text:
\section{Introduction}

The very definition of a spinor in dealing with physics may be treated as a matter of some importance itself whatsoever. In fact, from simple quaternionic compositions revealing a definite rotation \cite{ALT} to the fermionic quantum internal structure \cite{WEI}, the spinorial approach reveals its richness. Among these possible systematizations concerning spinors, there is a particularly relevant one that encodes all the algebraic necessary information and the important relativistic construction as well, namely, the multivector spinor representation. When represented as a section of a bundle comprised by the $SL(2,\mathbb{C})$ group and $\mathbb{C}^4$, it is possible to understand several spinor properties by inspecting the multivector part constructed out specific $SL(2,\mathbb{C})$ objects. These objects are nothing but the bilinear covariants associated to the regarded spinor  \cite{LOU,Cra}.

Following this reasoning, it is not surprising the usefulness of such a representation, since the bilinear covariants are, at least in principle, related to a set of fermionic observables. Our aim in this paper is to delineate the importance of the representation space, by studying its properties, and then relating them to their physical  consequences. As one will realize, the representation space is quite complicated due to the constraints coming out the Fierz-Pauli-Kofink identities. However, a systematic study of the space properties ends up being useful to relate different domains (subspaces) to the corresponding physics. Moreover, this study allows to face the fermions from a different and useful perspective.

This paper is organized as follows: in the next section the standard framework and the three equivalent definitions of spinors are revisited for the Minkowski spacetime, emphasizing the most relevant aspects concerning our purposes. Sect. III is devoted to approach the Lounesto's spinors classification and related issues. In Sect. IV we construct and study the spinor representation space and explore the topological and physical consequences. In the final section we conclude.

\section{The three equivalent definitions of spinors}

Consider the Minkowski spacetime $(M\simeq\mathbb{R}^4, \eta_{\mu\nu})$ and its tangent bundle $
TM$, where $\eta$ denotes the Minkowski metric and Greek (spacetime) indexes run from 0 to 3. Denoting sections of the exterior bundle by
$\sec\Omega (M)$,  the spacetime Clifford algebra shall be denoted by $\mt{C}\ell_{1,3}$.  The set $\{{e}_{\mu }\}$ represents sections of the frame bundle
$\mathbf{P}_{\mathrm{SO}_{1,3}^{e}}(M)$, whereas the set  $\{\gamma^{\mu }\}$ can be further thought as being  the dual
basis, $\gamma^{\mu }({e}_{\nu})=\delta^\mu_{\;\nu}$.
 Classical spinors are objects of the space that carries the usual
$\tau=(1/2,0)\oplus (0,1/2)$ representation of the Lorentz group, that  can be thought as being sections of the vector bundle
$\mathbf{P}_{\mathrm{Spin}_{1,3}^{e}}(M)\times _{\tau }\mathbb{C}^{4}$ \cite{Mosna:2002fr,50}.

 The underlying idea that can join the three definitions of spinors relies on a quite straightforward root, and was inspired by the spacetime algebra, whose elements satisfy $e_\mu e_\nu + e_\nu e_\mu = 2\eta_{\mu\nu}\mathbf{1}$. Indeed,  any arbitrary element
$\Pi = s+s^{\mu}{e}_{\mu}+s^{\mu\nu}{e}_{\mu}e_{\nu}+s^{\mu\nu\tau}{e}_{\mu}e_{\nu}e_{\tau}+p{e}_{0}e_{1}e_{2}e_{3}\in {\mathcal{C}}\ell _{1,3}
$
has a quaternionic representation.  By denoting $\mathbb{H}$ the quaternionic ring, a spinor representation of the Clifford algebra ${\mathcal{C}}\ell _{1,3}\simeq {\mathcal{M}}(2,\mathbb{H})$ can be derived. A primitive idempotent
$f=\frac{1}{2}(1+{e}_{0})$  defines a minimal left ideal ${\mathcal{C}}\ell _{1,3}f$, whose arbitrary element can be expressed as \cite{jayme,jayme1}\begin{eqnarray}
 \upxi &=&\left[\left(s+s^{0}\right)+\left(s^{23}+s^{023}\right){e}_{2}e_{3}-\left(s^{13}+s^{013}\right){e}_{3}e_{1}+\left(s^{12}+s^{012}\right){e}_{1}e_{2}\right]f\nonumber\\&&+\left[\left(
p -s^{123}\right)+\left(s^{1}-s^{01}\right){e}_{2}e_{3}+\left(s^{2}-s^{02}\right){e}_{3}e_{1}+\left(s^{3}-s^{03}\right)\right]{e}_{0}e_{1}e_{2}e_{3}f,
\label{kkk}
\end{eqnarray}
\noindent
constituting then an algebraic spinor $\upxi \in \mt{C}\ell_{1,3}f$.
The set comprised by the units $\textgoth{i}={e}_{2}{e}_{3},\;\textgoth{j}={e}_{3}{e}_{1},$ and $\textgoth{k}={e}_{1}{e}_{2}$ settles a basis for the quaternionic algebra~$\mathbb{H}$.

Representations of the $\{{e}_{\mu}\}$ in $\mathcal{M}(2,\mathbb{H})$ read
 \cite{abla}:
\begin{equation}\label{quatrn}
\left[{e}_{1}\right]=%
{\scriptstyle\begin{pmatrix}
0 & \textgoth{i} \\
\textgoth{i} & 0%
\end{pmatrix}}%
,\quad\left[{e}_{2}\right]=%
{\scriptstyle\begin{pmatrix}
0 & \textgoth{j} \\
\textgoth{j} & 0%
\end{pmatrix}}%
,\quad\left[{e}_{3}\right]=%
{\scriptstyle\begin{pmatrix}
0 & \textgoth{k} \\
\textgoth{k} & 0%
\end{pmatrix}},\quad \left[{e}_{0}\right]=%
{\scriptstyle\begin{pmatrix}
1 & 0 \\
0 & -1%
\end{pmatrix}}\,.%
\end{equation}%
\noindent
Then, the elements $f$ and ${e}_{0}e_{1}e_{2}e_{3}f$ have, respectively, the  representations
$
[f]={\footnotesize\left(\begin{matrix}
1 & 0 \\
0 & 0%
\end{matrix}\right)}$ and $[{e}_{0}e_{1}e_{2}e_{3}f]=%
{\footnotesize\begin{pmatrix}
0 & 0 \\
1 & 0%
\end{pmatrix}}
$   \cite{oxford}.
Hence, an arbitrary element $\Psi \in \mt{C}\ell_{1,3}^{+}$ in the even subalgebra, corresponds  to the so called {spinor operator} \cite{LOU}
\begin{equation}
\Psi=s+s^{\mu\nu}{e}_{\mu}e_{\nu}+p{e}_{0}e_{1}e_{2}e_{3}
\label{400}\simeq
{\scriptstyle\begin{pmatrix}
q_{1} & -q_{2} \\
q_{2} & q_{1}%
\end{pmatrix}}\in\mathcal{M}(2,\mathbb{H}),
\end{equation}
where, according to Eq. (\ref{kkk}), it yields
\begin{eqnarray}
q_{1}&=&s+s^{23}\textgoth{i}+s^{31}\textgoth{j}+s^{12}\textgoth{k}\,,\\
q_2&=&-p+s^{01}\textgoth{i}+s^{02}\textgoth{j}+s^{03}\textgoth{k}\,.
\end{eqnarray}
The vector space isomorphisms $\mathbb{H}^{2}\simeq{\mathcal{C}}\ell _{1,3}^{+} \simeq \mathbb{C}^{4}\simeq {\mathcal{C}}\ell _{1,3}f$ constitute the landmark for the relationship among the spinor operator, the algebraic, and the classical definitions of a spinor \cite{50,Figueiredo:1990bw}.  Hence, it is possible to alternatively write the Dirac algebraic spinor field as an element ${\scriptstyle\begin{pmatrix} q_1\\q_2 \end{pmatrix}}$ of the ring $\mathbb{H}\oplus \mathbb{H}$, as \cite{jayme,jayme1,abla}
\begin{equation}\begin{pmatrix}
q_{1} & -q_{2} \\
q_{2} & q_{1}%
\end{pmatrix} [f]\in \mt{C}\ell_{1,3}f.  \label{hh}
\end{equation}%
\noindent
Returning to Eq.~(\ref{400}), and using for instance the standard representation,
the complex matrix associated to the spinor operator $\Psi$ in~(\ref{400})  reads \begin{equation}
[\Psi]=\begin{pmatrix}
\mathbb{A}&\mathbb{-B}\\
\mathbb{B}&\mathbb{A}\end{pmatrix},\quad\qquad \text{for} \quad
\mathbb{A}=
\begin{pmatrix}
\psi_1 & -\psi_2^{*} \\
\psi_2 & \phantom{-}\psi_1^{*}
\end{pmatrix},\quad\mathbb{B}=
\begin{pmatrix}
\psi_3 & -\psi_4^{*}\\
\psi_4 & \phantom{-}\psi_3^{*}
\end{pmatrix},
\end{equation}
where
\begin{eqnarray}
\psi_1=s+s^{23}i,\qquad
\psi_2=s^{13}+s^{12}i\qquad
\psi_3=p+s^{10}i,\qquad
\psi_4=s^{02}+s^{30}i
\end{eqnarray}
The standard Dirac spinor $\psi$ was identified, e. g., in Ref. \cite{LOU} as an element of the minimal left ideal $(\mathbb{C}\otimes \mt{C}\ell _{1,3}){\rm f}$ associated to the complexified spacetime algebra $(\mathbb{C}\otimes \mt{C}\ell _{1,3})$, generated by the primitive idempotent \cite{LOU}
\begin{eqnarray}
{\rm f}=\frac{1}{4}(1+e_{0})(1+ie_{1}e_{2})=\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix},\end{eqnarray} yielding $\psi =\Pi \frac{1}{2}(1+i\gamma _{1}\gamma_{2})\in (\mathbb{C}\otimes \mt{C}\ell _{1,3})f$, with the identification $e_\mu\mapsto \gamma_\mu$. It yields the  bijection between the algebraic spinor \cite{Mosna:2002fr,50,hestenes}
\begin{equation}
\psi =
\begin{pmatrix}
\psi_1 & 0 & 0 & 0 \\
\psi_2 & 0 & 0 & 0 \\
\psi_3 & 0 & 0 & 0 \\
\psi_4 & 0 & 0 &
\end{pmatrix}\in (\mathbb{C}\otimes \mt{C}\ell _{1,3})f \simeq \mathcal{M}(4,\mathbb{C}), \end{equation} and the classical one $\psi=(\psi_1, \psi_2, \psi_3, \psi_4)^\intercal
\in \mathbb{C}^{4}.$

Given a representation $\rho: \mathbb{C}\otimes \mt{C}\ell _{1,3}\to \mathcal{M}(4,\mathbb{C})$, the adjoint of $A\in\mathbb{C}\otimes \mt{C}\ell _{1,3}$, defined by $A^\dagger = \rho^{-1}(\rho(A)^\dagger)$ (where $\rho(A)^\dagger$ denotes the standard Hermitian conjugation in $\mathcal{M}(4,\mathbb{C})$),
reads $A^\dagger = e_0\tilde{A}^* e_0$, where $\tilde{A}$ stands for the reversion of $A$ and $(\,\cdot\,)^*$ denotes the complex conjugation. Besides, its trace is given by Tr$(\rho(\psi)) = 4\langle \psi\rangle_0$, where this notation is used to indicate the projection of a multivector onto
its scalar part.

