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#  Dynamics, quantum states and Compton scattering in nonlinear gravitational
waves

G. Audagnotto<EMAIL_ADDRESS>Max Planck Institute for Nuclear Physics,
Saupfercheckweg 1, D-69117 Heidelberg, Germany  A. Di Piazza
<EMAIL_ADDRESS>Max Planck Institute for Nuclear Physics,
Saupfercheckweg 1, D-69117 Heidelberg, Germany  Department of Physics and
Astronomy, University of Rochester, Rochester, New York 14627, USA  Laboratory
for Laser Energetics, University of Rochester, Rochester, New York 14623, USA

###### Abstract

The classical dynamics and the construction of quantum states in a plane wave
curved spacetime are examined, paying particular attention to the similarities
with the case of an electromagnetic plane wave in flat spacetime. A natural
map connecting the dynamics of a particle in the Rosen metric and the motion
of a charged particle in an electromagnetic plane wave is unveiled. We then
discuss how this map can be translated into the quantum description by
exploiting the large number of underlying symmetries. We examine the complete
analogy between Volkov solutions and fermion states in the Rosen chart and
properly extend this to massive vector bosons. We finally report the squared
$S$-matrix element of Compton scattering in a sandwich plane wave spacetime in
the form of a two-dimensional integral.

##  I Introduction

Gravitational waves that reach the Earth are generally weak. To have an idea,
the typical fractional deformation $h$ coming from astrophysical sources is of
the order $h\sim 10^{-21}$ or less. This means that Earth-based detectors like
LIGO, Virgo or KAGRA, which have sizes of the order of $10^{3}\text{m}$, need
the incredible displacement sensitivity of $\sim 10^{-18}\text{m}$ in order to
detect a gravitational perturbation [1, 2, 3]. This is the reason why
gravitational waves are usually treated in the weak field approximation [4],
in which the spacetime metric $g_{\mu\nu}$ is approximated as the flat metric
$\eta_{\mu\nu}$ plus a samll correction $\kappa h_{\mu\nu}$ and a first-order
treatment of the latter is in most cases enough for any measurable prediction.
Despite the weakness of the perturbation amplitudes typically measured on
Earth, nonlinear gravitational plane waves, as exact solutions of Einstein’s
equations, can become an interesting subject for different reasons. One
motivation comes from the fact that higher order corrections can grow
substantially with the distance between the source and the observer, making
nonlinear effects eventually not negligible. This has been pointed out, for
example, in Refs. [5, 6]. This is due to the fact that the dynamics can be
expressed in terms of a matrix $e_{ij}$ satisfying the harmonic equation
$\ddot{e}_{ij}=H_{ik}e^{k}_{\\-\ j}$, where the profile $H_{ij}$ encodes the
spacetime curvature as a function of the wave phase (see Sec. III). For small
amplitudes one can identify $2H_{ij}=\kappa\ddot{h}_{ij}$. However, this does
not necessarily imply that only the first-order term in $e_{ij}$ has to be
considered, in fact the second-order correction involves an integration of
$(\dot{h}^{i}_{\\-\ i})^{2}$ and this generally grows with the phase length of
the wave [5]. These large-scale effects could in principle affect pulsar
timing measurements [7, 5], which very recently evidenced the presence of low-
frequency background gravitational waves [8]. Nonlinear effects are
interesting also because they can be of a different nature than the linear
ones, position and velocity memory effects provide an example that attracted a
lot of attention in the last years as well [9, 5, 6, 10, 11]. Another reason
for which exact gravitational plane waves are worth to be studied comes from
the so-called Penrose limit [12, 13]. Penrose proved that “any spacetime has a
plane wave as a limit”, namely the spacetime in a small region around a null
geodesic assumes a plane wave form. This is the gravitational analog of a well
known fact in electromagnetism, indeed an observer moving at ultrarelativistic
velocities perceives an arbitrary electromagnetic field approximately as a
plane wave and the faster the observer moves the more accurate is the
similarity [14]. The generality of this property suggests that despite being
idealizations exact plane waves can provide an interesting scenario for
studying limiting cases.

The paper is organized as follows, in Sec. II we discuss how plane waves in
flat spacetime can be defined through their symmetries and the role that these
play in the dynamics. Section III is devoted to the description of Brinkmann
and Rosen charts, which provide two somehow complementary ways of describing a
nonlinear gravitational wave. In Sec. IV we examine the construction of
quantum states in a plane wave spacetime for scalar, spinor, and vector
particles, underlining the similarities with the flat spacetime case deriving
from shared symmetries and extending the results found in Ref. [15]. Finally,
in Sec. V we provide the full spin and polarization summed squared $S$-matrix
element for Compton scattering in a nonlinear gravitational wave background.

###  I.1 Conventions

The signature of the metric chosen in this work is -2 and we assume
$\hbar=c=1$, $\kappa=\sqrt{32\pi G}$. Referring to the Misner-Thorne-Wheeler
systematization [16] we adopt the convention $[-,-,-]$. Plane waves
propagating along $\bm{z}$ depend on the variable $t-z$ only. For this reason
we will choose to work with light-cone coordinates $\\{{x^{+},x^{i},x^{-}}\\}$
defined as $x^{-}=t-z\equiv\phi$, $x^{+}=\frac{1}{2}(t+z)$, $x^{i}=\\{x,y\\}$.
The Latin indices will be used throughout this paper referring to the two
transverse coordinates. In general, we can introduce the four quantities
$\\{{n^{\mu},\delta_{i}^{\mu},\tilde{n}^{\mu}}\\}$ with $n^{\mu}=(1,0,0,1)$,
$\tilde{n}^{\mu}=\frac{1}{2}(1,0,0,-1)$ such that any vector $v^{\mu}$ can be
decomposed as
$v^{\mu}=v^{-}\tilde{n}^{\mu}+v^{+}n^{\mu}+v^{i}\delta_{i}^{\mu}$ with
$v^{-}=n\cdot v$, and $v^{+}=\tilde{n}\cdot v$. Letters belonging to the first
half of the Greek alphabet $\alpha$, $\beta$,… will refer to the flat
spacetime metric unless otherwise stated, whereas $\mu$, $\nu$,… will be used
as curved spacetime indices. The dot symbol will only be used for flat
spacetime products such that $v\cdot w=v^{\alpha}w^{\beta}\eta_{\alpha\beta}$
but $v^{\mu}w^{\nu}g_{\mu\nu}\neq v\cdot w$. Overdots $\dot{f}$ will represent
derivatives with respect to plane wave phase $\phi$, as already done in the
introduction. Vierbein projected quantities are indicated with an overbar
$\bar{v}^{\alpha}=e^{\alpha}_{\\-\ \mu}v^{\mu}$, consequently
$\bar{\gamma}^{\alpha}$ are the usual Dirac matrices. Symmetrization and
antisymmetrization are defined as
$T_{(\mu_{1}..\mu_{n})}=\frac{1}{n!}\sum_{\sigma\in
S_{n}}T_{\mu_{\sigma(1)}...\mu_{\sigma(n)}}$ and
$T_{[\mu_{1}..\mu_{n}]}=\frac{1}{n!}\sum_{\sigma\in
S_{n}}\text{sgn}(\sigma)T_{\mu_{\sigma(1)}...\mu_{\sigma(n)}}$, respectively,
where $S_{n}$ indicates the permutations of $n$ indices and sgn is the sign
function.

##  II Plane waves and their symmetries

What is a plane wave? We are used to think about a plane wave $\Phi(\phi)$ as
a field that depends only on the combination $\phi=n\cdot x=t-z$ of spacetime
coordinates, with $\bm{z}$ identifying the propagation direction of the plane
wave itself. This intuitive idea unveils three symmetries, namely translations
along all the spacetime coordinates except $\phi$, i.e. $x^{+}=(t+z)/2$ and
$x^{i}$, where $i=1,2$ refers to the coordinates transverse to $\bm{z}$. A
more careful investigation reveals that plane waves feature two more
symmetries $\chi_{i}$ belonging to the Lorentz group. These Killing vector
fields correspond to the infinitesimal Lorentz transformations
$\Lambda^{\alpha}{}_{\beta}=\delta^{\alpha}{}_{\beta}+\omega^{\alpha}{}_{\beta}$
leaving $n^{\alpha}$ unchanged, namely $\omega_{\alpha\beta}n^{\beta}=0$. This
equation defines the massless Wigner little group associated to the null
vector $n^{\alpha}$ (see e.g. [17, 18, 19]). The Lorentz coefficients
satisfying this equation are a linear combination of the two antisymmetric
tensors
$\mathscr{F}_{i}^{\alpha\beta}=n^{\alpha}\delta_{i}^{\beta}-n^{\beta}\delta_{i}^{\alpha}$,
which are the natural transverse tensors associated to the wave. Recalling the
scalar field representation of the Lorentz generators
$\Sigma_{\Phi}^{\alpha\beta}=2ix^{[\alpha}\partial^{\beta]}$ we conclude that
the two symmetries $\chi_{i}$ leaving unchanged the scalar wave
$\chi_{i}\Phi(\phi)=0$ are

$\begin{split}\chi_{i}=2i\mathscr{F}_{i}^{\alpha\beta}x_{\alpha}\partial_{\beta}.\end{split}$
(1)

The five Killing vector fields $(\partial_{+},\partial_{i},\chi_{j})$ have
been reported by Bondi, Pirani and Robinson in one of the first treatments of
exact gravitational waves [20], along with the coordinate isometries they
generate. Above, we considered a scalar wave but the same symmetries preserve
electromagnetic waves as well, i.e., the vector field $A^{\alpha}(\phi)$. To
be more precise, the Lorentz transformations $\chi_{i}$ in this case perform a
gauge transformation on $A^{\alpha}(\phi)$ [21, 22] and therefore
$F_{\alpha\beta}$ is precisely invariant, we will discuss this in more detail
in the following. It has been observed that these generators define a
Heisenberg algebra (see e.g. [23], Eq. 2.3), which is substantially the same
as the one satisfied by space and momentum operators in quantum mechanics but
with $\partial_{+}$ replacing the identity. One way to extend plane waves to
curved spacetime is to require the latter to exhibit at least five Killing
vectors. This is what has been accomplished in Ref. [20], where it is observed
that such a metric can always be locally put in a Rosen-type form
$ds^{2}=2dx^{+}d\phi+\gamma_{ij}(\phi)dx^{i}dx^{j}$ [24, 25]. We will see that
this chart exhibits coordinate singularities, however other charts describing
a gravitational plane wave and covering the whole spacetime are available. For
the sake of clarity, it is useful to discuss here in more detail the familiar
case of an electromagnetic plane wave in flat spacetime $A^{\alpha}(\phi)$. As
previously observed, this field is gauge transformed by the Lorentz-like
generators $\mathscr{F}_{i}^{\alpha\beta}$. This being said, there is a
particular combination of these generators which is the key to solve the
dynamics in this background, namely
$\omega^{\alpha\beta}(\phi,A)=eA^{i}(\phi)\mathscr{F}_{i}^{\alpha\beta}$ or
equivalently
$\omega^{\alpha\beta}(\phi,A)=e\int^{\phi}d\tilde{\phi}F^{\alpha\beta}(\tilde{\phi})$.
As already mentioned, for generic functions $A^{i}(\phi)$ these define the
local little group $E_{2}(n)$ associated to the vector $n^{\alpha}$ [22, 21,
26]. In addition, when these functions are the plane wave components it turns
out that these Lorentz-like generators completely solve the motion. Namely,
one can easily show that the momentum of a charged particle in a plane wave is
described by [21, 27]

$\begin{split}\pi_{p}^{\alpha}(\phi)=\Lambda^{\alpha}_{p,\beta}(\phi,A)p^{\beta}\quad,\qquad\Lambda_{p}(\phi,A)=\exp\left(e\int^{\phi}\frac{d\tilde{\phi}}{p^{-}}F(\tilde{\phi})\right)\equiv
e^{\frac{\omega(\phi,A)}{p^{-}}},\end{split}$ (2)

where the transformation $\Lambda^{\alpha}_{p,\beta}$ has the following
properties [26, 22, 21]:

$\begin{split}\Lambda^{\alpha}_{p,\beta}\Lambda^{\gamma\beta}_{p}=\eta^{\alpha\gamma}\qquad,\qquad\Lambda^{\alpha}_{p,\beta}n^{\beta}=n^{\alpha}\quad\Rightarrow\quad\partial^{\beta}\Lambda^{\alpha}_{p,\beta}=0\qquad,\qquad\Lambda^{\alpha}_{p,\beta}A^{\beta}=A^{\alpha}+\partial^{\alpha}f,\end{split}$
(3)

with
$f(\phi)=\frac{e}{p^{-}}\int^{\phi}d\tilde{\phi}A^{\alpha}(\tilde{\phi})A_{\alpha}(\tilde{\phi})$.
The physical meaning of the above equations is that
$\Lambda^{\alpha}_{p,\beta}$ is a Lorentz-like transformation, it describes
the Wigner little group of the null vector $n^{\alpha}$ and finally it defines
a $U(1)$ gauge transformation. The momentum evolution tells us something
interesting. The motion of a particle in a plane wave can be described by
local Lorentz-like transformations of the initial momentum on the
constant-$\phi$ hypersurfaces. These transformations depend on the plane wave
components. They do not alter the background because they act as gauge
transformations and they do not change the phase direction due to the fact
that they belong to its little group. These features will turn out to be very
general and surprisingly useful also in the gravitational generalization. In
fact, we will show that they are manifestly present in the Rosen metric. The
underlying connection between gauge and spacetime symmetries basically turns
the problem of finding quantum field states in a plane wave background into a
simple procedure. This construction will be discussed in detail later.

