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\hyphenation{midi-super-space}

\begin{document}
\begin{titlepage}
\vspace{.5in}
\begin{flushright}
June 2021\\
revised October 2021\\
 \end{flushright}
\vspace{.5in}
\begin{center}
{\Large\bf
 Midisuperspace foam\\[1ex]
  and the cosmological constant}\\
\vspace{.4in}
{S.~C{\sc arlip}\footnote{\it email: carlip@physics.ucdavis.edu}\\
       {\small\it Department of Physics}\\
       {\small\it University of California}\\
       {\small\it Davis, CA 95616}\\{\small\it USA}}
\end{center}

\vspace{.5in}
\begin{center}
{\large\bf Abstract}
\end{center}
\begin{center}
\begin{minipage}{4.5in}
{\small
Wheeler's conjectured ``spacetime foam''---large quantum fluctuations of spacetime
at the Planck scale---could have important implications for quantum gravity, perhaps
even explaining why the cosmological constant seems so small.  Here I explore this
problem in a midisuperspace model consisting of metrics with local spherical symmetry.
Classically,  an infinite class of ``foamy'' initial data can be constructed, in which
cancellations between expanding and contracting regions lead to a small average
expansion even if $\Lambda$ is large.  Quantum mechanically, the model admits corresponding
stationary states, for which the probability current is also nearly zero.  These states appear
to describe a self-reproducing spacetime foam with very small average expansion,
effectively hiding the cosmological constant.}
\end{minipage}
\end{center}
\end{titlepage}
\addtocounter{footnote}{-1}

\section{Spacetime foam and the cosmological constant}

More than 65 years ago, Wheeler suggested that quantum uncertainties in the metric should
be of order one at the Planck scale, leading to large fluctuations in geometry and (perhaps)
topology,  which he called ``spacetime foam'' \cite{Wheeler}.   While this idea has
continued to generate  interest,
it has been notoriously hard to investigate  quantitatively.  My goal here is to develop a
model simple enough to allow calculations, but still complex enough to describe a large
assortment of foam-like quantum states.

More specifically, one aim is to test a proposal  that spacetime foam might help solve
the cosmological constant problem.  In \cite{Carlip1}, I suggested that our Universe
might in fact have a large cosmological constant $\Lambda$, whose effects are
``hidden''  in Planck scale fluctuations.   I showed that for a large class of initial data,
the expansion and shear---the observational signature of a cosmological constant---average
to zero over small regions even if $\Lambda$ is large.  A slightly oversimplified picture
is that a cosmological constant can cause either expansion or contraction; using a
construction of Chrusciel, Isenberg, and Pollack \cite{Chrusciel,Chruscielb}, one can sew
together Planck-scale regions with random initial data in a way that naturally leads to
large cancellations.

A key question is whether such cancellations persist during evolution.  This is fundamentally
a matter for quantum gravity, and a full treatment lies beyond our current capabilities.
There is, however, a partial step: we can explore mini- and midisuperspace
models,\footnote{Conventionally a minisuperspace keeps only finitely many degrees of
freedom, while a midisuperspace has an infinite, although still restricted, set.}
in which many of the degrees of freedom are frozen out.   Such models have
been used to study a range of issues in quantum gravity, and while
the results are rarely conclusive, they can be strongly suggestive.

Here I will focus on a midisuperspace consisting of geometries with local spherical symmetry.
When combined with the ``dust time'' of Brown and Kucha{\v r} \cite{Brown,Marolf,Husain},
the Wheeler-DeWitt equation takes a Schr{\"o}dinger-like form, and can be solved by WKB
methods.  We shall see that there are stationary states in which a foamlike structure and
small average expansion are preserved in time.  Moreover, while regularization ambiguities
exist,  ``foamy'' spacetimes seem to occur with high probabilities.  In this setting,
at least, a cosmological constant may indeed be hidden in spacetime foam.

\section{Local spherical symmetry}

The setting for locally spherically symmetric midisuperspace has been studied
extensively by Morrow-Jones, Witt, and Schleich \cite{Witt1,Witt2,Witt3}.  At first sight, this
symmetry requirement may seem too strong: it is widely believed that  spherical symmetry
with $\Lambda>0$ leads inevitably to  Schwarzschild--de Sitter space.  But while Birkhoff's
theorem implies that a spherically symmetric vacuum spacetime must be \emph{locally} isometric
to some region of Schwarzschild-de Sitter space, local patches can be sewn together to form a
spacetime that looks  drastically different  \cite{Witt3}.  In particular, we will be able to construct
explicit initial data containing both expanding and contracting regions.

Here, for simplicity, I will assume a positive cosmological constant,
 and specialize to the spatial topology $S^1\times S^2$.  Recall that to construct a
manifold with this topology, we start with a solid three-ball; cut out a ball from its center to form a
manifold $[0,1]\times S^2$; and identify the boundaries $\{0\}\times S^2$ and $\{1\}\times S^2$.
To build ``foamy'' spacetimes,
we will take this construction a step further, splitting space into an onion-like
sequence of concentric shells, each with its own geometry.

Following \cite{Witt1}, let us start with the initial value formalism, with
geometric data consisting of a spatial metric $q_{ij}$ and its conjugate momentum $\pi^i{}_j$.
These cannot be chosen arbitrarily, but must solve the momentum and Hamiltonian constraints
\begin{subequations}
\begin{align}
&\mathscr{P}_j = {}^{\scriptscriptstyle(3)}\nabla_i\pi ^i{}_j = 0 \label{a0a}\\
&\mathscr{H} = \frac{2\kappa^2}{\sqrt{q}}\left(\pi^i{}_j\pi^j{}_i - \frac{1}{2}\pi^2\right)
    - \frac{1}{2\kappa^2}\sqrt{q}\left({}^{\scriptscriptstyle(3)}\!R - 2\Lambda\right) = 0 .
 \label{a0b}
 \end{align}
 \end{subequations}
 (I use the conventions of \cite{Carlip3}; in particular, $\kappa^2=8\pi G$.)  On a hypersurface of
 constant time, a general locally spherically symmetric metric may be written in the form
\begin{align}
ds^2 = h^2d\psi^2 + f^2(d\theta^2 + \sin^2\theta\, d\varphi^2)
\label{a1}
\end{align}
where $h$ and $f$ are functions of $\psi$ and $t$.  The spatial scalar curvature is
\begin{align}
{}^{\scriptscriptstyle(3)}\!R
   = \frac{1}{f^2}\left[ -\frac{4f}{h}\left(\frac{f'}{h}\right)^\prime -\frac{2f^{\prime2}}{h^2} + 2\right] ,
\label{a2}
\end{align}
where a prime denotes a derivative with respect to $\psi$.  Ref.\ \cite{Witt1} makes an added
coordinate choice $h(\psi,t)=a(t)$, $f(\psi,t)=a(t){\tilde f}(\psi)$; this simplifies the classical solutions,
but complicates the quantum treatment by gauge-fixing one of the diffeomorphism constraints.

The nonvanishing canonical momenta  are
\begin{align}
\pi^\theta{}_\theta = \pi^\varphi{}_\varphi = \sin\theta\,Q, \quad \pi^\psi{}_\psi = \sin\theta\,P ,
\label{a3}
\end{align}
where $P$ and $Q$ are functions of $\psi$ and $t$.  (The $\sin\theta$ factors appear because
the momenta are tensor densities).  Geometrically, $P$ and $Q$ are extrinsic curvatures, fractional
rates of expansion; explicitly,
\begin{align}
Q = \frac{1}{2\kappa^2}hf^2\left(K^\theta{}_\theta + K^\psi{}_\psi\right), \quad
P = \frac{1}{\kappa^2}hf^2K^\theta{}_\theta .
\label{a3a}
\end{align}
The only nontrivial momentum constraint is
 \begin{align}
 \mathscr{P} = P' - \frac{h'}{h}P -\frac{2f'}{f}Q = 0 \ \Rightarrow\ Q = \frac{f}{2f'}\left(P' - \frac{h'}{h}P\right) ,
 \label{a4}
 \end{align}
 while the Hamiltonian constraint becomes, after a little algebra,
 \begin{align}
 \mathscr{H} = \frac{h}{f'}\frac{d\ }{d\psi}\left[-\kappa^2\frac{P^2}{fh^2}
   + \frac{1}{\kappa^2}\left(\frac{ff^{\prime2}}{h^2} - f\right) + \frac{\Lambda}{3\kappa^2}f^3\right]  = 0.
\label{a5}
\end{align}
This simple total derivative structure was first noted in \cite{Witt1}.

Given a metric $(h,f)$, we can now solve (\ref{a5}) for $P$,
\begin{align}
P = \pm \frac{1}{\kappa^2}\left[f^2h^2\left(\frac{f^{\prime2}}{h^2}-1\right)
   + \frac{\Lambda}{3}f^4h^2 + \gamma fh^2\right]^{1/2}
\label{a6}
\end{align}
where $\gamma$ is an integration constant.  As also noted in \cite{Witt1}, this constant is a mass:
if we look at a slice $P=0$ and gauge fix to $f=\psi$ (allowed locally, though not globally), we see that
\begin{align}
h^2 = \left(1-\frac{\gamma}{\psi} - \frac{\Lambda}{3}\psi^2\right)^{-1},
\label{a7}
\end{align}
a piece of the usual Schwarzschild-de Sitter metric with $\gamma=2Gm$.

Note that the value of the integration constant $\gamma$ can have a drastic effect on the spacetime
geometry \cite{Lake,SW}.  Here I will assume that
\begin{align}
0< \gamma < \frac{2}{3\sqrt{\Lambda}}  .
\label {a7x}
\end{align}
For the complete Schwarzschild-de Sitter metric, this is the condition for the existence of two horizons,
that is, for a black hole that is smaller than the cosmological horizon.  As we will see shortly, it is also
necessary for the existence of a Cauchy surface with vanishing extrinsic curvature, a requirement for
attaching expanding and contracting regions to realize the ``spacetime foam'' of \cite{Carlip1}.

By standard existence theorems, the initial data
$(h,f,P,Q)$ can now be evolved to form a maximal globally hyperbolic spacetime.  As expected,
the system has a time reversal symmetry: if  the data $(h,f,P,Q)$ are admissible, so are
$(h,f,-P,-Q)$.  Since $P$ and $Q$ determine the expansion and shear, for any expanding solution
there is a corresponding contracting one.

Of course, there is no guarantee that the resulting spacetimes are geodesically complete.  On
the contrary, regions of the initial surface can be isometric to regions of the interior of a
Schwarzschild-de Sitter black hole, so one might generically expect singularities to form, as
happens in the more general setting \cite{Burkhartb}.  Evolution thus requires a quantum
treatment.

