# Hilbert Bundles and Holographic Space-time Models

T. Banks  NHETC and Dept. of Physics and Astronomy, Rutgers University,
Piscataway, NJ

###### Abstract

We reformulate Holographic Space-time Models in terms of Hilbert bundles over
the space of time-like geodesics in a Lorentzian manifold. This reformulation
resolves the issue of the action of non-compact isometry groups on finite
dimensional Hilbert spaces. Following Jacobsonted1995  we view the background
geometry as a hydrodynamic flow, whose connection to an underlying quantum
system follows from the Bekenstein-Hawking relation between area and entropy,
generalized to arbitrary causal diamonds. Time-like geodesics are equivalent
to nested sequences of causal diamonds and the area of the holoscreen111The
holoscreen is the maximal $d-2$ volume (”area”) leaf of a null foliation of
the diamond boundary. We use the term area to refer to its volume. encodes the
entropy of a certain density matrix on a finite dimensional Hilbert space. We
review argumentscarlip solo bz  that the modular Hamiltonian of a diamond is a
cutoff version of the Virasoro generator $L_{0}$ of a $1+1$ dimensional CFT of
large central charge, living on an interval in longitudinal coordinate on the
diamond boundary. The cutoff is chosen so that the von Neumann entropy is
${\rm lnD_{\diamond}},$ up to subleading corrections, in the limit of large
dimension diamond Hilbert space. We also connect those arguments to the
derivationV291  of the ’t Hooft commutation relations for horizon
fluctuations. We present a tentative connection between the ’t Hooft relations
and $U(1)$ currents in the CFT’s on the past and future diamond boundaries.
The ’t Hooft relations are related to the Schwinger term in the commutator of
vector and axial currents. The paperV291  can be read as a proof that near
horizon dynamics for causal diamonds much larger than Planck scale is
equivalent to a topological field theory of the ’t Hooft CR, plus small
fluctuations of transverse geometry. Connes’connes  demonstration that
Riemannian geometry is encoded in the Dirac operator leads one to a completely
finite theory of transverse geometry fluctuations, in which the variables are
fermionic generators of a superalgebra, which are the expansion coefficients
of sections of the spinor bundle in Dirac eigenfunctions. A finite cutoff on
the Dirac spectrum gives rise to the area law for entropy, and makes the
geometry both ”fuzzy”tbjk  and quantum. Following the analysis of Carlip and
Solodukhin, we model the expansion coefficients as two dimensional fermionic
fields. We argue that local excitations in the interior of a diamond are
constrained states where the spinor variables vanish in regions of small area
on the holoscreen. This leads to an argument that quantum gravity in
asymptotically flat space must be exactly supersymmetric.

###### Contents

  1. I Introduction
    1. I.1 $Verlinde^{2}\rightarrow Carlip+Solodukhin$ in a General Causal Diamond
  2. II Transverse Geometry in Terms of Fluctuating Fuzzy Spinors
    1. II.1 Time Evolution
  3. III The Quantum Principle of Relativity
    1. III.1 Space-time Locality and the Origin of Quantum Field Theory
  4. IV Representation of Non-compact Isometry Groups
    1. IV.1 Summary
  5. V The ’t Hooft Commutation Relations
  6. VI Conclusions, Speculations and (Ugh) Philosophy
  7. VII Appendix The Curious Case of Anti de Sitter Space
  8. VIII Appendix II Some Remarks about Black Holes in Minkowski Space

##  I Introduction

Quantum Field Theory (QFT) is our most successful model of the physics of the
universe and enjoys a remarkable degree of agreement with experiment. Every
experiment ever done, or likely to be done in the forseeable future, is
performed by a detector moving for a finite amount of proper time along a near
geodesic time-like trajectory in a Lorentzian manifold with curvature that is
small in Planck units. Such an experiment defines a causal diamond. The
authors ofCKN  showed that the agreement of QFT with current experiments was
unaffected if we omitted from the QFT Hilbert space all states whose semi-
classical gravitational back reaction would create a black hole larger than
the causal diamond. With the UV/IR cutoffs imposed inCKN  current experiments
were near the limits implied by these constraints, butpdetal  showed that
other forms of cut off put experimental tests of these a priori constraints on
QFT off into the distant future. The total entropy allowed by these
constraints is $\sim S_{BH}^{\frac{d-1}{d}}$, in $d$ dimensional space-time.
$S_{BH}$ is the area ($d-2$ volume) of the maximal area $d-2$ surface on the
null boundary of the diamond, divided by $4G_{N}$. We will call this the area
of the holographic screen.

Although the experimental success of QFT does not require us to believe this,
the theoretical structure of QFT gives us a compelling reason to suspect that
there is something special about an area law for the entropy of a causal
diamond. In the 1960s, algebraic quantum field theoristshaagbrandeis
discovered that the algebra of operators localized in a diamond was always
Type III in the Murray von Neumann classification. This means that the algebra
has no density matrices of finite entropy. In 1983, Sorkinsorkin  computed the
vacuum entanglement entropy of a diamond and found that it was proportional to
the area of the holographic screen, with a UV divergent proportionality
constant. This calculation languished in obscurity until it was repeated by
more sophisticated methods in the 1990ssrednicki cw . It is clear that these
calculations are unchanged in any state obtained by acting with a finite
number of smeared local operators on the vacuum, as long as their support is
spacelike separated from the boundary of the diamond. The infinite
entanglement comes from divergent light cone commutators of fields on either
side of the diamond boundary.

Entanglement entropy of one system with another is a lower bound on the
maximal entropy of the system. The fact that the finite temperature entropy of
the portion of a Cauchy slice whose causal boundary is the diamond, is finite
after vacuum entropy subtraction, combined with the Cohen Kaplan Nelson (CKN)
argument, is a strong hint that this divergent boundary entropy has
fundamental significance. Susskind and Uglumsussug  and Jacobsonjacobson
suggested that this divergence should be viewed as a renormalization of
Newton’s constant in the Bekenstein Hawking entropy law. None of these authors
commented on the revolutionary nature of this suggestion. The Bekenstein-
Hawking law was a proposal about black holes. The calculations ofsorkin
srednicki cw  were all done in empty flat space. At the time, the present
author found this confusing, but was not clever enough to follow up on the
hint.

The most convincing evidence that we should take the area law proposal for the
entropy of a causal diamond seriously came one year later in the work of
Jacobsonted1995 . This paper was phrased in terms of local changes of entropy
and so did not explicitly invoke the Covariant Entropy Principle(CEP): In
models of quantum gravity, the entropy of a causal diamond is equal to the
area of its holographic screen divided by $4G_{N}$.  Entropy should be thought
of as the expectation value of the modular Hamiltonian associated with the
”empty diamond” state. The CEP is meant as the first term in an asymptotic
expansion of the entropy as a function of the area. The term ”area” makes
sense only in the context of this expansion. It was not until 1998 that the
CEP was formulated explicitly, first in the special case of FRW cosmology by
Fischler and Susskindfs  and then in the masterful follow up work by
Boussobousso .

Implicit in the CEP is the central notion proven inted1995 : the equations

$k^{m}(x)k^{n}(x)(R_{mn}(x)-\frac{1}{2}g_{mn}(x)R(x)-8\pi G_{N}T_{mn}(x))=0,$
(1)

are the hydrodynamic equations of the CEP. Here $k^{m}(x)$ is an arbitrary
null vector in space-time. That is, these equations follow from the definition
of Lorentzian geometry (which includes the Raychaudhuri equation for the local
spreading of a congruence of geodesics) and the covariant conservation of the
stress tensor, applied to the thermodynamic equation $dE=TdS$, assuming that
entropy is given by the CEP. One also uses the local relation
$E=k^{m}k^{n}T_{mn}$ and chooses a congruence of trajectories centered around
a maximally accelerated Unruh trajectory, which grazes a point on the
holographic screen of the diamond. The fact that the CEP is only an asymptotic
relation follows from the fact that the derivation assumes that everything can
be treated as locally flat.

A few comments are in order. The CEP can be ”derived” from Euclidean path
integralsBDF  just like black hole or dS entropyGH . The relation between
Euclidean path integrals and hydrodynamics has been clarified recently
intbwormhole . For a system consisting of large independent subsystems $X_{i}$
connected by small ”interfaces” with many fewer degrees of freedom,
hydrodynamics can be derived from quantum mechanics as a Markov equation for
the diagonal matrix elements of the density matrix in a basis of states in
which the local values $C(X_{i})$ of conserved quantities are simultaneously
diagonaltbaletal . Kac’s path integral solutionKac  of this Markov equation is
the Euclidean path integral for hydrodynamics.

The hydrodynamic view of Einstein’s equations means that they are valid in
contexts where there is no systematic weak coupling or large N expansion.
Strongly coupled condensed matter systems have high entropy states whose
coarse grained properties are described by the Navier-Stokes equations, even
though the underlying quantum mechanics is not well approximated by
perturbative quasi-particle physics. As a bonus, Jacobson’s derivation makes
it clear that the cosmological constant in Einstein’s equations should not be
thought of as an energy density since it does not appear in the equations of
hydrodynamics. String theorists should have known this since the invention of
the AdS/CFT correspondence, where the c.c. plays the role of a parameter in
the high energy density of states in a CFT and is determined by discrete
parameters like the gauge group $SU(N)$.

Further evidence for the CEP, and for Jacobson’s view of GR as hydrodynamics,
comes from remarkable work done by Carlipcarlip  and Solodukhinsolo  in
1998222Carlip had begun exploring these ideas earliercarlipearly  but the
fully developed realization of his ideas appeared simultaneously with the
independent paper of Solodukhin.. These authors proposed a state counting
interpretation of black hole entropy for quite general black holes. More
recently, K. Zurek and the present author realized that their derivation
applied to a general causal diamondbz . We will explore these ideas using an
argument introduced by the VerlindesV291  in order to provide a formal
derivation of ’t Hooft’s commutation relations between light ray operators in
high energy scattering. The same argument is precisely suited for studying the
near horizon limit of any causal diamond. In a later section, we will return
to provide a microscopic model of the ’t Hooft commutation relations.

###  I.1 $Verlinde^{2}\rightarrow Carlip+Solodukhin$ in a General Causal
Diamond

The observation of the Verlindes is that one can analyze high energy
scattering at large transverse distance by taking length scales in a two
dimensional subspace of Minkowski signature to be Planck scale, while those in
the transverse dimensions are of order $L\gg L_{P}$. The same is true in the
near horizon limit of any causal diamond, since the two dimensional length
scales are nearly null and the transverse length scale $L$, will generally be
much larger than $L_{P}$. The Einstein Hilbert action splits into three terms
of very different size.

$I_{EH}=(L/L_{P})^{d-4}I_{tH}+(L/L_{P})^{d-2}I_{\perp}+I_{3}.$ (2)
$I_{\perp}=\int\sqrt{-g}\sqrt{h}[-R(g)-g^{ab}\partial_{a}h_{mn}\partial_{b}h_{kl}(h^{ml}h^{nk}-h^{hm}h^{nl})].$
(3)
$I_{tH}=\int\sqrt{-g}\sqrt{h}[-R(h)-h^{mn}\partial_{m}g_{ab}\partial_{n}g_{cd}(g^{ad}g^{bc}-g^{ac}g^{bd})].$
(4)

We will not need the formula for the third term. Here
$g_{ab}=\hat{g}_{ab}-\hat{g}_{ai}h^{ij}\hat{g}_{jb}$, where $\hat{g}$ is the
original metric. The equations of motion of $I_{\perp}$ say that $g_{ab}$ is
flat, and that $h_{mn}$ is independent of $y$. The authors ofV291  then show
that $I_{tH}$ becomes a purely topological action for the boundary values of
the coordinate transformation that takes $g_{ab}$ to $\eta_{ab}$. Those
boundary values satisfy the ’t Hooft commutation relations. We will return to
a microscopic model of those commutation relations below.

A quick and dirty way to understand the argument ofV291  and to see that it is
valid in any dimension, is to note that we can view the near horizon limit of
any causal diamond as an asymmetric rescaling of coordinates in which the
transverse coordinates are taken much larger than those in the two dimensional
Minkowksi signature directions. Under this rescaling, the covariant components
of the Ricci tensor transform as a tensor and its clear that the dominant term
for large $(L/L_{P})$ is $R_{ab}$ . The leading terms in Einstein’s equations,
for a stress tensor that consists of pure traceless boundary waves (as
suggested by the CEP) are precisely those following from $I_{\perp}$.

The fact that the fluctuations of the transverse metric become small whenever
length scales are large compared to the Planck scale is another indication
that Jacobson’s hydrodynamic view of gravity is correct. One can construct a
very general derivation of hydrodynamics for lattice quantum systems, using
only the assumption that large blocks of the lattice have Hamiltonians that
commute with each other up to subleading surface termstbaletal . The
Schrodinger equation leads directly to classical stochastic equations for the
hydrodynamic variables, which have a standard Euclidean path integral
solution. Fluctuations of transverse geometry are similarly suppressed simply
by the size of the region they describe. No assumptions of ”weak string
coupling” are necessary.

Solodukhin’s analysis of the linearized fluctuations around the classical
solution is particularly transparent. One defines a scalar field $\phi$ in the
Lorentzian dimensions in terms of the fluctuations of the conformal factor of
the metric $h$ around its classical value. After a $\phi$ dependent Weyl
transformation of the two dimensional metric (which, to linear order in $\phi$
is just a constant rescaling) the action takes the form

$I[\phi]=-\int
d^{2}y\sqrt{-g}[\frac{1}{2}g^{ab}\partial_{a}\phi\partial_{b}\phi+q\phi\sqrt{\frac{(d-3)S}{(d-2)8\pi}}R(g)+U(\phi)].$
(5)

The metric is set to be Minkowski by the equations of motion of $I_{\perp}$,
but we have kept it general to show that the field $\phi$ has a stress tensor
with a large classical central charge. The first two terms define a classical
conformal field theory, which was shown inBPZ  to have a stress tensor whose
correlation functions satisfy all the Ward identities of a general CFT with
the same value of the central charge, because stress tensor correlators in two
dimensional CFT are completely determined by the central charge. This is often
called the Liouville theory, though it does not have the usual two dimensional
cosmological constant term that gives Liouville’s equation for constant
curvature metrics. An illuminating way to state the result ofBPZ  is that the
”Liouville” theory is the solution of the hydrodynamic equations of two
dimensional CFT with a given value of $c$. Solodukhin shows that the potential
term $U(\phi)$ contains only ”classically irrelevant” perturbations of this
CFT, in the near horizon limit.

