The multiverse interpretation of quantum mechanics

Preprint typeset in JHEP style - HYPER VERSION

The Multiverse Interpretation of Quantum
Mechanics

Raphael Boussoa,b and Leonard Susskindc
arXiv:1105.3796v1 [hep-th] 19 May 2011

a Center for Theoretical Physics, Department of Physics
University of California, Berkeley, CA 94720, U.S.A.
b Lawrence Berkeley National Laboratory, Berkeley, CA 94720, U.S.A.
c Stanford Institute for Theoretical Physics and
Department of Physics, Stanford University, Stanford, CA 94305, U.S.A.

Abstract: We argue that the many-worlds of quantum mechanics and the many
worlds of the multiverse are the same thing, and that the multiverse is necessary to
give exact operational meaning to probabilistic predictions from quantum mechanics.
Decoherence—the modern version of wave-function collapse—is subjective in that it
depends on the choice of a set of unmonitored degrees of freedom, the “environment”.
In fact decoherence is absent in the complete description of any region larger than
the future light-cone of a measurement event. However, if one restricts to the causal
diamond—the largest region that can be causally probed—then the boundary of the
diamond acts as a one-way membrane and thus provides a preferred choice of environ-
ment. We argue that the global multiverse is a representation of the many-worlds (all
possible decoherent causal diamond histories) in a single geometry.
We propose that it must be possible in principle to verify quantum-mechanical pre-
dictions exactly. This requires not only the existence of exact observables but two
additional postulates: a single observer within the universe can access infinitely many
identical experiments; and the outcome of each experiment must be completely definite.
In causal diamonds with finite surface area, holographic entropy bounds imply that no
exact observables exist, and both postulates fail: experiments cannot be repeated in-
finitely many times; and decoherence is not completely irreversible, so outcomes are
not definite. We argue that our postulates can be satisfied in “hats” (supersymmetric
multiverse regions with vanishing cosmological constant). We propose a complemen-
tarity principle that relates the approximate observables associated with finite causal
diamonds to exact observables in the hat.

Contents
1. Introduction 1

2. Building the multiverse from the many worlds of causal diamonds 6
2.1 Decoherence and causality 6
2.2 Failure to decohere: A problem with the global multiverse 7
2.3 Simplicio’s proposal 10
2.4 Objective decoherence from the causal diamond 12
2.5 Global-local measure duality 17
2.6 Constructing a global multiverse from many causal diamond worlds 20

3. The many worlds of the census taker 24
3.1 Decoherence and recoherence 24
3.2 Failure to irreversibly decohere: A limitation of finite systems 26
3.3 Sagredo’s postulates 27
3.4 Irreversible decoherence and infinite repetition in the hat 33
3.5 Black hole complementarity and hat complementarity 36
3.6 The global multiverse in a hat 41

1. Introduction
According to an older view of quantum mechanics, objective phenomena only occur
when an observation is made and, as a result, the wave function collapses. A more
modern view called decoherence considers the effects of an inaccessible environment
that becomes entangled with the system of interest (including the observer). But at
what point, precisely, do the virtual realities described by a quantum mechanical wave
function turn into objective realities?
This question is not about philosophy. Without a precise form of decoherence,
one cannot claim that anything really “happened”, including the specific outcomes of
experiments. And without the ability to causally access an infinite number of pre-
cisely decohered outcomes, one cannot reliably verify the probabilistic predictions of a
quantum-mechanical theory.

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The purpose of this paper is to argue that these questions may be resolved by
cosmology. We will offer some principles that we believe are necessary for a consistent
interpretation of quantum mechanics, and we will argue that eternal inflation is the
only cosmology which satisfies those principles. There are two views of an eternally
inflating multiverse: global (or parallel) vs. local (or series). The parallel view is like
looking at a tree and seeing all its branches and twigs simultaneously. The series view
is what is seen by an insect climbing from the base of the tree to a particular twig along
a specific route.
In both the many-worlds interpretation of quantum mechanics and the multiverse
of eternal inflation the world is viewed as an unbounded collection of parallel universes.
A view that has been expressed in the past by both of us is that there is no need to
add an additional layer of parallelism to the multiverse in order to interpret quantum
mechanics. To put it succinctly, the many-worlds and the multiverse are the same
thing [1].

Decoherence Decoherence1 explains why observers do not experience superpositions
of macroscopically distinct quantum states, such as a superposition of an alive and a
dead cat. The key insight is that macroscopic objects tend to quickly become entangled
with a large number of “environmental” degrees of freedom, E, such as thermal photons.
In practice these degrees of freedom cannot be monitored by the observer. Whenever
a subsystem E is not monitored, all expectation values behave as if the remaining
system is in a density matrix obtained by a partial trace over the Hilbert space of E.
The density matrix will be diagonal in a preferred basis determined by the nature of
the interaction with the environment.
As an example, consider an isolated quantum system S with a two-dimensional
Hilbert space, in the general state a|0 S + b|1 S . Suppose a measurement takes place in
a small spacetime region, which we may idealize as an event M . By this we mean that
at M , the system S interacts and becomes correlated with the pointer of an apparatus
A:
(a|0 S + b|1 S ) ⊗ |0 A → a |0 S ⊗ |0 A + b |1 S ⊗ |1 A , (1.1)

This process is unitary and is referred to as a pre-measurement.
We assume that the apparatus is not a closed system. (This is certainly the case
in practice for a macroscopic apparatus.) Thus, shortly afterwards (still at M in our
idealization), environmental degrees of freedom E scatter off of the apparatus and
become entangled with it. By unitarity, the system SAE as a whole remains in a pure
1
For reviews, see [2, 3]. For a pedagogical introduction, see [4].

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quantum state,2

|ψ = a |0 S ⊗ |0 A ⊗ |0 E + b |1 S ⊗ |1 A ⊗ |1 E . (1.2)

We assume that the observer does not monitor the environment; therefore, he will
describe the state of SA by a density matrix obtained by a partial trace over the
Hilbert space factor representing the environment:

ρSA = TrE |ψ ψ| (1.3)

This matrix is diagonal in the basis {|0 S ⊗ |0 A , |0 S ⊗ |1 A , |1 S ⊗ |0 A , |1 S ⊗ |1 A } of
the Hilbert space of SA:
ρSA = diag(|a|2 , 0, 0, |b|2 ) . (1.4)
This corresponds to a classical ensemble in which the pure state |0 S ⊗ |0 A has prob-
ability |a|2 and the state |1 S ⊗ |1 A has probability |b|2 .
Decoherence explains the “collapse of the wave function” of the Copenhagen in-
terpretation as the non-unitary evolution from a pure to a mixed state, resulting from
ignorance about an entangled subsystem E. It also explains the very special quantum
states of macroscopic objects we experience, as the elements of the basis in which the
density matrix ρSA is diagonal. This preferred basis is picked out by the apparatus
configurations that scatter the environment into orthogonal states. Because interac-
tions are usually local in space, ρSA will be diagonal with respect to a basis consisting
of approximate position space eigenstates. This explains why we perceive apparatus
states |0 A (pointer up) or |1 A (pointer down), but never the equally valid basis states
|± A ≡ 2−1/2 (|0 A ±|1 A ), which would correspond to superpositions of different pointer
positions.
The entangled state obtained after premeasurement, Eq. (1.1) is a superposition
of two unentangled pure states or “branches”. In each branch, the observer sees a
definite outcome: |0 or |1 . This in itself does not explain, however, why a definite
outcome is seen with respect to the basis {|0 , |1 } rather than {|+ , |− }. Because
the decomposition of Eq. (1.1) is not unique [5], the interaction with an inaccessible
environment and the resulting density matrix are essential to the selection of a preferred
basis of macroscopic states.
Decoherence has two important limitations: it is subjective, and it is in principle
reversible. This is a problem if we rely on decoherence for precise tests of quantum
2
We could explicitly include an observer who becomes correlated to the apparatus through inter-
action with the environment, resulting in an entangled pure state of the form a|0 S ⊗ |0 A ⊗ |0 E ⊗
|0 O + b|1 S ⊗ |1 A ⊗ |1 E ⊗ |1 O . For notational simplicity we will subsume the observer into A.

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mechanical predictions. We argue in Sec. 2 that causal diamonds provide a natural
definition of environment in the multiverse, leading to an observer-independent notion
of decoherent histories. In Sec. 3 we argue that these histories have precise, irreversible
counterparts in the “hat”-regions of the multiverse. We now give a more detailed
overview of this paper.

Outline In Sec. 2 we address the first limitation of decoherence, its subjectivity. Be-
cause coherence is never lost in the full Hilbert space SAE, the speed, extent, and
possible outcomes of decoherence depend on the definition of the environment E. This
choice is made implicitly by an observer, based on practical circumstances: the envi-
ronment consists of degrees of freedom that have become entangled with the system
and apparatus but remain unobserved. It is impractical, for example, to keep track of
every thermal photon emitted by a table, of all of its interactions with light and air
particles, and so on. But if we did, then we would find that the entire system SAE
behaves as a pure state |ψ , which may be a “cat state” involving the superposition
of macroscopically different matter configurations. Decoherence thus arises from the
description of the world by an observer who has access only to a subsystem. To the
extent that the environment is defined by what a given observer cannot measure in
practice, decoherence is subjective.
The subjectivity of decoherence is not a problem as long as we are content to
explain our own experience, i.e., that of an observer immersed in a much larger system.
But the lack of any environment implies that decoherence cannot occur in a complete
unitary description of the whole universe. It is possible that no such description exists
for our universe. In Sec. 2.1 we will argue, however, that causality places restrictions
on decoherence in much smaller regions, in which the applicability of unitary quantum-
mechanical evolution seems beyond doubt.
In Sec. 2.2, we apply our analysis of decoherence and causality to eternal infla-
tion. We will obtain a straightforward but perhaps surprising consequence: in a global
description of an eternally inflating spacetime, decoherence cannot occur; so it is in-
consistent to imagine that pocket universes or vacuum bubbles nucleate at particular
locations and times. In Sec. 2.3, we discuss a number of attempts to rescue a unitary
global description and conclude that they do not succeed.
In Sec. 2.4, we review the “causal diamond” description of the multiverse. The
causal diamond is the largest spacetime region that can be causally probed, and it can
be thought of as the past light-cone from a point on the future conformal boundary.
We argue that the causal diamond description leads to a natural, observer-independent
choice of environment: because its boundary is light-like, it acts as a one-way membrane
and degrees of freedom that leave the diamond do not return except in very special

