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Our Leaders, Our Institutions, and Our Media Have Failed Us. Here's How.
September 27, 2020 1320Elements
It is difficult to write about Covid-19 without feeling that you are simply adding to the cesspool of unsolicited opinion and the ceaseless stream of obfuscating pseudo-facts. It is also hard to not write about Covid without appearing self-absorbed and insensible of the crisis. Yet, it might also be argued that it is an abdication… Continue reading Our Leaders, Our Institutions, and Our Media Have Failed Us. Here's How.
Communicating Information to the Public during the Coronavirus Crisis
March 24, 2020 March 24, 2020 1320Elements
The manner of communicating the coronavirus crisis to the public has been woeful, if not dismal. The message about just how serious the outbreak will be hasn't been getting through. Arguably, it had no chance of ever doing so. The immediate confusion that has arisen from the UK government's "lockdown that isn't a lockdown" is… Continue reading Communicating Information to the Public during the Coronavirus Crisis
Book Review: 'The Rockblaster' by Henning Mankell
March 20, 2020 October 15, 2020 1320Elements
A lot of things will make me pick up a novel. I like a title that infers a great deal without being deliberately obscure. The cover should be similarly suggestive without duping the reader into believing a promise that isn't going to be fulfilled. The name of the author is less important than the idea… Continue reading Book Review: 'The Rockblaster' by Henning Mankell
Frank Buckland's Visit to the 21st Century
March 1, 2020 March 21, 2020 1320Elements
Frank Buckland's Visit To The 21st Century is the novel that I always wanted to write but struggled to find time to begin, much less complete. It was always going to be an onerous undertaking given the scope of the material. Our relationship with other livings things touches on every significant aspect of our being,… Continue reading Frank Buckland's Visit to the 21st Century
1320BOOKS AT THE POET'S BOX, DUNDEE
November 16, 2019 1320Elements
1320Books has found a perfect niche with The Poets's Box, a pop-up shop in Dundee's Wellgate Centre specially created as part of the nationwide Being Human Festival. Orginally conceived as a platform for research, understanding and discussion of the humanities, the festival has grown into a national phenomenon and a major winter festival of 'a… Continue reading 1320BOOKS AT THE POET'S BOX, DUNDEE


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The Santiago General Cemetery () in Santiago, Chile, is one of the largest cemeteries in Latin America with an estimated two million burials. The cemetery was established in 1821 after Chile's independence when Bernardo O'Higgins inaugurated the Alameda de las Delicias along the old course of the Mapocho River. O'Higgins set aside more than 85 hectares of land for the foundation of what became a magnificent grounds filled with ornate mausoleums surrounded by palm and leaf trees set amidst lush gardens and numerous sculptures, which have been estimated be 237. The cemetery, which is located northwest of Cerro Blanco, serves as a true urban park for Santiago located in the municipality of Recoleta.

This cemetery is the final resting place for at least 172 of the most influential people in Chile, including all but two of the deceased Presidents of Chile, the exceptions being Gabriel González Videla and Augusto Pinochet. One of the most visited memorials is that of former President Salvador Allende who had been buried in the Santa Ines cemetery at Viña del Mar following his suicide in the 1973 coup d'état. With the democratic changes that began in the 1990s, Allende was exhumed and his remains were transported in a solemn procession through the streets of Santiago to a place of honor in the Cementerio General de Santiago. The cemetery also has a memorial to the people that were 'disappeared' during the dictatorship of Augusto Pinochet that ousted President Allende.

The gatehouse that serves as the main entrance to the cemetery is crowned by a dome, which acts as a terminating vista for La Paz Avenue. This entrance is preceded by the Plaza La Paz, a semicircular plaza whose curved portion is framed by two exposed brick arcades. At the center of the square stands a monument dedicated to the people who died in the Church of the Company Fire. The cemetery can be accessed via Cementerios metro station.

The historical portion of the cemetery was designated as a national monument in 2010, reaching the same status as Patio 29.

Notable interments
 Salvador Allende (1908–1973), President of Chile
 Arturo Alessandri (1868–1950), President of Chile
 Jorge Alessandri (1896–1986), President of Chile
 Eduardo Frei Montalva (1911–1982), President of Chile
 Eduardo Alquinta (1945–2003), Musician, leader of the Chilean band Los Jaivas
 Patricio Aylwin (1918–2016), President of Chile
 Jaime Guzmán (1946–1991), Senator
 Erich Honecker (1912–1994), General Secretary of the Socialist Unity Party of Germany
 Víctor Jara (1932–1973), poet, folk singer
 Orlando Letelier (1932–1976), statesman
 Carlos Prats (1915–1974), general, statesman
 Violeta Parra (1917–1967), folk singer
 Miguel Enríquez Espinosa (1944–1974), General Secretary of the Revolutionary Left Movement
 Max Westenhofer (1871–1957), German scientist and professor of pathology at the University of Berlin and the University of Chile
 Daniel Zamudio (1987-2012), murder victim

See also
 Memorial for the Disappeared
 Patio 29

References

External links

Cemeteries in Chile
Buildings and structures in Santiago
1821 establishments in Chile
Tourist attractions in Santiago, Chile
Parks in Santiago, Chile
National Monuments of Chile

redpajama_set_name: RedPajamaWikipedia


text:
\section{Introduction}

Space-time curvature leads to particle production.  This effect arises in the study of quantum field theory in curved geometries and has profound observational and theoretical consequences.   In an eternally inflating de Sitter geometry the particle production takes a particularly simple form; in the standard vacuum state an inertial observer observes a bath of thermal radiation emitted by the cosmological horizon \cite{GH}.
During inflation the universe was approximately de Sitter, and the effects of this particle production have been observed indirectly in the cosmic microwave background.

In this note we investigate free quantum field theory in de Sitter space and discover that this apparently staid and hoary subject still has a few surprises left in store.  We will describe a remarkable  structure arises when one studies de Sitter field theory in global coordinates.  In every case which we have studied, the field equations reduce to an effective one dimensional quantum mechanics problem which takes a special algebraic form.  For fields of integer spin the quantum mechanical potential is a multi-soliton solution to the KdV equation known as the Poschl-Teller potential.  When the number of space-time dimensions is odd, the potentials describe an integer number of coincident solitons.  This potential has the unusual property that it is reflectionless, i.e. the reflection coefficients vanish for all values of the frequency. For the case of half-integer spins the equations of motion  reduce to a similar one-dimensional Schrodinger problem which can be regarded as a complexified cousin of the Poschl-Teller potential.\footnote{In fact, precisely this potential also arises in a somewhat different context,  in the study of non-Hermitian Hamiltonians with PT symmetry.}  In each case the quantum spectrum can be solved using algebraic techniques related to supersymmetry.  When the number of space-time dimensions is even -- including the most important case of $D=4$ -- a similar structure arises.  The potentials can be thought of as describing a half-integer number of coincident KdV solitons and, while they are not reflectionless, can be solved using similar algebraic techniques based on supersymmetry.

The physical implications of this are remarkable.  In de Sitter space there are three natural sets of modes that one can use to define Fock space operators for free field theory.  There are those with no physical excitations at ${\cal I}^+$, those with no excitations at ${\cal I}^-$ and those which are obtained by analytic continuation from Euclidean signature.   Each of these choices form a complete basis of solutions to the wave equation.  Hence the modes are linearly related; the coefficients in the linear transformation between these different modes are known as Bogoliubov coefficients.

The reflection and transmission coefficients of the auxilliary one dimensional quantum mechanics problem described above are precisely the Bogoliubov coefficients relating the modes at ${\cal I}^-$ with those at ${\cal I}^+$.  The reflectionless property implies that the mode solutions are, up to a phase factor, equal.  Thus, in an odd number of space-time dimensions de Sitter space is transparent, in the sense that a state with no particles in the far past will evolve into one with no particles in the far future.  At any intermediate time the state will contain particles; only when measured at future infinity do these excitations disappear.

