Generalized Entanglement Capacity of de Sitter Space

Tom<EMAIL_ADDRESS>and Patrick<EMAIL_ADDRESS>

(a)NHETC and Department of Physics &

Rutgers University, Piscataway, NJ 08854-8019

(b)Illinois Center for Advanced Studies of the Universe &

Department of Physics, University of Illinois, Urbana, IL 61801

Near horizons, quantum fields of low spin exhibit densities of states that
behave asymptotically like 1+1 dimensional conformal field theories. In
effective field theory, imposing some short-distance cutoff, one can compute
thermodynamic quantities associated with the horizon, and the leading cutoff
sensitivity of the heat capacity is found to equal to the leading cutoff
sensitivity of the entropy. One can also compute contributions to the
thermodynamic quantities from the gravitational path integral. For the
cosmological horizon of the static patch of de Sitter space, a natural
conjecture for the relevant heat capacity is shown to equal the Bekenstein-
Hawking entropy. These observations allow us to extend the well-known notion
of the generalized entropy to a generalized heat capacity for the static patch
of dS. The finiteness of the entropy and the nonvanishing of the generalized
heat capacity suggests it is useful to think about dS as a state in a finite
dimensional quantum gravity model that is not maximally uncertain.

##  1 Introduction

The quantum theory of de Sitter (dS) space remains obscure. We do not know the
nature or even the quantity of the microscopic degrees of freedom, or their
interactions.

In [2, 1] it was proposed that the Gibbons-Hawking entropy of dS space could
be interpreted as the logarithm of the dimension of a finite-dimensional
quantum system describing a static patch of the spacetime. This hypothesis
implies that empty dS space corresponds to the maximally uncertain density
matrix of the quantum system. Within semiclassical physics, it is supported by
the Schwarzschild-de Sitter (SdS) entropy formula, which shows that objects
localized near a geodesic reduce the entropy, and by the fact that generic
localized excitations melt into the near-horizon region of any given static
patch in a time of order $R_{dS}$. Furthermore, attempts to introduce higher-
entropy initial states in the remote past of the global dS manifold lead to
singular space-times111This was originally a conjecture made by TB and was
later proven in [3]. rather than future asymptotically-dS space.

The “maximum entropy” proposal is simple and compelling, and it has received
support of a kind from recent progress in the theory of operator algebras. It
would be interesting to establish whether it is strictly true, or only holds
in some approximation. A natural quantity to examine is the second cumulant of
the modular Hamiltonian $K$, where $\rho=e^{-K}$ is the normalized density
matrix of the microscopic state corresponding to empty dS. The entropy
satisfies $S=\langle K\rangle$. For a state of maximum entropy,
$\rho=\mathbf{I}/\dim H$, and the second cumulant (also referred to as the
“capacity of entanglement” [4, 5, 6, 7]) vanishes, $\langle(K-\langle
K\rangle)^{2}\rangle=0$. Thus the magnitude of the modular fluctuation can
discriminate the maximum entropy state from other possibilities.

This note is a follow-up to previous work [8], in which it was argued that the
semiclassical state of empty de Sitter space need not correspond to a
microscopic maximum entropy state. Random projections also reproduce the
entropy deficit properties of the Schwarzschild-de Sitter (SdS) geometry, for
a general class of density matrices, not just the maximally uncertain one.
Furthermore there exist density matrices with the properties that $\langle
K\rangle=\log\dim H+{\cal O}(1)$ and $\langle(K-\langle
K\rangle)^{2}\rangle\sim\langle K\rangle$.

Here we consider two other probes of the modular fluctuation in the dS static
patch. The first is the behavior of effective quantum field theory (EFT) on
the static patch background (Sec. 2). EFTs are presumed to have a physical
short-distance cutoff. Without knowing the UV completion, one can estimate the
sensitivity of quantities like the entropy and capacity of entanglement to the
physical UV scales by putting general, unphysical cutoffs on the EFT.

The second approach we consider is the semi-classical Euclidean functional
integral for gravity, where we make a conjecture in Sec. 3 about how one
should identify and vary the replica index. A modular fluctuation analog to
the classical Bekenstein-Hawking entropy can be computed from a derivative
with respect to the replica index. This kind of semi-classical replica trick
has been successful in a wide variety of contexts in the AdS/CFT
correspondence. Our proposal remains consistent with the idea that states with
localized matter in the bulk correspond to constrained states, essentially
because thermal states may either be viewed as minimizing the free energy at
fixed temperature, or maximizing the entropy with a constraint on the
expectation value of the energy.

Both EFT and semiclassical gravity are consistent with a proportionality of
the form $\langle(K-\langle K\rangle)^{2}\rangle=a\langle K\rangle$, with
coefficient $a$ of order one (and in fact equal to one in the cases considered
here.) It is straightforward to explain why the SdS argument described above
did not “see” the large modular fluctuation, which we also discuss in Sec. 2.
We may define a “generalized capacity of entanglement” $C_{gen}$ as the sum of
the EFT and leading-order semiclassical gravity contributions, and in cases
where the value of $a$ is the same in both terms, $C_{gen}$ is proportional to
the generalized entropy.

In Sec. 4 we discuss the question in the context of operator algebras and the
kinds of modifications to the crossed product construction needed to produce a
large nonzero $\langle(K-\langle K\rangle)^{2}\rangle$. In Sec. 5 we conclude
with a discussion of the relation between the modular fluctuation and other
ideas in static patch holography.

