# Gravitational Decoherence: A Thematic Overview

Charis<EMAIL_ADDRESS>
Department of Physics, University of Patras, 26500 Patras, Greece
and
Bei-Lok<EMAIL_ADDRESS>
Maryland Center for Fundamental Physics and Joint Quantum Institute,
University of Maryland, College Park, Maryland 20742-4111, USA

(November 3, 2021)

###### Abstract

Gravitational decoherence (GD) refers to the effects of gravity in actuating
the classical appearance of a quantum system. Because the underlying processes
involve issues in general relativity (GR), quantum field theory (QFT) and
quantum information, GD has fundamental theoretical significance. There is a
great variety of GD models, many of them involving physics that diverge from
GR and/or QFT. This overview has two specific goals along one central theme:
(i) present theories of GD based on GR and QFT and explore their experimental
predictions; (ii) place other theories of GD under the scrutiny of GR and QFT,
and point out their theoretical differences. We also describe how GD
experiments in space in the coming decades can provide evidences at two
levels: a) discriminate alternative quantum theories and non-GR theories; b)
discern whether gravity is a fundamental or an effective theory.

— Invited paper in a Special Issue of AVS-QS in celebration of Sir Roger
Penrose’s 2020 Physics Nobel Prize Award.

##  I Introduction

Gravitational decoherence (GD) is a fundamental issue of theoretical physics
because gravitation is a universal interaction and decoherence is an essential
factor in how a quantum system assumes some classical behavior. GD encompasses
the issues of both the quantum-to-classical and the microscopic-to-macroscopic
transitions. The quantum-to-classical transition is the description of a
quantum system in terms of classical physics. The microscopic-to-macroscopic
transition is the accurate description of systems with $10^{23}$ degrees of
freedom in terms of a few observables, as in thermodynamics.

By their very nature, GD and gravitational entanglement (GE) also involve
basic quantum information issues, starting with the effects of gravity on the
quantum-to-classical transition [1, 2, 3], and continuing to the construction
and probing of gravitational cat states [4]. The possibility of direct
experimental feedback about a non-trivial interplay between gravity and
quantum theory is what makes those phenomena so interesting from a theoretical
viewpoint.

Note that one does not need to appeal to theories of quantum gravity at the
Planck scale to see the contradictions between the quantum and gravitation, to
ask questions about the quantum nature of gravity and how the quantum-
informational features show up. Their manifestations are already present in
low energy physics, namely, in the nonrelativistic (NR) and weak field (WF)
limits of general relativity (GR) and quantum field theory (QFT). This is the
regime we live in where laboratory experiments on Earth and in space can offer
invaluable observational data to cross-examine the above-mentioned theoretical
issues of fundamental importance.

###  I.1 Quantum Decoherence

For a long time since the advent of quantum mechanics, the issues of the
quantum-to-classical transition and of the emergence of the classical world
have been a subject of curiosity for debates among the philosophers of
science. Physicists reclaimed the subject in the 80s with full rigor in three
lines of development.

A) Consistent/ Decoherent histories. This program was developed by Griffiths
[5], Omnés [6] and Gell-Mann & Hartle [7]. The key idea is that quantum theory
must be expressed in terms of histories, i.e., sequences of properties of a
system at different moments of time, rather than in terms of evolving single-
time quantum states. The decoherent histories (DH) program places great
emphasis on the logical structure of propositions about histories, which leads
to a rigorous implementation of the crucial notion of coarse-graining. As a
result, the DH program offers the most sophisticated account of the quantum-
to-classical transition in a unified way with the microscopic-to-macroscopic
transition. However, DH guarantees neither the uniqueness nor the stability of
the classical world of our experience [8], while it offers only minor
improvements over Copenhagen quantum mechanics on the measurement problem [9].

B) Environment-Induced Decoherence (EID). This program originates from Zurek
[10], and Joos and Zeh [11], following an earlier work by Zeh [12]. The key
idea is that quantum systems classicalize as a result of their interaction
with the environment. This program went hand in hand with the development of
the theory of open quantum systems. Most prominent are the master equations
that built on the work of Schwinger [13] and Feynman and Vernon [14] in the
1960s for the ubiquitous quantum Brownian motion, to the Caldeira-Leggett
Markovian master equation [15] of the 1980s which is valid for Ohmic
environments at high temperatures, and to the Hu-Paz-Zhang non-Markovian
master equation of the 1990s [16] that is valid for a general environment at
all temperatures. The applicability of the program is limited to setups that
admit a natural system-environment split. And, despite its many successes in
matching with experimental outcomes, it does not provide, by itself, a
satisfactory solution to the quantum measurement problem [17].

C) Wavefunction Collapse models. This program introduces a scale that
separates the microscopic from the macroscopic and postulates that quantum
mechanics as we know it which applies to a plethora of natural phenomena from
subatomic scales up, no longer holds for macroscopic phenomena. There are two
major directions. The first direction originates from Ghirardi, Rimini and
Weber [18] and Pearle [19] (hence, GRWP), and it culminates in the formulation
of dynamical reduction or continuous state localization (CSL) models [20]. The
second direction, exemplified by Diosi [2] and Penrose [3] (DP model), focuses
on wave-function collapse due to gravity, and we will discuss it separately.

