11institutetext: School of Mathematics and Maxwell Institute for Mathematical
Sciences
University of Edinburgh, EH9 3FD, United Kingdom22institutetext: Higgs Centre,
School of Physics and Astronomy
University of Edinburgh, EH9 3FD, United Kingdom

# Classical physics from amplitudes on curved backgrounds

Tim Adamo 1 Andrea Cristofoli 2 Anton Ilderton<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We generalise the Kosower-Maybee-O’Connell (KMOC) formalism relating classical
observables and scattering amplitudes to curved backgrounds. We show how to
compute the final semiclassical state for a particle moving in a curved
background in terms of scattering amplitudes on that background. Two-point
amplitudes in this framework correspond to conservative physics with
background-dependent memory effects. As an application, we consider plane wave
and shock wave backgrounds both in electromagnetism and general relativity. We
determine the final semiclassical state, showing it satisfies a notion of
double copy on curved backgrounds. We then conclude by computing the impulse
of a particle on such backgrounds, deriving exact results and velocity memory
effects.

##  1 Introduction

The Kosower-Maybee-O’Connell (KMOC) formalism is a successful framework in
which to study how classical physics arises from on-shell scattering
amplitudes on flat space-time Kosower:2018adc . Over the past few years, it
has demonstrated remarkable and deep relations between classical observables
and the classical limit of scattering amplitudes. For instance, $6$-point
amplitudes have been used to study the zero variance properties of observables
in the classical limit Cristofoli:2021jas ; Britto:2021pud , while $5$-point
amplitudes and soft emission encode leading-order waveforms and the radiative
Newman-Penrose scalar in scattering processes Cristofoli:2021vyo ;
Bautista:2021inx ; Bautista:2021llr ; Bautista:2019tdr . Similarly, tree-level
$4$-point amplitudes have been used to study conservative physics such as the
impulse in general relativity, Yang-Mills and modified gravity Maybee:2019jus
; Guevara:2019fsj ; Aoude:2021oqj ; delaCruz:2020bbn ; delaCruz:2021gjp ;
Burger:2019wkq , and cuts of two-loop, $4$-point amplitudes relate to
radiative physics such as the total radiated momentum in classical scattering
Herrmann:2021lqe ; Herrmann:2021tct . More recently, $3$-point amplitudes for
massive spinning particles have been used to elucidate the Newman-Janis shift
Arkani-Hamed:2019ymq , while in split signature it has been shown that they
correspond to the Maxwell and Weyl curvature spinors of linear solutions
Monteiro:2020plf ; Monteiro:2021ztt .

Considering these successes, one is tempted to ask if lower-point amplitudes
could play any role in this setup, but the answer would seem to be negative:
$2$-point amplitudes are trivial on a flat space-time. However, when computed
on curved backgrounds, such amplitudes can reveal an incredibly rich structure
such as memory effects, eikonal behaviour and even stringy factorization
properties (cf., Gibbons:1975jb ; tHooft ; Garriga:1990dp ; Verlinde:1991iu ;
Jackiw:1991ck ; Kabat:1992tb ; Adamo:2017nia ; Adamo:2021jxz ; Adamo:2021hno ;
Adamo:2021rfq ). A natural question is then: is it possible to generalise the
KMOC formalism to extract classical physics from amplitudes, at 2-points and
beyond, on a curved background? It seems natural to expect that such a
generalization would provide insight into the classical limit of non-
perturbative observables that are typically studied in strong field QED
Fedotov:2022ely  and QFT on curved spacetime DeWitt:1975ys .

Besides broadening the range of observables accessible to KMOC Kosower:2018adc
, these questions also relate to the fact that when considering the dynamics
of classical particles, we have implicitly fixed a background in which those
particles are moving. Viewing KMOC as essentially a framework for extracting
classical information from quantum field theory (QFT), this arises from fixing
a background around which to perform the perturbative expansion of QFT.
However, at the level of first principles there is nothing (besides
computational simplicity) which requires that the background be trivial:
indeed, any classical solution to the equations of motion can be used as a
background for perturbation theory in the background field formulation of QFT
(cf., Furry:1951zz ; DeWitt:1967ub ; tHooft:1975uxh ; Abbott:1981ke ). Using a
curved background in QFT should then translate, in the classical limit, into
classical particle dynamics in a curved background: we will see that this is
indeed the case, thus generalizing the main idea presented in Kosower:2018adc
beyond a trivial background.

This paper is organized as follows. In section 2, we generalise the KMOC
formalism by considering QFT on a curved background, assumed to be an
asymptotically flat solution of the Maxwell or Einstein equations which admits
an S-matrix. We show how to express the final semiclassical state in terms of
$n$-point amplitudes on a curved background, focusing for the rest of the
paper on the contributions arising from $2$-points. In sections 3 and 4 we
consider the examples of (vacuum) plane wave and shockwave solutions,
respectively; in both cases we look at electromagnetism and gravity. We give
expressions for the final semiclassical state describing a single point
particle crossing such backgrounds – in the same spirit of Cristofoli:2021jas
– and provide a notion of double copy for final states. We will show in
particular that the final semiclassical state on a plane wave or shockwave
background assumes a universal form where all the classical information
carried by the background is dressed in a shifted momentum of the wavepacket,
both in electromagnetism and gravity. Finally, we extract the physical
observable of the impulse (without radiation) from the final semiclassical
state. For shockwaves, we find agreement with the geodesics on such
backgrounds (e.g., tHooft ; Klimcik:1988az ). For plane waves, we recover the
‘velocity memory effect’ for a point particle crossing a plane wave (e.g.,
Zhang:2017rno ; Zhang:2018srn ; Shore:2018kmt ; Steinbauer:2018iis ;
Bieri:2020zki ; DiVecchia:2022owy ), which so far has not been derived in KMOC
on a flat space-time.

