# Spacetime Quantum Reference Frames and superpositions of proper times

Flaminia Giacomini<EMAIL_ADDRESS>Perimeter Institute for
Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, N2L 2Y5, Canada

###### Abstract

In general relativity, the description of spacetime relies on idealised rods
and clocks, which identify a reference frame. In any concrete scenario,
reference frames are associated to physical systems, which are ultimately
quantum in nature. A relativistic description of the laws of physics hence
needs to take into account such quantum reference frames (QRFs), through which
spacetime can be given an operational meaning.

Here, we introduce the notion of a spacetime quantum reference frame,
associated to a quantum particle in spacetime. Such formulation has the
advantage of treating space and time on equal footing, and of allowing us to
describe the dynamical evolution of a set of quantum systems from the
perspective of another quantum system, where the parameter in which the rest
of the physical systems evolves coincides with the proper time of the particle
taken as the QRF. Crucially, the proper times in two different QRFs are not
related by a standard transformation, but they might be in a quantum
superposition one with respect to the other.

Concretely, we consider a system of $N$ relativistic quantum particles in a
weak gravitational field, and introduce a timeless formulation in which the
global state of the $N$ particles appears “frozen”, but the dynamical
evolution is recovered in terms of relational quantities. The position and
momentum Hilbert space of the particles is used to fix the QRF via a
transformation to the local frame of the particle such that the metric is
locally inertial at the origin of the QRF. The internal Hilbert space
corresponds to the clock space, which keeps the proper time in the local frame
of the particle. Thanks to this fully relational construction we show how the
remaining particles evolve dynamically in the relational variables from the
perspective of the QRF. The construction proposed here includes the Page-
Wootters mechanism for non interacting clocks when the external degrees of
freedom are neglected. Finally, we find that a quantum superposition of
gravitational redshifts and a quantum superposition of special-relativistic
time dilations can be observed in the QRF.

##  1 Introduction

In both quantum theory and general relativity, spacetime is treated as an
abstract entity. In quantum theory, spacetime is considered as a fixed, non-
dynamical background providing an arena for physical phenomena. In general
relativity, spacetime is a dynamical quantity, which is influenced by a change
in the configuration of massive bodies. In both cases, spacetime coordinates
acquire meaning via rods and clocks, which identify a reference frame.
However, these rods and clocks, and hence reference frames, are typically
treated as idealised classical systems. In this sense, the description of
spacetime is not fully operational.

Reference frames are very useful to specify the point of view from which
observations are carried out. Although measurements are usually made in a
specific reference frame, all physical laws are formulated in a way that is
independent of the reference frame chosen, thanks to the principle of general
covariance. This means that there is no preferred reference frame.

The rods and clocks which specify the reference frames are physical systems,
and ultimately quantum in nature. Hence, an operational formulation of the
laws of physics requires being able to describe physical phenomena from the
point of view of such quantum systems, which can be in a superposition or
entangled with other physical systems. This is the main intuition behind the
notion of Quantum Reference Frames (QRFs).

QRFs have been studied in quantum information, quantum foundations, and
quantum gravity since 1967. In the quantum information literature [1, 2, 3, 4,
5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], they have been
used to overcome superselection rules, applied to quantum communication
scenarios, and related to invariance properties and symmetries of quantum
systems. In the quantum gravity literature, it has been conjectured that QRFs
are needed to formulate a quantum theory of gravity [22, 23], especially when
taking a relational perspective on spacetime [24]. In particular, a relational
approach to QRFs is common to both the quantum gravity and quantum information
approaches. Recently, Ref. [25] introduced a formalism to describe physics
from the point of view of a specific QRF and transformations to change the
description between different QRFs. Subsequent work developed this approach
further in different contexts. Refs. [26, 27, 28] adopt a symmetry principle
to describe a Galilean system in a fully relational way, and Refs. [29, 30]
extend the formalism to the special relativistic regime, applying it to
concrete problems in Relativistic Quantum Information. Other works [31, 32,
33] have focussed on the group properties of QRFs both in discrete and in
continuous-variable systems. A time version of QRFs, also related to the Page-
Wootters mechanism [34, 35], has been introduced in Ref. [36] for interacting
clocks. Recently, a generalisation of the formalism has been applied to to
quantum systems living on a superposition of gravitational fields, and showed
that the Einstein Equivalence Principle can be extended to QRFs [37].

The work on QRFs is related to research on quantum clocks, which have been
studied with a relational approach, both in the interacting and non-
interacting case, in Refs. [38, 39, 40, 41, 42, 43]. In particular, Ref. [43]
studied relativistic clocks by extending the Page-Wootters approach to a
relativistic particle with internal and external degrees of freedom. A
conceptually similar approach to QRFs, but with some important differences, is
the quantum coordinate systems [44, 45].

All these works have studied QRFs either in space or in time, and separately
considered the role of internal and external variables (such as, respectively,
position or momentum and internal quantities acting as clocks) in identifying
the QRF. So far, no unifying approach to QRFs in spacetime has been
formulated. A spacetime formulation of QRFs is a crucial step towards a fully
relational and covariant description of physics from the point of view of a
quantum system, and could also provide a natural setting for a quantum
approach to time. However, achieving such a formulation faces some important
challenges, such as describing space and time in quantum theory on an equal
footing. This task requires the incorporation of a time operator into the
description of QRFs in space, and is conceptually related to the problem of
time in quantum gravity [46, 47, 48]. The methodology and the scope of the
present work are different to those employed in the canonical approaches to
the problem of time.

In this work, we overcome the difficulty of incorporating a time operator into
QRFs and develop a model for spacetime QRFs. We introduce a timeless and fully
relational formulation, which methodologically adopts techniques from
different approaches to QRFs [25, 26, 27, 36, 37], unifying them by adopting
aspects of Covariant Quantum Mechanics [49, 50, 51] and the Page-Wootters
mechanism [34, 35]. In particular, we consider a set of $N$ relativistic
quantum particles in a weak gravitational field, each of which has a quantum
state living in the tensor product Hibert space of position/momentum and some
internal “clock” Hilbert space (see Refs. [52, 43] for a similar analysis in a
different context). In this timeless formulation, the global state of the $N$
particles is “frozen”, but the dynamical evolution is recovered in terms of
the relational variables to one of the particles, which is chosen as the QRF.
In order to obtain this relational description on the reduced set of $N-1$
particles, we build a transformation to some relational variables (e.g.,
relative positions) of the particle chosen as the QRF, and then fix the origin
of the QRF at the location of this particle. The evolution of the remaining
$N-1$ particles is parametrised in terms of the proper time of the particle
serving as the QRF. Concretely, the proper time of each particle is encoded in
its internal variables, which serve as quantum clocks. Hence, both the
external and the internal variables play an important role in buiding the
spacetime QRF: the former allow us to transform to the local frame of a
quantum particle, whose spacetime position can be in a superposition or
entangled with other systems, and the latter allow us to describe the
dynamical evolution of the remaining particles in terms of the proper time in
each QRF. Notice that, although proper time is a standard evolution parameter
in each QRF, the proper times of two different quantum particles are not in a
classical relation one with respect to the other, but can be in a quantum
superposition. This joint use of the external variables to fix the QRF and the
internal variables as clocks is one of the novel aspects of this work.

In addition, we show that we can always find a transformation to the QRF of
the particle such that the metric is locally inertial at the origin of the
QRF. This is an instance of a Quantum Locally Inertial Frame (QLIF), as
introduced in Ref. [37]. We find the relational dynamics of the remaining
particles from the point of view of the chosen QRF, which coincides, in the
appropriate limit, with the dynamical evolution predicted with different
methods (for instance, see Refs. [53, 54, 55, 56]). The expression of this
relational dynamics has the same functional form in any chosen QRF, and is
thus symmetric under QRF transformations, as defined in Ref. [25].

Finally, we find that this description of QRFs in spacetime allows us to
observe a _superposition of special relativistic time dilations_ and a
_superposition of gravitational redshifts_ , which could be measured
experimentally in the future.

Overall, our model paves the way for a full extension of the formalism of QRFs
to arbitrary spacetimes, and superpositions thereof. This is an important step
to achieve a fully relational formulation of physics on a nonclassical
spacetime, which is a key requirement to formulate a quantum theory of
gravity.

The structure of the paper is as follows. In Section 2, we introduce the model
for spacetime QRFs: we start from a fully constrained description of the
system, which is neutral to the specific perspective, and is encoded in a set
of first-class constraints. We then show how to recover a relational
description from the perspective of one of the particles. In Section 3, we
show that this construction leads to an extension of the spacetime symmetries
to the set of QRF transformations. This result generalises to the spacetime
picture and to quantum clocks the notion of extended symmetry transformation
introduced in Ref. [25]. In Section 4, we introduce a measurement model and
show how the phenomenon of _superposition of special relativistic time
dilation_ and _superposition of gravitational redshift arises_. For clarity,
the technicalities are kept to the essential in the main text, however the
relevant calculations are detailed in the Appendices, which are referred to
where appropriate.

##  2 From a covariant timeless model to relational dynamics in a locally
inertial quantum reference frame

[scale=0.35]GlobalModel

Figure 1:  We consider a system of $N$ relativistic quantum particles in a
weak gravitational field produced by a mass $M$. Each particle has a quantum
state representing its position in the spacetime diagram, as well as a clock
state (the hands of the clocks) which keeps the proper time in the particle’s
frame. We introduce a timeless model, in which the global state of the $N$
particles and of the mass $M$ is “frozen”, and is described by an $N$-particle
quantum state in spacetime. Concretely, this means that the quantum state of
the external variables can be, for instance, in a quantum superposition of
coordinate times $x^{0}$ and spatial coordinates $\mathbf{x}$. Classically,
the position in spacetime and the velocity of a particle influence its proper
time due to relativistic time dilation. When the particle is in a quantum
superposition of positions or velocities, the proper time displayed by its
internal clock is also in a quantum superposition from an external
perspective.
While, at the global level, the system does not evolve, the dynamical
evolution is recovered in terms of the relational variables between one of the
particles, which is chosen as the quantum reference frame (QRF), and the rest
of the particles. To transform to the QRF of one of the particles, we first
map the initial spacetime coordinates to the relational spacetime coordinates
from the perspective of the particle chosen as the QRF. We then find a
transformation which makes the metric locally inertial at the origin of the
QRF. Finally, we use the proper time of the particle to parametrise the
dynamics of the remaining particles from the perspective of the chosen QRF. In
such QRF, the dynamical evolution of the remaining particles is described in
terms of a Hamiltonian operator and is parametrised by the clock’s proper
time, which is just a classical parameter in the clock’s rest frame.

In this section, we introduce a fully relational model for $N$ quantum
particles in spacetime, where

  * •

Space and time are treated on equal footing, as in special and general
relativity, also when the systems under study are quantum, and not classical;

  * •

Time evolution emerges from internal correlations of physical clocks, which
are described as living in the internal Hilbert spaces of the particles. The
way in which the dynamical laws are recovered from the perspective of such
clocks incorporates the Page-Wootters formulation [34, 35];

  * •

The formalism is fully relational, i.e., all external spacetime structure is
eliminated.

Furthermore, we show that it is possible to take the perspective of one of
these quantum particles, and how the transformation to such QRF is
constructed.

We consider a system of N particles of mass $m_{I}$, for $I=1,\cdots,N$, each
of which has a quantum state living in the tensor product of an external
Hilbert space, representing its position in spacetime or momentum, and an
internal Hilbert space (the clock), and a mass $M$ which is the source of the
gravitational field, as illustrated in Fig. 1. The ideas we present in this
paper are described in their simplest form in the time and radial components
of a model with spherical symmetry. To simplify the presentation, we work in
$1+1$ dimensions, which allows us to neglect the angular part of the model. We
adopt a timeless formulation [34, 49, 50, 51, 35] describing the full state of
the particles. The intuitive idea behind these formulations is that the
dynamics of particles is an emergent phenomenon stemming from the relational
degrees of freedom. Formally, this means that the state of the $N$ particles
and of the mass $M$ does not evolve, but the dynamics is recovered by
“conditioning” on one of the systems considered. The global state of the $N$
particles and of the mass $M$ — called the _physical state_ — is then a
“frozen” (i.e., non-evolving) state $\ket{\Psi}_{ph}$ satisfying the relation

$\hat{C}\ket{\Psi}_{ph}=0,$ (1)

where $\hat{C}$ is a set of first-class constraints defined on the Hilbert
space of the $N$ particles and of the mass $M$. This fully constrained model
is not reference-frame specific, but corresponds to a neutral perspective. The
set of constraints encodes both the dynamics of the particles in spacetime and
the global symmetries (global translations in space and time) of the model. In
this Section, we discuss in detail the form of the constraints and their
physical meaning.

Differently to other formulations for QRFs, we here describe both the quantum
state associated to the position/momentum of the particle and the clock state,
and they both play a role in the identification of the QRF. In particular,
when one of the particles, say particle $1$, acts as a QRF, the external part
of the quantum state is used to fix the QRF (in our case, the origin of the
coordinate system in spacetime), while the internal part serves as a clock,
ticking according to the proper time in each particle’s frame. We will see
that imposing the constraints corresponds to removing absolute (or external)
variables from the description, and that the relational variables, defined
from the point of view of one of the physical systems considered, evolve
dynamically in a non-trivial way. In particular, the presence of the internal
clocks is crucial to recover the relational dynamics of the rest of the
particles from the point of view of the particle chosen as the QRF.

Throughout the paper, we use the following notation: Greek letters indicate
spacetime labels, i.e., $\mu=0,1$, while capital Latin letters label the
particles from $1$ to $N$. Vectors with no indices are, unless differently
specified, two-vectors in $1+1$ spacetime, i.e., $v=(v^{0},\mathbf{v})$, where
$v^{0}$ is the time component of the vector and $\mathbf{v}$ denotes the
spatial component of the vector. We use the Einstein’s convention on all sums,
unless explicitly stated.

We take the particles to be special-relativistic and in a Newtonian field
generated by a mass $M$. We describe the latter as the weak-field limit of a
more general gravitational field, corresponding to a metric

$\begin{split}&g_{00}=1+\frac{2\Phi(\mathbf{x}-\mathbf{x}_{M})}{c^{2}};\\\
&g_{01}=g_{10}=0;\\\ &g_{11}=-1,\end{split}$ (2)

where $\Phi(\mathbf{x})$ is the Newtonian potential due to the mass $m_{M}$ of
the system $M$ and $|\Phi(\mathbf{x})|/c^{2}\ll 1$ in the spacetime region
considered.

The external Hilbert space of each particle $I=1,\cdots,N$ is
$\mathcal{H}_{I}\simeq L^{2}(\mathbb{R}^{2})$. The mass $M$ is also assigned a
Hilbert space $\mathcal{H}_{M}\simeq L^{2}(\mathbb{R}^{2})$. The motion of any
of the $N$ particles in this external, weak gravitational field sourced by the
mass $M$ can be encoded in the expression

$\hat{C}_{I}=\sqrt{g^{00}(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M})}\hat{p}^{I}_{0}-\hat{\omega}_{p}^{I},$
(3)

where
$[\hat{x}^{\mu}_{I},\hat{p}_{\nu}^{I}]=\mathrm{i}\hbar\delta^{\mu}_{\nu}$,
$\hat{\omega}_{p}^{I}=\sqrt{m_{I}^{2}c^{2}+\hat{\mathbf{p}}^{2}_{I}}$ with
$I=1,\cdots,N$, and
$[\hat{x}^{\mu}_{M},\hat{p}_{\nu}^{M}]=\mathrm{i}\hbar\delta^{\mu}_{\nu}$. The
constraints $\hat{C}_{I}$ of Eq. (3) straightforwardly follow from the
quantisation of the (classical) general-relativistic dispersion relation
$g^{\mu\nu}(x)p_{\mu}p_{\nu}-m^{2}c^{2}=0$ thanks to the weak-field
approximation, which is crucial to avoid ordering ambiguities. Notice that,
for simplicity, we assume that the state of the mass $M$ does not undergo any
dynamical evolution in this initial description, hence we do not associate a
dynamical constraint to it. In the following we show that the mass acquires a
dynamical evolution as a result of taking the perspective of one of the
particles.

