As consumers move their financial activities online, applying online for a payday loan may seem like the natural thing for a cash-strapped person to do.
But you could be setting yourself up for a world of hurt, from paying exorbitant interest rates to having funds swiped from your bank account to being threatened by debt collectors. Just filling out an application could be enough to begin the harassment and thievery.
"Absolutely the worst thing you can do is apply for an online payday loan," says Jay Speer, executive director of the Virginia Poverty Law Center.
Most online payday loan sites aren't even operated by lenders. They're run by "lead generators," who seek your personal information, such as Social Security number, driver's license number and bank account details. They then sell that information to lenders.
"Your email and telephone explode after that," Speer says, as lenders vie to offer you cash. That can happen even if you live in one of the 15 states where payday loans are illegal.
Lenders aren't the only ones in the market for your personal information. "There's a good chance they sell to fraudsters — people who come after you months or years later," he says.
Sandra Green (not her real name) has experienced this firsthand. The Virginia woman turned to online payday loans after her husband was injured and couldn't work for two years. Their credit was damaged and they couldn't get cash to pay their bills from traditional financial institutions.
Green took out several loans totaling $3,000 to $4,000 starting around 2010. The lenders that she received cash from took their payments from her bank account — but they weren't the only ones. A company she had never heard of swiped money from her account, creating an overdraft.
She filled out a request for the bank to stop payment. That worked for about six months, and then the withdrawals started again. "People will change the identity of the company and then they'll hit it (the account) again. Once they do this it's a never-ending cycle," she says.
Companies she'd never done business with would call her at work and at home, harassing her. One threatened to file papers with the local sheriff's office if she didn't pay immediately.
"They get really belligerent when you don't do what they want you to do," Green recalls.
She feared she'd wind up in bankruptcy because of the loans and finally sought help from Blue Ridge Legal Services, a Virginia legal aid society, in 2013. Blue Ridge connected her with the Virginia Poverty Law Center.
Her experiences are not uncommon. "Fraud and Abuse Online: Harmful Practices in Internet Payday Lending," a 2014 study by the Pew Charitable Trusts, found online installment payday loans typically have an APR of 300 percent to more than 700 percent. Online lump-sum payday loans have a typical APR of 650 percent, or $25 per $100 borrowed per pay period. Exorbitant fees are also charged, and initial payments might not be applied to the loan's principal.
"Harassment and fraud are really concentrated in the online lending market," says Nick Bourke, project director for Pew's study on payday loans.
Part of the problem stems from the fact that there's no control over who can get your information once you apply for an online payday loan. "People's personal information can be spread far and wide," Bourke says.
Even if the loans are fraudulent, a consumer's failure to pay them may be reported to one of the three main credit bureaus, Speer says, which can impact a consumer's ability to rent an apartment or land a job.
"These unlawful lenders roam the Internet trolling for customers. They are scammers. They are fraudsters," says Amy Cantu, spokeswoman for the Community Financial Services Association of America, which represents more than half of the country's storefront payday lenders.
Association members vow to adhere to the organization's best practices, which include complying with state and federal laws, being licensed in each state in which they do business and adhering to acceptable debt collection practices.
Some of the association's larger members also have an online presence, she says, but those sites also adhere to the organization's best practices.
Her organization wants the federal consumer watchdog agency, the Consumer Financial Protection Bureau, to crack down on illegal lenders.
Already the CFPB and the Federal Trade Commission are stepping up action against fraudsters. In a joint news conference in September, the agencies announced they'd filed suit against two online payday lenders.
The CFPB received more than 1,300 consumer complaints about the Hydra Group.
Both the Hydra Group and CWB Services were accused of buying personal information, including bank account numbers, from lead generators. The companies would deposit money into consumers' bank accounts without any signed agreements, and then make unauthorized withdrawals from the accounts. If a consumer complained, the companies would produce false loan documents.
A federal judge temporarily shut down the Hydra Group, freezing its assets. The CFPB is requesting a permanent shutdown, along with penalties imposed upon the company and refunds made to consumers.
With CWB Services, the federal court froze the company's assets and appointed a receivership and the FTC is requesting consumers' money be refunded. The company had raked in $46 million in 11 months, said Jessica Rich, the FTC's director of the Bureau of Consumer Protection.
Bourke says the CFPB should ensure that small loans are tailored to the borrower's ability to pay them off and should provide more protection to consumers, particularly against illegal debt collection practices.
"The core of the problem is that payday loans don't help people. They drive people further into debt and distress," he says.


Willy Brandt Center for German and European Studies, Academia Europaea Knowledge Hub, College of Eastern Europe and Institute of Political Science at University of Wrocław invited to a public lecture by Yale Ferguson on The United States, NATO, and Ukraine: European Security in a Changing Context.
Russia's military seizure of Crimea was the first invasion of a country and forced change of sovereign borders in Europe since World War II, and a ruthless campaign of violence and intimidation continued against Eastern Ukraine. These shocking events caught the United States and its NATO allies woefully unprepared. In 2008, President Vladimir Putin had signaled his revanchist intentions and tested Western resolve by intervening and engineering "frozen conflicts" in Georgia's separatist Abhazia and South Ossetia. Meanwhile, NATO's involvement in Libya and bickering over Syria had highlighted the alliance's internal faultlines as well as organizational and military weaknesses. Concern was also rising over ISIS gains in the Middle East and Chinese assertiveness in the East and South China Sea.
The West's response to Russian aggression proved to be slow, divided, largely symbolic at the outset, and minimal as the crisis evolved. Sanctions were too little and too late, and NATO's establishment of a relatively small rapid response force is an insufficient military capability for what appears certain to be a renewed Cold War. Much greater resolve, military build-up, and new NATO strategy will be required to curb Putin's expansive ultranationalism and clear design to re-establish the Russian empire as far as opportunity permits.
Yale H. Ferguson, is Professorial Fellow in the Rutgers University graduate Division of Global Affairs at Rutgers University-Newark and Emeritus Professor of Global and International Affairs. Previously he was Professor II of Political Science and from 2002-08 was Co-Director of the Division of Global Affairs. He also is Honorary Professor at the University of Salzburg (Austria).
Dr. Ferguson' publications include 12 books and over 60 book chapters and articles. Books with frequent co-author Richard W. Mansbach are Globalization: The Return of Borders to a Borderless World?; A World of Polities; Remapping Global Politics; The Elusive Quest Continues; Polities; The Elusive Quest; The State, Conceptual Chaos, and the Future of International Relations Theory; and The Web of World Politics. Others are On the Cutting Edge of Globalization (with James N. Rosenau, David C. Earnest, and Ole R. Holsti); Political Space (with R. J. Barry Jones);Continuing Issues in Global Politics (with Walter Weiker); and Contemporary Inter-American Relations. He and Mansbach are currently working on a sequel to Polities, with the working title of Preinternational Polities: Political Evolution in the Ancient Mediterranean.
Dr. Ferguson is on the editorial advisory board of Global Governance. From 2002-2010 he served on the editorial advisory board for International Studies Review; from 1999-2003, on the editorial advisory board of International Studies Quarterly; and from 1995-2000, on the international advisory board of the European Journal of International Relations.


Hollywood, CA (November 21, 2017) -- Fender Musical Instruments Corporation (FMIC) today released the JMJ Road Worn Mustang Bass, the latest addition to the Fender Artist Signature Series that debuted at Summer NAMM 2017. Evocative of JMJ's (Justin Meldal-Johnsen) versatile music background, the bass is modeled after his own vintage 1966 Daphne Blue 4-string Mustang Bass. Often working behind the scenes, JMJ has used his prized Mustang Bass and diverse talents – from playing to producing – to make a big impact on the modern rock genre in collaboration with countless acclaimed artists. The JMJ Road Worn Mustang Bass project was inspired by JMJ's constant pursuit of new ideas and sounds, as well as an instinct to piece together multiple components to build something meaningful. The result is an artist signature model that epitomizes the features and tone JMJ loves about his original Mustang, giving him the chance to share it with bass buffs everywhere.
The JMJ Road Worn Mustang Bass is equipped for a variety of musical settings, and features a vintage-style look and feel. The road worn aspect recreates the wear of the original, while the old school hardware, thumb rest and lollypop oval-shaped tuners all evoke 60s Fender nostalgia. Designed for comfort and speed, the bass is powered by a custom-spec split-coil pickup, 30" scale, and modern 9.5"- radius fingerboard.
In true tradition, Fender's Artist Signature Series models honor popular and iconic musicians through product progression and storytelling, creating instruments inspired by the unique specifications of the world's greatest guitarists and bassists.


text:
\section{Introduction}

In a previous paper \cite{kerr1} we laid out the framework for
Ashtekar--Barbero variables for general relativity with one
axisymmetric Killing vector field. Here we would like to discuss the
quantization of the model. We will implement Thiemann's Hamiltonian
\cite{thiemann} and show that its algebra is consistent. This provides
a framework to discuss other implementations of the Hamiltonian like
those proposed by Varadarajan \cite{varadarajan} or Lewandowski and
collaborators \cite{lewandowski}. It provides the simplest environment
where those types of dynamics can be implemented capturing key
elements of the full three dimensional implementation. Simpler
examples like spherical symmetry or cosmologies do not allow to
implement all the features needed in the full theory. From a
phy\-si\-cal point of view, the axisymmetric case is of great
relevance in astrophysics. Our treatment opens the possibility of
considering the quantum space-times associated with black holes with
angular momentum, for example.

The organization of this paper is as follows. In section 2 we present
a summary of the classical theory. In section 3 we recall some
properties of loop quantization for the full theory. In section 4
we introduce the kinematics of the axisymmetric case. In section 5 we
discuss the implementation of the Euclidean portion of the Hamiltonian
constraint. In section 6 we analyze the Lorentzian portion. In section
7 we prove cylindrical consistency, diffeomorphism covariance and
freedom from anomalies. We end with conclusions in section 8.

\section{Summary of the classical theory}

Following our previous treatment \cite{kerr1},
we choose adapted spatial coordinates $(x,y,\phi)$ and the Killing
vector is $\partial_\phi$. In this framework one can decompose the
triads and the connections in terms of reduced variables,
\begin{align}
 A&=&A_a^i\tau_i \mathrm{d}x^a= \left((\cos(\phi)\tau_1 + \sin(\phi) \tau_2)  \a_a^1 + (-\sin(\phi) \tau_1 + \cos(\phi) \tau_2) \a_a^2 + \a_a^3 \tau_3 \right)\mathrm{d}x^a \\
 E&=&E^a_i \tau^i \partial_a= \left((\cos(\phi)\tau^1 + \sin(\phi) \tau^2) \e^a_1 + (-\sin(\phi) \tau^1 + \cos(\phi) \tau^2) \e^a_2 + \e^a_3 \tau^3 \right)\partial_a,
\end{align}
where $\tau_i=-\frac{i}{2} \sigma_i$ with $\sigma_i$ the Pauli matrices.

The symmetry adapted variables $(\a_a^i,\e^b_j)$ do not depend
on the angular coordinate $\phi$, i.e. only on $(x,y)$, and are
canonically conjugate, i.e.,
\begin{equation}
\label{poisson_bracket_axi}
\{\a^i_a(\vec x),\e^b_j(\vec x')\}=4 G\beta\,\delta^i_j\delta^b_a\delta^{(2)}(\vec x-\vec x'),
\end{equation}
where $\beta$ is the Immirzi parameter. It should be noted that
although the symmetry adapted variables only depend on $x,y$ their
indices run from 1 to 3.

The total Hamiltonian in terms of the reduced variables is,
\begin{equation}
h_T=\frac{1}{8 G}\left[g(\vec\Lambda)+ d(\vec N)+c(N)\right],
\end{equation}
and the (smeared) constraints are,
\begin{align}
g(\vec\lambda)&=\frac{1}{\beta}\int dxdy \lambda^i\left(\partial_a\e^a_i+\varepsilon_{ijk}\a^j_a\e^a_k+\varepsilon_{ijk}\delta^j_3\delta^\phi_a \e^a_k\right),\\
d(\vec N)&=\frac{1}{\beta}\int dxdy N^a\left(\e^b_i\partial_a\a^i_b-\partial_b(\e^b_i\a^i_a)+\delta_a^\phi\delta^i_3\varepsilon_{ijk}\a^j_b\e^b_
\right), \\
h_E(N)&=-2\int dxdy \frac{N}{\sqrt{\e}}\left[(\a^{i}_{b,a} - \a^{i}_{a,b} + \epsilon_{ilm}\a^{l}_{a} \a^{m}_{b})\epsilon_{ijk}\e_{j}^{a} \e_{k}^{b} +2\delta^j_3 \delta_b^\phi\left(\a^i_a \e_i^a\e_j^b - \a^i_a\e_i^b \e_j^a\right)\right] ,\\
h_L(N)&=2\int dxdy \frac{N}{\sqrt{\e}} (1+\beta^2)\epsilon_{ijk} \epsilon_{ilm} \e_{j}^{a} \e_{k}^{b} \k^{l}_{a} \k^{m}_{b},
\end{align}
where  we have written the scalar constraint as
$c(N)=h_E(N) + h_L(N)$.