##  2 KMOC on curved backgrounds

Consider the classical dynamics of a point particle following the geodesics on
an exact, asymptotically flat solution to the Einstein field equations, or the
classical dynamics under the Lorentz force of a charged point particle in an
exact, asymptotically flat solution to the Maxwell equations. Our aim is to
develop a formalism, analogous to KMOC Kosower:2018adc , relating the
observables of this classical system to on-shell scattering amplitudes in a
curved background; as such, we assume that the background solutions admit an
S-matrix111In the sense that there is unitary evolution between the
asymptotically flat regions and no spontaneous pair production in the free
theory Hawking:1975vcx ; Gibbons:1975jb ; Gibbons:1975kk ; Woodhouse:1976fe ..
To this end, we start by considering the initial state of the system in the
asymptotically flat region

$\ket{\Psi}=\int\mathrm{d}\Phi(p)\,\phi_{b}(p)\,\ket{p}\ ,$ (1)

where $\phi_{b}(p):=\phi(p)\,\mathrm{e}^{\mathrm{i}p\cdot b/\hbar}$, $\phi(p)$
is a properly normalized wavepacket, $b^{\mu}$ is a four-vector encoding
information on the origin of the reference frame, and

${\mathrm{d}}\Phi(p):=\frac{\mathrm{d}^{4}p}{(2\pi)^{4}}\>2\pi\,\Theta\\!\left(p^{0}\right)\,\delta\\!\left(p^{2}-m^{2}\right)\,.$
(2)

Within this region of space-time, the wavepacket $\phi(p)$ is chosen to
provide a well-defined notion of classical particle dynamics. More precisely,
its Compton wavelength $\ell_{c}$, wavepacket spread $\ell_{w}$ and scattering
length $b^{\mu}$ obey the ‘Goldilocks’ relations
$\ell_{c}\ll\ell_{w}\ll\sqrt{-b^{2}}$ Kosower:2018adc .

We denote the $S$-matrix on the electromagnetic or gravitational background by
$\mathcal{S}$; since the background is treated exactly and without
approximation (i.e., non-perturbatively), $\mathcal{S}$ itself depends on and
contains the background fields. Interactions between the initial state (1) and
the exact background can be expressed in terms of the final state

$\mathcal{S}\ket{\Psi}=\int\mathrm{d}\Phi(p)\,\phi_{b}(p)\,\mathcal{S}\ket{p}\
.$ (3)

Following Cristofoli:2021jas , we introduce a completeness relation in the
integrand of (3), so that the final state is fixed in terms of $n$-point
scattering amplitudes on the curved background222We use a shorthand notation
for the integration measure where
$\mathrm{d}\Phi(p_{1},..p_{n}):=\prod_{j=1}^{n}\mathrm{d}\Phi(p_{j})$.

$\mathcal{S}\ket{\Psi}=\sum_{\mathrm{states}}\int\mathrm{d}\Phi(p,p^{\prime},k_{1},\ldots,k_{n})\>\phi_{b}(p)\>\sum_{\eta_{1},..\eta_{n}}\ket{p^{\prime},k_{1}^{\eta_{1}},\ldots,k_{n}^{\eta_{n}}}\bra{p^{\prime},k_{1}^{\eta_{1}},\ldots,k_{n}^{\eta_{n}}}\mathcal{S}\ket{p}\;.$
(4)

Here, the first sum runs over all massless states that can be emitted during
the scattering process (i.e., photons or gravitons), while the second sum is
over the quantum numbers $\eta$ (e.g., polarizations) of each emitted state.

Neglecting the emission of radiation, we have

$\displaystyle\mathcal{S}\ket{\Psi}=\int\mathrm{d}\Phi(p,p^{\prime})\>\phi(p)\>\mathrm{e}^{\mathrm{i}\,p\cdot
b/\hbar}\,\bra{p^{\prime}}\mathcal{S}\ket{p}\,\ket{p^{\prime}}\,,$ (5)

which implies that the conservative dynamics of a point particle on a curved
background is fully captured by the $2$-point amplitude on the background.
While these 2-point amplitudes vanish in trivial backgrounds, they are
generically non-zero in curved backgrounds. To provide some concrete examples,
we now evaluate the final semiclassical state (5) on exact plane wave and
shockwave backgrounds. This amounts to the evaluation of the associated
$2$-point amplitude, which we consider both in electromagnetism and in general
relativity.

##  3 Plane wave backgrounds

Vacuum plane waves are highly-symmetric solutions of electromagnetism and
gravity which have a natural physical interpretation as coherent states of
photons or gravitons. Furthermore, asymptotically flat ‘sandwich’ plane waves
admit an S-matrix Schwinger:1951nm ; Gibbons:1975jb ; Garriga:1990dp ;
Adamo:2017nia  as well as WKB-exact solutions to background coupled wave
equations Wolkow:1935zz ; Ward:1987ws ; Mason:1989 ; Adamo:2017nia .