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cases. These degrees of freedom can be traced over, leading to a branching tree of
causal diamond histories.
Next, we turn to the question of whether the global picture of the multiverse can
be recovered from the decoherent causal diamonds. In Sec. 2.5, we review a known
duality between the causal diamond and a particular foliation of the global geometry
known as light-cone time: both give the same probabilities. This duality took the
standard global picture as a starting point, but in Sec. 2.6, we reinterpret it as a way of
reconstructing the global viewpoint from the local one. If the causal diamond histories
are the many-worlds, this construction shows that the multiverse is the many-worlds
pieced together in a single geometry.
In Sec. 3 we turn to the second limitation associated with decoherence, its re-
versibility. Consider a causal diamond with finite maximal boundary area Amax . En-
tropy bounds imply that such diamonds can be described by a Hilbert space with finite
dimension no greater than exp(Amax /2). This means that no observables in such dia-
monds can be defined with infinite precision. In Sec. 3.1 and 3.2, we will discuss another
implication of this finiteness: there is a tiny but nonzero probability that decoherence
will be undone. This means that the decoherent histories of causal diamonds, and the
reconstruction of a global spacetime from such diamonds, is not completely exact.
No matter how good an approximation is, it is important to understand the precise
statement that it is an approximation to. In Sec. 3.3, we will develop two postulates
that should be satisfied by a fundamental quantum-mechanical theory if decoherence
is to be sharp and the associated probabilities operationally meaningful: decoherence
must be irreversible, and it must occur infinitely many times for a given experiment in
a single causally connected region.
The string landscape contains supersymmetric vacua with exactly vanishing cosmo-
logical constant. Causal diamonds which enter such vacua have infinite boundary area
at late times. We argue in Sec. 3.4 that in these “hat” regions, all our postulates can
be satisfied. Exact observables can exist and decoherence by the mechanism of Sec. 2.4
can be truly irreversible. Moreover, because the hat is a spatially open, statistically
homogeneous universe, anything that happens in the hat will happen infinitely many
times.
In Sec. 3.5 we review black hole complementarity, and we conjecture an analogous
“hat complementarity” for the multiverse. It ensures that the approximate observables
and approximate decoherence of causal diamonds with finite area (Sec. 2.4) have precise
counterparts in the hat. In Sec. 3.6 we propose a relation between the global multiverse
reconstruction of Sec. 2.6, and the Census Taker cutoff [6] on the hat geometry.
Two interesting papers have recently explored relations between the many-worlds
interpretation and the multiverse [7, 8]. The present work differs substantially in a

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E S+A

M

Figure 1: Decoherence and causality. At the event M , a macroscopic apparatus A becomes
correlated with a quantum system S. Thereafter, environmental degrees of freedom E interact
with the apparatus. In practice, an observer viewing the apparatus is ignorant of the exact
state of the environment and so must trace over this Hilbert space factor. This results in a
mixed state which is diagonal in a particular “pointer” basis picked out by the interaction
between E and A. The state of the full system SAE, however, remains pure. In particular,
decoherence does not take place, and no preferred bases arises, in a complete description of
any region larger than the future lightcone of M .

number of aspects. Among them is the notion that causal diamonds provide a pre-
ferred environment for decoherence, our view of the global multiverse as a patchwork
of decoherent causal diamonds, our postulates requiring irreversible entanglement and
inﬁnite repetition, and the associated role we ascribe to hat regions of the multiverse.

2. Building the multiverse from the many worlds of causal di-
amonds
2.1 Decoherence and causality
The decoherence mechanism reviewed above relies on ignoring the degrees of freedom
that a given observer fails to monitor, which is ﬁne if our goal is to explain the ex-
periences of that observer. But this subjective viewpoint clashes with the impersonal,
unitary description of large spacetime regions—the viewpoint usually adopted in cos-
mology. We are free, of course, to pick any subsystem and trace over it. But the

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outcome will depend on this choice. The usual choices implicitly involve locality but
not in a unique way.
For example, we might choose S to be an electron and E to be the inanimate
laboratory. The system’s wave function collapses when the electron becomes entangled
with some detector. But we may also include in S everything out to the edge of the
solar system. The environment is whatever is out beyond the orbit of Pluto. In that
case the collapse of the system wavefunction cannot take place until a photon from the
detector has passed Pluto’s orbit. This would take about a five hours during which the
system wavefunction is coherent.
In particular, decoherence cannot occur in the complete quantum description of
any region larger than the future light-cone of the measurement event M (Fig. 1). All
environmental degrees of freedom that could have become entangled with the apparatus
since the measurement took place must lie within this lightcone and hence are included,
not traced over, in a complete description of the state. An example of such a region
is the whole universe, i.e., any Cauchy surface to the future of M . But at least at
sufficiently early times, the future light-cone of M will be much smaller than the whole
universe. Already on this scale, the system SAE will be coherent.
In our earlier example, suppose that we measure the spin of an electron that is
initially prepared in a superposition of spin-up and spin-down, a|0 S + b|1 S , resulting
in the state |ψ of Eq. (1.2). A complete description of the solar system (defined as
the interior of a sphere the size of Pluto’s orbit, with a light-crossing time of about
10 hours) by a local quantum field theory contains every particle that could possibly
have interacted with the apparatus after the measurement, for about 5 hours. This
description would maintain the coherence of the macroscopic superpositions implicit in
the state |ψ , such as apparatus-up with apparatus-down, until the first photons that
are entangled with the apparatus leave the solar system.
Of course, a detailed knowledge of the quantum state in such large regions is
unavailable to a realistic observer. (Indeed, if the region is larger than a cosmological
event horizon, then its quantum state is cannot be probed at all without violating
causality.) Yet, our theoretical description of matter fields in spacetime retains, in
principle, all degrees of freedom and full coherence of the quantum state. In theoretical
cosmology, this can lead to inconsistencies, if we describe regions that are larger than the
future light-cones of events that we nevertheless treat as decohered. We now consider
an important example.

2.2 Failure to decohere: A problem with the global multiverse
The above analysis undermines what we will call the “standard global picture” of an
eternally inflating spacetime. Consider an effective potential containing at least one

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inflating false vacuum, i.e. a metastable de Sitter vacuum with decay rate much less
than one decay per Hubble volume and Hubble time. We will also assume that there
is at least one terminal vacuum, with nonpositive cosmological constant. (The string
theory landscape is believed to have at least 10100 s of vacua of both types [9–12].)
According to the standard description of eternal inflation, an inflating vacuum nu-
cleates bubble-universes in a statistical manner similar to the way superheated water
nucleates bubbles of steam. That process is described by classical stochastic production
of bubbles which occurs randomly but the randomness is classical. The bubbles nucle-
ate at definite locations and coherent quantum mechanical interference plays no role.
The conventional description of eternal inflation similarly based on classical stochastic
processes. However, this picture is not consistent with a complete quantum-mechanical
description of a global region of the multiverse.
To explain why this is so, consider the future domain of dependence D(Σ0 ) of a
sufficiently large hypersurface Σ0 , which need not be a Cauchy surface. D(Σ0 ) consists
of all events that can be predicted from data on Σ0 ; see Fig. 2. If Σ0 contains suffi-
ciently large and long-lived metastable de Sitter regions, then bubbles of vacua of lower
energy do not consume the parent de Sitter vacua in which they nucleate [13]. Hence,
the de Sitter vacua are said to inflate eternally, producing an unbounded number of
bubble universes. The resulting spacetime is said to have the structure shown in the
conformal diagram in Fig. 2, with bubbles nucleating at definite spacetime events. The
future conformal boundary is spacelike in regions with negative cosmological constant,
corresponding to a local big crunch. The boundary contains null “hats” in regions
occupied by vacua with Λ = 0.
But this picture does not arise in a complete quantum description of D(Σ0 ). The
future light-cones of events at late times are much smaller than D(Σ0 ). In any state
that describes the entire spacetime region D(Σ0 ), decoherence can only take place at
the margin of D(Σ0 ) (shown light shaded in Fig. 2), in the region from which particles
can escape into the complement of D(Σ0 ) in the full spacetime. No decoherence can
take place in the infinite spacetime region defined by the past domain of dependence
of the future boundary of D(Σ0 ). In this region, quantum evolution remains coherent
even if it results in the superposition of macroscopically distinct matter or spacetime
configurations.
An important example is the superposition of vacuum decays taking place at dif-
ferent places. Without decoherence, it makes no sense to say that bubbles nucleate at
particular times and locations; rather, a wavefunction with initial support only in the
parent vacuum develops into a superposition of parent and daughter vacua. Bubbles
nucleating at all places at times are “quantum superimposed”. With the gravitational
backreaction included, the metric, too, would remain in a quantum-mechanical super-

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future boundary

Σ0

Figure 2: The future domain of dependence, D(Σ0 ), (light or dark shaded) is the spacetime
region that can be predicted from data on the timeslice Σ0 . If the future conformal boundary
contains spacelike portions, as in eternal inflation or inside a black hole, then the future
light-cones of events in the dark shaded region remain entirely within D(Σ0 ). Pure quantum
states do not decohere in this region, in a complete description of D(Σ0 ). This is true even for
states that involve macroscopic superpositions, such as the locations of pocket universes in
eternal inflation (dashed lines), calling into question the self-consistency of the global picture
of eternal inflation.

position. This contradicts the standard global picture of eternal inflation, in which
domain walls, vacua, and the spacetime metric take on definite values, as if drawn from
a density matrix obtained by tracing over some degrees of freedom, and as if the inter-
action with these degrees of freedom had picked out a preferred basis that eliminates
the quantum superposition of bubbles and vacua.
Let us quickly get rid of one red herring: Can the standard geometry of eternal
inflation be recovered by using so-called semi-classical gravity in which the metric is
sourced by the expectation value of the energy-momentum tensor,

Gµν = 8π Tµν ? (2.1)

This does not work because the matter quantum fields would still remain coherent. At
the level of the quantum fields, the wavefunction initially has support only in the false
vacuum. Over time it evolves to a superposition of the false vacuum (with decreasing
amplitude), with the true vacuum (with increasing amplitude), plus a superposition
of expanding and colliding domain walls. This state is quite complicated but the
expectation value of its stress tensor should remain spatially homogeneous if it was so
initially. The net effect, over time, would be a continuous conversion of vacuum energy
into ordinary matter or radiation (from the collision of bubbles and motion of the scalar
field). By Eq. (2.1), the geometry spacetime would respond to the homogeneous glide
of the vacuum energy to negative values. This would result in a global crunch after
finite time, in stark contrast to the standard picture of global eternal inflation. In
any case, it seems implausible that semi-classical gravity should apply in a situation in

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which coherent branches of the wavefunction have radically different gravitational back-
reaction. The AdS/CFT correspondence provides an explicit counterexample, since the
superposition of two CFT states that correspond to different classical geometries must
correspond to a quantum superposition of the two metrics.
The conclusion that we come to from these considerations is not that the global
multiverse is meaningless, but that the parallel view should not be implemented by
unitary quantum mechanics. But is there an alternative? Can the standard global
picture be recovered by considering an observer who has access only to some of the
degrees of freedom of the multiverse, and appealing to decoherence? We debate this
question in the following section.