\section{Summary of loop quantization}

We review some general concepts of loop quantum gravity, more details
can be found in \cite{thiemann}
In the full theory one considers densitized triads $E^a_i$ and
SU(2) connections $A_a^i$. One can construct a quantum connection
representation in which the actions of the basic variables are,
\begin{eqnarray}
 A_a^i(x) \psi[A] &=&  A_a^i (x) \psi[A], \\
  E^a_i(x) \psi[A] &=& -i 8\pi G \hbar \beta \frac{\delta}{\delta A_a^i (x)} \psi[A].
\end{eqnarray}
It is useful to consider smeared versions of the variables, for
instance the holonomy of the connection along a closed curve $\gamma$,
\begin{align}
 H[A,\gamma]=P\left( e^{\int_\gamma A^i_a \tau_i \mathrm{d}x^a} \right),
\end{align}
and the flux of the triad through a surface $S$ parametrized by
$\sigma_1$ and $\sigma_2$,
\begin{align}
\label{flujo_triada}
 E_i= \int_S n_a E^a_i = \int_S \mathrm{d}\sigma_1 \mathrm{d} \sigma_2 \varepsilon_{abc} \frac{\partial x^b (\vec{\sigma})}{\partial \sigma_1} \frac{\partial x^c (\vec{\sigma})}{\partial \sigma_2} E^a_i.
\end{align}
We will need to consider the algebra of these quantities given that
the triad and connection are canonically conjugate,
\begin{align}
 \left\lbrace A_a^i (x) , E^b_j (y) \right\rbrace = 8\pi G \beta \delta_a^b \delta_i^j \delta^3(x-y).
\end{align}
This leads to,
\begin{align}
\label{flujo}
 \left\lbrace E_i (S) , H[A,\gamma] \right\rbrace = \pm 8\pi G \beta H[A,\gamma_{1s}] \tau_i H[A,\gamma_{s0}],
\end{align}
where we introduced the notation $H[A,\gamma_{s_1 s_2}]=P\left(
  e^{\left( \int_{s_1}^{s_2} A_a^i \tau_i \dot{\gamma}^a(t)
      \mathrm{d}t \right)} \right)$ for the parallel transport operator, $\gamma(s)$ is the intersection
point of $S$ and $\gamma$ and $1$, $0$ are the values of the parameter
of the origin of the loop. The expression also holds for open paths,
in that case $1$ would be the origin and $0$ the endpoint. The sign is given by the relative
orientation between them and can be computed through the integral,
\begin{align}
 \int_S \int_\gamma \mathrm{d}\sigma^1 \mathrm{d}\sigma^2 \mathrm{d} s
  \varepsilon_{abc} \frac{\partial x^a (\sigma)}{\partial \sigma^1} \frac{\partial x^b (\sigma)}{\partial \sigma^2} \frac{\partial x^c (s)}{\partial s} \delta^3 (\vec{x}(\vec{\sigma}),\vec{x}(s))=\pm 1.
\end{align}

\section{Kinematical quantum theory for axisymmetry}

We would now like to repeat some of these constructions for the
axisymmetric theory in terms of the reduced variables. As we
mentioned, the reduced variables are ${\bf a}_a^i(x,y)$ and ${\bf
  e}^a_i(x,y)$. With $a=x,y,\phi$. The spatial Killing vector field is perpendicular to a family of two-surfaces which contain the rotation axis. The coordinates $(x,y)$ are defined on one of these surfaces (which we will denote by $\Sigma$) and can be carried to the rest of the spacetime along the integral curves of the Killing vector field (see for example \cite{wald_book}, sec. 7.1). We will
consider holonomies along curves either contained in $\Sigma$ or transverse to it. Also the surfaces $S$ we will
consider can either be contained in $\Sigma$ or are transverse and
of the form
$\gamma_a\times [0,\mu]$ where $\gamma_a$ is a curve contained in
$\Sigma$ and $\mu$ is a real number, an interval in the  $\phi$ coordinate.

We start by considering the Poisson bracket of the flux through one of
these latter surfaces
with a holonomy along a curve in $\Sigma$,
\begin{align}
 \left\{E_{i}(S), H[A, \gamma]\right\}=\int_{S} \int_{\gamma} \mathrm{d} \sigma^{1} \mathrm{d} \sigma^{2} \mathrm{d} s \varepsilon_{abc} \frac{\partial x^{a}}{\partial \sigma^{1}} \frac{\partial x^{b}}{\partial \sigma^{2}} \frac{\partial x^{c}}{\partial s} \delta^{3}(\vec{x}(\vec{\sigma}), \vec{x}(s))\left[H\left[A, \gamma_{1 s}\right] \tau_{i} H\left[A, \gamma_{s 0}\right]\right].
\end{align}
One of the coordinates that parametrizes the surface (say, $\sigma_2$)
corresponds to the direction transverse to $\Sigma$; we can
therefore identify it with the coordinate $\phi$. Integrating in it
just picks the value of $\phi$ in which the curve and the surface
intersect, which is just the value of $\phi$ corresponding to the two-surface transverse to $\partial_\phi$ that we have chosen
\begin{align}
 \left\{E_{i}(S), H[A, \gamma]\right\}=\int_{\gamma_\alpha} \int_{\gamma} \mathrm{d} \sigma \mathrm{d} s \varepsilon_{\alpha\beta} \frac{\partial x^{\alpha}}{\partial \sigma} \frac{\partial x^{\beta}}{\partial s} \delta^{2}(\vec{x}(\sigma), \vec{x}(s))\left[H\left[A, \gamma_{1 s}\right] \tau_{i} H\left[A, \gamma_{s 0}\right]\right].
\end{align}
One can carry out the integrals more or less like in the general case
and obtain an analogous result.
\begin{align}
\label{flujo_axi}
 \left\{E_{i}(S), H[A, \gamma]\right\}=\pm H\left[A, \gamma_{1 s}\right] \tau_{i} H\left[A, \gamma_{s 0}\right].
\end{align}
(Where we have switched to units such that the factor $4G\beta$ in \eqref{poisson_bracket_axi} equals one).
We will now consider holonomies along curves $\gamma^\phi$ generated by the Killing vector field,
\begin{align}
 H\left[A, \gamma^{\phi}\right]=e^{\int \mathrm{d} \phi \a_{\phi}^{i}(x, y) \tau_{i}}=e^{ \mu \a_{\phi}^{i}(x, y) \tau_{i}}
\end{align}
where $\mu$ is the length of the curve. These are actually point holonomies since under a gauge transformation $g$:
\begin{align}
H\left[A, \gamma_{\phi}(x, y)\right] \rightarrow U(g(x, y)) H\left[A, \gamma_{\phi}(x, y)\right] U^{-1}(g(x, y))
\end{align}
with $U(g)$ the gauge transformation's representation. It is easy to
see that the Poisson bracket between one of these holonomies and the
flux on a surface whose normal vector is of the form $n_\alpha\in \Sigma$ vanishes due to the $\varepsilon$ factor.
Let us now consider the case of  a surface contained in the plane and
a curve perpendicular to it of arbitrary origin and length, but such that they intersect at one point. Calling
the intersection point $s$ one has that,
\begin{align}
 \left\{E_{i}(S), H[A, \gamma^\phi]\right\}=\pm H\left[A,
  \gamma^{\phi}_{1s}\right] \tau_{i} H\left[A, \gamma^{\phi}_{
  s0}\right].
\end{align}

Later, in order to build gauge invariant spin networks states we will
need to consider transverse holonomies along complete circles that
start and end in the plane. In that case the previous result reduces
to,
\begin{align}
 \left\{E_{i}(S), H[A, \gamma^\phi]\right\}=\pm H\left[A, \gamma^{\phi}\right] \tau_{i}.
\end{align}

As is usual in loop quantum gravity we wish to consider kinematical quantum states
that are cylindrical functions of parallel transports of the
connection. Given a series of open paths $\gamma^n$ we have,
\begin{align}
 \Psi_{\Gamma,f}=f\left( H[A,\gamma^1] , \dots , H[A, \gamma^N] \right).
\end{align}

A basis of gauge invariant states for this space is given by spin networks. The latter
can be defined as a triplet $S=(\Gamma,j_l, i_n)$ with $\Gamma$ a
graph with $L$ edges and $N$ nodes.  The quantity $j_l$ indicates the
representation of the parallel transport matrix along
$\gamma_l$. The spin network state has the form,
\begin{align}
 \Psi_S[A]=\left\langle A | S \right\rangle = \left( \bigotimes_{l=1}^L \left( H^{j_l}[A,\gamma^l] \right) \right) \cdot \left( \bigotimes_{n=1}^N i_n \right).
\end{align}
\indent The intertwiners $i_n$ have a set of indices dual to that of the
product of parallel transport matrices and their contraction with them
ensures gauge invariance of the state.

For the symmetry reduced theory we will consider curves
$\gamma^\alpha$ contained in the $x,y$ two-surface and $\gamma^\phi$ curves
perpendicular to it. The spin network states take the form,
\begin{align}
 \Psi_S[A]=\left\langle A | S \right\rangle = \left( \bigotimes_l \left( H^{j_l}[A,\gamma^{\alpha_l}] \right) \right) \cdot \left( \bigotimes_n i_n \right) \cdot \left( \bigotimes_k \left( H^{j_k}[A,\gamma^{\phi_k}] \right) \right).
\end{align}
The parallel transports along the curves $\gamma^\phi$ take the form
of point holonomies, as we discussed,
\begin{align}
\label{grasp_transversal}
H^{j_l}[A,\gamma_\phi]=P\left( e^{\int \mathrm{d}\phi \a_\phi^i (\tau_i)^{j_l}} \right)= e^{2 \pi n \a_\phi^i (\tau_i)^{j_l}},
\end{align}
where ${\bf a}_\phi^i$ is evaluated in a point $x,y$ of the two-surface and
$n$ is a natural number that corresponds to the number of turns of the
curve $\gamma^\phi$ around the axis of symmetry.\\[10pt]
\begin{figure}[h]

{\includegraphics[width=9cm]{spin_network_axi}}

\caption{
Example of a graph on which the spin-networks in the reduced
theory are based. The plane actually represents any two-surface
orthogonal to the Killing vector field (we chose a plane for
simplicity). The edges orthogonal to the plane start and end at the
same vertex.}
\end{figure}

Let us turn our attention to the area operator. The classical
ex\-pre\-ssion for the area of a surface in terms of the triads is,
\begin{align}
 A(S)=\int_S \sqrt{n_a E^a_i n_b E^b_i} \mathrm{d}^2 \sigma.
\end{align}
This can be promoted to an operator in the quantum theory using the
ex\-pre\-ssions we discussed for the action of the flux of the triads on
parallel transport matrices. The end result on a spin network state
based on a graph $\Gamma$ is well known,
\begin{align}
 \hat{A}(S) \psi_\Gamma[A] = \sum\limits_{P\in S\cap \Gamma} \sqrt{j_p (j_p+1)} \psi[A],
\end{align}
with $j_p$ the valence of the edge of $\Gamma$ that intersects the
surface $S$ at the point $P$.

In the reduced theory it only makes sense to consider surfaces
compatible with the symmetry. That is, either surfaces $\Sigma$ perpendicular to the Killing vector field or surfaces of the form $\gamma\times
\Phi$ where $\gamma$ is an edge contained in the two-surface $\Sigma$ and
$\Phi$ is an interval of the real line along $\phi$. It is relatively
straightforward to see that the expression
of the area operator is the same as in the full theory, only with the
spin network living in the two-surface $\Sigma$ with possible transverse
insertions of curves $\gamma^\phi$. For areas lying in the $x,y$ plane one will only have
non-vanishing contributions from the insertions.

For the definition of the Hamiltonian constraint it is important to
have the definition of the volume operator. In the full theory it is
given by \cite{asle_volume},
\begin{align}
 \hat{V} \Psi_\Gamma = \kappa_0 \sum\limits_v \sqrt{\hat{q}_v} \Psi_\Gamma,
\end{align}
where the sum is over all the vertices of the graph $\Gamma$ and $\kappa_0$ is an arbitrary constant. The
operator $\hat{q}_v$ is given by
\begin{align}
 \hat{q}_v  = \frac{1}{48} \varepsilon_{ijk} \sum\limits_{e,e',e''}\epsilon \left(e,e',e''\right) J^i_{v,e} J^j_{v,e'} J^k_{v,e''},
\end{align}
with the $J$'s the angular momentum associated to a point $x$ in the
manifold and an edge $e$ that emerges from it,
 \begin{align}
 \left(J_{x, e}^{i} \Psi_{\Gamma}\right) &=i\left(H[A,\Gamma] \tau^{i}\right)_{B}^{A} \frac{\partial \psi_\Gamma}{\partial\left(H[A,\Gamma]\right)_{B}^{A}} \nonumber \\ &=i \frac{d}{d t}\left[\psi_\Gamma \left( \ldots, H[A,\gamma] \exp \left(t \tau^{i}\right), \ldots \right)\right], \label{momento_angular}
 \end{align}
 with $A,B$ indices in any representation of SU(2).
In the above expression the sum extends to all triplets of edges $e,e',e''$ that intersect in a
vertex. The quantity $\epsilon(e,e',e'')$ is $\pm 1$ if the edges are
linearly independent and zero otherwise.

