###### Abstract

We study algebras of observables for semiclassical quantum gravity in
cosmological backgrounds, focusing on two key examples: slow-roll inflation
and evaporating Schwarzschild-de Sitter black holes. In both cases, we
demonstrate the existence of a nontrivial algebra of diffeomorphism-invariant
observables _without_ the introduction of an explicit clock system or the
presence of any asymptotic gravitational charges. Instead, the rolling
inflaton field and the evaporating black hole act as physical clocks that
allow a definition of gauge-invariant observables at $G=0$. The resulting
algebras are both Type II∞ factors, but neither is manifestly a crossed
product algebra. For appropriate states, we establish a connection between the
Type II entropy of these algebras and generalized entropies. Our work extends
previous results on Type II gravitational algebras and highlights the crucial
role of out-of-equilibrium dynamics when defining gauge-invariant observables
in semiclassical canonically quantised gravity. We also briefly discuss the
construction of gauge-invariant algebras for compact wedges bounded by
extremal surfaces in generic spacetimes (i.e. in the absence of any Killing
symmetry). In contrast to the inflaton and black hole cases, this algebra does
end up being a simple crossed product. No clock or asymptotic charges are
required because of the absence of any symmetry in the classical background.

A clock is just a way to tell the time: gravitational algebras in cosmological
spacetimes

Chang-Han Chen and Geoff Penington

Department of Physics, University of California, Berkeley, CA 94720, USA

<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 A slow rolling inflaton in quasi-de Sitter space
    1. 2.1 The Bunch-Davies state for a light scalar field
    2. 2.2 The Bunch-Davies weight for an inflaton field
    3. 2.3 The algebra of observables
    4. 2.4 Density matrices and entropies
  3. 3 Black holes
    1. 3.1 Black holes in asymptotically flat spacetimes
    2. 3.2 Schwarzschild-de Sitter black holes
    3. 3.3 The algebras of observables
    4. 3.4 The gauge constraints
    5. 3.5 The trace, density matrices and entropies

##  1 Introduction

Recent work has shown that the horizon entropy $A_{\rm hor}/4G$ associated
both to black holes and to the cosmological horizon of an observer in de
Sitter space can be accounted for by considering algebras of observables in
the $G\to 0$ limit of canonical quantum gravity [Leutheusser:2021frk,
Leutheusser:2021qhd, Leutheusser:2022bgi, Witten_2022, CLPW, CPW,
kudlerflam2024generalized, Jensen_2023, faulkner2024gravitational]. For a
black hole, the relevant algebra consists of semiclassical observables at
asymptotic spatial or null infinity; for de Sitter space, it is generated by
observables localised along the observer’s worldline.

In both cases, the algebras in question are Type II von Neumann factors.111See
[entropy1, entropy2, entropy3, entropy4] for other recent work on the role
played by Type II factors in quantum gravity and quantum field theory. Such
algebras describe degrees of freedom that are infinitely entangled with their
environment, reflecting the fact that horizon entropies diverge as $G\to 0$.
However there is a natural notion of the renormalised entropy of a state on a
Type II factor that is unique up to a state-independent additive constant.
Roughly, this is because _fluctuations_ in the entanglement spectrum on a Type
II algebra are finite, even though the total entanglement diverges. For an
appropriate class of states the entropy of the Type II algebra exactly matches
the generalised entropy [Bek, QES1, QES5]

$\displaystyle S_{\rm gen}=A_{\rm hor}/4G+S_{\rm QFT}$ (1.1)

of the observer’s causal diamond.222Here $S_{\rm QFT}$ is the entropy of
quantum fields in the causal diamond. This entropy is UV-divergent but the
divergence can be absorbed into the renormalisation of Newton’s constant $G$
so that the generalised entropy $S_{\rm gen}$ is UV-finite [Susskind:1994sm].
The primary difference between the algebras for black holes and for de Sitter
is that black hole algebras are Type II∞ whereas the de Sitter algebra is Type
II1. This reflects the fact that the entropy of a black hole is unbounded and
can be increased arbitrarily by adding mass. On the other hand, the
generalised entropy of the static patch is maximised by empty de Sitter space;
any perturbation thereof can only decrease entropy.

There is, nonetheless, an important subtlety here that is only present in the
de Sitter context. Because Cauchy slices in de Sitter space are compact, the
isometries of de Sitter space act as gauge constraints on the gravitational
theory. In particular, gauge-invariant observables need to commute with the
boost Hamiltonian $H$ that generates time translations of the observer’s
static patch. This is, in fact, the only reason that the algebra of a de
Sitter observer in quantum gravity is different from that of the same observer
in quantum field theory, even in the limit $G\to 0$.

Unfortunately or otherwise, however, it is a well established fact about
quantum field theory in de Sitter space that any operator that is both a)
localised to the static patch and b) time translation invariant is necessarily
a $c$-number, i.e. a multiple of the identity. As a result, the observer’s
algebra is apparently trivial. One way to see why this is the case is the
following. To construct a gauge-invariant observable that is independent of
coordinate time, the obvious approach is to integrate a time-dependent
observable $a(t)$ over the observer’s worldline to obtain an apparently boost-
invariant observable

$\displaystyle\widetilde{a}\stackrel{{\scriptstyle?}}{{=}}\int dt\,a(t)$ (1.2)

However, any state of quantum fields in de Sitter space thermalises on the
static patch at sufficiently early and late times. Since the thermal state is
separating (i.e. it is not annihilated by any nonzero operator $a$) the
integral in (1.2) diverges when acting on any state $\ket{\Phi}$. As a result,
the operator $\widetilde{a}$ does not exist.

This explanation also makes clear why the physically relevant algebra of
observables should _not_ be trivial. It is true that physical observations
cannot depend on a choice of coordinate time. However, neither do they require
averaging an observation over all times from past to future infinity. Instead,
in practice, physical observations occur at a particular “clock time”, defined
with respect to the state of a physical system or clock. In this context, a
clock can mean anything from a literal device used to tell the time to the
state of the observer’s brain or the state of some external dynamical system.
Crucially, since the universe is quantum mechanical, the clock, whatever it
might be, should itself be a quantum system.

In [CLPW], a minimal model of a quantum clock was used to define the algebra
of an observer in de Sitter space. To enable nontrivial observables, the clock
needed to be able to measure arbitrarily long times $\Delta t\gg\ell_{dS}$ (in
units of the de Sitter radius $\ell_{dS}$) using finite energy. This required
it to have a continuous spectrum in order to prevent Poincaré
recurrences.333More precisely, it requires the typical spectral gap $\delta E$
to satisfy $\delta E\ll 1/\ell_{dS}$ as $G\to 0$. This is easy enough to
engineer, even with the size of the clock remaining much smaller than the de
Sitter scale, because the typical spectral gap of a quantum system is
exponentially small in its entropy. Additionally, it was assumed on physical
grounds that the energy of the clock was expected to be bounded from below.

The simplest model of a clock satisfying those properties is the Hilbert space
$L^{2}({{\mathbb{R}}}^{+})$ of wavefunctions with position $x\geq 0$ and
Hamiltonian $H_{\rm clock}=x$. Assuming no coupling between the clock and the
background quantum fields, so that the boost gauge constraint is simply
$H+x=0$, the inclusion of this clock was shown in [CLPW] to lead to a Type II1
algebra of observables with the properties described above.

It is important to emphasize that the lesson of [CLPW] (or, at the very least,
the lesson as interpreted by one of its authors) was not that in practice
physical observers are equipped with clocks of exactly the form just
described. Instead, it is that a clock of that form provides a simple model
with features that all good physical clocks should have. The hope was that the
lessons drawn from studying such a clock, and in particular the conclusion
that the entropy of the observer’s algebra includes a contribution from the
cosmological horizon area, would generalise fairly universally to any
reasonable physical system that can act as a clock.

In this paper, we provide strong evidence that this is indeed the case, by
considering two natural classes of “clocks” that are not associated to the
observer themselves, but rather to a dynamical quantum evolution of the
observer’s environment that remains out-of-equilibrium for parametrically long
times. The first such example is quasi-de Sitter space in the presence of a
slow-rolling inflaton field. The second is an evaporating Schwarzschild-de
Sitter black hole.

The technical details of how the algebra works in each case is somewhat
different, but the overall conclusions end up being very similar. Firstly, the
out-of-equilibrium nature of the solutions mean that there exist nontrivial
gauge-invariant observables even if we don’t equip the observer with an
explicit clock; the dynamics of the quantum system already provides one for
us. In the inflaton case, the scalar field wants to roll downhill as we
approach asymptotic future (or past) infinity. The value of the inflaton field
therefore acts as the physical clock distinguishing different times. This
claim will of course come as no surprise to cosmologists, since treated the
inflaton value as a clock is standard practice in inflationary cosmology.
Similarly, because the temperature of the black hole is higher than the
cosmological horizon, a Schwarzschild-de Sitter black hole will tend to
evaporate as we approach past or future infinity. The energy of the black hole
can therefore be used to distinguish different times.

Secondly, in both cases the algebra becomes a Type II∞ von Neumann factor.
This is because, unlike for perturbations around pure de Sitter space, the
entropy of these solutions can be arbitrarily increased simply by simply
shifting the state further along the road to equilibrium. For example, given
any state of the inflaton field we can produce a state with smaller
cosmological constant, and hence larger generalised entropy, by shifting the
inflaton wavefunction down its potential gradient. Similarly, we can
arbitrarily increase generalised entropy in the presence of a Schwarzschild-de
Sitter black hole by decreasing the energy of the black hole and hence
increasing the area of the cosmological horizon. Of course, at finite $G$,
neither process can be continued indefinitely: eventually an inflaton will
reach the bottom of its potential and a black hole will reach zero mass. But,
as $G\to 0$, the increase in generalised entropy ends up diverging
parametrically sooner than either of these events will occur.

An interesting feature of our constructions is that – to the best of our
current understanding – neither algebra has a description as a crossed product
of a Type III von Neumann factor by a modular automorphism group. (The black
hole algebra does involve such a crossed product as an intermediate step in
its construction.) This is in contrast to the Type II algebras that describe
de Sitter space in the presence of a worldline clock or semiclassical black
holes in asymptotically flat/AdS spacetimes and is instead more similar to the
boundary algebras describing quantum JT gravity (beyond the semiclassical
limit) [penington2023algebras, Kolchmeyer:2023gwa].

However, like in the de Sitter algebra introduced in [CLPW], the traces on
both algebras we will define do both have a natural interpretation as
expectation values of operators in a no-boundary Hartle-Hawking state. This is
consistent with the proposal of [witten2023background] that the no-boundary
Hartle-Hawking state should be viewed as a maximum entropy state in
background-independent quantum gravity.

Like previous gravitational Type II algebras, the algebras we construct have
traces that are proportional to the number of horizon microstates $\exp(A_{\rm
hor}/4G)$. In particular, the inflaton algebra has a symmetry that shifts the
inflaton field by a constant. We show that this symmetry rescales the trace by
$\exp(\delta A_{\rm hor}/4G)$ where $\delta A_{\rm hor}$ is the perturbative
change in cosmological horizon area induced by the shift in the inflaton
potential. Similarly, perturbatively increasing the mass of a Schwarzschild-de
Sitter black hole rescales the trace on its gravitational algebra by
$\exp((\delta A_{BH}+\delta A_{CH})/4G)$ where $\delta A_{BH}>0$ and $\delta
A_{CH}<0$ are the respective changes in the black hole and cosmological
horizon areas.

We also obtain a precise match between entropies on our algebras and
generalised entropies of semiclassical states that have been formally averaged
over the gauge group. In the inflaton case, this correspondence holds for any
QFT state. For Schwarzschild-de Sitter black holes, there is a timeshift mode
associated to travelling in a nontrivial spacelike cycle through the black
hole and cosmological horizons that cannot be described by perturbative
quantum field theory. The correspondence between the entropy of the Type II
algebra and generalised entropy holds for states with sufficiently small
fluctuations in this timeshift.

The structure of this paper is as follows. In Section 2, we describe the
semiclassical gravitational algebra associated to a slow-rolling inflaton in
quasi de Sitter space. In Section 3, we describe the algebras for a
Schwarzschild-de Sitter black hole, after first pausing briefly to discuss
asymptotically flat black holes. Finally in Section LABEL:sec:generic, we
briefly discuss the gravitational algebra of a wedge bounded by an extremal
surface in a generic cosmological spacetime (i.e. in the absence of any
background symmetry). In Appendices, we provide some technical details on
Tomita-Takesaki theory and higher-dimensional spherical harmonics. We also
include a detailed glossary of notation used throughout the paper.

Note added: The submission of this paper to arXiv has been coordinated with
the submission of [KLS], which contains related discussion of von Neumann
algebras in the presence of a slow-rolling inflaton field. A significant
difference between our work and [KLS] is that [KLS] makes use of an explicit
gravitating clock, while we do not.

##  2 A slow rolling inflaton in quasi-de Sitter space

We consider a single scalar inflaton field $\phi$ with potential $V(\phi)$.
This is a well-studied inflation model that has produced successful
predictions for primordial curvature perturbations [inflation0, inflation,
inflation1, inflation2, inflation3]. See [Maldacena_2003,
maldacena2024comments] for excellent reviews. For simplicity, we assume four
spacetime dimensions, although the generalisation to other dimensions is
straightforward. In units where $\hbar=1$, the slow-roll approximation assumes
that

$\displaystyle\varepsilon=\frac{G}{2}\left(\frac{V^{\prime}}{V}\right)^{2}\ll
1~{}~{}~{}~{}\text{and}~{}~{}~{}~{}|\eta|=G\frac{|V^{\prime\prime}|}{V}\ll 1.$
(2.1)

In this limit, symmetric vacuum solutions to the Einstein equations can be
well approximated by de Sitter space with cosmological constant $\Lambda=8\pi
GV$ over scales that are large compared to the de Sitter length
$\ell_{dS}=\sqrt{3/\Lambda}$.

In cosmology, one usually takes $\varepsilon$ and $|\eta|$ to be small but
finite so that the exit from inflation occurs at finite time. Mathematically,
however, giving a precise algebraic description of gravitational observables
at finite $\varepsilon,|\eta|$, or indeed even in a formal perturbative
expansion in $\varepsilon,|\eta|$, would be very challenging. It is much
easier to take a strict $G\to 0$ limit where $V\sim O(1/G)$, $V^{\prime}\sim
O(1)$ and $V^{\prime\prime}\sim o(1)$. In this limit, the cosmological
constant $\Lambda$ remains finite while dynamical backreaction from the matter
fields goes to zero. Meanwhile – if we include the cosmological constant in
the Einstein-Hilbert action – the inflaton action becomes

$\displaystyle S=-\int
d^{4}x\sqrt{-g}\;\bigg{[}\frac{1}{2}\nabla_{a}\phi\nabla^{a}\phi+V^{\prime}\phi\bigg{]}.$
(2.2)

which is just the action for massless scalar field with a potential
$V(\phi)=V^{\prime}\phi$ that has constant linear slope $V^{\prime}$. Without
loss of generality we can assume $V^{\prime}>0$. There also exist graviton
fluctuations that are described by a free massless spin-2 quantum field
theory, along with quantum perturbations of any other quantum fields that we
choose to include in our theory. We assume that, at least in the limit $G\to
0$, all such fields are either free or described by a UV-complete interacting
theory. In the $G\to 0$ limit, the full gravitational theory then reduces to a
continuum quantum field theory in a fixed background de Sitter spacetime.

In global coordinates, the metric for $dS_{4}$ is

$\displaystyle
ds^{2}=-\ell_{dS}^{2}\big{(}d\tau^{2}+\cosh^{2}(\tau)d\Omega_{3}^{2}\big{)},$
(2.3)

where $\Omega_{3}=(\chi,\theta,\phi)$ describes angular coordinates on
$S^{3}$. The Penrose diagram is shown in Figure 1.

$\mathcal{A}$$\mathcal{A}^{\prime}$$\tau=0$$\chi=0$$\chi=\pi$ Figure 1:  A
Penrose diagram for global de Sitter space. The compact $\tau=0$ Cauchy slice
is shown in red. We consider the QFT algebra $\mathcal{A}$ of operators
localised in the static patch centered around $\chi=0$. The commutant algebra
$\mathcal{A}^{\prime}$ describes operators in the opposite static patch.

For notational convenience, we will mostly set $\ell_{dS}=1$ for the remainder
of this section. Since global de Sitter space has compact spatial Cauchy
slices, there exists a unique natural Hilbert space $\mathcal{H}$ describing
any set of free quantum fields propagating within it; this result is expected,
but is not known, to extend to UV-complete interacting theories [Wald:1995yp,
witten2022does]. Essentially, this is because, on a compact spatial slice,
there are a finite number of long-wavelength modes, which consequently have a
unique representation by the Stone-von Neumann theorem, together with an
asymptotically large number of high-energy short-wavelength modes that have an
unambiguous decomposition into positive and negative frequencies because
locally the spacetime looks like Minkowski space. Our goal in this section
will be to study this Hilbert space for the slow rolling inflaton (and other
fields) and show that it contains a Type II∞ von Neumann subalgebra describing
gauge-invariant observables in any given static patch.

Working with global de Sitter space is again slightly unusual in inflationary
cosmology, where typically only the inflating patch, covered by a flat slicing
of de Sitter, is considered. We make this choice for two reasons. First, at a
technical level, quantising an inflaton field in a flat slicing of de Sitter
space leads to a number of mathematical difficulties related to the fact that
the inflationary patch does not contain compact Cauchy slices. In particular,
in the flat slicing of de Sitter, there exist, at any finite time, infinitely
many long wavelength modes that have already exited the cosmological horizon.
These contribute divergent fluctuations to the value of the inflaton at sub-
horizon scales. Cosmologists typically get around this issue by treating such
modes classically, which works well for predicting cosmological observables
since only a finite number of these IR modes have so far reentered our
particle horizon. However, to describe the mathematical structure of the
inflaton algebra of observables, we find it simplest to avoid IR divergences
in the first place by working in global de Sitter space, where Cauchy slices
are compact and hence (angular) momentum is quantised.

A second, somewhat related, point is that the Hartle-Hawking proposal
[Hartle:1983ai, HH2, hh3, hh4] for the no-boundary state in quantum gravity
defines a natural choice of wavefunctional at finite $G$ for spatially compact
spacetimes like global de Sitter. An excellent recent review of the Hartle-
Hawking state and its (incorrect) predictions for inflationary cosmology, was
given in [maldacena2024comments]. The Hartle-Hawking state will play a very
important role in our story: indeed, the trace that we construct to show that
our algebra is Type II∞ is essentially the $G\to 0$ limit of the Hartle-
Hawking state. Again, all of this is simplest to describe when working in
global de Sitter space.