In fact, this property has been noted previously in discussions of de Sitter invariant vacuum states referred to as Motolla-Allen or $\alpha$ vacua \cite{Mottola:1984ar, Allen:1985ux}.  Each choice of mode decomposition defines a different quantum vacuum state; the three sets of modes described above define the $|in\rangle$ (no particles at ${\cal I}^-$), $|out\rangle$ (no particles at ${\cal I}^+$) and $|E\rangle$ (Euclidean or Bunch-Davies) states, respectively.   It was observed in \cite{Bousso:2001mw} that for scalar fields $|in\rangle=|out\rangle$ in an odd number of space-time dimensions.  The goal of the present paper is to generalize this result to higher spin fields using the algebraic techniques described above.

Before proceeding, we note that the Euclidean vacuum is the only de Sitter invariant vacuum state with the same short-distance structure as the usual Minkowksi vacuum.
If all modes are placed in the $|in\rangle$ or $|out\rangle$ vacuum then the resulting state is de Sitter invariant but has a non-standard short-distance structure which affects physics at length scales much shorter than the de Sitter radius.  This leads to problems when considering interacting field theory (see e.g. \cite{ Einhorn:2002nu, Kaloper:2002cs, Banks:2002nv}).  Nevertheless, the structure of the $|in\rangle$ and $|out\rangle$ wave functions still contains important physical information.  For example, while it seems unphysical to place arbitrarily short distance modes in the $|in\rangle$ and $|out\rangle$ states, it is certainly  reasonable to study states in which certain finite wavelength modes are placed in these states as opposed to the standard Euclidean vacuum.  Such states are essentially finite norm excitations of the Bunch-Davies vacuum and have been proposed as candidate states for quantum fields during inflation \cite{Einhorn:2003xb, deBoer:2004nd}.

Moreover, the properties of the $|in\rangle$ and $|out\rangle$ wave functions may be relevant for quantum gravity in an eternally inflating universe.  For example, they (and other similar states) have appeared in various discussions of the dS/CFT correspondence  \cite{Strominger:2001pn} (see e.g. \cite{Anninos:2011jp} for a recent discussion of related issues).

In the present paper we will restrict ourselves to the study of formal properties of free wave equations in de Sitter space, leaving potential applications to cosmology for the future.  In the following section we describe in detail the reflectionless property of the scalar wave equation.  The physical content of this section has appeared in the literature before, but we will emphasize a somewhat different approach based on algebraic techniques.  In
section 3 the discussion is extended to Dirac fields, before turning
to higher spin fields in section 4. Section 5 contains a brief
discussion of open questions.

\section{Scalar Fields in de Sitter Space}

\subsection{Wave Equations in Global Coordinates}

In this section we consider a free scalar field $\phi$ in $n$ dimensional de Sitter space. In global coordinates the metric is
\begin{align}
  {ds^2\over \ell^2}=-d\tau^2+\cosh^2\tau ~ d\Omega_{n-1}^2~
\end{align}
where $d\Omega_{n-1}^2$ is the metric on the sphere $S^{n-1}$.
This space-time is the maximally symmetric solution
of Einstein's equations with a positive cosmological constant $\Lambda=(n-2)(n-1)/(2\ell^2)$.
We henceforth use units where the de Sitter radius is $\ell=1$.  In these coordinates asymptotic past/future infinity are given by $\tau\to\pm\infty$.

It was noticed in \cite{Bousso:2001mw} that when $n$ is odd there is a sense in which
there is no production of quantum excitations of $\phi$ due to the curvature of de Sitter space.
In this section, we will reproduce this result and emphasize
the connection with the theory of reflectionless potentials.  The physical content of this section is not particularly new and we suspect that
the appearance of KdV solitons in the scalar wave equation has been noticed  by many people
over the years (see e.g. \cite{polyakov}).  There is a vast literature on the subject of quantum field theory in de Sitter space; we refer the reader to \cite{BD} for an introduction.

To quantize the field $\phi$, we start by choosing a complete basis of solutions to the wave equation
\begin{align}
  (\nabla^2-m^2) \phi = 0 ~,
\end{align}
where $m$ is the mass of the field.  In this paper we will, for simplicity, focus on massive fields in order to avoid the various subtleties which arise for massless fields in de Sitter space.

We denote by $\{\phi_{\vec \ell}\}$ a complete basis of solutions to the wave equation, where
${\vec \ell}$ is a set of parameters labelling the modes.  We expand $\phi$ in terms of raising and
lowering operators as
$$
\phi = \sum_{\vec \ell} a_{\vec \ell} \phi_{\vec \ell} + {a_{\vec \ell}}^\dagger \phi_{\vec \ell}^*
$$
In order to guarantee that
$a_{\vec \ell}$ and $a_{\vec \ell}^\dagger$ obey the usual algebra of raising and lowering operators, we must require the $\phi_{\vec \ell}$ to be normalized with respect to the standard Klein-Gordon inner product
$$
\langle f | g\rangle = i \int_\Sigma d\Sigma^\mu (f^* \partial\mu g - \partial_\mu f^* g)
$$
Here the integral is over an arbitrary spacelike slice $\Sigma$ through spacetime, which we can take to be a sphere $S^{n-1}$ at constant time.
 Associated to the mode decomposition $\phi_{\vec \ell}$ we can define a vacuum state, which is the state in the Fock space annihilated by $a_{\vec \ell}$ for all values of $\vec \ell$.

In curved space-time there is no canonical choice of mode decomposition $\phi_{\vec \ell}$, and hence no canonical choice of vacuum state.
One natural choice of modes are those $\phi_{\vec \ell}$ which are purely positive frequency at $\tau\to -\infty$.  These modes
will oscillate like $e^{-i \omega \tau}$ at $\tau\to-\infty$, and define a state $|in\rangle$.  From the point of view of a local observer at ${\cal I}^-$ this state will appear to have no particles.  Likewise,
the modes which go like $e^{-i\omega\tau}$ at $\tau \to +\infty$ define a state $|out\rangle$ with no particles in the
far future.  Neither of these states are equivalent to the Euclidean vacuum $|E\rangle$ which is defined using the Hartle-Hawking construction.  This vacuum is constructed using modes which are smooth on the lower half of sphere $S^{n}$ obtained by Wick rotation of the de Sitter metric to Euclidean signature.
 The Euclidean modes have both positive and negative frequency parts at ${\cal I}^\pm$.

It is instructive to work out the form of the modes explicitly.
In global coordinates, the de Sitter laplacian is
\begin{align}
  \nabla^2=-(\cosh\tau)^{1-n}\partial_\tau\left[(\cosh\tau)^{n-1}\partial_\tau\right]
  +(\cosh\tau)^{-2}\Delta_{S^{n-1}}~,
\end{align}
where $\Delta_{S^{n-1}}$ is the Laplacian on $S^{n-1}$.
We can then expand these wave functions in terms of spherical harmonics on $S^{n-1}$,
which are unit normalized and obey
$$
\Delta_{S^{n-1}}Y_{\vec \ell}=-l(l+n-2)Y_{\vec \ell}.
$$
Here the index ${\vec \ell} = (l,m_1,m_2\dots)$ is a collective index which labels the various spherical harmonics, and $l$ is a non-negative integer.
The full mode solution can be expanded as
\begin{align}\label{modes}
  \phi_{\vec \ell}=
  u_{\vec \ell}(\tau)~(\cosh\tau)^{-(n-1)/2} ~Y_{l,m_1,m_2,\ldots}(\Omega_{n-1})
\end{align}
where $\Omega_{n-1}$ is a point on the sphere.
We have chosen the normalization factor ${\cosh^{(n-1)/2}\tau}$ in $\phi_{\vec \ell}$ for future convenience.  With this choice,  the
Klein-Gordon normalization condition is simply
\begin{equation}\label{norm}
u{\partial_\tau u}^* - u^*\partial_\tau u=i
\end{equation}

The equation of motion then takes the form
\begin{align}\label{equivQM}
\left(  \partial_\tau^2 +
    \left(\frac{2l+n-3}{2}\right)\left(\frac{2l+n-1}{2}\right)
    \frac{1}{\cosh^2\tau}
    +\left(m^2-\frac{(n-1)^2}{4}\right)
  \right) u=0~.
\end{align}
This is the time-independent Schrodinger equation for a one dimensional quantum mechanical system, where $u(\tau)$ is regarded as the wave function and $\tau$ as the position variable.  The Klein-Gordon normalization condition \rref{norm} is just the statement that the probability current of the wave function $u(\tau)$ is conserved.\footnote{With a different choice of prefactor in \rref{modes} we would obtain a different equation for the probability current, along with a friction term proportional to $\partial_\tau u$ in the equation of motion \rref{equivQM}.}  The  potential appearing in this effective Schrodinger equation is known as the Poschl-Teller potential \cite{pt}
\begin{equation}\label{pt}
V = - {L(L+1)\over \cosh^2 \tau},~~~~~L={2l+n-3\over 2}
\end{equation}
We note that the parameter $L$ is an integer when the number of space-time dimensions is odd.