By applying Eq. (3) to a quantum state $\ket{\psi_{I}}_{ph}$ that solves the
constraint, i.e., $\hat{C}_{I}\ket{\psi_{I}}_{ph}=0$ and conditioning on the
time coordinate $\ket{x^{0}_{I}}_{I}$ via the procedure
$\braket{x^{0}_{I}}{\hat{C}_{I}}{\psi_{I}}_{ph}=0$, it is possible to recover
the Schrödinger equation for a quantum particle in the Newtonian field (see
Appendix A for details). Hence, the dynamical evolution emerges from the
entanglement, at the level of $\ket{\psi_{I}}_{ph}$, between the time and
spatial degrees of freedom.

We are working here in a perturbative regime in which the commutator
$\left[\sqrt{g_{00}(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M})},\hat{\omega}_{p}^{I}\right]$
is negligible. This is equivalent to demanding that all terms which are at
least of the order
$\frac{\Phi(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M})\hat{\mathbf{p}}^{2}_{I}}{m_{I}^{2}c^{4}}$
are negligible whenever they are applied to the quantum state of each particle
$I$. Notice that we retain terms of the order
$\left[\hat{\mathbf{p}}_{I}/(m_{I}c)\right]^{4}$, corresponding to special-
relativistic corrections in the velocity of the particles.

The constraint $\hat{C}_{I}$ encoding the dynamical evolution of the particle,
which holds for each particle of the set we consider, is not yet cast in
relational terms. The introduction, compared to the standard quantum-
mechanical formalism, of the $\hat{x}_{I}^{0}$ and $\hat{p}_{0}^{I}$
operators, such that
$[\hat{x}_{I}^{0},\hat{p}_{0}^{J}]=\mathrm{i}\hbar\delta_{I}^{J}$ is necessary
to treat space and time on the same footing, as in the case of Covariant
Quantum Mechanics [49, 50]. In order to enforce the relational character of
the formalism, we need to eliminate the external structure. In Appendix B we
review and compare different relational approaches.

In the following, we show how to build a fully relational model for quantum
particles in spacetime and in a weak gravitational field. Methodologically,
this construction adopts the tools of QRFs, and unifies the framework for
spatial QRFs [25, 26, 27] with time reference frames [36], the latter in the
non-interacting case. The resulting model utilises techniques of Covariant
Quantum Mechanics in order to recover a full spacetime covariance, and
incorporates both external and internal degrees of freedom.

In particular, we consider two constraints on top of the $N$ constraints of
Eq. (3), one for each spacetime dimension, which encode the conservation of
the total energy and the total momentum of our $N$-particle model. Thus, we
write

$\begin{split}&\hat{f}^{0}=\sum_{I=1}^{N}\left[\hat{p}^{I}_{0}+\Delta(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M},\hat{\mathbf{p}}_{I})\frac{\hat{H}_{I}}{c}\right]+\hat{p}_{0}^{M};\\\
&\hat{f}^{1}=\sum_{I=1}^{N}\hat{\mathbf{p}}_{I}+\hat{\mathbf{p}}_{M},\end{split}$
(4)

where
$\Delta(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M},\hat{\mathbf{p}}_{I})=\sqrt{g_{00}(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M})}\left(1+\frac{\hat{\mathbf{p}}^{2}_{I}}{m_{I}^{2}c^{2}}\right)^{-1/2}$.
The constraint $\hat{f}^{1}$ corresponds to the global translational
invariance required in Refs. [26, 27]. The constraint $\hat{f}^{0}$
corresponds to the conservation of the total energy. Notice that particle $M$
counts towards the total energy and momentum balance because, although its
state does not obey a dynamical constraint, it is a quantum system in its own
right, with energy and momentum associated to it. Such quantum state in
spacetime should be chosen so that it corresponds to a physical configuration
of the mass $M$ which does not dynamically evolve in the initial coordinates.

The constraint $\hat{f}^{0}$ is derived by summing all the contributions to
the energies (up to a multiplicative factor of $c$) coming from the motion of
the particles in spacetime, from the internal Hamiltonian $\hat{H}_{I}$,
living in the Hilbert space $\mathcal{H}_{C_{I}}\simeq L^{2}(\mathbb{R})$ of
the $N$ particles, and from the system $M$. In particular, the factor in front
of the internal energy Hamiltonian corresponds to the relativistic time
dilation from the rest frame to an arbitrary frame. The relation between the
rest frame and an external frame to the particle is quantum, because the
particles have a quantum state associated to the position/momentum, which can
be in a quantum superposition of positions and velocities. QRF techniques
justify operationally the change between the rest frame of a quantum system
and an arbitrary frame [25, 29, 30]. The relation between the internal
energies in the rest frame and in an arbitrary frame is encoded in the
operator
$\Delta(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M},\hat{\mathbf{p}}_{I})$,
which coincides with the worldline element of particle $I$. In Appendix C we
provide an explicit derivation of the operator
$\Delta(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M},\hat{\mathbf{p}}_{I})$.

The two constraints $\hat{f}^{0}$ and $\hat{f}^{1}$, being first-class
constraints, generate a gauge transformation on the canonically conjugated
variables. These constraints have the effect of reducing the number of degrees
of freedom of the full system to obtain the dynamics of the relative positions
between the particles. In the following we show that, by picking one of the
particles as the preferred point of view from which we describe the dynamics
of the remaining particles, the resulting dynamics can be seen as the dynamics
of the $N-1$ particles and the mass $M$ from the point of view of the chosen
particle.

We then see that the introduction of the zero component of the coordinate and
momentum operators not only allows us to treat the $N$-particle system in a
covariant way, but allows us to generalise the symmetry principle of Refs.
[26, 27] to all the spacetime components and to show the connection with the
Page-Wootters inspired approach of Ref. [36]. When only the internal state of
the particles is considered, the constraint $\hat{f}^{0}$ reduces to the Page-
Wootters constraint of the clocks. We stress here that the constraint
$\hat{f}^{0}$ can be equivalently seen as arising from a Page-Wootters
construction, which emphasises the relational approach to time, or from a
relational formulation of physics, which eliminates the background structure
by enforcing that the total energy is zero.

As a result of this construction, we have a fully constrained model describing
the $N$ particle system, where the full constraint is

$\hat{C}=\sum_{I=1}^{N}\mathcal{N}_{I}\hat{C}_{I}+z_{\mu}\hat{f}^{\mu},$ (5)

where $\mathcal{N}_{I}$, with $I=1,\cdots,N$ and $z_{\mu}$, with $\mu=0,1$ are
Lagrange multipliers. Notice that, to our order of approximation, all
constraints are first-class, i.e., they at least weakly commute with each
other. In the two cases $\left[\hat{C}_{I},\hat{f}^{1}\right]=0$ and
$\left[\hat{f}^{0},\hat{f}^{1}\right]=0$, the relation holds exactly. For the
commutator $\left[\hat{C}_{I},\hat{f}^{0}\right]=0$, the result can be easily
obtained by remembering that we are working in a regime where
$\left[g_{00}(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M}),\hat{\omega}_{p}^{I}\right]=0$
for all $I=1,\cdots,N$. Notice that, thanks to the fact that all the
constraints commute, it is possible, as we will show, to straightforwardly
build a state that satisfies all of them.

Here, we only consider non-interacting clocks in order to maintain the
commutative property of the constraints. Had we allowed for gravitational
interaction between the particles, we would have had non-commutative
constraints (namely, second-class constraints), which require a different
treatment to what is outlined here. We thus leave this question for future
work.

The physical state of Eq. (1) with the constraint $\hat{C}$ of Eq. (5) can
equivalently be written as

$\ket{\Psi}_{ph}\propto\int
d^{N}\mathcal{N}d^{2}ze^{\frac{\mathrm{i}}{\hbar}\mathcal{N}_{I}\hat{C}_{I}}e^{\frac{\mathrm{i}}{\hbar}z_{\mu}\hat{f}^{\mu}}\ket{\phi},$
(6)

where $\ket{\phi}$ is an arbitrary state that can be expanded in a basis as

$\ket{\phi}=\int\Pi_{I}\left[d\mu(x_{I})dE_{I}\right]d^{2}x_{M}\phi(x_{1},\cdots,x_{N},x_{M},E_{1},\cdots,E_{N})\ket{x_{1},\cdots,x_{N},x_{M}}\ket{E_{1},\cdots,E_{N}},$
(7)

with $d\mu(x_{I})=\sqrt{g_{00}(\mathbf{x}_{I}-\mathbf{x}_{M})}d^{2}x_{I}$
being the covariant integration measure and
$\ket{x_{I}}=\ket{x_{I}^{0},\mathbf{x}_{I}}$. The states $\ket{E_{I}}$ are the
eigenstates of the internal hamiltonian of each particle $I$, such that
$\hat{H}_{I}\ket{E_{I}}=E_{I}\ket{E_{I}}$. The state $\ket{\phi}$, in general,
is not a solution of the constraint. However, the state that solves the
constraint $\hat{C}\ket{\Psi}_{ph}=0$ can be obtained via the improper
projection $\hat{P}\ket{\phi}=\ket{\Psi}_{ph}$ defined in Eq. (6), where
$\hat{P}=\int
d^{N}\mathcal{N}d^{2}ze^{\frac{\mathrm{i}}{\hbar}\mathcal{N}_{I}\hat{C}_{I}}e^{\frac{\mathrm{i}}{\hbar}z_{\mu}\hat{f}^{\mu}}$.
The procedure to obtain the state $\ket{\Psi}_{ph}$ involves a technical
subtlety, due to the fact that $\hat{P}^{2}\neq\hat{P}$. For instance, in
situations in which the spectrum of the constraint is continuous around zero,
the solution the set of states satisfying the constraint is empty. However,
this problem can be solved, e.g., by redefining the inner product in order to
renormalise the state $\ket{\Psi}_{ph}$ (for details and different approaches
to the solution see, e.g., [57, 58, 59, 60]).

In order to take the perspective of a specific particle, e.g., particle $1$,
we need to first map the phase-space observables to the relational observables
to particle $1$, and then enforce that particle $1$ is in the origin of the
reference frame via a projection operation. This procedure extends the
technique introduced in Ref. [26, 27] to our model for $N$ relativistic
particles in a weak gravitational field. Overall, we obtain the relational
state from the perspective of particle $1$ to be

$\ket{\psi}^{(1)}={}_{1}\bra{q_{1}=0}\hat{\mathcal{T}}_{1}\ket{\Psi}_{ph},$
(8)

where we choose

$\hat{\mathcal{T}}_{1}=e^{\frac{i}{\hbar}\frac{\log\sqrt{g_{00}(\hat{\mathbf{x}}_{M})}}{2}\sum_{I=1}^{N}(\hat{x}^{0}_{I}\hat{p}_{0}^{I}+\hat{p}_{0}^{I}\hat{x}^{0}_{I})}e^{\frac{\mathrm{i}}{\hbar}\hat{\mathbf{x}}_{1}\left(\hat{f}^{1}-\hat{\mathbf{p}}^{1}\right)}e^{\frac{\mathrm{i}}{\hbar}\hat{x}_{1}^{0}\left(\hat{f}^{0}-\hat{p}^{1}_{0}\right)},$
(9)

and we have defined the positions after the transformation
$\hat{\mathcal{T}}_{1}$ as $q_{I}$.

The operator $\hat{\mathcal{T}}_{1}$ maps the initial positions to the
relative positions to particle $1$. Its construction can be intuitively
explained by distinguishing two parts of the operator. The two rightmost terms
map the spacetime coordinates $\hat{x}_{\ell}$, with $\ell=2,\cdots,N,M$ to
the relative coordinates to particle $1$, i.e.,
$\hat{x}_{\ell}-\hat{x}_{1}\mapsto\hat{x}_{\ell}$. The last term on the left,
which is a function of the metric, sets the metric field to a locally inertial
metric field at the location of particle $1$. In order to achieve this, the
metric at the location of particle $1$ is mapped to the Minkowski metric, and
the motion of the other particles is encoded in a new constraint

$\hat{C}^{\prime}_{i}=\hat{\mathcal{T}}_{1}\hat{C}_{i}\hat{\mathcal{T}}_{1}^{\dagger}=\sqrt{\frac{g_{00}(\hat{\mathbf{q}}_{i}-\hat{\mathbf{q}}_{M})}{g_{00}(\hat{\mathbf{q}}_{M})}}\hat{k}_{0}^{i}-\hat{\omega}_{k}^{i},$
(10)

where $i=2,\cdots,N$, $\hat{q}_{i}$, $\hat{q}_{M}$ being the (spacetime)
position operators of the particles relative to particle $1$ in the new
coordinate system and $\hat{k}_{i}$, $\hat{k}_{M}$ the momentum operators that
are canonically conjugated respectively to $\hat{q}_{i}$ and $\hat{q}_{M}$,
i.e.,
$\left[\hat{q}_{i}^{\mu},\hat{k}_{\nu}^{j}\right]=\mathrm{i}\hbar\,\delta^{\mu}_{\nu}\,\delta_{ij}$
and
$\left[\hat{q}_{M}^{\mu},\hat{k}_{\nu}^{M}\right]=\mathrm{i}\hbar\,\delta^{\mu}_{\nu}$
(see Appendix D for the complete action of the operator
$\hat{\mathcal{T}}_{1}$ on the phase-space operators). We hence find that

$g^{\prime}_{00}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M})=\hat{\mathcal{T}}_{1}\frac{g_{00}(\hat{\mathbf{x}}_{i}-\hat{\mathbf{x}}_{M})}{g_{00}(\hat{\mathbf{x}}_{1}-\hat{\mathbf{x}}_{M})}\hat{\mathcal{T}}_{1}^{\dagger},$
(11)

is the new metric field in the perspective of particle $1$. In summary, the
transformation $\hat{\mathcal{T}}_{1}$ realises a transformation to the
Quantum Locally Inertial Frame (QLIF) centred in particle $1$, in the spirit
of Ref. [37]. A different choice of the operator $\hat{\mathcal{T}}_{1}$ is
possible, and would correspond to a different set of relational variables (and
coordinate system).

By performing a lengthy calculation (which is detailed in Appendix E), it is
possible to show that the state of Eq. (8) can be cast as

$\ket{\psi}^{(1)}=\int
d\tau_{1}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}\tau_{1}}\ket{\psi^{(1)}_{0}}\ket{\tau_{1}},$
(12)

where $\ket{\tau_{1}}$ is the state of the internal clock $1$ such that, given
the internal time operator $\hat{T}_{I}$,
$\hat{T}_{I}\ket{\tau_{I}}=\tau_{I}\ket{\tau_{I}}$ and
$\left[\hat{T}_{I},\hat{H}_{J}\right]=\mathrm{i}\hbar\delta_{IJ}$, the
Hamiltonian $\hat{H}^{(1)}$ is

$\hat{H}^{(1)}=\hat{\gamma}_{\Sigma
p,1}\sum_{i}\sqrt{g^{\prime}_{00}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M})}\left\\{c\hat{\omega}_{k}^{i}+\hat{\gamma}_{i}^{-1}\hat{H}_{i}\right\\}+c\hat{\gamma}_{\Sigma
k,1}\sqrt{g^{00}(\hat{\mathbf{q}}_{M})}\hat{k}_{0}^{M}+m_{1}c^{2}\hat{\gamma}_{\Sigma
k,1}^{2},$ (13)

with $\hat{\gamma}_{\Sigma
k,1}=\sqrt{1+\frac{\left(\sum_{i}\hat{\mathbf{k}}_{i}+\hat{\mathbf{k}}_{M}\right)^{2}}{m_{1}^{2}c^{2}}}$,
$\hat{\gamma}_{i}=\sqrt{1+\frac{\hat{\mathbf{k}}_{i}^{2}}{m_{i}^{2}c^{2}}}$.
Finally, the state $\ket{\psi^{(1)}_{0}}$ formally plays the role of an
initial state, whose relation with the state $\ket{\phi}$ is given in Appendix
E. The state of Eq. (12) can be interpreted as a “history state”, namely a
state that associates to any arbitrary time eigenstate $\ket{\tau_{1}}$ read
in the QRF of particle $1$ a state
$\ket{\psi^{(1)}_{\tau_{1}}}=e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}\tau_{1}}\ket{\psi^{(1)}_{0}}$
of the remaining particles at the time $\tau_{1}$ read by the clock. A further
projection on the state of the clock $1$, i.e.,
$\braket{\tau_{1}}{\psi^{(1)}}$, recovers the standard description of quantum
mechanics.

