text:
\section{Introduction}

   Recently we embarked on the investigation of the bosonic ${\mathbb CP}^{N-1}$ model \cite{D'Adda:1978uc,Witten:1978bc}, defined on finite space interval $L$,  i.e.,   on a finite width worldstrip,  in the large $N$ approximation \cite{BKO}. Such a system could provide a useful model for various physical situations.  For instance,
it appears as the low-energy effective theory describing the quantum
excitations of the monopole-vortex soliton complex  \cite{MVComplex1,MVComplex2,MVComplex3,MVComplex4} in hierarchically broken gauge symmetries, such as  $SU(N+1) \to SU(N) \times U(1) \to {\bf 1}$ in a color-flavor locked $SU(N)$ symmetric vacuum.   The ${\mathbb CP}^{N-1}$  model  describes the nonAbelian orientational zeromodes of the nonAbelian vortex (string) \cite{Hanany:2003hp,Auzzi:2003fs,Shifman:2004dr},  whereas its boundaries represent the monopoles arising from a higher-scale gauge-symmetry breaking, carrying the same orientational ${\mathbb CP}^{N-1}$
moduli. NonAbelian monopoles, not plagued by the well-known difficulties, could  emerge in such a context.
The fate of the nonAbelian monopoles as a quantum mechanical entity is then linked to the phase of the low-energy  ${\mathbb CP}^{N-1} $  effective action attached to it.

In \cite{BKO}  it was found that the quantum saddle-point equations
describing the ${\mathbb CP}^{N-1}$ model with Dirichlet or Neumann
boundary conditions has a unique solution under certain conditions.   In the large-$L$ limit,
this solution approaches smoothly the well-known confining phase of
the standard 2D ${\mathbb CP}^{N-1}$ system.  A phase transition
between a Higgs-like phase and the confining phase for a shorter $L$, which was claimed   to be present in the literature \cite{Milekhin:2012ca},
was shown  not to exist in the system.\footnote{For periodic boundary
  conditions (and large-$N$), however, such a phase transition
  does occur \cite{Monin:2015xwa}.  See also  \cite{Milekhin:2016fai}. }

The model is interesting also from a formal point of view, as it
provides a prototype model of a quantum system of varying
dimensions in the presence of dynamical mass generation: it interpolates between  a  2D QFT  (in the $L\to \infty$ limit)  with all well-known phenomena such as asymptotic freedom and
confinement  and a 1D system in the $L \to 0$ limit - quantum mechanics.
For shorter strings of length $L \le  1/\Lambda$, quantum fluctuations
of the ${\mathbb CP}^{N-1}$ fields $n_i$,  ($i=1,2, \ldots,  N$)
remain weakly coupled,  as they lack sufficient 2D spacetime
``volume'' in which the fields fluctuate. With the Dirichlet
condition, the system reduces effectively to a classical system in the
$L \ll 1/\Lambda$ limit.

In this paper, we delve in more detail into the properties of the large-$N$ ${\mathbb CP}^{N-1}$ model on a finite-width worldsheet.
First, with a more refined numerical method we improve the precision
of the solution to the generalized gap equation. This enables us to explore a larger region of the parameter space and in particular the limit of large $L$.
The second problem is to understand
 the energy density of the string itself as a function of $x$, computed at the functional saddle point,  completing the analysis presented  in  \cite{BKO}.
The third problem is to
 clarify the approach to the QFT ($L \to \infty$) limit of our system;
 this involves the question of certain consistency with the known
 field-theory limit, as well as of figuring out interesting $L$-dependent effects. It will be seen that power-behaved corrections such as the L\"uscher term are absent.  This is consistent  as all fields acquire dynamically generated mass; at the same time no spontaneous breakdown of the global $SU(N)$ symmetry takes place.

The paper is organized as follows.
In Section \ref{two} we review the ${\mathbb CP}^{N-1}$  model on a finite strip and also present new numerically improved results which allows to reach higher values of $L$ than before.
In Section \ref{sec:gapeq}  the generalized gap equation is re-derived,   paying special attention to the anomalous term that arises in the functional variation, which is analogous to the axial anomaly.
In Section \ref{four} we consider the energy density in detail, its various contributions, and its  $L \to \infty$ limit.
In Section \ref{sec:Analytic} we study an analytical Ansatz that describes the large but finite  $L \gg 1/\Lambda$ and the approach to the $L\to \infty$ limit.
The numerical results for the renormalized energy density are presented in Section \ref{six}.
In Section \ref{seven} we discuss the Casimir force.  Our conclusion is in
Section \ref{eight}.   Some details of our analysis are given in Appendices \ref{app:prop} $\sim$ \ref{app:algorithm_test}.

\section{Review of the ${\mathbb CP}^{N-1}$  model on a finite width worldsheet}
\label{two}

The classical  action for the ${\mathbb CP}^{N-1}$ sigma model is defined  by
\begin{eqnarray}
S= \int dx dt \left( (D_{\mu} n_i)^*D^{\mu} n_i -  \lambda (n_{i}^* n_i - r)   \right) \;,\qquad  r= \frac{4 \pi}{g^2}\;,
\end{eqnarray}
where $n_i$ with $i=1,\dots,N$ are $N$ complex scalar fields and the
covariant derivative is given by $D_\mu=\partial_\mu-i g A_\mu$.
Configurations related by $U(1)$ gauge transformations
$n_i \to e^{i\alpha} n_i$ are not only gauge-equivalent, but
  are equivalent because the $U(1)$ gauge field
$A_{\mu} $ does not have a kinetic term in the classical action.
$\lambda$ is a Lagrange multiplier field that enforces the classical
gap equation
\begin{eqnarray}
\label{classicalconstrain}
n_{i}^* n_i  = r\;,     \end{eqnarray}
where $r= \tfrac{4\pi}{g^2}$ is related to the gauge coupling and can
be thought of as the ``size'' of the ${\mathbb CP}^{N-1}$ manifold.

For the ${\mathbb CP}^{N-1}$ theory on a finite interval of length $L$,  $x \in[-\tfrac{L}{2},\tfrac{L}{2}]$,\footnote{With the aim of studying the $L \to \infty $
limit of the string at fixed  $x$ (and $\Lambda$)  in mind,  we take the space interval to be  $[-\tfrac{L}{2},\tfrac{L}{2}]$, rather than $[0,   L]$   as done  in \cite{BKO},  by a trivial shift of the
spatial coordinate.}
    the boundary conditions must be specified. One possibility is the
    Dirichlet-Dirichlet boundary condition which -- up to a $U(N)$
    transformation -- is
\begin{eqnarray}
\hbox{D-D}: \qquad
n_1\!\left(-\tfrac{L}{2}\right)=n_1\!\left(\tfrac{L}{2}\right) =
\sqrt{r}\;,   \qquad
n_{i}\!\left(-\tfrac{L}{2}\right)=  n_{i}\!\left(\tfrac{L}{2}\right) =0\;,  \quad i>1\;.  \label{DDbc}
\end{eqnarray}
For the moment we take the boundary conditions for the $n_i$ fields in the same direction in the ${\mathbb CP}^{N-1}$ space at the two boundaries.
Another possibility is the Neumann-Neumann  boundary condition\footnote{
Mixed conditions can be chosen
where one of the boundaries takes the Dirichlet condition and the other the Neumann condition.
}
\begin{eqnarray}
\hbox{N-N}: \qquad \partial_x n_i\!\left(-\tfrac{L}{2}\right) =
\partial_x n_i\!\left(\tfrac{L}{2}\right) =0\;, \quad \forall i. \label{NNbc}
\end{eqnarray}
In this paper, we will focus on the Dirichlet-Dirichlet boundary
condition. Thus with this condition the $N$ fields can naturally be
separated into a classical component $\sigma \equiv n_1$ and the rest,
$n_i$    ($i=2,\dots,N$).
Integrating out the $n_i$ fields yields the effective action:
\begin{eqnarray}
{S}_{{\rm eff}} = \int d^2 x \left( (N-1) \, {\rm tr}\, {\rm log} (- D_{\mu}D^{\mu}  + \lambda) + (D_{\mu} \sigma)^*D^{\mu} \sigma -  \lambda (|\sigma|^2  - r)  \right)\;.\label{from}
\end{eqnarray}
Because the effective action only depends on $|\sigma|^2$ and
$|\partial_\mu\sigma|^2$ and the boundary conditions take real
positive values on both sides, one can take $\sigma$ to be a real
field and set the gauge field to zero.
Finally, we will consider the leading contribution at large $N$ only.

The generalized gap equations following from Eq.~(\ref{from})  (see also Section~\ref{sec:gapeq} below)
\begin{eqnarray}
\frac{N}{2} \, \sum_n\frac{f_n(x)^2}{\omega_n} e^{-\epsilon \omega_n}  + \sigma(x)^2 - r_{\epsilon} = 0  \,,\qquad  \partial_x^2 \sigma(x) - \lambda(x) \sigma(x) = 0  \,,
\label{gapeqbb}
\end{eqnarray}
where
\begin{eqnarray}
r_\epsilon \equiv
\frac{N}{2\pi}\left(\log\left(\frac{2}{\Lambda  \epsilon}\right)
-\gamma \right)\;,
 \label{uvbeta}
\end{eqnarray}
have been solved numerically  in \cite{BKO}, by a Hartree-like self-consistent method.
The renormalized, finite functions of $x, \Lambda, L$, $\lambda(x)$ and $\sigma(x)$
have been obtained numerically for various  values of $L $ and $\Lambda$.
The calculations of  \cite{BKO} have  been extended to larger values
of $L$, with a considerably improved method.
The  weak point of the
Hartree-like method  -- from a numerical point of view --  is the need  to determine $\lambda(x)$ from the second
equation in Eq.~\eqref{gapeqbb}, where  $\sigma$ tends to zero in the
middle of the string.

It is sometimes convenient to rewrite the first equation in Eq.~(\ref{gapeqbb}) as
\begin{eqnarray}
N D(x,\epsilon; x,0)  + \sigma(x)^2 - r_{\epsilon} = 0 \;,  \label{Dgapeq}
\end{eqnarray}
  in terms of the two-point function
\begin{eqnarray}
 D(x,\tau; x',\tau')\equiv \sum_n\frac{e^{-|\tau-\tau'| \omega_n}}{2\omega_n} f_n(x) f_n(x')\;.   \label{thetwo}
\end{eqnarray}
The latter satisfies,  for the D-D boundary condition, an equation
\begin{eqnarray}
&& \left(-\partial_\tau^2 -\partial_x^2+\lambda(x) \right) D(x,\tau;x',\tau')\nonumber  \\
&=& \sum_{n\in \mathbb Z}
\delta(\tau-\tau')\left\{ \delta(x-x'+2n L)-\delta(x+x'+(2n+1) L) \right\}. \label{Dequation}
\end{eqnarray}
Note that the infinite number of mirror poles are required to satisfy the D-D boundary condition.
See Appendix \ref{derivation} for more details.