InAS , Strominger argued that the boundary ”Liouville” theory discovered by
Brown and Henneauxbrownhenneaux  in $2+1$ dimensional AdS gravity, was an
avatar of the AdS/CFT correspondence. In our Jacobsonian language: asymptotic
3d gravity in AdS space is the hydrodynamics of $1+1$ CFT. Strominger argued,
as a consequence, that quantum gravity should be quantum CFT. This was an
after the fact justification for the $AdS_{3}/CFT_{2}$ correspondence. Carlip
and Solodukhin advocate using the same logic for all black holes and show that
Cardy’s entropy formula reproduces the Bekenstein-Hawking formula in every
case. Inbz  we argued that these arguments applied to every causal diamond,
giving further justification for the CEP. In addition we argued that they
implied the universal fluctuation formula $(\Delta K)^{2}=\langle K\rangle$
for the modular Hamiltonian333The sole known exceptions are the horizons of
large stable black holes in AdS space. These are also the only horizons that
propagate sound modes. Both facts can be understood in terms of the tensor
network model of AdS/CFT. The universal fluctuation formula is valid in nodes
of the tensor network but the entropy of a large black hole is dominated by
sound modes propagating between the nodes, which have a different capacity of
entanglement. See the Appendix for more detail.. Solodukhin remarks that there
is no apparent reason for neglecting fluctuations of the unimodular part of
the transverse metric $h_{mn}$. In the next section we will use insights from
the Holographic Space-time formalism to incorporate these into the Carlip-
Solodukhin picture.

When combined with the CKN argument, the work of Carlip and Solodukhin leads
one to a remarkable conclusion. Most of the quantum degrees of freedom of a
model of quantum gravity are inaccessible to local experiments done on near
geodesic trajectories through the diamond. They reside on the boundaries of
causal diamonds and we will only be able to get very indirect experimental
information about them. What is more, so far the CS picture does not give us a
clue about the nature of the local excitations, to which the usual rules of
QFT apply. These are the excitations explored by experiments done along
geodesics inside a causal diamond. Fortunately, the answer to that question
was found by Fiol, Fischler and the present author some time ago, and we will
return to it below.

##  II Transverse Geometry in Terms of Fluctuating Fuzzy Spinors

The Holographic Space-time (HST) formalism was introduced in order to
construct models of quantum gravity with a closer connection to local physics
than that achieved in perturbative string theory or the AdS/CFT
correspondence. Its most basic postulate is the existence of a net of local
subalgebras of the operator algebra describing an entire space-time. These
subalgebras are finite dimensional and are in one to one correspondence with
finite area causal diamonds in the space-time. When the area is much larger
than any microscopic scale the dimension of the Hilbert space associated with
a diamond is approximately444A more precise statement is that the area is the
expectation value of the modular Hamiltonian of the empty diamond state in the
Hilbert space. However, in models constructed so far, this coincides with the
log of the dimension up to subleading corrections.

$d({\cal H}_{\diamond})\approx e^{\frac{A}{4G_{N}}}.$ (6)

A nested sequence of intervals along any time-like trajectory defines a nested
sequence of causal diamonds. Every diamond contains a unique timelike geodesic
that connects its past and future tips. Given a choice of nested intervals on
a timelike geodesic we have a sequence of causal diamonds. The relations
between proper time interval and area for all diamonds along all geodesics
completely determines the space-time metric $g_{mn}$. The relation between
area and Hilbert space dimension thus allows us to construct a map from
quantum mechanics to geometry: we consider a Hilbert bundle over the space of
time-like geodesics that connect two Cauchy slices in the space-time555For
space-times that are asymptotically AdS, we must restrict the maximal proper
time between points on the two Cauchy surfaces, so that the maximal causal
diamonds have finite area, in order to treat the infinite area limit with
proper care. Note that for non-negative c.c. we either have no restriction on
the proper time between past and future surfaces or only the restriction that
it be finite. . We can make a synchronized choice of proper time
intervals,starting from the past Cauchy surface or going backwards from the
future surface. In time symmetric space-times we can nest outwards from a
small diamond surrounding a point of time symmetry (Figure 1).

Figure 1: Future directed, time symmetric, and past directed nested coverings
of a causal diamond.

Specifying the dimensions of Hilbert spaces for each time interval is
equivalent to specifying the space-time metric, in the limit of exponentially
large dimensions. The basic hypothesis of HST is that Hilbert space dimension
is the correct extension of the concept of area of a causal diamond’s
holographic screen, to short distances. This idea lines up perfectly with
Jacobson’sted1995  claim that Einstein’s Equations are the hydrodynamic
equations of the area law, and provides an immediate resolution of space-like
singularities in solutions of GR. Space-like singularitiesHawking penrose  are
places where causal diamonds shrink to zero area and it’s precisely in low
entropy situations that we can expect the hydrodynamic approximation to a
quantum system to break down.

The hydrodynamic view of Einstein’s equations casts doubt on a number of
assumptions common among researchers in all branches of quantum gravity. The
first of these is that there should be a single ”background independent”
formulation of quantum gravity, with all particular instances of the theory
some sort of ”vacuum states”. Among string theorists, this myth should have
been dispelled long ago by the success of the AdS/CFT correspondence. From the
hydrodynamic perspective it is tantamount to saying that because almost all
systems have states that satisfy the Navier-Stokes equation, then they are
described by the same microscopic quantum Hamiltonian.

The second erroneous idea, most common among string theorists, is that the
Einstein equations should always be viewed as the first term in some kind of
semi-classical expansion, which is in some way related to a ”path integral
over metrics”. For condensed matter systems with a gapless ground state, it is
often the case that one can view the low lying excitations as quantized fields
obeying the linearized hydrodynamic equations, with interactions that can be
understood by Wilsonian renormalization group analysis and symmetry
considerations. The classical hydrodynamic equations have a much broader
validity than this and apply to states far removed from the ground state, in
which it is inappropriate to quantize them. Instead, one searches for a
microscopic system that reproduces the same coarse grained hydrodynamic flow.
This is the principle that we will be following. Our models of quantum gravity
always begin with a fixed classical background, which we view as a
hydrodynamic flow to which we must match a quantum theory, following the clues
outlined in the previous section. Hydrodynamic equations have to be
supplemented by stochastic ”stirring” terms, in order to agree with
observation. There is a long history of representing the solutions of these
stochastic equations by Euclidean path integrals, following Kac’ original
treatment of the diffusion equation. This is the way one should view Euclidean
quantum gravity.

As noted, the big lacuna in the discussion of the previous section was a model
for the fluctuations in the transverse geometry. Here we follow the analysis
of HST. Finite entropy implies that the algebra of operators in a diamond be
Type II in the Murray-von Neumann classification, but it is essentially
impossible for experimental physics to distinguish between infinite Type II
algebras and finite dimensional algebras, so HST has always been restricted to
finite dimensional Hilbert spaces. Any such Hilbert space is a representation
of a fermionically generated superalgebra, or equivalently of a system of
canonical fermionic oscillators with constraints. Alain Connes pointed out
long agoconnes  that all of Riemannian geometry was encoded in the Dirac
operator. Indeed, on page 544 ofconnes  one can find a (rather abstract)
formula for the geodesic distance between two points in a manifold in terms of
properties of the Dirac operator. Physicists are more familiar with the fact
that the short time expansion of the heat kernel of the square of the Dirac
operator can be written in terms of curvature invariants, and that topological
properties of the manifold are also related to the index of the Dirac
operator. This means that a cut off on the spectrum of the Dirac operator
provides a method of ”fuzzifying” a Riemannian geometry even if the space does
not have a symplectic structuretbjk . Furthermore, if we postulate that the
expansion coefficients of a generic section of the spinor bundle into Dirac
eigenfunctions of a fixed background metric, are $1+1$ dimensional (cut-off)
quantum fields, then each quantum field theory state will define a probability
distribution for the curvature invariants of the transverse geometry. More
precisely, the quantum field

$\Psi(x,y)=\sum_{E}e^{-iEt}\psi_{E}(y)\chi_{E}(x),$ (7)

satisfies the Dirac equation, with Hamiltonian $H=[L_{0},]+D$ , where $D$ is
the transverse Dirac operator in the background geometry, and $E$ its cutoff
spectrum. $[L_{0},]$ is the action of the $L_{0}$ generator on operators in
the CFT. Its spectrum is determined by that of $L_{0}$. Thus, we can view the
fluctuations in $L_{0}$ as being fluctuations in the transverse Dirac
spectrum, and thus of the transverse geometry.

This is not the only possible definition of a fluctuating transverse geometry.
The fermion bilinears

$\bar{\psi}(\sigma,x)\Gamma_{(A_{1}\ldots A_{k})}e_{m(1)}^{A_{1}}\ldots
e_{m(k)}^{A_{k}}\psi(\sigma,x),$ (8)

are rank $k$ differential forms, and the bundle of all these forms satisfies
the Kahler-Dirac equation

$(d-d^{*})F=0,$ (9)

where $d$ is the exterior derivative on the transverse manifold and $d*$ its
Hodge dual. This equation implies that there is a linear combination of the
bilinear

$\bar{\psi}(\sigma,x)\Gamma_{(A_{1}\ldots
A_{4})}e_{m(1)}^{A_{1}}e_{m(2)}^{A_{2}}\psi(\sigma,x),$ (10)

and the divergence of the rank $5$ bilinear, which has the symmetry properties
and satisfies the Bianchi identity of the field strength of an $O(d-2)$
connection on the spinor bundle. This is an operator quantum field and we can
view it as a fluctuating curvature tensor for the holoscreen. It’s not clear
which, if any of these definitions is correct, or whether there is any way of
testing either hypothesis. We’ve emphasized that the kinds of measurements
available to a local observer use detectors with far fewer q-bits than the
number that actually describe the holoscreen. We should not be surprised to
discover that our experiments will never be able to probe the details of
”quantum geometry”.

If we arrange that the quantum density matrix for each causal diamond
satisfies

$\langle K\rangle=\frac{A}{4G_{N}},$ (11) $\langle(K-\langle
K\rangle)^{2}\rangle=\frac{A}{4G_{N}},$ (12)

then we will have a quantum model whose hydrodynamic equations coincide with
the solution of the Einstein equations from which we began, according to the
analysis of Carlip and Solodukhin.

Another immediate benefit of adopting the Connes-Dirac formulation of
Riemannian geometry and the quantization rules of HST is that it builds in the
spin statistics connection. The connection between spin and statistics is a
theorem in local quantum field theory, but even more importantly, it is a
property of the real world, and local quantum field theory is not. We’ll see
below that in asymptotically flat space-time it also (almost) proves the
necessity for Supersymmetry (SUSY) in asymptotically flat space: a
”phenomenological” fact about string theory with no obvious explanation.

We begin to see the outlines of a quantum theory of gravity, applicable on
scales much larger than the Planck scale. One starts from a classical space-
time, obeying Einstein’s equations with a stress tensor satisfying the null
energy condition666The null energy condition follows from the second law of
thermodynamics in Jacobson’s derivation of the Einstein equations.. For each
time-like geodesic in the space-time one introduces a nested sequence of
proper time intervals and constructs a corresponding Hilbert bundle of cutoff
$1+1$ dimensional fermion fields. The UV and IR cutoffs in two dimensions are
related to the minimal size of causal diamond (in Planck units) for which one
believes the Carlip/Solodukhin arguments are valid. At the moment, we have no
principled way of deciding how large the minimal diamond is. This choice of
course removes any Big Bang singularities from the background space-time. Once
this choice is made, one increases the number of fermion fields according to
the Bekenstein-Hawking-Carlip-Solodukhin area/entropy law. The quantum state
of an empty diamond has modular Hamiltonian equal to the $L_{0}$ generator of
a $1+1$ CFT constructed from the fermions, with the UV/IR cutoffs described
above. The cutoff is imposed on the spectrum of $L_{0}$, with a central value
chosen by the solution of the CS Liouville equation. In general (see below)
there appears to be a conformal manifold of CFTs to choose from and it is not
clear whether these correspond to different consistent models for quantum
fluctuations around the same classical geometry.

###  II.1 Time Evolution

For causal diamonds in maximally symmetric spaces, the $L_{0}$ generator used
by Carlip and Solodukhin showed up in more recent work of Casini, Huerta and
MyersCHM  and Jacobson and VisserJV  on the modular Hamiltonian of conformal
field theory on these space-times. $L_{0}$ is a quantum operator, which
implements the action of a certain space-time vector field on the time-slice
through the holographic screen of the diamond. The vector field preserves the
diamond and the flows associated with it are time-like inside it and define a
set of inextensible Diamond Universe (DU) coordinates inside the diamond. In
fact this is a conformal Killing vector of the maximally symmetric space-time,
so the same statements will be true for any geometry conformal to the
maximally symmetric space-times. The DU coordinates will be different for
different conformal factors. The Virasoro algebra fixes the normalization of
the CKV, which is of course ambiguous as a geometrical object.

Now let’s consider two causal diamonds in a future directed nesting, with
future tips separated by a single Planck unit. We have argued that each should
be described by a $1+1$ dimensional CFT built from fermion fields, with the
larger diamond containing more fermion fields than the smaller one. The
density matrix for the empty diamond state is $e^{-L_{0}(\tau)}$ for the small
diamond and $e^{-L_{0}(\tau+L_{P})}$ for the larger one. Geometrically, at
least for geometries conformal to maximally symmetric ones, it would seem that
the time evolution operator for the Planck scale time slice between the two
diamonds, in DU coordinates, should be $e^{-iL_{0}(\tau+L_{P})}$. By Planck
scale time slice, we mean one Planck time along the geodesic. Since this
operator is a unitary in the larger Hilbert space it will, in general,
entangle the smaller diamond’s fermions with those of the larger diamond.