The expression of the volume for the axisymmetric case is essentially
the same as in the full theory, except that one of the three edges is
required to be in the transverse direction in order to have a
non-vanishing volume.

The Gauss constraint is automatically satisfied by considering spin
networks, as they are gauge invariant. The diffeomorphism constraint is discussed in the following section.

\section{Implementation of the diffeomorphism constraint}

When written in terms of our reduced variables (2.1, 2.2), the diffeomorphism constraint takes the following form:
\begin{align}
 d(\vec{N})=\frac{2}{\beta} \int d x d y N^{a}\left(\mathbf{e}_{i}^{b} \partial_{a} \mathbf{a}_{b}^{i}-\partial_{b}\left(\mathbf{e}_{i}^{b} \mathbf{a}_{a}^{i}\right)+\delta_{a}^{\phi} \delta_{3}^{i} \varepsilon_{i j k} \mathbf{a}_{b}^{j} \mathbf{e}_{k}^{b}\right)
\end{align}
and its action on the canonical variables is given by \cite{kerr1}:
\begin{align}
 \left\{\mathbf{a}_{a}^{i}(\vec{x}), d(\vec{N})\right\}=\left.\left(\mathbf{a}_{b}^{i} N_{, a}^{b}+N^{b} \mathbf{a}_{a, b}^{i}+\varepsilon_{i j k} N^{d} \delta_{d}^{\phi} \delta_{3}^{j} \mathbf{a}_{a}^{k}\right)\right|_{\vec{x}}
\end{align}
Let us first analyze the diffeomorphisms in the direction orthogonal
to the Killing vector field, that is, shifts of the form
$ N=N^\alpha \partial_\alpha $, with $\alpha=x,y$ (we will focus on
the action of the constraint on the connection, everything we argue
below applies equally to the triad). The previous action reduces to:
\begin{align}
 \left\{\mathbf{a}_{a}^{i}(\vec{x}), d(N^\alpha \partial_\alpha)\right\}=\left.\left(\mathbf{a}_{\alpha}^{i} N_{, a}^{\alpha}+N^{\alpha} \mathbf{a}_{a, \alpha}^{i}

 \right)\right|_{\vec{x}}
\end{align}
The right hand side is just the Lie derivative of the connection in the direction of $N^\alpha \partial_\alpha$, the action of the diffeomorphism constraint in a direction transverse to the Killing Vector Field on the reduced variables is thus completely analogous to the one on the old variables. \\
The case of the diffeomorphisms in the direction of the Killing vector field, although it is also analogous, is a little more subtle and requires some discussion.

Let us begin by noting that the action of the constraint
\begin{align}
 \left\{\mathbf{a}_{a}^{i}(\vec{x}), d(N^\phi \partial_\phi)\right\}=\left.\left(\mathbf{a}_{\phi}^{i} N_{, a}^{\phi}+N^{\phi} \underbrace{\mathbf{a}_{a, \phi}^{i}}_{0}+\varepsilon_{i j k} N^{d} \delta_{d}^{\phi} \delta_{3}^{j} \mathbf{a}_{a}^{k}\right)\right|_{\vec{x}}
\end{align}
does not even look like a Lie derivative, so it would appear that it is more complicated than the previous case. However, it is easy to show that after considering the connection $\tilde{\a}_a^i \coloneqq \a_a^i + \delta_a^\phi $ and redefining a multiplier in the Gauss constraint, we obtain
\begin{align}
 \left\{\tilde{\mathbf{a}}_{a}^{i}(\vec{x}), d(N^\phi \partial_\phi)\right\}=\left.\left(\tilde{\mathbf{a}}_{\phi}^{i} N_{, a}^{\phi}+N^{\phi} \underbrace{\tilde{\mathbf{a}}_{a, \phi}^{i}}_{0} \right)\right|_{\vec{x}}
\end{align}
which is just the Lie derivative in the direction of
$N^\phi \partial_\phi$. So in reality the action of the constraint was
just obscured by the use of $\a_a^i$ instead of $\tilde{\a}_a^i$,
which should be expected since the former doesn't even transform as a
connection under the action of the Gauss constraint while the later
does (see \cite{kerr1}). The redefinition of the Lagrange multiplier
corresponding to a Gauss constraint should also be expected since we
arrived at the reduced variables by following a typical reduction
procedure for connections:
\begin{align}
 \mathcal{L}_{\tilde{K}} A_a^i = \epsilon_{i j k} \lambda^{j} A_{a}^{k}
\end{align}
being $\tilde{K}=\lambda \partial_\phi$ and $\vec{\lambda}=(0,0,\lambda)$, with $\lambda$ a constant (the simplest choice possible). The previous equation amounts to
\begin{align}
 \partial_{\phi} A_{a}^{i}=\epsilon_{i 3 k} A_{a}^{k}
\end{align}
so we see that our requirement actually mixes a term corresponding to
the Lie derivative with the action of the Gauss constraint.

In summary, the action of the diffeomorphism constraint in the
direction of the Killing vector field is analogous to that of the full
theory (with the only exception that there is homogeneity in that
direction so that scalar functions would be left invariant by the
constraint (but not one-forms and vectors). The constraint should
therefore be implemented in the same way as in the full theory, two
2-surfaces which can be deformed into each other by dragging points
along the Killing direction (the finite action of the constraint)
should be considered equivalent.

Imposing the diffeomorphism constraints in the directions
orthogonal to the Killing orbits has an analogue effect to that of the full theory,
namely, we obtain that all graphs which are knotted in the same way in
the two-surface are equivalent to each other.

\section{The Euclidean portion of the Hamiltonian constraint}

The Hamiltonian constraint of general relativity written in terms of
Ashtekar--Barbero variables contains two terms, the Euclidean and
Lorentzian ones,
\begin{align}
C(N) &= H_E(N)+H_L(N).
\end{align}
We will discuss the Lorentzian part later. For the Euclidean portion
Thiemann \cite{thiemann} showed that it can be written as,
\begin{align}
\label{hamiltoniano_reescrito}
 H_E(N) = 2 \int_{\Sigma} \mathrm{d}^3 x N(x) \varepsilon^{abc} \operatorname{tr}\left( F_{ab} \left\lbrace A_c , V \right\rbrace \right).
\end{align}

Thiemann also showed how a discretization of  this classical expression can be written in terms of
holonomies. Let us review his construction in general before
proceeding to the reduced theory. One partitions the manifold in elementary tetrahedra
$\Delta$. In each tetrahedron we choose a vertex and call it
$v(\Delta)$. Let $s_i(\Delta)$ with $i=1,2,3$ the edges that arrive at
$v$ and let $\alpha_{ij}=s_i(\Delta)\circ a_{ij} \circ s^{-1}_j(\Delta)$
a loop based in $v(\Delta)$ with $a_{ij}$ an edge that links the
endpoints of $s_i$ and $s_j$ that do not coincide with $v(\Delta)$.
One considers the quantity
\begin{align}
\label{h_triangulacion}
 H_T^E (N) = \sum\limits_{\Delta \in T} H_\Delta^E (N),
\end{align}
with,
\begin{align}
\label{h_triangulacion_2}
 H_\Delta^E (N) \coloneqq -\frac{2}{3} N_v \varepsilon^{ijk} \operatorname{tr} \left( h_{\alpha_{ij}(\Delta)} h_{s_k (\Delta)} \left\lbrace h^{-1}_{s_k (\Delta)} , V \right\rbrace \right).
\end{align}
It can be readily seen that this quantity tends to $2\int_{\Delta} N
{\rm tr}(F\wedge \{A,V\})$ when the tetrahedron $\Delta$ shrinks to
its base point $v(\Delta)$. All of the quantities involved in that
expression (holonomies, volume
operator) can be promoted to quantum operators. Replacing each
quantity by its corresponding operator and the Poisson bracket by
$i\hbar$ the commutator we get,
\begin{align}
 \hat{H}_{T}^{E}(N) :=\sum_{\Delta \in T} \hat{H}_{\Delta}^{E}(N), \hat{H}_{\Delta}^{E}(N) & :=-2 \frac{N(v(\Delta))}{3 i \ell_{p}^{2}} \epsilon^{i j k} \operatorname{tr}\left(h_{\alpha_{i j}(\Delta)} h_{s_{k}(\Delta)}\left[h_{s_{k}(\Delta)}^{-1}, \hat{V}\right]\right) \nonumber \\ & =: N_{v} \hat{H}_{\Delta}^{E},
\end{align}
with $\ell_{p}$ Planck's length. Although we have chosen $4 G\beta=1$
we have kept $\beta$ and $\ell_{p}$ explicit in various places without
replacing their values for clarity.

It can be shown that when applied to a function $\Psi_\Gamma$ cylindrical with respect to the graph $\Gamma$, the Hamiltonian acts only on the vertices of $\Gamma$:
\begin{align}
 \hat{H}^E_T (N) \Psi_\Gamma = \sum\limits_{v\in \Gamma} N_v \hat{H}^E_v \Psi_\Gamma,
 \end{align}
with $ \hat{H}^E_v \coloneqq \sum\limits_{v(\Delta)=v} \hat{H}^E_\Delta $. This quantity is finite as long as the number of tetrahedra that
adjoin the vertices of the graph $\Gamma$ is maintained finite when one
refines the partition. There are many possibilities when choosing a partition. One possible prescription is the following: given an edge
$e_I$ of $\Gamma$ with origin at $v$ and outgoing from it, we choose
as an edge of $\Delta$ a segment $s_I$ with an end in $v$  which is
contained in $e_I$, the other end does not coincide with the end of
$e_I$ that is not $v$. Given an unordered pair of edges $s_I$ and
$s_J$ of $\Delta$, let $a_{IJ}$ be a curve that joins the end of those
edges that are not $v$; $a_{IJ}$ does not intersect $\Gamma$ at any other
point. For each unordered triplet of edges $s_I$, $s_J$, $s_K$ that
intersect at $v$ and define a tetrahedron $\Delta$  \cite{thiemann}, one can define in a
diffeomorphism invariant way seven other tetrahedra (called mirror
tetrahedra) that together with $\Delta$ saturate the vertex $v$ \cite{thiemann}. With
this one can complete the regularization of the operator. We define
$D(\Delta)$ a closed region covered by the tetrahedra $\Delta$ and its seven mirror images. The union of the regions corresponding to the tetrahedra formed with all the possible unordered triplets of edges converging in $v$
is denoted by $D(v)$, that is $D(v)\coloneqq \cup_{ v(\Delta)=v }
D(\Delta) $. Taking this into account and assuming that the vertex $v$ has
valence $n$, then the number of possible unordered triplets is
$E(v)=n(n-1)(n-2)/6$, and the relevant contribution to the integral will
be,
\begin{align}
\sum\limits_{v} \frac{1}{E(v)} \sum\limits_{v(\Delta)=v} \sum\limits_{\Delta' \in D(\Delta)} \int_{\Delta'}.
\end{align}
 Where we already took into account the fact that the Hamiltonian acts only on the vertices of the graph. The first sum is over all vertices. The second in the tetrahedra
formed by all the possible triplets of edges that arrive at the
vertex. The third sum is in the tetrahedron $\Delta$ and its mirror
images, which produces a factor eight. Then, using the triangulation adapted to the graph of the spin
network $\Gamma$ we mentioned (we call it $T(\Gamma)$) we obtain the
quantum expression,
\begin{align}
 \hat{H}^E_T (N) = \sum\limits_{v(\Delta)=v} N_v \frac{8}{E(v)} \sum\limits_{v(\Delta)=v} \hat{H}^E_\Delta = \sum\limits_{v\in V(\gamma)} N_v \hat{H}^E_v .
\end{align}

For the reduced theory the form of the Euclidean parts of the
Hamiltonian are just particular cases of the expression we just
discussed, but we would like to have their expressions in terms of the
reduced variables. The starting point is the classical expression,
\begin{align}
h_E&=\int dxdy \frac{N}{\sqrt{\e}}\left[(\a^{i}_{b,a} - \a^{i}_{a,b} + \epsilon_{ilm}\a^{l}_{a} \a^{m}_{b})\epsilon_{ijk}\e_{j}^{a} \e_{k}^{b} +2\delta^j_3 \delta_b^\phi\left(\a^i_a \e_i^a\e_j^b - \a^i_a\e_i^b \e_j^a\right)\right] \nonumber \\
\label{h_E_axi}
&=2 \int dxdy {N}  \left[\varepsilon^{abc} f_{ab}^i \left\lbrace \a_c^i , V \right\rbrace + 2 \delta_3^j \delta_b^\phi \a_a^i \varepsilon_{ijk} \varepsilon^{a b c} \left\lbrace \a_c^k , V \right\rbrace \right].
\end{align}
It should be noted that the volume that appears here is that of the
complete theory, which can be written in terms of the reduced
variables as $V=\int d^3x \sqrt{{\rm det}({\bf e})}$.

