text:
\section{Introduction and summary}

The proposed duality \cite{Klebanov:2002ja} of the singlet sector of
the $O(N)$ vector model in three space-time dimensions and Vasiliev's
higher spin gauge theory in AdS$_4$ \cite{Vasiliev:2012vf} has received a definite verification\cite{Giombi:2009wh,Sezgin:2003pt}
and has also thrown  valuable light on the origins of holography. Since the field theory is
solvable in the large-N limit, one might hope that there is an
explicit derivation of the higher spin gauge theory from the vector
model, thus providing an explicit understanding of the emergence of
the holographic direction. Indeed, the singlet sector of the $O(N)$
model can be expressed in terms of a Hamiltonian for the bi-local
collective field, $\sigma(\vx,\vy) = \phi^i (\vx) \phi^i (\vy)$
where $\phi^i(\vx), i= 1 \cdots N$ is the $O(N)$ vector field. In \cite{Das:2003vw} it was proposed that
Vasiliev's fields are in fact components of $\sigma(\vx,\vy)$. The
precise connection between the bi-local and HS bulk fields was written explicitly  in the
light cone frame\cite{Koch:2010cy, Jevicki:2011aa, Jevicki:2011ss}:
the correspondence in general  involves a nonlocal transformation corresponding to
a canonical transformation in phase space\footnote{See also reference \cite{Douglas:2010rc}.}. This provides
a direct understanding of the emergence of a holographic direction
from the large-N degrees of freedom, in a way similar to the well
known example of the $c=1$ Matrix model \cite{Das:1990kaa}. In both
these models, the large-N degrees of freedom gave rise to an additional
dimension which had to be interpreted as a {\em spatial} dimension \footnote{Other instances
of emergence of dimensions from large-N degrees of freedom, e.g. Eguchi-Kawai
models \cite{Eguchi:1982nm}, Matrix Theory \cite{Banks:1996vh,Ishibashi:1996xs}
also lead to spatial directions in Lorentzian signature or Euclidean theories.}.

In contrast to AdS/CFT correspondence, any dS/CFT correspondence \cite{dS}
involves an emergent holographic direction which is {\em timelike}. It is then of interest
to understand how a {\em timelike} dimension is generated from large-N degrees of freedom. Recently,
Anninos, Hartman and Strominger \cite{Anninos:2011ui} put forward a conjecture that the {\em euclidean} $Sp(2N)$ vector
model in three dimensions is dual to Vasiliev higher spin theory in
four dimensional de Sitter space.

In this work we construct a collective field theory of the {\em Lorentzian} $Sp(2N)$ model which captures the singlet state dynamics of
the $Sp(2N)$ vector model. Using the results of \cite{Das:2003vw} and
\cite{Koch:2010cy} we then argue that a natural interpretation of the
resulting action is by double analytic continuation which makes the emergent direction time-like, relating this to  higher spin theory in dS$_4$, in a way
reminiscent of the way the Louiville mode in worldsheet string theory has to be interpreted as a time beyond critical
dimensions \cite{Das:1988ds}. Our map establishes the bi-local theory as the bulk space-time  representation of de Sitter higher spin gravity.

The bilocal collective field is a composite of two Grassmann variables and therefore might not appear to be a genuine bosonic field. In particular  for finite $N$ a sufficiently large power of the field operator vanishes, reflecting its Grassmannian origin \footnote{This property of higher spin currents has been already recognized in \cite{Ng:2012xp}}. This is further reflected on the size of its Hilbert space. The bulk theory cannot be a usual bosonic theory defined on dS space, though it may be regarded as such in a perturbative $1/N$ expansion.

The implementation of the Grassmann origin of the Hilbert space will be given a central attention  in the present work. For this we will describe a geometric (pseudo-spin) version of the collective theory  which will be seen to incorporate these effects.  For dS/CFT, this implies that the true number of degrees of freedom in the dual higher spin theory in dS is in this framework reduced from what is  seen perturbatively (with $G=R_{dS}^2/N$ being the coupling constant squared).
The issue of the size of the Hilbert space is of central relevance for possible accounting of entropy of de Sitter space.
For pure Gravity in de Sitter space, it has argued that the Entropy being $S=A/4G$  with a finite area of the horizon requires a finite dimensional Hilbert space \cite{witten,Balasubramanian:2001rb,Banks:2003cg}. Interesting quantum mechanical models have been  proposed \cite{Balasubramanian:2001rb,Volovich:2001rt,Parikh:2004wh,Lowe:2004nw}  to account for this. But apparent conflicts  between a finite entropy of de Sitter space with the usual formulations of dS/CFT have been discussed for example in in \cite{Dyson:2002nt}. In the present case of dS/CFT we are dealing with N-component quantum field theory with d=3 dimensional space so clearly the number of degrees of freedom must be infinite. Consequently the question of Entropy remains open and is an interesting  topic for further investigations.

\section{The $Sp(2N)$ vector model}

The $Sp(2N)$ vector model in $d$ spacetime dimensions is defined by the action
\ben
S = i\int dt d^{d-1}x~\left[ \{ \partial_t \phi^i_1\partial_t
  \phi^i_2 - \nabla  \phi^i_1 \nabla \phi^i_2\} -V(i\phi^i_1\phi^i_2)
  \right]
\label{one}
\een
where $\phi^i_1,\phi^i_2$ with $i=1\cdots N$ are $N$ pairs of
Grassmann fields.  This is of course a model of ghosts.

In this section we will quantize this model following \cite{Henneaux:1982ma} and \cite{Finkelstein:1985cx}.
In this quantization, the fields $\phi_1^i$ and $\phi_2^i$ are hermitian operators, while the canonically conjugate momenta
\ben
P_1^i = i\partial_t \phi^i_2~~,~~~~~~~~~P_2^i = - i\partial_t \phi^i_1
\label{two}
\een
are anti-hermitian. The Hamiltonian $H$ is hermitian
\ben
H =i \int d^{d-1}x \left[ P^i_2 P^i_1 + \nabla \phi^i_1 \nabla \phi^i_2 + V(i\phi^i_1\phi^i_2) \right]
\label{three}
\een
The (equal time) canonical anticommutation relations are
\bea
\{ \phi^a_i (\vx) , P^b_j (\vy) \} & = & -i \delta_{ij}\delta^{ab}\delta^{d-1}(\vx - \vx^\prime) \nn \\
\{ \phi^i_a (\vx) , \phi^j_b (\vy) \} & = & \{ P^i_a (\vx) , P^j_b (\vy) \} = 0~~,~~~~~~~~~~(a,b=1,2)
\label{four}
\eea
with all other anticommutators vanishing.  With these anticommutators the equations of motion for the corresponding Heisenberg picture operators
\ben
\partial_t^2 \phi^i_a - \nabla^2 \phi^i_a + V^\prime = 0
\label{five}
\een
follow.
The operator relations (\ref{four}) allow a representation of the operators as follows
\ben
\phi^a_i (\vx) \rightarrow \phi^a_i (\vx)~~,~~~~~~P^a_i \rightarrow -i \frac{\delta}{\delta \phi^a_i(\vx)}
\label{six}
\een
where $\phi^i_a$ are now Grassmann fields.

For the free theory, the solution to the equation of motion is
\ben
\phi^i_a (\vx,t) = \int\frac{d^{d-1}k}{(2\pi)^{d-1}\sqrt{2|k|}} \left[
  \alpha_a^i (\vk) e^{-i(|k|t - \vk \cdot \vx)} + \alpha_a^{i\dagger}(\vk)e^{i(|k|t - \vk
    \cdot \vx)} \right]
\label{seven}
\een
and the operators $\alpha^i_a$ satisfy
\ben \{ \alpha_1^i (\vk), \alpha_2^{\dagger j} (\vk^\prime) \} = i
\delta^{ij} \delta (\vk - \vk^\prime) ~~,~~~~~\{ \alpha_1^{\dagger
  i}(\vk), \alpha_2^j (\vk^\prime) \} = -i \delta^{ij}\delta (\vk -
\vk^\prime)
\label{eight}
\een
with all the other anticommutators vanishing.
The Hamiltonian is given by
\ben
H = i\int [d\vk]~|\vk| \left[\alpha_1(\vk)^\dagger \alpha_2 (\vk) - \alpha_2 (\vk)^\dagger \alpha_1 (\vk) \right]
\een
The basic commutators lead to
\ben
[ H , \alpha^i_a (k)] = - k \alpha^i_a (k)~~,~~~~
[ H , \alpha^{i\dagger}_a] = k \alpha^{i\dagger}_a (k)
\label{aone}
\een
To discuss the quantization of the free theory it is useful to review
the quantization of the $Sp(2N)$ oscillator,
following \cite{Finkelstein:1985cx} \footnote{Note that our
  notation is different from that of \cite{Finkelstein:1985cx}}. The Hamiltonian is
\ben
H = i(-\frac{\partial^2}{\partial \phi_2^i \partial \phi_1^i}+k^2 \phi^i_1 \phi^i_2)
\label{four-one}
\een
where $\phi^i_1,\phi^i_2$ are $N$ pairs of Grassmann numbers.
Because of the Grassmann nature of the variables the
spectrum of the theory is bounded both from below and from above.
The oscillators are defined by (in the Schrodinger picture) \ben \phi_a^i =
\frac{1}{\sqrt{2k}} [ \alpha_a^i + \alpha_a^{i\dagger} ]
\label{asix}
\een
while the momenta are
\ben
P_a^i = \epsilon_{ab} \sqrt{\frac{k}{2}}(\alpha_b^i - \alpha_b^{i\dagger})
\label{aseven}
\een
The ground state $|0\rangle$ and the highest state $|2N\rangle$ are then given by the conditions
\ben
\alpha^i_a |0\rangle = 0
~~,~~~~~~~~~~~~~~~\alpha^{i\dagger}_a |2N\rangle = 0
\label{atwo}
\een
with the wavefunctions
\ben
\Psi_0 = {\rm exp}[-ik \phi_1^i \phi_2^i]~~,~~~~~~~~~~~~~
\Psi_{2N} = {\rm exp}[ik \phi_1^i \phi_2^i]
\label{athree}
\een
and the energy spectrum is given by
\ben
E_n = k [n - N]~~,~~~~~~~~~~~~~~n = 0,1,\cdots, 2N
\label{afour}
\een
Finally, the Feynman correlator of the Grassmann coordinates may be easily seen
to be
\ben
\langle 0|T[\phi^i_1(t)\phi^j_2(t^\prime)]|0\rangle= \frac{i\delta^{ij}}{2k}~e^{-ik|t-t^\prime|}
\label{afive}
\een
Extension of these results to the free field theory is straight forward: for each momentum $\vk$, we have a fock space with a finite number of states.

\section{Collective Field Theory for the $Sp(2N)$ model}

In the representation (\ref{six}) a general wavefunctional is given by $\Psi[\phi^i_a(\vx),t]$. Our aim is to obtain a description of the singlet sector of the theory, i.e. wavefunctionals which are invariant under the $Sp(2N)$ rotations of the fields $\phi^i_a(\vx)$. All the invariants in field space are functions of the bilocal collective fields
\ben
\rho(\vx,\vy) \equiv i \epsilon^{ab}\phi^i_a (\vx) \phi^i_b (\vy)
\label{twelve}
\een

We have defined this collective field to be hermitian (which is why there is a $i$ in the definition). Clearly $\rho(\vx,\vy) = \rho(\vy,\vx)$. The aim now is to rewrite the theory in terms of a Hamiltonian which is a functional of $\rho (\vx,\vy)$ and its canonical conjugate $-i \frac{\delta}{\delta \rho(\vx,\vy)}$ which acts on wavefunctionals which are functionals of $\rho (\vx,\vy)$.

