$\displaystyle{\rm d}^{3}P({\bf p}_{B})={\rm tr}\left[\ket{{\bf
p}_{B}}\bra{{\bf p}_{B}}_{B}\hat{\tilde{\rho}}_{B}^{(A)}\right]{\rm d}^{3}{\rm
x}_{B}=|\tilde{\psi}_{O}|^{2}*|\tilde{\phi}_{B}|^{2}({\bf p}_{B}){\rm
d}^{3}{\rm p}_{B}\,,$ (3.90)

and

$\displaystyle{\rm d}^{3}P({\bf p}_{O})={\rm tr}\left[\ket{\phi_{B}({\bf
p}_{B})}\bra{\phi_{B}({\bf p}_{B})}_{B}\hat{\tilde{\rho}}_{B}^{(A)}\right]{\rm
d}^{3}{\rm p}_{B}=|\tilde{\psi}_{O}(-{\bf p}_{B})|^{2}{\rm d}^{3}{\rm
p}_{O}\,,$ (3.91)

as required. The required GUR for momentum measurements,

$\displaystyle\Delta_{\rho}p_{Bj}=\sqrt{(\Delta_{\psi}p_{Aj})^{2}+(\Delta_{\phi}p_{Bj})^{2}}\,,$
(3.92)

is obtained from the trace

$\displaystyle\langle(\hat{p}_{Bj})^{n}\rangle_{\rho}={\rm
tr}\left[(\hat{p}_{Bj})^{n}\hat{\rho}_{B}^{(A)}\right]\,,$ (3.93)

where

$\displaystyle\hat{p}_{Bj}=\int p_{Bj}\ket{{\bf p}_{B}}\bra{{\bf
p}_{B}}_{B}{\rm d}^{3}{\rm p}_{B}$ (3.94)

is the canonical one-particle momentum operator. Again, in this formulation of
our QRF model, we have no need to to refer to the fictitious quantum
subsystem, $\mathcal{H}_{O}^{(A)}:=\hat{\mathcal{P}}_{O\leftrightarrow
A}\mathcal{H}_{A}\equiv\mathcal{H}_{A}$. Unlike the operator $\hat{P}_{Bj}$
(3.58), which acts on the state
$\ket{\Psi_{OB}^{(A)}}\in\mathcal{H}_{OB}^{(A)}:=\mathcal{H}_{O}^{(A)}\otimes\mathcal{H}_{B}$
(3.46), or, equivalently, $\hat{\rho}_{OB}^{(A)}$ (3.47), the canonical
momentum operator $\hat{p}_{Bj}$ (3.94) acts directly on Bob’s effective mixed
state in momentum space, $\hat{\tilde{\rho}}_{B}^{(A)}$ (3.86), where
$\ket{\tilde{\phi}_{B}({\bf p}_{O})}_{B}\in\mathcal{H}_{B}$ (3.75). The exact
relation between $\hat{\tilde{\rho}}_{OB}^{(A)}$ (3.47) and
$\hat{\tilde{\rho}}_{B}^{(A)}$ (3.86) is

$\displaystyle\hat{\tilde{\rho}}_{B}^{(A)}:=\int{\rm
tr}_{O}\left[\ket{\mathbb{p}_{O}}\bra{\mathbb{p}_{O}}_{O}\otimes\hat{\tilde{U}}_{B}({\bf
p}_{O})\,\hat{\rho}_{OB}^{(A)}\right]{\rm d}^{3}{\rm p}_{O}\,.$ (3.95)

Yet again, we appear to have a similar problem to that encountered in the GCB
formalism. We have associated two inequivalent mixed states,
$\hat{\rho}_{B}^{(A)}$ (3.77) and $\hat{\tilde{\rho}}_{B}^{(A)}$ (3.86), which
are related by non-unitary transformations, with the same QRF $A$, as opposed
to associating this with two inequivalent pure states, which are related by
unitary transformations, as in [14]. However, in reality, there is no
ambiguity. As we have already shown, the mixed states $\hat{\rho}_{B}^{(A)}$
(3.77) and $\hat{\tilde{\rho}}_{B}^{(A)}$ (3.86) yield the GURs (3.81) and
(3.92), respectively. In Sec. 3.2.5, we showed how, by combining these
relations, Alice is in principle able to determine that she, like Bob, is
quantum mechanical in nature. To do this, she must combine the statistical
results obtained by performing repeated position measurements, on an ensemble
of identically prepared systems, with the statistics obtained from repeated
momentum measurements, performed on a second ensemble of states.

Performing only position measurements leads Alice to conclude that Bob’s state
is the mixed state $\hat{\rho}_{B}^{(A)}$ (3.77), whereas performing only
momentum measurements leads her to identify his state as
$\hat{\tilde{\rho}}_{B}^{(A)}$ (3.86). Only when she combines these two sets
of complimentary results does she notice an apparent contradiction; the
inequivalence of these two viewpoints alerts her to the fact that, in order to
explain the observational facts, she must account, also, for the quantum
mechanical nature of her own system. This suggests an observational protocol,
capable of distinguishing between competing definitions of the term ‘quantum
reference frame’.

In summary, if the states $\hat{\rho}_{B}^{(A)}$ (3.77) and
$\hat{\tilde{\rho}}_{B}^{(A)}$ (3.86) are dissimilar enough for Alice to be
able to detect a difference between the two, within the limits of her
experimental precision, then she is able to discern the fact that she, like
Bob, is a quantum system. At the level of the uncertainty relations, this is
equivalent to observing the non-Heisenberg statistics implied by Eqs. (3.2.5).
Such deviations from the canonical HUP do not appear to be possible within the
standard QRF formalism [14], so that designing an experiment to detect them
should enable us to determine, once and for all, whether wave functions
objectively exist in the physical background space [91], and, hence, what the
correct definition of a QRF should be.

These remarks complete our picture of Alice’s QRF, but the findings above beg
an important question: If all physical observers are QRFs, and if all QRFs,
like Alice, perceive pure states as mixed, then how can we reconcile this with
the observation of pure states in physical experiments? The answer, of course,
is that the mixed nature of the states $\hat{\rho}_{B}^{(A)}$ (3.77) and
$\hat{\tilde{\rho}}_{B}^{(A)}$ (3.86) is obscured for observers with
macroscopic proportions. To see this, it is first easier to rewrite (3.86) in
terms of velocities rather than momenta, using the relations

$\displaystyle{\bf v}_{O}={\bf p}_{O}/M_{O}\,,\quad{\bf v}_{B}={\bf
p}_{B}/m_{B}\,,$ (3.96)

which are valid in the nonrelativistic approximation. The corresponding
uncertainty relations are then

$\displaystyle\Delta_{\psi}x_{O}^{i}\,\Delta_{\psi}v_{Oj}\geq\frac{\hbar}{2M_{O}}\,,\quad\Delta_{\psi}x_{B}^{i}\,\Delta_{\psi}v_{Bj}\geq\frac{\hbar}{2m_{B}}\,.$
(3.97)

Here, we assume that the mass of the observed system, Bob, is very small
compared to that of the observer, Alice, $m_{B}\ll M_{O}\equiv M_{A}$, and
note that it is possible to obtain the limit

$\displaystyle\Delta_{\psi}x_{O}^{i}\,\Delta_{\psi}v_{Oj}\rightarrow
0\,,\quad{\rm as}\quad M_{O}\rightarrow\infty\,.$ (3.98)

This is compatible with the approximations

$\displaystyle|\psi_{O}(-{\bf x}_{O})|^{2}\simeq\delta^{3}(-{\bf
x}_{O})\quad{\rm and}\quad|\tilde{\psi}_{O}(-{\bf
v}_{O})|^{2}\simeq\delta^{3}(-{\bf v}_{O})\,,$ (3.99)

giving

$\displaystyle\Delta_{\psi}x_{O}^{i}\simeq 0\quad{\rm
and}\quad\Delta_{\psi}v_{Oj}\simeq 0\,,$ (3.100)

individually, for large enough $M_{O}$.

Next consider, for example, the scenario in which the system corresponding to
Alice’s QRF is not a microscopic particle, but a massive composite body
consisting of many individual particles. In this case, her position coordinate
${\bf r}_{A}=-{\bf x}_{O}$ does not represent the position of a single
particle, but the position of the centre-of-mass of the composite body, and
$\bm{\kappa}_{A}=-{\bf p}_{O}=-M_{O}{\bf v}_{O}$ is its conjugate momentum.
Now let us assume, for the sake of argument, that Alice’s wave function has a
positional uncertainty of the order of an angstrom and that her mass is of the
order of $\sim 100{\rm kg}$, which would not be unreasonable for a realistic
piece of laboratory equipment, or for the human observer operating it. Her
corresponding velocity uncertainty is of the order of $\sim 10^{-26}$ ${\rm
ms^{-1}}$. 141414For context, we note that, if this were a true velocity,
rather than the standard deviation of a statistical distribution, it would
take an object travelling at this speed roughly the age of the Universe,
$t_{\rm U}\simeq 10^{17}$ s, to traverse the diameter of a single atom. In
short, her wave function is extremely narrow, in both position space and
velocity space, due to the large value of her mass, so that both
$|\psi_{O}({\bf x}_{O})|^{2}$ and $|\tilde{\psi}_{O}({\bf v}_{O})|^{2}$ have
extremely narrow and effectively compact support. In this limit, both the
mixed states $\hat{\rho}_{B}^{(A)}$ (3.77) and $\hat{\tilde{\rho}}_{B}^{(A)}$
(3.86) reduce, approximately, to the same density matrix, which is simply the
density matrix of the canonical pure state of Bob’s system, as seen by the CRF
$O$, $\ket{\phi_{B}}\bra{\phi_{B}}_{B}$, though just how good this
approximation is will depend on how tight Alice’s positional uncertainty is,
and whether or not she is heavy enough to compensate for the corresponding
spread in momentum. 151515Specifically, we obtain
$\hat{\rho}_{B}^{(A)}\,,\,\hat{\tilde{\rho}}_{B}^{(A)}\rightarrow\ket{\phi_{B}}\bra{\phi_{B}}_{B}$
as $|\psi_{O}({\bf x}_{O})|^{2}\rightarrow\delta^{3}({\bf x}_{O})$ and
$|\tilde{\psi}_{O}({\bf v}_{O})|^{2}\rightarrow\delta^{3}({\bf v}_{O})$
($M_{O}\rightarrow\infty$), but the correct pure state is obtained from the
limits of the corresponding state vectors, $\ket{\phi_{B}({\bf
x}_{O})}_{B}\,,\,\ket{\phi_{B}({\bf
v}_{O})}_{B}\rightarrow\ket{\phi_{B}}_{B}$, as integration over the
aforementioned probability distributions, in Eqs. (3.77) and (3.86), sets
${\bf x}_{O}=0$ and ${\bf v}_{O}=0$ in Eqs. (3.75) and (3.87), respectively.

In our QRF model, pure quantum states exist objectively in the classical
background geometry but these can only be perceived as pure, with complete
accuracy, by classical observers (i.e., by CRFs defined by the background
itself [62, 91]). By contrast, all physical observers perceive pure states as
mixed. However, as the quantum subsystem that acts as a QRF becomes ‘more
classical’ (i.e., larger and heavier), it more accurately perceives the nature
of its fellow subsystems. In this sense, taking the $M_{O}\rightarrow\infty$
limit of Eqs. (3.77) and (3.86) provides a model of the effective quantum-to-
classical transition, for a designated ‘observer’, and shows how this
observer’s perception of the ‘observed’ quantum subsystem changes,
accordingly.

###  3.3 Gedanken experiment with a tripartite system

We may construct a model in which a single subsystem of a tripartite state
acts as a QRF by complete analogy with the bipartite state analysis, given in
Sec. 3.2. Here, we use the bold face letters $({\bf
r}_{A/B/C},\bm{\kappa}_{A/B/C})$ to denote the phase space coordinates of
Alice, Bob and Charlie, as seen by an arbitrary CRF $O$, $({\bf
x}_{O/B/C},{\bf p}_{O/B/C})$ to denote the coordinates of $O$, Bob and
Charlie, as seen by Alice, $({\bf q}_{O/A/C},\bm{\pi}_{O/A/C})$ to denote the
coordinates of $O$, Alice and Charlie, as seen by Bob, and $({\bf
a}_{O/A/B},\bm{\alpha}_{O/A/B})$ to denote the coordinates of $O$, Alice and
Bob, as seen by Charlie, as shown in Fig. 6. The transition CRF $O$
$\rightarrow$ QRF $A$ corresponds to the coordinate transformation

$\displaystyle{\bf r}_{A}\mapsto{\bf x}_{O}=-{\bf r}_{A}\,,\quad{\bf
r}_{B}\mapsto{\bf x}_{B}={\bf r}_{B}-{\bf r}_{A}\,,\quad{\bf r}_{C}\mapsto{\bf
x}_{C}={\bf r}_{C}-{\bf r}_{A}\,,$ $\displaystyle\bm{\kappa}_{A}\mapsto{\bf
p}_{O}=-\bm{\kappa}_{A}\,,\quad\bm{\kappa}_{B}\mapsto{\bf
p}_{B}=\bm{\kappa}_{B}-\bm{\kappa}_{A}\,,\quad\bm{\kappa}_{C}\mapsto{\bf
p}_{C}=\bm{\kappa}_{C}-\bm{\kappa}_{A}\,,$ (3.101)

the transition CRF $O$ $\rightarrow$ QRF $B$ corresponds to

$\displaystyle{\bf r}_{A}\mapsto{\bf q}_{A}=-({\bf r}_{B}-{\bf
r}_{A})\,,\quad{\bf r}_{B}\mapsto{\bf q}_{O}=-{\bf r}_{B}\,,\quad{\bf
r}_{C}\mapsto{\bf q}_{C}={\bf r}_{C}-{\bf r}_{AB}\,,$
$\displaystyle\bm{\kappa}_{A}\mapsto\bm{\pi}_{A}=-(\bm{\kappa}_{B}-\bm{\kappa}_{A})\,,\quad\bm{\kappa}_{B}\mapsto\bm{\pi}_{O}=-\bm{\kappa}_{B}\,,\quad\bm{\kappa}_{C}\mapsto\bm{\pi}_{C}=\bm{\kappa}_{C}-\bm{\kappa}_{B}\,,$
(3.102)

and the transition CRF $O$ $\rightarrow$ QRF $C$ corresponds to

$\displaystyle{\bf r}_{A}\mapsto{\bf a}_{A}=-({\bf r}_{C}-{\bf
r}_{A})\,,\quad{\bf r}_{B}\mapsto{\bf a}_{B}=-({\bf r}_{C}-{\bf
r}_{B})\,,\quad{\bf r}_{C}\mapsto{\bf a}_{O}=-{\bf r}_{C}\,,$
$\displaystyle\bm{\kappa}_{A}\mapsto\bm{\alpha}_{A}=-(\bm{\kappa}_{C}-\bm{\kappa}_{A})\,,\quad\bm{\kappa}_{B}\mapsto\bm{\alpha}_{B}=-(\bm{\kappa}_{C}-\bm{\kappa}_{B})\,,\quad\bm{\kappa}_{C}\mapsto\bm{\alpha}_{O}=-\bm{\kappa}_{C}\,.$
(3.103)

Figure 6: A tripartite state, consisting of quantum subsystems $A$, $B$ and
$C$, viewed from the perspective of an arbitrary classical observer, $O$, in
both position and momentum space.

The composite state of the ABC system, from $O$’s perspective, is

$\displaystyle\ket{\Psi_{ABC}^{(O)}}_{ABC}=\int\int\int\psi_{A}({\bf
r}_{A})\phi_{B}({\bf r}_{B})\xi_{C}({\bf r}_{C})\ket{{\bf r}_{A}}_{A}\ket{{\bf
r}_{B}}_{B}\ket{{\bf r}_{C}}_{C}{\rm d}^{3}{\rm r}_{A}{\rm d}^{3}{\rm
r}_{B}{\rm d}^{3}{\rm r}_{C}$
$\displaystyle=\int\int\int\tilde{\psi}_{A}(\bm{\kappa}_{A})\tilde{\phi}_{B}(\bm{\kappa}_{B})\tilde{\xi}_{C}(\bm{\kappa}_{C})\ket{\bm{\kappa}_{A}}_{A}\ket{\bm{\kappa}_{B}}_{B}\ket{\bm{\kappa}_{C}}_{C}{\rm
d}^{3}{\rm\kappa}_{A}{\rm d}^{3}{\rm\kappa}_{B}{\rm d}^{3}{\rm\kappa}_{C}\,,$
(3.104)

which is equivalent to the density operator

$\displaystyle\hat{\rho}_{ABC}^{(O)}:=\ket{\Psi_{ABC}^{(O)}}\bra{\Psi_{ABC}^{(O)}}_{ABC}\,,$
(3.105)

where we again assume separability, for the sake of simplicity, and the states
$\ket{\Psi_{OBC}^{(A)}}_{OBC}$, $\ket{\Psi_{AOC}^{(B)}}_{AOC}$ and
$\ket{\Psi_{ABO}^{(C)}}_{ABO}$ are obtained by acting on (3.3) with the
relevant parity swap operators. The probabilities associated with the outcomes
of physical measurements, from the perspective of the QRFs $A$, $B$ or $C$,
are then determined by performing an appropriate projection, followed by the
relevant partial trace,

$\displaystyle{\rm
d}^{6}P(\Phi_{BC}^{(A)}|\Psi_{ABC}^{(O)}):=S_{OABC}^{O\rightarrow
A}(\Phi_{BC}^{(A)})\ket{\Psi_{ABC}^{(O)}}_{ABC}:={\rm
tr}_{O}\left[\hat{\Pi}_{\ket{{\rm Bob-
Charlie}=\Phi_{BC}^{(A)}}_{OBC}}\hat{\rho}_{OBC}^{(A)}\right]{\rm d}^{6}{\rm
V}_{\Phi}\,,$ $\displaystyle{\rm
d}^{6}P(\Phi_{AC}^{(B)}|\Psi_{ABC}^{(O)}):=S_{OABC}^{O\rightarrow
B}(\Phi_{AC}^{(B)})\ket{\Psi_{ABC}^{(O)}}_{ABC}:={\rm
tr}_{O}\left[\hat{\Pi}_{\ket{{\rm Bob-
Charlie}=\Phi_{AC}^{(B)}}_{AOC}}\hat{\rho}_{AOC}^{(B)}\right]{\rm d}^{6}{\rm
V}_{\Phi}\,,$ $\displaystyle{\rm
d}^{6}P(\Phi_{AB}^{(C)}|\Psi_{ABC}^{(O)}):=S_{OABC}^{O\rightarrow
C}(\Phi_{AB}^{(C)})\ket{\Psi_{ABC}^{(O)}}_{ABC}:={\rm
tr}_{O}\left[\hat{\Pi}_{\ket{{\rm Bob-
Charlie}=\Phi_{AB}^{(C)}}_{ABO}}\hat{\rho}_{ABO}^{(C)}\right]{\rm d}^{6}{\rm
V}_{\Phi}\,.$

We may determine the probabilities associated with the results of simultaneous
position (momentum) measurements, on the composite state of Bob and Charlie,
from Alice’s perspective, by projecting onto the bases

$\displaystyle\left\\{\ket{{\bf x}_{O}}_{O}\ket{{\bf x}_{B}-{\bf
x}_{O}}_{B}\ket{{\bf x}_{C}-{\bf x}_{O}}_{C}\right\\}\quad(\left\\{\ket{{\bf
p}_{O}}_{O}\ket{{\bf p}_{B}-{\bf p}_{O}}_{B}\ket{{\bf p}_{C}-{\bf
p}_{O}}_{C}\right\\})\,.$ (3.107)

Likewise, we may determine the probabilities associated with simultaneous
position (momentum) measurements, on the composite state of Alice and Charlie,
from Bob’s perspective, by projecting onto

$\displaystyle\left\\{\ket{{\bf q}_{A}-{\bf q}_{O}}_{A}\ket{{\bf
q}_{O}}_{O}\ket{{\bf q}_{C}-{\bf
q}_{O}}_{C}\right\\}\quad(\left\\{\ket{\bm{\pi}_{A}-\bm{\pi}_{O}}_{A}\ket{\bm{\pi}_{O}}_{O}\ket{\bm{\pi}_{C}-\bm{\pi}_{O}}_{C}\right\\})\,.$
(3.108)

Lastly, we may determine the probabilities associated with simultaneous
position (momentum) measurements, on the composite state of Alice and Bob,
from Charlie’s perspective, by projecting onto

$\displaystyle\left\\{\ket{{\bf a}_{A}-{\bf a}_{O}}_{A}\ket{{\bf a}_{B}-{\bf
a}_{O}}_{B}\ket{{\bf
a}_{O}}_{O}\right\\}\quad(\left\\{\ket{\bm{\alpha}_{A}-\bm{\alpha}_{O}}_{A}\ket{\bm{\alpha}_{B}-\bm{\alpha}_{O}}_{B}\ket{\bm{\alpha}_{O}}_{O}\right\\})\,.$
(3.109)

The explicit calculation for the determination of ${\rm d}^{6}P({\bf
x}_{B},{\bf x}_{C}|\Psi_{OBC}^{(A)})$, for which we introduce the shorthand
notation ${\rm d}^{6}P({\bf x}_{B},{\bf x}_{C})$, for simplicity, is

$\displaystyle{\rm d}^{6}P({\bf x}_{B},{\bf x}_{C})$ (3.110) $\displaystyle:=$
$\displaystyle{\rm tr}_{O}\left[\ket{{\bf x}_{O}}\bra{{\bf
x}_{O}}_{O}\otimes\ket{{\bf x}_{B}-{\bf x}_{O}}\bra{{\bf x}_{B}-{\bf
x}_{O}}_{B}\otimes\ket{{\bf x}_{C}-{\bf x}_{O}}\bra{{\bf x}_{C}-{\bf
x}_{O}}_{C}\hat{\rho}_{OBC}^{(A)}\right]{\rm d}^{3}{\rm x}_{B}{\rm d}^{3}{\rm
x}_{C}$ $\displaystyle=$ $\displaystyle\left(\int|\psi_{O}(-{\bf
x}_{O})|^{2}|\phi_{B}({\bf x}_{B}-{\bf x}_{O})|^{2}|\xi_{C}({\bf x}_{C}-{\bf
x}_{O})|^{2}{\rm d}^{3}{\rm x}_{O}\right){\rm d}^{3}{\rm x}_{B}{\rm d}^{3}{\rm
x}_{C}$ $\displaystyle=$ $\displaystyle|\psi_{O}|^{2}*(|\phi_{B}({\bf
x}_{B})|^{2}|\xi_{C}({\bf x}_{C})|^{2})\,{\rm d}^{3}{\rm x}_{B}{\rm d}^{3}{\rm
x}_{C}\,,$

where the notation $|\psi_{O}|^{2}*(|\phi_{B}({\bf x}_{B})|^{2}|\xi_{C}({\bf
x}_{C})|^{2})$ indicates that $|\psi_{O}|^{2}$ is convolved with the joint
distribution $|\phi_{B}({\bf x}_{B})|^{2}|\xi_{C}({\bf x}_{C})|^{2}$, through
integration over a single dummy variable, but that $|\phi_{B}({\bf
x}_{B})|^{2}$ and $|\xi_{C}({\bf x}_{C})|^{2}$ are not convolved with each
other, since each is translated by the same parameter in the convolution, in
this case ${\bf x}_{O}$. 161616This should not be confused with either
$|\psi_{O}|^{2}*|\phi_{B}|^{2}*|\xi_{C}|^{2}({\bf x}_{B})$ or
$|\psi_{O}|^{2}*|\phi_{B}|^{2}*|\xi_{C}|^{2}({\bf x}_{C})$. The former
requires integration over both ${\bf x}_{O}$ and ${\bf x}_{C}$, whereas the
latter requires integration over both ${\bf x}_{O}$ and ${\bf x}_{B}$.