The potential \rref{pt} has several truly remarkable properties when $L$ is an integer.
First, its reflection coefficients vanish identically.  Second, it possesses exactly $L$ normalizable bound states.
 These two results can be derived in one of three ways.  The first is by direct solution of the wave equation in terms of Legendre or hypergeometric functions.  The second uses algebraic techniques related to supersymmetry.     The third (which is overkill in the present situation) uses inverse scattering techniques and relies on the fact that the potential \rref{pt} is the solution of the KdV equation  describing L coincident solitons.  We review these results in the next subsection, where we use algebraic techniques.

However, before giving a mathematical proof of the reflectionless properties of the potential \rref{pt},  we note that in the present context this property has a simple physical interpretation.  As $\tau\to\pm\infty$ the solutions to the wave equation (\ref{equivQM}) are plane waves\footnote{For simplicity we focus here on the case of massive fields $m^2>\frac{(n-1)^2}{4}$ so that $\omega$ is real.  Thus we are considering massive fields whose Compton wavelength is smaller than the Hubble scale.}
$$
u \sim e^{\pm i \omega \tau}~~~~~\omega = \sqrt{m^2 - \frac{(n-1)^2}{4}  }
$$
The modes which define the $|in\rangle$ and $|out\rangle$ state behaves as $e^{-i \omega \tau}$ at $\tau \to -\infty$ and $\tau \to +\infty$, respectively.  The reflectionless property is simply the statement that the $|in\rangle$ modes $u^{in}$ have the following behaviour at $\tau \to \pm \infty$:
$$
u^{in} \sim \left\{{e^{-i \omega \tau}~~~~~~~~\tau\to -\infty \atop T(\omega) e^{-i \omega \tau}~~~~~\tau\to\infty} \right.
$$
where $T(\omega)$ is a phase which  will be computed in the next section.   Thus the state which is purely positive frequency in the far past remains purely positive frequency in the far future.  This means that  $|in\rangle$ and $|out\rangle$ modes are related by
$$
u^{in} = T(\omega) u^{out}
$$
and the $|in\rangle$ and $|out\rangle$ vacua are equal.
Thus the scalar wave equation in odd-dimensional de Sitter space is transparent.

\subsection{Algebraic Approach to Reflectionless Potentials}

We now wish to demonstrate that potential \rref{pt} is reflectionless.
In fact \rref{pt} is a member of a remarkable family of quantum mechanical potentials which have the property that any incoming wave will be transmitted through with unit
probability, regardless of the frequency of the incoming wave.  These are known variously as reflectionless or transparent potentials, and are sometimes discussed in textbooks on elementary quantum mechanics (see e.g. \cite{LL}).  We will describe here a slightly more sophisticated approach based on supersymmetry, which allows for an elegant algebraic
computation of the spectra and scattering data of certain reflectionless potentials.  The advantage of this technique is that it allows us to demonstrate the reflectionless property, and to compute the phase $T(\omega)$, without needing to solve any differential equations. Our discussion here is not new and can be found in any number of references; we refer the reader to \cite{SQM}  for a  pedagogical treatment.

We start by considering a Hamiltonian
\begin{equation}
H = p^2 + V~~~~~p=-i{\partial \over \partial\tau}
\end{equation}
where $V(\tau)$ approaches a constant at $\tau\to\pm \infty$.  We would like to ask under what circumstances the scattering solutions
\begin{equation}
H \psi = \omega^2 \psi
\end{equation}
are reflectionless, in the sense that
\begin{equation}
\psi \sim \left\{{e^{i \omega \tau}~~~~\tau \to - \infty \atop T(\omega) e^{i \omega \tau}~~~~~\tau \to +\infty}\right.
\end{equation}
for all values of the frequency $\omega$.

At first sight one might guess that the only potential whose reflection coefficients vanish identically is the trivial
potential $V= {\rm constant}$.  To see why this is not the case, it is useful to consider the following pair of one dimensional Hamiltonians
$$
H_+= A_+A_-~,\qquad H_-=A_- A_+~
$$
where we have defined the two operators
$$
A_\pm = p \pm i W(\tau)
$$
The function $W(\tau)$, referred to as a superpotential, will be assumed to approach a constant as $\tau\to \pm \infty$.
Writing out the two Hamiltonians $H_\pm$ explicitly
$$
H_\pm = p^2 + V_\pm ~,\qquad V_\pm = W^2 \mp W'~.
$$
We see that the potentials $V_\pm$ approach constants as  $\tau\to\pm\infty$.  The spectra of $H_\pm$ will include scattering states as well as possible bound states.

In fact, for each eigenstate $|\psi\rangle$ of $H_\pm$ one can construct a corresponding eigenstate of $H_\mp$ with the same energy by acting with the operator $A_\mp$:
$$H_\pm|\psi\rangle = E |\psi\rangle ~~~~\implies~~~~H_\mp \left(A_\mp |\psi\rangle \right) = E \left(A_\mp |\psi\rangle\right)$$
Thus the spectra of $H_+$ and $H_-$ are nearly identical.  The only exception is if $H_\pm$ has a zero energy state $|\psi\rangle$ which is annihilated by $A_\mp$. In this case there is no corresponding non-trivial eigenstate of $H_\mp$.

This implies that the reflection and transmission coefficients of $H_\pm$ are related in a simple manner.
To see this, consider a scattering solution of $H_+$, i.e. an eigenfunction $\psi(\tau)$ of $H_+$ obeying
$$
\psi\sim\left\{{e^{i\omega\tau} + R_+(\omega) e^{-i\omega\tau},~~~~~\tau\to-\infty\atop T_+(\omega) e^{i \omega \tau},~~~~~~~~~~~~~ \tau\to \infty}\right.
$$
where $R_+(\omega)$ and $T_+(\omega)$ are the reflection and transmission coefficients of $H_+$.
 The operator $A_-$ which converts this into a eigenfunction of $H_-$ takes a simple form at $\tau\to\pm\infty$ where $W(\tau)$ is constant, allowing us to determine the scattering coefficients of $H_-$:
 $$
 R_-(\omega) = \frac{\omega+i W(-\infty)}{-\omega+i W(-\infty)} R_+(\omega)~,\qquad T_-=\frac{\omega-i W(\infty)}{ \omega-i W(-\infty)} T_+(\omega)~.
 $$

This construction can be described in terms of supersymmetry.  To see this, note that the matrix operators
 $$
 H = \left(
 \begin{array}{cc}
   H_-&0 \\0&H_+
 \end{array}\right)~, \qquad
Q_+ = \left(
 \begin{array}{cc}
   0&0 \\A_+&0
 \end{array}\right)~, \qquad
Q_- = \left(
 \begin{array}{cc}
   0&A_- \\0&0
 \end{array}\right)~, \qquad
 $$
 obey the usual supersymmetry commutation relations
 $$
 [Q_\pm, H]=0~,\qquad \{Q_+,Q_-\}=H~.
 $$
For this reason, the Hamiltonians $H_\pm$ and potentials $V_\pm$ will be referred to as superpartners.
Indeed, the spectral degeneracy described above is nothing more than the usual supersymmetric relation between even and odd spin states.  Of course, as we are working with quantum mechanics rather than quantum field theory, the use of the word ``spin" here does not represent the transformation property of the states under rotation, but simply reflects a grading of the Hilbert space into even and odd parts.  States which are annihilated by $A_\pm$ are the analogues of BPS states, and the mismatch described above between the zero energy spectra of $H_\pm$ is the statement that the index $Tr (-1)^F$ may be non-zero.