##  2 Effective Field Theory

Effective quantum field theory on a static patch of rigid dS, or any spacetime
with a horizon, contains a density of near-horizon states that is sensitive to
the ultraviolet cutoff. For free theories the sensitivity can be easily
calculated with a brick wall regulator [9], or covariant regulators like
Pauli-Villars [10] or character expansions [11]. For example, for $N$ free
bosons in $D$ spacetime dimensions on a spacetime
$ds^{2}=-fdt^{2}+f^{-1}dr^{2}+r^{2}d\Omega_{D-2}$ one finds a density of near-
horizon single-particle modes of the form

$\displaystyle
g(\omega)=N\left(\frac{r_{H}}{\rho_{c}}\right)^{D-2}X(r_{H}T_{H})+\dots$ (1)

Here $r_{H}$ is a horizon radius, $f(r_{H})=0$; $T_{H}$ is the surface gravity
temperature with respect to the timelike Killing vector $\partial_{t}$,
$T_{H}=|f^{\prime}(r_{H})|/4\pi$; $X$ is a function; $\rho_{c}$ is a short-
distance cutoff; and $\dots$ denotes terms that are subleading for small
$\rho_{c}$. The key point is that the single-particle density of near-horizon
states is independent of energy $\omega$. This sector of the theory behaves
like a 1+1-dimensional CFT of $N\left(\frac{r_{H}}{\rho_{c}}\right)^{D-2}$
free bosons. The factor $\left(\frac{r_{H}}{\rho_{c}}\right)^{D-2}$ counts
angular modes. In the Hartle-Hawking (HH) or Bunch-Davies (BD) state, these
modes are thermally excited at a temperature $T_{H}$ and make a large
contribution to the thermal entropy of order $T_{H}g(T_{H})\sim
A/\rho_{c}^{D-2}$. However, they are also strongly coupled to gravity, in the
sense that one cannot excite very many of them away from the HH/BD state at
once before gravitational backreaction becomes large [12]. One is left with
the unsatisfactory conclusion that there is a large collection of near-horizon
modes, some but not all of which are perhaps well modeled by effective quantum
field theory on a rigid background.

In any case, we cannot really say that just because low-entropy bulk
excitations lower the cosmological horizon area, the state describing empty de
Sitter in the relevant microscopic theory of quantum gravity is of maximum
entropy. The near-horizon degrees of freedom do not have to be in this state.

Suppose, for example, that we could neglect the backreaction argument of [12]
and take effective QFT seriously up to a cutoff scale of order the Planck
length, $\rho_{c}\sim\ell_{p}$. At inverse temperature $\beta$, because the
density of states is independent of energy, the near-horizon states contribute
to the partition function

$\displaystyle\Delta(\beta
F)\propto\int\log\left(1-e^{-\beta\omega}\right)d\omega$ (2)

which on dimensional grounds scales as $1/\beta$. ($\beta$ must be kept
independent of $T_{H}$ in the density of states until we are done taking
$\beta$ derivatives.) So the thermal entropy

$\displaystyle S=\beta^{2}\partial F/\partial\beta$ (3)

and the heat capacity

$\displaystyle C=\beta^{2}\partial^{2}(-\beta F)/\partial\beta^{2}$ (4)

are equal to each other and proportional to $A/G_{N}$. In a thermal state,

$\displaystyle C=\langle(K-\langle K\rangle)^{2}\rangle$ (5)

so the large heat capacity indicates that the state is far from the maximally
uncertain density matrix.

Furthermore, since $\langle K\rangle=S$, $C=S$ implies

$\displaystyle\langle(K-\langle K\rangle)^{2}=a\langle
K\rangle\;,\;\;\;\;\;a=1.$ (6)

The conclusion that $a=1$ actually holds whenever the near-horizon physics is
approximately that of a 1+1-dimensional CFT with a cutoff, which is the case
for free massless scalars and spin-$1/2$ fields, among others. $C$ and $S$ are
both sensitive to the UV cutoff, but the ratio, $a$, is finite. In general the
value of $a$ may depend on the details of the QFT.

One way of understanding the apparent $1+1$ dimensional nature of the near
horizon density of states is the observation of [13] and [14] that the near
horizon geometry of dS space, similarly to non-extremal causal diamonds, has
an approximate Virasoro symmetry. Any regularization procedure that respects
this symmetry will appear to have the density of states of a $1+1$ CFT. Of
course, this might be misleading since any higher dimensional field theory
will produce an infinite number of two dimensional free fields. This of course
reflects the fact that the entanglement entropy has a power law UV divergence,
with a power that depends on the dimension of space-time. Our point however is
that the same divergence shows up in both the entanglement entropy and the
capacity of entanglement, and that the ratio is finite, with $a=1$.

An alternative regulator for the entanglement entropy of QFT has been proposed
by [15]. This elegant method preserves the entire conformal group of the
higher dimensional field theory, but explicitly breaks the near horizon
Virasoro symmetry. It converts the UV divergence of the entanglement entropy
into the volume divergence of the thermal entropy on a hyperbolic space. Again
the ratio between the entanglement entropy and the capacity of entanglement is
finite, but now it depends on the particular higher dimensional CFT one is
studying. Curiously, whenever that CFT has an Einstein-Hilbert dual in the
AdS/CFT correspondence, the CHM procedure predicts $a=1$, confirming the
calculation of [16] for boundary anchored Ryu-Takayanagi diamonds in quantum
gravity.

