# Watching the Clocks: Interpreting the Page-Wootters Formalism and the
Internal Quantum Reference Frame Programme

Emily Adlam  University of Western Ontario

(August 27, 2024; August 27, 2024)

###### Contents

  1. 1 Introduction
  2. 2 Background
    1. 2.1 Internal Quantum Reference Frames
    2. 2.2 The Page-Wootters Formalism
  3. 3 The Page-Wootters account of time
  4. 4 The meaning of the PW probabilities
    1. 4.1 The final-measurement interpretation
      1. 4.1.1 Comparison to the perspective-neutral collapse
      2. 4.1.2 Example: Temporal Nonlocality
  5. 5 Quantum Reference Frames
    1. 5.1 The Quantum Equivalence Principle
    2. 5.2 Summary
      1. 5.2.1 Less operational alternatives
    3. 5.3 Example: Temporal nonlocality again
  6. 6 Relational superposition
  7. 7 Conclusion
  8. A Other Relational Formalisms
  9. B Final-measurement interpretation
  10. C Penrose
  11. D Relational superposition
  12. E Acknowledgements

##  1 Introduction

Recently the field of quantum foundations has seen a surge of interest in the
Page-Wootters (PW) formalism, as well as the development of a new research
programme based on the related notion of an internal quantum reference frame
(IQRF). A number of exciting results have come out of this research, giving
new insight into deep questions around issues like the nature of time,
reference frames, and the equivalence principle. These concerns put the PW and
IQRF research programmes right at the interface of the foundations of quantum
mechanics and the ongoing search for a theory of quantum gravity, so it is of
great interest to understand what exactly the results of these programmes mean
for our understanding of these fields. In this article we aim to clarify the
foundational consequences of some of the main themes of the PW and IQRF
programmes, although of course we will not be able to cover everything that
researchers in these fields have have achieved.

One feature of these research programmes which raises a number of questions is
the fact that the PW formalism, and the IQRF research programme more
generally, don’t obviously seem to make room for a mechanism of wavefunction
collapse or any other such prescription which would ensure that measurements
have unique outcomes. So one might think that in order to take these research
programmes seriously we will have to adopt the Everett interpretation, a neo-
Copenhagen interpretation, or some other such interpretation which does not
insist on unique measurement outcomes. Thus in order to understand the
foundational significance of this research, it is important to establish
whether or not the PW and IQRF formalisms implicitly depend on a certain
interpretation of quantum mechanics, and whether there is any possibility of
making sense of their results in the context of a single-world realist
interpretation.

There are also important questions surrounding the operational credentials of
the PW and IQRF formalisms. It is common for proponents of both approaches to
motivate their research in operational terms - for example, ref [1] argues
that ‘ _Extending the operational view to quantum theory, one is led to define
time through measurements of quantum systems serving as clocks.’_ This
operational approach to time sounds very reasonable (and indeed it is
inherited from Einstein’s approach to time in special and general relativity),
but it’s important to keep in mind that these frameworks do not often
explicitly model observers and therefore there remains a gap to be bridged
between the formal results and the operations performed by actual observers.
Closing that gap will likely require us to adopt some standpoint on the role
of observers and the nature of the probabilities that the PW and IQRF
formalisms ascribe to measurement outcomes.

Thus after giving a brief introduction to the PW and IQRF research programmes,
we will divide our enquiry up into four questions, which in fact will all turn
out to be interrelated:

  1. 1.

What account does the PW formalism give of the nature of time?

  2. 2.

What do the probabilities predicted by the PW formalism refer to?

  3. 3.

Is a ‘quantum reference frame’ really a reference frame in an operationally
meaningful sense?

  4. 4.

What does it mean to say that superposition is relative to a reference frame,
and what consequences does that have for the interpretation of quantum
mechanics?

We will discuss some specific difficulties that arise in attempting to
interpret the PW formalism in particular, and then suggest one possible
interpretational approach based on an approach advocated in other contexts in
refs [2, 3, 4]. This ‘final-measurement’ interpretation demonstrates that it
may be possible to make sense of the PW formalism within a single-world
realist context; it preserves all of the mathematical results of the PW and
IQRF research programmes, but does alter the conclusions we can draw from some
of these results. It is not our intention to claim that the final-measurement
interpretation is the only possible interpretation of the PW and IQRF
formalisms, but the broader point we wish to make is that _some_
interpretation of the framework is necessary if we are to use it to make
foundational claims, and moreover the choice of interpretation will inevitably
change our understanding of some of the mathematical results.

We demonstrate this by arguing that if reference frames are required to have
operational significance such that they can be associated with measurement
procedures, different interpretations of quantum mechanics will lead to
different conclusions about what constitutes a quantum reference frame.
Similarly, we argue that the notion of superposition should take into account
the way in which an instantaneous state is embedded in ongoing dynamical
evolution, which leads to a more nuanced way of thinking about the relativity
of superposition. Thus this paper serves to emphasize the fact that
conclusions drawn from mathematical frameworks like the PW and IQRF formalisms
will not typically be interpretation-neutral, so it is important for research
programmes addressing foundational issues to explicitly address questions
about the interpretation of quantum mechanics.

##  2 Background

###  2.1 Internal Quantum Reference Frames

Typically in physics we describe the behaviour of systems relative to some
fixed reference frame which is understood as being external to the systems we
aim to describe. However, there are presumably no reference frames external to
the whole universe, so ultimately we must always define the behaviour of
subsystems of the universe relative to other subsystems of the universe.
Moreover, reference frames are often treated as abstract mathematical
constructions in physics, but taking this kind of internal, relative view
requires us to decide what the physical correlate of a reference frame is. The
Internal Quantum Reference Frames (IQRF) programme[5, 6, 7, 8, 9] offers one
possible answer to this question: the physical correlate of a reference frame
is simply an individual physical system. Thus to arrive at a reference frame,
we fix a reference system and rewrite our models such that _‘the position and
momentum of the reference frame are not dynamical variables when considered
from the reference frame itself … the reference frame is not a degree of
freedom in its own description, but external systems to it are.’_[5] The IQRF
programme studies the problem of establishing and switching between such
internal reference frames. A variety of techniques have been applied to this
problem, but the field seems to have largely converged on using the framework
of constraint quantisation first formulated by Dirac[10].

In constraint quantisation, we start with a ‘kinematical Hilbert space’ that
defines all the possible quantum degrees of freedom, and then we impose
constraints encoding the idea that real physical states should be invariant
under certain transformations deemed to be physically meaningless - for
example, diffeomorphisms and global time translations of the whole
universe[11]. Mathematically this is done by requiring that a certain set of
constraints should vanish on allowed states: for example, since we require
that physical states should be invariant under translations of the whole
universe, the energy of the whole universe obeys a superselection rule
expressed by the constraint $H||\Psi\rangle=0$, where $H$ is the Hamiltonian
of the whole universe and $||\Psi\rangle$ is the universal quantum state. This
constraint is known as the Hamiltonian constraint, and its constraint equation
is known as the Wheeler DeWitt equation[12]. The solutions to this equation
may not be normalizable with respect to the inner product defined on the
kinematical Hilbert space, so we introduce a new inner product which is used
to complete the solution space of the constraint equation, giving us what is
known as the ‘physical Hilbert space,’ to reinforce the fact that only states
satisfying all the constraints are considered physically possible. Similarly,
only operators which commute with the constraints on the physical state space
are considered to be genuinely observable, and such quantities are known as
Dirac observables[13].

In this framework, the fact that the universe is constrained to lie in a
subspace of the full kinematic Hilbert space entails that there is redundancy
associated with standard descriptions of physical systems relative to a fixed
background coordinate system. The IQRF research programme seeks to remove this
redundancy (i.e. to perform gauge-fixing) by relativizing such descriptions to
some chosen physical system. To see how this works, consider a particle $P$
and system $S$ which are in some joint state $\psi$, and suppose that the
system as a whole is invariant under symmetry transformations belonging to
some group $G$ (e.g. diffeomorphisms and/or global time translations), so any
member of the equivalence class of states related by transformations in $G$ is
a legitimate description of the composite system. We can remove the redundancy
by choosing to describe the system relative to the fixed reference frame of
$P$: we simply look for a state $\psi^{\prime}$ which is related to $\psi$ by
a symmetry transformation $G$ such that $\psi^{\prime}$ can be written in the
product state $|e\rangle_{P}\otimes\phi_{S}$, where $|e\rangle_{P}$ is the
unit element of the symmetry group and $\phi_{S}$ is any state of $S$: then we
say that $\phi_{S}$ is the state of $S$ relative to $P$, or in the reference
frame of $P$[14]. States $\psi$ for which this transformation is possible are
sometimes known as ‘alignable,’ and in the case where the transformation is
possible it is unique up to a global phase. Transformations between such
reference frames are elements of $G_{sym}$, which includes the transformations
in $G$ and also the additional set of transformations obtained by selecting a
transformation in $G$ in a way that depends on gauge-independent relative
facts about the configuration of the system[14]. That is to say,
transformations between reference frames are symmetry transformations.

We note that there are two different ways of interpreting this procedure. The
kinematical Hilbert space is the set of states distinguishable relative to an
external reference frame, while the physical Hilbert space is the set of
states distinguishable from the point of view of an observer inside the
universe. So one option is to take a realist approach to the whole kinematical
Hilbert space and thus suppose that transformations like global time
translations of the whole universe are in principle possible, although they
would not have any effects that would be observable from inside the universe.
For example, this would be the appropriate approach to adopt if we were
dealing with a constrained system constructed in a lab, in which case the
laboratory provides the external reference frame and thus translations of the
whole system are physically meaningful. The alternative is to take a
deflationary approach to the kinematical Hilbert space where we hold that
there is no reference frame outside of the universe as a whole and thus
transformations like global time translations of the whole universe are
physically meaningless (assuming that the universe has no boundaries). From
this point of view, the kinematical space is not literally a set of possible
states for the universe; it should simply be regarded as a mathematical
abstraction which helps us construct the true set of possible states for the
universe, i.e. the physical Hilbert space.

Recent work[15, 16] has shown that the set of all internal quantum reference
frames can be unified by a ‘perspective-neutral’ universal quantum state
defined on the physical Hilbert space of gauge-invariant states. We can obtain
‘quantum coordinate maps’ which take us from the universal quantum state into
the perspective of an individual quantum reference system, and then in order
to transform between reference frames we can always transform to the
perspective-neutral universal state using the inverse of the quantum
coordinate map, and then apply another quantum coordinate map to arrive at a
different reference frame. Gauge-invariant relational observables can be
constructed in this framework to encode physical properties relative to an
individual reference frame, so the perspective-neutral framework still upholds
the basic idea that ultimately all physical quantities in our universe are
relational.

In appendix A we explain the relation between the IQRF formalism and several
other relevant research programmes dealing with quantum reference frames.

###  2.2 The Page-Wootters Formalism

The internal reference frame approach is closely related to the Page-Wootters
(PW) formalism[17], which was originally proposed by Page and Wootters[18] as
a response to the so-called ‘problem of time’ that follows from the constraint
quantisation procedure[19]: as a result of the constraint equation
$H||\psi\rangle=0$, all Dirac observables must commute with $H$ and therefore
no observable quantities in the theory can have any time-dependence. That is
to say, in this picture time itself is treated as a form of gauge. This may
sound troubling, and has been questioned by several authors on the grounds
that states of the world at different times are obviously physically
distinct[20]. However, this feature makes sense if one regards the state
$||\psi\rangle$ as a global description which represents the whole history of
the universe, as naturally a representation of the whole of history cannot
undergo any change. Moreover, because we are dealing with a theory which is
deterministic in both directions of time, the state at any one time determines
the whole course of history, so there is a sense in which states at different
times are genuinely physically equivalent in the sense that they can be
regarded as different descriptions of one and the same global configuration
$||\psi\rangle$. The ‘problem of time’ then reduces to the problem of
explaining how it is that observers within this block universe have temporal
experiences.

