11institutetext: John F. Donoghue, Department of Physics, University of
Massachusetts, Amherst, Amherst, MA 01003 , 11email<EMAIL_ADDRESS>
# Quantum General Relativity and Effective Field Theory

John F. Donoghue corresponding author

###### Abstract

This is a review of some of the concepts and results of the effective field
theory treatment of quantum General Relativity. Included are lessons of low
energy quantum gravity, and a discussion of the limits of effective field
theory techniques.

## Keywords

Effective Field Theory, General Relativity, Quantum Gravity, Quantum Field
Theory

##  1 Introduction

Perhaps without realizing it, we have lived through a paradigm change in the
way that we understand the fundamental interactions. Historically we started
out uncovering what we call classical physics, and then found awkward ways to
describe quantum physics. The quantum techniques themselves have evolved and
are now different than even 30 years ago. The logic of effective field theory
now permeates the field. We also now give primacy to the quantum theory, and
have some modest understanding of how the classical world emerges from the
quantum world. This shift in our understanding is particularly important in
the case of General Relativity. The earliest quantum techniques did not work
well for General Relativity. However, the modern viewpoint is well-suited for
gravity. We have a theory of quantum General Relativity which treats it,
within various limits, as an effective field theory. This brief review is a
chance to reflect on the basic ideas and lessons of this effective field
theory.

Our fundamental theory is now defined by a path integral over the dynamical
degrees of freedom guided by a local Lagrangian. Physics is an experimental
science and through much effort we have determined the particles and the
structure of their interactions. In a compact notation (and hiding about 26
parameters - masses, couplings, etc.), our core theory is presently given by

$Z^{core}=\int[d\phi d\psi dAdg]_{\Lambda}~{}e^{i\int
d^{4}x\left[-\frac{1}{4}F^{2}+{\bar{\psi}}i\not{D}\psi+\frac{1}{2}\partial\phi\partial\phi-V(\phi)-\Gamma{\bar{\psi}}\phi\psi-\Lambda_{cc}+\frac{2}{\kappa^{2}}R+...\right]}\
\ .$ (1)

We use the path integral because canonical quantization is not a useful way to
treat the gauge interactions of the Standard Model, while path integrals are
simple and direct. Here the subscript $\Lambda$ is meant to indicate that the
path integral is to be restricted to those energies which we have
experimentally explored - i.e. those below some scale $\Lambda$. We do not,
and need not, pretend that these degrees of freedom are alway the correct
ones. The ellipses indicate that we expect further local terms in the action,
suppressed by heavy mass scales, even if we have not identified them yet. That
is, we think of all of our core theory as an effective field theory valid at
ordinary energies.

From this starting point, General Relativity is also fundamentally a quantum
theory. The metric degrees in freedom need to be dynamical and they need to be
included in the path integral because otherwise we could not obtain the
classical physics such as gravitational waves.

The idea of effective field theory is that the low energy degrees of freedom
organize themselves as quantum fields, governed by a local Lagrangian, in
general containing so-called nonrenormalizable terms suppressed by powers of a
heavy scale. Nevertheless, one can make predictions without knowledge of the
full high energy theory.

The basic theme of this exposition is that General Relativity is a quantum
field theory which is not much different than the other interactions in our
core theory. It is explored using the tools of effective field theory. In the
region of validity of the effective field theory, it can be studied in
perturbation theory and the quantum corrections are small - it is the best
perturbative theory ever. However the lessons learned are nevertheless
interesting.

In other reviews, I have presented more of the technical details of the
general relativistic effective field theory (GREFT) Donoghue:1994dn ;
Donoghue:1995cz ; Donoghue:2012zc ; Donoghue:2017ovt ; Donoghue:2017pgk , see
also Burgess:2003jk ; Burgess:2020tbq , as well as other expositions on
effective field theory Weinberg:2016kyd ; Weinberg:2021exr ; Petrov:2016azi ;
Meissner  Here I attempt a broader overview.

##  2 Does the graviton exist?

The reader who finds this question annoying or boring is free to skip this
section. However I include it because it bears on the rationale for effective
field theory.

One often hears the question of whether the graviton exists. Some people will
never be satisfied until they see the clicks in a detector caused by a
graviton. With this criterium, there will not be a resolution in our
lifetimes, or perhaps ever. But for some the question means to ask whether it
would be consistent to have everything but gravity be described by quantum
fields, but to have gravity be classical. As we presently understand quantum
theory, this is not possible. The following brief comments illustrate how the
existence of quantum fields is required.

Steven Weinberg in particular has presented the argument that any quantum
theory satisfying Lorentz invariance, causality, crossing symmetry and cluster
decomposition will be described by a quantum field theory Weinberg:2016kyd ;
Weinberg:2021exr  This is part of the reasoning that all of our theories are
effective field theories. Let us see how this could be applied to the
gravitational interaction111 This form of the argument comes from D. Carney
Carney:2021vvt  . The Newtonian potential has the Fourier transform

$-\frac{1}{\bf{q}^{2}}=-\frac{1}{(\bf{p}_{1}-\bf{p}_{2})^{2}}\ \ .$ (2)

But for this to be consistent with special relativity, this must be made into
a Lorentz invariant.

$-\frac{1}{(\bf{p}_{1}-\bf{p}_{2})^{2}}\to\frac{1}{(p_{1}-p_{2})_{\mu}(p_{1}-p_{2})^{\mu}}=\frac{1}{(p_{1}-p_{2})^{2}}=\frac{1}{q^{2}}\
\ .$ (3)

We now need to specify how we deal with the pole in this function. Of course
we know that the correct answer is the Feynman propagator

$\frac{1}{q^{2}+i\epsilon}$ (4)

and it is this which emerges from the path integral treatments. However, if we
are trying to be more general, Carney Carney:2021vvt  has shown that alternate
prescriptions with retarded or advanced Green functions, i.e. with $(q_{0}\pm
i\epsilon)^{2}-{\bf q}^{2}$ does not satisfy unitarity, nor do these forms
satisfy causality. The propagator using $-i\epsilon$ is equivalent to the one
with $+i\epsilon$, although the arrow of causality is reversed
Donoghue:2019ecz ; Donoghue:2020mdd . But now we are basically done. Because
of the identity

$\frac{1}{q^{2}+i\epsilon}=P\frac{1}{q^{2}}-i\pi\delta(q^{2})$ (5)

we see that what originally was the potential comes accompanied by massless
on-shell radiation - the graviton. This combination yields graviton exchange
and graviton emission. The general principles of relativity, unitarity and
causality have turned the potential into a quantum propagator.

This conclusion can also be addressed in different ways. As a recent example,
the authors of Ref. Belenchia:2018szb ; Danielson:2021egj  show that in order
to resolve a gedanken experiment involving superposed charges or masses, in
electromagnetism or gravity, requires the existence of radiation - photons or
gravitons. There are several previous arguments which say that for
consistency, the gravitational potential must also be accompanied by gravitons
Page:1981aj ; Carlip:2008zf ; Giampaolo:2018yql .

We have also partially addressed this issue using the path integral mentioned
in the introduction. The metric appears in the Lagrangian because we need to
describe particles in curved space. With the rest of the interactions
described by the path integral, could we leave the graviton out of the
integration variables? If we did, we would not obtain the graviton propagator.
This in turn would not allow the interaction of two masses, which occurs due
to one graviton exchange. With the integration over the gravitational field we
obtain Einstein’s equation as the equation of motion as well as the causal
graviton propagator222Using $e^{-iS}$ in the path integral instead of $e^{iS}$
results in the time reversed propagator with the $-i\epsilon$ prescription
Donoghue:2019ecz . In Section 5, we will see explicit examples of how further
classical physics emerges from the path integral. So our starting point for
the other fields also points to the quantum nature of the graviton. It is
worthwhile to note although we often refer to the classical limit as
$\hslash\to 0$, in fact $\hslash$ is a fixed number (here often set equal to
unity) and classical physics emerges in the appropriate kinematic regions. It
is not that we have classical physics and then treat quantization as an
optional extra step. Rather the modern view is that our starting point is
quantum and that the understanding of the classical limit is the extra step.

##  3 Detour into QED

The effective field theory treatment is not a change in quantum field theory.
It is rather using regular QFT with a careful attention to the scales in the
problem. The common features of EFT are also seen in other theories in certain
limits. In other publications, I have used the sigma model to motivate EFT
techniques Donoghue:1995cz ; Donoghue:2012zc ; Donoghue:2017ovt ;
Donoghue:2017pgk , and indeed that analogy is very useful for gravity. In
order to here give a different example, we can use QED for this purpose.

