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\title{Anomaly in conformal quantum mechanics:
\\
From molecular physics to black holes}


\author{ Horacio E. Camblong$^{1}$
and
Carlos R. Ord\'{o}\~{n}ez$^{2,3}$}

\affiliation{
$^{1}$
Department of Physics, University of San Francisco, San
Francisco, CA 94117-1080
\\
$^{2}$ Department of Physics, University of Houston, Houston,
TX 77204-5506
\\
$^{3}$
World Laboratory Center for Pan-American Collaboration in Science and
Technology,
\\
University of Houston Center, Houston, TX 77204-5506
}


\begin{abstract}
A number of physical systems exhibit asymptotic conformal invariance.
Within a particular range of distances, they are characterized by an inverse square 
potential, the absence of dimensional scales, and an SO(2,1) symmetry algebra. Examples 
ranging from molecular physics to black holes are provided. For such systems, 
 quantum symmetry breaking---revealed by the failure of the symmetry 
generators to close the algebra---is shown
to be independent of the renormalization procedure.
\end{abstract}
\pacs{11.10.Gh, 11.30.Qc, 03.65.Fd, 11.25.Hf}
\maketitle


{\em Introduction.\/}---An
anomaly is the symmetry breaking of a classical invariance at the 
quantum level. This intriguing phenomenon has
played a crucial role in quantum field theory and particle 
phenomenology since its  discovery in the 1960s~\cite{tre:85},
as well as in the study of conformal invariance in  string theory.
Surprisingly,
the presence of
 an infinite number of degrees of freedom
does not appear to be a prerequisite for the emergence of anomalies.
This fact was first recognized 
within a model with conformal invariance:
the two-dimensional contact interaction
 in quantum mechanics~\cite{jackiw:91-beg}.
In conformal quantum mechanics,
a physical system is classically
invariant under the most general 
combination of the following time reparametrizations:
time translations,
generated by the Hamiltonian $H$;
 scale transformations, generated by the
 dilation operator
$
D
\equiv
tH
- 
\left( {\bf p} \cdot {\bf r}
+  {\bf r} \cdot {\bf p}
\right)/4$; and 
translations of reciprocal time,
generated by the 
special conformal operator
$
 K
\equiv
2t D -
t^{2}H
+ 
m r^{2}/2
$.
These generators define
the noncompact
SO(2,1) $\approx$ SL(2,R) Lie algebra~\cite{wyb:74} 
\begin{equation}
[D,H]
= - i \hbar H
 \;  ,
\; \;  
[K,H]
= - 2 i \hbar D
\;  ,
\; \;  
[D, K]
=  i \hbar K
\;  .
\label{eq:naive_commutators}
\end{equation}
This symmetry algebra has also been recognized in:
 the free nonrelativistic particle~\cite{niederer-hagen:nr-SO(21)},
 the inverse square 
potential~\cite{jackiw:72,alfaro_fubini_forlan:76},
the magnetic
monopole~\cite{jackiw:80},  the magnetic vortex~\cite{jackiw:90},
and  various nonrelativistic quantum field 
theories~\cite{niederer-hagen:nr-SO(21),jackiw-bergman:nr-SO(21)}.
Furthermore,
conformal quantum mechanics has 
been fertile ground for the study of 
singular potentials and
renormalization,
using Hamiltonian~\cite{gup:93, camblong:isp-dt,beane:00}
as well as path integral methods~\cite{pi_collective}.


The main goals of this paper are: 
(i)  to illustrate the relevance 
of conformal quantum mechanics
to several physical problems,
from  molecular physics to black holes;
(ii)
to show the details  of the breakdown of the 
commutator algebra~(\ref{eq:naive_commutators})
for the inverse square potential.