###  3.1 Final states and double copy

We start by considering a vacuum plane wave solution to Maxwell equations. It
is convenient to work in lightfront coordinates for which the line element is

$\mathrm{d}s^{2}=2\mathrm{~{}d}x^{+}\mathrm{d}x^{-}-\mathrm{d}x^{\perp}\mathrm{d}x^{\perp}\
,$ (6)

using $\perp$ as a shorthand notation for $b=1,2$. In these coordinates, plane
wave solutions are given by electromagnetic 4-potentials of the form

$\mathrm{A}_{\mu}(x)=-x^{\perp}\,{E}_{\perp}(x^{-})\,n_{\mu}\ ,$ (7)

where $n_{\mu}=\delta_{\mu}^{-}$ and ${E}_{\perp}(x^{-})$ are the $2$
(transverse) electric field components, depending on the lightfront variable
$x^{-}$. We take the electric fields to be ‘sandwich’: supported on a finite
range $x_{i}^{-}<x^{-}<x_{f}^{-}$ but otherwise of arbitrary profile. We will
also need the quantity

$a_{\perp}(x^{-}):=\int_{-\infty}^{x^{-}}\mathrm{d}s\>E_{\perp}(s)\,,$ (8)

which is the ‘work done’ on a charged particle as it crosses the plane wave,
entering from the asymptotic past.

The scalar 2-point amplitude is computed by a boundary term in the classical
action (see Section 2 of Adamo:2021rfq  for a recent review); from
Adamo:2017nia , the tree-level charged scalar $2$-point amplitude on an
electromagnetic plane wave background is, in terms of (8),

$\bra{p^{\prime}}\mathcal{S}\ket{p}=2p_{+}\,\hat{\delta}\\!\left(p^{\prime}_{+}-p_{+}\right)\,\hat{\delta}^{2}\\!\left(p^{\prime}_{\perp}-p_{\perp}+ea_{\perp}(\infty)\right)\,\mathrm{e}^{\mathrm{i}\,\tilde{s}_{p}/\hbar}\;,$
(9)

where $\hat{\delta}(x):=2\pi\delta(x)$, $e$ is the charge of the scalar
particle (incoming and outgoing, due to charge conservation) and

$\tilde{s}_{p}:=\int_{x_{i}^{-}}^{x^{-}_{f}}\mathrm{d}s\,s\frac{{\mathrm{d}}}{{\mathrm{d}}s}\frac{e^{2}a^{2}_{\perp}(s)-2ep_{\perp}a_{\perp}(s)}{2\,p_{+}}\,.$
(10)

The quantity $a_{\perp}(\infty)$ is related to the zero-frequency modes of the
electric field Dinu:2012tj ,

$a_{\perp}(\infty):=\int_{-\infty}^{+\infty}\mathrm{d}x^{-}\>E_{\perp}(x^{-})=\tilde{E}_{\perp}(0)\,,$
(11)

where $\tilde{E}(\omega)$ denotes the Fourier transform. Note that this can be
non-zero even though the physical fields vanish outside the sandwich region.
The implication, given that $a_{\perp}$ is in some sense the work done, is
that there can be a net change in momentum delivered to a classical particle
crossing a plane wave; this is of course an example of the electromagnetic
(velocity) memory effect Dinu:2012tj ; Bieri:2013hqa ; Susskind:2015hpa ;
Pasterski:2015zua , as we will investigate below.

The final state (5) can then be expressed in the compact but explicit form

$\mathcal{S}\ket{\Psi}=\int{\mathrm{d}}\Phi(p^{\prime})\>e^{\mathrm{i}\tilde{s}_{l}/\hbar}\,\phi_{b}(p^{\prime}_{+},p^{\prime}_{\perp}+ea_{\perp}(\infty))\,\ket{p^{\prime}}\quad\
,$ (12)

in which we have highlighted that, because of the on-shell condition, the
wavepacket $\phi_{b}(p)$ is a function of only three independent momentum
variables, which it is natural to choose to be $p_{+}$ and $p_{\perp}$.

We turn now to the analogue final state on a gravitational plane wave. These
are vacuum solutions to the Einstein equations which can be represented, in
Brinkmann coordinates, as

$\mathrm{d}s^{2}=2\mathrm{~{}d}x^{+}\mathrm{~{}d}x^{-}-H_{ab}(x^{-})\,x^{a}x^{b}\,(\mathrm{d}x^{-})^{2}-\mathrm{d}x^{\perp}\mathrm{~{}d}x^{\perp}\;,$
(13)

in which $H_{ab}(x^{-})$ is a trace-free $2\times 2$ matrix with support on a
compact ‘sandwich’ region $[x^{-}_{i},x^{-}_{f}]$ but otherwise arbitrary.

Of particular relevance in the study of gravitational memory effects
Strominger:2017zoo ; Shore:2018kmt  are the transverse zweibeins which satisfy
the geodesic deviation equations $\ddot{E}_{ai}=H_{ab}E_{i}^{b}$. Just like
the electromagnetic case, constants of integration for these equations are
fixed by boundary conditions; we impose

$E^{a}_{i}(x^{-}<x_{i}^{-})=\delta^{a}_{i}\,,\qquad
E^{a}_{i}(x^{-}>x_{f}^{-})=\delta^{a}_{i}+x^{-}\,c^{a}_{i}\,,$ (14)

where $c^{a}_{i}$ is a non-vanishing constant matrix fixed by the boundary
conditions and $H_{ab}(x^{-})$; this is the gravitational analogy of
$a_{\perp}(\infty)$. It is easy to see that despite linear $x^{-}$-dependence,
$E^{a}_{i}$ does indeed describe flat Minkowski space in the $x^{-}>x^{-}_{f}$
region.