It is now straightforward to construct reflectionless potentials.
We start by seeking the potential which is the superpartner of the constant potential.  That is, we seek a superpotential $W(\tau)$ for which $V_-=1$\footnote{Here we have restricted our attention to case where the potential is a positive constant, which can be set equal to one by rescaling $\tau$.  If we replace the right hand side of this equation by a non-positive constant we will obtain another solution which is formally reflectionless, but which diverges at finite $\tau$ so is less interesting for our purposes.}
$$
W^2 + W' = 1
$$
The solution is
$$
W = \tanh \tau
$$
and the corresponding superpartner potential is
\begin{equation}\label{v2}
V = 1 - {2\over \cosh^2  {\tau}}~.
\end{equation}
We therefore conclude that this potential is reflectionless.  Note that if we shift $V$ by a constant the potential will remain reflectionless, so we can freely omit the constant term.

We can now generate a second reflectionless potential by identifying this $V$ as the $V_-$ of another superpotential and then solving for the corresponding $V_+$.  Iterating this procedure will yield an infinite series of reflectionless potentials of the form
\begin{equation}
V_L = - {L(L+1) \over \cosh^2 \tau}
\end{equation}
for any integer value of $L$.   This is proven by noting that the potentials $V_{L}$ and $V_{L-1}$ are superpartners connected by the superpotential
\begin{equation}
W_L = L \tanh \tau~.
\end{equation}
In this computation, as above, we have omitted constant terms in $V_L$ which do not affect the reflectionless property.
The potentials $V_L$ described above are the famous Poschl-Teller potentials, whose reflectionless properties have been known since the early days of quantum mechanics.

Using the algebraic technique described above it is straightforward to obtain expressions for the scattering states along with the transmission coefficients $T(\omega)$.  Given a solution $\psi_{L-1}(\tau)$ of the potential $V_{L-1}$, a solution of $V_L$ is obtained by applying the operator $A_L = p + i W_L$.  So starting with the plane wave solution $e^{i\omega\tau}$ of the constant potential $V_0$ we obtain the scattering solution of the potential $V_L$
\begin{equation}\label{psiL}
\psi_L = A_L \dots A_1 e^{i\omega\tau}~.
\end{equation}
Since the $A_L$ take a simple form at $\tau\to \infty$ we can easily extract the transmission coefficients
$$
T(\omega) = \prod_{n=1}^L {\omega+in \over \omega-in}~.
$$
This is pure phase, as it must be since the reflection coefficient vanishes.

A similar argument can be used to show that the potential $V_L$ has $L$ bound states.  Indeed, this is reflected in the fact that the
transmission coefficient has $L$ poles at the complex momenta $\omega=i,2i,\dots,Li$.  When $L$ is not integer one can still use algebraic methods to simplify the construction of scattering and bound states, even though the potential will not be reflectionless.  For example, when $L$ is half integer (the case of interest for even dimensional de Sitter space) the scattering states are given by acting with $A_L \dots A_{3/2}$ on the basic scattering solutions of the $L=1/2$ potential.

Finally, we note that the algebraic procedure described above does not give the most general form of a reflectionless potential.  More general examples can be constructed using inverse scattering techniques; we refer the reader to the literature for a discussion of these techniques (see for example  \cite{Drazin:1989qi}).

\section{Spinor Fields in de Sitter Space}

In this section we consider free spin $1/2$ fields in de Sitter space. We will show that, as in the scalar case, odd
dimensional de Sitter spaces is transparent.  Rather than solving the Dirac equation explicitly, we will simply show that the
corresponding effective potential which appears in global coordinates is reflectionless, following the argument of the previous section.

\subsection{Wave Equation in Global Coordinates}

The Dirac equation is
\begin{align}\label{DDirac}
  (\Gamma^\mu D_\mu+m)\Psi = 0 ~.
\end{align}
Our conventions for Gamma matrices and spin connections are given in Appendix A.  We will now write out the Dirac equation
in global coordinates.

We start by considering the case where the number of spacetime
dimensions $n$ is odd, with $n\geq 3$.  In this
case a Dirac spinor in $n$ dimensions has the same number of components as a Dirac spinor in $n-1$ dimensions.
It is then just a matter of working out the components of the spin connection.  In global coordinates, the Dirac equation is \begin{align}
  \left(\partial_\tau+\frac{n-1}{2}\tanh\tau\right)
 \gamma_0
  \Psi
  + \frac{\tilde{\slashed\nabla}\Psi}{\cosh\tau}
  + m\Psi = 0 ~.
\end{align}
where $\tilde{\slashed\nabla}$ is the Dirac operator on $S^{n-1}$.

We now expand in terms of a complete basis of eigenfunctions of the Dirac operator on $S^{n-1}$.
We will follow the conventions of Camporesi and Higuchi \cite{Camporesi:1995fb}, who defined a complete orthonormal basis of spinors ${\hat \chi}^{(\pm)}_{lm}$ on the sphere. They obey the eigenvalue equation
\begin{align}
  -i\gamma^0 \tilde{\slashed\nabla}\hat\chi_{lm}^{(\pm)}
  = \pm \frac{2l+n-1}{2} \hat\chi_{lm}^{(\pm)} ~,~~~~~
-i \gamma^0
  \hat\chi_{lm}^{(\pm)}=\hat\chi_{lm}^{(\mp)}~.
\end{align}
Here $l$ and $m$ are integers.

We now expand
\begin{align}
    \Psi = \sum
       f_{lm}^{(\pm)}(\tau) (\cosh\tau)^{-(n-1)/2}~ \hat\chi_{lm}^{(\pm)}(\Omega^{n-1})~,
\end{align}
where $f_{lm}^\pm$ are some functions of $\tau$ and we have introduced a factor of $(\cosh\tau)^{-(n-1)/2}$ for future convenience.
The Dirac equation then takes the form\footnote{As in the scalar case, our prefactor in has been chosen so that there is no term linear in $\partial_\tau$.}
\begin{align}\label{dirac}
  \left(\partial_\tau^2
    \pm\frac{iL \sinh\tau}{\cosh^2\tau}
    +\frac{L^2}{\cosh^2\tau} + m^2
  \right) f_{lm}^{(\pm)} =0 ~.
\end{align}
where $L$ is defined by
\begin{align}\label{Lis}
  L=\frac{(2l+n-1)}{2}
\end{align}
We note that $L$ is an integer, as $n$ is odd.
For reasons that will become clear below, this potential should be regarded as a complex version of the Poschl-Teller potential.

It is straightforward to show that in even dimensions the Dirac equation reduces to the same one dimensional equation \rref{dirac}, with $L$ again given by \rref{Lis}.
We start by recalling that in this case an $n$ dimensional Dirac spinor has twice the number of components as an $n-1$ dimensional
spinor.
So writing out the Dirac equation in global coordinates, we find
\begin{align}
  \left\{
    \left(\partial_\tau + \frac{n-1}{2}\tanh\tau\right)
    \left(
    \begin{array}{cc}
      { 0}&{\bf 1} \\{\bf 1}&{ 0}
    \end{array}
  \right)
  +
  \frac{1}{\cosh\tau}
  \left(
    \begin{array}{cc}
      0& \tilde{\slashed\nabla} \\ -\tilde{\slashed\nabla}&0
    \end{array}
  \right)
  -im
\right\} \Psi = 0 ~,
\end{align}
where $\tilde{\slashed\nabla}$ is the Dirac operator for $S^{n-1}$.  Following \cite{Camporesi:1995fb}, we use the complete basis of spinors $\chi_{lm}^{(\pm)}$ on $S^{n-1}$ which obey
\begin{align}
  \tilde{\slashed\nabla} \chi_{lm}^{(\pm)}(\Omega^{n-1}) = \pm \frac{2l+n-1}{2} \chi_{lm}^{(\pm)}(\Omega^{n-1})~.
\end{align}
We then expand in terms of these fields
\begin{equation}
\Psi=\sum \left(f_{lm}^{(\pm)} \atop g_{lm}^{(\pm)} \right)
(\cosh\tau)^{-(n-1)/2} \chi_{lm}^{(\pm)}~,
\end{equation}
where $f_{lm}^{(\pm)}$ and $g_{lm}^{(\pm)}$  are functions of $\tau$.
The Dirac equation reduces then reduces to equation \rref{dirac}, along with a similar equation for $g_{lm}^{(\pm)}$.  We note that in the even dimensional case the parameter $L$ appearing in the equation of motion is half-integer rather than an integer.