{\em Relevant Physical Applications.\/}---In recent years,  
diverse examples of systems have been studied from the viewpoint
of the conformal algebra~(\ref{eq:naive_commutators}),
assumed to be a representation of
an approximate symmetry
within specific scale domains.
In the applicable conformally-invariant domain, the relevant physics is
described by a $d$-dimensional  effective Hamiltonian
\begin{equation}
H= \frac{ p^{2}}{2m}  - \frac{g}{r^{2}}
\; . 
\label{eq:ISP_Hamiltonian_unregularized}
\end{equation}
In the problems discussed below,
$\lambda = 2m g/\hbar^{2}$
is the dimensionless form of the coupling constant and
$\nu = (d-2)/2$; furthermore,
 the choice $\hbar = 1 = m$ will be made for the problem involving black holes.
In all cases,
the strong coupling regime is defined by the condition $g \geq g^{(*)}$,
with a critical dimensionless coupling 
$\lambda^{(*)} = (l + \nu)^{2}$ (for 
angular momentum $l$)~\cite{camblong:isp-dt}.

A generic class of applications 
arises from the near-horizon conformal 
invariance of black holes,  its impact on
their  thermodynamics~\cite{carlip_solodukhin:near_horizon},
and its extension to superconformal 
quantum mechanics~\cite{superconformal}.
In particular, analyses  based on the interaction~(\ref{eq:ISP_Hamiltonian_unregularized})
have been used to  explore horizon states~\cite{gov:BH_states,gupta:BH}
and to shed light on black hole thermodynamics~\cite{gupta:BH}.
Another class of current
applications~\cite{gupta:calogero} involves a many-body generalization
of Eq.~(\ref{eq:ISP_Hamiltonian_unregularized}):
the Calogero model,
which has also been directly linked to
black holes~\cite{calogero_black_holes}. 
These remarkable
 connections
seem to confirm the conjecture that it is the horizon itself
that encodes the quantum properties of a black hole~\cite{thooft:85}.
In this context, 
we consider the spherically symmetric 
Reissner-Nordstr\"{o}m geometry 
 in $D$ spacetime dimensions,
whose metric
\begin{equation}
 ds^{2}
=
- f (r) \,  dt^{2}
+
\left[ f(r) \right]^{-1} \, dr^{2}
+ r^{2} \,
 d \Omega_{D-2}
\; 
\label{eq:RN_metric}
\end{equation}
is minimally
coupled to a
 massless scalar field $\phi(x)$ 
with action ($c=1$ and $\hbar=1$) 
\begin{equation}
S
=
-
\frac{1}{2}
\int
d^{D} x
\,
\sqrt{-g}
\,
g^{\mu \nu}
\,
\partial_{\mu} \phi
\, 
\partial_{\nu} \phi
\; .
\label{eq:massless_scalar_action}
\end{equation}
In Eq.~(\ref{eq:RN_metric}),
$ d \Omega_{D-2}$
stands for the metric on the unit $(D-2)$-sphere,
$
f (r)  =
1
-
2
\left( a_{M}/r \right)^{D-3}
 +
\left( b_{Q}/r \right)^{2(D-3)}
$,
and the lengths $a_{M}$ and $b_{Q}$ are determined from the
 mass $M$ and charge $Q$
of the black hole respectively~\cite{mye:86}.
In this approach, the conformal structure is
 revealed by a two-step procedure consisting of:
(a) a reduction to an effective 
Schr\"{o}dinger-like equation,
to be analyzed in its frequency ($\omega$)
components;  (b)
the introduction of a near-horizon 
expansion in the variable
$x= r -r_{+}$ 
[with $r=r_{\pm}$ being the roots of $f(r)=0$, and $r_{+} \geq r_{-}$].
Two fundamental
facts arise from this reduction for the nonextremal case ($r_{+} \neq r_{-}$).
First,
the ensuing effective problem is described by the
interaction $V(x) \sim -\lambda/x^{2}$, which is conformally invariant
with respect to the near-horizon coordinate $x$ and represents
 a one-dimensional $(d=1)$ realization of the inverse square potential.
Second,
 the coupling constant 
$\lambda =
1/4 + [\omega/ 
 f'(r_{+})]^{2} 
$ 
is  supercritical because
$\lambda^{(*)} = 1/4$ for $d=1$. Then, the relevant physics occurs 
in the strong coupling regime~\cite{camblong:isp-dt},
in  which the framework  discussed in this paper
can be applied.