Under the assumption
\begin{eqnarray}
\lim_{x\to \pm \frac{L}2} \left(x\pm \frac{L}2\right)^2 \lambda(x) =0\;, \label{asymptotic-free}
\end{eqnarray}
 the near-the-boundary behavior of the fields turns out to be \cite{BKO} (see Section~\ref{section:near} below):
\begin{eqnarray}
 \sigma^2 \simeq   \frac{N}{2 \pi}  \log{\frac{1}{|x\pm L/2|} }  \;;
\qquad
\lambda(x) \simeq    \frac{1}{2  \,(x\pm L/2)^2 \log{1/|x\pm L/2|} } \;. \label{satisfying}
\end{eqnarray}

\subsection{Numerical method and solutions}

The new method is based on a random-walk algorithm and is reversed in
some sense with respect to the old method.
A guess can be made for the function $\lambda(x)$, but the precise
starting point is not important.
The algorithm has two assumptions built in; basically just for saving
computational costs; i.e.~$\lambda(x)$ is a symmetric function in
$x$; the second is that $\lambda(x)$ is a monotonically increasing
function from the midpoint of the string to the boundary (both
assumptions are indeed consistent with the results of \cite{BKO}).
Now the algorithm makes a random change to a part of the function
$\lambda(x)$ (viz.~on an interval that is a randomly chosen subset of
the full string interval) yielding $\tilde{\lambda}(x)$.
Now the new $\tilde{\lambda}(x)$ function is tested in the following
way. $\sigma(x)$ is calculated from its equation of motion (second
equation of Eq.~\eqref{gapeqbb}) with the appropriate Dirichlet
boundary conditions as well as from the generalized gap
equation (the first equation of Eq.~\eqref{gapeqbb}); these two are
compared.
If the new $\tilde{\lambda}(x)$ makes the two $\sigma$s move closer to
each other, then the new $\tilde{\lambda}(x)$ function is accepted as
the new improved $\lambda(x)$, otherwise it is rejected.
Then the cycle repeats until the precision is good enough (for the
solutions we found, that is
$\int_{-L/2}^{L/2} dx\;|\sigma^2-\tilde{\sigma}^2|<10^{-5}$).
See Appendix~\ref{sec:walk} and Appendix~\ref{app:algorithm_test}  for more details.

Some  examples  are shown in Figure \ref{fig1} where the approach to the confined phase at large $L$ is evident.
The values of fields in the middle of the interval $\sigma^2(0)$ and $\lambda^2(0)-\Lambda^2$ are shown in a logarithmic plot in Figure \ref{fig2}.
The approach of $\sigma^2(0)$ to zero is clearly exponential and consistent with its mass.
\begin{figure}[!ht]
\begin{center}
\mbox{\subfloat{\includegraphics[width=0.49\linewidth]{rw_lambdaLs}}
\subfloat{\includegraphics[width=0.49\linewidth]{rw_sigmasqLs}}}\vspace{1em}\\
\caption{The functions  $\lambda(x)$ (left) and   $\sigma^2(x)$ (right)  which are solutions to the gap equation, Eq.~(\ref{gapeqbb}), for
various values of $L$  ranging  $L=1 \sim 12$.   $\Lambda=1$ in this figure.
The innermost (outermost) curve corresponds to $L=1$ ($L=12$).
}
\label{fig1}
\end{center}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\mbox{\subfloat{\includegraphics[width=0.49\linewidth]{rw_sigmahalf_log2}}
\subfloat{\includegraphics[width=0.49\linewidth]{rw_lambdahalf_log2}}}\vspace{1em}\\
\caption{The values of fields in the middle of the interval
  $\sigma^2(0)$ and $\lambda^2(0)-\Lambda^2$ in a logarithmic plot. $\Lambda=1$ in this figure.
  These figures are just an illustration of the exponential behavior of $\sigma^2(0)$ and $\lambda^2(0)-\Lambda^2$ as functions of $L$, and are not intended
  to be precise fits to theory. Especially the numerical errors are quite large at large $L$ in the right figure (but still much smaller than $\Lambda^2=1$).
}
\label{fig2}
\end{center}
\end{figure}
In Figure \ref{fig3} we show the solutions in the interval $(-\frac{L}{2},0)$ by keeping one boundary fixed at $-\frac{L}{2}$.  This clearly shows the convergence at large $L$ to the half-line solution $(-\infty,0]$.
\begin{figure}[!ht]
\begin{center}
\mbox{\subfloat{\includegraphics[width=0.49\linewidth]{rw_lambdaLs_semiretta}}
\subfloat{\includegraphics[width=0.49\linewidth]{rw_sigmasqLs_semiretta}}} \vspace{1em} \\
\caption{The function  $\lambda(x)$ (left) and   $\sigma^2(x)$ (right)  which are the solutions of the gap equation, Eq.~(\ref{gapeqbb}) in the interval $(-\frac{L}{2},0]$ for
various values of $L$  ranging  $L=1 \sim 12$. $\Lambda=1$ in this figure.
}
\label{fig3}
\end{center}
\end{figure}

As already noted in \cite{BKO},  one can clearly see  that the
asymptotic ($L\to \infty$) regime  (with $\lambda(x)\sim \Lambda^2$
and $\sigma(x)\sim 0$   except near the boundaries)  has already set
in  at $L\simeq  4$, which is quite reasonable ($L \Lambda=4 \gg 1$).
The effect of the boundaries is seen to propagate only for $\Delta x
\sim  1/\Lambda$ from the latter:   the system effectively reduces to
the standard  2D ${\mathbb CP}^{N-1}$   model in an infinite spacetime,  as one moves away from the boundaries by $\sim1/\Lambda$ or more,   as expected on general grounds.

 \subsection{$L$ dependence, near-the-boundary  behavior of $\lambda(x), \sigma(x)$, and the classical limit   \label{section:near}}

Our system has two parameters, the dynamically generated mass scale $\Lambda$ and the interval length $L$.  But as $\Lambda$ fixes the physical unit of length, the model actually possesses only one parameter: what distinguishes two physically distinct systems is the product $\Lambda L$.
The crucial point is that the UV divergences
are  short-distance effects around any fixed space point, and are universal. They do not depend on  the presence or absence of the boundaries.
This is what allows us to define unambiguously systems of ``different space width $L$".

The near-the-boundary behavior (\ref{satisfying}) can be obtained  \cite{BKO} as follows.  Under the assumption (\ref{asymptotic-free}),
the large $n$ modes are given at any fixed $x$  by
 \begin{eqnarray}
\label{largen}
f_n \simeq
\sqrt{\frac2L} \sin\left(\frac{n \pi (x+ L/2)}L\right)\;,  \qquad
\omega_n \simeq \frac{n\pi}L\;, \qquad n \gg 1\;.
\end{eqnarray}
The finite part of the sum over modes in the gap equation behaves then as
\begin{equation}
\frac{N}{2} \, \sum_n\frac{1}{\omega_n}\left( f_n(x)^2 -\frac1L \right)  =
 \frac{N}{2 \pi} \log
\left(2 \sin{\left(\frac{\pi (x+L/2)}{L} \right)}  \right)\simeq     \frac{N}{2 \pi}   \log \left(\frac{2\pi}{L}(x+L/2)\right)\; ,  \label{massless}
\end{equation}
near the left boundary.  This singularity can only be compensated by $\sigma(x)^2$ in the gap equation, hence  Eq.~(\ref{satisfying}).  The numerical solutions found in \cite{BKO} and here clearly exhibit this logarithmic behavior.

There is an alternative way of understanding  the behavior of $\sigma(x)$ and $\lambda(x)$ near the boundaries. Consider the regularized but un-renormalized form of the gap equation (\ref{gapeqbb}).  By keeping the UV regularization parameter $\epsilon $ fixed and by going to $x=\mp  \tfrac{L}{2}$, one finds   that
\begin{eqnarray}   \sigma\!\left(\mp \tfrac{L}{2}\right)^2 =    r_{\epsilon} =  \frac{N}{2\pi}   \log\left(\frac{1}{ \epsilon} \right) +\cdots \;,  \label{this}
\end{eqnarray}
as $f_n=0$.   This is nothing but the classical ${\mathbb CP}^{N-1}$ model (\ref{classicalconstrain}) with the Dirichlet boundary condition
    \begin{eqnarray}
   n_1\!\left(-\tfrac{L}{2}\right)=n_1\!\left(\tfrac{L}{2}\right) = \sqrt{r}\;,   \qquad   n_{i}\!\left(-\tfrac{L}{2}\right)=  n_{i}\!\left(\tfrac{L}{2}\right) =0\;,  \quad i>1\;.
\end{eqnarray}
With $x$  close to but not exactly  at  a boundary, $\epsilon$  in (\ref{this})  is replaced by
 $\left|x\pm\tfrac{L}{2}\right|$ and one finds Eq.~(\ref{satisfying}).  This statement requires an explanation.
 What really happens is that   in the gap equation (\ref{gapeqbb}),      $\log 1/\epsilon$  which is in  $\sigma(x)$ at  exactly $x=-L/2$,    is moved  at small but nonvanishing  $|x+L/2|$  to  the first term involving the sum over the modes.  After $\log 1/\epsilon$
 is eliminated by the bare coupling constant term
 $r_{\epsilon}$ and the gap equation is renormalized and made finite,
    it produces $- \log 1/ |x+L/2| $,    which can only be compensated by
    $\sigma(x)^2$, as  in   (\ref{satisfying}).  See
    Eq.~(\ref{both3})  and Eq.~(\ref{both4}).

This discussion clearly shows that the origin of the singular behavior of the mass gap  $\lambda(x)$  and of the $\sigma(x)$ field is the fact that  the system reduces to its classical limit \footnote{We thank Misha Shifman for useful discussions on this point.} near the boundaries, not having sufficient 2D spacetime volume for the $n_i$ fields to fluctuate.

The same reasoning  explains \footnote{These two issues are indeed one
  and the same:  in the small-$L$ limit, the system consists of its boundaries only, so to speak.}   the behavior of  the value of $\lambda(x)$ and $\sigma(x)$ at the midpoint of the string  at small  $L \ll 1/\Lambda$,  found in  \cite{BKO} (see Fig. 3 there),
\begin{equation}  \lambda(0)   \sim   \frac{4}{L^2}  \log\frac{1}{L}\;, \qquad      \sigma^2(0) \sim   \frac{N}{2\pi}\log\frac{1}{L}\;. \label{decrease}
\end{equation}

\section{Anomalous functional variation and the generalized gap equation  \label{sec:gapeq}}

We now re-derive the generalized gap equation.
Our starting point is  the  energy density
\begin{eqnarray}
 {\cal E}(x)&=&\frac{N}2
\sum_{n}\left(\omega_n f_n(x)^2+\frac{1}{\omega_n}\left(f'_n(x)^2+\lambda
	 f_n(x)^2\right)\right)e^{-\epsilon \omega_n} \nonumber   \\
&&\mathop+\sigma'(x)^2+\lambda(x)\left(\sigma(x)^2-r_\epsilon^0\right)+{\cal
           E}_{\rm uv} \; ,  \label{density}
\end{eqnarray}
where $\epsilon$ has been introduced as a regulator of the UV divergences coming from higher modes,
${\cal E}_{\rm uv}$ is a subtraction constant and $f_n(x)$, $\omega_n$
are the eigenmodes of the $n_i$ field equations
\begin{equation}
 -f''_n(x)+\lambda(x)f_n(x)=\omega_n^2 f_n(x)\;,\qquad
\int_{-L/2}^{L/2} dx \; f_n(x)f_m(x)=\delta_{n,m} \;.    \label{using}
\end{equation}
By integrating Eq.~\eqref{density} over
$x \in\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$  and by using  (\ref{using}),  one has the expression for the integrated energy,
\begin{eqnarray}
 E   &\equiv & \int_{-L/2}^{L/2} d x\;  {\cal E}(x)\nonumber  \\
&=&  N \sum_{n}\omega_n e^{-\epsilon \omega_n}
+\int_{-L/2}^{L/2}  d x \left[\sigma'(x)^2+\lambda(x)(\sigma(x)^2-r_\epsilon^0 )+{\cal E}_{\rm uv}   \label{total}
\right]\;.
\end{eqnarray}
For instance the total derivative terms vanish as  \cite{BKO}
\begin{eqnarray}
  \lim_{x\to - L/2} f_n(x)f_n'(x)
  = \lim_{x\to -  L/2} \left(\frac{x+L/2}{-\log(x+L/2)}
  + {\cal O}\left(\frac{x+L/2}{2\log^2(x+L/2)}\right) \right)=0\;, \quad \forall n\;.
\end{eqnarray}
By varying (\ref{total})  with respect to  $\lambda(x)$, and by using
\begin{eqnarray}
\delta \omega_n^2 = \int_{-L/2}^{L/2}  dx \;  \delta \lambda(x)  f_n(x)^2 \,,\qquad    \frac{\delta \omega_n } {\delta \lambda(x) }=  \frac{f_n(x)^2}{2 \omega_n}\,,
\end{eqnarray}
  one gets  the generalized gap equation
\begin{eqnarray}
\frac{N}{2} \, \sum_n\frac{f_n(x)^2}{\omega_n} e^{-\epsilon \omega_n}  + \sigma(x)^2 - r_{\epsilon} = 0  \,,
\label{gapeq}
\end{eqnarray}
where
\begin{equation}      r_{\epsilon}  = r_{\epsilon}^0
 +\frac{N}{2\pi}  \;,   \label{anomaly}
\end{equation}
whereas the variation with respect to $\sigma$  gives
\begin{eqnarray}
\partial_x^2 \sigma(x) - \lambda(x) \sigma(x) = 0  \ . \label{lambda}
\end{eqnarray}