2.3 Simplicio’s proposal

Simplicio and Sagredo have studied Sections 2.1 and 2.2, supplied to them by Salviati.
They meet at Sagredo’s house for a discussion.
Simplicio: You have convinced me that a complete description of eternal inflation
by unitary quantum evolution on global slices will not lead to a picture in which bubbles
form at definite places and times. But all I need is an observer somewhere! Then I
can take this observer’s point of view and trace over the degrees of freedom that are
inaccessible to him. This decoheres events, such as bubble nucleations, in the entire
global multiverse. It actually helps that some regions are causally disconnected from the
observer: this makes his environment—the degrees of freedom he fails to access—really
huge.
Sagredo: An interesting idea. But you seem to include everything outside the
observer’s horizon region in what you call the environment. Once you trace over it, it
is gone from your description and you could not possibly recover a global spacetime.
Simplicio: Your objection is valid, but it also shows me how to fix my proposal.
The observer should only trace over environmental degrees in his own horizon. Deco-
herence is very efficient, so this should suffice.
Sagredo: I wonder what would happen if there were two observers in widely
separated regions. If one observer’s environment is enough to decohere the whole
universe, which one should we pick?
Simplicio: I have not done a calculation but it seems to me that it shouldn’t
matter. The outcome of an experiment by one of the observers should be the same, no
matter which observer’s environment I trace over. That is certainly how it works when
you and I both stare at the same apparatus.
Sagredo: Something is different about the multiverse. When you and I both
observe Salviati, we all become correlated by interactions with a common environment.

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future boundary
O
P
Σ0

Figure 3: Environmental degrees of freedom entangled with an observer at O remain within
the causal future of the causal past of O, J + [J − (O)] (cyan/shaded). They are not entangled
with distant regions of the multiverse. Tracing over them will not lead to decoherence of
a bubble nucleated at P , for example, and hence will fail to reproduce the standard global
picture of eternal inflation.

But how does an observer in one horizon volume become correlated with an object in
another horizon volume far away?
Salviati: Sagredo, you hit the nail on the head. Decoherence requires the inter-
action of environmental degrees of freedom with the apparatus and the observer. This
entangles them, and it leads to a density matrix once the environment is ignored by the
observer. But an observer cannot have interacted with degrees of freedom that were
never present in his past light-cone.
Sagredo: Thank you for articulating so clearly what to me was only a vague
concern. Simplicio, you look puzzled, so let me summarize our objection in my own
words. You proposed a method for obtaining the standard global picture of eternal
inflation: you claim that we need only identify an arbitrary observer in the multiverse
and trace over his environment. If we defined the environment as all degrees of freedom
the observer fails to monitor, then it would include the causally disconnected regions
outside his horizon. With this definition, these regions will disappear entirely from your
description, in conflict with the global picture. So we agreed to define the environment
as the degrees of freedom that have interacted with the observer and which he cannot
access in practice. But in this case, the environment includes no degrees of freedom
outside the causal future of the observer’s causal past. I have drawn this region in
Fig. 3. But tracing over an environment can only decohere degrees of freedom that it
is entangled with. In this case, it can decohere some events that lie in the observer’s
past light-cone. But it cannot affect quantum coherence in far-away horizon regions,
because the environment you have picked is not entangled with these regions. In those
regions, bubble walls and vacua will remain in superposition, which again conflicts with
the standard global picture of eternal inflation.

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Simplicio: I see that my idea still has some problems. I will need to identify more
than one observer-environment pair. In fact, if I wish to preserve the global picture
of the multiverse, I will have to assume that an observer is present in every horizon
volume, at all times! Otherwise, there will be horizon regions where no one is around
to decide which degrees of freedom are hard to keep track of, so there is no way to
identify and trace over an environment. In such regions, bubbles would not form at
particular places and times, in conflict with the standard global picture.
Sagredo: But this assumption is clearly violated in many landscape models. Most
de Sitter vacua have large cosmological constant, so that a single horizon volume is too
small to contain the large number of degrees of freedom required for an observer. And
regions with small vacuum energy may be very long lived, so the corresponding bubbles
contain many horizon volumes that are completely empty. I’m afraid, Simplicio, that
your efforts to rescue the global multiverse are destined to fail.
Salviati: Why don’t we back up a little and return to Simplicio’s initial sug-
gestion. Sagredo, you objected that everything outside an observer’s horizon would
naturally be part of his environment and would be gone from our description if we
trace over it...
Sagredo: ...which means that the whole global description would be gone...
Salviati: ...but why is that a problem? No observer inside the universe can ever
see more than what is in their past light-cone at late times, or more precisely, in their
causal diamond. We may not be able to recover the global picture by tracing over
the region behind an observer’s horizon, but the same procedure might well achieve
decoherence in the region the observer can actually access. In fact, we don’t even
need an actual observer: we can get decoherence by tracing over degrees of freedom
that leave the causal horizon of any worldline! This will allow us to say that a bubble
formed in one place and not another. So why don’t we give up on the global description
for a moment. Later on, we can check whether a global picture can be recovered in
some way from the decoherent causal diamonds.
Salviati hands out Sections 2.4–2.6.

2.4 Objective decoherence from the causal diamond
If Hawking radiation contains the full information about the quantum state of a star
that collapsed to form a black hole, then there is an apparent paradox. The star is
located inside the black hole at spacelike separation from the Hawking cloud; hence, two
copies of the original quantum information are present simultaneously. The xeroxing of
quantum information, however, conflicts with the linearity of quantum mechanics [14].
The paradox is resolved by “black hole complementarity” [15]. By causality, no observer
can see both copies of the information. A theory of everything should be able to describe

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future boundary q

Σ0 p

Figure 4: The causal diamond (pink/shaded) spanned by two events p and q is the set of
points that lie on causal curves from p to q. p is called the origin and q the tip of the causal
diamond. In the example shown, p lies on the initial surface and q on the future conformal
boundary of the spacetime. The causal diamond is largest spacetime region that can be
causally probed by an observer travelling from p to q.

any experiment that can actually be performed by some observer in the universe, but it
need not describe the global viewpoint of a “superobserver”, who sees both the interior
and the exterior of a black hole. Evidently, the global description is inconsistent and
must be rejected.
If the global viewpoint fails in a black hole geometry, then it must be abandoned
in any spacetime. Hence, it is useful to characterize generally what spacetime regions
can be causally probed. An experiment beginning at a spacetime event p and ending at
the event q in the future of p can probe the causal diamond I + (p) ∩ I − (q) (Fig. 4). By
starting earlier or ﬁnishing later, the causal diamond can be enlarged. In spacetimes
with a spacelike future boundary, such as black holes and many cosmological solutions,
the global universe is much larger than any causal diamond it contains. Here we will be
interested in diamonds that are as large as possible, in the sense that p and q correspond
to the past and future endpoints of an inextendible worldline.
We will now argue that the causal diamond can play a useful role in making decoher-
ence more objective. Our discussion will be completely general, though for concreteness
it can be useful to think of causal diamonds in a landscape, which start in a de Sitter
vacuum and end up, after a number of decays, in a crunching Λ < 0 vacuum.
Consider a causal diamond, C, with future boundary B and past boundary B, as ˜
˜
shown in Fig. 5. For simplicity, suppose that the initial state on B is pure. Matter
degrees of freedom that leave the diamond by crossing B become inaccessible to any
experiment within C, by causality. Therefore they must be traced over.
In practice, there will be many other degrees of freedom that an observer fails

– 13 –

Figure 5: Causal diamond spanned by the world-line (green) of an observer. Environmental
degrees of freedom (purple dashed line) that leave the observer’s past light-cone (blue) at
some finite time can be recovered using mirrors.

to control, including most degrees of freedom that have exited his past light-cone at
any finite time along his worldline. But such degrees of freedom can be reflected by
mirrors, or in some other way change their direction of motion back towards the observer
(Fig. 5). Thus, at least in principle, the observer could later be brought into contact
again with any degrees of freedom that remain within the causal diamond C, restoring
coherence. Also, the observer at finite time has not had an opportunity to observe
˜
degrees of freedom coming from the portion outside his past lightcone on B; but those
he might observe by waiting longer. Hence, we will be interested only in degrees of
freedom that leave C by crossing the boundary B.
The boundary B may contain components that are the event horizons of black holes.
If black hole evaporation is unitary, then such degrees of freedom will be returned to
the interior of the causal diamond in the form of Hawking radiation. We can treat
this formally by replacing the black hole with a membrane that contains the relevant
degrees of freedom at the stretched horizon and releases them as it shrinks to zero
size [15]. However, we insist that degrees of freedom crossing the outermost component
of B (which corresponds to the event horizon in de Sitter universes) are traced over. It
does not matter for this purpose whether we regard these degrees of freedom as being
absorbed by the boundary or as crossing through the boundary, as long as we assume
that they are inaccessible to any experiment performed within C. This assumption
seems reasonable, since there is no compelling argument that the unitarity evaporation

– 14 –

Figure 6: The surface β divides the future boundary of the causal diamond into two portions
B± . Degrees of freedom that passed through B− are forever inaccessible from within the
diamond. Tracing over them deﬁnes a density matrix at the time γ. The pure states that
diagonalize this matrix can be represented as branches. As more degrees of freedom leave the
causal diamond, a branching tree is generated that represents all possible decoherent histories
within the diamond.

of black holes should extend to cosmological event horizons. Indeed, it is unclear how
the statement of unitarity would be formulated in that context. (A contrary viewpoint,
which ascribes unitarity even to non-Killing horizons, is explored in Ref. [6].)
The boundary B is a null hypersurface. Consider a cross-section β of B, i.e., a
spacelike two-dimensional surface that divides B into two portions: the upper portion,
B+ , which contains the tip of the causal diamond, and the lower portion B− . We
may trace over degrees of freedom on B− ; this corresponds to the matter that has
left the causal diamond by the time β and hence has become inaccessible from within
the diamond. Thus we obtain a density matrix ρ(β) on the portion B+ . Assuming
unitary evolution of closed systems, the same density matrix also determines the state
on any spacelike surface bounded by β; and it determines the state on the portion of
the boundary of the past of β that lies within C, γ. Note that γ is a null hypersurface.
In fact, γ can be chosen to be a future lightcone from an event inside C (more precisely,
the portion of that light-cone that lies within C); the intersection of γ with B then
deﬁnes β.
A useful way of thinking about ρ(β) is as follows. The boundary of the causal past