In a similar manner, the interaction between an electron and a polar molecule
can be effectively described with an anisotropic 
three-dimensional ($d=3$) inverse square potential 
$V({\bf r}) = -g \cos \theta /r^{2}$.
After an appropriate separation of variables,
an effective isotropic inverse square potential 
yields nontrivial physics~\cite{molecular_dipole_anomaly}.
The central issue in this analysis is
the existence of a conformally-invariant
domain whose ultraviolet boundary leads to the anomalous
emergence of bound states via renormalization. This 
simple fact alone captures the essence 
of the observed critical
dipole moment in polar molecules.

In short, the essential feature shared by the problems discussed above
is the existence of an {\em effective\/} description in terms of  
SO(2,1) conformal invariance,
which results from a prescribed {\em reduction\/} framework.


{\em Conformal Anomaly and Short-Distance Physics.\/}---Conformal 
symmetry is 
guaranteed at the quantum level when  
the naive scaling of operators, described by the
algebra~(\ref{eq:naive_commutators}), is maintained.
A measure of the deviation from this scaling is afforded by the
 ``anomaly''~\cite{camblong:anom_delta}
\begin{eqnarray}
{\mathcal A} ({\bf r})
& \equiv &
\frac{1}{i \hbar}
[D,H] +  H
=
\left[
\openone
+
\frac{1}{2}
\,
{\mathcal E}_{\bf r}
 \right]
V ({\bf r})
\label{eq:time_rate_of_dilation_op}
\\
& =& 
\frac{r^{d-2}}{2}
\;
{\bf \nabla}
\!
\cdot
\!
\left[
\frac{ {\bf r} \,
 V ({\bf r})}{r^{d-2}} 
\right]
\;  
\label{eq:time_rate_of_dilation_op_ddim}
\end{eqnarray}
(valid for arbitrary $d$ spatial dimensions),  in which  $\openone$
is the identity operator and
 ${\mathcal E}_{\bf r} = {\bf r} \cdot 
{\bf \nabla} $.
At first sight,
the right-hand side of
Eq.~(\ref{eq:time_rate_of_dilation_op})
appears to be zero for any scale-invariant potential; however,
upon closer examination,
this apparent cancellation may break down at 
$r=0$,
where the interaction is singular.
Equations~(\ref{eq:time_rate_of_dilation_op}) and (\ref{eq:time_rate_of_dilation_op_ddim})
can be directly applied to any of the interactions within the conformal
quantum mechanics class, and reduce to the familiar results
known for the two-dimensional contact 
interaction~\cite{camblong:anom_delta,esteve:anom_delta}.
However, the most interesting case is provided by the 
Hamiltonian~(\ref{eq:ISP_Hamiltonian_unregularized}),
whose symmetry breaking can be made apparent 
by means of the
formal $d$-dimensional identity
$
{\bf \nabla}
\!
\cdot
\!
\left[
%\frac{
\hat{\bf r}/
%}{ 
r^{d-1} %}
\right]
=
\Omega_{d-1}
\,
 \delta^{(d)} ({\bf r}) $,
in which $\Omega_{d-1}$ is the 
surface area of the unit $(d-1)$-sphere $S^{d-1}$;
then,
\begin{equation}
{\mathcal A} ({\bf r})=
- g \,
\frac{ \Omega_{d-1}}{2}
\,
r^{d-2}
\,
\delta^{(d)} ({\bf r})
\;  .
\label{eq:time_rate_of_dilation_op_ddim_2D_ISP}
\end{equation}
Despite its misleading appearance,
this term is {\em not\/} identically equal to zero,
 due to the singular nature of the interaction at $r=0$.
However, two important points should be  clarified.
First, 
Eq.~(\ref{eq:time_rate_of_dilation_op_ddim_2D_ISP}) is merely a formal identity,
whose physical meaning can only be manifested through appropriate 
integral expressions.
Second,  the coordinate singularity
 highlights the need to
determine the behavior of the wave function near $r=0$.
Therefore,  nontrivial consequences of
Eq.~(\ref{eq:time_rate_of_dilation_op_ddim_2D_ISP})
can  only be displayed by the expectation value with a
normalized state $\left| \Psi \right\rangle$,
\begin{equation}
\frac{d}{dt}
\left\langle
D
\right\rangle_{\scriptstyle \!  \Psi}
=
\left\langle
{\mathcal A} ({\bf r})
\right\rangle_{\scriptstyle \!  \Psi}
=
- g \, 
\frac{ \Omega_{d-1}}{2}
\,
\int
d^{d} {\bf r}
\,
\delta^{(d)} ({\bf r})
\left| r^{\nu} \Psi ({\bf r})
\right|^{2}
\;   .
\label{eq:time_rate_of_dilation_op_ddim-EV_2D_ISP}
\end{equation}
For the unregularized inverse square potential,
the integral in Eq.~(\ref{eq:time_rate_of_dilation_op_ddim-EV_2D_ISP}) selects the
limit $r \rightarrow 0$ 
of the product $r^{\nu  } \Psi ({\bf r})$, which is known to be
proportional to
a Bessel function of  order $i \Theta$, where
$\Theta  \equiv \Theta_{l+\nu}  
= \sqrt{\lambda - (l+\nu)^{2}}
$.
This limit ill defined in the strong coupling regime,
due to the uncontrolled oscillatory behavior of
the Bessel functions of imaginary order.
Consequently, a regularization procedure is called for.