The same probability density can also be obtained, by complete analogy with
our previous analysis of the bipartite state, by acting with the canonical
projection $\ket{{\bf x}_{B},{\bf x}_{C}}\bra{{\bf x}_{B},{\bf x}_{C}}_{BC}$
on the density operator of the effective $BC$ state, as seen by $A$, which is
defined as

$\displaystyle\hat{\rho}_{BC}^{(A)}:=\int\frac{{\rm d}^{3}P({\bf x}_{O})}{{\rm
d}^{3}{\rm x}_{O}}\ket{\phi_{B}({\bf x}_{O})}\bra{\phi_{B}({\bf
x}_{O})}_{B}\otimes\ket{\xi_{C}({\bf x}_{O})}\bra{\xi_{C}({\bf
x}_{O})}_{C}{\rm d}^{3}{\rm x}_{O}\,,$ (3.111)

where ${\rm d}^{3}P({\bf x}_{O})=|\psi_{O}(-{\bf x}_{O})|^{2}$ (3.78),
$\ket{\phi_{B}({\bf x}_{O})}_{B}$ is given by (3.75), and

$\displaystyle\ket{\xi_{C}({\bf x}_{O})}_{C}:=\hat{U}_{C}({\bf
x}_{O})\ket{\xi_{C}}_{C}:=\int\xi_{C}({\bf x}_{C}-{\bf x}_{O})\ket{{\bf
x}_{C}}_{C}{\rm d}^{3}{\rm x}_{C}\,.$ (3.112)

However, here, an additional subtlety arises that was not present in the case
of the bipartite quantum system. If Alice performs separate measurements of
${\bf x}_{B}$ and ${\bf x}_{C}$, on different copies of identically prepared
states in an ensemble, then the tripartite system reduces, in effect, to two
distinct bipartite systems, $AB$ and $AC$. The former is analysed in Sec. 3.2,
above, and gives rise to the mixed state $\hat{\rho}_{B}^{(A)}$ (3.77),
together with the corresponding GUR (3.81). The $AC$ system then gives rise to
completely analogous expressions, namely

$\displaystyle\hat{\rho}_{C}^{(A)}:=\int\frac{{\rm d}^{3}P({\bf x}_{O})}{{\rm
d}^{3}{\rm x}_{O}}\ket{\xi_{C}^{(A)}({\bf x}_{O})}\bra{\xi_{C}^{(A)}({\bf
x}_{O})}_{C}{\rm d}^{3}{\rm x}_{O}\,,$ (3.113)

and

$\displaystyle\Delta_{\rho}x_{C}^{i}=\sqrt{(\Delta_{\psi}x_{C}^{i})^{2}+(\Delta_{\xi}x_{C}^{i})^{2}}\,.$
(3.114)

In this case, the probabilities of obtaining the relative displacements ${\bf
x}_{B}$ and ${\bf x}_{C}$ are independent,

$\displaystyle{\rm d}^{3}P({\bf x}_{B})$ $\displaystyle=$
$\displaystyle|\psi_{O}|^{2}*|\phi_{B}|^{2}({\bf x}_{B})\,{\rm d}^{3}{\rm
x}_{B}\,,$ $\displaystyle{\rm d}^{3}P({\bf x}_{C})$ $\displaystyle=$
$\displaystyle|\psi_{O}|^{2}*|\xi_{C}|^{2}({\bf x}_{C})\,{\rm d}^{3}{\rm
x}_{C}\,.$ (3.115)

We then see, immediately, that

$\displaystyle{\rm d}^{6}P({\bf x}_{B},{\bf x}_{C})\neq{\rm d}^{3}P({\bf
x}_{B}){\rm d}^{3}P({\bf x}_{C})\,,$ (3.116)

where ${\rm d}^{6}P({\bf x}_{B},{\bf x}_{C})$ is given by Eq. (3.110).

This result can be compared with the analogous results for both a classical
observer in canonical QM and for a QRF in the GCB model [14]. First, we note
that in the canonical theory it is always possible to prepare a bipartite pure
state that is separable, from the perspective of a given CRF $O$. In this
case, the joint-probability distribution ${\rm d}^{6}P({\bf x}_{B},{\bf
x}_{C})$, which represents the probability of obtaining the measured values
${\bf x}_{B}$ and ${\bf x}_{C}$, is related to the individual (marginal)
probability distributions via

$\displaystyle{\rm d}^{6}P({\bf x}_{B},{\bf x}_{C})={\rm d}^{3}P({\bf
x}_{B}){\rm d}^{3}P({\bf x}_{C})\,.$ (3.117)

In particular, we note that this relation holds true regardless of whether the
measurements are performed simultaneously, on a single state in an ensemble,
or separately, on different states therein. Second, we note that in the GCB
model a tripartite quantum state is described by only two wave functions.
Hence, every tripartite state in the GCB formalism has the same mathematical
form as some bipartite state in canonical QM, and vice versa. It is,
therefore, always possible to prepare a state $BC$ that is separable, from the
perspective of a given QRF $A$. For such states, Eq. (3.117) also holds,
regardless of whether the measurements are performed simultaneously, on a
single state in an ensemble, or separately on different states.

These two inequivalent measurement procedures are illustrated in Figs. 7(a)
and 7(b), respectively, and it is clear that they suggest a simple
experimental procedure to discriminate between competing QRF models. In the
GCB formalism, if the composite state of the $BC$ system, as seen by Alice, is
separable, then the relevant probabilities are ${\rm d}^{6}P({\bf x}_{B},{\bf
x}_{C})=|\psi({\bf x}_{C})|^{2}|\phi({\bf x}_{B})|^{2}{\rm d}^{3}{\rm
x}_{B}{\rm d}^{3}{\rm x}_{C}$ and ${\rm d}^{3}P({\bf x}_{B})=|\phi({\bf
x}_{B})|^{2}{\rm d}^{3}{\rm x}_{B}$, ${\rm d}^{3}P({\bf x}_{C})=|\psi({\bf
x}_{C})|^{2}{\rm d}^{3}{\rm x}_{C}$, which clearly obey the relation (3.117).
By contrast, in our model, Eq. (3.117) holds only for separate measurements,
on different states in an ensemble, and cannot hold for simultaneous
measurements on a single tripartite state, regardless of how we prepare the
initial states of the subsystems $A$, $B$ and $C$.

If the wave functions of Alice, Bob and Charlie exist objectively in the
classical background space [62, 91], as we propose here, these different
experimental procedures give rise to different joint-probability densities
because the former projects pairs of wave functions onto pairs of position
eigenstates, whereas the latter projects three wave functions onto three
position eigenstates, at the same time. The relation (3.117) is then
satisfied, for simultaneous measurements, only in the limit $M_{O}\equiv
M_{A}\rightarrow\infty$. In this limit, there is again no distinction between
separate and simultaneous measurements, since Alice’s positional uncertainty
tends to zero ($|\psi_{O}|^{2}\equiv|\psi_{A}|^{2}\rightarrow\delta^{3}(-{\bf
x}_{O})$) and the effective mixed-state density matrix $\hat{\rho}_{BC}^{(A)}$
(3.111) reduces to canonical pure-state form,
$\hat{\rho}_{BC}^{(A)}\rightarrow\hat{\rho}_{BC}=\ket{\phi_{B}}\bra{\phi_{B}}_{B}\otimes\ket{\xi_{C}}\bra{\xi_{C}}_{C}$,
which is valid for a classical observer.

In the momentum space representation, the explicit calculation for the
determination of the probability density ${\rm d}^{6}P({\bf p}_{B},{\bf
p}_{C}|\Psi_{OBC}^{(A)})$, for which we introduce the shorthand notation ${\rm
d}^{6}P({\bf p}_{B},{\bf p}_{C})$, proceeds as

$\displaystyle{\rm d}^{6}P({\bf p}_{B},{\bf p}_{C})$ (3.118) $\displaystyle:=$
$\displaystyle{\rm tr}_{O}\left[\ket{{\bf p}_{O}}\bra{{\bf
p}_{O}}_{O}\otimes\ket{{\bf p}_{B}-{\bf p}_{O}}\bra{{\bf p}_{B}-{\bf
p}_{O}}_{B}\otimes\ket{{\bf p}_{C}-{\bf p}_{O}}\bra{{\bf p}_{C}-{\bf
p}_{O}}_{C}\hat{\rho}_{OBC}^{(A)}\right]{\rm d}^{3}{\rm p}_{B}{\rm d}^{3}{\rm
p}_{C}$ $\displaystyle=$ $\displaystyle\left(\int|\tilde{\psi}_{O}(-{\bf
p}_{O})|^{2}|\tilde{\phi}_{B}({\bf p}_{B}-{\bf
p}_{O})|^{2}|\tilde{\xi}_{C}({\bf p}_{C}-{\bf p}_{O})|^{2}{\rm d}^{3}{\rm
p}_{O}\right){\rm d}^{3}{\rm p}_{B}{\rm d}^{3}{\rm p}_{C}$ $\displaystyle=$
$\displaystyle|\tilde{\psi}_{O}|^{2}*(|\tilde{\phi}_{B}({\bf
p}_{B})|^{2}|\tilde{\xi}_{C}({\bf p}_{C})|^{2})\,{\rm d}^{3}{\rm p}_{B}{\rm
d}^{3}{\rm p}_{C}\,.$

The same probability density can be obtained by acting with the canonical
projection operator $\ket{{\bf p}_{B},{\bf p}_{C}}\bra{{\bf p}_{B},{\bf
p}_{C}}_{BC}$ on the density matrix

$\displaystyle\hat{\tilde{\rho}}_{BC}^{(A)}:=\int\frac{{\rm d}^{3}P({\bf
p}_{O})}{{\rm d}^{3}{\rm p}_{O}}\ket{\phi_{B}({\bf p}_{O})}\bra{\phi_{B}({\bf
p}_{O})}_{B}\otimes\ket{\xi_{C}({\bf p}_{O})}\bra{\xi_{C}({\bf
p}_{O})}_{C}{\rm d}^{3}{\rm p}_{O}\,,$ (3.119)

where ${\rm d}^{3}P({\bf p}_{O})=|\tilde{\psi}_{O}(-{\bf p}_{O})|^{2}$ (3.89),
$\ket{\phi_{B}({\bf p}_{O})}_{B}$ is given by (3.87), and

$\displaystyle\ket{\xi_{C}({\bf p}_{O})}_{C}:=\hat{\tilde{U}}_{C}({\bf
p}_{O})\ket{\xi_{C}}_{C}:=\int\xi_{C}({\bf p}_{C}-{\bf p}_{O})\ket{{\bf
p}_{C}}_{C}{\rm d}^{3}{\rm p}_{C}\,,$ (3.120)

by complete analogy with our previous analysis for position measurements.

If, instead, Alice performs separate measurements of ${\bf p}_{B}$ and ${\bf
p}_{C}$, on different systems in an ensemble of identically prepared states,
the tripartite system again reduces, in effect, to two bipartite systems, $AB$
and $AC$. Performing measurements of ${\bf p}_{B}$ on the former gives rise to
the mixed state $\hat{\tilde{\rho}}_{B}^{(A)}$ (3.86), together with the
corresponding GUR (3.92), and perfomring measurements of ${\bf p}_{C}$ on the
latter generates completely analogous expressions,

$\displaystyle\hat{\tilde{\rho}}_{C}^{(A)}:=\int\frac{{\rm d}^{3}P({\bf
p}_{O})}{{\rm d}^{3}{\rm p}_{O}}\ket{\xi_{C}^{(A)}({\bf
p}_{O})}\bra{\xi_{C}^{(A)}({\bf p}_{O})}_{C}{\rm d}^{3}{\rm p}_{O}\,,$ (3.121)

and

$\displaystyle\Delta_{\rho}p_{C}^{i}=\sqrt{(\Delta_{\psi}p_{C}^{i})^{2}+(\Delta_{\xi}p_{C}^{i})^{2}}\,.$
(3.122)

In the is case, the probabilities of obtaining the relative momenta ${\bf
p}_{B}$ and ${\bf p}_{C}$ are independent,

$\displaystyle{\rm d}^{3}P({\bf p}_{B})$ $\displaystyle=$
$\displaystyle|\tilde{\psi}_{O}|^{2}*|\tilde{\phi}_{B}|^{2}({\bf p}_{B})\,{\rm
d}^{3}{\rm p}_{B}\,,$ $\displaystyle{\rm d}^{3}P({\bf p}_{C})$
$\displaystyle=$
$\displaystyle|\tilde{\psi}_{O}|^{2}*|\tilde{\xi}_{C}|^{2}({\bf p}_{C})\,{\rm
d}^{3}{\rm p}_{C}\,.$ (3.123)

It follows immediately that

$\displaystyle{\rm d}^{6}P({\bf p}_{B},{\bf p}_{C})\neq{\rm d}^{3}P({\bf
p}_{B}){\rm d}^{3}P({\bf p}_{C})\,,$ (3.124)

where ${\rm d}^{6}P({\bf p}_{B},{\bf p}_{C})$ is given by Eq. (3.118).

This result should also be compared with the analogous results for a classical
observer in canonical QM and for a QRF in the GCB model [14]. In the canonical
theory the relevant probability distributions for a bipartite, separable, pure
quantum state $BC$ obey the relation

$\displaystyle{\rm d}^{6}P({\bf p}_{B},{\bf p}_{C})={\rm d}^{3}P({\bf
p}_{B}){\rm d}^{3}P({\bf p}_{C})\,,$ (3.125)

regardless of whether the measurements are performed separately, on different
states in an ensemble, or simultaneously on a single state. A completely
analogous relation therefore holds, in the GCB formalism, for all $BC$ states
that appear separable from the perspective of the QRF $A$.

In our QRF model, the relation (3.125) can only satisfied be satisfied in the
limit $M_{O}\equiv M_{A}\rightarrow\infty$, when the mixed state density
matrix for momentum-measurements, $\hat{\tilde{\rho}}_{BC}^{(A)}$ (3.119),
reduces to the density matrix for the objective pure state,
$\hat{\tilde{\rho}}_{BC}^{(A)}\rightarrow\hat{\rho}_{BC}=\ket{\phi_{B}}\bra{\phi_{B}}_{B}\otimes\ket{\xi_{B}}\bra{\xi_{B}}_{B}$.
To see this, we need only rewrite (3.119) as an integral over the velocity-
space volume element, ${\rm d}^{3}{\rm v}_{O}$, and note that ${\rm
d}^{3}P({\bf v}_{O})/{\rm d}^{3}{\rm v}_{O}=|\tilde{\psi}_{O}(-{\bf
v}_{O})|^{2}\rightarrow\delta^{3}(-{\bf v}_{O})$, for
$M_{O}\rightarrow\infty$. Hence, in this limit, there is no distinction
between separate and simultaneous measurements, of either position or
momentum, since Alice’s position and velocity uncertainties both tend to zero,
and she becomes ‘fully classical’, as expected.

Finally, we note that the generalised position and momentum operators,
representing Alice’s measurements on the $BC$ subsystem of the objective
tripartite state $\ket{\Psi_{ABC}^{(O)}}_{ABC}$ (3.3), can be denoted as
$\hat{X}_{B/C}^{i}:=\hat{\mathcal{X}}_{B/C}^{i}+\hat{\mathcal{X}}_{B/C}^{\prime
i}$ and
$\hat{P}_{B/Cj}:=\hat{\mathcal{P}}_{B/Cj}+\hat{\mathcal{P}}_{B/Cj}^{\prime}$,
respectively. These are constructed, explicitly, via a straightforward
extension of the definitions (3.57) and (3.58), given in Sec. 3.2. By
exponentiating commuting pairs of the sub-operators
$\left\\{\hat{{\bf\mathcal{X}}}_{B/C},\hat{{\bf\mathcal{X}}}_{B/C}^{\prime},\hat{{\bf\mathcal{P}}}_{B/C},\hat{{\bf\mathcal{P}}}_{B/C}^{\prime}\right\\}$
we can again construct useful unitary operators that shuffle between entangled
and unentangled bases of $\ket{\Psi_{ABC}^{(O)}}_{ABC}$ (3.3), in whatever
coordinate system we wish to employ, $({\bf r}_{A/B/C},\bm{\kappa}_{A/B/C})$,
$({\bf x}_{O/B/C},{\bf p}_{O/B/C})$, $({\bf q}_{O/A/C},\bm{\pi}_{O/A/C})$ or
$({\bf a}_{O/A/B},\bm{\alpha}_{O/A/B})$.

We stress, however, that such transformations do not change the fundamental
relations between these coordinate systems. All such unitarily equivalent
bases are consistent with the perspective of the CRF $O$, and with the
relevant coordinate transformation, (3.3), (3.3) or (3.3), for whichever
coordinates we choose to use. Generalised operators representing Bob’s
measurements on the $AC$ subsystem and Charlie’s measurements on the $AB$
subsystem can also be constructed in like manner, although the corresponding
proliferation of notation becomes rather cumbersome. We address this issue in
our treatment of the general $N$-partite system, given in Sec. 3.4.

Figure 7: Fig. 7(a) (left) illustrates an experimental procedure in which
separate measurements of ${\bf x}_{B}$ and ${\bf x}_{C}$ are performed on
different states in an ensemble, causing Alice’s wave function to collapse, in
general, to different position eigenstates. Fig. 7(b) (right) illustrates the
simultaneous measurement of ${\bf x}_{B}$ and ${\bf x}_{C}$ on a single state
of the tripartite system. These procedures generate different joint-
probability distribtutions, which can be used to differentiate between
competing definitions of the term ‘quantum reference frame’. An analogous
diagram can also be constructed for measurements of the relative momenta,
${\bf p}_{B}$ and ${\bf p}_{C}$. In these figures, we indicate the regions in
which the wave functions of $A$, $B$ and $C$ are nonzero with three closed
curves, but neglect to shade their interiors, so that the different
configurations to which the subsystems collapse can be seen more clearly.

###  3.4 Generalisation to $N$-particle states

In generalising our treatment of the tripartite state to a general $N$-partite
system, for arbitrarily large $N$, the biggest challenge we face is that of
choosing an appropriate notation. For $N\gtrsim 10^{2}$, it is not only
cumbersome, but also impracticable, to use different Latin or Greek letters
(and variants thereof) to denote the different coordinates employed by
different subsystems.

We therefore choose, instead, to label the coordinates used by an arbitrary
CRF with a zero, rather than an $O$, and to label those employed by quantum
subsystems numerically as $1,2,\dots N$. The corresponding phase space
coordinates are written explicitly as $({\bf x}_{(0)}^{[m]},{\bf
p}_{(0)}^{[m]})$ and $({\bf x}_{(m)}^{[n]},{\bf p}_{(m)}^{[n]})$,
respectively, for $m\in\left\\{1,2,\dots N\right\\}$ and
$n\in\left\\{0,1,2,\dots N\right\\}$, with $n\neq m$. In other words, we use
round brackets to indicate the subsystem that is acting as the QRF and square
brackets to label all other subsystems. The individual components of these
displacements and momenta can then be labelled using unbracketed suffixes, as
usual, i.e., as $({\bf x}_{(0)}^{[m]i},{\bf p}_{(0)j}^{[m]})$ and $({\bf
x}_{(m)}^{[n]i},{\bf p}_{(m)}^{[n]j})$, where $i,j\in\left\\{1,2,3\right\\}$.
In this notation, the objective state of the $N$-partite system, from the
perspective of CRF-0, is written as

$\displaystyle\ket{\Psi_{1,2,\dots N}^{(0)}}_{1,2,\dots
N}=\int\int\dots\int\bigotimes_{n=1}^{N}\psi_{n}({\bf x}_{(0)}^{[n]})\ket{{\bf
x}_{(0)}^{[n]}}_{n}{\rm d}^{3}{\rm x}_{(0)}^{[n]}\,.$ (3.126)

Again, we assume the state is separable, from the perspective of the classical
observer, for simplicity.