The PW formalism aims to answer this question by employing an internal
reference frame, as in the IQRF programme; specifically, we choose a reference
frame that can be regarded as a clock, such that we can understand time in
terms of states of the universe relative to different clock readings. To do
this, we choose a subsystem $C$ of the universe to act as a clock and then
define a ‘time observable’ $E_{C}(t)=|t\rangle\langle t|$ which acts on the
clock system $C$ and which transforms covariantly with respect to the group
generated by the Hamiltonian $H_{C}$ of that subsystem, i.e.
$E(t+t^{\prime})=e^{-iH_{C}t^{\prime}}E(t)(e^{-iH_{C}t^{\prime}})^{\dagger}$.
For simplicity it is common to work largely with ‘ideal clocks’ in which the
time observable is sharp, i.e. the clock states are orthogonal, though it
should be recognised that most real clocks are unlikely to have this feature;
non-ideal clocks can also be used in the PW framework but in that case we must
take a little more care in defining the covariant time observable. Using the
covariant time observable and the universal quantum state $||\Psi\rangle$ we
can define a state for the rest of the world, $W$, conditional on $C$ reading
time $t$: $|\psi_{W}(t)\rangle=\langle t|\otimes I_{W}||\Psi\rangle$. We
define an inner product by requiring that this relative state always has norm
one. Finally, by differentiating with respect to time and invoking the
Hamiltonian constraint, we infer that
$i\frac{d}{dt}|\psi_{W}(t)\rangle=H_{S}|\psi_{W}(t)\rangle$, i.e. relative to
$C$ the system $W$ undergoes the standard Schrodinger evolution[1]. In
essence, this works because the constraint $H||\Psi\rangle$ entails that the
total energy is conserved and therefore since the time variables are defined
in terms of the Hamiltonians encoding the energy, it follows that if the time
variable for one system changes by $+t$ then the other variable changes by
$-t$, so the clock and the system are necessarily correlated. According to the
PW approach, these sorts of relational facts are supposed to be _constitutive_
of time - that is, absolute time has been eliminated in favour of a relational
picture.

##  3 The Page-Wootters account of time

The principal foundational claim of proponents of the PW formalism is that it
provides a novel account of time suitable for the prima facie ‘timeless’
setting of canonical quantum gravity, so an obvious starting point for our
enquiry is to ask what precisely is the content of that account. To make
things more concrete, consider the following question: given a state of the
world like $|t\rangle_{C}\otimes|\psi\rangle_{S}$, what is it that makes it
the case that this is the correct state, rather than some other state
$|t\rangle_{C}\otimes|\psi^{\prime}\rangle_{S}$? This question might seem
trivial, but keep in mind that if we were working in the context of standard
quantum mechanics, the link between $C$ and $S$ which determines that the
correct state is $|t\rangle_{C}\otimes|\psi\rangle_{S}$ rather than
$|t\rangle_{C}\otimes|\psi^{\prime}\rangle_{S}$ is of course the fixed
background time metric: system $S$ is in the state $|\psi\rangle$ and not
$|\psi^{\prime}\rangle$ _at the same time_ as system $C$ is in the state
$|t\rangle_{S}$. But in the PW formalism we are not allowed to appeal to a
fixed background time metric, so we need some alternative way of justifying
the claim that the correct state is $|t\rangle_{C}\otimes|\psi\rangle_{S}$ and
not $|t\rangle_{C}\otimes|\psi^{\prime}\rangle_{S}$ \- that is to say, we need
an alternative account of the physical meaning of the tensor product which
doesn’t require a fixed background time metric. If such an explanation can’t
be given, then we may have to conclude that the account of time offered by the
PW formalism in fact derives its plausibility from intuitions developed in the
context of standard quantum mechanics which are no longer legitimate in the
absence of a fixed background metric, and thus this account may not offer a
satisfactory solution to the problem of time in the absence of that background
temporal metric.

One common way of answering this question is to suggest that the PW approach
shows us that the temporal relations between clock and system are based on
_entanglement_[21, 22, 23]. Prima facie this claim seems surprising, because
macroscopic objects do not typically exhibit entanglement, so it would seem
that entanglement should be irrelevant to the problem of making sense of our
macroscopic temporal experiences. But in fact, as explained in ref [24] the
relevant entanglement between clocks and systems is not the usual sort of
entanglement which we can detect by measurement. The entanglement in question
is defined on the kinematical Hilbert space, not the physical Hilbert space -
and kinematical entanglement can’t be probed using gauge-invariant Dirac
observables, which means that it can’t be detected from within the universe.
Therefore it could be the case that clocks and systems are generically
entangled in the kinematical Hilbert space, even though we can’t observe that
entanglement by means of any measurements. So one might try to argue that
despite being unobservable, kinematical entanglement plays some kind of
coordinating role and thus is ultimately responsible for the relationships
between clocks and systems expressed by states like
$|t\rangle_{C}\otimes|\psi\rangle_{S}$.

However, recall that according to the second interpretation of the constraint
framework discussed above, the kinematical Hilbert space is considered to be
not only unobservable but _unphysical_ \- it does not represent a real state
space, just a mathematical abstraction. This interpretation of the constraint
framework is quite common in the literature and evidently adherents of it have
good reason to question the idea that entanglement in the kinematical Hilbert
space is a crucial feature of the relationship between clocks and systems,
since entanglement in a wholly abstract space presumably cannot lead to
substantive consequences like our experiences of time. It should also be noted
that the physical Hilbert space does not in general inherit the tensor product
structure of the kinematical Hilbert space[13], so the decomposition into
‘clock’ and ‘system’ may not even exist in the kinematical Hilbert space. In
general, if the kinematical Hilbert space has two subsystems $A,B$, when we
transform to the physical Hilbert space it is possible that we will end up
with a ‘clock’ made up of some of the degrees of freedom of $A$ and some of
$B$, and likewise a ‘system’ also made up of some of the degrees of freedom of
$A$ and some of $B$. Moreover we also generally find that tensor product
decompositions change when transforming between different internal references
frames, so it may also be the case that the tensor product structure of the
kinematical Hilbert space is preserved in some possible internal references
frames but is not preserved in others. So generically it will not be correct
to say that a ‘clock’ and ‘system’ in the physical Hilbert space are entangled
in the kinematical Hilbert space, because they may not even exist as distinct
subsystems in the kinematical Hilbert space.

Moreover, there are a number of formalisms which achieve a relational
description very similar to the PW approach without employing entanglement at
all. For example, ref [24] has shown that one can also obtain the same
relational dynamics without even postulating any _kinematical_ entanglement
between clock and system degrees of freedom: first we employ the Heisenberg
trivialisation which transforms the clock and system state into a product
state (see appendix A) and only after that do we employ the PW reduction
scheme. So it certainly can’t be kinematical entanglement which plays the role
of linking clock and system within this particular version of the PW approach.
Likewise, Poulin[25] has shown that if we employ statistical group averaging
(rather than the coherent group averaging that features in the usual PW
approach) we arrive at a description very similar to the PW account except
that the universal state is mixed and thus the correlations between clocks and
systems are entirely classical. And Vedral[26] has recently set out a
classical version of the Page-Wootters construction and shown that ‘ _the time
variables of the two subsystems are correlated, not because the subsystems are
coupled, but because the overall energy is conserved._ ’ Vedral’s account
requires us to suppose that the classical state of the whole universe is
mixed, and in a classical setting a mixed state must be understood either as a
characterisation of our lack of knowledge or as a description of an ensemble
of universes, so it’s unclear that this picture is compatible with the
standard classical picture of a single universe with a well-defined state, but
nonetheless it does demonstrate that the conservation of energy is already
enough to derive relationships between time variables without any need to
appeal to entanglement. These frameworks show that we can get relational
dynamics very similar to the PW picture without any entanglement at all,
kinematical or otherwise, and thus they seem to undermine the argument that
kinematical entanglement is playing some crucial role in the account of time
offered by the PW formalism.

So if it’s not entanglement which links the state of the clock to the state of
the system in the PW formalism, what _is_ the nature of this link? Another
popular idea is that the PW formalism has reduced time down to _correlations_
between the states of clocks and the states of systems: for example, ref [6]
presents ‘ _a “timeless” approach according to which time emerges though
correlations between “what a clock shows” and the state of the system._ ’ But
there are several obstacles to making physical sense of these sorts of
correlations in the PW framework. First, in order to observe correlations
between events we must already be provided with a way of putting those
entities into pairs - for example, we can’t observe Bell correlations if we
are simply given a set of 1000 distinct measurement settings and the
corresponding outcomes, we must also be told which pairs of measurements
belong together, i.e. are associated with the same Bell pair. So in the PW
case, how are we supposed to identify pairings of a ‘state of a clock’ and a
‘state of the rest of the universe,’ in order to observe these supposed
correlations? The literature on the topic frequently uses locutions like ‘
_given that clock A shows a time $t$_’[6] or ‘ _$W$ measures outcome w at
clock time $t_{2}$_’[17] or ‘ _the probability that one clock reads a given
proper time conditioned on another clock reading a different proper time_
,’[1] but all of these phrases seem a little suspect, because in ordinary
language they have a temporal connotation: ‘the state of the system given that
the clock reads $t$’ usually means something like ‘the state of the system _at
the same time as_ the clock reads $t$.’ But if we are hoping to _reduce_ time
to correlations then of course we can’t understand phrases like ‘given that,’
‘at,’ ‘conditional on’ and so on in this temporal sense, and yet the
literature does not obviously seem to offer any alternative way of
understanding them. So in order to interpret the account of time offered by
the Page-Wootters formalism in terms of correlations we will need to come up
with some more concrete way of saying what it means to condition the outcome
of one measurement on another in a context where we are not able to appeal to
a temporal background metric.

Second, we typically think of correlations as being exhibited between _events_
such as measurement outcomes and clock readings, and it’s a little unclear
what the ‘events’ are in the PW picture. Ref [6] offers ‘ _a local,
operational definition of an “event”, in which an operation is applied to a
quantum system conditioned on a clock reading a specific “time”._ However,
events of this kind are not in and of themselves adequate to play a role in an
account of our temporal experiences, because the IQRF formalism typically
avoids explicitly modelling observers and hence the formalism does not
describe events of this kind as actually being observed. Nor is it
straightforward to say what it would mean for these events to be observed,
because the relative state $|\psi_{W}(t)\rangle=\langle t|\otimes
I_{W}||\Psi\rangle$ is a state of the whole world minus a single clock, so
real observers will always be subsystems of the ‘rest of the world,’ which
means we can’t model their observations using the usual quantum measurement
formalism which presumes external observers. So in order to translate the PW
approach into statements about the temporal experiences of actual observers,
we will need some prescription about how the state $|\psi_{W}(t)\rangle$
relates to observations made by observers who are subsystems of $W$. Moreover,
even if we can find a way to identify a representation of an observer having
some experiences within the state $|\psi_{W}(t)\rangle$, it doesn’t seem
obvious that we can conclude that the observer is having that experience _at
time $t$_, because the state $|\psi_{W}(t)\rangle$ doesn’t include a
representation of the observer, or indeed anyone else, measuring the clock and
getting the result $t$. Indeed it is common for applications of the PW
formalism to represent the clock as non-interacting, and when interactions are
considered they are usually gravitational interactions[27], so in most of this
research there is no provision made for observers to actually obtain time
readings from the clock. Some such provision must be made if we are to
understand how the PW account of time in terms of correlations can explain our
temporal experiences.

Third, in order to define correlations between clocks and systems we obviously
have to calculate some probabilities (or at least relative frequencies).
Usually the probabilities prescribed by the IQRF formalism are for measurement
outcomes conditioned on various clock readings; but if these measurements are
actually performed and the standard formalism of wavefunction collapse is
applied, the collapse mechanism would take the state outside the physical
Hilbert space[11], so prima facie it seems like the IQRF formalism must forbid
any such measurements to actually be performed. Page himself observed that the
collapse of the wavefunction is only _‘part of one’s computational process,
not as a physical process, so it should not be surprising that the result is
rather unphysical,’_[18] but this response on its own doesn’t seem entirely
adequate - we may well accept that the collapse of the wavefunction is just a
way of computing probabilities, but what exactly are these probabilities _for_
if not for unique measurement outcomes as defined by a wavefunction collapse?
Without a mechanism for selecting a unique measurement outcome we seem obliged
to conclude that _all_ of the outcomes of these potential measurements
actually occur, which is problematic for the account of time in terms of
correlations, as we can’t define any meaningful correlations between clock
readings and outcomes if all of the outcomes always occur - the correlations
will all be trivially equal to one.