\section{Sewing \label{sew}}

Ref.\ \cite{Carlip1} focused on manifolds formed by ``sewing'' elementary pieces
to build a model of spacetime foam.   An almost identical procedure exists in the spherically
symmetric setting \cite{Witt1}.  Choose coordinates in which $h$ is constant
in a small region around $\psi=\psi_0$.  Pick two manifolds $M_1$ and $M_2$,
with arbitrary functions $f_1(\psi)$ and $f_2(\psi)$.  Let $u_\epsilon$ be a smoothed step
function, interpolating between $u_\epsilon(\psi)=0$ when $\psi<-\epsilon$ and
$u_\epsilon(\psi)=1$ when $\psi>\epsilon$.

Now define
\begin{align}
f(\psi) = [1-u_\epsilon(\psi-\psi_0)]f_1(\psi) + u_\epsilon(\psi-\psi_0)f_2(\psi)  .
\label{b1}
\end{align}
This ``sewn''  $f$ looks like $f_1$ for $\psi\in(-\infty,\psi_0-\epsilon)$ and like $f_2$ for
$\psi\in(\psi_0+\epsilon,\infty)$, with a smooth interpolation between.  Since the momenta $P$
and $Q$ are determined locally from $f$ and $h$, the full initial data will sew together in the same way.
Topologically, this process is a connected sum $M_1\#M_2$; geometrically, it is very close to the
general construction of \cite{Chrusciel,Chruscielb}.

This isn't yet quite good enough.  The functions $f_1$ and $f_2$ each
determine two sets of data, $(h,f_1,\pm P_1,\pm Q_1)$ and $(h,f_2,\pm P_2,\pm Q_2)$.
We would like to sew any combination, for instance attaching an expanding region to a contracting
region.  But for $P$ and $Q$ to change sign, they must go through zero, a condition that
is not automatic.

To allow such a sign change, we construct an extra ``neck'' $N$, an intermediate manifold
in which $P$ and $Q$ go to zero on a central two-sphere.  We can then sew $M_1$ and
$M_2$ via this neck, $M_1\#N\#M_2$, with either choice of momenta on each side.
We specify $N$, say with $\psi\in(-\delta,\delta)$, by demanding that
 \begin{align}
 {f}'(0) =0, \  f^{\prime\prime}(0) >0 ,\  P(0) = Q(0) = 0 .
\label{b2}
\end{align}

 The condition $P(0)=0$ is easy: by (\ref{a6}), it simply requires that
\begin{align}
\frac{\Lambda}{3}f_0{}^3 = f_0-\gamma ,
\label{b3}
\end{align}
where $f_0 = f(0)$.   Note, though, that (\ref{b3}) has real, positive roots only if the inequality
(\ref{a7x}) is satisfied \cite{Lake}.  If the inequality is violated, $P$ can never go through zero,
and it becomes impossible to join expanding and contracting regions, at least within the constraint
of local spherical symmetry.

Setting $Q(0)=0$ is harder: eqn.\ (\ref{a4}) for $Q$ has a factor $f'$ in the denominator, which
goes to zero at the center of the neck. To cancel this term, we must demand that
$P(\psi)$ be of order $\psi^3$ near $\psi=0$.  A  straightforward calculation yields
the requirement
\begin{align}
f(\psi)
&= f_0 + \left( 1 - \Lambda f_0^2\right)f_0%
     \left(\frac{h^2\psi^2}{4f_0{}^2}\right)\left(1 - \frac{h^2\psi^2}{12f_0{}^2}\right)
    + \mathcal{O}\left(\psi^6\right)
    \quad \hbox{with}\  1-\Lambda f_0^2 > 0  .
\label{b4}
\end{align}
This restriction is not quite as stringent as it
may seem: the $\mathcal{O}\left(1\right) $ and $\mathcal{O}\left(\psi^2\right)$ terms are
already fixed by the demand that $Q$ remain finite at $f'=0$.

By the constraint (\ref{a0b}), the scalar curvature
${}^{\scriptscriptstyle(3)}\!R$ is large at $\psi=0$, since the extrinsic curvature vanishes there.  But
the curvature falls off quickly away from the center of the neck:
\begin{multline}
{}^{\scriptscriptstyle(3)}\!R = 2\Lambda - \frac{8\kappa^4}{h^2f^4}\left(PQ-\frac{1}{4}P^2\right)
   = 2\Lambda - \frac{2\kappa^4}{3h^2}\frac{1}{f_0^3f^{\prime\prime}(0)}P_3{}^2\psi^4
   + \mathcal{O}\left(\psi^5\right)\\[.4ex]
   \hbox{where}\ P = \frac{1}{6}P_3\psi^3 + \mathcal{O}\left(\psi^4\right) .
\label{b4a}
\end{multline}

We can now construct the promised ``onion'' initial data, with a mix of expanding and
contracting layers.  Choose a $K$-layer sequence $\psi_0<\psi_1<\dots<\psi_K$, where $\psi_K$ will be
identified with $\psi_0$ to create a closed $S^1\times S^2$.  In each layer $\psi\in(\psi_n,\psi_{n+1})$,
choose functions $(h_n,f_n)$, and randomly pick the sign of the corresponding $\pm(P_n,Q_n)$.
Sew successive layers along necks as described above.  For a configuration with enough layers,
the random choice of signs means that $P$ and $Q$ will typically average to near zero, even if
$\Lambda$ is large.  Thus, as in \cite{Carlip1}, we have hidden the initial cosmological constant, but
now in a setting in which we can address the quantum theory much more concretely.

\section{The problem of time \label{time}}

To investigate the quantum evolution of this system, we must first confront a basic
puzzle, the notorious ``problem of time'' \cite{Kuchar,Isham}.   Quantum gravity is a relational theory,
and time evolution can only be described relative to other degrees of freedom.  To describe
evolution, we must first choose a ``clock.''

In some expanding minisuperspace models \cite{Misner}, and perhaps more generally \cite{expand}, one
can use spatial volume as a clock.  Here, though, our spacetimes have some regions that expand
while others contract.  One can often use ``York time'' \cite{York}, in which the trace of
the extrinsic curvature---the local Hubble constant---serves as a clock.  But here, again,
we are interested in cases in which the extrinsic curvature doesn't even have a fixed sign at constant
time.  A more general choice of time, the mean curvature flow \cite{MCF}, fails for the same reason.
In some cosmological spacetimes, ``time since the big bang'' can serve as a clock \cite{costime},
but it is not obvious that the spacetimes considered here have a suitable initial big bang singularity.

An alternative,  developed by Brown and Kucha{\v r} \cite{Brown,Marolf}, uses
congruences of test particles, ``dust,'' as a reference system.\footnote{Footnote 1 of \cite{Torre} gives
a history of this idea, which dates back to Einstein's ``mollusc of reference.''}   This is not ideal---it
would be preferable to describe the evolution purely in terms of a gravitational variable---but in the
present setting I don't know how to define such a ``clock.''
Here I will use a slightly simpler version of dust time, due to Husain and Paw{\l}owski \cite{Husain}
(see also \cite{Torre}), in which a space-filling congruence of irrotational dust is used to specify
time evolution.  The ``cloud of clocks'' is introduced with an action
\begin{align}
I_{\hbox{\tiny dust}} = \frac{1}{2}\int d^4x\sqrt{-g}\,\rho\left(g^{ab}\partial_aT\partial_bT -1\right) ,
\label{c1}
\end{align}
where $\rho$ is a Lagrange multiplier.  It is easy to see that the variation of $I_{\hbox{\tiny dust}}$
yields the usual stress-energy tensor for noninteracting irrotational matter.  Carrying out a
standard canonical decomposition of $I_{\hbox{\tiny grav}}+I_{\hbox{\tiny dust}}$, one finds
constraints
\begin{subequations}
\begin{align}
&\mathscr{P}_i = \mathscr{P}_i^{\hbox{\tiny grav}} + \mathscr{P}_i^{\hbox{\tiny dust}} = 0  \label{c2a}\\
&\mathscr{H} = \mathscr{H}^{\hbox{\tiny grav}} + \mathscr{H}^{\hbox{\tiny dust}} = 0  ,
\label{c2b}
\end{align}
\end{subequations}
where $\mathscr{H}^{\hbox{\tiny grav}}$ and $\mathscr{P}_i^{\hbox{\tiny grav}}$ are given by
(\ref{a0a})--(\ref{a0b}) and
\begin{subequations}
\begin{align}
&\mathscr{P}_i^{\hbox{\tiny dust}} = -p_T\partial_iT  \label{c3a}  ,\\
&\mathscr{H}^{\hbox{\tiny dust}}
  = \left(p_T{}^2 +  q^{ij}\mathscr{P}_i^{\hbox{\tiny dust}}\mathscr{P}_j^{\hbox{\tiny dust}}\right)^{1/2} ,
\label{c3b}
\end{align}
\end{subequations}
where $p_T$ is the momentum conjugate to $T$.

As shown in \cite{Brown}, one can use these constraints in a ``many-fingered time'' quantization,
in which the wave function evolves as a functional of $T$.  But one can also gauge fix time by choosing
$T=t$.  Physically, this amounts to using proper time along the dust worldlines as a time coordinate.
With this choice, $\mathscr{P}_i^{\hbox{\tiny dust}} = 0$, and the constraints reduce to
\begin{subequations}
\begin{align}
&\mathscr{P}_i^{\hbox{\tiny grav}} = 0 \label{c4a}\\
&p_T = \mathscr{H}^{\hbox{\tiny grav}} \label{c4b}    .
\end{align}
\end{subequations}

Eqn.\ (\ref{c4b}) has a slightly peculiar structure.  The left-hand side is independent of position, since
we have chosen a gauge in which $T$ depends only on time, but the right-hand side is certainly
position-dependent.  This is actually sensible: the zero mode of $\mathscr{H}^{\hbox{\tiny grav}}$ is now a
true Hamiltonian, while the remaining position-dependent modes continue to act as constraints, much
like the situation in unimodular gravity \cite{unimodular}.

\section{Wheeler-DeWitt quantization \label{WQ}}

To quantize this system a la Wheeler and DeWitt \cite{WdW}, we rewrite the momentum
and  Hamiltonian constraints as operators, with canonical momenta replaced by
(functional) derivatives.  To find the appropriate conjugates, we start with the symplectic current
\begin{align}
\omega = \int\!d^3x\,\pi^{ij}\delta q_{ij} = 8\pi\int\!d\psi\,\left(\frac{2}{f}Q\,\delta f + \frac{1}{h}P\,\delta h\right) ,
\label{d0}
\end{align}
from which can we read off
\begin{subequations}
\begin{align}
P = \frac{i}{8\pi}h\frac{\delta\ }{\delta h}, \quad  Q =  \frac{i}{16\pi}f\frac{\delta\ }{\delta f} .
\label{d1a}
\end{align}
\end{subequations}
The spatial volume $\hat V$ and the mean curvature $\hat K$ (the trace of the extrinsic curvature) become
\begin{subequations}
\begin{align}
&{\hat V} = \int\!d^3x\,\sqrt{q} = 4\pi\int\!d\psi\,hf^2  \label{d2a}\\[.5ex]
&{\hat K} = -\frac{\kappa^2}{\sqrt{q}}\,{\hat\pi}
     = -\frac{i\kappa^2}{8\pi}\frac{1}{hf^2}\left( h\frac{\delta\ }{\delta h} + f\frac{\delta\ }{\delta f}\right) ,\label{d2b}
\end{align}
\end{subequations}
and satisfy $\displaystyle [{\hat K},{\hat V}] = -{3i\kappa^2}/{2}$ .