– 15 –

of β consists of two portions, γ and B− . The degrees of freedom that cross B− are
analogous to the environment in the usual discussion of decoherence, except in that
they are inaccessible from within the causal diamond C not just in practice but in
principle. The remaining degrees of freedom in the past of β cross through γ and thus
stay inside the causal diamond. They are analogous to the system and apparatus, which
are now in one of the states represented in the density matrix ρ(β). A measurement
is an interaction between degrees of freedom that later pass through γ and degrees of
freedom that later pass through B− . The basis in which ρ(β) is diagonal consists of the
different pure states that could result from the outcome of measurements in the causal
past of β.
We can now go further and consider foliations of the boundary B. Each member of
the foliation is a two-dimensional spatial surface β dividing the boundary into portions
B+ and B− . We can regard β as a time variable. For example, any a foliation of the
causal diamond C into three-dimensional spacelike hypersurfaces of equal time β will
induce a foliation of the boundary B into two-dimensional spacelike surfaces. Another
example, on which we will focus, is shown in Fig. 6: consider an arbitrary time-like
worldline that ends at the tip of the causal diamond. Now construct the future light-
cone from every point on the worldline. This will induce a foliation of B into slices β.
It is convenient to identify β with the proper time along the worldline.
The sequence of density matrices ρ(β1 ), ρ(β2 ), . . . , ρ(βn ) describes a branching tree,
in which any path from the root to one of the final twigs represents a particular history of
the entire causal diamond, coarse-grained on the appropriate timescale. These histories
are “minimally decoherent” in the sense that the only degrees of freedom that are traced
over are those that cannot be accessed even in principle. In practice, an observer at the
time β may well already assign a definite outcome to an observation even though no
particles correlated with the apparatus have yet crossed B− (β). There is a negligible
but nonzero probability of recoherence until the first particles cross the boundary; only
then is coherence irreversibly lost.
Strictly speaking, the above analysis should be expanded to allow for the different
gravitational backreaction of different branches. The exact location of the boundary B
at the time β depends on what happens at later times. (This suggests that ultimately,
it may be more natural to construct the decoherent causal diamonds from the top
down, starting in the future and ending in the past.) Here we will be interested mainly
in the application to the eternally inflating multiverse,3 where we can sidestep this
3
However, the above discussion has implications for any global geometry in which observers fall
out of causal contact at late times, including crunching universes and black hole interiors. Suppose all
observers were originally causally connected, i.e., their past light-cones substantially overlap at early
times. Then the different classes of decoherent histories that may be experienced by different observers

– 16 –

issue by choosing large enough timesteps. In de Sitter vacua, on timescales of order
tΛ ∼ |Λ|−1/2 , the apparent horizon, which is locally defined, approaches the event
horizon B at an exponential rate. Mathematically, the difference between the two
depends on the future evolution, but it is exponentially small and thus is irrelevant
physically. Vacua with negative cosmological constant crunch on the timescale tΛ [16]
and so will not be resolved in detail at this level of coarse-graining.
We expect that the distinction between causal diamond bulk and its boundary is
precise only to order e−A(β) , where A is the area of the boundary at the time β. Because
of entropy bounds [17–20], no observables in any finite spacetime region can be defined
to better accuracy than this. A related limitation applies to the objective notion of
decoherence we have given, and it will be inherited by the reconstruction of global
geometry we propose below. This will play an imporant role in Sec. 3, where we will
argue that the hat regions with Λ = 0 provide an exact counterpart to the approximate
observables and approximate decoherence described here.

2.5 Global-local measure duality
In this section, we will review the duality that relates the causal diamond to a global
time cutoff called light-cone time: both define the same probabilities when they are
used as regulators for eternal inflation. As originally derived, the duality assumed the
standard global picture as a starting point, a viewpoint we have criticized in Sec. 2.2.
Here we will take the opposite viewpoint: the local picture is the starting point, and
the duality suggests that a global spacetime can be reconstructed from the more funda-
mental structure of decoherent causal diamond histories. Indeed, light-cone time will
play a central role in the construction proposed in Sec. 2.6.
By restricting to a causal diamond, we obtained a natural choice of environment:
the degrees of freedom that exit from the diamond. Tracing over this environment leads
to a branching tree of objective, observer-independent decoherent histories—precisely
the kind of notion that was lacking in the global description. In the causal diamond,
bubbles of different vacua really do nucleate at specific times and places. They decohere
when the bubble wall leaves the diamond.
Consider a large landscape of vacua. Starting, say, in a vacuum with very large
cosmological constant, a typical diamond contains a sequence of bubble nucleations
(perhaps hundreds in some toy models [9, 21]), which ends in a vacuum with negative
cosmological constant (and thus a crunch), or with vanishing cosmological constant (a
supersymmetric open universe, or “hat”). Different paths through the landscape are

arise from differences in the amount, the identity, and the order of the degrees of freedom that the
observer must trace over.

– 17 –

future boundary

Q

!0 "( Q)

Figure 7: In this diagram, the standard global picture of eternal inflation is taken as a
starting point. Geodesics (thin vertical lines) emanating from an initial surface Σ0 define
an ensemble of causal patches (the leftmost is shaded grey/light) with a particular mix of
initial conditions. In the limit of a dense family of geodesics, the global spacetime is thus
deconstructed into overlapping causal patches. The number of patches overlapping at Q is
determined by the number of geodesics entering the future light-cone of Q. In the continuum
limit, this becomes the volume (Q) from which the relevant geodesics originate, which in turn
defines the light-cone time at Q. This relation implies an exact duality between the causal
patch measure and the light-cone time cut-off for the purpose of regulating divergences and
computing probabilities in eternal inflation.

followed with probabilities determined by branching ratios. Some of these paths will
pass through long-lived vacua with anomalously small cosmological constant, such as
ours.
The causal diamond has already received some attention in the context of the mul-
tiverse. It was proposed [22] as a probability measure: a method for cutting off infinities
and obtaining well-defined amplitudes. Phenomenologically, the causal diamond mea-
sure is among the most successful proposals extant [23–26]. From the viewpoint of
economy, it is attractive since it merely exploits a restriction that was already imposed
on us by black hole complementarity and uses it to solve another problem. And con-
ceptually, our contrasting analyses of decoherence in the global and causal diamond
viewpoints suggests that the causal diamond is the more fundamental of the two.
This argument is independent of black hole complementarity, though both point at
the same conclusion. It is also independent of the context of eternal inflation. However,
if we assume that the universe is eternally inflating, then it may be possible to merge
all possible causal diamond histories into a single global geometry.
If we are content to take the standard global picture as a starting point, then it

– 18 –

is straightforward to deconstruct it into overlapping causal diamonds or patches4 [8,
27] (see Fig. 7, taken from Ref. [27]). Indeed, a stronger statement is possible: as
far as any prediction goes, the causal diamond viewpoint is indistinguishable from a
particular time cutoff on the eternally inflating global spacetime. An exact duality [27]
dictates that relative probabilities computed from the causal diamond agree exactly
with the probabilities computed from the light-cone time cutoff.5 The duality picks
out particular initial conditions for the causal diamond: it holds only if one starts in
the “dominant vacuum”, which is the de Sitter vacuum with the longest lifetime.
The light-cone time of an event Q is defined [28] in terms of the volume (Q) of
the future light-cone of Q on the future conformal boundary of the global spacetime;
see Fig. 7:
1
t(Q) ≡ − log (Q) . (2.2)
3
The volume , in turn, is defined as the proper volume occupied by those geodesics
orthogonal to an initial hypersurface Σ0 that eventually enter the future of Q. (For an
alternative definition directly in terms of an intrinsic boundary metric, see Ref. [30].)
We emphasize again that in these definitions, the standard global picture is taken for
granted; we disregard for now the objections of Sec. 2.2.
The light-cone time of an event tells us the factor by which that event is overcounted
in the overlapping ensemble of diamonds. This follows from the simple fact that the
geodesics whose causal diamonds includes Q are precisely the ones that enter the causal
future of Q. Consider a discretization of the family of geodesics orthogonal to Σ0 into
a set of geodesics at constant, finite density, as shown in Fig. 7. The definition of
light-cone time ensures that the number of diamonds that contain a given event Q is
proportional to = exp[−3t(Q)]. Now we take the limit as the density of geodesics
on Σ0 tends to infinity. In this limit, the entire global spacetime becomes covered by
the causal diamonds spanned by the geodesics. The relative overcounting of events at
two different light-cone times is still given by a factor exp(−3∆t). (To show that this
implies the exact equivalence of the causal diamond and the light-conetime cutoff, one
must also demonstrate that the rate at which events of any type I occur depends only
on t. This is indeed the case if Σ0 is chosen sufficiently late, i.e., if initial conditions
on the patches conform to the global attractor regime. Since we are not primarily
4
A causal patch can be viewed as the upper half of a causal diamond. In practice the difference is
negligible but strictly, the global-local duality holds for the patch, not the diamond. Here we use it
merely to motivate the construction of the following subsection, which does use diamonds.
5
It is worth noting that the light-cone time cutoff was not constructed with this equivalence in
mind. Motivated by an analogy with the UV/IR relation of the AdS/CFT correspondence [29], light-
cone time was formulated as a measure proposal [28] before the exact duality with the causal diamond
was discovered [27].

– 19 –

interested in the measure problem here, we will not review this aspect of the proof; see
Ref. [27] for details.)
Given the above deconstruction of the global spacetime, it is tempting to identify
the eternally inflating multiverse with the many worlds of quantum mechanics, if the
latter could somehow be related to branches in the wavefunction of the causal diamonds.
Without decoherence, however, there is neither a consistent global picture (as shown
in Sec. 2.2) nor a sensible way of picking out a preferred basis that would associate
“many worlds” to the pure quantum state of a causal diamond (Sec. 2.4).6
We have already shown that decoherence at the causal diamond boundary leads
to distinct causal diamond histories or “worlds”. To recover the global multiverse and
demonstrate that it can be viewed as a representation of the many causal diamond
worlds, one must show that it is possible to join together these histories consistently
into a single spacetime. This task is nontrivial. In the following section we offer a
solution in a very simple setting; we leave generalizations to more realistic models to
future work. Our construction will not be precisely the inverse of the deconstruction
shown in Fig. 7; for example, there will be no overlaps. However, it is closely related;
in particular, we will reconstruct the global spacetime in discrete steps of light-cone
time.

2.6 Constructing a global multiverse from many causal diamond worlds
In this section, we will sketch a construction in 1 + 1 dimensions by which a global
picture emerges in constant increments of light-cone time (Fig. 8). For simplicity, we
will work on a fixed de Sitter background metric,

ds2
2
= −(log 2 dt)2 + 22t dx2 , (2.3)

where is an arbitrary length scale. A future light-cone from an event at the time t
grows to comoving size = 21−t , so t represents light-cone time up to a trivial shift
and rescaling: t = 1 − log2 . We take the spatial coordinate to be noncompact, though
our construction would not change significantly if x was compactified by identifying
x ∼ x + n for some integer n.
=
The fixed background assumption allows us to separate the geometric problem—
building the above global metric from individual causal diamonds—from the problem of
matching matter configurations across the seams. We will be interested in constructing
a global spacetime of infinite four-volume in the future but not int the past. Therefore,
we take the geodesic generating each diamond to be maximally extended towards the
6
In this aspect our viewpoint differs from [8].

– 20 –

t
3

2

A C D E
1

F G

0 x
−1/2 0 1/2 1 3/2

Figure 8: Construction of a global spacetime from decoherent causal diamond histories.
Green squares indicate the origins of causal diamonds that tile the global de Sitter space. Red
dots indicate discrete time steps of order tΛ at which decoherence occurs within individual
causal diamonds. For example, a nonzero amplitude for bubble nucleation in the region
bounded by ACDEF G entangles the null segment A with C: either both will contain a
bubble or neither will. Similarly, D is entangled with E. The density matrix for CD is
diagonal in a basis consisting of pure states in which a bubble either forms or does not
form. This eliminates superpositions of the false (white) and true (yellow) vacuum. Initial
conditions for new causal diamonds at green squares are controlled by entanglements such as
that of A with C. Further details are described in the text.

future, but ﬁnite towards the past. This means that the lower tips do not lie on the
past conformal boundary of de Sitter space. Note that all such semi-inﬁnite diamonds
are isometric.