Unlike the previous two, this approach postulates a fundamental modification
of quantum theory, and for this reason we will refer to the theories in this
class as alternative quantum theories (AQT). CSL models start from the
pragmatic aim of introducing a scale where macroscopic phenomena overtake the
microscopic quantum mechanical behavior through a mathematically consistent
modification of quantum dynamics. This mostly works in non-relativistic
physics. Still, the whole procedure is ad hoc, with no fundamental
explanation, and one encounters grave problems when attempting to formulate a
relativistic version of CSL models appropriate for QFT. Despite significant
progress in recent years [21], a CSL theory that is fully consistent with
special relativity has yet to appear [22].

###  I.2 Gravitational Decoherence

The main body of work on gravitational decoherence began a decade later. We
shall mention the main lines of development in the next section but give an
overall perspective here.

One special feature in this research subfield is a widened scope of
possibilities, invoking novel features from gravity as well as quantum theory.
We observe that both the DH and the EID programs do not call on alteration of
GR or QM. DH allows for generalizations of quantum theory for systems that are
difficult to treat with usual methods, or for the definition of more elaborate
observables, but the physical predictions in the domain of current theories
are the same with the standard ones. EID highlights the role played by the
environment in determining the classical outcomes of a quantum system. As
such, it requires tools and concepts in open quantum systems and non-
equilibrium statistical mechanics, but it does not need to alter existing
theories at the fundamental level.

With gravitational decoherence, we see proposals which call for the alteration
of quantum mechanics, such as in the GRWP and DP theories. The motivation
ranges from pragmatic considerations as in CSL to the ‘gravitisation’ of
quantum mechanics, proposed by Penrose [24]. Now that gravity enters the
picture many proponents find it convenient to use GR, even altering it, to
explain the quantum-to-classical transition. This is done by invoking
fluctuations either in the manifold structure or in the spacetime metric,
under the labels of ‘intrinsic’, ‘fundamental’, or ‘quantum gravity’
decoherence. We shall point out how such models infringe on GR in subtle ways.

###  I.3 Features of this overview

An excellent review is available on this subject [23] which covers a broad
range of topics with prominent representation of CSL and AQTs. For this
reason, we shall narrow our goals in this overview to two: (i) Present
theories of gravitational decoherence based on GR and QFT and their
experimental predictions. (ii) Place other theories of GD under the scrutiny
of GR and QFT, and point out the theoretical differences. In fact, we would
urge all proponents of new and old alternative theories to present, alongside
with what they consider as their proposals’ attractive features, also a self-
scrutiny against GR and QFT as benchmarks. The reason is simply that, these
two theories are the pillars of modern physics, and they serve as the
yardstick by which to distinguish between proposals based on ”known physics”
or conservative extensions thereof and proposals based on ”new physics” which
violate either GR or QFT or both, in ways big or small.

In Sec. 2, we present the main theories of gravitational decoherence
classified according to the physical origins of decoherence. Decoherence may
originate from the fundamentally classical nature of gravity, from quantum
gravity processes or from spacetime fluctuations. In Sec. 3, we present the
Anastopoulos-Blencowe-Hu (ABH) theory of gravitational decoherence [25, 26]
that fully lies within the scope of GR and QFT. In Sec. 4, we briefly review
estimates for gravitational decoherence effects, mainly comparing the
predictions of the DP and the ABH model.

##  II Theories of gravitational decoherence

###  II.1 Decoherence Vs. dephasing

The term “gravitational decoherence” is sometimes used in the literature when
the authors actually refer to dephasing due to gravity [27]. Decoherence is an
irreversible process, while dephasing may also occur without irreversibility.
A classic case of mere dephasing without decoherence is provided by the spin
echo experiments [28]. In these experiments, phase information is apparently
lost, but a simple operation on the system will recover this information. In
absence of dissipation (an irreversible process), the recovery of information
is complete [29]. In decoherence, the phase information is lost forever, or at
least for a very long time (of the order of the Poincaré recurrence time of
the environment). Another analogy in terms of energy is the following. Mere
dephasing is in the nature of Landau ‘damping’ a la the Vlasov equation which
is of the Hamiltonian type, decoherence is in the nature of the nonunitary
Boltzmann equation with real physical dissipation.

Here, we will only be concerned with models of actual irreversible
decoherence.

###  II.2  Decoherence from classical gravity

This is the oldest approach and most influential approach to gravitational
decoherence. It originates with Karolyhazy [1] in the 1960s. An underlying
idea is that gravity is a fundamentally classical channel of interaction. This
implies that gravity can act as an agent of decoherence in quantum system.

####  II.2.1 The Diosi-Penrose theory

According to Penrose [3], decoherence is a plausible consequence of the
fundamental incompatibility between GR and quantum theory, especially in their
respective treatments of time. Time in quantum theory is essentially classical
and forms a background to all quantum phenomena, while in GR time is
fundamentally dynamical as it is determined by the spacetime metric that is a
dynamical observable.