The wave equation \rref{dirac} is reflectionless when $L$ is an integer, just as in the scalar case.  We give a simple proof of its reflectionless property in the following subsection using algebraic methods.\footnote{The reflectionless property of this potential has been noted in a different context by \cite{Levai:2001zs, CDV,Ahmed:2005zz}.}
This implies  that, as in the scalar case, odd dimensional de Sitter space is transparent for Dirac fields.  In the even dimensional case there will be particle production, as the parameter $L$ is now half-integer.

Finally, it is worth noting that the case of two dimensional de Sitter space is rather special, since in this case there are two possible spin structures.  They correspond to taking the fermions to have either periodic or antiperiodic boundary conditions as we go around the spatial circle.  The effective potential described above corresponds to the anti-periodic case, which is the natural spin structure which generalizes to the higher dimensional spheres.  With periodic boundary conditions, however, the parameter $l$ appearing in \rref{Lis} is half-integer.  Therefore  $L$ is an integer and the potential is reflectionless.  Thus with the appropriate choice of fermion boundary conditions two dimensional de Sitter space is transparent as well.

\subsection{Complexified Reflectionless Potentials}

We will now generalize the discussion of section 2.2 to show that the equation of motion \rref{dirac} is reflectionless, just as in the scalar case.

We first note that if equation \rref{dirac} is regarded a one dimensional Schrodinger equation then the corresponding  potential is complex and Hamiltonian is non-Hermitian.  This  does not reflect any underlying sickness in our theory; the Dirac equation \rref{DDirac} describes Hermitian time evolution as usual.  It is only means that, when written in terms of an auxilliary one dimensional potential, the Dirac equation can be formulated in terms of a  non-Hermitian Hamiltonian.\footnote{As an aside we note that while this auxilliary system is non-Hermitian it is PT symmetric in the sense of \cite{Bender:1998gh}.}  We note that this potential approaches a constant at $\tau\to\pm\infty$, so one can compute its scattering solutions just as in section 2.2.  It is these scattering solutions which are reflectionless.

The proof of this is a straightforward generalization of the discussion in section 2.2.  In that case $W$ was taken to be real, which is the usual case of interest in quantum mechanics.  The operators $A_+$ and $A_-$ were Hermitian conjugates of one another and the $H_\pm$ were Hermitian.  However, when $W$ is allowed to be complex an additional family of complex reflectionless potentials can be constructed. The potentials are of the form
\begin{equation}
V_L = - {L^2 \over  \cosh^2 \tau} + {i L } {\sinh \tau \over \cosh^2 \tau}
\end{equation}
for integer values of $L$.
To show that these potentials are reflectionless, we note that (up to constant terms which may be dropped) the potentials $V_L$ and $V_{L-1}$ are superpartners, connected by the complex superpotential
\begin{equation}
W_L= (L-\frac{1}{2})\tanh \tau + \frac{i}{2 \cosh \tau}~.
\end{equation}
When $L=0$ this gives the trivial constant potential; thus the whole family of potentials is reflectionless for integer values of $L$.

As before, we can now obtain expressions for the scattering states along with the transmission coefficients $T(\omega)$.  Defining $A_L = p + i W_L$ the scattering solutions are again
\begin{equation}\label{psiL}
\psi_L = A_L \dots A_1 e^{i\omega\tau}~.
\end{equation}
The transmission coefficients are
$$
T(\omega) = \prod_{n=1}^L {\omega+i(n-1/2) \over \omega-i(n-1/2)}~.
$$

\section{Higher Spin Fields in de Sitter space}

We will now generalize the previous discussion to free fields of arbitrary integer spin, focusing on the case of differential form fields and symmetric tensor fields.  In both cases the wave equations will be reflectionless in an odd number of space-time dimensions.

\subsection{Differential Form Fields}

We first consider the case of massive form fields of arbitrary rank.

The action of a free $p$-form field $A_p$ is
\begin{align}
  S=\int d^nx \sqrt{-g}
  \left[
    -\frac{1}{2(p+1)!}F_{\lambda_1\cdots\lambda_{p+1}}F^{\lambda_1\cdots\lambda_{p+1}}
    -\frac{m^2}{2p!}A_{\lambda_1\cdots\lambda_p}A^{\lambda_1\cdots\lambda_p}
  \right]~.
\end{align}
where
\begin{align}
  F_{\lambda_1\cdots\lambda_{p+1}}

  = (p+1)\partial_{[\lambda_1}A_{\lambda_2\cdots\lambda_{p+1}]} ~.
\end{align}
The equations of motion are
\begin{align}\label{formEOM}
  (p+1)\partial_{\lambda_1}
  \left\{
    \sqrt{-g}~g^{\lambda_1\rho_1}\cdots g^{\lambda_{p+1}\rho_{p+1}}
    \partial_{[\rho_1}A_{\rho_2\cdots\rho_{p+1}]}
  \right\}
  -m^2 \sqrt{-g} A^{\lambda_2\cdots\lambda_{p+1}}=0~.
\end{align}
Note that we may act with $\partial_{\lambda_2}$ on equation (\ref{formEOM}) to obtain the  constraint equation
\begin{align}\label{formConstraint}
  \partial_{\lambda_1}\left(\sqrt{-g}A^{\lambda_1\cdots\lambda_p}\right)=0~.
\end{align}

Our strategy will be to divide the $p$-form field $A_p$ into two components: a component $A^{i_1\dots i_{p}}$ with indices in the $S^{n-1}$ directions and a component $A^{0i_1\dots i_{p-1}}$ with one time-like index.  Roman indices $i_1, \dots$ denote sphere directions.

Let us start with the components $A^{0i_1\dots i_{p-1}}$, which may be regarded as a $p-1$ form on the sphere.  Setting $\lambda_p=0$ in equation (\ref{formConstraint})
\begin{align}
  \partial_{i_2}\left( \sqrt{-g}A^{0i_2\cdots i_p} \right) =0~.
\end{align}
we see that $A^{0i_2\cdots i_p}$ is a co-closed $(p-1)$-form on sphere.
On the other hand, setting $\lambda_a\neq 0$, $a=2,\ldots, p$,
equation (\ref{formConstraint}) becomes
\begin{align}\label{constraint1}
  \sqrt{-g}\left[\partial_\tau A^{0i_2\cdots i_p}
  +(n-1)\tanh\tau A^{0i_2\cdots i_p}\right]
  +\partial_{i_1}\left( \sqrt{-g}A^{i_1\cdots
      i_p}\right)
  =0~.
\end{align}
Using equation (\ref{constraint1}), the $\lambda_2=0$ part of
equation (\ref{formEOM}) is
\begin{align}
  \left[
    \partial_\tau^2 + (n-1+2p)\tanh\tau~\partial_\tau+2p(n-1)+m^2
    -\frac{(2p-1)(n-1)}{\cosh^2\tau}-\frac{\tilde\Delta}{\cosh^2\tau}
  \right]A^{0i_2\cdots i_p}=0~,
\end{align}
where $\tilde\Delta=d\delta+\delta d$ is the  Hodge-de Rham Laplacian on $S^{n-1}$.
In order to write this in Schrodinger form we first introduce the $p-1$ form $f^{i_2\cdots i_{p-1}}$
defined by
\begin{align}
  A^{0i_2\cdots i_p} = (\cosh\tau)^{-(n-1+2p)/2} f^{i_2\cdots i_{p}}~.
\end{align}
We then expand in a basis of eigenforms of the Laplacian on the sphere, which were described in \cite{CH}.  The operator $\tilde\Delta$ acting on co-closed $(p-1)$-forms has eigenvalues $-(l+p-1)(l+n-p-1)$ where $l$ is a non-negative integer.  Expanding in this basis, a component of the form field $f$ will obey the Schrodinger equation
\begin{align}
  \partial_\tau^2 f +
  \left\{
    \frac{L(L+1)}{\cosh^2\tau}
    +\left[m^2-\frac{1}{4}(n-1-2p)^2\right]
  \right\}f=0~.
\end{align}
where
\begin{equation}
L = \frac{2l+n-3}{2}
\end{equation}
Up to a shift of mass, this equation is identical to the scalar case
(\ref{equivQM}).  In particular, it is reflectionless when $n$ is odd since in that case  $L$ is an integer.