In terms of these quantities, the $2$-point amplitude is Garriga:1990dp ;
Adamo:2017nia

$\left\langle
p^{\prime}|\mathcal{S}|p\right\rangle=2p_{+}\,\hat{\delta}\\!\left(p_{+}^{\prime}-p_{+}\right)\,F\\!\left(\mathrm{p}^{\prime}_{\perp}-\mathrm{p}_{\perp},p_{+}\right)\quad,$
(15)
$F\\!\left(\mathrm{q}_{\perp},p_{+}\right):=\frac{2\pi}{p_{+}\>\sqrt{\left|\operatorname{det}(c)\right|}\>\hbar}\,\exp\\!\left(-\frac{\mathrm{i}}{2p_{+}\,\hbar}\>\mathrm{q}_{\perp}\cdot
c^{-1}\cdot\mathrm{q}_{\perp}\right)\ ,$ (16)

where $c^{a}_{b}$ is shorthand for $c^{a}_{i}\,\delta^{i}_{b}$. This yields
the following expression for the final semiclassical state (5) in gravity

$\mathcal{S}|\Psi\rangle=\int{\mathrm{d}}\Phi(p,p^{\prime})\,2\,p_{+}\,\phi_{b}(p)\,\hat{\delta}\\!\left(p_{+}^{\prime}-p_{+}\right)\,F\\!\left(\mathrm{p}^{\prime}_{\perp}-\mathrm{p}_{\perp},p_{+}\right)\,\left|p^{\prime}\right\rangle\,.$
(17)

This expression can be further simplified by changing variable $p\to
q=p-p^{\prime}$, which gives

$\mathcal{S}|\Psi\rangle=\int{\mathrm{d}}\Phi(p^{\prime})\,2\,p_{+}^{\prime}\,e^{ip^{\prime}\cdot
b/\hbar}\int\hat{d}^{4}q\,\hat{\delta}\\!\left(q_{+}\right)\hat{\delta}(2q\cdot
p^{\prime})\>\phi(p^{\prime}+q)e^{iq\cdot
b/\hbar}F\\!\left(\mathrm{q}_{\perp},p_{+}\right)\,\left|p^{\prime}\right\rangle\,,$
(18)

the benefit being that the integrals over $q_{+}$ and $q_{-}$ become
straightforward. Following Cristofoli:2021jas , we can perform the remaining
integral over $q_{\perp}$ via a stationary phase approximation – since we are
considering our expressions in the limit of $\hbar\rightarrow 0$, this
approximation effectively becomes exact. The stationary point $q^{*}_{\perp}$
is given by

$q^{*}_{a}=p_{+}\>c_{a}^{i}b_{i}\ ,$ (19)

which yields an expression for the final semiclassical state in gravity that
is remarkably close to the electromagnetic case (12),

$\mathcal{S}|\Psi\rangle=\int{\mathrm{d}}\Phi(p^{\prime})\,e^{i\delta(q^{*})/\hbar}\phi(p^{\prime}_{+},p^{\prime}_{\perp}+p_{+}^{\prime}\>c_{\perp}^{i}b_{i})\,\left|p^{\prime}\right\rangle\,.$
(20)

where $\delta(q^{*})$ is an $\hbar$ independent phase which arises after
stationary phase.

#### Double copy for final states:

Before proceeding to analyse the physics contained in the 2-point amplitudes
and associated final states, we point out that the final semiclassical state
in electromagnetism and gravity can be understood from the point of view of
the double copy on plane wave backgrounds introduced in Adamo:2017nia . The
starting point is given by the amplitudes (9) and (15) on electromagnetic and
gravitational plane wave backgrounds, respectively. The most obvious
difference between the two is the number of momentum-conserving delta
functions; this is despite the waves having the same symmetry group. Another
puzzle is what to do with the asymptotic gauge field appearing inside the
delta functions in the electromagnetic case. These two points are related; to
see how, we must first undo the integral over the 2-dimensional transverse
coordinates which was part of the 2-point amplitude, writing the delta
function as

$\hat{\delta}^{2}\\!\left(p^{\prime}_{\perp}-p_{\perp}+ea_{\perp}(\infty)\right)=\int\\!{\mathrm{d}}^{2}x^{\perp}\,\exp\left[\mathrm{i}\,x^{\perp}\,(p^{\prime}-p+e\,a(\infty))_{\perp}\right]\,.$
(21)

The double copy rules of Adamo:2017nia  can now be applied, in particular the
replacement

$\left.e\,a^{i}(x^{-})\right|_{x^{-}>x^{-}_{f}}\longrightarrow\frac{p_{+}}{2}\,c_{a}^{i}\,x^{a}\;,$
(22)

which is motivated by the observation that the non-trivial component of the
electromagnetic plane wave is a linear function of $x^{a}$ (7) while the non-
trivial component of a gravitational plane wave is quadratic (13).

As a consequence of this double copy procedure, we obtain in the gravitational
amplitude

$\int\\!{\mathrm{d}}^{2}x^{\perp}\,\exp\left[\mathrm{i}\,x^{\perp}\,(p^{\prime}-p)_{\perp}+\mathrm{i}\,p_{+}\,\frac{c_{ab}}{2}\,x^{a}x^{b}\right]\;.$
(23)

It is easy to see that the resulting Gaussian integral yields the desired
result (15) for gravity, thus providing a double copy explanation for the
structure of this amplitude.