##  III Two complementary charts: Brinkmann and Rosen coordinates

There are two charts particularly useful to describe a plane wave spacetime:
the Brinkmann [28] and the Rosen [25] coordinates. These metrics have the form
$\mathscr{G}_{\mu\nu}=\eta_{\mu\nu}+H_{ij}(\phi)X^{i}X^{j}n_{\mu}n_{\nu}$ and
$g_{\mu\nu}=2n_{(\mu}\tilde{n}_{\nu)}+\gamma_{ij}(\phi)\delta^{i}_{\mu}\delta^{j}_{\nu}$,
respectively. While the Rosen metric shows manifestly three of the five
symmetries of plane waves, it has the drawback of being not global: in
general, at least two Rosen charts are needed in order to cover the whole
spacetime. On the other hand, the Brinkmann chart has only one manifest
symmetry but it is global and the Einstein equations have a trivial form when
described in these coordinates. In the following, we will describe the main
properties and the geodesic motion in both these charts, underlying the
natural interplay between them. It is convenient to introduce from the
beginning the key element connecting these metrics, namely the Rosen vierbein
$e_{\alpha\mu}$ defined by

$\begin{split}e_{\alpha\mu}e_{\beta\nu}\eta^{\alpha\beta}=g_{\mu\nu}.\end{split}$
(4)

The matrix $e_{\beta\nu}$ will appear in the discussions of both the Rosen and
the Brinkmann charts, the link between the two metrics being encoded in the
evolution of its transverse part $\ddot{e}_{ij}=H_{ik}e^{k}_{\\-\ j}$. This
equation will be properly derived in the following. Note that the Rosen
vierbein’s first index will be assumed to be raised and lowered by the
Minkowski metric while the second one by the Rosen metric. Thus,
$e_{\beta\nu}$ is not a tensor in Brinkmann coordinates but just a matrix.
Moreover, we will assume without losing generality the symmetry condition
$\dot{e}_{\alpha}^{\\-\ \mu}e_{\beta\mu}=\dot{e}_{\beta}^{\\-\
\mu}e_{\alpha\mu}$ [13].

###  III.1 The Brinkmann chart

A plane wave is described throughout all the spacetime by the Brinkmann metric
$\mathscr{G}_{\mu\nu}(X)$ [28]

$\begin{split}\mathscr{G}_{\mu\nu}=\eta_{\mu\nu}+H_{ij}(\phi)X^{i}X^{j}n_{\mu}n_{\nu},\end{split}$
(5)

where $\phi=n\cdot X=T-Z$ is the same light-cone coordinate introduced in the
previous sections. The independence of $\mathscr{G}_{\mu\nu}$ on
$X^{+}=(T+Z)/2$ makes manifest the symmetry associated to the Killing vector
field $\partial_{+}$. It is worth noting that only one of the five symmetries
of plane waves is explicit in this metric, for this reason the equations of
motion in this chart are not as trivial as in the Rosen one, where three
symmetries out of five are manifest. The fact that this metric is in the Kerr-
Schild form assures three easily verified facts [29]: the inverse metric is
simply
$\mathscr{G}^{\mu\nu}=\eta^{\mu\nu}-H_{ij}(\phi)X^{i}X^{j}n^{\mu}n^{\nu}$, the
vector $n^{\mu}$ is null with respects to both $\mathscr{G}_{\mu\nu}$ and
$\eta_{\mu\nu}$, and, finally, the metric determinant is constant and equal to
the Minkowskian one $\det\mathscr{G}_{\mu\nu}=\det\eta_{\mu\nu}=-1$. The
physical information is encoded in the matrix $H_{ij}$, which is connected to
the Ricci tensor by the simple relation [30]

$\begin{split}R_{\mu\nu}=H_{i}^{\\-\ i}n_{\mu}n_{\nu}.\end{split}$ (6)

The Einstein’s equations in this spacetime read

$\begin{split}H_{i}^{\\-\ i}n_{\mu}n_{\nu}=-8\pi GT_{\mu\nu}\end{split}$ (7)

and therefore the trace of $H_{ij}$ represents the energy density of the
source which induces the gravitational perturbation. This is clearly zero for
vacuum solutions, the Brinkmann profile $H_{ij}$ in this case satisfies the
same traceless condition as the weak-field perturbation $h_{ij}$ in the
physical transverse traceless (TT)-gauge. The pure gravitational wave is thus
represented by a $2\times 2$ traceless matrix $H_{ij}$ and we can identify the
two polarizations usually introduced under weak-field approximation [13, 6]

$\begin{split}H_{ij}=\begin{pmatrix}H_{+}&H_{\times}\\\
H_{\times}&-H_{+}\end{pmatrix}.\end{split}$ (8)

Going back to the general case, where $T_{\mu\nu}\neq 0$, the weak energy
condition $T_{00}\geq 0$ implies $H_{i}^{\\-\ i}\leq 0$ [31]. Exploiting the
simple form of the Riemann tensor it is possible to verify that its traceless
part, the Weyl tensor, depends on the traceless part of the matrix $H_{ij}$.
It thus follows that the spacetime is conformally flat if and only if the
Brinkmann profile is a pure trace $H_{ij}\propto\delta_{ij}$. This is the case
for an electromagnetic wave perturbating the spacetime [32, 13, 33], while
pure gravitational waves are represented by the traceless components and are
Ricci-flat so their Weyl tensor is equal to the Riemann one. This reflects the
nature of the tidal forces acting on geodesics: in the electromagnetic case
these are acting on the volume of a body while in the gravitational case they
deform its shape along the two polarizations $+,\times$. Studying the
geodesics we will see that the actual connection between the Brinkmann profile
$H_{ij}$ and the usual weak-field perturbation is given by
$H_{ij}=\frac{\kappa}{2}\ddot{h}_{ij}+\mathscr{O}(h^{2})$, where the dot
represents a phase derivative. Despite the similarity between $H_{ij}$ and
$h_{ij}$, the Brinkmann chart is not the best candidate to generalize the TT-
gauge because of its dependence on the transverse coordinates, indeed the
Rosen chart will be more suitable for this purpose. While the Einstein’s
equations are algebraic, the equations of motion in this metric are not
completely trivial, as anticipated. The trajectories can be extracted from the
Lagrangian $\mathscr{L}=m^{-1}\Pi^{\mu}\Pi^{\nu}\mathscr{G}_{\mu\nu}$, the
fact that $n^{\mu}\partial_{\mu}$ is a Killing vector implies that $\Pi^{-}$
is conserved such that $\Pi^{-}=p^{-}$, where $p$ is the initial momentum. The
transverse components are easily found to follow the harmonic equation

$\begin{split}\ddot{X}^{i}=H^{i}_{\\-\ j}X^{j}.\end{split}$ (9)

The last component $\Pi^{+}$ can be found algebraically from the on-shell
condition $\Pi^{\mu}\Pi^{\nu}\mathscr{G}_{\mu\nu}=m^{2}$. A study of the
geodesic equation shows that there are groups of null trajectories in plane
wave spacetimes that converge on two nonintersecting null geodesics. This was
first pointed out by Penrose [32], who deduced from this feature the
impossibility to embed the spacetime into a globally hyperbolic one. A key
point to study the geodesics in this chart is to find its Killing vectors.
Studying the Killing equation $\mathscr{K}_{(\mu;\nu)}=0$, it is possible to
identify two important symmetries. By introducing the matrix $e_{ij}$
satisfying $\ddot{e}_{ij}=H_{ik}e^{k}_{\\-\ j}$ and $\dot{e}_{[i}^{\\-\
j}e_{k]j}=0$, the Killing vectors take the form [23, 34]

$\begin{split}\mathscr{K}^{\mu}_{j}=\gamma_{jk}\partial^{\mu}e_{i}{}^{k}X^{i},\end{split}$
(10)

where $\gamma_{ij}=e_{li}e^{l}{}_{j}$ is a Rosen metric. The introduction of
the matrix $e_{ij}$, which as anticipated at the beginning of this section is
just the transverse part of the Rosen natural vierbein, seems unjustified at
this point. For now this matrix can just be considered a useful placeholder
for the transverse coordinate evolution and its role will become clearer when
studying the Rosen metric. From these two Killing vectors follow two
conservation laws: let $\dot{X}^{\mu}$ be a geodesics, then the identities
$\mathscr{K}_{j}^{\mu}\dot{X}_{\mu}$ are conserved. It is immediate to see
that these are equivalent to the quantities
$\partial_{\phi}e_{i}{}^{j}X^{i}=c_{k}\gamma^{kj}$ with $c_{k}$ being
constant. Let us now suppose that the gravitational wave belongs to the
sandwich-type, such that $H_{ij}(\phi)\neq 0$ only in the interval
$\phi_{0}<\phi<\phi_{1}$ for arbitrary $\phi_{0,1}$. If we assume boundary
conditions in the past flat region $\phi<\phi_{0}$, then we can choose
$e_{ij}=\eta_{ij}$ initially and write these conservation laws as
$p^{-}\partial_{\phi}e_{i}{}^{j}X^{i}=p_{k}\gamma^{kj}$, where $p_{k}$ is the
particle initial momentum before the interaction with the wave. Expanding the
derivative and introducing the symmetric tensor
$\sigma_{ij}=\dot{e}_{ik}e_{j}^{\\-\ k}$ we get the following expression for
the transverse momentum

$\begin{split}\Pi^{i}_{p}=e^{ij}p_{j}+p^{-}\sigma^{i}_{\\-\ j}X^{j}\equiv
p^{i}+\Delta_{p}^{i},\end{split}$ (11)

where for future convenience we have introduced the symbol
$\Delta^{i}_{p}=\Pi_{p}^{i}-p^{i}=\Delta e^{ij}p_{j}+p^{-}\sigma^{i}_{\\-\
j}X^{j}$ to represent the correction to the transverse constant momentum one
would have in flat spacetime. It is worth noting that the knowledge of
$\Pi^{i}$ still presumes the ability to solve the second-order differential
equation $\ddot{e}_{ij}=H_{ik}e^{k}_{\\-\ j}$ and the number of wave profiles
$H_{ij}$ allowing to find analytical solutions is extremely limited. However,
this expression can be useful to study the geodesic congruences behavior.
Exploiting the on-shell condition we can write the full momentum as [35, 36]

$\begin{split}\Pi^{\mu}_{p}=\eta^{\mu\alpha}p_{\alpha}+\Delta^{\mu}_{p}-\frac{1}{2p^{-}}\bigg{[}2p_{i}\Delta^{i}_{p}+\Delta_{p,i}\Delta_{p}^{i}+(p^{-})^{2}H\bigg{]}n^{\mu},\end{split}$
(12)

where $\Delta^{\mu}_{p}=\delta_{i}^{\mu}\Delta^{i}_{p}$ and
$H=H_{ij}X^{i}X^{j}$. While by definition of plane waves $n_{\mu;\nu}=0$, the
geodesic congruence generated by $\partial_{-}$ has the non trivial
deformation tensor $\sigma_{ij}$ [37, 23]. Observing that