It is important to remember that $\rho(\vx,\vy)$ is not a genuine bosonic field. This will have important consequences at finite $N$. In a perturbative expansion in $1/N$, however,
there is no problem \cite{deMelloKoch:1996mj} in  treating $\rho(\vx,\vy)$ as a bosonic field.

Before dealing with the $Sp(2N)$ field theory, it is useful to review some aspects of the collective theory for the usual $O(N)$ model, starting with the $O(N)$ oscillator.

\subsection{Collective fields for the $O(N)$  theory}

In this section we review the bi-local collective field theory construction for the $O(N)$ field theory, starting with the $O(N)$ oscillator. This has a Hamiltonian
\ben
H = \frac{1}{2} [ P^iP^i + k^2 X^i X^i]
\label{fourteen}
\een
The collective variable is the square of the radial coordinate
$\sigma = X^i X^i$ and the Jacobian for transformation from $X^i$ to $\sigma$ and the angles is
\ben
J(\sigma) = \frac{1}{2}t  \sigma^{(N-2)/2}\Omega_{N-1}
\label{eighteen}
\een
where $\Omega_{N-1}$ is the volume of unit $S^{N-1}$.
The idea is to find the Hamiltonain $H(\sigma, \frac{\partial}{\partial \sigma})$ which acts on wavefunctions $[J(\sigma)]^{1/2} \Psi (\sigma)$.
The key observation of \cite{Jevicki:1979mb} is that this can also be obtained by requiring that $H(\sigma, \frac{\partial}{\partial \sigma})$ acting on wavefunctions $[J(\sigma)]^{1/2} \Psi (\sigma)$ is hermitian with the trivial measure $d\sigma$. This determines both the Jacobian and the Hamiltonian and the technique generalizes to higher dimensional field theory. The final result is well known,
\ben
H_{coll} = -2\frac{\partial}{\partial \sigma} \sigma \frac{\partial}{\partial \sigma} + \frac{(N-2)^2}{8\sigma} + \frac{1}{2} k^2 \sigma
\label{twothree}
\een
The large-$N$ expansion then proceeds as usual by expanding around the saddle point solution $\sigma_0$ which minimizes the potential \footnote{To see why the saddle point approximation is valid, rescale
$\sigma \rightarrow N\sigma$ and $\Pi_\sigma \rightarrow \frac{1}{N} \Pi_\sigma$ so that there is an overall factor of $N$ in front of the potential energy term. We will, however, stick to the unrescaled fields.},
\ben
\sigma_0^2 = \frac{N^2}{4k^2}
\label{twofive}
\een
Clearly, we have to choose the positive sign since in this case $\sigma$ is a {\em positive} real quantity,
\ben
\sigma_0 = \frac{N}{2k}
\label{twosix}
\een
which reproduces the coincident time two point function $\langle 0|X^i(t)X^i(t)|0\rangle$ and the correct ground state energy, $
E_0 = \frac{N}{2}k $. The subleading contributions are then obtained by expanding around the saddle point,
\ben
\sigma = \sigma_0 + \sqrt{\frac{2N}{k}}\eta~~,~~~~~\Pi_\sigma =
\sqrt{\frac{k}{2N}}
\pi_\eta
\label{twoeight}
\een
The quadratic part of the Hamiltonian becomes
\ben
H^{(2)} = \frac{1}{2} \left[\pi_\eta^2+4k^2 \eta^2 \right]
\label{bone}
\een
This leads to the excitation spectrum to $O(1)$,
$E_n = 2n k$ with $n = 0,1,\cdots ,\infty$.
The Hamiltonian of course contains all powers of $\eta$.  {\em
Terms with even number of the fluctuations $(\pi_\eta,\eta)$ come with odd factors of $\sigma_0$. } This fact will play a key role in the following.

In the following it will be necessary to consider wavefunctions. It follows directly from (\ref{fourteen}) that the ground state wavefunction is given by (up to a normalization which is not important for our purposes)
\ben
\Psi_0 (X^i) = {\rm exp} [ -\frac{k}{2} \sigma ] \sim {\rm exp} [ -\sqrt{\frac{Nk}{2}} \eta ]
\label{bthree}
\een
where we have expanded $\sigma$ as in (\ref{twoeight}), used (\ref{twosix}) and ignored an overall constant.
We should get the same result from the collective theory. Recalling that the collective wavefunction is related to the original wavefunction by a Jacobian factor, the ground state wavefunction follows from (\ref{bone})
\ben
\Psi_0^\prime(\eta) = [J(\sigma)]^{-\frac{1}{2}}~{\rm exp} [-k \eta^2]
\label{bfive}
\een
The presence of the Jacobian is crucial in obtaining agreement with (\ref{bthree}) \cite{Jackiw:1980xj}.
Expanding the argument in the Jacobian in powers of $\eta$ according to (\ref{twoeight}) it is easy to see that the quadratic term in $\eta$ coming from the Jacobian exactly cancels the explicit quadratic term in (\ref{bfive}) and the linear term in $\eta$ is in exact agreement with (\ref{bthree}). The expression (\ref{bfive}) of course contain all powers of $\eta$ once exponentiated - these should also cancel once one takes into account the cubic and higher terms in the collective Hamiltonian as well as finite $N$ corrections which we have ignored to begin with.
The above formalism can be easily generalized to an additional invariant potential, since the latter would be a function of $\sigma$.

The collective theory for $O(N)$ field theory can be constructed along identical lines. We reproduce the relevant formulae
from \cite{Jevicki:1979mb} which are direct generalizations of
the formulae for the oscillator. The $O(N)$ model has the Hamiltonian
\ben
H = \frac{1}{2} \int d^{d-1}x \left[ -\frac{\delta^2}{\delta
    \phi^i(\vx) \delta \phi^i(\vx)}+ \nabla \phi^i (\vx) \nabla \phi^i
  (\vx) + U[\phi^i(\vx) \phi^i(\vx)] \right]
\label{twonine}
\een
The singlet sector Hamiltonian in terms of the bi-local collective field
$\sigma(\vx,\vy) = \phi^i(\vx)\phi^i(\vy)$ and its canonically conjugate momentum $\Pi_\sigma(\vx,\vy)$ is, to leading order in $1/N$ \footnote{To subleading order there are singular terms which are crucial for reproducing the correct $1/N$ contributions.}
\ben
H^{O(N)}_{coll} = 2 {\rm{Tr}} \left[ (\Pi_\sigma \sigma \Pi_\sigma) +\frac{N^2}{16} \sigma^{-1} \right] - \frac{1}{2}\int d\vx \nabla_x^2 \sigma (\vx,\vy) |_{\vy = \vx} + U(\sigma(\vx,\vx))
\label{three-three}
\een
where  the spatial coordinates are treated as matrix indices.

So far our considerations are valid for an arbitrary interaction potential $U$. Let us now restrict ourselves to the free theory, $U = 0$ to discuss the large-$N$ solution explicitly.
In momentum space
the saddle point solution is
\ben
\sigma(\vk_1,\vk_2) = \frac{N}{2|\vk_1|}\delta (\vk_1-\vk_2)
\een
Once again we have chosen the positive sign in the solution of the saddle point equation, and the saddle point value of the collective field agrees with the two point correlation function of the basic vector field, which should be positive. The $1/N$ expansion is generated in a fashion identical to the single oscillator,
\ben
\sigma(\vk_1,\vk_2) = \sigma_0 (\vk_1,\vk_2)+ \left( \frac{|\vk_1||\vk_2|}{N(|\vk_1|+|\vk_2|)} \right)^{-\frac{1}{2}}\eta(\vk_1,\vk_2)~~,~~\Pi_\sigma = \left( \frac{|\vk_1||\vk_2|}{N(|\vk_1|+|\vk_2|)} \right)^{\frac{1}{2}}\pi_\eta (\vk_1,\vk_2)
\label{three-five}
\een
the quadratic piece becomes
\ben
H^{(2)} = \frac{1}{2}\int d\vk_1 d\vk_2 \left[ \pi_\eta (\vk_1,\vk_2) \pi_\eta (\vk_1,\vk_2)+ (|\vk_1|+|\vk_2|)^2 \eta (\vk_1,\vk_2)\eta (\vk_1,\vk_2) \right]
\label{three-eight}
\een
so that the energy spectrum is given by
\ben
E(\vk_1, \vk_2) = |\vk_1|+|\vk_2|
\label{three-nine}
\een
as it should be. It is easy to check that the unequal time two point function of the fluctuations reproduces the connected part of the two point function of the full collective field as calculated from the free field theory.
A nontrivial $U$ can be reinstated easily (see e.g. the treatment of the $(\vphi^2)^2$ model in \cite{Das:2003vw}, which discusses the RG flow to the nontrivial IR fixed point).

\subsection{ Collective theory for the $Sp(2N)$ oscillator}

Since there is a representation of the field operator and the
conjugate momentum operator of the $Sp(2N)$ theory in terms of
Grassmann fields, (\ref{six}), it is clear that the derivation of the
collective field theory of the $Sp(2N)$ model closely parallels that of
the $O(N)$ theory. In this subsection we consider the $Sp(2N)$
oscillator. The Hamiltonian is given by (\ref{four-one}.
The collective variable is
\ben
\rho = i \epsilon^{ab}\phi^i_a\phi^i_b
\label{four-two}
\een
The fully connected correlators
of this collective variable have a simple relationship with those of
the $O(2N)$ harmonic oscillator,
\ben
\langle \rho(t_1)\rho(t_2)\cdots\rho(t_n)\rangle^{conn}_{Sp(2N)} = -
\langle \sigma(t_1)\sigma(t_2) \cdots \sigma(t_n)\rangle^{conn}_{SO(2N)}
\label{four-two1}
\een
This result follows from (\ref{afive}) and the application of Wick's
theorem for Grassmann variables.

The collective variable $\rho$ is a Grassmann even variable - it is not an usual bosonic variable. This key fact is intimately related to the finite number of states of the $Sp(2N)$ oscillator. In this section we will show that in a $1/N$ expansion we can nevertheless proceed, defering a proper discussion of this point to a later section.

The Hamiltonian for the collective theory is obtained by the same
method used to obtain the collective theory in the bosonic case, with various negative sign coming from the Grassmann nature of the variables.
Using the chain rule and taking care of negative signs coming because of Grassmann numbers, one gets the Jacobian
$J^\prime(\rho)$ (determined by requiring the hermicity of $J^{-1/2} H J^{1/2}$)
\ben J^\prime(\rho) = A^\prime~\rho^{-(N+1)}
\label{four-five}
\een
where $A^\prime$ is a constant.
The negative power of $\rho$ of course reflects the Grassmann nature
of the variables \footnote{This $\rho$ dependence of the Jacobian
follows from a direct calculation $
J^\prime(\rho) =  \int
d\phi_1^i d\phi_2^i \delta (\rho - i\phi_1^i\phi_2^i) = \int d\lambda
e^{i\lambda \rho}\int d\phi_1^i
d\phi_2^i~e^{-i\lambda\phi_1^i\phi_2^i} \sim \rho^{-(N+1)} $}
Despite this difference, the final collective Hamiltonian is in fact
{\em identical} to the $O(2N)$ oscillator collective Hamiltonian
\ben
H_{coll}^{Sp(2N)} = -2\frac{\partial}{\partial \rho} \rho  \frac{\partial}{\partial \rho} + \frac{N^2}{2\rho}+\frac{1}{2}k^2\rho
\label{four-seven}
\een This leads to the same saddle point equation, and the solutions
satisfy the same equation as (\ref{twofive}) with $N \rightarrow 2N$.

In the $O(2N)$ oscillator, we had to choose the positive sign, since
$\sigma$ is by definition a real {\em positive} variable. In this case, there is no reason for $\rho$ to be positive. In fact we need
to choose the negative
sign, since (\ref{four-two1}) requires that the one point function of
$\rho$ must be the negative of the one point function of $\sigma$.
\ben \rho_0 =
-\frac{N}{k}
\label{four-eight}
\een
It is interesting that the singlet sectors of the $O(2N)$ and $Sp(2N)$ models are described by two different solutions of the {\em same} collective theory.

