# On inference of quantization from gravitationally induced entanglement

Vasileios Fragkos∗<EMAIL_ADDRESS>Department of Physics,
Stockholm University, SE-106 91 Stockholm, Sweden  Michael Kopp∗
<EMAIL_ADDRESS>Department of Physics, Stockholm University, SE-106 91
Stockholm, Sweden  Nordita, KTH Royal Institute of Technology and Stockholm
University, Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden  Igor Pikovski
<EMAIL_ADDRESS>Department of Physics, Stockholm University, SE-106
91 Stockholm, Sweden  Department of Physics, Stevens Institute of Technology,
Hoboken, NJ 07030, USA

###### Abstract

Observable signatures of the quantum nature of gravity at low energies have
recently emerged as a promising new research field. One prominent avenue is to
test for gravitationally induced entanglement between two mesoscopic masses
prepared in spatial superposition. Here we analyze such proposals and what one
can infer from them about the quantum nature of gravity, as well as the
electromagnetic analogues of such tests. We show that it is not possible to
draw conclusions about mediators: even within relativistic physics,
entanglement generation can equally be described in terms of mediators or in
terms of non-local processes – relativity does not dictate a local channel.
Such indirect tests therefore have limited ability to probe the nature of the
process establishing the entanglement as their interpretation is inherently
ambiguous. We also show that cosmological observations already demonstrate
some aspects of quantization that these proposals aim to test. Nevertheless,
the proposed experiments would probe how gravity is sourced by spatial
superpositions of matter, an untested new regime of quantum physics.

##  I Introduction

Quantum gravity remains one of the main challenges of modern physics. Many
promising theoretical avenues are being pursued Green  _et al._ (2012);
Rovelli (2004) but to date no complete theory of quantum gravity is known. An
alternative route has been proposed by Roger Penrose Penrose (1996, 1998) and
others Diósi (1987, 1989); Bassi  _et al._ (2013) in which quantum mechanical
unitarity breaks down at sufficiently large mass scales. Such proposals have
ushered in the exciting new field of experimental searches for quantum gravity
at low energies, as they provide distinct signatures that can be observed if
sufficiently large systems can be controlled and prepared in quantum
superpositions.Bose  _et al._ (1999); Marshall  _et al._ (2003); Kleckner  _et
al._ (2008); Carney  _et al._ (2019) These proposals and the rapid progress in
quantum control of larger and larger systems Fein  _et al._ (2019); Delić  _et
al._ (2020) has sparked an interest in experimental signatures of the
interface between quantum mechanics and gravity at low energies both in terms
of testing speculative new models Penrose (1996, 1998); Diósi (1987, 1989);
Bassi  _et al._ (2013); Pikovski  _et al._ (2012); Bekenstein (2012); Kafri
_et al._ (2014); Belenchia  _et al._ (2016) and expected new signatures from
gravitational and quantum physics as we know them. Zych  _et al._ (2011);
Pikovski  _et al._ (2015); Margalit  _et al._ (2015); Brodutch  _et al._
(2015) To date, no experiment exists that either proves any of the new physics
or the quantum nature of gravitational degrees of freedom, and it also seems
that Gedankenexperiments in support of quantization are inconclusive. Dyson
(2013); Baym and Ozawa (2009); Belenchia  _et al._ (2018); Rydving  _et al._
(2021) Thus, any experimental signature that would suggest the quantum nature
of gravity would be of extreme interest and shed light on uncharted territory.

One promising experimental avenue that aims to infer the quantum nature of
gravity is based on creating gravitationally induced entanglement between two
source massesBose  _et al._ (2017); Marletto and Vedral (2017)
(GIE).d)d)d)Some manuscripts refer to such setups as BMV, referring to some of
the authors of the proposals. In this variation of the well-known Colella-
Overhauser-Werner setup,Colella  _et al._ (1975) two matter-wave
interferometers are placed very close to each other (see Figure 1). The two
interferometers create the superposition states
$(\ket{\Psi_{L}}+\ket{\Psi_{R}})/\sqrt{2}$ and
$(\ket{\chi_{L}}+\ket{\chi_{R}})/\sqrt{2}$, respectively, each of size $\Delta
x$ and their centers a distance $d$ apart. If the matter waves are
sufficiently heavy they will source observable gravitational effects
themselves, and the Newtonian interaction will generate entanglement between
the two systems. After time $t$ the state becomes
$(\ket{\Psi_{L}}\ket{\chi_{L}}+e^{i\varphi_{+}}\ket{\Psi_{L}}\ket{\chi_{R}}+e^{i\varphi_{-}}\ket{\Psi_{R}}\ket{\chi_{L}}+\ket{\Psi_{R}}\ket{\chi_{R}})/2$
. The two relative phases are induced by the Newtonian interaction and are
given by $\varphi_{\pm}=\frac{Gm_{1}m_{2}t}{\hbar}\left(\frac{1}{d\pm\Delta
x}-\frac{1}{d}\right)$. In this idealized case (neglecting noise) the state is
entangled for any phases $\varphi_{+}+\varphi_{-}\neq 2\pi n$,
$n\in\mathbb{Z}$, and maximally entangled for
$\varphi_{+}+\varphi_{-}=(2n+1)\pi$, which can be measured through an
entanglement witness.Bose  _et al._ (2017); Chevalier  _et al._ (2020); Tilly
_et al._ (2021); Guff  _et al._ (2022) The authors of Refs.Bose  _et al._
(2017); Marletto and Vedral (2017) argue that this entanglement generation is
an indirect witness of the quantization of gravity. The essence of the
argument is that the interaction is not instantaneous as suggested by the
Newtonian description, but mediated by some local degrees of freedom, and
since entanglement cannot be generated by local operations and classical
communication (LOCC)Peres and Wootters (1991); Chitambar  _et al._ (2014),
these degrees of freedom have to be quantized. Since the original proposals,
several variations of this concept have been put forward. Margalit  _et al._
(2021); van de Kamp  _et al._ (2020); Marshman  _et al._ (2020); Pedernales
_et al._ (2022); Marletto and Vedral (2020); Kent and Pitalúa-García (2021);
Taylor  _et al._ (2021); Ma  _et al._ (2022); Hosten (2022); Streltsov  _et
al._ (2022) However, the very question if such proposals can test the quantum
nature of gravity has remained open for debate. Some have argued against it
Anastopoulos and Hu (2018); Hall and Reginatto (2018, 2021) while others put
arguments forward in favor. Marletto and Vedral (2018); Carney (2022);
Belenchia  _et al._ (2018); Marshman  _et al._ (2020); Christodoulou and
Rovelli (2019); Christodoulou  _et al._ (2022); Danielson  _et al._ (2022) The
main argument in favor of inferring the quantum nature of the mediating
gravitational field stems from the quantum information perspective that limits
entanglement generation from LOCC and the assumption that the interaction is
induced by local mediators. In contrast, the major argument against the
inference of quantum gravity is the use of the Newtonian limit in which the
interaction is mediated non-locally.

Figure 1: Schematic of an experimental setup to test for gravitationally
induced entanglement. Two system are brought into a spatial superposition,
denoted by $\ket{\Psi}$ and $\ket{\chi}$, respectively. Due to their mutual,
position-dependent interaction, here the Newtonian gravitational potential
$\hat{\Phi}=-Gm_{1}m_{2}/|\hat{x}_{1}-\hat{x}_{2}|$, the two systems become
entangled. The authors of Refs. Bose  _et al._ (2017); Marletto and Vedral
(2017) proposed such a setup to prove quantization of gravity, based on the
argument that local, classical operations (LOCC) cannot generate entanglement.
This conclusion, however, depends on what prior assumptions one is willing to
take for granted.

Here we clarify what conclusions one can draw from the experiment by
critically exploring the assumptions that go into it. By definition, if
entanglement is observed in the experiment, then this demonstrates a non-LOCC
type channel. This alone is of course of great interest. Nevertheless, the
open question around these proposals – and what we address in this work – is
if one can learn more about the nature of the underlying system that mediates
entanglement, namely of the gravitational field. As we will show below, the
question remains ambiguous and depends on what prior assumptions one takes for
granted – in some cases even what metaphysical view one prefers. We show that
while inferring the existence of gravitons is reasonable if one assumes the
validity of relativistic field theory, this interpretation is not unique. It
relies on the assumption of local mediators for entanglement, which in
relativity are not the physical gravitons for this case, but particles that
are otherwise unmeasurable. Moreover, models with inherent non-locality can
reproduce the entanglement without the existence of any mediators.
Importantly, while the term locality is often interchangeably used with the
relativistic notion of Lorentz invariance, it needn’t be: even within
relativistic physics as we know it, entanglement generation can be described
non-locally by eliminating redundant gauge degrees of freedom. And more
generally, a fully non-local interpretation exists – the Wheeler-Feynman
absorber formulation.Tetrode (1922); Lewis (1926); Fokker (1929); Wheeler and
Feynman (1945, 1949) When applied to quantized charges Hoyle and Narlikar
(1969, 1971); Davies (1971, 1972); Pegg (1975, 1979); Durrant (1980); Hoyle
and Narlikar (1995) it reproduces all physical predictions of Quantum
Electrodynamics (QED) including entanglement, but the theory does not contain
quantized, local degrees of freedom that create the entanglement. If such an
absorber theory can be formulated for gravity, it would explain
gravitationally induced entanglement without introducing gravitons, while
still being consistent with classical general relativity. With a viable non-
local explanation for the generation of entanglement, its observation alone is
therefore insufficient to infer the existence of quantized mediators. However,
in contrast to the electromagnetic case, it remains an open question if an
absorber-type formulation exists for full general relativity although it
exists in the relativistic weak field limit. Regardless of the question how
entanglement is mediated, the experiment involves superpositions of source
masses. As such, observation of entanglement thus supports the view that
gravitational fields are sourced coherently by a superposition of sources, if
quantum physics is not modified. This is independent of the question of what
mediates entanglement and does not rely on the LOCC argument. The experiment,
if successful, would prove this superposition of Newtonian fields, without the
above mentioned ambiguities, as also highlighted in Refs. Christodoulou and
Rovelli (2019); Christodoulou  _et al._ (2022). Nonetheless, we argue that it
can already be inferred from cosmological observations that quantum states
coherently source Newtonian fields.

In this manuscript we explore systematically what conclusions one can draw
from a GIE experiment, depending on what prior assumptions one takes for
granted. Our work is structured as follows: In Section II we discuss the most
conservative case, the Newtonian limit without any additional assumptions. In
Section III we focus on QED as an analogous case, and show that the
conclusions are ambiguous and depend on (equivalent) formulations of the field
theory. Section IV discusses the case of relativistic weak-field quantum
gravity relevant to the GIE experiment. Section V revisits all previously
discussed formulations from the perspective of path integral quantisation,
providing a unified picture. We focus in section VI on cosmology and show that
current observations already give evidence of quantum weak-field gravity as in
the GIE proposal. Our discussion in Section VII gives a brief comparison to
other works and provides an overview of the different interpretations and
conclusions that one can draw from GIE.

##  II Non-relativistic case

We first focus on the physics in the non-relativistic limit, which is entirely
sufficient to explain the setup. At this level, there is no assumption of any
underlying field theory or additional ontology. It is the minimal
interpretation, as no additional assumptions about the underlying physics
beyond what is experimentally measured have to be imposed.

In the non-relativistic limit there is a direct analogy between the Newtonian
gravitational interaction and the electrostatic Coulomb law. To clarify the
relevant physics, we briefly review how these enter the quantum theory of non-
relativistic particles. This will highlight the non-locality of the non-
relativistic theory, and to what extent one can say that the Coulomb and
Newtonian potentials “are quantised”.

The non-relativistic quantum mechanics of $N=2$ interacting charged particles
of charge $q$ and mass $m$ is given by the Hamiltonian

$\displaystyle\hat{H}$
$\displaystyle=\sum_{i=1}^{N}\left(\frac{(\hat{\vec{p}}_{i})^{2}}{2m}+\frac{q^{2}}{8\pi\epsilon_{0}}\sum_{j=1,j\neq
i}^{N}\frac{1}{|\hat{\vec{x}}_{i}-\hat{\vec{x}}_{j}|}\right)\,.$ (1)

For opposite charges this comprises the simplest (“textbook”) model of the
hydrogen atom. The analogous Hamiltonian of gravitationally interacting
particles is given by

$\displaystyle\hat{H}=\sum_{i=1}^{N}\left(\frac{(\hat{\vec{p}}_{i})^{2}}{2m}-\frac{m^{2}G}{2}\sum_{j=1,j\neq
i}^{N}\frac{1}{|\hat{\vec{x}}_{i}-\hat{\vec{x}}_{j}|}\right)\,.$ (2)

The latter Hamiltonian is sufficient to fully account for the proposed GIE
experiments and is used to derive all the results in the proposals.Bose  _et
al._ (2017); Marletto and Vedral (2017) Since (2) does not contain any
quantized (or unquantized for that matter) gravitational degrees of freedom,
this theory of gravitionally interacting particles has no Hilbert space
associated with gravity.