The probabilities associated with the outcomes of physical measurements, from
the perspective of the QRF-$m$, for an arbitrary value of $m$ in the range
$2<m<N$, are then given by

$\displaystyle{\rm d}^{3(N-1)}P(\Phi_{1,2,\dots m-1,m+1,\dots
N}^{(m)}|\Psi_{1,2,\dots N}^{(0)})$ (3.127) $\displaystyle=$ $\displaystyle
S_{0,1,2,\dots N}^{(0\rightarrow m)}(\Phi_{1,2,\dots m-1,m+1,\dots
N}^{(m)})\,\ket{\Psi_{1,2,\dots N}^{(0)}}_{1,2,\dots N}$ $\displaystyle:=$
$\displaystyle{\rm tr}_{0}\left[\hat{\Pi}_{\ket{\Phi_{1,2,\dots m-1,m+1,\dots
N}^{(m)}}_{1,2,\dots N}}\hat{\rho}_{1,2,\dots m-1,0,m+1,\dots
N}^{(m)}\right]{\rm d}^{3(N-1)}{\rm V}_{\Phi}\,,$

where the partial trace is taken over the subspace
$\mathcal{H}_{0}^{(m)}:=\hat{\mathcal{P}}_{0\leftrightarrow
m}\,\mathcal{H}_{m}^{(0)}\equiv\mathcal{H}_{m}$ of the total Hilbert space of
the $N$-partite system. Those for $m=1,2$ and $N$ are defined in like manner,
with index addition taken modulo $N$, but we select $m$ in the range $2<m<N$,
to illustrate our procedure. Hence, in order to determine probability density
associated with simultaneous measurements of the $N-1$ relative displacements,
${\bf x}_{(m)}^{[1]}$, ${\bf x}_{(m)}^{[2]}$, . . , we must project onto the
bases

$\displaystyle\left\\{\ket{{\bf x}_{(m)}^{[1]}-{\bf
x}_{(m)}^{[0]}}_{1}\dots\otimes\ket{{\bf x}_{(m)}^{[m-1]}-{\bf
x}_{(m)}^{[0]}}_{m-1}\otimes\ket{{\bf x}_{(m)}^{[0]}}_{0}\otimes\ket{{\bf
x}_{(m)}^{[m+1]}-{\bf x}_{(m)}^{[0]}}_{m+1}\dots\right\\}\,,$ (3.128)

whereas, to determine the probability density associated with simultaneous
measurements of the $N-1$ relative momenta, ${\bf p}_{(m)}^{[1]}$, ${\bf
p}_{(m)}^{[2]}$, . . , we must project onto the bases

$\displaystyle\left\\{\ket{{\bf p}_{(m)}^{[1]}-{\bf
p}_{(m)}^{[0]}}_{1}\dots\otimes\ket{{\bf p}_{(m)}^{[m-1]}-{\bf
p}_{(m)}^{[0]}}_{m-1}\otimes\ket{{\bf p}_{(m)}^{[0]}}_{0}\otimes\ket{{\bf
p}_{(m)}^{[m+1]}-{\bf p}_{(m)}^{[0]}}_{m+1}\dots\right\\}\,.$ (3.129)

The generalised operators, corresponding to measurements of relative position
and momentum, performed by the QRF-$m$, are denoted as

$\displaystyle\hat{X}_{(m)}^{[n]i}:=\hat{\mathcal{X}}_{(m)}^{[n]i}+\hat{\mathcal{X}}_{(m)}^{\prime[n]i}\,,\quad\hat{P}_{(m)j}^{[n]}:=\hat{\mathcal{P}}_{(m)j}^{[n]}+\hat{\mathcal{P}}_{(m)j}^{\prime[n]}\,,$
(3.130)

and the definitions of the sub-operators
$\left\\{\hat{\mathcal{X}}_{(m)}^{[n]i},\hat{\mathcal{X}}_{(m)}^{\prime[n]i},\hat{\mathcal{P}}_{(m)j}^{[n]},\hat{\mathcal{P}}_{(m)j}^{\prime[n]}\right\\}$
should be obvious to the reader who has patiently followed our arguments thus
far. Using these basic ingredients, the density operators of the effective
mixed states seen by an arbitrary QRF, labelled QRF-$m$, in both position and
momentum space, can be constructed, and it is straightforward to show that
these reduce to the density operator of the objective pure state seen by
CRF-$0$ \- absent the $\mathcal{H}_{m}$ subsystem, which has been traced out -
in the limit $M_{0}\equiv M_{m}\rightarrow\infty$.

##  4 Advantages of the alternative formalism

###  4.1 The spatial background and CRFs

One can view the QRF formalism developed in [14], and in subsequent follow-up
works [15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 23, 26, 27, 28, 29, 30, 31, 32,
33, 34, 35, 39, 38, 37, 36], as encoding the idea that even classical space is
relational, and that there is no ‘objective’ background spacetime. 171717We
thank Flaminia Giacomini for personal communications, clarifying this point.
In our view, this is at least potentially inconsistent, since even QRF
formalisms must rely on some version of the Born rule, as discussed in Sec.
2.1.

By contrast, our QRF formalism pre-supposes the existence of a classical
spacetime background, in which material quantum systems ‘live’, as
superpositions of position (momentum) eigenstates, relative to an arbitrary
CRF. Our justification for this assumption is the formal equivalence between
geometries and the groups that define their symmetries, according to the
Erlangen program [54, 55, 56, 57, 58, 59, 60, 61]. These symmetries also
define the relationships between arbitrary CRFs, so that a one-to-one
correspondence exists between the set of all possible classical frames of
reference, in a given geometry, and the geometry itself. For three-dimensional
Euclidean space, the corresponding momentum space is also Euclidean, and the
relevant symmetry group is the three-dimensional Galilean group, ${\rm G}^{3}$
[63, 62].

We regard this as a distinct advantage of our model, since the relationship
between the wave functions of material quantum systems and the classical
geometric space, on which they are defined as complex-valued scalars, is also
clearly defined [91]. The correct definition of ‘relational’ space, on the
other hand, appears to us to be fraught with metaphysical difficulties, and
the debate regarding its validity goes all the way back to the debate between
Newton and Leibniz [92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104,
105, 106]. The formalism developed here places us very much within the
Newtonian camp, for better or worse.

In the remainder of this subsection, we investigate the consequences of our
metaphysical position for the time evolution of quantum systems, from the
perspective of an arbitrary QRF. For concreteness, we consider the tripartite
state analysed in Sec. 3.3, though our results generalise naturally to higher-
order systems.

From the perspective of an arbitrary CRF $O$, the Hamiltonian of the
tripartite system $ABC$ is

$\displaystyle\hat{H}_{ABC}^{(O)}=\frac{\hat{\kappa}_{A}^{2}}{2m_{A}}+\frac{\hat{\kappa}_{B}^{2}}{2m_{B}}+\frac{\hat{\kappa}_{C}^{2}}{2m_{C}}+\hat{V}_{ABC}(\hat{{\bf
r}}_{A},\hat{{\bf r}}_{B},\hat{{\bf r}}_{C})\,,$ (4.1)

where

$\displaystyle\hat{V}_{ABC}(\hat{{\bf r}}_{A},\hat{{\bf r}}_{B},\hat{{\bf
r}}_{C}):=\hat{V}_{AB}(|\hat{{\bf r}}_{A}-\hat{{\bf
r}}_{B}|)+\hat{V}_{BC}(|\hat{{\bf r}}_{B}-\hat{{\bf
r}}_{C}|)+\hat{V}_{CA}(|\hat{{\bf r}}_{C}-\hat{{\bf r}}_{A}|)\,.$ (4.2)

This generates time evolution of the total state according to

$\displaystyle\ket{\Psi_{ABC}^{(O)}(t)}_{ABC}:=\exp\left(-\frac{i}{\hbar}\hat{H}_{ABC}^{(O)}t\right)\ket{\Psi_{ABC}^{(O)}(0)}_{ABC}\,,$
(4.3)

where $\ket{\Psi_{ABC}^{(O)}(0)}_{ABC}\equiv\ket{\Psi_{ABC}^{(O)}}_{ABC}$ and
the latter is defined in Eqs. (3.3). (Here, we again assume that the initial
state is separable, from $O$’s perspective, for simplicity.)

However, there is, of course, no absolute measure of the energy of a system,
since only relative energies are physically observable [68, 107, 108]. We must
therefore construct three relative Hamiltonians,
$\hat{\mathcal{H}}_{OBC}^{(A)}$, $\hat{\mathcal{H}}_{AOC}^{(B)}$ and
$\hat{\mathcal{H}}_{ABO}^{(C)}$, for the QRFs $A$, $B$ and $C$. The
eigenvalues of these operators represent, respectively, the possible measured
values of the relative energies between (a) Alice’s subsystem and the
composite state of Bob and Charlie, (b) Bob’s subsystem and the composite
state of Charlie and Alice, and (c) Charlie’s subsystem and the composite
state of Alice and Bob. These are defined as

$\displaystyle\hat{\mathcal{H}}_{OBC}^{(A)}:=\frac{\hat{P}_{B}^{2}}{2m_{B}}+\frac{\hat{P}_{C}^{2}}{2m_{C}}+\hat{V}_{OB}(|\hat{{\bf
X}}_{B}|)+\hat{V}_{CO}(|\hat{{\bf X}}_{C}|)+\hat{V}_{BC}(|\hat{{\bf
X}}_{B}-\hat{{\bf X}}_{C}|)\,,$ (4.4)

where

$\displaystyle\hat{{\bf X}}_{B}$ $\displaystyle:=$ $\displaystyle\hat{{\bf
x}}_{O}\otimes\hat{\mathbb{I}}_{B}\otimes\hat{\mathbb{I}}_{C}+\hat{\mathbb{I}}_{O}\otimes(\widehat{{\bf
x}_{B}-{\bf x}_{O}})_{B}\otimes\hat{\mathbb{I}}_{C}\,,$
$\displaystyle\hat{{\bf P}}_{B}$ $\displaystyle:=$ $\displaystyle\hat{{\bf
p}}_{O}\otimes\hat{\mathbb{I}}_{B}\otimes\hat{\mathbb{I}}_{C}+\hat{\mathbb{I}}_{O}\otimes(\widehat{{\bf
p}_{B}-{\bf p}_{O}})_{B}\otimes\hat{\mathbb{I}}_{C}\,,$
$\displaystyle\hat{{\bf X}}_{C}$ $\displaystyle:=$ $\displaystyle\hat{{\bf
x}}_{O}\otimes\hat{\mathbb{I}}_{B}\otimes\hat{\mathbb{I}}_{C}+\hat{\mathbb{I}}_{O}\otimes\hat{\mathbb{I}}_{B}\otimes(\widehat{{\bf
x}_{C}-{\bf x}_{O}})_{C}\,,$ $\displaystyle\hat{{\bf P}}_{C}$
$\displaystyle:=$ $\displaystyle\hat{{\bf
p}}_{O}\otimes\hat{\mathbb{I}}_{B}\otimes\hat{\mathbb{I}}_{C}+\hat{\mathbb{I}}_{O}\otimes\hat{\mathbb{I}}_{B}\otimes(\widehat{{\bf
p}_{C}-{\bf p}_{O}})_{C}\,;$ (4.5)
$\displaystyle\hat{\mathcal{H}}_{AOC}^{(B)}:=\frac{\hat{\Pi}_{C}^{2}}{2m_{C}}+\frac{\hat{\Pi}_{A}^{2}}{2m_{A}}+\hat{V}_{OC}(|\hat{{\bf
Q}}_{C}|)+\hat{V}_{AO}(|\hat{{\bf Q}}_{A}|)+\hat{V}_{CA}(|\hat{{\bf
Q}}_{C}-\hat{{\bf Q}}_{A}|)\,,$ (4.6)

where

$\displaystyle\hat{{\bf Q}}_{C}$ $\displaystyle:=$
$\displaystyle\hat{\mathbb{I}}_{A}\otimes\hat{{\bf
q}}_{O}\otimes\hat{\mathbb{I}}_{C}+\hat{\mathbb{I}}_{A}\otimes\hat{\mathbb{I}}_{O}\otimes(\widehat{{\bf
q}_{C}-{\bf q}_{O}})_{C}\,,$ $\displaystyle\hat{\bm{\Pi}}_{C}$
$\displaystyle:=$
$\displaystyle\hat{\mathbb{I}}_{A}\otimes\hat{\bm{\pi}}_{O}\otimes\hat{\mathbb{I}}_{C}+\hat{\mathbb{I}}_{A}\otimes\hat{\mathbb{I}}_{O}\otimes(\widehat{\bm{\pi}_{C}-\bm{\pi}_{O}})_{C}\,,$
$\displaystyle\hat{{\bf Q}}_{A}$ $\displaystyle:=$
$\displaystyle\hat{\mathbb{I}}_{A}\otimes\hat{{\bf
q}}_{O}\otimes\hat{\mathbb{I}}_{C}+(\widehat{{\bf q}_{A}-{\bf
q}_{O}})_{A}\otimes\hat{\mathbb{I}}_{O}\otimes\hat{\mathbb{I}}_{C}\,,$
$\displaystyle\hat{\bm{\Pi}}_{A}$ $\displaystyle:=$
$\displaystyle\hat{\mathbb{I}}_{A}\otimes\hat{\bm{\pi}}_{O}\otimes\hat{\mathbb{I}}_{C}+(\widehat{\bm{\pi}_{A}-\bm{\pi}_{O}})_{A}\otimes\hat{\mathbb{I}}_{O}\otimes\hat{\mathbb{I}}_{C}\,;$
(4.7)

and

$\displaystyle\hat{\mathcal{H}}_{ABO}^{(C)}:=\frac{\hat{{{{\alpha}}}}_{A}^{2}}{2m_{A}}+\frac{\hat{{{{\alpha}}}}_{B}^{2}}{2m_{B}}+\hat{V}_{OA}(|\hat{{\bf
A}}_{A}|)+\hat{V}_{BO}(|\hat{{\bf A}}_{B}|)+\hat{V}_{AB}(|\hat{{\bf
A}}_{A}-\hat{{\bf A}}_{B}|)\,,$ (4.8)

where

$\displaystyle\hat{{\bf A}}_{A}$ $\displaystyle:=$
$\displaystyle\hat{\mathbb{I}}_{A}\otimes\hat{\mathbb{I}}_{B}\otimes\hat{{\bf
a}}_{O}+(\widehat{{\bf a}_{A}-{\bf
a}_{O}})_{A}\otimes\hat{\mathbb{I}}_{B}\otimes\hat{\mathbb{I}}_{O}\,,$
$\displaystyle\hat{{{\bm{\alpha}}}}_{A}$ $\displaystyle:=$
$\displaystyle\hat{\mathbb{I}}_{A}\otimes\hat{\mathbb{I}}_{B}\otimes\hat{\bm{\alpha}}_{O}+(\widehat{\bm{\alpha}_{A}-\bm{\alpha}_{O}})_{A}\otimes\hat{\mathbb{I}}_{B}\otimes\hat{\mathbb{I}}_{O}\,,$
$\displaystyle\hat{{\bf A}}_{B}$ $\displaystyle:=$
$\displaystyle\hat{\mathbb{I}}_{A}\otimes\hat{\mathbb{I}}_{B}\otimes\hat{{\bf
a}}_{O}+\hat{\mathbb{I}}_{A}\otimes(\widehat{{\bf a}_{B}-{\bf
a}_{O}})_{B}\otimes\hat{\mathbb{I}}_{O}\,,$
$\displaystyle\hat{{{\bm{\alpha}}}}_{B}$ $\displaystyle:=$
$\displaystyle\hat{\mathbb{I}}_{A}\otimes\hat{\mathbb{I}}_{B}\otimes\hat{\bm{\alpha}}_{O}+\hat{\mathbb{I}}_{A}\otimes(\widehat{\bm{\alpha}_{B}-\bm{\alpha}_{O}})_{B}\otimes\hat{\mathbb{I}}_{O}\,.$
(4.9)

Note that the definitions in Eqs. (4.4), (4.6) and (4.8) suppose that the
‘observing’ QRF is able to detect the presence of an interaction potential
between itself and the ‘observed’ subsystems. This is physically reasonable.
For example, suppose that all three subsystems are charge-neutral and that the
masses of the observed particles are small compared to the mass of the
designated QRF. In this case, the gravitational interactions between the QRF
and the two observed particles will dominate the dynamics of the tripartite
system, while the interaction between the observed particles themselves may be
considered negligible. 181818The ability to account for physical interactions
between the ‘observer’ and the ‘observed’ subsystems is, we believe, another
advantage of our QRF model, over the relational models proposed by GCB and
others [14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 23, 26, 27, 28, 29, 30,
31, 32, 33, 34, 35, 39, 38, 37, 36]. In these models, it is impossible to
account for such interactions, since the QRF is not equipped with quantum
degrees of freedom of its own. In our model, an interesting question then
arises, namely, ‘what happens when these interactions generate entanglement,
from the perspective of a classical observer, between the QRF and the observed
subsystems?’. In this case, there is no clear distinction between the quantum-
observer and the quantum-observed, even from the ‘objective’ viewpoint defined
by the classical background geometry. Throughout this work, we have considered
only separable initial states, from the perspective of the CRF $O$, but we
will consider this question in detail in a future analysis. On the other hand,
there is no physical mechanism by which the designated observer can measure
the value of their own kinetic energy, relative to an arbitrary CRF that is
defined by the classical background space. The kinetic terms in these
Hamiltonians therefore contain only the relative momenta between physical
subsystems.

The three relative Hamiltonians generate three distinct relative time-
evolutions, according to

$\displaystyle\ket{\varPsi_{OBC}^{(A)}(t)}_{OBC}:=\exp\left(-\frac{i}{\hbar}\hat{\mathcal{H}}_{OBC}^{(A)}t\right)\hat{\mathcal{P}}_{O\leftrightarrow
A}\ket{\Psi_{ABC}^{(O)}(0)}_{ABC}$ (4.10) $\displaystyle\neq$
$\displaystyle\ket{\Psi_{OBC}^{(A)}(t)}_{OBC}:=\hat{\mathcal{P}}_{O\leftrightarrow
A}\exp\left(-\frac{i}{\hbar}\hat{H}_{ABC}^{(O)}t\right)\ket{\Psi_{ABC}^{(O)}(0)}_{ABC}\,,$
$\displaystyle\ket{\varPsi_{AOC}^{(B)}(t)}_{AOC}:=\exp\left(-\frac{i}{\hbar}\hat{\mathcal{H}}_{AOC}^{(B)}t\right)\hat{\mathcal{P}}_{O\leftrightarrow
B}\ket{\Psi_{ABC}^{(O)}(0)}_{ABC}$ (4.11) $\displaystyle\neq$
$\displaystyle\ket{\Psi_{AOC}^{(B)}(t)}_{AOC}:=\hat{\mathcal{P}}_{O\leftrightarrow
B}\exp\left(-\frac{i}{\hbar}\hat{H}_{ABC}^{(O)}t\right)\ket{\Psi_{ABC}^{(O)}(0)}_{ABC}\,,$

and

$\displaystyle\ket{\varPsi_{ABO}^{(C)}(t)}_{ABO}:=\exp\left(-\frac{i}{\hbar}\hat{\mathcal{H}}_{ABO}^{(C)}t\right)\hat{\mathcal{P}}_{O\leftrightarrow
C}\ket{\Psi_{ABC}^{(O)}(0)}_{ABC}$ (4.12) $\displaystyle\neq$
$\displaystyle\ket{\Psi_{ABO}^{(C)}(t)}_{ABO}:=\hat{\mathcal{P}}_{O\leftrightarrow
C}\exp\left(-\frac{i}{\hbar}\hat{H}_{ABC}^{(O)}t\right)\ket{\Psi_{ABC}^{(O)}(0)}_{ABC}\,.$

The expressions on the bottom lines of Eqs. (4.10), (4.11) and (4.12) are
equivalent to each other, and are each equivalent to the time-evolution given
by Eq. (4.1), where it is understood that, when implementing the relevant
parity swap, we must also implement the relevant coordinate transformation,
given in Eqs. (3.3), (3.3) or (3.3). Note, also, the subtle change in notation
between the top and bottom lines,
$\ket{\varPsi_{OBC}^{(A)}(t)}_{OBC}\neq\ket{\Psi_{OBC}^{(A)}(t)}_{OBC}$, etc.,
where the latter is defined as
$\ket{\Psi_{OBC}^{(A)}(t)}_{OBC}:=\hat{\mathcal{P}}_{O\leftrightarrow
A}\ket{\Psi_{ABC}^{(O)}(t)}_{ABC}$. In other words,
$\ket{\Psi_{OBC}^{(A)}(t)}_{OBC}$ is equivalent to
$\ket{\Psi_{ABC}^{(O)}(t)}_{ABC}$, since the two states differ by only a
parity swap, plus the corresponding change of coordinates.