The QED path integral is given by

$Z[J]=\int[dA_{\mu}d\psi]_{\Lambda}~{}e^{i\int d^{4}x[{\cal
L}_{QED}(A,\psi)-J_{\mu}A^{\mu}]}$ (6)

The subscript $\Lambda$, implying a limited range of this theory, is also
appropriate here. This is because we know that the photon is not the correct
degree of freedom at all energies. Above the scale of electroweak symmetry
breaking, it is replaced by linear combinations of the $SU(2)_{L}$ gauge field
$W^{3}_{\mu}$, the hypercharge field $B_{\mu}$ and the Higgs boson. Treating
the photon as a separate field is only valid below the electroweak scale. But
we do not need to know that fact for QED to work at low energies. Similar
comments apply to the charged fermion here. At high energy, the fermion mass
eigenstates are decomposed into different fields (the weak eigenstates) and
also involve the Higgs field.

We can explore this at even lower energies - below the mass of the fermion. In
this case, in situations where the external fermions are not present, they
still propagate in loops in the original theory but can be removed from the
effective field theory for photons. That is, we can form an effective action
for the photons by integrating out the massive fermion. We do this by
performing the path integral over the fermion field to define an effective
Lagrangian involving only the photon

$e^{i\int d^{4}x{\cal L}_{eff}(A)}=\int[d\psi]_{\Lambda}e^{i\int d^{4}x[{\cal
L}_{QED}(A,\psi)}\ \ .$ (7)

This leaves behind the effective field theory defined by

$Z[J]=\int[dA_{\mu}]_{m}~{}e^{i\int d^{4}x[{\cal L}_{eff}(A)-J_{\mu}A^{\mu}]}\
\ .$ (8)

The subscript on the path integration is now the mass $m$ rather than the
electroweak scale, because the effective field theory is only valid below that
mass.

In practice we can do this in perturbation theory by matching the full theory
to the effective theory. At leading order in the electric charge, this
involves the vacuum polarization diagram,
$\Pi_{\mu\nu}i(q)=\left(q_{\mu}q_{\nu}-\eta_{\mu\nu}q^{2}\right){\Pi}(q^{2})$
which is described in momentum space using dimensional regularization

$\displaystyle{\Pi}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{\alpha}{3\pi}\left[\frac{1}{\epsilon}+\log
4\pi-\gamma-\log\frac{m^{2}}{\mu^{2}}-\frac{q^{2}}{5m^{2}}\right]~{}~{}~{}~{}~{}~{}~{}(q^{2}<<m^{2})$
(9) $\displaystyle=$
$\displaystyle\frac{\alpha}{3\pi}\left[\frac{1}{\epsilon}+\log
4\pi-\gamma-\log\frac{-q^{2}}{\mu^{2}}\right]~{}~{}~{}~{}~{}~{}~{}(q^{2}>>m^{2})$

where $\epsilon=(d-4)/2$. At low energy only the top line is relevant. We know
what to do with the divergence - it goes into the renormalization of the
electric charge. However we should also note the $\log m^{2}$ term. This is
present even for the heaviest masses. However, if we are to measure the
electric charge at $q^{2}=0$, it also goes into the definition of the charge.
The residual describes the deviation from the result at $q^{2}=0$,

$\displaystyle{\hat{\Pi}}_{\mu\nu}(q)={\Pi}_{\mu\nu}(q)-{\Pi}_{\mu\nu}(0)=\frac{\alpha}{15\pi}\left(q_{\mu}q_{\nu}-\eta_{\mu\nu}q^{2}\right)\frac{q^{2}}{m^{2}}~{}~{}~{}~{}~{}~{}~{}(q^{2}<<m^{2})\
\ .$ (10)

This result can be described by an effective Lagrangian containing only the
photon field333Here I have chosen the $1/4e^{2}$ normalization to underscore
the renormalization of the electric charge is applied at $q^{2}=0$.

${\cal
L}_{eff}(A)=-\frac{1}{4e^{2}(0)}F_{\mu\nu}F^{\mu\nu}-\frac{1}{240\pi^{2}m^{2}}F_{\mu\nu}\Box
F^{\mu\nu}\ \ .$ (11)

The form of this Lagrangian has been chosen to match with the vacuum
polarization amplitude when a matrix element is taken.

The most important point here is that the new term in the effective Lagrangian
is local. This follows from the uncertainty principle. Effects from high
energy/momentum appear only at short distance in coordinate space. Such
effects can be Taylor expanded in the the light momenta and are then
represented by a local derivative expansion for the effective Lagrangian. The
takeaway is that effects from high energy appear local when viewed at low
energy, and that they can be represented by local Lagrangians.

The other important property demonstrated here is the decoupling of the heavy
mass. You can see through Eq. 9 that the vacuum polarization depends on the
logarithm of the heavy mass. However, this is absorbed into the definition of
the renormalized coupling. In this sense, the electric charge depends on the
masses of charged particles, no matter how heavy. However, there is no physics
in this dependence - we only use the measured value of the charge. This is the
Appelquist-Carazzone theorem at work Appelquist:1974tg . The effects of a
heavy particle appears either in the renormalization of the coupling constants
of the theory, or are suppressed by powers of the heavy mass.444An exception
is when integrating out the heavy particle violates the symmetry of the
theory..

This is perhaps a good place to note that despite our early discussion which
emphasized that we treat the path integral as correct below some scale
$\Lambda$, we generally do not use a cutoff in calculations. Dimensional
regularization respects the symmetries of many theories where a cutoff often
does not, and it is easy to use. However the loop integration does run over
all scales including those beyond the applicability of the effective field
theory. This is nevertheless acceptable. The ”wrong” behavior at high energy
appears as a local effect and satisfies the Appelquist-Carazzone theorem. It
then disappears into the renormalization and identification of the parameters
of the local effective Lagrangian. Those parameters will be the appropriate
ones as long as the EFT is not applied outside of its range of validity.

Another feature that can be seen here is that there is no Ostrogradsky
instability Ostrogradsky ; Woodard:2015zca  associated with the higher
derivatives. This refers to the result in classical mechanics where theories
with higher time derivatives the Hamiltonian, calculated by canonical methods,
exhibits an instability. While the second term has the higher time
derivatives, in practice it does not lead to any instability. At low energy
its effect is small compared to the usual energy. The higher derivatives can
become comparable, and potentially trigger an instability, only at energies
which are far higher than those appropriate for the effective field theory
Simon:1990jn . Despite the extra derivatives, the classical limit of the
effective field theory is usual E&M.

There are corrections to this result suppressed by more powers of the mass.
The next term in the effective Lagrangian is that of Euler and Heisenberg

${\cal
L}_{EH}=\frac{\alpha^{2}}{90m^{4}}\left[(F_{\mu\nu}F^{\mu\nu})^{2}+\frac{7}{4}(F_{\mu\nu}\tilde{F}^{\mu\nu})^{2}\right]$
(12)

which occurs due to the box diagram. This mediates interactions of photons.

It is also instructive to look at the opposite extreme, where the mass of the
fermion goes to zero or the relative momentum transfer is large compared to
the mass. In this case there can be no expansion in inverse powers of the
mass. In the vacuum polarization diagram, the logarithm becomes more
important. This is non-analytic and cannot be Taylor expanded in derivatives.
If we try to represent it in position space it would be non-local, represented
by the non-local function

$\langle
x|\log\Box|y\rangle\equiv\int\frac{d^{4}q}{(2\pi)^{4}}e^{iq\cdot(x-y)}\log(-q^{2})$
(13)

Physically, this is non-local because massless fields can propagate long
distances. If we were to try to match this to an effective action, it would
also have to be non-local, schematically represented by Donoghue:2015xla

$S_{light}=\int
d^{4}x~{}-\frac{1}{4}F_{\mu\nu}\left[\frac{1}{e^{2}(\mu)}-\frac{1}{12\pi^{2}}\log\frac{\Box}{\mu^{2}}\right]F^{\mu\nu}\
\ ,$ (14)

with the shorthand notation

$\int d^{4}xA\log\Box~{}B\equiv\int d^{4}xd^{4}y~{}A(x)\langle
x|\log\Box|y\rangle B(y)\ \ .$ (15)

In Eq. 14 we note the appearance of the running coupling constant. However,
the main point here is that massless fields yield non-analytic terms such as
$\log q^{2}$ in momentum space, and non-localities in position space.

The effective field theory is a full quantum field theory. This can be seen by
the fact that the effective action, Eq. 11, still contains the integration
over the photon field.