It is instructive to reformulate the quantum mechanical $N$-body problem as a
quantum field theory. This is sometimes known as “second quantised” theory,
but it is fully equivalent to the conventional formulation in terms of
particle position operators. The quantum field (operator) in the Schrödinger
picture $\hat{\psi}(\vec{x})$ has to property
$\hat{\psi}^{\dagger}(\vec{x}_{1})|0\rangle=|\vec{x}_{1}\rangle$, that is, it
creates a particle at the position $\vec{x}_{1}$. A general $N$-particle state
is constructed by acting with $\int
d^{3}x_{1}...d^{3}x_{N}\Psi(\vec{x}_{1},...,\vec{x}_{N})\hat{\psi}^{\dagger}(\vec{x}_{1})...\psi^{\dagger}(\vec{x}_{N})$
on the vacuum state, where $\Psi$ is the $N$-particle wave function. The
Hamiltonian (2) then takes the equivalent form

$\hat{H}=\int
d^{3}x\frac{\hbar^{2}}{2m}\nabla\hat{\psi}^{\dagger}(\vec{x})\nabla\hat{\psi}(\vec{x})-\\\
-\frac{m^{2}G}{2}\int
d^{3}xd^{3}x^{\prime}\frac{\hat{\psi}^{\dagger}(\vec{x})\hat{\psi}(\vec{x})\hat{\psi}^{\dagger}(\vec{x}^{\prime})\hat{\psi}(\vec{x}^{\prime})}{|\vec{x}-\vec{x}^{\prime}|}\,,$
(3)

and analogously for the Coulomb case. We can rewrite this Hamiltonian in the
following way

$\hat{H}=\int
d^{3}x\Big{(}\frac{\hbar^{2}}{2m}\nabla\hat{\psi}^{\dagger}(\vec{x})\nabla\hat{\psi}(\vec{x})+m\hat{\Phi}(\vec{x})\hat{\psi}^{\dagger}(\vec{x})\hat{\psi}(\vec{x})\Big{)}$
(4)

where the Newtonian potential satisfies

$\nabla^{2}\hat{\Phi}(\vec{x})=4\pi
Gm\hat{\psi}^{\dagger}(\vec{x})\hat{\psi}(\vec{x})\,.$ (5)

Thus, non-relativistic quantum mechanics implies that the Newtonian
gravitational field is quantized, in the sense that it is a field operator.
There is no option to treat $\hat{\Phi}(\vec{x})$ as a c-number field since
the right hand side of the defining equation of $\hat{\Phi}(\vec{x})$ is an
operator. One way to insist on the classicality of the potential is to modify
this equation to $\nabla^{2}\Phi(\vec{x})=4\pi
Gm\langle\Psi(t)|\hat{\psi}^{\dagger}(\vec{x})\hat{\psi}(\vec{x})|\Psi(t)\rangle,$
which is the non-relativistic limit of the so-called semi-classical
gravity.Singh and Padmanabhan (1989) This theory is expected to be incorrect
Page and Geilker (1981) in situations where the quantum state of matter
corresponds to density distribution in large spatial superpositions, where a
mean field description in terms of
$\langle\Psi(t)|\hat{\psi}^{\dagger}(\vec{x})\hat{\psi}(\vec{x})|\Psi(t)\rangle$
does not capture the quantum state $|\Psi(t)\rangle$. More importantly,
though, this theory is not quantum mechanics since the Schrödinger equation is
modified. Any proposed theory in which the Newtonian potential field is a
c-number field (and thus classical in this sense), is not equivalent to
“textbook” quantum mechanics, and thus would predict different outcomes for
the GIE setup. Conversely, standard quantum mechanics implies that the
Newtonian gravitational potential field is quantized in the sense that it is a
field operator

$\displaystyle\hat{\Phi}(\vec{x},t)=-mG\int
d^{3}x^{\prime}\frac{\hat{\psi}^{\dagger}(\vec{x}^{\prime})\hat{\psi}(\vec{x}^{\prime})}{|\vec{x}-\vec{x}^{\prime}|}\,.$
(6)

The quantisation of the source masses implies the operator nature of the
potential. Only in situations where a mean field description for the sources
is adequate can the operator nature of $\hat{\Phi}$ be neglected (such as for
quantum test bodies falling evolving in the gravitational field of Earth
Colella  _et al._ (1975); Zych  _et al._ (2011); Pikovski  _et al._ (2015)).

The novelty of the GIE experiment is thus that the Newtonian field as in eq.
(6) has to be treated quantum mechanically, and cannot be approximated by a
mean field description. The sources for gravity are themselves in
superposition. While this regime of physics has never been directly tested,
the predictions assume regular quantum mechanics (2). The gravitationally
active density is in a spatial superposition and thus the operator nature of
the potential cannot be neglected. However, nothing in this description
involves mediators and thus nothing can be said whether gravitons are
quantized, at this level of the description. Such claims rely on extra
assumptions, such as the assumption that $\hat{\Phi}(\vec{x})$ is the
component of a more fundamental relativistic quantum field that has local
degrees of freedom.

The following sections will answer under which conditions/assumptions our
quantized Newtonian potential also implies the existence of quantized
gravitons, and deals with questions of non-locality using the simpler but very
analogous example of electromagnetism  (summarized in Table 1).

##  III Electromagnetic Field Theory

We now proceed to analyze possible conclusions beyond the Newtonian limit,
under the assumption that one takes relativistic field theory for granted.
Again, we note that the non-relativistic limit is sufficient to describe the
experiment as in the section above. Now, however, we take the view that from
other experiments and established physics we can additionally assume that
there is an underlying field theory that mediates the entanglement, even if it
is not directly probed in the experiment. The question is, can one conclude
something more than what has been established in the previous section?

To simplify the discussion and gain intuition we will assume that the
entanglement production in a GIE-like experiment is solely due to electric
charge (thus technically it is electromagnetically induced entanglement),
before discussing the case of gravity in the next section. Our discussion
within this section is then guaranteed to be based on established physics. Our
task is thus to find out what we would learn about the quantum nature of
electromagnetism in the hypothetical scenario where we only had the
electromagnetic analogue of the GIE experiment at our disposal. In particular,
we investigate which assumptions about electromagnetism are needed to infer
the existence of (quantized) photons from witnessing entanglement from this
experiment alone.

In this section we therefore slightly modify the experiment shown Fig. 1 and
assume that two objects that are put into superposition have electric charge
$q$ with $q^{2}/\epsilon_{0}\gg Gm^{2}$, and interact via the Coulomb
potential $\Phi_{em}$.

Formulation of Interaction | Equation | LOCC?
---|---|---
non-relativistic Coulomb | (1) |  ~~L~~ OCC
QED (Lorentz gauge) | (9) | LO~~C~~ C
QED (Coulomb gauge) | (17) |  ~~L~~ OCC
QED (Absorber, no EM field) | (23) |  ~~L~~ OCC
Table 1: An LOCC channel cannot generate entanglement. But if entanglement is
observed, can one deduce more about the physics of the underlying process,
such as quantized mediators? Comparison of quantum theories (first column) of
electrically charged particles and their defining equations (second column),
that are expected to produce identical entanglement, and what aspect of LOCC
they violate (third column) such that they can generate entanglement. Only the
formulation of QED in the (local) Lorentz gauge includes quantized mediators
relevant for the electromagnetic analog of the GIE experiment. The other
formulations of QED remain equally viable, from which no quantization of the
mediators can be inferred from the LOCC argument, and the experiment cannot
discern them. Even beyond this ambiguity related to the gauge freedom, the
absorber formulation of QED by Hoyle, NarlikarHoyle and Narlikar (1995) and
DaviesDavies (1971, 1972) (based on quantizing the classical Wheeler-Feynman
formulation of electrodynamics) is experimentally indistinguishable from
standard QED even in the fully relativistic regime. Thus the GIE experiment if
applied to the electromagnetic interaction would not be able to tell us
whether there is a Hilbert space associated with the electromagnetic field or
not. As non-local entanglement generation cannot be ruled out, we therefore
cannot conclude based on the LOCC argument whether underlying mediators are
quantized.

###  III.1 Local mediators: Lorentz invariant formulation of QED

The dynamics of the photon field in any Lorentz invariant gauge is local, as
captured by the local interaction Lagrangian

$\mathcal{L}_{int}=j^{\mu}(x)A_{\mu}(x)\,,$ (7)

where $A_{\mu}(x)=(\Phi_{em},\vec{A})$ is the electromagnetic four-potential,
$j_{\mu}(x)=(c\rho,\vec{j})$ the four-current and $x=(t,\vec{x})$. In the
following we set $c=1$. By choosing a particular gauge, namely the Lorentz
gauge, locality becomes manifest even on the level of the mediators
themselves, giving a very intuitive physical picture.

In this formulation of QED,Gupta (1950); Bleuler (1950) all four components of
the gauge field $\hat{A}_{\mu}$ are quantized. Therefore, the theory contains
4 types of photons, each for every component of the four-potential. We expand
the gauge field $\hat{A}_{\mu}$ in terms of creation and annihilation
operators

$\hat{A}_{\mu}(\vec{r})=\int
d^{3}k\sqrt{\frac{\hbar}{2\epsilon_{0}(2\pi)^{3}\omega_{k}}}\biggl{[}\hat{a}_{\mu}(\vec{k})e^{i\vec{k}\cdot\vec{r}}+\\\
+\hat{a}_{\mu}^{\dagger}(\vec{k})e^{-i\vec{k}\cdot\vec{r}}\biggr{]}$ (8)

where $\mu=(0,1,2,3)$. The expression for the Hamiltonian isCohen-Tannoudji
_et al._ (1992)

$\hat{H}=\hat{H}_{P}+\hat{H}_{R}+\hat{H}_{I}$ (9a) where $\hat{H}_{P}$ is the
free particle Hamiltonian, $\hat{H}_{R}=\int
d^{3}k\hbar\omega\biggl{[}\sum_{\lambda=\pm,l}\biggl{(}\hat{a}^{\dagger}_{\lambda}(\vec{k})\hat{a}_{\lambda}(\vec{k})\biggr{)}-\\\
-\hat{a}^{\dagger}_{s}(\vec{k})\hat{a}_{s}(\vec{k})+1\biggr{]}$ (9b)

is the field Hamiltonian, which contains creation and annihilation operators
for scalar (s), longitudinal (l) and transverse ($\pm$) photons. The
interaction Hamiltonian between the particles and the field is

$\hat{H}_{I}=\int
d^{3}r\bigl{[}-\hat{\vec{j}}(\vec{r})\cdot\hat{\vec{A}}(\vec{r})+\hat{\rho}(\vec{r})\hat{\Phi}_{em}(\vec{r})\bigr{]}\,.$
(10)

Note that in (9b), the term which corresponds to scalar photons comes with a
minus sign. Quantization of QED in Lorentz gauge, requires the use of the so-
called indefinite Hilbert space metric. Gupta (1950) A pedagogical
presentation of the method can be found in Ref. Cohen-Tannoudji  _et al._
(1992).

As seen in eq. (9b), this method of quantization introduces fictitious
longitudinal and scalar photons that are not directly observable in any
experiment (in the sense that expectation values of any observable, as well as
in- and out-states in scattering theory can only involve quantum states where
the scalar and longitudinal photons are in the vacuum state). In presence of
matter sources, scalar and longitudinal photons can appear in virtual states
Franson (2011) but have no observable consequences. Only the transverse
photons are physical. Thus a subsidiary condition on the Hilbert space of
physical states is imposed Gupta (1950)

$\bigl{(}\hat{a}_{l}(\vec{k})-\hat{a}_{s}(\vec{k})+\hat{\lambda}(\vec{k})\bigr{)}|\chi\rangle=0$
(11)

$\forall\vec{k}$ and where
$\hat{\lambda}(\vec{k})=\frac{1}{\omega\sqrt{2\epsilon_{0}\hbar\omega}}\hat{\rho}(\vec{k})$.

In the free theory we have $\hat{\lambda}(\vec{k})=0$ so the condition (11)
takes the form

$\bigl{(}\hat{a}_{l}(\vec{k})-\hat{a}_{s}(\vec{k})\bigr{)}|\chi\rangle=0$ (12)

$\forall\vec{k}$. This relation guarantees that, for every physical state
$|\chi\rangle$, the effect of longitudinal and scalar photons cancel.e)e)e)One
can formally show, by making use of the so-called Ward identity, that the
effect of scalar and longitudinal photons can be neglected in any QED
scattering process. For a standard treatment, the reader is referred to
Ref.Peskin and Schroeder (1995).

What is crucial is that entanglement is generated in a local and causal way,
but it is mediated not by the physical photons but by the scalar and
longitudinal ones.Franson (2011) This can be easily seen in a toy example
(analogous to the GIE setup) of a system of two harmonic oscillators $(A,B)$
which are initially in a product state

$|\Psi_{0}\rangle\equiv|0\rangle_{A}\otimes|0\rangle_{B}\otimes|\vec{0}\rangle_{\gamma}$
(13)

where $|0\rangle_{i}$ for $i=(A,B)$, corresponds to the ground state of the
oscillators and $|\vec{0}\rangle_{\gamma}$ is a state which contains no
photons. Coupling the system of the two oscillators to the electromagnetic
field via the interaction Hamiltonian (10), the perturbed state can be
calculated using time independent second order perturbation theory and it gets
entangled between $A$ and $B$ Franson (2011)

$|\Psi\rangle\simeq|\Psi_{0}\rangle+\varepsilon_{L}|\Psi_{2}\rangle$ (14)

where

$|\Psi_{2}\rangle=|1\rangle_{A}\otimes|1\rangle_{B}\otimes|\vec{0}\rangle_{\gamma}$
(15)

corresponds to the second order $(\mathcal{O}(q^{2}))$ change in
$|\Psi\rangle_{0}$ and $\varepsilon_{L}$ is given by

$\varepsilon_{L}=\sum_{l}\frac{\langle\Psi_{2}|\hat{H}_{I}|l\rangle\langle
l|\hat{H}_{I}|\Psi_{0}\rangle}{(E_{0}-E_{1})(E_{0}-E_{l})}\,.$ (16)

We have used a compact notation for the intermediate states
$|l\rangle\equiv|l\rangle_{A}\otimes|l\rangle_{B}\otimes|\vec{k}\rangle_{\gamma}$f)f)f)In
principle, the intermediate photon states $|\vec{r}\rangle_{\gamma}$ contain
scalar, longitudinal and transverse photons. For the electromagnetic analogue
of the GIE proposal, we omit the contribution of transverse photons since the
latter play no role in establishing the entanglement.. $E_{0}$ corresponds to
energy of the state (13), where the oscillators $(A,B)$ are in the ground
state whereas $E_{1}$ is the energy of the perturbed state (15) where the
oscillators occupy the first excited state $|1\rangle_{(A,B)}$. Both states
contain no photons. On the other hand, $E_{l}$ corresponds to the energy of
the intermediate states $|l\rangle$, where virtual photons do exist. It is
important to notice that since $E_{l}\equiv
E_{({l_{A},l_{B},l_{\gamma}})}=E_{l_{A}}+E_{l_{B}}+E_{l_{\gamma}}$ appears in
the denominator of (16), one cannot trivially perform the summation over the
oscillator states.

