# Matter relative to quantum hypersurfaces

Philipp A. Höhn<EMAIL_ADDRESS>Okinawa Institute of Science and
Technology Graduate University, Onna, Okinawa 904 0495, Japan  Department of
Physics and Astronomy, University College London, London, United Kingdom
Andrea Russo<EMAIL_ADDRESS>Department of Physics and
Astronomy, University College London, London, United Kingdom  Alexander R. H.
Smith<EMAIL_ADDRESS>Department of Physics, Saint Anselm College,
Manchester, New Hampshire 03102, USA  Department of Physics and Astronomy,
Dartmouth College, Hanover, New Hampshire 03755, USA

###### Abstract

We explore the canonical description of a scalar field as a parameterized
field theory on an extended phase space that includes additional embedding
fields that characterize spacetime hypersurfaces $\mathsf{X}$ relative to
which the scalar field is described. This theory is quantized via the Dirac
prescription and physical states of the theory are used to define conditional
wave functionals $\ket{\psi_{\phi}[\mathsf{X}]}$ interpreted as the state of
the field relative to the hypersurface $\mathsf{X}$, thereby extending the
Page-Wootters formalism to quantum field theory. It is shown that this
conditional wave functional satisfies the Tomonaga-Schwinger equation, thus
demonstrating the formal equivalence between this extended Page-Wootters
formalism and standard quantum field theory. We also construct relational
Dirac observables and define a quantum deparameterization of the physical
Hilbert space leading to a relational Heisenberg picture, which are both shown
to be unitarily equivalent to the Page-Wootters formalism. Moreover, by
treating hypersurfaces as quantum reference frames, we extend recently
developed quantum frame transformations to changes between classical and
nonclassical hypersurfaces. This allows us to exhibit the transformation
properties of a quantum field under a larger class of transformations, which
leads to a frame-dependent particle creation effect.

##  I Introduction

Physical theories that are independent of background spacetime structure are
typically characterized by constraints. For example, in the absence of a
boundary, the canonical Hamiltonian of general relativity vanishes on-shell.
In such theories, the Hamiltonian does not generate an evolution of the
physical degrees of freedom; instead, dynamics emerges from relations between
internal subsystems. As a consequence, a relational notion of dynamics has
long been recognized as a conservative expectation of a quantum theory of
gravity Kuchař (2011); Isham (1993); Smolin (2006); Rovelli (1991a, 2004, b).
Often mechanical models with a finite number of degrees of freedom are studied
in relational scenarios to exhibit mathematical and conceptual subtleties
within a fully relational quantum theory DeWitt (1967); Rovelli (1991b, 2004);
Ashtekar and Lewandowski (2004); Banerjee  _et al._ (2012); Bojowald (2010);
Gielen and Menéndez-Pidal (2022); Poulin (2006); Bartlett  _et al._ (2007);
Angelo  _et al._ (2011). However, given that general relativity is a field
theory, a natural stepping stone is a relational theory of quantum fields.

With this as an aim, Dirac Dirac (1964) and later Kuchař Kuchař (1989a);
*kuchar_dirac_1989 (see also Isham and Kuchař (1985a, b)) developed what is
known as parameterized field theory (PFT), in which a matter field $\phi$ is
described relative to arbitrary curvilinear coordinates $X^{\mu}(t,x)$
associated with a foliation of Minkowski space. For constant $t$, these
coordinate functions are treated as dynamical fields that describe
hypersurfaces $\Sigma$ as embeddings into Minkowski space,
$\mathsf{X}:\Sigma\to\mathcal{M}$. This leads to an extended phase space
description of the matter field $\phi$ and the spacetime embedding fields
characterizing $\mathsf{X}$. The resulting phase space can then be quantized
via the Dirac procedure, leading to a consistent quantum theory that treats
$\phi$ and $\mathsf{X}$ on equal footing Ashtekar  _et al._ (1994); Torre and
Varadarajan (1998, 1999); Ashtekar  _et al._ (2002); Ashtekar and Lewandowski
(2004); Varadarajan (2007); Kiefer (2012); Anastopoulos  _et al._ (2021). This
approach to field theory is distinct from standard treatments (e.g., Ref.
Weinberg (1995)) in which position is treated as a classical background
parameter alongside the time variable. In contrast, PFT promotes both the time
variable and the position operator to embedding fields, which are then
quantized.

Prior to the development of PFT, Tomonaga Tomonaga (1946); Koba  _et al._
(1947) and Schwinger Schwinger (1948); *schwinger_quantum_1949 put forward an
alternative approach to a manifestly covariant formulation of quantum field
theory. Inspired by the desire for a Lorentz invariant framework and the need
to address divergences in quantum electrodynamics, they promoted the time-
dependent quantum mechanical wave function to a wave functional
$\ket{\psi_{\phi}[\mathsf{X}]}$ by replacing the time variable with a
spacelike hypersurface $\mathsf{X}$ in Minkowski space. The dynamics of the
theory is governed by a functional differential equation known as the
Tomonaga-Schwinger equation, which characterizes changes of
$\ket{\psi_{\phi}[\mathsf{X}]}$ under deformations of $\mathsf{X}$, analogous
to how the Schdrödinger equation governs time evolution.

In this article, we develop a novel approach to a relational formulation of
quantum field theory along the lines envisaged by Page and Wootters Page and
Wootters (1983); Wootters (1984). We begin with the Dirac quantization of PFT,
which requires constructing Hilbert spaces characterizing both a matter field
and the embedding fields. We then introduce sets of coherent states
$\\{\ket{\mathsf{X}}\\}$ relative to subgroups of spacetime diffeomorphisms.
These coherent states are eigenstates of the quantized embedding field
operators with eigenfunctions corresponding to coordinate functions defining
definite hypersurfaces. These coherent states form a basis for the embedding
Hilbert space whose elements are connected to one another by a group action,
which is crucial for implementing the Page-Wootters formalism Wootters (1984);
Smith and Ahmadi (2019, 2020); Höhn  _et al._ (2021a, b); de la Hamette and
Galley (2020); de la Hamette  _et al._ (2021). Physical states $\ket{\Psi_{\rm
phys}}$ characterizing the scalar field and embedding fields are constructed
by promoting the classical phase space constraints to operators that
annihilate $\ket{\Psi_{\rm phys}}$. Conditioning physical states on a specific
coherent state $\ket{\mathsf{X}}$ of the embedding fields defines a state of
the scalar field
$\ket{\psi_{\phi}[\mathsf{X}]}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\bra{\mathsf{X}}\otimes\hat{I}_{\phi}\ket{\Psi_{\rm
phys}}$, which is to be interpreted as the state of $\phi$ relative to
$\mathsf{X}$. The conditional state is a wave functional that is shown to
satisfy the Tomonaga-Schwinger equation, thus demonstrating the equivalence of
the Page-Wootters formalism to standard formulations of wave functional
dynamics in quantum field theory. In particular, we show that the foliation-
independence of the integrated Tomonaga-Schwinger equation, which is usually
attributed to the microcausality condition satisfied by the stress-energy
tensor of the scalar field, is seen to be independently derived from the gauge
invariance of physical states (see Fig. 3).

Moreover, the coherent state construction is used to build relational Dirac
observables and a relational Heisenberg picture equivalent to a quantum
deparametrization of the physical Hilbert space Höhn and Vanrietvelde (2020);
Vanrietvelde  _et al._ (2023); Höhn (2019); Vanrietvelde  _et al._ (2020);
Höhn  _et al._ (2021a, b); de la Hamette  _et al._ (2021), which give
unitarily equivalent formulations to the Page-Wootters formalism analogous to
the non-interacting quantum mechanical scenario Höhn  _et al._ (2021a, b).
Finally, we present a field-theoretic analog of the quantum reference frame
changes introduced by Giacomini et al. Giacomini  _et al._ (2019), which
allows us to describe a quantum field theory relative to a superposition of
classical embeddings. We show that when transforming the conditional state of
the scalar field to one defined relative to an embedding in a state that has
support on hypersurfaces related by Bogoliubov transformations that are non-
trivial combinations of creation and annihilation operators, changing
embedding fields that characterize a reference frame can lead to a particle
creation effect. This effect is a field-theoretic analog of the conversion
between spatial superposition and entanglement under a change of quantum
reference frame Giacomini  _et al._ (2019).

Note that the embedding fields constitute a field-theoretic quantum reference
frame; specifically, the embedding fields are quantum reference frame
associated with the infinite dimensional diffeomorphism group, just like the
quantum clocks in the Page-Wootters formalism are temporal quantum reference
frames associated with the one-dimensional group of time reparametrizations.
Our work thus constitutes a field-theoretic extension of recent results about
quantum reference frames, which rather focused on finite-dimensional groups
and thereby a mechanical setting or symmetry reduced cosmological models Smith
and Ahmadi (2019); Höhn and Vanrietvelde (2020); Vanrietvelde  _et al._
(2023); Höhn (2019); Loveridge  _et al._ (2018); Giacomini  _et al._ (2019);
Vanrietvelde  _et al._ (2020); de la Hamette and Galley (2020); Castro-Ruiz
_et al._ (2020); Smith and Ahmadi (2020); Höhn  _et al._ (2021a, b); Giacomini
(2021); de la Hamette  _et al._ (2021); Ballesteros  _et al._ (2021); A. Ahmad
_et al._ (2022); Carette  _et al._ (2023); J. Głowacki (2023); Barbado  _et
al._ (2020) (see also Giacomini and Kempf (2022); Kabel  _et al._ (2022,
2023); Goeller  _et al._ (2022); Carrozza  _et al._ (2022)). We also note that
a complementary approach Kaya (2023) using parameterized field theory has
recently been shown to yield a Schrödinger-like equation describing the
dynamics of a matter field relative to embedding fields.

We begin in Sec. II by reviewing parameterized field theory and derive a
family of constraints satisfied by the scalar field and a set of embedding
fields. In Sec. III, we construct the kinematical Hilbert spaces of the scalar
field and embedding fields (the latter at a somewhat formal level) and carry
out the Dirac quantization of PFT. In doing so, we introduce group coherent
states associated with diffeomorphisms of Minkowski space. In Sec. IV, we
develop the field-theoretic generalization of the Page-Wootters formalism and
show that the conditional state of the scalar field satisfies the Tomonaga-
Schwinger equation, and construct relational Dirac observables encoding the
same dynamics. We also develop a quantum deparametrization of the physical
Hilbert space leading to a relational Heisenberg picture. As an application of
the developed framework, in Sec. V we analyze changes of quantum reference
frames, induced Bogoliubov transformations on a scalar field, and a frame-
dependent particle creation effect. We conclude in Sec. VI with a summary of
our results and an outlook on future work.

Throughout, we adopt units such that $\hbar=c=1$, employing the $(-,+)$
convention for the metric signature. We use round brackets ( ) for the
arguments of functions and square brackets [ ] for the arguments of
functionals.

##  II Parametrized field theory

Consider a real scalar field $\phi(X^{\mu})$ living on 1+1 dimensional
Minkowski space $\mathcal{M}$ equipped with the metric
$\eta_{\mu\nu}=\operatorname{diag}(-1,1)$, defined in terms of the inertial
coordinates $X^{\mu}=(T,X)$. The dynamics of the theory is specified by the
action

$S[\phi]=\int dTdX\,\mathcal{L}(\phi,\partial\phi/\partial X^{\mu}),$ (1)

where $\mathcal{L}(\phi,\partial\phi/\partial X^{\mu})$ is the Lagrangian
density of the field. One can vary this action and obtain the dynamics of the
field with respect to the inertial coordinates $X^{\mu}$.

However, the dynamics of the theory may be described with respect to an
arbitrary set of curvilinear coordinates $x^{\mu}=(t,x)$ defined by the
coordinate functions $X^{\mu}(t,x)$. Such a coordinate system can be chosen so
that for each $t\in[t_{1},t_{2}]$, the functions $X^{\mu}(t,x)$ parameterize a
family of spacelike embeddings of the 1-manifold $\Sigma\simeq\mathbb{R}$ in
$\mathcal{M}$:

$\begin{split}\mathsf{X}(t):\Sigma&\to\mathcal{M},\\\ x&\mapsto
X^{\mu}(t,x).\end{split}$

A foliation of Minkowski space is defined to be a smooth, one-parameter family
$\mathcal{F}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\\{\mathsf{X}(t),\
\forall t\in[t_{1},t_{2}]\\}$ of spacelike hypersurfaces $\mathsf{X}(t)$ such
that each spacetime point is located on precisely one hypersurface of the
family; see Fig. 1. A spacelike foliation defines a timelike deformation
vector field Isham and Kuchař (1985b); Thiemann (2008); Wald (2010)

$\displaystyle
t^{\mu}(t,x)\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\partial_{t}X^{\mu}(t,x)=N(t,x)n^{\mu}(t,x)+N^{\mu}(t,x),$
(2)

where we have decomposed this vector field into its components orthogonal and
parallel to the embedding $\mathsf{X}(t)$. Here, $n^{\mu}(t,x)$ is the unique
future-pointing unit normal vector to $\mathsf{X}(t)$, $N(t,x)$ is known as
the lapse function, and
$N^{\mu}(t,x)\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}N_{x}(t,x)\partial_{x}X^{\mu}(t,x)$
is the shift vector, which is orthogonal to $n^{\mu}(t,x)$. The flow generated
by the deformation vector field defines a timelike congruence of curves in
$\mathcal{M}$, which can be interpreted as a family of world lines
corresponding to a set of observers relative to which the dynamics of the
field is to be described. Also note that the timelike deformation vector field
is a complete vector field.111To see this, consider the integral curve
$\gamma(t):(a,b)\rightarrow\mathcal{M}$ given by integrating the timelike
deformation vector. Then, $\forall p\in\mathcal{M}$ and any integral curve
$\gamma$ passing through the point $p$, $\gamma$ can be extended to an
integral curve defined on $\mathbb{R}$,
$\Gamma:\mathbb{R}\rightarrow\mathcal{M}$, and $\Gamma$ has a global flow.

The action $S[\phi]$ can be re-cast on an extended configuration space
characterizing both the field $\phi(t,x)$ and the embedding fields
${X}^{\mu}(t,x)$ as dynamical degrees of freedom. This is accomplished by
reparameterizing the action in terms of the coordinates $x^{\mu}$ as

$\displaystyle S[\phi,X^{\mu}]=\int
dtdx\,J\mathcal{L}\left(\phi,\frac{\partial\phi}{\partial
x^{\nu}}\frac{\partial x^{\nu}}{\partial X^{\mu}}\right),$ (3)

where
$J\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\frac{\partial
T}{\partial t}\frac{\partial X}{\partial x}-\frac{\partial X}{\partial
t}\frac{\partial T}{\partial x}$ denotes the determinant of the Jacobian
associated with the coordinate transformation $X^{\mu}(t,x)$. When cast in
Hamiltonian form, the Lagrangian density in Eq. (3) yields the family of
constraints

$\displaystyle
C_{\mu}(t,x)\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\Pi_{\mu}(t,x)+h_{\mu}\left[\phi,\pi,X^{\mu}\right](t,x)\approx
0,$ (4)

where $\Pi_{\mu}(t,x)$ is the momentum conjugate to the inertial coordinate
$X^{\mu}(t,x)$ and

$\begin{split}h_{\mu}\left[\phi,\pi,X^{\mu}\right](t,x)&\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}J\frac{\partial
t(t,x)}{\partial X^{\nu}}{T^{\nu}}_{\mu}(t,x)\\\
&=n_{\nu}(t,x){T^{\nu}}_{\mu}(t,x),\end{split}$ (5)

where
$n_{\nu}(t,x)\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\epsilon_{\nu\rho}\partial
X^{\rho}(t,x)/\partial x$ and ${T^{\mu}}_{\nu}(t,x)$ denotes the stress-energy
tensor of the field $\phi$.