From the expression in Eq. (13), we see that the Hamiltonian of the particles
$i=2,\cdots,N$ is affected by the presence of particle $1$. On the one hand,
this is because the metric field at the location of the other particles, in
the QLIF of particle $1$, is a quantum operator depending also on the initial
value of the metric field at the location of particle $1$, as it is clear by
looking at Eq. (11). This is an expression of the relational character of
general relativity, extended to when we consider quantum particles as QRFs:
the metric field in each QRF is defined purely in terms of relational
quantities (operators) of the particles and the reference frame. On the other
hand, there is also a contribution from the special-relativistic time-dilation
operator. This contribution is due to the motion of particle $1$ and appears
as the operator $\hat{\gamma}_{\Sigma k,1}$. The interpretation and the
operational consequences of this result are discussed in Section 4.

In this Section, we have outlined the method to reduce to the perspective of
one of the particle in the general model that we have introduced. However,
this model has a well-defined limit to the Galilean free-particle case, the
special relativistic case, and the Newtonian case of slowly-moving particles.
In all these cases, a similar procedure to what is described in this Section
can be defined. All these limiting cases are detailed in Appendix F.

##  3 Extended symmetries and QRF changes

The construction we have introduced in the previous section is compatible with
an extended notion of spacetime symmetries. This means that every quantum
particle can equally serve as a QRF, and that no preferred QRF is singled out.
This is easy to see by noticing that the procedure we have followed to
transform to the QLIF of particle $1$ can be repeated if we take another
particle, i.e., particle $2$, as the QRF. In this case, we obtain the same
result, but with all indices $1$ and $2$ swapped. The “history state” from the
point of view of particle $2$ is then

$\ket{\psi}^{(2)}=\int
d\tau_{2}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(2)}\tau_{2}}\ket{\psi^{(2)}_{0}}\ket{\tau_{2}},$
(14)

where all quantities are defined as in the previous section, and

$\hat{H}^{(2)}=\hat{\gamma}_{\Sigma u,2}\sum_{k\neq
2}\sqrt{g^{\prime}_{00}(\hat{\mathbf{r}}_{k},\hat{\mathbf{r}}_{M})}\left\\{c\hat{\omega}_{u}^{k}+\hat{\gamma}_{u}^{-1}\hat{H}_{k}\right\\}+c\hat{\gamma}_{\Sigma
u,2}\sqrt{g^{00}(\hat{\mathbf{r}}_{M})}\hat{u}_{0}^{M}+m_{2}c^{2}\hat{\gamma}_{\Sigma
u,2}^{2},$ (15)

with $\hat{\gamma}_{\Sigma u,2}=\sqrt{1+\frac{\left(\sum_{k\neq
2}\hat{\mathbf{u}}_{k}+\hat{\mathbf{u}}_{M}\right)^{2}}{m_{2}^{2}c^{2}}}$.
Here, $\hat{r}_{k}$, with $k=1,3,\cdots,N$, and $\hat{r}_{M}$ are the position
operators in the QRF of particle $2$, and $\hat{u}_{k}$, $\hat{u}_{M}$ their
conjugate momenta.

This construction then induces an invertible transformation from the reduced
“history state” of Eq. (8) to the reduced “history state” of Eq. (14) which
amounts to changing the QRF from the particle $1$ to the particle $2$. It is
then clear by construction, and can also be checked by direct computation,
that the state of Eq. (14) is obtained from Eq. (8) via the operation

$\ket{\psi}^{(2)}={}_{2}\bra{r_{2}=0}\hat{\mathcal{T}}_{12}\ket{\psi}^{(1)}\otimes\ket{p_{1}=0}_{1},$
(16)

where
$\hat{\mathcal{T}}_{12}=\hat{\mathcal{T}}_{2}\hat{\mathcal{T}}_{1}^{\dagger}$
(see Appendix D for the action of this operator on the phase space operators).
This construction parallels the one introduced in Ref. [26].

We have thus found that the Hamiltonian from the perspective of particle $1$
is mapped to a Hamiltonian that is form-invariant, but where all labels $1$
and $2$ are swapped, by a reversible QRF transformation. Thus, we have
generalised the notion of an extended symmetry introduced in Ref. [25] to the
full spacetime picture and quantum relativistic particles in a weak
gravitational field.

##  4 Gravitational redshift and relativistic time-dilation from a QRF

a) [scale=0.4]MeasurementSup b) [scale=0.4]SupTime

Figure 2:  We depict the particular situation in which particle $2$ ($C_{2}$
in the picture) evolves in a superposition of two semi-classical trajectories
from the perspective of particle $1$ ($C_{1}$ in the picture). In the most
general case, the state of particle $2$ from the point of view of particle $1$
is arbitrary. We describe a measurement performed at time $\tau_{2}^{*}$ in
the frame of clock $2$ from the point of view of $C_{1}$. a) The measurement,
which is localised in time in the frame of $C_{2}$, appears delocalised in
time in the frame $C_{1}$. In particular, it is performed in a superposition
of proper times $\tau_{1}^{\prime}$ and $\tau_{1}^{\prime\prime}$ in the
proper time of $C_{1}$. The two times $\tau_{1}^{\prime}$ and
$\tau_{1}^{\prime\prime}$ are related to $\tau_{2}^{*}$ by the expressions
$\tau_{1}^{\prime}=\Delta_{12}^{-1}(W_{1})\tau_{2}^{*}$ and
$\tau_{1}^{\prime\prime}=\Delta_{12}^{-1}(W_{2})\tau_{2}^{*}$, where
$\Delta_{12}^{-1}(W_{i})$, $i=1,2$, encodes the special-relativistic time
dilation or the gravitational redshift evaluated on the worldline $W_{i}$. b)
In general, $C_{1}$ “sees” the clock $C_{2}$ as ticking in a superposition of
times, depending on the state of the external degrees of freedom.
Specifically, this effect can be understood as a superposition of special-
relativistic time dilations, when the particles move at relativistic
velocities in a Minkowski background, or as a superposition of gravitational
redshifts, when the particles move slowly in a weak gravitational field.

We now restrict our consideration to $N=2$ particles and a source mass $M$ and
introduce the possibility of performing a measurement in the QRF of particle
$2$. Analogously to the measurement procedure of Ref. [36], we modify the
constraint $\hat{f}^{0}$ as

$\begin{split}&\hat{C}_{I}=\sqrt{g^{00}(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M})}\hat{p}^{I}_{0}-\hat{\omega}_{p}^{I}\qquad\text{for}\qquad
I=1,2;\\\
&\hat{f}_{Q}^{0}=\hat{p}^{1}_{0}+\hat{p}^{2}_{0}+\hat{p}_{0}^{M}+\Delta(\hat{\mathbf{x}}_{1}-\hat{\mathbf{x}}_{M},\hat{\mathbf{p}}_{1})\frac{\hat{H}_{1}}{c}+\Delta(\hat{\mathbf{x}}_{2}-\hat{\mathbf{x}}_{M},\hat{\mathbf{p}}_{2})\left[\frac{\hat{H}_{2}}{c}+\delta(\hat{T}_{2}-\tau_{2}*)\frac{\hat{Q}_{2}}{c}\right];\\\
&\hat{f}^{1}=\hat{\mathbf{p}}_{1}+\hat{\mathbf{p}}_{2}+\hat{\mathbf{p}}_{M},\end{split}$
(17)

where $\hat{Q}_{2}$ is an observable commuting with every constraint111Notice
that this condition can be relaxed by adding an auxiliary quantum state as a
measurement device. For the sake of simplicity, we do not consider this case..
The same procedure adopted in Section 2 can be used to calculate the history
state, with the only _caveat_ that the operator $\hat{\mathcal{T}}_{1}$ needs
to be modified to

$\hat{\mathcal{T}}_{Q,1}=e^{\frac{i}{\hbar}\frac{\log\sqrt{g_{00}(\hat{\mathbf{x}}_{M})}}{2}\sum_{I=1}^{N}(\hat{x}^{0}_{I}\hat{p}_{0}^{I}+\hat{p}_{0}^{I}\hat{x}^{0}_{I})}e^{\frac{\mathrm{i}}{\hbar}\hat{\mathbf{x}}_{1}\left(\hat{f}^{1}-\hat{\mathbf{p}}^{1}\right)}e^{\frac{\mathrm{i}}{\hbar}\hat{x}_{1}^{0}\left(\hat{f}_{Q}^{0}-\hat{p}^{1}_{0}\right)}.$
(18)

With a similar procedure to the one adopted in Section 2, we can calculate the
physical state $\ket{\Psi}_{ph}$ and the history state
$\ket{\psi}^{(1)}={}_{1}\bra{q_{1}=0}\hat{\mathcal{T}}_{Q,1}\ket{\Psi}_{ph}$.
The calculations are detailed in Appendix G, and the resulting history state
is

$\ket{\psi}^{(1)}=\int
d\tau_{1}\overleftarrow{T}\left\\{e^{-\frac{\mathrm{i}}{\hbar}\int_{0}^{\tau_{1}}ds\left[\hat{H}^{(1)}+\Delta_{12}\delta(\hat{T}_{2}+\Delta_{12}s-\tau_{2}^{*})\hat{Q}_{2}\right]}\right\\}\ket{\psi^{(1)}_{0}}\ket{\tau_{1}},$
(19)

where $\overleftarrow{T}$ denotes the time-ordering operator, defined as
$\overleftarrow{T}[\hat{O}(t_{1})\hat{O}(t_{2})]=\hat{O}(t_{1})\hat{O}(t_{2})$
if $t_{1}>t_{2}$ and
$\overleftarrow{T}[\hat{O}(t_{1})\hat{O}(t_{2})]=\hat{O}(t_{2})\hat{O}(t_{2})$
if $t_{2}>t_{1}$ for $\hat{O}(t_{1})$, $\hat{O}(t_{2})$ being any two
arbitrary operators. We have also defined

$\Delta_{12}=\frac{\Delta(\hat{\mathbf{q}}_{2}-\hat{\mathbf{q}}_{M},\hat{\mathbf{k}}_{2})}{\sqrt{g_{00}(\hat{\mathbf{q}}_{M})}\hat{\gamma}_{\Sigma
k,1}^{-1}}=\sqrt{\frac{g_{00}(\hat{\mathbf{q}}_{2}-\hat{\mathbf{q}}_{M})}{g_{00}(\hat{\mathbf{q}}_{M})}}\frac{\hat{\gamma}_{2}^{-1}}{\hat{\gamma}_{\Sigma
k,1}^{-1}},$ (20)

as the “worldline operator” of particle $2$ in the perspective of clock $1$.
Notice that the metric field in Eq. (20) coincides with the metric
$g^{\prime}_{00}(\hat{\mathbf{q}}_{2},\hat{\mathbf{q}}_{M})$ in the QLIF of
particle $1$, which was derived in Section 2. The Hamiltonian $\hat{H}^{(1)}$
was defined in Eq. (13), and $\ket{\psi^{(1)}_{0}}$ is explicitly calculated
in Appendix G.

In order to understand how a measurement happening at time $\tau_{2}^{*}$ in
the frame of particle $2$ is seen in the QLIF of clock $1$, we restrict our
attention to the Galilean, special relativistic, and Newtonian cases, detailed
in Appendix F. We want to study the case in which the clock $2$ is an ideal
clock, which corresponds to the requirement that the initial state
$\ket{\psi_{0}^{(1)}}$ is sharp in the clock $2$ time variable.
Mathematically, this condition can be expressed as
$\psi_{0}^{(1)}(\mathbf{p}_{2},p_{M},t_{2})\propto\delta(t_{2}-t_{2}^{*})\psi_{0}^{(1)}(\mathbf{p}_{2},p_{M})$.
In the following, we choose $t_{2}^{*}=0$, equivalent to the initial
synchronisation of clocks $1$ and $2$. Our results are summarised in Fig. 2.

In the Galilean case of Appendix F.1, there is no mass $M$ sourcing the
gravitational field, and the special relativistic effects are not present.
Hence, $\Delta_{12}$ reduces to the identity operator. We can rewrite the
state of Eq. (19) as

$\ket{\psi}^{(1)}=\int_{-\infty}^{\tau_{2}^{*}}d\tau_{1}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}\tau_{1}}\ket{\psi^{(1)}_{0}}\ket{\tau_{1}}+\int^{\infty}_{\tau_{2}^{*}}d\tau_{1}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}(\tau_{1}-\tau_{2}^{*})}e^{-\frac{\mathrm{i}}{\hbar}\hat{Q}_{2}}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}\tau_{2}^{*}}\ket{\psi^{(1)}_{0}}\ket{\tau_{1}}.$
(21)

In this case we see that, in the QRF of clock $1$, the measurement happens at
the same time $\tau_{2}^{*}$ as it would in the QRF of clock $2$.

Let us consider now the special-relativistic case described in Appendix F.2.
In this case, we take into account the special relativistic effects, but we do
not have a mass $M$ sourcing the gravitational field. The operator
$\Delta_{12}$ reduces to $\Delta_{12}=\hat{\gamma}_{\Sigma
k,1}/\hat{\gamma}_{2}\approx\sqrt{1+\frac{\hat{\mathbf{k}}_{2}^{2}}{m_{1}^{2}c^{2}}-\frac{\hat{\mathbf{k}}_{2}^{2}}{m_{2}^{2}c^{2}}}$.

The history state is

$\begin{split}\ket{\psi}^{(1)}=\int
d\mathbf{k}_{2}dt_{2}&\left\\{\int_{-\infty}^{\tau_{k}}d\tau_{1}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}\tau_{1}}\psi_{0}^{(1)}(\mathbf{k}_{2},t_{2})\ket{\mathbf{k}_{2},t_{2}}+\right.\\\
+&\left.\int^{\infty}_{\tau_{k}}d\tau_{1}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}(\tau_{1}-\tau_{k})}e^{-\frac{\mathrm{i}}{\hbar}\hat{Q}_{2}}\psi_{\tau_{k}}^{(1)}(\mathbf{k}_{2},t_{2})\ket{\mathbf{k}_{2},t_{2}}\right\\}\ket{\tau_{1}},\end{split}$
(22)

where $\ket{\psi_{\tau_{k}}^{(1)}}=\int
dt_{2}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}\tau_{k}}\psi_{0}^{(1)}(\mathbf{k}_{2},t_{2})\ket{\mathbf{k}_{2},t_{2}}$
and $\tau_{k}=\Delta_{12}^{-1}(\mathbf{k}_{2})\tau_{2}^{*}$.

From the previous equation we see that the measurement happening at time
$\tau_{2}^{*}$ in the QRF of clock $2$ is seen by clock $1$ in a quantum
superposition of special-relativistic time dilations. The relation to the
usual time-dilation can easily be understood by comparison with the classical
case by considering two particles, $1$ and $2$, which measure the same
interval in coordinate time with their own proper time. In this case, the
relation $dx^{0}=d\tau_{1}\gamma_{1}=d\tau_{2}\gamma_{2}$ is precisely the
classical equivalent of what we found when we consider two clocks $1$ and $2$
in their own QRF.

Here, for each value $\mathbf{k}_{2}$ of the momentum of particle $2$, there
is a well-defined value of the time dilation, which coincides with the
classical one. However, since the relational degrees of freedom between the
QRF and particle $2$ are in a quantum superposition of momentum eigenstates,
we obtain a coherent superposition of special-relativistic time dilations,
which implies that clock $1$ sees the measurement as a delocalised event in
time.

These results are in agreement with those presented in Refs. [40, 43] in the
case of a quantum relativistic particle with a sharp momentum, or in a
superposition of two momenta222Formally, each sharp momentum state
$\ket{p^{*}}$ is described in Refs. [40, 43] by a gaussian wavepacket centred
in the momentum $p^{*}$, and following a semi-classical trajectory., and
generalise them to arbitrary quantum states of the particles.