##  4 General Relativity as an effective field theory

While most pedagogic treatments of General Relativity emphasize geometry and
curved spacetime, it is also possible to develop it as a gauge field theory
Kibble:1961ba ; Utiyama:1956sy ; Donoghue:2017pgk . If we want to obtain a
field theory coupled to energy and momentum, we will gauge spacetime
translations, which are the corresponding symmetries. This leads to general
covariance, the metric as a dynamical field and covariant derivatives. The
action then must be an invariant, with the simplest terms being the
cosmological constant ant the Einstein action. The geometric treatment is
exceptionally powerful for the classical theory. The field theory treatment is
conceptually closer to the development of the Standard Model, and is more
useful for the quantum theory. Effective field theory then helps make sense of
the quantum field theory.

The ultimate origin of the gravitational interactions may not be known, but we
do know the symmetry of the theory - general covariance. The unknown physics
from high energy will produce a local Lagrangian at low energy. The curvatures
are second order in derivatives of the metric, so the action can be ordered in
the derivative/energy expansion, with the first several terms being

$S_{grav}=\int
d^{4}x\sqrt{-g}\left[-\Lambda+\frac{2}{\kappa^{2}}R+c_{1}R^{2}+c_{2}R_{\mu\nu}R^{\mu\nu}+...\right]$
(16)

with $\Lambda$ being the cosmological constant and $\kappa^{2}=32\pi G$. The
effective field theory by itself says nothing about the magnitude of the
various constants. If we do not know the underlying theory we need to measure
these parameters. At ordinary energies, the effects of the curvature squared
terms are negligible if the constants $c_{1},c_{2}$ have any normal size555In
particular there would only be noticeable effects in the gravitational
interaction at a millimeter if the coefficients were greater than about
$10^{65}$. If the derivative expansion of General Relativity is scaled by the
Planck mass, we would expect these dimensionless coefficients to be of order
unity.. But, given the unexpectedly small value of the cosmological constant
and the possibility of new physics as we increase the energy, we should be
open to the possibility that this latter expectation is not correct.. Moreover
these do not trigger an Ostrogradsky instability when used as effective
interactions Simon:1990jn  in the same way that we saw in the QED example
above.

Including extra terms in the action beyond the Einstein term is not a big deal
by itself. But doing this allows one to perform renormalization of the quantum
theory. By using the most general Lagrangian consistent with general
covariance, we can be sure that all the UV divergences can be renormalized
into the various coefficients as long as we use a regularization which does
not break general covariance. The quantum field is the fluctuation in the
metric which deviates from a given background metric
$g_{\mu\nu}={\bar{g}}_{\mu\nu}+\kappa h_{\mu\nu}$. There is a residual
covariance associated with the background field. The divergences are at short
distance, which by the equivalence principle can be treated as almost flat, so
that we know the behavior of propagators at short distance. The one-loop
renormalization was carried out beautifully by ’t Hooft and Veltman
tHooft:1974toh . The divergences due to graviton loops can be represented by
an effective Lagrangian of the form

${\cal
L}_{div}=\frac{1}{16\pi^{2}}\frac{1}{\epsilon}\left(\frac{1}{120}R^{2}+\frac{7}{20}R_{\mu\nu}R^{\mu\nu}\right)$
(17)

while those for other matter fields are similar but with different
coefficients. One can see that these divergences can be absorbed into the
renormalized values of the coefficients $c_{1},~{}c_{2}$. Power counting in
powers of $G$ Donoghue:2017pgk  reveals that higher order graviton loops yield
divergences at higher order in the derivative expansion, i.e two loops
divergences are of order $R^{3}$ Goroff:1985th . In contrast, higher order
loops of matter fields from renormalizable field theories remain at order
$R^{2}$. This also can be seen by power counting, because the divergences in
such theories do not have any inverse powers of the mass needed to compensate
for extra powers of the derivatives in the numerator.

However, the renormalization of divergences is also not that big of a deal,
although it was the focus of this subject for many years. The divergences
themselves come from the high energy end of the theory, which we know is not
reliable. The ultimate UV completion will eventually tell us the correct way
to treat this domain, and will predict the value of the coefficients. So
renormalization is a necessary step, but one without much content.

The real power of the effective field theory is that it shifts our attention
from the UV (where we do not know the physics) to the IR (where we do). There,
EFT techniques allow one to make real predictions. This is because we know the
light degrees of freedom active there and we know their interactions. At a
given order in the energy expansion, we have a small number of coefficients,
such as $\Lambda,~{}G,~{}c_{1},~{}c_{2}$, so that we have reduced our
ignorance of the full theory of quantum gravity to a few constants. However,
there are dynamical effects which are independent of these coefficients. This
comes from the fact that massless fields like the graviton can propagate large
distances, so that this propagation is distinct from any term in a local
Lagrangian.

As an example, let us display what happens with the gravitational interaction
of non-relativistic particles, which will be discussed with more specificity
in the next section. At tree level, one graviton exchange gives an amplitude
which behaves as $1/q^{2}$ much like photon exchange in QED. When Fourier
transformed this gives the Newtonian potential. At one loop level the
amplitude picks up non-analytic behavior from the loops of gravitons.
Schematically at the next order in $G$, we see that this has the form

${\cal
M}\sim\frac{GMm}{q^{2}}\left[1+aG\sqrt{-m^{2}q^{2}}+bGq^{2}\log(-q^{2})+cGq^{2}+....\right]$
(18)

with $a,~{}b,~{}c$ being constants to be calculated. The analytic term
$cGq^{2}$ will in practice contain effects from the coefficients
$c_{1},c_{2}$, which come from the extra derivatives when we consider the
squares of curvatures in the local action. However the non-analytic terms in
this matrix element will be independent of these parameters. If we are to
Fourier transform the matrix element to obtain a position-space potential, it
will have the form

$V(r)=-\frac{GMm}{r}\left[1+a^{\prime}\frac{GM}{r}+b^{\prime}\frac{G\hslash}{r^{2}}+c^{\prime}G\delta^{3}(x)\right]\
\ .$ (19)

The power-law corrections come from the non-analytic terms in momentum space
and are independent of the coefficients $c_{1},c_{2}$. When we restore powers
of $\hslash$ we can see that the first correction $\sim GM/r$ is classical,
and the second one $\sim G\hslash/r^{2}$ is a quantum correction.

The form of the low energy dynamics is similar to what happens in other
quantum field theories. One does not need to know what happens at extremely
high energies in order to make predictions at ordinary energies - this is the
basic message of effective field theory. Knowledge of the low energy particles
and interactions are sufficient. General Relativity fits beautifully into this
paradigm.

##  5 Lessons of Quantum General Relativity

Perturbative quantum field theory is best at calculating transitions and
scattering amplitudes. From these we can learn some features about low energy
quantum gravity. We have waited a long time for a quantum theory of General
Relativity. While the EFT does not answer all of our questions, let us see
what we can do with it.

###  5.1 There is a universal quantum correction to the non-relativistic
potential

We can calculate the gravitational scattering amplitude for two massive
particles at one loop order. Because all diagrams are included, this is the
complete quantum amplitude and it is a gauge invariant. While the full
amplitude is a function of all the kinematic variables, we can take the non-
relativistic limit in which the only important variable is the three momentum
transfer $-q^{2}=\mathbf{q}^{2}$. For display purposes, one can then Fourier
transform this function to obtain a position space potential. As explained in
the previous section, the power law corrections in $r$ follow from the non-
analytic momentum dependence and are independent of any divergences or unknown
coefficients. The result for two particles of mass $M$ and $m$ is Bjerrum-
Bohr:2002gqz ; Khriplovich:2002bt

$V(r)=-\frac{GMm}{r}\left[1+3\frac{G(M+m)}{r}+\frac{41}{10\pi}\frac{G\hslash}{r^{2}}\right]\
\ .$ (20)

It is obviously the third term which is the quantum correction.

It is interesting that result has now been calculated using three methods. The
original calculations used the usual Feynman diagram methods. However, the
same result can be obtained by modern unitarity-based methods .Bjerrum-
Bohr:2013bxa ; Holstein:2016cyq  Here only the on-shell gravitational Compton
amplitudes (those involving two on-shell gravitons) are required. These are
related to the unitarity cut in the crossed channel. By evaluating this cut
and mapping it onto the cuts of the master Feynman integrals, one can obtain
the final result more simply. In addition, because of the property that the
graviton amplitude is related to the square of a gauge theory amplitude (the
gauge-gravity correspondence or “double copy” Bern:2002kj ; Bern:2019prr ) one
really only has to evaluate the QED Compton amplitude in this way of
calculating Choi:1994ax . This avoids needing to use the very messy triple-
graviton vertex. Because in this method only on-shell physical gravitons are
used, there are no Fadeev-Popov ghosts involved. The original Feynman diagram
calculation was done in harmonic (deDonder) gauge, while the on-shell results
use a form of an axial gauge, confirming the gauge invariance of the result.
The result has also been obtained via dispersion relations Bjerrum-
Bohr:2013bxa , where the spectral function also is calculated from the cut, in
both harmonic gauge and using the double copy axial gauge.