One way to address this issue would be to move to a version of the Everett
interpretation[28] and employ one of the strategies used by Everettians to
make sense of probabilities in a context without unique measurement outcomes.
However, the interpretation of probability is perhaps the most contentious
feature of the Everett picture, and there remain serious doubts as to whether
any of these strategies succeed[29, 30, 31]. We will not rehearse these
debates here; suffice it to say that we are not convinced that the Everettian
probability problem can be solved, and therefore we consider it important to
establish whether the IQRF framework can be understood without appeal to the
Everett interpretation or similar. That is to say, we would like to know
whether there is any way to extract a _single-world realist_ picture out of
the IQRF framework - i.e. a picture in which measurements have unique outcomes
and at least the macroscopic parts of reality exist in a single definite way.
In such a picture, ‘correlations between clocks and systems’ could be
understood in a straightforward way by appeal to relative frequencies across
the whole of history, and therefore we would have a clear physical
interpretation for the account of time offered by the IQRF framework.111We
also note that even if one is satisfied by the Everett account of
probabilities, problems remain for an Everettian account of the PW formalism.
It would still be necessary to add something to the PW picture representing
some kind of interaction in which clocks are actually observed, and it would
still be necessary to explain how to pair measurements on systems with
measurements on clocks in order to calculate correlations - the standard
Everett interpretation assumes we have unitary evolution on a background
spacetime, so in an Everettian version of the IQRF formalism we would have to
take care to avoid illegitimately appealing to a background time metric which
is no longer available. We will not discuss here how these problems might be
solved in the Everett picture, but we would invite proponents of the Everett
interpretation to consider the question.

##  4 The meaning of the PW probabilities

Extracting a single-world realist picture out of the IQRF framework is not
trivial, because if measurements have unique outcomes then necessarily there
exist some beables associated with those outcomes, and these beables will
single out some preferred ‘quantum reference frames’ which represent the
beables as being in sharp states in some physically relevant basis (e.g. in
the literature, typically ‘beables’ are expected to have sharp states in the
position basis). Thus in order to obtain a single-world realist account must
find some way of breaking the symmetry between quantum reference frames, which
are all considered on an equal footing in the standard IQRF formalism. The
symmetry breaking does not have to be based on any objective difference
between IQRFs - it could simply be a random process akin to the collapse of
the wavefunction. However it is not straightforward to employ a literal
collapse of the wavefunction in the IQRF framework, because as noted, such a
collapse would take the system outside of the physical Hilbert space.
Moreover, since there is no universal background time in the IQRF framework,
it would seem difficult to say anything sensible about the time at which the
collapse of the wavefunction occurs if indeed it is supposed to apply across
different reference frames - in particular, there will in general be no frame-
independent way of determining the order in which measurements take place, and
since changing the order of wavefunction collapses will typically change the
predictions, it’s hard to see how we could arrive at a frame-independent
approach under these circumstances.

Ref [16] argues that the universal wavefunction in the perspective-neutral
approach ‘ _encodes all the descriptions of the universe relative to all
possible choices of reference system at once and constitutes the crucial link
between these internal perspectives,_ ’ so one might naturally hope to obtain
a single-world realist description by collapsing the universal wavefunction
directly at the level of the physical Hilbert space in order to obtain a
definite course of history. This collapse would not have to be thought of as
literally occurring at any particular time; it would simply be understood as
the process by which a course of history is selected and actualised all at
once in an atemporal manner. However, there are several obstacles standing in
the way of extracting concrete predictions out of this kind of collapse. The
first obstacle is just the usual measurement problem in a new guise: we know
that the collapse must resolve macroscopic features of reality into a single
definite history, but at the same time we know that it must leave some
microscopic features of reality uncollapsed in order to allow characteristic
quantum phenomena to occur for microscopic particles, and quantum mechanics
itself provides no precise prescription for where to put the line between the
microscopic and the macroscopic other than the vague suggestion that it should
be associated with measurements. So in order to make this work we would have
to give some prescription in precise terms (i.e. not referring to vague,
anthropocentric concepts like ‘measurement’) to determine exactly which parts
of the universal wavefunction are supposed to collapse in this atemporal
collapse.

The second problem is that the universal wavefunction is ‘ _without immediate
physical interpretation_ ’ [16] \- that is to say, because its primary role is
to encode a large number of different perspectives, it does not directly
encode any definite histories which we could collapse to. If we tried to
proceed by simply collapsing the wavefunction separately inside each
perspective according to the probabilities given by the Born rule, then the
outcomes of measurements in one perspective would not correspond in any
systematic way to the outcomes of the same measurements viewed from other
perspectives, and thus we would all be trapped inside our own perspective,
unable to find out what other observers have witnessed. Not only would this
lead towards solipsism, it would also prevent us from being able to obtain
empirical confirmation for a theory like quantum mechanics which purports to
describe not just a single perspective but reality as a whole, and therefore
our epistemic justication for believing quantum mechanics in the first place
would be undermined. So it would seem that if the perspective-neutral collapse
is to be made viable, some additional structure will have to be added to the
IQRF approach.

We will return to the perspective-neutral approach shortly, but first let us
take a closer look at the way in which probabilities are actually calculated
in the Page-Wootters formulation. This issue has given rise to a vigorous
debate, as it was pointed out by Kuchar[20] that since applying the effect
operator for the measurement which is used to define the Page-Wootters
probabilities throws the state out of the physical Hilbert space, it appears
that the formalism gives the wrong answer to two-time questions like ‘if I
perform measurement $O_{1}$ at time $t_{1}$ and obtain outcome $X$, what is
the probability that when I perform measurement $O_{2}$ at time $t_{2}$ I will
obtain outcome $Y$?’ 222Of course, one possible way of responding to this
problem is to move to a version of the framework which postulates neither
probabilities nor relative frequencies - i.e. perhaps regard the framework as
simply a description of the universal wavefunction, understood in something
like an Everettian picture with no attempt to attach probabilities to
branches. However, it would seem very hard to understand the physical meaning
of the framework’s predictions in the absence of any probabilities or relative
frequencies, and without predictions which can be linked to observations it
would seem impossible for us to obtain evidence for or learn anything about
the putative ‘universal wavefunction’ in the first place, so this approach
would lead to serious epistemic problems similar to those which have been
raised for more vanilla versions of the Everett interpretation[29]. Also, as
noted above, we do need either probabilities or relative frequencies if we are
to have correlations in the usual mathematical sense, so if the physical
meaning of time in these frameworks is supposed to come down to correlations,
then we don’t have the option of simply refraining from ascribing
probabilities. There are a few different ways of arriving at the correct two-
time probabilities. Dolby proposes that we should apply the usual measurement
operator, taking us out of the physical Hilbert space, and then immediately
apply another operator which takes us back into the physical Hilbert
space[32]. Hoehn et al[24] have shown that the correct two-time probabilities
can be obtained as manifestly gauge-invariant relational Dirac observables
within the constraint picture - this approach actually turns out to be
formally equivalent to the Dolby proposal. Both Baumann et al [17] and Craig
and Singh [33] discuss a consistent histories picture where we are only
allowed to calculate two-time probabilities in the case where the two
measurements belong to the same consistent history, i.e. the measurement
operators commute when compared at the same instant of time. Finally,
Giovannetti et al propose employing generalized measurements[22] \- instead of
applying the usual measurement operator which collapses the wavefunction into
a unique basis state, we apply a unitary transformation which has the effect
of recording the result of the measurement in a quantum register without any
measurement collapse, so the register itself ends up in a superposition of
possible outcomes. We can then do the same with the second measurement at the
later time, thus recording both results in the register, and then subsequently
we can perform a measurement on the register to determine the actual pair of
outcomes.

All of these methods reproduce the results predicted by standard quantum
mechanics (except possibly in special cases like Wigner’s friend experiments,
where even standard quantum mechanics does not yield an unambiguous result).
But do any of them point us to a viable interpretation of the PW
probabilities? The approach based on relational observables has the appealing
feature that it does not require us to add anything to the PW framework, and
hence the results of ref [24] show that at least mathematically speaking
Kuchar’s objection can be addressed without going outside the original
framework. However, because this approach adds nothing to the PW picture it
also doesn’t offer any new physical insight which could help solve the
interpretational problems discussed in the last section - that is to say,
although ref [15] emphasizes that the relational observables encode gauge-
invariant conditional probabilities of the form ‘the probability that the
operator $F$ has some value $f$, given that the frame has orientation $g$,’ it
remains unclear what those conditional probabilities are probabilities _for_
and what the locution ‘given that’ means, since there is no mechanism to
select and actualise one particular measurement outcome within this picture.

Dolby’s approach, although formally equivalent, seems a little more physically
transparent in that it postulates a collapse which could possibly represent a
measurement. However, it’s unclear whether we’re supposed to take this
collapse literally, or whether we should say that the measurement operator
plus additional operator are to be understood all in one as some sort of
nonphysical transformation; and either way, there does not seem to be any time
at which this whole process could take place, given that the PW formalism
presents us with an eternal picture in which the state of the whole universe
is given once and forever. As argued by Callender and Weingard in regard to
the problem of projection operators within quantum cosmology, ‘ _the physical
meaning of these projection operators needs to be clarified. Why is the wave
function projected, and what selects the associated eigenvalues? … If
‘something else’ is involved in the answers, we wish to know what it is. Until
then, the purely formal application of projection operators to regain time is
not physically justified._ ’[34] So although this method does work
mathematically, it doesn’t offer us any natural way to understand what the
probabilities it predicts actually describe.

The consistent histories picture seems a little easier to interpret because it
allows us to think of the PW probabilities as being probabilities for one
course of history or another to occur. However, this approach prevents us from
calculating probabilities at all for a large range of two-time measurements -
for example, ref [17] shows that when applied to Wigner’s friend experiments
it would allow us to calculate probabilities only for the comparatively
uninteresting case where Wigner’s measurement is non-disturbing. Moreover, the
consistent-histories approach does not yield just _one_ set of consistent
histories: the formalism defines a large number of consistent sets of
histories, together with a probability distribution across the histories in
each set, so even if we select one history from each set according to the
prescribed probability distribution, we are left with a large number of
mutually exclusive histories[35]. If we take the ‘many-histories’ view where
all of these histories are supposed to occur, then we will presumably find
that there are some histories in which some observer measures outcome $O$ when
the clock $C$ reads time $t$, and others where the observer measures some
other outcome $O^{\prime}$ when the clock $C$ reads time $t$’ and others where
no measurement occurs at all at this time, so we will have nothing definite to
say about what the observer actually observes, thus undermining the attempt to
define any ‘correlations’ between clocks and systems. Moreover, we don’t have
amplitudes or probability distributions over consistent sets (probabilities
are assigned only _within_ sets), so we don’t even have access to the
strategies employed in the Everett case to make sense of probabilities under
circumstances where all outcomes occur. And finally, it doesn’t seem that the
consistent histories picture provides us with any representation of observers
measuring clocks - in order to say anything about the temporal experiences of
observers within the universe it will be necessary to include interactions in
which observers actually measure the clocks and systems in question, which of
course is exactly what the proposal of Giovanetti et al[22] offers us.

###  4.1 The final-measurement interpretation

The Giovanetti proposal requires us to add extra structure to the PW
formalism, as we must include some interactions in which clock and system
states are written to a register. However, from an interpretational standpoint
something of this nature would probably have been necessary in any case, as
evidently it cannot be operationally meaningful to define time in terms of
correlations between clocks and systems if nobody ever observes the clocks.
Indeed, from the interpretational standpoint Giovanetti proposal has an
obvious advantage over the other approaches to two-time probabilities, as it
is the only one of these calculation methods which straightforwardly fulfils
the desideratum of providing a representation of observers actually observing
or measuring clocks. We reinforce that although the proposal is based on a
notion of ‘generalized measurement’ it does not commit the sin of ascribing a
special privileged role to observers or measurements, as other approaches to
quantum mechanics have been criticised for doing; for a ‘generalized
measurement,’ unlike the standard notion of quantum measurement, is simply a
feature of the usual unitary quantum dynamics, and thus we are not required to
postulate an additional mechanism of physical collapse which occurs only when
an observer gets involved. Any physical interaction which maps information in
certain degrees of freedom to other degrees of freedom counts as a measurement
in this sense, and thus although observers may get involved by deliberately
setting up measuring devices which map information about a quantum system to
the ‘register’ represented by their own brain, observers aren’t necessary,
because natural processes like decoherence are perfectly capable of performing
these sorts of mappings without any help from observers.