Note the extra $\frac{N}{2\pi} $  term  in (\ref{anomaly}).  It  arises when the variation $\delta/\delta \lambda(x)$  acts on the regulator  factor $e^{-\epsilon \omega_n}$:
\begin{equation}   N \sum_{n}\omega_n  \frac{\delta }{\delta \lambda(x)}  e^{-\epsilon \omega_n} =
 -  \frac{\epsilon    N}{2}   \sum_{n}  f_n(x)^2 e^{-\epsilon \omega_n} :
\label{anvar}\end{equation}
this term is superficially of the order of  $\mathcal{O}(\epsilon)$:   however the sum in the last expression diverges as $\frac{1}{\epsilon}$, so it gives
a nonvanishing
contribution \footnote{This is  exactly as the axial anomaly arises when the spacetime derivatives act on the string bit in the point-split axial current operator.}.
 As the divergence comes from large $n$ it may be
calculated as
 \begin{equation} -  \frac{\epsilon    N}{2}   \sum_{n}  f_n^2(x)  e^{-\epsilon \omega_n} \simeq -  \frac {\epsilon  N}{ 2 L}   \sum_{n} \left[  1 -  \cos \left(\frac{ 2  n \pi (x+L/2)  }L\right) \right]   e^{-\epsilon \pi n /L} \;=  - \frac{N}{2\pi} +  \mathcal{O}(\epsilon)\;,    \label{both1}
\end{equation}
for
 \begin{equation}   - \tfrac{L}{2} < x < \tfrac{L}{2}\;,     \end{equation}
where use was made of an approximate form for the
eigenmodes
 \begin{eqnarray}
\label{largenBis}
f_n \simeq
\sqrt{\frac2L} \sin\left(\frac{n \pi (x+ L/2)}L\right)\;,  \qquad
\omega_n \simeq \frac{n\pi}L\;, \qquad n \gg 1\;.
\end{eqnarray}
The same result  can be found
by using the propagator representation  (\ref{thetwo}).
See Appendix \ref{derivation}.

\section{Energy  density \label{four}}

We now go back to the density itself, and rewrite (\ref{density})  as
\begin{eqnarray}
    {\cal E}(x)  &=&    {\cal E}_0(x)
    +  \lambda(x)  \left( \frac{N}{2}  \sum_{n} \frac{1}{\omega_n}   f_n(x)^2       +     \sigma(x)^2-r_\epsilon^0   \right)   +{\cal E}_{\rm uv}\;  \nonumber \\
 &=&     {\cal E}_0(x)   + \frac{N}{2\pi} \lambda(x) +  {\cal E}_{\rm uv}\;,   \label{Edensity}
\end{eqnarray}
where
\begin{equation}   {\cal E}_0(x)  \equiv  \frac{N}{2}  \sum_{n}\left(\omega_n f_n(x)^2+\frac{1}{\omega_n}f'_n(x)^2
	 \right)e^{-\epsilon \omega_n}   +\sigma'(x)^2  \;,    \label{EdensityConst}
\end{equation}
by collecting terms proportional to $\lambda(x)$ and by using Eqs.~(\ref{gapeq}) and (\ref{anomaly}).
Note that the anomaly (\ref{anomaly}) is crucial to give the term proportional to $\lambda(x)$ in the energy density, after using the gap equation.

It turns out that
$ {\cal E}_0(x)  $  is  a constant.
The space derivative of $ {\cal E}_0(x)  $ is:
\begin{eqnarray}     \frac {d {\cal E}_{0} (x) }{dx}  &=&   N    \sum_{n=1}^\infty
\left( \omega_n
 f_n  f'_n+\frac{1}{\omega_n}   f'_n  f''_n \right)  e^{-\epsilon  \omega_n}    + 2  \sigma' \sigma''\; \nonumber \\
 &=&    N    \sum_{n=1}^\infty
\left( \omega_n
 f_n  f'_n+\frac{1}{\omega_n}   f'_n  (\lambda - \omega_n^2) f_n \right)  e^{-\epsilon  \omega_n}    + 2  \sigma' ( \lambda \sigma ) \; \nonumber \\
&=& \lambda(x) \left[     N    \sum_{n=1}^\infty    \frac{1}{\omega_n}  f_n  f'_n   e^{-\epsilon  \omega_n}    +  2  \sigma  \sigma'     \right] \;.
\end{eqnarray}
Noting that the expression in the square bracket above is the derivative of the gap equation  (\ref{gapeq}),  we conclude that
\begin{eqnarray}
 \frac {d {\cal E}_{0} (x) }{dx}  &=&  0 \ .
\end{eqnarray}
We thus find that the energy density is
\begin{equation}    {\cal E}(x, \Lambda, L)=   {\cal E}_0(\Lambda, L)  +   \frac{N}{2\pi}  \lambda(x, \Lambda, L)  +  {\cal E}_{\rm uv}\;,  \label{result0}
\end{equation}
where the only dependence on $x$ is through the  function $\lambda(x)$.

Let us study the   $L \to \infty$   limit of this expression.
The value of the constant part of the energy density,  ${\cal E}_{0}$,  in the  $L\to \infty$  limit  may be calculated
by noting that $\lambda(x) \to \Lambda^2$ and  the fact that the spectrum is exactly known in that limit.
See Appendix \ref{sec:infiniteL}.  The result is
\begin{equation}    {\cal E}_0 (\Lambda,L=\infty)    = \frac{N}{\pi \epsilon^2}   -  \frac{N  \Lambda^2  }{  4  \pi } \;.
\label{divergence}   \end{equation}
This shows that the energy density of the system, after the standard regularization and renormalization of the coupling constant has been  made  to render   the
gap equation finite,  still contains a quadratic divergence. This is a little similar to the vacuum density in QCD:  the theory can be renormalized
 and all physical quantities can be calculated order by order in perturbation theory,   but the  vacuum energy density  (a contribution to the cosmological constant)
 is still divergent, and requires a further subtraction.
The result  (\ref{divergence}) however  suggests that we take the vacuum energy subtraction constant simply  as
\begin{equation}        {\cal E}_{\rm uv}      = -  \frac{N}{\pi \epsilon^2} \;, \label{subtract}
\end{equation}
 and the constant part of the energy density is thus
\begin{equation}    {\cal E}_{0} (\Lambda, L=\infty) + {\cal E}_{\rm uv} =    -  \frac{N  \Lambda^2  }{  4  \pi } \;.
\end{equation}
As \footnote{This follows both from the numerical results given in
  \cite{BKO}, and analytical calculations such as in the Appendices, as well as from the general observation that the generalized gap equation itself  reduces to the known  equation of the standard 2D ${\mathbb CP}^{N-1}$ model.
}
 \begin{equation} \lambda(x, \Lambda, L)    \to  \Lambda^2,    \qquad    \forall x, \quad  x \ne \pm \frac{L}{2}, \qquad  L \to \infty\;,   \label{lambdaL}    \end{equation}
one finds that the total  energy density approaches a constant
\begin{equation}     {\cal E}(x, \Lambda, L=\infty)   =       \frac{N}{ 4  \pi }  \Lambda^2 \; ,  \label{analogous}
\end{equation}
at any fixed finite  $x$.    This gives the quantum corrections \footnote{Our system can be  interpreted either as
a low-energy  effective action of the monopole-vortex soliton complex,
or  as just an ad hoc  ${\mathbb CP}^{N-1}$ model  defined on a finite worldstrip. In the first case,  the vortex energy scale (or  the vortex classical tension), plays the role of the UV cutoff.
One is interested in the effects of the  quantum fluctuations of the orientational zeromodes at lower energies, i.e., at length scales larger than the vortex width.  In the latter case, a UV cutoff is introduced
to renormalize the gap equation (the coupling constant renormalization) and to renormalize the vacuum energy.   From this latter point of view
(\ref{analogous}) is analogous to the vacuum energy density in QCD. }
due to the fluctuations of the $n_i$ fields to the classical  "tension" of the vortex,  $\xi\,$,   where   $\Lambda^2  \ll  \xi\,.$
This result is in agreement with the one \cite{Monin:2015xwa}, found in a finite worldstrip ${\mathbb CP}^{N-1}$ model with periodic boundary conditions.

As we shall see in the next section, and as can be verified by a WKB analysis done  in Appendix \ref{sec:WKB},  the divergence in the energy density  remains purely quadratic: the only subtraction needed is
(\ref{subtract}),  in the case of finite  ($L$)  string also.   No linear or logarithmic divergences are present.  This is reasonable as the divergences due to the fluctuations of the $n_i$ fields
is a short-distance effect, local in $x$, and  cannot depend on the presence of  the boundaries.

\section{Large but finite  $L$: an Ansatz and analytic calculation   \label{sec:Analytic} }

We now study the corrections to (\ref{lambdaL}) and (\ref{analogous}) (and $\sigma(0) =0$)  for  large but finite  $L$.
In order to do that, it is clearly necessary to analyze the large-$L$   behavior of (the solution of)  the gap equation, (\ref{gapeq}), (\ref{lambda}) itself.
To start with,   $\lambda(x, \Lambda, L)$ can be  approximately  taken to be a constant for $|x|\ll L/2$;  we parametrize its asymptotic approach to $\Lambda^2$ as
\begin{eqnarray}
 \lambda(0, \Lambda, L) =  \tilde \Lambda^2
\equiv  \Lambda^2 e^{2a},\qquad \lim_{L\to \infty}a(\Lambda L) =0\;. \label{tostart}   \end{eqnarray}
The factor $a$ represents  the nonlocal effect due to the boundary condition.
To estimate this, consider the propagator,  Eq.~(\ref{thetwo}).
For $x,x' \sim 0 $ (i.e., far from the boundaries), it satisfies locally
\begin{eqnarray}
 \left(\partial_\epsilon^2+\partial_x^2 -\tilde \Lambda^2\right)
D(x,\epsilon;x',0)\sim \delta(\epsilon)\delta(x-x')\;,   \label{byusing}
\end{eqnarray}
thus  its solution can be assumed to have the form
\begin{eqnarray}
 D(x,\epsilon;x',0)  &\sim&    \frac1{2\pi}
K_0\big(\tilde \Lambda \sqrt{(x-x')^2+\epsilon^2}\big)       \nonumber  \\
&&
-\frac{A}{2\pi} \left(K_0\big(\tilde \Lambda \sqrt{(x+x' +L)^2+\epsilon^2}\big)
+K_0\big(\tilde \Lambda \sqrt{ (L-x-x')^2+\epsilon^2}\big)\right)   \nonumber  \\
&& +\cdots   \label{usethis}
\end{eqnarray}
with an unknown constant $A=A(\Lambda L)$.\footnote{$K_0$ and $K_{1,2}$ below are the modified Bessel functions of the second kind.}  The subleading  terms in the second line come from
the nearest mirror poles in  Eq.~(\ref{Dequation}).\footnote{
Strictly speaking,  far from the boundaries,  the positions of the mirror poles might also be  effectively  shifted.
Dominant effects of their shift, however,  just rescale factor $A$ and we omit such shifts here for simplicity.
}
  It is expected that  $A=A(\Lambda L)\sim \mathcal{O}(1)$, but due to the effects of the boundaries where $\lambda(x)$ is non constant and singular \cite{BKO}
it will not coincide  with the exact value
$A=1$  (for the Dirichlet boundary condition at $x=\pm L/2$)  or  $A=-1$   (for the Neumann boundary condition)   (cfr.  Eq.~(\ref{both2}))  \cite{BKO}.