The state (14) corresponds to an entangled state of the two oscillators
$(A,B)$ for $\varepsilon_{L}\neq 0$. It’s crucial that in (16), apart from
summing over intermediate oscillator states, there is a summation over
intermediate photon states $|\vec{k}\rangle_{\gamma}$. In a recent work, a
similar computation was performed for the gravitational case but the summation
over intermediate oscillator states has been omitted.Bose  _et al._ (2022)
Thus we conclude that in this picture, the intermediate photons establish the
entanglement between the two oscillators. In the near field regime in which
the transverse photons do not contribute, the oscillators get entangled due to
the exchange of scalar and longitudinal photons – mediators that cannot be
directly observed.

When using a local gauge such as the Lorentz gauge as the underlying ontology,
the LOCC argument therefore implies that entanglement is created by quantized
mediators, since locality is pre-assumed and thus classicality of the channel
cannot hold if entanglement is generated. The process is a particular
combination of the scalar and longitudinal photons and does not directly
reveal the quantum nature of transverse (physical) photons. If QED was not
pre-assumed, but instead, we attempted to infer the quantum nature of the
electromagnetic field from the GIE experiment and the LOCC argument, we would
still not be able to directly infer quantization of physically measurable
mediators. However, consistency with Lorentz invariance and locality of the
quantum fields will then also require these transverse degrees of freedom to
exist and be quantized. This is a matter of theoretical consistency, but not
something that can be directly inferred from the entanglement experiment, even
in this picture. The experiment is not sensitive to actual physical mediators
– photons (or gravitons in the GIE case) – and entanglement is created by
other local mediators that can never be directly observed. Nevertheless LOCC
then implies that these must be quantized.

###  III.2 No mediators: QED in Coulomb gauge

We now turn to a different, but physically equivalent description of the
process. The fundamental locality of eq. (7) does not necessarily imply
locality of mediators of the interaction. Contrary to the covariant
quantization of QED,Gupta (1950); Bleuler (1950) in the Coulomb gauge no
scalar and longitudinal photons appear at any stage. The expression for the
QED Hamiltonian in the Coulomb gauge (assuming spinless particles for
simplicity) is Cohen-Tannoudji  _et al._ (1992)

$\begin{split}\hat{H}=\sum_{i}\frac{\bigl{(}\hat{\vec{p}}_{i}\bigr{)}^{2}}{2m_{i}}&+\int
d^{3}r\hat{\vec{j}}(\vec{r})\cdot\hat{\vec{A}}_{\perp}(\vec{r})+\\\
&+\frac{1}{8\pi\epsilon_{0}}\int\int
d^{3}rd^{3}r^{\prime}\frac{\hat{\rho}(\vec{r})\hat{\rho}(\vec{r}^{\prime})}{|\vec{r}-\vec{r}^{\prime}|}+\\\
&+\int
d^{3}k\hbar\omega\sum_{\lambda=\pm}\bigl{(}\hat{a}^{\dagger}_{\lambda}(\vec{k})\hat{a}_{\lambda}(\vec{k})+\frac{1}{2}\bigr{)}\end{split}$
(17)

where the first line describes the kinetic energy of particles as well as
their interaction with the field, the second term is the Coulomb interaction
energy and the third term corresponds to the the Hamiltonian of the radiation
field. The operators $\hat{a}^{\dagger}_{\lambda}(\vec{k})$ and
$\hat{a}_{\lambda}(\vec{k})$ create and annihilate only transverse (i.e.
physical) photons with momentum $\vec{k}$ whereas $\hat{\vec{r}}_{i}$ and
$\hat{\vec{p}}_{i}$ correspond to the $i$’s particle position and momentum
operators.

The Hilbert space of the whole system (particles+EM field) is
$\mathcal{H}=\mathcal{H}_{P}\otimes\mathcal{H}_{R}$, where $\mathcal{H}_{P}$
is the state space of the particles in which $\hat{\vec{r}}_{i}$ and
$\hat{\vec{p}}_{i}$ act and $\mathcal{H}_{R}$ corresponds to the state space
of the photon-radiation field in which $\hat{a}^{\dagger}_{\pm}(\vec{k})$ and
$\hat{a}_{\pm}(\vec{k})$ act. Note that predominantly the third term in (17)
generates the direct interaction between the two non-relativistic and quasi-
static systems, and it is fundamentally non-local. Since radiation
(retardation effects) are not important in the GIE experiment, one has thus to
conclude that, in the formulation of QED in the Coulomb gauge, the
entanglement between the charges is established non-locally, via the Coulomb
term (as also pointed out in Ref. Anastopoulos and Hu (2018)). In this case,
the LOCC argument alone to prove that the underlying process that generates
entanglement is itself a quantum system does not apply and nothing about the
mediators can be inferred. For further details regarding QED in Coulomb gauge,
the reader is referred to Appendix IX.1.

In the toy example of the two oscillators $(A,B)$, the entangled state is
computed using first order time independent perturbation theory Franson (2011)

$|\Psi\rangle=|\Psi_{0}\rangle+\varepsilon_{C}|\Psi_{1}\rangle$ (18)

where initially the oscillators are in the ground state

$|\Psi_{0}\rangle\equiv|0\rangle_{A}\otimes|0\rangle_{B}$ (19)

and the perturbed state is

$|\Psi_{1}\rangle\equiv|1\rangle_{A}\otimes|1\rangle_{B}$ (20)

where $\varepsilon_{C}$ is given by

$\varepsilon_{C}=\frac{\langle\Psi_{1}|\hat{H}_{C}|\Psi_{0}\rangle}{(E_{0}-E_{1})}$
(21)

and

$\hat{H}_{C}=\frac{1}{8\pi\epsilon_{0}}\int\int
d^{3}rd^{3}r^{\prime}\frac{\hat{\rho}(\vec{r})\hat{\rho}(\vec{r}^{\prime})}{|\vec{r}-\vec{r}^{\prime}|}$
(22)

is the Coulomb interaction Hamiltonian. The state (18) corresponds to an
entangled state of the two oscillators. However, in (21), there are no
intermediate or final photon states. Therefore, in that formulation one can
explicitly see that nothing can be inferred about photons, physical or
unphysical, mediating the entanglement. We thus conclude that depending on the
preferred picture within the full scope of QED, the LOCC argument can
equivalently imply either quantized mediators as in the previous section, or a
non-local interaction. Locality is therefore not a necessary principle, but a
choice. While the Coulomb formulation of QED is fundamentally non-local,
causality is preserved and is the fundamental principle that holds in all
formulations, as discussed in more detail in section III.4. Note that manifest
Lorentz invariance is not necessarily lost in a non-local formulation of QED,
see for example Ref.DeWitt (1962) in which the electromagnetic field is
quantised using the field strength tensor instead of the vector potential. It
is thus clear that non-locality is not a gauge artefact but a property of any
formulation of QED that only quantizes the physical degrees of freedom of the
electromagnetic field.

In fact all interactions of the standard model and gravity are gauge theories
and thus if the formulation of these theories is restricted to involve only
the dynamics of the physical degrees of freedom all these theories are non-
local (in that the Lagrangian density is not a sum of local products of
operators). Only with the help of auxiliary gauge degrees of freedom are these
theories local. It thus is a matter of taste if locality is pre-assumed or
not, and thus whether the LOCC argument implies the existence of quantum
mediators or not.

###  III.3 No mediators: QED as absorber theory

Above we discussed different interpretations of the experiment based on
choosing different gauges. This ambiguity was also pointed out in
Ref.Anastopoulos and Hu (2018). It shows that the notion of local mediators is
not necessary to explain the experiment, and thus no conclusion about its
quantization can be drawn (see also Table 1).

However, there is a deeper issue in the context of electromagnetism. One
general loophole to conclude that a relativistic theory of interacting
particles has to involve exchange bosons is that one can remove the mediators
from the theory. An example of such a drastic approach is provided by the
Wheeler-Feynman (WF) absorber theory of electromagnetism.Schwarzschild (1903);
Tetrode (1922); Fokker (1929); Wheeler and Feynman (1945, 1949) In the
absorber theory, the interaction between charges is non-local, captured by the
Lagrangian

$\displaystyle\mathcal{L}_{int}=j^{\mu}(x)\int
d^{4}x^{\prime}\,j_{\mu}(x^{\prime})\delta_{\rm
D}\big{[}(x-x^{\prime})^{2}\big{]}$ (23a) $\displaystyle\delta_{\rm
D}\big{[}(x-x^{\prime})^{2}\big{]}=\frac{1}{2}\big{(}G_{-}(x-x^{\prime})+G_{+}(x-x^{\prime})\big{)}$
(23b) where
$(x-x^{\prime})^{2}=c^{2}(t-t^{\prime})^{2}-|\vec{x}-\vec{x}^{\prime}|^{2}$ is
the Minkowski distance between $x=(t,\vec{x})$ and
$x^{\prime}=(t^{\prime},\vec{x}^{\prime})$, $\delta_{\rm D}$ the Dirac delta
function, $G_{-}$ is the retarded and $G_{+}$ the advanced Green’s function of
the d’Alembert operator $\Box$:
$G_{\pm}=|\vec{x}-\vec{x}^{\prime}|^{-1}\delta_{\rm
D}\big{[}c(t-t^{\prime})\pm|\vec{x}-\vec{x}^{\prime}|\big{]}\,.$ (23c)

The Lagrangian density (23a) is manifestly causal due to $\delta_{\rm
D}\big{[}(x-x^{\prime})^{2}\big{]}$ restricting the spatial non-locality to
the future and past lightcone. It is manifestly non-local because the delta
function has in general support arbitrarily far away from the position
$\vec{x}$. Despite the intrinsic non-locality and seemingly drastic departure
from our usual intuition for physics, it was shown that all results of
classical electromagnetism, in particular Dirac’s half-advanced-half-retarded
solution to the radiation reaction problem, can be derived from such a theory
if suitable absorbing boundary conditions are chosen. Within standard
electromagnetism this corresponds to the assumption that all emitted radiation
is eventually completely absorbed by some charge in the future. When and how
this absorption happens is irrelevant. These boundary conditions are thus
cosmological in nature and might appear contrived or unintuitive. But this is
just a matter of taste and cannot be refuted logically or by any observation –
it is just a reformulation of regular electromagnetism but with a different
ontology.g)g)g)In Sec. IV.3 we discuss that in the case of non-perturbative
gravity, an absorber theory is likely different from the conventional field
theory formulation. However, in the weak-field regime relevant to GIE,
gravitational absorber theory and standard weak-field GR are reformulations of
each other.  Furthermore, all results of QED can be recovered in such a theory
as well by simply quantizing the charges, thereby obtaining a viable
reformulation of QED without any electromagnetic degrees of freedom, as first
shown by Hoyle and Narlikar in 1969 and further worked out by Davies, Pegg and
others. Hoyle and Narlikar (1969, 1971); Davies (1971, 1972); Pegg (1975,
1979); Durrant (1980); Hoyle and Narlikar (1995),h)h)h) While all predictions
for experimental outcomes seem to be the same for the Hoyle-Narlikar-Davies
quantized version of the absorber theory, it should be pointed out that only
path integral quantisation has been possible so far. Canonical quantisation is
complicated by the fact that the Lagrangian contains a double integral over
time. See however Ref.Lienert (2018) for a canonical approach.

In particular, Feynman suggested in 1948 Feynman (1948) that one can replace
(7) with (23a) for any QED process which contains no external, real photons,
without any change in the experimental results. According to this theorem (see
Ref.Davies (1971) for a proof), one can eliminate all virtual photons from the
mathematical description of the QED processes. This theorem directly applies
to the EM analogue of the GIE proposal in which on-shell photons are not
involved.i)i)i)Note the there is technically a difference between an “absorber
theory of radiation”, Wheeler and Feynman (1945) in which fields are an
element of the theory but absorbing boundary conditions eliminate free fields,
and an action-at-a-distance theory in which absorbing boundary conditions need
to be assumed for the theory to be phenomenologically viable.Wheeler and
Feynman (1949) In this paper we lump together these two two types of absorber
theories since they make the same predictions for the electromagnetic case,
and more importantly for our discussion, upon quantisation photons do not
exist in physical states in both these theories.

While the physical predictions are the same, the interpretation of such a
theory is different. In this action-at-a-distance formulation the
electromagnetic field does not exist and particles interact directly, without
mediators, on the lightcone, see (23a). Such an ontology in the context of the
GIE experiment has profound consequences: Since the direct interaction on the
lightcone is non-local in space, the LOCC argument does not imply a quantised
mediating system for entanglement. This means that witnessing entanglement in
the electromagnetic version of the GIE experiment could not unambiguously tell
us if the electromagnetic field needs to be quantised. Thus entanglement alone
only proves source mass quantisation, since there is an experimentally
equivalent alternative formulation, where no mediators are involved in
producing the entanglement.

In conclusion, the electromagnetic analogue of the GIE experiment would tell
us nothing about mediators beyond our prior believes about which formulation
of electromagnetism is more likely to be true. One either believes that
physics is local which means firstly that longitudinal and scalar photons
establish the entanglement – and not the physical transverse photons –, and
secondly that transverse photons must exist and be quantized as well, since
only the combination of these three types of photons can build a local field
theory. Or one doesn’t manually exclude non-local establishment of
entanglement, which is the case in the Coulomb formulation of QED.
Alternatively, in an absorber framework only the sources are real and the
entire electromagnetic field is a mathematical fiction and should not be
quantized. The entanglement in this picture does not stem from mediators but
from the charges interacting non-locally. All formulations, however, yield the
exact same experimental outcomes. It is important to note that the case of
~~L~~ OCC (where locality does not hold, see Table 1) does not necessarily
imply a classical channel. It simply means that a local quantum channel is not
necessary to produce the entanglement. In the case of non-local formulations
of QED, entanglement is produced through a quantum channel which can be
directly inferred from the Hamiltonian (17) or Lagrangian (23). But in the
case of Coulomb gauge or absorber theory that quantum channel is not a
mediating degree of freedom but a non-local interaction between the charges.

###  III.4 Locality vs. Causality

As we have discussed above, locality is not a necessary condition even in a
relativistic theory. One is free to choose a local and manifestly covariant
formulation of QED, but a non-local description is also always possible.
Nevertheless causality is preserved in all of the formulations. There is thus
a fundamental difference between non-locality and causality, and only the
latter is fundamentally required by Lorentz invariance.