By contrast, the three states denoted with the ‘varPsi’ symbol, $\varPsi$, are
not equivalent to $\ket{\Psi_{ABC}^{(O)}(t)}_{ABC}$. These represent the
apparent time-evolutions of the tripartite state, as perceived by each of the
three QRFs, Alice, Bob and Charlie. They encode only relative changes, which
are physically observable (i.e., measurable), to the relevant designated
observer. This description is clearly compatible with a Page-Wooters type
scenario [109, 110, 111, 112, 113], in which the wave function of the entire
Universe exists in an energy eigenstate of the total, Universal Hamiltonian,
but apparent time-evolution occurs between subsystems, due to their restricted
access to information about the global state. The relation between the Page-
Wooters model and the QRF model presented here is interesting, and worthy of
further study, but a detailed analysis of this problem lies beyond the scope
of the present work.

###  4.2 Single particles (wave functions) objectively exist

In our formalism, a collection of $N$ wave functions,
$\left\\{\psi_{I}\right\\}_{I=0}^{N-1}$, can be associated with a system of
$N$ quantum particles, whenever the state of the system is separable in some
basis. Although, for entangled states, the notion of a single-particle wave
function - that is, a single function of three quantum mechanical degrees of
freedom - breaks down, the wave function of a general entangled state can
still be expressed as a function of $3N$ degrees of freedom, $\Psi_{N}$. In
terms of the linguistic shorthand introduced in Sec. 2.2, this is equivalent
to ‘an $N$-wave function description of an $N$-particle state’, and the fact
that such a description is not only possible, but necessary, is a direct
consequence of the objective existence of the classical background geometry
[62, 91].

The key point is that, due to the objective existence of the classical spatial
background, in which the wave functions of material particles are defined as
complex-valued scalars [91], it is impossible for situations like the
following to occur. Consider, for example, the three possible pairings of two
particles out of a system of three, $(A,B)$, $(B,C)$ and $(C,A)$. In the GCB
formalism [14], the total state of the tripartite system is given by some
functional, $F_{AB}[\psi\phi]$, from $C$’s perspective, by the functional
$F_{BC}[\psi\phi]$, from $A$’s perspective, and by the functional
$F_{CA}[\psi\phi]$, from $B$’s perspective. These therefore represent the
effective wave functions of the pairs $(A,B)$, $(B,C)$ and $(C,A)$,
respectively.

Thus, even if there exists a basis in which one of these functionals is
separable, e.g.
$F_{AB}[\psi\phi]:=\ket{\psi_{A}}_{A}\otimes\ket{\phi_{B}}_{B}$, we still
cannot associate the function $\psi$ with the particle $A$, nor $\phi$ with
the particle $B$, objectively, since ‘jumping’ to a different QRF changes the
labels on these functions, as well as the labels on the corresponding basis
vectors. In our view, this leads to a paradoxical situation, in which the same
two complex-valued functions of three variables describe, say, a proton and a
neutron from an electron’s perspective, a proton and an electron from a
neutron’s perspective, and an electron and a neutron from a proton’s
perspective.

By contrast, in our formalism, it is always possible to prepare the initial
state of such a composite system in the form
$\ket{\Psi_{pne}}_{pne}=\ket{\psi_{p}}_{p}\otimes\ket{\psi_{n}}_{n}\otimes\ket{\psi_{e}}_{e}$,
in some basis. Furthermore, although subsequent operations may generate
entanglement between the subsystems, the labels associated with the individual
functions, $(\psi_{p1},\psi_{p2},\dots)$, $(\psi_{n1},\psi_{n2},\dots)$ and
$(\psi_{e1},\psi_{e2},\dots)$, which combine to make the total wave function
of entangled state, are physically meaningful, since these do not change under
CRF-to-QRF transitions, or under the transition from one QRF to another. One
simply traces out the degrees of freedom associated with a particular subspace
of the total Hilbert space, $\mathcal{H}_{p}$, $\mathcal{H}_{n}$ or
$\mathcal{H}_{e}$, which is associated with the designated ‘observer’,
according to the procedure outlined in Sec. 3.

###  4.3 No ambiguities in transformations between QRFs

As explained in Sec. 2.3, there exists an ambiguity in the definition of the
final state, when ‘jumping’ from one QRF to another, in the GCB formalism
[14]. There is, therefore, an inherent ambiguity in the definition of all QRF-
to-QRF transitions, since the final state depends on which basis we choose to
express the initial state in.

In our formalism, however, there is only an apparent contradiction. In a
bipartite system, in which the composite state of the two subsystems, $A$ and
$B$, is both pure and separable from the perspective of a given CRF, $O$,
Alice will perceive Bob’s state as mixed. However, the mixed state she ‘sees’
depends on the measurements she uses to probe Bob’s state, with, for example,
position and momentum measurements yielding different density matrices. But
this is not a genuine ambiguity. In the limit $M_{A}\rightarrow\infty$, the
two mixed states reduce to the same ‘objective’ pure state, seen by $O$, and
the measurement-sensitivity that Alice requires, in order to observe a
difference between the two, is exactly the sensitivity she requires to resolve
the non-Heisenberg terms in the GURs that characterise measurements of the
relational variables. Thus, by resolving these terms, Alice may demonstrate
that she, too, is quantum mechanical in nature, like Bob.

In this sense, what appears at first sight to be a bug in our model, is in
fact a feature, since it suggests experimental protocols by which competing
definitions of the term ‘quantum reference frame’ can be distinguished,
empirically. Furthermore, we are able to describe both the initial and final
states in a given QRF-to-QRF transition, as well as the corresponding position
and momentum measurements, in whatever basis we choose, without ambiguity or
contradiction. In short, our results are measurement-dependent, in an
operational sense, but not basis-dependent, mathematically, like those
presented in [14]. Our contention is that basis-dependent predictions, for the
outcomes of physical measurements, do not make sense.

###  4.4 Angular momentum and the spin degrees of freedom

In this section, we again consider the bipartite quantum state, studied in
Sec. 3.2, as the simplest nontrivial example of a QRF model. Our aim is to
show that QRF-to-QRF transitions between states with different relative
angular momenta, and different relative spins, can be rigorously and
unambiguously treated within our proposed formalism. Our conclusions then
generalise, naturally, to higher-order systems.

We begin by notating that, were Alice and Bob classical systems, their angular
momenta, relative to the coordinate origin of the CRF $O$, would be ${\bf
L}_{A}^{(O)}={\bf r}_{A}\times\bm{\kappa}_{A}$ and ${\bf L}_{B}^{(O)}={\bf
r}_{B}\times\bm{\kappa}_{B}$, respectively. The corresponding quantum
operators are $\hat{{\bf L}}_{A}^{(O)}:=(\hat{{\bf
r}}_{A}\otimes\hat{\mathbb{I}}_{B})\times(\hat{\bm{\kappa}}_{A}\otimes\hat{\mathbb{I}}_{B})$
and $\hat{{\bf L}}_{B}^{(O)}:=(\hat{\mathbb{I}}_{A}\otimes\hat{{\bf
r}}_{B})\times(\hat{\mathbb{I}}_{A}\otimes\hat{\bm{\kappa}}_{B})$, where
$\hat{\bm{\kappa}}_{A(B)}$ generate real space translations of Alice’s (Bob’s)
state, relative to $O$, and $\hat{{\bf r}}_{A(B)}$ generate the corresponding
translations in momentum space. The operators $\hat{{\bf L}}_{A}^{(O)}$ and
$\hat{{\bf L}}_{B}^{(O)}$ then generate infinitesimal rotations of $A$ and
$B$, about $O$, respectively.

The question therefore arises, what is the angular momentum of Alice, relative
to Bob? For a classical system, this would clearly be
${\bf\mathfrak{L}}_{B}^{(A)}\equiv{\bf\mathfrak{L}}_{B}:={\bf x}_{B}\times{\bf
p}_{B}$, in our chosen notation, where ${\bf x}_{B}$ and ${\bf p}_{B}$ denote
the relative displacement and momentum of the two states [114]. The
corresponding quantum operator is

$\displaystyle\hat{{\bf\mathfrak{L}}}_{B}:=\hat{{\bf X}}_{B}\times\hat{{\bf
P}}_{B}\,,$ (4.13)

where the components of $\hat{{\bf X}}_{B}$ and $\hat{{\bf P}}_{B}$ are
defined in Eqs. (3.57) and (3.58). Furthermore, since the definitions of
$\hat{{\bf X}}_{B}$ and $\hat{{\bf P}}_{B}$ include the parity swap operation,
$\hat{\mathcal{P}}_{O\leftrightarrow A}$, the eigenvalues of
$\hat{{\bf\mathfrak{L}}}_{B}$ represent ‘the angular momentum of Bob’, as
perceived by Alice.

To confirm that the definition (4.13) makes sense, we now show that $\hat{{\bf
X}}_{B}$ and $\hat{{\bf P}}_{B}$ generate infinitesimal translations of Bob’s
state, relative to Alice’s subsytem, in the relevant dual space. Using these
results, it is straightforward to show that $\hat{{\bf\mathfrak{L}}}_{B}$
generates rotations of $B$’s state, relative to $A$, in both the position and
momentum space representations. Thus, we have

$\displaystyle\exp\left[\frac{i}{2\hbar}\,\hat{{\bf P}}_{B}\,.\,{\bf
a}\right]\,\ket{{\bf x}_{O}}_{O}\otimes\ket{{\bf x}_{B}-{\bf x}_{O}}_{B}$
(4.14) $\displaystyle=$
$\displaystyle\exp\left[\frac{i}{2\hbar}\,\hat{{\bf\mathcal{P}}}_{B}\,.\,{\bf
a}\right]\,\ket{{\bf
x}_{O}}_{O}\otimes\exp\left[\frac{i}{2\hbar}\,\hat{{\bf\mathcal{P}}}^{\prime}_{B}\,.\,{\bf
a}\right]\,\ket{{\bf x}_{B}-{\bf x}_{O}}_{B}$ $\displaystyle=$
$\displaystyle\ket{{\bf x}_{O}+{\bf a}/2}_{O}\otimes\ket{{\bf x}_{B}-{\bf
x}_{O}+{\bf a}/2}_{B}\,,$
$\displaystyle\exp\left[-\frac{i}{2\hbar}\,\bm{\alpha}\,.\,\hat{{\bf
X}}_{B}\right]\,\ket{{\bf p}_{O}}_{O}\otimes\ket{{\bf p}_{B}-{\bf p}_{O}}_{B}$
(4.15) $\displaystyle=$
$\displaystyle\exp\left[-\frac{i}{2\hbar}\,\bm{\alpha}\,.\,\hat{{\bf\mathcal{X}}}_{B}\right]\,\ket{{\bf
p}_{O}}_{O}\otimes\exp\left[-\frac{i}{2\hbar}\,\bm{\alpha}\,.\,\hat{{\bf\mathcal{X}}}^{\prime}_{B}\right]\,\ket{{\bf
p}_{B}-{\bf p}_{O}}_{B}$ $\displaystyle=$ $\displaystyle\ket{{\bf
p}_{O}+\bm{\alpha}/2}_{O}\otimes\ket{{\bf p}_{B}-{\bf
p}_{O}+\bm{\alpha}/2}_{B}\,,$

which generates the transformations

$\displaystyle{\bf x}_{O}\mapsto{\bf x}^{\prime}_{O}={\bf x}_{O}+{\bf
a}/2\,,\quad{\bf x}_{B}-{\bf x}_{O}\mapsto{\bf x}^{\prime}_{B}-{\bf
x}^{\prime}_{O}=({\bf x}_{B}-{\bf x}_{O})+{\bf a}/2\,,$ $\displaystyle{\bf
x}_{B}\mapsto{\bf x}^{\prime}_{B}={\bf x}_{B}+{\bf a}\,,$ (4.16)
$\displaystyle{\bf p}_{O}\mapsto{\bf p}^{\prime}_{O}={\bf
p}_{O}+\bm{\alpha}/2\,,\quad{\bf p}_{B}-{\bf p}_{O}\mapsto{\bf
p}^{\prime}_{B}-{\bf p}^{\prime}_{O}=({\bf p}_{B}-{\bf
p}_{O})+\bm{\alpha}/2\,,$ $\displaystyle{\bf p}_{B}\mapsto{\bf
p}^{\prime}_{B}={\bf p}_{B}+\bm{\alpha}\,,$ (4.17)

or, equivalently,

$\displaystyle-{\bf r}_{A}\mapsto-{\bf r}^{\prime}_{A}=-{\bf r}_{A}+{\bf
a}/2\,,\quad{\bf r}_{B}\mapsto{\bf r}^{\prime}_{B}={\bf r}_{B}+{\bf a}/2\,,$
$\displaystyle{\bf r}_{B}-{\bf r}_{A}\mapsto{\bf r}^{\prime}_{B}-{\bf
r}^{\prime}_{A}={\bf r}_{B}-{\bf r}_{A}+{\bf a}\,,$ (4.18)
$\displaystyle-\bm{\kappa}_{A}\mapsto-\bm{\kappa}^{\prime}_{A}=-\bm{\kappa}_{A}+{\bf
a}/2\,,\quad\bm{\kappa}_{B}\mapsto\bm{\kappa}^{\prime}_{B}=\bm{\kappa}_{B}+{\bf
a}/2\,,$
$\displaystyle\bm{\kappa}_{B}-\bm{\kappa}_{A}\mapsto\bm{\kappa}^{\prime}_{B}-\bm{\kappa}^{\prime}_{A}=\bm{\kappa}_{B}-\bm{\kappa}_{A}+{\bf
a}\,.$ (4.19)

(See Fig. 8.)

Figure 8: The operator $\exp\left[\frac{i}{2\hbar}\,\hat{{\bf P}}_{B}\,.\,{\bf
a}\right]$ translates ${\bf r}_{B}$ by ${\bf a}/2$ and $-{\bf r}_{A}$ by ${\bf
a}/2$. The latter is equivalent to shifting ${\bf r}_{A}$ by $-{\bf a}/2$, so
that the net result is a shift of $B$’s state, relative to $A$, by ${\bf a}$.
The analogous diagram in momentum space corresponds to the action of the
operator $\exp\left[-\frac{i}{2\hbar}\,\bm{\alpha}\,.\,\hat{{\bf
X}}_{B}\right]$.

Note that, in Eqs. (4.14) and (4.15), the factor of $(2\hbar)^{-1}$ in the
exponent is included due to the rescaled Heisenberg algebra,

$\displaystyle[\hat{X}_{B}^{i},\hat{P}_{Bj}]=2i\hbar\,\delta^{i}{}_{j}\hat{\mathbb{I}}_{OB}\,,$
(4.20) $\displaystyle[\hat{X}_{B}^{i},\hat{X}_{B}^{j}]=0\,,$ (4.21)
$\displaystyle[\hat{P}_{Bi},\hat{P}_{Bj}]=0\,,$ (4.22)

which, in turn, is generated by the algebra of the subcomponents
$\left\\{\hat{\mathcal{X}}_{B}^{i},\hat{\mathcal{X}}_{B}^{\prime
i},\hat{\mathcal{P}}_{Bj},\hat{\mathcal{P}}^{\prime}_{Bj}\right\\}$,

$\displaystyle[\hat{\mathcal{X}}_{B}^{i},\hat{\mathcal{P}}_{Bj}]=i\hbar\delta^{i}{}_{j}\hat{\mathbb{I}}_{OB}\,,\quad[\hat{\mathcal{X}}_{B}^{\prime
i},\hat{\mathcal{P}}^{\prime}_{Bj}]=i\hbar\delta^{i}{}_{j}\hat{\mathbb{I}}_{OB}\,,$
(4.23)
$\displaystyle[\hat{\mathcal{X}}_{B}^{i},\hat{\mathcal{P}}^{\prime}_{Bj}]=[\hat{\mathcal{X}}_{B}^{i},\hat{\mathcal{P}}^{\prime}_{Bj}]=0\,,$
(4.24)
$\displaystyle[\hat{\mathcal{X}}_{B}^{i},\hat{\mathcal{X}}_{B}^{j}]=[\hat{\mathcal{X}}_{B}^{\prime
i},\hat{\mathcal{X}}_{B}^{\prime j}]=0\,,$ (4.25)
$\displaystyle[\hat{\mathcal{X}}_{B}^{i},\hat{\mathcal{X}}_{B}^{\prime
j}]=[\hat{\mathcal{X}}_{B}^{\prime i},\hat{\mathcal{X}}_{B}^{j}]=0\,,$ (4.26)
$\displaystyle[\hat{\mathcal{P}}_{Bi},\hat{\mathcal{P}}_{Bj}]=[\hat{\mathcal{P}}^{\prime}_{Bi},\hat{\mathcal{P}}^{\prime}_{Bj}]=0\,,$
(4.27)
$\displaystyle[\hat{\mathcal{P}}_{Bi},\hat{\mathcal{P}}^{\prime}_{Bj}]=[\hat{\mathcal{P}}^{\prime}_{Bi},\hat{\mathcal{P}}_{Bj}]=0\,.$
(4.28)

Equations (4.23)-(4.28) define two commuting copies of the canonical
Heisenberg algebra, the first of which acts nontrivially only on the subspace
$\mathcal{H}_{O}^{(A)}:=\hat{\mathcal{P}}_{A\rightarrow
O}\mathcal{H}_{A}\equiv\mathcal{H}_{A}$ of the total Hilbert space of the
bipartite system, while the second acts nontrivially only on the
$\mathcal{H}_{B}$ subspace. However, as we will now show, the resulting
algebra for the subcomponents of the relative angular momentum operators,
$\left\\{\hat{\mathfrak{L}}_{Bi}\right\\}_{i=1}^{3}$, is not so simple. First,
we define these via the relations

$\displaystyle\hat{\mathfrak{L}}_{Bi}:=\epsilon_{ij}{}^{k}\hat{X}_{B}^{j}\hat{P}_{Bk}:=\hat{\mathcal{L}}_{Bi}+\hat{\mathcal{L}}^{\prime}_{Bi}+\hat{\mathbb{L}}_{Bi}\,,$
(4.29)

where

$\displaystyle\hat{\mathcal{L}}_{Bi}:=\epsilon_{ij}{}^{k}\hat{\mathcal{X}}_{B}^{j}\hat{\mathcal{P}}_{Bk}\,,\quad\hat{\mathcal{L}}^{\prime}_{Bi}:=\epsilon_{ij}{}^{k}\hat{\mathcal{X}}_{B}^{\prime
j}\hat{\mathcal{P}}^{\prime}_{Bk}\,,\quad\hat{\mathbb{L}}_{Bi}:=\epsilon_{ij}{}^{k}(\hat{\mathcal{X}}_{B}^{j}\hat{\mathcal{P}}^{\prime}_{Bk}+\hat{\mathcal{X}}_{B}^{\prime
j}\hat{\mathcal{P}}_{Bk})\,.$ (4.30)

Following our previous convention, the unprimed subcomponents
$\hat{\mathcal{L}}_{Bi}$ act nontrivially only on the
$\mathcal{H}_{O}^{(A)}:=\hat{\mathcal{P}}_{A\rightarrow
O}\mathcal{H}_{A}\equiv\mathcal{H}_{A}$ subspace, whereas the primed
subcomponents $\hat{\mathcal{L}}^{\prime}_{Bi}$ act nontrivially only on
$\mathcal{H}_{B}$. In addition, there exists a third subcomponent,
$\hat{\mathbb{L}}_{Bi}$, which acts nontrivially on both subspaces, and the
resulting algebra for the subcomponents
$\left\\{\hat{\mathcal{L}}_{Bi},\hat{\mathcal{L}}^{\prime}_{Bi},\hat{\mathbb{L}}_{Bi}\right\\}$
is

$\displaystyle[\hat{\mathcal{L}}_{Bi},\hat{\mathcal{L}}_{Bj}]=i\hbar\epsilon_{ij}{}^{k}\,\hat{\mathcal{L}}_{Bk}\,,\quad[\hat{\mathcal{L}}^{\prime}_{Bi},\hat{\mathcal{L}}^{\prime}_{Bj}]=i\hbar\epsilon_{ij}{}^{k}\,\hat{\mathcal{L}}^{\prime}_{Bk}\,,$
(4.31)
$\displaystyle[\hat{\mathcal{L}}_{Bi},\hat{\mathcal{L}}^{\prime}_{Bj}]=[\hat{\mathcal{L}}^{\prime}_{Bi},\hat{\mathcal{L}}_{Bj}]=0\,,$
(4.32)
$\displaystyle[\hat{\mathcal{L}}_{Bi},\hat{\mathbb{L}}_{Bj}]-[\hat{\mathcal{L}}_{Bj},\hat{\mathbb{L}}_{Bi}]=i\hbar\epsilon_{ij}{}^{k}\,\hat{\mathbb{L}}_{Bk}\,,$
(4.33)
$\displaystyle[\hat{\mathcal{L}}^{\prime}_{Bi},\hat{\mathbb{L}}_{Bj}]-[\hat{\mathcal{L}}^{\prime}_{Bj},\hat{\mathbb{L}}_{Bi}]=i\hbar\epsilon_{ij}{}^{k}\,\hat{\mathbb{L}}_{Bk}\,,$
(4.34)
$\displaystyle[\hat{\mathbb{L}}_{Bi},\hat{\mathbb{L}}_{Bj}]=i\hbar\epsilon_{ij}{}^{k}\,\hat{\mathcal{L}}_{Bk}+i\hbar\epsilon_{ij}{}^{l}\,\hat{\mathcal{L}}^{\prime}_{Bl}\,.$
(4.35)

Together, Eqs. (4.31)-(4.35) yield

$\displaystyle[\hat{\mathfrak{L}}_{Bi},\hat{\mathfrak{L}}_{Bj}]=2i\hbar\epsilon_{ij}{}^{k}\,\hat{\mathfrak{L}}_{Bk}\,,$
(4.36)

and, hence,

$\displaystyle[\hat{\mathfrak{L}}_{B}^{2},\hat{\mathfrak{L}}_{Bi}]=0\,.$
(4.37)

The corresponding uncertainty relations take the form

$\displaystyle(\Delta_{\Psi}\mathfrak{L}_{Bi})^{2}\,(\Delta_{\Psi}\mathfrak{L}_{Bj})^{2}\geq\dots\geq\hbar^{2}(\epsilon_{ij}{}^{k})^{2}\,\langle\hat{\mathfrak{L}}_{Bk}\rangle_{\Psi}^{2}\,,$
(4.38)

where the absolute limit on the far right-hand side follows from the
Schrödinger-Robertson bound [68, 69], in conjunction with (4.36), and the dots
in the middle represent a sum of terms that are generically greater than or
equal to this. There are $7^{2}=49$ such terms, due to the generalised
uncertainties

$\displaystyle(\Delta_{\Psi}\mathfrak{L}_{Bi})^{2}$ $\displaystyle=$
$\displaystyle(\Delta_{\Psi}\mathcal{L}_{Bi})^{2}+(\Delta_{\Psi}\mathcal{L}^{\prime}_{Bi})^{2}+(\Delta_{\Psi}\mathbb{L}_{Bi})^{2}$
(4.39) $\displaystyle+$ $\displaystyle{\rm
cov}(\hat{\mathcal{L}}_{Bi},\hat{\mathbb{L}}_{Bi})+{\rm
cov}(\hat{\mathbb{L}}_{Bi},\hat{\mathcal{L}}_{Bi})$ $\displaystyle+$
$\displaystyle{\rm
cov}(\hat{\mathcal{L}}^{\prime}_{Bi},\hat{\mathbb{L}}_{Bi})+{\rm
cov}(\hat{\mathbb{L}}_{Bi},\hat{\mathcal{L}}^{\prime}_{Bi})\,,$

where the subscript $\Psi$ again refers to the composite state of the
bipartite system, $\ket{\Psi_{OB}^{(A)}}_{OB}$ (3.46), and the covariance of
two random variables is defined as ${\rm cov}(X,Y):=\langle XY\rangle-\langle
X\rangle\langle Y\rangle$.