The gap equation (\ref{gapeq})  then  yields near $x=0$
\begin{eqnarray}
 \sigma^2(x)&\sim&  \frac{N}{2\pi}\log \frac{\tilde \Lambda }{\Lambda }
+\frac{NA}{2\pi} \left(K_0\big(2\tilde \Lambda (x+L/2) \big)+K_0\big(2\tilde \Lambda (L/2 -x)\big)\right)\nonumber  \\
&\simeq  & \frac{N}{2\pi}a+\frac{NA}{\pi} K_0(\tilde \Lambda L)
+\frac{2NA \tilde \Lambda^2}{\pi}K_0''(\tilde \Lambda L) \,
x^2+\cdots
\end{eqnarray}
Equation (\ref{lambda})  gives   $\lambda(x)$  near $x=0$    ($\sigma'\simeq 0$):
\begin{eqnarray}
 \lambda(x)= \frac{\sigma''(x)}{\sigma(x)}\sim         \frac{(\sigma(x)^2)''}{  2 \,   \sigma(x)^2}  \sim
\tilde \Lambda^2
\frac{2 AK_0''}{a/2+A K_0}\;.
\end{eqnarray}
By requiring the consistency  of this  with the initial Ansatz (\ref{tostart}) and making use of the identity
\begin{equation}  K_0(x)+K_2(x)=2K_0''(x)\;, \end{equation}
 one finds that
\begin{eqnarray}
 a= 2 A\, K_2(\tilde \Lambda L) =  2 A\, K_2(\Lambda L  e^a  )  \simeq      2 A\, K_2( \Lambda L)    \; ,
\end{eqnarray}
and
\begin{equation}      \tilde \Lambda^2
\equiv  \Lambda^2 e^{2a}  \simeq    \Lambda^2  ( 1+ 2a) \sim    \Lambda^2   \left(1+4 A K_2(\Lambda L)\right)\;.  \label{Lamdtild}
\end{equation}
Thus  one finds at  large $L \gg \tfrac{1}{\Lambda}$   that, around $x=0$,
\begin{eqnarray} \lambda(x, \Lambda, L) &\sim&  \lambda(0, \Lambda, L) \sim
\Lambda^2 \left(1+4 A K_2(\Lambda L)\right),\nonumber  \\
  \sigma^2(x, \Lambda, L) &\sim&
\frac{2NA}{\pi}K_0''(\Lambda L)\left(1+\Lambda^2  x^2   \right) \;.  \label{reflect1}
\end{eqnarray}

With these results in hand, one can now calculate the asymptotic behavior of the energy density   itself:
\begin{equation}    {\cal E}(x, \Lambda, L)=   {\cal E}_0(\Lambda, L)  +   \frac{N}{2\pi}  \lambda(x, \Lambda, L)  +  {\cal E}_{\rm uv}\;.  \label{result1}
\end{equation}
It turns out under the same approximation (\ref{usethis}) that the constant part of the energy density, ${\cal E}_0(\Lambda, L)$, is given at finite large $L$  by
\begin{equation}
 {\cal E}_0(\Lambda, L)
\sim     \frac{N}{\pi\epsilon^2}
-\frac{N \Lambda^2}{4\pi}
-\frac{N A \Lambda^2}{\pi}\left(K_0(\Lambda L) +K_2( \Lambda L)\right)  +{\cal O}(e^{-2\Lambda L})\;.
\end{equation}
The derivation of this result is given in Appendix~\ref{ConstEnergy}.    Note that the  divergence  in the energy density is just the  purely quadratic
one, $ \frac{N}{\pi\epsilon^2}$, the same as in the $L \to \infty$ case  discussed in the previous section.  This is correct, as  the divergences arise from the UV fluctuations of the
$n_i$ fields which is a local effect, independent of the boundaries, or of the value of $L$.

Finally one finds,  by adding the $\lambda(0, \Lambda, L)$ term and by
making the same subtraction as before, viz.~(\ref{subtract}),
\begin{eqnarray}
 {\cal E}(0, \Lambda, L)&=&\frac{N}{2\pi}\lambda(0, \Lambda, L)+{\cal
  E}_0(\Lambda, L)     +  {\cal E}_{\rm uv}   \nonumber  \\
&\sim& \frac{N\Lambda^2}{4\pi}
+\frac{NA \Lambda^2 }{\pi}(K_2(\Lambda L)-K_0(\Lambda L))+{\cal O}(e^{-2\Lambda L})\nonumber  \\
 &=&  \frac{N \Lambda^2 }{4\pi}  +
\frac{2N A \Lambda }{\pi L} K_1(\Lambda L)+{\cal O}(e^{-2\Lambda L})  \;,    \label{reflect2}
\end{eqnarray}
where another  identity
\begin{equation}  K_2(x)-K_0(x)=2K_1(x)/x \; , \end{equation}
 has been used.

To conclude, we find that the approach to the asymptotic value of the energy density   is  exponential: no pure power
corrections in  $1/L$  (i.e.~the L\"uscher  term)  are present.  This is perfectly consistent with the general result found in \cite{BKO} that our system has a unique phase,
which smoothly matches -- in the large $L$ limit --  the ``confinement
phase'' of the standard 2D ${\mathbb CP}^{N-1}$ model.  All $n_i$
($i\ne 1$)   fields gain a dynamically generated mass $\sim \Lambda$;  at the same time $\sigma \sim 0$
except at the boundaries. In other words, no dynamical breaking of the isometry group $SU(N)$  takes pace. No Nambu-Goldstone modes associated with the internal, orientational modes  are generated.   The absence of a long-range correlation in  the large-$L$ corrections in Eqs.~(\ref{reflect1}), (\ref{reflect2}) is a simple reflection of this fact.

\section{Numerical results}
\label{six}

Due to the quadratic divergence present in the sum
(\ref{EdensityConst})  the numerical calculation turns out to present
quite a bit of a challenge.  Any tiny errors in the eigenmodes and in the energy
levels will introduce   linear or logarithmic divergences in the sum,  and the finite answer  for
${\cal E}(x, \Lambda, L)$ one gets (including its dependence on $x$, $L$ and $\Lambda$)
depends on how these fake divergences are appropriately subtracted, together with the genuine quadratic divergences.   Because of this,  even the best results so far do not have
a precision comparable to  the solution of the generalized gap
equation discussed in Section~\ref{sec:gapeq}.

The check of the  constancy of  ${\cal E}_0$, is shown in
Fig.~\ref{E0const}. Note that it  is  found indeed to be constant
everywhere,
including values of $x$  very close to the boundaries,  where the distances from the latter  are much smaller than $1/\Lambda$.
Fig.~\ref{E0}  shows  the value of  ${\cal E}_0$ as a function of $L$ ($\Lambda=1$).
The total energy density, including the $N\lambda(x)/2\pi$ term,  calculated at the midpoint, $x=0$, is shown in Fig.~\ref{E}.
The numerical results are nicely consistent with the exact result at
$L=\infty$, and with the analytic behavior  for large but finite $L$, found  in the previous section, with $A \sim 1$.
 ${\cal E}_0$ and  ${\cal E}(0, \Lambda, L)$  are plotted against $\Lambda$ at fixed $L$,  in  Fig.~\ref{E0bis} and in Fig.~\ref{Ebis}.
In particular we see that as $\Lambda \to 0$ the energy density
converges, although quite slowly (i.e.~logarithmically),  to the free-field value $-\pi/12 L^2$.

\begin{figure}[!htp]
\begin{center}
\includegraphics[width=3.5in]{rw_E0xL5}\vspace{1em}\\
\caption{Verification of the constancy of
  $({\cal E}_0+{\cal E}_{\rm uv})$ with respect to
  $x$. $\Lambda=1$, $L=5$ in this figure. }
\label{E0const}
\end{center}
\end{figure}

\begin{figure}[!htp]
\begin{minipage}{0.49\linewidth}
\begin{center}
\includegraphics[width=3in]{rw_E0_ss}\vspace{1em}\\
\caption{The constant part of the energy density,
  ${\cal E}_0+{\cal E}_{\rm uv}$, is plotted as a function of $L$ for
  $\Lambda=1$ fixed.  }
\label{E0}
\end{center}
\end{minipage}\ \ \
\begin{minipage}{0.49\linewidth}
\begin{center}
\includegraphics[width=3in]{rw_E_ss}\vspace{1em}\\
\caption{The total energy density at the midpoint, ${\cal E}$, is plotted against $L$ for $\Lambda=1$ fixed. }
\label{E}
\end{center}
\end{minipage}
\end{figure}

\begin{figure}[!htp]
\begin{minipage}{0.49\linewidth}
\begin{center}
\includegraphics[width=3in]{rw_E0_ss_smallLambda}\vspace{1em}\\
\caption{The constant part of the energy density,
  ${\cal E}_0+{\cal E}_{\rm uv}$, is plotted as a function of $\Lambda$ for $L=1$ fixed.  }
\label{E0bis}
\end{center}
\end{minipage}\ \ \
\begin{minipage}{0.49\linewidth}
\begin{center}
\includegraphics[width=3in]{rw_E_ss_smallLambda}\vspace{1em}\\
\caption{The total energy density at the midpoint, ${\cal E}$, is plotted against $\Lambda$ for $L=1$ fixed. }
\label{Ebis}
\end{center}
\end{minipage}
\end{figure}

\newpage
\section{Boundary divergence and the Casimir force}
\label{seven}

The energy density
\begin{equation}    {\cal E}(x, \Lambda, L)=   {\cal E}_0(\Lambda, L)  +
\frac{N}{2\pi}  \lambda(x, \Lambda, L) + {\cal E}_{\rm uv} \;,  \label{result2}
\end{equation}
after renormalization is  a finite function of $x, \Lambda, L$.
When integrated over $x\in \left[-\frac{L}{2}, \frac{L}{2}\right]$,  it  gives the total energy of the string
\begin{eqnarray}   E(\Lambda, L) &=&   \int_{-L/2}^{L/2}  dx \;   {\cal E}(x, \Lambda, L) =
\int_{-L/2}^{L/2}  dx \, \left[  {\cal E}_0(\Lambda, L) +  \frac{\lambda(x, \Lambda, L)}{2\pi}\right] \nonumber \\
&=&  L\,  {\cal E}_0(\Lambda, L)    +     \int_0^{L/2} dx \;  \frac{\tilde \lambda(x, \Lambda, L)}{\pi}  \;, \label{totenergy}
\end{eqnarray}
where we introduced the mass gap function defined on the interval
$[0,L]$,
\begin{equation}
\tilde \lambda(x,\Lambda, L) \equiv \lambda(x-L/2,\Lambda,L)\,.
\end{equation}
The Casimir force is defined as \footnote{With this definition a positive ({\it vis a vis},  a negative)  $F$ corresponds to an attractive  (repulsive) force.}
\begin{equation}   F= \frac{\partial E(\Lambda, L)}{\partial L} \;.    \label{Casimir}
\end{equation}