###  3.2 Memory effects from amplitudes on plane wave backgrounds

Equipped with the knowledge of the conservative final state in both the
electromagnetic and gravitational cases, we can proceed to compute
observables. A key example for conservative physics is the change in momentum
– or _impulse_ – experienced by a particle crossing a plane wave background

$\Delta
p^{\mu}:=\bra{\Psi}\mathcal{S}^{{\dagger}}\mathbb{P}^{\mu}\mathcal{S}\ket{\Psi}-\bra{\Psi}\mathbb{P}^{\mu}\ket{\Psi}\,,$
(24)

where $\mathbb{P}_{\mu}$ is the momentum operator. Let us start with the
electromagnetic case. Using the final state (12), the impulse becomes

$\Delta
p^{\mu}=\int{\mathrm{d}}\Phi(p^{\prime})\,\left|\phi(p^{\prime}_{+},\,p^{\prime}_{\perp}+e\,a_{\perp}(\infty))\right|^{2}\,p^{\prime\mu}-p^{\mu}\,,$
(25)

where $p^{\mu}$ is the incoming momentum. Taking the classical limit, one
obtains

$\Delta p^{\mu}=-e\,a^{\mu}(\infty)+n^{\mu}\bigg{(}\frac{2e\>p\cdot
a(\infty)-e^{2}\,a^{2}(\infty)}{2\,p_{+}}\bigg{)}\,\ ,$ (26)

where $a_{\mu}=(0,0,a_{\perp})$.

This result is in agreement with the literature Taub:1948zza ; Sengupta , and
has previously been derived, including $\mathcal{O}(e^{2})$ radiative
corrections, from the classical limit of QED amplitudes in plane wave
backgrounds Ilderton:2013dba ; Ilderton:2013tb . For reference, a purely
classical derivation is given in Appendix A. The same observable has also been
studied in the flat background formulation of KMOC Cristofoli:2021vyo  by
considering the interactions between a coherent state and a semiclassical
state describing a classical point particle. Interestingly, it was found that
the leading order non-vanishing contribution to the impulse started at
$e^{2}$, in contrast with the linear contribution appearing in (26). In our
framework, the non-vanishing impulse of the particle at order $e$ is related
to the non-zero net work done (11) by the electric field and it is known as
the electromagnetic velocity memory effect Campoleoni:2019ptc ;
Pasterski:2015zua ; Susskind:2015hpa . From the perspective of scattering
amplitudes on a flat background Cristofoli:2021vyo , one can derive this
result by noticing that the ‘in’ and ‘out’ vacua are related by a large gauge
transformation, which must be incorporated into the amplitude by modifications
of the LSZ formulae Kibble:1965zza ; Dinu:2012tj 333Similar asymptotic
dressings by large gauge transformations arise in the study of IR divergenes
in QED; see Mirbabayi:2016axw ; Gabai:2016kuf ; Kapec:2017tkm  for recent
investigations.. These issues are avoided here by treating the background non-
perturbatively444For alternative ways to obtain such memory effects, see also
Manohar:2022dea ; DiVecchia:2022owy ..

Let us now turn our attention to gravitational plane waves. Using the final
state derived in (20) we have that the impulse experienced by a particle
crossing a gravitational plane wave background - in absence of emitted
radiation - is

$\Delta
p^{\mu}=\int\mathrm{d}\Phi\left(p^{\prime}\right)\left|\phi\left(p_{+}^{\prime},p_{\perp}^{\prime}+p_{+}c_{\perp}^{i}b_{i}\right)\right|^{2}p^{\prime\mu}-p^{\mu}$
(27)

A straightforward calculation shows that the final result is

$\Delta
p^{\mu}=p_{+}\,\delta^{\mu}_{a}\,c^{a}_{i}\,b^{i}-n^{\mu}\,\bigg{(}\frac{2p_{+}\>p_{a}c^{a}_{i}b^{i}-p^{2}_{+}c^{a}_{i}b^{i}c^{b}_{j}b^{j}\delta_{ab}}{2p_{+}}\bigg{)}\,,$
(28)

where we recall from (14) that $c^{a}_{i}$ corresponds to $\dot{E}^{a}_{i}$
for the transverse zweibein, evaluated in the far future, $x^{-}>x_{f}^{-}$.

Removing the masses from both sides of this relation, this result is
equivalent to the change in the geodesic velocity along the $\perp$ direction

$\Delta
u^{a}=\frac{p_{+}}{m}\,\left.\dot{E}^{a}_{j}\right|_{x^{-}>x_{f}^{-}}\,b^{j}\,,$
(29)

in agreement with the so called velocity memory effect555We refer to Appendix
A for a classical derivation of this effect. in general relativity (e.g.,
Shore:2018kmt ). The dependence on the origin of the frame $b^{\mu}$, absent
in the electromagnetic case, is a consequence of the lack of translation
invariance in a gravitational plane wave.

##  4 Shock wave backgrounds

We now turn our attention to shock wave backgrounds. In the electromagnetic
case, shock waves represent exact solutions of the Maxwell equations sourced
by a massless charged particle. In lightfront coordinates (6), the shock wave
potential can be written as Jackiw:1991ck

$\mathrm{A}_{\mu}(x)=-\frac{e}{2\pi}\,\delta(x^{-})\,n_{\mu}\,\log(|x^{\perp}|)\,,$
(30)

where $e$ is the charge of the shock wave background and, as before,
$n_{\mu}=\delta^{-}_{\mu}$. Physically, these fields can be understood as
arising from the ultra-relativistic boost of a Coulomb field. The natural
analogue in gravity is the Aichelburg-Sexl metric describing the gravitational
field of an ultra-boosted Schwarzschild black hole Aichelburg ; Dray:1984ha :

${\mathrm{d}}s^{2}=2\,{\mathrm{d}}x^{+}\,{\mathrm{d}}x^{-}-{\mathrm{d}}x^{\perp}{\mathrm{d}}x^{\perp}+8G_{N}\,P_{-}\,\delta(x^{-})\,\log(|x^{\perp}|)\,({\mathrm{d}}x^{-})^{2}\
,$ (31)

where $P_{-}$ denotes the lightfront energy of the shock wave background.

