# A model of quantum spacetime

Tommaso Favalli<EMAIL_ADDRESS>QSTAR, INO-CNR and LENS, Largo Enrico
Fermi 2, Firenze, Italy  Universitá degli Studi di Napoli Federico II, Via
Cinthia 21, Napoli, Italy  Augusto Smerzi  QSTAR, INO-CNR and LENS, Largo
Enrico Fermi 2, Firenze, Italy

###### Abstract

We consider a global quantum system (the “Universe”) satisfying a double
constraint, both on total energy and total momentum. Generalizing the Page and
Wootters quantum clock formalism, we provide a model of 3+1 dimensional, non-
relativistic, quantum spacetime emerging from entanglement among different
subsystems in a globally “timeless" and “positionless" Universe.

††preprint: AIP/123-QED

##  I Introduction

The idea that time may emerge from quantum entanglement was first proposed in
1983 by Don Page and William Wootters pagewootters ; wootters ; pagearrow  to
solve the, so called, “problem of time”. This arises in the context of
canonical quantization of gravity where, according to the Wheeler-DeWitt
equation, the whole universe should be in an eigenfunction of the total
Hamiltonian dewitt ; isham . Page and Wootters (PaW) theory splits the total
Hilbert space into two entangled subsystems, $C$ and $S$, where $C$ is the
clock subspace of an appropriately chosen clock observable. Looking at the
relative state everett  of the subsystem $S$, PaW show that the dynamical
Schrödinger evolution can be recovered with respect to the clock values. This
approach has recently attracted a large interest and has stimulated several
generalizations (see for example Refs. lloydmaccone, ; esp2, ; vedral, ;
vedraltemperature, ; macconeoptempoarrivo, ; interacting, ; simile, ; simile2,
; leonmaccone, ; review, ; review2, ; nostro, ; nostro2, ; timedilation, ;
scalarparticles, ; dirac, ; foti, ; brukner, ; wigner, ; indefinite, ;
asimmetry, ; pinto1, ; pinto2, ), including an experimental illustration esp1
. The PaW framework can be read as an internalization of the time reference
frame, with the clock being an appropriately chosen physical system and time
is considered as “what is shown on a quantum clock”. We study here the
possibility to extend this protocol in order to internalize, together, the
temporal and the spatial reference frames. In this approach also space becomes
an emerging property of entangled subsystems and the concept of position is
recovered relative to “what is shown on a quantum rod”.

Quantum reference frames for the spatial degree of freedom have been
extensively studied in quantum information and quantum foundations (see for
example Refs. burnett, ; QRF1, ; QRF2, ; QRF3, ; QRF4, ; QRF5, ; QRF6, ; QRF7,
; QRF8, ; QRF9, ; QRF10, ; QRF11, ; QRF12, ; QRF13, ; QRF14, ; QRF15, ; QRF16,
; QRF17, ; QRF18, ; QRF19, ; QRF20, ; QRF21, ). In the quantum gravity
literature, it has been suggested that quantum reference frames are needed to
formulate a quantum theory of gravity dewitt ; afundamental ; QG1 ; QG2 . In
Ref. change1,  (see also Refs. change2, ; change3, ; change4, ; change5, ;
change6, ; change7, ; change8, ; change9, ) has also been introduced a
formalism for describing transformations to change the description between
different quantum reference frames in various contexts. Just as the PaW
mechanism has been extensively studied in order to describe the temporal
degree of freedom, all these works have dealt only with internalization of the
space reference frame. Only recently, in Ref. giacomini, , has been introduced
a fully relational formulation of a $1+1$ dimensional spacetime for the case
of a system of $N$ relativistic quantum particles in a weak gravitational
field.

In this work we first focus on space and we divide the total Hilbert space in
two entangled subsystems, $R$ and $S$, where $R$ is the quatum rod that acts
as a spatial reference frame for $S$. A generalization of the PaW mechanism
for the spatial degree of freedom has already been addressed in Refs. hoehn1,
; hoehn2, ; hoehn3,  (see also Refs. change2, ; change3, ). Here we give our
own version adopting and generalizing the approach outlined in Ref. pegg,
(see also Ref. peggbar, ). We consider indeed discrete spectra for the
momentum operators and we take the spatial degree of freedom described by
POVMs. This choice allow us to recover continuous values for the spatial
degrees of freedom even if the momenta have discrete spectra (the
generalization to the case of a continuous spectrum is also disussed). We
therefore assume the Universe satisfying a constraint on total momentum
$\hat{P}_{tot}\ket{\Psi}=0$. Even if the global position is completely
undetermined, a well-defined relative position emerges from the entanglement
between the two subsystems $R$ and $S$. Finally we introduce an additional
subsystem $C$ acting as a clock and we consider the Universe satisfying a
double constraint: both on total momentum and total energy, that is
$\hat{P}_{tot}\ket{\Psi}=0$ and $\hat{H}_{tot}\ket{\Psi}=0$. We show that this
framework can be implemented consistently and we thus provide a model of non-
relativistic quantum spacetime emerging from entanglement. In order to
facilitate the reading and simplify the notation, in Sections II, III and VI
we consider a single spatial degree of freedom for the subsystems $R$ and $S$.
In Section V we generalize the results to the case of $3+1$ dimensional
spacetime and we discuss some examples.

##  II Emergent Relative Position from Entanglement

###  II.1 General Framework

We divide the total Hilbert space (the “Universe”) in two subsystems $R$ and
$S$ where $R$ acts as quantum reference frame for $S$. We consider the two
subsystems non-interacting and entangled with global Hamiltonian

$\hat{H}=\hat{H}_{R}+\hat{H}_{S}$ (1)

where $\hat{H}_{R}$ and $\hat{H}_{S}$ act on $R$ and $S$ respectively. We
consider the momenta of $R$ and $S$ having a discrete, bounded, non-degenerate
spectra and introduce the spatial degrees of freedom as POVMs. The case of
momenta with continuous, unbounded spectrum will be discussed in Section
III.E.

We begin by first considering the $R$ subspace. In introducing the POVMs of
space we follow a generalization of the framework outlined in Ref. pegg,  (see
also Refs. nostro, ; nostro2, ), namely we assume the momentum operator
$\hat{P}_{R}$ of $R$ with point-like spectrum, equally-spaced eigenvalues and
non-degenerate eigenstates. It can be illustrated by taking $d_{R}$ momentum
states $\ket{p_{k}}$ and $p_{k}$ momentum levels with $k=0,1,2,...,d_{R}-1$
such that (we set $\hslash=1$):

$p_{k}=p^{(R)}_{0}+\frac{2\pi}{L_{R}}k.$ (2)

In this space we can define the states

$\ket{x_{j}}_{R}=\frac{1}{\sqrt{d_{R}}}\sum_{k=0}^{d_{R}-1}e^{-ip_{k}x_{j}}\ket{p_{k}}_{R}$
(3)

and the values

$x_{j}=x_{0}+j\frac{L_{R}}{D_{R}}$ (4)

with $j=0,1,2,...,z=D_{R}-1$ and with the constraint $z+1=D_{R}\geq d_{R}$. If
we take $D_{R}=d_{R}$ the states (3) are orthogonal (with the $D_{R}=d_{R}$
values of $x_{j}$ uniformly distributed over $L_{R}$) and the spatial degree
of freedom is described by the Hermitian operator
$\hat{X}_{R}=\sum_{j}x_{j}\ket{x_{j}}\bra{x_{j}}$ complement of $\hat{P}_{R}$.
If, instead, we take $D_{R}>d_{R}$ the number of the states $\ket{x_{j}}_{R}$
is greater than the number of momenta states in $R$ (with the $D_{R}$ values
of $x_{j}$ again uniformly distributed over $L_{R}$). These states still
satisfy the key property
$\ket{x_{j}}_{R}=e^{-i\hat{P}_{R}(x_{j}-x_{0})}\ket{x_{0}}_{R}$ and it can
furthermore be used for writing the resolution of the identity:

$\mathbb{1}_{R}=\frac{d_{R}}{D_{R}}\sum_{j=0}^{D_{R}-1}\ket{x_{j}}\bra{x_{j}}.$
(5)

Thanks to (5) the spatial degree of freedom is represented by a POVM, being
$d_{R}D^{-1}_{R}\ket{x_{j}}\bra{x_{j}}$ the $D_{R}$ non-orthogonal elements.
In order to obtain a continuous representation of the coordinate $x$
(maintaining a discrete momentum spectrum), we can now consider the limit
$z\longrightarrow\infty$, defining

$\ket{x}_{R}=\sum_{k=0}^{d_{R}-1}e^{-ip_{k}x}\ket{p_{k}}_{R}$ (6)

where $x$ can now take any real value from $x_{0}$ to $x_{0}+L_{R}$. In this
limiting case the resolution of the identity (5) becomes

$\mathbb{1}_{R}=\frac{1}{L_{R}}\int_{x_{0}}^{x_{0}+L_{R}}dx\ket{x}\bra{x}.$
(7)

The states $\ket{x_{j}}$ and $\ket{x}$ are not orthogonal, but we will see in
the following that this will not constitute a problem in our derivation of
spacetime.

As mentioned, also the subspace $S$ can be equipped with the POVMs of space
considering $\hat{P}_{S}$ with discrete, bounded spectrum and applying the
same formalsim adopted in the subspace $R$. So we assume that also in $S$ all
the momentum eigenvalues can be written as multiples of a minimum step, that
is $p_{k}=p^{(S)}_{0}+\frac{2\pi}{L_{S}}k$ with $k=0,1,2,...,d_{S}-1$. We thus
define the states
$\ket{y_{l}}_{S}=\frac{1}{\sqrt{d_{S}}}\sum_{k=0}^{d_{S}-1}e^{-ip_{k}y_{l}}\ket{p_{k}}_{S}$
and the $z+1=D_{S}$ values $y_{l}=y_{0}+l\frac{L_{S}}{D_{S}}$ or, in the
limiting case ($z\longrightarrow\infty$) in which $y$ take any real value from
$y_{0}$ to $y_{0}+L_{S}$, we consider the states
$\ket{y}_{S}=\sum_{k=0}^{d_{S}-1}e^{-ip_{k}y}\ket{p_{k}}_{S}$. Also in this
case, when taking $D_{S}=d_{S}$ we can define the operator
$\hat{Y}_{S}=\sum_{l}y_{l}\ket{y_{l}}\bra{y_{l}}$ (complement of
$\hat{P}_{S}$) which is an Hermitian operator.

###  II.2 Emergent Relative Distance

In order to obtain the emergence of space from entanglement we consider now
the following constraint on the total momentum:

$\hat{P}\ket{\Psi}=(\hat{P}_{R}+\hat{P}_{S})\ket{\Psi}=0$ (8)

where $\hat{P}_{R}$ and $\hat{P}_{S}$ act on $R$ and $S$ respectively.
Assuming $d_{R}\gg d_{S}$ the global state $\ket{\Psi}$ satisfying (8) can be
writen as

$\ket{\Psi}=\sum_{k=0}^{d_{S}-1}c_{k}\ket{p=-p_{k}}_{R}\otimes\ket{p_{k}}_{S}.$
(9)

We can now expand $\ket{\Psi}$ in the $\left\\{\ket{x_{j}}\right\\}$ basis on
$R$ through (5), thus obtaining

$\begin{split}\ket{\Psi}&=\frac{d_{R}}{D_{R}}\sum_{j=0}^{D_{R}-1}\ket{x_{j}}\braket{x_{j}}{\Psi}=\\\
&=\frac{\sqrt{d_{R}}}{D_{R}}\sum_{j=0}^{D_{R}-1}\ket{x_{j}}_{R}\otimes\sum_{k=0}^{d_{S}-1}c_{k}e^{-ip_{k}x_{j}}\ket{p_{k}}_{S}=\\\
&=\frac{\sqrt{d_{R}}}{D_{R}}\sum_{j=0}^{D_{R}-1}\ket{x_{j}}_{R}\otimes\ket{\phi(x_{j})}_{S}\end{split}$
(10)

where in last step we have defined
$\ket{\phi(x_{j})}_{S}=\sum_{k=0}^{d_{S}-1}c_{k}e^{-ip_{k}x_{j}}\ket{p_{k}}_{S}$.
This state can be obtained from the global state $\ket{\Psi}$ through the
relative state definition everett  of the subsystem $S$ with respect to $R$
nostro :

$\ket{\phi(x_{j})}_{S}=\frac{\braket{x_{j}}{\Psi}}{1/\sqrt{d_{R}}}.$ (11)

Now using the fact that
$\ket{x_{j}}_{R}=e^{-i\hat{P}_{R}(x_{j}-x_{0})}\ket{x_{0}}_{R}$ and equations
(8) and (11), we obtain

$\begin{split}\ket{\phi(x_{j})}_{S}&=\sqrt{d_{R}}\braket{x_{j}}{\Psi}=\\\
&=\sqrt{d_{R}}\bra{x_{0}}e^{i\hat{P}_{R}(x_{j}-x_{0})}\ket{\Psi}=\\\
&=\sqrt{d_{R}}\bra{x_{0}}e^{i(\hat{P}-\hat{P}_{S})(x_{j}-x_{0})}\ket{\Psi}=\\\
&=e^{-i\hat{P}_{S}(x_{j}-x_{0})}\ket{\phi(x_{0})}_{S}\end{split}$ (12)

that is, we have that the operator $\hat{P}_{S}$ is the generator of spatial
traslations in the coordinate $x_{j}$. In this framework it is evident that
the translation moves the system with respect to the coordinate of the
reference frame $R$ and therefore “external”to $S$. Furthermore we want to
discuss briefly the consequences of using states $\ket{x_{j}}_{R}$ in $R$ that
are not orthogonal. Clearly this fact introduces a possible conceptual warning
because considering $\ket{x_{j}}_{R}$ that are not orthogonal implies that
these are partially indistinguishable with a single measurement, the
probability of indistinguishability being proportional to
$\left|\braket{x_{i}}{x_{j}}\right|^{2}$. Nevertheless, as mentioned
previously, this does not constitute a problem in our framework. Indeed, even
if the $\ket{x_{j}}_{R}$ are partially indistinguishable, the state of the
system $S$ conditioned on a given $x_{j}$ does not dependes on $x_{i}\neq
x_{j}$ (as we can clearly see in equations (10) and (12)) and so interference
phenomena are not present even if the coordinates in $R$ are not orthogonal.

These results can easily extended in the limiting case
$z\longrightarrow\infty$. Indeed in this case the global state satisfying the
constraint (8) can be written:

$\begin{split}\ket{\Psi}&=\frac{1}{L_{R}}\int_{x_{0}}^{x_{0}+L_{R}}dx\ket{x}\braket{x}{\Psi}=\\\
&=\frac{1}{L_{R}}\int_{x_{0}}^{x_{0}+L_{R}}dx\ket{x}_{R}\otimes\sum_{k=0}^{d_{S}-1}c_{k}e^{-ip_{k}x}\ket{p_{k}}_{S}=\\\
&=\frac{1}{L_{R}}\int_{x_{0}}^{x_{0}+L_{R}}dx\ket{x}_{R}\otimes\ket{\phi(x)}_{S}\end{split}$
(13)

and, for the relative state $\ket{\phi(x)}_{S}=\braket{x}{\Psi}$, can be
easily obtained

$\begin{split}\hat{P}_{S}\ket{\phi(x)}_{S}&=\bra{x}\hat{P}_{S}\ket{\Psi}=\bra{x}(\hat{P}-\hat{P}_{R})\ket{\Psi}=\\\
&=-\bra{x}\hat{P}_{R}\ket{\Psi}=-\sum_{k=0}^{d_{R}-1}\bra{p_{k}}p_{k}e^{ip_{k}x}\ket{\Psi}=\\\
&=i\frac{\partial}{\partial x}\braket{x}{\Psi}=i\frac{\partial}{\partial
x}\ket{\phi(x)}_{S}\end{split}$ (14)

that is the same of equation (12), showing that the momentum $\hat{P}_{S}$ is
the generator of translations in the coordinates $x$, but written through the
differential expression
$\hat{P}_{S}\ket{\phi(x)}_{S}=i\frac{\partial}{\partial x}\ket{\phi(x)}_{S}$.

In this framework the absolute position of $R+S$ is totally indeterminate.
However, considering discrete values for the coordinates in $R$ and $S$, we
can look for the conditional probability of obtaining $y_{l}$ on $S$
conditioned of having $x_{j}$ on $R$, where
$y_{l}=y_{0}+l\frac{L_{S}}{D_{S}}$, $x_{j}=x_{0}+j\frac{L_{R}}{D_{R}}$. We
have (see Appendix A for the proof)

$P(y_{l}\>on\>S\>|\>x_{j}\>on\>R)=\\\
=\frac{d_{S}}{D_{S}}\left|\braket{y_{l}}{\phi(x_{j})}\right|^{2}=\frac{1}{D_{S}}\left|\sum_{k=0}^{d_{S}-1}c_{k}e^{ip_{k}(y_{l}-x_{j})}\right|^{2}$
(15)

that is a well-defined probability distribution, where
$\sum_{l=0}^{D_{S}-1}P(y_{l}\>on\>S\>|\>x_{j}\>on\>R)=1$. Working instead in
the limit $z\longrightarrow\infty$, the probability for a value of $y$ in the
small range between $y$ and $y+dy$ is given by $P(y\>on\>S\>|\>x\>on\>R)dy$,
where the probability density $P(y\>on\>S\>|\>x\>on\>R)$ is (the proof is
given in Appendix B):

$P(y\>on\>S\>|\>x\>on\>R)=\\\
=\frac{1}{L_{S}}\left|\braket{y}{\phi(x)}\right|^{2}=\frac{1}{L_{S}}\left|\sum_{k=0}^{d_{S}-1}c_{k}e^{ip_{k}(y-x)}\right|^{2}$
(16)

that is again a well-defined probability density distribution depending on the
distance between $S$ and $R$. Indeed, also in this case, if we consider the
integral over all possible values of $y$ we obtain

$\int_{y_{0}}^{y_{0}+L_{S}}dyP(y\>on\>S\>|\>x\>on\>R)=\\\
=\frac{1}{L_{S}}\int_{y_{0}}^{y_{0}+L_{S}}dy\sum_{k}\sum_{n}c_{k}c^{*}_{n}e^{i(p_{k}-p_{n})(y-x)}=1$
(17)

where we have used (see Appendix C):

$\int_{y_{0}}^{y_{0}+L_{S}}dye^{iy(p_{k}-p_{n})}=L_{S}\delta_{p_{k},p_{n}}.$
(18)

Equations (15) and (16) display an essential feature of the complementarity
between positions and momenta. Indeed, if the system $S$ is in an eigenstate
of the momentum, in the right-hand side of (15) and (16) there is only one
term of modulus unity and we have $P(y_{l}\>on\>S\>|\>x_{j}\>on\>R)=1/D_{S}$
and $P(y\>on\>S\>|\>x\>on\>R)=1/L_{S}$. So, in this case, the probability
$P(y_{l}\>on\>S\>|\>x_{j}\>on\>R)$ and the probability density
$P(y\>on\>S\>|\>x\>on\>R)$ are constant across the whole interval
$\left[y_{0},y_{0}+L_{S}\right]$ indicating that, when the momentum of the
system $S$ can be determined exactly, thus the position with respect to the
reference frame $R$ is completely random.

We have so considered a “positionless”Universe, satisfying the constraint (8)
on the total momentum (where the absolute position is totally indeterminate)
and we found the well-defined conditional probability
$P(y_{l}\>on\>S\>|\>x_{j}\>on\>R)$ (for the case of discrete coordinates) and
the probability density $P(y\>on\>S\>|\>x\>on\>R)$ (for the limiting case
$z\longrightarrow\infty$) where appears the relative distance between the two
entangled subsystems.

###  II.3 On the Position-Momentum Uncertainty Relation

In this paragraph we want to focus on the quantity $\delta x$, namely the
minimum interval in the $x$ values of $R$ over which the state of the system
$\ket{\phi(x)}_{S}$ varies significantly. We will show that, if the momentum
spread in the expansion (9) is $\Delta p$, than $\delta x\geq\hslash/\Delta p$
(re-introducing $\hslash\neq 1$). This means that will be impossible to
distinguish states of the system $S$ conditioned to $x$ values on $R$ which
are close to each other less than $\simeq\hslash/\Delta p$, in accordance with
position-momentum uncertainty relation.