Figure 1: Depicted is a smooth one-parameter family of spacelike hypersurfaces
$\mathsf{X}(t)$ that define a foliation of Minkowski space,
$\mathcal{F}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\\{\mathsf{X}(t),\
\forall t\in[t_{1},t_{2}]\\}$. The parameter $x$ is a spatial coordinate
parameterizing each hypersurface (blue), and $t^{\mu}(t,x)$ denotes the
deformation vector field (green), defined in Eq. (2), which is tangent to
integral curves (red) associated with the $\mathcal{F}$.

##  III Dirac Quantization of PFT

While it is true that the gauge-invariant dynamics defined by the action in
Eq. (3) coincides with the dynamics defined by the action Eq. (1), the former
is defined on an extended configuration space that explicitly includes degrees
of freedom associated with an embedding relative to which the scalar field is
to be described. However, the embedding degrees of freedom are not independent
of the matter field degrees of freedom due to the constraints in Eq. (4).

Quantization proceeds via the Dirac prescription by promoting the
configuration variables, $X^{\mu}(x)$ and $\phi(x)$, and their conjugate
momenta, $\Pi_{\mu}(x)$ and $\pi(x)$, to operators satisfying canonical
commutation relations on the kinematical Hilbert space $\mathcal{H}_{\rm
kin}\simeq\mathcal{H}_{\rm Emb}\otimes\mathcal{H}_{\phi}$, where
$\mathcal{H}_{\rm Emb}$ and $\mathcal{H}_{\phi}$ are the kinematical embedding
and matter field Hilbert spaces, respectively. These canonical phase space
operators then define a quantization of the constraint operator in Eq. (4)
upon a choice of factor ordering and regularization. Physical states are
defined as those that are annihilated by the constraint operator and used to
construct the physical Hilbert space $\mathcal{H}_{\rm phys}$. We outline the
formal construction of $\mathcal{H}_{\phi}$, $\mathcal{H}_{\rm Emb}$, and
$\mathcal{H}_{\rm phys}$, and introduce structures necessary for a field-
theoretic generalization of the Page-Wootters formalism.

###  III.1 The matter field Hilbert space $\mathcal{H}_{\phi}$

As a minimal assumption, take the classical configuration space of the matter
field $\mathcal{C}_{\phi}=\\{\phi(x)\\}$ to be the space of twice
differentiable functions that decay rapidly at infinity on the hypersurface
$\mathsf{X}$. By analogy with the quantum description of a nonrelativistic
free particle, where the Hilbert space is the space of square-integrable
functions over the configuration space $\mathbb{R}^{3}$, one would naïvely
expect that the Hilbert space of the matter field to be
$\mathcal{H}_{\phi}\simeq L^{2}(\mathcal{C}_{\phi},\mathcal{D}\phi)$ for some
measure $\mathcal{D}\phi$. However, the field configuration space
$\mathcal{C}_{\phi}$ is infinite-dimensional, complicating the integration
theory necessary to define $\mathcal{H}_{\phi}$. To address these issues, we
follow the construction of $\mathcal{H}_{\phi}$ presented in Refs. Glimm and
Jaffe (1981); Ashtekar  _et al._ (1994); Ashtekar and Lewandowski (2004);
Ticciati (1999).

One proceeds by introducing an arbitrary, linearly independent set of probe
functions $V=\\{v_{1}(x),\dots,v_{n}(x)\\}$ that are elements of the Schwartz
space $\mathcal{S}$ of smooth, rapidly decreasing functions on $\mathbb{R}$.
These functions probe the structure of the field configuration space through
the linear functionals constructed by smearing the field operators on $\Sigma$

$\phi_{i}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\int
d\Sigma(x)\,v_{i}(x)\phi(x),$

where
$d\Sigma(x)\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}dx\sqrt{\gamma(x)}$
is the invariant measure on $\Sigma$ and
$\gamma(x)\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\eta_{\mu\nu}\partial_{x}X^{\mu}(x)\partial_{x}X^{\nu}(x)$
is both the induced metric on $\Sigma$ and its determinant; the numbers
$\phi_{i}$ can be interpreted as the component of the field along $v_{i}$. One
then defines the set $\mathrm{Cyl}_{V}$ of so-called cylindrical functions on
$\mathcal{C}_{\phi}$ with respect to $V$ as those that can be expressed as
functions of the coordinates $\phi_{i}$,

$f(\phi)=f(\phi_{1},\dots,\phi_{n})\in\mathrm{Cyl}_{V},$ (6)

i.e. functions that are constant along the other directions in
$\mathcal{C}_{\phi}$ (not encompassed by the probes), and introduces an inner
product on $\mathrm{Cyl}_{V}$

$\braket{f,g}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\int
d\mu_{n}\,f^{*}(\phi_{1},\dots,\phi_{n})g(\phi_{1},\dots,\phi_{n}),$ (7)

where $d\mu_{n}$ is a measure on $\mathbb{R}^{n}$. The space
$\mathrm{Cyl}_{V}$ is then extended to the space $\mathrm{Cyl}$ of all
functions that are cylindrical with respect to some set of probes, not
necessarily the set $V$. For the inner product in Eq. (7) to be well-defined,
the value of the integral cannot depend on the specific set of probes used to
represent the functions. This imposes nontrivial consistency requirements on
the measure $d\mu_{n}$ on $\mathrm{Cyl}$ that can be met, for example, by a
normalized Gaussian measure on $\mathbb{R}^{n}$. The Cauchy completion of
$\mathrm{Cyl}$ with respect to the norm induced by Eq. (7) is taken to be the
Hilbert space of the field $\mathcal{H}_{\phi}$. One can then extend the
measure
$\mathcal{D}\phi\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\lim_{n\to\infty}d\mu_{n}$
onto the space of tempered distributions $\mathcal{S}^{\prime}$, which is the
topological dual of the Schwartz space of probes $\mathcal{S}$ (i.e., the
space of linear functionals on $\mathcal{S}$), and show that
$\mathcal{H}_{\phi}\simeq L^{2}(\mathcal{S}^{\prime},\mathcal{D}\phi)$; for
this reason, $\mathcal{S}^{\prime}\supset\mathcal{C}_{\phi}$ is known as the
quantum configuration space, which is larger than $\mathcal{C}_{\phi}$
Ashtekar  _et al._ (1994); Ashtekar and Lewandowski (2004). In that way, it
encodes the distributional character of quantum field theory.

Having defined the Hilbert space $\mathcal{H}_{\phi}$, we seek a
representation of the field operator, its conjugate momentum, and their
canonical commutation relations. Analogous to the generalized eigenstates of
the position operator in nonrelativistic quantum mechanics, we can define
generalized field eigenstates $\ket{\phi}$ associated with configurations of
the field $\phi(x)\in\mathcal{S}^{\prime}$. More specifically, consider the
distributional field operator $\hat{\phi}(x)$ on a hypersurface $\mathsf{X}$
satisfying

$\hat{\phi}(x)\ket{\phi}=\phi(x)\ket{\phi}.$

The states $\ket{\phi}$ can be represented as a delta functional on
$\mathcal{S}^{\prime}$,

$\ket{\phi}=\int_{\mathcal{S}^{\prime}}\mathcal{D}\phi^{\prime}\,\delta[\phi-\phi^{\prime}]\ket{\phi^{\prime}},$

and are orthogonal to one another
$\braket{\phi}{\phi^{\prime}}=\delta[\phi-\phi^{\prime}]$. It follows that
these states form a basis for $\mathcal{H}_{\phi}$, so that any field state on
$\mathsf{X}$ may be expanded as

$\ket{\Psi_{\phi}}=\int_{\mathcal{S}^{\prime}}\mathcal{D}\phi\,\Psi_{\phi}[\phi]\ket{\phi},$

where $\Psi_{\phi}[\phi]=\braket{\phi}{\Psi_{\phi}}\in\mathcal{H}_{\phi}$.

It will be useful to smear the distributional field operator with a probe
function $f\in\mathcal{S}$,

$\hat{\phi}[f]\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\int
d\Sigma(x)\,f(x)\hat{\phi}(x),$

and define its conjugate momentum

$\hat{\pi}[g]\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}-i\int_{\mathcal{S}^{\prime}}\mathcal{D}\phi\int
d\Sigma(x)\,g(x)\ket{\phi}\frac{\delta}{\delta\phi(x)}\bra{\phi},$

which together furnish a representation of the canonical commutation relations

$\left[\hat{\phi}[f],\hat{\pi}[g]\right]\ket{\Psi_{\phi}}=i\left(\int
d\Sigma(x)f(x)g(x)\right)\ket{\Psi_{\phi}}.$

###  III.2 The embedding field Hilbert space $\mathcal{H}_{\rm Emb}$

####  III.2.1 General framework

The embedding field Hilbert space $\mathcal{H}_{\rm Emb}$ ought to be defined
as the space of square-integrable functions over the configuration space of
the embedding fields $Q$ containing all possible hypersurfaces,
$\mathsf{X}:\Sigma\to\mathcal{M}\in\text{Emb}(\Sigma,\mathcal{M})$. This is a
mathematically subtle topic because $\text{Emb}(\Sigma,\mathcal{M})$ is not a
vector space (one cannot add arbitrary embeddings to form another one), and we
shall invoke a few assumptions to proceed in analogy to the construction of
the scalar field Hilbert space.222For an alternative construction using
polymer quantization, see Varadarajan (2007). Following Isham and Kuchař Isham
and Kuchař (1985b, a), we first consider the space of all (not necessarily
spacelike) embeddings $\text{Emb}:\Sigma\to\mathcal{M}$, where
$\Sigma=\mathbb{R}$. We then take the configuration space to be

$Q\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\text{Emb}(\Sigma,\mathcal{M})/\text{Diff}(\Sigma)\simeq
G/H,$

where
$G\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\text{Diff}(\mathcal{M})$
is the group of diffeomorphisms on $\mathcal{M}$,
$H\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\text{Diff}(\mathcal{M},\mathsf{X}_{0})$
is the subgroup of diffeomorphisms that leave invariant a fiducial embedding
$\mathsf{X}_{0}:\Sigma\to\mathcal{M}$, and $\text{Diff}(\Sigma)$ is the group
of diffeomorphisms on $\Sigma$. The embedding field Hilbert space will then be
$\mathcal{H}_{\rm emb}\simeq L^{2}(\tilde{Q},\mathcal{D}q)$ for an appropriate
measure $\mathcal{D}q$ on the quantum configuration space $\tilde{Q}\supset
Q$.333In arriving at this point, we assume that it is possible to define a
complete orthonormal set of probe one-forms
$B_{q}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\\{q_{1\mu}(x),q_{2\mu}(x),\dots,q_{n\mu}(x)\\}$
as elements of the Schwartz space $\mathcal{S}(\Sigma)$ of rapidly decreasing
functions on $\Sigma$. Proceeding analogously to Sec. III.1, let us introduce
linear functions that probe the configuration space  $q_{i}:=\int
d\Sigma(x)\,q_{i\mu}(x)X^{\mu}(x)\in\mathbb{R}.$ We assume that one can again
define the set of cylindrical functions $\mathrm{Cyl}_{B_{q}}$ with respect to
$B_{q}$,  $\psi(q)=\psi(q_{1},q_{2},\dots,q_{n})\in\mathrm{Cyl}_{B_{q}},$
analogous to Eq. (6), and introduce an inner product on $\mathrm{Cyl}_{B_{q}}$
analogous to Eq. (7), which requires the introduction of a cylindrical measure
$d\mu_{n}=\prod_{i=1}^{n}d\mu$, which we take to be constructed as a product
measure. We assume that this space can be extended to the space of all
functions on $Q$ that are cylindrical with respect to some set of probe
functions, not necessarily $B_{q}$, and then take the Cauchy completion of
this space to form the embedding Hilbert space $\mathcal{H}_{\rm Emb}$. We
further assume that the embedding Hilbert space can be represented as
$\mathcal{H}_{\rm Emb}\simeq L^{2}(\tilde{Q},\mathcal{D}q)$, where
$\mathcal{D}q\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\lim_{n\to\infty}d\mu_{n}$
and $\tilde{Q}=\mathcal{S}_{2}^{\prime}(\Sigma)$ is a suitable space of two-
component tempered distributions on $\Sigma$.

The embedding fields are assumed to be promoted to self-adjoint operators on
$\mathcal{H}_{\rm Emb}$ defined by the eigenvalue-eigenvector equation

$\hat{X}^{\mu}(x)\ket{\mathsf{X}_{q}}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}X_{q}^{\mu}(x)\ket{\mathsf{X}_{q}},$

where $X_{q}^{\mu}(x):\Sigma\to\mathcal{M}$ is the eigenfunction corresponding
to the generalized eigenstate $\ket{\mathsf{X}_{q}}$ and equal to the
coordinate functions describing the embedding $\mathsf{X}_{q}$.

Given that the embedding field operators are taken to be self-adjoint, the
embedding operator $\hat{X}^{\mu}(x)$ is densely defined on the basis
$\ket{\mathsf{X}_{q}}$, which are generalized eigenstates of the embedding
operator $\hat{X}^{\mu}(x)$ with real eigenvalue functions $X^{\mu}_{q}(x)$.
This is consistent with the demand that $\hat{X}^{\mu}(x)$ is self-adjoint and
thus $\ket{\mathsf{X}_{q}}$ are orthogonal for different $q$, so that we may
represent hypersurface states as Dirac delta functionals

$\ket{\mathsf{X}_{q}}=\int\mathcal{D}q^{\prime}\,\delta[\mathsf{X}_{q^{\prime}}-\mathsf{X}_{q}]\ket{\mathsf{X}_{q^{\prime}}}.$

It follows that $\\{\ket{\mathsf{X}_{q}},\ \forall q\in\tilde{Q}\\}$ forms a
basis for $\mathcal{H}_{\rm Emb}$, and thus we have a resolution of the
identity

$\hat{I}_{\rm
Emb}=\int\mathcal{D}q\,\ket{\mathsf{X}_{q}}\\!\bra{\mathsf{X}_{q}},$ (8)

and we can expand embedding states as

$\ket{\psi}=\int\mathcal{D}q\,\psi[\mathsf{X}_{q}]\ket{\mathsf{X}_{q}},$

where
$\psi[\mathsf{X}_{q}]\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\braket{\mathsf{X}_{q}}{\psi}$.