As before, we want to write down the final state for a particle crossing a
shock wave, neglecting radiation; by (5) this is fixed by the 2-point
amplitude on the background. Exploiting a double copy for shock wave space-
times, the electromagnetic and gravitational cases can be treated uniformly by
writing the 2-point amplitude as tHooft ; Jackiw:1991ck ; Kabat:1992tb ;
Adamo:2021rfq ; Adamo:2021jxz :

$\bra{p^{\prime}}\mathcal{S}\ket{p}=\frac{p_{+}}{\hbar^{2}}\hat{\delta}\left(p_{+}-p_{+}^{\prime}\right)\int\mathrm{d}^{2}x^{\perp}\,\mathrm{e}^{\mathrm{i}\,q_{\perp}x^{\perp}/\hbar+\mathrm{i}\,\chi(x^{\perp})/\hbar}\,,$
(32)

where $q_{\perp}:=p_{\perp}^{\prime}-p_{\perp}$ and $\chi(x^{\perp})$ is a
theory-dependent eikonal phase whose explicit form for electromagnetism and
gravity will be given below. The final semi-classical state for a particle
crossing a generic shock wave background is thus

$\mathcal{S}\ket{\Psi}=\int{\mathrm{d}}\Phi(p,p^{\prime})\,\phi_{b}(p)\,\frac{p_{+}}{\hbar^{2}}\,\hat{\delta}\\!\left(p_{+}^{\prime}-p_{+}\right)\int\mathrm{d}^{2}x^{\perp}\,\mathrm{e}^{\mathrm{i}\,q_{\perp}x^{\perp}/\hbar+\mathrm{i}\,\chi(x^{\perp})/\hbar}\,\ket{p^{\prime}}\
.$ (33)

We can integrate over $p_{+}$ directly while the $p_{-}$ integral is fixed by
the on-shell delta function in the measure. Changing variables from
$p_{\perp}$ to $q_{\perp}$ leaves

$\mathcal{S}\ket{\Psi}=\frac{1}{\hbar^{2}}\int\\!{\mathrm{d}}\Phi(p^{\prime})\,\mathrm{e}^{\mathrm{i}\,p\cdot
b/\hbar}\int\\!\mathrm{d}^{2}x^{\perp}\\!\\!\int\\!\mathrm{d}^{2}q_{\perp}\>\phi(p_{+}^{\prime},p^{\prime}_{\perp}+q_{\perp})\,\mathrm{e}^{\mathrm{i}\,q_{\perp}(x-b)^{\perp}/\hbar+\mathrm{i}\,\chi(x^{\perp})/\hbar}\,\ket{p^{\prime}}\
.$ (34)

Expressed in this form, the final semi-classical state agrees, in the
conservative sector and in the massless limit of one particle, with the final
state describing two point particles,recently studied from the point of view
of on-shell amplitudes in Cristofoli:2021jas . Following the same strategy as
employed there, we can evaluate the remaining $x$ and $q$ integrals using a
stationary phase approximation. This is motivated by the fact that in the
classical limit the eikonal phase (equal to the radial action to this order)
is large compared to $\hbar$.

The stationary phase conditions in the $(x,q)$ plane are given by

$x_{*}^{\perp}=b^{\perp}-\frac{\partial}{\partial{q_{\perp}}}\chi(x_{*})\quad,\quad
q_{*\perp}=-\partial_{\perp}\chi(x_{*})\ .$ (35)

The first equation defines the so called eikonal impact parameter, while the
second localizes the exchange momentum to the derivative of the phase666By
considering the eikonal phase at LO in the coupling one can safely neglect the
derivative with respect to $q_{\perp}$. However, at higher orders this term is
non vanishing and is crucial in deriving the correct observables. For further
details, we refer the reader to Section 4 of Cristofoli:2021jas .. As a
result, we have that the final _semi-classical_ state on a shock wave
background may be written as

$\mathcal{S}|\Psi\rangle=\int{\mathrm{d}}\Phi(p^{\prime})\,\mathrm{e}^{\mathrm{i}\,\delta(x_{*},q_{*})/\hbar}\phi(p_{+}^{\prime},p^{\prime}_{\perp}-\partial_{\perp}\chi(b))\,\left|p^{\prime}\right\rangle\
,$ (36)

where $\delta(x_{*},q_{*})$ is the remaining contribution in the exponent from
the stationary phase argument; it will not contribute to the observables of
interest.

Note that (36) is formally equivalent to the corresponding final state (12)
for a particle crossing an electromagnetic plane wave, if we replace the
derivative of $\chi$ with $a(\infty)$. This suggests that the non-vanishing
value of the eikonal phase can be interpreted as a velocity memory effect in a
shock wave background. As for the explicit eikonal phases in electromagnetism
and gravity, these are related by a double copy Cristofoli:2020hnk ;
Shi:2021qsb  and are given by

$\chi_{\mathrm{EM}}(x^{\perp})=\frac{e^{2}\,\log(|x^{\perp}|)}{2\pi}\,,\qquad\chi_{\mathrm{GR}}(x^{\perp})=-4G_{N}\,p\cdot
P\,\log(|x^{\perp}|)\,.$ (37)

The calculation of the classical impulse proceeds by inserting (36) into (24),
and is equivalent to that in a plane wave. (In fact one can read off the
result from the argument of the wavepacket in (36).) For the case of an
electromagnetic shock wave we obtain, using (37),

$\Delta
p^{\mu}=-\frac{e^{2}\,b^{\mu}}{2\pi\,|b^{2}|}+\frac{e^{4}\,n^{\mu}}{8\pi^{2}\,p_{+}\,|b^{2}|}\,,$
(38)

in which we have used $p\cdot b=0$ and defined $b\cdot b:=-|b|^{2}$. Finally,
the impulse in the gravitational case is

$\Delta p^{\mu}=\frac{4G_{N}\,p\cdot
P\,b^{\mu}}{|b^{2}|}+\frac{8\,G^{2}_{N}\,(p\cdot
P)^{2}\,n^{\mu}}{p_{+}\,|b^{2}|}\;.$ (39)

We find in both cases agreement with the literature Adamo:2021jxz ;
Shore:2018kmt . Incidentally, both (38) and (39) uniquely determines the
geodesics on such backgrounds.