We now observe that the key term
\begin{align}
\label{key_term}
\varepsilon^{abc} f_{ab}^i \left\lbrace \a_c^i , V \right\rbrace
\end{align}
can be divided into two sums,
\begin{itemize}
 \item $a=\alpha,b=\beta,c=\phi$
 \item $a=\alpha,b=\phi,c=\beta ~~~ (\text{and} ~ a=\phi,b=\alpha,c=\beta) $.
 \end{itemize}
 The first sum works as in the full theory except that the term with
 the Poisson bracket will have a point holonomy.
  As in the full theory, we now proceed to discretize \eqref{h_E_axi} in terms of holonomies. Let us first partition the two-surface $\Sigma$ in elementary triangles $\Delta$. Analogously to the full theory, in each triangle we choose one of its vertices and call it $v(\Delta)$. Let $s_i(\Delta)$ with $i=1,2$ be the edges that arrive at $v$ and $\alpha_{ij}=s_i(\Delta)\circ a_{ij}\circ s_j^{-1}(\Delta) $, with $a_{ij}$ the edge that links the endpoints of $s_i$ and $s_j$ which do not coincide with $v(\Delta)$. Consider now the quantity
 \begin{align}
\label{h_triangulacion_axi_sigma}
 h_T^{E,\Sigma} (N) = \sum\limits_{\Delta \in T} h_\Delta^{E,\Sigma} (N),
\end{align}
where we have defined
\begin{align}
\label{h_triangulacion_2_axi_sigma}
 h_\Delta^{E,\Sigma} (N) \coloneqq -\frac{1}{2\pi} N_v \varepsilon^{ijk} \operatorname{tr} \left( h_{\alpha_{ij}(\Delta)} h_{s_k (\Delta)} \left\lbrace h^{-1}_{s_k (\Delta)} , V \right\rbrace \right).
\end{align}
being $s_k$ an edge perpendicular to $\Sigma$ such that one of its endpoints is $v(\Delta)$.
 The presence of the factor of $2\pi$ will become clear shortly . It can be readily seen that when the triangle $\Delta$ shrinks to its base point, the previous quantity tends to
\begin{align}
\label{h_E_axi_discretized}
 2 \frac{\mu}{2\pi} \int_\Delta N \left( \varepsilon^{\alpha\beta\phi} f^i_{\alpha\beta}\{\a_\phi^i , V \} \right)= \frac{1}{2\pi}\int_{\Delta \times \left[ 0,\mu \right]} 2  N \left( \varepsilon^{\alpha\beta\phi} f^i_{\alpha\beta}\{\a_\phi^i , V \} \right)
\end{align}
where we have renamed the indices to facilitate comparison with \eqref{key_term}. This integral is performed in the volume $\Delta\times \left[0,\mu\right]$ instead of a tetrahedron. Recall that when obtaining \eqref{h_E_axi}, one has to perform an integral in the coordinate corresponding to the Killing vector field \cite{kerr1}, which simply yields a factor of $2\pi$ since none of the quantities depend on such variable, leaving us with a two dimensional integral on a surface perpendicular to the k.v.f. Since we partitioned the two surface $\Sigma$ in the triangles $\Delta$, and these surfaces can be \lq\lq carried\rq\rq around the whole space by moving them along the orbits of the Killing vector field (whose orbits are closed spacelike curves), it follows that we can partition our three dimensional space by elemental tori obtained from carrying the elemental triangles $\Delta$ around the orbits of $\partial_\phi$ of length $2\pi$. Partitioning those tori in regions of the form $\Delta\times[0,\mu_i]$ such that $\sum\limits_i \mu_i = 2\pi$ and summing for all $\mu_i$ in \eqref{h_E_axi_discretized} cancels the $2\pi$ factor in the denominator and is the equivalent to performing the integral in $\phi$ that leaves us with the two dimensional integral in \eqref{h_E_axi}. After doing this and taking the sum in \eqref{h_E_axi_discretized} for all $\Delta$ we obtain,
\begin{align}
 \int_{\Sigma} 2  N \left( \varepsilon^{\alpha\beta\phi} f^i_{\alpha\beta}\{\a_\phi^i , V \} \right).
\end{align}

We conclude therefore that \eqref{h_triangulacion_2_axi_sigma} is the
discretized or ``holonomized''  version of the $a=\alpha,b=\beta,c=\phi$ part of the first term in \eqref{h_E_axi}.
\\[30pt]
\begin{figure}
{\includegraphics[width=12cm]{celda_ambas_2}} \\
\caption{In the full theory, one partitions the space into elementary tetrahedra (left), while in the axisymmetric theory we can partition it into elementary triangular tori (right).}
\end{figure}

A discretization for the rest of the euclidean Hamiltonian constraint, namely
\begin{align}
\label{h_E_axi_2}
 h^{E,\phi} &\coloneqq 2\int dxdy {N}  \left[\left(\varepsilon^{\alpha\phi\beta} f_{\alpha\phi}^i \left\lbrace \a_\phi^i , V \right\rbrace + (\alpha \leftrightarrow \phi) \right) + 2 \delta_3^j \delta_b^\phi \a_a^i \varepsilon_{ijk} \varepsilon^{a b c} \left\lbrace \a_c^k , V \right\rbrace \right],
\end{align}
can be obtained in a similar fashion.
Let us first consider the following product of holonomies:

\begin{align*}
 h^{-1}_{\mu v}(x) h^{-1}_{\varepsilon u}(x+\varepsilon u) h_{\mu v} (x+\varepsilon u) h_{\varepsilon u} (x)
\end{align*}
with $u$ a unit vector in $\Sigma$ and $v$ also unit in the Killing
direction. $x$ is an arbitrary point in $\Sigma$ and dependence in the parentheses denotes the starting point of the holonomy.
Up to now wherever it says $\a_a^i$ we could have substituted the variable that
transforms properly as a connection \cite{kerr1},
$\tilde{\a}_a^i \coloneqq \a_a^i + \delta_a^\phi \delta^i_3$.  It
makes no difference as it appears inside a Poisson bracket and the
extra term does not contribute. Similarly, the components of $f_{ab}$ considered do not involve the extra term. From now on, however,
it will be better to consider holonomies built explicitly with $\tilde{\a}_a^i$.
Both variables are canonically conjugate to $\e^a_i$.

Taking now the limit $\varepsilon\to 0$ and $\mu\to 0$
\begin{align*}
 h^{-1}_{\mu v}(x) h^{-1}_{\varepsilon u}(x+\varepsilon u) h_{\mu v} (x+\varepsilon u) h_{\varepsilon u} (x) \approx & \left( 1 - \mu \tilde{\a}_a(x) v^a \right)\left( 1 - \varepsilon \tilde{\a}_a(x+\varepsilon u) u^a \right) \\
 &\left( 1 + \mu \tilde{\a}_a(x+\varepsilon u) v^a \right) \left( 1 + \varepsilon \tilde{\a}_a(x) u^a \right) \\
 = & \left( 1 - \mu \tilde{\a}_v(x)  \right)\left( 1 - \varepsilon \tilde{\a}_u(x+\varepsilon u) \right) \\
 &\left( 1 + \mu \tilde{\a}_v(x+\varepsilon u) \right) \left( 1 + \varepsilon \tilde{\a}_u(x) \right).
\end{align*}
Where to keep things brief we defined $\tilde{\a}_a \coloneqq
\tilde{\a}_a^i \tau_i$. Keeping terms up to order $\varepsilon \mu$ (and dropping the dependence in the starting point):
\begin{align}
 h^{-1}_{\mu v} h^{-1}_{\varepsilon u} h_{\mu v} h_{\varepsilon u} \approx & 1+\varepsilon\mu \left( \partial_u \tilde{\a}_v^i + \tilde{\a}_v^j \tilde{\a}^k_u \varepsilon_{jki} \right) \tau_i \nonumber \\
 & 1 + \varepsilon \mu \left( \partial_u {\a}_v^i + \left( {\a}_v^j + \delta^j_3 \right) \a_u^k \varepsilon_{jki} \right) \tau_i.
\end{align}
Now consider the term
\begin{align}
 h^{-1}_{\varepsilon_2 w} \left\lbrace h_{\varepsilon_2 w} , V \right\rbrace \approx \varepsilon_2 \left\lbrace \a_w^i , V \right\rbrace \tau_i,
\end{align}
with $w$ a unit vector in the two-surface, and the limit $\varepsilon_2\to 0$ was taken.
Using these last two results (and substituting $v\to \phi$, and $u,w\to
\alpha,\beta$) we have that,

\begin{align*}
 \varepsilon^{\alpha\phi\beta} \operatorname{tr}\left(h^{-1}_{\mu\phi} h^{-1}_{\varepsilon \alpha} h_{\mu\phi} h_{\varepsilon\alpha} h^{-1}_{\varepsilon_2 \beta} \left\lbrace h_{\varepsilon_2 \beta} , V \right\rbrace \right\rbrace \approx \varepsilon \varepsilon_2 \mu ~ \varepsilon^{\alpha\phi\beta} \left( f_{\alpha\phi}^i \left\lbrace \a_\beta^i , V \right\rbrace + \varepsilon_{3ji} \a_\alpha^j \left\lbrace \a_\beta^i , V \right\rbrace \right),
\end{align*}
that is,
\begin{align*}
\label{haxi_infinitesimal_angular}
 \varepsilon^{\alpha\phi\beta} \operatorname{tr}\left(h^{-1}_{\mu\phi} h^{-1}_{\varepsilon \alpha} h_{\mu\phi} h_{\varepsilon\alpha} h^{-1}_{\varepsilon_2 \beta} \left\lbrace h_{\varepsilon_2 \beta} , V \right\rbrace \right\rbrace \rightarrow \frac{1}{2} \int_{\Delta\times [0,\mu]} ~ \varepsilon^{\alpha\phi\beta} \left( f_{\alpha\phi}^i \left\lbrace \a_\beta^i , V \right\rbrace + \varepsilon_{3ji} \a_\alpha^j \left\lbrace \a_\beta^i , V \right\rbrace \right).
\end{align*}
Where $\Delta$ is a triangle with one of its vertices in $x$ and two of its sides $ \varepsilon u $ and $ \varepsilon_2 w $.
It is straightforward to check that by interchanging $\alpha$ and $\phi$ we obtain
\begin{align}
 &\frac{1}{2} \int_{\Delta\times [0,\mu]} ~ \varepsilon^{\phi\alpha\beta} \left( f_{\phi\alpha}^i \left\lbrace \a_\beta^i , V \right\rbrace - \varepsilon_{3ji} \a_\alpha^j \left\lbrace \a_\beta^i , V \right\rbrace \right)\nonumber \\
 =& \frac{1}{2} \int_{\Delta\times [0,\mu]} ~ \varepsilon^{\phi\alpha\beta} f_{\phi\alpha}^i \left\lbrace \a_\beta^i , V \right\rbrace + \varepsilon^{\alpha\phi\beta} \varepsilon_{3ji} \a_\alpha^j \left\lbrace \a_\beta^i , V \right\rbrace.
\end{align}
Putting these two together we obtain the term corresponding to
$a=\alpha (\phi), b=\phi (\alpha), c=\beta$ in \eqref{key_term} plus
the last term in \eqref{h_E_axi}.

We are now able to find the Euclidean Hamiltonian operator. Let us
partition the two-surface $\Sigma$ in elementary triangles $\Delta$ as
before. For each triangle we pick one of its vertices and name it
$\Delta(v)$. Let the edges $s_\alpha$ and $s_\beta$ be the two sides
that start at $v$, and $s_\phi$ an edge starting at $v$ and
perpendicular to $\Sigma$.  With the results obtained so far, it is
straightforward to see that the following operator results from the discretization of \eqref{h_E_axi},
 \begin{align}
 \hat{H}_{T}^{E}(N) :=\sum_{\Delta \in T} \hat{H}_{\Delta}^{E}(N) = \sum\limits_{\Delta\in T} \left( \hat{h}_\Delta^\alpha (N) + \hat{h}^\phi_\Delta (N) \right)
\end{align}
with the terms in the sum,
\begin{align}
\label{h_E_alpha_delta}
 \hat{h}^{\alpha}_{\Delta} (N) &:=-2 \frac{N(v(\Delta))}{2 i \ell_{p}^{2}} \epsilon^{\alpha \beta \phi} \operatorname{tr}\left(h_{\alpha_{\alpha \beta}(\Delta)} h_{s_{\phi}(\Delta)}\left[h_{s_{\phi}(\Delta)}^{-1}, \hat{V}\right]\right), \\
 \label{h_E_phi_delta}
 \hat{h}^\phi_\Delta &:=-2 \frac{N(v(\Delta))}{2 i \ell_{p}^{2}} \epsilon^{\alpha \phi \beta} \operatorname{tr}\left( h^{-1}_{s_\phi} h^{-1}_{s_\alpha} h_{s_\phi} h_{s_\alpha} h^{-1}_{s_\beta} \left[ h_{s_\beta} , \hat{V} \right] \right) + \left( \alpha \leftrightarrow \phi \right).
\end{align}
Again this operator only acts on the vertices of the graph $\gamma$ of the
state it is acting on. We conclude this section by (roughly) noting
how do \eqref{h_E_alpha_delta} and \eqref{h_E_phi_delta} act on the
simplest non trivial vertex possible. Let $v\in \gamma$ be a vertex
such that the edges $e_\alpha \supset s_\alpha$ and $e_\beta \supset
s_\beta$ are both outgoing at it, while the edge $e_\phi \supset
s_\phi$ (which is transverse to $\Sigma$) starts and ends at $v$. It
is easy to see that $\hat{V}$ annihilates such a vertex so this
selects the term in the commutator in both \eqref{h_E_alpha_delta} and
\eqref{h_E_phi_delta} in which it acts after the holonomy. We will see
in more detail in a future work that the action of the holonomy which
acts before the volume and its inverse after it can be \lq\lq factored
out\rq\rq; basically, after the first holonomy and the volume have
both acted, the second holonomy gets contracted with the first one
(its inverse) giving as a result a Kronecker delta which simply gives
us the trace of the product of rest of the holonomies. We are
therefore left with the action of the holonomies along the loops
$\alpha_{\alpha\beta}$ and $ s^{-1}_\phi \circ s^{-1}_\alpha \circ
s_\phi \circ s_\alpha  $. Their action on the graph $\gamma$ can be
represented pictorially, as shown in figure 3.