The embedding operators $\hat{X}^{\mu}(x)$ may be smeared with a one-form
$w_{\mu}(x)$ on $\Sigma$, yielding

$\displaystyle\hat{\mathsf{X}}[w]$
$\displaystyle\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\int
d\Sigma(x)\,w_{\mu}(x)\hat{X}^{\mu}(x)$ $\displaystyle=\int
d\Sigma(x)\,w_{\mu}(x)\int\mathcal{D}q\,\ket{\mathsf{X}_{q}}{X}_{q}^{\mu}(x)\bra{\mathsf{X}_{q}}.$

The momentum operator canonically conjugate to $\hat{\mathsf{X}}[w]$ can be
represented as

$\displaystyle\hat{\mathsf{\Pi}}[v]$
$\displaystyle\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\int
d\Sigma(x)\,v^{\mu}(x)\hat{\Pi}_{\mu}(x)$ $\displaystyle=-i\int
d\Sigma(x)\,v^{\mu}(x)\int\mathcal{D}q\ket{\mathsf{X}_{q}}\frac{\delta}{\delta
X_{q}^{\mu}(x)}\bra{\mathsf{X}_{q}},$

where $v^{\mu}(x)$ is a vector field on $\Sigma$. Together, these operators
furnish a representation of the canonical commutation relations

$\left[\hat{\mathsf{X}}[w],\hat{\mathsf{\Pi}}[v]\right]=i\int
d\Sigma(x)\,w_{\mu}(x)v^{\mu}(x).$

Note that $\hat{\mathsf{\Pi}}[v]$ cannot be a self-adjoint operator in
general. If this were the case, then the canonical commutation relation would
imply that the embedding configuration space has a linear structure and would
thus be a vector space. However, this is not true, as one cannot add arbitrary
embeddings to produce another embedding (see Isham and Kuchař (1985a, b) for
further discussion). Nevertheless, because $\hat{\mathsf{\Pi}}[v]$ is
symmetric, for different vector fields $v^{\mu}$ it will have non-trivial
domains in $\mathcal{H}_{\rm emb}$, where it may effectively act like a self-
adjoint operator. On these domains it will act like a translation operator for
the embedding fields, however, only locally in the embedding configuration
space.

####  III.2.2 Coherent states relative to subgroups of spacetime
diffeomorphisms

Consider a representation of a one-dimensional subgroup
$G_{v}\subset\mathrm{Diff}(\mathcal{M})$ generated by the vector field
$v^{\mu}(x)\in\mathrm{Vec}(\mathcal{M})$, which is taken to vanish
asymptotically, given by

$\hat{U}_{v}(s)\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}e^{-is\hat{\mathsf{\Pi}}[v]}=e^{-is\int
d\Sigma(x)\,v^{\mu}(x)\hat{\Pi}_{\mu}(x)},$ (9)

for all $s\in[s_{0},s_{1}]\subset\mathbb{R}$, where this interval will depend
on the states this operator acts on. For some (possibly distributional) states
it may even be defined for all $s\in\mathbb{R}$. On the domain of
$\hat{\mathsf{\Pi}}[v]$, this group representation will effectively act
unitarily. Let us now consider a seed state $\ket{\mathsf{X}_{0}}$, taken to
be a generalized eigenstate of the embedding field operator which thus
corresponds to a definite hypersurface configuration, and define the set of
group coherent states Perelomov (1986) generated by the action of
$\hat{U}_{v}(s)$:

$\left\\{\ket{\mathsf{X}_{v}(s)}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\hat{U}_{v}(s)\ket{\mathsf{X}_{0}}\mbox{
for all }s\in[s_{0},s_{1}]\subset\mathbb{R}\right\\}\,,$ (10)

where the interval $[s_{0},s_{1}]$ depends on $\mathsf{X}_{0}$. Similarly
defined coherent states relative to subgroups of the diffeomorphism group of
Euclidean space have found application in general quantization methods
Twareque Ali and Goldin (1991). As shown in Appendix A.1, these coherent
states satisfy

$\displaystyle\hat{X}^{\mu}(x)\ket{\mathsf{X}_{v}(s)}$
$\displaystyle=\Big{(}X_{0}^{\mu}(x)+sv^{\mu}(x)\Big{)}\ket{\mathsf{X}_{v}(s)}.$
(11)

This yields an interpretation of $\ket{\mathsf{X}_{v}(s)}$ as a state of the
embedding fields associated with a hypersurface described by the coordinate
functions $X^{\mu}(x)=X_{0}^{\mu}(x)+sv^{\mu}(x)$, where $X_{0}^{\mu}(x)$ are
the coordinate functions of the seed hypersurface $\mathsf{X}_{0}$; see Fig.
2.

Figure 2: Hypersurfaces corresponding the seed state $\ket{\mathsf{X}_{0}}$
and the state
$\ket{\mathsf{X}_{v}(s)}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\hat{U}_{v}(s)\ket{\mathsf{X}_{0}}$,
which is connected to the seed embedding by a unitary representation
$\hat{U}_{v}(s)$ of the one-dimensional subgroup
$G_{v}\subset\mathrm{Diff}(\mathcal{M})$ generated by the vector field
$v^{\mu}(x)\in\mathrm{Vec}(\mathcal{M})$.

###  III.3 The physical Hilbert space $\mathcal{H}_{\rm phys}$

Physical states $\ket{\Psi_{\rm phys}}$ are defined as those living in the
kernel of the constraint operators resulting from the quantization of Eq.
(4):444Note that $\hat{C}_{\mu}$ can in general not be a self-adjoint operator
on $\mathcal{H}_{\rm kin}$ due to $\hat{\Pi}_{\mu}$ not being self-adjoint.
However, one can still impose it to construct its space of solutions.

$\hat{C}_{\mu}(x)\ket{\Psi_{\rm
phys}}=\left(\hat{\Pi}_{\mu}(x)+\hat{h}_{\mu}(x)\right)\ket{\Psi_{\rm
phys}}=0,$ (12)

where

$\hat{h}_{\mu}(x)\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\hat{n}_{\nu}(x)\otimes{\hat{T}^{\nu}}_{\
\mu}(x)+\hat{A}_{\mu}(x)$ (13)

is the quantization of the Hamiltonian flux in Eq. (5), which picks up an
anomalous term

$\hat{A}_{\mu}(x)\\!=\\!-\\!\int\mathcal{D}q\,\ket{\mathsf{X}_{q}}\sqrt{\gamma(x)}\\!\left(\\!\eta_{\mu\nu}X_{q}^{\nu}(x)\frac{\partial\mathcal{K}}{\partial
x}\\!-\\!\mathcal{K}^{2}n_{\mu}\\!\right)\\!\bra{\mathsf{X}_{q}},$

if we consider a free scalar field theory, i.e. one with mass $m=0$. In that
case, we are dealing with a conformal field theory (CFT) which leads to
anomalies in the quantum setting. Adding the anomaly to the Hamiltonian
ensures the constraint algebra closes Kuchař (1989b), and is further discussed
in Appendix C. In the expression above, $\mathcal{K}$ represents the trace of
the extrinsic curvature of the embedding $\mathsf{X}_{q}$. The effect of this
anomalous term is to redistribute the energy content of the slice based on how
it is embedded in the higher dimensional spacetime Kuchař (1989b). Notice that
when acting on states with support only on flat embeddings, the anomalous term
vanishes, as such embeddings have constant zero extrinsic curvature. One
should also keep in mind that if the embeddings are taken to be flat at
infinity, the anomaly vanishes when integrated over the entire spacelike slice
(in any coordinate frame)555In Kuchař (1989b), this was shown for cylindrical
Minkowski space, but from the equations in that work it follows that this also
holds for open Minkowski on hypersurfaces that become flat towards infinity.

$\int d\Sigma_{q}(x)\,\hat{A}_{\mu}(x)=0.$ (14)

If the embedding states $\ket{\mathsf{X}_{v}(s)}$ in Eq. (10) are generated
from an asymptotically flat embedding $\mathsf{X}_{0}$, then the anomaly
vanishes when integrated over the associated hypersurface $\mathsf{X}_{v}(s)$.

We assume that the set of states satisfying Eq. (12) can be equipped with an
inner product and completed to form the physical Hilbert space
$\mathcal{H}_{\rm phys}$. Gauge transformations are generated by
$\hat{\mathsf{C}}[v]\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\hat{\mathsf{\Pi}}[v]+\hat{\mathsf{h}}[v]$
and leave physical states invariant, $e^{i\hat{\mathsf{C}}[v]}\ket{\Psi_{\rm
phys}}=\ket{\Psi_{\rm phys}}$, for all $\ket{\Psi_{\rm
phys}}\in\mathcal{H}_{\rm phys}$.

##  IV Matter relative to quantum fields

When reparametrization-invariant theories are canonically quantized, it is
found that the Hamiltonian of the theory is proportional to a constraint
operator. This implies that physical states do not evolve under the action
generated by the quantized Hamiltonian. This is a generic feature of
relativistic theories and, ultimately, a manifestation of background
independence. This means that there is no physical evolution relative to the
external background.

Nonetheless, we are tasked with recovering a notion of dynamics from the
quantized theory. One approach is to identify a subsystem after quantization
to serve as a reference frame relative to which another subsystem evolves.
Three approaches that accomplish this task are the construction of so-called
relational Dirac observables (a.k.a., evolving constants of motion) Rovelli
(2004); Dittrich (2006, 2007); Thiemann (2008), the Page and Wootters
formalism Page and Wootters (1983); Wootters (1984), and a quantum
deparametrization procedure Höhn and Vanrietvelde (2020); Höhn (2019). For
simple mechanical systems, these approaches have been shown to be equivalent
in Höhn  _et al._ (2021a, b).

In this section, we introduce the Page-Wootters formulation of the
parameterized field theory reviewed in Sec. II. A conditional state of the
matter field is defined by conditioning physical states on configurations of
the embedding fields. This conditional state is then shown to formally satisfy
the covariant Tomonaga-Schwinger equation and an appropriate Schrödinger
equation describing evolution along arbitrary spacelike foliations of
Minkowski space, thus demonstrating the formal equivalence of the Page-
Wootters formalism with standard formulations of quantum field theory. We also
introduce relational Dirac observables and a quantum deparameterization of the
physical Hilbert space that results in a relational Heisenberg picture. These
relational formulations are shown to be unitarily equivalent to the Page-
Wootters formalism.

###  IV.1 The Page-Wootters formalism and Tomonaga-Schwinger equation

The typical starting point of the Page-Wootters formalism applied to a
mechanical system is a physical state $\ket{\Psi_{\rm phys}}$ describing a
non-interacting clock $C$ and system of interest $S$ satisfying a Hamiltonian
constraint

$\hat{C}_{H}\ket{\Psi_{\rm
phys}}=\left(\hat{H}_{C}+\hat{H}_{S}\right)\ket{\Psi_{\rm phys}}=0.$ (15)

One then considers a time observable

$T_{C}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\\{\ket{t}\\!\bra{t},\
\forall t\in G\subseteq\mathbb{R}\\},$

defined by a set of rank-1 effect density operators $\ket{t}\\!\bra{t}$
constructed from the outer product of so-called clock states $\ket{t}$ for all
$t\in\mathbb{R}$ that parameterize the one-dimensional time evolution group
$G$ generated by the clock Hamiltonian $\hat{H}_{C}$. What distinguishes
$T_{C}$ as a time observable is that it transforms covariantly under the
action of $G$, which amounts to the following relation between the clock
states666We note that in the case when the spectrum of $\hat{H}_{C}$ is
unbounded, then $T_{C}$ can be associated with a self-adjoint time operator
$\hat{T}_{C}$ that is canonically conjugate to $\hat{H}_{C}$, ensuring that
the clock Hamiltonian generates time translations. However, in the case when
$\hat{H}_{C}$ is bounded from below, as in any physical system, then such a
construction is not possible. Nonetheless, the covariant POVM $T_{C}$ is still
well-defined and yields the optimal estimate of parameter time Holevo (1982);
Busch and P. J. Lahti (1991); *buschOperationalQuantumPhysics1995;
*buschQuantumMeasurement2016; Smith and Ahmadi (2019, 2020); Höhn  _et al._
(2021a, b).

$\ket{t^{\prime}}=e^{-i\hat{H}_{C}\left(t^{\prime}-t\right)}\ket{t}.$ (16)

One then defines the conditional state of the system $\ket{\psi_{S}(t)}$ at a
time $t$ as

$\ket{\psi_{S}(t)}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\left(\bra{t}\otimes\hat{I}_{S}\right)\ket{\Psi_{\rm
phys}}.$ (17)

It then follows from Eqs. (15) and (16) that the conditional wave function
satisfies the Schrödinger equation,
$i\frac{d}{dt}\ket{\psi_{S}(t)}=\hat{H}_{S}\ket{\psi_{S}(t)}$, and yields the
correct one- and two-time probabilities Höhn  _et al._ (2021a). Thus, we
conclude that the conditional state $\ket{\psi_{S}(t)}$ ought to be
interpreted as the usual time-dependent solution to the Schrödinger equation,
which constitutes the recovery of the standard quantum mechanical framework
from the Page-Wootters formalism.

However, Page and Wootters never intended these ideas to be limited to
mechanical systems, which is made clear when Page writes Page (1989):

> ‘For simplicity, one can think of the […] system as a “particle” with
> “position”, which is a convenient model to have in mind, but my discussion
> is not intended to be limited to such a simple model. For example, the
> closed system could consist of relativistic quantum fields within a given
> classical background spacetime.’

Taking this seriously, we develop a field-theoretic generalization of the
Page-Wootters formalism for PFT. We begin by considering a physical state
$\ket{\Psi_{\rm phys}}$ satisfying Eq. (12), and introducing the so-called
reduction map $\mathcal{R}_{\mathbf{S}}[\mathsf{X}_{q}]$, which plays the same
role as Eq. (17). The reduction map is defined as

$\begin{split}\mathcal{R}_{\mathbf{S}}[\mathsf{X}_{q}]:\mathcal{H}_{\rm
phys}&\rightarrow\mathcal{H}_{\mathsf{X}_{q}},\\\ \ \ket{\Psi_{\rm
phys}}&\mapsto\ket{\psi_{\phi}[\mathsf{X}_{q}]}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\mathcal{R}_{\mathbf{S}}[\mathsf{X}_{q}]\ket{\Psi_{\rm
phys}},\end{split}$

where

$\mathcal{R}_{\mathbf{S}}[\mathsf{X}_{q}]\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\bra{\mathsf{X}_{q}}\otimes\hat{I}_{\phi},$
(18)

is used to condition a physical state $\ket{\Psi_{\rm phys}}$ on an embedding
state $\ket{\mathsf{X}_{q}}$. We refer to $\ket{\psi_{\phi}[\mathsf{X}_{q}]}$
as a conditional state of the field $\phi$ relative to the hypersurface
$\mathsf{X}_{q}$.