Finally, we consider the Newtonian case described in Appendix F.3. In this
case, we take into account the gravitational effects due to the presence of
the mass $M$, but we do not consider the special-relativistic effects. Hence,
$\Delta_{12}=\sqrt{g^{\prime}_{00}(\hat{\mathbf{q}}_{2},\hat{\mathbf{q}}_{M})}=\sqrt{\frac{g_{00}(\hat{\mathbf{q}}_{2}-\hat{\mathbf{q}}_{M})}{g_{00}(\hat{\mathbf{q}}_{M})}}$
and the history state can be written as

$\begin{split}\ket{\psi}^{(1)}=\int
d\mathbf{q}_{2}d^{2}q_{M}dt_{2}&\left\\{\int_{-\infty}^{\tau_{q}}d\tau_{1}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}\tau_{1}}\psi_{0}^{(1)}(\mathbf{q}_{2},q_{M},t_{2})\ket{\mathbf{q}_{2},q_{M},t_{2}}+\right.\\\
+&\left.\int^{\infty}_{\tau_{q}}d\tau_{1}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}(\tau_{1}-\tau_{q})}e^{-\frac{\mathrm{i}}{\hbar}\hat{Q}_{2}}\psi_{\tau_{q}}^{(1)}(\mathbf{q}_{2},q_{M},t_{2})\ket{\mathbf{q}_{2},q_{M},t_{2}}\right\\}\ket{\tau_{1}},\end{split}$
(23)

where $\ket{\psi_{\tau_{q}}^{(1)}}=\int
dt_{2}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}\tau_{q}}\psi_{0}^{(1)}(\mathbf{q}_{2},q_{M}t_{2})\ket{\mathbf{q}_{2},q_{M},t_{2}}$
and $\tau_{q}=\Delta_{12}^{-1}(\mathbf{q}_{2},\mathbf{q}_{M})\tau_{2}^{*}$.

Analogously to the special-relativistic case, we have that the measurement
happening at time $\tau_{2}^{*}$ in the QRF of clock $2$ is delocalised in the
proper time of clock $1$. However, the effect in this case is a quantum
superposition of gravitational redshifts, as can be easily seen by recalling
the standard expression of the gravitational redshift
$\tau_{obs}=\sqrt{\frac{g_{00}(x_{obs})}{g_{00}(x_{em})}}\tau_{em}$, where
$\tau_{obs}$ and $x_{obs}$ are respectively the proper time and spacetime
coordinates of the observer, and $\tau_{em}$ and $x_{em}$ are respectively the
proper time and spacetime coordinates of the emitter. If we identify the
observer with clock $1$ and the emitter with clock $2$, we recover the
relation
$\tau_{q}=\Delta_{12}^{-1}(\mathbf{q}_{2},\mathbf{q}_{M})\tau_{2}^{*}$.
Interestingly, this result can be equivalently obtained in the initial set of
coordinates, where
$\Delta_{12}=\sqrt{\frac{g_{00}(\hat{\mathbf{x}}_{2}-\hat{\mathbf{x}}_{M})}{g_{00}(\hat{\mathbf{x}}_{1}-\hat{\mathbf{x}}_{M})}}$,
or knowing that it is always possible to transform to a QLIF where the metric
is locally inertial at the location of clock $1$. In this case, it is enough
to calculate $\Delta_{12}$ using the metric in the QLIF, i.e.,
$\Delta_{12}=\sqrt{\frac{g_{00}(x_{em})}{g_{00}(x_{obs})}}=\sqrt{g^{\prime}_{00}(\hat{\mathbf{q}}_{2},\hat{\mathbf{q}}_{M})}$,
because $g_{00}(x_{obs})=1$ in the QLIF.

The two effects discussed here, namely the _superposition of special-
relativistic time dilations_ and the _superposition of gravitational
redshifts_ are, in principle observable. A promising experimental system to
carry out tests on the generalisation of these concepts could be atom
interferometry [61].

An interesting consequence of the delocalisation of the measurement that we
have described is that, if we fix some time $\tau_{1}=\tau_{1}^{*}$, clock $1$
in general “sees” a superposition of the measurement having taken place or
not. Hence, it seems that whether a measurement has occurred or not is a QRF-
dependent feature. It is, however, an open question whether this would allow
for a change of the causal relations. Previous work on QRFs [36] and on causal
reference frames [62] suggests that the causal relations are preserved under
change of time reference frame, but other approaches [44] might allow for a
change of the causal relations.

The time delocalisation of events has been studied in Ref. [36]. However, in
that case the mechanism causing the delocalisation was the interaction between
the internal degrees of freedom of the clocks, or the choice of a clock with
an unsharp initial state. Here, the delocalisation happens due to the fact
that the external degrees of freedom of the particle are in a quantum state,
which is unsharp in the relevant basis (momentum in the special-relativistic
case, and position in the Newtonian case). Other authors have also studied,
with different techniques to the one that is developed here, the consequences
of proper time running in a superposition due to the external degrees of
freedom being in a quantum superposition of positions or momenta [53, 54, 56,
43, 63]. However, these results rely on an external and classical reference
frame. Here, we generalise these effects to QRFs in a quantum relationship
with each other, and show that they are purely due to the relation between the
QRF and a physical system that is described.

Furthermore, we show here that regardless of the state of the particle taken
as a QRF in the global model, it is always possible to find a QLIF in which,
locally, the metric field is flat and the particle is at rest. This result
corroborates the generalisation of the Einstein Equivalence Principle of Ref.
[37] and provides a concrete model which describes the QRF change including
multiple particles. In particular, this model is suitable to describe an
experimental test, such as an interferometric experiment [64], to verify the
superposition of gravitational redshifts and the extension of the Einstein
Equivalence Principle. It would be interesting to generalise this result to
more general metric fields and to a superposition of metric fields as
considered in Ref. [37], where only the dynamics of a single system was
considered.

##  5 Discussion

In this work, we have introduced the notion of a spacetime quantum reference
frame, i.e., a reference frame in spacetime associated to a quantum system
which can be in a superposition, or entangled from the point of view of
another quantum system. We have developed a formulation to describe a set of
relativistic quantum particles in a weak gravitational field from the
perspective of such spacetime quantum reference frame. In order to achieve
this, we developed a covariant formulation describing both the quantum state
associated to the position/momentum of the particles and its internal state.
The former is used to fix the spacetime quantum reference frame, and the
latter is used as the internal clock of the quantum reference frame ticking
according to its own proper time. This formulation allows us to compare the
dynamical evolution in the proper time of different quantum clocks, when the
state of their external variables (position or momentum) is in a quantum
superposition from the perspective of one of them. We describe such clocks as
being attached to a quantum particle evolving with a Hamiltonian operator, and
not having a (semi)classical worldline. We find a transformation to the
Quantum Locally Inertial Frame of such clocks, introduced in Ref. [37], and we
describe a _superposition of special-relativistic time dilations_ and a
_superposition of gravitational redshifts_ from the perspective of such
spacetime quantum reference frames.

We adopt a timeless formulation, according to which the global state of $N$
particles appears “frozen”, but the dynamical evolution is recovered in terms
of relational quantities of the particles and the quantum reference frame.
Formally, we achieve this by imposing a set of constraints on the model, some
of which encode the free evolution of the particles and treat space and time
on the same footing, and others impose total energy and momentum conservation.

The model that we derive also includes the Page-Wootters mechanism for non-
interacting clocks in Refs. [34, 35, 36] as a particular case, when the
external degrees of freedom are neglected.

This model is suitable to be extended to more general situations. Here, we
have only considered a description in $1+1$ dimensions, and a more general
model in $3+1$ dimensions will probably have to face technical difficulties
such as those solved in Ref. [27]. For instance, it would be interesting to
explore the possibility of imposing as a constraint the total angular momentum
tensor in spacetime, i.e., $M_{\mu\nu}$, with $\mu,\nu=0,\cdots,3$, rather
than just its spatial components, as done in Ref. [27].

Another possible generalisation involves the description of (gravitationally)
interacting systems. This will likely require a different way of handling the
constraints, which would no longer commute (and would thus be second-class
constraints). This generalisation, if successful, would allow us to fully
incorporate the interacting clocks model, i.e., the description of time
reference frames [36] in the spacetime description of quantum reference
frames.

In addition, it would be interesting to generalise the model to arbitrary
spacetimes and superpositions thereof and build the transformation to a
Quantum Locally Inertial Frame as in Ref. [37], in the general case.

Finally, we notice that the internal energies enter the model presented here
in the total energy balance. This means that they are considered as part of
the zero component of the total momentum. In other works, for instance in
Refs. [53, 56, 65, 43], the internal energies contribute instead to the total
mass, via the relativistic mass-energy equivalence. These two ways of
considering the role of the internal energies, as the zero-component of a
vector and as a relativistically invariant quantity are, in principle,
different. However, the results presented here are in agreement, to our order
of approximation, to those one obtains by considering the internal energy as
part of the total mass via the mass-energy equivalence. It would be
interesting to investigate the connection between these two methods further,
in order to gain a deeper insight on the role of the internal degrees of
freedom when both quantum and gravitational effects are relevant.

At the interface of quantum theory and gravity, the notion of a classical
spacetime is no longer adequate to describe physical phenomena. It is then
crucial to generalise the formulation of the laws of physics to situations in
which spacetime is non-classical. One of the main goals of the research on
quantum reference frames is to generalise the laws of physics to when
reference frames are associated to quantum systems. Here, we have taken a step
further and we have associated a reference frame in spacetime to a quantum
system. Generalisations of this approach to more general scenarios could pave
the way for formulating physics on a quantum spacetime.

I would like to thank Lin-Qing Chen, Thomas D. Galley, and Lee Smolin for
useful discussions at the early stages of this work. I am also grateful to
Achim Kempf, Lorenzo Maccone, and Alexander R. H. Smith for helpful comments
on the draft. Research at Perimeter Institute is supported in part by the
Government of Canada through the Department of Innovation, Science and
Economic Development and by the Province of Ontario through the Ministry of
Colleges and Universities.

##  Appendix A Mass dispersion relation in a weak gravitational field

In the Newtonian limit, the metric field is described as in Eq. (2), i.e.,

$\begin{split}&g_{00}=1+\frac{2\Phi(\mathbf{x}-\mathbf{x}_{M})}{c^{2}};\\\
&g_{01}=g_{10}=0;\\\ &g_{11}=-1,\end{split}$ (24)

where $\Phi(\mathbf{x})$ is the Newtonian potential due to the mass $m_{M}$ of
the system $M$ and $|\Phi(\mathbf{x})|/c^{2}\ll 1$ in the spacetime region
considered. Hence, the general-relativistic dispersion relation
$\tilde{C}_{I}=g^{\mu\nu}p_{\mu}p_{\nu}-m_{I}^{2}c^{2}$ of a classical
particle $I$ with mass $m_{I}$ takes the form

$\tilde{C}_{I}=g^{00}(\mathbf{x}_{I}-\mathbf{x}_{M})p_{0}^{I2}-\mathbf{p}_{I}^{2}-m_{I}^{2}c^{2}.$
(25)

We impose the positive energy condition by enforcing that $p^{I}_{0}\geq
0$333Notice that this condition can only be enforced in the weak-field limit
of the gravitational field, i.e., when $|\Phi(\mathbf{x})|/c^{2}\ll 1$.. By
noting that $\tilde{C}_{I}$ can be decomposed as

$\tilde{C}_{I}=\left[\sqrt{g^{00}(\mathbf{x}_{I}-\mathbf{x}_{M})}p^{I}_{0}-\sqrt{\mathbf{p}_{I}^{2}+m_{I}^{2}c^{2}}\right]\left[\sqrt{g^{00}(\mathbf{x}_{I}-\mathbf{x}_{M})}p^{I}_{0}+\sqrt{\mathbf{p}_{I}^{2}+m_{I}^{2}c^{2}}\right],$
(26)

we see that, if the energies are to be positive, it is enough to consider the
new dispersion relation

$C_{I}=\sqrt{g^{00}(\mathbf{x}_{I}-\mathbf{x}_{M})}p^{I}_{0}-\sqrt{\mathbf{p}_{I}^{2}+m_{I}^{2}c^{2}}.$
(27)

Clearly, this relation only holds in the low-energy regime of the particle,
hence it holds perturbatively in $\frac{\mathbf{p}_{I}^{2}}{m_{I}^{2}c^{2}}$.
It is now straightforward to quantise Eq. (27) and obtain the constraint

$\hat{C}_{I}=\sqrt{g^{00}(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M})}\hat{p}^{I}_{0}-\sqrt{\hat{\mathbf{p}}_{I}^{2}+m_{I}^{2}c^{2}}.$
(28)

If we now define
$\hat{\omega}^{I}_{p}=\sqrt{\hat{\mathbf{p}}_{I}^{2}+m_{I}^{2}c^{2}}$, we find
the constraint $\hat{C}_{I}$ of Eq. (3). Notice that the weak-field
approximation $|\Phi(\mathbf{x})|/c^{2}\ll 1$ played a crucial role here to
avoid ordering ambiguities in the quantisation procedure, as a more general
form of the metric field would also have position-dependent spatial
components, which, when quantised, do not commute with their conjugate
momentum.

It is easy to show that the expression
$\bra{x_{I}^{0}}\hat{C}_{I}\ket{\psi_{I}}_{ph}=0$, where $\ket{\psi_{I}}_{ph}$
is the quantum state of particle $I$ solving the constraint $\hat{C}_{I}$, is
equivalent to

$\mathrm{i}\hbar\frac{d}{dx_{I}^{0}}\ket{\psi_{I}(x^{0}_{I})}=\sqrt{g_{00}(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M})}\hat{\omega}_{p}^{I}\ket{\psi_{I}(x^{0}_{I})},$
(29)

where $\ket{\psi_{I}(x^{0}_{I})}=\braket{x_{I}^{0}}{\psi_{I}}_{ph}$. By
defining $x^{0}_{I}=ct_{I}$ and expanding perturbatively, we find

$\mathrm{i}\hbar\frac{d}{dt_{I}}\ket{\psi_{I}(t_{I})}=\left[m_{I}c^{2}+\frac{\hat{\mathbf{p}}_{I}^{2}}{2m_{I}}-\frac{\hat{\mathbf{p}}_{I}^{4}}{8m_{I}^{3}c^{2}}+m_{I}\Phi(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M})\right]\ket{\psi_{I}(t_{I})},$
(30)

where we do not consider higher-order terms. The equation above corresponds to
Schrödinger equation of a particle in a Newtonian field, with first-order
special-relativistic corrections.

##  Appendix B Comparison between relational approaches

In the literature, there are several, and in principle different, ways of
eliminating an external, absolute structure. For instance, the Page-Wootters
mechanism [34, 35] considers a set of clocks which, at the global level,
satisfy a Hamiltonian constraint as in Eq. (1). Upon conditioning on the state
of one of the clocks, one recovers the dynamics of the other clocks. This is
usually interpreted as the internal perspective of the clock on which one
conditions. The Page-Wootters mechanism was applied to interacting clocks in
Refs. [40, 36]. In particular, in the case of gravitationally interacting
clocks studied in Ref. [36] it was shown that the interaction term between the
internal degrees of freedom of the clocks leads to a relative temporal
localisation of events, due to the fact that different internal energy states
contribute differently to the total mass of the system. In addition, the
limits to the measurability of time studied in Ref. [65] could be recovered.
In a different approach to relational dynamics, it was shown in Refs. [26, 27]
that the formalism for QRFs introduced in Ref. [25] can be derived by imposing
a symmetry principle on a set of $N$ particles, and specifically by imposing
invariance under global translations and global rotations. This symmetry
principle is reminiscent of the construction of Shape Dynamics [66], which
also inspired the construction of a classical model in which a gravitational
arrow of time emerges [67]. The above-mentioned techniques have been derived
in different frameworks and are, in principle, different. However, a
connection between different approaches to relational dynamics can be found,
as shown in Refs. [41, 42] for the case of the Page-Wootters mechanism,
relational Dirac observables, and quantum deparametrisation.