For the sake of simplicity we consider here $D_{R}=D_{S}=d_{R}=d_{S}=d$ and
$L_{R}=L_{S}=L$. We assume also discrete values for the space, so we have
$\ket{x_{j}}_{R}=\frac{1}{\sqrt{d}}\sum_{k}e^{-ip_{k}x_{j}}\ket{p_{k}}_{R}$
and
$\ket{y_{l}}_{S}=\frac{1}{\sqrt{d}}\sum_{k}e^{-ip_{k}y_{l}}\ket{p_{k}}_{S}$
with $x_{j}=x_{0}+j\frac{L}{d}$ and $y_{l}=y_{0}+l\frac{L}{d}$. As already
mentioned, in this particular case we can define the operators
$\hat{X}_{R}=\sum_{j}x_{j}\ket{x_{j}}\bra{x_{j}}$ and
$\hat{Y}_{S}=\sum_{l}y_{l}\ket{y_{l}}\bra{y_{l}}$ (complement of $\hat{P}_{R}$
and $\hat{P}_{S}$) which are Hermitian operators.

The crucial point for our argument here is to understand that in this
framework, since the reference frame $R$ and the system $S$ are entangled in
the global state $\ket{\Psi}$, the $\Delta p$ related to the spread of the
coefficients in the expansion (9), that is
$\ket{\Psi}=\sum_{k=0}^{d_{S}-1}c_{k}\ket{p=-p_{k}}_{R}\otimes\ket{p_{k}}_{S}$,
does not refer exclusively to $R$, but to $R$ and $S$ together. For this
reason what we will find is that a limited spread in the expansion in the
momentum eigenbasis will reduce the distinguishability of states of $S$
conditioned on proximal values of $R$. So, starting from equation

$\ket{\phi(x_{j})}_{S}=\frac{\braket{x_{j}}{\Psi}}{1/\sqrt{d}}=\sum_{k=0}^{d-1}c_{k}e^{-i\hslash^{-1}p_{k}x_{j}}\ket{p_{k}}_{S},$
(19)

we can calculate in the space $S$:

$\begin{split}\braket{\phi(x_{i})}{\phi(x_{j})}=\sum_{k=0}^{d-1}\left|c_{k}\right|^{2}e^{-i\hslash^{-1}p_{k}(x_{j}-x_{i})}.\end{split}$
(20)

Equations (20) indicates that, if $\left|c_{k}\right|^{2}$ has a spread
$\simeq\Delta p$, than the scalar product
$f(x_{j}-x_{i})=\braket{\phi(x_{i})}{\phi(x_{j})}$ will have a spread of the
order $\simeq\hslash/\Delta p$. This means that the state
$\ket{\phi(x_{j})}_{S}$ of the subsystem $S$ varies significantly for
intervals

$\delta x\geq\hslash/\Delta p$ (21)

where $\Delta p$ is indeed the uncertainty in momentum related to the spread
of the coefficients $c_{k}$. This means that it is impossible to distinguish
states of the system $S$ conditioned to $x$ values on $R$ which are closer
than $\simeq\hslash/\Delta p$ to each other, in accordance with the position-
momentum uncertainty relation.

Expression (20) for $f(x_{j}-x_{i})$ holds also in the case of non-orthogonal
space states where the spatial degrees of freedom are described by POVMs. So
we notice here that the function $f(x_{j}-x_{i})$, and consequently the scale
on which the system $S$ varies significantly, is not related to the overlap of
the states in $R$. Namely the fact of using coordinates states that are not
orthogonal does not have an effect on the derivation of the function
$f(x_{j}-x_{i})$. As already mentioned, this is because, calculating the
conditioned state of $S$ to a certain value $x_{j}$ on $R$ through (19), we
find no contributions from different positions $x_{i}\neq x_{j}$, and so
interference phenomena are not present even if the position states are not
orthogonal. Rather $f(x_{j}-x_{i})$ is related to the spread of the
coefficients appearing in the global state $\ket{\Psi}$ and this fact shows us
a condition for a good functioning of our framework: a large spread in the
coefficients within the global state in the expansion (9) is needed in order
to distinguish states of $S$ projected to closer values of $R$ asimmetry .

##  III Spacetime from Entanglement

###  III.1 Introducing the $C$ Subspace

In order to introduce the temporal degree of freedom we have to consider an
additional Hilbert space and assign it to time. We so assume that the Universe
is divided into three subsystems, that is we work in
$\mathcal{H}=\mathcal{H}_{C}\otimes\mathcal{H}_{R}\otimes\mathcal{H}_{S}$,
where $\mathcal{H}_{C}$ is the time Hilbert space. The three subsystems are
non-interacting but still entangled, so we have

$\hat{H}=\hat{H}_{C}+\hat{H}_{R}+\hat{H}_{S}$ (22)

where $\hat{H}_{C}$, $\hat{H}_{R}$ and $\hat{H}_{S}$ act on $C$, $R$ and $S$
respectively. To be as general as possible, we consider the clock Hamiltonian
$\hat{H}_{C}$ with bounded, discrete spectrum with unequally-spaced energy
levels and we introduce also the time observable described by a POVM nostro ;
nostro2 .

We start assuming $\hat{H}_{C}$ with point-like spectrum, non-degenerate
eigenstates and having rational energy differences. The framewowrk can be
illustrated by taking $p+1=d_{C}$ energy states $\ket{E_{i}}$ and $E_{i}$
energy levels with $i=0,1,2,...,d_{C}-1$ such that
$\frac{E_{i}-E_{0}}{E_{1}-E_{0}}=\frac{A_{i}}{B_{i}}$, where $A_{i}$ and
$B_{i}$ are integers with no common factors. Doing this we obtain (we set
again $\hslash=1$):

$E_{i}=E_{0}+r_{i}\frac{2\pi}{T}$ (23)

where $T=\frac{2\pi r_{1}}{E_{1}}$, $r_{i}=r_{1}\frac{A_{i}}{B_{i}}$ for $i>1$
(with $r_{0}=0$) and $r_{1}$ equal to the lowest common multiple of the values
of $B_{i}$. In this space we define the states

$\ket{t_{m}}_{C}=\frac{1}{\sqrt{d_{C}}}\sum_{i=0}^{d_{C}-1}e^{-iE_{i}t_{m}}\ket{E_{i}}_{C}$
(24)

where

$t_{m}=t_{0}+m\frac{T}{D_{C}}$ (25)

with $m=0,1,2,...,s=D_{C}-1$ and $s+1\geq r_{p}$ (we notice here that also in
this case, by assuming the energy spectrum with equally-spaced eigenvalues and
taking $D_{C}=d_{C}$, the time states (24) are orthogonal and the temporal
degree of freedom is described by the Hermitian operator
$\hat{T}_{C}=\sum_{m}t_{m}\ket{t_{m}}\bra{t_{m}}$). The number of
$\ket{t_{m}}_{C}$ states is therefore greater than the number of energy states
in $\mathcal{H}_{C}$ and the $D_{C}$ values of $t_{m}$ are uniformly
distributed over $T$. These states satisfy the key property
$\ket{t_{m}}_{C}=e^{-i\hat{H}_{C}(t_{m}-t_{0})}\ket{t_{0}}_{C}$ and
furthermore can be used for writing the resolution of the identity in the $C$
subspace:

$\mathbb{1}_{C}=\frac{d_{C}}{D_{C}}\sum_{m=0}^{D_{C}-1}\ket{t_{m}}\bra{t_{m}}.$
(26)

Thanks to property (26) also time is represented by a POVM, being
$d_{C}D^{-1}_{C}\ket{t_{m}}\bra{t_{m}}$ the $D_{C}$ non-orthogonal elements.
As we did for space, in order to obtain a continuous flow of time in the PaW
framework, we can now consider the limit $s\longrightarrow\infty$, defining

$\ket{t}_{C}=\sum_{i=0}^{d_{C}-1}e^{-iE_{i}t}\ket{E_{i}}_{C}$ (27)

where $t$ can now take any real value from $t_{0}$ to $t_{0}+T$. In this
limiting case the resolution of the identity (26) becomes

$\mathbb{1}_{C}=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}dt\ket{t}\bra{t}.$ (28)

As in the case of space, the states $\ket{t_{m}}_{C}$ and $\ket{t}_{C}$ are
not orthogonal, but thanks to the properties (26) and (28) they can be used as
time states for introducing time through the PaW mechanism. This framework
allows us to use a generic Hamiltonian as a clock Hamiltonian, with the only
constraint of considering rational ratios of energy levels. However, we
emphasize that this limitation can be opportunely relaxed. Indeed, in the case
of non-rational ratios of energy levels, the resolutions of the identity (26)
and (28) are no longer exact but, since any real number can be approximated
with arbitrary precision by a ratio between two rational numbers, the residual
terms in the resolutions of the identity and consequent small corrections can
be arbitrarily reduced. In this way we can consider that the mechanism works,
at least approximately, for every generic Hamiltonian with no restrictions
nostro ; nostro2 .

###  III.2 Emergent $1+1$ Dimensional Spacetime

We want now to obtain a model of spacetime emerging from entanglement. So we
consider the global state
$\ket{\Psi}\in\mathcal{H}_{C}\otimes\mathcal{H}_{R}\otimes\mathcal{H}_{S}$
simultaneously satisfying

$\hat{H}\ket{\Psi}=(\hat{H}_{C}+\hat{H}_{R}+\hat{H}_{S})\ket{\Psi}=0$ (29)

and

$\hat{P}\ket{\Psi}=(\hat{P}_{R}+\hat{P}_{S})\ket{\Psi}=0$ (30)

where we have assumed $\hat{P}_{C}=0$. We notice that the mechanism works also
with $\hat{P}_{C}\neq 0$ but, in this case, there could be limitations in the
allowed momenta to ensure that (29) and (30) are simultaneously satisfied (we
discuss the case of $\hat{P}_{C}\neq 0$ in Appendix D). The framework with
$\hat{P}_{C}=0$ can be implemented for example by assuming the subspace $C$
describing the internal degree of freedom (i.e. the spin) of the system $R$.
If we want to look at the explicit form of the state $\ket{\Psi}$ we need to
know the relation between the momenta and the energy of $R$ and $S$.
Nevertheless, assuming again $d_{C},d_{R}\gg d_{S}$, we can write in general:

$\ket{\Psi}=\sum_{k=0}^{d_{S}-1}c_{k}\ket{E=-\epsilon_{k}}_{C}\otimes\ket{p=-p_{k}}_{R}\otimes\ket{p_{k}}_{S}$
(31)

where

$\epsilon_{k}=E^{(R)}(-p_{k})+E^{(S)}(p_{k})$ (32)

is the energy function related to the momenta of $R$ and $S$. For simplicity
we consider here that the energy function depends only on the momenta and not
on the coordinates. Clearly the model works also considering the presence of
external potentials in $R$ and $S$ (equations (34), (36) and (38) that we will
find in the following would be indeed still valid) but, in this case, the
state $\ket{\Psi}$ can not be written in the simple form (31) and we can not
explicitly calculate the conditional probabilities (41) and (43). For this
reason we prefer to simplify the model, as we believe this choice helps to
capture the essence of the mechanism. In Section III.E we give a
generalization to the case where potentials are present in $R$ and $S$. So,
considering for example $R$ and $S$ as free particles (with mass $M$ and $m$
respectively), we have $\hat{H}_{R}=\frac{\hat{P}^{2}_{R}}{2M}$ and
$\hat{H}_{S}=\frac{\hat{P}^{2}_{S}}{2m}$, which implies
$\epsilon_{k}=\frac{p^{2}_{k}}{2M}+\frac{p^{2}_{k}}{2m}$.

Starting from the state $\ket{\Psi}$ satisfying (29) and (30), we can now
expand it on the basis $\left\\{\ket{t_{m}}_{C}\right\\}$ in $C$ thanks to
(26), thus obtaining

$\begin{split}\ket{\Psi}&=\frac{d_{C}}{D_{C}}\sum_{m=0}^{D_{C}-1}\ket{t_{m}}\braket{t_{m}}{\Psi}=\\\
&=\frac{\sqrt{d_{C}}}{D_{C}}\sum_{m=0}^{D_{C}-1}\ket{t_{m}}_{C}\otimes\ket{\phi(t_{m})}_{R,S}\end{split}$
(33)

where $\ket{\phi(t_{m})}_{R,S}=\sqrt{d_{C}}\braket{t_{m}}{\Psi}$ is state of
the composite system $R+S$ at time $t_{m}$, namely it is the relative state
everett  of $R+S$ conditioned on having the value $t_{m}$ on $C$. For such a
state, through (29) and the relative state definition, it is easy to find the
time evolution with respect to the clock $C$:

$\begin{split}\ket{\phi(t_{m})}_{R,S}&=\sqrt{d_{C}}\braket{t_{m}}{\Psi}=\\\
&=\sqrt{d_{C}}\bra{t_{0}}e^{i\hat{H}_{C}(t_{m}-t_{0})}\ket{\Psi}=\\\
&=\sqrt{d_{C}}\bra{t_{0}}e^{-i(\hat{H}_{R}+\hat{H}_{S}-\hat{H})(t_{m}-t_{0})}\ket{\Psi}=\\\
&=e^{-i(\hat{H}_{R}+\hat{H}_{S})(t_{m}-t_{0})}\ket{\phi(t_{0})}_{R,S}\end{split}$
(34)

where $\ket{\phi(t_{0})}_{R,S}=\sqrt{d_{C}}\braket{t_{0}}{\Psi}$ is the state
of $R+S$ conditioned on $t_{0}$ that is the value of the clock taken as
initial time. Equation (34) shows, as expected, the simultaneous evolution of
$R$ and $S$ over time. Having indeed considered a quantum spatial reference
frame, it was reasonable to expect that it also evolves in time together with
the subsystem $S$. We can consider then the limiting case
$s\longrightarrow\infty$ where $t$ takes all the real values between $t_{0}$
and $t_{0}+T$. The global state can now be written

$\begin{split}\ket{\Psi}=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}dt\ket{t}\braket{t}{\Psi}=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}dt\ket{t}_{C}\otimes\ket{\phi(t)}_{R,S}\end{split}$
(35)

and defining the relative state of $R+S$ as
$\ket{\phi(t)}_{R,S}=\braket{t}{\Psi}$ we obtain nostro :

$i\frac{\partial}{\partial
t}\ket{\phi(t)}_{R,S}=\left(\hat{H}_{R}+\hat{H}_{S}\right)\ket{\phi(t)}_{R,S}$
(36)

that is the Schrödinger evolution for the state of $R+S$ with respect to the
clock time $t$, written in the usual differential form.

We can therefore expand the state $\ket{\Psi}$ in the coordinates
$\left\\{\ket{x_{j}}_{R}\right\\}$ in $R$, thus obtaining:

$\begin{split}\ket{\Psi}&=\frac{d_{R}}{D_{R}}\sum_{j=0}^{D_{R}-1}\ket{x_{j}}\braket{x_{j}}{\Psi}=\\\
&=\frac{\sqrt{d_{R}}}{D_{R}}\sum_{j=0}^{D_{R}-1}\ket{x_{j}}_{R}\otimes\ket{\varphi(x_{j})}_{C,S}\end{split}$
(37)

where $\ket{\varphi(x_{j})}_{C,S}=\sqrt{d_{R}}\braket{x_{j}}{\Psi}$ is the
relative state of $C+S$ conditioned to the value $x_{j}$ on the reference
frame $R$. All the results found in the previous Section apply to the state
$\ket{\varphi(x_{j})}_{C,S}$. Indeed we have also in this case

$\ket{\varphi(x_{j})}_{C,S}=e^{-i\hat{P}_{S}(x_{j}-x_{0})}\ket{\varphi(x_{0})}_{C,S}$
(38)

where the momentum of the clock $C$ does not appear since we have chosen
$\hat{P}_{C}=0$. Also here we consider the limit $z\longrightarrow\infty$,
where again $x$ can take all the real values between $x_{0}$ and
$x_{0}+L_{R}$. In this case the global state can be written

$\ket{\Psi}=\frac{1}{L_{R}}\int_{x_{0}}^{x_{0}+L_{R}}dx\ket{x}\braket{x}{\Psi}=\\\
=\frac{1}{L_{R}}\int_{x_{0}}^{x_{0}+L_{R}}dx\ket{x}_{R}\otimes\ket{\varphi(x)}_{C,S}$
(39)

and defining the relative state of $C+S$ as
$\ket{\varphi(x)}_{C,S}=\braket{x}{\Psi}$ we obtain
$\hat{P}_{S}\ket{\varphi(x)}_{C,S}=i\frac{\partial}{\partial
x}\ket{\varphi(x)}_{C,S}$. Through this latter equation and (38) we can see
again that $\hat{P}_{S}$ is the generator of translations in the coordinate
values $x$ for the relative state $\ket{\varphi(x)}_{C,S}$.

Finally we can expand the state $\ket{\Psi}$ simultaneously on the coordinates
$\left\\{\ket{x_{j}}_{R}\right\\}$ in $R$ and on the time basis
$\left\\{\ket{t_{m}}_{C}\right\\}$ in $C$. We have for the global state:

$\begin{split}\ket{\Psi}&=\left(\frac{d_{C}}{D_{C}}\sum_{m=0}^{D_{C}-1}\ket{t_{m}}\bra{t_{m}}\otimes\frac{d_{R}}{D_{R}}\sum_{j=0}^{D_{R}-1}\ket{x_{j}}\bra{x_{j}}\right)\ket{\Psi}=\\\
&=\frac{\sqrt{d_{C}}}{D_{C}}\frac{\sqrt{d_{R}}}{D_{R}}\sum_{m=0}^{D_{C}-1}\sum_{j=0}^{D_{R}-1}\ket{t_{m}}_{C}\otimes\ket{x_{j}}_{R}\otimes\ket{\psi(t_{m},x_{j})}_{S}\end{split}$
(40)

where
$\ket{\psi(t_{m},x_{j})}_{S}=\sqrt{d_{C}}\sqrt{d_{R}}(\bra{t_{m}}\otimes\bra{x_{j}})\ket{\Psi}$
is the relative state of the system $S$ at time $t_{m}$ conditioned on the
value $x_{j}$ for the reference frame $R$. With the state
$\ket{\psi(t_{m},x_{j})}_{S}$ we have not yet defined the position of the
system $S$. This state indeed gives us the value of the time that enters as a
parameter thanks to the entanglement with the subspace $C$ and indicates the
position of the reference frame $R$. What we can now search is the conditional
probability of having a certain position $y_{l}$ in $S$ at time $t_{m}$ and
knowing that the reference frame is in $x_{j}$, that is (see Appendix E):

$P(y_{l}\>on\>S\>|\>x_{j}\>on\>R,\>t_{m}\>on\>C)=\\\
=\frac{d_{S}}{D_{S}}|\braket{y_{l}}{\psi(t_{m},x_{j})}|^{2}=\frac{1}{D_{S}}\left|\sum_{k=0}^{d_{S}-1}c_{k}e^{-i\epsilon_{k}t_{m}}e^{ip_{k}(y_{l}-x_{j})}\right|^{2}$
(41)

where, we recall, $\epsilon_{k}$ is the energy function related to the momenta
$p_{k}$ of $R$ and $S$ and where it is easy to verify that
$\sum_{l=0}^{D_{S}-1}P(y_{l}\>on\>S\>|\>x_{j}\>on\>R,\>t_{m}\>on\>C)=1$ given
each $x_{j}$ and $t_{m}$. Clearly we can extend these results also to the
limiting cases $z,s\longrightarrow\infty$. Indeed we can write the global
state $\ket{\Psi}$ as

$\begin{split}\ket{\Psi}&=\left(\frac{1}{T}\int_{t_{0}}^{t_{0}+T}dt\ket{t}\bra{t}\otimes\frac{1}{L_{R}}\int_{x_{0}}^{x_{0}+L_{R}}dx\ket{x}\bra{x}\right)\ket{\Psi}=\\\
&=\frac{1}{T}\frac{1}{L_{R}}\int_{t_{0}}^{t_{0}+T}dt\int_{x_{0}}^{x_{0}+L_{R}}dx\ket{t}_{C}\otimes\ket{x}_{R}\otimes\ket{\psi(t,x)}_{S}\end{split}$
(42)

where again $\ket{\psi(t,x)}_{S}=(\bra{t}\otimes\bra{x})\ket{\Psi}$ is the
relative state of the system $S$ at time $t$ conditioned on the value $x$ for
the reference frame $R$. The conditional probability density of having a
certain position $y$ in $S$ at time $t$ and knowing that the reference $R$ is
in $x$ is (see Appendix F):

$P(y\>on\>S\>|\>x\>on\>R,\>t\>on\>C)=\\\
=\frac{1}{L_{S}}\left|\braket{y}{\psi(t,x)}\right|^{2}=\frac{1}{L_{S}}\left|\sum_{k=0}^{d_{S}-1}c_{k}e^{-i\epsilon_{k}t}e^{ip_{k}(y-x)}\right|^{2}.$
(43)

We notice that also the probability density (43) is well-defined for each time
(indeed it is easy to verify that
$\int_{y_{0}}^{y_{0}+L_{S}}dyP(y\>on\>S\>|\>x\>on\>R,\>t\>on\>C)=1$ for all
$x$ and $t$) and it depends on time $t$ as well as on the distance $y-x$
between $S$ and $R$.