This correspondence provides an immediate identification between $\psi$ and the classical Dirac spinor field. Having recovered the equivalence between these current spinor definitions, we undergo to the building blocks of the spinorial representation space, namely the Fierz aggregate and the bilinear identities, after what we define the space itself, allowing for the connection of its points to a physical spinor, regardless the chosen classical definition.

\section{Lounesto's spinors classification, Pauli-Fierz-Kofink identities, and the Fierz aggregate}

Any spinor field
$\psi \in \sec \mathbf{P}_{\mathrm{Spin}_{1,3}^{e}}(M)\times_{\tau
}\mathbb{C}^{4}$ can be employed to construct its bilinear covariants  as section of bundle $\Omega(M)$,  reading \cite{LOU,Cra,bj}
\begin{subequations}\begin{eqnarray}
\label{cova}
\sigma&=&\bar\psi\psi\in\Omega^0(M),\\
 \omega&=&-\bar\psi\gamma_0\gamma_1\gamma_2\gamma_3\psi\in\Omega^4(M),\\
 \mathbf{J}&=&(\bar\psi\gamma_{\mu}\psi)\;\gamma^{\mu}\in\Omega^1(M), \\
\mathbf{K}&=&i(\bar\psi\mathrm{\gamma_0\gamma_1\gamma_2\gamma_3}\gamma_{\mu}\psi)\;\gamma^{\mu}\in\Omega^1(M),\\
 \mathbf{S}&=&\left(\bar\psi\,[\gamma_{\mu}, \gamma_{\nu}]\,\psi\right)\,\gamma^{\mu}\wedge\gamma^{\nu}\in\Omega^2(M)\,.
\end{eqnarray} \end{subequations}
 Equivalently, the components of the bilinear covariants are, respectively,
denoted by
\begin{subequations}
\begin{eqnarray}
J_\mu &=& \bar\psi\gamma_{\mu}\psi,\\
K_\mu &=&i\bar\psi\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{\mu}\psi,\\
S_{\mu\nu}&=&\bar\psi\,[\gamma_{\mu}, \gamma_{\nu}]\,\psi\,.
\end{eqnarray}
\end{subequations}

The bilinear covariants in the Dirac theory are interpreted respectively  as the mass term (or invariant length) in the Lagrangian of the electron ($\sigma$), the pseudo-scalar ($\omega$) relevant for parity-coupling, the current of probability density ($\mathbf{J}$), the chiral current density ($\mathbf{K}$), and the probability density of the (intrinsic) electromagnetic moment ($\mathbf{S}$) \cite{LOU,Cra}.
A prominent requirement for the Lounesto's spinors classification is that the bilinear covariants satisfy quadratic algebraic relations, namely, the so-called Fierz-Pauli-Kofink (FPK) identities, which read
\begin{subequations}
\begin{eqnarray}
\label{Fierz}
\mathbf{J}^{2}&\equiv& J^\mu J_\mu =\omega^{2}+\sigma^{2},\\
\mathbf{K}^{2}&\equiv& K^\mu K_\mu =-\mathbf{J}^{2},\\
\mathbf{J}\cdot\mathbf{K}&\equiv& J^\mu K_\mu =0,\\
\mathbf{J}\wedge\mathbf{K}&
=&-(\omega+\sigma\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3})\mathbf{S}.  \label{fi}
\end{eqnarray}
\end{subequations}
When an arbitrary spinor $\xi$ satisfies $\gamma_0\xi^{\dagger}\gamma_0\psi\neq 0$, the original spinor $\psi\neq 0$ can be reconstructed, using the aggregate
\begin{eqnarray}
\mathbf{Z}=\sigma + \mathbf{J} +i\mathbf{S}+i\mathbf{K}\gamma_0\gamma_1\gamma_2\gamma_3+\omega \gamma_0\gamma_1\gamma_2\gamma_3\, \label{Z}
\end{eqnarray}
by (a version) of the inversion theorem, $
\psi=\frac{1}{2\sqrt{\xi^{\dagger}\gamma_{0}\mathbf{Z}\xi}}\;e^{-i\theta}\mathbf{Z}\xi, \label{3}%
$  where
$\theta=i\log\left({2}(\xi^{\dagger}\gamma_{0}\psi{\xi^{\dagger}\gamma_{0}\mathbf{Z}\xi})^{-1/2}\right)$ is  \cite{Cra,Holland:1986wj,mosna}.
   Moreover, when  $\sigma,\omega,\mathbf{J},\mathbf{S}%
,\mathbf{K}$ satisfy the Fierz identities, then the complex multivector
operator $\mathbf{Z}$ is named a {Fierz aggregate}, which can be self-adjoint, being called a
 boomerang in this case \cite{LOU}.
 The regular spinors are those whose at least one of the  bilinear covariants $\sigma$ and $\omega$ do not vanish. On the other hand, singular spinors present $\sigma=0=\omega$, and, in this case, the Fierz identities, given in Eq. \eqref{Fierz},  are in general replaced
by the most general conditions \cite{Cra}:
\begin{subequations}
\begin{align}
\!\!\!\!\!\frac14\mathbf{Z}^{2}    =\sigma \mathbf{Z},\qquad\qquad \frac14\mathbf{Z}\gamma_{\mu}\mathbf{Z}=J_{\mu}\mathbf{Z},\qquad\qquad \frac14\mathbf{Z}i[\gamma_{\mu},\gamma_{\nu
}]\mathbf{Z}=S_{\mu\nu}\mathbf{Z},\\
\frac14\mathbf{Z}i\gamma_0\gamma_1\gamma_2\gamma_3\gamma_{\mu}\mathbf{Z}    =K_{\mu}\mathbf{Z},\qquad\qquad -\frac14\mathbf{Z}\gamma_0\gamma_1\gamma_2\gamma_3\mathbf{Z}=\omega \mathbf{Z}.
\end{align}
\end{subequations}
The conditions are satisfied also by regular spinors. Insomuch, such relations are called by more general Fierz-Pauli-Kofink identities, however, written based on the Fierz aggregate.

Moreover, the inversion theorem (to be further regarded in the next section) is inspired by this spinor representation. More significantly here, the aggregate plays a central role within the Lounesto's classification since, in order to complete the classification itself, $\mathbf{Z}$ has to be promoted to a boomerang, satisfying $\mathbf{Z}^{2}=4\sigma\mathbf{Z}$. Obviously, for regular spinors the above condition is satisfied and $\mathbf{Z}$ is automatically a boomerang. However, for singular spinors case it is not so direct. Indeed, for singular spinors we must envisage the underlying geometric structure  to the multivector. From the geometric point of view, the following relations between the bilinear covariants must be fulfilled, in order to ensure that the aggregate is a boomerang: $\mathbf{J}$ field must be parallel to $\mathbf{K}$ and both are elements in the plane formed by the bivector field $\mathbf{S}$. Hence, using the Eq. (\ref{Z}) and taking into account singular spinors, it is straightforward to see that the aggregate can be recast as \cite{LOU}
\begin{eqnarray}
\mathbf{Z}=\mathbf{J}(1+i\mathbf{s}+ih\gamma_0\gamma_1\gamma_2\gamma_3), \label{ZB}
\end{eqnarray} where $\mathbf{s}$ is a space-like vector orthogonal to $\mathbf{J}$, and $h$ is a real scalar that is related to the spinor helicity. The multivector as expressed in Eq. (\ref{ZB}) is a boomerang \cite{Julio1}. The condition $\mathbf{Z}^{2}=4\sigma\mathbf{Z}$ yields $\mathbf{Z}^2=0$, for singular spinors, namely, the Fierz aggregate is nilpotent. However, for the FPK identities to hold,  the vector field $J$ must be light-like (isotropic) and the multivector $h\gamma_0\gamma_1\gamma_2\gamma_3+\mathbf{s}$ must be a pure imaginary \cite{LOU,Julio1}.

With these ingredients, it is possible to envisage six different classes of spinors, according to the following classification:
\begin{center}
\begin{tabular}[c]{||c|| c| c| c| c| c||}
\hline\hline
Class & $\sigma$ & $\omega$ & $\bf{K}$ & $\bf{S}$& $\bf{J}$\\
\hline\hline
1& $\neq 0$ & $\neq 0$ & $\neq 0$ & $\neq 0$& $\neq 0$\\
\hline
2& $\neq 0$ & $0$ & $\neq 0$ & $\neq 0$& $\neq 0$\\
\hline
3& 0 & $\neq 0$ & $\neq 0$ & $\neq 0$& $\neq 0$\\
\hline
4& 0 & 0 & $\neq 0$ & $\neq 0$& $\neq 0$\\
\hline
5& 0 & 0 & 0 & $\neq 0$& $\neq 0$\\
\hline
6& 0 & 0 & $\neq 0$ & 0& $\neq 0$\\
\hline\hline
\end{tabular}\medbreak
\footnotesize{Table 1:  Lounesto's spinor field classification.}
\end{center}  The three first classes are composed by regular spinors, that comprise the standard textbook Dirac spinor. As stated in the literature, the representation spaces for the mentioned spinors are linked by the parity symmetry, however, quite recently regular spinors have been shown to be built without reference to this symmetry \cite{crd}. The elements of the fifth class are also called flag-pole spinors, represented by particular cases as Majorana and Elko  spinors,  whereas the sixth class comprises Weyl spinors. The fourth class, the flag-dipole, has had its first physical example recently discovered \cite{fabri}. For later reference we stress that $\mathbf{J}$ is always non null within this context.
The Lounesto's classification has been explored in a comprehensive range of context, comprising field theory \cite{5307,daSilva:2012wp}, cosmology \cite{Fabbri:2010ws},  gravitation \cite{Ahluwalia} and formal aspects as well \cite{bonora,daRocha:2011yr,Julio1}. The general form of spinors in each one of the above classes was derived in Refs. \cite{Cavalcanti:2014wia,fabri}, and a classification that encodes gauge aspects was established in Ref. \cite{Fabbri:2016msm}.

\section{The Representation Space}

Bearing in mind that a given spinor can be written as a section of the bundle $\mathbf{P}_{\mathrm{Spin}_{1,3}^{e}}(M)\times _{\tau }\mathbb{C}^{4}$ we shall envisage the spinor space structure adopting a bottom-up, and somewhat pragmatic, approach by defining the regarded manifolds and spaces with respect to their points and elements. Notice that, as reinforced throughout Sect. II, the understanding of spinors as sections of the aforementioned bundle are not strictly necessary, although highly convenient as we shall see.