###  2.1 The Bunch-Davies state for a light scalar field

For a massive scalar field theory (and more generally for any gapped QFT) in
de Sitter space, there exists a unique normalisable state in $\mathcal{H}$,
called the Bunch-Davies state, that is invariant under the isometry group
$SO(4,1)$.

One way to construct the Bunch-Davies state is via a path integral on a
Euclidean hemisphere as shown in Figure 2.

$\phi(\Omega_{3})$ Figure 2:  The Bunch-Davies wavefunctional
$\Psi_{BD}(\phi(\Omega_{3}))$ is defined by a path integral on a Euclidean
hemisphere with Dirichlet boundary conditions $\phi(\Omega_{3})$.

The Bunch-Davies wavefunctional $\Psi(\phi(\Omega_{3}))$ is proportional to
the value of the path integral with boundary conditions $\phi(\Omega_{3})$.

For free theories, there is a very useful trick where the path integral can be
evaluated by considering the classical action of a single saddle point
solution with the correct boundary conditions. Consider e.g. a single harmonic
oscillator, with Euclidean action

$\displaystyle S=\int
dt\,\frac{{{m_{0}}}}{2}\dot{x}^{2}-\frac{1}{2}{{m_{0}}}\omega^{2}x^{2}.$ (2.4)

The vacuum wavefunction $\psi(x_{0})$ for the harmonic oscillator can be
evaluated using a Euclidean path integral on the half-line $it<0$ with
boundary conditions $x(it=-\infty)=0$ and $x(it=0)=x$. The classical solution
with these boundary conditions is $x(it)=xe^{\omega it}$ which leads to a
saddle-point contribution to the partition function of

$\displaystyle\psi(x)\sim\exp(iS)\sim\exp(-\frac{1}{2}{{m_{0}}}\omega x^{2}).$
(2.5)

Crucially, because the theory is free, this answer is exact. The one-loop
determinant depends only on the second derivatives of the action around the
classical saddle, which are constant because the action is quadratic in $x$.
And, for the same reason, no higher-order corrections to (2.5) exist.

Let us recall how to implement the same procedure for a massive scalar field
in the de Sitter space. Partially for technical simplicity, and partially
because it means that our result will be useful later when we come to consider
an inflaton field, we assume that the mass ${{m_{0}}}\ll\ell_{dS}$ of the
scalar field is parametrically small and hence can be treated perturbatively.
The action for this field is

$\displaystyle S=-\int
d^{4}x\sqrt{-g}\;\frac{1}{2}\,\nabla_{a}\phi\nabla^{a}\phi+\frac{1}{2}{{m_{0}}}^{2}\phi^{2}.$
(2.6)

In the limit where the mass ${{m_{0}}}$ is small, this leads to a classical
equation of motion

$\displaystyle\frac{1}{\sqrt{-g}}\partial_{a}\big{(}\sqrt{-g}g^{ab}\partial_{b}\phi\big{)}=0,\qquad\sqrt{-g}=\cosh^{3}\tau\cdot\sin^{2}\chi\cdot\sin\theta.$
(2.7)

With respect to the metric (2.3), this becomes

$\displaystyle\frac{1}{\cosh^{3}\tau}\partial_{\tau}\big{(}\cosh^{3}\tau\partial_{\tau}\phi\big{)}-\frac{\Delta_{S^{3}}\phi}{\cosh^{2}\tau}=0,$
(2.8)

where $\Delta_{S^{3}}$ is the Laplacian for the unit $3-$sphere. This can be
solved using separation of variables,

$\displaystyle\phi(\tau,\Omega_{3})=A(\tau)B(\Omega_{3}).$ (2.9)

The solutions for $B$ are the standard spherical harmonics on $S^{3}$, namely
$Y^{k\ell m}(\Omega_{3})$ with $k\geq\ell\geq 0$ and $\ell\geq m\geq-\ell;$
see Appx.LABEL:sh for details. The two independent solutions for $A$ with non-
zero angular momentum, $k\neq 0,$ are known [maldacena2024comments,
Bousso_2002]444Compared to [maldacena2024comments], we normalize the modes
with respect to the Klein-Gordon inner product
$\displaystyle\cosh^{3}\tau(\dot{A}_{k}A_{k}^{\ast}-\dot{A}_{k}^{\ast}A_{k})\overset{!}{=}-\mathrm{i}.$
This is not obvious but can checked straightforwardly for (2.10). As usual,
the normalization is guaranteed to be a time-independent constant by the
equation of motion,

$\displaystyle A_{k}(\tau)$
$\displaystyle=(-1)^{\frac{k+1}{2}}\frac{\Gamma(k+3)\,\Gamma(k+\frac{3}{2})}{\Gamma(\frac{3}{2})\,\Gamma(2k+3)}\frac{1}{\sqrt{2k(k+1)(k+2)}}\;e^{3\tau}\,(1+e^{2\tau})^{k}\;{}_{2}F_{1}\big{(}\frac{3}{2}+k,3+k,3+2k;1+e^{2\tau}\big{)},$
(2.10)

and its complex conjugate

$\displaystyle A_{k}^{\ast}(\tau)$ $\displaystyle=A_{k}(-\tau).$ (2.11)

Taking a formal limit $k\to 0$, (2.10) and (2.11) become constant, which is
one of the two independent solutions to (2.8) with $k=0$. The other is

$\displaystyle
A_{0}(\tau)=\frac{1}{2}\left[\frac{\sinh\tau}{\cosh^{2}\tau}+\tan^{-1}(\sinh\tau)\right].$
(2.12)

The most general real solution to (2.8) is therefore

$\displaystyle\phi$
$\displaystyle=\phi_{0}\frac{1}{\sqrt{2}\pi}+\pi_{0}A_{0}(\tau)\frac{1}{\sqrt{2}\pi}+\sum\phi_{k\ell
m}Y_{k\ell m}$ (2.13)
$\displaystyle=\phi_{0}\frac{1}{\sqrt{2}\pi}+\pi_{0}A_{0}(\tau)\frac{1}{\sqrt{2}\pi}+\sum\bigg{[}a_{k\ell
m}A_{k}(\tau)+(-1)^{m}a^{\ast}_{k\ell-m}A_{k}^{\ast}(\tau)\bigg{]}Y_{k\ell m}$
(2.14)
$\displaystyle=\phi_{0}\frac{1}{\sqrt{2}\pi}+\pi_{0}\,A_{0}(\tau)\frac{1}{\sqrt{2}\pi}+\sum
a_{k\ell m}A_{k}Y_{k\ell m}+a_{k\ell m}^{\ast}A_{k}^{\ast}Y^{\ast}_{k\ell m},$
(2.15)

for arbitrary real coefficients $\phi_{0}$, $\pi_{0}$ and complex coefficients
$a_{k\ell m}$. The conjugate momentum field is

$\displaystyle\pi_{\phi}$ $\displaystyle=\cosh^{3}\tau\,\partial_{\tau}\phi.$
(2.16) $\displaystyle=\pi_{0}\frac{1}{\sqrt{2}\pi}+\sum_{k\ell m}\pi_{k\ell
m}Y_{k\ell m};$ (2.17)

the second line defines the modes for conjugate momentum field. Clearly,
$\pi_{0}$ is the conjugate momentum of $\phi_{0}$.

To compute the wavefunctional $\Psi(\phi(\Omega_{3}))$ of the Bunch-Davies
state, we need to look for (complex) solutions to the classical equations of
motion with boundary conditions $\phi(0,\Omega_{3})=\phi(\Omega_{3})$ that are
regular at the Euclidean south pole $\tau=i\pi/2$. It turns out that analytic
continuations of the solutions (2.10) and the constant solution are regular,
while (2.11) and (2.12) are not. Since

$\displaystyle A_{k}(\tau=0)=-\sqrt{\frac{k+1}{2\,k\big{(}k+2\big{)}}}.$
(2.18)

the regular solution with boundary conditions

$\displaystyle\phi(0,\Omega_{3})=\phi_{0}\frac{1}{\sqrt{2}\pi}+\sum\phi_{k\ell
m}Y_{k\ell m}(\Omega_{3})$ (2.19)

is therefore

$\displaystyle\phi(\tau,\Omega_{3})=\phi_{0}\frac{1}{\sqrt{2}\pi}-\sum\sqrt{\frac{2\,k\big{(}k+2\big{)}}{k+1}}\phi_{k\ell
m}A_{k}(\tau)Y_{k\ell m}(\Omega_{3}).$ (2.20)

Plugging (2.20) back into the action (2.6), we have

$\displaystyle\mathrm{i}S$ $\displaystyle=\mathrm{i}\int
d\tau\cosh^{3}\tau\Bigg{\\{}-\frac{1}{2}m_{0}^{2}\,\phi_{0}^{2}+\frac{k(k+2)}{k+1}\bigg{[}\sum(\dot{A_{k}})^{2}\phi_{k\ell
m}\phi_{k\ell-m}-\frac{1}{\cosh^{2}\tau}\sum k(k+2)\,A_{k}^{2}\phi_{k\ell
m}\phi_{k\ell-m}\bigg{]}\Bigg{\\}}$ (2.21)
$\displaystyle=\mathrm{i}\sum\phi_{k\ell-m}\bigg{[}\int
d\tau\frac{k(k+2)}{k+1}\bigg{(}(\dot{A_{k}})^{2}\cosh^{3}\tau-k(k+2)A_{k}^{2}\cosh\tau\bigg{)}\bigg{]}\phi_{k\ell
m}-\mathrm{i}\int d\tau\cosh^{3}\tau\frac{1}{2}m_{0}^{2}\,\phi_{0}^{2},$
(2.22)

and we are left with the time integral inside the large bracket to evaluate.
To find the Bunch-Davies wavefunctional, we integrate $\tau$ along the
imaginary axis from $\mathrm{i}\frac{\pi}{2}$ to $0$ to obtain

$\displaystyle\Psi_{BD}[\phi(\Omega_{3})]\sim\exp\Bigg{\\{}-\frac{1}{3}{{m_{0}}}^{2}\phi_{0}^{2}-\sum_{\begin{subarray}{c}k\neq
0\end{subarray}}\frac{k\big{(}k+2\big{)}}{2\big{(}k+1)}\phi_{k\ell-m}\phi_{k\ell
m}\Bigg{\\}}.$ (2.23)

###  2.2 The Bunch-Davies weight for an inflaton field

In the limit ${{m_{0}}}\to 0$, the state (2.23) stops being normalisable and
if therefore no longer contained in the Hilbert space $\mathcal{H}$. However
the expectation values of a dense set of observables on the unnormalized state
(2.23) converge to finite limits as ${{m_{0}}}\to 0$. For example, consider
the smeared field operator

$\displaystyle\phi_{f}=\int d^{4}xf(x)\phi(x)$ (2.24)

for some suitable smearing function $f(x)$. This is a densely defined
unbounded self-adjoint operator, and so we can define the bounded operator
$F(\phi_{f})$ for any bounded function $F$. If $F$ vanishes sufficiently fast
as $\phi_{f}\to\pm\infty$, e.g.

$\displaystyle F(\phi_{f})=\exp(-\phi_{f}^{2})$ (2.25)

the expectation value $\braket{\Psi_{BD}}{F(\phi_{f})}{\Psi_{BD}}$ will remain
finite ${{m_{0}}}\to 0$. As a result, the ${{m_{0}}}\to 0$ limit of (2.23)
defines a faithful, normal semifinite weight on $\mathcal{H}$, which we will
call the Bunch-Davies weight. Importantly, since in the argument above
$\phi_{f}$ could be localised on an arbitrarily small region of spacetime, the
Bunch-Davies weight is semifinite on the von Neumann subalgebra
$\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$ associated to any causal
diamond.

Our goal is to construct the analogous Bunch-Davies weight for the inflaton
action (2.2). We do so by an identical procedure to the one above. We first
define the unnormalizable Bunch-Davies wavefunctional by a path integral over
the Euclidean hemisphere with appropriate boundary conditions and then use the
fact that the action (2.2) is free to evaluate this path integral by
evaluating the action of the regular Euclidean solution to the classical
equation of motion with appropriate boundary conditions. In this case, the
classical equations of motion are

$\displaystyle\frac{1}{\cosh^{3}\tau}\partial_{\tau}\big{(}\cosh^{3}\tau\partial_{\tau}\phi\big{)}-\frac{\Delta_{S^{3}}\phi}{\cosh^{2}\tau}=V^{\prime},$
(2.26)

which is a linear inhomogeneous equation with homogeneous part equal to (2.8).
A solution to (2.26) is

$\displaystyle\phi(\tau,\Omega_{3})=C_{0}(\tau)=\frac{V^{\prime}}{3}\left(\frac{1}{\cosh^{2}\tau}-\log\left[2\cosh\tau\right]\right).$
(2.27)

In the limit $\tau\to\pm\infty$, this becomes the standard solution
$C_{0}(\tau)=-V^{\prime}|\tau|/3$ for a slow-rolling inflaton field in a flat
slicing with Hubble constant $H=\mathrm{sgn}(\tau)$. The inflaton field
naturally rolls towards lower value of the potential, at a terminal velocity
determined by “friction” from the expansion of space in the asymptotic past
and future.

The most general real solution to $\eqref{eq:eommasslessscalar}$ is a sum of
(2.27) and the general solution (2.15) to the homogeneous equation (2.8). Note
that there is no freedom in rescale the solution (2.27); its coefficient is
fixed by the slope of the linear potential.

As in Section 2.1, to solve for the Bunch-Davies wavefunctional we need to
look for solutions with given boundary conditions at $\tau=0$ that are regular
at the Euclidean south pole $\tau=i\pi/2$. The regular solution with boundary
conditions $\phi(0,\Omega_{3})=0$ is

$\displaystyle\phi(\tau,\Omega_{3})=C_{0}(\tau)+\frac{V^{\prime}}{3}\left[2iA_{0}(\tau)+(\log
2-1)\right]=\frac{V^{\prime}}{3}\Bigg{(}\frac{1+\mathrm{i}\sinh\tau}{\cosh^{2}\tau}-\log[1-\mathrm{i}\sinh\tau]-1\Bigg{)}.$
(2.28)

The regular solution with general boundary conditions is given by a linear
combination of (2.28) and the solution (2.20) to the homogeneous equation
(2.8) with those boundary conditions.

Plugging this solution into the action (2.2), we obtain

$\displaystyle\mathrm{i}S$ $\displaystyle=\mathrm{i}\int
d\tau\cosh^{3}\tau\Bigg{\\{}\frac{k(k+2)}{k+1}\bigg{[}\sum(\dot{A_{k}})^{2}\phi_{k\ell
m}\phi_{k\ell-m}-\frac{1}{\cosh^{2}\tau}\sum k(k+2)\,A_{k}^{2}\phi_{k\ell
m}\phi_{k\ell-m}\bigg{]}+2\pi^{2}\frac{1}{\sqrt{2}\pi}V^{\prime}\phi_{0}\Bigg{\\}}.$
(2.29)

The Bunch-Davies weight is therefore

$\displaystyle\Psi_{BD}[\phi(\Omega_{3})]\sim\exp\Bigg{\\{}-\frac{4\pi}{3\sqrt{2}}V^{\prime}\phi_{0}-\sum_{\begin{subarray}{c}k\neq
0\end{subarray}}\frac{k\big{(}k+2\big{)}}{2\big{(}k+1)}\phi_{k\ell-m}\phi_{k\ell
m}\Bigg{\\}}.$ (2.30)

So far, we have considered only the inflaton field and not any other fields,
including e.g. graviton excitations that may be present in the spacetime.
However this is easy to rectify. Because the different fields do not interact,
the full Bunch-Davies weight for a theory containing an inflaton and
additional quantum fields is simply a tensor product of the Bunch-Davies
weights for the individual decoupled sectors. For example, in the presence of
additional light scalar field the Bunch-Davies weight will be a tensor product
of (2.30) with (2.23).

The formula (2.30) was derived purely using quantum field theory. However,
because gravitational effects decouple in the limit we have taken, one obtains
the same result by taking the $G\to 0$ limit of the no-boundary Hartle-Hawking
state in quantum gravity. More precisely, given an observable that e.g.
projects onto an $O(1)$ range of inflaton values with potential
$V=\Lambda/8\pi G$, the $G\to 0$ limit of the expectation value of that
observable for the Hartle-Hawking state agrees with the corresponding
expectation value for the Bunch-Davies weight, up to an overall divergent
factor. In particular, the exponential factor proportional to $V^{\prime}$ in
(2.30) means that the weight is dominated by field values with a small
inflaton potential. This is the same phenomenon reviewed for the Hartle-
Hawking state in [maldacena2024comments]: the inflaton wants to already start
in equilibrium at the bottom of the potential rather than rolling down for a
long time.

An important check of our results is that the Bunch-Davies weight is invariant
under the $SO(4,1)$ isometry group of de Sitter space. Rotation invariance is
easy to check, and follows from the fact that (2.30) only depends explicitly
on $k$ and not on $\ell$ or $m$. To demonstrate full de Sitter invariance it
therefore suffices to check invariance under a single boost generator, which
for simplicity we take to be the generator that preserves the quantum numbers
$\ell$ and $m$.

To find the boost generator, it is easiest to use the embedding coordinate
$\\{X_{0},X_{1},\cdots,X_{4}\\},$ in which the de Sitter space is a
hyperboloid that satisfies $-X_{0}^{2}+\sum_{i=1}^{4}X_{i}^{2}=1.$ The boost
generator in embedding coordinate in the $i$-th direction is

$\displaystyle K_{i}=X_{i}\frac{\partial}{\partial
X_{0}}+X_{0}\frac{\partial}{\partial X_{i}}.$ (2.31)

The global slicing is given by

$\displaystyle X_{0}$ $\displaystyle=\sinh{\tau},$ (2.32) $\displaystyle
X_{i}$ $\displaystyle=\cosh{\tau}z_{i},$ (2.33)

where $\sum_{i=1}^{4}z_{i}=1,$ parametrizing the three-sphere. With this
parametrization, the boost generator becomes

$\displaystyle
K_{i}=z_{i}\,\partial_{\tau}+\sum_{j=1}^{4}\tanh{\tau}\,\Big{(}\delta_{ij}-z_{i}z_{j}\Big{)}\partial_{j}.$
(2.34)

In quantum theory, the boost Hamiltonian can be obtained from the boost
generator by substituting in the stress energy tensor and integrating over a
Cauchy slice so that

$\displaystyle H_{i}=\int
d^{3}x\sqrt{-h}z_{i}\,T^{0}{}_{0}+\sum_{j=1}^{4}\tanh{\tau}\,\Big{(}\delta_{ij}-z_{i}z_{j}\Big{)}T^{0}{}_{j}.$
(2.35)

with $d^{3}x\sqrt{-h}$ the volume form on the Cauchy slice.