So, through entanglement, we have found for the subsystem $S$ a conditional
probability density that give us informations about the evolution of $S$ in
time and space, where for time we consider the clock time and for space we
consider the relative distance between $S$ and the quantum reference frame
$R$. All these results are obtained within a globally static and
“positionless”Universe.

To conclude this paragraph we notice that a good spatial reference frame is a
reference that moves only slightly in time. If a good spatial reference frame
is considered, one can look at the evolution of $S$ by itself. We show this
point with an example: assuming $R$ and $S$ as free particles (with mass $M$
and $m$ respectively), we could take $M\gg m$ thus obtaining
$\frac{\hat{P}^{2}_{R}}{2M}\ll\frac{\hat{P}^{2}_{S}}{2m}$. Starting from the
state (42) we can consider the relative state
$\ket{\psi(t,x)}_{S}=(\bra{t}\otimes\bra{x})\ket{\Psi}$ and investigate its
evolution. If the mass $M$ is sufficiently large, we have:

$i\frac{\partial}{\partial
t}\ket{\psi(t,x)}_{S}\simeq\hat{H}_{S}\ket{\psi(t,x)}_{S}.$ (44)

Equation (44) shows that if $M$ is sufficiently large, the evolution of $S$
alone can be recovered with respect to time $t$ and with respect to a spatial
reference frame that does not evolve (or that move negligibly in time).
Furthermore, equation (44) together with the property
$\hat{P}_{S}\ket{\psi(t,x)}_{S}=i\frac{\partial}{\partial
x}\ket{\psi(t,x)}_{S}$ lead to:

$i\frac{\partial}{\partial
t}\ket{\psi(t,x)}_{S}\simeq-\frac{1}{2m}\frac{\partial^{2}}{\partial
x^{2}}\ket{\psi(t,x)}_{S}$ (45)

which clearly describes the dynamics of the particle in $S$ with respect to
the coordinates of the $1+1$ dimensional quantum reference frame. We emphasize
here that, in this case, we can write the equation (45) for the state
$\ket{\psi(t,x)}_{S}$ because the values of time and space of the subspaces
$C$ and $R$ enter as parameters in $S$ thanks to the entanglement present in
the global state $\ket{\Psi}$. We will return to this point later, in Section
V, when we also discuss the example of relativistic particles.

###  III.3 A simple Example

We consider here a simple example assuming $R$ and $S$ as free particles with
mass $M$ and $m$ respectively and $d_{R}=d_{S}=3$. We start assuming
$D_{R}=D_{S}=d_{R}=d_{S}$, $L_{R}=L_{S}=L$ and discrete values of space and
time. We have therefore: $p^{(R)}_{k}=p^{(S)}_{k}=p_{0}+\frac{2\pi}{L}k$, with
$p_{0}=-\frac{2\pi}{L}$ (that implies $p_{1}=0$, $p_{2}=\frac{2\pi}{L}$),
$x^{(R)}_{j}=x_{0}+j\frac{L}{3}=0,\frac{L}{3},\frac{2L}{3}$ and
$y^{(S)}_{l}=y_{0}+l\frac{L}{3}=0,\frac{L}{3},\frac{2L}{3}$. The global state
satisfying the constraints on total energy and total momentum can be written
as:

$\ket{\Psi}=c_{0}\ket{E_{2,0}}_{C}\ket{p_{2}}_{R}\ket{p_{0}}_{S}+c_{1}\ket{E_{1,1}}_{C}\ket{p_{1}}_{R}\ket{p_{1}}_{S}+c_{2}\ket{E_{0,2}}_{C}\ket{p_{0}}_{R}\ket{p_{2}}_{S}$
(46)

where $E_{k,n}=-(\frac{p^{2}_{k}}{2M}+\frac{p^{2}_{n}}{2m})$ and where we
assume, for simplicity, the coefficients $c_{i}$ to be real. Furthermore we
have
$E_{2,0}=E_{0,2}=\epsilon=\left(\frac{2\pi}{L}\right)^{2}(\frac{1}{2M}+\frac{1}{2m})$
and $E_{1,1}=0$. We can now expand the global state $\ket{\Psi}$
simultaneously on the coordinates $\left\\{\ket{x_{j}}_{R}\right\\}$ in $R$
and on the time basis $\left\\{\ket{t_{m}}_{C}\right\\}$ in $C$, and then we
search the state
$\ket{\psi(t_{m},x_{j})}_{S}=\sqrt{d_{C}}\sqrt{3}(\bra{t_{m}}\otimes\bra{x_{j}})\ket{\Psi}$,
thus obtaining:

$\begin{split}\ket{\psi(t_{m},x_{j})}_{S}&=\sqrt{3}\left[c_{0}e^{-i\epsilon
t_{m}}\braket{x_{j}}{p_{2}}\ket{p_{0}}_{S}+c_{1}\braket{x_{j}}{p_{1}}\ket{p_{1}}_{S}+c_{2}e^{-i\epsilon
t_{m}}\braket{x_{j}}{p_{0}}\ket{p_{2}}_{S}\right]=\\\ &=c_{0}e^{-i\epsilon
t_{m}}e^{i\frac{2\pi}{L}x_{j}}\ket{-\frac{2\pi}{L}}_{S}+c_{1}\ket{0}_{S}+c_{2}e^{-i\epsilon
t_{m}}e^{-i\frac{2\pi}{L}x_{j}}\ket{\frac{2\pi}{L}}_{S}.\end{split}$ (47)

We can now calculate the conditional probability of obtaining $y_{l}$ on $S$
conditioned on having $x_{j}$ on $R$ and $t_{m}$ on $C$. Considering
$d_{S}=D_{S}=3$, we have:

$P(y_{l}\>on\>S\>|\>x_{j}\>on\>R,\>t_{m}\>on\>C)=\left|\braket{y_{l}}{\psi(t_{m},x_{j})}\right|^{2}=\frac{1}{3}\left|c_{0}e^{-i\epsilon
t_{m}}e^{-i\frac{2\pi}{L}(y_{l}-x_{j})}+c_{1}+c_{2}e^{-i\epsilon
t_{m}}e^{i\frac{2\pi}{L}(y_{l}-x_{j})}\right|^{2}.$ (48)

Proceeding with the calculations from equation (48) and remembering we have
real coefficients, we obtain

$P(y_{l}\>on\>S\>|\>x_{j}\>on\>R,\>t_{m}\>on\>C)=\frac{1}{3}+\frac{2}{3}c_{0}c_{1}\cos(\epsilon
t_{m}+\frac{2\pi}{L}(y_{l}-x_{j}))+\\\ +\frac{2}{3}c_{1}c_{2}\cos(\epsilon
t_{m}-\frac{2\pi}{L}(y_{l}-x_{j}))+\frac{2}{3}c_{0}c_{2}\left(1-2\sin^{2}(\frac{2\pi}{L}(y_{l}-x_{j}))\right)$
(49)

that is the expression of how the probability of having a certain relative
distance between the particles $S$ and $R$ given $x_{j}$ for the reference
frame $R$ at time $t_{m}$.

If we consider the limiting cases $s,z\longrightarrow\infty$ (mantaining the
assumption $L_{R}=L_{S}=L$), we obtain for the probability density:

$P(y\>on\>S\>|\>x\>on\>R,\>t\>on\>C)=\frac{1}{L}+\frac{2}{L}c_{0}c_{1}\cos(\epsilon
t+\frac{2\pi}{L}(y-x))+\\\ +\frac{2}{L}c_{1}c_{2}\cos(\epsilon
t-\frac{2\pi}{L}(y-x))+\frac{2}{L}c_{0}c_{2}\left(1-2\sin^{2}(\frac{2\pi}{L}(y-x))\right).$
(50)

###  III.4 On the Quantum Speed Limit Time

An important question is the time scale over which the conditioned state
$\ket{\phi(t)}_{R,S}$ evolve into an othogonal configuration, thus becoming
fully distinguishable. In dealing with this topic it is useful what was
already said in Section II.C. For the sake of simplicity we consider again the
case of $d_{R}=D_{R}$, $d_{S}=D_{S}$ that leads to discrete values of space
and orthogonal states. For the temporal degree of freedom we consider the
limiting case $s\longrightarrow\infty$. The conditioned state of $R+S$ can be
obtained from (31), calculating $\ket{\phi(t)}_{R,S}=\braket{t}{\Psi}$. We
have ($\hslash\neq 1$):

$\ket{\phi(t)}_{R,S}=\sum_{k=0}^{d_{S}-1}c_{k}e^{-i\hslash^{-1}\epsilon_{k}t}\ket{p=-p_{k}}_{R}\otimes\ket{p_{k}}_{S}$
(51)

where $\epsilon_{k}=E^{(R)}(-p_{k})+E^{(S)}(p_{k})$ is the energy function
related to the momenta $p_{k}$ of $R$ and $S$. Starting from equation (51) we
can calculate in the space $R+S$:

$\begin{split}\braket{\phi(t_{0})}{\phi(t)}=\sum_{k=0}^{d_{S}-1}|c_{k}|^{2}e^{-i\hslash^{-1}\epsilon_{k}(t-t_{0})}.\end{split}$
(52)

Looking at equation (52), we can now consider the quantum speed limit time
$\delta t$ which gives us the minimum time needed for $R+S$ to evolve to an
orthogonal configuration. We have speedlimit ; margolus :

$\delta t\geq max\left(\frac{\pi\hslash}{2E_{R,S}},\frac{\pi\hslash}{2\Delta
E}\right)$ (53)

where $E_{R,S}=\bra{\phi(t_{0})}\hat{H}_{R}+\hat{H}_{S}\ket{\phi(t_{0})}$ and
$\Delta E$ is the spread in energy related to the coefficients $c_{k}$ through

$\Delta
E=\sqrt{\bra{\phi(t_{0})}(\hat{H}_{R}+\hat{H}_{S}-E_{R,S})^{2}\ket{\phi(t_{0})}}.$
(54)

The aspect we emphasize is that, as in the case of space, the function
$f(t-t_{0})=\braket{\phi(t_{0})}{\phi(t)}$, and consequently the time scale on
which $R+S$ varies significantly, is not related to the overlap of the states
of the clock. This can be seen considering that in $f(t-t_{0})$ do not enter
time values different from $t,t_{0}$ and $f(t-t_{0})$ takes the form expressed
in Ref. speedlimit, . So the fact that our time states are not orthogonal does
not have a consequence on the speed at which the state $\ket{\phi(t)}_{R,S}$
evolves with respect to $t$. Rather $f(t-t_{0})$ is related to the spread of
the coefficients $c_{k}$ appearing in the state (51). These considerations,
together with equation (53), indicate the key point in this framework: a large
spread in the coefficients $c_{k}$ within state $\ket{\phi(t)}_{R,S}$, and so
in the global state
$\ket{\Psi}=\sum_{k=0}^{d_{S}-1}c_{k}\ket{E=-\epsilon_{k}}_{C}\otimes\ket{p=-p_{k}}_{R}\otimes\ket{p_{k}}_{S}$,
is needed to make the time evolution of the subsystem $R+S$ faster asimmetry .

###  III.5 Introducing Potentials in $R$ and $S$

We illustrate here the case in which potentials are present within $R$ and $S$
by considering these two subspaces consisting of two harmonic oscillators. We
use this example also to extend our framework assuming momentum operators with
continuous spectra. So far we have indeed always used momentum operators in
$R$ and $S$ with discrete spectra, but the entire discussion can be easily
generalized to the case of continuous values for momenta and coordinates, with
orthogonal position states. Such a framework allows us to more easily address
the problem of considering the presence of potentials in $R$ and $S$. What we
need is indeed to have Hermitian operators $\hat{X}$ (within $R$) and
$\hat{Y}$ (within $S$) in order to describe the potentials of the harmonic
oscillators. In the subspace $C$ we assume the framework described so far
working in the limiting case $s\longrightarrow\infty$.

We therefore start considering the Hamiltonians of $R$ and $S$ written as:

$\begin{split}&\hat{H}_{R}=\frac{\hat{P}^{2}_{R}}{2M}+V_{R}(\hat{X})=\frac{\hat{P}^{(2)}_{R}}{2M}+\frac{1}{2}M\omega_{R}\hat{X}^{2}\\\
&\hat{H}_{S}=\frac{\hat{P}^{2}_{S}}{2m}+V_{S}(\hat{Y})=\frac{\hat{P}^{(2)}_{S}}{2m}+\frac{1}{2}m\omega_{S}\hat{Y}^{2}.\end{split}$
(55)

In this framework, the global state $\ket{\Psi}$ satisfying the constraint on
total momentum $\left(\hat{P}_{R}+\hat{P}_{S}\right)\ket{\Psi}=0$ is:

$\ket{\Psi}=\sum_{n=0}^{d_{C}-1}\int
dp~{}c_{n}\psi(p)\ket{E_{n}}_{C}\otimes\ket{-p}_{R}\otimes\ket{p}_{S}$ (56)

where $\sum_{n=0}^{d_{C}-1}|c_{n}|^{2}=\int dp|\psi(p)|^{2}=1$. The state (56)
can be rewritten in the energy eigenbasis for $R$ and $S$ as

$\ket{\Psi}=\sum_{n=0}^{d_{C}-1}\sum_{k}\sum_{l}\int dp~{}c_{n}\psi(p)\\\
\times\beta(-p,E_{k})\beta(p,E_{l})\ket{E_{n}}_{C}\otimes\ket{E_{k}}_{R}\otimes\ket{E_{l}}_{S}$
(57)

where $\beta(a,b)=\braket{b}{a}$. Through the state (57) we can now impose the
constraint on total energy
$\hat{H}\ket{\Psi}=\left(\hat{H}_{C}+\hat{H}_{R}+\hat{H}_{S}\right)\ket{\Psi}=0$
where the Hamiltonians of $R$ and $S$ can be rewritten ($\hslash=1$):
$\hat{H}_{R}=\omega_{R}\left(\hat{a}^{\dagger}_{R}\hat{a}_{R}+\frac{1}{2}\right)$
and
$\hat{H}_{S}=\omega_{S}\left(\hat{a}^{\dagger}_{S}\hat{a}_{S}+\frac{1}{2}\right)$,
being $\hat{a}^{\dagger}_{R}$, $\hat{a}^{\dagger}_{S}$, $\hat{a}_{R}$ and
$\hat{a}_{S}$ the usual rising and lowering operators for the subsystems $R$
and $S$. For the global state $\ket{\Psi}$ we thus find:

$\ket{\Psi}=\sum_{k}\sum_{l}\int dp~{}\tilde{c}_{kl}\psi(p)\\\
\times\beta(-p,E_{k})\beta(p,E_{l})\ket{E=-\epsilon_{kl}}_{C}\otimes\ket{E_{k}}_{R}\otimes\ket{E_{l}}_{S}$
(58)

with

$\epsilon_{kl}=\omega_{R}\left(k+\frac{1}{2}\right)+\omega_{S}\left(l+\frac{1}{2}\right).$
(59)

The state (58) provides the explicit form of the global state of our quantum
Universe simultaneously satisfying both the constraints: on total energy and
total momentum. We can then expand again the subspaces $R$ and $S$ back on the
momentum eigenbasis and rewriting (58) as

$\ket{\Psi}=\int dp^{\prime}\sum_{k}\sum_{l}\int
dp~{}\tilde{c}_{kl}\psi(p)\beta(-p,E_{k})\beta(p,E_{l})\\\
\times\beta^{*}(-p^{\prime},E_{k})\beta^{*}(p^{\prime},E_{l})\ket{E=-\epsilon_{kl}}_{C}\otimes\ket{-p^{\prime}}_{R}\otimes\ket{p^{\prime}}_{S}.$
(60)

Through the states (58) and (60) we can find again all the results shown in
Section III.B. For the relative state $\ket{\phi(t)}_{R,S}=\braket{t}{\Psi}$
we have:

$i\frac{\partial}{\partial
t}\ket{\phi(t)}_{R,S}=\left(\hat{H}_{R}+\hat{H}_{S}\right)\ket{\phi(t)}_{R,S}=\\\
=\left(\frac{\hat{P}^{(2)}_{R}}{2M}+\frac{1}{2}M\omega_{R}\hat{X}^{2}+\frac{\hat{P}^{(2)}_{S}}{2m}+\frac{1}{2}m\omega_{S}\hat{Y}^{2}\right)\ket{\phi(t)}_{R,S}.$
(61)

which provides the Schrödinger evolution for the subsystem $R+S$. In the same
way, for the relative state
$\ket{\varphi(x)}_{C,S}=\sqrt{2\pi}\braket{x}{\Psi}$ we have:
$\ket{\varphi(x+a)}_{C,S}=e^{-ia\hat{P}_{S}}\ket{\varphi(x)}_{C,S}$ which
shows how the operator $\hat{P}_{S}$ is the generator of translations in the
values of the reference frame $R$ for the relative state of the subsystem
$C+S$. Finally we expand the state $\ket{\Psi}$ simultaneously on the
coordinates in $R$ and on the time basis in $C$ thus finding:

$\begin{split}\ket{\Psi}=\frac{1}{T\sqrt{2\pi}}\int dt\int
dx\ket{t}_{C}\otimes\ket{x}_{R}\otimes\ket{\psi(t,x)}_{S}\end{split}$ (62)

where the integral on $dt$ is evaluated from $t_{0}$ and $t_{0}+T$ and where
now the relative state
$\ket{\psi(t,x)}_{S}=\sqrt{2\pi}\left(\bra{t}\otimes\bra{x}\right)\ket{\Psi}$
reads

$\ket{\psi(t,x)}_{S}=\int dp^{\prime}\sum_{k}\sum_{l}\int
dp~{}\tilde{c}_{kl}\psi(p)\beta(-p,E_{k})\beta(p,E_{l})\\\
\times\beta^{*}(-p^{\prime},E_{k})\beta^{*}(p^{\prime},E_{l})e^{-i\epsilon_{kl}t}e^{-ixp^{\prime}}\ket{p^{\prime}}_{S}.$
(63)

Through the state (63) we can search also in this case the probability density
$P(y\>on\>S\>|\>x\>on\>R,\>t\>on\>C)=\left|\braket{y}{\psi(t,x)}\right|^{2}$
obtaining:

$P(y\>on\>S\>|\>x\>on\>R,\>t\>on\>C)=\\\ =\frac{1}{2\pi}|\int
dp^{\prime}\sum_{k}\sum_{l}\int
dp~{}\tilde{c}_{kl}\psi(p)\beta(-p,E_{k})\beta(p,E_{l})\\\
\times\beta^{*}(-p^{\prime},E_{k})\beta^{*}(p^{\prime},E_{l})e^{-i\epsilon_{kl}t}e^{ip^{\prime}(y-x)}|^{2}$
(64)

which still depends on time $t$ and on the relative distance $y-x$ between $S$
and $R$.