Before analyzing the behavior of $F$ for various values of $L$,
let us note that the second term in Eq.~(\ref{totenergy})
 gives rise to a new divergence in the integrated energy  due to the singular behavior near the boundaries, e.g., near $x=0$,
\begin{equation}
\tilde \lambda(x) \sim \frac{1}{x^2\log 1/(x\Lambda)}\;, \qquad
L \gg  1/\Lambda  \gg  x \;.
\label{leaddiv}
\end{equation}
 As $\tilde \lambda(x)$  quickly approaches the constant value  $\Lambda^2$  beyond $x \simeq  1/\Lambda$, for sufficiently large $L$,  the divergent part can be extracted by considering the finite  integral
\begin{equation}    E_1  = \frac{N}{2\pi}\int_{\epsilon} dx\;    (\tilde \lambda(x)- \Lambda^2) \sim   \frac{N}{2\pi}    \frac{1}{\epsilon \log{1/\epsilon  \Lambda} }\simeq    \frac{N}{2\pi \epsilon  }   \frac{N g(\epsilon)^2}{8 \pi^2}  \;,  \label{FITWITH}
\end{equation}
where a UV cutoff  ($x=\epsilon$) has been introduced.  Similarly $E_2$ for the contribution from  the right boundary.
$E_{1}$  ($E_{2}$) is an energy concentrated at the left (right) boundary:  it can be  interpreted as the quantum corrections to the monopole  (antimonopole) mass,
due to the $n_i$ field fluctuations (the factor $N$).   $E_{1,2}$  can be subtracted (i.e., compensated with the bare mass terms)   from the total  energy,   leaving finite, renormalized monopole masses. They do not affect the discussion on the $L$ dependent Casimir effect below.

The Casimir force can be rewritten, by differentiating Eq.~(\ref{totenergy}),   as
\begin{eqnarray}   F &=&  {\cal E}_0(\Lambda, L) + L\, \frac{\partial{\cal E}_0(\Lambda, L)  }{\partial L}   +     \frac {\lambda(0, \Lambda, L)}{2\pi} + \frac{1}{\pi} \int_0^{L/2} dx \,   \frac{\partial\tilde \lambda(x, \Lambda, L)}{\partial L}  \nonumber \\
&=&  {\cal E}(0, \Lambda, L) + L\, \frac{\partial{\cal E}_0(\Lambda, L)  }{\partial L}  + \frac{1}{\pi} \int_0^{L/2} dx \,   \frac{\partial\tilde \lambda(x, \Lambda, L)}{\partial L} \;.\label{casforce}
\end{eqnarray}
Note that this is a finite quantity since it is not sensitive to the leading divergence of $\tilde \lambda$, viz.~(\ref{leaddiv}).

At very small $L$ (i.e.~$L\Lambda\ll1$) one expects the dominant effect to come  from the second term of (\ref{casforce})   (see Fig.~\ref{Ebis}):
\begin{equation}     F \simeq  L\, \frac{\partial{\cal E}_0(\Lambda, L)  }{\partial L}
\simeq \frac{N\pi}{6 L^2} \;:
\end{equation}
 it is an attractive free-field Casimir force.

At intermediate values,  $L\sim \mathcal{O}(1/\Lambda)$, instead, we find that the force is dominated by the third term in  (\ref{casforce}).  The strong decrease ($\sim 1/ L^2 \log (1/L)$)  of the mass gap $\tilde {\lambda}(x)$ with $L$ for all $x$ (see e.g., Eq.~(\ref{decrease})) cannot be compensated by a linear effect of integration, therefore  there is an effective repulsive force at work.  The total energy of the system is lowered when the space interval $L$ gets larger.

At sufficiently large $L$,  $L  \gg1/\Lambda$, where the $2D$ regime sets in,   the leading contribution comes from the  first term in (\ref{casforce}):
\begin{equation}   F\simeq  {\cal E}(0, \Lambda, L)  \simeq  \frac{N \Lambda^2
}{4\pi} \;,
\end{equation}
corresponding to an approximately constant string tension (Eq.~(\ref{analogous})).  An external observer who attempts to pull the boundaries further apart will experience an attractive, constant force countering her/him.

A precise numerical verification of this nontrivial behavior of the force turned out to be
exceedingly difficult because of the singular behavior of $\lambda(x)$
near the boundaries.  Our preliminary result (not shown)  however clearly confirms the change from a repulsive regime at $L =  \mathcal{O}(1/\Lambda)$ to an attractive force at
 $L \gg 1 / \Lambda$.

\section{Conclusion}
\label{eight}

In this paper we have examined the  energy density  function  ${\cal E}(x, \Lambda, L)$  of the  large-$N$  $\mathbb CP^{N-1}$ sigma model on a finite string, defined with the Dirichlet boundary conditions. We find that it is a sum of  two terms, the first  expressed as a sum over fluctuation modes, which turns out to be constant in $x$, and the second term proportional to the mass gap
$\lambda(x)$. The only $x$ dependence arises from the second.  The first term is quadratically divergent, analogous to the  vacuum energy in QCD.
The $L$-dependence of ${\cal E}(x, \Lambda, L)$  at fixed $\Lambda$
shows that the effect of the boundaries are limited to their vicinity of width $\sim 1/\Lambda$:  the system approaches quickly the standard 2D large-$N$  $\mathbb CP^{N-1}$ sigma model,
with dynamical generation  of the mass gap, and with no dynamical breaking of the  isometry group $SU(N)$.  In the small $L\Lambda$ limit, the system approaches
the classical  weakly-coupled $\mathbb CP^{N-1}$  model, as appropriate for the Dirichlet boundary conditions.

 The approach to  the limit
$ {\cal E}(x, \Lambda, L=\infty)   =       \tfrac{N}{ 4  \pi }  \Lambda^2 $   is  found to be  purely  exponential: no   power
corrections in  $1/L$  such as   the L\"uscher  term  are present.  This is perfectly consistent with the general result found in \cite{BKO} that our system has a unique phase,
which smoothly matches the ``confinement phase'' in the large-$L$ limit
of the standard 2D  ${\mathbb CP}^{N-1}$ model.  All $n_i$ ($i\ne 1$)
fields gain a dynamically generated mass $\sim \Lambda$;  at the same time $\sigma \sim 0$
except at the boundaries. In other words, no dynamical breaking of the isometry group $SU(N)$  takes pace, and no associated Nambu-Goldstone modes are generated.   The absence of the  long-range correlation reflects this fact.

Recently a  paper appeared \cite{Nitta:2017uog} in which
 some  analytical large-$N$  $\mathbb CP^{N-1}$ sigma model solutions
 for  inhomogeneous condensates are presented by a mapping to the Gross-Neveu model \cite{Flachi:2017cdo}. These solutions correspond to
 {\it periodic} boundary conditions and, as far as we can see,
  none of the solutions proposed there correspond to our system defined with the Dirichlet boundary conditions. It would certainly be very interesting if our type of solution could be found analytically with developments of these techniques in the future.

\subsection*{Acknowledgments}

We thank Jarah Evslin, Muneto Nitta, Misha Shifman and Ryosuke Yoshii
for discussions.
The work of S.~B.~is funded by the grant ``Rientro dei Cervelli Rita
Levi Montalcini'' of the Italian government.
The work of S.~B.~G.~was supported by the National Natural Science
Foundation of China (Grant No.~11675223).
K.~O.~is supported by the Ministry of Education, Culture, Sports,
Science (MEXT)-Supported Program for the Strategic Research Foundation
at Private Universities ``Topological Science'' (Grant No.~S1511006)
and by the Japan Society for the Promotion of Science (JSPS)
Grant-in-Aid for Scientific Research (KAKENHI Grant No.~16H03984).
The present research work is supported by the INFN special research
project grant, GAST (``Gauge and String Theories'').

When this paper was being prepared for submission we were informed by Muneto Nitta and Ryosuke Yoshii of their paper
\cite{NittaYoshiiNew}, which deals with similar problems as ours, and with some overlap.
 The boundary behavior for the gap function
and some other qualitative aspects of their solutions are different from ours.  Also, another paper  \cite{Pavshinkin}  just appeared,
 discussing a Grassmannian sigma model on finite-width world sheets.

\subsection*{Comments on Ref. \cite{NittaYoshiiNew}}

In Ref.~\cite{NittaYoshiiNew}, submitted to the ArXiv on the same day
as ours, the same system is analyzed in a different approach, and the
authors there claim to find analytic solutions for mass gap function
$\lambda(x)$ and for $\sigma(x)$, both in confinement and in Higgs
phases.

By imposing Dirichlet boundary conditions and solving the generalized
gap equations, we instead find the solutions (e.g., illustrated in
Fig.~\ref{fig1}) in a unique phase with mass gap for all values of
$L$, which smoothly approaches the well-known solution in the infinite
$L$  limit (the standard $2D$ ${\mathbbm C}P^{N-1}$ model). Our
solutions are moreover consistent with the classical
${\mathbbm C}P^{N-1}$ model in the $L \ll 1/\Lambda$ limit as
discussed in Sec.~\ref{section:near}.

It is possible that, if a Higgs-like solution (as in
Fig.~1(b) of \cite{NittaYoshiiNew}) would exist, it represents
an unstable solution, whereas our procedure necessarily picks up the
stable solution, and that actually the confinement-type solution is
always the stable one.

However, we find it difficult to make a proper comparison, as the
renormalization of the gap equation and the generation of the mass
scale $\Lambda$ are not explained in \cite{NittaYoshiiNew}.

The boundary behavior of the mass gap function and the field $\sigma$
given in \cite{NittaYoshiiNew} is powerlike, whereas the logarithmic
behavior found by us reflects the situation characteristic of a
finite-space-width system.  The system must compromise between the
$2D$ physics at $L \ge  1/\Lambda$ - the divergences of the $n_i$
field fluctuations and the generation of the mass scale $\Lambda$ -
and the classical limit to which the model must reduce correctly in
the $L \ll 1/\Lambda$ region, as explained in Sec.~\ref{section:near}.

As the physical values of $L$ (the space width) are not given in
reference to $1/\Lambda$ in \cite{NittaYoshiiNew}, in contrast to what
is done in the present paper, it is not clear to us which physical
values of $L$ their solutions in Fig.~1(b) or Fig.~1(a) refer to, for
instance.