Without loss of generality, we can set $z_{1}=\cos\chi$ and choose to focus on
the static patch centered at $\chi=0.$ Plugging in the expression for the
energy density $T^{0}{}_{0}$ and evaluating $H$ on the $\tau=0$ slice, the
corresponding boost Hamiltonian 555The boost Hamiltonian $H$ should not be
confused with the global Hamiltonian $H_{\rm gl}(\tau)$, which generates
translations in $\tau$. The latter is explicitly time dependent since global
time translations are not a symmetry of de Sitter space. is

$\displaystyle H$
$\displaystyle=\int\frac{1}{2}\,\cos\chi\bigg{(}(\partial_{\tau}\phi)^{2}+(\nabla_{S^{3}}\phi)^{2}+V^{\prime}\phi\bigg{)}$
(2.36)
$\displaystyle=\int\frac{1}{2}\,\cos\chi\bigg{(}\pi_{\phi}^{2}+(\nabla_{S^{3}}\phi)^{2}+V^{\prime}\phi\bigg{)}.$
(2.37)

In the second line we replaced $\partial_{\tau}\phi$ by the conjugate momentum
$\pi_{\phi}$. Expanding the boost Hamiltonian in terms of angular momentum
modes for the field $\phi$ as in (2.13) and the conjugate momentum field $\pi$
as in (2.17), this becomes

$\displaystyle H=$
$\displaystyle-\frac{1}{2}\Bigg{[}\pi_{0}\pi_{100}+\sum_{k>0}\sqrt{\frac{(k+\ell+2)(k-\ell+1)}{(k+1)(k+2)}}\bigg{(}\pi_{k\ell
m}\pi_{k+1\ell-m}+k(k+3)\phi_{k\ell
m}\phi_{k+1\ell-m}\bigg{)}\Bigg{]}-\frac{\pi}{\sqrt{2}}V_{\ast}^{\prime}\phi_{100},$
(2.38)

where we have used the orthogonality relations of the spherical harmonics
(LABEL:ortho).

In canonical quantization, the conjugate momenta get promoted to the
functional derivative of fields,

$\displaystyle\pi_{k\ell
m}\rightarrow-\mathrm{i}\frac{\delta}{\delta\phi_{k\ell m}}.$ (2.39)

The boost Hamiltonian then becomes

$\displaystyle H=$
$\displaystyle\frac{1}{2}\Bigg{[}\frac{\delta^{2}}{\delta\phi_{0}\delta\phi_{100}}+\sum_{k>0}\sqrt{\frac{(k+\ell+2)(k-\ell+1)}{(k+1)(k+2)}}\bigg{(}\frac{\delta^{2}}{\delta\phi_{k\ell
m}\delta\phi_{k+1\ell-m}}-k(k+3)\phi_{k\ell
m}\phi_{k+1\ell-m}\bigg{)}\Bigg{]}-\frac{\pi}{\sqrt{2}}V_{\ast}^{\prime}\phi_{100}.$
(2.40)

Applying (2.40) to (2.30), we find that

$\displaystyle H\Psi_{BD}=\Bigg{(}\bigg{[}\frac{1}{2}$
$\displaystyle\left(-\frac{4\pi
V^{\prime}}{\sqrt{2}\,3}\right)\left(-\frac{1(1+2)}{1+1}\right)-\frac{\pi}{\sqrt{2}}V^{\prime}\bigg{]}\phi_{100}$
$\displaystyle+\sum_{k>0}\bigg{[}\frac{k\big{(}k+2\big{)}}{k+1}\frac{\big{(}k+1\big{)}\big{(}k+3\big{)}}{k+2}-k(k+3)\bigg{]}\phi_{k\ell
m}\phi_{k+1\ell{-m}}\Bigg{)}\Psi_{BD}=0,$ (2.41)

so the Bunch-Davies weight is indeed boost invariant.

###  2.3 The algebra of observables

So far all of our discussion has not really been about quantum gravity at all,
but about quantum field theory in de Sitter space. However, despite the fact
that we took a strict $G\to 0$ limit, there continue to exist residual effects
of quantum gravity that go beyond the existence of a free massless spin-2
field. Specifically, the isometry group of the background de Sitter space acts
not as the physical symmetries of a quantum field theory but as quantum
gravity gauge constraints.

Our goal in this section is to explain how those constraints impact the
physical algebra of observables $\widetilde{\mathcal{A}}$ associated to a
single static patch $P$. In pure de Sitter space, there is no gauge-invariant
way to pick out a particular static patch $P$. Instead, we will assume that
$P$ is defined in a gauge-invariant way relative to the location of the
observer. This observer could either be present during inflationary era or,
perhaps more physically given the role inflation seems to have played in our
universe, they could exist only at some asymptotically late time. Regardless,
the location and velocity of the observer can then be used to define the patch
$P$. Crucially, unlike in [CLPW] we will only make use of the observer’s
(classical) location to define $P$ and will not additionally equip them with
any quantum Hilbert space to act as their clock.

The choice of $P$ breaks the isometric group of de Sitter space down to the
group ${{\mathbb{R}}}\times SO(3)$ of the boost- and rotation-isometries of
the patch $P$. The algebra $\widetilde{\mathcal{A}}$ is then the boost- and
rotation-invariant subalgebra of the QFT static patch algebra $\mathcal{A}$.

While there exist many rotation-invariant observables in the algebra
$\mathcal{A}$, an important part of the story of [CLPW] is that for quantum
fields in de Sitter space with a potential bounded from below the only boost-
invariant operators in the algebra $\mathcal{A}$ are $c$-numbers. This led to
the need to include an explicit model of a clock in order to obtain a
nontrivial algebra $\widetilde{\mathcal{A}}$.

This is not true, however, in the presence of an inflaton field. As we saw in
Section 2.2, a classical solution to the inflaton equations of motion will
tend to roll linearly down the potential gradient at early and late times as
described by (2.27). Perturbations to this solution are described by (2.15)
and have finite limits as $\tau\to\pm\infty$. So this behaviour is universal.

The late- and early-time behaviour of the quantum theory is related but
slightly more complicated. At arbitrarily late times short wavelength modes
with nonzero vacuum fluctuations continue to exit the horizon and contribute
to the effective classical background $\phi_{\rm cl}$ – the sum over all modes
that have exited the horizon – within the static patch. As a result, the
evolution of $\phi_{\rm cl}$ at long times becomes a Wiener process, with a
drift proportional to $V^{\prime}$ and stochastic noise from modes exiting the
horizon. The probability distribution $p(\phi_{\rm cl},t)$ as a function of
the time $t$ is therefore described by the Fokker-Planck equation [chaos1,
chaos2, chaos3]

$\displaystyle\frac{\partial}{\partial t}p(\phi_{\rm
cl})=\frac{V^{\prime}}{3}\frac{\partial}{\partial\phi}p(\phi_{\rm
cl},\tau)+\frac{1}{8\pi^{2}}\frac{\partial^{2}}{\partial\phi^{2}}p(\phi_{\rm
cl},\tau).$ (2.42)

It follows from (2.42) by integrating by parts that the expectation value
$\langle\phi_{\rm cl}\rangle$ satisfies

$\displaystyle\frac{\partial}{\partial t}\langle\phi_{\rm
cl}\rangle=-\frac{V^{\prime}}{3}$ (2.43)

while the variance $\delta\phi_{\rm cl}^{2}$ satisfies

$\displaystyle\frac{\partial}{\partial t}\delta\phi_{\rm
cl}^{2}=\frac{\partial}{\partial t}\left(\langle\phi_{\rm
cl}^{2}\rangle-\langle\phi_{\rm cl}\rangle^{2}\right)=\frac{1}{4\pi^{2}}.$
(2.44)

Since the late-time probability distribution is Gaussian by the central limit
theorem, this means that the probability $p(\phi_{\rm cl}>\phi_{0})$ that the
contribution to the inflaton field from modes that have exited the horizon
remains above fixed finite value $\phi_{0}$ decays exponentially at late
times. By a time-reversed version of the same argument, this is also true at
early times. In particular, this means that our argument in the introduction
that the operator (1.2) did not exist is no longer valid in the presence of an
inflaton. In the inflaton quantum field theory, the static patch never
equilibrates and so there is no reason that the integral over time in (1.2)
needs to diverge when acting on a normalisable state.

Naively, the evolution of $\phi_{\rm cl}$ seems in tension with the boost-
invariance of the Bunch-Davies weight. However, the above argument applies
only to normalisable probability distributions $p(\phi_{\rm cl})$ where we can
integrate by parts without running into boundary terms. In contrast, under the
field redefinition $\phi\to\phi+\kappa$ for some constant $\kappa$ (or
equivalently under $\phi_{0}\to\phi_{0}+\sqrt{2}\pi\kappa$), we see from
(2.30) that

$\displaystyle\ket{\Psi_{BD}}\to\exp\left(-\frac{4\pi^{2}}{3}\kappa\right)\ket{\Psi_{BD}}.$
(2.45)

It follows that the Bunch-Davies weight must have

$\displaystyle p(\phi_{\rm cl})\sim\exp(-\frac{8\pi^{2}V^{\prime}}{3}\phi_{\rm
cl}),$ (2.46)

which is of course unnormalisable. It is easy to check that (2.46) is in fact
preserved by the Fokker-Planck equation (2.42), consistent with the boost
invariance of $\ket{\Psi_{BD}}$.

Suppose we consider a QFT operator $a\in\mathcal{A}$ that decays to zero
faster than exponentially as $\phi_{\rm cl}\to-\infty$ (assuming modes with
subhorizon wavelength are in the Bunch-Davies state). For example, $a$ might
be the operator $\exp(-\phi_{f}^{2})$ defined in (2.25). Let
$a(t)=e^{iHt}ae^{-iHt}$ be the same operator but boosted by time $t$. Then for
any fixed state $\ket{\Phi}\in\mathcal{H}$, we have $\lVert
a(t)\ket{\Phi}\rVert\to 0$ exponentially fast as $|t|\to\infty$. This means
that the operator

$\displaystyle\widetilde{a}=\int dt\,a(t)$ (2.47)

is densely defined, boost invariant and has nontrivial action on the Hilbert
space $\mathcal{H}$. It will also have finite expectation value
$\braket{\Psi_{BD}}{a}{\Psi_{BD}}$ in the Bunch-Davies weight. Since operators
that behave like $a$ are dense in $\mathcal{A}$ with respect to the strong
operator topology, this leads to a large algebra
$\widetilde{\mathcal{A}}\subseteq\mathcal{A}$ of boost- (and rotation-)
invariant operators.

We will now argue that the Bunch-Davies weight can be used to construct a
normal semifinite trace for the algebra $\widetilde{\mathcal{A}}$. Suppose we
tried to construct a formal density matrix for the Bunch-Davies weight on
$\mathcal{A}$. To do so, we would take the Bunch-Davies wavefunctional and
partial trace over fields at $\chi>\pi/2$. This leads to the path integral
shown in Figure 3,

$e^{-\varepsilon\boldsymbol{h}}$$\chi=0$$\chi=\pi/2$$\chi=\pi$ Figure 3:  The
formal density matrix of the Bunch-Davies weight on the algebra $\mathcal{A}$
is computed by a Euclidean path integral over the full sphere, with fixed
boundary conditions on either side of a cut at the static patch centered on
$\chi=0$. This can be reinterpreted as a sequence of operators
$\exp(-\varepsilon\boldsymbol{h})$ generating Euclidean rotations (i.e. the
analytic continuation of Lorentzian boosts). The full density matrix describes
(up to normalisation) a Euclidean rotation by $2\pi$.

which can be reinterpreted as the path integral for a boost of the static
patch by Euclidean time $2\pi$. In other words we have

$\displaystyle\boldsymbol{\rho}\sim\exp(-2\pi\boldsymbol{h}+{\rm const})$
(2.48)

where

$\displaystyle H=\boldsymbol{h}-\boldsymbol{h^{\prime}}$ (2.49)

is a formal splitting of the boost Hamiltonian $H$ into singular Hamiltonians
$\boldsymbol{h}$ and $\boldsymbol{h^{\prime}}$ on the static patch and its
commutant respectively. A constant, that turns out to be divergent, has been
included because the density matrix needs to be normalised to have trace one.

The splitting (2.49) certainly exists with $\boldsymbol{h}$ and
$\boldsymbol{h}^{\prime}$ defined as integrals over the stress-energy tensor.
We can therefore also define $\boldsymbol{h}$ and $\boldsymbol{h}^{\prime}$ as
sesquilinear forms mapping dense subsets of bras and kets to their
correspoding matrix elements. However $\boldsymbol{h}$ and
$\boldsymbol{h}^{\prime}$ do not exist as densely defined operators on
$\mathcal{H}$ because $\boldsymbol{h}\ket{\Phi}$ is not normalisable for a
dense set of states $\ket{\Phi}\in\mathcal{H}$.666If a sesquilinear form
$\boldsymbol{h}$ is not actually an operator, it might be unclear what it
means to say that it is localised in the static patch $P$. What we mean by
this is that $\boldsymbol{h}$ is invariant under conjugation by any unitary
$U_{\ell}\in\mathcal{A}_{\ell}$. Equivalently, $\boldsymbol{h}$ defines a
linear functional on a dense set of states on $\mathcal{A}_{r}$. (Recall that
states on $\mathcal{A}$ are really themselves linear functionals on the
algebra $\mathcal{A}$. So $\boldsymbol{h}$ is a linear functional on the space
of linear functionals. To indicate this, we write them in bold font. As a
result, it does not make sense, except at a very formal level, to talk about
nonlinear functions of $\boldsymbol{h}$. However, one can consider the effect
of the action generated by the full boost Hamiltonian $H$ on the algebra
$\mathcal{A}$. It turns out that $H$ generates an (outer) automorphism of
$\mathcal{A}$, which is another sense in which $H$ morally splits as (2.49),
without literally doing so.

In a similar manner to (2.48), the density matrix $\boldsymbol{\rho^{\prime}}$
on the commutant static patch can be written as

$\displaystyle\boldsymbol{\rho^{\prime}}\sim\exp(-2\pi\boldsymbol{h^{\prime}}+{\rm
const}).$ (2.50)

By symmetry, we expect the divergent constants in (2.48) and (2.50) should be
the same. While the density matrices $\boldsymbol{\rho}$ and
$\boldsymbol{\rho^{\prime}}$ do not actually exist as we explained above, the
formal combination
$\Delta_{\Psi}=\boldsymbol{\rho}\otimes\boldsymbol{\rho^{\prime}}^{-1}$
describes a densely defined operator called the modular operator for the
semifinite weight $\ket{\Psi_{BD}}$. This operator can be rigorously defined
using Tomita-Takesaki theory, briefly reviewed in Appendix LABEL:appx:a. By
the formal manipulations above, we have

$\displaystyle\Delta_{\Psi}=\exp(-2\pi H).$ (2.51)

More rigorously, one can prove (2.51) using KMS condition. Given
$a,b\in\mathcal{A}$, the KMS condition says that

$\displaystyle\braket{\Psi_{BD}}{a(t)b}{\Psi_{BD}}=\braket{\Psi_{BD}}{b\,a(t+2\pi
i)}{\Psi_{BD}},$ (2.52)

where the right-hand side is defined by analytic continuation. (2.52) follows
immediately from the definition of $\ket{\Psi_{BD}}$ via a Euclidean path
integral because the left- and right-hand sides of (2.52) are defined by the
same path integral. Consider a path integral on the Euclidean sphere with
operators $b$ and $a$ inserted separated by a Euclidean angle $0>it>-2\pi$. By
analytic continuation, this correlation function can be extended to the strip
$0>{\rm Im}\,(t)>-2\pi$. In the limits ${\rm Im}\,(t)\to 0$ and ${\rm
Im}\,(t)\to-2\pi$, the difference in Euclidean time between the two insertions
goes to zero. But, as ${\rm Im}\,(t)\to 0$, the operator $b$ is inserted
slightly before $a$ in Euclidean time and the correlation function limits to
the left-hand side of (2.52). On the other hand, as ${\rm Im}\,(t)\to-2\pi$
the operator $a$ is inserted slightly prior to $b$ and the correlation
function limits to the right-hand side of (2.52). But, by Theorem 16 in
[Sorce_2023], (2.51) follows from the four conditions that a)
$\ket{\Psi_{BD}}$ is cyclic-separating on $\mathcal{A}$, b)
$H\ket{\Psi_{BD}}=0$, c) $H$ generates an automorphism of $\mathcal{A}$ and d)
(2.52).

Modulo a minor subtlety about normalisation, the Bunch-Davies weight will act
as a trace on the boost-invariant algebra $\widetilde{\mathcal{A}}$ because,
for boost-invariant states, the KMS condition (2.52) is identical to the
tracial condition

$\displaystyle\text{Tr}[\widetilde{a}\widetilde{b}]=\text{Tr}[\widetilde{b}\widetilde{a}].$
(2.53)

More precisely, suppose we have an operator
$\widetilde{a}\in\widetilde{\mathcal{A}}$ that can be written as (2.47). We
define the trace

$\displaystyle\text{Tr}(\widetilde{a})=\braket{\Psi_{BD}}{a(t)}{\Psi_{BD}}$
(2.54)

Since the Bunch-Davies weight is boost invariant, this is independent of the
time $t$ that we pick. Furthermore, suppose we have the operator equality

$\displaystyle\widetilde{a}=\int dt\,a_{0}(t)=\int dt\,a_{1}(t).$ (2.55)

If

$\displaystyle 0=\int dt\,\braket{\Psi_{BD}}{(a_{0}(t)-a_{1}(t))}{\Psi_{BD}},$
(2.56)

then the integrand must itself vanish because the Bunch-Davies weight is boost
invariant. Therefore (2.54) is not only independent of $t$ but also of the
choice of expansion (2.47).