$\begin{split}\dot{\sigma}_{ij}=H_{ij}-\sigma_{ik}\sigma_{j}^{\\-\
k}\end{split}$ (13)

and exploiting the weak energy condition we can deduce by Schwarz inequality
that $\dot{\sigma}_{i}^{\\-\ i}+(\sigma_{i}^{\\-\ i})^{2}\leq 0$, which is
equivalent to $\partial^{2}_{-}\exp{\int d\phi\sigma_{i}^{\\-\ i}}\leq 0$
[32]. Now, if we choose the initial conditions at $\phi_{0}$ in an
asymptotically flat region we have $\sigma_{ij}(\phi_{0})=0$, and therefore
$\partial_{-}\exp{\int d\phi\sigma_{i}^{\\-\ i}}|_{\phi_{0}}=0$. It follows
then that at some point $\phi^{*}$ we will find $\exp{\int
d\phi\sigma_{i}^{\\-\ i}}|_{\phi^{*}}=0$, and there some components of
$\sigma_{ij}$ have to be infinite. That point is at the same time a null
geodesic focusing point and a singularity of Rosen coordinates, in fact the
Rosen metric $g_{\mu\nu}$ inherits the singularity from its vierbein $e_{ij}$.
Another important property of plane wave spacetimes concerns the imprint the
wave leaves on particles after its passage. These features are known as memory
effects [9, 5, 10, 11]. Let us consider a pair of particles in a sandwich-
wave, which we know being Minkowskian in the in $\phi<\phi_{0}$ and out-region
$\phi>\phi_{1}$, both initially at rest $p^{\nu}=(m,0,0,0)$ for simplicity. If
their initial perpendicular displacement is $\Delta X_{0}^{i}$ then,
exploiting the fact that $e_{i}^{\\-\ j}\Delta X^{i}=\Delta X_{0}^{j}$ due to
the symmetries discussed above Eq. (11), we get for the momentum difference in
the out-region the following expression [11]

$\begin{split}\Delta\Pi_{\nu}\big{|}_{\text{out-region}}=m\dot{e}_{\nu
i}(\phi_{1})\Delta X_{0,\perp}^{i}-\frac{m}{2}\left(\Delta
X_{0,\perp}^{i}\dot{e}_{ji}(\phi_{1})\dot{e}^{j}_{\\-\ k}(\phi_{1})\Delta
X_{0,\perp}^{k}\right)n_{\nu}.\end{split}$ (14)

This is an example of velocity memory effect: the vierbein is trivial in the
in-region $e_{ij}(\phi_{0})=\eta_{ij}$ but it is constrained by the
differential equation $\ddot{e}_{\nu i}(\phi_{1})=0$ in the out-region, such
that its first derivative will generally be different from zero. This means
that two particles initially at rest but displaced acquire a relative momentum
after the passage of the wave. As we will see this property is also connected
to the classical and quantum scattering of a particle by the gravitational
wave itself.

###  III.2 The Rosen chart

The other chart commonly used to describe plane wave spacetimes is the Rosen
one $g_{\mu\nu}(\phi)$ [25]

$\begin{split}g_{\mu\nu}(\phi)=2n_{(\mu}\tilde{n}_{\nu)}+\gamma_{ij}(\phi)\delta^{i}_{\mu}\delta^{j}_{\nu},\end{split}$
(15)

where $\phi$ is the same as in Brinkmann coordinates $\phi=n\cdot x$ and
$\gamma_{ij}$ is a $2\times 2$ matrix. In this chart three of the five Killing
vector fields are clearly manifest: $\partial_{+},\partial_{i}$. This will be
reflected in the simplicity of the equations of motion, as we will see in the
following. An important feature of this chart is that, unlike the Brinkmann
one, it does not cover the whole spacetime. Indeed, it exhibits spurious
coordinates singularities, as anticipated in the previous paragraph. This will
become clear studying the geodesics. For this reason it is generally necessary
to work in Brinkmann coordinates whenever global problems such as scattering
processes are studied. Nonetheless this chart is very important in order to
exploit the symmetries of plane waves and it will be essential in solving
field equations. Moreover the Rosen chart can be chosen to generalize the
concept of TT-gauge usually introduced in the weak-field approximation. In
fact, the Rosen profile $\gamma_{ij}$ is transverse and depends on $\phi$
only, as the perturbation $h_{ij}$. The traceless property, which the
Brinkmann profile shares with $h_{ij}$, is not satisfied by $\gamma_{ij}$.
However, a perturbative expansion in vacuum shows that all the odd orders of
$\gamma_{ij}$ are traceless while all the even ones are pure traces [5]. Thus,
in the linear limit $\gamma_{ij}=\eta_{ij}+\kappa h_{ij}$, with the
perturbation in the TT-gauge. The link between the Rosen and the Brinkmann
charts is easily found observing that the Brinkmann transverse coordinates,
being flat, are just the Rosen vierbein projection of the Rosen ones
$X^{i}=e^{i}_{\\-\ j}x^{j}$ , with $e_{\alpha\mu}e^{\alpha}_{\\-\
\nu}=g_{\mu\nu}$. In order to connect the profiles $\gamma_{ij}$ and $H_{ij}$
one can for example compare the Rosen and Brinkmann Riemann tensors in the
Rosen chart. With the usual symmetries understood, one finds (see App. A)

$\begin{split}\frac{1}{4}\dot{\gamma}^{kl}\dot{\gamma}_{lm}+\frac{1}{2}\gamma^{kl}\ddot{\gamma}_{lm}=e^{ik}e^{j}_{\\-\
m}H_{ij}.\end{split}$ (16)

We can rewrite this relation in terms of the vierbein only as
$\ddot{e}_{ij}=H_{ik}e^{k}_{\\-\ j}$, recalling that we assumed without losing
generality that the symmetry condition $\dot{e}_{\alpha}^{\\-\
\mu}e_{\beta\mu}=\dot{e}_{\beta}^{\\-\ \mu}e_{\alpha\mu}$ is fulfilled [13].
This finally exhibits the connection between the Rosen and the Brinkmann
charts in a clear way. For completeness we report here the full coordinate
transformation that connects the two systems

$\begin{split}x^{i}=e_{j}^{\\-\ i}X^{j}\quad,\quad
x^{+}=X^{+}+\frac{1}{2}\sigma_{ij}X^{i}X^{j}\quad,\quad\ddot{e}_{ij}=H_{ik}e^{k}_{\\-\
j}.\end{split}$ (17)

We recall from the previous discussion that at some focusing point $\phi^{*}$
the Rosen metric has a singulairty and therefore this coordinate
transformation has to be singular as well in order to compensate for it and
produce a globally defined Brinkmann chart. It is now time to exploit the full
power of manifest symmetries in these coordinates to find the geodesics. From
the three Killing vector fields $\partial_{+},\partial_{i}$ we deduce the
corresponding conserved momenta $\pi_{p}^{-}=p^{-},\\-\ \pi_{p,i}=p_{i}$. The
last component $\pi^{+}_{p}$ can be derived from the on-shell condition
$\pi^{\mu}_{p}\pi^{\nu}_{p}g_{\mu\nu}=m^{2}$. The complete momentum with
initial conditions $p_{\mu}$ can be conveniently written as

$\begin{split}\pi_{p,\mu}=p_{\mu}-\frac{1}{2p^{-}}\left(g^{\rho\nu}p_{\rho}p_{\nu}-m^{2}\right)n_{\mu}=p_{\mu}-\frac{p_{i}p_{j}}{2p^{-}}\int_{-\infty}^{\phi}d\tilde{\phi}\dot{\gamma}^{ij}(\tilde{\phi})n_{\mu}.\end{split}$
(18)

It is interesting to observe that the vierbein projection of geodesics in
Rosen coordinates is in a one-to-one correspondence with the motion of a
charged particle in an electromagnetic plane wave. To show this let us define
$e_{\alpha}^{\\-\ \mu}=\delta_{\alpha}^{\\-\ \mu}+\Delta e_{\alpha}^{\\-\
\mu}$, where $\Delta e_{\alpha}^{\\-\ \mu}$ is a $2\times 2$ matrix with
transverse indices, moreover we introduce the notation

$\begin{split}\Delta e_{\alpha}^{\\-\ \mu}p_{\mu}=-\kappa
P_{\alpha}.\end{split}$ (19)

The vierbein projected momentum $\bar{\pi}^{\alpha}_{p}=e^{\alpha}_{\\-\
\mu}\pi_{p}^{\mu}$ is easily found to have the form

$\begin{split}\bar{\pi}^{\alpha}_{p}=p^{\alpha}-\kappa
P^{\alpha}+\frac{\kappa}{p^{-}}\left(p_{\beta}P^{\beta}-\frac{\kappa}{2}P_{\beta}P^{\beta}\right)n^{\alpha},\end{split}$
(20)

with $p_{\alpha}=\delta_{\alpha}^{\mu}p_{\mu}$. This is exactly the momentum
of a charged particle with charge $e$ moving in an electromagnetic plane wave
$A^{\alpha}(\phi)$ in flat spacetime, with the substitution $\kappa
P^{\alpha}\leftrightarrow eA^{\alpha}$. It is worth noting that the vector
field $P^{\alpha}$ has only transverse degrees of freedom by definition, as
the electromagnetic wave. This formal equivalence is a consequence of the
symmetries shared between the Rosen metric and a plane wave in flat spacetime.
In particular, one can consider the vierbein projected geodesic equation
$p^{-}\dot{\bar{\pi}}_{\beta}+\bar{\pi}^{\alpha}\omega_{\alpha\beta\delta}\bar{\pi}^{\delta}=0$
where $\omega_{\alpha\beta\delta}$ are the spin connection coefficients
$\omega_{\alpha\beta\delta}=e_{\alpha}^{\\-\ \mu}e_{\beta}^{\\-\
\nu}e_{\delta\nu;\mu}$. The coefficients depend on $\phi$ only, moreover the
contraction $\bar{\pi}^{\alpha}\omega_{\alpha\beta\delta}\bar{\pi}^{\delta}$
is actually linear in the geodesic momentum because of the conservation of
$p_{i}$ and $p^{-}$. It is easy to show that
$\bar{\pi}^{\alpha}\omega_{\alpha\beta\delta}=\bar{p}^{\alpha}\omega_{\alpha\beta\delta}=-\kappa\mathscr{P}_{\beta\delta}$,
where we introduced the analog of the plane wave Maxwell tensor
$\mathscr{P}_{\beta\delta}=2\bar{\partial}_{[\beta}P_{\delta]}$. Exploiting
this result we obtain a Lorentz equation for the vierbein projected geodesics

$\begin{split}\dot{\bar{\pi}}^{\alpha}_{p}=\frac{\kappa}{p^{-}}\mathscr{P}^{\alpha\beta}\bar{\pi}_{p,\beta}.\end{split}$
(21)

As before this is formally equivalent to the Lorentz equation of a particle
with charge $\kappa$ in flat spacetime in presence of an electromagnetic plane
wave $P^{\alpha}(\phi)$. It follows from the analysis developed in Sec. II
that we can write the momentum as a local Lorentz transformation of the
initial one, now depending on the vierbein components

$\begin{split}\bar{\pi}_{p}^{\alpha}(\phi)=\Lambda^{\alpha}_{p,\beta}(\phi,P)p^{\beta}\quad,\qquad\Lambda_{p}(\phi,P)=\exp\left(\kappa\int^{\phi}_{-\infty}\frac{d\tilde{\phi}}{p^{-}}\bm{\mathscr{P}}(\tilde{\phi})\right).\end{split}$
(22)

The transformation $\Lambda^{\alpha}_{p,\beta}(\phi,P)$ is a gauge
transformation for the vierbein, therefore $\tilde{e}_{\alpha}^{\\-\
\mu}=e_{\beta}^{\\-\ \mu}\Lambda^{\beta}_{p,\alpha}$ defines a local reference
system in which the particle has a constant momentum $p^{\alpha}$ throughout
all the trajectory. This map is not only a formal interesting feature, it is
very useful for translating some known results in electromagnetism and QED to
systems in plane wave spacetimes and offers a very direct way to compare the
linear results in these different contexts. One example is the motion of a
charged particle moving in an electromagnetic plane wave keeping into account
the spacetime curvature produced by the latter. The vierbein projected motion
in this case is described by
$p^{-}\dot{\bar{\pi}}^{\alpha}=\left(\kappa\mathscr{P}^{\alpha\beta}+e\bar{F}^{\alpha\beta}\right)\bar{\pi}_{\beta}$,
therefore the momentum is formally identical to the one of a free-falling
particle with the substitution $\kappa P^{\alpha}\rightarrow\kappa
P^{\alpha}+e\bar{A}^{\alpha}$. While the formal analogy is as simple as clear,
in the physical interpretation one has to be more careful. In fact, the
connection between $P_{\alpha}$ and the physical complete gravitational wave
profile $H_{ij}$ is $\ddot{P}_{i}=H_{i}^{\\-\ j}(p_{j}+P_{j})$. Nonetheless,
in the linear limit it is $P_{\alpha}=\frac{1}{2}p^{\beta}h_{\beta\alpha}$ and
this is the most natural representative of the gravitational perturbation in
the weak-field limit.