As a consequence of the resolution of the identity in Eq. (8), the reduction
map is formally invertible on physical states, with inverse

$\displaystyle\mathcal{R}^{-1}_{\mathbf{S}}[\mathsf{X}_{q}]:\mathcal{H}_{\mathsf{X}_{q}}$
$\displaystyle\rightarrow\mathcal{H}_{\rm phys};$ $\displaystyle\
\ket{\psi_{\phi}[\mathsf{X}_{q}]}$ $\displaystyle\mapsto\ket{\Psi_{\rm
phys}}=\mathcal{R}^{-1}_{\mathbf{S}}[\mathsf{X}_{q}]\ket{\psi_{\phi}[\mathsf{X}_{q}]},$

where

$\displaystyle\mathcal{R}_{\mathbf{S}}^{-1}[\mathsf{X}_{q}]$
$\displaystyle=\int\mathcal{D}q^{\prime}\,\ket{\mathsf{X}_{q^{\prime}}}\otimes\hat{U}_{\phi}[\mathsf{X}_{q^{\prime}}-\mathsf{X}_{q}]$
(19)
$\displaystyle=\int\mathcal{D}q^{\prime}\,\hat{U}_{\mathsf{X}}[\mathsf{X}_{q^{\prime}}-\mathsf{X}_{q}]\ket{\mathsf{X}_{q}}\otimes\hat{U}_{\phi}[\mathsf{X}_{q^{\prime}}-\mathsf{X}_{q}],$

and

$\displaystyle\hat{U}_{\mathsf{X}}[\mathsf{X}_{q^{\prime}}-\mathsf{X}_{q}]$
$\displaystyle\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}e^{-i\int
d\Sigma_{q}(x)\,\left(X^{\mu}_{q^{\prime}}(x)-X^{\mu}_{q}(x)\right)\hat{\Pi}_{\mu}(x)},$
$\displaystyle\hat{U}_{\phi}[\mathsf{X}_{q^{\prime}}-\mathsf{X}_{q}]$
$\displaystyle\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}e^{-i\int
d\Sigma_{q}(x)\,\left(X^{\mu}_{q^{\prime}}(x)-X^{\mu}_{q}(x)\right)n_{\nu}(x){\hat{T}^{\nu}}_{\
\mu}(x)}.$ (20)

Note that the surface integrals above are defined on the embedded hypersurface
$\mathsf{X}_{q}$ in Minkowski space and that the difference
$X^{\mu}_{q^{\prime}}(x)-X^{\mu}_{q}(x)$ defines a vector field on (but not
necessarily tangential to) this surface. In the line above, $n_{\nu}(x)$ is no
longer an operator as it has already been evaluated on the embedding
$\mathsf{X}_{q}$. The action of the inverse map on a conditional state of the
field is then

$\mathcal{R}_{\mathbf{S}}^{-1}[\mathsf{X}_{q}]\ket{\psi_{\phi}[\mathsf{X}_{q}]}=\int\mathcal{D}q\,\ket{\mathsf{X}_{q}}\ket{\psi_{\phi}[\mathsf{X}_{q}]}=\ket{\Psi_{\rm
phys}}.$ (21)

Given that $\mathcal{R}_{\mathbf{S}}[\mathsf{X}_{q}]$ is invertible, it
follows that $\mathcal{H}_{\rm phys}$ is isomorphic to
$\mathcal{H}_{\mathsf{X}_{q}}$.

We will now consider the conditional state of the field

$\ket{\psi_{\phi}[\mathsf{X}_{v}(s)]}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\mathcal{R}_{\mathbf{S}}[\mathsf{X}_{v}(s)]\ket{\Psi_{\rm
phys}}.$ (22)

associated with a one-dimensional subgroup $G_{v}\subset G$ defined by the
vector field $v^{\mu}(x)$ and group element $s$. In what follows, we will
derive the variation of $\ket{\psi_{\phi}[\mathsf{X}_{v}(s)]}$ in the
parameter $s$, parameterizing the subgroup $G_{v}$, and then its variation in
the vector field $v^{\mu}(x)$, characterizing deformations of the hypersurface
$\mathsf{X}_{0}$. We then consider the evolution of the conditional state
along a foliation $\mathcal{F}$ of $\mathcal{M}$.

####  IV.1.1 Evolution in the group parameter $s$

One notable difference to the quantum mechanical case in Eq. (15) is that in
PFT the embedding fields, serving as a dynamical reference frame for the
scalar field, interacts with the scalar field via the embedding-dependent
Hamiltonian in Eq. (13) (cf., Ref. Smith and Ahmadi (2019)). Nevertheless, we
can proceed similarly.

The evolution of the conditional state $\ket{\psi_{\phi}[\mathsf{X}_{v}(s)]}$
in the parameter $s\in[s_{0},s_{1}]$ satisfies:

$\begin{split}i\frac{d}{ds}\ket{\psi_{\phi}[\mathsf{X}_{v}(s)]}&=i\frac{d}{ds}\braket{\mathsf{X}_{v}(s)}{\Psi_{\rm
phys}}\\\
&=i\frac{d}{ds}\bra{\mathsf{X}_{0}}e^{is\hat{\mathsf{\Pi}}[v]}\ket{\Psi_{\rm
phys}}\\\
&=i\bra{\mathsf{X}_{0}}e^{is\hat{\mathsf{\Pi}}[v]}i\hat{\mathsf{\Pi}}[v]\ket{\Psi_{\rm
phys}}\\\ &=-\bra{\mathsf{X}_{v}(s)}\hat{\mathsf{\Pi}}[v]\ket{\Psi_{\rm
phys}}\\\ &=\bra{\mathsf{X}_{v}(s)}\hat{\mathsf{h}}[v]\ket{\Psi_{\rm phys}}\\\
&=\hat{\mathsf{h}}[v,\mathsf{X}_{v}(s)]\braket{\mathsf{X}_{v}(s)}{\Psi_{\rm
phys}}\\\
&=\hat{\mathsf{h}}[v,\mathsf{X}_{v}(s)]\ket{\psi_{\phi}[\mathsf{X}_{v}(s)]},\end{split}$
(23)

where the dependence on $\mathsf{X}_{v}(s)$ is through $n_{\mu}(x)$, and we
have defined

$\hat{\mathsf{h}}[v,\mathsf{X}_{v}(s)]\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\int
d\Sigma(x)\,v^{\mu}(x)n_{\nu}(x){\hat{T}^{\nu}}_{\ \mu}(x),$ (24)

and made use of Eqs. (9), (12), (22). It is thus seen that the operator
$\hat{\mathsf{h}}[v,\mathsf{X}_{v}(s)]$ generates the flow in $s$. We remind
the reader that the anomaly (when present) has vanished because we have
integrated over asymptotically flat spatial hypersurfaces as in Eq. (14).

As an example, let us consider an inertial foliation of Minkowski space by
flat spacelike hypersurfaces defined by the one-parameter family of
hypersurfaces through the coordinate functions

$X^{\mu}(s,x)=X^{\mu}(0,x)+sv^{\mu}(x),$

where the fiducial embedding is taken to be $X^{\mu}(0,x)=\left(-x\sinh
w,x\cosh w\right)$, the vector field $v^{\mu}(x)=\left(\cosh w,-\sinh
w\right)$, and $w$ denotes the rapidity of the inertial frame moving at a
relative speed $\beta$ defined as $\cosh
w\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}1/\sqrt{1-\beta^{2}}$.
It then follows that

$n_{\nu}(x)=\epsilon_{\nu\rho}\left.\frac{\partial{X}^{\rho}(x)}{\partial
x}\right|_{\mathsf{X}=\mathsf{X}_{v}(s)}=(\cosh w,\sinh w),$

which is normal to each hypersurface of the foliation. For such a foliation,
Eq. (23) reduces to the Schödinger equation

$i\frac{d}{dt}\ket{\psi_{\phi}[\mathsf{X}_{v}(t)]}=\hat{H}_{{\rm
flat},\beta}\ket{\psi_{\phi}[\mathsf{X}_{v}(t)]},$

where Eq. (24) simplifies to the Hamiltonian

$\hat{H}_{{\rm
flat},\beta}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\int
d\Sigma(x)\,v^{\mu}(x)n_{\nu}(x){\hat{T}^{\nu}}_{\ \mu}(x),$

which generates an evolution in the group parameter $s=t$, interpreted as the
proper time of an observer moving along the timelike congruence defined by the
inertial foliation.

####  IV.1.2 Variation of the conditional hypersurface

Instead of considering the conditional state
$\ket{\psi_{\phi}[\mathsf{X}_{v}(s)]}$ to be a function of $s$, we may instead
consider it to be a functional depending on the coordinate functions
$X^{\mu}(x)=X_{0}^{\mu}(s)+sv^{\mu}(x)$, where $v^{\mu}(x)$ defines a one-
dimensional subgroup $G_{v}\subset\mathrm{Diff}(\mathcal{M})$ characterizing a
set of particular deformations of the fiducial hypersurface $\mathsf{X}_{0}$.
Let us introduce the notation
$\ket{\psi_{\phi}[{X}^{\mu}(x)]}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\ket{\psi_{\phi}[\mathsf{X}_{v}(s)]}$
to emphasize the functional dependence of the conditional state on the
hypersurface $\mathsf{X}_{v}(s)$ through its coordinate functions
${X}^{\mu}(x)$.

Now consider variations of the conditional state under deformations of the
hypersurface $\mathsf{X}_{v}(s)$ by taking the functional derivative with
respect to the coordinate functions $X^{\mu}(x)$, while keeping
$X_{0}^{\mu}(x)$ fixed for $s\in[s_{0},s_{1}]$:

$\displaystyle i\frac{\delta\ket{\psi_{\phi}[{X}^{\mu}(x)]}}{\delta
X^{\mu}(x)}\\!$ $\displaystyle=i\frac{\delta}{\delta
X^{\mu}(x)}\braket{\mathsf{X}_{v}(s)}{\Psi_{\rm phys}}$
$\displaystyle=i\frac{\delta}{\delta
X^{\mu}(x)}\bra{\mathsf{X}_{0}}e^{is\hat{\mathsf{\Pi}}[v]}\ket{\Psi_{\rm
phys}}$ $\displaystyle=i\bra{\mathsf{X}_{0}}\\!\frac{\delta
e^{i\\!\int\\!d\Sigma(x)\,\left(X^{\mu}(x)-X_{0}^{\mu}(x)\right)\hat{\Pi}_{\mu}(x)}}{\delta
X^{\mu}(x)}\\!\ket{\Psi_{\rm phys}}$
$\displaystyle=i\bra{\mathsf{X}_{0}}e^{is\hat{\mathsf{\Pi}}[v]}i\hat{\Pi}_{\mu}(x)\ket{\Psi_{\rm
phys}}$
$\displaystyle=-\bra{\mathsf{X}_{v}(s)}\hat{\Pi}_{\mu}(x)\ket{\Psi_{\rm
phys}}$ $\displaystyle=\bra{\mathsf{X}_{v}(s)}\hat{h}_{\mu}(x)\ket{\Psi_{\rm
phys}}$
$\displaystyle=\hat{h}_{\mu}(x,X^{\mu}(x))\ket{\psi_{\phi}[{X}^{\mu}(x)]}.$
(25)

Let us define the surface variation as the normal projection of the functional
derivative with respect to the coordinate functions Doplicher (2004)

$\frac{\delta}{\delta\mathsf{X}(x)}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}n^{\mu}(x)\frac{\delta}{\delta
X^{\mu}(x)},$

where $n^{\mu}(x)$ is the normal to $\mathsf{X}_{v}(s)$; this derivative
characterizes normal deformations of the conditional wave functional. By
contracting Eq. (25) with $n^{\mu}(x)$, we recover the Tomonaga-Schwinger
equation:

$i\frac{\delta}{\delta\mathsf{X}(x)}\ket{\psi_{\phi}[{X}^{\mu}(x)]}=n^{\mu}(x)\hat{h}_{\mu}(x,X^{\mu}(x))\ket{\psi_{\phi}[{X}^{\mu}(x)]}.$
(26)

The recovery of the Tomonaga-Schwinger equation equation Tomonaga (1946);
Schwinger (1948); Wakita (1976); Breuer and Petruccione (2002) using the Page-
Wootters formalism indicates its formal equivalence with standard functional
formulations of quantum field theory. One should note that the conformal
anomaly potential $\hat{A}_{\mu}(x)$ is embedded in the Hamiltonian flux (for
the $m=0$ case), underlying a redistribution of the energy flux over the
embedding for hypersurfaces with non-vanishing extrinsic curvature. However,
this once again does not affect the integrated version of the Tomonaga-
Schwinger equation, as the anomaly vanishes for asymptotically flat
embeddings.

####  IV.1.3 Schrödinger evolution along a one-parameter family of
hypersurfaces

Consider a one-parameter family
$\mathcal{F}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\\{\mathsf{X}(t),\
\forall t\in[t_{1},t_{2}]\\}$ of spacelike hypersurfaces $\mathsf{X}(t)$ in
Minkowski space (which may or may not constitute a foliation). Each
hypersurface in the family is associated with a conditional state
$\ket{\psi_{\phi}[\mathsf{X}(t)]}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\mathcal{R}_{\mathbf{S}}[\mathsf{X}(t)]\ket{\Psi_{\rm
phys}}$, which evolves along the family according to:

$\displaystyle i\frac{d}{dt}\ket{\psi_{\phi}[\mathsf{X}(t)]}$
$\displaystyle=i\lim_{\varepsilon\to
0}\frac{\ket{\psi_{\phi}[\mathsf{X}(t+\varepsilon)]}-\ket{\psi_{\phi}[\mathsf{X}(t)]}}{\varepsilon}$
$\displaystyle=i\int
d\Sigma(t,x)\,t^{\mu}(t,x)\frac{\delta}{\delta{X}^{\mu}(x)}\ket{\psi_{\phi}[\mathsf{X}(t)]}$
$\displaystyle=\hat{H}(t)\,\ket{\psi_{\phi}[\mathsf{X}(t)]},$ (27)

where we have used similar reasoning that led to Eq. (25) to arrive at the
last equality and defined the Hamiltonian

$\hat{H}(t)\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\int
d\Sigma(t,x)\,t^{\mu}(t,x)\,\hat{h}_{\mu}(x,X^{\mu}(t,x)),$

by projecting the energy-momentum density $\hat{h}_{\mu}(x,X^{\mu}(t,x))$
along the deformation vector field $t^{\mu}$ and integrating over the Cauchy
surface $\Sigma$. In particular, we can decompose the Hamiltonian into normal
and parallel deformation generators:

$\hat{H}(t)=\hat{H}_{\bot}(t)+\hat{H}_{\parallel}(t),$

where

$\displaystyle\hat{H}_{\bot}(t)$
$\displaystyle\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\int
d\Sigma(x)\,N(t)\,n^{\mu}(t,x)\,\hat{h}_{\mu}(x,X^{\mu}(t,x)),$
$\displaystyle\hat{H}_{\parallel}(t)$
$\displaystyle\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\int
d\Sigma(x)\,N^{\mu}(t,x)\,\hat{h}_{\mu}(x,X^{\mu}(t,x)).$

It is seen that Eq. (27) constitutes a generalized Schrödinger equation that
accounts for normal deformations of the hypersurface $\mathsf{X}(t)$ generated
by $\hat{H}_{\perp}(t)$, and spatial redistribution of points generated by
$\hat{H}_{\parallel}(t)$ (i.e., spatial diffeomorphisms of $\mathsf{X}(t)$).
Moreover, the standard Schrödinger equation in Minkowski space is recovered
when restricting to inertial embeddings, such as $T(t,x)=t$ and $X(t,x)=x$. It
then follows that $t^{\mu}(t,x)=n^{\mu}(t,x)=(1,0)$, so that

$i\frac{d}{dt}\ket{\psi_{\phi}[\mathsf{X}(t)]}=\hat{H}_{{\rm
flat},0}\ket{\psi_{\phi}[\mathsf{X}(t)]},$

where $\hat{H}_{{\rm flat},0}=\int d\Sigma(x)\,\hat{T}^{0}{}_{0}(x)$, is the
standard Hamiltonian on a flat embedding and $\hat{T}^{0}{}_{0}(x)$ is the
standard energy density in the employed coordinate frame.