In what follows let us denote by $\mathring{N}$ the five-dimensional manifold whose points are section  in sec $\Omega^a(M)$, with $a=0,\ldots,4$. The space $\mathring{N}$ is  isomorphic to the exterior bundle $\Omega(M)=\oplus_{a=0}^4\Omega(M)$.
 Let us denote by $P=(p^0,p^1,p^2,p^3,p^4)$ an arbitrary point of the manifold $\mathring{N}$, and the function $Z$, that establishes such a canonical isomorphism $\mathring{N}\overset{Z}{\simeq}\Omega(M)$. Obviously  $Z(P)\in \Omega(M)$.
 \medbreak
{\bf Definition 1:} \textit{$\mathring{\Sigma}$ is  the space whose elements are given by $Z\upeta$, where $\upeta \in \mathbb{C}^4$.}
\medbreak
Notice that as long as $Z$ is restricted to the bilinear covariants, namely,  we impose that it acts only upon points of $N$ satisfying the FPK identities, then the Fierz aggregate is straightforwardly obtained. Equivalently, however more generally, we proceed with the following direct construction:
\medbreak
{\bf Definition 2:} \textit{$N$ is a submanifold of $\mathring{N}$ whose points are such that $Z(P)$ obeys the FPK identities.}
\medbreak
When acting upon elements of $M$ it is convenient to write $Z(P)$ as
\begin{equation}
Z(P)=\sigma+{\bf J} + {\bf K} + {\bf S}+\omega,\nonumber
\end{equation} making explicit the multivector structure in terms of the bilinear covariants, just as to express $P=(\sigma, {\bf J}, {\bf K}, {\bf S},\omega)$.
\medbreak
{\bf Definition 3:} \textit{The representation space $\Sigma(N)$ is performed by elements given by $Z(N)\eta$, where $Z(N)$ stands for $Z(P)$ with $P\in N$ only. Therefore $Z\upeta\cong \Psi\in \Sigma(N)$ and the elements of $\Sigma$ are, thus, physical spinors.}
\medbreak
It is worth to emphasize that since the bilinears are invariant with respect to Lorentz transformation, the elements of $\Sigma(N)$ respect a relativistic dynamics. Clearly $\Sigma(N)\subset\mathring\Sigma$, i. e., the representation space is contained in the broader spinorial space. Therefore, the complement space $\mathring\Sigma\setminus\Sigma(N)$ comprises  points corresponding to spinors which \emph{do not} obey the FPK identities, the so-called anomalous spinors.

The underlying idea to this construction regards the possibility to change from one physical spinor configuration to another one, by covering a given continuous path in the representation space. Differently of what happens to $\mathring{N}$, however, the submanifold $N$ must have a quite constraint topology inherited from the validity of the FPK identities.

Let us make this point clearer by considering merely regular spinors for a moment. In this case, it is possible to attain the appropriate subspace of $\Sigma(N)$ by defining the following canonical projector $\xi_{\rm reg}$:
\begin{eqnarray}
\xi_{\rm reg}: N&\rightarrow&\Omega(M)\nonumber\\(\sigma, {\bf J}, {\bf K}, {\bf S},\omega)&\mapsto& (\sigma, {\bf J}, 0, 0, \omega),\nonumber
\end{eqnarray}
with image $\Omega^0(M)\oplus\Omega^1(M)\oplus\Omega^4(M)=\xi_{\rm reg}(N)$.
Within the space $\xi_{\rm reg}(N) \subset \Sigma$, taking into account the identity ${\bf J}^2=\sigma^2+\omega^2$, that holds for regular spinors, it is always possible to associate a topological invariant for every regular state. Moreover, taking into account the usual mass dimension $3/2$ fermion, for which ${\bf J}$ represents the conserved current, the regarded topological invariant  must be related to the electric charge. Before proceeding, let us make two parenthetical remarks. First, it is straightforward to realize that $\xi_{\rm reg}$ may be naturally adapted for a lower dimension projection leading to elements as either $(\sigma, {\bf J}, 0, 0, 0)$ or $(0, {\bf J}, 0, 0, \omega)$. Second, when we refer to mass dimension of a given spinor, we mean the canonical mass dimension which shall be inherited by the quantum field from the dynamic respected by the expansion coefficients. Particular cases of the expansion coefficients are the objects treated here.

It is worth to emphasize that the space $\xi_{\rm reg}(N)$ has a rich underlying geometric structure. Indeed, it consists not merely a of submanifold, but furthermore it manifests a intriguing structure arising from the monopole construction of the Hopf
fibration $S^1\ldots S^3\rightarrow S^2$, where $S^1$ is homeomorphic to the
Lie gauge group U(1) of the electromagnetism \cite{jayme}. Using a similar construction,   the instanton is related to a principal bundle with structure
Lie group SU(2), homeomorphic to the 3-sphere $S^3$. The instanton
was described in Refs. \cite{jayme,jayme1} using the Hopf fibration $S^3\ldots
S^7\rightarrow S^4$, in the context of the Witten's monopole equations, by means of the bilinear covariants associated with regular spinor fields, under the Lounesto's spinor field classification \cite{Julio1}.

Let us make this point clear, working with a slightly different $\xi_{\rm reg}(N)$ space, after what the general case shall be regarded. Regular spinor fields in either class 1 or class 2  in Lounesto's classification can be thought of as satisfying $%
\sigma=1$ without loss of generality, defining the manifold $S^7$, when the Dirac spinor field is
classically described by an element of $\mathbb{C}^4\simeq\mathbb{H}^2$. Considering $\mathcal{C}\ell_4$ be the Clifford algebra of the 4-dimensional Euclidean vector space $\mathbb{R}^4$, i.e, the algebra generated by the set of vectors $\lbrace {\rm e}_{\mu}\rbrace$, subjected to the relations $
{\rm e}_{\mu}^2 = 1$, and ${\rm e}_{\mu}{\rm e}_{\nu} +{\rm e}_{\nu}{\rm e}_{\mu}=0$, with $ \mu = 0,1,2,3$.

The quaternionic representation can be constructed by using Eq. (\ref{quatrn}), by
$
 {\rm e}_i \mapsto [e_i][e_0].
$ One defines the observables of $\psi \in \mathbb{C}^4$ in a 4-dimensional Euclidean space by the following expressions:
\begin{eqnarray}
\label{bili1}\!\!\!\!\!\!\sigma = \bar\psi\psi, \qquad \omega = \bar\psi{\rm e}_5\psi, \qquad J_{\mu} = \bar\psi{\rm e}_{\mu}\psi, \qquad K_{\mu} = i\bar\psi{\rm e}_5{\rm e}_{\mu}\psi, \qquad S_{\mu\nu} = \bar\psi[{\rm e}_{\mu},{\rm e}_{\nu}]\psi,
\end{eqnarray}
where ${\rm e}_5 = {\rm e}_0{\rm e}_1{\rm e}_2{\rm e}_3$.
Note that the only different relations due to the Minkowski space case is the expression for $\omega$ (see Eq.\eqref{cova}). In fact, in the $\mathcal{C}\ell_{1,3}$ algebra one has ${\rm e}_5^2 = -1$,  whereas considering $\mathcal{C}\ell_{4}$ yields  ${\rm e}_5^2=1$.
In this way, the Fierz identities must be modified \cite{jayme,mosna}, yielding
\begin{eqnarray}\label{if}
\boldsymbol{{\rm J}}^2 = \sigma^2 - \omega^2, \qquad \boldsymbol{{\rm J}}^2=\boldsymbol{{\rm K}}^2, \qquad \boldsymbol{{\rm J}}\wedge\boldsymbol{{\rm K}} = (\sigma - {\rm e}_5\omega)\boldsymbol{{\rm S}}, \qquad J^\mu K_\mu = \boldsymbol{{\rm J}}\cdot\boldsymbol{{\rm K}} = 0.
\end{eqnarray}
In this context, the first Fierz identity \eqref{if} provides the expression $\mathbf{J}^2 + \omega^2 =1$ defining the 4-sphere $S^4$,  with
coordinates $(J_\mu,\omega)$.

Now, taking into account Eqs. (\ref{hh})  and  Eq.(\ref{cova}) yields \cite{jayme,jayme1}
\begin{eqnarray}
\sigma &=& q_1\cdot q_1 \!+\! q_2\cdot q_2,\qquad \omega = 2\,\Re(q_1^\ast
q_2),\qquad J_0 = q_1\cdot q_1 \!-\! q_2\cdot q_2, \qquad
   J_i = 2\,\epsilon_i^{\;jk}\Re(q_1^\ast\,{\rm e}_j{\rm e}_k\, q_2), \qquad
\end{eqnarray}
\noindent \text{for $i,j,k=1,2,3$} and $\epsilon_i^{\;jk}$ is the Levi-Civita symbol. Hence, the representation in Eq. (\ref{bili1}) reads
\cite{jayme,jayme1}
\begin{eqnarray}
\sigma &=& |\,\psi_1|\,^2 + |\,\psi_2\,|^2 + |\,\psi_3\,|^2 + |\,\psi_4\,|^2\,,\qquad \omega = 2 \Re(\psi_1\psi_3^\ast + \psi_2\psi_4^\ast), \\
J_0 &=& |\,\psi_1\,|^2 + |\,\psi_2\,|^2 - |\,\psi_3\,|^2 - |\,\psi_4\,|^2  \notag, \qquad
J_1 = 2 \Im(\psi_1\psi_4^\ast +\psi_2\psi_3^\ast)\,,\\
J_2 &=& 2 \Re(\psi_2\psi_3^\ast - \psi_1\psi_4^\ast),
\notag \qquad J_3 = 2 \Im(\psi_3\psi_1^\ast +\psi_2\psi_4^\ast)\,.  \label{s44}
\end{eqnarray}%
\noindent
This important geometric feature reveals that
the underlying geometry induced by spinors classes can further point to more structures. Indeed, the following proposition regards the topological invariants associated to regular spinors.
\medbreak

{\bf Proposition 1}: \textit{Let $\xi_{\rm reg}(N)$ be the space consisting of regular spinors. Then $$\pi_{[\dim(\xi_{\rm reg}(N))-1]}(\xi_{\rm reg}(N))=n\in\mathbb{Z},$$ where $\pi_i(R)$ stands for the $i^{\rm th}$-homotopy group of a given $R$ space. In the case of a mass dimension $3/2$ fermion described by the regular spinor, the topological invariant is associated to the electric charge.}
\medbreak
{\bf Proof}: The first assertion directly follows  from the above discussion. In fact, a regular spinor is an element of $\xi_{\rm reg}(N)$ and therefore there exists just two possibilities:
\begin{itemize}
\item $\dim\xi_{\rm reg}(N)=3$, and the spinor belongs to class 1 of Lounesto's classification. In this case the point $(0,0,0)\in\xi_{\rm reg}(N)$ can not be attained, implying that $\pi_2(\xi_{\rm reg}(N))=n\in\mathbb{Z}$;

\item $\dim\xi_{\rm reg}(N)=2$, and thus the spinor belongs to either class 2 or class 3 of Lounesto's classification. This is the case when $\pi_1(\xi_{\rm reg}(N))=n\in\mathbb{Z}$.
\end{itemize}
In general, ${\bf J}$ is not necessary related to the conserved current associated to a given fermion. Nevertheless, when it does -- what is the case for mass dimension $3/2$ fermions -- then the conserved charge is the electric charge itself. Since a vanishing charge is forbidden in these situations, one has the physical counterpart of the above topological constraint. Indeed the $(0,0,0)$ point for  regular spinors can never be reached.   \hspace{1cm}$\Box$
\medbreak
As it is readily verified, in the case of the three dimensional $\xi_{\rm reg}(N)$ space the conserved current may also be seen as the generator of cohomology, for $H^1(\xi_{\rm reg}(N))\simeq H^1(\mathbb{R}^2\setminus\{0\})$ and the usual closed form
\begin{eqnarray}
\frac{\sigma d\omega-\omega d\sigma}{\sigma^2+\omega^2},
\end{eqnarray} is not everywhere exact. The usefulness of the representation space construction is now evident. By treating physical spinor (states) as points of a given space, constrained by the algebraic bilinear relations, it is possible to work in the interplay of topology, multivector algebra, and physics. As a matter of fact, the very existence of a relativistic spinor is related to a topological invariant in the representation space.

Nevertheless, one shall not be so optimistic just by looking at the example just studied, since we were dealing only with a projection. The $\Sigma$ space, where not only regular spinors are taken into account, is certainly very difficult to be analysed. There are, however, some interesting points that we shall report on the study of $\Sigma$ in its general form. In fact, Lounesto's classification provides six classes of spinors, wherein a continuous path in the representation space allows access to different configuration states. Let us make this idea more clear and precise.