##  IV Multiple Time Measurements

Throughout this work, we have seen how conditional probabilities lie at the
heart of the PaW mechanism. This aspect of the theory has been criticised by
K. V. Kuchar kuchar  who emphasized that the PaW mechanism is not able to
provide the correct propagators when considering multiple measurements. Indeed
measurements performed at two times return wrong statistics because the first
measurement “collapses”the time state and freezes the rest of the Universe
(namely $R$ and $S$ in our framework). Two possible ways out of this problem
have been proposed: the first by R. Gambini, R. A. Porto, J. Pullin, and S.
Torterolo (GPPT) in Ref. gppt,  (see also Ref. esp1, ) and the second by V.
Giovannetti, S. Lloyd and L. Maccone (GLM) in Ref. lloydmaccone, .

We will now explore how these proposals can be applied to our framework of
emerging spacetime. For the sake of simplicity we consider again a simple
case: we start from the framework described in Sections II.A and III.A, we
work with discrete values of space and time and we choose $d_{C}=D_{C}$,
$d_{R}=D_{R}$, $d_{S}=D_{S}$. This latter assumption implies that we are
considering an equally-spaced spectrum for $\hat{H}_{C}$ and we emphasize that
this choice leads to orthogonal states of time and position, namely
$\braket{t_{m}}{t_{m^{\prime}}}=\delta_{m,m^{\prime}}$ on $C$,
$\braket{x_{i}}{x_{j}}=\delta_{i,j}$ on $R$ and
$\braket{y_{l}}{y_{k}}=\delta_{l,k}$ on $S$. In the case of GPPT theory, this
simplified framework can then be readily generalized to the case of unequally-
spaced levels for $\hat{H}_{C}$ and to the limiting cases
$z,s\longrightarrow\infty$, where time and position are represented by POVMs.
Conversely, for the GLM proposal the assumption of orthogonal time states in
$C$ and orthogonal position states in $R$ and $S$ will be necessary.

###  IV.1 The GPPT proposal

As pointed out in Ref. esp1,  one of the main ingredients of the GPPT theroy
is the averaging over the abstract coordinate time (the “external time”) in
order to eliminate any external time dependence in the observables. With the
perspective of calculating the probabilities for multiple time measurements,
we look in this section at the probability
$P(y_{l}\>on\>S,\>x_{j}\>on\>R\>|\>t_{m}\>on\>C)$ of having $y_{l}$ on $S$ and
$x_{j}$ on $R$ conditioned to having $t_{m}$ on $C$. This probability
essentially is not different from the probabilities we have calculated so far
(apart from a numerical factor). We will see indeed that it will depend on the
relative distance between $R$ and $S$ in the same way as we found in the
previous Section. What clearly is different is the interpretation of this
probability, where the value of the reference frame $R$ is not given and can
vary. Following GPPT this probability is given by gppt :

$P(y_{l}\>on\>S,\>x_{j}\>on\>R\>|\>t_{m}\>on\>C)=\frac{\int
d\theta\>Tr\left[\hat{\Pi}_{t_{m},x_{j},y_{l}}(\theta)\hat{\rho}\right]}{\int
d\theta\>Tr\left[\hat{\Pi}_{t_{m}}(\theta)\hat{\rho}\right]}=\\\
=\frac{1}{d_{R}}\frac{1}{d_{S}}\left|\sum_{k=0}^{d_{S}-1}c_{k}e^{-i\epsilon_{k}t_{m}}e^{ip_{k}(y_{l}-x_{j})}\right|^{2}$
(65)

where
$\hat{\rho}=\ket{\Psi}\bra{\Psi}=\sum_{k}\sum_{k^{\prime}}c_{k}c^{*}_{k^{\prime}}\ket{E=-\epsilon_{k}}\bra{E=-\epsilon_{k^{\prime}}}\otimes\ket{p=-p_{k}}\bra{p=-p_{k^{\prime}}}\otimes\ket{p_{k}}\bra{p_{k^{\prime}}}$
is the global state of the Universe, $\theta$ is the external time,
$\hat{\Pi}_{t_{m}}(\theta)=\hat{U}^{\dagger}(\theta)\hat{\Pi}_{t_{m}}\hat{U}(\theta)$
(with $\hat{U}(\theta)=e^{-i\hat{H}\theta}$) is the projector relative to the
result $t_{m}$ for a clock measurement at external time $\theta$ and
$\hat{\Pi}_{t_{m},x_{j},y_{l}}(\theta)=\hat{U}^{\dagger}(\theta)\hat{\Pi}_{t_{m},x_{j},y_{l}}\hat{U}(\theta)$
is the projector relative to the result $y_{l}$ for a measurement on $S$,
$x_{j}$ for a measurement on $R$ and $t_{m}$ for a measurement on $C$ at
external time $\theta$ (we are working here in the Heisenberg picture with
respect to the external time $\theta$). Equation (65) now takes the place of
equations (41) and (43) and, as expected, depends on time $t_{m}$ and on the
distance $y_{l}-x_{j}$ between the two subsystems $S$ and $R$.

Equation (65) can be generalized to the case of multiple time measurements.
For two measurements at times $t_{m}$ and $t_{m^{\prime}}$ (with
$t_{m^{\prime}}>t_{m}$) we have:

$P(y_{l^{\prime}}\>on\>S,x_{j^{\prime}}\>on\>R\>|\>t_{m^{\prime}}\>on\>C,y_{l},x_{j},t_{m})=\\\
=\frac{\int d\theta\int
d\theta^{\prime}\>Tr\left[\hat{\Pi}_{t_{m^{\prime}},x_{j^{\prime}},y_{l^{\prime}}}(\theta)\hat{\Pi}_{t_{m},x_{j},y_{l}}(\theta^{\prime})\hat{\rho}\hat{\Pi}_{t_{m},x_{j},y_{l}}(\theta^{\prime})\right]}{\int
d\theta\int
d\theta^{\prime}\>Tr\left[\hat{\Pi}_{t_{m^{\prime}}}(\theta)\hat{\Pi}_{t_{m},x_{j},y_{l}}(\theta^{\prime})\hat{\rho}\hat{\Pi}_{t_{m},x_{j},y_{l}}(\theta^{\prime})\right]}$
(66)

which provides the conditional probability of obtaining $y_{l^{\prime}}$ and
$x_{j^{\prime}}$ on $S$ and $R$ at clock time $t_{m^{\prime}}$, given that a
“previous”joint measurement of $S$, $R$ and $C$ returns $y_{l}$, $x_{j}$ and
$t_{m}$. Proceeding with the calculations from the equation (66) we obtain:

$P(y_{l^{\prime}}\>on\>S,x_{j^{\prime}}\>on\>R\>|\>t_{m^{\prime}}\>on\>C,y_{l},x_{j},t_{m})=\\\
=\frac{1}{d^{2}_{R}}\frac{1}{d^{2}_{S}}\left|\sum_{k=0}^{d_{S}-1}e^{-i\epsilon_{k}(t_{m^{\prime}}-t_{m})}e^{ip_{k}(\Delta_{f}-\Delta_{i})}\right|^{2}$
(67)

where we have written $\Delta_{i}=y_{l}-x_{j}$ and
$\Delta_{f}=y_{l^{\prime}}-x_{j^{\prime}}$ respectively the initial distance
and the final distance (namely the distance at the first measurement at time
$t_{m}$ and the distance at the second measurement at time $t_{m^{\prime}}$)
between the particle $S$ and $R$. This equation provides the the correct
propagator (the same result can be indeed obtained calculating
$\left|\bra{x_{j^{\prime}}}\bra{y_{l^{\prime}}}e^{-i(\hat{H}_{R}+\hat{H}_{S})(t_{m^{\prime}}-t_{m})}\ket{x_{j}}\ket{y_{l}}\right|^{2}$)
and again depends on the initial and the final relative distances between $S$
and $R$.

As previously mentioned, these results can be readily generalized to the case
of unequally-spaced levels for $\hat{H}_{C}$ and to the limiting cases
$z,s\longrightarrow\infty$. Indeed, in our framework, the fact of using POVMs
in describing time and space does not constitute a problem in the application
of the GPPT theory because when we calculate the probability
$P(y_{l^{\prime}}\>on\>S,x_{j^{\prime}}\>on\>R\>|\>t_{m^{\prime}}\>on\>C,y_{l},x_{j},t_{m})$
through the (66) we do not find terms related to the overlap between the
states and therefore interference phenomena are not present even if the states
are not orthogonal nostro2 .

###  IV.2 GLM’s Multiple Measurements

We focus now on the GLM proposal applying it directly to our case of emergent
spacetime. In this Section we consider the global state of the Universe
written as in (33), that in our particular case of orthogonal states of time
and space ($d_{C}=D_{C}$, $d_{R}=D_{R}$, $d_{S}=D_{S}$) becomes

$\begin{split}\ket{\Psi}=\frac{1}{\sqrt{d_{C}}}\sum_{m=0}^{d_{C}-1}\ket{t_{m}}_{C}\otimes\hat{U}_{R,S}(t_{m}-t_{0})\ket{\phi(t_{0})}_{R,S}\end{split}$
(68)

where $\ket{\phi(t_{0})}_{R,S}$ is the state of $R+S$ conditioned on $t_{0}$
that is the value of the clock taken as initial time and where thanks to the
(34) we have defined
$\hat{U}_{R,S}(t_{m}-t_{0})=e^{-i(\hat{H}_{R}+\hat{H}_{S})(t_{m}-t_{0})}$.

Again we start considering to perform a single measurement within $R+S$ at
time $t_{m^{\prime}}$. We divide the spaces $R$ and $S$ into two subsystems
respectively: $\mathcal{H}_{R}=\mathcal{H}_{Q_{R}}\otimes\mathcal{H}_{M_{R}}$
and $\mathcal{H}_{S}=\mathcal{H}_{Q_{S}}\otimes\mathcal{H}_{M_{S}}$ where
$Q_{R}$, $Q_{S}$ are the systems to be measured (the observed) and $M_{R}$,
$M_{S}$ are the ancillary memory systems (the observers). In this framework
GLM use von Neumann’s prescription for measurements vonneumann , where a
measurement apparatus essentially consists in an (ideally instantaneous)
interaction between the observed and the observers. The interaction correlates
$Q_{R}$ with $M_{R}$ and $Q_{S}$ with $M_{S}$ along the eigenbasis
$\\{\ket{x_{j},y_{l}}\\}$ of the observables
$\hat{X}=\sum_{j}x_{j}\ket{x_{j}}\bra{x_{j}}$ and
$\hat{Y}=\sum_{l}y_{l}\ket{y_{l}}\bra{y_{l}}$ to be measured, that is

$\ket{\phi(t_{m})}_{Q_{R},Q_{S}}\otimes\ket{r,r}_{M_{R},M_{S}}\rightarrow\sum_{j=0}^{d_{R}-1}\sum_{l=0}^{d_{S}-1}\left(\braket{x_{j},y_{l}}{\phi(t_{m})}\right)\ket{x_{j},y_{l}}_{Q_{R},Q_{S}}\otimes\ket{x_{j},y_{l}}_{M_{R},M_{S}}$
(69)

where $\ket{r,r}_{M_{R},M_{S}}$ is the stete of the memories before the
interaction and $\braket{x_{j},y_{l}}{\phi(t_{m})}$ is the probabiliy
amplitude of obtaining $x_{j}$ and $y_{l}$ when measuring the observables
$\hat{X}$ and $\hat{Y}$. In this framework the Hamiltonian of $R+S$ can be
written as

$\hat{H}_{R}(t_{m})+\hat{H}_{S}(t_{m})=\hat{H}_{Q_{R}}+\hat{H}_{Q_{S}}+\delta_{m,m^{\prime}}\left(\hat{h}_{Q_{R},M_{R}}+\hat{h}_{Q_{S},M_{S}}\right)$
(70)

where $\hat{h}_{Q_{i},M_{i}}$ are responsible for the mapping equation (69).
So we can write the global state (68) including the measurement thus obtaining
lloydmaccone ; esp2

$\ket{\Psi}=\frac{1}{\sqrt{d_{C}}}\sum_{m<m^{\prime}}\ket{t_{m}}_{C}\otimes\hat{U}_{R,S}(t_{m}-t_{0})\ket{\phi(t_{0})}_{Q_{R},Q_{S}}\otimes\ket{r,r}_{M_{R},M_{S}}+\\\
+\frac{1}{\sqrt{d_{C}}}\sum_{m\geq
m^{\prime}}\ket{t_{m}}_{C}\otimes\sum_{j=0}^{d_{R}-1}\sum_{l=0}^{d_{S}-1}\left(\braket{x_{j},y_{l}}{\phi(t_{m})}\right)\hat{U}_{Q_{R},Q_{S}}(t_{m}-t_{m^{\prime}})\ket{x_{j},y_{l}}_{Q_{R},Q_{S}}\otimes\ket{x_{j},y_{l}}_{M_{R},M_{S}}$
(71)

where the first summation describes the evolution of $R+S$ prior to the
measurement, when the memories are in the state $\ket{r,r}_{M_{R},M_{S}}$,
whereas the second summation describes the evolution after the measurement,
when the memories are correlated with the subsystems $Q_{R}$ and $Q_{S}$. Now
the probability that, at a given time $t_{m^{\prime}}$, the values
$x_{j^{\prime}}$ and $y_{l^{\prime}}$ will be registered by the memories
$M_{R}$ and $M_{S}$ respectively can be expressed as lloydmaccone :

$P(x_{j^{\prime}},y_{l^{\prime}}\>|\>t_{m^{\prime}})=\frac{||\left({}_{C}\bra{t_{m^{\prime}}}\otimes_{M_{R}}\bra{x_{j^{\prime}}}\otimes_{M_{S}}\bra{y_{l^{\prime}}}\right)\ket{\Psi}||^{2}}{1/d_{C}}$
(72)

where we use the norm of a vector as $||\ket{v}||^{2}=\braket{v}{v}$. Equation
(72) returns the correct result

$P(x_{j^{\prime}},y_{l^{\prime}}\>|\>t_{m^{\prime}})=\left|\braket{x_{j^{\prime}},y_{l^{\prime}}}{\phi(t_{m^{\prime}})}\right|^{2}=\frac{1}{d_{R}}\frac{1}{d_{S}}\left|\sum_{k=0}^{d_{S}-1}c_{k}e^{-i\epsilon_{k}t_{m^{\prime}}}e^{ip_{k}(y_{l^{\prime}}-x_{j^{\prime}})}\right|^{2}.$
(73)

The GLM formalism allows us easily to calculate also the probability
$P(y_{l^{\prime}}\>|\>x_{j^{\prime}},t_{m^{\prime}})$ simply by dividing
equation (72) by $1/d_{R}$. In this case we obtain:
$P(y_{l^{\prime}}\>|\>x_{j^{\prime}},t_{m^{\prime}})=\frac{1}{d_{S}}\left|\sum_{k=0}^{d_{S}-1}c_{k}e^{-i\epsilon_{k}t_{m^{\prime}}}e^{ip_{k}(y_{l^{\prime}}-x_{j^{\prime}})}\right|^{2}$
in accordance with our previous results.

It is now possible to extend equations (70), (71) and (72) in order to perform
multiple measurements. The framework allows an arbitrary number of
measurements but we consider here only the simple case of a double measurement
at times $t_{m^{\prime}}$ and $t_{m^{\prime\prime}}$ (with
$t_{m^{\prime\prime}}>t_{m^{\prime}}$). What we have to do is simply consider
a larger set of memories $M^{(1)}_{R}$, $M^{(1)}_{S}$, $M^{(2)}_{R}$ and
$M^{(2)}_{S}$ (where $M^{(1)}_{R}$, $M^{(1)}_{S}$ refer to the first
measurement and $M^{(2)}_{R}$, $M^{(2)}_{S}$ refer to the second measurement)
which couple with $Q_{R}$ and $Q_{S}$ through the time-dependent Hamiltonian

$\hat{H}_{R}(t_{m})+\hat{H}_{S}(t_{m})=\hat{H}_{Q_{R}}+\hat{H}_{Q_{S}}+\delta_{m,m^{\prime}}\left(\hat{h}_{Q_{R},M^{(1)}_{R}}+\hat{h}_{Q_{S},M^{(1)}_{S}}\right)+\delta_{m,m^{\prime\prime}}\left(\hat{h}_{Q_{R},M^{(2)}_{R}}+\hat{h}_{Q_{S},M^{(2)}_{S}}\right).$
(74)

The global state (68) including the double measurement can then be written
lloydmaccone ; esp2

$\ket{\Psi}=\frac{1}{\sqrt{d_{C}}}\sum_{m<m^{\prime}}\ket{t_{m}}_{C}\otimes\hat{U}_{R,S}(t_{m}-t_{0})\ket{\phi(t_{0})}_{Q_{R},Q_{S}}\otimes\ket{r,r}_{M^{(1)}_{R},M^{(1)}_{S}}\otimes\ket{r,r}_{M^{(2)}_{R},M^{(2)}_{S}}+\\\
+\frac{1}{\sqrt{d_{C}}}\sum_{m^{\prime}\leq
m<m^{\prime\prime}}\ket{t_{m}}_{C}\otimes\sum_{j=0}^{d_{R}-1}\sum_{l=0}^{d_{S}-1}\left(\braket{x_{j},y_{l}}{\phi(t_{m})}\right)\\\
\times\hat{U}_{Q_{R},Q_{S}}(t_{m}-t_{m^{\prime}})\ket{x_{j},y_{l}}_{Q_{R},Q_{S}}\otimes\ket{x_{j},y_{l}}_{M^{(1)}_{R},M^{(1)}_{S}}\otimes\ket{r,r}_{M^{(2)}_{R},M^{(2)}_{S}}+\\\
+\frac{1}{\sqrt{d_{C}}}\sum_{m\geq
m^{\prime\prime}}\ket{t_{m}}_{C}\otimes\sum_{i=0}^{d_{R}-1}\sum_{k=0}^{d_{S}-1}\sum_{j=0}^{d_{R}-1}\sum_{l=0}^{d_{S}-1}\left(\bra{x_{i},y_{k}}\hat{U}_{Q_{R},Q_{S}}(t_{m^{\prime\prime}}-t_{m^{\prime}})\ket{x_{j},y_{l}}\right)\left(\braket{x_{j},y_{l}}{\phi(t_{m})}\right)\\\
\times\hat{U}_{Q_{R},Q_{S}}(t_{m}-t_{m^{\prime\prime}})\ket{x_{i},y_{k}}_{Q_{R},Q_{S}}\otimes\ket{x_{j},y_{l}}_{M^{(1)}_{R},M^{(1)}_{S}}\otimes\ket{x_{i},y_{k}}_{M^{(2)}_{R},M^{(2)}_{S}}.$
(75)

Through the state (75) we search now the probability of obtaining
$x_{j^{\prime\prime}}$ and $y_{l^{\prime\prime}}$ on $R$ and $S$ at time
$t_{m^{\prime\prime}}$, given that a “previous”measurement at time
$t_{m^{\prime}}$ returns $x_{j^{\prime}}$ and $y_{l^{\prime}}$. This can be
formally expressed as follows lloydmaccone

$P\left(\left(x_{j^{\prime\prime}},y_{l^{\prime\prime}}\>|\>t_{m^{\prime\prime}}\right)\>|\>\left(x_{j^{\prime}},y_{l^{\prime}}\>|\>t_{m^{\prime}}\right)\right)=\frac{P(x_{j^{\prime\prime}},y_{l^{\prime\prime}},x_{j^{\prime}},y_{l^{\prime}}\>|\>t_{m^{\prime\prime}})}{P(x_{j^{\prime}},y_{l^{\prime}}\>|\>t_{m^{\prime}})}=\\\
=\frac{||(_{C}\bra{t_{m^{\prime\prime}}}\otimes_{M^{(1)}_{R}}\bra{x_{j^{\prime}}}\otimes_{M^{(1)}_{S}}\bra{y_{l^{\prime}}}\otimes_{M^{(2)}_{R}}\bra{x_{j^{\prime\prime}}}\otimes_{M^{(2)}_{S}}\bra{y_{l^{\prime\prime}}})\ket{\Psi}||^{2}}{||(_{C}\bra{t_{m^{\prime}}}\otimes_{M^{(1)}_{R}}\bra{x_{j^{\prime}}}\otimes_{M^{(1)}_{S}}\bra{y_{l^{\prime}}})\ket{\Psi}||^{2}}$
(76)

which returns the correct result for a two-times measurement:

$P\left(\left(x_{j^{\prime\prime}},y_{l^{\prime\prime}}\>|\>t_{m^{\prime\prime}}\right)\>|\>\left(x_{j^{\prime}},y_{l^{\prime}}\>|\>t_{m^{\prime}}\right)\right)=\\\
=\left|\bra{x_{j^{\prime\prime}},y_{l^{\prime\prime}}}\hat{U}_{Q_{R},Q_{S}}(t_{m^{\prime\prime}}-t_{m^{\prime}})\ket{x_{j^{\prime}},y_{l^{\prime}}}\right|^{2}=\frac{1}{d^{2}_{R}}\frac{1}{d^{2}_{S}}\left|\sum_{k=0}^{d_{S}-1}e^{-i\epsilon_{k}(t_{m^{\prime\prime}}-t_{m^{\prime}})}e^{ip_{k}(\Delta_{f}-\Delta_{i})}\right|^{2}$
(77)

where $\Delta_{i}=y_{l^{\prime}}-x_{j^{\prime}}$ and
$\Delta_{f}=y_{l^{\prime\prime}}-x_{j^{\prime\prime}}$ are again the distances
between $R$ and $S$ at times $t_{m^{\prime}}$ and $t_{m^{\prime\prime}}$
respectively. The result (77) follows from the fact that

$P(x_{j^{\prime\prime}},y_{l^{\prime\prime}},x_{j^{\prime}},y_{l^{\prime}}\>|\>t_{m^{\prime\prime}})=|\left(\bra{x_{j^{\prime\prime}},y_{l^{\prime\prime}}}\hat{U}_{Q_{R},Q_{S}}(t_{m^{\prime\prime}}-t_{m^{\prime}})\ket{x_{j^{\prime}},y_{l^{\prime}}}\right)\times\left(\braket{x_{j^{\prime}},y_{l^{\prime}}}{\phi(t_{m^{\prime}})}\right)|^{2}$
(78)

obtained from the third summation in (75) and
$P(x_{j^{\prime}},y_{l^{\prime}}\>|\>t_{m^{\prime}})=\left|\braket{x_{j^{\prime}},y_{l^{\prime}}}{\phi(t_{m^{\prime}})}\right|^{2}$
obtained from the second summation in (75). It is finally easy to verify that
the probability
$P\left(\left(y_{l^{\prime\prime}}\>|\>x_{j^{\prime\prime}},t_{m^{\prime\prime}}\right)\>|\>\left(y_{l^{\prime}}\>|\>x_{j^{\prime}},t_{m^{\prime}}\right)\right)$
can be obtained dividing (77) by $1/d^{2}_{R}$. So we find:

$P\left(\left(y_{l^{\prime\prime}}\>|\>x_{j^{\prime\prime}},t_{m^{\prime\prime}}\right)\>|\>\left(y_{l^{\prime}}\>|\>x_{j^{\prime}},t_{m^{\prime}}\right)\right)=\frac{1}{d^{2}_{S}}\left|\sum_{k=0}^{d_{S}-1}e^{-i\epsilon_{k}(t_{m^{\prime\prime}}-t_{m^{\prime}})}e^{ip_{k}(\Delta_{f}-\Delta_{i})}\right|^{2}.$
(79)

As we mentioned previously the GLM proposal can not be generalized, in our
framework, to the case of non-orthogonal states of time and space and
consequently in applying the GLM theory we are forced to assume $D_{R}=d_{R}$,
$D_{S}=d_{S}$ and equally-spaced energy levels in $\hat{H}_{C}$.

##  V Generalization to $3+1$ Dimensional Spacetime

We generalize here the results of Section III.B to the case of $3+1$
dimensional spacetime, meaning that we have now three degrees of freedom
within the subspaces $R$ and $S$. The constraint on total energy reads again:

$\hat{H}\ket{\Psi}=(\hat{H}_{C}+\hat{H}_{R}+\hat{H}_{S})\ket{\Psi}=0.$ (80)

The Hamiltonians $\hat{H}_{R}$ and $\hat{H}_{S}$ in (80) depend respectively
on the operators $\hat{P}^{(1)}_{R}$, $\hat{P}^{(2)}_{R}$, $\hat{P}^{(3)}_{R}$
and $\hat{P}^{(1)}_{S}$, $\hat{P}^{(2)}_{S}$, $\hat{P}^{(3)}_{S}$ where
$1,2,3$ are the three degrees of freedom identifying three orthogonal
directions in space (also in this case, for simplicity, we assume that no
potentials are present in $R$ and $S$). The constraint on the total momentum
reads now

$\vec{P}\ket{\Psi}=(\vec{P}_{R}+\vec{P}_{S})\ket{\Psi}=0,$ (81)

where we consider

$\begin{split}(\hat{P}^{(J)}_{R}+\hat{P}^{(J)}_{S})\ket{\Psi}=0\>\>\>\>\>\>for\>\>J=1,2,3.\end{split}$
(82)

Also in this case, for simplicity we have chosen $\vec{P}_{C}=0$ (with
$\hat{P}^{(1)}_{C}=\hat{P}^{(2)}_{C}=\hat{P}^{(3)}_{C}=0$). So, assuming
$d_{C},d^{(1)}_{R},d^{(2)}_{R},d^{(3)}_{R}\gg
d^{(1)}_{S},d^{(2)}_{S},d^{(3)}_{S}$ and simplifying the notation as much as
possible, the global state $\ket{\Psi}$ satisfying the constraints (80) and
(82) can be written as

$\ket{\Psi}=\sum_{i=0}^{d^{(1)}_{S}-1}\sum_{j=0}^{d^{(2)}_{S}-1}\sum_{k=0}^{d^{(3)}_{S}-1}c_{ijk}\ket{-\epsilon_{ijk}}_{C}\otimes\ket{-p^{(1)}_{i},-p^{(2)}_{j},-p^{(3)}_{k}}_{R}\otimes\ket{p^{(1)}_{i},p^{(2)}_{j},p^{(3)}_{k}}_{S}$
(83)

where $d^{(1)}_{R},d^{(2)}_{R},d^{(3)}_{R}$ and
$d^{(1)}_{S},d^{(2)}_{S},d^{(3)}_{S}$ are the dimensions of the subspaces of
$R$ and $S$ (associated with the three spatial directions 1,2,3) and
$\epsilon_{ijk}$ is the energy function related with the momenta of $R$ and
$S$. In the next two paragraphs we will study separately the case of discrete
values for space and time and then the case of continuous values for space and
time.

###  V.1 Discrete Values for Space and Time

Starting from the state $\ket{\Psi}$ satisfying (80) and (82), we can now
expand it on the basis $\left\\{\ket{t_{a}}_{C}\right\\}$ in $C$ thanks to
(26), thus obtaining

$\begin{split}\ket{\Psi}=\frac{d_{C}}{D_{C}}\sum_{a=0}^{D_{C}-1}\ket{t_{a}}\braket{t_{a}}{\Psi}=\frac{\sqrt{d_{C}}}{D_{C}}\sum_{a=0}^{D_{C}-1}\ket{t_{a}}_{C}\otimes\ket{\phi(t_{a})}_{R,S}\end{split}$
(84)

where the relative state
$\ket{\phi(t_{a})}_{R,S}=\sqrt{d_{C}}\braket{t_{a}}{\Psi}$ takes now the form

$\ket{\phi(t_{a})}_{R,S}=\sum_{i=0}^{d^{(1)}_{S}-1}\sum_{j=0}^{d^{(2)}_{S}-1}\sum_{k=0}^{d^{(3)}_{S}-1}c_{ijk}e^{-it_{a}\epsilon_{ijk}}\ket{-p^{(1)}_{i},-p^{(2)}_{j},-p^{(3)}_{k}}_{R}\otimes\ket{p^{(1)}_{i},p^{(2)}_{j},p^{(3)}_{k}}_{S}.$
(85)

For this relative state we again easily find the Schrödiger evolution with
respect to the clock values, namely

$\ket{\phi(t_{a})}_{R,S}=e^{-i(\hat{H}_{R}+\hat{H}_{S})(t_{a}-t_{0})}\ket{\phi(t_{0})}_{R,S}$
(86)

where $\ket{\phi(t_{0})}_{R,S}=\sqrt{d_{C}}\braket{t_{0}}{\Psi}$ is the state
of $R+S$ conditioned on $t_{0}$ that is the value of the clock taken as
initial time. All the consideration made in Section III.B are clearly still
valid in this case and we do not repeat them.

We emphasize here that in each subspace of $R$, related to the three degrees
of freedom, we apply the formalism described in Section II. This means
considering the states
$\ket{x^{(J)}_{n}}=1/\sqrt{d^{(J)}_{R}}\sum_{k=0}^{d^{(J)}_{R}-1}e^{-ip^{(J)}_{k}x^{(J)}_{n}}\ket{p^{(J)}_{k}}$
and the values $x^{(J)}_{n}=x^{(J)}_{0}+nL^{(J)}_{R}/D^{(J)}_{R}$ with
$n=0,1,2,...,z=D^{(J)}_{R}-1$ for $J=1,2,3$. The same clearly holds for the
subspaces within the system $S$ that we equip with the states
$\ket{y^{(J)}_{q}}=1/\sqrt{d^{(J)}_{S}}\sum_{k=0}^{d^{(J)}_{S}-1}e^{-ip^{(J)}_{k}y^{(J)}_{q}}\ket{p^{(J)}_{k}}$
and with the values $y^{(J)}_{q}=y^{(J)}_{0}+qL^{(J)}_{S}/D^{(J)}_{S}$ with
$q=0,1,2,...,z=D^{(J)}_{S}-1$ (for semplicity we choose
$D^{(J)}_{R}=D^{(J)}_{S}=z+1~{}\forall J$). We can now expand the state
$\ket{\Psi}$ in the coordinates
$\left\\{\ket{x^{(1)}_{l},x^{(2)}_{m},x^{(3)}_{n}}_{R}\right\\}$ in $R$:

$\begin{split}\ket{\Psi}&=\frac{d^{(1)}_{R}}{D^{(1)}_{R}}\frac{d^{(2)}_{R}}{D^{(2)}_{R}}\frac{d^{(3)}_{R}}{D^{(3)}_{R}}\sum_{l=0}^{D^{(1)}_{R}-1}\sum_{m=0}^{D^{(2)}_{R}-1}\sum_{n=0}^{D^{(3)}_{R}-1}\ket{x^{(1)}_{l},x^{(2)}_{m},x^{(3)}_{n}}\braket{x^{(1)}_{l},x^{(2)}_{m},x^{(3)}_{n}}{\Psi}=\\\
&=\frac{\sqrt{d^{(1)}_{R}}}{D^{(1)}_{R}}\frac{\sqrt{d^{(2)}_{R}}}{D^{(2)}_{R}}\frac{\sqrt{d^{(3)}_{R}}}{D^{(3)}_{R}}\sum_{l=0}^{D^{(1)}_{R}-1}\sum_{m=0}^{D^{(2)}_{R}-1}\sum_{n=0}^{D^{(3)}_{R}-1}\ket{x^{(1)}_{l},x^{(2)}_{m},x^{(3)}_{n}}_{R}\otimes\ket{\varphi(x^{(1)}_{l},x^{(2)}_{m},x^{(3)}_{n})}_{C,S}\end{split}$
(87)

where
$\ket{\varphi(x^{(1)}_{l},x^{(2)}_{m},x^{(3)}_{n})}_{C,S}=\sqrt{d^{(1)}_{R}d^{(2)}_{R}d^{(3)}_{R}}\braket{x^{(1)}_{l},x^{(2)}_{m},x^{(3)}_{n}}{\Psi}$
is the relative state of $C+S$ conditioned to the value
$(x^{(1)}_{l},x^{(2)}_{m},x^{(3)}_{n})$ of the reference frame $R$. For this
relative state we find:

$\ket{\varphi(x^{(1)}_{l},x^{(2)}_{m},x^{(3)}_{n})}_{C,S}=\sqrt{d^{(1)}_{R}d^{(2)}_{R}d^{(3)}_{R}}\braket{x^{(1)}_{l},x^{(2)}_{m},x^{(3)}_{n}}{\Psi}=\\\
=\sqrt{d^{(1)}_{R}d^{(2)}_{R}d^{(3)}_{R}}\bra{x^{(1)}_{l},x^{(2)}_{m},x^{(3)}_{n}}e^{i\hat{P}^{(1)}_{R}(x^{(1)}_{l}-x^{(1)}_{0})}e^{i\hat{P}^{(2)}_{R}(x^{(2)}_{m}-x^{(2)}_{0})}e^{i\hat{P}^{(3)}_{R}(x^{(3)}_{n}-x^{(3)}_{0})}\ket{\Psi}=\\\
=e^{-i\hat{P}^{(1)}_{S}(x^{(1)}_{l}-x^{(1)}_{0})}e^{-i\hat{P}^{(2)}_{S}(x^{(2)}_{m}-x^{(2)}_{0})}e^{-i\hat{P}^{(3)}_{S}(x^{(3)}_{n}-x^{(3)}_{0})}\ket{\varphi(x^{(1)}_{0},x^{(2)}_{0},x^{(3)}_{0})}_{C,S}$
(88)

where we used the relative state definition and the constraint (82). Equation
(88) can be rewritten in a more compact form as

$\ket{\varphi(\vec{x_{0}}+\vec{a})}_{C,S}=e^{-i\vec{a}\cdot\vec{P}_{S}}\ket{\varphi(\vec{x_{0}})}_{C,S}$
(89)

where $\vec{P}_{S}=(\hat{P}^{(1)}_{S},\hat{P}^{(2)}_{S},\hat{P}^{(3)}_{S})$,
$\vec{x_{0}}=(x^{(1)}_{0},x^{(2)}_{0},x^{(3)}_{0})$ is the initial position of
the reference frame and the translation vector is
$\vec{a}=(x^{(1)}_{l}-x^{(1)}_{0},x^{(2)}_{m}-x^{(2)}_{0},x^{(3)}_{n}-x^{(3)}_{0})$.

Finally we can expand the state $\ket{\Psi}$ simultaneously on time
$\left\\{\ket{t_{a}}_{C}\right\\}$ and on the coordinates
$\left\\{\ket{x^{(1)}_{l},x^{(2)}_{m},x^{(3)}_{n}}_{R}\right\\}$, thus
obtaining:

$\begin{split}\ket{\Psi}&=\left(\frac{d_{C}}{D_{C}}\sum_{a}\ket{t_{a}}\bra{t_{a}}\otimes\frac{d^{(1)}_{R}}{D^{(1)}_{R}}\frac{d^{(2)}_{R}}{D^{(2)}_{R}}\frac{d^{(3)}_{R}}{D^{(3)}_{R}}\sum_{l,m,n}\ket{x^{(1)}_{l},x^{(2)}_{m},x^{(3)}_{n}}\bra{x^{(1)}_{l},x^{(2)}_{m},x^{(3)}_{n}}\right)\ket{\Psi}=\\\
&=\frac{\sqrt{d_{C}}}{D_{C}}\frac{\sqrt{d^{(1)}_{R}}}{D^{(1)}_{R}}\frac{\sqrt{d^{(2)}_{R}}}{D^{(2)}_{R}}\frac{\sqrt{d^{(3)}_{R}}}{D^{(3)}_{R}}\sum_{a}\sum_{l,m,n}\ket{t_{a}}_{C}\otimes\ket{x^{(1)}_{l},x^{(2)}_{m},x^{(3)}_{n}}_{R}\otimes\ket{\psi(t_{a},x^{(1)}_{l},x^{(2)}_{m},x^{(3)}_{n})}_{S}\end{split}$
(90)

where the summation on time runs from $0$ to $D_{C}-1$, the summations on
$l,m,n$ run from $0$ to $D^{(1)}_{R}-1$, $D^{(2)}_{R}-1$, $D^{(3)}_{R}-1$
respectively and where
$\ket{\psi(t_{a},x^{(1)}_{l},x^{(2)}_{m},x^{(3)}_{n})}_{S}=\sqrt{d_{C}}\sqrt{d^{(1)}_{R}d^{(2)}_{R}d^{(3)}_{R}}(\bra{t_{a}}\otimes\bra{x^{(1)}_{l},x^{(2)}_{m},x^{(3)}_{n}})\ket{\Psi}$
is the relative state of the system $S$ at time $t_{a}$ and conditioned on the
value $(x^{(1)}_{l},x^{(2)}_{m},x^{(3)}_{n})$ for the spatial reference frame
$R$. Through this state, extending the formalism of Section III.B, we can
search the conditional probability:

$P(y^{(1)}_{p},y^{(2)}_{q},y^{(3)}_{r}\>|\>x^{(1)}_{l},x^{(2)}_{m},x^{(3)}_{n},t_{a})=\frac{d^{(1)}_{S}}{D^{(1)}_{S}}\frac{d^{(2)}_{S}}{D^{(2)}_{S}}\frac{d^{(3)}_{S}}{D^{(3)}_{S}}\left|\braket{y^{(1)}_{p},y^{(2)}_{q},y^{(3)}_{r}}{\psi(t_{a},x^{(1)}_{l},x^{(2)}_{m},x^{(3)}_{n})}\right|^{2}=\\\
=\frac{1}{D^{(1)}_{S}}\frac{1}{D^{(2)}_{S}}\frac{1}{D^{(3)}_{S}}\left|\sum_{i,j,k}c_{ijk}e^{-it_{a}\epsilon_{ijk}}e^{ip^{(1)}_{i}(y^{(1)}_{p}-x^{(1)}_{l})}e^{ip^{(2)}_{j}(y^{(2)}_{q}-x^{(2)}_{m})}e^{ip^{(3)}_{k}(y^{(3)}_{r}-x^{(3)}_{n})}\right|^{2}$
(91)

where the summations on $i,j,k$ run from $0$ to $d^{(1)}_{S}-1$,
$d^{(2)}_{S}-1$ and $d^{(3)}_{S}-1$ respectively. Equation (91) provides the
probability of having a certain position
$(y^{(1)}_{p},y^{(2)}_{q},y^{(3)}_{r})$ on $S$ at time $t_{a}$ and knowing
that the spatial reference frame $R$ is in
$(x^{(1)}_{l},x^{(2)}_{m},x^{(3)}_{n})$. This conditional probability is well-
defined (indeed it is easy to verify that we have
$\sum_{p=0}^{D^{(1)}_{S}-1}\sum_{q=0}^{D^{(2)}_{S}-1}\sum_{r=0}^{D^{(3)}_{S}-1}P(y^{(1)}_{p},y^{(2)}_{q},y^{(3)}_{r}\>|\>x^{(1)}_{l},x^{(2)}_{m},x^{(3)}_{n},t_{a})=1$)
and, as expected depends on the relative distance between $R$ and $S$.

###  V.2 Continuous Values for Space and Time

We consider now the case of continuous values for space and time. We therefore
assume $s\longrightarrow\infty$ for the space $C$ and $z\longrightarrow\infty$
within the subspaces of $R$ and $S$. This implies that we have the states
$\ket{x^{(J)}}=\sum_{k=0}^{d^{(J)}_{R}-1}e^{-ip^{(J)}_{k}x^{(J)}}\ket{p^{(J)}_{k}}$
with $x^{(J)}\in\left[x^{(J)}_{0},x^{(J)}_{0}+L^{(J)}_{R}\right]$ within $R$
and the states
$\ket{y^{(J)}}=\sum_{k=0}^{d^{(J)}_{S}-1}e^{-ip^{(J)}_{k}y^{(J)}}\ket{p^{(J)}_{k}}$
with $y^{(J)}\in\left[y^{(J)}_{0},y^{(J)}_{0}+L^{(J)}_{S}\right]$ within $S$
for $J=1,2,3$. Starting from the state (83) we can write:

$\ket{\Psi}=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}dt\ket{t}\braket{t}{\Psi}=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}dt\ket{t}_{C}\otimes\ket{\phi(t)}_{R,S}$
(92)

where the relative state $\ket{\phi(t)}_{R,S}=\braket{t}{\Psi}$ takes now the
form

$\ket{\phi(t)}_{R,S}=\sum_{i=0}^{d^{(1)}_{S}-1}\sum_{j=0}^{d^{(2)}_{S}-1}\sum_{k=0}^{d^{(3)}_{S}-1}c_{ijk}e^{-it\epsilon_{ijk}}\ket{-p^{(1)}_{i},-p^{(2)}_{j},-p^{(3)}_{k}}_{R}\otimes\ket{p^{(1)}_{i},p^{(2)}_{j},p^{(3)}_{k}}_{S}.$
(93)

As in the $1+1$ dimensional case, using (80) and (93), for this relative state
we find:

$i\frac{\partial}{\partial
t}\ket{\phi(t)}_{R,S}=(\hat{H}_{R}+\hat{H}_{S})\ket{\phi(t)}_{R,S}$ (94)

which provides the Scrhödinger evolution of $R+S$ with respect to the clock
time $t$.