Integrating Eq.(27), we arrive at

$\ket{\psi_{\phi}[\mathsf{X}_{2}]}=\hat{U}_{\mathcal{F}}(t_{2},t_{1})\ket{\psi_{\phi}[\mathsf{X}_{1}]},$
(28)

where

$\hat{U}_{\mathcal{F}}(t_{2},t_{1})\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\mathcal{P}e^{-i\\!\int_{{t_{1}}}^{{t_{2}}}\\!dt\,\hat{H}(t)},$
(29)

is formally a unitary operator that propagates the conditional state of the
matter field from $\mathsf{X}_{1}$ to $\mathsf{X}_{2}$, and $\mathcal{P}$
denotes the path-ordering operator in the foliation parameter $t$. Note that
the Hamiltonian density satisfies the microcausality condition,
$[\hat{h}^{\mu}(t,x),\hat{h}^{\nu}(t,x^{\prime})]=0$ for spacelike separated
points labeled by the coordinates $(t,x)$ and $(t,x^{\prime})$. This provides
the integrability requirement for the integral in Eq. (29) to be well-defined
and independent of the spacetime foliation of the region being integrated over
Tomonaga (1946); Schwinger (1948); Koba (1950); Wakita (1976); Breuer and
Petruccione (2002). If the Hamiltonian generating
$\hat{U}_{\mathcal{F}}(t_{2},t_{1})$ is constructed by smearing the
Hamiltonian flux with a deformation vector field associated with a complete
spacelike foliation, then it is a complete vector field in Minkowski space
$\mathcal{M}$. The deformation vector fields lifts to a complete vector field
on the classical scalar field configuration space $\mathcal{C}_{\phi}$ (see
Sec. III) because the deformation vector field generates a gauge
diffeomorphism, which cannot change the $C^{2}$ nature of a classical field
configuration, nor its asymptotic drop-off property, given that also gauge
diffeomorphisms have to vanish asymptotically. In fact, this should extend to
the quantum configuration space $\mathcal{S}^{\prime}$, where the action of
the Lie derivative on distributions can be understood via partial integration
and the properties of the gauge diffeomorphisms should ensure the preservation
of the properties of the permitted set of distributions. Making this rigorous
is somewhat challenging; however, it suggests that van Hove’s theorem in
quantum mechanics Hove (1951); Gotay  _et al._ (1996); Gotay (1999) could be
extrapolated to field theory, in which case it would imply that the
Hamiltonian is formally self-adjoint.

Figure 3: The composition of the inverse reduction map relative to
$\mathsf{X}_{1}$ and reduction map relative to $\mathsf{X}_{2}$ (upper path),
which maps the conditional state $\ket{\psi_{\phi}[\mathsf{X}_{1}]}$ to
$\ket{\psi_{\phi}[\mathsf{X}_{2}]}$, is shown in Eq. (31) to be equivalent to
integrating the Tomonaga-Schwinger equation along an arbitrary spacelike
foliation of Minkowski space connecting $\mathsf{X}_{1}$ and $\mathsf{X}_{2}$
(lower path). Foliation-independence of the evolution of the conditional state
from $\mathsf{X}_{1}$ and $\mathsf{X}_{2}$ arises in the former as a
consequence of the diffeomorphisim invariance of the physical states from
which the conditional states of the scalar field are constructed, while in the
latter it arises as a consequence of the microcausality condition satisfied by
the stress-energy tensor of the scaler field.

Alternatively, we may evolve $\ket{\psi_{\phi}[\mathsf{X}_{1}]}$ to
$\ket{\psi_{\phi}[\mathsf{X}_{2}]}$ by first applying the inverse of the
reduction map evaluated at $\mathsf{X}_{1}$ and then the reduction map
evaluated at $\mathsf{X}_{2}$, yielding:

$\displaystyle\ket{\psi_{\phi}[\mathsf{X}_{2}]}$
$\displaystyle=\mathcal{R}_{\mathbf{S}}[\mathsf{X}_{2}]\circ\mathcal{R}_{\mathbf{S}}^{-1}[\mathsf{X}_{1}]\ket{\psi_{\phi}[\mathsf{X}_{1}]}$
$\displaystyle=\bra{\mathsf{X}_{2}}\otimes\hat{I}_{\phi}\int\mathcal{D}q\,\ket{\mathsf{X}_{q}}\otimes\hat{U}_{\phi}[\mathsf{X}_{q}-\mathsf{X}_{1}]\ket{\psi[\mathsf{X}_{1}]}$
$\displaystyle=\hat{U}_{\phi}[\mathsf{X}_{2}-\mathsf{X}_{1}]\ket{\psi[\mathsf{X}_{1}]}.$
(30)

Assuming the family of hypersurfaces constitutes a foliation, applying Eq.
(30) between successive leaves, as is done in Appendix B, together with the
fact that
$\mathcal{R}_{\mathbf{S}}^{-1}[\mathsf{X}_{q}]\circ\mathcal{R}_{\mathbf{S}}[\mathsf{X}_{q}]=\hat{I}_{\rm
phys}$, which stems from the gauge invariance of physical states imposed by
Eq. (12), it follows that we must have

$\ket{\psi_{\phi}[\mathsf{X}_{2}]}=\hat{U}_{\mathcal{F}}(t_{2},t_{1})\ket{\psi_{\phi}[\mathsf{X}_{1}]}=\hat{U}_{\phi}[\mathsf{X}_{2}-\mathsf{X}_{1}]\ket{\psi_{\phi}[\mathsf{X}_{1}]}.$
(31)

This shows that the evolution between the two hypersurfaces is formally
unitary and foliation independent since neither the reduction map nor its
inverse depends on $\mathcal{F}$. This is a different way of demonstrating the
foliation-independence of the Tomonaga-Schwinger equation that does not use
the microcausality condition but instead uses the fact that physical states
are invariant under diffeomorphisms, which is a gauge symmetry of PFT. See
Fig. 3 for more detail. This is an instantiation of the fact that foliation-
independence and diffeomorphism invariance are deeply intertwined in canonical
formulations of generally covariant theories, e.g. see Kiefer (2012); Hojman
_et al._ (1976); Kuchar (1976a, b); Thiemann (2008).

###  IV.2 Changes of embedding configuration

Having constructed the conditional states $\ket{\psi_{\phi}[\mathsf{X}_{q}]}$
of the scalar field relative to an embedding $\mathsf{X}_{q}$, we now consider
how the conditional state changes under a change of embedding,
$\mathsf{X}_{q}\to\mathsf{X}_{q^{\prime}}$. Since the embedding fields
constitute a dynamical reference frame, each configuration can be regarded as
a reference frame orientation. The action of a generic change in frame
configuration, $\mathsf{X}_{q}\to\mathsf{X}_{q^{\prime}}$, on the conditional
state $\ket{\psi_{\phi}[\mathsf{X}_{q}]}$ is given in terms of the reduction
map and its inverse, and is seen to induce a unitary map on conditional
states:

$\begin{split}\ket{\psi_{\phi}[\mathsf{X}_{q^{\prime}}]}&=\mathcal{R}_{\mathbf{S}}[\mathsf{X}_{q^{\prime}}]\circ\mathcal{R}_{\mathbf{S}}^{-1}[\mathsf{X}_{q}]\ket{\psi_{\phi}[\mathsf{X}_{q}]}\\\
&=\hat{U}_{\phi}[\mathsf{X}_{q^{\prime}}-\mathsf{X}_{q}]\ket{\psi_{\phi}[\mathsf{X}_{q}]}.\end{split}$
(32)

As depicted in Fig. 3, this map is used to transform conditional states
between two embedding configurations associated with different leaves of a
foliation. However, the map in Eq. (32) is more general as it maps the
conditional state between two embedding configurations that are not
necessarily part of a foliation. For example, consider two embedding
configurations related by a finite Lorentz boost $\Lambda$. The coordinate
functions characterizing these embeddings are related by $X^{\mu}_{\Lambda
q}(x)={\Lambda^{\mu}}_{\nu}X^{\nu}_{q}(x)$. Inserting this in Eq. (32), we
arrive at:

$\begin{split}\ket{\psi_{\phi}[\mathsf{X}_{\Lambda
q}]}&=\hat{U}_{\phi}[\mathsf{X}_{\Lambda
q}-\mathsf{X}_{q}]\ket{\psi_{\phi}[\mathsf{X}_{q}]}\\\ &=e^{-i\int
d\Sigma(x)\,({\Lambda^{\mu}}_{\nu}-{\delta^{\mu}}_{\nu})X_{q}^{\nu}(x)\hat{h}_{\mu}(x)}\ket{\psi_{\phi}[\mathsf{X}_{q}]}.\end{split}$

Furthermore, covariance requires that field observables $\hat{A}_{\phi}(q)$
transform between hypersurfaces $q$ and $\Lambda q$ as

$\hat{A}_{\phi}(\Lambda q)=\hat{U}_{\phi}[\mathsf{X}_{\Lambda
q}-\mathsf{X}_{q}]\hat{A}_{\phi}(q)\hat{U}^{\dagger}_{\phi}[\mathsf{X}_{\Lambda
q}-\mathsf{X}_{q}].$

We may also consider how the conditional state transforms under a
transformation from an inertial to a uniformly accelerating frame Fulton  _et
al._ (1962); Wood  _et al._ (1989). In particular, if
$b^{\mu}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\left(0,\frac{1}{\alpha}\right)$
is a constant vector, the embedding configuration obtained through the finite
special conformal transformation
$X^{\mu}_{q^{\prime}}(x)=\frac{X^{\mu}_{q}(x)-b^{\mu}X^{2}_{q}(x)}{\beta(x)}$
experiences a uniform acceleration $\alpha$ relative to $X_{q}$, where
$\beta(x)\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}2X^{\sigma}(x)b_{\sigma}-X_{q}^{\sigma}(x){X_{q}}_{\sigma}(x)/\alpha^{2}$.
Using Eq. (32), the conditional state of the field transforms as

$\begin{split}\ket{\psi_{\phi}[\mathsf{X}_{q^{\prime}}]}&=\hat{U}_{\phi}[\mathsf{X}_{q^{\prime}}-\mathsf{X}_{q}]\ket{\psi_{\phi}[\mathsf{X}_{q}]}\\\
&=e^{-i\int
d\Sigma(x)\,\left[\frac{\beta(x)X_{q}^{\mu}(x)-b^{\mu}X_{q}^{2}(x)}{1-\beta(x)}\right]\hat{h}_{\mu}(x)}\ket{\psi_{\phi}[\mathsf{X}_{q}]}.\end{split}$

Note that this operator may also a non-trivial domain, as
$X_{q^{\prime}}^{\mu}(x)$ defined via the acceleration above is not in general
guaranteed to be an embedding itself.

###  IV.3 Construction of relational Dirac observables

We may construct gauge-invariant observables directly on the physical Hilbert
space that encode relations between the matter field and embedding fields;
such observables are known as relational Dirac observables Rovelli (2004);
Thiemann (2008); Dittrich (2006, 2007); Goeller  _et al._ (2022). Consider an
observable on the field Hilbert space

$\hat{A}_{0}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\int
d\Sigma_{0}(x)\,A[\hat{\phi}(x),\hat{\pi}(x)]\in\mathcal{B}(\mathcal{H}_{\phi}).$

Using the reduction map and its inverse in Eqs. (18) and (19), we can
construct a relational Dirac observable $\hat{F}_{A_{0}}[\mathsf{X}_{q}]$ on
the physical Hilbert space Höhn  _et al._ (2021a), such that

$\bra{\psi_{\phi}[\mathsf{X}_{q}]}\hat{A}_{0}\ket{\psi_{\phi}[\mathsf{X}_{q}]}=\bra{\Psi_{\rm
phys}}\hat{F}_{A_{0}}[\mathsf{X}_{q}]\ket{\Psi_{\rm phys}},$

where

$\displaystyle\hat{F}_{A_{0}}[\mathsf{X}_{q}]$
$\displaystyle\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\mathcal{R}^{-1}_{\bf
S}[\mathsf{X}_{q}]\hat{A}_{0}\mathcal{R}_{\bf S}[\mathsf{X}_{q}]$ (33)
$\displaystyle=\int\mathcal{D}q^{\prime}\,\hat{U}_{\mathsf{X}}[\mathsf{X}_{q^{\prime}}\\!-\\!\mathsf{X}_{q}]\ket{\mathsf{X}_{q}}\\!\bra{\mathsf{X}_{q}}\otimes\hat{U}_{\phi}[\mathsf{X}_{q^{\prime}}\\!-\\!\mathsf{X}_{q}]\hat{A}_{0}.$

As shown in Appendix A.2, $\hat{F}_{A_{0}}[\mathsf{X}_{q}]$ commutes with the
constraint $\hat{\mathsf{C}}[g]$ on physical states,

$\big{[}\hat{\mathsf{C}}[g],\hat{F}_{A_{0}}[\mathsf{X}_{q}]\big{]}\ket{\Psi_{\rm
phys}}=0,$

and thus constitutes a gauge invariant relational Dirac observable.
$\hat{F}_{A_{0}}[\mathsf{X}_{q}]$ encodes the outcome of the observable
$\hat{A}_{0}$, conditioned on the embedding being in configuration
$\mathsf{X}_{q}$.

###  IV.4 Quantum deparametrization to a relational Heisenberg picture

Conditioning physical states on embedding states yielded a relational
functional Schrödinger picture for the free scalar field that formally
encompasses the standard wave functional formulation in Minkowski space. As a
next step, we apply a quantum deparametrization procedure related to the
method developed in Refs. Höhn and Vanrietvelde (2020); Höhn (2019); Höhn  _et
al._ (2021a, b); de la Hamette  _et al._ (2021) to give a unitarily
equivalent, functional Heisenberg picture for the scalar field. Such a
procedure is modeled on the classical analog of deparameterizing a classical
reparametrization-invariant theory.

Let us consider the following deparametrization map relative to a fiducial
hypersurface $\mathsf{X}_{0}$777The analog in Höhn and Vanrietvelde (2020);
Höhn (2019); Höhn  _et al._ (2021a, b); de la Hamette  _et al._ (2021)
trivializes the constraints to the reference system, thereby effectively
disentangling (relative to the kinematical tensor product structure) the
latter and the degrees of freedom to be described relative to it. In the
present case, the map does not achieve this for arbitrarily smeared
constraints.

$\mathcal{T}_{\mathsf{X}_{0}}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\int\mathcal{D}q\,\ket{\mathsf{X}_{q}}\\!\bra{\mathsf{X}_{q}}\otimes\hat{U}_{\phi}[\mathsf{X}_{0}-\mathsf{X}_{q}],$
(34)

with inverse

$\mathcal{T}^{-1}_{\mathsf{X}_{0}}=\int\mathcal{D}q\,\ket{\mathsf{X}_{q}}\\!\bra{\mathsf{X}_{q}}\otimes\hat{U}_{\phi}^{\dagger}[\mathsf{X}_{0}-\mathsf{X}_{q}],$
(35)

such that
$\mathcal{T}^{-1}_{\mathsf{X}_{0}}\circ\mathcal{T}_{\mathsf{X}_{0}}=\hat{I}_{\rm
phys}$.