The contradiction between the role of time in GR and in quantum theory is most
emphatically manifested when considering superposition of macroscopically
distinct states for matter. By Einstein’s equations, each component of the
superposition generates a different spacetime. Since there is no canonical way
of relating time parameters in different spacetime manifolds, there is a
fundamental ambiguity in the time parameter of the evolved quantum state.
According to Penrose, this ambiguity is manifested even at low energies, when
the gravitational interaction can be effectively described by the Newtonian
theory. It may be expressed as a mechanism of gravity-induced decoherence for
superpositions of states with different mass densities $\mu_{1}(x)$ and
$\mu_{2}(x)$. Penrose proposed that the time-scale of decoherence is of the
order of the gravitational self-energy of the difference in the two mass
densities.

Diosi [2] constructed a master equation in which he postulated a collapse
term, with noise correlator proportional to gravitational potential. The
master equation is

$\displaystyle\frac{\partial\hat{\rho}}{\partial
t}=-i[\hat{H},\hat{\rho}]-\frac{G}{2}\int
d{\pmb{r}}d{\pmb{r}^{\prime}}\frac{[\hat{\mu}({\pmb{r}}),[\hat{\mu}({\pmb{r}}^{\prime}),\hat{\rho}]]}{|{\pmb{r}}-{\pmb{r}}^{\prime}|},$
(1)

where $\hat{\mu}({\pmb{r}})$ is the mass density operator. Key value in this
master equation is that predictions do not involve any free parameters, except
for a high-frequency cut-off $\Lambda$ that is necessary for the definition of
the mass density operator. The natural scale of $\Lambda$ for non-relativistic
particles is the nuclear scale, in the sense that $\Lambda^{-1}$ corresponds
to a nucleon’s radius, nuclei being the simplest particles most affected by
this decoherence mechanics. The decoherence rate is typically of the order of
$\frac{1}{\Delta E}$ where $\Delta E$ is a regularized gravitational self-
energy difference associated to a macroscopic superposition of mass densities.

Penrose’s arguments for gravitational decoherence are not model-specific, but
his proposal for the decoherence timescale is the same with the one obtained
from Diosi’s model, hence the name Diosi-Penrose for this theory.

Diosi’s model predicts a small violation of energy conservation, which can be
used to test the theory. A recent experiment rules out the most natural
version of the model where $\Lambda$ is the nuclear scale [30]. There is no
obvious physical justification for the smaller value of the cut-off.

There are also some theoretical problems in the D-P theory. A recent analysis
[31] of the quantization of weak gravity interacting with matter has shown
that the ambiguity pointed out by Penrose is not related to the gravitational
self-energy. The latter appears the same in all gauges / reference frames. The
frame-dependent terms on the Hamiltonian are different, and they have no
obvious physical interpretation.

Moreover, the Diosi master equation implies a coupling of quantum matter to
gravity that differs from that of GR, in effect, through a non-unitary
channel. This channel should also persist at the macroscopic level (for
example, via non-conservation of energy). As a result, the resulting theory
for matter interacting with gravity cannot be a Hamiltonian theory, hence, it
cannot be GR. Even if the avowed aim of Penrose is to gravitize quantum
mechanics, this specific approach apparently requires strong modifications in
our current theory of gravity, and not only in quantum mechanics.

####  II.2.2 The Newton-Schrödinger equation

An early candidate for the coupling of classical gravity to quantum matter is
the so-called Moller-Rosenfeld (MR) theory [32, 33], according to which the
source in Einstein equations in the expectation value of the stress-energy
tensor: $G_{\mu\nu}=\frac{\kappa}{2}\langle\hat{T}_{\mu\nu}\rangle$ is taken
as fundamental in this theory, rather than an approximation to a quantum
gravity theory. (See Ref. 35 for how the MR theory, at least the way it is
interpreted by recent followers, differs from semiclassical gravity theory
proper [46], based solely on GR+QFT). It is far from obvious that the MR
theory is consistent, or even that it makes mathematical sense, but at least
in the weak field, nonrelativistic regimes, one can subject it to tests
against theories based on GR+QFT. In these regimes, adherents of the M-R
theory like to work with the Newton-Schrödinger equation (NSE) [34], which is
a non-linear equation for the wave-function

$\displaystyle i\frac{{\partial}\psi}{{\partial
t}}=-\frac{1}{2m}\nabla^{2}\psi+m^{2}V_{N}[\psi]\psi$ (2)

where $V_{N}(\bf r)$ is the (normalized) gravitational (Newtonian) potential
given by

$\displaystyle V_{N}({\bf r},t)=-G\int d{\bf r^{\prime}}\frac{|\psi({\bf
r^{\prime}},t)|^{2}}{|{\bf r}-{\bf r^{\prime}}|}.$ (3)

Note, as we have emphasized in Ref. 37, such a single (N=1) particle NS
equation is not derivable from GR+QM. It is emphatically not a representative
of semiclassical gravity based on GR and QFT, which can be viewed as the large
N limit of quantum gravity [36].