The leading order ground state energy is the Hamiltonian evaluated on the saddle point,
\ben
E_{gs} = -Nk
\label{cone}
\een
in agreement with (\ref{afour}).
The fluctuation Hamiltonian is obtained as usual by expanding
\ben
\rho = \rho_0 +\sqrt{\frac{4N}{k}}\xi~~,~~~~~
\Pi_\rho = \sqrt{\frac{k}{4N}}\pi_\xi
\label{four-nine}
\een
The quadratic Hamiltonian is now {\em negative}, essentially because
of the negative sign in the saddle point,
\ben
H^{(2)}_\xi = -\frac{1}{2} \left[ \pi_\xi^2 + 4 k^2 \xi^2 \right]
\label{ctwo}
\een
A standard quantization of this theory leads to a spectrum which is unbounded from below. We will now argue that we need to quantize this theory rather differently, in a way similar to the treatment of
\cite{shimo}. This involves defining
annihilation and creation operators $a_\xi, a_\xi^\dagger$
\ben
\xi = \frac{1}{\sqrt{4k}}[a_\xi + a^\dagger_\xi]~~,~~~~~~\pi_\xi =
i\sqrt{k}[a_\xi - a^\dagger_\xi]
\label{cthree}
\een
which now
satisfy
\ben
[ a_\xi, a_\xi^\dagger ] = -1~~,~~~[ H, a_\xi] = - 2k a_\xi~~,~~~[H,
  a^\dagger_\xi] = 2k a^\dagger_\xi
\label{cfour}
\een
Because of the negative sign of the first commutator in (\ref{cfour}) a standard quantization will lead to a {\em highest energy} state annihilated by $a^\dagger_\xi$, and then the action of powers of $a_\xi$ leads to an infinite tower of states with lower and lower energies. The highest state has a normalizable wavefunction of the standard form $e^{-k\xi^2}$ (Note that the expression for $\pi_\xi$ has a negative sign compared to the usual harmonic oscillator). It is easy to see that this standard quantization does not reproduce the correct two-point function of the $Sp(2N)$ theory, does not lead to the correct spectrum (\ref{afour}) and, as shown below, does not lead to the correct wavefunction.

All this happens because $\rho$ and hence $\xi$ is not really a bosonic variable, and this allows other possibilities.
Consider now a state
$|0\rangle_\xi$ which is annihilated by the annihilation operator $a_\xi$. This leads to a wavefunction ${\rm exp}[k \xi^2]$, which is inadmissible if $\xi$ is really a bosonic variable since it would be non-normalizable. However the true integration is over the Grassmann partons of these collective fields, and in terms of Grassmann integration this wavefunction is perfectly fine.
This is in fact the state which has to be identified
with the ground state of the $Sp(2N)$ oscillator. Including the factor
of the Jacobian, the full wavefunction is (at large $N$)
\ben
\Psi_{0\xi}^\prime[\xi] = [J^\prime(\rho)]^{-1/2}{\rm exp}[k\xi^2]
= [-\frac{N}{k} +2 \sqrt{\frac{N}{k}}\xi]^{N/2}{\rm
  exp}[k\xi^2]
\label{cfive}
\een
Expanding the Jacobian factor in powers of $\xi$ one now sees that the
term which is quadratic in $\xi$ cancels exactly, leaving with
\ben
\Psi_{0\xi}^\prime[\xi] = {\rm exp}[-\sqrt{Nk} \xi + O(\xi^3)]
\label{csix}
\een
This is easily seen to exactly agree with $\Psi_0$ in (\ref{athree})
\ben
\Psi_0 \sim {\rm exp}[-\frac{1}{2}k\rho] \sim {\rm exp}[-\sqrt{Nk}\xi]
\label{cseven}
\een
up to a constant. Once again we need to take into account the
interaction terms in the collective Hamiltonian to  check that the
$O(\xi^3)$ terms cancel.  It can be easily verified that the propagator of fluctuations
$\xi$ will now be {\em negative} of the usual harmonic oscillator
propagator. Furthermore the action of $a^\dagger_\xi$ now generates a tower of states with the energies (\ref{afour}) - except that the integer $n$ is not bounded by $N$.

The fact that we get an unbounded (from above) spectrum from the collective theory is not a surprise. This is an expansion around $N=\infty$ and at $N=\infty$ the spectrum of $Sp(2N)$ is also unbounded.
At finite $N$ a change of variables to $\rho$ is not useful because of the constraints coming from the Grassmann origin of $\rho$.
Nevertheless, even in the $1/N$ expansion, the Grassmann origin allows us to consider wavefunctions which would be otherwise considered inadmissible.

The negative propagator ensures that the relationship
(\ref{four-two1}) is satisfied for the 2 point functions.
Once this choice is made, the relationship (\ref{four-two1}) holds for
all $m$-point functions to the
leading order in the large-$N$ limit.
As commented earlier, a term with even
number of $\pi_\xi$ or $\xi$ would have an odd number of factors of
$\rho_0$. Therefore a $n$-point vertex in the theory will differ from
the corresponding $n$-point vertex of the $O(N)$ theory by a factor of
$(-1)^{n+1}$.
The connected correlator which appears in (\ref{four-two1}) is the sum of
all connected tree diagrams with $n$ external legs.
The collective theory gives us the following Feynman rules
\begin{itemize}
    \item [1] Every propagator contributes to a negative sign.
    \item [2] A $p$ point vertex has a factor of $(-1)^{p+1}$
\end{itemize}
We now argue that these rules ensure the validity of the basic relation (\ref{four-two}).
We do it by the following simple diagrammatic method:
\begin{figure}[h]
\centering
\includegraphics[scale=0.50]{trees.eps}
\caption{Connected tree level correlators of the collective theory}
\label{fig:tree}
\end{figure}

Consider first the simplest diagram for a $n$-point function, figure
A, which is a star graph. The net sign of the diagram is $(-1)^{n+1}
\times (-1)^n = -1$, where the first factor is from the vertex $a_0$
and the second one from the number of lines. Now we proceed to construct all
other tree level diagrams from A, by pulling `$r$' lines resulting in
figure B, which now has vertices, $a_1$ and $b_1$ joined by a new
line. It is easy to see, that the sign of figure A is not changed by
this operation. The net sign of figure B is $(-1)^{(n-r+1)+1} \times
(-1)^{(r+1)+1}\times(-1)^{(n+1)} = -1$, where the 3 factors are from
$a_1$, $b_1$ and the number of lines respectively. In figure C we
repeat this method for the substar diagrams until we exhaust all
possibilities. It is easy to see that the sign stays
invariant. Assigning a sign $\alpha$ to the blob, we first find the
net sign of the left diagram in figure C. It turns out to be, $\alpha
\times (-1)^{(k+1)+1} \times (-1)^{k+1} = - \alpha$. After the
``pulling" operation we get $\alpha \times (-1)^{(k-r+2)+1} \times
(-1)^{(r+1)+1} \times (-1)^{k+1+1} = - \alpha$. Thus it is proved that
in every move the sign is preserved. This proves the relationship
(\ref{four-two1}) for all correlation functions.

\subsection{$Sp(2N)$ Correlators}

Our discussion of the bosonic $O(N)$ collective field theory shows
that the $Sp(2N)$ collective field theory in momentum space is a
straightforward generalization. In this subsection we discuss the
relevant features of the collective theory for the free $Sp(2N)$ model.

The collective Hamiltonian is again exactly the same as in the $O(N)$
theory, given by (\ref{three-three}) with $\sigma \rightarrow
\rho$. Since the connected correlators of the collective fields
satisfy
\ben
\langle\rho(\vk_1,\vk_1^\prime, t_1)\rho(\vk_2, \vk_2^\prime
,t_2)\cdots\rho(\vk_n,\vk_n^\prime,t_n)\rangle^{conn}_{Sp(2N)} = -
\langle\sigma(\vk_1,\vk_1^\prime,t_1)\sigma(\vk_2,\vk_2^\prime,t_2) \cdots
\sigma(\vk_n,\vk_n^\prime, t_n)\rangle^{conn}_{SO(2N)}
\label{fifty}
\een
we now need to choose the negative saddle point,
\ben
\rho_0(\vk ,\vk^\prime,t)= -\frac{N}{|\vk|}\delta(\vk -\vk^\prime)
\label{five-one}
\een
The fluctuation Hamiltonian once again has a factor of $(-1)^{n+1}$
for the $n$-point vertex. In particular, the propagator of the
collective field is negative of that of the $O(N)$ collective field -
the quadratic Hamiltonian has an overall negative sign! This is
required - the diagramatic argument for the $Sp(2N)$ oscillator
generalizes in a straightforward fashion, ensuring that (\ref{fifty})
holds.

\section{Bulk Dual of the $Sp(2N)$ model}

In \cite{Das:2003vw}, it was proposed that the collective field theory
for the $d$ dimensional free $O(N)$ theory is in fact Vasiliev's higher spin theory in
AdS$_{d+1}$.  It is easy to see that the collective field has the
right collection of fields.  Consider for example $d=3$. The field
depends on four spatial variables, which may be reorganized as three
spatial coordinates one of which is restricted to be positive and an
angle. A fourier series in the angle then gives rise to a set of
fields $\chi_{\pm n}$ which depend on three spatial variables, with
the integer $n$ denoting the conjugate to the angle. Symmetry under
interchange of the arguments of the collective field then requires $n$
to be even integers. But this is precisely the content of a theory of
massless even spin fields in four space-time dimensions, with $n$
labelling the spin and the two signs corresponding to the two
helicities. (Recall that in four space-time dimensions massless fields
with any spin have just two helicity states).