##  IV Quantum fields in a plane wave

###  IV.1 In flat spacetime

In Sec. II we noticed that in an electromagnetic background the Lorentz-like
transformation generated by
$\omega^{\alpha\beta}(\phi,A)=e\int^{\phi}d\tilde{\phi}F^{\alpha\beta}(\tilde{\phi})$
plays a very important role. Indeed,
$\Lambda_{p}=\exp\left(\omega/p^{-}\right)$ completely solves the dynamics
acting as an evolution operator on the initial four-momentum:
$\pi_{p}^{\alpha}=\Lambda^{\alpha}_{p,\beta}p^{\beta}$. The importance of this
transformation is not limited to particle dynamics but it is essential in
solving fields equations. Let us introduce the Hamilton-Jacobi action $S_{p}$
such that

$\begin{split}e^{-iS_{p}}iD_{\alpha}e^{iS_{p}}=e^{-iS_{p}}(i\partial_{\alpha}-eA_{\alpha})e^{iS_{p}}=\pi_{p,\alpha}.\end{split}$
(23)

From the momentum evolution Eq. (22) we find that
$e^{-iS_{p}}iD_{\alpha}e^{iS_{p}}=\Lambda^{\alpha}_{p,\beta}p^{\beta}$. This
means that the quantum operator $\mathscr{U}=\exp i\left(S_{p}+p\cdot
x\right)|_{p\rightarrow i\partial}$ constructed from the difference between
the dressed and the free action is a unitary transformation that changes the
covariant derivative into the locally Lorentz transformed free derivative [22,
21]. If we define the operator $\Lambda=\Lambda_{p}|_{p\rightarrow i\partial}$
we find the useful relation

$\begin{split}\mathscr{U}^{-1}D^{\alpha}\mathscr{U}=\Lambda^{\alpha}_{\\-\
\beta}\partial^{\beta}.\end{split}$ (24)

This can be seen as an evolution operator for the free field
$\varphi_{p}=e^{-ip\cdot x}$ such that $\varPhi_{p}=\mathscr{U}\varphi_{p}$
solves the wave equation
$\left(D^{\alpha}D_{\alpha}+m^{2}\right)\varPhi_{p}=0$. It is immediate to
check this exploiting the identity
$\mathscr{U}^{-1}D^{2}\mathscr{U}=\partial^{2}$, which in turn follows from
the properties $\Lambda^{\alpha}_{\\-\
\beta}\Lambda^{\gamma\beta}=\eta^{\alpha\gamma}$ and $\Lambda^{\alpha}_{\\-\
\beta}n^{\beta}=n^{\alpha}$, recalling that $\Lambda^{\alpha}{}_{\beta}$
depends only on $\phi$. We now proceed considering Dirac spinors and massive
charged vector fields [22, 38], as we will see there is a very natural way to
exploit $\Lambda$ in order to find these quantum states. Let us introduce the
generalized quantum operator

$\begin{split}\mathbf{\Lambda}(\phi,A)=e^{\frac{e}{2\partial_{+}}\int^{\phi}d\tilde{\phi}F^{\alpha\beta}(\tilde{\phi})\mathbf{\Sigma}_{\alpha\beta}},\end{split}$
(25)

where now $\mathbf{\Sigma}_{\alpha\beta}$ are the generators of the Lorentz
algebra in a general representation. Above, the four-vector representation was
considered, where
$(\mathbf{\Sigma}_{\alpha\beta})^{\gamma\delta}=-i(\delta_{\alpha}{}^{\gamma}\delta_{\beta}{}^{\delta}-\delta_{\alpha}{}^{\delta}\delta_{\beta}{}^{\gamma})$.
The wave equations for fields $\bm{\varPhi}_{p}$ of spin
$\left\\{0,\frac{1}{2},1\right\\}$ in an electromagnetic plane wave can be
written as

$\begin{split}(\tilde{\nabla}^{\alpha}\tilde{\nabla}_{\alpha}+m^{2})\bm{\varPhi}_{p}=0\qquad,\quad\text{with}\qquad\tilde{\mathbf{\nabla}}_{\alpha}=D_{\alpha}\mathbf{1}-\frac{e}{2\partial_{+}}n_{\alpha}F^{\gamma\beta}\mathbf{\Sigma}_{\gamma\beta}=\mathbf{\Lambda}D_{\alpha}\mathbf{\Lambda}^{-1}.\end{split}$
(26)

Note that in order to prove the last equality one has to exploit the identity
$[F^{\alpha\beta}(\phi)\mathbf{\Sigma}_{\alpha\beta},F^{\gamma\delta}(\phi^{\prime})\mathbf{\Sigma}_{\gamma\delta}]=0$
for arbitrary values of $\phi$ and $\phi^{\prime}$, which derives from the
fact that $\mathbf{\Sigma}_{\alpha\beta}$ are the generators of the Lorentz
algebra. One can expand
$\tilde{\nabla}^{\alpha}\tilde{\nabla}_{\alpha}=D_{\alpha}D^{\alpha}-eF^{\gamma\beta}\mathbf{\Sigma}_{\gamma\beta}$
and check that this reproduces the equations studied for example in [22, 38]
for the scalar, Dirac and vector fields

$\begin{split}&(D_{\alpha}D^{\alpha}+m^{2})\varPhi_{p}=0,\\\
&(D_{\alpha}D^{\alpha}+ie\not{n}\not{A}+m^{2})\varPsi_{p}=0,\\\
&[(D_{\alpha}D^{\alpha}+m^{2})\delta^{\beta}_{\gamma}+2ieF^{\beta}_{\\-\
\gamma}]\varPhi^{\gamma}_{p}=0\quad,\quad
D_{\alpha}\varPhi^{\alpha}_{p}=0.\end{split}$ (27)

Making use of the definition
$\tilde{\mathbf{\nabla}}_{\alpha}=\mathbf{\Lambda}D_{\alpha}\mathbf{\Lambda}^{-1}$,
we can write the equations
$(\tilde{\nabla}^{\alpha}\tilde{\nabla}_{\alpha}+m^{2})\bm{\varPhi}_{p}=0$ as
$(D^{\alpha}D_{\alpha}+m^{2})\mathbf{\Lambda}^{-1}\bm{\varPhi}_{p}=0$, such
that the problem is reduced to finding the scalar field solution. This problem
has already been solved with $\varPhi_{p}=\mathscr{U}\varphi_{p}$ and we thus
conclude that $\mathscr{U}\mathbf{\Lambda}$ is the evolution operator for
fields up to spin one

$\begin{split}\bm{\varPhi}_{p}=\mathbf{\Lambda}\mathscr{U}\bm{\varphi}_{p},\end{split}$
(28)

where $\bm{\varphi}_{p}$ is the initial condition in absence of the background
wave. This clearly corresponds to the fact that $\mathbf{\Lambda}$ satisfies
the equation
$\mathbf{\Lambda}^{-1}\mathscr{U}^{-1}\tilde{\mathbf{\nabla}}^{\alpha}\mathscr{U}\mathbf{\Lambda}=\Lambda^{\alpha}_{\\-\
\beta}\partial^{\beta}$ and from this it follows
$\mathbf{\Lambda}^{-1}\mathscr{U}^{-1}\tilde{\mathbf{\nabla}}^{2}\mathscr{U}\mathbf{\Lambda}=\partial^{2}$.
For the sake of clarity, let us introduce the $\phi$-dependent polarization
$\mathbf{S}_{p}(\phi)$ such that

$\begin{split}\bm{\varPhi}_{p}=\mathbf{S}_{p}e^{iS_{p}}.\end{split}$ (29)

Defining $\mathbf{\Lambda}_{p}=\mathbf{\Lambda}|_{\partial\rightarrow-ip}$ it
is easy to check that this can be explicitly written as
$\mathbf{S}_{p}=\mathbf{\Lambda}_{p}\bm{s}_{p}$, where
$\bm{s}_{p}=\\{1,u_{p},\varepsilon_{p}^{\alpha}\\}$ are the free polarizations
for scalar, Dirac, and vector fields, respectively. For example, in the well-
known Volkov solution [39] $\psi_{p}=U_{p}e^{iS_{p}}$ the spinorial term can
be found as $U_{p}^{a}=\bm{\Lambda}^{a}_{p,b}u_{p}^{b}$, such that

$\begin{split}\psi_{p}=e^{iS_{p}}\left(1+e\frac{\not{n}\not{A}}{2p^{-}}\right)u_{p}.\end{split}$
(30)

In the same way, the spin-one solution
$\varPhi_{p}^{\alpha}=\mathscr{E}^{\alpha}_{p}e^{iS_{p}}$ is found to be [22]

$\begin{split}\varPhi_{p}^{\alpha}=e^{iS_{p}}\left[\varepsilon^{\alpha}_{p}-\frac{\varepsilon^{-}_{p}}{p^{-}}eA^{\alpha}+\frac{e}{p^{-}}\left(\varepsilon_{p}^{\beta}A_{\beta}-\frac{\varepsilon^{-}_{p}}{2p^{-}}eA_{\beta}A^{\beta}\right)n^{\alpha}\right].\end{split}$
(31)

Being in the vector representation,
$\mathscr{E}^{\alpha}_{p}=\Lambda^{\alpha}_{p,\beta}\varepsilon^{\beta}_{p}$
reduces to the particle classical momentum if $\varepsilon_{p}^{\alpha}$ is
replaced by $p^{\alpha}$. The Lorenz condition corresponds to the natural
requirement
$\mathscr{E}^{\alpha}_{p}\pi_{p,\alpha}=\varepsilon_{p}^{\alpha}p_{\alpha}=0$.
As a side note, let us mention that the symbol
$\tilde{\mathbf{\nabla}}_{\alpha}$ was not randomly chosen. In fact, it
defines a sort of parallel transport rule for the field. From the properties
of the operator $\tilde{\nabla}_{\alpha}$ it follows that
$\tilde{\nabla}^{\alpha}\bm{\varPhi}_{p}=-i\Lambda_{p,\beta}^{\alpha}p^{\beta}\bm{\varPhi}_{p}$.
Thus, a contraction with the momentum gives
$\pi_{p}^{\alpha}\tilde{\nabla}_{\alpha}\bm{\varPhi}_{p}=-im^{2}\bm{\varPhi}_{p}$.
If we insert the expression $\bm{\varPhi}_{p}=\mathbf{S}_{p}e^{iS_{p}}$ this
equation reduces to a condition on the polarization function, which is
formally equivalent to a parallel transport
$\pi_{p}^{\alpha}\tilde{\nabla}^{\prime}_{p,\alpha}\mathbf{S}_{p}=0$, where
$\tilde{\nabla}^{\prime}_{p,\alpha}=\partial_{\alpha}\mathbf{1}-i\frac{e}{2p^{-}}n_{\alpha}F^{\gamma\beta}\mathbf{\Sigma}_{\gamma\beta}=\mathbf{\Lambda}_{p}\partial_{\alpha}\mathbf{\Lambda}^{-1}_{p}$.
It is also worth noting that if we choose the vector representation then
$\pi_{p}^{\alpha}\tilde{\nabla}^{\prime}_{p,\alpha}\pi_{p}^{\beta}=0$ is
actually equivalent to the Lorentz equation. We could rephrase the preceding
discussion as follows: to construct the fields previously considered one can
first operate a unitary transformation $\mathscr{U}$ that turns the gauge
derivative $D_{\alpha}$ into $\Lambda^{\alpha}_{\\-\ \beta}\partial^{\beta}$,
then transport the polarization function through the operator
$\tilde{\nabla}^{\prime}_{p,\alpha}$. We will now see how these considerations
can be translated in a plane wave spacetime.