The geometric problem is particularly simple in 1+1 dimensions because it is pos-
sible to tile the global geometry precisely with causal diamonds, with no overlaps. We
will demonstrate this by listing explicitly the locations of the causal diamond tiles.
All diamonds in our construction are generated by geodesics that are comoving in the
metric of Eq. (2.3): if the origin of the diamond is at (t, x), then its tip is at (∞, x).
Hence we will label diamonds by the location of their origins, shown as green squares

– 21 –

in Fig. 8. They are located at the events

(x, t) = (m, 0) (2.4)
2m + 1
(x, t) = ,n , (2.5)
2n

where n runs over the positive integers and m runs over all integers. From Fig. 8, it
is easy to see that these diamonds tile the global spacetime, covering every point with
t ≥ 1 precisely once, except on the edges. We now turn to the task of matching the
quantum states of matter at these seams.
We will assume that there exists a metastable vacuum F which decays with a rate
Γ per unit Hubble time and Hubble volume to a terminal vacuum T which does not
decay. We will work to order Γ, neglecting collisions between bubbles of the T vacuum.
A number of further simpliﬁcations are made below.
Our construction will be iterative. We begin by considering the causal diamond
with origin at (x, t) = (0, 0). In the spirit of Sec. 2.4, we follow the generating geodesic
(green vertical line) for a time ∆t = 1 (corresponding to a proper time of order tΛ ) to
the event t = 1, x = 0, marked by a red circle in Fig. 8. The future light-cone of this
event divides the boundary of the (0, 0) diamond into two portions, B± , as in Fig. 6.
B− itself consists of two disconnected portions, which we label A and E. Together
with segments C and D of the future light-cone, ACDE forms a Cauchy surface of
the diamond. The evolution from the bottom boundary F G to the surface ACDE is
unitary. For deﬁniteness, we will assume that the state on F G is the false vacuum,
though other initial conditions can easily be considered.
The pure state on ACDE can be thought of as a superposition of the false vac-
uum with bubbles of true vacuum that nucleate somewhere in the region delimited by
ACDEF G. To keep things simple, we imagine that decays can occur only at three
¯
points, at each with probability Γ ∼ Γ/3: at the origin, (0, 0); at the midpoint of
the edge F , (− 1 , 1 ), and at the midpoint of G, ( 4 , 1 ). We assume, moreover, that
4 2
1
2
the true vacuum occupies the entire future light-cone of a nucleation point. In this
approximation the pure state on ACDE takes the form

¯
(1 − 3Γ)1/2 |F A |F C |F D |F E
¯
+Γ1/2 |T A |T C |F D |F E
¯
+ Γ1/2 |T A |T C |T D |T E
¯
+Γ1/2 |F A |F C |T D |T E , (2.6)

where the last three terms correspond to the possible nucleation points, from left to
right.
From the point of view of an observer in the (0, 0) diamond, the Hilbert space
factor AD should be traced out. This results in a density matrix on the slice CD,

– 22 –

which can be regarded as an initial condition for the smaller diamond beginning at the
point (x, t) = (0, 1):

¯
ρ(0, 1) = (1 − 3Γ) |F C |F D D F| C F|
+Γ ¯ |F C |T D D T | C F|
+Γ ¯ |T C |F D D F| C T |
¯
+Γ |T C |T D D T |C T | (2.7)

The density matrix can be regarded as an ensemble of four pure states: |F C |F D with
¯ ¯
probability (1 − 3Γ); and |F C |T D , |T C |F D , |T C |T D , each with probability Γ.
The same construction can be applied to every “zeroth generation” causal diamond:
the diamonds with origin at (m, 0), with m integer. Since their number is infinite, we
can realize the ensemble of Eq. (2.7) precisely, in the emerging global spacetime, by
assigning appropriate initial conditions to the “first generation sub-diamonds” (m, 1).
¯
The state |F C |T D is assigned to a fraction 1 − 3Γ of the (m, 1) diamonds; and
¯
each of the states |F C |T D , |T C |F D , |T C |T D is assigned to a fraction Γ of (m, 1)
7
diamonds.
So far, we have merely carried out the process described in Fig. 6 for one time step
in each of the (m, 0) diamonds, resulting in initial conditions for the subdiamonds that
start at the red circles at (m, 1). In order to obtain a global description, we must also
“fill in the gaps” between the (m, 1) diamonds by specifying initial conditions for the
“first generation new diamonds” that start at the green squares at (m + 1 , 1). But their
2
initial conditions are completely determined by the entangled pure state on ACDE,
Eq. (2.6), and the identical pure states on the analogous Cauchy surfaces of the other
(m, 0) diamonds. Because of entanglement, the state on C is the same as on A. If
C is in the true vacuum, then so is A; and if C is in the false vacuum, then so is A.
The edges D and E are similarly entangled. Thus, the assignment of definite initial
conditions to the (m, 1) diamonds completely determines the initial conditions on the
(m + 1 ) diamonds. We have thus generated initial conditions for all first-generation
2
diamonds (those with n = 1). Now we simply repeat the entire procedure to obtain
initial conditions for the second generation (n = 2), and so on.8
7
We defer to future work the interesting question of whether further constraints should be imposed
¯
on the statistics of this distribution. For example, for rational values of Γ, the assignment of pure
states to diamonds could be made in a periodic fashion, or at random subject only to the above
constraint on relative fractions.
8
Note that this construction determines the initial conditions for all but a finite number of diamonds.
In a more general setting, it would select initial conditions in the dominant vacuum. We thank Ben
Freivogel and I-Sheng Yang for pointing out a closely related observation.

– 23 –

This will generate a fractal distribution of true vacuum bubbles, of the type that is
usually associated with a global description (Fig. 8). The manner in which this picture
arises is totally distinct from a naive unitary evolution of global time slices, in which a
full superposition of false and true vacuum would persist (with time-dependent ampli-
tudes). The standard global picture can only be obtained by exploiting the decoherence
of causal diamonds while proliferating their number. The multiverse is a patchwork of
infinitely many branching histories, the many worlds of causal diamonds.
The construction we have given is only a toy model. The causal diamonds in higher
dimensions do not fit together neatly to fill the space-time as they do in 1+1 dimensions,
so overlaps will have to be taken into account. Moreover, we have not considered the
backreaction on the gravitational field. Finally, in general the future boundary is
not everywhere spacelike but contains hats corresponding to supersymmetric vacua
with Λ = 0. Our view will be that the hat regions play a very special role that is
complementary, both in the colloquial and in the technical sense, to the construction
we have given here. Any construction involving finite causal diamonds in necessarily
approximate. We now turn to the potentially precise world of the Census Taker, a
fictitious observer living in a hat region.

3. The many worlds of the census taker

3.1 Decoherence and recoherence

In Sec. 2.1, we noted that decoherence is subjective to the extent that the choice of
environment is based merely on the practical considerations of an actual observer. We
then argued in Sec. 2.4 that the boundary of a causal patch can be regarded as a
preferred environment that leads to a more objective form of decoherence. However,
there is another element of subjectivity in decoherence which we have not yet addressed:
decoherence is reversible. Whether, and how soon, coherence is restored depends on
the dynamical evolution governing the system and environment.
Consider an optical interference experiment (shown in Fig. 9) in which a light beam
reflects off two mirrors, m1 and m2 , and then illuminates a screen S. There is no sense
in which one can say that a given photon takes one or the other routes.
On the other hand if an observer or simple detector D interacts with the photon
and records which mirror the photon bounced off, the interference is destroyed. One
of the two possibilities is made real by the interaction and thereafter the observer may
ignore the branch of the wave function which does not agree with the observation.
Moreover, if a second observer describes the whole experiment ( including the first

– 24 –

Figure 9: Three optical interference experiments: m’s denote mirrors, S is a screen, and D
means detector. In the first setup, the coherence of the two paths is maintained. In the second
setup, a detector is placed at one mirror. If the detector is treated as an environment and its
Hilbert space is traced over, then the coherence of the superposition of photon paths is lost
and the interference pattern disappears. In the third setup, the detector reverts to its original
state when the photon passes it a second time. During the (arbitrarily long) time when the
photon travels between m1 D and m3 , tracing over the detector leads to decoherence. But
after the second pass through m1 D, coherence is restored, so the screen will show the same
pattern as in the first setup.

observer) as a single quantum system, the second observer’s observation will entangle
it in a consistent way.
Now let’s consider an unusual version of the experiment in which the upper arm
of the interferometer is replaced by a mirror-detector m1 D which detects a photon and
deflects it toward mirror m3 . From m3 the photon is reflected back to the detector
and then to the screen. The detector is constructed so that if a single photon passes
through it, it flips to a new state (it detects) but the next time a photon passes through,
it flips back to the original state. The detector is well-insulated from any environment.
The lower arm of the interferometer also has the path length increased but without a
detector.
Since if the photon goes through the detector, it goes through twice, at the end of
the experiment the detector is left in the original state of no-detection. It is obvious

– 25 –

that in this case the interference is restored. But there is something unusual going
on. During an intermediate time interval the photon was entangled with the detector.
The event of passing through the upper arm has been recorded and the photon’s wave
function has collapsed to an incoherent superposition. But eventually the photon and
the detector are disentangled. What happened was made to unhappen.
This illustrates that in order to give objective meaning to an event such as the
detection of a photon, it is not enough that the system becomes entangled with the
environment: the system must become irreversibly entangled with the environment, or
more precisely, with some environment.

3.2 Failure to irreversibly decohere: A limitation of finite systems

The above example may seem contrived, since it relied on the perfect isolation of
the detector from any larger environment, and on the mirror m3 that ensured that
the detected photon cannot escape. It would be impractical to arrange in a similar
manner for the recoherence of macroscopic superpositions, since an enormous number
of particles would have to be carefully controlled by mirrors. However, if we are willing
to wait, then recoherence is actually inevitable in any system that is dynamically closed,
i.e., a system with finite maximum entropy at all times.
For example consider a world inside a finite box, with finite energy and perfectly
reflecting walls. If the box is big enough and is initially out of thermal equilibrium,
then during the return to equilibrium structures can form, including galaxies, planets,
and observers. Entanglements can form between subsystems, but it is not hard to
see that they cannot be irreversible. Such closed systems with finite energy have a
finite maximum entropy, and for that reason the state vector will undergo quantum
recurrences. Whatever state the system finds itself in, after a suitably long time it
will return to the same state or to an arbitrarily close state. The recurrence time is
bounded by exp(N ) = exp(eSmax ), where N is the dimension of the Hilbert space that
is explored and Smax is the maximum entropy.
This has an important implication for the causal diamonds of Sec. 2.4. We argued
that the diamond bulk decoheres when degrees of freedom cross the boundary B of
the diamond. But consider a potential that just contains a single, stable de Sitter
vacuum with cosmological constant Λ. Then the maximum area of the boundary of
the diamond is the horizon area of empty de Sitter space, and the maximum entropy is
Smax = Amax /4 = 3π/Λ. This is the maximum total entropy [31], which is the sum of
the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary.
Assuming unitarity and ergodicity [32], this system is dynamically closed, and periodic
recurrences are inevitable.