##  Appendix C Worldline operator of a quantum relativistic particle in a weak
gravitational field

Let us consider a classical relativistic particle of mass $m$ in a weak
gravitational field. The classical equivalent of the constraint considered in
the main text (up to factors of order $O(c^{-2})$, is

$C_{I}=\sqrt{g^{00}(\mathbf{x}_{I})}p_{0}^{I}-m_{I}c\sqrt{1+\frac{\mathbf{p}_{I}^{2}}{m^{2}_{I}c^{2}}},$
(31)

where $\mathbf{p}_{I}^{2}$ is the spatial norm of the momentum. Via the
Hamilton’s equations of motion we have
$\dot{x}_{I}^{\mu}=\frac{dx_{I}^{\mu}}{d\tau}=\frac{\partial C_{I}}{\partial
p^{I}_{\mu}}$, which yields

$\begin{split}&\dot{x}_{I}^{0}=\sqrt{g^{00}(\mathbf{x}_{I})},\\\
&\dot{\mathbf{x}}_{I}=\frac{\mathbf{p}_{I}}{m_{I}c}\left(1+\frac{\mathbf{p}_{I}^{2}}{m_{I}^{2}c^{2}}\right)^{-1/2}.\end{split}$
(32)

Hence, the line element is, to order $O(c^{-2})$,

$\begin{split}ds&=\sqrt{g_{00}(\mathbf{x}_{I})-\frac{1}{c^{2}}\left(\frac{d\mathbf{x}_{I}}{d\tau}\right)^{2}\left(\frac{d\tau}{dx^{0}_{I}}\right)^{2}}dx^{0}_{I}=\\\
&=\sqrt{g_{00}(\mathbf{x}_{I})}\sqrt{1+\frac{\mathbf{p_{I}}^{2}}{m^{2}_{I}c^{2}}}dx_{I}^{0}.\end{split}$
(33)

The quantisation of this expression is straightforward, given that in the
regime we consider we have
$\left[g_{00}(\hat{\mathbf{x}}),\sqrt{1+\frac{\hat{\mathbf{p}}_{I}^{2}}{m_{I}^{2}c^{2}}}\right]=0$.
We then write the “worldline operator” as

$\Delta(\hat{\mathbf{x}}_{I},\hat{\mathbf{p}}_{I})=\sqrt{g_{00}(\mathbf{\hat{x}}_{I})}\sqrt{1+\frac{\hat{\mathbf{p}}_{I}^{2}}{m_{I}^{2}c^{2}}}.$
(34)

##  Appendix D Action of the $\mathcal{\hat{T}}_{1}$ operator on the position
and momentum operators

In this Appendix, we calculate the action of the operator
$\mathcal{\hat{T}}_{1}$ on the phase space operators of the quantum particle
in spacetime. We have

$\begin{split}&\mathcal{\hat{T}}_{1}\hat{x}_{1}^{0}\mathcal{\hat{T}}_{1}^{\dagger}=\sqrt{g^{00}(\hat{\mathbf{q}}_{M})}\hat{q}_{1}^{0},\hskip
10000.0pt\\\
&\mathcal{\hat{T}}_{1}\hat{\mathbf{x}}_{1}\mathcal{\hat{T}}_{1}^{\dagger}=\hat{\mathbf{q}}_{1},\\\
&\mathcal{\hat{T}}_{1}\hat{x}_{i}^{0}\mathcal{\hat{T}}_{1}^{\dagger}=\sqrt{g^{00}(\hat{\mathbf{q}}_{M})}\left(\hat{q}_{i}^{0}+\hat{q}_{1}^{0}\right),\\\
&\mathcal{\hat{T}}_{1}\hat{\mathbf{x}}_{i}\mathcal{\hat{T}}_{1}^{\dagger}=\hat{\mathbf{q}}_{i}+\hat{\mathbf{q}}_{1},\\\
&\mathcal{\hat{T}}_{1}\hat{x}_{M}^{0}\mathcal{\hat{T}}_{1}^{\dagger}=\hat{q}_{M}^{0}+\sqrt{g^{00}(\hat{\mathbf{q}}_{M})}\hat{q}_{1}^{0},\\\
&\mathcal{\hat{T}}_{1}\hat{\mathbf{x}}_{M}\mathcal{\hat{T}}_{1}^{\dagger}=\hat{\mathbf{q}}_{M}+\hat{\mathbf{q}}_{1},\end{split}$
$\begin{split}&\mathcal{\hat{T}}_{1}\hat{p}^{1}_{0}\mathcal{\hat{T}}_{1}^{\dagger}=\sqrt{g_{00}(\hat{\mathbf{q}}_{M})}\Big{(}\hat{k}^{1}_{0}-\sum_{i\neq
1}\hat{k}_{0}^{i}\Big{)}-\sum_{i\neq
1}\Delta(\hat{\mathbf{q}}_{i}-\hat{\mathbf{q}}_{M},\hat{\mathbf{k}}_{i})\frac{\hat{H}_{i}}{c}-\hat{k}_{0}^{M}-\Delta\Big{(}\hat{\mathbf{q}}_{M},\hat{\mathbf{k}}_{1}-\sum_{i\neq
1}\hat{\mathbf{k}}_{i}-\hat{\mathbf{k}}_{M}\Big{)}\frac{\hat{H}_{1}}{c},\\\
&\mathcal{\hat{T}}_{1}\hat{\mathbf{p}}_{1}\mathcal{\hat{T}}_{1}^{\dagger}=\hat{\mathbf{k}}_{1}-\sum_{i\neq
1}\hat{\mathbf{k}}_{i}-\hat{\mathbf{k}}_{M},\\\
&\mathcal{\hat{T}}_{1}\hat{p}^{i}_{0}\mathcal{\hat{T}}_{1}^{\dagger}=\sqrt{g_{00}(\hat{\mathbf{q}}_{M})}\hat{k}^{i}_{0},\\\
&\mathcal{\hat{T}}_{1}\hat{\mathbf{p}}_{i}\mathcal{\hat{T}}_{1}^{\dagger}=\hat{\mathbf{k}}_{i},\\\
&\mathcal{\hat{T}}_{1}\hat{p}^{M}_{0}\mathcal{\hat{T}}_{1}^{\dagger}=\hat{k}^{M}_{0},\\\
&\mathcal{\hat{T}}_{1}\hat{\mathbf{p}}_{M}\mathcal{\hat{T}}_{1}^{\dagger}=\hat{\mathbf{k}}_{M},\end{split}$
(35)

where $i=2,\cdots,N$.

We can also define a new transformation $\mathcal{\hat{T}}_{2}$ to the
relational variables of particle $2$ by swapping the labels $1$ and $2$ in the
transformation $\mathcal{\hat{T}}_{1}$. We define the relational phase-space
operators to particle $2$ as $\hat{r}_{j}$ (position operator) and
$\hat{u}_{j}$ (momentum operator), for $j=1,3,\cdots,N,M$, and we obtain

$\begin{split}&\mathcal{\hat{T}}_{2}\hat{x}_{2}^{0}\mathcal{\hat{T}}_{2}^{\dagger}=\sqrt{g^{00}(\hat{\mathbf{r}}_{M})}\hat{r}_{2}^{0},\hskip
10000.0pt\\\
&\mathcal{\hat{T}}_{2}\hat{\mathbf{x}}_{2}\mathcal{\hat{T}}_{2}^{\dagger}=\hat{\mathbf{r}}_{2},\\\
&\mathcal{\hat{T}}_{2}\hat{x}_{j}^{0}\mathcal{\hat{T}}_{2}^{\dagger}=\sqrt{g^{00}(\hat{\mathbf{r}}_{M})}\left(\hat{r}_{j}^{0}+\hat{r}_{2}^{0}\right),\\\
&\mathcal{\hat{T}}_{2}\hat{\mathbf{x}}_{j}\mathcal{\hat{T}}_{2}^{\dagger}=\hat{\mathbf{r}}_{j}+\hat{\mathbf{r}}_{2},\\\
&\mathcal{\hat{T}}_{2}\hat{x}_{M}^{0}\mathcal{\hat{T}}_{2}^{\dagger}=\hat{r}_{M}^{0}+\sqrt{g^{00}(\hat{\mathbf{r}}_{M})}\hat{r}_{2}^{0},\\\
&\mathcal{\hat{T}}_{2}\hat{\mathbf{x}}_{M}\mathcal{\hat{T}}_{2}^{\dagger}=\hat{\mathbf{r}}_{M}+\hat{\mathbf{r}}_{2},\\\
\end{split}$
$\begin{split}&\mathcal{\hat{T}}_{2}\hat{p}^{2}_{0}\mathcal{\hat{T}}_{2}^{\dagger}=\sqrt{g_{00}(\hat{\mathbf{r}}_{M})}\Big{(}\hat{u}^{2}_{0}-\sum_{j\neq
2}\hat{u}^{j}_{0}\Big{)}-\sum_{j\neq
2}\Delta(\hat{\mathbf{r}}_{j}-\hat{\mathbf{r}}_{M},\hat{\mathbf{u}}_{j})\frac{\hat{H}_{j}}{c}-\hat{u}_{0}^{M}-\Delta\Big{(}\hat{\mathbf{r}}_{M},\hat{\mathbf{u}}_{2}-\sum_{j\neq
2}\hat{\mathbf{u}}_{j}-\hat{\mathbf{u}}_{M}\Big{)}\frac{\hat{H}_{1}}{c}\\\
&\mathcal{\hat{T}}_{2}\hat{\mathbf{p}}_{2}\mathcal{\hat{T}}_{2}^{\dagger}=\hat{\mathbf{u}}_{2}-\sum_{j\neq
2}\hat{\mathbf{u}}_{j}-\hat{\mathbf{u}}_{M}\\\
&\mathcal{\hat{T}}_{2}\hat{p}^{j}_{0}\mathcal{\hat{T}}_{2}^{\dagger}=\sqrt{g_{00}(\hat{\mathbf{r}}_{M})}\hat{u}^{j}_{0}\\\
&\mathcal{\hat{T}}_{2}\hat{\mathbf{p}}_{j}\mathcal{\hat{T}}_{2}^{\dagger}=\hat{\mathbf{u}}_{j}\\\
&\mathcal{\hat{T}}_{2}\hat{p}^{M}_{0}\mathcal{\hat{T}}_{2}^{\dagger}=\hat{u}^{M}_{0}\\\
&\mathcal{\hat{T}}_{2}\hat{\mathbf{p}}_{M}\mathcal{\hat{T}}_{2}^{\dagger}=\hat{\mathbf{u}}_{M},\end{split}$
(36)

where $j=1,3,\cdots,N$.

We can also transform from one set of relational phase-space operators to
another, e.g., from particle $1$ to particle $2$. In order to do this, we
define
$\mathcal{\hat{T}}_{12}=\mathcal{\hat{T}}_{2}\mathcal{\hat{T}}_{1}^{\dagger}$
and find

$\begin{split}&\mathcal{\hat{T}}_{12}\hat{q}_{1}^{0}\mathcal{\hat{T}}_{12}^{\dagger}=\sqrt{\frac{g_{00}(\hat{\mathbf{r}}_{1}-\hat{\mathbf{r}}_{M})}{g_{00}(\hat{\mathbf{r}}_{M})}}\left(\hat{r}_{1}^{0}+\hat{r}_{2}^{0}\right),\hskip
10000.0pt\\\
&\mathcal{\hat{T}}_{12}\hat{\mathbf{q}}_{1}\mathcal{\hat{T}}_{12}^{\dagger}=\hat{\mathbf{r}}_{1}+\hat{\mathbf{r}}_{2},\\\
&\mathcal{\hat{T}}_{12}\hat{q}_{2}^{0}\mathcal{\hat{T}}_{12}^{\dagger}=-\sqrt{\frac{g_{00}(\hat{\mathbf{r}}_{1}-\hat{\mathbf{r}}_{M})}{g_{00}(\hat{\mathbf{r}}_{M})}}\hat{r}_{1}^{0},\\\
&\mathcal{\hat{T}}_{12}\hat{\mathbf{q}}_{2}\mathcal{\hat{T}}_{12}^{\dagger}=-\hat{\mathbf{r}}_{1},\\\
&\mathcal{\hat{T}}_{12}\hat{q}_{\ell}^{0}\mathcal{\hat{T}}_{12}^{\dagger}=\sqrt{\frac{g_{00}(\hat{\mathbf{r}}_{1}-\hat{\mathbf{r}}_{M})}{g_{00}(\hat{\mathbf{r}}_{M})}}(\hat{r}_{\ell}^{0}-\hat{r}_{1}^{0}),\\\
&\mathcal{\hat{T}}_{12}\hat{\mathbf{q}}_{\ell}\mathcal{\hat{T}}_{12}^{\dagger}=\hat{\mathbf{r}}_{\ell}-\hat{\mathbf{r}}_{1},\\\
&\mathcal{\hat{T}}_{12}\hat{q}_{M}^{0}\mathcal{\hat{T}}_{12}^{\dagger}=\hat{r}_{M}^{0}-\sqrt{g^{00}(\hat{\mathbf{r}}_{M})}\hat{r}_{1}^{0},\\\
&\mathcal{\hat{T}}_{12}\hat{\mathbf{q}}_{M}\mathcal{\hat{T}}_{12}^{\dagger}=\hat{\mathbf{r}}_{M}-\hat{\mathbf{r}}_{1},\end{split}$
$\begin{split}&\mathcal{\hat{T}}_{12}\hat{k}^{1}_{0}\mathcal{\hat{T}}_{12}^{\dagger}=\sqrt{g^{00}(\hat{\mathbf{r}}_{1}-\hat{\mathbf{r}}_{M})}\sqrt{g_{00}(\hat{\mathbf{r}}_{M})}\hat{u}^{2}_{0},\hskip
10000.0pt\\\
&\mathcal{\hat{T}}_{12}\hat{\mathbf{k}}_{1}\mathcal{\hat{T}}_{12}^{\dagger}=\hat{\mathbf{u}}_{2},\\\
&\mathcal{\hat{T}}_{12}\hat{k}^{2}_{0}\mathcal{\hat{T}}_{12}^{\dagger}=\sqrt{\frac{g^{00}(\hat{\mathbf{r}}_{1}-\hat{\mathbf{r}}_{M})}{g^{00}(\hat{\mathbf{r}}_{M})}}\Big{(}\hat{u}^{2}_{0}-\sum_{j\neq
2}\hat{u}_{0}^{j}\Big{)}-\sqrt{g^{00}(\hat{\mathbf{r}}_{1}-\hat{\mathbf{r}}_{M})}\left[\sum_{j\neq
2}\Delta_{j}\frac{\hat{H}_{j}}{c}+\hat{u}_{0}^{M}+\Delta_{\Sigma
u,2}\frac{\hat{H}_{2}}{c}\right],\\\
&\mathcal{\hat{T}}_{12}\hat{\mathbf{k}}_{2}\mathcal{\hat{T}}_{12}^{\dagger}=\hat{\mathbf{u}}_{2}-\sum_{j\neq
2}\hat{\mathbf{u}}_{j}-\hat{\mathbf{u}}_{M},\\\
&\mathcal{\hat{T}}_{12}\hat{k}^{\ell}_{0}\mathcal{\hat{T}}_{12}^{\dagger}=\sqrt{\frac{g^{00}(\hat{\mathbf{r}}_{1}-\hat{\mathbf{r}}_{M})}{g^{00}(\hat{\mathbf{r}}_{M})}}\hat{u}^{\ell}_{0},\\\
&\mathcal{\hat{T}}_{12}\hat{\mathbf{k}}_{\ell}\mathcal{\hat{T}}_{12}^{\dagger}=\hat{\mathbf{u}}_{\ell},\\\
&\mathcal{\hat{T}}_{12}\hat{k}^{M}_{0}\mathcal{\hat{T}}_{12}^{\dagger}=\hat{u}^{M}_{0},\\\
&\mathcal{\hat{T}}_{12}\hat{\mathbf{k}}_{M}\mathcal{\hat{T}}_{12}^{\dagger}=\hat{\mathbf{u}}_{M},\end{split}$
(37)

where $\ell=3,\cdots,N$ and
$\Delta_{j}=\Delta(\hat{\mathbf{r}}_{j}-\hat{\mathbf{r}}_{M},\hat{\mathbf{u}}_{j})$
and $\Delta_{\Sigma
u,2}=\Delta\Big{(}\hat{\mathbf{r}}_{M},\hat{\mathbf{u}}_{2}-\sum_{j\neq
2}\hat{\mathbf{u}}_{j}-\hat{\mathbf{u}}_{M}\Big{)}$. Notice that the relevant
relational operators are those of the particles $2,\cdots,N,M$ from the
perspective of particle $1$, which are mapped into quantities which only
depend on the operators of particle $2$ via $\hat{u}_{2}$, which is the
transformed constraint, since
$\hat{\mathcal{T}}_{2}\hat{f}^{0}\hat{\mathcal{T}}_{2}^{\dagger}=\sqrt{g_{00}(\hat{\mathbf{r}}_{M})}\hat{u}^{0}_{2}$
and
$\hat{\mathcal{T}}_{2}\hat{f}^{1}\hat{\mathcal{T}}_{2}^{\dagger}=\hat{\mathbf{u}}_{2}$.