##  3 Euclidean Gravity and the Replica Trick

The contribution to the entropy from near-horizon QFT modes, $S_{QFT}$, has a
gravitational analog $A/4G_{N}$. The leading UV-cutoff sensitivity of
$S_{QFT}$ generally coincides with the perturbative renormalization of
$G_{N}$, so that the sum of the two contributions, at leading order, is just
$A/4G_{N}$ with the renormalized Newton constant. We would like to know if a
similar picture holds for the modular fluctuation / heat capacity, for which
we need a computation of this quantity in pure Euclidean gravity.

One approach is to use replica methods. Let the microscopic density matrix of
empty dS space be $\rho=e^{-K}$, with ${\rm Tr}(\rho)=1$, and define
$\rho_{n}=e^{-nK}/{\rm Tr}(e^{-nK})$. Then

$\displaystyle S_{n}\equiv-{\rm Tr}\left(\rho_{n}\log\rho_{n}\right)$ (7)

The modular fluctuation can be shown to satisfy

$\displaystyle\langle(K-\langle K\rangle)^{2}\rangle=-S^{\prime}_{1}.$ (8)

The angle brackets on the left-hand side denote expectation values with
respect to $\rho$.

Most applications of replica methods to quantum gravity begin with a boundary
in Euclidean signature. First one constructs $n$ copies of the boundary, and
the boundary data, glued together in a chain. Then, for each $n$, one looks
for smooth interior solutions to the semiclassical gravitational field
equations [17]. The solution ${\cal M}(n)$ of the lowest action ${\cal I}(n)$
provides an approximation to the $n$-fold partition function of the
microscopic quantum gravity theory, $-\log Z(n)\approx{\cal I}(n)$. This is
because the boundary is where a nongravitational holographic dual theory
lives, and replicating the boundary geometry $n$ times corresponds to
constructing $Z(n)$ in the dual theory [17].

It is not immediately obvious how to apply similar techniques to learn about
quantum gravity “in de Sitter space.” First, the Euclidean continuation of de
Sitter is a sphere, which has no boundary. Second, if a holographic dual
description lives anywhere, it probably lives at the horizon of the static
patch, and the Euclidean continuation of a null surface is of codimension two.
One might attempt to place some boundary data on a “stretched horizon” a small
fixed distance above the horizon. There is a problem with the most naïve
application of this idea, however, which will be discussed below.

Nonetheless it is widely believed that there is some microscopic theory of
quantum gravity with a density matrix $\rho$, for which various properties of
$\rho$ – like the von Neumann entropy – are computed in some approximation by
the Euclidean de Sitter saddle point of the gravitational path integral. One
would like to know whether other suitable saddle points compute $\rho_{n}$.
Here we make a conjecture based on the Schwarzschild de Sitter (SdS)
geometries.

The SdS metric in $D$ space-time dimensions is

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega_{D-2}^{2},$ (9)
$f(r)=1-\left(\frac{r_{s}}{r}\right)^{D-3}-\frac{r^{2}}{L^{2}}.$ (10)

$f(r)$ has two positive zeros, and for $r_{s}\ll L$ one is approximately at
$r_{b}\approx r_{s}$ and the other at $r_{c}\approx
L-\frac{r_{s}^{D-3}}{2L^{D-4}}$. As a consequence, the effect of inserting a
localized mass distribution near the geodesic, whether a black hole or a
“star,” is to shrink the area of the cosmological horizon. Thus, according to
the Bekenstein-Hawking-Gibbons law, the entropy of empty dS space is also
reduced. This was interpreted [2, 1, 18] as an explanation of the Gibbons-
Hawking temperature of dS space as a purely entropic effect, arising from a
constraint on the holographic degrees of freedom on the horizon.
Semiclassically, the constraint takes the form of a projection onto fields of
fixed quasilocal energy [19, 20, 21]. This is one bit of an enormous amount of
evidence that most of the quantum states of dS space are localized near the
horizon, relatively inaccessible to experimental probes, and have static
energies of order $1/L$, as indicated by the redshift in static coordinates.

As described above, the conventional idea of the gravitational replica trick
in AdS space is to find a Euclidean manifold, which is a smooth solution of
Einstein’s equations and has periodicity around a thermal cycle on the
boundary that is $n$ times as large as the original manifold on which the
boundary field theory lives. Our proposal for dS space is motivated by the
fact that the Euclidean SdS manifold changes the periodicity of the Euclidean
static time coordinate as it changes the entropy. We thus define the replica
index as the ratio of cosmological horizon surface gravity temperatures (for
$r_{s}\ll L$),

$\displaystyle n\equiv 1+\frac{D-2}{2}\left(\frac{r_{s}}{L}\right)^{D-3}.$
(11)

The entropy of the cosmological horizon of SdS is

$\displaystyle
S=S_{dS}\left[1-\frac{D-2}{2}\left(\frac{r_{s}}{L}\right)^{D-3}\right].$ (12)

Thus, with the identification of $r_{s}/L$ and the replica index in Eq. (11),
and identifying the SdS entropy in Eq. (12) with $S_{n}$ in Eq. (7), we find

$\displaystyle S_{n}$ $\displaystyle=S_{dS}[1-(n-1)],$
$\displaystyle-S^{\prime}_{1}$ $\displaystyle=S_{dS}.$ (13)

Thus we find $a=1$, independent of $D$. This extends the semiclassical
approach of [16] for RT diamonds in AdS to dS, and is in agreement with the
results of [22] (arrived at by other means, extrapolating the ideas of [13,
14]). Note that we could have multiplied our definition of $n$ by any function
$h(r_{s}/L)$ such that $h(1)=1$ and $h^{\prime}(1)=0$ without changing the
result. This is in keeping with expectations from AdS/CFT that higher order
fluctuations do not necessarily take on a universal form.