We can write the product of two operators $\widetilde{a}$ and $\widetilde{b}$
as

$\displaystyle\widetilde{a}\widetilde{b}=\int
dt_{1}dt_{2}\,a(t_{1})b(t_{2})=\int dt\int dt^{\prime}\,a(t)b(t+t^{\prime}).$
(2.57)

But then

$\displaystyle\text{Tr}(\widetilde{a}\widetilde{b})$ $\displaystyle=\int
dt^{\prime}\braket{\Psi_{BD}}{a(t)b(t+t^{\prime})}{\Psi_{BD}}$ (2.58)
$\displaystyle=\int dt^{\prime}\braket{\Psi_{BD}}{b(t+t^{\prime}-2\pi
i)a(t)}{\Psi_{BD}}$ (2.59) $\displaystyle=\int
dt^{\prime}\braket{\Psi_{BD}}{b(t-t^{\prime\prime})a(t)}{\Psi_{BD}}$ (2.60)
$\displaystyle=\int
dt^{\prime\prime}\braket{\Psi_{BD}}{b(t)a(t+t^{\prime\prime})}{\Psi_{BD}}$
(2.61) $\displaystyle=\text{Tr}(\widetilde{b}\widetilde{a}).$ (2.62)

In the second equality we used the KMS condition, while in the third equality
we substituted $t^{\prime\prime}=2\pi i-t^{\prime}$ and then shifted the
contour of integration to real $t^{\prime\prime}$. An interesting feature of
this proof is that we never needed to assume rotation invariance of
$\widetilde{a}$ or $a(t)$. It is only the boost symmetry, i.e. the noncompact
part of the gauge group, that plays a nontrivial role.

It would be nice to express $\text{Tr}(\widetilde{a})$ directly as a linear
functional of the operator $\widetilde{a}$ rather than indirectly via $a(t)$.
The problem is that the boost invariance of $\ket{\Psi_{BD}}$ means that the
integral over $t$ in

$\displaystyle\braket{\Psi_{BD}}{\widetilde{a}}{\Psi_{BD}}=\int
dt\braket{\Psi_{BD}}{a(t)}{\Psi_{BD}}$ (2.63)

will always diverge. Since this divergent factor is independent of the
operator $\widetilde{a}$, we could hope remove it by a formal infinite
rescaling of $\ket{\Psi_{BD}}$. Indeed, this is exactly what the definition
(2.54) achieves.

However, we can also achieve the same result by regulating $\ket{\Psi_{BD}}$
to produce a normalisable state and then dividing through by a regulator-
dependent factor that goes to infinity as the regulator is removed. Explicitly
let $P_{\phi_{0}>\phi_{\rm min}}$ be the projector onto the spatial zero mode
$\phi_{0}>\phi_{\rm min}$ for some constant $\phi_{\rm min}$.
$P_{\phi_{0}>\phi_{\rm min}}\ket{\Psi_{BD}}$ is then a normalisable state. We
obviously have

$\displaystyle\lim_{\phi_{\rm
min}\to-\infty}\braket{\Psi_{BD}}{P_{\phi_{0}>\phi_{\rm
min}}\,a(t)\,P_{\phi_{0}>\phi_{\rm
min}}}{\Psi_{BD}}=\braket{\Psi_{BD}}{a(t)}{\Psi_{BD}}$ (2.64)

for any $a\in\mathcal{A}$. However, for any finite $\phi_{\rm min}$, the
expectation values $\braket{\Psi_{BD}}{P_{\phi_{0}>\phi_{\rm
min}}a(t)P_{\phi_{0}>\phi_{\rm min}}}{\Psi_{BD}}$ and
$\braket{\Psi_{BD}}{a(t)}{\Psi_{BD}}$ will be very different whenever $|t|$ is
sufficiently large. Since we are only interested in the behaviour of the
inflaton field at very late times, the relevant physics is captured by the
Fokker-Planck equation (2.42). A standard property of Wiener processes, which
can be obtained by solving the Fokker-Planck equation with a delta-function
initial condition, says that the conditional probability distribution

$\displaystyle p(\phi_{\rm cl},t|\phi_{{\rm
cl},0})=\frac{(2\pi)^{1/2}}{t}\exp(-\frac{2\pi^{2}(\phi_{\rm cl}-\phi_{{\rm
cl},0}+V^{\prime}t/3)^{2}}{t}),$ (2.65)

where the initial probability distribution is $p(\phi_{{\rm cl},0})$. For the
state $P_{\phi_{0}>\phi_{\rm min}}\ket{\Psi_{BD}}$ we have

$\displaystyle p(\phi_{{\rm
cl},0})=\exp(-\frac{8\pi^{2}V^{\prime}}{3}\phi_{{\rm cl},0})\theta(\phi_{{\rm
cl},0}-\phi_{\rm min}).$ (2.66)

We therefore obtain

$\displaystyle p(\phi_{\rm cl},t)$ $\displaystyle=\int d\phi_{{\rm
cl},0}\,p(\phi_{\rm cl},t|\phi_{{\rm cl},0})p(\phi_{{\rm cl},0})$
$\displaystyle=\exp(-\frac{8\pi^{2}V^{\prime}}{3}\phi_{\rm cl})\int
d\phi_{{\rm cl},0}\frac{(2\pi)^{1/2}}{t}\exp(-\frac{2\pi^{2}(\phi_{\rm
cl}-\phi_{{\rm cl},0}-V^{\prime}t/3)^{2}}{t})\theta(\phi_{{\rm
cl},0}-\phi_{\rm min})$
$\displaystyle=\exp(-\frac{8\pi^{2}V^{\prime}}{3}\phi_{\rm
cl})\,\Phi(\frac{2\pi(\phi_{\rm cl}-V^{\prime}t/3-\phi_{\rm min})}{\sqrt{t}})$
(2.67)

where $\Phi(z)=\int_{-\infty}^{z}dt\exp(-t^{2}/2)/\sqrt{2\pi}$ is the
cumulative probability distribution for the standard Gaussian. In other words,
we see that for any fixed $\phi_{\rm cl}$ and large negative $\phi_{\rm min}$,
the probability distribution $p(\phi_{\rm cl},t)$ looks like (2.46) for
$t\lesssim 3|\phi_{\rm min}|/V^{\prime}$ and then quickly goes to zero over a
comparatively short time $\Delta t\sim t^{1/2}$. By symmetry, the same
behaviour also occurs at very early times. We therefore have

$\displaystyle\text{Tr}(\widetilde{a})=\lim_{\phi_{\rm
min}\to-\infty}\frac{V^{\prime}}{6\,|\phi_{\rm
min}|}\braket{\Psi_{BD}}{P_{\phi_{0}>\phi_{\rm
min}}\widetilde{a}P_{\phi_{0}>\phi_{\rm min}}}{\Psi_{BD}},$ (2.68)

which gives our desired definition of $\text{Tr}(\widetilde{a})$ as an
explicit linear functional of $\widetilde{a}$. Note that (2.68) only defines a
semifinite normal weight on $\widetilde{\mathcal{A}}$; unlike the Bunch-Davies
weight $\ket{\Psi_{BD}}$ it does not form a semifinite, normal weight on
$\mathcal{A}$.

On general grounds, we expect that the algebra $\widetilde{\mathcal{A}}$ does
not contain any nontrivial centre and so is a von Neumann factor. Since it has
a trace, it cannot be Type III. Since the trace of the identity is infinite
(as can be easily verified from (2.68)), $\widetilde{\mathcal{A}}$ is not
finite-dimensional or Type II1. The remaining possibilities are Type I∞ or
Type II∞. Finally, we observe that the boost Hamiltonian $H$ is invariant
under the field transformation $\phi\to\phi+\kappa$ for constant $\kappa$. So
$\phi\to\phi+\kappa$ generates an automorphism of the algebra
$\widetilde{\mathcal{A}}$. As we already saw, that automorphism rescales the
trace Tr by a factor of $\exp(-8\pi^{2}V^{\prime}\kappa/3)$, which is
inconsistent with the existence of a minimal trace for projectors and hence
with a Type I∞ factor. We conclude that $\widetilde{\mathcal{A}}$ is Type II∞.

We have so far not talked about imposing the gauge constraints on the
commutant algebra $\mathcal{A}^{\prime}$. However it is obvious by symmetry
that this works in exactly the same way. The gauge-invariant algebras
$\widetilde{\mathcal{A}}\subseteq\mathcal{A}$ and
$\widetilde{\mathcal{A}}^{\prime}\subseteq\mathcal{A}^{\prime}$ commute
because they are contained in commuting algebras. However, as algebras acting
on the Hilbert space $\mathcal{H}$, they are not commutants, because e.g.
gauge transformations are nontrivial operators on $\mathcal{H}$ that commute
with both.

The example of gauge transformations makes it clear that the reason (or at
least one reason) $\widetilde{\mathcal{A}}$ and
$\widetilde{\mathcal{A}}^{\prime}$ are not commutants is that we have imposed
the gauge constraints on the algebras but not on the Hilbert space
$\mathcal{H}$. Because the gauge group is not compact, the constraints on
$\mathcal{H}$ need to be imposed using the method of coinvariants; see e.g.
Appendix B of [CLPW] for a review. This leads to a gauge-invariant Hilbert
space $\widetilde{\mathcal{H}}$ with natural actions of
$\widetilde{\mathcal{A}}$ and $\widetilde{\mathcal{A}}^{\prime}$.777We again
assume that only the symmetries of the static patch $P$ need to be imposed as
constraints, because the remaining symmetries of de Sitter space are broken by
the presence of an observer. It is easy to check that these actions still
commute. We would like to conjecture that in fact $\widetilde{\mathcal{A}}$
and $\widetilde{\mathcal{A}}^{\prime}$ are commutants when acting on
$\widetilde{\mathcal{H}}$. We certainly don’t know of any operators that
commute with both. But it is not true in general that if two algebras
$\mathcal{A}$ and $\mathcal{A}^{\prime}$ are commutants on a Hilbert space
$\mathcal{H}$ and we gauge an automorphism group of those algebras that the
invariant subalgebras $\widetilde{\mathcal{A}}\subseteq\mathcal{A}$ and
$\widetilde{\mathcal{A}}^{\prime}\subseteq\mathcal{A}^{\prime}$ act as
commutants on the coinvariant Hilbert space $\widetilde{\mathcal{H}}$. A
counterexample is given by stable quantum fields in de Sitter space, where
$\widetilde{\mathcal{A}}$ and $\widetilde{\mathcal{A}}^{\prime}$ are trivial
but the coinvariant Hilbert space $\widetilde{\mathcal{H}}$ and hence the
algebra of operators $\mathcal{B}(\widetilde{\mathcal{H}})$ are not.

###  2.4 Density matrices and entropies

In previous work, the trace of gravitational Type II von Neumann algebras
behaves as if there is a (divergent) density of states proportional to
$\exp(A_{\rm hor}/4G)$ with $A_{\rm hor}$ the area of the black hole or
cosmological horizon. We find the same result for the algebra
$\widetilde{\mathcal{A}}$.

We have already seen that shifting $\phi\to\phi+\kappa$ rescales Tr by
$\exp(-8\pi^{2}V^{\prime}\kappa/3)$. Changes in the de Sitter horizon area
$A_{\rm hor}$ are related to the changes in the cosmological constant
$\Lambda=3/\ell_{dS}^{2}$ by

$\displaystyle\delta A_{\rm
hor}=8\pi\delta\ell_{dS}=-\frac{4\pi}{3}\delta\Lambda=-\frac{32\pi^{2}}{3}G\delta
V.$ (2.69)

To absorb the constant shift in the QFT Lagrangian into the Einstein-Hilbert
action, and leave the total action invariant under the field redefinition, we
need to shift the cosmological constant $\Lambda$ by

$\displaystyle\delta\Lambda=8\pi G\delta V=8\pi GV^{\prime}\kappa.$ (2.70)

So the trace will be rescaled by

$\displaystyle\exp(-8\pi^{2}V^{\prime}\kappa/3)=\exp(-\delta A_{\rm hor}/4G),$
(2.71)

consistent with our claim above.

Furthermore, as we explained in Section 2.2, the Bunch-Davies weight can be
thought of as a $G\to 0$ limit of the no boundary Hartle-Hawking state (up to
an overall normalisation factor). Our results are therefore consistent with
the proposal of [witten2023background] that, beyond the strict $G\to 0$ limit,
the no-boundary Hartle-Hawking state acts as a trace on the gauge-invariant
algebra associated to an observer.

To obtain a more precise relationship between the Type II algebra
$\widetilde{\mathcal{A}}$ and the generalised entropy of its static patch, we
need to find the density matrix $\rho_{\Phi}$ associated with some state
$\ket{\Phi}$ for the gauge-invariant algebra $\widetilde{\mathcal{A}}$.

We assume for simplicity that the state $\ket{\Phi}$ is already rotation-
invariant. If not, then one can produce a rotation invariant state with the
same expectation values for all operators in $\widetilde{\mathcal{A}}$ by
simply integrating the state (i.e. the positive linear functional) on
$\mathcal{A}$ defined by $\ket{\Phi}$ over the rotation group and dividing by
its volume, which is finite because the rotation group is compact. This
defines a normalised rotation-invariant positive linear functional on
$\mathcal{A}$ which can then always be purified (for a Type III1 algebra) to a
rotation-invariant pure state on $\mathcal{H}$.

By definition, $\rho_{\Phi}$ is the unique operator affiliated to the algebra
$\widetilde{\mathcal{A}}$ such that

$\displaystyle\bra{\Phi}\widetilde{a}\ket{\Phi}=\text{Tr}(\rho_{\Phi}\,\widetilde{a}),\quad\forall\,\widetilde{a}\in\widetilde{\mathcal{A}}.$
(2.72)

We claim that

$\displaystyle\rho_{\Phi}=\int
dt\;e^{(\pi-\mathrm{i}t)H}\,\Delta_{\Phi|\Psi}\,e^{(\pi+\mathrm{i}t)H}=\int
dt\;\Delta_{\Psi}^{-\frac{1}{2}-\mathrm{i}\frac{t}{2\pi}}\,\Delta_{\Phi|\Psi}\,\Delta_{\Psi}^{-\frac{1}{2}+\mathrm{i}\frac{t}{2\pi}},$
(2.73)

where $\Delta_{\Phi|\Psi}$ is the relative modular operator of $\ket{\Phi}$
relative to $\ket{\Psi_{BD}}$ on the algebra $\mathcal{A}$. See Appendix
LABEL:appx:a for a brief review. Given an operator $a^{\prime}$ in the
commutant algebra $\mathcal{A}^{\prime}$, we have

$\displaystyle[\log\Delta_{\Phi|\Psi},a^{\prime}]=[\log\Delta_{\Psi},a^{\prime}]=-2\pi[H,a^{\prime}]$
(2.74)

It follows immediately that $[\rho_{\Phi},a^{\prime}]=0$ and hence
$\rho_{\Phi}$ affiliated to $\mathcal{A}$. Since

$\displaystyle[H,\rho_{\Phi}]=i\int dt\;\frac{\partial}{\partial
t}\left[e^{(\pi-\mathrm{i}t)H}\,\Delta_{\Phi|\Psi}\,e^{(\pi+\mathrm{i}t)H}\right]=0,$
(2.75)

$\rho_{\Phi}$ is also affiliated to the boost-invariant algebra
$\widetilde{\mathcal{A}}$. Finally, we have

$\displaystyle\text{Tr}(\rho_{\Phi}a)$ $\displaystyle=$ $\displaystyle=$
$\displaystyle=\braket{\Phi}{a}{\Phi}.$ (2.76)

In the second equality we used $[H,a]=0$ and $H\ket{\Psi_{BD}}=0$, and in the
last step we used (LABEL:eq:appxeq).

In Appendix LABEL:appx:a, we review proofs of two other standard properties of
the relative modular operator that we will need: a) given a reference system
$\mathcal{H}_{R}$ and a state
$\ket{\Phi}=\ket{\Phi_{0}}\ket{0}+\ket{\Phi_{1}}\ket{1}\in\mathcal{H}\otimes\mathcal{H}_{R}$,
the relative modular operator
$\Delta_{\Phi|\Psi}=\Delta_{\Phi_{0}|\Psi}+\Delta_{\Phi_{1}|\Psi}$ and b) if
we define the state $\ket{\Phi(s)}=\Delta_{\Psi}^{-is}\ket{\Phi}$, then

$\displaystyle\Delta_{\Phi(s)|\Psi}=\Delta_{\Psi}^{-is}\Delta_{\Phi|\Psi}\Delta_{\Psi}^{is}.$
(2.77)

We therefore have

$\displaystyle\rho_{\Phi}=\int dte^{\pi H}\Delta_{\Phi(t)|\Psi}e^{\pi
H}=e^{\pi H}\Delta_{\widetilde{\Phi}|\Psi}e^{\pi H},$ (2.78)

where $\ket{\Phi(t)}=e^{-itH}\ket{\Phi}$ and the weight
$\ket{\widetilde{\Phi}}$ is defined such that

$\displaystyle\braket{\widetilde{\Phi}}{a}{\widetilde{\Phi}}=\int
dt\braket{\Phi(t)}{a}{\Phi(t)}\int dt\braket{\Phi}{a(t)}{\Phi},$ (2.79)

The functional (2.79) is finite whenever $a$ defines a boost-invariant
operator $a$ by (2.47). We already argued that this was true for a dense set
of operators $a$, which means the weight $\ket{\widetilde{\Phi}}$ is
semifinite.

Since the expectation value (2.79) is boost invariant, we have
$[\Delta_{\widetilde{\Phi}|\Psi},H]=0$. We therefore have

$\displaystyle\log\rho_{\Phi}=\log\Delta_{\widetilde{\Phi}|\Psi}-\log\Delta_{\Psi}=\log\Delta_{\widetilde{\Phi}}-\log\Delta_{\Psi|\widetilde{\Phi}},$
(2.80)

where the second equality is a standard identity that follows from taking a
derivative of Connes’ cocycle flow. The entropy of the state $\ket{\Phi}$ on
the algebra $\widetilde{\mathcal{A}}$ is therefore

$\displaystyle
S(\Phi)=-\braket{\Phi}{\log\rho_{\Phi}}{\Phi}=\braket{\Phi}{\log\Delta_{\Psi}{\widetilde{\Phi}}|\Phi}{,}$
(2.81)

where the expectation of the first term from (2.80) vanishes because
$\log\Delta_{\widetilde{\Phi}}$ is boost invariant and
$\braket{\widetilde{\Phi}}{\log\Delta_{\widetilde{\Phi}}}{\widetilde{\Phi}}$
vanishes by definition.