All spinors in an arbitrary class  are connected by a simple rescaling. From the point of view of elements in $\Sigma(N)$, two different elements $\Psi'$ and $\Psi$ are connected by an usual transformation along the same class by $\Psi'=S\Psi$. In this context, it is possible to assert, in a manner akin to Wigner \cite{Wigner:1939cj}, the following proposition.
\medbreak
{\bf Proposition 2}: \textit{Let $D_{\lambda}$ be an 1-parameter infinitesimal operator acting on the spinor space of a given class according Lounesto's classification. Suppose that it is an homomorphism, $D_\lambda D_{\lambda'}=D_{\lambda+\lambda'}$, with $\lambda \in \mathbb{R}$. If there exists a physical state on which the application of $D_\lambda$ is well defined, then there exists a dense set of such states in the respective class, with respect to the Lounesto's classification.}
\medbreak
{\bf Proof}: If the application of $D_\lambda$ is well defined for a given state, there exists the limit $ \lim_{\lambda\rightarrow 0} \lambda^{-1}(D_{\lambda}-1)\Psi,$ what implies that $\lim_{\lambda\rightarrow 0} \lambda^{-1}(D_{\lambda}S^{-1}S-S^{-1}S)\Psi$. Since  the rescaling commutes with $D_\lambda$, it follows that  there exists the limit $ \lim_{\lambda\rightarrow 0} \lambda^{-1}(D_{\lambda}-1)S\Psi.$ Hence, within an arbitrary but fixed class in Lounesto's classification, it is possible to operate with infinitesimal operators in a rather usual way. Moreover, in view of the above result, physical spinors are indeed points of $\Sigma(N)$. \hspace{1cm}$\Box$

It is important to remark that an arbitrary class in Lounesto's classification is invariant under $S$. It is furthermore possible, however, to connect two different classes by algebraic transformation. More specifically, it was shown in Ref. \cite{JMP2007} that there exists a subset of spinors in class 1, 2, and 3 which can be mapped into a subclass of class 5 spinors. Let us denote this transformation,  between different classes, by $S_{C}$. It turns out that $\det S_C\neq 0$ \cite{JMP2007}. Hence, the alluded algebraic bridge, in a manner of speaking, is also dense. In fact, as far as we restrict ourselves to the subset of states which can be mapped, the proof of Proposition 2 holds, in this switching class case.

A given algebraic bridge, however, is not always necessarily well behaved. In the sequel we give a (counter-) example, presenting a mapping between a subset of spinors in class 1, 2, and 3, and a subset in class 4, which is neither Hermitian nor invertible. As one shall see, this example is quite severe in the constraints it imposes,  and it is not discarded the possibility of a more manageable mapping. It is worth to  stress that the mapping, from regular spinors to class 4 spinors, is chosen as a particular case. Class 4 spinors are understood as the most unvoiced class in Lounesto's classification, having just a rare single example in the literature \cite{fabri} as a physical solution of the Dirac equation in a Riemann-Cartan Bianchi-I, $f(R)$, background. Lounesto  describes such class as the only one that, at that time, had not corresponded to any type of spinor already found in Nature \cite{LOU}.  Except for such solution, neither other types of flag-dipole spinors nor their respective dynamics as well have been found, yet. The algebraic mapping between regular and class 4 spinors can be parenthetically seen, then, as an attempt to put forward a bottom-up approach, embracing flag-dipoles spinors into the standard setup of high energy physics.

We start by introducing a matrix $\textit{M}={\small\begin{pmatrix}
\mathbb{M}_{\mathrm{11}} & \mathbb{M}_{\mathrm{12}}\\
\mathbb{M}_{\mathrm{21}} & \mathbb{M}_{\mathrm{22}}
\end{pmatrix}} \in \mathcal{M}(4,\mathbb{C})$, defining the mapping:
\begin{eqnarray}
M: \mathcal{D} &\rightarrow& \mathcal{T}_{\mathrm{4}}\nonumber\\
\Phi_{D}&\mapsto& \Psi_{4}=\textit{M}\Phi_{D},\label{MAPA}
\end{eqnarray} where $\mathcal{D}$ and $\mathcal{T}_{\mathrm{4}}$ stands for the sets comprising regular and flag-dipole spinors, respectively. The formalism is clearly
representation-independent, however the Weyl representation
$\footnotesize{\gamma_{0}={}
\begin{pmatrix}
\mathbb{O} && \mathbb{I} \\
\mathbb{I} && \mathbb{O}
\end{pmatrix}}$, $
\footnotesize{
\gamma_{k}=
\begin{pmatrix}
\mathbb{O} && \sigma_{k} \\
-\sigma_{k} && \mathbb{O}
\end{pmatrix}}$ shall be used, being $\sigma_k$ the usual Pauli matrices, to fix the notation hereon.

According to the Lounesto's spinor classification, using the mapping defined in (\ref{MAPA}), class-4 spinors satisfy
\begin{eqnarray}
\label{16}
\sigma=\Phi_{D}^{\dag}\textit{M}^{\dag}\gamma_{0}\textit{M}\Phi_{D}=0, \;\;\;\;\;\; \qquad \omega=-\Phi_{D}^{\dag}\textit{M}^{\dag}\gamma_{123}\textit{M}\Phi_{D}=0.
\end{eqnarray} Let us investigate here the constraints exclusively on the $\textit{M}$ matrix. The conditions (\ref{16}) imply
\begin{eqnarray}
\label{12}
\textit{M}^{\dag}\gamma_{0}\textit{M}=0,\qquad\textit{M}^{\dag}\gamma_{123}\textit{M}=0,
\end{eqnarray} which, in the Weyl representation,  yield
\begin{eqnarray}
\label{7}
\pm\mathbb{M}^{\mathrm{\dag}}_{\mathrm{11}}\mathbb{M}_{\mathrm{21}}+\mathbb{M}^{\mathrm{\dag}}_{\mathrm{21}}\mathbb{M}_{\mathrm{11}}=\mathbb{O}=
\pm\mathbb{M}^{\mathrm{\dag}}_{\mathrm{11}}\mathbb{M}_{\mathrm{22}}+\mathbb{M}^{\mathrm{\dag}}_{\mathrm{21}}\mathbb{M}_{\mathrm{12}},\\
\label{9}
\pm\mathbb{M}^{\mathrm{\dag}}_{\mathrm{12}}\mathbb{M}_{\mathrm{21}}+\mathbb{M}^{\mathrm{\dag}}_{\mathrm{22}}\mathbb{M}_{\mathrm{11}}=\mathbb{O}=
\pm\mathbb{M}^{\mathrm{\dag}}_{\mathrm{12}}\mathbb{M}_{\mathrm{22}}+\mathbb{M}^{\mathrm{\dag}}_{\mathrm{22}}\mathbb{M}_{\mathrm{12}}\,.
\end{eqnarray}
The system (\ref{7})-(\ref{9}) is satisfied when
\begin{eqnarray}
\label{17}
\mathbb{M}^{\dag}_{\mathrm{11}}\mathbb{M}_{\mathrm{21}}=
\mathbb{M}^{\dag}_{\mathrm{11}}\mathbb{M}_{\mathrm{22}}=
\mathbb{M}^{\dag}_{\mathrm{12}}\mathbb{M}_{\mathrm{21}}=
\mathbb{M}^{\dag}_{\mathrm{12}}\mathbb{M}_{\mathrm{22}}=\mathbb{O}.
\end{eqnarray} Therefore, by considering a general representation,
\begin{center}
$\mathbb{M}$$_{11}={}
\begin{pmatrix}
m_{11} && m_{12}\\
m_{21} &&m_{22}
\end{pmatrix}
{},\;\;$
$\mathbb{M}$$_{12}={}
\begin{pmatrix}
m_{13} && m_{14}\\
m_{23} &&m_{24}
\end{pmatrix}
{},\;\;\mathbb{M}$$_{21}={}
\begin{pmatrix}
m_{31} && m_{32}\\
m_{41} &&m_{42}
\end{pmatrix}
{},$\;\;
$\mathbb{M}$$_{22}={}
\begin{pmatrix}
m_{33} && m_{34}\\
m_{43} &&m_{44}
\end{pmatrix}
{},$
\end{center}
it is possible to rewrite
\begin{equation}
\label{13}
\textit{M}={}
\begin{pmatrix}
m_{11} & m_{12} & m_{13} & m_{14}\\
\frac{m_{11}m_{22}}{m_{12}} & m_{22} & \frac{m_{13}m_{22}}{m_{12}} &\frac{m_{14}m_{22}}{m_{12}}\\
\frac{-m_{22}^{*}m_{41}}{m_{12}^{*}} & \frac{-m_{22}^{*}m_{42}}{m_{12}^{*}} & \frac{-m_{22}^{*}m_{43}}{m_{12}^{*}} &\frac{-m_{22}^{*}m_{44}}{m_{12}^{*}}\\
m_{41} & m_{42} & m_{43} &m_{44}
\end{pmatrix}
{}.
\end{equation}

The $\textit{M}$ matrix is responsible to perform the mapping between regular and class-4 spinors\footnote{A simple, but tedious calculation show that the other bilinears behave in such a way that the mapping (\ref{13}) works well, ensuring a final class-4 spinor.}. There are, however, important restrictions on such a mapping which must be highlighted.

Firstly, the mapping performed cannot occur from class-4 to regular spinors. In fact, supposing the existence of ${\mathring{M}}$ such that $\mathring{{M}}\Psi_{4}=\Phi_{D}$, then ${M}\mathring{{M}}\Psi_{4}=\textit{M}\Psi_4$, leading to $\mathring{{M}}={M}^{-1}$. Nevertheless, as it can be explicitly calculated from (\ref{13}), $\det{{M}}=0$ and there is no such a $\mathring{{M}}$ matrix. Secondly, the mapping (\ref{13}) cannot be Hermitian. Indeed, the requirement $\textit{M}=\textit{M}^\dagger$ yields
\begin{eqnarray}
m_{11}&=&m_{11}^{*}, \;\;\;\;\;\;\; m_{12}=m_{12}^{*},\;\;\;\;\;\;\;m_{22}=m_{22}^{*},\;\;\;\;\;\;\;\ m_{14}=m_{41}^{*}, \\
m_{13}&=&\frac{-m_{22}m_{14}}{m_{12}}=-m_{42}^{*},\;\;\;\;\;\;\; m_{44}=\frac{-m_{12}m_{43}}{m_{22}}.
\end{eqnarray} Nevertheless, the other bilinear invariants do not behave as the ones for class-4 spinors. In fact, it can be verified that ${\bf K}=0=\bf{S}$ for this case, rendering a spinor different from a class-4 one and, then, the hermiticity is forbidden.

It is significant to stress that the above mapping was performed using class-1 regular spinors, as it is clear from (\ref{MAPA}) and (\ref{16}). Nevertheless, as far as we implement the additional constraints coming from class-2 and 3 Dirac-like spinors, the final form of the bilinear invariants are slightly modified, but the net result is the same. While property one is sound in prospecting possible information about the representation space (in the context of Proposition 2), the study of the hermicity property  may be relevant in a quantum mechanical context.

The counter-example just studied indicates an elaborated representation space, whose non-triviality deserves further exploration. It must be once again emphasized, however, that the constraints (\ref{12}) are too restrictive, since it extends the kernel of the transformation to the whole $(\sigma, \omega)$-plane.