Let us address some technical points. The Euclidean SdS manifold is singular
on the continued black hole horizon. If we add any sort of matter fields that
produce a spherically symmetric “star” solution, the singularity disappears.
We can deal with it simply by cutting out an artificial boundary around the
singularity and imposing Brown-York microcanonical boundary conditions[25, 23,
24], so that nothing depends on the details of what is going on near the
geodesic, and the on-shell action simply computes the cosmological horizon
entropy. Of course, since we take $r_{s}\rightarrow 0$ at the end of the
calculation, this is not a serious issue in any event.

A more subtle point is the normalization of the Killing vector $\partial_{t}$
in a uniform, $r_{s}$ independent way. The traditional way to do this [26]
extrapolates the static coordinates to an asymptotic region far outside the
static patch, where it is “naturally” normalized to $1$. We prefer to think
about the $L\rightarrow\infty$ limit, where $r_{s}$ is naturally defined in
terms of the asymptotic geometry, and the derivative of the entropy with
respect to $r_{s}$ has a well defined normalization. Another point of view is
to regard the mass parameter $M$ as distinguished by the fact that it is the
asymptotic energy in the $L\rightarrow\infty$ limit, and the surface gravity
temperature of the cosmological horizon with the usual normalization is
distinguished by the SdS first law $dS_{c}/dM=-1/T_{c}$.

Let us briefly explore other possible replica tricks using the SdS geometry.
We could introduce a boundary at some radius $r_{0}$ and try to define

$\displaystyle
n\overset{?}{=}\frac{\beta_{SdS}\sqrt{f_{SdS}(r_{0})}}{\beta_{dS}\sqrt{f_{dS}(r_{0})}}.$
(14)

where the $\beta$’s are the inverse cosmological horizon temperatures. This is
essentially taking the ratio of the local (Tolman) temperatures of the
artificial boundary at $r_{0}$, as one might imagine one should do if “the
quantum system was sitting at this boundary.” It is easy to see that this
proposal is not sensible, because the putative modular fluctuation computed in
this manner is negative for all $0<r_{0}<L$.

Similarly, motivated by the idea that “the hologram lives on the stretched
horizon,” we might try to perform a replica trick around a circle a Planck
distance from the horizon,

$\displaystyle
n\overset{?}{=}\frac{\beta_{SdS}\sqrt{f_{SdS}(\ell_{p})}}{\beta_{dS}\sqrt{f_{dS}(\ell_{p})}}.$
(15)

This is similar to the previous case, except we hold the stretched horizon at
a fixed proper radius as we vary the replica index. Because the local
temperature close to the horizon is nearly infinite, the heat capacity is
small; however, it is still negative. One finds
$-S^{\prime}_{1}\approx-2\pi\ell_{p}^{2}$, and again the hypothesized map from
$r_{s}$ to the replica index is inconsistent.

We are encountering the familiar fact that if we think of any of these
temperatures in the conventional way, then lowering the temperature increases
the entropy, so that the specific heat is negative. The interpretation of
energy in terms of constraints on a large number of degrees of freedom in a
high entropy system explains this apparent paradox222Parenthetically, the same
mechanism can also be applied to the negative specific heat of black holes for
non-negative c.c... The relevant constraint fixes the expectation value of the
energy. With a fixed $\langle E\rangle$ constraint, the maximum entropy
density matrix is the canonical ensemble at finite temperature $T=dS/d\langle
E\rangle$. Our identification of the replica index corresponds to the ansatz
that the energy of the microscopic theory describing the horizon is
proportional to $-M$. In this way the constraint varies the surface gravity
temperature, or equivalently the replica index. Then $-S^{\prime}_{1}$ is
equivalent to $T_{c}dS_{c}/dT_{c}=-dM/dT_{c}$, and the $-M$ can be thought of
as removing energy from the cosmological horizon degrees of freedom.

We may define a generalized heat capacity which is the sum of the heat
capacity of quantum fields and the contribution from gravity,

$\displaystyle C_{gen}=C_{QFT}-S^{\prime}_{1}.$ (16)

When the value of $a$ obtained in effective field theory matches the value of
$a$ obtained from semiclassical gravity, the generalized heat capacity so
defined is proportional to the generalized entropy, $C_{gen}=aS_{gen}$. Thus
it shares the properties of the generalized entropy, for example, in many
cases the cutoff dependence of the quantum field contribution may be absorbed
into the renormalized Newton constant. This can be anticipated from the
renormalization of the Euclidean gravitational effective action on smooth
manifolds.