We can now expand the global state $\ket{\Psi}$ in the coordinate basis
$\left\\{\ket{x^{(1)},x^{(2)},x^{(3)}}_{R}\right\\}$ in $R$, that is

$\begin{split}\ket{\Psi}&=\frac{1}{L^{(1)}_{R}}\frac{1}{L^{(2)}_{R}}\frac{1}{L^{(3)}_{R}}\int
dx^{(1)}\int dx^{(2)}\int
dx^{(3)}\ket{x^{(1)},x^{(2)},x^{(3)}}\braket{x^{(1)},x^{(2)},x^{(3)}}{\Psi}=\\\
&=\frac{1}{L^{(1)}_{R}}\frac{1}{L^{(2)}_{R}}\frac{1}{L^{(3)}_{R}}\int
dx^{(1)}\int dx^{(2)}\int
dx^{(3)}\ket{x^{(1)},x^{(2)},x^{(3)}}_{R}\otimes\ket{\varphi(x^{(1)},x^{(2)},x^{(3)})}_{C,S}\end{split}$
(95)

where each integral on $dx^{(J)}$ is evaluated from $x^{(J)}_{0}$ to
$x^{(J)}_{0}+L^{(J)}_{R}$ and where the state
$\ket{\varphi(x^{(1)},x^{(2)},x^{(3)})}_{C,S}$ takes the form:

$\ket{\varphi(x^{(1)},x^{(2)},x^{(3)})}_{C,S}=\sum_{i,j,k}c_{ijk}e^{-ip^{(1)}_{i}x^{(1)}}e^{-ip^{(2)}_{j}x^{(2)}}e^{-ip^{(3)}_{k}x^{(3)}}\ket{-\epsilon_{ijk}}_{C}\otimes\ket{p^{(1)}_{i},p^{(2)}_{j},p^{(3)}_{k}}_{S}$
(96)

where the summations on $i,j,k$ run from $0$ to $d^{(1)}_{S}-1$,
$d^{(2)}_{S}-1$ and $d^{(3)}_{S}-1$ respectively. Using the definition (96)
and the constraint (82), for the relative state
$\ket{\varphi(x^{(1)},x^{(2)},x^{(3)})}_{C,S}=\braket{x^{(1)},x^{(2)},x^{(3)}}{\Psi}$
we can now obtain:

$i\left(\frac{\partial}{\partial x^{(1)}}+\frac{\partial}{\partial
x^{(2)}}+\frac{\partial}{\partial
x^{(3)}}\right)\ket{\varphi(x^{(1)},x^{(2)},x^{(3)})}_{C,S}=\\\
=\sum_{i,j,k}c_{ijk}p^{(1)}_{i}e^{ip^{(1)}_{i}x^{(1)}}e^{ip^{(2)}_{j}x^{(2)}}e^{ip^{(3)}_{k}x^{(3)}}\ket{-\epsilon_{ijk}}_{C}\otimes\ket{p^{(1)}_{i},p^{(2)}_{j},p^{(3)}_{k}}_{S}+\\\
+\sum_{i,j,k}c_{ijk}p^{(2)}_{j}e^{ip^{(1)}_{i}x^{(1)}}e^{ip^{(2)}_{j}x^{(2)}}e^{ip^{(3)}_{k}x^{(3)}}\ket{-\epsilon_{ijk}}_{C}\otimes\ket{p^{(1)}_{i},p^{(2)}_{j},p^{(3)}_{k}}_{S}+\\\
+\sum_{i,j,k}c_{ijk}p^{(3)}_{k}e^{ip^{(1)}_{i}x^{(1)}}e^{ip^{(2)}_{j}x^{(2)}}e^{ip^{(3)}_{k}x^{(3)}}\ket{-\epsilon_{ijk}}_{C}\otimes\ket{p^{(1)}_{i},p^{(2)}_{j},p^{(3)}_{k}}_{S}=\\\
=\left(\hat{P}^{(1)}_{S}+\hat{P}^{(2)}_{S}+\hat{P}^{(3)}_{S}\right)\ket{\varphi(x^{(1)},x^{(2)},x^{(3)})}_{C,S}$
(97)

which shows
$\vec{P}_{S}=(\hat{P}^{(1)}_{S},\hat{P}^{(2)}_{S},\hat{P}^{(3)}_{S})$ to be
the generator of translations for the states of $C+S$ acting on the
coordinates of the spatial reference frame
$\vec{x}=(x^{(1)},x^{(2)},x^{(3)})$.

Expanding the state $\ket{\Psi}$ simultaneously on the coordinates basis
$\left\\{\ket{x^{(1)},x^{(2)},x^{(3)}}_{R}\right\\}$ and on the time basis
$\left\\{\ket{t}_{C}\right\\}$ we find:

$\begin{split}\ket{\Psi}&=A\left(\int dt\ket{t}\bra{t}\otimes\int dx^{(1)}\int
dx^{(2)}\int
dx^{(3)}\ket{x^{(1)},x^{(2)},x^{(3)}}\bra{x^{(1)},x^{(2)},x^{(3)}}\right)\ket{\Psi}=\\\
&=A\int dt\int dx^{(1)}\int dx^{(2)}\int
dx^{(3)}\ket{t}_{C}\otimes\ket{x^{(1)},x^{(2)},x^{(3)}}_{R}\otimes\ket{\psi(t,x^{(1)},x^{(2)},x^{(3)})}_{S}\end{split}$
(98)

where
$A=\frac{1}{T}\frac{1}{L^{(1)}_{R}}\frac{1}{L^{(2)}_{R}}\frac{1}{L^{(3)}_{R}}$.
In equation (98) the integral on time is evaluated from $t_{0}$ to $t_{0}+T$
and each integral on $dx^{(J)}$ is evaluated from $x^{(J)}_{0}$ to
$x^{(J)}_{0}+L^{(J)}_{R}$. The state
$\ket{\psi(t,x^{(1)},x^{(2)},x^{(3)})}_{S}=(\bra{t}\otimes\bra{x^{(1)},x^{(2)},x^{(3)}})\ket{\Psi}$
takes now the form

$\ket{\psi(t,x^{(1)},x^{(2)},x^{(3)})}_{S}=\sum_{i,j,k}c_{ijk}e^{-it\epsilon_{ijk}}e^{-ip^{(1)}_{i}x^{(1)}}e^{-ip^{(2)}_{j}x^{(2)}}e^{-ip^{(3)}_{k}x^{(3)}}\ket{p^{(1)}_{i},p^{(2)}_{j},p^{(3)}_{k}}_{S}$
(99)

where the summations on $i,j,k$ run again from $0$ to $d^{(1)}_{S}-1$,
$d^{(2)}_{S}-1$ and $d^{(3)}_{S}-1$ respectively. Through this state we can
calculate the conditional probability density of having the position
$(y^{(1)},y^{(2)},y^{(3)})$ on $S$ at time $t$ and knowing that the spatial
reference frame $R$ is in $(x^{(1)},x^{(2)},x^{(3)})$. We have:

$P(y^{(1)},y^{(2)},y^{(3)}\>|\>x^{(1)},x^{(2)},x^{(3)},t)=\frac{1}{L^{(1)}_{S}}\frac{1}{L^{(2)}_{S}}\frac{1}{L^{(3)}_{S}}\left|\braket{y^{(1)},y^{(2)},y^{(3)}}{\psi(t,x^{(1)},x^{(2)},x^{(3)})}\right|^{2}=\\\
=\frac{1}{L^{(1)}_{S}}\frac{1}{L^{(2)}_{S}}\frac{1}{L^{(3)}_{S}}\left|\sum_{i,j,k}c_{ijk}e^{-it\epsilon_{ijk}}e^{ip^{(1)}_{i}(y^{(1)}-x^{(1)})}e^{ip^{(2)}_{j}(y^{(2)}-x^{(2)})}e^{ip^{(3)}_{k}(y^{(3)}-x^{(3)})}\right|^{2}.$
(100)

As in the previous cases, this probability density is well-defined (indeed we
have $\int dy^{(1)}\int dy^{(2)}\int
dy^{(3)}P(y^{(1)},y^{(2)},y^{(3)}\>|\>x^{(1)},x^{(2)},x^{(3)},t)=1$ with each
integral on $dy^{(J)}$ evaluated from $y^{(J)}_{0}$ to
$y^{(J)}_{0}+L^{(J)}_{S}$) and depends on time and on the relative distance
between $R$ and $S$ in a 3-dimensional continuous space.

###  V.3 Free Particles (with $M\gg m$) in $3+1$ Spacetime

We give here a simple example of an emerging $3+1$ dimensional spacetime using
continuous values of space and time, by starting from the framework described
in Section V.B. In doing so we adopt a slightly different formalism, which
allows us to emphasize how space and time are treated here on equal footing.

We consider two systems that we call $\mathfrak{R}$ and $S$. The system
$\mathfrak{R}$ acts as spacetime reference frame for $S$ and it is composed of
a free particle of mass $M$ together with an additional degree of freedom
(with zero momentum) that acts as a clock. We choose also $S$ as free particle
of mass $m$ and we assume $M\gg m$ (as mentioned in Section III.B this choice
implies that we will be able to neglect the kinetic energy term of
$\mathfrak{R}$ with respect to the kinetic energy of $S$, namely
$\mathfrak{R}$ is a good reference frame and moves very slightly in time). The
global Hamiltonian can be written:

$\hat{H}=\hat{H}^{(0)}_{\mathfrak{R}}+\hat{H}^{(1)}_{\mathfrak{R}}+\hat{H}^{(2)}_{\mathfrak{R}}+\hat{H}^{(3)}_{\mathfrak{R}}+\hat{H}_{S}$
(101)

where $\hat{H}^{(0)}_{\mathfrak{R}}$ is the Hamiltonian of the temporal
reference frame (which takes the place of what was $\hat{H}_{C}$ in the
previous discussion), $\hat{H}^{(1)}_{\mathfrak{R}}$,
$\hat{H}^{(2)}_{\mathfrak{R}}$, $\hat{H}^{(3)}_{\mathfrak{R}}$ are the
Hamiltonians depending on the momenta of the reference frame in the three
spatial directions through
$\hat{H}^{(1)}_{\mathfrak{R}}=\left(\hat{P}^{(1)}_{\mathfrak{R}}\right)^{2}/2M$,
$\hat{H}^{(2)}_{\mathfrak{R}}=\left(\hat{P}^{(2)}_{\mathfrak{R}}\right)^{2}/2M$,
$\hat{H}^{(3)}_{\mathfrak{R}}=\left(\hat{P}^{(3)}_{\mathfrak{R}}\right)^{2}/2M$
and $\hat{H}_{S}=\hat{H}^{(1)}_{S}+\hat{H}^{(2)}_{S}+\hat{H}^{(3)}_{S}$ is the
Hamiltonian of $S$ depending on the momenta $\hat{P}^{(1)}_{S}$,
$\hat{P}^{(2)}_{S}$, $\hat{P}^{(3)}_{S}$ through
$\hat{H}^{(1)}_{S}=\left(\hat{P}^{(1)}_{S}\right)^{2}/2m$,
$\hat{H}^{(2)}_{S}=\left(\hat{P}^{(2)}_{S}\right)^{2}/2m$,
$\hat{H}^{(3)}_{S}=\left(\hat{P}^{(3)}_{S}\right)^{2}/2m$. With the intent of
treating space and time on equal footing, within the time subspace we rewrite
the time states as
$\ket{x^{(0)}}=\sum_{k=0}^{d^{(0)}_{\mathfrak{R}}-1}e^{-ix^{(0)}p^{(0)}_{k}}\ket{p^{(0)}_{k}}$
and the time values as
$x^{(0)}\in\left[x^{(0)}_{0},x^{(0)}_{0}+L^{(0)}_{\mathfrak{R}}\right]$, where
$d^{(0)}_{\mathfrak{R}}$ is the dimension of the time subspace, $p^{(0)}_{k}$
are the eigenvalues of $\hat{H}^{(0)}_{\mathfrak{R}}$ and where
$L^{(0)}_{\mathfrak{R}}$ takes now the place of what was $T$ in the previous
discussion. Furthermore we redefine the position states as
$\ket{x^{(J)}}=\sum_{k=0}^{d^{(J)}_{\mathfrak{R}}-1}e^{ix^{(J)}p^{(J)}_{k}}\ket{p^{(J)}_{k}}$
in $\mathfrak{R}$ and
$\ket{y^{(J)}}=\sum_{k=0}^{d^{(J)}_{S}-1}e^{iy^{(J)}p^{(J)}_{k}}\ket{p^{(J)}_{k}}$
in $S$ for $J=1,2,3$.

The constraints (80) and (82) read now:

$\hat{H}\ket{\Psi}=\left(\hat{H}^{(0)}_{\mathfrak{R}}+\hat{H}^{(1)}_{\mathfrak{R}}+\hat{H}^{(2)}_{\mathfrak{R}}+\hat{H}^{(3)}_{\mathfrak{R}}+\hat{H}_{S}\right)\ket{\Psi}=0$
(102)

and

$\begin{split}\vec{P}\ket{\Psi}=\left(\vec{P}_{\mathfrak{R}}+\vec{P}_{S}\right)\ket{\Psi}=0\end{split}$
(103)

with
$(\hat{P}^{(1)}_{\mathfrak{R}}+\hat{P}^{(1)}_{S})\ket{\Psi}=(\hat{P}^{(2)}_{\mathfrak{R}}+\hat{P}^{(2)}_{S})\ket{\Psi}=(\hat{P}^{(3)}_{\mathfrak{R}}+\hat{P}^{(3)}_{S})\ket{\Psi}=0$.
Assuming again
$d^{(0)}_{\mathfrak{R}},d^{(1)}_{\mathfrak{R}},d^{(2)}_{\mathfrak{R}},d^{(3)}_{\mathfrak{R}}\gg
d^{(1)}_{S},d^{(2)}_{S},d^{(3)}_{S}$, the global state $\ket{\Psi}$
simultaneously satisfying (102) and (103) can be written as

$\ket{\Psi}=\sum_{i=0}^{d^{(1)}_{S}-1}\sum_{j=0}^{d^{(2)}_{S}-1}\sum_{k=0}^{d^{(3)}_{S}-1}c_{ijk}\ket{p^{(0)}=-\epsilon_{ijk},-p^{(1)}_{i},-p^{(2)}_{j},-p^{(3)}_{k}}_{\mathfrak{R}}\otimes\ket{p^{(1)}_{i},p^{(2)}_{j},p^{(3)}_{k}}_{S}$
(104)

where $p^{(0)}$ is the value for the energy of the time reference and the
energy function $\epsilon_{ijk}$ is:
$\epsilon_{ijk}=\left(\frac{1}{2M}+\frac{1}{2m}\right)\left(\left(p^{(1)}_{i}\right)^{2}+\left(p^{(2)}_{j}\right)^{2}+\left(p^{(3)}_{k}\right)^{2}\right)\simeq\frac{1}{2m}\left(\left(p^{(1)}_{i}\right)^{2}+\left(p^{(2)}_{j}\right)^{2}+\left(p^{(3)}_{k}\right)^{2}\right)$.

We can now expand the global state $\ket{\Psi}$ in the basis
$\left\\{\ket{x^{(0)},x^{(1)},x^{(2)},x^{(3)}}_{\mathfrak{R}}\right\\}$ in the
space $\mathfrak{R}$, thus obtaining:

$\begin{split}\ket{\Psi}&=A\int dx^{(0)}\int dx^{(1)}\int dx^{(2)}\int
dx^{(3)}\ket{x^{(0)},x^{(1)},x^{(2)},x^{(3)}}\braket{x^{(0)},x^{(1)},x^{(2)},x^{(3)}}{\Psi}=\\\
&=A\int dx^{(0)}\int dx^{(1)}\int dx^{(2)}\int
dx^{(3)}\ket{x^{(0)},x^{(1)},x^{(2)},x^{(3)}}_{\mathfrak{R}}\otimes\ket{\psi(x^{(0)},x^{(1)},x^{(2)},x^{(3)})}_{S}\end{split}$
(105)

where
$A=\frac{1}{L^{(0)}_{\mathfrak{R}}}\frac{1}{L^{(1)}_{\mathfrak{R}}}\frac{1}{L^{(2)}_{\mathfrak{R}}}\frac{1}{L^{(3)}_{\mathfrak{R}}}$
and the integrals on $dx^{(J)}$ are evaluated from $x^{(J)}_{0}$ to
$x^{(J)}_{0}+L^{(J)}_{\mathfrak{R}}$ for $J=0,1,2,3$. The state
$\ket{\psi(x^{(0)},x^{(1)},x^{(2)},x^{(3)})}_{S}=\braket{x^{(0)},x^{(1)},x^{(2)},x^{(3)}}{\Psi}$
is the relative state of the system $S$ conditioned on the value
$(x^{(0)},x^{(1)},x^{(2)},x^{(3)})$ for the spacetime reference frame
$\mathfrak{R}$ and it takes the form:

$\ket{\psi(x^{(0)},x^{(1)},x^{(2)},x^{(3)})}_{S}\simeq\sum_{i=0}^{d^{(1)}_{S}-1}\sum_{j=0}^{d^{(2)}_{S}-1}\sum_{k=0}^{d^{(3)}_{S}-1}c_{ijk}e^{-i\frac{1}{2m}\left((p^{(1)}_{i})^{2}+(p^{(2)}_{j})^{2}+(p^{(3)}_{k})^{2}\right)x^{(0)}}\\\
\times
e^{ip^{(1)}_{i}x^{(1)}}e^{ip^{(2)}_{j}x^{(2)}}e^{ip^{(3)}_{k}x^{(3)}}\ket{p^{(1)}_{i},p^{(2)}_{j},p^{(3)}_{k}}_{S}.$
(106)

For the relative state (106), through (102) and (103), we easily find:

$i\frac{\partial}{\partial
x^{(0)}}\ket{\psi(x^{(0)},x^{(1)},x^{(2)},x^{(3)})}_{S}\simeq\hat{H}_{S}\ket{\psi(x^{(0)},x^{(1)},x^{(2)},x^{(3)})}_{S}$
(107)

and

$-i\frac{\partial}{\partial
x^{(J)}}\ket{\psi(x^{(0)},x^{(1)},x^{(2)},x^{(3)})}_{S}=\hat{P}^{(J)}_{S}\ket{\psi(x^{(0)},x^{(1)},x^{(2)},x^{(3)})}_{S}$
(108)

for $J=1,2,3$. Equations (106), (107) and (108) lead to (writing
$\vec{x}=(x^{(1)},x^{(2)},x^{(3)})$):

$i\frac{\partial}{\partial
x^{(0)}}\ket{\psi(x^{(0)},\vec{x})}_{S}\simeq-\frac{1}{2m}\left(\frac{\partial^{2}}{\left(\partial
x^{(1)}\right)^{2}}+\frac{\partial^{2}}{\left(\partial
x^{(2)}\right)^{2}}+\frac{\partial^{2}}{\left(\partial
x^{(3)}\right)^{2}}\right)\ket{\psi(x^{(0)},\vec{x})}_{S}$ (109)

which describes the dynamics of the particle in $S$ with respect to the
coordinates of the $3+1$ dimensional quantum reference frame. We emphasize
here that the formalism adopted allows us to write the equation (109) for the
state $\ket{\psi(x^{(0)},x^{(1)},x^{(2)},x^{(3)})}_{S}$ because the values of
time and space of the subspace $\mathfrak{R}$ enter as parameters in $S$
thanks to the entanglement present in the global state $\ket{\Psi}$.