 We would like to finalize this work by pointing out three new classes of spinors which also reside in the spinor representation space. These spinors were obtained in the operatorial and algebraic form in Ref. \cite{Villalobos:2015xca}, having, by construction, $\mathbf{J}=0$. Therefore, their dynamics cannot be described by the Dirac operator. The nontrivial topology of the representation space, as already remarked, is inherited from the constraints imposed by the FPK identities. For the sector of $\Sigma(N)$ comprised by regular spinors, $\mathbf{J}$ is the generator of cohomology and cannot vanish. The sector of $\Sigma(N)$ encompassing singular spinors, nevertheless, may also accommodate the spinors of \cite{Villalobos:2015xca}. Notice that a vanishing $\mathbf{J}$ does not lead to a contradiction, and the FPK identities still hold in this case. Hence,  these spinors are also physical in the sense previously discussed. The spinors founded in \cite{Villalobos:2015xca}, assuming that ${\bf
 J}=0$,  may be called pole (only $\mathbf{K}\neq 0$), flag (only $\mathbf{S}\neq 0$), or flag-pole\footnote{It is worth to mention
 that these flag-poles are essentially different
 of the standard flag-poles characterized by the flag $\mathbf{S}\neq 0$) and the pole ($\mathbf{J}\neq 0$), since in this case $(\mathbf{K}= 0)$ \cite{LOU}.}. They live in a special subspace of $\Sigma(N)$ whose topology also deserve further attention.

\section{Concluding Remarks}

The formalization of a spinor representation space, whose points can be faced as physical spinors, has been constructed. These spinors have been shown to behave into dense paths of the representations space which, in view of the FPK identities, perform highly topologically constrained subsets. Some of these subsets have topological properties intrinsically connected to physical relevant quantities. The representation space show itself as an adequate tool to explore dynamics and interactions usually by means of using infinitesimal operators.

It should be emphasized that along this work we took advantage from dealing with spinors as elements of $\mathcal{C}\ell_{1,3}\frac{1}{2}(1+{e}_0)$ in Sec. II. Similar constraints in the representation space, coming from the FPK identities in the $\mathcal{C}l_4$ isomorphic case, are expected. However, our main interest here is the study of the representation space taking into account the Clifford algebra constructed upon the Minkowski space.

In showing that type-4 spinors can not be led into regular ones, we asserted about  the mapping connecting different physical spinors -- spinors of different sectors do $\Sigma(N)$, belonging to different classes. However, no reference has been made about quantum mechanics. It is time to elaborate this a little further. The very possibility of crossing over different classes, by means of a well defined algebraic transformation connecting different sectors of $\Sigma(N)$ could, in principle, be related to some type of swapping spinor class due to a specific physical process. In fact, bearing in mind the existence of a dense set in between different classes in the light of proposition two, this switching could be performed by a specific (unknown) scattering matrix modeling the physical process.  Apart from unitarity concerns\footnote{Typically, the $S_C$ matrix have enough symmetry to be recast into a specific form allowing for an unitary scattering process}, it is difficult to envisage how this attemptive process can duplicate the helicity states in going from regular spinors to type-5 spinors, these last spinors with known dual property helicity. Perhaps, and here we are entering in the fancy ground of speculation, a comprehensive transformation performed in the quantum operator as a whole may seed a precise light into the formal aspect of this possible swapping. It turns out, however, that the physical process would still be lacking. In the opposite way of this expectation, we delivered a mapping which is neither invertible nor Hermitian, evincing the high degree of topological constraint presented on the representation space. Further investigation on the algebraic/topological relationship concerning singular spinors are under current investigation.

\section*{Acknowledgments}
JMHS thanks to CNPq (304629/2015-4; 445385/2014-6) for partial financial support. CHCV thanks to CAPES (PEC-PG) for financial support. RJBR thanks to CAPES for financial support, and RdR~is grateful to CNPq (Grant No. 303293/2015-2),
and to FAPESP (Grant No.~2015/10270-0), for partial financial support.

redpajama_set_name: RedPajamaArXiv


CALGARY, ALBERTA--(Marketwire - Feb. 22, 2010) - NuVista Energy Ltd. ("NuVista" or the "Company") is pleased to provide the following operational update on its winter drilling program and 2009 year end reserves information. NuVista's annual audit of its consolidated financial statements is not yet complete and accordingly all financial amounts referred to in this press release are unaudited and represent management's best estimates.
The difficult business environment in 2009 created both challenges and opportunities for NuVista. When presented with the opportunity to acquire stable high quality assets at low purchase price metrics, NuVista increased its total 2009 capital program to $310 million, with approximately 73% allocated to acquisitions. In order to accommodate the increased acquisition capital expenditures, while maintaining financial strength in an uncertain commodity price environment, NuVista's exploration and development program was reduced from a planned level of $225 million to approximately $84 million. During this period of lower exploration and development expenditures, NuVista focused on expanding its prospect inventory by evaluating the use of new horizontal drilling and completion technology on its land base.
Over the past few years NuVista's asset base has changed significantly. NuVista's oil properties and long life W5/W6 natural gas properties now represent more than 75% of NuVista's production base, increasing our proved and probable reserve life index to 9.5 years and creating a stable platform for future growth. In addition, approximately 50% of NuVista's production now resides in the deep basin where a significant portion of NuVista's land base, both developed and undeveloped, contains high reserve-in-place oil and natural gas deposits that are currently experiencing dramatic improvements in recovery from the application of new technology. In the fourth quarter of 2009, NuVista embarked on the highest impact drilling program in its history including the testing of a number of repeatable high reserve-in-place plays using horizontal wells with multi-frac completion technology.
NuVista's unaudited fourth quarter results press release is expected to be released on March 8, 2010 and its audited consolidated financial statements are expected to be filed on SEDAR (www.sedar.com) on or before March 31, 2010. In addition to the reserves information disclosed in this press release, more detailed reserves information will be included in NuVista's Annual Information Form in addition to the full NI 51-101 disclosure for the year ended December 31, 2009, which is also expected to be filed on SEDAR on or before March 31, 2010. NuVista anticipates providing an updated corporate presentation on its website (www.nuvistaenergy.com) on March 9, 2010.
NuVista is continuing to test recovery concepts in high reserve-in-place formations where NuVista's substantial acreage position and success could have a material impact on its future growth plans. The use of horizontal wells with multi-frac completion technology will be tested in six different formations in the Wapiti core area during 2010. Horizontal wells are planned to be drilled during the first quarter in the Cardium, Dunvegan, Cadomin and Montney formations, with additional follow up wells planned prior to year end. Horizontal wells are also expected to be drilled in the Falher and Nikanassin formations later in 2010.
NuVista completed drilling a horizontal well (100% working interest) into a 20 metre thick Dunvegan D channel sand in January 2010. A multi-stage packer assembly has been run into the horizontal section with a multi-stage frac involving eleven intervals planned prior to the end of February. In addition to this horizontal well, over the past six months, NuVista has drilled and is now completing five vertical Dunvegan wells. A vertical test well into a 20 metre thick Dunvegan E sand flowed over 0.5 mmcf/d, adding the Dunvegan E formation to the list of formations in the area that could benefit from multi-stage frac completion technology. A horizontal well targeting the Dunvegan E sand is planned for later in 2010. NuVista has over 105 net sections of deep basin natural gas charged Dunvegan lands prospective for both vertical and horizontal development.
NuVista has extensive Montney landholdings in Wapiti with approximately 125 net prospective sections with upper Montney porosity in a deep basin gas setting. In the greater Wapiti area, another operator has drilled and completed two Montney horizontal wells. Both wells continue to produce at rates in excess of 2 mmcf/d and the first well has now been on production for a full year. Encouraged by these initial test rates and considering the size of the Montney resource-in-place, NuVista plans to conduct two horizontal well tests on our own Montney lands in 2010. NuVista's Wapiti Montney lands are situated in two large contiguous blocks. The north block will be tested by a 100% working interest horizontal well which is currently drilling and the south block is anticipated to be tested in the third quarter.
NuVista's Wapiti core area vertical drilling program of high impact wells continues to reward the Company. NuVista drilled a successful multi-zone natural gas well (97% working interest) that has been completed in multiple zones, including the Falher C, which has tested at a combined rate of over 7 mmcf/d. This well extends the prospectivity for the Falher C trend encountered in a well drilled by NuVista in the third quarter of 2009 which tested at a combined rate of over 10 mmcf/d from multiple zones. Three additional multi-zone vertical wells are planned in prior to spring break up.
NuVista is also currently drilling its first horizontal natural gas well into the Cadomin formation (70% working interest) and plans to use multi-stage frac completion technology to complete the well. NuVista has identified up to 100 gross (30 net) Cadomin infill drilling locations in the Wapiti area, many of which will have stacked multi-zone zone potential.
NuVista has approximately 85 net sections of contiguous Cardium lands in the Wapiti core area with similar characteristics to oil wells in the Pembina and Pine Creek area that have been successfully completed using horizontal drilling and multi-stage frac completion technology. NuVista has farmed-out five low working interest Cardium sections to a third-party who has committed to begin drilling a Cardium horizontal well prior to spring break up. NuVista plans to drill its first Cardium horizontal well (100% working interest) during the summer and has plans to drill up to three Cardium horizontal wells in the Wapiti core area prior to year end.
NuVista continues to increase its land position in its Wapiti core area and has a multi-year prospect inventory which continues to grow with the ongoing evaluation of its lands by its technical team and the use of new drilling and completion technology. NuVista plans to spend over one third of its corporate exploration and development capital in the Wapiti core area during 2010.
NuVista has approximately 200 gross (125 net) sections of Cardium rights in its Pembina/Ferrier core area. In the Pembina area, NuVista has approximately 70 net sections of Cardium rights. NuVista has high graded its inventory of Cardium lands in the central Pembina area where it has 34 net prospective sections and plans to drill up to 10 (average 75% working interest) Cardium horizontal wells during 2010. Three Cardium horizontal wells are planned prior to spring break-up. Most of the wells drilled during 2010 will be the first wells on sections, potentially leading to follow up development locations for 2011. NuVista has identified up to 100 net potential Cardium locations on its high graded 34 net sections. As the Cardium play fairway is expanded, some portion of the remaining 70 net sections in Pembina/Ferrier may ultimately become prospective for Cardium horizontal drilling.
NuVista participated in its first Cardium horizontal well (30% working interest) in the Pembina area during the fourth quarter of 2009. This well was operated by a third-party and encountered 1,050 metres of Cardium reservoir rock in the horizontal section. The well was completed with an 11 stage frac and is expected to be on production prior to the end of February. The second Cardium horizontal well (46% working interest) in the same region was drilled by NuVista and is currently being completed. Test results are expected on this well by early March. The third Cardium horizontal well (100% working interest) is planned to be drilled late in the first quarter of 2010 depending on weather.
In addition to Cardium potential, NuVista's lands in the Pembina/Ferrier area are also highly prospective for the emerging Notikewin resource play. During 2010, NuVista's primary focus in the Ferrier area will be testing the Notikewin formation using horizontal drilling and multi-stage frac completion technology. NuVista has approximately 190 net sections of Notikewin rights in the Pembina/Ferrier core area of which over 110 are prospective for Notikewin horizontal natural gas drilling. Most of NuVista's Notikewin lands are concentrated in high working interest contiguous blocks and NuVista has planned up to three Notikewin horizontal wells for the second half of 2010.
NuVista recently drilled its first Kaybob Montney horizontal well (100% working interest) at its Fir property and completed the well with nine 10 tonne fracs. The well flowed at a rate of approximately 6 mmcf/d during a three day in-line flow test and is now on production. A second Montney horizontal well (100% working interest) was drilled in the first quarter and is now waiting on completion. Up to five additional horizontal wells (100% working interest) are planned for the remainder of 2010. Based upon well spacing of three horizontal wells per section, NuVista has identified 23 gross (19.4 net) potential horizontal well locations at its Fir property. Facilities are being expanded from 7 mmcf/d to 10 mmcf/d by break-up and an additional expansion is being reviewed for the second half of 2010.
NuVista has also drilled three high working interest vertical wells in its Kaybob core area. Two 100% working interest wells flowed at combined rates of 7 mmcf/d and are now on stream at approximately 6 mmcf/d. The third well is on production at over 2 mmcf/d. NuVista is evaluating the drilling of additional vertical wells and one Notikewin horizontal natural gas test well in the Kaybob core area during the second half of 2010.
NuVista participated in three successful Dunvegan delineation oil wells (30% working interest) in its Kaybob core area during the first quarter of 2010. The wells are follow-ups to a third-party well drilled in 2009 that tested at over 200 bbls/day. Further locations are planned for winter 2010/2011.
At Hallam, in West Central Saskatchewan, NuVista drilled and completed four horizontal Birdbear oil wells (76% working interest) in the fourth quarter of 2009 as part of its ongoing pool development. Initial production rates from these wells ranged from 160 bbls/d to 240 bbls/d. 15 additional locations have been identified with four wells planned to be drilled during the first quarter of 2010.
The independent engineering evaluation of NuVista's corporate reserves effective December 31, 2009, has been completed by GLJ Petroleum Consultants Ltd. ("GLJ") (the "GLJ Report"). NuVista is pleased to disclose that its 2009 capital expenditure program totaling approximately $310 million achieved proved plus probable finding, development and acquisition costs of $11.77/boe (after revisions and including the change in future development costs and Alberta drilling royalty credits).
Unless otherwise indicated, the reserves information set forth in this press release are "company interest" reserves. "Company interest" means, in relation to NuVista's' interest in reserves, its working interest (operating or non-operating) share before deduction of royalties, plus NuVista's royalty interests in production or reserves. "Company interest" is not a term defined or recognized under National Instrument 51-101 - Standards of Disclosure for Oil and Gas Activities ("NI 51-101") and does not have a standardized meaning under NI 51-101. Therefore, the "company interest" reserves of NuVista may not be comparable to similar measures presented by other issuers, and investors are cautioned that "company interest" reserves should not be construed as an alternative to "gross" or "net" reserves calculated in accordance with NI 51-101.
- Despite a substantial reduction in year end pricing for natural gas, the proved plus probable net present value before tax at a 10% discount rate increased by $235 million to $1,586 million, implying a favourable return on capital expended in a very challenging pricing environment.
development costs related to reserve additions for the year.
(2) Drilling credits of $10.7 million were recorded during 2009.
and $20.9 million for proved plus probable reserves.
The estimated future net revenue contained in the following table does not necessarily represent the fair market value of the reserves. There is no assurance that the forecast price and cost assumptions contained in the GLJ 2009 Report will be attained and variations could be material. The recovery and reserve estimates described herein are estimates only. Actual reserves may be greater or less than those calculated.
Based on the success of our first quarter drilling program, the extent of our prospect inventory, and increasing stability in commodity prices, in particular for natural gas, NuVista has sufficient opportunities to target the upper end of our previously approved 2010 capital program of $240 million to $280 million. Approximately 75% of 2010 capital expenditures are expected to be spent on internally generated exploration and development activities. Based on a capital program of $280 million, 2010 production is forecast to increase from 28,345 boe/d in the fourth quarter of 2009, to between 32,000 boe/d and 33,000 boe/d by the end of 2010. Production for 2010 is forecast to average between 30,000 boe/d and 31,000 boe/d depending on the timing of exploration and development activities and downtime.
As NuVista continues to implement the largest high impact drilling program in our history, we look forward to continuing to update our shareholders on our drilling results.
This press release contains forward-looking statements and forward-looking information within the meaning of applicable securities laws. The use of any of the words "will", "expects", "believe", "plans", "potential" and similar expressions are intended to identify forward-looking statements or information. More particularly and without limitation, this press release contains forward looking statements and information concerning: NuVista's planned exploration and development program, the timing of the release and/or filing of NuVista's fourth quarter results, audited financial statements, Annual Information Form and updated corporate presentation, NuVista's drilling, testing and completion plans, the timing thereof and the results therefrom, anticipated drilling and development locations, facility expansion plans, expected future development capital, 2010 capital expenditure program and allocation thereof and estimated 2010 production and NuVista's ongoing strategy and growth plans.
In addition, please note that statements relating to "reserves" or "resources" are deemed to be forward-looking statements, as they involve the implied assessment, based on certain estimates and assumptions, that the resources and reserves described can be profitably produced in the future.
The forward-looking statements and information in this press release are based on certain key expectations and assumptions made by NuVista, including prevailing commodity prices and exchange rates; applicable royalty rates and tax laws; future well production rates; reserve and resource volumes; the performance of existing wells; the success obtained in drilling new wells; the sufficiency of budgeted capital expenditures in carrying out planned activities; the availability and cost of labour and services; and the receipt, in a timely manner, of regulatory and other required approvals. Although NuVista believes that the expectations and assumptions on which such forward-looking statements and information are based are reasonable, undue reliance should not be placed on the forward-looking statements and information because NuVista can give no assurance that they will prove to be correct. There is no certainty that NuVista will achieve commercially viable production from its undeveloped lands and prospects.
Since forward-looking statements and information address future events and conditions, by their very nature they involve inherent risks and uncertainties. Actual results could differ materially from those currently anticipated due to a number of factors and risks. These include, but are not limited to the risks associated with the oil and gas industry in general such as: operational risks in development, exploration and production; delays or changes in plans with respect to exploration or development projects or capital expenditures; the uncertainty of reserve estimates; the uncertainty of estimates and projections relating to reserves, production, costs and expenses; health, safety and environmental risks; commodity price and exchange rate fluctuations; marketing and transportation of petroleum and natural gas and loss of markets; environmental risks; competition; incorrect assessment of the value of acquisitions; failure to realize the anticipated benefits of acquisitions; ability to access sufficient capital from internal and external sources; stock market volatility; and changes in legislation, including but not limited to tax laws, royalty rates and environmental regulations.
The term "boe" or barrels of oil equivalent may be misleading, particularly if used in isolation. A Boe conversion ratio of six thousand cubic feet per barrel (6 Mcf: 1 Bbl) of natural gas to barrels of oil equivalence is based on an energy equivalency conversion method primarily applicable at the burner tip and does not represent a value equivalency at the wellhead.
This press release contains test results for various NuVista wells. Actual production from these wells could differ materially from these test results.