We now consider the component $A^{i_1\cdots i_p}$, which can be regarded as a $p$-form living on the sphere.  We will denote this component ${\hat A}$, and denote by $d$ and $\delta $ the differential and codifferential on $S^{n-1}$.    To show that $\hat A$ is reflectionless, we first note that $\delta \hat A$ is completely fixed by  the constraint equation (\ref{constraint1}), so can be expressed in terms of  $A^{0i_2\cdots i_p}$.  Therefore $\delta \hat A$ is reflectionless.
Likewise, acting with $d$ on (\ref{formEOM})  we obtain

\begin{align}
  \left[\partial_\tau^2+(n-1-2p)\tanh\tau\partial_\tau
    -\frac{\tilde \Delta }{\cosh^2\tau}+m^2\right](d \hat A) =0~.
\end{align}
We then remove the $\partial_\tau$ term by defining
\begin{align}
  d\hat A = (\cosh\tau)^{-(n-1-2p)/2}f~
\end{align}
Using the fact that the Laplacian $\tilde \Delta $ has eigenvalues $-(l+p)(l+n-p-2)$ the equation becomes
\begin{align}\label{ddeltabeta}
  \partial_\tau^2 f+
  \left\{
    \frac{L(L+1)}{\cosh^2\tau}
    +\left[m^2-\frac{1}{4}(n-2p-1)^2\right]
  \right\}f=0~~~~~~L=\frac{2l+n-3}{2}.
\end{align}
Again, the equation is identical to (\ref{equivQM}) and is reflectionless in an odd number of space-time dimensions when $L$ is an integer.
We have now shown that both $d\hat A$ and $\delta \hat A$ are reflectionless.  It follows that $\tilde \Delta \hat A$ is reflectionless.  As the sphere has no non-trivial cohomology, there are no non-trivial harmonic forms on the sphere and the operator $\tilde \Delta$ is invertible.  Thus $\hat A$ is reflectionless.

We conclude that all of the components of a differential form field are reflectionless in odd dimensional de Sitter space.

\subsection{Symmetric Tensor Fields}

We now consider a massive symmetric tensor field $h_{\mu_1\dots \mu_r}$ of rank $r$ which  obeys the equation of motion
\begin{equation}
(\nabla^2-m^2) h_{\mu_1\mu_2\cdots\mu_r}=0~.
\end{equation}
and is divergenceless and traceless
\begin{align}
  g^{\mu_1\mu_2} h_{\mu_1\mu_2\cdots\mu_r}=0~,~~~~~  \nabla^{\mu_1}h_{\mu_1\mu_2\cdots\mu_r}=0~,
\end{align}
As in the previous cases, the reflectionless behaviour can be derived in a straightforward way from the wave equation.  We will proceed by relating the equations to those for symmetric tensor harmonics on the sphere described by \cite{Higuchi:1986wu}.

To write out the wave equation, we start by noting that
for any $p$ between $0$ and $r$, the component $h_{\tau\cdots\tau i_1\cdots i_{p}}$ can be regarded as a symmetric tensor of rank $p$ on the sphere $S^{n-1}$.
We can then define the $p^{th}$ component of $h$ as
\begin{align}\label{hp}
  h_{\tau\cdots\tau i_1\cdots i_{p}} =
  (\cosh\tau)^{2p-r}f_{i_1\cdots i_p}~
\end{align}
where the $\tau$ dependent factor has been introduced for future
convenience.  We will abbreviate $f_{i_1\dots i_p}$ by $f_p$.  It is
then straightforward (but tedious) to write out the wave equation in
global coordinates.  We find
\begin{align}
    \left[
    \partial_\tau^2+(n-1)\tanh\tau\partial_\tau
    -\frac{\tilde\nabla^2-p}{\cosh^2\tau}+m^2
  \right]f_p &
  \nonumber\\
  - 2ip \frac{\sinh\tau}{\cosh^3\tau}\tilde\nabla f_{p-1}&
  +p(p-1)\frac{\sinh^2\tau}{\cosh^4\tau}\eta f_{p-2}=0 ~,
  \label{eq:wave-tensor}
\end{align}
Here $\eta$ and $\tilde\nabla$ denote the metric and covariant derivative
on the sphere.  We have not written out the indices explicitly; $\eta f_{p-2}$ and $\tilde \nabla f_{p-1}$ are the rank $p$ tensors on the sphere which are symmetrized over their indices.
The primary subtlety is that the wave equation (\ref{eq:wave-tensor}) couples components \rref{hp} with different values of $p$.

To solve this equation, we note that for every solution $h_{\mu_1\dots
  \mu_r}$ there is a value of $p$ (possibly zero) such that
$h_{\tau\cdots\tau  i_1\cdots i_{q}}$ vanishes for all $q<p$.  We can
then expand $f_p$ in terms of a basis of symmetric tensor spherical
harmonics on $S^{n-1}$ as
\begin{align}
  f_p = \sum_{l, m} u^{(p)}_l(\tau) Y^{(p)}_{l m}(\Omega_{n-1})~.
\end{align}
This basis of spherical harmonics was described explicitly in \cite{Higuchi:1986wu}.  Here $l, m$ are integers.
These tensors obey the eigenvalue equation
\begin{equation}
\tilde\nabla^2 Y^{(p)}_{l m} = [-l(l+n-2)+p] Y^{(p)}_{l m} ~.
\end{equation}
Then the equation of motion for $u_l^{(p)}$ is
\begin{align}
  &
  \left[
    \partial_\tau^2+(n-1)\tanh\tau\partial_\tau
    +\frac{l(l+n-2)}{\cosh^2\tau}+m^2
  \right]u_l^{(p)} =0 ~,
  \label{eq:wave-tensor-lowest}
\end{align}
From this it is easy to see that $u_l^{(p)} (\cosh\tau)^{(n-1)/2}$ satisfies the Poschl-Teller
equation (\ref{equivQM}). Thus $f_p$ (and hence $h_{\tau\cdots\tau i_1\cdots i_{p}}$) is reflectionless in odd dimensions.

The equations for the higher rank components $f_{p+1}$, $\ldots$,
$f_r$ are more
complicated and involve couplings between the fields.  However the
wave equation \rref{eq:wave-tensor} allows us to solve for these
higher rank components in terms of $f_p$.  The computation is
essentially identitical to that in \cite{Higuchi:1986wu} so we will
just summarize the answer.  We find that
 \begin{align} \label{eq:f-T}
   f_{p+q}=
    c_0^{(q)} T_{p+q}^{(q)}+\cdots+
    (-1)^k c_k^{(q)} \eta^k
    T^{(q-2k)}_{p+q-2k}+\cdots
\end{align}
solves the equation of motion (\ref{eq:wave-tensor}),
where $\eta_{ij}$ is metric on the sphere and the $T$'s are functions on
$S^{n-1}$, defined as
\begin{align}
  T^{(q)}_{p+q}  = \sum_{k=0}^{[q/2]} \alpha(n,k,p,q) \eta^k
  \tilde\nabla^{q-2k} Y_{p}~,
\end{align}
where $[q/2]$ is the integer part of $q/2$. Here $\alpha(k,n,p,q,l)$ is a
number, recursively defined in \cite{Higuchi:1986wu}. Again all
indices are symmetrized.