In the Coulomb gauge, despite the various non-local (instantaneous) pieces
that appear in (17), only causal physical processes are described.Rohrlich
(2002); Gardiner and Drummond (1988) In particular, it was shown in Refs.
Jackson (2002); Gardiner and Drummond (1988) that, in Coulomb gauge, the
instantaneous Coulomb potential cancels exactly the instantaneous piece that
appears in the vector field $\hat{\vec{A}}_{\perp}$ to give rise to causal
(retarded) EM fields, the so-called Jefimenko fields.Jefimenko (1989)

From a Quantum field theory point of view, the photon propagator contains a
piece which violates Lorentz invariance. This term in the propagator exactly
cancels the acausal, Lorentz violating Coulomb potential that appears in the
Hamiltonian (17), to give a perfectly Lorentz invariant, causal QED. A
detailed discussion can also be found in Ref.Weinberg (2005).

The absorber theory is manifestly causal due to the lightcone integral in eq
(23a). The absorbing boundary conditions are not required for causality, but
they ensure approximate forward causation, see Ref.Wheeler and Feynman (1945).

Overall, since all these formulations are just different ways of interpreting
Maxwell’s equations, it is clear that relativistic causality is preserved in
any physical process described in these frameworks. Nevertheless, even while
being causal, a non-local ontology is admitted by relativity. This peculiarity
is at the core of other seeming puzzles of classical electromagnetism, such as
the question how the Jefimenko equations Jefimenko (1989) for $\vec{E}$ and
$\vec{B}$ can be causal despite the Coulomb equation,
$\nabla\cdot\vec{E}=\rho/\epsilon_{0}$, suggesting the opposite.Wong (2010);
Jackson (2002) The Coulomb equation seems acausal because
$\nabla\cdot\vec{E}$, and thus the longitudinal part of $\vec{E}$, depend
instantaneously on the charge density $\rho$. To see explicitly that this
instantaneous dependence is consistent with causality one can use the
dynamical Maxwell equations to write
$\Box(\nabla\cdot\vec{E})=\frac{1}{\epsilon_{0}}\nabla^{2}\rho+\mu_{0}\nabla\partial_{t}\vec{J}$.
All relativistic field theories with first class constraints have the property
that instantaneous constraint equations are compatible with the causal
evolution.

As already discussed, the GIE proposal as well as its EM analogue, heavily
rely on the notion of locality, which lies at the core of the LOCC argument.
In the preceding subsection, we have concluded that locality is not a
prerequisite, instead it’s a matter of choice to formulate physics in a local
or non-local way. This discussion addresses similar questions and shares some
similarities with the correspondence between DeWitt and Aharonov and Bohm
DeWitt (1962); Aharonov and Bohm (1962) (see also Ref.Belinfante (1962))
shortly after their seminal work on the “Significance of the EM potentials in
quantum theory”.Aharonov and Bohm (1959) In particular, in Ref.DeWitt (1962),
DeWitt asks: _Which is more significant, the fact that nonlocal formulations
of causal theories exist which deal only with observables, or the fact that in
all known cases local formulations in terms of potentials also exist?_ And
continues with the following statement: _In a similar vein the author
disagrees with the assertion of Aharonov and Bohm that quantum electrodynamics
is ultimately determined by the requirement that it be expressible in a local
form. QED is really determined by experiment._

This assessment aligns with our interpretation of the electromagnetic analogue
of the GIE proposal. QED is determined by experiment and it cannot discern a
local from a non-local formulation. The underlying ontology corresponds to a
_metaphysical statement_ which by no means is tested. Thus, any formulation of
QED, local or non-local are equally viable, as long as the experimental
outcomes are indistinguishable. For the EM analogue of GIE, this implies that
one cannot _unambiguously_ infer the existence of local mediators.

Figure 2: Left: Bronstein cubeBronstein (1933); Okun (2002) organising various
theories depending on their dependence on $\hbar,G$ and $c$. The GIE
experiment tests “non-relativistic quantum gravity”, which is a non-local and
non-causal theory (due to Newtonian gravity being instantaneous action-at-a-
distance). Since in the $\hbar\rightarrow 0$ limit General relativity is
geometric, causal and admits a local formulation, it is plausible that quantum
gravity itself should share these features – but a complete theory is unknown
as indicated by a grey circle. We note that quantum field theory and special
relativity admit non-local causal formulations, such that it is plausible that
also quantum gravity might admit a non-local formulation. Therefore
entanglement may also be generated non-locally in the fully relativistic
theory, and LOCC does not imply quantization of mediators. Right: The yellow
region indicates the direct experimental access of the GIE where gravitational
fields are weak and matter non-relativistic. An unexpected result (dashed
circles) would rule out non-relativistic quantum gravity. For an expected
result, at the bottom, we outline the “standard” view how non-relativistic
quantum gravity is connected (black arrows) to quantum gravity. However, this
is merely a reasonable extrapolation and it is conceivable that the
fundamental theory deviates from these properties (grey circles with question
marks that may have conventional theories as limiting cases, indicated by
dashed arrows). Thus observing the expected outcome of the GIE experiment only
directly confirms non-relativistic quantum gravity and everything else is an
extrapolation based on ones’ theoretical priors.

##  IV The gravitational case: quantized mediators or non-local alternatives?

Above we discussed the electromagnetic version of the GIE experiment. We
showed that conclusions beyond the non-relativistic limit are ambiguous
because there is always an alternative explanation possible that generates
entanglement non-locally, thus the LOCC argument does not unambiguously imply
quantization of mediators. It is conceivable that similar dual arguments based
on either a local mediator theory or a non-local theory could apply to quantum
gravity. This is indeed the case for weak gravity.

###  IV.1 Local mediators: Weak-field quantum gravity in Lorentz gauge

In the weak-field limit, covariant quantization of gravity has been developed
in Refs.Gupta (1952, 1968) by making use of the so-called indefinite metric.
This procedure is very similar to the covariant quantization discussed
previously for the case of QED. In this formulation, all the components of the
(linearized) metric tensor $h_{\mu\nu}$ are quantized. The fictitious-gauge
degrees of freedom are eliminated by imposing supplementary conditions on the
Hilbert space. Classically the Lorentz gauge (or de Donder gauge) is defined
as $\partial^{\mu}\bar{h}_{\mu\nu}=0$ with $\bar{h}_{\mu\nu}\equiv
h_{\mu\nu}-1/2\,h_{\mu}^{\mu}$. The procedure in brief is the following: We
expand the gravitational field in Fourier basis,

$\hat{h}_{\mu\nu}(\vec{r})=\int d^{3}k\sqrt{\frac{\hbar
G}{\pi^{2}\omega_{k}}}\biggl{[}\hat{a}_{\mu\nu}(\vec{k})e^{i\vec{k}\cdot\vec{r}}+\\\
+\hat{a}_{\mu\nu}^{\dagger}(\vec{k})e^{-i\vec{k}\cdot\vec{r}}\biggr{]}\,.$
(24)

The Hamiltonian of the gravitational field has the form Gupta (1952)

$\hat{H}_{g}=\frac{1}{2}\int
d^{3}k\hbar\omega_{k}\biggl{[}\hat{a}^{\dagger}_{\mu\nu}(\vec{k})\hat{a}^{\mu\nu}(\vec{k})-\frac{1}{2}\hat{a}^{\dagger\mu}_{\mu}(\vec{k})\hat{a}^{\nu}_{\nu}(\vec{k})\biggr{]}$
(25)

where the graviton creation and annihilation operators satisfy the commutation
relations

$[\hat{a}_{\mu\nu}(\vec{k}),\hat{a}^{\dagger}_{\lambda\rho}(\vec{k}^{\prime})]=\bigl{(}\eta_{\mu\lambda}\eta_{\nu\rho}+\eta_{\mu\rho}\eta_{\nu\lambda}\\\
-\eta_{\mu\nu}\eta_{\lambda\rho}\bigr{)}\delta(\vec{k}-\vec{k}^{\prime}).$
(26)

where $\eta_{\mu\nu}$ is the Minkowski metric. Note that the
$\hat{a}_{0i}(\vec{k})$ for $i=1,2,3$ components as well as the trace
$\hat{a}^{\mu}_{\mu}(\vec{k})\equiv\hat{a}(\vec{k})$ satisfy the commutation
relations with a negative sign:

$[\hat{a}_{0i}(\vec{k}),\hat{a}^{\dagger}_{0i}(\vec{k}^{\prime})]=[\hat{a}(\vec{k}),\hat{a}^{\dagger}(\vec{k}^{\prime})]=-\delta(\vec{k}-\vec{k}^{\prime})$
(27)

thus an indefinite Hilbert space metric has to be used.Gupta (1952, 1968)
Matching up the components of this metric which have negative commutation
relations along with components of the metric with positive commutation
relation, one can eliminate the eight redundant graviton polarizations to end
up with the two physical degrees of freedom.

In presence of interactions, the supplementary condition needs to take into
account the presence of matter degrees of freedom Gupta (1952)

$\begin{split}\biggl{[}&\hat{a}_{3\nu}(\vec{k})-\hat{a}_{0\nu}(\vec{k})+\\\
&+\kappa\int_{t=t^{\prime}}d^{3}x^{\prime}D^{(+)}(\vec{x}-\vec{x}^{\prime})\hat{T}_{0\nu}(\vec{x}^{\prime})\biggr{]}|\chi\rangle=0\end{split}$
(28)

for $\nu=0,1,2,3$ where $D^{(+)}(\vec{x}-\vec{x}^{\prime})$ is the positive
frequency part of the graviton propagator and
$\hat{T}_{0\nu}(\vec{x}^{\prime})$ corresponds to the matter stress energy
tensor. In this case, the redundant gravitons are still absent in all physical
processes, but they can exist in virtual states. In connection to GIE, this
quantization procedure has been adopted in Ref.Bose  _et al._ (2022) in order
to argue that the masses get entangled via the exchange of virtual gravitons.

###  IV.2 No mediators: Weak-field quantum gravity in Poisson gauge

Up to this point, we have discussed the covariant quantization of weak-field
gravity. In this picture, the virtual gravitons that correspond to the many
components of the metric tensor, even though they are not the usual physical
gravitons and are unobservable, exist in the mathematical formulation of the
theory. It is possible however, to formulate the theory in a picture where no
redundant and unphysical gravitons appear at any stage in the theory. In this
formulation, one eliminates the redundant degrees of freedom completely by
fixing the gauge classically, and then canonically quantizes the remaining
physical degrees of freedom $\hat{s}_{ij}^{TT}$ which are the spatial and
transverse traceless components of $\hat{h}_{\mu\nu}$ (as defined in eq.
(24)). See App. IX.2 for further details on the isolation of the physical
degrees of freedom. The Hamiltonian in this so-called Poisson gauge –
analogous to the Coulomb gauge in QED – is given by

$\displaystyle\hat{H}_{int}=$ $\displaystyle-\int
d^{3}r\bigl{(}\hat{s}_{ij}^{TT}(\vec{r})\hat{T}^{ij}(\vec{r})\bigr{)}-$ (29)
$\displaystyle-\frac{G}{2}\int\frac{d^{3}rd^{3}r^{\prime}}{|\vec{r}-\vec{r}^{\prime}|}\biggl{[}-4\hat{f}_{\perp,i}(\vec{r}^{\prime})\hat{T}^{0i}(\vec{r})+$
$\displaystyle+\hat{T}_{00}(\vec{r}^{\prime})\bigl{(}\hat{T}^{00}(\vec{r})+\hat{T}^{k}_{k}(\vec{r})-2\hat{\Pi}^{||}(\vec{r})\bigr{)}\biggr{]}\,.$

All terms except the first one are spatially non-local functions of the
stress-energy tensor. Note that $\hat{f}_{\perp,i}(\vec{r})$ and
$\hat{\Pi}^{||}(\vec{r})$ are defined via the stress energy tensor. For
further details and derivation of (29), the reader is referred to Appendix
IX.2.

The Hamiltonian in Poisson gauge (29) can generate entanglement between two
masses through the exchange of the physical spin-2 gravitons (the term
$\hat{s}^{TT}_{ij}$) and/or the non-local interaction between the quantized
matter degrees of freedom (2nd and 3rd line, which are both non-local). In the
regime in which gravitational radiation is irrelevant, as it is the case for
the GIE proposal, the entanglement is generated non-locally via direct matter
interactions. In direct analogy to the toy model for QED entanglement
generation, as discussed in section III, one can now directly compute how the
two masses entangle in this non-local formulation. The computation carries
over one-to-one: the non-local term in (29) that involves
$-\frac{G}{2}\int\frac{d^{3}rd^{3}r^{\prime}}{|\vec{r}-\vec{r}^{\prime}|}\hat{T}_{00}(\vec{r}^{\prime})\hat{T}^{00}(\vec{r})$
generates the entanglement, since all other terms are negligible. No mediators
appear at any stage of the calculation. In contrast, the authors of Ref.Bose
_et al._ (2022) arrive at this same interaction term above in the local
Lorentz gauge, where quantized virtual mediators appear as discussed in the
previous section – the two physical predictions are thus equivalent, as
expected, but the ontology can be local or non-local. Additionally, also in
the Lorentz gauge the authors show explicitly that no virtual spin-2 gravitons
appear in the non-relativistic limit relevant for GIE, only the auxiliary,
unobservable mediators. Thus neither in the Lorentz gauge nor in the Poisson
gauge do the spin-2 gravitons contribute (their contribution is gauge
invariant), not even in virtual processes.

Overall, from the non-local picture presented here in the Poisson gauge, the
relevant interaction term appears without the involvement of any local
mediators. Thus one cannot use the LOCC argument to infer the existence of
quantized mediators in the gravitational case either.

In this context, we also refer to Zel’dovich and Novikov Zel’dovich and
Novikov (2014), who say: _[a] popular but untrue assertion is that the
gravitational interaction is due to an “exchange of gravitons.” […] The
difference in character between the longitudinal and the transverse fields
clearly illustrates the absurdity of a literal interpretation of the phrase,
“Interaction is an exchange of quanta.”_ While this quote highlights the
caveats of a strictly local interpretation and emphasizes the equivalent non-
local one, in our view the two are simply equally viable. Any preference boils
down to imposing additional assumptions about the nature of physics that are
not distinguishable in relativity, such as imposing that only physical degrees
of freedom are quantised, or alternatively that physics is strictly local.
Only if the latter is imposed, such that the equivalent non-local description
is simply excluded by hand, can the LOCC argument be used to indirectly infer
quantized mediators of gravity.