That said, there are a couple of technical caveats to be made about the
Giovanetti proposal. First, ref [22] considers only the case of ideal clocks
(for which the time observable is sharp); it is yet to be seen how the
approach would work in the context of non-ideal clocks. However, since it
seems inevitable that a fully-developed version of the PW formalism will
include representations of clocks being observed, we think it is reasonable to
assume that some generalisation of this method will also work for non-ideal
clocks - if a non-ideal clock couldn’t be ‘read’ in this way, we are doubtful
that it could be considered a clock at all in any operationally meaningful
sense.

Second, although deferring the measurement of the register until after both
experiments have been completed ensures that we will get the correct
probabilities for that pair of experiments, if we wish to also get correct
probabilities for any subsequent experiments conditioned on the results of
earlier experiments we will have to defer the collapse process until after
those experiments as well, maintaining the register in its superposition state
until all the experiments we’re interested in have been performed. It follows
that if we want this scheme to produce correct joint probability distributions
for all measurements and measurement-like events across the whole of history,
we will have to defer the collapse process until after all such events have
taken place.

Third, in the original Giovanetti proposal the conditional probability for a
measurement at time $t_{2}$ to have outcome $n$ given that a measurement at
time $t_{1}$ had outcome $m$ is given by the following expression:

$\frac{\langle\langle\Psi|t_{2}\rangle\langle
t_{2}|\otimes\Pi^{n}\otimes\Pi^{m}|\Psi\rangle\rangle}{\langle\langle\Psi|t_{1}\rangle\langle
t_{1}|\otimes\Pi^{m}|\Psi\rangle\rangle}$

As noted by ref [17], it’s somewhat puzzling that this expression involves
dividing by the individual modulus for the first measurement at time $t_{1}$
when this measurement didn’t really take place at time $t_{1}$. Moreover,
although this method does indeed reproduce the predictions of standard quantum
mechanics in all the cases where standard quantum mechanics makes an
unambiguous prediction, ref [17] shows that if we try to apply it to a
Wigner’s friend scenario the result of this prescription isn’t always a well-
defined probability. Ref [17] observes that we can solve both of these
problems by replacing $t_{1}$ with $t_{2}$ everywhere in the expression, but
then of course it isn’t a two-time probability any more. This might look
discouraging given that the initial intention here was to define probabilities
across events taking place at different times, but in fact it may simply be
regarded as an indication that we should take more literally the idea that
this equation describes not two measurements performed at different times, but
rather a single measurement which resolves both of the results of the
individual experiments at once.

These technical considerations point to an alternative way of thinking about
these probabilities: we should simply defer _all_ measurements until the end
of time. That is to say, every interaction which has the form of a measurement
should be written as a generalized measurement operator which maps the outcome
of the measurement and also the reading on a suitable clock to a register,
without actually collapsing the wavefunction to select a unique outcome and
time. We then simply evolve the quantum state forwards in time, applying more
generalized measurements of this form for every measurement interaction, and
writing the results and corresponding times to a blank section of the
register. Thus when the register is finally measured it will contain a variety
of entries of the form ‘outcome $O$ was observed at time $t$.’ The
probabilities ascribed by the PW formalism to pairings of measurement outcomes
and times can then be interpreted as probabilities for the register to display
the corresponding readings at the end of time, and the experiences of
conscious agents can be understood as supervening not on the quantum state but
on the actual course of history recorded in the register. We reinforce there
is no need to imagine there exists an observer external to the universe who
performs the final measurement: the term ‘measurement’ is simply a way of
describing a process in which some particular state of the register is
selected and the course of history it records actualised, so the result of the
measurement dictates the content of reality - in Kent’s words, ‘ _an event
occurs if and only if it leaves effective records in the final time …
measurement_.’[3] Thus the probability distribution over the register is
simply a set of probabilities for different courses of history to be
actualised, and the universal quantum state is not itself an element of
physical reality; it is simply a mathematical tool that we can use to
calculate this probability distribution.

There are precedents for this kind of approach in the literature. Kent [3, 4,
36] has proposed a ‘final-measurement’ approach to solve the Lorentzian
quantum reality problem for relativistic quantum mechanics - for example ref
[4] postulates a final measurement of the density of mass-energy across
spacetime, and then stipulates that the actual contents of reality (‘the
beables’) are given by a mass-energy distribution over spacetime such that the
value at a given point $x$ is equal to the expectation value of the stress-
energy tensor at $x$, conditional on the outcome of the final measurement
outside the future lightcone of $x$. Similarly, ref [2] reaches a very similar
conclusion with respect to the problem of assigning probabilities to sequences
of events in general-relativistic quantum mechanics: the authors envision the
outcomes of sequences of measurements being recorded in the state of the
measuring apparatus, which plays the role of the ‘register’. It is shown in
ref [2] that the probabilities for sequences of measurements thus defined will
always be a Dirac observable, and thus these probabilities can legitimately be
employed in the context of the PW formalism.

Formally, this way of thinking about two-time measurements can be represented
in the constraint formulation by adding terms of the form
$K_{SRC_{x}}\delta(T_{C_{y}}-t_{m})$333We note that the use of a delta
function in this kind of expression, while common in the literature, is
technically problematic: trying to extract relational variables from such a
constraint requires us to combine a distribution with a function in a way that
is not generally well-defined. Thus it seems likely that in fact the delta
functions should be replaced with something like a Gaussian, sharply peaked
around the time $t_{m}$. This is in any case more physically realistic, as the
delta function is being used here to indicate that an event occurs when a
clock reads a certain value, but since clocks generally cannot be read with
infinite precision, any real implementation would necessary have some spread
in the exact time of the event. However, since we will not be solving the
constraints here we will continue using delta functions for simplicity. to the
constraint, with $K_{SRC_{x}}$ a generalized measurement operator defined such
that:

$e^{-iK_{SRC_{x}}}|\psi\rangle|t\rangle_{C_{x}}|\phi\rangle_{R}=\sum_{i}M_{i}\int_{t}|t^{\prime}\rangle\langle
t^{\prime}||\psi\rangle|t\rangle_{C_{x}}|i\rangle_{R_{1}}|t^{\prime}\rangle_{R_{2}}=\sum_{i}M_{i}|\psi\rangle|t\rangle_{C_{x}}|i\rangle_{R_{1}}|t\rangle_{R_{2}}$

where $|\phi\rangle_{R}$ describes the blank register prior to this operation,
and we split the register $R$ into two parts $R_{1},R_{2}$ in order to record
the result of the measurement on $S$ in $R_{1}$ and the time according to
clock $C_{x}$ in $R_{2}$. Here $\delta(T_{C_{y}}-t_{m})$ is a delta function
acting on the time operator $T_{C_{y}}$ for some clock $C_{y}$; the
generalized measurement thus occurs at the time when the clock $C_{y}$ reads
$t_{m}$. We could choose to have $C_{x}=C_{y}$, in which case the time reading
written to the register $R_{2}$ will always be $t_{m}$, but we could also
choose to make $C_{x}$ and $C_{y}$ different clocks such that the resulting
reading on the register represents information about the time reading on the
second clock $C_{x}$ relative to clock $C_{y}$. We then suppose that the
states of $R_{1}$ and $R_{2}$ are preserved until the end of time so that the
final measurement on the register resolves a definite reading $(i,t)$ which
represents the measurement outcome $M_{i}$ being obtained at the time that
clock $C_{x}$ reads time $t$.

Now, there is a complication associated with applying a final-measurement
approach in the context of the Page-Wootters formalism. Namely, the whole
point of the PW formalism is that there exists no universal time, just a range
of different measures of time associated with different possible internal
clocks. So if there is no universal time, how can we make sense of the notion
of ‘the end of time’? Won’t we find that different choices of internal clock
give us a different result for the state of the universe at the end of time?
In fact, a problem of this kind arises even if we are dealing with the simpler
case of quantum mechanics in Minkowskian spacetime with some well-defined
notion of universal unitary evolution, because different choices of reference
frame will give us different final time-slices and hence different results for
the final measurement. Kent’s solution to this problem is essentially to put
the measurement on some time-slice in the distant future and then take a limit
as that time-slice tends to the infinite future[3]. This limit will be well-
defined provided that at some point in the distant future the universe ceases
to undergo any meaningful change or interaction. For example, Kent envisions a
scenario in which, due to the expansion of space, eventually all particles and
fields become effectively non-interacting. Beyond this point in the history of
the universe there are no further interactions, just unitary time evolution,
and since unitary time evolution will not alter or destroy any of the
information stored in the ‘register,’ it doesn’t matter on what time-slice the
final measurement takes place as long as it is past the time at which all
particles cease to interact.

In the PW picture, we can’t simply put the measurement on some time-slice
since different clocks will say different things about the state on that time-
slice, but what we can do is take a limit as $t\rightarrow\infty$ for every
individual clock in the universe, disregarding any clocks which only exist for
a finite amount of time. (Ref [3] provides a technical description of how
these limits can be constructed in a relativistically covariant way). We will
assume that each of these limits is well-defined and that furthermore they
each give rise to the same probability distribution over readings in the
‘register.’444Note that in order to adopt this strategy we do not need to
assume that all the clocks in the universe have some prima facie orientation
which distinguishes between the two possible directions of time, because if we
take it that there is some beginning of time, as indicated by our current
understanding of cosmology, it follows that the readings on the clocks can go
to infinity in only one of the two possible temporal directions, so the limit
$t\rightarrow\infty$ will necessarily single out the same temporal direction
for all the clocks. We do have to assume that the set of all clocks is at
least well enough coordinated that there are two distinguishable time
orientations; but clearly in order for the PW approach to be empirically
adequate it must reproduce the dimensionality of spacetime at least at an
emergent level, so this assumption is reasonable. As in Kent’s example, we
conjecture that this will be the case provided that at some point in history
all particles and fields become non-interacting: beyond this point we will get
the same probability distribution over register readings regardless of time,
since unitary time evolution will not alter or destroy any of the information
stored in the register, so in the limit as $t\rightarrow\infty$ all of the
possible relational states relative to all of the different clocks in the
universe will give rise to the same probability distribution. As noted by
Kent[3], there are some indications in current cosmology that eventually all
particles and fields will indeed become non-interacting, but we would not go
so far as to claim that this has been proven; we regard it as an assumption
which we adopt in order to arrive at a transparent operational interpretation
of the PW probabilities, or perhaps even as a novel prediction of this
interpretational approach.

This approach offers solutions to the problems we identified in connection
with defining clock-system correlations in the PW framework. First, this
interpretation allows us to understand the meaning of ‘given that’ in
statements like ‘given that clock A shows time $t$,’ because the clock and
system are now linked together by the operation which writes the measurement
outcome and time onto the register: hence rather than appealing to
illegitimate quasi-temporal notions to explain the relationship between states
of the clock state and states of the system state, we can make use of pairs of
times and measurement outcomes recorded together in the register. Second,
there is a clear sense in which events can actually be observed, since the
operation associated with an event is now a generalized measurement operator
which writes the result of a measurement onto a register, and that register
might sometimes be the brain of an observer. Finally, we no longer have to say
that every outcome of a measurement always occurs, as the measurement on the
‘register’ at the end of time selects out a state of the register which
represents a single observer observing a single outcome, in accordance with
the common-sense notion of ‘event.’ So we don’t have to appeal to decision
theory or any kind of subjective considerations: we can understand the PW
probabilities as just the usual sorts of probabilities which describe the
outcome of some uncertain event. It is then straightforward to define
correlations between clock readings and measurement outcomes based on the
probability distribution over register readings defined by the universal
quantum state. Since the experiences of conscious agents supervene on the
actual course of history which is selected in the final measurement, we now
have a clear picture of the way in which the PW probabilities constrain the
experiences of agents inside the universe, as they help to determine which
course of history is made actual.