{\em Regularization and Renormalization.\/}---The 
Hamiltonian~(\ref{eq:ISP_Hamiltonian_unregularized}),
in the strong coupling regime,
describes an effective system with
singular behavior in the ultraviolet domain. In this interpretation, 
regularization and renormalization are mandatory; physically, 
this behavior  reveals the onset of ``new'' physics for 
sufficiently short distances. Consequently, in this paper we propose
a generic class of regularization schemes that explicitly modify the 
ultraviolet physics; each scheme is described by a potential $V^{(<)}({\bf r})  $,
for $ r \alt a$, where $a$ is a small real-space regulator.
After a complete redefinition of all relevant physical quantities,
including the anomaly, the limit  $a \rightarrow 0$ should be applied.

An appropriate procedure for the  selection of 
solutions of this singular interaction 
was proposed in Ref.~\cite{landau:77},
using a constant potential for
$ r \alt a$.
Our approach is based on a 
generalization of this procedure to an arbitrary  
 $V^{(<)}({\bf r}) $ 
subject to the conditions that:
(i) $V_{0} \equiv {\rm min} \left[  V^{(<)}({\bf r}) 
 \right]$ be finite; (ii) 
$V^{(<)}({\bf r})  $ be joined continuously 
with the external inverse square potential at $r=a$.
In addition,
all the energies for the interior problem
are conveniently
redefined from the minimum value $V_{0}$;  thus,
$U(r) \equiv V^{(<)}(r) -V_{0}$
and
 the wave number 
$\tilde{k} \equiv \tilde{\xi}/a$
is implicitly defined from
 the relation
\begin{equation}
\tilde{\xi}^2
+ \xi^{2} 
= - \frac{2 mV_{0} a^{2} }{ \hbar^{2}}
\; ,
\label{eq:pythagorean}
\end{equation}
in which $\xi
 = 
\kappa a
$.
Spherical symmetry
leads to the separable solution
 $\Psi ({\bf r})= \check{Y}_{lm}( {\bf \Omega})  \, v (r)/r^{\nu }$,
in which  $\check{Y}_{lm}( {\bf \Omega})  $ stands for the hyperspherical harmonics on $S^{d-1}$
(redefined with a normalization integral equal to $\Omega_{d-1}$). Then, the
 corresponding effective radial Schr\"odinger equation for bound states
provides solutions of the form:

(i)  $v^{(<)}(r)= B_{l,\nu} \,
w_{l+\nu}( \tilde{k} r; \tilde{k})$ 
for $r<a$, 
[for example, $w_{l+\nu}( \tilde{k} r; \tilde{k})$ is
  a Bessel function of order $l+\nu$ when the potential
$V^{(<)}({\bf r})  $ is a constant].

(ii)
 $v^{(>)}(r)= A_{l,\nu} \,
K_{i \Theta } (\kappa r)$ for $r>a$,
in which 
$K_{i \Theta } (z)$ 
is the Macdonald function
and
$ \kappa^2
= - 2m E/\hbar^{2}$.