Another interesting feature of this result is that it is universal. The
universality was first found by Holstein and Ross Holstein:2008sx  by redoing
the Feynman diagram calculations for particles of different spins. In such a
method the universality is remarkable because different Feynman diagrams are
involved in the various cases, and the universality is only seen when adding
all diagrams together. However, by the unitarity and dispersive methods the
result can be understood to be the consequence of tree-level soft theorems.
Electromagnetic Compton amplitudes and gravitational Compton amplitudes are
universal in the limit of small momenta Low:1954kd ; Weinberg:1965nx ;
Gross:1968in . This applies not only to elementary particles but to composite
macroscopic objects. In the unitary and dispersive methods, the non-analytic
terms arise from multiplying together the universal tree amplitudes. The
result is then a one-loop soft theorem for quantum gravity.

###  5.2 Both classical and quantum effects come from loop diagrams

We are often told that the loop expansion is an expansion in $\hslash$. If
this were the case, we would not expect to obtain the classical correction
seen in Eq. 20 from one loop Feynman diagrams. However the folk theorem is in
fact not true. For the gravitational interaction, this has been known since
the work of Ishikawa Iwasaki:1971vb  and Gupta and Radford Gupta:1980zu , but
the insight is more general Holstein:2004dn  . At a technical level, when one
is counting powers of $\hslash$ by pulling out overall factors of this
quantity, there are residual factors left behind. For example in the Dirac
equation one has $\hslash~{}\bar{\psi}(i\not{\partial}-m/\hslash)\psi$. The
$m/\hslash$ factor can compensate an overall factor of $\hslash$, as in
$G\sqrt{q^{2}m^{2}}=G\hslash\sqrt{q^{2}m^{2}/\hslash^{2}}$. On a more
philosophical level, if we are to reconstruct the world, including the
classical limit, from the path integral treatment, then the classical results
need to be contained somewhere in the Feynman diagram expansion.

This insight was first developed into a calculational program for classical
gravitational wave physics by Goldberger and Rothstein Goldberger:2004jt .
With some further developments it has now become a subfield for obtaining
classical results from QFT techniques. For recent reviews, see Bjerrum-
Bohr:2022blt ; Bjerrum-Bohr:2022ows ; Goldberger:2022rqf

###  5.3 There is no “test mass” limit for quantum effects

It is common to imagine a test mass of vanishing size moving along a geodesic
without itself distorting the spacetime. Perhaps surprisingly this does not
work for quantum effects. The quantum effects sample more than just the
geodesic even for small masses. The test mass limit for the classical
interaction can be seen in the non-relativistic potential, Eq. 20, where the
classical correction depends on $M+m$. If $m$ is the smaller mass of the two,
then sending $m$ to zero leaves just the larger mass determining the
correction. But the quantum effect is independent of the relative masses, and
both masses contribute equally.

Figure 1: Two diagrams which contribute to the gravitational scattering of two
masses. The solid lines represent particles with a large mass $M$ and a small
mass $m$. The dashed lines are gravitons.

To see this in more detail, let us look at two of the diagrams which
contribute to the corrections of the Newtonian potential, shown in Fig. 1.
These diagrams contribute to both classical and quantum effects. For the
classical portion, Fig. 1(a) gives a correction which is proportional to $M$
and that of Fig.1(b) yields one proportional to $m$. So in the test mass limit
Fig. 1(b) is negligible, and it is reasonable to interpret that of Fig. 1(a)
as a correction to the metric that the test body moves in. However, for the
quantum effects, both diagrams 1(a) and 1(b) give equal quantum corrections,
so the idea of a test mass does not work for these diagrams. In Fig. 1(b), the
graviton propagates for a long distance and samples the gravitational field
not only along the geodesic but also at different points. It is an irreducible
tidal effect. As a correction to the Newtonian potential, it does not vanish
as the mass is taken to zero. There are other diagrams with this property
also.

###  5.4 The bending of massless particles is not universal

One can address the bending of light also by a scattering calculation. This
should not be done by calculating the quantum cross-section and then using the
classical relation to the bending angle - that procedure only works for $1/r$
potentials. However, the eikonal approximation is designed to recover the
geometric optics result at large impact parameter $b$. By calculating the peak
of eikonal phase one reproduces the classical bending angle to impressively
high order in inverse powers of the impact parameter Akhoury:2013yua ;
Bjerrum-Bohr:2016hpa . One can add quantum corrections to this result. The use
of untiarity methods, where the diagrams are reconstructed from the on-shell
cuts, simplifies the calculation greatly.

As usual, the result is unobservably small at any reasonable impact parameter.
However the interesting aspect is that it is not the same for all massless
particles Bjerrum-Bohr:2014zsa ; Bai:2016ivl ; Chi:2019owc . One finds

$\theta=\frac{4GM}{b}+\frac{15}{4}\frac{G^{2}M^{2}\pi}{b^{2}}+\frac{8c_{b}-47+64\log(2r_{0}/b)}{\pi}\frac{G^{2}M\hslash}{b^{3}}$
(21)

where $c_{b}=(371/120,~{}113/120,~{}-29/8)$ for scalars, photons and gravitons
respectively666The original calculation only included gravitons in the cuts.
Subsequent calculations Bai:2016ivl ; Chi:2019owc . correctly also include
massless matter particles. and $r_{0}$ is an infrared cutoff. We should not be
surprised at this lack of universality, as there are no low energy theorems
for massless particles as there were in the non-relativistic limit. The tidal
effects include the long distance propagation of gravitons and the massless
particles themselves. There are entirely different diagrams involved for the
different cases.

This result has interesting implications for the concepts of light cones and
geodesics, as will be discussed below.

###  5.5 $G$ and $\Lambda$ are not running couplings in physical processes

We are used to having our coupling constants depend on the energy scale at
which one measures it. The idea of a running coupling captures the effects of
quantum processes relevant for that scale. These are universal because they
come along with the renormalization of the couplings. In a mass independent
scheme, the $1/\epsilon$ in dimensional regularization are always accompanied
by $\log({\rm Energy})^{2}/\mu^{2}$, where $({\rm Energy})$ represents some of
the kinematic variable in the process under investigation 777The use of a mass
independent scheme is useful to avoid confusion with $1/\epsilon-\log
m^{2}/\mu^{2}$ which does not indicate kinematic running..

In the gravitational action the coefficients quadratic in the curvature, i.e.
$c_{1},~{}c_{2}$, obey this paradigm. We have seen that loops of massless
particles, including gravitons, renormalize the coefficients of operators of
order $R^{2}$. This can be seen most simply in the gravitational vacuum
polarization diagram, and here the divergences do come along with factors
$\log q^{2}/\mu^{2}$. In analogy with the electromagnetic case in Eq. 14 , we
can represent this physical effect to an action such as

$\sim\int
d^{4}x\sqrt{-g}\left[c_{1}(\mu)R^{2}+bR\log(\Box/\mu^{2})R+...\right]\ \ .$
(22)

This is exactly what is done in the formalism developed by Barvinsky and
Vilkovisky Barvinsky:1985an ; Barvinsky:1995jv ; Barvinsky:1993en ;
Satz:2010uu . It is straightforward to convert this into a renormalization
group equation for $c_{1}$.

But $\Lambda$ and $G$ are not like this. When we calculate loops of gravitons
or other massless particles using dimensional regularization, $\Lambda$ and
$G$ are not renormalized - only the curvature squared terms are. So the
logarithms which accompany renormalization are not present.

However, there is renormalization of $\Lambda$ and $G$ when one has loops of
massive particles, although these have a different character. For example for
a scalar of mass $m$ the one loop divergence relevant for the cosmological
constant is Donoghue:2022chi

$\delta\Lambda=-\frac{m^{4}}{32\pi^{2}}\left[\frac{1}{\epsilon}-\gamma+\log(4\pi)+\log\frac{\mu^{2}}{m^{2}}+\frac{3}{2}\right]\
\ .$ (23)

Note that the logarithm here is $\log m^{2}$. It has nothing to do with the
external scales of the problem. Once one measures the cosmological constant at
one scale, it does not change when working at a different scale. Following the
$\log\mu$ dependence does not signify running in physical reactions when
masses are present because $\log m^{2}/\mu^{2}$ does not change with scale.
Another example of this sort is in the QED vacuum polarization for
$q^{2}<<m^{2}$ , i.e. Eq 9. At low energy the $\log m_{t}^{2}/\mu^{2}$ from a
top quark in the loop does not lead to running. For the reader who would like
to see this realized in a EFT setting close to gravity, but with real
comparison to experiment, can study the renormalization of the pion decay
constant $F_{\pi}^{2}$ in chiral perturbation theory Gasser:1984gg ;
Donoghue:1992dd , which plays the same role in the chiral EFT as $1/G$ does in
General Relativity.