As a consequence, it is unclear how and when (at which value of $L$)
the Higgs phase vacuum disappears, as $L$ is increased. Or, vice
versa, at which $L$, if $L$ decreases toward zero, the Higgs vacuum
takes over, if it does at all. The authors of \cite{NittaYoshiiNew} do
not give the criteria to decide which solutions should be chosen at
any given $L$.  As far as we can see, the analysis of the vacuum
energy density, as made in the present paper, has not been done yet
there.

redpajama_set_name: RedPajamaArXiv


text:
Boeing 777 was chartered to fly Meng Wanzhou home, Canada court hears as she seeks to have bail relaxed
A Boeing 777 was chartered by China's government to fly Huawei Technologies executive Meng Wanzhou back home last May, in the mistaken expectation that she was about to be freed, a Canadian court heard on Tuesday, in a dramatic hearing that featured testimony from Meng's husband for the first time.
Liu Xiaozong said he feared his wife might contract Covid-19 from the security guards assigned to prevent her fleeing while her extradition battle continues. As Meng's lawyers sought to relax her bail conditions by releasing her from their custody whenever she leaves her Vancouver mansion, Liu also said that their two children feared being identified by the public because of the guards' presence.
But a Canadian government lawyer tried to undermine Liu's professed concerns about the virus by highlighting aspects of Meng's lifestyle in Vancouver on bail that have previously gone unreported.
Under cross-examination, Liu described a Christmas dinner in which Meng's party of 14 people booked out an entire restaurant in apparent defiance of local pandemic rules, as well as shopping trips and gatherings at a home rented by Huawei employees.
The president of the security firm guarding Meng later described visits from a masseuse and an art teacher, and elaborated on Meng's "numerous" private shopping trips to high-end Vancouver boutiques.
Liu said he was aware that a large jet had been chartered from China Southern Airlines in May, ahead of a key ruling that Meng's lawyers believed would result in her freedom; she and her colleagues had even posed in a group for victory photos on the court steps, he said. But the May 27 ruling, on the issue of "double criminality", went against Meng and her extradition case was ordered to continue.
Meng Wanzhou, centre, leaves the Supreme Court of British Columbia during a break from a hearing in Vancouver on Tuesday. Photo: AP
Meng, the chief financial officer of Huawei and the daughter of company founder Ren Zhengfei, was arrested at Vancouver's airport on December 1, 2018.
The United States wants her to face trial in New York on charges she defrauded HSBC by lying to the bank about Huawei's business dealings in Iran, thus putting the bank at risk of breaking US sanctions on the Middle Eastern country.
Meng denies the allegations, and argues that the case is politically motivated.
Under terms of the C$10 million bail (US$7.8 million) granted in December 2018, Meng lives in a C$13.6 million home, one of two properties she owns in Vancouver, while she continues her battle against extradition.
Meng must abide by an 11pm-to-6am curfew, but is otherwise allowed to roam most of the city, so long as she stays away from the airport. Meng has to wear a GPS tracker on her ankle, and is accompanied at all times by the private guards from Lions Gate Risk Management. Meng pays for the guards, who chauffeur her around town in large black SUVs, but they are acting as officers of the court to prevent her escape.
Meng's GPS ankle monitor, seen as she leaves her home to attend a hearing at British Columbia Supreme Court, in Vancouver on November 27, 2020. Photo: The Canadian Press via AP
A lawyer for Meng, William Smart, said he was seeking the lifting of two bail conditions that require her to be accompanied by the guards outside the home.
But Canadian authorities consider Meng a flight risk, and say she has great resources at her disposal.
Canadian government lawyer John Gibb-Carsley, representing US interests in the case, said the private plane booked for Meng's would-be victory flight was a Boeing 777, capable of seating more than 360 people, and that it had been chartered with the assistance of the Chinese consulate. Liu said he did not know these details.
Liu, testifying in the Supreme Court of British Columbia, before Mr Justice William Ehrcke, said he and the couple's two children lived in Hong Kong. Canada's immigration department granted them exemptions from Covid-19 travel restrictions that allowed Liu to travel to Vancouver in October, and the two children in December.
But Liu said he objected to the presence of the guards, saying "I believe my wife is at increased risk of Covid-19" because they travelled in the same vehicle with her. He noted in an affidavit that was read in court that his wife was a thyroid cancer survivor who suffered from hypertension.
Additionally, Liu said, the presence of the guards troubles their 12-year-old daughter and 18-year-old son. They had previously enjoyed going to coffee shops, malls and playgrounds together, but the presence of the guards made this a "challenge".
Gibb-Carsley noted that even if it was challenging, there was nothing to prevent the family doing these activities under guard, and in any case, the pandemic made such activities less likely. Liu agreed to both points.
US deal for Meng Wanzhou could be a trap, says extradition expert
Liu said he did not like travelling outside with Meng and the Lions Gate guards because they attracted public attention. "Sometimes, I walk together with Sabrina on the street … it is easy to identify [us]," he said, using Meng's English name.
Passers-by tried to take their photo. "That's why the kids try to avoid being outside with Sabrina," he added.
But Gibb-Carsley questioned whether Liu was "truly concerned" about Meng catching Covid-19, since both he and the children had spent their 14-day quarantine periods in the same house as her. Liu said his concern was genuine.
Under further questioning, Liu said he, Meng and their children had attended a 14-person dinner at a Richmond restaurant on Christmas Day with her "team".
Gibb-Carsley suggested this was in breach of BC guidelines limiting gatherings to six people. "The restaurant was reserved only for Sabrina's team," said Liu. "There is no other people except Lions Gate."
Liu later testified that he and Meng had gone out for only four meals while she was on bail, three at the same Richmond restaurant and one at a Vancouver coffee shop, that was likewise closed to other customers.
Gibb-Carsley asked Liu if he was aware that Meng sometimes did "private shopping". Liu said he had been on one shopping excursion with her, in which dresses were booked for Meng to try on and she was served in "guest rooms special for her".
Liu agreed with Gibb-Carsley, who said Liu and Meng had also visited a home rented by Huawei employees. Under further questioning by Smart, Liu said he did not worry about Meng contracting Covid-19 from members of her Huawei team because "they come here, they only work with Ms Meng, and they don't have local families … they don't have other contacts." Some had also been vaccinated against Covid-19 he said.
Private security guards from Lions Gate Risk Management escort Meng Wanzhou as she exits her vehicle at her home on January 17, 2020. Photo: AFP
In a statement released on Monday, Canada's Justice Department said it opposed the bail relaxation.
"It is Canada's position that the conditions imposed on Ms Meng by the British Columbia Supreme Court at her initial bail application in December 2018 were necessary and appropriate, and there is no reason to vary the conditions at this time," the statement said.
Meng's home is in Shaughnessy, one of Canada's most expensive neighbourhoods. The 8,351-square-foot home – just two doors down from the official residence of the US consul-general – was recently renovated and features four bedrooms, eight bathrooms and a circular driveway.
The mansion is one of two homes Meng owns in Vancouver, a city where she once lived.
Meng Wanzhou's private jailers were hired to guard China's consulate too
The next witness at the hearing was Douglas Maynard, president of Lions Gate.
He said the primary risks involved in guarding Meng were that she would try to escape; that another nation or organised criminals would attempt her "extraction"; that she would be targeted for the purposes of extortion; or that she would be targeted for harm.
"We believe the risks for some of these factors is actually rising," Maynard said. As the end of the scheduled proceedings neared, the window for nefarious intervention was narrowing, increasing the relative risks, he argued.
Maynard said two employees of Lions Gate had tested positive for Covid-19. One played no role in the Meng deployment. The other worked in a control van outside Meng's home and did not have contact with her.
Asked if Meng had engaged in behaviour that concerned him regarding the risk of Covid-19, Maynard cited a restaurant dinner in Richmond on December 4 with 10 people in attendance, sharing food from common plates. On December 5, "nine or 10 people" attended a coffee shop, where some drank from the same cup.
He also mentioned the May 23 photo shoot, involving unmasked people.
Asked for a general account of Meng's activities on bail, Maynard said: "There's shopping downtown, there's an art teacher who comes to the residence, there is a masseuse who comes to the residence."
Meng's shopping trips to downtown Vancouver were "to some of the higher end stores. I won't name them…They provide private shopping opportunities."
Food was sometimes served and shared on these excursions, he said, which had once been "numerous", but not in recent months, under pandemic conditions.
Maynard also elaborated on the 777 charter. He said it had been booked by the Chinese government; although it was a private flight, the plane was too big to leave from the south side of the Vancouver airport terminal from where private jets typically depart. Instead, it was parked in readiness on the commercial north side of the terminal.
The bail hearing was adjourned by Ehrcke until Wednesday morning, while the extradition hearings are due to resume on March 1 before Associate Chief Justice Heather Holmes.
The current schedule for the extradition case includes hearings in the BC Supreme Court until May 14; appeals could extend the process for years.
Meng's arrest infuriated China and upended its relations with both the US and Canada. Two Canadians, Michael Kovrig and Michael Spavor, were arrested in China days later and accused of espionage, although Canada considers their arrests to be retaliatory and regards them as hostages.
Members of the media standing outside Meng's home on January 20, 2020, in Vancouver. Photo: AFP
Kovrig is being held in a Beijing detention centre, while Spavor is imprisoned in Shenyang, near the North Korean border. For most of their two years behind bars, Kovrig and Spavor were denied consular visits, but since October, Canadian Ambassador Dominic Barton has been granted "on-site virtual consular access" to the men, according to Global Affairs Canada.
In a handwritten letter published by The Globe and Mail newspaper last month, Spavor purportedly requested quick-drying clothes, deodorant and a sleep mask, as well as a long list of books, including Chinese-language instructional texts and "prison biographies".
Kovrig's wife told The Globe that he walked 7,000 steps a day by pacing around his three-metre-by-three-metre cell.
US may be trying to trap Huawei's Meng Wanzhou with deal, warns Canadian extradition expert
After weeks of courtroom drama, curtain falls on witness phase of Meng Wanzhou extradition battle
Police and FBI feared Huawei's Meng Wanzhou would escape via China's consulate in Vancouver. Then the consulate hired her private jailers too
This article Boeing 777 was chartered to fly Meng Wanzhou home, Canada court hears as she seeks to have bail relaxed first appeared on South China Morning Post

redpajama_set_name: RedPajamaCommonCrawl


Our new friend Omar came to the clinic hoping to get help with obtaining a prosthetic leg. He is a diabetic patient and lost his leg in 2013 as a result of a wound on his left foot. Sadly, this is an all too common scenario when a patient receives their amputation to save their life (which is fantastic) but are left with little hope of ever walking again. Many times these patients cannot afford a wheelchair as well.
We met him at the clinic and thought he would be a great candidate for a leg. He received his leg October 19, 2015 and has been walking ever since. You can see him in the photo with his daughter and his smile says it all. Experiencing their joy with them is one of the best parts of serving with Hope To Walk. From now on, Omar will not be limited by immobility or confined to a wheel chair.