Since $\log\Delta_{\Psi|\widetilde{\Phi}}$ is boost invariant, the formula
(2.81) can be formally interpreted as the relative entropy of
$\ket{\widetilde{\Phi}}$ with respect to $\ket{\Psi_{BD}}$, divided by a
divergent factor equal to the squared norm of $\ket{\widetilde{\Phi}}$. There
is a general relationship between relative entropy and generalised entropy
that was first developed by Casini [Casini_2008] and then later Wall
[Wall_2012]. We can formally write

$\displaystyle\log\Delta_{\Psi|\widetilde{\Phi}}=\log\boldsymbol{\rho}_{\Psi}-\log\boldsymbol{\rho}_{\widetilde{\Phi}}=2\pi\boldsymbol{h}-\log\boldsymbol{\rho}_{\widetilde{\Phi}}^{\prime}+{\rm
const},$ (2.82)

where $\boldsymbol{\rho}_{\Psi}$ and
$\boldsymbol{\rho}_{\widetilde{\Phi}}^{\prime}$ after formal density matrices
for $\ket{\Psi_{BD}}$ and $\ket{\widetilde{\Phi}}$ on $\mathcal{A}$ and
$\mathcal{A}^{\prime}$ respectively and the constant is again divergent. It
follows that

$\displaystyle\braket{\Phi}{\log\Delta_{\Psi}{\widetilde{\Phi}}|\Phi}{=}2\pi\braket{\Phi}{\boldsymbol{h}}{\Phi}-\braket{\Phi}{\log\boldsymbol{\rho}_{\widetilde{\Phi}}^{\prime}}{\Phi}+{\rm
const},$ (2.83)

Up to a divergent additive constant, the second term in (2.83) is the entropy
of $\ket{\widetilde{\Phi}}$ divided by its divergent normalisation. Meanwhile,
comparing the surface integral and volume formulas for the Komar mass of the
static patch, we have

$\displaystyle\boldsymbol{\hat{A}}_{\rm hor}-4\pi=8\pi G\boldsymbol{h}.$
(2.84)

The first term in (2.83) is therefore equal to
$\braket{\Phi}{\boldsymbol{\hat{A}}_{\rm hor}/4G}{\Phi}$ minus another
divergent additive constant. We conclude that the Type II∞ entropy $S(\Phi)$
is equal to the generalised entropy for the boost-invariant weight
$\ket{\widetilde{\Phi}}$ up to the divergent constant needed to render Type II
entropies finite.

##  3 Black holes

###  3.1 Black holes in asymptotically flat spacetimes

We now turn from slow-rolling inflaton to our second example of a cosmological
clock – an evaporating black hole. As a warm up, we first consider black holes
in asymptotically flat spacetimes. Our discussion will closely follow
[Witten_2022, CPW], although some details are new. Our construction can be
applied to any stationary two-sided black hole solution including black holes
that break spherical symmetry (i.e. Kerr or Kerr-Newman metrics), which were
first studied in this context in [kudlerflam2024generalized].

As in Section 2, we take a strict $G\to 0$ limit where fluctuations of free
quantum fields, including fluctuations in the semiclassical gravitational
field $h_{\mu\nu}=\sqrt{G}g_{\mu\nu}$ are described in the $G\to 0$ limit by
local quantum field theory on a fixed black hole background.

Unlike in Section 2, we are now working with quantum field theory in a curved
spacetime with noncompact Cauchy slices. As a result, we have to be somewhat
more careful when defining our Hilbert space. Because the spacetime is
asymptotically flat, there exists a particularly natural choice of Hilbert
space that we will call $\mathcal{H}_{\rm QFT}$. Heuristically, this Hilbert
space consists of states that look like the Minkowski vacuum both close to the
black hole bifurcation surface and near to spatial infinity. Like the Hilbert
space for an inflaton in Section 2, the Hilbert space $\mathcal{H}_{\rm QFT}$
does not contain any normalisable preferred states that are invariant under
the black hole isometry group. The Hartle-Hawking state, for example,
satisfies the first requirement but not the second, because it looks like a
thermal state far from the black hole. The Boulwaré vacuum on the other hand
satisfies the second requirement but not the first. Instead all states in
$\mathcal{H}_{\rm QFT}$ will look like an evaporating black hole at
sufficiently late times. At sufficiently early times, meanwhile, they look
like a white hole that is steadily absorbing thermal radiation and thereby
growing in size. Since this is the time reverse of an evaporating black hole,
we will call it an evaporating white hole; it is evaporating in the direction
of the thermodynamic arrow of time (i.e. towards the past).

A somewhat more careful definition of the Hilbert space $\mathcal{H}_{\rm
QFT}$, using the constructions of [Wald:1995yp, witten2022does], is as
follows. We pick a Cauchy slice $\Sigma$ of the asymptotically flat black hole
and deform the spacetime outside of a small neighbourhood of $\Sigma$ so that
it is given by a time-independent metric $g_{+}$ sufficiently far in the
future and by a potentially different metric $g_{-}$ sufficiently far in the
past. See Figure 4. Specifically, we choose the metrics $g_{-}$ and $g_{+}$ to
agree with the time-independent metrics of the left and right black hole
outside of some finite radius, while connecting those regions by some smooth
geometry so that there exists a globally timelike Killing vector.

$\Sigma$ (a)

$\Sigma$$g_{+}$$g_{-}$ (b)

Figure 4:  The metric near a Cauchy slice $\Sigma$ (the dark-blue region) of a
Schwarzschild (or other asymptotically flat) black hole (left) can be smoothly
connected to time-independent metrics on an asymptotically flat space (the
light-blue region on the right). This allows us to construct an unambiguous
Hilbert space $\mathcal{H}_{\rm QFT}$ for the black hole.

Since the metrics $g_{-}$ and $g_{+}$ are time independent, we can separate
solutions to the equations of motion in those regions into positive and
negative frequencies. This allows us to define Hilbert spaces
$\mathcal{H}_{g_{-}}$ and $\mathcal{H}_{g_{+}}$ as Fock spaces. Moreover, the
transition amplitudes $\mathcal{H}_{g_{-}}\to\mathcal{H}_{g_{+}}$ are well
defined and unitary. This is because the full deformed spacetime is smooth and
time independent outside of a finite region. The former property ensures that
the transition amplitude is not UV divergent from particle production at
asymptotically high energies, while the latter means that there are no IR
issues from particle production at asymptotic infinity. As a result, one
obtains the same Hilbert space $\mathcal{H}_{\rm QFT}$ in a neighbourhood of
$\Sigma$ by either evolving states forwards from $\mathcal{H}_{g_{-}}$ or
backwards from $\mathcal{H}_{g_{+}}$. Since $g_{+}$ and $g_{-}$ can be
independently varied, it follows that the Hilbert space constructed in this
way is independent of the choice of $g_{\pm}$.

To be clear, the Hilbert space constructed in this way is not the only Hilbert
space that one might reasonably consider. For Schwarzschild black holes,
another obvious choice is to construct a GNS Hilbert space using the Hartle-
Hawking state. This does not contain states that look like the Minkowski
vacuum far from the black hole. Instead all states will asymptote to a thermal
state at the Hawking temperature, purified by thermal modes in the other
asymptotic region. In other words, far from the black hole the Hartle-Hawking
GNS Hilbert space becomes the thermofield double Hilbert space described in
Section 3 of [witten2022does]. In quantum gravity, thermal states with
infinite extent cannot exist in an asymptotically flat spacetime at finite $G$
because of backreaction issues: if the radius $r_{max}$ of the thermal bath
becomes sufficiently large, the system will collapse to form a larger black
hole. To construct a semiclassical limit of quantum gravity that reproduces
the Hartle-Hawking GNS Hilbert space, one would therefore need to take a
double scaling limit where at finite $G$ states only look like the Hartle-
Hawking state at radii $r_{BH}\ll r\ll r_{max}$. After first taking $G\to 0$
one could then safely take $r_{max}\to\infty$.

A related issue is that even after taking $G\to 0$, both the expectation value
and fluctuations of the ADM mass of a spacetime with quantum fields in the
Hartle-Hawking state receive divergent contributions from thermal modes far
from the black hole. To obtain a Type II von Neumann algebra, we need the
fluctuations in the ADM mass to be finite. But, if the quantum fields are in
the Hartle-Hawking state, this can only happen if the divergent fluctuations
in the energy of thermal modes far from the black hole are cancelled by
divergent fluctuations in the quasilocal mass of the black hole at finite
distances from the horizon. This is not only somewhat physically unnatural
condition but also means that means that the ADM mass cannot be measured (even
approximately) by an observer at finite radius.

For Kerr black holes, the analogue of the Hartle-Hawking state, sometimes
called the Frolov-Thorne state, cannot exist at all as a normalisable state.
This is because the near-horizon boost symmetry generator becomes spacelike
sufficiently far from the black hole. On the other hand, the Unruh state,
where ingoing modes from infinity are in the Boulwarè vacuum while outgoing
near horizon modes are in the Hartle-Hawking state, does exist and one can
construct a GNS Hilbert space from it. This Hilbert space was used in
[kudlerflam2024generalized] to construct gravitational algebras for Kerr black
holes.

Unlike perturbations of the Hartle-Hawking state, perturbations of the Unruh
state do not run into gravitational backreaction issues at large radius,
because only a small number of low angular momentum Hawking modes escape the
near-horizon region and radiate to infinity. However, the Unruh state still
has the divergent contributions to the ADM mass from modes asymptotically far
from the black hole. Additionally, the Unruh state, and perturbations thereof,
are singular at the white hole horizon. As a result, the Unruh state is only
expected to describe a late-time approximation to equilibrating states such as
black holes formed from collapse; the _global_ Unruh state is not expected to
exist as a well defined state in nonperturbative quantum gravity. To obtain
the Hilbert space described in [kudlerflam2024generalized] in a $G\to 0$ limit
of quantum gravity, one would need to e.g. a) fix a semclassical method of
forming a black hole from collapse, b) take the coupling $G\to 0$, and then c)
take the time at which the collapse occurred to the infinite past. This can
certainly be done: indeed a similar limit was described explicitly in Section
4 of [CPW] in the context of asymptotically-AdS black holes. However, it is a
somewhat convoluted procedure.

In contrast to the issues above, we expect that any semiclassical state
$\ket{\Phi}$ in $\mathcal{H}_{\rm QFT}$ should exist in nonperturbative
quantum gravity at any sufficiently small but finite $G$. By this, we mean
that there should exist a state $\ket{\widetilde{\Phi}}$ in the
nonperturbative quantum gravity Hilbert space that is defined uniquely up to
perturbatively small corrections and that has the same expectation values
$\ket{\Phi}$ for all semiclassical observables that do not probe either a) the
singularities or b) parametrically early or late times where backreaction from
the evaporation becomes large. In particular, there is no need to take any
double scaling limit like the ones above. Finally, for states in
$\mathcal{H}_{\rm QFT}$, the total energy of modes far from the black hole
will be finite. As a result the ADM mass of the spacetime can be obtained as a
limit (in the strong operator topology) of quasilocal observables.

The Hilbert space $\mathcal{H}_{\rm QFT}$ describes a perfectly good
semiclassical limit of quantum gravity where all metric fluctuations are
$O(\sqrt{G})$ and hence vanish as $G\to 0$. However, there is a particular
mode of the metric, called the timeshift mode $T$, that describes the
asymptotic right boundary time reached by starting at $t=0$ on the left
boundary and travelling along a spacelike geodesic orthogonal to the time-
translation Killing vector through the Einstein-Rosen bridge.888For Kerr (or
Kerr-Newman) black holes, the time translation here should be replace by the
isometry of the Kerr black hole that looks like a boost generator near the
horizon. As described above, at asymptotic infinity this is actually becomes a
combination of a time translation and a rotation and so is actually spacelike.
The operator $G^{-1/2}T$ has a finite $G\to 0$ limit acting on the Hilbert
space $\mathcal{H}_{\rm QFT}$. We can therefore write

$\displaystyle\mathcal{H}_{\rm QFT}\cong\mathcal{H}_{0}\otimes
L^{2}({{\mathbb{R}}}),$ (3.1)

where $L^{2}({{\mathbb{R}}})$ describes the timeshift (with $G^{-1/2}T$ acting
as the position operator), while ${\mathcal{H}}_{0}$ describes the Hilbert
space of quantum fluctuations of matter and graviton fields with the timeshift
frozen at zero.999There are also other similar modes associated to the
rotational symmetry of the black hole solution. Those modes turn out to be
comparatively unimportant. In particular their treatment does not affect the
type classification of the algebra of observables; see [Witten_2022, CPW] for
a detailed discussion. For simplicity of presentation, we will assume that,
like the timeshift mode, the rotational modes are frozen when defining the
Hilbert space $\mathcal{H}_{0}$ and we will not include operators that excite
them in our algebra. (Since the rotational modes are included in
$\mathcal{H}_{\rm QFT}$, there should therefore really be an additional factor
of $L^{2}({{\mathbb{R}}}^{k})$ (with $k$ the dimension of the rotation group)
on the right-hand side of (3.1).) The timeshift mode is locally pure gauge; it
only affects the relationship between times at the left and right boundaries.
As a result, we can consider a Hilbert space where the timeshift mode has
$O(1)$ fluctuations without having to running into issues from nonperturbative
backreaction on the spacetime.

The timeshift mode is canonically conjugate to both the left and right ADM
masses $H_{L/R}$.101010Again, for Kerr black holes, the ADM masses here should
be replaced by the asymptotic charges associated to the boost isometry. At
leading order in $G$, the left and right ADM mass are equal to a constant
value $E_{0}=O(1/G)$ determined by the mass of the black hole background.
Generically, $O(G^{1/2})$ metric fluctuations lead to $O(G^{-1/2})$
fluctuations in the ADM mass. Since backreaction from matter fields and
graviton excitations only changes the ADM mass at $O(1)$, this leads to an
operator $X=\sqrt{G}(H_{L}-E_{0})=\sqrt{G}(H_{R}-E_{0})$, defined by rescaling
and shifting either the left or right ADM mass, that is finite in the $G\to 0$
limit and observable at both the left and right boundaries. In fact, it is in
the joint center of the left and right boundary algebras acting on the Hilbert
space $\mathcal{H}$.

However, if fluctuations $\Delta T$ in the timeshift $T$ are $O(1)$, the
fluctuations $\scalebox{0.8}{$\Delta$}H_{L/R}$ can also be $O(1)$ without
violating the uncertainty principle $\scalebox{0.8}{$\Delta$}H_{L/R}\,\Delta
T\geq 1/2$. We can then define renormalised ADM masses $h_{L/R}=H_{L/R}-E_{0}$
that have a finite $G\to 0$ limit. These act on a (different!) semiclassical
Hilbert space $\mathcal{H}$ that can again be written as

$\displaystyle\mathcal{H}\cong\mathcal{H}_{0}\otimes L^{2}({{\mathbb{R}}})$
(3.2)

but where the position operator on $L^{2}({{\mathbb{R}}})$ is now the
timeshift $T$ without any rescaling. It is convenient to choose a gauge where
bulk coordinates, and in particular the location of bulk QFT operators acting
on $\mathcal{H}_{0}$, are defined relative to right boundary coordinates. This
is commonly described as operators on $\mathcal{H}_{0}$ being “dressed” to the
right boundary. In this gauge, the timeshift $T$ is defined as the bulk time
at which the left boundary time is zero. The (renormalised) left ADM mass
$h_{R}$ then acts only on $L^{2}({{\mathbb{R}}})$ as the momentum operator
$-i\partial_{T}.$

In contrast, the right ADM mass acts nontrivially on the Hilbert space
$\mathcal{H}_{0}$. The difference between the two ADM masses can be written as

$\displaystyle H_{R}-H_{L}=h_{R}-h_{L}=H$ (3.3)

where $H$ acts on the quantum fields $\mathcal{H}_{0}$ as the boost generator
(i.e. as the generator of Schwarzschild time translations in the case of a
Schwarzschild black holes). For AdS-Schwarzschild black holes, where the
Hartle-Hawking state is contained in the natural QFT Hilbert space and exists
in quantum gravity at small but finite $G$, we have

$\displaystyle\beta H=-\log\Delta_{\Psi}$ (3.4)

where $\Delta_{\Psi}$ is the modular operator for the Hartle-Hawking state on
the right exterior QFT algebra and $\beta$ is the inverse temperature of the
black hole.

For an asymptotically flat black hole, $H$ is _not_ directly related to
$\log\Delta_{\Phi}$ for any normalisable state in $\ket{\Psi}\in\mathcal{H}$.
However, given any state $\ket{\Phi}\in\mathcal{H}$ that is cyclic and
separating on the right exterior, we _can_ write

$\displaystyle\beta
H=-\log\Delta_{\Phi}+\beta\scalebox{0.8}{$\Delta$}H_{\Phi,r}-\beta\scalebox{0.8}{$\Delta$}H_{\Phi,\ell},$
(3.5)

where $\scalebox{0.8}{$\Delta$}H_{\Phi,r}$ and
$\scalebox{0.8}{$\Delta$}H_{\Phi,\ell}$ are densely defined operators (and not
just sesquilinear forms!) on $\mathcal{H}$ localised in the left and right
exteriors respectively. The operators $\scalebox{0.8}{$\Delta$}H_{\Phi,r}$ and
$\scalebox{0.8}{$\Delta$}H_{\Phi,\ell}$ are unique up to the addition of a
common $c$-number. The splitting (3.5) is possible because the obstruction to
writing $H$ directly as a sum of operators in the left and right exteriors
comes from local physics near the horizon. And near the horizon all states
$\ket{\Phi}\in\mathcal{H}$ look like the Hartle-Hawking state, so that
$\log\Delta_{\Phi}$ and $\beta H$ act in the same way. Far from the horizon,
$\beta H$ and $\log\Delta_{\Phi}$ of course act very differently; these
differences are captured by the operators $\scalebox{0.8}{$\Delta$}H_{\Phi,r}$
and $\scalebox{0.8}{$\Delta$}H_{\Phi,\ell}$.