We define the Heisenberg reduction map as

$\displaystyle\mathcal{R}_{\mathbf{H}}:\mathcal{H}_{\rm phys}$
$\displaystyle\rightarrow\mathcal{H}_{\mathsf{X}_{q}};$ $\displaystyle\
\ket{\Psi_{\rm phys}}$
$\displaystyle\mapsto\mathcal{R}_{\mathbf{H}}\ket{\Psi_{\rm phys}},$

where

$\mathcal{R}_{\mathbf{H}}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\bra{\mathsf{X}_{q}}\mathcal{T}_{\mathsf{X}_{0}}=\hat{U}_{\phi}[\mathsf{X}_{0}-\mathsf{X}_{q}]\mathcal{R}_{\mathbf{S}}[\mathsf{X}_{q}]=\mathcal{R}_{\mathbf{S}}\left[\mathsf{X}_{0}\right],$
(36)

with inverse

$\displaystyle\mathcal{R}_{\mathbf{H}}^{-1}$
$\displaystyle\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\mathcal{T}^{-1}_{\mathsf{X}_{0}}\\!\int\mathcal{D}q\,\ket{\mathsf{X}_{q}}\otimes\hat{I}_{\phi}=\\!\int\mathcal{D}q\,\ket{\mathsf{X}_{q}}\otimes\hat{U}_{\phi}^{\dagger}[\mathsf{X}_{0}-\mathsf{X}_{q}].$

Given a physical state $\ket{\Psi_{\rm phys}}$, the Heisenberg reduction map
and its inverse can be used to construct a relational Heisenberg picture.
Analogous to the conditional state defined in Eq. (22), we define a
conditional Heisenberg state relative to a fiducial embedding $\mathsf{X}_{0}$
as

$\mathcal{R}_{\mathbf{H}}\ket{\Psi_{\rm
phys}}=\mathcal{R}_{\mathbf{S}}[\mathsf{X}_{0}]\ket{\Psi_{\rm
phys}}=\ket{\psi_{\phi}[\mathsf{X}_{0}]}.$

In addition, the relational Dirac observables defined in Eq. (33) can be
mapped to Heisenberg picture observables (cf., Theorem 5 of Ref. Höhn  _et
al._ (2021a))

$\displaystyle\hat{A}_{\phi}[\mathsf{X}_{q}]$
$\displaystyle\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\mathcal{R}_{\mathbf{H}}\hat{F}_{A_{0}}[\mathsf{X}_{q}]\mathcal{R}_{\mathbf{H}}^{-1}$
$\displaystyle=\hat{U}_{\phi}[\mathsf{X}_{0}-\mathsf{X}_{q}]\hat{A}_{0}\,\hat{U}_{\phi}^{\dagger}[\mathsf{X}_{0}-\mathsf{X}_{q}],$

which satisfies the Heisenberg functional equation of motion

$\frac{\delta}{\delta
X^{\mu}_{q}(x)}\hat{A}_{\phi}[\mathsf{X}_{q}]=i\left[h_{\mu}[\mathsf{X}_{q},x],\hat{A}_{\phi}\right],$
(37)

as we show in Appendix A. Thus, together with the Tomonaga-Schwinger equation
in Eq. (26), the relation between the Heisenberg and Schrödinger reduction
maps in Eq. (36), and the definition of the Dirac observables in Eq. (33), it
is seen that the description of a PFT in terms of the Page-Wootters formalism
developed in Sec. IV.1, the quantum deparameterization procedure introduced
immediately above, and the relational Dirac observable prescription presented
in Sec. IV.3 yield relational formalisms that are unitarily equivalent. Thus,
as in the case of non-interacting mechanical systems with vanishing
Hamiltonian constraints Höhn  _et al._ (2021a, b), there exists a field-
theoretic trinity of relational quantum dynamics for PFT.

##  V Quantum embedding transformations

Analogous to the usual interpretation of the conditional state in the Page-
Wootters formalism, the conditional state $\ket{\psi_{\phi}[\mathsf{X}]}$ is a
functional describing a matter field $\phi$ relative to a hypersurface
associated with the embedding $\mathsf{X}$. Operationally, the embedding
fields serve as abstractions of a reference frame consisting of the rods and
clocks of a congruence of observers.

Given this interpretation, an important task is to examine how the conditional
wave functional $\ket{\psi_{\phi}[\mathsf{X}]}$ transforms under a change of
embedding $\mathsf{X}$, and in doing so, recover classical frame
transformations in a certain approximation. Moreover, the formalism introduced
above treats embeddings as genuine quantum degrees of freedom associated with
a Hilbert space, allowing us to go beyond classical frame transformations and
examine the field from the perspective of a nonclassical embedding. Such a
generalization is in the spirit of Ref. Giacomini  _et al._ (2019) in which
spatial superposition and entanglement were interchangeable under an analogous
quantum frame transformation. Moreover, such quantum frame transformations
provide the foundation on which to build a quantum theory of general
covariance Höhn (2019); Höhn and Vanrietvelde (2020); Vanrietvelde  _et al._
(2020); de la Hamette and Galley (2020); Höhn  _et al._ (2021a, b); de la
Hamette  _et al._ (2021); Vanrietvelde  _et al._ (2023), which is expected to
play an important role in a quantum theory of gravity.

###  V.1 Changes of embedding fields

In the remainder, we shall now assume that there exist two dynamically
distinct embedding fields, either of which we can employ as a reference frame.
For example, one of them might be given by some additional matter fields,
though we shall not specify how this effects the Hamiltonian constraint. In
this way, while being more formal, we are also more general.

Consider then two embedding Hilbert spaces $\mathcal{H}_{A}$ and
$\mathcal{H}_{B}$ with their respective embedding states denoted as
$\ket{\mathsf{X}_{A}}$ and $\ket{\mathsf{Y}_{B}}$. Suppose the conditional
wave functional relative to $A$ is $\ket{\psi_{\phi}[\mathsf{X}_{A}]}$, and
the state of embedding field $B$ is

$\ket{\psi_{B|A}[\mathsf{X}_{A}]}=\int\mathcal{D}q\,\psi_{B}[\mathsf{Y}_{q},\mathsf{X}_{A}]\ket{\mathsf{Y}_{q}},$

so that their joint state is

$\ket{\psi_{B,\phi}[\mathsf{X}_{A}]}=\ket{\psi_{B|A}}\ket{\psi_{\phi}[\mathsf{X}_{A}]}.$
(38)

The change of embedding map $\Lambda_{A\to
B}:\mathcal{H}_{B}\otimes\mathcal{H}_{\phi}\to\mathcal{H}_{A}\otimes\mathcal{H}_{\phi}$
transforms a conditional state relative to embedding $A$ to the conditional
state relative to embedding $B$

$\displaystyle\Lambda_{A\to B}$
$\displaystyle\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\mathcal{R}_{\mathbf{S}}[\mathsf{Y}_{B}]\circ\mathcal{R}_{\mathbf{S}}^{-1}[\mathsf{X}_{A}]$
$\displaystyle=\int\mathcal{D}q\,\ket{\mathsf{X}_{q}}\otimes\bra{\mathsf{Y}_{B}}\hat{U}_{\mathsf{Y}}[\mathsf{X}_{q}-\mathsf{X}_{A}]\otimes\hat{U}_{\phi}[\mathsf{X}_{q}-\mathsf{X}_{A}].$

Transforming the conditional state in Eq. (38), defined as the state of $B$
and $\phi$ relative to $A$, to a conditional state of $A$ and $\phi$ relative
to $B$:

$\displaystyle\ket{\psi_{A,\phi}[\mathsf{Y}_{B}]}$
$\displaystyle=\Lambda_{A\to B}\ket{\psi_{B,\phi}[\mathsf{X}_{A}]}$
$\displaystyle=\int\mathcal{D}q\,\psi_{B|A}\left[\mathsf{Y}_{B}-\mathsf{Y}_{q}\right]\ket{\mathsf{X}_{q}}\ket{\psi_{\phi}[\mathsf{X}_{q}]}.$

It is seen that even though the matter field $\phi$ and embedding fields
associated with $B$ are seperable relative to $A$, from the perspective of
$B$, they are entangled. This is analogous to the frame-dependence of
superposition and entanglement exhibited in Ref. Giacomini  _et al._ (2019);
A. Ahmad  _et al._ (2022).

###  V.2 Particle creation due to reference frame coherence

Having defined the change of embedding map $\Lambda_{A\to B}$, its action can
transform observables between foliations of spacetime associated with distinct
embedding frame fields. In particular, consider two foliations
$\mathcal{F}_{A}=\\{\mathsf{X}_{A}(\tau)\\}$ and
$\mathcal{F}_{B}=\\{\mathsf{X}_{B}(t)\\}$ of $\mathcal{M}$, each associated
with a timelike killing vector field, $\partial_{\tau}$ and $\partial_{t}$,
respectively. Suppose the field is expanded in creation and annihilation
operators along $\mathcal{F}_{A}$ as

$\hat{\phi}(\tau,x)=\int
dk\,\left(u_{k}(\tau,x)\hat{a}_{k}+u_{k}^{*}(\tau,x)]\hat{a}_{k}^{\dagger}\right),$

and along $\mathcal{F}_{B}$ as

$\hat{\phi}(t,x)=\int
dk\,\left(v_{k}(t,x)\hat{b}_{k}+v_{k}^{*}(t,x)\hat{b}_{k}^{\dagger}\right),$

where $u_{k}(\tau,x)$ and $u_{k}(\tau,x)$, together with their complex
conjugate, form two complete sets of orthonormal solutions to the classical
field equation.

The number operator
$\hat{N}_{k,A}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}a_{k}^{\dagger}a_{k}$,
which counts the number of particles relative to $\mathcal{F}_{A}$, can be
expressed relative to $\mathcal{F}_{B}$, as follows:

$\displaystyle\hat{N}_{k,A}$ $\displaystyle\mapsto\Lambda_{A\to
B}\big{(}\hat{I}_{B}\otimes\hat{N}_{k,A}\big{)}\Lambda_{A\to B}^{\dagger}$
$\displaystyle=\int\mathcal{D}q\ket{\mathsf{X}_{q}}\\!\bra{\mathsf{X}_{q}}\otimes\hat{N}_{k,q},$

where
$\hat{N}_{k,q}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\hat{c}_{k,q}^{\dagger}\hat{c}_{k,q}$
is the number operator on the hypersurface $\mathsf{X}_{q}$, and
$\hat{c}_{k,q}$ and $\hat{c}_{k,q}^{\dagger}$ are the associated creation and
annihilation operators with mode functions $w_{i}(\tilde{t},x)$, where
$\tilde{t}$ is a time coordinate along a foliation between $\mathsf{X}_{A}$
and $\mathsf{X}_{q}$. In particular, observe that the transformation maps a
local operator into a non-local one. We note in passing that the transformed
number operator has the same form as the encoding map that appears in the
context of quantum communication Bartlett  _et al._ (2007, 2009); Smith  _et
al._ (2016); Smith (2019), as a quantization of Dirac observables using
covariant POVMs Höhn  _et al._ (2021a); Chataignier (2020, 2021); de la
Hamette  _et al._ (2021), and as a relativization map in the context of Refs.
Loveridge  _et al._ (2018); Carette  _et al._ (2023); J. Głowacki (2023). The
operators $\hat{c}_{k,q}$ can be expressed as a linear combination of the
creation and annihilation operators $\hat{b}_{k}$ and $\hat{b}_{k}^{\dagger}$
via the Bogoliubov transformation

$\hat{c}_{k,q}=\int
dj\left(\alpha_{jk}(q)\hat{b}_{j}+\beta^{*}_{jk}(q)\hat{b}_{j}^{\dagger}\right),$

where
$\alpha_{jk}(q)\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\left(v_{j},w_{k}\right)$
and
$\beta_{jk}(q)\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}-\left(v_{j},w_{k}^{*}\right)$
are Bogoliubov coefficients and $(\,,\,)$ is the usual Klein-Gordon inner
product on $\mathsf{X}_{q}$.

Now suppose that relative to $\mathcal{F}_{B}$ the scalar field is in its
vacuum state $\ket{0_{\phi|B}}$, and the state of embedding $A$ is
$\ket{\psi_{A|B}}$. Then, the expected number of particles relative to $A$ is

$\displaystyle\bra{\psi_{A|B}}\bra{0_{\phi|B}}$
$\displaystyle\hat{N}_{k,A}\ket{\psi_{A|B}}\ket{0_{\phi|B}}$
$\displaystyle=\int\mathcal{D}q\left|\psi_{A|B}\left[\mathsf{X}_{q}\right]\right|^{2}\int
dj\,\left|\beta_{jk}(q)\right|^{2}.$ (39)

Supposing that $|\psi_{A|B}\left[\mathsf{X}_{q}\right]|^{2}$ has support on
$\mathsf{X}_{q}$ that have non-vanishing $\beta_{jk}(q)$, the vacuum state of
$B$ will be seen to have particles relative to $A$. A ‘classical’ Bogoliubov
transformation is recovered when
$|\psi_{A|B}\left[\mathsf{X}_{q}\right]|^{2}\to\delta\left[\mathsf{X}_{q}-\mathsf{X}_{B}\right]$
becomes sharply localized around $\mathsf{X}_{B}$.

A possible interpretation of Eq. (39) is that standard treatments of quantum
field theory miscount the number of field quanta by assuming an infinitely
precise localization of the laboratory frame. In contrast, Eq. (39) weighs the
expected particle number for each embedding by the distribution
$\left|\psi_{A|B}\left[\mathsf{X}_{q}\right]\right|^{2}$. This particle
creation effect is a field-theoretic analogue of the transformation between
superposition and entanglement that occurs in general quantum reference frame
changes discussed in Ref. Giacomini  _et al._ (2019); A. Ahmad  _et al._
(2022), and the interplay between clock state localization and temporally
localized dynamics discussed Castro-Ruiz  _et al._ (2020); Höhn  _et al._
(2021a). We note that related, though qualitatively different, particle
creation effects have been discussed in the context of superpositions of
accelerated observers Barbado  _et al._ (2020) and superpositions of spacetime
structure Foo  _et al._ (2023, 2022); Kabel  _et al._ (2022).

##  VI Discussion and outlook

Beginning with the Dirac quantization of PFT, we have developed a field-
theoretic generalization of the Page-Wootters approach to relational dynamics.
We began by formally constructing the kinematical Hilbert spaces of the scalar
matter field and embedding fields. We then introduced group coherent states of
the embedding fields relative to subgroups of the diffeomorphisms of Minkowski
space, and used them to construct states of the scalar field relative to
hypersurface configurations. Exploiting the group properties of these coherent
states, we showed that the conditional state of the scalar field satisfies the
Tomonaga-Schwinger equation, and thus demonstrated that the field-theoretic
generalization of the Page-Wootters formally coincides with standard
functional formulations of quantum field theory. As depicted in Fig. 3, what
is notable in this framework is the appearance of a dual picture connecting
the foliation-independence of the conditional state’s evolution between two
spacelike hypersurfaces emerging as a consequence of the invertibility of the
Page-Wootters reduction maps, and ultimately the gauge-invariance of physical
states. Alternatively, foliation-independence of the integrated Tomonaga-
Schwinger is usually derived as a consequence of the microcauslity condition
satisfied by the scalar field’s stress-energy tensor.

In addition, we constructed relational Dirac observables corresponding to
field observables relative to configurations of the embedding fields and a
quantum deparameterization of the physical Hilbert space, which led to a
relational Heisenberg picture Höhn and Vanrietvelde (2020); Höhn (2019); Höhn
_et al._ (2021a, b, a, b). Both of these relational frameworks were shown to
be unitarily equivalent to the Page-Wootters formalism, extending the
equivalence established in Refs. Höhn  _et al._ (2021a, b); de la Hamette  _et
al._ (2021). Finally, we introduced a field-theoretic extension of quantum
reference frame changes Giacomini  _et al._ (2019), which can be used to
transform the number operator between different quantum reference frames
characterized by a quantum state of the embedding fields. In doing so, we
showed that the particle content of a field depends on the quantum state of
the embedding it is being described relative to, highlighting a consequence of
treating embeddings as quantum reference frames.