On the other hand, if we think about time evolution between time $\tau-L_{P}$
and $\tau$ and time $\tau$ , then the evolution operator in that time slice is
$e^{iL_{0}(\tau)}$, but the extra degrees of freedom that are added between
$\tau$ and $\tau+L_{P}$ must remain decoupled during evolution from $\tau-
L_{P}$ to $\tau$ . Combined with conformal invariance this motivates us
strongly to add the new fermions as free fermions. That is, between time
$\tau$ and $\tau+L_{P}$ the full evolution operator on the entire Hilbert
space is

$e^{-iL_{0}(\tau+L_{P})}\otimes e^{-iL_{0}^{out}}.$ (13)

$L_{0}^{out}$ acts on the tensor complement of the diamond Hilbert space of
the interval $[-T,\tau+L_{P}]$ in the Hilbert space of $[-T,T]$. The tensor
complement can be generated by fermions that are added in each of the
intervals $\tau+nL_{p}$ to $\tau+(n+1)L_{P}$. Choosing their Hamiltonian to be
that of free fermions in $L_{0}^{out}$ guarantees that the time evolution
preserves causality and also ”prepares” them for the interaction with the
expanding nest of prior diamonds.

The interaction, to leading order in $L_{P}/L$, is largely determined by two
constraints: $1+1$ dimensional conformal invariance and fast scramblinglshpss
. The evidence for fast scrambling comes predominantly from the rapid
homogenization of perturbations on the horizon of a causal diamond, as viewed
from an accelerated trajectory that avoids penetrating the horizon. For non-
negative c.c. or for boundary anchored RT diamonds in AdS space, perturbations
become homogeneous in times of order the horizon curvature scale. Intbwf17  we
argued that this could be explained if the dynamics was invariant under area
preserving diffeomorphisms in $2$ or more dimensions. This is a global, rather
than a gauge symmetry. It implies that the dynamics has no respect for
geodesic distance in the transverse space. A small spherical cap is equivalent
to a many fingered amoeba of the same area, which touches points that are
arbitrarily far away on the manifold.

Figure 2: Holoscreen dynamics invariant under area preserving diffeomorphisms
can propagate quantum information without regard to geometric distance.
Consistent with exponentially fast homogenization of charge and mass on
horizons.

Fast scrambling is often defined by the statement that a perturbation of a
single q-bit operator at time zero, will fail to commute with every other
q-bit in a time that scales like the number of q-bits. It is characteristic of
models that involve ”all to all” couplings of a set of q-bits. The
homogenization of mass and charge in logarithmic time certainly does not
require non-locality which is quite so drastic. A paradox pointed out by
Hayden and Preskillhp  is the strongest argument for the logarithmic time
scale. It’s not clear, at least to this author, whether it implies the kind of
”all to all” coupling that is often assumed.

Since our fundamental variables are spinors, we can easily construct
differential forms

$J_{(m(1)\ldots m(k))}^{a}=\bar{\psi}(\sigma,x)\Gamma_{(A_{1}\ldots
A_{k})}\gamma^{a}e_{m(1)}^{A_{1}}\ldots e_{m(k)}^{A_{k}}\psi(\sigma,x),$ (14)

where $e$ is the veilbein on the transverse manifold, the $\Gamma$ matrices
are $d-2$ dimensional Euclidean signature and the $\gamma$ matrices $2$
dimensional Lorentzian signature. Recalling that our manifold has fixed
volume, we can make area preserving diffeomorphism invariant Hamiltonians by
integrating products of differential forms of total rank $d-2$. If the
fermions at each $x$ were independent canonical variables then we could
construct independent $U(1)$ currents for differential forms of each rank $p$

$J_{a}^{p}(x,\sigma)=\bar{\psi}(x,\sigma)\gamma_{a}\Gamma_{(A(1)\ldots
A(p))}e_{m(1)}^{A(1)}(x)\ldots e_{m(p)}^{A(p)}(x)\psi(x,\sigma).$ (15)

If $d$ is even the currents with rank $p$ and $d-2-p$ commute with each other
and we can construct a fixed line of conformally invariant Thirring
interactions from products of these currents. For $p$ odd, we would have to
use the reducible representation of the $d-2$ dimensional Clifford algebra and
insert the extra anti-commuting element $\Gamma$ into either the $p$ or
$d-p-2$ rank current to construct a similar interaction. These interactions
are formally invariant under volume preserving diffeomorphisms. However, if
the $\psi(x,\sigma)$ for different $x$ were truly independent two dimensional
fermion fields, the interactions would be ultra-local in $x$ and could not
propagate information along the horizon at all. It’s the fixed cutoff on
transverse Dirac eigenvalues that leads to fast scrambling.

This same cutoff will violate $1+1$ dimension conformal invariance, but recall
that the arguments of Carlip and Solodukhin are only valid to leading order in
an expansion in $L_{P}/L$. The finiteness of the entropy of the causal diamond
already told us that the CFT had to be cut off, violating conformal
invariance. A central conjecture of the current paper is that for large
$L/L_{P}$ the system is fast scrambling, but also close enough to a CFT that
the Cardy formula for the spectral density is valid in the vicinity of the
value of $L_{0}$ picked out by the classical argument of Carlip and
Solodukhin.

Our proposal now is that we choose the CFT cutoff once and for all by the
criterion that for each causal diamond along a nested sequence of proper time
intervals the modular Hamiltonian is the $L_{0}$ generator of the cutoff CFT,
where the cutoff is imposed by choosing an $L_{0}$ value closest to the
classical ”Liouville” value computed by Carlip and Solodukhin’s methods, and
one chooses an interval of discrete eigenvalues around that value such that

$\langle L_{0}\rangle=\langle(L_{0}-\langle
L_{0}\rangle)^{2}\rangle=\frac{A}{4G_{N}}.$ (16)

This condition is enforced for some minimal area diamond, the smallest area
for which we believe the CS arguments are valid. It determines the cutoff on
the width of the band of $L_{0}$ eigenvalues once and for all. Only the number
of fermion fields is allowed to vary as we change the proper time interval
along the geodesic, and that changes with the area of the transverse manifold
in order to continue obeying the above equation. The change is implemented by
changing the eigenvalue cutoff on the transverse Dirac operator for the
appropriate transverse manifold at each time slice. Time evolution in the
nested sequence of diamonds is given by $e^{iL_{0}(\tau+L_{P})}$ in the wedge
between the diamond whose future tip is at $\tau$ and the next one in the
sequence. $L_{0}(\tau+L_{P})$ is the modular Hamiltonian of the empty diamond
state of that large diamond.

This procedure produces a finite quantum model for a given choice of a
solution of Einstein’s equations with a stress tensor satisfying the null
energy condition, at least if the geometry is conformal to a maximally
symmetric space-time. In higher dimension there seems to be a bit of ambiguity
in the choice of Thirring couplings for the various allowed values of $p$.
These models are certainly well defined and have the hydrodynamic behavior
that Jacobson’s argument tells us to expect from Einstein’s equations. By
construction they obey a primitive notion of causality: the (even functions of
the) degrees of freedom in a given causal diamond form a factor in the algebra
of all operators in the Hilbert space of a large finite proper time
interval777As always, we inserted a proper time cutoff in order to deal only
with finite dimensional algebras. We know how to take the limit for AdS/CFT,
and it involves a drastic change in the formalism in which we introduce a
tensor network of the systems discussed here in order to obtain a QFT limit.
This is discussed in the appendix. Below we’ll discuss the subtleties of the
infinite entropy limit in asymptotically flat space., and time evolution
preserves that factor throughout the proper time in the diamond. To generalize
these models to arbitrary geometries, one would have to find the analog of DU
coordinates for each space-time. That is, the geometric transformation that
acts as $L_{0}=\partial_{t}$ on the holographic screen of each diamond, should
be the boundary value of a vector field whose flow lines inside the diamond
define a set of time-like trajectories (including the geodesic) which start
and end at the tips of the diamond.

The formalism so far is missing at least three different key elements: it is
tied to a particular geodesic and does not obey a ”principle of relativity”.
The correspondence with local quantum field theory is obscure. The connection
to the AdS/CFT correspondence, our best understood model of quantum gravity,
is also absent. We will deal with the last of these issues in an appendix. The
other two will be discussed in the following sections.

##  III The Quantum Principle of Relativity

Given two proper time intervals along geodesics $G_{1,2}$, the corresponding
causal diamonds have an intersection , which might be empty. The intersection
is not itself a diamond, but contains a maximal area diamond, $D_{12}$, unique
up to symmetries. The Quantum Principle of Relativity (QPR) is the statement
that $D_{12}$ corresponds to an isomorphism between a tensor factor in the
Hilbert space of $D_{1}$ and that of $D_{2}$. Given initial states on the past
boundaries of $D_{1,2}$, there are density matrices $\rho_{1,2}$ on this
tensor factor. The QPR further states that these density matrices have the
same entanglement spectra. The QPR is the dynamical principle that makes the
Hilbert bundle over the space of geodesics into a quantum version of
Einstein’s equations. It has, unfortunately, proven extremely difficult to
find efficient machinery for implementing this principle. What has been done
is, I believe, non-trivial, but is mostly at the level of pictures and
stories.

The simplest complete HST model is one in which space-time is a flat
Friedmann-Robertson-Walker (FRW) universe with scale factor

$a(t)=\sinh^{1/3}(3t/R).$ (17)

This is a universe that begins its life with equation of state

$p=\rho,$ (18)

and asymptotes to dS space with Hubble radius $R$. The quantum model consists
of fermions fields $\psi(\Omega,\sigma)$ on an interval $\times S^{2}$ with an
angular momentum cutoff that increases linearly with time. The fermion
Hamiltonian is the sum of a free $1+1$ dimensional Dirac Hamiltonian for every
transverse angular momentum mode and a term

$g\int d\sigma\ J_{a}^{0}(\sigma,x)J^{a2}(\sigma,x),$ (19)

built from the product of the 0-form and 2-form currents. The integral
includes an integral of the two form over the transverse two sphere. When the
expectation value of $L_{0}(t_{max})$ reaches the dS entropy $\pi R^{2}$, we
stop increasing the number of fermion fields and continue the time evolution
of the system with $L_{0}(t_{max})$ as the Hamiltonian.

Now consider another geodesic in the same space-time and the causal diamond
corresponding to the interval $[0,\tau]$ in proper time along that diamond.
The intersection with the diamond $[0,\tau]$ along the original geodesic
contains a maximal diamond of area $A(\tau,D)$ where $D$ is the space-like
distance between the two geodesics on some fiducial surface of fixed proper
time. As long as the rule for the dynamics in any diamond is that for the
empty diamond initial state, the modular Hamiltonian in any subdiamond is
unitarily equivalent to $L_{0}(A)$ for the same CFT described above, with the
appropriate value of the central charge. Then the QPR will be satisfied, if
the initial state is chosen to be such that, for each choice of geodesic, the
initial state is a tensor product of $e^{-L_{0}^{min}}$, where $L_{0}^{min}$
is the Virasoro generator of the CFT corresponding to the minimal diamond for
which the Carlip-Solodukhin analysis is valid. This is close888The CS analysis
provides a finite quantum model of the space-time down to diamond areas of
Planck scale. The classical equations clearly break down in the vicinity of
the Planck scale, while the CS model does not. The question is whether there
is any reason to believe this particular finite quantum mechanical model. A
physicist’s answer should be that only experiment could decide. to the minimal
diamond for which a description of the space-time as a flat FRW model with
equation of state $p=\rho$ makes sense. Once the total entropy gets small
enough, the description of the density of states by Cardy’s formula will
become less accurate. In our model of the system as fermion fields with
Thirring interactions, the detailed values of the couplings and the precise
nature of the cutoff will become more important. Although the quantum model
that Carlip and Solodukhin guessed from hydrodynamics is well defined, we have
no particular reason to consider it to be the only correct answer when the
diamond becomes small. If one were really probing Planck scale distances in
the laboratory, only experiment would be the arbiter of what the right
microscopic theory was.

From the point of view of a space-time picture, we can think of this as a
restriction on how useful it is to think of two different FRW geodesics as
being spatially close to each other on a fixed time slice. The initial state
above, with a given choice of the minimal entropy, prescribes a minimal
initial distance between FRW geodesics. If evolution along each geodesic is
chosen to be identical (this is the definition of our Hilbert bundle model,
not a fine tuning of the initial state), then the QPR will be satisfied. We
have described elsewhereholoinflation  how to construct more interesting,
inflationary, cosmologies based on these ideas. In those models the QPR
implies that inflationary horizon volumes along distant geodesics appear in
the DU coordinates of a given geodesic as a dilute gas of black holes. Some
mild assumptions about the properties of those black holes produce a very
economical theory of early universe cosmology, which accounts for CMB
fluctuations999and makes predictions that are in principle distinguishable
from those of QFT models of inflation., the Hot Big Bang, baryogenesis, and
dark matter. The model has no Transplanckian problem, and is completely
consistent with quantum mechanics, causality and unitarity.

The QPR is also essential for establishing that our Hilbert bundle formulation
of quantum gravity actually corresponds to unitary evolution on a Hilbert
space. The prescription that the DU evolution operator on the time interval
$[\tau,\tau+L_{P}]$ is $e^{-iL_{0}(\tau+L_{P})}$ defines time evolution in
terms of embedding maps of the Hilbert space of a smaller diamond into that of
a larger one. When we compare two diamonds along the same geodesic, whose
future tips differ by one Planck unit, we add new degrees of freedom only
along the new section of null boundary. These degrees of freedom are however
already included in the diamonds of smaller proper time, along other
geodesics. So the QPR puts constraints on the evolution of these ”out of
diamond” degrees of freedom. Applying the QPR along every pair of geodesics we
conclude that on the full Hilbert space along each geodesic there is a unitary
operator $U(t)=U_{in}(t)\otimes U_{out}(t)$, where $U_{in}(t)$ operators only
on the Hilbert space of the diamond at time $t$, and $U_{out}$ only on its
tensor complement. Strictly speaking, this argument only applies to the dS
case, where the Hilbert space of a complete geodesic is finite dimensional.
For non-positive c.c. we must be more careful about infinite dimensional
limits. We’ll discuss some aspects of this in the next section and in the
appendix. Above we have proposed that $U_{out}$ be defined in terms of the
$L_{0}$ of a collection of free fermion fields. An important question that we
have not yet studied is whether this proposal is consistent with the QPR. The
real force of the QPR comes into play for states describing localized
excitations in a diamond, to which we now turn.