The operators appearing in (3.5) are unbounded and hence are only defined on a
dense subset of Hilbert space. To make the above discussion completely
rigorous we would need to specify those domains explicitly. Instead, however,
we can rephrase things entirely in terms of bounded operators as follows. As
in Section 2, the Hilbert space $\mathcal{H}_{0}$ is acted on by two Type III
von Neumann factors $\mathcal{A}_{\ell}$ and its commutant
$\mathcal{A}_{r}=\mathcal{A}_{\ell}^{\prime}$ that describe bounded operators
localised in the left and right exteriors respectively.111111If we had not
frozen the modes associated to relative rotations of the left and right
boundaries when defining $\mathcal{H}_{0}$, these would form a common center
to left and right exterior algebras. The center would disappear if we took a
limit where the fluctuations in the relative rotations were $O(1)$, which
leads to algebras that are the crossed product of the factors
$\mathcal{A}_{\ell}$ and $\mathcal{A}_{r}$ by the rotation group [Witten_2022,
CPW]. One can also obtain nontrivial center if the QFT Hilbert space contains
multiple superselection sectors. We claim that one can write

$\displaystyle
e^{-iHt}\Delta_{\Phi}^{-it/\beta}=U_{\Phi,\ell}(t)\,U_{\Phi,r}(t)\,$ (3.6)

where $U_{\Phi,\ell}(t)\in\mathcal{A}_{\ell}$ and
$U_{\Phi,r}(t)\in\mathcal{A}_{r}$ are unitary operators in the left and right
exteriors respectively. Because the intersection
$\mathcal{A}_{\ell}\cap\mathcal{A}_{r}$ is trivial, the unitaries
$U_{\Phi,\ell}(t)$ and $U_{\Phi,r}(t)$ (if they exist) must be unique up to a
shift of a phase between the two.

If the Hartle-Hawking state $\ket{\Psi}$ (i.e. a state satisfying (3.4))
existed as a normalisable state in the Hilbert space $\mathcal{H}$, the
operators

$\displaystyle
U_{\Phi,\ell}(t)=\Delta_{\Psi}^{it/\beta}\Delta_{\Psi|\Phi}^{-it/\beta}=\Delta_{\Phi|\Psi}^{it/\beta}\Delta_{\Phi}^{-it/\beta}~{}~{}~{}~{}~{}~{}\text{and}~{}~{}~{}~{}~{}~{}U_{\Phi,r}(t)=\Delta_{\Psi}^{it/\beta}\Delta_{\Phi|\Psi}^{-it/\beta}=\Delta_{\Psi|\Phi}^{it/\beta}\Delta_{\Phi}^{-it/\beta}$
(3.7)

would be Connes’ cocycle flows for $\ket{\Phi}$ relative to $\ket{\Psi}$. In
that case it is a standard result of Tomita-Takesaki theory that
$U_{\Phi,\ell}(t)\in\mathcal{A}_{\ell}$ and $U_{\Phi,r}(t)\in\mathcal{A}_{r}$
and that all the definitions given in (3.7) agree. Since the Hartle-Hawking
state is not contained in $\mathcal{H}_{0}$, we cannot directly use those
results. However, the reason the Hartle-Hawking state fails to be contained in
$\mathcal{H}_{0}$ only involves its properties at asymptotic infinity, and not
the behaviour of $\ket{\Psi}$ (or of the boost Hamiltonian $H$) in the near-
horizon region. The latter is all we need to determine whether
$e^{-iHt}\Delta_{\Phi}^{-it/\beta}$ splits into a product of operators in the
left and right algebras. So we can still write (3.6) even though
$\ket{\Psi}\not\in\mathcal{H}_{0}$. Taking the derivative of (3.6) at $t=0$
leads to our original claim (3.5), with

$\displaystyle\scalebox{0.8}{$\Delta$}H_{\Phi,\ell}=\lim_{t\to
0}\frac{U_{\Phi,\ell}(t)-1}{it}~{}~{}~{}\text{and}~{}~{}~{}\scalebox{0.8}{$\Delta$}H_{\Phi,r}=\lim_{t\to
0}\frac{U_{\Phi,r}(t)-1}{-it}$ (3.8)

which are unbounded operators affiliated to $\mathcal{A}_{\ell}$ and
$\mathcal{A}_{r}$ respectively. One can then easily check that

$\displaystyle U_{\Phi,\ell}(t)$ $\displaystyle=e^{-iHt}e^{i\Lambda_{\Phi,\ell
t}}=e^{-i\Lambda_{\Phi,r}t}\Delta_{\Phi}^{-it/\beta}$ (3.9) $\displaystyle
U_{\Phi,r}(t)$
$\displaystyle=e^{-iHt}e^{i\Lambda_{\Phi,r}t)}=e^{-i\Lambda_{\Phi,\ell}t}\Delta_{\Phi}^{-it/\beta},$
(3.10)

where $\Lambda_{\Phi,\ell}=H+\scalebox{0.8}{$\Delta$}H_{\Phi,\ell}$ generates
modular flows of $\mathcal{A}_{\ell}$ and boosts of $\mathcal{A}_{r}$, while
$\Lambda_{\Phi,r}=H-\scalebox{0.8}{$\Delta$}H_{\Phi,r}$ generates boosts of
$\mathcal{A}_{\ell}$ and modular flows of $\mathcal{A}_{r}$. To see this, note
that both formulas for $U_{\Phi,\ell}(t)$ given in (3.9) commute with
$\mathcal{A}_{r}$ (and hence are contained in $\mathcal{A}_{\ell}$) and
generate the same automorphism for $\mathcal{A}_{\ell}$ as (3.6). Meanwhile
the formulas for $U_{\Phi,r}(t)$ given in (3.9) commute with
$\mathcal{A}_{\ell}$ and generate the correct automorphism for
$\mathcal{A}_{r}$.

In our later discussion of Schwarzschild-de Sitter black holes, we will often,
out of practical convenience, manipulate unbounded densely defined operators
without being careful to define their domains. However, those manipulations
should always be possible to make rigorous by similar methods to those we have
just used.

The full gravitational algebra $\mathcal{A}_{R}$ of right exterior observables
is generated by $\mathcal{A}_{r}$ along with (bounded functions of) $h_{R}$.
But, since we have already established (3.6), this is the same as the crossed
product algebra $\mathcal{A}_{R}$ generated by $\mathcal{A}_{r}$ along with
(bounded functions of) $\Delta_{\Phi}$.

An alternative way to understand this result is the following. Classically,
the horizon area $A_{\rm hor}$ generates timeshifts between the left and right
boundaries via the Hamiltonian flow $\exp(\\{A_{\rm hor},\cdot\\}t/\beta)$ but
preserves the left and right exteriors separately. Quantum mechanically, the
operator $\boldsymbol{\hat{A}}_{\rm hor}/4G$ is UV-divergent. However the
operator $\boldsymbol{\hat{A}}_{\rm hor}/4G-\log\boldsymbol{\rho}_{\Phi,r}$,
with $\boldsymbol{\rho}_{\Phi,r}$ a (formal) QFT density matrix on the right
exterior, is UV finite. In the $G\to 0$ limit we need to subtract a divergent
constant to obtain a finite operator

$\displaystyle\beta X_{\Phi,R}=\frac{\boldsymbol{\hat{A}}_{\rm
hor}}{4G}-\log\boldsymbol{\rho}_{\Phi,r}-\mathrm{const}.$ (3.11)

We only need to add this operator for one state $\ket{\Phi}$ because, for any
pair of states $\ket{\Phi_{1}}$, $\ket{\Phi_{2}}$, the difference
$(X_{\Phi_{1},R}-X_{\Phi_{2},R})$ is already contained in the QFT algebra
$\mathcal{A}_{r}$. We therefore obtain the modular crossed product algebra
$\mathcal{A}_{R}$ that we just described. In fact the operator $X_{\Phi,R}$
can be written (up to an arbitrary constant) as

$\displaystyle X_{\Phi,R}=h_{R}-\scalebox{0.8}{$\Delta$}H_{\Phi,r}.$ (3.12)

Interesting, the ADM mass itself did not show up anywhere in this version of
the derivation. Instead, we just used the fact that (3.11) is a UV-finite
operator that is localised in the right exterior. An advantage of this sort of
approach is therefore that it can be applied in cosmological spacetimes where
there are no asymptotic charges.

Since the crossed product algebra $\mathcal{A}_{R}$ is a Type II∞ von Neumann
factor, it has a unique semifinite trace. One way to describe this trace is as
follows. We have

$\displaystyle
U_{\Phi,\ell}(T)^{\dagger}X_{\Phi,R}U_{\Phi,\ell}(T)=U_{\Phi,\ell}(T)^{\dagger}\left(h_{L}+\Lambda_{\Phi,r}\right)U_{\Phi,\ell}(T)=\Delta_{\Phi}^{iT/\beta}h_{L}\Delta_{\Phi}^{-iT/\beta}=h_{L}-\frac{1}{\beta}\log\Delta_{\Phi}.$
(3.13)

We also have
$U_{\Phi,\ell}(T)^{\dagger}\mathcal{A}_{r}U_{\Phi,\ell}(T)=\mathcal{A}_{r}$.
The algebra $U_{\Phi,\ell}(T)^{\dagger}\mathcal{A}_{R}U_{\Phi,\ell}(T)$ is
therefore generated by $\mathcal{A}_{r}$ and $\beta h_{L}-\log\Delta_{\Phi}$.
Since $h_{L}$ acts only on $L^{2}({{\mathbb{R}}})$, this is a canonical
description of a modular crossed product algebra. A standard formula then says
that

$\displaystyle\text{Tr}(a)=\bra{0}_{T}e^{\beta
h_{L}/2}\braket{\Phi}{\,U_{\Phi,\ell}(T)^{\dagger}\,a\,U_{\Phi,\ell}(T)\,}{\Phi}e^{\beta
h_{L}/2}\ket{0}_{T},$ (3.14)

where the states $\ket{T_{0}}_{T}$ are delta-function normalisable position
eigenstates for $L^{2}({{\mathbb{R}}})$ satisfying
$\braket{T_{0}}{T_{1}}_{T}=\delta(T_{0}-T_{1})$. In terms of the conjugate
momentum $h_{L}$, the state $\ket{0}_{T}$ is the constant wavefunction.

It turns out that there is a much more natural expression that we can instead
use. There is a converse theorem [connes1973classification,
takesaki2003theory, Jensen_2023] to the statement that Connes’ cocycle flow
for a von Neumann algebra is contained in that algebra: given any operator
$\beta H$ generating automorphisms of $\mathcal{A}_{r}$ for which it is
possible to write (3.6) there must exist a faithful normal semifinite weight
$\ket{\Psi_{0}}$ for $\mathcal{A}_{r}$ such that

$\displaystyle\beta H=-\log\Delta_{\Psi_{0}}.$ (3.15)

In other words, while the normalisable Hartle-Hawking state $\ket{\Psi}$ does
not exist as a normalisable state in $\mathcal{H}_{0}$, there does exist an
unnormalisable weight $\ket{\Psi_{0}}$ that generates boosts as modular flows.
All of our work above was therefore really just reproducing the standard
result that a modular crossed product algebra, despite appearances, is
independent of the choice of normalisable state, or unnormalisable semifinite
weight, used to define it [Witten_2022].

The fact that $\ket{\Psi_{0}}$ exists as a weight on $\mathcal{A}_{r}$ may be
unfamiliar to most readers (it certainly was to the authors of this paper!)
but it should not in fact be surprising. The reason that the Hartle-Hawking
state cannot exist as a normalisable state on $\mathcal{H}_{0}$ is the same
reason that the thermofield double state cannot exist as a normalisable state
on two copies of the (vacuum sector) Minkowski Hilbert space $\mathcal{H}_{\rm
Mink}$, namely that the Minkowski partition function
$Z(\beta)=\text{Tr}(e^{-\beta H})$ (with $H$ the Minkowski Hamiltonian) is
infinite. However the linear functional $\text{Tr}(e^{-\beta H}[\cdot])$ (just
like the trace Tr itself) is still a semifinite weight on $\mathcal{H}_{\rm
Mink}$. Indeed, because $H\geq 0$, any trace-class operator $a$ has finite
expectation value $\text{Tr}(e^{-\beta H}a)$. Furthermore, the linear
functional $\text{Tr}(e^{-P_{i}}[\cdot])$ is also semifinite, even though the
momentum $P_{i}$ is not bounded from below, because trace-class operators that
vanish exponentially fast as $P_{i}\to-\infty$ above some minimal value are
also s.o.t. dense in $\mathcal{B}(\mathcal{H}_{\rm Mink})$. So the existence
of $\ket{\Psi_{0}}$ as a semifinite weight on $\mathcal{A}_{r}$ for Kerr black
holes with a modular Hamiltonian that generates boosts at the horizon (and
hence generates a spacelike isometry at infinity) should also not be
surprising.

It is important not to confuse the Hartle-Hawking weight $\ket{\Psi_{0}}$ on
the algebra $\mathcal{A}_{r}$ with the normalised Hartle-Hawking state
$\ket{\Psi}$ viewed as an element of its own GNS Hilbert space
$\mathcal{H}_{\Psi}$. The algebra $\mathcal{A}_{\Psi,r}$ of right exterior
observables on $\mathcal{H}_{\Psi}$ is fundamentally different from
$\mathcal{A}_{r}$, just like the Type III algebra $\mathcal{A}_{TFD}$ of
operators on one copy of Minkowski space in the thermofield double Hilbert
space $\mathcal{H}_{\rm TFD}$ is fundamentally different from the Type I
algebra of operators $\mathcal{B}(\mathcal{H}_{\rm Mink})$. Even when an
operator like the identity is apparently in both algebras, we have e.g.
$\braket{\Psi_{0}}{\mathds{1}}{\Psi_{0}}=\infty$, while
$\braket{\Psi}{\mathds{1}}{\Psi}=1$. In fact, we do not expect that any
operators $a\in\mathcal{A}_{r}$ with $\braket{\Psi_{0}}{a}{\Psi_{0}}$ finite
make sense as operators on $\mathcal{H}_{\Psi}$.

Given the existence of the semifinite weight $\ket{\Psi_{0}}$, we can write
the trace on $\mathcal{A}_{R}$ is a more natural form as

$\displaystyle\text{Tr}(a)=\int dh_{L}e^{\beta
h_{L}}\braket{\Psi_{0}}{a}{\Psi_{0}},$ (3.16)

where $\braket{\Psi}{a}{\Psi}$ is interpreted as a function of $h_{L}$. Since
the trace on a Type II∞ von Neumann factor is unique up to rescaling, (3.16)
must be proportional to (3.14). In fact, one can show that they are the same
[CPW].

Given that the final conclusion of this section was that the only difference
between asymptotically flat and asymptotically AdS black holes is whether the
Hartle-Hawking state is a normalisable state or merely a semifinite weight, it
might seem peculiar that we included such a long discussion of it at all. The
reason is that our strategy above will play a crucial role in understanding
the gravitational algebras for black holes in de Sitter space, to which we now
turn. And for black holes in de Sitter space, there does not exist even a
semifinite weight whose modular Hamiltonian generates boosts, because the
black hole and cosmological horizons have different temperatures.

###  3.2 Schwarzschild-de Sitter black holes

Roughly speaking, black holes in de Sitter space combine all the ingredients
from our studies of both slow-rolling inflatons in Section 2 and
asymptotically flat black holes in Section 3.1. They feature a boost symmetry
that needs to be imposed as a gauge constraint that remains out-of-equilbrium
for parametrically long times. But they also feature a timeshift mode that can
have $O(1)$ fluctuations which are not described by quantum field theory in a
fixed spacetime background. As a result, their analysis is considerably more
technical than either; the rest of this section should probably not be read
without first reading those. Black holes in de Sitter space were previously
considered in [kudlerflam2024generalized], but with significant differences
that we discuss further below. In particular, the analysis in
[kudlerflam2024generalized] was based on the GNS Hilbert space for the Unruh
state, and made use of both asymptotic gravitational charges and of an
observer with a clock.

We focus for simplicity on the case of Schwarzschild-de Sitter (SdS) black
holes but the analysis can be easily extended to e.g. Kerr-Newman-de Sitter
black holes. Specifically, we focus on a spatially compact, two-sided
Schwarzschild-de Sitter solution where the left and right black hole exteriors
$U_{\ell}$ and $U_{r}$ are glued together at the cosmological horizon. This is
shown in Figure 5. The metric of the SdS spacetime in Schwarzschild
coordinates is

$\displaystyle ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}d\Omega_{2}^{2}$ (3.17)

where $f(r)=1-r_{s}/r-r^{2}/\ell_{dS}^{2}$. These coordinates are singular at
the black hole and cosmological horizons.

(a)

(b)

Figure 5: Penrose diagrams for Schwarzschild-de Sitter black hole, with the
red slice being a spacelike geodesics that winds around the closed universe.
The left and right edges are periodically identified as shown. The dashed
lines indicate the direction of the boosts. On the left, the black hole
bifurcation surface is placed in the middle, while on the right the
cosmological bifurcation surface is placed there.

As usual, there is a $G\to 0$ limit of an SdS black hole where the quantum
gravity Hilbert space becomes, up to gauge constraints, the QFT Hilbert space
describing perturbations of matter fields and gravitons on a fixed classical
SdS spacetime background. This Hilbert space is unambiguous because the SdS
background has compact Cauchy slices. However, as in Section 3.1, we can
consider an alternative limit, where a locally pure-gauge metric mode has
$O(1)$, rather than $O(\sqrt{G})$, fluctuations. In this case, that mode is
the shift $T$ in boost time $t$ from travelling on a static cycle through the
black hole and cosmological horizons; see Figure 6.121212As in Section 3.1,
there are additional locally pure-gauge modes associated to rotations from
travelling around the nontrivial cycle. The treatment of these modes is again
inessential to our discussion, and we will assume for simplicity that they are
frozen out. In a Kerr-de Sitter black hole, however, the rotational mode would
need to be treated more carefully, however, because the boost generators at
the black hole and cosmological horizons differ by a rotation generator.