Natural generalizations of the above analysis are possible, though they do not
come without difficulty. It would be natural to extend the Page-Wootters
formalism to higher spacetime dimensions; however, the canonical quantization
of such a PFT leads to difficulties in constructing unitary evolution along
arbitrary foliations Torre and Varadarajan (1998, 1999). Nevertheless,
extensions to higher spacetime dimensions have been developed either through
the use of loop quantization techniques Varadarajan (2007) or through
algebraic methods Torre and Varadarajan (1999), the latter of which implements
evolution between spacelike hypersurfaces via a map that has a similar
compositional structure as the evolution map depicted in Fig. 3.
Alternatively, one could imagine generalizing the framework by replacing the
Minkowski background considered here with a curved spacetime, though the
simple difference of vector fields used to connect points on different
hypersurfaces would have to be suitably generalized (e.g., Fermi-Walker
transport). Instead of modifying the background spacetime, another avenue to
explore could be replacing the ‘ideal’ embedding fields with physical matter
that possesses its own independent dynamics. Such a generalization is directly
relevant to the canonical quantization program given that the canonical
quantization of general relativity with dust is deparamterizatble like PFT
Brown and Kuchař (1995); Tambornino (2012); Goeller  _et al._ (2022); Giesel
_et al._ (2010); Giesel and Thiemann (2010); Husain and Pawlowski (2012);
Rovelli (1991a) (see also Ref. Kaya (2023)). In principle, a Page-Wootters
formulation of geometrodynamics may be possible, even if such a goal is
ambitious. Related, connections with the recently proposed geometric event-
based relativistic quantum mechanics Giovannetti  _et al._ (2023) are expected
since this extension of quantum mechanics is based on a similar conditioning
procedure that was employed in this work.

We further mention that phase space extensions by embedding fields also play a
crucial role in the recent discussions on finite subregions and edge modes in
gravity Donnelly and Freidel (2016); Speranza (2018, 2019); Freidel  _et al._
(2020, 2023); Ciambelli  _et al._ (2022); Carrozza  _et al._ (2022); Kabel
_et al._ (2023). In that context, these embedding fields rather define the
spatial boundaries of the subregion in a relational manner. Again, a
conditional state formulation may also be applicable to such a setting.

Aside from the dynamical model provided by PFT, our analysis relies on
coherent states of the embedding field introduced in Eq. (10), which
correspond to classical configurations characterizing hypersurfaces in
Minkowski space. Group coherent states play an important role in many
applications of quantum theory Perelomov (1986), and often serve to indicate
the orientation of reference frame Busch and P. J. Lahti (1991);
*buschOperationalQuantumPhysics1995; *buschQuantumMeasurement2016; Bartlett
_et al._ (2007); Smith and Ahmadi (2019, 2020); Höhn  _et al._ (2021a, b); de
la Hamette and Galley (2020); de la Hamette  _et al._ (2021); Loveridge  _et
al._ (2018); Carette  _et al._ (2023); J. Głowacki (2023); Chataignier (2020,
2021). However, the literature has focused on coherent states associated with
finite-dimensional Lie groups, whereas the coherent states of the embedding
fields considered above are associated with subgroups of the infinite-
dimensional Lie group of spacetime diffeomorphisms, and further study of their
mathematical structure is of interest Twareque Ali and Goldin (1991).

###### Acknowledgements.

We thank S. Ali Ahmad for comments on an early version of this manuscript. PH
is grateful for the hospitality of the high-energy physics group at EPF
Lausanne, where the final stages of this work were carried out. This work was
supported by funding from Okinawa Institute of Science and Technology Graduate
University and initially by an “It-from-Qubit” Fellowship of the Simons
Foundation (awarded to PH). This project/publication was also made possible
through the support of the ID# 62312 grant from the John Templeton Foundation,
as part of the ‘The Quantum Information Structure of Spacetime’ Project
(QISS). The opinions expressed in this project/publication are those of the
author(s) and do not necessarily reflect the views of the John Templeton
Foundation. ARHS is grateful for financial support through a Saint Anselm
College Summer Research Grant.

##  Appendix A Explicit calculations

In this appendix, we collect explicit calculations leading to some of the
results in the main body. In particular, we describe the connection between
eigenstates of the embedding operator and coherent states, and we prove the
commutation of lifted Dirac observables with the constraint operator.

###  A.1 Generalized eigenstates of the embedding operator as group coherent
states

Here we show that generalized eigenstates $\ket{\mathsf{X}_{v}(s)}$ of the
embedding operator $\hat{X}^{\mu}(x)$ are coherent states associated with one-
dimensional subgroups $G_{v}\subset G$ of the group
$G\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\text{Diff}(\mathcal{M})$
with eigenfunctions corresponding to the coordinate functions of the
associated embedding $\mathsf{X}_{v}(s):\Sigma\to\mathcal{M}$, where
$s\in[s_{0},s_{1}]$.

Consider the coherent state system relative to the group $G_{v}$ introduced in
Eq. (10):

$\left\\{\ket{\mathsf{X}_{v}(s)}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\hat{U}_{v}(s)\ket{\mathsf{X_{0}}}\mbox{
for all }s\in[s_{0},s_{1}]\subset\mathbb{R}\right\\},$

where

$\hat{U}_{v}(s)\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}e^{-is\hat{\mathsf{\Pi}}[v]}=e^{-is\int
d\Sigma(x)\,v^{\mu}(x)\hat{\Pi}_{\mu}(x)}$

is a unitary representation of $G_{v}$. Now consider that

$\displaystyle\hat{\mathsf{X}}[w]\ket{\mathsf{X}_{v}(s)]}$
$\displaystyle=\hat{\mathsf{X}}[w]e^{-is\hat{\mathsf{\Pi}}[v]}\ket{\mathsf{X}_{0}}=\sum_{k=0}^{\infty}\frac{(-is)^{k}}{k!}\hat{\mathsf{X}}[w]\hat{\mathsf{\Pi}}[v]^{k}\ket{\mathsf{X}}.$
(40)

To simplify the sum, observe that

$\displaystyle\left[\hat{\mathsf{X}}[w],\hat{\mathsf{\Pi}}[v]^{2}\right]$
$\displaystyle=\left[\hat{\mathsf{X}}[w],\hat{\mathsf{\Pi}}[v]\right]\hat{\mathsf{\Pi}}[v]+\hat{\mathsf{\Pi}}[v]\left[\hat{\mathsf{X}}[w],\hat{\mathsf{\Pi}}[v]\right]=2i(w,v)\hat{\mathsf{\Pi}}[v].$

Suppose that this holds for the $m$th case, so that

$\displaystyle\left[\hat{\mathsf{X}}[w],\hat{\mathsf{\Pi}}[v]^{m}\right]=mi(w,v)\hat{\mathsf{\Pi}}[v]^{m-1}.$

Then it also holds for the $m+1$th case:

$\displaystyle\left[\hat{\mathsf{X}}[w],\hat{\mathsf{\Pi}}[v]^{m+1}\right]$
$\displaystyle=\left[\hat{\mathsf{X}}[w],\hat{\mathsf{\Pi}}[v]^{m}\right]\hat{\mathsf{\Pi}}[v]+\hat{\mathsf{\Pi}}[v]^{m}\left[\hat{\mathsf{X}}[w],\hat{\mathsf{\Pi}}[v]\right]=(m+1)i(w,v)\hat{\mathsf{\Pi}}[v]^{m}.$

By induction it follows that
$[\hat{\mathsf{X}}[w],\hat{\mathsf{\Pi}}[v]^{k}]=ki(w,v)\hat{\mathsf{\Pi}}[v]^{k-1}$,
which allows for a further simplification of Eq. (40):

$\displaystyle\hat{\mathsf{X}}[w]\ket{\mathsf{X}_{v}(s)}$
$\displaystyle=\sum_{k=0}^{\infty}\frac{(-is)^{k}}{k!}\Big{(}\hat{\mathsf{\Pi}}[v]^{k}\hat{\mathsf{X}}[w]+ki(w,v)\hat{\mathsf{\Pi}}[v]^{k-1}\Big{)}\ket{\mathsf{X}_{0}}$
$\displaystyle=\Big{(}e^{-is\hat{\mathsf{\Pi}}[v]}\hat{\mathsf{X}}[w]+i(w,v)\sum_{k=1}^{\infty}\frac{(-is)^{k}}{(k-1)!}\hat{\mathsf{\Pi}}[v]^{k-1}\Big{)}\ket{\mathsf{X}_{0}}$
$\displaystyle=\Big{(}e^{-is\hat{\mathsf{\Pi}}[v]}{\mathsf{X}}_{0}[w]+i(w,v)\sum_{m=0}^{\infty}\frac{(-is)^{m+1}}{m!}\hat{\mathsf{\Pi}}[v]^{m}\Big{)}\ket{\mathsf{X}_{0}}$
$\displaystyle=\left(\mathsf{X}_{0}[w]e^{-is\hat{\mathsf{\Pi}}[v]}+(w,v)se^{-is\hat{\mathsf{\Pi}}[v]}\right)\ket{\mathsf{X}_{0}}$
$\displaystyle=\big{(}\mathsf{X}_{0}[w]+(w,v)s\big{)}\ket{\mathsf{X}_{g_{s}^{v}}}$
$\displaystyle=\int
d\Sigma(x)\,w_{\mu}(x)\big{(}X_{0}^{\mu}(x)+sv^{\mu}(x)\big{)}\ket{\mathsf{X}_{g_{s}^{v}}}.$

The above equality must hold for all $w_{\mu}(x)$, which implies

$\displaystyle\hat{X}^{\mu}(x)\ket{\mathsf{X}_{v}(s)}$
$\displaystyle=\Big{(}X_{0}^{\mu}(x)+sv^{\mu}(x)\Big{)}\ket{\mathsf{X}_{v}(s)},$

as stated in Eq. (11).

###  A.2 Commutation of $\hat{F}_{A}[\mathsf{X}_{q}]$ with the smeared
constraint operator

We can observe that $\hat{F}_{A}[\mathsf{X}_{q}]$ formally commutes with the
constraint:

$\displaystyle\big{[}\hat{\mathsf{C}}[g],\hat{F}_{A}[\mathsf{X}_{q}]\big{]}\ket{\Psi_{\rm
phys}}$
$\displaystyle=\hat{\mathsf{C}}[g]\hat{F}_{A}[\mathsf{X}_{q}]\ket{\Psi_{\rm
phys}}$
$\displaystyle=\hat{\mathsf{C}}[g]\int\mathcal{D}q\,\hat{U}_{\mathsf{X}}[\mathsf{X}_{\bar{q}}-\mathsf{X}_{q}]\otimes\hat{U}_{\phi}[\mathsf{X}_{\bar{q}}-\mathsf{X}_{q}]\ket{\mathsf{X}_{q}}\\!\bra{\mathsf{X}_{q}}\otimes\hat{A}\ket{\Psi_{\rm
phys}}$
$\displaystyle=\hat{\mathsf{C}}[g]\int\mathcal{D}q\,\hat{U}_{\mathsf{X}}[\mathsf{X}_{\bar{q}}-\mathsf{X}_{q}]\otimes\hat{U}_{\phi}[\mathsf{X}_{\bar{q}}-\mathsf{X}_{q}]\ket{\mathsf{X}_{q}}\otimes\hat{A}\ket{\psi_{\phi}[\mathsf{X}_{q}]}.$

Let us now expand the conditional state in the eigenbasis of $\hat{A}$, so
that
$\ket{\psi_{\phi}[\mathsf{X}_{q}]}=\sum_{k}\psi_{k}[\mathsf{X}_{q}]\ket{a_{k}}$,
where $\hat{A}\ket{a_{k}}=a_{k}\ket{a_{k}}$ and
$\psi_{k}[\mathsf{X}_{q}]\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\braket{a_{k}}{\psi_{\phi}[\mathsf{X}_{q}]}$.
This allows us to express the above commutator as

$\displaystyle\big{[}\hat{\mathsf{C}}[g],\hat{F}_{A}[\mathsf{X}_{q}]\big{]}\ket{\Psi_{\rm
phys}}$
$\displaystyle=\hat{\mathsf{C}}[g]\int\mathcal{D}q\,\hat{U}_{\mathsf{X}}[\mathsf{X}_{\bar{q}}-\mathsf{X}_{q}]\otimes\hat{U}_{\phi}[\mathsf{X}_{\bar{q}}-\mathsf{X}_{q}]\ket{\mathsf{X}_{q}}\otimes\hat{A}\sum_{k}\psi_{k}\ket{a_{k}[\mathsf{X}_{q}]}$
$\displaystyle=\hat{\mathsf{C}}[g]\int\mathcal{D}q\,\hat{U}_{\mathsf{X}}[\mathsf{X}_{\bar{q}}-\mathsf{X}_{q}]\otimes\hat{U}_{\phi}[\mathsf{X}_{\bar{q}}-\mathsf{X}_{q}]\ket{\mathsf{X}_{q}}\otimes\sum_{k}\psi_{k}a_{k}\ket{a_{k}[\mathsf{X}_{q}]}$
$\displaystyle=\sum_{k}\psi_{k}a_{k}\hat{\mathsf{C}}[g]\left(\int\mathcal{D}q\,\hat{U}_{\mathsf{X}}[\mathsf{X}_{\bar{q}}-\mathsf{X}_{q}]\otimes\hat{U}_{\phi}[\mathsf{X}_{\bar{q}}-\mathsf{X}_{q}]\ket{\mathsf{X}_{q}}\otimes
I_{\phi}\right)\ket{a_{k}[\mathsf{X}_{q}]}$
$\displaystyle=\sum_{k}\psi_{k}a_{k}\hat{\mathsf{C}}[g]\mathcal{R}^{-1}_{\bf
S}[\mathsf{X}_{q}]\ket{a_{k}[\mathsf{X}_{q}]}$
$\displaystyle=\sum_{k}\psi_{k}a_{k}\hat{\mathsf{C}}[g]\ket{\bar{\Psi}_{{\rm
phys},k}}$ $\displaystyle=0,$

where going from the third to fourth equality we have made use of the
definition of the inverse of the reduction map given in Eq. (19).