##  Appendix E Explicit calculation of the “history state” in the frame of
particle 1

We recover in this Appendix the dynamical evolution of the relational degrees
of freedom from the state of Eq. (6)

$\ket{\Psi}_{ph}\propto\int
d^{N}\mathcal{N}d^{2}ze^{\frac{\mathrm{i}}{\hbar}\mathcal{N}_{i}\hat{C}_{i}}e^{\frac{\mathrm{i}}{\hbar}z_{\mu}\hat{f}^{\mu}}\ket{\phi},$
(38)

where $\ket{\phi}$ is represented as

$\ket{\phi}=\int\Pi_{I}\left[d\mu(x_{I})dE_{I}\right]d^{2}x_{M}\phi(x_{1},\cdots,x_{N},x_{M},E_{1},\cdots,E_{N})\ket{x_{1},\cdots,x_{N},x_{M}}\ket{E_{1},\cdots,E_{N}},$
(39)

and $d\mu(x_{i})=\sqrt{g_{00}(\mathbf{x}_{I}-\mathbf{x}_{M})}d^{2}x_{I}$ is
the covariant integration measure. The set of constraints that we enforce in
this model is

$\begin{split}&\hat{C}_{I}=\sqrt{g^{00}(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M})}\hat{p}^{I}_{0}-\hat{\omega}_{p}^{I},\qquad\qquad
I=1,\cdots,N\\\
&\hat{f}^{0}=\sum_{I=1}^{N}\left[\hat{p}^{I}_{0}+\Delta(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M},\hat{\mathbf{p}}_{I})\frac{\hat{H}_{I}}{c}\right]+\hat{p}_{0}^{M};\\\
&\hat{f}^{1}=\sum_{I=1}^{N}\hat{\mathbf{p}}_{I}+\hat{\mathbf{p}}_{M},\end{split}$
(40)

where
$\Delta(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M},\hat{\mathbf{p}}_{I})=\sqrt{g_{00}(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M})}\left(1+\frac{\hat{\mathbf{p}}^{2}_{I}}{m_{I}^{2}c^{2}}\right)^{-1/2}$.

In order to find the dynamical evolution, we define an operator
$\hat{\mathcal{T}}_{1}$ which maps the observables to the relational
observables from the perspective of particle $1$. We define the operator
$\hat{\mathcal{T}}_{1}$ as

$\hat{\mathcal{T}}_{1}=e^{\frac{i}{\hbar}\frac{\log\sqrt{g_{00}(\hat{\mathbf{x}}_{M})}}{2}\sum_{I=1}^{N}(\hat{x}^{0}_{I}\hat{p}_{0}^{I}+\hat{p}_{0}^{I}\hat{x}^{0}_{I})}e^{\frac{\mathrm{i}}{\hbar}\hat{\mathbf{x}}_{1}\left(\hat{f}^{1}-\hat{\mathbf{p}}^{1}\right)}e^{\frac{\mathrm{i}}{\hbar}\hat{x}_{1}^{0}\left(\hat{f}^{0}-\hat{p}^{1}_{0}\right)},$
(41)

where
$\hat{f}^{1}-\hat{\mathbf{p}}^{1}=\sum_{i}\hat{\mathbf{p}}_{i}+\hat{\mathbf{p}}_{M}$,
$\hat{f}^{0}-\hat{p}^{1}_{0}=\sum_{i}\left[\hat{p}_{0}^{i}+\Delta(\hat{\mathbf{x}}_{i}-\hat{\mathbf{x}}_{M},\hat{\mathbf{p}}_{i})\frac{\hat{H}_{i}}{c}\right]+\Delta(\hat{\mathbf{x}}_{1}-\hat{\mathbf{x}}_{M},\hat{\mathbf{p}}_{1})\frac{\hat{H}_{1}}{c}+\hat{p}_{0}^{M}$
and the lowercase latin letters label all particles except for the particle
serving as the QRF, i.e., $i=2,\cdots,N$. The action of the operator
$\hat{\mathcal{T}}_{1}$ on the constraints is

$\begin{split}\hat{\mathcal{T}}_{1}\hat{f}^{0}\hat{\mathcal{T}}_{1}^{\dagger}=&\sqrt{g_{00}(\hat{\mathbf{q}}_{M})}\hat{k}^{1}_{0};\qquad\hat{\mathcal{T}}_{1}\hat{f}^{1}\hat{\mathcal{T}}_{1}^{\dagger}=\hat{\mathbf{k}}_{1};\qquad\hat{\mathcal{T}}_{1}\hat{C}_{i}\hat{\mathcal{T}}_{1}^{\dagger}=\hat{C}^{\prime}_{i}\,\,\,i\neq
1;\\\
\hat{\mathcal{T}}_{1}\hat{C}_{1}\hat{\mathcal{T}}_{1}^{\dagger}=&\hat{k}_{0}^{1}-\left\\{\sum_{i}\left[\hat{k}_{0}^{i}+\Delta^{\prime}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M},\hat{\mathbf{k}}_{i})\frac{\hat{H}_{i}}{c}\right]+\sqrt{g^{00}(\hat{\mathbf{q}}_{M})}\hat{k}_{0}^{M}\right\\}+\\\
&-m_{1}c\sqrt{1+\frac{\left(\hat{\mathbf{k}}_{1}-\sum_{i}\hat{\mathbf{k}}_{i}-\hat{\mathbf{k}}_{M}\right)^{2}}{m_{1}^{2}c^{2}}}-\left(1+\frac{\left(\hat{\mathbf{k}}_{1}-\sum_{i}\hat{\mathbf{k}}_{i}-\hat{\mathbf{k}}_{M}\right)^{2}}{m_{1}^{2}c^{2}}\right)^{-1/2}\frac{\hat{H}_{1}}{c},\end{split}$
(42)

where $\hat{C}^{\prime}_{i}$ has been defined in the main text as
$\hat{C}^{\prime}_{i}=\sqrt{g^{\prime
00}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M})}\hat{k}_{0}^{i}-\hat{\omega}_{k}^{i}$,
$\Delta^{\prime}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M},\hat{\mathbf{k}}_{i})=\sqrt{g^{\prime}_{00}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M})}\left(1+\frac{\hat{\mathbf{k}}_{i}^{2}}{m_{i}^{2}c^{2}}\right)^{-1/2}$,
and the transformed metric field is

$g^{\prime}_{00}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M})=\frac{g_{00}(\hat{\mathbf{q}}_{i}-\hat{\mathbf{q}}_{M})}{g_{00}(\hat{\mathbf{q}}_{M})}.$
(43)

The relational state from the perspective of system $1$ is then obtained as
$\ket{\psi}^{(1)}=\bra{q_{1}=0}\hat{\mathcal{T}}_{1}\ket{\Psi}_{ph}$

By defining the internal time operator $\hat{T}_{I}$ such that
$\left[\hat{T}_{I},\hat{H}_{J}\right]=i\hbar\delta_{IJ}$ and the state of the
internal clock of each particle $\ket{\tau_{I}}$ as the one satisfying the
relation $\hat{T}_{I}\ket{\tau_{I}}=\tau_{I}\ket{\tau_{I}}$, we find

$\ket{\psi}^{(1)}=2\pi\hbar\left(1-\frac{\Phi(\mathbf{\hat{q}}_{M})}{c^{2}}\right)\int
d^{N}\mathcal{N}e^{\frac{\mathrm{i}}{\hbar}\mathcal{N}_{i}\hat{C}^{\prime}_{i}}e^{-\frac{\mathrm{i}}{\hbar}\frac{\hat{\gamma}_{\Sigma
k,1}^{-1}}{c}\left(\hat{K}^{(1)}+\hat{H}_{1}\right)\mathcal{N}_{1}}\bra{q_{1}=0}\hat{\Pi}_{0}\mathcal{\hat{T}}_{1}\ket{\phi},$
(44)

where $\hat{K}^{(1)}$ is the operator encoding the relational dynamics of
particles $2,\cdots,N$ from the point of view of particle $1$, i.e.,

$\hat{K}^{(1)}=\hat{\gamma}_{\Sigma
k,1}\left[\sum_{i}c\hat{k}_{0}^{i}+c\sqrt{g^{00}(\hat{\mathbf{q}}_{M})}\hat{k}_{0}^{M}\right]+\sum_{i}\sqrt{g^{\prime}_{00}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M})}\frac{\hat{\gamma}_{i}^{-1}}{\hat{\gamma}_{\Sigma
k,1}^{-1}}\hat{H}_{i}+m_{1}c^{2}\hat{\gamma}_{\Sigma k,1}^{2},$ (45)

$\hat{\gamma}_{i}=\sqrt{1+\frac{\hat{\mathbf{k}}_{i}^{2}}{m_{i}^{2}c^{2}}}$,
$\hat{\gamma}_{\Sigma
k,1}=\sqrt{1+\frac{\left(\sum_{i}\hat{\mathbf{k}}_{i}+\hat{\mathbf{k}}_{M}\right)^{2}}{m_{1}^{2}c^{2}}}$,
and $\hat{\Pi}_{0}=\ket{k_{1}=0}\bra{k_{1}=0}$. We now define

$\ket{\phi_{/1}}=2\pi\hbar\left(1-\frac{\Phi(\mathbf{\hat{q}}_{M})}{c^{2}}\right)\bra{q_{1}=0}\hat{\Pi}_{0}\mathcal{\hat{T}}_{1}\ket{\phi}$
(46)

and act with the operator $\hat{H}_{1}$ on the internal state of clock $1$. We
then rewrite

$\ket{\psi}^{(1)}=\int
d\tau_{1}d^{N-1}\mathcal{N}e^{\frac{\mathrm{i}}{\hbar}\mathcal{N}_{i}\hat{C}^{\prime}_{i}}e^{-\frac{\mathrm{i}}{\hbar}\hat{K}^{(1)}\tau_{1}}\ket{\phi^{(1)}_{0}}\ket{\tau_{1}},$
(47)

where

$\begin{split}\ket{\phi^{(1)}_{0}}=&\int
dt_{1}\Pi_{i}[d^{2}k_{i}dE_{i}]d^{2}k_{M}e^{\frac{\mathrm{i}}{\hbar}\hat{K}^{(1)}t_{1}}\phi_{/1}(k_{2},\cdots,k_{N},k_{M},t_{1},E_{2},\cdots,E_{N})\times\\\
&\times\ket{k_{2},\cdots,k_{N},k_{M}}\ket{E_{2},\cdots,E_{N}}.\end{split}$
(48)

Following a similar procedure to the one outlined above, it is possible solve
all the constraints and to cast the previous expression as a “history state”
in the sense of Refs. [36, 37] evolving unitarily as

$\ket{\psi}^{(1)}=\int
d\tau_{1}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}\tau_{1}}\ket{\psi^{(1)}_{0}}\ket{\tau_{1}},$
(49)

where the Hamiltonian $\hat{H}^{(1)}$ is

$\hat{H}^{(1)}=\hat{\gamma}_{\Sigma
k,1}\sum_{i}\sqrt{g^{\prime}_{00}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M})}\left\\{c\hat{\omega}_{k}^{i}+\hat{\gamma}_{i}^{-1}\hat{H}_{i}\right\\}+c\hat{\gamma}_{\Sigma
k,1}\sqrt{g^{00}(\hat{\mathbf{q}}_{M})}\hat{k}_{0}^{M}+m_{1}c^{2}\hat{\gamma}_{\Sigma
k,1}^{2},$ (50)

and

$\begin{split}\ket{\psi^{(1)}_{0}}=&\int\Pi_{i}\left(d\mu(q_{i})dq^{\prime
0}_{i}dE_{i}\sqrt{g^{\prime}_{00}(\mathbf{q}_{i},\mathbf{q}_{M})}\right)d^{2}q_{M}e^{\frac{\mathrm{i}}{\hbar}\sum_{i}(q^{\prime
0}_{i}-q^{0}_{i})\sqrt{g^{\prime}_{00}(\mathbf{q}_{i},\mathbf{q}_{M})}\hat{\omega}_{k}^{i}}\times\\\
&\times\phi^{(1)}_{0}\left(q_{2},\cdots,q_{N},q_{M},E_{2},\cdots,E_{N}\right)\ket{q^{\prime
0}_{2},\mathbf{q}_{2},\cdots,q^{\prime
0}_{N},\mathbf{q}_{N}}\ket{q_{M}}\ket{E_{2}\cdots E_{N}},\end{split}$ (51)

which reproduces the result of the main text.

The generalisation of this technique to general gravitational fields requires
dealing with second-class constraints, and is thus beyond the scope of this
work.

##  Appendix F Limiting cases of the general $N$-particle model

In the following, we are going to study some relevant limits of the model
introduced in the main text. In particular, we will focus on a set of quantum
free particles moving slowly compared to the speed of light (Galilean case),
on a set of quantum special-relativistic particles, and on a set of quantum
Galilean particles in an external weak gravitational field (Newtonian case).

We remind the reader that operators and vectors with no indices are, unless
differently specified, operators and two-vectors in spacetime, while the
spatial component of operators and vectors is boldface. Greek letters label
spacetime indices, capital Latin letters label all the particles, e.g.,
$I=1,\cdots,N$, and lowercase Latin letters label all the particles except the
one serving as the QRF, e.g., $i=2,\cdots,N$.

###  F.1 Galilean case

When the particles move slowly compared to the speed of light and are free,
the contraints introduced in the main text reduce to

$\displaystyle\hat{C}_{I}=\hat{p}_{I}^{0}-\frac{\hat{\mathbf{p}}^{2}_{I}}{2m_{I}c},$
(52)
$\displaystyle\hat{f}^{0}=\sum_{I=1}^{N}\left(\hat{p}_{I}^{0}+\frac{\hat{H}_{I}}{c}\right),$
(53) $\displaystyle\hat{f}^{1}=\sum_{I=1}^{N}\hat{\mathbf{p}}_{I},$ (54)

where here we have neglected particle $M$ because it plays no role. In
addition, we eliminated the constant term $m_{I}c$ in the expansion of
$\hat{C}_{I}$, because it simply amounts to a rescaling of the zero component
of the momentum.