If we allow $O(1)$ fluctuations in the timeshift, the Hilbert space becomes
the direct integral

$\displaystyle\mathcal{H}=\int^{\oplus}_{{{\mathbb{R}}}}dT\,\mathcal{H}_{T}$
(3.18)

where $\mathcal{H}_{T}$ is the QFT Hilbert space for an SdS black hole
background with fixed timeshift $T$. Unlike for asymptotically flat black
holes, there is no (global) isometry between SdS black holes with different
timeshifts that can be used to (almost) canonically identify the Hilbert
spaces $\mathcal{H}_{T}$.131313There were two (or perhaps three) natural
identifications we could have made: fixing the bulk coordinates to match the
right boundary coordinates, the left boundary coordinates or possibly an
average of the two. All three choices of identification, in slightly different
language, were discussed in [Witten_2022]. Of course, since they are all
separable infinite-dimensional Hilbert spaces, the Hilbert spaces
$\mathcal{H}_{T}$ are in principle isomorphic: we simply choose a countable
basis for each and define a unitary to map one basis to the other. For
practical purposes, it is convenient to pick a (hopefully somewhat less
arbitrary) choice of identification $\mathcal{H}_{T}\cong\mathcal{H}_{0}$ so
that we can write

$\displaystyle\mathcal{H}\cong\mathcal{H}_{0}\otimes L^{2}({{\mathbb{R}}}),$
(3.19)

with the timeshift $T$ again acting as the position operator on
$L^{2}({{\mathbb{R}}})$.

In classical general relativity, a natural choice would be to identify the
(perturbative) phase spaces for different timeshifts $T$ as follows. First
define a Cauchy slice $\Sigma_{0}$ for the zero-timeshift spacetime that is
defined by $t=0$ for some choice of boost time $t$ on each exterior and that
is smooth at the bifurcation surfaces. Then for $T\neq 0$, we pick a slice
$\Sigma_{T}$ that is locally diffeomorphic to $\Sigma_{0}$ everywhere except
at the cosmological horizon, where there will necessarily be a kink with boost
angle $T/\beta_{CH}$ (with $\beta_{CH}$ the inverse temperature of
cosmological horizon). (We will similarly denote the inverse temperature of
the black hole horizon by $\beta_{BH}$.) We then identify classical states of
the matter (and graviton) fields for different timeshifts $T$ and $T^{\prime}$
if their field configurations on the respective Cauchy slices $\Sigma_{T}$ and
$\Sigma_{T^{\prime}}$ are the same. With this identification, the area
$A_{CH}$ of the cosmological horizon generates a Hamiltonian flow
$\exp(\\{A_{CH}/4G,\cdot\\}\Delta T/\beta_{CH})$ on the phase space of general
relativity that increases the timeshift by $\Delta T$ while leaving the matter
fields unchanged.

(a)

(b)

Figure 6: Penrose diagrams for time-shifted Schwarzschild-de Sitter black
hole. This time, the left and right edges are periodically identified up to a
time-shift. The effect is that a geodesic, such as the red slice, gets boosted
at the bifurcation surface at the edges, i.e., picks up a non-zero boost
angle.

In quantum field theory, however, a transformation that creates a kink at the
cosmological bifurcation surface while leaving the state of the quantum fields
unchanged is singular because it locally acts like a one-sided boost. As we
discussed in Section 3.1, the singular behaviour of a one-sided boost is
closely related to the fact that the quantum operator
$\boldsymbol{\hat{A}}_{CH}/4G$ is UV-divergent. To make a nonsingular operator
that we can use to identify the Hilbert spaces $\mathcal{H}_{t}$ we need to
add $-\log\boldsymbol{\rho}$ with $\boldsymbol{\rho}$ a formal density matrix
on an appropriate algebra.

But what is the appropriate algebra? A naive choice would be e.g. the algebra
$\mathcal{A}_{r}$ of operators on the right exterior $U_{r}$. But density
matrices on this algebra have divergences both from modes near the
cosmological horizon and from modes near the black hole horizon. The latter
cannot be cancelled solely by adding $\boldsymbol{\hat{A}}_{CH}/4G$. So
$\mathcal{A}_{r}$ (or the algebra $\mathcal{A}_{\ell}$ of operators in the
left exterior) won’t work.

Instead, we first use the split property of quantum field theory [split1,
split2, split3, split4, split5] to write the zero-timeshift Hilbert space as

$\displaystyle\mathcal{H}_{0}\cong\mathcal{H}_{BH}\otimes\mathcal{H}_{CH}$
(3.20)

such that operators within a causal diamond $U_{BH}$ containing the black hole
bifurcation surface act only on $\mathcal{H}_{BH}$ while operators acting
within an almost complementary causal diamond $U_{CH}$ containing the
cosmological bifurcation surface act only on $\mathcal{H}_{CH}$; see Figure 7.
We call $U_{BH}$ and $U_{CH}$ the black hole and cosmological wedges
respectively. A decomposition (3.20) can be found in reasonable quantum field
theories so long as $U_{BH}$ and $U_{CH}$ are spacelike separated, even if
there is an arbitrarily small gap between them. For convenience, we can choose
this splitting to be invariant under rotations so that we can write the total
angular momentum $L_{i}$ of the quantum fields as

$\displaystyle L_{i}=L_{BH,i}+L_{CH,i}$ (3.21)

with $L_{BH,i}\in\mathcal{B}(\mathcal{H}_{BH})$ and
$L_{CH,i}\in\mathcal{B}(\mathcal{H}_{CH})$. We can also choose it to be
invariant under the discrete antiunitary symmetry

$\displaystyle J=J_{BH}+J_{CH}$ (3.22)

that exchanges the left and right exteriors $U_{\ell}$ and $U_{r}$ while also
reflecting time about $t=0$. (This symmetry is the de Sitter version of the
CRT symmetry of Minkowski space.) It cannot however be made boost invariant,
because the only boost-invariant causal diamonds in the SdS spacetime are
$U_{\ell}$ and $U_{r}$.

$U_{BH}$$U_{CH}$ (a)

$U_{CH}$$U_{BH}$ (b)

Figure 7:  The black hole and cosmological wedges with zero timeshift. The
black hole wedge $U_{BH}$ is a causal diamond with edges in the Cauchy slice
$\Sigma_{0}$. The cosmological wedge $U_{CH}$ is spacelike separated from
$U_{BH}$ but contains almost its entire causal complement. We can factorise
the Hilbert space
$\mathcal{H}_{0}\cong\mathcal{H}_{BH}\otimes\mathcal{H}_{CH}$ so that
operators in $U_{BH}$ act only on $\mathcal{H}_{BH}$ while operators in
$U_{CH}$ act only on $\mathcal{H}_{CH}$.

There are Type I von Neumann factors $\mathcal{B}(\mathcal{H}_{BH})$ and
$\mathcal{B}(\mathcal{H}_{CH})$ that contain all bounded operators acting on
only $\mathcal{H}_{BH}$ or $\mathcal{H}_{CH}$ respectively. Each of these
algebras contains Type III subfactors $\mathcal{A}_{BH,r}$ and
$\mathcal{A}_{CH,r}$ of operators localised within the right black hole
exterior $U_{r}$. The commutants of these algebras, acting respectively on
$\mathcal{H}_{BH}$ and $\mathcal{H}_{CH}$, are the algebras
$\mathcal{A}_{BH,\ell}$ and $\mathcal{A}_{CH,\ell}$ consist of all operators
on their respect Hilbert spaces that are localised in the left black hole
exterior $U_{\ell}$. The full algebras $\mathcal{A}_{\ell}$ and
$\mathcal{A}_{r}$ associated to the left and right black hole exteriors
$U_{\ell}$ and $U_{r}$ can then be written as

$\displaystyle\mathcal{A}_{\ell}\cong\mathcal{A}_{BH,\ell}\otimes\mathcal{A}_{CH,\ell}~{}~{}~{}\text{and}~{}~{}~{}\mathcal{A}_{r}\cong\mathcal{A}_{BH,r}\otimes\mathcal{A}_{CH,r}.$
(3.23)

Unlike density matrices on $\mathcal{A}_{\ell}$ and $\mathcal{A}_{r}$, density
matrices on $\mathcal{A}_{CH,\ell}$ and $\mathcal{A}_{CH,r}$ only feature
divergences from modes near the cosmological bifurcation surface. They are
therefore much more hopeful candidates to regularise the area operator.

Importantly, because the left and the right wedges are boost-invariant, while
the black hole and the cosmological wedges $U_{BH}$ and $U_{CH}$ are not, the
boost Hamiltonian $H_{0}$ generates automorphisms of $\mathcal{A}_{\ell}$ and
$\mathcal{A}_{r}$ but couples $\mathcal{H}_{BH}$ and $\mathcal{H}_{CH}$ so
that e.g. $\mathcal{A}_{BH,r}$ and $\mathcal{A}_{CH,r}$ mix. We denote the
image of those algebras under a boost by

$\displaystyle\mathcal{A}_{BH,r}^{(t)}\cong
e^{iH_{0}t}\mathcal{A}_{BH,r}e^{-iH_{0}t}~{}~{}~{}\text{ and
}~{}~{}~{}\mathcal{A}_{CH,r}^{(t)}\cong
e^{iH_{0}t}\mathcal{A}_{CH,r}e^{-iH_{0}t}$ (3.24)

respectively.141414Note that we have labelled the boost Hamiltonian $H_{0}$
rather than $H$ as in Section 3.1. This is because, once we identify
$\mathcal{H}_{T}$ with $\mathcal{H}_{0}$, the operator $H_{0}$ will generate
boosts on the zero-timeshift Hilbert space $\mathcal{H}_{0}$ but not on the
Hilbert spaces $\mathcal{H}_{T}$ for nonzero $T$.

What about the Hilbert spaces $\mathcal{H}_{T}$ for timeshifts $T\neq 0$?
Locally, the time shift is not detectible. So identifying Cauchy slices
$\Sigma_{T}$ for different $T$ allows us identify the causal complement of the
cosmological bifurcation surface, and in particular to identify diffeomorphic
wedges $U_{BH}$ for SdS spacetimes with different timeshifts $T$. In turn,
this leads to an action of the Type I von Neumann factor
$\mathcal{B}(\mathcal{H}_{BH})$ on the Hilbert space $\mathcal{H}_{T}$ for any
$T$. However $\mathcal{H}_{T}\not\cong\mathcal{H}_{BH}\otimes\mathcal{H}_{CH}$
because any causal diamond in the timeshifted SdS spacetime that we could
locally identify with $U_{CH}$ will end up partially timelike-separated from
$U_{BH}$. As a result, operators in $\mathcal{B}(\mathcal{H}_{BH})$ would not
commute with operators in $\mathcal{B}(\mathcal{H}_{CH})$. Suppose that we
instead define a region $U_{CH,T}$ in the timeshifted spacetime by fixing the
location of its edges in $\Sigma_{T}$, as shown in Figure 8, so that by
construction it is spacelike separated from $U_{BH}$. Then locally $U_{CH,T}$
will look like we have taken the right edge of $U_{CH}$ and boosted forwards
in time by $T$ relative to the left edge. We therefore have a natural
identification

$\displaystyle\mathcal{H}_{T}\cong\mathcal{H}_{BH}\otimes\mathcal{H}_{CH,T}$
(3.25)

where $\mathcal{B}(\mathcal{H}_{CH,T})$ can be identified with the algebra of
operators generated by $\mathcal{A}_{CH,\ell}$ and
$\mathcal{A}_{CH,r}^{(T)}$.151515The algebra generated by
$\mathcal{A}_{CH,\ell}$ and $\mathcal{A}_{CH,r}^{(T)}$ must be a Type I factor
because we can identify it as the commutant of the Type I factor
$\mathcal{B}(\mathcal{H}_{BH})$ on the Hilbert space $\mathcal{H}_{T}$.

$U_{BH}$$U_{CH,T}$$\Sigma_{T}$$U_{CH,T}$$U_{BH}$$\Sigma_{T}$ Figure 8:  The
black hole and cosmological wedges in a spacetime with non-zero timeshift. We
define the black hole wedge $U_{BH}$ to be locally indistinguishable from the
black hole wedge with zero-timeshift. Consequently, the cosmological wedge
$U_{CH}(t)$ needs to locally look like its right edge has been boosted
forwards in time by the timeshift $T$ relative to the zero-timeshift
cosmological wedge $U_{CH}$ so that its edges are both spacelike separated
from $U_{BH}$.

To construct an isomorphism $\mathcal{H}_{T}\cong\mathcal{H}_{0}$ it therefore
remains to construct an isomorphism $\mathcal{H}_{CH,T}\cong\mathcal{H}_{CH}$.
Suppose there existed a splitting of the boost Hamiltonian

$\displaystyle H_{0}=\boldsymbol{h}_{r,0}-\boldsymbol{h}_{\ell,0}$ (3.26)

into one-sided boost generators $\boldsymbol{h}_{r,0}$ and
$\boldsymbol{h}_{\ell,0}$ that are localised to the right and left exteriors
respectively. Then, since

$\displaystyle\mathcal{A}_{CH,r}^{(T)}=e^{\mathrm{i}H_{0}T}\mathcal{A}_{CH,r}e^{-\mathrm{i}H_{0}T}=e^{\mathrm{i}\boldsymbol{h}_{r,0}T}\mathcal{A}_{CH,r}e^{-\mathrm{i}\boldsymbol{h}_{r,0}T},$
(3.27)

we would have

$\displaystyle\mathcal{B}(\mathcal{H}_{CH,T})\cong
e^{\mathrm{i}\boldsymbol{h}_{r,0}T}\mathcal{B}(\mathcal{H}_{CH})e^{-\mathrm{i}\boldsymbol{h}_{r,0}T}.$
(3.28)

However, as discussed around (2.49), $\boldsymbol{h}_{r,0}$ and
$\boldsymbol{h}_{\ell,0}$ only exist as sesquilinear forms mapping pairs of
bras and kets to expectation values and not as densely defined operators. They
therefore cannot be exponentiated. Indeed, attempting to identify
$\mathcal{H}_{CH,T}\cong\mathcal{H}_{CH}$ via (3.28) is exactly the same
strategy that we already tried, and failed, to use to identify
$\mathcal{H}_{T}\cong\mathcal{H}_{0}$ – namely localising the timeshift at the
cosmological horizon while leaving the QFT state otherwise unchanged.

What we need is a nonsingular unitary $U(T)$ such that

$\displaystyle U(T)^{\dagger}\mathcal{A}_{CH,r}U(T)=\mathcal{A}_{CH,r}^{(T)}$
(3.29)

while

$\displaystyle U(T)^{\dagger}\mathcal{A}_{CH,\ell}U(T)=\mathcal{A}_{CH,\ell}.$
(3.30)

The choice $U(T)=e^{-\mathrm{i}\boldsymbol{h}_{r,0}T}$ didn’t work because it
was singular. Meanwhile the choice $U(T)=e^{-\mathrm{i}H_{0}T}$ doesn’t work
because it doesn’t satisfy (3.30). Suppose we fix some choice of states
$\ket{\Phi_{BH}}\in\mathcal{H}_{BH}$ and $\ket{\Phi_{CH}}\in\mathcal{H}_{CH}$.
Let $\Delta_{\Phi_{BH}}$ and $\Delta_{\Phi_{CH}}$ be the corresponding modular
operators on $\mathcal{A}_{BH,r}$ and $\mathcal{A}_{CH,r}$ respectively. As in
our choice of $\mathcal{H}_{BH}$ and $\mathcal{H}_{CH}$, it is convenient to
choose $\ket{\Phi_{BH}}$ and $\ket{\Phi_{CH}}$ to be rotation invariant, so
that

$\displaystyle L_{BH,i}\ket{\Phi_{BH}}=L_{CH,i}\ket{\Phi_{CH}}=0$ (3.31)

and to be canonically purifications with respect to the symmetry $J$ so that
their respective modular conjugation operators are $J_{\Phi_{BH}}=J_{BH}$ and
$J_{\Phi_{CH}}=J_{CH}$.

While the splitting (3.28) does not exist, we can, in close analogy with
(3.5), write

$\displaystyle
H_{0}=-\frac{1}{\beta_{BH}}\log\Delta_{\Phi_{BH}}-\frac{1}{\beta_{CH}}\log\Delta_{\Phi_{CH}}+\scalebox{0.8}{$\Delta$}H_{\Phi,r}-\scalebox{0.8}{$\Delta$}H_{\Phi,\ell}$
(3.32)

with $\scalebox{0.8}{$\Delta$}H_{\Phi,\ell}$ and
$\scalebox{0.8}{$\Delta$}H_{\Phi,r}$ densely defined unbounded operators
affiliated to $\mathcal{A}_{\ell}$ and $\mathcal{A}_{r}$ respectively. As in
(3.5), $\scalebox{0.8}{$\Delta$}H_{\Phi,\ell}$ and
$\scalebox{0.8}{$\Delta$}H_{\Phi,r}$ are uniquely defined up to the addition
of a common $c$-number. Equivalently, there exist densely defined operators

$\displaystyle\Lambda_{\Phi,\ell}=H_{0}+\scalebox{0.8}{$\Delta$}H_{\Phi,\ell}~{}~{}~{}\text{and}~{}~{}~{}\Lambda_{\Phi,r}=H_{0}-\scalebox{0.8}{$\Delta$}H_{\Phi,r}$
(3.33)

that generate boosts of $\mathcal{A}_{r}$ (respectively $\mathcal{A}_{\ell}$)
and modular flows of $\mathcal{A}_{BH,\ell}$ and $\mathcal{A}_{CH,\ell}$
(respectively $\mathcal{A}_{BH,r}$ and $\mathcal{A}_{CH,r}$).

As we did for the boost Hamiltonian $H_{0}$ in (3.26), we can write

$\displaystyle-\log\Delta_{\Phi_{CH}}=\beta_{CH}\boldsymbol{h}_{\Phi_{CH},r}-\beta_{CH}\boldsymbol{h}_{\Phi_{CH},\ell}$
(3.34)

with $\boldsymbol{h}_{\Phi_{CH},\ell}$ and $\boldsymbol{h}_{\Phi_{CH},r}$
sesquilinear forms on $\mathcal{H}_{CH}$ that are localised in the left and
right exteriors respectively.161616One way to define
$\boldsymbol{h}_{\Phi_{CH},\ell}$ and $\boldsymbol{h}_{\Phi_{CH},r}$ is by
$\displaystyle\beta_{CH}\braket{\Phi^{\prime}}{\boldsymbol{h}_{\Phi_{CH},r}}{\Phi^{\prime}}=-\braket{\Phi^{\prime}}{\log\Delta_{\Phi_{CH}}{\Phi^{\prime}}|\Phi^{\prime}}{~{}}~{}~{}\text{and}~{}~{}~{}\beta_{CH}\braket{\Phi^{\prime}}{\boldsymbol{h}_{\Phi_{CH},\ell}}{\Phi^{\prime}}=\braket{\Phi^{\prime}}{\log\Delta_{\Phi^{\prime}}{\Phi_{CH}}|\Phi^{\prime}}{.}$
(3.35) Since taking a derivative of Connes’ cocycle flow leads to
$\displaystyle\log\Delta_{\Phi_{CH}|\Phi^{\prime}}+\log\Delta_{\Phi^{\prime}|\Phi_{CH}}=\log\Delta_{\Phi_{CH}}+\log\Delta_{\Phi^{\prime}},$
(3.36) we can then obtain the desired relation (3.34) via the standard
identity
$\braket{\Phi^{\prime}}{\log\Delta_{\Phi^{\prime}}}{\Phi^{\prime}}=0$.
Similarly, we can write

$\displaystyle-\log\Delta_{\Phi_{BH}}=\beta_{BH}\boldsymbol{h}_{\Phi_{BH},r}-\beta_{BH}\boldsymbol{h}_{\Phi_{BH},\ell}.$
(3.37)

We can then write e.g.