##  4 Comments on Operator Algebras

The construction of Type $II$ crossed product algebras from the Type $III_{1}$
algebras of local quantum field theory and their modular automorphism groups
has no a priori connection to gravitational physics. To more easily appreciate
this fact, imagine that relativistic quantum field theory had been discovered
long before general relativity and no one had proposed a connection between
gravitation and the geometry of space-time. Nonetheless, the Euclidean
continuation of QFT would be well-known, and physical mathematicians would
have appreciated the possibility and utility of putting QFTs on a variety of
Riemannian manifolds. One can put a QFT on Euclidean Schwarzschild and observe
that the Lorentzian continuation leads to the Hartle-Hawking state on the
Kruskal manifold. Noting that observables on the right causal patch have
thermal expectation values, as expected from the Euclidean periodicity, naive
field theorists will interpret this as a geometric realization of the Thermo-
field Double construction. More sophisticated theorists will note that the
algebra of the right asymptotic region is Type $III_{1}$ and has no density
matrices. The crossed product construction can be viewed as a way of
justifying this purely field theoretic conundrum of an apparent geometric
picture of the TFD. It leads to a way of calculating thermal traces, but not
the total entropy, because the trace in Type $II$ algebras has a
multiplicative ambiguity. The same construction is applicable to QFT on any
Riemannian manifold whose Lorentzian continuation has two causally
disconnected diamonds that meet in a surface of co-dimension 2. The point that
the modular crossed product is not intrinsically gravitational, and that the
crossed product should be interpreted as a generic tool for regulating the
entanglement entropy in arbitrary QFTs, was emphasized in [27, 28].

Indeed, when one tries to calculate the entropy in any of these situations,
one finds a divergence [29] coming from short wavelength states with very
small eigenvalues of the would-be modular Hamiltonian of the Type $III$
algebra. Going back to the Riemannian calculation, this divergence can be
canceled by local counterterms involving the background geometry. The most
important of these [30] is proportional to $\int\sqrt{g}R$ and associated
boundary terms [31, 32]. If one interprets the Euclidean path integral as a
free energy, with the time periodicity interpreted as a temperature, then
varying the temperature to compute the entropy introduces singularities on the
fixed surface of the time translation generator. One way to regulate it is to
introduce a new boundary near that surface with associated GHY term. Thus this
counterterm tells us that both the entropy and the specific heat are cutoff-
dependent and proportional to each other. As reviewed above, direct Hilbert
space calculations lead to the same conclusion.

In non-gravitational QFT, the crossed product construction leads to Type
$II_{\infty}$ algebras. In general [33] the entropies assigned to the naive
density matrix induced on a causal diamond by a global field theory state will
have infinite entropy in this algebra, but regulated, finite entropy, density
matrices can be defined, which allow one to make rigorous constructions of
relative entropies and prove a number of folk theorems in the QFT literature.
The finite entropies have a finite ambiguity stemming from the multiplicative
ambiguity of the trace in Type $II$ algebras. Using the regulator in [33] the
entropy of QFT density matrices for local regions diverges like ${\rm ln}\
\epsilon$, while the modular fluctuation diverges like $\epsilon^{-2}$, so the
ratio we have denoted by “$a$” diverges. This disagrees with all standard
cutoff field theory calculations and with the CHM/Perlmutter prescription for
CFT. We are not sure whether this is generic or a feature of the particular
regularization scheme introduced in[33], or whether one should use the
ambiguity in the entropy to add a term of order $\epsilon^{-2}$ to it.

The re-interpretation of the cross-product construction as a partial solution
to the gravitational constraint equations in the bulk [34, 35, 36, 37, 38]
does not change any of these facts about the relation of cross products to
renormalization of Euclidean path integrals for non-gravitational QFT in
background geometries.333We also note that it is important to distinguish the
role of the crossed product in systems with genuine symmetries, e.g. in the
presence of boundaries, from its role in enforcing constraints [39]. The
observations of[40] show that within the AdS/CFT correspondence, the TFD
interpretation of the AdS black hole geometries are rigorously correct. In
that context, for large $AdS_{d}$ black holes we know that the modular
fluctuation is proportional to the expectation value of the entropy, with
$a=d-2$. In[22] the authors explained, using the tensor network model of AdS,
how this was consistent with $a=1$ on scales small compared to the AdS radius.

We conclude that the crossed product construction applied to QFT in curved
space-time eliminates the largest contribution to the entropy and on
geometries that have a naive TFD interpretation, but does not regulate the
modular fluctuation. It does not lead to an automatic prescription for
calculating gravitational entropies and modular fluctuations and any such
prescription is put in by hand based on other assumptions. In particular, it
appears that [35] made essentially the same assumptions about the density
matrix of dS space as[2, 1, 18], except that the latter authors made the
stronger assumption that the full Hilbert space has finite dimension. The QFT
and gravitational heat capacity arguments described in the current note add to
the evidence[16, 22, 8] that the maximal entropy assumption about the form of
the density matrix may have been too strong.

There is an enormous amount of evidence that black holes in Minkowski space
are finite entropy meta-stable equilibria with extremely long lifetimes and
universal properties. The energy levels of these systems cannot have finer
spacing than their inverse lifetime, which is power law in $1/S$. Yet in order
to account for the entropy, the level spacing of the modular Hamiltonian must
be of order $e^{-S}\ll 1/S$. The specific heat is obviously not the modular
fluctuation in this case, but it is reasonable to conjecture that there is
some universal formula for it. The Carlip-Solodukhin ansatz provides such a
universal formula.