###  IV.3 No mediators: Weak-field quantum gravity as absorber theory

Additionally, in analogy to electromagnetism, in the weak-field regime of
gravity there also exists an absorber formulation,Rosen (1979); Louis-Martinez
(2012a) which would comprise a weak-field quantum gravity after source mass
quantisation. Thus the same ambiguity that we highlighted in the
electromagnetic case also exists in the weak-field limit of General
Relativity: Mediators can be fully removed from the theory by changing to a
fundamentally non-local source and absorber ontology. Within this framework
alone, it is therefore not possible to conclude that entanglement is generated
through quantized, local mediators from the outcome of the GIE experiment and
validity of Post-Newtonian gravity alone.

However, no absorber formulation of full general relativity has been found so
far. It is currently not known if a classical absorber theory for gravity can
exist at all. Attempts to construct absorber theories of non-Abelian gauge
theories and gravity are incomplete,Hoyle and Narlikar (1972); Burgers and Van
Dam (1987); Turygin (1986); Vladimirov (2008); Louis-Martinez (2012b, a) or
failed.Gibbons and Will (2006) Note that the gravitational theory developed by
Hoyle and Narlikar Hoyle and Narlikar (1964) is not an absorber theory of
gravity since the metric appears as degree of freedom in the action.Deser and
Pirani (1965); Hoyle and Narlikar (1966) There are gravitational absorber
theories that correctly reproduce the Post-Newtonian limit of GR for N
particles, the Einstein-Infeld-Hoffman equations,Einstein  _et al._ (1938) see
Refs.Louis-Martinez (2012a); Bolokhov and Klenitsky (2013). This limit however
does not include the effects of gravitational radiation. Thus, whether a
loophole to the LOCC argument in form of a non-local absorber theory applies
also to gravitational interactions depends on whether one can construct a
complete absorber theory for gravity, which is a fascinating open question.
Wesley and Wheeler (2003)

It is nevertheless conceivable that any hypothetical quantum theory of gravity
that is theoretically acceptable and experimentally viable could share the
ambiguity of QED of whether entanglement is produced through virtual
longitudinal and scalar gravitons, or through a direct non-local interaction
of quantized source masses. In the absence of such a theory of quantum
gravity, the conceivable possibilities for interpreting gravitationally
induced entanglement can only be broader. There are even arguments that
suggest that any quantum gravity might exhibit fundamental non-locality.
Giddings and Lippert (2004); Giddings  _et al._ (2006); Giddings (2006) Thus,
observing gravitationally induced entanglement does not tell us whether
gravity is quantized unless we insist on assuming that quantum gravity is a
local quantum field theory – and only the local ontology is permissible. The
resilience of gravity to quantization and certain properties of gravity that
suggest that it would have a locally finite dimensional Hilbert space, could
be an indication that quantum gravity is at odds with being fundamentally a
local QFT. We thus have two possible sets of assumptions for general
relativity: either being fundamentally a local quantum field theory or having
some non-local interpretation. Neither our theoretical understanding today nor
the GIE experiment can distinguish these two. Thus also for gravity the
conclusion from the GIE experiment about quantized mediators depends on what
prior assumptions we are willing to accept, but that are not imposed on us
from our current understanding of physics.

##  V Path integrals

###  V.1 Electromagnetic case

To further clarify the differences in the various interpretations, here we
consider in the electromagnetic case the path integrals involved in the non-
relativistic Coulomb case, Coulomb gauge QED, Lorenz gauge QED, and absorber
theory. This formulation helps highlight the apparent paradoxes and their
resolutions and fully clarifies the relation between all 4 cases in the
electromagnetic case. Interestingly, the mathematical starting points of the
various QED formulations are very different, but for the case of QED they lead
to identical physical outcomes.

The path integral approach to QED assumes that any amplitude between quantum
states of particle configurations and the electromagnetic field can be
calculated from the path integral

$\int\mathcal{D}\mathcal{F}^{\prime}\mathcal{D}x^{\prime}\exp\left(\frac{iS}{\hbar}\right)$
(30)

with the action $S=S[x^{\prime}_{a}(t),\mathcal{F}^{\prime}(x,t)]$ being a
functional of the particle trajectories $x^{\prime}_{a}(t)$, where $a=1,2$
labels the two particles, and the field configuration “trajectories”
$\mathcal{F}^{\prime}(x,t)$. With the measures
$\mathcal{D}\mathcal{F}^{\prime}$ and
$\mathcal{D}x^{\prime}=\Pi_{a}\mathcal{D}x^{\prime}_{a}$ the path integral
sums independently over all conceivable field configuration trajectories
$\mathcal{F}^{\prime}$ and particle trajectories $x^{\prime}_{a}(t)$.

Since the electromagnetic field is a gauge field with unphysical degrees of
freedom care needs to be taken when performing the path integral
$\mathcal{D}\mathcal{F}^{\prime}$. There are several options. One can fix the
gauge completely before performing the path integral, such as in Coulomb
gauge, or one can include a gauge fixing term in the path integral. The latter
can be effectively achieved by quantizing in Lorenz gauge following Gupta and
Bleuler.Gupta (1950); Bleuler (1950)

The theory is called local if the Lagrangian density
$\mathcal{L}[x^{\prime}_{a}(t),\mathcal{F}^{\prime}(x,t)](x,t)$ consists only
of local polynomials of the current $J[x^{\prime}_{a}(t)](x,t)$ and fields
$\mathcal{F}^{\prime}(x,t)$ (and its first and second derivatives). An example
for a local Lagrangian is
$\mathcal{L}(x,t)=J[x^{\prime}_{a}(t)](x,t)\mathcal{F}^{\prime}(x,t)$, whereas
an example for a non-local Lagrangian density is $\int
d^{4}x^{\prime}J(x^{\prime},t^{\prime})D(x^{\prime}-x,t^{\prime}-t)J(x,t)$
with some kernel $D$. In any case the action is given by $S=\int
d^{4}x\mathcal{L}(x,t)$.

####  V.1.1 QED in Lorenz gauge

In the Lorenz gauge the path integral involves the two physical (transverse)
photon polarizations and two unphysical auxiliary photons, the so-called
scalar and longitudinal photon, see Ref.Cohen-Tannoudji  _et al._ (1992), so
that

$\int\mathcal{D}\mathcal{F}_{\rm phys}^{\prime}\mathcal{D}\mathcal{F}_{\rm
aux}^{\prime}\mathcal{D}x^{\prime}\exp\left(\frac{iS}{\hbar}\right)$ (31)

where $S=S[x^{\prime}_{a}(t),\mathcal{F}^{\prime}_{\rm
phys}(x,t),\mathcal{F}^{\prime}_{\rm aux}(x,t)]$. Keeping these auxiliary
photons in the path integral allows the Lagrangian to remain manifestly
Lorentz invariant and local.

####  V.1.2 QED in Coulomb gauge

If we chose to integrate only over physical degrees of freedom, which can be
conveniently done by fixing the gauge completely to Coulomb gauge, we obtain
the path integral

$\int\mathcal{D}\mathcal{F}_{\rm
phys}^{\prime}\mathcal{D}x^{\prime}\exp\left(\frac{iS}{\hbar}\right)$ (32)

where $S=S[x^{\prime}_{a}(t),\mathcal{F}^{\prime}_{\rm
phys}(x,t),\mathcal{F}_{\rm aux}[x^{\prime}_{a}(t)]]$ and with
$\mathcal{F}_{\rm aux}[x^{\prime}_{a}(t)]$ denoting the on-shell, or
classical, solution of the auxiliary field sourced by the charges at
$x^{\prime}_{a}(t)$. The Lagrangian density lost its manifest Lorentz
invariance and its locality. More importantly since $\mathcal{F}_{\rm
aux}[x^{\prime}_{a}(t)]$ is instantaneously determined by the charge
distribution $x^{\prime}_{a}(t)$ at the same time, physics appears to have
become acausal in Coulomb gauge. But this is not the case as discussed further
below.

####  V.1.3 QED absorber formulation

In the absorber theory the path integral only involves the charges:

$\int\mathcal{D}x^{\prime}\exp\left(\frac{iS}{\hbar}\right)$ (33)

with $S=S[x^{\prime}_{a}(t),\mathcal{F}[x^{\prime}_{a}(t_{\rm ret})]]$. Note
in contrast to (34) where the dependence of $\mathcal{F}$ on the trajectories
is instantaneous it is here at the retarded time and thus causal, see
Ref.Davies (1972). Note that we have here omitted the half-advanced-half-
retarded contribution since radiation reaction effects can be neglected in the
GIE experiment. Relevant to the GIE experiment is the case of weak-field
gravity where an equivalent absorber action can be easily constructed which
has been used in Christodoulou  _et al._ (2022) to simplify calculations of
the GIE. Equation 9 in Ref.Christodoulou  _et al._ (2022) is the same as the
absorber action with the radiation reaction contribution neglected, so that a
fully retarded action and the absorber action lead to indistinguishable
results. The crucial point is that the expression only involves the charges,
and not the fields. While this was an approximation in Ref.Christodoulou  _et
al._ (2022), we argue here that it could instead be used as the starting point
of the whole analysis.

####  V.1.4 Non-relativistic Coulomb interactions

The analog of Newtonian gravity is Coulomb theory which is electrodynamics
with everything but the scalar photon discarded. This leaves us with

$\int\mathcal{D}x^{\prime}\exp\left(\frac{iS}{\hbar}\right)$ (34)

where $S=S[x^{\prime}_{a}(t),\mathcal{F}_{\rm aux}[x^{\prime}_{a}(t)]]$ and
$\mathcal{F}_{\rm aux}[x^{\prime}_{a}(t)]$ is the solution to the Coulomb
equation given the charge distribution of $x^{\prime}_{a}(t)$. Like in the
Coulomb gauge of QED $\mathcal{F}_{\rm aux}[x^{\prime}_{a}(t)]$ is
instantaneously determined by the $x^{\prime}_{a}(t)$. As we will discuss
further below, the absence of $\mathcal{F}^{\prime}_{\rm phys}(x,t)$ is
responsible for the physics to be genuinely acausal in this theory.

####  V.1.5 Equivalent predictions

Interestingly, the path integrals for QED in Lorenz gauge, QED in Coulomb
gauge and the absorber theory, all lead to identical predictions for
amplitudes after renormalization. Hoyle and Narlikar (1995) It thus becomes
obvious that the question of “what produces the entanglement in the
electromagnetic GIE experiment?” is ambiguous, even in the relativistic
descriptions. To answer the question in all these cases we just need to look
at what determines the dominant part of phase factors of the four different
semi-classical paths of the two particles (See Figure 1, where the four
distances are LL, LR, RL, RR). In Lorenz gauge QED the phase is due to the
auxiliary fields a.k.a “virtual photons”, in Coulomb gauge QED and the Coulomb
theory it is due to auxiliary fields slaved to the particles due to the
constraint equation, and in the absorber theory it is due to direct particle
interactions on the lightcone in the complete absence of any field. Coulomb
gauge QED, non-relativistic Coulomb theory and absorber theory thus share the
property that entanglement is established without any mediators.

####  V.1.6 Causality

The path integral formulation again suggests that apart from the picture in
the Lorentz gauge, entanglement in QED seems to be established in the same
acausal manner as in Coulomb theory, even though QED is a fundamentally causal
theory. This arises because in QED the physical photons do not contribute to
the entanglement in the proposed experiment in the limit of very slow wave-
packet splitting and merging, and thus the entangling phase factor in the path
integral is dominated by the Coulomb field that is sourced instantaneously
from the charge distribution rather than the retarded charge distribution.
Nevertheless the QED results are fully causal in all formulations: the on-
shell $\mathcal{F}^{\prime}_{\rm phys}[x_{a}](x^{\prime},t)$ combined with
$\mathcal{F}_{\rm aux}[x^{\prime}_{a}(t)](x,t)$ leads to a manifestly causal
expression, which coincides with the expression appearing in the absorber
theory. Interestingly, this precise expression has been explicitly used in
Ref. Christodoulou  _et al._ (2022) (not in the context of absorber theories)
to establish that no entanglement would be produced in a GIE type experiment
if the splitting and merging of the wave packets happened at spacelike
separations. We thus see that while $\mathcal{F}_{\rm aux}$ dominates the
phase factor in the GIE experiment, one needs to include $\mathcal{F}_{\rm
phys}$ to restore causality, or generally whenever a near field approximation
is not a good approximation and retardation effects become important. At
spacelike distances the would-be entanglement of $\mathcal{F}_{\rm aux}$ is
exactly cancelled by $\mathcal{F}_{\rm phys}$.

###  V.2 Gravitational path integrals

The discussion carries over analogously to linearized gravity. In the case of
nonlinear gravity, the equivalence between a field path integral and an
absorber theory will likely break down. However one could simply postulate a
theory of quantum gravity in which the path integral only involves matter and
the gravitational field is “slaved” to the matter configuration like in an
absorber theory. In other words, one would associate with each quantum state
of matter a correspondingly sourced gravitational field and allow for coherent
superpositions of such source-gravity entangled states. This type of
“simplified” quantum gravity model can be a useful approach, as in e.g. Refs.
Zych  _et al._ (2019); Giacomini and Brukner (2020) and would correctly
predict the GIE experiment because it automatically has the same linearized
weak-field limit that was assumed to hold for a “proper” quantum gravity
theory in which geometries are summed over in the path integral independently
of the matter.Christodoulou  _et al._ (2022) The two situations would likely
only be distinguishable in the strong field regime where graviton loops
contribute in the latter case but not in the former. As long a we limit
ourselves to weak gravity, however, the equivalence between the local and non-
local path integral formulations as discussed in the QED case remains.

##  VI Cosmology and signatures of quantum gravity

In the previous chapters we have considered the GIE experiment from the point
of view of different theoretical (or even meta-physical) assumptions. Here we
discuss how the same assumptions applied to cosmology can provide similar
conclusions from already existing observations.