text:
Q: Google drive for desktop - how can I remove the virtual drives? I just transitioned from Backup & Sync to Google Drive for Desktop, and now the drive app has created a virtual drive in my desktop. Within this virtual drive there is a link to my drive folder in the other drive. I chose the Mirror option, not the Stream, so there is nothing in the virtual drive except that link.
Is there a way to remove this virtual drive, and just have the drive folder synced, like it did before?

redpajama_set_name: RedPajamaStackExchange


I was lucky enough to be sent the multi award winning sun cream Tancream. I had read some great reviews about this product so was excited to try. The sun cream has 50SPF and a instant golden glow so this cream is perfect to use for a day in the sun.
The packaging is stunning. It's purple and pink with clear writing down the side and the sizing is perfect for carrying in your handbag or beach bag on holiday. This sun cream is also a 5-in-1 product meaning you have a moisturizer, anti ageing cream, SPF, self tan and bronzer all in one bottle, packing for holiday has never been easier!
I applied the Tancream in the morning before a day out in this lovely weather we've been having. I noticed that there is no smell on the sun cream which is ideal especially because with products with instant glow you sometimes get the fake tan smell.
The application went on smoothly and there was no streaks at all. I was amazed how easy the sun cream applied and left a beautiful, natural glow on my skin. This is perfect for the face too as it has a high SPF and the cream will cover any blemishes or scaring, like a CC cream, meaning you can go makeup free on those hot days.
As the day was continuing I noticed there wasn't any patches where the gradual tan was and the skin hadn't gone dry.
I can understand why this product has featured in magazines including You and Heat as it's the ideal travel buddy. Thank you Tancream and Luxuria Lifestyle for letting me try and fall in love with this product.


Let the local experts at Travel The Rockies design your custom, stress-free vacation. Lodging, activities & daily itinerary. Each detail of your vacation perfectly tailored to suite your travel needs. Our backyard is your playground… Get Out & Explore. Yellowstone Your Way!


This week, New York society stepped out for the Park Avenue event "Save Venice," a super-fancy masquerade ball to raise money for the restoration and preservation of Venetian artworks. The difference between big-time parties like these and the Academy Awards is that socialites actually pay for and own their couture.
These guys on the left with Michelle Harper (whose mask was created by Björk's makeup artist, FYI) are Peter and Harry Brant, model Stephanie Seymour's sons. Can you believe she has adult sons? To the right is Charlotte Ronson in a simpler netted mask.
Purple-haired writer/artist Beatrix Ost [far left] is my favorite with her everything-but-the-kitchen-sink/Beale look. Although Susan Shin [far right] is a close second in an elaborate muumuu accompanied by white fur. The rest of the ladies, from second from the left: Anne Fitzpatrick Cucchiaro, Heather Georges, Kate Peck, and Pamela Ludwick.
Some of the more elaborate masks were worn by Louis Corello, Donna De Cruz, and Christine Schwartzman's breasts.
Coralie Charriol Paul, Cornelia Guest and Tinsley Mortimer weren't as married to the masked theme as other attendees.
Full of skirt: Chiu-Ti Jansen, Alexandra Lebenthal, and Amanda Hearst.
They'll see you in your nightmares.