###  II.3 Decoherence from quantum gravity processes

This involves proposals about decoherence from Planck scale processes that
cannot be identified by low energy physics. Early models attempted to find a
connection with non-unitarity suggested by the black hole information paradox
[38]. Now, we are well aware that there is no direct relation: the presumed
non-unitarity in black hole evaporation is accompanied with the loss of Cauchy
surfaces, hence, with the loss of the global notion of a state. Therefore, it
is incompatible also with the non-unitarity of Markovian master equations that
describe the evolution of a globally defined state [39].

Gravitational decoherence of this type is presumed to originate from specific
features of quantum gravity theories, for example, string-theory couplings to
massive-string states [40] or emergent non-locality from spacetime foam [41].
The key critique of this type of models is presented in Ref. 25.

Any entity of the quantum gravity realm, such as spacetime foam, exists at the
Planck scale, before spacetime with a Lorentz structure emerges. One needs a
strong case to show that some Planck-scale properties can escape the coarse-
graining and scaling which subsumes their effect to that of the average as the
large scale manifold structure of spacetime emerges, and emerge at low
energies. The most reasonable assumption is that their average behavior is
contained in the effective description that survives at the low energy limit,
namely, GR; we can think of no counter-example in all fields of physics. Even
if some Planck-scale effects survive at low energies, they must be much
smaller than the effects that originate from the gravitational dynamics of GR.
Hence, decoherence from quantum gravity effects will be subdominant compared
to all decoherence effects that originate from GR, or are postulated on the
basis of GR.

###  II.4 Decoherence from Spacetime fluctuations

In these models decoherence originates from some fundamental imprecision in
the measuring devices (starting with clocks and rulers) [42] or uncertainties
in the dynamics [43], or treating time as a statistical variable [44].
Usually, the fluctuations are assumed to originate from Planck-scale physics.

These approaches make very strong assumptions about the physics of the Planck
scale. There is absolutely no reason to expect that such uncertainties can be
modeled by stochastic processes which are intrinsically classical, as is
commonly the case in this approach. The modeling of uncertainties by classical
noise is a valid assumption for randomness at the macroscopic scale. In
contrast, quantum uncertainties are different in nature as they involve non-
localities and correlations with no analogues in the classical theory of
stochastic processes.

To explain this point, we note that the limitations posed by the Planck length
are not a priori different from those placed by the scale
$\sqrt{\hbar/c^{3}}q/m$ in quantum electrodynamics: At this scale quantum
field effects are strong, and the fluctuations from these effects are fully
quantum. Any effect they cause at low energy is also inherently quantum. One
needs to specify the conditions (e.g., Gaussian systems) or the regime for the
quantum field, and justify the means by which they could be treated like
classical fluctuations described by a stochastic process. In particular, the
effects of the fluctuations of the EMF at low energies ($E<<mc^{2}$) have been
well studied. It has been shown that the ‘noise’ induced by these fluctuations
is non-Markovian and does not cause significant decoherence effects in the
microscopic regime [45]. In other words, the coherence of the EMF vacuum does
not allow for the a priori generation of classical (i.e., decohering)
fluctuations in the quantum motion of the particle. The assumption that the
gravitational field exhibits a different behavior is completely ad hoc, with
no justification unless one postulates that gravity is fundamentally
classical.

Note that the theory of semiclassical stochastic gravity (ScStG) [46] can
introduce stochasticity associated solely with the quantum fluctuations of
matter fields. The backreaction of the mean and the fluctuations of the stress
energy tensor of a quantum matter field is taken into account by way of the
Einstein-Langevin equation [47] governing the dynamics of the induced metric
fluctuations. Simple dimensional analysis shows that these induced metric
fluctuations are dominant only at the Planck scale, and thus are on the same
footing as spacetime foams, albeit Wheeler wanted spacetime foams to admit
topology changes. In ScStG the noise is fundamentally quantum, they are not
put in by hand, and the theory is entirely based on GR +QFT. The drop off
behavior of metric fluctuations in Minkowski spacetime has been derived in 48.
Indeed, the strength of quantum noise in ScStG provides a useful measure of
the validity of semiclassical gravity[49]). In contrast, the noise employed by
the models discussed above is defined by a purely classical stochastic
process.

A second problem in these approaches is that they often contradict the
symmetries of GR. Fluctuations in the time or space coordinates of an event
are indistinguishable mathematically from time and space reparameterizations
of the system, only such reparameterizations are viewed as stochastic.
However, time and space reparameterizations are pure gauge variables in
classical relativity (even in the linearized approximations); they do not have
any dynamical content. The invariance of the theory under space and time
reparameterizations follows from the diffeomorphism invariance of the
classical action, a fundamental symmetry of general relativity.

The assignment of dynamical contents to such fluctuations implies that they
are not treated as gauge variables. Doing so violates the fundamental symmetry
of classical GR. Any theory with this property would have far-reaching
implications which goes beyond the gravitationally-induced decoherence
effects. The diffeomorphism symmetry affects both the dynamics and the
kinematics of general relativity, and its abandonment ought to be manifested
in other gravitational phenomena.

We also note that time and space reparameterizations decouple from the terms
describing Newtonian interaction, already at the classical level. Hence, there
is no reason for Newton’s constant in GR to modulate the strength of
decoherence effects.