###  IV.2 In the Rosen metric

The Rosen chart is a particularly appropriated choice to solve fields
equations. Let us start with the scalar field satisfying
$(\nabla_{\mu}\nabla^{\mu}+m^{2})\Phi=0$. In vacuum flat spacetime we are used
to expand fields in terms of momentum eigenstates but in a general curved
spacetime this is not possible because there is no clear global notion of
Fourier components. Nonetheless, the plane wave spacetime is quite peculiar
because in this space the scalar wave satisfies the Huygens’ principle [40,
41]. By exploiting the identity
$\nabla_{\mu}\nabla^{\mu}\Phi=\frac{1}{\sqrt{g}}\partial_{\mu}(\sqrt{g}\partial^{\mu}\Phi)$
we can rewrite the wave equation as
$(\dot{\log}(\sqrt{g})\partial_{+}+\partial^{\mu}\partial_{\mu}+m^{2})\Phi=0$.
Now, being $\partial_{i},\partial_{+}$ symmetries of the spacetime under
consideration, we will assume the solution to be an eigenstate of these
operators. It is then easy to reduce the equation to

$\begin{split}(\partial^{\mu}\partial_{\mu}+m^{2})g^{\frac{1}{4}}\Phi=0.\end{split}$
(32)

From this equation one immediately sees that
$g^{\frac{1}{4}}\Phi_{p}=e^{iS_{p}}$ is a solution with initial momentum $p$
if $S_{p}$ is the action satisfying
$g^{\mu\nu}\partial_{\mu}S_{p}\partial_{\nu}S_{p}=m^{2}$ and
$\partial^{\mu}\partial_{\mu}S_{p}=0$. Introducing the notation
$\Omega=g^{-\frac{1}{4}}$ we can write the scalar wave as

$\begin{split}\Phi_{p}=\Omega e^{iS_{p}}.\end{split}$ (33)

The action can be easily computed exploiting the symmetries of the metric, but
it can also be borrowed from electromagnetism. If we project the derivatives
onto the vierbein and we split it as in the previous section $e_{\alpha}^{\\-\
\mu}=\delta_{\alpha}^{\\-\ \mu}+\Delta e_{\alpha}^{\\-\ \mu}$ we obtain
$[\eta^{\alpha\beta}(\partial_{\alpha}+\Delta e_{\alpha}^{\\-\
i}\partial_{i})(\partial_{\beta}+\Delta e_{\beta}^{\\-\
j}\partial_{j})+m^{2}]\Omega^{-1}\Phi_{p}=0$. By observing that
$\eta^{\alpha\beta}[\Delta e_{\alpha}^{\\-\
\mu}\partial_{\mu},\partial_{\beta}+\Delta e_{\beta}^{\\-\
\nu}\partial_{\nu}]=0$ and by recalling that $\Phi_{p}$ is eigenstate of
$\partial_{i}$, this equation can be written as

$\begin{split}[\eta^{\alpha\beta}(\partial_{\alpha}+i\kappa
P_{\alpha})(\partial_{\beta}+i\kappa
P_{\beta})+m^{2}]\Omega^{-1}\Phi_{p}=0.\end{split}$ (34)

Now, this is formally the scalar wave equation for the field
$\Omega^{-1}\Phi_{p}$ in an electromagnetic plane wave with the potential and
charge product replaced by $\kappa P_{\beta}$. It then follows that the action
$S_{p}(x)$ is just the one of a charged particle moving in an electromagnetic
plane wave under the same substitution:

$\begin{split}S_{p}(x)=-p_{\mu}x^{\mu}-\frac{\kappa}{p^{-}}\int^{\phi}d\tilde{\phi}\left(p_{\alpha}P^{\alpha}-\frac{\kappa}{2}P_{\alpha}P^{\alpha}\right).\end{split}$
(35)

To summarize, we found that the solution of the Klein-Gordon equation in the
Rosen metric is $\Phi_{p}=\Omega\varPhi_{p}$, where $\varPhi_{p}$ is formally
the field with charge $\kappa$ satisfying the Klein-Gordon equation in flat
spacetime with an electromagnetic plane wave potential $P_{\beta}$.

The same is true for Dirac fields, such that the spinor in a plane wave
spacetime is $\Psi_{p}=\Omega\varPsi_{p}$, where $\varPsi_{p}$ is the Volkov
state [39] of charge $\kappa$ in a wave potential $P_{\beta}$

$\begin{split}\Psi_{p}=\Omega
e^{iS_{p}}\left(1+\kappa\frac{\not{n}\not{P}}{2p^{-}}\right)u_{p}.\end{split}$
(36)

This can be verified by observing that the Dirac equation
$(i\gamma^{\mu}\nabla_{\mu}-m)\Psi_{p}=0$ can be reduced to
$[i(\not{\partial}+i\kappa\not{P})-m]\Omega^{-1}\Psi_{p}$ and this is once
again formally the Dirac equation in flat spacetime in presence of a plane
wave field $\kappa P^{\alpha}$. In this case, the spin connection term
$\gamma^{\mu}\Gamma_{\mu}$ is exactly compensated by the $\Omega$ prefactor.
The spin-one massive case is slightly more involved. Let us introduce the
notation $\bar{A}^{\alpha}=\Omega\bar{\varPhi}^{\alpha}$, where
$\bar{A}^{\alpha}=e^{\alpha}_{\\-\ \mu}A^{\mu}$ is the vierbein-projected
vector field. Keeping into account the condition $\nabla_{\mu}A^{\mu}=0$ it is
possible to reduce the field equation
$(\nabla_{\mu}\nabla^{\mu}+m^{2})A^{\nu}+R^{\nu}_{\\-\ \mu}A^{\mu}=0$ to the
form

$\begin{split}(\bar{\partial}^{\beta}\bar{\partial}_{\beta}+m^{2})\bar{\varPhi}^{\alpha}+2i\kappa\hat{\mathscr{P}}^{\alpha}_{\\-\
\gamma}\bar{\varPhi}^{\gamma}+\frac{1}{2}\ddot{\log}g\bar{\varPhi}^{-}n^{\alpha}=0,\end{split}$
(37)

where we introduced the quantum analog of the Maxwell-like tensor
$\mathscr{P}_{\alpha\beta}$ substituting $p_{i}\rightarrow i\partial_{i}$,
namely
$\kappa\hat{\mathscr{P}}^{\alpha\beta}=-\mathscr{F}_{j}^{\alpha\beta}\dot{e}^{jk}i\partial_{k}$.
Here, we see that the last term in the l.h.s., which comes from the
contraction of two spin connections, is a nonlinear term absent in flat
spacetime. However, contracting with $n^{\alpha}$ one finds that
$(\bar{\partial}^{\beta}\bar{\partial}_{\beta}+m^{2})\bar{\varPhi}^{-}=0$ and
therefore assuming initial momentum $p$ one finds
$\bar{\varPhi}^{-}_{p}=\varepsilon^{-}_{p}e^{iS_{p}}$, with
$\varepsilon^{-}_{p}$ constant. Exploiting the property
$\bar{\partial}_{\beta}\bar{\varPhi}^{-}_{p}=-i\bar{\pi}_{p,\beta}\bar{\varPhi}^{-}_{p}$,
one can check that the shift
$\bar{\varPhi}^{\prime\alpha}_{p}=\bar{\varPhi}^{\alpha}_{p}+\frac{i}{4p^{-}}\dot{\log}g\bar{\varPhi}^{-}_{p}n^{\alpha}$
restores the form of a flat spacetime field equation in an electromagnetic
wave background

$\begin{split}(\bar{\partial}^{\beta}\bar{\partial}_{\beta}+m^{2})\bar{\varPhi}^{\prime\alpha}_{p}+2i\kappa\mathscr{P}^{\alpha}_{\\-\
\gamma}\bar{\varPhi}^{\prime\gamma}_{p}=0.\end{split}$ (38)

Indeed, this field satisfies the condition
$\bar{\partial}_{\alpha}\bar{\varPhi}_{p}^{\prime\alpha}=(\partial_{\alpha}+i\kappa
P_{\alpha})\bar{\varPhi}^{\prime\alpha}_{p}=0$, which is the analog of
$D_{\alpha}\varPhi^{\alpha}_{p}=0$ found in flat spacetime in the previous
section.

Finally, we can report the solution for the massive vector field in the form
$\bar{A}_{p}^{\alpha}=\Omega e^{iS_{p}}\bar{\mathscr{E}}^{\alpha}_{p}$, namely

$\begin{split}\bar{A}^{\alpha}_{p}=\Omega
e^{iS_{p}}\left[\varepsilon^{\alpha}_{p}-\frac{\varepsilon^{-}_{p}}{p^{-}}\kappa
P^{\alpha}+\frac{\kappa}{p^{-}}\left(\varepsilon_{p}^{\beta}P_{\beta}-\frac{\varepsilon^{-}_{p}}{2p^{-}}\kappa
P_{\beta}P^{\beta}\right)n^{\alpha}+i\dot{\log}{\Omega}\frac{\varepsilon^{-}_{p}}{p^{-}}n^{\alpha}\right].\end{split}$
(39)

Here one can explicitly observe once again the formal analogy with the case of
an electromagnetic background in flat spacetime. Once the shift previously
described is operated the polarization $\bar{\mathscr{E}}^{\alpha}_{p}$ can be
found substituting $eA^{\alpha}\rightarrow\kappa P^{\alpha}$ in the flat
spacetime solution. If the massless case is considered one can exploit the
gauge freedom to fix
$\bar{\partial}_{\alpha}\bar{\varPhi}^{\alpha}_{k}=0=\bar{\Phi}^{-}_{k}$,
which implies $\nabla_{\mu}A_{k}^{\mu}=0$ as well. In this case the additional
term proportional to $\ddot{\log}g$ in Eq. (37) vanishes and the solution is
once again in direct correspondence with the one in flat spacetime without any
shift needed. We will deal with this choice studying the field in Brinkmann
coordinates. The massive spin-two field is not of our interest in the present
work and we will confine ourselves to the massless case, which is easily
treated in Brinkmann coordinates once the proper gauge is chosen [23].

We can now retrace the discussion carried out in flat spacetime, i.e., that
the operator constructed from the difference between the free and the curved
spacetime actions $\mathscr{U}=\exp
i\left(S_{p}+p_{\mu}x^{\mu}\right)|_{p\rightarrow i\partial}$ is a unitary
transformation that changes the vierbein projected derivative into the Lorentz
transformed curved one
$\mathscr{U}^{-1}\bar{\partial}_{\alpha}\mathscr{U}=\Lambda_{\alpha}^{\\-\
\beta}\partial_{\beta}$, with
$\partial_{\alpha}=\delta_{\alpha}^{\mu}\partial_{\mu}$. We can introduce the
gravitational analog of the Lorentz-like transformation exploited in the
electromagnetic wave background treatment, which can be obtained through the
simple substitution emerged in the study of the dynamics
$e\dot{A}^{j}\rightarrow-i\kappa\Delta\dot{e}^{jk}\partial_{k}$

$\begin{split}\mathbf{\Lambda}=e^{\frac{\kappa}{2\partial_{+}}\int^{\phi}_{-\infty}d\tilde{\phi}\hat{\mathscr{P}}^{\alpha\beta}(\tilde{\phi})\mathbf{\Sigma}_{\alpha\beta}}.\end{split}$
(40)

The wave equation for the fields
$\\{\Phi,\Psi,\bar{A}^{\prime\alpha}\\}\equiv\bar{\bm{\Phi}}_{p}$ can be
written as

$\begin{split}(\tilde{\nabla}^{\alpha}\tilde{\nabla}_{\alpha}+m^{2})\Omega^{-1}\bar{\bm{\Phi}}_{p}=0\qquad,\quad\text{with}\qquad\tilde{\mathbf{\nabla}}_{\alpha}=\bar{\partial}_{\alpha}\mathbf{1}-\frac{\kappa}{2\partial_{+}}n_{\alpha}\hat{\mathscr{P}}^{\gamma\beta}\mathbf{\Sigma}_{\gamma\beta},\end{split}$
(41)

where we recall that the spin-one field involves the shift
$\bar{A}^{\prime\alpha}=\bar{A}^{\alpha}-\frac{i}{p^{-}}\dot{\log}\Omega\bar{A}^{-}n^{\alpha}$.
Let us point out that $\tilde{\nabla}_{\alpha}$ is not the vierbein projected
covariant derivative. This operator is defined by the representation of the
considered field and not by the object it is acting on. This is where its
relevance comes from as it allows one to write the wave equation as the actual
square of an operator. However, when taken along a geodesic this operator is
actually equivalent to the vierbein-projected covariant derivative for the
representations here discussed, namely
$\bar{\pi}^{\alpha}_{p}\tilde{\nabla}_{p,\alpha}=\bar{\pi}^{\alpha}_{p}\bar{\nabla}_{\alpha}$,
with
$\tilde{\mathbf{\nabla}}_{p,\alpha}=\bar{\partial}_{\alpha}\mathbf{1}-i\frac{\kappa}{2p^{-}}n_{\alpha}\mathscr{P}^{\gamma\beta}\mathbf{\Sigma}_{\gamma\beta}$.
This is quickly verified exploiting the spin coefficients property
$\bar{\pi}^{\alpha}\omega_{\alpha\beta\delta}=-\kappa\mathscr{P}_{\beta\delta}$.
To complete the correspondence we have the properties