Thus, within the crossed product construction, as in the earlier conjecture
[2, 1] of a finite dimensional Hilbert space for quantum theory in dS space,
one must search for other principles to determine the density matrix of empty
dS space. We have shown in [8] that one can retain the idea of a purely
entropic definition of static energy for objects near the geodesic with
essentially any finite value for the modular fluctuation coefficient $a$. The
Carlip-Solodukhin ansatz $a=1$ has a clear but conjectural derivation from the
Einstein equations near a generic causal diamond boundary, and we have
provided a derivation of it from a version of the replica trick in this paper.
Quantum field theory in curved space-time gives regulator dependent answers
for $a$. All are independent of the state of the QFT. An elegant form of
regulator for CFTs gives model dependent answers, which all converge [4] to
$a=1$ for models with Einstein-Hilbert AdS duals, in agreement with [16].
Pauli-Villars regulators for free fields of low spin always reduce to $1+1$
dimensional CFTs near the boundary, again giving $a=1$. No field theory
calculation gives $a=0$.

We conclude these comments with an observation about the crossed product as an
approximation to quantum gravity in finite space-time regions.

Much of the recent work on crossed product constructions began [34] as a
method of finding systematic $1/N$ corrections to the $N=\infty$ description
of “finite regions of AdS space-time” in terms of a Type $III_{1}$ algebra of
local quantum fields[41]. As such, one would expect the crossed product to
incorporate some of the well known intuitive restrictions that gravity imposes
on local quantum field theory in finite regions. There is, as far as we can
see, a fundamental problem with that idea. The standard description of the
crossed product algebra is that it consists of all linear combinations of
elements of the form

$f_{i}(q)e^{ipH}a_{i}e^{-ipH},$ (17)

where $H$ is the generator of the outer automorphism group of the Type $III$
algebra, $a_{i}$ is an element of the algebra and $f_{i}(q)$ is a bounded
function that maps square integrable functions of $q$ into themselves. $p$ is
the canonical conjugate of $q$. It is clear that the Type $III$ algebra is a
proper sub-space of the crossed product algebra. The Type $II_{1}$ algebra
that has been suggested for dS space simply projects this algebra onto the
positive spectrum of $q$ and still contains the QFT algebra as a proper sub-
space. It is not a sub-algebra because the projection operator distorts the
operator product.

It has long been argued that the gravitational backreaction produced by the
expectation value of the stress tensor

$\langle 0|a^{\dagger}T_{mn}(x)a|0\rangle,$ (18)

for most of the operators $a$ in the algebra of a finite region in Minkowski
space, would distort that geometry and create large black holes. Such states,
however, can be excised from the theory without causing any conflict with
experiment [42, 43, 44]. This suggests that the correct quantum gravity
algebra of a finite space time region is smaller, rather than larger, than the
Type $III$ algebra assigned to the region by QFT.

##  5 Discussion

We have argued that a reasonable interpretation of semiclassical evidence is
that empty de Sitter space is described by a finite dimensional quantum theory
in a state with a large but not maximal entropy. Deviations from maximal
uncertainty are associated with a nonzero modular fluctuation, or a heat
capacity in thermal states, which in cutoff quantum field theory is of order
the entropy. In semiclassical gravity a natural conjecture for the relevant
heat capacity provides the same result.

Of course, this is only an observation, not a proof. Even if each term in the
generalized heat capacity accurately computes an energy fluctuation in the
corresponding subsystem, the sum may not share this interpretation for the
whole system. An elementary example is an isolated box of gas in the
microcanonical state, divided into two subsystems by a heat-conducting
membrane.

The apparent tension with crossed product proposals is not fully resolved. Of
course, it may be that they are simply computing approximations to different
quantities.

Let us conclude by describing connections to related ideas in holography. The
Covariant Entropy Bound (CEB) for causal diamonds suggests a conjecture that
physics inside diamonds can be described by models with finite dimensional
Hilbert spaces obeying an area law [45]. A “natural” state is the maximal
entropy density matrix on that Hilbert space, and other guiding principles are
not immediately apparent in the absence of space-time isometries. Indeed,
various prior works by Banks and Fischler viewed the maximal entropy
conjecture for de Sitter space as a special case of this reasoning for general
causal diamonds. However, such a principle had already been found [13, 14]
although its validity for general causal diamonds was not recognized until
much later [22]. According to this principle, the modular Hamiltonian of a
causal diamond is the Virasoro generator $L_{0}$ of a cut-off $1+1$
dimensional CFT living on an interval. The CFT has a central charge
proportional to the area of the diamond’s holographic screen, and Cardy’s
formula reproduces the Bekenstein-Hawking area law for entropy of black holes
and causal diamonds. Ref. [22] interpreted [13, 14] as a principle about the
microscopic theory and argued that it implies $a=1$. The present work
establishes a level of semiclassical compatibility.

The Carlip-Solodukhin ansatz also arose in an entirely different context[46],
in an attempt to model the dynamics of an FRW cosmology that saturated the CEB
at all times. This is a flat FRW model with equation of state that is a
mixture of $p=\pm\rho$. It asymptotes to dS space and predicts that the dS
modular Hamiltonian is not a c-number. In this context, the fact that the
modular fluctuations are proportional to the entropy is important for
explaining the Cosmic Microwave Background fluctuations[47][48].