text:
Driverless trucks show promise for the American supply chain, a possible threat to trucker jobs
By Thomas Fudge / Science and Technology Reporter
One day soon, you may be driving on the interstate and pull past a truck and notice there's no one driving it. A similar thing took place late last month. That's when a San Diego company called two simple announced what it called the world's first semi truck run on public roads with no driver in the cab. K P B S science technology reporter Thomas fudge tells us how robo trucks could strengthen the American supply chain and maybe threaten the livelihood of truckers
Robotics. Professor Henrick Christensen stands next to what looks like a golf cart. That's been delivering mail at UC San Diego without a driver to guided computer program, driverless vehicles like this one are becoming more common and Christensen says the new technology coupled with overloaded supply lines and a shortage of truck drivers are putting driverless trucks in the sites of a lot of companies. We have
This massive need for getting more out of the ports, getting it on the roads. That's where we are seeing these new. So we've seen multiple new companies come up that are actually working on this technology. Not only too simple, Aurora GM, uh, Teslas promising
This, but let's stick with that. San Diego startup simple it's driverless run last month from a rail yard in Tucson to a distribution center in Phoenix was a first for the company and they say a first for the industry. But working with partners like ups, the company is already delivering goods with its autonomous trucks provided a human safety driver is present. Pat Dillon is the chief financial officer of too simple. The
Vision of too simple is that for, from a major freight, uh, location, like a port like a rail yard, uh, we can take our custom built, uh, trucks that will have no driver in it to take a journey that could be a hundred miles or it could be a thousand miles and go from one freight location to another freight location.
The idea that Rob trucks would be the spear point of self-driving technology. Make sense to U C S D professor Christensen. He says, driving on highways is the easiest application for self-driving vehicles. And given the choice between trusting a human or trusting a robot at the wheel of a truck, the professor's view is pretty clear. The
Drivers get tired from these long hauls. Uh, and at the same time, you might not be aware of this, but when you have autonomous driving trucks, they actually save fuel. It's no surprise, really, but if you think about it, computer are actually better drivers than people,
Trucks park side by side, in many rows at this stop in O time Mesa, truckers, stop here for fuel a shower and some rest. Some of these drivers are resigned to the future of automation in the trucking industry, but they don't share the view that computers are better drivers than they are. Um, I think it's the way the
Industry's gonna go because of the lack of drivers that we do have in it. But I don't particularly
Care for it. Ron caplet, who hails from Pennsylvania says he's been driving a truck for 30 years.
I just think it's gonna be unsafe for a while. You know, they, you know, not being able to, you know, deal with the traffic conditions, the weather conditions, there's too many variables out here that maybe a computer won't be able to adjust to the same way. It just seems like it won't be able to adjust. Like we can adjust. They're
Gonna ask you if you need a professional about 10 miles away, students working to become registered truck drivers, listen to an instructor at the United truck driving school and mission valley. The school training coordinator is Phil Harris, who says he thinks automation is great. Will it replace all truckers? No way he says, but it'll change the nature of the job, making it necessary for truck drivers to understand the technology. We're gonna
Have someone in there as a technician, probably. So I can see us being called pilots, navigators, technicians, overseers, or maybe we'll be called a driver, but that's why we'll always be in that truck.
Driverless technology still needs some attention. Again, here's pat Dillon of too simple things
Like redundant breaking and steering so that, uh, you always have the ability to control the vehicle. Even if there is some type of AEN, uh, a system
Degradation, one student at the United truck driving school, Antoine Rackley says if the job of being a truck driver changes, that's okay with him advancement.
I'm trying to stay on top of, so it's, you know, you, you either keep up or you get left behind. So if that's the way the world change and you
Gotta change it with it, Thomas fudge, K PBS news.
One day soon you may be driving on the Interstate and pull past a truck, and notice there's no one driving it. A similar thing took place late last month. That's when a San Diego computer tech company called TuSimple announced what it said was the world's first semi-truck run on public roads with no driver in the cab. KPBS Science and technology reporter Thomas Fudge tells us how robotrucks could strengthen the American supply chain… and maybe threaten the livelihoods of truckers.
Professor Henrik Christensen is the director of robotics at UC San Diego, and he stands next to what looks like a golf cart that's been delivering mail at UCSD ... without a driver to guide it.
Tall, trim and spectacled, Chistensen speaks with the accent of his once-native Denmark. And he's a good evangelist for driverless technology who says children, born today, will never have to learn to drive a car.
But he's not focused just on the needs of daily commuters. Christensen says the new technology, paired with America's overloaded supply chains and a shortage of truck drivers have put driverless trucks in the sites of a lot of auto and computer tech companies.
Become a KPBS sponsor
"We have this massive need for getting more out of the ports. Getting it on the roads. So we've seen multiple new companies come up that are actually working on this technology. Not just TuSimple. Aurora. GM. Tesla is promising this," Christensen said.
The San Diego company he mentions, TuSimple, announced late last month what it said made the world's first semi-truck run on public roads without a driver in the cab or human intervention … done with its technology. It was another step toward fully automating freight hauling in trucks.
It's something that promises to strengthen the American supply chain for products and materials. It may also threaten the livelihoods of a lot of truck drivers.
TuSimple's driverless run last month, from a railyard in Tucson, Arizona to a distribution center in Phoenix, was a first for the company. But working with partners like UPS, it is already delivering goods with it's autonomous truck computer programs, provided a human safety driver is present. TuSimple's Chief Financial Officer is Pat Dillon .
Mike Damron
Robotics professor Henrik Christensen at UCSD stands in front of a driverless vehicle that's been delivering mail on campus. Dec 7, 2021.
"The vision of TuSimple is that from a major freight location like a port, like a railyard, we can take our custom-built trucks that will have no driver in it to take a journey that could be a hundred miles or a thousand miles and go from one freight location to another freight location," Dillon said.
TuSimple went public in April. Dillon said their initial public offering, or IPO, raised $1 billion in cash. The markets put the company's value at $8.5 billion.
The idea that trucks would be the spearpoint of self-driving technology makes sense to Christensen. He says driving on highways is the easiest application for self-driving vehicles. And given the choice between trusting a human or trusting a robot at the wheel of a truck, the professor's view is pretty clear.
"Drivers get tired from these long hauls. And at the same time ... when you have autonomous driving trucks, they actually save fuel," he said. "It's no surprise, really, but when you think about it computers are better drivers than people."
Truckers today see it coming
Trucks park side by side in many rows at this truck stop in Otay Mesa, where truckers stop for fuel, a shower and some rest. One trucker who lives in Bakersfield said he was waiting for permission to enter Mexico, where he will deliver his cargo.
Truck driver Juan Martinez, from Bakersfield, sits in his cab while stopping for lunch at an Otay Mesa truck stop. Jan 5, 2022.
He and the other drivers at the truck stop seem resigned to the future automation of the trucking industry. But they don't share the view that robotrucks can do their jobs better.
"I think it's the way the industry is going to go because of the lack of drivers we do have in it. But I don't particularly care for it," said Ron Caplette, from Pennsylvania. He said he's been driving a truck for 30 years.
"I just think it's going to be unsafe for a while." he said. "You know, not being able to deal with the traffic conditions and the weather conditions. There are too many variables out there that maybe a computer won't be able to adjust to the same way."
About 10 miles away, students working to become registered as truck drivers listen to an instructor at the United Truck Driving Driving school in Mission Valley. The school training coordinator is Phil Harris, and he says he thinks automation is great, to a point.
He spoke of a trucking company called Werner, based in Omaha, Nebraska, that already employs a lot of automation in its fleet.
"Werner trucks will (automatically) stay in their lanes. It will turn on the brakes by itself." Harris said. "And it's great for us drivers. There are times when we have a lapse of attention. We get distracted. And when you're hauling weight, you need something else to help you out. So if you can set a car length distance, that could be good."
Will automation get rid of truck drivers? No way, Harris says. But it'll change the nature of the job, making it necessary for them to understand the technology.
"How are you going to maintain that? How are you going to upkeep it? Then if it breaks down," he said, "you're going to need somebody in there as a technician, probably. So I can see us being called navigators, pilots, technicians, overseers, whatever! We might not be called the driver but that's why we'll always be in that truck."
Phil Harris is the training coordinator for United Truck Driving School in San Diego. He sits at his desk in the school's Mission Valley office. Jan 5, 2022
The driverless technology for trucks still needs some attention, according to TuSimple's Dillon.
One day soon, you may be driving on the interstate and pull past a truck and notice there's no one driving it. A similar thing took place late last month. That's when a San Diego company called too simple, announced what it called the world's first semi truck run on public roads with no driver in the cab. K P B a science and technology reporter. Thomas fudge tells us how robo trucks could strengthen the American supply chain and maybe threaten the livelihoods of truckers
Robotics. Professor Henrick Christensen stands next to what looks like a golf cart. That's been delivering mail at UC San Diego without a driver to guy. I did computer program, driverless vehicles like this one are becoming more common and Christensen says the new technology coupled with overloaded supply lines and a shortage of truck drivers are putting driverless trucks in the sites of a lot of companies. We have this
Massive need for getting more out the ports, getting it on the roads. That's where we are seeing these new. So we've seen multiple new companies come up that are actually working on this technology. Not only too simple, Aurora GM, uh, Tesla promising this,
But let's stick with that. San Diego startup simple it's driverless run last month from a rail yard in Tucson to a distribu center in Phoenix was a first for the company and they say a first for the industry. But working with partners like ups, the company is already delivering goods with its autonomous trucks provided a human safety driver is present. Pat Dillon is the chief financial officer of two simple.
The vision of two simple is that from a major freight, uh, location, like a port like a rail yard, uh, we can take our custom built, uh, trucks that will have no driver in it to take a journey that could be a hundred miles or it could be a thousand miles and go from one freight location to another freight location. The
Idea that robo trucks would be the spear point of self-driving technology makes sense to U C S D professor Christensen. He says, driving on highways is the easiest application for self-driving vehicles. And given the choice between trusting a human or trusting a robot at the wheel of a truck, the professor's view is pretty clear. The drive
Get tired from these long hauls. Uh, and at the same time, you might not be aware of this, but when you have autonomous driving trucks, they actually save fuel. It's no surprise, really, but if you think about it, computer are actually better drivers than people,
Trucks park side by side, in many roads at this truck, stop in O time Mesa, truckers, stop here for fuel a shower and some rest. Some of these drivers are resigned to the future of automation in the trucking industry, but they don't share the view that computers are better drivers than they are. Um, I think it's the
Way the industry's gonna go because of the lack of drivers that we do have in it. But I don't particularly care
For it. Ron caplet, who hails from Pennsylvania says he's been driving a truck for 30 years. I
Just think it's gonna be unsafe for a while. You know, they, you know, not being able to, you know, deal with the traffic Condit, the weather conditions, there's too many variables out here that maybe a computer won't be able to adjust to the same way. It just seems like it won't be able to adjust. Like we can adjust. They're gonna ask you
If you need a professional about 10 miles away, students working to become registered truck drivers, listen to an instructor at the United truck driving school and mission valley. The school training coordinator is Phil Harris, who says he thinks automation is great. Will it replace all truckers? No way he says, but it'll change the nature of the job, making it necessary for truck drivers to understand the technology. We're gonna have someone in
There as a technician, probably. So I can see us being called pilots, navigators, technicians, either whatever. Maybe we won't be called a driver, but that's why we'll always be
In that truck. Driverless technology still needs some attention. Again, here's pat Dillon of too simple things like
Redundant breaking and steering so that, uh, you always have the ability to control the vehicle. Even if there is some type of AEN, uh, a system degradation,
One student at the United truck driving school, Antoine Rackley says if the job of being a truck driver changes, that's okay with him
Advancement. I'm trying to stay on top of, so it's, you know, you, you either keep up or you get left behind. So if that's the way the world change and you gotta change it with it.
Joining me is K P V S science and technology reporter, Tom and fudge, and Tom, welcome to the program.
Hi Maureen.
Now the first truck run completely without a driver took place last month in Arizona. And in your report, ups is already using some driverless trucks.
They are according to, uh, my sources of two simple, which is a San Diego ago, startup, which started up in 2015. They have a partnership with ups and they are delivering goods, you know, as we speak with automated trucks, but it's not entirely automated. There's still a steering wheel in the cab of the truck. And there is a driver. There are sort of a safety driver, I guess you would say.
So what's the aim is, is the two simple company planning to put a fleet of trucks on the road without any drivers at all, or are they planning to keep it the way it is now to keep backup drivers in the truck?
Well, there will be backup drivers for a while. When I asked, uh, the CFO of too simple, sort of what their goal is, he told me they want to be selling driverless trucks in about five years. And I think their goal is to have trucks run without a driver in them, until that is something that is acceptable and safe. And there are still some issues that they need to iron out. Uh, one thing he talked about was redundant breaking systems, just to make sure that the breaks still work, if there's some degradation in the existing system. And so there's still some safety concerns. There is also a li and this is the case, not just with trucks, but with driverless vehicles all, all together, uh, when a driverless vehicle gets into an accident, whose fault is it? Is it the fault of the owner of the vehicle? Is it the fault of the computer system? So there are still some stuff they need, some things that they need to work out.
What is the underlying reason for the development of robo trucks without drivers? Is it for the trunking companies to save money? Is it to cut down on accidents? What, why do they wanna do this?
Well, I think both of those things are true. Although we should point out that, uh, these computerized trucks are going to be quite expensive. I can't give you a figure, but, uh, they will save a lot of money on labor because it won't require at least the way they look at it. It won't require a driver to be there. So it should be a cost savings. And, uh, a robotics professor that I spoke to at U C S D said to me, let's face it. Computers are better drivers than he humans are. Now. That's not an opinion that all truckers agree with, but that's what he said.
Is there a benefit for the public though? Tom are, I don't know, prices supposed to come down. Is there any reason that the public should wanna see these driverless trucks?
Well, I think, uh, we have seen in recent years, there is a shortage of truck drivers, and that means that our supply lines get bogged down with automated trucks. You're automating a system where there's a labor shortage, which is what industry very often does. And, uh, when we get into this business, uh, the supply lines will move more quickly. There won't be as great a need for drivers eventually that will get products to consumers more quickly, it'll get, uh, materials to builders more quickly, and there will be less in terms of labor cost.
And what do the truckers unions have to say about driverless trucks?
Well, well, I didn't interview anybody with, uh, the unions, but I did dig up a statement from the Teamsters union. And, uh, as you might expect, they are not crazy about it. Uh, it says right here at the top, autonomous trucks threatened the livelihoods of millions of truckers across the country. It goes on to say, this would be devastating in this well paying field where more than 93% of the workers have less than a college degree. So the Teamsters are clearly, if not opposed to it very concerned about it. And they are lobbying the government to, if not slow down the system, at least make sure it's very well regulated.
Despite the optimism you heard from the truck driving students that you spoke with, this does sound like it will decrease the need for drivers. And can you tell us again what the timeframe is that too simple and other companies like it would like to see for the automation of the industry?
Well, what the timeframe is when this is going to happen, kind of depends on who you talk to. A couple of truck drivers. I talked to at a truck stop in O time Mesa expected it to come in. About 10 years, the CFO of two simple said that they hoped to be selling their trucks to company is like ups. In about five years, the job of a truck driver is going to change and they are going to have to start to understand this technology better. What their role will be is kind of hard to say at this point. Um, maybe they won't be drivers. Maybe they will be computer technicians, but the guy I talked to at the school there will always be in his view. There will always have to be a person in the cab of that truck.
And I've been speaking with KPBS science and technology reporter, Thomas fudge, and Tom, thank you so much.
Thank you very much, Maureen.
"Things like redundant braking and steering," he said. "So you always have the ability to control the vehicle, even if there is a system degradation. Making sure you have a supercomputer that controls all elements of the sensor input and actually controlling the vehicle."
And if the nature of being a truck driver changes, Antowain Rackley, one of the students at the United Truck Driving School, said that's alright with him.
"I'm always open to learning. And all new advances, I'm trying to stay on top of," Rackley said. "You either keep up, or you get left behind. So if that's the way the world changes, you've got to change with it."
Thomas Fudge
A journalist with 30 years of experience, Tom covers science and technology stories for KPBS' platforms. He has received recognition for his outstanding work in hosting and public affairs reporting from the Unity Awards, the Northwest Broadcast News Association, and the San Diego chapter of the Society of Professional Journalists.
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text:
Q: How do I apply int to every string using map and lambda in python? I want to avoid doing ugly int to each element in a list.
dictionary[l[0]] = [(int(l[1])+ int(l[2])+ int(l[3]))/3] #avg score of 3 grades
    sub1 += int(l[1])
    sub2 += int(l[2])
    sub3 += int(l[3])