$\begin{split}\mathbf{\Lambda}^{-1}\mathscr{U}^{-1}\tilde{\mathbf{\nabla}}_{\alpha}\mathscr{U}\mathbf{\Lambda}=\Lambda_{\alpha}^{\\-\
\beta}\partial_{\beta}\qquad,\qquad\mathbf{\Lambda}^{-1}\mathscr{U}^{-1}\tilde{\mathbf{\nabla}}^{2}\mathscr{U}\mathbf{\Lambda}=\eta^{\alpha\beta}\partial_{\alpha}\partial_{\beta}.\end{split}$
(42)

This allows us to easily find the aforementioned fields as

$\begin{split}\bar{\bm{\Phi}}_{p}=\Omega\mathbf{\Lambda}\mathscr{U}\bm{\varphi}_{p},\end{split}$
(43)

where $\bm{\varphi}_{p}$ is the flat initial condition. Once again we can
introduce the polarization $\mathbf{S}_{p}(\phi)$ such that
$\bar{\mathbf{\Phi}}_{p}=\Omega\bar{\mathbf{S}}_{p}(\phi)e^{iS_{p}}$. As
before one can check that this can be written as
$\bar{\mathbf{S}}_{p}=\mathbf{\Lambda}_{p}\bm{s}_{p}$ where
$\bm{s}_{p}=\\{1,u_{p},\varepsilon_{p}^{\alpha}\\}$ are the flat-spacetime
free polarizations for scalar, Dirac and vector fields respectively.
By adapting the discussion in an electromagnetic wave background we find that
$\tilde{\mathbf{\nabla}}_{\alpha}\Omega^{-1}\bar{\mathbf{\Phi}}_{p}=-i\Lambda_{\alpha}^{\\-\
\beta}p_{\beta}\Omega^{-1}\bar{\mathbf{\Phi}}_{p}$ and from this
$\bar{\pi}^{\alpha}\tilde{\mathbf{\nabla}}_{\alpha}\Omega^{-1}\bar{\mathbf{\Phi}}_{p}=-im^{2}\Omega^{-1}\bar{\mathbf{\Phi}}_{p}$.
Inserting into this equation the form
$\bar{\mathbf{\Phi}}_{p}=\Omega\bar{\mathbf{S}}_{p}(\phi)e^{iS_{p}}$ we obtain
a condition for the polarization which reads
$\bar{\pi}_{p}^{\alpha}\tilde{\nabla}^{\prime}_{p,\alpha}\bar{\mathbf{S}}_{p}=0$,
where
$\tilde{\mathbf{\nabla}}^{\prime}_{p,\alpha}=\partial_{\alpha}\mathbf{1}-i\frac{\kappa}{2p^{-}}n_{\alpha}\mathscr{P}^{\gamma\beta}\mathbf{\Sigma}_{\gamma\beta}$.
Now, observing that
$\bar{\pi}_{p}^{\alpha}\tilde{\nabla}^{\prime}_{p,\alpha}\bar{\mathbf{S}}_{p}=\bar{\pi}_{p}^{\alpha}\tilde{\nabla}_{p,\alpha}\bar{\mathbf{S}}_{p}=\bar{\pi}_{p}^{\alpha}\bar{\nabla}_{p,\alpha}\bar{\mathbf{S}}_{p}$,
we finally find that the polarization function is defined by a proper parallel
transportation $\pi^{\mu}_{p}\nabla_{\mu}\mathbf{S}_{p}=0$, or

$\begin{split}\frac{D\mathbf{S}_{p}}{D\phi}=0.\end{split}$ (44)

This is a covariant equation and it holds in other charts as well. In
particular, it can be used as an alternative way to find the states in
Brinkmann coordinates. Moreover, this equation underlines once again the
semiclassical behaviour of quantum states in a plane wave background and
offers an intuitive geometric picture. For example, the polarization of the
shifted spin-one field evolves as a free falling gyroscope [4] being just
parallel transported.

###  IV.3 In the Brinkmann metric

It is needless to say that we could just transform the fields obtained in the
Rosen chart and find the states in Brinkmann coordinates. However here we want
to underline the role of the operator $\mathbf{\Lambda}$ in this chart and
also the form this takes in a particular gauge reproducing the spin-raising
operator studied in [23] for massless fields. The scalar field can be clearly
expressed as $\Phi_{p}=\Omega e^{iS_{p}}$ where $S_{p}$ is the classical
action for a particle in Brinkmann coordinates [23, 31]

$\begin{split}S_{p}(X)=-p^{-}X^{+}-X^{i}e_{i}^{\\-\
j}p_{j}-\frac{p^{-}}{2}\sigma_{ij}X^{i}X^{j}+\frac{p_{i}p_{j}}{2p^{-}}\int^{\phi}d\tilde{\phi}\gamma^{ij}.\end{split}$
(45)

The operator $\mathbf{\Lambda}$ can be written in this chart as

$\begin{split}\mathbf{\Lambda}=e^{-\frac{i}{2\partial_{+}}\mathbf{\Sigma}_{\alpha\beta}\mathscr{F}_{i}^{\alpha\beta}\eta^{ij}(\partial_{X^{j}}-e^{k}_{\\-\
j}\partial_{X^{k}})}.\end{split}$ (46)

If we choose the natural vierbein associated to this chart
$E_{\alpha\mu}=\eta_{\alpha\mu}+\frac{H}{2}n_{\alpha}n_{\mu}$ we can
immediately find the spinor solutions, in fact $E_{ij}=e^{\prime}_{ij}$ where
$e^{\prime}_{ij}=e_{ik}\frac{\partial x^{k}}{\partial X^{j}}$ is the Rosen
transverse vierbein in Brinkmann coordinates. We can thus expand the operator
to obtain

$\begin{split}\Psi_{p}=\Omega
e^{iS_{p}}\left(1+\frac{\not{\Delta}_{p}\not{n}}{2p^{-}},\right)u_{p},\end{split}$
(47)

where we recall that $\Delta^{i}_{p}=\Pi_{p}^{i}-p^{i}$. Once again the
solution is similar to the Volkov one, with the substantial difference that
now $\Delta^{i}_{p}$ depends on the three coordinates $\phi,X^{i}$ and not
just on $\phi$. This will be a major difference as compared to the
electromagnetic case when we will calculate $S$-matrix elements, along with
the definition of in- and out- states discussed in the next section. This
solution can be verified to match the one found in the Rosen chart, in fact
the transformation of the spinor field is generated by the Lorentz
transformation connecting the Brinkmann chosen vierbein and the Rosen one
expressed in Brinkmann coordinates $E^{\alpha}_{\\-\
\mu}=\tilde{\Lambda}^{\alpha}_{\\-\ \beta}e^{\beta}_{\\-\ \nu}\frac{\partial
x^{\nu}}{\partial X^{\mu}}=\tilde{\Lambda}^{\alpha}_{\\-\
\beta}e^{\prime\beta}_{\\-\ \mu}$. Expanding
$\tilde{\Lambda}_{\alpha\beta}=\eta_{\alpha\beta}-\omega_{\alpha\beta}$ the
infinitesimal Lorentz transformation is found to be
$\omega_{\alpha\beta}=2n_{[\alpha}\sigma_{\beta]i}X^{i}$ and accordingly the
transformed spinor follows as

$\begin{split}\Psi_{p}=e^{-\frac{i}{2}\omega_{\alpha\beta}\bm{\Sigma}^{\alpha\beta}}\Omega
e^{iS_{p}}\left(1+\kappa\frac{\not{n}\not{P}}{2p^{-}}\right)u_{p}=\Omega
e^{iS_{p}}e^{2\sigma_{\alpha
i}X^{i}\bar{\gamma}^{\alpha}\not{n}}\left(1+\kappa\frac{\not{n}\not{P}}{2p^{-}}\right)u_{p}=\Omega
e^{iS_{p}}\left(1+\frac{\not{\Delta}_{p}\not{n}}{2p^{-}}\right)u_{p}.\end{split}$
(48)

One can also verify that the spinorial matrix satisfies the parallel transport
equation, as discussed in the context of the Rosen metric. Let us now consider
the photon field. If we fix the gauge such that only the transverse components
of the flat polarization are left
$\varepsilon^{\mu}=\delta^{\mu}_{i}\varepsilon^{i}$ and they obey the Lorentz
condition $k_{i}\varepsilon^{i}=0$, we can reduce the operator action to
$A_{k}^{\mu}=\Omega\mathbf{\Lambda}^{\mu}_{\\-\
i}\varepsilon^{i}e^{iS_{k}}\equiv\Omega R^{\mu}e^{iS_{k}}$, where we
introduced the spin raising operator studied in [23, 42]
$R^{\mu}=\mathbf{\Lambda}^{\mu}_{\\-\
i}\varepsilon^{i}=\frac{1}{\partial_{+}}\left[\delta_{i}^{\mu}\partial_{+}-\partial_{i}n^{\mu}\right]\varepsilon^{i}=-\frac{1}{\partial_{+}}\varepsilon^{i}\mathscr{F}^{\mu\nu}_{i}\partial_{\nu}$.
In this gauge the massless vector field is thus

$\begin{split}A_{k}^{\mu}=\Omega
e^{iS_{k}}\left(\delta_{i}^{\mu}-\frac{1}{k^{-}}\Delta_{k,i}n^{\mu}\right)\varepsilon^{i}.\end{split}$
(49)

This is clearly in agreement with the field found in the Rosen chart once the
same gauge is imposed.
As noticed in [23], in vacuum we can choose for a massless spin-2 field the
covariant TT-gauge and fix $n^{\mu}h_{\mu\nu}=0$. As a result the field can be
found applying two times the spin raising operator, such that
$h_{k}^{\mu\nu}=\Omega R^{\mu}R^{\nu}\Phi_{k}$. The polarization tensor is
easily found to be
$\mathscr{E}_{k}^{\mu\nu}=\mathscr{E}_{k}^{\mu}\mathscr{E}_{k}^{\nu}-\frac{i}{k^{-}}\sigma_{ij}\varepsilon^{i}\varepsilon^{j}n^{\mu}n^{\nu}$
such that

$\begin{split}h_{k}^{\mu\nu}=\Omega
e^{iS_{k}}\left[\left(\delta_{i}^{\mu}-\frac{\Delta_{k,i}n^{\mu}}{k^{-}}\right)\left(\delta_{j}^{\nu}-\frac{\Delta_{k,j}n^{\nu}}{k^{-}}\right)-\frac{i}{k^{-}}\sigma_{ij}n^{\mu}n^{\nu}\right]\varepsilon^{i}\varepsilon^{j}.\end{split}$
(50)

##  V Compton scattering

###  V.1 Definition of scattering states

When dealing with scattering problems we have to work in Brinkmann
coordinates, because the regularity of fields over the whole spacetime is
essential. Due to the presence of a gravitational wave in and out states are
generally different. Let us consider for simplicity the scalar case
$\Phi(X)=\Omega(\phi)e^{iS(X)}$. Moreover we will assume the spacetime to be
flat at $\phi=\pm\infty$. We can impose the free plane wave boundary condition
in the in- or out-region and define the positive energy in- and out-states,
respectively $\Phi^{\uparrow},\Phi^{\downarrow}$ as [31, 23]

$\begin{split}\Phi_{p}^{\updownarrow}=\Omega^{\updownarrow}e^{iS_{p}^{\updownarrow}}\qquad,\qquad\lim_{\phi\rightarrow\mp\infty}\Phi_{p}^{\uparrow,\downarrow}=e^{-ip\cdot
X}.\end{split}$ (51)