text:
Clinical Sub Dean - Nottinghamshire Hospitals Hub (Part time)
School of Medicine - Education Centre
Location: UK Other
Salary: £88,364 to £119,133 per annum pro-rata depending on skills and experience.
Closing Date: Monday 13 February 2023
Reference: MED616222
Applications are invited for the above post based at the University of Nottingham and our Nottingham Hub placement sites at the University Hospitals of Derby and Burton, Chesterfield Royal Hospital NHS Foundation Trust, and Derbyshire based primary care practices.
The Clinical Sub Dean (CSD) will work with the existing CSD team to oversee the pastoral care and developmental progress of students in the Nottinghamshire hub by providing advice, overseeing special circumstances and monitoring progression in students who are experiencing difficulty. Students who have problems or who request support will be seen individually by the CSD for review and development of an appropriate action plan and follow-up.
This exciting role is crucial to the delivery of our Support and Wellbeing strategy and our medical student experience during the final two full time clinical years of our BMBS programme and transition into the foundation programme.
You will work with the other CSDs at our other two hubs (Derbyshire and Lincolnshire) and the wider support and wellbeing team and share responsibilities to ensure that requirements specified in the University of Nottingham Quality Manual and GMC document "Outcomes for Graduates" are met.
This role is available on a permanent basis. Whilst this is a University academic appointment at 0.2 FTE (7.7 hours per week), it is a requirement that you maintain regular NHS clinical consultant practice in Speciality or General Practice, GMC registration, a licence to practice and entry on the relevant GMC Specialist or GP Register and be revalidated with the GMC.
As with all roles in the University, the School would welcome expressions of interest from individuals whose backgrounds and identity broadens and enhances the diversity of our team and community.
Informal enquiries may be addressed to Professor Pam Hagan, pamela.hagan@nottingham.ac.uk. Please note that applications sent directly to this email address will not be accepted.
It is a condition of this post that satisfactory enhanced disclosure is obtained from the 'Disclosure and Barring Service'.
View Additional Information (this will open a new window)

redpajama_set_name: RedPajamaCommonCrawl


Well, if you are among those lucky people who have managed to bag a ticket to the summer's biggest festival, then here are some tips on what to wear at Glastonbury.
One of the most appropriate times to wear the boho look is definitely during the festival season. And with luxe hippy being such a huge trend this season you got so much to look forward to and flaunt in the best of ways!
If you are set to make it to Glastonbury for the weekend, then you do not have to look beyond Alexa Chung, the ultimate trendsetter. Definitely there are other famous figures to follow too- Kate Moss, Glasto's all time fav and well Millie Mackintosh is also there!
These accessories will definitely look super trendy. They will help to hide those dark circles especially. And they will also save you as people will not get to know about those puffy eyes and drunken look! Rayban Aviators and Wayfarers are simply classic for almost every face shapes.
These are kinds of the standard style statement at Glastonbury. Definitely when you look up to Kate Moss she has worn them ample number of times, and well so has Sienna Miller. These are one of the bests of ways to give your willies a more edgy and stylish look!
Go for bangles, multiple neck pieces, hair accessories, belts and so on.
When it comes to fringed Minnetonka we have Kate Moss, who is a big time fan of these. These boots provide you with a perfect Glasto appearance. However wear them in case it's not raining!
At the spring/summer fashion runway 2015, the big festival look seemed to be all over the place. Hence go for boots, bags and tops with studs and fringes.


For more than six years, I've been lucky to work with someone I admire a lot. Professor Sujatha Jesudason is an innovative leader in design and racial justice and teaches in practice in management at the Milano School for Public Policy at The New School. I was a fellow in the first generation of her Generative Fellowship, where we learned design thinking and how to test our ideas and to take them from contemplation to planning to action. Last year, we were both offered positions at The New School and moved to NYC where we've continued to build our relationship and our work together. Now, I have the privilege of serving her successor, Judy Pryor-Ramirez as an advisor and communications consultant for Social Movements + Innovation @ The New School. This week, I wrote on fear for the SM+I blog. Check it out.
On Tuesday, poll workers will brave the cold autumn daybreak, set up their folding tables and chairs, and prepare to facilitate the democratic process. Many of us, though not nearly enough, will cast our ballots in the hard-fought, and not entirely won, custom of choosing the candidates and issues we believe matter to us. We're casting our ballot for what we want, but we're also casting our vote against what we don't want—against what makes us fearful.
Fear is something we all experience. It is among the few innate emotions humans have. The "fear center" of our brains is called the amygdala, and it supports survival because when we experience fear, our reptilian brain (aka our amygdala) signals to us that we should fight, flee, or freeze. Voting is one way we fight back against the things that scare us.
Since September, we've been on the road hearing from CoreAlign alum and new SM+I collaborators about what they fear. They're afraid of rising conservatism, of power dynamics between philanthropists and their grantees, and of losing sight of the big picture in favor of small, incremental wins. I'm afraid of those things, too.
We fear many of the same things, and sometimes we're afraid of the opposite things. Some of the folks we talked to are afraid that we are losing momentum and on the precipice of burnout as a movement cohort, and others fear that we are not doing nearly as much as we could. Some people believe we are over-reliant on technology, and others don't think we have access to enough of it. A lot of people fear loss: loss of opportunity, time, and political power.
Fear intersects with our lives all the time. And when we fear things, we feel it in our body, too. We experience fear in our vagus nerve, which psychologist and racialized trauma expert Resmaa Menakem calls "the soul nerve"—this is where we feel hope, dread, fear, empathy, anxiety, disgust, and despair. This nerve interfaces with parasympathetic (relating to the part of the automatic nervous system that counterbalances the action of the sympathetic nerves) control of the heart, lungs, and digestive tract. "It's the largest organ in your body's autonomic nervous system which regulates all of your body's basic functions," Menakem writes in his book, My Grandmother's Hands.
Whether our fear is real or perceived, it motivates us. Right now, less than a week before the midterm elections, there is a lot of talk about fear. On Monday, President Trump sent more than 5,200 active-duty troops to the U.S.-Mexico border to escalate already-rising fear of the Latin American men, women, and children fleeing terror in their homelands in search of a better, safer life. Those families are also fearful: fearful of what will happen to their children if they stay and undoubtedly afraid of the dangerous trek toward a hostile country. Fear is everywhere, and we can use it to make things better or worse; because although fear is innate, we have control over how we wield it.
The first step is recognizing fear when you feel it. Recognizing our fear is challenging because the reptilian brain cannot distinguish between real and perceived danger. This makes moving outside our comfort zone scary. Harnessing our fear for good requires us to question what we think we know about the world around us and to ask ourselves, "What kind of fear is this—real or perceived?" Yes, this adds one more step to our already complicated lives, and it can also help us all make more rational decisions about how we engage with one another and the systems to which we are all bound, such as democracy.
As adults, we're not provided with many spaces to talk about what we're afraid of, but since fear is innate and something we all experience, we knew it was important to ask people to give us specific answers to the question, "What do you fear?" on our six-city innovation pop-up listening tour. As we finish up our rounds, we are documenting our fears so we can have an honest discussion of which fears are real and which are perceived, and so we can find innovative and mutually beneficial ways to confront them.
Happy Halloween. Vote if you can.


Robert A Emmons; Guilford Press.
This volume makes a powerful case for the inclusion of ultimate concerns - spiritual and religious themes in personal strivings - in an attempt to build a motivational theory of personality.
Druk na podstawie: "Paperback edition 2003."
Part I: Personal Goals: An Approach to Personality and Subjective Well-Being. Introduction: Motivation and Spirituality in Personality. Personal Goals as Units of Analysis. Personal Goals and Subjective Well-Being. Goal Conflict and Personality Fragmentation. Part II: Spirituality, Goals, and Intelligence. Spiritual Strivings as Ultimate Concerns. From Goal to Whole: Spiritual Strivings and Personality Integration. Personal Goals and Life Meaning. Spiritual Intelligence: Toward a Theory of Personality and Spirituality. Appendix A: Personal Striving Assessment Packet. Appendix B: Personal Striving Coding Manual.
Add tags for "The psychology of ultimate concerns : motivation and spirituality in personality". Be the first.
schema:publisher <http://viaf.org/viaf/123113876> ; # Guilford Press.


Merriam-Webster Dictionary defines 'cheerleader' as, "One that calls for and directs organized cheering (as at a football game)." Lovejoy cheer team encompasses that definition, and so much more.
Lovejoy Cheer, under the direction of Shelly Wiggins, has increased in size since last year. There are currently 57 students on the freshman, JV and varsity cheer teams.
Varsity cheerleaders, (from left) Lacie Gager and Madison Kohl, decorate the school with spirit posters.
Students must tryout to become a Lovejoy High Cheerleader. Contenders are scored on spirit and crowd appeal, tumbling, toe touch, front hurdler, pike, combo jumps, cheer technique, projection points, fight song technique and overall impression. The same score sheet is used in middle school. Everyone who tries out makes the team in middle school.
Wiggins said that being a member of Lovejoy Cheer brings multiple benefits to the students. She said that cheerleaders get to know their peers and people in the community. They are strong athletes. They improve their leadership and time management skills and learn to balance their priorities.
Members of the junior varsity cheerleaders during practice.
In addition to cheering at games, Lovejoy Cheerleaders compete at UIL Wiggins said, "Last year, over 125 teams competed in our division, we finished in 11thplace. This was a big improvement over the previous year." They perform at football, volleyball and basketball games. They attend other school sports and offer their support. They also perform at pep rallies. They participate in Reading with Leopards where they read to students at elementary schools. They host mini-cheer camps which conclude with the younger students performing with them at a football game. They decorate the community for homecoming. They decorate football players' lockers. They log many volunteer hours at school and within the community. They are currently working on programs which will include breast cancer research fundraising and a special needs cheer team.
Wiggins has been a cheer coach for 23 years. This is her third year at Lovejoy High School.
Lovejoy Varsity Cheer Team. Photo by Southern Images Photography.


Brewed to showcase the best of Irish malted barley, the Full Irish Single Malt IPA is a beer that's very close to our hearts.
We were delighted to have it acknowledged at the 2017 Dublin Beer Cup, when it won a lovely, heavy silver medal.
It also gave *some people* a chance to pull out the photo that we took to launch it, way way back in 2014 (still makes me smile too!).
Find the tasting notes for The Full Irish Single Malt IPA here.


If you're passionate about the bygone days of two-wheeled motorsport, then the MRN forum is a haven where you'll be among friends! Mingle with former riders, journalists, mechanics and fans as motorcycle racing of the past is enjoyed once again.
Here's the venue for you to discuss your favourite genre of motor sporting memorabilia, or to enquire about it, or maybe to simply show off your collection. Whether your hobby is collecting models, photos, stickers, autographs, artwork, team clothing, badges or any other form of racing memorabilia & ephemera - this is the place!
Here's the place to discuss motorsport mysteries from the past, to raise questions and debate issues of a purely historical nature. Share talk about events of the past that might puzzle you, about your pet research projects, and perhaps explode some myths. WARNING: this is a strictly 'fluff-free' zone!
Threads that have been inactive since December 31 2013 are found here.
Le Mans 89 - which C9 broke the 400?
The eBay thread - post your news here!
Historic F1 fan? Get to Mallory Park every Wednesday morning!