\begin{figure}[h]
{\includegraphics[width=12cm]{accion_h_simple}}
\caption{Action of the Hamiltonian on the graph $\gamma$ when applied
  to a simple vertex. Both terms add a loop to the graph. Although the
  lines should intersect, they have been separated for clarity. In the
diagram, the straight lines live in $\Sigma$ and the ovals represent
circles in the $\phi$ direction of the Killing vector field.}
\end{figure}

\section{Lorentzian part of the Hamiltonian cons\-traint}

In the full theory the Lorentzian part of the Hamiltonian constraint
is given by \cite{thiemann},

\begin{align}
\label{H_L}
 H_{L}(N)&=2 \int d^{3} x N\left(1+\beta^{2}\right) \frac{1}{\sqrt{E}} \epsilon_{i j k} \epsilon_{i l m} E_{j}^{a} E_{k}^{b} K_{a}^{l} K_{b}^{m} \nonumber
 \\
 &= 4 \int d^{3} x N\left(1+\beta^{2}\right) \varepsilon^{a b c} \varepsilon_{i j k}\left\{A_{a}^{j}, K\right\}\left\{A_{b}^{k}, K\right\}\left\{A_{c}^{i}, V\right\},
\end{align}
with $K\coloneqq E^a_i K_a^i$. This quantity can in turn be written as
\begin{align}
\label{K}
 K=-\left\{V, \int_{\Sigma} d^{3} x {\cal H}_{E}(x)\right\}=-\left\{V, H_{E}(1)\right\},
\end{align}
where $\cal H$ is the Hamiltonian density.
An analogous expression holds for the axisymmetric case:
\begin{align}
\label{h_L}
h_{L}(N) &=2 \int d^{3} x N\left(1+\beta^{2}\right) \frac{1}{\sqrt{\e}} \epsilon_{i j k} \epsilon_{i l m} \e_{j}^{a} \e_{k}^{b} k_{a}^{l} k_{b}^{m} \nonumber
\\
&=4 \int d^{3} x N\left(1+\beta^{2}\right) \varepsilon^{a b c} \varepsilon_{i j k}\left\{ \a_{a}^{j}, k\right\}\left\{ \a_{b}^{k}, k\right\}\left\{ \a_{c}^{i}, V\right\} .
\end{align}
Again we could have used $\tilde{\a}_a^i$, or ${\a}_a^i$.
It is also easy to show the following identity holds:
\begin{align}
\label{k_identity}
 k=-\left\{V, \int_{\Sigma} d^{3} x \mathcal{ h}_{E}(x)\right\},
\end{align}
with $\mathcal{h}$ the Hamiltonian density.
To see this, we first note that
\begin{align}
\label{k}
 k\coloneqq k_a^i \e^a_i.
\end{align}
Now recall that $\beta k_a^i= \a_a^i - \gamma_a^i$, with $\gamma_a^i$
the spin connection,
\begin{align}
 \gamma_{a}^{i}=\frac{1}{2} \epsilon_{i j k} \mathbf{e}_{j}^{b}\left(\mathbf{e}_{a, b}^{k}-\mathbf{e}_{b, a}^{k}+\mathbf{e}_{k}^{c} \mathbf{e}_{a}^{l} \mathbf{e}_{c, b}^{l}+\mathbf{e}_{a}^{k} \mathbf{e}_{c}^{l} \mathbf{e}_{l, b}^{c}\right)-\delta_{3}^{i} \delta_{a}^{\phi}:=\tilde{\gamma}_{a}^{i}-\delta_{3}^{i} \delta_{a}^{\phi},
\end{align}
using this result we define
$\beta k_a^i=\a_a^i-\tilde{\gamma}_a^i+\delta^i_a \delta^\phi_a \coloneqq
\beta \tilde{k}_a^i + \delta^i_a \delta^\phi_a $ so that \eqref{k} can be
written as,
\begin{align}
\beta k=\beta k_{a}^{i} \mathbf{e}_{i}^{a}=\beta \tilde{k}_{a}^{i} \mathbf{e}_{i}^{a}+\mathbf{e}_{3}^{\phi}.
\end{align}
Now if we write the Euclidean Hamiltonian as $h_E = h_\alpha  +h_\phi $ so that,
\begin{align}
\label{v_h_E_axi}
 -\left\{V, \int_{\sigma} d^{2} x \mathcal{h}_{E}(x)\right\}=-\left\{V, \int_{\sigma} d^{2} x \mathcal{h}_{\alpha}(x)\right\}-\left\{V, \int_{\sigma} d^{2} x \mathcal{h}_{\phi}(x)\right\}.
\end{align}
It is evident that the first term corresponds to $\tilde{k}\coloneqq
\tilde{k}_a^i \e^a_i$ since it is completely analogous to
\eqref{K}. Now for the second term let us write $h_\phi$ explicitly:
\begin{align}
 h_{\phi}(N)=\frac{1}{8 G} \int \mathrm{d}^{2} x \frac{N}{\sqrt{\mathrm{e}}} 2 \delta_{3}^{j} \delta_{b}^{\phi}\left(\mathbf{a}_{a}^{i} \mathbf{e}_{i}^{a} \mathbf{e}_{j}^{b}-\mathbf{a}_{a}^{i} \mathbf{e}_{i}^{b} \mathbf{e}_{j}^{a}\right).
\end{align}

From the following identity
\begin{align}
 \frac{1}{\sqrt{\e}} \epsilon^{i j k} \e_{j}^{a} \e_{k}^{b}=2 \epsilon^{a b c}\left\{ \a_{c}^{i}, V\right\},
\end{align}
we can obtain
\begin{align}
 \left\{V, \mathbf{a}_{a}^{i}\right\}=- \frac{1}{2} \sqrt{\mathbf{e}} \mathbf{e}_{a}^{i},
\end{align}
and using this, the second term in \eqref{v_h_E_axi} is:
\begin{align}
 -\left\{V, h_{\phi}[1]\right\}=\frac{1}{2} \mathbf{e}_{a}^{i} \delta_{3}^{j} \delta_{b}^{\phi}\left(\mathbf{e}_{i}^{a} \mathbf{e}_{j}^{b}-\mathbf{e}_{i}^{b} \mathbf{e}_{j}^{a}\right)=\frac{1}{2}\left(3 e_{3}^{\phi}-\mathbf{e}_{i}^{\phi} \delta_{3}^{i}\right)=\mathbf{e}_{3}^{\phi}
\end{align}
so \eqref{k_identity} holds as claimed.

The steps for finding a discretization for the Lorentzian portion of
the Hamiltonian constraint are identical to those followed in the Euclidean case so we will not repeat them here. We partition the two-surface $\Sigma$ as before in elementary triangles $\Delta$ and define,
\begin{align}
\label{h_L_Delta}
 h_{\Delta}^{L}(N)=\frac{4}{2} N(v(\Delta)) \left(1+\beta^{2}\right) \varepsilon^{i j k} \operatorname{tr}\left(h_{s_{i}}^{-1}\left\{h_{s_{i}}, k\right\} h_{s_{j}}^{-1}\left\{h_{s_{j}}, k\right\} h_{s_{k}}^{-1}\left\{h_{s_{k}}, V\right\}\right),
\end{align}
where two of the edges $s_i,s_j,s_k$ are sides of the triangle joining
at $v(\Delta)$ while the other one is perpendicular to $\Sigma$. It is
straightforward to check that when $v(\Delta)$ shrinks to its base
point $v$, this previous expression tends to $ 4 (1+\beta^2) \int_{\Delta\times
  [0,\mu]} N \operatorname{tr}\left( \left\lbrace {\bf a},k \right\rbrace
  \left\lbrace {\bf a},k \right\rbrace \left\lbrace {\bf a},V \right\rbrace
\right) $, therefore a discretization for \eqref{h_L} is given by:
\begin{align}
 h^L(N)=\sum\limits_{\Delta} h^L_\Delta (N).
\end{align}
The quantization of this expression is obtained by adapting the triangulation to a graph $\gamma$ as was done for the Euclidean case, promoting functions to operators and replacing the Poisson brackets by commutators. The resulting operator has the following action on a function $\Psi$ cylindrical with respect to $\gamma$:
\begin{eqnarray}
\hat{h}^L(N) \Psi&=&-\frac{16}{2\left(i \ell_{p}^{2}\right)^{3}} \sum_{v \in V(\gamma)} N_{v} \frac{1}{E(v)} \sum_{v(\Delta)=v} \epsilon^{i j k} \times\nonumber \\ &&\times \operatorname{tr}\left(h_{s_{i}(\Delta)}\left[h_{s_{i}(\Delta)}^{-1}, k_{v}\right] h_{s_{j}(\Delta)}\left[h_{s_{j}(\Delta)}^{-1}, k_{v}\right] h_{s_{k}}(\Delta)\left[h_{s_{k}(\Delta)}^{-1}, V_{v}\right]\right) \Psi \nonumber \\ &=:& \sum_{v \in V(\gamma)} \frac{N_{v}}{E(v)} \sum_{v(\Delta)=v} \hat{h}^L_{\Delta} \Psi =: \sum_{v \in V(\gamma)} \frac{N_{v}}{E(v)} \hat{h}^L_{v} \Psi,
\end{eqnarray}
where $k_v$ and $V_v$ indicates they are evaluated at the vertex $v$ and
we have already used that the operator only acts on the vertices
of $\gamma$ (the argument used in the Euclidean case can be repeated
here) and also the fact that the commutators with holonomies along
edges adjacent to $v$ are non vanishing only when $k$ and $V$ are
evaluated on the same $v$ \cite{thiemann}. The vertex multiplicity in
this case is given by $E(v)=n(v)(n(v)-1) / 2$.

\section{Requirements for a triangulation adapted to a graph}
\subsection{Cylindrical consistency}
All the operators we have defined above depend on a choice of
partition. In the chosen prescription, the partition $T$ is adapted to
a graph $\gamma$, so if $\gamma \subset \gamma'$, $T(\gamma)$ and
$T(\gamma')$ will be different from each other. We should require that
$T$ is consistently defined (possibly up to a diffeomorphism), that
is, if $f$ is a cylindrical function of
$\gamma$, then $\hat{h}_{T(\gamma)} f$ and $ \hat{h}_{T(\gamma')} f$
should be diffeomorphic to each other. This is easily accomplished by
introducing the following operator $\hat{p}_e$ associated with an edge
$e$ whose action on a function $f$ cylindrical with respect to a graph
$\gamma$ is to annihilate it if $\gamma$ and $e$ do not intersect in a
finite segment having the same endpoint as $e$ and to leave
it invariant otherwise. We call this operator the edge projector. With
this definition we can construct a projector associated with a
triangle (in $\Sigma$) from the partition chosen:
$\hat{p}_{\Delta}=\hat{p}_{s_1(\Delta)}
\hat{p}_{s_2(\Delta)}\hat{p}_{s_{\phi}(\Delta)}$ ($s_1(\Delta)$ and
$s_2(\Delta)$ are edges from the partition which converge at
$v(\Delta)$, while $s_\phi(\Delta)$ is an edge in the direction of the
Killing vector field outgoing at $v(\Delta)$) and with it a vertex
projector: $\hat{E}(v)= \sum\limits_{v(\Delta)=v} \hat{p}_\Delta $. We
can now define the following self-consistent operator (up to a
diffeomorphism):
\begin{align}
\label{h_E_consistent}
\hat{h}^E_\gamma \coloneqq \sum\limits_{v\in V(\gamma)} 4 N_v \sum\limits_{v(\Delta)=v}\hat{h}^E_\Delta \frac{\hat{p}_\Delta}{E(v)} .
\end{align}
Given two graphs $\gamma$ and $\gamma'$ such that $\gamma \subset
\gamma'$ and a function $f$ cylindrical with respect to $\gamma'$, it
is clear from \eqref{h_E_consistent} that the terms in
$\hat{h}^E_\gamma$ corresponding to triangles formed with at least an
edge not present in $\gamma$, or such that the edge perpendicular to
the 2-surface based at the corresponding vertex is not present in
$\gamma$ will vanish when applied to $f$. Also, the factor $E(v)$
reduces to the correct value on $f$. That $\hat{h}^E_{\gamma'} f$ and
$\hat{h}^E_{\gamma} f$ are related by a diffeomorphism follows from
the manifestly diffeomorphism invariant prescription for
loop-assigments (see \cite{thiemann} for details) when acting on a
function $f$ cylindrical in $\gamma$.