– 26 –

Next, consider a landscape that contains vacua with positive and negative cosmo-
logical constant. We assume, for now, that there are no vacua with Λ = 0. Then the
maximum area Amax of the causal diamond boundary B is given by the greater of Λ−1 +
and Λ−2 , where Λ+ is the smallest positive value of Λ among all landscape vacua, and
−
Λ− is the largest negative value [33]. B is a null hypersurface with two-dimensional
spatial cross-sections, and it can be thought of as the union of two light-sheets that
emanate from the cross-section of maximum area. Therefore, the entropy that passes
through B is bounded by Amax /2 and hence is finite.
What does finite entropy imply for the decoherence mechanism of Sec. 2.4? If the
causal diamond were a unitary, ergodic quantum system, then it would again follow
that recurrences, including the restoration of coherence, are inevitable. This is plausible
for causal diamonds that remain forever in inflating vacua, but such diamonds form
a set of measure zero. Generically, causal diamonds will end up in terminal vacua
with negative cosmological constant, hitting a big crunch singularity after finite time.
In this case it is not clear that they admit a unitary quantum mechanical description
over arbitrarily long timescales, so recurrences are not mandatory. However, since
ergodicity cannot be assumed in this case, it seems plausible to us that there exists a
tiny non-zero probability for recurrences. We expect that in each metastable de Sitter
vacuum, the probability for re-coherence is given by the ratio of the decay timescale to
the recurrence timescale. Typically, this ratio is super-exponentially small [34], but it
does not vanish. In this sense, the objective decoherence of causal diamond histories
described in Sec. 2.4 is not completely sharp.

3.3 Sagredo’s postulates
The next morning, Simplicio and Salviati visit Sagredo to continue their discussion.
Simplicio: I have been pondering the idea we came up with yesterday, and I am
convinced that we have completely solved the problem. Causal diamonds have definite
histories, obtained by tracing over their boundary, which we treat as an observer-
independent environment. This gets rid of superpositions of different macroscopic ob-
jects, such as bubbles of different vacua, without the need to appeal to actual observers
inside the diamond. Each causal diamond history corresponds to a sequence of things
that “happen”. And the global picture of the multiverse is just a representation of
all the possible diamond histories in a single geometry: the many worlds of causal
diamonds!
Sagredo: I wish I could share in your satisfaction, but I am uncomfortable. Let
me describe my concerns, and perhaps you will be able to address them.
Salviati: I, too, have been unable to shake off a sense that that this is not
the whole story—that we should do better. I would be most interested to hear your

– 27 –

thoughts, Sagredo.
Sagredo: It’s true, as Simplicio says, that things “happen” when we trace over
degrees of freedom that leave the causal diamond. Pure states become a density matrix,
or to put it in Bohr’s language, the wavefunction describing the interior of the diamond
collapses. But how do we know that the coherence will not be restored? What prevents
things from “unhappening” later on?
Simplicio: According to Bohr, the irreversible character of observation is due to
the large classical nature of the apparatus.
Salviati: And decoherence allows us to understand this rather vague statement
more precisely: the apparatus becomes entangled with an enormous environment, which
is infinite for all practical purposes.
Sagredo: But even a large apparatus is a quantum system, and in principle,
the entanglement can be undone. The irreversibility of decoherence is often conflated
with the irreversibility of thermodynamics. A large system of many degrees of freedom
is very unlikely to find its way back to a re-cohered state. However, thermodynamic
irreversibility is an idealization that is only true for infinite systems. Irreversibility is
an approximation, true only for infinite systems.
Simplicio: But think how exquisite the approximation is! In a causal diamond
containing our own history, the boundary area becomes as large as billions of light
years, squared, or 10123 in fundamental units. As you know, I have studied all the
ancients. I learned that the maximum area along the boundary of a past light-cone
provides a bound on the size N of the Hilbert space describing everything within the
light-cone: N ∼ exp(10123 ) [31]. And elsewhere I found that re-coherence occurs on a
timescale N N ∼ exp[exp(10123 )] [2]. This is much longer than the time it will take for
our vacuum to decay [10,35] and the world to end in a crunch. So why worry about it?
Sagredo: It’s true that re-coherence is overwhelmingly unlikely in a causal di-
amond as large as ours. But nothing you said convinces me that the probability for
things to “unhappen” is exactly zero.
Salviati: To me, it is very important to be able to say that some things really
do happen, irreversibly and without any uncertainty. If this were not true, then how
could we ever make sense of predictions of a physical theory? If we cannot be sure that
something happened, how can we ever test the prediction that it should or should not
happen?
Sagredo: That’s it—this is what bothered me. The notion that things really
happen should be a fundamental principle, and the implementation of fundamental
principles in a physical theory should not rely solely on approximations. So let me
define this more carefully in terms of a definition and a postulate.

– 28 –

Definition I Consider an instance of decoherence (or “collapse of a wave function”)
in a Hilbert space HS , which occurs as a result of entanglement with another Hilbert
space HE . The event will be said to happen if the entanglement between HE and HS is
irreversible; and the system S can then be treated as if it was in one of the pure states
that constitute the basis that diagonalizes the density matrix obtained by tracing over
the Hilbert space HE .

Postulate I Things happen.

In other words, there exist some entanglements in Nature that will not be reversed with
any finite probability.
Simplicio: Let me see if I can find an example that satisfies your postulate.
Suppose that an apparatus is in continuous interaction with some environment. Even
an interaction with an single environmental photon can record the event. If the photon
disappears to infinity so that no mirror can ever reflect it back, then the event has
happened.
Sagredo: Your example makes it sound like irreversible decoherence is easy, but
I don’t think this is true. For example, in the finite causal diamonds we considered,
there is no “infinity” and so nothing can get to it.
Salviati: Sagredo’s postulate is in fact surprisingly strong! Any dynamically
closed system (a system with finite entropy) cannot satisfy the postulate, because
recurrences are inevitable. This not a trivial point, since it immediately rules out
certain cosmologies. Stable de Sitter space is a closed system. More generally, if we
consider a landscape with only positive energy local minima, the recurrence time is
controlled by the minimum with the smallest cosmological constant. So the recurrence
time is finite, and nothing happens. Anti de Sitter space is no better. As is well known,
global AdS is a box with reflecting walls. At any finite energy it has finite maximum
entropy and also gives rise to recurrences.
Simplicio: I can think of a cosmology that satisfies Postulate I, along the lines
of my previous example. I will call it “S-matrix cosmology”. It takes place in an
asymptotically flat spacetime and can described as a scattering event. The initial
state is a large number of incoming stable particles. The particles could be atoms of
hydrogen, oxygen, carbon, etc. The atoms come together and form a gas cloud that
contracts due to gravity. Eventually it forms a solar system that may have observers
doing experiments. Photons scatter or are emitted from the apparatuses and become
irreversibly entangled as they propagate to infinity in the final state. The central star
collapses to a black hole and evaporates into Hawking radiation. Eventually everything
becomes outgoing stable particles.

– 29 –

Sagredo: It is true that there are some things that happen in your S-matrix
cosmology. But I am not satisfied. I think there is a larger point that Salviati made:
we would like a cosmology in which it is possible to give a precise operational meaning
to quantum mechanical predictions. The notion that things unambiguously happen is
necessary for this, but I now realize that it is not sufficient.
Salviati: Now, I’m afraid, you have us puzzled. What is wrong with the S-matrix
cosmology?
Sagredo: Quantum mechanics makes probabilistic predictions; when something
happens, each possible outcome has probability given by the corresponding diagonal
entry in the density matrix. But how do we verify that this outcome really happens
with predicted probability?
Salviati: Probabilities are frequencies; they can be measured only by performing
an experiment many times. For example, to test the assertion that the probability
for “heads” is one half, you flip a coin a large number of times and see if, within the
margin of error, it comes up heads half of the time. If for some reason it were only
possible to flip a coin once there would be no way to test the assertion reliably. And to
be completely certain of the probability distribution, it would be necessary to flip the
coin infinitely many times.
Simplicio: Well, my S-matrix cosmology can be quite large. For example, it might
contain a planet on which someone flips a quantum coin a trillion times. Photons record
this information and travel to infinity. A trillion outcomes happen, and you can do your
statistics. Are you happy now?
Sagredo: A trillion is a good enough approximation to infinity for all practical
purposes. But as I said before, the operational testability of quantum-mechanical pre-
dictions should be a fundamental principle. And the implementation of a fundamental
principle should not depend on approximations.
Salviati: I agree. No matter how large and long-lived Simplicio makes his S-
matrix cosmology, there will only be a finite number of coin flips. And the cosmology
contains many “larger” experiments that are repeated even fewer times, like the explo-
sion of stars. So the situation is not much better than in real observational cosmology.
For example, inflation tells us that the quadrupole anisotropy of the CMB has a gaus-
sian probability distribution with a variance of a few times 10−5 . But it can only
be measured once, so we are very far from being able to confirm this prediction with
complete precision.
Sagredo: Let me try to state this more precisely. Quantum mechanics makes
probabilistic predictions, which have the following operational definition:

Definition II Let P (i) be the theoretical probability that outcome i happens (i.e., i

– 30 –

arises as a result of irreversible decoherence), given by a diagonal entry in the density
matrix. Let N be the number of times the corresponding experiment is repeated, and
let Ni be the number of times the outcome i happens. The sharp prediction of quantum
mechanics is that
Ni
P (i) = lim . (3.1)
N →∞ N

Salviati: Do you see the problem now, Simplicio? What bothers us about your
S-matrix cosmology is that N is finite for any experiment, so it is impossible to verify
quantum-mechanical predictions with arbitrary precision.
Simplicio: Why not proliferate the S-matrix cosmology? Instead of just one
asymptotically flat spacetime, I will give you infinitely many replicas with identical
in-states. They are completely disconnected from one another; their only purpose is to
allow you to take N → ∞.
Salviati: Before we get to the problems, let me say that there is one thing I like
about this proposal: it provides a well-defined setting in which the many worlds inter-
pretation of quantum mechanics appears naturally. For example, suppose we measure
the z-component of the spin of an electron. I have sometimes heard it said that when
decoherence occurs, the world into two equally real branches. But there are problems
with taking this viewpoint literally. For example, one might conclude that the proba-
bilities for the two branches are necessarily equal, when we know that in general they
are not.
Simplicio: Yes, I have always found this confusing. So how does my proposal
help?
Salviati: The point is that in your setup, there are an infinite number of worlds
to start with, all with the same initial conditions. Each world within this collection
does not split; the collection S itself splits into two subsets. In a fraction pi of worlds,
the outcome i happens when a spin is measured. There is no reason to add another
layer of replication.
Simplicio: Are you saying that you reject the many worlds interpretation?
Salviati: That depends on what you mean by it. Some say that the many worlds
interpretation is a theory in which reality is the many-branched wavefunction itself. I
dislike this idea, because quantum mechanics is about observables like position, mo-
mentum, or spin. The wavefunction is merely an auxiliary quantity that tells you
how to compute the probability for an actual observation. The wavefunction itself is
not a measurable thing. For example the wavefunction ψ(x) of a particle cannot be
measured, so in particular one cannot measure that it has split. But suppose we had a
system composed of an infinite number of particles, all prepared in an identical manner.