One can also make this theoretical argument by noting that the running of
these parameters with external scales would need to match on to some effective
action, most likely non-local. For logarithmic running we have seen now how
this works at the curvature squared order in gravity, Eq. 22, and in QED, Eq.
14. For the cosmological constant and for the Einstein term, general
covariance says that there are no non-local operators which share the form of
the local operator. The leading non-local operator closest to the cosmological
constant has been calculated Donoghue:2022chi , but it has a different
structure. In particular in an expansion about flat space, the cosmological
constant has a term linear in the gravitational field, while any non-local
partner must have at least two fields.

When using a cutoff regularization, one finds divergences depending on powers
of the cutoff such as ${\rm(cutoff)}^{4}$ or ${\rm(cutoff)}^{2}$ for $\Lambda$
and $G$888However, some of the naive dependences often quoted are in fact
absent due to cutoff dependent terms in the path integral Jacobean
Donoghue:2020hoh .. It needs to be emphasized that these do not define running
parameters in physical processes. On one hand, physical results do not depend
on regularization scheme, and the absence of these dependences in dimensional
regularization implies that they are not physical. They are absorbed when
measuring $\Lambda$ or $G$ at a given scale and do not change when the
external scales do. In an effective action framework, powers of external
scales would be represented in the local Lagrangian by powers of derivatives
or curvatures. They are already included in the local terms in the derivative
expansion of the EFT, but with fixed coefficients. Again, the related case of
chiral perturbation theory is an experimentally tested EFT where one verifies
these comments.

Finally, one could just try to identify a running coupling in the calculation
of physical processes. Perhaps there turns out to be some useful way to
capture the quantum effects as a function of scale. There are multiple
calculations which have been done. In any one process, one could identify some
energy dependence which describes the quantum effects for that process. But
there is no useful or universal identification that works with multiple
processes Anber:2011ut

The subfield of Asymptotic Safety Percacci:2007sz ; Niedermaier:2006wt ;
Reuter:2019byg  uses a IR cutoff and defines the theory by using Wilsonian
ideas for running that cutoff from a UV fixed point down to zero energy.
However, this is just used to define the physical theory once the full range
of the cutoff is included. It does not mean that the physical parameters run,
and is not intrinsically in conflict with the above discussion. But it should
be recognized that it is incorrect to use that power-law cutoff dependence in
physical settings as if it were running in phenomenological applications
Donoghue:2019clr . More modern treatments use the derivative expansion to
describe energy behavior in physical processes Knorr:2022dsx , and are more in
line with the effective field theory treatment.

###  5.6 There is no “quantum metric”

It is tempting to look for quantum effects modifying the metric describing
various classical solutions. We do not do this for QED (there is no “quantum
corrected electric field” surrounding a charge) but classical solutions play
such a foundational role in General Relativity that we are certainly
interested in this question.

For cases where gravity remains classical and the quantum effects are due to
matter fields, such as those involving photons in the Reissner-Nordstrom
metric, this question appears well defined Donoghue:2001qc . For example, the
quantum correction due to photon loops in $g_{00}$ is999In this equation I
have removed the factor of $\hslash$ from the usual definition of $\alpha$ so
that $\alpha=e^{2}/4\pi$.

$g_{00}=1-\frac{2GM}{r}+\frac{G\alpha}{r^{2}}-\frac{8}{3\pi}\frac{G\alpha\hslash}{Mr^{3}}\
\ .$ (24)

Since gravity is classical here, this result just follows from loop
corrections to the energy momentum tensor. Here again, loop diagrams reproduce
both the leading classical correction and a quantum correction.

However, when gravity itself is treated in QFT, the result is more
problematic. Part of the problem is that the metric itself is not a well
defined quantum concept. Quantum physics traditionally describes transition
amplitudes and the like, and the results described earlier in this section
have been derived from these. But the traditional class of well-defined
quantum objects does not include the field variable itself.

Along these lines, there have been attempts Radkowski ; Duff:1974ud ; Bjerrum-
Bohr:2002fji ; Khriplovich:2004cx  (including one by the present author) to
calculate the quantum corrections to the Schwarzschild metric. These have been
criticized by Kirilin Kirilin:2006en  as not being invariant under the
reparametrization of the gravitational field, i.e. the way that one choses to
expand about a background metric. In QFT, is it not supposed to make a
difference if we perform field redefinitions as long as the identification of
the on-shell free fields is maintained. This is commonly used and is referred
to as Haag’s theorem Haag . However, even in non-gravitational QFT the theorem
only applies to on-shell matrix elements. Off-shell quantities and
intermediate results are not similarly invariant. A given set of quantum
corrections to a metric are intermediate results to a full calculation and not
only depend on the gauge (which is to be expected) but also on the field
parameterization (which is harder to overcome).

The semiclassical idea of the expectation value of the metric $\langle
g_{\mu\nu}\rangle$ is also not a valid quantum object when the gravitational
field itself is quantum. It is subject to Kirilin’s criticism. One would need
to understand how this expectation value would be measured in a quantum
process for it to be well defined.

This problem as described above is one of perturbation theory. The idea of a
covariant non-local effective actions seems well defined. If treated fully,
this presumably could reveal the nature of quantum solutions. But in practice
such actions are only approximately known. The method of Barvinsky and
Vilkovisky called the expansion in the curvature Barvinsky:1985an ;
Barvinsky:1995jv ; Barvinsky:1993en , and referred to briefly in Eq. 22 is an
example. It appears expressed in terms of curvatures, so that might be
independent of field redefinitions. However, there is not a unique
understanding of what is meant by $\log\Box$ \- this part of the calculation
is not expressed in terms of curvatures. The correction to the logarithmic
term in the effective action involves structures which are generically of the
form $R^{2}\frac{1}{\Box}R$. These are of the same order in both the loop
expansion and in the derivative expansion as the logarithmic correction.
Dimensional analysis can be used to show that in the Schwarzschild case, near
the horizon these corrections are of the same magnitude as the logarithmic
ones. So even here we see some limits of perturbation theory.

In a less formal context, we can also see in the calculations above that there
is not a universal quantum metric. We can look at the calculations and see if
there is a corrected metric which would describe these situations. One can
easily see that there is no single metric around a massive body which would
recover the behavior for both the non-relativistic and massless particles.
This result also implied indirectly in the discussion of test particles. The
diagrams that go into the calculation of a change in the metric, which would
include Fig. 1a, are not the final result, which would also include Fig. 1b.

###  5.7 Light cones/Penrose diagrams appear as uncontrolled approximations

It is perhaps redundant to point out that the results described above call
into question many of our standard tools. If massless particles follow
different trajectories, which one is it that defines the light cone? If there
is an intrinsic fuzziness to this concept, how does one draw a proper Penrose
diagram? To be sure, these effects are small in the limits where the EFT is
valid. Using the standard tools in that regime will be approximately valid,
with calculable corrections. But the effects get larger and more important as
one approaches the limits of the EFT, where many of the interesting quantum
gravity questions are posed. These tools are not controlled approximations
outside of the EFT region.

##  6 Limits of the gravitational effective field theory

The heart of effective field theory is the idea that only the degrees of
freedom which are active at a given energy need to be included and we only
need to know their interactions near that scale. Physics is an experimental
science and there are boundaries to what we know. Here are some comments on
the limits of the effective field theory for gravity.

###  6.1 High energy

The most obvious limit to the effective field theory is at high energy, or
large curvature. At some energy our knowledge of the right degrees of freedom
or of their interactions fails. We then need a more complete theory. Or
perhaps if the same ingredients remain valid, near the Planck scale we would
enter a strongly coupled regime where EFT techniques would be useless.

Many of the most interesting question in the study of quantum gravity are
sensitive to the high energy limit. In EFT jargon, this is referred to as the
need for a UV completion. Many of the other contributions to this volume
describe these theories.

###  6.2 The extreme infrared limit of the theory

Effective field theory is meant to be best in the infrared, and there is no
indication that this is not correct. However, there are technical limitations
on what we can do with present techniques, which become most obvious in the
extreme infrared. These are perhaps more interesting than the high energy
limitations, as they may lead to new techniques and perhaps new insights.