The Lorentzian Hamiltonian involves only $\hat{k}$ and holonomies with respect to the edges of the original graph $\gamma$. It follows that if $\hat{h}^L$ is cylindrically consistently defined up to a diffeomorphism provided we introduce as before the projectors $\hat{p}_\Delta$:
\begin{align}
 \hat{h}^L_v \coloneqq \sum\limits_{v(\Delta)=v} \hat{h}^L_\Delta \frac{\hat{p}_\Delta}{E(v)}
\end{align}
\subsection{Diffeomorphism Covariance}
When looking for solutions to the Euclidean Hamiltonian constraint, we will consider distributions $\Psi$ in the space of cylindrical functions such that,
\begin{align}
\label{dif_cov_1}
 \Psi [\hat{h}^E (N) f] = 0 ~ \forall N ~ \forall f \in \operatorname{Cyl}^\infty (\mathcal{A}/\mathcal{G}).
\end{align}
 We want this quantity to depend only on the diffeomorphism invariant
 pro\-per\-ties of the triangulation assignment, which can be
 accomplished if we can make the quantum Hamiltonian be diffeomorphism
 covariant. If $f$ is cylindrical with respect to a graph $\gamma$,
 then $\hat{h}^E f$ will be a linear combination of cylindrical
 functions with respect to the graph $T(\gamma)$, now if $f'$ is
 cylindrical with respect to a graph $\gamma'$ related to $\gamma$ by
 a diffeomorphism $\gamma'=\phi(\gamma)$, the quantity $\hat{h}^E f'$ will consist of cylindrical functions with respect to the graph $T(\phi(\gamma))$. We will require that $T(\gamma)$ and $T(\phi(\gamma))$ be diffeomorphic to each other. Let us begin by observing that \eqref{dif_cov_1} can be written as
\begin{align}
\label{dif_cov_2}
 \Psi [\sum\limits_{v} \hat{h}^E_v f]= \sum\limits_{v} \Psi [ \hat{h}^E_v f] = 0 ~ \forall \gamma ~ \forall f\in\operatorname{Cyl}^\infty (\mathcal{A}/\mathcal{G})
\end{align}
so that the requirement of diffeomorphism covariance can be formulated
in terms of the Hamiltonian evaluated at each vertex $v$. It is easy to
see that for every vertex, each triangle $\Delta$ (with
$v(\Delta)=v$) of the corresponding partition can be treated
separately. Moreover, each term $\hat{h}^E_\Delta$ is the sum of
$\hat{h}^{E,\alpha}_\Delta$ and $ \hat{h}^{E,\phi}_\Delta $ which can
also be treated separately. The projectors are already covariantly
defined so we will focus on the rest of the operator.

Let's start by making $ \hat{h}^{E,\alpha} $ covariantly defined. This
case is analogous to that of the full theory \cite{thiemann}. If two
graphs $\gamma$ and $\gamma'$ are related by a diffeomorphism $\phi$,
then the triangles $\Delta(\phi(\gamma)), \phi(\Delta(\gamma))$ will
in general be different. More specifically, if $s_1$ and $s_2$ are
segments of the edges $e_1 \subset \gamma$ and $e_2 \subset\gamma$
both outgoing at $v$, then the graph $T(\gamma)$ is formed by adding
to $\gamma$ an arc $a$ joins the endpoints of $s_1$ and $s_2$ which do
not coincide with $v$. When acting with $\phi$ on $T(\gamma)$, the
edges $s_i, e_i ~ (i=1,2)$ and $a_{12}$ will be mapped to $s'_i, e'_i
~ (i=1,2)$ and $a'_{12}$ respectively. Now consider the graph
$\phi(\gamma)$, to obtain $T(\phi(\gamma))$ we add an arc $
\tilde{a}_{12} $ joining two segments $\tilde{s}_i \subset e'_i, ~
i=1,2$, which are not necessarily equal to $s'_i, ~ i=1,2$, therefore,
the arcs $a'_{12}$ and $\tilde{a}_{12}$ will in general differ. Recall
however that the arcs depend only on the topology of the vertex, and
the graphs $\gamma, ~ \gamma'$ are diffeomorphic to each other. What
we need is a diffeomorphism $\phi'$ that leaves $\gamma'$ invariant
while $\phi' (\Delta(\gamma')) = \phi(\Delta(\gamma))$. Such $\phi'$
clearly exists: It is the diffeomorphism that leaves the image of the
graph invariant while moving its points within itself it can put the
arc $\tilde{a}_{12}$ into any diffeomorphic shape. By using the
$\phi'$ corresponding to each $\Delta$ in \eqref{dif_cov_2}, we can
make $\hat{h}^{E,\alpha}$ diffeomorphism covariant. All of the above
can be summed up in figure 4.

\begin{figure}[h]
{\includegraphics[width=12cm]{diffeo_covariance}} \\
\caption{Simple graphical explanation of the paragraph above. The perpendicular edge has been omitted for simplicity. The diffeomorphism $\phi^{\prime}$ doesn't change the original graph but moves its points within itself, thus being able to put the triangles into any diffeomorphic shape.}
\end{figure}

The procedure we can follow for the term $\hat{h}^{E,\phi}$ is very
similar, we begin by noting that it involves holonomy terms out of
which only one is taken along an edge not belonging to the original
graph $\gamma$. Applying $\hat{h}^{E,\phi}_\Delta$ to the same
$f_\gamma$ as before, we obtain a linear combination of functions
cylindrical with respect to a graph $T(\gamma)$ which will differ from
the original graph by a segment $s_\phi$, transverse to the two
surface. (There is a subtlety that will be discussed later, but it
doesn't affect the present discussion: basically, $T(\gamma)$ differs
from $\gamma$ by a closed loop \lq\lq with a nose\rq\rq that
intersects with it everywhere except at the segment $s_\phi$). This
new segment intersects $\gamma$ at a point, which is the endpoint of a
segment $s_\alpha\subset e$, where $e \subset \gamma$ is an edge
contained in the two surface and outgoing at $v(\Delta)$. Acting with
a diffeomorphism $\phi$ on the graph $T(\gamma)$ yields
$\phi(T(\gamma))$ while $s, s_\phi$ and $e$ where mapped to $s',
s'_\phi$ and $e'$ respectively. Now consider the function
$f_{\phi(\gamma)}$, after acting with $h^{E,\phi}_{\phi(\Delta)}$ we
obtain functions cylindrical with respect to $\Delta(\phi(\gamma))$,
which differs from $ \phi(\Delta(\gamma)) $ by an edge $s''_\phi$,
which is transverse to the two surface and intersects $\phi(\gamma)$
at the endpoint of $s'' \subset e'$ that doesn't coincide with the
vertex. Neither $s''$ nor $s''_\phi$ are necessarily equal to $s'$ and
$s'_\phi$ so $ T(\phi(\gamma)) $ and $ \phi(T(\gamma)) $ will in
general differ. Similar to the previous case, we would like to have a
diffeomorphism such that $\phi'(\phi(\gamma))=\phi(\gamma)$ and
$\phi'(\Delta(\phi(\gamma)))=\phi(\Delta(\gamma))$. The required
diffeomorphism again exists and its action on $\phi(\gamma)$ is to
leave the graph image invariant but move its points such that the
intersection point $s'' \cap s''_\phi$ can be moved within $e'$ until
it coincides with $s' \cap s'_\phi$. Pictorially, this would
correspond, in the diagram on the lower right of figure 3, to shorten
the edges that connect the two ovals. This is all we need to make the
term $\hat{h}^{E,\phi}$ diffeomorphism covariant.

Finally, since we have managed to make the entire Euclidean
Hamiltonian diffeomorphism covariant, the Lorentzian Hamiltonian's
covariance follows tri\-via\-lly since it is defined in terms of
$\hat{k}$ and holonomies along edges of the original graph, so it will
be covariantly defined if and only if $\hat{h}^{E}$ is.

\subsection{Anomaly freeness}

The considerations in the previous sections have allowed us to redefine the Hamiltonian operator in a cylindrically consistent, diffeomorphism covariant way. There is still some freedom in our prescription for a triangulation, the operator depends on the diffeomorphism class of the triangulation assignment. We should check now if our Hamiltonian is anomaly free, that is, if
\begin{align}
\label{anomaly_freeness}
 \Psi \left[ \left[\hat{h}(N),\hat{h}(M)\right] f \right] = 0.
\end{align}
For any diffeomorphism invariant distribution $\Psi$, cylindrical function $f$ and lapses $N$ and $M$ \cite{thiemann}. Since
\begin{align}
\label{anomaly_freeness2}
\hat{h}(N)=\hat{h}^E(N) + \hat{h}^L(N),
\end{align}
and moreover,
\begin{align}
\label{anomaly_freeness3}
\hat{h}^E(N)=\hat{h}^{E,\alpha}(N)+\hat{h}^{E,\phi}(N),
\end{align}
trying to verify anomaly freeness directly in \eqref{anomaly_freeness} would be (although straightforward) very tedious, even more so than in the case of the full theory. Out of all the terms that would appear in \eqref{anomaly_freeness} if we substituted \eqref{anomaly_freeness2} and \eqref{anomaly_freeness3}, we will focus just on one of them, namely,
\begin{align}
\label{anomaly_freeness4}
 \Psi \left[ \left[\hat{h^{E,\alpha}}(N),\hat{h}^{E,\alpha}(M)\right] f \right],
\end{align}
and show that it equals zero for every distribution $\Psi$ and every cylindrical function $f$. This particular term is perhaps the easiest to analyze since it resembles
\begin{align}
\label{anomaly_freeness4}
 \Psi \left[ \left[\hat{H^{E}}(N),\hat{H}^{E}(M)\right] f \right] ,
\end{align}
where $\hat{H}^E$ is the Euclidean Hamiltonian operator from the full
theory \cite{thiemann}. Moreover, after proving that the operator
$\hat{h}^{E,\alpha}$ is anomaly free, a moment of reflexion can
convince one that every single term appearing in the decomposition of
\eqref{anomaly_freeness} mentioned above can be shown to vanish
separately, thus proving that the entire Hamiltonian operator is
anomaly free. To see this, we will use the following key observation:
Every term in the linear combination forming the Hamiltonian operator,
when acting on a function cylindrical with respect to a certain graph,
produces a linear combination of functions cylindrical with respect to
a graph with additional vertices; however, none of the terms of the
Hamiltonian acts on these new vertices.

Let us begin by recalling the definition of $\hat{h}^{E,\alpha}$. Given a graph $\gamma$ and a triangulation $T(\gamma)$:
\begin{align}
 \hat{h}^{E,\alpha}=\sum\limits_{v=V(\gamma)} \frac{N_v}{E(v,\gamma)} \sum\limits_{v(\Delta)=v}\hat{h}^{E,\alpha}_{\Delta} \hat{p}_{\Delta}.
\end{align}

Now, if $f$ is a cylindrical function based on $\gamma$, and we have
that $v= v(\Delta) \neq v(\Delta')=v'$, then
$\hat{p}_{\Delta'} \hat{h}^{E,\alpha}_\Delta f =
\hat{h}^{E,\alpha}_\Delta f $ since the only possible removed edges
in $\gamma$ are in $v$. If follows that
$\hat{E}(v') \hat{h}^{E,\alpha}_\Delta f = E(v',\gamma)
\hat{h}^{E,\alpha}_\Delta f$, that is, all the vertices adjacent to
$v'$ in the original graph $\gamma$ are still present, so their
multiplicity remains the same after acting with
$\hat{h}^{E,\alpha}_\Delta$. If however $v'=v$, then
$ \hat{h}^{E,\alpha}_\Delta f $ is a linear combination of cylindrical
functions whose graphs may have missing edges in $v$, in those cases
$\hat{E}(v') \hat{h}^{E,\alpha}_\Delta f \neq E(v',\gamma)
\hat{h}^{E,\alpha}_\Delta f$.