Finally, we are able to demonstrate that the operator
$\exp\left[\frac{i}{2\hbar}\,\theta\hat{\underline{n}}\,.\,\hat{{\bf\mathfrak{L}}}_{B}\right]$
generates rotations of Bob’s state, by an angle $\theta$, about an arbitrary
unit vector $\hat{\underline{n}}$ centred on the origin of Alice’s ‘coordinate
system’. We recall, also, that the latter is really a superposition of
coordinate systems, all of which have the same relative orientation, as shown
in Sec. 3. It is therefore sufficient to show that the operator
$\exp\left[\frac{i}{2\hbar}\,\delta\theta_{i}\hat{\mathfrak{L}}_{Bi}\right]$
generates infinitesimal rotations of $B$’s state, relative to each possible
coordinate system in $A$’s superposition, for all $i\in\left\\{1,2,3\right\\}$
$\equiv\left\\{x,y,z\right\\}$. Thus, we have

$\displaystyle\exp\left[\frac{i}{2\hbar}\,\delta\theta_{z}\hat{{\bf\mathfrak{L}}}_{Bz}\right]\,\ket{{\bf
x}_{O}}_{O}\otimes\ket{{\bf x}_{B}-{\bf x}_{O}}_{B}$ (4.40) $\displaystyle=$
$\displaystyle\exp\left[\frac{i}{2\hbar}\,\delta\theta_{z}(\hat{{\bf\mathcal{L}}}_{Bz}+\hat{{\bf\mathcal{L}}}^{\prime}_{Bz})\right]\,\exp\left[\frac{i}{2\hbar}\,\delta\theta_{z}\hat{{\bf\mathbb{L}}}_{Bz}\right]\,\ket{{\bf
x}_{O}}_{O}\otimes\ket{{\bf x}_{B}-{\bf x}_{O}}_{B}$ $\displaystyle=$
$\displaystyle\exp\left[\frac{i}{2\hbar}\,\delta\theta_{z}\left(\hat{\mathcal{X}}_{B}\hat{\mathcal{P}}_{By}-\hat{\mathcal{Y}}_{B}\hat{\mathcal{P}}_{Bx}+\hat{\mathcal{X}}^{\prime}_{B}\hat{\mathcal{P}}^{\prime}_{By}-\hat{\mathcal{Y}}^{\prime}_{B}\hat{\mathcal{P}}^{\prime}_{Bx}\right)\right]$
$\displaystyle\times$
$\displaystyle\exp\left[\frac{i}{2\hbar}\,\delta\theta_{z}\left(\hat{\mathcal{X}}_{B}\hat{\mathcal{P}}^{\prime}_{By}-\hat{\mathcal{Y}}_{B}\hat{\mathcal{P}}^{\prime}_{Bx}+\hat{\mathcal{X}}^{\prime}_{B}\hat{\mathcal{P}}_{By}-\hat{\mathcal{Y}}^{\prime}_{B}\hat{\mathcal{P}}_{Bx}\right)\right]$
$\displaystyle\times$
$\displaystyle\ket{x_{O},\,y_{O},\,z_{O}}_{O}\otimes\ket{x_{B}-x_{O},\,y_{B}-y_{O},\,z_{B}-z_{O}}_{B}$
$\displaystyle=$
$\displaystyle\exp\left[\frac{i}{2\hbar}\,\delta\theta_{z}\left(\hat{\mathcal{X}}_{B}\hat{\mathcal{P}}_{By}-\hat{\mathcal{Y}}_{B}\hat{\mathcal{P}}_{Bx}+\hat{\mathcal{X}}^{\prime}_{B}\hat{\mathcal{P}}^{\prime}_{By}-\hat{\mathcal{Y}}^{\prime}_{B}\hat{\mathcal{P}}^{\prime}_{Bx}\right)\right]$
$\displaystyle\times$
$\displaystyle\bigg{|}x_{O}-\frac{\delta\theta_{z}}{2}(y_{B}-y_{O}),\,y_{O}+\frac{\delta\theta_{z}}{2}(x_{B}-x_{O}),\,x_{O}\bigg{\rangle}_{O}$
$\displaystyle\otimes$
$\displaystyle\bigg{|}(x_{B}-x_{O})-\frac{\delta\theta_{z}}{2}y_{O},\,(y_{B}-y_{O})+\frac{\delta\theta_{z}}{2}x_{O},\,z_{B}-z_{O}\bigg{\rangle}_{B}$
$\displaystyle=$
$\displaystyle\bigg{|}x_{O}-\frac{\delta\theta_{z}}{2}y_{B},\,y_{O}+\frac{\delta\theta_{z}}{2}x_{B},\,z_{O}\bigg{\rangle}_{O}$
$\displaystyle\otimes$
$\displaystyle\bigg{|}(x_{B}-x_{O})-\frac{\delta\theta_{z}}{2}y_{B},\,(y_{B}-y_{O})+\frac{\delta\theta_{z}}{2}x_{B},\,z_{B}-z_{O}\bigg{\rangle}_{B}\,,$

plus an analogous expression in momentum space, to first order in
$\delta\theta_{z}\ll 1$. Note that, in this derivation, we have used the fact
that, although $[\hat{{\bf\mathcal{L}}}_{Bi},\hat{{\bf\mathbb{L}}}_{Bi}]\neq
0$ and $[\hat{{\bf\mathcal{L}}}^{\prime}_{Bi},\hat{{\bf\mathbb{L}}}_{Bi}]\neq
0$, the relevant commutators for the application of the Baker-Campbell-
Hausdorff formula [115] are
$[\delta\theta_{z}\hat{{\bf\mathcal{L}}}_{Bi},\delta\theta_{z}\hat{{\bf\mathbb{L}}}_{Bi}]$
and
$[\delta\theta_{z}\hat{{\bf\mathcal{L}}}^{\prime}_{Bi},\delta\theta_{z}\hat{{\bf\mathbb{L}}}_{Bi}]$.
These are of order $(\delta\theta_{z})^{2}$ and can therefore be neglected.
Equation (4.40), plus its momentum space counterpart, yield the
transformations

$\displaystyle x_{O}\mapsto
x^{\prime}_{O}=x_{O}-\frac{\delta\theta_{z}}{2}y_{B}\,,\quad p_{Ox}\mapsto
p^{\prime}_{Ox}=p_{Ox}-\frac{\delta\theta_{z}}{2}p_{By}\,,$ $\displaystyle
y_{O}\mapsto y^{\prime}_{O}=y_{O}+\frac{\delta\theta_{z}}{2}x_{B}\,,\quad
p_{Oy}\mapsto p^{\prime}_{Oy}=p_{Oy}+\frac{\delta\theta_{z}}{2}p_{Bx}\,,$
$\displaystyle z_{O}\mapsto
z^{\prime}_{O}=z_{O}\,,\quad\quad\quad\quad\quad\,p_{Oz}\mapsto
p^{\prime}_{Oz}=p_{Oz}\,,$ (4.41)

and

$\displaystyle x_{B}-x_{O}\mapsto
x^{\prime}_{B}-x^{\prime}_{O}=x_{B}-x_{O}-\frac{\delta\theta_{z}}{2}y_{B}\,,$
$\displaystyle y_{B}-y_{O}\mapsto
y^{\prime}_{B}-y^{\prime}_{O}=y_{B}-y_{O}+\frac{\delta\theta_{z}}{2}x_{B}\,,$
$\displaystyle z_{B}-z_{O}\mapsto
z^{\prime}_{B}-z^{\prime}_{O}=z_{B}-z_{O}\,;$ $\displaystyle
p_{xB}-p_{xO}\mapsto
p^{\prime}_{Bx}-p^{\prime}_{Ox}=p_{Bx}-p_{Ox}-\frac{\delta\theta_{z}}{2}p_{By}\,,$
$\displaystyle p_{yB}-p_{yO}\mapsto
p^{\prime}_{By}-p^{\prime}_{Oy}=p_{By}-p_{Oy}+\frac{\delta\theta_{z}}{2}p_{Bx}\,,$
$\displaystyle p_{zB}-p_{zO}\mapsto
p^{\prime}_{Bz}-p^{\prime}_{Oz}=p_{Bz}-p_{Oz}\,,$ (4.42)

giving

$\displaystyle x_{B}\mapsto
x^{\prime}_{B}=x_{B}-\delta\theta_{z}\,y_{B}\,,\quad p_{Bx}\mapsto
p^{\prime}_{Bx}=p_{Bx}-\delta\theta_{z}\,p_{By}\,,$ $\displaystyle
y_{B}\mapsto y^{\prime}_{B}=y_{B}+\delta\theta_{z}\,x_{B}\,,\quad
p_{By}\mapsto p^{\prime}_{By}=p_{By}+\delta\theta_{z}\,p_{By}\,,$
$\displaystyle z_{B}\mapsto
z^{\prime}_{B}=z_{B}\,,\quad\quad\quad\quad\quad\,p_{Bz}\mapsto
p^{\prime}_{Bz}=p_{Bz}\,,$ (4.43)

and it is straightforward to show that similar expressions also hold for
$\exp\left[\frac{i}{2\hbar}\,\delta\theta_{y}\hat{\mathfrak{L}}_{By}\right]$
and
$\exp\left[\frac{i}{2\hbar}\,\delta\theta_{x}\hat{\mathfrak{L}}_{Bx}\right]$.
This validates our interpretation of $\hat{{\bf\mathfrak{L}}}_{B}$ as the
operator representing Bob’s angular momentum, as seen by Alice, and the
generator of ‘rotations’ of his state, relative to hers, in both the position
and momentum space representations. 191919Note that, by simply interpreting
the coordinates ${\bf x}_{B}$ and ${\bf p}_{B}$ as relative coordinates, but
leaving their mathematical forms, and the state spaces on which they act,
unchanged from the canonical theory, the GCB formalism is unable to capture
behaviour of this kind. Here, we have shown that, when both Alice and Bob
exist in a superposition of position (momentum) eigenstates, a ‘rotation of
$A$ relative to $B$’ really represents infinitely many rotations, of each
possible set of axes in a superposition of coordinate systems, $\psi_{A}$
($\tilde{\psi}_{A}$), with respect to each possible set of axes in second
superposition of coordinate systems, $\phi_{B}$ ($\tilde{\phi}_{B}$).
Physically, this situation is different to that in which $B$’s superposition
of states is rotated around a fixed point in classical space, corresponding to
the centre of classical-$A$’s fixed coordinate system. (See Fig. 9, for
illustrations of rotation, with respect to both CRFs and QRFs.) If we wish to
describe this using standard position, momentum and angular momentum
operators, with their canonical mathematical forms unchanged, we must first
construct the appropriate space of effective mixed states, as outlined in Sec.
3.

Next, we consider the spin degrees of freedom within the QRF paradigm. In
canonical QM, the Pauli matrices were originally discovered by imposing two
conditions on the would-be spin operators; (a), that they be independent of
the canonical phase space variables, position and momentum, as befits an
‘internal’ property of a quantum state, and (b) that they satisfy the same
algebra as the operators for orbital angular momentum [68]. Following the same
logic here, we are inspired to search for a set of generalised spin operators,
$\left\\{\hat{S}_{Bi}\right\\}_{i=1}^{3}$, which can each be split into the
sum of three terms,

$\displaystyle\hat{S}_{Bi}:=\hat{\mathcal{S}}_{Bi}+\hat{\mathcal{S}}^{\prime}_{Bi}+\hat{\mathbb{S}}_{Bi}\,.$
(4.44)

The first term, $\hat{\mathcal{S}}_{Bi}$, should act nontrivially only on the
Hilbert space corresponding to Alice’s spin, after the parity swap
$O\leftrightarrow A$, whereas the second term,
$\hat{\mathcal{S}}^{\prime}_{Bi}$, should act nontrivially only on Bob’s spin
space, giving

$\displaystyle\hat{\mathcal{S}}_{Bi}:=\hat{s}_{Oi}\otimes\hat{\mathbb{I}}_{B}\,,\quad\hat{\mathcal{S}}^{\prime}_{Bi}:=\hat{\mathbb{I}}_{O}\otimes\hat{s}_{Bi}\,,$
(4.45)

where

$\displaystyle\hat{s}_{i}=\frac{\hbar}{2}\,\sigma_{i}\,,$ (4.46)

and $\left\\{\sigma_{i}\right\\}_{i=1}^{3}$ are the canonical Pauli matrices.
These satisfy the ${\rm su}(2)$ Lie algebra, giving

$\displaystyle[\sigma_{i},\sigma_{j}]=2i\epsilon_{ij}{}^{k}\sigma_{k}\,,\quad[\hat{s}_{i},\hat{s}_{j}]=i\hbar\epsilon_{ij}{}^{k}\hat{s}_{k}\,,$
(4.47)

together with

$\displaystyle[\sigma^{2},\sigma_{i}]=0\,,\quad[\hat{s}^{2},\hat{s}_{i}]=0\,,$
(4.48)

where

$\displaystyle\hat{s}^{2}=\sum_{i=1}^{3}\hat{s}_{i}^{2}=\left(\frac{\hbar}{2}\right)^{2}\sum_{i=1}^{3}\sigma_{i}^{2}=\frac{\hbar^{2}}{4}\sigma^{2}=\frac{3\hbar^{2}}{4}\,\hat{\mathbb{I}}$
(4.49)

is the Casimir operator, and we have used the fact that
$\sigma_{i}^{2}=\hat{\mathbb{I}}$ for all $i$ [68, 62]. In the fundamental
spin-$1/2$ representation the Pauli matrices also obey the Clifford algebra,

$\displaystyle[\sigma_{i},\sigma_{j}]_{+}=2\delta_{ij}\hat{\mathbb{I}}\,,\quad[\hat{s}_{i},\hat{s}_{j}]_{+}=\frac{\hbar^{2}}{2}\delta_{ij}\hat{\mathbb{I}}\,,$
(4.50)

where $[A,B]_{+}=AB+BA$ denotes the anti-commutator, and combining Eqs. (4.47)
and (4.50) gives

$\displaystyle\sigma_{i}\sigma_{j}=\delta_{ij}\hat{\mathbb{I}}+i\epsilon_{ij}{}^{k}\sigma_{k}\,,\quad\hat{s}_{i}\hat{s}_{j}=\left(\frac{\hbar}{2}\right)^{2}\delta_{ij}\hat{\mathbb{I}}+i\left(\frac{\hbar}{2}\right)\epsilon_{ij}{}^{k}\hat{s}_{k}\,.$
(4.51)

Let us also recall that the canonical spin operators generate generalised
rotations via the formula

$\displaystyle\exp\left[\frac{i}{\hbar}\theta\,\hat{\underline{n}}\,.\,\hat{{\bf
s}}\right]=\exp\left[\frac{i\theta}{2}\,\hat{\underline{n}}\,.\,\bm{\sigma}\right]=\cos(\theta/2)\,\hat{\mathbb{I}}+i\sin(\theta/2)\,\hat{\underline{n}}\,.\,\bm{\sigma}\,,$
(4.52)

where we have used the fact that
$(\hat{\underline{n}}\,.\,\bm{\sigma})^{2m}=\hat{\mathbb{I}}$, for
$m\in\mathbb{Z}^{+}$ [116]. Physically, the operator
$\exp\left[\frac{i}{\hbar}\theta\,\hat{\underline{n}}\,.\,\hat{{\bf
s}}\right]$ generates rotations of the spin-measurement axes, by the angle
$\theta$, about the unit vector $\hat{\underline{n}}$, which is centred on the
coordinate origin of $O$. The operators representing the total angular
momentum, including both the orbital angular momentum and spin angular
momentum components, may be constructed as

$\displaystyle\hat{J}_{i}=\hat{L}_{i}+\hat{s}_{i}\,,$ (4.53)

and the operator
$\exp\left[\frac{i}{\hbar}\theta\,\hat{\underline{n}}\,.\,\hat{{\bf
J}}\right]=\exp\left[\frac{i}{\hbar}\theta\,\hat{\underline{n}}\,.\,(\hat{{\bf
L}}+\hat{{\bf s}})\right]$ generates rotations of the axes along which both
spin and orbital angular momentum are measured. These two sets of measurements
therefore remain aligned.

In our QRF formalism, the third subcomponent of the generalised spin operator
(4.44), $\hat{\mathbb{S}}_{Bi}$, must therefore be constructed so that is
satisfies the relevant terms in the algebra

$\displaystyle[\hat{\mathcal{S}}_{Bi},\hat{\mathcal{S}}_{Bj}]=i\hbar\epsilon_{ij}{}^{k}\,\hat{\mathcal{S}}_{Bk}\,,\quad[\hat{\mathcal{S}}^{\prime}_{Bi},\hat{\mathcal{S}}^{\prime}_{Bj}]=i\hbar\epsilon_{ij}{}^{k}\,\hat{\mathcal{S}}^{\prime}_{Bk}\,,$
(4.54)
$\displaystyle[\hat{\mathcal{S}}_{Bi},\hat{\mathcal{S}}^{\prime}_{Bj}]=[\hat{\mathcal{S}}^{\prime}_{Bi},\hat{\mathcal{S}}_{Bj}]=0\,,$
(4.55)
$\displaystyle[\hat{\mathcal{S}}_{Bi},\hat{\mathbb{S}}_{Bj}]-[\hat{\mathcal{S}}_{Bj},\hat{\mathbb{S}}_{Bi}]=i\hbar\epsilon_{ij}{}^{k}\,\hat{\mathbb{S}}_{Bk}\,,$
(4.56)
$\displaystyle[\hat{\mathcal{S}}^{\prime}_{Bi},\hat{\mathbb{S}}_{Bj}]-[\hat{\mathcal{S}}^{\prime}_{Bj},\hat{\mathbb{S}}_{Bi}]=i\hbar\epsilon_{ij}{}^{k}\,\hat{\mathbb{S}}_{Bk}\,,$
(4.57)
$\displaystyle[\hat{\mathbb{S}}_{Bi},\hat{\mathbb{S}}_{Bj}]=i\hbar\epsilon_{ij}{}^{k}\,\hat{\mathcal{S}}_{Bk}+i\hbar\epsilon_{ij}{}^{l}\,\hat{\mathcal{S}}^{\prime}_{Bl}\,,$
(4.58)

by analogy with Eqs. (4.31)-(4.35). Assuming that Alice and Bob are both
spin-$1/2$ particles, these requirements yield the unique definition

$\displaystyle\hat{\mathbb{S}}_{Bi}:=\frac{2}{\hbar}\epsilon_{i}{}^{jk}\,\hat{\mathcal{S}}_{Bj}\,\hat{\mathcal{S}}^{\prime}_{Bk}:=\frac{\hbar}{2}\epsilon_{i}{}^{jk}\,\sigma_{Oj}\otimes\sigma_{Bk}\,,$
(4.59)

where we have made use of both the Lie algebra and Clifford algebra properties
of the fundamental representation, (4.47) and (4.50), which also imply
202020By making this assumption, we verify that our formalism holds for all
fundamental particles that are not force-transmitting bosons, including all
known leptons and quarks [107, 108]. It is clear that all such particles can,
potentially, be accorded the status of an ‘observer’ within the QRF paradigm,
but the status of force-transmitting bosons, which mediate interactions
between them, is less clear. In a future study, we will analyse composite
systems including spin-$0$, spin-$1/2$, spin-$1$, spin-$3/2$ and spin-$2$
particles, and will consider whether measures of the relative spin between
these subsystems can be reasonably defined. However, such an analysis lies
beyond the scope of the present work.