The momentum constraint (\ref{a4}) now becomes
\begin{align}
\mathscr{\hat P} = \frac{i}{8\pi}\left(h\partial_\psi\frac{\delta\ }{\delta h} - f'\frac{\delta\ }{\delta f}\right) .
\label{d2}
\end{align}
Acting on $h$ and $f$,  $\mathscr{\hat P} $ generates spatial diffeomorphisms, and invariant
wave functions can be built from integrals of the form
\begin{align}
F[h,f] = \int\!d\psi\,h L[f,Df,D^2f,\dots] \qquad\hbox{with $\displaystyle D = \frac{1}{h}\frac{d\,}{d\psi}$}.
\label{d3}
\end{align}
The Hamiltonian constraint (\ref{a5}) is a bit more complicated, but can be reduced to the form
\begin{align}
\mathscr{\hat H} = \frac{h}{f'}\frac{d\ }{d\psi}\left[
    \frac{\kappa^2}{64\pi^2}\frac{1}{f}\frac{\delta^2\ }{\delta h^2}
    + \frac{1}{\kappa^2}f\left(\frac{f^{\prime 2}}{h^2} -1\right) + \frac{\Lambda}{3\kappa^2}f^3\right] ,
\label{d4}
\end{align}
while the momentum $p_T$ of (\ref{c4b}) becomes
\begin{align}
p_T = i hf^2\frac{d\ }{dT}
\label{d5}
\end{align}
($hf^2$ is the volume measure $\sqrt{q}$).  The Wheeler-DeWitt equation---the operator version of
(\ref{c4b}), acting on a wave function $\Psi$---is thus
\begin{align}
if^2f'\frac{d\Psi}{dT} = \frac{d\ }{d\psi}\left[
    \frac{\kappa^2}{64\pi^2}\frac{1}{f}\frac{\delta^2\ }{\delta h^2}
    + \frac{1}{\kappa^2}f\left(\frac{f^{\prime 2}}{h^2} -1\right) + \frac{\Lambda}{3\kappa^2}f^3\right] \Psi .
\label{d6}
\end{align}

As in the classical case, (\ref{d6}) can be integrated.   The derivative $d\Psi/dT$ is
independent of position---$\Psi$ is a functional of $h$ and $f$, not a function of $\psi$---so
the left-hand side of (\ref{d6}) is a total derivative.  Hence
\begin{align}
i\frac{d\Psi}{dT} = \left[ \frac{3\kappa^2}{64\pi^2}\left(\frac{1}{f^2}\frac{\delta\ }{\delta h}\right)^2
    + \frac{3}{\kappa^2}\frac{1}{f^2}\left(\frac{f^{\prime 2}}{h^2} -1\right) + \frac{\Lambda}{\kappa^2}
    + \frac{3\gamma}{\kappa^2 f^3}\right]\Psi
    = {\hat H}\Psi ,
\label{d7}
\end{align}
where $\gamma$ is again an integration constant.  This is the fundamental equation we must solve
to understand quantum midisuperspace.

\subsection{The wave function}

Eqn.\ (\ref{d7}) is a Schr{\"o}dinger-type equation, and there is no difficulty in principle
in interpreting $|\Psi|^2$ as a probability density on midisuperspace \cite{Kuchar}.  In particular,
the time derivative
\begin{align}
 \frac{d(\Psi^*\Psi)}{dT} = - \frac{3i\kappa^2}{64\pi^2}\frac{\delta\ }{\delta h}%
    \left[\frac{1}{f^4}\left(\Psi^*\frac{\delta\Psi}{\delta h} - \Psi\frac{\delta\Psi^*}{\delta h}\right)\right]
\label{da1}
\end{align}
is a total (functional) derivative, so formally, the norm
$$\int [df][dh]\Psi^*\Psi$$
is conserved, at least for suitable boundary conditions for midisuperspace.

There is a subtlety, though.  The wave function $\Psi$ should really be defined on ``reduced midi\-superspace,''
the space of symmetric metrics modulo diffeomorphisms.  To   properly define a norm on this quotient
space, one must gauge fix the inner product to avoid overcounting diffeomorphism-equivalent
configurations \cite{Woodard}.  This is a difficult task, since one must account for not only the
spatial diffeomorphisms, but also the symmetries generated by the non-zero modes of the Hamiltonian
constraint.  Partial results are given in the Appendix; for now, I will avoid this issue, and merely use
$|\Psi|^2$  to determine relative probabilities, with the understanding that the inner product measure
may give corrections.

It is worth reiterating that $\Psi$ is a functional of the metric, not a function of space.  Its role is to
determine probabilities of configurations $(h,f)$.  Of course, the probability of any given
metric occurring will depend on its spatial form, and we shall see that a wave function can imply ``a
high probability for metrics with spatial characteristic X.''   But this will only make sense if the ``spatial
characteristic X'' is expressed in a diffeomorphism-invariant way, one that does not refer to any
particular values of the coordinates.

\subsection{The probability current}

As a Schr{\"o}dinger-type equation, (\ref{d7}) should also admit a probability current.
Recall that in the WKB approximation in ordinary quantum
mechanics, the probability current distinguishes genuinely time-independent configurations
(e.g., bound states) from steady state descriptions of a secretly time-dependent configurations
(e.g., plane waves scattering off a potential barrier).  We shall see that the same is true here.

It is easy to see that the current
\begin{align}
J[f,h;x] = \frac{3i\kappa^2}{64\pi^2}%
    \left(\Psi^*\frac{1}{f^2}\frac{\delta\Psi}{\delta h} - \Psi\frac{1}{f^2}\frac{\delta\Psi^*}{\delta h}\right) .
\label{d8}
\end{align}
obeys a sort of continuity equation on midisuperspace,
\begin{align}
 \frac{d(\Psi^*\Psi)}{dT} + \frac{1}{f^2}\frac{\delta J}{\delta h} = 0 .
\label{d9}
\end{align}
But $J$ is not an observable: it is not annihilated by the momentum constraint $\mathscr{\hat P}$.
We can project $J$ onto reduced midisuperspace by taking a  volume average, or equivalently
forming a group average over the spatial diffeomorphisms \cite{Marolfb},
\begin{align}
\langle J\rangle = \frac{1}{V}\int\!d^3x\,hf^2J ,
\label{d10}
\end{align}
but this average no longer obeys an obvious continuity equation.

To do better, consider the functions
\begin{align}
\varphi_n(\psi) = \exp\left\{\frac{8\pi^2 i n}{V}\int^\psi\!\! f^2h\,d\psi'\right\}  .
\label{d11}
\end{align}
These are orthogonal and complete:
\begin{align}
& \int hf^2\varphi_m^*\varphi_n\,d\psi = \frac{V}{4\pi}\delta_{mn} , \nonumber\\[1ex]
&\sum_n \varphi_n^*(\psi')\varphi_n(\psi) = \frac{V}{4\pi hf^2}\,\delta(\psi-\psi')  .
\label{d12}
\end{align}
If we now project
\begin{align}
D_n = \frac{1}{V}\int\!d^3x\,hf^2\varphi^*_n\left(\frac{1}{f^2}\frac{\delta\ }{\delta h}\right) , \ \
J_n = \int\!d^3x\,hf^2\varphi_n\,J ,
\label{d13}
 \end{align}
 a bit of work shows that
 \begin{align}
 \sum_n D_nJ_n = \frac{1}{V}\int\!d^3x\,hf^2\left(\frac{1}{f^2}\frac{\delta J}{\delta h}\right) = -  \frac{d(\Psi^*\Psi)}{dT} ,
 \label{d14}
 \end{align}
where the last equality comes from (\ref{d9}).
$D_n$ and $J_n$ are spatial invariants, that is, that they are annihilated by the
momentum constraint.\footnote{$D_n$ and $J_n$ are not annihilated by the Hamiltonian constraint; a further
projection may be needed to construct spacetime invariants, but that lies beyond the scope of this paper.}
The $D_n$ may be viewed as the components of the gradient in our (infinite-dimensional) midisuperspace,
making (\ref{d14}) a standard continuity equation.

\section{Stationary states and the WKB approximation}

We next look for stationary states
\begin{align}
\Psi[f,h;T] = {\tilde\Psi}[f,h]e^{-iET}  .
\label{e1}
\end{align}
For states of this form, the Wheeler-DeWitt equation (\ref{d7}) becomes
\begin{align}
\left[ \frac{\kappa^2}{64\pi^2}\left(\frac{1}{f^2}\frac{\delta\ }{\delta h}\right)^2
    + \frac{1}{\kappa^2}\frac{1}{f^2}\left(\frac{f^{\prime 2}}{h^2} -1\right) + \frac{\tilde\Lambda}{3\kappa^2}
    + \frac{\gamma}{\kappa^2 f^3}\right]{\tilde\Psi} = 0  ,
\label{e2}
\end{align}
where
\begin{align}
{\tilde\Lambda} = \Lambda - \kappa^2 E .
\label{e2a}
\end{align}
This equation is identical to the original gravitational Hamiltonian constraint, except that the
cosmological constant is shifted by the energy, again reminiscent of unimodular gravity
\cite{unimodular}.  This shift in $\Lambda$ is physical---it is a backreaction of our ``cloud of clocks'' on
the spacetime---and to see purely gravitational properties, we should limit ourselves to states with relatively
small energies.   For a Planck-scale $\Lambda$, though, this is a rather mild restriction.

To better understand these states, let us consider a WKB approximation,
\begin{align}
{\tilde\Psi} = A e^{iS}  .
\label{e3}
\end{align}
The first order WKB equation,
\begin{align}
\frac{\kappa^2}{64\pi^2}\left(\frac{\delta S}{\delta h}\right)^2
    = \frac{1}{\kappa^2} f^2\left(\frac{f^{\prime 2}}{h^2} -1\right) + \frac{\tilde\Lambda}{3\kappa^2} f^4
    + \frac{\gamma}{\kappa^2} f ,
\label{e4}
\end{align}
can be solved exactly:
\begin{align}
S = \frac{8\pi}{\kappa^2}\int\!d\psi\,\sigma[h,f;\psi]ff'\left\{\sqrt{1+\beta h^2} - \tanh^{-1}\sqrt{1+\beta h^2} \right\} ,
\label{e5}
\end{align}
where
\begin{align}
\beta = \frac{f^2}{f^{\prime2}}\left(\frac{\tilde\Lambda}{3} - \frac{1}{f^2} + \frac{\gamma}{f^3}\right)
\label{e6}
\end{align}
and $\sigma$ is a functional of $h$ and $f$ and a function of $\psi$ such that
\begin{align}
\left\{ \begin{array}{ll}\sigma^2 = 1 \qquad&\hbox{almost everywhere}\\[.5ex]
  \partial_\psi\sigma =0 &\hbox{unless}\  1+\beta h^2 = 0  .
   \end{array}\right.
\label{e7}
\end{align}

The factor $\sigma$ requires a bit of explanation.   As always in the WKB approximation, the
phase $S$ is determined only up to sign. But  eqn.\ (\ref{e4}) holds pointwise, and thus admits solutions
in which this  sign, $\sigma$, can vary with $f$, $h$, and $\psi$.  The choice is not completely
free, though: we must still require that $S$ be annihilated by the momentum constraint.  If the position
dependence of $\sigma$  only arose implicitly from its dependence on $h$ and $f$, then $S$
would be of the form (\ref{d3}),  and we would automatically have $\mathscr{\hat P}S = 0$.  But any
explicit dependence of $\sigma$ on $\psi$ introduces a new term in $\mathscr{\hat P}S$,   proportional to
$(1+\beta h^2)^{1/2}\,\partial_\psi\sigma$.  The constraints require this term to vanish, leading to the
second condition in (\ref{e7}).