Next let us compute,
\begin{align}
 & \hat{h}^{E,\alpha}(N) \hat{h}^{E,\alpha}(N) f = \sum_{v \in V(\gamma)} \frac{N_{v}}{E(v, \gamma)} \sum_{v(\Delta(\gamma))=v} \hat{h}^{E,\alpha}(N) \hat{h}_{\Delta(\gamma)}^{E,\alpha} f \nonumber
 \\
 &=\sum_{v \in V(\gamma)} \frac{N_{v}}{E(v, \gamma)} \sum_{v(\Delta(\gamma))=v} \sum_{v^{\prime} \in V(\gamma \cup \Delta(\gamma))} M_v \sum_{v(\Delta(\gamma \cup \Delta(\gamma)))=v^{\prime}} \hat{h}_{\Delta(\gamma \cup \Delta(\gamma))}^{E,\alpha} \frac{1}{\hat{E}\left(v^{\prime}\right)} \hat{h}_{\Delta(\gamma)}^{E,\alpha} f.
\end{align}
If we compute the same quantity with the roles of $N$ and $M$ exchanged and then subtract:
 \begin{align}
&\left[\hat{h}^{E,\alpha}(N), \hat{h}^{E,\alpha}(N)\right] f \nonumber
\\
=&\sum_{v, v^{\prime} \in V(\gamma)}\left(M_{v^{\prime}} N_{v}-M_{v} N_{v^{\prime}}\right) \frac{1}{E(v, \gamma)}  \sum_{v(\Delta(\gamma))=v, v(\Delta(\gamma \cup \Delta(\gamma)))=v^{\prime}} \hat{h}_{\Delta(\gamma \cup \Delta(\gamma))=v^{\prime}}^{E,\alpha} \frac{1}{\hat{E}(v')} \hat{h}_{\Delta(\gamma)}^{E,\alpha} f \nonumber
\\
=&\sum_{v, v^{\prime} \in V(\gamma), v<v^{\prime}} \frac{M_{v^{\prime}} N_{v}-M_{v} N_{v^{\prime}}}{E(v) E\left(v^{\prime}\right)} \times \nonumber \\
&\times \quad \sum_{v\left(\Delta\left(\gamma \cup \Delta^{\prime}(\gamma)\right)\right)=v, v\left(\Delta^{\prime}(\gamma)\right)=v\left(\Delta^{\prime}(\gamma \cup \Delta(\gamma))\right)=v^{\prime}}\left[\hat{h}_{\Delta^{\prime}(\gamma \cup \Delta(\gamma))}^{E,\alpha} \hat{h}_{\Delta(\gamma)}^{E,\alpha}-\hat{h}_{\Delta\left(\gamma \cup \Delta^{\prime}(\gamma)\right)}^{E,\alpha} \hat{h}_{\Delta^{\prime}(\gamma)}^{E,\alpha}\right] f
\end{align}
where we have used the fact that the second Hamiltonian term doesn't
act on the vertices added to the graph by the first. We were able to
substitute $\hat{E}({v'})$ by $ E(v') \coloneqq E(v',\gamma) $, since
the only possible case in which $\hat{E}(v') \hat{h}^{E,\alpha} f \neq
E(v',\gamma) \hat{h}^{E,\alpha}_{\Delta(\gamma)} f$ is when $v'=v$
(there may be some missing edges in the graph of some of the terms in
the linear combination $\hat{h}^{E,\alpha}_{\Delta(\gamma)} f$), but
those terms do not contribute thanks to the antisymmetrized lapses
product.

To verify anomaly freeness, it will be sufficient to show that,
\begin{align}
 \label{anomaly_freeness5}
\Psi\left[\left(\hat{h}_{\Delta^{\prime}(\gamma \cup \Delta(\gamma))}^{E,\alpha} \hat{h}_{\Delta(\gamma)}^{E,\alpha}-\hat{h}_{\Delta\left(\gamma \cup \Delta^{\prime}(\gamma)\right)}^{E,\alpha} \hat{h}_{\Delta^{\prime}(\gamma)}^{E,\alpha}\right) f\right]=0,
\end{align}
holds for each choice of $v(\Delta(\gamma))=v\left(\Delta\left(\gamma \cup \Delta^{\prime}(\gamma)\right)\right)=v, v\left(\Delta^{\prime}(\gamma)\right)=v\left(\Delta^{\prime}(\gamma \cup \Delta(\gamma))\right)=v^{\prime}$. Now, since $\hat{h}_{\Delta^{\prime}(\gamma \cup \Delta(\gamma))}$ doesn't really act on the vertices added by $ \hat{h}^{E,\alpha}_{\Delta(\gamma)} $ and besides $v\neq v'$, we can find a diffeomorphism $\phi^{\prime}$ such that $\phi^{\prime} \left( \Delta' (\gamma \cup \Delta(\gamma)) \right) = \Delta^{\prime}(\gamma) $ and $\phi'(\Delta(\gamma))= \Delta(\gamma) $. Similarly, we can find another diffeomorphism $\phi$ such that $\phi(\Delta(\gamma \cup \Delta' (\gamma)))=\Delta(\gamma)$ while $\Delta' (\gamma)$ is left invariant. It follows that we can write or previous claim \eqref{anomaly_freeness5} as
\begin{align}
\label{anomaly_freeness6}
\Psi\left[\left(\hat{U}(\phi') \hat{h}_{\Delta^{\prime}(\gamma)}^{E,\alpha} \hat{h}_{\Delta(\gamma)}^{E,\alpha}- \hat{U}(\phi) \hat{h}_{\Delta\left(\gamma\right)}^{E,\alpha} \hat{h}_{\Delta^{\prime}(\gamma)}^{E,\alpha}\right) f\right]=0.
\end{align}

The state $\Psi$ is diffeomorphism invariant so both $\hat{U}(\phi')$ and $ \hat{U}(\phi) $ can be removed and since $\hat{h}^{E,\alpha}_{(\Delta(\gamma))}$ and $ \hat{h}^{E,\alpha}_{(\Delta'(\gamma))} $ act on different vertices they commute. We have thus verified \eqref{anomaly_freeness5}. \\
The reasoning followed to analyze the rest of the terms in \eqref{anomaly_freeness} involving $\hat{h}^{E,\alpha}$ and $\hat{h}^L$ is essentially the same. Repeating the same steps, we will arrive at a term analogous to \eqref{anomaly_freeness5}, in which the operator to the left will correspond to a triangle with vertex $v'$ in the triangulation of a graph $\gamma'$ bigger than the original $\gamma$. The operator to the right acts on a different vertex $v$ and the triangulation is adapted to the original graph. Removing those diffeomorphisms and taking into account the fact that the operators in each term act on different vertices and therefore commute, we obtain de desired result. Following this same line of reasoning for all the remaining terms, we are able to verify our initial claim \eqref{anomaly_freeness}.

\section{Conclusion}

We have constructed a quantum theory for axisymmetric space-times
based on a symmetry reduction of the Ashtekar formulation of gravity
we introduced in a previous paper. We defined the kinematical states,
and the actions of the basic operators, area and volume. With them we
proceeded to construct the quantum version of the Euclidean and
Lorentzian parts of the Hamiltonian constraint. We showed their
cylindrical consistency and anomaly-freeness of the constraint
algebra. In a further paper we will present the explicit evaluation of
the Hamiltonian constraint and discuss its space of solutions.

The parallelism with the full theory is remarkable. This has
advantages and disadvantages. On the one hand, it does not appear that
working in one dimension less due to the symmetry reduces the
complexities of the action of the Hamiltonian. In fact, it requires
treating two different kinds of terms of comparable complexity to
those of the full theory. On the other hand, it appears that the steps
taken to include matter in a anomaly free way in the full theory could
be implemented in this case. It should be noted that in the case of
even one dimension less than the case considered in this paper,
spherically symmetric reductions, up to today
there is no known way to include matter without modifying the
constraint algebra \cite{bojo}

\section*{Acknowledgment}
We wish to thank Javier Olmedo
for discussions. This work was supported in part by Grants NSF-PHY-1603630,
NSF-PHY-1903799, funds of the Hearne Institute for Theoretical
Physics, CCT-LSU, fqxi.org, Pedeciba, and CAP-CSIC.

redpajama_set_name: RedPajamaArXiv


text:
PHOTO: NICK HELDERMAN
Suuns: Hold/Still
Suuns made a career-focusing album reflective of, and worthy of, our scattered brains.
By Ryan Dieringer / 19 April 2016
Hold/Still
Suuns have always had an original take on rock music, their records marked by electronic-leaning applications of guitar, synth and drum sounds that placed their music just off the rails of any rock sub-genre. But if those great albums — Images du Futur, Zeroes QC — ultimately sounded at home in your living room next to other rock songs, Hold/Still is more of a conversation piece. Across the 13 tracks, they've removed many of the sticky sinews from their music (vocal harmonies threaded through the tracks, consistent rock drumming), but left the robust skeleton of their core ideas standing. As a result, it's an album you have to sit down and listen to. Which shouldn't feel revolutionary, but these days, feels like it.
Hold/Still doesn't ask fundamentally different questions of their listeners, it just poses them in a more unique way. The elements which remain in their sound, though markedly more ambient and tonal in presentation, are fundamentally sworn to the same basic rock 'n' roll tenets: there is a song at the center of the track; the bass drum is designed to line up with the bass; guitars fill out the higher frequencies. And yet, few tracks contain these elements all lined up at once. Instead, they are shifted in time, deleted, or withheld such that their unity is contained more within tension than in its release. To locate Suuns' sound, you have to listen closely, or else the otherwise beautiful elements can fall through your fingers.
This is all not to say that this record isn't chock full of stunning and visceral beauty. In fact, musically it's they're most accomplished. There are tracks where the full band's thrilling post-post-punk sound is mainlined, in particular via the early singles, "Paralyzer" and "Translate" (really the only two songs that could have been released as singles). On the album, their fully realized grooves are shiver-inducing.
Ultimately, Hold/Still is a headphone rock album, existing comfortably alongside fellow Canadians, Viet Cong, and filling out a contemporary indie rock scene that is increasingly competing in that arena. The album's broad reaching taste turns on a dime from minimal techno to balladry and contains tones throughout that should live up to even metal heads' taste for novel ambience. The production effort by John Congleton (St. Vincent, Modest Mouse, the Walkmen) is stacked with ingenious vocal effects, creative drum treatments, and sensory intense textures that feel utterly congruous with the versatile and mind-bending synth and guitar playing.
On "Infinity", singer Ben Shemie opines "four thoughts at the same time / ten thoughts at the same time". Suuns have created a record which doesn't thumb its nose at the existential problems of listener in enjoying and listening to music, but identifies with it and mounts the only truly acceptable response: make music reflective of, and worthy of, our scattered brains.
Persona and Modern Music in 17 Songs
Zmei3 – 'Rough Romanian Soul' (album stream) (premiere)

redpajama_set_name: RedPajamaCommonCrawl


text:
is the fifth single by Japanese singer/songwriter Chisato Moritaka. The lyrics were written by Chisato Moritaka, the music was composed and arranged by Shinji Yasuda. The single was released by Warner Pioneer on October 25, 1988.

Music video
The music video features Moritaka walking on her way home after work. During her commute, she rents a VHS copy of Aitsu ni Koishite (which she also starred in) and buys a bento meal. Upon arriving in her apartment, she has her dinner and watches the movie before looking at a ring in a jar, which is presumably an engagement ring from a former boyfriend.

Chart performance
"Alone" peaked at No. 25 on Oricon's singles chart and sold 26,000 copies.

Other versions
"Alone" was remixed for the 1989 greatest hits album Moritaka Land.

Moritaka re-recorded the song and uploaded the video on her YouTube channel on December 19, 2014. This version is also included in Moritaka's 2015 self-covers DVD album Love Vol. 8.

Track listing

Personnel
 Chisato Moritaka – vocals
 Naoki Suzuki – synthesizer programming
 Shinji Yasuda – backing vocals

Chart positions

References

External links

1988 singles
1988 songs
Japanese-language songs
Chisato Moritaka songs
Songs with lyrics by Chisato Moritaka
Warner Music Japan singles