###  A.3 Heisenberg equation from deparametrization

Starting with a Heisenberg picture of the form

$\hat{A}_{\phi}(q)\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\mathcal{R}_{\mathbf{H}}\hat{F}_{A_{0}}[\mathsf{X}_{q}]\mathcal{R}_{\mathbf{H}}^{-1}=\hat{U}_{\phi}[\mathsf{X}_{0}-\mathsf{X}_{q}]\hat{A}_{0}\,\hat{U}^{\dagger}_{\phi}[\mathsf{X}_{0}-\mathsf{X}_{q}],$

where, as defined in the text:

$\hat{U}_{\phi}[\mathsf{X}_{0}-\mathsf{X}_{q}]\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}e^{-i\\!\int\\!d\Sigma(x)\left(X_{0}^{\mu}(x)-X_{q}^{\mu}(x)\right)\hat{h}_{\mu}[X_{q}^{\mu}(x)]}.$

Taking the functional derivative of the Heisenberg-picture operator:

$\displaystyle\int
d\Sigma(y)\frac{\delta\hat{A}_{\phi}[\mathsf{X}_{q}]}{\delta X^{\mu}_{q}(y)}$
$\displaystyle=\int d\Sigma(y)\frac{\delta}{\delta
X^{\mu}_{q}(y)}\left(e^{-i\\!\int\\!d\Sigma(x)\left(X_{0}^{\nu}(x)-X_{q}^{\nu}(x)\right)\hat{h}_{\nu}[\mathsf{X}_{q},x]}\hat{A}_{0}e^{i\\!\int\\!d\Sigma(x)\left(X_{0}^{\nu}(x)-X_{q}^{\nu}(x)\right)\hat{h}_{\nu}[\mathsf{X}_{q},x]}\right)$
$\displaystyle=\int
d\Sigma(x)d\Sigma(y)\left[i\delta(x-y)\hat{h}_{\mu}[\mathsf{X}_{q},x]-i\left(X_{0}^{\nu}(x)-X_{q}^{\nu}(x)\right)\frac{\delta\hat{h}_{\nu}[\mathsf{X}_{q},x]}{\delta
X^{\mu}_{q}(y)}\right]\hat{A}_{\phi}[\mathsf{X}_{q}]$ $\displaystyle\quad+\int
d\Sigma(x)d\Sigma(y)\hat{A}_{\phi}[\mathsf{X}_{q}]\left[-i\delta(x-y)\hat{h}_{\mu}[X_{q},x]+i\left(X_{0}^{\nu}(x)-X_{q}^{\nu}(x)\right)\frac{\delta\hat{h}_{\nu}[\mathsf{X}_{q},x]}{\delta
X^{\mu}_{q}(y)}\right]$
$\displaystyle=i\left[\hat{H}_{\mu}[\mathsf{X}_{q}],\hat{A}_{\phi}[\mathsf{X}_{q}]\right]-i\int
d\Sigma(x)d\Sigma(y)\left(X_{0}^{\nu}(x)-X_{q}^{\nu}(x)\right)\left[\frac{\delta\hat{h}_{\nu}[\mathsf{X}_{q},x]}{\delta
X^{\mu}(y)},\hat{A}_{\phi}[\mathsf{X}_{q}]\right]$
$\displaystyle=i\left[\hat{H}_{\mu}[\mathsf{X}_{q}],\hat{A}_{\phi}[\mathsf{X}_{q}]\right]-i\int
d\Sigma(x)d\Sigma(y)\left(X_{0}^{\nu}(x)-X_{q}^{\nu}(x)\right)\left[\frac{\delta
n_{\rho}(x)}{\delta
X^{\mu}(y)}{T^{\rho}}_{\nu},\hat{A}_{\phi}[\mathsf{X}_{q}]\right]$
$\displaystyle=i\left[\hat{H}_{\mu}[\mathsf{X}_{q}],\hat{A}_{\phi}[\mathsf{X}_{q}]\right]-i\int
d\Sigma(x)d\Sigma(y)\left(X_{0}^{\nu}(x)-X_{q}^{\nu}(x)\right)\left[\epsilon_{\rho\sigma}\left(\partial_{x}\frac{\delta
X^{\sigma}(x)}{\delta
X^{\mu}(y)}\right){T^{\rho}}_{\nu},\hat{A}_{\phi}[\mathsf{X}_{q}]\right]$
$\displaystyle=i\left[\hat{H}_{\mu}[\mathsf{X}_{q}],\hat{A}_{\phi}[\mathsf{X}_{q}]\right]-i\int
d\Sigma(x)d\Sigma(y)\left(X_{0}^{\nu}(x)-X_{q}^{\nu}(x)\right)\left[\epsilon_{\rho\mu}\left(\partial_{x}\delta(x-y)\right){{\hat{T}}^{\rho}}_{\
\nu}(x),\hat{A}_{\phi}[\mathsf{X}_{q}]\right]$
$\displaystyle=i\left[\hat{H}_{\mu}[\mathsf{X}_{q}],\hat{A}_{\phi}[\mathsf{X}_{q}]\right]-i\int
dxd\Sigma(y)\delta(x-y)\left[\epsilon_{\rho\mu}\partial_{x}\left(\sqrt{\gamma(x)}\left(X_{q}^{\nu}(x)-X_{0}^{\nu}(x)\right){{\hat{T}}^{\rho}}_{\
\nu}(x)\right),\hat{A}_{\phi}[\mathsf{X}_{q}]\right]$
$\displaystyle=i\left[\hat{H}_{\mu}[\mathsf{X}_{q}],\hat{A}_{\phi}[\mathsf{X}_{q}]\right]-i\left[\int
dx\,\epsilon_{\rho\mu}\partial_{x}\left(\sqrt{\gamma(x)}\left(X_{q}^{\nu}(x)-X_{0}^{\nu}(x)\right){{\hat{T}}^{\rho}}_{\
\nu}(x)\right),\hat{A}_{\phi}[\mathsf{X}_{q}]\right]$
$\displaystyle=i\left[\hat{H}_{\mu}[\mathsf{X}_{q}],\hat{A}_{\phi}[\mathsf{X}_{q}]\right],$

where the integral appearing in the penultimate equality vanishes given that
it is equal to a boundary term that vanishes under the assumption that the
matter field stress-energy tensor vanishes at infinity. We thus recover the
Heisenberg equation as stated in Eq. (37).

##  Appendix B Foliation independence through equivalence of evolutions of the
conditioned state

We give here the derivation of Eq. (31), which illustrates the foliation
independence of the evolution of the field degrees of freedom from an initial
to a final hypersurface. The unitary operator evolves the conditioned state
degrees of freedom through the integrated Schrödinger equation. It is also
equivalent to mapping back the conditioned state to the perspective-neutral
physical Hilbert space and then projecting back to the final embedding through
a composition of the Schrödinger reduction maps. We start with the general
map:

$\hat{U}_{\phi}[\mathsf{X}_{q^{\prime}}-\mathsf{X}_{q}]\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}e^{-i\int
d\Sigma(x)\,\left(X^{\mu}_{q^{\prime}}(x)-X^{\mu}_{q}(x)\right)h_{\mu}(x)}.$

We will now show that given a foliation that connects two embeddings, mapping
from the initial embedding to the perspective-neutral space and back into the
final embedding is equivalent to projecting the field on the initial embedding
and then evolving it through an arbitrary foliation using the Hamiltonian
flux. This is only possible due to the fact that physical states are zero
eigenvalue states of the constraint and are hence gauge-invariant.

Consider a discrete one-parameter family
$\mathcal{F}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\\{\mathsf{X}(t),\forall
t\in[t_{i},t_{f}]\\}$ consisting of $n+1$ spacelike hypersurfaces
$\mathsf{X}(t)$ “foliating” Minkowski space such that
$\mathsf{X}(t_{i})=\mathsf{X}_{q}$ and
$\mathsf{X}(t_{f})=\mathsf{X}_{q^{\prime}}$. This foliation is highly non-
unique and each surface is separated from the previous and successive one by a
small finite interval $\Delta t$ of the time coordinate $t$ such that:

$\Delta t=\frac{t_{f}-t_{i}}{n+1}$

and $t_{k+1}=t_{k}+\Delta t$. Let us now consider the unitary evolution
operator of the conditioned state between the $k$-th and the $(k+1)$-th
surfaces of the foliation:

$U_{\phi}[\mathsf{X}(t_{k}+\Delta t)-\mathsf{X}(t_{k})]=e^{-i\int
d\Sigma(t_{k},x)\,\Delta t\left(\frac{X^{\mu}(t_{k}+\Delta
t,x)-X^{\mu}(t_{k},x)}{\Delta t}\right)h_{\mu}(t_{k},x)}.$

The evolution between the initial and final leaves of the foliation is now
obtained by applying the unitary evolution between two adjacent slices $n$
amount of times:

$\displaystyle\hat{U}_{\phi}[\mathsf{X}(t_{f})-\mathsf{X}(t_{i})]$
$\displaystyle=\mathcal{P}\prod_{k=1}^{n}\hat{U}_{\phi}[\mathsf{X}(t_{k}+\Delta
t)-\mathsf{X}(t_{k})]$
$\displaystyle=\mathcal{P}\prod_{k=1}^{n}\exp\left[-i\int
d\Sigma(t_{k},x)\,\Delta
t\left(\frac{X^{\mu}(t_{k+1},x)-X^{\mu}(t_{k},x)}{\Delta
t}\right)\hat{h}_{\mu}(t_{k},x)\right]$
$\displaystyle=\mathcal{P}\exp\left[-i\sum_{k=1}^{n}\Delta t\int
d\Sigma(t_{k},x)\,\left(\frac{X^{\mu}(t_{k+1},x)-X^{\mu}(t_{k},x)}{\Delta
t}\right)\hat{h}_{\mu}(t_{k},x)\right].$

The path-ordering operator ensures that the unitary evolution acts in the
correct order on the leaves of the foliation, and allows us not to worry about
whether the Hamiltonian flux commutes at different points of the foliation. We
now take the limit for an infinite number of infinitesimally close slices, we
recover Eq. (29):

$\displaystyle\hat{U}(t_{f},t_{i})$
$\displaystyle=\lim_{\begin{subarray}{c}\Delta t\to 0\\\
n\rightarrow\infty\end{subarray}}\hat{U}_{\phi}[\mathsf{X}(t_{f})-\mathsf{X}(t_{i})]$
$\displaystyle=\lim_{\begin{subarray}{c}\Delta t\to 0\\\
n\rightarrow\infty\end{subarray}}\mathcal{P}\exp\left[-i\sum_{k=1}^{n}\Delta
t\int
d\Sigma(t_{k},x)\,\left(\frac{X^{\mu}(t_{k+1},x)-X^{\mu}(t_{k},x)}{\Delta
t}\right)\hat{h}_{\mu}(t_{k},x)\right]$
$\displaystyle=\mathcal{P}\exp\left(-i\int_{\mathsf{X}(t_{i})}^{\mathsf{X}(t_{f})}dtd\Sigma(t,x)\,t^{\mu}(t,x)\hat{h}_{\mu}(t,x)\right),$
(41)

where
$t^{\mu}(t,x)\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\partial_{t}X^{\mu}=n^{\mu}(t,x)N(t,x)+N^{\mu}(t,x)$
is the deformation vector field.

Alternatively, before taking the continuum limit, the recovery map and its
inverse, defined in Eqs. (18) and (19), can be used to map between successive
leaves:

$\displaystyle\hat{U}_{\phi}[\mathsf{X}(t_{f})-\mathsf{X}(t_{i})]$
$\displaystyle=\mathcal{P}\prod_{k=1}^{n}\hat{U}_{\phi}[\mathsf{X}(t_{k}+\Delta
t)-\mathsf{X}(t_{k})]$
$\displaystyle=\mathcal{P}\prod_{k=1}^{n}\mathcal{R}_{\mathbf{S}}[\mathsf{X}(t_{k+1})]\circ\mathcal{R}_{\mathbf{S}}^{-1}[\mathsf{X}(t_{k})]$
$\displaystyle=\mathcal{R}_{\mathbf{S}}[\mathsf{X}(t_{f})]\left(\mathcal{P}\prod_{k=1}^{n-1}\mathcal{R}_{\mathbf{S}}^{-1}[\mathsf{X}(t_{k+1})]\circ\mathcal{R}_{\mathbf{S}}[\mathsf{X}(t_{k})]\right)\mathcal{R}_{\mathbf{S}}^{-1}[\mathsf{X}(t_{i})]$
$\displaystyle=\mathcal{R}_{\mathbf{S}}[\mathsf{X}(t_{f})]\circ\mathcal{R}_{\mathbf{S}}^{-1}[\mathsf{X}(t_{i})],$

where in moving from the third line, we have used the fact that
$\mathcal{R}_{\mathbf{S}}^{-1}[\mathsf{X}(t_{k+1})]\circ\mathcal{R}_{\mathbf{S}}[\mathsf{X}(t_{k})]=\hat{I}_{\phi}$.
The final expression depends only on the initial and final hypersurfaces, not
discrete foliation. Thus the limit $\Delta t\to 0,\,n\to\infty$ can be taken
with the result that

$\hat{U}(t_{f},t_{i})=\mathcal{R}_{\mathbf{S}}[\mathsf{X}(t_{f})]\circ\mathcal{R}_{\mathbf{S}}^{-1}[\mathsf{X}(t_{i})].$
(42)

Given that Eqs. (41) and (42) must be equal, it follows that the evolution
between $\mathsf{X}(t_{i})$ and $\mathsf{X}(t_{f})$ is independent of the
choice of foliation, despite the foliation dependence of the integrand in Eq.
(41).

##  Appendix C Discussion of the anomaly

In this paper, we have considered the quantization of a scalar field on a two-
dimensional spacetime. In the massless case, this constitutes a CFT. Hence, we
need to address the conformal anomaly arising in the process. Since the
conformal group is a subgroup of the spacelike preserving diffeomorphisms that
evolve the embeddings, it carries the Virasoro algebra into the Hamiltonian
flux when the constraint $C_{\mu}(x)$ is smeared by a conformal vector.
Following Kuchař Kuchař (1989b), we see how the anomaly arising from the
commutation relation of the Hamiltonian flux can be expressed in terms of
derivatives of an anomaly potential $A_{\mu}$. This potential needs to be
incorporated with the Hamiltonian flux, such that the right commutation
relations will be ensured. Following Kuchař’s notation, the anomaly potential
can be spilt into two parts:

$A_{\mu}(x)={{}_{I}}A_{\mu}(x)+{{}_{III}}A_{\mu}(x).$

The first term is related to a soft breaking of the conformal symmetry by the
introduction of a macroscopic length scale. For example, in Kuchař, it is
given for the choice of manifold being $\mathbb{R}\times S^{1}$. If we assume
the stress-energy tensor of the field to be normally ordered such that its
vacuum expectation value on the plane vanishes, we can interpret the first
part of the anomaly potential as a Casimir energy given by the periodic
boundary condition Kuchař (1989b). It is embedding-independent and vanishes
for the unbounded $\mathbb{R}^{2}$ manifold considered in this paper.

${{}_{I}}A_{\mu}(x)=0$

The second part of the anomaly is instead embedding-dependent. It relates the
distribution of the energy along the embedding to the trace of its extrinsic
curvature $\mathcal{K}(x)={\mathcal{K}^{\mu}}_{\mu}(x)$:

${{}_{III}}A_{\mu}(x)=\sqrt{\gamma(x)}\left(X_{\mu}(x)\frac{\partial\mathcal{K}(x)}{\partial
x}-\mathcal{K}(x)^{2}n_{\mu}(x)\right).$

As explained in Sec. II, this vanishes for embeddings with zero extrinsic
curvature and when integrated over asymptotically flat embeddings. Given a
spacetime embedding $X^{\mu}(t,x)=(T(t,x),X(t,x))$. We can define the oriented
Minkowski basis vectors as $t_{\mu}=-\partial_{\mu}T$,
$s_{\mu}=\partial_{\mu}X$ such that $t_{\mu}t^{\mu}=-1$, $s_{\mu}s^{\mu}=1$
and $t^{\mu}s_{\mu}=0$. Following Kuchař (1989b), we can rewrite this part of
the anomaly such that when integrated

$\displaystyle\int d\Sigma(x)\,{{}_{III}}A_{\mu}(x)t^{\mu}[X^{\mu}(x)]=-\int
d\Sigma(x)\,\partial_{x}(\sqrt{\gamma}\mathcal{K}(x)t^{0})=0,$
$\displaystyle\int d\Sigma(x)\,{{}_{III}}A_{\mu}(x)s^{\mu}[X^{\mu}(x)]=-\int
d\Sigma(x)\,\partial_{x}(\sqrt{\gamma}\mathcal{K}(x)s^{1})=0,$

and we see that the anomaly vanishes as the extrinsic curvature vanishes at
infinity.

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