As promised, a MeMe, but thankfully I've got mother.wife.me holding my hand to help me on my way.
A thought: Our society doesn't value or support mothers as well as it could.
I've blogged a couple of times on this issue, as has mother.wife.me. So, we've decided we'd like to team up and make a bit more noise in the blogosphere about it.
Oh and finally….Little Blue Feet, Japanese Mummy UK A Mum and More, A Twenty Something Mum, Multi Layer Mummy you're tagged!!
Did you work before becoming a mum?
Yes, I decided to build up my career before having children as I was aware it would not be easy afterwards. I got as many promotions as I could and made sure my salary was locked before I went on maternity leave.
I now work three days a week in a "professional job". I am constantly subject to derisive comments from others about being "part-time" despite the fact I am always a full time mum. We are lucky in that my parents look after our children three days a week although they are getting older and it is not easy for them.
To get my part time position was very difficult and I really had to jump through hoops, I felt there was nothing to really protect me from a flat refusal apart from my own wit. I am constantly exhausted from trying to be the best worker and best mum and don't know how long I can sustain it for. I feel I am being pulled apart and every time I leave my children to go to work my heart breaks a little.
A lot of my friends have had their part time work request refused, so they have been forced to leave their jobs to seek (lower paid) temporary contracts. A lot of people seem to sneer at the idea of a woman wanting to spend time with her children if she previously had a career, like it's some sort of consolation prize. The fact is that I physically could not work full time and give 100%. As it stands, being part time has benefited my employer as I never take sick days and am much fresher coming to work for three days.
I don't want to leave work completely as in the current job market I fear I will never find work again.
Here we go.... lovely to see our respective tagees are on the case with their A Mothers Work Meme contributions. It is going to be really interesting to collate everyone's thoughts and experiences from this.
Oh I have lots to say about this! Do I have to be tagged to participate or can I just wade in and add my thoughts on the matter?!
My addition to the #amothersworkmeme is a go go! Lots of interesting reading on the others already out there.
Super! Have read and commented.
I hear your pain about work/life balance. I returned to work part-time when I had baby #1. My boss at the time, said he got more form me in 3 days than he did from most of the so-called "full-timers" (but maybe he was just being nice):-) I guess, smoking and tea breaks soon take a chunk out of a working week.
During this phase of 2 years, I felt dreadful that #1 had to go to a childminder from 4 months old. When I was at work I fretted about #1 and when I was with #1, I fretted about work. Feeling that there was no "win-win" situation, we took steps to ensure that #2 would not have to go to another childminder.
Having left the place of "employee" some 8 years ago now, (time flies), my few attempts to return to the "professional" roles I had BC (before children) have failed. I'll never know if it's becasue I'm a mum or other shortcomings. I have kept my professional qualifications up to date throughout that time and still do self-employed work to keep my hand in. Part of me would like a happy medium of a part-time, employed role because on top of everything else mums have on their plate, generating an income yourself is b****y hard work!!
PS: In my experience , it's never the dads who get the call from school or nursery asking for said child to be picked up because of illness or injury.
Great comment about dads never getting the call from school or nursery. So true. Why is it always the Mum? I was the bread winner, and my husband never worked and I always got the call, despite them telling them to call him. Because I was the bread winner, I was looked on as a mum not doing her job.. being a 'proper' mum!
I took time off to be with my kids, and no-one looks at me seriously now. The big fancy jobs I had don't fit in with my life, and the other jobs barely pay my petrol!
I now have 3 part time jobs and run around ragged so I can pay a few bills, and be there for my children. Have it all? Crikey!
I'm in the same boat now, I have three part time jobs too, I'm ridiculously busy and feel that I am glued to the computer, it's not the best thing for the children really but it allows us to keep a roof over our heads. Having said that, I could never go back to being a normal employee or a stay at home mum, it just doesn't stimulate me enough and it makes me unhappy. That may be selfish but being a mum is unselfish for the majority of the time so sometimes you have to claw back what you can!


text:
Q: Erro: NodeJs, javascript "belongsTo called with something that's not a subclass of Sequelize.Model" Estou tentando usar a função belongsTo e recebo este erro.
Connection DATABASE has been established successfully.
(node:14119) UnhandledPromiseRejectionWarning: Error: User.belongsTo called with something that's not a subclass of Sequelize.Model
    at Function.<anonymous> (/home/pedro/Documentos/projetos/node/gostack/goBarber/node_modules/sequelize/lib/associations/mixin.js:93:13)
    at Function.associate (/home/pedro/Documentos/projetos/node/gostack/goBarber/src/app/models/User.js:38:10)
    at models.map.map (/home/pedro/Documentos/projetos/node/gostack/goBarber/src/database/index.js:25:48)
    at Array.map (<anonymous>)
    at Database.init (/home/pedro/Documentos/projetos/node/gostack/goBarber/src/database/index.js:25:8)
(node:14119) UnhandledPromiseRejectionWarning: Unhandled promise rejection. This error originated either by throwing inside of an async function without a catch block, or by rejecting a promise which was not handled with .catch(). (rejection id: 1)
(node:14119) [DEP0018] DeprecationWarning: Unhandled promise rejections are deprecated. In the future, promise rejections that are not handled will terminate the Node.js process with a non-zero exit code.

model/User
import Sequelize, { Model } from 'sequelize';
import bcrypt from 'bcryptjs';

const saltRounds = 4;

/* virtual significa que este campo nunca vai existir */

class User extends Model {
  static init(sequelize) {
    super.init({
      name: Sequelize.STRING,
      email: Sequelize.STRING,
      phone: Sequelize.STRING,
      password: Sequelize.VIRTUAL,
      password_hash: Sequelize.STRING,
      provider: Sequelize.BOOLEAN,
      dd: Sequelize.STRING,
    },
    {
      sequelize,
    });

    /* addHook é uma propiedade do sequelize para executar uma bloco de código em uma
     * operação seja ela antes depois etc */

    this.addHook('beforeSave', async (user) => {
      if (user.password) {
        user.password_hash = await bcrypt.hash(user.password, saltRounds);
      }
    });
    return this;
  }

  /* Associa o avatar_id do model file com o model usuario */

  static associate(models) {
    this.belongsTo(models.File, { foreignKey: 'avatar_id' });
  }
  /* Dentro do model pode fazer funções referentes a ele, muito interessante */

  checkPassword(password) {
    return bcrypt.compare(password, this.password_hash);
  }
}

export default User;

e então instancio no index
import Sequelize from 'sequelize';
import DbConfig from '../config/database';
import User from '../app/models/User';
import File from '../app/models/File';

const models = [User, File];

class Database {
  constructor() {
    this.init();
  }

  async init() {
    this.connection = new Sequelize(DbConfig);

    try {
      await this.connection.authenticate();
      console.log('Connection DATABASE has been established successfully.');
    } catch (err) {
      console.error('connection to database ERROR ERROR ERROR:', err);
    }

    models
      .map((model) => model.init(this.connection))
      .map((model) => model.associate && model.associate(this.connection.model));
  }
}

export default new Database();

A: Por algum motivo misterioso foi apenas trocar a palavra 'model' dentro do map por qualqer outra coisa que funciona exemplo.
models
  .map((exemplo) => model.init(this.connection))
  .map((exemplo) => model.associate && model.associate(this.connection.model));


text:
Q: Shebang pointing to script (also having shebang) is effectively ignored Consider following code:
#!/usr/bin/env python

import sys

print "Hello! I've got %r as input." % sys.stdin.read()

This is chmod +xed script in /usr/local/bin/my_interpreter. And this:
#!/usr/local/bin/my_interpreter

This is intended to be passed "as is" to python script.

Is chmod +xed script that tries to make use of it. If I echo something | /usr/local/bin/my_interpreter, it works fine, but once I try to execute script above, it fails with
/Users/modchan/test_interpreter/foo.bar: line 3: This: command not found

Seems that foo.bar is silently redirected to bash instead of my script. What am I doing wrong? How to make this work?

A: This depends on the program loader of the operating system you're running, which I take to be OS X from your tags. Many UNIX-like operating systems require the shebang interpreter to be a compiled executable binary, not another script with another shebang.
http://en.wikipedia.org/wiki/Shebang_(Unix)
Linux has supported this since 2.6.27.9, but the author of this article suggests that there probably aren't any Berkeley-derived Unixen (which would probably include OS X) that do:
http://www.in-ulm.de/~mascheck/various/shebang/#interpreter-script
One way to accomplish what you want would be something like this:
$!/bin/sh
exec /usr/local/bin/my_interpreter <<EOM

... content to be executed ...
EOM

Another way would be something like this:
$!/usr/bin/env /usr/local/bin/my_interpreter
... content to be executed ...

A: Looks like Mac OS X requires interpreter to be binary, not another script. To make it work, change the second script's interpreter to
#!/usr/bin/env /usr/local/bin/my_interpreter

But you've got a second problem here: the contents of the second script will not go to stdin of its interpreter, but the script pathname will be passed as command line argument, i.e.
/usr/bin/env /usr/local/bin/my_interpreter /Users/modchan/test_interpreter/foo.bar

You shall read the file by name sys.argv[1] rather than from sys.stdin.


text:
Category: Milwaukee Bucks
Basketball, Brooklyn Nets, Charlotte Hornets, Detroit Pistons, Houston Rockets, Milwaukee Bucks, NBA, NBA Draft, Oklahoma City Thunder, Philadelphia 76ers, Podcasts, Trade Rumours
The Sportcodex Show: Ep.38 NBA Trade Frenzy & Draft with Tuaine
Basketball, Boston Celtics, Houston Rockets, los angeles clippers, Los Angeles Lakers, Miami Heat, Milwaukee Bucks, NBA, NFL, Playoffs, Podcasts, Sports Cards, Toronto Raptors
The Sportcodex Show: Ep.29 NFL, Chronicles Cards, Steve Nash & NBA
Basketball, Dallas Mavericks, Final Fantasy, Gaming, Los Angeles Lakers, Milwaukee Bucks, Movie Reviews, Movies, NBA, Podcasts, Portland Trail Blazers
The Sportcodex Show: Ep.26 NBA, Tax Collector & Final Fantasy Rankings
Basketball, Dallas Mavericks, Denver Nuggets, General, Houston Rockets, Indiana Pacers, Memphis Grizzlies, Miami Heat, Milwaukee Bucks, NBA, New Orleans Pelicans, Oklahoma City Thunder, Philadelphia 76ers, Podcasts, San Antonio Spurs, Toronto Raptors
The Sportcodex Show: Ep.25 The NBA Restart
Basketball, Betting, Gaming, General, los angeles clippers, Los Angeles Lakers, Milwaukee Bucks, Music, NBA, NRL, Podcasts, Rugby League
The Sportcodex Show: Ep.8 FFVII Remake, Ranking Tool's Albums, LeBron MVP Rant & Twitter Questions
Basketball, Chicago Bulls, Cleveland Cavaliers, Detroit Pistons, Indiana Pacers, Milwaukee Bucks, NBA
NBA Season Preview – Central Division
Basketball, Boston Celtics, Brooklyn Nets, Detroit Pistons, Golden State Warriors, Houston Rockets, Indiana Pacers, los angeles clippers, Milwaukee Bucks, NBA, Oklahoma City Thunder, Orlando Magic, Philadelphia 76ers, Playoffs, Podcasts, Portland Trail Blazers, San Antonio Spurs, Toronto Raptors, Utah Jazz
On Point Podcast: Playoff Edition 2018-19
Basketball, Brooklyn Nets, Denver Nuggets, Detroit Pistons, Milwaukee Bucks, NBA, New Orleans Pelicans, Philadelphia 76ers, Phoenix Suns, Portland Trail Blazers, Trade Rumours
Five non-Lakers/Celtics trades for Anthony Davis
Basketball, Boston Celtics, Chicago Bulls, los angeles clippers, Milwaukee Bucks, NBA, Playoffs, Toronto Raptors, Utah Jazz
NBA Wrap Up Playoffs Day 4
Basketball, Boston Celtics, Cleveland Cavaliers, Golden State Warriors, Houston Rockets, Indiana Pacers, los angeles clippers, Memphis Grizzlies, Miami Heat, Milwaukee Bucks, NBA, Oklahoma City Thunder, Podcasts, San Antonio Spurs, Toronto Raptors, Utah Jazz, Washington Wizards
On Point Podcast: Ep.8 The NBA Playoff's Potential Series & Matchups
Basketball, Boston Celtics, Cleveland Cavaliers, Denver Nuggets, Golden State Warriors, Houston Rockets, los angeles clippers, Miami Heat, Milwaukee Bucks, NBA, Oklahoma City Thunder, Podcasts, San Antonio Spurs, Toronto Raptors, Utah Jazz, Washington Wizards
The On Point Podcast: Ep.7 Resting, styles of play for Playoff Teams and Award Races
Basketball, Charlotte Hornets, Detroit Pistons, Milwaukee Bucks, Minnesota Timberwolves, NBA, Podcasts
On Point Podcast: Ep.4 LaVine & Timberwolves, East Overview and Non-Allstar Teams
Basketball, Detroit Pistons, Memphis Grizzlies, Milwaukee Bucks, NBA, New Orleans Pelicans, Oklahoma City Thunder, Portland Trail Blazers
NBA Point Spread Results: Day 4 & 5