The precise relationship between collective fields and higher spin
fields in AdS was found in \cite{Koch:2010cy} which we now summarize
for $d=3$. The correspondence is formulated in the  light front
quantization. Denote the usual Minkowski coordinates on the space-time
on which the $O(N)$ fields live by $t,y,x$ and define light cone
coordinates
\ben
x^\pm = \frac{1}{\sqrt{2}}(t \pm y)
\label{five-one1}
\een
The conjugate momenta to $x^+,x^-$ are denoted by $p^-,p^+$. Then in
light front quantization where $x^+$ is treated as time, the
Schrodinger picture fields are $\phi^i (x^-,x)$ while the momentum
space fields are given by $\phi^i (p^+,p)$. The corresponding
collective field is then defined as
\ben
\sigma(p^+_1,p_1;p^+_2,p_2)=\phi^i(p^+_1,p_1)\phi^i(p^+_2,p_2)
\label{five-two}
\een The fluctuation of this field around the saddle point is denoted
by $\Psi(p^+_1,p_1;p^+_2,p_2)$. Now define the following bilocal field
\ben
\Phi(p^+,p^x,z,\theta) = \int dp^z
dp_1^+dp_2^+dp_1dp_2~K(p^+,p^x,z,\theta; p^+_1,p_1,p^+_2,p_2)
\Psi(p^+_1,p_1;p^+_2,p_2)
\label{five-three}
\een
where the kernel is given by
\bea
K(p^+,p^x,z,\theta; p^+_1,p_1,p^+_2,p_2)
&= & z~e^{izp_z}~ \delta (p^+_1+p^+_2-p^+) ~\delta(p_1+p_2-p) \nn \\ &
& \delta(p_1\sqrt{\frac{p_2^+}{p_1^+}}-
p_2\sqrt{\frac{p_1^+}{p_2^+}}-p^z)~
\delta(2\tan^{-1}\sqrt{\frac{p_2^+}{p_1^+}}-\theta) \nn
\label{five-four}
\eea
In \cite{Koch:2010cy} it was shown that the Fourier transforms of the field
$\Phi(p^+,p^x,z,\theta)$ with respect to $\theta$ satisfy the same linearized equation of motion as the physical helicity modes of
higher spin gauge fields in AdS$_4$ in light cone gauge. The metric of this AdS$_4$ is given by the standard Poincare form
\ben
ds^2 = \frac{1}{z^2}[-2dx^+dx^-+dx^2+dz^2]=\frac{1}{z^2}[-dt^2+dy^2+dx^2+dz^2]
\label{five-five}
\een
The momenta $p^+,p$ are conjugate to $x^-,x$. The additional dimension generated from the large-N degrees of freedom is $z$, which is canonically conjugate to $p^z$ and is given in terms of the phase space coordinate of the bi-locals by
\ben
z = \frac{(x_1-x_2)\sqrt{p^+_1p^+_2}}{p^+_1+p^+_2}
\label{five-six}
\een
In particular, the linearized equation for the spin zero field, $\varphi (x^-,x,z)$, follows from the quadratic action
\ben
S = \frac{1}{2} \int dx^+dx^-dzdx~\left[\frac{1}{z^2}\left( -2\partial_+\varphi\partial_-\varphi -(\partial_x\varphi)^2-(\partial_z\varphi)^2 \right)+\frac{2}{z^4}\varphi^2 \right]
\label{five-seven1}
\een
which is of course the action of a conformally coupled scalar in the AdS$_4$ with coordinates given by (\ref{five-six}). The actions for the spin-$2s$ fields can be similarly written down. Even though these actions are derived using light cone coordinates, they can be covariantized easily since these are free actions. In terms of the coordinates $t,y,x,z$ the scalar action is given by
\ben
S = \frac{1}{2} \int dt dz dx dy~\left[\frac{1}{z^2}\left( (\partial_t \varphi)^2 -(\partial_y \varphi)^2 -(\partial_x\varphi)^2-(\partial_z\varphi)^2 \right)+\frac{2}{z^4}\varphi^2 \right]
\label{five-seven}
\een

Let us now turn to the $Sp(2N)$ collective theory. One can define once
again the fields as in (\ref{five-three}) and (\ref{five-four}). The
coordinates $(x^+,x^-,x,z)$ will continue to transform appropriately
under AdS isometries. However, we saw earlier that the quadratic
part of the Hamiltonian, and therefore the quadratic part of the
action will have an overall {\em negative} sign.

A negative kinetic term signifies a pathology. Indeed we derived this
theory with the Lorentzian signature $Sp(2N)$ model, which has negative
norm states. The negative kinetic term of the collective theory is
possibly intimately related to this lack of unitarity.

However, the form of the action (\ref{five-seven}) cries out for a {\em analytic continuation}
\ben
z = i\tau~~,~~~~~~t = -iw
\label{five-eight}
\een
Under this continuation the action, $S$  becomes
\ben
S^\prime = \frac{1}{2} \int d\tau dw dx dy~\left[\frac{1}{\tau^2}\left( (\partial_\tau \varphi)^2 -(\partial_y \varphi)^2 -(\partial_x\varphi)^2-(\partial_w\varphi)^2 \right)-\frac{2}{\tau^4}\varphi^2 \right]
\label{five-nine}
\een
The sign of the mass term {\em has not changed} in this analytic continuation, and this action has become the action of a conformally coupled scalar field in de Sitter space with the metric
\ben
ds^2 = \frac{1}{\tau^2} [ -d\tau^2 + dx^2 + dy^2 + dw^2]
\label{sixty}
\een
This mechanism works for all {\em even} higher spin fields at the
quadratic level.

To summarize, the collective field theory of the three dimensional
Lorentzian $Sp(2N)$ model can be written as a theory of massless even
spin fields in AdS$_4$, but with negative kinetic terms. Under a
double analytic continuation this becomes the action in dS$_4$ with
positive kinetic terms.  This is consistent with the conjecture of
\cite{Anninos:2011ui} that the {\em euclidean} $Sp(N)$ model is dual
to Vasiliev theory in dS$_4$. It is interesting to note that the way
an emergent holographic direction is similar to the way the Liouville
mode has to be interpeted as a time dimension in worldsheet
supercritical string theory \cite{Das:1988ds}. In this latter case,
the sign of the kinetic term for the Liouville mode is negative for $d
> d_{cr}$.

Even for the $O(N)$ model, the collective field is an represents seemingly an overcomplete description, since for a finite number of points in space $K$, one  replaces at most $NK$ variables by $K^2$ variables, which is much larger in the thermodynamic and continuum limit. However, in the perturbative $1/N$ expansion this is not an issue and the collective theory is known to reproduce the standard results of the $O(N)$ model. The issue becomes of significance at  finite $N$ level.The relevance of incorporating  for such  features  has been noted in \cite{Ng:2012xp,Shenker:2011zf}.

For the fermionic  $Sp(2N)$ model, there appears potentially an even more important redundancy related to the Grassmannian origin of the construction.  Consequently the fields are to obey nontrivial constraint relationships and the Hilbert space is subject to a cutoff of highly excited states. This `exclusion principle'
 was noted already in the AdS correspondence involving $S_N$ orbifolds\cite{Maldacena:1998bw,Jevicki:1998rr,Jevicki:1998bm}.

In an expansion around $N = \infty$ most effects of this are invisible and our discussion shows that this can be regarded as a theory of higher spin fields in dS is insensitive to these effects. However, as we saw above, the Grassmann origin was already  of importance  in choosing the correct saddle point and the correct quanization of the quadratic hamiltonian. In the next section we will address the question of finite $N$ and the Hilbert space of the bi-local theory. In the framework of  geometric (pseudospin) representation we will give evidence that the bi-local theory is non-perturbatively satisfactory at the finite $N$ level.

\section{Geometric Representation and The Hilbert Space}

The bi-local collective field representation is seen to give a bulk description dS space and the Higher Spin fields. It provides an interacting theory with  vertices governed by $G=1/N$ as the coupling constant.  We would now to show that the collective theory  has an equivalent   geometric (Pseudo) Spin variable description
appropriate for nonperturbative considerations. The essence of this (geometric) description is in reinterpreting  the bi-local collective fields (and their canonical conjugates) as   matrix variables (of infinite dimensionality) endowed with a Kahler structure.This geometric description  will provide a tractable framework for quantization and non-perturbative definition of the bi-local and HS  de Sitter  theory. It will be seen capable  to incorporate non-perturbative features related to the Grassmannian origin of bi-local fields and  its Hilbert space.
Pseudo-spin collective variables  represent all $Sp(2N)$ invariant variables of the theory (both commuting  and non-commuting). These  close  a compact algebra and at large $N$ are constrained by the corresponding Casimir operator. One therefore has an algebraic pseudo-spin system whose nonlinearity is governed by the coupling constant $G=1/N$. As such they have been employed earlier for developing a large $N$ expansion \cite{Jevicki:1980mj}  and  as a model for  quantization \cite{Berezin:1978sn}. This version of the theory is in its perturbative ($1/N$) expansion identical to the bi-local collective representation. It therefore has the same map to and correspondence with Higher Spin  dS$_4$ at perturbative level. We will see however  that the geometric  representation becomes of use for defining
(and evaluating)  the Hilbert space and its quantization.

To describe the pseudo-spin description of the  $Sp(2N)$ theory we will follow the quantization procedure  of \cite{LeClair:2007iy}. In this approach one starts from  the action:
\begin{equation}
S=\int d^dx\; dt(\partial^\mu\eta^i_1\partial_\mu\eta^i_2)
\end{equation}
and deduces the canonical anti-commutation relations
\begin{equation}
\{\eta^i_1(x,t)\partial_t\eta^j_2(x',t)\}=-\{\eta^i_2(x,t)\partial_t\eta^j_1(x',t)\}=i\delta^d(x-x')\delta^{ij}
\end{equation}
The quantization based on the mode expansion
\begin{eqnarray}
\eta^i_1(x)&=&\int \frac{d^dk}{(2\pi)^{d/2}\sqrt{2\omega_k}}(a_{k+}^{i\dagger} e^{-ikx}+a^i_{k-}e^{ikx})\cr
\eta^i_2(x)&=&\int \frac{d^dk}{(2\pi)^{d/2}\sqrt{2\omega_k}}(-a_{k-}^{i\dagger} e^{-ikx}+a^i_{k+}e^{ikx})
\end{eqnarray}
with
\begin{eqnarray}
\{a^i_{k-},a_{k'-}^{j\dagger}\}=\{a^i_{k+},a_{k'+}^{j\dagger}\}=\delta^d(k-k')\delta^{ij}
\end{eqnarray}
Note that in this approach the operators $\eta^i_a$ are not hermitian, but pseudo-hermitian in the sense of \cite{Bender:2007nj}.

Pseudo-spin bi-local variables will be introduced based on  $Sp(2N)$ invariance, we have the  vectors:
\begin{eqnarray}
\eta&=&(\eta_1^1,\eta_2^1,\eta_1^2,\eta_2^2,\cdots,\eta_1^N,\eta_2^N)\cr
a(k)&=&(a^1_{k-},a_{k+}^1,a^2_{k-},a_{k+}^2,\cdots ,a^N_{k-},a_{k+}^N)\cr
\tilde{a}(k)&=&(a_{k+}^{1\dagger},-a^{1\dagger}_{k-},a_{k+}^{2\dagger},-a^{2\dagger}_{k-},\cdots ,a_{k\dagger}^{N+},-a^{N\dagger}_{k-})
\end{eqnarray}
and the notation:
\begin{equation}
\eta(x)=\int \frac{d^dk}{(2\pi)^{d/2}\sqrt{2\omega_k}}(\tilde{a}(k)e^{-ikx}+a(k)e^{ikx})
\end{equation}
so that  a complete set of $Sp(2N)$ invariant operators now follows:
\begin{eqnarray}
S(p_1,p_2)&=&\frac{-i}{2\sqrt{N}}a^T(p_1)\epsilon_N a(p_2)=\frac{i}{2\sqrt{N}}\sum_{i=1}^N(a_{p_1+}^i a_{p_2-}^i+a_{p_2+}^i a_{p_1-}^i)\cr
S^\dagger(p_1,p_2)&=&\frac{-i}{2\sqrt{N}}\tilde{a}^T(p_1)\epsilon_N \tilde{a}(p_2)=\frac{i}{2\sqrt{N}}\sum_{i=1}^N(a_{p_1+}^{i\dagger} a_{p_2-}^{i\dagger}+a_{p_2+}^{i\dagger} a_{p_1-}^{i\dagger})\cr
B(p_1,p_2)&=&\tilde{a}^T(p_1)\epsilon_N a(p_2)=\sum_{i=1}^Na_{p_1+}^{i\dagger}a_{p_2+}^{i}+a_{p_1-}^{i\dagger}a_{p_2-}^{i}
\end{eqnarray}
and  $\epsilon_N=\epsilon \otimes \mathbb{I}_N$, $\epsilon=\left(    \begin{array} {cc} 0&1 \\  -1&0  \end{array}  \right)$

These invariant operators close an invariant algebra. The  commutation relations are found to equal:
\begin{eqnarray}
\big[ S(\vec{p}_1,\vec{p}_2),S^\dagger(\vec{p}_3,\vec{p}_4)\big]=&&{1\over 2}\left(
\delta_{\vec{p}_2,\vec{p}_3}\delta_{\vec{p}_4,\vec{p}_1}+\delta_{\vec{p}_2,\vec{p}_4}\delta_{\vec{p}_3,\vec{p}_1}\right)
-{1\over {4N}}\left[\delta_{\vec{p}_2,\vec{p}_3}B(\vec{p}_4,\vec{p}_1)+\delta_{\vec{p}_2,\vec{p}_4}B(\vec{p}_3,\vec{p}_1)\right.\cr
&&+\left.
\delta_{\vec{p}_1,\vec{p}_3}B(\vec{p}_4,\vec{p}_2)+
\delta_{\vec{p}_1,\vec{p}_4}B(\vec{p}_3,\vec{p}_2)\right]\cr
\big[ B(\vec{p}_1,\vec{p}_2),S^\dagger(\vec{p}_3,\vec{p}_4)\big]=&&
\delta_{\vec{p}_2,\vec{p}_3}S^\dagger(\vec{p}_1,\vec{p}_4)+
\delta_{\vec{p}_2,\vec{p}_4}S^\dagger(\vec{p}_1,\vec{p}_3)\cr
\big[ B(\vec{p}_1,\vec{p}_2),S(\vec{p}_3,\vec{p}_4)\big]=&&-
\delta_{\vec{p}_1,\vec{p}_3}S(\vec{p}_2,\vec{p}_4)-
\delta_{\vec{p}_1,\vec{p}_4}S(\vec{p}_2,\vec{p}_3)
\end{eqnarray}

The singlet sector of the original $Sp(2N)$ theory is characterized by a further constraint. This constraint is is associated with the Casimir operator of of the algebra and can be shown  to take the form:
\begin{eqnarray}
\frac{4}{N}S^\dagger\star S+(1-\frac{1}{N}B)\star (1-\frac{1}{N}B)=\text{$\mathbb{I}$}
\end{eqnarray}
Here we have used the matrix star product  notation: $\star$ product as: with $A\star B=\int d\vec{p}_2A(\vec{p}_1\vec{p}_2)B(\vec{p}_2\vec{p}_3)$.

