rotterdam.
 In België bij de Binnenvaartloketten van Antwerpen en Luik

Verklaring van afgifte vaartijdenboek
Een verklaring van afgifte vaartijdenboek is een registratiekaart die hoort bij het vaartijdenboek. Als het boek vol is, kan er een nieuw worden aangevraagd, dat wordt voorzien van een vervolgnummer. Op deze registratiekaart staat hoeveel vaartijdenboeken er in de afgelopen jaren voor het schip zijn afgegeven.

Uitvoering
Een vaartijdenboek omvat 200 bladzijden, genummerd van 1 tot en met 200. De aantekeningen in dit boek dienen met inkt en duidelijk leesbaar (bijvoorbeeld in drukletters) te worden aangebracht.  Naam van het schip: ...... Officieel scheepsnummer: ......

Aantekeningen in het vaartijdenboek
De aantekeningen die de schipper in dit vaartijdenboek moet aanbrengen dienen te voldoen aan het Reglement onderzoek schepen op de Rijn. Daaraan wordt geacht te zijn voldaan, wanneer de aantekeningen het tijdvak van 48 uur bestrijken dat onmiddellijk voorafgaat aan het binnenvaren van het toepassingsgebied van het Reglement onderzoek schepen op de Rijn.

De functies van de bemanningsleden kunnen als volgt worden aangeduid:
   Cd-Sch 	 = Conducteur-Schiffsführer-Schipper
  Ti-St  	 = Timonier-Steuermann-Stuurman
   mMt-Bm-vMt 	 = Maître-matelot-Bootsmann-Volmatroos
 Mm  	 = Matelot garde-moteur-Matrose-Motorwart-Matroos-motordrijver
 Mt  	 = Matelot-Matrose-Matroos
  Hp-Dm  	 = Homme de pont-Decksmann-Deksman
 Ms-Sj  	 = Mousse-Schiffsjunge-Scheepsjongen
   Mc  	 = Mecanicien-Maschinist-Machinist

Op iedere bladzijde dient het volgende te worden aangetekend:
de exploitatiewijze (aantekeningen in verband met een wijziging van exploitatiewijze dienen steeds op een nieuwe bladzijde te worden aangebracht)
zodra het schip de vaart begint:

1e kolom- de datum (dag en maand)
2e kolom - de tijd (uur en minuten)
3e kolom - de plaats waar de vaart begint
4e kolom - de kilometerraai van die plaats

zodra het schip de vaart onderbreekt:
1e kolom - de datum (dag en maand) indien deze afwijkt van de begindatum
5e kolom - de tijd (uur en minuten)
6e kolom - de plaats waar het schip stilligt
7e kolom - de kilometerraai van die plaats

zodra het schip de vaart voortzet: dezelfde aantekeningen als bij het begin van de vaart
zodra het schip de vaart beëindigt: dezelfde aantekeningen als bij een onderbreking van de vaart
de 8e kolom moet worden ingevuld wanneer de bemanning voor de eerste keer aan boord komt en vervolgens telkens wanneer deze van samenstelling verandert
in de kolommen 9 t/m 11 moet het begin en het einde van de rusttijd van elk bemanningslid worden aangetekend. Deze aantekeningen dienen uiterlijk om 08 uur de volgende ochtend te worden aangebracht. Ingeval de bemanningsleden hun rust nemen volgens een regelmatig rooster, kan per reis met één schema worden volstaan
in de kolommen 12 en 13 moet bij wisseling van de bemanning telkens het tijdstip van aan boord komen of van boord gaan worden vermeld

Sancties
Overtreding van de bemanningsvoorschriften van het Reglement onderzoek schepen op de Rijn is strafbaar. Hetzelfde geldt voor het niet bijhouden, dan wel het niet volgens de voorschriften bijhouden van het vaartijdenboek. In het boek wordt deze notitie gevolgd door de van kracht zijnde tekst van hoofdstuk 23 van het Reglement onderzoek schepen op de Rijn in de drie talen.

Binnenvaart


This week on Calling All Catholics, Fr. Chip Hines and Dom Bettinelli discuss Pope Francis' new encyclical letter "Laudato Si'", On Care for Creation, and debunk the myths you may have heard about it. Listen online here or subscribe.
Calling All Catholics with Fr. Chip and Dom airs on Mondays at 5pm on WQOM 1060AM Boston and across the Station of the Cross radio network.