Quantizing the weak-field linearized gravitational field is standard procedure
within inflationary cosmology.Mukhanov and Chibisov (1981, 1982); Mukhanov
(2005) In this scenario the Universe’s energy density is initially dominated
by a slowly evolving scalar field $\varphi$ in an approximately homogeneous
and isotropic state $\bar{\varphi}$, which according to General Relativity
leads to an approximate de Sitter state of spacetime in which vacuum
fluctuations of the metric and the scalar field are amplified and stretched to
superhorizon scales. These fluctuations then seed the cosmic large scale
structure after the end of inflation by giving rise to local variations in the
energy density of radiation. This in turn is perceived as temperature
anisotropies in the Cosmic Microwave Background today. The full Einstein-
Hilbert and scalar field action is expanded around the time-dependent
Friedmann-Robertson-Walker background with scale factor $a(t)$ so that the
scalar fluctuations $v$ are governed by the action

$S=\frac{1}{2}\int d^{4}x(v^{\prime
2}-\partial_{i}v\partial_{i}v+\frac{z^{\prime\prime}}{z}v^{2})\,.$ (35)

Here ′ denotes a derivative with respect to conformal time,
$\mathcal{H}=a^{\prime}/a$ is the conformal Hubble parameter and
$v(t,\vec{x})=a(\delta\varphi(t,\vec{x})+\psi(t,\vec{x})\bar{\varphi}^{\prime}/\mathcal{H})$
is a combination of the scalar field fluctuation
$\delta\varphi=\varphi-\bar{\varphi}$ and the Newtonian scalar perturbation of
the spatial metric $g_{ij}=a^{2}(1-2\psi)\delta_{ij}$.Mukhanov (1988) The
function $z=a\bar{\varphi}^{\prime}/\mathcal{H}$ is purely time dependent, and
thus classical. The quantity $v(t,\vec{x})$, however, is quantized. The gauge
invariantj)j)j)Note that while a gauge transformation could remove the metric
fluctuation $\psi$ from the variable $v$, quantisation of $v$ still implies
the quantisation of scalar metric perturbations via the perturbed Einstein
equation (36). quantized gravitational field $\hat{\Phi}(t,\vec{x})$ is then
given by

$\nabla^{2}\hat{\Phi}(t,\vec{x})=4\pi
G\frac{\bar{\varphi}^{\prime}{}^{2}}{\mathcal{H}^{2}}\Big{(}\frac{\hat{v}(t,\vec{x})}{z}\Big{)}^{\prime}.$
(36)

This is the quantized Newtonian gravitational potential as we discussed in
section II, sourced by the quantized perturbations of the inflaton field. Only
if the gravitational field is quantized in this way, and in
essentiallyk)k)k)The only difference is that the weak-field is quantized
around a Friedmann-Robertson-Walker background instead of a Minkowski
background and instead of particles the source is a scalar field. Apart from
this, equations (36) and (5) have the same physical meaning. the same way as
required for the GIE experiment to produce the expected entanglement, does the
inflationary paradigm predict metric fluctuations with variance $\langle
0|\hat{\Phi}(t,\vec{x})\hat{\Phi}(t,\vec{x}+\vec{r})|0\rangle$ with an
amplitude and $r$-dependence in accordance with that observed in cosmological
observations.Planck Collaboration  _et al._ (2020) This variance is directly
linked to the observed angular dependence of the temperature anisotropy
variance of the cosmic microwave background. More specifically, on large
scales the observed fluctuation in temperature
$\hat{T}(t_{0},\vec{x}_{0},\vec{n})/\bar{T}(t_{0})-1$ in a given direction
$\vec{n}$ is proportional to the Newtonian potential
$\hat{\Phi}(t_{r},\vec{x}_{0}-\vec{n}t_{0})$ on the corresponding location at
the last scattering surface, when photons last scattered at time of
recombination $t_{r}$. This is predominantly due to the gravitational red-
shift, known as Sachs-Wolfe effect, of photons climbing out of gravitational
potential wells.Mukhanov (2005) The quantum nature of the sources for the
gravitational Newtonian field are essential, as their quantum fluctuations
(uncertainty) produce the observed signal. Thus, the observed angular
dependence of temperature fluctuations – and some of its qualitative features
on the largest scales, such as anti-correlations of temperature and
polarisation fluctuations on super horizon scales, as well as the small
increase of power towards the largest scales (“red tilt”) – can be considered
evidence for the necessity of quantized gravity.

The prediction of the CMB temperature fluctuation correlation from inflation
thus required the quantisation of the weak-field gravitational field many
decades before the GIE proposal. Several generic a priori predictions based on
canonical single field inflation made 40 years ago Mukhanov and Chibisov
(1981, 1982) have been confirmed observationally, such that one might say that
evidence for quantizing weak-field gravity already exist today. Indeed, mean-
field theory as mentioned in Sec. II would not be adequate to predict the
temperature fluctuations of the Cosmic Microwave Background within the
inflationary paradigm, as no fluctuations would be produced at all during
inflation.

In principle the inflationary paradigm also predicts the amplification and
stretching of gravitational radiation which leaves a distinctive imprint in
the polarisation of the temperature fluctuations, so-called primordial B-mode
polarisation. Currently there are only upper bounds on the amplitude of this
type of anisotropy which has already ruled out certain simple shapes of the
inflaton potential.Planck Collaboration  _et al._ (2020) Observation of
primordial B-modes would be evidence for the quantum nature of the spin-2
graviton. But as discussed above, within the assumptions of the GIE proposal,
already the CMB fluctuations as observed indicate the quantisation of the
Newtonian part in similar spirit to the GIE proposal.

Finally, we note that fields evolving under their self-gravity or on an
expanding space can exhibit more intricate quantum features. For instance, the
quantum states of the inflaton or of a light scalar field comprising the dark
matter, as well as the quantum state of gravitational radiation can get
squeezed.Grishchuk and Sidorov (1990); Albrecht  _et al._ (1994); Kopp  _et
al._ (2022); Kuß and Marsh (2021) Cosmological observations might thus provide
unique tests of weak-field quantum gravity,Kanno and Soda (2019); Martin
(2019); Green and Porto (2020) or even of genuine superpositions in more
speculative scenarios, Pikovski and Loeb (2016) should these scalars fields
exist.

##  VII Discussion

###  VII.1 Newtonian quantization in GIE and cosmology

What then does the GIE experiment teach us in addition to current cosmological
observations, as discussed above? First of all, not everyone might believe in
the inflationary paradigm: the conclusion that gravity is quantized requires a
prior belief in the current standard inflation paradigm where it is sourced by
quantum fields. Furthermore, while cosmological observations of the cosmic
microwave background are very accurate, the object that is used to study the
quantisation of gravity has been created more than 13 billion years ago and
modulated by unknown components such as dark matter and dark energy. Any
independent and more direct evidence for the quantisation of weak-field
gravity is thus still very valuable. Furthermore, the GIE experiment probes
more directly the fact that the gravitational field can be put into
superposition, which is necessary for entanglement between source masses to
arise. In contrast, in the cosmological setting it is quantum fluctuations of
masses that source fluctuations in the gravitational field. While this is a
direct consequence of quantizing the gravitational field, it is conceivable
that this effect can be mimicked by an entirely classical theory, e.g. the
standard classical weak-field Einstein equations augmented by a stochastic yet
classical noise term that sources fluctuations maybe during a (classical)
inflation. In other words it is conceivable that the gravitational field is
never put into superposition or any quantum state, albeit considered
implausible in the cosmology community. The strength of the GIE experiment
lies in the difficulty of alternative explanations for producing
gravitationally induced entanglement that would not require the gravitational
field to exist in superposition during the experiment – there are however some
suggestions for alternative models Hall and Reginatto (2018); Kemal Döner and
Großardt (2022) in which LOCC does not necessarily hold since quantum physics
is modified. Overall, observing gravitationally induced entanglement would
give more direct support to the idea that the gravitational field can exist in
superposition.

###  VII.2 Ambiguous and unambiguous conclusions from the GIE experiment

As we have shown, the conclusions from GIE vary depending on what set of
assumptions one uses as a starting point. At the core, the experiment probes
what has been discussed in section II: The expected result is based on non-
relativistic quantum mechanics with gravitational interactions, that is
quantized source masses interacting through a Newtonian potential
$\hat{\Phi}=-Gm_{1}m_{2}/|\hat{\vec{x}}_{1}-\hat{\vec{x}}_{2}|$. Gravity is
introduced into the non-relativistic quantum theory the same way as the
electric Coulomb potential. The experiment, if successful, thus unambiguously
confirms that gravity is sourced coherently by a mass in superposition, from
which one can conclude that Newtonian gravity is coherently associated with a
spatial superposition of the source. To date such a superposition of Newtonian
potentials remains unverified and untested, thus GIE would provide a test of
physics in a new regime. The experiment can also rule out alternative models,
such as a mean-field source for the gravitational potential or a classical
channel model.Kafri  _et al._ (2014) And of course any unexpected result would
hint at new physics.

Going beyond these conclusions, however, is ambiguous. The original aim of the
GIE proposals was to draw conclusions about the nature of quantum gravity by
assuming that this is a relativistic and local theory of which the non-
relativistic limit with its inherent non-locality is merely an artefact. But
as we have shown, even if one accepts this reasonable starting point, still no
unambiguous conclusion about the underlying mediators can be made. This is
because non-local entanglement generation is perfectly allowed even within
relativistic physics. QED is a good example that locality is not a necessary
assumption and does not need to be imposed on such a relativistic parent
theory – as opposed to causality. This is not simply an issue of gauge, but
that locality is not necessarily satisfied for the underlying ontology even in
relativity. This is important here, as the proposed tests are of indirect
nature and the underlying ontology is central for the interpretation of the
outcomes. If a non-local explanation is possible, then LOCC is simply not able
to distinguish between a quantum and a classical mediator. Thus, while the
absorber formulation may be considered as a “quirk” of the theory, it still
shows that locality needn’t be satisfied at the fundamental level, even if all
observations are causal. For gravity, it remains an open question if an
absorber theory can capture all of general relativity, but it exists for the
weak-field limit. Moreover, it is conceivable that quantum gravity is
fundamentally non-local in some way. In all these cases the LOCC argument
breaks down, since the locality condition is no longer satisfied and thus
observation of entanglement generation does not necessarily imply quantised
gravitational mediators.

Thus it is only when locality is imposed as an additional condition that one
can conclude quantization of mediators in such indirect tests. While we agree
that the additional assumption of locality is not just allowed but also
plausible and intuitive,Vedral (2020); Marletto  _et al._ (2021) we have no
indication that it must be satisfied in nature at the fundamental level. A
serious drawback is that imposing locality introduces additional, fictitious
degrees of freedom such as scalar and longitudinal photons in QED, and
analogous types of new gravitons in weak-field quantum gravity, that can never
be directly observed. Yet they are responsible for mediating entanglement in
this picture. The other picture, which is non-local, gets rid of these
unphysical degrees of freedom, and thus has its own merits. Fundamentally,
causality seems to be the underlying principle in all these formulations, and
locality only a convenient consequence in one of them. Thus, the only
unambiguous statement one can draw from the GIE experiments with regards to
mediators is that if one pre-assumes an underlying local field theory, then
such a field theory would need to include quantized mediators. But this
assumption is very restrictive in that it artificially excludes perfectly
valid non-local formulations. Our cosmology example shows that if the same
assumptions about the nature of gravity are accepted, in addition to the
standard inflationary scenario, then already the observed CMB fluctuations are
sufficient to deduce quantum fluctuations of the gravitational field.

###  VII.3 Agents and locality

From a quantum information perspective the GIE proposal involves two localised
agents (Alice and Bob) situated close to their respective particle. Each agent
directly manipulates only their respective nearby particle locally (through
the particle’s spin in the GIE proposalBose  _et al._ (2017)) so as to create
the spatial superposition and subsequently interfere the particle’s wave
function to infer the creation of entanglement. If, however, the two particles
interact non-locally, then Alice’s local interaction with her particle will
inadvertently and unavoidably also affect Bob’s particle non-locally, and vice
versa. Thus the GIE proposal’s experimental setup does not guarantee that the
two agents perform only local operations because the gravitational interaction
between the particles is not necessarily local. The LOCC theorem in
combination with witnessing entanglement does therefore not imply the
existence of quantized mediators since a necessary condition for the theorem
to apply – locality – is not necessarily present and cannot be tested for.

As we discussed in Sec.III.3, in the specific non-local theories studied in
this paper in which quantum physics is unmodified, entanglement is necessarily
and by construction produced quantum mechanically. Our point is that the
existence of the quantum nature of the underlying system that generates
entanglement could not have been inferred by merely invoking the LOCC argument
alone. Even if entanglement is observed, it may be produced by some process
that may correspond to some underlying system that is not quantum mechanical
in the usual sense.Popescu and Rohrlich (1994); Pal  _et al._ (2021) We note
here that it is therefore also interesting to exactly quantify the expected
entanglement, and probe for possible deviations.

###  VII.4 Superpositions of geometries

Can one draw conclusions about the quantum nature of gravity beyond mediators?
It was pointed out in several previous works Christodoulou and Rovelli (2019);
Christodoulou  _et al._ (2022) that it is not the quantum nature of mediators
the GIE proposal is sensitive to, but the ability of the gravitational field
to exist in a quantum superposition. Can one gain insights on quantum gravity
from that alone?