The form of the  Casimir, which commutes with the above pseudo-spin fields points to the compact nature of the bi-local pseudo-spin algebra associated with the $Sp(2N)$ theory. This will have major consequences which we will highlight later.

Indeed it is interesting to compare the algebra with the  bosonic case, where we have:
\begin{eqnarray}
S(p_1,p_2)&=&\frac{1}{2\sqrt{N}}\sum_{i=1}^{2N} a_i(p_1)a_i(p_2)\cr
S^\dagger(p_1,p_2)&=&\frac{1}{2\sqrt{N}}\sum_{i=1}^{2N} a_i^\dagger(p_1)a_i^\dagger(p_2)\cr
B(p_1,p_2)&=&\sum_{i=1}^{2N} a_i^\dagger(p_1)a_i(p_2)
\end{eqnarray}
with the  commutation relations:
\begin{eqnarray}
\big[ S(\vec{p}_1,\vec{p}_2),S^\dagger(\vec{p}_3,\vec{p}_4)\big]=&&{1\over 2}\left(
\delta_{\vec{p}_2,\vec{p}_3}\delta_{\vec{p}_4,\vec{p}_1}+\delta_{\vec{p}_2,\vec{p}_4}\delta_{\vec{p}_3,\vec{p}_1}\right)
+{1\over {4N}}\left[\delta_{\vec{p}_2,\vec{p}_3}B(\vec{p}_4,\vec{p}_1)+\delta_{\vec{p}_2,\vec{p}_4}B(\vec{p}_3,\vec{p}_1)\right.\cr
&&+\left.
\delta_{\vec{p}_1,\vec{p}_3}B(\vec{p}_4,\vec{p}_2)+
\delta_{\vec{p}_1,\vec{p}_4}B(\vec{p}_3,\vec{p}_2)\right]\cr
\big[ B(\vec{p}_1,\vec{p}_2),S^\dagger(\vec{p}_3,\vec{p}_4)\big]=&&
\delta_{\vec{p}_2,\vec{p}_3}S^\dagger(\vec{p}_1,\vec{p}_4)+
\delta_{\vec{p}_2,\vec{p}_4}S^\dagger(\vec{p}_1,\vec{p}_3)\cr
\big[ B(\vec{p}_1,\vec{p}_2),S(\vec{p}_3,\vec{p}_4)\big]=&&-
\delta_{\vec{p}_1,\vec{p}_3}S(\vec{p}_2,\vec{p}_4)-
\delta_{\vec{p}_1,\vec{p}_4}S(\vec{p}_2,\vec{p}_3)
\end{eqnarray}
In this case the Casimir constraint is found to equal:
\begin{eqnarray}
-\frac{4}{N}S^\dagger\star S+(1+\frac{1}{N}B)\star (1+\frac{1}{N}B)=\text{$\mathbb{I}$}
\end{eqnarray}
featuring the non-compact nature of the bosonic problem.

We can see therefore that the singlet sectors of the  fermionic $Sp(2N)$ theory and the bosonic $O(2N)$ theory can be described in analogous a bi-local pseudo-spin algebraic formulations with a quadratic Casimir taking the form:
\begin{eqnarray}
4\gamma S^\dagger\star S+(1-\gamma B)\star (1-\gamma B) =\text{$\mathbb{I}$}
\end{eqnarray}
the difference being that  with $\gamma=\frac{1}{N}(-\frac{1}{N})$ for the fermionic (bosonic) case respectively. This signifies the compact versus the non-compact nature of the algebra, but also exhibits the relationship obtained through the $N \leftrightarrow -N$  switch that was central in the argument for de Sitter correspondence in \cite{Anninos:2011ui}.

From this algebraic bi-local formulation one can easily see the the Collective field representation(s) that we have discussed in sections 2 and 3.  Very simply, the Casimir constraints can be solved, and the algebra implemented in terms of a canonical pair of bi-local fields:
\begin{eqnarray}
S(p_1p_2)&=& \frac{\sqrt{-\gamma}}{2}\int dy_1dy_2e^{-i(p_1y_2+p_2y_2)}\{-\frac{2}{\kappa_{p_1}\kappa_{p_2}}\Pi\star\Psi\star\Pi(y_1y_2)-\frac{1}{2\gamma^2\kappa_{p_1}\kappa_{p_2}}\frac{1}{\Psi}(y_1y_2)\cr
&&+\frac{\kappa_{p_1}\kappa_{p_2}}{2}\Psi(y_1y_2)-i\frac{\kappa_{p_1}}{\kappa_{p_2}}\Psi\star\Pi(y_1y_2)-i\frac{\kappa_{p_2}}{\kappa_{p_1}}\Pi\star\Psi(y_1y_2)\}\cr
S^\dagger(p_1p_2)&=& \frac{\sqrt{-\gamma}}{2}\int dy_1dy_2e^{-i(p_1y_2+p_2y_2)}\{-\frac{2}{\kappa_{p_1}\kappa_{p_2}}\Pi\star\Psi\star\Pi(y_1y_2)-\frac{1}{2\gamma^2\kappa_{p_1}\kappa_{p_2}}\frac{1}{\Psi}(y_1y_2)\cr
&&+\frac{\kappa_{p_1}\kappa_{p_2}}{2}\Psi(y_1y_2)+i\frac{\kappa_{p_1}}{\kappa_{p_2}}\Psi\star\Pi(y_1y_2)+i\frac{\kappa_{p_2}}{\kappa_{p_1}}\Pi\star\Psi(y_1y_2)\}\cr
B(p_1p_2)&=&\frac{1}{\gamma}+\int dy_1dy_2e^{-i(p_1y_2+p_2y_2)}\{\frac{2}{\kappa_{p_1}\kappa_{p_2}}\Pi\star\Psi\star\Pi(y_1y_2)+\frac{1}{2\gamma^2\kappa_{p_1}\kappa_{p_2}}\frac{1}{\Psi}(y_1y_2)\cr
&&+\frac{\kappa_{p_1}\kappa_{p_2}}{2}\Psi(y_1y_2)-i\frac{\kappa_{p_1}}{\kappa_{p_2}}\Psi\star\Pi(y_1y_2)+i\frac{\kappa_{p_2}}{\kappa_{p_1}}\Pi\star\Psi(y_1y_2)\}
\end{eqnarray}
where $\kappa_{p}=\sqrt{\omega_{p}}$.

Recalling that the Hamiltonian is given in terms of  $B$  we now see that its bi-local form is  the same in the fermionic and the bosonic case. This explains the feature that we have established by direct construction in Sec. 2,3. While the bi-local field representation of $B$ is the same in the fermionic and bosonic cases, the difference is seen in the representations of operators $S$ and $S^\dagger$. These operators create singlet states in the Hilbert space and the difference contained in the sign of gamma  implies the opposite shifts for the background fields that we have identified in Sec. 2,3. The algebraic pseudo spin reformulation is therefore seen to account for all the perturbative ($1/N$) features of the the bi-local theory that we have identified in Sec. 2,3. However, in addition and we would like to emphasize that, the algebraic  formulation provides a proper framework  for defining the bi-local   Hilbert space.

\subsection{ Quantization and the Hilbert Space}

The bi-local pseudo-spin algebra has several equivalent representations that turn out to be  useful.  Beside that collective representation that we have explained above, one has the simple oscillator representation:
\begin{eqnarray}
S(p_1,p_2)&=&\alpha \star(1-\frac{1}{N}\alpha^\dagger\star \alpha)^{\frac{1}{2}}(p_1,p_2)\cr
S^\dagger(p_1,p_2)&=&(1-\frac{1}{N}\alpha^\dagger\star \alpha)^{\frac{1}{2}}\star \alpha^\dagger(p_1,p_2)\cr
B(p_1,p_2)&=&2\; \alpha^\dagger\star \alpha(p_1,p_2)
\end{eqnarray}
with standard canonical canonical commutators (or Poisson brackets).

A more relevant  geometric  representation is obtained through a change:
\begin{eqnarray}
\alpha&=&Z(1+\frac{1}{N}\bar{Z}Z)^{-\frac{1}{2}}\cr
\alpha^\dagger&=&(1+\frac{1}{N}\bar{Z}Z)^{-\frac{1}{2}}\bar{Z}
\end{eqnarray}

The pseudo-spins  in the  $Z$ representation are given by:
\begin{eqnarray}
S(p_1,p_2)&=&Z\star(1+\frac{1}{N}\bar{Z}\star Z)^{-1}(p_1,p_2)\cr
S^\dagger(p_1,p_2)&=&(1+\frac{1}{N}\bar{Z}\star Z)^{-1}\star \bar{Z}(p_1,p_2)\cr
B(p_1,p_2)&=&2\;Z\star (1+\frac{1}{N}\bar{Z} \star Z)^{-1}\star \bar{Z}(p_1,p_2)
\end{eqnarray}

It's easy to see that this satisfy the Casimir constraint: $\frac{4}{N}S^\dagger\star S+(1-\frac{1}{N}B)^2=1$

One can write the Lagrangian in this  $Z$ representation as:
\begin{eqnarray}
\text{$\mathcal{L}$}=i\int dt \;\text{tr}[Z(1+\frac{1}{N}\bar{Z}Z)^{-1}\dot{\bar{Z}}-\dot{Z}(1+\frac{1}{N}\bar{Z}Z)^{-1}\bar{Z}]-\text{$\mathcal{H}$}
\end{eqnarray}

For regularization purposes, it is useful to consider putting $\vec{x}$ in a box and limiting the momenta by a cutoff $\Lambda$: this makes the bi-local fields into finite dimensional matrices (which we will take to be a size $K$). For $Sp(2N)$ one deals with a $K\times K$ dimensional complex matrix $Z$ and we have obtained in the above a compact symmetric (Kahler) space :

\begin{equation}
ds^2=\text{tr}[dZ(1-\bar{Z}Z)^{-1}d\bar{Z}(1-Z\bar{Z})^{-1}]
\end{equation}

According to the classification of \cite{Berezin:1975}, this would correspond to manifold $M_I(K,K)$. We note that the standard fermionic problem which was considered in detail in \cite{Berezin:1978sn} corresponds to manifold $M_{\MakeUppercase{\romannumeral 3}}(K,K)$ of complex antisymmetric matrices.