We may also place in this category the so-called event operator formalism of
Ralph, Milburn and Downes [50, 51]. This model originates from a modification
of standard QFT that allows for unitary evolution in presence of closed
timelike curves. This model does not lead to decoherence of individual
particles, but it predicts rather strong decorrelation of entangled photon
pairs.

The presumed physics in this model is rather implausible from the perspective
of gravity theory. The event operator formalism relies on the presence of
closed-timelike curves [52] without accompanying quantum gravity effects. This
strongly contradicts the well-motivated chronology protection conjecture [53]
which asserts that the laws of physics, including quantum phenomena, do not
allow for the appearance of closed time-like curves. Chronology protection is
valid also for semi-classical gravity [54]. All this strongly suggests that
closed timelike curves can emerge only as Planck scale quantum gravity effects
(like Wheeler’s spacetime foam [55]), if at all.

##  III ABH theory: based on GR and QM

###  III.1 Key ideas and results

In the theory of gravitational decoherence of Anastopoulos & Hu [25] and
Blencowe [26] (ABH) the source of decoherence comes as noise (fluctuations)
from gravitational waves (classical perturbations) or gravitons (quantized
linear perturbations), or simply, metric fluctuations. These fluctuations
satisfy Einstein’s equations and as such they are to be identified with
transverse-traceless perturbations. The source of these fluctuations may be
cosmological [26] (stochastic gravitons produced in the early universe near
the big bang or from inflation pre- or post), astrophysical [56] or structural
to GR as an emergent theory—see Ref. 57 for an explanation. Note also the
later work, Ref. 58, which derives a related theory of gravitational
decoherence for matter and light.

There is one parameter in the master equation of the ABH theory which is the
noise temperature $\Theta$. It coincides with the graviton temperature if the
origin of perturbations is cosmological. Early works in this direction
includes the study of decoherence due to the graviton vacuum [59], and of open
system dynamics of particles in graviton baths [60, 61]. The Power-Persival
[62] collapse model also falls in the same category, but the perturbations
there are restricted to conformal waves.

It is important to distinguish the ABH theory from models of gravitational
decoherence from metric perturbations that do not require the perturbations to
satisfy the linearized Einstein equations. Such models have been developed by
Kok and Yurtsever [63]; Breuer, Göklü and Lämmerzahl [64]; and Asprea,
Gasbari, Ulbricht and Bassi [65]. These models describe the perturbations of
the metric by a stochastic process, but they do not distinguish between true
and pure gauge degrees of freedom in the perturbations. Treating variables
that are pure gauge in GR as stochastic is not compatible with GR, because
they do not implement the diffeomorphism symmetry. (Unless one assumes a
stochastic matter source of unknown origin and physics, to which the pure
gauge stochastic perturbations are slaved.) They are intermediate between the
ABH model and the models of Sec. II.4, in that they postulate stochastic
behavior of pure-gauge quantities, but they also include a contribution from
true degrees of freedom.

The ABH type of analysis applies to any type of quantum matter fields in
addition to scalar, any number of particles, in the ultra-relativistic as well
as the non-relativistic regimes. The specific methodology is the following.

The system under consideration is a quantum field (a massive scalar field in
the simplest case), interacting with a gravitational field as its environment.
Gravity is described by classical general relativity. In the weak field limit,
we describe gravitational perturbations in the linearized approximation.
Hence, one starts from a linearization of the Einstein-Hilbert action around
Minkowski spacetime, and constructs the associated Hamiltonian through a 3+1
decomposition. The constraints of the system are solved classically, thereby
allowing us to express the Hamiltonian in terms of the true physical degrees
of freedom of the theory, namely, the transverse-traceless perturbations for
gravity and the scalar field. We then quantize the scalar field and the
gravitational perturbations and trace- out the contribution of the latter.

A key input in this stage is the specification of an initial state for the
gravitational perturbations. We consider an initial condition that
interpolates between the regime of negligible (vacuum) perturbations and
strong classicalized perturbations. The initial state is defined in terms of a
free parameter $\Theta$ that can be loosely interpreted as the noise
temperature of the perturbations. $\Theta$ conveys coarse-grained information
reflective of the micro-structures of spacetime, similar to temperature with
regard to molecular motion, or the spectral density function of the
environment in Brownian motion. It is in this sense that we think
gravitational decoherence may reveal the underlying ‘textures’ of spacetime
beneath that described by classical general relativity.

Following the standard methodology of open quantum systems a 2nd order
(perturbative) master equation for the matter field is obtained. This master
equation applies to configurations with any number of particles. We project
the master equation to the single-particle subspace and we derive a master
equation for a single particle. The latter simplifies significantly in the
non-relativistic regime, leading to the ABH master equation

$\displaystyle\frac{\partial\hat{\rho}}{\partial
t}=-i[\hat{H},\hat{\rho}]-\frac{\tau}{16m^{2}}(\delta^{ij}\delta^{kl}+\delta^{ik}\delta^{jl})[\hat{p}_{i}\hat{p}_{j},[\hat{p}_{k}\hat{p}_{l},\hat{\rho}]]$
(4)

where $\tau$ is a constant of dimension time and
$\hat{H}=\frac{\hat{p}^{2}}{2m}$.