Evidently, interpreting the probabilities in this way does require us to add
some additional structure and additional physical hypotheses to the
straightforward PW formalism, and this might be regarded as a disadvantage of
the approach. However, it seems more or less inevitable that some additional
structure must be added if a single-world realist picture is to be extracted
from the PW framework, and thus we are willing to accept this sort of
additional structure. Obviously we do not claim that this is the only possible
interpretation of the PW framework, or even the only possible single-world
realist approach - for example, we are also optimistic that there would be a
feasible approach based on the framework of relational quantum mechanics. But
what we particularly want to reinforce is that because the final measurement
occurs at the ‘end of time,’ the full formal structure of the PW framework is
preserved unchanged under this interpretation. Other interpretational
approaches do not necessarily have this consequence: for example, the Bohmian
approach advocated by Callender and Weingard[34] and Goldstein and Teufel[37]
requires us to accept a preferred foliation or place additional constraints on
the set of allowed wave-functions to counteract the fact that Bohmian
canonical quantisation leads to a spacetime which depends on our choice of
lapse function. So there seem to be very good reasons for proponents of the PW
framework to adopt the final-measurement interpretation in order to ground the
operational significance of their results: in the final-measurement
interpretation, the _interpretation_ of some of the results produced by the
research programme may be different, but their mathematical correctness is not
in doubt, and therefore those who favour a single-world realist approach can
still make sense of the PW framework and the various interesting results that
have been obtained within it.

####  4.1.1 Comparison to the perspective-neutral collapse

The reason that a measurement at ‘the end of time’ makes sense as a way of
extracting a single-world realist picture out of the universal quantum state
is because, as discussed by Kent in refs [3, 4], our brains and all the
objects which feature in our macroscopic experience leave a large number of
traces on the environment via the mechanism of decoherence and thus even when
we are long dead and all the historical records of our lives have been
destroyed, the memories once stored in our brains will still be recorded in
the state of some part of the universe, so in principle these memories can be
resolved to a definite state by a final measurement on the relevant
‘register.’ Obviously our use of the term ‘register’ is overly simplistic -
there is not literally a book somewhere recording pairs of times and
measurement outcomes - but any physical system can in principle play the role
of the register provided that it does not undergo too much disturbance after
the interactions in which the relevant record is formed. Therefore the final-
measurement approach ensures that our brains, measuring instruments and other
macroscopic objects will be resolved by the final measurement to have unique,
definite histories, as required for a single-world realist approach.

In particular, the final measurement mechanism provides a simple and yet well-
defined and physically motivated way of drawing the line between macroscopic
features of reality which must be resolved to a definite state in the final
measurement and microscopic features which need not be resolved: ‘macroscopic’
features are those which are recorded in the final state, and ‘microscopic’
features are those which are not recorded in the final state and thus can
participate in quantum phenomena like interference and superposition. Roughly
speaking this means that systems which undergo decoherence will be resolved to
a definite state while systems which do not undergo decoherence will
participate in quantum phenomena555We reinforce that decoherence alone is not
enough to explain the emergence of a single definite macroreality; decoherence
makes density matrices diagonal, thus suppressing interference effects, but
the measurement mechanism is still needed to select one of the diagonal
elements so we get a unique macroscopic history.. Indeed it would seem that
any empirically adequate single-world realist approach must draw this line in
roughly the same place, and therefore we conjecture that most other single-
world realist approaches which succeed in retaining the mathematical structure
of the PW formalism will ultimately look quite similar to the final-
measurement approach, so the conclusions we draw using the final-measurement
approach in the following sections are also likely to be valid in other
similar approaches.

In this spirit, we noted earlier that one might hope to extract a single-world
realist picture from the perspective-neutral approach by collapsing the
quantum state of the universe at the level of the physical Hilbert space. We
now observe that the final-measurement approach has obvious structural
similarities to the perspective-neutral collapse approach; both strategies
postulate a single wavefunction collapse which is required to select the whole
course of history in an all-at-once manner. The key difference is that by
making use of records, the final-measurement approach arrives at a unifying,
asymptotically perspective-neutral object which _directly_ encodes histories
that we can collapse to, thus sidestepping the problem of fixing the
relationships between collapses which occur in different reference frames. As
we have noted, it also straightforwardly provides a solution to the
measurement problem by answering the question about where to put the line
between the microscopic and the macroscopic. So at least for the moment the
final-measurement approach seems like a more complete answer to the
intepretational problems of the PW framework than the perspective-neutral
collapse. That said, the two approaches are not necessarily in competition;
for proponents of a perspective-neutral collapse, the final-measurement
interpretation could be regarded as a heuristic tool which we may employ in
the short term to get some idea of what the perspective-neutral collapse might
look like, since we do not yet yet have a formal framework showing how to
implement that collapse.

We discuss a few more interpretative points in appendix B.

####  4.1.2 Example: Temporal Nonlocality

This is all good news for the PW programme and more broadly the IQRF research
programme. First off, the final-measurement interpretation goes some way
towards vindicating the claim that the PW formalism provides an adequate
account of time in the block universe, for it provides us with a clear
operational connection between the PW relative states and the temporal
experiences of actual observers. Moreover, we also see that the PW account of
time does not in fact depend on the adoption of the Everett interpretation or
some kind of non-realist view - those who prefer a ‘single-world realist’
account which allows for unique, intersubjective measurement outcomes also
have a way of making sense of the PW formalism. We will now verify this claim
by examining the interpretation of a specific result of the IQRF framework
within the final-measurement interpretation.

Ref [6] discusses ‘temporal non-locality’ in the IQRF approach. For example,
consider the case where an event $E$ occurs in the reference frame associated
with a clock $C_{B}$, i.e. the event $E$ is fixed to occur when clock $C_{B}$
reads $t$. Suppose now that clock $C_{B}$ is in a delocalised state relative
to another clock $C_{A}$ \- i.e. the state of clock $C_{B}$ relative to clock
$C_{A}$ represents it as being in a superposition of time readings. Ref [6]
demonstrates that an event $E$ will therefore be delocalised in time relative
to clock $C_{A}$. But what does that actually mean? Is it just a mathematical
statement about the relation between $E$ and $C_{A}$ or does it have
operational consequences for real temporal experiences?

We can answer these questions by appeal to the final-measurement
interpretation. In order to represent the event $E$ in the reference frame of
clock $A$, we will use a generalised measurement which is scheduled to occur
when clock $B$ reads $t$. This can be expressed using the constraint equation
$(H_{A}+H_{B}+\delta(t-T_{B})K_{RC_{A}})|\psi\rangle=0$, which encodes the
fact that when the clock $C_{B}$ reads $t$ the operator $K_{RC_{A}}$ copies
the time from clock $C_{A}$ over to the register. Here $T_{B}$ is the time
operator associated with clock $C_{B}$ as defined in section 2.1; in this case
we are only interested in the time of the event and not in any additional
measurement result, so we don’t need the operator $K$ to also copy over a
measurement outcome as defined in section 4.1. Let $|\psi_{A}(0)\rangle_{\lnot
A}$ be the state of all systems except clock $A$ conditioned on $A$ reading
$t=0$; then as in ref [6], $|\psi_{A}(0)\rangle_{\lnot A}$ can be written as
$\int dt_{B}\phi(t_{B})|t_{B}\rangle\otimes|\chi\rangle_{R}$, where the
normalised function $\phi$ encodes the localisation of clock $C_{B}$ relative
to clock $C_{A}$: if it’s a delta function then the two clocks have sharp
states relative to one another, whereas if it isn’t a delta function then
there is no definite fact about what clock $B$ reads when $A$ reads $0$.

Now let us suppose that experiments of this kind are carried out many times
and in each experiment the clocks $C_{A},C_{B}$ are reset to some appropriate
initial state at the start, so we can expect to see numerically consistent
time readings across these different experiments. Then if $\phi$ is a delta
function, the generalized measurement operator $K_{RC_{A}}$ mapping the
reading of $C_{A}$ to the register will always occur when clock $C_{A}$ reads
some fixed value $t^{\prime}$ and therefore when the register is measured we
will find that all of the registers for the various copies of this experiment
read $t^{\prime}$. On the other hand, if $\phi$ is not a delta function then
the state of the register $R_{1}$ after the generalized measurement operator
$K_{RC_{A}}$ has been applied will be some superposition of different time
readings, and and therefore when the register is measured we will find that
the records for the various copies of this experiment contain a variety of
different time readings. So the final-measurement interpretation wholly
vindicates the claims of the IQRF framework in this instance and indeed puts
those claims on a more solid operational framework: what it is for one event
to be delocalised in the reference frame of a given clock is for the time of
the event to exhibit some spread across an ensemble of identical experiments
as recorded in the register at the end of time, rather than always showing the
same time.

##  5 Quantum Reference Frames

We will now consider how the foundational and conceptual claims that have been
associated with the PW and IQRF frameworks may be affected by a choice of
interpretation for those frameworks. We will begin by evaluating the claim
that the IQRF research programme can be understood as a way of thinking about
reference frames in the quantum world. As we have seen, the results of the PW
and IQRF research programmes are often expressed using language like ‘ _the
time reference frame of a clock_ ’ and ‘ _In this situation, how would another
clock, say A, describe the time evolution of a quantum experiment?_ ’[6] and
the process of selecting a quantum references frame is often described as ‘
_jumping into the perspective_ ’ of a quantum system[7]. Prima facie this
language may seem a little odd, as clocks and individual quantum particles do
not literally have a perspective on the world and they do not literally
provide descriptions of experiments. So this language must be understood as
shorthand for something else - but what?

The IQRF idea that reference frames are simply individual physical systems is
motivated by Einstein’s treatment of reference frames, where we are encouraged
to think about the operational procedures used to establish the values of
physical quantities in different reference frames using physical objects like
rods and clocks. For example, ref [9] suggests that we should ‘ _take the
view, already advocated by Einstein, that coordinate systems correspond to
physical rods and clocks._ ’ But note that Einstein does not tell us that rods
and clocks _are_ reference frames - rather a number of rods and clocks are
used jointly to operationally establish a reference frame[38]. An alternative
way of thinking about reference frames is suggested by the fact that rods and
clocks are not just any old physical objects - they are _measuring
instruments_ which take part in operational procedures that are used to
characterise physical events relative to the reference frames. This feature of
rods and clocks is obviously crucial to Einstein’s use of them, as the whole
point of a reference frame is to serve as a background against which we can
describe physics, and typically a ‘description of physics’ involves
characterising the results of some actual or possible observations. If a
description of physics ‘relative to’ some reference frame is not anchored to
actual or possible observations then there will inevitably be a worry that it
is merely a mathematical abstraction which does not represent anything in
physical reality - and this worry is particularly pressing when the physics we
are trying to describe is quantum-mechanical, since there remains a great deal
of uncertainty about the ontology of the theory and therefore it’s by no means
certain that descriptions of the evolution of a wavefunction have any physical
meaning outside of the role they play in characterising possible observations.

Of course, in classical physics the measurements can largely be left implicit
when we characterise a reference frame, because classical measurements can be
regarded as straightforwardly probing classical reality without disturbing it
in any way. But this is no longer the case when we are dealing with quantum
physics, because measurements and records can’t be taken for granted in the
quantum world - performing quantum measurements will generally disturb the
evolution of the relevant systems, and no-cloning constraints place limits on
the degree to which records can be made[39], so in a quantum setting it is not
necessarily the case that abstract statements about the evolution of one
system relative to another system can be converted in any straightforward way
to operational statements about the results of measurements. Therefore it is
important to explicitly include measurements and records in a theory of
quantum reference frames, because otherwise we risk ending up with
descriptions which can’t possibly correspond to any actual observations.

These considerations seem to indicate that in order for a IQRF to play the
same kind of role as a classical reference frame, we must be able to make
sense of the idea that it could have measuring instruments which are ‘carried
along’ with it, like the co-moving rods and clocks employed in the
construction of an idealised classical reference frame. Describing the physics
inside an IQRF would then amount to saying what the measuring instruments
associated with the reference frame would read under various circumstances (if
there were in fact such measuring instruments present). This would allow us to
make physical sense of the notion of ‘the evolution of some system $S$
relative to the reference frame $F$,’ since it could be cashed out in terms of
operational procedures, such that if we are in reference frame $F$ and we
perform and record a sequence of measurements on the system $S$, the recorded
results will be consistent with the specified evolution law.

We will henceforth take it that the distinctive feature of IQRFs as compared
to classical references frames is that they can be in superpositions relative
to one another: thus if IQRFs are in general to be operationally accessible we
will have to postulate that measuring instruments or measurements or the
physical systems recording the results of those measurements can in some sense
be in superposition relative to one another. For example, if a particle is in
a superposition relative to a laboratory, the outcomes of experiments
performed using the laboratory instruments will appear to be in a
superposition relative to the particle reference frame, and conversely if we
could somehow manage to associate some measuring instruments with the
particle, then the outcomes of experiments performed using those instruments
would appear to be in a superposition relative to the laboratory reference
frame.