$\displaystyle\Lambda_{\Phi,\ell}=\boldsymbol{h}_{r,0}-\boldsymbol{h}_{\Phi_{BH},\ell}-\boldsymbol{h}_{\Phi_{CH},\ell}.$
(3.38)

Naively one might think that $\Lambda_{\Phi,\ell}$ is also only a sesquilinear
form. However, in a small neighbourhood of the black hole bifurcation surface,
with size $L\ll r_{s}$, there locally exists a Minkowski vacuum state
$\ket{\Psi_{BH}}$ with modular operator
$\Delta_{\Psi_{BH}}=e^{-\beta_{BH}H_{0}}$. Near this horizon, the operator
$\beta_{BH}\Lambda_{\Phi,\ell}$ acts like the relative modular Hamiltonian
$-\log\Delta_{\Psi_{BH}|\Phi_{BH}}$ which is a densely defined operator.
Similarly, in a small neighbourhood of the cosmological horizon with size
$L\ll\ell_{dS}$, there locally exists a Minkowski vacuum $\ket{\Psi_{CH}}$
with $\Delta_{\Psi_{CH}}=e^{-\beta_{CH}H_{0}}$ and the operator
$\beta_{CH}\Lambda_{\Phi,\ell}$ acts like the relative modular Hamiltonian
$-\log\Delta_{\Psi_{CH}|\Phi_{CH}}$. In other words, the singular behaviour of
the one-sided boost of $\mathcal{A}_{r}$ is cancelled out by the singular one-
sided modular flows on $\mathcal{A}_{BH,\ell}$ and $\mathcal{A}_{CH,\ell}$.
Since the two horizons are the only locations where potential divergences in
(3.38) could occur, it follows that $\Lambda_{\Phi,\ell}$, and hence also
$\scalebox{0.8}{$\Delta$}H_{\Phi,\ell}$ via (3.33), exist as densely defined
operators. Similar arguments apply to $\Lambda_{\Phi,r}$ and
$\scalebox{0.8}{$\Delta$}H_{\Phi,r}$.

Since the modular flow $\Delta_{\Phi_{CH}}^{iT/\beta_{CH}}$ preserves
$\mathcal{A}_{CH,\ell}$, we have

$\displaystyle\mathcal{B}(\mathcal{H}_{CH,T})\cong
e^{i\Lambda_{\Phi,\ell}T}\mathcal{B}(\mathcal{H}_{CH})e^{-i\Lambda_{\Phi,\ell}T}.$
(3.39)

This gives our desired identification of $\mathcal{H}_{CH,T}$ with
$\mathcal{H}_{CH}$ and hence trivialisation of the direct integral (3.18) as

$\displaystyle\mathcal{H}\cong\mathcal{H}_{0}\otimes L^{2}({{\mathbb{R}}}),$
(3.40)

with the timeshift $T$ acting as the position operator on
$L^{2}({{\mathbb{R}}})$. Note that this trivialisation only depends on the
choice of state $\ket{\Phi_{CH}}\in\mathcal{H}_{CH}$ and not on
$\ket{\Phi_{BH}}\in\mathcal{H}_{BH}$. In effect, the identification (3.39)
cancels the singular behaviour of (3.28) with an equally one-sided modular
flow of $\mathcal{A}_{CH,\ell}$.

We already saw that the classical area of the cosmological bifurcation surface
generates a timeshift localised at that surface via Hamiltonian flow.
Consequently, if we had used the singular identification
$\mathcal{H}_{T}\cong\mathcal{H}_{0}$ induced by (3.28), the quantum operator
$\boldsymbol{\hat{A}}_{CH}$ describing that area would generate timeshifts
while leaving the state of the quantum fields unchanged. More precisely,
(3.28) would lead to an identification of
$X=-i\partial_{T}\in\mathcal{B}(L^{2}({{\mathbb{R}}}))$ with
$(\boldsymbol{\hat{A}}_{CH}-A_{CH,0})/4G\beta_{CH}$, where we have subtracted
the area $A_{CH,0}$ of the cosmological horizon in the classical background to
remove the divergence as $G\to 0$. Because we instead used the identification
(3.39), which featured an additional one-sided modular flow of
$\mathcal{A}_{CH,\ell}$ to make the isomorphism nonsingular, we actually have

$\displaystyle\beta_{CH}\,X_{\Phi}=\frac{\boldsymbol{\hat{A}}_{CH}-A_{CH,0}}{4G}+\beta_{CH}\boldsymbol{h}_{\Phi_{CH},\ell}=\frac{\boldsymbol{\hat{A}}_{CH}}{4G}-\log\boldsymbol{\rho}_{\Phi_{CH},\ell}-\mathrm{const}.$
(3.41)

We have added a subscript to
$X_{\Phi}=-i\partial_{T}\in\mathcal{B}(L^{2}({{\mathbb{R}}}))$ to indicate
that the identification (3.39), which was crucial to its physical
interpretation, depended on the arbitrary choice of state $\ket{\Phi_{CH}}$.
In the last equality of (3.41), we used the fact that the formal density
matrix $\boldsymbol{\rho}_{\Phi_{CH},\ell}$ satisfies

$\displaystyle-\log\boldsymbol{\rho}_{\Phi_{CH},\ell}=\beta_{CH}\boldsymbol{h}_{\Phi_{CH},\ell}+S({\Phi_{CH}}),$
(3.42)

with $S(\Phi_{CH})$ the divergent entropy of $\ket{\Phi_{CH}}$ on
$\mathcal{A}_{CH,\ell}$, in order to write (3.41) in a form analogous to
(3.11).

###  3.3 The algebras of observables

We now turn to describe the algebra $\widetilde{\mathcal{A}}_{R}$ of quantum
gravity operators localised in the right black hole exterior $U_{r}$. In order
to be gauge-invariant, operators in $\widetilde{\mathcal{A}}_{R}$ need to
commute with the boosts and rotations of the SdS background. First, however,
we describe the algebra $\mathcal{A}_{R}$ of operators acting on $\mathcal{H}$
that are localised within the right black hole exterior but that are not
necessarily gauge-invariant. This algebra contains a subalgebra of timeshift-
preserving right exterior operators that, for each possible timeshift $T$, is
isomorphic to the algebra $\mathcal{A}_{r}$ of right-exterior operators on
$\mathcal{H}_{0}$. In fact, the isomorphism
$\mathcal{H}_{T}\cong\mathcal{H}_{0}$ described around (3.39) allows us to
identify this algebra directly with $\mathcal{A}_{r}$.

However there also exist operators that are localised in the right exterior
but that do not commute with the timeshift $T$. These include, in particular,
the operator

$\displaystyle\beta_{CH}\,X_{\Phi,R}$
$\displaystyle=\frac{\boldsymbol{\hat{A}}_{CH}-A_{CH,0}}{4G}+\beta_{CH}\,\boldsymbol{h}_{\Phi_{CH},r}$
(3.43) $\displaystyle=\beta_{CH}\,X_{\Phi}-\log\Delta_{\Phi_{CH}}$ (3.44)

In fact, the algebra $\mathcal{A}_{r}$ and (bounded functions of)
$X_{\Phi,BH,R}$ will turn out to generate the full algebra $\mathcal{A}_{R}$.
We therefore have

$\displaystyle\mathcal{A}_{R}\cong\mathcal{A}_{BH,r}\otimes\left(\mathcal{A}_{CH,r}\rtimes{{\mathbb{R}}}\right)$
(3.45)

where $\left(\mathcal{A}_{CH,r}\rtimes{{\mathbb{R}}}\right)$ is the modular
crossed product of $\mathcal{A}_{BH,r}$. Since $\mathcal{A}_{BH,r}$ is a Type
III1 factor while $\left(\mathcal{A}_{CH,r}\rtimes{{\mathbb{R}}}\right)$ is a
Type II∞ factor, the algebra $\mathcal{A}_{R}$ is a Type III1 factor.

One might think that additionally we have e.g. the operator

$\displaystyle\beta_{BH}\,X_{\Phi,BH,R}$
$\displaystyle=\frac{\boldsymbol{\hat{A}}_{BH}-A_{BH,0}}{4G}+\beta_{BH}\,\boldsymbol{h}_{\Phi_{BH},r}$
(3.46)

However it follows from the Einstein equations – specifically the conservation
of the Komar mass associated to boost time translations – that

$\displaystyle 0$
$\displaystyle=\frac{1}{\beta_{BH}}\frac{\boldsymbol{\hat{A}}_{BH}-A_{BH,0}}{4G}+\frac{1}{\beta_{CH}}\frac{\boldsymbol{\hat{A}}_{CH}-A_{CH,0}}{4G}+\boldsymbol{h}_{r}$
(3.47)
$\displaystyle=X_{\Phi,BH,R}+X_{\Phi,R}+\scalebox{0.8}{$\Delta$}H_{\Phi,r}.$
(3.48)

Since $\scalebox{0.8}{$\Delta$}H_{\Phi,r}$ is affiliated to $\mathcal{A}_{r}$,
(3.47) tells us that the operator $X_{\Phi,BH,R}$ is _already_ affiliated to
$\mathcal{A}_{R}$ and does not need to be added separately.171717There is a
subtlety here that should be briefly commented on. Conservation of the Komar
mass also tells us that
$\displaystyle(\boldsymbol{\hat{A}}_{BH}-A_{BH,0})/4G\beta_{BH}+(\boldsymbol{\hat{A}}_{CH}-A_{CH,0})/4G\beta_{CH}+\boldsymbol{h}_{l}=0.$
(3.49) After imposing the gauge constraint
$H=\boldsymbol{h}_{r}-\boldsymbol{h}_{\ell}=0$, (3.47) and (3.49) are
equivalent. But before doing so they lead to different definitions of
$X_{\Phi,BH,R}$ as an operator on $\mathcal{H}$. Adding $X_{\Phi,BH,R}$ as
defined via (3.49) to $\mathcal{A}_{R}$ is therefore equivalent to adding
functions of the boost generator $H$. Doing so would not change the final
gauge-invariant algebra $\widetilde{\mathcal{A}}_{R}$. Similarly, operators
analogous to (3.43) but with the state $\ket{\Phi_{CH}}$ replaced by a
different state $\ket{\Phi^{\prime}_{CH}}$ are already affiliated to
$\mathcal{A}_{R,0}$ because Connes cocycle flow on $\mathcal{A}_{CH,r}$ is an
inner automorphism.

What about operators in the left black hole exterior? The identification
$\mathcal{H}_{T}\cong\mathcal{H}_{0}$ not only created a kink at the
cosmological bifurcation surface, when identifying $\Sigma T$ with
$\Sigma_{0}$, but also conjugated $\mathcal{A}_{CH,\ell}$ by
$e^{i\Lambda_{\Phi,\ell}t}$, or equivalently by
$\Delta_{\Phi_{CH}}^{-iT/\beta_{CH}}$. As a result, the algebra of timeshift-
preserving left exterior operators is

$\displaystyle\mathcal{A}_{BH,\ell}\otimes\Delta_{\Phi_{CH}}^{iT/\beta_{CH}}\,\mathcal{A}_{CH,\ell}\,\Delta_{\Phi_{CH}}^{-iT/\beta_{CH}}\cong\Delta_{\Phi_{CH}}^{iT/\beta_{CH}}\,\mathcal{A}_{\ell}\,\Delta_{\Phi_{CH}}^{-iT/\beta_{CH}}.$
(3.50)

A useful check of (3.50) is that any operator $a^{\prime}$ in (3.50) satisfies

$\displaystyle\beta_{CH}[X_{\Phi,R},a^{\prime}]$
$\displaystyle=\beta_{CH}[X_{\Phi},a^{\prime}]-[\log\Delta_{\Phi_{CH}},a^{\prime}]=0$
(3.51)

and so commutes with the right exterior operator $X_{\Phi,R}$. In addition, we
know from (3.41) that the left exterior algebra should also bounded functions
of $X_{\Phi}=-i\partial_{T}$. Together these generate the commutant algebra

$\displaystyle\mathcal{A}_{L}\cong\mathcal{A}_{R}^{\prime}\cong\mathcal{A}_{BH,\ell}\otimes\left(\mathcal{A}_{CH,r}\rtimes{{\mathbb{R}}}\right)^{\prime}$
(3.52)

where $\left(\mathcal{A}_{CH,r}\rtimes{{\mathbb{R}}}\right)^{\prime}$, which
is generated by $X_{\Phi}$ and
$\Delta_{\Phi_{CH}}^{iT/\beta_{CH}}\mathcal{A}_{CH,\ell}\Delta_{\Phi_{CH}}^{-iT/\beta_{CH}}$,
is the commutant of $\left(\mathcal{A}_{CH,r}\rtimes{{\mathbb{R}}}\right)$ on
$\mathcal{H}_{CH}$ and is also a modular crossed product algebra.

As one might hope, the algebras $\mathcal{A}_{L}$ and $\mathcal{A}_{R}$ are
isomorphic, as can be easily verified by conjugating by
$\Delta_{\Phi_{CH}}^{iT/\beta_{CH}}$. In effect, this conjugation switches
from the identification (3.39) to an identification
$\mathcal{B}(\mathcal{H}_{CH,T}\cong
e^{-i\Lambda_{\Phi,r}t}\mathcal{B}(\mathcal{H}_{CH})e^{i\Lambda_{\Phi,r}t}$.
They are also independent of the choice of state $\ket{\Phi_{CH}}$. For
example, conjugation by the Connes’ cocycle flow
$\Delta^{iT/\beta_{CH}}_{\Phi_{CH}}\Delta^{-iT/\beta_{CH}}_{\Phi|\Phi^{\prime}_{CH}}$
maps the algebras $\mathcal{A}_{L}$ and $\mathcal{A}_{R}$ to the corresponding
algebras with $\ket{\Phi_{CH}}$ replaced by $\ket{\Phi^{\prime}_{CH}}$. Again,
this can be physically interpreted as a replacement of $\ket{\Phi_{CH}}$ by
$\ket{\Phi_{CH}^{\prime}}$ in the identification (3.39).

###  3.4 The gauge constraints

We now turn to the question of how to impose the gauge constraints associated
to boosts and rotations of the SdS background. On the Hilbert space
$\mathcal{H}_{0}$ boosts are generated by the operator $H_{0}$ while rotations
are generated by $L_{i}$. However, when acting on the timeshifted SdS Hilbert
space $\mathcal{H}_{T}$ (identified with $\mathcal{H}_{0}$ via (3.39)) we
needed to conjugate operators in $\mathcal{A}_{CH,\ell}$ – but not operators
in $\mathcal{A}_{CH,r}$ – by $\Delta_{\Phi_{CH}}^{-iT/\beta_{CH}}$. Formally,
we could describe this by saying that all operators acting on
$\mathcal{H}_{T}$ need to be conjuated by
$\exp(-i\boldsymbol{h}_{\Phi_{CH},\ell}T)$. In general, it does not make sense
to conjugate an operator in $\mathcal{B}(\mathcal{H}_{0})$ (that isn’t e.g.
contained in $\mathcal{A}_{\ell}$ or $\mathcal{A}_{r}$) by a singular one-
sided modular flow, which is why the algebras of operators
$\mathcal{B}(\mathcal{H}_{0})$ and $\mathcal{B}(\mathcal{H}_{T})$ cannot be
naturally identified. However, in this case, we know that boosts and rotations
_are_ isometries of the timeshifted SdS background, and so the operators
generating them need to exist. So somehow things have to work out. Because we
chose the state $\ket{\Phi_{CH}}$ to be rotation invariant, we have

$\displaystyle[\boldsymbol{h}_{\Phi_{CH},\ell},L_{i}]=[h_{\Phi_{CH},\ell},L_{CH,i}]=0.$
(3.53)

Rotations on $\mathcal{H}_{T}$, like rotations on $\mathcal{H}_{0}$ are
therefore generated by $L_{i}$.

On the other hand, the generator

$\displaystyle
H=e^{i\boldsymbol{h}_{\Phi_{CH},\ell}T}H_{0}e^{-i\boldsymbol{h}_{\Phi_{CH},\ell}T}$
(3.54)

of boosts on $\mathcal{H}_{T}$ is not equal to the generator $H_{0}$ of boosts
on $\mathcal{H}_{0}$ because $\ket{\Phi_{CH}}\ket{\Phi_{BH}}$ and, for that
matter, the Hilbert space $\mathcal{H}_{CH}$ itself are not boost invariant.
However, it follows from (3.38) that the operator $\Lambda_{\Phi,\ell}$ does
commute with $h_{\Phi_{CH},\ell}$. Meanwhile

$\displaystyle
H_{0}-\Lambda_{\Phi,\ell}=-\scalebox{0.8}{$\Delta$}H_{\Phi,\ell}$ (3.55)

is affiliated with $\mathcal{A}_{\ell}$ so that

$\displaystyle
e^{ih_{\Phi_{CH},\ell}T}\scalebox{0.8}{$\Delta$}H_{\Phi,\ell}e^{-ih_{\Phi_{CH},\ell}T}=\Delta_{\Phi_{CH}}^{iT/\beta_{CH}}\scalebox{0.8}{$\Delta$}H_{\Phi,\ell}\Delta_{\Phi_{CH}}^{-iT/\beta_{CH}}.$
(3.56)

We therefore have

$\displaystyle
H=\Lambda_{\Phi,\ell}-\Delta_{\Phi_{CH}}^{iT/\beta_{CH}}\scalebox{0.8}{$\Delta$}H_{\Phi,\ell}\Delta_{\Phi_{CH}}^{-iT/\beta_{CH}},$
(3.57)

which is manifestly densely defined.

It is important to check that $H$ generates automorphisms of the algebra
$\mathcal{A}_{R}$. For any $a\in\mathcal{A}_{r}$, we have