###  V.4 System $S$ as a Relativistic Particle

The formalism adopted in the previous paragraph is particularly well suited in
describing the behavior of a relativistic particle. Considering indeed the
system $S$ to be a relativistic particle with no internal degree of freedom
(namely with spin $0$), we have for the energy function ($c=1$):

$\epsilon_{ijk}\simeq\pm\sqrt{(p^{(1)}_{i})^{2}+(p^{(2)}_{j})^{2}+(p^{(3)}_{k})^{2}+m^{2}}$
(110)

which can be obtained from the energy constraint
$\left(\left(\hat{H}^{(0)}_{\mathfrak{R}}\right)^{2}-\left|\vec{P}_{S}\right|^{2}-m^{2}\right)\ket{\Psi}\simeq
0$ (where
$\vec{P}_{S}=\left(\hat{P}^{(1)}_{S},\hat{P}^{(2)}_{S},\hat{P}^{(3)}_{S}\right)$
and the kinetic energy term of $\mathfrak{R}$ has been neglected). The
relative state (106) of the system $S$ conditioned on the value
$(x^{(0)},x^{(1)},x^{(2)},x^{(3)})$ of the spacetime reference frame
$\mathfrak{R}$ reads now:

$\ket{\psi_{\pm}(x^{(0)},x^{(1)},x^{(2)},x^{(3)})}_{S}\simeq\sum_{i=0}^{d^{(1)}_{S}-1}\sum_{j=0}^{d^{(2)}_{S}-1}\sum_{k=0}^{d^{(3)}_{S}-1}c_{ijk}e^{\mp
ix^{(0)}\sqrt{(p^{(1)}_{i})^{2}+(p^{(2)}_{j})^{2}+(p^{(3)}_{k})^{2}+m^{2}}}\\\
\times
e^{ip^{(1)}_{i}x^{(1)}}e^{ip^{(2)}_{j}x^{(2)}}e^{ip^{(3)}_{k}x^{(3)}}\ket{p^{(1)}_{i},p^{(2)}_{j},p^{(3)}_{k}}_{S}$
(111)

where we have considered together both the results for the energy function
(110). For the relative state (111) we find:

$\frac{\partial^{2}}{\partial(x^{(0)})^{2}}\ket{\psi_{\pm}(x^{(0)},x^{(1)},x^{(2)},x^{(3)})}_{S}\simeq-\left(\left(\hat{P}^{(1)}_{S}\right)^{2}+\left(\hat{P}^{(2)}_{S}\right)^{2}+\left(\hat{P}^{(3)}_{S}\right)^{2}+m^{2}\right)\ket{\psi_{\pm}(x^{(0)},x^{(1)},x^{(2)},x^{(3)})}_{S}$
(112)

and

$\begin{split}\frac{\partial^{2}}{\partial(x^{(J)})^{2}}\ket{\psi_{\pm}(x^{(0)},x^{(1)},x^{(2)},x^{(3)})}_{S}=-\left(\hat{P}^{(J)}_{S}\right)^{2}\ket{\psi_{\pm}(x^{(0)},x^{(1)},x^{(2)},x^{(3)})}_{S}\end{split}$
(113)

with $J=1,2,3$. Through equations (112) and (113) we easily obtain:

$\left(\frac{\partial^{2}}{\partial(x^{(0)})^{2}}-\frac{\partial^{2}}{\partial(x^{(1)})^{2}}-\frac{\partial^{2}}{\partial(x^{(2)})^{2}}-\frac{\partial^{2}}{\partial(x^{(3)})^{2}}+m^{2}\right)\ket{\psi_{\pm}(x^{(0)},\vec{x})}_{S}\simeq
0$ (114)

which describes the dynamics of the particle in $S$ with respect to the
coordinates of the $3+1$ dimensional quantum reference frame. Equation (114)
has the form of the Klein-Gordon equation but differs from it being the
derivatives applied to the state
$\ket{\psi_{\pm}(x^{(0)},x^{(1)},x^{(2)},x^{(3)})}_{S}$ and not to the wave
function. Also in this case, this is possible since the time and space values
of $\mathfrak{R}$ enter as parameters in the state of the subsystem $S$ thanks
to the entanglement present in the global state $\ket{\Psi}$.

A similar result can be obtained considering the system $S$ as a relativistic
particle with spin $1/2$. In this case the global state of the Universe can be
written:

$\ket{\Psi}=\sum_{\sigma=0}^{3}\sum_{n=0}^{d^{(0)}_{\mathfrak{R}}-1}\sum_{i=0}^{d^{(1)}_{S}-1}\sum_{j=0}^{d^{(2)}_{S}-1}\sum_{k=0}^{d^{(3)}_{S}-1}c_{nijk}\ket{p^{(0)}_{n},-p^{(1)}_{i},-p^{(2)}_{j},-p^{(3)}_{k}}_{\mathfrak{R}}\otimes\ket{p^{(1)}_{i},p^{(2)}_{j},p^{(3)}_{k},\sigma}_{S}$
(115)

where we have introduced the spin degree of freedom within the subspace $S$ in
accordance to Refs. dirac, ; librodirac,  and where the value of $p^{(0)}_{n}$
is constrained through

$\left(\hat{H}^{(0)}_{\mathfrak{R}}+\vec{\alpha}\cdot\vec{P}_{S}+\beta
m\right)\ket{\Psi}\simeq 0.$ (116)

In equation (116) we have written for the system $S$ the free Dirac
Hamiltonian as $\hat{H}_{S}=\vec{\alpha}\cdot\vec{P}_{S}+\beta m$ dirac  and
we have again neglected the kinetic energy term of $\mathfrak{R}$. The state
$\ket{\psi(x^{(0)},x^{(1)},x^{(2)},x^{(3)})}_{S}=\braket{x^{(0)},x^{(1)},x^{(2)},x^{(3)}}{\Psi}$
reads now:

$\ket{\psi(x^{(0)},x^{(1)},x^{(2)},x^{(3)})}_{S}=\sum_{\sigma=0}^{3}\sum_{n=0}^{d^{(0)}_{\mathfrak{R}}-1}\sum_{i=0}^{d^{(1)}_{S}-1}\sum_{j=0}^{d^{(2)}_{S}-1}\sum_{k=0}^{d^{(3)}_{S}-1}c_{nijk}e^{-ip^{(0)}_{n}x^{(0)}}\\\
\times
e^{ip^{(1)}_{i}x^{(1)}}e^{ip^{(2)}_{j}x^{(2)}}e^{ip^{(3)}_{k}x^{(3)}}\ket{p^{(1)}_{i},p^{(2)}_{j},p^{(3)}_{k},\sigma}_{S}.$
(117)

For the relative state (117) still holds

$i\frac{\partial}{\partial
x^{(0)}}\ket{\psi(x^{(0)},x^{(1)},x^{(2)},x^{(3)})}_{S}\simeq\hat{H}_{S}\ket{\psi(x^{(0)},x^{(1)},x^{(2)},x^{(3)})}_{S}$
(118)

and

$-i\frac{\partial}{\partial
x^{(J)}}\ket{\psi(x^{(0)},x^{(1)},x^{(2)},x^{(3)})}_{S}=\hat{P}^{(J)}_{S}\ket{\psi(x^{(0)},x^{(1)},x^{(2)},x^{(3)})}_{S}.$
(119)

So, starting from equations (118) and (119), writing
$\vec{\alpha}=(\alpha^{(1)},\alpha^{(2)},\alpha^{(3)})$ and remembering that
$\hat{H}_{S}=\vec{\alpha}\cdot\vec{P}_{S}+\beta m$, we obtain:

$i\frac{\partial}{\partial
x^{(0)}}\ket{\psi(x^{(0)},\vec{x})}_{S}\simeq\left(-i\alpha^{(1)}\frac{\partial}{\partial
x^{(1)}}-i\alpha^{(2)}\frac{\partial}{\partial
x^{(2)}}-i\alpha^{(3)}\frac{\partial}{\partial x^{(3)}}+\beta
m\right)\ket{\psi(x^{(0)},\vec{x})}_{S}$ (120)

which has the form of the Dirac equation and again describes the dynamics of
the particle in $S$ with respect to the coordinates of the $3+1$ dimensional
quantum reference frame. All the considerations made for equation (114) still
apply in this case. Clearly, in order to give a complete relativistic
generalization of the model, in addition to this discussion, we need to
consider relativistic reference frames and a protocol that allows to change
the point of view between different observers in different reference frames
(so that dilation of times and contraction of lengths can be derived), but
this is beyond the scope of the present work.

##  VI Conclusions

The PaW mechanism was originally introduced in order to describe the emergence
of time from entanglement. In this work we first extended the PaW mechanism at
the spatial degree of freedom and then we provide a description of a model of
non-relativistic quantum spacetime. In doing this we started focusing on space
and we showed that, in a closed quantum system satisfying a global constraint
on total momentum (and therefore with the absolute position totally
indeterminate), a well-defined relative position emerge between the subsystems
$S$ and $R$, where the latter is taken as quantum spatial reference frame. In
the spaces $R$ and $S$, generalizing the approach outlined in Refs. nostro, ;
nostro2, ; pegg, , we considered non-degenerate, discrete spectra for the
momentum operators and we introduce POVMs in describing the spatial degrees of
freedom. In this way we recovered continuous values of space in $S$ and $R$
also for a discrete momentum spectra (the case of momentum with continuous
spectrum was then also treated in Section III.E). Finally we introduced in the
Universe an additional subsystem $C$ acting as a clock and we considered the
Universe satisfying a double constraint: both on total momentum and total
energy. We showed how this framework can be implemented without contradiction
in the simple case of one spatial degree of freedom (considering also the case
of multiple time measurements) and in the “more physical”case of three spatial
degrees of freedom thus providing a $3+1$ dimensional quantum spacetime
emerging from entanglement.

###### Acknowledgements.

We acknowledge funding from the H2020 QuantERA ERA-NET Cofund in Quantum
Technologies projects QCLOCKS.

##  Appendix A Proof of Equation (15)

We start considering the global state written as

$\ket{\Psi}=\frac{\sqrt{d_{R}}}{D_{R}}\sum_{j=0}^{D_{R}-1}\ket{x_{j}}_{R}\otimes\ket{\phi(x_{j})}_{S}$
(121)

where
$\ket{\phi(x_{j})}_{S}=\sum_{k=0}^{d_{S}-1}c_{k}e^{-ip_{k}x_{j}}\ket{p_{k}}_{S}$.
We can now calculate the conditional probability as follows

$\begin{split}&P(y_{l}\>on\>S\>|\>x_{j}\>on\>R)=\frac{d_{S}}{D_{S}}\frac{\braket{\Psi}{x_{j}}\bra{x_{j}}\otimes\ket{y_{l}}\braket{y_{l}}{\Psi}}{\braket{\Psi}{x_{j}}\braket{x_{j}}{\Psi}}=\\\
\\\
&=\frac{d_{S}}{D_{S}}\frac{\frac{d_{R}}{D^{2}_{R}}\sum_{n}\sum_{m}\braket{x_{n}}{x_{j}}\braket{x_{j}}{x_{m}}\braket{\phi(x_{n})}{y_{l}}\braket{y_{l}}{\phi(x_{m})}}{\frac{d_{R}}{D^{2}_{R}}\sum_{n^{\prime}}\sum_{m^{\prime}}\braket{x_{n^{\prime}}}{x_{j}}\braket{x_{j}}{x_{m^{\prime}}}\braket{\phi(x_{n^{\prime}})}{\phi(x_{m^{\prime}})}}=\\\
\\\
&=\frac{d_{S}}{D_{S}}\frac{\sum_{n}\sum_{m}\braket{x_{n}}{x_{j}}\braket{x_{j}}{x_{m}}\sum_{k}\sum_{a}c^{*}_{k}c_{a}e^{ip_{k}x_{n}}e^{-ip_{a}x_{m}}\braket{p_{k}}{y_{l}}\braket{y_{l}}{p_{a}}}{\sum_{n^{\prime}}\sum_{m^{\prime}}\braket{x_{n^{\prime}}}{x_{j}}\braket{x_{j}}{x_{m^{\prime}}}\sum_{b}\sum_{k^{\prime}}c^{*}_{b}c_{k^{\prime}}e^{ip_{b}x_{n^{\prime}}}e^{-ip_{k^{\prime}}x_{m^{\prime}}}\braket{p_{b}}{p_{k^{\prime}}}}=\\\
\\\
&=\frac{1}{D_{S}}\frac{\sum_{n}\sum_{m}\sum_{i}\sum_{g}e^{-p_{i}(x_{j}-x_{n})}e^{ip_{g}(x_{j}-x_{m})}\sum_{k}\sum_{a}c^{*}_{k}c_{a}e^{ip_{k}x_{n}}e^{-ip_{a}x_{m}}e^{-iy_{l}p_{k}}e^{iy_{l}p_{a}}}{\sum_{n^{\prime}}\sum_{m^{\prime}}\sum_{i^{\prime}}\sum_{g^{\prime}}e^{-p_{i^{\prime}}(x_{j}-x_{n^{\prime}})}e^{ip_{g^{\prime}}(x_{j}-x_{m^{\prime}})}\sum_{k^{\prime}}|c_{k^{\prime}}|^{2}e^{ip_{k^{\prime}}(x_{n^{\prime}}-x_{m^{\prime}})}}\end{split}$
(122)

We use now the fact that (see Appendix C for the proof)

$\sum_{j=0}^{D_{R}-1}e^{-ix_{j}(p_{k}-p_{n})}=D_{R}\delta_{p_{k},p_{n}}$ (123)

and so we obtain

$P(y_{l}\>on\>S\>|\>x_{j}\>on\>R)=\frac{1}{D_{S}}\frac{\sum_{k}\sum_{a}c^{*}_{k}c_{a}e^{i(p_{a}-p_{k})(y_{l}-x_{j})}}{\sum_{k^{\prime}}|c_{k^{\prime}}|^{2}}=\frac{d_{S}}{D_{S}}\left|\braket{y_{l}}{\phi(x_{j})}\right|^{2}=\frac{1}{D_{S}}\left|\sum_{k=0}^{d_{S}-1}c_{k}e^{ip_{k}(y_{l}-x_{j})}\right|^{2}$
(124)

where we have considered that $\sum_{k^{\prime}}|c_{k^{\prime}}|^{2}=1$.

##  Appendix B Proof of Equation (16)

We start considering the global state written as

$\ket{\Psi}=\frac{1}{L_{R}}\int_{x_{0}}^{x_{0}+L_{R}}dx\ket{x}_{R}\otimes\ket{\phi(x)}_{S}$
(125)

where
$\ket{\phi(x)}_{S}=\sum_{k=0}^{d_{S}-1}c_{k}e^{-ip_{k}x}\ket{p_{k}}_{S}$. We
can now calculate the conditional probability density as follows (all the
integrals are evaluated between $x_{0}$ and $x_{0}+L_{R}$):

$\begin{split}&P(y\>on\>S\>|\>x\>on\>R)=\frac{1}{L_{S}}\frac{\braket{\Psi}{x}\bra{x}\otimes\ket{y}\braket{y}{\Psi}}{\braket{\Psi}{x}\braket{x}{\Psi}}=\\\
\\\ &=\frac{1}{L_{S}}\frac{\frac{1}{L^{2}_{R}}\int dx^{\prime}\int
dx^{{}^{\prime\prime}}\braket{x^{\prime}}{x}\braket{x}{x^{{}^{\prime\prime}}}\braket{\phi(x^{\prime})}{y}\braket{y}{\phi(x^{{}^{\prime\prime}})}}{\frac{1}{L^{2}_{R}}\int
dx^{\prime}\int
dx^{{}^{\prime\prime}}\braket{x^{\prime}}{x}\braket{x}{x^{{}^{\prime\prime}}}\braket{\phi(x^{\prime})}{\phi(x^{{}^{\prime\prime}})}}=\\\
\\\ &=\frac{1}{L_{S}}\frac{\int dx^{\prime}\int
dx^{{}^{\prime\prime}}\sum_{m}\sum_{l}e^{-ip_{m}(x-x^{\prime})}e^{ip_{l}(x-x^{{}^{\prime\prime}})}\sum_{k}\sum_{n}c^{*}_{k}c_{n}e^{ip_{k}x^{\prime}}e^{-ip_{n}x^{{}^{\prime\prime}}}\braket{p_{k}}{y}\braket{y}{p_{n}}}{\int
dx^{\prime}\int
dx^{{}^{\prime\prime}}\sum_{m^{\prime}}\sum_{l^{\prime}}e^{-ip_{m^{\prime}}(x-x^{\prime})}e^{ip_{l^{\prime}}(x-x^{{}^{\prime\prime}})}\sum_{k^{\prime}}\sum_{n^{\prime}}c^{*}_{k^{\prime}}c_{n^{\prime}}e^{ip_{k^{\prime}}x^{\prime}}e^{-ip_{n^{\prime}}x^{{}^{\prime\prime}}}\braket{p_{k^{\prime}}}{p_{n^{\prime}}}}=\\\
\\\ &=\frac{1}{L_{S}}\frac{\int dx^{\prime}\int
dx^{{}^{\prime\prime}}\sum_{m}\sum_{l}e^{-ip_{m}(x-x^{\prime})}e^{ip_{l}(x-x^{{}^{\prime\prime}})}\sum_{k}\sum_{n}c^{*}_{k}c_{n}e^{ip_{k}x^{\prime}}e^{-ip_{n}x^{{}^{\prime\prime}}}e^{-ip_{k}y}e^{ip_{n}y}}{\int
dx^{\prime}\int
dx^{{}^{\prime\prime}}\sum_{m^{\prime}}\sum_{l^{\prime}}e^{-ip_{m^{\prime}}(x-x^{\prime})}e^{ip_{l^{\prime}}(x-x^{{}^{\prime\prime}})}\sum_{k^{\prime}}|c_{k^{\prime}}|^{2}e^{ip_{k^{\prime}}(x^{\prime}-x^{{}^{\prime\prime}})}}.\end{split}$
(126)

We use now the fact that (see Appendix C for the proof)

$\int_{x_{0}}^{x_{0}+L_{R}}dx^{\prime}e^{-ix^{\prime}(p_{k}-p_{n})}=L_{R}\delta_{p_{k},p_{n}},$
(127)

thus we obtain:

$\begin{split}P(y\>on\>S\>|\>x\>on\>R)=\frac{1}{L_{S}}\frac{\sum_{k}\sum_{n}c^{*}_{k}c_{n}e^{i(p_{n}-p_{k})(y-x)}}{\sum_{k^{\prime}}|c_{k^{\prime}}|^{2}}=\frac{1}{L_{S}}\left|\braket{y}{\phi(x)}\right|^{2}=\frac{1}{L_{S}}\left|\sum_{k=0}^{d_{S}-1}c_{k}e^{ip_{k}(y-x)}\right|^{2}\end{split}$
(128)

where again we have considered that $\sum_{k^{\prime}}|c_{k^{\prime}}|^{2}=1$.