One might be tempted to suggest that instead of having a distinct set of
measuring instruments for each frame, as with co-moving rods and clocks in
general relativity, we could employ just a single measuring device such that
when we transform to a particular IQRF, we also transform the state and
observables of the measuring device. So for example, in the particle-
laboratory case, rather than associating measuring instruments with the
particle itself, we could simply characterize how the outcomes obtained from
the laboratory instruments will look in the particle IQRF - i.e. they will be
in superpositions. However, it seems quite difficult to arrive at an
operational understanding of such measurement results, as we ourselves do not
ever find measuring instruments or records of measurement outcomes in
superpositions and indeed we do not even seem to have a coherent notion of
what it would be like to find a measuring instrument or record in a
superposition. Moreover, as we will shortly see, the IQRF research programme
tells us that within each reference frame the metric is locally Minkowskian (
i.e. the metric is flat and it is not in a superposition relative to the
reference frame), and if this statement is to be understood operationally,
then it must be the case that within each reference frame we can perform local
measurements probing the metric which produce results that are not in a
superposition relative to that reference frame. So it is reasonable to
conclude that if IQRFs are in general to be operationally accessible, we will
sometimes have to postulate cases where two distinct measuring instruments and
the physical records of their results are in superposition relative to one
another.

We note that our emphasis on the operational features of reference frames is
not predicated on the philosophical view sometimes known as ‘operationalism’
which holds that a concept cannot be meaningful unless we have a method of
measurement for it[40]. It is not our intention to claim that ‘quantum
reference frames’ which are not associated with measurements are in virtue of
that fact not _meaningful_ \- we merely contend that such constructions are
not well suited to play the role of _reference frames_ , because the function
of reference frames in our physics is intimately tied to operations of
observation and measurement, so a theory of reference frames must ultimately
say something about measurements and observations. Moreover, we recall that
the original motivation for the PW framework was to define time in operational
terms through measurements of quantum clocks, so if the framework is to fulfil
this promise, the reference frames that it employs must be given some
operational interpretation.

###  5.1 The Quantum Equivalence Principle

Against these arguments, one might object that as noted in section 2.1,
choosing a reference frame can be regarded as a choice of gauge-fixing
procedure, and for other types of gauge-fixing procedures we don’t typically
insist that a gauge must have operational significance in terms of actual or
possible measurements. Thus for example ref [41] argues that matters of this
kind can be left until later: _‘One may finally ask whether observers who are
part of the theory may in fact have good operational access to that chosen
frame of reference, but this is an additional (though important) question that
we here regard as secondary.’_ And of course we have no objection to the IQRF
methodology if it is merely regarding as gauge-fixing. There are certainly
good reasons to wish to have access to these sorts of alternative
descriptions: for example, ref [42] notes that this may help for certain sorts
of calculations: ‘ _if we do not know how to handle the quantum case, let us
perform a IQRF transformation that renders some of the subsystems classical._
’ However, the use of the terminology ‘reference frame’ for this gauge-fixing
procedure is not entirely innocuous, because it gives the impression that
‘quantum reference frames’ play a similar role in the theory to classical
reference frames, and thus it it risks giving undue weight to descriptions of
the world that are not in fact operationally or physically meaningful. In
particular, reference frames play a key role in certain formulations of
Einstein’s equivalence principle, and therefore proponents of the IQRF
reference programme have drawn connections between their results and the
equivalence principle: for example, ref [9] argues that ‘ _there is no
fundamental incompatibility between the principles of (quantum mechanics and
general relativity), provided that we revise the formulation of the
Equivalence Principle in such a way that it holds in all reference frames that
quantum theory allows for._ ’ Given the well-known difficulties associated
with the attempt to unify quantum mechanics and general relativity, this is a
very strong claim, and one might worry that to some degree the proposed
solution relies on equivocation around what a reference frame is.

The ‘quantum equivalence principle’ proposed by refs[9, 43] states that ‘ _In
any and every Quantum Locally Inertial Frame, anywhere and anytime in the
universe, all the (nongravitational) laws of physics must take on their
familiar non-relativistic form._ ’ Here, a Quantum Locally Intertial Frame is
obtained by implementing a unitary IQRF transformation such that the
gravitational field, and the superposition of gravitational fields, is locally
Minkowskian in the quantum reference frame associated to the position of some
quantum particle. That is to say, a QLIF is defined with respect to an
individual physical system rather than in terms of actual or possible
measurement operations. On the other hand, the ‘reference frames’ employed in
general relativity typically _are_ defined in terms of actual or possible
measurements, so some further work is needed before we can verify that the
quantum equivalence principle is suited to play the same kind of role in
quantum gravity as the ordinary equivalence principle plays in general
relativity.

At this juncture we should acknowledge that even if we confine our attention
to General Relativity, there are a number of different ideas going by the name
of ‘equivalence principle,’ - ref [44] demonstrates that there are at least
four different classes of equivalence principle. However, it seems clear that
the version of the equivalence principle invoked by the ‘quantum equivalence
principle’ above is what ref [44] describes as the ‘small enough regions’
formulation: i.e. the notion that in any general relativistic spacetime, the
metric in a sufficiently small region of a given point will look Minkowskian.
Thus we will henceforth focus on this specific version of the equivalence
principle, which is sometimes known as the ‘strong equivalence principle’
(SEP)[45]. What, then, is the function of the SEP in general relativity?
666It’s worth nothing that there has been considerable debate about whether
the SEP is universally true - it has been pointed out that tidal effects may
spoil the Lorentz invariance unless the ‘region’ is just a point[46], and more
recently it has been noted that minimal coupling may also lead to apparent
violations of the equivalence principle[47]; the fact that the SEP can
apparently be violated even in the classical regime casts some doubt on the
idea that we should expect it to be universally true in the quantum regime.

In Einstein’s own writings, the SEP is understood in operational terms as a
description of possible measurement procedures: he famously characterised the
equivalence principle in terms of what an observer would experience: _‘for an
observer in free-fall from the roof of a house there is during the fall—at
least in his immediate vicinity—no gravitational field.’_[48] Expanding on the
important role of these operational considerations in the theory, he wrote in
1921, ‘ _According to the special theory of relativity the coordinates
$x,y,z,t$ are directly measurable via clocks at rest with respect to the
coordinate system. Thus, the invariant $ds$, which is defined via the equation
$ds^{2}=dt^{2}-dx^{2}-dy^{2}-dz^{2}$, likewise corresponds to a measurement
result. The general theory of relativity rests entirely on the premise that
each infinitesimal line element of the spacetime manifold physically behaves
like the four-dimensional manifold of the special theory of relativity._’[49]
This line of thought has been taken up in particular by Brown[50], who argues
that the crucial function of the strong equivalence principle in General
Relativity is to make a link between the metric and the behaviour of actual
clocks and rods: the metric _‘attains its chronogeometric significance in GR …
by way of the strong equivalence principle.’_[47] Similarly Knox writes that
the SEP _‘expresses just that fact about our matter theories that must be true
if systems formed from appropriate matter are to reflect the structure of the
metric field, that is, if phenomenological geometry is to reflect the geometry
of the metric field,’_ and further that _‘the whole point of the SEP is that
it’s meant to tell you what it means for experiments to pick out a
connection.’[51]_ That is to say, the SEP is important because without it the
mathematical framework of GR can’t be understood as describing the behaviour
of actual rods and clocks, which would largely prevent it from having
meaningful observational consequences.777We reinforce that although it is
particularly common for proponents of the dynamical perspective on
relativity[50] to emphasize the role of the SEP in defining chronogeometric
behaviour, this understanding of the SEP isn’t predicated on acceptance of the
dynamical approach: the SEP plays a very similar operational role in the
competing geometric approach[52]. If we adopt the dynamical perspective, we
would say that the equivalence principle is true because the dynamics of
matter fields locally exhibit the same symmetries as the metric; whereas if we
adopt the geometric perspective, we would say that the equivalence principle
is a consequence of the fact that the metric constrains the matter fields to
be invariant with respect to its own local symmetries: either way, it is clear
that the equivalence principle is important in large part because it encodes a
practical connection between the metric and the dynamics of matter. The two
approaches differ on the direction of explanation, but agree that the SEP is
significant because of this practical connection.

So it seems that the role played by the SEP in general relativity, or at least
one particularly crucial role that it plays, is founded on the fact that the
‘reference frames’ to which it refers have an operational significance in
terms of actual or possible measurements which can be used to survey the
metric. These operational features are not incidental - they are at the heart
of the equivalence principle’s relevance and importance. Therefore if a
‘quantum equivalence principle’ is to play a similar role in a theory of
quantum gravity, it should likewise be defined with respect to reference
frames which can be understood in terms of actual or possible measurements
which give us operational access to the spacetime metric.

We reinforce that the IQRF research programme is still under very active
development, so although the connection between the IQRF formalism and actual
or possible observations is not currently very clear, this is a topic which
researchers are still working on, and we are confident that interesting
answers will emerge in due course. It is not our intention to anticipate these
conclusions, so we will limit ourselves to making some general observations
about the way in which the status of the quantum equivalence principle will
necessarily depend on a choice of interpretation of quantum mechanics. The
crucial point is that different interpretations of quantum mechanics differ in
what they say about the possibility of performing distinct measurements within
different IQRFs.

##### Collapse approaches

In the case of interpretations which involve wavefunction collapses there is a
very concrete sense in which it is not possible for two reference frames each
containing its own set of measurements to be in a superposition relative to
one another, because measuring instruments are macroscopic objects and hence
they are, according to these approaches, not able to be in superpositions. In
particular, the GRW collapse and gravitational collapse interpretations
predict that superpositions inevitably collapse when the objects involved in
them get too large. That is to say, because these approaches define a set of
beables which by stipulation are in a definite state, they necessarily single
out a set of preferred quantum reference frames relative to which the beables
are not in superpositions, and from that point of view quantum descriptions
relativized to any other frame of reference are strictly false. Thus these
approaches tell us that many of the reference frames that we might extract
from the perspective-neutral universal wavefunction don’t correspond to
anything physically real or meaningful. Moreover by construction it is
guaranteed that in these approaches macroscopic objects will never be in
superpositions relative to the preferred reference frames, and therefore
collapse approaches will never allow us to have two distinct macroscopic
measuring instruments or observers in superposition relative to one another,
so we will never be able to perform distinct measurements surveying distinct
spacetime metrics in IQRFs which are in superposition relative to one another.
Therefore it would seem that collapse approaches don’t admit the possibility
of operationally meaningful quantum reference frames (except insofar as they
coincide with classical reference frames).

##### No-collapse single-world realist approaches

The final-measurement interpretation, on the other hand, tells us that the
universal wavefunction may indeed contain representations of measuring
instruments and observers which are in superposition relative to one another.
Thus all of the different reference frames that can be extracted from the
perspective-neutral universal wavefunction are in some sense physically
meaningful, and we can understand transformations between quantum reference
frames as gauge transformations relating different possible ways of describing
the universal wavefunction. On the other hand, the final-measurement
interpretation tells us that when we have two reference frames superposed
relative to one another and we attempt to perform measurements surveying the
metric individually in each of these reference frames, if it is the case that
the metrics are macroscopically different then the records of the two sets of
measurements will be orthogonal, meaning that the final measurement will
necessarily reveal a set of register readings which actualises only one set of
measurement outcomes. Therefore the final-measurement approach tells us that
superpositions of reference frames relative to one another are a real feature
of the universal wavefunction, but nonetheless only a subset of these quantum
reference frames will have concrete operational significance in terms of
actual measurements, and in particular we will never find that two reference
frames in a superposition relative to one another both come to have direct
operational significance.

Similarly, in the de Broglie-Bohm interpretation there are no wavefunction
collapses and hence in a de Broglie-Bohm version of the IQRF framework it
follows that all of the different reference frames that can be extracted from
the perspective-neutral universal wavefunction are in some sense physically
meaningful. However, the de Broglie-Bohm approach tells us that measuring
instruments and measurement outcomes supervene on the arrangement of the de
Broglie-Bohm variables (e.g. positions of particles in the simplest version of
de Broglie-Bohm), and since the de Broglie-Bohm variables cannot be in
superpositions relative to one another, once again we will never find that two
reference frames in a superposition relative to one another both have direct
operational significance.