A compelling interpretation is that the successful experiment shows that
geometries of space-times exist in superposition. This is based on additional
assumptions, that we have not yet discussed: The extrapolation of classical
general relativity (GR) and its ontology into the quantum domain. In this
view, the Newtonian limit is just a limiting case of what is space-time
geometry, and any superpositions of the Newtonian field correspond to
superpositions of geometries in the context of GR. One way to see this is in
the path integral formulation of the gravitational case: The phase difference
between the semi-classical paths of the particles is dominated by solutions of
the classical Einstein equations sourced by the semi-classical matter
configurations entering the superposition.Christodoulou  _et al._ (2022) In
this sense the GIE experiment would show that geometry can be put in
superposition. This is true no matter which gauge one adopts, or if one thinks
of the geometry as fundamental or arising from virtual gravitons, the
constraint equation, or direct particle interaction on the light cone. In all
cases superposition of effective geometry is the origin of the entanglement.
The only way to avoid this conclusion is to find a viable gravitational theory
that is not geometrical in the quantum regime and/or to replace the path
integral, and thus quantum mechanics, by some other theory in this limit.
While we are not aware of any theory of gravity consistent with observations
that is not geometrical, it is conceivable that the path integral formula is
not correct if contributions of different geometries would lead to an
observable effect. For instance in Penrose’s gravitational collapse mechanism,
if the wave packets in the GIE experiment were kept a sufficiently long time
apart before they are attempted to be merged, only one of the four semi-
classical contributions to the path integral (LL, LR, RL, RR) would survive
according to Penrose (the other branches are “dying” before the matter wave
packets are attempted to be merged, and thus no merging actually happens), in
which case we might expect the complete absence of entanglement and any other
interference effect of the particles. In general, a modification of the path
integral that would lead to the expected entanglement without having
geometries in superposition would have to be drastically different from
standard quantum mechanics. For instance we could imagine that each path was
weighted with a phase produced by the gravitational mean field (thus geometry
is not in a superposition) but that there is an additional contribution to the
phases of the four semi-classical paths (LL, LR, RL, RR) that involves non-
local interactions between particles that have no geometric interpretation.
While this is highly contrived, we mention this possibility only to highlight
that GIE does not provide any direct evidence of geometries in superposition
beyond the Newtonian limit, and possible deviations beyond this limit are at
least conceivable, such as in Refs.Hall and Reginatto (2018); Kemal Döner and
Großardt (2022). Indeed, some recent works highlight the possible difference
between superposed geometries that have drastically different curvatures and
those that differ only by a displaced source mass, as in the GIE case. Foo
_et al._ (2021, 2022)

Overall, the various conclusions and possibilities for new physics are
summarized in Figure 2.

###  VII.5 Comparison to other works on interpretations of GIE

Various recent works have considered the implications and interpretations of
the GIE experiment. Here we briefly compare them to our results.

While not directly addressing the GIE proposal, an interesting recent related
line of research has emerged that focuses on Gedankenexperiments and what they
imply for the quantization of gravity.Baym and Ozawa (2009); Mari  _et al._
(2016); Belenchia  _et al._ (2018, 2019); Danielson  _et al._ (2022) Belenchia
et al.Belenchia  _et al._ (2018) show how a seeming paradox arises for the
interferometric which-way information between two space-like separated masses,
but that it can be avoided when quantizing gravitational radiation. Such
arguments relate to GIE, as they operate in the same Newtonian limit and
support the view of the need to quantize gravity. Nevertheless, the paradox in
such Gedankenexperiments can also be resolved without quantized radiation, and
thus the question of quantization from these arguments remains
inconclusive.Rydving  _et al._ (2021); Großardt (2021)

Shortly after the GIE proposals, Anastopoulos and Hu pointed out in Ref.
Anastopoulos and Hu (2018) that due to the static Newtonian limit in GIE, the
physical mediators do not play a role and that the mediator-interpretation is
misguided. The authors argued that no quantization of the mediators can
therefore be inferred from the Newtonian limit, and discussed it further in
Refs. Anastopoulos and Hu (2020); Anastopoulos  _et al._ (2021) This aligns
with what we discuss in the present manuscript as well, mainly in section III
where we show that (unphysical) mediators only appear when a specific gauge
choice is made. We make the additional observation that both local and non-
local formulations are not gauge artefacts, but viable interpretations of
known relativistic physics.

In a recent work, Bose et al.Bose  _et al._ (2022) show how gravitational
entanglement is explicitly generated in a local picture, in analogy with a
previous work by Franson, Franson (2011) as we also use in section III.1. The
toy model involves virtual mediators, albeit not the physical ones. However,
this demonstration alone is insufficient to infer their existence, it merely
shows that it is possible to describe entanglement in this form. One can
equivalently describe the creation non-locally, as we show in Sec. III.2 and
Appendix IX.1. This invalidates somewhat the claims in Ref.Bose  _et al._
(2022), where it is argued that the presence of virtual gravitons is essential
for the masses to get entangled. The same can be achieved non-locally without
gravitons in this proposal, as we showed in Sec. IV and Appendix IX.2.

Providing a complementary perspective on GIE and inferred quantization, Carney
has proven the existence of quantized physical gravitons based on three
assumptions:Carney (2022) that quantized source particles interact via a $1/r$
potential in the non-relativistic limit, that the $S$ matrix theory is unitary
and that it is Lorentz invariant. This seems in contradiction with the
discussion in our work here, which highlights that a quantized absorber theory
of weak-field gravity provides an alternative theory without gravitons. The
resolution of this seeming contradiction is a loophole in Carney’s proof. If
one is willing to make the cosmological assumption that the far future
contains a complete gravitational absorber, then the $1/r$ potential, the
unitarity and the Lorentz invariance do not require the existence of physical
graviton states, or a Hilbert space associated with such states. This is
discussed in more detail in Appendix IX.3.

It was pointed out by Rovelli and Christodoulou that the fundamental mediators
of gravity are not the central objects of interest in the GIE setup. The
authors instead highlight the quantum nature of the sources and conclude that
an expected outcome in the GIE experiment would imply that gravitational
fields can exist in superposition.Christodoulou and Rovelli (2019)
Superposition of geometries, as discussed in Sec. VII.4 are then inferred from
an extension of the Newtonian superpositions to general relativistic
superpositions.

In a recent paper, Christodoulou et al.Christodoulou  _et al._ (2022) analyze
GIE starting from a relativistic description based on path integrals, in order
to highlight the inherent locality of entanglement generation in the
relativistic setting. But as we have discussed in Sections III andV,
relativistic Lorentz invariance does not necessarily mean that physics has to
be local, but that causality has to be preserved. The local interpretation is
only true for the formulation in the Lorentz gauge, which is what is
exclusively used in Ref.Christodoulou  _et al._ (2022). The same can be
described in terms of non-local processes in the path integral formulation,
both in a Poisson gauge and in terms of absorber theory, as we showed in
section V. While a modified setup that allows for retardation effects as
proposed in Ref.Christodoulou  _et al._ (2022) would involve local terms also
in the Poisson gauge, the absorber non-local loophole still remains.

Finally, we mention that some alternative models have been proposed that do
not involve the expected quantized Newtonian fields but that can still explain
a positive outcome of the GIE experiment.Hall and Reginatto (2018, 2021);
Kemal Döner and Großardt (2022) These unorthodox semi-classical models are
interesting test theories that warrant further study, at the very least as
possible test theories against which the expected predictions can be probed
and to design schemes that may distinguish them.

##  VIII Conclusions

In this paper we have critically assessed possible conclusions that one can
draw from observing gravitationally induced entanglement. We have shown that
entanglement can be generated by quantized mediators, but that such an
interpretation is not unique. The possibility of non-local generation of
entanglement is also viable even within known relativistic physics (absorber
theory being one example), and thus no conclusion can be drawn about mediators
unless the locality condition is additionally imposed. While locality is a
choice and not fundamental in both QED and general relativity, causality is
always preserved even in non-local formulations. The experiment nevertheless
probes quantum aspects of gravity independently of the LOCC argument, as it
involves superpositions of gravitational source masses, demonstrating the
creation of superpositions of the Newtonian gravitational field.

Any interpretation beyond this relies on assumptions about the nature of
quantum gravity which may or may not be true, and thus remain ambiguous. If
locality is elevated to a fundamental assumption, then LOCC indeed predicts
the existence of quantized mediators if GIE is witnessed. Intense debates
about the proposal thus reach different conclusions, depending on what prior
assumptions one is willing to accept as “most natural”. Since the proposed
tests are of indirect nature, the choice for the underlying ontology is
crucial for the conclusions one can draw. The electromagnetic case differs
somewhat from the gravitational one, as the former can always be cast in a
non-local absorber formulation, while gravity is only known in the Newtonian
and post-Newtonian limits to have a non-local absorber picture. In both cases,
also an interpretation with local mediators of entanglement is permitted,
although the involved mediators are not the observable physical bosons. But
since non-local formulations are permitted as well, observing the expected
entanglement would be compatible with both the non-existence and existence of
local quantized mediators. We also showed that under reasonable prior
assumptions, one can already claim tests of the quantum nature of the
Newtonian gravitational field in cosmological observations. Such a claim,
however, relies additionally on the validity of the usual inflationary
paradigm.

In sum, there are three types of conclusions one can draw from GIE: i) The
experiment can probe some alternative theories that would yield an unexpected
outcome, such as Penrose collapse, classical local channel models or mean
field theory. ii) An expected result will show that spatial superpositions of
matter coherently source Newtonian gravity, yielding superpositions of
Newtonian fields. The caveat is that if one assumes the usual inflationary
paradigm, matter in quantum states sourcing Newtonian gravity is already shown
in current CMB observations. iii) If one assumes an underlying relativistic
theory for gravity, different interpretations exist on how entanglement is
generated. One of them includes local mediators, which however are not the
usual observable bosons. In such an interpretation these mediators have to be
quantized. But entanglement generation can equally be described in terms of
non-local processes in the relativistic setting, thus induced entanglement is
insufficient to conclude the quantization of mediators. Overall, our results
clarify that while quantum superpositions of gravitational sources are probed
in such proposals, claims about mediators remain ambiguous, both in the
electromagnetic and in the gravitational setups.

## Acknowledgements

We thank Erik Aurell, Fawad Hassan, Myungshik Kim, Jonas Larson, Florian
Niedermann, Germaine Tobar, Frank Wilczek and Ting Yu for useful discussions
and feedback. This work was supported by the Swedish Research Council under
grant no. 2019-05615, the European Research Council under grant no. 742104 and
The Branco Weiss Fellowship – Society in Science.

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Authors contributions

∗ VF and MK contributed equally to this work.

##  IX Appendix

###  IX.1 QED in Coulomb gauge

For the sake of completeness, in this section we will explicitly show how
entanglement between two charged quantum particles, is generated non-locally
yet causally, in the Coulomb gauge. The interaction Hamiltonian between
charged particles and EM field is

$H_{int}=\int d^{3}rj_{\mu}A^{\mu}=\int
d^{3}r\bigl{(}-\rho\Phi_{em}+\vec{j}\cdot\vec{A}\bigr{)}\,.$ (37)

Writing the electric field in terms of the potentials
$\vec{E}=-\nabla\Phi_{em}-\partial_{t}\vec{A}$ and substituting into the Gauss
law

$\nabla\cdot\vec{E}=\frac{\rho}{\epsilon_{0}}$ (38)

we end up with a Poisson equation for $\Phi_{em}$

$\nabla^{2}\Phi_{em}=-\nabla\cdot\frac{\partial\vec{A}}{\partial
t}-\frac{\rho}{\epsilon_{0}}\,.$ (39)

The solution of (39) is

$\Phi_{em}(\vec{r})=\int
d^{3}r^{\prime}\frac{\nabla\cdot\partial_{t}\vec{A}+\frac{\rho}{\epsilon_{0}}}{4\pi|\vec{r}-\vec{r}^{\prime}|}\,.$
(40)

From (40), it’s clear that the $0$ component of the four vector $A_{\mu}$,
namely $\Phi_{em}$ , is not an independent degree of freedom. Instead, it’s
completely determined by $\vec{A}$ and the presence of matter $\rho$. It
therefore means that the four vector $A_{\mu}$ contains three degrees of
freedom. Moreover, once the $U(1)$ gauge symmetry of QED is fixed, we will end
up with two physical degrees of freedom which correspond to the two
polarizations of the photon. The Coulomb gauge condition reads

$\nabla\cdot\vec{A}=0\,.$ (41)

This condition is leaving behind 2 degrees of freedom and implies that the
vector field $\vec{A}$ is transverse, namely $\vec{A}\equiv\vec{A}_{\perp}$.
Relation (40) now takes the form

$\Phi_{em}(\vec{r})=\frac{1}{4\pi\epsilon_{0}}\int
d^{3}r^{\prime}\frac{\rho(\vec{r}^{\prime})}{|\vec{r}-\vec{r}^{\prime}|}$ (42)

which corresponds to the Coulomb potential. The resulting interaction
Hamiltonian in Coulomb gauge is therefore

$H_{int}=-\frac{1}{8\pi\epsilon_{0}}\int
d^{3}rd^{3}r^{\prime}\frac{\rho(\vec{r})\rho(\vec{r}^{\prime})}{|\vec{r}-\vec{r}^{\prime}|}+\int
d^{3}r\vec{j}\cdot\vec{A}_{\perp}\,.$ (43)

Upon quantization, this Hamiltonian is responsible for entangling two charged
quantum particles via the non-local, instantaneous Coulomb interaction and/or
via virtual spin-1 photons. In the non-relativistic regime in which the latter
is irrelevantFranson (2011), namely for slowly moving particles, as it is the
case for electromagnetic analogue of the GIE proposal,Bose  _et al._ (2017);
Marletto and Vedral (2017) we have to conclude that entanglement generation is
due to the Coulomb term, which corresponds just to quantized source masses.
This conclusion had also been pointed out in Ref. Anastopoulos and Hu (2018).