As observed in the previous sections, these fields depend on a matrix
$e_{\alpha\mu}$, which is also the vierbein of a Rosen metric. This is the
reason why in- and out-states are different, they depend on vierebeins which
do not represent the physical wave profile $H_{ij}$ but they are connected to
it through the differential equation $\ddot{e}_{ik}=H_{ij}e^{j}_{\\-\ k}$. A
vierbein does not have to be trivial $e_{\alpha\mu}=\eta_{\alpha\mu}$ in order
to describe a flat region, but it can be at most linear in $\phi$ such that
$\ddot{e}_{ik}=0$. In other words a Rosen metric does not have to be
Minkowskian in a flat region and if we require it to be so, for example in the
in-region, it will not be Minkowskian in the out-region. The boundary
conditions are thus transferred to the vierbeins, such that

$\begin{split}\lim_{\phi\rightarrow\mp\infty}e_{\alpha}^{\uparrow,\downarrow\mu}=\delta_{\alpha}^{\mu}\qquad,\qquad
e_{\alpha}^{\uparrow,\downarrow\mu}=b_{\alpha}^{\uparrow,\downarrow\mu}\phi+c_{\alpha}^{\uparrow,\downarrow\mu}\quad\text{for}\quad\phi\rightarrow\pm\infty.\end{split}$
(52)

There are only two constrains on these coefficients, one comes from the
symmetry condition $\dot{e}_{[\beta}^{\\-\ \mu}e_{\alpha]\mu}=0$ and reads
[23]
$b_{[\beta}^{\uparrow,\downarrow\mu}c^{\uparrow,\downarrow}_{\alpha]\mu}=0$,
the other one comes from the conservation of the Wronskian related to the
harmonic oscillator equation and links in and out-states [31]
$b_{\alpha\mu}^{\downarrow}\delta^{\alpha}_{\nu}=-b_{\alpha\nu}^{\uparrow}\delta^{\alpha}_{\nu}$.
It is worth recalling that the coefficients $b^{\updownarrow}$ represent the
velocity memory effect induced by the passage of the wave. Now, the states are
well defined we should introduce a scalar product, however the standard
definition [43] needs a Cauchy surface $\Sigma$ to be defined

$\begin{split}\langle\Psi_{2}|\Psi_{1}\rangle=-i\int_{\Sigma}\sqrt{\mathscr{G}_{\Sigma}}d\Sigma^{\mu}\Psi_{2}\overleftrightarrow{\partial}_{\mu}\Psi_{1}^{*}.\end{split}$
(53)

As we already discussed, it is impossible to define a Cauchy surface $\Sigma$
in this spacetime [32], nevertheless the surfaces of constant $\phi$ are
intersected by almost all the geodesics except the ones with constant $\phi$
[44], therefore they can be chosen to construct a foliation of the spacetime.
We can thus choose an arbitrary $\phi$ and rewrite the the product definition
as

$\begin{split}\langle\Psi_{2}|\Psi_{1}\rangle=-i\int_{\phi=const.}dX^{+}d^{2}X_{\perp}\Psi_{2}\overleftrightarrow{\partial}_{+}\Psi_{1}^{*},\end{split}$
(54)

where we recall that the absolute value of the Brinkmann determinant is always
equal to one due to its Kerr-Schild form. A straightforward calculation shows
that the Bogoliubov coefficient encoding particle creation is zero. This
corresponds to the amplitude for a positive frequency in-state to develop a
negative frequency out-state component:
$\langle\Phi^{\downarrow}_{1}|\Phi^{\uparrow*}_{2}\rangle=0$ [44, 31, 23]. It
follows that the in- and out-vacua can be identified, as in the
electromagnetic case no particle is created by the background. On the other
hand, one can show that the in-to-out scattering probability depends on the
determinant of $\dot{e}^{\uparrow}_{ij}$ in the out-region [31]

$\begin{split}\left.\left|\langle\Phi^{\downarrow}_{1}|\Phi^{\uparrow}_{2}\rangle\right|^{2}\propto\frac{\delta(p_{1}^{-}-p_{2}^{-})}{|\det\dot{e}^{\uparrow}_{ij}|}\right|_{\text{out-
region}}.\end{split}$ (55)

Thus the probability for a scalar field to be scattered from the gravitational
wave background is directly correlated to the velocity memory effect. This is
what one would expect, in fact the difference between the initial and final
momenta induced by the background corresponds to this effect. The same is true
for the classical cross section describing the interaction between a particle
and the gravitational wave [31].

###  V.2 S-matrix element

In the previous sections we prepared all the tools needed in order investigate
the emission of a single photon by an electron (Compton scattering) in a
sandwich plane wave spacetime. The amplitude for the process $e(p)\rightarrow
e(p^{\prime})+\gamma(k^{\prime})$ in this background is

$\begin{split}M=-ie\int
d^{4}X\bar{\Psi}^{\downarrow}_{p^{\prime}}\gamma^{\mu}\Psi^{\uparrow}_{p}A^{\downarrow*}_{k^{\prime},\mu}.\end{split}$
(56)

As a first consistency check it is interesting to consider its linear order in
$\kappa$ in the monochromatic assumption. This has to coincide with the known
results for the inverse graviton photoproduction $g(k)+e(p)\rightarrow
e(p^{\prime})+\gamma(k^{\prime})$ in vacuum quantum field theory, where
$k^{\mu}=\omega n^{\mu}$ is the momentum of the gravitational wave. In this
approximation we are free to use the quantum states in Rosen coordinates, in
fact the difference between the in and the out-vacua emerges at the second
order. We will thus drop the in and out labels and consider $M=-ie\int
d^{4}x\bar{\Psi}_{p^{\prime}}\bar{\gamma}^{\alpha}\Psi_{p}\bar{A}^{*}_{k^{\prime},\alpha}+\mathscr{O}(\kappa^{2})$,
where the states will be defined below and where the metric determinant does
not contribute at this order. Everything we need for this approximation is
encoded in the vierbein, which is now expanded as

$\begin{split}e_{\alpha\mu}=\eta_{\alpha\mu}+\frac{\kappa}{2}h_{\alpha\mu}\quad,\quad
e_{\alpha}^{\\-\ \mu}=\delta_{\alpha}^{\mu}-\frac{\kappa}{2}h_{\alpha}^{\\-\
\mu},\end{split}$ (57)

where $h_{\alpha\mu}$ is the graviton field. We can now introduce the
monochromatic assumption defining
$h_{\mu\nu}=\varepsilon_{\mu}\varepsilon_{\nu}e^{-ik\cdot x}$, where the
graviton polarization tensor has been written as a product of two photon
polarization vectors
$\varepsilon_{\mu\nu}(k)=\varepsilon_{\mu}(k)\varepsilon_{\nu}(k)$ satisfying
$\varepsilon\cdot\varepsilon=0$, $k\cdot\varepsilon=0$ and
$\varepsilon\cdot\varepsilon^{*}=-1$ [45]. This together with the energy-
momentum conservation leads to the simple substitution rule

$\begin{split}P_{\alpha}\rightarrow\frac{1}{2}e^{-ik\cdot x}\varepsilon\cdot
p\varepsilon_{\alpha}\end{split}$ (58)

in $\Psi_{p}$, in $\bar{\Psi}_{p^{\prime}}$ (with $p\to p^{\prime}$), and in
$\bar{A}^{*}_{k^{\prime},\alpha}$ (with $p\to k^{\prime}$). With this in mind
we can easily find the states under these assumptions in the form

$\begin{split}&\Psi_{p}\simeq e^{-ip\cdot x}\left[1+e^{-ik\cdot
x}\kappa\frac{p\cdot\varepsilon}{4p\cdot
k}\left(2p\cdot\varepsilon+\not{k}\not{\varepsilon}\right)\right]u_{p}+\mathscr{O}(\kappa^{2}),\\\
&\bar{A}^{*}_{k^{\prime},\alpha}\simeq e^{ik^{\prime}\cdot
x}\left[\varepsilon^{{}^{\prime}*}_{\alpha}+e^{-ik\cdot
x}\kappa\frac{k^{\prime}\cdot\varepsilon}{2k\cdot
k^{\prime}}\left(\varepsilon^{{}^{\prime}*}\cdot\varepsilon
k_{\alpha}-k^{\prime}\cdot\varepsilon\varepsilon^{{}^{\prime}*}_{\alpha}-\varepsilon^{{}^{\prime}*}\cdot
k\varepsilon_{\alpha}\right)\right]+\mathscr{O}(\kappa^{2}).\end{split}$ (59)

Now, one can plug these expressions into the amplitude previously defined.
Introducing the notation
$M=(2\pi)^{4}\delta^{(4)}(p+k-p^{\prime}-k^{\prime})\mathscr{A}+\mathscr{O}(\kappa^{2})$,
the following result is found after a proper rearrangement of the various
terms

$\begin{split}\mathscr{A}=-ie\frac{\kappa}{2}\bar{u}_{p^{\prime}}&\left[\frac{p^{\prime}\cdot\varepsilon}{2p^{\prime}\cdot
k}\left(-2p^{\prime}\cdot\varepsilon+\not{\varepsilon}\not{k}\right)\not{\varepsilon}^{{}^{\prime}*}+\frac{p\cdot\varepsilon}{2p\cdot
k}\not{\varepsilon}^{{}^{\prime}*}\left(2p\cdot\varepsilon+\not{k}\not{\varepsilon}\right)\right.\\\
&\qquad\left.+\frac{\varepsilon\cdot k^{\prime}}{k\cdot
k^{\prime}}\left(\varepsilon^{{}^{\prime}*}\cdot\varepsilon\not{k}-\varepsilon\cdot
k^{\prime}\not{\varepsilon}^{{}^{\prime}*}-\varepsilon^{{}^{\prime}*}\cdot
k\not{\varepsilon}\right)\right]u_{p},\end{split}$ (60)

which agrees with the vacuum QFT result as expected (see e.g. [46, 47]). The
first-order expansion of the dressed fermions naturally generates the $s$ and
$u$ channels contributions, while the expansion of the photon state reduces to
the sum of the $\gamma$-pole and the seagull diagrams [46]. The contact term
never shows up explicitly in our calculation as a consequence of graviton
gauge invariance [48]: the seagull diagram only provides a momentum-
independent contribution which cancels out a term of the same kind in the
$\gamma$-pole diagram.

Let us now go back to the original amplitude. In the fully nonlinear case we
have to work in the Brinkmann chart in order to have a well defined matrix
element. Choosing the gauge $\nabla_{\mu}A^{\mu}=0=A^{-}$ for the photon
field, the amplitude gets the following form

$\begin{split}M=-ie\int
d^{4}X(\Omega^{\downarrow})^{2}\Omega^{\uparrow}e^{i(S^{\uparrow}_{p}-S^{\downarrow}_{p^{\prime}}-S^{\downarrow}_{k^{\prime}})}\bar{u}_{p^{\prime}}\left(1-\frac{\not{\Delta}^{\downarrow}_{p^{\prime}}\not{k}}{2k\cdot
p^{\prime}}\right)\left(\not{\varepsilon}^{*}-\frac{\varepsilon^{i*}\Delta^{\downarrow}_{k^{\prime},i}\not{k}}{k\cdot
k^{\prime}}\right)\left(1-\frac{\not{k}\not{\Delta}^{\uparrow}_{p}}{2k\cdot
p}\right)u_{p},\end{split}$ (61)

The unpolarized squared amplitude averaged over the initial spin reads

$\begin{split}&\frac{1}{2}\sum_{s,s^{\prime},\varepsilon^{\prime}}M^{\dagger}M=\frac{e^{2}}{2}\int
d^{4}X\mathfrak{I}(X)\int d^{4}X^{\prime}\mathfrak{I}^{*}(X^{\prime})\times\\\
&\quad\times\text{Tr}\left[(\not{p}^{\prime}+m)\left(1-\frac{\not{\Delta}^{\downarrow}_{p^{\prime}}(X)\not{k}}{2k\cdot
p^{\prime}}\right)\left(\bar{\gamma}^{\alpha}-\frac{\Delta^{\downarrow,\alpha}_{k^{\prime}}(X)\not{k}}{k\cdot
k^{\prime}}\right)\left(1-\frac{\not{k}\not{\Delta}^{\uparrow}_{p}(X)}{2k\cdot
p}\right)\right.\\\
&\quad\times\left.(\not{p}+m)\left(1-\frac{\not{\Delta}_{p}^{\uparrow}(X^{\prime})\not{k}}{2k\cdot
p}\right)\left(\bar{\gamma}^{\beta}-\frac{\Delta^{\downarrow,\beta}_{k^{\prime}}(X^{\prime})\not{k}}{k\cdot
k^{\prime}}\right)\left(1-\frac{\not{k}\not{\Delta}_{p^{\prime}}^{\downarrow}(X^{\prime})}{2k\cdot
p^{\prime}}\right)\right]\left(-\eta_{\alpha\beta}+\frac{k_{\alpha}k^{\prime}_{\beta}+k^{\prime}_{\alpha}k_{\beta}}{k\cdot
k^{\prime}}\right),\end{split}$ (62)

where the prefactor is defined as
$\mathfrak{I}=(\Omega^{\downarrow})^{2}\Omega^{\uparrow}e^{i(S^{\uparrow}_{p}-S^{\downarrow}_{p^{\prime}}-S^{\downarrow}_{k^{\prime}})}$.
Once the trace is computed this expression reduces to the following form