$\displaystyle[\hat{\mathcal{S}}_{Bi},\hat{\mathcal{S}}_{Bj}]_{+}=\frac{\hbar^{2}}{2}\delta_{ij}\,\hat{\mathbb{I}}_{OB}\,,\quad[\hat{\mathcal{S}}^{\prime}_{Bi},\hat{\mathcal{S}}^{\prime}_{Bj}]_{+}=\frac{\hbar^{2}}{2}\delta_{ij}\,\hat{\mathbb{I}}_{OB}\,,$
(4.60)
$\displaystyle[\hat{\mathcal{S}}_{Bi},\hat{\mathbb{S}}_{Bj}]_{+}+[\hat{\mathcal{S}}_{Bj},\hat{\mathbb{S}}_{Bi}]_{+}=0\,,$
(4.61)
$\displaystyle[\hat{\mathcal{S}}^{\prime}_{Bi},\hat{\mathbb{S}}_{Bj}]_{+}+[\hat{\mathcal{S}}^{\prime}_{Bj},\hat{\mathbb{S}}_{Bi}]_{+}=0\,,$
(4.62)
$\displaystyle[\hat{\mathbb{S}}_{Bi},\hat{\mathbb{S}}_{Bj}]_{+}=\hbar^{2}\delta_{ij}\,\hat{\mathbb{I}}_{OB}-[\hat{\mathcal{S}}_{Bi},\hat{\mathcal{S}}^{\prime}_{Bj}]_{+}-[\hat{\mathcal{S}}_{Bj},\hat{\mathcal{S}}^{\prime}_{Bi}]_{+}\,.$
(4.63)

Next, we define the generalised Pauli matrices via the relation

$\displaystyle\hat{S}_{Bi}:=\hbar\,\Sigma_{Bi}\,.$ (4.64)

It follows immediately that these satisfy canonical-type Lie and Clifford
algebras, yielding

$\displaystyle[\Sigma_{Bi},\Sigma_{Bj}]=2i\epsilon_{ij}{}^{k}\,\Sigma_{Bk}\,,\quad[\hat{S}_{Bi},\hat{S}_{Bj}]=2i\hbar\epsilon_{ij}{}^{k}\,\hat{S}_{Bk}$
(4.65)

and

$\displaystyle[\Sigma_{Bi},\Sigma_{Bj}]_{+}=2\delta_{ij}\,\hat{\mathbb{I}}_{OB}\,,\quad[\hat{S}_{Bi},\hat{S}_{Bj}]_{+}=2\hbar^{2}\delta_{ij}\,\hat{\mathbb{I}}_{OB}\,,$
(4.66)

so that

$\displaystyle\Sigma_{Bi}\Sigma_{Bj}=\delta_{ij}\,\hat{\mathbb{I}}_{OB}+i\epsilon_{ij}{}^{k}\,\Sigma_{Bk}\,,\quad\hat{S}_{Bi}\hat{S}_{Bj}=\hbar^{2}\delta_{ij}\,\hat{\mathbb{I}}_{OB}+i\hbar\epsilon_{ij}{}^{k}\,\hat{S}_{Bk}\,.$
(4.67)

We then have

$\displaystyle[\Sigma_{B}^{2},\Sigma_{Bi}]=0\,,\quad[\hat{S}_{B}^{2},\hat{S}_{Bi}]=0\,,$
(4.68)

where

$\displaystyle\hat{S}_{B}^{2}=\sum_{i=1}^{3}\hat{S}_{Bi}^{2}=\hbar^{2}\sum_{i=1}^{3}\Sigma_{Bi}^{2}=\hbar^{2}\Sigma_{B}^{2}=3\hbar^{2}\,\hat{\mathbb{I}}_{OB}\,,$
(4.69)

and where we have used the fact that $\Sigma_{Bi}^{2}=\hat{\mathbb{I}}_{OB}$,
for all $i$.

Exponentiation of the generalised spin operators yields

$\displaystyle\exp\left[\frac{i}{2\hbar}\theta\,\hat{\underline{n}}\,.\,\hat{{\bf
S}}_{B}\right]=\exp\left[\frac{i\theta}{2}\,\hat{\underline{n}}\,.\,\bm{\Sigma}_{B}\right]=\cos(\theta/2)\,\hat{\mathbb{I}}_{OB}+i\sin(\theta/2)\,\hat{\underline{n}}\,.\,\bm{\Sigma}\,,$
(4.70)

where $(\hat{\underline{n}}\,.\,\bm{\Sigma}_{B})^{2m}=\hat{\mathbb{I}}_{OB}$
for $m\in\mathbb{Z}^{+}$. Note that, in Eqs. (4.70), the factor of
$(2\hbar)^{-1}$ arises due to the rescaled Lie algebra, given in Eqs. (4.65).
Its inclusion in the exponential is consistent with the definition of the
generalised Pauli matrices, Eqs. (4.64), and ensures that a rotation by
$720^{{\rm o}}$ is needed to bring a generalised spin vector - that is, an
arbitrary superposition of the eigenstates of $\hat{S}_{Bi}$ \- back to its
original state. The generalised uncertainty relations for the operators
$\left\\{\hat{S}_{Bi}\right\\}_{i=1}^{3}$ are

$\displaystyle(\Delta_{\Psi}S_{Bi})^{2}\,(\Delta_{\Psi}S_{Bj})^{2}\geq\dots\geq\hbar^{2}(\epsilon_{ij}{}^{k})^{2}\,\langle\hat{S}_{Bk}\rangle_{\Psi}^{2}\,,$
(4.71)

where

$\displaystyle(\Delta_{\Psi}S_{Bi})^{2}$ $\displaystyle=$
$\displaystyle(\Delta_{\Psi}\mathcal{S}_{Bi})^{2}+(\Delta_{\Psi}\mathcal{S}^{\prime}_{Bi})^{2}+(\Delta_{\Psi}\mathbb{S}_{Bi})^{2}$
(4.72) $\displaystyle+$ $\displaystyle{\rm
cov}(\hat{\mathcal{S}}_{Bi},\hat{\mathbb{S}}_{Bi})+{\rm
cov}(\hat{\mathbb{S}}_{Bi},\hat{\mathcal{S}}_{Bi})$ $\displaystyle+$
$\displaystyle{\rm
cov}(\hat{\mathcal{S}}^{\prime}_{Bi},\hat{\mathbb{S}}_{Bi})+{\rm
cov}(\hat{\mathbb{S}}_{Bi},\hat{\mathcal{S}}^{\prime}_{Bi})\,,$

by complete analogy with Eqs. (4.38) and (4.39).

It is straightforward to find a unitary similarity transform such that

$\displaystyle\hat{S}_{Bi}\mapsto\hat{U}\,\hat{S}_{Bi}\,\hat{U}^{-1}=\hbar\,(\sigma_{Oi}\oplus\sigma_{Bi})\,,$
(4.73)

for all $i$, and its existence follows directly from the Lie group $-$ Lie
algebra correspondence, which holds for all compact groups [117]. The
generalised Pauli matrices are, therefore, reducible representations of the
${\rm SU}(2)$ group generators, equivalent to the direct sum of two
fundamental representations,
$\left\\{\hat{U}\Sigma_{Bi}\hat{U}^{-1}=\sigma_{Oi}\oplus\sigma_{Bi}\right\\}_{i=1}^{3}$.
However, it would be incorrect to think that each of these acts, as a
generator of ${\rm SU}(2)$, on an individual subspace of the bipartite system,
$\mathcal{H}_{O}^{(A)}$ or $\mathcal{H}_{B}$. Instead, each irreducible
representation acts in a distributed way, across both subspaces, so that the
$\Sigma_{Bi}$ do not generate rotations of the spin-measurement axes used by
the CRF $O$, relative to either Alice or Bob. The latter are generated by
$\hat{\mathcal{S}}_{Bi}$ and $\hat{\mathcal{S}}^{\prime}_{Bi}$, respectively,
but the action of the third subcomponent, $\hat{\mathbb{S}}_{Bi}$, ensures
that the total action of the operators $\hat{S}_{Bi}=\hbar\Sigma_{Bi}$ is
subtler, and more complex.

Nonetheless, the canonical-type expression for the operator
$\exp\left[\frac{i\theta}{2}\,\hat{\underline{n}}\,.\,\bm{\Sigma}_{B}\right]$
(4.71) suggests that $\hat{{\bf S}}_{B}=\hbar\bm{\Sigma}_{B}$ does, in fact,
rotate some set of spin-measurement axes, by an angle $\theta$, about the
vector $\hat{\underline{n}}$. But what axes are these? In fact, the
generalised spin operators (4.44) generate rotations of each and every set of
spin-measurement axes, in Alice’s superposition of spin-measurement axes,
relative to each and every set of spin-measurement axes in Bob’s
superposition. (See Figs. 9 and 10.)

We therefore interpret the eigenvalues of
$\left\\{\hat{S}_{Bi}\right\\}_{i=1}^{3}$ as measures of the relative spin
between the subsystems $A$ and $B$. Equivalently, these are possible the
values of Bob’s spin, measured by Alice, relative to her superposition of
spin-measurement axes. These are coincident with the superposition of axes
that Alice uses to determine the components of Bob’s relative displacement,
${\bf x}_{B}$, and relative linear momentum, ${\bf p}_{B}$, as well as his
relative angular momentum, $\mathbb{\mathfrak{L}}_{B}$. This scenario is
illustrated in Fig. 10 and may be compared with the standard notion of
rotation in canonical QM, depicted in Fig. 9. We stress, again, that the two
scenarios are physically distinct and require different mathematical
descriptions, which standard relational formalisms fail to capture [14, 15,
16, 17, 18, 19, 20, 21, 22, 24, 25, 23, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35, 39, 38, 37, 36, 40]. 212121See [118, 119, 120, 121, 122, 123, 124, 125],
for extensions of the GCB formalism to include QRFs with spin, as well as
alternative approaches to this problem. To the best of our knowledge, no
existing works arrive at the same conclusions reached here, namely, that the
operator representing the relative angular momentum between two quantum
subsystems should be defined as in Eqs. (4.29)-(4.30), and that the operator
representing the relative spin between these systems can be defined by
requiring the algebraic consistency of the latter with the former.

For the sake of compactness, we do not quote the explicit forms of the
relative-spin operators here, but it is straightforward to show that each
possesses 4 distinct eigenvectors, corresponding to 2 distinct eigenvalues.
Each eigenvalue, $+\hbar$ and $-\hbar$, therefore possesses a 2-fold
degeneracy. This is an important point: were some eigenvectors associated with
different eigenvalues, Bob’s subsystem would no longer appear, to Alice, as a
spin-1/2 particle. For example, if at least some eigenvectors were associated
with the eigenvalue zero, Bob’s state would appear, instead, as a spin-1
particle. Although the eigenvalues $\pm\hbar$ are not exactly equal to the
canonical values, $\pm\hbar/2$, there are still two of them, indicating binary
spin ‘up’ and spin ‘down’ states.

To avoid confusion with the canonical spin vectors
$\left\\{\ket{\uparrow_{i}}_{O},\ket{\downarrow_{i}}_{O}\right\\}$ and
$\left\\{\ket{\uparrow_{i}}_{B},\ket{\downarrow_{i}}_{B}\right\\}$, we may
denote the generalised eigenstates as
$\left\\{\ket{\uparrow_{i}}\rangle_{OB},\ket{\downarrow_{i}}\rangle_{OB};\ket{\uparrow^{\prime}_{i}}\rangle_{OB},\ket{\downarrow^{\prime}_{i}}\rangle_{OB}\right\\}$.
In general, these correspond to entangled states of the canonical spins. The
interested reader is referred to [82, 84, 85], for further details.

The above results then allow us to define the components of the total relative
angular momentum operator, $\hat{{\bf\mathfrak{J}}}_{B}$, as

$\displaystyle\hat{\mathfrak{J}}_{Bi}:=\hat{\mathfrak{L}}_{Bi}+\hat{S}_{Bi}\,,$
(4.74)

or, equivalently,

$\displaystyle\hat{\mathfrak{J}}_{Bi}:=\hat{\mathcal{J}}_{Bi}+\hat{\mathcal{J}}^{\prime}_{Bi}+\hat{\mathbb{J}}_{Bi}\,,$
(4.75)

where

$\displaystyle\hat{\mathcal{J}}_{Bi}:=\hat{\mathcal{L}}_{Bi}+\hat{\mathcal{S}}_{Bi}\,,\quad\hat{\mathcal{J}}^{\prime}_{Bi}:=\hat{\mathcal{L}}^{\prime}_{Bi}+\hat{\mathcal{S}}^{\prime}_{Bi}\,,\quad\hat{\mathbb{J}}_{Bi}:=\hat{\mathbb{L}}_{Bi}+\hat{\mathbb{S}}_{Bi}\,.$
(4.76)

The algebra of the subcomponents
$\left\\{\hat{\mathcal{J}}_{Bi},\hat{\mathcal{J}}^{\prime}_{Bi},\hat{\mathbb{J}}_{Bi}\right\\}$
is completely analogous to Eqs. (4.31)-(4.35) and (4.54)-(4.58), yielding the
GURs

$\displaystyle(\Delta_{\Psi}\mathfrak{J}_{Bi})^{2}\,(\Delta_{\Psi}\mathfrak{J}_{Bj})^{2}\geq\dots\geq\hbar^{2}(\epsilon_{ij}{}^{k})^{2}\,\langle\hat{\mathfrak{J}}_{Bk}\rangle_{\Psi}^{2}\,,$
(4.77)

where

$\displaystyle(\Delta_{\Psi}\mathfrak{J}_{Bi})^{2}$ $\displaystyle=$
$\displaystyle(\Delta_{\Psi}\mathcal{J}_{Bi})^{2}+(\Delta_{\Psi}\mathcal{J}^{\prime}_{Bi})^{2}+(\Delta_{\Psi}\mathbb{J}_{Bi})^{2}$
(4.78) $\displaystyle+$ $\displaystyle{\rm
cov}(\hat{\mathcal{J}}_{Bi},\hat{\mathbb{J}}_{Bi})+{\rm
cov}(\hat{\mathbb{J}}_{Bi},\hat{\mathcal{J}}_{Bi})$ $\displaystyle+$
$\displaystyle{\rm
cov}(\hat{\mathcal{J}}^{\prime}_{Bi},\hat{\mathbb{J}}_{Bi})+{\rm
cov}(\hat{\mathbb{J}}_{Bi},\hat{\mathcal{J}}^{\prime}_{Bi})\,,$

as expected. The operator
$\exp\left[\frac{i}{2\hbar}\theta\,\hat{\underline{n}}\,.\,\hat{{\bf\mathfrak{J}}}_{B}\right]=\exp\left[\frac{i}{2\hbar}\theta\,\hat{\underline{n}}\,.\,(\hat{{\bf\mathfrak{L}}}_{B}+\hat{{\bf
S}}_{B})\right]$ rotates every possible set of axes in Alice’s superposition
of coordinate systems, relative to Bob’s superposition of coordinate systems.
In another example of (somewhat inaccurate) linguistic shorthand, we may say
that it ‘rotates Bob’s wave function, relative to Alice’s wave function’. The
more precise meaning of this operation is depicted, schematically, in Fig. 10.

Figure 9: Exponentiating the total angular momentum operator of canonical QM,
$\exp\left[i\theta\hat{\underline{n}/\hbar}\,.\,\hat{{\bf J}}\right]$, where
$\hat{{\bf J}}=\hat{{\bf L}}+\hat{{\bf s}}$ and $\hat{{\bf L}}$, $\hat{{\bf
s}}$ are the canonical orbital angular momentum and spin angular momentum
operators, respectively, generates rotations of Bob’s quantum state about the
fixed origin of a classical reference frame. For the purposes of this diagram,
we will call this CRF ‘classical Alice’, instead of the usual ‘$O$’. This
transformation may be viewed in two equivalent ways: (i) We may say that
$\exp\left[i\theta\hat{\underline{n}/\hbar}\,.\,\hat{{\bf J}}\right]$ rotates
Alice’s classical coordinate system, relative to every possible coordinate
system in Bob’s superposition. This is depicted in Fig. 9(a). (ii) We may say
that $\exp\left[i\theta\hat{\underline{n}/\hbar}\,.\,\hat{{\bf J}}\right]$
rotates every possible coordinate system in Bob’s superposition, relative to
classical-Alice’s fixed coordinate system. This is depicted in Fig. 9(b).
Figure 10: In the QRF scenario, the situation is physically different to that
presented in Fig. 9. Exponentiating the relative angular momentum operator,
$\exp\left[i\theta\hat{\underline{n}/\hbar}\,.\,\hat{{\bf\mathfrak{J}}}_{B}\right]$,
where $\hat{\mathfrak{J}}_{Bi}:=\hat{\mathfrak{L}}_{Bi}+\hat{S}_{Bi}$ and
$\hat{\mathfrak{L}}_{Bi}$, $\hat{S}_{Bi}$ denote the relative orbital angular
momentum and relative spin operators, defined in Eqs. (4.29)-(4.30) and
(4.44)-(4.45) plus (4.59), respectively, generates rotations of Bob’s quantum
state, about every possible coordinate system in a superposition of
coordinates (a QRF). For the purposes of this diagram, we call this QRF
‘quantum Alice’. This transformation may also be viewed in two equivalent
ways: (i) We may say that
$\exp\left[i\theta\hat{\underline{n}/\hbar}\,.\,\hat{{\bf\mathfrak{J}}}_{B}\right]$
rotates every possible coordinate system in quantum-Alice’s superposition of
coordinates, relative to every possible coordinate system in Bob’s
superposition. This is depicted in Fig. 10(a). (ii) We may say that
$\exp\left[i\theta\hat{\underline{n}/\hbar}\,.\,\hat{{\bf\mathfrak{J}}}_{B}\right]$
rotates every possible coordinate system in Bob’s superposition of
coordinates, relative to every possible coordinate system in quantum-Alice’s
superposition. This is depicted in Fig. 10(b). By simply interpreting the
canonical operator $\hat{{\bf J}}_{B}:=\hat{{\bf L}}_{B}+\hat{{\bf s}}_{B}$ as
the relative angular momentum between two quantum subsystems, but leaving its
mathematical form unchanged from the canonical theory, we believe that
existing relational formalisms, including the GCB formalism [14], fail to
capture the important physical distinction between the two scenarios depicted
in Figs. 9 and 10.

Finally, before concluding this subsection, we note that operators of the form
(4.29) must, in fact, be included in any QRF model, including the GCB
formalism and related models. However, in this case, their physical
interpretation is problematic. For example, let us consider the tripartite
state defined in [14]. Mathematically, this very much resembles our bipartite
state, in the sense that it depends on only two wave functions, or,
equivalently, $6$ quantum mechanical degrees of freedom. In this scenario, the
orbital angular momentum of Bob’s state, relative to Alice’s, from Charlie’s
perspective, is evidently described by an operator with the same basic
structure as (4.29), i.e., one that can be split into $3$ subcomponents
obeying the algebra (4.31)-(4.35). By contrast, both Bob’s angular momentum
and Alice’s angular momentum, relative to Charlie’s frame of reference, are
described by canonical-type operators, $\hat{\mathbb{L}}_{B}^{(C)}$ and
$\hat{\mathbb{L}}_{A}^{(C)}$, in this formalism. This is the case in our model
only when $M_{C}\rightarrow\infty$, that is, in the limit where Charlie
becomes, effectively, a classical observer. The mathematical distinction
between these operators, in this limit, therefore corresponds to a real
physical distinction between the observer and the observed. In the GCB model,
however, there is no such distinction, as all three subsystems are supposed to
be quantum mechanical in nature. It is, therefore, very difficult to explain
the necessity of using operators with different mathematical structures, to
describe apparently similar measurements, all of which, supposedly, represent
measurements of relative variables.

##  5 The significance of QRFs for a future theory of quantum gravity

In this section, we show how the conceptual and mathematical machinery of the
QRF formalism, developed in Secs. 1-4, can be adapted and applied to the
background geometry in which material quantum systems live. This is the only
part of our work that goes beyond the basic postulates of canonical quantum
theory, which are formulated for material particles inhabiting a classical
three-dimensional Euclidean space, $\mathbb{E}^{3}$ [62, 64, 68, 69, 91].

###  5.1 Quantised spacetime as a QRF

Einstein’s theory of general relativity equates the gravitational force with
the curvature of the spacetime geometry [126]. It is therefore expected that
the quantisation of the gravitational field should be equivalent to the
quantisation of the spacetime background, in which material bodies live [127,
128]. In addition, dimensional analysis and a host of thought experiments [77,
129], plus a variety of specific approaches to the problem of quantum gravity,
including string theory [130] and loop quantum gravity [131], suggest that the
Planck length, $l_{\rm Pl}\simeq\sqrt{\hbar G/c^{3}}\simeq 10^{-33}$ ${\rm
cm}$, should act as a fundamental limit to the resolution of spacetime
distances. Although the theoretical status of minimum momentum scale is less
clear, it is noteworthy that, in a dark energy Universe, there exists an upper
bound on the positional uncertainty of any particle, equal to radius of the de
Sitter horizon, $l_{\rm dS}\simeq\sqrt{3/\Lambda}\simeq 10^{-28}$ ${\rm cm}$,
where $\Lambda\simeq 10^{-56}$ ${\rm cm}^{-2}$ is the observed value of the
cosmological constant [132, 133, 134, 135, 136, 137]. By the Heisenberg
uncertainty principle (HUP), this immediately implies the existence of a
minimum measurable momentum, of order $\sim\hbar\sqrt{\Lambda/3}\simeq
10^{-61}$ ${\rm kg\,ms^{-1}}$.