##  5 Conclusion

We have generalized the KMOC framework Kosower:2018adc  by providing a
systematic way to extract classical observables from the classical limit of
scattering amplitudes on curved backgrounds. We considered asymptotically flat
gravitational backgrounds which are exact solutions of the Einstein equations
admitting an $S$-matrix, and electromagnetic backgrounds which are exact
solutions to the Maxwell equations, both sourced and source-free. To relate
observables to amplitudes computed on such backgrounds, we followed the same
strategy as adopted in Cristofoli:2021jas , which consists of first
determining the final state in terms of amplitudes and only afterwards
evaluating the mean value of an operator by extracting classical
observables.As applications, we considered two non-trivial backgrounds: vacuum
plane waves and shock waves, both in electromagnetism and in gravity. In the
plane wave case, we saw that $2$-point amplitudes provide classical velocity
memory effects, finding agreement with the literature, including the
contribution coming from zero frequency modes of the emitted field that has
not previously been obtained from KMOC. In the case of a shockwave background,
we traced the form of the final state back to a recent conjecture for the
final semiclassical state of two interacting particles Cristofoli:2021jas  and
derived the impulse by reconstructing all the geodesics on such space-times.
Interestingly, we have seen that the final semiclassical state assumes a
universal form given by a simple shift in the exchanged momentum of a free
wavepacket, regardless of its definition on a plane wave or shockwave
background, either electromagnetic or gravitational. Based on Adamo:2021rfq ,
we expect the same to hold also for a particle on a linearized Schwarzschild
and Kerr background since their respective $2$-points will still be given by
(32) but for a different eikonal phase. For plane waves, this is due to the
high degree of symmetry present, which renders the conservative physics
semiclassical-exact. For Schwarzschild, Kerr and shockwaves, the result can be
interpreted as the statement that the final states describe conservative
semiclassical dynamics for $2$ particles on a flat space time, _in the probe
limit_ of one of the two Cristofoli:2021jas .

While our focus here was on conservative physics, the general formalism we
developed relates classical radiative physics to higher-point scattering
amplitudes on curved backgrounds. This is apparent through the general
expression (4) for the final state. A clear next step would be to compute
leading radiative effects on plane wave and shockwave backgrounds using
3-point amplitudes; many of the ingredients for this have already been found
in both electromagnetism Adamo:2021jxz ; Fedotov:2022ely  and gravity
Lodone:2009qe ; Adamo:2017nia ; Adamo:2020qru . Beyond 3-points, explicit
calculations for amplitudes in the background field formalism become
increasingly difficult. However, _all-multiplicity_ formulae for scattering
amplitudes in chiral plane wave backgrounds have recently been found with
twistor methods Adamo:2020syc ; Adamo:2020yzi ; Adamo:2022mev , showing that
there is scope for studying high-order radiation effects with this formalism.

We note that for many years an obstacle to pushing plane wave impulse
calculations in QED Ilderton:2013dba ; Ilderton:2013tb  beyond
$\mathcal{O}(e^{2})$ was the difficultly in establishing the full quantum
result, corresponding to higher point and higher loop amplitudes, before the
classical limit was taken. By instead reconsidering such calculations within
the framework proposed here, we hope to reduce this complexity. While the
ultimate goal is gravitational observables, recent _all_ -orders classical and
quantum results in QED Torgrimsson:2021wcj  can provide a benchmark for this
program.

Another generalization would be to investigate other backgrounds, particularly
black hole space-times in gravity. While these do not have a well-defined
$S$-matrix, scattering ‘at large distances’ still makes sense (cf.,
Adamo:2021rfq ). However, wavefunctions on these backgrounds are not WKB-
exact, so calculations will be substantially more involved. A broad class of
WKB-exact backgrounds is still available in terms of ‘impulsive pp-waves,’
which have a physical interpretation in terms of ultra-boosted beams. It would
also be interesting to explore in more detail the relation between this
formulation of KMOC on curved backgrounds, and recent works based on the
exponentiation of the $S$-matrix defined on a flat background
Cristofoli:2021jas ; Damgaard:2021ipf ; DiVecchia:2022owy .

More generally, we hope that this formalism will enable an interaction between
QFT on strong backgrounds and the modern ‘amplitudes program,’ much as the
original KMOC formalism has leveraged the successes of the amplitudes program
into classical physics.

###### Acknowledgements.

We are grateful to Riccardo Gonzo, Donal O’Connell and Matteo Sergola for
useful conversations and comments on the draft. The authors are supported by a
Royal Society University Research Fellowship (TA) and by the Leverhulme Trust,
RPG-2020-386 (TA and AC).