redpajama_set_name: RedPajamaWikipedia


text:
Healthy Beauty Guide
Healthy Nails
Skin Problems & Treatments
Psoriasis: Simple Skin Care Tips
Lasers Offer Many Options to Improve Appearance
By Jane Schwanke
March 15, 2000 (San Francisco) -- Lasers have taken center stage here at the 58th Annual Meeting of the American Academy of Dermatology, where dermatologists from the world are gathering to discuss the latest medical and cosmetic advances in skin care.
Revolutionary advances in laser surgery have enabled physicians to safely and effectively eliminate a number of skin conditions such as wrinkles, scars, excessive body hair, birthmarks, tattoos, and facial and leg veins. Today's laser procedures produce fewer side effects and offer shorter recovery times than earlier lasers. In addition, recent improvements in laser technology have resulted in more successful outcomes in fair-skinned patients as well as darker skin types.
Described by some experts as "magic bullets of bright light," lasers work by generating heat that tightens the skin. The result is a more uniform, smooth appearance. While laser surgery is generally not painful, patients may experience mild discomfort afterwards.
The best candidates for laser treatments are those with skin that has been damaged by too much exposure to sunlight, according to dermatologist Arielle N.B. Kauvar, MD. "Most wrinkles around the eyes and mouth are associated with sun exposure," she tells WebMD. "[These] include all the changes you'd expect -- wrinkling, brown spots, dryness -- and they all respond well to laser resurfacing." Kauvar is associate director of the Laser & Skin Center of New York and is a clinical associate professor of New York University.
Unwanted, excess hair is a concern for men and women, particularly among ethnic populations. Recently, laser improvements have made it possible to effectively treat patients with darker skin types, such as African-American, Native American, and Asian patients.
"Everybody has the right to be treated for cosmetic reasons," Eliot F. Battle Jr., MD, tells WebMD. "Now, lasers are being introduced to a population that needs the benefits but has been ignored." He stresses that it's important to choose a clinician who is knowledgeable in the use of new lasers for ethnic skin types. Battle is with the department of dermatology at Wellman Laboratories of Photomedicine in Boston, and the Harvard Medical School.
Until recently, lasers were primarily used for facial veins. Now, leg veins can be effectively -- and comfortably -- treated with a variety of lasers, depending on the size, diameter, and color of the vein, according to Melanie C. Grossman, MD, clinical assistant professor of dermatology at Cornell University Medical School in New York.
New lasers allow more energy to be released into the skin so that the overall treatment is more effective with fewer side effects. "Even larger veins can now be treated, thanks to newer technologies," she says.
Costs for laser surgery range from several hundred to several thousand dollars, and because it is cosmetic, insurance companies generally don't pay for it. Still, insurance coverage varies according to the carrier and the nature of the medical problem. Patients should consult their insurance carrier to determine if a procedure is covered by their policy.
Although new lasers are relatively risk-free and highly efficient, their smaller size and portability mean that many more physicians have access to them. "An increasing number of physicians ... are sharing or renting laser resurfacing machines, and the number of complications ... is increasing," according to presenter Mitchel P. Goldman, MD, associate clinical professor at the University of California, San Diego.
The take-home message for consumers, says Grossman, is "do your homework and become educated before undergoing any procedure."
How do patients feel after their laser surgery? "Patients are uniformly happy," Kauvar says. "Most patients say they look younger and that people ask them what they had done."
In the future, lasers will result in shorter recovery times, fewer side effects, and better patient outcomes. The goal, Kauvar says, is to design lasers that are extremely safe and gentle for the patient.
Vital Information:
Advances in laser surgery are allowing physicians to safely and effectively eliminate a number of skin conditions, including wrinkles, scars, excess body hair, and facial and leg veins.
Improvements are also being made in laser surgery for different ethnic skin types, a population that used to be difficult to treat.
With the increased ease of performing laser surgery, more physicians are doing it, and the rate of complications is going up, so patients should educate themselves before having any surgery.
WebMD Health News
© 2000 WebMD, Inc. All rights reserved.
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text:
<?php

namespace Zend\ModuleManager\Listener;

use Generator;
use Zend\ModuleManager\ModuleEvent;

/**
 * Module resolver listener
 */
class ModuleResolverListener extends AbstractListener
{
    /**
     * Class names that are invalid as module classes, due to inability to instantiate.
     *
     * @var string[]
     */
    protected $invalidClassNames = [
        Generator::class,
    ];

    /**
     * @param  ModuleEvent $e
     * @return object|false False if module class does not exist
     */
    public function __invoke(ModuleEvent $e)
    {
        $moduleName = $e->getModuleName();

        $class = sprintf('%s\Module', $moduleName);
        if (class_exists($class)) {
            return new $class;
        }

        if (class_exists($moduleName)
            && ! in_array($moduleName, $this->invalidClassNames, true)
        ) {
            return new $moduleName;
        }

        return false;
    }
}

redpajama_set_name: RedPajamaGithub


FIRST LOOK : My Big Fat Greek Wedding 2 – Nia Vardalos, John Corbett, Lainie Kazan, Michael Constantine, Andrea Martin, Ian Gomez & Elena Kampouris.
Fans of the hugely successful Canadian-American comedy "My Big Fat Greek Wedding", released over 10 YEARS AGO via Universal Pictures, will be delighted to find out that a long awaited sequel will be out nationwide on 25th March starring Nia Vardalos & John Corbett as well as Lainie Kazan, Michael Constantine, Andrea Martin, Ian Gomez and Elena Kampouris.
"My Big Fat Greek Wedding 2" continues off where first film left off reintroducing us to Toula & Ian as a married couple and also their newest addition their teenage daughter Paris. Unfortunately for our favourite comedy couple, Toula and Ian have started to lose the romance in their relationship while dealing with an a teenage daughter who is off to university & not completely proud of her greek heritage and let's not even spoil the film by giving you details on the wedding scandal.
This is a classic light hearted comedy film that will leave you and your friends laughing through-out. The much anticipated "My Big Fat Greek Wedding 2" is released in UK cinemas on March 25th 2016.


text:
Q: How to activate base env on conda prompt Conda works as expected and activates the base env on every new installation when I run conda in my miniconda prompt. However, it does not activate the base env anymore at when I rerun after installation, I have added paths to system environment and conda activate doesn't work

A: Don't add miniconda/python.exe to the path. It won't help you, since more sophisticated packages like numpy would also need the paths to the C-libraries. This is exactly what is achieved by activating an environment via
conda activate

in the command shell. (Check PATH before and after.)
For further information read the docs: https://conda.io/projects/conda/en/latest/user-guide/tasks/manage-environments.html#activating-an-environment

redpajama_set_name: RedPajamaStackExchange


High quality and high capacity machine ideal for large joineries and furniture manufacturers. Allows rapid production of mortises on door and window posts of different lengths and depths.
The working head slides on ground round bars with the assistance of precision self-lubricating bushes and self-cleaning scrapers. This robust sliding table provides high accuracy and repeatability.


The Pradhan Mantri Sahaj Bijli Har Ghar Yojana ('Saubhagya') launched in September, which claims to ensureelectrification of all willing households in the country, is too ambitious a project. While it makes grandiosepromises to provide a free electricity connection to all willing Below Poverty Line households and to all others on a payment of ₹500 (which shall be recovered by the power distribution companies/power departments in 10 instalments along with electricity bills), it expects the poor to pay the bills without providing any subsidy to ease their burden. Even to the best of their abilities the poor would often not be in a position to pay regular electricity bills, which in turn could result in disconnection. The government has conveniently overlooked the fact that for the poor in some States, the inability to pay an electricity bill is a big impediment. This new scheme is just a way of refurbishing the Deen Dayal Upadhyaya Gram Jyoti Yojana (DDUGJY), the earlier scheme of rural electrification launched in July 2015, which aimed to electrify all un-electrified villages by May 2018.
Under DDUGJY, the government managed to electrify 14,701 villages while 2,760 villages remain un-electrified; out of these, work is still in progress in a total of 2,611 villages. However, out of the 14,701 villages, only in 8%, i.e. 1,198 villages, do all households have connectivity. Even if we take into consideration the fact that so many villages have been "electrified", the next point of contention is the definition used. According to the definition, a village is considered to be electrified if 10% households have an electricity connection and related basic infrastructure. Furthermore, even in these 10% of households, there is no promise of minimum hours of supply. The question we then need to ask is this: given that 90% of households may not have power supply and of those 10% with electricity not having a regular supply, can we still consider such a village to be electrified in a meaningful way? The objective of the Saubhagya scheme is to "provide energy access to all by last mile connectivity and electricity connections to all remaining un-electrified households in rural as well as urban areas to achieve universal household electrification in the country." On the face of it, the scheme may only be able to plug the gaps and address the issues of entry barrier, last mile connectivity and release of connections, but it can guarantee neither regular electricity supply nor continuation of those connections in case of non-payment. A free electricity connection may provide some relief as far as the financial burden is concerned. However, expecting poor households to bear the recurring burden of bills as per the prevailing tariff of DISCOMs is unimaginable. Even if the programme is successful, hypothetically, and all households are provided a connection, there would still be the problem of regular supply. Industry estimates suggest that this scheme would potentially require an additional 28,000 MW and additional energy of about 80,000 million units per annum, which is roughly 7% of India's current installed power capacity. There is a power shortage even at this moment leading to scheduled and unscheduled load shedding, often up to 10 hours or more. The problem is graver still in interior rural India. Considering the huge lapses as far as electricity availability is concerned, managing this additional demand would prove to be challenging. We should also not forget that the provision of providing one-two hours' supply a day is not the same as provision of regular supply. In the past three years, we have seen a series of policies and promises urging us to ponder over the type of welfare politics India is witnessing. Symbolically, all such attempts have a lot of significance as far as the bid to secure popular support is concerned. However, there has been a lot of debate in the past over many of the government's policies which it claims to be based on the primary goal of "ushering in development". Nevertheless, it must be a cause of worry that the government has embraced the slogans of welfare politics without being able to deliver substantial and meaningful results. Irrespective of the poor track record as far as meaningful change is concerned, such policies have aided the government in building an image of being people-centric. Certainly, Saubhagya has some positives such as provision for households outside the reach of grid lines. However, our contention is that the policy has set a standard for itself without enough focus on its capacity to deliver results. The policy statement echoes the commitment to facilitate economic growth and social development, but we still need to ask whether this is another instance of messaging for an electoral purpose.
The Centre is clearly stepping up its bid to address concerns about the fledgling goods and services tax regime. In what is being seen as official recognition that drastic action is called for, Revenue Secretary Hasmukh Adhia conceded in an interview over the weekend that it is possible that differential treatment to some similar items may not necessarily be fair. The rates will be revised, he indicated, wherever small and medium businesses and the common man face 'a big burden'. While GST has brought 27 lakh new registered entities into formal tax coverage in its first three months, just over half of those in the GST net are paying taxes and filing tax returns. September GST collections stood at ₹92,150 crore by Monday — compared to over ₹95,000 crore in July, its first month. Separately, Prime Minister Narendra Modi on Sunday sought to assure traders that anyone joining the GST net will not be harassed by taxmen seeking scrutiny of their past records. These latest statements of intent cap the overtures the Centre has made over the past three weeks to display its commitment to fix the flaws in the new tax regime, starting with the Prime Minister's October 4 promise to address the woes of small businesses and exporters. With Mr. Adhia now saying that the GST regime will take a year to stabilise and the government signalling a major revamp in the coming days, we may still be some distance away from seeing the final shape of India's one-nation, one-tax plan. The commitment to correct course is welcome. But the real question is the timing and sequencing of changes. At the last GST Council meeting, rates of over two dozen items were reset, taking such rate changes since July 1 to over 100 — while some procedural and compliance-related norms were eased. But many measures announced on October 6 haven't been implemented swiftly enough — refunds to exporters, for instance, have been slow due to cumbersome norms for reconciling records. This should be a cue for the government to consider doing away with the stringent requirement for matching invoices in order to allow input tax credits, which is impeding the uptake of the new tax. On November 10, the Council is expected to take up more rate revisions, process simplification and the inclusion of real estate in the GST regime. Excessive tinkering with rates indicates that the initial levies were not thought through. While at one level, high or incongruent rate structures have driven small and medium enterprises to the wall, at another level, entire product categories are under threat of disappearing, such as eco-friendly hybrid cars. Lastly, given the amount of damage control the government has embarked on, in comparison with its spirited defence of the preparedness for a July 1 roll-out, there needs to be some real introspection about why those at the top were led to believe all GST systems were ready to fire.
Example: Legislation to ensure equal opportunities for all.
Example: Grandiose plans to reform the world.
Example: The premises have been completely refurbished in our corporate style.
Example: The captured territory was the main area of contention between the two countries.
Meaning: Mention (a product, event, or establishment) publicly in order to promote it.
Example: During the show he plugged his new record.
Meaning: Occur again periodically or repeatedly.
Example: When the symptoms recurred, the doctor diagnosed something different.
Meaning: Existing at a particular time; current.
Example: The unfavourable prevailing economic conditions.
Example: The reduction of trade barriers and import tariffs.
Meaning: Action to reduce the load on something, especially the interruption of an electricity supply to avoid excessive load on the generating plant.
Meaning: An offer of a price, especially at an auction.
Example: At the fur tables, several buyers make bids for the pelts.
Example: Besides traditional methods, artists are embracing new technology.
Meaning: Of considerable importance, size, or worth.
Example: A substantial amount of cash.
Meaning: An increase in the amount, speed, or intensity of something.
Example: The recent stepping-up of the campaign.
Example: The country's fledgling democracy.
Meaning: Likely to have a strong or far-reaching effect; radical and extreme.
Example: A drastic reduction of staffing levels.
Meaning: Admit or agree that something is true after first denying or resisting it.
Example: I had to concede that I'd overreacted.
Example: He sought help from the police.
Meaning: Subject to aggressive pressure or intimidation.
Example: If someone is being harassed at work because of their sexuality they should contact the police.
Meaning: A mistake or shortcoming in a plan, theory, etc. which causes it to fail or reduces its effectiveness.
Example: There were fundamental flaws in the case for reforming local government.
Meaning: Great sorrow or distress (often used hyperbolically).
Example: The Everton tale of woe continued.
Example: An attempt to revamp the museum's image.
Example: The king and the archbishop were publicly reconciled.
Example: Stringent guidelines on air pollution.
Meaning: A list of things provided or work done together with their cost, for payment at a later time.
Example: Invoices must be submitted by the 24th of every month.
Meaning: Delay or prevent (someone or something) by obstructing them; hinder.
Example: The sap causes swelling which can impede breathing.
Example: They have been selected for inclusion in the scheme.
Meaning: Not suitable or not fitting well with something else.
Example: Violence is incongruent with our values and legal system.
Meaning: A person or thing likely to cause damage or danger.
Example: Hurricane damage poses a major threat to many coastal communities.
Meaning: Full of energy, enthusiasm, and determination.
Example: A spirited campaigner for women's rights.
Meaning: The examination or observation of one's own mental and emotional processes.
Example: Quiet introspection can be extremely valuable.