###  III.1 Space-time Locality and the Origin of Quantum Field Theory

The formalism described so far allows us to localize quantum information on
the boundaries of causal diamonds in space-time but has not given us any clue
about the description of the quintessential object of experimental physics: a
localized excitation traveling on a time-like or null trajectory through
space-time, particularly a timelike geodesic. The essential hint about the
description of localized objects came from the study of black holes in de
Sitter (dS) space. Before explaining that, we should comment on the
unfortunate fact that localization in AdS space is quite a different story,
which we will mostly relegate to an appendix. In brief, AdS space is best
described by the Error Correcting Code/Tensor Network formalism, in which a
cutoff $1+d-2$ dimensional conformal field theory is described as a sequence
of lattice field theories arranged as shells in $d-1$ dimensional space, with
embedding maps (called a Tensor Network Renormalization Group) connecting
consecutive shells in the sequenceswingleetal . This is a manifestly local
presentation of an approximation to $AdS_{d}$ geometry, which also manifests
Maldacena’s scale radius duality. When the radius of AdS space is much larger
than all microscopic scales, the number of degrees of freedom inside each node
of the tensor network is very large and space-time locality has to do with
what is going on inside the node, rather than with communication between the
nodes. The ECC/TN picture becomes irrelevant. Thus, our claim is that locality
on scales smaller than the AdS radius is a property of the cutoff lattice
models of the tensor network, and cannot be addressed easily by computations
in the continuum boundary field theory.

In the year 2000, Fischler and the present author independently
conjecturedtbwfds  that the finite Gibbons-Hawking entropy of dS space implied
that the Hilbert space of a quantum model of dS space was finite dimensional.
An essential part of that conjecture was the fact that there is a maximal mass
black hole in dS space, so that it is extremely plausible that a quantum model
of dS space has a UV cutoff. In discussing this at the Festschrift for L.
Susskind at Stanford, Fischler and I were struck by the fact that the
Bekenstein-Hawking area law for black holes much smaller than the maximal size
in dS space showed that the total entropy of the space-time with a black hole
in it was less than the entropy of empty dS space! Moreover, this gave an
almost classical derivation of the Gibbons-Hawking temperature of dS space,
which Gibbons and Hawking had derived from quantum field theory in a de Sitter
backgroundGH . We conjectured that the density matrix of empty dS space was
maximally uncertain (and that the same was true for any empty causal diamond
of finite area101010A similar claim was made by R. Boussobousso  at about the
same time.) and that energy in the static patch was defined by saying the
number of ”constraints” on the states was of order $ER$, where $R$ was the dS
radius. We did not have a very clear model of this in mind but in the summer
of 2006 Tomeu Fiol walked into my office in Santa Cruz and presented me with
such a model.

Suppose one has a matrix, with matrix elements that are fermionic oscillators

$[\psi_{i}^{j},\psi^{\dagger\ k}_{l}]_{+}=\delta_{i}^{k}\delta^{j}_{l}.$ (20)

Then if we insist on being in a subspace of the Hilbert space of fermions
where we set the off diagonal blocks between a block of size $n$ and one of
size $N$ to zero, the statistical probability of being in a quantum state that
has order one probability of being in that subspace is $P\sim e^{-nN},$ which
looks like a Boltzmann factor, with $N^{-1}$ proportional to the temperature
and $n$ proportional to the energy. Something we did not realize at the time,
was that if we only require this when the probability is small, then it
remains true for a wide variety of density matrices which are not maximally
uncertaintbpd22 . In particular, even if dS space follows the Carlip-
Solodukhin law, the explanation of dS temperature can be valid for the states
in dS space that, according to CKN, are well described by QFT.

Fiol’s matrix model idea needs a little tweaking to make it compatible with
the ideas presented in the present paper. Some of that tweaking was already
done in 2006bfm . First of all, the fermions should be spinors. In four
dimensions, we can think of the cut-off spinor bundle as $N\times N+1$
matrices, since these can be thought of as transforming in all half integer
spin representations of $SU(2)$ up to $J=N-\frac{1}{2}$. We can then construct
square $N\times N$ matrices $M_{i}^{j}=\psi_{i}^{A}\psi_{A}^{\dagger\ j}$ and
impose the constraints on states in terms of these. These bilinears in spinors
can be thought of as ”fuzzy differential forms on the two sphere”. The trace
of products of these matrices is the fuzzy analog of the integral of two forms
over the sphere. When a matrix is block diagonal its trace is the sum of the
traces of the blocks. We can think of this as writing the sphere as a line
bundle over an interval and doing a one dimensional sum over slices of two
volume which are then discretized into matrices with matrix elements that are
bounded operators in a fermionic Hilbert space. Block diagonal matrices
correspond to bands of vanishing volume form. This does not quite capture the
full structure of a trace however, since we can do unitary transformations,
which change the order of the blocks. A better picture then is regions of
finite area surrounded by bands where the variables vanish. 3 is a cartoon of
what we mean.

Figure 3: The holoscreen of a diamond with localized excitations penetrating
it. Red are regions where jet momentum flows into or out of the diamond. White
regions have strictly zero momentum, while black regions have momentum flows
of magnitude less than the inverse diamond radius.

Readers will note the large black region on the figure, the white bands of
vanishing variables, and the smaller red regions. The latter represent small
blocks of the matrix of size $n_{i}$ while the black region has size $N-\sum
n_{i}\gg 1$. Constraints setting the off diagonal blocks between different
small blocks to zero can easily be removed by finite time dynamics, while it’s
plausible that with an appropriate scaling law for the Hamiltonian, the $o(N)$
constraints between the large block and each of the small ones will never be
removed. In the strict $N\rightarrow\infty$ limit, $\sum n_{i}$ would become
an asymptotic conservation law if the proper time was also scaling like $N$.

On the other hand, in our model for dS space, $N$ is kept finite while the
proper time goes to infinity, so the quantity $\sum n_{i}$ is not conserved,
but controls the Boltzmann factor of low probability states in the density
matrix. $\sum n_{i}$ is thus proportional to the static energy, which will
become the conserved Minkowski energy if we let $N$ go to infinity with proper
time. The measure theoretic cartoon of 3 tells us how to generalize these
ideas to higher dimensions, for large $N$. Isolated localized excitations
entering a causal diamond with area large in microscopic units, but small
compared to the scale defined by the c.c,, are defined by red regions of
finite area $A_{i}$ on the holographic screen, whose area scales like
$N^{d-2}$. The energy carried by those excitations is proportional to
$A_{i}^{\frac{d-3}{d-2}}$, i.e. the volume of its bounding surface. The empty
region surrounding the red region has an area $\sim NA_{i}^{\frac{d-3}{d-2}}$.
To implement this picture in a matrix model, we need to write the matrices as

$M_{i}^{j}=\psi_{i}^{A}\psi^{\dagger\ j}_{A},$ (21)

where the index $A$ runs over of order $N^{d-3}$ values. Fischler and the
authortbwfnewton , using the facttbjk  that the number of solutions of the
Dirac equation with angular momentum cutoff $N$ is equal to the number of
components of a $d-2$ form in $N$ dimensions, tried to write $\psi_{i}^{A}$ as
a $d-2$ form. However, one would also like to be able to impose constraints on
of order $n_{i}^{d-3}N$ fermion number operators in order to make all the off
diagonal matrix elements between $n_{i}$ and $N-n_{i}$ indices vanish. It is
not yet clear how to do this. Alternatively, we could try to implement the
constraints directly from the figure in terms of the ”local” spinor field
expanded in a basis of solutions of the Dirac equation with an eigenvalue
cutoff. Fuzzily localized fields cannot reproduce exact characteristic
functions of regions but for large values of the cutoff the approximation
should be good enough for statements about non-interaction of the localized
objects with the horizon to be valid. Indeed, a Hamiltonian written in terms
of integrals of products of differential forms constructed from truly local
fermions would be ultra-local and could not propagate information around the
horizon. Fuzzy localization is essential if we are to reproduce the fast
scrambling properties of horizons. In four dimensions there is a well known
connection between the matrix model formulation and area preserving
diffeomorphisms, with unitary conjugations of the matrices playing the role of
a regularization of the infinite dimensional group of area preserving
diffeomorphisms. We need a d dimensional generalization of that formalism.

It is unlikely that there is an interesting quantum theory of dS space in
dimension greater than $4$. Since 2001tbwf99 , Fischler and the present author
have argued that dS space should be thought of as a regulated version of a
scattering theory of ”particles” in Minkowski space111111The reason for the
scare quotes will become clear in the next section.. There is no S-matrix in
dS space: every state returns to a maximal entropy equilibrium with no local
excitations in a time in any static coordinate system which is not much longer
than the dS radius121212This statement is irrelevant for sufficiently complex
objects like a galaxy. The constituents of the complex object decohere its
center of mass trajectory and define a classical reference frame independent
of any mathematically defined coordinate system. . However, if the
$R_{dS}/L_{P}\rightarrow\infty$ limit of finite time transition operators in
dS space defines a Minkowski scattering operator, then we can view this as a
mathematical definition of dS space that is independent of particular complex
localized objects (see previous footnote) . However, we have an enormous
amount of evidence from string theory that Poincare invariance implies exact
super Poincare invariance in models of quantum gravity. Rigorous theorems in
classical SUGRA show us that there are no dS solutions of SUGRA for $d\geq 5$.

The real reason to be interested in this description of localized excitations
for $d>4$ is that it applies equally well to the Minkowski limit. Indeed it
provides the crucial link between the formalism we have been discussing and
QFT. Let us begin by noting the analogy between 3 and a layer in a modern
particle detector. We can think of the red regions as the pixels in the
detector that have been ”lit up” by a particular scattering event and the
white regions as ”jet isolation zones” that are used to define particle jets.
In a real detector these are defined by including all soft particles that we
think should be included in the jet, leaving an empty region around it. The
dark regions are pixels that don’t light up, which means that anything that
flowed through them was below the detector threshold. Here we are insisting
that not even ”asymptotically zero energy” flows through the annular regions
surrounding the jets. The dark region has energy of order $1/N$ as measured
along the geodesic in a finite area diamond of area $N^{d-2}$ in Planck units,
Now imagine a time symmetric nesting of diamonds as a very very finely layered
particle detector. Then following the red regions from layer to layer, we get
a picture of the trajectory of the jet through the bulk of space-time4.

Figure 4: The holoscreens of two consecutive nested causal diamonds, showing
how following the constraints leads to a picture of the trajectory of a jet of
particles through space-time.

In the description of this trajectory from the point of view of the Hilbert
fiber over a given geodesic, all we learn about this trajectory is how much
area it takes up on each holoscreen in the nest of diamonds, but the QPR
relates this to the Hilbert fibers over all other trajectories and consistency
between them determines the geometric shape of the jet trajectory through
space-time.

Thus, given an initial state on a very large diamond, containing some number
of well separated jets on its past boundary, we can compute the amplitudes for
Feynman like diagrams describing their evolution into a different collection
of well separated jets on the future boundary (5).

Figure 5: Decomposition of amplitudes in HST models into time ordered Feynman-
like diagrams, describing jets of particles propagating between causal
diamonds in the background space-time.

These are not exactly Feynman diagrams because each picture in which energy
$E$ propagates between two points contains amplitudes for this energy to split
up into distinct pieces carrying fractions of the total energy. The reason for
this is that each jet contains all fermion operators contributing to the
matrix $M_{i}^{j}$ with indices running from $1$ to $n$ with $n^{d-3}\propto
E$.

It is also very easy to see where the particle/jet picture characteristic of
QFT breaks down in these models. Suppose that for some macroscopically large
$N$ we find that $\sum n_{i}^{d-3}\sim N^{d-3}$ . Then clearly the
approximation that we have a small deviation from the equilibrium state has
broken down, and we can expect a rapid evolution back to equilibrium. This is
precisely the criterion that the Schwarzschild radius is of order the size of
the diamond, here derived from a quantum model, rather than general
relativity.

There are other ways in which the diagrams derived from the Hilbert bundle
formalism differ from Feynman diagrams. They are definitely causally ordered.
The meaning of a diagram like that of figure 5 is that a jet that exits the
future boundary of the diamond $1$ enters the past boundary of diamond $2$ and
interacts with other jets entering that diamond, producing some final state,
for which the diagram computes the amplitude131313Of course another big
difference is that we do not have a nice closed formula for the amplitude,
just a sort of Trotter product formula in terms of amplitudes in $1+1$
dimensional CFTs with time varying central charge. . The resemblance to time
ordered Feynman diagrams then is that these pictures give a bulk space-time
decomposition of an amplitude into processes that occur in different causal
diamonds. Unlike Feynman diagrams they include both small diamonds, which
resemble localized vertices in which a number of particles come into a diamond
and a generally different number emerge and ”black hole formation processes”
in which particles enter a diamond, which then retains its identity as a meta-
stable object of fixed energy, emitting jets of energy in a random thermal
manner for an extended period. As far as scaling laws are concerned, these
objects behave like the black holes of general relativity, but are excitations
of a unitary quantum mechanical system. In the next section, we will indicate
how these models can lead to a Lorentz invariant scattering theory in
Minkowski space.

##  IV Representation of Non-compact Isometry Groups

All of the maximally symmetric space-times have non-compact groups of
isometries, which we will denote collectively by $G$. Finite causal diamonds
in these space-times are invariant only under an $SO(d-1)$ subgroup and
geodesics only under $SO(d-1)\otimes R$. Transformations in $G/(SO(d-1)\otimes
R)$ map one fiber of the Hilbert bundle onto another and so are not in general
represented on any fixed Hilbert space. The static time translation $U_{S}$
maps one interval along a geodesic to another one. It is not the same as the
time dependent Hamiltonian that describes propagation in a diamond. In terms
of conventional coordinate systems $U_{S}$ uses the global time slicing, while
the Hamiltonian $H(t)=iU(t+1)U^{\dagger}(t)-1$ describes propagation between
slices within the diamond, as in Figure 1. For maximally symmetric space-times
these are the diamond universe coordinates ofCHM JV . In the case of dS space,
$H(t)$ approaches the global Hamiltonian $i\partial_{t^{s}}$ as the static
time goes to infinity. The asymptotic dimension of the Hilbert space is
$e^{A_{d-2}R_{dS}^{d-2}/4G_{N}}$ up to subleading corrections in the
exponential. $A_{d-2}$ is the area of the unit sphere. Strictly speaking, the
area is just the expectation value of the modular Hamiltonian, but intbpd22
we’ve shown that the conditions $\langle K\rangle\rightarrow{\rm ln\ dim}{\cal
H}$ and $(\Delta K)^{2}=\langle K\rangle$ are compatible in the limit of a
Hilbert space of large dimension.