The local effective action which we start with is an expansion in the local
curvature. In general the nonlocal quantum effects are calculated by a
perturbative expansion around a background metric,
$g_{\mu\nu}=\bar{g}_{\mu\nu}+\kappa h_{\mu\nu}$. However, gravitational
effects build up and the metric can get large even when the local curvature is
small. By the equivalence principle, we can alway chose the background metric
to be almost flat in a neighborhood of any point. This is most clearly seen
using normal coordinates, which expand about a position in spacetime

$g_{\mu\nu}(x^{\prime})=\eta_{\mu\nu}+\frac{1}{3}R_{\mu\alpha\mu\beta}(x)y^{\alpha}y^{\beta}+...~{}~{}~{}~{}~{},~{}~{}~{}~{}~{}y=(x^{\prime}-x)\
\ .$ (25)

However, if the distance away is far enough, the metric in this expansion will
become large unless higher order terms in the expansion, of order $R^{2}$ and
higher, become important. An extreme example of this is the standard
Schwarzchild coordinates for a black hole. Coordinates which are smooth at
infinity have a metric which blows up at the horizon. If we choose coordinates
which are smooth near a point on the horizon, they will blow up somewhere
else. This happens even though the curvature itself will be small outside and
on the horizon for massive black holes.

This problem also permeates classical perturbation theory. One can use post-
Newtonian expansions to calculate gravitational radiation for the far-field
part of the inspiral phase, but we need numerical techniques to capture the
results near the horizon scale even when the curvature there is small.

However, the problem is more severe in quantum perturbation theory. We have
seen that the quantum gravity effects are non-local because the massless
propagators can probe large distances. The non-locality implies that even if
we are in a region of small metric deviation, the propagators can probe the
larger metrics further away. Most of our field theoretic techniques are
adaptions from flat space methods. For a specific example, consider the
logarithmic non-locality such as we have discussed frequently above. In flat
space, dimensional analysis tells us that for time independent problems

$\langle\mathbf{x}|\log\Box|\mathbf{y}\rangle=\int\frac{d^{3}q}{(2\pi)^{3}}e^{-i\mathbf{q}\cdot(\mathbf{x}-\mathbf{y})}\log(\mathbf{q}^{2})\sim\frac{1}{(\mathbf{x}-\mathbf{y})^{3}}$
(26)

The resulting $1/r^{3}$ dependence is seen in the non-relativisitic potential
Eq. 20, the bending angle Eq. 21, and the metric correction Eq. 24. However,
this means that if you were to use this form in the analysis of the
Schwarzschild solution, even if you were at a large distance from the center,
you would be sensitive to the horizon where the metric blows up and sensitive
even to the curvature singularity at the origin.

Again this seems to be a technical problem, which could potentially be solved
by numerical relativity. Intuitively we expect such effects to be small - we
should not need to know about the black hole at the center of our galaxy in
order to do weak field calculations on earth. But we presently do not have a
universal estimator for how big such corrections are. And we do not have any
proof that the quantum effects do not build up over long distances like the
classical effects do 101010Yang Mills theory also is a good perturbation
theory at short distances but, for very different reasons, at large distance
becomes non-perturbative. At the very largest distances, QCD and QED in the
real world have neutral states so that the growth with distance is not
important. Gravitational charges are all the same sign, so effects can build
up.. So this becomes a limitation on the EFT techniques.

Perhaps the issue could be addressed by combining patches using different
coordinates and matching on the boundary. In each patch we can use the
equivalence principle to make the coordinates nearly flat. Then matching at
the boundaries would convey the information from one patch to another.
However, this program has not yet been carried out.

###  6.3 What are the right quantum questions?

Quantum field theory in curved spacetime is a challenging subject even for
non-gravitational particles and interactions. It is not even clear how to
rigorously define a particle in general curved spacetimes. However, in lightly
curved worlds we can approximately use Minkowski field theory techniques. We
do this all the time since we live in a lightly curved spacetime. In curved
spacetime, the gravitational effective field theory shares these challenges,
and adds an extra one because the metric is now a dynamical variable.

The perturbative solution is to expand around a background metric and treat
the fluctuating field quantum-mechanically. We have seen that this can yield
leading predictions at low energy and curvature. But the effective theory has
also demonstrated that some of our usual techniques have some limitations.
Some of these can be traced to the perturbative expansion, as we have seen the
difficulty of identifying the backreaction on the background metric itself.

The larger challenge is to identify valid quantum questions for gravity which
are also able to be answered with techniques which we know how to apply. This
may require the adoption of nonperturbative methods, at least in cases where
the metric field becomes strong. Even in nonperturbative computational methods
such as Causal Dynamical Triangulations, it is also a challenge to identify
sensible objects to be calculated Loll:2019rdj . The limits of perturbation
theory itself may prove to be one of the limits of the effective field theory.
In any case, there still are interesting field theory developments possible
even within the effective field theory.

###  6.4 Other potential limits

Effective field theory is a “humble” theory in that it attempts to work within
the framework of existing knowledge, and to acknowledge its limits. In this
regard, we should recognize that there are other limits on our understanding
besides just the energy/distance variable.

In particular, quantum mechanics itself has only been tested within some
limits. In reactions, there are generally only a few particles involved. Even
in condensed matter physics, with enormous numbers of particles, the
interactions are primarily two-body or sometimes three-body. We presently have
no experimental need for modifications to quantum mechanics, as we do for
other interactions outside their known frontiers. However there could be a
“macroscopicity” frontier. Many issues in the transition from quantum physics
to classical physics and decoherence remain poorly understood. Gravity could
be the place where these effects become important, as gravity is not screened
and very large masses are available. Recent work on stochastic effects and
decoherence, reviewed by C. Burgess in this volume Burgess , provides some new
techniques within quantum mechanics. But perhaps there could be real changes
with quantum physics for objects that are macroscopic enough. This point of
view has been put forward by Penrose Penrose:1976js  and by Stamp and
collaborators Stamp:2015vxa .

##  7 Conclusion

Claims that General Relativity clashes with Quantum Mechanics are wrong, or at
least misleadingly simplistic. The covariant quantization of General
Relativity is by now an old topic Feynman:1963ax ; DeWitt:1967ub ;
Donoghue:2017pgk ; Buchbinder:2021wzv . Modern quantum field theory techniques
- effective field theory - help us to extract physical quantum predictions.
These techniques are completely normal quantum field theory methods which are
applied routinely in other settings. So the quantum field theory of quantum
General Relativity is quite normal.

Indeed, General Relativity is perhaps the best example of the effective field
theory paradigm. It appears to be valid over many orders of magnitude in
distance or energy, and it overlaps with quantum physics in regions where the
latter is relevant. We have tested the low energy degrees of freedom, and
their interactions follow from a simple action. Plus, since we do not know the
ultimate UV completion for quantum gravity, the effective field theory is all
that we can be sure about.

There is no reason to be misleadingly simplistic about quantum gravity. We
have made strides in quantizing, renormalizing and applying quantum General
Relativity. The effective field theory results are interesting in themselves.
And we can readily motivate further quantum gravity work without being
misleading. What we should be saying is that quantum General Relativity points
unmistakably to our lack of understanding of the full theory.

Quantum gravity is not optional. While we likely will not know the
experimental outcome in our lifetimes regarding the various options, there is
still much to learn about the consistency and structure of the theories.
Perhaps the exploration will change our understanding of the world. But we do
know that all of the theories of quantum gravity need to reduce to the
effective field theory of General Relativity in the appropriate limit. This
effective field theory provides a foundation for our exploration of quantum
gravity.

## Acknowledgement

The writing of this review has been supported in part by the US National
Science Foundation grant NSF-PHY-21-12800.