##  Appendix A Classical motion in plane waves

We briefly review how contributions linear and quadratic in the coupling, and
related to memory effects in the impulse, can be derived from the classical
equations of motion of a particle on a given plane wave, first in
electromagnetism and then in gravity. The field strength for a given
electromagnetic plane wave is
$F_{\mu\nu}(x)=n_{\mu}\,E_{\nu}(x^{-})-{E}_{\mu}(x^{-})\,n_{\nu}$, in which
$E_{\mu}=(0,0,E_{\perp})$ are the compactly supported electric field
components of the wave. A particle of charge $e$ and mass $m$, moving in the
wave obeys the Lorentz force equation

$\dot{P}_{\mu}(\tau)=\frac{e}{m}\,F_{\mu\nu}(x(\tau))\,P^{\nu}(\tau)\;,$ (40)

in which $x^{\mu}(\tau)$ is the orbit parameterised by the proper time $\tau$,
$P_{\mu}=m{\dot{x}}_{\mu}$ is the on-shell physical momentum, and a dot
denotes a derivative with respect to the proper time. Assuming the plane wave
background is fixed, one could study the dynamics of a charged particle by
inserting the ansatz $x=x_{(0)}+ex_{(1)}+e^{2}x_{(2)}+\ldots$ in the equations
of motions and by solving order by order in the coupling, without expanding in
the background field. One would find in this approach that the
$\mathcal{O}(e^{2})$ results are in fact exact. To see this in a simple, and
manifestly gauge-invariant, manner we dot $n_{\mu}$ into (40), finding
$n\cdot\ddot{x}=0\rightarrow x^{-}=p_{+}\tau/m$, in which $p_{\mu}$ is the
free momentum in the asymptotic past. One can thus change variables from
$\tau$ to lightfront time $x^{-}$, upon which (40) becomes _an ODE_ in $x^{-}$
which is solved by exponentiation:

$\frac{\mathrm{d}P_{\mu}}{{\mathrm{d}}x^{{-}}}(x^{-})=\frac{e}{p_{+}}F_{\mu\nu}(x^{-})P^{\nu}(x^{-})\quad\rightarrow\quad
P_{\mu}(x^{-})=\mathcal{T}\exp\bigg{[}\frac{e}{p_{+}}\int_{x^{-}_{i}}^{x^{-}}\\!{\mathrm{d}}s\,F(s)\bigg{]}_{\mu\nu}p^{\nu}\;.$
(41)

The time-ordered exponential truncates at second order since contractions with
more than two field strengths are vanishing. Hence the integrals are easily
evaluated and the exact result is

$P_{\mu}(x^{-})=p_{\mu}-ea_{\mu}(x^{-})+n_{\mu}\frac{2ea(x^{-})\cdot
p-e^{2}a(x^{-})\cdot a(x^{-})}{2p_{+}}\;,$ (42)

in which $a_{\mu}$ is the work done on the charged particle as it crosses the
plane wave, given by (8). Evaluating the final momenta at $x^{-}=+\infty$, one
immediately recovers the expression (26) for the impulse as a function of
$a_{\mu}(\infty)$ which was derived from a $2$ point amplitude in
electromagnetism. For further examples of classical particle dynamics in
strong electromagnetic backgrounds see Gonoskov:2021hwf .

We now consider the corresponding change in momentum for a particle on the
gravitational plane wave background (13). The classical dynamics is governed
by the geodesic equation

$\dot{P}^{\mu}(\tau)=-\frac{1}{m}\Gamma_{\nu\rho}^{\mu}(x(\tau))P^{\nu}(\tau)P^{\rho}(\tau)\
.$ (43)

On a plane wave, the only nonvanishing Christoffel symbols are

$\Gamma_{--}^{a}=-H_{\>b}^{a}\left(x^{-}\right)x^{b},\quad\Gamma_{--}^{+}=-\frac{1}{2}\dot{H}_{ab}\left(x^{-}\right)x^{a}x^{b},\quad\Gamma_{-a}^{+}=-H_{ab}\left(x^{-}\right)x^{b}.$
(44)

The vanishing of $\Gamma^{-}_{\mu\nu}$ implies ${\dot{P}}^{-}=0$ as in the
electromagnetic case, so that proper time can again be traded for lightfront
time $x^{-}$. Despite the additional dependence on the transverse coordinates
$x^{a}$ in the geodesic equation (43), the formal exponential solution

$P^{\mu}(x^{-})=\mathcal{P}\exp\bigg{(}-\frac{1}{p_{+}}\int_{-\infty}^{x^{-}}\mathrm{d}y^{-}\>\Gamma_{\nu\rho}^{\mu}\dot{x}^{\rho}(y^{-})\bigg{)}p^{\nu}\;,$
(45)

still truncates and leads to the _explicit_ solution

$\begin{split}P_{+}(x^{-})&=p_{+}\;,\\\
P_{a}(x^{-})&=p_{i}E_{a}^{i}(x^{-})+p_{+}\sigma_{ab}(x^{-})x^{b}(x^{-})\ ,\\\
P_{-}(x^{-})&=\frac{m^{2}}{2p_{+}}+\gamma^{ij}(x^{-})\frac{p_{i}p_{j}}{2p_{+}}+\frac{p_{+}}{2}\dot{\sigma}_{bc}(x^{-})x^{b}(x^{-})x^{c}(x^{-})+p_{i}\dot{E}_{b}^{i}(x^{-})x^{b}(x^{-})\
,\end{split}$ (46)

where $\sigma_{ab}\left(x^{-}\right):=\dot{E}_{a}^{i}E_{ib}$ and
$\gamma^{ij}(x^{-}):=E^{a(i}E^{j)}_{\>a}$. It is easy to see that
$P^{2}=g^{\mu\nu}P_{\mu}P_{\nu}=0$ for all $x^{-}$, thanks to the identity
$\dot{\sigma}_{ab}=\dot{E}^{i}_{a}\dot{E}_{ib}-H_{ab}$. Using (14) in the
limit $x^{-}\to\infty$ one easily recovers (28), in agreement with the impulse
derived from a $2$-point amplitude in gravity.

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