Then,
the  problem is completely determined by the following three conditions:

(a) Continuity at  $r=a$
of the wave function,
$
 B_{l,\nu} \, 
w_{l+\nu}( \tilde{\xi}; \tilde{k})
= A_{l,\nu} \, K_{i \Theta}  ( \xi)
$.

(b)
Continuity at  $r=a$
of the logarithmic derivatives,
$
{\cal L}_{l+\nu}^{(<)} (\tilde{\xi}; \tilde{k})
=
{\cal L}_{i \Theta}^{(>)} (\xi)
$,
which are defined here
in the form
${\cal L}_{i \Theta}^{(>)} (\xi) \equiv 
{\mathcal E}_{\xi} 
\left[
\ln K_{i \Theta} (\xi) \right]$, with
${\mathcal E}_{\xi} = \xi \partial/\partial \xi$,
and similarly for 
$
{\cal L}_{l+\nu}^{(<)} (\tilde{\xi}; \tilde{k})
$ 
in terms of
$w_{l+\nu}( \tilde{\xi}; \tilde{k})$.

(c) Normalization of the wave function, which
provides the values of the constants  $ A_{l,\nu} $ and $ B_{l,\nu}$ 
%\begin{equation}
%B_{l,\nu}^{2}
%=
%\frac{ \kappa^{2}}{ \Omega_{d-1} }
%\,
%\left\{
%\frac{ \xi^{2} }{ \tilde{\xi}^{2} } 
%\,
%{\mathcal J}_{l+\nu}(\tilde{\xi}; \tilde{k} ) 
%\!
%+
%\!
%\left[
%\frac{
% w_{l+\nu}  ( \tilde{\xi} ; \tilde{k}) }{  K_{i \Theta}  (\xi) }
%\right]^{2} \!
%\!
%{\mathcal K}_{i \Theta }(\xi)
%\right\}^{-1}
%\; 
%,
%\label{eq:coeff_B_value}
%\end{equation}
%where
in terms of the integrals
$
{\mathcal K_{i \Theta}} (\xi)
 \equiv
\int_{\xi}^{\infty} 
d z \, z
 \left[ K_{i \Theta} (z) \right]^{2} 
$
and
\begin{equation}
{\mathcal J_{l+\nu}} ( \tilde{\xi};  \tilde{k}  )
  \equiv  
\int_{0}^{  \tilde{\xi}  } 
d z \, z
 \left[ 
w_{l+\nu}( z; \tilde{k})
\right]^{2} 
\;  .
\label{eq:mathcal_J_integral_def}
\end{equation}
Remarkably, 
these  integrals can be evaluated in closed form~\cite{wat:44};
the first one becomes
\begin{equation}
{\mathcal K_{i \Theta}} (\xi)
=
\frac{ 1}{2}
\left[K_{i \Theta} (\xi) \right]^{2}
\,
{\mathcal M}^{(>)}_{i \Theta} ( \xi ) 
\; ,
\label{eq:mathcal_K_integral_Lommel_2nd_useful}
\end{equation}
with
${\mathcal M}_{i\Theta}^{(>)} (\xi) 
\equiv 
\left[ {\mathcal L}^{(>)}_{i \Theta} (\xi)     \right]^{2} 
+ \Theta^{2} - \xi^{2}$,
while the second one takes the form
\begin{equation}
{\mathcal J_{l+\nu}} ( \tilde{\xi};  \tilde{k}  )
=
\frac{1}{2}  \left[ w_{l+\nu} (\tilde{\xi};  \tilde{k} ) \right]^{2} 
{\mathcal M}^{(<)}_{l+\nu} ( \tilde{\xi};  \tilde{k}  ) 
+
{\mathcal U}_{l+\nu} ( \tilde{\xi};  \tilde{k}  ) 
\; ,
\label{eq:mathcal_J_integral_useful}
\end{equation}
where
\begin{equation}
{\mathcal U}_{l+\nu} ( \tilde{\xi};  \tilde{k}  ) 
\!
 \equiv 
\!
\int_{0}^{  \tilde{\xi}  } 
d z \, z
%\,
 \left[ w_{l+\nu} (z;  \tilde{k} ) \right]^{2} 
%\,
\!
\left[
\!
\left(
\openone
+
\frac{1}{2}
\,
{\mathcal E}_{z}
 \right)
\!
\check{U} (z; \tilde{k})
\right]
%\; 
,
\label{eq:mathcal_U_integral_def}
\end{equation}
 $\check{U} \equiv 2 mU/(\hbar^{2} \tilde{k}^{2})$,
${\mathcal E}_{z} = z \partial/\partial z$,
and
\begin{equation}
{\mathcal M}^{(<)}_{l+\nu} ( \tilde{\xi};  \tilde{k}  ) 
\!
 \equiv 
\!
\left[
 {\mathcal L}^{(<)}_{l+\nu}  (\tilde{\xi}; \tilde{k}) \right]^{2} 
+ \left[
\tilde{\xi}^{2} - 
\!
(l+\nu)^{2} 
-
\tilde{\xi}^{2} 
\check{U} (\tilde{\xi}; \tilde{k})
\right]
%\; 
.
\label{eq:mathcal_M_integral_def}
\end{equation}
Equations~(\ref{eq:pythagorean})--(\ref{eq:mathcal_M_integral_def})
permit the exact evaluation of all relevant expectation values,
when applied concurrently with the conditions (a)-(c) listed above;
in addition,
the continuity (at $r=a$)
 of the matching functions
$
{\mathcal M}^{(<)}_{l+\nu} ( \tilde{\xi};  \tilde{k}  ) 
=
{\mathcal M}^{(>)}_{i \Theta} ( \xi ) 
$
is satisfied.