The distinction between reference frames which have concrete operational
significance and reference frames which do not is important, because in the
case of classical relativity, one of the main motivations for studying
transformations between reference frames is the idea that all reference frames
are equally valid: no reference frame is more ‘real’ than any other reference
frame, and therefore we expect that observers should have roughly similar
kinds of experiences in different reference frames. But this is not
necessarily the case when we are dealing with IQRFs in a single-world realist
approach. For example, the de Broglie-Bohm interpretation and the final-
measurement interpretation both tell us that when a particle is superposed
relative to a laboratory, _only_ the reference frame associated with the
laboratory is genuinely an element of physical reality: a reference frame
defined with respect to a superposed quantum particle is not ‘real’ in the
same way as ordinary physical objects, it is simply an artefact of some
mathematical structure that plays a role in defining the behaviour of the
physical world. So adherents of these sorts of single-world realist approaches
might reasonably feel that physics should be described relative to the real
reference frames composed of physical objects, rather than relative to
‘virtual’ reference frames defined with respect to superposed particles which
are not elements of ordinary physical reality. And thus adherents of single-
world approaches might not necessarily think it is correct to expect that real
reference frames should be ‘equivalent’ to these virtual reference frames in
the way defined by the equivalence principle.

##### Everett

In the Everett interpretation, on the other hand, in principle there could be
IQRFs in superposition relative to one another such that each IQRF is
operationally accessible to a distinct set of observers inside it. For
example, we could have a ‘Wigner’s friend’ type scenario where an observer
performs a measurement on a superposed quantum particle and thus aligns
herself with the reference frame of the particle, such that both the observer
and her measurement results are in a superposition relative to the reference
frame of some other external observer. However, recall that decoherence has
the effect of ensuring that states involving macroscopic objects very rapidly
become diagonal, so even in an Everettian world it would be extremely
difficult to actually implement a scenario where we have two IQRFs associated
with macroscopic objects in superposition relative to one another for any
nontrivial length of time.

Moreover, even if we did manage to implement such a scenario, recall that it
is crucial to the empirical success of the Everett view that due to
decoherence, observers within one branch of a superposition will have exactly
the same experiences as they would have had if the wavefunction had in fact
collapsed and there were no other branches, so it is unclear to what extent
those observers will need a special quantum version of the equivalence
principle to describe their experiences. For example, consider the QEP as
formulated in ref [53]: _‘The local effects of (quantum) motion in a
superposition of uniform gravitational fields are indistinguishable from those
of an observer in flat spacetime that undergoes a quantum superposition of
accelerations.’_ In order to say that these two experiences are
‘indistinguishable’ we must first say something about the subjective
experience of the observer in both cases: so what does it feel like to
experience ‘a quantum superposition of accelerations?’ The Everett
interpretation tells us that ‘an observer that undergoes a quantum
superposition of accelerations’ will in fact become a set of distinct
observers within distinct branches of the wavefunction, each experiencing a
different single-valued acceleration, so it would seem that in the Everett
picture there is no difference between the subjective experience of
‘undergoing a quantum superposition of accelerations’ and the subjective
experience of undergoing a single acceleration. Therefore it’s unclear what it
could mean in the Everett picture for the experience of undergoing a
superposition of accelerations to be indistinguishable from the local effects
of (quantum) motion in a superposition of uniform gravitational fields, unless
this is simply to be cashed out in terms of a mapping between the branches in
the two cases, such that each observer undergoing motion in some branch of a
superposition of uniform gravitational fields has experiences
indistinguishable from those of some observer in a branch of a quantum
superposition of accelerations in flat spacetime. But that is just the same as
saying that ‘the local effects of motion in a uniform gravitational field are
indistinguishable from the effects of acceleration in flat spacetime,’ which
already follows from the classical SEP. There don’t seem to be any distinctive
experiences associated with being in a IQRF which set IQRFs apart in
_operational terms_ from classical reference frames, and hence there is no
extra work for an operationally construed quantum equivalence principle to do
here.

Given that it is decoherence which appears to block Everettian observers from
having subjective experiences whilst in superpositions which differ from the
experiences they have when not in superpositions, one might be inclined to
suggest that the operational significance of the quantum equivalence principle
could be understood in a counterfactual sense: in some theory like Everettian
quantum mechanics in which observers were _not_ subject to decoherence, the
experience of being in a superposition of accelerations would be
indistinguishable from the experience of being in a superposition of uniform
gravitational fields. But it’s very hard to make sense of this because, as
often observed in the literature on interpretations of quantum mechanics, it’s
very hard to even imagine what the subjective experience of ‘being in a
superposition of accelerations’ would feel like[54]: the consensus seems to be
that the only way we can make sense of this notion is to suppose that each
branch is associated with a distinct conscious experience. So it’s unclear who
the subject of experience could possibly be in the ‘indistinguishabilility’
claim made in ref [53], even if we allow ourselves to consider an alternative
decoherence-free quantum world. (Indeed, it seems unlikely that a decoherence-
free quantum world would contain any conscious observers at all!)

###  5.2 Summary

Thus we see roughly three possible cases for the operational status of quantum
reference frames: 1) reference frames can’t be in superpositions relative to
one another at all; 2) the universal quantum state may contain representations
of reference frames which are in superposition relative to one another, but in
such cases only one of the reference frames can be operationally accessible;
and 3) operationally accessible reference frames can be in superposition
relative to one another, but from a subjective point of view being in a IQRF
is just the same as being in a classical reference frame, so in most cases the
classical SEP will already be enough to characterise the experiences of
observers in such reference frames. Evidently the status of the ‘quantum
equivalence principle’ will be different in each of these cases: however, in
none of these cases does the quantum equivalence principle appear to play the
same kind of role as the classical SEP. (In appendix C we discuss a particular
case where the difference between the classical SEP and quantum equivalence
principle becomes important.)

We reinforce that the points made in this section are not intended as a
criticism of the IQRF framework itself. The basic reasoning that we should be
able to describe physics in a way that does not depend on the existence of a
hypothetical reference frame outside the universe seems impeccable, and the
IQRF formalism is doing important work to show how this can be achieved using
a gauge-fixing process to deal with the redundancy arising from the absence of
a universal frame. However, it seems that in many cases it is not really
viable to understand this form of gauge-fixing in operational terms, so it is
not particularly helpful to think of the resulting description as being
relativized to a ‘reference frame’ in the usual operational sense. Not every
meaningful statement about physics needs to be characterised directly in
operational terms: it is enough that we can see how relevant operational
features would appear in some appropriate limit.

Indeed, instead of thinking of the IQRF framework as characterising reference
frames directly, we can potentially regard the formalism as a route to showing
how ‘reference frames’ and other operational features emerge from the
perspective-neutral view in the limit as the size of the systems involved
becomes large. In this limit, and _only_ in this limit, we can think of the
gauge-fixing procedure as ‘jumping into the perspective’ of an internal
subsystem, because if a system is macroscopic we can ascribe to it a
‘perspective’ which corresponds to an operationally meaningful description.
This suggests that rather than postulating a distinctive ‘quantum equivalence
principle,’ the IQRF framework should perhaps instead be seeking to show how
the ordinary classical equivalence principle emerges in the appropriate limit.
In this connection we recall Knox’s verdict on the SEP: in ref [55] she argues
that the SEP has to be understood as ‘ _contextual and approximate_ ’ and that
‘ _phenomenological geometry is exactly the kind of thing that could be
emergent_ ,’ which leads her to question whether it makes sense to think of
the SEP as holding in the context of quantum gravity, given that quantum
gravitational theories are likely to involve a characterisation of reality
somewhere below the level at which phenomenological geometry emerges. The
conclusions we have reached here on the ‘quantum equivalence principle’ seem
to uphold Knox’s arguments: we have seen that ‘reference frames’ themselves
are necessarily an emergent phenomenon, and thus any equivalence principle
making essential use of reference frames must also be emergent.

Poulin demonstrates this kind of emergence in ref [25], which offers a
description of a ‘measurement’ in relational terms: a quantum particle is
coupled to a ‘gyroscope’ and then we check whether the spins of the system and
gyroscope are parallel or antiparallel, averaging over all possible directions
to eliminate non-relational facts. Poulin shows that in the limit as the
gyroscope becomes very large, the probabilities for the two outcomes just
reduce to the amplitude squared for each of the two possible spin directions
in the particle’s non-relational quantum state, i.e. in this limit we arrive
at the standard quantum description. Poulin’s work is done in the context of
the quantum information approach to reference frames (see appendix A) where we
use statistical averaging rather than coherent averaging as in the IQRF
framework, but nonetheless the general principle that as a system becomes
macroscopic the physics relative to it approaches the standard non-relational
quantum description is presumably also applicable in the IQRF approach, and
this provides an important constraint on the operational credentials of that
framework. Any ‘operational’ approach to physics must take account of the fact
that the way in which we ourselves are embedded in reality places certain
limits on the operations we can perform; in this case the key point is that we
are macroscopic objects who are always subject to decoherence effects, and
therefore we will never be able to directly observe relational effects which
hold only when all the systems involved are microscopic. The act of
observation necessarily introduces a macroscopic reference frame, and as shown
by Poulin, that macroscopic reference frame introduces qualitatively different
features into the relational description, which inevitably makes it difficult
to determine concrete operational features of quantum reference frames which
differ from the operational features of classical reference frames.

####  5.2.1 Less operational alternatives

Given the considerations discussed in the previous section, potentially the
gauge-fixing procedure of the IQRF framework would be better understood as
characterising ‘quantum coordinate systems,’ rather than reference frames,
since a coordinate system is not required to have any particular operational
significance. Then rather than a quantum generalisation of the SEP we should
perhaps be aiming for a quantum generalisation of the principle of general
covariance, which would presumably say something like ‘The equations
describing the laws of physics should have the same form in all (quantum)
coordinate systems, or IQRFs,’ - for example, in ref [56] Hardy embarks on the
project of writing the laws of quantum mechanics in a tensorial form and
expressing them in a bundle of four-dimensional Riemannian spacetimes, so a
principle of general covariance can be applied. However we reinforce that as
noted by Kretschmann[57], the principle of general covariance is in some ways
trivial - it is possible to write virtually any classical spacetime theory in
such a way that it ‘has the same form in all coordinate systems,’ since this
can be guaranteed by writing the laws in a tensorial form and expressing them
in a four-dimensional Riemannian spacetime. The SEP, on the other hand,
certainly has non-trivial content: as noted by ref [50] the SEP does not even
follow from the Einstein Equations of general relativity, it is a further
assumption needed to give operational meaning to the theory. So the
equivalence principle and the principle of general covariance play distinctly
different roles in general relativity, and thus it is important to distinguish
clearly between a quantum principle of general covariance and a quantum
equivalence principle, which would presumably have quite different roles to
play in quantum gravity.

If we do still want to postulate a specific quantum equivalence principle, we
might perhaps move to a ‘quantum equivalence principle’ defined using
analogues of measuring instruments which are not macroscopic. For example,
although ref [53] initially describes the three aspects of the quantum
equivalence principle in terms of the experiences of (presumably macroscopic)
observers, the way in which it is proposed that we test the quantum
equivalence principle is by employing superpositions involving particles with
an internal degree of freedom which can be treated as a clock. It is then
argued that we will have a violation of the quantum equivalence principle if
_‘in the QLIF of system A, the clock behaves differently according to where it
is placed in the gravitational field.’_

Now, in order to use the internal degree of freedom as a measuring instrument
we must make some observation of it, and obviously any single-world realist
interpretation will insist that we always observe exactly one out of the set
of possible superposed clock readings, so such interpretations must still
insist that in this case it is impossible to actually perform distinct sets of
measurements surveying different metrics in the different IQRFs. However,
instead of observing the clocks directly, we might consider performing some
kind of interference experiment between the branches of the clock and then
observing the result of that experiment. This is exactly what ref [53]
proposes, and it is shown that various kinds of violations of the quantum
equivalence principle can be expected to have characteristic effects on the
results of the interference experiment. So it seems that according to ref [53]
the quantum equivalence principle does not in fact have direct operational
significance, but rather it is to be accessed indirectly by performing
measurements which give us information about what the relationship between the
measurement results _would have been_ if we could have performed distinct
measurements for each of the frames. The operational meaning of the quantum
equivalence principle then reduces to the claim that the results of this
interference experiment will be consistent with what we would expect if the
internal degrees of freedom of the particles each individually surveyed the
metric inside their own IQRF in the same way that clocks are expected to
survey the metric in ordinary classical GR.