 Quantization on Kahler manifolds in general has been formulated  in detail by Berezin \cite{Berezin:1978sn}. We also note that the usefullnes of Kahler quantization for discretizing de Sitter space was pointed out by  A. Volovich in a quantum mechanical scenario\cite{Volovich:2001rt}.
 In the present Quantization we are dealing with a field theory with infinitely many degrees of freedom and infinite Khaler matrix variables.
 We will now summarize some of the results of quantization which are directly relevant  to the $Sp(2N)$ bi-local collective fields theory. Commutation relations of this system follow from the Poisson Brackets associated with the Lagrangian $\mathcal{L}(\bar{Z},Z)$. States in the Hilbert space are represented  by (holomorphic) functions (functionals) of the bi-locals $Z(k,l)$. A Kahler scalar product defining the bi-local Hilbert space reads:
\begin{equation}
(F_1,F_2)=C(N,K)\int d\mu(\bar{Z},Z)F_1(Z)F_2(\bar{Z})\det [1+\bar{Z}Z]^{-N}
\end{equation}
with the (Kahler) integration measure:
\begin{eqnarray}
d\mu=\det[1+\bar{Z}Z]^{-2K}d\bar{Z}dZ
\end{eqnarray}

The normalization constant is found from requiring $(F_1,F_1)=1$ for $F=1$. Let:
\begin{equation}
a(N,K)=\frac{1}{C(N,K)}=\int d\mu(\bar{Z},Z)\det [1+\bar{Z}Z]^{-N}
\end{equation}

This leads to the matrix integral (complex Penner Model)
\begin{equation}
a(N,K)=\frac{1}{C(N,K)}=\int \prod_{k,l=1}^K d\bar{Z}(k,l)dZ(k,l)\det [1+\bar{Z}Z]^{-2K-N}
\end{equation}
which determines $C(N,K)$.

The following results on quantization of this type of Kahler system are of note: First, the parameter $N$: much like for ordinary spin, one can show that $N$ (and therefore $G$ in Higher Spin Theory) can only take integer values, i.e. $N=0,1,2,3,\cdots$. Next, one has question about the total  number of states in the above Hilbert space. Naively, bi-local theory would seem to grossly overcount the number of states of the original fermionic theory. Originally one essentially had $2NK$ fermionic degrees of freedom with a finite Hilbert space. The bi-local description is based on (complex) bosonic variables of dimensions $K^2$ and the corresponding Hilbert space would appear to be  much larger. But due to the compact nature of the phase space, the number of states much smaller.

We will now evaluate this number (at finite $N$ and $K$) for the present case of $Sp(2N)$ (in \cite{Berezin:1978sn} ordinary fermions were studied) and show that the exact  dimension of the bi-local Hilbert space in geometric (Kahler) quantization agrees with the dimension of the singlet Hilbert space of the $Sp(2N)$ fermionic theory.

The dimension of quantized Hilbert space is found as follows: Considering the operator $\hat{O}=I$ one has that:
\begin{equation}
\text{Tr}(I)=C(N,K)\int \prod_{k,l=1}^K d\bar{Z}(k,l)dZ(k,l)\det [1+\bar{Z}Z]^{-2K}
\end{equation}
Consequently the dimension of the bi-local Hilbert space is given by:
\begin{equation}
\text{Dim  $\mathcal{H}$}_B=\frac{C(N,K)}{C(0,K)}=\frac{a(0,K)}{a(N,K)}
\end{equation}

The evaluation of the matrix (Penner) integral therefore also determines the dimension of the bi-local Hilbert space. Since this evaluation is a little bit involved, we present it in the following. Evaluation of matrix integrals (for real matrices) is given in \cite{Brezin:1977sv} the extension to the complex case was considered
in\cite{Ginibre:1965}.

We will use results of \cite{Berezin:1975}, whereby every (complex) matrix can be reduced through (symmetry) transformations to a diagonal form:
\begin{eqnarray}
Z(k,l)\rightarrow \begin{bmatrix}
\omega_1 & & & & \\
& \omega_2 & &  \text{\huge{0}} & \\
& & \omega_3 &  & \\
&  \text{\huge{0}} & & \ddots &\\
& & & & \omega_K
\end{bmatrix}
\end{eqnarray}

and the matrix integration measure becomes:
\begin{eqnarray}
[d\bar{Z}dZ]=\vert \Delta(\omega) \vert^2 \prod_{l=1}^K d\omega_l d\Omega
\end{eqnarray}

where $d\Omega$ denotes ``angular" parts of the integration and $\Delta(x_1,\cdots,x_K)=\prod_{k<l}(x_k-x_l)$ is a Vandermonde determinant, with $x_i=\omega_i^2$. Consequently the matrix integral for $a(N,K)$ (and $C(N,K)$) becomes:

\begin{eqnarray}
a(N,K)=\frac{\text{Vol} \;\;\Omega}{K!}\int \Delta(x_1,\cdots,x_K)^2 \prod_l (1+\omega_l^2)^{-2K-N}\prod_l d\omega_l
\end{eqnarray}

changing variables:  $x_i=-\frac{y_i}{1-y_i}$, we get:
\begin{eqnarray}
a(N,K)=\frac{\text{Vol} \;\;\Omega}{2^K K!}\int_0^\Lambda \prod_i^K dy_i \Delta(y_1,\cdots,y_K)^2 \prod_i (1-y_i)^{N}
\end{eqnarray}

This integral can be evaluated exactly. It belongs to a class of integrals evaluated by Selberg in 1944 \cite{Selberg:1944}:
\begin{eqnarray}
I(\alpha,\beta,\gamma,n)&=&\int_0^1 dx_1\cdots \int_0^1dx_n \vert\Delta(x)\vert^{2\gamma} \prod_{j=1}^n x_j^{\alpha-1}(1-x_j)^{\beta-1}\cr
&=&\prod_{j=0}^{n-1}\frac{\Gamma(1+\gamma+j\gamma)\Gamma(\alpha+j\gamma)\Gamma(\beta+j\gamma)}{\Gamma(1+\gamma)\Gamma(\alpha+\beta+(n+j-1)\gamma)}
\end{eqnarray}

we have the case with $\alpha=1,\;\;\beta=N+1,\;\;\gamma=1,\;\;n=K$ and
\begin{eqnarray}
I(1,N+1,1,K)&=&\prod_{j=0}^{K-1}\frac{\Gamma(2+j)\Gamma(1+j)\Gamma(N+1+j)}{\Gamma(2)\Gamma(N+K+j+1)}
\end{eqnarray}

We therefore obtain the following formula for the number of states in our Bi-local $Sp(2N)$ Hilbert space:

\begin{equation}
\text{Dim  $\mathcal{H}$}_B=\prod_{j=0}^{K-1}\frac{\Gamma(j+1)\Gamma(N+K+j+1)}{\Gamma(K+j+1)\Gamma(N+j+1)} \label{counting}
\end{equation}

We have compared this number with explicit enumeration of $Sp(2N)$ invariant states in the fermionic Hilbert space (for low values of $N$ and $K$) and found complete agreement. It is probably not that difficult to prove agreement for all $N,K$.
This settles however the potential problem of overcompletness of the bi-local representation. Since the $Sp(2N)$ counting uses the fermionic nature of creation operators and features exclusion when occupation numbers grow above certain limit it is seen that bi-local geometric quantization elegantly incorporates these effects. The compact nature of the associated infinite dimensional Kahler manifold secures the correct dimensionality of the the singlet  Hilbert space. By using Stirling's approximation for the number of states in the bi-local Hilbert space \eqref{counting}, we see the dimension growing linearly in $N$ (with $K \gg N$):
\begin{equation}
\ln (\text{Dim  $\mathcal{H}$}_B)\sim 2NK\;\ln 2     \qquad\text{at the leading order}
\end{equation}
This is a clear demonstration of the presence of an $N$-dependent cutoff in agreement with the fermionic nature of the original $Sp(2N)$ Hilbert space. So in the nonlinear bi-local theory with $G=1/N$ as coupling constant, we have the desired effect that the Hilbert space is cutoff through $1/G$ effects. Consequently we conclude  that the geometric bi-local representation  with infinite dimensional matrices $Z (k,l)$  provides a complete framework for quantization of the bi-local theory and  of de Sitter HS Gravity.

\section{Comments}
We have motivated the use of double analytic continuation and hence the connection between the $Sp(2N)$ model and de Sitter higher field theory for the quadratic action for the collective field. To establish this connection one of course needs to establish this for the interaction terms. This is of course highly nontrivial, and in fact the connection between the collective theory for the $O(N)$ model and the AdS higher spin theory is only beginning to be understood. We believe that once this is understood well enough one can address the question for the $Sp(2N)$-dS connection.

In this paper we have dealt mostly with the {\em  free} $Sp(2N)$ vector model. As the parallel  $O(N)$/AdS case this theory is characterized  with  an infinite sequence of conserved higher spin currents and associated conserved charges. The question regarding the  implementation of the Coleman-Mandula theorem then arises, this question was discussed recently in \cite{Maldacena:2011jn,Maldacena:2012sf,Koch:2012vc}. One can expected that identical conclusions hold for the present $Sp(2N)$ case.
 The bi-local collective field theory technqiue is trivially extendible to the linear sigma model based on $Sp(2N)$, as commented in section (4.2). Of particular interest is the IR behavior of the theory which presumably takes the theory from the Gaussian fixed point to a nontrivial fixed point.

It is well known that dS/CFT correspondence is quite different from AdS/CFT correspondence, particularly in the interpretation of bulk correlation functions \cite{dS, dS2}. We have not addressed these  issues in this paper. Recently it has been proposed that the $Sp(2N)$/dS connection can be used to understand subtle points about dS/CFT \cite{Ng:2012xp}. We hope that an explicit construction as described in this paper will be valuable for a deeper understanding of these issues.

The bi-local formulation  that we have presented was  cast in a geometric, pseudo-spin framework. We have suggested that this representation  offers the best framework for quantization of the bi-local  theory and consequently  the Hilbert space in dS/CFT. We have demonstrated through counting of the size  of the
Hilbert space that it incorporates finite $N$ effects through  a cutoff  which depends on the coupling constant of the theory: $G=1/N$. Most importantly it incorporates the finite $N$ exclusion principle and provides an explanation on the quantization of $G=1/N$ from the bulk point of view. These features are obviously of definite relevance for understanding  quantization of Gravity in de Sitter space-time. Nevertheless the question of understanding de Sitter Entropy from this   3 dimensional CFT remains an interesting and challenging problem.

It would be interesting  to consider the analogues of $Sp(2N)$/dS correspondence in the  CFT$_2$/Chern-Simons version\cite{Gaberdiel:2010pz,Henneaux:2010xg,Campoleoni:2010zq}, as well as to three dimensional conformal theories which have a line of fixed points, as in \cite{Giombi:2011kc}. Finally higher spin theories arise as limits of string theory in several contexts, e.g. \cite{Sezgin:2002rt} and \cite{Giombi:2011kc}. It would be interesting to see if these models can be modified to realize a dS/CFT correspondence in string theory.

\section{Acknowledgements}

The work of D.D. and S.R.D. is partially supported by National Science Foundation grants PHY-0970069 and PHY-0855614. The work of A.J. and Q.Y. is supported by the Department of Energy under contract DE-FG-02-91ER40688. We would like to thank  Matthew Dodelson, Misha Eides, Igor Klebanov, Gautam Mandal,  Shinji Mukhoyama, Marcus Spradlin, Alfred Shapere and Anastasia Volovich for discussions and Luis Alvarez-Gaume for a comment. S.R.D. thanks PCTS at Princeton University, Theory Group at CERN and University of Barcelona for hospitality during the final stages of the preparation of this manuscript.