For AdS space, as $t$ approaches $\pi R_{AdS}/2$ the Hilbert space dimension
goes to infinity and the Hamiltonian approaches the generator of the AdS
isometry group that leaves a particular time-like geodesic invariant. The
AdS/CFT correspondence tells us that this is the Hamiltonian of a CFT, and the
c.c. is determined by the asymptotic density of states in that CFT. The
transformations that do not leave that particular geodesic invariant act as
unitary transformations on the CFT Hilbert space. So in this case, the full
isometry group acts on a single Hilbert space. We can think of the CFT Hilbert
space as being a trivial Hilbert bundle over the space of time-like geodesics
in AdS space. A choice of the Hamiltonian $K_{0}+P_{0}$ within its conjugacy
class in the conformal group is equivalent to a choice of timelike geodesic
and we can think of the fiber over that geodesic as the Hilbert space
expressed in the basis where $K_{0}+P_{0}$ is diagonal. There is no holonomy
when we traverse a closed loop in the conjugacy class, so the bundle is
trivial. We’ll discuss the case of negative c.c. in more detail in the
appendix.

The case of vanishing c.c. is more subtle and is intertwined with the
existence (or not) of the Bondi-Metzner-Sachs algebra on the Hilbert space on
which the S-matrix acts. There is also considerable confusion about whether
that Hilbert space is a traditional Fock space. It is clear that it is not in
four dimensions, and the present author has argued for some timetbir  that
Fock space is not the correct arena for asymptotically flat quantum gravity in
any dimension. That conviction grew out of the picture of particle/jets as
constrained states of holographic degrees of freedom on the boundaries of a
diamond, which we described in the previous section. In the appendix, we’ll
see that the same conclusion follows if we follow the approach of Polchinski
and Susskindpolchsuss  and derive the Minkowski S matrix as a limit of CFT
correlators.

Let us then consider the variables in a finite causal diamond in $d$
dimensional Minkowski space times a compact $11-d$ manifold ${\cal K}$ of
fixed size much larger than the $11$ dimensional Planck scale. The magic
number $11$ appears because we anticipate the result of our investigation,
namely that the limiting theory must be super-Poincare invariant. Decades of
investigation into string theory make it fairly clear that every
supersymmetric model of quantum gravity can be viewed as a compactification
plus duality transform of eleven dimensional SUGRA.

The variables are fields $\psi_{a}^{I,M}({\bf\Omega},\sigma)$ . $a$ is a two
dimensional Dirac index and $I$ a $d-2$ dimensional Dirac index, while $M$
labels a finite set of eigenfunctions of the Dirac operator on ${\cal K}$. We
will omit the indices $I,M$ in future equations. ${\bf\Omega}$ is a coordinate
on the $d-2$ sphere. The spherical harmonic expansion on this sphere is cut
off, but we’re considering the limit of an infinite diamond where the cutoff
goes to infinity. Recall that the volume of the sphere scales like $N^{d-2}$,
where $\pm N$ is the scaling of the cutoff on the Dirac eigenvalue.

Now we want to define in(out) scattering states. We consider a sequence of
diamonds with $N_{i}$ going to infinity, in a future(past) oriented nested
covering of the Penrose diagram of Minkowski space (Figure 1). Pick a finite
set of points ${\bf\Omega}_{k}$ on the sphere. Around each point we impose a
constraint that the variables $\psi_{a}({\bf\Omega},\sigma)$ ”vanish” in a
shell of volume $N_{i}A({\bf\Omega}_{k})^{\frac{d-3}{d-2}}\equiv N_{i}E_{k}$
surrounding an area $A({\bf\Omega}_{k})$, which includes that point. By
independent area preserving maps on each diamond boundary, we can turn each of
these regions into a spherical cap surrounded by an annulus of ”vanishing”
variables. Of course, with only a finite number of spinor spherical harmonics,
we cannot make a function that vanishes exactly in an annulus. We mean the
closest approximation to such a half-measure with the available basis set.

For finite $N_{i}$ the ”areas” are discrete numbers, but in the limit they
become continuous and the $E_{k}$ are continuous variables, which we associate
with the magnitudes of null momenta in Planck units141414Stable massive
particles are always associated with BPS states or bound states of them and
require a generalization of the SUSY algebras we will write below.. In other
words, as we let $N_{i}\rightarrow\infty$ we are free to rescale the $E_{k}$
at will since we can absorb the rescaling into $N$. As usual, in order to
obtain finite results in the limit, one must impose some kind of scaling
symmetry, and the obvious one is the conformal group of the $d-2$ sphere,
which is the Lorentz group of the null cone $P^{2}=0$ in $d$ dimensions. Our
spinor variables thus become spinors $Q_{\alpha}(P)$ on the null cone and they
satisfy the Lorentz covariant condition

$P_{m}(\Gamma^{m})_{\alpha\beta}Q_{\beta}(P)=0.$ (22)

which reduces the number of components to the transverse ones that existed
before we took the limit. In the limit, the $Q_{\alpha}(P)$, are generalized
functions on the null cone. For non-zero values of $P$, these generalized
functions only exist in a finite number of spherical caps, surrounded by
annuli in which $Q_{\alpha}(0)=0$.

Note that we have dropped the two dimensional structure that was a crucial
part of the description of the boundary of finite causal diamonds, which we’ve
emphasized by also changing the names of the operators. That structure was
used by Carlip and Solodukhin to understand the finite boundary entropy and
bybz  to understand the finite fluctuations of that entropy. In the limit,
both of these quantities become infinite and must be ”renormalized away” if
there is to be a finite limiting scattering operator. We have interpreted the
fluctuations implied by this two dimensional structure as fluctuations in the
transverse geometry, so its disappearance in the asymptotic limit is the
statement that the geometry at null infinity is frozen. This implies an
asymptotic decoupling of the dynamics of the emergent zero energy
”topological” sector of the theory in the infinite diamond limit. The various
”soft theorems”soft  in the extensive literature on BMS symmetries prove this
kind of decoupling, but there is as yet no clear picture of what the Hilbert
space of asymptotically flat quantum gravity looks like. Above four
dimensions, most authors would just claim that it is Fock space, but this is
an incredibly ambitious claim. Initial states consisting of a few particles
with large sub energies, can create large black holes which orbit around each
other for extended periods of time, merge, emit Hawking radiation and so on.
To claim that the final state of any such process is a normalizable state in
Fock space, even in 11 dimensional SUGRA where there are no apparent infrared
problems in perturbation theory, is the height of hubris. In the BFSS matrix
modelBFSS , which is a manifestly unitary scattering theory with the particle
content of $11$ D SUGRA (in the large N limit), the claim is equivalent to
having control over processes in which large block diagonal matrices come in
from remote regions of the moduli space and go out into regions in which other
large blocks plus an arbitrary number of small blocks with transverse momenta
of order $N^{-1/2}$ are emitted. In four dimensions we know that Fock space is
not the right answer and there is recent evidenceWaldetal  that the Fadeev-
Kulish proposal does not work either.

I have proposedtbir  that the proper framework for scattering theory in
quantum gravity is a map between representations of the algebras of the
$Q_{\alpha}^{M}(P)$ operators on the positive and negative energy parts of the
null cone. These algebras are

$[Q_{\alpha}^{M\ \pm}(P),Q_{\beta}^{N\ \pm}(P^{\prime}]_{+}=\pm\delta(P\cdot
P^{\prime})\delta^{MN}P_{m}(\Gamma^{m})_{\alpha\beta}.$ (23)

The angular delta function follows from the fact that the fermion fields
corresponding to individual angular momenta anti-commute. The delta function
in the ”internal” indices is achieved by taking appropriate linear
combinations. The rest follows from Lorentz invariance and the fact that the
objects we are describing are localized in angle. They cannot carry tensor
charges at infinity.

The delicate question of what the Hilbert space of asymptotically flat quantum
gravity is, has no experimental relevance. For practical purposes, the
inclusive cross sections defined by Weinberg are all we could ever hope to
measure if we lived in asymptotically flat space, even in four dimensions,
where IR effects are most important. In eleven dimensions these IR questions
could effect scattering amplitudes only via terms exponential in ratios of
Mandelstam invariants to the Planck mass. In ten dimensional superstring
theory they might be of order $e^{-g_{S}^{-2}}$, exponentially subdominant
compared to the leading non-perturbative $e^{-1/g_{S}}$ contributions.

We should note that while Lorentz invariance seems necessary in order to get
any kind of sensible infinite diamond limit, it also follows from the QPR. Our
formalism has a built in requirement that everything must be invariant under
passage to the fiber over any geodesic of the background space-time. We’ve
described the infinite diamond limit along a particular geodesic, and
discovered an apparent need for Lorentz invariance in the infinite diamond
limit, which is consistent with the requirement that the scattering operator
should be identical to that obtained by working over a different geodesic.

Something similar happens for space and time translation invariance. Our
Hamiltonian description of time evolution in a causal diamond uses a time
dependent Hamiltonian because it is evolving in DU coordinates. However, when
we define asymptotic energy in terms of constraints on initial conditions on
the past boundary of the diamond we can see that it is asymptotically
conserved. The argument has two parts and its form depends on whether we use
the past or future directed nest of diamonds to cover the Penrose diagram. Let
us consider the future directed nesting. Then, at early times it is unlikely
that the constraints are imposed on the degrees of freedom in the small nested
diamond. Those outside the diamond are described as free fermions, so the
constraints are preserved by time evolution. However, even after the
constrained DOF are engulfed by the growing diamond, the size of the
Hamiltonian is of order $1/R_{diamond}$, the Hamiltonian is $4-local$ in the
fermion variables, and the number of constraints is of order the radius of the
Penrose sphere, which we are taking to infinity. The asymptotic energy is
proportional to the coefficient of that infinite number of constraints, so it
cannot be changed. The same conclusion follows from the QPR applied to causal
diamonds that are shifted by a rigid time translation along the same geodesic.

The situation for conservation of spatial momentum is a bit different. Here we
use the QPR to impose constraints on the tensor complement of the Hilbert
space corresponding to proper time interval $[-T,\tau]$ along a particular
geodesic $G_{1}$ in Minkowski space in terms of the constraints on the Hilbert
spaces corresponding to that same time interval along geodesics related to
$G_{1}$ by a rigid space translation. In words, these relations tell us when
and where a certain amount of energy enters into the causal diamond of a
detector traveling along $G_{1}$. Thus, although the constraints themselves
are defined only in terms of areas, the QPR and the background geometry tell
us about the trajectories followed by the centers of jets of particles through
the background space-time. One must recall that the jets are not particles,
but actually systems that have (asymptotically) infinite dimensional Hilbert
spaces151515For finite size causal diamond the jet Hilbert spaces always have
dimension exponentially smaller than that of the diamond, because of the bound
that says that the jet energies must be much smaller than the mass of a black
hole that would fill the diamond. But as the diamond approaches infinite size,
this allows the jet Hilbert spaces to grow without bound.. The jet trajectory
is a decoherent semi-classical observable of these large quantum systems.

Using the area preserving diffeomorphism invariance of the large $N$
Hamiltonians, we can always view the jets as living in spherical caps,
surrounded by symmetric annuli where the spinor variables vanish. The map from
the past boundary to the future boundary is invariant under relative rotations
of the centers of the jets, since this is a residual symmetry of the area
preserving group after we have fixed the jet profiles to be symmetric.
Translation invariance then follows from Lorentz invariance and energy
conservation. It can be derived independently by insisting that the scattering
operator obtained at one fiber of the Hilbert bundle be independent of the
fiber. As in the case of energy conservation and boost invariance, the first
derivation involves dynamics of the time evolution operator for a given
geodesic, while the second is a symmetry constraint on the Hilbert bundle as a
whole.

While these arguments are not rigorous, their conclusion is remarkable, and
explains an empirical fact. What we have found is that the zero c.c. limit of
our Hilbert bundle theory of dS space, has stable localized excitations. If we
enforce the Poincare symmetry of Minkowski space, the asymptotic algebra
automatically has the form of the supersymmetric generalization of the Bondi-
Metzner-Sachs algebra (actually its Fourier transform) derived from SUGRA by
Awada, Gibbons and Shawags . Since the invention of string theory in the late
1960s, every attempt to formulate a non-supersymmetric theory of quantum
gravity in asymptotically flat space has met with failure. If we accept the
premise of this paper, that quantum gravity is a Hilbert bundle over the space
of timelike geodesics on its hydrodynamic space-time manifold, and the the
variables of quantum gravity are Connes-Carlip-Solodukhin conformal fields on
causal diamond boundaries, then we understand that failure.

###  IV.1 Summary

The Hilbert bundle formulation of HST resolves all problems with
representations of non-compact isometry groups on finite dimensional Hilbert
spaces. In AdS space, the proper time $\rightarrow\frac{\pi R}{2}$ limit,
leads to a trivial Hilbert bundle and a unitary representation of the isometry
group on the Hilbert space of the CFT on $R\times S^{d-2}$. In asymptotically
flat space we have suggested that one obtains a unitary representation of the
semi-direct product of the Lorentz group and the AGS superalgebra (or its
generalization to include massive BPS states), but the nature of the
asymptotic Hilbert space remains unresolved. In dS space, which probably only
makes sense in dimensions below $5$161616Seesatbpdwf  for a model of
$dS_{3}$., the isometry group acts only on the Hilbert bundle as a whole, not
on any single Hilbert space.