An additional choice needs to be made in order to extend the semiclassical
heat capacity of de Sitter space studied here to smaller causal diamonds. This
is because the first law of causal diamonds contains a term proportional to
$kdV$ [49], where $V$ is a spatial volume and $k$ is a codimension-2 extrinsic
curvature trace. $k$ vanishes for maximal diamonds like the static patch of de
Sitter, but not for more general diamonds. When we compute $TdS/dT$ for de
Sitter, we are effectively defining the diamond for different $T$ by fixing
the c.c. and fixing $k=0$. Thus a natural extension to non-maximal diamonds
seems to be holding $k$ (the Brown-York surface energy density) fixed but
nonzero. This question will be explored elsewhere.

Acknowledgments
We thank Shadi Ali Ahmad, Ro Jefferson, Marc Klinger, Rob Leigh, Horacio
Casini, and Antony Speranza for discussions and comments on the manuscript.
The work of TB is partially supported by the Department of Energy under grant
DOE SC0010008. PD acknowledges support from the U.S. Department of Energy,
Office of Science, Office of High Energy Physics under award number DE-
SC0015655.

## References

  * [1] W. Fischler, Taking de Sitter Seriously, Talk at the Festschrift for Geoffrey West, Taos, New Mexico, July 2000.
  * [2] T. Banks, Talk at the Festschrift for L. Susskind, Stanford University, May 2000. T. Banks, “Cosmological breaking of supersymmetry?,” Int. J. Mod. Phys. A 16, 910-921 (2001) doi:10.1142/S0217751X01003998 [arXiv:hep-th/0007146 [hep-th]].
  * [3] R. Bousso and B. Freivogel, Phys. Rev. D 73, 083507 (2006) doi:10.1103/PhysRevD.73.083507 [arXiv:hep-th/0511084 [hep-th]].
  * [4] J. De Boer, J. Järvelä and E. Keski-Vakkuri, Phys. Rev. D 99, no.6, 066012 (2019) doi:10.1103/PhysRevD.99.066012 [arXiv:1807.07357 [hep-th]].
  * [5] Y. Nakaguchi and T. Nishioka, JHEP 12, 129 (2016) doi:10.1007/JHEP12(2016)129 [arXiv:1606.08443 [hep-th]].
  * [6] Y. O. Nakagawa and S. Furukawa, Phys. Rev. B 96, no.20, 205108 (2017) doi:10.1103/PhysRevB.96.205108 [arXiv:1708.08924 [cond-mat.str-el]].
  * [7] P. Nandy, JHEP 07, 019 (2021) doi:10.1007/JHEP07(2021)019 [arXiv:2106.00228 [hep-th]].
  * [8] T. Banks and P. Draper, JHEP 01, 135 (2023) doi:10.1007/JHEP01(2023)135 [arXiv:2209.08991 [hep-th]].
  * [9] G. ’t Hooft, Nucl. Phys. B 256, 727-745 (1985) doi:10.1016/0550-3213(85)90418-3
  * [10] J. G. Demers, R. Lafrance and R. C. Myers, Phys. Rev. D 52, 2245-2253 (1995) doi:10.1103/PhysRevD.52.2245 [arXiv:gr-qc/9503003 [gr-qc]].
  * [11] D. Anninos, F. Denef, Y. T. A. Law and Z. Sun, JHEP 01, 088 (2022) doi:10.1007/JHEP01(2022)088 [arXiv:2009.12464 [hep-th]].
  * [12] T. Banks, P. Draper and M. Karydas, [arXiv:2401.03572 [hep-th]].
  * [13] S. Carlip, Phys. Rev. D 51, 632 (1995), arXiv:gr-qc/9409052, S. Carlip, Phys. Rev. Lett. 82, 2828 (1999), arXiv:hep-th/9812013, S. Carlip, Class. Quant. Grav. 15, 3609 (1998), arXiv:hep-th/9806026, S. Carlip, AIP Conf. Proc. 1483, 54 (2012), arXiv:1207.1488 [gr-qc].
  * [14] S. N. Solodukhin, Phys. Lett. B 454, 213 (1999), arXiv:hep-th/9812056.
  * [15] H. Casini, M. Huerta and R. C. Myers, “Towards a derivation of holographic entanglement entropy,” JHEP 05, 036 (2011) doi:10.1007/JHEP05(2011)036 [arXiv:1102.0440 [hep-th]].
  * [16] E. Verlinde and K. M. Zurek, “Spacetime Fluctuations in AdS/CFT,” JHEP 04, 209 (2020) doi:10.1007/JHEP04(2020)209 [arXiv:1911.02018 [hep-th]].
  * [17] A. Lewkowycz and J. Maldacena, JHEP 08, 090 (2013) doi:10.1007/JHEP08(2013)090 [arXiv:1304.4926 [hep-th]].
  * [18] T. Banks, B. Fiol and A. Morisse, “Towards a quantum theory of de Sitter space,” JHEP 12, 004 (2006) doi:10.1088/1126-6708/2006/12/004 [arXiv:hep-th/0609062 [hep-th]].
  * [19] P. Draper and S. Farkas, “de Sitter black holes as constrained states in the Euclidean path integral,” Phys. Rev. D 105, no.12, 126022 (2022) doi:10.1103/PhysRevD.105.126022 [arXiv:2203.02426 [hep-th]].
  * [20] E. K. Morvan, J. P. van der Schaar and M. R. Visser, [arXiv:2203.06155 [hep-th]].