The coefficients $c_0^{(q)}$ in equation (\ref{eq:f-T}) vanish for $q<0$.  For $q=0$ we have
\begin{align}
 c^{(0)}_{0}= u_{l}^{(p)}(\tau)~,
\end{align}
The wave equation, along with the divergenceless and traceless conditions, then reduces to the following equation
\begin{align}
  c^{(q)}_k = \frac{1}{2^k
    k!}\frac{(p+q)!}{(p+q-2k)!}
  \left[\prod_{a=1}^{k}\frac{1}{n+2(p+q-2k+a-1)}\right]\frac{c_0^{(q-2k)}}{(\cosh\tau)^{2k}}~,
\end{align}
\begin{align}
  &
  \frac{(q+1)(n+2p+q-3)
    \left[
      (p+q)(p+q+n-2)-l(l+n-2)
    \right]
  }{(p+q+1)(n+2p+2q-3)} c_0^{(q+1)}
  \nonumber\\
  & - \frac{(p+q)c_0^{(q-1)}}{(n+2p+2q-3)\cosh^2\tau}
  +i\left[\partial_\tau +(n+p+q-2)\tanh\tau\right] c_0^{(q)}=0
\end{align}
This allows us to solve recursively for $c_k^{(q)}$ in terms of $u_{l}^{(p)}$. As
$u_{l}^{(p)}$ is reflectionless for odd $n$, all of these coefficients will similarly be
reflectionless.

We conclude that massive symmetric tensor fields are
reflectionless in odd dimensional de Sitter space.

\section{Conclusion and Discussion}

We have learned that the presence of curvature does not necessarily lead to particle production in cosmological settings.
This observation has relied on the relationship between free wave equations in de Sitter space in global coordinates and the theory of reflectionless potentials.  This relationship has largely escaped notice in the literature because most work has focused on wave equations in the Poincare patch where the spatial slices are flat.

In the present paper we have considered various types of massive free fields and given proofs of transparency on a case by case basis. It is possible that there are other types of fields which do not have this reflectionless property; this would be nice to investigate.  For example, it would be interesting to study massless or very light (Compton wavelength of order Hubble scale) fields in de Sitter space, which exhibit well known subtleties at the free and interacting level.  To this end it may be useful to rephrase the results presented in this paper in a more unified manner based on the representation theory of the de Sitter group.

We conclude by mentioning three important open questions.  The first is whether other cosmologies are also transparent in the sense described in this paper.  The answer to this question is almost certainly yes, as the classes of potentials described above are but the simplest examples of reflectionless potentials. The second question is whether this transparency remains once interactions are present.  In this case the answer is not clear; in the absence of a deeper symmetry explanation for the present results we see no reason why the transparency property should be preserved.  The final, and most important, question is whether these results are of observational relevance for theories of the early universe.  We leave this as a challenge for the future.

\section*{Acknowledgments}
We thank A. Adams, D. Anninos, R. Bousso, A. Higuchi, D. Marolf, I. Morrison, P. Horava, L. Leblond and E. Silverstein
for discussions.  We thank the participants of the McGill workshop
``Infrared effects in de Sitter Space" for encouraging the authors to
write up these results.  A. M. thanks his loyal undergraduate researchers M. Emelin,  N. Lvov and P. Sabella-Garnier for stimulating discussions.  This work is supported by the
Fonds Qu\'eb\'ecois de la Recherche sur la Nature et les Technologies, the
Natural Sciences and Engineering Research Council of Canada and the
Institute of Particle Physics.

redpajama_set_name: RedPajamaArXiv


I am vaguely aware that there is, coming up, some kind of "graduation" ceremony for my older daughter's middle school. I'm still waiting to hear the details, such as when and what this actually is. One thing I know for certain: nobody has asked me to give a speech at the event.
The nice thing about being a blogger is that nobody has to ask. Here is my unsolicited commencement speech for all eighth-graders—and their parents—everywhere. I encourage you to give this speech at the dinner table tonight, like I'm going to do.
Of course I've never given a speech like this before. I'm not even sure why middle schools have such ceremonies, since middle school is just a stepping-stone anyway. But I could say the same thing about high school, which has itself become a mere waypoint on the way to college.
As I hope you've already heard, middle school (or junior high, as I knew it) is actually the hard part. If you've done a halfway decent job of making the transition from largely irresponsible, clueless child to basically self-sufficient (albeit exasperated) teenager, high school shouldn't be a difficult transition. Moreover, kids seem to mellow out in high school, as their brains catch up to their hormone-ravaged bodies. So, since you matriculating middle-schoolers are all going to be just fine, I'm going to focus this speech on what you can do for your poor parents over the next four years.
Look, a lot of fuss is about to be made over all of you. You'll wear this fancy cap and gown and attend this big ceremony with all these people watching. But don't get too excited. In the big scheme of things, graduating from high school isn't that big an achievement. It basically means you showed up for four years. And if you really think about it, most of you could figure out a more fun way to spend your afternoon than dressing up and standing around through a bunch of speeches. Try to appreciate that this ceremony isn't actually for you. It's for your parents, and your grandparents, so please honor their ceremony by not decorating your mortarboard with stickers and whatnot, and justifying this misbehavior by saying it's your ceremony. Instead, behave respectfully, go where you're told, and try not to do anything to disturb the event. You have the rest of the summer to screw around and do as you please.
Of course, there aren't any caps or gowns at today's ceremony, and since kids your age aren't organized or devious enough to smuggle in beer or inflatable dolls, like the students at my high school graduation did [wait for laughter], I'm going to give you some guidelines on how to advance your development as thinking, considerate, and mature beings in your day-to-day life. If you follow this advice you might start to feel a bit less anxious about grades, popularity, life changes, and the ten other things that I, as a parent, could never understand.
Guideline #1: Honor your commitments! No excuses!
This does not mean "Prioritize your commitments and honor the most important ones." It means you should not commit to doing something unless you know you will have the time, resources, ability, discipline, and commitment to do it, as promised and on time. You must honor every commitment you make, no matter whom you made it to.
When you're in high school, you will have all kinds of distractions—some of them of the "hanging out" variety, but also a great number of worthy extracurricular activities. That's great—but you shouldn't, say, skip practicing the piano because you have to write a student election speech. If you cannot honor all your obligations, you are overscheduled and need to scale back. Commitments come first; leisure comes second. That said, before committing to something, ensure you will still have time to set aside for leisure. It's important, too.
Remember that extracurricular activities shouldn't be sacrificed at the altar of good grades. If you're on a sports team, your teammates rightfully expect a certain level of participation, even if you have a big test coming up (and after all, who doesn't?). It is more important to learn to be reliable, balanced, and responsible than to get straight As. Many great scholars totally wash out in real life, and not all successful people were top students.
There is almost never a good excuse for failing to honor a commitment. For example, to say "I ran out of time" means a) "I did not manage my time well," b) "I did not prioritize this commitment to you," and c) "By making this excuse I am demonstrating a willingness to let this happen again." Remember, nobody wants to hear an excuse. Often, an excuse functions as the opposite of an apology. It suggests that you're not holding yourself responsible for your failure.
You must recognize when something is your problem, and don't try to make it into your parents' problem. Say, for example, you forgot to get up extra early for a student council meeting, and now you're about to be late for it. Don't yell at your mom or dad, "You gotta drive me! You gotta!" After all, you made the commitment to attend that early meeting, not your parents. It is unpleasant and offensive for them to be shrieked at by somebody who has made a mistake and behaves as though it's his or her parent's job to fix it.
Sometimes you will fail. Everybody does. Fail gracefully, without dragging down the people around you, particularly your parents who suffer (if only silently) whenever you do. Yes, of course you can always go to your parents to be consoled, but resist the impulse to implicate them in your mistake. Learn how to accept a setback as stoically as you can; try to learn from it; then move on.
Most of you kids probably dream of having the cool kind of parents, who are a combination of personal assistant, chauffeur, purveyor of booze, and ATM. Well, I hate to break it to you, but that's not what your parents signed up for. And all those hundreds of loaded diapers you stuck them with have probably exhausted your parents' patience.
There are all kinds of ways your parents can support your efforts, but not all of them are actually helpful. Should they be available to help with homework? Of course (unless it's something like calculus that your parents put behind them decades ago). Is it right for your mom or dad to bend over backwards to get you out of a tough spot, when you're behind on an assignment due to procrastination and/or disorganization?
Perhaps not. If they help you out of such a bind, perhaps they're sending the wrong message, such as "It's okay to fly by the seat of your pants, since you'll always have a parent to pick up the slack." (You won't.) As important as that school assignment seems now, the bigger education is in managing your time. This never seems obvious in the moment, but it's better for your mom or dad to let you fail now—and thus learn self-sufficiency the hard way—than for them to leave you stranded later, when you're truly on your own.
There is more to parenting than helping a child succeed. A good parent strives to teach his or her child to be helpful to others. Many of us parents are trying to instill character, which is rarer and more valuable than the ability to succeed.
Of course this development takes time. Infants are practically useless; grade-school kids are often clueless; teenagers often feel overwhelmed. Self-sufficiency is an evolution. You are now at an age where you can and should be taking on as much as you can: setting and adhering to your routines, cleaning up after yourself, getting yourself where you need to go, and knowing what to do and when to do it without being reminded.
I hope that most of you have already managed to climb well into the purple tier of the developmental triangle I've presented here. This is a simplistic schematic; the colors should blend m ore, to convey that even before being totally self-sufficient, a person can still be helpful to others.
Remember that you are not yet an independent adult, free to pursue only what is important to you. So long as you are dependent on your parents, you should expect them to be involved in setting your priorities. When you set goals for yourself, work to keep them aligned with the goals your parents have for you. And when you ask your mom or dad for support, remember that you do not have the big picture; gracefully accept it when the answer is no.
Not all the help your parents give you is solicited; for example, they are justified in giving you unsolicited advice. That's part of their job. And even though you didn't ask for advice, you're expected to follow it. It's exasperating to your parents when you don't, and then they have to deal with the consequences ... which brings me to my next guideline, which is a sore point with so many parents.
Guideline #4: GET ORGANIZED! STAY ORGANIZED!
Why don't the greatest mathematicians do everything in their heads? It's because they need pencil and paper to organize their thoughts. Why was written language invented? It's because oral communication is too imprecise, unreliable, and ungainly. Writing something down frees your brain up for bigger, better thoughts, while keeping tasks from being forgotten. As your life continues to get more complicated it will be impossible for you to honor all your commitments without systematically recording what is due and when. So get a pocket calendar you can bring to school with you. Record your commitments in it. Keep it handy. Look it over frequently. Document deadlines well in advance. Check things off as you complete them.
This calendar is only one of your tools. Figure out everything you'll need and keep it handy. I'm talking about your wristwatch, your textbook, your homework, that permission slip, your house key, your gym shorts, your glasses, and the ten other things that I as a parent hope to be blissfully unaware of because they're your problem, not mine.
I don't claim to be some world authority on all this. Perhaps the most important skill for you to develop is a sense of being in charge of your own life, becoming self-sufficient, and figuring out—usually when something doesn't work—what you need to do to become more effective. Periodically step back, look at your life, and ask the question, "How can I make this work better? What is falling through the cracks? How can I keep that from happening?" If you create the right habits and systems, you will get more done and avoid all kinds of strife as you go along.
I'll leave you with this refreshingly non-didactic thought: as systematic and prescriptive as all this seems, everybody is just winging it, and that's okay. You're all amateurs. There's no such thing as a professional high-schooler, and if there were, we'd see all kinds of malpractice suits.