Even though renormalization is a necessary condition for 
the emergence of the conformal anomaly,
 the actual details of the renormalization procedure are not 
explicitly required for the computation and conclusions
 shown below. It suffices to know that
these details are to be consistently derived from the relation
$ {\cal L}_{l+\nu}^{(<)} (\tilde{\xi}; \tilde{k})
=
{\cal L}_{i \Theta}^{(>)} (\xi)
$, 
 by enforcing the finiteness of
a  particular bound state energy
and finding the corresponding behavior of the running coupling constant.
Incidentally, the renormalization framework of Ref.~\cite{camblong:isp-dt}
reproduces almost identically all the steps leading to the thermodynamics 
of black holes---a similar treatment,
using the {\em alternative\/} method of self-adjoint extensions,
 has been recently discussed in 
Refs.~\cite{gov:BH_states,gupta:BH}.
An analysis of renormalization schemes,   
a comparison of various  frameworks,
and additional applications to  black holes,
will be presented elsewhere.

{\em Calculation of the Conformal Anomaly.\/}--Finally, the
 regularized anomaly can be computed from
\begin{eqnarray}
%\begin{equation}
{\mathcal A}_{a} ({\bf r})
& = &
%=
\left[
\left(
\openone
+
\frac{1}{2}
\,
{\mathcal E}_{\bf r}
 \right)
\,
V^{(<)} ({\bf r})
\right]
 \,
\theta (a-r) 
\nonumber
\\
& &
- g \,
\frac{ \Omega_{d-1}}{2}
\,
r^{d-2}
\,
\delta^{(d)} ({\bf r})
 \,
 \theta (r-a) 
\;  ,
\label{eq:time_rate_of_dilation_op_ddim_2D_ISP_reg}
\end{eqnarray}
%\end{equation}
which replaces Eq.~(\ref{eq:time_rate_of_dilation_op_ddim_2D_ISP}).
In Eq.~(\ref{eq:time_rate_of_dilation_op_ddim_2D_ISP_reg}),
 $\theta (z)$ stands for the Heaviside function; then
\begin{equation}
\frac{d}{dt}
\left\langle
D
\right\rangle_{\scriptstyle \!  \Psi}
=
\lim_{a \rightarrow 0}
\left[
\left\langle
{\mathcal A}_{a}  ({\bf r})
\right\rangle_{\scriptstyle \!  \Psi_{a} }^{(<)}
+
\left\langle
{\mathcal A}_{a}  ({\bf r})
\right\rangle_{\scriptstyle \!  \Psi_{a} }^{(>)}
\right]
\;  ,
\label{eq:time_rate_of_dilation_op_ddim-EV_2D_ISP_reg}
\end{equation}
where the integration range
defining the expectation value 
is split into two regions: $0<r<a$ and $r>a$.
Moreover,
 the identically vanishing term
$\left\langle
{\mathcal A}_{a} 
({\bf r})
\right\rangle_{\scriptstyle \!  \Psi_{a} }^{(>)}
= 0 
$
in Eq.~(\ref{eq:time_rate_of_dilation_op_ddim_2D_ISP_reg})  shows that the source 
of the conformal anomaly is confined to an arbitrarily small region about the origin.
%and
%\begin{eqnarray}
%\! \! \! \! \! \!
%& & \frac{d}{dt}
%\left\langle
%     D
%\right\rangle_{\scriptstyle \!  \Psi}
%  = 
%\lim_{a \rightarrow 0}