– 31 –

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-
tem. For example, the single particle probability density ψ ∗ (x)ψ(x) is a many-particle
observable:
N
∗ 1
ψ (x)ψ(x) = lim δ(xi − x) (3.2)
N →∞ N
i=1

Simplicio: I see. Now if an identical measurement is performed on each particle,
each single particle wavefunction splits, and this split wavefunction can be measured
in the many-particle system.
Salviati: To make this explicit, we could make the individual systems more com-
plicated by adding a detector d for each particle. Each particle-detector system can be
started in the product state ψ0 (x)χ0 (d). Allowing the particle to interact with the detec-
tor would create entanglement, i.e., a wavefunction of the form ψ1 (x)χ1 (d)+ψ2 (x)χ2 (d).
But the branched wave function cannot be measured any better than the original one.
Now consider an unbounded number of particle-detector pairs, all starting in the same
product state, and all winding up in entangled states. It is easy to construct operators
analogous to Eq. (3.2) in the product system that correspond to the single system’s
wave function. So you see, that in the improved S-matrix cosmology there is no reason
to add another layer of many-worlds.
Sagredo: That’s all very nice, but Simplicio’s proposal does not help with mak-
ing quantum mechanical predictions operationally well-defined. You talk about many
disconnected worlds, so by definition it is impossible to collect the N → ∞ results and
compare their statistics to the theoretical prediction.
Simplicio: I see. By saying that quantum mechanical predictions should be op-
erationally meaningful, you mean not only that infinitely many outcomes happen, but
that they are all accessible to an observer in a single universe.
Sagredo: Yes, it seems to me that this requirement follows from Salviati’s fun-
damental principle that predictions should have precise operational meaning. Let me
enshrine this in another postulate:

Postulate II Observables are observable.

By observables, I mean any Hermitian operators whose probability distribution is pre-
cisely predicted by a quantum mechanical theory from given initial conditions. And by
observable, I mean that the world is big enough that the observable can be measured
infinitely many times by irreversible entanglement.
Salviati: Like your first postulate, this one seems quite restrictive. It obviously
rules out the infinite set of S-matrix cosmologies, since products of field operators in
different replicas are predicted but cannot be measured by any observers in any one

– 32 –

cosmology. And it gives us additional reasons to reject Simplicio’s earlier suggestion:
the single S-matrix cosmology contains observables such as the total number of outgoing
particles, which cannot even happen, since there is no environment to measure them.
Simplicio: Well, we were quite happy yesterday with the progress we had made
on making decoherence objective in the causal diamond. But these postulates of yours
clearly cannot be rigorously satisfied in any causal diamond, no matter how large.
Perhaps it’s time to compromise? Are you really sure that fundamental principles have
to have a completely sharp implementation in physical theory? What if your postulates
simply cannot be satisfied?
Salviati: Yesterday, we did not pay much attention to causal diamonds that end
in regions with vanishing cosmological constant. Perhaps this was a mistake? In such
regions, the boundary area and entropy are both infinite. I may not have not thought
about it hard enough, but I see no reason why our postulates could not be satisfied in
these “hat” regions.
Sagredo: Even if they can, there would still be the question of whether this helps
make sense of the finite causal diamonds. I care about this, because I think we live in
one of them.
Salviati: I understand that, but let us begin by asking whether our postulates
might be satisfied in the hat.

3.4 Irreversible decoherence and infinite repetition in the hat
In the eternally inflating multiverse there are three types of causal diamonds. The first
constitute a set of measure zero and remain forever in inflating regions. The entropy
bound for such diamonds is the same as for the lowest-Λ de Sitter space they access,
and hence is finite. The second type are the diamonds who end on singular crunches. If
the crunches originated from the decay of a de Sitter vacuum then the entropy bound
is again finite [36]. Finally there are causal diamonds that end up in a supersymmetric
bubble of zero cosmological constant. A timelike geodesic that enters such a bubble
will remain within the diamond for an infinite time. It is convenient to associate an
observer with one of these geodesics, called the Census Taker.
The Census Taker’s past light-cone becomes arbitrarily large at late times. It
asymptotes to the null portion of future infinity called the “hat”, and from there extends
down into the non-supersymmetric bulk of the multiverse (Fig. 10). If the CT’s bubble
nucleates very late, the entire hat appears very small on the conformal diagram and
the CT’s past light-cone looks similar to the past light-cone of a point on the future
boundary of ordinary de Sitter space. However, this is misleading; the entropy bound
for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space. It is
infinite.

– 33 –

Figure 10: The shaded region is the late time portion of a causal diamond that ends in a
supersymmetric vacuum with vanishing cosmological constant (left). We will think of this
region as the asymptotic causal past of an “observer” in this vacuum, called the Census Taker.
A portion of the boundary of such causal diamonds has infinite cross-sectional area. This
portion, called a “hat”, coincides with a light-like portion of the future conformal boundary
in the standard global picture of the multiverse. Λ = 0 bubbles that form late look small on
the conformal diagram (right). But they always have infinite area. Hence the hat regions can
admit exact observables and can allow things to happen, with enough room for arbitrarily
good statistics.

According to the criterion of Ref. [36], the existence of light-sheets with unbounded
area implies the existence of precise observables. A holographic dual should exist
by which these observables are defined. The conjecture of Ref. [37]—the FRW/CFT
duality—exhibits close similarities to AdS/CFT. In particular, the spatial hypersur-
faces of the FRW geometry under the hat are open and thus have the form of three-
dimensional Euclidean anti-de Sitter space. Their shared conformal boundary—the
space-like infinity of the FRW geometry—is the two-dimensional “rim” of the hat.
As time progresses, the past light-cone of the Census Taker intersects any fixed earlier
time slice closer and closer to the boundary. For this reason the evolution of the Census
Taker’s observations has the form of a renormalization group flow in a two-dimensional
Euclidean conformal field theory [38].
We will return to the holographic dual later, but for now we will simply adopt the

– 34 –

conjecture that exact versions of observables exist in the hat. We can then ask whether
hat observables are observable, in the sense of Postulate II above. In particular, this
would require that things happen in the hat, in the sense of Postulate I. We do not
understand the fundamental description of the hat well enough to prove anything rigor-
ously, but we will give some arguments that make it plausible that both postulates can
indeed be satisfied. Our arguments will be based on conventional (bulk) observables.
Consider an interaction at some event X in the hat region that entangles an ap-
paratus with a photon. If the photon gets out to the null conformal boundary, then
a measurement has happened at X, and postulate I is satisfied. In general the photon
will interact with other matter, but unless this interaction takes the form of carefully
arranged mirrors, it will not lead to recoherence. Instead will enlarge the environment,
and sooner or later a massless particle that is entangled with the event at X will reach
the null boundary. For postulate I to be satisfied it suffices that there exist some events
for which this is the case; this seems like a rather weak assumption.
By the FRW symmetry [16] of the hat geometry, the same measurement happens,
with the same initial conditions, at an infinite number of other events Xi . Moreover,
any event in the hat region eventually enters the Census Taker’s past light-cone. Let
N (t) be the number of equivalent measurements that are in the Census Taker’s past
light-cone at the time t along his worldline, and let Ni (t) be the number of times the
outcome i happens in the same region. Then the limit in Eq. (3.1) can be realized as

Ni (t)
pi = lim , (3.3)
t→∞ N (t)

and Postulate II is satisfied for the particular observable measured.
A crucial difference to the S-matrix cosmology discussed in the previous subsection
is that the above argument applies to any observable: if it happens once, it will happen
infinitely many times. It does not matter how “large” the systems are that participate
in the interaction. Because the total number of particles in the hat is infinite, there
is no analogue of observables such as the total number of outgoing particles, which
would be predicted by the theory but could not be measured. This holds as long as
the fundamental theory predicts directly only observables in the hat, which we assume.
Thus, we conclude that Postulate II is satisfied for all observables in the hat.
Since both postulates are satisfied, quantum mechanical predictions can be op-
erationally verified by the Census Taker to infinite precision. But how do the hat
observables relate to the approximate observables in causal diamonds that do not enter
hats, and thus to the constructions of Sec. 2? We will now argue that the non-hat
observables are approximations that have exact counterparts in the hat.

– 35 –

Figure 11: A black hole formed by gravitational collapse. The horizonal shaded region is
the causal diamond of an observer (red) who remains forever outside the black hole. The
vertical shaded region is the causal diamond of an observer (blue) who falls into the black
hole. Note that there are infinitely many inequivalent infalling observers, whose diamonds
have different endpoints on the conformal boundary (the singularity inside the black hole);
on the other hand, all outside observers have the same causal diamond.

3.5 Black hole complementarity and hat complementarity

In this subsection, we will propose a complementarity principle that relates exact Cen-
sus Taker observables defined in the hat to the approximate observables that can be
defined in other types of causal diamonds, which end in a crunch and have finite max-
imal area. To motivate this proposal, we will first discuss black hole complementarity
[15] in some detail.
Can an observer outside the horizon of a black hole recover information about the
interior? Consider a black hole that forms by the gravitational collapse of a star in
asymptotically flat space. Fig. 11) shows the spacetime region that can be probed by
an infalling observer, and the region accessible to an observer who remains outside the
horizon. At late times the two observers are out of causal contact, but in the remote
past their causal diamonds have considerable overlap.

– 36 –

Figure 12: An observable A inside the black hole can be approximately defined and evolved
using local field theory. In this way, it can be propagated back out of the black hole to an
operator Ain defined on the asymptotic past boundary. From there, it is related by an exact
S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed
line) by a late time observer.