To understand the physical significance of this condition, we can rewrite (\ref{e6}) as
\begin{align}
1+\beta h^2 = \frac{\kappa^4{\tilde P}^2}{f^2f^{\prime2}} ,
\label{e9}
\end{align}
where
\begin{align}
{\tilde P} = \frac{1}{\kappa^2}\left[f^2h^2\left(\frac{f^{\prime2}}{h^2}-1\right)
   + \frac{{\tilde\Lambda}}{3}f^4h^2 + \gamma fh^2\right]^{1/2}   .
\label{e9a}
\end{align}
Comparing to (\ref{a6}), we see that ${\tilde P}$ is essentially the classical momentum
$\pi^\psi{}_\psi$, though with a shifted cosmological constant.  Hence the sign
of $S$ can change only when the ${\tilde P}$ goes through zero, the quantum version of the the
classical sewing condition of section \ref{sew}.  There is also an analog of the inequality (\ref{a7x})
for $\gamma$.  For large $f$, $\beta$ is positive, while for $\tilde P$ to go through zero, $\beta$
must be negative.  But  (\ref{e6}) involves the same cubic as (\ref{b3}), and $\beta$
can change sign only if the inequality (\ref{a7x}) is satisfied, albeit again with a
shifted $\Lambda$.

This means, in particular, that for suitable $\gamma$, the quantum theory allows such sewing.   More
precisely, let $(h,f)$ be a metric configuration that, as a classical metric, describes layers joined by
necks in which the extrinsic curvature goes through zero.  A choice of $\sigma$ for this configuration is
then equivalent to a choice of the sign of the extrinsic curvature in each layer.  Just as this sign
is not determined classically by the constraints, it is not determined quantum mechanically by the
 Wheeler-DeWitt equation: different choices of $\sigma$ give different wave functions $\Psi_\sigma$.

We can now explore our WKB wave functions in various regions of midisuperspace.  First, consider
the case of large $f$---that is, specialize to metrics in which $f$ is large in some region of space,
and look at the contribution of that spatial region to the integral (\ref{e5}).  In such a region,
${\tilde P}\gg0$, so the sign $\sigma$ is fixed, and $\beta$ is dominated by the cosmological
constant, yielding
\begin{align}
S \sim \frac{2}{\kappa^2}\,\sigma\biggl(\frac{\tilde\Lambda}{3}\biggr)^{1/2}V  .
\label{e8}
\end{align}
The mean curvature operator (\ref{d2b})---the local Hubble constant---then has a simple action,
\begin{align}
{\hat K}e^{iS} \approx 3\sigma \biggl(\frac{\tilde\Lambda}{3}\biggr)^{1/2}\,e^{iS}  .
\label{e8a}
\end{align}
This is ordinary de Sitter behavior, and, as expected, the sign $\sigma$ determines whether a spatial
region is expanding or contracting.  The probability current (\ref{d13}) provides similar information:
\begin{align}
J_n = -\frac{3V}{4\pi}\sigma\biggl(\frac{\tilde\Lambda}{3}\biggr)^{1/2}\delta_{n0}  ,
\label{e8b}
\end{align}
so $\sigma$ determines the direction of flow of probability.  This is not unlike the WKB approximation
in ordinary quantum mechanics, where a plane wave $e^{ikx}$ is formally a stationary state, but the
probability current reveals a hidden dynamics.

But the wave function does not have its support only on such de Sitter-like regions.  Consider the
contribution of a multilayered ``foamy'' region.  From (\ref{d2b}) and (\ref{d2}),  the mean curvature acts
on diffeomorphism-invariant states as
\begin{align}
{\hat K} = \frac{i\kappa^2}{8\pi}\frac{1}{f^2f'}\partial_\psi\left(f\frac{\delta\ }{\delta h}\right)  ,
\label{e11}
\end{align}
yielding
\begin{align}
{\hat K}e^{iS} = \frac{1}{f^2f'}\partial_\psi\left(\frac{f^2f'}{h}\sigma\sqrt{1+\beta h^2}\right)e^{iS}
  =   \frac{\kappa^2}{f^2f'}\partial_\psi\left(\frac{f}{h}\sigma{\tilde P}\right)e^{iS}   .
\label{e12}
\end{align}
As expected, the WKB wave function is not an eigenfunction of ${\hat K}$.  But  in for foamy
regions of midisuperspace, it is almost an eigenfunction of the spatially averaged mean
curvature.  Indeed,
\begin{align}
\left(\frac{1}{V}\int\!d^3x\,\sqrt{q}{\hat K}\right) e^{iS}
   = -\frac{4\pi\kappa^2}{V}\int\!d\psi\, \frac{f}{h}\left(\frac{h}{f'}\right)^\prime\sigma{\tilde P}\, e^{iS} .
\label{e13}
\end{align}
For a typical wave function, evaluated at a typical multilayered metric, the contribution of each layer will
come with a random sign $\sigma$.  Hence for a configuration with many layers, we expect extensive
cancellation---just as in the classical case, the averaged mean curvature will be much smaller than it
would be for a single de Sitter region.

The evaluation of the probability current leads to an identical conclusion, perhaps even more clearly.
The current is
\begin{align}
J_n = -\frac{3\kappa^2}{4}\int\!d^3x\,\sigma {\tilde P}\varphi_n ,
\label{e10}
\end{align}
and for foamy metrics the right-hand side will again average to a very small number.   Our
wave functions are thus stationary not only in the sense that they are independent of $T$, but
also in the sense that there is very little flow of probability within midisuperspace, so initial foamy
structures will tend to be preserved.

Of course, this leaves open the question of how common such multilayered foamy configurations
are.  An answer requires the next order WKB approximation, which we will turn to
shortly.  Meanwhile, there are a few other features of (\ref{e5}) that  deserve future exploration:
\begin{enumerate}
\item The integrand in (\ref{e5}) diverges when $\beta h^2=0$.  Comparing (\ref{e6}) to (\ref{a7}), we
might suspect this to occur at some sort of horizon.  This is correct: the
vanishing of $\beta h^2$ marks the location of a marginally trapped sphere, a midisuperspace
version of the trapped surfaces that appear in the more general connected sum analysis \cite{Burkhartb}.
\item The integrand vanishes at $1+\beta h^2=0$, the classical neck.  But it also has zeroes at
$1+\beta h^2 = -z_n^2$, where $z_n=\tan z_n$.  These are in a classically forbidden region,
but in principle they are additional sites at which expanding and contracting regions can join.
\item When $1+\beta h^2>0$, $\tanh^{-1}\sqrt{1+\beta h^2}$ has a constant imaginary part.  In a
region between two successive zeroes of $1+\beta h^2$, though, $\sigma$ is constant, and the
imaginary part of $S$ is
$$\frac{4\pi^2 i}{\kappa^2}\int_{\psi_1}^{\psi_2}\!d\psi\,\sigma ff' = \frac{2\pi^2 i}{\kappa^2}\sigma[f^2(\psi_2)-f^2(\psi_1)]$$
with $1+\beta h^2 = 0$ at $\psi_1$ and $\psi_2$.  We saw earlier that at a ``classical'' zero of $1+\beta h^2$,
the value of $f$ is fixed by (\ref{b3}), so the contributions of the two endpoints cancel.  It is less clear
what happens at the nonclassical zeroes $1+\beta h^2 = -z_n^2$.
\end{enumerate}

\section{Next order WKB}

To say more about probabilities for ``foamy'' spacetimes, we will need to go to the next order
in the WKB approximation (\ref{e3}),
\begin{align}
2\frac{\delta S}{\delta h}\frac{\delta A}{\delta h} + A\frac{\delta^2S}{\delta h^2} = 0  .
\label{f1}
\end{align}
This equation involves two functional derivatives at a single point, leading to a well known
divergence.  For our midisuperspace model, we can write
\begin{align}
S = \int\!d\psi\,L ,
\label{f2}
\end{align}
where $L$ depends on $h$ but not $h'$.  Then
\begin{align}
\frac{\delta S}{\delta h(\psi)} = \frac{\partial L}{\partial h}(\psi), \ \
\frac{\delta^2 S}{\delta h(\psi)\delta h(\psi')} = \frac{\partial^2 L}{\partial h^2}(\psi)\delta(\psi-\psi') ,
\label{f3}
\end{align}
giving a factor $\delta(0)$ at $\psi=\psi'$.

There are several proposals for regulating this infinite factor.  In the original formulation of the
Wheeler-DeWitt equation \cite{WdW}, DeWitt suggested that such $\delta(0)$ factors should
be set to zero.  Here, that choice would make the first order WKB approximation exact.
A ``volume average regularization'' \cite{Feng}, applied to this simple case, replaces
$\delta(\psi-\psi')$ by
$$\frac{h}{\ell}\int d\psi'\delta(\psi-\psi')$$
where $\ell = \int\!d\psi\, h$ is the length of the spatial $S^1$ factor.
Heat kernel regularization \cite{Horiguchi} replaces the factor $\delta(\psi-\psi')$
by a heat kernel $K(\psi,\psi';s)$.  In the $s\rightarrow0$ limit,  $K(\psi,\psi';s)$ becomes a delta function,
but if one first takes $\psi\rightarrow\psi'$, the divergences appear as a set of terms involving
inverse powers of $s^{1/2}$, which can be individually regularized.  In the present context, the
outcome is almost the same as the volume average: $\delta(\psi-\psi')$ becomes
$$\alpha h \int d\psi'\delta(\psi-\psi')$$
where $\alpha$ is an undetermined constant with dimensions of inverse length.