%\left\langle
%{\mathcal A}_{a}  ({\bf r})
%\right\rangle_{\scriptstyle \!  \Psi_{a} }^{(<)}
%=
%\Omega_{d-1} 
%\,
%\lim_{a \rightarrow 0}
%\left(
%\frac{B_{l,\nu}^{2} }{\tilde{k}^{2} }
%\, \int_{0}^{  \tilde{\xi}  } 
%d z \, z
%\right.
%\,
%\nonumber
%\\
%\! \! \! \! \! \!
%&  &
%\left.
%\times
%\,
% \left[ 
%w_{l+\nu}( z; \tilde{k})
% \right]^{2} 
%\,
%\left\{
%\left(
%\openone
%+
%\frac{1}{2}
%\,
%{\mathcal E}_{z}
% \right)
%\left[  V_{0} + U \left(  \frac{z}{\tilde{k}}  \right)
%\right]
%\right\}
%\right)
%\,  .
%\label{eq:anomaly_internal_region}
%\end{eqnarray}
Thus, when 
Eqs.~(\ref{eq:pythagorean})--(\ref{eq:mathcal_M_integral_def})---together 
with the conditions (a)-(c) and
the limit prescribed in Eq.~(\ref{eq:time_rate_of_dilation_op_ddim-EV_2D_ISP_reg})---are 
applied to the first term on the right-hand side of
Eq.~(\ref{eq:time_rate_of_dilation_op_ddim_2D_ISP_reg}), the anomaly becomes
\begin{equation}
\frac{d}{dt}
\left\langle
D
\right\rangle_{\scriptstyle \!  \Psi}
= 
E
\;  ,
\label{eq:anomaly_2D_ISP_circular_well}
\end{equation}
where
$E$ is the energy of the corresponding stationary normalized state.
This interpretation, which agrees with the prediction
from properties of expectation values~\cite{camblong:anom_delta},
is now validated for a generic regularization procedure. Therefore, 
regardless of the renormalization framework used, when the limit $a \rightarrow 0$
is applied, an {\em anomaly\/} is generated.
The generality of Eq.~(\ref{eq:anomaly_2D_ISP_circular_well})
 makes it available for a variety of physical applications,
and is a {\em necessary condition when the theory is renormalized\/}.

{\em Conclusions.\/}---Realizations of the conformal anomaly
involve a breakdown of the 
associated SO(2,1) algebra.
In this paper,
 we have shown
that the actual emergence and value of the conformal
anomaly rely 
upon the application of a renormalization procedure, but are otherwise
 independent of the details of the ultraviolet physics.
In this sense, the  results derived herein 
are robust and totally general.
As such, they are 
intended to shed light on the physics of 
any system
with
a conformally-invariant
domain
for which the short-distance physics dictates the
existence of bound states.

{\em Acknowledgements.\/}---
This research was supported in part by
the University of San Francisco Faculty Development Fund.
We also thank Dean Stanley Nel for generous travel support
and
Professors Luis N. Epele,  Huner Fanchiotti,
and Carlos A. Garc\'{\i}a Canal
for early discussions of
 this work.


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\end{document}