We write the state in the Physical Hilbert space as

$\ket{\Psi}_{ph}\propto\int
d^{N}\mathcal{N}d^{2}ze^{\frac{\mathrm{i}}{\hbar}\mathcal{N}_{i}\hat{C}_{i}}e^{\frac{\mathrm{i}}{\hbar}z_{\mu}\hat{f}^{\mu}}\ket{\phi},$
(55)

where $\ket{\phi}$ is expanded in momentum basis as

$\ket{\phi}=\int\Pi_{I}\left[d^{2}p_{I}dE_{I}\right]\phi(p_{1},\cdots,p_{N},E_{1},\cdots,E_{N})\ket{p_{1},\cdots,p_{N}}\ket{E_{1},\cdots,E_{N}}.$
(56)

In this case, the operator $\hat{\mathcal{T}}_{1}$ simplifies to

$\hat{\mathcal{T}}_{1}=e^{\frac{\mathrm{i}}{\hbar}\hat{\mathbf{x}}_{1}\sum_{i}\hat{\mathbf{p}}_{i}}e^{\frac{\mathrm{i}}{\hbar}\hat{x}_{1}^{0}\left[\sum_{i}\left(\hat{p}_{0}^{i}+\frac{\hat{H}_{i}}{c}\right)+\frac{\hat{H}_{1}}{c}\right]},$
(57)

and the constraints are mapped to

$\begin{split}&\hat{\mathcal{T}}_{1}\hat{f}^{0}\hat{\mathcal{T}}_{1}^{\dagger}=\hat{k}^{1}_{0};\qquad\hat{\mathcal{T}}_{1}\hat{f}^{1}\hat{\mathcal{T}}_{1}^{\dagger}=\hat{\mathbf{k}}_{1};\qquad\hat{\mathcal{T}}_{1}\hat{C}_{i}\hat{\mathcal{T}}_{1}^{\dagger}=\hat{C}_{i}\,\,\,i\neq
1;\\\
&\hat{\mathcal{T}}_{1}\hat{C}_{1}\hat{\mathcal{T}}_{1}^{\dagger}=\hat{k}_{0}^{1}-\sum_{i}\left(\hat{k}_{0}^{i}+\frac{\hat{H}_{i}}{c}\right)-\frac{\hat{H}_{1}}{c}-\frac{\left(\hat{\mathbf{k}}_{1}-\sum_{i}\hat{\mathbf{k}}_{i}\right)^{2}}{2m_{1}}.\end{split}$
(58)

Defining the relational state from the perspective of particle $1$ to be
$\ket{\psi^{(1)}}={}_{1}\bra{q_{1}=0}\mathcal{T}_{1}\ket{\Psi}_{ph}$, we find

$\ket{\psi}^{(1)}=\int
d\tau_{1}d^{N-1}\mathcal{N}\ket{\tau_{1}}e^{\frac{\mathrm{i}}{\hbar}\mathcal{N}_{i}\hat{C}_{i}}e^{-\frac{\mathrm{i}}{\hbar}\hat{K}^{(1)}\tau_{1}}\ket{\phi^{(1)}_{0}},$
(59)

where

$\hat{K}^{(1)}=\sum_{i}\left(\hat{k}_{0}^{i}+\frac{\hat{H}_{i}}{c}\right)+\frac{\left(\sum_{i}\hat{\mathbf{k}}_{i}\right)^{2}}{2m_{1}},$
(60)

and

$\begin{split}&\ket{\phi^{(1)}_{0}}=\frac{1}{\sqrt{2\pi\hbar}}\int
dt_{1}dE_{1}\Pi_{i}(d^{2}k_{i}dE_{i})e^{\frac{\mathrm{i}}{\hbar}(\hat{K}^{(1)}+E_{1})t_{1}}\phi\left(\bar{k}_{1},k_{2},\cdots,k_{N},E_{1},\cdots,E_{N}\right)\times\\\
&\hskip 56.9055pt\times\ket{k_{2}\cdots k_{N}}\ket{E_{2}\cdots
E_{N}}.\end{split}$ (61)

By solving the constraint we have rewritten the two-vector
$\bar{k}_{1}=\left(-\sum_{i}\left(k_{0}^{i}+\frac{E_{i}}{c}\right)-\frac{E_{1}}{c},-\sum_{i}\mathbf{k}_{i}\right)$.
Notice that we can easily recover the framework of standard quantum mechanics
by integrating over the $N_{i}$ variables. After this operation, the state
$\ket{\psi^{(1)}}$ can be rewritten as the “history state”

$\ket{\psi}^{(1)}=\int
d\tau_{1}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}\tau_{1}}\ket{\psi^{(1)}_{0}}\ket{\tau_{1}},$
(62)

where

$\displaystyle\hat{H}^{(1)}=\sum_{i}\frac{\hat{\mathbf{k}}_{i}^{2}}{2m_{i}}+\frac{(\sum_{{i}}\hat{\mathbf{k}}_{i})^{2}}{2m_{1}}+\sum_{i}\hat{H}_{i},$
$\displaystyle\begin{split}&\ket{\psi^{(1)}_{0}}=\int\Pi_{i}(d\mathbf{k}_{i}dE_{i})\phi^{(1)}_{0}\left(\frac{\mathbf{k}_{2}^{2}}{2m_{2}},\mathbf{k}_{2},\cdots,\frac{\mathbf{k}_{N}^{2}}{2m_{N}},\mathbf{k}_{N},E_{1},\cdots,E_{N}\right)\times\\\
&\hskip
56.9055pt\times\ket{\frac{\mathbf{k}_{2}^{2}}{2m_{2}},\mathbf{k}_{2},\cdots,\frac{\mathbf{k}_{N}^{2}}{2m_{N}},\mathbf{k}_{N}}\ket{E_{2}\cdots
E_{N}}.\end{split}$

###  F.2 Special-relativistic case

In this section, we consider a set of N special-relativistic particles moving
freely. The set of constraints is

$\displaystyle\hat{C}_{I}=\hat{p}_{I}^{0}-\hat{\omega}_{p}^{I},$ (63)
$\displaystyle\hat{f}^{0}=\sum_{I=1}^{N}\left(\hat{p}_{I}^{0}+\hat{\gamma}_{I}^{-1}\frac{\hat{H}_{I}}{c}\right),$
(64) $\displaystyle\hat{f}^{1}=\sum_{I=1}^{N}\hat{\mathbf{p}}_{I},$ (65)

where we have defined
$\hat{\omega}_{p}^{I}=m_{I}c^{2}\sqrt{1+\frac{\hat{\mathbf{p}}_{I}^{2}}{m_{I}^{2}c^{2}}}$
and
$\hat{\gamma}_{I}=\sqrt{1+\frac{\hat{\mathbf{p}}_{I}^{2}}{m_{I}^{2}c^{2}}}$.
We write the state in the Physical Hilbert space as

$\ket{\Psi}_{ph}\propto\int
d^{N}\mathcal{N}d^{2}ze^{\frac{\mathrm{i}}{\hbar}\mathcal{N}_{i}\hat{C}_{i}}e^{\frac{\mathrm{i}}{\hbar}z_{\mu}\hat{f}^{\mu}}\ket{\phi},$
(66)

where $\ket{\phi}$ is expanded in momentum basis as

$\ket{\phi}=\int\Pi_{I}\left[d^{2}p_{I}dE_{I}\right]\phi(p_{1},\cdots,p_{N},E_{1},\cdots,E_{N})\ket{p_{1},\cdots,p_{N}}\ket{E_{1},\cdots,E_{N}},$
(67)

In this case, the trivialisation operator is

$\hat{\mathcal{T}}_{1}=e^{\frac{\mathrm{i}}{\hbar}\hat{\mathbf{x}}_{1}\sum_{i}\hat{\mathbf{p}}_{i}}e^{\frac{\mathrm{i}}{\hbar}\hat{x}_{1}^{0}\left[\sum_{i}\left(\hat{p}_{0}^{i}+\hat{\gamma}_{i}^{-1}\frac{\hat{H}_{i}}{c}\right)+\,\hat{\gamma}_{1}^{-1}\frac{\hat{H}_{1}}{c}\right]},$
(68)

and the constraints are mapped to

$\begin{split}&\hat{\mathcal{T}}_{1}\hat{f}^{0}\hat{\mathcal{T}}_{1}^{\dagger}=\hat{k}^{1}_{0};\qquad\hat{\mathcal{T}}_{1}\hat{f}^{1}\hat{\mathcal{T}}_{1}^{\dagger}=\hat{\mathbf{k}}_{1};\qquad\hat{\mathcal{T}}_{1}\hat{C}_{i}\hat{\mathcal{T}}_{1}^{\dagger}=\hat{C}_{i}\,\,\,i\neq
1;\\\
&\hat{\mathcal{T}}_{1}\hat{C}_{1}\hat{\mathcal{T}}_{1}^{\dagger}=\hat{k}_{0}^{1}-\sum_{i}\left(\hat{k}_{0}^{i}+\hat{\gamma}_{i}^{-1}\frac{\hat{H}_{i}}{c}\right)-\left(1+\frac{(\hat{\mathbf{k}}_{1}-\sum_{i}\hat{\mathbf{k}}_{i})^{2}}{m_{1}^{2}c^{2}}\right)^{-1/2}\frac{\hat{H}_{1}}{c}-m_{1}c\sqrt{1+\frac{(\hat{\mathbf{k}}_{1}-\sum_{i}\hat{\mathbf{k}}_{i})^{2}}{m_{1}^{2}c^{2}}}.\end{split}$
(69)

Following analogous steps to the Galilean case, we define
$\ket{\psi}^{(1)}=\bra{q_{1}=0}\hat{\mathcal{T}}_{1}\ket{\Psi}_{ph}$, where

$\ket{\psi}^{(1)}=\int
d\tau_{1}d^{N-1}\mathcal{N}\ket{\tau_{1}}e^{\frac{\mathrm{i}}{\hbar}\mathcal{N}_{i}\hat{C}_{i}}e^{-\frac{\mathrm{i}}{\hbar}\hat{K}^{(1)}\tau_{1}}\ket{\phi^{(1)}_{0}},$
(70)

where

$\hat{K}^{(1)}=\hat{\gamma}_{\Sigma
k,1}\left[\sum_{i}\left(c\hat{k}_{0}^{i}+\hat{\gamma}_{i}^{-1}\hat{H}_{i}\right)+c\hat{\omega}_{\Sigma
k}^{1}\right],$ (71)

and

$\begin{split}&\ket{\phi^{(1)}_{0}}=\frac{1}{\sqrt{2\pi\hbar}}\int\Pi_{i}(d^{2}k_{i}dE_{i})dt_{1}dE_{1}\gamma_{\Sigma
k,1}e^{\frac{\mathrm{i}}{\hbar}(\hat{K}^{(1)}+E_{1})t_{1}}\phi\left(\bar{k}_{1},k_{2},\cdots,k_{N},E_{1},\cdots,E_{N}\right)\times\\\
&\hskip 56.9055pt\times\ket{k_{2}\cdots k_{N}}\ket{E_{2}\cdots
E_{N}}.\end{split}$ (72)

Here, we have defined $\hat{\gamma}_{\Sigma
k,1}=\sqrt{1+\frac{(\sum_{i}\hat{\mathbf{k}}_{i})^{2}}{m_{1}^{2}c^{2}}}$ and
$\hat{\omega}_{\Sigma
k}^{1}=m_{1}c\sqrt{1+\frac{(\sum_{i}\hat{\mathbf{k}}_{i})^{2}}{m_{1}^{2}c^{2}}}$
and the two-vector
$\bar{k}_{1}=\left(-\sum_{i}\left(k_{0}^{i}+\gamma_{i}^{-1}\frac{E_{i}}{c}\right)-\gamma_{\Sigma
k,1}^{-1}\frac{E_{1}}{c},-\sum_{i}\mathbf{k}_{i}\right)$.

It is again possible, with a similar procedure to the Galilean case, to
recover the dynamical evolution of a quantum relativistic particle. We find

$\ket{\psi}^{(1)}=\int
d\tau_{1}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}\tau_{1}}\ket{\psi^{(1)}_{0}}\ket{\tau_{1}},$
(73)

where

$\displaystyle\hat{H}^{(1)}=\hat{\gamma}_{\Sigma
k,1}\left[\sum_{i}\left(c\hat{\omega}_{k}^{i}+\hat{\gamma}_{i}^{-1}\hat{H}_{i}\right)+c\hat{\omega}_{\Sigma
k}^{1}\right],$
$\displaystyle\begin{split}&\ket{\psi^{(1)}_{0}}=\int\Pi_{i}(d\mathbf{k}_{i}dE_{i})\phi^{(1)}_{0}\left(\omega_{k}^{2},\mathbf{k}_{2},\cdots,\omega_{k}^{N},\mathbf{k}_{N},E_{2},\cdots,E_{N}\right)\times\\\
&\hskip
56.9055pt\times\ket{\omega_{k}^{2},\mathbf{k}_{2},\cdots,\omega_{k}^{N},\mathbf{k}_{N}}\ket{E_{2}\cdots
E_{N}}.\end{split}$

Note that the factor $\sqrt{1+\frac{\left(\sum_{i\neq
1}\hat{k}_{i}\right)^{2}}{m_{1}^{2}c^{2}}}$ multiplying the hamiltonian of the
$N-1$ particles from the point of view of particle $1$ is the special-
relativistic time dilation due to the fact that the QRF is moving in a
superposition of special relativistic velocities.

###  F.3 Newtonian case

Finally, we consider the case in which a set of $N$ particles move in a
Newtonian gravitational field produced by a mass $M$. The gravitational field
in the Newtonian limit is the same as in the main text

$\begin{split}&g_{00}=1+\frac{2\Phi(\mathbf{x}-\mathbf{x}_{M})}{c^{2}};\\\
&g_{01}=g_{10}=0;\\\ &g_{11}=-1,\end{split}$ (74)

where $\Phi(\mathbf{x})$ is the Newtonian potential due to the mass $m_{M}$ of
the system $M$ and $|\Phi(\mathbf{x})|/c^{2}\ll 1$ in the spacetime region
considered.

The set of constraints for this situation are

$\begin{split}&\hat{C}_{I}=\sqrt{g^{00}(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M})}\hat{p}^{I}_{0}-m_{I}c-\frac{\hat{\mathbf{p}}^{2}_{I}}{2m_{I}c}\qquad\text{for}\qquad
I=1,\cdots,N;\\\
&\hat{f}^{0}=\sum_{I=1}^{N}\left[\hat{p}^{I}_{0}+\sqrt{g_{00}(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M})}\frac{\hat{H}_{I}}{c}\right]+\hat{p}_{0}^{M};\\\
&\hat{f}^{1}=\sum_{I=1}^{N}\hat{\mathbf{p}}_{I}+\hat{\mathbf{p}}_{M}.\end{split}$
(75)

Notice that, since this is a special case of the case considered in the main
text, the constraints are still first-class to our order of approximation.

The state in the Physical Hilbert space is

$\ket{\Psi}_{ph}\propto\int
d^{N}\mathcal{N}d^{2}ze^{\frac{\mathrm{i}}{\hbar}\mathcal{N}_{i}\hat{C}_{i}}e^{\frac{\mathrm{i}}{\hbar}z_{\mu}\hat{f}^{\mu}}\ket{\phi},$
(76)

where, in this case, it is convenient to expand $\ket{\phi}$ in position basis
as

$\ket{\phi}=\int\Pi_{I}\left[d\mu(x_{I})dE_{I}\right]d^{2}x_{M}\phi(x_{1},\cdots,x_{N},x_{M},E_{1},\cdots,E_{N})\ket{x_{1},\cdots,x_{N},x_{M}}\ket{E_{1},\cdots,E_{N}},$
(77)

where $d\mu(x_{I})=\sqrt{g_{00}(\mathbf{x}_{I}-\mathbf{x}_{M})}d^{2}x_{I}$ is
the covariant integration measure.

We define the operator $\hat{\mathcal{T}}_{1}$ as

$\hat{\mathcal{T}}_{1}=e^{\frac{i}{\hbar}\frac{\log\sqrt{g_{00}(\hat{\mathbf{x}}_{M})}}{2}\sum_{I=1}^{N}(\hat{x}^{0}_{I}\hat{p}_{0}^{I}+\hat{p}_{0}^{I}\hat{x}^{0}_{I})}e^{\frac{\mathrm{i}}{\hbar}\hat{\mathbf{x}}_{1}\left(\hat{f}^{1}-\hat{\mathbf{p}}^{1}\right)}e^{\frac{\mathrm{i}}{\hbar}\hat{x}_{1}^{0}\left(\hat{f}^{0}-\hat{p}^{1}_{0}\right)},$
(78)

where in this case
$\hat{f}^{1}-\hat{\mathbf{p}}^{1}=\sum_{i}\hat{\mathbf{p}}_{i}+\hat{\mathbf{p}}_{M}$
and
$\hat{f}^{0}-\hat{p}^{1}_{0}=\sum_{i}\left[\hat{p}_{0}^{i}+\sqrt{g_{00}(\hat{\mathbf{x}}_{i}-\hat{\mathbf{x}}_{M})}\frac{\hat{H}_{i}}{c}\right]+\sqrt{g_{00}(\hat{\mathbf{x}}_{1}-\hat{\mathbf{x}}_{M})}\frac{\hat{H}_{1}}{c}+\hat{p}_{0}^{M}$.