## References

  * (1) J. F. Donoghue, “General Relativity As An Effective Field Theory: The Leading Quantum Corrections,” Phys. Rev. D 50, 3874 (1994)
  * (2) J. F. Donoghue, “Introduction to the effective field theory description of gravity,” [arXiv:gr-qc/9512024 [gr-qc]].
  * (3) J. F. Donoghue, “The effective field theory treatment of quantum gravity,” AIP Conf. Proc. 1483, no.1, 73-94 (2012) doi:10.1063/1.4756964 [arXiv:1209.3511 [gr-qc]].
  * (4) J. Donoghue, “Quantum gravity as a low energy effective field theory,” Scholarpedia 12, no.4, 32997 (2017) doi:10.4249/scholarpedia.32997
  * (5) J. F. Donoghue, M. M. Ivanov and A. Shkerin, “EPFL Lectures on General Relativity as a Quantum Field Theory,” [arXiv:1702.00319 [hep-th]
  * (6) C. P. Burgess, “Quantum gravity in everyday life: General relativity as an effective field theory,” Living Rev. Rel. 7, 5-56 (2004) doi:10.12942/lrr-2004-5 [arXiv:gr-qc/0311082 [gr-qc]].
  * (7) C. P. Burgess, “Introduction to Effective Field Theory,” Cambridge University Press, 2020, ISBN 978-1-139-04804-0, 978-0-521-19547-8 doi:10.1017/9781139048040 iteWeinberg:2016kyd
  * (8) S. Weinberg, “Effective field theory, past and future,” Int. J. Mod. Phys. A 31, no.06, 1630007 (2016) doi:10.1142/S0217751X16300076
  * (9) S. Weinberg, “On the Development of Effective Field Theory,” Eur. Phys. J. H 46, no.1, 6 (2021) doi:10.1140/epjh/s13129-021-00004-x [arXiv:2101.04241 [hep-th]].
  * (10) A. A. Petrov and A. E. Blechman, “Effective Field Theories,” World Scientific Press, 2016, ISBN 978-981-4434-92-8, 978-981-4434-94-2 doi:10.1142/8619
  * (11) U. G. Meißner and A. Rusetsky, “Effective Field Theories” Cambridge University Press, 2022
  * (12) D. Carney, “Newton, entanglement, and the graviton,” Phys. Rev. D 105, no.2, 024029 (2022) doi:10.1103/PhysRevD.105.024029 [arXiv:2108.06320 [quant-ph]].
  * (13) J. F. Donoghue and G. Menezes, “Arrow of Causality and Quantum Gravity,” Phys. Rev. Lett. 123, no.17, 171601 (2019) doi:10.1103/PhysRevLett.123.171601 [arXiv:1908.04170 [hep-th]].
  * (14) J. F. Donoghue and G. Menezes, “Quantum causality and the arrows of time and thermodynamics,” Prog. Part. Nucl. Phys. 115, 103812 (2020) doi:10.1016/j.ppnp.2020.103812 [arXiv:2003.09047 [quant-ph]].
  * (15) A. Belenchia, R. M. Wald, F. Giacomini, E. Castro-Ruiz, Č. Brukner and M. Aspelmeyer, ‘Quantum Superposition of Massive Objects and the Quantization of Gravity,” Phys. Rev. D 98, no.12, 126009 (2018) doi:10.1103/PhysRevD.98.126009 [arXiv:1807.07015 [quant-ph]].
  * (16) D. L. Danielson, G. Satishchandran and R. M. Wald, “Gravitationally mediated entanglement: Newtonian field versus gravitons,” Phys. Rev. D 105, no.8, 086001 (2022) doi:10.1103/PhysRevD.105.086001 [arXiv:2112.10798 [quant-ph]].
  * (17) D. N. Page and C. D. Geilker, “Indirect Evidence for Quantum Gravity,” Phys. Rev. Lett. 47, 979-982 (1981) doi:10.1103/PhysRevLett.47.979
  * (18) S. Carlip, “Is Quantum Gravity Necessary?,” Class. Quant. Grav. 25, 154010 (2008) doi:10.1088/0264-9381/25/15/154010 [arXiv:0803.3456 [gr-qc]].
  * (19) S. M. Giampaolo and T. Macrì, “Entanglement, holonomic constraints, and the quantization of fundamental interactions,” Sci. Rep. 9, no.1, 11362 (2019) doi:10.1038/s41598-019-47844-8 [arXiv:1806.08383 [quant-ph]].
  * (20) T. Appelquist and J. Carazzone, “Infrared Singularities and Massive Fields,” Phys. Rev. D 11, 2856 (1975) doi:10.1103/PhysRevD.11.2856
  * (21) M. Ostrogradsky, “Memoires sur les equations differentielles, relatives au probleme des isoperimetres,” Mem. Acad. St. Petersbourg 6, no.4, 385-517 (1850)
  * (22) R. P. Woodard, “Ostrogradsky’s theorem on Hamiltonian instability,” Scholarpedia 10, no.8, 32243 (2015) doi:10.4249/scholarpedia.32243 [arXiv:1506.02210 [hep-th]].
  * (23) J. Z. Simon, “The Stability of flat space, semiclassical gravity, and higher derivatives,” Phys. Rev. D 43, 3308-3316 (1991) doi:10.1103/PhysRevD.43.3308
  * (24) J. F. Donoghue and B. K. El-Menoufi, “QED trace anomaly, non-local Lagrangians and quantum Equivalence Principle violations,” JHEP 05, 118 (2015) doi:10.1007/JHEP05(2015)118 [arXiv:1503.06099 [hep-th]].
  * (25) T. W. B. Kibble, “Lorentz invariance and the gravitational field,” J. Math. Phys. 2, 212-221 (1961) doi:10.1063/1.1703702
  * (26) R. Utiyama, “Invariant theoretical interpretation of interaction,” Phys. Rev. 101, 1597-1607 (1956) doi:10.1103/PhysRev.101.1597
  * (27) G. ’t Hooft and M. J. G. Veltman, “One loop divergencies in the theory of gravitation,” Ann. Inst. H. Poincare Phys. Theor. A 20, 69-94 (1974)
  * (28) M. H. Goroff and A. Sagnotti, “The Ultraviolet Behavior of Einstein Gravity,” Nucl. Phys. B 266, 709-736 (1986) doi:10.1016/0550-3213(86)90193-8
  * (29) N. E. J. Bjerrum-Bohr, J. F. Donoghue and B. R. Holstein, “Quantum gravitational corrections to the nonrelativistic scattering potential of two masses,” Phys. Rev. D 67, 084033 (2003) [erratum: Phys. Rev. D 71, 069903 (2005)] doi:10.1103/PhysRevD.71.069903 [arXiv:hep-th/0211072 [hep-th]].
  * (30) I. B. Khriplovich and G. G. Kirilin, “Quantum power correction to the Newton law,” J. Exp. Theor. Phys. 95, no.6, 981-986 (2002) doi:10.1134/1.1537290 [arXiv:gr-qc/0207118 [gr-qc]].
  * (31) N. E. J. Bjerrum-Bohr, J. F. Donoghue and P. Vanhove, “On-shell Techniques and Universal Results in Quantum Gravity,” JHEP 02, 111 (2014) doi:10.1007/JHEP02(2014)111 [arXiv:1309.0804 [hep-th]].
  * (32) B. R. Holstein, “Analytical On-shell Calculation of Low Energy Higher Order Scattering,” J. Phys. G 44, no.1, 01LT01 (2017) doi:10.1088/0954-3899/44/1/01LT01 [arXiv:1609.00714 [hep-ph]].
  * (33) Z. Bern, “Perturbative quantum gravity and its relation to gauge theory,” Living Rev. Rel. 5, 5 (2002) doi:10.12942/lrr-2002-5 [arXiv:gr-qc/0206071 [gr-qc]].
  * (34) Z. Bern, J. J. Carrasco, M. Chiodaroli, H. Johansson and R. Roiban, “The Duality Between Color and Kinematics and its Applications,” [arXiv:1909.01358 [hep-th]].
  * (35) S. Y. Choi, J. S. Shim and H. S. Song, “Factorization and polarization in linearized gravity,” Phys. Rev. D 51, 2751-2769 (1995) doi:10.1103/PhysRevD.51.2751 [arXiv:hep-th/9411092 [hep-th]].
  * (36) B. R. Holstein and A. Ross, “Spin Effects in Long Range Gravitational Scattering,” [arXiv:0802.0716 [hep-ph]].
  * (37) F. E. Low, “Scattering of light of very low frequency by systems of spin 1/2,” Phys. Rev. 96, 1428-1432 (1954) doi:10.1103/PhysRev.96.1428
  * (38) S. Weinberg, “Infrared photons and gravitons,” Phys. Rev. 140, B516-B524 (1965) doi:10.1103/PhysRev.140.B516
  * (39) D. J. Gross and R. Jackiw, “Low-Energy Theorem for Graviton Scattering,” Phys. Rev. 166, 1287-1292 (1968) doi:10.1103/PhysRev.166.1287
  * (40) Y. Iwasaki, “Quantum theory of gravitation vs. classical theory. - fourth-order potential,” Prog. Theor. Phys. 46, 1587-1609 (1971) doi:10.1143/PTP.46.1587