Thornton's Budgens came to JooJoo Creative looking for a new website. Specifically, Thornton's Budgens wanted a website that they could manage themselves in order to keep the community abreast of news and events.
Unlike your 'run of the mill', generic supermarket, Thornton's Budgens is very much at the heart of the community and brings together suppliers and producers of wonderful food whilst promoting sustainable living.
I went to JooJoo Creative for help with our new company website as I knew that they were easy to communicate with but also creative. I am not particularly IT literate but knew what I wanted as an idea, and needed someone who wouldn't think me stupid, but wouldn't take over either!
We needed to combine our supermarket proposition with our environmental/sustainability agenda and our BIG part in the community - getting the message across clearly and easily. Jennie got this straight away, understanding the feel of the site required and our vision and values. I would happily recommend JooJoo!


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HomeResearch Grants & Themes Mapping A Colony
www.mappingacolony.org
ETHOS Lab is part of a funded Europeana research project called Mapping A Colony. The project is being conducted in collaboration with The Royal Library, the Uncertain Archives project, the Past's Future project, and lead by the author Lene Asp.
The objective of the Mapping A Colony project is to create an interactive map to highlight and investigate Danish colonial heritage. Since 2017 marks the centennial of the sale of the Danish West Indies (present day US Virgin Islands) to the United States, there will be a special emphasis on locations and information connected to the Danish West Indies. This collection was digitized by the Royal Library and used for the interdisciplinary datasprint 'Representing History Through Data'. Read more about the specific aims of the project below.
Status of the research project
Phase 1 is completed with the generous assistance of a community of contributors and the technical work of Mace Ojala, resulting in the prototype website available on www.mappingacolony.org
Read about the reflections of ETHOS Lab's involvement in the blog post written by Lab Manager Marie Blønd.
Mapping a Colony aims to:
Create interactive maps of Denmark and the US Virgin Islands, which highlight and investigate the colonial heritage, with a specific focus on the Danish West Indies, as it manifests itself historically and/or presently in these areas.
Disseminate Europeana API archival material to a wider audience.
Create a deeper understanding of how the colonial past is still very much present in our contemporary societies and landscapes.
Connect archival resources, map and photographic collections to places of historical interest in order to facilitate a conversation about them between citizens of Denmark, the US Virgin Islands and the rest of the world.
Develop a dynamic digital platform for the presentation of a hitherto underexposed, perhaps even suppressed, part of Danish cultural heritage.
In the first phase of the project we focused on establishing a website prototype that shows the various possibilities for spatial/geographic mapping of history. This has only been possible with the help of a network of professionals, researchers and artists, who have generously contributed valuable material and information about colonial heritage. A special thanks goes to the team behind the Datasprint series at the Royal Library, which formed the basis for this project: namely Mette Kia Krabbe Meyer from the Royal Library and Michael Hockenhull from the IT University of Copenhagen.
Presently, the map features many of the well-known places and prominent historical figures of the colonial era, but as the work continues we would like to see the site expand to include accounts and stories about the lesser-known people of the era. For example, we aim to include a better representation of black West Indian communities living in Copenhagen, as well as more detailed insight into the lives of the enslaved and Free Colored people in the colonies, as well as documenting where slave-owners lived (inspired by this British project: http://www.ucl.ac.uk/lbs/). Another potential for widening the scope is to include significant places and stories from the African coast to visualize the full global scale of the triangular trade.
As of November 2017 we are looking into the possibility of creating an augmented reality app for the second phase of the project.
One of the first things the Danish colonizers did when they came to the West Indies was to document land acquisition in detail through map-making. Obviously, the layout of the land, and their knowledge and control of it, would determine their success in this unknown place. For around almost 250 years Denmark colonized what is now the US Virgin Islands (St. Croix, St. Thomas and St. John), until the territory was sold to the US in 1917.
The Danish West Indies have – as part of the Atlantic slave trade between Denmark, Ghana, and the Virgin Islands – contributed immensely to the growth and prosperity in and around Golden Age Copenhagen. The period is also known in Danish art history as the "Florissant Age:" parts of Copenhagen and numerous estates north of Copenhagen (including Marienborg, the present official residence of the Danish prime minister) were funded by and built on 'sugar fortunes', relying primarily on the extensive slave trade and slave economy.
This element of the Danish national consciousness is subdued. Not many think of themselves as citizens of a former colonial power. The Virgin Islands have asked for acknowledgement of this shared history, while some movements on the islands have also asked for a formal apology from the Danish state. Yet, even in the Virgin Islands, knowledge about the Danish past remains inadequate due to several factors, the most important being lack of access to archival material, language barriers and also – in many cases – traumatic suppression.
When the islands were sold to the US in 1917, and the Danish administration left the islands, all official archives, including images, photographs and maps went to Copenhagen, where the documents became part of the National Archives as well as the Image and Maps Collection at the Royal Library. These archives are presently being digitized, and The Royal Library confirms that the Danish-Virgin Islands' West Indies Archives will be integrated with the European cultural heritage portal Europeana, through European Library, by the beginning of 2017. This project makes use of Europeana Collections while extending the collaborative contributions to include other underrepresented voices. The mapping exercise once practiced to document landownership and Danish colonization is now employed to counteract this same gesture by bringing forward the people, places and stories that have remained hidden or underrepresented. Mapping a Colony addresses this important issue by employing mapping tools and methods from digital humanities.

redpajama_set_name: RedPajamaCommonCrawl