The action of the operator $\hat{\mathcal{T}}_{1}$ on the constraints is

$\begin{split}&\hat{\mathcal{T}}_{1}\hat{f}^{0}\hat{\mathcal{T}}_{1}^{\dagger}=\sqrt{g_{00}(\hat{\mathbf{q}}_{M})}\hat{k}^{1}_{0};\qquad\hat{\mathcal{T}}_{1}\hat{f}^{1}\hat{\mathcal{T}}_{1}^{\dagger}=\hat{\mathbf{k}}^{1};\qquad\hat{\mathcal{T}}_{1}\hat{C}_{i}\hat{\mathcal{T}}_{1}^{\dagger}=\hat{C}^{\prime}_{i}\,\,\,i\neq
1;\\\
&\hat{\mathcal{T}}_{1}\hat{C}_{1}\hat{\mathcal{T}}_{1}^{\dagger}=\hat{k}_{0}^{1}-\left[\sum_{i}\left(\hat{k}_{0}^{i}+\sqrt{g^{\prime}_{00}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M})}\frac{\hat{H}_{i}}{c}\right)+\sqrt{g^{00}(\hat{\mathbf{q}}_{M})}\hat{k}_{0}^{M}\right]+\\\
&\hskip 56.9055pt-
m_{1}c-\frac{(\hat{\mathbf{k}}_{1}-\sum_{i}\hat{\mathbf{k}}_{i}-\hat{\mathbf{k}}_{M})^{2}}{2m_{1}c}-\frac{\hat{H}_{1}}{c},\end{split}$
(79)

where
$\hat{C}^{\prime}_{i}=\sqrt{g^{\prime}_{00}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M})}\hat{k}^{i}_{0}-m_{i}c-\frac{\hat{\mathbf{k}}^{2}_{i}}{2m_{i}c}$
and $g^{\prime
00}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M})=\frac{g_{00}(\hat{\mathbf{q}}_{i}-\hat{\mathbf{q}}_{M})}{g_{00}(\hat{\mathbf{q}}_{M})}$
as in the main text.

Similarly to the other cases, the relational state from the perspective of
system $1$ is
$\ket{\psi}^{(1)}=\bra{q_{1}=0}\hat{\mathcal{T}}_{1}\ket{\Psi}_{ph}$. In order
to find its explicit expression, we here define

$\begin{split}\ket{\tilde{\phi}}=\hat{\mathcal{T}}_{1}\ket{\phi}=&\int
d^{2}k_{1}\Pi_{i}\left[d\mu(q_{i})dE_{i}\right]dE_{1}d^{2}q_{M}\tilde{\phi}(k_{1},q_{2},\cdots,q_{N},q_{M},t_{1},E_{2},\cdots,E_{N})\times\\\
&\times\ket{k_{1},q_{2},\cdots,q_{N},q_{M}}\ket{t_{1}}\ket{E_{2},\cdots,E_{N}}.\end{split}$
(80)

We then find

$\ket{\psi}^{(1)}=\int
d\tau_{1}d^{N-1}\mathcal{N}e^{\frac{\mathrm{i}}{\hbar}\mathcal{N}_{i}\hat{C}^{\prime}_{i}}e^{-\frac{\mathrm{i}}{\hbar}\hat{K}^{(1)}\tau_{1}}\ket{\phi^{(1)}_{0}}\ket{\tau_{1}},$
(81)

where

$\begin{split}\ket{\phi^{(1)}_{0}}&=c\int
dt_{1}\Pi_{i}[d\mu(q_{i})dE_{i}]d^{2}q_{M}\sqrt{g^{00}(\mathbf{q}_{M})}e^{\frac{\mathrm{i}}{\hbar}\hat{K}^{(1)}t_{1}}\times\\\
&\times\tilde{\phi}(k_{1}=0,q_{2},\cdots,q_{N},q_{M},t_{1},E_{2}\cdots,E_{N})\ket{q_{2},\cdots,q_{N},q_{M}}\ket{E_{2},\cdots,E_{N}},\end{split}$
(82)

and $\hat{K}^{(1)}$ is the operator encoding the relational dynamics of
particles $2,\cdots,N$ from the point of view of particle $1$, i.e.,

$\hat{K}^{(1)}=\sum_{i}c\hat{k}_{0}^{i}+c\sqrt{g^{00}(\hat{\mathbf{q}}_{M})}\hat{k}_{0}^{M}+\sum_{i}\sqrt{g^{\prime}_{00}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M})}\hat{H}_{i}+m_{1}c^{2}+\frac{(\sum_{i}\hat{\mathbf{k}}_{i}+\hat{\mathbf{k}}_{M})^{2}}{2m_{1}}.$
(83)

With a similar procedure to the previous cases, we find that we can recover
the usual dynamical evolution as

$\ket{\psi}^{(1)}=\int
d\tau_{1}\ket{\tau_{1}}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}\tau_{1}}\ket{\psi^{(1)}_{0}},$
(84)

where the relational Hamiltonian in the perspective of particle $1$ is

$\hat{H}^{(1)}=\sqrt{g^{00}(\hat{\mathbf{q}}_{M})}c\hat{k}_{0}^{M}+\sum_{i}\sqrt{g^{\prime}_{00}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M})}\left[m_{i}c^{2}+\frac{\hat{\mathbf{k}}_{i}^{2}}{2m_{i}}+\hat{H}_{i}\right]+m_{1}c^{2}+\frac{(\sum_{i}\hat{\mathbf{k}}_{i}+\hat{\mathbf{k}}_{M})^{2}}{2m_{1}},$
(85)

and

$\begin{split}&\ket{\psi^{(1)}_{0}}=\int\Pi_{i}\left[d\mu(q_{i})dq^{\prime
0}_{i}\sqrt{g^{\prime}_{00}(\mathbf{q}_{i}-\mathbf{q}_{M})}dE_{i}\right]d^{2}q_{M}e^{\frac{\mathrm{i}}{\hbar}\sum_{i}(q^{\prime
0}_{i}-q^{0}_{i})\sqrt{g^{\prime}_{00}(\mathbf{q}_{i}-\mathbf{q}_{M})}\left[m_{i}c+\frac{\hat{\mathbf{k}}_{i}^{2}}{2m_{i}c}\right]}\times\\\
&\times\phi^{(1)}_{0}\left(q_{2},\cdots,q_{N},q_{M},E_{2},\cdots,E_{N}\right)\ket{q^{\prime
0}_{2},\mathbf{q}_{2},\cdots,q^{\prime
0}_{N},\mathbf{q}_{N}}\ket{q_{M}}\ket{E_{2}\cdots E_{N}}.\end{split}$ (86)

Notice that, if we expand the Hamiltonian to lowest order in the general
relativistic corrections for the external degrees of freedom, while still
allowing the internal degrees of freedom to have relativistic corrections, we
obtain

$\begin{split}\hat{H}^{(1)}=&\sum_{i}\left\\{m_{i}c^{2}+\frac{\hat{\mathbf{k}}_{i}^{2}}{2m_{i}}+m_{i}[\Phi(\hat{\mathbf{q}}_{i}-\hat{\mathbf{q}}_{M})-\Phi(\hat{\mathbf{q}}_{M})]+\left[1+\frac{[\Phi(\hat{\mathbf{q}}_{i}-\hat{\mathbf{q}}_{M})-\Phi(\hat{\mathbf{q}}_{M})]}{c^{2}}\right]\hat{H}_{i}\right\\}+\\\
&+m_{1}c^{2}+\frac{(\sum_{i}\hat{\mathbf{k}}_{i}+\hat{\mathbf{k}}_{M})^{2}}{2m_{1}}+\sqrt{g^{00}(\hat{\mathbf{q}}_{M})}c\hat{k}_{0}^{M}.\end{split}$
(87)

As a result, we see that the potential in the QRF of particle $1$ is the
difference between the gravitational potential between particle $i$ and the
mass $M$ and particle $1$ and the mass $M$. This is the straightforward
generalisation of the standard reference frame description.

##  Appendix G Explicit calculation of the “history state” with a quantum
measurement

The calculation of the history state in the case with a measurement parallels
the one outlined in Appendix E. With a fully analogous procedure, fixing
$N=2$, and choosing the constraints of Eq. (17)

$\begin{split}&\hat{C}_{I}=\sqrt{g^{00}(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M})}\hat{p}^{I}_{0}-\hat{\omega}_{p}^{I}\qquad\text{for}\qquad
I=1,2;\\\
&\hat{f}_{Q}^{0}=\hat{p}^{1}_{0}+\hat{p}^{2}_{0}+\hat{p}_{0}^{M}+\Delta(\hat{\mathbf{x}}_{1}-\hat{\mathbf{x}}_{M},\hat{\mathbf{p}}_{1})\frac{\hat{H}_{1}}{c}+\Delta(\hat{\mathbf{x}}_{2}-\hat{\mathbf{x}}_{M},\hat{\mathbf{p}}_{2})\left[\frac{\hat{H}_{2}}{c}+\delta(\hat{T}_{2}-\tau_{2}*)\frac{\hat{Q}_{2}}{c}\right];\\\
&\hat{f}^{1}=\hat{\mathbf{p}}_{1}+\hat{\mathbf{p}}_{2}+\hat{\mathbf{p}}_{M},\end{split}$
(88)

where $\hat{Q}_{2}$ is an observable commuting with every constraint, and the
operator

$\hat{\mathcal{T}}_{Q,1}=e^{\frac{i}{\hbar}\frac{\log\sqrt{g_{00}(\hat{\mathbf{x}}_{M})}}{2}\sum_{I=1}^{N}(\hat{x}^{0}_{I}\hat{p}_{0}^{I}+\hat{p}_{0}^{I}\hat{x}^{0}_{I})}e^{\frac{\mathrm{i}}{\hbar}\hat{\mathbf{x}}_{1}\left(\hat{f}^{1}-\hat{\mathbf{p}}^{1}\right)}e^{\frac{\mathrm{i}}{\hbar}\hat{x}_{1}^{0}\left(\hat{f}_{Q}^{0}-\hat{p}^{1}_{0}\right)},$
(89)

we find that the history state can be cast in the form

$\ket{\psi}^{(1)}=\int
d\tau_{1}d^{N-1}\mathcal{N}e^{\frac{\mathrm{i}}{\hbar}\mathcal{N}_{i}\hat{C}^{\prime}_{i}}e^{-\frac{\mathrm{i}}{\hbar}\hat{K}_{Q}^{(1)}\tau_{1}}\ket{\phi^{(1)}_{0}}\ket{\tau_{1}},$
(90)

where $\hat{C}^{\prime}_{i}$ is the same as in the main text and in Appendix
E. In this case, we have that

$\hat{K}_{Q}^{(1)}=\hat{K}^{(1)}+\sqrt{g^{\prime}_{00}(\hat{\mathbf{q}}_{2},\hat{\mathbf{q}}_{M})}\frac{\hat{\gamma}_{2}^{-1}}{\hat{\gamma}_{\Sigma
k,1}^{-1}}\delta(\hat{T}_{2}-\tau_{2}^{*})\hat{Q}_{2},$ (91)

where $\hat{K}^{(1)}$ was defined in Eq. (45) for a general number of
particles $N$ (which here we have fixed to $N=2$). We also recall that we have
defined
$\hat{\gamma}_{2}=\sqrt{1+\frac{\hat{\mathbf{k}}_{2}^{2}}{m_{2}^{2}c^{2}}}$
and $\hat{\gamma}_{\Sigma
k,1}=\sqrt{1+\frac{\left(\hat{\mathbf{k}}_{2}+\hat{\mathbf{k}}_{M}\right)^{2}}{m_{1}^{2}c^{2}}}$.
Finally, we also write

$\ket{\phi^{(1)}_{0}}=c\int dt_{1}d^{2}k_{2}dt_{2}d^{2}k_{M}\gamma_{\Sigma
k,1}e^{\frac{\mathrm{i}}{\hbar}\hat{K}_{Q}^{(1)}t_{1}}\phi_{/1}(k_{2},k_{M},t_{1},t_{2})\ket{k_{2},k_{M},t_{2}}$
(92)

where the state $\ket{\phi_{/1}}$ has the same expression as in Eq. (46) with
the operator $\hat{\mathcal{T}}_{Q,1}$ instead of $\hat{\mathcal{T}}_{1}$. We
now consider the following equality, proved in Ref. [36],

$e^{-\frac{\mathrm{i}}{\hbar}\alpha[\hat{H}_{I}+\hat{f}(\hat{T}_{I})]}=e^{-\frac{\mathrm{i}}{\hbar}\alpha\hat{H}_{I}}\overleftarrow{T}\left\\{e^{-\frac{\mathrm{i}}{\hbar}\int_{0}^{1}ds\alpha\hat{f}(\hat{T}_{I}+\alpha
s)}\right\\},$ (93)

where $[\hat{T}_{I},\hat{H}_{I}]=\mathrm{i}\hbar$ and $\hat{f}(\hat{T}_{I})$
is an operator-valued function, and we slightly generalise it to

$e^{-\frac{\mathrm{i}}{\hbar}\alpha[\hat{K}_{R}+\hat{H}_{I}+\hat{f}(\hat{T}_{I})]}=e^{-\frac{\mathrm{i}}{\hbar}\alpha\hat{H}_{I}}\overleftarrow{T}\left\\{e^{-\frac{\mathrm{i}}{\hbar}\int_{0}^{1}ds\alpha\left[\hat{K}_{R}+\hat{f}(\hat{T}_{I}+\alpha
s)\right]}\right\\},$ (94)

where $\hat{K}_{R}$ is an arbitrary hermitian operator which does not
necessarily commute with $\hat{f}(\hat{T}_{I})$. Then, the expression of Eq:
(90) can be cast as

$\ket{\psi}^{(1)}=\int
d\tau_{1}\overleftarrow{T}\left\\{e^{-\frac{\mathrm{i}}{\hbar}\int_{0}^{\tau_{1}}ds\left[\hat{H}^{(1)}+\Delta_{12}\delta(\hat{T}_{2}+\Delta_{12}s-\tau_{2}^{*})\hat{Q}_{2}\right]}\right\\}\ket{\psi^{(1)}_{0}}\ket{\tau_{1}},$
(95)

where
$\Delta_{12}=\sqrt{g^{\prime}_{00}(\hat{\mathbf{q}}_{2},\hat{\mathbf{q}}_{M})}\frac{\hat{\gamma}_{2}^{-1}}{\hat{\gamma}_{\Sigma
k,1}^{-1}}$. The Hamiltonian $\hat{H}^{(1)}$ is the same as in the main text

$\hat{H}^{(1)}=\hat{\gamma}_{\Sigma
k,1}\sum_{i}\sqrt{g^{\prime}_{00}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M})}\left\\{c\hat{\omega}_{k}^{i}+\hat{\gamma}_{i}^{-1}\hat{H}_{i}\right\\}+c\hat{\gamma}_{\Sigma
k,1}\sqrt{g^{00}(\hat{\mathbf{q}}_{M})}\hat{k}_{0}^{M}+m_{1}c^{2}\hat{\gamma}_{\Sigma
k,1}^{2},$ (96)

and

$\begin{split}\ket{\psi^{(1)}_{0}}=&\int d\mu(q_{2})dq^{\prime
0}_{2}dt_{2}d^{2}q_{M}\sqrt{g^{\prime}_{00}(\mathbf{q}_{2}-\mathbf{q}_{M})}e^{\frac{\mathrm{i}}{\hbar}(q^{\prime
0}_{2}-q^{0}_{2})\sqrt{g^{\prime}_{00}(\mathbf{q}_{2}-\mathbf{q}_{M})}\hat{\omega}_{k}^{2}}\times\\\
&\times\phi^{(1)}_{0}\left(q_{2},q_{M},t_{2}\right)\ket{q^{\prime
0}_{2},\mathbf{q}_{2}}\ket{q_{M}}\ket{t_{2}},\end{split}$ (97)

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