##  V The ’t Hooft Commutation Relations

The basic postulate of our approach to quantum gravity is that the Einstein
equations are the hydrodynamic equations of a collection of quantum systems
living on the boundaries of causal diamonds in space-time, whose equilibrium
density matrix $\rho=e^{-K}$ satisfies $\langle K\rangle=\langle(K-\langle
K\rangle)^{2}\rangle=\frac{A}{4G_{N}}$. When we use the Verlinde analysis and
examine the near boundary limit of a causal diamond, the EH action splits into
two parts. Our discussion so far has dealt with the large part of the action,
when the transverse size of the diamond $L$ satisfies $L\gg L_{P}$. We have
argued that it is given by a cut off quantum field theory of $1+1$ dimensional
fermions, which is a particular kind of abelian Thirring model. The fermion
fields encode fluctuations of the transverse geometry around a classical
background metric $h_{mn}$.

The authors ofV291  showed that the small term in the action was purely
topological (once one imposed the classical equations of motion of the large
term), and that its entire content was the ’t Hooft commutation relations

$[X^{+}(\Omega),X^{-}(\Omega^{\prime})]=(\frac{L_{P}}{L})^{d-4}(\triangle_{h}-R_{h})^{-1}(\Omega,\Omega^{\prime}).$
(24)

Our purpose in this section is to provide a model for these commutation
relations in terms of the $1+1$ CFT that we have constructed to describe the
transverse fluctuations171717For another approach to the ’t Hooft commutation
relations, seeVZ3 . The first of these papers gives a derivation of $(\Delta
K)^{2}=\langle K\rangle$ directly from the ’t Hooft commutation relations. The
variables $X^{\pm}$ are defined on the future/past boundary of the causal
diamond, and the equal time surface on which the commutation relations are
evaluated is the bifurcation surface of the diamond. The reason that there is
such a relation at every point along the past/future boundary is because every
point is on the bifurcation surface of some diamond in a future or past
directed nested cover of the diamond.

Our model of transverse dynamics has separate CFTs on the past and future
boundaries of a diamond, related by the time evolution operator. Note that in
going from past to future boundary, the role of the time and space coordinates
of the CFT are switched. Thus, the space and time components of a $U(1)$
current are also exchanged. The space and time components of a conserved
$U(1)$ current in $1+1$ CFT have a c-number Schwinger term in their
commutator181818A previous relation between the ’t Hooft commutation relations
and Schwinger terms in 1+1 dimensional electrodynamics was pointed out inpeet
.. Thus we postulate

$X^{+}({\bf\Omega})=\int d\sigma f^{+}(\sigma)J^{0}({\bf\Omega},\sigma).$ (25)
$X^{-}({\bf\Omega})=\int d\sigma f^{-}(\sigma)J^{1}({\bf\Omega},\sigma).$ (26)

Here

$\int d\sigma f^{+}\partial_{\sigma}f^{-}=1,$ (27)

and the Schwinger term is

$[J^{0}({\bf\Omega},\sigma),J^{1}({\bf\Omega^{\prime}},\sigma^{\prime})]=(\triangle_{h}-R_{h})^{-1}({\bf\Omega},{\bf\Omega^{\prime}})\partial_{\sigma}\delta(\sigma-\sigma^{\prime}).$
(28)

$\triangle_{h}$ is the scalar Laplacian of the transverse manifold, and
$R_{h}$ its scalar curvature. To understand how to obtain the last equation,
we consider a set of $1+1$ dimensional fermion fields labelled by the
eigenspinors of the transverse Dirac operator $D$ and write

$J^{\mu}=\bar{\psi}\gamma^{\mu}D^{-1}\psi.$ (29)

There are two ways in which this formula deviates from that of ’t Hooft.
First, the currents, and thus the light ray operators $X^{\pm}$ are bilocal
rather than local operators on the transverse manifold. The second is that the
transverse integral operator we obtain from this ansatz is

$D^{-2}=(\triangle_{h}-\frac{1}{4}R_{h})^{-1}.$ (30)

The discrepancy in the coefficient of $R_{h}$ can perhaps be fixed by choosing
a non-Riemannian connection for the Dirac operator, though one would clearly
like to have a deep physical justification for such a replacement. Another
possibility, which should be explored is to replace the Dirac operator by the
Rarita-Schwinger operator as the proper representation of transverse geometry.
If we do choose such a connection, or if the Rarita-Schwinger idea works, then
the equations of motionV291  show that the null energy fluxes, which enter
into actual scattering amplitudes are still local operators. The very
preliminary nature of this analysis should be evident.

##  VI Conclusions, Speculations and (Ugh) Philosophy

For many researchers in quantum gravity there is much in the present work that
will appear negative. Quite frankly, it is probably because some of the
conclusions I’ve been drawn to were so distasteful that it has taken a long
time to come to the present formulation of HST, despite the fact that they
were implied by the basic postulates of the formalism. I’ve long believed that
the deepest insight into the problem of quantum gravity was Jacobson’s
demonstration that Einstein’s equations, with an arbitrary stress tensor
satisfying the null energy condition, were the hydrodynamic equations of the
Bekenstein-Hawking entropy law, applied to the boundary of an arbitrary
diamond in space-time. If one thinks about the role the Navier-Stokes
equations play in the physics of ordinary matter, Jacobson’s paper should have
made us all extremely suspicious of the idea that quantum gravity was all
about quantizing Einstein’s equations. Extremely suspicious as well of the
idea that there was some kind of ”background independent” formulation of QG,
just because Einstein’s equations were universal. Thinking about this in
condensed matter language, this is like saying that there’s a ”substance
independent” theory of condensed matter, just because the NS equations are so
universal. The real clue in Jacobson’s work, in a phrase that J.A. Wheeler
invented, but clearly did not understand, was that space-time was the IT in
”IT from BIT”. The areas of the holoscreens of all causal diamonds determine
space-time geometry and these are determined by the von Neumann entropies of
quantum density matrices associated with those diamonds.

It follows that one should investigate QG by choosing a background geometry
satisfying Einstein’s equations with a stress tensor obeying the null energy
condition (the second law of thermodynamics), and try to find a quantum system
obeying Jacobson’s principle in that geometry. This leads directly to the
Hilbert bundle formulation of QG because there is a one to one correspondence
between the set of causal diamonds, and the set of choices of nested proper
time intervals along time-like geodesics. The QPR is the natural consistency
condition to impose on this system of Hilbert spaces and density matrices.

We next face the question of what to choose for the density matrix of a
diamond. Obviously there can’t be a unique choice for this because we know
from experiment that there can be many different states inside a causal
diamond. Here we get two clues from quantum field theory and another big boost
from the hydrodynamic interpretation of Einstein’s equations. Paradoxically,
one of the most important clues is the demonstration by CKN that all of our
experiments probe an entropy that scales at most like $(A/G_{N})^{3/4}$ (in
four dimensions), and cannot account for the entropy used by Jacobson to
derive Einstein’s equations. On the other hand, QFT, the theoretical framework
we use to explain our experiments, predicts area law entropy for an empty
diamond, roughly independent of the state in the QFT Hilbert space. The QFT
prediction for the coefficient of area is UV divergent and was conjectured to
be ”absorbed into Newton’s constant” in the Bekenstein-Hawking law. So we
should be looking for a universal prediction for the density matrix of an
empty diamond and try to understand the tiny corrections due to excitations
localized inside the diamond after we’ve understood ”nothing”.

That step was taken by Carlip and Solodukhin in 1998, whose analysis is best
understood in an analysis done by the Verlindes, where to leading order in
$L/L_{P}$ a redefined metric is block diagonal between a two dimensional
Lorentzian and a $d-2$ dimensional Riemannian block, with the Riemannian
length scales $L\gg L_{P}^{(d)}$ while those of the Lorentzian block are of
order the Planck length in the near horizon limit. The equations of motion of
the large part of the EH action imply that the two dimensional metric is flat
and the Riemannian metric is independent of the two dimensional coordinates.
The two dimensional stress tensor fluctuations of the conformal factor of the
Riemannian metric satisfy the Ward identities of a stress tensor of a CFT of
large central charge. That is, they are the hydrodynamic equations of such a
CFT. Carlip and Solodukhin thus postulate that the actual near horizon quantum
theory is such a CFT, and show that that postulate reproduces most known black
hole entropy formulae. Zurek and the present author generalized their
hypotheses to an arbitrary causal diamond and pointed out that it implied the
universal fluctuation formula $\langle(K-\langle K\rangle)^{2}\rangle=\langle
K\rangle.$

In the present paper we have taken this analysis one step further, using the
insight gained from my work with Fischler on HST. Carlip and Solodukhin do not
deal with fluctuations of the unimodular part of the transverse geometry, and
Solodukhin comments explicitly that he does not know why it is less important
than the conformal factor. The resolution of this is to imagine that the
transverse geometry is encoded in the target space of the CFT. The fact that
the finite diamond has finite entropy suggests immediately that the target
space fields be fermions191919It’s already implicit in the Carlip/Solodukhin
work that the CFT lives on an interval and has a cut off spectrum of the
Virasoro generator $L_{0}$.. Connes’ work on Riemannian geometry and the Dirac
equation tells us that there is a natural set of fermion variables lying
around, once we remember the spin statistics connection. This relation was
long ago incorporated into HST. We update it here by turning the spinor
fermionic oscillators into $1+1$ dimensional spinor fields. They’re actually
Dirac spinors both in the Lorentzian and Riemannian geometries, and so can in
a sense be thought of as fields on the full $d$ dimensional space-time.
However, the UV cutoff in the transverse space depends on the proper time in
two dimensions in a way that a $d$ dimensional field theorist would find
bizarre. The UV cutoff on the $L_{0}$ generator is also a bit bizarre. One
takes a central value determined by the classical formula of Carlip and
Solodukhin and chooses a width in discrete $L_{0}$ eigenvalue space such that
the Carlip-Solodukhin formula for the entropy is satisfied exactly, for ”the
smallest causal diamond for which one believes the Carlip-Solodukhin
analysis”. That last phrase has unfortunate philosophical implications, to
which we will return at the conclusion of these conclusions.

We are almost done with empty causal diamonds. To proceed we have to switch
our focus from the density matrices of empty diamonds to the time evolution
operator between two successive diamonds in a nested covering of a given
diamond. Let us choose a future oriented one 1 for definiteness. Here we will
restrict attention to geometries that are conformal to maximally symmetric
spaces. For such geometries, the modular Hamiltonian $L_{0}$ is relatedCHM JV
to a conformal Killing transformation which leaves the diamond invariant. More
precisely, if we consider a conformal quantum field theory on the same
background space-time, then the modular Hamilton of that field theory is
proportional to the quantum generator of the action of the conformal Killing
vector (CKV) acting on the bifurcation surface of the diamond. The
proportionality constant is fixed by the Virasoro algebra. The vector field
associated with that CKV defines a set of inextensible coordinates inside the
diamond, and the flows associated with the vector field are timelike lines,
one of which is the geodesic in the diamond. For these space-times at least,
it seems plausible that also in QG we should associate time evolution between
two diamonds whose future tips are $\tau$ and $\tau+L_{P}$ to be

$U(\tau+L_{P},\tau)=e^{-iL_{0}(\tau+L_{P})}.$ (31)

This unitary operator acts in the Hilbert space of the diamond corresponding
to the time interval $[-T,\tau+L_{P}]$ along the geodesic and we expect it to
entangle degrees of freedom in the Hilbert space corresponding to $[-T,\tau]$
with the new degrees of freedom that are added in the larger diamond. It
should be tensored with a unitary that acts in the tensor complement of
$[-T,\tau+L_{P}]$ in the full Hilbert space $[-T,T]$202020$T$ is a proper time
cutoff inserted to make sure that we are always dealing with finite
dimensional Hilbert spaces. Its value depends on the background space-time.
The limiting cases where the Hilbert space dimension goes to infinity have to
be treated with great care.

We have argued in the previous section that the twin requirements of
approximate two dimensional conformal invariance and fast transverse
scrambling of information restrict the form of $L_{0}$ to be that of free
fermions plus a limited set of abelian Thirring interactions constructed from
quartic fermion operators that would be invariant under transverse area
preserving diffeomorphisms if the transverse cutoff were removed. More work is
required to see whether those arguments are indeed correct, but if they are,
we have constructed a consistent set of models of quantum gravity for a
certain class of space-times.

The most disturbing thing about these models is the fact that we have had to
insert an arbitrary restriction to a ”smallest causal diamond for which we
believe the Carlip/Solodukhin analysis” into our construction. This is
inevitable in an approach that tries to bootstrap quantum theory from
hydrodynamics. By definition, hydrodynamics contains only coarse grained
information about an underlying quantum theory. In principle one should not
expect to find the correct description of the microscopic theory applying to
the real world, without comparing to microscopic experiments. This raises the
question about models in asymptotically flat or AdS space, where we have a
mathematically precise description of asymptotic observables, at least in
principle. In the case of AdS space there seems to be a pretty definitive
negative answer to the question of whether we could have an absolutely precise
and unambiguous description of the physics inside a causal diamond whose
radius was much smaller than the cosmological radius of curvature. According
to the description we have given of that physics in the appendix to this
paper, that physics is described by a single node in a tensor network. But a
tensor network is a cutoff quantum field theory and we have known since the
work of Wilson that many different cutoff theories converge to the same CFT.
Which one gives the ”correct” physics in a small causal diamond, with
arbitrary precision? One can try to impose symmetries to refine the allowed
set of cutoff theories, but it seems doubtful that one could come up with a
unique definition of time evolution in a sequence of small causal diamonds in
terms of unambiguous CFT data.

The same issue arises in asymptotically flat space. In quantum field theory,
we’re familiar with the fact that the S-matrix does not determine local
correlation functions. Even in integrable theories in two dimensions one has
to supply extra information to determine them, and we can do so only because
we know their mathematical definition and properties. The problem for
asymptotically flat quantum gravity is even worse. For the most part we only
know it as a perturbation expansion which is not Borel summable. There are a
few cases where it can be treated by BFSS Matrix Theory, but the large $N$
limit is not under control. In no case do we have a rigorous understanding of
the Hilbert space in which the scattering operator acts. It seems unlikely
that this will lead us to an unambiguous definition of finite time physics in
a local region of space-time.