\subsection{Heat kernel regularization \label{heat}}

Let us first consider heat kernel regularization.  The second  order WKB equation becomes
\begin{align}
\frac{1}{A}\frac{\delta A}{\delta h}
   = -\frac{\alpha h}{2}\left(\frac{\partial L}{\partial h}\right)^{-1} \frac{\partial^2L}{\partial h^2} ,
\label{f4}
\end{align}
or with $S$ as in (\ref{e5}),
\begin{align}
\frac{1}{A}\frac{\delta A}{\delta h} = \frac{\alpha}{2}\,\frac{1}{1+\beta h^2} .
\label{f5}
\end{align}
This can be solved in closed form:
\begin{align}
A=\exp\left\{ \frac{\alpha}{2}\int\!d\psi\,\frac{1}{\sqrt{\beta}}\tan^{-1}(\sqrt{\beta}h)\right\}  .
\label{f6}
\end{align}

We can now look at relative probabilities for behaviors of the metric---that is,  at features
of the functions $(h,f)$ that lead to comparatively small or large vales of the amplitude $A$.

\begin{itemize}
\item
When $\beta h^2$ is large, the integrand in (\ref{f6}) is small, since $\tan^{-1}(\sqrt{\beta}h)$
is bounded by $\pi/2$.  By (\ref{e9}), this implies that metrics with large regions of high extrinsic
curvature are suppressed.  If we abbreviate
\begin{align}
\left|\frac{\kappa^2{\tilde P}}{ff'}\right| =  \sqrt{1+\beta h^2} = \Delta
\label{f7x}
\end{align}
then for a region of length $\ell_1$,
\begin{align}
\ln A\bigl|_{\Delta\gg1} \
   \sim\ \frac{\alpha\pi}{4}\left\langle\Delta^{-1}\right\rangle\ell_1 \ll \frac{\alpha\ell_1}{2} ,
\label{f7a}
\end{align}
where the angle brackets denote a spatial average.
\item
When $\beta h^2$ is small but positive, it is still the case that $\tan^{-1}(\sqrt{\beta}h) < \sqrt{\beta}h$,
so
\begin{align}
\ln A\bigl|_{\Delta\sim1}\  \lesssim \ \frac{\alpha\ell_1}{2}  .
\label{f7b}
\end{align}
\item
When $\beta h^2<0$,
\begin{align}
\frac{\alpha}{2}\int\!d\psi\,\frac{1}{\sqrt{\beta}}\tan^{-1}(\sqrt{\beta}h)
    = \frac{\alpha}{4}\int\!d\psi\,\frac{1}{\sqrt{|\beta|}}\ln\left(\frac{1 + \sqrt{|\beta|}h}{1 - \sqrt{|\beta|}h}\right) ,
 \label{f7c}
 \end{align}
and the integral receives large contributions when $1+\beta h^2\sim 0$, that is, from the ``necks'' of
the preceding section.  Again by (\ref{e9}), in regions of small $\tilde P$,
\begin{align}
\frac{\alpha}{2}\int\!d\psi\,\frac{1}{\sqrt{\beta}}\tan^{-1}(\sqrt{\beta}h) \approx
   \frac{\alpha}{2}\int\!d\psi\,h\ln\left(\frac{2ff'}{\kappa^2|{\tilde P}|}\right) .
\label{f8}
\end{align}
Classically, we saw in section \ref{sew} that ${\tilde P}\sim\psi^3$ near a neck, while $f'\sim\psi$, so the
integrand goes as $h\ln(a/h\psi)$ for some parameter $a$.  The integral is maximum when integrated
over a region of length $\ell_1 \sim a$, and yields
\begin{align}
\ln A\bigl|_{\Delta\ll1}\  \lesssim \  \frac{\alpha\ell_1}{2}  .
\label{f7cc}
\end{align}
(The dependence on the parameter $a$ imposes no further limits.   By section \ref{sew},
$$a \sim (f^2 D^3K)^{-1/2}\bigl|_{\psi=0}$$
where $K$ is a component of the extrinsic curvature, $D$ is the invariant derivative (\ref{d3}),
and $f(0)$ is fixed by (\ref{b3}).   If the next order of the expansion (\ref{b4}) involves only constants
of order one, then $a\sim f_0$, the characteristic size of the neck.  But $f_0$ itself depends on
the integration constant $\gamma$, and can range from $0$ to $\sqrt{{3}/{\Lambda}}$.  Moreover,
there is no strong reason to demand that the expansion (\ref{b4}) have coefficients of order one;
 if extrinsic curvatures remain small near a neck, $a$ can be large.)
 \end{itemize}

Combining these results, we see that the wave function strongly disfavors spaces with large
regions of high extrinsic curvature (large $\Delta$), while giving roughly equal weight to metrics
with regions of relatively small intrinsic curvature near the start of a ``neck'' (trapped surfaces
$\Delta = 1$) and regions of very small extrinsic curvature where the sign of the expansion can
change ($\Delta=0$).   Note that at this order, there is no limit to the proper length $\ell$.
Since the integrand in (\ref{f6}) is positive, this suggests an infrared divergence.  As
observed in section \ref{WQ}, though, we have not yet accounted for the measure in the inner
product, so it is premature to draw too firm a conclusion.

We can also ask how this higher order WKB term affects the probability current.  For $\beta h^2>-1$,
$A$ is real, so it merely multiplies the lowest order current (\ref{d8}) by $A^2$.  For $\beta h^2<-1$,
the exponent (\ref{f7c}) becomes complex, but the imaginary part is independent of $h$, and does
not contribute to the current.  The moments (\ref{e10}) thus become
\begin{align}
J_n = -\frac{3\kappa^2}{4}\int\!d^3x\,\sigma |A|^2 {\tilde P}\varphi_n .
\label{fx1}
\end{align}

 \subsection{Volume regularization}

 We can now repeat the argument using volume regularization.  Eqn.\ (\ref{f5}) becomes
 \begin{align}
\frac{1}{A}\frac{\delta A}{\delta h} = \frac{1}{2\ell}\,\frac{1}{1+\beta h^2} ,
\label{f9}
\end{align}
and it is tempting to guess a solution
\begin{align}
A=\exp\left\{ \frac{1}{2\ell}\int\!d\psi\,\frac{1}{\sqrt{\beta}}\tan^{-1}(\sqrt{\beta}h)\right\}  .
\label{f10}
\end{align}
This doesn't quite work, though: because of the metric dependence of $\ell$,
 \begin{align}
\frac{1}{A}\frac{\delta A}{\delta h} = \frac{1}{2\ell}\,\frac{1}{1+\beta h^2}
    - \frac{1}{2\ell^2}\int\!d\psi\,\frac{1}{\sqrt{\beta}}\tan^{-1}(\sqrt{\beta}h) .
\label{f11}
\end{align}
In fact, (\ref{f9}) is not integrable:  for any putative solution,
$$\frac{\delta^2A}{\delta h(\psi)\delta h(\psi')} \ne \frac{\delta^2A}{\delta h(\psi')\delta h(\psi)}  .$$
The amplitude (\ref{f10}) is, however, a good \emph{approximate} solution, since it follows from
the preceding section that the extra term in (\ref{f11}) is of order $1/\ell$.   It might be possible to view
this extra piece as a counterterm in the WKB equation (\ref{f1})---it amounts to adding a term proportional
to $\frac{\delta S}{\delta h}$ to the divergent $\frac{\delta^2 S}{\delta h^2}$---%
but it would be a rather peculiar one.

If we ignore this issue, the conclusions from volume regularization are almost identical
to those from heat kernel regularization.  The only difference is that the prefactor $\alpha$
now becomes $1/\ell$, providing a natural infrared cutoff.

\section{Conclusions and next steps}

The space of locally spherically symmetric three-geometries is surprisingly rich.
It includes configurations that exhibit spacetime foam: spaces with multiple layers with
different geometries, joined by connected sums.  The constraints determine the extrinsic curvature
only up to a sign, and a typical multilayered configuration includes both expanding and contracting
regions.  While the these spacetimes are certainly not realistic models of our Universe, they
are qualitatively very similar to the more general foamy geometries discussed in \cite{Carlip1}.

This structure persists in the quantum theory.  By using the ```dust time'' of Brown and Kucha{\v r}, in
which physical time is proper time along a congruence of timelike geodesics, we can reduce the
Wheeler-DeWitt equation to a Schr{\"o}dinger-type equation, and search for stationary states.  These
states have a sign ambiguity in their phase that closely mimics the classical ambiguity in the sign
of the extrinsic curvature.  When evaluated on a multilayer metric, this sign $\sigma$ can change
at precisely the locations that the sign of the extrinsic curvature can change classically.

As we know from quantum mechanics, some care must be taken in interpreting such states.
A stationary state may imply time independence, but it may instead signify a steady state
flow of probability, as in a scattering state.  Indeed, here one can construct a state in which the sign
$\sigma$ in (\ref{e5}) is the same for every layer.  The probability current is then large, proportional
to the de Sitter expansion $\sqrt{{\Lambda}/{3}}$.  But for a typical state, the current (\ref{fx1}),
evaluated at a typical multilayered geometry, will be very small:  the foamy structure will tend
to reproduce itself in a nearly steady state.  This lends support to the proposal of \cite{Carlip1} that
a cosmological constant might be ``hidden'' in spacetime foam.

How common are multilayered foamy configurations?  The second order WKB approximation
has regularization ambiguities, but standard choices such as heat kernel regularization indicate
that they are at least fairly probable.  Indeed, geometries with large regions of high extrinsic curvature
are strongly disfavored, while necks that occur in connected sums of layers are favored.
 Note, though, that the quantum version of the inequality (\ref{a7x}) depends, via (\ref{e2a}), on
 the state, so for fixed $\gamma$ the prevalence of foamy configurations is state dependent.

There are, of course, further questions that must be answered before we can be confident in
these conclusions.  Perhaps the most important is the problem of the measure for the inner
product on midisuperspace.  I have been using the ``naive Schr{\"o}dinger interpretation'' for
$|\Psi|^2$ \cite{Kuchar}, which implicitly assumes that the correct inner product is just a functional
integral over $h$ and $f$.  But this functional integral must be gauge fixed to account for the
invariances generated by the constraints, and the resulting Faddeev-Popov determinants may
affect probabilities \cite{Woodard}.   Partial results in this direction are described in the  Appendix.
Crucially, the Faddeev-Popov determinant is independent of the sign of the extrinsic curvature,
and will not change the cancellation between expanding and contracting regions.

It would also be useful to move beyond stationary states and look at the behavior of wave packets.
Our choice of time makes this a bit difficult: our ``cloud of clocks'' back-reacts on the
geometry, and for this to be unimportant we must restrict the energy to be small (on the scale set by
$\Lambda$).  This means wave packets cannot be too sharply peaked.  Ideally, one would
avoid this by choosing a purely gravitational ``internal time,'' but such a parameter seems very
hard to find in a spacetime containing both expanding and contracting regions.  A more careful study
of the role of the integration constant $\gamma$ would also be valuable.  In particular, it is not
currently clear whether different choices of $\gamma$ form superselection sectors.

A number of straightforward extensions of this work should be possible.  One could apply the same
techniques to a spatial topology $S^3$, by again starting with a manifold $[0,1]\times S^2$
but now shrinking each boundary to a point.  Beyond that, though, Morrow-Jones and Witt
have shown that local spherical symmetry allows a huge variety of topologies \cite{Witt1},
although a detailed analysis would require much more complicated boundary conditions.
An extension to $\Lambda<0$ should also be easy; most of the computations will be
unchanged, though the qualitative results are likely to be quite different.