Let A be an observable behind the horizon as shown in Fig. 12. A might be a
slightly smeared field operator or product of such field operators. To the freely falling
observer the observable A is a low energy operator that can be described by conventional
physics, for example quantum electrodynamics.
The question we want to ask is whether there is an operator outside the horizon
on future light-like infinity, that has the same information as A. Call it Aout . By that
we mean an operator in the Hilbert space of the outgoing Hawking radiation that can
be measured by the outside observer, and that has the same probability distribution
as the original operator A when measured by the in-falling observer.
First, we will show that there is an operator in the remote past, Ain that has the
same probability distribution as A. We work in the causal diamond of the infalling
observer, in which all of the evolution leading to A is low energy physics. Consider an
arbitrary foliation of the infalling causal diamond into Cauchy surfaces, and let each
slice be labeled by a coordinate t. We may choose t = 0 on the slice containing A.
Let U (t) be the Schr¨dinger picture time-evolution operator, and let |Ψ(t) be the
o

– 37 –

state on the Cauchy surface t. We can write |Ψ(0) in terms of the state at a time −T
in the remote past,
|Ψ(0) = U (T )|Ψ(−T ) . (3.4)
The expectation value of A can be written in terms of this early-time state as

Ψ(−T )|U † (T )AU (T )|Ψ(−T ) . (3.5)

Thus the operator
Ain = U † (T )AU (T ) (3.6)
has the same expectation value as A. More generally, the entire probability distributions
for A and Ain are the same. Let us take the limit T → ∞ so that Ain becomes an
operator in the Hilbert space of incoming scattering states.
Since the two diamonds overlap in the remote past, Ain may also be thought of as
an operator in the space of states of the outside observer. Now let us run the operator
forward in time by the same trick, except working in the causal diamond of the outside
observer. The connection between incoming and outgoing scattering states is through
the S-matrix. Thus we define
Aout = SAin S † (3.7)
or
Aout = lim SU † (T )AU (T )S † (3.8)
T →∞

The operator Aout , when measured by an observer at asymptotically late time, has the
same statistical properties as A if measured behind the horizon at time zero.
The low energy time development operator U (T ) is relatively easy to compute, since
it is determined by integrating the equations of motion of a conventional low energy
system such as QED. However, this part of the calculation will not be completely
precise, because it involves states in the interior if the black hole, which have finite
entropy bound. The S-matrix should have a completely precise definition but is hard
to compute in practice. Information that falls onto the horizon is radiated back out in
a completely scrambled form. The black hole horizon is the most efficient scrambler of
any system in nature.
This is the content of black hole complementarity: observables behind the horizon
are not independent variables. They are related to observables in front of the horizon by
unitary transformation. The transformation matrix is limT →∞ [U (T )S † ]. It is probably
not useful to say that measuring Aout tells us what happened behind the horizon [39].
It is not operationally possible to check whether a measurement of A and Aout agree.
It is enough for us that the every (approximate) observable behind the horizon has a

– 38 –

(precise) complementary image among the degrees of freedom of the Hawking radiation
that preserves expectation values and probabilities.
What is the most general form of operators A inside the black hole that can be
written in the form of Eq. (3.8), as an operator Aout on the outside? Naively, we might
say any operator with support inside the black hole can be so represented, since any
operator can be evolved back to the asymptotic past using local field theory. But this
method is not completely exact, and we know that there must be situations where
it breaks down completely. For example, by the same argument we would be free to
consider operators with support both in the Hawking radiation and in the collapsing
star and evolve them back; this would lead us to conclude that either information
was xeroxed or lost to the outside. This paradox is what led to the proposal that
only operators with support inside someone’s causal patch make any sense. But that
conclusion has to apply whether we are inside or outside the black hole; the infalling
observer is not excepted. For example, we should not be allowed to consider operators at
large spacelike separation near the future singularity of the black hole. The semiclassical
evolution back to the asymptotic past must be totally unreliable in this case.9
We conclude that there are infinitely many inequivalent infalling observers, with
different endpoints on the black hole singularity. Approximate observables inside the
black hole must have the property that they can be represented by an operator with
support within the causal diamond of some infalling observer. Any such operator can
be represented as an (exact) observable of the outside observer, i.e., as an operator Aout
acting on the Hawking radiation on the outside.
This completes our discussion of black hole complementarity. Following Refs. [38,
40], we will now conjecture a similar relation for the multiverse. The role of the ob-
server who remains outside the black hole will be played by the Census Taker; note
that both have causal diamonds with infinite area. The role of the many inequivalent
infalling observers will be played by the causal diamonds that end in crunches, which
we considered in Sec. 2.4, and which have finite area. The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal
diamonds of the multiverse can be represented by an exact observable in the Census
Taker’s hat.

9
The notion of causality itself may become approximate inside the black hole. However, this does
not give us licence to consider operators at large spacelike separation inside the black hole. Large
black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside
any single causal patch. By the equivalence principle, if we were permitted to violate the restriction
to causal patches inside a black hole, then we would have to be allowed to violate it in any spacetime.

– 39 –

t
4
3

2

1

Figure 13: Hat complementarity states that operators in finite causal diamonds (right)
have an exact counterpart in the hat. Over time, the census taker receives an unbounded
amount of information. The past light-cones of the Census Taker at different times along his
worldline define a natural cutoff on his information. Because there are only finitely many
different diamonds, there must be infinitely many copies of every piece of information in the
hat. They are related by hat complementarity to the infinite number of causal diamonds that
make up the global spacetime; see also Fig. 8. At late times, increasing the Census Taker’s
cutoff is related to increasing the light-cone time cutoff, which adds a new (redundant) layer
of diamonds.

More precisely, we assume that for every observable A in a finite causal diamond, there
exists an operator Ahat with the same statistics as A. A and Ahat are related the same
way that A and Aout are in the black hole case. See Fig. 13 for a schematic illustration.

Hat complementarity is motivated by black hole complementarity, but it does not
follow from it rigorously. The settings differ in some respect. For example, a black
hole horizon is a quasi-static Killing horizon, whereas the Census Taker’s causal hori-
zon rapidly grows to infinite area and then becomes part of the conformal boundary.
Correspondingly, the radiation in the hat need not be approximately thermal, unlike
Hawking radiation. And the argument that any operator behind the horizon can be
evolved back into the outside observer’s past has no obvious analogue for the Census
Taker, since his horizon can have arbitrarily small area and hence contains very little
information at early times; and since the global multiverse will generally contain re-
gions which are not in the causal future of the causal past of any Census Taker. Here,
we adopt hat complementarity as a conjecture.
Next, we turn to the question of how the information about finite causal diamonds

– 40 –

shows up in the precise description of the outside observer or the Census Taker.

3.6 The global multiverse in a hat
An important difference between the black hole and the multiverse is that the infor-
mation in the Hawking radiation is finite, whereas the number of particles in the hat
is infinite. The observer outside the black hole, therefore, receives just enough infor-
mation to be able to recover the initial state, and thus the approximate observables
inside the black hole. On the other hand, the O(3, 1) symmetry of the FRW universe
implies that the entropy accessible to the Census Taker is infinite. Since the number
of quantum states in the finite causal diamonds that end in crunches is bounded, this
implies an infinite redundancy in the Census Taker’s information. We will now argue
that this redundancy is related to the reconstruction of a global multiverse from causal
diamonds, described in Sec. 2.6, in which an infinite number of finite causal diamonds
are used to build the global geometry.
Black holes scramble information. Therefore, an observer outside a black hole has
to wait until half the black hole has evaporated before the first bit of information can be
decoded in the Hawking radiation [15, 41]. After the whole black hole has evaporated,
the outside observer has all the information, and his information does not increase
further. Can we make an analogous statement about the Census Taker?
If the radiation visible to the Census Taker really is complementary to the causal
diamonds in the rest of the multiverse, it will surely be in a similarly scrambled form.
For hat complementarity to be operationally meaningful, this information must be
organized in a “friendly” way, i.e., not maximally scrambled over an infinite-dimensional
Hilbert space. Otherwise, the census taker would have to wait infinitely long to extract
information about any causal patch outside his horizon.10 This would be analogous to
the impossibility of extracting information from less than half of the Hawking radiation.
An example of friendly packaging of information is not hard to come by (see also [42]).
Imagine forever feeding a black hole with information at the same average rate that
it evaporates. Since the entropy of the black hole is bounded it can never accumulate
more than S bits of information. Any entering bit will be emitted in a finite time even
if the total number of emitted photons is infinite.
10
We are thus claiming that things happen in the past light-cone of the Census Taken at finite time.
The corresponding irreversible entanglement leading to decoherence must be that between the interior
and the exterior of the Census Taker’s past light-cone on a suitable time-slice such as that shown in
Fig. 13. Because the timeslice is an infinite FRW universe, the environment is always infinite. The
size of the “system”, on the other hand, grows without bound at late times, allowing for an infinite
number of decoherent events to take place in the Census Taker’s past. Thus the postulates of Sec. 3.3
can be satisfied.

– 41 –

We will assume that this is also the case in the hat. Then we can ask whether the
Census Taker’s cutoff translates to a multiverse cutoff, hopefully something like the
light-cone time cutoff of Sec. 2.5.
To understand the Census Taker’s cutoff, we start with the metric of open FRW
space with a vanishing cosmological constant. We assume the FRW bubble nucleated
from some ancestor de Sitter vacuum.

ds2 = a(T )2 (−dT 2 + dH3 )
2
(3.9)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH3 = dR2 + sinh2 R dΩ2 ,
2
2 (3.10)

and T is conformal time. The spatial hypersurfaces are homogeneous with symmetry
O(3, 1) which acts on the two-dimensional boundary as special conformal transforma-
tions. Matter in the hat fills the noncompact spatial slices uniformly and therefore
carries an infinite entropy.
If there is no period of slow-roll inflation in the hat then the density of photons
will be about one per comoving volume. We take the Census Taker to be a comoving
worldline (see Fig. 13). The number of photons in the Census Taker’s causal past is

Nγ ∼ e2TCT . (3.11)

In this formula TCT is the conformal time from which the Census Taker looks back. Nγ
represents the maximum number of photons that the Census Taker can detect by the
time TCT .
The Census Taker’s cutoff is an information cutoff: after time TCT a diligent Census
Taker can have gathered about e2TCT bits of information. If the de Sitter entropy of
the ancestor is Sa , then after a conformal time TCT ∼ log Sa the Census Taker will
have accessed an amount of information equal to the entropy of the causal patch of the
ancestor. Any information gathered after that must be about causal diamonds in the
rest of the multiverse, in the sense that it concerns operators like A that are beyond
the horizon.
Over time, the Census Taker receives an unbounded amount of information, larger
than the entropy bound on any of the finite causal diamonds beyond the hat. This
means that the Census Taker will receive information about each patch history over
and over again, redundantly. This is reminiscent of the fact that in our reconstruction
of the global picture, every type of causal diamond history occurs over and over again
as the new diamonds are inserted in between the old ones. This is obvious because we

– 42 –

used infinitely many diamonds to cover the global geometry, and there are only finitely
many histories that end in a crunch or on an eternal endpoint.
A particular history of a causal diamond has a larger or smaller maximum area,
not a fixed amount of information. But in our reconstruction of the global multiverse,
each generation of new causal diamonds (the set of diamonds starting at the green
squares at equal time t in Fig. 8) contains all possible histories (at least for sufficiently
late generations, where the number of diamonds is large). Therefore the amount of
information in the reconstructed global geometry grows very simply, like 2t . This
redundant information should show up in the hat organized in the same manner. Thus,
it is natural to conjecture that the information cutoff of the Census Taker is dual, by
complementarity, to the discretized light-cone time cutoff implicit in our reconstruction
of a global spacetime from finite causal diamonds.

Acknowledgments
We would like to thank B. Freivogel, A. Guth, D. Harlow, P. Hayden, S. Leichenauer,
V. Rosenhaus, S. Shenker, D. Stanford, E. Witten, and I. Yang for helpful discussions.
This work was supported by the Berkeley Center for Theoretical Physics, by the Na-
tional Science Foundation (award numbers 0855653 and 0756174), by fqxi grant RFP2-
08-06, and by the US Department of Energy under Contract DE-AC02-05CH11231.

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