$\begin{split}\frac{1}{2}\sum_{s,s^{\prime},\varepsilon^{\prime}}M^{\dagger}M=&\\-\
\frac{e^{2}}{2}\int d^{4}X\mathfrak{I}(X)\int
d^{4}X^{\prime}\mathfrak{I}^{*}(X^{\prime})\\\ &\times\bigg{\\{}\frac{k\cdot
p^{\prime}}{k\cdot k^{\prime}}\left(\Pi^{\uparrow
i}_{p}(X)\Pi^{\downarrow}_{k^{\prime},i}(X^{\prime})+\Pi^{\downarrow
i}_{k^{\prime}}(X)\Pi^{\uparrow}_{p,i}(X^{\prime})+2p^{+}k^{\prime-}+2p^{-}k^{\prime+}\right)\\\
&+\frac{k\cdot p}{k\cdot k^{\prime}}\left(\Pi^{\downarrow
i}_{p^{\prime}}(X)\Pi^{\downarrow}_{k^{\prime},i}(X^{\prime})+\Pi^{\downarrow
i}_{k^{\prime}}(X)\Pi^{\uparrow}_{p^{\prime},i}(X^{\prime})+2p^{\prime+}k^{\prime-}+2p^{\prime-}k^{\prime+}\right)\\\
&-\frac{k\cdot p^{\prime}}{k\cdot p}\left(\Pi^{\uparrow
i}_{p}(X)\Pi^{\uparrow}_{p,i}(X^{\prime})+2p^{+}p^{-}\right)-\frac{k\cdot
p}{k\cdot p^{\prime}}\left(\Pi^{\downarrow
i}_{p^{\prime}}(X)\Pi^{\downarrow}_{p^{\prime},i}(X^{\prime})+2p^{\prime+}p^{\prime-}\right)\\\
&-2\frac{k\cdot pk\cdot p^{\prime}}{(k\cdot
k^{\prime})^{2}}\left(\Pi^{\downarrow
i}_{k^{\prime}}(X)\Pi^{\downarrow}_{k^{\prime},i}(X^{\prime})+2k^{\prime+}k^{\prime-}\right)+\frac{m^{2}(k\cdot
k^{\prime})^{2}}{k\cdot pk\cdot p^{\prime}}\bigg{\\}}.\end{split}$ (63)

It is interesting to observe that the terms in the amplitude quadratic in the
gravitational field disappear in the unpolarized sum. For this reason, the
integrals in the three coordinates $X^{+},X^{i}$ are easily carried out by
observing that the prefactor is Gaussian in $X^{i}$. Namely

$\begin{split}\int d^{4}X\mathfrak{I}(X)=&\\-\
-i(2\pi)^{2}\delta(p^{-}-p^{\prime-}-k^{\prime-})\\\ &\times\int
d\phi\frac{(\Omega^{\downarrow})^{2}\Omega^{\uparrow}}{p^{-}\sqrt{C}}e^{\frac{i}{2p^{-}}B^{i}(C^{-1})_{ij}B^{j}+i\frac{p_{i}p_{j}}{2p^{-}}\int^{\phi}_{-\infty}d\tilde{\phi}\gamma^{\uparrow
ij}-i\left(\frac{p^{\prime}_{i}p^{\prime}_{j}}{2p^{\prime-}}+\frac{k^{\prime}_{i}k^{\prime}_{j}}{2k^{\prime-}}\right)\int^{\phi}_{\infty}d\tilde{\phi}\gamma^{\downarrow
ij}},\end{split}$ (64)

where $C_{ij}=\sigma^{\uparrow}_{ij}-\sigma^{\downarrow}_{ij}$, $C=\det
C_{ij}$ and $B_{i}=e^{\uparrow j}_{i}p_{j}-e^{\downarrow
j}_{i}(p^{\prime}_{j}+k^{\prime}_{j})$. By setting $\int
d^{4}X\mathfrak{I}(X)=i(2\pi)^{2}\delta(p^{-}-p^{\prime-}-k^{\prime-})\int
d\phi\tilde{\mathfrak{I}}(\phi)$, the only other integral needed is the one
with a prefactor linear in the transverse coordinate $X^{i}$, namely

$\begin{split}\int
d^{4}X\mathfrak{I}(X)X^{i}=i(2\pi)^{2}\delta(p^{-}-p^{\prime-}-k^{\prime-})\int
d\phi\tilde{\mathfrak{I}}(\phi)\frac{(C^{-1})^{i}_{\\-\
j}B^{j}}{p^{-}}.\end{split}$ (65)

We can finally write down the partially integrated transverse momentum as

$\begin{split}\int d^{4}X\mathfrak{I}(X)\Pi^{\uparrow
i}_{p}(X)=-i(2\pi)^{2}\delta(p^{-}-p^{\prime-}-k^{\prime-})\int
d\phi\tilde{\mathfrak{I}}(\phi)\left(e^{\uparrow ij}p_{j}-\sigma^{\uparrow
i}_{j}(C^{-1})^{j}_{\\-\ k}B^{k}\right),\end{split}$ (66)

with this substitution rule the squared amplitude can be written as an
integral in $\phi,\phi^{\prime}$ only. A further simplification can be
achieved exploiting the identity $C^{-1}_{ij}=\frac{1}{C}(C_{k}^{\\-\
k}\eta_{ij}-C_{ij})$ for rank-two matrices. Having derived the full nonlinear
squared amplitude it would be natural to consider some physically interesting
examples, however one runs very soon into the mathematical complexity of the
problem. In particular we recall that once the Brinkmann profile is chosen one
has to solve the differential equation $\ddot{e}_{ij}=H_{ik}e^{k}_{\\-\ j}$,
in fact the knowledge of the in and out vierbein is a key element. There are
very few profiles $H_{ij}$ for which an analytical solution of this equation
is available. These comprend for example impulsive waves
$H_{ij}=\delta(\phi)d_{ij}$ where $d_{ij}$ is a constant matrix, these
solutions are relevant as they represent the gravitational field produced by a
massless particle moving at the speed of light [49, 50, 51]. Other
possibilities are profiles with conformal symmetry of the form
$H_{ij}=a(\phi^{2}+b^{2})^{-1}\epsilon_{ij}$ studied in [52]. However
physically interesting scenarios for the process here considered would require
proper approximations and numerical evaluations. Also the concept of formation
length, of great importance in electromagnetism, could be worth to be studied
in this context. In fact, despite the non-locality of the interaction and the
clear implications of a curved spacetime background, the high amount of
symmetries of the problem could lead to some interesting developments. A
detailed study of this and similar processes in the aforementioned directions
will be left for future works.

##  VI Conclusions and outlook

We examined the one-to-one map connecting the dynamics of a charged particle
in an electromagnetic plane wave and a particle in a nonlinear gravitational
plane wave. This map enables to translate interesting results known in
electromagnetism to the context of plane wave spacetimes. The dynamics in an
electromagnetic plane wave is completely determined by a specific Lorentz-like
transformation $\Lambda^{\alpha}_{\\-\ \beta}$, which provides a way to
transport the initial momentum of a free particle through the wave such that
$\pi^{\alpha}=\Lambda^{\alpha}_{\\-\ \beta}p^{\beta}$. This transformation has
a number of interesting properties. Besides belonging to the Lorentz group, it
operates on constant-$\phi$ hypersurfaces and it acts as a gauge
transformation on the background wave. For these reasons its proper
generalization as a quantum operator $\mathbf{\Lambda}$ turns out to be a key
element to construct quantum states in an electromagnetic plane wave
background. Exploiting the aforementioned analogy between electromagnetic
waves in flat spacetime and gravitational nonlinear waves we showed that the
curved spacetime generalization of $\mathbf{\Lambda}$ can be used to find
quantum states in gravitational plane waves as well. In particular we showed
how scalar, spinor, and vector fields can be naturally mapped from one context
to the other. Finally, in the last part of the paper we applied the found
states for spin 1/2 and spin 1 particles to consider Compton scattering in a
nonlinear sandwich gravitational wave. Here we provide the spin- and
polarization-summed squared amplitude for the process, exact in the
gravitational plane wave. A detailed study of the Compton cross section for
physically interesting plane wave gravitational backgrounds is left for a
future work. Other processes could be naturally investigated as well in the
future, an interesting example being the graviton emission by a moving
particle. This could provide insights about its relation with photon emission,
a relation which has been studied in vacuum QFT [46, 53] and in the presence
of a background electromagnetic plane wave within strong-field QED [54, 55].

## Acknowledgments

The authors would like to thank C. H. Keitel for reading the paper and for the
valuable comments.

##  Appendix A Metrics handbook

In this appendix, we collect a few known but useful formulas about the
Brinkmann and the Rosen metrics. We refer to the main text for the notation.

  * •

Brinkmann

    * –

The metric: $ds^{2}=\eta_{\mu\nu}dX^{\mu}dX^{\nu}+Hd\phi^{2}$

    * –

Christoffel symbols:
$\Gamma^{\lambda}_{\mu\nu}=n^{\lambda}\left(\frac{1}{2}\dot{H}n_{\mu}n_{\nu}+2H_{(\mu}n_{\nu)}\right)-H^{\lambda}n_{\mu}n_{\nu}$

    * –

Riemann tensor:
$R_{\rho\mu\nu\delta}=\left[H_{\rho\nu}n_{\mu}n_{\delta}-(\rho\leftrightarrow\mu)\right]-(\nu\leftrightarrow\delta)$

    * –

Ricci tensor: $R_{\mu\delta}=H^{i}_{\\-\ i}n_{\mu}n_{\delta}$

    * –

Vierbein: $E_{\alpha\mu}=\eta_{\alpha\mu}+\frac{H}{2}n_{\alpha}n_{\mu}$

    * –

Spin connection: $\omega_{\mu\alpha\beta}=2n_{\mu}H_{[\beta}n_{\alpha]}$

where $H=H_{ij}dX^{i}dX^{j}$, $H_{\mu}=H_{\mu i}X^{i}$.

  * •

Rosen

    * –

The metric: $ds^{2}=2dx^{+}d\phi+\gamma_{ij}dx^{i}dx^{j}$

    * –

Christoffel symbols:
$\Gamma^{\lambda}_{\mu\nu}=\frac{1}{2}g^{\lambda\rho}\left(n_{\mu}\dot{\gamma}_{\rho\nu}+n_{\nu}\dot{\gamma}_{\rho\mu}-n_{\rho}\dot{\gamma}_{\mu\nu}\right)$

    * –

Riemann tensor:
$R_{\rho\mu\nu\delta}=\left[\left(\frac{1}{4}\gamma_{\rho\sigma}\dot{\gamma}^{\sigma\lambda}\dot{\gamma}_{\lambda\nu}+\frac{1}{2}\ddot{\gamma}_{\rho\nu}\right)n_{\mu}n_{\delta}-(\rho\leftrightarrow\mu)\right]-(\nu\leftrightarrow\delta)$

    * –

Ricci tensor:
$R_{\mu\delta}=\left(\frac{1}{4}\dot{\gamma}^{\rho\lambda}\dot{\gamma}_{\lambda\rho}+\frac{1}{2}\gamma^{\rho\lambda}\ddot{\gamma}_{\lambda\rho}\right)n_{\mu}n_{\delta}$

    * –

Vierbein: $e_{\alpha\mu}$

    * –

Spin connection: $\omega_{\mu\alpha\beta}=n_{[\alpha}\dot{e}_{\beta]\mu}$

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