The list l has 4 elements, 0 to 3 indexes. I need to apply int starting with the 1st-element, not 0th element.
I'm trying to write something like:
   map lambda x: int(x) for i in l[1:]

Do I need lambda here, since map can already do the job?

A: map calls a function on each element of a list. Since int is a function you can use
map(int, lst)

directly.
Lambda functions just give you more flexibility, say if you wanted to do:
map(lambda x: 5*int(x), lst)

A more Pythonic way to do the above would be to use list comprehensions
[int(x) for x in lst]
[5*int(x) for x in lst]

In Python3 map returns a generator. If you wish to have a list, you'd want to do:
list(map(int, lst))

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Energy > Our Reviews
Co-operative Energy Review
Robin Bowman
Another relatively new energy supplier, launched to take a slice of the supply cake from the Big Six.
Co-operative Energy is part of the Midcounties Co-operative – the country's third biggest co-operative. It now has more than 504,000 members.
COULD YOU BE SAVING WITH CO-OPERATIVE ENERGY?
The organisation offers energy nationwide and has no shareholders – instead members get a share of its profits.
As with the other smaller operators, the Co-op's offering says it wants to keep costs to a minimum and tariffs simple, all to benefit customers.
Sound good? Maybe.
But Co-operative Energy also bills itself as something else.
It claims to put principles before profit.
If your supplier isn't treating you well, ditch them today
Co-operative Energy say, "That means only making decisions with you in mind, always offering fair prices, and doing our bit to combat climate change too."
That's a bold claim. Certainly being all ethical is a good thing, especially for our Co-operative Energy review. But it's not so great if your service is poor.
The fact is that Co-operative Energy has a lot of work to do to win back the trust of the UK public – and it's going to take a lot more than the long list of glowing reviews and testimonials on its website.
Most complained about Energy company
Because it has the distinction of being an award winner – it has actually outshone the Big Six energy suppliers on at least one front.
Last year it was the most complained-about energy supplier in the country – the first of the new guys to beat the Big Six in this regard.
The company was plagued by complaints and bad reviews from customers about inaccurate bills, mistakes over energy switches and problems with logging into its website. The IT failure was caused after it brought in a new billing system last summer.
And it was warned that if it didn't sort things out it would be banned from recruiting new customers.
Data released by the Ombudsman showed that Co-op Energy had received a market leading 136 complaints for every 100,000 customers – and these were only those complaints that had been escalated to ombudsman level after the company had failed to resolve them in a six-week time-frame.
Billing problems are one the most common energy complaints
A word of caution – and not just about Co-op Energy
But it's worth pointing out that many industry experts do not accept that such a rise in complaints can be wholly put down to IT glitches.
The real reason, they argue, is one that can apply to any of the new energy companies and it's about capacity.
As these smaller companies try to grow fast, they naturally try and compete hard on price. One of the consequences is that when they market an especially attractive price, they receive a sudden influx of new custom.
And the stark truth is that they are not equipped to cope with the sudden increase in volume.
Chief ombudsman Lewis Shand Smith said: "Small firms are growing at a pace that is faster than they expect, and as a result find it difficult to cope when they do receive a complaint.
"Meanwhile the Big Six have improved significantly after investing heavily in complaints handling."
Did you know? An energy provider that finds itself at the top of the best price league table can expect to bring in 50,000 new customers a week.
That might be worth bearing in mind when making your choice of energy provider.
Don't put up with bad service. Switch and save
Ethical credentials
If you care about more than just landing the best price for your energy, and if you believe ethics is not just the next county to Sussex, then Co-operative Energy may well score highly when you come to choosing supplier.
Because one of its big selling points is its ethical credentials.
As Co-operative Energy says: "What other energy company can say their decisions are made in your best interest? The Big 6 energy suppliers are owned by shareholders; when they make a profit, it is the shareholders who benefit, not the customers. However, when we make profit, it is you, our members, and your community who benefits.
"As members, you are the owners of The Co-operative. You have a say in how the business is run, decide how much profit is shared and reinvested, as well as share any profits that we make. And, to say thank you, we reward our members once a year with a share of the profits in proportion to how much business they do with us."
On top of that, this is an outfit that actually believes in paying a fair share of taxes!
"As a values led and ethical organisation, we believe that we should pay the taxes which are due and not engage in aggressive tax avoidance schemes, even if legal."
COMPARE QUOTES TO SEE WHAT YOU CAN SAVE
What about green credentials?
Co-operative Energy doesn't make a big sell of how green it is – it focuses on ethics instead.
But it does say that it commits to ensuring that the renewable content of the energy it provides is at least better that the Big Six.
So, where does its energy actually come from?
* 36.6% Coal
* 34.24% Natural Gas
* 13.43% Nuclear
* 9.77% Renewables
* 5.96% Other
So, under 10% from green sources, or over 23% if you count nuclear as green.
Co-operative Energy says it is "committed to a sustainable world, and this is reflected in our ambition to obtain 75% of our energy from renewable sources within the next three years."
Under 10% of the Co-operative's energy is from renewals
Co-op Energy Customer Service
We've already mentioned the problems Co-operative Energy faced last summer, problems that made it the most complained about energy company.
But the number of bad reviews to the Ombudsman it attracted is one thing, what about other aspects of customer service?
Not great unfortunately. It came 20th out of 22 energy companies in the Which? customer survey.
For customer service it scored just two stars out of five, two out of five for clarity and accuracy of bills and just one star for the way it dealt with complaints. Its best ratings, of three stars each, came for helping customers save money and for value for money.
It's overall rating was a lacklustre 45%.
Update: In late October, Co-op energy agreed to pay £1.8m in compensation to customers for shocking customer service. Here's the full story.
Sick of not being treated fairly? Switch today
Co-op Energy Tariffs
A number of options are currently offered.
Two fixed tariffs until March 2017 – Prices, as the name suggests, are guaranteed until the end of March 2017. A cancellation fee of £30 applies per fuel – so £60 for both gas and electricity – for the paperless billing version.
There's also a fixed tariff until the end of April 2017 without exit fees and with a choice of paper bills or paper free.
The Pioneer Variable is equivalent of a standard tariff with variable prices.
To find prices offered you'll need to answer a series of questions about energy use and the type of property you live in.
Generally speaking, you're unlikely to find Co-operative Energy to be the cheapest energy provider on the market.
UPDATE: In September 2016, Co-op Energy announced a 3% gas and electricity price rise (6% if you have a prepayment meter) would kick in from October. It's the only supplier so far this autumn to raise prices.
What about its online offering?
We rate its website as pretty average. It's reasonably clear and easy to navigate and it's mobile version is very accessible.
You should have no trouble finding your way around the site to pay bills, offer readings, or move home.
Its guides on ways to pay, reading your meter, providing a meter reading and so on are well laid out and easy to understand.
A Spokesman Said's view
Co-operative Energy has, in some ways, in the past few months been a victim of its own success, being unable to deal with the tidal wave of new customers last year.
This led to a certain amount of chaos that has reflected badly on the Co-op's reviews and customer service rankings. Whether this has been fully sorted out is unclear.
Generally probably not the cheapest offering on the energy market.
Like the other smaller players, tariffs are relatively simple and clear.
While there are no extra incentives to switch to Co-operative Energy, other than finding it offers you a better deal than other providers, it does at least attempt to behave ethically.
For many customers, this could be a serious consideration. The bottom line is it's a co-operative not a company, so if that appeals this may be the provider for you.
They claim they won't do what pretty much every other energy provider does - offer great cut-price promotions for a limited time and then increase prices. Instead, they claim, their intention is to save the customer money in the long term.
SEE HOW CO-OPERATIVE'S PRICES COMAPRE TO THE REST
This guide was updated on 28.10.16 to include news of Co-op Energy paying compensation to customers for poor service.
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redpajama_set_name: RedPajamaGithub