It would also be interesting to look at the ``reduced phase space'' quantization of this model,
in which the reparametrization invariance of $\psi$ is gauge fixed and the momentum constraint
is eliminated classically.  Whether such a quantization is equivalent to the Dirac quantization method used
here is an open question, involving subtleties in operator ordering and the choice of inner
product  \cite{Kunstatter,Epp}, but it is  certainly worth exploring.  Unfortunately, it is also a bit
harder than it might appear.  Following \cite{Witt1}, we can \emph{nearly} fix the reparametrization invariance
by redefining $\psi$ to make $h$ constant on the initial slice.  But the integral $\int\!hd\psi$ is invariant,
and leaves us with an extra variable (which is also classically time dependent).  The resulting Hamiltonian
constraint then contains a mixture of ordinary and functional derivatives, and becomes technically quite
complicated.

Finally, there may be another somewhat orthogonal approach to these questions.
One way to view spacetime foam in a universe
with large $\Lambda$ is to consider the nucleation of contracting bubbles in expanding regions,
and expanding bubbles in contracting regions.  It is then natural to ask whether
there are instantons that mediate such processes.  In the context of our locally spherically
symmetric midisuperspace, the question is whether there are Euclidean solutions joining
a space with $n$ layers to one with $n+1$ layers.  Some very preliminary calculations
show no obstruction to such solutions, but much more work is needed.

\begin{flushleft}
\large\bf Acknowledgments
\end{flushleft}

 This work was supported in part by Department of Energy grant DE-FG02-91ER40674.
 I would also like to thank the Quantum Gravity Unit of the Okinawa Institute of Science and
 Technology (OIST), where part of the work was completed, for their hospitality.

redpajama_set_name: RedPajamaArXiv


San Diego, CA - TDK-Lambda October 2010 -- Following the successful launch of its LS Series of low-cost, convection cooled power supplies rated from 25 to 150-watts; TDK-Lambda has now announced its 200-watt models. The new LS200 Series feature a universal input range of 88 to 264VAC (no select switch), with an input surge rating of 300VAC for five seconds, and is available in either an enclosed fan-cooled version or in an open-frame U-channel configuration.
Developed to provide low-cost and small size power solutions, the LS200 Series of AC-DC single-output power supplies are available now with the most popular DC output voltages, including: 3.3V, 5V, 7.5V, 12V, 15V, 24V, 36V and 48VDC. All DC outputs are adjustable up to ±10% via a front panel mounted adjustment potentiometer. In addition, a front panel green LED is "On" when the output is good and remote On/Off control is standard.
With its internal fan-cooling the LS200 Series will provide up to 200W of output power. The open-frame U-channel version is rated at 150W with convection cooling, and up to 200W with user provided air-flow. These power supplies not only carry the CE Mark but are certified to UL, CSA, and IEC 60950-1 safety specifications. The units have a with-stand voltage of up to 1.5kVAC input to ground, 3kVAC input to output, and 500VDC output to ground. This series conforms to the immunity standards of EN61000-4-X and the EMC/EMI standards of EN55011-B, EN55022-B and FCC-Class B.
In many applications, the AC input line-quality is poor and factory equipment can cause input voltages to fluctuate and generate high voltage spikes. Such line transients and disturbances can drastically reduce the life of many low-cost power supplies that are only designed for use in office environments. "TDK-Lambda's LS Series offers a new solution for these applications," says David Norton, VP of Marketing for TDK-Lambda Americas. "At a low cost this series still retains the specifications needed for many industrial, automation and test & measurement applications."
All models meet the Power Factor and Harmonic Correction requirements of EN61000-3-2 and are RoHS compliant. Also included are overcurrent and overvoltage protection circuits and all units carry a three year warranty.


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Mostly enjoyed this story. I appreciated the character development and their background stories, particularly Robert's relationship with his Dad. It was a different perspective for a dying parent scenario and one that I hadn't really considered before.
My issue with this book is that it romanticised the student/teacher relationship between Robert and Andrew, whilst also making me feel uncomfortable with it. A few times, I felt Drew took advantage of his position as Robert's teacher and I felt he didn't get the seriousness of what he was doing, by starting a relationship with Robert before he graduated. He seemed more worried about the potential harm to his career, than his being involved with Robert, who was a vulnerable seventeen year old.
I get the appeal of student teacher stories, they're taboo and in this case the age difference was minimal (6 years I think). Yet Andrew crossed the line of what is acceptable behaviour from a teacher for me, and that held me back from supporting their relationship.


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text:
"May We Quote You?"
A Look Back at Commencement 2018
Many ceremonies, big and small, in three days. Over 1,450 degree recipients. More than 10,000 visitors on campus.
Graduation ceremonies are often described by the numbers, especially those held at schools with pi in their names. While the numbers are impressive, something that makes Commencement (well, Commencements) is the stories. The celebrations, the relief, the happiness that come with finally being able to grasp years of hard work in hand. From the students, faculty, parents, friends, and staff present over the past few days, here's a glimpse at Commencement in their own words.
"Graduation day always strikes me with the possibility of rebirth—the hope that our old, broken world can be remade in a more generous spirit by a new generation empowered with knowledge, confident in their abilities, and inspired with a sense of purpose. I love WPI graduation day!" —Kristin Boudreau, Paris Fletcher Distinguished Professor of Humanities; Head, Department of Humanities and Arts, and 2018 recipient of the Trustees' Award for Outstanding Research and Creative Scholarship
"This is so exciting, and I am so proud of him." —Maryam Razavi (whose husband, Payam Razavi, earned a PhD in mechanical engineering) with their children Melody (1) and Nika (6)
"I'm excited to see the looks of my family and friends when they see me in uniform as I continue on with the next step of life." —Anderson DeRosier, chemical engineering, submarine warfare officer. At the ROTC Commissioning ceremony, he received his first salute from his best friend from high school, Hayden Bellamy.
"This is a later-in-life accomplishment for me, and I was a little hesitant to even come today. But this ties a nice bow on this. We ended a month ago and it didn't feel final. This does." —Janna Muhlhausen, MBA through Biogen (left)
"Graduation is a combination of all the hard work we put in and the relief that we are done." —Megan Goodnow, MBA through Biogen (right)
"Commencement is about interacting with the students today. We just had six different students present six different versions of WPI. Seeing each of them tell their stories of the whole WPI experience—that is today." —Wole Soboyejo, Bernard M. Gordon Dean of Engineering and Professor of Engineering Leadership
"It's still exciting to see the students we worked with over the years. You grow a bond. It's a family kind of thing." —Kelly Foley, Facilities (middle)
"I'm looking forward to celebrating everything that I've worked for and dreamed of over the past four years; it's all finally happening." —Kenedi Heather, computer science, cyber second lieutenant U.S. Army. At the ROTC Commissioning ceremony, she received her first salute from her pastor, gunnery sergeant James Hopkins.
"Looking around and seeing all the people you went to class with, you did clubs with. You see that this could be the final time with them, but you're happy you get to finish it with the people you started it with." —Connor Willgress, mechanical engineering
"It's such a tremendous honor to lead our students over Earle Bridge and into the next phase of their careers and lives. It's a highlight of my time at WPI and a moment I'll certainly never forget!" —Natalie Farny, assistant teaching professor of biology & biotechnology and 2018 recipient of the Board of Trustees' Award for Outstanding Teaching
"I'll remember being here and accomplishing my degree as an engineer. My children (ages 5 and 10) are here. This gives them motivation and maybe someday they will be here." —Saida Elfaquir, MS, power systems engineering
"We're so excited ... we can feel the energy—the kids have worked so hard to get here. That's what I'll remember most, and I brought my son's beanie from freshman year; we're going to get a picture of him wearing it, with Gompei. Every year when he moves in, we've taken a picture next to Gompei, showing the progression, so this is going to be the conclusion." —Shelley Murphy (whose son, Patrick, earned a double major in computer science and robotics engineering)
"I remember when I graduated and I know for me it was a really amazing experience, so just being able to help facilitate that for other families and students is really why I enjoy doing this." —Jake Goldblum, complex coordinator, Residential Services
"I have a lot of friends who are seniors and are graduating. I'd rather be doing something instead of just watching the ceremony. I am happy for all my friends, and I know this will be me next year. I only have one year left, and I have to take advantage of it." —Celeste Marsan '19, usher (chemical engineering)
"The feeling of accomplishment." —Kelsey Leigher, architectural engineering
"Being done with four years of hard work and getting to see it all pay off." —April Locke, environmental engineering
"I'll always think of 2 am in the library and that winter storm! Today, it's hard for me to imagine I have my master's in math from WPI, one of the hardest schools. I am proud." —Zhenyan Lin, MS, financial mathematics
"I'm excited to celebrate with my friends and roommates. We are all in the same program. It has been a great opportunity to bond and learn together. But I will still be here. I am continuing here for my PhD." —Kenechukwu Mbanisi, MS, robotics engineering
"This is just a really exciting conclusion for the last 5½ years. And to make it official. It's taken a lot to get to this point." —Hannah Strobel, PhD, biomedical engineering
"We're proud of the students, we hope they excel, and it's an honor to keep everyone safe today." —WPI Police Officer Brian Lavallee
"Ruff." —Bella

redpajama_set_name: RedPajamaCommonCrawl


text:
Blu-ray Review: Hocus Pocus 25th Anniversary Edition
By Dr. Zaius | @ | September 17th, 2018 at 10:00 am
Blu-ray | DVD
Directed by Kenny Ortega
Starring Bette Midler, Thora Birch, Kathy Najimy, Sarah Jessica Parker, Stephanie Faracy
Walt Disney Studios Home Entertainment
It is the 25th anniversary of this cult favorite and to celebrate, there is a spanking brand new Blu-ray for sale everywhere. Just for fans, you can watch the We Love Hocus Pocus Trivia and Treats Edition of Hocus Pocus, which plays the movie "with fun facts, trivia, concept art, storyboards, behind-the-scenes magic, and more!"
In 1993, Hocus Pocus was written off by critics as a silly Halloween-themed kids movie. While it grossed $39.5 million, it barely cracked the top 40 box office for the year, under such "classics" as The Beverly Hillbillies and Dennis the Menace. However, as often is the case with movies aimed at children, the kids grow up and with them come large pairs of nostalgia goggles that, in this case, have turned Disney's witch comedy into a cult classic, especially for those now in their mid-30s. Capitalizing on this nostalgia and the fact that most of these 30-somethings have children of their own now, Disney released a new Anniversary Edition of the film on Blu-ray and digital right in time for its 25th birthday and Halloween season.
More below on Hocus Pocus 25th Anniversary Edition.
Topics: Blu-ray Review, Movies, Reviews
Tags: Bette Midler, David Kirschner, Hocus Pocus, Kathy Najimy, Kenny Ortega, Mick Garris, Sarah Jessica Parker, Stephanie Faracy, Thora Birch