In this section, we show that by promoting individual points in the classical
spatial background, $\mathbb{E}^{3}$, to quantum superpositions of points,
i.e., to non-material QRFs, we are able to re-derive some of the key
phenomenological consequences of the existence of minimum length and / or
momentum scales, in a new way. These include the most widely studied
generalised uncertainty relations, considered in the phenomenological quantum
gravity literature, namely, the generalised uncertainty principle (GUP) [70,
71, 72], extended uncertainty principle (EUP) [73, 74, 75] and extended
generalised uncertainty principle (EGUP) [76, 77, 78, 79, 80]. The new
derivations are then compared with standard derivations in the literature,
based on modified commutation relations [76, 77, 78, 79, 80], and several
advantages of the new formalism are identified [81, 82, 84, 90]. 222222The
results presented in this section were, in fact, first derived in a series of
previous papers [81, 82, 83, 84, 85, 86, 87, 88, 89, 90]. However, the
analysis presented here, as well as in Sec. 4.4, goes beyond that provided in
these works by clarifying the explicit connection between the QRF formalism
developed in Secs. 2-4 and the GURs commonly studied in the quantum gravity
literature. That such a connection may exist was hinted at, and discussed
previously [81, 82, 84], but the exact correspondence between these models had
not been clarified, until now.

For later convenience, we define the Planck mass and length scales, and the de
Sitter mass and length scales, as

$\displaystyle m_{\rm Pl}:=\sqrt{\frac{\hbar c}{G}}\simeq 10^{-5}\,{\rm
g}\,,\quad l_{\rm Pl}:=\sqrt{\frac{\hbar G}{c^{3}}}\simeq 10^{-33}\,{\rm
cm}\,,$ (5.1)

and

$\displaystyle m_{\rm dS}:=\frac{\hbar}{c}\sqrt{\frac{\Lambda}{3}}\simeq
10^{-66}\,{\rm g}\,,\quad l_{\rm dS}:=\sqrt{\frac{3}{\Lambda}}\simeq
10^{28}\,{\rm cm}\,,$ (5.2)

respectively. The corresponding densities are

$\displaystyle\rho_{\rm Pl}:=\frac{3}{4\pi}\frac{c^{5}}{\hbar G^{2}}\simeq
10^{93}\,{\rm g\,.\,cm}^{-3}\,,$ (5.3)

and

$\displaystyle\rho_{\Lambda}:=\frac{\Lambda c^{2}}{8\pi G}\simeq
10^{-29}\,{\rm g\,.\,cm}^{-3}\,,$ (5.4)

where $\rho_{\Lambda}$ is approximately equal to the critical density of the
present-day Universe, $\rho_{\Lambda}\simeq\rho_{\rm crit}(t_{0})$ [126].

###  5.2 GUP and EUP

When searching for phenomenological consequences of a quantised gravitational
field, with or without a cosmological constant term, it is useful to note that
the HUP can be ‘derived’ in two different ways. The first ‘derivation’ is
heuristic, and is based on the so-called Heisenberg microscope thought
experiment, in which an experimenter attempts to localise the position of a
particle, by means of a scattering event, which inevitably disturbs its
momentum [68]. The resulting trade-off between the position and momentum
‘uncertainties’, $\Delta x$ and $\Delta p$, result in the relation

$\displaystyle\Delta x^{i}\,\Delta
p_{j}\gtrsim\frac{\hbar}{2}\delta^{i}{}_{j}\,.$ (5.5)

The inequality in Eqs. (5.5) is imprecise because the associated
‘uncertainties’ are not mathematically well defined. Alternatively, a more
rigorous version of the HUP can be derived from abstract Hilbert space
structure of canonical QM. By combining the Schrödinger-Robertson inequality,

$\displaystyle\Delta_{\psi}O_{1}\,\Delta_{\psi}O_{2}\geq\frac{1}{2}|\braket{\psi}{[\hat{O}_{1},\hat{O}_{2}]}{\psi}|\,,$
(5.6)

which holds for any two Hermitian operators, $\hat{O}_{1}$ and $\hat{O}_{2}$
[69], with the canonical commutation relation,

$\displaystyle[\hat{x}^{i},\hat{p}_{j}]=i\hbar\delta^{i}{}_{j}\
\hat{\mathbb{I}}\,,$ (5.7)

we obtain

$\displaystyle\Delta_{\psi}x^{i}\,\Delta_{\psi}p_{j}\geq\frac{\hbar}{2}\delta^{i}{}_{j}\,.$
(5.8)

The inequality in Eqs. (5.8) is precise and the uncertainties
$\Delta_{\psi}x^{i}$ and $\Delta_{\psi}p_{j}$ are well defined, as the
standard deviations of the probability distributions $|\psi({\bf x})|^{2}$ and
$|\tilde{\psi}({\bf p})|^{2}$, respectively. From here on, we use an
appropriate subscript, attached to the symbol $\Delta$, to distinguish well-
defined standard deviations from heuristic ‘uncertainties’, introduced via
thought experiments.

Despite being heuristic, the thought experiment leading to Eqs. (5.5) has its
uses. For example, one can reanalyse this experimental setup, accounting also
for the gravitational interaction between the observed particle and the probe
particle. This leads to an additional non-Heisenberg term in the positional
uncertainty, of order $\sim G\Delta p/c^{3}$ [70, 71, 72], and, hence, to the
GUP,

$\displaystyle\Delta x^{i}\gtrsim\frac{\hbar}{2\Delta
p_{j}}\delta^{i}{}_{j}\left[1+\alpha_{0}\frac{2G}{\hbar c^{3}}(\Delta
p_{j})^{2}\right]\,,$ (5.9)

where $\alpha_{0}$ is a numerical constant of order unity. The GUP implies the
existence of a minimum resolvable position uncertainty, of the order of the
Planck length, but no minimum momentum scale.

Similarly, one can reanalyse the thought experiment, accounting for the
presence of a background dark energy density (5.4). This yields an additional
non-Heisenberg term in the momentum uncertainty, of order
$\sim\hbar\sqrt{\Lambda/3}\Delta x$ [73, 74, 75], giving rise to the EUP,

$\displaystyle\Delta p_{j}\gtrsim\frac{\hbar}{2\Delta
x^{i}}\delta^{i}{}_{j}\left[1+2\eta_{0}\Lambda(\Delta x^{i})^{2}\right]\,,$
(5.10)

where $\eta_{0}$ is of order unity. The EUP implies the existence of a minimum
resolvable momentum uncertainty, of the order of the de Sitter momentum $\sim
m_{\rm dS}c$, but no minimum length scale. Accounting for both canonical
gravitational attraction, and the effects of repulsive dark energy, then leads
to the EGUP,

$\displaystyle\Delta x^{i}\Delta
p_{j}\gtrsim\frac{\hbar}{2}\delta^{i}{}_{j}\left[1+\alpha_{0}\frac{2G}{\hbar
c^{3}}(\Delta p_{j})^{2}+2\eta_{0}\Lambda(\Delta x^{i})^{2}\right]\,,$ (5.11)

which implies the existence of both minimum length and momentum scales in
nature.

The great advantage of these thought experiment arguments is their model-
independence. If we trust our intuition, regarding the way in which the
gravitational field and dark energy density can interact with microscopic
quantum systems, the EGUP offers a phenomenological guide, which, arguably,
any would-be theory of quantum gravity, including a dark energy term, should
be able to reproduce. It is also their greatest disadvantage. The
‘uncertainties’ in Eqs. (5.9), (5.10) and (5.11) are heuristic, like those in
Eqs. (5.5), and it is not clear how to derive them, rigorously, as the
standard deviations of probability distributions, obtained from a modified
quantum formalism.

Until recently, the only approach considered in the literature was to modify
the canonical commutator (5.7), such that
$[\hat{x}^{i},\hat{p}_{j}]=i\hbar\hat{F}^{i}{}_{j}(\hat{{\bf x}},\hat{{\bf
p}})$ [76, 77, 78, 79, 80]. This immediately leads to modified uncertainty
relations, due to the modification of the Schrödinger-Robertson bound (5.6).
Specifically, setting $\hat{F}^{i}{}_{j}(\hat{{\bf x}},\hat{{\bf
p}})=1+\hat{p}_{j}^{2}+\hat{x}^{i2}$ leads to an EGUP-type expression, though
it may be shown that the resulting formalism exhibits various pathologies.
These include, but are not limited to, the following [77, 78, 79, 84, 90]:

  1. 1.

The generation of mass-dependent accelerations, even in the classical limit.
Clearly, this violates any sensible definition of the equivalence principle
(EP), in the quantum regime [138, 139, 140, 141, 142, 143, 144, 145].

  2. 2.

The fact that the background geometry, though not Euclidean, is still
manifestly classical in nature. This an undesirable trait in a would-be model
of quantum gravity, but, worse still, the exact metric of the underlying
classical geometry is unknown. It cannot be inferred from the form of the
uncertainty relations, alone, so that it is not clear ‘where in space’ the
eigenvalues of the modified position operators,
$\left\\{\hat{x}^{i}\right\\}_{i=1}^{3}$, refer to [84, 90]. Similar remarks
hold for $\left\\{\hat{p}_{j}\right\\}_{i=1}^{3}$, with respect to momentum
space.

  3. 3.

The dependence of the position and momentum uncertainties on the classical
frame of reference. For example, including a GUP-type term via the function
$\hat{F}^{i}{}_{j}(\hat{{\bf p}})=1+\hat{p}_{j}^{2}$ generates velocity-
dependent positional uncertainties, whereas including an EUP-type term using
$\hat{F}^{i}{}_{j}(\hat{{\bf x}})=1+\hat{x}^{i2}$ generates momentum
uncertainties that depend of the shift-translation parameter [84, 90].

  4. 4.

The so-called ‘soccer ball problem’, which refers to the inability to define
sensible multi-particle states, with energies exceeding the Planck-energy.
This limit arises as an effective upper bound for the energies of all systems,
including macroscopic bodies, due to the way in which modified commutators
implement the Planck length as a minimum length scale [77, 78, 79, 83, 84,
90].

Recently, an alternative basis for EGUP was developed, by a research team that
includes the present authors, which neatly avoids these problems [81, 82, 83,
84, 85, 86, 87, 88, 89, 90]. In the remainder of this section, we review the
basic formalism of this model and show how it is equivalent to applying the
QRF formalism, developed in Secs. 2-4, to spatial points in the background
geometry, as opposed to point-like quantum particles in classical Euclidean
space.

To begin, we note that in canonical QM the eigenstate $\ket{{\bf x}}$
represents the localisation of a quantum particle at the point ‘${\bf x}$’ in
the background geometry. It does not correspond to a quantum state of this
geometry, since the latter remains purely classical. To associate quantum
degrees of freedom with the spatial background, therefore, we first ‘square’
the quantum mechanical phase space (in the position space representation), via
the map $\ket{{\bf x}}\mapsto\ket{{\bf x}}\otimes\ket{{\bf x^{\prime}}}$. The
new eigenstates $\ket{{\bf x^{\prime}}}$ are interpreted, from here on, as
representing the perfect localisation of a spatial ‘point’ in the quantum
mechanical background geometry. Generically, however, we expect such points to
be delocalised, in the quantum gravity regime, over a region comparable to the
Planck volume. We therefore implement the map

$\displaystyle\hat{S}_{\mathbb{x}}:\ket{{\bf x}}\mapsto\ket{{\bf
x}}\otimes\ket{g_{{\bf x}}}\,.$ (5.12)

where

$\displaystyle\ket{g_{{\bf x}}}=\int g({\bf x}^{\prime}-{\bf x})\ket{{\bf
x}^{\prime}}{\rm d}^{3}{\rm x}^{\prime}\,,$ (5.13)

and $g(\bf{x}^{\prime}-\bf{x})$ is any normalised function,

$\displaystyle\int|g({\bf x}^{\prime}-{\bf x})|^{2}{\rm d}^{3}{\rm
x}^{\prime}=\int|g({\bf x}^{\prime}-{\bf x})|^{2}{\rm d}^{3}{\rm x}=1\,.$
(5.14)

For convenience, we will visualise $|g(\mathbb{x}^{\prime}-\mathbb{x})|^{2}$
as a Planck-scale Gaussian distribution,

$\displaystyle|g({\bf x}^{\prime}-{\bf
x})|^{2}=\left(\frac{1}{\sqrt{2\pi}\sigma_{g}}\right)^{3}\exp\left[-\frac{({\bf
x}^{\prime}-{\bf x})^{2}}{2\sigma_{g}^{2}}\right]\,,$ (5.15)

where $\sigma_{g}\simeq l_{\rm Pl}$, from here on.

The associated amplitude, $g(\bf{x}^{\prime}-\bf{x})$, is interpreted as the
quantum probability amplitude for the coherent transition
$\bf{x}\leftrightarrow\bf{x}^{\prime}$, where it is assumed that this
transition also interchanges the amplitude(s) of any material-particle wave
functions associated with the classical point ‘x’,
$\psi(\bf{x})\leftrightarrow\psi(\bf{x}^{\prime})$ [81, 82, 83, 84, 85, 86,
87, 88, 89, 90]. By ‘smearing’ each spatial point in the classical Euclidean
space, $\mathbb{E}^{3}$, we map the classical background geometry to a
superposition of Euclidean geometries, which is formally equivalent to the
Cartesian product $\mathbb{E}^{3}\times\mathbb{E}^{3}$. Each geometry in the
superposition is equivalent to every other geometry, except for the pair-wise
exchange of at least two points, $\bf{x}\leftrightarrow\bf{x}^{\prime}$. For
isotropic smearing functions, including (5.15), the most probable position of
each point, relative to all others, is precisely the classical value, ${\bf
x}^{\prime}={\bf x}$, but fluctuations within a volume of order $\sim l_{\rm
Pl}^{3}$ remain relatively likely. We now show how these fluctuations generate
additional Planck-scale contributions to the positional uncertainties of
material particles.

Acting with $\hat{S}_{\mathbb{x}}$ (5.12) on the canonical one-particle wave
function $\ket{\psi}=\int\psi({\bf x})\ket{{\bf x}}{\rm d}^{3}{\rm x}$, which,
we recall, ‘lives’ in classical Euclidean space $\mathbb{E}^{3}$, we have
$\hat{S}_{\mathbb{x}}:\ket{\psi}\mapsto\ket{\Psi^{\prime}}$, where

$\displaystyle\ket{\Psi^{\prime}}:=\int\psi(\mathbb{x})\ket{\mathbb{x}}\ket{g_{{\mathbb{x}}}}{\rm
d}^{3}{\rm x}=\int\int\psi(\mathbb{x})g(\mathbb{x}^{\prime}-{\bf
x})\ket{\mathbb{x}}\ket{\mathbb{x}^{\prime}}{\rm d}^{3}{\rm x}{\rm d}^{3}{\rm
x}^{\prime}\,.$ (5.16)

This represents the wave function of a single particle that ‘lives’ in the
smeared superposition of geometries, $\mathbb{E}^{3}\times\mathbb{E}^{3}$. For
simplicity, we will analyse only smeared one-particle wave functions, in this
paper. The interested reader is refereed to [81] for a preliminary analysis of
multi-particle states.

From (5.30), it is clear that, since an observed value of the particle’s
position, ‘${\bf x}^{\prime}$’, cannot determine which point(s) in the smeared
superposition of geometries underwent the transition ${\bf
x}\leftrightarrow{\bf x}^{\prime}$, we must sum over all possibilities by
integrating the joint probability distribution, $|\Psi({\bf x},{\bf
x}^{\prime})|^{2}:=|\psi({\bf x})|^{2}|g({\bf x}^{\prime}-{\bf x})|^{2}$, over
${\rm d}^{3}{\rm x}$. This yields

$\displaystyle\frac{{\rm d}^{3}P(\mathbb{x}^{\prime}|\Psi)}{{\rm d}{\rm
x}^{\prime 3}}=\int|\Psi(\mathbb{x},\mathbb{x}^{\prime})|^{2}{\rm d}^{3}{\rm
x}=|\psi|^{2}*|g|^{2}(\mathbb{x}^{\prime})\,,$ (5.17)

where the star again denotes a convolution. Thus, in this formalism, only
primed degrees of freedom represent measurable quantities, whereas the
unprimed degrees of freedom, inherited from canonical QM, are physically
inaccessible [81, 82, 83, 84, 85, 86, 87, 88, 89, 90]. They are recovered only
in the limit
$|g(\mathbb{x}^{\prime}-\mathbb{x})|^{2}\rightarrow\delta^{3}(\mathbb{x}^{\prime}-\mathbb{x})$
($\sigma_{g}\rightarrow 0$), for which $\mathbb{x}^{\prime}=\mathbb{x}$.

However, we may apply exactly the same logic to the momentum space
representation of canonical QM, yielding

$\displaystyle\hat{S}_{\mathbb{p}}:\ket{{\bf x}}\mapsto\ket{{\bf
p}}\otimes\ket{g_{{\bf p}}}\,.$ (5.18)

where

$\displaystyle\ket{g_{{\bf p}}}=\int\tilde{g}({\bf p}^{\prime}-{\bf
p})\ket{{\bf p}^{\prime}}{\rm d}^{3}{\rm p}^{\prime}\,,$ (5.19)

and

$\displaystyle\int|\tilde{g}({\bf p}^{\prime}-{\bf p})|^{2}{\rm d}^{3}{\rm
p}^{\prime}=\int|\tilde{g}({\bf p}^{\prime}-{\bf p})|^{2}{\rm d}^{3}{\rm
p}=1\,.$ (5.20)

Again, for simplicity, we may visualise $|\tilde{g}({\bf p}^{\prime}-{\bf
p})|^{2}$ as a Gaussian,

$\displaystyle|\tilde{g}({\bf p}^{\prime}-{\bf
p})|^{2}=\left(\frac{1}{\sqrt{2\pi}\tilde{\sigma}_{g}}\right)^{3}\exp\left[-\frac{({\bf
p}^{\prime}-{\bf p})^{2}}{2\tilde{\sigma}_{g}^{2}}\right]\,,$ (5.21)

where $\tilde{\sigma}_{g}$ represents the minimum momentum space smearing and
$\tilde{g}({\bf p}^{\prime}-{\bf p})$ is the quantum probability amplitude for
the transition ${\bf p}\leftrightarrow{\bf p}^{\prime}$,
$\tilde{\psi}(\bf{p})\leftrightarrow\tilde{\psi}(\bf{p}^{\prime})$.

Applying the map $\hat{S}_{\mathbb{p}}$ (5.18) to the spectral representation
of the canonical one-particle wave vector in the momentum basis,
$\ket{\psi}=\int\tilde{\psi}({\bf p})\ket{{\bf p}}{\rm d}^{3}{\rm p}$, gives
$\hat{S}_{\mathbb{p}}:\ket{\psi}\mapsto\ket{\Psi^{\prime\prime}}$, where

$\displaystyle\ket{\Psi^{\prime\prime}}:=\int\psi(\mathbb{p})\ket{\mathbb{p}}\ket{g_{{\mathbb{p}}}}{\rm
d}^{3}{\rm p}=\int\int\psi(\mathbb{p})g(\mathbb{p}^{\prime}-{\bf
p})\ket{\mathbb{p}}\ket{\mathbb{p}^{\prime}}{\rm d}^{3}{\rm p}{\rm d}^{3}{\rm
p}^{\prime}\,.$ (5.22)

The joint probability distribution, associated with the measured value ${\bf
p}^{\prime}$, is
$|\tilde{\Psi}(\mathbb{p},\mathbb{p}^{\prime})|^{2}:=|\tilde{\psi}({\bf
p})|^{2}|\tilde{g}({\bf p}^{\prime}-{\bf p})|^{2}$, giving

$\displaystyle\frac{{\rm d}^{3}P(\mathbb{p}^{\prime}|\tilde{\Psi})}{{\rm
d}{\rm p}^{\prime
3}}=\int|\tilde{\Psi}(\mathbb{p},\mathbb{p}^{\prime})|^{2}{\rm d}^{3}{\rm
x}=|\psi|^{2}*|\tilde{g}|^{2}(\mathbb{p}^{\prime})\,.$ (5.23)

Here too, the unprimed degrees of freedom, which are inherited from canonical
QM, are physically inaccessible and are recovered only in the limit
$|\tilde{g}(\mathbb{p}^{\prime}-\mathbb{p})|^{2}\rightarrow\delta^{3}(\mathbb{p}^{\prime}-\mathbb{p})$
($\tilde{\sigma}_{g}\rightarrow 0$), for which
$\mathbb{p}^{\prime}=\mathbb{p}$.

Using our previous terminology, we may say that both $\hat{S}_{\mathbb{x}}$
(5.12) and $\hat{S}_{\mathbb{p}}$ (5.18) represent CRF-to-QRF transitions, and
the physical process of ‘smearing’ the background geometry is illustrated,
heuristically, in Fig. 11. (See also Fig. 12, for a simplified, but
phenomenologically almost equivalent picture). Yet again, we appear to have
run into a similar problem to that faced by the GCB formalism [14], namely, an
ambiguity in the definition of such a transition, due to the fact that
$\hat{S}_{\mathbb{x}}\ket{\psi}\neq\hat{S}_{\mathbb{p}}\ket{\psi}$
($\ket{\Psi^{\prime}}\neq\ket{\Psi^{\prime\prime}}$). Which choice is correct?