It’s crucial to notice that despite the various non-local (instantaneous)
pieces that appear in the formulation, QED in Coulomb gauge describes causal
physical processes.Jackson (2002); Wong (2010); Jefimenko (1989); Rohrlich
(2002); Gardiner and Drummond (1988)

###  IX.2 Weak-field gravity in Poisson gauge

The aim of this section is to show how entanglement between the masses can be
generated non-locally without any reference to gravitons. We expand on the
exposition in Ref. Bertschinger (1999). In the linear regime of general
relativity, the metric tensor is written as
$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, where $|h_{\mu\nu}|\ll 1$ and
$\eta_{\mu\nu}$ corresponds to the flat Minkowski metric tensor. Decomposing
the linear field $h_{\mu\nu}$ according to its transformation properties under
spatial rotations in terms of scalar, vector and tensor components
$(\phi,\psi,w_{i},s_{ij})$, we have

$\displaystyle h_{00}$ $\displaystyle=-2\phi$ (44) $\displaystyle h_{0i}$
$\displaystyle=w_{i}$ (45) $\displaystyle h_{ij}$
$\displaystyle=-2\psi\delta_{ij}+2s_{ij}$ (46)

where $s_{ij}$ is the so-called traceless strain, $s_{j}^{j}=0$. So far, no
gauge condition has been imposed. The interaction Hamiltonian $H_{int}$
between the weak gravitational field and matter is given

$H_{int}=-\frac{1}{2}\int d^{3}rh_{\mu\nu}(\vec{r})T^{\mu\nu}(\vec{r})$ (47)

or equivalently

$H_{int}=\int d^{3}r\bigl{[}\phi
T^{00}+(\psi\delta_{ij}-s_{ij})T^{ij}-w_{(i}T^{i)0}\bigr{]}\,.$ (48)

The Poisson gauge condition

$\displaystyle\partial_{j}s^{j}_{i}=0\quad\textrm{and}\quad\partial_{i}w^{i}=0$
(49)

implies that the traceless strain $s_{ij}$ as well as the vector $w_{i}$ are
spatially transverse. An $s_{ij}$ that satisfies eq. (49) is denoted by
$s^{TT}_{ij}$. Similarly we denote a $w_{i}$ satisfying eq. (49) by
$w_{i}^{\perp}$. In this gauge, the interaction Hamiltonian is written as

$H_{int}=\int d^{3}r\bigl{[}\phi
T^{00}+(\psi\delta_{ij}-s_{ij}^{TT})T^{ij}-w_{(i}^{\perp}T^{i)0}\bigr{]}\,.$
(50)

The dynamics of remaining six degrees of freedom
$(\phi,\psi,s_{ij}^{TT},w_{i}^{\perp})$ are dictated by Einstein’s equation,
which in the Poisson gauge takes the following form Bertschinger (1999)

$\nabla^{2}\psi=4\pi GT_{00}$ (51)
$\nabla\cdot(-\nabla\phi-\partial_{t}\vec{w}_{\perp})-3\partial_{t}^{2}\psi=-4\pi
G(T_{00}+T^{i}_{i})$ (52) $\nabla\times\vec{H}=-16\pi G\vec{f}_{\perp}$ (53)
$\displaystyle\bigl{(}\partial_{i}\partial_{j}-\frac{1}{3}\delta_{ij}\nabla^{2}\bigr{)}(\psi-\phi)$
$\displaystyle=8\pi G\Pi_{ij}^{||}$ (54) $\Box s_{ij}^{TT}=8\pi
G\Pi_{ij}^{TT}$ (55)

where $\vec{H}=\nabla\times\vec{w}_{\perp}$ and $\vec{f}\equiv
T^{0i}\vec{e}_{i}$.

$\Pi^{TT}_{ij}$ corresponds to the transverse and traceless part of the stress
energy tensor. Making use of the so-called Helmholtz decomposition, the three-
vector $\vec{f}$ can be decomposed into longitudinal and transverse parts as
follows

$\vec{f}=\vec{f}_{||}+\vec{f}_{\perp}$ (56)

where

$\vec{f}_{\perp}=\frac{1}{4\pi}\nabla\times\nabla\cdot\int
d^{3}r^{\prime}\frac{\vec{f}(\vec{r}^{\prime})}{|\vec{r}-\vec{r}^{\prime}|}$
(57)

and

$\vec{f}_{||}=-\frac{1}{4\pi}\nabla\int
d^{3}r^{\prime}\frac{\nabla^{\prime}\cdot\vec{f}(\vec{r}^{\prime})}{|\vec{r}-\vec{r}^{\prime}|}\,.$
(58)

The function $\Pi_{ij}$ is the traceless spatial part of the stress energy
tensor

$\Pi_{ij}\equiv T_{ij}-\frac{1}{3}\delta_{ij}T^{k}_{k}\,.$ (59)

In a similar way, it can be decomposed as

$\Pi_{ij}=\Pi_{ij}^{||}+\Pi_{ij}^{\perp}+\Pi_{ij}^{TT}$ (60)

where its longitudinal part is defined as

$\displaystyle\nabla^{2}\Pi_{ij}^{||}(\vec{r})$
$\displaystyle=\bigr{(}\partial_{i}\partial_{j}-\frac{1}{3}\delta_{ij}\nabla^{2}\bigl{)}\Pi_{||}(\vec{r})$
(61) $\displaystyle\Pi_{||}(\vec{r})$
$\displaystyle=-\frac{1}{4\pi}\frac{3}{2}\int
d^{3}r^{\prime}\frac{{\partial^{i}}^{\prime}{\partial^{j}}^{\prime}\Pi_{ij}(\vec{r}^{\prime})}{|\vec{r}-\vec{r}^{\prime}|}$
(62)

and its rotational part

$\Pi_{ij}^{\perp}(\vec{r})=\partial_{i}\Pi^{\perp}_{j}+\partial_{j}\Pi^{\perp}_{i}$
(63)

in terms of a divergence-less spatial vector $\Pi^{\perp}_{j}$.
$\Pi^{\perp}_{j}$ in turn is defined in terms of $\partial^{j}\Pi_{ij}$
analogously to $\vec{f}_{\perp}$. The last term in (60), $\Pi^{TT}_{ij}$,
corresponds to the gauge invariant transverse part. Applying the same steps to
the metric perturbations provides a procedure for the extraction of
$s_{ij}^{TT}$ from $h_{ij}$.

We note in passing that, in analogy with electrodynamics, where the Poisson
(constraint) equation $\nabla\cdot\vec{E}=\rho/\epsilon_{0}$ enforces charge
conservation $\partial_{\mu}J^{\mu}=0$, in the weak-field of GR, gauge
invariance, namely invariance under infinitesimal change of the coordinates
$x^{\mu}\rightarrow x^{\mu}-\xi^{\mu}$, implies four conserved quantities
$\partial_{\mu}T^{\mu\nu}=0$. This equation shows the limitations of weak-
field gravity since it contradicts the equations of motion of matter. This
inconsistency can only be avoided after resumming all perturbation orders to
get back full general relativity.

The set of equations (51)-(55) describe the causal dynamics of the metric
components in Poisson gauge. Neglecting some pieces, or focusing only on the
constraint equations, would erroneously lead to the conclusion that GR in the
weak-field limit describes non-causal physics.

Generic boundary
condition:${p}^{\prime}_{1}$${p}^{\prime}_{2}$${p}_{1}$${p}_{2}$${k}$${k}^{\prime}$$\tilde{k}$$\Rightarrow$${p}^{\prime}_{1}$${p}_{2}$${p}_{1}$${p}_{2}$${k}$$\tilde{{k}}$

Absorbing boundary
condition:${p}^{\prime}_{1}$${p}^{\prime}_{2}$${p}_{1}$${p}_{2}$${k}$${k}^{\prime}$$\tilde{k}$${p}_{3}$${p}_{3}$$\Rightarrow$${p}^{\prime}_{1}$${p}_{2}$${p}_{1}$${p}_{2}$${k}$$\tilde{{k}}$${p}^{\prime}_{3}$${p}_{3}$

Figure 3: Upper panel: Unitarity at the tree-level of a six-point amplitude
(left) implies the existence of diagrams with on-shell graviton lines (right),
as described in Ref.Carney (2022). Lower panel: Absorbing boundary conditions
can be implemented by having one additional absorber particle $p_{3}$ with
trivial propagator as disconnected line “participating” in the scattering on
the left. In this case the pole can be removed from the scattering amplitude
and the gravitons do not need to exist.

The solutions of (51) and (53) are easily obtained

$\psi=-G\int
d^{3}r^{\prime}\frac{T_{00}(\vec{r}^{\prime})}{|\vec{r}-\vec{r}^{\prime}|}$
(64) $\vec{w}_{\perp}=-4G\int
d^{3}r^{\prime}\frac{\vec{f}_{\perp}(\vec{r}^{\prime})}{|\vec{r}-\vec{r}^{\prime}|}\,.$
(65)

Inserting (61) into $\eqref{constraints}$ one obtains the Poisson-like
equation

$\nabla^{2}(\psi-\phi)=8\pi G\Pi^{||}\,$ (66)

which can be solved to give

$\phi(\vec{r})=\psi(\vec{r})+2G\int
d^{3}r^{\prime}\frac{\Pi^{||}(\vec{r}^{\prime})}{|\vec{r}-\vec{r}^{\prime}|}$
(67)

where the scalar function $\Pi_{||}(\vec{r})$ is defined in (62).

Inserting the solution to constraint equations (64),(65) and (67) into the
(50) we end up with an expression for the interaction Hamiltonian which only
depends on the gauge invariant spin-2 gravitons $s_{ij}^{TT}$ and the matter
degrees of freedom

$\displaystyle H_{int}=$ $\displaystyle-\int
d^{3}r\bigl{(}s_{ij}^{TT}(\vec{r})T^{ij}(\vec{r})\bigr{)}-$ (68)
$\displaystyle-\frac{G}{2}\int\frac{d^{3}rd^{3}r^{\prime}}{|\vec{r}-\vec{r}^{\prime}|}\biggl{[}-4f_{\perp,i}(\vec{r}^{\prime})T^{0i}(\vec{r})+$
$\displaystyle+T_{00}(\vec{r}^{\prime})\bigl{(}T^{00}(\vec{r})+T^{k}_{k}(\vec{r})-2\Pi^{||}(\vec{r})\bigr{)}\biggr{]}$

where $f_{\perp,i}\propto T_{\perp}^{0i}$. The first term in (68) describes
gravitational radiation. The second and third line correspond to non-local
terms. The third term contains the standard Newtonian instantaneous
interaction between mass densities. Note that in the fully nonperturbative
case the Hamiltonian can be decomposed in a similar fashion by reinserting the
solution of algebraic and elliptic equations in a fully constrained
gauge.Bonazzola  _et al._ (2004) Thus, slowly moving charges entangle
predominantly at short distances without the physical gravitational degrees of
freedom even if spacetime curvature is strong. Quantization of the Hamiltonian
(68) is straightforward, as all redundant gravitational degrees of freedom
have already been removed and only the term $s_{ij}^{TT}$ remains, which
corresponds to the spin-2 gravitons. Quantization then yields eq. (29) in the
main text. One can see that entanglement is generated either via the physical
gravitons (1st term $\propto s_{ij}^{TT}$ in (68))) and/or via the non-local
matter terms (the remaining terms).

After quantization, and  since the experiment is quasi-static with all $v\ll
c$, the only relevant component is the $\hat{T}_{00}(\vec{r})$ term.
Therefore, (68) takes the form

$\hat{H}_{int}\approx-\frac{G}{2}\int\frac{d^{3}rd^{3}r^{\prime}}{|\vec{r}-\vec{r}^{\prime}|}\hat{T}_{00}(\vec{r}^{\prime})\hat{T}^{00}(\vec{r})$
(69)

For
$\hat{T}_{00}(\vec{r})=m_{1}\delta(\vec{r}-\hat{\vec{r}}_{1})+m_{2}\delta(\vec{r}-\hat{\vec{r}}_{2})$,
we get

$\hat{H}_{int}=-\frac{Gm_{1}m_{2}}{|\hat{\vec{r}}_{1}-\hat{\vec{r}}_{2}|}$
(70)

which corresponds to the Newtonian interaction, as discussed in Section II.

###  IX.3 Graviton existence theorem and the absorber loophole

It was recently proven in Ref.Carney (2022) that unitarity and Lorentz
invariance, together with the assumption that non-relativistically gravity is
a $1/r$ potential, imply the existence of physical gravitons, or in other
words the existence of quantum states of gravitational degrees of freedom. As
we discussed in the main text, assuming locality and the LOCC argument does
not directly allow to conclude the existence of physical graviton states, only
the existence of auxiliary or “unphysical” gravitons that can only appear in
virtual processes. These gravitons are the analog of the scalar and
longitudinal photons of QED.

It therefore seems that Carney’s existence proof of physical gravitons is not
only stronger than the LOCC argument but also already shows the existence of
the graviton without the need to perform the GIE experiment. More strikingly,
Carney’s proof which is also formulated in the weak-field regime of gravity,
seems to be in contradiction with the fact that an absorber theory of weak-
field quantum gravity is unitary, Lorentz invariant and has a classical $1/r$
potential. Thus the properties that Carney assumed are present in an absorber
theory, but that by construction has no gravitons!

In this appendix we point out the hidden additional assumption that goes into
the proof and allows one to infer the existence of the graviton, and that
dropping this extra assumption is then consistent with the existence of a
gravitational absorber theory that has no gravitons. To summarise our result:
unitarity and Lorentz invariance, and the $1/r$ potential only imply the
graviton if absorbing boundary conditions are excluded. This exclusion was the
additional hidden assumption in Ref.Carney (2022)

Carney considered a tree-level process in which two masses are controlled
electromagnetically, for instance to produce the superpositions in the GIE
experiment, and then interact through the $1/r$ potential. Considering a six-
point amplitude, see upper left diagram in Fig. 3, involving the photon (curvy
lines), massive particles (full lines) and gravitational interaction (dashed
line), the existence of diagrams with on-shell graviton lines is derived, see
the upper right diagram. The pole at $\tilde{k}^{2}\rightarrow 0$ has an
imaginary residue. This residue is equal to the product of a pair of
amplitudes, one shown on the upper right of Fig. 3, where the physical
graviton (dashed line) is emitted into the final state. The disconnected line
represents a trivial propagator. If this diagram, and thus physical graviton
states did not exist, unitarity is violated. This was in a nutshell Carney’s
argument.

Now we come to the absorber loophole. In an absorber ontology the diagram in
the upper left of Fig. 3 is incomplete and misses the existence of absorber
particles. While most absorber particles are not involved in any given
scattering event, some always will be. This is indicated by the lower left
diagram in Fig. 3. This “spectator” absorber allows then to restore unitarity
through the diagram shown in the lower right. It absorbs the would-be
graviton. In the absorber ontology, there are no on-shell gravitons, so the
graviton pole does not exist in the absorber theory, and thus restoration of
unitarity only requires proper consideration of absorbing boundary condition,
and no introduction of graviton quantum states.

Note that absorbing boundary conditions do not exclude the existence of a
physical graviton state, but they do not require such a state either. Thus
gravitons cannot be inferred from Carney’s argument if absorbing boundary
conditions are imposed.

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