What is common between Bernie Sanders and Donald Trump?
There has been huge attendance in the recent Democratic rallies featuring the Vermont Senator Mr. Bernie Sanders. On the other hand Mr. Donald Trump has been sucking up all the oxygen and attention with his harsh language against immigrants and his fellow presidential candidates in the Republican primaries. The popular support for these candidates may be seen as the pent up anger in the population against the Wall Street and the perceived syphoning of jobs and economic opportunities by the illegal immigrants from the south of the border respectively. This kind of anecdotal explanations by the reporters and political pundits may not serve the nation well for the long term.
In a recent article published in the Washington Post titled: Sorry, but the jobless future isn't a luddite fallacy the author makes the point: Policy makers have a big new problem to deal with: the disappearance of human jobs. Not only will there be fewer jobs for people doing manual work, the jobs of knowledge workers will also be replaced by computers. Almost every industry and profession will be impacted and this will create a new set of social problems — because most people can't adapt to such dramatic change.
The author makes it look like it might be a problem developing into the future. But, reality suggests that the problem started a decade before Regan years, got established through the Clinton years. When the very survival of the economic order was in question due to the financial crisis along with protracted engagements in wars, when Mr. Obama took office, gradual erosion of employable opportunities for the large cross sections of the society has continued but seen merely a side issue. Now as these two national concerns are addressed to some extent, the pent up fear and anger and despair in the population is finding its voice through the emotional appeals of Mr. Sanders and Mr. Trump, even if their tones are completely different.
We have documented this erosion in the meaningful opportunities for employment and economic well being as the Binary Economy well documented in our recent book. Instead of waiting for some one else to fix the problem we have suggested that technical professionals, students and others should systematically cultivate specific skills – Transformational Skills – to enhance their job and career opportunities. These individual centered initiatives will go only so far. It requires national leaders to comprehend the evolving Binary Economy and develop policies and national consensus for coping with its impact. Merely cloaking the Digital Technology as "high tech" and glorifying the thief using DT tools as "hackers" does not do well for societies that have succeeded through the use of all sorts of technologies as the backbone for their success.
Absent such policies we see continued isolation and layering of job opportunities. This leaves people in the lower levels of jobs (largely involving physical work or information tasks) with a sense of despair that unless you are born in a well to do family and go to school in a better neighborhood, your opportunities are permanently shut off. The knowledge economy and the skill based economy have become code words for the success of a few while others stagnate. These issues may be the under currents supporting the fan base for Mr. Sanders and Mr. Trump.
Clearly the answer is not the anger and abuse against the immigrants unleashed by Mr. Trump. Raising minimum wage and such social policies for re-distribution of wealth by Mr. Sanders may be temporary fix. But they may not be the sufficient or sustainable long term answers as well. As noted in the Washington Post article: If we can develop the economic structures necessary to distribute the prosperity we are creating, most people will no longer have to work to sustain themselves. They will be free to pursue other creative endeavors. The problem, however, is that without jobs, they will not have the dignity, social engagement, and sense of fulfillment that comes from work. The life, liberty and pursuit of happiness that the constitution entitles us to won't be through labor, it will have to be through other means. Is this the way of life we want?
What are the other means for such living, where anything a human can do – read, write, think, analyze, infer, conclude, decide and control – can all be done by computers and IT solutions? Mr. Steven Hawkins has raised this specter and fear one can envision due to irrational deployment of Artificial Intelligence in the future. But, the irrational deployment of digital technology for incremental gains in cost reduction and elimination of human employment has started at least four decades ago. This has been happening in parallel with unprecedented gains in many new areas thanks to advancements in Digital Technology. Are these two sides of the lever balanced? If not how to maintain the balance? These are the questions that have not been addressed for the past four decades.
There is also a need for discussion on another balancing act: Investors – those with capital or money – can work and use all resources across the globe and we call that as global capitalism. Yet national economic policies – of the countries in which the same investors live – are constrained by what the government can do within its shores. When human beings as employable resource migrate from rural areas of China and India to the urban centers it is considered legal. The same migration of people for the same reasons from poor countries to more affluent regions in their neighborhoods we decry them as illegal labor.
Such balancing act in several fronts are entirely new. This is an open space for any national leader from either party to develop a dialogue and national mandate for future actions. Whoever fills this void may emerge as our next President. Or we may go through another election without any consensus and more shouting and name calling. After all any social system under great stress needs to find the vent to release its anger and frustration. We see this release of frustration as the common outcome of the campaigns of Mr. Sanders and Mr. Trump at this time.
Some might say "the train has already left the station and it is too late to do anything". Such pessimism and acceptance of systematic loss of opportunity for large segments of work force and relegate them as un-employable will only fuel the pent up anger. Then we can expect to see even larger crowds at Bernie Sanders rallies or higher poll numbers for Mr. Trump, even though neither of them may win the presidency. But it will also weaken any future president without a mandate to strike the right balance required, as mentioned above.
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Sometimes, there may be a waiting list for child care services and to ensure the system is fair, the Australian Government has "Priority of Access Guidelines" for allocating places in these circumstances. The guidelines only apply to Child Care Benefit approved child care.They are used when there is a waiting list for a child care service or when a number of parents are applying for a limited number of vacant places.
Every Child Care Benefit approved child care service has to abide by the guidelines and tell you about them when you enrol your child into care.
When Exploring Hub is at Capacity, before putting you on a waiting list, the 'Priority of Access Guides are followed' After these guides have been followed and there are still no positions available, you will be put on a waiting list which requires an Administration fee of $50.00. This fee is non-refundable, however your child will stay on Exploring Hub's waiting list until a spot becomes available. Once a spot becomes available and you wish to take up that spot, a bond of $250.00 and $50.00 administration fee is to be paid to secure the spot.
For more information on waiting lists please contact the Director of Teaching during business hours.


text:
Malala Yousafzai Just Joined Twitter: 'I'm Fighting for Girls — Will You Join Me?'
The ultimate #FollowFriday.
By Andrea Park
Photo by Volkan Furuncu/Anadolu Agency/Getty Images
Malala Yousafzai has helped improve conditions for Syrian refugees, underprivileged girls, and countless others around the world, and now, she's making Twitter a better place as well.
The 19-year-old humanitarian tweeted for the first time on Friday, although, according to Twitter, the @Malala account was made and set aside in November 2012. "Hi, Twitter," she wrote in her first-ever post. She followed up the short and sweet intro with a powerful thread of six tweets in which she asked her followers to join her in increasing educational opportunities for girls around the world.
"Today is my last day of school and my first day on @Twitter," she wrote. "Graduating from secondary school (high school) is bittersweet for me. I'm excited about my future, but I know that millions of girls around the world are out of school and may never get the opportunity to complete their education." Malala continued, "Next week, I will be back on my #GirlPowerTrip to meet girls in Middle East, Africa & Latin America. Each girl's story is unique — and girls' voices are our most powerful weapons in the fight for education and equality." According to the Malala Fund website, the Girl Power Trip started in April and is a way for Malala to "spend as much time as possible meeting with girls around the world and fighting for their futures" before she heads to Oxford University in the fall.
Malala finished her first (and, may I say, expertly crafted) Twitter thread with a call to action: "On and off Twitter, I'm fighting for girls — will you join me?✋🏾" TBH, where Malala leads, we will follow.
https://twitter.com/Malala/status/883324902186635266
Malala has accomplished more before her high school graduation than most adults do in their entire lives. She won the Nobel Peace Prize in 2014, at age 17, has repeatedly called out Donald Trump for racism and mean-spiritedness, has raised millions of dollars through her Malala Fund to improve educational opportunities for young women and, earlier this year, became the youngest-ever United Nations Messenger of Peace.
Although the teen, who turns 20 next week, has dreams of becoming the prime minister of Pakistan, right now she's focusing on going off to college. "It is quite a good moment because you live without your parents and you live in a college and that's the exciting part," she told Teen Vogue earlier this year. "After that, I'm not sure what I'm going to do in terms of career, but I'm really sure that I'm going to be focused on the Malala Fund and the work we do for girls' education, so that's going to be my mission."
Related: These Are Malala Yousafzai's Next Steps for Changing the World
Keywordssnd malala yousafzai