Yet again, the ambiguity is only apparent. We require the ‘objective’ state of
the composite quantum-particle-plus-quantum-background system to be
independent of the chosen representation, so that we must define the momentum
space representation of $\ket{\Psi^{\prime}}$ as

$\displaystyle\ket{\Psi^{\prime}}:=\int\int\tilde{\psi}(\mathbb{p})\tilde{g}(\mathbb{p}^{\prime}-{\bf
p})\ket{\mathbb{p}\,\mathbb{p}^{\prime}}{\rm d}^{3}{\rm p}{\rm d}^{3}{\rm
p}^{\prime}\,,$ (5.24)

where

$\displaystyle\bra{\mathbb{x}}\braket{\mathbb{x}^{\prime}}{\mathbb{p}\,\mathbb{p}^{\prime}}:=\left(\frac{1}{\sqrt{2\pi\hbar\beta}}\right)^{6}\exp\left[\frac{i}{\hbar}\mathbb{p}.\mathbb{x}+\frac{i}{\beta}(\mathbb{p}^{\prime}-\mathbb{p}).(\mathbb{x}^{\prime}-\mathbb{x})\right]\,,$
$\displaystyle\braket{\mathbb{p}\,\mathbb{p}^{\prime}}{\bm{\alpha}\,\bm{\alpha}^{\prime}}:=\delta^{3}(\mathbb{p}-\bm{\alpha})\,\delta^{3}(\mathbb{p}^{\prime}-\bm{\alpha}^{\prime})\,,$
(5.25)

and the position space representation of $\ket{\Psi^{\prime\prime}}$ as

$\displaystyle\ket{\Psi^{\prime\prime}}:=\int\int\psi(\mathbb{x})g(\mathbb{x}^{\prime}-{\bf
x})\ket{\mathbb{x}\,\mathbb{x}^{\prime}}{\rm d}^{3}{\rm x}{\rm d}^{3}{\rm
x}^{\prime}\,,$ (5.26)

where

$\displaystyle\bra{\mathbb{p}}\braket{\mathbb{p}^{\prime}}{\mathbb{x}\,\mathbb{x}^{\prime}}:=\left(\frac{1}{\sqrt{2\pi\hbar\beta}}\right)^{6}\exp\left[\frac{i}{\hbar}\mathbb{p}.\mathbb{x}+\frac{i}{\beta}(\mathbb{p}^{\prime}-\mathbb{p}).(\mathbb{x}^{\prime}-\mathbb{x})\right]\,,$
$\displaystyle\braket{\mathbb{x}\,\mathbb{x}^{\prime}}{\mathbb{a}\,\mathbb{a}^{\prime}}:=\delta^{3}(\mathbb{p}-\mathbb{a})\,\delta^{3}(\mathbb{p}^{\prime}-\mathbb{a}^{\prime})\,,$
(5.27)

together with

$\displaystyle\tilde{\psi}(\mathbb{p}):=\int\psi(\mathbb{x})\exp\left[-\frac{i}{\hbar}\mathbb{p}.\mathbb{x}\right]{\rm
d}^{3}{\rm x}\,,$ (5.28)

as in canonical QM, and

$\displaystyle\tilde{g}(\mathbb{p}^{\prime}-\mathbb{p}):=\int
g(\mathbb{x}^{\prime}-\mathbb{x})\exp\left[-\frac{i}{\beta}(\mathbb{p}^{\prime}-\mathbb{p}).(\mathbb{x}^{\prime}-\mathbb{x})\right]{\rm
d}^{3}{\rm x}^{\prime}\,.$ (5.29)

The physical interpretation of the constant $\beta$, introduced in Eqs. (5.2),
(5.2) and (5.29), will be considered shortly. Before that, the careful reader
will easily anticipate our next step. It is most convenient to work in the
symmetric bases, for which we define the state

$\displaystyle\ket{\Psi}$ $\displaystyle:=$
$\displaystyle\int\int\psi(\mathbb{x})g(\mathbb{x}^{\prime}-\mathbb{x})\ket{\mathbb{x}}\ket{\mathbb{x}^{\prime}-\mathbb{x}}{\rm
d}^{3}{\rm x}{\rm d}^{3}{\rm x}^{\prime}$ (5.30) $\displaystyle:=$
$\displaystyle\int\int\tilde{\psi}(\mathbb{p})\tilde{g}(\mathbb{p}^{\prime}-\mathbb{p})\ket{\mathbb{p}}\ket{\mathbb{p}^{\prime}-\mathbb{p}}{\rm
d}^{3}{\rm p}{\rm d}^{3}{\rm p}^{\prime}$ $\displaystyle:=$
$\displaystyle\ket{\psi}\otimes\ket{g}\,,$

which corresponds to the application of the map

$\displaystyle\hat{S}:\ket{\psi}\mapsto\ket{\psi}\otimes\ket{g}\,.$ (5.31)

Note that, here,

$\displaystyle\ket{g}:=\int
g(\mathbb{x}^{\prime}-\mathbb{x})\ket{\mathbb{x}^{\prime}-\mathbb{x}}{\rm
d}^{3}{\rm
x}^{\prime}:=\int\tilde{g}(\mathbb{p}^{\prime}-\mathbb{p})\ket{\mathbb{p}^{\prime}-\mathbb{p}}{\rm
d}^{3}{\rm p}^{\prime}\,,$ (5.32)

which is not equivalent to either $\ket{g_{{\bf x}}}$ (5.13) or $\ket{g_{{\bf
p}}}$ (5.19), corresponds to the quantum state of the whole background
geometry, rather than the quantum state of a specific ‘point’ in either
physical space or momentum space.

Next, we construct the generalised position and momentum operators,

$\displaystyle\hat{X}^{i}$ $\displaystyle:=$
$\displaystyle\hat{\mathcal{X}}^{i}+\hat{\mathcal{X}}^{\prime i}$ (5.33)
$\displaystyle:=$ $\displaystyle\int\int
x^{i}\ket{\mathbb{x}}\bra{\mathbb{x}}\otimes\ket{\mathbb{x}^{\prime}-\mathbb{x}}\bra{\mathbb{x}^{\prime}-\mathbb{x}}{\rm
d}^{3}{\rm x}{\rm d}^{3}{\rm x}^{\prime}$ $\displaystyle+$
$\displaystyle\int\int(x^{\prime
i}-x^{i})\ket{\mathbb{x}}\bra{\mathbb{x}}\otimes\ket{\mathbb{x}^{\prime}-\mathbb{x}}\bra{\mathbb{x}^{\prime}-\mathbb{x}}{\rm
d}^{3}{\rm x}{\rm d}^{3}{\rm x}^{\prime}\,,$

and

$\displaystyle\hat{P}_{j}$ $\displaystyle:=$
$\displaystyle\hat{\mathcal{P}}_{j}+\hat{\mathcal{P}}^{\prime}_{j}$ (5.34)
$\displaystyle:=$ $\displaystyle\int\int
p_{j}\ket{\mathbb{p}}\bra{\mathbb{p}}\otimes\ket{\mathbb{p}^{\prime}-\mathbb{p}}\bra{\mathbb{p}^{\prime}-\mathbb{p}}{\rm
d}^{3}{\rm p}{\rm d}^{3}{\rm p}^{\prime}$ $\displaystyle+$
$\displaystyle\int\int(p^{\prime}_{j}-p_{j})\ket{\mathbb{p}}\bra{\mathbb{p}}\otimes\ket{\mathbb{p}^{\prime}-\mathbb{p}}\bra{\mathbb{p}^{\prime}-\mathbb{p}}{\rm
d}^{3}{\rm p}{\rm d}^{3}{\rm p}^{\prime}\,,$

whose eigenvalues are $x^{\prime i}$ and $p^{\prime}_{j}$, respectively.
Physically, these represent the possible measured values of the particle’s
position and momentum relative to the fluctuations of the quantum background
geometry, in accordance with our interpretation of the latter as a non-
material QRF. In this scenario, the sub-operators
$\left\\{\hat{\mathcal{X}}^{i},\hat{\mathcal{X}}^{\prime
i};\hat{\mathcal{P}}_{j},\hat{\mathcal{P}}^{\prime}_{j}\right\\}$ do not
represent observable quantities. We are restricted to performing measurements
on material particles, whose motion is determined by a combination of
canonical quantum diffusion and fluctuations of the quantum background
geometry they inhabit, but there is no physical measurement that enables us to
isolate either of these effects. Nonetheless, we can again use the position
and momentum sub-operators to construct the useful unitaries

$\displaystyle\hat{\mathcal{U}}^{\prime}:=\exp\left[\frac{i}{\beta}\hat{\mathbb{\mathcal{P}}}^{\prime}.\hat{\mathbb{\mathcal{X}}}\right]\,,\quad\hat{\mathcal{U}}^{\prime\prime}:=\exp\left[-\frac{i}{\beta}\hat{\mathbb{\mathcal{P}}}.\hat{\mathbb{\mathcal{X}}}^{\prime}\right]\,.$
(5.35)

These act on the entangled bases according to

$\displaystyle\ket{\mathbb{x}}\ket{\mathbb{x}^{\prime}},\,\ket{\mathbb{p}\,\mathbb{p}^{\prime}}\stackrel{{\scriptstyle\hat{\mathcal{U}}^{\prime\dagger}}}{{\longrightarrow}}\ket{\mathbb{x}}\ket{\mathbb{x}^{\prime}-\mathbb{x}},\,\ket{\mathbb{p}}\ket{\mathbb{p}^{\prime}-\mathbb{p}}\stackrel{{\scriptstyle\hat{\mathcal{U}}^{\prime\prime\dagger}}}{{\longleftarrow}}\ket{\mathbb{x}\,\mathbb{x}^{\prime}},\,\ket{\mathbb{p}}\ket{\mathbb{p}^{\prime}}\,,$
(5.36)

and, hence, on the states $\ket{\Psi^{\prime}}$ (5.16),
$\ket{\Psi^{\prime\prime}}$ (5.22) and $\ket{\Psi}$ (5.30), according to

$\displaystyle\ket{\Psi^{\prime}}\stackrel{{\scriptstyle\hat{\mathcal{U}}^{\prime\dagger}}}{{\longrightarrow}}\ket{\Psi}\stackrel{{\scriptstyle\hat{\mathcal{U}}^{\prime\prime\dagger}}}{{\longleftarrow}}\ket{\Psi^{\prime\prime}}\,.$
(5.37)

By analogy with our previous results, presented in Sec. 3, the unitary
equivalence of these states indicates their physical equivalence, as
representations of the canonical state $\ket{\psi}$, ‘viewed by’ the smeared
geometry $\ket{g}$ (5.32), which acts as a non-material QRF.

The different but equivalent representations of ‘non-material QRF states’,
$\ket{g_{\mathbb{x}}}$ (5.13), $\ket{g_{\mathbb{p}}}$ (5.19) and $\ket{g}$
(5.30), are also related according to

$\displaystyle\ket{g_{\mathbb{x}}}\stackrel{{\scriptstyle\hat{\mathcal{U}}^{\prime\dagger}}}{{\longrightarrow}}\ket{g}\stackrel{{\scriptstyle\hat{\mathcal{U}}^{\prime\prime\dagger}}}{{\longleftarrow}}\ket{g_{\mathbb{p}}}\,.$
(5.38)

Hence, the smearing maps $\hat{S}_{\mathbb{x}}$ (5.12), $\hat{S}_{\mathbb{p}}$
(5.18) and $\hat{S}$ (5.31) are related via

$\displaystyle\hat{\mathcal{U}}^{\prime}\hat{S}_{\mathbb{x}}\,\hat{\mathcal{U}}^{\prime\dagger}=\hat{S}=\hat{\mathcal{U}}^{\prime\prime}\hat{S}_{\mathbb{p}}\,\hat{\mathcal{U}}^{\prime\prime\dagger}\,.$
(5.39)

This again demonstrates the fact that the CRF-to-QRF transition, which
represents the transition from classical Euclidean space $\mathbb{E}^{3}$ to
the smeared superposition of geometries $\mathbb{E}^{3}\times\mathbb{E}^{3}$,
is, in fact, basis-independent, as required. In other words, we could just as
easily define $\hat{S}_{\mathbb{x}}$ via its action on the momentum space
eigenstates, $\ket{\mathbb{p}}$, and $\hat{S}_{\mathbb{p}}$ via its action on
the position space eigenstates, $\ket{\mathbb{x}}$, and we may act with either
operator on either representation of the canonical state $\ket{\psi}$. This
indicates that, despite initial appearances to the contrary,
$\hat{S}_{\mathbb{x}}$ and $\hat{S}_{\mathbb{p}}$ are not really basis-
dependent operations, unlike the operators $\hat{S}_{x}^{(C\rightarrow A)}$
and $\hat{S}_{p}^{(C\rightarrow A)}$, considered in [14].

It is straightforward to show that $\hat{X}^{i}$ and $\hat{P}_{j}$ generate
GURs, via their action on the smeared-state $\ket{\Psi}$, according to

$\displaystyle(\Delta_{\Psi}X^{i})^{2}$ $\displaystyle=$
$\displaystyle\braket{\Psi}{(\hat{X}^{i})^{2}}{\Psi}-\braket{\Psi}{\hat{X}^{i}}{\Psi}^{2}$
(5.40) $\displaystyle=$
$\displaystyle(\Delta_{\Psi}\mathcal{X}^{i})^{2}+(\Delta_{\Psi}\mathcal{X}^{\prime
i})^{2}$ $\displaystyle=$ $\displaystyle(\Delta_{\psi}x^{\prime
i})^{2}+(\Delta_{g}x^{\prime i})^{2}\,,$

and

$\displaystyle(\Delta_{\Psi}P_{j})^{2}$ $\displaystyle=$
$\displaystyle\braket{\Psi}{(\hat{P}_{j})^{2}}{\Psi}-\braket{\Psi}{\hat{P}_{j}}{\Psi}^{2}$
(5.41) $\displaystyle=$
$\displaystyle(\Delta_{\Psi}\mathcal{P}_{j})^{2}+(\Delta_{\Psi}\mathcal{P}^{\prime}_{j})^{2}$
$\displaystyle=$
$\displaystyle(\Delta_{\psi}p^{\prime}_{j})^{2}+(\Delta_{g}p^{\prime}_{j})^{2}\,.$

Alternatively, we may generate the same relations by acting with canonical-
type position and momentum operators on the density matrices of the effective
mixed states, corresponding to the probability distributions (5.17) and
(5.23). These are constructed by complete analogy with the mixed states (3.77)
and (3.86), obtained in Sec. 3.2.6.

However, when treating the background geometry as a non-material QRF, we
encounter a subtle issue that was not present in the canonical QRF model,
developed in Secs. 3-4, namely, that the canonical displacement vector
experiences both ‘front-end’ and ‘back-end’ smearing. This is depicted,
schematically, in Fig. 11, in which the the canonical displacement is labelled
as ${\bf r}$. Strictly, every point along ${\bf r}$ is smeared into a Planck-
width superposition of points, but, due to the vector-equivalence between
curved lines and straight lines in flat Euclidean space [52, 53], only the
smearing of the start and end points of the vector need concern us.

Thus, the back-end smearing is simply the smearing of the fixed classical
origin, $O$, which is mapped to a superposition of origins, $O$’, as in the
case of a canonical QRF depicted in Fig. 5. In addition to this, every
possible end point of ${\bf r}$ is also smeared, when $|{\bf r}|>0$, according
to the map $\ket{\bf r}\mapsto\ket{\bf r}\otimes\ket{g_{\bf r}}$ (2.1). The
back-end smearing therefore has a double effect; in addition to mapping
$\psi({\bf r})$ to $\psi({\bf r}-{\bf x}^{\prime}_{O})$ (in the notation of
Fig. 11), with probability $|g(-{\bf x}^{\prime}_{O})|^{2}$, it also maps the
origin of each and every smearing function, $g({\bf r}^{\prime}-{\bf r})$, to
$g({\bf r}^{\prime}-{\bf r}-{\bf x}^{\prime}_{O})$, for $|{\bf r}|>0$. In
other words, it maps the composite matter-plus-geometry wave function,
$\Psi({\bf r},{\bf r}^{\prime}):=\psi({\bf r})g({\bf r}^{\prime}-{\bf r})$, to
the shifted composite matter-plus-geometry wave function, $\Psi({\bf r},{\bf
r}^{\prime};{\bf x}^{\prime}_{O}):=\psi({\bf r}-{\bf x}^{\prime}_{O})g({\bf
r}^{\prime}-{\bf r}-{\bf x}^{\prime}_{O})$, with probability $|g({\bf
x}^{\prime}_{O})|^{2}$.

This represents the perspective of the ‘observer’, who would have been located
at $O$, if fluctuations of the background geometry hadn’t also smeared out
his/her position, in addition to smearing out the canonical wave function of
the observed particle, $\psi({\bf r})$. The resulting density operator is

$\displaystyle\hat{\varrho}_{\Psi}$ $\displaystyle:=$
$\displaystyle\int\frac{{\rm d}^{3}P({\bf x}^{\prime}_{O})}{{\rm d}^{3}{\rm
x}^{\prime}_{O}}\ket{\Psi({\bf x}_{O})}\bra{\Psi({\bf x}^{\prime}_{O})}{\rm
d}^{3}{\rm x}^{\prime}_{O}$ (5.42) $\displaystyle:=$
$\displaystyle\int\frac{{\rm d}^{3}P({\bf x}^{\prime}_{O})}{{\rm d}^{3}{\rm
x}^{\prime}_{O}}\ket{\psi({\bf x}^{\prime}_{O})}\bra{\psi({\bf
x}^{\prime}_{O})}\otimes\ket{g({\bf x}^{\prime}_{O})}\bra{g({\bf
x}^{\prime}_{O})}{\rm d}^{3}{\rm x}^{\prime}_{O}$

where

$\displaystyle\ket{\Psi({\bf x}^{\prime}_{O})}:=\ket{\psi({\bf
x}^{\prime}_{O})}\ket{g({\bf x}^{\prime}_{O})}:=\int\int\psi({\bf r})g({\bf
r}^{\prime}-{\bf r})\ket{{\bf r}+{\bf x}^{\prime}_{O}}\ket{{\bf
r}^{\prime}-{\bf r}+{\bf x}^{\prime}_{O}}{\rm d}^{3}{\rm r}{\rm d}^{3}{\rm
r}^{\prime}\,,$ (5.43)

and ${\rm d}^{3}P({\bf x}^{\prime}_{O}):=|g(-{\bf x}^{\prime}_{O})|^{2}{\rm
d}^{3}{\rm x}^{\prime}_{O}$.

Equations (5.42)-(5.43) give ${\rm
d}^{3}P(\mathbb{x}^{\prime}|\Psi)=|\psi|^{2}*|\mathfrak{g}|^{2}(\mathbb{x}^{\prime}){\rm
d}^{3}{\rm x}^{\prime}$, where $|\mathfrak{g}|^{2}=|g|^{2}*|g|^{2}$ and ${\bf
x}^{\prime}={\bf r}^{\prime}-{\bf x}^{\prime}_{O}$ is the measured value of
the displacement, which incorporates the effects of both the front-end and
back-end smearing of the canonical displacement vector, as described above
(see Fig. 11). Nonetheless, if $|g|^{2}$ is a Planck-width Gaussian, as
assumed in Eq. (5.15), the only measurable difference between this scenario
and the simplified picture shown in Fig. 12, in which only the front-end
smearing of the canonical displacement is accounted for (for single-particle
states), is a change in the effective smearing scale such that
$\Delta_{g}x^{\prime i}=\sigma_{g}\simeq l_{\rm Pl}\rightarrow
2\sigma_{g}\simeq 2l_{\rm Pl}$ in the GUR (5.40).

For momentum measurements, the corresponding density operator is

$\displaystyle\hat{\tilde{\varrho}}_{\Psi}$ $\displaystyle:=$
$\displaystyle\int\frac{{\rm d}^{3}P({\bf p}^{\prime}_{O})}{{\rm d}^{3}{\rm
p}^{\prime}_{O}}\ket{\Psi({\bf p}_{O})}\bra{\Psi({\bf p}^{\prime}_{O})}{\rm
d}^{3}{\rm p}^{\prime}_{O}$ (5.44) $\displaystyle:=$
$\displaystyle\int\frac{{\rm d}^{3}P({\bf p}^{\prime}_{O})}{{\rm d}^{3}{\rm
p}^{\prime}_{O}}\ket{\psi({\bf p}^{\prime}_{O})}\bra{\psi({\bf
p}^{\prime}_{O})}\otimes\ket{g({\bf p}^{\prime}_{O})}\bra{g({\bf
p}^{\prime}_{O})}{\rm d}^{3}{\rm p}^{\prime}_{O}$

where

$\displaystyle\ket{\Psi({\bf p}^{\prime}_{O})}:=\ket{\psi({\bf
p}^{\prime}_{O})}\ket{g({\bf
p}^{\prime}_{O})}:=\int\int\tilde{\psi}(\bm{\kappa})\tilde{g}(\bm{\kappa}^{\prime}-\bm{\kappa})\ket{\bm{\kappa}+{\bf
p}^{\prime}_{O}}\ket{\bm{\kappa}^{\prime}-\bm{\kappa}+{\bf
p}^{\prime}_{O}}{\rm d}^{3}{\rm\kappa}{\rm d}^{3}{\rm\kappa}^{\prime}\,,$
(5.45)

and ${\rm d}^{3}P({\bf p}^{\prime}_{O}):=|\tilde{g}(-{\bf
p}^{\prime}_{O})|^{2}{\rm d}^{3}{\rm p}^{\prime}_{O}$. Again, the only
measurable difference between the statistical predictions of Eqs.
(5.44)-(5.45) and those given by the pure-state density matrix
$\hat{\rho}_{\Psi}:=\ket{\Psi}\bra{\Psi}$, for single-particle states, is
accounted for by substituting
$\Delta_{g}p^{\prime}_{j}=\tilde{\sigma}_{g}\simeq m_{\rm dS}c\rightarrow
2\tilde{\sigma}_{g}\simeq 2m_{\rm dS}c$ in the GUR (5.41). Because both the