For motion in one dimension, the ABH master equation simplifies to

$\displaystyle\frac{\partial\hat{\rho}}{\partial
t}=-i[\hat{H},\hat{\rho}]-\frac{\tau}{2}[\hat{H},[\hat{H},\hat{\rho}]],$ (5)

where $\tau$ is a constant of dimension time and
$\hat{H}=\frac{\hat{p}^{2}}{2m}$.

In the ABH model

$\displaystyle\tau=\frac{32\pi
G\Theta}{9}=\frac{32\pi}{9}\tau_{P}(\Theta/T_{P}),$ (6)

where $T_{P}=1.4\times 10^{32}\;K$ is the Planck temperature.

The master equation (5) appears in models by Milburn [43], Adler [67], Diosi
[68]and Breuer et al [64] where $\tau$ is obtained from postulated stochastic
fluctuations of time, discreteness of time, stochastic fluctuations of the
metric, or even stochastic fluctuations of $\hbar$. In these models, the value
of $\tau$ is not fixed, but the natural candidate is the Planck-time
$\tau_{P}=10^{-43}s$. In Ref. 25, it is argued that $\tau$ need not be
restricted to $\tau_{P}$ but can be a free parameter depending on the
underlying structures of spacetime at different scales which may vary in
different theories.

The ABH model can be generalized to photons, where we can obtain a master
equation for a general photon state [69]. For a single photon,

$\displaystyle\frac{\partial\hat{\rho}}{\partial t}$
$\displaystyle=-i\,[\hat{H},\hat{\rho}]-\frac{\tau_{ph}}{2}\left(\delta^{in}\delta^{jm}-\frac{1}{3}\delta^{ij}\delta^{nm}\right)\left[\frac{\hat{p}_{i}\,\hat{p}_{j}}{\hat{p}_{0}},\left[\frac{\hat{p}_{n}\,\hat{p}_{m}}{\hat{p}_{0}},\hat{\rho}\right]\right]\,,$
(7)

where $\hat{H}=|\hat{\bf p}|$ and $\tau_{ph}=4G\Theta$.

###  III.2 Observational constraints to the ABH

If we regard the parameter $\Theta$ in the ABH theory as a noise
temperature—originating from some emergent gravity theory—, then $\Theta$ need
not be related to the Planck length and even $\Theta>>T_{P}$ is perfectly
acceptable from a theoretical point of view. $\Theta$ is only a measure of the
power $P$ carried by the noise, $P\sim\Theta\Delta\omega$, where
$\Delta\omega$ is the band-width of the noise.

Some bounds to $\Theta$ can be estimated from the non-relativistic analysis of
59. This paper employs the Feymnan-Vernon influence functional method, which
has the benefit of providing a simple stochastic equation for the semi-
classical evolution of a particle interacting with a heat bath. The results
that we will present are new, but they follow from the direct use of the
‘thermal’ noise of the ABH model to the analysis of Ref. 59.

The effective semiclassical equation for a non-relativistic particle in
presence of classical gravitational perturbations of noise temperature
$\Theta$ turns out to be

$\displaystyle\ddot{x}+\frac{2G}{15}\ddot{x}^{2}\dot{x}=2\ddot{x}\xi$ (8)

where $\xi(t)$ is Gaussian noise with
$\langle\xi(t)\xi(t^{\prime})\rangle=\eta(t-t^{\prime})$, where $\eta$ is
known as the noise kernel. The dissipative term is relatively weak at it
corresponds to energy loss due to gravitational wave radiation, so we can
ignore it.

The noise kernel for the ABH model is

$\displaystyle\eta(s)=\frac{G}{2}\int_{0}^{\Lambda}dkk\cos(ks)\coth\frac{k}{2\Theta},$
(9)

where $\Lambda$ is a cut-off and $\Theta$ the noise temperature. The
physically relevant regime corresponds to $\Theta>>\Lambda$, whence,

$\displaystyle\eta(s)=\pi G\Theta\delta(s).$ (10)

By Eq. (8), the noise behaves like a stochastic fluctuations of the particle’s
inertial mass of order

$\displaystyle\frac{\delta m}{m}\sim\sqrt{\xi^{2}(t)}$ (11)

With the noise kernel (10) we find

$\displaystyle\frac{\delta m}{m}\sim\sqrt{\frac{\Theta}{T_{P}}},$ (12)

where $T_{P}$ is the Planck length. We can use Eq. (12) to establish bounds to
$\Theta$ from cosmological and solar-system measurements.

Cosmology does not lead to strong constraints in $\Theta$. If we assume that
the $\Lambda$CDM model holds we can identify the maximum value of
$\frac{\delta m}{m}$ with the relative error in the determination of baryon
mass density, hence $\frac{\delta m}{m}\sim 10^{-2}$, which implies that
$\Theta<10^{-4}T_{P}$. However, if we take into account the changes in the
values of the baryon mass density in different dark energy models, a bound
$\frac{\delta m}{m}\sim 10^{-1}$ is more plausible, but even this may be too
restrictive. We need a model that includes intrinsic stochastic gravitational
perturbations in the evolution of the Universe, in order to estimate a proper
bound to the size of these fluctuations.