  * [21] P. Draper, S. Farkas and M. Karydas, Class. Quant. Grav. 41, no.2, 025004 (2024) doi:10.1088/1361-6382/ad1449 [arXiv:2310.02101 [hep-th]].
  * [22] 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]].
  * [23] S. Miyashita, JHEP 09, 121 (2021) doi:10.1007/JHEP09(2021)121 [arXiv:2106.12273 [gr-qc]].
  * [24] P. Draper and S. Farkas, Phys. Rev. D 105, no.12, 126021 (2022) doi:10.1103/PhysRevD.105.126021 [arXiv:2203.01871 [hep-th]].
  * [25] J. D. Brown and J. W. York, Jr., Phys. Rev. D 47, 1420-1431 (1993) doi:10.1103/PhysRevD.47.1420 [arXiv:gr-qc/9209014 [gr-qc]].
  * [26] G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2738-2751 (1977) doi:10.1103/PhysRevD.15.2738
  * [27] S. Ali Ahmad and R. Jefferson, [arXiv:2306.07323 [hep-th]].
  * [28] M. S. Klinger and R. G. Leigh, Nucl. Phys. B 999, 116453 (2024) doi:10.1016/j.nuclphysb.2024.116453 [arXiv:2306.09314 [hep-th]].
  * [29] R. D. Sorkin, “1983 paper on entanglement entropy: ”On the Entropy of the Vacuum outside a Horizon”,” [arXiv:1402.3589 [gr-qc]]; M. Srednicki,“Entropy and area,” Phys. Rev. Lett. 71, 666-669 (1993) doi:10.1103/PhysRevLett.71.666 [arXiv:hep-th/9303048 [hep-th]]; 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]].
  * [30] 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]]; T. Jacobson, “Black hole entropy and induced gravity,” [arXiv:gr-qc/9404039 [gr-qc]].
  * [31] J. W. York, Jr., Phys. Rev. Lett. 28, 1082-1085 (1972) doi:10.1103/PhysRevLett.28.1082
  * [32] G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2752-2756 (1977) doi:10.1103/PhysRevD.15.2752
  * [33] J. Kudler-Flam, S. Leutheusser, A. A. Rahman, G. Satishchandran and A. J. Speranza, “A covariant regulator for entanglement entropy: proofs of the Bekenstein bound and QNEC,” [arXiv:2312.07646 [hep-th]].
  * [34] E. Witten, “Gravity and the crossed product,” JHEP 10, 008 (2022) doi:10.1007/JHEP10(2022)008 [arXiv:2112.12828 [hep-th]].
  * [35] V. Chandrasekaran, R. Longo, G. Penington and E. Witten, “An Algebra of Observables for de Sitter Space,” [arXiv:2206.10780 [hep-th]].
  * [36] V. Chandrasekaran, G. Penington and E. Witten, “Large N algebras and generalized entropy,” JHEP 04, 009 (2023) doi:10.1007/JHEP04(2023)009 [arXiv:2209.10454 [hep-th]].
  * [37] K. Jensen, J. Sorce and A. J. Speranza, “Generalized entropy for general subregions in quantum gravity,” JHEP 12, 020 (2023) doi:10.1007/JHEP12(2023)020 [arXiv:2306.01837 [hep-th]].
  * [38] J. Kudler-Flam, S. Leutheusser and G. Satishchandran, [arXiv:2309.15897 [hep-th]].
  * [39] M. S. Klinger and R. G. Leigh, [arXiv:2312.16678 [hep-th]].
  * [40] J. M. Maldacena, “Eternal black holes in anti-de Sitter,” JHEP 04, 021 (2003) doi:10.1088/1126-6708/2003/04/021 [arXiv:hep-th/0106112 [hep-th]].
  * [41] S. Leutheusser and H. Liu, “Causal connectability between quantum systems and the black hole interior in holographic duality,” Phys. Rev. D 108, no.8, 086019 (2023) doi:10.1103/PhysRevD.108.086019 [arXiv:2110.05497 [hep-th]]; S. Leutheusser and H. Liu, “Subalgebra-subregion duality: emergence of space and time in holography,” [arXiv:2212.13266 [hep-th]].
  * [42] A. G. Cohen, D. B. Kaplan and A. E. Nelson, Phys. Rev. Lett. 82, 4971-4974 (1999) doi:10.1103/PhysRevLett.82.4971 [arXiv:hep-th/9803132 [hep-th]].
  * [43] T. Banks and P. Draper, Phys. Rev. D 101, no.12, 126010 (2020) doi:10.1103/PhysRevD.101.126010 [arXiv:1911.05778 [hep-th]].
  * [44] N. Blinov and P. Draper, Phys. Rev. D 104, no.7, 076024 (2021) doi:10.1103/PhysRevD.104.076024 [arXiv:2107.03530 [hep-ph]].
  * [45] R. Bousso, Rev. Mod. Phys. 74, 825-874 (2002) doi:10.1103/RevModPhys.74.825 [arXiv:hep-th/0203101 [hep-th]].
  * [46] T. Banks, W. Fischler and L. Mannelli, Phys. Rev. D 71, 123514 (2005) doi:10.1103/PhysRevD.71.123514 [arXiv:hep-th/0408076 [hep-th]].
  * [47] T. Banks and W. Fischler, “Holographic Inflation Revised,” doi:10.1017/9781316535783.013 [arXiv:1501.01686 [hep-th]].
  * [48] T. Banks and Sidan A., In progress
  * [49] T. Jacobson and M. Visser, SciPost Phys. 7, no.6, 079 (2019) doi:10.21468/SciPostPhys.7.6.079 [arXiv:1812.01596 [hep-th]].