##  Appendix C Proof of Equations (18) and (123)

We first approach the case with discrete space values (namely we first
demonstrate equation (123)). We start considering the global state
$\ket{\Psi}$ written as
$\ket{\Psi}=\sum_{k=0}^{d_{S}-1}c_{k}\ket{p=-p_{k}}_{R}\otimes\ket{p_{k}}_{S}$
and we apply in sequence the resolutions of the identity
$\mathbb{1}_{R}=\frac{d_{R}}{D_{R}}\sum_{j=0}^{D_{R}-1}\ket{x_{j}}\bra{x_{j}}$
and $\mathbb{1}_{R}=\sum_{n=0}^{d_{R}-1}\ket{p_{n}}\bra{p_{n}}$. We obtain

$\begin{split}\ket{\Psi}&=\frac{d_{R}}{D_{R}}\sum_{j=0}^{D_{R}-1}\ket{x_{j}}\braket{x_{j}}{\Psi}=\\\
&=\frac{\sqrt{d_{R}}}{D_{R}}\sum_{j=0}^{D_{R}-1}\ket{x_{j}}_{R}\otimes\sum_{k=0}^{d_{S}-1}c_{k}e^{-ip_{k}x_{j}}\ket{p_{k}}_{S}=\\\
&=\sum_{n=0}^{d_{R}-1}\ket{p_{n}}\bra{p_{n}}\frac{\sqrt{d_{R}}}{D_{R}}\sum_{j=0}^{D_{R}-1}\ket{x_{j}}_{R}\otimes\sum_{k=0}^{d_{S}-1}c_{k}e^{-ip_{k}x_{j}}\ket{p_{k}}_{S}=\\\
&=\sum_{n=0}^{d_{R}-1}\sum_{k=0}^{d_{S}-1}c_{k}\frac{1}{D_{R}}\sum_{j=0}^{D_{R}-1}e^{-ix_{j}(p_{n}+p_{k})}\ket{p_{n}}_{R}\otimes\ket{p_{k}}_{S}\end{split}$
(129)

from which we have

$\sum_{j=0}^{D_{R}-1}e^{-ix_{j}(p_{k}+p_{n})}=D_{R}\delta_{p_{n},-p_{k}}.$
(130)

Let us now consider the limiting case $z\longrightarrow\infty$, i.e. we assume
the continuous representation of the coordinate $x$ which can now take any
real value from $x_{0}$ to $x_{0}+L_{R}$. We start again from the global state
written as (9) and we apply in sequence the resolutions of the identity

$\mathbb{1}_{R}=\frac{1}{L_{R}}\int_{x_{0}}^{x_{0}+L_{R}}dx\ket{x}\bra{x}\>\>\>and\>\>\>\mathbb{1}_{R}=\sum_{n=0}^{d_{R}-1}\ket{p_{n}}\bra{p_{n}}.$
(131)

We have:

$\begin{split}\ket{\Psi}&=\frac{1}{L_{R}}\int_{x_{0}}^{x_{0}+L_{R}}dx\ket{x}\braket{x}{\Psi}=\\\
&=\frac{1}{L_{R}}\int_{x_{0}}^{x_{0}+L_{R}}dx\ket{x}_{R}\otimes\sum_{k=0}^{d_{S}-1}c_{k}e^{-ip_{k}x}\ket{p_{k}}_{S}=\\\
&=\sum_{n=0}^{d_{R}-1}\ket{p_{n}}\bra{p_{n}}\frac{1}{L_{R}}\int_{x_{0}}^{x_{0}+L_{R}}dx\ket{x}_{R}\otimes\sum_{k=0}^{d_{S}-1}c_{k}e^{-ip_{k}x}\ket{p_{k}}_{S}=\\\
&=\sum_{n=0}^{d_{R}-1}\sum_{k=0}^{d_{S}-1}c_{k}\frac{1}{L_{R}}\int_{x_{0}}^{x_{0}+L_{R}}dxe^{-ix(p_{n}+p_{k})}\ket{p_{n}}_{R}\otimes\ket{p_{k}}_{S}\end{split}$
(132)

from which we can easily see that
$\int_{x_{0}}^{x_{0}+L_{R}}dxe^{-ix(p_{n}+p_{k})}=L_{R}\delta_{p_{n},-p_{k}}$.

##  Appendix D Emergent Spacetime with $\hat{P}_{C}\neq 0$

We discuss here our model of $1+1$ dimensional spacetime in the case of
$\hat{P}_{C}\neq 0$. The contraints for the global state
$\ket{\Psi}\in\mathcal{H}_{C}\otimes\mathcal{H}_{R}\otimes\mathcal{H}_{S}$ are
now:

$\hat{H}\ket{\Psi}=(\hat{H}_{C}+\hat{H}_{R}+\hat{H}_{S})\ket{\Psi}=0$ (133)

and

$\hat{P}\ket{\Psi}=(\hat{P}_{C}+\hat{P}_{R}+\hat{P}_{S})\ket{\Psi}=0.$ (134)

As we mentioned in the main text, in this case, there could be limitations in
the allowed momenta to ensure that (133) and (134) are together satisfied and
the global state $\ket{\Psi}$ can not be written here in the simple form (31).
The limitations could arise from the fact that the energies and the momenta of
$C$, $R$ and $S$ simultaneously have to sum to zero and the difficulty of the
problem depends on the type of dispersion relations in the three subspaces.
However, assuming $\ket{\Psi}$ satisfying the constraints, the discussion
follows as in the case of $\hat{P}_{C}=0$ that we have seen in Section III.B.

We expand also here the global state on the basis
$\left\\{\ket{t_{m}}_{C}\right\\}$ in $C$ thanks to (26), thus obtaining

$\begin{split}\ket{\Psi}&=\frac{d_{C}}{D_{C}}\sum_{m=0}^{D_{C}-1}\ket{t_{m}}\braket{t_{m}}{\Psi}=\\\
&=\frac{\sqrt{d_{C}}}{D_{C}}\sum_{m=0}^{D_{C}-1}\ket{t_{m}}_{C}\otimes\ket{\phi(t_{m})}_{R,S}\end{split}$
(135)

where $\ket{\phi(t_{m})}_{R,S}=\sqrt{d_{C}}\braket{t_{m}}{\Psi}$ is state of
the composite system $R+S$ at time $t_{m}$. For such a state, through (133)
and the relative state definition, it is easy to find again the time evolution
with respect to the clock $C$:

$\begin{split}\ket{\phi(t_{m})}_{R,S}=e^{-i(\hat{H}_{R}+\hat{H}_{S})(t_{m}-t_{0})}\ket{\phi(t_{0})}_{R,S}\end{split}$
(136)

where $\ket{\phi(t_{0})}_{R,S}=\sqrt{d_{C}}\braket{t_{0}}{\Psi}$ is the state
of $R+S$ conditioned on $t_{0}$ that is the value of the clock taken as
initial time. Equation (136) shows, as expected, the simultaneous evolution of
$R$ and $S$ over time. In the limiting case $s\longrightarrow\infty$ where $t$
takes all the real values between $t_{0}$ and $t_{0}+T$ the global state can
again be written

$\ket{\Psi}=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}dt\ket{t}\braket{t}{\Psi}=\\\
=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}dt\ket{t}_{C}\otimes\ket{\phi(t)}_{R,S}$
(137)

and defining the relative state of $R+S$ as
$\ket{\phi(t)}_{R,S}=\braket{t}{\Psi}$ we obtain:

$i\frac{\partial}{\partial
t}\ket{\phi(t)}_{R,S}=\left(\hat{H}_{R}+\hat{H}_{S}\right)\ket{\phi(t)}_{R,S}$
(138)

that is the Schrödinger evolution for $R+S$ with respect to the clock time
$t$, written in the differential form.

We can therefore expand the state $\ket{\Psi}$ in the coordinates
$\left\\{\ket{x_{j}}_{R}\right\\}$ in $R$, thus obtaining:

$\begin{split}\ket{\Psi}&=\frac{d_{R}}{D_{R}}\sum_{j=0}^{D_{R}-1}\ket{x_{j}}\braket{x_{j}}{\Psi}=\\\
&=\frac{\sqrt{d_{R}}}{D_{R}}\sum_{j=0}^{D_{R}-1}\ket{x_{j}}_{R}\otimes\ket{\varphi(x_{j})}_{C,S}\end{split}$
(139)

where $\ket{\varphi(x_{j})}_{C,S}=\sqrt{d_{R}}\braket{x_{j}}{\Psi}$ is the
relative state of $C+S$ conditioned to the value $x_{j}$ on the reference
frame $R$. For the state $\ket{\varphi(x_{j})}_{C,S}$ we find now:

$\begin{split}\ket{\varphi(x_{j})}_{C,S}&=\sqrt{d_{R}}\braket{x_{j}}{\Psi}=\\\
&=\sqrt{d_{R}}\bra{x_{0}}e^{i\hat{P}_{R}(x_{j}-x_{0})}\ket{\Psi}=\\\
&=\sqrt{d_{R}}\bra{x_{0}}e^{i(\hat{P}-\hat{P}_{C}-\hat{P}_{S})(x_{j}-x_{0})}\ket{\Psi}=\\\
&=e^{-i(\hat{P}_{C}+\hat{P}_{S})(x_{j}-x_{0})}\ket{\phi(x_{0})}_{C,S}\end{split}$
(140)

where the momentum of the clock $C$ appear in the equation since we have
$\hat{P}_{C}\neq 0$. Also here we consider the limit $z\longrightarrow\infty$,
where again $x$ can take all the real values between $x_{0}$ and
$x_{0}+L_{R}$. In this case the global state can be written

$\ket{\Psi}=\frac{1}{L_{R}}\int_{x_{0}}^{x_{0}+L_{R}}dx\ket{x}\braket{x}{\Psi}=\\\
=\frac{1}{L_{R}}\int_{x_{0}}^{x_{0}+L_{R}}dx\ket{x}_{R}\otimes\ket{\varphi(x)}_{C,S}$
(141)

and, defining the relative state of $C+S$ as
$\ket{\varphi(x)}_{C,S}=\braket{x}{\Psi}$, we obtain

$\begin{split}\left(\hat{P}_{C}+\hat{P}_{S}\right)\ket{\varphi(x)}_{C,S}&=\bra{x}\left(\hat{P}_{C}+\hat{P}_{S}\right)\ket{\Psi}=\\\
&=\bra{x}\left(\hat{P}-\hat{P}_{R}\right)\ket{\Psi}=\\\
&=-\sum_{k=0}^{d_{R}-1}e^{ip_{k}x}\bra{p_{k}}\hat{P}_{R}\ket{\Psi}=\\\
&=-\left(\sum_{k=0}^{d_{R}-1}p_{k}e^{ip_{k}x}\bra{p_{k}}\right)\ket{\Psi}=\\\
&=i\frac{\partial}{\partial x}\ket{\varphi(x)}_{C,S}.\end{split}$ (142)

Through this latter equation and (140) we can see again that now the generator
of translations in the coordinate values $x$ for the state
$\ket{\varphi(x)}_{C,S}$ is the operator $\hat{P}_{C}+\hat{P}_{S}$.

Finally we can expand the state $\ket{\Psi}$ simultaneously on the coordinates
$\left\\{\ket{x_{j}}_{R}\right\\}$ in $R$ and on the time basis
$\left\\{\ket{t_{m}}_{C}\right\\}$ in $C$. We have for the global state:

$\begin{split}\ket{\Psi}&=\left(\frac{d_{C}}{D_{C}}\sum_{m=0}^{D_{C}-1}\ket{t_{m}}\bra{t_{m}}\otimes\frac{d_{R}}{D_{R}}\sum_{j=0}^{D_{R}-1}\ket{x_{j}}\bra{x_{j}}\right)\ket{\Psi}=\\\
&=\frac{\sqrt{d_{C}}}{D_{C}}\frac{\sqrt{d_{R}}}{D_{R}}\sum_{m=0}^{D_{C}-1}\sum_{j=0}^{D_{R}-1}\ket{t_{m}}_{C}\otimes\ket{x_{j}}_{R}\otimes\ket{\psi(t_{m},x_{j})}_{S}\end{split}$
(143)

where
$\ket{\psi(t_{m},x_{j})}_{S}=\sqrt{d_{C}}\sqrt{d_{R}}(\bra{t_{m}}\otimes\bra{x_{j}})\ket{\Psi}$
is the relative state of the system $S$ at time $t_{m}$ conditioned on the
value $x_{j}$ for the reference frame $R$. Also here we can search the
conditional probability of having a certain position $y_{l}$ in $S$ at time
$t_{m}$ and knowing that the reference frame is in $x_{j}$, that is:

$P(y_{l}\>on\>S\>|\>x_{j}\>on\>R,\>t_{m}\>on\>C)=\frac{d_{S}}{D_{S}}|\braket{y_{l}}{\psi(t_{m},x_{j})}|^{2}$
(144)

that will depend on time $t_{m}$ and on the values $x_{j}$ and $y_{l}$ of $R$
and $S$ respectively. Clearly we can extend these results also to the limiting
cases $z,s\longrightarrow\infty$. Indeed we can write the global state
$\ket{\Psi}$ as

$\begin{split}\ket{\Psi}&=\left(\frac{1}{T}\int_{t_{0}}^{t_{0}+T}dt\ket{t}\bra{t}\otimes\frac{1}{L_{R}}\int_{x_{0}}^{x_{0}+L_{R}}dx\ket{x}\bra{x}\right)\ket{\Psi}=\\\
&=\frac{1}{T}\frac{1}{L_{R}}\int_{t_{0}}^{t_{0}+T}dt\int_{x_{0}}^{x_{0}+L_{R}}dx\ket{t}_{C}\otimes\ket{x}_{R}\otimes\ket{\psi(t,x)}_{S}\end{split}$
(145)

where again $\ket{\psi(t,x)}_{S}=(\bra{t}\otimes\bra{x})\ket{\Psi}$ is the
relative state of the system $S$ at time $t$ conditioned on the value $x$ for
the reference frame $R$. The conditional probability density of having a
certain position $y$ in $S$ at time $t$ and knowing that the reference frame
is in $x$ is:

$P(y\>on\>S\>|\>x\>on\>R,\>t\>on\>C)=\frac{1}{L_{S}}\left|\braket{y}{\psi(t,x)}\right|^{2}$
(146)

which will depend on time $t$ as well as on the position values of $x$ and
$y$. We can not explicitly calculate the probabilities (144) and (146) here
since we can not write the state $\ket{\Psi}$ explicitly in the simple form
(31).

In summary we assume as a “good clock”a system with $\hat{P}_{C}=0$, a
framework that can be easily implemented using an internal degree of freedom
in describing the clock. If we want to use $\hat{P}_{C}\neq 0$ we have to find
a state $\ket{\Psi}$ that satisfies (133) and (134), undergoing the
limitations that this choice imposes and the discussion then follows as shown
in this Appendix. A different (less elegant) framework might be to take
$(\hat{P}_{R}+\hat{P}_{S})\ket{\Psi}=0$ as a constraint for the theory even in
the case of a clock with non-zero momentum, but we believe that this choice
(even if functional) does not have a good physical justification. In this
latter case indeed the Universe would not be in an eigenstate of the global
momentum and thus the symmetry with respect to the temporal degree of freedom
(derived from a Universe in an eigenstate of the global Hamiltonian) would be
lost.

##  Appendix E Proof of Equation (41)

We start here considering the global state $\ket{\Psi}$ written as in equation
31, that is

$\ket{\Psi}=\sum_{k=0}^{d_{S}-1}c_{k}\ket{E=-\epsilon_{k}}_{C}\otimes\ket{p=-p_{k}}_{R}\otimes\ket{p_{k}}_{S}$
(147)

where $\epsilon_{k}=E^{(R)}(-p_{k})+E^{(S)}(p_{k})$ is the energy function
related to the momenta $p_{k}$ of $R$ and $S$. Now we write the conditional
probability as:

$\begin{split}&P(y_{l}\>on\>S\>|\>x_{j}\>on\>R,\>t_{m}\>on\>C)=\frac{d_{S}}{D_{S}}\frac{\braket{\Psi}{t_{m}}\bra{t_{m}}\otimes\ket{x_{j}}\bra{x_{j}}\otimes\ket{y_{l}}\braket{y_{l}}{\Psi}}{\braket{\Psi}{t_{m}}\bra{t_{m}}\otimes\ket{x_{j}}\braket{x_{j}}{\Psi}}=\\\
\\\
&=\frac{d_{S}}{D_{S}}\frac{\sum_{k}\sum_{n}c_{k}c^{*}_{n}\braket{E=-\epsilon_{n}}{t_{m}}\braket{t_{m}}{E=-\epsilon_{k}}\braket{p=-p_{n}}{x_{j}}\braket{x_{j}}{p=-p_{k}}\braket{p_{n}}{y_{l}}\braket{y_{l}}{p_{k}}}{\sum_{k^{\prime}}\sum_{n^{\prime}}c_{k^{\prime}}c^{*}_{n^{\prime}}\braket{E=-\epsilon_{n^{\prime}}}{t_{m}}\braket{t_{m}}{E=-\epsilon_{k^{\prime}}}\braket{p=-p_{n^{\prime}}}{x_{j}}\braket{x_{j}}{p=-p_{k^{\prime}}}\braket{p_{n^{\prime}}}{p_{k^{\prime}}}}=\\\
\\\
&=\frac{1}{D_{S}}\frac{\sum_{k}\sum_{n}c_{k}c^{*}_{n}e^{-it_{m}(\epsilon_{k}-\epsilon_{n})}e^{i(p_{k}-p_{n})(y_{l}-x_{j})}}{\sum_{k^{\prime}}|c_{k^{\prime}}|^{2}}=\frac{1}{D_{S}}\left|\sum_{k=0}^{d_{S}-1}c_{k}e^{-i\epsilon_{k}t_{m}}e^{ip_{k}(y_{l}-x_{j})}\right|^{2}\end{split}$
(148)

where, in the last step, we have considered
$\sum_{k^{\prime}=0}^{d_{S}-1}|c_{k^{\prime}}|^{2}=1$. From equation (148) and
considering
$\ket{\psi(t_{m},x_{j})}_{S}=\sum_{k=0}^{d_{S}-1}c_{k}e^{-i\epsilon_{k}t_{m}}e^{-ip_{k}x_{j}}\ket{p_{k}}$,
we can also see that the probability
$P(y_{l}\>on\>S\>|\>x_{j}\>on\>R,\>t_{m}\>on\>C)$ can be written as
$\frac{d_{S}}{D_{S}}|\braket{y_{l}}{\psi(t_{m},x_{j})}|^{2}$.

##  Appendix F Proof of Equation (43)

We start also here considering the global state written as in 31, that is

$\ket{\Psi}=\sum_{k=0}^{d_{S}-1}c_{k}\ket{E=-\epsilon_{k}}_{C}\otimes\ket{p=-p_{k}}_{R}\otimes\ket{p_{k}}_{S}$
(149)

where $\epsilon_{k}=E^{(R)}(-p_{k})+E^{(S)}(p_{k})$ is the energy function
related to the momenta $p_{k}$ of $R$ and $S$. Now we write the probability
density as:

$\begin{split}&P(y\>on\>S\>|\>x\>on\>R,\>t\>on\>C)=\frac{1}{L_{S}}\frac{\braket{\Psi}{t}\bra{t}\otimes\ket{x}\bra{x}\otimes\ket{y}\braket{y}{\Psi}}{\braket{\Psi}{t}\bra{t}\otimes\ket{x}\braket{x}{\Psi}}=\\\
\\\
&=\frac{1}{L_{S}}\frac{\sum_{k}\sum_{n}c_{k}c^{*}_{n}\braket{E=-\epsilon_{n}}{t}\braket{t}{E=-\epsilon_{k}}\braket{p=-p_{n}}{x}\braket{x}{p=-p_{k}}\braket{p_{n}}{y}\braket{y}{p_{k}}}{\sum_{m}\sum_{l}c_{m}c^{*}_{l}\braket{E=-\epsilon_{l}}{t}\braket{t}{E=-\epsilon_{m}}\braket{p=-p_{l}}{x}\braket{x}{p=-p_{m}}\braket{p_{l}}{p_{m}}}=\\\
\\\
&=\frac{1}{L_{S}}\frac{\sum_{k}\sum_{n}c_{k}c^{*}_{n}e^{-it(\epsilon_{k}-\epsilon_{n})}e^{i(p_{k}-p_{n})(y-x)}}{\sum_{m}|c_{m}|^{2}}=\frac{1}{L_{S}}\left|\sum_{k=0}^{d_{S}-1}c_{k}e^{-i\epsilon_{k}t}e^{ip_{k}(y-x)}\right|^{2}\end{split}$
(150)

where, in the last step, we have considered again
$\sum_{m=0}^{d_{S}-1}|c_{m}|^{2}=1$. Also in this case, from equation (150)
and considering
$\ket{\psi(t,x)}_{S}=\sum_{k=0}^{d_{S}-1}c_{k}e^{-i\epsilon_{k}}e^{-ip_{k}}\ket{p_{k}}$,
we can see that
$P(y\>on\>S\>|\>x\>on\>R,\>t\>on\>C)=\frac{1}{L_{S}}\left|\braket{y}{\psi(t,x)}\right|^{2}$.

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