Solar system measurements provide a better constraint to $\Theta$, if we
assume that Eq. (12) also applies to large astronomical bodies like planets.
This assumption is by no means evident, because the derivation of Eq. (8)
treats particles as pointlike, or at least, much smaller than the typical
wavelength of gravitational perturbations. Eq. (12) overestimates the effects
of the gravitational noise. In an extended system, part of the noise would be
expended on the moment of inertia and on higher moments of the mass density,
leading to weaker effects on the center of mass motion.

In any case, if we take Eq. (12) to apply to planets, we can use the relative
accuracy in the measurement of Earth’s mass to place an upper bound
$\frac{\delta m}{m}<10^{-4}$, which implies that $\Theta<10^{-8}T_{P}$. This
estimate is probably too stringent, because the most accurate measurements of
mass in astronomical bodies comes from the measurement of the gravitational
acceleration on its surface, not from its orbit in the solar system, as would
be required for comparison with Eq. (8). An exact bound on $\Theta$ will
require an analysis of the motion of planet-sized bodies under ABH-type noise.
However, our simple analysis shows that under pessimal assumptions,
sufficiently large values of $\Theta$ that are compatible with observable
decoherence effects cannot be ruled out on the basis of existing measurements.

##  IV Types of experiments

In this section, we provide order-of-magnitude estimates for ABH-type
decoherence, in comparison with the D-P model.

###  IV.1 Optomechanical experiments

Consider a body brought into a superposition of a zero momentum and a finite
momentum state, corresponding to an energy difference $\Delta E$. For the ABH
model, the decoherence rate for the center of mass is then

$\displaystyle\Gamma_{ABH}=\frac{(\Delta E)^{2}\tau}{\hbar^{2}},$ (13)

where $\tau$ is the free parameter in the master equation (4). A value for
$\Gamma_{ABH}$ of the order of $10^{-3}s$ may be observable in optomechanical
systems, as it is competitive with current environment-induced-decoherence
timescales. Hence, to exclude values of $\tau>\tau_{P}$, we must prepare a
quantum state with $\Delta E\sim 10^{-14}$J.

In the DP model, the decoherence rate for a sphere of mass $M$ of radius $R$
in a quantum superposition of states with different center of mass position
(though the predicted decoherence rate is largely independent of the details
of the prepared state) is of the order of

$\displaystyle\Gamma_{DP}=\frac{GM^{2}}{\hbar\sqrt{R^{2}+\ell^{2}}},$ (14)

where $\ell$ is a cut-off length, originally postulated to be of the order of
the size of the nucleus, but recently constrained to $\ell>0.5\cdot 10^{-10}m$
(see Ref. 30). Alternative models postulate $\ell$ up to a scale of
$10^{-7}$m. For an optomechanical nanosphere with $M\sim 10^{10}$ amu and
$R\sim 100$ nm, $\Gamma_{DP}\sim 10^{-3}s^{-1}$, a value that is in principle
measurable in optomechanical experiments.

###  IV.2  Matter wave interferometry.

In far field interferometry, the ABH model (but not the 1-d master equation
(5)) leads to loss of phase coherence of the order of
$(\Delta\Phi)^{2}=m^{2}v^{3}\tau L/\hbar^{2}$, where $L$ is the propagation
distance inside the interferometer. While the exact derivation of
$(\Delta\Phi)^{2}$ requires a dynamical analysis, it is of the order of
$\Gamma_{ABH}t_{int}$, where $t_{int}=L/v$ is the average time of the particle
in the interferometer. Setting an upper limit of $L=100$ km, and $v=10^{4}$
m/s, decoherence due to cosmological gravitons requires particles with masses
of the order of $10^{16}$amu. If $\Theta$ is a free parameter, experiments
with particles at $10^{10}$amu will test up to $\Theta\sim 10^{-5}T_{P}$. For
comparison, the heaviest molecules used to date in quantum mechanical
interference experiments are oligoporphyrines with mass of “only” $2.6\cdot
10^{4}\,$amu [66]. ftfc🙌 The Diosi-Penrose model and other models that lead to
decoherence in the position basis can also be tested by near-field [70] and
far-field [71] matter-wave interferometry. A rough estimation for the loss of
phase coherence is $(\Delta\Phi)^{2}\simeq\Gamma_{DP}L/v=\frac{Gm^{2}L}{\hbar
Rv}$, where $R$ is the radius of the particles. In contrast to the ABH model,
this loss of coherence is enhanced at low velocities. Assuming $L=100$ km,
$v=10$ m/s, and $R=100$ nm, an experiment would require a mass $M\sim
10^{9}-10^{10}$ amu to observe decoherence according to the DP model.