There is a ”philosophical” issue at stake here, in which the nasty questions
of the interpretation of quantum mechanics raise their heads. The present
author adheres to the modern version of the Copenhagen interpretation of QM.
Certain quantum systems have collective variables whose dynamics takes place
on a time scale slower than typical micro-scales. Hydrodynamic flows are prime
examples of such variables. For collective variables the uncertainties implied
by QM are small, down by powers of some ratio of a small length scale to a
large one. More importantly, the violations of Bayes’ sum over histories rule
for probabilities in the stochastic evolution equations for collective
variables are exponentially small in the ratio of the large length scale over
the small one. This means that we can treat the quantum fluctuations in these
quantities like measurement errors or errors caused by random disturbances.
They form the basis for what Bohr and Heisenberg called the Classical World.
In our models, space-time geometry is part of this classical world.

In contemplating the physics that takes place inside a causal diamond, we’re
imagining a measuring apparatus traveling along some time-like trajectory and
”doing experiments”. The meaning of a mathematically precise theory of some
set of physical phenomena is that we can account for everything that measuring
apparatus can measure, with arbitrary precision. If the total Hilbert space of
the causal diamond is finite dimensional this is simply impossible. There is
some point at which the quantum fluctuations in the measuring device will
limit the precision. Furthermore, in a theory of quantum gravity, when we make
the measuring device have more q-bits, in order to endow it with more robust
collective observables, it uses up more of the entropy of the diamond,
eventually creating a black hole. So the claim is that there is no such thing
as a mathematically precise theory of the physics of a local region of space-
time in quantum gravity, and the smaller the region we want to study, the more
ill defined the theory becomes. At a certain point the physics in a ”causal
diamond” will depend on the nature of the device one uses to probe it. In
language that is probably too classical we can say that the device’s
gravitational field will distort the local space-time geometry, so that one no
longer knows what one was probing.

For this reason, it seems to me that the phrase ”the smallest causal diamond
for which we can trust the approximation of Carlip and Solodukhin” represents
a fundamental barrier to the construction of local theories of quantum
gravity, which can only be removed, if at all, by experimental probes. That
is, the models described in the present paper might one day be tested, and
found to apply at large enough scales. Experiments at smaller scale might then
provide guidance about how to construct a more fine grained quantum theory.
Hydrodynamics, as embodied in Einstein’s equations and the QPR, can only take
us so far.

I want to end this rather ambitious paper with an outline of how a general
theory of quantum gravity looks from the perspective advanced here.

  * •

Einstein’s equations with a smooth stress tensor satisfying the null energy
condition are the hydrodynamic equations of the CEP for a general causal
diamond. The c.c. is not included in these equations but is an asymptotic
boundary condition which controls the high entropy limit.

  * •

Generically, causal diamonds have only surface excitations. Bulk excitations
are constrained states. The empty diamond state has maximal entropy. In space-
times which are asymptotically AdS, these statements are all true on scales
below the AdS radius. The full space-time is a tensor network made up of
locally coupled subsystems satisfying these principles.

  * •

Given a background space-time/hydrodynamic flow, we construct a Hilbert bundle
over the space of time-like geodesics on that space-time. A choice of nested
proper time intervals along each geodesic corresponds to a set of nested
causal diamonds. For large enough causal diamonds the modular Hamiltonian of
each diamond is the $L_{0}$ generator of a model of $1+1$ dimensional fermion
fields, labeled by the eigenspinors of the Dirac operator on the holoscreen of
the diamond. The interactions between these fields is determined by a small
number of abelian Thirring couplings, described above. For space-times
conformal to maximally symmetric ones, $L_{0}$ is the quantum realization of
the action of a smooth vector field, restricted to the holographic screen of
the diamond. The flows of that field inside the diamond are time-like and
define a set of inextensible coordinates on the interior. This leads to the
natural conjecture that the time evolution operator along those flows, between
proper times $\tau$ and $\tau+L_{P}$ along the geodesic, is just
$e^{-iL_{0}(\tau+L_{P})}$. The proper times along other time-like trajectories
in the coordinate system are shorter than those along the geodesic, so this
ansatz properly incorporates the ”redshift” between different observers. A
full unitary operator on the Hilbert space over the geodesic is the tensor
product of this operator with the $L_{0}$ of free fermions for those degrees
of freedom that are not causally connected to the diamond $[-T,\tau+L_{P}]$.
For empty diamond states, this ansatz satisfies the QPR. For constrained
states, with localized excitations, the QPR leads, at the level of pictures,
to the Feynman-like diagrams we have discussed in the text. A more
quantitative verification of the QPR for localized states is an important
unsolved problem.

  * •

In the limit of asymptotically flat space-time, all of these models are
exactly supersymmetric. That is, their spectrum contains supergravitons and
their unitary S-matrix is non-trivial and Poincare invariant. This implies
well known results. The maximum space-time dimension is $11$. There are a
variety of exactly stable BPS branes, and in particular, various
compactifications have stable strings with tension much smaller than the
Planck scale, so that a systematic perturbation expansion is possible. Thus,
the Hilbert bundle formulation implies conventional string perturbation
theory.

  * •

The most important features implied by this formalism are those not covered in
this paper: the connection with experiment. The fluctuation formula
$\langle(K-\langle K\rangle)^{2}\rangle=\langle K\rangle$, confirms the
arguments of Verlinde and ZurekVZ123etal  that there should be observable
quantum gravity fluctuations in interferometer experiments. The crucial
calculation that remains to be done is the power spectral density (PSD), for
which only an ad hoc modelpixellon  exists at present. I have also argued for
years that the connection between the finite horizon of dS space and the
breaking of Supersymmetry should lead to insight into the SUSY particle
spectrum. An update on those ideas will appear soonoldideas2 .

  * •

In the appendix I emphasize that the tensor network construction is applicable
to quite general CFTs, and that the local picture of AdS space that it
provides is not a good guide to the way that bulk locality arises at scales
small compared to the AdS radius. $AdS_{d}/CFT_{d-1}$ models with EH duals
have tensor networks whose nodes are Hilbert spaces with entropy that scales
like $R^{d+p}$ where $p\geq 2$. The reduced density matrices in the node
Hilbert spaces, in typical states obtained by acting on the ground state with
local lattice operators near the boundary of the (cut off) network, also have
large entropy, and Polchinski-Susskind scattering states in the ”arena” are
constrained, atypical states of the network field theory. Locality on scales
small compared to the AdS radius arises in much the same way as it does in the
HST formalism.

I want to end this on a personal/philosophical note. Like most working
physicists, I have spent most of my career regarding the problems of the
foundations of quantum mechanics in the ”shut up and calculate” mode. We know
how to use the theory to make stunningly accurate predictions about
observations. That is all we know on earth and all we need to know.
Controversies that arose during my tenure at UCSC forced me to take a closer
look at these issues, as a consequence of which I came away convinced that
Bohr and Heisenberg had it essentially right but needed the more modern work
on decoherence to justify their somewhat mystical language. I’ve explained
this in great detail in my textbook, but in brief: certain quantum systems
with many DOF and ”local” interactions, have a large number of collective
variables. These are operators whose uncertainties scale like inverse powers
of a ratio of microscopic and macroscopic length scale, and whose quantum
probability distributions satisfy linear stochastic equations up to
corrections that are exponentially small in the ratio of the macro to micro
length scale. These variables are the ”classical measuring devices” to which
the Copenhagen interpretation refers.

The CKN bound on the validity of local field theory in a causal diamond,
combined with the apparent inability of black hole dynamics to produce long
lived complex collective variables (fast scrambling/incompressible
hydrodynamics), suggests that the Copenhagen interpretation is not applicable
to the detailed quantum mechanics of gravitational systems. A local detector
can be a complex measuring device, but does not have enough q-bits to store
information about the diamond in which it resides. This implies that
mathematically precise descriptions of the physics measureable by that
detector are in principle impossible to construct, because they are impossible
to test. For those of us living in an asymptotically de Sitter universe this
is somewhat disappointing.

Our mathematically precise theories of AdS space do not help, as discussed in
the appendix. For asymptotically flat space the picture is less clear because
we do not have a definition, outside of perturbation theory or the finite N
BFSS matrix models of the Hilbert space in which the scattering operator is
supposed to be unitary. Conservatively, one might assume that ambiguities
could show up in non-perturbative corrections to the string perturbation
series, which is not Borel summable. As a dramatic example, in lindil  I
proposed modifications of a model of N copies of the minimal Type 0B string
theory in $1+1$ dimensions. These are non-interacting, integrable systems,
which have a string perturbation series that is not Borel summable. Despite
the fact that the low energy scattering resembles linear dilaton gravity,
there are no linear dilaton black holes in the exact fermionic field theory
that gives rise to the perturbation expansion. The modified models do not
effect the perturbation series, but all have meta-stable excitations with the
qualitative properties of black holes. There are a large number of such models
(for large $N$), whose detailed predictions are all different.

##  VII Appendix The Curious Case of Anti de Sitter Space

For researchers whose interest in quantum gravity started in String Theory,
the focus of interest for 25 years has been the AdS/CFT correspondence. There
are many excellent reasons for this, the principle one being that quantum
field theory is our best understood theoretical tool in physics, and this
correspondence reduces quantum gravity to the study of a particular limiting
case of quantum field theory. I have nothing but admiration for the work in
this field and the insights it has brought us, but I believe it has also led
to a number of serious misconceptions.

As indicated in the main text, the most serious of these is a completely
incorrect notion of the emergence of locality in models of quantum gravity.
This error is both a technical one, and also obscures a fundamental conceptual
ambiguity of the notion of local physics in any mathematical theory of quantum
gravity. We will leave the conceptual issue to the end of this appendix and
focus on the technical. The CFT of the AdS/CFT correspondence lives on the
boundary $R\times S^{d-2}$ of the universal cover of $AdS_{d}$, and enjoys the
usual locality properties of field theory on that boundary. One of the
earliest and most celebrated properties of the correspondence is Maldacena’s
scale radius dualitymalda1 . This implies in particular that if we consider
any finite region of $AdS_{d}$ , below some radius $R*$ in a fixed global
coordinate system, we are dealing with some kind of cut off version of the
CFTsusswit .

A very explicit version of the cutoff, which displays the local structure of
$AdS_{d}$ is given by the tensor network (TN) or error correcting code (ECC)
models of the AdS/CFT correspondenceswingleetal . Commonly referred to as ”toy
models” these should instead be thought of as sequences of cut off lattice
field theories, which converge on the CFT, in a manner that explicitly
displays scale/radius duality. The tensor network renormalization group (TNRG)
of Evenbly and Vidalswingleetal  makes this even more explicit. These authors
construct unitary embedding maps using a variational principle, which map the
lattice model on each shell of the tensor network into that on the next shell.
Remarkably, the eigenvalues of the ”Hamiltonians” for small shells, are, in
simple soluble CFTs, close to the dimensions of low dimension operators in the
CFT on the boundary.

Intbwfads  Fischler and the present author interpreted the embedding maps of
the TNRG as the maps between successive nested causal diamonds along the
geodesic that runs through the center of the tensor network. The shells of the
network are the holographic screens of successive nested diamonds. The TN/ECC
picture of CFT thus gives a rather remarkable local picture of the structure
of AdS space. The observant reader will have noticed however that we have
never really used the limiting behavior of CFTs that is required in order to
make holographic duality work. This is particularly evident in the work of
Evenbly and Vidal, whose numerical studies are done for soluble models with
low central charge. The TN/ECC formalism is a property of QFT, independent of
the particular limits which guarantee that a QFT has an ”Einstein-Hilbert (EH)
dual”.

There is a universal feature of all known explicit examples of CFTs with an EH
dual, namely that one always finds that when the AdS radius $R$ is much larger
than all microscopic scales, there are also at least $2$ compact dimensions,
whose Kaluza-Klein modes have masses that scale like $R^{-1}$. Note that it’s
important here that we say all microscopic scales. The free energy of a CFT at
temperature $T$ on a $d-2$ sphere of radius $R$ is

$F=cT(TR)^{d-2}.$ (32)

Comparing this to the Bekenstein-Hawking formula for the free energy of a
stable black hole in $AdS_{d}$ we conclude that the AdS radius is large
compared to the Planck scale whenever the constant $c$ is large. A model has
an EH dual only if, in addition to large $c$ there is a large gap in dimension
between the energy momentum tensor and the typical exponentially growing
density of states of a CFT. In all known examples, the density of states grows
too rapidly to be accounted for by a finite number of fields in $AdS_{d}\times
S^{1}$.

It has been pointed out in various places, by the present author, as well as
by Susskind, that the way to account for this in the tensor network picture is
that the nodes of the tensor network must be large quantum systems, and that
local physics on scales below the AdS radius, is connected to the behavior of
the Hamiltonian within a single node, rather than the local lattice physics of
the network. There are a number of examples of AdS/CFT with tunable
parameters, such that for small values of the parameter a Lagrangian picture
of the CFT is valid. Although this is the opposite limit from the one in which
the model has an EH dual, one can examine the ”single node” physics in this
limit. It is always a highly non-local ”matrix model” (in $1+1$ dimensions
this means a permutation orbifold) in which the AdS and compact dimensions
appear on a roughly equal footing. The model exhibits fast scrambling. On
larger scales in the tensor network we find ballistic scrambling on the $d-2$
sphere, obeying a Lieb-Robinson bound, while scrambling in the compact
dimensions is still fast. These properties are mirrored in the hydrodynamics
of large black holes in AdS space, in the opposite limit where the system is
well described by an EH dual. On scales short compared to the AdS radius,
hydrodynamics is incompressible and identical to the hydrodynamics of flat
space black hole horizons. There is no hydrodynamic propagation of entropy
because information is scrambled at a much faster rate than transverse
momentum. On scales large compared to the AdS radius we find sound modes,
which propagate information ballistically. These dominate the modular
fluctuations of large black holes.

These distinctions also show up in the relation between temperature and infall
time to the singularity for large and small black holes. For small black
holes, the Schwarzschild radius determines the time scale of infall and also
the change in entropy when a particle falls into the black hole. For a large
black hole the particle is falling into one particular node of the tensor
network, so the infall time is determined by the timescale of that subsystem,
which is the AdS radius. The temperature on the other hand is a measure of the
equilibrium response of the entire network, after the ballistic scrambling
process has taken place, and has a very different value.