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Community Spotlight – Matt Stultz on 3D Printing Materials #3DThursday #3DPrinting « Adafruit Industries – Makers, hackers, artists, designers and engineers!
Matt Stultz on 3D Printing Materials!
Q: Most of us who are using desktop 3D printers are using ones of the filament extruding variety (FFF and FDM are both accepted names for the type). Can you tell us a little bit about the composition of the filament itself?
Most FFF filaments are all part of class of materials known as Thermoplastics. These materials lend themselves to 3D printing because they can be transformed into a viscous liquid with heat and will reconstitute themselves back into a solid as they cool. PLA, ABS, Polystyrene, and Nylon are all thermoplastics.
Of course plastics have been around for quite a while now and we are starting to see more and more of them coming to market for 3D printing. One of the bigger trends is starting to see blends of plastics and plastics imbued with other materials such as wood fiber, carbon fiber, and metal particles.
Q: Now that affordable stereolithography (SLA, UV laser based) and digital light processing (DLP, typically optically visible light) desktop 3D printers and DIY kits, many of us are looking to learn more about photopolymer "liquid resins" — what are they made of?
SLA resins are typically a proprietary formula of some kind of acrylate resin mixed with a photosensitive hardener. Companies are spending a lot of time and money into finding better recipes for resins but because of this they are largely keeping them secret.
Q: What is the first step you take when you are looking to try a new experimental material?
Well I mostly work with materials that are coming in as filaments because these are what are going to be accessible to users. I do have a small filament extruder that I plan on starting to run experiments on materials that are just in pellet form soon. So the first thing I will do is look at the documentation from the company. This has really been improving as of late. Then if I know the base polymer, I will try to look at any specs I can find for it online. Wikipedia is actually a great source to find out more about the characteristics of different plastics.
One of the biggest things I am interested in is material interoperability. How can I use a multi head machine to mix two or more plastics in a single print and use the unique characteristics of both to make better printed objects. I got started with this when I was hunting for a material to use as a support material. I finally stumbled upon the hips limonene solution. I published my findings (here) and it's now being adopted by many users and vendors. So any new material I will start to try to print with other materials and see how they bond to each other and what advantages the two might have.
Q: What are some counterintuitive discoveries you have made about working with ABS, PLA, TPE, etc that a beginner to 3D printing needs to know?
Well we found that PLA is actually stronger than ABS. This was counterintuitive because PLA prints tend to break more than ABS prints. The actual fact is that on pure strength, PLA can take more force but it's far more brittle so shocks to it will result in breaks where ABS can bend and flow more. TPE (Thermo Plastic Elastomers) are great but are not for a beginner in my opinion, they just want to worm their way out of your extruder anyway they can find.
Q: Let's talk ABS — in what contexts do you prefer it, how do you like to finish with it, where is it a limitation?
ABS is my go to filament I prefer it for almost everything. There are many finishing options left to the user from sanding to acetone processes. Again, it doesn't tend to break as easily as PLA prints. The downside is when you need a really strong rigid part, then I turn to PLA or Nylon.
Q: Let's talk PLA — in what contexts do you prefer it, how do you like to finish with it, where is it a limitation?
I use PLA on machines I own that don't have heated beds. I also use it for big flat prints when I'm worried about the possibility of curling and warping. I don't like the finishing options available for it although I did just get the Smooth On XTC 3D print finishing kit that I look forward to checking out.
Q: Let's talk flexible filament — in what contexts do you prefer it, how do you like to finish with it, where is it a limitation?
Well I prefer it anytime I want something flexible of course! Really, I have been using it as a secondary material in prints. It's great for making hinges in your designs or shock absorbers. I'm less of a designer of pretty things and mostly make functional items. Giving by prints the option to bend and move without breaking has been huge.
Q: Let's talk metal filament — in what contexts do you prefer it, how do you like to finish with it, where is it a limitation?
Q: Let's talk dissolvable filament — in what contexts do you prefer it, how do you like to finish with it, where is it a limitation? Let's focus on HIPS — and your discoveries with this material!
Dissolvable filaments really allow you to take your prints to new heights. Breakaway support is great for places where you can reach it. Being able to get in and remove support cleanly by dissolving it from mechanisms that are printed in one go is really a game changer. The downside right now is it requires a multi-extruder machine and users are just not adopting them because they are being marketed more towards printing multiple colors instead of multiple materials. Hopefully we will start to see more users trying multi-material prints and demanding the machines from the manufacturers (as well as better mechanisms).
HIPS is a great material though just for printing with, it has a really nice surface quality right out of the printer. When I first found it, only one vendor was selling it for 3D Printing, Filaco. They didn't even know you could use it for support material, they were just selling it as an alternative. When I started digging into the properties of HIPS and found it could dissolve in limonene, I ordered a bunch of it. After my work was released, filaco sold a bunch of it and many vendors started carrying it also. I have been asked many times why I didn't patent the process before releasing it, I guess I care more about giving back to the community that chasing patents and making things more difficult for everyone.
Q: What resources exist to learn more about filament and how it works?
Well, again, wikipedia will tell you a lot but can really bog you down too. Matweb (matweb.com) is a great place to find out about all kinds of materials and their properties. If I can ever get things organized enough and get my team on track (we are busy people and this isn't our job), we are working on a wiki that will hopefully be a one stop place for information about materials that are on the market for 3D printing.
Q: what are you excited about in 3D printing materials?
I'm really excited about not just having filament materials at our disposal. Paste extrusion apparatuses were attempted in the early days of desktop 3D printing but they were very inelegant solutions. Now at least two companies (Structur3d and Printrbot) have released new paste extruders that may fix some of the problems of the past. They tend to show these off by printing food products out of them, but in actuality it opens a wide range of other materials to users. I'm super excited to play around with printing ceramics that are ready to be fired and turned into usable pieces!
Thanks again, Matt, for sharing your expertise!


text:
Home :: Military :: Library :: News :: 2020 :: March ::
US military sails another warship through Taiwan Strait
Thursday, 26 March 2020 10:11 AM
The US Navy says one of its warships has sailed through the strategic Taiwan Strait, which separates Taiwan from China, a provocative move that is certain to anger Beijing.
In a tweet on Thursday, the US Pacific Fleet confirmed that the USS McCampbell, a guided-missile destroyer of the US Seventh Fleet, had passed through the strait on Wednesday.
Anthony Junco, a spokesman for the US Seventh Fleet, also claimed that the combat vessel had conducted "a routine Taiwan Strait transit March 25 (local time) in accordance with international law."
He further said that the "the US Navy will continue to fly, sail and operate anywhere international law allows."
Taiwan's defense ministry confirmed the transit in a statement, calling the sailing an "ordinary mission" through the sensitive waterway and no cause for alarm.
There was no immediate comment from China, but it is expected to object. While China says it does not oppose the "normal passage" of foreign vessels through the strait, it has censured trips aimed at sending a geopolitical signal.
China has sovereignty over self-ruled Taiwan, and under the "One China" policy, almost all world countries recognize that sovereignty, saying the island is part of China's territory awaiting reunification.
The US, too, recognizes Chinese sovereignty over the island but has long courted Taipei in an attempt to counter Beijing.
Washington has no formal diplomatic relations with Taipei by law, but remains the island's largest weapons supplier and an international backer of Taiwanese president Tsai Ing-wen's government, causing increasing tensions with Beijing over trade and a host of other issues.
The US regularly carries out so-called "freedom of navigation" operations through the narrow strait, angering Beijing, which has never ruled out the use of force to bring the island under its full control.
Meanwhile, Washington and Beijing have also been trading accusations of who has been responsible for the coronavirus pandemic.

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In the March edition of Cleanroom Technology, Jason Kelly, VP of Services, Lighthouse Worldwide Solutions, talks about particle counter management for better data integrity and reliability. » Lighthouse Worldwide Solutions, Technology That Counts!
In this article, Jason Kelly reveals the best housekeeping methods that users of particle counters can apply to help mitigate against loss of data integrity and accuracy.
Click the file below to download and read a copy of the actual article.


text:
Skip to Main Content / Passer au contenu
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What is Inquiry?
Inquiry is an approach to learning which uses a series of processes to explore the natural, material and/or social worlds. Inquiry engages students in asking and answering meaningful questions. Some of these questions will be posed by the teacher, while others will be generated by the students. As often as possible, students should have the opportunity to participate in practical activities that generate answers to questions and solutions to problems and may lead to further inquiry.
Scientific inquiry involves the following processes. These processes are not always a linear progression but often more of a back and forth, or cyclical series of events.
The process of inquiry usually starts with an Initiating and Planning phase, and then moves to the phase of Performing and Recording to conduct fair tests or protocols to produce data and evidence. The data and evidence gathered is then examined in an Analyzing and Interpreting phase, which can lead to further questions and start the cycle all over again with the initiation and planning of a new inquiry. Communication and Teamwork are important to the success and efficiency of all phases of the inquiry cycle.
The familiar 5Es instructional model, which includes the cyclical steps of Engage, Explore, Explain, Extend and Evaluate can be superimposed over the basic cycle of inquiry. The Engage phase corresponds to initiating and planning. Explore and Explain encompass the performing and recording phase of inquiry, and Extend and Evaluate encompassing the analyzing and interpreting phase of inquiry.
Features of Scientific Inquiry
Scientific Inquiry: Exploring and Investigating
Scientific inquiry is driven by curiosity, wonder, interest, need or passion to answer a question within the context of science
This process begins with a question to answer
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From: Let's Talk Science Framework – The Goals of Science and Technology Education
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Jordan Davis scored his very first No. 1 this week with "Singles You Up," which topped both the Country Aircheck and Billboard country charts. The song, which he co-wrote with Justin Ebach and Steven Dale Jones, is the first release from Jordan's debut album, Home State.
Jordan will join Chris Janson on Jake Owen's Life's Whatcha Make It Tour, which kicks off May 18th in Chattanooga, Tennessee and is currently scheduled to wrap September 22nd in Ogden, Utah.


Worldwide Books: (Exhibition of Drawings by Jacopo Tintoretto and His School). Anna Forlani.
OP	 MOSTRA DI DISEGNI DI JACOPO TINTORETTO E DELLA SUA SCUOLA (Exhibition of Drawings by Jacopo Tintoretto and His School). Anna Forlani. Gabinetto Disegni e Stampe degli Uffizi, Florence, 1956. Published by Leo S. Olschki Editore, Florence. 76 pp. with 27 ills. 25 x 17cm. In Italian.