Note that it seems crucial to this formulation of the quantum equivalence
principle that the IQRFs in question are not accessed by direct measurement -
for example, we don’t actually perform readings in the QLIF of system A to see
if the clock behaves differently according to where it is placed in the
gravitational field. For if we did that, then the ordinary classical SEP would
come into force, mandating that the clock must not behave differently
according to where it is placed in the gravitational field. Thus if the
quantum equivalence principle is to have any content over and above the
classical SEP, it must be understood as characterising readings on measuring
devices at times when they are not actually being read. This immediately
indicates that a quantum equivalence principle formulated in these terms
cannot play the same kind of role as the SEP, which as we have seen is
particularly important to GR because of the connection it makes between
_actual_ observations and the underlying spacetime metric. Moreover, this way
of thinking about IQRFs takes an unusually robust view of the properties of
systems at times when they are not being measured. For recall that the
contextuality theorems in quantum mechanics[58, 59] have demonstrated that it
is not always possible to think of quantum systems as having definite well-
defined properties at times in between measurements: in view of this feature
of quantum mechanics, the claim that we can meaningfully think of an
individual quantum particle as having a well-defined ‘perspective’ or
‘reference frame’ and a well-defined reading on its internal clock at times in
between measurements is a highly nontrivial interpretational assumption. And
if we don’t think of particles as having well-defined states, properties or
perspectives at all times, then it seems we must take a somewhat deflationary
view of the quantum equivalence principle - i.e. we must regard it as a
heuristic principle which guides the predictions we make for actual
measurements, rather than as a literal description of physics within
inaccessible reference frames.

###  5.3 Example: Temporal nonlocality again

Returning to the case of the final-measurement interpretation, let us consider
what would be required in that picture in order for a IQRF to correspond to a
physically and operationally meaningful reference frame. At the very least,
the IQRF must be equipped with some kind of register, such that we can perform
generalized measurements as discussed in section 4.1 which map the results of
measurements on the systems to records on the register. So for example in the
case where the reference frame is a quantum clock $C$, we should add a quantum
register $R$ to our description, and then measurements performed on some
system $S$ in order to describe its evolution relative to the reference frame
associated with clock $C$ can be represented as generalized measurements which
are added to the Hamiltonian constraint as terms of the form
$\delta(T_{C}-t)K_{SC}$, where as in section 4.1 the operator $K_{SC}$ copies
the reading on the clock $C$ plus the result of some measurement on the system
$S$ onto the register $R$. Such operations may be carried out at several
different times throughout history, so the full history state represents the
register associated with clock $C$ as being in a superposition of various
different histories, and what is written on the register when it is measured
at the end of time can be regarded as the observed time evolution of system
$S$ relative to clock $C$. Talk about the ‘evolution of system $S$ relative to
frame $F$’ then has a transparent operational interpretation: it can be
understood in terms of statements about what the register would be expected to
read if appropriate generalized measurements were performed and the register
were then measured at the end of time. Whether or not those measurements are
actually made, including the register explicitly in our description ensures
that descriptions of physics in the relevant reference frame have some
physical content.

One may then ask whether the altered definition of a ‘reference frame’
suggested by the final-measurement interpretation substantially changes any of
the results of the IQRF research programme. For example, consider the time
localisation scenario discussed in section 4.1.2. There we discussed the case
where only clock $C_{A}$ is associated with a reference frame, but ref [6]
also discusses a putative case where we have one reference frame associated
with each of the two clocks $C_{A}$ and $C_{B}$, and each reference frame is
delocalised relative to the other. For example, this can be done by having the
first clock $C_{A}$ be an atomic clock and the second clock $C_{B}$ be a clock
associated with the laboratory, where $C_{A}$ is entangled with some other
atomic clock from the perspective of the laboratory and the two atomic clocks
are sent through an interferometer in a gravitational field, thus providing
what ref [53] regards as a test of the ‘quantum equivalence principle.’ Now,
we take it that both clocks must be associated with registers if they are to
be regarded as reference frames, and for the moment let us also take it that
both clock systems are associated with _macroscopic_ registers $R_{A},R_{B}$.
We will also adopt the plausible assumption that the relation ‘is temporally
localised relative to’ is transitive and symmetric. Clearly it is also
reflexive, and therefore it defines an equivalence relation which partitions
events and objects into equivalence classes within which everything is
temporally localised relative to everything else. Now suppose event $1$ is
temporally localised relative to register $R_{A}$ and event $2$ is temporally
localised relative to register $R_{B}$. Since registers $R_{A}$ and $R_{B}$
are macroscopic they will presumably be temporally localised relative to one
another, as decoherence will very quickly destroy any coherent superpositions;
it follows that event $1$ is temporally localised relative to event $2$, and
vice versa. Thus while it certainly can be the case that some quantum event is
not temporally localised relative to some reference frame, as we saw in
section 4.1.2, it is less clear that one reference frame $R_{1}$ can be
delocalised relative to a second reference frame $R_{2}$ \- if we require that
both reference frames are equipped with _macroscopic_ registers, then they
will necessarily be localised with respect to one another, and therefore
events which are localised relative to one reference frame will necessarily
also be localised relative to the other.

Of course this logic does not necessarily follow if we allow that the
reference frames are associated with _quantum_ registers, which may indeed be
delocalised relative to one another. Arguably that is what is going on in ref
[53], where the internal degrees of freedom of the atomic clocks may in some
sense be regarded as registers. But recall that the final-measurement
interpretation requires that in order to be physically meaningful the
information stored in the registers must be stored in stable ongoing records,
and certainly the easiest way to achieve that is to make the registers
macroscopic. That said, it’s possible that the registers could be quantum
systems at the time of the generalized measurements and the information stored
in them could subsequently be transferred into a macroscopic register, so the
reference frames could be delocalised relative to one another at the time of
the measurement and obtain definite states only later on. Because the
relational variables describing events in frames which are not temporally
localised relative to one another will typically not commute, it will not in
general be possible to find a choice of basis for the final measurement such
that over many repetitions of this setup both events are always recorded as
occurring at a well-defined sharp time, but what we _will_ find is that if we
repeat this setup many times, when we measure in a basis which reveals a sharp
time for event $1$ then the time for event $2$ will exhibit some spread, and
when we measure in a basis which reveals a sharp time for event $2$ then the
time for event $1$ will exhibit some spread. Thus we conclude that within the
final-measurement interpretation, it is indeed possible to have two reference
frames delocalised relative to one another as argued by refs [6, 53], but in
this case the reference frames in question must be set up very carefully such
that their registers are quantum at the relevant time but the information
about time localisation is subsequently transferred into macroscopic degrees
of freedom.

This also suggests a way of expressing the quantum equivalence principle which
allows it to have indirect operational significance without requiring
measurements performed by actual observers to be in superposition relative to
one another. That is to say, rather than expressing the equivalence principle
in terms of observations, observers or measurements, we should formulate it
with respect to records in a register which are understood to be observed at a
time later than they were recorded. For example, rather than asserting that ‘
_The outcome of any local nongravitational experiment is independent of the
position and velocity of the freely falling quantum reference frame in which
it is performed_ ’[53] one could instead assert that the records expected to
result from performing generalised measurements on local non-gravitational
degrees of freedom within a IQRF are independent of the position and velocity
of the freely falling IQRF. This gives a clear (albeit indirect) operational
meaning to the assertion of equivalence which may indeed be applied even in
the case where the IQRFs in question are in superpositions relative to one
another, meaning that under this construal the quantum equivalence principle
can indeed be regarded as having content that goes beyond the classical
equivalence principle.

##  6 Relational superposition

We now move to an evaluation of one particularly intriguing outcome of the
IQRF research programme - the conclusion that superposition is _relational_.
That is to say, a system may be in a superposition in some fixed basis
relative to one reference frame but not another[5]. (Note that there are
several other ways in which superposition might be said to be relative; we
discuss how these types of relational superposition differ to the IQRF
approach in appendix D, but here we will focus on the IQRF notion that
superposition _in a fixed basis_ may be relative to the choice of internal
quantum reference frame). Similarly, the IQRF approach also tells us that
entanglement is relational - systems that are not entangled in one reference
frame may appear to be entangled in another reference frame, or the amount of
entanglement may change. The underlying explanation for this behaviour is the
fact that tensor decompositions are not in general invariant under quantum
reference frame transformations[15], so degrees of freedom may be shifted from
one subsystem to another under such transformations: therefore each IQRF
actually contains a different set of subsystems and thus inevitably these
different subsystems also exhibit different amounts of superposition and
entanglement.

Now, it may be tempting to draw strong foundational conclusions from the fact
that superposition in a fixed basis is relative to an internal reference
frame. For example, consider a laboratory containing a quantum particle which
has been prepared in a superposition of position basis states. The IQRF
framework tells us that systems don’t have a quantum state relative to
themselves, and hence by default the laboratory reference frame says that the
laboratory has a well-defined position; but in the quantum reference frame of
the _particle_ the laboratory is in a superposition of position basis states.
This would seem to have serious consequences for a range of interpretations of
quantum mechanics, because many interpretational approaches are founded
specifically on the conviction that macroscopic objects can’t be in
superpositions; for example, the de Broglie-Bohm interpretation[60],
spontaneous collapse models[61] and gravitational collapse models[62] are all
designed to avoid macroscopic superpositions. Of course, superposition is
always relative to a basis, so we should be more specific: what these models
actually aim to do is avoid superpositions in bases corresponding to
macroscopically distinct measurement outcomes. Thus these interpretations can
be jointly characterised by the mantra that ‘measurements have unique
outcomes,’ by contrast to Everett, QBist and Copenhagen-style interpretations,
which don’t always insist on unique measurement outcomes.

The notion that superpositions are relative to reference frames seems to pose
a threat to this mantra, because it now appears that there can be no fact of
the matter about whether measurements have unique outcomes, since uniqueness
of outcomes is defined only relative to a reference frame. One might therefore
begin to worry that any interpretation of quantum mechanics which insists on
unique measurement outcomes must break some kind of covariance principle. On
the plus side, one might begin to wonder if there is some novel interpretation
of quantum mechanics lurking in this vicinity - perhaps the fact that no
quantum system is ever in a superposition relative to itself is enough to
explain why we experience measurements as having unique outcomes, even though
from the perspective of other frames their outcomes may not be unique? For
example, Hardy seems to be suggesting something of this kind when he writes, ‘
_This offers some hope that macroscopic superpositions and entanglement based
nonlocality are gauge artefacts and will not truly be present amongst the
beables._ ’ [56]. However, an interpretation of quantum mechanics based on
this observation would necessarily have some of the same problems as the
Everett interpretation with regard to quantum probabilities. For suppose we
say that an observer who measures a particle superposed relative to the lab
reference frame ends up in a superposition relative to that reference frame,
but she herself perceives herself as making a single definite observation,
since she can’t be in a superposition relative to her own reference frame. But
then it seems hard to say anything concrete about which measurement outcome
the observer sees, and in particular it’s difficult to make sense of the
probability distribution over her measurement outcomes, since whatever she
does observe, it seems she must observe the same thing every time she performs
this measurement. Since we are not convinced that the Everettian probability
problem can be solved, we also have doubts about whether these similar
probability problems could be solved; so let us instead try to determine how
these assertions would fare in the context of a single-world realist
interpretation.

To start with let us use the final-measurement approach as an example.
Consider the case where a laboratory ‘sees’ a particle as being in a
superposition state relative to the position basis, whereas the particle
‘sees’ the laboratory as being in a superposition state relative to the
position basis. In the final-measurement approach, this means that there is no
possible choice of measurement on the register at the end of time which will
select a definite state both for the laboratory in the position basis and for
the particle in the position basis during this period. (Another way to see
this is to note that the relational observables associated with the position
states for the particle and the laboratory are non-commuting). Thus far, the
description looks entirely neutral between the particle and the laboratory,
which might in some sense be considered to vindicate the claim that
superposition is a relative matter, since it seems that either of the two
descriptions is valid up until to the point far in the future where the final-
measurement is actually performed and one of these options is selected.

But in fact this is not correct, because we do not have freedom to select
either a final measurement which resolves a definite state for the laboratory