text:
Andrew Bacevich
(1947-07-05) July 5, 1947 (age 73)
Normal, Illinois, U.S.
United States Military Academy (BS)
Princeton University (MA, PhD)
Historian, writer, professor; Colonel, U.S. Army (Retired)
Analysis of U.S. foreign policy
4 (1 deceased)
Military career
Service/branch
Battles/wars
Andrew J. Bacevich Jr. (/ˈbeɪsəvɪtʃ/ BAY-sə-vitch); born July 5, 1947) is an American historian specializing in international relations, security studies, American foreign policy, and American diplomatic and military history. He is a Professor Emeritus of International Relations and History at the Boston University Frederick S. Pardee School of Global Studies.[1] He is also a retired career officer in the Armor Branch of the United States Army, retiring with the rank of colonel. He is a former director of Boston University's Center for International Relations (from 1998 to 2005), now part of the Pardee School of Global Studies.[1] Bacevich is the co-founder and president of the Quincy Institute for Responsible Statecraft.
Bacevich has been "a persistent, vocal critic of the U.S. occupation of Iraq, calling the conflict a catastrophic failure."[2] In March 2007, he described George W. Bush's endorsement of such "preventive wars" as "immoral, illicit, and imprudent."[2][3] His son, Andrew John Bacevich, also an Army officer, died fighting in the Iraq War in May 2007.[2]
2 Writings
4.2 Essays and reporting
Bacevich was born in Normal, Illinois, the son of Martha Ellen (née Bulfer; later Greenis) and Andrew Bacevich, Sr.[4] His father was of Lithuanian descent and his mother was of Irish, German, and English ancestry.[5] He graduated from the United States Military Academy at West Point in 1969 and served in the United States Army during the Vietnam War, serving in Vietnam from the summer of 1970 to the summer of 1971.[6]
Later he held posts in Germany, including in the 11th Armored Cavalry Regiment; the United States; and the Persian Gulf up to his retirement from the service with the rank of colonel in the early 1990s. His early retirement is thought to be a result of his taking responsibility for the Camp Doha (Kuwait) explosion[6] in 1991 in command of the 11th Armored Cavalry Regiment.[7] He holds a Ph.D. in American Diplomatic History from Princeton University, and taught at West Point and Johns Hopkins University before joining the faculty at Boston University in 1998.[citation needed]
Bacevich has described himself as a "Catholic conservative"[8] and initially published writings in a number of politically oriented magazines, including The Wilson Quarterly. He advocates for a non-interventionist foreign policy.[9] His recent writings have professed a dissatisfaction with the Bush Administration and many of its intellectual supporters on matters of U.S. foreign policy.
On August 15, 2008, Bacevich appeared as the guest of Bill Moyers Journal on PBS to promote his book, The Limits of Power. As in both of his previous books, The Long War (2007) and The New American Militarism: How Americans are Seduced by War (2005), Bacevich is critical of U.S. foreign policy in the post-Cold War era, maintaining the United States has developed an over-reliance on military power, in contrast to diplomacy, to achieve its foreign policy aims. He also asserts that policymakers in particular, and the U.S. people in general, overestimate the usefulness of military force in foreign affairs. Bacevich believes romanticized images of war in popular culture (especially films) interact with the lack of actual military service among most of the U.S. population to produce in the U.S. people a highly unrealistic, even dangerous notion of what combat and military service are really like.
Bacevich conceived The New American Militarism as "a corrective to what has become the conventional critique of U.S. policies since 9/11 but [also] as a challenge to the orthodox historical context employed to justify those policies."
Finally, he attempts to place current policies in historical context, as part of a U.S. tradition going back to the Presidency of Woodrow Wilson, a tradition (of an interventionist, militarized foreign policy) which has strong bi-partisan roots. To lay an intellectual foundation for this argument, he cites two influential historians from the 20th century: Charles A. Beard and William Appleman Williams.
Ultimately, Bacevich eschews the partisanship of current debate about U.S. foreign policy as short-sighted and ahistorical. Instead of blaming only one president (or his advisors) for contemporary policies, Bacevich sees both Republicans and Democrats as sharing responsibility for policies which may not be in the nation's best interest.
In March 2003, at the time of the U.S. invasion of Iraq, Bacevich wrote in the Los Angeles Times that "if, as seems probable, the effort encounters greater resistance than its architects imagine, our way of life may find itself tested in ways that will make the Vietnam War look like a mere blip in American history."[2]
Bacevich's book American Empire: The Realities and Consequences of US Diplomacy, published in 2004, was highly praised by Professor of International Relations and author Peter Gowan for being "a tonic to read: crisp, vivid, pungent, with a dry sense of humour and sharp sense of hypocrisies." Gowan describes Bacevich as a "conservative, who explains that he believed in the justice of America's war against Communism, and continues to do so, but once it was over came to the conclusion that U.S. expansionism both preceded and exceeded the logic of the Cold War, and needed to be understood in a longer, more continuous historical durée."[10]
Bacevich wrote an editorial about the Bush Doctrine published in The Boston Globe in March 2007.[3]
In an article of The American Conservative dated March 24, 2008, Bacevich depicts Democratic Presidential candidate Barack Obama as the best choice for conservatives in the fall. Part of his argument includes the fact that "this liberal Democrat has promised to end the U.S. combat role in Iraq. Contained within that promise, if fulfilled, lies some modest prospect of a conservative revival."[11] He also goes on to mention that "For conservatives to hope the election of yet another Republican will set things right is surely in vain. To believe that President John McCain will reduce the scope and intrusiveness of federal authority, cut the imperial presidency down to size, and put the government on a pay-as-you-go basis is to succumb to a great delusion."
In the October 11, 2009, issue of The Boston Globe,[12] he wrote that the decision to commit more troops to Afghanistan may be the most fateful choice of the Obama administration. "If the Afghan war then becomes the consuming issue of Obama's presidency – as Iraq became for his predecessor, as Vietnam did for Lyndon Johnson, and as Korea did for Harry Truman – the inevitable effect will be to compromise the prospects of reform more broadly," Bacevich wrote.
In his article "Non Believer" in the July 7, 2010, issue of The New Republic, Bacevich compared President George W. Bush, characterized as wrong-headed but sincere, with President Obama, who, he says, has no belief in the Afghanistan war but pursues it for his own politically cynical reasons: "Who is more deserving of contempt? The commander-in-chief who sends young Americans to die for a cause, however misguided, in which he sincerely believes? Or the commander-in-chief who sends young Americans to die for a cause in which he manifestly does not believe and yet refuses to forsake?"[13]
In an October 2010 interview with Guernica Magazine, Bacevich addressed his seemingly contradictory stance on Obama. While Bacevich supported Obama during the 2008 presidential race in which Obama repeatedly said he believed in the Afghanistan war, Bacevich has become increasingly critical of Obama's decision to commit additional troops to that war: "I interpreted his campaign rhetoric about Afghanistan as an effort to insulate him from the charge of being a national security wimp. His decision to escalate was certainly not a decision his supporters were clamoring for."[14]
Regarding nuclear policy in particular, Bacevich noted in The Limits of Power that there is no feasible scenario under which nuclear weapons could sensibly be used and keeping them entails many other risks: "For the United States, they are becoming unnecessary, even as a deterrent. Certainly, they are unlikely to dissuade the adversaries most likely to employ such weapons against us – Islamic extremists intent on acquiring their own nuclear capability. If anything, the opposite is true. By retaining a strategic arsenal in readiness (and by insisting without qualification that the dropping of atomic bombs on two Japanese cities in 1945 was justified), the United States continues tacitly to sustain the view that nuclear weapons play a legitimate role in international politics ... ."[15]
Bacevich's papers are archived at the Howard Gotlieb Archival Research Center at Boston University.
On May 13, 2007, Bacevich's son, Andrew John Bacevich, was killed in action in Iraq by an improvised explosive device south of Samarra in Saladin Governorate.[16] The younger Bacevich, 27, was a first lieutenant in the U.S. Army,[17] assigned to the 3rd Battalion, 8th U.S. Cavalry Regiment, 1st Cavalry Division. Bacevich also has three daughters.[17]
This list is incomplete; you can help by adding missing items with reliable sources.
Bacevich, Andrew J. (1986). The pentomic era : the US Army between Korea and Vietnam. Washington DC: National Defense University Press. OCLC 13525013.
Diplomat in Khaki: Frank Ross McCoy and American Foreign Policy, 1898–1949 (University Press of Kansas, 1989) ISBN 0700604014.
American Empire: The Realities and Consequences of US Diplomacy (Harvard University Press, 2004) ISBN 0-674-01375-1.
The New American Militarism: How Americans Are Seduced by War (Oxford University Press Inc, U.S., 2005) ISBN 0-19-517338-4.
The Long War: A New History of U.S. National Security Policy Since World War II (Columbia University Press, U.S., 2007) ISBN 0-231-13158-5.
The Limits of Power: The End of American Exceptionalism (Macmillan, New York, U.S., 2008) ISBN 0-8050-8815-6.
Washington Rules: America's Path to Permanent War (Macmillan, U.S., 2010) ISBN 0-8050-9141-6.[18][19]
Breach of Trust: How Americans Failed Their Soldiers and Their Country (Henry Holt and Co., 2013) ISBN 978-0-8050-8296-8.
America's War for the Greater Middle East: A Military History (Random House, 2016) ISBN 978-0553393934.
Twilight of the American Century (University of Notre Dame Press, 2018) ISBN 9780268104856
The Age of Illusions: How America Squandered Its Cold War Victory (Metropolitan Books, 2020) ISBN 9781250175083
Essays and reporting
Bacevich, Andrew (April 27, 2016). "The battle for truth over Saudi Arabia's Role in 9/11". Retrieved 2 May 2016.
"America Decides: Is Change in the Heir?". The Diplomat. 7 (3): 28–30. Sep–Oct 2008.
"Breaking Washington's Rules". The American Conservative. Vol. 10 no. 1. January 2011. pp. 23–26.
Bacevich, Andrew J. (September 2012). "How we became Israel : peace means dominion for Netanyahu—and now for us". The American Conservative. Retrieved 2015-07-20.
Reprinted: Bacevich, Andrew J. (November 2012). "How we became Israel". Readings. Harper's Magazine. 325 (1950): 17–20.
Andrew J. Bacevich, "The Old Normal: Why we can't beat our addiction to war", Harper's Magazine, vol. 340, no. 2038 (March 2020), pp. 25–32. "In 2010, Admiral Michael Mullen, chairman of the Joint Chiefs of Staff, declared that the national debt, the prime expression of American profligacy, had become 'the most significant threat to our national security.' In 2017, General Paul Selva, Joint Chiefs vice chair, stated bluntly that 'the dynamics that are happening in our climate will drive uncertainty and will drive conflict." (p. 31.)
Biography portal
Republican and conservative support for Barack Obama in 2008
The Imperial Presidency
^ a b "Boston University – Andrew J. Bacevich – The Frederick S. Pardee School of Global Studies". bu.edu.
^ a b c d MacQuarrie, Brian (2007-05-15). "Son of professor opposed to war is killed in Iraq". The Boston Globe.
^ a b Bacevich, Andrew J. (2007-03-01). "Rescinding the Bush Doctrine". The Boston Globe. Retrieved 2007-05-01.
^ "Martha Greenis Obituary". The Times.
^ "OralHistory". westpointcoh.org. Archived from the original on 2013-09-25.
^ a b "TAB I – The Camp Doha Explosion and Fires (July 1991)". Environmental Exposure Report – Depleted Uranium in the Gulf (II). United States Department of Defense. December 13, 2000.
^ "About Andy Bacevich". The Atlantic. August 16, 2008.
^ Barlow, Rich (November 22, 2010). "Are Americans God's Chosen People?". BU Today. Retrieved February 18, 2013.
^ "Andrew Bacevich: 'Christian realism' for foreign affairs | NewBostonPost". newbostonpost.com. Retrieved 2017-12-12.
^ Gowan, Peter (May–June 2003). "Instruments of Empire". New Left Review.
^ "The American Conservative". The American Conservative. Archived from the original on 2008-06-12. Retrieved 2017-12-12.
^ Bacevich, Andrew J. (October 11, 2009). "Afghanistan – the proxy war". The Boston Globe.
^ Bacevich, Andrew, "Non-Believer", The New Republic, July 7, 2010. Retrieved 2010-09-05. Referenced in Frank Rich, "Freedom's just another word", The New York Times, September 4, 2010 (September 5, 2010 p. WK8, NY ed.).
^ Bacevich, Andrew J. (October 1, 2010). "Blood Without Guts". Guernica Magazine. Archived from the original on October 6, 2010.
^ pp. 178-179
^ "Honor the Fallen Army 1st Lt. Andrew J. Bacevich". Military Times.
^ a b "Soldier from Fort Hood killed in Iraq". Associated Press. May 14, 2007. Archived from the original on May 17, 2007. Retrieved May 15, 2007.
^ "Search". Mises Institute.
^ Review of Washington Rules at NY Times
Wikiquote has quotations related to: Andrew Bacevich
Wikimedia Commons has media related to Andrew Bacevich.
Academic profile of Prof. Andrew Bacevich at the Pardee School of Global Studies, Boston University
"Is the war in Afghanistan worth fighting?" Great Debate at Boston University, November 4, 2009
Appearances on C-SPAN
Democracy Now! appearances
"I Lost My Son to a War I Oppose", The Washington Post, May 27, 2007
Interview with Andrew Bacevich in Bostonia, alumni magazine of Boston University, Seduced by War
Extensive excerpts from The New American Militarism
Conversations with Andrew Bacevich
Andrew Bacevich bloggings at HuffPost
Bill Moyers Journal interview of Andrew Bacevich
Author reading with Q&A at Politics and Prose on 8 January 2020


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