A final piece of evidence for the same fact is the difference between the
behavior of infinite RT diamonds (AdS Rindler space) and large black holes,
both with regard to the coefficient in their modular fluctuation formulae, and
the question of the existence of sound modes. From the bulk point of view this
might appear a little puzzling, because the infinite RT diamond has a horizon
much larger than the AdS radius, yet it has no sound modes and its modular
fluctuations are like those of a flat space diamond. The resolution of this
puzzle is again found by appealing to scale/radius duality and the tensor
network construction. The volume divergence in the entropy of an RT diamond is
a UV divergence in the CFT, and UV divergences have to do with modes of the
field theory that are very short wavelength on $S^{d-2}$. In the tensor
network picture, these are modes restricted to a single node, and therefore it
is no surprise that they behave like flat space horizon modes. This example
also makes it clear that the sphere at infinity in the CFT is not the same as
the sphere seen by a local observer in a diamond of radius smaller than the
AdS radius. The latter sphere, like the compact dimensions, appears in the
target space of the field theory, rather than the base space over which the
fields are defined.

Thus, the emergence of locality at sub AdS radius scales in AdS models with EH
duals is not connected with the tensor network and entanglement structure of
the boundary CFT.  Instead, we claim the picture that emerges is the same as
the one we abstracted from the zero c.c. limit of dS space. The key
observation is the ”arena” picture of the emergence of the flat space S-matrix
from CFT correlators proposed by Polchinski and by Susskindpolchsuss . These
authors focus on a single causal diamond called the arena whose size is larger
than all microscopic scales but smaller than the AdS radius, and construct CFT
operators which, when a few of them act on the vacuum, create Witten diagrams
that are approximately non-interacting until all the lines enter into the
arena. The resulting CFT correlators are argued to be directly related to flat
space S-matrix elements.

There are two properties that follow from this picture. The first assumes the
CEP, namely that the arena is a finite dimensional quantum subsystem of the
CFT Hilbert space. By Page’s theorempage  this would seem to imply that for
most states of the CFT, the density matrix of the subsystem is maximally
uncertain. It’s unclear whether the CEP is true in the exact CFT, but if we
identify the arena with the central node in a tensor network approximation to
the cut off CFT then it is certainly true and the tensor network shows us that
Page’s argument is correct for this system. For a finite tensor network the
probability that the density matrix is maximally uncertain is less than one.
It’s clear that a lot of care has to be taken when applying Page’s theorem to
infinite dimensional systems.

The second property was first pointed out intbwfads2  and has to do with the
origin of Minkowski amplitudes with large numbers of soft massless particles.
Because of the gap in the AdS energy spectrum for massless particles, one
cannot construct Polchinski-Susskind operators that insert an arbitrary number
of massless particles into the arena, without creating black holes inside the
arena. Instead, amplitudes with large numbers of Minkowski massless particles
arise from AdS correlators of the form

$\langle PS(1)PS(2)PS(3)PS(4)O(y_{1})\ldots O(y_{n})\rangle_{c},$ (33)

where the $4$ Polchinski-Susskind operators prepare a hard $2\rightarrow 2$
scattering process in the arena, and the other operators create particles
whose amplitude to be in the arena (as computed from Witten diagrams) is very
small because they are equally probable to be anywhere in AdS space. We
conjecture that it is linear combinations of such amplitudes that converge to
S-matrix amplitudes for emission and absorption of arbitrary numbers of soft
particles with arbitrarily low energy in the $R\rightarrow\infty$, Minkowski
limit. It’s the non-perturbative behavior of those amplitudes for large $n$,
which determines whether the resulting states are normalizable in Fock space.
At the moment, it does not appear that the AdS/CFT correspondence sheds
additional light on these questions, but the fact that it leads to the same
questions as the approach from dS space suggests that the issues raised in the
text are real.

We conclude that the issue of local physics on scales smaller than the AdS
radius involves a cut-off version of the CFT. This means that it is not
encoded unambiguously in CFT correlators, for many different cut-off systems
converge to the same CFT. In the text we have pointed out a possible deep
reason for this ambiguity. The phrase ”physics in a causal diamond of finite
area” is shorthand for ”a mathematical theory that explains the results of
measurements performed by a detector traveling along a time-like trajectory
between the tips of the diamond”. However, because the information gathering
capacity of a detector is limited by the CKN bound, no mathematically precise
theory is testable, so some ambiguity in the definition of local physics is
inevitable. In the real world we could (in principle, though probably not in
practice) probe this ambiguity experimentally by showing that the results of
experiments depended on the type of detector used. In the AdS/CFT paradigm the
ambiguity seems to be related to the well know insensitivity of continuum
field theory to UV details. This means that the question of precisely what
goes on in finite area causal diamonds is not well defined in AdS/CFT.
Different theorists can choose different sequences of cut off models, all of
which converge to the same CFT, but differ in their predictions for local
physics in finite area diamonds.

##  VIII Appendix II Some Remarks about Black Holes in Minkowski Space

Large black holes in AdS space are conventional thermal ensembles. Their
modular Hamiltonian is just the Hamiltonian of the CFT. Black holes in
Minkowski space have negative specific heat. One cannot relate their thermal
properties to their modular Hamiltonian. Indeed, the black hole entropy
formula would predict a energy level spacing of order $e^{-S}$, while the
width of black hole levels (the inverse lifetime) scales like a power of the
entropy. The level spacing of the modular Hamiltonian has nothing to do with
the energy of the black hole as measured by an observer at infinity. We have
conjectured instead that it’s related to the spacing of levels seen by a near
horizon detector that measures the operator $L_{0}$.

We should note that the classical properties of black holes beg for an
interpretation as some kind of universal equilibrium ensemble. They are
independent of the black hole’s formation history and small perturbations of a
black hole just lead us to another black hole differing in a small number of
macroscopic parameters. Black holes also admit hydrodynamic flows, which, in
the Minkowski case, do not allow for entropy transport. Our conjecture is a
model for this equilibrium state.

Finally, we note again that negative specific heat gives the simplest
resolution of the so called ”firewall paradox”. It implies that a large amount
of entropy is created when a localized, low entropy, object is dropped onto a
large black hole. The explanation for this is that, in the Hilbert space of
the final state black hole, there must have been a large number of frozen
q-bits, which must be activated in order for the object to come into
equilibrium with the rest of the black hole q-bits. If the time scale (in
proper time of the infalling object) for activation of these q-bits is the
Schwarzschild radius, then the infalling object will not begin to feel the
effect of being inside the black hole for that amount of time. Furthermore,
the consequence of equilibration is that more and more q-bits with which the
object could have interacted if it had been in empty space, will instead be
coming into equilibrium. Using the connection between entropy and area, the
effective area ”felt” by the object shrinks.

Acknowledgments
I would like to thank W. Fischler for collaboration over more than 20 years on
material that led to this paper. Thanks also to B. Fiol for the clue to
understanding localized states in quantum gravity. Special thanks to K. Zurek
for showing me the work of the Verlindes and of Solodukhin, which were so
important to the formulation of this paper, and for insisting on the
universality of the modular fluctuation formula. Thanks also to E. Witten and
E. and H. Verlinde for pointing out an error in my description of the
Verlindes’ work and the way to correct it. This work was supported in part by
the U.S. Department of Energy under grant DE-SC0010008

## References

  * (1) T. Jacobson, “Thermodynamics of space-time: The Einstein equation of state,” Phys. Rev. Lett. 75, 1260-1263 (1995) doi:10.1103/PhysRevLett.75.1260 [arXiv:gr-qc/9504004 [gr-qc]].
  * (2) S. Carlip,“Black hole entropy from conformal field theory in any dimension,” Phys. Rev. Lett. 82, 2828-2831 (1999) doi:10.1103/PhysRevLett.82.2828 [arXiv:hep-th/9812013 [hep-th]].
  * (3) S. N. Solodukhin, “Conformal description of horizon’s states,” Phys. Lett. B 454, 213-222 (1999) doi:10.1016/S0370-2693(99)00398-6 [arXiv:hep-th/9812056 [hep-th]].
  * (4) T. Banks and K. M. Zurek, “Conformal description of near-horizon vacuum states,” Phys. Rev. D 104, no.12, 126026 (2021) doi:10.1103/PhysRevD.104.126026 [arXiv:2108.04806 [hep-th]].
  * (5) H. L. Verlinde and E. P. Verlinde, “Scattering at Planckian energies,” Nucl. Phys. B 371, 246-268 (1992) doi:10.1016/0550-3213(92)90236-5 [arXiv:hep-th/9110017 [hep-th]]; E. P. Verlinde and H. L. Verlinde, “High-energy scattering in quantum gravity,” Class. Quant. Grav. 10, S175-S184 (1993) doi:10.1088/0264-9381/10/S/018
  * (6) A. Connes, Noncommutative Geometry, ISBN-13 978-0121858605, First Edition , Academic Press, 1995.
  * (7) T. Banks and J. Kehayias, “Fuzzy Geometry via the Spinor Bundle, with Applications to Holographic Space-time and Matrix Theory,” Phys. Rev. D 84, 086008 (2011) doi:10.1103/PhysRevD.84.086008 [arXiv:1106.1179 [hep-th]].
  * (8) A. G. Cohen, D. B. Kaplan and A. E. Nelson, “Effective field theory, black holes, and the cosmological constant,” Phys. Rev. Lett. 82, 4971-4974 (1999) doi:10.1103/PhysRevLett.82.4971 [arXiv:hep-th/9803132 [hep-th]].
  * (9) N. Blinov and P. Draper, “Densities of states and the Cohen-Kaplan-Nelson bound,” Phys. Rev. D 104, no.7, 076024 (2021) doi:10.1103/PhysRevD.104.076024 [arXiv:2107.03530 [hep-ph]]; T. Banks and P. Draper, “Remarks on the Cohen-Kaplan-Nelson bound,” Phys. Rev. D 101, no.12, 126010 (2020) doi:10.1103/PhysRevD.101.126010 [arXiv:1911.05778 [hep-th]].
  * (10) R. Haag, in Brandeis Lectures on Elementary Particles and Quantum Field Theory, Volume 2, 1970; S. Deser, editor. MIT Press.
  * (11) R. D. Sorkin, “1983 paper on entanglement entropy: ”On the Entropy of the Vacuum outside a Horizon”,” [arXiv:1402.3589 [gr-qc]].
  * (12) M. Srednicki,“Entropy and area,” Phys. Rev. Lett. 71, 666-669 (1993) doi:10.1103/PhysRevLett.71.666 [arXiv:hep-th/9303048 [hep-th]].
  * (13) C. G. Callan, Jr. and F. Wilczek, “On geometric entropy,” Phys. Lett. B 333, 55-61 (1994) doi:10.1016/0370-2693(94)91007-3 [arXiv:hep-th/9401072 [hep-th]].
  * (14) L. Susskind and J. Uglum, “Black hole entropy in canonical quantum gravity and superstring theory,” Phys. Rev. D 50, 2700-2711 (1994) doi:10.1103/PhysRevD.50.2700 [arXiv:hep-th/9401070 [hep-th]].
  * (15) T. Jacobson, “Black hole entropy and induced gravity,” [arXiv:gr-qc/9404039 [gr-qc]].
  * (16) W. Fischler and L. Susskind, “Holography and cosmology,” [arXiv:hep-th/9806039 [hep-th]].
  * (17) R. Bousso, “The Holographic principle for general backgrounds,” Class. Quant. Grav. 17, 997-1005 (2000) doi:10.1088/0264-9381/17/5/309 [arXiv:hep-th/9911002 [hep-th]]; R. Bousso, “Holography in general space-times,” JHEP 06, 028 (1999) doi:10.1088/1126-6708/1999/06/028 [arXiv:hep-th/9906022 [hep-th]]; R. Bousso, “A Covariant entropy conjecture,” JHEP 07, 004 (1999) doi:10.1088/1126-6708/1999/07/004 [arXiv:hep-th/9905177 [hep-th]].
  * (18) S. Carlip, “Statistical mechanics and black hole thermodynamics,” Nucl. Phys. B Proc. Suppl. 57, 8-12 (1997) doi:10.1016/S0920-5632(97)00348-4 [arXiv:gr-qc/9702017 [gr-qc]];
  * (19) S. Carlip,“Statistical mechanics and black hole entropy,” [arXiv:gr-qc/9509024 [gr-qc]]; S. Carlip, “The Statistical mechanics of the (2+1)-dimensional black hole,” Phys. Rev. D 51, 632-637 (1995) doi:10.1103/PhysRevD.51.632 [arXiv:gr-qc/9409052 [gr-qc]].
  * (20) A. Strominger, JHEP 02, 009 (1998), arXiv:hep-th/9712251.
  * (21) J. D. Brown, M. Henneaux, Commun. Math. Phys. 104, 207 (1986).
  * (22) T. Banks, P. Draper and S. Farkas, “Path Integrals for Causal Diamonds and the Covariant Entropy Principle,” Phys. Rev. D 103, no.10, 106022 (2021) doi:10.1103/PhysRevD.103.106022 [arXiv:2008.03449 [hep-th]].
  * (23) G. W. Gibbons and S. W. Hawking,“Euclidean quantum gravity,” G. W. Gibbons and S. W. Hawking, “Cosmological Event Horizons, Thermodynamics, and Particle Creation,” Phys. Rev. D 15, 2738-2751 (1977) doi:10.1103/PhysRevD.15.2738 G. W. Gibbons and S. W. Hawking, “Action Integrals and Partition Functions in Quantum Gravity,” Phys. Rev. D 15, 2752-2756 (1977) doi:10.1103/PhysRevD.15.2752
  * (24) T. Banks and P. Draper, “Comments on the entanglement spectrum of de Sitter space,” JHEP 01, 135 (2023) doi:10.1007/JHEP01(2023)135 [arXiv:2209.08991 [hep-th]].
  * (25) T. Banks,“Fluctuating Hydrodynamics and Wormholes,” [arXiv:2203.08855 [hep-th]].
  * (26) T. Banks and A. Lucas, “Emergent entropy production and hydrodynamics in quantum many-body systems,” Phys. Rev. E 99, no.2, 022105 (2019) doi:10.1103/PhysRevE.99.022105 [arXiv:1810.11024 [cond-mat.stat-mech]].
  * (27) M. Kac, ”On Distributions of Certain Wiener Functionals”, Transactions of the American Mathatical Society. 65 (1), doi:10.2307/1990512.