  * (41) S. N. Gupta and S. F. Radford, “Quantum field theoretic electromagnetic and gravitational two particle potentials,” Phys. Rev. D 21, 2213-2225 (1980) doi:10.1103/PhysRevD.21.2213
  * (42) B. R. Holstein and J. F. Donoghue, “Classical physics and quantum loops,” Phys. Rev. Lett. 93, 201602 (2004) doi:10.1103/PhysRevLett.93.201602 [arXiv:hep-th/0405239 [hep-th]].
  * (43) W. D. Goldberger and I. Z. Rothstein, “An Effective field theory of gravity for extended objects,” Phys. Rev. D 73, 104029 (2006) doi:10.1103/PhysRevD.73.104029 [arXiv:hep-th/0409156 [hep-th]].
  * (44) N. E. J. Bjerrum-Bohr, P. H. Damgaard, L. Plante and P. Vanhove, “The SAGEX Review on Scattering Amplitudes, Chapter 13: Post-Minkowskian expansion from Scattering Amplitudes,” [arXiv:2203.13024 [hep-th]].
  * (45) N. E. J. Bjerrum-Bohr, L. Planté and P. Vanhove, “Effective Field Theory and Applications: Weak Field Observables from Scattering Amplitudes in Quantum Field Theory,” [arXiv:2212.08957 [hep-th]].
  * (46) W. D. Goldberger, “Effective Field Theory for Compact Binary Dynamics,” [arXiv:2212.06677 [hep-th]].
  * (47) R. Akhoury, R. Saotome and G. Sterman, “High Energy Scattering in Perturbative Quantum Gravity at Next to Leading Power,” Phys. Rev. D 103, no.6, 064036 (2021) doi:10.1103/PhysRevD.103.064036 [arXiv:1308.5204 [hep-th]].
  * (48) N. E. J. Bjerrum-Bohr, J. F. Donoghue, B. R. Holstein, L. Plante and P. Vanhove, “Light-like Scattering in Quantum Gravity,” JHEP 11, 117 (2016) doi:10.1007/JHEP11(2016)117 [arXiv:1609.07477 [hep-th]].
  * (49) N. E. J. Bjerrum-Bohr, J. F. Donoghue, B. R. Holstein, L. Planté and P. Vanhove, “Bending of Light in Quantum Gravity,” Phys. Rev. Lett. 114, no.6, 061301 (2015) doi:10.1103/PhysRevLett.114.061301 [arXiv:1410.7590 [hep-th]].
  * (50) D. Bai and Y. Huang, “More on the Bending of Light in Quantum Gravity,” Phys. Rev. D 95, no.6, 064045 (2017) doi:10.1103/PhysRevD.95.064045 [arXiv:1612.07629 [hep-th]].
  * (51) H. H. Chi, “Graviton Bending in Quantum Gravity from One-Loop Amplitudes,” Phys. Rev. D 99, no.12, 126008 (2019) doi:10.1103/PhysRevD.99.126008 [arXiv:1903.07944 [hep-th]].
  * (52) A. O. Barvinsky and G. A. Vilkovisky, “The Generalized Schwinger-Dewitt Technique in Gauge Theories and Quantum Gravity,” Phys. Rept. 119, 1-74 (1985) doi:10.1016/0370-1573(85)90148-6
  * (53) A. O. Barvinsky, Y. V. Gusev, V. V. Zhytnikov and G. A. Vilkovisky, “Asymptotic behaviors of one loop vertices in the gravitational effective action,” Class. Quant. Grav. 12, 2157-2172 (1995) doi:10.1088/0264-9381/12/9/005
  * (54) A. O. Barvinsky, Y. V. Gusev, V. V. Zhytnikov and G. A. Vilkovisky, “Covariant perturbation theory. 4. Third order in the curvature,” [arXiv:0911.1168 [hep-th]].
  * (55) A. Satz, A. Codello and F. D. Mazzitelli, “Low energy Quantum Gravity from the Effective Average Action,” Phys. Rev. D 82, 084011 (2010) doi:10.1103/PhysRevD.82.084011 [arXiv:1006.3808 [hep-th]].
  * (56) J. F. Donoghue, “Nonlocal partner to the cosmological constant,” Phys. Rev. D 105, no.10, 105025 (2022) doi:10.1103/PhysRevD.105.105025 [arXiv:2201.12217 [hep-th]].
  * (57) J. Gasser and H. Leutwyler, “Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark,” Nucl. Phys. B 250, 465-516 (1985) doi:10.1016/0550-3213(85)90492-4
  * (58) J. F. Donoghue, E. Golowich and B. R. Holstein, “Dynamics of the standard model,” Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 2, 1-540 (1992) CUP, 2014, doi:10.1017/CBO9780511524370
  * (59) J. F. Donoghue, “Cosmological constant and the use of cutoffs,” Phys. Rev. D 104, no.4, 045005 (2021) doi:10.1103/PhysRevD.104.045005 [arXiv:2009.00728 [hep-th]].
  * (60) M. M. Anber and J. F. Donoghue, “On the running of the gravitational constant,” Phys. Rev. D 85, 104016 (2012) doi:10.1103/PhysRevD.85.104016 [arXiv:1111.2875 [hep-th]].
  * (61) R. Percacci, “Asymptotic Safety,” [arXiv:0709.3851 [hep-th]].
  * (62) M. Niedermaier and M. Reuter, “The Asymptotic Safety Scenario in Quantum Gravity,” Living Rev. Rel. 9, 5-173 (2006) doi:10.12942/lrr-2006-5
  * (63) M. Reuter and F. Saueressig, “Quantum Gravity and the Functional Renormalization Group: The Road towards Asymptotic Safety,” Cambridge University Press, 2019, ISBN 978-1-107-10732-8, 978-1-108-67074-6
  * (64) J. F. Donoghue, “A Critique of the Asymptotic Safety Program,” Front. in Phys. 8, 56 (2020) doi:10.3389/fphy.2020.00056 [arXiv:1911.02967 [hep-th]].
  * (65) B. Knorr, C. Ripken and F. Saueressig, “Form Factors in Asymptotically Safe Quantum Gravity,” [arXiv:2210.16072 [hep-th]].
  * (66) J. F. Donoghue, B. R. Holstein, B. Garbrecht and T. Konstandin, “Quantum corrections to the Reissner-Nordström and Kerr-Newman metrics,” Phys. Lett. B 529, 132-142 (2002) [erratum: Phys. Lett. B 612, 311-312 (2005)] doi:10.1016/S0370-2693(02)01246-7 [arXiv:hep-th/0112237 [hep-th]].
  * (67) A.F. Radkowski, “Some aspects of the source description of gravitation”, Ann. of Phys. 56, 319 (1970)
  * (68) M. J. Duff, “Quantum corrections to the Schwarzschild solution,” Phys. Rev. D 9, 1837-1839 (1974) doi:10.1103/PhysRevD.9.1837
  * (69) N. E. J. Bjerrum-Bohr, J. F. Donoghue and B. R. Holstein, “Quantum corrections to the Schwarzschild and Kerr metrics,” Phys. Rev. D 68, 084005 (2003) [erratum: Phys. Rev. D 71, 069904 (2005)] doi:10.1103/PhysRevD.68.084005 [arXiv:hep-th/0211071 [hep-th]].
  * (70) I. B. Khriplovich and G. G. Kirilin, “Quantum long range interactions in general relativity,” J. Exp. Theor. Phys. 98, 1063-1072 (2004) doi:10.1134/1.1777618 [arXiv:gr-qc/0402018 [gr-qc]].
  * (71) G. G. Kirilin, “Quantum corrections to the Schwarzschild metric and reparametrization transformations,” Phys. Rev. D 75, 108501 (2007) doi:10.1103/PhysRevD.75.108501 [arXiv:gr-qc/0601020 [gr-qc]].
  * (72) R. Haag, “Quantum field theories with composite particles and asymptotic conditions,” Phys. Rev. 112, 669-673 (1958) doi:10.1103/PhysRev.112.669
  * (73) R. Loll, “Quantum Gravity from Causal Dynamical Triangulations: A Review,” Class. Quant. Grav. 37, no.1, 013002 (2020) doi:10.1088/1361-6382/ab57c7 [arXiv:1905.08669 [hep-th]].
  * (74) C. P. Burgess and G. Kaplanek, “Gravity, Horizons and Open EFTs,” [arXiv:2212.09157 [hep-th]].
  * (75) R. Penrose, “Nonlinear Gravitons and Curved Twistor Theory,” Gen. Rel. Grav. 7, 31-52 (1976) doi:10.1007/BF00762011
  * (76) P. C. E. Stamp, “Rationale for a Correlated Worldline Theory of Quantum Gravity,” New J. Phys. 17, no.6, 065017 (2015) doi:10.1088/1367-2630/17/6/065017 [arXiv:1506.05065 [gr-qc]].
  * (77) R. P. Feynman, “Quantum theory of gravitation,” Acta Phys. Polon. 24, 697-722 (1963)
  * (78) B. S. DeWitt, “Quantum Theory of Gravity. 2. The Manifestly Covariant Theory,” Phys. Rev. 162, 1195-1239 (1967) doi:10.1103/PhysRev.162.1195
  * (79) I. L. Buchbinder and I. Shapiro, Oxford University Press, 2021, ISBN 978-0-19-883831-9