###  IV.3 Wave-packet spread.

The intrinsic spreading of a matter wave-packet in free space is a hallmark of
Schrödinger evolution. ABH-type models predict negligible deviations in the
wave-packet spread from that of unitary evolution. The DP model and all other
models that involve decoherence in the position basis predict a wave packet
spread of the form

$\displaystyle(\Delta x)^{2}(t)=(\Delta
x)_{S}^{2}(t)+\frac{\Lambda}{2m^{2}}t^{3},$ (15)

where $(\Delta x)_{S}^{2}(t)$ is the usual Schrödinger spreading, and
$\Lambda$ depends on the model. The changes from free Schrödinger evolution
become significant at later times. An exact estimation of this effect depends
on properties of the initially prepared state, and is rather involved. The
MAQRO proposal estimates that for a free-propagation time equal to $100$s
(accessible in their setup) it is possible to constrain CSL-type models, some
models of quantum gravity decoherence, but not decoherence of the D-P type.

In contrast, the Newton-Schrödinger Equation predicts a retraction of the
wave-packet spread for masses around $10^{10}$ amu [72]. An osmium nanosphere
of radius $R\simeq 100$ nm would require a couple of hours of free propagation
in order to observe significant deviation from Schrödinger spreading [73].

###  IV.4 Decoherence of photons

Only the ABH model has been generalized for photons [58, 69]. For
interferometer experiments with arm length $L$, the model predicts loss of
visibility of order $(\Delta\Phi)^{2}=\frac{8G\Theta E^{2}L}{\hbar^{2}c^{6}}$.
For $L=10^{5}$km, $\Theta\sim T_{P}$ and photon energies $E$ of the order of
1eV, this implies a loss of coherence of the order of $\Delta\Phi=10^{-8}$. In
principle, this would be discernible with EM-field coherent states with mean
photon number $\bar{N}>10^{16}$, though it would be very challenging to
suppress all other systematic errors to this degree.

The linear dependence of $\Delta\Phi$ on energy implies that decoherence is
significantly stronger at high frequencies. For interferometry in the extreme
UV, $\Delta\Phi$ may increase by two orders of magnitude or more. Alternative
set-ups, such as the formation of effective Fabry-Perot ‘cavities’ with
mirrors could increase the effective propagation length by many orders of
magnitude, and hence, lead to stronger signatures of ABH-predicted, photon
gravitational decoherence.

##  V Conclusion

Quantum gravity research, the quest for theories of the microscopic
constituents of spacetime, in achieving the fusion of quantum and gravity
(Q$\times$G) at the Planck scale ($10^{-35}m$), has occupied the attention of
a significant number of theoretical physicists for the past seven decades. Yet
the lack of observable experimental data has prevented any of the resulting
theories to claim success. Instead of chasing this lofty yet unattainable
goal, with little chance of finding directly verifiable evidence at today’s
energy, we should set our targets at the union of quantum and gravity (Q+G)
because the contradictions and inconsistencies between quantum theory and
general relativity (GR) already show up acutely at today’s accessible low
energy scales and there are earthbound and space experiments which can probe
into issues at the joining of these two fundamental theories.

In particular, taking advantage of long baselines and small environmental
influence, deep space experiments [74, 75, 70, 71] can provide significant
novel information about the coexistence of quantum and gravity – no matter how
precarious it is – and separate them from the alternative theories . In this
context gravitational decoherence experiments have far-reaching theoretical
significance in at least two respects: i) discriminating alternative quantum
or gravity theories, such as those mentioned earlier, based on their
predictions, against that from theories based on general relativity and
quantum field theory, such as the ABH theory described above; ii) the
possibility of discerning the nature of gravity, whether it is a fundamental
theory or an effective theory.

Concerning the first aspect, it is easy to make the demarcation because most
proposals for gravitational decoherence involve a violation of quantum theory,
or of GR, or (usually) of both, whereas the ABH theory respects both QFT and
GR. Precision experimental data can quickly discern these two categories of
theories. Concerning the second aspect, why it is important and how can one
make such a distinction, we offer some background perspective in the
following.

GR is commonly accepted as the best theory for the description of macroscopic
spacetime, but whether quantizing GR will yield the true theory of the
microscopic structure of spacetime at the Planck scale remains an open
question. GR could well be an effective theory emergent from some fundamental
theory of quantum gravity, valid only at the macroscopic scale we are familiar
with. The ABH model distinguishes these two alternatives. In the fundamental
theory view, Minkowski spacetime is the ground state of a quantum gravity
theory. In the emergent theory view, Minkowski spacetime is a low energy
collective state or macrostate of quantum gravity, whereby one could associate
a thermodynamic description. The key difference between a ground state and a
macrostate is not energy, but the strength of fluctuations: thermodynamic
fluctuations are much stronger than quantum fluctuations in spacetime, and
they can cause significantly stronger decoherence.

Therefore, if we can see evidences of gravitational decoherence, then, using a
theory based on GR+QFT such as the ABH theory, even crude orders of magnitude
differences in the observation data could provide a strong discriminant
separating gravity as a fundamental theory from an effective one.

Acknowledgment On the theoretical side, the ABH theory presented here refers
to work done not only by us but also independently by Miles Blencowe, with
whom we enjoy synergetic interactions. The experimental considerations
presented here are assimilated in the grand review paper produced by the DSQL
team led and coordinated by Makan Mohageg of JPL, whose persistent interest in
the “texture of spacetime” served as an stimulant to our investigations. This
work is supported in part by a Schwinger foundation grant JSF-19-07-0001; BLH
is supported in part by NASA/JPL grant 301699-00001.

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