text:
Wednesday, July 3, 2013 by Moshe Koppel | Jewish Ideas Daily » Daily Features
I want to make an argument for limiting the role of the Israeli state in maintaining Jewish institutions. I do so, however, as one who wishes to see an expansion of the influence of traditional Judaism in the Israeli public square. Read in full on Mosaic.
Orthosexuality
Wednesday, May 29, 2013 by Elli Fischer | Jewish Ideas Daily » Daily Features
Addressing a trend toward greater openness about sexuality in the Modern Orthodox community, Elli Fischer reminds us, in an article first published December 19, 2011, that Judaism has never treated sex as a taboo subject.
High Concept in Dialogue With Tradition
Thursday, May 9, 2013 by Diane Cole | Jewish Ideas Daily » Daily Features
The artifacts of Jewish cultural history have never looked so freshly inviting or unexpectedly contemporary as in a provocative new exhibition at New York's Jewish Museum.
Wednesday, May 8, 2013 by Judah Bellin | Jewish Ideas Daily » Daily Features
In his new book, Rod Dreher insists that communities are difficult to forge in America's largest cities. But for traditional Jews, the demands of modern capitalism and community are not as incommensurate as Dreher assumes.
The Black-Hat Underground
Monday, May 6, 2013 by Yoel Finkelman | Jewish Ideas Daily » Daily Features
The likely closure of Aderaba, the magazine by, for, and about frustrated Israeli ba'alei teshuvah, demonstrates that the mainstream haredi community is too great to overcome—for now.
The Last Books
Wednesday, May 1, 2013 by Jonathan Brent | Jewish Ideas Daily » Daily Features
The invisible structures created by the Jewish people of Eastern Europe over a thousand years were given shape and transmitted through the books and the documents collected by YIVO. These structures still move us. If we do not know what they are, we do not know ourselves.
Faith and Matrimony
Friday, April 19, 2013 by Dana Evan Kaplan | Jewish Ideas Daily » Daily Features
An initiative to admit intermarried students to Reform rabbinical schools is yet another indication that Reform Judaism is swiftly becoming not so much a religious movement as a Jewish activities club.
Enigmas of Modern Jewish Identity
Tuesday, April 9, 2013 by Stephen J. Whitfield | Jewish Ideas Daily » Daily Features
Over the course of a lifetime facing modernity's conflicting demands, how might Jewishness affect the struggle to sustain identity, and how might this struggle mark the contours of Diaspora history?
Eizenstat on the Jewish Future
Friday, March 15, 2013 by Jerome A. Chanes | Jewish Ideas Daily » Daily Features
In his new book on the Jewish future, Jewish diplomat Stuart Eizenstat sees Jewish destiny evolving in the friendly competition between the sovereignty of Israel and the pluralism of America.
The Voice That Speaks in My Soul
Friday, March 8, 2013 by Susan Taubes | Jewish Ideas Daily » Daily Features
Echoing Kafka in this 1949 letter of protest to a domineering male, Susan Taubes writes: "I can no more keep to the laws of the Bible than I can cross myself or take the sacrament."
Star and Stripes Michael Freund, Jerusalem Post. Theodor Herzl had proposed a white banner with gold stars as the Zionist flag. But David Wolfssohn disagreed: "The tallit with which we wrap ourselves when we pray: That is our symbol."
Israel's High Holidays Donniel Hartman, eJewish Philanthropy. Once scorned as a representation of the Old Jew's weakness, Yom Hashoah is now as much a Zionist commemoration as Yom Hazikaron and Yom Ha'atzma'ut. But is that right?
Purim vs. Passover Yossi Klein Halevi, Hartman Institute. "'Passover Jews' are motivated by empathy with the oppressed; 'Purim Jews' are motivated by alertness to threat."
Evading the Synagogue Tax Rachel S. Harris, Sh'ma. Young Jews are increasingly reluctant to pay synagogue membership fees. But they are paying for other kinds of Jewish affiliation.
Unifying Three Communities Jonathan Sacks, Algemeiner. "To preserve the diversity of a tsibbur with the unity of purpose of an edah–that is the challenge of kehillah-formation, community-building, itself the greatest task of a great leader."
On a Mission to Save Kaifeng Anson Laynter, Algemeiner. After centuries of decline, Kaifeng's Jewish community has been rejuvenated by an American Christian philanthropist. Is he a benefactor—or a missionary?
The Indonesian Temple Mount Zakir Hussain, Jakarta Globe. His mother is Muslim, his father Protestant. But Yaakov Baruch, who runs Indonesia's only Jewish community, says, "I feel like I'm in Jerusalem."
Access Denied Daniel Estrin, Atlantic. Israel's Russian immigrants must prove their Jewish ancestry to obtain marriage licenses from the government. But Russia is making it increasingly difficult to access the Soviet archives.
The Conversion Crisis Marc Angel, Haaretz. "The halakhic Jewishness of an Orthodox convert is not decided by the Rabbanut—but by God. The Rabbanut has no right whatever to question or deny the validity of halakhic conversions."
Who Is A Jew? Gil Student, Torah Musings. In 1958, David Ben-Gurion consulted with Jewish scholars and leaders around the world in hopes of finding a universal definition. He received 45 different answers, of course.


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If you like to spend time outdoors and you have space a pond of garden Is a great idea to make your garden a more natural place. There are a lot of ideas for ponds in different styles.
With a pond you will turn the garden into an oasis. Decorating it with plants around, marsh plants, wetlands, mold, even fish or turtles and aquatic plants in the pond will turn your garden into the favorite place for everyone.
Choose carefully the flora and fauna that surrounds your garden pond and also the aquatic vegetation. Your friends and family will enjoy the walk through the garden crossing the pond by a bridge decorative Of wood for example to make it more striking can incorporate a source. The variety of complements is very wide as we said.
Decorative bridge, fountain, waterfall, vegetation or lighting. The size and size of the pond will depend on the space you have available. There are infinite forms for your garden pond geometric shapes that highlight the water instead the free give you naturalness you will only have to choose the shape that you consider most appropriate.
The depth of the garden pond is limited if it is less deep it will be easier to clean it and change the water continuously because in winter it can freeze. But if you want to have fish or turtles in the staque then it will have to be deeper. The image of the pond is very important to define the style of your garden.
To your liking you will find the best place in your garden where you will incorporate the pond and the decorations that you like best to create a space where you can feel comfortable and comfortable. If the weather does not allow you to be in the garden by just peeking out the window the colors and water will reassure you. Other similar ideas can be found in Pinterest .
Next post Hammock: a piece in our patio that invites to the pleasure!


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The Community Grid Project website terms of use.
These Terms of Use, together with any and all other documents referred to herein, set out the terms of use under which you may use this website www.thecommunitygrid.com.au ("Our Site"). By accessing and/or using Our Site, you agree to these terms, which include our Privacy Policy ("Terms").
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Q: Windows XP Scheduler - Error Adding Another Task (data area passed to a system call is too small) I'm getting an error "data area passed to a system call is too small" when trying to add a task to the Windows Scheduler. We currently have about 20 tasks in it.
Googling the issue, we found to stop/start the task service, wait, and some kind of clean-up would be run. We've done this multiple times and it's not fixing anything. Even if we delete a task, we can not re-add one.
Has anyone seen this problem and how did you fix it?

A: Task Scheduler returns a "HRESULT: 0x8007007A" error message after you save many jobs:

To work around this behavior, stop and start the task scheduler
  service, wait for 10 to 15 minutes, and then schedule jobs.
There is a scavenger tool in task scheduler that runs 10 minutes after
  the service is started. This tool frees the memory that is still being
  used by deleted or scheduled jobs.

How long did you wait?
Alternatively, try to delete the following reg key:
HKEY_USERS\.DEFAULT\Software\Microsoft\Cryptography\UserKeys\Schedule

redpajama_set_name: RedPajamaStackExchange


OSSSC Junior Clerk Admit Card 2017 – Odisha Sub-Ordinates Staff Selection Commission which is also known as OSSSC recently issued online application forms for various Junior Clerk Posts on its official website which were filled by numerous candidates across Odisha State. All such candidates are now seeking for their OSSSC Junior Clerk Admit Card 2017 online on the internet. It is hereby to inform all such candidates that the OSSSC Jr Clerk Admit Card 2017 will shortly available online on its official website. Though soon you will be able to get your examination admit card very soon. Till then you are highly advised to keep in touch with us to get all the latest and updated notifications regarding the exam.
So many high volume of candidates across the State of Odisha are now seeking for- OSSSC Junior Clerk Admit Card 2017, OSSSC Jr Clerk Hall Ticket 2017, Odisha SSSC Junior Clerk Exam Call Letter 2017, OSSSC Jr. Clerk Exam Dates 2017, Syllabus, Exam Pattern, Previous Year Papers Download Online @ www.osssc.gov.in.
Name of Organization: Odisha Subordinate Staff Selection Commission(OSSSC).
Name of The Post: Junior Clerk.
As per the official recruitment notification for Junior Clerk Posts so many large number of candidates across Odisha filled the online application form the last date was 8th Sep 2017 and now seeking for their examination dates/ call letter/ hall ticket information online on the internet. All such candidates will be able to get their examination admit card for the written examination going to held on very soon online on its official website. There is no official notification released yet by the commission against admit card and exam dates. But as per some sources it is clear that the Commission will shortly announce the examination dates and admit card.
OSSSC Junior Clerk Result 2016 is out on www.osssc.gov.in site. Odisha Junior Stenographer Exam 2016 was held on 20th March 2016 and now the results have been announced by the official board members. Here we have shared the Orissa Jr. clerk answer sheet, paper solutions, merit lists on this page. Do go through this post for more details related to OSSSC Junior Clerk Results 2016.
Few days back Odisha Sub-Ordinates Staff Selection Commission has conducted an exam for recruiting Junior Clerk posts. Many candidates who are interested appeared for this test and done well. So those candidates may be eagerly waiting for their results and now they got it. As per the official notification, the OSSSC Result of Jr Clerk Exam 2016 was declared on their official web portal. Candidates can check out their OSSS Exam results from 2016 by visiting through the direct link given below.
OSSSC Recruitment 2016 notification was announced last month for filling around 875 posts on Junior Clerk and Junior Steno. And the exam was conducted few days back in the month of March 2016 and now the candidates are looking for OSSSC Junior Clerk Official Answer Key 2016 along with OSSC Jr Clerk and Jr Assistant Preliminary Results Cut off Mark.
Candidates looking for Odisha Staff Selection Commission (OSSSC) Junior Clerk/Assistant Pre Exam Result 2016 will be able to check their result very soon only on the official website of OSSSC which is osssc.gov.in. Here on this website we are sharing all the latest and updated news and information about Odisha SSC Junior Clerk Result 2016, OSSSC Jr. Clerk Answer Key 2016, OSSSC Junior Clerk Results 2016, OSSSC Answer Key 2016|Merit List|Cut Off Marks 2016 Download, OSSSC Stenographer Exam Result 2016, www.osssc.gov.in, OSSSC Junior Clerk Official Answer Key 2016. So, you are highly advised to keep in touch with us to stay tuned.
Fink a link 'OSSSC Stenographer Exam Result 2016' and click on it.
Enter Hall ticket number and required details.
The result will appear soon and take a hard copy of it.
As per now, there is no any official announcement for downloading the OSSSC Answer Key 2016 and OSSSC Merit List 2016. Do stay tuned here for getting latest updates on OSSSC Jr. Clerk Answer Key 2016 once they are announced officially it will be updated here.
When osssc Jr clerk Jr sten result will be publis?
sir, which date diclared osssc rerult?
which date declared osssc junior clerk stenographer result ?pl z say ,How much late to declare the result. we R worried to know the result.plz plz plz say.
sir , when the osssc junior clerk result will b published ???how many days to go i mean with in one month or more….
respected osssc how many days are require to published the jr clerk result ?because aplicants are so worried! plz inform imedeatly.
osssc junior clerk result aau kete dina pare janapadiba sir…..
jaldi bahariba naa late heba….
Kono sir mane apana mananku result kadhibaro icha nahi ki. samastank career sahit kahinki kheluchanti. odisha BJD govt hei 17 barsa purilani abe to tike nidaru utho.
sir, why the osssc junior clerk result is being late ????
how many days left for this result ????
Osssc just a rubbish thing odisha govt requitment team such a big incompatible members plz. Government requitment give some new members than so satisfied new job seekers.
Tete team plz inform us ,why result is not pupublished?
junior clerk 2nd result kebe asiba ?
Computer practical test was cheating !
On notification it was given 1 hour.
But they only gave 30 minutes.
If they counted password opening and submitting hard copy time.
Then what is the marks for password opening and documents submitting.
For RI/ARI/AMIN they gave 1 hour.
As syllabus was same, so time must be same for Junior clerk and RI/ARI/AMIN.
Computer practical test time was also bungle at exam centre.
mr a k b ur absolutely right.
we have to must take any action for this.
co operate please all of candidates.
are vai mane imidiate sty kara.
Respectively Chairman of OSSSC. When the final result of jr clerk announced.